The Geography of Competition
John R. Miron
The Geography of Competition Firms, Prices, and Localization
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John R. Miron Department of Social Sciences University of Toronto Scarborough 1265 Military Trail Toronto ON M1C 1A4 Canada
[email protected]
ISBN 978-1-4419-5625-5 e-ISBN 978-1-4419-5626-2 DOI 10.1007/978-1-4419-5626-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010920156 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Carol, my fellow traveler, for the spiritedness, love, companionship, counsel, enthusiasm, and forbearance that made this long journey possible. To all my family—tolerant and understanding though they be—from whom I have been inexcusably absent for far too long: “Don’t disturb Dad; he’s working.” To my father, whose keenness, curiosity, and relentlessness in seeking a good answer has been a lifelong inspiration.
Preface
This book focuses on three questions. First, what are the impacts of location decisions of firms—by affecting prices in various markets—on the locations of their customers, suppliers, and competitors in a market economy? Second, how, when, and why does this result in the clustering of firms in geographic space? Third, when and how is society—overall or by sector within it—made better or worse off as a result? At its heart, this book is in an area of scholarship that I label competitive location theory. Broadly, this is the study of how competition among firms leads to localization: geographic patterns of concentration among firms in markets in equilibrium. Competitive location theory takes the form of a set of interesting problems or stories. In this book, each story is constituted as a model, and placed within a set of models for the purposes of comparison. Methods of analysis in this book take the form of model construction and interpretation. What is meant by these terms? Dorfman (1960, p. 579) suggests three aspects to model building: (1) inventing symbols for the components and writing down the relationships connecting them; (2) creative hypothesizing wherein behavioral and technological assumptions are introduced; (3) quantification and statistical estimation. Location theory, as presented in this book, emphasizes primarily the first two aspects. In this book, I reinterpret 11 sets of basic models in competitive location theory focusing on these three questions. From this reinterpretation, I conclude that (1) competitive location theory offers diverse, rich, and profound ideas about the nature of a regional economy and that (2) the conceptualization of geography is central to economic analysis. Answers to the three questions above are of great interest to students and scholars in a variety of disciplines: e.g., City Studies, Civil Engineering, Development Studies, Economics, Geography, Housing Studies, Management, Public Finance, Public Policy, Real Estate, Regional Science, Regional Studies, Transportation, and Urban Planning. Much of competitive location theory is drawn from Economics. It is mathematical and logically rigorous. As such, students in other disciplines who would benefit from its insights and could contribute to its debates do not readily grasp it. A geographer myself, I have written this book in part to make this area of scholarship more accessible to students from outside Economics. My goal here is to enable the kind of intellectual breakthroughs that are made possible by a broader discussion of ideas. What is it that makes study in this area problematic for students outside Economics? These students quickly discover that the discipline is perhaps vii
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unique among the social sciences. More than any other discipline, Economics starts from a core body of theory. As this book is focused on microeconomic applications, I take this to include neoclassical theory of consumer demand, theory of the firm, and welfare economics. Students begin to learn this theory from their first course in the subject. As I remember, it was a rush; students are challenged to use the theory from their first course and often feel empowered after just a few courses. Just as their professors, they are soon able to assess and critique the work of others. Contrast this with other social sciences such as Geography, Anthropology, Sociology, or Political Science where the absence of a common integrated core means students often have to learn multiple perspectives over a longer period of time before they can begin to critique work in a meaningful way. This has the further implication that observation and measurement play a more important role in the early stages of study in social science disciplines other than Economics.1 Arguments are made there partly on the basis of evidence (whatever the intellectual lens used to see this evidence) and partly on the basis of theory. The same is true in Economics, but here there is an emphasis from the outset on the idea that evidence and core theory must be jointly consistent. Economic reasoning has a distinctive logic. As a child, I remember a particular set of economic stories my father told me. They were fascinating accounts of the nature of money and banking, risk, insurance, and investment. In each case, as appropriate to a storyteller, the tale was cast in the simplest of terms to enable the listener to see how and why something worked. My father’s purpose was to get me to better understand the working of business. Later, as an undergraduate student studying Economics, I came across similar stories told in lectures. Of course, at the University level, these stories were more sophisticated and disciplined, heavily graphical and mathematical, and even more intriguing. Strangely, though, my father always was uncomfortable with what might be thought to be straightforward extensions of economic reasoning. When I came home from university for a holiday, he would have assembled newspaper articles or quotes from radio or television interviews in which an economist had made a prediction about the future based on economic reasoning. He would pounce on cases where he thought the prediction was incorrect or improbable. He could not understand that the economist was not actually predicting the future per se. What the economist was doing was to show economic analysis could be relevant to a matter of popular concern or public policy. The caveat, always implicit in the economist’s argument, is that a certain economic outcome will follow based on a certain set of assumptions. But, my father scoffed, if the assumptions are not tenable, what is the value of the prediction? It took me a while to realize his own economic stories also were based on assumptions and that the conclusions to his stories were actually predictions: as flawed as those of the economists he scorned.
1 See, for example, the classic statement by Sauer (1924) on the survey method in geographic research. I am not arguing here that the teaching of economics ignores empirical evidence; it does not. What I am arguing is the relative importance of core theory early on in a student’s study of economics.
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I will return to that strand of thought in a moment. First, let me make a connected argument. My minor subject at university was Mathematics. On the one hand, my studies in Mathematics helped a lot with my studies in Economics. Areas of Mathematics provide tools economists use to advance their thinking. However, it quickly becomes apparent to students of Mathematics at the University level that the subject is much more than that. Mathematics enables a kind of reasoning that is pure; it is tied only to assumptions (axioms) from which a set of deductions can then be derived. This is pure reasoning; there is no requirement these deductions be applicable to anything in the real world. At the same time, even a mathematician recognizes that it is the application that makes pure reasoning relevant to a broader audience. They too have to worry about a reader’s suspension of disbelief and the criticism that they are otherwise being self-indulgent. In any science, mathematical tools—and the stories we build with them—must be appropriate to the real world in which they are to be applied. It is never enough simply to lay out some assumptions and then draw conclusions from them only in the abstract. Adding to the confusion, some economists do not lay out all of their assumptions at the outset of their model: perhaps because it might make the story too dry or formal. Instead, they reveal assumptions—sometimes explicitly, sometimes not—at appropriate points in the course of telling the story. This can be maddening to readers from outside Economics: my sense is that they like to know what is being assumed at the outset of an argument. However, this raises a fundamental problem in economic analysis. Is it ever possible to outline fully the assumptions that underlie a real-world problem in economics? Take one simple economic problem. A market consists of N identical individuals. Some are endowed only with one unit of commodity A. The rest of them, otherwise identical, each have only one unit of commodity B. Assume the proportion endowed with commodity A is pa and that the two commodities are each indispensable. These N individuals somehow gather in a market to exchange commodity A for commodity B. However it is determined, what will be the market equilibrium rate of exchange between A and B? Economists here usually assume in such cases an auctioneer and motivated individuals who have identical preferences characterized by a well-behaved utility function. Stated as such, the problem is simple, even elegant. Notice, however, I say nothing here about how a legal system, social institutions, and other mechanisms make the market work. The reluctance of economists to define even the notion of a market adds to the discomfort. Other things have to be assumed in order for there to be a predictable market rate of exchange. Therein is the source of a key shortcoming in any attempt to build economic models. Compared to Mathematics where the axioms constitute all that can be assumed, economic models assume more than the author typically states at the outset. Strangely, it can be viewed also as a source of richness. After all, what makes economics and the other social sciences so intriguing (especially to those of us who are relativists2 ) is the nuance: the ambiguity about underlying motivations
2 An essentialist sees explanations as ultimate truths. A relativist sees explanations simply as bettering current understanding.
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and processes. The differing simplifications we invoke to understand behavior (here the set of assumptions underlying each model) each give insights into the human condition. By varying the model (i.e., the set of assumptions), we come to better understand the importance of each assumption and, indeed, the complexity of the human condition. In analyzing models, economists emphasize comparative statics: a comparison of outcomes (endogenous values) predicted by a model when a given (exogenous variable or parameter) is changed by a small amount. Some models describe market equilibrium: here comparative statics details the changes in equilibrium when a given is changed by a small amount. In other cases, models describe optimal outcomes: here comparative statics detail changes in optimal outcome when a given is changed. Basically, comparative statics are predictions of how—based on core theory—equilibrium or optimal outcomes will change conditional upon a one-time change in a single given. Critics (including my father) would question the value of the comparative statics exercise if in fact the future turns out differently from what is predicted here. However, the economist would argue that we learn from this that the assumptions were not appropriate to a given situation. After all, they might argue that comparative statics results are simply derived from the set of assumptions that underlie the model. For my students, this is typically a higher and deeper understanding of the use of models. It goes well beyond simple exposition of a model. In my experience, some students are captivated initially by the elegant structuring of models: others by model predictions that are unexpected, even startling. For most students, the idea that we need to focus on assumptions and their effects on location outcomes comes later.3 From my relativist perspective, the objective of this book is to bring the reader to understand the key role of assumptions. Of course, the reader might well ask, “why bother?” If the reader’s interest ultimately is in using location theory to study problems of interest to them, what is to be made of the idea that assumptions are everything? My advice here is that practitioners need to work with a model that best approximates the situation they currently confront. Even so, they need to be mindful of how the assumptions of that model color the conclusions they seek to draw. It is that level of facility with location models that I aspire to instill in the reader. For readers from other disciplines, the reasoning that underlies models in competitive location theory is something relatively new and thus requires introduction. Four approaches to reasoning come to mind: descriptivist, functionalist, explanatory, and instrumentalist.4
3 See,
for example, the discussion of assumptions in Dantzig (1991, p. 22).
4 Although the focus of this book is on theory, there is a close relationship here to the three notions
from exploratory data analysis as discussed in Mosteller and Tukey (1977, pp. 259–262): association, dependence, and causation. Imagine we have a sample of observations, for each of which we have values for two variables: Y (dependent or endogenous) and X (independent or exogenous). Association is a statistical relationship evidenced between two variables (Y and X) observed from a sample of data. Dependence is a statistical relationship backed up by a theory that explains how the
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In a descriptivist approach, a theory or model is simply a summary of observed experience: this is aspect one in Dorfman’s view of model building. The approach here is inductive and sometimes labeled atheoretic.5 However, critics argue that observations (what is observed and how measured) are, in fact, theory laden. That is, our selectivity in observation and measurement (i.e., the intellectual lens through which we look at the world) means some things are not being observed or measured: presumably, these are not important. The decision as to what and how to measure already involves implicitly some notion of theory, however rudimentary. A second critique of theory as summary is that this is in itself a theory: a criticism that I raise in the context of Barnes (2003) in Chapter 1 below. In my view, descriptivists necessarily invoke theory (i.e., they are not atheoretic) even if that theory is itself ill defined. I mention this argument because parts of the social sciences do adhere to descriptivist principles. Whatever the attraction of this perspective, I exclude it from further consideration in this book, as it does not help us think about competitive location theory other than as antithesis. In a functionalist approach, models are constructed using “as if” and analogies. Typically, here we borrow a model from elsewhere in sciences, show that the outcomes of the model are like the outcome of some problem in locational competition, and then assert the belief that the model provides insight into what will happen in the process of interest.6 As a rudimentary model of the spatial structure and organization of a city, we might argue, for example, that a city is like the human body; it has traffic (blood flow), a central business district (brain), and a variety of specialized land uses (corresponding to body organs). We might then try to model the growth and decline of a city in terms of maturing and aging of the human body. Why do we use these kinds of analogies? Usually, the reason is that there has been a breakthrough in method or conceptualization in another discipline, and we want to explore whether this might lead to a breakthrough in our discipline as well. Although critics can be dismissive of such work, I think the use of analogies can be an important tool in the scientist’s kit when thinking about how to advance a discipline.7 In an explanatory approach, we build directly a theory that explains the process. In Economics, this is usually based on an agency perspective wherein individual economic actors (be they firms or consumers, suppliers or demanders) each act to
variation in Y across the sample is explained by variations in X. Causation is a statistical relationship, a theory as described above, plus the belief that were we to change X for any one observation in our sample, the value of Y for that observation would change correspondingly. 5 In the terminology of Mosteller and Tukey, if we had a sample of data, there would be only the possibility of association here: not dependence or causation. 6 In economics, Morehouse, Strotz, and Horwitz (1950) used an electrical system analogy to study inventory accumulation. Enke (1951) also used an electrical system analogy to study interregional price differences. Fisher (1925) used a number of hydraulic analogies to study economic behavior. 7 Can a functionalist approach ever imply dependence or causation? At first glance, the answer might seem to be “no”; what we have here is at best statistical association. However, a fuller answer would be that dependence or causation might be possible depending on how close the analog is to the process under study.
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maximize the benefit of their participation in a market. To avoid ad hoc modeling, economists prefer that their assumptions include at least one universal law drawn from their core theory. Here is an important way in which building of theory differs from other social sciences. In parts of the social sciences, for example, it is common to begin with a process under study then ask what kind of theory might explain it regardless of whether that theory is core or contains a universal law. The economic approach more typically is to ask what core theory has to say about a process under study.8 Now let us consider the instrumentalist approach. One matter that troubles some scholars outside Economics is the reliance core theory that puts on relatively simple forms of rational behavior by firms and consumers. After all, critics argue, is it not true that people can be motivated by considerations other than the consumption of commodities that typically underlies the neoclassical conceptualization of consumer utility; if so, what is the relevance of core theory? The instrumentalist approach argues that we should judge core theory by what it predicts about behavior not by what it assumes.9 This concern over the fundamental assumptions of core theory has led economists to propose an instrumentalist approach to reasoning sometimes called modeling for prediction.10 It specifically argues the assumptions of models are not to be tested. The only possible test of a model concerns whether the model predicts correctly. Critics ask just how different a prediction has to be from an assumption in order for it to be testable. Put differently, they argue that people behave as if they maximized utility even if their motivation is something else. However, in my view, this now blurs the distinction between functionalist and explanatory approaches.11 In this book, I focus on models that explain location as an outcome behavior by market participants that can be thought to be rational.12 In talking about explanation, my view is relativist rather than essentialist. An essentialist sees explanations
8I
should be careful here. Not all economists concur with all aspects of the core theory. Nonetheless, a widespread acceptance of the relevance of core theory within the economics discipline is sometimes thought by readers from other social science disciplines as verging on the ideological. 9 See Roth (1989) 10 See Friedman (1953, pp. 3–43). 11 It also instances the question, raised above, of how close the analog has to be to generate dependence or causality. 12 By rational, I mean simply that an economic actor makes choices consistently: see Becker (1962). A consumer is deemed to be rational if, in choosing between alternatives, he or she exhibits preferences among these alternatives that are (1) well-ordered and (2) stable over time. Here wellordered means that if the consumer prefers alternative A to B and also prefers B to C, then that consumer will prefer A to C. The phrase “stable over time” is to suggest that if the consumer preferred A to B yesterday, then other things being equal, will still prefer A to B today. However, economists typically impose additional constraints on rationality, including diminishing marginal utility, limitations on separability, and notions of expected utility. For an interesting discussion of the nature and paradoxes of rational choice, see Sugden (1991).
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as ultimate truths. A relativist like me sees explanations simply as bettering current understanding. Of course, current understanding will differ from one person to the next. Therefore, a relativist must pitch his or her argument in terms of an intended audience: the people for whom the arguments presented will see it as an improved understanding. In my case, I have aimed this book primarily at graduate and advanced undergraduate students in Geography, Planning, Business, and Urban Studies as well as Economics. However, I hope there is much in each chapter that will be novel even for seasoned readers of location theory. There are still other respects in which Economic methodology might seem puzzling to students from elsewhere in the social sciences. Six come to mind. • Deductive approach. Broadly speaking, there are two kinds of methodologies in the social sciences: Case Study13 and Deductive.14 Both are valuable.15 While the advantage of the Case Study approach is the rich detail it can offer about an occurrence of the phenomenon, the significant disadvantage is the risk of ad hoc explanation: we just do not know whether the story is equally applicable to what might be thought to be other occurrences of the same phenomenon. Ad hoc explanation is less likely in the Deductive approach (although not necessarily eliminated) because we are looking at whether many different cases are consistent with the theory. This book is concerned with the construction of theory applicable to a Deductive approach. In a Deductive approach, one tells a special kind of story to readers: one that explains a phenomenon while clarifying the generality of application.16 Be it fiction or nonfiction, Case Study or Deductive, any story to be told successfully must engage readers to the extent that they, the readers, are prepared to suspend their disbelief; that is the craft of the storyteller. Otherwise, any writer risks being seen by the reader as simply self-indulgent: i.e., without appeal to a broader readership. The onus on the scholarly storyteller (especially in the Deductive approach) is perhaps even greater here. We do not have quite the same access to fantasies, ambiguities, subtleties, and other tricks as do writers
13 A
Case Study approach makes use of in-depth analysis involving one or a few examples. Here typically—in a process sometimes called process tracing—proponents construct an explanation for the case study and then show that the explanation is consistent with all information available about the process. 14 The Deductive approach is based on a theory (definitions, classifications, and logic) and makes use potentially of data for a large number of examples (alternatively, observations or cases) thought to illustrate the phenomenon. Here proponents typically identify a theory to be evaluated (the so-called alternative hypothesis) and a null hypothesis (its negation), a statistical population (the set of cases generated by the same process), a sufficiently large sample of cases from that population, and then a statistical test using the sample data to see if the null hypothesis can be rejected in favor of the alternative hypothesis. 15 Rodwin (1945) nicely compares the two approaches with respect to industrial location analysis. 16 The exercise is similar in a Case Study.
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of fiction. We are more like writers of “how to” books than books of fiction.17 This book is no different; my objective is to use a deductive approach to tell compelling stories: i.e., stories structured to pique the curiosity of readers while leaving no escape from their sometimes startling conclusions. • Reductionism. Reductionism is at the core of this book. A reductionist like me believes the best way to improve our understanding is to focus on an aspect of the process under study and simplify the problem enough to permit us to represent it as a model (often mathematical). Inherently, this involves a suspension of disbelief (is the model informative even if it is unrealistic): i.e., the sense we have assumed away something in fact important.18 In using a model, we hope to better our understanding of the process under study: i.e., our ability to describe, simulate, explain, predict, or control it. In an idealized (mathematical) setting, a model is built from a set of assumptions (axioms); logical deduction then leads to a set of outcomes (or hypotheses). The outcomes themselves sometimes initially surprise: as in “I did not know these assumptions would necessarily give this outcome!” The derivation of such outcomes can be boring, tedious and mechanical but is always necessary; altogether too often in my own humbling experience, errors arise when we take shortcuts in the derivations. At the same time, reductionists have an obligation to consider the broader implications of their thinking, not just the immediate conclusions. To avoid the label self-indulgent, reductionists need to consider just how generalizable are their findings.19 To a greater extent than in the other social sciences, reductionism is at the heart of economic reasoning. Economic reasoning consists of applying notions from core theory to the sub-discipline. To do this, questions in the sub-discipline must be recast in, or reduced to, a form consistent with core theory. There are several goals here. One is to confirm that core theory is applicable to each sub-discipline: otherwise the theory is not core. A second is to garner insights about the process. A third is to create testable hypotheses that can then be refuted on the basis of data analysis. If the data lead to rejection of a hypothesis consistent with core theory, this may raise important questions about the core. In an explanatory approach, the model can generally be subjected to an independent test. For instance, the model to be statistically estimated allows for the possibility that either X affects Y or it does not. Economists are fond of such models; (human) geographers—more
17 Worse
than that for the wary reader, scholarly writers overtly proselytize; they seek to convert readers from disbelievers into believers. The Case Study approach seeks to convince the reader through immersion in detail applicable to that one case; the Deductive approach seeks that suspension of the reader’s disbelief through an appeal to generalization (an escape from detail). Truth be told, the drive for compellingness typically means scholarly writing often has the subtlety of a sledgehammer. 18 I include here the translation problem raised by Dennis (1982a, 1982b, 2002), wherein economists do not prove by deductive exercises what they claim to prove about their economic subject matter. 19 This well-known point is made in Edgeworth (1888, p. 348, Footnote 6) and in Solow (1956, p. 65).
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skeptical of reductionism—see a tradeoff here between thick analysis (detailed consideration of a small number of cases) and thin (statistical analysis of a large number of cases); to them, there is a choice here between statistical testing and comprehensiveness. Put more crudely, they do not share the idea of core theory in Economics as a holy grail. As a geographer and a relativist, I prefer to think core theory is useful even if there are respects in which I might prefer something better. • Agency. In the social sciences, there are broadly two complementary categories of explanation: structural and agency. An agency explanation focuses on how individuals (e.g., persons or firms) make choices from among alternatives coping with the exigencies of daily life. These choices then explain outcomes. An agency explanation typically does not dwell on the question of how the set of alternatives gets structured, determined, or limited. A structural explanation, in contrast, sees social science outcomes as determined by the institutional and structural constraints that limit or deny choice. Why these constraints exist, how they operate, and how they change over time are central questions addressed in a structural explanation.20 While we find both agency and structural explanations in all disciplines, agency explanations are more prevalent in Economics than in other social science disciplines. This is especially true in competitive location theory, which—almost by definition—focuses on the choices made by rational firms. This book similarly focuses on agency explanations. • Theory and model. Every area of social science may be said to deal in theory, but only a small subset are typically thought to deal with models. A mathematical model usually takes a simple form: givens (exogenous values and parameters), relationships (behavioral and identity), and outcomes (endogenous). Implicit here is a notion of causality; the givens determine the outcomes (and not vice versa). Implicit in both the givens and the relationships is an underlying theory. There are two ways to think about the theory here. One is that the theory identifies parameters and exogenous values and provides the basis for the relationships used to solve for the endogenous variables. The other is that the theory is a set of deductions arising from a set of assumptions. Surprisingly, it is not easy to move back and forth between these two ways of viewing theory. It would be nice for example if assumptions corresponded to givens and deductions to relationships. However, there is no particular reason to expect this. This makes for some interpretation of models; we cannot just look at a mathematical structure. How do I make use of models in this book? What I do is to compare the outcomes of models with similar assumptions. I want to show how, when one assumption is altered, the outcomes of the model change. After we have repeated this process from one model to the next, we begin to see patterns (a characterization) to the effect of a particular assumption. Economic analysis is often typified by use of a theoretical
20 I intend this description to include structuralist, poststructuralist, and structurationist approaches.
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model,21 something readers from Geography and other social science disciplines can find mystifying, impenetrable, inappropriate, unrealistic, or too deterministic. Economists favor a parsimonious writing style—at its best, breathtakingly insightful—that other kinds of readers may see as either narrowly focused ways of thinking about behavior or simply as too dense or terse. This book attempts to overcome aspects of that by casting models in terms readers from other disciplines can more easily understand. In general, the objective here is to make some of the tools of competitive location theory better understood in broad swaths of the social sciences to which it should be of interest.22 • The origins of economic models. Where do models come from? Put differently, when a location theorist first identifies a problem (i.e., a question to be addressed), how does he or she come to represent it in the form of a model? Sometimes the model is spawned directly from a body of theory; sometimes it is an extension to an already-existing model. And, sometimes, as Murphy and Panchanadam (1997, p. 342) state, it is an analog (as in the functionalist approach), wherein we abstract features of our problem, look for similar problems that have been solved elsewhere, and then apply those solution methods to our own problem. Part of the unease that other scholars have with neoclassical economics is the way in which that area of study has borrowed models from Newtonian physics. Put differently, these critics see microeconomic explanations as functionalist rather than as explanatory or instrumentalist. In this book, I hope to convince skeptical readers that the benefit to be derived from use of these models outweighs the limitations. • Refutability. Another source of bewilderment to some scholars is the emphasis placed by economists on the refutability of scientific propositions.23 A model is generally deemed inappropriate in Economics if it does not permit the possibility of being refuted. In areas of the social sciences where the culture is one of inclusiveness of ideas rather than exclusiveness, this emphasis on refutability may seem unwarranted. However, I think that every scholar would agree that it is important to cull arguments or ideas that can be shown to be incorrect or inappropriate; in my view, refutability is valuable in that exercise.
21 Here,
I use model to mean a simplified representation of the real world whose purpose is to describe, simulate, explain, predict, or control a process under study. Although models can take various forms from physical to chart to mathematical, I use the term in this book to mean only something that can be expressed mathematically. In this situation, a model includes values that are exogenously given (typically parameters and exogenous or lagged variables) and equations (both behavioral and identities) that link these to the values of endogenous variables. 22 In large part, this is due to the way economists practicing competitive location theory look at the world in general and location in particular; their ideas and methods, however insightful, do not come easily to students in other disciplines. Of course, every discipline has its own sensibilities, and the starting assumptions that characterize one discipline may well be the antithesis of another discipline’s creed. Indeed, one might ask, why else have disciplines? 23 See, for example, Ellsberg (1954, p. 529).
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Let me conclude this preface with a few statements on the background assumed and the pedagogical approach of this book. In this book, I assume the reader has completed first-year university calculus. Even though I do not explain calculus concepts and derivations here, readers without that background should still be able to follow the main ideas presented in each chapter. The book uses other mathematical methods as well: notably vector analysis, fixed point analysis, variational inequalities, and mathematical programming. In such cases, the text does not assume readers are knowledgeable: I therefore take time to explain important concepts and procedures where needed. In keeping with the emphasis on mathematical basics, this book uses elementary representations of geography. In every chapter, the importance of distance is emphasized, typically for its effect on transaction cost. This generally includes a shipping cost or commuting cost. Usually here, I assume unit transportation cost is strictly proportional to distance. The book also assumes familiarity with introductory economics that includes basic elements in the theory of the consumer, theory of the firm, and welfare economics. This is the kind of material students acquire in their first two undergraduate semester courses in microeconomics. I make no attempt to derive microeconomic fundamentals here, but I try to provide enough details about concepts for readers to follow the main arguments in case their background in microeconomics is rusty. The book is written to accommodate different styles of learning: mathematical, graphical, and verbal. The body of each chapter is a verbal exposition of a model and its outcomes. Mathematical renditions of the model are presented in tables at the end of each chapter. Graphical renditions are similarly presented as a set of figures. Finally, let me end on a personal note regarding conclusions. In each chapter that follows, I present one or more models: 44 in total. As befitting the idea that each model is a story, the chapter is in part an exposition on structure and in part deductions or derivations that necessarily follow from this. In this, I strive for parsimony. Some authors (and editors) like to end each chapter with a restatement of conclusions. I do not. The chapters that follow end with a section titled “Final comments” but nothing titled “Conclusions.” In part, my objection is philosophical. To a relativist like me, a conclusion is just the point in your analysis that you reach when you decide to stop thinking any further. In part, my objection is in terms of parsimony. If a conclusion I reach midway through a chapter, say on p. 6, required understanding the derivation and qualification on the pages that preceded it, how do I then write a conclusions section at the end of the chapter without having to repeat derivations and qualifications. To me, a model is like a well-told story. It is something to be appreciated for the craft in telling it, not something with which to bludgeon the reader at the end. I apologize now to the readers who get to the end of each succeeding chapter and wonder why the conclusions are not restated. Lake Sesekinika, Ontario, Canada
John R. Miron
Acknowledgments
To all the scholars who, through imagination, creativity, and questioning, make it possible for others, including me, to better understand and think more critically about this subject. To students in my courses in location theory over the years, whose surprise, wonderment, and enthusiasm reminds me constantly that competitive location theory is a special part of social science thought. To my graduate students over the years who, like me, frittered away their youth lost in the puzzles that are competitive location theory only to find those ideas, methods, and perspectives coming back to their aid wherever their careers have since taken them. To my university for fostering an environment where scholars can reach for their highest goals.
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1 The Craft of the Story Teller . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1.2 Geographic and Other Perspectives on Localization 1.3 Marshall’s Perspective as a Starting Point . . . . . . 1.4 The Development of Economics Since Marshall . . 1.5 Why Is Competitive Location Theory Problematic? 1.6 Location Theory and Geography . . . . . . . . . . 1.7 My Approach . . . . . . . . . . . . . . . . . . . . 1.8 What This Book Is About . . . . . . . . . . . . . . 1.9 What This Book Is Not About . . . . . . . . . . . .
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2 The Firm at Home and Abroad . . . . . . . . . . . . . . . . . 2.1 The Greenhut–Manne Problem . . . . . . . . . . . . . . 2.2 Model 2A: Non-spatial Monopolist . . . . . . . . . . . . 2.3 Model 2B: Monopolist Selling at Two Places; Factory at Place 1 Only . . . . . . . . . . . . . . . . . . . . . . . 2.4 One Market or Two? . . . . . . . . . . . . . . . . . . . . 2.5 Pricing Strategies . . . . . . . . . . . . . . . . . . . . . 2.6 Model 2C: Factory at Each Place . . . . . . . . . . . . . 2.7 Model 2D: Choice of Sites and Localization . . . . . . . 2.8 Two Markets Identical . . . . . . . . . . . . . . . . . . . 2.9 Differing Markets . . . . . . . . . . . . . . . . . . . . . 2.10 Comparative Statics in Model 2D . . . . . . . . . . . . . 2.11 Risk Aversion and Multiple Plants . . . . . . . . . . . . . 2.12 Model 2E: Contestability and Preemption of Competitors 2.13 Final Comments . . . . . . . . . . . . . . . . . . . . . .
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3 Logistics and Programming . . . . . . . . . . . . 3.1 The Hitchcock–Koopmans Problem . . . . . 3.2 An Illustrative Example . . . . . . . . . . . 3.3 Model 3A: Non-spatial Version of the Model 3.4 The Example in a Non-spatial Version . . . 3.5 Model 3B: Spatial Version of the Model . . 3.6 The Example: A Spatial Version . . . . . . .
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3.7 3.8
What Is a Market? . . . . . . . . . . . . . . . . . . . . . . . Final Comments . . . . . . . . . . . . . . . . . . . . . . . .
92 93
4 The Struggling Masses . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Cournot–Samuelson–Enke Problem . . . . . . . . . . 4.2 Model 4A: Autarky . . . . . . . . . . . . . . . . . . . . . 4.3 Model 4B: Integrated Market Solution: Zero Shipping Cost 4.4 Model 4C: Spatial Price Equilibrium with Shipping Costs . 4.5 Final Comments . . . . . . . . . . . . . . . . . . . . . . .
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97 97 100 107 113 122
5 Arbitrage in the Grand Scheme . . . . . . . . . . . . . . . . 5.1 The Samuelson–Takayama–Judge Problem . . . . . . . 5.2 Model 5A . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Social Welfare at Place i . . . . . . . . . . . . . . . . . 5.4 Net Social Payoff and Global Net Social Welfare . . . . 5.5 A Special Case: Horizontal Supply Curve at Each Place 5.6 Three Examples of Multiregional Shipment . . . . . . . 5.7 Application . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Case Study . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Final Comments . . . . . . . . . . . . . . . . . . . . .
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6 Ferrying Inputs and Outputs . . . . . . . . . . . . . . . . 6.1 The Weber–Launhardt Problem . . . . . . . . . . . 6.2 Model 6A: I = 2 Input Places, J = 1 Output Place; Location on a Line . . . . . . . . . . . . . . . . . . 6.3 Model 6B: I = 2 Input Places, J = 1 Output Place: Location on a Two-Dimensional Plane . . . . . . . 6.4 Model 6C: Substitutability, Scale, and Location . . 6.5 Model 6D: Price Elasticity . . . . . . . . . . . . . . 6.6 Model 6E: More Than 2 Input Places and/or More Than 1 Output Place . . . . . . . . . . . . . . . . . 6.7 Model 6F: Location on a Transportation Network . 6.8 Final Comments . . . . . . . . . . . . . . . . . . .
7 What the Firm Does On-Site . . . . . . . . . . . . . . . . . 7.1 The Marshall–Lentnek–MacPherson–Phillips Problem 7.2 Inventory Models in Management . . . . . . . . . . . 7.3 Model 7A: The Firm Doing Repairs In-House . . . . 7.4 Model 7B: Outsourced Repairs . . . . . . . . . . . . 7.5 Model 7C: The Decision to Outsource . . . . . . . . 7.6 Model 7D: The Advantage of Agglomeration . . . . . 7.7 How Far Away Can the Contractor Be? . . . . . . . . 7.8 Final Comments . . . . . . . . . . . . . . . . . . . .
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177 177 181 183 188 191 193 197 197
8 Staking Out the Firm’s Market . . . . . . . . . . . . . . . . . . . 8.1 The Market Area Problem . . . . . . . . . . . . . . . . . . . 8.2 Range and Geographic Size of Market . . . . . . . . . . . .
201 201 205
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8.3 8.4 8.5 8.6 8.7 8.8
8.9
8.10 8.11 8.12
xxiii
Trade Area and Market Area in Retailing . . . . . . . . . Model 8A: Two Firms Selling Commodity at Same f.o.b. Price . . . . . . . . . . . . . . . . . . . . . . . . . Model 8B: Market Area Boundary Between Two Firms Selling Same Commodity at Different f.o.b. Prices Model 8C: Why Do Prices Differ Among Firms? . . . . . Model 8D: Market Area Boundary Between Two Firms with Different Capacities . . . . . . . . . . . . . . Model 8E: Market Area Boundary Between Two Firms with Different, but Perfectly Substitutable, Commodities . . . . . . . . . . . . . . . . . . . . . . . . Model 8F: Market Area Boundary Between Two Firms with Different, but Perfectly Substitutable, Commodities When Customers Are of Two Types . . . . Model 8G: Market Area Boundary Between Two Firms Supplying Different Commodities . . . . . . . . . Model 8H: Destination Choice Under Uncertainty . . . . Final Comments . . . . . . . . . . . . . . . . . . . . . .
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9 The Cautious Farmer and the Local Market . . . . . . . 9.1 The Economides–Siow Problem . . . . . . . . . . . 9.2 The Barter Market . . . . . . . . . . . . . . . . . . 9.3 Uncertainty and Rationality . . . . . . . . . . . . . 9.4 Model 9A: Non-spatial Market . . . . . . . . . . . 9.5 Model 9B: Cooperation in a Spatial Market . . . . . 9.6 Model 9C: Competition for Land in a Spatial Market 9.7 Final Comments . . . . . . . . . . . . . . . . . . .
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10 Farming for Cash . . . . . . . . . . . . . . . . . . . . . . . 10.1 The Thünen–Lee–Averous Problem . . . . . . . . . . 10.2 Model 10A: Farms Producing Wheat Along a Line . . 10.3 Model 10B: Farms Producing Wheat on a Rectangular Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Model 10C: Farms Producing Two Independently Demanded Crops Along a Line . . . . . . . . . . . . 10.5 Model 10D: Farms Producing Two Independently Demanded Crops on a Rectangular Plane . . . . . . . 10.6 Final Comments . . . . . . . . . . . . . . . . . . . . 11 The City and Its Hinterland . . . . . . . . . . . . . . 11.1 The Thünen–Beckmann–Samuelson Problem . . 11.2 Model 11A: Factor Substitution with One Crop and in the Absence of Shipping Cost . . . . . . 11.3 Model 11B: Factor Substitution with One Crop and in Presence of Shipping Cost . . . . . . . .
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11.4 11.5 11.6 11.7
Model 11C: Factor Substitution with Two Crops in Presence of Shipping Costs . . . . . . . . . . . Model 11D: Non-spatial Version of Samuelson’s Model of a Thünen Economy . . . . . . . . . . . Model 11E: Spatial Version of Samuelson Model . Final Comments . . . . . . . . . . . . . . . . . .
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318 323 331
12 Local Production and Consumption . . . . . . . . . . . . . . . . 12.1 The Thünen–Miron Problem . . . . . . . . . . . . . . . . . 12.2 Local Production in the Farm Economy . . . . . . . . . . . 12.3 Model 12A: Farm in Autarky . . . . . . . . . . . . . . . . 12.4 Model 12B: Farm Purchasing Soap from the Factory . . . . 12.5 Comments on Model 12B . . . . . . . . . . . . . . . . . . 12.6 The Soap Factory as Profit Maximizer . . . . . . . . . . . 12.7 Model 12C: The Factory as Monopolist Using f.o.b. Pricing 12.8 Model 12C: The Factory Using Discriminatory Pricing . . . 12.9 Model 12D: The Factory as Bilateral Monopolist . . . . . . 12.10 Final Comments About This Chapter . . . . . . . . . . . . 12.11 The Connecting Topics . . . . . . . . . . . . . . . . . . .
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335 335 337 339 344 355 356 358 362 366 373 378
Appendix A: Assumptions and Rationale for Localization . . . . . . . .
385
Appendix B: Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
393
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
411
First Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
445
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
451
Chapter 1
The Craft of the Story Teller Economic Reasoning in a Geographic Setting
This book is in an area of scholarship practiced mainly by neoclassical economists. The principal method of analysis is interpretation of models that are deductive, theoretical, and usually mathematical. These models often seem intimidating. However, a model is just economic reasoning: an intricate story to help us better understand important ideas. This chapter explores the perspectives, rationales, and limitations of economic reasoning in a geographic setting. The chapter introduces concepts central to competitive location theory: localization; shipping cost, congestion, and risk; organization of firm and market; division of labor, indivisibilities, economies of scale, and other local cost advantages; Walrasian equilibrium; regional economy and autarky; monopoly and geographic space; and the nature of a geography. For readers from outside Economics, I consider other perspectives on localization, relate competitive location theory to these, summarize relevant neoclassical economic thought from Alfred Marshall to the present day, discuss the nature of explanation as used in this book, and assess the problematic nature of competitive location theory.
1.1 Introduction In 1776 Adam Smith proposed—in his famous book, An Inquiry into the Nature and Causes of the Wealth of Nations—that division of labor is limited by the extent of the market. Put simply, his idea was that the greater the output (scale) of a firm the better able it is to take advantage of the added productivity from having specialized labor.1 In my view, this important notion has five corollaries of particular significance for this book. First, it implies ferocious competition2 among firms, each only 1 This
idea is explored further in Stigler (1951).
2 I follow the practice of economists here in using competition to refer generally to rivalrous behav-
ior of firms: e.g., in choosing prices, commodities, strategies, or locations. Elsewhere in the social sciences, contestation has come into vogue as a term to connote a kind of conflict between parties. In Economics, however, a distinction is drawn between a competitive market and a contestable market. The latter is a market served by only one or a few firms that nonetheless (because of the potential for entry by new competitors) behaves similar to a competitive market: see Baumol, Panzar, and Willig (1982). Where contestation enters the economics literature is usually in political economy: see Bowles and Gintis (1988).
J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_1, C Springer Science+Business Media, LLC 2010
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1 The Craft of the Story Teller
too well aware that improved efficiency and profit—even survivorship itself—hinge on increased size and market share. Second, it implies that the firm has a keen interest in anything—say an improvement in transportation infrastructure—that enables it to grow its market. Third, it raises the question of the economic design of a firm; what activities of a firm are central to profitability and why, where, and how does it undertake these activities? Fourth, the division of labor implies that something—an inefficiency I label congestion within or outside the firm—limits just how large a firm can become. Fifth, and in my view most significant, is the implied simultaneity of price and location; just as firms select locations where prices allow them to maximize their profits so also do the locations of firms come to affect those very prices and profits. In a market economy, how do firms get organized at the sites where they produce, distribute, or sell their commodities or services? Put differently, what are the locational strategies used by firms to compete? In a sense, almost any strategy by which a firm survives and prospers in competition has a locational aspect. Among those commonly used are the following: siting to increase quantity sold, to increase price obtained, to increase market share, to reduce cost of commodities in production, to better access business (producer) services, to better access sites where inputs are more productive, to better access information about new technologies and market innovations, or to reduce risks. All these strategies share the idea that the firm is using the organization of markets to help it cope and prosper. Not all of these strategies are equally important to every firm and every industry.3 Taking into account competition among firms as well as the impact of firm siting on the siting of their suppliers and customers, how when and why does this locational behavior lead to localization (clustering) of firms in geographic space, the growth of some places (e.g., some cities or districts), and the decline of others? How do these locational strategies themselves shape the operation of local markets? More broadly, if we assume a local economy made up of workers, capitalists, and land owners whose incomes are wages, interest and profit, and land rents, respectively, how is the distribution of such incomes affected by the spatial configuration of economic activity.4 Equally important, how do changes in price locally change the translation of these incomes into levels of economic well-being? The wrenching local changes (e.g., economic, social, cultural, political, and environmental) that locational behavior appears to inflict on particular places (e.g., cities or districts) are the stuff of public policy and practice. Explaining locational behavior is the stuff of the social sciences in general and Economics in particular. Economists typically argue that localization is driven by the efficiencies (more correctly, the enhanced profit) made possible through agglomeration of production 3 Aji (1995)—studying intra-urban variations in productivity of manufacturing and business service sectors in Los Angeles—finds that the influence of skilled labor access on firm productivity is stronger than the influence of access to other firms in the metropolitan area. 4 The partitioning of factors into land, labor, and capital is widespread in both Economics and Geography, For an alternative perspective, see Curry (1985b). For an example of a model of distribution that ignores location, see Whitaker (1982).
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Geographic and Other Perspectives on Localization
3
and tempered by costs of transportation and congestion.5 As noted above, Smith saw the advantage of agglomeration in terms of division of labor. In a related fashion, others argue that localization is driven in general by economies of scale6 or by external economies. In trade models associated with Heckscher (1919) and Ohlin (1933), localization arises because firms specialize in commodities that make use of resources that are plentiful nearby: I refer to these as local cost advantage. As these efficiencies and costs wax and wane over time, so too would we expect the impetus to localization.7
1.2 Geographic and Other Perspectives on Localization When geographers see farms in a locale all producing the same crop, say celery, they often think the cause is an advantageous soil, hydrology, or climate, or perhaps circumstances that have produced a culture, institutions, and/or practices that result in celery production.8 When urban planners see suppliers to the interior redecorating industry localized in one area of a city, they may understand it as the outcome of amenities and services that attract those suppliers. When businesses plan the location of a new factory, warehouse, or store, they typically take prices and costs as 5A
consequence arising from an economic landscape whereby the firm finds it relatively more costly or less profitable to marginally increase the quantity of a commodity that it supplies. Congestion may arise, for example, because of (1) limitations in the firm’s ability to manage a larger output, (2) limitations at the factory or in the supply chain (supplier, warehouse, mode transfer station, or transportation network), or (3) a deterioration in the firm’s profit from the response of competitors. Usually, congestion is measured over the short run: i.e., before the firm has the opportunity to adjust its investment in land, plant, and equipment. 6 An attribute of a production function whereby the firm can produce more efficiently at a higher level of output. In the usual conceptualization, indivisibilities in production technology are thought to make possible a lower unit production cost possible when the firm achieves a particular scale of output. In some cases, a firm is thought to experience economies of scale through all relevant levels of output. In other cases, the firm is thought to have a most efficient scale of output (lowest possible unit production cost) above which the firm begins to experience diseconomies of scale. Economies and diseconomies of scale are generally measured over the long run: i.e., giving the firm sufficient time to adjust its investment in land, plant, and equipment. 7 Dividing the US into 9 geographic regions, Kim (1995) concludes that, after a decline between 1860 and 1890, localization rose. Localization stopped increasing during the interwar period and then fell substantially after the 1930s. US regions were less specialized by 1987 (the final year examined) than they were in 1860. Kim argues that long-term trends in localization are more consistent with production scale economies and Heckscher–Ohlin models than with explanations based on external economies. See also Belleflamme, Picard, and Thisse (2000), Bennett, Graham, and Bratton (1999), Cook, Pandit, Beaverstock, Taylor, and Pain (2007), Giarratani, Gruver, and Jackson (2007), Head, Ries, and Swenson (1995), Nachum and Keeble (2003), Perry and Hui (1998), Pinch, Henry, Jenkins, and Tallman (2003), and Rugman and Verbeke (2004). 8 Studies of this type include Atwood (1928), Lawrence (1934), and Hammond (1942) on cotton and textiles, Birkett (1930), Carlson and Gow (1936), and Alexandersson (1961) on iron and steel production, Breedlove (1932) on the ice industry, Landon (1935) on chewing gum, Brand (1937) on fertilizer production, and Barnes (1958) on milk production.
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1 The Craft of the Story Teller
given. When regional planners today strategize with respect to trade liberalization and globalization, they look to support localizations (clusters) of kindred businesses, often with a focus on the kind of jobs that require creative skills and pay well. When such scholars or policy analysts first become interested in phenomena like the four above, they might well start by collecting information. Take the case of the celery farmers. We might want to collect information that helps us answer questions like the following. Who are these farmers? What is their history? When, why, where, and how did they get into the business? How important has celery production been to them over the years? Why this locale? Why did particular farmers lead in the adoption of celery production? Why did others follow? What kinds of social, business, and professional ties link these farmers? Such information is helpful in developing or assessing explanations (ideas or theories) about the phenomenon. This kind of research, sometimes labeled Economic Sociology, is the stuff of what geographers and others have also sometimes called behavioral geography, locational analysis, or simply economic geography, among other labels. Here, a range of methods—from exploratory data analysis to process tracing—help identify explanations that appear consistent with the evidence collected. Along the way, of course, there always is the risk of arriving at an explanation that is ad hoc (idiosyncratic): i.e., consistent only with the evidence from the case study at hand—celery farming today at that locale—and not generalizable (e.g., to this locale at another time, to other locales, or to other kinds of agricultural production). Let me now critique such approaches. A central theme of this book is that we are looking at the same process in all these cases: be it celery farming, redecorating suppliers, factory location, or regional redevelopment. In each case, we look at the localization of businesses in geographic space. To me, it is only natural to ask whether and how these processes might in fact be the same. In so doing, we begin to elaborate a theory of location general enough to explain a range of phenomena.9 What is the similarity that binds the four phenomena outlined here? In my view, none of the descriptions above mentioned an important process in common: namely how the location of a firm or farm affects prices locally, including the prices of commodities sold, the wages paid to labor, the market rents paid by firms for their sites, or prices of other inputs. Even if we assume that each firm individually is in a competitive market—that is, the firm is a price-taker—as more firms join a geographic cluster, they do collectively affect prices locally. At the core of this book is the idea that the common process underlying all of these is the way in which localization shape prices which in turn impact on localization itself. The notion that prices in an economy are interdependent is widely attributed to Walras.10 The idea here is simple. Assume a demand curve for a consumer good in which the quantity demanded in the region depends upon its price, the prices of 9 In an early paper advocating quantitative geography, Huntingdon (1927, p. 289) makes a similar argument. In the language of Curry (1967, p. 265), I am a modeler: my focus is a pre-determined set of relationships existing in a portion of reality, not the portion of reality as a whole. 10 Marie-Ésprit Léon Walras (born 1834), a French (Swiss) economist, first published his Élements d’économie politique pure in 1874.
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Geographic and Other Perspectives on Localization
5
other goods (complements and substitutes), the income of the consumer, and tastes. Assume a supply curve for the same good in which quantity supplied depends on its price, the prices of factor inputs (e.g., wages, land rents, interest rates) and other inputs used to manufacture the good, and technology. Now, also envisage a similar demand and supply curve for every other good in the economy. Assume there are N goods in total (inclusive of factors) and that we have competitive equilibrium in every market in the economy. In that situation, there must be a set of N prices, one for each good, such that the excess demand (the amount by which quantity demanded at that price exceeds the quantity supplied) is simultaneously zero for all goods (markets) in the economy.11 Prices here are linked in two important respects: (1) goods are substitutes or complements for one another and (2) goods must compete for inputs in the same factor markets. Such an economy is then said to be in Walrasian equilibrium. Let me express this differently. In the context of a geography, the Walrasian assumption is that, in competitive equilibrium, all firms are price takers and the good sells for the same price everywhere. Location theorists say that such a good is ubiquitous . However, once we take into account shipping cost and commuting cost, it is possible that a good is non-ubiquitous: that is, priced differently between locations. As such, geography poses three problems for Walrasian equilibrium. • How do we handle the idea that in competitive equilibrium price may vary from one location to the next? One possibility here is to assume that the economy can be portioned into geographic regions: say 2 regions. Then, instead of solving for one price for each of the N goods, we solve for 2 prices: one for each region. There may be some amount of the good being shipped from one region to the other. Nonetheless, a gap in equilibrium price between the two regions implies that it is no longer worthwhile for a trader to purchase any more in the region where price is lower for resale in the other region. If there are more than 2 regions, we could imagine repeating this process for every pair of regions. Of course, we would also have to take into account situations where there might be only 1 geographic region (spanning the economy) for some commodities, 2 for others, and still more, say k, for other commodities. In this case, the problem posed by geography is that it means we generally need to find not just N prices but some multiple of N prices. • How do we take into account the idea that geography might create a local monopoly for each firm? Inherent in Walrasian equilibrium is the idea that the market for each good is competitive in the sense that each buyer and each seller is a price taker. However, given shipping cost and commuting cost, any one firm finds that it can affect the price paid by nearby customers when its competitors are located sufficiently far away. Rather than the traditional models of perfect competition on which Walrasian equilibrium is normally predicated, geography implies the role of imperfect competition across space. 11 Walras
cast this as an elegant mathematical problem: N implicit equations in N unknowns.
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1 The Craft of the Story Teller
• How does geography affect the way in which factors (inputs) such as land, labor, and capital get priced? Wage rates, net of the cost of commuting, can easily differ across the landscape. Differences in shipping and commuting costs by location also may have the implication that land rents differ with location. A geographic setting makes Walrasian equilibrium more problematic as a concept. My perspective here is Walrasian in origin. I want to know (1) how the location decision of a firm changes prices across markets, (2) how these changes then influence the locational behavior of other firms, and (3) who is made better off or worse off by these changes.12 However, I seek to do this in ways that recognize explicitly the roles played by geography. This book is about the development of a theory of location focused on prices, applicable to any competitive firm, and that therefore might be empirically tested as part of a Deductive approach (see Preface). Specifically, it is about the consequences of competition on the locations of firms generally in a geographic setting in a market economy. From an economic perspective, a decision to locate or relocate is inherently a decision to invest. In so doing, the firm is thought to weigh the cost of an investment (the initial outlay) against return (the present value of the stream of future benefits net of recurring costs) taking into account a planning horizon, riskiness and risk aversion, opportunity cost of capital, and scrap value. Firms sense that some places might be more profitable or less risky for their factory, warehouse, or store than are others. To them, relocation is also an investment; its cost is the expense of relocation, and the future annual benefit is the net increase in profit or reduction in risk associated with the new location. Where such an investment is attractive, we can expect firms to compete for best locations. The use of geographic location as part of a firm’s competitive strategy, for example, has been extensively evidenced and described elsewhere.13 This book is part of an area of scholarship popularly known as location theory. In fact, location theory broadly consists of two overlapping subareas: optimal location theory and competitive location theory. Optimal location theory deals with how firms, industries, or societies might be positioned in geographic space: e.g., to be most efficient, most profitable or to make society best off.14 Optimal location theory is not the focus of this book. Instead, this book is concerned with competitive location theory, which looks at how competition among firms leads to geographic
12 Walras
(1954 [originally 1900], p. 47) was unequivocal, indeed stirring, on the importance of mathematics to the study of Economics. 13 See, for example, Ghosh and McLafferty (1987) and Netz and Taylor (2002). 14 See, for example, Bigman and ReVelle (1978, 1979), De Felice (1972), de Smith (1981), Eiselt and Sandblom (2004), Hamacher and Nickel (1998), Hurter and Martinich (1989), Mirchandani and Francis (1990), Teitz (1968a), and Thomas (2002).
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Geographic and Other Perspectives on Localization
7
patterns in markets in equilibrium (otherwise known as spatial equilibrium15 ).16 The two areas of location theory have common roots. This will be evident in several models in this book; nonetheless, the focus here is on how such models help us think about the linkages between localization and prices. Competitive location theory has applications in the study of business, urban planning, economic geography, regional science, and urban studies. Businesses need to understand how competition will drive them and their competitors into particular spatial patterns even where the patterns are not necessarily the most profitable, or otherwise best, for an individual firm. Urban planners similarly need to anticipate how markets might react to a particular plan or policy initiative, how any why growth might be linked to inequality, and issues of environmental degradation and sustainability. Economic geographers and regional scientists need to understand how and why the geography of markets might change as a result of trade liberalization or other initiative. Urban studies scholars are interested in how and why an urban economy (i.e., the production and allocation of commodities and services, and distribution of income) changes. This book has been written primarily with these audiences in mind. Economists, for their part, are interested in what location theory might tell them about the core of economic theory. Although not an economist myself, I hope this reinterpretation of location theory also provides insights that help economists to advance their own thinking.17 This is a book about the symbiosis of Geography and Economics.18 Via competitive location theory, readers in Geography get to understand and apply economic perspectives on issues of concern to their discipline. For their part, readers in Economics get a geographical perspective that—perhaps surprisingly—helps clarify issues and questions central to their discipline.19 Let me give two instances here. First, when economists draw a supply curve, they imagine suppliers arrayed from least costly to most costly. Why does every unit not cost the same? There are several possible explanations here in which geography or spatial arrangement may play an important role. For one, as the quantity to be supplied increases, the market must rely on suppliers who are further away and incur added shipping cost. For another, as
15 An
equilibrium in which market participants at different locations have no incentive to change the way they participate in a market: e.g., their location, type of good produced, price, or quantity produced. 16 I include an extensive glossary at the end of the book. The first time that a concept or name is introduced in the book, I present it in italics and explain its significance there. That text is then repeated in the glossary entry. The first time that the concept or name reappears in a succeeding chapter, I also present it in bold italics; in such cases, the reader should refer to the Glossary for further details. 17 To me, an important indicator of the failure to integrate location theory into Economics is its sparse application to date in the theory of international trade. Well-regarded texts like Dunning (1993) contain only tangential references to location theory. 18 For other examples, see Dicken and Lloyd (1990), Krugman (1995), Ohlin, Hesselborn, and Wijkman. (1977), Ottaviano, Tabuchi, and Thisse (2002), and Scott (1998). 19 See Garrison (1959a, p. 238).
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quantity is increased, individual firms encounter congestion (including capacity limitations) at the production site or in the supply chain. Put differently, they encounter problems at the geographic micro-scale (factory, warehouse, mode transfer station, or transportation network) that make it impossible to further increase the quantity supplied at the same unit cost. Incorporating location theory helps economists think about the meaning of basic concepts like a supply curve. My second instance is the way that geography—by introducing “lumpiness”—helps economists rethink the marginal analysis that is at the heart of microeconomic theory.20 In exploring the linkages between localization and prices, my aim in this book is to reinterpret competitive location theory in a way that (1) makes it relevant and accessible to a broader audience of readers and (2) focuses attention on central problems and questions in this area of scholarship to a broader audience. Economists have done much work in this area. While competitive location theory is more familiar overall to economists, location theorists are, even there, sometimes viewed as a fringe. The same is true of geographers working on this subject. A focus of this book is to show how the linkages between localization and prices can be made more central to both disciplines.
1.3 Marshall’s Perspective as a Starting Point More than a century ago, Alfred Marshall21 crafted an insightful prologue to thought on this question.22 Looking back historically, he emphasized the rationale and limitations to localization: (1) heavy wares and the relative cost of shipping; (2) changing wants and communication between customer and producer; (3) the costs and risks of trade; and (4) the creation of wants in the promotion of trade. Marshall emphasized broadly the role of the market in organizing and informing both production and consumption activity. From his vantage point at the start of the twentieth century, Marshall then asked why production of some commodities comes to be localized at just a few select places. He began with an obvious short list: (1) local physical conditions, wherein he includes climate, soil, mineral deposits, and access to waterborne transportation, (2) variations in local demand,23 and (3) economies of
20 A
similar point is made in Samuelson (1952: p. 283). Marshall (born 1842), an English economist (and a founder of the neoclassical school in which economists study wealth and human behavior to understand why we make the choices we do), published the first edition of his Principles of Economics in 1890. Revised extensively over the years in a succession of eight editions, Principles of Economics postulated important arguments about implications of geography for the nature and pattern of economic activity. 22 See Marshall (1907, 267–268). 23 Marshall (1907, pp. 268–269). For a contemporary view of the effect of local demand on industry location, see Justman (1994). 21 Alfred
1.3
Marshall’s Perspective as a Starting Point
9
scale.24 To this, he added the role played by skilled labor nearby. What advantages arise because workers in a skilled trade are localized? Marshall saw two mechanisms at work here: (1) children learn the trade from their parents; (2) workers implement innovations that they hear about from friends and neighbors. Marshall also saw advantages to the firm. A local market25 for skill means that employers have a choice of workers with the skill they require nearby, and workers find plentiful employers.26 Nowadays, we tend to label all such advantages localization economies.27 Included under localization economies as well are the advantages to customers from being able to efficiently do comparison shopping when suppliers are localized.28 Marshall also discussed what we now call urbanization economies.29 For some reason, having a diversity of firms in other businesses nearby makes your firm more efficient.30 Marshall mentioned (1) the development of subsidiary trades locally that improve efficiency in the supply of materials and services, (2) the insurance principle31 in the local labor and consumer markets, (3) joint demand by firms
24 Marshall (1907, pp. 278–279) saw three kinds of economies of scale: economy of skill (division
of labor enabled by a larger scale of production), economy of machinery (indivisibilities enabled by a larger scale of production), and economy of materials (less wastage in a larger operation). 25 A local market is a set of agents (suppliers and demanders) engaged in the sale and purchase of a good wherein market price—not a single price set in a global market—varies from one local market to the next because of some impediment (characterized by a unit shipping cost between markets). Where a mechanism links participants so that market price varies systematically from local market to local market (i.e., local markets are not in autarky), local markets can be termed submarkets. A domestic market or home market each instance a local market. The counterargument to Marshall’s labor pooling is the notion of labor poaching: see Combes and Duranton (2006). 26 Strangely, Marshall did not observe the flip side to this argument. If a firm has an advantage in technology or process, would it not want to keep knowledge of the advantage away from its competitors and their employees? 27 Localization economies are reductions in unit production cost that arise when several firms in the same industry locate in close proximity. For some reason, having other firms in the same business in close proximity allows your firm to be more efficient. Glaeser, Kallal, Scheinkman, and Shleifer (1992) refer to these as Marshall-Arrow-Romer externalities to take into account the contributions of Arrow (1962) and Romer (1986). 28 Keir (1919, p. 48) concurs with Marshall overall but cites some disadvantages of localization: distance from markets for materials and outputs, the localization of suffering in hard times, the strength of labor unions, and the creation of a labor class. 29 Urbanization economies are reductions in unit production cost made possible when firms in different industries locate in close proximity. Glaeser, Kallal, Scheinkman, and Shleifer (1992) refers to these as Jacobs’ economies after Jacobs (1969, 1984). 30 Javorcik (2004)—looking at foreign direct investment in Lithuania in the form of joint ventures—found urbanization economies in the form of productivity spillovers from the foreign affiliates to local suppliers. 31 With insurance, a consumer or firm incurs a small upfront cost (the premium) now to protect themselves against the possibility of a substantial, though unlikely, loss during some future period. What enables a market in insurance is the prospect of a profit by the insurer; that total payouts to the insured plus other costs of business do not exceed the revenue earned from premiums paid by consumers willingly insured. This is the insurance principle. In general, it requires that the risk of loss for each customer be small and that the occurrences of loss be statistically independent.
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for common services (e.g., transportation, telegraph, and printing press), and (4) the benefit from having a variety of jobs (e.g., for men and women)32 to better address the complementary work needs of households. At the same time, Marshall noted the offsetting tendency of land rents to rise the more industry localizes at one place. At the same time, he was careful to avoid oversimplification. For one, Marshall (1907, pp. 273–274) saw the tradeoffs here between inputs and outputs: a lessening of shipping cost or tariffs tends to increase the range and market area, of a given factory but on the other hand also increases people’s readiness to migrate from one place to another and thereby bring production closer to places of consumption. For another, he saw the importance of technology and its transmission across geographic space but also pointed out (p. 270) that technology is easily spread where favored by the character, cultures, and ideals of the people, and by religious, social, economic, and political institutions. Finally, he explicitly recognized the importance of great political events and strong personalities historically on the development of regional economies over time. Over the last century, economists have added to Marshall’s ideas. Keir (1919, p. 48) discussed the branching of new plants from old, the subdivision of production, the acquisition of skills by labor locally, and the emergence of new factories that make use of production waste. Stigler (1951, p. 192) builds on the ideas that (1) localization allows individual firms to achieve the gains of specialization and (2) auxiliary and complementary industries that must be nearby to operate efficiently. In recent years, we have come to see the connections between the economics of agglomeration and networks. Feldman and Florida (1994, p. 210) adds that innovation is increasingly dependent on geographic networks of firms that relay knowledge and expertise, channel new scientific discoveries and applications, and give access to expertise in product marketing. Networks of economic actors—perhaps not even in close physical proximity—can create benefits that might otherwise be available only to actors clustered at the same place.33
1.4 The Development of Economics Since Marshall Much has happened in economic thought since the time of Marshall. Marchionatti (2004, p. 443) summarizes early critiques of Marshall: loose definitions and generic concepts; a failure to use mathematics; no general economic equilibrium analysis; and no use of systematic statistical analysis. There are a variety of views as to how economic thought evolved in the twentieth century in light of these criticisms. Marchionatti (2004, p. 454) focuses on the increased use of mathematics.34
32 Reflecting
his Victorian sensibilities, Marshall also includes here jobs for children. Johansson and Quigley (2004). 34 Others have argued that economists borrowed much from Physics in this process. See Mirowski (1984) and Turk (2006). In recent years, there has been a growing critique of the mathematical approach to economic thought. See, for example, Velupillai (2007). 33 See
1.4
The Development of Economics Since Marshall
11
Mindful of the significance of key works such as Arrow and Debreu (1954), Bowles and Gintis (2000, p. 1411) argues that major streams of economic thought in the twentieth century—behavior based on self-interest, exogenous preferences, and complete and costless contracting—built more closely on a Walrasian approach than a Marshallian approach. Baumol (2000, pp. 2–3) suggests two major advances in the discipline since Marshall: (1) new empirical tools and the insights they give and (2) the widespread use of theory and econometric analysis in application.35 Stiglitz (2000, p. 1441) focuses on the importance of innovations in how we think about information in modern economic thought.36 Other scholars today critique Marshall for not paying enough attention to geography in the dissemination of knowledge.37 Finally, economic thought has been significantly shaped since Marshall by the idea that it is valuable to break down production by firms (and possibly consumption by consumers too) into processes and distinct activities that can be analyzed separately and therefore packaged in different ways (i.e., differences in vertical integration) at the level of the firm or geographic location.38 At the same time, Bowles and Gintis (2000, p. 1411) argue that Marshall’s ideas continue to find application in new areas within Economics: e.g., endogenous growth theory, behavioral and experimental economics, and evolutionary game theory.39 From the perspective of location theory, also missing from Marshall’s analysis is an emphasis on the implication of distance—and the associated shipping cost—for the notion of competitive behavior in general and perfect competition in particular.40 Also missing from Marshall’s account was any anticipation as to how economic conditions would change over the course of the twentieth century: e.g., establishment of a national power grid, better transportation systems, improved functioning of the markets for capital as well as for raw materials, technological change and the reallocation of labor,41 and the second industrial revolution that characterized the twentieth century.42 These changes were seen to be conducive to the emergence of great clusters of factories that came to form manufacturing belts: e.g., the US 35 Baumol
had a third major advance—the formalization of macroeconomics—that I do not think is relevant to the subject matter of this book. 36 See also Storper (2000). 37 Keller (2002) presents interesting evidence, for example, that the geographical diffusion of technology has been imperfect. 38 See Coase (1937), Marschak (1959), Hurwicz (1973), and McLaren (2000). 39 Important contributions to economic thought have also begun to emerge in other social sciences. In recent decades, sociologists such as Harrison C. White have begun to think about markets and firms as social organizations. To me, reminiscent of Marshall, White (2002, p. 1) sees markets as networks of firms operating to mobilize production while reducing the risks associated with uncertainty. 40 Enke (1942) and Isard (1949) note this shortcoming in Marshall. Scotchmer and Thisse (1992) also emphasize the role played by distance in blurring the economist’s distinction between public commodities (commodities characterized by non-rivalrous consumption) and other (i.e., private) commodities. 41 See Slichter (1932) for an early discussion of issues related to job displacement. 42 For early perspectives on these changes, see White (1924) and Jevons (1931).
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Midwest.43 Keir (1921, p. 83) argued that firms became increasingly better able to overcome restrictions in materials, labor, power or fuel, or capital that otherwise would have limited them to specific locations. At the same time, Keir concludes that market and transportation considerations each continued to be important in the placement of factories. Building on Marshall and consistent with the directions of economic thought and changes over the twentieth century listed above, the subject matter of competitive location theory today is generally thought to include the following: market formation and areas; trade and spatial equilibrium; location of a firm (Weber-Launhardt problem); spatial pricing (Cournot-Enke-Samuelson problem); transportation pricing (Dupuit problem); locational competition (Hotelling problem); equilibrium for an industry (Lösch problem); locational interdependence (Koopmans-Beckmann problem); agricultural location (Thünen problem); urban spatial equilibrium (Alonso problem), and the existence and growth of cities.44 The breadth and variety of subjects here is indicative of the diverse ways in which prices and localization have been linked. Because the purpose of this book is to explore these linkages in a way that helps readers think more comprehensively and creatively about prices and localization, I have strived to cover a variety of the most promising of these topics. As a field of study, competitive location theory has a long history. It dates back at least to early work by Thünen first published in German in 1826.45 Early contributors include Cournot in 1838 Dupuit46 beginning about 1842, Launhardt47 in 1885, Marshall starting in 1890, Weber48 in 1909, Predohl49 in 1928, Palander50 in
43 See
De Geer (1927). from this list are topics less connected to the microeconomic foundations emphasized in this book: e.g., central places, social physics, and destination choice. 45 Chisholm (1979b) argues that important ideas often thought to have originated with Thünen appear to have been stated already by Adam Smith. 46 Arsène Jules Etienne Dupuit (born 1804), a French engineer and economist, published an important set of articles between 1842 and 1865. Several of these papers are republished in De Bernardi (1933). See also Ekelund and Hébert (1999) and Ekelund (2000). 47 Wilhelm Launhardt (born 1832), a German engineer and economist, was an early advocate of mathematics in Economics. His Mathematische Begründung der Volkswirtschaftslehre was published in 1885. An English translation of this book, Mathematical Foundations of Economics, was first published in 1993. His main economic contribution lies in founding location theory. See Backhaus (2000, 2002). 48 Alfred Weber (born 1868), a German economist, published Bei den Standort der Industrie in 1909. An English translation of this book, Theory of the Location of Industries, was first published in 1929. 49 Andreas Predohl, born 1893, published “The theory of location and its relation to general economics” in the Journal of Political Economy in 1928. 50 Tord Palander (born 1902), a Swedish economist, completed his PhD thesis, entitled Beiträge zur Standortstheorie (Contributions to Location Theory) at Stockholm University in 1935. 44 Omitted
1.4
The Development of Economics Since Marshall
13
1935, and Lösch51 in 1939.52 Among the early American scholars writing in the area are Fetter53 in 1924, Hotelling54 in 1929, Hoover55 in 1937, Hitchcock56 in 1941, Koopmans57 in 1947, Samuelson58 in 1952, and Alonso59 in 1964.60 Since then, economists have written extensively on competitive location theory.61 Geographers and others have also done important work in location theory.62 51 August
Lösch (born 1906), a German economist, worked on equilibrium in a spatial economy. His main contribution was Die Räumliche Ordnung der Wirtschaft , published in 1939. An English translation of this book, The Economics of Location, was first published in 1954. 52 See Krzyzanowski (1927) and Blaug (1979) on the German origins of location theory. 53 Frank Fetter (born 1863), an American economist, authored “The economic law of market areas” published in the Quarterly Journal of Economics in 1924. 54 Harold Hotelling (born 1895), an American economist and statistician, published a seminal article on locational competition entitled “Stability in Competition” in the Economic Journal in 1929. See also Samuelson (1960). 55 Edgar Malone Hoover (born 1907), an American economist, published Location Theory and the Shoe Leather Industries in 1937. 56 Frank Lauren Hitchcock (born 1875), an American mathematician, published “The distribution of a product from several sources to numerous localities” in Journal of Mathematics and Physics 20 (1941): pp. 224–230. 57 Tjalling Charles Koopmans (born 1910), a Dutch-born American economist and Nobel Laureate in 1975, published “Optimum utilization of the transportation system” in D.H. Leavens (ed.). The Econometric Society Meeting (Washington, D.C., September 6{18, 1947; Proceedings of the International Statistical Conferences, Volume V, 1948, pp. 136–146. 58 Paul Anthony Samuelson (born 1915), an American economist and Nobel Laureate (1970), published “Spatial price equilibrium and linear programming” in the American Economic Review in 1952. 59 William Alonso (born 1933), an American regional scientist, published Location and Land Use: Toward a General Theory of Land Rent in 1964. 60 For a historical review, see Krzyzanowski (1927) and Ponsard (1983). 61 Over the last six decades or so, important markers among writings by economists for me include Arnott (1986), Berliant and Dunz (1995, 2004), Berliant and ten Raa (1988, 1992), Beckmann (1968, 1999), Beckmann and Thisse (1987), Borts and Stein (1964), Bressler and King (1970), Capozza and Van Order (1978), Combes, Mayer, and Thisse (2008), Dunn (1954), Duranton and Puga (2001), Eaton and Lipsey (1977, 1997), Economides and Siow (1988), Fujita, Krugman, and Venables (1999), Gabszewicz and Thisse (1986a), Greenhut (1970, 1995), Greenhut and Norman (1992, 1995), Greenhut and Ohta (1975), Greenhut, Norman, and Hung (1987), Harris (1973), Harris and Nadji (1987), Henderson (1988), Herbert and Stevens (1960), Huriot and Thisse (2000), Isard (1954), Johansson, Karlsson, and Stough (2001), Koopmans and Beckmann (1957), Lee and Averous (1973), Markusen (1986, 2002), Mathur (1979, 1982, 1989), Meardon (2002), Mills and McDonald (1992), Moses (1958), Nickel and Puerto (2005), North (1955), Peneder (2001), Puu (2003), Rossi-Hansberg (2005), Sakashita (1968), Sasaki (1996), Siebert (1969), Smithies (1941), Stahl and Varaiya (1978), Steininger (2001), Stern (1972a), Takayama and Judge (1971), Takayama and Labys (1986), Weinschenck, Henrichsmeyer, and Aldinger (1969), Wheaton (1974), and Wilson (1987). Seminal related work thought to be more in the area of economics, but of importance to location theory, include Dixit and Stiglitz (1977), Myrdal (1957), Ohlin (1933), Ricardo (1821), and Scherer (1975). 62 Here, I include Alao (1974), Batty (1978), Boots (1980), Bunge (1966), Berry, Parr, Epstein, Ghosh, and Smith (1988), Curry (1964, 1972, 1976a, 1976b, 1978, 1984a, 1984b, 1985a, 1985b, 1985c, 1986, 1989), Daly and Webber (1973), Chisholm (1970, 1979a), Dicken and Lloyd (1990),
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Modern (neoclassical) microeconomics—with its focus on optimizing behavior of the firm and the household (hence marginal analysis)—is popularly thought to have originated with the works of William Stanley Jevons (writing in English), Carl Menger (in German), and Léon Walras (in French) in the last half of the nineteenth century. However, these three were preceded by three others, each working alone, whose contributions marked the way: Thünen63 (writing in German), Cournot64 (in French), and Dupuit (also in French).65 Interestingly, the latter three, as noted above, each have an important connection to location theory: Thünen wrote on agricultural location and production, Cournot on spatial price equilibrium, and Dupuit on the efficient provision of transportation. To me, this is evidence of the importance of geography as a concept right from the early days of modern economic thought. In my view, competitive location theory starts from the idea that geography has the potential to shape price or cost locally. It then asks how geography, technology, endowments, business strategy, and preferences shape industries, prices, and costs across the landscape. There are two overlapping streams in this argument. One stream draws a distinction between what is popularly called a home, domestic, or local market—the geographic place where the firm’s production facility is located—and a remote or foreign market in which it comes to sell its product in competition with local producers. Later in the book, I argue the need to carefully unpack our usage of the term market here. Nonetheless, the idea here is that some home markets—because of resources, size, institutions, networks, or technological advantages—nurture firms to become efficient producers that subsequently are
Epping (1982), Erickson (1989), Fik (1988), Fik and Mulligan (1991), Fotheringham (1979), Garrison (1959a, 1959b), Garrison and Marble (1957), Ghosh and McLafferty (1987), Golledge (1967, 1970, 1996), Golledge and Amedeo (1968), Griffith (1986), Hartshorne (1927), Huff and Jenks (1968), Jones (1984a, 1984b, 1988), Jones and Krummel (1987), Jones and O’Neill (1993, 1994), Kellerman (1989a, 1989b), Lentnek, Harwitz, and Narula (1988), Lentnek, MacPherson, and Phillips (1992), Lo (1990, 1991a, 1991b, 1992), McCann (1993, 1995), McCann and Shefer (2004), McCann and Sheppard (2003), Miller and Finco (1995) Miron (1975, 1976, 1978a, 1978b, 1982, 2002), Miron and Lo (1997), Miron and Skarke (1981), Mu (2004), Mulligan (1981), Mulligan and Reeves (1983), Nijkamp and Paelinck (1973), O’Kelly and Miller (1989), Papageorgiou (1973, 1976, 1978, 1979, 1980, 1990), Papageorgiou and Pines (1999), Parr (1993, 1995a, 1995b, 1997a, 1997b), Peet (1969), Penfold (2002), Pitts and Boardman (1998), Pred (1964, 1969a, 1969b), Rushton (1971a), Scott (1988, 1998), Sheppard and Curry (1982), Solomon and Pyrdol (1986), Stimson (1981), Teitz (1968a, 1968b), Thrall (1987), Wagner (1974), Webber (1984), Wendell and McKelvey (1981), Wilson (1967), and Zhang (2007). 63 Johann Heinrich von Thünen (born 1783), a German economist, published the original version of his Der Isolierte Staat in 1826. An English translation of this book, Isolated State: An English Edition of Der Isolierte Staat, was first published in 1966. See Blaug (1992). 64 Antoine Augustin Cournot (born 1801), a French economist and mathematician, published the original version of his Recherches sur les principes mathématiques de la théorie des richesses in 1838. 65 See Blaug (1992).
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15
better able to compete in foreign markets.66 The second stream draws a distinction between price (the net amount received by the supplier selling the commodity) and effective price (the amount paid by the purchaser). An effective price includes price plus any transaction costs paid by the purchaser related to search and information gathering, negotiation, and acquisition (including freight and transfer, storage and inventory, agency and brokerage fees, credit, cost of insurance67 and other loss risks,68 installation and removal, warranty and service, and taxes and tariffs), inclusive of normal profit (the profit attributable to an unpriced factor of production such as entrepreneurial skill or owner equity), with respect to the commodity.69 The supplier firm is also seen to modify effective price through its choice of pricing strategies (e.g., f.o.b. price,70 basing point pricing, uniform pricing, and discriminatory pricing).71 The net amount received by the supplier, as described above, is net of any such cost alternatively borne by the supplier. For ease of exposition, I refer to the transaction costs collectively as unit shipping cost throughout this book.72 Unit shipping cost can make a firm less able to compete when it is badly located. Put differently, to compete from a distance, there must be some advantage to the firm at least sufficient to offset unit shipping cost. Ideas about the distinction between price and effective price are applicable to commodities or services sold by the firm as an output, or purchased by the firm as an input. Specifically, they apply to the labor market where the firm obtains the workers needed to produce its output; here, we can substitute wage for price and commuting for shipping. The kinds of
66 Davis
(1998), Davis and Weinstein (1999, 2003), Head and Ries (2001), and Hanson and Xiang (2004) discuss and present evidence on the home market effect. 67 For an early analysis of insurance in shipping, see Bernoulli (1954, pp. 29–30) originally published in 1738. 68 For simplicity of exposition, I imagine here that there are competitive markets in which vendors and purchasers purchase protection against some risks (i.e., insurable risks), but other risks (i.e., uninsurable risks) must be borne by the vendor or purchaser as appropriate. 69 Cournot in 1838 argued similarly for a broad view of transaction cost in this regard. See Cournot (1960, p. 117). Launhardt—in mentioning costs for packing, freight, storage, taxes and tariffs, interest losses, insurances, provisions for agents, advertising, and profits of the merchants— employs a notion similar to a transaction cost. McCann and Sheppard (2003) also puts an emphasis on the role of transaction costs in rethinking location theory. Caves (2007, p. 2) uses a transaction cost approach in his analysis of multinational enterprise. 70 Abbreviation for “free on board.” Firm sets price at factory, warehouse, or store; customer pays freight from that place–also known as mill pricing. According to Marshall (1907, p. 325), the label “f.o.b.” arises from the practice of merchants to quote a price for their commodity on board a vessel in port, each purchaser then incurring any shipping cost from there. 71 For further discussion of spatial pricing policies, see Anderson and Ginsberg (1999), Capozza and Attaran (1976), Chisholm (1970, chap. 7), DeCanio (1984), Deutsch (1965), Furlong and Slotsve (1983), Gilligan (1992), Greenhut and Ohta (1975), Haddock (1982), Hughes and Barbezat (1996), Levy and Reitzes (1993), Mulligan (1982), Needham (1964), Ohta, Lin, and Naito (2005), Soper, Norman, Greenhut, and Benson (1991), and Thisse and Vives (1992). 72 Among others, Louveaux, Thisse, and Beguin (1982) discuss the role of transportation costs in location theory.
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transaction costs involved in the labor market may be different from those in an output market (e.g., the cost of delay in commuting may be more important than the cost of delay in shipping), but they are costs nonetheless. The two streams are connected; what differentiates the foreign market from the home market is generally a substantial unit shipping cost. This book explores location theory using the perspectives of these two streams of thought. What has motivated scholars to think about location theory? Two principal motivations come to mind. One group of scholars appears to have used geographic space as a way of evaluating the reasonableness of emerging economic theory.73 Thünen, Dupuit, and Cournot fit into this category and so too do scholars working in the new economic geography that began with Krugman (1995).74 Thünen, for example, wanted to understand whether arguments about the efficiency of the then-new English system of crop rotation were correct. He also wanted to develop a marginal productivity theory of income distribution to see if it could explain the relatively low wage paid to agricultural workers in Germany at his time. From the time of Walras, economists have been interested in the notion of market price equilibrium across markets75 : for example, if the price of bread were to rise, what would happen to the price of butter? Location theory adds a spatial dimension; what is the effect of a rise in the price of bread in city A on the price of bread (or, for that matter, the price of butter) in city B? Location theory also adds mechanisms for this price interaction: e.g., if butter and bread producers require production sites, they compete with each other for available land. Location theory also contributes to economic thought about the nature and organization of the firm. From the perspective of Coase (1937), production happens inside a firm when nonmarket allocation is more efficient than allocation through a market (price) mechanism. Usually, this happens when market allocation imposes a substantial transaction cost. Location theory provides a useful venue for thinking about the role of transaction costs in this regard. The second motivation has been to better understand, from a economic perspective, modern policy issues such as industrialization and postindustrialization,76 transportation investment and finance,77 the growth and decline of
73 In seminal works in Economics, Coase (1937, pp. 402–403) and Solow (1956, p. 65) each exem-
plify their ideas about Economics using a locational example. Spengler (1974, p. 532) suggests that the development of modern location theory was among the great achievements of Economics after 1945. 74 See also Baldwin, Forslid, Martin, Ottaviano, and Robert-Nicoud (2003), Fan, Treyz, and Treyz (2000), Fujita and Thisse (2009), Krugman (1998a, 1998b), Meardon (2002), and Neary (2001). 75 And the process (tâtonnement ) by which market equilibrium comes to exist. 76 Back to at least Clark (1887), there was concern whether industry would be competitive in the same way as the market for agricultural commodities. In the early days of industrialization, there was also concern over monopoly pricing by railroads: see Macdonell (1891). 77 An early source here is Cooley (1894).
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17
cities,78 economic unions,79 industrial reorganization and globalization, privacy and property rights, national security, and the environment. In the English language literature, much of this was focused on the scale and scope of the American experience: from trust formation and trust-busting at the start of the twentieth century to globalized regional production systems80 and environmental sustainability at the start of the twenty-first century.81 For his part, Marshall saw an economic landscape in the process of being transformed (localized anew) by an emerging industrial economy. Whatever the motivation, the development of location theory can be understood as a response by economists to newly emerging conditions at the time.82 Since Marshall’s time, some major economic events have had important impacts on the development of economic thought in general and location theory in particular. The growth of big business (a combination of oligopoly and vertical integration) and mass production in the twentieth century led to theories of monopolistic competition,83 including Lösch’s notion of spatial competition and free entry. The Great Depression and the Second World War led to modeling of the overall economy in input-output (activity) analysis84 and other linear models and sectorization and regionalization helped to simplify this. The postwar expansion in the Global North, 1945–1975 led to theories about regional development such as cumulative causation. The industrial restructuring (a combination of contracting, leaner management, and vertical disintegration) and globalization after about 1980 encouraged new thinking about the nature of networks and communications (including strategies for command and control). In each of these, there have been corresponding developments in location theory. This is nicely instanced in the use of mathematical programming85 (optimization) to find the best locations for economic activities. Early work86 includes Dantzig (1951b, originally 1947), Dorfman (1951), and Charnes, Cooper, and
78 To
illustrate the range of thinking by economists in this area over the years, see Hart (1890), Gras (1922), Haig (1926a, 1926b), Lampard (1955), Harris (1973), Quigley (1998), Anas, Arnott, and Small (1998), and Aranya (2008). 79 An early example here is Giersch (1949). 80 A system of global production wherein firms in industrialized nations do product design, distribution and marketing, and other aspects of good production and delivery but assembly to plants (branch or contractor) in emerging nations. See Hanson (1996). 81 An early source here is Jones (1905). 82 I take to heart here the critique by Massey (1973) that sees as impossible the idea of an autonomous location theory. In my view, it is inextricably linked to economic thought. At the same time, it is useful to scholars and practitioners in a wide variety of disciplines. 83 See Chamberlin (1962). 84 See Leontief (1951). 85 In its initial application—which was to scheduling problems—programming was an appropriate descriptor. Since then, the same label has been used to refer generally to optimization subject to constraints. 86 John von Neumann, the mathematician, was important here. He wrote extensively on the subject although nothing was published at the time. Dantzig (1991, p. 24) acknowledges his contributions.
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1 The Craft of the Story Teller
Henderson (1955) on linear programming (maximization of a linear objective function subject to inequality constraints),87 and Kuhn and Tucker (1951, originally 1950) on the optimization conditions that underlie nonlinear programming.88 The early history of mathematical programming includes important overlaps among mathematical programming, game theory, and fixed point (homotropy-based) analysis.89 Mathematical programming was introduced to me as a graduate student as a set of methods that could be applied in location theory. What I did not appreciate at the time was that mathematical programming also originated in part to address important questions in location theory.90
1.5 Why Is Competitive Location Theory Problematic? Unfortunately, competitive location theory has too often been analyzed and presented in ways that will seem complex or difficult—even esoteric or impenetrable— to many readers who might otherwise benefit from it. Why is that? What makes competitive location theory so problematic? My sense is that part of the problem is any discipline needs scholarship in terms of both appeal to a broader audience (breadth) and appeal to specialists in the field (depth), but that competitive location theory has focused on the latter.91 In part, the purpose of this book is to redress that. Why has it been difficult to present location theory to a broader audience? I think the problem here is the complexity that location theory brings to even the elementary
87 For
a brief introduction to linear programming, see Spivey (1962). Early applications pertinent to location theory include Garrison (1959b). 88 The Kuhn-Tucker conditions generalize the classical Lagrange method (optimization subject to equality constraints) to allow for inequalities in the constraint set. See also John (1948). An even-earlier unpublished analysis of the same problem is to be found in the MSc thesis (Minima of functions of several variables with inequalities as side conditions) of W. Karush at University of Chicago in 1939. In the 1950s, applications were mainly linear programs, and the main constraint was computing power. Later, the focus shifted to nonlinear programming and a variety of computational techniques emerged: e.g., recursive programming, separable programming, gradient methods, dynamic programming, integer programming, and quadratic programming. Bodington and Baker (1990) review the history of application in management science. Day (1961) initiated recursive programming. 89 The early history of mathematical programming is described in Hadley (1962, pp. 17–21, 1964, pp. 14–16), Dantzig (1963, chap. 2), Lenstra, Rinnooy Kan, and Schrijver (1991), and Murphy and Panchanadam (1997). Important work in game theory includes von Neumann and Morgenstern (1947), Marschak (1950), Hurwicz (1953), Simon (1955, 1959), Koo (1959), Bishop (1963), Harsanyi (1965, 1966), Loomes and Sugden (1982), and Sugden (1991). 90 On the relationship between location theory and mathematical programming, see Takayama and Judge (1971), Lahr and Miller (2001, pp. xxv–xxviii), and Fernandez Lopez (2002). 91 In my view, the books on competitive location theory with broad appeal include Beckmann (1968), Greenhut (1956, 1970), Isard (1956), Lefeber (1958), Richardson (1969), and Siebert (1969). Anthologies in the area include Thisse, Button, and Nijkamp (1996).
1.5
Why Is Competitive Location Theory Problematic?
19
concept of a market.92 Marshall (1907, pp. 112, 324–325) illustrates the conundrum here. Early on in his book, Marshall envisaged a market as a district—containing many traders (buyers and sellers) all keenly on the alert and well acquainted with one another’s affairs—wherein as a result the price of a commodity is effectively everywhere the same. He then undermines this definition with three caveats: (1) in practice, it is difficult for any trader to know exactly what price was paid in a transaction among others; (2) the geographical limits of a market are seldom clearly drawn; and (3) if the market is large, allowance must be made for the fact that each purchaser might pay a different price on account of the cost of delivery. Later in his book, Marshall muddies the water further in asserting that the central point of a market is a mechanism (e.g., a public auction or published price list) whereby traders are able to be in close communication with each other. What is not clear here is how traders are to be “keenly on the alert and well acquainted with one another’s affairs” when their knowledge of the market is limited to a published price list. How do I propose to conceptualize a market in this book? In a market in equilibrium where I (for the moment) ignore geography, there is neither an incentive for either another firm (supplier) to add to the quantity supplied, nor for a new customer (demander) to add to quantity demanded in that market. We can imagine a supply schedule that shows the amount of the commodity offered at any particular price and a demand schedule that shows the amount of the commodity demanded at any particular price. Market equilibrium is then associated with the price where demand equals supply. I take that to be the price at which the market clears.93 This is the stuff of a first undergraduate course in Economics. Now, let me introduce geography into this. To this point, I have left unstated the question of how a market is defined. As a rule, economists are loath to define a concept like a market: and understandably so. However, geographic distance is often thought to separate one market from the next: whether it is because of shipping, commuting, search or communication costs, or externality effects. The notion of a market itself might be endogenous94 to the notion of spatial equilibrium. This is an added complexity addressed in this book. At least three additional complications arise. First, to the extent a market has a geographic limit, a spatial equilibrium must incorporate the notion that there is no incentive for a potential supplier or demander either to move into this market or to leave it. Second, to the
92 A
market can be described as succinctly as a locus where buyers and sellers intersect. The purchase or sale of a good typically involves search activity on the part of both buyers and sellers— activity that is costly and time-consuming. The classical notion of a market is a combination of institutions and mechanisms operating at a site at which (or portal through which) offerings of a commodity are on view and sales are recorded. In this way, markets are seen to facilitate the efficient exchange of the commodity. 93 For a further discussion of market-clearing versus equilibrium price, see Fehr, Kirchsteiger, and Riedl (1993) and De Vroey (2007). The modeling of market clearing has been of interest in areas ranging from environmental pricing to automated exchanges: see Flaam and Godal (2008). 94 In a model, an endogenous value is an outcome; a value predicted by the model based on other (endogenous) values. Endogenous variables may have a stochastic component.
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extent firms can vary both what they produce and how they produce it, spatial equilibrium means there must not be any further incentive to change what is produced or how. Third, the existence of transportation costs inherent in a spatial economy makes necessary the assumption of something akin to increasing returns to scale; after all, in the presence of transportation costs, would not the geographic size of a market otherwise be very small?95 However, this need not (indeed, in this book, typically does not) take the form of assuming increasing returns to scale.96
1.6 Location Theory and Geography Notwithstanding a famous early example in Garrison and Marble (1957), the reductionist approach is less commonly applied by geographers. In Geography, reductionism has (perhaps always) been treated with suspicion. My sense is that the many non-reductionist geographers have the following four principal objections to reductionism. First, the skepticism of geographers arises because of a conflict between the notions of being a realist and making realistic assumptions.97 Economists and geographers differ in this regard. The realism practiced by economists focuses on the usefulness that arises from unambiguous logical deduction (i.e., the testing of hypotheses) even if that involves unrealistic assumptions. Maki (2004) expands on this idea in arguing that Thünen was a realist who deliberately used unrealistic assumptions to model agricultural land-use patterns. Those assumptions allowed Thünen to ignore the impact of other factors (e.g., differences in soil fertility, sun exposure, hydrology) on agricultural land use and instead focus on the causal contribution of one factor: distance from the market. To a reductionist, the validity of Thünen’s model does not require that its assumptions be true or that the representation of the resulting land-use pattern be accurate.98 Second, many geographers see knowledge building as arising from a fuller, more realistic description of the process under study. In part, this is because the discipline of Geography focuses on the concept of region (place and space) and the importance of understanding human–environment relations holistically. Barnes (2003) exemplifies this idea in arguing that the justification for location theory is limited to its basis in logical and mathematical reasoning. Barnes argues the importance of local knowledge, the idea that knowledge is shaped not by universal laws but by the particular
95 See
Eaton and Lipsey (1977) and Fujita and Thisse (2009).
96 Instead, many models in this book use constant returns to scale. I follow the approach of Nerlove
and Sadka (1991, p. 100) here. See, for example, McCann (1999). See the counter-argument in Alao (1974, p. 59) who decries the efforts spent on criticizing assumptions as opposed to amplifying, refining, and evaluating the conclusions or theorems so that we might better extend and improve our understanding. 98 Geography students in particular might benefit from the parallel argument in Ekelund and Hébert (1999, p. 6). 97
1.6
Location Theory and Geography
21
historical and geographical context of its production. This argument itself sounds— to people like me—strangely like a universal law. Nonetheless, it leads Barnes and others to argue in favor of a kind of nuanced analysis (including triangulation99 ) wherein social, political, economic, and cultural aspects are interwoven. Advocates point out that such an approach diminishes the problem of suspension of disbelief inherent in the reductionist approach. From a reductionist’s perspective however, it can be difficult to know whether any one hypothesis, or part of the overall story, is more correct than some alternative. The fear here is we wind up with no way of testing among competing hypotheses. Further, reductionists would argue that any attempt to write about (i.e., to encapsulate) the world around us, regardless of the degree of richness or nuance, of necessity requires a suspension of disbelief. A third objection is that, to some geographers, theirs is a discipline of difference.100 They see different things happening locally around the globe and seek local explanations for this diversity. In contrast, most economists see theirs as a discipline of generalization. To them (and to me), competition leads other firms to adopt (copy) the behavior of those who are successful, and this copying leads firms inexorably to behave similarly, not differently.101 Further, the market tends to drive out firms that do not conform to best practice. So important are these concepts of adaptation and adoption that models in Economics typically start by assuming that economic actors in a group are identical. To some outside Economics, the relentless assumption of otherwise-similar economic agents may make the discipline seem colorless, irrelevant, or evidence its status as the dismal science. To economists (and to me), the strength of this perspective is the unrelenting emphasis it puts on the idea of how economic actors behave under competition and the clarity this brings to the conclusions that follow. A fourth objection of some geographers is the undue emphasis put by location theory on the role of unit shipping costs. In the 1960s and 1970s, for example, it was commonly argued the marginal cost of transportation shrank because of heavy investment (both private and public) in transportation infrastructure.102 At the time, critics argued that the marginal cost of transportation would steadily drop in the future and that locational models based on transportation costs were therefore of little use. However, this argument misses several key points. First, unit shipping cost—envisaged in this book as the cost of arbitraging between two geographic markets—is broader in scope than just the cost of physically moving a commodity from one place to another. Second, while the marginal cost to users may well have declined, the overall cost of construction, operation, and repair of transportation
99
See, for example, England (1993), Tarrow (1995), and Downward and Mearman (2007). See, for example, Sauer (1924, p. 17). 101 See Alchian (1950). 102 In an early study in this area, Chisholm (1961) concludes that reductions in shipping cost began in the nineteenth century and that refrigerated freight also became of great importance in the twentieth century. A similar argument is made in Peet (1969, see Fig. 3 on p. 293). 100
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facilities has remained substantial.103 Further, the social and environmental costs of transportation facilities are large. That transportation services may have been one of the great mispriced commodities of the twentieth century104 does not take away from the costliness of trips and shipments to society. In recent decades, as municipalities, regions, and nations consider how to implement road pricing and other rationing schemes, the substantial marginal costs of transportation have become more apparent. Third, we need to keep in mind how public discourse is shaped by private interests. In Chapter 2, I present a model (2E) in which a monopolist, facing the possibility of entry by a competitor into a remote market, engages in uniform pricing across local markets despite incurring shipping costs. In Chapter 12, I present another model (12C) in which—given an inelastic demand for its product—a monopolist firm finds uniform pricing to be most profitable within a given geographic market area despite incurring shipping costs. In such situations—to avoid the connotations associated with entry deterrence or discriminatory pricing and give the appearance of being a good corporate citizen—the firm has an incentive to say that its uniform pricing is simply the result of a negligible marginal shipping cost.
1.7 My Approach As a field of study, competitive location theory is littered with work that is incorrect. I am referring here to errors of deduction, not the relativist’s problem where a new explanation renders previous explanations obsolete and hence incorrect. Although this is perhaps true of any field of study, I suspect competitive location theory may be a special case. In part, I think the problem is that even the simplest model in competitive location theory must, of necessity, be more complicated than other related models in Economics for the reasons listed above. In part, the problem is that, in their search for generalizations, scholars have tried to use complex mathematical forms that sometimes have unexpected consequences. In part, the problem also is that scholars have not always been able to cross-check their understanding of a model against numerical simulations. As with all intrepid authors in the area, I hope to have avoided such errors but, of course, readers may yet prove me wrong as well.
103 Supporting
evidence can be found in Bukenya and Labys (2005) who examine convergence in world commodity prices over much of the twentieth century for coffee, cotton, wheat, lead, copper, and tin. In addition to reductions in unit shipping cost, the authors argue that convergence has been aided by (1) improvements in communication and information technology, (2) centralization of commodity markets (e.g., the London Metal Exchange), (3) the rise of coordinating central bank activities in developing countries, (4) the globalization of enterprise, and (5) trade liberalization policies at the national level. Overall, their empirical results do not support the convergence hypothesis but rather a pattern of fluctuating divergences. They suggest the following might explain the lack of convergence: (1) political unrest, wars, and climate change, (2) international business cycle conditions, oil price shocks over the years, and (4) the persistence of substantial unit shipping cost. 104 See the opening comment in Vickrey (1963, p. 452).
1.7
My Approach
23
The purpose of this book is to present and interpret applications in the form of models that illustrate the linkages between localization and prices. I present models of general locational problems that are starting points for analysis. Much work remains to be done on this topic. I leave it to readers—indeed encourage them—to think further about how to extend or generalize from the models presented here. Let me now explain how and why my approach in this book is different from past work in location theory. First, I rely on relatively simple notions related to the operation and clearing of markets. A market is a locus of buyers and sellers that facilitates efficient exchange of the commodity. Inherent in that conceptualization is a process (like auctioning) wherein a price gets established such that demand equals supply (i.e., a price that clears the market). I treat that process as instantaneous and ignore issues related to the timing of consumption or production. The market process results in a single price in common to all suppliers and all demanders. Of course, if the market were not to clear, there would be potential vendors with product they would like to have sold at that price (i.e., inventory) and/or potential consumers with unmet demand at that price. Of course, it is possible to have markets that do not clear, or are slow to adjust, but this would add unduly to the complexity of our models here. Unless otherwise stated, I ignore uncertainty.105 I also ignore here the impacts of legislation that may affect the operation of markets: e.g., anti-trust and environmental regulation.106 Second, I rely mainly on simple versions of the models developed by others. On the spectrum of approaches from complex to simpler models, I think that this book is most valuable for its intended audience if it focuses on simpler models.107 I use simple demand functions where quantity demanded depends only on own price (and possibly income); to keep the models simple, I generally ignore direct cross price effects. As well, where others might use a generalized utility function108 or one that presumes only homothetic preferences, I use a log-linear utility function.109
105
On the treatment of uncertainty in location theory, see Alperovich and Katz (1983), Asami and Isard (1989), Cromley (1982), Dean and Carroll (1977), Hsu and Tan (1999), Mai (1984, 1987), Mai, Yeh, and Suwanakul (1993), and Mathur (1983, 1985). 106 For an example of the incorporation of antitrust regulation into location theory, see Bobst and Waananen (1968). Ulph and Valentini (1997) model the role of environmental policy. 107 See the parallel treatment of complexity based versus evidence-based models in Coelho and McClure (2005, 2008). 108 Following neoclassical theory, I use a utility function here simply to mean an ordered (i.e., ordinal) scoring of choices (bundles of commodities whose consumption is desirable)—that reflect consumer preferences—and that are transitive and evidence diminishing marginal utility: see Becker (1962). As a ranking, a given utility function transfor is said to be unique up to a monotonic mation. For example, the utility functions f x, y = xb y1−b , where 1 < b < 0 and g x, y = axb yc , where b > 0, c > 0, and b + c < 1, calculated at consumption of x units of good 1 and y units of good 2, generate the same rank ordering: i.e., g [x, y] is a monotonic transformation of f [x, y]. The two utility functions above exhibit diminishing marginal utility. I find Gorman (1976) useful in thinking about the use of utility functions. 109 A log-linear utility function in two commodities takes the form ln [U] = a ln q 1 + , where q is the quantity of good 1 and q is the (1 − a) ln q2 or, equivalently U = qa1 q1−a 1 2 2
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Something is lost here in terms of the generality of results. Economists are driven by a desire to generalize their conclusions. They do not want to be limited to results that are peculiar to the particular mathematical model employed. However, at the same time, something also is gained by use of a specific mathematical form: tractability and explicit solutions. With specific and simple mathematical forms, I can undertake numerical simulations using standard spreadsheet software. While this does not eliminate every error, it enables me to catch mistakes that might otherwise go unnoticed. I also avoid extensive use of game theory, largely because these typically rely on a specific conjectural variation110 and a rigidly structured decision-making process. I cannot entirely avoid game theory since there is much in location theory that is interesting to analyze in this way. However, for most of the readers I would hope to reach with this book, game-theoretic approaches are too abstract, mechanical, contrived or unwieldy. Similarly, I avoid the use of fixed-point (homotropy based) analysis for similar reasons. Fixed-point analysis is useful in game theory; it is also useful in studying Walrasian multimarket equilibrium. However, in this book, I do present models for thinking about multimarket equilibrium that do not necessitate fixed-point analysis. Third, my focus is on interpretation rather than on proof. I leave it to others, for example, to consider second-order conditions for optimization and the uniqueness of solutions (typically, equilibria) to the models. These are important considerations, but my sense is that they can quickly turn into insurmountable obstacles for the intended audience. My focus instead is on understanding what the models can and cannot tell us. I apologize here to more-advanced readers who correctly understand the importance of such considerations and are surprised by their absence here.
1.8 What This Book Is About The objective of this book is to reinterpret competitive location theory by focusing on the relationship between Walrasian price setting and the localization of firms. In the exposition of models in each chapter, I emphasize eight sets of connecting topics. • Localization, clustering, or geographic concentration. What exactly do we mean by localization? Different models presented in this book conceptualize localization, clustering, or geographic concentration in different ways. Is one of these the
quantity of good 2. This utility function has the desirable properties that marginal utility is positive for each commodity and that there is diminishing marginal utility. At the same time, a log-linear utility function has the special properties that each good is indispensable, that the proportion of the consumer budget spent on each good is fixed (a for good 1, 1 – a for good 2), and that the cross price elasticity of demand is zero (in other words, the demand for one good is independent of the price of the other good). 110 A conjectural variation is a player’s perception (assumption) about the reaction of another player to the first player’s choice of action. See Greenhut and Norman (1992).
1.8
•
•
•
•
•
•
What This Book Is About
25
correct or best way of thinking about localization? Alternatively, do differences in conceptualization across chapters suggest different dimensions to localization that make it useful to have a variety of models? Shipping cost, congestion, and risk. Although shipping cost, congestion, and risk sound at first like distinct concepts, they overlap in practice, in part because of their linkages to geography. Models in this book differ in conceptualization of shipping cost, congestion, and risk. So, when we argue, for example, that risk is important in one model but not another, what does this actually mean? Is there a way of thinking about these three concepts that enables us to better distinguish among the roles played by them? Organization of firm and market. In recent decades, it has become fashionable in Economics and elsewhere to focus on the organization of the firm and market. This can be done from perspectives that range from sociological to evolutionary. In this book, several location models give us insights into the organization of the firm and market. What are these insights? How do they build on the work of Coase and others on the economics of organization? Division of labor, indivisibilities, economies of scale, urbanization economies, agglomeration economies, and local cost advantage. Some firms are more efficient than others. The models in this book make use of concepts like division of labor, indivisibilities, economies of scale, localization economies, urbanization economies, and local cost advantage to account for this efficiency. Are these concepts in fact descriptions of the same thing? If not, when, how, and why are the differences among them important in the models in this book? Commodity prices and Walrasian equilibrium. Location theory introduces the idea that the same commodity is transacted in several distinct markets: one for each region (regional economy) and in which some markets are competitive and others monopolistic. One view is that the number of markets is therefore a multiple of the number of commodities. Another view is that each commodity constitutes one market wherein regionalization takes the form of a partitioning into submarkets. Whatever the view, what is the nature of Walrasian equilibrium in the presence of geography? Regional economy, social welfare, factor incomes, and autarky. Another way to think about location models is that each typically models the shipment of goods or services. In the absence of shipments (usually on the assumption of prohibitive unit shipping costs), we have a condition of “no trade” that economists call autarky. In autarky, we can imagine a society with particular levels of wellbeing (social welfare) and incomes. I begin most chapters with a model in which autarky is assumed, and then show how the introduction of endogenous shipments creates a region in which a local economy takes shape and affects factor incomes and social welfare. What does each of our models tell us about the impacts of shipments on the regional economy? Monopoly and space. In general, unit shipping cost implies that the effective price of a firm’s product need not necessarily be the same as its competitors for customers at any given location. The implication of this is that the firm is therefore
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1 The Craft of the Story Teller
not necessarily a price taker with respect to customers nearby even if other competitors elsewhere mean that it is a price taker for customers further away. The diverse links between geographical space and a monopoly that is at least local in extent make it difficult to know when or how to apply basic economic ideas about the consequences of competition. How, and to what extent, do the models in this book allow us to do this? • Geography. Location models represent geography in a variety of ways: e.g., as (1) a bounded or unbounded line or ribbon, (2) a ring, (3) a bounded or unbounded rectangular plane, or (4) a transportation network. In parts of the book, assuming the world is stretched out along a line or ribbon allows me to represent geography as one-dimensional (rather than the two-dimensional representation inherent in the rectangular plane). Generally, this readily yields a solution that is easily grasped. While it is true there may be particular geographies in the real world that can be thought to be one-dimensional, my main purpose in presenting such models is to then contrast them with the same problem in two-dimensional space. In the two-dimensional case, the solution is generally not so simple. Where I assume geography takes the form of a plane or line that is boundless, my purpose is to put aside complications that arise because of boundedness. Given an emphasis on the efficient firm, I therefore focus on the shortest distance between two places (two points on a map). In a spherical world, the calculation of shortest distance (great circle route) is a challenging mathematical exercise. For better or for worse, location theory has instead largely assumed that the world can be represented as a rectangular plane (i.e., as flat) and that Euclidean distance measures the shortest path. These models typically assume competitors, customers, and suppliers are uniformly spread out across this geography and, economic activities are seen to be either points on a map (punctiform) or to completely occupy space in simple ways (tessellations). How important is the specific geography chosen to the outcomes or conclusions of the model? What might have been the consequences of using other geographical representations? Why represent geography in the ways we do?
1.9 What This Book Is Not About This book is about competitive location theory. It is microeconomic in focus (i.e., about the behavior of individual firms) rather than macroeconomic.111 It is not centrally about what is otherwise known as locational analysis or behavioral geography. Locational analysis (also termed spatial analysis) is an area of Geography and other disciplines concerned with the statistical and mathematical analysis of spatial
111 It
therefore excludes a large literature on aggregate regional economic growth that begins with Borts and Stein (1964), Grether (1948), Nourse (1968), Richardson (1969), Siebert (1969), and Vining (1946).
1.9
What This Book Is Not About
27
outcomes.112 Locational analysis overlaps with this book but only to the extent that locational analysis makes use of competitive location theory.113 Behavioral geography is another area of study concerned with the ways in which human behavior might be studied in the context of a given physical and social setting.114 Competitive location theory provides one set of tools that might be used in locational behavior. Behavioral geography might be thought to include locational behavior The book presents, extends, and reinterprets 11 sets of models that have been developed in the literature: 44 models in all. I make no apologies for my selection of the 11; my purpose here is to provide an introduction and indicate the scope and potential of competitive location theory, not necessarily either to identify the 11 most important sets of models or exhaust the range of models.115 I have designed the book so that these models are all largely static.116 I constructed this book using the firm—consisting of one or more establishments—that produces a commodity or service for profit as starting point. I did this for simplicity of presentation. I could have used other starting points: e.g., the household, the consumer, or the local, regional, or national government. I find the perspective afforded by the firm is especially useful because it allows us to address a wide range of issues important to location theory. I do not look explicitly at issues that arise when firms are multinational enterprises.117 This includes issues related to corporate organization, foreign direct investment, and international production. This book does consider a firm with more than one factory but does not take into account the range of considerations—from currency hedging to trade restrictions—thought to be central to a multinational enterprise.118
112 Examples
include Curry (1998), Ghosh and Rushton (1987), Haggett, Cliff, and Frey (1977), Raina (1989), and Wolpert (1964, 1970). To illustrate the breadth of application of locational analysis across sociology, management science, ecology, and Geography, see Alba and Logan (1993), Francis, McGinnis, and White (1983), Minta (1992), and Scott and Angel (1987), respectively. 113 See, for example, Harris and Hopkins (1972), Pred (1969a, 1969b), and Reggiani (1998). 114 Examples in this area include Aitken and Bjorklund (1988), Clark (1992), Couclelis and Golledge (1983), Gale (1972), Gold (1980), Golledge (1970), Golledge and Stimson (1987), Gould (1963), Hart (1980), Harvey (1966), Krumme (1969), Mark, Freksa, Hirtle, Lloyd, and Tversky (1999), Pipkin (1977), Rushton (1971b), Scott (2000), Walmsley and Lewis (1984, 1993), and Wolpert (1964). See also the collection of papers in Cox and Gollledge (1969, 1981) and Golledge and Timmermans (1988). 115 Interestingly, 2 of the 12 sets of models included might not, I think, be regarded by the originating authors as centrally in the area of competitive location theory. Rather, they represent work with fascinating implications for the central theme of this book. 116 For readers interested in dynamic models, Duranton and Puga (2001) present a model of localization economies that emphasizes how product life cycle and process innovation interact with localization economies. 117 See Buckley and Casson (1991), Caves (2007), Dunning (1981), Dunning and Lundan (2008), and Markusen (1995). 118 Leading works in the field include Buckley and Casson (1991), Caves (2007), Dunning (1981), and Markusen (1995, 2002). See also McCann and Mudambi (1984).
28
1 The Craft of the Story Teller
Undergraduate students often say that they become interested initially in competitive location theory because they want to start a business and would like to know where best to locate. This question—interesting on its own—is not centrally the subject of this book.119 In this book, I present tools that might be used to look at such a question, but the principal focus is on the question of how competition drives firms into spatial equilibrium. Put differently, my interest is not to explain where someone should locate their business; it is to say something about the locations of firms once market equilibrium is established. In this book, unit shipping costs are treated as exogenous.120 However, they could also be viewed as endogenous. For both producers and consumers, an investment in transportation facilities that brings about a reduction in the marginal cost of transportation may be seen as mutually beneficial. This book ignores locational questions that arise because of congestion on transportation networks. In reality, network congestion can lead a firm to a location that it would not otherwise choose. In effect, network congestion makes the location of firms interdependent. This can be modeled, but the level of complexity is substantial. I therefore ignore it here and assume unit transportation cost is constant (a given) to generate models that are readily solved. I also ignore here the fact that commodities are typically shipped in a vessel (or vehicle). If shipments occur only from Place A to Place B, the shipper is then saddled with the cost of returning the empty vessel to Point A for its next shipment. In such cases, a firm in the business of providing shipping services might well find it profitable to cross-subsidize cross-hauling from Place B to Place A. Finally, this book is focused on equilibrium, not on dynamics. I am not trying to argue here that processes of change are not important; they are.121 However, in trying to make this book accessible to as wide an audience as possible, a focus on equilibrium models only is helpful.
119 Instead,
see Schmenner (1982).
120 In a model, an exogenous value is a given—a value used in the model to predict other (endoge-
nous) values. Usually, exogenous values include parameters, independent (or predictor) variables, and lagged or nearby values of endogenous variables. Usually, exogenous variables are not thought to be stochastic. For discussion of the role of transport demand in spatial equilibrium, see Smart (2008). 121 There is an extensive literature in locational dynamics. See, for example, Ishikawa and Toda (1995), Walker and Homma (1996), Phipps and Langlois (1997).
Chapter 2
The Firm at Home and Abroad Monopoly at Two Places (Greenhut–Manne Problem)
A monopolist sells a product at home and remote places. There is a cost to ship the product to the remote place or, alternatively, to build a second factory there. In Model 2A, a monopolist with one factory sells its product only at the home place. In Model 2B (localization), the firm also supplies the remote place from the same factory. In Model 2C (no localization), the firm builds a second factory at the remote place and serves customers at each place from their own factory. Model 2D considers a monopolist choosing whether to build one factory or two. Where the number of customers is sufficiently large—which in turn may require that the two places be sufficiently close together—there will be at least one factory and it will be located at the place with the larger demand. If the unit shipping cost is sufficiently low, there will be localization (i.e., one factory serves both places), and soap will be priced differently at the two places (partial freight absorption). Model 2E shows how one might think about entry deterrence at the remote place. If unit shipping cost is sufficiently high, and the smaller place has enough customers, the firm builds a second factory there and prices will be the same at the two places. In this chapter, localization and prices (one for each place) are joint outcomes of profit-maximizing behavior.
2.1 The Greenhut–Manne Problem This book looks at the localization of firms in geographic space. This chapter begins with a model of a firm that serves customers at up to two places and has to decide whether to serve (1) one or both places from one factory or (2) each place from its own factory. The models presented in this chapter are adapted from Markusen (2002, Chap. 2). There is an extensive literature in this area that goes back to Manne (1967) in operations research and to Greenhut (1956) in Economics; hence, I call it the Greenhut–Manne problem. In the first of these alternatives, all production is localized in one factory; in the second alternative, production is spread across two distinct places. Since this model is concerned with just one firm, it allows us to look at localization of production rather than localization of firms. Nonetheless, it is a useful starting point for thinking about why concentration of production arises. I start the book with this model because it builds on the stuff of a first course J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_2, C Springer Science+Business Media, LLC 2010
29
30
2
The Firm at Home and Abroad
in microeconomics so familiar to students whose only background is a first-year undergraduate course in Economics. To begin, I review a conventional non-spatial microeconomic model of a monopolist who maximizes profit.1 Is it strange to begin a book on competition and location with the case of a firm that is both non-spatial and a monopolist? After all, you might ask, “is not a monopolist uncompetitive by definition?” or “why non-spatial when we are interested in geography?” In part, the problem here is with the interpretation of monopolist in popular literature as opposed to Economics. Popularly, monopolist is taken to mean an industry with just one firm while, to an economist, it means simply that the firm can affect the price it receives or pays for a commodity. In this book, I use the economic interpretation and envisage the possibility of competitors. Even a firm that is alone in its industry faces competition to the extent a potential customer can forego the firm’s product in favor of substitutes. To me, such a firm is non-spatial2 ; it sells its product—which I hereinafter will call soap for ease of exposition—only at an adjacent (local) place3 (Place 1). For the firm, Place 1 is its home market. Throughout the book, I use “place” and “customer point” interchangeably. More narrowly, suppose Place 1 is an isolated market.4 I want to ignore shipping costs for the moment. Please be patient here. I begin with a non-spatial firm because I want to use it—later in this chapter—to contrast with a firm in a spatial setting otherwise similar. A final note is in order as we get underway. In Chapter 1, I distinguish between competitive location theory (the subject of this book) and optimal location theory (not the subject of this book). However, cannot this chapter also be construed as a set of models in optimal location theory? After all, we are looking at a firm that maximizes profit. I agree, but my focus is different. What I do in this book is to use such models to address questions related to the consequences of competition.
2.2 Model 2A: Non-spatial Monopolist I begin with the notion of a firm which I take to be a person or group of persons engaged in the production of soap for the purpose of earning a profit. For the moment, I don’t distinguish between a firm and the narrower concept of an establishment; that is, a branch of the firm that carries on business at a particular site. I don’t address here how the firm is organized (e.g., in terms of research, development, production, distribution, finance, and marketing) or about vertical integration and outsourcing because what the firm does and how it does it may
1 The amount by which a firm’s revenue for a period exceeds its costs inclusive of a normal return on any unpriced factors such as owner equity or management skill. Also known as excess profit. 2 A feature of a model wherein geography plays no role. Typically, shipping cost and/or commuting costs are assumed zero or are otherwise ignored. 3 A market sufficiently small in area that we can ignore shipping costs on shipments of the good within that market. 4 For this good, local producers and demanders transact in this and only this market. No one else transacts product there: e.g., no one purchases there for the purpose of reselling elsewhere or resells there a good purchased elsewhere.
2.2
Model 2A: Non-spatial Monopolist
31
depend on where it locates its chosen activities. Put another way, in the perspective of Coase (1937), any firm can be thought to use a combination of market prices for outputs and inputs together with some hierarchical (command) allocation and the balancing of these, in the context of location, is at the heart of this book. However, I ignore such balancing at this early stage of the book. Assume the firm operates in a (paper) fiat money economy5 and plans to build a factory at Place 1 for which K dollars of capital are needed.6 Assume this factory will last forever with regular annual maintenance. Suppose the annual opportunity cost of capital7 is a yield rate r. In this case, the opportunity cost of the firm’s investment is the profit that normally could otherwise be earned on this capital: rK. I use a period of 1 year here, but the analysis would be similar if I used say a month, a week, or a day.8 Note that rK (the opportunity cost of capital) is a fixed cost9 ; the firm incurs it regardless of the level of production. I assume here that the other possible investments by the firm all have a comparable level of risk—an assumption made to avoid the complexities of comparing investments with different yields and risks. For simplicity, assume the firm has no other fixed costs. As well, assume the firm has no choice as to the size of factory. The model does not permit the firm to build either a smaller factory (presumably at less cost) or a larger factory (presumably at higher cost). In this sense, I can imagine an indivisibility10 in scale of factory. In making its investment decision, I imagine the firm in two distinct scenarios. In the capital-constrained scenario, the firm has a fixed amount of capital available, say K, and chooses where or how to invest it. In the unconstrained scenario, the firm can obtain as much capital as it wants at a given opportunity cost of r annually per dollar invested. These scenarios are similar in that a profit-maximizing firm
5 Ritter (1995, p. 134–135) describes the emergence of paper (fiat) money economies in the twenti-
eth century and its relationship to seigniorage (the profit earned by a government on the difference between the nominal value of a coin and the cost of minting it). I assume here that the paper money supply is maintained by an authority (central bank) to ensure that the currency is a good store of value (or, equivalently, that the level of inflation is low). 6 Economists might prefer to measure capital physically (e.g., number of machines) rather than in dollars. Capital—measured in dollars—is a given physical amount of capital times the price per unit of capital. In turn, the price of a unit of capital can be thought to be determined in a market for plant and equipment where suppliers of capital commodities interact with demanders including our firm. A problem with using dollars, as done here, is that a change the price of a capital asset will have no direct effect on the amount of output that can be produced, say daily, from a given physical amount of capital. However, such productivity concerns are not relevant in this model since, as I comment below, the model assumes an unlimited wellspring of production once the factory is built. 7 The return on the best alternative investment opportunity available to the firm at a similar level of risk. 8 A caveat is in order here. Later in the chapter, I argue that the market clears. Depending on the nature of the production technology, that might be difficult to ensure if the period were very short, say one second to the next. 9 A cost incurred by the firm for a period of operation that does not depend on the quantity of output produced. 10 An attribute of a production process such that production cannot be replicated at a smaller scale with the same efficiency.
32
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The Firm at Home and Abroad
would seek to use capital efficiently. However, looking at the two scenarios gives us complementary perspectives on the behavior of the firm. Assume the marginal cost is a constant C dollars per unit of soap produced. There is no congestion here. The marginal cost curve11 —the schedule of unit cost arrayed by quantity produced—is a horizontal line: see DF in Fig. 2.1. This includes the cost of manufacturing (e.g., commodity inputs, labor, and energy), the regular maintenance required to offset the wear and tear of production, advertising and promotion, and normal profit.12 The notion of a constant marginal cost is easiest to envision if I assume, over the relevant range of output considered, the firm has a Leontief technology13 (and therefore a linear expansion path),14 purchases its inputs in competitive markets (so the purchase price of each unit of an input is the same), regardless of the number of units purchased, and experiences no constraints on capacity,15 no congestion in its factory (that causes unit cost to rise as the scale of output is increased), and no economies of scale. Put differently, having invested K to construct the factory, and thereby incurring a fixed annual opportunity cost rK, the firm finds itself with an unlimited wellspring of production16 at a constant marginal cost of production17 (C). Assume here the firm is a price taker18 in markets for its inputs. As far as the firm is concerned, r and C are each simply a fixed expenditure required (per unit of K and Q, respectively) that reduce profit. At the same time, r and C reflect the prices of inputs. In purchasing them, the firm competes in markets for inputs along with
11 A
schedule showing the marginal (additional) cost incurred by the firm (usually over the short run wherein capital invested is held constant) as a function of the quantity to be supplied. 12 The profit attributable to an unpriced factor of production such as entrepreneurial skill or owner equity. 13 A production technology characterized by the fact that each unit of output produced requires exactly the same amount of an input regardless of the relative prices of inputs. In other words, there is no substitutability among inputs in production. Put differently, a doubling of output requires a doubling of each and every input used in production. It is named after Wassily Leontief, an American Economist and Nobel Laureate (1973), whose main area of research was input–output analysis. 14 As used here, a condition of a production function wherein, if as firm expenditure (output) is increased by a fixed proportion holding prices of inputs constant, the efficient firm purchases the same proportion more of each input. 15 The amount of output that can be produced by a given factory over a stated period of time. In a simple case, we imagine that the unit variable cost of production is fixed per unit for any quantity up to capacity (marginal cost of production is a horizontal line up to capacity). To allow for the concept that no quantity greater than capacity is possible, we typically assume the marginal cost curve then becomes vertical. In other words, no matter how much the firm might want to further increase quantity, it cannot produce a quantity in excess of capacity. 16 In effect, rK is akin to a lump-sum licensing fee paid annually by the firm that gives it authority to produce as much of the commodity as it wants during that period. 17 The increment to the firm’s total cost incurred by the last unit produced. 18 A condition under which a market participant (supplier or demander) is unable to affect the price they receive or pay for a unit of the product by varying the quantity that they supply or demand. The supplier (demander) sees the demand (supply) for its product as horizontal: i.e., infinitely elastic at the given market price.
2.2
Model 2A: Non-spatial Monopolist
33 Model 2A: Monopoly price at one place
G
Price
A
K
L
AB Aggregate inverse demand curve: see (2.1.4) AC Marginal revenue curve: see (2.1.5) AN Elastic segment of the aggregate inverse demand curve DF Marginal cost curve DMHE Fixed cost (rK) GHI Average cost curve: see (2.1.2) LAKL Consumer surplus: see (2.1.12) MLKH Excess profit due to monopoly: see (2.1.8) NB Inelastic segment of the aggregate inverse demand curve OA Intercept of inverse demand curve (α) OAKJO Consumer benefit: see (2.1.10) OD Marginal cost (C ) OJ Profit-maximizing output (Q1): see (2.1.7) OL Profit-maximizing price (P1): see (2.1.6) OM Average cost OMHJO Producer cost including fixed cost: see (2.1.11)
N
P
H
M
0
I
E
D
J
F
C
Quantity
B
Fig. 2.1 Model 2A: monopolist located and selling at Place 1 only. Note: α = 15; β = 1; C = 3; K = 50,000; N1 = 200; r = 0.05. To maximize profit, firm sets P1 = 9 and Q1 = 1200. Horizontal axis scaled from 0 to 4,000; vertical from 0 to 16
others demanding those inputs. To the extent it affects the price of r or the prices of other inputs making up C, such competition may affect the location of our firm. Suppose the firm plans to produce quantity Q1 annually with this factory. Since I assume that markets clear throughout this book, I use Q1 to refer interchangeably to the quantity produced by the firm and the quantity demanded by customers. Therefore, inventory is zero (ignored). The firm’s total cost19 and average cost,20 both inclusive of the opportunity cost of capital, are now given by (2.1.1) and (2.1.2). See Table 2.1, wherein I summarize the equations, assumptions, notation, and rationale for localization in Model 2A. Average cost is the sum of marginal cost (assumed constant above) and average fixed cost (which drops the greater the quantity of soap over which to spread the total fixed cost, here only rK). This firm has a declining average cost; the greater the output, the lower the average cost of units produced. See the average cost curve and marginal cost curve: labeled GHI and DF, respectively, in Fig. 2.1.
19 For a firm, the sum of variable and fixed costs of production inclusive of any unpriced resources
such as entrepreneurial talent. a firm, total cost divided by the amount produced.
20 For
34
2
The Firm at Home and Abroad
Table 2.1 Model 2A: monopolist located and selling at local Place 1 only Total cost rK + CQ1
(2.1.1)
Average cost rK/Q1 + C
(2.1.2)
Individual inverse demand curve P = α − βq
(2.1.3)
Aggregate inverse demand curve P1 = α − βQ1 /N1
(2.1.4)
Marginal revenue α − 2βQ1 /N1
(2.1.5)
Profit-maximizing price, assuming α > C P1 = 0.5(α + C)
(2.1.6)
Profit-maximizing quantity Q1 = 0.5N1 (α − C)/β
(2.1.7)
Monopoly excess profit (MP) 0.25(α − C)2 N1 /β − rK
(2.1.8)
Minimum number of customers required N1∗ ≥ 4βrK/(α − C)2
(2.1.9)
Consumer benefit (CB) 0.25(1.5α + 0.5C)(α − C)N1 /β
(2.1.10)
Producer cost including fixed costs (PC) rK + 0.5(α − C)CN1 /β
(2.1.11)
Consumer surplus (CS) 0.125(α − C)2 N1 /β
(2.1.12)
Producer surplus (PS) 0 Social welfare (SW): SW = CS + PS + MP or SW = CB − PC 0.375(α − C)2 N1 /β) − rK Price elasticity of demand at Place 1 ε11 = − (P1 /Q1 )(dQ1 /dP1 ) = (α + C)/(α − C)
(2.1.13) (2.1.14) (2.1.15)
Notes: Rationale for localization (see Appendix A): Z1—Presence of a fixed cost; Givens (parameter or exogenous): a—Intercept of individual linear inverse demand curve: maximum price; b—Negative of slope of individual linear inverse demand curve: marginal effect of quantity on price received; C—Marginal unit production cost; K—Capital required to build factory; N1 — Number of consumers at Place 1; r—Opportunity cost of capital. Outcomes (endogenous): N1 ∗ —Minimum number of consumers required; P1 —Price of unit of soap at Place 1; q—Per capita consumption of soap; Q1 —Quantity of soap supplied to Place 1; ε11 —Price elasticity at Place 1 at market equilibrium.
2.2
Model 2A: Non-spatial Monopolist
35
I implicitly assume this is an efficient firm.21 This should not be surprising given I have assumed that the firm maximizes profit. However, let me illustrate the pervasiveness of the concept of efficiency here by focusing on four assumptions I make. First, there is no other way of building this production facility at Place 1 that would require less than K units of capital. Second, the lowest possible opportunity cost of capital is r annually per dollar invested assuming the firm draws capital away from its worst (more correctly the least-best) performing investment of similar risk and that the latter is nonetheless better than any other investment at that level of risk that the firm might make. Third, there is no other production technology more efficient than that which underlies the fixed marginal cost C. Fourth, the average cost given by (2.1.2) is the least cost possible for any given level of output. Throughout the book, I assume firms are efficient in all respects. In reality—as any business manager might be quick to point out—a firm will typically have to make much effort simply to get to this stage. As far as the economist is concerned, the firm can be thought to have already done a kind of economic due diligence. Assume N1 identical individuals at Place 1 who each purchase soap for their own consumption. Assume unit shipping cost is zero; hence effective price is simply price. Put differently, the home market customer is assumed to bear no costs related to search and information gathering, negotiation, and acquisition (including freight and transfer, storage and inventory, agency and brokerage fees, credit, cost of insurance and other loss risks, installation and removal, warranty and service, and taxes and tariffs). For simplicity of analysis, I assume such costs—where they occur—are borne by the firm and therefore included in the marginal cost of production, C, per unit of soap produced. Assume as well each customer has the same individual linear inverse demand curve22 for the firm’s product where P is price, q is individual quantity consumed over the year, α is the intercept (i.e., the price above which a customer demands zero), and β is the slope (the amount by which the price the customer is willing to pay drops for each unit of soap the firm wants to sell to the customer in the time period under consideration). In responding to this demand, the firm agrees to exchange units of soap for money. Why does it want to do this? In large part, the answer is that—to the extent inflation is low—money is what economists call “a store of value” that the firm can then use to pay its suppliers and to distribute 21 An
efficient firm (1) incurs the least possible cost in achieving its desired output and (2) seeks the maximum revenue possible from that output. It adopts an organizational structure than enables it to be efficient. It has a production function which shows, for each combination of inputs, the maximum possible output that can be produced with those inputs. The firm also has a cost function which shows, for each level of output (Q), the minimum possible cost of achieving that Q. Finally, the firm knows the demand for its product and exploits that information along with the knowledge of its cost and production functions to maximize its own profit. 22 See (2.1.3). A demand function is generally expressed as a schedule of quantity demanded (Q) at various prices (P): i.e., Q = f [P]. An inverse demand function rearranges this as the price consumers are willing to pay as a function of the quantity supplied to the market: that is, P = f −1 [Q].
36
2
The Firm at Home and Abroad
profits to its owners. More generally, we can think of money as just another desirable commodity used by the firm (1) to barter with suppliers or (2) for whatever other purpose ownership of that commodity might be used. What kinds of assumptions are implicit in Equation (2.1.3)? • There is no multiyear setting here. The customer demands q units each year unless something (i.e., P, α, or β) changes from 1 year to the next. This is consistent with the idea that the individual starts the year with no inventory of soap and consumes all q units by the end of the year. Implicitly, soap is therefore perishable—perhaps because it has a “best before” date—and cannot itself serve as a store of value.23 Person
Demand
1 2 ... N
q = α/β − P/β q = α/β − P/β ... q = α/β − P/β
All
Nq = Nα/β − NP/β
• The demand curve tells us nothing about how frequently the customer shops for soap during the year or how much is purchased at a time. A commodity may be consumed on the spot at time of purchase (e.g., an ice cream cone). In other situations (e.g., a bar of soap), the commodity might be brought home and consumed slowly over time. In the latter case, the customer maintains a stock (e.g., a partial bar of soap), but I presume this disappears by the end of the year. • Equation (2.1.3) is a particularly simple demand curve; it ignores income and the prices of complements and substitutes. Where does a demand curve come from? I find it helpful to think that (1) the customer is simultaneously participating in markets for commodities, (2) these commodities are, to varying degrees, pairwise substitutable or complementary, and (3) the demand for any one product is downward sloping in price in part because of diminishing marginal utility and in part because of substitution effects.24 When I see a linear inverse demand curve, I understand it to be an approximation to a process in which the customer is trading off the price of that commodity compared to the prices of other commodities
23 A commodity would be a store of value if it was nonperishable and could be readily resold in the
future: i.e., either converted back to cash at a low transaction cost or offer the possibility of capital gains. 24 Economists in general moved away the notion of diminishing marginal utility of commodities because it was seen to be based on a cardinal measurement of utility. Instead, the ordinal assumption of a diminishing marginal rate of substitution between commodities was invoked. In this book, I retain the use of diminishing marginal utility because students find it helpful. Where it becomes problematic (e.g., in Chapter 9), I will discuss the matter further.
2.2
Model 2A: Non-spatial Monopolist
37
subject to a budget constraint. By explicitly ignoring income and the prices of other commodities, (2.1.3) subsumes them into α and β. I assume here that the firm knows—and is able to exploit—the individual demand curve.25 The firm is not a price taker in the market for soap (as is assumed in a perfectly competitive market); by varying the quantity supplied, the firm can affect price. After all, that is what makes it a monopolist. The intercept may be interpreted as a maximum price26 ; the price paid rises to α as quantity supplied to the customer annually approaches zero. This corresponds to OA in Fig. 2.1. The customer views soap as an expendable commodity27 ; it would sooner go without soap at all than pay a price in excess of α. In a corresponding sense and with apologies to economists who may take affront with my casual usage here, I think of the marginal cost, C, as like a minimum price28 ; a firm presumably could not sustain selling at a price below marginal cost indefinitely. Rewrite this as a demand curve (q = α/β − P/β) once for each customer in a list as shown above. Summing both sides of this set of individual demand equations, I get an aggregate demand equation: N1 q − N1 (α/β) − N1 P/β. Aggregate demand is Q = N1 q. Because demand depends on the number of customers at Place 1, this is best envisaged strictly as a local demand29 , that is, assuming no demand (consumption) by nonresidents. After substitution and rearrangement, I get the aggregate linear inverse demand curve (2.1.4): see curve AB in Fig. 2.1. Of course, since all customers are identical, q = Q1 /N1 is simply consumption per customer. Here, the intercept (α) and slope ( − β/N1 ) are the parameters of this aggregate demand curve. By definition, the larger the market (N1 ), the less quickly price drops for a given increase in 25 A
skeptical reader here might well ask how a firm can know the demand curve of its customer when in fact it can, at best, try varying the price it charges to see how much of the good is demanded. This raises more generally the problem of whether economic agents ever have the full information about the market that is presumed by economists in these models. For an important paper on this topic, see Alchian (1950, pp. 220–221) on the famous adoption-versus-adaptation argument. 26 In a linear inverse demand curve, maximum price is the Y-intercept: the price at and above which the quantity demanded is zero. 27 An attribute of consumer demand for a product such that there is a price above which the consumer demands none of it. The opposite of an expendable good is an indispensable good: i.e., a good which must be consumed in some quantity, however small, even when its price is high. 28 The lowest price at which a profit-maximizing firm would participate in the market. Revenue just covers the variable cost of production. Minimum price is not sustainable over the longer run because fixed costs of production are not covered. 29 In a region, this is the demand for a product by consumers for local consumption and firms for local production. Specifically, local demand does not include demand by arbitrageurs for resale in another region. Marshall (1907, p. 112) invokes a similar notion when he refers to those who buy for their own consumption, and not for the purposes of trade. Larch (2007) uses a similar approach to local demand in international trade theory. In practice, there is no generally accepted standard in Economics for measuring local demand. In empirical practice, local demand is often taken to mean simply that the quantity demanded varies from one location to the next: see, for example, Megdal (1984) and Justman (1994).
38
2
The Firm at Home and Abroad
quantity. As N1 becomes large, the aggregate inverse demand curve approaches a horizontal line regardless of the slope of the individual inverse demand curve. If the firm supplies a small quantity for this market, customers are willing to pay a price approaching α; as quantity supplied rises, price drops. Before making use of this aggregate inverse demand curve, let me add a caveat. I assumed above that all customers at Place 1 are identical. Specifically—thinking of the customer as consumer—each consumer has the same individual demand curve and each pays the same price. In this non-spatial model, the latter assumption is consistent with the view that unit shipping cost is zero. If instead customers at Place 1 were spread out over a geographic area and incurred a shipping cost for each unit of soap consumed, customers further from the firm would have a higher effective price—compared to customers closer to the firm—and therefore demand less. For the moment, I continue to ignore the complexities that arise in demand aggregation in a spatial economy because of shipping cost. At this point, let me also juxtapose the firm’s demand curve beside its decision to invest capital in the construction of a plant. The demand curve shows the quantity the firm expects to sell in a period of time as a function of price. However, K is the money amount that has to be invested now to produce a quantity of soap each period from now until forever. Our model is so simple that it does not envisage a termination date for production or a scrap value at that time for the investment. How does a firm know what the demand will be for soap next year or at some other future period? Is not the firm somehow balancing the risk of a future downturn in demand against the return (profit) that it hopes to make in future years? To ease our way into location theory, I ignore such considerations in this chapter. My preference here is to think that the firm has some advantage over its competitors—e.g., better management, a process innovation, a good reputation in the industry, or social, political, and economic ties built up over the years—that it expects will enable it to adapt and survive in the future. To accountants, this corresponds to the value of goodwill; the amount a firm might be sold for over and above the value of its physical assets net of liabilities. To me, goodwill is a form of capital that the firm accumulates over time through investments of time (effort) and money. As such, it is different from the capital invested to build a factory in a particular location. In this chapter, I assume that the returns to this capital are part of the normal profit included in C. Put differently, the firm invests in goodwill as a means of ensuring that its demand curve in future years will correspond to (2.1.4). I assume the firm maximizes total profit (net of the opportunity cost of capital): PQ − CQ − rk. We can imagine the firm, starting from an output of zero, increases Q until the additional revenue generated from the last unit of soap (marginal revenue) is no larger than the additional cost incurred (marginal cost). Marginal cost is easy to calculate here: it is just C. The marginal revenue30 (MR) curve shows how 30 The addition to the firm’s total revenue created by the last unit supplied by the firm to the market.
If the firm’s demand curve is horizontal, it is a price taker (i.e., in a perfectly competitive market) and therefore its marginal revenue is simply the price. On the other hand, if the firm’s demand curve is downward sloping, marginal revenue is the price of the last unit sold minus the revenue
2.2
Model 2A: Non-spatial Monopolist
39
much total revenue increases for the last unit produced at different levels of output: MR = d(PQ)/dQ = P + Q(dP/dQ). See curve AC in Fig. 2.1. Because the aggregate inverse demand curve (2.1.4) is linear in quantity in this model, application of calculus shows us that marginal revenue is also linear in quantity and has the same Y-intercept (α) and a slope twice as steep ( − 2β/N1 ) as the demand curve.31 As the firm increases quantity supplied, its semi-net revenue32 continues to increase as long as price stays above marginal cost. In Fig. 2.1, semi-net revenue is the rectangle DLKED.33 To maximize excess profit34 (semi-net revenue minus fixed cost), the firm chooses the quantity where marginal revenue equals marginal cost.35 As is well known to Economics students, the firm does not maximize profit by setting price as high as possible (i.e., P = α); instead, the firm takes into account how profit is also affected by the quantity sold. Note here also that excess profit will be larger when C is smaller; to the firm, C is a drain on profit. In Fig. 2.1, total revenue of the firm is the rectangle OLKJO, and excess profit is the rectangle MLKHM at the profit-maximizing quantity Q. From another perspective, excess profit arises because competitors for some reason cannot or do not produce the same commodity. If enough competitors were to produce the same commodity, the firm’s own demand curve would become horizontal and excess profit would disappear as new firms enter the market. Why does a firm have such an advantage? Often, the advantage can be tied to an input. It might be attributable, for example, to a technology not available to other firms (because of a patent for instance) or to a managerial skill unique to the owner of that firm. In such cases, the excess profit can be thought to be a Ricardian Rent36 associated with that advantageous input. As a consequence of assuming the aggregate inverse demand curve is linear in quantity, the profit-maximizing price (OL in Fig. 2.1) is halfway between the minimum price (C) and the maximum price (α): OD and OA, respectively, in Fig. 2.1. Since C, α, and P can be measured in dollars, (2.1.6) has the desirable feature that lost on all other units sold because the market price has now been reduced because of the marginal unit of product supplied. 31 See (2.1.5). For ease of exposition throughout this book, I do not discuss the second-order conditions for profit maximization. 32 For a firm, revenue minus variable cost. In this book, variable cost includes both production and shipment. The firm’s net revenue (profit) is semi-net revenue minus fixed cost. 33 Throughout this book, I use the convention that variables (algebraic labels), equations, and expressions are italicized while other figure labels (representing points, lines, and areas) are not. All figures are labeled by their vertexes, and I indicate a polygon by repeating the first vertex again at the end of the label. 34 Revenue of the firm in excess of all costs, including normal profit on unpriced inputs like the firm’s capital and entrepreneurial skill. 35 See (2.1.6) and (2.1.7) and the corresponding excess profit (2.1.8). For those readers whose microeconomics is rusty, to the left of Q in Fig. 2.1, the firm finds that increasing Q also increases semi-net revenue. To the right of Q, semi-net revenue falls if we further increase Q. 36 An excess profit that arises because of an asset or market situation unique to a firm that prevents competitors from entering the market and/or earning the same profit.
40
2
The Firm at Home and Abroad
there is no money illusion.37 Also, price is the same for every customer (since they are all identical and the marginal cost curve is the same for every one of them); it is unaffected therefore by size of market. In turn, the profit-maximizing quantity is positive as long as maximum price (α) is higher than minimum price (C). This implies soap is expendable when P1 exceeds α in that, if price is or above α, customers at Place 1 would eschew soap. This itself is a consequence of the linearity of demand: i.e., because the demand curve crosses the Y-axis. Note also that although the number of customers at Place 1 (N1 ) does not affect the profit-maximizing price (2.1.6), it does affect aggregate quantity demanded (2.1.7) and profit (2.1.8). Finally, as is well known to students of Economics, the monopolist always chooses a point (price and quantity) along the aggregate demand curve where the price elasticity of demand is larger than one (along segment AN in Fig. 2.1): i.e., where demand is price elastic.38 Here, in thinking about the behavior of the firm, I distinguish between the short and long term. To survive in the short term, the firm must usually sell its product at a price sufficient to cover the variable cost39 of production. In other words, P1 is greater than or equal to C. However, to be profitable over the long term, the firm must also recoup its fixed costs (that is, the opportunity cost of capital): i.e., (P1 − C)Q1 − rK ≥ 0. There is an implication here for size of market. If the minimum required level of excess profit in (2.1.8) is zero, then the minimum number of customers (N1 ∗ ) needed to achieve this is given by (2.1.9). Put differently, satisfying (2.1.9) creates a kind of home market effect in the sense that the market is large enough to enable production at Place 1. Given a demand curve and a supply curve for a market, we know that as we hold the supply curve constant and shift the demand curve up or down, the equilibrium market outcomes (P1 and Q1 ) change in a way that has the effect of tracing out the supply curve. Now, imagine a thought experiment in which we alter the intercept of the demand curve (α) in Model 2A. As we do, we know from (2.1.6) that P1 will be higher if α is made larger. We also know from (2.1.7) that Q1 will also be higher if α is made larger. In Fig. 2.2, I show—as curve A3 DEFC—the locus of price and quantity that result from different values for α. In this sense, we can think of curve A3 DEFC as a quasi supply curve. To be clear, however, curve A3 DEFC is upward sloped not because the firm’s unit costs are increasing with the scale of output (if anything, they are decreasing); instead, this supply curve is upward sloped because the monopolist captures the higher price now profitable because of the increase in α. More generally, in this model, there are six givens (α, β, C, K, N1 , and r) and three outcomes (P1 , Q1 , and N1 ∗ ). How would the outcomes change here were the 37 In
a demand model, “no money illusion” means that price is relative to the units in which other money variables are measured. Put differently, if money quantities were all to double, quantity demanded would be unchanged. 38 See (2.1.15). 39 A cost incurred by the firm for a period of operation that varies with the quantity of output produced.
2.2
Model 2A: Non-spatial Monopolist
AvBv A 3C D E F
A21
41
Model 2A: Tracing a local supply curve Aggregate inverse demand curve where v = α Local supply curve Profit-maximizing Q1 and P1 at α = 9 Profit-maximizing Q1 and P1 at α = 15 Profit-maximizing Q1 and P1 at α = 21
C
Price
A15 F A9
E D
A3 0
B3
B9 B15 Quantity
B21
Fig. 2.2 Model 2A: tracing out a local supply curve by varying α. Note: β = 1; C = 3; K = 50,000; N1 = 200 r = 0.05. Here, α is set to 3, 9, 15, or 21. Curve A3 DEFC is the local supply curve: i.e., the locus of combinations of P1 and Q1 that maximize profit as α is varied. Horizontal axis scaled from 0 to 4,500; vertical from 0 to 25
givens different? In Economics, such questions are the stuff of comparative statics.40 In this model, comparative statics can be derived from inspection of (2.1.6), (2.1.7), and (2.1.9): see Table 2.2. C
K
N1
If C is increased, the firm’s profit margin is squeezed. The firm increases its price P1 ; however, Q1 decreases. Because each customer is purchasing less, N1 ∗ must become larger. If K is increased, P1 and Q1 are unchanged. However, N1 ∗ increases because the firm needs more customers to make enough net revenue to cover the opportunity cost of capital. When N1 is increased, the aggregate demand curve becomes flatter; that is, sweeps counterclockwise about (0, α). P1 is unaffected, but Q1 increases. N1 ∗ is unchanged.
40 A comparison of outcomes
(endogenous values) predicted by a model when a given (exogenous variable or parameter) is changed by a small amount. Some models describe market equilibrium; here comparative statics details the changes in equilibrium when a given is changed by a small amount. In other cases, models describe optimal outcomes; here comparative statics details changes in optimal outcome when a given is changed.
42
2
The Firm at Home and Abroad
Table 2.2 Model 2A: comparative statics of an increase in exogenous variable Outcome Given
P1 [1]
Q1 [2]
N1 ∗ [3]
C K N1 r α β
+ 0 0 0 + 0
− 0 + 0 + −
+ + 0 + − +
Notes: See also Table 2.1; +, Effect on outcome of change in given is positive; –, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome.
r
When r is increased, P1 and Q1 are unchanged. However, N1 ∗ increases because the firm once again needs more customers to make enough net revenue to cover the opportunity cost of capital.
α
If α is increased, the demand curve shifts upward and the price-equantity combination runs up the supply curve. Price rises: so too does the quantity transacted. Because customers individually purchase more than before, N1 ∗ becomes smaller.
β
If β is increased, the individual inverse demand curve sweeps clockwise about (0, α) and the price-quantity combination runs down the supply curve. At any price below α, consumer demand is now lower than it was before. For the profit-maximizing firm, P1 is unaffected but Q1 decreases. Because each customer is purchasing less, N1 ∗ must become larger.
Before leaving this model, let me add four thoughts. First, the market here clears. On the supply side, the firm starts the year with no stock and retains no stock at year-end; in fact, there is no incentive to accumulate stock any time during the year. It sells all that it produces during the course of the year; there is no profit to be earned by producing more than the market demands over the year. On the demand side, customers demand exactly the quantity given by (2.1.7) when market price is (2.1.6); there is no unmet demand. Second, I have said nothing so far about how the firm sells soap to customers at Place 1. Does it, for example, sell to a retailer who then operates a store selling soap to the customer? Costs are incurred in retailing: e.g., advertising and promotion, inventory and sales, and service and warranty replacement. In my view, it is easiest here to assume that (1) the factory operates a retail outlet on-site, (2) the unit cost C includes the costs of retailing, and (3) customers purchase soap there. If instead I assume a separate retail establishment that sells soap, I need to consider the
2.3
Model 2B: Monopolist Selling at Two Places; Factory at Place 1 Only
43
wholesale price received by the factory, the markup used by the retail establishment, and its inventory holding.41 Third, the firm earns its semi-net revenue by producing at unit cost C a commodity that it then sells at a price P1 . Put differently, it earns a profit to the extent the prices of its inputs are low relative to the price at which it sells soap. In general, therefore, the firm—indeed any firm—can be thought to engage in a kind of arbitrage: buying in (input) markets where prices are low for resale in a (output) market where price is high. What the firm does is transform inputs from low-priced markets into a commodity sold in a higher priced market. Fourth, for some reason, the ability to produce soap at a marginal cost of C is restricted to just the firm or a small number of rival firms. This might be, for instance, because of proprietary knowledge, a particular managerial skill, purchasing power, or some economy of scale or division of labor possible in production. Assume, instead, K= 0 and every customer at Place 1 could produce soap themselves at the same unit cost, C, as the firm. I label this home production. Faced with paying a price P1 in excess of C to acquire soap why would not customers simply produce soap themselves? In this case, the competition confronting the firm is home production. Of course, in the real world, the firm may have any of the advantages listed above that make it more efficient than home production. However, this idea gives us an interesting way (though not the only way) to think about α. Since α is the maximum price, presumably it might correspond to the unit total cost (i.e., C plus unit capital cost) customers incur in home production. Suppose the firm sets its price just below α. On the assumption customers have diminishing marginal productivity in the activities in which they allocate their time, capital, and other household resources, they will give up some home production in favor of product purchased from the firm. If the firm were to further lower its price, the customer might give up more home production (a production substitution effect) as well as consume more of the commodity in total (a price substitution effect and an income effect). Presumably, such changes are reflected in the shape (slope) of the individual’s inverse demand curve.
2.3 Model 2B: Monopolist Selling at Two Places; Factory at Place 1 Only Now, suppose this same firm has the possibility of selling soap at a remote customer point (Place 2) with a common currency and no restrictions on the shipment of commodities other than a unit shipping cost. Other than for the possibility that the firm sells in both Places 1 and 2, assume Place 2 is also isolated. Let Q1 and Q2 be the amounts the firm supplies annually to Places 1 and 2, respectively, from its factory at Place 1 where Q2 = 0 if the firm does not sell there.
41 See,
for example, Hsu and Tan (1999).
44
2
The Firm at Home and Abroad
Suppose the distance from Place 1 to Place 2 is x km. Let s be the (constant by assumption) cost per unit of soap shipped 1 km; hereinafter, I call s the unit shipping rate. In total, therefore, it costs sx to ship one unit from the factory at Place 1 to a customer at Place 2. In Chapter 1, I introduced the notion of a unit shipping cost. There, I defined it to be “transaction costs paid by the purchaser related to search, negotiation, and acquisition (including freight and transfer, storage and inventory, agency and brokerage fees, credit, cost of insurance and other loss risks, installation and removal, warranty and service, and taxes and tariffs) with respect to the commodity”. As noted in Chapter 1, there are two ways to think about sx. One is that sx is exogenous: simply a resource cost for overcoming distance: e.g., fuel consumption, storage, and or costs related to information gathering. The other is that sx is endogenous to a regional economy: it includes services provided by shippers, brokers, service agents, retailer, and others, as well as the prices of these. Assume here that the firm is a price taker in these input markets. In this model, as far as the firm is concerned, sx is simply an exogenous aggregate fixed expenditure required (per unit shipped) that reduces profit.42 Before proceeding, let us consider the components of unit shipping cost. Specifically, when might these be proportional to distance as assumed here? Search, information gathering, and negotiation: These expenditures are usually thought to be greater, the further apart are vendor and purchaser. However, it is not clear that these need be strictly proportional to distance. Also, these expenditures might best be regarded as an investment, with the cost itself being the opportunity cost of the capital invested. In an extreme case—search once then purchase frequently and forever—the opportunity cost involved may be small. Freight, transfer, agency, brokerage, credit, and cost of insurance and other loss risks: These expenditures are usually thought to have two components: one related to distance and the other related to time. The latter includes both travel time and time spent handling the commodity at either end of the trip, at transfer points along the way, and in administration. Storage and inventory: This includes costs related to the space required to hold inventory as well as to the opportunity cost of capital tied up in the commodity being store. It includes costs incurred by the producer, the shipper, and the customer. In each case, there is typically a tradeoff between inventory cost and freight cost; each agent holds inventory to reduce overall cost. In that respect, storage and inventory costs may well increase with the length of the trip. Installation and removal: This includes costs related to the installation of the commodity where it is to be used by the customer and the cost of removing and disposing of product that it replaces. To the extent that installers must 42 In assuming sx fixed, I ignore the possibility of congestion on transportation networks that might
cause sx to vary with the level of shipments. In this book, except where otherwise noted, I also ignore the idea that unit shipping cost might somehow vary directly with price: see Azar (2008) for a model where consumers perceive unit shipping cost to be relative to price.
2.3
Model 2B: Monopolist Selling at Two Places; Factory at Place 1 Only
45
be deployed and recycling/waste hauled away, such costs may increase with distance shipped. Warranty and service: This includes costs incurred by the vendor or purchaser related to warranty coverage and/or service. To the extent that service agents must be deployed and product must be shipped for repair or replacement, such costs may increase with distance. Taxes and tariffs: This includes sales and import taxes paid by the customer on purchases of the commodity. These can be related to distance as, when, and where tax or tariff is ad valorem based on a selling price that includes the cost of shipping. The dichotomization of geographic space that I use here usually is called punctiform. This is a commonly used abstraction of geographic space wherein economic activity is portrayed as clustered at distinct geographic points (places). Put differently, economic activities themselves do not use land or otherwise occupy space. In this abstraction, shipping between activities located at the same place incurs a negligible (zero) cost; shipping between activities at two different places incurs a nonzero cost that is invariant with respect to the number or volume of economic activities there. Throughout this chapter, I assume that all costs are borne by the firm; customers at Places 1 and 2 incur no unit shipping costs. In this section, I assume that the incremental unit shipping cost to a customer at Place 2 (over and above the cost incurred in supplying a customer at Place 1) is strictly proportional to distance. I assume here that the firm absorbs the unit shipping cost; it is free to set a delivered price43 at each customer point. Put differently, the firm engages in discriminatory pricing.44 I assume also that there are no impediments to shipment (e.g., quotas) other than shipping cost. The total cost to the firm is now given by (2.3.1). See Table 2.3. At the same time, assume that the firm can charge a distinct delivered price at each place (here, P1 and P2 ) without fearing an arbitrageur might purchase soap at the lower-price place for resale at the higher-price place. To the extent the firm sets a higher delivered price at remote Place 2, it can recoup some, all, or perhaps even more than all, of the cost of shipping incurred.45 Suppose each customer at Place 2 has the same individual linear inverse demand curve as at Place 1. In other words, α is the same at the two places: so too is β. Assume here as well that transaction costs are zero so that effective price equals price. Place 2 differs only in the number of customers there (N2 ). Its aggregate inverse demand curve is shown in (2.3.2). Here, once again, α is the maximum price the firm could expect at Place 2; β/N2 is how much the price at Place 2 falls as the firm increases the amount supplied annually by one unit. 43 Firm
sets price for good delivered to customer; consumer does not pay a separate shipping charge. 44 A pricing scheme used by a monopolist to enhance profit that results in different prices for different markets or submarkets, it is sometimes called third-degree price discrimination. 45 Depending on local price elasticities, there is even the possibility that the firm could set a price in the remote market that is lower than the price in the home market.
46
2
The Firm at Home and Abroad
Table 2.3 Model 2B: monopolist also selling at remote Place 2 Total cost rK + CQ1 + (C + sx)Q2
(2.3.1)
Aggregate linear inverse demand curve at Place 2 P2 = α − βQ2 /N2
(2.3.2)
Marginal revenue at Place 2 α − 2βQ2 /N2
(2.3.3)
Profit-maximizing price at Place 2, assuming α > C+sx P2 = 0.5(α + (C + sx))
(2.3.4)
Profit-maximizing quantity at Place 2 Q2 = 0.5(α − (C + sx))N2 /β
(2.3.5)
Monopoly excess profit (MP) 0.25((α − C)2 N1 + (α − C − sx)2 N2 )/β − rK
(2.3.6)
Range X = (α − C)/s
(2.3.7)
Minimum number of customers at Place 2 needed given N1 fails N2∗ ≥ {4βrK/(α − C)2 − N1 }{(α − C)2 /(α − C − sx)2 }
(2.3.8)
Consumer benefit (CB) 0.25(1.5α + 0.5C)(α − C)N1 /β + 0.25(1.5α + 0.5(C + sx))(α − C − sx)N2 /β
(2.3.9)
Producer cost (PC) 0.5((α − C)CN1 + (α − C − sx))(C + sx)N2 )/β + vrK
(2.3.10)
Consumer surplus (CS) 0.125((α − C)2 N1 + (a − C − sx)2 N2 )/β
(2.3.11)
Producer surplus (PS) 0
(2.3.12)
Social welfare (SW): SW = CS+PS+MP or SW = CB − PC 0.375(α − C)2 N1 /β + 0.375(α − C − sx)2 N2 /β − rK
(2.3.13)
Price elasticity of demand at Place 2 ε12 = − (P2 /Q2 )(dQ2 /dP2 ) = (α + C + sx)/(α − C − sx)
(2.3.14)
Notes: Rationale for localization (see Appendix A): Z1—Presence of a fixed cost; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Marginal unit production cost; K—Capital required to build factory; Ni —Number of consumers at Place i; r—Opportunity cost of capital; s—Cost of shipping one unit of product one km; x—Geographic distance from Place 1 to Place 2; a—Intercept of individual linear inverse demand curve: maximum price; b—Negative of slope of individual linear inverse demand curve: marginal effect of quantity on price received. Outcomes (endogenous): N2 ∗ —Minimum number of consumers required at Place 2; Pi —Price of unit of soap at Place i; Qi —Quantity of soap supplied to Place i; X—Range of soap (kilometers); ε 12 — Price elasticity at Place 2 at market equilibrium.
2.3
Model 2B: Monopolist Selling at Two Places; Factory at Place 1 Only
47
As at Place 1, the intercept here is assumed positive and the slope (here β/N2 ) is negative. Here I implicitly assume that the demands at Places 1 and 2 are separable; the firm can set quantity and price separately at the two places. For some reason, the firm does not have to worry whether a difference in price between the two places might lead customers at the place with the higher price to purchase soap instead where the price is lower. I return to this matter shortly. Suppose the firm now chooses Q1 and Q2 to maximize excess profit. My assumption that marginal cost is the same regardless of the quantity supplied to either Place 1 or Place 2 makes the problem separable: I can solve for Q1 without knowing Q2 and vice versa. In this case, the solutions for Q1 and hence P1 remain as in Table 2.1. With respect to Q2 , the firm then simply equates marginal revenue (2.3.3) with marginal cost (C + sx) for customers at Place 2. The firm will find it profitable to ship to customers at Place 2 as long as the maximum price at Place 2 (α) is larger than the marginal cost of serving a customer there (C + sx) The firm will not find it profitable to supply Place 2 if C + sx > α. If profitable, Place 2 will be part of the market supplied by the firm with a profit-maximizing combination of price and quantity there46 and a corresponding semi-net revenue (or, equivalently, added excess profit) earned from customers there (2.3.6). This is in addition to the excess profit (2.1.8) earned from customers at Place 1. Finally, note that the price elasticity of demand is larger here47 compared to (2.1.15) because unit shipping cost pushes the firm higher up the demand curve (i.e., up segment AN in Fig. 2.1) where demand becomes more elastic. Comparing profit-maximizing prices at the two places, (2.1.6) and (2.3.4), we see the two prices differ by half the cost of shipping one unit of product from local Place 1 to remote Place 2. This principle—termed the half-freight48 rule—arises as a special case because I assume linear individual demand curves at the two places. If I had used another demand function (e.g., q = aP−P ), the half-freight rule would not hold. Nonetheless, the model illustrates an important principle. For a monopolist, the difference in selling price between two places is not necessarily equal to the difference in shipping cost involved. Now, imagine a thought experiment in which Place 2 is pushed further away from Place 1 and soap thereby becomes more costly to ship. Eventually, given the linear demand curve at Place 2, there would come a distance X at which the most profitable price (P2 ) would equal the maximum price (α), and hence annual demand
46 See
(2.3.4) and (2.3.5). (2.3.14). 48 A market outcome in which the prices at which a monopolist sells the same good in two markets differ by half the difference in shipping costs of shipping to the two markets. In general, this arises when consumers in the two markets are identical and have linear demand curves. 47 See
48
2
The Firm at Home and Abroad
there would drop to zero.49 In the view of the firm, that distance would constitute what is called the range50 of soap. Of course, not every commodity need have a range.51 Some commodities might be indispensable52 : that is, customers need them regardless of price. In that case, the linear demand curve used here would be inappropriate. As indicated in (2.3.6), Place 2 may add to the profit of the factory built at Place 1. Since the firm maximizes profit, it will ignore the second place unless it adds to the total excess profit that would otherwise be earned from Place 1 alone. From (2.1.8) and (2.3.6), the ratio of the semi-net revenue generated by a customer at Place 1 to the semi-net revenue of a customer at Place 2 is (α − C)2 /(α − C − sx)2 : put differently this gives the number of Place 2 customers equivalent to one Place 1 customer. Given sufficient customers at Place 2, a firm would have built the factory at Place 1 even if Place 1 alone would not support a factory in the sense of (2.1.9). In (2.3.8), the minimum number of customers at Place 2 needed to do this is calculated. On the right-hand side of (2.3.8), the left-hand pair of {} braces enclose the amount by which N1 falls short of the requirement for profitability. The right-hand pair of {} braces enclose the Place 2 equivalent. Important here too is the conclusion that the delivered price at Place 2 does not depend on the number of customers (N2 ) there. If the number of customers is insufficient, the firm may choose not to build a factory at all. However, if it is profitable to build a factory, the price charged at each place is insensitive to the relative numbers of customers at the two places. Are there circumstances in which there would be arbitrage? Would a trader have the incentive to purchase soap at Place 1 and resell it at Place 2? Comparison of (2.1.6) and (2.3.4) tells us the price difference the monopolist would choose is P2 − P1 = 0.5sx. In this case, on the presumption that the unit cost of shipping for the arbitrageur is also sx, there is no incentive to purchase at Place 1 for resale at Place 2. I have already characterized Place 1 as the home market and Place 2 as the remote market. Potentially, there is a home market effect here. If the population of Place 1 were large enough to enable (2.1.9), the firm would build a plant there. If C + sx is smaller than α, the firm would also sell product at Place 2 that it had produced in its plant in the home market. If the population at Place 1 is not large enough to enable (2.1.9), the firm would nonetheless build a plant there to serve both the home and remote markets as long as the population at Place 2 is large enough to enable (2.3.8).
49 See
(2.3.7).
50 For an expendable
good sold at an f.o.b. price, the distance at which shipping cost is sufficiently high to cause demand to drop to zero. 51 The starting point for this book is the firm producing a good or service. The book therefore ignores firms whose business is the construction of a network. De Fraja and Manenti (2003) study the extension of local telephone calling areas—within which long distance charges do not apply— as a strategic variable chosen to maximize the carrier’s profit. 52 An expendable good is such that there is a price above which the consumer demands none of it. The opposite of an expendable good is an indispensable good: i.e., a good which must be consumed in some quantity, however small, even when its price is high.
2.3
Model 2B: Monopolist Selling at Two Places; Factory at Place 1 Only
49
In this model, there are eight givens (α, β, C, K, N1 , N2 , r, and s) and eight outcomes (N1∗ , N2∗ , P1 , P2 , Q1 , Q2 , X). How would the outcomes change here were the givens to change? In this model, comparative statics are readily derived from inspection of (2.1.6), (2.1.7), (2.1.9), (2.3.4), (2.3.5), (2.3.7), and (2.3.8). See Table 2.4. Table 2.4 Model 2B: comparative statics of an increase in exogenous variable Outcome Given
P1 [1]
Q1 [2]
N1∗ [3]
P2 [4]
Q2 [5]
N2∗ [6]
X [7]
C K N1 N2 r s α β
+ 0 0 0 0 0 + 0
− 0 + 0 0 0 + −
+ + 0 0 + 0 − +
+ 0 0 0 0 + + 0
− 0 0 + 0 − + −
+ + 0 0 + + − +
− 0 0 0 0 − + 0
Notes: See also Table 2.3; +, Effect on outcome of change in given is positive; –, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome; ?, Effect on outcome of change in given is unknown.
C
If C is increased, the firm’s profit margin is squeezed. The firm increases price; however, quantity decreases as a result at both customer points. Because each customer is purchasing less, N1 ∗ and N2 ∗ must become larger. Finally, X becomes smaller.
K
If K is increased, price and quantity are unchanged at both customer points. However, N1 ∗ and N2 ∗ increase because the firm needs more customers to make enough net revenue to cover the opportunity cost of capital. X is unchanged. If N1 is increased, the aggregate demand curve at Place 1 becomes flatter, that is, sweeps counterclockwise about (0, α). P1 is unaffected, but Q1 increases. N1 ∗ is unchanged. P2 , Q2 , and N2 ∗ are unchanged. X is unchanged. If N2 is increased, the aggregate demand curve at Place 2 becomes flatter. P2 is unaffected, but Q2 increases. N2 ∗ is unchanged. P1 , Q1 , and N1 ∗ are unchanged. X is unchanged. If r is increased, price and quantity are unchanged at each location. However, N1 ∗ and N2 ∗ increase because the firm once again needs more customers to make enough net revenue to cover the opportunity cost of capital. X is unchanged. If s is increased, P1 , Q1 , and N1 ∗ are unchanged. However, at the remote Place 2, the firm will increase P2 and therefore see a drop in Q2 . Therefore,
N1
N2
r
s
50
α
β
2
The Firm at Home and Abroad
N2 ∗ increases because the firm once again needs more customers there to make enough net revenue to cover the opportunity cost of capital. Finally, X becomes smaller as s is increased. If α is increased, the individual demand curve shifts upward. The firm now finds that consumers at both customer points are willing to pay more for any given quantity than before. Both price and quantity increase in each market. Because customers individually purchase more than before, both N1 ∗ and N2 ∗ become smaller. Finally, as α becomes larger, it becomes possible for Place 2 to be further away from the firm, that is, X becomes larger. If β is increased, the individual inverse demand curve sweeps clockwise about (0, α). At any price below α, customer demand is now lower than it was before at both customer points. For the profit-maximizing firm, price in each market is unaffected but quantity decreases. Because each customer is purchasing less, N1 ∗ and N2 ∗ must become larger. X is unchanged.
In this model, how does the firm retail to a customer at Place 2? One strategy is to sell to both customers at Places 1 and 2 from the same retail outlet on-site at Place 1. A second strategy is to have a retailer at Place 2 sell soap there at an agreedupon markup. In Model 2A, the firm’s marginal cost of production, C, presumably includes the cost of retailing per unit sold to a customer at local Place 1. In Model 2B, the unit shipping cost, sx, presumably includes any extra cost of retailing per unit sold to a customer at remote Place 2.
2.4 One Market or Two? From the firm’s perspective, Places 1 and 2 can be said to constitute its market; they are the customers and places where the firm sells soap. To an economist however, a market is something else. In Chapter 1, I describe a market as a locus of buyers and sellers that facilitates efficient exchange of soap. Inherent in that conceptualization is a process (like auctioning) wherein a price gets established such that demand equals supply (i.e., a price that clears the market). The central idea is that a market process results in a single price in common to all suppliers and all demanders.53 In practice, it is rare to find markets that operate exactly like this. Nonetheless, we commonly think of consumer commodities (e.g., for housing, automobiles, clothing, or bread) in a metropolitan area as each being provided in a market. How do such notion of markets diverge from the economic model? Two popular critiques come to mind. Let me now respond to each of these. One critique is that consumer commodities are rarely auctioned in the way envisaged: i.e., in a process that clears a market. In the short run—from minute to
53 A
similar point is made in McChesney, Shughart, and Haddock (2004).
2.4
One Market or Two?
51
minute for example—there may indeed be a divergence between demand and supply. However, this does not in itself negate the conceptualization of markets for consumer commodities. When supply exceeds demand in the short term, inventory accumulates in the hands of the supplier. When demand exceeds supply, inventory gets drawn down or demand goes unmet. Each of these possibilities—which can arise from the vagaries of demand and supply from minute to minute—imposes costs on the vendor: costs for maintaining inventory and opportunity costs of unmet demand. Auctioning too has its costs. Firms use strategies—from auctioning to price discounting to supplier and customer contracting—to maximize their profits. Put differently, firms use these strategies in a way that assists efficient clearing of markets even in the absence of a formal auction process. A second critique—using the market for bread as an example—is that the price paid for a loaf will differ from white bread to whole wheat, from one neighborhood of the city to the next, and from one type of retail outlet (e.g., convenience store or supermarket) to the next. Doesn’t this mean that the market for white bread is different from the market for whole wheat, different from neighborhood to neighborhood, or different from one type of retail outlet to the next? Perhaps. However, another way to think about these is that they are submarkets54 within a larger market for bread. Submarkets are used to refer to the markets for commodities that are alike, substitutable for one another, and whose prices therefore tend to change similarly; when the price of one commodity increases, consumers tend to substitute for a less-expensive alternative, and thereby normally drive up the price of the substitute. Prices can differ among submarkets. After all, consumers might prefer whole wheat bread, or it might be more costly to produce. As such, there might be a price premium for whole wheat bread. However, the idea behind submarkets is that such premiums are thought to be relatively stable over time, and factors that cause the price of bread generally to rise or fall need not necessarily affect the price premium in a particular submarket. In this chapter, we have only one supplier (the firm) and demanders at Places 1 and 2. Consider first the case where sx = 0. In this case, the economist’s requirements for a market are met because there is a common price (since P1 = P2 ). Alternatively, suppose sx = 0. Production and allocation appears to conform to a market process, but with one notable difference. Here, the firm sets different prices at Places 1 and 2. Does this difference in price mean that the two places are necessarily each in a market of their own? Note here though that the difference in price between the two markets (in other words, the price premium at remote Place 2) is a constant amount: in this case, half-freight. If C were to rise by 1.00 for example, then P1 and P2 would each rise by 0.50. In this respect, Places 1 and 2 appear to be like two submarkets. At the same time, a profit-maximizing solution does not 54 To
me, a market for identical, or similar, commodities is said to be formed of submarkets when prices in the submarkets differ but are linked in some respect. In a strong version of submarkets, the price in one submarket is a fixed premium on the price in another submarket. In a weak version, a rise in price in one submarket causes the price in the other submarket to change, but there is no fixed premium.
52
2
The Firm at Home and Abroad
concur with Marshall’s idea of a common market price wherein remote customers pay a special charge on account of delivery; here remote customers are paying only a half freight charge.
2.5 Pricing Strategies Because this is a book about location theory, it is appropriate to begin with a model where the firm varies its price from one customer point to the next in response to differences in unit shipping cost. However, a skeptical reader might well point out that some monopolists in reality have other pricing practices in addition to discriminatory pricing. Four popular alternatives come to mind. F.o.b. price: The monopolist sets the same price at the factory gate for all customers. Customers at Place 2 pay the same price as customers at Place 1 but then incur a shipping cost; the effective price is higher at Place 2, compared to Place 1, by the amount of the unit shipping cost. Put differently, the customer absorbs the unit shipping cost. Uniform pricing: The monopolist makes the price the same for a unit of the commodity delivered at different customer points even though the marginal costs of serving customers there may vary. The firm absorbs the unit shipping cost. Here, the firm must set both a price and a maximum distance beyond which the customer is too costly to serve. Pickup or delivery pricing: The monopolist sets two prices—an f.o.b. price and a delivered price—and allows the customer to choose between them.55 In this case, the unit shipping cost is absorbed either by the firm or by the customer depending on the option chosen by the customer. In the case of delivered price, the firm must set both the delivered price and a maximum distance beyond which the customer is too costly to serve. Basing point pricing: The monopolist chooses some customer points (so-called “basing points”) and sets a delivered price at each of them.56 Customers elsewhere then pay the unit shipping cost to shipping home the commodity from the basing point. The firm in general here absorbs part of the unit shipping cost. It is not difficult to show that, under simple assumptions, discriminatory pricing—use of the half-freight rule as shown above—maximizes profit compared
55 See
Furlong and Slotsve (1983).
56 Classic works in this field include Smithies (1942), Stigler (1949), and Machlup (1949). See also
DeCanio (1984), Deutsch (1965), Faminow and Benson (1990), Gilligan (1992), Haddock (1982), Hughes and Barbezat (1996), Levy and Reitzes (1993), Lord and Farr (2003), Needham (1964), Soper, Norman, Greenhut, and Benson (1991), and Thisse and Vives (1992).
2.6
Model 2C: Factory at Each Place
53
to the other options.57 Why then might a monopolist choose f.o.b. or some other pricing scheme and forego the extra profit possible with discriminatory pricing?58 Usually, the problem with discriminatory pricing is that it is difficult to enforce. Why, for example, wouldn’t customers at Place 2 purchase soap at Place 1 for the lower price and either consume it there or bring it back home with them? One might initially want to argue that such travel or shipment can be ignored because the firm, due to its scale of operation will already have found the least costly way to ship to Place 2 and that its price is higher at Place 2 by only one-half of that amount. However, this argument disregards the possibility customers might have to travel from Place 2 to Place 1 for another purpose and the effective price of purchasing soap at Place 1 for them is only P1 . Another aspect disregarded here is the relationship between inventory acquired by the traveling consumer and the unit cost of shipping. The customer who makes a trip to Place 1 to purchase soap can reduce the per unit cost of the trip by choosing to stock up on this trip: in effect, the customer trades off the opportunity cost of holding an inventory of soap at home against the cost of traveling to Place 1 to purchase it. I do not consider such inventory holding explicitly in this chapter.59
2.6 Model 2C: Factory at Each Place Now, suppose the firm has already built a factory at the larger Place 1 and is now considering an identical second factory (therefore, another establishment) at a smaller Place 2 to service customers there. Assume the opportunity cost of capital for this second factory is the same (r) as for the factory built at Place 1. Assume the capital cost here is also K, and this new factory also has a marginal cost curve also constant at C dollars per unit soap. The total cost to the firm is now given by (2.5.1). See Table 2.5. The second factory possibly gives the firm a more efficient way of supplying customers at Place 2 because it eliminates shipping cost. With the second factory, the firm would never ship soap from one place to the other; to do so would involve an unnecessary cost.60 Under these assumptions, the profit-maximizing price and quantity at Place 1 remain as in Table 2.1. However, the profit-maximizing price and quantity for Place 2 are now given by (2.5.2) and (2.5.3) in lieu of (2.3.4) and
57 Among
papers making the same argument, see Gee (1985). for example, Ohta, Lin, and Naito (2005). 59 Also, in this chapter, I assume that all consumers are the same. Suppose however that consumers at Place 2 differed in that some regularly visited Place 1 and could stock up on the good, while others did not. The firm might then be able to use this information to better price discriminate between the two markets. See Anderson and Ginsberg (1999). 60 Because the model assumes no uncertainty, the firm does not need to think about the possibility of a breakdown, strike, or temporary reduction in the flow of inputs at one of its plants. If it did have such uncertainties, the firm might well ship output from one place to the other. 58 See,
54
2
The Firm at Home and Abroad
Table 2.5 Model 2C: monopolist with a factory at each place Total cost 2rK + C(Q1 + Q2 )
(2.5.1)
Profit-maximizing price at Place 2 P2 = 0.5(α + C)
(2.5.2)
Profit-maximizing quantity at Place 2 Q2 = 0.5N2 (α − C)/β
(2.5.3)
Monopoly excess profit (MP) 0.25((α − C)2 N1 + (α − C)2 N2 )/β − 2rK
(2.5.4)
Condition for second factory at a smaller Place 2 to be more profitable than single factory at larger Place 1 N2 ≥ 4βrK/((α − C)2 − (α − C − sx)2 )
(2.5.5)
Consumer benefit (CB) 0.25(1.5α + 0.5C)(α − C)(N1 + N2 )/β
(2.5.6)
Producer cost (PC) 0.5(α − C)C(N1 + N2 )/β + 2rK
(2.5.7)
Consumer surplus (CS) 0.125(α − C)2 (N1 + N2 )/β
(2.5.8)
Producer surplus (PS) 0
(2.5.9)
Social welfare (SW): SW = CS+PS+MP or SW = CB-PC 0.375(α − C)2 (N1 + N2 )/β − 2rK Price elasticity of demand ε11 = − (P1 /Q1 )(dQ1 /dP1 ) = (α + C)/(α − C) ε22 = − (P2 /Q2 )(dQ2 /dP2 ) = (α + C)/(α − C)
(2.5.10) (2.5.11)
Notes: Rationale for localization (see Appendix A): Z1—Presence of fixed cost. Givens (parameter or exogenous): C—Marginal unit production cost; K—Capital required to build each factory; Ni —Number of consumers at Place i; r—Opportunity cost of capital; α—Intercept of individual linear inverse demand curve: maximum price; β—Negative of slope of individual linear inverse demand curve: marginal effect of quantity on price received. Outcomes (endogenous): Pi —Price of unit of soap at Place i; Qi —Quantity of soap supplied to Place i; ε 11 —Price elasticity at Place 1 at market equilibrium; ε22 —Price elasticity at Place 2 at market equilibrium.
(2.3.5), respectively. The contribution to firm profit arising from sales at Place 2 is shown in (2.5.4). Comparing (2.5.2) with (2.3.4), the price at Place 2 will be lower than in the onefactory solution in Table 2.3. The elimination of the unit shipping cost allows the firm to lower its price at Place 2 and still further increase semi-net revenue since the quantity sold at Place 2 will be greater. Further, assume demand at each place is constant from 1 year to the next; Q1 and Q2 do not change if P1 and P2 remain unchanged. Here, we can think the firm makes an investment now of K to build the second factory, the return for which is an improvement in semi-net revenue that arises partly because of
2.7
Model 2D: Choice of Sites and Localization
55
shipping cost savings and partly because the firm can increase its quantity sold at Place 2. The firm would find it profitable to build the factory at Place 2 where the gain in semi-net revenue from local production at Place 2 exceeds the opportunity cost of the capital required for the second factory: i.e., where (2.5.4) is non-negative. In part, this is driven by the number of customers at Place 2 (i.e., N2 ) since the larger the market the larger the semi-net revenue. Equation (2.5.5) indicates that the larger the opportunity cost of capital (rK) or the price sensitivity of demand (β), or the smaller the shipping rate (sx) the larger Place 2 must be to make the second factory profitable. The comparative statics of Model 2C are the same as in Model 2B except for two implications that arise because shipments are no longer necessary. The first implication is that the notion of a range (X) is not relevant in Model 2C. The second is that variations in the unit shipping rate, s, will have no effect on price, quantity, or minimum size of market. See Table 2.6. Table 2.6 Model 2C: comparative statics of an increase in exogenous variable. Outcome Given
P1 [1]
Q1 [2]
N1∗ [3]
P2 [4]
Q2 [5]
N2∗ [6]
C K N1 N2 r s α β
+ 0 0 0 0 0 + 0
− 0 + 0 0 0 + −
+ + 0 0 + 0 − +
+ 0 0 0 0 0 + 0
− 0 0 + 0 0 + −
+ + 0 0 + − − +
Notes: See also Table 2.5; +, Effect on outcome of change in given is positive; –, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome; ?, Effect on outcome of change in given is unknown.
In Model 2A and Model 2C, we might assume that the firm has an outlet counter at the front of its factory from which it sells product and that the marginal cost of production, C, includes the unit cost of retailing soap and servicing a customer there. In Model 2B, however, the firm needs to have a way of retailing to customer at Place 2. In this respect, Model 2C gives the firm a less-costly option for selling soap that is reflected in the absence of s in Model 2C.
2.7 Model 2D: Choice of Sites and Localization Finally, where, if at all, should the firm put a factory or factories? For example, rather than assuming that the first factory is at Place 1 and looking at whether to put a second factory at Place 2, how does our problem change if the firm has the option also of building one factory only and putting this factory at Place 2? In all, the firm has four options in the static world envisioned here: (i) build no factories, (ii) build
56
2
The Firm at Home and Abroad
only one factory and put it at Place 1, (iii) build only one factory and put it at Place 2, or (iv) build one factory each at both Places 1 and 2. The excess profit that arises under each option is specified in Table 2.7. Here, Vij is the semi-net revenue from having the factory at Place i supply customers at Place j where feasible: i.e., where maximum price (α) is higher than minimum price (C plus any shipping cost). Of course, the do-nothing option i has a zero excess profit. Also shown in Table 2.7 is the rate of return for each of options ii, iii, and iv. In these, I have assumed symmetric shipping costs.61 Using Table 2.7, I can now deduce whether and where the firm will build its factories. However, since this model contains 8 parameters—α, β, N1 , N2 , C, K, r and sx—it is easiest to envisage solutions if I make some simplifying assumptions. Table 2.7 Model 2D: the four options Profit under option i: no factories at all 0
(2.7.1)
Profit under option ii: one factory built, at Place 1 only V11 + V12 − rK
(2.7.2)
Profit under option iii: one factory built, at Place 2 only V21 + V22 − rK
(2.7.3)
Profit under option iv: factories built at both Places 1 and 2 V11 + V22 − 2rK
(2.7.4)
Rate of return for option ii (V11 + V12 )/K
(2.7.5)
Rate of return for option iii (V21 + V22 )/K
(2.7.6)
Rate of return for incremental option iv (V11 + V22 − Max[V11 + V12 ,V21 + V22 ])/K
(2.7.7)
where semi-net revenue of serving customers at j from factory at i, Vij , is given by the following (2.7.8) V11 = 0.25N1 (α − C)2 /β if α > C, else 0 (2.7.9) V12 = 0.25N2 (α − C − sx)2 /β if α > C + sx, else 0 (2.7.10) V21 = 0.25N1 (α − C − sx)2 /β if α > C + sx, else 0 (2.7.11) V22 = 0.25N2 (α − C)2 /β if α > C, else 0 Notes: Rationale for localization (see Appendix A): Z1—Presence of fixed cost; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Marginal unit production cost; K—Capital required to build factory; Ni —Number of consumers at Place i; r—Opportunity cost of capital; s—Cost of shipping one unit of product one km; x—Geographic distance from Place 1 to Place 2; α—Intercept of individual linear inverse demand curve: maximum price; β—Negative of slope of individual linear inverse demand curve: marginal effect of quantity on price received. Outcomes (endogenous): Pi —Price of unit of soap at Place i; Qi —Quantity of soap supplied to Place i; Vij — Semi-net revenue from serving customers at Place j from factory at Place i; X—Range of soap (kilometers).
61 A
feature of shipping rates such that the cost of shipping a unit of product from Place i to Place j is the same as the cost of shipping a unit from j to i.
2.8
Two Markets Identical
57
2.8 Two Markets Identical To begin, assume the two markets are identical in size: i.e., N1 = N2 . Assume also, for the moment, sx = 0. In this case, the firm would build at most one factory (because there is no shipping cost to be saved by building a second factory) and would be indifferent between putting it at Place 1 or Place 2.62 Without loss of generality, I presume the factory is at Place 1. However, if the rate of return were inadequate even here, the firm would choose option i: no factory at all. The firm would never choose option iv because that would be less profitable (i.e., generate less profit and a lower rate of return) than serving both places from the one factory at Place 1. Suppose instead that sx is greater than zero. Assume initially that there is only one factory and it is at Place 1. Again, imagine a thought experiment in which sx is made progressively larger, perhaps because I imagine Place 2 further away from Place 1. As this happens, V11 remains constant, but V12 shrinks. Eventually as sx is increased still more, C + sx approaches α, V12 drops to zero, and the firm restricts itself to the home market. For sx still larger, V12 remains zero because the firm would no longer find it profitable to supply Place 2. Now assume there is a factory at each place. Here, the profit of option iv remains the same at every level of sx since the firm no longer incurs a shipping cost. The preferable option here depends on the magnitude of the shipping rate, sx. As an example, suppose α = 15, β = 1, K = 50,000, C = 3, r = 0.05 N1 = 200, and N2 = 200. In this case, when sx is greater than 12 (in other words, α − C), the firm’s marginal cost of supplying customers at Place 2 from a factory at Place 1 (C + sx) is greater than the maximum price customers there are willing to pay (α). Curve AB in Fig. 2.3 shows the profit under option iii at various levels of shipping rate (sx); curve CD shows profit under option iv. I do not show option ii here because it has the same excess profit as option iii since N1 = N2 . Starting from zero, as I increase sx, the excess profit from option iii declines. At a stated unit shipping cost, draw a vertical line (line EG in Fig. 2.3) up to the higher of the profit schedules (in the case of EG, this is option iv) and read across to the y-axis to get the excess profit earned (OC in Fig. 2.3). For sx sufficiently low (below OH in Fig. 2.3), option iii—or equally option ii—becomes the most profitable choice. The same example is displayed in Fig. 2.4 in terms of rate of return. Here it helps to think of two investment decisions in sequence. The first is to invest in the more profitable of either option ii or iii. Here, the two relevant rates of return are given by (2.7.5) and (2.7.6). The firm will choose option ii or iii if, at the stated unit shipping cost, curve AB lies above the opportunity cost of capital (see the horizontal line DE in Fig. 2.3). The second decision is whether to then add a factory at the other place; here I want to know if a second factory adds sufficient (incremental) profit to make the investment there worthwhile. The relevant rate of return here is (2.7.7) and is
fact, if sx = 0 , the firm is indifferent between having its factory at Place 1 and Place 2 even if N1 = N2 .
62 In
58
2
The Firm at Home and Abroad
Model 2D: Profit, locational choice, and unit shipping cost AB CD EF EG HI OE OH
A
I
Profit schedule under options ii or iii Profit schedule under option iv: see (2.7.4) Profit under options ii or iii at stated unit shipping cost Profit under option iv at stated unit shipping cost Profit earned (options ii, iii, or iv) at critical shipping cost Stated unit shipping cost (sx ) Critical unit shipping cost
G
D
C Profit
F
B
0
H
E
Unit shipping cost
Fig. 2.3 Model 2D: profit as a function of shipping rate. Note: α = 15, β = 1; C = 3; K = 50,000; N1 = 200 N2 = 200, r = 0.05, sx = 4. Profit is 7,900 under option iii; 9,400 under option iv. Critical unit shipping cost is 2.30. Horizontal axis scaled from 0 to 14; vertical axis from 0 to 14,000
shown as the curve OB in Fig. 2.4. The firm will choose option iv if, at the stated unit shipping cost, curve OB lies above DE. Where the stated unit shipping cost is OF in Fig. 2.4, the firm finds that the rate of return (FH) on the factory at one place is larger than the opportunity cost of capital (OD), as is the incremental rate of return (FG) on the factory at the other place. Why does the rate of return on option iv take the shape of curve OB? To me, there are four points of interest about OB. First, when unit shipping cost is zero, the incremental rate of return on option iv is zero. In graphical terms, OB passes through the origin of Fig. 2.4. Why? When unit shipping cost is zero, there is no excess profit to be made by building a second factory. Second, the firm will always find either or both options ii and iii have a rate of return that, at every sx, is greater than or equal to the rate of return on the incremental investment in option iv. This is because the firm chooses first the more profitable of options ii and iii and therefore in option iv is looking at investing in a second factory, this time at the less profitable place. Third, whether the firm builds the second factory depends on the position of curve OB in relation to the line DE. As unit shipping cost becomes large enough, it is uneconomic to supply a remote market. The greater is sx, the more attractive it becomes to build a factory at the second place. Fourth, the rate of return on option iv converges with the rate of return on one plant: at sx sufficiently large, each plant serves only its local market
2.9
Differing Markets
A
59
Model 2D: Rate of return, locational choice, and unit shipping cost AB Rate of return schedule under options ii or iii DE Opportunity cost of capital (r) FG Rate of return under option iv at stated unit shipping cost FH Rate of return under options ii or iii at stated unit shipping cost OB Rate of return schedule under option iv OF Stated unit shipping cost
Rate of return
H
B
G D
E
0 F
Unit shipping cost
Fig. 2.4 Model 2D: rate of return as a function of shipping rate. Note: α = 15, β = 1; C = 3; K = 50,000; N1 = 200 N2 = 200, r = 0.05, sx = 4. Rate of return is 0.208 under options ii or iii; 0.080 under option iv. Horizontal axis scaled from 0 to 14; vertical from 0 to 0.35
2.9 Differing Markets Suppose instead that there are more customers at Place 2 than at Place 1. In that situation, the profit and rate of return associated with option iii increases compared to option ii. Figures 2.5 and 2.6 show profit and rates of return, respectively, for case where α = 15, β = 1, C = 3, r = 0.05, K = 50,000, and N1 = 200 as before, but now N2 = 250. The larger Place 2 now makes all options more profitable. When sx = 0, the profit of options ii and iii are the same. At larger values of sx, the larger size of Place 2 means that the profit associated with option iii falls off less quickly than does option ii. Compare curves AH, AB, and CD in Fig. 2.5. Comparing Figs. 2.6 and 2.4, the changes in rates of return are broadly similar to the changes in profit. The main difference here is that, when sx is large, the rate of return in option iii remains above the rate of return for option iv. The rate of return for options ii and iv eventually converge for sx sufficiently large, as these both then become ways of serving Place 1 alone. The implication of Fig. 2.6 is that when Place 2 is larger than Place 1, option iii is everywhere preferable to option ii, but when sx is sufficiently large, the monopolist still builds a factory at the smaller place if the market there is large enough. The rates of return graphed in Fig. 2.6 tell us that, as mutually exclusive alternatives, option iii is always preferable to option ii when N2 > N1 , and that while option iv may generate greater total excess profit
60
2
A
C
The Firm at Home and Abroad
Model 2D: Profit, locational choice, and unit shipping cost AB Profit schedule under option ii: see (2.7.2) AH Profit schedule under option iii: see (2.7.3) CD Profit schedule under option iv: see (2.7.4) EF Profit under option ii at stated unit shipping cost EG Profit under option iii at stated unit shipping cost EI Profit under option iv at stated unit shipping cost OE Stated unit shipping cost (sx)
I
D
Profit
G F
H B
0
E Unit shipping cost
Fig. 2.5 Model 2D: profit as a function of shipping rate. Note: α = 15, β = 1; C = 3; K = 50,000; N1 = 200 N2 = 250, r = 0.05, sx = 4. Profit is 8,700 under option ii, 9,700 under option iii, and 11,200 under option iv. Horizontal axis is scaled from 0 to 14; vertical from 0 to 16,000
than iii, the opportunity cost of capital has to be sufficiently low to make option iv attractive.63 Finally, because I have assumed that the two places have a common currency and no restrictions on the shipment of commodities other than a unit shipping cost, I do not have to consider here the strategy of tariff jumping, where a firm builds a second factory to avoid tariffs or other restrictions on trade. This is a rich literature on its own.64 Let me close this section with an observation about the use of shipping here. Our firm will ship either (1) nothing at all or (2) from the home to the remote market only. Since the remote market would then normally have a price higher than the home market by half-freight, the shipment is said to flow up the price gradient. There will never be a shipment in the opposite direction: i.e., a cross haul or shipment down the price gradient. This is not surprising. To have both shipments from A to B and
63 Backhaus
(2002) makes a similar point about the importance of opportunity costs in location theory. 64 The topic of tariff jumping appears to have begun with Brander and Spencer (1987), Motta (1992), and Neven and Slotis (1996). See also Belderbos (1997), Belderbos and Sleuwaegen (1998), Horn and Persson (2001), Hwang and Mai (2002), Neary (2002), and Norback and Persson (2004).
2.10
Comparative Statics in Model 2D
A
61
Model 2D: Rate of return, locational choice, and unit shipping cost AB Rate of return schedule under option iii AC Rate of return schedule under option ii DE Opportunity cost of capital (r) FG Rate of return under option iv at stated unit shipping cost FH Rate of return under option ii at stated unit shipping cost FI Rate of return under option iii at stated unit shipping cost OC Rate of return schedule under option iv OF Stated unit shipping cost
I
Rate of return
H
B
C
G E
D
0 F
Unit shipping cost
Fig. 2.6 Model 2D: rate of return as a function of shipping rate. Note: α = 15, β = 1; C = 3; K = 50,000; N1 = 200 N2 = 250, r = 0.05, sx = 4.00. Rate of return is 0.224 under options ii, 0.244 under option iii, and 0.080 under option iv. Horizontal axis scaled from 0 to 14; vertical from 0 to 0.35
from B to A happening at the same time would presumably require plants at both places; in such cases, why would the firm ship at all?
2.10 Comparative Statics in Model 2D Comparative statics are more difficult to assess in Model 2D than was the case earlier in Models 2A through 2C. The reason for this is that the outcome in Model 2D is sometimes Model 2A, sometimes Model 2B (or its equivalent assuming one factory only at Place 2), and sometimes Model 2C. As long as the change in the exogenous variable or parameter can be solved with the same model, the comparative statics are shown there. However, if the change in the exogenous variable or parameter
62
2
The Firm at Home and Abroad
causes us to switch from one of Models 2A through 2C to another, the comparative statics are more complicated. I will not belabor that analysis here. I make only one comment. Fortunately, as evidenced in Fig. 2.3 through Fig. 2.6 in the case of the shipping cost, there appear to be at most three switches possible in a regional economy where N1 and N2 grow over time: a switch from no production to a factory at one place, a switch from a factory at one place to a factory at the other place, and a switch from one factory to two factories.
2.11 Risk Aversion and Multiple Plants In Models 2A through 2D, I assume that all outcomes are known with certainty. It is beyond the scope of this chapter to model uncertainty here as exemplified in the implicit treatment of goodwill above. However, I can speculate how some particular kinds of uncertainty might affect our conclusions. Suppose a factory suddenly were to become unusable or more costly to operate, customer demand at either Place 1 or Place 2 were to grow unexpectedly, or the cost of shipping between the two places were to change or even become prohibitive. Option iv gives the firm flexibility to cope with such changes and therefore might be a more attractive to a firm averse to risks. Put differently, where options ii or iii are modestly more profitable than option iv, a risk-averse firm might be willing to forego that profit to guarantee that it has the ability to participate profitably.65 Of course, we should keep in mind that these are only a few among many risks facing a firm.
2.12 Model 2E: Contestability and Preemption of Competitors Is there a case to be made here for behavior by a firm that we might label market preemption or entry deterrence?66 We have seen here that building just one factory (by spreading fixed cost over two markets) is more profitable than building two separate factories (each with that same fixed cost) unless the shipping rate is too high. However, this raises the question of how the firm behaves given the risk a competitor builds a factory at the second (smaller) market in the firm’s absence to serve just that market. For the firm with a factory only at the larger Place 1, its profit at risk is (2.3.6). In effect here, I am treating Place 2 as a contestable market.67 We could set this up as a sequential game played by the firm (the incumbent) and a competitor (the entrant). I do not do that here. My purpose is simply to show that the models developed in this chapter give an interesting insight into the conditions under which the incumbent might want to preempt: i.e., build a factory of its own at
65 See
Penfold (2002).
66 For related work in this area, see Aguirre, Espinosa, and Macho-Stadler (1998), Eaton and Lipsey
(1979), Hohenbalken and West (1986), Schmitt (1993), Serra and ReVelle (1994), West (1981), and Ziss (1993). 67 See Baumol, Panzar, and Willig (1982).
2.12
Model 2E: Contestability and Preemption of Competitors
63
Place 2 to forestall the entrant even though that would be less profitable than option ii absent competition. In what follows, assume the following specific scenario. An entrant with the same costs (i.e., same r, C, and K) and product plans a factory at the smaller Place 2 to sell soap there only. Prompting the entrant here is the gap (0.5sx) between the delivered price of the incumbent (absent the entrant) and the price the entrant intends to set. Despite the fact that the incumbent finds it profitable to ship to Place 2, the entrant does not plan to ship soap to Place 1. The entrant is myopic.68 Its conjectural variation is that it does not expect the incumbent to respond by changing its price at Place 2 or by building a second factory there. That entrant sets a price (2.8.1) and expects to earn an excess profit (2.8.2) similar to the one-market solutions (2.1.6) and (2.1.8), respectively. See Table 2.8. Here, price and profit are invariant with respect to sx because I have assumed the entrant does not ship product to Place 1. There are two implications here for the incumbent. First, the entrant will find Place 2 profitable only if demand there is sufficient. The necessary condition can be inferred from (2.8.3): akin to the one-market solution (2.1.9). If Place 2 is smaller than this, the incumbent need not worry about an entrant. If Place 2 is larger than this, consider a second implication: the entrant’s profit-maximizing price (2.8.1) at Place 2 would be lower than that of the incumbent (2.3.4). Absent any difference between their products, the entrant would therefore capture all the market at Place 2. However, is there a catch here? Since I have assumed that the entrant is like the incumbent in the sense of facing the same costs, whenever it is profitable for Table 2.8 Model 2E: market pre-emption Profit-maximizing price of competitor with factory at Place 2 P2 = 0.5(α + C)
(2.8.1)
Maximized profit of competitor with factory at Place 2 0.25N2 (α − C)2 /β − rK
(2.8.2)
Minimum number of customers required by competitor with factory at Place 2 N2 ≥ 4βrK/(α − C)2
(2.8.3)
Rate of return for incremental option iv (V22 − V12 )/K
(2.8.4)
Reduced form for rate of return for incremental option iv (0.25N2 /(βK))( − sx + 2(α − C))sx
(2.8.5)
Notes: Rationale for localization (see Appendix A): Z1—Presence of fixed cost; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Marginal unit production cost; K—Capital required to build factory; Ni —Number of consumers at Place i; r—Opportunity cost of capital; s—Cost of shipping one unit of soap one kilometer; x—Geographic distance from Place 1 to Place 2; α—Intercept of individual linear inverse demand curve: maximum price; β—Negative of slope of individual linear inverse demand curve: marginal effect of quantity on price received. Outcomes (endogenous): Pi —Price of unit of soap at Place i; Qi —Quantity of soap supplied to Place i; Vij —Semi-net revenue from serving customers at Place j from factory at Place i. 68 Any behavior of a firm such that it does not foresee any reaction by its competitors to its choices:
e.g., with respect to price, range, or quality of commodities sold, or geographic location.
64
2
The Firm at Home and Abroad
an entrant to locate at Place 2, wouldn’t it also be profitable for the incumbent to use option iv: i.e., also have a factory at Place 2? If so, market preemption would not be a concern for the incumbent. Interestingly, there is indeed a catch. Where N2 < N1 , the rate of return for incremental option iv in (2.7.7) reduces to (2.8.4) and then to (2.8.5). For the incumbent, the profit of building a separate factory at Place 2 is based on a differential in semi-net revenues (V22 relative to V12 ), whereas for the entrant it is based on V22 alone. When sx is prohibitively high, V12 is zero and the incumbent and entrant would experience the same gain in profit from having a factory at Place 2. However, when sx is smaller, the rate of return on incremental option iv is less than the rate of return to the entrant. Put differently, an entrant may find it profitable to build at the smaller market even when it is unprofitable for the incumbent to do so. What is the potential loss in profit to the incumbent under option ii that arises should the entrant take over the market at Place 2? Assume here N2 satisfies at least (2.8.3). If sx is sufficiently large, the incumbent finds its rate of return on incremental option iv69 is at or above r; it therefore invests in a separate factory at Place 2 and incurs no loss. In Fig. 2.7, I show the opportunity cost of capital as the horizontal line HI: i.e., the same regardless of shipping rate. When the rate of return on incremental option iv is larger than r (i.e., for sx larger than OS in Fig. 2.7), the firm builds a second factory at Place 2 and there is no risk. However, at a unit shipping cost smaller than OS, there is an incentive for the entrant to build at Place 2. See the curve DQR showing the option ii rate of return with the entrant at Place 2. The curve UVS in Fig. 2.7 shows the loss to the incumbent because of the entrant. The loss here is greatest (amount OU) at sx near 0 and smallest (amount SV) when sx approaches the amount above which the incumbent would build a second factory. At the same time, remember that the entrant is less interested in investing in a factory when sx is close to zero because the price it would set would be almost the same as the incumbent’s; the entrant would be happier investing where it would a substantial price advantage: i.e., when unit shipping cost is closer to OJ. What exactly is the loss here? Suppose the unit shipping cost is somewhere in the interval between O and S in Fig. 2.7: say at point J. Absent the entrant, the incumbent earns a return of JN based on option ii; the incremental return on option iv is JK which is below the opportunity cost of capital OH. If the entrant now builds at Place 2, the return to the incumbent on option ii drops to JP and its loss is JY: note amount JY equals amount PN. Suppose the incumbent wants to mitigate the risk, in advance of the arrival of the entrant (i.e., preempt the competitor), by building a second factory at Place 2. The incumbent’s returns would then be JP on option ii and an incremental JT on option iv. As shown, both are above OH (the opportunity cost of capital). Preemption is profitable here, although not as profitable as option ii. Nonetheless, the incumbent might prefer the lower profit of preemption because it reduces the risk of loss. In Chapter 9, I return to the topic of locational decision making under uncertainty. What does this model tell us about prices at the two places? If the firm preempts a competitor by building a factory at each place, the firm will sell the good at the
69 See
(2.7.7)
2.13
Final Comments
65 Model 2E: Rate of return and presence of competitor Rate of return schedule under option ii if no competitor Rate of return schedule under option ii if competitor Opportunity cost of capital schedule Rate of return under option iv at stated unit shipping cost if no competitor Rate of return under option ii at stated unit shipping cost if no competitor Rate of return under option ii at stated unit shipping cost if competitor Rate of return under option iv at stated unit shipping cost if competitor Rate of return schedule under option iv if no competitor Opportunity cost of capital Stated unit shipping cost Unit shipping cost above which return on option iv is above r Risk (loss) if competitor Rate of return on option iv if competitor
Rate of return
A
AB DQR HI JK JN JP JT OC OH OJ OS UVS UWX
D U H
N R P T
X
K
Y 0
Q W
B C I
V J
S Unit shipping cost
Fig. 2.7 Model 2E: rate of return on option ii as a function of shipping rate and presence of competitor at Place 2. Note: α = 15, β = 1; C = 3; K = 50,000; N1 = 200 N2 = 180, r = 0.05, sx = 4. Rate of return is 0.045 on option iv, 0.78 on option 3, 0.083 on option 2 without competitor, and 0.068 on option 2 with competitor. Horizontal axis is scaled from 0 to 9; vertical from 0 to 0.16
same price at both locations. Put differently, despite shipping costs that are not high enough to warrant option iv, preemption drives the firm to price as though they were.
2.13 Final Comments In this chapter, the principal model has been 2D. I included Models 2A through 2C to help readers better understand aspects of Model 2D. I included Model 2E to show how one thinks about entry deterrence at a remote place. In Table 2.9, I summarize the assumptions that underlie Model 2A through 2E. Many assumptions are common to all these models: see panel (a) of Table 2.9. The models differ in that (i) a remote place (market) is assumed in Models 2B through 2E and with it the possibility of price discrimination, (ii) even then, shipping cost can still be ignored in Model 2C because the firm produces at both places, (iii) Models 2D and 2E give the firm the choice of how many factories to build and where to build them, and (iv)
66
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The Firm at Home and Abroad
Table 2.9 Assumptions in Models 2A–2E Assumptions (a) Assumptions in common A1 Closed regional market economy A3 Punctiform landscape B1 Exchange of soap for money B4 Local demand C2 Fixed local customers C4 Identical customers C5 Identical individual linear demand curve D1 Monopolist D6 Fixed cost in the form of interest on fixed capital requirements D7 Horizontal marginal cost curve (b) Assumptions specific to particular models H2 Location(s) of firm given C3 Fixed remote customers M2 Price discrimination E5 Firm bears shipping cost to market E4 Unit shipping cost symmetric D4 Choice of factory locations F1 Competitor sells same product F4 Competitor sells at price that ignores competition F7 Location of competitor is given (Place 2) F9 Peer competitor in remote market F10 Customers shared
2A [1]
2B [2]
2C [3]
2D [4]
2E [5]
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x
x
x
x
x
x
x x x x
x x x x x x
x x x x x x x x x
Model 2E allows for competition from a new entrant at the remote place and the uncertainty that creates. Given the assumptions of the model introduced here, where the number of customers is sufficiently large—which in turn may require that the two places be sufficiently close together—there will be at least one factory, and it will be located at the place with the larger number of customers (the home market). If the unit shipping cost is sufficiently low, there will be localization (i.e., only one factory) and soap will be priced differently at the two places (the half-freight rule). If unit shipping cost is sufficiently high, and the smaller place has enough customers, the firm will decentralize production (i.e., build a second factory there) and prices will be the same at the two places. Put differently, localization (of production) and price differentiation are joint outcomes in Model 2D. In Chapter 1, I argue that prices are important in shaping the location of firms. In this chapter, those prices are r, the price of a unit of capital inherent in our measure of K, various input prices subsumed in C and s, as well as consumer income (typically derived from a labor market) and the prices of other consumer commodities that are presumably incorporated in α and β. These other prices are all determined in markets outside the scope of this model where the firm competes against other firms and demanders. These input prices in turn help determine which of the four
2.13
Final Comments
67
options (each a combination of locations and prices) is most profitable to the firm and thereby the extent of localization. To this point, I think Walras would argue that the analysis has been only partial in the sense that we have not looked explicitly at the simultaneity among prices in these markets. In later chapters, there will be opportunities to do that. Nonetheless, even here, the decision of the firm to choose option ii or iii implies that the price of soap will differ between the two places. To the extent that the firm does not choose option iv, this price effect has the potential (not explored in this chapter) to create an incentive for customers to relocate to a place where the firm has built a factory. What about the regional economy here? There are two problems. First, what is the region here? Second, what is the nature and extent of economic activity within it? The models in this chapter paint a picture of only one part of regional economy: one firm and its customers. We don’t know anything about overall regional product or regional income (factor payments); however that region might be defined. The models in this chapter say little about the distribution of income in society among factors (labor, capital, and land). We know how much profit the firm will earn. We know also that the firm incurs costs, but (aside from the opportunity cost of capital) these are not related explicitly to factor payments such as the hiring of labor or the rental of land. Therefore, the only recognizable income gains that arise accrue to the firm owner in the form of increased profit. We also know that the price set by the firm affects the well-being of customers; in this chapter, such changes have been measured by consumer surplus. Suppose Place 1 is large enough by itself to make production profitable for the firm, Place 2 is not, and the firm therefore builds a factory at Place 1. Assume also initially that the unit shipping cost is prohibitive in the sense that C + sx > α. The firm is limited to its home market. In the absence of information about other firms and their shipments, we might think of the region here simply as the firm and its customers at Place 1. However, we do know that the firm benefits from this market (as indicated by its monopoly profit) as do consumers at Place 1 (as indicated by their consumer surplus). Now suppose the unit shipping cost were to drop to a level where it becomes profitable for the firm to serve the Place 2 as well. In this case, we might think that the region now includes Places 1 and 2. We still don’t know anything about overall regional product or regional income. However, we do know that there has been an addition to the firm’s monopoly profit, a new consumer surplus for customers at Place 2, and no reduction in consumer surplus for customers at Place 1. Finally, this chapter has put much emphasis on the role of the opportunity cost of capital as a fixed cost. That the firm has to invest capital in one or two factories and incurs an opportunity cost thereby implies that unit cost falls the greater the scale of production. That is the one and only rationale for localization envisaged in this chapter. There is not, for example, any explicit consideration of the efficiencies that might arise from division of labor in accounting for localization here.
Chapter 3
Logistics and Programming Getting the Commodity to Customers (Hitchcock–Koopmans Problem)
A firm has several factories that differ in terms of capacity and unit cost of production. The firm sells its product at customer places, each with its own fixed quantity demanded. How much does it ship from each factory to each customer place? To solve this problem, linear programming is introduced. In Model 3A, the firm incurs production costs, but unit shipping cost is zero everywhere. Model 3B includes a unit production cost and a unit shipping cost that varies from one customer place to the next for each factory. In addition to allocating production to customers, linear programs like 3A and 3B generate shadow prices: one for each constraint. The shadow price on the capacity constraint at factory i tells us how much money the firm could save (i.e., additional profit it could earn) if only it had one more unit of capacity there. This is a kind of opportunity cost. There was no congestion (i.e., restriction on production) in Chapter 2. The models in Chapter 3 help us to better understand congestion and its impact on locational decisions (how much, if any, to produce at each factory). In this chapter, localization and the shadow prices on capacity (one for each factory) are joint outcomes of cost-minimizing behavior.
3.1 The Hitchcock–Koopmans Problem In Chapter 2, I considered an efficient firm that sells at two places and operates either a factory at one place only or alternatively a factory at each place. At each factory, the firm manufactures soap—from materials purchased elsewhere and processed onsite—and packages it for sale to customers. The firm there was assumed—having invested K dollars in land, plant, and equipment—to have an unlimited wellspring of production: it can expand its output as needed to meet customer demand. In this chapter, I envisage a firm that might start business with one factory and then, over the years, add factories (establishments) at other places by purchase of assets or by construction and/or reorganize production activity including the division of labor across establishments. In so doing, I assume that the firm has maximized profit in setting the price for its product at each customer place. In this chapter, let me again take a single firm producing soap and operating in a fiat money economy but model it now differently. First, let us assume that there are potentially more than two places (customer points) served by our firm. Second, J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_3, C Springer Science+Business Media, LLC 2010
69
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3 Logistics and Programming
let us now take into account production capacity. In this chapter, I assume each factory—including land, plant (buildings, improvements, facilities, and fittings), and equipment—has been optimized for a particular production technology and scale of production (capacity). Implicit here is a notion of economies of scale; that is, the firm constructs a factory to take advantage of the indivisibilities possible at that scale of production. Once the factory is built, we might reasonably expect a firm to experience congestion at a scale of production above capacity in the short term: that is, to be unable to further increase output without substantially increasing its marginal cost of production. In this chapter, I present what is generally labeled the Transportation Problem (also known as the Hitchcock–Koopmans Problem).1 Koopmans initially solved this model during the Second World War. Hitchcock developed an early formulation of the problem. In the 1950s, Dantzig recast this as the total cost, which allowed for it to be solved using newly developed simplex algorithm.2 Nowadays, such problems are readily solved using off-the-shelf optimization software.3 In this problem, imagine you manage logistics4 for this firm over the short run. Assume you have I factories in total, each producing soap. Assume each factory has a fixed capacity (rate of output), say Si units of soap weekly for factory i. As manager, you can set any level of production at factory i for the week from zero up to but not exceeding Si .5 In this chapter—as in Chapter 2—assume a punctiform landscape wherein the firm’s customers are distributed across a given set of J places. 1 In this Linear Program, the firm seeks to minimize the sum of the costs of production at a set of factories, subject to capacity constraints (one for each factory), and the shipments from these factories to meet the demand requirements of customers at each distinct places. See also Appa (1973, p. 79) on the early history here. It was originally solved using a Stepping Stone Algorithm. In general, linear programs are solved using the Simplex Algorithm. Hitchcock (1941) laid out the problem and sketched its solution. Among others who also contributed in important ways to the Transportation Linear Program and its solution were Boldyre, Dantzig, Ford and Fulkerson, Kantorovich, Robinson, and Tolstoi. Publication of solutions came shortly after the Second World War. Koopmans (1949) gave an economic interpretation of the problem and its solution that was further elaborated in Koopmans and Reiter (1951). Orden (1952) was the first to use the conventional notation now used for the model. An early application of the Hitchcock-Koopmans Problem is found in Henderson (1955). Modern applications include Bullard and Engelbrecht-Wiggans (1988), Rautman, Reid, and Ryder (1993), and Cunha and Mutarelli (2007). 2 The use of simplex here is confusing. Dantzig had originally explored the use of simplexes to solve linear programs, but the algorithm he eventually used (the simplex algorithm) did not incorporate simplexes. However, others used simplexes in the fixed point approach to estimating economic equilibrium. See Kakutani (1941). 3 A well-known early application is found in Harris and Hopkins (1972) and Harris (1973). This model forecasts investment, production, and employment at the county level and uses the shadow price on capacity constraints for each industry sector to help predict future investment locally. See also Harris (1980), Harris and Nadji (1987), and the critique by Fjeldsted and South (1979). 4 The management of production, inventory, and shipments so as to enable a firm to achieve its objectives. 5 Green, Cromley, and Semple (1980) extend the Hitchcock–Koopmans model to allow for the possibility that there might be a minimum level of capacity utilization at each factory. In other
3.1
The Hitchcock–Koopmans Problem
71
Suppose further customers at each place demand a fixed amount (say Dj units at Place j) of soap for the coming week. Further assume, aggregated across all demand places, total demand (j Dj ) for next week is less than total capacity (i Si ). In the models in Chapter 2, we did not know what determined the level of demand. I assumed a downward sloped demand curve: see curve AB in panel (a) of Fig. 3.1. Here in Chapter 3, I assume a vertical demand curve (also labeled AB) shown in panel (b) of Fig. 3.1. Unlike Chapter 2, demand is fixed; it does not depend on price. The objective here simply is to find the least costly way of meeting the given demands of these customers.6 Note here the implication that the firm is assumed to receive a given but unstated price at the destination for its soap. Perhaps this is because the firm is a monopolist as in Chapter 2, has calculated the delivered price using discriminatory pricing that maximizes profit for that part of the market, and now seeks the least-costly way of supplying customers there.7 Alternatively, the firm might be in a perfectly competitive market at the destination; it cannot affect the price it receives but nonetheless wants to minimize the cost of serving customers there.8 In passing, note the contrast
A
0
Price
Price
A
B Quantity
(a) Demand curve (AB) in Chapter 2
0
B Quantity
(b) Demand curve (AB) in Chapter 3
Fig. 3.1 Aggregate demand curves in Chapters 2 and 3 contrasted
words, they allow for a level of production at each factory that has both a maximum (the capacity constraint) and a minimum (to ensure efficient operation). 6 For discussion of the transportation problem in continuous space which includes the transportation problem on a line, see Beckmann (1952) and McCann (1999). 7 In practice, it is not clear how this might be done. A monopolist sets price so that marginal revenue equals marginal cost. However, we don’t know marginal cost here until we determine which factory is to provide the shipments to a given customer. 8 The model is silent on the possibility of arbitrage; however, the assumption of a fixed price is perhaps easier to imagine under the assumption that there is only a local demand at each destination.
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3 Logistics and Programming
here in disciplinary perspectives: the absence of price makes economists uncomfortable; management scientists and logisticians, on the other hand, see the model as helpful. The firm’s objective therefore is not to find the most profitable set of prices at the customer places. Rather, the firm’s problem is to find a physical solution—the quantity to ship from each factory to each customer’s place—such that all demands are met, and the level of shipments do not exceed the capacity of each factory. In this chapter, I look at how constraints on factory capacity affect production and shipments. This is cast and solved as a quantity problem—what to produce and where to ship. This is in contrast to Chapter 2 where production and shipments were solved in terms of the price at each location that would bring about a particular level of supply locally. That is not to say that price is not important here: rather, it is just not part of this model. This may seem strange at first, since the book is about prices and geography. However, I will make clear the connection to prices shortly. Because I look at this firm in the short run, I assume the marginal cost of production is fixed for factory i at Ci per unit produced for any quantity between zero and Si . Put differently, the firm ignores any fixed cost associated with each factory. The marginal cost curve for factory i is a horizontal line between quantities zero and Si . There are no economies of scale here and fixed costs are ignored. To allow for the concept that no weekly quantity greater than Si is possible, assume the marginal cost of production for that factory then becomes vertical at quantity Si . Thus, the firm here differs from a firm in Chapter 2. In Chapter 2, the firm is assumed to have a constant marginal cost of production (no congestion in production): see the horizontal line AB in panel (a) of Fig. 3.2. In the ensuing Chapter 4, the industry is assumed to have an increasing marginal cost of production (some congestion in production). In the present chapter, the factory has a constant marginal cost of production (zero congestion) up to its capacity S—see the horizontal curve segment AB in panel (b) of Fig. 3.2—that becomes a vertical curve segment BC, at S; congestion at the factory makes it impossible to produce any more quantity than S there regardless of the customer’s willingness to pay. With normal wear-and-tear, plant and equipment typically become increasingly costly to maintain as the years go by.9 As well, obsolescence arises because of changes in manufacturing technology, scale of production, production and distribution efficiencies, market for soap, and consumer preferences. Presumably, the firm designs, locates, and builds new factories with these changes in mind, and closes older factories. Why does the firm have more than one factory at a time? • Because of considerations like those in discussed in Chapter 2, the saving in shipping costs exceeds the opportunity cost of the capital required from having an additional factory;
9 Perhaps, land too becomes depleted (e.g., as a result of quarrying associated with production) or contaminated.
3.1
The Hitchcock–Koopmans Problem
73
Marginal cost
Marginal cost
C
A
B B
A
0
Output
(a) Marginal cost curve (AB) in Chapter 2
0 S
Output
(b) Marginal cost curve (ABC) in Chapter 3
Fig. 3.2 Marginal cost curves in Chapters 2 and 3 contrasted
• The firm might expect ordinarily to meet demand entirely from its new, more efficient, factories but retains older factories in case it unexpectedly needs more capacity or loses use of a newer factory temporarily; • Unlike Chapter 2, each factory is designed to operate most efficiently at a particular scale of production—its design capacity—and is simply not capable of producing any and all output needed. As a result, the firm may have factories available that range from more efficient (state of the art) to less efficient (obsolete). In the short run, say one week or one month, the firm can do little to alter a particular factory. Largely, it can adjust only the amounts of materials and labor used in production at the factory or not produce there at all. These variations are reflected in Ci for each factory. At an obsolescent or older factory, unit production cost might, therefore, typically be higher than at a newer factory. In a longer run beyond the scope of this model, the firm may have the opportunity to repair, upgrade, or remediate land, plant, and equipment at an existing factory. Assume further the firm incurs a unit shipping cost for each unit of soap shipped from factory i to customers at j. Assume this is a constant unit cost, sij . By constant, I mean once again that the cost for each unit shipped does not depend on how much is shipped: e.g., no economies of the large haul.10 I assume here that (1) there is a minimum efficient size of shipment, (2) the firm ships annually multiples of this amount, and (3) the firm decides how many multiples (one or more) to ship at any one time. Let qij be the amount of soap produced weekly at i for shipment to 10 In
an early empirical study of this kind, Chisholm (1959) finds no evidence of economies of scale in road goods transport in his case study of off-farm milk collection in England.
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3 Logistics and Programming
customers at j. The shipping cost for this particular shipment is sij qij and the total shipping cost for all shipments by the firm over the week is i j sij qij .11 The total cost of production and shipment weekly is i j (Ci + sij )qij . Your job as manager is to find the least-cost allocation of factory production to customer demand under the assumptions that demand at every customer place is satisfied and that capacity at every factory is not exceeded. The problem can now be written as in Table 3.1: there, I summarize the equations, assumptions, notation, and rationale for localization in Model 3B. Table 3.1 Model 3B: Hitchcock–Koopmans problem Minimize production plus shipping costs Z = Σi Σj (Ci + sij )qij
(3.1.1)
Subject to the following conditions Shipments must satisfy demand at each customer place Σi qij ≥ Dj at each j = 1, 2, . . . , J
(3.1.2)
Shipments from each factory must not exceed capacity Σj qij ≤ Si at each i = 1, 2, . . . , I
(3.1.3)
Each possible shipment must be either zero or positive qij ≥ 0 for each combination of i and j
(3.1.4)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z5—Capacity constraints and congestion. Givens (parameter or exogenous): Ci —Unit production (variable) cost for Factory i; Dj —Amount of soap demanded at Place j in period of time; I—Number of factories (supply points); J—Number of places (customer points); Si — Amount of soap that can be produced at Factory i in period of time. Outcomes (endogenous): qij —Amount of soap shipped to Place j from Factory i in period of time; Z—Total variable cost of producing and shipping soap.
Above, I introduced the concept of the short run vs. the long run and then stated that the logistics manager is concerned mainly with the short run. The short run is the focus of the remainder of this chapter. However, before I consider specific versions of the model, it is helpful to think how much different the analysis would be in a longer run perspective. In the short run, assume the firm cannot alter anything: buildings, improvements, facilities, equipment, and fittings, or a factory design built around a particular technology and scale of production. Therefore, there may be a substantial difference between unit production costs at any two factories. In the longer run, the firm can make an investment: e.g., replace or add equipment, facilities, and buildings, and/or redesign a factory to increase its capacity. This should bring unit production cost (i.e., reduce Ci ) at the more costly factory down closer to that of the more efficient factory, or reduce shipping cost by enabling more production at a more efficient factory (i.e., increase Si ). In making that investment decision, the firm presumably uses a criterion similar to that discussed in Chapter 2,
11 Note the asymmetric treatment of congestion here; the model takes into account factory capacity
but otherwise assumes no congestion over a transportation network.
3.2
An Illustrative Example
75
namely, “Is the reduction in production and shipping cost sufficiently large to offset the opportunity cost of the investment.” However, even in the long run, differences among factories arise, for example, because inputs are less expensive for a factory at one place compared to another (i.e., one or more inputs are not ubiquitous). See, for example, Chapter 6. Because I look at this firm in the short run, differences in marginal cost curve from factory to factory and differences in costs of shipping to a consumer are givens; that is, cannot be affected by the firm. Under this interpretation, more efficient factories generate excess profits compared to the marginal factory12 employed. To the extent such excess profits arise because of the uniqueness of an input (e.g., an advantageous location), these are also Ricardian Rents. In allocating output from factories to the demand at a particular customer place therefore involves considerations of opportunity cost. In this chapter, I introduce the concept of a shadow price—a mathematical term—to implement the idea of opportunity cost here. As in Chapter 2, this chapter includes material that would normally be part of optimal location theory. Later in this chapter, I show how optimization by a firm produces geographic outcomes—the allocation of its factories to customer demand—that appear to be as though the firm’s factories compete with one another. Put differently, in trying to be efficient, the monopolist allocates production and shipments in a way that replicates a competitive market.
3.2 An Illustrative Example To illustrate the problem with a specific example, suppose you have customers at three distinct customer places (labeled A, B, and C) that you can serve from any combination of four factories (labeled 1, 2, 3, and 4) laid out on a rectangular plane: see the map in Fig. 3.3. The pertinent details on demand, capacity, production costs, and shipping costs are shown in Table 3.3. I have assumed here that shipping cost is proportional to distance measured as the crow flies. I have constructed this example to help clarify the application of the Hitchcock–Koopmans model. I intentionally placed the factory with the lowest unit production cost (factory 1) far away from the customer place where much of the demand is concentrated (namely, Place C). After all, you might rightly ask why the most efficient factory is so far away from Place C; presumably the firm might have wanted to build the factory closer to C to save on shipping costs. Where inputs are ubiquitous, the firm would locate close to C on the line joining C, B, and A that could be drawn on the map in Fig. 3.3. Presumably, the reason here might be that either (i) there is a non-ubiquitous input that is less costly near the site of factory i or (ii) the spatial pattern of demand has shifted over time since factory 1 was built so that now customer demand is typically further away.
12 For
a firm with multiple factories or an industry composed of multiple firms each with its own factory, the marginal factory is the factory, which produces the last (most costly) unit to the market.
76 Fig. 3.3 Model 3B: map of example problem showing factories (labeled 1–4) and customer places (labeled A–C)
3 Logistics and Programming 1
2
A B 3
C
4
0 km
50 km
100 km
I constructed this example with specific ideas in mind. First, I deliberately put the two factories with the lowest unit production costs far away from customer places B and C where demand is relatively large. I did this to exemplify the tradeoff for the firm between production and shipping costs. Second, for ease of exposition, I have set the factory capacities in this example problem so that the firm will always need three factories to meet the aggregate demand (4,200 units of soap weekly). Therefore, one factory will always be unused. Also, one of the three factories used will be marginal: that is, the factory whose output would be reduced by a unit if aggregate demand was reduced by one unit (alternatively, in general, the factory is producing, but with slack; that is, producing below its capacity).13
3.3 Model 3A: Non-spatial Version of the Model Let us begin to analyze this problem by considering a special case in which unit shipping cost (sij ) is everywhere zero. See Table 3.2. In that situation, our model becomes non-spatial since it does not matter where our customers or our factories happen to be. Sum the demand at each customer place to give an aggregate quantity demanded, QD = j Dj . I can also derive an aggregate supply curve for the firm. Without loss of generality, index factories from lowest unit production cost to highest so that C1 ≤ C2 ≤ . . . ≤ CI . Thus, for any aggregate quantity demanded up to S1 , the marginal cost is a constant C1 . Beyond that and up to a quantity of S1 + S2 , the marginal cost is C2 . Beyond that and up to a quantity of S1 + S2 + S3 , the marginal cost is C3 , and so on. This yields a kinked14, 15 supply curve, also known 13 I
ignore here the knife-edge possibility that two or more factories have the same unit cost (production plus shipping) and may therefore both qualify as marginal factory. 14 A condition wherein a function (e.g., demand schedule, supply schedule, excess supply, excess demand, or Price Difference Curve) takes the form of a polyline or piecewise-linear spline function: i.e., forms a continuous function with a discontinuous derivative. 15 Samuelson (1952, p. 286) presents a diagram in which local demand and local supply curves are each kinked, but offers no explanation or interpretation of this.
3.3
Model 3A: Non-spatial Version of the Model
77
Table 3.2 Model 3A; production problem Minimize production costs Z = Σi Σj Ci qij
(3.2.1)
Subject to the following conditions Shipments must satisfy demand at each customer place. Σi qij ≥ Dj at each j = 1, 2, . . ., J
(3.2.2)
Shipments from each factory must not exceed capacity. Σj qij ≤ Si at each i =1, 2, . . ., I
(3.2.3)
Each possible shipment must be either zero or positive. qij ≥ 0 for each combination of i and j
(3.2.4)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z5—Capacity constraints and congestion; Z8—Limitation of shipping cost. Givens (parameter or exogenous): Ci —Unit production (variable) cost for Factory i; Dj —Amount of soap demanded at Place j in period of time; I—Number of factories (supply points); J—Number of places (customer points); Si —Amount of soap that can be produced at Factory i in period of time; sij —Cost of shipping each unit of commodity from Factory i to Place j. Outcomes (endogenous): qij —Amount of soap shipped to Place j from Factory i in period of time; Z—Total variable cost of producing and shipping soap.
as a supply step function. A kinked line is also known as a polyline.16 I then simply move to the right along the horizontal axis until I reach quantity QD , then travel vertically until I intersect the supply step function. Suppose this happens along the horizontal segment of the step function corresponding to factory m. I then know all the more efficient factories (factories 1, 2, . . . , m − 1) will be fully used along with some or all of factory m. Factory m is the marginal factory.17 The amount of production is now clear. I use all of the output available from the m − 1 most efficient factories with the remainder constituting QD coming from the marginal factory m. However, the pattern of shipments, which output to ship to which customers, is indeterminate. Because shipping costs are zero, it does not matter which of the most efficient m factories are used to supply a given customer. So far, we have thought about the firm’s problem from the perspective of logistics: i.e., quantities to be produced and shipped. However, an economist usually finds it helpful to think about prices and/or opportunity costs. How are we to do this however when the problem is set out ignoring price setting by the firm? The answer lies in the idea that each choice of shipment by the firm has a measurable opportunity cost. Compared to the marginal factory, factory 1 is more efficient by the amount Cm − C1 . Let us now calculate how much money the firm could save if only it had one more unit of capacity at any one factory. If the firm had one more unit of capacity at the most efficient factory 1, that is, S1 + 1 instead of S1 , it could
16 A
piecewise-linear spline function composed of two or more segments. Each segment (piece) is itself a linear function. The notion of a spline is that adjacent pieces share an endpoint: i.e., forms a continuous function with a discontinuous derivative. 17 I assume here for simplicity of exposition that C m − 1 < Cm < Cm + 1 .
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3 Logistics and Programming
save Cm − C1 . If the firm had one more unit of capacity at factory 2, it could save Cm − C2 , and so on up to factory m where the savings would be zero (Cm − Cm ) by definition. For any factory from m + 1 to I, the savings would be zero, since each such factory is not currently in use, and therefore added capacity would not help at all. Let us refer to such savings as the shadow price18 of capacity at each factory. These shadow prices implement the idea of an opportunity cost. Note the similarities here among shadow price, excess profit, and Ricardian Rent. If excess profit is arising because of an input unique to factory 1, then that excess profit is also the Ricardian Rent. If the quantity Qd can be sold at unit price P: EP SV RV
Excess profit attributable to factory: e.g., EP1 = (P − C1 )S1 . Aggregate shadow value attributable to factory: e.g., SV1 = (Cm − C1 )S1 . Residual difference between EP and CV (profit earnable on the marginal factory and standardized to the capacity of factory): e.g., EP1 − SV1 = (P − Cm )S1 .
Note that EP = SV + RV. Further RV is the component of excess profit earned by every factory used in production regardless of its own unit production cost. On the other hand, SV is the portion of excess profit that arises because the factory is more efficient than the marginal factory. SV is a Ricardian Rent; the excess profit attributable to the fact that C1 for some reason is lower than the unit production cost at the marginal factory. Of course, a similar calculation can be made for any other factory up to the marginal factory. RV is the excess profitable attributable to the firm as a whole: some advantage that enables to reap a profit even on its marginal factory. This too might be a Ricardian rent depending on why the profit occurs.
3.4 The Example in a Non-spatial Version To illustrate the ideas so far, consider a non-spatial version of the example problem presented in Table 3.3. Here I use all data from Table 3.3 except to assume sij is zero for all flows. In Fig. 3.4, I show the outputs forthcoming from the four factories in ascending order of production cost: see the polyline ABCDEFG. I also show the aggregate quantity demanded in the three customer places as a vertical line at 4,200 units: see PQ in Fig. 3.4. The intersection of this curve with the cumulative step function comes at unit production cost (Cm ) equal to 22: see OM in Fig. 3.4. Therefore, 22 is the shadow price on demand at any of the three places: put differently, if we reduce demand at any place by 1 unit, the firm would save 22. As well, 22 is the unit cost of production at the marginal factory. Therefore, factory 1—the most efficient factory—has a shadow price on capacity of 22−20 = 2: see AM in Fig. 3.4. Factory 2 has a shadow price of 22−21 = 1 : see LM in 18 A standard Linear Programming problem maximizes an objective function linear in endogenous
variables subject to linear constraints and non-negativity constraints on the endogenous variables. A shadow price, one for each linear constraint, is the amount by which the value of the objective function could be increased if only the constraint were relaxed by one unit (made one unit less binding).
3.4
The Example in a Non-spatial Version
79
Table 3.3 Model 3B: example problem Factory i [1]
Unit production cost (Ci ) [2]
Unit shipping cost sij j=A j=B [3] [4]
j=C [5]
Factory capacity weekly Si [6]
1 2 3 4
20 21 23 22
6 6 6 6
7 7 4 4
9 9 4 4
1,000 2,000 2,500 1,500
j=A 200
j=B 1,000
j=C 3,000
Weekly demand Dj
Minimize production plus shipping costs Z = (20 + 6)q1a + (20 + 7)q1b + (20 + 9)q1c + (21 + 6)q2a + (21 + 7)q2b + (21 + 9)q2c + (23 + 6)q3a + (23 + 4)q3b + (23 + 4)q3c + (22 + 6)q4a + (22 + 4)q4b + (22 + 4)q4c
(3.3.1)
Subject to the following conditions Shipments must satisfy demand at each customer place q1a + q2a + q3a + q4a ≥ 200 q1b + q2b + q3b + q4b ≥ 1,000 q1c + q2c + q3c + q4c ≥ 3,000
(3.3.2) (3.3.3) (3.3.4)
Shipments from each factory must not exceed capacity q1a + q1b + q1c ≤ 1,000 q2a + q2b + q2c ≤ 2,000 q3a + q3b + q3c ≤ 2,500 q4a + q4b + q4c ≤ 1,500
(3.3.5) (3.3.6) (3.3.7) (3.3.8)
Each possible shipment must be either zero or positive qij ≥ 0 for i = 1, 2, 3, or 4 and j = a, b, or c
(3.3.9)
Note: See also Table 3.1.
Fig. 3.4. Factory 4 (the marginal factory) and factory 3 (unused) each have a shadow price of zero. In this non-spatial model, total cost of meeting customer demand is 1,000(20) + 2,000(21) + 1,200(22) = 88,400. In Fig. 3.4, the box AMNBA shows the cost savings from Factory 1 and the box CNEDC is the cost savings from Factory 2. I have just determined the shadow prices when Qd is 4,200 units. How would these shadow prices differ if Qd were something else? The idea here is to see how shadow prices change as I go from a relatively low level of aggregate demand to a relatively high level. Again, Fig. 3.4 is helpful here. Outcomes are summarized in Table 3.4 • If Qd is under 1,000 units, use only factory 1. Factory 1 is the marginal factory. The shadow price for each factory is zero. The shadow price on demand anywhere is 20.
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3 Logistics and Programming
Cost
Model 3A: Firm's use of its four factories ABCDEFG AM AMNBA CNEDC HI IJ JK LM OA OH OL OM OQ PQ
Firm's supply schedule from its four factories Shadow price on capacity at factory 1 Cost savings from capacity at factory 1 compared to marginal factory Cost savings from capacity at factory 2 compared to marginal factory Production available from factory 2 Production available from factory 3 Production available from factory 4 Shadow price on capacity at factory 2 Unit production cost at factory 1 Production available from factory 1 Unit production cost at factory 2 Unit production cost at marginal factory 3 Quantity demanded Vertical line
P G
M
L
A
0
N
E
F
D
C B
H
I
Q J
Quantity
K
Fig. 3.4 Model 3A: firm’s use of its four factories. Notes: Horizontal axis scaled from 0 to 8,000; vertical from 18 to 27
• If Qd is between 1,000 and 3,000 units, use both Factories 1 and 2. The marginal factory is factory 2. Capacity at factory 1 now has a shadow price of 1. The shadow price at each other factory is zero. The shadow price on demand anywhere is now 21. • If Qd is between 3,000 and 4,500 units, use Factories 1, 2, and 4. The marginal factory is factory 4. Capacities at Factories 1 and 2 now have shadow price of 2 and 1, respectively. The shadow prices at other factories are each zero. The shadow price on demand anywhere is now 22. • If Qd is between 4,500 and 7,500 units, use Factories 1, 2, 4, and 3. The marginal factory is factory 3. Capacities at Factories 1, 2, and 4 now have shadow price of 3, 2, and 1, respectively. The shadow price at the now-marginal factory 3 is zero. The shadow price on demand anywhere is now 23. What I have just done here is to illustrate how the solution to a problem changes if I were to change a given (in this case, Qd ). This kind of exercise, based on shadow
3.5
Model 3B: Spatial Version of the Model
81
Table 3.4 Model 3B: shadow prices and capacity surplus as Qd is varied Aggregate demand (Qd ) Qd ≤ 1,000
[1]
Min. slack [2]
(a) Capacity at factory 1 0 2 2,000 3 2,500 4 1,500
1,000
3,000
Shadow price (ri ) [3]
Min. slack [4]
Shadow price (ri ) [5]
Min. slack [6]
Shadow price (ri ) [7]
0 0 0 0
0 0 2,500 1,500
1 0 0 0
0 0 0 1,500
2 1 0 0
(b) Demand at Place
A B C
Shadow price (vj )
Shadow price (vj )
Shadow price (vj )
20 20 20
2 21 21
22 22 22
Notes: Min. slack is excess capacity at factory when QD is at maximum of range for that column. Parameter values given in Table 3.3 with the following modifications: DA =(200/4,200)QD ; DB =(1,000/4,200)QD ; DC =(3,000/4,200)QD . Calculations by author.
prices, is what a mathematician would call a sensitivity analysis. To an economist, this is equivalently an exercise in comparative statics. Although shadow prices are not the same as the market prices that have been at the core of this book so far, they are central to a comparative statics analysis here and give valuable insights into how the firm’s production decisions are shaped by the level of aggregate demand.
3.5 Model 3B: Spatial Version of the Model How different is the solution where shipping costs are not zero. This is another possible exercise in comparative statics. Put simply, the answer is that the solution can change a lot! With shipping costs, it now matters which factory is used to supply a given customer. A factory with a low production cost may no longer be an efficient source if shipping costs from it are relatively high.19
19 Transportation
linear programs have been applied widely. Illustrative of the breadth of application, Kuznar (1991) uses such a model to look at herd management among Aymara alpaca herders in the south central Andes in response to improved transportation systems. The focus here is on how herders choose the size and type of herd constrained by the land and labor resources available to them. The study concludes that herders are optimizing herd value, that sheep are a poor option for Andean herders, and that this explains the reluctance of herders to adopt sheep herding.
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3 Logistics and Programming Table 3.5 Primal linear program Maximize Z = Σi ai Xi
(3.5.1)
Subject to Σi bij Xi ≤ cj at each j =1, 2, . . ., J Xi ≥ 0 for each i = 1, 2, . . ., I
(3.5.2) (3.5.3)
Notes: Z is the objective: endogenous. X1 , X2 , . . ., XI are endogenous variables whose values are selected to maximize Z. J is the number of constraints: exogenous. ai , bij , and cj are constants: exogenous.
Linear programming20 can be used to solve any problem (the so-called primal21 ) that can be written as shown in Table 3.5. There, Z is an objective to be maximized, X1 , X2 , . . . , XI are each an instrument variable22 whose value is selected to maximize Z, there are J less-than-or-equal-to constraints on linear combinations of Xi , and each Xi is nonnegative. The remaining terms (ai , bij , and cj ) are constants. This is said to be a Linear Program because both the objective function (3.3.1) and each of the J constraints can be written as a linear function of the instrument variables. Linear programs are solved using numerical algorithms.23 The solution to the primal gives the values of the instrument variables— X1 , X2 , . . . , XI —that optimize the value of Z. In the case of the Hitchcock– Koopmans problem, the instrument variables are the shipments from factories to customer places where customers demand soap. Given those shipments, the firm can then readily calculate the total cost of production and shipment across all factories. There is an important caveat here. A linear program is said to be solved when we find a combination of instrument variable values—X1 , X2 , . . . , XI —that optimize the value of Z. However, such a combination may not be unique. Typically, there are other possible combinations of instrument variable values that also produce the same optimimal value of Z. When we solve a linear program, it is therefore more
20 The
solving of optimization problems in which a linear objective function is maximized or minimized subject to linear inequality and nonnegativity constraints. 21 In Linear Programming, a problem in general cast as a maximization subject to less-than-orequal-to and non-negativity constraints. 22 In a model to be solved as an optimization (i.e., which specifies an objective function to be optimized), an instrument variable is a given whose value is chosen for the purpose of that optimization. For example, given the problem Maxx Y = f (X), Y is the objective function and X is the instrument variable. 23 Several numerical methods have been derived to solve the Transportation Problem. Dantzig (1951a) solved the transportation problem using the simplex algorithm.23 Charnes and Cooper (1954) present an alternative solution using the stepping stone algorithm. Other solution methods are discussed in Houthakker (1955), Ford and Fulkerson (1956), Wagener (1965), Arsham and Kahn (1989), and Gass (1990). Vidale (1956) and Hall (1989) present graphical (map) solutions. See also the review in Charnes and Cooper (1957). In numerical examples in this chapter, I use the Solver procedure in Microsoft Excel 2003/2004.
3.5
Model 3B: Spatial Version of the Model
83
correct to say no other combination of instrument variable values could generate a still better value of Z. There are four sets of givens (parameters and exogenous values) in this model: Ci , sij , Dj , and Si .24 To do comparative statics here, we need to understand how and why the solution changes when we change any of these givens. Imagine, for the moment, a solution to the Linear Program in (3.5.1), (3.5.2), and (3.5.3) and the corresponding value of Z that results. Now, let us go back and solve a second Linear Program, otherwise identical to (3.5.1), (3.5.2), and (3.5.3) except that the right-hand side of the first constraint in (3.5.2) is now one unit larger: c1 + 1 instead of c1 . Put differently, constraint 1 is now one unit less binding. Now, suppose I solve this slightly altered Linear Program. One possible solution is that the instrument variables and the value of the objective remain the same as before. In this case, we say that the shadow price of the constraint is zero; nothing is gained by making the constraint one unit less binding. The second possible solution is that the instrument variables change in value such that the value of the objective is larger than before. In this case, the shadow price of the constraint is positive; here we can improve on Z if the constraint is one unit less binding. You might ask whether it is possible alternatively that Z decreases in value. However, if Z is to be maximized, we can always use the same solution as before; therefore Z cannot decrease when a constraint becomes less binding. The implication here is that a shadow price is never negative. I have described above how to calculate the shadow price on the first constraint; a similar approach can be used to calculate the shadow price on each of the other constraints. In total, to find the optimal solution (X1 , X2 , . . . , XI ) to the original problem plus find the shadow prices for each of the I constraints, I would have to solve I + 1 linear programs. Fortunately, I can reduce the amount of computation here. At the optimal solution (X1 , X2 , . . . , XI ), I may find constraint j has slack25 (i.e., is not binding): b1j X1 + b2j X2 + . . . + bIj XI < cj . In such cases, releasing the constraint by one unit does not change the optimal solution, and hence the shadow price is zero. The complementary slackness theorem states at least one of the slack and/or the shadow price must be zero for each constraint. Therefore, we know every nonbinding constraint must have a shadow price of zero.26 The Duality theorem in linear programming says that for every primal, there is a corresponding dual Linear Program that can be written as shown in Table 3.6.27 Here, Z is the objective, v1 , v1 , . . . , vJ are the instrument variables selected to minimize Z , and there are I constraints on the values of vj , and each vj is nonnegative. The remaining terms (ai , bij , and cj ) are the same constants as in the primal. The 24 In
this chapter, sij is a unit shipping cost. In Chapter 2, s was a unit shipping rate. Linear Programming, the amount by which the left-hand-side of a less-than-or-equal-to inequality is less than the right-hand-side. 26 This has the further implication that the number of variables in the solution to the primal that are nonzero cannot exceed the number of constraints. 27 Among early work on duality, see Gale, Kuhn, and Tucker (1951), Charnes, Cooper, and Henderson (1953), and Wagner (1958). 25 In
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3 Logistics and Programming Table 3.6 Dual linear program Minimize Z = Σ j cj vj
(3.6.1)
Subject to Σj bij vi ≤ ai at each i = 1, 2, . . ., I vj ≥ 0 for each j =1, 2, . . ., J
(3.6.2) (3.6.3)
Note: See also Table 3.5.
Duality theorem has two remarkable conclusions. First, the optimized value of the objective function in the dual, Z , is the same as the optimal value of the objective function in the primal, Z. In the Hitchcock–Koopmans primal, Z is the negative sum of production and shipping costs. So too therefore is Z . The total cost (Z) to be minimized in the primal is the same as aggregating the opportunity cost of each unit of demand served (j Dj vj ) net of the savings made possible by each unit of capacity (i Si ri ). The primal is about quantities (shipments) that are efficient, while the dual is about the shadow prices that underlie this. Put differently, the dual reinterprets firm expenditure as a sum of opportunity costs. Second, the optimal values of the instrument variables in the dual (v1 , v1 , . . . , vJ ) are the shadow prices of the primal. The Duality theorem is important here for two reasons. The lesser of the two reasons is that it sharply reduces the amount of computation. Instead of computing up to J + 1 linear programs to calculate a primal solution and its shadow prices, I need to compute only 2: the primal and the dual. From the perspective of this book, the more important of the two reasons is that the dual gives us an insight into the comparative statics of the allocation process that are not evident from the primal. Let me now explain why in more detail. To see the significance of this latter reason in our case, I need to write the dual of the Hitchcock–Koopmans problem. As a first step, I must rewrite the Problem as a primal. See Table 3.7. Two changes are necessary here. One, as is done in (3.7.1) is to convert the objective from a minimization to a maximization by multiplying through by –1. Minimizing a positive amount is like maximizing its negative; (3.7.1) is equivalent in structure to (3.5.1). The capacity constraints (3.2.3) in the Hitchcock–Koopmans problem already look like the inequalities (3.5.2) in the primal. However, demand constraints (3.2.2), which take the form of greater-than-or equal-to constraints do not. This is remedied by multiplying both sides of (3.2.2) by –1; this automatically converts the inequalities to a less-than-or-equal-to form.28 With these two changes, the Hitchcock–Koopmans problem is now in the form of a primal. The dual to this Problem can then be constructed as shown in Table 3.8. Note here two sets of shadow prices: one set of shadow prices (ri ) for the capacity constraints and another set (vj ) for the demand constraints. I have already shown ri is the cost saving that would arise were the capacity of factory i one unit larger. What 28 See
(3.7.6), (3.7.7), and (3.7.8).
3.5
Model 3B: Spatial Version of the Model
85
Table 3.7 Model 3B as a Primal Linear Program Maximize − Z = − Σi Σj (Ci + sij )qij
(3.7.1)
Subject to − Σi qij ≤ − Dj at each j Σj qij ≤ Si at each i qij ≥ 0 for each combination of iand j
(3.7.2) (3.7.3) (3.7.4)
Example problem Maximize negative of production plus shipping costs Z = − (20 + 6)q1a − (20 + 7)q1b − (20 + 9)q1c − (21 + 6)q2a − (21 + 7)q2b − (21 + 9)q2c − (23 + 6)q3a − (23 + 4)q3b − (23 + 4)q3c − (22 + 6)q4a − (22 + 4)q4b − (22 + 4)q4c
(3.7.5)
Subject to the following conditions Shipments must satisfy demand at each customer place − q1a − q2a − q3a − q4a ≤ − 200 − q1b − q2b − q3b − q4b ≤ − 1,000 − q1c − q2c − q3c − q4c ≤ − 3,000
(3.7.6) (3.7.7) (3.7.8)
Shipments from each factory must not exceed capacity q1a + q1b + q1c ≤ 1,000 q2a + q2b + q2c ≤ 2,000 q3a + q3b + q3c ≤ 2,500 q4a + q4b + q4c ≤ 1,500
(3.7.9) (3.7.10) (3.7.11) (3.7.12)
Each possible shipment must be either zero or positive qij ≥ 0 for i =1, 2, 3, or 4 and j = a, b, or c
(3.7.13)
Note: See also Table 3.2.
about vj ? If the demand constraint at Place j were one unit less binding, the demand there would be only Dj −1. In that case, the firm would be able to save production and shipping costs. That cost saving is vj . It is not really a cost saving; the firm presumably makes a profit by selling that unit of soap and therefore would not want to forego it simply to avoid a cost. Rather, I think of vj as the marginal cost of the last (most costly) unit produced for customers at that place. The second thing the dual tells us is something about the relationship between the two kinds of shadow prices. Key here is the role of (3.8.2). To begin thinking, suppose I have more than sufficient capacity at every factory to satisfy all demands that might be placed on it. In that special case, ri would be zero for every factory by the Complementary Slackness Theorem. As shown, (3.8.2) states ri ≥ vj − (Ci + sij ). Given the objective function in (3.8.1), I want to make each ri as small as possible which means, given the nonnegativity of shadow prices, ri = max [0, vj − (Ci + sij )] for the customer place with the largest vj − (Ci + sij ). Intuitively, this should not be surprising; if there are no capacity constraints, the opportunity cost of supplying the last unit demanded at Place j (vj ) is the cost of acquiring soap from the lowest cost factory (Ci + sij ) and this should imply ri = 0.
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3 Logistics and Programming Table 3.8 Model 3B as a dual linear program Minimize Z = Σi Si ri − Σj Dj vj
(3.8.1)
Subject to ri ≥ vj − (Ci + sij ) at each i and j ri ≥ 0 at each i vj ≥ 0 for each j
(3.8.2) (3.8.3) (3.8.4)
Example problem Minimize Z = 1,000r1 + 2,000r2 + 2,500r3 + 1,500r4 − 2,00va − 1,000vb − 3,000vc Subject to r1 − va ≥ − (20 + 6) r1 − vb ≥ − (20 + 7) r1 − vc ≥ − (20 + 9) r2 − va ≥ − (21 + 6) r2 − vb ≥ − (21 + 7) r2 − vc ≥ − (21 + 9) r3 − va ≥ − (23 + 6) r3 − vb ≥ − (23 + 4) r3 − vc ≥ − (23 + 4) r4 − va ≥ − (22 + 6) r4 − vb ≥ − (22 + 4) r4 − vc ≥ − (22 + 4) ri ≥ 0 for i = 1, 2, 3, or 4 vj ≥ 0 for j = a, b, or c
(3.8.5)
(3.8.6) (3.8.7) (3.8.8) (3.8.9) (3.8.10) (3.8.11) (3.8.12) (3.8.13) (3.8.14) (3.8.15) (3.8.16) (3.8.17) (3.8.18) (3.8.19)
Notes: See also Table 3.2. Here, ri is shadow price on the capacity constraint at factory i. vj is shadow price on the demand at market j.
Alternatively, suppose there are binding supply constraints. In this case, even though factory i may be the lowest cost supplier to Place j, the entire output of factory i may be more efficiently committed to other customer places, and the opportunity cost of supplying Place j may be larger than Ci + sij . Thus, the factory’s opportunity cost of capacity (ri ) is the largest difference vj − (Ci + sij ) among the customer places. This is a remarkable result.29 In Chapter 2, I looked at the outcomes in competitive markets. Assume here that each factory was an independent, competitive supplier and that shipping costs were either zero or sufficiently low for all suppliers to compete for all customers. Then, based on the models presented in Chapters 2 and 4, I might expect market prices at each pair of geographic demand places to differ by no more than a fraction of the shipping cost between the two places. The Hitchcock–Koopmans dual tells us that something akin happens—substituting
29 Boventer
(1961) came up with a similar interpretation by looking at the numerical procedure used to solve the primal.
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The Example: A Spatial Version
87
“price” for “opportunity cost”—when customers are supplied by a monopolist from different factories. At first glance, this should be both surprising and not surprising.
• On the one hand, this should not be surprising. After all, the monopolist is simply trying to be efficient here and hence mirrors the operation of a competitive market. • On the other hand, this might appear at first glance to be surprising. In Chapter 2 a monopolist sets delivered prices at two customer places, where the local demand curves are linear in price, to differ by one-half the unit shipping cost. In the Hitchcock–Koopmans dual, where the firm uses the same factory to serve customers at two different places, the firm sets the two opportunity costs to differ by the full amount of the unit shipping cost. Remember here though that opportunity cost and price are not the same thing and that the Hitchcock–Koopmans model takes quantity demanded as given and is not designed to solve for the profit-maximizing price charged at each demand place.
3.6 The Example: A Spatial Version I return now to the example problem. The solution shipments and shadow prices are shown in panel (a) of Table 3.9. In the least cost solution, the firm supplies customers at A from factory 1, customers at B from factory 4, and customers at C from factories 3 and 4. Factory 3 is marginal and factory 2 is unused. The total cost of supplying all customers is 200(20 + 6) + 2,500(23 + 4) + 1,000(22 + 4) + 500(22 + 4) = 111,700. That this is greater than the total cost (88,400) in the non-spatial version above reflects the shipping costs incurred here. These shipping costs are sufficient to make the firm give up use of factory 2, with its relatively low unit production cost, in favor of factories 3 and 4 which are closer to the customer Place C where much demand is concentrated. Shadow price on capacity is 1 at factory 4 and zero elsewhere. Shadow price on demand is 26 at A and 27 at either B or C. In panel (b) of Table 3.9, I present the example problem cast in terms of the dual inequality (3.8.2). Column [2] shows the 4 shadow prices on capacity: one row for each factory. In any row, Columns [3], [4], and [5] show the value of vj − (Ci + sij ) for each of the three sets of customers. For none of the customer places does an extra unit of capacity at factory 1 help reduce costs: hence r1 = 0. The same is true in the next two rows for factories 2 and 3. However, in the case of factory 4, an extra unit of capacity would reduce the cost of serving customers at B by 1. The same is true for customers at C. Therefore, r4 = 1. Even though this is a model to solve logistics, the firm allocates production as though it were responding to customers willing to bid the most: i.e., the places with the highest opportunity cost. In the example problem, I consider the case where Qd = 4,200. Suppose I experiment by trying a different Qd while holding the demands at the three places in the same relative shares (200: 1,000: 3,000). Here, I use Qd = 4,200v where v ranges from 0 to 1. Put another way, I maintain the relative spatial pattern of
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3 Logistics and Programming Table 3.9 Model 3B: solution to example problem
(a) Solution to primal Shipment qij Factory i [1]
Capacity Sj [2]
1 1,000 2 2,000 3 2,500 4 1,500 Surplus Demand Shadow price vj Total cost Z
Aj=a [3]
Bj=b [4]
Cj=c [5]
Slack [6]
Shadow price ri [7]
200 0 0 0 0 200 26
800 0 200 0 0 1,000 27
0 0 1,500 1,500 0 3,000 27
0 2,000 800 0
0 0 0 1
111,700
(b) Dual inequalities (3.6.2) vj − (Ci + sij ) i [1] 1 2 3 4
ri [2] 0.00 0.00 0.00 1.00
≥ ≥ ≥ ≥
j=A [3]
j=B [4]
j=C [5]
0.00 −1.00 −3.00 −2.00
0.00 −1.00 0.00 1.00
−2.00 −3.00 0.00 1.00
Note: Calculations by author.
consumers (concentrated mainly at Place C) but set the aggregate number of consumers either higher or lower. I do not look here at a region growing over the longer run because that would require us to think about how the firm foresees growth and invests accordingly in new plant and equipment. Instead, what I am doing here is a kind of comparative statics analysis; we treat the populations of consumers at Places A, B, and C at the stated Qd as given and solve for the least cost shipments. See Table 3.10. If aggregate demand is set low, say at Qd = 420, the least cost solution is to supply the 300 units demanded at A from factory 1 and the 100 units demanded at B as well as the 20 at C both from factory 4. Since there is slack in every factory, r1 = r2 = r3 = r4 = 0. The shadow prices on demand are 26 for each of the three customer places because factory 1 supplies customers at A at that cost as does factory 4 for B and C. As I increase Qd , the shadow prices on capacity stay at zero up until capacity is reached at one of the factories. First to run out is factory 4, when Qd = 1,575. At that point, DB = 375 and DC = 1,125, which exhausts the capacity at factory 4. Note that DA = 75 only, so there is much spare capacity at factory 1. Factory 2 and 3 are unused here.
22.00 22.00 22.00
A B C
22.20 22.40 22.80
Shadow price (vj )
1.00 0.00 0.00 0.00
shadow price (ri ) [5] 0 300 2,500 0
Min. slack [6]
23.00 23.33 24.00
Shadow price (vj )
1.00 0.00 0.00 0.67
shadow price (ri ) [7]
0.20 < v <0.33
0 1,000 1,800 0
Min. slack [8]
23.40 23.80 24.60
Shadow price (vj )
1.00 0.00 0.00 1.00
shadow price (ri ) [9]
0.33 < v <0.40
0 1,800 1,000 0
Min. slack [10]
25.00 25.67 25.67
Shadow price (vj )
1.00 0.00 0.00 1.00
shadow price (ri ) [11]
0.40 < v <0.67
0 2,000 800 0
Min. slack [12]
26.00 27.00 27.00
Shadow price (vj )
0.00 0.00 0.00 1.00
shadow price (ri ) [13]
0.67 < v <1
Note: Min. slack is excess capacity at factory when v is at maximum of range for that column. Parameter values given in Table 3.3. Calculations by author.
Shadow price (vj )
j
(b) Demand at Place j
0 0 2,500 300
Min. slack [4]
shadow price (ri ) [3]
Min. slack [2]
(a) Capacity at factory i 1 0 2.00 2 0 1.00 3 2,500 0.00 4 300 0.00
i [1]
0 < v <0.2
v =0
Aggregate demand (Qd )
Table 3.10 Model 3B: shadow prices and capacity surplus as Qd is varied
3.6 The Example: A Spatial Version 89
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When Qd rises above 1,575, B and C have to be supplied from the next best alternative. There are two options here that are equally efficient. I Use factory 3. Factory 3 can supply either B or C at a next best cost of 23 + 4 = 27. II Use factory 1. Factory 1 can also supply B (but not C) at 27. Basically, Option II supplies B from factory 1, and continues to supply C from factory 4. To make the latter possible, the firm redirects output that had been going from factory 4 to B with an augmented shipment from factory 1 to B. The firm keeps expanding use of factories 1 and 3 to meet the growth in Qd until it runs out of capacity at both factories. The shadow price on capacity on factory 4 is now 1 (because it costs us now 1 more to supply the marginal unit at B or C); the shadow price on capacity at each other factory is still zero. The shadow price on demand at A remains at 26; for both B and C, shadow price on demand at B and C is now 27 because the marginal customer in each case is being supplied on a more costly basis. As Qd is increased, the solution retains these characteristics up to the level where the factories 1 and 3 run out of capacity. This happens when Qd = 5,000. At that point, factory 1 supplies 238 units to A and the remainder of its capacity to B. Factory 3 supplies 429 units of its capacity to B, and the remainder of its capacity to C. Factory 2 is still unused here. When Qd rises above 5,000 units, shipments begin from the least efficient factory 2. The shadow price on demand rises to 27 at A, 28 at B, and 30 at C because the marginal customer in each case is now being supplied on a still more costly basis. The shadow prices on capacity for the more efficient factories are now 4 at factory 4, 3 at factory 3, and 1 at factory 1. This simple experiment illustrates the idea that shadow prices are tied fundamentally to the notion of the marginal factory. As the increase in demand exhausts the capacity of the marginal factory, the firm must switch to a new marginal factory and then the shadow prices rise for factories already used to capacity. Finally, I do one more experiment: this time varying the relative cost of shipping. As in the previous experiment, I use the example problem from Table 3.3 (including Qd = 4,200). This time, however, assume that shipping cost from any factory to a customer place is a fixed fraction (v) of the shipment cost shown in Table 3.3. For example, if I set v = 0.5, s1A would now be 3.00, s4C would be 2.00, and so on. If I set v = 0, I get the non-spatial version of the model. If I set v = 1.0, I get the solution that we have already discussed in Table 3.9. If I set < v < 1, I get a problem in which shipping cost is everywhere proportionally less than in Table 3.9. To understand these outcomes, look at Fig. 3.4 and keep in mind that capacity constraints may mean the firm cannot always choose the least cost factory. First, I consider what happens when v > 1. If shipping cost became sufficiently high, that is, v much larger than 1, the firm would still find it least costly, capacity permitting, to serve customers at A from factory 1; all factories are equidistant from A, but factory 1 still has an advantageous unit production cost. This is like when v = 1. Since DA = 200 and S1 = 1,000, there is sufficient capacity at factory
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The Example: A Spatial Version
91
1. When v is large, the firm would want to serve customers at B and C—capacity permitting—from factory 3 because it is closer than factories 1 or 2 and has a unit production cost advantage over factory 4. Since DB + DC + 4,000, the capacity of factory 3 is insufficient. Given v sufficiently high, the firm finds it less costly to meet the excess demand at B and C from factory 4 than from either factory 1 or factory 2. Hence, factory 2 will be unused and factory 4 will be the marginal factory. This is like when v = 1. In other words, when v > 1, we get a similar solution—in terms of the pattern of shipments—as when v = 1. This is akin to, but not the same as, the situation of autarky30 to be outlined in Chapter 4. In Chapter 4, I assume shipping costs large enough to prohibit any shipments. Here, we assume v is sufficiently large to make unattractive all shipments except from the nearest supplier. Panel (a) of Table 3.10 shows the minimum slack and shadow price on capacity for each factory as v is varied from 0 to 1. Panel (b) of Table 3.10 shows the shadow price on demand for consumers at each place. What happens to the least cost solution as v is varied from 0.0 to 1.0? • At v = 0, we get the non-spatial version of the model. For completeness, let me restate the outcomes here. Shadow price on capacity is 2 at factory 1, 1 at factory 2, and zero at the unused and marginal factories. Since v = 0, it does not matter which factory supplies which customer place; the shadow price on demand at each customer place is 22 (the unit cost of production at factory 4). The two most efficient factories are used to capacity: factory 1 supplies 1,000 units and factory 2 supplies 2,000 units. Factory 4 supplies the balance (1,200 units) to satisfy QD . Factory 4 is marginal; factory 3 is unused. • At 0 < v ≤ 0.20, shipping costs are sufficiently low that the firm can do no better than maintain the same level of production at each factory as in the non-spatial version. Put another way, shipping costs are not yet high enough to make it attractive to the firm to use either factory 3, or to switch factory 4 out of its marginal status. Shipping costs, no matter how small, now make a difference which factory supplies a given customer place. Over this interval of v, the shadow price on capacity declines linearly from 2 to 1 at factory 1, declines linearly from 1 to 0 at factory 2, and remains at zero for factories 3 and 4. The shadow prices on demand rise linearly everywhere: from 22 to 22.20 at A, from 22 to 22.40 at B, and from 22 to 22.80 at C. At v = 0.20, the firm can do no better than to supply: A entirely from factory 1 at a cost of 21.20, B from factory 2 at a cost of 22.40, and C from a blend of factories 1, 2, and 4 at a unit cost of 21.80, 22.80, and 22.80, respectively. Factory 4 is marginal; factory 3 is unused. • At 0.20 < v ≤ 0.33, shipping costs are sufficiently high for the proximity of factory 4 to C to offset its higher unit production cost. Over this interval of v, the shadow price on capacity remains at 1 at factory 1, zero at factories 2 and 3,
30 A condition wherein regions do not engage in trade. Local supply in one geographic market is not available to meet local demand in another geographic market for the same product. Autarky arises when shipping costs are too high to permit arbitrage.
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and rises linearly from 0 to 0.67 at factory 4. The shadow prices on demand rise linearly everywhere, from 22.20 to 23.00 at A, 22.40 to 23.33 at B, and 22.80 to 24.00 at C. At v = 0.33, the firm can do no better than to supply A from factory 2 at a cost of 23.00, B from factory 2 at a cost of 23.33, and C from a blend of factories 1, 2, and 4 at 23.00, 24.00, and 23.33, respectively. Factory 2 is marginal; factory 3 is still unused. • At 0.33 < v ≤ 0.40, shipping costs are sufficiently high for the proximity of factory 4 to Place B to offset its higher unit production cost. It is now less costly to supply customers at B from factory 4 than it is from factory 2. However, this is of no value to the firm since factory 4 is already fully used. Over this interval of v, the shadow price on capacity remains at 1 at factory 1, zero at factories 2 and 3, and rises from 0.67 to 1 at factory 4. The shadow price on demand rises linearly everywhere: from 23.00 to 23.40 at A, 23.33 to 23.80 at B, and 24.00 to 24.60 at C. At v = 0.40, the firm can do no better than to supply B entirely from factory 2 now at a cost of 23.80, A from factory 1 at a cost of 22.40, and C from a mix of factories 1, 3, and 4 at unit costs of 23.60, 24.60, and 23.60, respectively. Again, factory 2 is marginal; factory 3 remains unused. • At 0.40 < v < 0.67. The least cost supplier for customers at C is factory 4 when v > 0.40. At v > 0.60. the second least cost supplier for C is factory 3. Over this interval of v, the shadow prices on capacity are 1 at factory 1, zero at factories 2 and 3, and 1 at factory 4. The shadow price on demand rises linearly everywhere: from 23.40 to 25.00 at A, 23.80 to 25.67 at B, and 24.60 to 25.67 at C. At v = 0.67, the firm can do no better than to ship 1,500 units each from factories 4 and 3 to C at a unit cost of 24.67 and 25.67, respectively, 1,000 units from factory 1 to B at a cost of 24.67, and 200 units to A from factory B at a cost of 25.00. Factory 2 and now factory 3 are marginal; there is no unused factory. • At 0.67 < v < 1.00, for customers at B, factory 4 becomes a lower cost alternative to factory 1. Over this interval of v, the shadow price on capacity falls linearly from 1 to 0 at factory 1, remains at zero for factories 2 and 3, and remains at 1 for factory 4. The shadow price on demand rises linearly everywhere, from 25 to 26 at A, and 25.67 to 27.00 at B and C. At v = 1.00, the firm can do no better than to ship 3,000 units to C—1,700 from factory 3 and 1,300 from factory 4—at a unit cost of 27.00 and 26.00, respectively, 1,000 units to B—800 units from factory 1 and 200 units from factory 4—at a unit cost of 27.00 and 26.00, respectively, and 200 units to A from factory 1 at a cost of 26.00. Factory 3 is marginal; factory 2 is unused.
3.7 What Is a Market? The Hitchcock–Koopmans Model gives us a tantalizing look at the geographic delineation of a market. What makes it so promising is that it gives us a glimpse as to the places that will be supplied by one factory as opposed to another. That might tempt us into thinking of a geographic market for a factory as a set of the customers it serves. In earlier chapters, I have thought about how factories get located, and
3.8
Final Comments
93
shipments get determined. By incorporating local capacity constraints into the analysis, the Hitchcock–Koopmans Model introduces a new dimension to thinking about the geographic structuring of markets. Indeed, because it is possible that the costminimizing firm may be able to forego production altogether at one or more of its factories, the Model, like that presented in Chapter 2, makes it possible to begin to think about where to locate as well as how much to produce and ship. At the same time, the model reminds us of the importance of opportunity costs: the shadow price of the capacity constraint at each factory. That shadow price reminds us that we cannot, in general, look at one factory in isolation. Which customers will be served by a factory depends in part on the other factories the firm has available to serve these and other customer places. At best then, the existence of shadow prices implies that the set of customers served by just one factory constitutes a submarket. I return to this matter in Chapter 8. Unfortunately, the Hitchcock–Koopmans Model is also of limited value in thinking about markets for three additional reasons. First is the fact that it considers just the factories that belong to one firm (supplier) whereas a competitive market might better be thought to be composed of numerous buyers and sellers. The second is that the Hitchcock–Koopmans model, unlike the models to be presented in Chapter 4, does not directly consider price. Without reference to the price of soap being sold, it is difficult to know how one might delineate a market for soap. A different way of posing the same question is to ask whether there is spatial price equilibrium across places in this Model? I simply cannot answer such a question with this Model. The third reason is that, because it ignores fixed costs including the investment in plant and equipment, the Hitchcock–Koopmans Model cannot be used to look at the prospects for building a new factory; it looks only at the efficient use of already-existing factories.
3.8 Final Comments In this chapter, the principal model has been 3B. I included Model 3A to help readers better understand aspects of Model 3B. In Table 3.11, I summarize the assumptions that underlie Models 3A and 3B. Many assumptions are in common to both models: see panel (a) of Table 3.11. Model 3B differs from 3A in that the firm potentially incurs a unit shipping cost for every unit shipped from a factory to a customer. In Chapter 1, I argue that prices are important in shaping the location of firms. In this chapter, those prices are the input prices embedded in Ci and sij . As in Chapter 2, these other prices are all determined in markets outside the scope of this model where the firm competes against other firms and demanders for production inputs and shipping services. These input prices in turn help determine which patterns of production and shipment are least costly to the firm and thereby the extent of localization. Once again, I think Walras would have argued that the analysis has been only partial in the sense that we have not looked explicitly at the simultaneity among prices in these markets. In later chapters, we will have opportunities to do that.
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3 Logistics and Programming Table 3.11 Assumptions in Models 3A and 3B Assumptions
3A [1]
3B [2]
(a) Assumptions in common A1 Closed regional market economy A3 Punctiform landscape B1 Exchange of soap for money (price implicitly given) C1 Fixed demand (quantity given at each location) D2 Firm minimizes cost of production and shipping D3 No capacity exceeded D5 I factories D8 Reverse-L marginal cost curve H2 Locations of firm given
x x x x x x x x x
x x x x x x x x x
(b) Assumptions specific to particular models E1 Zero shipping cost everywhere E5 Firm bears shipping cost to market
x x
Through its focus on capacity, the Hitchcock–Koopmans Model gives us a new tool for thinking about congestion and the nature of the local supply curve. In the model, the firm incurs the same marginal cost of production as long as the level of output does not exceed factory capacity. At the same time, there is something lumpy about the cost curve shown in panel (c) of Fig. 3.2. Would it not be more reasonable to assume that the firm finds its marginal cost rises, slowly at first and then increases rapidly the more output it wants to produce at a factory? The Hitchcock–Koopmans Model cannot do that; we need a different model for that purpose. This idea leads to a second limitation of the Hitchcock–Koopmans Model as presented here. The model assumes that every unit of soap shipped from a factory at i to customers at j incurs the same cost. In effect, the Model assumes that the transportation network has an unlimited capacity. In fact, a transportation network is usually composed of links, nodes, and mode transfer points, each of which may have capacity limitations. Although not done in this book for ease of exposition, it turns out that the Hitchcock–Koopmans Model can be extended to take into account capacitated networks.31 In an important sense, these two kinds of congestion are parallel. When we think of why a factory becomes congested, it is often because the site does not have room to store more inventory or the ability to achieve even greater levels of production. This is like the constraints facing shippers trying to handle a greater volume of shipments. The models in this chapter do not say anything about the distribution of income in society among labor, capital, and land. The firm is envisaged to incur costs, but these are not related explicitly to the hiring of labor or to the rental of land. In any case, all benefits that can arise (e.g., because of a decrease in Ci or sij ) accrue to the firm owner in the form of reduced cost (and therefore, if implicitly, increased profit). The models in this chapter are silent on the price paid 31 See,
for example, Magnanti, Mirchandani, and Vachani (1995) and Mayer and Sinai (2003).
3.8
Final Comments
95
by their customers and therefore have nothing to say about the well-being of consumers. Finally, the models in this chapter are silent on local cost advantage. We do know that if Factory i has a relatively low Ci it will be efficient to use that factory to supply customers nearby. Place i might, for example, have a low Ci because it is larger and takes advantage of an economy of scale or division of labor (e.g., when the firm assembles its most efficient staff at the factory sites that will service the bulk of demand). However, there are other possibilities too. Place i might, for example, have access to inexpensive inputs for production. Until we know what it is that makes some factories more efficient than others, it is difficult to infer how localization and prices in general are linked.
Chapter 4
The Struggling Masses Perfect Competition at Two Places (Cournot–Samuelson–Enke Problem)
Two places, in isolation from the rest of the world, each meet the requirements of a perfectly competitive market. In Model 4A, the unit shipping cost is prohibitive. Each place is in autarky. Price locally reflects only local demand and local supply. However, if the unit shipping cost is low, arbitrageurs purchase where price is low for resale at the other place. In Model 4B, unit shipping cost is zero everywhere, and there is a common equilibrium price at the two places. A change in any parameter of local demand or local supply at either place can affect this price. In Model 4C, shipping cost is neither prohibitive nor zero. Here, shipping occurs up the price gradient. Because of the actions of arbitrageurs, the price difference between the two places shrinks to the unit shipping cost. Corner solutions—in which either demand or supply drops to zero in one or the other of the two places—are solved and interpreted. The models in this chapter are the competitive market equivalent of the models of a monopolist in Chapter 2. However, as in Chapter 3, congestion in production means that unit cost rises the more output the industry produces. Usually, we imagine that competition causes excess profit to disappear. However, the notion of congestion (an upward sloped supply curve) here in Chapter 4 means that some producers are less efficient than others. More efficient suppliers earn a monopoly profit even in competitive markets. In this chapter, localization of production, prices (one for each place), and excess profit (for all but the marginal producer) are joint outcomes of a competitive market.
4.1 The Cournot–Samuelson–Enke Problem In this chapter, I explore the localization of firms from the perspective of demand curve, supply curve, and competitive market. In Chapter 2, I introduced local demand; in this chapter, I now consider demand by arbitrageurs buying in a market (or submarket) where the price is low for resale elsewhere where the price is high; I treat this external demand as endogenously determined. This chapter comes early in this book because it, as does Chapter 2, starts from material familiar to anyone who has taken a first course in Economics. J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_4, C Springer Science+Business Media, LLC 2010
97
98
4
The Struggling Masses
When we look at a competitive market, our focus shifts from the firm to the industry as a whole. With the monopolist in Chapter 2, I could examine directly the potential return to capital invested in a factory at each location. In this chapter, I rely instead on the notion of local supply.1 The local supply curve tells us how much more local producers would make available in total if the local price were higher. Put differently, the local supply curve is an ordering of local production from least costly to more costly. See the right-hand side of Fig. 4.1. A
A
H
D
Price
Price
I
G G
D
F
E C 0
0
F
C Quantity
E
B
Chapter 2: Firm maximizes profit AB AC DE DIHGD ODGFO OF OI OIHFO
Demand curve Marginal revenue Firm’s marginal cost curve Firm’s semi-net revenue Variable cost incurred by firm Quantity supplied by the firm Price received by firm Firm’s revenue
Quantity
B
Chapter 4: Industry in competitive equilibrium AB CD CGFC GAFG OCFEO OE
Demand curve Supply curve Producer surplus Consumer surplus Producer cost Quantity supplied by the industry OG Equilibrium price
Fig. 4.1 Models in Chapter 2 and 4 compared
This chapter distinguishes between local supply (the quantity offered by local producers) and external supply (the quantity offered by traders who bring a commodity produced elsewhere into the local market). The chapter then uses the relative size of external supply and external demand (the quantity demanded by traders for resale elsewhere) to draw conclusions about the impacts of a change in market conditions on the localization of production. In Chapter 2, I looked at how a monopolist prices product and decides where to produce (i.e., invest in factories). My treatment of market in that chapter was asymmetric: there were many demanders (customers) but only one supplier (the firm). The firm gets a lower price per unit the more the quantity it supplies to the market. To maximize profit, the firm there chose the quantity Q where marginal 1 In a region, the supply of a product by firms from local production. Local supply does not include supply offered by arbitrageurs importing from another region.
4.1
The Cournot–Samuelson–Enke Problem
99
revenue is equal to marginal cost, then sells it in the market at price P. See the left-hand side of Fig. 4.1. In this chapter, I consider a local industry consisting of a large number of firms each supplying the same product. I say large here to invoke the notion of a competitive market. Economists describe a competitive market as one with a sufficient number of participants so that no one individually affects price: neither a demander the price paid by varying the quantity they individually demand, nor suppliers the price received by varying the quantity they individually supply. Each firm (and consumer) individually is a price taker; equivalently the market exemplifies perfect competition.2 Customers simply bypass suppliers with a higher price in favor of a supplier at the market price. In the same way, suppliers ignore a customer willing to pay only a lower price in favor of a customer willing to pay the market price. I think of the local supply curve as the industry marginal cost curve3 there. Of course, I can measure the industry marginal cost curve over either the short run or the longer run. I assume in what follows that we are looking at the industry marginal cost curve over a longer run in which capacity adjustments are possible. We want to think about situations where unit shipping cost changes and affects the amount of a commodity produced at a given place. Presumably new firms start up at some places while other firms die off at others. However, we should be careful here. Over the longer term, other things might be happening that also affect the local supply curve: prices of inputs, technological change, and firm reorganization to take advantage of economies of scale or Adam Smith’s idea that the division of labor is limited by the extent of the market. Put simply, his idea was that the greater the output (scale) of a firm the better able it is to take advantage of the added productivity from having specialized labor. These factors can cause the supply curve to shift or twist. In the relatively simple models used in this chapter, I don’t want to complicate the analysis with such considerations. In drawing a local supply curve, I am envisaging free entry of firms. There is nothing to prevent new firms from entering the industry other than for considerations of profit. If demand shifts so as to cause a rise in market price, new firms will keep entering the industry until the price is driven down to the point where a potential new firm finds that it is no longer profitable to get into that industry. In competitive location theory, the concept of free entry is often associated with the work of Lösch on imperfect competition in space. However, free entry is also central to perfect competition in general and to the supply curve over the long run in particular. In this chapter, I introduce models of spatial price equilibrium. In these models, assume all firms are competitive. There is no monopolist here pricing among these markets to maximize profit. Instead, the emphasis is on the role of competition 2 An
attribute of a market wherein each supplier and each demander is a price taker. schedule showing the marginal cost to an industry (usually over the longer run where capital invested adjusts as needed) as a function of the quantity to be supplied. The market supply curve is thought to be the same as the industry marginal cost curve. We can measure the industry marginal cost curve over either the short run (no additional competitors or factories) or the long run (additional competitors and/or factories possible). 3A
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The Struggling Masses
among firms and traders (arbitrageurs) who are each a price taker in markets for their commodity. Location theory includes a substantial subfield with a long history that deals with spatial price equilibrium.4 Early work in this area starts with Cournot5 in 1838 and includes Enke (1951) and Samuelson (1952).6 I first came across the price difference curve solution method in Samuelson (1952) and hence refer to it here as the Cournot–Samuelson–Enke Model: Model 4C later in this chapter. However, let me first introduce two models as foils. Model 4B shows the solutions when unit shipping rate is zero. Model 4A shows spatial price equilibrium in autarky; we can think of autarky as a situation arising when unit shipping cost is prohibitive. Model 4C can be thought therefore to lie between Models 4A and 4B: unit shipping cost larger than zero but not large enough to bring about autarky.
4.2 Model 4A: Autarky As in Chapter 2, I continue the assumption of a punctiform landscape made up of two places, Places 1 and 2, sharing a common fiat money economy. Once again, assume N1 consumers at Place 1 and N2 consumers at Place 2, and consumers at the two places each have identical individual linear inverse demand curves: P = α − βq. As before, assume β > 0. The aggregate local inverse demand curves at Places 1 and 2 are now given by (4.1.1). See Table 4.1 wherein I summarize equations, assumptions, notation, and rationale for localization in Model 4A. To start, I assume the two places are each a market in autarky. The modeling of such markets is non-spatial; there is no shipping cost involved between the places because the cost of shipment is prohibitive in autarky. As in Chapter 2, a non-spatial model helps us think about the effect the subsequent introduction of geography has on model outcomes. Suppose these local producers in aggregate at each place have an inverse supply curve7 showing the price required to ensure suppliers generate a given quantity of the commodity.8 Assume the slope coefficient here (i.e., δ1 in the case of Place 1) is positive. It tells us how much higher the autarky price would have to be to attract one more unit of the commodity from local suppliers. Assume here initially demand is sufficient to ensure production of the commodity: e.g., α > C1 at Place 1. In Fig. 4.2, see the demand curve (AB1 ), the supply curve (C1 D1 ), equilibrium quantity (OF1 ), and equilibrium price (OG1 ). 4 In the case of multiple places, a condition in which arbitrageurs have no further incentive to purchase in a low-price market for resale in a high-price market. 5 See Cournot (1960, chap. 10). 6 Early writers in the field also mention an unpublished paper by William Baumol—dated 1952 and entitled Spatial Equilibrium With Supply Points Separated From Markets and Supplies Predetermined—that might be similar to Samuelson (1952). 7 A supply function is generally expressed as a schedule of quantity supplied (Q) at various prices (P): i.e., Q = g[P]. An inverse demand function rearranges this as the price needed by suppliers at the margin in order to ensure that a given quantity is supplied to the market: i.e., P = g−1 [Q]. 8 See (4.1.2) for Place 1.
4.2
Model 4A: Autarky
101 Table 4.1 Model 4A: autarky in two markets
Local inverse demand in each market Pi = α − βQi /Ni
i = 1,2
(4.1.1)
Local inverse supply in each market Pi = Ci + δi Qi
i = 1,2
(4.1.2)
Autarky price in each market Pi = (αδi Ni + βCi )/(δi Ni + β)
i = 1,2
(4.1.3)
Autarky quantity in each market Qi = (α − Ci )Ni /(δi Ni + β)
i = 1,2
(4.1.4)
Monopoly excess profit (MP) in each market 0
(4.1.5)
Consumer benefit (CB) in each market 0.5(2αδi Ni + (α + Ci )β)(α − Ci )Ni /(δi Ni + β)2
i = 1,2
(4.1.6)
Producer surplus (PS) in each market 0.5δi Ni2 (α − Ci )2 /(δi Ni + β)2
i = 1,22
(4.1.7)
Producer cost (PC) in each market (0.5δi Ni (Ci + α) + Ci β)(α − Ci )Ni /(δi Ni + β)2
i = 1,2
(4.1.8)
Consumer surplus (CS) in each market 0.5βNi (α − Ci )2 /(δi Ni + β)2
i = 1,2
(4.1.9)
Social welfare (SW) in each market 0.5Ni (α − Ci )2 /(δi Ni + β)
i = 1,2
(4.1.10)
Price elasticity of demand in each market at autarky (αδi Ni + βCi )/(β(α − Ci ))
i = 1,2
(4.1.11)
Notes: Rationale for localization (see Appendix A): Z3—Implicit unit price advantage at some locales; Z8—Prohibitive shipping cost. Givens (parameter or exogenous): Ci —Intercept of inverse supply curve at place i; Ni —Population of place i; α—Intercept of individual linear inverse demand curve; β—Slope of individual inverse demand curve; δ i —Slope of inverse supply curve at place i. Outcomes (endogenous): Pi —Price at Place i; Qi —Equilibrium quantity transacted at Place i.
In this chapter, each market is thought to be competitive; both consumers and producers are price takers. As such, local producers increase the quantity supplied to the market until rising unit cost exhausts the excess profit on the marginal unit supplied: i.e., where price equals marginal cost (inclusive of the opportunity cost of any unpriced inputs, such as entrepreneurial talent). This is the familiar demand–supply equilibrium (which is different from the monopolist in Chapter 2 who chooses a quantity such that marginal revenue equals marginal cost). In autarky, prices at the two places must satisfy the inverse demand equations in (4.1.1) and long run marginal cost is implied by the local supply curves in (4.1.2). The resulting marketclearing price and quantity at each Place under autarky are given by (4.1.3) and (4.1.4). How does this differ from Chapter 2? In Model 2A, I considered the case of a monopolist with a constant marginal cost curve in a single market that yielded
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4
Fig. 4.2 Model 4A: Place 1 in equilibrium in autarky. Notes: In this example, α = 15, β = 1 C1 = 2, δ1 = 0.001, and N1 = 1,000. At market equilibrium, Q1 = 6,500 and P1 = 8.50. Horizontal axis scaled from 0 to 30,000; vertical from 0 to 22
The Struggling Masses
Model 4A Autarky equilibrium at Place 1 AB1 C1AE1C1 C1D1 C1G1E1C1 G1AE1G1 OAE1F1O1 OC1E1F1O OF1 OG1
Local demand curve at Place 1 Social welfare at Place 1 Local supply curve at Place 1 Producer surplus at Place 1 Consumer surplus at Place 1 Consumer benefit at Place 1 Producer cost at Place 1 Autarky equilibrium quantity at Place 1 Autarky equilibrium price at Place 1
D1
Price
A
G1
E1
C1 0
F1
B1
Quantity
a price given by (2.1.5) and a quantity given by (2.1.6). For ease of comparison, assume now a horizontal supply curve here (e.g., δ1 = 0 in the case of Place 1). There, autarky price in competitive equilibrium reduces to P1 = C1 , which is lower than the price (2.1.5) charged by the monopolist. I have indicated this in Fig. 4.1 using a competitive price (OG) lower than the monopolist’s price, (0l). By implication, as shown in Fig. 4.1, the quantity transacted in market equilibrium (OE) is greater than the quantity sold by the monopolist (OF) of Chapter 2. From (4.1.4), I get Q1 = N1 (α − C1 )/β when δ 1 approaches zero, which is twice the quantity a profit-maximizing monopolist would supply to the market (2.1.6). The competitive market envisaged here is different from the monopolist of Chapter 2. The monopolist exploits the downward sloping demand curve to set marginal revenue equal to marginal cost, and so supplies a smaller quantity and gets a higher price than would a competitive firm. That the monopolist supplies exactly one-half of the amount supplied in perfect competition is an outcome of the assumed linearity of demand; nonetheless it illustrates the idea that the monopolist will supply less and get a higher price than will firms that are perfectly competitive. In what ways does the treatment of costs in this chapter differ from Chapter 2? • In Chapter 2, I used C to denote the (constant) marginal cost. I also argued there the firm would not normally supply product to the market, even in the short run,
4.2
Model 4A: Autarky
103
if the price of that product was below C. In the long run, price would have to be higher to provide a return on invested capital as well. In this chapter, I interpret C1 as the price below which no quantity would be supplied to the market. Because I think of the local supply curve as a long run marginal cost curve, C1 presumably sums a marginal cost like C and a normal return on invested capital and any other unpriced factors of production for the most efficient location. • The industry marginal cost curve is not the same as a firm’s marginal cost curve. What are key differences? For one, the firm’s marginal cost curve is usually thought of in the short term only. Above, I argue that we look at the industry marginal cost curve over the long run.9 A second difference is that, in Chapter 2, the firm’s marginal cost does not include unit fixed costs. However, when I draw the industry supply curve here, I must assume a price sufficient to keep firms in the industry over the long term. Therefore, the industry supply curve, and hence the industry marginal cost curve, must include unit fixed cost (at least over the long term). A third difference is that, unlike Chapter 2, the link between profit and costs is not clear. In Chapter 2, I assumed the firm was a price taker10 in the markets for its inputs: the opportunity cost of capital r and the marginal cost of production C were each simply a fixed expenditure required (per unit of K and Q, respectively) that reduces profit. Here in Chapter 4, the supply curve tells us about the quantity forthcoming from local suppliers at any given price, but this does not say anything directly about input or unit costs like r and C. Nonetheless, in interpreting the supply curve, it is helpful to assume that each supplier is an efficient firm. • In this chapter, I assume the supply curve (hence costs) may differ from one local market to the next. In Chapter 2, I had assumed the firm faced the same costs of production at each place. • In this chapter, I assume the local supply curve is upward (positively) sloped. In Chapter 2, I had assumed that the firm, by investing K, obtained an unlimited wellspring of production at a constant marginal cost, C. The upward sloped supply curve is different from this. For some reason, it now costs more per unit to induce a greater supply. To me, the approach of this chapter raises three questions. • Why is the local supply curve upward sloped? In this chapter, I assume unit cost increases with level of output in the market. This is in contrast to the constant
9 Over the short term, the firm is not able to adjust its capital stock. In the short term, therefore, the firm’s marginal cost rises as it attempts to increase the level of production, as the firm encounters congestion and capacity limitations. Of course, this is not what happens in Chapter 2 where I assumed marginal cost is constant; the unlimited wellspring assumption allowed us to ignore questions about the relationship between output and capital stock. 10 A condition under which a market participant (supplier or demander) is unable to affect the price they receive or pay for a unit of the product by varying the quantity that they supply or demand. The supplier (demander) sees the demand (supply) for its product as horizontal: i.e., infinitely elastic at the given market price.
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The Struggling Masses
marginal cost assumed in Chapter 2. In effect, I now allow for the possibility of (1) congestion at the firm level that causes unit cost to rise eventually the more output it produces, (2) competition among firms for the scarce resources they need to produce more output, and (3) differences among firms that make some more efficient than others. Why does this happen? In effect, the firms individually or the industry collectively experience congestion. That is, they are unable to increase production as an industry without incurring a higher marginal cost of production. In the standard introduction to competitive equilibrium in a first course in Economics, it is typically argued that an industry is made up of identical firms with identical costs. As demand for the industry’s product increases, the standard introduction envisages additional identical firms entering the industry. In that case, the long run supply curve (industry marginal cost curve) is a horizontal line. If every firm looked like the firm in Chapter 2 for example, then local market price, P, would approach the marginal cost of production, C. In other words, over the long term, the models in Chapter 2 ignore the congestion (within the firm due to its structure and physical plant) and the competition for scarce resources among firms that arises from producing more output from a given set of firms using their existing factories and assume that all output needed can be produced at the same unit cost. • Where does a supply curve come from? In this chapter, I assume a local supply curve that shows the amount forthcoming from local suppliers at each place contingent on the market price there alone. This is an assumption made to simplify the model. In fact, when an entrepreneur—like the firm in Chapter 2—is deciding when and where to build a factory, it takes into account the price of the commodity locally as well as the price that might be possible for a factory built elsewhere. Put differently, in a world of two places, the supply forthcoming locally from any place will depend on the prices at both places. In Chapter 10, I present a model, which incorporates joint aspects of supply. • Why do costs differ from market to market? Are costs that differ from market to market inherently geographic (as in variations in labor or other factor costs) or related to the efficiency or productivity of the firms that just happen to be found in that market? In competitive equilibrium, as the number of consumers (N1 ) increases, autarky price increases. This is because more output will be demanded at any price (below α) when the market is larger, but in autarky this can only be obtained by moving to the right along the local supply curve and therefore inducing more costly suppliers to enter the market.11 What makes the solution differ here in part is that, in Chapter 2, the monopolist had a constant marginal cost whereas here marginal cost increases with the level of output.12 In a competitive market, excess profit disappears for the marginal supplier in equilibrium. In 11 This
is different from the case of the monopolist in Chapter 2 where, in the case of a single market, price was not affected by market size: see (2.1.5). 12 Here, I can approximate the condition of constant marginal costs by letting δ approach zero. 1 In that case as noted above, (3.2.3) implies P1 approaches C1 , and market size indeed no longer affects price in a perfectly competitive market.
4.2
Model 4A: Autarky
105
other words, the supplier in the market with the highest unit cost—in this sense the marginal producer—earns only normal profit. However, the upward slope of the supply curve tells us some producers are more efficient and therefore do earn excess profit even though the marginal producer does not. Whatever the particular advantage of the more efficient firm, that advantage gives rise to a Ricardian rent. In this model, I have assumed all consumers in the two markets have the same individual demand curve for the commodity. Because of that assumption, differences in equilibria between the two markets under autarky do not arise because consumers are different. However, differences can arise because of differences in market sizes (N1 vs. N2 ) or the shapes of the supply curves (C1 vs. C2 or δ 1 vs. δ 2 ). We can now imagine an autarky solution for Place 2. See the example in Fig. 4.3 where I have assumed Place 2 is smaller (demand curve AB2 ) and less efficient (supply curve C2 D2 ) than Place 1 (reproduced from Fig. 4.2). Suppose, for example, these two markets in autarky are similar in that C1 = C2 and δ1 = δ2 , but that N2 > N1 . Here, at any given market price, the quantity
Model 4A Autarky equilibrium at Place 2 AB2 C2AE2C2 C2D2 C2G2E2C2 G2AE2G2 OAE2F2O OC2E2F2O OF2 OG2
Price
A
G2
Local demand curve at Place 2 Social welfare at Place 2 Local supply curve at Place 2 Producer surplus at Place 2 Consumer surplus at Place 2 Consumer benefit at Place 2 Producer cost at Place 2 Autarky equilibrium quantity at Place 2 Autarky equilibrium price at Place 2
D2
E2
C2
0 F2
B2
Quantity
Fig. 4.3 Model 4A: Place 2 in equilibrium in autarky. Notes: In this example, α = 15, β = 1 C2 = 7, δ2 = 0.002, and N1 = 900. At market equilibrium, Q2 = 2,571 and P2 = 12.14. Horizontal axis scaled from 0 to 30,000; vertical from 0 to 22
106
4
The Struggling Masses
demanded at Place 2 is greater than at Place 1. To coax that larger supply locally, the autarky price at Place 2 would have to be larger than at Place 1 on the assumption that δ1 = δ2 > 0. Put differently, the diseconomies of scale (i.e., the upward-sloping supply curves) assumed in this model means that the price of the commodity will be higher in a large market compared to a small market. Now, instead, suppose the two markets are similar in that δ1 = δ2 and N1 = N2 , but that now C2 > C1 . In this case, the supply curve for Place 2 will be parallel to, and lie above, the supply curve for Place 1. Therefore, P2 will be higher than P1 in equilibrium under autarky. Finally, suppose the two markets are similar in that C1 = C2 and N1 = N2 , but that now δ2 > δ1 . In this case, the supply curve for Place 2, will be steeper than the supply curve for Place 1 and the two will intersect at the Y-axis (since C1 = C2 ) Therefore, in equilibrium, P2 will once again be higher than P1 under autarky. When prices differ between the two markets in autarky, the possibility of shipments arises. Shipment of the commodity will occur, if at all, only from the market with the lower price in autarky to the market with the higher price. There would be no incentive to ship in the opposite direction. Note the importance here of our assumption that all firms produce an identical product. If suppliers at Place 2 were to produce a similar, but not identical, product then consumers at Place 1 might still want to consume some of it even if P2 > P1 in autarky. In principle, shipment is not the only way in which the two markets might be affected by this difference in autarky prices. To the extent that firms are more efficient than others because of an internal advantage (e.g., more skilled management), they might be able to increase profits by shifting production into the market with the higher autarky price. Similarly, consumers might be attracted to relocate into the market with the lower price. However, in this chapter, we ignore movements of firms or consumers between the two places to focus on the role that might be played by the shipment of commodities. Before I do this, what about comparative statics in Model 4A? Here, I present results for Place 1 only; the story would be the same for Place 2. In this autarky model, there are five givens (α, β, C1 , δ 1 , and N1 ) and two outcomes (P1 and Q1 ). Remember here the implicit role of prices everywhere : consumer income and the prices of consumer commodities shape α and β; input prices help shape the local supply curve, hence C1 and δ 1 .13 As in Model 2A, comparative statics are easy to do because I have explicit solutions for P1 and Q1 . See (4.1.3) and (4.1.4) and Table 4.2. C1 N1
If C1 is increased, the supply curve shifts upward. Therefore, market equilibrium runs up the demand curve: P1 increases and Q1 decreases. When N1 is increased, the aggregate demand curve becomes flatter, that is, sweeps counterclockwise about (0, α). Therefore, market equilibrium runs up the supply curve: P1 and Q1 both increase.
13 To the extent that population adjusts so as enable each resident to be best off, N may also be 1 shaped by prices. I do not pursue that idea further here. See Chapters 11 and 12.
4.3
Model 4B: Integrated Market Solution: Zero Shipping Cost
Table 4.2 Model 4A: comparative statics of an increase in exogenous variable
107 Outcome
Given
P1 [1]
Q1 [2]
C1 N1 α β δ1
+ + + − +
− + + − −
Notes: See also Table 4.1; +, Effect on outcome of change in given is positive; –, Effect on outcome of change in given is negative.
α β
δ1
If α is increased, the demand curve shifts upward. Therefore, market equilibrium runs up the supply curve: P1 and Q1 both increase. If β is increased, the individual inverse demand curve sweeps clockwise about (0, α) and market equilibrium runs down the supply curve. At any price below a, consumer demand is now lower than it was before. For the profit-maximizing firm, P1 declines. Q1 also decreases. If δ 1 is increased, the supply curve sweeps counterclockwise about (0, C1 ). Therefore, market equilibrium again runs up the demand curve: P1 increases and Q1 decreases.
4.3 Model 4B: Integrated Market Solution: Zero Shipping Cost In autarky, the price of a product may vary from one local market to the next. As a thought experiment, imagine that the local demand and local supply curves are identical in the two markets, so that the equilibrium prices are the same. Now imagine a shift in either the demand or supply curve in one of the two markets. That shift alone would generally be sufficient to cause the equilibrium price in that market to differ from the other market. If the difference in prices is sufficiently large, arbitrageurs will be encouraged to purchase in the lower price market for resale in the other market. Commonly, we think of an arbitrageur as a professional who trades only in a commodity: i.e., neither produces the commodity nor purchases it for own consumption or use. However, when producers look for a higher priced market in which to sell their commodity, or consumers scour markets looking for the lowest price, their behavior is similar in effect to that of the professional trader. In keeping with the treatment in Chapter 2, I use arbitrage to refer to behavior by traders, producers, and consumers that leads to the flow of a commodity from one local market to another. As in Chapter 22, assume Place 1 is x km away from Place 2, and the cost of shipping a unit of product a distance of 1 km is a constant and directionally symmetric s dollars. Therefore, the unit shipping cost from Place 1 to Place 2, or vice versa, is sx
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The Struggling Masses
per unit shipped. In keeping with the treatment of cost elsewhere in this book, I take the cost of shipping to include any normal profit associated with arbitrage. In this chapter, and throughout the book, I assume for simplicity that the cost of arbitrage is the same for every potential trader.14 For the moment, assume an integrated market.15 In effect, the action of arbitrageurs is to convert two markets into a single unified market on the assumption that sx = 0. To see the consequences of this supposition, ignore the arbitrageurs for the moment, and instead simply aggregate first the demand curves and then the supply curves for the two markets by summing horizontally. In fact, this is the consequence of arbitrage. See Fig. 4.4. In the case of the two demand curves, horizontal aggregation is relatively easy since I have assumed that the demand curves in the two markets have the same Y-intercept (i.e., α). Rearranging (4.1.1) under the assumption that P1 = P yields Q1 = aN1 /β + N1 P/β. A similar rearrangement of the demand at Place 2 yields Q2 = aN2 /β + N2 P/β. Summing these two equations yields Q1 + Q2 = α(N1 + N2 )/β + (N1 + N2 )P/β. Upon rearrangement, the combined demand at Places 1 and 2 is given by (4.3.1). See Table 4.3. In Fig. 4.4, the two local demand curves (AB1 from Fig. 4.2 and AB2 from Fig. 4.3) sum to the aggregate demand curve ABt . The two supply curves can also be aggregated horizontally. However, computational complexity arises here because the two local supply curves need not have the same Y-intercept. In general, the aggregate supply curve will be kinked. Without loss of generality, let Place 1 be such that C1 < C2 . The aggregate supply curve can now be written as (4.3.2), (4.3.3), and (4.3.4) depending on the level of P. Under (4.3.2), there is no production since P is not high enough for production even among the lower cost suppliers at Place 1. If P is high enough to enable production at Place 1, but not high enough for production at Place 2, (4.3.3) ensues. Equation (4.3.4) applies when it is feasible to produce at both Places 1 and 2. Of these, only (4.3.3) and (4.3.4) need concern us here. In Fig. 4.4, the two local supply curves (C1 D1 from Fig. 4.2 and C2 D2 from Fig. 4.3) sum to the aggregate kinked supply curve C1 Ct Dt . The integrated market equilibrium price corresponding to (4.3.1) and (4.3.3)— using only suppliers from Place 1—is given by (4.3.5) and that corresponding to (4.3.1) and (4.3.4)—using suppliers from both markets—is given by (4.3.6). In Fig. 4.4, equilibrium quantity is OFt and equilibrium price is OGt . Local suppliers at Place 1 supply OH units and local customers there demand OI units. Consider a numerical example wherein the two markets differ only in that C2 > C1 : namely, α = 15, β = 1, N1 = N2 = 1,000, C1 = 10, C2 = 11, and δ1 = δ2 = 0.001. See first the autarky solutions presented in Table 4.5. From
14 See
Anderson and Ginsburgh (1999) for an analysis of the case where the cost of arbitrage is different between firms and consumers. 15 A condition of two places arising when the cost of shipping product from one market to the other is zero.
4.3
Model 4B: Integrated Market Solution: Zero Shipping Cost
109
Model 4B Integrated market AB2 Demand curve for integrated market C1CtDt Supply curve for integrated market C1GtEtCtC1 Producer surplus GtAEtGt Consumer surplus HFt Quantity supplied by firms at Place 2 IFt Quantity demanded locally at Place 2 OAEtFtO Consumer benefit OC1CtEtFtO Producer cost OFt Equilibrium quantity in integrated market OGt Equilibrium price in integrated market OH Quantity supplied by firms at Place 1 OI Quantity demanded locally at Place 1
D2
D1
Dt
Price
A
Et
Gt
C2
Ct
C1 0 I
H Ft
B2 B1
Quantity
Bt
Fig. 4.4 Model 4B: equilibrium in integrated market with zero shipping cost. Notes: This example assumes same market parameters as Figs. 4.2 and 4.3. At equilibrium in integrated market, Pc = 10.00 and Qc = 9,500. Of this total, consumers at Place 1 demand 5,000 units (D1c ), and firms at Place 1 supply 8,000 units (S1c ); the remainder is demanded or supplied at Place 2. Therefore, 3,000 units are shipped from the lower priced Place 1 to the higher priced Place 2. Horizontal axis scaled from 0 to 30,000; vertical from 0 to 22
(4.1.3), the autarky price at Place 1 is 12.50. From (4.3.6), the integrated market price (P) is 12.75; since P here is larger than C2 , production occurs at both places. In the integrated market, as calculated from (4.1.2), firms at Place 1 produce 22,750 units: up from the 22,500 units produced under the autarky price. They (including any arbitrageurs who buy from them) are better off because they can now sell their product in the higher price market. Consumers at Place 1 are worse off in the integrated market since they now pay 12.75 a unit compared to 12.50 previously. As calculated from (4.1.1) their demand in the integrated market is only 2,250 units: down from 2,500 units under autarky. The excess supply16 at Place 1 (the amount
16 In a market at a given price P, excess supply is the amount if any by which local supply exceeds
local demand. Local here excludes demand or supply by arbitrageurs.
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The Struggling Masses
Table 4.3 Model 4B: competitive equilibrium in integrated markets (zero shipping costs) assuming C1 < C2 Integrated-market linear inverse demand (where Q = Q1 + Q2 ) P = α − βQ/(N1 + N2 )
(4.3.1)
Integrated-market supply (assuming C1 < C2 ) Q = Q1 + Q2 = 0 if P < C1 Q = Q1 + Q2 = − C1 /δ1 + (1/δ1 )P if C2 > P > C1 Q = Q1 + Q2 = − C1 /δ1 − C2 /δ2 + (1/δ1 + 1/δ2 )P if P > C2
(4.3.2) (4.3.3) (4.3.4)
Integrated-market price P = (αδ1 (N1 + N2 ) + βC1 )/(δ1 (N1 + N2 ) + β) if C1 < P < C2 P = (αδ1 δ2 (N1 + N2 ) + βC1 d2 + βC2 d1 )/(δ1 δ2 (N1 + N2) + β(δ1 + δ2 )) if P > C2
(4.3.5) (4.3.6)
Condition under which all production is at Place 1 w ≤ (C2 − C1 )/(α − C1 )
(4.3.7)
where w = δ1 (N1 + N2 )/(δ1 (N1 + N2 ) + β)
(4.3.8)
Price elasticity of demand ((N1 + N2 )/β)(P/Q)
(4.3.9)
Notes: Rationale for localization (see Appendix A): Z3—Implicit unit price advantage at some locales. Givens (parameter or exogenous): Ci —Intercept of inverse supply curve at place i; Ni —Population of place i; α—Intercept of individual linear inverse demand curve; β—Slope of individual inverse demand curve; δ i —Slope of inverse supply curve at Place i. Outcomes (endogenous): P—Price (same at both places); Qi —Equilibrium quantity supplied by Place i; Q—Aggregate quantity.
by which local supply exceeds local demand, local here excluding demand or supply by arbitrageurs) is 2,750 − 2,250 = 500 units. This is the amount that arbitrageurs purchase at Place 1 for resale at Place 2. At the more costly Place 2, the autarky price is 13.00. In the integrated market, the firms of Place 2 produce 1,750 units: down from 2,000 units under autarky. They are worse off in the integrated market compared to autarky because they have lost customers to the more efficient producers from Place 1. At the same time, consumers at Place 2 benefit because, for them, the integrated market price is lower than in autarky. Consumers now demand 2,250 units: up from 2,000 units under autarky. The excess demand17 at Place 2 (the amount by which local demand exceeds local supply, local here excluding demand or supply by arbitrageurs), 2,250 − 1,750 = 500, of course, is exactly the amount reallocated from Place 1 to Place 2. I label demand by consumers at a place as local demand in Table 4.5; supply by producers at that market is called local supply. This serves to differentiate such demand and supply from the (nonlocal) demand and supply attributable to arbitrageurs. It is interesting to note that I have drawn neither a demand curve for
17 In
a market at a given price P, excess demand is the amount, if any, by which local demand exceeds local supply. Local here excludes demand or supply by arbitrageurs.
4.3
Model 4B: Integrated Market Solution: Zero Shipping Cost
111
arbitrageurs in the market with the low autarky price (here Place 1) nor a supply curve for them in the market with the high autarky price (here Place 2). This is because the demand (and supply) of arbitrageurs is a derived demand (and supply); its level can be solved after netting out local demand and local supply at each place. Before leaving the integrated market solution, let me make a final comment about the kink in the supply curve noted above. To some, including many of my mathematical friends, kinks are ugly. At the least, having to solve both (4.3.5) and (4.3.6) before I can figure out the market price is tedious. In the standard versions of most of the basic models in Economics, kinks are nowhere to be found. So, what makes the integrated model so different? At the essence of the kink is the notion that geography here is both distinct and differentiated. Being punctiform, it is distinct; there are two markets between which consumers and producers themselves do not relocate. For example, the number of consumers at each place is assumed fixed (N1 and N2 ) even though you might think consumers would be motivated to reside in a market with a lower price, other things being equal. Second, geography is differentiated on the supply side here; the supply curve parameters at Place 1 are different from those at Place 2. Why should there be a difference? What prevents firms from carrying production technologies and efficiencies from one market to the next even though they can freely ship the commodity between markets? To help spur thinking here, assume for the moment that every producer in both markets was equally efficient. We can get as much output as we want in either market at the same unit cost: just as I did in the two-factory solution in Chapter 2. Under these circumstances, the supply curve at each place would be horizontal and the equilibrium price at each place would be the same regardless of the population in that market. In contrast, however, this chapter makes two different assumptions about the local supply curve. 1. Upward-sloped local supply curve at each place. As a result of congestion, the marginal unit becomes more costly to produce as quantity is produced in total locally. One aspect of this is an inputs argument that follows from the idea that a product is manufactured from material inputs. When the level of output is low, those material inputs can be obtained nearby. When the level of output is higher, the material inputs must be purchased (for some reason) from further afield, hence a higher unit cost. Implicit here is the idea that inputs are not ubiquitous. Another is an entrepreneur argument based on the idea that each unit of output is produced by an entrepreneur and that some entrepreneurs are more skilled than others. When quantity produced is small, only the most efficient entrepreneurs enter the market, and unit cost is low. As the demand for the commodity rises, less efficient entrepreneurs are drawn into the market, driving up marginal cost and hence price. 2. Supply curve intercepts, C1 and C2 , are not necessarily equal. Put differently, the most efficient firm at Place 1 and the most efficient firm at Place 2 have different unit costs. Perhaps this is because of the inputs or entrepreneur arguments above. In the integrated market version of the problem, these assumptions are not particularly bothersome. After all, in the absence of shipping costs on the commodity,
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The Struggling Masses
why should it matter where or how a firm produces? However, as we will see shortly, vexing questions come to the fore when shipping costs are nonzero. Before I do this, what about the comparative statics of Model 4B. In the integrated market solution, there are eight givens (α, β, C1 , C2 , δ1 , δ2 , N1 , and N2 ) and two endogenous variables (P and Q). Let us look here at the comparative statics only in the cases where C1 < P < C2 and where P > C2 ; in other words, I will ignore knife-edge cases (wherein P shifts from below C2 to above C2 ). The explicit solutions for P and Q in (4.3.3), (4.3.4), and (4.3.6) make it relatively easy to do comparative statics here. See Table 4.4. Table 4.4 Model 4B (integrated markets): comparative statics of an increase in exogenous variable
Outcome
Given
P [1]
Q [2]
C1 C2 N1 N2 α β δ1 δ2
+ + + + + − + +
− − + + + − − −
Notes: See also Table 4.3; +, Effect on outcome of change in given is positive; –, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome; ?, Effect on outcome of change in given is unknown.
C1
C2
N1
N2 α
If C1 is increased, the supply curve shifts upward parallel and market equilibrium runs up the demand curve. Price rises, while the quantity transacted drops. The results are the same whether C1 < P < C2 or P > C2 . If C1 < P < C2 , an increase in C2 has no effect on the outcome of the model, since production facilities at Place 2 are not being used. On the other hand, suppose. If C2 is increased here, the supply curve shifts upward parallel and market equilibrium runs up the demand curve. Price rises, while the quantity transacted drops. When N1 is increased, the aggregate demand curve becomes flatter; that is, sweeps counterclockwise about (0, α). It traces out the supply curve. Both P1 and Q1 increase. When N2 is increased, the aggregate demand curve becomes flatter—that is, sweeps counterclockwise about (0, α). It traces out the supply curve. Both P1 and Q1 increase. If α is increased, the demand curve shifts upward and market equilibrium runs up the supply curve. Price rises; so too does the quantity transacted. The results are the same whether C1 < P < C2 or P > C2 .
4.4
β
δ1 δ2
Model 4C: Spatial Price Equilibrium with Shipping Costs
113
If β is increased, the individual inverse demand curve sweeps clockwise about (0, α) and market equilibrium runs down the supply curve. Price drops, as does the quantity transacted. The results are the same whether C1 < P < C2 or P > C2 . If is increased, the supply curve sweeps counterclockwise about (0, C1 ), and market equilibrium runs up the demand curve. Price rises, while the quantity transacted drops. The results are the same whether C1 < P < C2 or P > C2 . If C1 < P < C2 , an increase in δ 2 has no effect on the outcome of the model, since production facilities at Place 2 are not being used. On the other hand, suppose P > C2 . Here, if δ 1 is increased, the supply curve sweeps counterclockwise about (0, C2 ), and market equilibrium runs up the demand curve. Price rises, while the quantity transacted drops.
Before leaving this model, one final comment is in order. When we draw a local supply curve for either Place 1 or Place 2, we did not ask ourselves where additional output comes from. In one sense, it does not matter; after all, we simply need to know what it will cost to produce a marginal unit, and the local supply curve tells us that. In another sense, however, how existing firms might reallocate production from one place to the other could be important. Inherent in the notion of an upward sloped supply curve is that each additional unit of product comes a higher cost supplier. Who is this marginal supplier and where does he come from? I will return to this question shortly.
4.4 Model 4C: Spatial Price Equilibrium with Shipping Costs How do we find the level of shipments when shipping cost is neither prohibitive nor zero. In the example presented in Table 4.5, we see a difference in autarky prices of 13.00 – 12.50 = 0.50. Now, as in Chapter 2, assume the unit shipping cost from Place 1 to Place 2 is sx. As in Chapter 2, assume either (1) freight is simply a cost of overcoming distance, or (2) the arbitrageur is a price taker in the market for shipping.18 As far as the arbitrageur is concerned, sx is simply a fixed expenditure required (per unit shipped) that reduces excess profit. In this chapter, sx is therefore the only cost considered explicitly; as noted above, other unit costs are implicit in the specification of the local supply curve. We know therefore that if the shipping cost (sx) is larger than 0.50 there will be no incentive for arbitrage and the two markets will be in autarky. However, if the shipping cost is less than 0.50, there is an incentive for arbitrageurs to purchase at Place 1 for resale at Place 2. We have already seen that, in the limiting case of zero shipping costs, arbitrageurs would demand 500 units at Place 1 to resell at Place 2. Given the linearity assumed in the local demand curves and supply curves in this chapter, it seems reasonable to expect that the derived demand by arbitrageurs will be broadly a linear function 18 As
in Chapter 2, I assume here no congestion over the shipping network. The firm can ship as much, or as little, as it likes for the same unit cost sx.
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The Struggling Masses
Table 4.5 Numerical example Model 4A autarky
Price Local demand Local supply Excess supply Consumer surplus Producer surplus Net social welfare Quantity shipped Shipping cost GNSW
Model 4C sx = 0.10
Model 4B integrated
Place 1 [1]
Place 2 [2]
Place 1 [3]
Place 2 [4]
Place 1 [5]
Place 2 [6]
12.50 2,500 2,500 0 3,125 3,125 6,250
13.00 2,000 2,000 0 2,000 2,000 4,000
12.70 2,300 2,700 400 2,645 3,645 6,290
12.80 2,200 1,800 −400 2,420 1,620 4,040
12.75 2,250 2,750 500 2,531 3,781 6,313
12.75 2,250 1,750 −500 2,531 1,531 4,063
0 0 10,250
400 40 10,290
500 0 10,326
Notes: Parameter values are α = 15, β = 1, N1 = N2 = 1,000, C1 = 10, C2 = 11, δ1 = δ2 = 0.001. Outcomes in autarky, sx = 0.10, and integrated market (sx = 0) are calculated from Tables 4.1, 4.5, and 4.2 respectively. GNSW is global net social welfare. As to be defined in Chapter 5, GNSW = NSW1 + NSW2 − Shipping cost. Calculations by author. Notation: sx – unit shipping cost.
of the shipping rate. Indeed that is the case. See the second numerical example presented in Fig. 4.5. The vertical difference between the excess supply curve for Place 1 and the excess demand curve for Place 2 is called a price difference curve19 (curve ABCD in Fig. 4.5), because it shows the quantity that must be traded by arbitrageurs to result in a given difference in prices between the two markets. For example, where the shipping rate in Fig. 4.5 is zero, the integrated market solution obtains. This would result in arbitrageurs purchasing 1,353 units at Place 1 (amount OJ in Fig. 4.5) for resale at Place 2. Such activity would cause the price at Place 1 to rise (because of increased demand) from OS to OL in Fig. 4.5 and cause the price at Place 2 to fall (because of increased supply) from OR to OL in Fig. 4.5 such that the price in both markets becomes 10. The linear nature of the price difference curve is easily confirmed by the following procedure 1. Derive the excess supply schedule for Place 1. See (4.6.1) in Table 4.6 and the polyline labeled E1 F1 G1 H1 in Fig. 4.5. This schedule shows the amount by 19 In a two-region model of trade, the Price Difference Curve shows the amount of the good shipped
from the lower priced to the higher priced region that would result in a given difference in prices between the two regions. Calculated as the vertical difference between the excess supply curve in the lower price region and the excess demand curve in the higher price region. In this text, I refer to this derivation as the Samuelson Model. There are variants of this approach that are essentially the same: see Siebert (1969, pp. 85–87) or Takayama and Judge (1971, pp. 135–137).
4.4
Model 4C: Spatial Price Equilibrium with Shipping Costs A
Model 4C Spatial price equilibrium
$
E2 F2
Q
B
R M L N S
ABCD E1F1G1H1
Price difference curve Excess supply curve for Place 1
E2F2G2H2 OI OJ OL OM ON OP
Excess demand curve for Place 2 Quantity shipped from Place 1 to Place 2 Quantity shipped in integrated market (Model 4B) Price in integrated market (Model 4B) Equilibrium price at Place 1 Equilibrium price at Place 2 Unit shipping cost
E1
H1
G1
G2
T
F1
115
V U P 0
H2
IJ
Shipment
C
D
Fig. 4.5 Model 4C: excess supply (or demand) curve and price difference curve. Notes: α = 15, β = 1, δ1 = 0.001, δ2 = 0.002, C1 = 2, C2 = 7, N1 = 1,000, N2 = 900, s = 0.013, x = 100. With arbitrage, P1 = 9.46, P2 = 10.76, and quantity shipped from Place 1 to Place 2 is 1,929. Horizontal axis scaled from –15,000 to 25,000; vertical from –8 to 22
which local supply exceeds demand (negative, if local demand larger than supply) at any given price. It is calculated by reexpressing Q1 as D1 in the demand equation (4.1.1), Q1 as S1 in the supply equation (4.1.2), rearranging the two equations to put quantity as a function of price, and then subtracting the demand equation from the supply equation. 2. For Place 2, calculate the excess demand schedule in a similar manner. See (4.6.5) and the polyline labeled E2 F2 G2 H2 in Fig. 4.5. 3. Rearrange the excess supply curve at Place 1 and excess demand curve at Place 2 each to put price on the left and excess supply (or demand) on the right. See (4.6.2) and (4.6.6). 4. Take (4.6.2) minus (4.6.6) under the assumption that ES1 = ED2 to get the price difference as a linear function of excess supply at Place 1. The result is (4.6.9). The price difference (P2 − P1 ) is now a linear function of excess supply (ES1 ).
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The Struggling Masses
Table 4.6 Model 4C: price difference curve (assuming C1 < P1 < α and C2 < P2 < α and product shipped only from Place 2 to Place 1 Excess supply at Place 2 ES2 = − (C2 /δ2 + αN2 /β) + (1/δ2 + N2 /β)P2
(4.6.1)
Rearranging (4.6.1) yields P2 = α2 + β2 ES2
(4.6.2)
Where a2 = (C2 β + αδ2 N2 )/(β + δ2 N2 ) b2 = (βδ2 )/(β + δ2 N2 )
(4.6.3) (4.6.4)
Excess demand at Place 1 ED1 = (C1 /δ1 + αN1 /β) − (1/δ1 + N1 /β)P1
(4.6.5)
Re-arranging (4.6.5) yields P1 = a1 − b1 ED1
(4.6.6)
Where a1 = (C1 β + αδ1 N1 )/(β + δ1 N1 ) b1 = (βδ1 )/(β + δ1 N1 )
(4.6.7) (4.6.8)
Equation (4.6.2) minus equation (4.6.6) yields P1 − P2 = (a1 − a2 ) − (b1 + b2 )ES1
(4.6.9)
Re-arranging (4.6.9) yields price difference curve ES1 = (a1 − a2 )/(b1 + b2 ) − (1/(b1 + b2 ))(P1 − P2 )
(4.6.10)
Notes: Rationale for localization (see Appendix A): Z3—Implicit unit price advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): Ci —Intercept of inverse supply curve at Place i; Ni —Population of Place i; α—Intercept of individual linear inverse demand curve; β—Slope of individual inverse demand curve; δ i —Slope of inverse supply curve at Place i. Outcomes (endogenous): EDi —Excess demand at Place i; ESj —Excess supply at Place j; Pi —Price at Place i; Qi —Equilibrium quantity transacted at Place i.
Operationally, I need to rearrange (4.6.9) to get the amount of the shipment on the left and the price difference on the right: see (4.6.10). See the polyline ABCD in Fig. 4.5. In Fig. 4.5, note some basic characteristics of the excess supply, excess demand, and price difference curves. • Since I have assumed a common value for α, the excess supply and demand curves have a kink in common at P = α, namely points F2 and G1 in Fig. 4.5. • The excess demand curve for place 1 has a kink at P = C1 : namely point F1 . Similarly, the excess supply curve for place 2 has a kink at P = C2 : namely point G2 . • Every kink in an excess demand (or excess supply) curve matches up vertically with a kink in the price difference curve: e.g., point F2 with point B and point G2 with point C in Fig. 4.5. In Fig. 4.5, we do not see the kinks in the price difference curve corresponding to points F1 and G1 simply because these are outside the area graphed.
4.4
Model 4C: Spatial Price Equilibrium with Shipping Costs
117
We are now able to predict the shipment, if any, between the two places. First, plot the unit shipping cost on the vertical axis of Fig. 4.5 (see point P): I use sx = 1.30 to illustrate here. Move horizontally to the right until you cross the price difference curve; if no intersection, then sx is too high to permit trade. Then, drop vertically down Fig. 4.5 to the point (I) where we cross the horizontal axis. This gives the amount (OI) shipped from Place 1 to Place 2. Then, extend the same vertical line upwards until it crosses the Excess Demand Place 2 curve or the Excess Supply Place 1 curve and move horizontally to the left until you reach the vertical axis; this gives the spatial equilibrium price, P2 (corresponding to OM in Fig. 4.5) or P1 (corresponding to ON), respectively. Equivalently, we can use (4.6.2) and (4.6.6) to predict the price at each place. Calculated in this way, these prices, therefore, differ at most by the unit shipping cost: amounts MN and OP are the same in Fig.4.5. Let us return to the example in Table 4.5. There, I show in columns [4] and [5] the solution to the problem when sx = 1.30, not zero as in the integrated market solution, but at least smaller than the sx that would bring on autarky. In this case, 1,929 units are shipped, and the equilibrium prices are P1 = 9.46 and P2 = 10.76. Compared to autarky, consumers at Place 1 are worse off while local producers are better off. Compared to autarky, consumers at Place 2 are better off while local producers are worse off. However, there is an important caveat here. The price difference curve, as shown in Fig. 4.5, is itself kinked, forming a polyline of up to 5 possible linear segments.20 These segments represent all feasible combinations of three levels of P1 —low (P1 < C1 ), medium (C1 < P1 < α), and high (P1 > α) —and three levels of P2 — low (P2 < C2 ), medium (C2 < P2 < α), and high (P2 > α). What is infeasible are the combinations of (1) P1 and P2 simultaneously low since there is no production anywhere, (2) P1 and P2 simultaneously high since there is no demand anywhere, or any other combination where p1 < C1 since we have assumed C1 < C2 . Table 4.6 illustrates the calculations for just one of five possible cases.
p2 < C2 C2 < p2 < α p2 > α
20 Early
p1 < C1 No supply C1 < C2 C1 < C 2
C1 < p1 < α Segment 1 Segment 3 Segment 5
p1 > α Segment 2 Segment 4 No demand
writers largely ignored this. Enke (1951, p. 42), for example, assumes that excess supply will always be linear in price, not piecewise linear as argued here. Samuelson (1952, pp. 286 and 288) draws excess supply curves and a price difference curve that are also kinked. However, Samuelson also draws local demand and supply curves that are kinked without any explanation and does not draw the kinks in the excess supply curves or price difference curves to correspond to situations, where either local demand or local supply have been driven to zero. Takayama and Judge (1971, p. 135) do incorporate corner solutions that arise because of kinks, but refer to such solutions as irregular.
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The Struggling Masses
The bewildered or bowed reader—there may be many at this stage—might say that Table 4.6 is enough. The thought of four more tables like it is enough to dampen the enthusiasm of anyone but the author and even then . . . After all, you might say, is not Table 4.6, wherein both markets have local demand and local supply, a case both commonplace and instructive. Instructive it may be, but commonplace is another question. An argument often made in international trade and location theory is that production of a commodity may increasingly become concentrated in just one place globally. Increasingly then, most places in the world would have local consumption but no local production. Viewed in light of that argument, Table 4.6 would then seem to represent a special case (both production and consumption at every place), not something commonplace. The five possible cases, of which Table 4.6 is one, reflect the existence of kinks in the price difference curve. A full set of solutions is shown in Table 4.7 for the case where shipments may arise from Place 1 to Place 2. The inverse excess demand curve at Place 2 takes one of the three forms shown in (4.7.1). The inverse excess supply curve at Place 1 takes one of the three forms shown in (4.7.2). From these, I can calculate the five possible segments of the price difference curve in (4.7.3) and then derive the equilibrium shipment for a given shipping cost in (4.7.4) for each of these segments. Consider an illustrative second example. If I make δ 1 smaller (δ 1 is just 0.0003 in Fig. 4.6) and C2 larger, a still greater production is possible at Place 1 at any given unit cost above C1 . The implication is that firms at Place 1 are better able to compete. In fact, if I make production at Place 1 sufficiently efficient, suppliers at Place 2 can no longer compete and production there ceases. This is the outcome illustrated in Fig. 4.6. Let us now return to the question of why the local supply curves differ between the two markets. Let me begin by assuming the entrepreneur argument: i.e., the local supply curve is upward sloping because some entrepreneurs are more efficient than others. Above, I have assumed that C1 < C2 which implies the most efficient entrepreneurs are at Place 1. However, Place 1 may also well be the low-price market. This begs the question of why the most efficient entrepreneur would want to supply there. Why not instead produce at Place 2 where the price and therefore the unit profit (assuming the same costs of production in the two markets) are higher. Is there not an incentive for efficient producers to move from the low-price market to the high-price market? In so doing, their unit profit would increase by the shipping rate. One possible counter-explanation is the inputs argument described above: i.e., the most efficient entrepreneur at Place 1 finds that he or she cannot achieve the same low unit cost producing at Place 2. This model requires us to think more about the nature and location of production efficiencies. Finally, as in Chapter 2, a price premium at Place 2 opens up the question of the happiness of consumers there compared with Place 1. Other things being equal, we might expect consumers at Place 2 to want to relocate to the less-costly Place 1, to go there to shop, or otherwise take advantage of the lower price at Place 1. For the time being, I ignore such considerations.
4.4
Model 4C: Spatial Price Equilibrium with Shipping Costs
119
Table 4.7 Model 4C: price difference curve (assuming product shipped only from Place 1 to Place 2) Excess demand at Place 2 when P2 is at level j (j = 1 when P2 < C2 , j = 2 when C2 < P2 < α, j =3 when P2 > α) P2 = aj20 + aj21 ED2
(4.7.1)
Excess supply at Place 1 when P1 is at level i (i = 1 when P1 < C1 , i = 2 when C1 < P1 < α, i = 3 when P1 > α) P1 = ai10 + ai11 ES1
(4.7.2)
Difference in price between markets in equilibrium (E = ES1 = ED2 ) P2 − P1 = (aj20 − ai10 ) + (aj21 − ai11 )E
(4.7.3)
Price difference curve E = − (aj20 − ai10 )/(aj21 − ai11 ) + (1/(aj21 − ai11 ))sx
(4.7.4)
where a110 = u10 /u11 a111 = 1/u11 a210 = −(v10 − u10 )/(v11 + u11 ) a211 = 1/(v11 + u11 ) a310 = −v10 /v11 a311 = 1/v11 a120 = u20 /u21 a121 = −1/u21 a220 = (u20 − v20 )/(u21 + v21 ) a221 = −1/(u21 + v21 ) a320 = − v20 /v21 a321 = −1/v21
(4.7.5) (4.7.6) (4.7.7) (4.7.8) (4.7.9) (4.7.10) (4.7.11) (4.7.12) (4.7.13) (4.7.14) (4.7.15) (4.7.16)
and u10 = αN1 /β u11 = N1 /β u20 = αN2 /β u21 = N2 /β v10 = − C1 /δ1 v11 = 1/δ1 v20 = −C2 /δ2 v21 = 1/δ2
(4.7.17) (4.7.18) (4.7.19) (4.7.20) (4.7.21) (4.7.22) (4.7.23) (4.7.24)
Note: See also Table 4.6.
What about comparative statics here? Because Table 4.7 does not permit explicit solutions for endogenous variables, I cannot derive general solutions for the comparative statics here. What I can do, however, is to take a base case and then show the magnitudes of changes in each of endogenous variables when I change a particular given. Here, I start from the following base case: α = 15, β = 1, C1 = 2, C2 = 7, δ1 = 0.001, δ2 = 0.002, N1 = 1,000, N2 = 900, and s = 0.02, and x = 100. In the base case, the equilibrium price in each market (P1 = 9.18 and P2 = 11.18) is above C1 and C2 , respectively, which implies that there will be local production at both places. In equilibrium, a shipment occurs from Place 1 to Place 2.
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The Struggling Masses
Model 4C Spatial price equilibrium
A $
E2
ABCD E1F1G1H1 E2F2G2H2 OI OJ OL OM ON OP
Price difference curve Excess supply curve for Place 1 Excess demand curve for Place 2 Quantity shipped from Place 1 to Place 2 Quantity shipped in integrated market (Model 4B) Price in integrated market (Model 4B) Equilibrium price at Place 1 Equilibrium price at Place 2 Unit shipping cost
H1 G1
F2
B
G2 M V L N S
C
P F1
E1
0
I J
H2
Shipment
D
Fig. 4.6 Model 4C: a second example showing demand by arbitrageurs at Place 1 for resale at Place 2. Note: α = 15, β = 1, δ2 = 0.0003, δ2 = 0.002, C1 = 2, C2 = 11, N1 = 1,000, N2 = 900, s = 0.04, x = 100. With arbitrage, P1 = 6.03, P2 = 10.03, zero quantity produced at Place 2, and quantity shipped from Place 1 to Place 2 is 7,452. Horizontal axis scaled from –15,000 to 25,000; vertical from –8 to 22
In Table 4.8, I show the comparative statics results. Each row is the result of changing one given value. In the first row, for example, I show what happens when α is increased to 15.1. C1
If C1 is increased, the supply curve shifts upward parallel among producers at the more efficient Place 1. At Place 1, market equilibrium runs up the demand
4.4
Model 4C: Spatial Price Equilibrium with Shipping Costs
121
Table 4.8 Model 4C: relative change in endogenous variable as percentage ratio of relative change in given Local price
Local demand
Local supply
Given
New value 1 [1] [2]
2 [3]
1 [4]
2 [5]
1 [6]
2 [7]
Shipment [8]
C1 C2 N1 N2 sx α β δ1 δ2
2.1 7.1 1,100 1,000 0.024 15.1 1.01 0.0011 0.0021
+ + + + + + − + +
− − + − + + − − −
− − − + − + − − −
− + + + − + − − +
+ − + + + + − + −
− + − + − + + − +
+ + + + − + − + +
Notes: See also Table 4.6. Calculated relative to base case where α = 15, β = 1, C1 = 7, C2 = 7, δ1 = 0.001, δ2 = 0.002, N1 = 1,000, N2 = 900, and s = 0.02. Calculations by author.
C2
N1
N2
s
curve: price rises, while the quantity demanded drops. The same thing happens at Place 2 because part of its supply comes from producers at Place 1. Shipments from Place 1 to Place 2 fall off because Place 1 now has a lesser cost advantage. The level of production drops at Place 1, but Place 2 producers, now relatively less inefficient, produce more. If C2 is increased, the supply curve shifts upward parallel among producers at the less-efficient Place 2. At Place 2, market equilibrium runs up the demand curve: price rises, while the quantity demanded drops. Because firms at Place 1 are now comparatively even more efficient, shipment from Place 1 rises, pushing up the price at Place 1 and cutting into the quantity demanded locally. When N1 is increased, local price increases at both places. The aggregate quantity demanded falls at Place 2 as a result; however, demand goes up at Place 1, despite the higher price, because of its now-larger population. Production rises at both places, and shipment falls off because Place 1 has a smaller price advantage than was the case before. When N2 is increased, local price increases at both places. The aggregate quantity demanded falls at Place 1 as a result; however, demand goes up at Place 2, despite the higher price, because of its now-larger population. Production rises at both places, and shipment increases because the more efficient firms that are at Place 1 can take advantage of the larger population now at Place 2. When s is increased, it becomes less attractive to ship the commodity from Place 1 to Place 2. This pushes up the price at Place 2 (less supply in total) and down at Place 1 (more supply). The reduced shipment means also that firms at Place 2 produce more, while firms at Place 1 produce less. The changes in price also imply that the quantity demanded will rise at Place 1 and fall at Place 2.
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α
β
δ1
δ2
4
The Struggling Masses
When α is increased, the demand curve at each of Places 1 and 2 shifts upward. Therefore, ignoring shipments, market equilibrium runs up the local supply curve. Therefore, the equilibrium price rises at each local market as does the quantity demanded locally and the quantity supplied locally. Since Place 1 producers are more efficient at any scale (i.e., C1 < C2 and δ1 < δ2 ), the shipment from Place 1 to Place 2 increases. When β is increased, the individual inverse demand curve sweeps clockwise about (0, α), and, again ignoring shipments for the moment, market equilibrium runs down the supply curve; hence lower price and lower quantity demanded locally and lower quantity supplied locally at each place. The decline in demand is felt mainly by the less-efficient producers at Place 2, and this is exacerbated by an increase in shipments from Place 1. If δ 1 is increased, the supply curve becomes steeper among producers at the more efficient Place 1. The effect here is similar to an increase in C1 . At Place 1, price rises, while local demand and local production drop. At Place 2, price rises and local demand drops; however, local production goes up. Shipments from Place 1 to Place 2 fall off. If δ 2 is increased, the supply curve becomes steeper among producers at the less-efficient Place 2. The effect here is similar to an increase in C2 . At Place 2, price rises while local demand and local production drop. At Place 1, price rises and local demand drops; however, local production goes up. Shipments from Place 1 to Place 2 go up.
4.5 Final Comments In this chapter, the principal model has been 4C. I included Models 4A and 4B to help readers better understand aspects of Model 4C. In Table 4.9, I summarize the assumptions that underlie Models 4A–4C. Many assumptions are in common to all models: see panel (a) of Table 4.9. In Model 4A, Places 1 and 2 are each in autarky. Each forms its own market for the commodity populated only by local suppliers and local demanders. The equilibrium price at Place 1 may or may not be equal to the equilibrium price at Place 2. What separates the two markets is the idea that—under autarky—a shift in any of the parameters in the market at Place 1 has no effect on the equilibrium price at Place 2 and vice versa. Further, since local production is just sufficient to meet local demand, there is no localization of production here. There are no shipments. In Model 4B, Places 1 and 2 now form a single market with a single equilibrium price. As a single market, a change in any parameter of local demand or local supply at one Place can affect equilibrium price at both Places 1 and 2. Where production of the commodity at one place is particularly advantageous, we see localization of production at that place. Model 4B says nothing about shipments. Since it is costless, there may be any amount of shipping; cross hauling is possible here. The extent of localization will depend on differences between the parameters of the local supply
4.5
Final Comments
123 Table 4.9 Assumptions in Models 4A through 4C
Assumptions
4A [1]
4B [2]
4C [3]
(a) Assumptions in common A1 Closed regional market economy A3 Punctiform landscape B1 Exchange of commodity for money B2 Upwardly sloped local supply curve B4 Local demand C2 Fixed local customers C3 Fixed remote customers C4 Identical customers C5 Identical linear demand
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
(b) Assumptions specific to particular models E3 Unit shipping cost is prohibitive E1 Zero shipping cost everywhere E4 Unit shipping costs symmetric
x x x
curves at the two places. Given C1 < C2 , production of the commodity will be solely at Place 1 if δ 1 —the slope of the local supply curve there—is small enough. See (4.3.7) and (4.3.8). Since there is only one market here, there is still no Walrasian process at work to link prices between markets. In Model 4C, Places 1 and 2 form two submarkets. As in Chapter 2, the place more costly to serve will have a price premium. Shipping occurs only up the price gradient. In this chapter, the price premium will be equal to the unit shipping cost. The price premium will be between zero (Model 4B solution) and the autarky price difference (Model 4A solution). As the price premium approaches the Model 4A solution, the extent of localization approaches zero. Model 4C exemplifies a Walrasian linkage of the prices at two places. Anything that causes the price to rise at the exporting place causes price to rise by the same amount in the importing place. Anything—other than a change in unit shipping cost—that causes the price to rise in the importing place will similarly cause the price to rise in the exporting place. And, when the unit shipping cost rises, price will increase in the importing place and drop in the exporting place. Note the difference in effect here vis-à-vis Model 2D; there, a constant marginal cost of production meant that a shift in demand at Place 1 had no effect on price at Place 2 and vice versa. At the same time, this Walrasian process is restricted; it looks at simultaneity only across two places; it does not consider the simultaneity between price of the commodity and the prices of other goods and services. In Chapter 1, I argue that prices are important in shaping the location of firms. In this chapter, the prices of inputs used by the firms in local production are implicit in the intercept and slope of the local supply curve. Arbitrageurs face input prices that are implicit in s. As in Chapter 2, consumers have income constraints and face prices for other goods and services that determine the intercept and slope for their demand
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curve. As in Chapter 2, these other prices are all determined in markets outside the scope of the models in Chapter 4. Once again, I think Walras would have argued that the analysis has been only partial in the sense that we have not looked explicitly at the simultaneity among prices in these markets. Once again, we must wait until a later chapter for the opportunity to do that. What is the source of localization in Model 4C? Presumably, it is the differences among places: i.e., in Ci , δ i , α, β, or Ni . A combination of these can make the autarky price sufficiently lower at Place i to give arbitrageurs the incentive to participate. However, Model 4C is silent on the reasons for this advantage. While it is true, for example, that economies of scale or division of labor might account for a lower Ci at Place i, there are other possible explanations: e.g., variations in input prices. The models in this chapter represent the outcome of a competitive market in which no one buyer or seller is able to influence the price at which the commodity is sold. As such, these models do not take into account the kind of monopoly power that arises in geographic space where, for example, competitors elsewhere find unit shipping cost makes it more costly for them to compete for customers near a given firm. I consider such effects later in this book. Finally, the models in this chapter do not say anything about the distribution of income in society among units of labor, capital, and land. As in Chapter 2, the firm is envisaged to incur costs, but these are not related explicitly to the hiring of labor or the rental of land. In any case, all income gains that can arise (e.g., because of an increase in α or a decrease in C1 , C2 , or s) accrue to firm owners in the form of increased profit. At the same time, any adjustment of market price affects the wellbeing of consumers; in this chapter, such changes are being measured by consumer surplus.
Chapter 5
Arbitrage in the Grand Scheme Perfect Competition at Many Places (Samuelson–Takayama–Judge Problem)
Imagine N places, isolated from the rest of the world. Assume perfect competition; no one supplier or demander can affect market price. If unit shipping costs are prohibitive, all places with both local demanders and local suppliers will be in autarky. However, if at least one unit shipping cost is sufficiently low, arbitrageurs will purchase the commodity where price is low for resale elsewhere. How does the amount shipped and the resulting prices at the N places depend on unit shipping costs? Following a conjecture by Samuelson as implemented by Takayama and Judge, Model 5A is a quadratic program in which Global Net Social Welfare— consumer benefit (under a linear demand curve) minus producer cost (under a linear supply curve) minus shipping costs—is maximized. This chapter builds on models presented in Chapters 3 and 4. Chapter 4 solves a similar problem graphically using price difference curves but that method limits us to two places. Model 5A solves the same problem for any number of places. Chapter 3 solves a logistics problem using linear programming where the quantities demanded and supplied at each place are fixed. Model 5A makes the quantities endogenous to the model and solves for the equilibrium prices (one for each place) that give rise to these outcomes.
5.1 The Samuelson–Takayama–Judge Problem Chapter 4 presents a graphical solution method—the price difference curve —with which I can calculate the equilibrium level of shipment, if any, from a place where price is low to a place where price is higher. We are then able to show how, if unit shipping cost were higher, the quantity shipped would be correspondingly smaller (or even zero). In the price difference curve approach, the actions of arbitrageurs— buying at one place, reselling at another—are outcomes derived endogenously from the price difference that would otherwise exist. Chapter 4 showed us how shipment benefited both consumers at the higher price place and efficient producers. The price difference curve approach is of considerable pedagogical merit. It is intuitive and can be presented graphically. It is also helpful in thinking about kinks that arise when production or consumption drops to zero at a given place: evidencing localization. However, it is not of much help in everyday life where typically we J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_5, C Springer Science+Business Media, LLC 2010
125
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have more than two places.1 At this point, it is also instructive to ask why, in this book, I want to look at the case of three or more places sharing a common fiat money economy. At the outset, I stated that the purpose of this book is reinterpretation. What motivates me in this chapter is the impact of geography on market formation in particular and on the insights this gives about economic reasoning in general. A spatial price equilibrium is just one example of a model of economic equilibrium.2 Harker and Pang (1990, pp. 161–162) argue that there have been three main approaches to computation of equilibrium: (1) the fixed point (homotropy-based) approach, (2) the variational inequalities—including complementarity—approach, and (3) the optimization approach. In my experience, the first two approaches are useful in computation but shed little added light on the nature and implications of spatial price equilibrium.3 To me, the first two approaches are also unsatisfactory; they assume spatial price equilibrium rather than have that equilibrium arise as an outcome of the model. I intend no offence to proponents of the other two approaches here; I am merely laying out my own preferences. In contrast, I find the optimization approach gives insights into the impact of geography on market formation and economic thinking. In an important and influential step forward, Samuelson (1952) invokes an optimization approach in thinking about spatial price equilibrium. Let me now describe Samuelson’s approach (the Samuelson conjecture) in more detail. He argued intuitively that economists would want to conceptualize trading activity as leading to the maximization of something at the level of the entire society. He conjectured that if shipment was to be desirable, it had to improve what he labeled net social payoff (NSP) in aggregate across the places in the same sense that an autarky solution maximizes well-being at one place. Specifically, he proposed (p. 288) a net social payoff with three components: Social Payoff in region 1 + Social Payoff in region 2 – Shipping Cost. Social payoff for a region is the algebraic area under its excess-demand curve. Samuelson (p. 288) was guarded in his use of terms here: being careful to say that social payoff was not the same as consumer surplus.4 At the same time, however reasonable Samuelson’s approach may seem to economists, it is not clear to noneconomists why markets should necessarily behave in this way. I return to this matter shortly.
1 See
also Enke (1951, p. 42). (1987) reformulates spatial price equilibrium as a game played by coalitions of participants each seeking to maximize their own benefits. 3 See Kakutani (1941), Lemke and Howson (1964), Scarf (1973, 1991), MacKinnon (1975, 1976), Kuhn and MacKinnon (1975), and Smith (1979, 1983) regarding the fixed point approach and Friesz, Harker, and Tobin (1984), Pang (1984), Harker (1985), Harker and Pang (1990), Güder, Morris, and Yoon (1992) and Wu, Florian, and Marcotte (1994) regarding the complementarity/variational inequality approach. 4 In his path-breaking treatise on economic analysis (Samuelson, 1947, pp. 195–202), Samuelson elaborates on his skepticism about the use of consumer surplus, producer surplus, and net social welfare. His use of net social payoff, despite this skepticism, suggests the usefulness of concepts like consumer surplus even though we might prefer something better. 2 Harker
5.1
The Samuelson–Takayama–Judge Problem
127
Samuelson had two hunches about this problem. First, he argued (p. 90) that one could experimentally vary exports, by trial and error, so as to maximize net social payoff. A number of early studies apparently took this advice.5 Fox (1953, p. 548) confirms Samuelson’s hunch: In this book, I do not pursue this trial-anderror process further because it does not contribute to the book’s focus on economic reasoning. His second hunch was based on the argument that part of the spatial price equilibrium problem was subsumed in the Hitchcock–Koopmans Problem (discussed in Chapter 3). The Hitchcock–Koopmans model is a linear (mathematical) program that solves for shipments given quantities demanded and supplied at each place and, therefore, does not solve for market prices.6 However, the Hitchcock– Koopmans model does solve for the opportunity costs (termed the shadow prices) of supply restrictions and demand requirements at each place. Samuelson argued that there should be a relationship between shadow price and market price, and several scholars use variants of that idea to solve the spatial price equilibrium problem.7 Another variant of this approach involves iterative techniques which move back and forth between predictions of shipment flows and predictions of prices until we find mutually sustainable solutions.8 In this chapter, I use a notion similar to Samuelson’s net social payoff that I label global net social welfare.9 I present and interpret the method used by Takayama and Judge (1971) to solve spatial price equilibrium and shipments when there are more than two places and that allows for the kinks that can arise when price locally becomes (1) low enough that there are no longer any local suppliers at one or more places, or (2) high enough that there is no longer any local demand for the commodity at one or more places.10 Because this chapter helps us think about an industry that could conceivably span nations or even the globe, I will discuss the possibilities and limitations of looking
5 See, for example, Fox (1953) and Fox and Taeuber (1955). Another early approach was to further simplify the problem by assuming that supply at each place was a given quantity (not a schedule varying by price). See Judge and Wallace (1958). 6 A similar approach is successive approximations each time using a linear objective function. See Marcotte, Marquis, and Zubieta (1992). 7 See Baker (1961), Henderson (1955), Judge and Wallace (1958), and Daniel and Goldberg (1981). 8 See the applications of reactive programming in Tramel and Seale (1959), Boyd (1983a, 1983b) and Seale, Seale, and Leng (2004). 9 Takayama and Judge (1971) prefer the term “net quasi-welfare”. 10 The model presented in this chapter is the simplest of the models presented in Takayama and Judge (1971). That book considers the case of (1) spatial price equilibrium across multiple commodities, rather than the spatial equilibrium for just one good considered here and (2) changes in inventory holding between periods so that demand need not equal supply each period as assumed in this chapter. Guise (1979) notes two limitations in the Takayama and Judge formulation: (1) commodities must be stored in the region where they are produced; and (2) a good cannot be stored for more than one time period. These are not applicable to this chapter because I ignore inventory holding altogether.
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at issues of globalization. In the meantime, there are immediate conceptual problems in making the transition from 2 places to n places. One of these problems is that, as we increase n, the model comes to span more than one nation or more than one currency area. In extending the model from Chapter 4, we now have to take into account currency fluctuations that might make places in one nation more attractive as a source. Second, in a related vein, national governments and trading blocs may have a tariff11 or nontariff barrier12 that make it difficult, costly, or unattractive to ship commodities from one country to another and may encourage foreign direct investment: i.e., tariff jumping. In keeping with the focus of this book on interregional—as opposed to international—trade, I do not dwell on these matters.
5.2 Model 5A When economists speak of the aggregate well-being of a society, what do they mean? Traditionally, they measure this by something like National Income: the total amount paid to labor, capital, and other factors of (i.e., inputs to) production. Anything that increases national income is seen to make the community better off. The advantage of national income as a measure is its relative clarity and simplicity. In economic cost–benefit analysis, additions to national income are benefits and reductions to national income are costs. In effect, a cost here means the allocation of a resource for any purpose other than consumption made possible by national income. The revenue earned by any firm is seen to be composed of two parts: one (cost of materials) incorporates payments by the firm for commodities consumed in production; the other incorporates explicit payments for labor, capital, entrepreneurial skill, and other factors of production plus any imputed payments such as retained earnings. From this perspective, any inefficiency that adds to costs (i.e., uses resources at the expense of contributing to factor payments) reduces national income.13 To critics, National Income is far from an ideal measure. Among other problems, it ignores issues related to environmental degradation, income inequality, unemployment, unpaid work, and other unpriced or mispriced impacts. It is possible to imagine a broader measure—call it quality of life—but then, in my view, it becomes less clear how to measure benefits and costs unambiguously. Nonetheless—and whatever its limitations—sometimes it is not possible for us to rise even to the level of a national income measure. To measure a change in national income, we need to trace how factor payments are affected throughout the economy. Consider here 11 A
tax charged on an imported good: often ad-valorem (i.e., a percentage of the value of the good). 12 A limitation, other than a tariff, such as a quota or other regulation that makes importing less attractive or feasible. 13 I leave aside here the debate—dating from Arrow (1951)—over whether a community, society, or nation can have a collective social welfare function by which members can be thought to rationally choose one outcome over another.
5.2
Model 5A
129
the monopolist in Model 2A. The firm’s revenue is P1 Q1 and Its semi-net revenue is (P1 − C)Q1 . How does the entry of our firm affect national income? If I assume, for simplicity, that there are no labor costs included in C, then semi-net revenue is the amount that the firm itself contributes to national income. If the firm then finds it profitable to serve customers at Place 2 also, either from the factory at Place 1 or from a new factory at Place 2, the firm’s semi net revenue would rise and its contribution to national income would increase.14 However, the firm’s own contribution to national income is not the entire story. When our firm opts to produce, consumers at Place 1 (and possibly Place 2) now start to purchase the commodity. As they do, they presumably switch from purchasing other commodities that they had been consuming before. In other words, other firms may experience a loss of revenue and semi-net revenue because of the entry of our firm. In practice, the story is complicated by the extent of substitutability—or complementarity—among consumer commodities. As well, there are income effects here; a consumer able to purchase the commodity at a lower price than before is now better off in real terms, and their consumption of commodities and services changes accordingly. In short, to fully assess the impacts of the entry of our firm on national income requires considerable data about all economic activity nationally. That has led some economists to look at approximations (short-cuts) to the measurement of economic well-being. That brings us to the concepts of consumer surplus, consumer benefit, producer surplus, producer cost, monopoly excess profit, and social welfare.15 Marshall is generally credited with introducing the term consumer surplus, although Dupuit had earlier labeled it relative utility.16 Consumer surplus is a money measure of the area below the demand curve and above the market price to the left of market quantity. Put differently, consumer surplus is the hypothetical amount that consumers might be thought to be willing to pay over and above the market price rather than go without the commodity. Among proponents, consumer surplus is thought to measure the benefit to consumers from participating in the market in excess of the amount P that they pay for each unit consumed. We can think of an aggregate demand curve as the ordering of consumers in descending order of the price they are willing to pay; in this sense, consumer surplus identifies the value placed on consumption by the most enthusiastic purchasers (i.e., those willing to pay the most for the commodity) relative to the marginal consumer. A related concept, consumer benefit, measures the entire area under the demand curve up to the
14 In
the alternative case where C includes labor costs, the firm’s contribution to National Income would be larger than its semi-net revenue by the wages paid (i.e., the amount of wage bill). 15 I do not consider other measures of well-being here. In general, economists have concerns with the use of consumer surplus as a measure of social benefit. See Winch (1965), Schmalensee (1972), Burns (1973), Willig (1976), Chipman and Moore (1980), Turnovsky, Shalit, and Schmitz (1980), Hausmann (1981), Blackorby, Donaldson, and Moloney (1984), and Hanemann (1991). Alternatives include the compensating variation and equivalent variation proposed by Hicks (1956, pp. 69–94). 16 Ekelund and Hébert (1999, p. 15).
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quantity demanded. Consumer surplus (CS) is equal to consumer benefit (CB) net of consumer expenditure (PQ) on the commodity: i.e., CS = CB − PQ. To fix ideas here, let me assume an individual consumer with an inverse demand curve as in Chapters 2 and 4: P = α − βq. At market price P, this consumer demands q = (α − P)/β. The consumer values the marginal unit as just worth P. Diminishing marginal utility leads us to expect that additional units of the commodity consumed are less and less valuable to the consumer. Put differently, if the consumer were restricted to a smaller quantity than q, the value of the marginal unit would be larger than P. Viewed in this way, the slope of the demand curve tells us something about the utility gained from additional units of consumption: going say from 0 to q. However, such a conclusion is problematic in at least two respects. First, consumer demand is typically thought to depend on income, the prices of other commodities, and tastes, as well as on the price of the commodity itself. When we write P = α − βq, we hide the effects of income, prices of other commodities, and tastes in α and β. This makes it difficult to interpret movement along an individual’s demand curve as corresponding to any particular change in utility (well-being). Second, when we move from an individual to an aggregate demand curve, it is not clear how to separate the effects of the distribution of income (the idea that different consumers have different incomes) on an aggregate measure of consumer surplus. In a similar way, producer surplus is a money measure of benefit to efficient producers (i.e., for all but the marginal unit supplied) from supplying to the market over and above their marginal cost of production.17 It is measured as the area under the market price and above the supply curve to the left of the market equilibrium. Producer surplus is the amount received by efficient producers over and above what is needed to secure their participation in the market. Producer surplus is a measure of monopoly excess profit that arises because some producers are more efficient than the marginal producer. Presumably, this profit accrues to the benefit of the owners of these efficient firms. It is in that sense that both producer surplus and consumer surplus can be thought to be the benefits that arise from having a competitive market that clears at price P. Related to this is the notion of producer cost. This is a money measure of the area under the supply curve to the left of quantity supplied. Producer surplus (PS) is consumer expenditure (PQ) net of producer cost (PC): PS = PQ − PC. In a competitive market, the sum of consumer surplus and producer surplus is a money measure labeled social welfare (SW): SW = CS + PS. Equivalently, social welfare is the amount by which consumer benefit exceeds producer cost: SW = CB + PC. However, in the case of a monopolist firm as in Chapter 2, we need to account also for the monopoly excess profit (MP) that arises because the firm exploits a downward sloping demand curve. In a monopoly, social welfare includes consumer surplus, producer surplus, and monopoly excess profit: SW = CS + PS + MP. This conceptualization of social welfare is simple minded;
17 Economists
in general have the same uneasiness about producer surplus as they do about consumer surplus. See Martin and Alston (1997).
5.2
Model 5A
131
it sees economic actors only as either consumers or producers. Specifically, it does not see consumers also as workers employed in production at that place. As efficient firms benefit from new opportunities for trade, they presumably hire more workers, and the total wages paid go up. Such income effects are ignored in this model because I have assumed the total number of consumers at each place is fixed, and each has the same individual demand curve (which depends on price, but not on income). Let me now review the models presented in Chapters 2 and 4 in terms of these concepts. In the case of Model 2A (monopolist selling only locally), I present surplus, cost, benefit, and welfare measures in Table 2.1 and Fig. 2.1. In this case, the industry consists of just one firm. Producer cost in this case is the area OMHJO in Fig. 2.1. In Model 2A, consumer benefit is given by (2.1.10) which corresponds to the area under the demand curve to the left of quantity Q1 (area OAKJO in Fig. 2.1.). Consumer benefit here can be partitioned into three components: consumer surplus is LAKL in Fig. 2.1; producer cost is OMHJO as noted above; and monopoly excess profit is MLKHM. See (2.1.12), (2.1.11), and (2.1.8), respectively. Producer surplus is 0 here because I assumed that the marginal cost of production is a constant C; the marginal unit increases total costs no more than does the first unit produced. Social welfare is given by (2.1.14). In Model 2B (wherein the firm also sells to remote customers) in Chapter 2, we can calculate similar measures for the market: see Table 2.3. Where α > C + sx, adding in the demand at Place 2 increases consumer benefit in (2.3.9) and two of its components—consumer surplus and monopoly excess profit as seen by comparing (2.3.11) and (2.3.6) with (2.1.12) and (2.1.8), respectively. Social welfare also rises compared to Model 2A, as seen in (2.3.13). Here producer surplus is still 0, for the same reason as in Model 2A. In Model 2C (firm has a factory at each place), we can again calculate similar measures for the market: see Table 2.5. Compared to Model 2B, adding a factory at Place 2 reduces price there and therefore increases consumer benefit in (2.5.6) and two of its components—consumer surplus and monopoly excess profit by (2.5.8) and (2.5.4), compared to (2.3.11) and (2.3.6), respectively. Where a second factory is profitable—that is, (2.5.5) holds—social welfare also rises compared to Model 2A: see (2.3.13). Here producer surplus is still 0: the same reason as in Model 2A. Finally, consider the autarky equilibrium model for Place 1 in Chapter 4. See Table 5.1 wherein I summarize the equations, assumptions, notation, and rationale for localization used in Model 5A. Because this is a competitive market, monopoly excess profit is zero: see (5.1.5). Producer surplus is shown in (4.1.7). It is interesting to compare the profit of a single-market monopolist in Chapter 2—see (2.1.7)—with producer surplus here: especially in the most similar case: i.e., where δ 1 approaches zero. With competition, producer surplus (4.1.7) drops to zero when and as δ 1 approaches zero. Put differently, if every producer were equally efficient, the supply curve would become horizontal in a competitive market and producer surplus would disappear altogether. While it is not clear why some producers are necessarily more efficient, δ 1 is an indicator of the extent of such differential efficiency, and therefore
132
5 Arbitrage in the Grand Scheme Table 5.1 Local demand and local supply at Place i in autarky
Local inverse demand at Place i Pi = α − βDi /Ni
(5.1.1)
Local inverse supply at Place i Pi = Ci + δi Si
(5.1.2)
Area under the demand curve at quantity Qi αQi − βQ2i /(2Ni )
(5.1.3)
Area under the supply curve at quantity Qi Ci Qi + δi Q2i /(2Ni )
(5.1.4)
Social welfare (SW) at quantity Qi SW = (α − Ci )Qi − (1/2)((β/Ni ) + δi )Q2i
(5.1.5)
Price that maximizes SW Pi = (Ni /β)α + (1/δi )Ci )/(Ni /β) + (1/δi ))
(5.1.6)
Notes: Assumptions (see Appendix A): A1—Closed regional market economy; A3—Punctiform landscape; B1—Exchange of soap for money; B2—Upwardly sloped local supply curve each supply place; B4—Local demand at each customer place; C4—Identical customers; C5—Identical linear demand; E4—Unit shipping costs symmetric. Rationale for localization (see Appendix A): Z3—Implicit unit price advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): Ci —Intercept of inverse supply curve at Place i; Ni —Population of Place i; α—Intercept of individual linear inverse demand curve; β—Slope of individual inverse demand curve; δ i —Slope of inverse supply curve at Place i. Outcomes (endogenous): Pi —Price at Place i; Qi —Equilibrium quantity transacted at Place i; Qij —Amount produced at Place i for sale to local demand at Place j.
a nonzero producer surplus. In Chapter 2, the monopolist had a horizontal marginal cost curve—i.e., corresponding to δ1 = 0—and earns a monopoly excess profit but zero producer surplus. In market equilibrium under autarky, social welfare at Place 1 is given by (4.1.10). Social welfare is monotonically increasing in number of consumers there (N1 ) and maximum price (α). It is monotonically decreasing in β, C1 , and δ 1 . Not surprisingly, where δ1 = 0, social welfare is higher in a competitive market than it would be under a corresponding monopoly. In the autarky example in Chapter 4, where local demand equals local supply, consumer surplus for Place 1 is shown in (4.1.9). Compare this with consumer surplus in the monopoly model of Chapter 2; we see that when δ1 = 0, the most comparable case, consumer surplus is four times larger under competitive equilibrium: an outcome of the lower price and greater quantity there.
5.3 Social Welfare at Place i In what follows, I assume a punctiform landscape that consists of n places. Here I am describing a world composed of places (geographic points), not a world composed of geographic regions. I use places rather than regions because I want to ignore local shipping costs that might occur within a region. Bear with me here. Starting in Chapter 9, I will introduce regions and the complexity that they introduce to location models.
5.3
Social Welfare at Place i
133
In Chapter 4, I assume there are Ni consumers at Place i, and consumers at each place each have identical individual linear inverse demand curve for the commodity: P = α − βq. Once again, assume β > 0 and Ni > 0. As in Chapters 2 and 4, I allow the number of consumers, Ni , to differ from one place to the next but assume for simplicity that consumers are otherwise identical. The aggregate local inverse demand curve at Place i is now (5.1.1): see Table 5.1. As in earlier chapters, I assume here that a consumer does not bear any transaction cost related to purchase. Assume further that the unit shipping cost to an arbitrageur, inclusive of normal profit (the profit attributable to an unpriced factor of production such as entrepreneurial skill or owner equity), per unit of product from Place i to Place j is a constant and directionally symmetric sij dollars.18 The unit shipping cost can differ for each pair of places. To start, let us think about Place i in isolation. At Place i, the price that local consumers are willing to pay for a given total quantity is given by an aggregate inverse demand curve: (5.1.1). Assume local producers, each an efficient firm, in aggregate at this place have an inverse supply curve given by (5.1.2). We can invert these inverse demand and supply curves to get the local demand and local supply curves. In Fig. 5.1, I have drawn local demand and local supply schedules for such a market. As in Chapter 4, I assume here that each place is perfectly competitive; consumers and producers everywhere are price-takers. Once again, assume here initially that demand is sufficient (or equivalently that the cost is low enough) to ensure local production of the commodity: i.e., α > Ci at Place i. For the moment, I leave aside the question of whether this is a local market in autarky: i.e., whether local demand (Di ) is equal to local supply (Si ) at the prevailing market price. How do we measure the level of well-being associated with this place? Economists use the notion of social welfare (SW) here. SW sums consumer surplus and producer surplus. The prevailing price need not necessarily be the autarky price. Consider panel (a) of Fig. 5.1, where a given quantity Q is available in the market; the marginal supplier is willing to supply at a price OI, while the marginal demander is willing to pay OH. Social welfare here is the area under the demand curve and above the supply curve: CAG1 FC. Because panel (a) uses a quantity Q that is smaller than the autarky equilibrium quantity where OH and OI would be the same, there is a monopoly profit. SW can be partitioned into consumer surplus (HAG1 H), monopoly excess profit (IHGFI), and producer surplus (CIFC). Equivalently, in panel (a), at a given Q, SW is the area under the demand curve, OAG1 EO minus the area under the supply curve, OCFEO. See (5.1.3) and (5.1.4). Panel (b) presents the case, where there is a single price (OI) in the local market and where quantity demanded (OE) and quantity supplied (OF) correspond to that price. In panel (b), consumer benefit is given by OAGEO, producer cost is OCHFO, and social welfare is the former minus the latter. SW is quadratic in quantity: see (5.1.5). As is clear from panel (a) of Fig. 5.1, if I had used a slightly smaller quantity, SW would be smaller. For a slightly larger Qi , SW would be larger. In fact, as I shift along the horizontal axis, SW keeps increasing
18 There
is no congestion over the transportation network here.
134
5 Arbitrage in the Grand Scheme
D
A A
D G
H I
F
C
0
H
G1 Price
Price
I
E
Quantity
B
Well-being when quantity is given AB CD CIFC HAGH IHGFI OAGEO OCFEO OE
Local demand curve Local supply curve Producer surplus Consumer surplus Excess profit Consumer benefit Producer cost Quantity given
(a) Quantity given
C
0
E
F
Quantity
B
Well-being when price is given AB CD CIHC IAGI OAGEO OCHFO OE OF
Local demand curve Local supply curve Producer surplus Consumer surplus Consumer benefit Producer cost Quantity demanded at given price Quantity supplies at given price
(b) Price given
Fig. 5.1 Consumer surplus, producer surplus, and social welfare at Place 1
up until the quantity where the price that the marginal consumer is willing to pay is just equal to the marginal cost of the last unit supplied by local producers. See (5.1.6). Of course, this is nothing more than the autarky solution: i.e., equilibrium of local demand and local supply. If we were to consider a still larger quantity, SW would in fact decline. In this sense, where the local demand curve and local supply curve intersect is a best solution for society in the case of autarky.
5.4 Net Social Payoff and Global Net Social Welfare I use here global net social welfare (GNSW), which I measure as the sum of social welfare at all places less shipping costs. Unit shipping cost, which I am viewing here as exogenous, can be envisaged as a loss that reduces the benefit to society overall: just as it lowered the profit to the firm in Chapter 2. Ignored here, for the sake of
5.4
Net Social Payoff and Global Net Social Welfare
135
simplicity, is the concept of an endogenous price of shipping, that is, the possibility of a shipping sector that generates earnings for shipping firm employees, profits for shippers, and other factor income. To me, the literature appears confused as to whether NSP (area under excess demand curves net of shipping cost) and GNSW (area under demand curve less area under supply curve net of shipping cost) proxy the same thing.19 To some, net social payoff has come to be seen as equal to social welfare net of shipping costs.20 What is clear is that NSP is zero in the absence of shipments (i.e., autarky) but that GNSW is generally positive even in autarky. Put differently, GNSW may proxy NSP in the sense that the shipments that maximize NSP also maximize GNSW, but GNSW is not equal to NSP. The implementation of a GNSW approach to spatial price equilibrium is evidenced in Table 4.5 from the previous chapter. There, I calculated producer and consumer surplus for each of two places (Places 1 and 2) in autarky. The two autarky prices were 12.50 and 13.00. The Table sums the two surpluses to get net social welfare at each place in autarky. Then, GNSW is calculated as the sum of these (I can ignore shipping costs here because there are no shipments in autarky). In contrast, when the shipping rate is only 0.10, the Table predicts 400 units shipped and a higher overall GNSW (10,290, up from 10,250 in autarky). If shipping rate had been zero (the integrated market solution), GNSW would have been even larger (10,326). I have already raised the question as to why a market economy would act to maximize GNSW and asserted that, to a non-economist, the underlying faith in the efficiency of markets might seem unwarranted.21 More interesting to me, however, is the question of whether maximizing GNSW will lead to a spatial price equilibrium of the kind that we have already seen in Chapter 4. That it does indeed do this further supports the idea that the maximization of GNSW is intuitively reasonable. The maximization of GNSW is relatively easy to implement numerically. Let Qij be the amount of output produced at Place i to meet a local demand at Place j. As a result Si = j Qij and Dj = i Qij . We can then write the local inverse demand and supply curves for Place i as (5.2.1) and (5.2.2). See Table 5.2. For Place i, the area under the demand curve is now given by (5.2.3), the area under the supply curve by (5.2.4), and social welfare by (5.2.5). Note that SWi is a quadratic function of quantities shipped (Qij ). We then get the GNSW in (5.2.6) by first aggregating social welfare for each place and then subtracting total shipping costs. I can then find the maximum value of GNSW (5.2.7) subject to the identities in (5.2.8) and (5.2.9) and the nonnegativity condition on every shipment (5.2.10). Since the constraints
19 Takayama
and Judge (1971, pp. 111–112) was among the earliest to model GNSW directly. and Schmitz (1971, p. 780) states that maximizing NSP is equivalent to maximizing net welfare gains. However, this argument is based on a comparison of areas on a graph—Currie, Murphy, and Schmitz (1971, Fig. 14)—that I find difficult to compare precisely. 21 Smith (1963) reinterprets Samuelson’s model as the minimization of global economic rent. This is an attractive idea because it suggests that the role of the arbitrageur is to reduce the consumer and producer surplus that would arise in the absence of shipments. For a discussion of the use of spatial price equilibrium models in development planning, see Luna (1978). 20 In a widely cited paper Currie, Murphy,
136
5 Arbitrage in the Grand Scheme Table 5.2 The GNSW formulation
Local inverse demand at Place i Pi = α − βDi /Ni
(5.2.1)
Local inverse supply at Place i Pi = Ci + δi Si
(5.2.2)
Area under the demand curve at quantity Di αDi − βD2i /(2Ni )
(5.2.3)
Area under the supply curve at quantity Si Ci Si + δi Si2 /2
(5.2.4)
Social welfare (SWi ) at Place i SWi = αDi − βD2i /(2Ni ) − (Ci Si + δi Si2 /2)
(5.2.5)
Global net social welfare (GNSW) GNSW = i SWi − i j sij Qij
(5.2.6)
Maximize Z = GNSW
(5.2.7)
Subject to Si = j Qij for each Place i Di = j Qju for each Place i Qij ≥ 0 for each Place i and j
(5.2.8) (5.2.9) (5.2.10)
Note: See also Table 5.1.
(5.2.8) and (5.2.9) are linear in shipments, this formulation exemplifies a quadratic programming22 problem. In recent decades, commercial software that solves various kinds of optimization problems, including quadratic programming, has become widely available.23 Quadratic programs are solved numerically in a set of repeated steps (each step is called an iteration): such a procedure is termed an algorithm. As in Chapter 3, I cannot derive an explicit (algebraic) solution as a general rule here; instead, I have to rely on numerical methods to solve specific problems of this type. In the version presented above, I have assumed a set of n places with local suppliers and local demanders in each. However, the model can be solved in a similar manner if some of these have suppliers only or demanders only. Interestingly, the GNSW approach can also be extended to cover situations where there is price
22 The solving of optimization problems in which a quadratic objective function is to be maximized
or minimized subject to linear inequality and nonnegativity constraints. See Tucker (1957). 23 Florian and Los (1982), Harker (1986), and Güder and Morris (1988) discuss alternative numer-
ical methods in nonlinear programming for solving the spatial price equilibrium problem. Rowse (1981) discusses the use of software that solves generalized programming problems in which there are linear constraints. I use the Solver routine in Microsoft Excel 2003/2004.
5.5
A Special Case: Horizontal Supply Curve at Each Place
137
rigidity (e.g., price controls) at one or more places that mean a local market is in disequilibrium.24
5.5 A Special Case: Horizontal Supply Curve at Each Place To begin, assume n = 3 places each with a horizontal supply curve. In other words, there is as much supply forthcoming at Place 1, at a price of C1 as consumers might demand. At Places 2 and 3, suppliers similarly supply as much as needed at a price of C2 or C3 , respectively. Put differently, δ1 = δ2 = δ3 = 0. Note the asymmetry here; firms are identical in terms of efficiency at each place but may differ otherwise from one place to the next. Assume initially that shipping rates are sufficiently high so as to deter any shipment of the commodity. In this autarky condition, local supply equals local demand at each place. In effect, there are three separate markets here. Since δi = 0 at each Place i and all markets are competitive, we get Pi = Cj . The quantity demanded at Place i is given by (5.3.3). See Table 5.3. Consumer surplus is given by (5.3.5). Since δ1 = δ2 = δ3 = 0, every producer at Place i is equally efficient. Therefore, no producer earns any excess profit and producer surplus is zero: see (5.3.7). Since Table 5.3 Horizontal supply curves Price at Place i Pi = Ci in autarky, P2 = C1 +s12 with shipment from 1 to 2 only
(5.3.1) (5.3.2)
Demand at Place i Di = (α − Ci )Ni /β in autarky, D2 = (α − C1 − s12 )N2 /β with shipment from 1 to 2 only
(5.3.3) (5.3.4)
Consumer surplus at Place i (α − Ci )2 Ni /(2β) in autarky, (α − minj=i [Cj + sji ])2 Ni /(2β) with shipment from 1 to 2 only
(5.3.5) (5.3.6)
Producer surplus at Place i 0 in every region under autarky 0 in every region even with shipment from 1 to 2 only
(5.3.7) (5.3.8)
GNSW i (α − Ci )2 Ni /(2β) in autarky ((α − C1 )2 N1 + {(α − C1 − s12 )2 − 2s12 (α − C1 − s12 )}N2 + (α − C3 )2 N3 )/(2β) with shipment from 1 to 2 only
(5.3.10)
Net gain in GNSW with shipment (α − C1 − s12 )2 − 2s12 (α − C1 − s12 )
(5.3.11)
Note: See also Table 5.1
24 See
Thore (1986).
(5.3.9)
138
5 Arbitrage in the Grand Scheme
there are no shipping costs and producer surplus is everywhere zero, GNSW is just the sum of consumer surplus at the three places. Now, let us undertake a thought experiment in which we reduce shipping rates everywhere proportionally until shipment now becomes attractive for just one pair of places. Without any loss of generality, assume shipping rates are low enough to attract shipment from Place 1 to Place 2. This happens when the shipping rate for one unit of output shipped from Place 1 to Place 2, s12 , satisfies C1 + s12 ≤ C2 . At this point, local production of the commodity at Place 2 ceases, and the firms at Place 1, being more efficient, now satisfy all the demand in both Places 1 and 2. At Place 1, consumer surplus is unchanged. As the shipping rate is still too high, Place 3 remains in autarky. Therefore, consumer surplus at Place 3 is unchanged. However, at Place 2, consumers are better off. The effective price drops: from a level of C2 in autarky to C1 + s12 . Consumers there are better off now that Place 2 is supplied from Place 1: GNSW is now larger. This is not surprising; after all, if shipments had not improved well-being why would anyone ship the commodity? In this special case where δ1 = δ2 = δ3 = 0, consumers—but not producers— benefit from shipments and only in those places where local supply is displaced by a lower cost supplier elsewhere. Producers do not benefit because all the producers at a given place are identical; no one is more efficient, and no one has access to a monopoly profit.
5.6 Three Examples of Multiregional Shipment Now, consider Example Problem 1 as detailed in Table 5.4. Here there are n = 3 places as before. The difference now is that the local supply curve at each place is upward sloped: no longer horizontal; for some reason, as in Chapter 4, there is congestion in production. The three places are identical except that N2 > N3 > N1 . Here, at any given price, the quantity demanded at Place 2 is greater than at Place 3, which in turn is greater than at Place 1. To coax that larger supply locally, the autarky price at Place 2 would have to be larger than at Place 3 and in turn higher than at Place 1: see (5.4.14). Where prices differ between places in autarky, the possibility of shipments arises. Given that shipping costs are positive, shipment of the commodity will occur only from a place with a lower price in autarky to a place with a higher price. In the case of Table 5.4, the shipping rates are too high to facilitate shipment. In (5.4.11), (5.4.12), and (5.4.13), producers ship only to local consumers, and therefore the prices (5.4.14) are autarky prices. For any pair of places, the difference in autarky prices is less than the corresponding shipping rate: (5.4.8), (5.4.9), and (5.4.10). Now, let us look at the components of GNSW in autarky. The absence of shipments means that shipping costs are zero: see (5.4.17). Since autarky price is highest there, the assumption that the supply curves are similar at the three places means that producer surplus is highest at Place 2: see (5.4.16). Despite the high autarky price at Place 2, the larger N there means that consumer surplus there—being an aggregate measure—is the greatest of the three places: see (5.4.15).
5.6
Three Examples of Multiregional Shipment
139
Table 5.4 Example problem 1 Local demand parameters α = 15 β = 1 N1 = 500 N2 = 1,200 N3 = 900
(5.4.1) (5.4.1) (5.4.3) (5.4.4) (5.4.5)
Local supply parameters C1 = C2 = C3 = 9 δ1 = δ2 = δ3 = 0.0008
(5.4.6) (5.4.7)
Shipping rate (assumed symmetric) s12 = s21 = 2.00 s13 = s31 = 2.00 s23 = s32 = 2.00
(5.4.8) (5.4.9) (5.4.10)
Solution Q11 = 2,143 Q21 = 0 Q31 = 0 P1 = 10.71 CS1 = 4,592 PS1 = 1,837 i j sij Qij = 0 GNSW = 26,868
Q12 = 0 Q22 = 3,674 Q32 = 0 P2 = 11.94 CS2 = 5,623 PS2 = 5,398
Q13 = 0 Q23 = 0 Q33 = 3,140 P3 = 11.51 CS3 = 5,476 PS3 = 3,943
(5.4.11) (5.4.12) (5.4.13) (5.4.14) (5.4.15) (5.4.16) (5.4.17) (5.4.18)
Note: See also Table 5.1. Calculations by author.
Now, consider Example Problem 2 described in Table 5.5. This is the same as Problem 1 except that unit shipping cost is everywhere now 1.00 instead of 2.00. Now, the solution that maximizes GNSW is one wherein suppliers at Place 1 supply all of the local demand at Place 1 and part of the local demand at Place 2. Price rises at Place 1 and falls at Place 2: compare (5.5.14) with (5.4.14). The price difference between the two places is 11.85 – 10.85, which is equal to the unit shipping cost; remember this is an endogenous outcome of maximizing GNSW, not an assumption of the model. Local suppliers at Place 2 supply the rest of demand of Place 2. Local suppliers at Place 3 supply all the local demand at Place 3; the price at Place 3 is therefore unchanged. We have now two markets: Places 1 and 2 form a single market; price may differ between the two places but only by an amount related to shipping cost. Place 3 is the other market. As in Chapter 4, market formation is now endogenous to the model. What happens to GNSW here? Shipments from Place 1 to Place 2 now create a shipping cost: see (5.5.17). At the same time, producer surplus rises at Place 1, drops at Place 2, and is unaffected at Place 3. This is not surprising. When unit shipping costs are lower, it becomes efficient to supply some of Place 2 from more efficient producers at Place 1 rather than the less-efficient producers at Place 2. This is because, even though the local supply curve is the same at both places, the larger number of consumers at Place 2 means that the marginal supplier at Place 2 in
140
5 Arbitrage in the Grand Scheme Table 5.5 Example problem 2
Local demand parameters α = 15 β = 1 N1 = 500 N2 = 1,200 N3 = 900
(5.5.1) (5.5.1) (5.5.3) (5.5.4) (5.5.5)
Local supply parameters C1 = C2 = C3 = 9 δ1 = δ2 = δ3 = 0.0008
(5.5.6) (5.5.7)
Shipping rate (assumed symmetric) s12 = s21 = 1.00 s13 = s31 = 1.00 s23 = s32 = 1.00
(5.5.8) (5.5.9) (5.5.10)
Solution Q11 = 2,077 Q21 = 0 Q31 = 0 P1 = 10.85 CS1 = 4,316 PS1 = 2,128 i j sij Qij = 229 GNSW = 26,893
Q12 = 229 Q22 = 3,557 Q32 = 0 P2 = 11.85 CS2 = 5,972 PS2 = 5,060
Q13 = 0 Q23 = 0 Q33 = 3,140 P3 = 11.51 CS3 = 5,476 PS3 = 3,943
(5.5.11) (5.5.12) (5.5.13) (5.5.14) (5.5.15) (5.5.16) (5.5.17) (5.5.18)
Note: See also Table 5.1. Calculations by author.
autarky is less efficient than the marginal supplier at Place 1. The lower unit shipping cost strips away the protection afforded to the less-efficient marginal producer at Place 2. In contrast, consumer surplus drops at Place 1 (since price rises relative to autarky), rises at Place 2 (since price has dropped), and stays the same at Place 3 (since price stays the same). So, the lower shipping rates in Example Problem 2 increase consumer benefit at Place 2 and increase in producer surplus at Place 1. Intuitively, GNSW is higher in Example Problem 2 because lower shipping rates allow us to better match demand and supply across places. Let us turn now to Example Problem 3 as described in Table 5.6. This is similar to Problems 1 and 2 except that unit shipping costs are everywhere now 0.19. Now, the solution that maximizes GNSW is one wherein suppliers at Place 1 supply all of the local demand at Place 1 and part of the local demand at each of Place 2 and Place 3. Price rises at Place 1 still further and falls at Places 2 and 3: compare (5.6.14) with (5.5.14) and (5.4.14). Because suppliers at Place 1 ship to both Place 2 and Place 3 and the unit shipping cost is 0.19 to both, the equilibrium prices in Places 2 and 3 are 0.19 higher than the price at Place 1; again, this is an endogenous outcome of a model that maximizes GNSW. We now have just one market, and it incorporates all three places. From Table 5.4, 5.5, and 5.6, unit shipping costs have decreased. As the cost of shipping declines, GNSW goes up. Looking back at Table 3.4, we saw a similar
5.6
Three Examples of Multiregional Shipment
141
Table 5.6 Example problem 3 Local demand parameters α = 15 β = 1 N1 = 500 N2 = 1,200 N3 = 900
(5.6.1) (5.6.1) (5.6.3) (5.6.4) (5.6.5)
Local supply parameters C1 = C2 = C3 = 9 δ1 = δ2 = δ3 = 0.0008
(5.6.6) (5.6.7)
Shipping rate (assumed symmetric) s12 = s21 = 0.19 s13 = s31 = 0.19 s23 = s32 = 0.19
(5.6.8) (5.6.9) (5.6.10)
Solution Q11 = 1,840 Q21 = 0 Q31 = 0 P1 = 11.32 CS1 = 3,387 PS1 = 3,361 i j sij Qij = 201 GNSW = 27,414
Q12 = 1,053 Q22 = 3,137 Q32 = 0 P2 = 11.51 CS2 = 7,312 PS2 = 3,935
Q13 = 6 Q23 = 0 Q33 = 3,136 P3 = 11.51 CS3 = 5,484 PS3 = 3,935
(5.6.11) (5.6.12) (5.6.13) (5.6.14) (5.6.15) (5.6.16) (5.6.17) (5.6.18)
Note: See also Table 5.1. Calculations by author.
result in the two-place case as we moved from autarky (high shipping rate) to the integrated market (zero shipping rate) solution. However, as I noted in Chapter 4, not everyone becomes better off when shipping costs are lower. Here, consumers at Place 1 are worse off (lower CS) when the shipping rate is lower because local suppliers are now drawn upon to supply other places as well, and this pushes up the marginal cost curve at Place 1. On the other hand, consumers in the larger Place 2 (and even Place 3 if shipping rate is low enough) benefit (higher CS) from a lower price. It is the opposite pattern among producers; those at Place 1 benefit from lower unit shipping costs, while those at Places 2 and 3 are worse off. Why these results? When unit shipping costs are high enough to result in autarky all around, consumers at Place 1 benefit because only they have access to the low cost producers there. When shipping rate is lower, those consumers at Place 1 lose the advantage of location and now pay a higher price for the commodity. The opposite is true for consumers at Places 2 and 3. What I have undertaken here can be thought of as a kind of sensitivity analysis. In other words, how and in what ways are shipments and local prices sensitive to shipping rates? Of course, unit shipping costs are just one set of parameters in this model. We could also look at the sensitivity of shipments and local prices to other parameters in the model: α, β, Ni , Cj , and δ j . However, I have already done this in the case of two places in Chapter 4; and the story is broadly similar in this chapter.
142
5 Arbitrage in the Grand Scheme
In this chapter, we have assumed that consumers everywhere have the same α and β. Why is that? Are consumer incomes and the prices of other commodities the same everywhere? In reality, we might expect incomes and the prices of other commodities to vary from place to place. This model in that sense is just a simplification of reality that helps us see the role of suppliers and customers in what is an endogenously defined market.
5.7 Application In location theory, the Samuelson–Enke problem is generally labeled as spatial price equilibrium because it seeks to explain how and why price differs from one geographic locale to the next in equilibrium. As modeled in the Samuelson– Enke approach, price differences reflect underlying differences in local supply and demand factors: differences that may be ameliorated by shipping rates. This conceptualization can be used to look at a broad range of spatial pricing25 problems from geographic variations in the price of lumber to levels of congestion on a network of urban roads. Spatial price equilibrium models have been widely used. Kennedy (1974) constructs a spatial price equilibrium model of four oil commodities (crude production, transportation, refining, and consumption) in global markets partitioned into 7 places (United States, Canada, Latin America, Europe, Middle East, Africa, and Asia). Furtran, Nagy, and Storey (1979) study the impact of Canadian rail freight policy and Japanese tariffs on the production of rapeseed.26 Their study divides the world into four regions (Canada, Japan, EEC, and rest of the world) in each of which there is a local demand and supply of rapeseed. Vaux and Howitt (1984) present an interregional trade model to assess demand and supply of water in California. Their model includes 5 demand regions and 8 sources of supply. Montgomery, Brown, and Adams (1994) use the US Forest Services’ Timber Assessment Market Model (TAMM)—a spatial price equilibrium model—in their assessment of the marginal cost of preserving the Northern Spotted Owl. Adams, Fleming, Chang, McCarl, and Rosenzweig (1995, pp. 149–150) describe an Agricultural Sector Model (ASM) that partitions the lower 48 US states into 10 producing regions; the model predicts the economic consequences of long-term climate change on US agriculture taking into account local supply and demand responses. Gabriel, Vikas, and Ribar (2000) presents a Gas Systems Analysis model (GSAM), which is an intertemporal spatial price equilibrium model of the North American natural gas market. On the supply side, GSAM models investment in—and production levels at—17,000 gas production reservoirs. On the demand side, it includes 4 sectors (residential, commercial, industrial, and electric power generation) spread over 16 broad
25 Furlong
and Slotsve (1983) consider a range of spatial pricing options. is used as a protein supplement and fertilizer as well as in margarines, shortenings, and salad oils.
26 Rapeseed
5.7
Application
143
geographic regions wherein demand varies seasonally. Models of both storage facilities and a pipeline network with 79 links are incorporated. Spatial price equilibrium models like these raise the question of whether, in practice, economic actors—be they buyers or sellers—succeed in integrating markets so as to give prices, which differ between places by no more than unit shipping cost. Baulch (1997a, p. 477) argues that market integration is key to market liberalization and parastatal reform in developing countries. Without integration, price signals are not correctly transmitted from sector to sector, agricultural producers do not specialize efficiently, and gains from trade are not realized. In practice, it is not easy to implement a spatial price equilibrium analysis. There are at least four principal problems. Heterogeneous commodities : The model presumes an industry in which the identical product is being produced and demanded in each region. Typically, industrial commodities are differentiated commodities; not exactly the same commodity is being produced or demanded at each place. It is, therefore, difficult to correctly identify quantities of an identical commodity and to fix the price of that commodity in each location. For that reason, spatial price equilibrium analysis is often restricted to basic agricultural commodities or raw resources that are thought to better approximate standardized commodities.27 Regions as places: The model presumes a set of places. In practice, each place represents a region within which there are shipping costs. The larger the region the more difficult it becomes to measure the shipping cost associated with a shipment because this varies with the precise origin or destination within the region. Shipping costs, currency differences, tariffs, and other barriers to shipment: In some cases, the places (regions) in these applications constitute a common fiat money economy: e.g., the United States of America. In other applications, however, the set of places spans different currency areas: e.g., Canada, United States of America, and Mexico. Within a nation, it may sometimes be reasonable to ignore trade restrictions that result from government policy. However, when shipments cross national boundaries, tariff, other nontariff barriers, and currency considerations may play an important role. In practice, it is difficult to quantify barriers to shipment so as to see whether spatial price equilibrium analysis is applicable. In practice, analysts tend to supplement shipping rates (often taken to be freight costs alone, ignoring other transaction costs) in modeling shipments until they get an equilibrium pattern of shipment that mirrors actual shipments at current currency exchange rates. While this practice is defensible, it is no longer clear whether spatial price equilibrium has been detected in the data or merely aped by an appropriate choice of parameters.
27 Baulch
(1997), for example, looks at the integration in the wholesale rice market around metropolitan Manila and its hinterland.
144
5 Arbitrage in the Grand Scheme
Assumption of perfect competition: In industries dominated by one supplier, the monopoly model described in Chapter 2 would be more appropriate. However, in such cases, the monopolist may well set a price at each place such that the difference between two places supplied by the same factory does not equal the difference in shipping costs. While we have not considered any model of oligopoly28 to this stage, we might expect there too that price differences between two places would be equal to shipping rates. So, the spatial price equilibrium analysis presented in this chapter is applicable only to those industries that can be thought of as perfectly competitive. Strangely, most empirical studies of market integration look only at whether prices converge over time from place to place and do not directly measure any unit shipping costs.29 In an exception, a careful study of rice markets across the Philippine islands, Baulch (1997a, p. 485) describes how he estimated unit shipping costs for markets connected by road by interviewing traders about rates they paid for trucking, loading, and unloading using typical truck/trailer combinations and then added a margin for arbitrageur profit. Baulch concludes that there is efficient arbitrage in the sense that price differences between places tend to reflect the unit shipping cost. At the same time, he notes (p. 477) that price spreads between markets may also reflect other factors such as government controls on produce flows, transportation bottlenecks, or oligopolistic pricing. The implication here is that any attempt to explain interregional shipments needs to take into account a variety of determining factors.
5.8 Case Study Boyd, Doroodian, and Abdul-Latif (1993) present a case study of the softwood lumber industry in North America when the North American Free Trade Agreement (NAFTA) first came into effect.30 The study divides America and Mexico broadly into 10 lumber-consuming regions centered in the following cities: Boston, New York City, Chicago, Minneapolis, Washington (DC), Dallas, Nashville, Los Angeles, Missoula (MT), and Mexico City. Note that this study area covers three currency areas and national sets of tariffs and nontariff barriers plus a variety of regional, state, and provincial regulations governing the logging and lumber industries.
28 A
market condition in which there only a few vendors. No vendor has a monopoly, but none of the vendors is strictly a price-taker. 29 See, for example, Boyd (1983a, 1983b), Buongiorno and Uusivuori (1992), Jung and Doroodian (1994), De Vany and Walls (1996), and Bukenya and Labys (2005). 30 Aspects of the market for softwood lumber in North America are the subject of Buongiorno and Uusivuori (1992), Gilligan (1992), Jung and Doroodian (1994), Montgomery, Brown, and Adams (1994), Mogus, Stennes, and van Kooten (2006).
5.8
Case Study
145
The study estimates a linear demand for lumber for each of these 10 regions: each treated as a place. The demand for lumber in Canada is ignored here because there were no reliable estimates of the price elasticity of demand31 for lumber there at the time. The demand equations in this study are linear and link the quantity of lumber demanded to the quantity of construction activity (the principal use of lumber) and to the price of lumber. Two kinds of lumber are modeled here: pine and fir. The study assumes that all pine lumber is perfectly substitutable, all fir lumber is perfectly substitutable, and that pine and fir are highly but not perfectly substitutable. In that study, unlike earlier in this chapter, demand per capita is not assumed to be necessarily the same in each region. For example, the demand for pine in region j (Dpj ) takes the form Dpj = αpj − βpj Ppj + βfj Pfj , where Pp and Pf are the prices of pine and fir, respectively; the intercept, own price coefficient, and cross price coefficients can each vary from one region to the next. On the supply side, the study identifies 18 regions (again treated as geographic points) that produce lumber: 12 in America, 5 in Canada, and 1 in Mexico. The study estimates a linear supply curve for each of the 18 supplier places. There are 28 places in total: 18 for customers and 10 for suppliers. The study, unlike earlier in this chapter, does not include any places that are both demanders and suppliers. It is not clear here whether the level of demand or supply is actually zero in each case, or just small enough to ignore. The study then combines estimated regional supply and demand equations with interregional freight and tariff rates to solve for a competitive spatial price equilibrium as of 1988. Discrepancies between actual and predicted shipments are presumed to be the result of nontariff barriers. As the model incorporates places in three distinct currency areas, presumably the demand functions, supply functions, and tariff and shipping rates were adjusted by prevailing exchange rates. The procedure is repeated under three alternative tariff regimes that might arise under NAFTA. Scenario (a) assumes status quo tariffs; scenario (b) assumes that the thencurrent 6.51% duty on Canadian softwood coming into the American market had been eliminated; scenario (c) includes also elimination of the 15% Mexican tariff on American softwood. The purpose of the study is to measure impacts of these three alternatives on shipment and economic welfare.32 The shipments of pine (in billions of board feet) that are predicted under each of the tariff regimes are summarized in Table 5.7.33 The scenario (a) projections
31 Price
elasticity of demand (e) is the percentage change in quantity demanded (q) for associated with a 1% change in price (p): e = (p/dp). Since dq/dp is typically negative, most treatments take the absolute value of e. However, for the benefit of non-economists, I leave the negative sign to remind the reader that when price goes up, we expect quantity demanded to drop. 32 The study solves the model in each scenario using reactive programming. 33 Surprisingly, in Table 5.7, total demand for pine in North America declines when tariffs are reduced. We might have expected the price of pine to drop and therefore the quantity demanded to rise when tariffs are reduced. Why that did not happen here is unclear from the study. My guess that the results also reflect a drop in the price of fir lumber; the strong substitutability between pine and fir may have caused the quantity of pine consumed to drop even though its price had decreased.
146
5 Arbitrage in the Grand Scheme Table 5.7 Shipments of softwood lumber (pine) in Bbf under three scenarios Supplying region US North
US South
US West
Mexico
Canada
Total demand∗
[1]
[2]
[3]
[4]
[5]
[6]
406 0 0 0 406
4,522 6,644 878 0 12,043
154 0 3,221 0 3,374
0 0 0 384 384
81 0 185 0 266
5,162 6,644 4,284 384 16,473
(b) Scenario 1 tariff reduction US North 406 US South 0 US West 0 Mexico 0 406 Total supplied∗
4,506 6,644 879 0 12,028
155 0 3,210 0 3,365
0 0 0 384 384
86 0 197 0 283
5,153 6,644 4,286 384 16,466
(c) Scenario 2 tariff reduction US North 406 US South 0 US West 0 Mexico 0 406 Total supplied∗
4,506 6,644 879 0 12,028
155 0 3,210 0 3,365
0 0 0 383 383
86 0 197 0 283
5,153 6,644 4,286 383 16,466
(a) With tariffs US North US South US West Mexico Total supplied∗
∗ Calculated
by author. Other data are from Boyd et al. (1993, p. 319).
should correspond to actual shipments of lumber in 1988. However, the report does not say whether these scenario (a) projections were close to the shipments actually observed in that year. The projections under scenarios (b) and (c) suggest the following interpretation. Suppliers in America North (Northeast and Central United States) are largely unaffected by the tariff reductions. However, suppliers in the US South see a reduction in shipments to the US North in scenario (b) compared to (a). Suppliers in the US West also see erosion in shipments with a reduced tariff on Canadian lumber. However, the elimination of the Mexican tariff in scenario (c) has no discernable effects on shipments anywhere in North America. Because the amount of lumber consumed changed by only a small amount in scenario (b) vs. (a), the study finds that overall change in economic well-being is small. The implication here is that NAFTA, as exemplified by scenarios (b) and (c), would have relatively little effect on well-being in the three countries.34 This study exemplifies all of the problems of application discussed above. First, the need to model both pine and fir shipments tells us that even a commodity as
34 The report describes the history of the trade dispute between Canada and America over softwood
lumber before the introduction of NAFTA. That dispute has continued to fester long after NAFTA came into effect. Considering the mild effects of NAFTA predicted by this study, it is hard to understand why this trade dispute has been so protracted.
5.9
Final Comments
147
basic as lumber is heterogeneous. Second, the representation of North America as a small number of places raises questions about how to measure shipping costs. Third, the absence of comparisons between scenario (a) and the actual shipments in 1988 makes me wonder whether these were indeed close: if not, were there other barriers to shipment (e.g., quotas) that prevented this. Finally, the report presents no information about the extent of competitiveness in the market for lumber. Boyd’s study breaks the link between a place and a local market that we have relied upon so far in this book. In Boyd’s analysis, some places are suppliers only, while other places represent demand only. In such cases, it does not make sense anymore to think about autarky as a starting condition for any one place. Instead, Boyd’s analysis presumes that the lumber market potentially spans North America. In effect, the market now becomes the set of places under consideration, or at least the set of places among which we observe shipments. This may be entirely reasonable in the case of pine and fir lumber shipments. However, it raises the question of how we know when the market is properly delineated. As this book is about location theory, I generally do not pay much attention to the empirical literature. In fact, Boyd’s study is the only empirical study that I discuss in detail. However, this is a useful point to think about how location theory gets applied in practice. I want to speak here specifically about shipping rates. Boyd, Doroodian, and Abdul-Latif (1993, p. 328) give the following information. Data include information on rail, truck, and waterborne shipping rates. This information is supplied by private US and Canadian shippers and transportation companies and supplemented by waybill data collected by the ICC. Two concerns come to mind here. First, shipping rates ($/kilometer/unit shipped) presumably vary with the length of the trip (economies of the long haul), modal mix, and size of an individual shipment (economies of the large haul).35 While the description above appears to incorporate some of these considerations, one can only speculate here as to whether or how the study incorporated economies of the large haul. Second, Boyd et al. appear to use shipping rates observed for current patterns of shipping. However, the study concludes that NAFTA could mean a substantial shift in the pattern and quantity of shipments. If there is a shift, would we not expect the shipping sector to respond with investments in shipping facilities that themselves lead to changes in shipping rates. My point here is not to criticize Boyd. Rather, I use Boyd here simply to illustrate the problems in the application of spatial price models in general.
5.9 Final Comments Spatial price equilibrium solves for quantities produced (hence localization) as well as for prices. In the three example problems in Tables 5.4, 5.5, and 5.6 production always occurs in all three regions because we have assumed that there was no
35 See
Inaba and Wallace (1989).
148
5 Arbitrage in the Grand Scheme
particular advantage to producing in any one region. In fact, in the example problems, the only reason why prices differ is because local demand (population) at Place 2 has pushed production further up the marginal cost curve for local producers than is the case for producers at Places 1 and 3. However, it is not difficult to envisage a different set of conditions wherein local production does not happen in some locales because those places are not efficient enough. In those locales, we can think that firms are not choosing to locate there. In that sense, localization can be an outcome of spatial price equilibrium models. Chapters 4 and 5 of this book include a discussion of the kinks that arise when equilibrium price is either too low to encourage local production or too high to permit local demand.36 These kinks are important in helping us understand the localization of production. Not surprisingly, this chapter reconfirms conclusions drawn in Chapter 4. The discussion in Section 5.3—where I assume horizontal local supply curves—illustrates the role of unit shipping costs in determining markets, submarkets, and the extent of localization. When unit shipping costs are everywhere prohibitive, each place is in autarky (i.e., its own market) even if autarky price differs from one place to the next and there is no localization. When unit shipping cost becomes low enough for the difference in autarky prices to induce shipment, production ceases at the place where it is costly and localizes where it is less costly. The place where production was costly ceases to be a market on its own and producers at the less costly place gain another submarket. The discussion in Section 5.4 extends this story by allowing local supply curves to be upward-sloped; we get localization but no longer necessarily the full cessation of production at a more costly place; depending on the slopes of the local supply curves, there may continue to be some local production even in the presence of shipments from less costly producers elsewhere. Can we use this kind of model to answer the question of where production might go in the future? The answer here depends on how we think of a local supply curve: i.e., on our rationale for localization. This chapter envisages that each place has its own unique place-intrinsic supply curve. Its supply curve is invariant with respect to the quantity supplied at any other place. This might be attributable to any of a number of factors that give advantage to production at that place: e.g., a favorable climate or proximity to inputs needed for production. In that case, if demand in total were to grow over time, upward-sloped supply curves imply that there would come a time when firms would start to produce at a place that had previously been inefficient. The problem here is that when we observe the production costs incurred by a given set of producers at their current locations, it is difficult to know whether new producers at those sites in the future would incur similar costs. What explicitly is the rationale for localization here? In Chapter 1, I argue that prices are important in shaping the location of firms. As in Chapter 4, I continue to assume that the prices of inputs used by the firms
36 Takayama and Judge (1971, p. 136), perhaps unaware of their significance, refer to these simply
as “irregular cases”.
5.9
Final Comments
149
in local production are implicit in the intercept and slope of local supply curves. Arbitrageurs face input prices that are implicit in s. As in Chapter 4, consumers have income constraints and face prices for other goods and services that determine the intercept and slope for their demand curve. As in Chapter 4, these other prices are all determined in markets outside the scope of the models in Chapter 5. What we have achieved in this chapter is the ability to analyze a simultaneity among multiple submarkets. This is an important step forward. However, I think Walras would have argued that the analysis has been only partial in the sense that we have not looked explicitly at the simultaneity among prices of distinct commodities. Because this chapter, like Chapter 4, looks at competitive markets only, it does not consider the issue of local monopoly in space arising from unit shipping costs. Once again, we must wait until a later chapter for opportunities to address these matters. As in Chapter 4, the models in this chapter do not say anything about the distribution of income in society among units of labor, capital, and land. As in Chapter 2, the firm incurs costs, but these are not related explicitly to labor or land inputs. All income gains that can arise (e.g., because of an increase in α or a decrease in Ci or s) accrue to firm owners in the form of increased profit. As in Chapter 4, any adjustment of market price affects the well-being of consumers; in this chapter, such changes are being measured by consumer surplus.
Chapter 6
Ferrying Inputs and Outputs Factory Location in a Non-ubiquitous World (Weber–Launhardt Problem)
A firm is considering where to locate a factory. The firm employs a given technology to make its product. The firm incurs a cost to ship product to a customer place. The firm uses two non-ubiquitous inputs—each available only at a specified location—that it then ships to the production site. All shipping costs are proportional to distance. All other inputs are ubiquitous. What location for the factory will maximize the firm’s unit profit? In Model 6B, geography takes the form of a rectangular plane. Model 6A (location on a line) is a simplification of Model 6B that illustrates important ideas about localization (here, clustering with a customer or supplier). Model 6A is also helpful in thinking about location on a network (Model 6F). Model 6C introduces substitutability of inputs and the possibility of economies of scale. Model 6D makes the prices of output and inputs endogenous. Model 6E expands on the number of inputs and the number of customer places. This chapter builds on Chapters 3 through 5. In Chapters 4 and 5, a local supply curve was assumed at each place that could have a different intercept. Chapter 3 made a similar assumption about unit production cost varying by location. Chapter 6 explains such differences in part as a result of the variation in the effective prices of non-ubiquitous inputs from place to place.
6.1 The Weber–Launhardt Problem In Chapters 4 and 5, I assumed a local supply curve that could vary from one place to the next. However, I did not ask why the supply curve differed among places. At the same time, we have seen how, in the market for an output, shipments arise that bring prices into a spatial equilibrium. Since the outputs of some firms are the inputs of others, the implication here is that the cost of production will vary from place to place for the firms’ customers. In Chapter 2, I assumed that the firm had the same costs wherever it chose to put up its factory. I presented there a model in which the firm chooses where to build a factory (or factories). That model compares the unit shipping cost to a market from elsewhere with the opportunity cost of the capital required to build a separate factory there. One of the assumptions of that model is that other costs of production J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_6, C Springer Science+Business Media, LLC 2010
151
152
6
Ferrying Inputs and Outputs
are the same wherever we construct a factory. However, costs of production may well vary from place to place: e.g., local tax rates may differ as may labor costs or energy costs. In this chapter, I focus on the role of shipping costs on inputs that the firm uses to produce its output. Sometimes, the firm can purchase a unit of the input at the same cost wherever they locate. Such inputs are ubiquitous.1 The model in Chapter 2 assumes that all inputs are like this. In this chapter, I assume some inputs are not ubiquitous; they are available only from particular places(e.g., supply points or basing points), and the firm incurs a unit shipping rate in transporting those inputs to the production site. In that case, under the assumption that the factory builds only one factory, what is the best location for the factory? Will there be conditions under which the firm would cluster with (i.e., choose a location beside) a supplier or customer? This chapter is also a first point to begin thinking about the role of substitutability. After all, if the prices of non-ubiquitous commodities vary from one site to the next, the firm may well want to substitute in favor of an input that is less costly. In the early models presented in this chapter, I assume—for simplicity—a Leontief technology wherein substitution is not possible. Later, in the chapter, I introduce a Cobb-Douglas production function2 that does allow for substitution. This chapter is also an appropriate spot to introduce ideas about conservation and sustainability. In this chapter, the firm gathers inputs from various places and uses them to manufacture a commodity then sold (possibly elsewhere). In so doing, the firm incurs shipping costs. One simple way to think about unit shipping cost is that it depends on the weight of material being shipped.3 If a manufacturing process creates a lot of waste material (i.e., the weight of the input materials greatly exceeds the weight of the output that the firm sells), then the firm might well locate the factory closer to the site of inputs. I have structured the models in this chapter to facilitate consideration of such matters. In thinking about the effect of shipping costs here, I assume in this chapter, for each commodity, the shipping rate per kilometer per unit shipped is constant; there are neither economies of the large haul nor economies of the long haul. To illustrate this point, consider the use of an input inventory by the firm. It is not uncommon for firms to find that, the larger the size of shipment, the cheaper it is to ship per unit of a commodity: i.e., economies of the large haul. The firm must, therefore, consider 1 A feature of an economic landscape wherein some input (factor of production) is available everywhere at the same effective price. 2 A firm using two inputs to produce an output is said to have a Cobb-Douglas production function if the maximum level of output, Q, obtainable from q1 units of input 1 and q2 units of input 2 can be expressed as Q = qab1 q2g where a, b, and g are parameters. This firm has returns to scale that are constant if b + g = 1, decreasing If b + g < 1, and increasing if b + g + 1. Assuming competitive markets for inputs and output and constant or diminishing returns to scale, a profit-maximizing firm will spend the proportion b of its revenue on purchases of input 1 and g on purchases of input 2, the cross price elasticity will be zero, and the firm’s expansion path will be linear. 3 Of course, unit shipping cost will also depend on perishability, breakability, and bulk (among other considerations); perishable/breakable commodities usually have a higher unit shipping cost because they have to be handled more carefully. I ignore such considerations here.
6.1
The Weber–Launhardt Problem
153
whether it is worthwhile to purchase an input in bulk and incur an inventory cost to hold the input until it can be used in production.4 In this chapter, assume the firm is efficient in that it has determined the least-cost strategy for input acquisition and that the shipping cost, as an inclusive measure, is a constant. In earlier chapters of this book, we saw price differentials emerge that might cause a customer to want to move to a place where price was lower. In this chapter, the firm, as the customer for its inputs, does exactly that. In this chapter, what holds a firm to a particular (most profitable) location are the tradeoffs to be made among shipping costs to its input and output places. In location theory, the factory location problem is named after the German economists, Alfred Weber and Wilhelm Launhardt, who independently first solved it.5 A modern version of the model is as follows. Assume this is an efficient firm and that production occurs using a Leontief technology in which ai (i = 1, 2, . . . , I) is the amount of non-ubiquitous input i (in cwt6 ) needed per cwt of output produced.7 Because of shipping costs, the effective price of any one non-ubiquitous input will vary depending on the location of the firm; nonetheless, the Leontief technology assumption means that the firm will always use a fixed amount of that input per unit output regardless of cost.8 Next, assume an otherwise uniform rectangular plane9 punctuated by I fixed places that represent sites of suppliers (one per non-ubiquitous input used in production) and J places where output is purchased. All these places share a common fiat money economy. On this plane, any location can be represented by Cartesian coordinates10 (X, Y)—where X and Y are the Easting and Northing
4 I assume here that all the costs of inventory are borne by the firm. Under certain circumstances, a supplier may have an incentive to get the customer firm to keep a larger inventory. In the context of inventory holding by retail outlets and resale price maintenance by a manufacturer, see Deneckere, Marvel, and Peck (1996). 5 Krzyzanowski (1927, pp. 281–284) is an early and thoughtful overview of Weber’s contributions to thought in this area. 6 A cwt (hundredweight) is 1/20 of a ton: in the American measurement system, it is 100 lb (45.4 kg); in the metric system, it is 50 kg. 7 The Leontief coefficients (a ) are here treated as givens; it is as though there is no other way to i produce the good other than through this fixed combination of inputs. Such an approach ignores the idea that, elsewhere, the same (or complementary) good may be produced using a different combination of inputs available locally. Further, the notion that a good is produced using other commodities as inputs raises the prospect that the output itself might be an input to the production of yet another good. This raises important questions about production chains: the sequencing of production of various components in an industrial economy. See McCann (1995). 8 An extensive literature that reworks the Weber–Launhardt model to incorporate substitutability of inputs is generally thought to have started with Moses (1958). 9 In this book, a representation of the world as though it were simply a two dimensional surface. Ignored here, for simplicity of exposition, are (1) spherical properties of a globe and (2) differences in elevation. On a rectangular plane, a geographic point can be represented simply by a pair of Cartesian coordinates. 10 Representation of a point in two-dimensional geographic space by a pair of distance coordinates: (x, y). An example of such a planar representations are is given by Universal Transverse Mercator (UTM) coordinates which take the form of easting and northing coordinates. Cartesian coordinates
154
6
Ferrying Inputs and Outputs
coordinates, respectively—and distance is given by the Pythagorean formula11 : i.e., Euclidean distance.12,13 Assume suppliers at input Place i are perfectly competitive so that the supply of input i is infinitely elastic at a given f.o.b. price. Assume also that the firm faces a perfectly elastic demand14 for its product in each of the J output markets. Assume the shipping rate for each input and the shipping cost of output (both in $/cwt/kilometer) are fixed so that total shipping costs are proportional— both distance and amount shipped. For the moment, I allow these unit shipping costs per cwt to differ from one commodity to the next; later in the chapter, I will look at the special case where unit shipping cost per cwt is the same for all commodities. Assume the costs of production include only the costs of ubiquitous and non-ubiquitous inputs including all shipping costs. One implication of these assumptions is that the scale of production is indeterminate within this model; the firm earns the same profit for each unit produced without limit. As in Chapter 2, there is no congestion in production here. This shares with Chapter 2 the notion of a constant unit cost of production: unlike Chapters 4 and 5. It differs from the model in Chapter 2 in that a downward-sloped demand curve there limits the total profit that a firm can earn. For that reason, I cannot now find a location that maximizes total profit here in Chapter 6; however, I can find a location that maximizes profit—or, equivalently, minimizes cost—per unit output. Later in this chapter, I consider an extension to the model where demand is sensitive to price. For much of this chapter however, there is something else (outside the model) that determines the quantity demanded; the Weber–Launhardt model solves only for the least costly way of meeting this demand. In that one respect, Chapter 6 is similar to Chapter 3. This model makes several other assumptions. For one, as with previous chapters in this book, I assume here no uncertainty. The firm knows exactly what the outcome will be given its decision about where to locate. I hold off the introduction of uncertainty until a later chapter.15 For another, the model assumes no agglomeration economies (other than possible savings in shipping cost)—e.g., urbanization economies or localization economies—that might make production more efficient or are an approximation to location in that they ignore both elevation and the fact that the Earth is a sphere. The advantage of Cartesian coordinates is that distance calculations are simplified compared to spherical coordinates. 11 The square of the length of the hypotenuse of a right angle triangle in two-dimensional space is the sum of the squares of the other two sides. 12 In two-dimensional space, the straight-line distance between two points with Cartesian coordi√ nates (x,y) and (x2 ,y2 ) is given by ((x1 − x2 )2 + (y1 − y2 )2 ). 13 As there is no transportation network here, we don’t have to worry about congestion on that network. 14 A condition of the demand curve facing a firm or an industry whereby consumers willingly demand any quantity at a given price but are unwilling to pay any more. Also known as a horizontal demand curve. 15 For the reader who wants a taste of the effects of uncertainty, see Asami and Isard (1989), Cromley (1982), Dean and Carroll (1977), Hsu and Tan (1999), Mai (1984, 1987), Mai, Yeh, and Suwanakul (1993b), Mathur (1983, 1985), and Stahl and Varaiya (1978).
6.2
Model 6A: I = 2 Input Places, J = 1 Output Place; Location on a Line
155
profitable at some places compared to others.16 To the restless reader, such assumptions may seem unrealistic. After all you might say, as in Chapter 2, is it reasonable to assume a firm at a given location can produce any quantity at all at the same unit cost? Doesn’t congestion set in when firms become large? Isn’t it possible that managerial inefficiency creeps in depending on the scale of production? When the firm becomes sufficiently large, isn’t there a scale where it is no longer a price taker in markets for its inputs and perhaps also for its output? The answer may well be yes to all of the above. In my defense, however, this chapter is an extension of the model presented in Chapter 2 (where all inputs were assumed ubiquitous). In this chapter, I explore the role of non-ubiquitous assumptions. In later chapters, I will release other assumptions. As I do, the analysis becomes more difficult. The reason for holding these assumptions at the moment is to enable a better understanding of the role played by the particular assumption that has just been released. Finally, this is the first chapter where I need to be specific about the geometry of geographic space. In earlier chapters, I refer to the distance between two places but have not been specific about how this distance might be calculated or measured. That changes in this chapter because I need to measure how a change in the firm’s location affects distances, and hence shipping costs, to its suppliers and customers. The impatient reader might say “why assume the world is flat as is done here when in fact the world is an oblate spheroid?” Good question! The reasons are twofold. First, it is much easier to solve the Weber–Launhardt problem using rectangular distances than spherical distances. Second, provided that the places overall are not too far apart, rectangular distances are a good approximation to spherical distances.
6.2 Model 6A: I = 2 Input Places, J = 1 Output Place; Location on a Line Consider the case where the firm uses 2 inputs each purchased from a single place to produce a commodity sold in a market located at a third place. The firm seeks a site for its factory that maximizes profit per unit output. See (6.1.1) in Table 6.1. The ubiquitous inputs, which might include labor and capital as well as normal profit, cost the firm a fixed f dollars per unit output regardless of location. Only inputs 1 and 2 are non-ubiquitous. In this section, I consider the case where the three places lie along a straight line with the market place xa kilometers east of input Place 1 and xb kilometers west of input Place 2.17 See the map in Fig. 6.1. Initially assume the firm considers putting its factory only at one of these three places. The firm’s profit per unit produced would then be as shown in (6.2.1), (6.2.2), and (6.2.3). See Table 6.2 wherein I summarize equations, assumptions, notation, and rationale for localization used in Model 6A. Between Places 1 and M, Place 1 is the more profitable if freight savings 16 For
an extension that incorporates external economies, see Mai and Hwang (1997). (1968) also used a line market in his modeling of Weber’s location problem.
17 Sakashita
156
6
Ferrying Inputs and Outputs
Table 6.1 The I = 2, J = 1 Weber–Launhardt factory location problem: Leontief production function Semi-net revenue per unit produced P − f − a1 P1 − a2 P2 − a1 x1 s1 − a2 x2 s2 − xm s
(6.1.1)
Euclidean √ distance from factory site (X, Y) to market place (Xm , Ym ) xm = ((X − Xm )2 + (Y − Ym )2)
(6.1.2)
Euclidean distance from input Place 1 (0, 0) to factory site (X, Y) x1 = (X 2 + Y 2 )
(6.1.3)
Euclidean distance from input Place 2 (X2 , 0) to factory site (X, Y) √ x2 = ((X − X2 )2 + Y 2 )
(6.1.4)
Force (freight saving per km) pulling toward the market wm = s
(6.1.5)
Force (freight saving per km) pulling toward input site 1 w1 = a1 s1
(6.1.6)
Force (freight saving per km) pulling toward input site 2 w2 = a2 s2
(6.1.7)
Note: See also Table 6.2 Fig. 6.1 Model 6A: map showing locations on a line. Notes: Input places are 1 and 2. Market place is M. Distance from market to Place 1 is x1 ; distance from market to Place 2 is x2
1
M
2
0 km
50 km
100 km
on input 1 is larger than the additional shipping cost for input 2 and the output: see (6.2.4). Now compare the profit if the factory locates at Place 2 vs. M. Place 2 is more profitable if the freight savings on input 2 exceed the additional shipping costs for input 1 and the output: see (6.2.5). Finally, compare Place 1 and 2. Place 2 is the more profitable if the freight savings on input 2 vis-à-vis Place 1 are larger than the additional shipping costs on input 1 taking into account the change (positive or negative) in output shipping cost: see (6.2.6). So far, I have considered a factory location only at Places 1, M, or 2. Might another place be even more profitable? • We can exclude sites off the line passing through places 1, M, and 2 because any such site would involve longer distances and a smaller unit profit than some point on the line 1M2. • By a similar argument, the firm would never choose a location on a line extended west of input Place 1 or east of input Place 2 because the firm could always improve on profit by choosing instead input Place 1 or 2, respectively. In other
6.2
Model 6A: I = 2 Input Places, J = 1 Output Place; Location on a Line
157
Table 6.2 Model 6A: factory location on straight line 1M2 Unit profit if factory locates at Place 1 P − f − a1 P1 − a2 P2 − a2 (xa + xb )s2 − xa s
(6.2.1)
Unit profit if factory locates at Place M P − f − a1 P1 − a2 P2 − a1 xa s1 − a2 xb s2
(6.2.2)
Unit profit if factory locates at Place 2 P − f − a1 P1 − a2 P2 − a1 (xa + xb )s1 − xb s
(6.2.3)
Place 1 more profitable than Place M if a1 xa s1 > xa s + a2 xa s2
(6.2.4)
Place 2 more profitable than Place M if a2 xb s2 > a1 xb s1 + xb s
(6.2.5)
Place 2 more profitable than Place 1 if a2 (xa + xb )s2 > a1 (xa + xb )s1 + (xb − xa )s
(6.2.6)
Freight savings on input 1 per unit output if 1 km closer to Place 1 a1 s1
(6.2.7)
Freight savings on input 2 per unit output if 1 km closer to Place 2 a2 s2
(6.2.8)
Freight savings on unit of output if 1 km closer to Place M s
(6.2.9)
Notes: See Fig. 6.1. The three places lie along a straight line with the market place xa kilometers east of input Place 1 and xb kilometer. Rationale for localization (see Appendix A): Z8—Limitation of shipping cost or travel delay; Z9—Shipping cost on nonubiquitous inputs. Givens (parameter or exogenous): ai —Amount of input i (cwt) needed per cwt of output; f—Cost per unit output of ubiquitous inputs; P—Price received per cwt of output delivered to the market place; Pi —F.o.b. price paid per cwt of input i; s—Cost to ship cwt of output a distance of 1 unit; si —Cost to ship cwt of input i a distance of 1 unit; xa —Distance east of input Place 1 to market; xb —Distance west of input Place 2 to market.
words, the most profitable location always has to be somewhere along the line segment from Place 1 to Place 2. • Finally, the firm would never find an intermediate location along the line segment more profitable than locating at one of 1, M, or 2. To see why, imagine that we calculate the unit profit for any intermediate location along the part of the line segment between Places 1 and M. Now imagine that we calculate the profit for a place 1 km further to the west but still on the line segment between Places 1 and M. The freight savings for input 1 per unit output each kilometer closer to Place 1 is a1 s1 , for input 2 per each kilometer closer to Place 2 is a2 s2 , and per unit output each kilometer closer to Place M is s. See (6.2.7), (6.2.8), and (6.2.9).18
18 An Isodapane is the name given to the locus of all geographic points on a map that have the same
total transport cost. See Isard (1951), Lindberg (1953), Pred (1965), Smith (1966), Koropeckyj (1967), Feldman (1976), Ohta and Thisse (1993), and Puu (2003).
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Since the move 1 km west means that we are 1 km closer to Place 1, we would save a1 s1 dollars on freight for input 1. Since we would then be 1 km further away from Places M and 2, we incur an additional cost of a2 s2 + s on freight for input 2 and the output, respectively. If a1 s1 > a2 s2 + s, we earn more profit moving 1 km closer to Place 1. However, since this would then be true for each and every kilometer, we would want to move the factory site even further west until we reach Place 1. If, on the other hand a1 s1 < a2 s2 + s, by a similar argument, we would keep moving east until we reached Place M. Where a1 s1 = a2 s2 + s, any location along the line segment from Place 1 to Place M is equally profitable; however, even here no intermediate location is more profitable than Places 1 or M. By analogy, the same argument works for factory sites on the segment from Place M to Place 2. Put differently, because unit profit is linear in distance along the line segment between 1 and M as well as the line segment from M to 2, I need look at only three sites (1, M, and 2) to find the maximum unit profit. That we can exclude any intermediate place is sometimes called the Exclusion Theorem.19 The unit profit is maximized at Place 1 if a1 s1 > a2 s2 + s; it is maximized at Place 2 if a2 s2 > a1 s1 + s; else (s > |a1 s1 − a2 s2 |) it is maximized at Place M. So far, I have assumed that Place M is between Places 1 and 2; by an analogous argument, conditions can be derived for location on a line when either Place 1 or Place 2 is the middle point. These results are summarized in Table 6.3. Table 6.3 Model 6A: conditions for localization Locate at the westerly place, Place 1, if a1 s1 > a2 s2 + s
(6.3.1)
Locate at the easterly place, Place 2, if a2 s2 > a1 s1 + s
(6.3.2)
Locate at the middle place, Place M, otherwise; that is, if s > |a1 s1 − a2 s2 |
(6.3.3)
Note: Corresponds to map in Fig. 6.1. See also Table 6.2.
Let me add a brief aside about the special case where the cost of shipping a cwt is the same for all commodities: i.e., s = s1 = s2 . In this case, we need only look at the weight of commodity 1 (a1 ) and of commodity 2 (a2 ) compared to the 1 unit of output they produce. The production process is said to be weight-losing (presumably involving solid waste) if a1 + a2 > 1 or weight gaining—as, for example, when ubiquitous local water is added to other inputs to make an output— when a1 + a2 < 1. Here (6.3.1) reduces to a1 > a2 + 1 which means that the factory is located at Place 1 when the weight of input 1 used to produce one unit of output is relatively large. Similarly, (6.3.2) implies that the factory is located at
19 The
first reference to this important idea appears to be Sakashita (1968).
6.3
Model 6B: I = 2 Input Places, J = 1 Output Place
159
Place 2 when the weight of input 2 is relatively large. In either case, this is generally consistent with the idea the production process is weight-losing. In contrast, (6.3.3) implies that when the weight of output (1 cwt) is large relative to the other two, the factory is sited at the market; this is consistent with the idea that the production process is weight-gaining. What about comparative statics here? How do variations in the givens (a1 , a2 , xa , xb , s1 , s2 , and s) affect the choice of most profitable location? From the previous paragraph, xa and xb have no effect at all on location. From Table 6.3, we see that a small change in a1 , a2 , s1 , s2 , or s has no effect on the efficient location of a factory. Put differently, these givens have no effect on the efficient location unless the change is sufficient to cross one of the thresholds in Table 6.3.
6.3 Model 6B: I = 2 Input Places, J = 1 Output Place: Location on a Two-Dimensional Plane Now, consider the case where Places 1, M, and 2 do not lie on a line. Instead, they are three points on a rectangular plane. In this case, the firm seeks a place for its factory—namely a Cartesian coordinate pair (X, Y)—that maximizes profit per unit output. Location (X, Y) determines distances to the market, input 1, and input 2: xm , x1 , and x2 , respectively. See (6.1.2), (6.1.3), and (6.1.4). These three places in general form what is referred to as the Weber–Launhardt map triangle.20 Without loss of generality, I assign Cartesian coordinates (0, 0) to Place 1, (X2 , 0) to Place 2, and (Xm , Ym ) to the market place. A simple and intuitive graphical analysis helps us narrow down the area of the map within which the firm might locate. See Fig. 6.2, which shows a map containing the map triangle M12M. Consider a place outside the map triangle as represented by point A. The shortest path21 from A to input Place 1 is the straight line A1, from A to input Place 2 is A2, and from A to the market is AM. Now, using M as the center, draw a circle that passes through A: i.e., has radius AM. As drawn, the circle crosses the map triangle at Place B. Place B, by construction, is the same distance from M as is Place A. However, Place B is closer to both input places, 1 and 2, than is A. Therefore, by (6.1.1), a factory at B has to be more profitable than a factory at A. Of course, a site inside the map triangle may be even more profitable. In conclusion, a site along the edge or inside the map triangle is always to be preferred to a site outside the map triangle. However, where within or on the map triangle is the most profitable site for our factory? To keep matters simple for the moment, assume the most profitable site is inside the map triangle (i.e., not at vertex 1, M, or 2). In that case, I can maximize
20 In the I = 2, J = 1 version of the Weber–Launhardt Model, the triangle formed when three map points on a rectangular plane—each with a pair of Cartesian coordinates—are drawn on a map. 21 Two definitions here: (i) on a two-dimensional plane, the Euclidean distance between two points; (ii) on a network, the path from one vertex to another that is the shortest.
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Fig. 6.2 Model 6B: map showing Weber–Launhardt map triangle. Notes: A, Site outside map triangle M12M. B, Site on boundary of map triangle same distance from M as is Place A
Ferrying Inputs and Outputs
M A
B 2
1
0 km
50 km
100 km
unit profit in (6.1.1) by partially differentiating with respect to the factory’s coordinates, X and Y, and then set the partial derivatives to zero. See (6.4.1) and (6.4.2) in Table 6.4. However, there are two problems with this approach. First, since (6.4.1) and (6.4.2) do not yield explicit equations for X and Y, it is hard to draw conclusions about how the factory site might shift in response to a given change in any one parameter. Second, (6.4.1) and (6.4.2) presume that the most profitable site is not at any of 1, M, or 2 but do not say anything about when this might happen. Table 6.4 Model 6B: first-order conditions to the Weber–Launhardt factory location problem assuming interior solution (i.e., factory not at 1, M, or 2) First-order conditions assuming interior solution s∂xm /∂X + a1 s1 ∂x1 /∂X + a2 s2 ∂x2 /∂X = 0 s∂xm /∂Y + a1 s1 ∂x1 /∂Y + a2 s2 ∂x2 /∂Y = 0
(6.4.1) (6.4.2)
where ∂xm /∂X = (X − Xm )/xm ∂xm /∂Y = (Y − Ym )/xm ∂x1 /∂X = X/x1 ∂x1 /∂Y = Y/x1 ∂x2 /∂X = (X − X2 )/x2 ∂x2 /∂Y = Y/x2
(6.4.3) (6.4.4) (6.4.5) (6.4.6) (6.4.7) (6.4.8)
Notes: Rationale for localization (see Appendix A): Z8—Limitation of shipping cost or travel delay; Z9—Shipping cost on nonubiquitous inputs. Givens (parameter or exogenous): ai —Amount of input i needed per unit of output; s—Cost to ship cwt of output a distance of 1 unit; si —Cost to ship cwt of input i a distance of 1 unit; Xi —X-coordinate of Place i; Ym —Y-coordinate of Place M. Outcomes (endogenous): X—X-coordinate of factory; xi —Distance from factory to Place i; Y—Y-coordinate of factory. See also Table 6.1.
6.3
Model 6B: I = 2 Input Places, J = 1 Output Place
161
Instead of calculus, Weber used an equilibrium of forces approach to solve for the most profitable factory site.22 In his approach, we imagine a most profitable site at (X, Y). Then, we think of the economic forces, the marginal shipping costs23 that pull us in the direction of the output market, the input market 1, and the input market 2: wm , w1 , and w2 , respectively. The marginal shipping cost is the amount of additional profit per unit output possible were we 1 km closer to that place. These are shown in (6.1.5) through (6.1.7) and are the same as the cost savings, (6.2.7), (6.2.8), and (6.2.9), we have already calculated for the case where Places 1, M, and 2 are on a line.24 In an equilibrium of forces approach (think back to high school physics here), we start by constructing a Weber–Launhardt weight triangle25 whose three sides are proportional to w1 , w2 , and wm . See Fig. 6.3. From this triangle, we obtain the three interior angles: β 1 , β 2 , and β 3 . Note here β 1 is the angle opposite the side w1 , β 2 is opposite w2 , and β 3 is opposite wm . If the Weber–Launhardt problem has an interior solution P (i.e., the production site lies strictly inside the map triangle), then the most profitable production site must be such that these three forces are in equilibrium. We can draw three interior angles on the map triangle each
β3
a1s1
a2s2
Fig. 6.3 Model 6B: Weber–Launhardt weight triangle. Note: See also Fig. 6.2
β1
β2 s
22 In this, he collaborated with Georg A. Pick, a professor of Mathematics at the German University
of Prague at the time. Figures like Fig. 5.4 are therefore sometimes called “Pick’s diagram.” See Ellison (1991). 23 In the I = 2, J = 1 Weber–Launhardt location problem, there are four points on a twodimensional map; the output point, the production point, and two input points. The marginal shipping cost is the extra profit per unit output that could be earned if only the production point were 1 km closer to another point. Marginal shipping costs are w1 , w2 , and wm . 24 In Mathematics, this is also known as the Steiner problem. 25 In the I = 2, J = 1 version of the Weber–Launhardt Model, the “equilibrium of forces” triangle created using the weights w1 , w2 , and wm . See Marginal Shipping Cost.
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M
wm
w1
γ2 P γ1 γ3 w2 2
1
0 km
50 km
100 km
Fig. 6.4 Model 6B: interior solution for sample Weber–Launhardt factory location problem. Notes: See also Fig. 6.3. γ 1 , Angle (g1 in diagram) formed at interior production place opposite input site 1. Angles γ 1 and β 1 are supplements: γ1 + β1 = 180◦ , γ 2 Angle (g2 in diagram) formed at interior production place opposite input site 2. Angles γ 2 and β 2 are supplements: γ2 + β2 = 180◦ γ 3 , Angle (g3 in diagram) formed at interior production place opposite market site M. Angles γ 3 and β 3 are supplements: γ3 + β3 = 180◦
with a vertex at P: γ1 = ∠MP2, γ2 = ∠MP1, and γ3 = ∠1P2. In an equilibrium of forces, γ 1 and β 1 are supplementary (i.e., γ1 + β1 = 180◦ ); the same is true of γ 2 and β 2 as well as γ 3 and β 3 . How are we to find the site that forms such angles? A geometric solution by Pick is one possibility. Referring to Fig. 6.5 to illustrate, the steps are as follows 1. Scale the weight triangle until the side of length s corresponds to the distance from input Place 1 to input Place 2. See (6.5.1) in Table 6.5. Draw the weight triangle upside down so that it hangs below the line joining input places 1 and 2 and such that the a1 s1 side is opposite input Place 1 and the a2 s2 side is opposite input Place 2. Label the bottom vertex of this weight triangle as W. In Fig. 6.5, the weight triangle is now the dotted W12W. 2. Draw a circle that passes through all three vertices of triangle W12. The easiest way to do this (thinking back to high school geometry) is to draw the perpendicular bisector to each of two sides of the triangle. The place where the two bisectors intersect is the center of this circle. The center of the circle is C in Fig. 6.5. See (6.5.4), (6.5.5), and (6.5.6) 3. Draw a straight line from W to the market site M. If there is an interior solution, it is the place where this straight line crosses the circle drawn in step 2. The most profitable production site is indicated by a P in Fig. 6.5. See (6.5.7) through (6.5.13)
6.3
Model 6B: I = 2 Input Places, J = 1 Output Place
163 M
Fig. 6.5 Model 6B: map showing interior solution for sample problem
γ2 P γ 1 γm 1
β1
wm
w2
β2
2
0 km
50 100 km km
w1
βm W
In general, these give the coordinates of the most profitable location as long as the solution is interior to the Weber–Launhardt map triangle. Why does this work? In essence, the geometric construction ensures that the angles formed about Place P are the supplements of the angles in the weight triangle. Consider ∠1P2. Since P12 and W12 have the same circumcircle and share the same chord 12 with P to one side and W to the other, ∠1P2 and ∠1W2 are supplements. Since ∠1W2 = βm , ∠1P2 = γm . As a consequence of this circle, the other two angles around P, ∠1PM, and ∠MP2 must be supplementary to β 2 and β 1 , respectively, and the Places M, P, and W must be collinear. However, under certain conditions, the most profitable location is at a vertex of the map triangle and not interior to it. Such cases constitute corner solutions.26 What are those conditions? Let ϕ1 = ∠M12 on the map triangle; let ϕ2 = ∠12M and let ϕ3 = ∠1M2. These are geographic angles because they each depend only on the locations of the three map points. In Fig. 6.4, γ1, is larger than the geographic angle at the Place 1 vertex. However, were I to change the weight triangle in such a way that the most profitable location is now closer to Place 1 (by increasing w1 relative to the other two weights), then γ1 would shrink. In fact, if w1 were to be increased enough relative to the other two weights, the most profitable location would be at
condition in which the solution to an optimization problem Y = Maxx [f (x)] occurs at a limiting value of x.
26 A
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Table 6.5 Model 6B: algebraic solution to the I = 2, J = 1 Weber–Launhardt factory location problem assuming interior location Weight triangle scaling factor v = X2 /s
(6.5.1)
Coordinates of Place W Xw = X2 − ((va1 s1 )2 + (X22 − (va2 s2 )2 )/(2X2 ) √ Yw = − ((va2 s2 )2 − Xw2 )
(6.5.2) (6.5.3)
Coordinates of Place C (Center of circle) Xc = 0.5X2 Yc = (Xw2 + Yw2 − X2 Xw )/(2Yw )
(6.5.4) (6.5.5)
Radius of circle √ r = ((Xc − Xw )2 + (Yc − Yw )2 )
(6.5.6)
Line (Y = mX+b) joining W and M. b = Ym − Xm (Yw − Ym )/(Xw − Xm ) m = (Yw − Ym )/(Xw − Xm )
(6.5.7) (6.5.8)
Quadratic to solve X:qa X 2 + qb X + qc = 0 qa = 1 + m2 qb = 2(bm − mYc − Xc ) qc = Xc2 + b2 + Yc2 − 2bYc − r2
(6.5.9) (6.5.10) (6.5.11)
Coordinates of most profitable production site √ X = ( − qb + (q2b − 4qa qc ))/(2qa ) Y = b + mX
(6.5.12) (6.5.13)
Note: See also Table 6.4.
Place 1: there, γ1 = ϕ1 . If I were to make w1 even larger, the most profitable location would still be at Place 1. A similar argument can be used to derive the conditions under which the most profitable location would be either Place 2 or Place M. The 3 sets of localization conditions are (6.6.1), (6.6.2), and (6.6.3). See Table 6.6. These three localization conditions each posit a relationship among the following parameters: technology (a1 , a2 ), shipping costs (s1 , s2 , and s), and relative geography (ϕ 1 , ϕ 2 , and ϕ 3 ). They each say production is drawn to that vertex when the Table 6.6 Model 6B: conditions under which factory is located at vertex in the I = 2, J = 1 Weber–Launhardt factory location problem General solution where 3 places on two-dimensional plane Site Locate here when √ Place 1: w1 ≥ (w22 + w2m + 2w2 wm cos [ϕ1 ]) √ Place 2: w2 ≥ (w21 + w2m + 2w1 wm cos [ϕ2 ]) √ Place M: wm ≥ (w21 + w22 + 2w1 w2 cos [ϕ3 ])
(6.6.1) (6.6.2) (6.6.3)
Corollary: when market place is on line between Input places 1 and 2 Site Locate here when Place 1: w 1 ≥ w2 + wm Place 2: w 2 ≥ w1 + wm Place M: wm ≥ |w1 − w2 |
(6.6.4) (6.6.5) (6.6.6)
Note: See also Table 6.4.
6.4
Model 6C: Substitutability, Scale, and Location
165
shipping cost associated with that vertex is sufficiently large relative to the shipping costs associated with other vertices. It is relatively easy to show at most that one of these conditions can be true. I refer above to relative geography. Note (6.6.1), (6.6.2), and (6.6.3) use the three map angles but take no account of the distances between places. The localization conditions do not depend on whether the places are 1 km or 1,000 km apart. This is different from the model in Chapter 2 where, given a shipping rate, there was a certain minimum distance beyond which the firm finds it profitable to construct a second factory (in that freight savings are larger than the opportunity cost of the capital needed to build the second factory). In this chapter, however, the conditions for localization depend on map angles rather than distances. In the preceding section of this chapter, I considered the case where Places 1, M, and 2 formed a line in geographic space. In that case, ϕ1 = 0◦ so cos [ϕ1 ] = 1; ϕ2 = 0◦ so cos [ϕ2 ] = 1, and ϕ3 = 180◦ so cos [ϕ3 ] = − 1. In this situation, the localization conditions (6.6.1), (6.6.2), and (6.6.3) reduce to the following. Locate at Place 1 if w1 ≥ w2 + wm ; at Place 2 if w2 ≥ w1 + wm ; at Place M if wm ≥ |w1 − w2 |. These are, of course, the same conditions I had derived earlier. What about comparative statics in this model? There are two possibilities here. One is when the least cost solution is at a vertex of the Map Triangle. In that case, a small change in any of the givens results in no change in the optimal location. The other possibility is that the least cost solution is an interior place within the Map Triangle. Here, an increase in any one given draws the least cost solution closer to the relevant vertex. An increase in a1 or s1 draws the firm closer to input Place 1; an increase in a2 or s2 draws it closer to Place 2; an increase in s draws it closer to M.
6.4 Model 6C: Substitutability, Scale, and Location To this point, I have envisaged a firm as making two decisions: (1) location of its factory and (2) scale of production there. As noted above, I set up the Weber–Launhardt model in such a way that scale itself is indeterminate. The linearity of a Leontief production technology separates those two decisions in the sense that the firm can decide where to locate irrespective of how much it will produce there. Students often say to me that this separation between the effects of location on marginal cost and quantity supplied on P (price) seems strange. Would it not be more appropriate, they say, to have a model where location and scale of production are jointly determined? To illustrate, let us consider here the case of a firm that is about to increase substantially its output. Two plausible situations come to mind here, wherein the firm’s choices of location and quantity will be interdependent. • One is that the firm will now find it profitable to alter the mix of its non-ubiquitous inputs.27 For some reason, it is more profitable to produce the firm’s commodity 27 Nijkamp
and Paelinck (1973) propose a grid search method to investigate the impact of substitutability among inputs.
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in a different way when the quantity being produced is larger. Put differently, the firm’s production function does not yield a linear expansion path. Typically, this is because of an indivisibility in production. If the scale of output is too small, the firm can’t take advantage of the indivisibility and so has to produce the commodity as best it can using some other mix of inputs. In this case, as the scale of production is increased, the firm seeks to locate closer to sources of inputs that will now be more important. • The other is that the amount of each non-ubiquitous input needed per unit of output shrinks when output is increased. This kind of economy of scale typically happens when the firm is better able to make use of its materials (i.e., reduce S wastage) the larger the scale of output. In this case, as scale of production is increased, the firm finds its optimal location now less dependent on the cost of shipping inputs and relatively more dependent on the cost of shipping product to market. These two situations illustrate the idea that separability of scale and location arises because I have so far assumed (1) linear expansion paths and (2) constant returns to scale in production. In part, what has driven location theorists here is the desire to integrate their theory with the neoclassical theory of the firm wherein input substitution and economies of scale are important. In neoclassical theory, much of the emphasis here has been on the substitution between labor and capital in production. However, we need to remember here that a Weberian model is necessarily focused on nonubiquitous inputs. If labor and capital are ubiquitous inputs, the substitution between them in itself will not affect the outcome of a Weberian model. Location is affected only when there is substitution between non-ubiquitous inputs. For that reason, here, I will not pursue further the implications of nonlinear expansion paths. However, the possibility of reduced wastage with larger production levels makes it useful to think about incorporating economies of scale into a Weberian model. Modern attempts to incorporate production functions other than a Leontief technology started with Moses (1958) and found a fuller statement in Khalili, Mathur, and Bodenhorn (1974).28 Let me here use a model (Model 6C) that emphasizes the role of economies of scale. As this model can be solved only numerically in general, I will use the numerical values shown in (6.7.7) through (6.7.12) of Table 6.7. Unless otherwise stated, I retain all assumptions from Model 6B. Assume here that there are N consumers at Place M and that each demands q units of the commodity at a given price P. The firm incurs an annual fixed cost (F). The firm’s total cost, to be minimized, is given by (6.7.1). Now, assume that the firm chooses an input combination (Q1 , Q2 ) sufficient—in terms of its Cobb-Douglas production function
28 See also Sakashita (1968), Bradfield (1971), Hurter and Wendell (1972), Tellier (1972). Emerson
(1973), Woodward (1973), Thisse and Perreur (1977), Miller and Jensen (1978), Mathur (1979), Eswaran, Kanemoto, and Ryan (1981), Alperovich and Katz (1983), Mai (1984), Hsu and Mai (1984), Kusumoto (1984, 1985), and Kilkenny and Thisse (1999).
6.4
Model 6C: Substitutability, Scale, and Location
167
Table 6.7 Model 6C: cost minimization in the I = 2, J = 1 Weber–Launhardt factory location problem: Cobb–Douglas production function Firm’s total cost Nqxm s + Q1 (P1 + x1 s1 ) + Q2 (P2 + x2 s2 ) + Nqf + F
(6.7.1)
Production function γ Q = Nq = AQε1 Q2
(6.7.2)
Distances from factory to input points 1 and 2 and market place M See (6.1.2), (6.1.3), and (6.1.4)
(6.7.3)
Minimization of firm’s total cost subject to (6.7.2) and (6.7.3) with respect to Q1 and Q2 yields {(P1 + s1 x1 )Q1 }/{(P2 + s2 x2 )Q2 } = ε/γ
(6.7.4)
Which yields ε+γ Q = A(γ /ε)γ {(P1 + s1 x1 )/(P2 + s2 x2 )}γ Q1
(6.7.5)
Under constant returns to scale (ε + γ = 1) Q1 /Q = (ε/γ )γ {(P2 + s2 x2 )/(P1 + s1 x1 )}γ /A
(6.7.6)
Givens held constant Production technology: A = 1.65, ε = 0.40 Costs: f = 2.00, F = 10,000 Prices: P = 9.00, P1 = 1.00, P2 = 1.50 Individual Quantity demanded: q = 6 Unit shipping rates: s = 0.90, s1 = 1.00, s2 = 1.00 Locations: X2 = 1, Xm = 0.50, Ym = 1.00
(6.7.7) (6.7.8) (6.7.9) (6.7.10) (6.7.11) (6.7.12)
Base case: constant returns to scale Production technology: γ = 0.60 Factory site: X = 0.51, Y = 0.76 Distances: x1 = 0.92, x2 = 0.90, xm = 0.24 Inputs: Q1 /Q = 0.54, Q2 /Q = 0.65
(6.7.13) (6.7.14) (6.7.15) (6.7.16)
Notes: Rationale for localization (see Appendix A): Z8—Limitation of shipping cost or travel delay; Z9—Shipping cost on nonubiquitous inputs. Givens (parameter or exogenous): A—Scalar in production (A > 0); f—Cost per unit output of ubiquitous inputs; F—Firm’s fixed cost; N— Number of consumers at market place; P—Price received per unit of output delivered to the market place; Pi —F.o.b. price per unit of nonubiquitous input i; q—Amount of commodity demanded by each consumer; s—Cost of shipping 1 unit of output a distance of 1 km; si —Cost of shipping 1 unit of input i a distance of 1 km; X2 —X-coordinate of Place 2; Xm —X-coordinate of market place; Ym —Y-coordinate of market place; γ —Exponent of input 2 in production function; ε— Exponent of input 1 in production function. Outcomes (endogenous): Q—Firm’s output (scale of production); Qi —Quantity of input i used in production; X—X-coordinate of factory; xi —Distance to supplier of input i; xm —Distance to market; Y—Y-coordinate of factory.
to—meet demand: see (6.7.2). Cost minimization means that it chooses a combination that satisfies (6.7.4). Finally, as in Model 6B, assume the firm also chooses a location for its factory to minimize total cost: see (6.7.3). The least cost site for production can be the same as in Model 6B. For a given quantity of output needed (Q), the solution to Model 6C consists of input levels (Q1 , Q2 ) and a factory location (X, Y). From these, estimate the Leontief production
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Fig. 6.6 Models 6B and 6C compared
Ferrying Inputs and Outputs
Models 6B and 6C compared
A
AB
C CGD EF
Iso-outlay linefor aggregate shipping expense on input 1 and input 2 atgiven site in Model 6C: slope is -s1x1/s2x2 Cobb-Douglas isoquant: Model 6C Leontief isoquant: Model 6B
Input 2
E
I
G
F
D
0
H
Input 1
B
coefficients: a1 = Q1 /Q and a2 = Q2 /Q. Using these estimates in Model B and solving for the least-cost production site (X, Y), gives the same site as previously obtained for Model 6B. See Fig. 6.6. What happens as we change the level of output? The firm experiences constant returns to scale when ε + γ = 1, decreasing returns when ε + γ < 1, and increasing returns when ε + γ > 1.29 Let us look at each of these cases in turn assuming the specific values for the givens shown in (6.7.7) through (6.7.12). Constant returns to scale: When ε + γ = 1, we get (6.7.13) and the linear expansion path implied by the Cobb-Douglas production function means that the cost-minimizing amount of input 1 (or input 2) used per unit of output produced remains constant if the site does not change: see (6.7.6) and (6.7.16). At the same time, there is no incentive for the firm to change location with scale here. The least cost location at every scale of output is given by (6.7.14) and the resulting distances in (6.7.15)
we can solve even the case where αβ > 1 here because we have assumed total quantity demanded (i.e., Nq) is fixed.
29 Here
6.4
Model 6C: Substitutability, Scale, and Location
169
Decreasing returns to scale: To illustrate the case of decreasing returns to scale (or, equivalently more wastage), I use γ = 0.59 in lieu of (6.7.13). In Fig. 6.7, I map the least cost production site for the factory. The least cost production site shifts south toward input places as N is varied from 1,000 up to 1,000,000. Locus AB shows expansion path; the least-cost location of the factory shifting with N. Increasing returns to scale: To illustrate the case of increasing returns to scale (or, equivalently, less wastage), I use γ = 0.61 in lieu of (6.7.13). In Fig. 6.7, the least cost production site shifts north toward input places as N is varied from 1,000 up to 1,000,000. Locus AC shows expansion path; the least-cost location of the factory shifting with N.30
M C A
B
1
2
0 km
50 km
100 km
Fig. 6.7 Map of I = 2, J = 1 Weber–Launhardt factory location problem: Cobb-Douglas production function. Notes: (1) See Table 6.7. (2) Input places 1 and 2 shown as diamonds on map; market place (M) shown as square. (3) Point A is least cost location for factory (at any level of output) under constant returns to scale (α = 0.40, β = 0.60). (4) To ensure profitability, number of customers (N) at market place must be sufficiently large. (5) Under increasing returns to scale (ε = 0.40, γ = 0.61)—equivalently less wastage—least cost production site shifts north toward market place as N is varied from 1,000 to 1,000,000. Locus AC shows expansion path; location of factory shifting with N. (6) Under decreasing returns to scale (ε = 0.40, γ = 0.59)—equivalently more wastage—least cost production site shifts south toward input places as N is varied from 1,000 to 1,000,000. Locus AB shows expansion path; location of factory shifting with N
30 Mai and Hwang (1997) look at Cournot-Nash competition among firms in the presence of exter-
nal economies. When external economies prevail, constant returns to scale at the firm level is not a sufficient condition for ensuring the separability of location and scale. Moreover, they show that when free entry is allowed and the production function exhibits decreasing returns to scale, but
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6.5 Model 6D: Price Elasticity Back in Chapter 3, I first introduced a model in which quantity demanded is given; that is, does not vary with price. In general, economists prefer to think of demand as a schedule of quantities demanded at different prices. Indeed, the purpose of this book is to show the interaction of price and location. Such considerations make one uneasy therefore with models that ignore price. It is straightforward to extend Model 6B in this chapter to allow prices of outputs and even inputs to be variable. The approaches used in Chapters 2, 4, and 5 are helpful here. As in Chapter 2, the firm can be thought to face a downward sloping demand curve for its product at the market place. As in Chapters 4 and 5, the firm can be thought to face also upward sloping supply curves for its inputs at the two input places. To the extent that the firm can affect both the price of its output and its inputs, it is a monopolist in its output market and a monopsonist in its input market: equivalently, a bilateral monopolist. I refer to this extension as Model 6D. I envisage the firm here maximizes profit: see (6.8.1) in Table 6.8. As in Chapter 2, I assume that the market place at M is populated by N consumers each with the same individual linear inverse demand curve: see (6.8.2). As in Chapter 4, I assume competitive suppliers at each of the input points and an upward sloped supply curve confronting our firm: see (6.8.4) and (6.8.5). Does extending the Weber-Launhardt model to incorporate variable prices for the firm’s product and inputs eliminate the separability of location and scale inherent in Model 6B? The answer is no. A profit maximizing firm facing a downward sloping demand curve and upward sloping supply curves for its inputs—but otherwise similar to the firm in Model 6B—chooses the same site for its factory as predicted by that model.31 Model 6D adjusts the quantity produced, but the most efficient location for that production remains the same. The implication of this is that Model 6B has a wider application than might first have been thought because we can now use it to find efficient locations even where price elasticity of demand is not zero.32 Let us now consider two more extensions to the I = 2, J = 1 Weber-Launhardt factory location problem.
with strong external economies, the optimal location moves toward or away from the market as demand increases according to whether the demand function is convex or concave. 31 Of course, I assume here that the profit is sufficiently large; otherwise a firm would choose no factory at all. 32 Before leaving Model 6D, it is instructive to think about how competition might shape the downward sloping demand curve faced by firm. I am thinking here of the effect of entry by competitors producing the same product.32 Another aspect concerns the effect of competition on the price and location of a firm. The key here is the conjectural variation. Under some assumptions of how a competitor might react to the presence of another, it is possible to get different locations for each competitor. See Mai and Hwang (1994).
6.6
Model 6E: More Than 2 Input Places and/or More Than 1 Output Place
171
Table 6.8 Model 6D: the I = 2, J = 1 Weber–Launhardt factory with a downward-sloping demand curve and upward-sloping supply curves for its inputs Profit (P − C)Q − F
(6.8.1)
Demand P = α − βQ/N
(6.8.2)
Unit cost C = f + a1 (P1 + s1 x1 ) + a2 (P2 + s2 x2 ) + sxm
(6.8.3)
Inverse supply curve for input 1 P1 = C1 + δ1 (a1 Q)
(6.8.4)
Inverse supply curve for input 2 P2 = C2 + δ2 (a2 Q)
(6.8.5)
Distances from factory to input points 1 and 2 and market place M See (6.1.2), (6.1.3), and (6.1.4)
(6.8.6)
Notes: See also Table 6.1. Rationale for localization (see Appendix A): Z8—Limitation of shipping cost or travel delay; Z9—Shipping cost on nonubiquitous inputs. Givens (parameter or exogenous): α—Intercept of individual demand curve; β—Slope of individual demand curve; ai —Amount of input i needed per unit of output; C—Unit cost (production and shipping); Ci —Intercept of supply curve for input i; f—Cost per unit output of ubiquitous inputs; F—Fixed cost; N—Number of customers at Place M; s—Cost of shipping 1 unit of output a distance of 1 km; si —Cost of shipping 1 unit of input i a distance of 1 km; Xm —X-coordinate of market place; X2 —X-coordinate of Place 2; Ym —Y-coordinate of market place; δ i —Slope of supply curve for input i. Outcomes (endogenous): C—Unit cost (production and shipping); P—Price received per unit of output delivered to the market place; Pi—F.o.b. price of input i; Q—Quantity of output produced; X—X-coordinate of factory; xm —Distance from factory to market place; xi —Distance from factory to Place i; Y—Y-coordinate of factory.
6.6 Model 6E: More Than 2 Input Places and/or More Than 1 Output Place Another extension is to consider more than 2 input places and/or more than one output place. In such a case, none of the methods developed above is of use. However, the Kuhn-Kuenne algorithm33 first published in 1962 solves this problem numerically. Where we have more than two input places, but just 1 output place, the problem remains one of minimizing unit cost. When more than 1 output place is involved, the problem becomes one in which we maximize unit profit. See Table 6.9. Maximize (6.9.1) subject to (6.9.2) and (6.9.3). For each input Place i, the algorithm requires knowledge of the Leontief coefficient, ai , and the shipping rate per kilometer, si . For each market place j, the algorithm requires the shipping rate, s, as well as the quantity to be supplied to that market, Qj . The same problem can now generally be solved also by standard optimization software. 33 A
numerical algorithm to solve the Weber-Launhardt factory location problem in twodimensional space for any number of input and output points.
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Table 6.9 Model 6E: the Weber–Launhardt factory location problem with I input places and J output places Profit j (Pmj − sxmj )Qj − {f + i ai (Pi + si xi )}j Qj − F
(6.9.1)
Euclidean distance from factory site to market place J √ xmj = ((X − Xmj )2 + (Y − Ymj )2 )
(6.9.2)
Euclidean distance from input Place i to factory site √ xi = ((X − Xi )2 + (Y − Yi )2 )
(6.9.3)
Notes: Rationale for localization (see Appendix A): Z8—Limitation of shipping cost or travel delay; Z9—Shipping cost on nonubiquitous inputs. Givens (parameter or exogenous): ai —Amount of input i needed per unit of output; f—Cost per unit output of ubiquitous inputs; F—Fixed cost; Pmj —Price received per unit output delivered to market place j; Qj —Quantity sold at market place j; Pi —F.o.b. price paid per unit of input i; s—Cost to ship unit of output a distance of 1 unit; si —Cost to ship unit of input i a distance of 1 unit; Xi —X-coordinate of input place i; Yi—Ycoordinate of input place i; Xmj —X-coordinate of market place j; Ymj —Y-coordinate of market place j. Outcomes (endogenous): X—X coordinate of factory site; xi —Distance from input place i to factory.; xmj —Distance from factory to market place j; Y—Y-coordinate of factory site.
In this extension, the firm will still locate near or at an input or output place if the unit cost saving by moving a kilometer closer to that place is substantially greater than the saving from moving toward any other place. Further, the comparative statics in this model are the same as in Model 6B. When the least-cost solution is not at an input or output place, a small change in any of the givens results in no change in the optimal location. When the least-cost solution is not at an input or output place, an increase in any one given draws the least-cost location closer to the relevant place. An increase in ai or si draws the firm closer to input place i; an increase in s draws it closer to an output place. As before, an increase in total output—specifically an increase spread proportionally among existing demand places—has no effect on location. However, the introduction of demand at a new place, or a relative shift in demands at existing places, can affect the most profitable site for the firm.
6.7 Model 6F: Location on a Transportation Network When first presented with the Weber–Launhardt factory location problem, students often express discomfort about the assumption of a uniform plane whereon shortest path routes as the crow flies34 are possible in any direction. After all, they say, factories must typically locate adjacent to a transportation facility: be it a highway exit, a railway siding, or a sea or air port facility. Therefore, they ask, can the Weber factory location be solved when we are considering location on a transportation network? Somewhat surprisingly, this turns out to be relatively easy to do.35 34 The
Euclidean distance between two points on a plane. the assumption used elsewhere in this book that there is no congestion over the network.
35 Under
6.7
Model 6F: Location on a Transportation Network
173
Earlier in this chapter, I considered the I = 2, J = 1 case where the input places and market place happen to lie along a straight line. Since all shipments can be shown to be along that line, this is a simple version of a transportation network. In that case, the conditions for location at Places 1, 2, and M were given in (6.6.4), (6.6.5), and (6.6.6). These conditions exhaust all the possible values for w1 , w2 , and wm : every combination of positive w1 , w2 , and wm satisfies exactly one of (6.6.4), (6.6.5), and (6.6.6). Therefore, when the three places lie on a line there is never an intermediate location (i.e., a site between two adjacent places) more profitable than locating at one of the places themselves.36 Now consider the general case of a transportation network. A network is a set of links each of which joins two vertices. In this treatment, every cul-de-sac by definition is a vertex as is every junction. On this transportation network, let us place a set of input places and output places. Without loss of generality, think of each input or output place also as a vertex since these either are to found at an existing vertex, or if along a link, redefine this as two links connecting a new vertex that is, the input or output place. Figure 6.8 is illustrative.
K
100
D
110
61
91
E
120
98
80 H
I J
131
87
G
91
91
100
C
99
100
91
F
B
94
65
96
A
103
Fig. 6.8 Model 6F: map illustrating location on a transportation network
14
9
L
M
91
161
0 km
N
50 km
100 km
Now, let us examine any one link on that transportation network: say for the sake of illustration the east–west link GH in Fig. 6.8. At either end of that link is a vertex. Imagine a production site at an intermediate location along that link. All inputs and outputs must be transported along part of the link because they all come from (or through) the vertex to the east (H) or the vertex to the west (G). We can readily calculate the profit per unit possible at this site from (6.1.1). Let us then consider a 36 See
Louveaux, Thisse, and Beguin (1982).
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second possible production site a little further west along this link. Recalculate the profit per unit output at this new location. There are three possibilities here: profit has (1) increased, (2) stayed the same, or (3) gone down. In fact, because (6.1.1) is linear in distances, each kilometer I move the factory from east to west changes the profit per unit output by exactly the same amount. The implication is that the profit per unit product is never higher in mid-link than it is at one or the other vertex (or possibly both). This is known in location theory as the Hakimi Theorem.37 To find the most profitable location on a network, therefore, I need look only at the profit at each vertex. In Table 6.10, I show an example using the network from Fig. 6.8. Assume I = 2 inputs: one available from vertex A, the other from C. Assume 1 market place; in this case, vertex N. At the weights (w1 , w2 , w3 ) given in Table 6.10, the most profitable site is vertex C since total shipping costs there are the lowest among all vertices. The calculation of unit profit site-by-site for the purpose of choosing the best location has long been known in location theory as
Table 6.10 Model 6F: total shipping cost per unit output at each vertex for location on network where I = 2 and J = 1 i Vertex [1]
wi [2]
Shipping cost [3]
A B C D E F G H I J K L M N
1
2.00
2
2.50
957 857 702 1,224 1,862 1,337 997 923 1,084 1,645 1,663 1,369 1,290 1,689
Market
1.00
Notes: Vertices and distances shown in Figure 6.8. Rationale for localization (see Appendix A): Z8—Limitation of shipping cost or travel delay; Z9—Shipping cost on nonubiquitous inputs. Givens (parameter or exogenous): None used. Outcomes (endogenous): None used.
37 On
a transportation network defined as consisting of M vertices (each vertex at least one of a customer location, a supplier location, or a point where two or more network segments intersect) with accompanying network segments, no location can be more efficient than one or more of the vertices. As a result of this theorem, we may use comparative cost analysis at each vertex to find the best location. See Hakimi (1964), Handler and Marchandani (1979), and Louveaux, Thisse, and Beguin (1982). The Hakimi Theorem can be seen as an extension of the Exclusion Theorem, even though it appears to predate it.
6.8
Final Comments
175
comparative cost analysis.38 The advantage created by the Hakimi Theorem is that I need do comparative cost analysis only at places that are vertices. What about comparative statics here? From the previous paragraph, distance along any one link has no effect at all on location unless we start from an initial condition where total cost is the same everywhere along the link. Except in that knife-edge situation, we see now as well that a small change in a given has no effect on the efficient location of factory.
6.8 Final Comments In this chapter, the principal model has been 6B; I like to think of this as the standard Weber–Launhardt model. I included Models 6A and 6C through 6F to help readers better understand aspects of Model 6B. In Table 6.11, I summarize the assumptions that underlie Models 6A through 6F. Many assumptions are in common to all models: see panel (a) of Table 6.11. I present Model 6A (location on a line) first because it is a simplification of Model 6B that illustrates similar Table 6.11 Assumptions in Models 6A 6F– Assumptions
6A [1]
6B [2]
6C [3]
6D [4]
6E [5]
6F [6]
(a) Assumptions in common A1 Closed regional market economy A3 Punctiform landscape B1 Exchange of commodity for money D4 Choice of factory location(s) E2 Fixed unit shipping rate E5 Firm bears shipping cost to market(s) H1 Some suppliers price f.o.b.
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x
x x x
x x x
x x x
x x x
x x
x
(b) Assumptions specific to particular models A5 Linear landscape I1 Leontief technology C1 Fixed demand locations D2 Firm minimizes cost of production and shipping D7 Horizontal marginal cost curve A4 Rectangular plane I2 Cobb–Douglas production function CRS B2 Upwardly sloped local supply curve C2 Fixed local customers C4 Identical customers C5 Identical linear demand D1 Monopolist and/or monopsonist
38 A
x
x x
x x x
x x x x x x
technique for finding the least cost location for a firm by enumerating possible locations and then calculating the unit cost of production at each.
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ideas; Model 6A is also helpful in thinking about location on a network (Model 6F). Models 6C through 6F extend the standard Weber–Launhardt model. Model 6C introduces substitutability of inputs and the possibility of economies of scale. Model 6D makes the prices of output and inputs endogenous. Model 6E expands on the number of inputs and the number of customer points. Finally, Model 6F allows for location and shipments to be constrained to a transportation network. In each case, these extensions give us new insights but also reinforce aspects of the standard model. What this chapter does—for the first time in this book—is to make explicit a rationale for localization based on local cost advantages arising from the nonubiquitousness of inputs. Chapters 3, 4, and 5 are built—at least in part—on the implicit assumption that some locations are less costly to produce at than others. Chapter 6 attributes differences in costliness specifically to the shipping cost of non-ubiquitous inputs needed in production. At the same time, however, this chapter continues to assume that the prices of inputs (including shipping services) used by the firm are exogenous. Put differently, input prices are all determined in markets outside the scope of the models in this chapter. As well, the models in this chapter are silent on prices in output markets. Overall I think Walras would have argued that the analysis in this chapter has been only partial in the sense that we have not looked explicitly at the simultaneity among prices in these markets. Once again, we must wait until a later chapter for the opportunity to do that. From Walras’ perspective, this chapter does not get us much further in understanding how price simultaneity arises with one important caveat. That caveat, as exemplified by Model 6B, is that the least cost location is not shaped by the f.o.b. prices set by suppliers of the firm’s inputs; instead, it is affected only by the shipping costs involved. The implication here is that the f.o.b. prices set by suppliers of these inputs are not important in localization: but the prices set by the shipping sector (the unit shipping costs) are. As in earlier chapters, this chapter does not say anything about the distribution of income in society among units of labor, capital, and land. As in Chapter 2, the firm incurs a ubiquitous cost, but this is not related explicitly to labor or land inputs. All income gains that can arise (e.g., because of a decrease in ai or s) accrue to the firm owner in the form of reduced cost (and, implicitly therefore, increased profit). As in Chapter 3, no adjustment of market price is envisaged, and there is no change in the well-being of consumers.
Chapter 7
What the Firm Does On-Site Agglomeration, Insurance, and the Organization of the Firm (Marshall–Lentnek–MacPherson–Phillips Problem)
A firm has a machine that breaks down periodically. In Model 7A, the firm does repairs in-house. The firm minimizes overall cost by balancing an inventory of product (to meet customer demand during downtime) and an inventory of repair staff. In Model 7B, the firm outsources repairs to a contractor as needed. There may now be a delay in starting repairs awaiting the arrival of the contractor’s repair crew. Outsourcing becomes attractive because of an insurance principle. If the client is in an industry using similar machines that break down stochastically over time, a contractor, by redeploying repair staff from client to client, may achieve lower unit costs than a firm doing repairs in-house. A clustering of clients with the same kind of machine enables the repair contractor to be more efficient. In Model 7C, the firm uses cost minimization to choose between in-house repair and outsourcing. Model 7D examines the implications of profit maximization by a repair contractor servicing client firms. This chapter builds on Chapters 4 and 5. In those chapters, a local supply curve was assumed at each place that could have a different intercept. In Chapter 6, I explained that difference in part as a result of the variation in the effective prices of non-ubiquitous inputs. Here in Chapter 7, it is the clustering of clients with the same kind of machine that enables an efficient repair contractor and thereby reduces unit cost for client firms. Model 7D parallels Chapter 6 in terms of how a firm and its supplier (in this case a repair contractor) co-locate. In this chapter, the repair contractor is a metaphor for any kind of producer service that can be outsourced, and the chapter suggests how we might similarly think about shippers and the endogeneity of the price of shipping and unit shipping cost.
7.1 The Marshall–Lentnek–MacPherson–Phillips Problem In the preceding chapter, I viewed the production process in terms that were relatively simple. For much of that chapter, a firm purchased inputs in fixed ratios and produced an output from them. However, in practice, firms choose how to make their product; in Chapter 6 we began to look at the role of substitution among inputs using a Cobb–Douglas production function. Important also, however, are decisions about the organization of a firm (specifically, what gets produced on-site versus J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_7, C Springer Science+Business Media, LLC 2010
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purchased from a supplier). In general, this is the outsourcing problem in manufacturing. In this chapter, I look at decisions about outsourcing, their relationship to agglomeration economies, and the impacts of firm organization on location. Why do firms localize? So far, we have considered stories that emphasize the importance of shipping costs. In Chapter 2, a firm builds a new factory beside a customer if the annual savings in shipping cost is larger than the annual cost of capital associated with construction and maintenance of the new factory. In a similar vein, the models in Chapters 4 and 5 indicate local suppliers will survive at a place only if they are sufficiently efficient compared to competitors located at remote places who must pay an additional shipping cost to compete here. In Chapter 6, a firm will locate beside either a supplier or a customer when the marginal shipping cost associated with that supplier or customer is large relative to the marginal shipping cost associated with other suppliers or customers. In a crude sense, Chapter 6 gives us an initial perspective on firm organization. If a firm decides to locate beside a particular supplier or a particular customer, what we see is colocation of two firms. Imagine, for example, that a bicycle manufacturer locates beside a supplier of bicycle tires. In terms of the cost of shipping, it is as though the firms have together formed a bicycle manufacturing operation with tire production on-site. As a perspective on firm organization, this is crude because we still have two firms (establishments), not one. However, that firms organize themselves (at least in part) to reduce shipping costs is a valuable idea. None of the chapters so far explains why particular organizations of firms might have a production cost advantage that might allow them to overcome a shipping cost disadvantage and, therefore, compete in a remote market. Of course, there might be a specific reason why a particular site is most profitable: e.g., a natural resource input that is not ubiquitous. The purpose of this chapter, however, is to explore a rationale for production cost advantage that arises simply through agglomeration in space. As discussed in Chapter 1, agglomeration economies are thought to be of two principal kinds: localization economies and urbanization economies.1 However persuasive the distinction between localization and urbanization economies in theory, it is difficult to separate these two kinds of economies in practice.2 Part of the problem here is the definition of an industry, which might loosely be thought to be the collection of firms producing the same product or commodities. We can always imagine a cluster of firms wherein each firm produces the same commodity. In the real world, however, firms tend to produce multiple and differentiated commodities, and this undermines the concept of an industry as outlined above and, therefore, the distinction between localization and urbanization economies. What underlies agglomeration economies? Three arguments dominate the literature.
1 See
McCann (1995). empirical evidence on localization economies, see Black and Henderson (1999) and Ó hUallacháin and Reid (1991). 2 For
7.1
The Marshall–Lentnek–MacPherson–Phillips Problem
179
One argument regards lower shipping costs. Agglomeration reduces shipping costs to the extent your customers and/or suppliers are now closer at hand. If the firm purchases inputs from many suppliers—or sells to many customers—nearby, this is tantamount to an urbanization economy. As the number of firms and households in an urban area increase, there is a tendency for a firm’s shipping costs to decrease thus lowering its unit cost. This is the kind of agglomeration economy considered in Chapter 6. A second argument regards indivisibilities and economies of scale. The argument is that economies of scale (including division of labor) in some cases make it desirable to operate with a large factory. Let us think of a particular manufacturing operation: say ballpoint pens. There are numerous elements to the production process, some of which can be done in-house and others that can be purchased (i.e., outsourced). If the firm wants to maximize the rate of return on its investment, it will do those things in-house that are most profitably done there (typically things that require that scale of production locally) and contract out those things that can be done more efficiently elsewhere (taking into account shipping costs). Suppose a simple ballpoint pen consists of a nib, an ink cartridge, a shaft and cap, and a supply of ink. Suppose the manufacturing of the nib and assembly of the pen are the elements that are subject to economies of scale at the size of plant envisaged. The firm would then purchase the other inputs, ship them to the site, manufacture the nib, and assemble the ballpoint pen. In economic analysis, an indivisibility arises whenever a production process cannot be scaled down at the same unit cost. For example, suppose a manufacturing firm relies on a rail terminal to ship its commodity. One firm by itself may not generate sufficient traffic to cover the cost of constructing and maintaining a rail terminal. However, with sufficient firms in the vicinity each demanding rail services, a rail terminal becomes profitable. Another example of an indivisibility was instanced in Chapter 2 where I assumed that the firm, having invested K dollars, could draw upon a limitless wellspring of production. In this sense, agglomeration in geographic space is fundamentally driven by indivisibilities in production. These result in what economists call an externality.3 In effect, each new firm arriving in the agglomeration helps further spread the cost of the railway terminal. In a profound sense, indivisibilities can be thought to be at the heart of economic geography.4 A third argument regards the insurance principle5 first identified by Marshall. The idea here is a firm faces uncertainty about its operations. The demand for its product, for instance, might be up 1 week and down the next. The firm might seek the most efficient, least risky, or most profitable way to operate given uncertainty. 3 A consequence on one economic actor (e.g., a firm) arising from the behavior of another that is not priced. 4 This argument suggests a simple principle. Because of the efficiencies made possible by indivisibilities, a firm will purchase inputs directly or indirectly from a supplier whose scale of production is much larger than their own and sell to firms whose scale of production is much smaller than their own. 5 See also Phillips, MacPherson, and Lentnek (1998).
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When a firm is geographically isolated (i.e., not in an agglomeration), its best recourse may be to create inventories of inputs, of product, and/or to have reserve capacity. When a firm is in an agglomeration, other possibilities arise. For example, a temporary employment agency might be able to supply labor as needed when the demand for a firm’s product increases suddenly. For firms in the locale, the agency operates like a kind of insurance scheme by pooling their individual short-term labor needs. If the demand for the product of one firm is statistically independent of (uncorrelated with) the demand for the product of another local firm, the temporary employment agency can be an efficient way of getting the labor needed in the short run. This chapter builds on a model first presented in Lentnek, MacPherson, and Phillips (1992), hereinafter the LMP Model. The firm in this model maximizes profit: see (7.1.1) in Table 7.1, wherein I summarize equations, assumptions, notation, and rationale for localization in Model 7A. The model assumes that a fixed Table 7.1 Model 7A: the firm doing repairs in-house Firm’s profit π = pqT − (f + cq)T − piIT − wnT
(7.1.1)
Cost to be minimized C = piIT + wnT
(7.1.2)
Average product inventory I = (1/2)qΔT
(7.1.3)
Downtime (fraction of T) Δ=δ
(7.1.4)
Repair time (fraction of T) δ = a + k/n where 1 > a > 0 and k > 0
(7.1.5)
Marginal effect of repair staff on cost dC/dn = −pi(1/2)q(k/n2 )T 2 + wT
(7.1.6)
Cost-minimizing repair staff (where dC/dn = 0) √ n = (kqpiT/(2w))
(7.1.7)
Minimum feasible repair staff (1=1) k/(1 − a)
(7.1.8)
Maximum wage if marginal effect of repair staff on cost is negative at minimum feasible repair staff w ≤ pi(1/2)q(1 − a)2 T/k
(7.1.9)
Notes: Rationale for localization (see Appendix A): Z4—Risk spreading and insurance. Givens (parameter or exogenous): a—Minimum repair time (fraction of production cycle); c—Unit variable production cost; f—Daily fixed cost; i—Interest rate (daily); k—Responsiveness of repair time to size of repair staff; p—Selling price of unit of output; q—Units sold daily; T—Length of production cycle (days) including downtime; w—Daily wage of repair worker. Outcomes (endogenous): C—Repair plus inventory cost to be minimized; l—Average stock of product inventory held-over period; n—Number of repair workers; δ—Time (fraction of T) required for repair; Δ—Downtime (fraction of T); π—Profit.
7.2
Inventory Models in Management
181
(exogenous) daily demand of q units must be met; the model does not allow for unsatisfied demand. Assume also that the fixed cost (f) and marginal cost (c) of production are fixed; therefore the firm’s total cost of production is fixed. The model assumes that the price (p) at which the firms sell its output is given; hence its sales revenue is also given. The LMP Model instances a classic inventory model. It assumes that the firm operates production machinery subject to periodic breakdown. The model assumes the machinery breaks down regularly each T days, ΔT days are required to repair it, and the remaining (1 − )T days is the length of the ensuing production period. Each time while the machinery is being repaired, the firm supplies customers from its inventory of product. Over the repair period, inventory is envisaged to drop steadily to zero see curve AB in Fig. 7.1. When repairs are complete, production resumes and output is used both to satisfy current customer demand and to rebuild inventory in anticipation of the next breakdown: see curve BC in Fig. 7.1. By the end of the production period (OT in Fig. 7.1), inventory has been returned to its original level, and the firm is ready for its next production cycle. As all other costs of production are fixed by assumption, it is only the inventory cost and the repair cost that the firm can vary. The two costs sum to an amount C as shown in (7.1.2). Fig. 7.1 The profit-maximizing firm
AB BC OA OB OT
Inventory holding Firm’s residual inventory during downtime Firm’s inventory accumulation in production Peak inventory (at start of downtime) Downtime Cycle length (T days)
C
Inventory
A
0
B
T Time
7.2 Inventory Models in Management In early studies of inventory within the field of Management, the problem was to derive a policy for inventory to enhance profit. Holding inventory is seen to be
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costly; typically in terms of storage costs (including rent, lighting, heating, cleaning) and of the working capital tied up (i.e., imputed interest cost) in holding inventory. To continue an earlier example, assume an efficient firm that manufactures bicycles. The firm purchases inputs: tires and valves, wheel rims and spokes, axles and bearings, fenders, seat, brake and gear systems, pedals and grips, and tubular steel. Suppose the firm fabricates the frame and handlebar from the steel and then assembles bicycles using purchased components. Suppose also that the firm uses a “just in time” production process wherein suppliers arrive at one factory door with an input exactly when and as needed, and a shipper arrives at another door to take the newly manufactured bicycle immediately to the firm’s customer. In this way, the firm eliminates inventory, minimizes its floor space requirements, and keeps its unit cost low. Compared to this “just in time” production process, why might a firm choose instead to order in advance and then hold inventory of its inputs and possibly of manufactured bicycles as well? Two arguments come to mind. One argument is that the fixed costs of placing an order are substantial so that the larger the order the marginal cost per unit ordered decreases. In an early study in the Harvard Business Review, Wilson (1934) emphasized the role of the ordering amount—an expenditure to restock inventory also known as the economic order quantity (EOQ)—and the ordering point (minimum inventory that triggers a replenishment order).6 Wilson apparently had in mind an inventory cost that summed two components: LA/C + IC/2 where A is annual consumption, C is the amount ordered each time (EOQ), L is the cost of placing an order, C/2 is the average inventory (which runs from C down to zero), and I is the cost of interest, insurance and other loss risks, depreciation, rent, lighting, cleaning, heating and other inventory holding costs per dollar of inventory held.7 The first component decreases as the firm increases its EOQ; the second component increases. To minimize inventory √ cost then, Wilson then set C = (2LA/I).8 However, this raises a problem for this book. If indeed the cost of placing an order is substantial, it is equivalent to saying that there are economies of the large haul. However, I have ignored economies of the large haul for simplicity of exposition elsewhere in this book. Nonetheless, to whoever bears the shipping cost (be it the bicycle manufacturer or its supplier), where I is small relative to L in Wilson’s model, economies of the large haul create
6 The confusion here over how something is labeled (in this case a quantity) and what is meant (in this case, an expenditure) is regrettable. To an economist, expenditure is price times quantity. EOQ predicts expenditure, not quantity. 7 In addition, the firm faces potential costs when customers are unable to get the product when and as needed: e.g., loss of business. Interestingly here, Wilson does not take into account the costs that can arise to a firm because a customer is not able to get the product as needed. Kahn (1992) presents evidence from the retailing of automobiles in the United States to support the argument that stockout avoidance is a major factor in retail inventory decision making. 8 Baumol (1952, pp. 545–547) uses a similar model to look at the cash holdings of households in a fiat money economy.
7.3
Model 7A: The Firm Doing Repairs In-House
183
an incentive to store product somewhere until there is an amount that can be shipped economically. A second argument has to do with uncertainty. The firm might be uncertain about the demand for its bicycles. If demand tomorrow were to suddenly rise, the firm might find its inventory (of fenders, say) gets depleted before the replenishing shipment arrives. A second uncertainty concerns how long it will take the fender supplier to deliver. In other words, the firm may run out fenders in inventory simply because the supplier takes longer to deliver than expected. Whenever the firm runs out of inventory, it cannot satisfy any more demand for bicycles until it gets a supply of the needed parts. Perhaps, customers will just patiently wait for their order to be completed. Perhaps, however, they will start purchasing from another bicycle manufacturer. In such cases, there is a cost to our firm: the present and future profit foregone because of the shortfall in inventory. The firm may choose from among several strategies for dealing with such uncertainty. One of these is to maintain larger inventories of fenders, other bicycle components, and assembled bicycles.9
7.3 Model 7A: The Firm Doing Repairs In-House The LMP model assumes that the firm has staff dedicated to equipment repair. They do nothing else: they sit idle while the machinery is functioning. The firm incurs two costs related to this cycle of repair and production. One is the cost of financing the commodity inventory necessary to bridge the downtime. The average amount of inventory is shown in (7.1.3). The other is the cost of maintaining the necessary repair staff. In this model, repair staff are paid at a daily wage w. As well, they are hired for the entire interval, T, even though they are needed only for the downtime interval ΔT. In the LMP Model, repair and production staff are apparently specialized so that neither can do the other’s job: production staff sit idle while repair staff are working and vice versa. In (7.1.4), the fraction of the time the machinery is down is entirely spent in repair (δ). In a sense, the choice of the firm is between two kinds of inventory—an inventory of product and an inventory of repair labor—each with its own cost. All other costs (e.g., wages for production staff) are assumed to be fixed (i.e., remain the same) regardless of how repairs are handled. The LMP model assumes repair time in (7.1.5) is inversely related to the number of repair workers (n). Here, it is assumed also that n is infinitely divisible: i.e., in addition to n = 1 or n = 2 for example, we can have fractional assignments like n = 0.1 or n = 1.7.10 Note that (7.1.5) is a production function. As such, it has two properties of interest here. First, the more repair staff are employed, the faster 9 Another strategy is to try to smooth out the demand for bicycles: e.g., by contracting regular deliveries to the principal customers in advance. Still another is to negotiate with current or prospective fender suppliers and shippers to ensure a more orderly flow or to share the risks (e.g., consignment). A third alternative is to negotiate consignment or right-to-return clauses in purchasing inventory. 10 Taken in conjunction with (7.1.2), this assumes no possibility that the firm might pay overtime wages to have a job completed more quickly.
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7 What the Firm Does On-Site
the repair. Second, each repair worker added reduces the time to repair by a smaller amount: i.e., (7.1.5) exhibits diminishing marginal productivity or congestion. The effect of the assumptions above is to make C U-shaped in n for the firm. After substituting (7.1.3), (7.1.4), and (7.1.5) back into (7.1.2), I get an expression for cost (C) in terms of the level of repair staff (n). Because of the decreasing returns to scale in repairs (7.1.5) in n, the marginal saving in inventory cost (from having one more repair worker) decreases as n is increased. On the other hand, repair cost rises steadily (proportionally) with n. The implication is that, when n is small enough, C will initially fall as n is increased because the marginal saving in inventory cost exceeds the marginal cost of a repair worker (wT); at a large enough n however, C will begin to rise because the marginal saving in inventory cost is now smaller than wT. C is U-shaped.11 See curve AB in Fig. 7.2. The same argument can be seen by differentiating C with respect to n as shown in (7.1.6): in Fig. 7.2, total cost is minimized at a repair staff of OG. In-house repair Total cost of repair Cost of inventory Cost of repair labor Amount of repair labor that minimizes total cost of repair OI Efficient cost of repair AB CD EF OG
B
A H
I
F
Cost
C
D
E 0
G
Repair staff
Fig. 7.2 Model 7A: in-house repair staff. Notes: Given a = 0.01, k = 1, w = 30, q = 1,000, p = 10, i = 0.00011, T = 200. Outcomes are n = 1.91, T = 106.45, C = 23,198. Horizontal axis scaled from 0 to 6; vertical axis from 0 to 35,000
Therefore, there is a cost-minimizing level of repair staff (n). We can see this as the bottom of the curve (C) labeled AB in Fig. 7.2. Using calculus to minimize C, I set the first derivative equal to zero and solve for n (see OG in Fig. 7.2); the result 11 A
neat trick has been performed here. In Wilson (1934), it takes an economy of the large haul to produce a U-shaped inventory cost curve. In the LMP model, the same result obtains simply by assuming diminishing marginal productivity to repair labor.
7.3
Model 7A: The Firm Doing Repairs In-House
185
is the firm’s demand equation for repair staff: (7.1.7). In this, the level of repair staff varies directly with k, q, p, i, and T and inversely with the wage w. The price elasticity of demand for repair staff here is 0.5; therefore, the demand for in-house repair staff is price-inelastic. Repair time and labor AB OC OT
T
Repair time Minimum amount of labor: k/(1–a) Maximum amount of downtime:T
A
Repair time (days)
Fig. 7.3 Model 7A: repair productivity. Notes: Parameters: a = 0.01, k = 1, T = 200. Horizontal axis scaled from 0 to 5: vertical axis from 0 to 250
B
0 C
Repair labor (n)
In the above, I presume that the amount of repair staff is sufficiently large that downtime is smaller than T (otherwise the firm could not rebuild its inventory before the next breakdown). Since this means δ < 1, therefore, no > k/(1 − a) is the minimum feasible repair staff: see (7.1.8) and amount OC in Fig. 7.3. For the minimum cost solution to be a labor input larger than k/(1 − a), C must be declining at n = k/(1 − a) ; otherwise least cost is the corner solution n = k/(1 − a) . The first derivative in (7.1.6) must be negative at no = k/(1 − a). See panel (a) of Fig. 7.4. In contrast, panel (b) there shows a corner solution. Since the firm’s price elasticity of demand for labor is negative, the derivative will be negative (i.e., no corner solution) provided that the wage (w) is small enough: see (7.1.9). Under these circumstances, as n rises above the minimum feasible repair staff, C initially falls before beginning to rise. Hence, given (7.1.9), the cost-minimizing repair staff is given by (7.1.7). The LMP Model to this point has 7 givens (a, i, k, p, q, T, w); the cost-minimizing n depends on 6 of them. What about comparative statics here? Let us focus here on the impacts on three outcomes: n, δT, and I. See Table 7.2. Assuming (7.1.9), I get: a
Where repair time is larger for any given n, repair staff is unaffected. However, repair time and inventory rise.
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7 What the Firm Does On-Site
B D
B
A
A
Cost
Cost
D
0 C
0
Repair staff
Repair staff
C
(a) No corner solution
(b) Corner solution
Fig. 7.4 Model 7A: corner solutions in the case of in-house repairs. Notes: AB is total cost of repair. OC is the minimum amount of repair labor: that is, k/(1 − a). In panel (a), (7.1.9) is satisfied; in panel (b), it is not Table 7.2 Model 7A: comparative statics of an increase in exogenous variable Outcome Given
n [1]
δT [2]
I [3]
a i k p q T w
0 + + + + + −
+ – + – – + +
+ – + – − + +
Notes: See also Table 7.1. +, Effect on outcome of change in given is positive; −, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome; ?, Effect on outcome of change in given is unknown.
i k
Where it is more expensive to hold inventory, the firm increases the amount of repair staff, while repair time and inventory shrink. Where k is larger, the firm increases the amount of repair staff and downtime and inventory rise.
7.3
p q T w
Model 7A: The Firm Doing Repairs In-House
187
Where the commodity sold is more valuable, the firm increases the amount of repair staff, repair time is reduced, and inventory shrinks. Where a greater quantity of the commodity is sold daily, the firm increases the amount of repair staff and repair time and inventory are reduced. Where the production period is longer, the firm increases the amount of repair staff and repair time and inventory rise. Where each unit of repair staff is more costly, the firm reduces the amount of repair staff. This causes repair time and inventory to rise.
Students sometimes voice concerns about the LMP model. In what respects does the LMP Model seem strange to them? • If a firm knew with certainty the machinery was going to break down every T days, why not keep inventory at zero until the last day of the production period, then rapidly restock inventory just before the breakdown? Wouldn’t that be less costly than slowly rebuilding inventory as assumed here? The reason is that, as is typical of inventory models generally, the LMP Model is a non-stochastic representation of what is essentially a stochastic problem. It is the randomness associated with breakdown that keeps the firm slowly rebuilding inventory; it simply does not know when the next breakdown will occur. • The LMP Model attempts to describe behavior in the face of uncertainty, where one might expect a risk aversion strategy rather than the simple cost minimization strategy used here. • The LMP model considers only a few options for the firm. The model assumes that the cost associated with delaying customer demand is infinite. While it may be true that customers might seek out another supplier if the firm is slow to deliver, it may also be the case that the firm has room to maneuver on the delivery schedule. The firm is assumed as well to have the option only of a specialized repair team and not be able to make use of overtime pay to get the repairs done sooner. • The firm can use other strategies to mitigate the risk associated with machine breakdown. One possibility is to engage in preventive maintenance. A second is to have multiple production lanes in the factory so that a machine breakdown in one lane does not stop production altogether. Finally, as seen in Chapter 3, the firm might have spare capacity in the form of multiple factories, some perhaps more efficient than others, that give it the possibility of still meeting customer needs. These concerns remind us that the LMP model is not without its shortcomings. However, the value of the LMP model is the way that it allows us to get a glimpse of an important role for agglomeration economies. Does Model 7A allow us to say something about the effects of division of labor? Suppose that a firm operates at a scale that, to maximize profit, requires 1 manufacturing machine, 20 production staff, and 2 repair staff. Suppose the firm then experiences a doubling of the demand for its product and finds that this is most
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profitably done with 2 machines, and 40 production staff. If the 2 machines break down independently, and downtime is a relatively small fraction of the production period, the firm might find the most profitable size of repair staff was still only 2 persons. In that case, the firm is operating more efficiently overall (because it did not need to double its repair staff). However, I think this is better seen as a consequence of the indivisibility of repair (wherein the worker is idle until the next breakdown) rather than a division of labor wherein the firm gets more productivity from workers by having them do more specialized work.
7.4 Model 7B: Outsourced Repairs Now, introduce the possibility that the firm can outsource this repair activity. The specific problem addressed by the firm in the LMP Model is the choice between (1) maintaining one’s own in-house repair staff as discussed above and (2) outsourcing of repair work. In the model, the choice is either-or; the firm chooses one and only one of them. This assumption simplifies the analysis. In reality, we can imagine a range of strategies. These might include (1) regular manufacturing employees also trained to do repairs, (2) having a small core repair staff that are complemented with outsourced workers when needed, and (3) renting repair staff to others at times when idle at the firm. Now, consider the firm as outsourcer purchasing repair services at a given daily rate per worker (wo ) from a contractor. See Table 7.3. I use a subscript “o” to denote variable and parameter values specific to the firm that outsources. The cost to be minimized, Co , is shown in (7.3.1). The average product inventory is shown in (7.3.2), the downtime in (7.3.3), and the repair time in (7.3.4). Assume here the same repair time function as for in-house repair staff. Taken in conjunction with (7.3.1), this assumes that the firm can choose how many repair workers to call up. Further, with outsourcing, the firm now finds that it has to wait ωT days for the contractor to arrive before repairs commence. This wait time is like a unit shipping cost; it is a direct cost of not having the repair staff in-house. In addition, the daily rate per repair worker (wo ) is presumably higher than the daily wage (w) paid by a firm doing in-house repairs. Presumably, these two disadvantages of outsourcing are offset by one major advantage: the outsourcing firm pays for repair labor only for the length of the repair (δ o T), not the whole production cycle (T) as does the firm doing repairs in-house. As in the case of the firm doing repairs in-house, I can derive an expression for the outsourcing firm’s repair plus inventory costs. After substituting (7.3.2), (7.3.3), and (7.3.4) back into (7.3.1), I get an expression for cost (Co ) in terms of the level of repair labor (no ). Co is u-shaped in no . The outsourcing firm finds that its repair costs rise, and its inventory costs fall, with the amount of repair services used daily (no ). See curves CD and EF Fig. 7.5. Because of the decreasing returns to scale (7.3.4) in no , eventually Co begins to rise following repair labor costs: see curve AB. Put differently, I can use calculus to differentiate Co with respect to no , yielding (7.3.5). Because the slope (dCo /dno ) is negative for small no and becomes positive for large
7.4
Model 7B: Outsourced Repairs
189
Table 7.3 Model 7B: the cost-minimizing firm: repairs outsourced Cost to be minimized Co = piIo T + wo no δo T
(7.3.1)
Average product inventory Io = (1/2)q o T
(7.3.2)
Downtime (proportion of T) Δo = w + δo
(7.3.3)
Repair time δo = a + k/no where 1 > a > 0 and k > 0
(7.3.4)
Marginal effect of repair labor on cost dCo /dno = −pi(1/2)qT 2 (k/n2o ) + wo Ta
(7.3.5)
Cost-minimizing repair labor (where dCo /dno = 0) √ no = (kqpiT/(2wo a))
(7.3.6)
Minimum feasible repair labor (Δ = 1) k/(1 − a − w)
(7.3.7)
Maximum daily rate if marginal effect of repair labor on cost is negative at minimum feasible repair labor wo < pi(1/2)qT(1 − a − w)2 /(ak)
(7.3.8)
Notes: Rationale for localization (see Appendix A): Z4—Risk spreading and insurance; Z8— Limitation of travel delay. Givens (parameter or exogenous): a—Minimum repair time (fraction of T); i—Interest rate (daily); k—Responsiveness of repair time to size of staff; p—Selling price of unit of output; q—Units sold daily; T—Length of production cycle (days) including downtime; wo —Daily fee per repair worker contracted; ω—Wait time until arrival of repair contractor (fraction of T). Outcomes (endogenous): Co —Repair plus inventory cost to be minimized; lo —Average stock of product inventory held-over period; no —Number of repair workers; δ o —Time (fraction of T) required for repair; Δo —Downtime (fraction of T).
no , Co is U-shaped. To minimize Co , I set the first derivative equal to zero and solve for no ; the result is the firm’s demand equation for repair labor: (7.3.6). In this, repair staffing varies directly with k, q, p, i, and T, and inversely with wo and a. The price elasticity of demand for repair labor here is 0.5: therefore, demand for outsourced repair labor is price inelastic. It is interesting to compare the least cost use of repair labor under outsourcing in (7.3.6) with the least cost use of labor for the in-house firm (7.1.7). The numerators are the same in the two equations. However, the denominator in (7.3.6) incorporates wo a as opposed to w in (7.1.7). This arises because w is what the firm pays daily for in-house repair workers; in contrast wo a is the minimum amount the firm pays daily on average for each outsourced worker. In the above, assume the amount of repair labor is sufficiently large that downtime is smaller than T. Since this means δ < 1, therefore, no > k/(1−a−ω) is the minimum feasible repair staff: see (7.3.7). For the minimum cost solution to be a labor input larger than k/(1−a−ω)—that is, no corner solution—the first derivative in (7.3.5) must be negative at no = k/(1−a−ω) . Since the firm’s price elasticity of
190
7 What the Firm Does On-Site Outsourced repair AB CD EF OG
A
Cost
OI
Total cost of repair Cost of inventory Cost of repair services Amount of repair labor that minimizes total cost of repair Efficient cost of repair
B
H
I
F C E
D 0 G
Repair labor
Fig. 7.5 Model 7B: outsourced repair staff. Notes: Given w = 0.05, a = 0.01, k = 1, wo = 100, q = 1,000, p = 10, i = 0.00011, T = 200. Outcomes are no = 10.49, o T = 31.07, Co = 25,515. Horizontal axis scaled from 0 to 25; vertical axis from 0 to 50,000
demand for labor is negative, the derivative will be negative provided the wage (wo ) is small enough: see (7.1.9). Under these circumstances, as no rises above the minimum feasible repair staff, Co initially falls before beginning to rise. Hence, given (7.3.8), the cost-minimizing repair labor is given by (7.3.6). What about comparative statics in Model 7B? The LMP Model to this point has 8 givens (a, i, k, p, q, T, w, ω). Comparative statics are largely the same as in Model 7A above. What is new here is the wait time parameter, ω. Assuming (7.3.8), I get the following additional comparative statics. ω
Although, when ω is increased, downtime is larger for any given no , repair labor is unaffected. Down time rises; hence a need for more inventory.
7.5
Model 7C: The Decision to Outsource
191
7.5 Model 7C: The Decision to Outsource Suppose a firm has the choice between outsourcing and in-house repairs. Assume here that the firm minimizes cost. It would, therefore, choose outsourcing if C − Co > 0, in-house otherwise. From (7.1.2), I can derive an expression for C at the cost-minimizing level of n: see (7.4.1) in Table 7.4. From (7.3.1), I can derive an expression for Co at the cost minimizing level of no : see (7.4.2). The difference between these two costs is given in (7.4.3). The cost saving from outsourcing will be larger when w is higher or when wo , a, or ω are lower. Table 7.4 Model 7C: in-house versus outsourced repairs assuming no corner solutions In-house cost, from (7.1.2), after substituting (7.1.5) and (7.1.7) √ √ C = (1/2)piqaT 2 + ( 2)T (kqpiTw)
(7.4.1)
Outsource cost, from (7.3.1) after substituting (7.3.4) and (7.3.6) √ √ Co = (1/2)piq(w + a)T 2 + wo Tk + ( 2)T (kpqiTwo a)
(7.4.2)
Cost savings from outsourcing (negative if in-house less expensive) √ √ √ C − Co = T (2kqpiT)( w − (wo a)) − (1/2)piqwT 2 − wo Tk
(7.4.3)
Cost advantage to outsourcing if √ w > ((1/2)piqwT + wo k + (2kqpiTwo a))2 /(2kqpiT)
(7.4.4)
Notes: See also Table 7.1. Rationale for localization (see Appendix A): Z4—Risk-spreading and insurance; Z8—Limitation of travel delay. Givens (parameter or exogenous): a—Minimum repair time (fraction of T); c—Unit variable production cost; f—Daily fixed cost; i—Interest rate (daily); k—Responsiveness of repair time to size of staff; p—Selling price of unit of output; q—Units sold daily; T—Length of production cycle (days) including downtime; w—Daily wage of repair worker (in-house); wo —Daily fee per repair worker contracted (outsourced); ω—Wait time until arrival of repair contractor (fraction of production cycle). Outcomes (endogenous): C—Repair plus inventory cost to be minimized (in-house); Co —Repair plus inventory cost to be minimized (outsourced); l—Average stock of product inventory held-over period; n—Number of repair workers; Δ—Downtime (fraction of T); δ—Time (fraction of T) required for repair; Δo —Downtime (fraction of T); δ o —Time (fraction of T) required for repair.
One way to think about C − Co > 0 is that for the in-house option to be rejected the daily wage for in-house staff (w) must be relatively high. I can rearrange (7.4.3) to show that C −Co > 0 implies (7.4.4). The tradeoff is illustrated in Fig. 7.6, which shows combinations of in-house daily wage (w) and outsourced daily rate (wo ). In the area below the curve AB are the combinations of w and wo where in-house repair is chosen because it is less costly (assuming wait time is zero); in the area above the curve, outsourcing is less costly. In Fig. 7.6, the curve CD shows the corresponding tradeoff between w and wo when wait time is 10% of T. In fact, there is a family of curves here: one for each possible value of ω. The greater the wait time, the lower the outsource rate (wo ) tolerable for any given ω. A caveat is in order here about the treatment of uncertainty in this model. As noted above, the LMP Model uses a cost-minimization approach. However, I have already argued above that the LMP model is an attempt to think about the firm
192
In-house versus outsourced repairs
Outsourced daily rate (wo)
Fig. 7.6 Model 7C: the decision to outsource. Notes: Given a = 0.01, k = 1, w = 30, wo = 100, q = 1,000, p = 10, i = 0.00011, T = 200. Horizontal axis scaled from 0 to 200; vertical axis from 0 to 275
7 What the Firm Does On-Site
AB Tradeoff curve when ω = 0.0 CD Tradeoff curve when ω = 0.1 EF Cost of repair services OE Given in-house wage rate OG Given outsourced labor rate
B D
In-house chosen
G
A C 0
F
E
Outsourcing chosen
In-house daily wage (w)
coping with uncertainty. How might our analysis have differed if I had assumed instead a strategy of risk aversion? In the LMP model, the only factor that the firm can vary is the size of the repair staff/labor. The firm presumably might be expected to choose a repair staff/labor to offset stochastic variation in any part of this problem (e.g., the time between breakdowns, the quantity demanded daily, or the time to complete repairs). Given a loss attached to each possible outcome, a risk-averse firm with either in-house repair staff or outsourcing might want to think about the relative losses associated with overshooting as opposed to undershooting. In this regard, it is instructive to compare Fig. 7.5 with Fig. 7.2. There are similarities. In both diagrams, inventory cost falls off quickly, asymptotically approaching zero, as repair staff/labor is increased. In both cases, total cost, be it C or Co , initially decreases with an increase in repair staff/labor; then begins to increase. However, there is one important difference here. In Fig. 7.2, as I increase or decrease in-house repair staff (n), the repair cost (wnT) changes proportionally. In Fig. 7.5, as I increase or decrease the amount of outsourced repair labor (no ), the repair cost (wo no δo T) changes only modestly. This is because an increase in no is offset in part by a decrease in δ o . There is no corresponding effect in the in-house repair wage bill. Although Co is U-shaped like C, it has a relatively modest upturn for repair labor above the least cost solution (7.3.6).
7.6
Model 7D: The Advantage of Agglomeration
193
Now, let us focus on the choice of n facing an in-house firm. Suppose the inhouse firm chooses a level of n using (7.1.7): based on expectations about time between breakdowns, quantity demand, time to repair, and so on. Suppose that one or more of these expectations turns out to be incorrect so that, in fact, some other level of repair staff is cost-minimizing. In its chosen n, the firm may incur a loss—a loss by either undershooting or overshooting the correct cost-minimizing n. In terms of Fig. 7.2, the chosen n lies either to the left or the right of the cost-minimizing n. Either way, the firm finds itself with a loss: spending more than it needs for inventory and repair costs. However, since C rises somewhat less quickly to the right of the cost-minimizing n than it does to the left, a risk-averse firm might prefer to err on the side of overshooting (as opposed to undershooting) n. Let us now look at the outsourcing firm. Compare Fig. 7.2 with Fig. 7.5. From Fig. 7.2, the loss from overshooting—that is, the additional cost arising because I had more than the efficient size of repair staff—is much higher than the loss from overshooting in Fig. 7.5. In Fig. 7.5, there is only a small increase in cost with having repair labor in excess of the cost-efficient amount. If the cost of outsourcing is only modestly greater than the cost of in-house repairs, the risk-averse firm would prefer outsourcing for this reason. In this section, I have referred to the ideas in terms of outsourcing. Cast differently, the ideas equivalently are about in-house production. When a firm decides to do something in-house, rather than use a contractor, it can be said to instance vertical integration. In the great industrial restructuring, downsizing, and globalization of the past three decades or so, large firms have looked to outsourcing to help them both to reduce costs or risks and focus on their core (most profitable) business activities.12
7.6 Model 7D: The Advantage of Agglomeration Suppose our firm is the only business enterprise in the region and that there is no contractor from whom repair services can be purchased as needed. In this case, the firm presumably must maintain its own in-house repair staff. Suppose now the manager of the repair team approaches the firm to suggest that the firm spin off the repair team and contract it to provide repair services when and as needed. But, the firm might reasonably ask, what is the advantage of this? If the repair team is efficiently managed within the firm, how might the firm be better off with a contractor here? “Ah”, students often say, “what about labor savings?” After all, if the firm needs to pay only for repair services when needed, won’t it save money? The problem with this argument is that the contractor would not be able to retain repair staff unless it was able to pay them appropriately. If the repair staff needed only 20% of
12 At
the same time, vertical integration thins markets for inputs. Every firm that chooses an inhouse repair staff risks undermining the efficient provision of contractor repairs for firms that prefer to outsource. See, for example, McLaren (2000).
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7 What the Firm Does On-Site
the time, for example, then the daily wage rate charged by the contractor would have to be at least 5 times the daily wage paid to in-house repair workers. You might be tempted to argue that the contractor might instead seek out part-time workers, people who would be happy to work only 20% of the time at the standard wage. Would it not then be feasible for a contractor to be more efficient than the firm at doing repair work? Of course, the answer here is “no”; after all, if a contractor can find part-time workers then what is to stop the firm from hiring part-time workers itself. The contractor has no inherent advantage here. Now assume that, in the vicinity, additional identical firms each require the same repair services under identical circumstances: m firms in all. Suppose further the machinery at each firm breaks down at a schedule uncorrelated with breakdowns at other firms. In this case, an efficiency arises to the extent that a repair contractor can better utilize the available resource staff than can any of the client firms in isolation. Rather than remain idle when not serving the first client, the contractor can now use staff to repair machinery at another client. Effectively, this means an efficiency advantage in that, for the repair contractor, the cost of its repair staff is being shared among the client firms. Under the assumption that C−Co > 0, the demand for repair services by each firm is given by (7.5.1): see Table 7.5. The aggregate demand for the m client firms is given by (7.5.2). Total contractor revenue can then be calculated using (7.5.3).
Table 7.5 Model 7D: the contractor with m identical cost-minimizing customers Demand for repairs by each firm, assuming (7.4.4) no δo T = no (a + k/no )T = kT + aTno
(7.5.1)
Aggregate demand for repairs from m identical firms mno δo T = mno T(a + k/no ) = kmT + amTno
(7.5.2)
Total revenue for repair contractor R = mTwo no δo = mTwo no (a + k/no ) = kmTwo + amTwo no
(7.5.3)
Repair production (delay) ω = 1 − e−(γ mno δo /N)
(7.5.4)
Net revenue NR = mTwo no δo − TwN
(7.5.5) ∗)
Profit-maximizing price (wo for monopolist repair contractor √ √ √ (k)wo∗+( (2kqpiTa)) wo∗ +((1/2)piqwT − (2kqpiTw)) = 0
(7.5.6)
Notes: See also Table 7.1. Rationale for localization (see Appendix A): Z4—Risk-spreading and insurance; Z8—Limitation of travel delay. Givens (parameter or exogenous): a—Minimum repair time (fraction of production cycle); i—Interest rate (daily); k—Responsiveness of repair time to size of staff; m—Number of client firms; p—Selling price of unit of output; q—Units sold daily by client firm; T—Length of production cycle (days) including downtime; w—Daily wage rate; γ —Congestion parameter for repair firm. Outcomes (endogenous): no —Number of repair workers; R—Total revenue of repair contractor; wo —Daily fee per repair worker contracted; δ o —Time (fraction of T) required for repair; ω—Wait time until arrival of repair contractor (fraction of T).
7.6
Model 7D: The Advantage of Agglomeration
195
If machine breakdowns occur stochastically, and are independent between pairs of firms, then in general the contractor could service several firms with just one repair team if δ o is sufficiently small: i.e., mδo T is much smaller than T. However, if machines break down simultaneously at two or more client firms, the contractor might have to respond by splitting a repair team or making one firm wait until repairs are finished at another. In either case, downtime will be longer. Each client firm must then assess whether the risk of such delay (the disadvantage of contracting) is sufficiently offset by the advantages of contracting. To incorporate this, assume now a simple production function for repair services in which the contractor hires N workers (no other inputs) and spreads them over time among the m firms such that the delay in servicing a client firm is the same amount, ω. See (7.5.4). Here, the parameter γ emulates the effect of congestion to the extent that client firms seek repair services at the same time. When γ is near zero, wait time is near zero regardless of N. When γ is large, ω is greater for any given N. With this, I am now able to write down an equation for the contractor’s profit: see (7.5.5).13 Suppose now that the repair contractor is a monopolist. Remember that a client firm here also has the option of doing repairs in-house. Because the client firms each have a price-inelastic demand, the contractor sets its profit-maximizing repair fee (w∗o ) just low enough to deter each of the m client firms from doing repairs inhouse: see the quadratic expression for w∗o in (7.5.6). Finally, the contractor then adjusts its use of labor, N, to maximize its profit. See quantity OC and the profit curve AB in Fig. 7.7. Presumably, client firms are spread out in two-dimensional geographic space. The repair firm’s problem then in part is to determine a service area to be served: an area outside of which servicing clients would necessitate travel by a repair team that would add a substantial delay in serving all clients. Because client demand is price inelastic, the repair contractor maximizes profit by setting the same effective price for repair service for all clients within the service area. So, in this case, uniform pricing within the service area is a consequence of profit maximization. Put differently, the repair contractor absorbs the costs of travel for the repair crew; because demand is price inelastic, the repair firm sets its price (hourly rate at a corner solution above which clients would do the repairs themselves. To an untrained eye, it might look here like geography (travel costs here) does not matter. That is true in the sense that the price of repair services is invariant over the service area. What geography (again, travel costs) does affect is the size of the repair contractor’s service area. The insurance principle implies that the production of repairs will become more efficient as the number of clients (m) increases. Because I have assumed a monopolist contractor, that efficiency takes the form of increased profit for the contractor rather than a reduction in the price of repair services to client firms. That increased
13 In
(7.5.5), I have assumed that the repair contractor has no fixed costs. I had assumed the same thing in the case of the firm doing repairs in-house. However, suppose instead that the repair contractor does incur a fixed cost. In that case, there would be a minimum number (m∗ ) of client firms below which contracting would not be profitable.
196
7 What the Firm Does On-Site Profit of repair contractor
D
Contractor's profit per client (NR/m)
E
AB Profit as function of size of workforce OC Workforce that maximizes profit of repair contractor OE Maximum profit of repair contractor
A
0 C
Contractor's workforce (N)
B
Fig. 7.7 Model 7D: contractor’s profit. Notes: Given a = 0.01., k = 1, w = 30, q = 1,000, p = 10, i = 0.00011, T = 200, m = 10, γ = 0.1. Outcomes are N = 7, NR/m = 16,68. Horizontal axis scaled from 0 to 60; vertical axis from 0 to 20,000
profit is best illustrated by setting γ =0 locally; i.e., where an increase in the number of client firms has no effect on wait time. The repair contractor provides the kind of insurance function Alfred Marshall envisaged and constitutes a kind of agglomeration economy. It is not necessary to have a contractor to realize this kind of efficiency. Instead, one firm might hire its own repair team and then rent out its team to other firms as available and needed. The important notion here is that the efficiency can arise regardless of the organization of production. The contractor merely exemplifies how the efficiency might be implemented. As well, note that the economies of scale implicit in (7.5.4) gradually run out. What happens then as the number (m) of client firms in the vicinity continues to rise: i.e., as the urban agglomeration grows in size. The repair contractor would like to retain all these client firms, but the possibility of a new contractor entering the market must now be considered. Presumably, a new contractor entering the market
7.8
Final Comments
197
reduces the profit of an existing contractor by stripping away customers and pushing down the price of repair services. It is at this stage that client firms begin to experience the advantages of agglomeration: i.e., a lower price for repair services. I consider further the impact of competitors on price in Chapter 8. In the model here, the contractor becomes ever more efficient. In that sense, the contractor is like the monopolist in Chapter 2 who experiences only a decline in average cost as output is increased. Presumably, however, some kind of congestion sets in above a particular level of output, and this opens the way for competitors to enter the business. I do not explore this idea further, as my objective is principally to show the tradeoff between in-house and outsourced production.
7.7 How Far Away Can the Contractor Be? As noted above, (7.4.4) must be satisfied for a contractor to be competitive with in-house repair. This effectively puts an upper limit on ω, the time the customer has to wait before the repair team arrives, and a lower limit on no , the size of the repair team to be then available. For the contractor, the time delay depends both on (1) the delay until the service team can be freed from the current job and (2) the time needed for a repair team to travel from their current location to the new customer. With the obvious implication for its wage bill, the contractor can overcome this time delay by increasing its total repair staff, N. Nonetheless, there will be a travel time beyond which the total profit of the contractor begins to decline. What combination of N and wo would the contractor choose? In addition to the model parameters discussed so far, the answer will depend on the likelihood function for machine breakdowns among customers. We also have to think about the nature of the contractor’s business. As I see it, the contractor makes a short-term investment in repair staff (by hiring them for a period of time: e.g., a day, week, month, or year) then rents them out as needed to clients. It makes its profit to the extent that the risk of machine breakdown across clients is not systematic (i.e., uncorrelated). However, it is beyond the scope of this book to undertake such a model here.14
7.8 Final Comments In this chapter, the principal models have been 7C (the client firm choosing in-house versus outsourcing) and 7D (the repair contractor). I included Models 7A and 7B to help readers better understand aspects of Model 7C which in turn helps us to understand 7D. In Table 7.6, I summarize the assumptions that underlie Models 7A through 7D. Many assumptions are in common to all models: see panel (a) of Table 7.6.
14 An interesting possibility here would be the approach of Dana and Spier (2001) in their study of
vertical control in the video rental industry.
198
7 What the Firm Does On-Site Table 7.6 Assumptions in Models 7A through 7D
Assumptions
7A [1]
7B [2]
7C [3]
7D [4]
(a) Assumptions in common A1 Closed regional market economy B1 Exchange of repair labor/services for money G1 Uncertainty implicit in model I4 Diminishing returns to labor I5 Delay for equipment repair
x x x x x
x x x x x
x x x x x
x x x x x
x x
x x x
x x x x
(b) Assumptions specific to particular models C1 Fixed demand locations D2 Firm minimizes cost of production and shipping I6 Production delay awaiting arrival of repair contractor I7 Choose in-house or outsourced repairs A3 Punctiform landscape B1 Market for the exchange of repair service for money C8 Other demand curve E5 Firm bears shipping cost to market(s) H2 Location(s) of firm given
x x x x x x x
In practice, a firm may use various kinds of contractors. Repair services are an interesting example, but just one of the possibilities here. Another possibility for example is the independent shipper. While some large firms do have their own inhouse fleets (be they planes, ships, rail, or trucks) to ship their own materials and commodities, many firms in practice rely on contractors in the shipping business. As the density of firms in need of shipping services increases within a particular geographic locale, presumably the typical shipper becomes more efficient for the same reasons as we saw above in the case of repair services. The shipper looks for the most profitable amount of transportation equipment and staff (e.g., drivers) and in so doing trades off the economies possible because customer demands are not perfectly correlated with one another against the cost of any delay experienced by customer firms. If the need for transportation services is sufficiently large to foster the establishment of a number of shippers in the vicinity, presumably, the economies achieved by the shippers will, as a result of competition among shippers, lead to lower shipping rates and form an urbanization economy for the client firms.15 In Chapter 1, I argue that prices are important in shaping localization. As in earlier chapters, I continue to assume that the prices of inputs (including shipping services) used by the firm are exogenous; input prices are all determined in markets outside the scope of the models in this chapter. As well, the models in this chapter are silent on prices in output markets. However, in beginning to think explicitly 15 Let
me add a cautionary note here. The advantage of focusing on the repair contractor in this chapter is that this is a clear example of an application of the insurance principle. In the case of shippers, there is some risk in shipping; however, my guess is that there are also substantial indivisibilities and economies of scale to the shipping business.
7.8
Final Comments
199
about the nature of contracting, I have introduced the idea that a sufficient number of client firms nearby makes it profitable for a contractor to locate there. In so doing, the contractor may well affect the price of repair services for firms. For the first time in this book, there is the possibility of simultaneous determination in the prices of two distinct commodities: the good manufactured by the firm and the service provided by the repair contractor. This is also the first chapter in the book to include specifically a (repair) labor input. In principle, this could mean that the chapter might be able to say something about the distribution of income in society among units of labor and capital, if not land. However, in several of the models presented in this chapter, the wage rate for repair staff is exogenous. Whether the firm decides to employ many repair staff or just a few, the wage paid remains the same. Therefore, all cost savings that can arise (e.g., because of a decrease in a or k) accrues to the firm owner: implicitly increasing profit. As in Chapters 3 and 5, no adjustment of market price is envisaged, and there is no change in the well-being of consumers. Also, this is the first chapter to address the question of what the firm does and where. Under certain circumstances, the firm finds it efficient to have its own repair staff; under other circumstances it does not. What gets done on-site is now endogenous. I characterized it as a tradeoff between the daily wage paid to in-house repair staff and the daily rate charged by the repair contractor. However, in important ways it is more than this. Eliminating in-house staff also means savings in floor space, in training costs, and in management of the repair staff. Nonetheless, the models in this chapter are helpful in thinking about issues confronting firms as they decide which activities to do in-house.
Chapter 8
Staking Out the Firm’s Market Price and the Geometry of Competition (Market Area Problem)
A firm typically has a market area: a geographic area wherein the firm dominates and does much of its sales. Within its market area, a firm is able to affect the price received to the extent that the effective prices from other suppliers make them uncompetitive. What determines the size and shape of a market area? How might the presence of a market area affect firm behavior? Model 8A considers market area when a firm and its competitor sell at the same f.o.b. price. Model 8B looks at market area when firms have different f.o.b. prices. In 8C, the firm sets a price that maximizes its profit assuming that competitors do not react. Model 8D studies how the firm’s market area boundary adjusts to capacity constraints. Model 8E shows how the firm’s market area boundary varies when it sells a different but perfectly substitutable good. Models 8F, 8G, and 8H introduce differences among consumers as well as imperfectly substitutable goods. Chapter 1 argues that there are important linkages between prices and localization. I illustrated that idea in Model 2D wherein price and localization were joint outcomes for the monopolist. However, since that chapter, the models in this book have been concerned only with how prices affect localization. This chapter considers how a firm sets its price in response to the proximity of competitors and the prices they set. This chapter explores how, as a consequence, localization and price are jointly determined.
8.1 The Market Area Problem My focus in this chapter is on the geometry of market areas in the presence of competition and its relationship to the linkage between price and localization.1 There is a long history of research in competitive location theory from the classical work of Fetter through contemporary work on destination choice modeling.2 As we have already discussed, in Economics, a market is a locus where buyers and sellers
1 As this book does not deal with optimal location theory, I exclude optimal market areas here. See, for example, Hsu (1997). 2 On contributions to market areas by economists, see Bacon (1992), Eswaran and Ware (1986), Fetter (1924), Greenhut (1952a, 1952b), Greenhut and Ohta (1975), Hartwick (1973a), Hyson
J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_8, C Springer Science+Business Media, LLC 2010
201
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Staking Out the Firm’s Market
intersect. Economists normally—and perhaps wisely—stop here. Those from other disciplines might prefer to augment this definition with additional concepts: e.g., including institutions and mechanisms operating at a site or portal through which offerings of a commodity can be viewed and information about prices is readily available. As has been shown in the past few chapters, a market may also have a geographic area insofar as buyers and/or sellers come from different places to participate in this market, or as a commodity gets shipped for sale elsewhere. The concept of a market area has been widely used: e.g., in Development Studies, Economics, Education Studies, Geography, Health Studies, Housing Studies, Management Science, Marketing and Retailing, Public Finance, Planning, Real Estate, Regional Science, Transportation, and Urban Studies.3 To exemplify, this includes studies that focus on (1) the customer area served by a firm, school, hospital, or other institution or facility; (2) the area from within which an employer draws its workers; and (3) the area within which dwellings or properties are thought to form a housing or property market or submarket.4 In keeping with this book’s focus on the firm, I emphasize here models that look at the customer area served by a firm. I start with the notion of a market area for a commodity produced by a monopolist and sold at an f.o.b. price. The firm is limited only by its range, that is, the market area as geographic area outside of which a prohibitive unit shipping cost means that the consumer would sooner go without the commodity altogether (whatever the reasons). Here, I retain the assumption of a linear individual inverse demand curve with its implication of a price above which consumers no longer demand the good. I then consider the monopolist who uses discriminatory pricing to extract more profit and show that here too market area is limited; it is not profitable to sell the commodity beyond some maximum distance. I then consider the market area for a firm—be it producer, supplier, or retailer— when there are other vendors nearby selling a good that is the same or similar (substitutable). In such cases, the market area of the firm may be further limited by the presence of competitors or other alternative establishments. Looking at where the firm locates and the price that it sets in response to the prices and locations of competitors gives us new insights into the linkage between the price of a commodity
and Hyson (1950), and Stern (1972a). On the modeling of destination choice, see Ghosh and McLafferty (1987), Lo (1990, 1991a, 1991b, 1992), and Miron and Lo (1997). 3 Geographers and other regional scientists who have written about the analysis of market area include Batty (1978), Boots (1980), Daly and Webber (1973). Epping (1982), Erickson (1989), Fotheringham (1988), Gillen and Guccione (1993), Golledge and Amedeo (1968), Golledge (1967, 1970, 1996), Huff and Jenks (1968), Lentnek, Harwitz, and Narula (1988), Lo (1990, 1991a, 1991b, 1992), McLafferty and Ghosh (1986), Miron and Lo (1997), Mu (2004), Mulligan (1982), O’Kelly and Miller (1989), Parr (1995a, 1995b, 1997a, 1997b), Pitts and Boardman (1998), Pred (1964), Rushton (1971a), Sheppard, Haining, and Plummer (1992), Solomon and Pyrdol (1986), Stimson (1981), and Wilson (1967). 4 See, for example, Pate and Loomis (1997).
8.1
The Market Area Problem
203
and localization. The literature in this part of location theory has a history that traces back to Hotelling (1929) and Lösch (1954) and that has benefited from advances in game theory.5 Models of the market area for a firm—in the presence of competitors—come in two varieties. In one variety (as in Models 8A through 8C below), the model assumes we know precisely which customers a firm will serve; in a simple case, the firm captures all potential customers within its market area and none outside. Where, for example, competitors price f.o.b. and lowest effective price is the sole determinant of customer choice, the firm’s market area is the geometric shape within which no other competitor is able to sell their product. In the other variety (as in Models 8D through 8G), the customer chooses a firm (supplier) based at least in part on considerations outside the realm of the model. Taking a model in which the only explanatory variable is the firm’s f.o.b. price, Hotelling (1929, p. 41) argues that (1) in spite of moderate differences of price, some purchasers of a commodity buy from one seller, some from another, and (2) If the supplier of a good gradually increases his price while his rivals keep theirs fixed, the diminution in his volume of sales will take place continuously rather than abruptly. In either variety, a range of models of competitive behavior is possible here. In a simple model, we might assume a firm and its competitors—with geographic locations given—each offering a given good (or variety of goods) at a given f.o.b. price and a given level of service. Customers who bear the unit shipping cost then choose from among the firms. In more sophisticated models, we might imagine the firm and its competitors adjusting their prices, geographic locations, the kinds of goods they offer for sale, and the quality of service they provide. Hotelling (1929, p. 44) argues that when a seller increases price he will only gradually lose business to his rivals. Some customers will still prefer to trade with him because he is more convenient, provides better service, or sells goods they desire. In Hotelling’s view, such customers make every vendor a monopolist within a limited geographic area, and there is no monopoly that is not confined to a limited geographic area. In any of
5 Important early work in the field also includes Lerner and Singer (1937) and Smithies (1941). Other work that largely predates the application of game theory include Capozza and Van Order (1978), Devletoglou (1965), Eaton and Lipsey (1975, 1979), Gannon (1972, 1973), Harker (1986), Mills and Lav (1964), Salop (1979), and Webber (1974). For game theory applications, see Anderson, de Palma, and Hong (1992), Anderson, Goeree, and Ramer (1997), Basu (1993, chap. 8), Benassi and Chirco (2008), Bester, de Palma, Leininger, Thomas, and von Thadden (1996), Boyer, Laffont, Mahene, and Moreaux (1995), Braid (2008), Damania (1994), De Frutos, Hamoudi, and Jarque (2002), Dorta-González, Santos-Peñate, and Suarez-Vega (2005), Economides (1993a, 1993b), Gabszewicz and Thisse (1986b, 1992), Goettler and Shachar (2001), Gupta, Pal, and Sarkar (1997), Gupta, Lai, Pal, Sarkar, and Yu (2004), Hamilton and Thisse (1992), Hay (1976), Huang (2009), Huck, Müller, and Vriend (2002), Irmen and Thisse (1998), Isard and Smith (1967), Iyer and Seetharaman (2008), Lambertini (1997), Lederer (1994), Lederer and Hurter (1986), Matsumura, Ohkawa, and Shimizu (2005), Osborne and Pitchik (1987), Pires (2005, 2009), Prescott and Visscher (1977), Seim (2006), Selten and Apesteguia (2005), Shaked and Sutton (1982), Stuart (2004), Tabuchi (1994), Tabuchi and Thisse (1995), and Zhu and Singh (2009).
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these models, imagine a spatial equilibrium exists in which no firm has an incentive to change its behavior. In the presence of unit shipping costs, effective price might vary across the landscape; so too might localization (the density of competitors nearby) as well as the kind of goods and level of service locally. In all of these models, the firm is competing in a world of differentiated goods, even if the differentiation is solely on the basis of unit shipping cost. Such ideas help us think about what might be causing Hotelling’s stability of competition. To exemplify stability, Hotelling uses a bounded linear market along which customers are spread at a given uniform density. Two competitors at given locations along this line compete for customers by each setting a f.o.b. price. In Hotelling’s model, there was no inherent reason for customers to patronize one particular store other than a lower effective price. However, in a linear market each competitor has a protected flank: the customers between him and the end of the line away from his competitor. Unless the competitor’s f.o.b. price is so low as to eliminate our firm’s market completely, the firm can always count on its protected flank. Critics might say that, in practice, bounded linear markets are uncommon in an unbounded rectangular plane—again assuming a uniform density of customers everywhere—the notion of a protected flank vanishes. However, I think that Hotelling would simply argue here that a linear market was just a representation that illustrates a source of stability. The purpose of this chapter is to give readers the flavor of this line of research through a sequence of models that emphasize aspects of the problem. Sometimes, these models are best cast in the context of a firm purchasing inputs from a supplier. Other times, these models are best cast in terms of a customer choosing a retail outlet (store) from which to purchase a commodity. Models 8A through 8H are presented in sequence.
8A Market areas when competitors charge the same f.o.b. price, shipping cost is everywhere proportional to distance, customers are identical, uniformly spread across geographic space, and purchase where shipping cost is lowest (Thiessen Polygons). 8B Market areas when competitors charge different f.o.b. prices for their commodity. Other assumptions remain the same as Model 8A. 8C Market areas when there is price competition among firms selling the same product. 8D Market areas when each establishment has a different capacity to supply the commodity. 8E Market areas when competitors supply different but perfectly substitutable commodities. Other assumptions remain the same as in Model 8B. 8F Market areas when customers are of two different types. This builds on Model 8E. 8G Market areas when competitors supply unrelated commodities (zero cross price elasticity).
8.2
Range and Geographic Size of Market
205
Why these seven models? These seven are not exhaustive but do illustrate factors that shape market area, a kind of spatial price equilibrium, and localization.6 Following these seven models, I briefly consider market areas when destination choice can be thought to be subject to uncertainty (discrete choice modeling). Each place can be thought of as a store selling the commodity at an f.o.b. price. In simple versions of this model, each store sells the same commodity and is similar in other respects such as store attractiveness, purchase warranty, and level of service. The stores are each just a place from which the commodity can be obtained. The only relevant consideration is the cost of purchasing a unit of the commodity: f.o.b. price plus any shipping cost. More realistic models can be envisaged in which some of these assumptions are relaxed. Ignored here are shopping safaris where a customer purchases commodities while traveling from place to place.7 Ignored also is the practice of bulk buying where the customer acquires a substantial inventory (i.e., stocks up on purchases) so as to minimize the effect of trip cost. Instead, the customer is presumed to incur a fixed shipping cost for each unit of the commodity consumed. Finally, the consumer (or analyst) may well have imperfect information, that is, does not know the price and availability of the commodity at all places. In this chapter, as elsewhere so far in this book, I take the locations of consumers as given. I do not ask why consumers have come to be where they are. Nor do I ask why they might not respond to differences in price by changing their location. Such considerations are left to later chapters in this book. I also assume that the firm, its customers, and competitors share a common fiat market economy. As in Chapter 2 and Chapter 3, price differences among places open up the question of the happiness of customers. Other things being equal, we might expect customers at places where the effective price is higher to want to relocate to places where the effective price is lower. In this chapter, I continue to ignore such inclinations.
8.2 Range and Geographic Size of Market To start here, suppose a monopolist is located on a rectangular plane with a uniform density of customers (g customers per unit area). Assume the monopolist sells its commodities at an f.o.b. price (P), that customers bear the shipping cost, that the shipping rate per kilometer is a constant (s), and that all distances are shortest paths (i.e., as the crow flies). This is different from Chapter 2 where I had assumed that the firm used discriminatory pricing; by setting a single f.o.b. price, the firm earns less profit than is possible with discriminatory pricing. As in earlier chapters, I assume
6 I do not, for example, include models based on Marxist perspectives. See Plummer, Sheppard, and Haining (1998). 7 See, for example, McLafferty and Ghosh (1986).
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each customer has a linear inverse demand curve. For a customer at distance x from the monopolist, the effective price is P + sx, and the individual demand curve is given by (8.1.1): see Table 8.1. Here total trip cost for a customer is, therefore, proportional to consumption. There is no notion here of the consumer having an
Table 8.1 The monopolist’s market in two-dimensional space using f.o.b. pricing Individual linear inverse demand curve for customer at x km P + sx = α − βq
(8.1.1)
Range of commodity (since q ≥ 0 ) X = (α − P)/s
(8.1.2)
Aggregate x demand for the firm’s product Q = 0 2πgxdx
(8.1.3)
Aggregate demand after accounting for (8.1.1) and integrating Q = 2πg[((α − P)/(2β))X 2 − (1/(3β))sX 3 ]
(8.1.4)
Aggregate demand after accounting for (8.1.2) Q = (1/3)πg(α − P)3 /(βs2 )
(8.1.5)
Marginal revenue α − (4/3)(3Qβs2 /π g)1/3
(8.1.6)
Profit of firm Z = (P − C)Q − F
(8.1.7)
Profit-maximizing f.o.b. price, assuming Z ≥ 0 P = (1/4)α + (3/4)C
(8.1.8)
Range of commodity at profit-maximizing price, assuming Z ≥ 0 (3/4)(α − C)/s
(8.1.9)
Maximized profit (9/326)πg(α − C)4 /(βs)2 − F
(8.1.10)
Consumer benefit (CB) αQ − (3βs2 /(πg))1/3 (3/4)Q4/3
(8.1.11)
Producer cost including F (PC) CQ + F
(8.1.12)
Consumer surplus (CS) CB − PQ
(8.1.13)
Producer surplus (PS) PQ − PC
(8.1.14)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Unit cost of production; F—Fixed cost; g—Density of consumers everywhere; P—F.o.b. price set by firm; s—Unit shipping rate; x—Distance from customer to firm; α—Intercept of individual inverse demand curve; β—Slope of individual inverse demand curve. Outcomes (endogenous): Q—Aggregate amount demanded; q—Quantity demanded by a consumer; X—Range of commodity.
8.2
Range and Geographic Size of Market
207
inventory of the commodity at home and making a tradeoff between trip frequency (trip cost) and household inventory (inventory cost).8 Lösch characterized models like (8.1.1) as giving rise to a demand cone9 over the rectangular plane.10 This implies that, given an f.o.b. price P, there is a maximum distance, range (X), beyond which customers will demand zero: see (8.1.2). Of course, this is also the radius (geographic size) of the market for this monopolist. The higher the firm sets its f.o.b. price P, the smaller the geographic area of the market. Aggregate quantity demanded is given by (8.1.3). After rearranging (8.1.1), substituting, and integrating, we get (8.1.4); I show this intermediate step to help the reader with the derivation. Finally, after substitution from (8.1.2), the aggregate quantity demanded reduces to an expression for the demand cone (8.1.5). As one might expect, quantity demanded is an increasing function of α and g, and a decreasing function of β, P, and s. This is the so-called free spatial demand curve; “free” in the sense that the firm does not here consider competitors. The aggregate demand equation (8.1.5) is not linear in price even though we have assumed each customer individually has a linear demand. This is a consequence of the fact that the firm’s demand curve is aggregated over customers: some close by and paying a low effective price, others further away and paying a high effective price. This is illustrated in Fig. 8.1 where I assume the market contains just 3 customers, each with the same linear inverse demand curve. Customer 1, nearest the firm, has the lowest effective price; Customer 2 is further away, and Customer 3 still more so. At a high f.o.b. price, the firm would see demand only from Customer 1. As it lowers its f.o.b. price, it eventually attracts demand from Customer 2 and at a still lower price from Customer 3. That the aggregate demand curve here is kinked, a polyline, is a result of assuming just three customers. However, (8.1.5) assumes customers are spread evenly across the market; thus, (8.1.5)—cubic function of P that it is—can be thought of as the continuous equivalent of a kinked aggregate demand curve. Put differently, when geographic space is continuous, so too is the aggregate demand curve; when geographic space composed of discrete places (punctuated), the aggregate demand curve is kinked. Now, let us look at the behavior of a firm that maximizes profit. Assume the firm has a fixed cost (F) and a marginal cost (C); profit is given by (8.1.7). After substituting from (8.1.5), we can find the first-order condition for profit maximization: marginal revenue equals marginal cost. See (8.1.6). This yields the f.o.b. price: (8.1.8). In Chapter 2, I argue it is helpful to think of C (marginal cost of production) as a minimum price to complement the idea of α as the maximum price. The profit-maximizing price is now one-quarter of the way from C to α. This is a lower price than is charged by the firm in the simplest version of the two-market
8 See, for example, Bacon (1992). Here, inventory models of the kind used in Chapter 7 allow us to look at the economics of buying in bulk and such phenomena as a buying safari or buying spree. 9 A term characterizing the tendency for individual quantity demanded to fall off with increasing distance from a supplier pricing f.o.b. because of the rise in effective price. 10 See Long (1971) for further discussion of the properties of a spatial demand curve.
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Staking Out the Firm’s Market
Individual and aggregate demand AiBi A1CDE OP 0Qi 0Q
A1
C
Price
A2
Demand schedule for individual i (i = 1, 2, or 3) Aggregate demand curve for the three individuals Given f.o.b. price Quantity demanded by individual i at given f.o.b. price (i = 1, 2, or 3) Aggregate demand at given f.o.b. price: sums 0Q1, 0Q2, and 0Q3
D
A3
P 0 Q3
Q B3 B2 B1 Q2 Q1 Quantity
E
Fig. 8.1 Aggregate demand curve in market with three customers, each with same linear inverse demand curve but paying different effective prices
model in Chapter 2; there, the firm sets its price halfway between C and α. This difference arises because I here assume f.o.b. pricing; in Chapter 2, I had assumed that the firm discriminates in pricing between customers at Place 1 and customers at Place 2. How large is the geographic area of the firm’s market here? After substituting (8.1.8) into (8.1.2), we get (8.1.9). From (8.1.2), we see that, were the firm to price at C, the range for its commodity would be (α − C)/s. This is the maximum possible range; therefore, the maximum possible geographic area is π ((α −C)/s)2 . However, because the firm maximizes profit, it sets the higher f.o.b. price in (8.1.8) that generates a range 3/4 the size of the maximum possible range. Put differently, the firm foregoes serving customers at radiuses from beyond (3/4)(α − C)/s to (α − C)/s because these remote customers are not profitable enough.11 I presume here that the market radius given by (8.1.9) is sufficiently large to make the firm profitable. As shown in (8.1.10), α and g must be sufficiently large and/or β, C, F, and s sufficiently small for this to happen. Finally, even though this is the market for the monopolist’s product and even though a single f.o.b. price is determined there, the effective price is different for customers depending on how far they are from the firm. Up until now, we have
11 See
also Alderighi and Piga (2008).
8.2
Range and Geographic Size of Market
209
thought of a market in geographic terms as consisting of one or more places. At each place, customers pay the same price for a commodity. While price may vary from one place to the next, this is still consistent with the notion of a market. Places might here be thought to be submarkets in that the price of the commodity is systemically higher in some of them compared to others. In this chapter, we imagine a market in which potentially every customer faces a different effective price. A firm may sell at the same f.o.b. price to everyone, but customers face different effective prices; each must pay a unique shipping cost to get the commodity home. It is perhaps easiest in this chapter to imagine a continuum of submarkets arrayed by distance ring from the firm, with a price premium (in terms of effective price) for rings further away from the firm. Here, we might be able to retain the notion of a market because, after all, firms are selling at the same f.o.b. price. However, we then turn to a model in which firms each might take advantage of a local monopoly to sell at different f.o.b. prices. In such a model, what does the notion of a market now mean? We return to this question below. Why start by assuming f.o.b. pricing? After all, as was seen in Chapter 2, a delivered pricing scheme is more profitable. Following the findings of Chapter 7, assume the firm sets its price so as to cover half freight; we then need find only the price (P0 ) set for a customer adjacent to the firm. Given the firm has a fixed cost (F), a marginal cost (C) of production and pays shipping cost, its marginal cost is C + sx for the customer at distance x. As seen in Chapter 2, marginal revenue for that customer is α − 2βq. Equate these: see (8.2.5) in Table 8.2. Remembering q = (α/β) − (1/β)P, where P is now delivered price, (8.2.5) yields the profit maximizing price: see (8.2.6). This is different from the f.o.b. pricing solution in the following respects. • Compare the delivered price in (8.2.6) with the f.o.b. price in (8.1.8). For customers nearby (small x), delivered price is higher than f.o.b. price; for customers further away, it is smaller. Were a firm to switch from f.o.b. pricing to delivered pricing, customers nearby would become worse off while more remote customers would become better off. • In Chapter 2, we saw the half-freight rule. Here, (8.2.6) instances it again. • Profit under delivered pricing, now given by (8.2.11), is higher than it was for f.o.b. pricing: compare (8.2.11) with (8.1.10). • The geographic size of market under delivered pricing—see (8.2.7)—is the largest possible; it is larger than it would be, for instance, under f.o.b. pricing. Figure 8.2 illustrates the two pricing schemes where α = 10, β = 1, s = 0.10, g = 20, C = 2, and F = 0. There, we see the effective price (under f.o.b. pricing) and the delivered price increase steadily the further away the customer, until we reach a distance at which demand drops to zero (i.e., effective or delivered price to the customer is α). That defines the range of the market under each pricing scheme.
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Table 8.2 The monopolist’s market using delivered pricing assuming P = P0 + 0.5sx, hence X = (α − P0 )/(0.5s) Aggregate demand (4/3)πg(α − P0 )3 /(βs2 )
(8.2.1)
Total revenue Q(α + P0 )/2
(8.2.2)
Marginal Revenue 0.5(a + P0 ) − (1/6 ){(3/4)βs2 Q/(π g)}1/3
(8.2.3)
Total cost F + Qd [C + α − P0 ]
(8.2.4)
Marginal cost equals marginal revenue for customer at distance x C + sx = α − 2βq
(8.2.5)
Profit maximizing price for customer at distance x P = 0.5α + 0.5(C + sx)
(8.2.6)
Radius of market (where P = α) X = (α − C)/s
(8.2.7)
Quantity at profit-maximizing price (1/6)πg(α − C)3 /(βs2 )
(8.2.8)
Revenue of firm x 0 2πgxqPdx
(8.2.9)
of firm defined Profit x 0 2πgx(P − C − sx)qdx − F
(8.2.10)
Maximized profit (1/24)(π g)(α − C)4 /(βs2 ) − F
(8.2.11)
Consumer benefit (CB) αQ − (3βs2 /(πg))1/3 (3/4)Q4/3
(8.2.12)
Producer cost including F (PC) CQ + F
(8.2.13)
Consumer surplus (CS) CB − PQ
(8.2.14)
Producer surplus (PS) PQ − PC
(8.2.15)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Unit cost of production; F—Fixed cost; g—Density of consumers everywhere; P—Delivered price set by firm; s—Unit shipping rate; x—Distance from customer to firm; α—Intercept of individual inverse demand curve; β—Slope of individual inverse demand curve. Outcomes (endogenous): P0 —Price set by firm for an adjacent customer; P0 —Price set by firm for an adjacent customer; Q—Aggregate amount demanded; q—Quantity demanded by a consumer; X—Range of commodity.
8.3
Trade Area and Market Area in Retailing
211
F.o.b. and delivered pricing on a rectangular plane AB CD OE
Effective price under most profitable f.o.b. pricing scheme Price under most profitable delivered pricing scheme Maximum extent of market under f.o.b. pricing
0F 0G 0H
Maximum extent of market under delivered pricing Price (α) above which consumer does not demand product Distance within which price to consumer is lower under f.o.b pricing
G
B
D
E
F
I
Price
J
C
A
0
Distance from firm
H
Fig. 8.2 f.o.b. and delivered pricing compared. Notes: a = 10; b = 1; s = 0.10; g = 20; C = 2, f = 100,000. With f.o.b. pricing: P = 4.00; R = 60; Q = 452,389; Z = 804,779. With delivered pricing: P(r = 0) = 6.00; R = 80, Q = 536,165, Z = 972,330. Horizontal axis scaled from 0 to 90; vertical axis scaled from 0 to 12
8.3 Trade Area and Market Area in Retailing Retail analysts use two distinct concepts in thinking about a store’s sales potential; trade area and market area. The trade area is the (larger) geographic area from within which most of the customers of a store and its competitors come from. The trade area represents most of the customers that one store might attract potentially were it to be highly successful against its competitors. If the trade area lies entirely inside a circle of radius X (the range), then potentially, the store could attract all of them. On the other hand, if you and your competitors are spread out so that some of the competitors’ customers are more than X kms away from you, you will not attract them even if customers might otherwise prefer your store. In contrast, the market area is the (smaller) geographic area within which you currently dominate and draw a substantial proportion of your customers. Put differently,
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the market area represents the core of your current customers. There are two critical numbers involved in this definition: (i) the firm’s share of customer purchases locally and (ii) sales within the market area as a percentage of the firm’s total sales. We can imagine that, in the retail sector containing a set of stores, there is a market. In a simple case, the trade area would be partitioned into a set of market areas, one for each store. In that sense, any one firm’s market area is a geographic subset of the market (i.e., the trade area) in which it and its competitors compete. We can characterize a market area of a firm as (u, v) where u is proportion of sales (or, alternatively, customers) by all firms in the area that accrue to the firm, and v is the proportion of the firm’s total sales (or customers) that originate in that area. For example, in a (60, 80) market area the firm captures 60% of purchases made by customers from there, and the market area accounts for 80% of the firm’s total sales. If a trade area contains just four firms, all clustered at a central location, and doing equal sales, each would share the same (25, 100) market area. If the absence of competitors means a firm’s market area is limited only by range, we could find the (100, 100) market area for this firm. In practice, there are, in general, many ways of drawing a (u, v) market area for any one firm. Usually, analysts choose a compact area: e.g., a circle around the firm’s site with a minimum radius that satisfies u and v.
8.4 Model 8A: Two Firms Selling Commodity at Same f.o.b. Price Assume the world can be represented as a rectangular plane. Assume only two firms (labeled 1 and 2): each firm sells the same product and at the same f.o.b. price (P). The two firms are at distinct places. Without loss of generality, assign Places 1 and 2 the Cartesian coordinates (0, 0) and (d, 0), respectively: Place 2 is, therefore, d kilometers due east of Place 1. See Table 8.3. Now, assume a customer firm purchases this product for use as an input in its own production. The customer pays a fixed unit shipping rate, s, for each unit of the input purchased. Suppose a firm is located x1 kilometers from Place 1 and x2 kilometers from Place 2. For the firm, the cost of a unit of the input is, therefore, P + sx1 purchased from Place 1, or P + sx2 purchased from Place 2. Assume the customer has a linear inverse demand function and P + sx1 and P + sx2 are each less than the maximum price, α. Assume also that each firm has sufficient capacity to meet all demand. Under these assumptions, the customer will be in the market area of the firm at Place 1 if P + sx1 < P + sx2 (i.e., if x1 < x2 ). and Place 2 if x2 < x1 . The customer will be on the boundary if x1 = x2 . Now, assume a customer on the boundary has Cartesian coordinates (X, Y). Application of the Pythagorean Formula implies the following: (X − 0)2 + (Y − 0)2 = (X − d)2 + (Y − 0)2 . Upon simplification, the boundary is given by (8.3.7). See Table 8.3 wherein I summarize equations, assumptions, notation, and rationale for localization in Model 8A. In other words, for any given Y, X is halfway between the x-coordinates of the two
8.4
Model 8A: Two Firms Selling Commodity at Same f.o.b. Price
213
Table 8.3 Model 8A: two suppliers on rectangular plane selling commodity at same f.o.b. price (P) Location of supplier 1 (0, 0)
(8.3.1)
Location of supplier 2 (d, 0)
(8.3.2)
Location of customer on boundary (X, Y)
(8.3.3)
Distance from supplier 1 to customer x1 = (X 2 + Y 2 )0.5
(8.3.4)
Distance from supplier 2 to customer x2 = ((X − d)2 + Y 2 )0.5
(8.3.5)
Equal effective prices on boundary P + sx1 = P + sx2
(8.3.6)
Boundary condition X = d/2 for any Y
(8.3.7)
Notes: Rationale for localization (see Appendix A): Z3—Implicit unit price advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): d—Distance (in kilometers) between firm and competitor; P—F.o.b. price set each by firm and by competitor; s—Unit shipping rate. Outcomes (endogenous): X—X-coordinate of customer on boundary between the two firms; x1 —Distance from customer to firm; x2 —Distance from customer to competitor; Y—Y-coordinate of customer on boundary between the two firms.
places. Therefore, viewed geometrically, the boundary between the two market areas is the perpendicular bisector between the two firms.12 See the line CE on the map in Fig. 8.3. Because of competition among firms, we might expect that the firm will not be able to realize the monopoly profit given in (8.1.10). After all, if the firm were to earn substantial profits, other firms would be attracted into the industry. Where they choose locations with a market radius that overlaps that of our firm, they cut into the profit and the market area of our firm. Imagine the following thought experiment. Two competitors—each pricing f.o.b. and producing the same product—locate d kilometers apart on a rectangular plane. Assume initially the competitors are too far apart to impinge on each other’s market: i.e., d > 2X where X is defined in (8.1.9). They, therefore, each earn the profit given by (8.1.10) and have a circular market whose radius is given by (8.1.9). Now, imagine we push the two competitors closer together until d < 2X. At this stage, the two markets intersect, and the 12 To
draw a boundary between the two suppliers, simply draw a straight line from Place s1 to 2 and then find a line that is perpendicular and bisects it. One method, familiar from high school mathematics, is to use a compass to draw a circle of radius r (where r > d/2) centered at Place 1, repeat at Place 2, then draw a straight line (the perpendicular bisector through the points of intersection of these two circles.
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Staking Out the Firm’s Market
Map of market area boundaries under f.o.b. pricing on a rectangular plane AB CE FH IK
Straight line joining the two firms Market boundary when P1 = P2 Market boundary when P1 = P2 +1 Market boundary when P1 = P2 +2
K
J GD
A(1)
I
E
H
F
B(2)
C
0 50 100 km km km
Fig. 8.3 Model 8B: map of boundary between market areas of two suppliers
boundary between the firms’ market areas is a perpendicular bisector. If customers purchase from the nearest firm, the market area for each firm shrinks the more the markets intersect. Continuing this thought experiment, as we push the two firms closer together, the revenue and profit earned by each firm shrinks. Thiessen13 polygons14 —also known as a Voronoi15 or Dirichlet Diagram—are a common tool in retail analysis.16 In this approach, a map of a trade area is drawn with each store represented as a point (dot). Thiessen polygons are created using an algorithm. For each store, a subset of the perpendicular bisectors with all other stores (as well as the map boundary) form the set of line segments that constitute a Thiessen polygon. The corresponding map showing the lines joining places sufficiently near to have a boundary in common is called a Delaunay triangulation.
13 Alfred
Henry Thiessen (born 1872), an American climatologist, defined polygons around individual rainfall stations to estimate total rainfall across a region. 14 A map polygon formed on a rectangular plane by constructing perpendicular bisectors to the straight lines joining a point (typically a store) to similar points nearby. The partitioning of a map in this way is called a Voronoi Diagram. 15 Georgy Fedoseevich Voronoi (born 1868), a Russian mathematician worked on polygonal partitioning of a two-dimensional plane. This follows on earlier work by Rene Descartes (writing around 1644) and Johann Peter Gustav Lejeune Dirchlet (writing around 1850). 16 See Boots (1980), Byers (1996), Graham and Yao (1990), Miles and Maillardet (1982), and Sibson (1980).
8.5
Model 8B: Market Area Boundary Between Two Firms Selling
215
Under the assumptions made and ignoring knife-edge cases (customer straddles the boundary), a Thiessen polygon is a (100, 100) market area. That is because there would be no incentive for a customer inside a polygon to purchase the commodity from a firm elsewhere, or for a customer outside the polygon to purchase from this firm. So far, we have assumed the firms each charge the same f.o.b. price for their input. Is that reasonable? Presumably, to the extent each firm can affect the price they receive, they would want to set a price where marginal revenue equals marginal cost as we saw earlier in (8.1.6). At the same time, one might imagine competition between the two firms should drive their prices to be similar if not identical. However, the two firms each have their own market area. This implies the firms are not perfectly competitive; is there any reason, therefore, why competition should lead to the same price for each firm?
8.5 Model 8B: Market Area Boundary Between Two Firms Selling Same Commodity at Different f.o.b. Prices Retain all of the assumptions above, except now assume the firm at Place 1 on a rectangular plane sells the commodity at one f.o.b. price (P1 ), while the firm at Place 2, located d kilometers from Place 1, sells for another, P2 .17 For a purchaser located x1 kilometers from Place 1 and x2 kilometers from Place 2, the effective price of a unit is, therefore, P1 + sx1 purchased from Place 1 or P2 + sx2 purchased from Place 2. See Table 8.4. For the moment, let us not ask why the two firms have different f.o.b. prices. Under these assumptions, a purchaser will be on the boundary if x1 − x2 = (P2 − P1 )/s. The implication here is that the boundary is the set of all places such that the difference in distance to two fixed places is a fixed amount (i.e., (P2 − P1 )/s. However, this is also the definition of a hyperbola. In the special case where the difference in distance is zero (i.e., P1 = P2 ), the hyperbola reduces to a straight line (the perpendicular bisector) as we have already seen. Under the assumptions made and again ignoring knife-edge cases where customers straddle a boundary, a market area constructed in this way is (100, 100). As in the case of the Thiessen polygon, there would be no incentive for a customer inside the market area to purchase the commodity from a firm elsewhere or for a customer outside the polygon to purchase from this firm. Imagine now a thought experiment in which the two competitors initially charge the same f.o.b. price. Assume the firms are close enough that the boundary between them incorporates a perpendicular bisector. Now, suppose the firm at Place 2 were to lower its price. The relevant portion of the market boundary would now become a hyperbola. It would generally lie closer to Place 1 than does the perpendicular bisector. See the boundaries FH (when P1 = P2 + 1) and IK (when P1 = P2 + 2) 17 A
similar model is studied in Parr (1995b).
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Table 8.4 Model 8B: two suppliers on rectangular plane selling commodity at different f.o.b. prices (P1 and P2 ) Location of supplier 1 (0, 0)
(8.4.1)
Location of supplier 2 (d, 0)
(8.4.2)
Location of customer on boundary (X, Y)
(8.4.3)
Distance from supplier 1 to customer x1 = (X 2 + Y 2 )0.5
(8.4.4)
Distance from supplier 2 to customer x2 = ((X − d)2 + Y 2 )0.5
(8.4.5)
Equal effective prices on boundary P1 + sx1 = P2 + sx2
(8.4.6)
Boundary condition x1 − x2 = (P2 − P1 )/s
(8.4.7)
(8.4.7) assumes shipping sufficiently costly to sustain differences in f.o.b. prices: i.e., P1 < P2 + sd and P2 < P1 + sd or s > |P1 − P|/d
(8.4.8)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): d—Distance (in kilometers) between firm and competitor; P1 —F.o.b. price set by firm; P2 —F.o.b. price set by competitor; s—Unit shipping rate. Outcomes (endogenous): X—X-coordinate of customer on boundary between the two firms; x1 —Distance from customer to firm; x2 —Distance from customer to competitor; Y—Y-coordinate of customer on boundary between the two firms.
in Fig. 8.3. As hyperbolae, these each bend back toward Place 1. Put intuitively, a firm holds onto to customers best when they are close by or behind it. There is an upper limit to the price the firm at Place 1 can set if it wants to have any market area at all: P1 < P2 + sd. Of course, if positions were reversed and firm 1 had the lower price, there would be a minimum price charged by the firm at Place 1 below which the firm at Place 2 would lose its markets: P1 + sd < P2 . I am not saying anything about the amount any one purchaser might demand; I am saying only that the purchaser will or will not patronize a particular firm. Nonetheless, there is an interesting insight here into the demand curve faced by a firm. To see this, suppose each purchaser has a price-inelastic demand for just one unit of the commodity per time period. Suppose further N purchasers in total. If firm 1 sets its price too high (P1 > P2 + sd), the demand for its product is zero. If firm 1 sets it price sufficiently low (P1 < P2 − sd) to undercut its competitor, its demand is N. If firm 1 sets a price anywhere in between these two, it gets a point on a negatively sloped demand curve. In other words, firm 1 has a demand curve kinked at both P1 = P2 − sd and P1 = P2 + sd. Imagine a thought experiment in which we push firms 1 and 2 closer together. As a polar case, d = 0 when Places
8.6
Model 8C: Why Do Prices Differ Among Firms?
217
1 and 2 are adjacent. Now, the demand curve for either firm reduces to the familiar perfect competition case: a horizontal line. In that sense, the kink is not inconsistent with neoclassical theory. However, we again have to ask why the kink arises. The answer once again is tied up inextricably with the discreteness of geography. It is a consequence of the idea that firm 2 is where it is, neither closer nor further away.
8.6 Model 8C: Why Do Prices Differ Among Firms? The purpose of this model is to better understand when and why competitors might charge different f.o.b. prices for the same product. Introducing the possibility that firms sell at different prices raises a question. What determines price in such instances? Making the assumptions that firms are price takers and everyone faces the same price is consistent with perfect competition. However, the notion implicit in a market area is that with f.o.b. pricing customers face different prices depending on location. Add to this the possibility that different firms set different prices, and the notion of market equilibrium is called further into question. To see how prices might be set, consider a simple geography wherein customers are spread uniformly along an east−west line of infinite length and that firms are also evenly spread out (one firm each d kilometers).18 Consider one firm with a competitor to the west (labeled w) and another to the east (labeled e). Assume all firms price f.o.b. Let 0 < x < d be the location (kilometers) of a customer to the west of the firm. For that customer, effective price of a unit purchased from the firm is P+sx and the effective price from competitor w is Pw +s(d − x). Assume Pw ≤ Pe and that w does not price e out of the market: i.e., Pw + 2sd > Pe (remembering w and e are 2d kilometers apart). Further, assume here P < Pw + sd and P + sd > Pe so the three firms coexist. At the boundary, Xw , between competitor w and the firm, we get (8.5.1). See Table 8.5. This yields a formula for the western boundary—see (8.5.2)—wherein our firm gets half the market to its west (d/2) adjusted for any difference in f.o.b. price with respect to its competitor there. Similar results apply in the market to the east of our firm: see (8.5.3). Therefore, the total length of market, X = Xw + Xe , for our firm is given by (8.5.4). Suppose further customers are spread uniformly along this line at density g and that each customer demands q units of the commodity regardless of price. In this case, aggregate demand for the firm’s product is given by (8.5.5). If the firm has a marginal cost of production, C, and a fixed cost of production, F, then its profit is given by (8.5.6). The firm then sets a price so as to maximize profit: see (8.5.7). Assume here the firm is myopic; it does not expect either firm w or firm e to change their price in response to the price chosen. This yields the profit-maximizing price shown in (8.5.8): halfway between the average of Pw and Pe on the one hand and a cost (C + sd) on the other hand. Note C + sd is greater than the actual cost (C + sd/2) of serving someone at a place midway between any pair of firms. 18
Other geographies have also been explored. See, for example, Sarkar, Gupta, and Pal (1997).
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Table 8.5 Model 8C: customers spread uniformly along a line market (zero price elasticity) with supplier every d kilometers and f.o.b. pricing Effective price at western boundary, assuming s > |Pw − P|/d P + sXw = Pw + s(d − Xw )
(8.5.1)
Western boundary, assuming s > |Pw − P|/d Xw = (1/2)(Pw − P)/s + (1/2)d
(8.5.2)
Eastern boundary, assuming s > |Pe − P|/d Xe = (1/2)(Pe − P)/s + (1/2)d
(8.5.3)
Total length of supplier’s market X = ((1/2)(Pw + Pe ) − P)/s + d
(8.5.4)
Aggregate quantity demanded Q = gqX
(8.5.5)
Profit (P − C)gqX − F
(8.5.6)
First-order condition for myopic profit maximization X − (P − C)/s = 0
(8.5.7)
Profit-maximizing price P = (1/2)(1/2)(Pw + Pe ) + (1/2)(C + sd)
(8.5.8)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Unit cost of production; d—Distance (in kilometers) between firm and competitor to East or West; F—Fixed cost; g—Density of consumers (assumed constant along line); Pe —Price of competitor to east; Pw —Price of competitor to west; q—Quantity demanded by a consumer; s—Unit shipping rate. Outcomes (endogenous): P—F.o.b. price set by firm; Q—Aggregate amount demanded; X—Total length of firm’s market; Xe —Distance to customer on boundary to east of firm; Xw —Distance to customer on boundary to west of firm.
There is something special about C + sd here. If all three firms were to each have an f.o.b. price of C+sd, there would be no incentive for our firm or, under symmetric conditions, for either competitor to change price. If the two competitors each set a price lower than C + sd, there would be an incentive for our firm to set its price higher than them and closer to C + sd. On the other hand, if the two competitors each set a price higher than C + sd, there would be an incentive for our firm to set its price lower and closer to C + sd. Therefore, C + sd is a kind of equilibrium price for all firms. I do not want to make too much of this idea, however, because of the following conundrum. If all firms in the market, not just the three examined here, were to conspire, they would want to set the price as high as possible since we have assumed demand is insensitive to price. However, in (8.5.8), there is no mechanism that would drive prices up in this way. Of course, you might say, this is a competitive market so why would we expect prices to be driven up? If it is indeed competitive, why isn’t profit being driven to zero? At price equal to C + sd each firm has the potential to some excess profit unless fixed cost is prohibitive. Why? Presumably, this is because the market is rich enough to support firms at a density at least equal
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Model 8C: Why Do Prices Differ Among Firms?
219
to one firm per d kilometers. This is the so-called Löschian equilibrium problem in which we imagine free entry of new firms, and the rearrangement of these firms in geographic space, and a shrinking of d until the excess profit earned by the marginal firm is effectively zero.19 As in Chapters 3 and 6, this chapter assumes that customers each demand a fixed amount of output, q units, regardless of price. Can we release this assumption? Is it possible, for instance, to imagine a linear inverse demand curve for each customer here? The firm’s aggregate demand Q on this linear market would then be defined by (8.6.1): see Table 8.6. Profit would then be given by the identity (8.6.3). Assuming again the firm is myopic in that it sets a price for its product ignoring the possibility that the two competitors might react by changing their prices implies (8.6.4). To solve this equation, we need to differentiate (8.6.2): see (8.6.5). As well, we need to remember Xw , Xe , and X here are determined by (8.5.2), (8.5.3), and (8.5.4). Unfortunately, substituting these equations back into (8.6.4) gives a quadratic equation (8.6.6) from which it is difficult to obtain insights into the determinants of P. That is why I presented the zero price elasticity of demand solution in Table 8.5; it is easier to solve and interpret. Table 8.6 Model 8C: customers spread uniformly along a line market (nonzero price elasticity) with supplier every d kilometers Demand x x for firm’s product Q = o w (α − (P + sx))(g/b)dx + o e (α − (P + sa))(g/b)dx
(8.6.1)
Which yields Q = (g/β)((α − P)X − (s/2)(Xw2 + Xe2 ))
(8.6.2)
Profit Z = (P − C)Q − F
(8.6.3)
Myopic first-order condition ∂Z/∂P = Q + (P − C)∂Q/∂P = 0
(8.6.4)
where ∂Q/∂P = (g/β)(−X/2 − (α − P)/s)
(8.6.5)
Yields a quadratic in P a1 P2 + b1 P + c1 = 0
(8.6.6)
where a1 = 6 − 3s b1 = 2s(α + C) − 2(2α + C) − 3(Pw + Pe + 2ds) c1 = (2α + C) (Pw + Pe + 2ds) + s(Pw + ds) (Pe + 2ds)− (s/2) (Pw + Pe + 2ds)2 − 2saC
(8.6.7) (8.6.8) (8.6.9)
Note: See also Table 8.5.
19 In
an industry where firms produce the same commodity and compete by choosing geographic locations, a long-run equilibrium wherein firms no longer have the incentive to enter or leave the industry. Every firm earns normal profit only. See, for example, Mai and Hwang (1994).
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8.7 Model 8D: Market Area Boundary Between Two Firms with Different Capacities In the previous chapter, I presented the Hitchcock–Koopmans problem in which a monopolist minimizes the cost of production and shipment in meeting a fixed schedule of demand. We saw there the economic insight that a firm will use a particular factory (establishment) to serve customers at a given place if the opportunity cost of using that factory is the lowest of all potential sites. In a related vein, we came to understand that the geographic area supplied by a factory—its market area—could be outlined from the pattern of shipments in the least cost solution. Where demand at a place is met entirely by shipments from one firm, that customer place is part of the firm’s market area. Suppose that where demand at a place is met by shipments from two or more firms we assign that place to the market area of the firm with the largest shipment. Constructed in this way, the market area of a firm may not necessarily include all of its customers but will include all those customer places where the firm is the exclusive or principal source of shipments. However, what the Hitchcock–Koopmans problem does not do is to give us an idea about the geography of the areas served by a given factory under the Hitchcock–Koopmans model. Inspired by Hall (1989), I show when and how these areas can be delimited. Suppose now firm 1, unlike firm 2, has insufficient capacity to meet the needs of all the purchasers who might otherwise want to purchase there. As in Chapter 7, I also assume demand for the commodity everywhere has a price elasticity of zero. There are two possible scenarios here. One scenario is where the firms are competitors. Assume the two firms initially intend to charge the same price. In this case, firm 1 finds too many customers to accommodate given its capacity. However, firm 1 may raise its price above that of its competitor in order to ration its output most profitably. As it does, some customers now find it less costly to purchase instead from the more distant firm 2. As firm 1 raises its price, the market boundary becomes a hyperbola bending back toward it as in Fig. 8.3. At whatever price firm 1 eventually settles, the market boundary will be a hyperbola. The second scenario, as used in the Hitchcock–Koopmans model in the preceding chapter, is where the suppliers are branch factories under common ownership. In that case, the firm faces a shadow price on capacity at each factory. The shadow price here is zero for the factory at Place 2 (excess capacity) and positive for the factory at Place 1 (no excess capacity). In this case, the firm can be thought to allocate customers to factories taking this opportunity cost into account. So, the boundary between the two market areas will therefore again follow a hyperbola. The primal and dual in the Hitchcock–Koopmans problem show us that quantity allocations (shipments) and prices (shadow prices) are just two ways of looking at the same problem. Therefore, we should not be surprised to see that a capacity limitation has the same effect on a market boundary as does a higher unit cost. This is a powerful conclusion because it extends our understanding of the primal in the Hitchcock–Koopmans problem. In Chapter 7, we saw a method (linear programming) for finding a solution to the factory allocation problem. However,
8.8
Model 8E: Market Area Boundary Between Two Firms
221
that solution did not give us much of an idea as to the geography of market areas. Now, we know that the boundary between two suppliers on a rectangular plane will be either a perpendicular bisector or a hyperbola bending back toward the supplier with the higher unit cost or the capacity constraint. So, the conclusion here is that a capacity constraint affects the market boundary exactly the same manner as does a price difference. It causes the market boundary to become hyperbolic and to bend back in the direction of the factory with the binding capacity constraint. Under the assumptions made (see Table 8.7) and once again ignoring knife-edge cases of customers straddling a boundary, a market area constructed in this way is (100, 100). As in the case of Models 8A and 8B, there would be no incentive for a customer inside the market area to purchase the commodity from a supplier elsewhere or for a customer outside the polygon to purchase from this supplier.
Table 8.7 Model 8D: assumptions and rationale for localization Assumptions (see Appendix A) A1 Closed regional market economy A3 Punctiform landscape A4 Rectangular plane B1 Exchange of commodity for money C1 Fixed demand locations D2 Firm minimizes cost of production and shipping D3 No capacity exceeded D5 I factories D8 Reverse-L marginal cost curve E2 Fixed unit shipping rate H2 Locations of the firm is given M1 Firm uses f.o.b. pricing
Rationale for localization (see Appendix A) Z2 Implicit unit cost advantage at some locales Z5 Capacity constraints Z8 Limitation of shipping cost
8.8 Model 8E: Market Area Boundary Between Two Firms with Different, but Perfectly Substitutable, Commodities Now assume the two firms each use f.o.b. pricing. See Table 8.8 for the list of assumptions and rationale for localization. Suppose that customers see the two firms as different but fully substitutable; each customer would purchase a quantity of only one of these two commodities. If the two firms are otherwise identical, the customer’s choice then depends strictly on the price they have to have to pay, inclusive of shipping costs. If customers see no inherent difference between the two commodities, they would choose the less-expensive commodity; Model 8E reduces to Model 8A if the commodities have the same f.o.b. price or Model 8B if their f.o.b. prices differ. On the other hand, suppose the commodities are inherently different in a way that customers prefer commodity from supplier 2 when effective prices are
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the same. Presumably, there is a difference in effective price at which the customer is indifferent between the two commodities. That difference therefore defines the boundary between the two firms. That difference also gives rise to a hyperbolic boundary as we have already seen in Model 8B. Under the assumptions made (see Table 8.8) and again ignoring straddling consumers, a market area constructed in this way is (100, 100). As in the case of Models 8A, 8B, and 8D, there would be no incentive for a customer inside the market area to purchase the commodity from a firm elsewhere.
Table 8.8 Model 8E: assumptions and rationale for localization Assumptions (see Appendix A) A1 Closed regional market economy A3 Punctiform landscape A4 Rectangular plane B1 Exchange of two commodities each for money C4 Identical customers E2 Fixed unit shipping rate F7 Location of competitor is given H2 Location of the firm is given M1 Firm uses f.o.b. pricing
Rationale for localization (see Appendix A) Z7 Variation in goods Z8 Limitation of shipping cost
8.9 Model 8F: Market Area Boundary Between Two Firms with Different, but Perfectly Substitutable, Commodities When Customers Are of Two Types Until now, we have assumed that customers are everywhere identical. How does customer heterogeneity add complexity to the determination of market areas? Assume two different kinds of customers: types 1 and 2. See the list of assumptions and rationale for localization in Table 8.9. Every customer of type 1 is identical. So too is every customer of type 2. Assume here that the two firms sell different but perfectly substitutable commodities and price f.o.b. See Table 8.9. It is then possible to imagine a market area for any given firm among customers of type 1 based on Model 8E above. Similarly, we can imagine a market area for the same firm among customers of type 2. These two market areas may well differ; for example, customers of type 1 may prefer the product of the first firm (where effective price is the same), whereas customers of type 2 prefer the second. In such cases, it would not be possible in general to construct a single (100, 100) market area that covers both types of customers. Here then is a situation where, no matter how we draw the market boundary we might expect some customers inside the market area for firm 1 to purchase from firm 2, and some customers outside the market area of firm 1 to purchase from that firm nonetheless.
8.10
Model 8G: Market Area Boundary Between Two Firms
223
Table 8.9 Model 8F: assumptions and rationale for localization Assumptions (see Appendix A) A1 Closed regional market economy A3 Punctiform landscape A4 Rectangular plane B1 Exchange of two commodities, each for money C10 Two kinds of customers E2 Fixed unit shipping rate F7 Location of competitor is given H2 Location of the firm is given M1 Firm uses f.o.b. pricing
Rationale for localization (see Appendix A) Z6 Differences among consumers Z8 Limitation of shipping cost
8.10 Model 8G: Market Area Boundary Between Two Firms Supplying Different Commodities This analysis generalizes from Model 8E. At this point in the book, we need something more general than a demand curve for a single product; we need to model how a consumer substitutes between two or more commodities. The economist does this by means of a utility function, U = f (q1 ,q2 ) that shows how the well-being (U) of the consumer varies depending on the amounts of commodities 1 and 2 consumed: q1 and q2 , respectively. An indifference curve traces combinations of quantities of commodity 1 and commodity 2 that generate the same level of well-being for the consumer. Maximizing utility subject to a budget constraint in turn yields the individual demand curves that we have used to this point in the book. In a world of differentiated commodities, it may seem strange to be focused here on the case where two commodities are perfectly substitutable. Is it not reasonable to expect that a given customer would purchase some amount of the commodity from firm 1 and some amount from supplier 2? Assume two suppliers on the map: d kilometers apart. They each sell one product. Each uses f.o.b. pricing. The prices set by the two firms are p1 and p2 , respectively. Consumers are identical. They each consume only the commodities provided by the two suppliers. Where q1 is the amount purchased by a customer from the first store monthly, and q2 is the amount purchased from the second store, each is rational; he or she maximizes the log-linearutility function given in (8.10.1).20 See Table 8.10. For ease of exposition,
20 A
ranking of alternative bundles that allows us to predict consumer choice. As a ranking, a given utility function is said to be unique up to a monotonic transformation. For example, the utility functions f (x,y) = xb y1−b where 1 < a < 0 and g(x,y) = axb yc where b > 0,c > 0, and b + c < 1, calculated at consumption of x units of good 1 and y units of good 2, generate the same rank ordering: i.e., g(x,y) is a monotonic transformation of f (x,y). At the same time, consumers are usually imagined to have diminishing marginal utility in that each additional unit of good consumed increases utility by a smaller amount. The two utility functions above exhibit diminishing marginal utility.
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Table 8.10 Model 8G: two stores selling different commodities; consumers with log-linear or CES utility functions Log-linear utility U = qν1 q1−ν 2
(8.10.1)
Budget constraint Y = (p1 + sx1 )q1 + (p2 + sx2 )q2
(8.10.2)
First-order conditions for log-linear utility maximization (p1 + sx1 )q1 = νY (p2 + sx2 )q2 = (1 − ν)Y
(8.10.3) (8.10.4)
Relative expenditures at store 1 (p1 q1 )/(p2 q2 ) = (ν/(1 − ν))(p1 /p2 )(p2 + sx2 )/(p1 + sx1 )
(8.10.5)
On market area boundary, where (p1 q1 )/(p2 q2 ) = 1 x2 = (1 − 2ν)p2 /(νs) + (1 − ν)p2 x1 /(νp1 )
(8.10.6)
CES (Constant elasticity of substitution) utility −ρ −ρ U = (δq1 + (1 − d)q2 )− 1/ρ
(8.10.7)
First-order condition for CES utility maximization q1 /q2 = (δ/(1 − δ))1 /(1+ρ) ((p1 + sx1 )/(p2 + sx2 )− 1/(1+ρ)
(8.10.8)
Boundary (fraction of distance) along line from Supplier 1 to Supplier 2 where budget shares are invariant X = ((1 − δ)P1 − δP2 + (1 − δ)s)/s
(8.10.9)
Notes: Rationale for localization (see Appendix A): Z7—Variation in goods; Z8—Limitation of shipping cost. Givens (parameter or exogenous): pi —F.o.b. price charged by firm i; s—Unit shipping rate; Y—Income of customer; ν—Relative preference for commodity 1: log-linear; d—Relative preference for commodity 1: CES; r—Substitutability of. Outcomes (endogenous): qi —Individual consumption of commodity 1; xi —Distance from firm i; X—Fraction of distance from Supplier 1 to Supplier 2; U—Utility of individual.
these are the only goods consumed and therefore exhaust consumer income. Assume also each consumer incurs a shipping cost of s dollars per kilometer for each unit of a commodity consumed. Consumers each have the same monthly income (Y). For a customer at distance x1 from one store and distance x2 from the other store, the budget constraint is (8.10.2). A utility-maximizing consumer can be shown to have demands for the two commodities as described by (8.10.3) and (8.10.4). These do not give the linear inverse demand curve that I have used elsewhere in the book so far. The log-linear utility function (8.10.1) used here has the implication that the consumer always spends fixed fractions (ν and 1 − ν) of income on the two goods.21 The demand for either
21 The literature includes some experimentation with alternative utility functions. See, for example,
Lo (1990, 1991a, 1991b, and 1992).
8.10
Model 8G: Market Area Boundary Between Two Firms
225
commodity does not depend on the price of the other commodity. This zero crossprice elasticity is another feature of the log-linear utility function (8.10.1) used here. While consumers exhaust their income on spending for the two commodities, (p1 + sx1 )q1 and (p2 + sx2 )q2 , respectively, the revenues received by each store from the consumer are only p1 q1 and p2 q2 , respectively. Therefore, the expenditure ratio for a customer at any given location is given by (8.10.5). I define the boundary of the market area to be the locations where (p1 q1 )/(p2 q2 ) = 1. Under the boundary condition (8.10.6), x2 is a linear function of x1 . Note here restrictions on (8.10.6): x1 ≥ 0, x2 ≥ 0, and x1 + x2 ≥ d. What does the market boundary look like? In the special case where ν = 0.5 and p1 = p2 , (8.10.5) reduces to x2 = x1 which means that the boundary is the perpendicular bisector. For other parameter values, the boundary is of an elliptical form. Under the assumptions made, a market area constructed in this way is never (100, 100). No matter how the market area for supplier 1 is drawn, customers inside it will still purchase some amount from supplier 2, and customers outside it will purchase some amount from supplier 1. As noted elsewhere in this book, a log-linear utility function implicitly assumes that the cross-price elasticity—e.g., dln[q1 ]/dln[p2 ]—is always zero and the elasticity of substitution dln[q1 /q2 ]/dln[p1 /p2 ] is always −1. The CES utility function shown in (8.10.7) is a more general utility function; it has two parameters (δ and ρ) where δ (like ν), constrained to lie between 0 and 1, is the relative preference for good 1 and ρ,constrained to lie between −1 and 0 is tied to the elasticity of substitution. From (8.10.8), dln[q1 /q2 ]/dln[p1 /p2 ] = −1/(1 + ρ). To illustrate ideas, consider now a customer located somewhere along the straight line joining Suppliers 1 and 2 on a rectangular plane. See Fig. 8.4. Let x denote the fraction of the way from Suppliers 1 to 2: x therefore ranges from 0 (customer located adjacent to Supplier 1) to 1 (customer located adjacent to supplier 2). When x is near 0, p1 + sx1 is relatively small and p2 + sx2 is relatively large; the opposite holds when x is near 1. In the case of a log-linear utility function, the budget share for a commodity is the same for customers everywhere along this line; however, for customers closer to Supplier 1, this in turn implies more quantity of that commodity. In the case of a CES utility function, both the budget share and quantity of commodity 1 increase as x approaches 0. As we increase x—move closer to Supplier 2—we find customers with a CES utility function reallocate budget share in favor of commodity 2. In fact, there is a distance (a fraction X of the length from Suppliers 1 to 2) where the budget share is the same regardless of substitutability (ρ). See (8.10.9) and distance OK in Fig. 8.4. We also see that as the elasticity of
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Fraction of budget spent on commodity 1
H
J
AB CD FG HI HJKI OK
Staking Out the Firm’s Market
Model 8G Budget share schedule by location for commodity 1 With log-linear utility function: ν = 0.45 With CES utility function: ρ = –0.60, δ = 0.45 With CES utility function: ρ = –0.78, δ = 0.45 With CES utility function: ρ = –0.92, δ = 0.45 With perfectly substitution Fraction of distance on line from Supplier 1 to Supplier 2 (X)
F
C
A
B
D
G 0
K I Fraction of distance from Firm 1 to Firm 2
Fig. 8.4 Model 8G: budget share schedule along line from Suppliers 1 to 2 using log-linear and CES utility functions. Notes: Horizontal axis scaled from 0 to 1; vertical from 0 to 1.2. Givens also include p1 = 11, p2 = 12, s = 10
substitution becomes larger (in absolute value) the budget share schedule transitions from AB to CD to FG to HI. The logical implication is that as the two goods approach perfect substitutes, the budget share approaches a Z shape (HJKI) where budget share is 1.0 when x < X, 0.0 for x > X, and in effect a vertical line at x = X. In such a case, Model 8H reduces to Model 8A (if p1 = p2 ) or 8B (if p1 = p2 .)
8.11 Model 8H: Destination Choice Under Uncertainty In recent decades, the discrete choice model22 has become popular as a means of explaining consumer shopping choices. Modeling of store choice is widely thought
22 A
statistical model used to predict consumer choice from among a set of discrete and denumerable alternatives. This includes the Multinomial Logit Model and Nested Logit Model.
8.12
Final Comments
227
to have begun with Reilly (1931).23 In its simplest form, discrete choice modeling imagines that from among all possible stores the consumer formulates a choice set. The consumer is then assumed to collect information about stores in the choice set and then choose one store to make the purchase.24 In so doing, there may be differences among consumers that lead them to formulate different choice sets and/or make different choices. The analyst is assumed to know only about some (not all) of the differences among consumers and some (again not all) of the differences among stores. Because of this imperfect information, we can never be sure that we have properly identified the consumer’s choice set nor the full set of factors that shape choice from within the choice set. Analysts typically use a variant of a multinomial logit model to predict shopping behavior. In a logit model, the log odds (reflecting the analyst’s uncertainty) of choosing a particular alternative are seen to gradually rise (fall) as conditions for that choice become more (less) favorable. By varying the magnitudes of the slope coefficients in the model, we can make consumers more sensitive (or less sensitive) to a particular factor. Earlier in this chapter, I presented several models in which there is a critical distance at which the customer suddenly and completely shifts from one supplier to another. In principle, it is possible to make the multinomial logit model reproduce a sudden and complete shift, but the model is most appropriate when change is gradual. What is not clear here, however, is just when and why change ought to be gradual. This brings us back to Hotelling’s thoughts on stability in competition.
8.12 Final Comments In this chapter, I have explored ways of defining the market area boundary for a firm. In Table 8.11, I summarize the assumptions that underlie selected models from 8A through 8G. Many assumptions are in common to all these models: see the list in panel (a) of Table 8.11. In 8A, the firm and its competitor sell the same commodity at the same given f.o.b. price. In 8B, the competitor sets an f.o.b. price that is different but given. In 8C, the firm gets to set a price that maximizes its profit assuming that competitors do not react. In 8D, I study how the firm’s market area boundary adjusts to capacity constraints. In 8E, I show how the firm’s market area boundary varies when it sells a different, but perfect substitutable good. In terms
23 Other
early work in the area includes Carrothers (1956), Brown (1957), and Huff (1963, 1964). Important contributions include Davis (2006), de Palma, Lindsey, von Hohenbalken, and West (1994), Iyer and Seetharaman (2008), Lee and Pace (2005), Miron and Lo (1997), Sheppard, Haining, and Plummer (1992), and Slade (2005). O’Kelly and Miller (1989) and Parr (1997b) are good summaries of work in this area. See also Berry, Parr, Epstein, Ghosh, and Smith (1988, Chap. 7). 24 Dudey (1990) considers the impact of consumer search behavior on the location choices of retailers. Also see Schulz and Stahl (1996).
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Table 8.11 Assumptions in Models 8A through 8G Assumptions (a) Assumptions in common A1 Closed regional market economy A3 Punctiform landscape B1 Exchange of commodity (or commodities) for money E2 Fixed unit shipping rate H2 Location(s) of firm given M1 Firm uses f.o.b. pricing (b) Assumptions specific to particular models F3 Competitor sells at same f.o.b. price A4 Rectangular plane F7 Location of competitor is given C4 Identical customers F1 Competitor sells same product F5 Competitor sells at a different f.o.b. price A5 Linear landscape D6 Fixed cost D7 Horizontal marginal cost curve F6 Firm is myopic C1 Fixed demand locations D2 Firm minimizes cost of production and shipping D3 No capacity exceeded D5 I factories D8 Reverse-L marginal cost curve C10 Two kinds of customers C7 Maximize same utility function F2 Competitor sells a different product
8A [1]
8B [2]
8C [3]
8D [4]
8E [5]
8F [6]
8G [7]
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x
x x x x x x x x
of the geometric shape of market areas, Models 8B, 8D, and 8E give similar outcomes. Models 8F, 8G, and 8H introduce differences among consumers, as well as imperfectly substitutable goods. In Chapter 1, I argue that there are important linkages between prices and localization. I illustrated that idea in Model 2D wherein price and localization were joint outcomes for the monopolist. In this chapter, I consider how a firm sets its price in response to the prices set by its competitors nearby and hence is affected by the geographic density of firms. The idea here has been to explore how, as a consequence, localization and price are jointly determined. Much more could be done here. There is an extensive game-theoretic literature in which the locations of firms and their prices are reactions to the actions of their competitors. By itself, that is the subject of another book. My modest aim in this chapter has been to provide pointers in that direction.
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Final Comments
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As in earlier chapters, I continue to assume that the prices of inputs (including shipping services) used by the firm are exogenous; input prices are all determined in markets outside the scope of the models in this chapter. As well, the models in this chapter are silent on prices in output markets. Once again, I think Walras would have argued that the analysis in this chapter has been only partial in the sense that we have not looked explicitly at the simultaneity among prices in these markets. Once again, we must wait until a later chapter for the opportunity to do that. The models in this chapter do not take into account the labor, capital, and land inputs used in production. As such, the models are of no use in looking at wages, interest and profit, and land rents. However, the models do provide some insight into the determination of commodity prices and thereby the well-being of consumers.
Chapter 9
The Cautious Farmer and the Local Market Market Participation Under Uncertainty (Economides–Siow Problem)
In market formation, risk-averse individuals choose larger markets to reduce price or liquidity risk. In a spatial economy, larger markets mean the marginal participant incurs added shipping costs. In this sense, the scale of the market may be limited by the tradeoff between the benefits and costs of participation. Model 9A assumes shipping costs are zero—absent a cost disadvantage, the firm (a farm here) is ever better off the larger the market. In Models 9B and 9C, nonzero shipping costs may limit size of market for a marginal participant. However, what about spatial equilibrium? If some farms are close to the market and others further away, there will be an incentive for farms at greater distance to relocate nearer the market. Model 9B assumes that farms reach spatial equilibrium by forming a cooperative in which members share the aggregate cost of shipping equally. Alternatively, Model 9C assumes that farms reach spatial equilibrium by bidding up the price (rent) for land at advantageous locations. Chapter 9 is the first in this book to look at firms as both producers and consumers. As we progress toward a model of location that characterizes the regional economy, it is important to integrate demand and supply. In Chapter 7, we began to think about the nature of a firm. There is a parallel here in the contrast between Models 9B and 9C. Model 9B redefines the nature of a firm because the cooperative internalizes a market transaction (for shipping) in the same way that a firm internalizes when it does some aspect of repair production in-house. This chapter therefore looks at how another aspect of localization (market organization) and price are jointly determined.
9.1 The Economides–Siow Problem So far, we have looked at a market mainly in terms of the constraint on participation posed by the existence of shipping costs. In the case of an expendable commodity, a customer is seen to participate in a market if the effective price (purchase price plus unit transaction cost) does not exceed the maximum price the customer would be willing to pay. As seen in Chapter 8, a unit shipping cost—which increases with distance shipped—limits the geographic extent of a market to its range. In the J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_9, C Springer Science+Business Media, LLC 2010
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case of an indispensable commodity, there is no corresponding notion of a market’s geographic extent. In Chapter 8, we considered the possibility of multiple suppliers of the same product at different places. In the presence of a unit shipping cost, the firm is thought to have a trade area that it shares with its competitors. A firm’s market area—the part of the trade area in which it dominates the competitors—can then be identified on the assumption that customers purchase from the supplier with the lowest effective price. However, this approach continues the assumption that a commodity is expendable without considering explicitly when, how, or why substitutes are available. Put differently, the models considered so far do not say anything about why or when someone would participate in a market for this particular product. The closest we have come so far is Model 8G where a utility function was introduced to model consumer choice between two commodities; in principle, this might mean that the consumer foregoes the market for one product in favor of the other. In this chapter, I begin to think about what motivates individuals to seek out a particular local market. Let me now cast the problem more generally. Why do local markets exist at various places across the landscape? So far, we have treated the local demand for a product and the local supply as exogenous. To this point, we have not asked what participants might do in the absence of a particular local market. In choosing to participate in a particular local market, a participant has more to gain than simply the ability to purchase a product. Presumably, to the extent that there is more than one possible market, participants might be attracted to one market over another because they think they are more likely to get a good price (i.e., a lower effective price) there. Underlying this idea is the notion that markets are inherently uncertain. Otherwise, if market outcomes were known with certainty, why would not otherwise-identical consumers always choose the same market? In this and the remaining chapters of this book, I look at models of agricultural markets. In each of them, I assume tenant farmers—as firms—produce a crop using rented land as an input and then ship all or part of that crop to a central place to exchange. I assume here initially that all land is equally fertile. I assume also that landlords maximize rents, are competitive (not collusive), and are numerous enough that each is a price taker in the local market for land. Such models can be used to describe a simple kind of regional economy. What makes these models interesting in part is that each firm occupies geographic space, and there is therefore a limit on the number of them that can be accommodated within a given land area.1 In later parts of this chapter, I use that feature of the model in thinking about how large a market might be in terms of the number of market participants. To the farmer, there are costs to exchange: e.g., the cost of shipping produce to a local market and possibly purchases back home. Presumably, a farmer plans a particular mix of produce taking into account the farm’s preferences, available resources, including skills of its workforce, and fertility of soil. Regardless of its plans, the harvest reaped by the farm depends also on events such as climatic variations that
1I
first considered density of customers in Chapter 7 and then again in Chapter 8 in more detail.
9.2
The Barter Market
233
are beyond the farm’s control. To maximize its utility, a farm whose harvest consists mainly of grain production might want to exchange with other farmers whose harvest was mainly other kinds of desirable produce: e.g., eggs, milk, meat, or vegetables. As a supplier, you decide the type and quantity of produce to bring to market without knowing in advance what will be brought by other farmers. As a consumer, in deciding whether to attend a local market, you cannot be sure in advance about the availability, quality, and exchange rate (price) of various produce. In the interest of simplicity, assume buyers and sellers have perfect information once they arrive at the local market and that this translates into a single Walrasian market-clearing exchange rate for the day; all those who want to exchange produce at that rate are able to do so. Under what conditions will a farm participate in a particular local market? If a farm has a choice between participating in a local market nearby or a larger market further away, which will it choose and why? This chapter is inspired by a pioneering spatial model of liquidity and market size in Economides and Siow (1988).2 At the same time, the E&S model is different from the one presented here. How? First, the E&S model assumes that actors maximize expected utility. Here, however, I assume behavior based on risk-return. A second difference from the E&S model is that I start with a version where geography plays no role. Later in the chapter, I introduce into this model a geography different from the E&S model. I will provide more detail on these and other differences later in this exposition.
9.2 The Barter Market Up to this point, we have looked only at markets in a fiat money economy wherein each commodity is exchanged for money. In this chapter, I introduce the notion of a barter market3 in which farmers meet to exchange commodities: i.e., they are both producers and consumers of produce. Put differently, buyers and sellers exchange commodities in barter rather than trade a commodity for money. Students typically think of a barter economy as primitive: something done in the absence of money. Another way to think about a barter economy is that it is essentially no different from a fiat money economy. Proponents here say that money itself is just one more
2 Economides and Siow (1988) are primarily concerned with the existence and size of financial markets. However, at the outset, that paper describes a simple locational model of trading by farmers, which is the focus of this chapter. Others who have made use of Economides and Siow (1988) to look at questions of location include Camacho and Persky (1990), Casella (2001), Gehrig (1998), Glazer, Gradstein, and Ranjan (2003), and Henkel, Stahl, and Walz (2000). 3 I use barter here in the economic sense of an exchange—a trade of some amount of one good in return solely for an amount of another good with no money involved—that takes place in the context of a market. This is seen here strictly as a matter of business; I exclude here any exchange (e.g., an exchange of gifts) where the motivation is, at least in part, something else.
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commodity. Among the reasons why people demand money is that it is a store of value; by a further transaction, you can readily get back most or all of what you gave up for it. Other commodities, such as gold, are also thought to be like money in the sense that they too are a store of value. If indeed money is a commodity, then in effect even exchanges in a fiat money economy are barter. While economic actors can meet up to match their needs on a pairwise basis, it is not always clear how an exchange rate (price ratio) gets established between every pair of commodities in a barter economy. Suppose, for example, we have a simple barter market that includes only two farmers. For the sake of argument, assume Farmer X arrives with an endowment of 0.4 units of wheat (commodity 1) and 0.6 units of corn (commodity 2): and that Farmer Y has an endowment of 0.60 units of wheat and 0.40 units of corn: qX1 = 0.40, qY2 = 0.40, qY1 = 0.60, and qX2 = 0.60. In Fig. 9.1, I draw a diagram—called an Edgeworth Box—for this problem. There,
A5 A3 A C3 C5
Good 2
de
f
c b Walrasian exchange rate
B5 B3 E
B D D3 D5
0
Good 1
Fig. 9.1 Barter and Walrasian price setting in market with two farmers (X and Y). Notes: ν = 0.3. Initial endowments are qX1 = 0.40, qX2 = 0.60, qY1 = 0.60, and qY2 = 0.40. Indifference curve reached in absence of barter: AbEB for person X; AfED for person Y. Walrasian exchange rate shown as dotted line passing through point A on the vertical axis and point d along the Marshallian Contract Curve bcdef. Horizontal axis scaled from 0.40 to 0.75; vertical axis from 0.40 to 0.65
9.2
The Barter Market
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point A represents the endowments before any barter. Through point A, I have drawn Person X’s indifference curve (AbEB) before barter. The idea of an Edgeworth Box is to show the indifference curves also for person Y; since total endowments are fixed, we can draw Y’s indifference maps from an origin at the upper right-hand corner of Fig. 9.1. In that case, Y’s indifference curve before barter would be AfED. The eye-shaped area above AbE and below AfE contains all the barter outcomes that would leave each of X and Y better off (or at least no worse off) than without barter. Where within this eye does bartering lead us? There are two answers to this question: Marshallian and Walrasian. Marshallian answer: Barter leads to an equilibrium in which neither party has an incentive to barter further; that is, the endowments after barter must be such that the indifference curves of X and Y are tangential at that point. In Fig. 9.1, I draw the locus of all points that satisfy this; it is labeled bcdef and generally called a Contract Curve. At point b on the Contract Curve, all the gains from barter accrue to Person Y, At point f, all the gains accrue to Person X. Points b and f are sometimes called the maximum concession points for that reason. At any point on the Contract Curve between b and f, both Person X and Person Y benefit. In the Marshallian view, where barter ends up along the Contract Curve depends on the bargaining strength of the two participants. Walrasian answer: Barter leads to the setting of an equilibrium exchange rate between the two commodities. At that exchange rate, the two participants trade commodities until neither party has the incentive to exchange further. Therefore, we must reach the point4 on the offer curve where the Walrasian exchange rate is tangential to the indifference curves of the two persons. In Fig. 9.1, I draw the Walrasian exchange rate as a dotted line that passes through the initial endowment (point A) and crosses the Contract Curve at point d where it is tangential to the indifference curves of both persons X and Y.5 In the Walrasian view, barter ends up at point d on the Contract Curve. In moving from a to d, Person X gives up 0.07 units of corn (the amount purchased by Person Y) to purchase 0.14 units of wheat (the amount given up by Person Y). For simplicity of exposition, I adopt the Walrasian view in the remainder of this chapter and find the exchange rate that will leave neither farmer wanting to trade any more.
4 Implicit in this description is an assertion that such a point exists and is unique. A determination of the conditions under which this assertion is valid is beyond the scope of this book. 5 The slope of the Walrasian exchange rate line is the negative of the exchange rate.
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9.3 Uncertainty and Rationality Economists often characterize markets as varying from thick6 to thin7 . As a thought experiment—with apologies in advance if the process appears mechanistic or vague—assume an asset market with the following characteristics: • A large number of potential buyers and a large number of potential sellers. Put differently, the asset is widely desired and widely held. • The asset is held at zero (storage) cost. However, buyers (sellers) incur transaction costs to acquire (dispose) of the asset. • Potential buyers and potential sellers each form a statistical population of individuals. Overall, these populations are both of the view that the market price of the asset is not expected to change in the future. However, individuals within either population randomly deviate from this. On any given market day, each potential buyer (seller) has a broad sense of (1) the current price and expectations about the future price of the commodity—expectations that differ randomly from person to person and from day to day—and of (2) the transaction costs associated with their participation in the market. • At the outset of any given market day, a subset of the potential buyers (call them market buyers) and a subset of the potential sellers (call them market sellers) engage in the market. By engage, I mean they undertake one or more of the following activities: search, gather, and analyze information; make contacts and establish relationships; negotiate price and terms; and acquire/dispose. Why only a subset? In my view, time and effort are required (i.e., transaction costs are incurred) to do these things. To acquire or dispose of a commodity, for example, these would—as noted earlier—include costs related to bank transactions and credit authorization, freight and transfer, storage and inventory, agency and brokerage fees, cost of insurance and other loss risks, installation and removal, warranty and service, and taxes and tariffs. As the day progresses, some market buyers/sellers will transact; others will—on the basis of the information obtained—choose not to transact. For simplicity, I assume that decisions to engage the market in previous days do not affect the transaction costs to be incurred today. • Other potential buyers and sellers do not participate in the market. I assume here that the perceived transaction costs are large enough relative to the gain expected keep such potential buyers or sellers out of the market that day. • Except insofar as transaction costs and price expectations in part vary randomly from one person to the next, market buyers and market sellers are no different
6 A market condition in which there are many buyers and sellers. From a search-theoretic perspective on markets, a seller in a thick market does not have to wait long to get a fair price for their good. 7 A market condition in which there are few buyers and sellers. From a search-theoretic perspective on markets, a seller in a thin market typically must wait longer to get a fair price for their goods.
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Uncertainty and Rationality
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statistically from other potential buyers and sellers; they can be thought of as just an independently drawn random sample. • Market sellers in aggregate have an upward-sloping supply schedule showing the amount they would sell at any given exchange rate. • Market buyers in aggregate have a downward-sloping demand schedule showing the amount they would purchase at any given exchange rate. • Following the Walrasian perspective, the market exchange rate that day settles at a level such that no market seller leaves with product that they would prefer to have sold at that exchange rate and no market buyer leaves without product they would have preferred to have bought at that exchange rate. Put differently, the market clears for market buyers and sellers that day. If the market did not clear, then either market suppliers would be left with inventory or market demanders would be left with unsatisfied demands. For an asset market that can be characterized this way, we would therefore expect to see some variation in market price from day to day even when no one expects price to change over the longer run. We also expect to see the number of market sellers or buyers rise one day and then perhaps fall the next on a random basis. I define the market to be thin when the numbers of market buyers and market sellers are small: thick when the numbers are relatively large. We might expect that the price of the asset in a thick market would be about the same from day to day because of the many participants. In a thin market under similar circumstances, however, we expect the exchange rate of a commodity to vary more from day to day. Put differently, there is more price risk8 in a thin market than in a thick market; the vendor in a thin market, for example, might get less than, or more than, either potential suppliers or potential buyers think is the expected price for the asset. Why engage in a market at all? In general, choosing a market can be seen as a means of reducing or spreading price risk. As a farmer, you do not necessarily need to participate in a weekly farmer’s market. You might have, for example, established relationships with one or more customers who travel to your farm weekly to purchase commodities. Why bother with the inconvenience of shipping to market if you can get a good exchange rate at the farm gate? However, if you do not attend the market, it is hard to know whether you are getting a good exchange rate; customers too might want to know if they are paying too much. For both farmer and customer, the market provides a means of assessing whether the exchange rate for a given transaction is fair. Even a thin market can be helpful here. However, the thicker the market, the less the price risk. 8 A loss (or increase in cost) arising because of an unforeseen change in market conditions that causes price to change over the short term, price risk is associated with price volatility. In a searchtheoretic perspective, sellers hold an asset until the price bid by a potential purchaser exceeds the vendor’s reservation price. Here, a distinction can be drawn between price risk and liquidity risk. Liquidity risk is the loss arising because of the delay in obtaining a bid at or above the reservation price. In practice, it is difficult to distinguish between price risk and liquidity risk. The approach in this book is to treat liquidity risk as simply an element of price risk.
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To implement the notion of price risk, we need to think about what it means to be rational under uncertainty.9 So far in the book, we have used the term rationality only in regard to choices made under certainty. On the one hand, we have assumed the firm maximizes profit or rate of return. On the other hand, we have assumed the individual maximizes utility, which in turn has been based on the consumption of commodities. In both situations, economists might argue that it is straightforward to model choice in the absence of uncertainty.10 In 1738, Daniel Bernoulli (a Swiss mathematician) developed a theory on the measurement of risk that set the stage for modern approaches.11 He starts from the notion of an expected value E[X]. For a discrete random variable (X), this is the sum of all possible occurrences of X each multiplied by the probability, p[X], of that occurrence: E[X] = x x(p[X = x]).12 However, he then assumes that 1. the value someone places on an outcome depends not on the expected money gain from a gamble but rather on the utility it yields; 2. the added utility, U[X], from a given money gain, [X], is more for a pauper than for someone who is rich; and 3. the expected value of the gain in utility, E[U] = x U[x]p[X = x], is what motivates individuals in a gamble. Note the implication here, drawn out by Bernoulli himself (p. 29), that no one would therefore rationally gamble in a fair game—one in which a dollar gain was as likely as a dollar loss—because the loss has a greater change in utility attached to it than does the gain. Bernoulli’s assertion (2) above is problematic with respect to the ordinality of utility in two regards. First, in effect, he assumes that individuals have a diminishing marginal utility of income. Why might this be problematic? Diminishing marginal utility of income itself need not be surprising since we commonly assume diminishing marginal utility in commodities consumption. However, in practice, we assume that a utility function is unique up to a monotonic transformation. For example, the utility functions f [x, y] = xb y1−b where 1 < b < 0 and g[x, y] = axb yc where b > 0, c > 0, and b + c < 1, calculated at consumption of x units of wheat and y units of corn, generate the same rank ordering: i.e., g[x, y] is a monotonic transformation of f [x, y]. The easiest way to think about diminishing marginal utility of
9 Important work in the area of utility and decision making under uncertainty includes von Neumann and Morgenstern (1947), Marschak (1950), Hurwicz (1953), Simon (1955, 1959), Koo (1959), Bishop (1963), Harsanyi (1965, 1966), Loomes and Sugden (1982), and Sugden (1991). 10 For the interested reader, Sugden (1991) discusses the philosophical limitations of neoclassical perspectives on rational choice. 11 Bernoulli (1954) is an English translation of that paper. 12 For example, if we toss a fair coin twice and let X be the number of times a head obtains. X can take on the values 0, 1, and 2. From the Binomial Theorem, we know that probabilities are 1/4, 1/2, and 1/4, respectively. Therefore, E(X) = 0(1/4) + (1/2)(1) + (1/4)2 = 1.0.
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Uncertainty and Rationality
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income is that b + c < 1, but then how would this differ from a monotonic transformation of b + c = 1? Second, Bernoulli assumes that the utility levels of consumers (the pauper and the rich person) are comparable which again violates the ordinality of utility. Although risk in the context of investment had long been of concern in Economics, it is Von Neumann and Morgenstern’s path-breaking book, Theory of Games and Economic Behavior, first published in 1944, that is widely credited with spawning the focus in Economics in general on game theory and in particular on the nature of rational behavior in the presence of uncertainty.13 That book builds on Bernoulli’s idea that economic actors maximize the expected value of the gain in utility. However, there are problems with expected utility maximization. (1) In practice, how do we find the required probabilities, especially when these may be subjective (Bayesian) in nature? (2) Do individuals have a taste for risk or an aversion to risk that leads them to prefer one gamble to another even when two gambles have the same expected utility? (3) More generally, why assume that rationality necessarily requires expected utility maximization? An alternative to analyze rational decision making under uncertainty is through a mean-variance (or, alternatively, risk-return) approach that originated with Markowitz (1952) and Sharpe (1963, 1964). Under this approach, we calculate two measures: (1) the expected utility ε—otherwise known as the mean or as the return—as per Bernoulli and (2) the variance ν—otherwise known as the risk— in utility.14 If two choices have the same return but different risks, the individual is thought to prefer the choice with the lower risk. If two choices have the same risk but different returns, the individual is thought to prefer the alternative with the higher return. If the two choices have different returns and different risks, then we need some way to measure the tradeoff between return and risk. Typically, this is done using what is termed a beta analysis.15 In this chapter, such a risk-return approach is used to characterize rational choice under uncertainty. Here, I distinguish between sub-utility and utility. Sub-utility is the level of happiness that arises from a choice when uncertainty is, or can be, ignored. Utility is the level of happiness after uncertainty has been taken into account; in a conventional beta analysis, utility is given by (9.1.1). Beta here is a parameter that measures the aversion of the individual to risk; when β = 0, the individual is indifferent to risk, for larger β, the individual is increasingly averse to choices with substantial risk.16 13 In my view, Georgescu-Roegen (1954, p. 503) is correct in pointing out that mathematicians and
statisticians dating back to Daniel Bernoulli and Gabriel Cramer had worked on similar ideas much before this. Harsanyi (1956) points out the similarities of game theory to earlier work by Zeuthen (1930). 14 That is, ε = U[x]p[X = x] and ν = (U[x] − ε)2 p[X = x]. x x 15 Beta is the increase in mean (return) required to offset a unit increase in variance if two alternatives are to be thought to be equally preferable. 16 This is an approach initially suggested by Markowitz (1976). See also Levy and Markowitz (1979) and Kroll, Levy, and Markowitz (1984).
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Many advances have been made using a risk-return approach in an area now known broadly as financial engineering. However, the approach is not without its critics. Among these are the following: • Some economists are not fond of the risk-return approach; they prefer an approach better grounded in neoclassical utility theory. To simplify the investment problem here, imagine an individual choosing between a risky investment with a higher expected return and a risk-free investment with a lower return. In effect, by declining the risky investment, individuals forego an amount (the amount by which the expected return is higher in the risky investment) to guarantee their wealth (the principal invested) at a future date. As such, deciding to invest in the risk-free alternative is like buying insurance and should be analyzable in that way. • The risk-return approach does not directly incorporate an asymmetry (skewedness) of gains and losses as proposed by Bernoulli. • The risk-return approach—as usually applied in investment analysis—assumes a continuity across investment choices: i.e., the ability to blend investments at differing levels of risk. The location problem that I consider in this chapter exhibits a kind of lumpiness that needs to be addressed specifically. On the other hand, students tell me that they, or their parents, deal with financial advisors who regularly cast investment portfolio choices in terms of risk versus return. Therefore, I find it helpful pedagogically to cast this problem using a riskreturn approach.
9.4 Model 9A: Non-spatial Market Assume an economy made up of otherwise-identical farmers. Each farmer has the same log-linear utility function for sub-utility defined over the consumption of two commodities: see (9.1.2) in Table 9.1, wherein I summarize equations, assumptions, notation, and rationale for localization in Model 9A.17 Here, let q1 and q2 be the amounts of wheat and corn, respectively, that the farmer consumes, and let U be the level of sub-utility achieved by the farmer. As q1 approaches zero in (9.1.2), so does U; the same is true for q2 . The model promotes trading by assuming that each farmer is randomly assigned—at harvest time—an initial endowment of one unit of one commodity and none of the other: i.e., (1, 0) or (0, 1). I refer to those with (1, 0) as having a wheat endowment and those with (0, 1) as having a corn endowment. In such circumstances, farmers have an incentive to exchange commodities. If they do not, their utility will be zero. An endowment here is production net of costs; for the moment, I leave unstated exactly how the agricultural commodity is produced except to say 17 Economides
and Siow (1988) also look at the case where the utility function is Constant Elasticity of Substitution (CES).
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Model 9A: Non-spatial Market
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Table 9.1 Model 9A: farmers and the market in a non-spatial economy Utility of farmer U − βV
(9.1.1)
Sub-utility of farmer U = q1ν q21 − ν
(9.1.2)
Initial endowment of farmer P[1, 0] = 1 − φ and P[0, 1] = φ
(9.1.3)
Mix of farmers in market N = N1 + N2
(9.1.4)
Probability of getting N1 wheat farmers in market N = N! / (N !(N − N )!) P[N1 ] = CNN1 (1 − φ)N1 φ N−N1 where CN1 1 1
(9.1.5)
Expected number of wheat and corn farmers in market E[N2 ] = φN E[N1 ] = (1 − φ)N
(9.1.6)
Wheat and corn expected to be offered in market E[Q2 ] = νφN E[Q1 ] = (1 − ν)(1 − φ)N
(9.1.7)
Ratio of expected offers (wheat per unit of corn) E[Q1 ] / E[Q2 ] = (1 − ν)(1 − φ) / (νφ)
(9.1.8)
Notes: C—Combination symbol: Cab = b!/(a!(b − a)!); P—Probability. Rationale for localization (see Appendix A): Z4—Risk-spreading and insurance; Z6—Differences among consumers; Z7— Variation in goods. Givens (parameter or exogenous): N—Number of farmers in market; β—Risk aversion parameter; ν—Exponent of wheat in utility function. Relative preference for wheat; φ— Probability farmer has (0,1) endowment. Outcomes (endogenous): N1 —Number of wheat farmers in market; N2 —Number of corn farmers in market; Q1 —Aggregate quantity of wheat offered in market; q1 —Quantity of wheat consumed by farmer; Q2 —Aggregate quantity of corn offered in market; q2 —Quantity of corn consumed by farmer; U—Sub-utility of farmer; V—Variance.
that each farmer is efficient and that those who harvest corn (as well as those who harvest wheat) produce an amount of 1 unit of the commodity net of costs including the opportunity cost of land (rent). Further, assume that these initial endowments obtain as though outcomes were random and statistically independent and that for each farmer there is a probability φ that he or she will be endowed with corn, and therefore 1 − φ probability of being endowed with wheat. See (9.1.3). In illustrating Model 9A (and again in Model 9B and Model 9C that follow), I use particular values for ν and φ. I assume ν = 0.3, which implies consumers prefer to consume relatively more wheat than corn. I assume φ = 0.4 which means that farmers are more likely to be endowed with wheat than corn. Together, these two values describe a world in which 60% of farmers are endowed with wheat, but where each farmer wants to spend only 30% of his or her endowment on wheat consumption. In that sense, our farmers would be happier if endowments of corn were more commonplace (in other words, if 1 − φ were closer to ν) and less happy otherwise. This further contributes to the imperative to trade. If they do participate in a market, they give up a portion of their initial endowment to get some amount of the other commodity to consume.
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Suppose N farmers constitute a market for this purpose: see (9.1.4). To simplify the subsequent analysis, I assume that farmers decide on their size of market in advance of knowing either their endowment or that of anyone around them (i.e., the endowments of other farmers who might be in the same market). This assumption may seem strange. After all, why not let farmers choose their market later. However, this assumption will make more sense later in this chapter when we introduce geography into the model. Having decided on a size of market and having subsequently harvested and realized their endowment, I assume that the farmers then meet in this market, transact as best they can, and get the utility that arises to all farmers with their endowment at the conclusion of the market. As stated above, the endowment for each farmer among these N is stochastic and independent of the endowment of any other farmer. Then, we can think of the mix of endowments at the marketplace as the outcome of a Bernoulli process18 : i.e., consists of N independent trials (one for each farmer) wherein there is a fixed probability φ that a particular outcome—a (0, 1) endowment—occurs.19 In that case, the market outcomes, as measured by N1 —the number of occurrences of a wheat farmer—follow a Binomial probability distribution: see (9.1.5). The number of corn farmers in this market is N2 = N − N1 and the fraction that they make of the market is k = N2 /N. Because N1 is a stochastic variable that is binomially distributed, it has a known expected value, E[N1 ]: see (9.1.6). Because N2 is a linear function of N1 , it too is a stochastic variable and has a binomial distribution with a calculable expected value. In a market of size N, we expect (on average) that farmers with a wheat endowment and farmers with a corn endowment will each offer in total the amounts of wheat and corn shown in (9.1.7) in exchange for the other crop. The ratio of these— the exchange rate of corn in terms of wheat—is shown in (9.1.8).20 However, the actual exchange rate will differ from the ratio of expected offers because k can (and often does) differ from φ. If we have N1 wheat farmers and N2 corn farmers in the market, wheat farmers will offer a total of (1 − ν)N1 units of wheat, corn farmers will offer a total of νN2 units of corn, and on average an equilibrium exchange rate of (1 − ν)(1 − k)/(νk) will therefore result. However, we do not always get this average exchange rate: as when farmers arrive at the market to discover to their chagrin that k = 0 or k = 1.21 Consider a 18 A
Bernoulli trial is a statistical experiment which can result in only one of two possible realizations. An experiment consisting of a series of independent Bernoulli trials is called a Bernoulli process. 19 We would not have a Bernoulli process if each individual could wait until harvest time to see his endowment and that of his or her neighbors before deciding in which local market to participate. 20 Economists usually say in this case that wheat is numéraire which means that other goods (corn in this case) are valued in units of wheat. 21 In an earlier footnote, I raised the question of whether a Walrasian outcome existed and was unique. In the case of a log-linear utility function, the answer intuitively is straightforward. Each farmer maximizes utility by allocating income so that the proportions spent on wheat and corn are α and 1 − a, respectively. At the equilibrium exchange rate, (1 − α)(1 − k)/(ak), a Walrasian solution exist; the market clears and farmers of each endowment are as well off as possible. The Walrasian solution is also unique; no other exchange rate clears the market.
9.4
Model 9A: Non-spatial Market
243
Table 9.2 Possible realizations in market of size 2 wherein ν = 0.3 and φ = 0.4 Wheat endowment
Corn endowment
q1 [5]
q2 [6]
U1 [7]
q1 [8]
q2 [9]
U2 [10]
W[k] [11]
Ex [12]
1.00 0.30 –
0.00 0.30 –
0.00 0.30 –
– 0.70 0.00
– 0.70 1.00
– 0.70 0.00
0.000 0.500 0.000 0.240 0.082
– 2.33 – 2.33
(b) Model 9B (Spatial: g = 30, r = 0, s = 0.40) 0 2 0.00 0.36 1.00 0.00 0.00 1 1 0.50 0.48 0.29 0.29 0.29 2 0 1.00 0.16 – – – Expected value Variance
– 0.67 0.00
– 0.67 1.00
– 0.67 0.00
0.000 0.480 0.000 0.230 0.075
– 2.33 – 2.33
(c) Model 9C (Spatial: g = 30, s = 0.40) 0 2 0.00 0.36 1.00 0.00 1 1 0.50 0.48 0.28 0.28 2 0 1.00 0.16 – – Expected value Variance
– 0.66 0.00
– 0.66 1.00
– 0.66 0.00
0.000 0.475 0.000 0.228 0.074
– 2.33 – 2.33
N2 [1]
N1 [2]
k [3]
P[k] [4]
(a) Model 9A (Non-spatial) 0 2 0.00 0.36 1 1 0.50 0.48 2 0 1.00 0.16 Expected value Variance
0.00 0.28 –
Notes: Calculations by author. See also Table 9.1. In panel (b), mean unit shipping cost is 0.0689. –, Indicates no one of that type present. See also Table 9.1. Ex, Exchange rate: units of wheat per unit of corn Endogenous: U[N], Utility of being in market of size N; V[N] Variance in utility in market of size N; W, Average utility weighted by number of farmers of each type.
market of just two farmers (N = 2). Remember here that I assume farmers choose a market in advance of knowing their endowment. There are three possible realizations for N1 : 0, 1, or 2. When φ = 0.4 and ν = 0.3, the ratio of expected offers (9.1.8) is 3.50. For each possible outcome of N1 , the corresponding binomial probability is shown in panel (a) of Table 9.2. In the event, N1 = 0 or N1 = 2 here, the market consists entirely of corn endowments or wheat endowments, respectively, and the utility for each farmer in the market is therefore zero. When N = 2, there is a probability of 0.5222 that the market participants will have a zero utility: i.e., return home without any of the other commodity. Suppose instead N1 = 1, which means we have one farmer of each type in the market. The farmer with a wheat endowment consumes 30% of his or her endowment of wheat, and trades the remaining 70% for corn. The farmer with a corn endowment consumes 70% and trades the remaining 30% for wheat. There are two possible utilities. With a probability (1 − k), the farmer will have an endowment of (1, 0), a consumption bundle
22 0.36
+ 0.16.
244
9 The Cautious Farmer and the Local Market
(0.3, 0.3), and a utility (U1 ) of 0.3. With a probability k, the farmer will have an endowment of (0, 1), a consumption bundle (0.7, 0.7), and a utility (U2 ) of 0.7. In this situation, the farmer with the corn endowment is better off after trade than the farmer with the wheat endowment: not surprising given that the corn is the commodity more strongly preferred. Put differently, the farmer is here guaranteed a utility of 0.3 with a 50% chance of getting 0.7 instead. By calculating W[k] = 0.5,23 I am simply taking an average of the two possible utilities weighted by the probabilities of the two outcomes. Considered over all the possible realizations of k, the weighted average utility of being in a market of N = 2, U[N], from columns [4] and [11] in panel (a) of Table 9.2, is 0.240.24,25 The variance in this utility, V[N], is also now calculable; V[N] = 0.082.26 What is the expected exchange rate across the three possible realizations of N1 from 0 to 2? We cannot calculate an exchange rate when k = 0 or k = 1 because no trade happens. In column [4] of panel (a) in Table 9.2, we see that the probability that k = 0 or k = 1 is 0.52. From column [12], we see that at the only other possibility, k = 0.50, the exchange rate is 2.33 units of wheat per unit of corn. I label this the Conditional Expected Exchange Rate (CEER); that is, the exchange rate we expect on average on the condition that k is neither 0 nor 1. This is different from the ratio of offers expected, (9.1.8), which incorporates the amounts offered when k = 0 and k = 1. In comparison, the ratio of offers expected is 3.50 when ν = 0.3 and φ = 0.4 as noted above. To begin thinking about what might happen if N were larger than 2, let us do similar calculations for a market of N = 8 participants. As before, I continue to illustrate using the case where ν = 0.3 and φ = 0.4. See panel (a) of Table 9.3. Consider first the case where N1 = 3. The three farmers with a wheat endowment each offer 0.7 units of wheat in exchange for corn. The remaining five farmers have a corn endowment; each offers 0.3 units of corn in exchange for wheat. The equilibrium exchange rate here is therefore 1.4027 units of wheat per unit of corn. Each farmer with a wheat endowment therefore consumes 0.3 units of wheat, and 0.528 units of corn for a utility of 0.43. Each farmer with a corn endowment consumes 0.7 units of corn and trades away the remaining 0.3 units in exchange for 0.4229 units of wheat to achieve a utility of 0.60. Therefore, the weighted average of utilities, W[k], is now 0.536.30 Considered over all the possible realizations of k as shown in Table 9.3, the weighted average utility of being in a market of N = 8, U[N], from columns [4] and [11] in panel (a) of Table 9.3, is 0.423 and the variance V[N] is now 0.087.
23 0.5(0.3) + 0.5(0.7). 24 0.52(0.00) + 0.48(0.50). 25 This
is another place where ordinalists might well cringe. If utility is indeed ordinal, what does it mean to take a linear combination of utilities as we do when we calculated U[N]. 26 0.52(0.00 − 0.24)2 + 0.48(0.5(0.30 − 0.24)2 + 0.5(0.70 − 0.24)2 ). 27 (3(0.7)/(5(0.3)). 28 0.7/1.40. 29 (0.3)(1.40). 30 (3/8)(0.43) + (5/8)(0.60).
9.4
Model 9A: Non-spatial Market
245
Table 9.3 Possible realizations in market of size 8 wherein ν = 0.3 and φ = 0.4 Wheat endowment
Corn endowment
q1 [5]
q2 [6]
U1 [7]
q1 [8]
q2 [9]
U2 [10]
W[k] [11]
Ex [12]
1.00 0.30 0.30 0.30 0.30 0.30 0.30 0.30 –
0.00 0.04 0.10 0.18 0.30 0.50 0.90 2.10 –
0.00 0.08 0.14 0.21 0.30 0.43 0.65 1.17 –
– 4.90 2.10 1.17 0.70 0.42 0.23 0.10 0.00
– 0.70 0.70 0.70 0.70 0.70 0.70 0.70 1.00
– 1.25 0.97 0.82 0.70 0.60 0.50 0.39 0.00
0.000 0.224 0.348 0.437 0.500 0.536 0.539 0.488 0.000 0.423 0.087
– 16.33 7.00 3.89 2.33 1.40 0.78 0.33 – 4.84
(b) Model 9B (Spatial: g = 30, r = 0, and s = 0.40) 0 8 0.00 0.02 1.00 0.00 0.00 1 7 0.13 0.09 0.28 0.04 0.07 2 6 0.25 0.21 0.28 0.09 0.13 3 5 0.38 0.28 0.28 0.17 0.19 4 4 0.50 0.23 0.28 0.28 0.28 5 3 0.63 0.12 0.28 0.46 0.40 6 2 0.75 0.04 0.28 0.83 0.60 7 1 0.88 0.01 0.26 1.94 1.08 8 0 1.00 0.00 – – – Expected value Variance
– 4.52 1.94 1.08 0.65 0.39 0.22 0.09 0.00
– 0.65 0.65 0.65 0.65 0.65 0.65 0.65 1.00
– 1.16 0.90 0.75 0.65 0.55 0.46 0.36 0.00
0.000 0.207 0.320 0.403 0.461 0.494 0.497 0.450 0.000 0.390 0.074
– 16.33 7.00 3.89 2.33 1.40 0.78 0.33 – 4.84
(c) Model 9C (Spatial: g = 30, and s = 0.40) 0 8 0.00 0.02 1.00 0.00 1 7 0.13 0.09 0.27 0.04 2 6 0.25 0.21 0.27 0.09 3 5 0.38 0.28 0.27 0.16 4 4 0.50 0.23 0.27 0.27 5 3 0.63 0.12 0.27 0.44 6 2 0.75 0.04 0.27 0.80 7 1 0.88 0.01 0.27 1.86 8 0 1.00 0.00 – – Expected value Variance
– 4.35 1.86 1.04 0.62 0.37 0.21 0.09 0.00
– 0.62 0.62 0.62 0.62 0.62 0.62 0.62 1.00
– 1.11 0.86 0.72 0.62 0.53 0.45 0.35 0.00
0.000 0.199 0.308 0.388 0.444 0.476 0.479 0.433 0.000 0.375 0.069
– 16.33 7.00 3.89 2.33 1.40 0.78 0.33 – 4.84
N2 [1]
N1 [2]
k [3]
P[k] [4]
(a) Model 9A (Non-spatial) 0 8 0.00 0.02 1 7 0.13 0.09 2 6 0.25 0.21 3 5 0.38 0.28 4 4 0.50 0.23 5 3 0.63 0.12 6 2 0.75 0.04 7 1 0.88 0.01 8 0 1.00 0.00 Expected value Variance
0.00 0.07 0.12 0.19 0.27 0.38 0.57 1.04 –
Note: Calculations by author. See also Table 9.2. In panel (b), mean unit shipping cost is 0.12.
What is CEER when N = 8? Here we see from column [4] in panel (a) of Table 9.3 that the probability that k = 0 or k = 1 is 0.018, and from column [12] that, for k in between, the exchange rate varies from 16.33 down to 0.33 which yields CEER = 4.84. This is higher than the 3.50 we noted above for the ratio of expected offers: the opposite of what we had found when N = 2. In this model, the farmer chooses the size of market in which to trade. This means the farmer will compare the combination of return, U[N], and risk, V[N], with those
246
9 The Cautious Farmer and the Local Market
achievable at other sizes of market.31 We are now able to compare the case of N = 2 and N = 8. When N = 8, the return is larger (0.423 vs. 0.240) but so too is the risk (0.087 vs. 0.082) compared to N = 2. In a risk-return analysis, the farmer would therefore prefer N = 8 over N = 2 if β is sufficiently small and prefer N = 2 otherwise. To understand what is happening to CEER, suppose we let N vary from 2 to 30 and calculate CEER at each N. The resulting estimates of CEER are plotted in Fig. 9.2. The dotted line is the ratio of expected offers (9.1.8); it is horizontal because this ratio is the same at every N.32 The solid curve shows CEER as a function of N. CEER is below the ratio of expected offers when N = 2, rises quickly, and peaks well above the ratio of expected offers at about N = 8, then begins to fall off asymptotically to the ratio of expected offers as N becomes large. We are now ready to answer some questions. • Why is the equilibrium exchange rate low at small N? This is because (1) the exchange rate is a declining function of k, (2) the equilibrium exchange rate cannot be calculated at k = 0 or k = 1, and (3) P(k = 0) is larger than P(k = 1)
Chapter 9: Conditional expected exchange rate (CEER) and ratio of expected offers
Exchange rate
AB CD
Fig. 9.2 Conditional expected exchange rate (CEER) and size of market. Notes: α = 0.30 and φ = 0.40. Horizontal axis scaled from 0 to 30; vertical axis from 0 to 6
31 In
Ratio of expected offers Conditional expected exchange rate
D B
A
C
0 Size of market (N)
Economics, utility is thought to be an index (or ordering) of preference among choice. As such, utility is an ordinal measure. However, when we calculate U[N], we appear to treat utility as though it were a cardinal measure. For a discussion of the issues raised, see Ellsberg (1954). 32 In this regard, Economides and Siow (1988, p. 110) appear to err in arguing that equilibrium price is independent of N because aggregate supply and demand for each commodity are proportional to N. My sense is that this confuses CEER and the ratio of expected offers.
9.4
Model 9A: Non-spatial Market
247
• Why is CEER then above the ratio of expected offers for N sufficiently large (above N = 4 in Fig. 9.2)? This happens because of the asymmetry of an exchange rate. To see this, suppose the quantities of the two commodities offered in the market are identical; the exchange rate here is 1.00. Now consider increasing the quantity of either the numerator or denominator. As we increase the denominator quantity, the exchange rate can drop from 1 to as low as 0. As we increase the numerator quantity, the exchange rate rises from 1 without limit. This asymmetry means that, ignoring the effect of the exclusion of k = 0 and k = 1 offers, CEER should be systematically higher than the ratio of offers expected in (9.1.8). However, this bias dissipates as the size of market becomes larger. Of course, CEER is not the key variable here. To understand how farmers choose markets, we must look at how market size affects utility. In Fig. 9.3, I use a solid line—the risk-return curve—to connect combinations of U[N] and V[N] attainable—for each level of N, again from N = 2 to N = 30. The attainable combination at each N is a black dot on this curve. Further, I have labeled the size of market at selected dots in Fig. 9.3. Here, we see that, as N is increased, the mean increases and the variance starts dropping above N = 4. In fact, the locus of points on this curve from N = 4 through N = 30 suggests that U[N] will continue to increase, and V[N] will decrease albeit both ever more slowly as N becomes still larger. In a beta analysis, we assume that the farmer is willing to accept a higher risk associated with a higher reward. One such tradeoff curve is shown as a dotted line in Fig. 9.3. We can imagine a family of such dotted lines, all parallel, such that the farmer is
30
turn
Risk-re
8 6 5 4
tradeoff
Risk: U[N]
3
Fig. 9.3 Risk, return, and size of market. Notes: α = 0.30 and φ = 0.40. Market size, N, shown as labeled dots for selected N from 2 to 30. Horizontal axis scaled from 0 to 0.1; vertical axis from 0 to 0.45
2
0 Risk: V[N]
248
9 The Cautious Farmer and the Local Market
happier the higher and to the left the tradeoff curve lies. Given the risk-return curve is negatively sloped above N = 4, the farmer would prefer to be in as large a market as possible. This is not surprising. After all, there is no disincentive here to join a larger market, and the larger the market the higher U[N] and the lower V[N]. In Fig. 9.3, I also include the case of autarky (N = 1) wherein farmers do not assemble into a market. In autarky under either endowment, the farmer gets a sub-utility of 0 with certainty: i.e., a zero variance. Autarky corresponds to the origin—that is, the intersection of the horizontal and vertical axes—in Fig. 9.3 since U[N] = 0 and V[N] = 0 there.33 If we were to draw two risk-return tradeoff lines, each parallel to the dotted line in Fig. 9.3, one drawn through the point where N = 2 and the other through the autarky point (the origin), we see that the utility of being in a market of size 2 exceeds the utility of not being in autarky. A similar analysis would lead us to conclude that utility is higher at N = 3 and still higher at N = 4. Put another way, the slope of the risk-return tradeoff drawn in Fig. 9.3 is sufficiently small (i.e., the risk premium, β, was low enough) to initiate an agglomeration process. Does the reverse argument hold true? Is there a β sufficiently large that farms might never switch from isolation (N = 1) to a market where N = 2, or from there to move to markets of N = 3 or N = 4? Consider the line in Fig. 9.3 joining autarky to N = 2. If β were sufficiently large to make the risk-return tradeoff line steeper than this, the utility of autarky would appear to be greater than the utility of N = 2. Suppose alternatively the risk-return tradeoff line passing through the origin cuts the line segment joining N = 2 and N = 3. Then the utility of being in a market of N = 3 would be greater than the utility of autarky. If so, does that imply that farms would never get to a market size of 3, even though it is advantageous, because no one would first have the incentive to form a market of 2? If so, this would be a disturbing feature of the model because it would imply that planners would have to do what a market could not: i.e., push myopic farmers, unable to see the benefits from further agglomeration, from autarky into a market of N = 2 to make it possible for farmers to then form a market of 3 or more. However, I think the problem here, specifically any reluctance to move from autarky to N = 2, when β is large, is actually indicative of a limitation of riskreturn analysis. In autarky, the farmer receives a sub-utility of zero with certainty. When N = 2, the farmer receives a sub-utility of zero if both farmers have the same endowment, and a larger sub-utility if they do not. In other words, at the worst, the farmer in an N = 2 market does as well as in autarky. In my view, the farmer in an N = 2 market is always therefore at least as well off as in autarky and has the possibility of being better off. Even if we assume that the farmer is myopic about the prospect of more farmers joining the agglomeration and pushing utility even higher, there is an incentive here to form a market of N = 2.34
33 CEER
is not calculable because there is no trade. my view, the shortcoming of risk-return analysis here is that the notion of variance loses generality when the number of realizations of a random variable (here, realizations of k) is small.
34 In
9.5
Model 9B: Cooperation in a Spatial Market
249
9.5 Model 9B: Cooperation in a Spatial Market Up until now, we have assumed a non-spatial market. How does space affect this model? In earlier chapters, I have shown how shipping costs shape market size. Let us now consider how shipping costs would affect the behavior of the bartering farmers here. In Economides and Siow (1988), farms are spread out (at fixed density) along a line left and right of the market point. As I understand the E&S model, this implies that farmers close to the market are better off than farmers who travel a longer distance to get to the market. To establish and maintain farmers in their locations in equilibrium, no farmer should be able to benefit by relocating. There must be some process to ensure that farmers are in equilibrium and do not have an incentive to further relocate. As I understand the E&S model, no such equilibrium process is specified. To correct this, I first add a cooperative process in this section (Model 9B) wherein farmers in a market form a club to share shipping costs in the market equally. In the next section (Model 9C), I introduce an alternative: a noncooperative process wherein farmers nearer the market bid a market rent premium for their plots of land compared to more remote farms. As it turns out, specifying such processes also helps us better rationalize the idea that farmers choose markets in advance of knowing their endowments. I will return to this subject shortly. In this section, I model farms forming a club or cooperative to share the cost of participating in the market (i.e., shipping costs).35 See Table 9.4. As used here, a club is an association whose purpose is to provide a benefit to its members. In return for paying a membership fee (f) each harvest, the club here provides free shipping services to the member. A club is like a firm except that it is not intended to make a profit: see (9.4.6).36 Usually, there is an optimal size of club—that is to say, a desired (in this case, most efficient) membership level. As well, implementation of a club also usually involves restrictions on nonmembers who might otherwise get a free ride—that is, benefit from the activities of the club without paying the membership fee. To keep the model simple, I assume that there are no costs to the formation or enforcement of a club other than the cost of shipping. I imagine here that each farmer envisages an optimal size of club and costlessly seeks out peers with a similar view until the optimal number of members has been obtained. If, say, many farmers see an optimal club size of 10 farmers, then additional clubs will form,
35 See
McGuire (1972) on economic models of club formation. For other modeling of cooperation in a geographic context, see Jayet (1997) and Soubeyran and Weber (2002). 36 Here I implicitly assume contingent shipping rates. That is, the cost of shipping wheat a kilometer is s units of wheat, and the cost of shipping corn a kilometer is s units of corn. Similarly, the coop fee is contingent; it is f units of wheat if the farmer has a wheat endowment, and f units of corn if a corn endowment. There is no adjustment here for the exchange rate between wheat and corn that will be obtained in the market. For the storyteller, the advantage of this scheme is that it simplifies decision making for the farmer who is still in anticipation of the harvest and does not yet know his or her endowment.
250
9 The Cautious Farmer and the Local Market Table 9.4 Model 9B: cooperative farmers in a spatial economy
Outer√boundary of farm at ring i from market xi = (i/(πg))for1 ≤ i ≤ N
(9.4.1)
Mid radius √ for farm at ring i mi = ((i − 0.5)/(πg))for1 ≤ i ≤ N
(9.4.2)
Endowment for farm at ring i net of shipping cost (1 − f , 0) or (0, 1 − f ) for 1 ≤ i ≤ N
(9.4.3)
Consumption of wheat and corn and sub-utility of farm with wheat endowment q11 = ν(1 − f ) q12 = (1 − ν))(1 − f ) / P U1 = (1 − f )ν ν (1 − ν)(1 − ν) / P1 − ν (9.4.4) Consumption of wheat and corn and sub-utility of farm with corn endowment U2 = (1 − f )ν ν (1 − ν)(1 − ν) Pν q21 = ν(1 − f )P q22 = (1 − ν)(1 − f ) (9.4.5) Coop’s balanced budget fN = i smi
(9.4.6)
Minimum β for farmers participating in this co-operative β ≥ (U[N − 1] − U[N]) / (V[N − 1] − V[N])
(9.4.7)
Maximum β for farmers participating in this co-operative β ≤ (U[N] − U[N + 1]) / (V[N] − V[N + 1])
(9.4.8)
Notes: See also (9.1.1) through (9.1.6) I and Table 9.1. Rationale for localization (see Appendix A): Z4—Risk spreading and insurance; Z6—Differences among consumers; Z7—Variation in goods; Z8—Limitation of shipping cost. Givens (parameter or exogenous): g—Density of farms (farms per square kilometer); N—Number of farmers in market; r—Opportunity cost of land (assumed zero); s—Unit shipping rate; β—Risk aversion parameter; ν—Exponent of wheat in utility function. Relative preference for wheat; φ—Probability farmer has (0,1) endowment. Outcomes (endogenous): f—Co-operative fee; mi —Mid-radius of farm i; P—Market exchange rate; Q1 — Aggregate quantity of wheat offered in market; q1i —Quantity of wheat consumed by farmer of type i; Q2 —Aggregate quantity of corn offered in market; q2i —Quantity of corn consumed by farmer of type i; U—Sub-utility of farmer; V—Variance; xi —Outer boundary of farm i.
each containing 10 farmers. Market formation here is an externality. The action of one farmer in choosing to join a particular market affects the well-being of others in a way that is unpriced. The club, as an organizational form, is a mechanism that captures (internalizes) this externality or spillover. Once formed, I assume that the club requires its members to locate so as to minimize the total shipping cost to be shared. Basically, the farms are tightly packed around the market point; this also helps control the free rider problem by keeping nonmembers further away from the market point. In what I characterize as an accretion process, assume that member farms are therefore required to form concentric rings around the market point. See the three panels in Fig. 9.4 where I map the locations of farms, each farm with the same area, when N = 6, N = 4, and N = 2, respectively. The farmer can be thought to ship from the mid-radius of the
9.5
Model 9B: Cooperation in a Spatial Market
(a) N = 6
(b) N = 4
251
(c) N = 2
Fig. 9.4 Models 9B and 9C: Maps of farms as rings in market of N = 6, N = 4, and N = 2
farm.37 The rationale for making each farm annular is that this shape minimizes the cost of shipping their endowments to the market. I ignore here other considerations that might shape the farm in a geographic sense.38 In effect, the market is at the center of the innermost circle: the center of farm 1. Since each farm occupies the same amount of land, the shipping cost associated with the marginal farmer (i.e., the farmer furthest from the market) and the fee (average cost) paid by each farmer in a coop increases at a decreasing rate with the number of farmers in the market. See Fig. 9.5. As I have already assumed land is plentiful, I do not need to worry about the possibility of clubs with overlapping market areas; a club would simply move to an unoccupied area so that it can achieve the same low total shipping cost as any other club of the same size. In this model, farmers may differ from one another in the following ways: (1) a randomly determined endowment not known in advance of club formation; (2) a given tradeoff between risk and return; and (3) a location vis-à-vis the market that the farmer can choose. Otherwise, I assume farmers are identical: same preferences for wheat and corn and same fee to join a given coop. Therefore, a club will be formed by farmers with similar tradeoffs between risk and return: i.e., similar βs. Once in a club, the farmer is indifferent to location because the club pays the marginal cost of shipping from that site to the market. Assume each farmer uses a fixed amount of land, 1/g, in agricultural production.39 Assume that land is not used for any other purpose (we ignore here any need for land for transportation, for a market site, for housing, or for the production of
37 Mid-radius
here is the distance which divides the farm into two equal areas. instance, while a ring might be the most efficient shape for getting the agricultural commodity to market, it may be inefficient for the daily chores of the farmer throughout the growing season. Thünen (1966, chaps. 11 and 13) discusses aspects of this problem. 39 This might be because each farmer has a Leontief technology that requires all inputs be in fixed proportion; however, the model to this stage is silent on other inputs to production. 38 For
252
Model 9B AB
Average cost (cooperative fee)
CD Marginal cost
D
Shipping cost per former
Fig. 9.5 Model 9B: Shipping cost for marginal farmer and average shipping cost (coop fee) as function of the number of farmers in the market (N) when farms arranged as concentric rings around market. Notes: g = 30 and s = 0.40. Graph shown for 2 ≤ N < 30. Horizontal axis scaled from 0 to 35; vertical axis from 0 to 0.25
9 The Cautious Farmer and the Local Market
B
C A 0 Size of Market (N)
any other commodity). On a two-dimensional plane, we can therefore assume farmers spread at a fixed density: g farmers per unit area.40 If N is the number of farmers participating in a market, then the outer radius of the market (i.e., the marginal farm) is X = (N/(π g))0.5 . Assume the coop incurs a constant unit shipping rate for each of its member farmers. Therefore, for the marginal farmer at distance x from the market, the cost of shipping to the market incurred by the coop is sx, where s is the unit shipping rate per kilometer shipped (assumed to be the same for both commodities). In the empirical modeling that follows, suppose g = 30 and s = 0.40. The first farm (area of 1/g = 0.033) stretches from an inner radius of 0 to an outer radius 0.1030; its mid-radius is 0.0728 and the shipping cost incurred by the coop for it is 0.0291.41 The second farm (also an area of 1/g = 0.033) stretches from radius 0.1030 to radius 0.1457; its mid-radius is 0.1262 and the marginal shipping cost for the coop is therefore 0.0505.42 See (9.4.1) and (9.4.2). I treat the membership fee (measured in units of the endowment and paid at the time of the harvest) as a constant per farmer.43 For the coop, this fee is simply cost 40 Economides
and Siow (1988) model the case where farms are spread out along a line in onedimensional space. 41 (0.40)(0.0728). 42 (0.40)(0.1262). 43 Since each farmer has an endowment of either (1, 0) or (0, 1), if I assume that each farmer carries his or her entire endowment to the market to trade (not unreasonable given that the farmer does not know the exchange rate that might be established), the shipping cost associated with each farmer is fixed whether measured per unit shipped or per farmer.
9.5
Model 9B: Cooperation in a Spatial Market
253
recovery: i.e., total shipping cost incurred by the coop is split equally among the N farmers in the coop: see (9.4.6). A feature of the log-linear utility function is that consumption has a linear expansion path.44 Even though the farmer’s income available for consumption of wheat and corn is now net of the membership fee, consumption changes proportionally with income; the marginal farmer still spends the same proportion (ν) of his or her net income on wheat and the remainder on corn. Here in Model 9B, I assume farmers cooperate by sharing equally the total shipping costs of all farmers in the market. If market size (N) is just 2 farms, the co-op fee (f) borne by each farmer would be 0.0398.45 Suppose N1 = 1. One farmer has an initial endowment (income net of production cost and land rent) of (1, 0). Since ν = 0.3, this person prefers to consume 30% of his or her endowment net of coop fee in wheat itself, and trade the remaining 70% away for corn. See panel (b) of Table 9.2. The other farmer has an initial endowment of (0, 1) of which he or she prefers to consume 70% (again net of coop fee) and trade away the remaining 30% for wheat. See (9.4.3), (9.4.4), and (9.4.5). Since each farmer pays a coop fee equal to the average shipping cost, that exchange ratio between wheat and corn in this market is 2.3346 units of wheat per unit of corn just as in Model 9A. If the N = 2 market, there are two possible utilities when k = 0.5: a (1, 0) endowment that yields a consumption bundle (0.29, 0.29) and a utility (U1 ) of 0.29 (compared to 0.30 in Model 9A) with a probability (k) of 0.5 or a consumption bundle (0.67, 0.67) and a utility (U2 ) of 0.67 (compared with 0.70 in Model 9A) with a probability (1 − k) of 0.5. Then, W[k] = 0.48047 (compared to 0.50 in Model 9A). The introduction of shipping cost here leaves the exchange rate unchanged but reduces the sub-utility in every outcome, not surprisingly given that the membership fee reduces the amount of wheat and corn available for consumption. As a result, U[N], from columns [4] and [11] in panel (b) of Table 9.2 is 0.230,48 down from 0.240 in Model 9A. Further, the variance in this utility, V[N], is also now calculable; V[N] = 0.07549 down from 0.082 in Model 9A. To conclude, in the case of N = 2, the introduction of shipping costs reduces both return and risk compared to Model 9A. Now let us do similar calculations for a market of N = 8 participants: see panel (b) of Table 9.3. CEER remains the same as in Model 9A. However, compared to Model 9A in panel (a), we find—as when N = 2—that U1 , U2 , and W[k] drop for any 0 < k < 1. As a result, U[N] = 0.390 is smaller than for Model 9A. V[N] too
44 As
used here, a condition of the utility function wherein, if as income is increased by a fixed proportion holding prices of commodities constant, the rational consumer purchases the same proportion more of each good. Put differently, each good has an income elasticity of +1.0. Such a utility function is also said to exhibit homotheticity. 45 (0.0505 + 0.0291)/2. 46 0.70(1−0.0398)/(0.30(1−0.0398)). 47 0.5(0.29) + 0.5(0.67). 48 0.52(0.00) + 0.48(0.480). 49 0.52(0.00 − 0.23)2 + 0.48(0.5(0.29 − 0.23)2 + 0.5(0.67 − 0.23)2 )
254
9 The Cautious Farmer and the Local Market
is smaller than in Model 9A. As in N = 2, return and risk are both smaller once we take shipping cost into account. We can then compare this combination of return and risk with those achievable at other sizes of market. I repeated the same process of calculating U[N] and V[N], as described above, for all market sizes from N = 2 to N = 30 in the spatial case. See Fig. 9.6. There, I show the risk-return curve as a faint line joining achievable combination at each market size in Model 9A (reproduced from Fig. 9.3) and as a solid line joining dots (labeled Model 9B).50 I have also reproduced the riskreturn tradeoff from Fig. 9.3. For the risk-return curves in Models 9A and 9B, I have labeled selected market sizes from 2 to 30. Here, we see the effects of introducing shipping cost.
Model A
Fig. 9.6 Model 9B: U[N], V[N], and size of market in a spatial market: α = 0.30, φ = 0.40, g = 30, and s = 0.40. Notes: Market size, N, shown as dots for selected N from 2 to 30. Gray line is the U[N] − V[N] curve when shipping costs are zero. Solid curve is the U[N] − V[N] curve for a marginal individual taking into account higher shipping costs to reach larger market. Horizontal axis scaled from 0.04 to 0.10; vertical axis scaled from 0.2 to 0.5
50 In
U[N] Expected sub-utility
10
deoff 1 turn tra Model B
Risk-re
deoff 2
turn tra
Risk-re
5 14
10
10
5 5
Model C
0 V[N] Variance in sub-utility
Model B, the shape of the risk-return curve is sensitive to the unit shipping rate, s. As s approaches zero, the risk-return curve approaches that for Model A in Fig. 9.6. On the other hand, as s is made larger, the risk-return curve for Model B is pulled even further back and down at larger N.
9.5
Model 9B: Cooperation in a Spatial Market
255
1. When N = 1, there is no effect since each farmer is in autarky. 2. For N ≥ 2, shipping cost pulls the risk-return curve downward and to the left: i.e., lower return and lower risk once shipping cost is incorporated. 3. For N ≥ 2 but small, the introduction of shipping cost reduces both risk and return by a relatively small amount. 4. For N ≥ 2 and large. Previously in Model 9A, a larger market meant unequivocally lower risk and higher return. In Model 9B however, this is no longer true. When market size is large, the reductions in both risk and return are relatively large. The geography of a larger market area may mean that the cost of shipping eventually reaches a magnitude where it undermines the advantage of further increasing market size. In the hypothetical example shown in Fig. 9.6, in the presence of shipping costs, the marginal farmer with a β associated with the risk-return tradeoff line displayed would find it best to participate in a cooperative of 14 farmers. In this way, shipping costs help explain why market size is not unlimited. Model 9A suggests farmers have an unlimited appetite for participating in large markets. However, shipping costs in the form of membership fees curbs this appetite. Size of market reflects the offsetting influences of return and risk on the one hand and average shipping costs on the other. If farmers have a higher β—that is, a greater aversion to risk—the risk-return tradeoff line in Fig. 9.6 would be steeper, and farmers would then opt for a coop with a larger N. See (9.4.7) and (9.4.8). In other words, farmers would then prefer a lower level of risk even though that might involve a substantially lower return. Earlier, I had said that farmers with a similar β would form a club. As is evident from Fig. 9.6, that is not always the case. If s is sufficiently small, every farmer would prefer an infinitely large market, and we might therefore find a great mix of βs among farmers in any club so formed. It is more correct to say that if s is sufficiently large, the risk-return curve under Model 9B will consist of a set of distinct market sizes each of which will make that size best for farmers within an interval of β. In that sense, Model 9B is lumpy. I draw the risk-return curve as a polyline in both Figs. 9.3 and 9.6; however, this is just for ease of exposition. In fact, the risk = return curve is just a set of combinations of U[N] and V[N]: one for each integer size of market; the line segments joining them have no particular meaning. When, in a beta analysis, we draw a risk-return tradeoff as an upward sloped line, we are asking simply which combination of U[N] and V[N] on the risk-return curve allows the farmer to reach the highest risk-return tradeoff. In the example shown in Fig. 9.7 for instance, farmers whose β is above 0.0873 would choose at least N = 12 (since that is the slope of risk and return joining N = 12 to N = 11); farmers whose β is below 0.3168 would choose no more than N = 12 (since that is the slope of risk and return joining N = 12 to N = 13). In Fig. 9.8, I show a step function from which we can read, for any given β, the appropriate size of market in this example; for instance, at β = 1.4, the farmer chooses a market of 20 in Model 9B. Others prefer to tell economic stories without such step functions (lumpiness); they would like something akin to Model 9B but wherein size of market, N, was a continuous
256
9 The Cautious Farmer and the Local Market Max β = 0.3168
11 U[N] Expected sub-utility
Fig. 9.7 Model 9B: Risk and return by size of market. Notes: α = 0.30, φ = 0.40, g = 30, and s = 0.40. Horizontal axis scaled from 0.065 to 0.072; vertical axis from 0.391 to 0.393
Min β = 0.0873
12 10
13
14
15 0 V[N] Variance in sub-utility
variable that could be analyzed more easily using calculus or other methods that rely on continuity.51 However, I like Model 9B from a pedagogical perspective because it clarifies just how a farm might decide in practice whether to join a given coop. What about comparative statics in this model? First note that compared to models elsewhere in this book Model 9B has relatively few parameters: ν, β, φ, g, and s. g In Model 9B, g appears to be measuring something similar to s. As s and thereby f is increased, the loss associated with the cost of shipping increases. As g is decreased, the density of farms declines and the marginal farmer has to travel further to participate in a market of size N and thereby f is increased. Put differently, the loss associated with shipping (and hence the coop fee) increases when g is decreased. s In the non-spatial world of Model 9A, a beta analysis of risk and return would lead farmers to congregate in a single global market since the risk-return curve bends up. If s is close enough to zero, that may also happen in Model 9B. In Model 9B, s must be sufficiently large to cause the risk-return curve to have a positive slope (bend back down) for sufficiently large N before we will observe farmers choosing to form a club—i.e., a local (smaller) market. Put differently, if we decrease s and thereby f, the farm has an incentive to
51 For
example, Models A and B here are built on binomial probabilities that arise because we have characterized market formation as a Bernoulli Process. It is well known that the Normal Distribution (which is continuous) can approximate a Binomial distribution as the number of trials in the Bernoulli process (in this case, the size of market) becomes sufficiently large.
Model 9B: Cooperation in a Spatial Market
Fig. 9.8 Risk aversion and size of market in Models 9B and 9C. Notes: α = 0.30, φ = 0.40, g = 30, and s = 0.40. Here, b and c are the lower and upper limits on β in Model 9B when the best size of market is N = 12: see Fig. 9.7. For farmers for whom β = a, best size of market is 10 in Model 9C and 14 in Model 9B. Horizontal axis scaled from 5 to 30; vertical axis from 0 to 3.5
257
AB CD
Models 9B and 9C: Size of market and risk aversion Risk aversion and size of market in Model C Risk aversion and size of market in Model B
B
Level of risk aversion (β)
9.5
D
a b c 0
A C Size of market (N) 12 14
travel further to benefit from a larger market, and for a sufficiently small s and thereby f, there will be a single global market. β If β is increased, farmers become more risk averse. They are willing to spend more to join a larger coop because they attach more importance to reducing risk. Provided s is sufficiently large to make the risk-return curve bend back down enough so that a risk-return tradeoff line can be tangential to it, an increase in b causes farmers to prefer a larger market. ν If ν is increased, each farmer prefers more wheat relative to corn. This causes CEER to fall since farmers now see wheat as more valuable. However, it has no effect on the efficient size of market, N. φ If φ is increased, a farmer is more likely to have a corn endowment. Effects on the model depend on the size of φ relative to 1 − ν. If φ is smaller than 1 − ν, an increase in φ brings the ratio of corn to wheat endowments closer to what consumers would prefer. If φ is larger than 1 − ν, an increase in φ pushes the ratio of corn to wheat endowments higher than what consumers would prefer. The size of market is unaffected. Model 9B is the first place in this book to make use directly of the idea of a club. Upon reflection, it might seem strange to introduce the idea of a club—whose members cooperate—into a book on the effects of competition (often thought to be the antithesis of cooperation). Why do farmers here seek to cooperate? What is driving them here is the risk associated with the randomness of the harvest— i.e., their endowment. To form a club, as is done in Model 9B, is just one possible response to that risk. What might be some other possibilities?
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9 The Cautious Farmer and the Local Market
• Crop insurance. If farmers need to consume both corn and wheat, and plant both crops, it would be reasonable for them to purchase insurance against crop failure. Model 9B does not include this possibility, so the alternative is to band together with other farmers hoping that there will be sufficient amounts of both wheat and corn at the market to enable a good trade. Presumably, crop insurance is structured to give farmers (every farmer in this case) an amount of the commodity they fail to harvest. Presumably the insurer collects a premium from each farmer, awards every farmer an insurance benefit, and still manages to keep something for himself or herself (e.g., profit and wages for the insurer and his or her workers). Under Model 9B, each farm pays its share of shipping costs (which is like an insurance premium). Each farm also receives back some amount of the commodity that they did not harvest (except when k = 0 or k = 1). In that sense, the farmers in Model 9B have greater risks than they would with crop insurance (since they could still arrive at a market where k was near 0 or near 1 but less risk than they would have in the absence of both a commodity market and crop insurance). • Arbitrage. Model 9B assumes there is no way to purchase commodities other than by traveling to a market. As envisaged here, there would potentially be a large number of local markets each with its own exchange rate between wheat and corn. Where exchange rates differ between two local markets by more than the unit shipping cost between markets, presumably arbitrageurs would have an incentive to enter. If the farmer knows that there are competitive traders lurking in every market looking for such opportunities, they will have an incentive to choose a smaller market and save on the cost of shipping to the extent that they know traders will act in a way that keeps the local exchange rate from becoming too unfavorable. At the outset of this book, I suggested that the organization of the firm was itself endogenous to locational competition. We have seen illustrations of that idea earlier in the book, most recently in the outsourcing decision in Chapter 7. If we think of farmers as firms, and the coop as an extension of the activities of a firm, Model 9B offers a new insight in this matter. With the creation of the coop, each farmer is agreeing to be bound by conditions and restrictions of that group. In effect, the farm’s pursuit of its own well-being is now spread across two establishments: one at the farm site and one at the level of the cooperative (market). In effect, the coop helps each farmer achieve a better market outcome than they might otherwise have. In the Marshallian sense, can Model 9B be thought of as a localization economy? I think the answer to that is yes. In Model 9B, the coop spreads (shipping) cost among participants and clusters farms. The process is endogenous; farmers participate in a cluster (or not) not by fiat but on the basis of their own assessment of costs and benefits. Here although farmers close to the market point pay a greater co-op fee than they recoup in shipping cost reimbursed, there is no incentive to leave the coop. Even though they are cross-subsidizing the more remote farmers, they know that the coop has pushed those farmers into concentric rings that are the most efficient and that they could not get the efficiencies of a market of this size without the coop.
9.6
Model 9C: Competition for Land in a Spatial Market
259
Let me expand on the idea of a club here. As presented, the coop internalizes the externality of market formation by sharing shipping cost and ensuring efficient location. In a more general sense, local government generally can be thought to (1) address local externalities through infrastructure investment, service provision, and activity regulation and to (2) redistribute costs among municipal revenue sources such as property tax, local sales tax, and other sources in addition to user fees. In this sense, a club can be thought of as a metaphor for local government. Put differently, coops serve to separate farmers by risk category; local governments can perform a similar function.
9.6 Model 9C: Competition for Land in a Spatial Market Now, consider a model in which there is no cooperation among farmers. Instead, assume farms choose a market and bid for a location in proximity to that market in the knowledge that each must pay their own shipping cost to get their endowment to the market. See (9.5.1) and (9.5.2) in Table 9.5. As in Model 9B, I assume that unit shipping cost is contingent on distance to the market chosen and is independent of the exchange rate that gets established for wheat and corn. In a competitive process, I imagine that market rent for land rises above the opportunity cost of land (paid by farmers implicitly in Model 9A and Model 9B) insofar as scarcity rents arise around market points. In this model, landlords are again absentee in the sense that rent payments to them disappear and do not affect the exchange rate in any local market. I also assume that all rent payments are contingent; the farmer’s rent is a fixed proportion of his or her endowment regardless of the exchange rate that gets
Table 9.5 Model 9C: competitive farmers and the market in a spatial economy where r is endogenous Outer√boundary of farm at ring i from market xi = (i/(πg))
(9.5.1)
Mid distance for farm at ring i from market √ mi = ((i − 0.5)/(πg))
(9.5.2)
Endowment for farm at ring i from market net of shipping cost (1 − r / g − smi ,0) or (0,1 − r / g − smi )
(9.5.3)
Rent at farm N at market boundary r[N] = 0
(9.5.4)
Rent at √ farm closer √ to market √ r[i] = s (g/π ){ (N − 0.5) − (i − 0.5)}
(9.5.5)
Notes: See also (9.1.1) through (9.1.6) I and Table 9.1. Rationale for localization (see Appendix A): Z4—Risk spreading and insurance; Z6—Differences among consumers; Z7—Variation in goods; Z8—Limitations of shipping cost. Givens (parameter or exogenous): g—Density of farms; i— Position of farm: rings away from the market; N—Number of farmers in market. Outcomes (endogenous): mi —Mid-radius of farm i; r—Market rent; xi —Outer boundary of farm i.
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9 The Cautious Farmer and the Local Market
established in any local market. In Model 9C, I retain the assumption that farms each form a concentric ring around the market point. After all, shipping cost now gives the farm an incentive to want to be near the market point. Between any pair of farm sites (rings), competitive bidding for land means the difference in rent in equilibrium must be just enough to make the farmer indifferent between the two sites. Rent for the farmer at the boundary of the market area is zero; see (9.5.4). Further, since I assume that the amount of land used by a farmer is fixed, the equilibrium difference in rent (per harvest) between the two sites must exactly offset any savings in the cost of shipping. That rent plus shipping cost must therefore total to a constant is the so-called Wingo condition.52 Rent for the farmer in ring i is given by (9.5.5). Once scarcity rents reach this level, there is no incentive for a farmer to prefer any one ring to another within a given market. However, there will still be differences among markets of different sizes. The implication of (9.5.5) is that the larger the N, the higher the rent in ring 1. Given a market of size N, there are important differences between Models 9C and 9B. In Model 9C, the farmer at ring N loses from his or her endowment a shipping √ cost of s ((N − 0.5)/(πg)), zero rent, and (needless to say here in Model 9C) there is no coop fee. In Model 9B, the √ same farmer incurs zero shipping cost, zero scarcity rent, and a coop fee of (1/N)i s ((i − 0.5)/(π g)). At ring N, the loss to endowment is smaller in Model 9B than in Model 9C. At any location closer to the market, the loss to endowment for the farmer in Model 9B stays the same and so too does the loss to endowment for the farmer in Model 9C since rent there increases to offset any savings in shipping cost. Therefore, the farmer in Model 9B is better off than the farmer in Model 9C by the same amount regardless of location in a market of a given size. Why is this? It is because absentee landlords are collecting scarcity rents in Model 9C that do not appear in Model 9B. To begin an interpretation of Model 9C, consider panel (c) of Table 9.2 wherein N = 2. Suppose N1 = 1. The wheat farmer has an initial endowment (income) of (1, 0). Continuing the assumption that ν = 0.3, this person prefers to consume 30% of their endowment (after rent and shipping cost) of wheat and trade the remaining 70% away for corn. The corn farmer has an initial endowment of (0, 1) of which he or she prefers to consume 70% (again after rent and shipping cost) and trade away the remaining 30% for wheat. Since each farmer, regardless of location, pays the same total of rent and shipping cost, that exchange ratio between wheat and corn in this market is still 2.33 units of wheat per unit of corn just as in Models 9A and 9B. There are two possible utilities when k = 0.5: a (1, 0) endowment that yields a consumption bundle (0.28, 0.28) and a utility (U1 ) of 0.28 (compared to 0.29 and 0.30 in Models 9B and 9A respectively) with a probability (k) of 0.5 or a consumption bundle (0.66, 0.66) and a utility (U2 ) of 0.66 (compared with 0.67 52 Wingo (1961) originated the idea that land rent offsets transportation cost savings. Later, Alonso
(1964) argued that the relationship between land rent and transportation cost savings was also affected by the elasticity of substitution between land and other commodities. Since I have here assumed that the amount of land used by each farmer is fixed, I do not have to take elasticity of substitution into account.
9.6
Model 9C: Competition for Land in a Spatial Market
261
or 0.70 in Models 9B or 9A respectively) with a probability (1 − k) of 0.5. Then, W[k] = 0.47553 (compared to 0.48 or 0.50 in Models 9B or 9A respectively). The introduction of shipping cost here leaves the exchange rate unchanged but reduces the sub-utility in every outcome, not surprising given that rent and shipping cost reduces the amount of wheat and corn available for consumption. As a result, U[N], from columns [4] and [11] in panel (b) of Table 9.2, is 0.228,54 down from 0.230 or 0.240 in Models 9B or 9A, respectively. Further, the variance in this utility, V[N], is also now calculable; V[N] = 0.07455 down from 0.075 or 0.082 in Models 9B or 9A, respectively. To conclude, in the case of N = 2, the introduction of rent and shipping costs reduces both return and risk compared to Models 9B and 9A. Now let us do similar calculations for a spatial market of N = 8 participants: see panel (c) of Table 9.3. CEER remains the same as in Models 9B and 9A. However, compared to Models 9A and 9B, we find—as when N = 2—that U1 , U2 , and W[k] drop for any 0 < k < 1. As a result, U[N] = 0.375 is smaller than it was in either Models 9B or 9A. V[N] too is smaller than in either Models 9B or 9A. As in N = 2, return and risk are both smaller once we take rent and shipping cost into account. A further implication is that, in a risk-return analysis, the farmer would prefer N = 8 over N = 2 in Model 9C. We can then compare this combination of return and risk with those achievable at other sizes of market. I repeated the same process of calculating U[N] and V[N], as described above, for all market sizes from N = 2 to N = 30 in Model 9C. See Fig. 9.6. There, I show the risk-return curves for Model 9C in addition to Models 9B and 9A. Here, we see that the effects of introducing rent and shipping cost is to cause the risk-return curve to bend back down (have a positive slope when N is sufficiently large) even more than in Model 9B. In the example shown in Fig. 9.6, in the presence of rent and shipping costs, the marginal farmer with a β associated with the risk-return tradeoff line displayed would find it best to participate in a coop a market of just 10 farmers. In this way, rent and shipping costs help explain why market size is not unlimited. Model 9A suggests farmers have an unlimited appetite for participating in large markets. Model 9B shows that shipping costs curb this appetite. In the presence of land rent, the efficient size of market shrinks even further in Model 9C where competition rather than cooperation is seen to underlie a firm’s behavior. In the example of Model 9C illustrated in Fig. 9.6, firms with the β illustrated by the slope of the risk-return tradeoff choose N = 10, compared to N = 14 in Model 9B. In Fig. 9.8, I show that the best size of market is consistently lower in Model 9C than in Model 9B at every level of β. The reason is simple; the marginal farmer in Model 9C finds it more costly to participate in a market of a given size (because they pay the full shipping cost from the edge of the market area) compared to the marginal farmer in Model 9B (who pays only the average shipping cost) and
53 0.5(0.27)
+ 0.5(0.66). + 0.48(0.475). 55 0.52(0.00 − 0.228)2 + 0.48(0.5(0.28 − 0.228)2 + 0.5(0.66 − 0.228)2 ). 54 0.52(0.00)
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therefore chooses a smaller size of market at any given level of risk aversion. In effect, the mechanism of land rent serves to separate farmers by risk category just as do coops. It is typically argued that competitive markets are efficient. Is that always the case here? The answer—perhaps surprisingly—is no. Specifically, a farm with a β corresponding to the risk-return tradeoff shown in Fig. 9.6 would be better off in Model 9B than in Model 9C. Put differently, the outcomes in the cooperative scheme in Model 9B appear to be more efficient than in the competitive scheme in Model 9C. Why is this? Two related answers come to mind. First, Model 9B may only appear to be more efficient because it ignores the costs of organizing cooperatives. If such costs were substantial, they could drag the risk-return curve for Model 9B below that for Model 9C, making the competitive solution the more efficient. Second, under the cooperative scheme, farms organize themselves into a larger unit to reap the advantages brought about by the reduction of risk. In this sense, the Coasian view would be that the firm is merely internalizing something (here, a reduced risk) that might otherwise be unavailable or costly to provide in a competitive market. Put differently, belonging to a collective is integral to firm organization: allowing the firm to be efficient. A similar idea arose in the modeling of repair services within or outside the firm back in Chapter 7. There too the firm sought to deal with risk (in that case, machine failure) efficiently through firm organization: be it in-house or through outsourcing. Models 9B and 9C say something new about why individuals participate in a local market. In earlier chapters, I presented the notion of range of good wherein size of market was limited only by the notion that a commodity was expendable. In Chapter 8, we saw a market boundary defined by the criterion that the presence of a competitor nearby limited the ability of a firm to attract customers. In this chapter, the farmer can be thought to make a choice between a smaller market and a larger market in terms of the of risk and return associated with each. This is the first model in the book wherein land rents vary by location within the regional economy. We can think here of a farm at the edge of its market area (i.e., the Nth distant farmer in a market of N farms) as a marginal farm. Compared to the marginal farm, the endowments of farms that are closer to the market (i.e., farms 1 through N − 1) now can be seen to include an excess utility that arises because of a savings in shipping cost. In Model 9C, the excess utility gets bid away as land rents so that the first N − 1 farmers end up only as well off as the marginal farmer in their market. Put differently, these land rents do not arise because of something that landlords have done to make their properties individually more attractive to tenants. Instead, they arise because of the clustering of N farms. To farms, they are a loss in consumption (a leakage to the farm economy) that happens because farms compete in the land market. The model is silent on how farms produce their endowments. However, we could readily imagine the farm as an enterprise that uses capital, labor, and land to produce its wheat or corn. If so, Model 9C hints at a fundamental underlying relationship among factor payments to these inputs. In Model 9C, the gain in utility from participating in a larger market is divvied up between (1) implicit
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Model 9C: Competition for Land in a Spatial Market
263
returns to labor and capital in the farm enterprise and (2) the return to land. I present models in later chapters of this book that further explore this idea. What about the comparative statics of Model 9C? Once we take into account that the coop fee in Model 9B is now replaced by shipping cost and land rent, the comparative statics are much the same as in Model 9B. Why is that? In part, it is because Models 9B and 9C have the same parameters: ν, β, φ, g, and s. In Model 9B, the farmer pays a fixed cost (the coop fee) regardless of location that increases with the size of market. In Model 9C, the farmer also pays a fixed total cost (for shipping plus land rent) that increases with the size of the market. g In Model 9C, as g is decreased, the density of farms declines and the marginal farmer has to travel further to participate in a market of size N, and thereby shipping cost is increased. Put differently, the loss associated with shipping and/or rent increases when g is decreased. s If s is near zero, farmers in Model 9C form a single global market because the risk-return curve bends up. If s is large enough to cause the risk-return curve to have a positive slope (bend back down) for sufficiently large N, we observe farmers choosing to form a local (smaller) club. Put differently, if we decrease s, the farm has an incentive to travel further to benefit from a larger market, and for a sufficiently small s, and there will be a single global market. β If β is increased, farmers become more risk averse. They are willing to spend more to join a larger coop because they attach more importance to reducing risk. Provided s is sufficiently large to make the risk-return curve bend back down enough so that a risk-return tradeoff line can be tangential to it, an increase in β causes farmers to prefer a larger market. ν If ν is increased, each farmer prefers more wheat relative to corn. This causes CEER to fall since farmers now see wheat as more valuable. However, it has no effect on the efficient size of market, N. φ If φ is increased, a farmer is more likely to have a corn endowment. Effects on the model depend on the size of φ relative to 1 − ν. If φ is smaller than 1 − ν, an increase in φ brings the ratio of corn to wheat endowments closer to what consumers would prefer. If φ is larger than 1 − ν, an increase in φ pushes the ratio of corn to wheat endowments still higher than what consumers would prefer. The size of market is unaffected. Before leaving this section, let me add one thought about landlords. As envisaged here, there will be a mix of markets across the landscape; some small and some large. Corresponding to these will be a mix of landlords; some will receive a high scarcity rent for their site because it happens to be near the center of a large market; others will receive only a rent equal to the opportunity cost of the land: i.e., a zero scarcity rent. If landlords themselves were competitive, why would a landlord with a zero scarcity rent not seek to entice farmers to relocate more advantageously? In the absence of collusion and given enough landlords (who each own only a small amount of land), there is no mechanism by which this could be achieved. At the
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same time, the potential for a higher land rent might in practice create an incentive for landlords to collude locally to ensure that they attract and hold the scarcity rents associated with a larger market.
9.7 Final Comments In this chapter, the principal model has been 9C. I included Models 9A and 9B to help readers better understand aspects of Model 9C. In Table 9.6, I summarize the assumptions that underlie Model 9A through 9C. Many assumptions are common to all these models: see the list in panel (a) of Table 9.6. The models differ in that (i) Model 9A assumes shipping costs are zero, (ii) Model 9B assumes that farms address differences in shipping costs with location by forming cooperatives, and (iii) Model 9C assumes that farms address differences in shipping costs by bidding up the price (rent) for land at advantageous locations. Table 9.6 Assumptions in Models 9A through 9C Assumptions (a) Assumptions in common A1 Closed regional market economy A2 Barter market B1 Exchange of wheat for corn C10 Two kinds of customers C7 Maximize same utility function F6 Firm is myopic G2 Uncertainty explicit in model G3 Risk incorporated into utility function (b) Assumptions specific to particular models E1 Zero shipping cost everywhere A4 Rectangular plane J3 Each farm occupies the same amount of land F8 Co-operative shares shipping costs among members J1 Zero opportunity cost (rent) for land J6 Competitive market for land
9A [1]
9B [2]
9C [3]
x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x
x x
x
x x
I end this chapter with six sets of thoughts. First, in Chapter 1, I introduced the idea of the importance of transaction cost in location theory. I defined there the concept of an effective price which includes the price paid plus the unit transaction costs incurred by the purchaser related to search and information gathering, negotiation, and acquisition, inclusive of normal profit. In Model 9B here, the coop fee can be thought of as a transaction cost. In Model 9C, the combination of shipping cost and rent can similarly be thought of as the transaction cost. In both models, the farmer is free to choose a level of transaction cost (i.e.,
9.7
Final Comments
265
a size of market or location). At the same time, there is nothing to guarantee that the farmer will always get a better price in a larger market. In these respects, choosing a larger market is like searching. The farmer in a larger market is getting to see a wider cross section of market participants. This is like (but not the same as) a consumer gathering information from different suppliers before deciding whether and from whom to purchase. The reader might therefore be tempted to apply this model to help in understanding search behavior. However, there are important differences between market formation as modeled here and conventional thought about search behavior. Principal among these in my mind is the idea that search is often seen as a sequential process. In my view, the consumer gathers information about another supplier or commodity, then makes a decision about whether to continue searching, to purchase, or give up on the idea of purchasing such a commodity. In contrast, the farmer in Models 9B and 9C is simply making a gamble among sizes of market; there is no sequential process at work here. Second, the farmer in this chapter is always assumed to use the same amount of land in production. However, in Model 9C, land becomes relatively more costly to rent, the closer the farm is to the market. Presumably, when an input like land becomes more costly, we might expect the farmer to change the way in which he or she produces agricultural commodities. One simple way to do this is to switch between production that is less labor intensive (when land rents are low) and production that is more labor intensive (when land rents are high). To do this, our model of the farm would have to include a production function that enables a substitution between land and labor. The models in this chapter assume a fixed amount of land per farm and are silent on labor input. From my perspective, the easiest way to think about the farmer in this chapter is that he or she constitutes one unit of labor and that the farm has a Leontief production technology that requires exactly 1 unit of labor and 1/g units of land to produce stochastically an endowment of either (1, 0) or (0, 1). What happens if we make amount of land used by a farm endogenous to the model? I return to this question over the next three chapters. Third, we began Model 9A with the apparently innocent assumption that each farm had an independently drawn random endowment of 1 unit of either wheat or corn. Implicit in this assumption is the idea that the scale of a farm is somehow fixed. Imagine how much different the model might look if we started with the assumption that the farm had an independently drawn random endowment of 1 unit of either wheat or corn for each unit of land farmed and was free to hire labor and rent land as needed. In that case, a farm could largely (if not entirely) eliminate the need to travel to a market by being of sufficient size to render insignificant the risk of being without suitable quantities of the two goods. In effect, the farm uses its own size to self-insure rather than rely on what is essentially an insurance function performed by the market. The gain to the farmer here is the higher level of utility now possible in the absence of scarcity rents. Fourth, I am able to solve the models in this chapter because I took a Walrasian perspective on exchange rates: i.e., I found the exchange rate that leaves neither kind of farmer feeling that there is still some amount of one commodity that they would prefer to trade for the other. However, as noted at the outset of the chapter,
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the Walrasian solution is just one point on a Marshallian Contract Curve. At one level, one might argue that this indeterminacy of outcome makes the Marshallian approach less useful than Walrasian approach. One might also argue that, as the number of market participants becomes large, the Marshallian solution converges on the Walrasian solution. Nonetheless, it seems to me unreasonable to believe that farmers will entirely ignore the prospect that in a small market their bargaining power (however derived) may leave them at an effective exchange rate different from that envisaged by Walras. The models in the chapter do not consider this. Fifth, what about Walrasian equilibrium across markets? In Model 9A, there is only one market: a global market for the exchange of wheat for corn. Walrasian equilibrium is irrelevant here. In Model 9B, unit shipping costs drive farms to organize themselves into local markets for the exchange of wheat for corn. Because the exchange rate realized in each local market is subject to random variation, we can say only that this particular process of choosing local markets leaves individual farms best off viewed over a longer term of repeated trials. The same is true of Model 9C with a twist; in 9C, there is a market for land as well as local markets for the exchange of wheat for soap. Sixth, Chapter 9 gives us some hints about the regional economy. However, much remains to be done to flesh this out. We need to know more about the process by which endowments of wheat and corn are generated. It would be helpful for example to know about the use of labor, capital, and land inputs in the production process. That would make it possible for us to better understand the regional economy and the impacts of changes in givens on regional well-being.
Chapter 10
Farming for Cash Market Participation and Demand (Thünen–Lee–Averous Problem)
An isolated place (the city) has aggregate demand schedules for up to two crops. Tenant farmers—each renting the same amount of land—supply the city: absent the uncertainties central to Chapter 9. This chapter considers landscapes that are either an unbounded ribbon or rectangular plane. A two-dimensional plane means that the local supply curve is a quadratic and the model becomes more cumbersome to solve without adding much in economic insight. Farms incur costs of production and shipping. This generates an endogenous supply curve, an equilibrium price and quantity in the market, and an outer radius for farm production. At the outer radius, the cost of production plus shipment is just equal to the price that the farmer receives for a unit of product. In a competitive market, for land these profits accrue to owners of the property; tenant farmers everywhere, in competing for the best sites, bid up the market rent for land until excess profits everywhere are driven to zero. Model 10A describes a regional economy wherein farms, spread along a line, produce one crop for the market. Model 10B considers the same problem except that farms locate in two-dimensional space. In Model 10C, I introduce demand for a second crop when farms are spread along a line. Even though Model 10C assumes a zero cross-price elasticity between the two crops, prices of the two are linked because both kinds of farmers compete for land around the demand place. Model 10D considers the twocrop case when farms are in two-dimensional space. For the first time in this book, a model solves jointly for prices of commodities in different markets (the markets for crops and the market for land).
10.1 The Thünen–Lee–Averous Problem Although the models in Chapter 9 are interesting, the treatment of endowment as a random occurrence leaves something to be desired. I want to tell a story about how individuals organize themselves in space through competition: some producing one commodity, others producing a second. Then, we can better understand how endowments come to exist and why trade ensues. In Chapter 9, I introduced the idea that land rent organizes production across space in a way that preserves locational equilibrium. There, however, the role of land rent is modest in the sense that it does not affect other prices in the model. J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_10, C Springer Science+Business Media, LLC 2010
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As will be seen in this chapter, the land market is central to contributions that location theory makes to Walrasian analysis. This should not be surprising. After all, if firms compete by choosing locations for their activities, it should affect the market rent for land and—since this can be an important component of cost overall for businesses—it should therefore also affect the prices at which they sell their products. The remainder of this book is devoted to that idea. In this, the second of four chapters that look at models of agricultural markets and a regional economy, I consider how shipping cost shapes the supply curve for an agricultural commodity and how the freight savings from being closer to a market result in competition for land which in turn impacts the prices of commodities. How land rents get determined in a market economy is a question that has long interested economists. In his definitive history of economic analysis, Schumpeter (1954, pp. 209–223) discusses early contributions to ideas of locational rent by William Petty (born 1625), Richard Cantillon (born around 1680), and Adam Smith (born 1723). However, I think many economists would agree that it is David Ricardo (born 1772) who is the first important figure here. Ricardo saw his Principles of Political Economy and Taxation first published in 1817.1 This work predated the development of neoclassical economics upon which modern location theory is built. Nonetheless, Ricardo presents a novel and relevant argument in his book about how income produced in an agricultural society gets distributed (attributed to factors of production). To Ricardo, the income generated by economic production is divided among three classes in the regional economy: rent to owners of land, profit to owners of capital, and wages to the laborers by whose industry it is cultivated. Ricardo argued that, in different societies, the proportions allotted to each of these classes would differ, depending mainly on the fertility of the soil, the accumulation of capital, and the skill, ingenuity, and instruments of its workforce. In Ricardo’s view, to determine the laws which regulate this distribution is the principal problem in Political Economy. Some economists argue that Ricardo’s emphasis is misplaced; Economics, they say, focuses on questions of efficiency rather than on questions of who gets what. In my view, Ricardo was trying to show how market competition (put differently, an efficient market) leads to a particular distribution of income. He builds an argument about the distribution of income on the notion that the excess profits that might otherwise be earned by firms get distributed as factor payments in a competitive economy. His argument assumes that: • Firms (be they farms or other economic enterprises) are numerous; • Firms require factors (i.e., labor, capital, and land) inputs to produce their outputs; • The quality of factors varies exogenously (e.g., some parcels of land are more fertile than others); • The amount of each factor is bounded (i.e., not inexhaustible) at any given level of quality (Ricardo treated the amount of each factor available as exogenous; he did 1 Second and third editions followed in 1819 and 1821, respectively. Ricardo died shortly thereafter (1823).
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not discuss how the owners of a factor of production might increase or decrease their holdings of a factor) in response to conditions in the factor market; • Firms compete in factor markets for labor, capital, and land for the purpose of production; • Factor owners, being numerous, are price takers too: they have no ability to extract factor payments over and above those determined in a competitive market. Ricardo (1821, pp. 33–45) argues that, in what later would come to be seen as part of a Walrasian equilibrium, firms bid up factor prices (payments) until all excess profit disappears (in this sense, factor payments reflect the scarcity of input of that quality). Put differently, the owners of the factors of production—be they workers, landlords, or capitalists—have the potential to earn scarcity rents; now known as Ricardian rents. Ricardo implicitly recognized the importance of location here,2 but presented neither a mathematical nor a graphical model to illustrate or accompany his discussion of rent and its impact on factor payments. Indeed, Ricardo’s focus was on the differences in fertility of soil from one site to the next. In his mind, the owner of a plot of land more fertile than the marginal plot would earn a corresponding rent. Model 9C—wherein farmers compete for farm sites—could also be thought of as similar to the process envisaged by Ricardo. Farm sites are not all the same. However, this is not because of differences in fertility. Rather, it is because some sites closer to the market have a lower shipping cost than do sites further away. In Model 9C, we assumed that farmers competed for sites closer to the market and, in the process, bid up the rent on land there until the advantage (the excess profit earnable by a farmer locating near the market) was reduced to zero. No farmer is better off than any other farmer in their market, and some landlords benefit to the extent that shipping costs make their sites more attractive. Model 9C follows more closely the ideas of Thünen than Ricardo. From his farm, Tellow, in Mecklenburg (Germany), Johann Heinrich von Thünen (born 1783)— I refer to him as Thünen, but he is also popularly labeled “Von Thünen” or “von Thünen”—advanced Ricardo’s thinking by marrying scientific experimentation in agricultural practice with economic theory and mathematical modeling. At the time, the leading scholar on agricultural practice in Germany, Albrecht Daniel Thaer (born 1752), was promoting the adoption of the then-modern English agricultural practices. Thünen took Ricardo’s conceptualization of rent, integrated it into the neoclassical approach that he helped devise and came up with a locational model to show the working and significance of rent. In so doing, he challenged some of Thaer’s key conclusions.3 This chapter, as well as the next two chapters, is based on analyses that originate with Thünen.
2 Ricardo (1821, p. 35) argues that no charge could be made for the use of land, if land everywhere had the same properties and were unlimited in quantity unless there were differences in the situation of that land. 3 See Murata (1959, pp. 38–56)
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To set the chapters up, let me explain here briefly his principal ideas. Clark (1967, pp. 370–371) argues that (1) Thünen’s complaint against Ricardo is that Ricardo developed his theory of rent in terms of an undifferentiated agricultural product, and (2) Thünen’s great contribution was to see transport costs as the cause, and rents the consequence, of important differentiations of agricultural, dairy, and forest production. Note here that Clark is not asking how shipping services come to be priced. Like Thünen, he treats transport cost simply as a resource use. Thünen’s model of the city and its rural hinterland makes the following assumptions: • A large city early in the 19th century—neither growing nor declining and viewed simply as a point in space4 —demands agricultural commodities and exchanges city-made commodities (including manure) for them; • The city is surrounded by limitless land of uniform quality (fertility) in which agricultural commodities can be produced; • No other use for land; • A competitive market for land with large numbers of landlords and tenant farmers, each a price taker; • A given unit shipping rate for each commodity (higher for perishable and heavy commodities, lower for nonperishable and lighter commodities) regardless of location; • Other inputs to agricultural production, notably labor, are ubiquitous and buildings, timber, fences, and other capital assets on the farm can be thought to be rented at a going rate of interest; • Farms experience diminishing returns to labor as they attempt to use farm land more intensively. Thünen analyzed price determination for agricultural commodities, land rent, the efficiency of various farming systems, labor costs, manure production in city and countryside and its use in farming, rings of production, and the impact of taxes. What Thünen brought to bear on these subjects was a development of marginal analysis (marginal productivity of labor and capital) that helped usher in neoclassical economics. He also emphasized the role of competition; for production of any one commodity to take place, the land rent bid for land by such farmers somewhere must exceed the rent bid by farmers producing other commodities. In the case of each commodity, he argues that the price must be sufficiently high to ensure adequate production. In Part I of The Isolated State, Thünen is interested in the most profitable use of land by farms. Organized by the relative cost of shipping, he envisaged six concentric rings around the city. From inner to outer, these were as follows: 4 In contrast, in a city that is growing, landlords and farmers just outside the current built-up area anticipate that land will become more valuable for urban uses in the near future and therefore reasonably make different decisions about investment in land improvements (e.g., soil drainage, fencing, barns, and other improvement to agricultural productivity) compared to landlords and farmers further away.
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Ring
Description
1
Cash cropping: delicate food (fruit and vegetable) production closest to the city; milk further away (largely from stall-fed cows), hay and straw furthest away5 : town manure is available and used only in this ring6 ; Forestry for timber and fuel needs Crop alternation system: regular alternation between grain crop and nongrain crop over time. Improved system (Koppelwirtschaft): cereal crops and a short grass ley7 on enclosed fields. Three-field system (spring grain, winter grain, fallow) Stock farming
2 3 4 5 6
It is noteworthy here that Thünen constructed his model without any specific reference to the demand for agricultural products in the city.8 In Part II (Section I) of The Isolated State—see Dempsey (1960, pp. 185–367)— Thünen, like Ricardo, was interested fundamentally in distribution (the allocation of society’s product between wages and profit including interest and rent) in an agrarian society.9 What determines how much income accrues to each factor of production? Thünen added two new insights into this question. He assumed away the part of the problem dealing with the return to capital by assuming a competitive market in which the price of (return on) capital is exogenous. In terms of the return to labor, Thünen assumed diminishing marginal productivity. In terms of the return to land, Thünen assumed that unit shipping rates limited the amount of agricultural commodity that could be produced as this would require more distant farm sites to add to production. More generally, the question of distribution—who gets what out of society’s production—had long been a concern of classical economics. A focus on what Thünen referred to as the natural wage (w), in work first published in 1850, was a part of this. Thünen imagined here a farmer—newly arrived at the frontier—whose activity generates annually product in the amount p. Of this, the worker is paid w, and in turn spends a on consumption and invests the remainder as capital in the farm. To maximize workers’ income from investment, Thünen finds that the natural
5 Thünen refers to this as “free” cropping in the sense that no crop is cultivated simply to nurse the soil by means of a bare fallow. 6 Of Ring 1, Clark (1967, p. 371) argues that, in the time of Thünen, much agricultural area had to be set aside to produce feed for horse. At the time, every town produced much horse manure (not to mention the cow manure from urban dairies) that had to be disposed of without incurring excessive shipping costs. This necessitated a small belt of market garden and milk-producing land in the immediate neighborhood of each town. 7 An arable field used temporarily as a pasture for grazing. 8 See Dickinson (1969, pp. 895–896). 9 See also Leigh (1946).
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wage must be set at the geometric √ mean of a and p. Famously inscribed on his tombstone is that equation: w = (ap). Nowadays, some see this as a model for profit sharing (i.e., the farm itself as a kind of club); others, perhaps not understanding the particular assumptions on which it is based, see it as a flaw in Thünen’s work. Marshall (1956, p. 360)—among others—rates Thünen highly despite a distaste for the natural wage.10 What is the genius of Thünen here? Schneider (1934, pp. 9–11) and Schumpeter (1954, pp. 466–467) see Thünen’s innovativeness in several areas: (1) his development of the ideas of marginal analysis and use of calculus; (2) his idea of the importance of static equilibrium for economic theory; (3) his emphasis on the roles of quantitative and mathematical analysis and his ingenuity in testing his ideas; (4) his idea of the interdependence among many markets; and (5) the need for methods of successive approximation to disentangle economic interrelations.11 Thünen saw that there must be equilibrium simultaneously across land, labor, capital, and commodity markets. Within this, his central idea is the differential advantage of a site. Through improved transportation, previously inaccessible areas take on economic value. Existing stocks of capital and labor in the region now get spread over a greater area of land. With more land in total, its marginal productivity declines. Competition eventually turns the newfound productivity of new sites into returns on capital and labor.12 Dickinson (1969, p. 897) argued that—based on his conceptualization of the natural wage—Thünen’s crowning achievement was as a social philosopher. At the time of Thünen, he argued, social revolution in Europe seemed imminent. Thünen sought to reconcile the claims of the workers and of the owners of land and capital. In seeking a scientific principle to determine wages. Thünen discovered his formula for the natural wage, a wage determined in a cooperative society composed of rational workers. From a modern perspective, one limitation of Thünen’s work, however, is that he does not develop a mathematical or graphical model of the kind that we have seen so far in this book. In the English language, Kryzmowski and Minneman (1928) was an early attempt to represent graphically Thünen’s argument about labor intensity in farming. In contrast, Dunn (1954)—initially holding labor intensity constant—was among the first to present a graph of location rent (bid rent) by a type of farmer (“industry” in his terminology) as a negatively sloped linear function of distance from the market. Dunn then added a second type of “industry” with its own bid rent for each location and solved graphically for amount of land assigned to each industry—on the assumption that land is allocated to the higher of the two bidders. Dunn concludes that the industry locating closest to the market will be the one for which the marginal cost of transporting the amount of crop produced on one unit
10 Samuelson
(1983, pp. 1468–1469) is suspicious of Marshall’s claim. the evident importance of Thünen in the development of neoclassical thought, Stigler (1937, p. 246, footnote 30) argues that Carl Menger was apparently unaware of his work. 12 See Johnson (1902, p. 112) who also argues that Thünen made important contributions to the idea of a normal profit. 11 Despite
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Model 10A: Farms Producing Wheat Along a Line
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of land (say, one hectare) is the highest. Garrison and Marble (1957) is among the first to provide a graphical integration of intensity, land rent, and shipping cost. Thünen’s work has also been extended to look at land uses within urban regions; Ullman (1941) is an early example. Indeed, I think of farm here simply as any firm that requires land as an input into its production process.
10.2 Model 10A: Farms Producing Wheat Along a Line In this chapter, I start the process of recasting Thünen’s analysis from a modern perspective. I build on the idea, first presented in Chapter 2, of an isolated market that can be thought to be a place. In that chapter, we began with a single monopolist located adjacent to the market. The place was where demand occurs. The customer point was thought to contain N consumers, each with the same linear inverse demand curve for a crop that I will call wheat. See (10.1.1) in Table 10.1, wherein I summarize equations, assumptions, notation, and rationale for localization in Model 10A. Therefore, the aggregate inverse demand curve for wheat is given by (10.1.2). In this chapter, let us retain the same assumptions about a customer point. So far in the book, we have assumed similarly an exogenous local supply curve. In Chapter 2 for example, we assumed that the firm had a variable cost and a fixed cost of production. Since the firm was a monopolist, it set its price where marginal revenue equals marginal cost of production. In effect here, the supply curve in this local market is the firm’s marginal cost curve even though the industry is not in a competitive equilibrium: i.e., not producing where demand equals supply. However, we might well want to understand how potential suppliers make decisions about when to supply wheat to this market. We saw elements of this in Chapters 2, 4, and 5 wherein producers elsewhere, or agents engaging in arbitrage, contribute to supply at a place by shipping product there when price makes it attractive. Is it possible to build a simple model to show when and how much producers spread across geographic space will supply to a given market? The models presented in this chapter do just that. They are inspired by a simple and insightful model in Lee and Averous (1973). However, I begin with an even simpler variant to illustrate the key mechanisms. The models presented in this chapter have assumptions in common. They each assume a common fiat money economy for all participants. They each assume consumers at a place demand wheat. The models assume the suppliers of wheat are farmers nearby and that farmers are all identical. The models assume the farm produces wheat at a fixed rate (yield) per unit land: to produce more wheat, the farm needs proportionally more land. In later chapters of this book, I introduce the possibility of varying labor intensity and thus yield; I assume a fixed yield in this chapter for simplicity of exposition. Since yield is fixed, we can measure the price of wheat, P, as the amount paid in dollars for the amount of wheat that can be produced on one unit of land. Assume each farm incurs no fixed cost of production (i.e., in the notation of earlier chapters, this means f = 0) and has a marginal cost of production cost, C, per unit of land farmed. As before, assume C includes normal
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Table 10.1 Model 10A: wheat produced on a ribbon landscape Individual inverse demand curve P = α − βq
(10.1.1)
Aggregate inverse demand curve P = α − βQ / N
(10.1.2)
Profit for farmer at distance x from customer point, also rent in competitive market for land R[x] = P − C − sx
(10.1.3)
Radius of market X = (P − C) / s if P ≥ C, 0 otherwise
(10.1.4)
Aggregate supply of wheat Q = 2X = 2(P − C) / s if P ≥ C, 0 otherwise
(10.1.5)
Equilibrium price P = (2β/ (2β + Ns))C + (Ns / (2β + Ns))α
(10.1.6)
Equilibrium quantity Q = 2 N(α − C) / (2β + Ns)
(10.1.7)
All farms’ revenue from wheat net of production costs 2(P − C)2 / s
(10.1.8)
All farms’ payments for shipping (P − C)2 / s
(10.1.9)
All farms’ payment for rent (P − C)2 s
(10.1.10)
Producer surplus: Farms’ revenue net of production cost, shipping, and rent (PS) 0
(10.1.11)
Consumer benefit 2 N(α − C)(βC + αβ + αNs))/(2β + Ns))2
(10.1.12)
Producer cost (including shipping) N(4αβC − 4βC2 − C2 Ns + α 2 Ns) / (2β + Ns)2
(10.1.13)
Consumer surplus (CS) 2 2βN(α − C)2 / (2β + Ns)
(10.1.14)
Social welfare (SW) N(α − C)2 / (2β + Ns)
(10.1.15)
Total market expenditure (PQ) 2 N(2βC + Nsα)(α − C) / (2β + Ns)2
(10.1.16)
Notes: A unit of the commodity is the amount yielded, assumed constant, by a unit of land. Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Production cost per unit of land; s—Unit shipping rate ($/kilometer) for wheat; N—Number of consumers; x— Distance (kilometers); α—Intercept of individual linear inverse demand curve: maximum price; β—Negative of slope of individual linear inverse demand curve: marginal effect of quantity on price received. Outcomes (endogenous): CS—Consumer surplus; P—Price of wheat (at market); PS—Producer surplus; Q—Quantity of wheat (square kilometers of yield); R[x]—Land rent bid by tenant farmers that leaves them with zero (excess) profit; SW—Social welfare; X—Radius of area supplying wheat (kilometers).
10.2
Model 10A: Farms Producing Wheat Along a Line
275
profit. The models assume further that the farmer bears the cost of shipping wheat to the market and that the shipping cost per unit shipped per kilometer, s, is the same for every farmer. I treat this shipping cost simply as a loss to the local economy: i.e., as resources used up (e.g., maintenance and fuel costs for the vehicle used to ship) and not as a source of factor payments (e.g., profit for shipping firms or wages paid to its workers). There is no uncertainty here. Put differently, the farmer knows exactly what amount of crop will be harvested and what price will be received.13 The idea, as in Chapter 9, that somehow there is a randomness associated with climate or environment is ignored.14 Assume a one-dimensional line (say, a road) of indefinite length down the center of a ribbon of land 1 km wide. The customer point lies on this road. All land in the ribbon is available to farmers. There is no alternative use for the land. Farmers incur no shipping cost in getting wheat from the field to the road itself; they incur a unit shipping rate only in getting wheat from there down the road to the customer point. Consider a farmer at a distance of x kilometers from the customer point.15 The farmer here earns an excess profit per unit land given by (10.1.3). By the definition of normal profit, the existence of excess profit means the farmer is earning more on his or her equity, skills, and so on, than would be the case in the next best alternative use. So, presumably, the farmer would want to engage in wheat production. The boundary of wheat production would then be the distance, X, at which P − C − sX = 0 which implies (10.1.4). As does Thünen, I assume here a frontier area sufficiently far away from the market wherein land is waste (i.e., unused) and therefore earns a rent of zero. In effect the farmed area surrounding the market is itself surrounded by wilderness in which there is no economic activity. The supply of wheat within radius X is simply 2X, taking into account land to either side of the customer point. Therefore, we get (10.1.5). Aggregate demand, derived from (10.1.2), can then be computed, and demand and supply equated. Upon simplification, this gives the equilibrium price in (10.1.6). Note here P is once again a weighted average of the “minimum price” (C) and the maximum price (α). As either s or N becomes large or β small, P approaches α. On the other hand, as β becomes large (i.e., price sensitive to demand) P approaches C. All other endogenous variables now can be solved by back substitution. For example, substituting this price back into (10.1.5) gives an expression for the equilibrium quantity transacted: (10.1.7). What about the well-being of producers and consumers in this market for wheat? The equilibrium quantity and price can be used to solve for farmers’ revenue net of production cost (i.e., (P − C)Q): see 10.1.8). Half of this net revenue is 13 Thünen models have been extended to look at cases where the farm faces uncertainty: see Okabe
and Kume (1983). Cromley (1982) for a discussion of the role of uncertainty in the Thünen model. 15 I measure distance here to be the length of the trip to ship the commodity to the market. I usually characterize this as the mid-distance in the sense that half of the farm’s parcel of land is closer to the customer point than this, and half is further away. 14 See
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expended on shipping: see (10.1.9). The other half is expended on land rent: see (10.1.10). Here land rent is not a cost (i.e., a use of resources) but rather a reallocation of income from farmer to landlord. Because of competition, the farmers have no revenue left over after production costs, shipping, and land rents have been deducted. Consumer benefit, (1/2) (P + α)Q, can also be determined by back substitution: see (10.1.12). Producer cost is given by (10.1.13). Consumer surplus, (1/2)(α − P)Q, reduces to (10.1.14), and social welfare is given by (10.1.5). A diagrammatic solution to market equilibrium is shown in Fig. 10.1. Shown there are aggregate demand curves Av Bv (where v = α); each of these is the same as the aggregate demand curve first introduced in Chapter 2. For the purposes of exposition, I have emphasized the demand curve where α = 4.5. Also shown is the supply curve (CD), which, while apparently similar to the conventional upwardsloped linear supply curve first used in Chapter 4, is in fact something different. Remember we have assumed all tenant farmers are equally efficient. In a non-spatial model, this would lead to a horizontal supply curve. However, CD in Fig. 10.1 is upward sloped because congestion arises; to produce more wheat, farms need to use land further away from the customer point and thereby incur greater shipping costs.
A7.6 D
Price
A0.8
A4.5
Pa
Model 10A: Market for wheat AvBv Aggregate demand for wheat at α = v CD Aggregate supply of wheat OAv Intercept of demand curve (α) OPa Equilibrium price at α = v OC Unit production cost (C) OQa Equilibrium quantity at α = v OPaEaQaO Market expenditure on wheat OCFaQaO Total production cost: excludes rent and shipping CEaFaC Total shipping cost CPaEaC Total land rent PaA4.50EaPa Consumer surplus OA4.50EaQaO Consumer benefit Producer cost including shipping OCEaQaO Social welfare CA4.50EaC
Ea
A2.2
A0.8
Fa
0 B0.85 Qa B2.20
B4.50
B6.00
B7.64
Quantity
Fig. 10.1 Model 10A: equilibrium in the market for wheat on a ribbon landscape. Notes: α = 4.5, β = 12, N = 1,000, C = 1, and s = 0.05. Equilibrium solution is P = 3.36, Q = 95. Price axis scaled from 0 to 8; quantity axis from 0 to 700
10.2
Model 10A: Farms Producing Wheat Along a Line
277
From (10.1.5), we can see the supply curve approaches a horizontal line—that is, congestion disappears—at P = C as the unit shipping rate (s) approaches zero. This has implications for the market rent for land. Assume tenant farmers bid for land for wheat production and that there are no other users for this land. The marginal farmer at distance X earns zero excess profit. However, farmers closer to the customer point have the potential to earn excess profits, compared to the marginal farmer, because they spend less on shipping. However, this excess profit is not arising because of some unique managerial skill or other monopoly advantage attributable to the farmer; instead, it arises because of the proximity of the site. The notion of Ricardian Rent is that, with competition, such an excess profit will accrue to the owner of the asset: in this case, the landowner. This is because the existence of excess profit implies other tenant farmers will try to bid away that land; the pressure to outbid other users continues right up to the point where excess profit drops to zero at every location. By this mechanism, Ricardian rent is translated into market rent and in this sense generates a spatial equilibrium. See Fig. 10.2.
Ga Ia
Rent
Ra
Fig. 10.2 Model 10A: equilibrium in the market for land on a ribbon landscape. Notes: α = 4.5, β = 12, N = 1,000, C = 1, and s = 0.05. Equilibrium solution is X = 47.30 and R[0] = 2.36. Rent axis scaled from 0.0 to 2.5; distance axis scaled from 0 to 50. See also Fig. 10.1
0
K
Distance from demand point
Ha
How about comparative statics here? See Table 10.2. C
When C is increased, the supply curve shifts upward parallel. As it does, price (P) increases and quantity (Q) decreases in equilibrium; the demand curve is traced out. Because P does not increase as quickly as C, market rent, R[x], falls everywhere by the same amount; this downward shift in the market rent curve in Fig. 10.2 has the implication that the radius of the market (X) must also decrease.
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Table 10.2 Model 10A: comparative statics of an increase in exogenous variable Outcome Given
P [1]
Q [2]
X [3]
R[0] [4]
R[X] [5]
C N s α β
+ + + + −
− + − + −
− + − + −
− + + + −
0 0 0 0 0
Notes: See also Table 10.1; 0, No change; +, An increase in value;−, A decrease in value.
N
s
α
β
When N is increased, the demand curve becomes flatter: sweeps counterclockwise around the quantity–price combination (0, α). As it does, both price (P) and quantity (Q) increase in equilibrium; once again the supply curve is traced out. As a result of the increase in P, market rent, R[x], rises everywhere by the same amount; this upward shift in the market rent curve in Fig. 10.2 has the implication that the radius of the market (X) must also increase. When s is increased, the supply curve becomes steeper: sweeps counterclockwise about (0, C). As it does, price (P) increases, and quantity (Q) decreases in equilibrium; once again the demand curve is traced out. As a result of the increase in P, market rent at the origin, R[0], rises. However, a larger s means also the slope of the market rent curve in Fig. 10.2 has increased. Since Q has decreased, the radius of the market (X) must also decrease. Because a larger s means the marginal supplier is more costly, market price increases. When α is increased, the demand curve shifts upward parallel. As it does, both price (P) and quantity (Q) increase in equilibrium; the supply curve is traced out. As a result of the increase in P, market rent, R[x], rises everywhere by the same amount; this upward shift in the market rent curve in Fig. 10.2 has the implication that the radius of the market (X) must also increase. When β is increased, the demand curve becomes steeper, that is, sweeps clockwise around the quantity-price combination (0, α). As it does, both price (P) and quantity (Q) decrease in equilibrium; once again the supply curve is traced out. As a result of the decrease in P, market rent, R[x], falls everywhere by the same amount; this downward shift in the market rent curve in Fig. 10.2 has the implication that the radius of the market (X) must also decrease.
Before leaving this model, it is interesting that, just as in Model 9C, the model solves for equilibrium in two commodity markets at the same time. In this chapter, one is the market for wheat to which Fig. 10.1 applies. The second is the market for land to which Fig. 10.2 applies. With Model 10A, we can begin to describe a regional economy. Admittedly, this description is only partial. We know little about (1) who is demanding wheat at
10.3
Model 10B
279
the market point, (2) how much the farm is spending on inputs (other than shipping services and rent) and how much wheat revenue is income in the farmer’s hands, and (3) what the farm uses its income for. Nonetheless, undertaking such a description is useful in starting to address Thünen’s focus on the distribution of regional income. As an example, suppose α = 4.5, β = 12, N = 1,000, C = 1, and s = 0.05. These are the values I used to generate Figs. 10.1 and 10.2. As shown in the notes to those two charts, the outcomes are P = 3.36, Q = 95, X = 47.30, and R[0] = 2.36. Column [1] of Table 10.4 provides more detail about this regional economy under these conditions. This example portrays a regional economy in which shipping costs are substantial. Land rent payments to absentee landlords are a significant drain on the regional economy. Out of an expenditure of $318 by demanders at the market point, farmers receive only $95 to purchase their other inputs (e.g., seed, fertilizer, workers) and to cover the normal profit on their own otherwise unpriced labor, management, and capital inputs. Anything that reduces the unit shipping rate (s) will reduce the leakages due to shipping costs and rents and thereby reduce price and increase the quantity of wheat transacted. We will also see shortly that constraining farms to a line (ribbon landscape) causes the price of the commodity to be substantially higher than it would be on a rectangular plane. Model 10A is useful in starting to think about Thünen. However, we need to remember here that Model 10A is simpler than the regional economy envisaged by Thünen. In Model 10A, we have considered only one crop, produced using fixed yields, on a strip of land; Thünen was thinking of many crops—produced on a rectangular plane—for each of which yield was linked to labor intensity. Model 10A, absent a two-dimensional geography and multiple crops, has no rings. Model 10A also does not include any direct discussion of labor or capital inputs.16 And, unlike Thünen’s derivation of nonlinear functions for transportation costs, crop prices, and land rents, the solution to Model 10A involves only linear functions. At the same time, what Model 10A adds that is new from the perspective of Thünen is a direct connection to demand theory so that the impact of a shift in demand on crop price and land rents can be assessed.
10.3 Model 10B: Farms Producing Wheat on a Rectangular Plane Now, let us make a simple extension that will make possible a model closer to Thünen’s vision. In this extension, I abandon the notion of a long strip of land with farmers spread along it and assume alternatively, as did Thünen, a rectangular plane in which all land is available for wheat production. This has a direct implication for
16 From Thünen’s perspective, Model 10A might also be unsatisfactory because it does not consider
crop rotation.
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Table 10.3 Model 10B: wheat produced on a rectangular plane Supply of wheat π ((P − C) / s)2 if P ≥ C, 0 otherwise
(10.3.1)
Aggregate demand (αN / β) − (N / β)P
(10.3.2)
Demand equals supply (π)P2 + (s2 N / β − 2π C)P + (π C2 − s2 Nα/ β) = 0
(10.3.3)
Equilibrium price: P = C + v where √ v = (s/(2πβ) (− sN + {s2 N 2 + 4πβN(α − C)}))
(10.3.4) (10.3.5)
Equilibrium quantity Q = (α − C − v)(N / β)
(10.3.6)
Market radius X = v/s
(10.3.7)
Rent at distance x R[x] = v − sx
(10.3.8)
All farms’ revenue from wheat net of production costs ((α − C)v − v2 )N / β
(10.3.9)
All farms’ payments for shipping 2πv3 / (3s2 )
(10.3.10)
All farms’ payment for rent π v3 / (3s2 )
(10.3.11)
Producer surplus: Farms’ net revenue (net of production cost, shipping, and rent) 0
(10.3.12)
Consumer benefit (1 / 2)(α 2 − (C + v)2 )N / β
(10.3.13)
Producer cost (including shipping) CQ + 2πv3 / 3s2
(10.3.14)
Consumer surplus (1 / 2)(α − C − v)2 N / β
(10.3.15)
Social welfare (1 / 2)(α − C − v)2 N / β + π v3 / (3s2 )
(10.3.16)
Total market expenditure (PQ) (α(C + v) − (C + v)2 )N / β
(10.3.17)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Production cost per unit of land; s—Unit shipping rate ($/kilometer) for wheat; N—Number of consumers; x— Distance (kilometers); α—Intercept of individual linear inverse demand curve: maximum price; β—Negative of slope of individual linear inverse demand curve: marginal effect of quantity on price received. Outcomes (endogenous): P—Price of wheat (at market); Q—Quantity of wheat (square kilometers of yield); R[x]—Land rent bid by tenant farmers that leaves them with zero (excess) profit; X—Radius of area supplying wheat (kilometers).
10.3
Model 10B
281
the quantity supplied. Since quantity supplied is equal to the land area under cultivation and since all land within distance X is used to produce wheat, the quantity supplied is given by (10.3.1). See Table 10.3. The supply curve is also drawn as CEb Db in Fig. 10.3.
Chapter 10, Model B, Market for wheat
A7.6 D
A6.0
CEbDb OPb OQb OPbEbQbO OCFbQbO CEbFbC CPbEbC PbA4.50EbPb OA4.50EbQbO OCEbQbO CA4.50EbCC
Aggregate supply of wheat Equilibrium price at α = 4.50 Equilibrium quantity at α = 4.50 Market expenditure on wheat Total production cost: excludes rent and shipping Total shipping cost Total land rent Consumer surplus Consumer benefit Producer cost including shipping Social welfare
Price
A4.5
Pa
Ea
A2.2 Eb
Pb C A0.8
0
Fa
B0.85 Qa
Db
Fb
B2.20
Qb
B4.50
B6.00 Quantity
B7.64
Fig. 10.3 Model 10B: equilibrium in the market for wheat on a rectangular plane. Notes: α = 4.5, β = 12, N = 1,000, C = 1, and s = 0.05. Equilibrium solution is P = 1.45, Q = 254. Price axis scaled from 0 to 8; quantity axis from 0 to 700
CEb Db is a conventional supply curve in the sense that Q is here an upwardsloped function of market price, P. We also usually like to think a supply curve reflects the prices of inputs. That is the case here too, where both C and s play a role in shaping supply. It also has a plausible shape. After all, the quantity supplied is zero when the price is less than or equal to C. The quantity supplied is larger when P is higher than C but is ultimately limited by s. The more costly is shipping, the smaller the area under cultivation and the less the total supplied at a given market price.
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In an important respect, however, CEb Db is not the usual supply curve. Typically, for ease of exposition in this book, I draw supply curves as straight lines. This was the case for the supply curve (10.1.5) in the one-dimensional model: reproduced as CD in Fig. 10.3. However, the supply curve in two-dimensional space is the upper half of a quadratic: the half where P is above C.17 Quantity supplied is zero when the price is less than or equal to C, then rises quickly at higher prices. Therefore, CEb Db in Fig. 10.3 lies below and to the right of the supply curve (CD) in Model 10A. This shift in the supply curve compared to Model 10A means that the twodimensional geography of Model 10B has immediate implications in terms of equilibrium price. Graphically, we can see one of these in Fig. 10.3; from Models 10A to 10B, equilibrium price drops substantially (from OPa to OPb when α = 4.5), and quantity increases (from OQa to OQb ). The nonlinear supply curve in (10.3.1) also implies that the equilibrium price in Model 10B is more complicated to solve. Rearranging the aggregate inverse demand equation in (10.1.2) yields aggregate demand (10.3.2). Equating demand (10.3.2) and supply (10.3.1) yields an expression for price that is quadratic in price. Ordinarily, a quadratic has two solutions (roots). However, one of these solutions is when P < C, and we can ignore this case since it is uneconomic. Therefore, the only solution to this problem is the larger root. That root is given by (10.3.4): see also (10.3.5). So far, you might think this is a simple model of two-dimensional geography. Yet, even this simple model generates a solution for price (10.3.4) that is messy to analyze. Why? The problem here appears to be one of geography since, in Model 10A, the solution for P (a weighted average of C and α) was simple by comparison. Nonetheless, the solution for price in Model 10B is similar to Model 10A in at least two important respects. We can see from (10.3.5) that since the term under the square root sign is always no smaller than s2 N 2 , v > 0 and therefore P must always be larger than C. This is similar to what we found in (10.1.6) for P in Model 10A. Further in 10A, we found that P was larger for larger α, C, N, or s, or smaller β. Again, the same results hold in Model 10B. So, although (10.3.5) is unwieldy, it does generate solutions for P similar to those in Model 10A. What happens to market rent? Market radius (X) is determined by (10.3.7), and market rent at any location out to X is given by (10.3.8). Market rent has to decline linearly with distance, just as it did in Model 10A. The rent at distance 0 is R[0] = P − C, also just as in Model 10A. In other respects too, Model 10B is similar to Model 10A. Model 10B also solves for equilibrium in two markets at the same time: the market for wheat and the market for land. With Model 10B, we can also partially envisage a regional economy in the sense of identifying aggregate farm receipts and disbursements. Finally, Model 10B too is based on just one agricultural commodity.
17 According
relationship.
to Backhaus (2002, p. 437), Launhardt was perhaps the first person to identify this
Model 10B
Fig. 10.4 Model 10B: equilibrium in the market for land on a rectangular plane. Notes: α = 4.5, β = 12, N = 1,000, C = 1, and s = 0.05. Equilibrium solution is X = 9 and R[0] = 0.45. Rent axis scaled from 0.0 to 2.5; distance axis scaled from 0 to 50. See also Fig. 10.3
283 Ga Ra
Ia
GbIbHb OGb OHb OK ORb
Model 10B: Market for land Rent bid by farmers at each distance Rent bid at distance 0: R[0]=P-C Distance (X) at which bid rent is zero Distance (x) Rent bid at distance x
Rent
10.3
Gb Rb 0
Ib Ha K Hb Distance from demand point
Despite the similarities, Model 10B describes a different world from that of Model 10A. I use the same parameter values to construct Figs. 10.3 and 10.4 in Model 10B as I had used for Figs. 10.1 and 10.2 in Model 10A. The only difference between these two pairs of charts is 1 dimension (ribbon) versus 2 (rectangular plane). In Model 10B (Figs. 10.3 and 10.4), the market draws out 254 km2 of wheat production within a radius of 9 km; that enables a market equilibrium price for wheat of just $1.45. In Model 10A (Figs. 10.1 and 10.2) by contrast, the market had to extend 47.3 km to generate just 95 km2 of wheat production, and the market equilibrium price there had to more than double ($3.36). In Fig. 10.3, the shift in supply curve (from CD in Model 10A to CEb Db in Model 10B) means that we slide from point Ea on the demand curve Av Bv all the way to point Eb . We can think of the ribbon of Model 10A and the rectangular plane of Model 10B as two extremes. In reality, we might expect a fraction of the land around a customer point to be unavailable or unsuitable for crop production; in that case, the supply curve in practice might well lie somewhere between CD and CEb Db . The regional economy is different in Model 10B compared to Model 10A. In Model 10B, production costs a more substantial proportion of farm receipts; shipping costs in total are much lower; rents are lower still. See column [2] of Table 10.4. How about comparative statics here? As the only difference between Models 10B and 10A here is the nonlinearity of the supply curve, the comparative statics are the same as for Model 10A above. To illustrate, imagine a thought experiment in which we decrease the unit shipping rate per kilometer, s, in Model 10B. As we do, the supply curve starts to flatten. Equilibrium price travels down the demand curve, and equilibrium quantity increases. Given the assumption of fixed yields, the land area
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Table 10.4 The farm economy in Models 10A through 10D One crop
Two crops
Ribbon of land 10A [1]
Rectangular plane 10B [2]
Ribbon of land 10C [3]
Rectangular plane 10D [4]
318 0 318
369 0 369
206 278 484
391 398 789
95 0 95
254 0 254
53 78 131
238 232 470
Shipping cost Wheat Corn Total
112 0 112
76 0 76
36 141 177
69 143 213
Land rent Wheat Corn Total
112 0 113
38 0 38
117 59 176
84 23 106
0 0 0
0 0 0
0 0 0
0 0 0
36 36
21 10
36 36
27 13
Farm Receipts Wheat Corn Total Farm Disbursements Production cost Wheat Corn Total
Excess profit Wheat Corn Total As percentage of total farm disbursements Shipping cost Land rent
Notes: Calculations by author. α1 = 4.5, α2 = 4.5, β1 = β2 = 12, N = 1000, C1 = C2 = 1, s1 = 0.05, and s2 = 0.039.
under cultivation (πX 2 ) must rise, hence market radius must increase. Since market rent falls linearly with distance at a slope of -s, the decline in s causes the market rent curve in Fig. 10.2 to flatten. As a limiting case when s approaches 0, the supply curve becomes a horizontal line of the form P = C. As we saw before in Model 10A, the case of s = 0 corresponds to a non-spatial world wherein the customer point can get as much of the product as it wants at a price corresponding to marginal cost. The same result would also hold in the case of the one-dimensional supply curve (10.1.5) as we let s approach zero. The comparative statics are summarized in Table 10.5.
10.4
Model 10C
285
Table 10.5 Model 10B: comparative statics of an increase in exogenous variable Outcome Given
P [1]
Q [2]
X [3]
R[0] [4]
R[X] [5]
C N s α β
+ + + + −
− + − + −
− + − + −
− + + + −
0 0 0 0 0
Notes: See also Table 10.3. From (10.3.1), Q and X always move in same direction; +, Effect on outcome of change in given is positive;−, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome.
10.4 Model 10C: Farms Producing Two Independently Demanded Crops Along a Line So far, we have looked at a market for wheat. Suppose, however, consumers at the customer point also purchase a second crop, say corn. I assume in Model 10C that the demands for each crop have a zero cross-price elasticity.18 The price of wheat in itself has no direct effect on the demand for corn and vice versa. Put differently, I ignore the possibility of corn and wheat being substitutes or complements for the consumer. The demand curves in (10.5.1) envisage that consumers will purchase less corn if the price of corn goes up but do not directly show this translating into a demand for more wheat. In this simple case, where and when would corn or wheat be produced? Geographers among others had long argued that the allocation of land to agricultural uses reflected the suitability of a site (e.g., soil, hydrology, or climate) for the production of a particular crop. Thünen showed that, even when there are no differences in land suitability from one site to the next, the operation of a land market acts to form rings of farmland, arranged by distance from the city, within which only one type of crop will be produced. This can be illustrated by an extension to Model 10A. Assume consumers at the customer point have a demand curve for each of the two commodities as given by (10.6.1): see Table 10.6. Without loss of generality, label as crop 1, the crop with the higher shipping rate per kilometer: s1 > s2 . Here, since I have assumed a fixed yield for each type of crop; si (where i = 1 or 2) is the cost of transporting the amount of crop i that can be produced on one unit of land; it is in that respect that s1 and s2 are comparable. There are now two categories
18 For
any two commodities, labeled 1 and 2, respectively, cross-price elasticity (c) is the percent change in quantity of good 1 demanded (q1 ) given a one percent change in the price of good 2 (p2 ):c = (p2 /q1 )(dq1 /dp2 ).
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Table 10.6 Model 10C: wheat and corn produced on ribbon landscape assuming s1 > s2 and both crops produced Aggregate inverse independent demand curves for the two crops P1 = α1 − β1 Q1 / N and P2 = α2 − β2 Q2 / N
(10.6.1)
Rent bid for land for the two crops R1 [x] = P1 − C1 − s1 x and R2 [x] = P2 − C2 − s2 x
(10.6.2)
Inner and outer radii X1 = ((P1 − C1 ) − (P2 − C2 )) / (s1 − s2 ) and X2 = (P2 − C2 ) / s2
(10.6.3)
Supply of crops Q1 = 2X1 and Q2 = 2(X2 − X1 )
(10.6.4)
Supply equals demand for crop 2 2((P1 − C1 ) − (P2 − C2 )) / (s1 − s2 ) = (α1 N / β1 ) − (N / β1 )P1
(10.6.5)
which reduces to P1 = a1 + b1 P2
(10.6.6)
a1 = ((α1 N / β1 ) + 2(C1 − C2 ) / (s1 − s2 )) / (2 / (s1 − s2 ) + (N / β1 ))
(10.6.7)
b1 = (2 / (s1 − s2 )) / (2 / (s1 − s2 ) + (N / β1 ))
(10.6.8)
Supply equals demand for crop 1 2((P2 − C2 ) / s2 ) − 2((P1 − C1 ) − (P2 − C2 )) / (s1 − s2 ) = (α2 N / β2 ) − (N / β2 )P2 (10.6.9) which reduces to P2 = a2 + b2 P1 Where a2 = ((α2 N / β2 ) + 2C2 ,/ s2 − 2C1 ,/ (s1 − s2 ) + 2C2 ,/ (s1 − s2 )) / (2 / s2 + N / β2 + 2 / (s1 − s2 )) b2 = (2 / (s1 − s2 )) / (2 / s2 + N / β2 + 2 / (s1 − s2 )) Substituting (10.6.6) into (10.6.10) P2 = (a2 + b2 a1 ) / (1 − b2 b1 )
(10.6.10) (10.6.11) (10.6.12) (10.6.13)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): Ci —Production cost per unit of land for crop i; si —Unit shipping rate ($/kilometer) for crop i; N—Number of consumers; x—Distance (kilometers); α i —Intercept of individual linear inverse demand curve for crop i: maximum price; β i —Negative of slope of individual linear inverse demand curve for crop i: marginal effect of quantity on price received. Outcomes (endogenous): Pi —Price of crop i (at market); Qi —Quantity of wheat (square kilometers of yield); Ri [x]—Bid rent by tenant farmers producing crop i that leaves them with zero (excess) profit; Xi —Radius of area supplying crop i (kilometers). A unit of either commodity is the amount yielded, assumed constant, by a unit of land.
of land users, wheat and corn farmers, bidding for land: each willing to bid up to the amount of the excess profit per unit land (i.e., their bid rent). Let R1 [x] be bid rent by farmers producing crop 1 for a unit of land at distance x from the market: R2 [x] for farmers producing crop 2. Assume that land is awarded to the highest bidder. Under the assumption of a competitive market for land, market rent, R[x],
10.4
Model 10C
287
will be the envelope curve of these bid rents: i.e., R[x] = Max{R1 [x],R2 [x]}. At each location, there are potentially both wheat farmers and corn farmers vying for land, and landlords accept the higher of the two bid rents. Without specifying the supply sides for the moment, let us imagine that an equilibrium price results for each product. From that, we can then imagine rents bid by the two kinds of farmers that look like (10.6.2); that is, a linear function of distance from the customer point with a slope of minus the shipping rate per kilometer for wheat (i.e., − s1 or − s2 ). There are two possibilities in thinking about the market rent that emerges here. One possibility, a corner solution, is that one kind of farmer (say crop 1 farmers to illustrate the idea here) outbids the other kind of farmer everywhere. This would be the case for example if, for crop 2 farmers, C2 > α2 . The other possibility is that both kinds of farmers are able to bid successfully for land somewhere. In this case, since s1 > s2 , farmers producing crop 1 would bid higher near the customer point while farmers producing crop 2 bid higher further away. To understand this important idea, see Fig. 10.5. There, curve G1 IH1 is the bid rent curve of crop 1 farmers, curve G2 IH2 is the bid rent curve of crop 2 farmers, and curve G1 IH2 is the resulting market rent (envelope) curve. As drawn here, curve Model 10C: Market for land
Rent
G1
Rent bid by farmers of crop 1 at each distance Rent bid by farmers of crop 2 at each distance Market rent (envelope curve) Rent bid at distance 0 by crop 1: R1[0]=P1-C1 Rent bid at distance 0 by crop 2: R2[0]=P2-C2 Distance at which bid rent of crop 1 is zero Distance (X2) at which bid rent of crop 2 is zero Distance (X1) at which bid rents of crops 1 and 2 same. Crop 1 outbids Crop 2 inside this distance; Crop 2 outbids Crop 1 outside this distance OJ Rent bid by both crops1 and 2
G1IH1 G2IH2 G1IH2 OG1 OG2 OH1 OH2 OK
G2
J
I
0 H1 K Distance from demand point
H2
Fig. 10.5 Models 10C and 10D: Equilibrium in the market for land when two crops are produced. Notes: P1 − C1 = 10; P2 − C2 = 6; s1 = 2.0; s2 = 0.60. Distance axis scaled from 0.0 to 12; rent axis scaled from 0 to 12
288
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G1 IH1 is steeper than curve G2 IH2 . If both crops are to be produced, there must be a point of intersection (I). However, since curve G1 IH1 is steeper, it must lie above curve G2 IH2 to the left of I and below curve G2 IH2 to the right. Therefore, since s1 > s2 , crop 1 is produced close to the customer point, while crop 2 is produced further away. Market rent, as the higher of the two bid rents at any location, is the envelope curve G1 IH2 .19 Ignoring for the moment the possibility of a corner solution, we are now ready to model supply and market equilibrium. Crop 1 is produced from distance 0 to distance X1 , while crop 2 is produced from distance X1 to distance X2 : see (10.6.3). The total land area under cultivation (and, given fixed yields, the amount of each crop produced in total) is given by (10.6.4). Equating aggregate demand and supply for crop 1 then yields (10.6.5) which reduces to a linear equation (10.6.6), wherein the price of crop 1 depends on exogenous parameters but also on the price of crop 2. Similarly, equating aggregate demand and supply for crop 2 then yields (10.6.9) which reduces to a linear equation (10.6.10), wherein the price of crop 2 depends on exogenous parameters but also on the price of crop 1. Note that this dependence on the price of the other crop is not arising because consumers substitute one for the other but rather because the two crops compete for land. These two linear equations—(10.6.6) and (10.6.10)—in two unknowns (P1 and P2 ) can then be solved. In (10.6.13), I show the resulting solution for P2 ; P1 can then be solved by back-substitution. At this point, the reader might look longingly back at Model 10A where equilibrium price was familiarly a weighted average of C and α for that crop alone. In contrast, (10.6.13) tells us that, in Model 10C, equilibrium price for each crop depends on parameters for both crops. The computational complexity in (10.6.13) should not be surprising. After all, this is the most sophisticated model of Walrasian equilibrium to this point in the book. Model 10C solves simultaneously for equilibria in the markets for wheat and corn, as well as in the market for rented land. Let us now return to the question of drawing supply curves for the two crops. The difficulty here is that the supply of each crop now depends on the prices of both crops because these affect bid rents and the amount of land available for production of a crop. Normally, when we draw a supply curve—because a graph is limited to two dimensions—we assume the quantity supplied is a function of own price only. However, in Model 10C, because of the competition for land, the amount supplied of one crop depends also on the price of the other crop. Therefore, when we switch from a one-crop model to a two-crop model we lose the ability to draw a simple supply curve to go with our aggregate demand curve for each product. However, despite that limitation, we have already seen (in Table 10.6) that we are able to solve the model algebraically in the absence of a corner solution. How about comparative statics here? See Table 10.7. In the following, I continue to presume an absence of corner solutions.
19 Stevens (1968) uses a mathematical programming approach to determine rents at a given location
in a regional economy with two crops.
10.4
Model 10C
289
Table 10.7 Model 10C: comparative statics of an increase in exogenous variable (assuming s1 > s2 )) Outcome Given
P1 [1]
Q1 [2]
X1 [3]
R1 [0] [4]
R1 [X1 ] [5]
P2 [6]
Q2 [7]
X2 [8]
R2 [X2 ] [9]
C1 C2 N s1 s2 α1 α2 β1 β2
+ − + + − + + − −
− + + − + + − − +
− − + − − + − − −
− − + + 0 + + − −
+ − + + − − + + −
− + + − + + + − −
+ − + + − − + + −
− − + − − + + − −
0 0 0 0 0 0 0 0 0
Notes: See also Table 10.6; +, Effect on outcome of change in given is positive; −, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome.
Cι
N
sι
αι
When Cι is increased, the supply curve for crop i shifts upward parallel. As it does, own price (Pι ) increases and quantity (Q) decreases in equilibrium. Because Pι does not increase as quickly as Cι , market rent, Rι [x], falls everywhere by the same amount; this downward shift in the market rent curve in Fig. 10.2 has the implication that the radius of the market (Xι ) must also decrease. On net, this increases the area available for production of the other crop j and causes its price (Pj ) to fall. When N is increased, each demand curve becomes flatter: sweeps counterclockwise around the quantity–price combination (0, αι ). As it does, both price (Pι ) and quantity (Qι ) increase in equilibrium. As a result of the increase in P, market rent, Rι [x], rises everywhere by the same amount; this upward shift in the market rent curve in Fig. 10.2 has the implication that the radius of each production area (X1 and X2 ) must also increase. When sι is increased, the supply curve for crop i becomes steeper; it sweeps counterclockwise about (0, Ci ). As it does, price (Pi ) increases and quantity (Qi ) decreases in equilibrium. As a result of the increase in Pi , bid rent at the origin, Ri [0], rises. However, a larger si means also the slope of the market rent curve in Fig. 10.2 has increased. Since Qi has decreased, the radius of the market (Xι ) must also decrease. On net, this increases the area available for production of the other crop j and causes its price (Pj ) to fall. When α ι is increased, the demand curve for crop i shifts upward parallel. As it does, both own price (Pι ) and quantity (Qι ) increase in equilibrium. As a result of the increase in Pι , market rent, Rι [x], rises everywhere by the same amount; this upward shift in the market rent curve in Fig. 10.2 has the implication that the radius of the market (Xι ) must also increase. On net, this reduces the area available for production of the other crop j and causes its price (Pj ) to rise.
290
βι
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Farming for Cash
When β ι is increased, the demand curve for crop i becomes steeper: sweeps clockwise around the quantity-price combination (0, αi ). As it does, both own price (Pι ) and quantity (Qι ) decrease in equilibrium. As a result of the decrease in Pι , market rent, Rι [x], falls everywhere by the same amount; this downward shift in the market rent curve in Fig. 10.2 has the implication the radius of the market (Xι ) must also decrease. On net, this increases the area available for production of the other crop j and causes its price (Pj ) to fall.
What about the possibility of corner solutions (kinks) here? Because the only difference between Models 10C and 10A is the presence or absence of crop 2, it is interesting to compare outcomes here in the two models as we shift the demand curve for crop 1 (by changing α 1 ). In Model 10A, where there is no production of crop 2, equilibrium price and quantity increase linearly with α 1 above the level required to make production profitable (i.e., α1 > C1 ). For Model 10A, I show a supply curve—the locus of equilibrium price and quantity traced out by parallel shifts in the demand curve—as the straight line CFD.20 At α1 = 7.64, for example, the equilibrium price for crop 1 would be P∗ and quantity supplied would be Q∗ in Fig. 10.6. In Model 10C, producers of crop 1 compete for land with producers of crop 2. Where crop 2 is profitable, a higher price for crop 1 is required to generate
Model 10C: Wheat and corn
A7.64 D
CcFD OPc OQc F
A6.0 P∗
Aggregate supply of crop 1 Equ ilibrium price at α1 = 4.50 Equilibrium quantity at α1= 4.50 Point where supply curves in Model C and Model A first converge; equilibrium price and quantity combinations same in Models C and A for α1 ≥ 7.64
F
A4.50 Pc
20 Normally,
Pa Cc
Ec Ea
Price
Fig. 10.6 Model 10C: equilibrium in the market for crop 1 of up to 2 crops on a ribbon landscape. Notes: α2 = 4.5, β1 = β2 = 12, N = 1,000, C1 = C2 = 1, s1 = 0.05, and s2 = 0.039. In constructing ADB and CDB, α 1 is varied from 2.0 to 200. Price axis scaled from 0 to 8; quantity axis from 0 to 700
A2.20 Pb
Db
Eb
C A0.85 0
B0.85 Qa Qc
B4.50 B2.20 Qb Q∗ Quantity
B6.00
B7.64
we think of a supply curve as the relationship between quantity and own price with other prices held constant. However, along the quasi supply curves CFD and Cc FD, the price of crop 2 is constantly changing.
10.4
Model 10C
291
a given quantity of crop 1 than would be the case in Model 10A. As shown in Fig. 10.6, this is a price of Pa in the case of Model 10A versus Pc in 10C. In Model 10C, I can draw something I call a quasi supply curve: the locus of equilibrium P1 and Q1 traced out by parallel shifts in the demand curve. The quasi supply curve for crop 1 is different from the supply curve for crop 1 in Model 10A because the latter assumes all other prices are held constant, whereas other prices (specifically the price of crop 2) may vary along the quasi supply curve. The quasi supply curve is given by a polyline Cc FD that consists of two straight-line segments Cc F and FD. In the segment FD, α 1 is so large that production of crop 2 is nowhere profitable. Put differently, in segment FD, Model 10C reduces to 10A. However, for α 1 < 7.64, it is still profitable to produce crop 2. So, in the segment Cc F, P1 and Q1 increase with α 1 , but the slope of Cc F is less than the slope of CF in Model 10A. In Model 10C, we can therefore envisage two critical values for α 1 that produce kinks in the relationship between P1 and Q1 ; one is when P1 is too low to encourage production of crop 1 (that is, P1 is below the amount OCc in Fig. 10.6); the other is the P1 above which there is no production of crop 2 (that is, α1 ≥ 7.64 in Fig. 10.6). In Model 10C, therefore, the supply curve of crop 1 is the polyline OCc FD with kinks at Cc and F; in Model 10A, it is OCFD with a kink at C only. The kink at point F complicates the interpretation of Model 10C. Let me now discuss a numerical method for solving Model 10C that specifically allows for kinks. The method is an adaptation of Samuelson’s programming method for finding a spatial price equilibrium discussed in Chapter 5. See Table 10.8. First, set the width of each ring: see (10.8.1). That in turn fixes the radii in (10.8.2), the supply of each
Table 10.8 Model 10C: numerical solution method wherein either or both crops produced Width of crop rings d1 ≥ 0 and d2 ≥ 0
(10.8.1)
Radii X1 = d1 and X2 = d1 + d2
(10.8.2)
Supply Q1 = 2d1 and Q2 = 2d2
(10.8.3)
Rent R[X2 ] = 0 and R[X1 ] = s2 (X2 − X1 ) R[0] = s2 (X2 − X1 ) + s1 X1
(10.8.4)
Price P1 = α1 − β1 Q1 / N and P2 = α2 − β2 Q2 / N
(10.8.5)
Consumer benefit CB1 = 0.5(P1 + α1 )Q1 and CB2 = 0.5(P2 + α2 )Q2
(10.8.6)
Production plus shipping cost (excludes land rent) PC1 = Q1 (C1 + 0.5 X1 s1 ) and PC2 = Q2 , (C2 + 0.5(X1 + X2 )s2 )
(10.8.7)
Net social welfare (to be maximized) NSW = CB1 + CB2 − (PC1 + PC2 )
(10.8.8)
Note: See also Table 10.6.
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Farming for Cash
Table 10.9 Model 10C: examples of numerical solution wherein NSW maximized α1
1.00 [1]
3.00 [2]
5.00 [3]
7.00 [4]
9.00 [5]
11.00 [6]
CB1 CB2 d1 d2 NSW P1 P2 PC1 PC2 Q1 Q2 R[0] R[X1 ] X1 X2
– 425.93 0.00 55.56 194.44 – 3.17 – 231.48 – 111.11 2.17 2.17 0.00 55.56
– 425.93 0.00 55.56 194.44 – 3.17 – 231.48 – 111.11 2.17 2.17 0.00 55.56
335.27 269.32 36.77 32.79 261.86 4.12 3.71 141.15 201.58 73.54 65.58 3.12 1.28 36.77 69.56
934.53 70.11 76.89 7.96 489.18 5.15 4.31 449.35 66.12 153.77 15.92 4.15 0.31 76.89 84.85
1,665.45 – 108.11 0.00 864.86 6.41 – 800.58 – 216.22 – 5.41 0.00 108.11 108.11
2,534.70 – 135.14 0.00 1,351.35 7.76 – 1,183.35 – 270.27 – 6.76 0.00 135.14 135.14
Notes: Calculations by author. See also Table 10.6. α2 = 4.5, β1 = β2 = 12, N = 1,000, C1 = C2 = 1, s1 = 0.05, and s2 = 0.039. CB consumer benefit; PC producer cost; NSW net social welfare (CB1 + CB2 − PC1 − PC2 ); – Not applicable. Neither production nor consumption of that crop. Numerical solution method is to find d1 and d2 that maximize Net Social Welfare (NSW).
crop in (10.8.3), and the prices that consumers are then willing to pay for the crops in (10.8.5). From that, we can then calculate consumer benefit (10.8.6), producer cost (10.8.7), and net social welfare (10.7.8). To summarize, the objective here is to find the d1 and d2 , each zero or positive, that maximizes net social welfare. As this is just a quadratic programming problem, it can be solved numerically just as we solved the Samuelson spatial price equilibrium problem. In Table 10.9, I show the solutions to Model 10C as α 1 is varied from 1.00 to 11.00 with other parameters given in the Note to that Table—the same numerical conditions as underlie Fig. 10.6. From Fig. 10.6, there is neither production nor consumption of crop 1 at α1 < 3.10 and neither production nor consumption of crop 2 at α1 > 7.64. Since α 2 is fixed here, consumer benefit regarding crop 2 (CB2 ) declines above α1 = 3.10 since increases in α 1 then cause P2 to increase. What happens when s1 = s2 or when s1 < s2 . We don’t have to worry about s1 < s2 because this means simply that we should relabel the crops so that the crop with the higher unit shipping rate is crop 1. However, we do have a special case when s1 = s2 . In this case, there are two possibilities. In one P1 − C1 = P2 − C2 ; here both types of farmers are able to bid the same amount for land at any given site. There are no rings here and the location of any one type of crop is indeterminate. In the other case, P1 − C1 = P2 − C2 . In this case, only the farmers producing the most profitable crop are able to bid successfully for land; there is no production of the crop for the lower of P1 − C1 or P2 − C2 .
10.5
Model 10D
293
What happens to the regional economy? See column [3] of Table 10.4. As in Model 10A, the ribbon landscape in Model 10C means that shipping cost and land rents will be substantial.
10.5 Model 10D: Farms Producing Two Independently Demanded Crops on a Rectangular Plane Finally, consider the case of tenant farmers producing two crops on a rectangular plane. Other than this, I use the same assumptions as in Model 10C. The model is summarized in Table 10.10. The two aggregate demand curves are shown in (10.10.1). The rent bid by each crop at a given location is shown in (10.10.2). The radius (X1 ) within which crop 1 is produced and the outer limit (X2 ) on crop 2 production are shown in (10.10.3). These are all the same as in Model 10C. Table 10.10 Model 10D: wheat and corn on rectangular plane: assuming s1 > s2 Aggregate demand for wheat and corn Q1 = (α1 N / β1 ) − (N / β1 )P1
Q2 = (α2 N / β2 ) − (N / β2 )P2
(10.10.1)
Rent bid for land for the two crops R1 [x] = P1 − C1 − s1 x
R2 [x] = P2 − C2 − s2 x
(10.10.2)
Inner and outer radii X1 = ((P1 − C1 ) − (P2 − C2 )) / (s1 − s2 )
X2 = (P2 − C2 ) / s2
(10.10.3)
Supply of crops Q1 = πX12
Q2 = π (X22 − X12 )
(10.10.4)
Note: See Table 10.11.
New to this model are the supply curves for the two crops. Crop 1 is produced in the inner ring as before, but the total land area here now is the area of a circle of radius X1 . Crop 2 is produced in the outer ring, from X1 to X2 kilometers from the customer point. The two quantities of crop production are shown in (10.10.4). The market equilibrium prices for the two crops, P1 and P2 , are those that make demand in (10.10.1) equal to the supply in (10.10.4) taking into account the radii in (10.10.3). This involves solving two simultaneous quadratic equations. As in Model 10B, there are two roots (solutions) each for P1 and P2 . One solution for Pi is smaller than Ci and can therefore be ignored. Therefore the solution is the larger root for each of P1 and P2 . To solve these two prices, I adapted the Net Social Welfare algorithm that I earlier used to solve Model 10C. See Table 10.11. To summarize, the algorithm invokes mathematical programming to maximize NSW. In each step of the algorithm carry out the following: 1. Assign a value each for d1 and d2 , consistent with (10.11.1); 2. Calculate the two radii in (10.11.2); 3. Calculate the supply of each crop from (10.11.3); 4. Calculate the price forthcoming for each crop from (10.11.5);
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Table 10.11 Model 10D: numerical solution method wherein either or both crops produced Width of crop rings d2 ≥ 0 d1 ≥ 0
(10.11.1)
Radii X1 = d1
(10.11.2)
Supply Q1 = πd12
X2 = d1 + d2 Q2 = π (X22 − d12 )
(10.11.3)
Rent R[X2 ] = 0 R[X1 ] = s2 (X2 − X1 ) R[0] = s2 (X − X1 ) + s1 X1
(10.11.4)
Price P1 = α1 − β1 Q1 ,/ N P2 = α2 − β2 Q2 ,/ N
(10.11.5)
Consumer benefit CB1 = 0.5(P1 + α1 ) Q1 CB2 = 0.5(P2 + α2 ) Q2
(10.11.6)
Production plus shipping cost (excludes land rent) PC1 = Q1 C1 + (2 / 3)π s1 X13 PC2 = Q2 C2 + (2/3) π s2 (X23 − X13 )
(10.11.7)
Net social welfare (to be maximized) NSW = CB1 + CB2 − (PC1 + PC2 )
(10.11.8)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): Ci —Production cost per unit of land for crop i; si —Unit shipping rate ($/kilometer) for crop i; N—Number of consumers; x—Distance (kilometers); α i —Intercept of individual linear inverse demand curve for crop i: maximum price; β i —Negative of slope of individual linear inverse demand curve for crop i: marginal effect of quantity on price received. Outcomes (endogenous): Pi —Price of crop i (at market); Qi —Quantity of wheat (square kilometers of yield); Ri [x]—Bid rent by tenant farmers producing crop i that leaves them with zero (excess) profit; Xi —Radius of area supplying crop i (kilometers).
5. Calculate consumer benefit for each crop in (10.11.6); 6. Calculate production plus shipping cost for each crop using (10.11.7); and 7. Calculate net social welfare from (10.11.8). In Table 10.12, I present solutions for a representative problem of this type. In Fig. 10.7, I present the corresponding outcomes graphically. The resulting quasi supply curve for crop 1 is the curve Cd Ed Dd . It lies just above the supply curve (CEb Db ) for Model 10B which implies, not surprisingly, that the price (P1 ) required to produce a given quantity of crop 1 is greater with the competition between crops in Model 10D than in the one-crop scenario of Model 10B. There is an important difference here between ribbon (Models 10A and 10C) and the rectangular plane (Models 10B and 10D). Above, I showed that a second
10.6
Final Comments
295
Table 10.12 Model 10D: examples of numerical solution wherein NSW maximized α1
1.00 [1]
3.00 [2]
5.00 [3]
7.00 [4]
9.00 [5]
11.00 [6]
CB1 CB2 d1 d2 NSW P1 P2 PC1 PC2 Q1 Q2 R[0] R[X1 ] X1 X2
– 747 0.00 8.89 413 – 1.52 – 334 – 248 0.35 0.35 0.00 8.89
272 734 6.15 4.53 502 1.57 1.62 143 360 119 240 0.48 0.18 6.15 10.68
926 717 9.41 3.31 899 1.66 1.74 365 379 278 230 0.60 0.13 9.41 12.71
1,915 702 11.80 2.68 1,614 1.75 1.85 610 392 438 221 0.69 0.10 11.80 14.48
3,237 687 13.80 2.27 2,650 1.82 1.94 873 400 598 213 0.78 0.09 13.80 16.07
4,892 673 15.54 1.98 4,007 1.89 2.03 1,152 406 759 206 0.85 0.08 15.54 17.53
Notes: Calculations by author. See also Tables 10.11. α2 = 4.5, β1 = beta2 = 12, N = 1,000, C1 = C2 = 1, s1 = 0.05, and s2 = 0.039. Numerical solution method is to find d1 and d2 that maximize Net Social Welfare (NSW).
crop introduced the possibility of a kinked supply curve Cc FD. In Model 10D, however, there is no similar kink. It is true that Cd Ed Dd in Model 10D and CEb Db in Model 10B do approach one another asymptotically as α 1 becomes sufficiently large. However, there is no kink in the supply curve Cd Ed Dd in the case of a rectangular plane. Given the important role that kinks play in competitive location—as evidenced throughout this book—the asymmetry between ribbon and rectangular plane geographies is an interesting oddity. What about comparative statics in Model 10D? In essence, Model 10D differs from Model 10C only in that the supply curve is now quadratic in distance instead of linear. However, despite the presence of a kink, that does not fundamentally affect the comparative statics; these remain the same as in Model 10C above. What happens to the regional economy? See Column [4] of Table 10.4. As with Model 10B, the farm economy on a rectangular plane in Model 10D spends relatively less on shipping cost and land rent than does the farm economy under either ribbon geography (Models 10A or 10C). Farms spend more on shipping and land rent in Model 10D than in Model 10B, presumably because of the competition between crops.
10.6 Final Comments In this chapter, the principal model has been 10D: two crops produced on a rectangular plane. I included Model 10B (one crop on a rectangular plane) to show how the introduction of a second crop affected model outcomes. I presented Models
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Model 10D: wheat and corn
A7.64
CdEdDd Aggregate supply curve of crop 1 OPd Equilibrium price at α1 = 4.50 OQd Equilibrium quantity at α1 = 4.50
D
A6.00 P∗
F
Price
A4.50 Pc Pa Cc
Ec Ea
A2.20
0
Dd
Ed
Pd Cd Pb C A0.8
Db
Eb
B0.85 Qc Qa
B2.20 Qd Q∗ Qb
B4.50 B6.00 Quantity
B7.64
Fig. 10.7 Model 10D: equilibrium in the market for crop 1 of up to 2 crops on a rectangular plane. Notes: α2 = 4.5, β1 = β2 = 12, N = 1,000, C1 = C2 = 1, s1 = 0.05, and s2 = 0.039. In constructing ADB and CDB, α 1 is varied from 2.0 to 200. Price axis scaled from 0 to 8; quantity axis from 0 to 700
10A and 10C to show the same results on a ribbon landscape. In Table 10.13, I summarize the assumptions that underlie Model 10A through 10D. Many assumptions are in common to all these models: see the list in panel (a) of Table 10.13.21 Given all other parameters, and allowing α or α 1 to vary gives us the opportunity to trace out a kind of supply curve for crop 1. The four models give distinctive (quasi) supply curves—CD, Cc FD, CEb Db , and Cd Ed Dd —depending on the combination of geography and number of crops. If we compare Cc FD with Cd Ed Dd in Fig. 10.7, it appears that geography does play a role in shaping the supply curve for 21 In
the models presented in this chapter, there is an exogenous demand for the agricultural product in the city, which I envisage as being for local consumption. Another alternative is to think of the demand at the city being by a firm that processes the agricultural commodity for resale to consumers including those living on farms. See Hsu (1997) for an extension of this type. Another extension might be to look at the setting of shipping rates by a profit-maximizer supplier of shipping services. See Harley (1988), Harris (1977), Hummels (2007), Macdonell (1891), and North (1958) for examples of work in this area.
10.6
Final Comments
297
Table 10.13 Assumptions in Models 10A through 10D
(a) Assumptions in common A1 Closed regional market economy B1 Exchange of crop (or crops) for money B3 All firms in the industry are identical C2 Fixed local customers C4 Identical customers C5 Identical linear demand D7 Horizontal marginal cost curve E2 Fixed unit shipping rate J1 Zero opportunity cost (rent) for land J5 Land is input to production J6 Competitive market for land (b) Assumptions specific to particular models A6 Ribbon landscape A4 Rectangular plane
10A [1]
10B [2]
10C [3]
10D [4]
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
x
x x
x
a crop. In the ribbon geography, there is a kink not found when geography takes the form of a rectangular plane. The various combinations of geography and number of crops also result in distinctively different farm economies. However, in all four models, land rents, the price of crop 1, and (in Models 10C and 10D) the price of crop 2 are jointly determined in Walrasian equilibrium. Early on in the chapter, I argued that the farm here can be thought of as any firm (factory) producing a good that requires land for production. There are parallels here between factory and farm in the sense that both typically use land, labor, and capital to produce an output that is then sold. The complication here is generally thought to be that the use of capital is more pervasive in the factory compared to farm. The parallels between factory and farm suggest that the notion of a Walrasian equilibrium based on competition in land, labor, and capital markets should be generalizable to a broad range of firms and production processes. Chapter 9 emphasized the importance of uncertainty in market formation. On the other hand, the models in Chapter 10 ignore uncertainty altogether. On the one hand, this is not unreasonable; Chapter 9 was concerned with the uncertainty that arises when farmers arrive to trade agricultural commodities among themselves, while Chapter 10 concerns farms supplying a city with a commodity for which the demand is known. On the other hand, Chapter 10 ignores plausible risks: e.g., the risk of crop loss or the risk of a sudden shift in demand. Let me wrap up this argument by saying simply that Chapter 9 gives some ideas about how to extend the models of Chapter 10 to incorporate uncertainty. While uncertainty is not a focus of this book, the simultaneity of prices and the linkage to localization are. What do the models in Chapter 10 tell us about Walrasian equilibrium and localization? In Models 10A and 10B, the price of the crop (wheat) and the rent gradient for land are jointly determined. Within the market radius, there
298
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Farming for Cash
is no linkage to localization; the geographic density of farms is everywhere the same. In Models 10C and 10D, the prices of the two crops (wheat and corn) and the rent gradient for land are jointly determined. Within the outer market radius, there is no linkage to localization overall in the sense that the geographic density of farms is everywhere the same. However, there is localization in the sense that, where both crops are produced, all the wheat farms are in one ring around the city while all the corn farms are in another. What about the regional economy? This is the first chapter in this book that has allowed us to look at receipts and disbursements of firms (in this case, farms) within the regional economy. This is an important step. However, we still don’t know much about factor incomes and well-being.
Chapter 11
The City and Its Hinterland A Regional Economy with Substitutability in Production (Thünen–Beckmann–Samuelson Problem)
A competitive industry in the city produces one commodity (soap) demanded by city dwellers and farmers alike. Farmers produce a second commodity (wheat) similarly demanded. Farmers ship wheat to the city in exchange for soap: again absent the uncertainties central to Chapter 9. That market is in competitive equilibrium. Tenant farmers bid up the market rent for land for sites close to the city until the Ricardian rent associated with proximity is entirely spent. Labor moves freely between city (soap production) and farm (wheat production). Labor market equilibrium is established wherein Model 11E incorporates land and labor as factors in agricultural production and therefore allows us to look at factor incomes. I include a non-spatial version, Model 11D, to show the consequences of adding unit shipping rates. I also included three versions of the simpler Beckmann Model because it too incorporates substitutability between land and labor in crop production: 11A (without shipping cost), 11B (one crop with shipping costs), and 11C (two crops with shipping costs). Chapter 11 extends the models in Chapter 10 first by incorporating labor and a production function for the agricultural commodity. In that respect, it can also be seen as akin to Chapter 7 with its repair labor and repair production function. By also introducing a good produced in the city and exchanged for the farm good, Chapter 11 builds on Chapters 9 and 10. As farms vary the intensity of land use, they also alter the geographic density of customers (farm workers) for the city’s product. In this way, the chapter considers how prices in commodity and input markets and the localization of farms are joint outcomes in a competitive market.
11.1 The Thünen–Beckmann–Samuelson Problem The models in Chapter 10 are instructive in thinking about how geography shapes a supply curve and how excess profits get attributed to landowners. In that chapter— with free entry—farms at different locations pay different rents for the use of land: rents that exhaust excess profits. However, the models in Chapter 10 have obvious shortcomings. This is the third of four chapters that look at models of agricultural markets. In this chapter, I introduce three new ideas to address shortcomings of the models in Chapter 10. J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_11, C Springer Science+Business Media, LLC 2010
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First, we might reasonably expect the farm to use land more intensively where it is relatively costly. In Chapter 10, I assumed the farmer always has a fixed yield per unit of land area. However, in Chapter 10, I also showed land rent would increase as we move closer to the customer point. In this chapter, I consider how the ability of farmers to substitute between land and labor in production affects the market for an agricultural product. Presumably, farmers use relatively less land if it is expensive and relatively more of other inputs (e.g., labor). This in turn affects rents as well as the market price required to elicit any given quantity of supply. To incorporate such substitution, this chapter moves beyond models that assume a fixed yield. Second, I look at how the allocation of labor between city production and farm production can be made endogenous in a model of the regional economy. The models in Chapter 10 do not say anything about how or why demand arises at a place. To think more about regional development and economic growth, we need to build a better understanding about what is generating demand here. I make a link in this chapter between individuals as participants in the labor market and as participants in the market for the exchange of commodities. In the later models in this chapter, I assume that city workers produce a good (soap) that they then trade for another good (wheat) produced on the farm. Farm and city then engage in trade because it is beneficial to both communities; demand becomes endogenous to the model. Third, the models in Chapter 10 do not say much about the distribution of income. Once we incorporate labor, we can begin to look at the individual wage rate and total wages paid to workers as part of the regional economy. In so doing, we get to link outcomes across markets in commodities, land, and labor. This is different from the approach used in Chapter 4, where we had assumed that consumption could be examined in the absence of labor force impacts. In the spatial price equilibrium model presented in Chapter 4, I argued that geography was treated as distinct and that it was important to think about situations in which consumers might be motivated to change their place of residence. The present chapter does that. I also argued that the spatial price equilibrium model in Chapter 4 differentiated locations on the supply side. For this chapter, I will continue to differentiate producers in that one group of firms (soap makers) cannot produce the commodity made by the other group of firms (wheat farmers). These three ideas are interconnected. Wheat workers who are further away from the city incur a greater shipping cost in consuming soap; presumably, they then need a higher wage to reach the same level of utility as farm workers closer to the city. At the same time, they substitute between soap and wheat as the exchange rate varies. As we move away from the city, we see land rents fall and wages rise; two trends that make the ability of farms to substitute between land and labor in production important. Equilibrium outcomes in the commodity and factor markets are then simultaneous and determined in part by the extent of substitution in both production and consumption. This chapter goes beyond Chapter 9 in two important respects. First, in Chapter 9, each individual had an endowment of one commodity and then seeks to trade it for another; there was no production there. There was a similar treatment in Chapter 10 where we assumed that the farmer had a fixed level of output (the yield or
11.1
The Thünen–Beckmann–Samuelson Problem
301
equivalently the endowment) for each square kilometer of land farmed. In this chapter, production is explicitly considered. Second, by introducing a labor market into a Thünen economy, we can get a fuller picture of the allocation of regional income among workers, capitalists, and landlords and the resulting interconnections among prices. In this chapter, I look in sequence at two interesting reformulations of Thünen: first, I present three models (11A, 11B, and 11C) inspired by Beckmann (1972) and then two (11D and 11E) based on Samuelson (1983).1 Both sets of reformulations help us to progressively better understand the operation of a regional labor market and its impacts on land and commodity markets. Before turning to Model 11A, let me comment briefly on the notion of a dual economy. Thünen models divvy up a regional economy into rural and urban sectors. Notions of dualization have permeated thinking in development economics since at least 1950.2 In thinking about the transition from underdevelopment nations to developed nations, many economists saw the importance of switching from an economy based on a traditional (often agrarian) sector to a modern (often technological, industrial, or manufacturing) sector in the national economy. In a classic in the field, Jorgenson (1961) describes the typical ingredients in a model of the dual economy in a closed region as follows: (1) output of the traditional sector depends on land and labor alone with diminishing returns and no capital accumulation; (2) the supply of land is fixed; (3) output of the modern sector depends on capital and labor alone with constant returns to scale and no land used; (4) production technology improves in both traditional and modern sectors over time; (5) population growth depends on the supply of food per capita; (6) the availability of labor for the modern sectors varies directly with agricultural surplus; and (7) wage rates are not necessarily the same in the two sectors. With such a model, development economists sought to explain differences in economic development around the globe. Related to this was research on the process of migration from rural to urban areas based on Harris and Todaro (1970).3 This chapter culminates in Model 11E wherein, unlike the standard dual economy model, the regional labor market is always in equilibrium. In that
1 See
also Nerlove and Sadka (1991). cited work in this field include Banerjee and Newman (1998), Cuddington (1993), Dixit (1971), Fields (1979), Hoare (1978), Hornby (1968), Jorgenson (1961, 1967), Kelley, Williamson, & Cheetham (1972), Lewis (1954, 1979), Nerlove and Sadka (1991), Oster (1979), Ranis and Fei (1961), Stern (1972b), and Zarembka (1970). For recent work, see Cai (2008), Chaudhuri (2008), Daitoh (2008), Islam and Yakota (2008), Landon–Lane and Robertson (2009), Le (2008), Satchi and Temple (2009), Skott and Ryoo (2008), Vollrath (2009), and Zenou (2008). See also the non-spatial perspective in Bowring (1986). 3 Commonly cited work here include Bhagwati and Srinivas (1974), Blomqvist (1978), Brueckner and Zenou (1999), Calvo (1978), Khan (1980), McCool (1982), Moene (1988), and Neary (1981). Recent work includes Chaudhuri (2008), Daitoh (2008), Edwards and Huskey (2008), Ghatak, Mulhern, & Watson (2008), Lee (2008), Martins (2008), Sarker, Gilbert, & Oladi (2008), Satchi and Temple (2009), and Zenou (2008). 2 Commonly
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The City and Its Hinterland
important respect, Model 11E does not fit into the dual economy literature. Nonetheless, I will demonstrate in this chapter that Model 11E is valuable in thinking about regional economic development.
11.2 Model 11A: Factor Substitution with One Crop and in the Absence of Shipping Cost Thünen emphasized the role of capital as an input in farm production. However, to simplify his model, Beckmann (1972) assumes a Cobb–Douglas production function for agriculture with just two inputs (labor and land) with constant returns to scale. Each input taken alone has diminishing marginal productivity. He assumes that labor is ubiquitous: available everywhere in any quantity at a given wage w. He is therefore thinking of the regional economy as a labor market either in part or in whole. In Model 11A, let me assume that only wheat is produced and that there are no shipping costs; I then introduce shipping costs in Model 11B. In Model 11A, I also introduce an opportunity cost for land: an annual rent (Ra ) per square kilometer of land that must be paid in order to gain use of the land for wheat production. In Chapters 9 and 10, in contrast, I assumed Ra = 0 (as did Thünen). The significance of Ra being nonzero becomes clear in later models of this chapter (Model 11D and Model 11E); I introduce it here merely for consistency of treatment. In order to emphasize the importance of the substitutability between land and labor, I set up Model 11A to be otherwise like Model 10B: farms on a rectangular plane producing only one crop (wheat) See Table 11.1 wherein I summarize equations, assumptions, notation, and rationale for localization in Model 11A. In terms of this book, the novelty of this model arises primarily from the production function for wheat: see (11.1.1). This production function assumes two inputs (labor and land) to crop production. It exhibits constant returns to scale.4 However, the farm can vary the amount of output produced by one unit of land by using labor more intensively (larger m) or less intensively (smaller m). Just how substitutable are the two inputs, labor and land, in this model? In a Cobb–Douglas production function, the elasticity of substitution is always −1.5 In reality, of course, elasticity of substitution need not be a constant (let alone −1); in that sense, Cobb–Douglas is a special case. However, I use a Cobb–Douglas production function here because it is a well-known example of a production function that incorporates a nonzero elasticity of substitution. The exponent, γ , of labor intensity in (11.1.1) is central to this model. For a profit-maximizing firm facing competitive markets for inputs and output, γ will be the fraction of firm
4 Were the farm to double both the amount of labor and land it uses, the ratio of labor to land used by the farm (n) would remain constant, the amount of output per unit land (q) would also remain the same, and therefore the total output of the farm would double. 5 The elasticity of substitution is the percentage change in ratio of land to labor when the ratio of factor prices is increased by 1%.
11.2
Model 11A: Factor Substitution with One Crop and in the Absence of Shipping Cost
303
Table 11.1 Model 11A: farms produce wheat with substitutability between land and labor when unit shipping cost is zero Production of wheat per square kilometer of land used where 0 < γ < 1 q = amγ
(11.1.1)
Profit per square kilometer of land for farm Pq − wm − Ra
(11.1.2)
Profit maximizing labor per square kilometer of land m = (γ aP/w)(1/(1 − γ ))
(11.1.3)
Profit per square kilometer (1 − γ )(γ /w)(γ /(1 − γ )) (aP)(1/(1 − γ ))
(11.1.4)
Equilibrium price (when free entry drives profit per square kilometer to zero) P = (Ra /(1 − γ ))1 − γ (w/γ )γ (1/a)
(11.1.5)
Intercept of supply curve (Ra /(1 − γ ))1 − γ (w/γ )γ (1/a)
(11.1.6)
Slope of supply curve 0 Aggregate demand for wheat Q = (α − P)N/β
(11.1.7) (11.1.8)
Land area occupied Q/q
(11.1.9)
Total labor in wheat production mQ/q
(11.1.10)
Producer cost (wm + Ra )Q/q
(11.1.11)
Consumer benefit (1/2)(α + P)Q
(11.1.12)
Net social welfare (1/2)(α + P)Q − (wm + Ra )Q/q
(11.1.13)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): a—Intercept of labor productivity equation for wheat; N—Population of consumers at customer point; Ra —Rent bid for square kilometer of land by alternative use everywhere; w—Wage rate; α—Intercept in individual linear inverse demand curve; β—Slope in individual linear inverse demand curve; γ —Exponent in labor productivity equations. Outcomes (endogenous): m—Labor used per unit land producing wheat; P—Market equilibrium price for wheat; Q—Aggregate quantity of wheat in market equilibrium; q—Wheat produced per unit land.
revenue spent on labor inputs. This limits the exponent: 0 < γ < 1.6 Within the interval from 0 to 1, the larger is γ , the more intensively the farm operates. 6 It is implausible to assume γ < 0 since this would mean gross output decreases as we increase the amount of labor used on a square kilometer of land. It is also implausible to assume γ = 0
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The farm’s profit per square kilometer of land is the amount by which revenue derived from the area exceeds the cost of labor and the opportunity cost of land: see (11.1.2). The farm is assumed to hire the amount of labor that maximizes this profit: see (11.1.3). Substituting this amount of labor, m, into (11.1.2) gives us the maximized profit (11.1.4). With competition in the wheat market, farms keep entering the market until excess profit is driven to zero; this happens when price (P) falls to
Model 11A Equilibrium in the market for wheat
A7.64 Price
A vBv CD CE aFC OQ a OA 4.50 OA 4.50 EQ aO OC OCE aQaO
A6.00
Demand curve (where v = a) Supply curve Consumer surplus (also net social welfare) Quantity demanded Intercept of demand curve (a = 4.50) Consumer benefit Intercept of supply curve (11.1.6) Producer cost
A4.50
A2.20 Ea
C
D
A0.85 0
B0.85
B2.20 Qa
B4.50
B6.00 B7.64 Quantity
Fig. 11.1 Model 11A: equilibrium in the market for wheat. Notes: α = 4.5, β = 12, a = 1, γ = 0.80, N = 1,000, w = 1, and Ra = 0.65. Equilibrium price is 1.51 and equilibrium quantity is 249. The horizontal axis is scaled from 0 to 700; the vertical axis is scaled from 0 to 8
since the farm then could produce the same output with any amount of labor on a unit of land; given the cost of labor (w), the profit-maximizing farmer would therefore choose zero labor and incur only land rent and shipping cost in production of wheat for sale at the market. Finally, the farm cannot have γ > 1 because this implies increasing returns which means the firm would maximize profit by shrinking the amount of land used to near zero, thereby making m very large.
11.2
Model 11A: Factor Substitution with One Crop and in the Absence of Shipping Cost
305
the level given in (11.1.5). Because I have simplified the model by ignoring inputs to production other than labor and land, there is no place here for normal profit arising from unpriced inputs of the farmer as entrepreneur; I therefore assume normal profit is zero. Further, that farms keep entering wheat production until (11.1.5) is reestablished implies that the supply curve for wheat is a horizontal line (11.1.7) that cuts the Y-axis at (11.1.6). We are now ready to solve for market equilibrium price and quantity. The familiar demand curve (11.1.8) can be laid on top of a supply curve that is horizontal here. Equilibrium price is therefore given by (11.1.5) and equilibrium quantity by (11.1.8). The total amounts of land and labor used for wheat production are then given by (11.1.9) and (11.1.10), respectively. As in earlier models, we are then able to calculate producer cost (11.1.11), consumer benefit (11.1.12), and net social welfare (11.1.13). I illustrate these ideas in Fig. 11.1. There, the supply curve is the horizontal line CD. Suppose the demand curve is A4.50 B4.50 . The intersection point Ea gives the equilibrium quantity of wheat (OQa ) and price (OC). In that diagram, I also show, using faint lines, the demand curve shifted by various amounts to the left or right. In all cases, because the supply curve is horizontal, the equilibrium price stays the same; only the quantity adjusts. What are the comparative statics here? There are two stories to be told: see Table 11.2. Table 11.2 Model 11A: comparative statics of an increase in exogenous variable
Outcome Given a N Ra w α β γ
P [1] − 0 + + 0 0 ?
Q [2] + + − − + − ?
Notes: See also Table 11.1; +, Effect on outcome of change in given is positive; −, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome; ?, Effect on outcome of change in given is unknown.
• One story is where the given (parameter value or exogenous variable) alters the position of the supply curve: this includes a, γ , Ra , and w. The supply curve remains horizontal but can shift either up or down. Anything (specifically an increase in Ra or w or a decrease in a) that causes the supply curve to shift
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up causes the equilibrium to run up the demand curve: i.e., P increases and Q decreases.7 • The other is where the parameter value shapes the position of the demand curve: this includes N, α„ and β. If α is increased as noted above, the demand curve shifts to the right, and the equilibrium travels to the right along the horizontal supply curve: P stays the same and Q increases. This kind of parallel shift in the demand curve is illustrated by various demand lines (Av Bv ) in Fig. 11.1. If β is increased or N decreased, the demand curve sweeps clockwise around the point (0, α), and the equilibrium point travels to the left along the horizontal supply curve: P stays the same and Q decreases. What does the regional economy look like? In column [1] of Table 11.3, I summarize the regional economy corresponding to equilibrium point Ea in Fig. 11.1. Here, 20% of all farm receipts are paid out to landlords in rents and the remaining 80% to workers as wages (since γ = 0.80 in this example). This is the first model in the book that has allowed us to separate out the wage bill in the regional economy.
11.3 Model 11B: Factor Substitution with One Crop and in Presence of Shipping Cost Now, let me introduce shipping costs into this regional economy. I retain the same production function for wheat—now labeled (11.4.1)—as in Model 11A. See Table 11.4. Given that the farm now incurs shipping costs as well, its profit per square kilometer of land is given by (11.4.2).8 I envisage the farm at any location choosing labor intensity (m) to maximize profit per square kilometer (r[x]) for wheat at any given distance: see (11.4.3). Substituting this optimized labor intensity back into (11.4.2), I show this profit for wheat, now its bid rent for land, decreasing with distance x from the market because of shipping cost: see (11.4.4). There is a radius X beyond which profit cannot possibly be positive regardless of how much (or little) labor is employed in production: see (11.4.5). As farms compete for sites that would otherwise be more profitable, they drive market rent at a given site up to the bid rent so that farmers everywhere earn only normal profits. How does this differ from Model 11A? In Model 11A, farmers enter the market until price is driven down to the point where excess profit is everywhere equal to Ra ; in Model 11B, price is driven down until excess profit at the boundary (X) is equal to Ra .
effects of an increase in γ on P and Q depends on values of the other givens. is a relationship of units here between s in (11.4.2) and a in (11.4.1). Typically, s is measured in terms of a count, a weight, or a volume: e.g., $0.02 per kilometer per 100 watermelons, or per cubic meter of watermelons, or per 100 kg of watermelons. The production scale parameter (a) must therefore be measured in the same unit of quantity. In his analysis, Beckmann appears to use weight as his measure. However, by allowing for a different unit shipping rate for each commodity, Beckmann is also able to incorporate commodity-specific costs like perishability. 7 The
8 There
11.3
Model 11B: Factor Substitution with One Crop and in Presence of Shipping Cost
307
Table 11.3 The regional economy in Models 11A through 11C One crop
Two crops
No shipping cost Model 11A [1]
With shipping cost Model 11B [2]
With shipping cost Model 11C [3]
377 0 377
402 0 402
408 402 810
Crop 1 (Wheat) Crop 2 Total
301 0 301
293 0 293
300 284 584
Shipping cost
Crop 1 (Wheat) Crop 2 Total
0 0 0
37 0 37
33 48 80
Land rent
Crop 1 (Wheat) Crop 2 Total
75 0 75
73 0 73
75 71 146
Excess profit
Crop 1 (Wheat) Crop 2 Total
0 0 0
0 0 0
0 0 0
As percentage of total farm disbursements
Shipping cost
0
9
10
20
18
18
Farm receipts
Crop 1 (Wheat) Crop 2 Total
Farm disbursements
Wage bill
Land rent Market price of crop Quantity of crop (produced or demanded)
Crop 1 (Wheat)
1.51
1.77
1.83
Crop 2
−
−
1.77
249
227
222
0
0
228
116
84
68
Crop 2
0
0
95
At x = 0
0.65
1.43
1.70
At x = X1 At x = X2 or X
0 0
0 0.65
0.87 0.65
Crop 1 (Wheat)
Crop 2 Amount of land used Land rent (per square kilometer)
Crop 1 (Wheat)
Notes: Model 11A: α = 4.5, β = 12, N = 1,000, a = 1, w = 1, γ = 0.80, Ra = 0.65; Model 11B: α = 4.5, β = 12, N = 1,000, a = 1, w = 1, γ = 0.80, Ra = 0.65, s = 0.05; Model 11C: α1 = α2 = 4.5, β1 = β2 = 12, N = 1,000, a1 = a2 = 1, w = 1, γ = 0.80, Ra = 0.65, s1 = 0.050, s2 = 0.035. Calculations by author.
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Table 11.4 Model 11B: farms producing wheat on a rectangular plane with substitutability between land and labor in presence of shipping cost Production of wheat per square kilometer of land used q = amγ where 0 < γ < 1
(11.4.1)
Schedule by location of profit per square kilometer of land for farm r[x] = (P − sx)q − wm
(11.4.2)
Schedule by location of profit maximizing labor per square kilometer of land m[x] = γ a(P − sx)/w)(1/(1 − γ ))
(11.4.3)
Schedule by location of bid rent by farm R[x] = (1 − γ )(γ /w)(γ /(1 − γ )) (a(P − sx))(1/(1 − γ ))
(11.4.4)
Distance at which crop is no longer profitable for farm X = (P − (Ra /(1 − γ ))(1 − γ (w/g)γ (1/a))/s
(11.4.5)
Total wheat supplied X Q = 2π o xq[x]dx
(11.4.6)
Total shipping cost X S = 2π o xq[x]sxdx
(11.4.7)
Total labor in wheat production X M = 2π o xm[x]dx
(11.4.8)
Producer cost PC = wM + S + Ra π X 2
(11.4.9)
Market clearing price V = α − βQ/N
(11.4.10)
Consumer benefit CB = (1/2)(α + V)Q
(11.4.11)
Net social welfare SW = CB − PC
(11.4.12)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): a—Intercept of labor productivity equation for wheat; N—Population of consumers at customer point; Ra —Rent bid for square kilometer of land by alternative use everywhere; s—Unit shipping rate; w—Wage rate; x— Distance; α—Intercept in individual linear inverse demand curve; β—Slope in individual linear inverse demand curve; γ —Exponent in labor productivity equations. Outcomes (endogenous): m—Labor used per unit land producing wheat; M—Total labor in wheat production; P—Market equilibrium price for wheat; Q—Aggregate quantity of wheat in market equilibrium; q—Wheat produced per unit land; r[x]—Profit per square kilometer from wheat; R[x]—Rent bid per unit land by wheat farmers; S—Total shipping cost; V—Price at which given quantity clears the market; X—Outer radius of area supplying wheat (kilometers).
We can now derive details about the market for wheat in equilibrium. I use numerical integration (specifically, the midpoint rule version of a Riemann sum9 )
9 In
calculations described here, I used N = 100 panels.
11.3
Model 11B: Factor Substitution with One Crop and in Presence of Shipping Cost
309
to solve for the total quantities of wheat produced (Q) and labor employed (M), and total expenditure on shipping (S): see (11.4.6), (11.4.7), and (11.4.8). From these, we can then calculate consumer benefit, producer cost, and net social welfare: see (11.4.11), (11.4.9), and (11.4.12). The equations in Table 11.4 do not permit an explicit algebraic solution for price. I use numerical methods to solve P. As in Chapter 10, I use quadratic programming to find the P that maximizes net social welfare in (11.4.12) subject to the constraints imposed by (11.4.1) through (11.4.11). As we did in Fig. 11.1, consider again the case where α = 4.5, β = 12, N = 1,000, a = 1, γ = 0.80, w = 1, and Ra = 0.65 and now let s = 0.05. The solution to this example of Model 11B is P = 1.77, Q = 227, X = 5.17, M = 293, L = 84, and R[0] = 1.43: see column [2] of Table 11.3. The empirical example of Model 10B that I used to generate Fig. 10.3 gave a lower P and higher Q compared to 11A at the same values of α, β, and N. Model 11B generates a smaller radius X and a higher central rent R[0]: in Model 10B, these were 9.00 and 0.45, respectively: see Fig. 11.3. These two differences reflect that (1) Ra was zero in Model 10B and (2) the lack of substitution in Model 10B denies the opportunity to seek a less costly mix of land and labor in producing wheat at any given distance. What does the supply curve for wheat look like? As in Chapter 10, I trace out a supply curve by systematically varying the intercept (α) of the demand curve. See the sample curve CEb Db in Fig. 11.2, but keep in mind that the exact position and shape of this curve will depend on the value we assume for s. The supply curve in Model A (the faint line CD now reproduced in Fig. 11.2) has the same vertical intercept as CEb Db : after all, if only a few farms are shipping, they will be all near the customer point, and hence shipping cost will be negligible. In general, the larger the P the faster the CEb Db rises, reflecting the fact that the area available for production increases with the square of the radius. What happens to land rent in this model? In Fig. 11.3, I sketch land rent (see curve Kb Lb ) by location corresponding to the case of point Eb in Fig. 11.2. For comparison purposes, I also show land rents (curve Gb Hb in Fig. 11.3) from Model 10B. The opportunity cost of land (Ra ) keeps rents higher everywhere in Model 11B compared to Model 10B. Further, by incorporating substitutability of labor and land, farms in Model 11A are able to bid rents up even more when nearer the market. The steeper rent gradient in Model 11B is what enables the decrease in maximum radius (X) in this model relative to Model 10B. What about the comparative statics here? There are basically three stories: see Table 11.5. • One story is where the given (parameter value or exogenous variable shapes the position of the supply curve: this includes a, γ , Ra , w, and s. The supply curve—now upward sloped—can shift either up or down. Anything (specifically an increase in Ra , s, or w or a decrease in a) that causes the supply curve to shift up causes the equilibrium to run up the demand curve: i.e., P increases and Q decreases.10 When P is increased, the bid rent curve shifts up. 10 The
effects of an increase in γ on P and Q depends on values of the other givens.
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Model 11B Equilibrium in the market for wheat
A7.64
Price
AvBv CEbDb CD CPbEbC OA4.50E bQ bO OCEbQ bO OQb OPb PbA4.50EbPb
A6.00
Demand curve (at a = v) Supply Curve for wheat Supply curve from Model 11A Producer surplus Consumer benefit Producer cost Equilibrium quantity Equilibrium price Consumer surplus
A4.50
A2.20
Eb
Pb C
Db D
A0.85 0
B0.85
B2.20Qb
B4.50
B6.00 B7.60 Quantity
Fig. 11.2 Model 11B: supply curve for wheat. Notes: a = 1, N = 1,000, s = 0.05, w = 1, β = 12, Ra = 0.65, and γ = 0.80. Equilibrium price is 1.77 and equilibrium quantity is 227. The horizontal axis is scaled from 0 to 700; vertical axis from 0 to 8
• The second story is where the parameter value shapes the position of the demand curve: this includes N, α, and β. If α is increased as noted above, the demand curve shifts to the right and the equilibrium travels to the right along the supply curve: P and Q increase. When P is increased, the bid rent curve shifts up. If β is increased or N decreased, the demand curve sweeps clockwise around the point (0, α), and the equilibrium point runs down the supply curve: P and Q decrease. When P is decreased, the bid rent curve shifts down. • The third story is about two parameters—γ and s—that affect the shape, not just the position, of the bid rent curve. When s is larger, the bid rent curve becomes steeper. When γ is larger, the farm is better able to substitute labor for land, and this too causes the bid rent curve to become steeper.
Model 11B: Factor Substitution with One Crop and in Presence of Shipping Cost
Fig. 11.3 Model 11B: bid rents and market rent as a function of distance from the market where both crops are sold. Notes: a = 1, N = 1,000, s = 0.05, w = 1 , α = 4.5, β = 12, and γ = 0.80. In Model 11B, R[0] = 1.43 and R[X] = 0.65. The horizontal axis is scaled from 0 to 10; the vertical axis is scaled from 0 to 1.6
311
Model 11B: Market for land
Kb
GbHb KbLb OK b ORa OX
Bid rent line for land in Model 10B (for comparison purposes) Bid rent curve for land by wheat farmers Bid rent for land adacent to market (R[0]) Opportunity cost of land (R a) Outer radius (X) enclosing farms producing wheat for this market
Rent
11.3
Lb
Ra Gb
0
Hb X Distance from market
Table 11.5 Model 11B: comparative statics of an increase in exogenous variable Outcome Given at x= a N Ra s w α β γ
P
Q
X
R[x]
[1]
[2]
[3]
0 [4]
− 0 + + + + 0 −
+ + − − − + − +
− + − − + + − −
+ + + + − + − +
X [5] 0 0 + 0 0 0 0 0
Notes: See also Table 11.4. Comparative statics numerically estimated from the base case: α = 4.5, β = 12, s = 0.05, a = 1, γ = 0.8, w = 1, N = 1,000, Ra = 0.65; +, Effect on outcome of change in given is positive; −, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome.
What happens to the regional economy? See column [2] of Table 11.3. Compared to Model 11A, the regional economy here produces less wheat in total, uses less land in total (in part also because of increased labor intensity), yet incurs only slightly lower rents in total (because of the Ricardian rents), and a substantial shipping cost.
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The equilibrium price for wheat is therefore higher; aggregate consumer expenditure on wheat is also higher. Aggregate farm disbursements for labor and land are slightly lower in 11B.
11.4 Model 11C: Factor Substitution with Two Crops in Presence of Shipping Costs So far, we have looked at a market for wheat. Suppose, however, consumers at the city also purchase a second crop, say corn. Let us imagine two concentric rings around the city; an inner ring of width d1 occupied by producers of crop 1 and an outer ring of width d2 occupied by producers of crop 2. X1 and X2 are derived from these: see (11.6.1) in Table 11.6. Cobb–Douglas production functions for the two crops are given in (11.6.2); note that the scalar differs (a1 vs. a2 ) but that the exponent (γ ) is assumed, for simplicity of exposition, to be the same for the two crops. The profits per square kilometer of both kinds of farms are given in (11.6.3). Using the condition that the profit-maximizing farm employs labor on each square kilometer of land at distance x up to the level where the marginal value product of labor is equal to its wage, we get (11.6.4). Substituting these expressions for labor intensity (m1 [x] and m2 [x]) back into the profit equations (11.6.3), we get the optimized profit for each type of farm at each possible site. Under the assumption of free entry by crop 2 farms, we can find the P2 that makes the excess profit (its bid rent) of a crop 2 farmer at distance X2 just equal to the alternative land rent (Ra ): see (11.6.5). Under the assumption of free entry by crop 1 farms, we can then find the P1 that makes the bid rent (excess profit) of a crop 1 farm just equal to the bid rent (excess profit) of a crop 2 farm at distance X1 : see (11.6.6). Note here a necessary condition for crop 1 to outbid crop 2 in the inner ring and be the lower bidder in ring 2 is that (11.6.7) be satisfied. I leave aside for the moment the question of whether P1 or P2 are high enough to permit any production of that crop. Ignoring for the moment the possibility of such a corner solution, we are now ready to model supply and market equilibrium using numerical integration (midpoint rule version of a Riemann sum11 ) for (11.6.8), (11.6.9), and (11.6.10). Total production levels for the two crops are given in (11.6.8), total labor inputs in (11.6.9), and shipping costs in (11.6.10). That allows us to calculate Producer Cost in each market (11.6.11). Given the quantities of crop supplied, we can calculate the market clearing prices, V1 and V2 : see (11.6.12). This then allows us to calculate Consumer Benefit for each crop (11.6.13) and then aggregate Net Social Welfare (11.6.14). As in Model 10C, we can then solve a quadratic program to find the d1 and d2 that maximize (11.6.14) subject to the constraints in (11.6.1) through (11.6.13). The consequence of this maximization is that we get market clearing prices in the two markets: P1 = V1 and P2 = V2 .
11 For
calculations reported here, I used N = 100 panels.
11.4
Model 11C: Factor Substitution with Two Crops in Presence of Shipping Costs
313
Table 11.6 Model 11C: Beckmann’s version of Thünen regional economy with 2 crops Radii X1 = d1 and X2 = d1 + d2
(11.6.1)
Production of crop per square kilometer of land used for that crop γ γ q1 = a1 m1 and q2 = a2 m2 where 0 < γ < 1
(11.6.2)
Profit per square kilometer of land for farm producing that crop r1 [x] = (P1 − s1 x)q1 − wm1 and r2 [x] = (P2 − s2 x)q2 − wm2
(11.6.3)
Profit maximizing labor per square kilometer of land for farm producing crop k m1 [x] = (γ a1 (P1 − s1 x)/w)(1/(1 − γ )) m2 [x] = (γ a2 (P2 − s2 x)/w)(1/(1 − γ ))
(11.6.4)
P2 required to make crop 2 no longer profitable for farm beyond distance X2 P2 = s2 X2 + (Ra / (1 − γ ))(1 − γ ) /(a2 (γ /w)γ )
(11.6.5)
P1 required to make crop 1 equally profitable as crop 2 for farm at distance X1 P1 = (a2 /a1 )P2 + ((a1 s1 − a2 s2 )/a1 )X1
(11.6.6)
Condition for crop 1 to be produced in inner ring, crop 2 in outer ring a1 s1 > a2 s2
(11.6.7)
Quantity of crops produced X X Q1 = 2π 0 1 xq1 [x]dx and Q2 = 2π X12 xq2 [x]dx
(11.6.8)
Quantity of labor used X1 X M1 = 2π 0 xm1 [x]dx and M2 = 2π X12 xm2 [x]dx
(11.6.9)
Shipping costs incurred X X S1 = 2π 0 1 xq1 [x]s1 xdx and S2 = 2π X12 xq2 [x]s2 xdx
(11.6.10)
Producer cost PC1 = wM1 + S1 + Ra π X12 and PC2 = wM2 + S2 + Ra π (X22 − X12 )
(11.6.11)
Market clearing price V1 = α1 − β1 Q1 /N and V2 = α2 − β2 Q2 /N
(11.6.12)
Consumer benefit CB1 = (1/2)(V1 + α1 )Q1 and CB2 = (1/2)(V2 + α2 )Q2
(11.6.13)
Net social welfare SW = CB1 + CB2 − S1 − S2
(11.6.14)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): ai —Intercept of labor productivity equation for crop i; N—Population of consumers at customer point; Ra —Rent bid for square kilometer of land by alternative use everywhere; si —Unit shipping rate for crop i; w—Wage rate; x—Distance; α i —Intercept in individual linear inverse demand curve for crop i; β i —Slope in individual linear inverse demand curve for crop i; γ —Exponent in labor productivity equations. Outcomes (endogenous): di —Width of ring in which crop i produced; mi —Labor used per unit land producing crop i; Mi —Total labor in production of crop i; Pi —Market equilibrium price for crop i; Qi —Aggregate quantity of crop i in market equilibrium; qi —Crop i produced per unit land; Ri [x]—Profit per square kilometer from crop i; Si —Total shipping cost; Vi —Price at which given quantity clears the market; Xi —Outer radius of area supplying crop 1.
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To illustrate, consider the case of two crops that are otherwise similar except that one is more costly to ship: a1 = a2 = 1, γ = 0.80, w = 1, Ra = 0.65, N = 1,000, α1 = α2 = 4.5, β1 = β2 = 12, s1 = 0.05, and s2 = 0.035. If two rings are present, crop 1 is therefore produced in the inside ring. I present details of the numerical solution in column [3] of Table 11.3. Note there that even though the demand and supply parameters are otherwise the same for the two crops, the higher s1 alone causes P1 to be higher than P2 . Less of crop 1 is demanded compared to crop 2. Because crop 1 is produced on the inside, where rents are higher, farms there compensate by using labor more intensively. In Fig. 11.4, I show the schedule of labor intensity (curve ABC) by distance. In Fig. 11.5, I show the schedule of market rents (as curve ABC) where the bid rents by crop 1 farms form AB while the bid rents by crop 2 farms form BC. As in Model 10C, we can now draw a quasi supply curve for crop 1: quasi again in the sense that we cannot hold the price of the other crop (P2 ) constant. As previously, I trace this supply curve by systematically varying α 1 ; Cc Ec Dc in Fig. 11.6 is the locus of equilibrium prices and quantities so obtained. The market equilibrium for the numerical example in the preceding paragraph is labeled Ec . For comparison purposes, I show the supply curve for the corresponding one-crop Model 11B:
Model 11C: Labor intensity
Labor intensity (m)
A
F
AB BC OA OD OE OF OG
Labor intensity for crop 1 farms Labor intensity for crop 2 farms Labor intensity for crop 1 farm adjacent to demand point Outer radius of crop 2 production (X 2) Outer radius of crop 1 production (X 1) Labor intensity at boundary between crop 1 and crop 2 farms Labor intensity at outer radius of crop 2 production
B
G
0
C
D E Distance from demand point
Fig. 11.4 Model 11C: labor intensity. Notes: a1 = a2 = 1, γ = 0.8, w = 1, Ra = 0.65, N = 1,000, α1 = α2 = 4.5, β1 = β2 = 12, s1 = 0.05 and s2 = 0.035. Here, X1 = 4.64, X2 = 7.20, m1 [0] = 6.80, m1 [X1 ] = m2 [X1 ] = 3.48, and m2 [X2 ] = 2.60. The horizontal axis is scaled from 0 to 8; the vertical axis from 0 to 8
11.4
Model 11C: Factor Substitution with Two Crops in Presence of Shipping Costs
315
Model 11C: Bid rents
Land rent
A
F
AB BC OA OD OE OF OG
Rent bid by crop 1 farms Rent bid crop 2 farms Rent bid by crop 1 farm adjacent to demand point Outer radius of crop 2 production (X 2) Outer radius of crop 1 production (X 1) Bid rent at boundary be tween crop 1 and crop 2 farms Bid rent at outer radius of crop 2 production
B
G
0
C
E Distance from demand point
D
Fig. 11.5 Model 11C: bid rents. Notes: a1 = a2 = 1, γ = 0.80, w = 1, Ra = 0.65, N = 1,000, α1 = α2 = 4.5, β1 = β2 = 12, s1 = 0.05, and s2 = 0.035. In Model 11C, X1 = 4.64, X2 = 7.20, R1 [0] = 1.70, R1 [X1 ] = R2 [X1 ] = 0.87, and R2 [X2 ] = 0.65. Horizontal axis is scaled from 0 to 8; vertical from 0 to 1.8
CEb Db from Fig. 11.2. Equilibrium shifts from Eb to Ec when we introduce the second crop. Competition for land at the margin of crop 1 production (i.e., near X1 ) pushes up the cost of the marginal unit of crop 1 supplied and thereby increases Ricardian rent for all sites still closer to the customer point. My interpretation of Cc Ec Dc in comparison to CEb Db is as follows. The introduction of crop 2 pushes the equilibrium point in market 1 back up its demand curve; for any given quantity of crop 1, the price at which that quantity is forthcoming has gone up. Overall, the effect of a second crop is to twist the supply curve so that Cc Ec Dc looks closer to a horizontal supply (the kind of supply curve consistent with a perfectly competitive market) than does CEb Db . Notice here that a corner solution wherein crop 1 is no longer produced occurs when α 1 is no larger than the amount OCc in Fig. 11.6. Similarly, we can imagine an α 1 so large that crop 2 is no longer able to compete for land anywhere. At that level of α 1 , Cc Ec Dc converges to CEb Db ; this would happen somewhere to the right of the graph area shown in Fig. 11.6. What about comparative statics in Model 11C? See Table 11.7.
316
11 A6.00
The City and Its Hinterland
Model 11C Equilibrium in market for crop 1 of 2 A vBv CcEcDc CEbDb CcPcEcCc OA4.50EcQcO OCcEcQcO OQc OPc Pc A4.50EcPc
Price
A4.50
Demand curve (at a1 = v) Supply Curve for wheat Supply curve from Model 11B Producer surplus Consumer benefit Producer cost Equilibrium quantity Equilibrium price Consumer surplus
A2.20 Pb
Ec Eb
Pc Cc C
Dc Db D
A0.85
0
B0.85
B2.20 Qb Qc
B4.50 B6.00 Quantity
Fig. 11.6 Model 11C: supply curve for crop 1. Notes: a1 = a2 = 1, γ = 0.10, w = 1, Ra = 0.65, N = 1,000, α2 = 4.5, β1 = β2 = 12, s1 = 0.05, and s2 = 0.035. In drawing Cc Ec Dc , α 1 is varied from 1.31 to 8. At Ec , P1 = 1.83, Q1 = 222, P2 = 1.77, and Q2 = 228. Horizontal axis is scaled from 0 to 600; vertical from 0 to 6
aι When aι is increased, more crop i can be produced with a given amount of labor per unit land. The amount of land and labor used in production of that crop declines, the price of the crop drops, and total quantity demanded rises. This reduces competition in the land market causing the price of the other crop to drop and its output (and use of labor and land) to rise. N When N is increased, there is more demand for both crops. The quantity of each crop produced rises: as does price. More labor and land are employed in the production of each crop. Ra When Ra is increased, land becomes more costly. In both crops, farms compensate in part by substituting labor for land. However, for both crops, price rises and quantity demanded drops. sι When sι is increased, crop i is more costly to ship. The price of crop i rises, the quantity demanded drops, and crop i farms employ less land and less labor. Because there is now less competition in the land market, farms producing the other crop now find it less costly; price drops in that market, quantity increases, and the amount of land and labor employed rises. w When w is increased, labor is more costly. All farms now find it more profitable to use less labor and more land in crop production. However, both
11.4
Model 11C: Factor Substitution with Two Crops in Presence of Shipping Costs
317
Table 11.7 Model 11C: comparative statics of an increase in exogenous variable Outcome Given
X1 [1]
P1 [2]
Q1 [3]
M1 [4]
R1 [0] [5]
R1 [X1 ] X2 [6] [7]
a1 a2 N Ra− s1 s2 w α1 α2 β1 β2 γ
− − + − − − + + − − + −
− + + + + + + + + − − −
+ − + − − − − + − − + +
− + + + − + − + + − − +
+ + + + + + − + + − − +
+ + + + + + − − + + − =
− − + − − − + + + − − −
P2 [8]
Q2 [9]
M2 [10]
R2 [X2 ] [11]
− − + + − + + + + − − −
+ + + − + − − − + + − +
+ − + + + − − − + + − +
0 0 0 + 0 0 0 0 0 0 0 0
Notes: See also Table 11.6. Comparative statics numerically estimated from the base case: α1 = α2 = 4.5, β1 = β2 = 12, s1 = 0.05, s2 = 0.035, a1 = a2 = 1, γ = 0.8, w = 1, N = 1,000, Ra = 0.65; +, Effect on outcome of change in given is positive; −, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome.
crops are now more costly to produce, so the price of each crop rises, the quantity demanded drops, and farms use less labor and land in total. αι When α ι is increased, the price each consumer is willing to pay for a given quantity of crop i goes up. The price of crop i rises; so too does the amount demanded of the crop and of the inputs (labor and land) used to produce it. The resulting competition in the land market means that the price of the other crop rises but that quantity demanded (and input usage) declines. βι When β ι is increased, the price each consumer is willing to pay drops faster the more quantity of crop i farms supply. The effect here is the opposite of an increase in α i . The price of crop i drops: so too does the amount demanded of the crop and of the inputs (labor and land) used to produce it. The resulting decrease in competition in the land market means that the price of the other crop drops and that quantity demanded (and input usage) both increase for that crop. γ When γ is increased, crop production becomes more dependent on labor input relative to land input. As farmers use less land and more labor to produce their crops, land rents decline. For the parameter values used to calculate the comparative statics here, each crop becomes less costly to produce, its price drops, and quantity demanded increases.12
12 One
can envisage here a counter-scenario in which the wage rate is high enough that the farms find it more costly (not less costly) to become more labor intensive in production of the two crops. In such a case, the price of each crop would rise, and the quantity demanded would decrease.
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I end this section with a final comment on the effect of including the substitutability of labor and land. The effect of incorporating substitutability is to increase the impact of proximity to the customer point on land rent. Put differently, substitutability makes locational advantage more important to the extent that producers can make more use of ubiquitous inputs to offset the higher land rents in closer proximity to the city.
11.5 Model 11D: Non-spatial Version of Samuelson’s Model of a Thünen Economy The three models above introduce the idea that substitutability of land and labor is important in location theory. However, to advance Thünen’s analysis of the distribution of income and to better understand the linkages between commodity and factor prices, we need to think more about outcomes in the labor market. In Models 11A through 11C, I assumed that the wage is fixed: i.e., labor is a ubiquitous input. However, Thünen had in mind the idea that the markets for labor, land, capital, and commodities had to be jointly in equilibrium. To incorporate a labor market into location theory, I use an approach derived from Samuelson (1983).13 To set the stage for this model, let us once again first look at a non-spatial version of the model—one wherein shipping costs are assumed to be zero everywhere. Samuelson envisages a competitive regional economy in which there is production of only two commodities.14 Labor is used to produce both commodities. Every unit of labor is also a consumer who purchases these commodities for consumption. In Models 11A through 11C, I used M to refer to the total amount of labor employed on farms. Earlier in the book, I used N to refer to the total number of consumers at a place. In the remainder of this book, I look at each person as both labor and consumer. I will therefore use N to refer to counts of either workers or consumers. For ease of exposition, Samuelson envisages a barter economy in which he treats wheat as numéraire. Farms produce wheat that they then use to pay their wage bill and land rent. Farm workers use some of their wages in wheat to pay for soap and for shipping. At the city, soap producers pay their workers in wheat too; which is the revenue they gain from selling soap to farm workers. Soap workers then use part of their wheat income to purchase soap. One commodity, that I call soap, is produced only at the customer point (which I characterize as the city). I assume that soap is produced with labor only, at constant returns to scale and by a perfectly competitive industry (efficient firms). In other words, every soap worker produces the same amount (l) of soap. This is a fixed coefficients technology. I also assume that fixed costs can be ignored. Since soap 13 Hartwick
(1973b) describes an alternative general equilibrium model of a farm economy.
14 Samuelson includes an analysis of multiple crops. For the sake of simplicity, I consider here only
the one-crop version of his model.
11.5
Model 11D: Non-spatial Version of Samuelson’s Model of a Thünen Economy
319
production takes up no land, Samuelson treats the city as a place: ignored here is the idea that city labor might also require land for housing, roads, or other uses. As did Beckmann, Samuelson ignores capital as a factor of production. See (11.8.1) in Table 11.8.
Table 11.8 Model 11D: Non-spatial version of the Samuelson model Soap production v2 = lN2
(11.8.1)
Wheat yield per square kilometer of land where 0 < γ < 1 q1 = a(N1 /L)γ
(11.8.2)
Wheat production net of rent v1 = L(q1 − Ra )
(11.8.3)
Land input that maximizes net wheat production L = ((1 − γ )a/Ra )1/γ N1
(11.8.4)
Maximized net wheat v1 = γ a1/γ ((1 − γ )/Ra )(1 − γ )/γ N1
(11.8.5)
Labor N = N 1 + N2
(11.8.6)
Production possibility frontier v1 = PlN − Pv2
(11.8.7)
Walrasian implicit exchange rate (units of wheat per unit soap) P = γ a1/γ ((1 − γ )/Ra )(1 − γ )/γ /l
(11.8.8)
Utility-maximizing allocation of labor N1 = (1 − n)N N2 = nN
(11.8.9)
Wage (in wheat) for farm labor in competitive labor market w1 = γ a1/γ ((1 − γ )/Ra )(1 − γ )/γ 1
(11.8.10)
Wage (in soap) for soap labor in competitive labor market w2 = l
(11.8.11)
Per capita consumption and utility q12 = nl q11 = (1 − n)l/P
(11.8.12)
U = q111− n qn12
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): a—Scalar in wheat production; l—Marginal product of labor in soap production; N—Total labor in region; Ra — Opportunity cost of land; ν—Preference for soap; γ —Exponent in wheat yield. Outcomes (endogenous): q1 —Wheat yield per square kilometer of land; L—Total amount of land used in wheat production; N1 —Regional labor allocated to wheat production; N2 —Regional labor allocated to soap production; P—Walrasian exchange rate in commodity market (units of wheat per unit soap); q11 —Amount of wheat consumed per laborer; q12 —Amount of soap consumed per laborer; U—Utility level of laborer; v1 —Amount of wheat produced in region net of land rent; v2 —Amount of soap produced in region; w1 —Wage rate (in units of wheat) for wheat labor; w2 —Wage rate (in units of soap) for soap labor.
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The other commodity, wheat, is produced only on farms outside the city, using only labor and land inputs. It is possible to incorporate two or more agricultural commodities here; I consider just one for the sake of exposition. Once again, capital inputs are ignored. Farms are also assumed to be efficient firms. Wheat production is assumed to be a perfectly competitive industry. Yield (wheat output per unit land) is a decreasing returns to scale function of labor intensity (labor per unit land), N1 : see the Cobb–Douglas production function in (11.8.2). Assume wheat production does not require capital or any other input (other than land and labor). Fixed costs are ignored. Note also that (11.8.2) implies constant returns to scale in wheat production overall; doubling the amount of land and labor leads to a doubling of wheat production. Wheat production net of land rent, v1 , is now given by (11.8.3). See the example in Fig. 11.7 where the most profitable intensity is OF, the wheat output per unit land is OG, the total wage paid per square kilometer of land is OH, and the excess profit (inclusive of land rent) is HG. Given we allocate N1 units of labor in total to wheat production, the amount of land used for wheat production is given by (11.8.4). The maximized level of net wheat production is then given by (11.8.5). Note the implication that Ra must be strictly greater than zero here. If Ra were zero
Wheat yeald per square kilometer
Model 11D: Wheat yield and labor intensity CD GH OA OB OF OG OG
Line parallel to OB Wheat yield net of wage bill at labor intensity OF Wheat yield as a function of labor intensity Wage bill per square kilometer of land farmed as function of labor intensity Labor intensity that maximizes profit Wheat yield at labor intensity OF Wage bill at labor intensity OF
A D
G
B
E
H
I
C 0
F
Labor intensity
Fig. 11.7 Model 11D: wheat yield and profit in the Samuelson model. Notes: ν = 0.5, l = 4, a = 2, γ = 0.8, N = 500, and Ra = 0.5. The profit-maximizing solution is a labor intensity of 1.84 and a wheat yield per square kilometer of 3.25. The horizontal axis is scaled from 0.0 to 4; vertical axis from 0 to 8
11.5
Model 11D: Non-spatial Version of Samuelson’s Model of a Thünen Economy
321
Net wheat production (Q1)
as was assumed by Thünen, there would be no limit to the amount of land used in wheat production. Indeed, this is why I introduced Ra at the start of this chapter. Because of the constant returns to scale in wheat production, wheat revenue is exhausted in factor payments for land and labor; put differently, no excess profits are earned by the farm. Assume land rent is paid to absentee landlords and therefore does not affect the demand for soap in the region. What happens to factor income in this regional economy? The only other factor of production is wheat labor. The labor market (which encompasses both city and farm areas) is assumed competitive.15 Wheat labor receives all the rest of the income generated. Soap production is assumed to be constant returns to scale and to require only one input (labor). Soap workers are paid their marginal product, l, in soap (and this exhausts soap revenue); in labor market equilibrium, wheat workers are each paid the equivalent amount in wheat, lP. We are now ready to draw the production possibility frontier for this regional economy. The total amount of labor in the region is fixed: see (11.8.6). Substituting soap production (11.8.1) and maximized net wheat (11.8.5) into (11.8.6) gives the region’s production possibility frontier, (11.8.7). This production possibility frontier takes the form of a straight line with a negative slope (the negative of the implicit exchange rate). See the production possibility frontier AB in Fig. 11.8. To understand the linearity, imagine initially all labor in the region is in soap production.
Model 11D: Production possibility frontier and community indifference curve
A C
F
AB CD OF OG
Production possibility frontier Highest indifference curve reachable on AB Quantity of wheat produced net of rent Quantity of soap produced
E
D
0
B G Soap production (Q2)
Fig. 11.8 Model 11D: production possibility frontier in a non-spatial Samuelson model. Notes: ν = 0.5, l = 4, a = 2, γ = 0.8, N = 500, and Ra = 0.5. At point E, v1 = 378 and v2 = 1,000. Horizontal axis is scaled from 0 to 2,500; vertical axis from 0 to 800 15 Among
others, Curry (1985a) considers the operation of geographical labor markets in the presence of imperfect information.
322
11
The City and Its Hinterland
The total amount of soap produced is lN, and the net amount of wheat produced is zero. Now, suppose 1 worker shifts from soap to wheat production. The amount of soap production declines by l units; from (11.8.5), the amount of wheat produced (β − 1)/β units. If we move a second unit net of rent rises by β(1 − β)(1 − β)/β a1/β Ra of labor from soap to wheat, we observe the same reduction in soap and the same gain in net wheat production. The reason for this linearity is that both soap and net wheat production have constant returns to scale. Where along this production possibility frontier does our region consume? In effect, each worker has the potential to earn l units of soap (by being a soap worker). Everyone in the model, whether wheat or soap worker, has the same log-linear utility function based on the consumption of two indispensable commodities: soap and wheat. See (11.8.9). As in Chapter 9, this kind of utility function means no one can live in autarky; wheat workers and soap workers must each trade with the other to consume both of these indispensable commodities. Utility maximization implies that each worker, as a consumer, will spend a fraction ν on soap (thereby consuming Nνl units of soap in total) and a fraction 1 − ν on wheat (thereby consuming N(1 − ν)lP units of wheat in total). In Fig. 11.8, this is shown as combination E on the indifference curve CED. This is a general equilibrium model.16 Up until Chapter 9 in this book, I had assumed at least one of either the demand curve or supply curve is exogenous. Samuelson’s version therefore is more ambitious because, like Chapter 9, it seeks to answer the question of why and how exchange, and thereby a regional economy, exists. There are three markets in this model: a market for labor, a market for land, and a market for the exchange of soap for wheat. In this model, there is a demand for labor and a demand for land. However, these are not exogenous to the model. Rather, as so nicely illustrated in Fig. 11.8, they are determined within the model as an outcome of the interaction between, on the one hand, the preferences for wheat vs. soap and, on the other hand, the tradeoff between production of wheat vs. soap. What about comparative statics here? How do the market outcomes change when one of the givens is changed? See Table 11.9. a When a is increased, the farm produces more wheat from a given level of land and labor inputs. Because wheat is now more plentiful, the exchange rate increases (more wheat is needed to purchase a unit of soap). Farms use more land in wheat production. The split of labor between soap and wheat remains unchanged, but utility rises because consumers have more wheat available (though not more soap). l When l is increased, soap workers each produce more output. Because soap is now more plentiful, the exchange rate falls (less wheat needed to purchase a unit of soap). The split of labor between soap and wheat remains unchanged. Utility rises because consumers have more soap available (though not more wheat).
16 A
representation of an economy in which supply and demand curves are endogenous.
11.6
Model 11E: Spatial Version of Samuelson Model
323
Table 11.9 Model 11D: comparative statics of an increase in exogenous variable Outcome Given
N1 [1]
N2 [2]
L [3]
v1 [4]
v2 [5]
w1 [6]
w2 [7]
P [8]
U [9]
a l N Ra γ ν
0 0 + 0 0 −
0 0 + 0 0 +
+ 0 + − − +
+ 0 + − − −
0 + + 0 0 +
0 0 0 0 0 0
+ 0 0 − − +
+ − 0 − − 0
+ + 0 − − +
Notes: See also Table 11.8; +, Effect on outcome of change in given is positive; −, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome.
N When N is increased, there are more workers available to produce both commodities. Production of both commodities is increased proportionally, and the exchange rate remains the same. However, per capita consumption and therefore utility remain the same. Ra When Ra is increased, farms must pay more for each unit of land rented. This makes wheat relatively more costly, pushing down the exchange rate, raising labor intensity, decreasing the amount of wheat consumed, and lowering utility. γ When γ is increased, wheat production becomes more reliant on labor relative to land. The amount of land used in wheat production drops. The exchange rate declines (less wheat needed to purchase a unit of soap). ν When ν is increased, consumers have a stronger preference for soap over wheat. Labor shifts from wheat into soap production, the amount of land used in wheat production decreases, and net wheat production declines. What does the regional economy look like? See the example in Table 11.10. Here, soap firms produce 1,000 units of soap; farms produce 473 units of wheat (equivalent to 1,250 units of soap at the going exchange rate of 0.3783 units of wheat per unit of soap). The labor force is split equally between soap and wheat production (since I have assumed ν = 0.50) and the total wage bill in city and farm is the same (when measured in units of soap). There are no shipping costs in Model 11D, and free entry ensures that excess profit is zero for both farms and soap firms. Soap firms pay no land rent (since they don’t use land as an input to production); farms pay 20% of their output on land rent (since 1 − γ = 0.20).
11.6 Model 11E: Spatial Version of Samuelson Model Now, let us recast this model to incorporate shipping cost. Assume an exchange rate—the amount of wheat paid for one unit of soap—gets established at the city, P[0]. This is the exchange rate confronting soap workers. For wheat workers further
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The City and Its Hinterland
Table 11.10 The regional economy in Models 11D and 11E Model 11D No shipping cost
Model 11E With shipping cost
s1 = s2 = 0 [1]
s1 = s2 = 0.01 [2]
s1 = s2 = 0.02 [3]
Firm disbursements Total wage bill Wheat producers Soap producers
354 1,000
354 979
352 962
Total land rent Wheat producers Soap producers
89 0
88 0
88 0
Land used Wheat producers Soap producers
136 0
125 0
114 0
Land rent (wheat per square kilometer) At x = 0 At x = X Wage rate for wheat labor (in kind) At x = 0 At x = X
0.65 –
0.83 0.65
1.05 0.65
1.42 –
1.33 1.42
1.26 1.42
Labor employed Wheat producers Soap producers
250 250
255 245
259 240
Exchange rate (units of wheat per unit soap) At x=0 At x=X
0.35 –
0.33 0.38
0.31 0.40
443 1,000
442 979
440 962
1.19
1.15
1.12
Firm production Wheat Soap Utility
Note: – indicates Model 11D has no outer radius. Parameter values in common to all 3 columns above: ν = 0.5, l = 4, a = 2, γ = 0.8, N = 500, and Ra = 0.65. Calculations by author.
away from the city, the effective exchange rate—P[x] at Euclidean distancex away from the city—is different because of the cost of shipping. Samuelson here thinks of an iceberg model where a proportion of the amount hauled evaporates, spoils, or is fed to the horse pulling the wagon on the way to and from the market.17 Here, once again, unit shipping cost is envisaged strictly as resource use. Samuelson uses 17 Samuelson
(1954) is an early use of the iceberg concept.
11.6
Model 11E: Spatial Version of Samuelson Model
325
Table 11.11 Model 11E: A spatial version of the Samuelson model Exchange ratio P[x] = e − s0 x − s1 x P[0]
(11.11.1)
Wheat yield q1 [x] = a(N1 [x])γ
(11.11.2)
Excess profit q1 [x] − w1 [x]N1 [x]
(11.11.3)
Labor X N = N2 + 2π 0 xN1 [x]dx
(11.11.4)
Soap wage w2 = l
(11.11.5)
Wheat wage w1 [x] = γ a(N1 [x])γ − 1
(11.11.6)
Bid rent R1 [x] = q1 [x] − w1 [x]N1 [x]
(11.11.7)
Radius of production R1 [X] = Ra
(11.11.8)
Utility-maximization for soap worker q21 = (1 − v)w2 P[0] q22 = vw2 U2 = qn22 q121− n
(11.11.9) (11.11.10)
Utility-maximization for wheat worker at distance x q11 [x] = (1 − v)w1 [x] q12 [x] = nw1 [x]/P[X] U1 [x] = (q12 [x])n (q11 [x])1 − n
(11.11.11) (11.11.12)
Equilibrium in market for exchange of wheat and soap X (1 − v)lN2 = 2π 0 N1 [x]q12 [x]dx
(11.11.13)
Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): a—Scalar in wheat production; l—Marginal product of labor in soap production; N—Total labor in region; Ra — Opportunity cost of land; s1 —Unit shipping rate for wheat (iceberg loss); s2 —Unit shipping rate for soap (iceberg loss); ν—Preference for soap; γ —Exponent in wheat yield. Outcomes (endogenous): q1 —Wheat yield per square kilometer of land; L—Total amount of land used in wheat production; N1 —Regional labor allocated to wheat production; N2 —Regional labor allocated to soap production; P—Walrasian exchange rate in commodity market (units of wheat per unit soap); q11 —Amount of wheat consumed per laborer; q12 —Amount of soap consumed per laborer; U—Utility level of laborer; v1 —Amount of wheat produced in region net of land rent; v2 —Amount of soap produced in region; w1 —Wage rate (in units of wheat) for wheat labor; w2 —Wage rate (in units of soap) for soap labor.
an iceberg model here, as opposed to the shipping rate in earlier chapters because he does not want to include a transport sector with its attendant factor incomes in his general equilibrium model. Following the idea from Chapter 3 with respect to a perfectly competitive industry that the price difference between two places should not exceed the unit shipping cost involved, Samuelson assumes (11.11.1): See Table 11.11.
326
11
The City and Its Hinterland
The existence of shipping costs has several implications in the model. First, consumers at different places each face a different exchange rate of soap for wheat and can therefore be expected to shift their consumption of the two commodities accordingly. Second, the wage rate for a farm worker therefore also has to vary by location as a result; if it did not, the utility of a worker might be higher in one location compared to another and this would contravene the assumption of equilibrium in the labor market. Third, the bid rent of farms will vary with location because profit is now affected by proximity to the city. Fourth, the farm can be expected to substitute between land and labor inputs as the prices of these inputs vary with location. In labor market equilibrium, all workers (N in total) are allocated to soap production in the city, or wheat production at each distance x from the city: see (11.11.4). Because the market for labor is competitive, each soap worker is paid his or her marginal product (l): see (11.11.5) in Table 11.11. The wage here is paid in soap and therefore all soap made is paid out as wages; there is no profit in soap production. Each wheat worker is also paid his or her marginal product: see (11.11.6). This wage is paid in wheat. In a market for land that is also competitive, landlords rent land to the highest bidder. In competition, tenant farmers continue to enter the market, raising their bid rents for a site until market rent absorbs excess profit: see (11.11.7). Samuelson assumes land is available in unlimited quantity: presumably there is an alternative use for land that offers the same bid rent (Ra ) for every location.18 Farms that produce wheat to exchange for soap in the city must be able to pay a rent at least equal to Ra : that defines the radius of wheat production (X): see (11.11.8). Tenant farmers bid themselves into a spatial equilibrium, just as they did in Chapter 10. Just as in Model 11D, the farms have no excess profits in equilibrium; free entry ensures that their entire production of wheat is expended in payments for labor and for land. Now, consider the market for the exchange of wheat for soap. It too is competitive. As in Chapter 9, trade occurs between individuals (in this case, farmers and city residents) who each start off with only one commodity, thereby a utility of zero, and must trade to achieve a positive utility. An exchange rate—the amount of wheat paid for one unit of soap—gets established at the city: P[0]. This is the exchange ratio confronting soap workers, and they make utility-maximizing choices based on this: see (11.11.9) and (11.11.10). For a wheat worker at distance x, the effective exchange rate is P[x] and the utility-maximizing choices are shown in (11.11.11) and (11.11.12). The total amount of labor (N) is exogenous (fixed): see (11.11.4). However, labor is allocated endogenously between city production and rural production. Put differently, labor is thought to seek out the sector (soap or wheat) and, in the case of wheat, location (i.e., distance from city) that maximizes utility. We are now ready to solve this model. For soap workers, there is an amount of soap they are willing to give up at the exchange rate, P[0]. For wheat workers at distance x,
18 Curiously, this is missing from Samuelson’s article. Samuelson appears to suggest that the alter-
native rent is zero. However, as is explained later in this chapter, the solution of the model requires Ra be larger than zero.
11.6
Model 11E: Spatial Version of Samuelson Model
327
there is an amount of soap they are willing to purchase at the exchange rate P[x]. What then is the exchange rate P[0]—which in turn translates into P[x] at each distance x—that clears the market: i.e., equalizes the amount of soap that soap workers are willing to give up with the amount wheat workers are willing to purchase. See (11.11.13). In Chapter 10, we saw that it is difficult to solve algebraically the agricultural location problem on a rectangular plane. Model 11E is even more difficult to solve because it also incorporates equilibrium in the labor market. To solve the model, I have found it helpful to use an algorithm to solve market radius (X) nested within an iterative process to solve the exchange rate at the city (P[0]). The steps are as follows. 1. Make new estimate of P[0]. 2. Make new estimate of X 3. Calculate exchange rate at boundary: P[X] = P[0]e(s2 + s1 )X 4. Calculate agricultural wage at boundary: w[X] = lP[0]1 − α P[X]α 5. Calculate labor intensity at boundary: N1 [X] = (aβ/w[X])1/(1 − β) 6. Calculate wheat yield at boundary: q1 [X] = a(N1 [X])β 7. Calculate unit profit at X: q1 [X] − w1 [X]N1 [X] 8. Calculate new estimate of X: ((q1 [X] − w1 [X]N1 [X])/Ra ) 9. If new estimate of X in step 8 is sufficiently similar to estimate in step 2, iteration has converged. If not, return to step 2 using X from step 8 as new estimate. 10. Divide X into intervals (distance rings). 11. For each distance ring, calculate wheat wage, labor intensity, yield, and wheat and soap consumption. 12. Total soap consumption over distance rings to get total wheat labor and soap demanded. 13. Calculate soap production needed to meet demand by wheat workers. 14. Calculate number of soap workers, wheat workers, and total labor (N). 15. If new estimate of N is equal to exogenous value, iteration has converged. If not, try new estimate of P[0] and return to step 1. Although Model 11E does not permit explicit (algebraic) solution,19 one can readily comprehend the patterns that arise in spatial equilibrium as derived from numerical solutions. See Fig. 11.9. P[x]
19 I
Exchange rate. In spatial equilibrium, the exchange rate (the amount of wheat given up to purchase one unit of soap) increases with distance from the city because of the cost of shipment: see curve AB in Fig. 11.9. Put differently, a wheat worker further away from the city must give up more
do not consider uniqueness of the equilibrium solution to this problem. However, Nerlove and Sadka (1991) prove uniqueness for a version of the Samuelson model that is similar to Model 11E.
328
The City and Its Hinterland
Model 11E: Outcomes by distance from city Exchange rate (wheat per unit soap) P[x] Land rent (in wheat) R[x] Wage paid (in wheat) to farm laborer w1[x] Wheat yield per square kilometer of land Soap consumed in ratio to farm wage q2[x]/w1[x] Opportunity cost of land Ra Land rent adjacent to city R[0] Outer radius of wheat production X
F
G E
H
Axis for AB and EF
AB CD EF GH IJ LD OC OL
Axis for CD, GH, and IJ
Fig. 11.9 Model 11E: equilibrium outcomes in a spatial version of the Samuelson model. Notes: ν = 0.5, l = 4, a = 2, γ = 0.8, s2 = 0.01, s1 = 0.01, N = 500, and Ra = 0.65 . Horizontal axis is scaled from 0 to 7; left vertical from 0 to 6; right vertical from 0 to 2
11
I J B
A C K
D
0 Distance from city
L
wheat per unit of soap to cover the cost of shipping the wheat in from the farm and soap back to the farm. q1 [x] Wheat yield. Wheat yield (per square kilometer) falls steadily with distance from the city in spatial equilibrium: see curve GH. This should not be surprising. After all, as noted above, the wheat wage increases with distance from the city. Therefore, farms further away from the city would want to use costly labor more sparingly. Because farming is less labor intensive there, wheat yield is lower. R[x] Market rent. Excess profit, dissipated as land rent, falls steadily with increased distance from the city in spatial equilibrium: see curve CD. This is also not surprising. After all, in the absence of any factor substitution, we expect market rent to fall with distance to the market because of shipping costs: that was a lesson learned in Chapter 10. As seen in Model 11B, the farm’s ability to substitute between land and labor allows it to switch to a lower labor intensity at more remote locations offsetting in part the impact of higher shipping costs. w1 [x] Wheat wage. That the exchange rate increases with distance from the city implies that the nominal wage paid to a wheat worker must also increase with distance in a spatial equilibrium: see curve EF. If wage did not increase
11.6
Model 11E: Spatial Version of Samuelson Model
329
with distance, workers there would not be able to achieve the same level of utility as wheat workers closer to the city. Wheat workers reach the same level of utility in part through a higher wage that permits them to offset the effect of higher shipping costs. In part also, wheat workers achieve the same level through their ability to substitute between soap and wheat. At locations where remoteness from the market makes soap more costly to consume, wheat workers substitute between consumption of soap and consumption of wheat. What happens in this model if shipping rates were to become very low? In Chapter 10, we saw that the Lee–Averous model predicts that the supply curve for the farm commodity becomes horizontal: i.e., suppliers will provide any quantity at a price equal to the (constant) marginal cost of production. Corresponding to this, there is no outer radius to production of the farm commodity. Thus, although the production area is unbounded, only a finite amount of land will be used for production. In this chapter, a similar result arises in Model 11D. Model 11E differs from the Lee–Averous model; agricultural yield is not fixed. Nonetheless, Model 11E is similar in that as shipping cost goes to zero, the location of farms becomes indeterminate. Farms are free to locate, unbounded by shipping costs. What about comparative statics here?20 How does a spatial equilibrium change when one of the givens is changed. See Table 11.12. Table 11.12 Model 11E: spatial version: comparative statics of an increase in exogenous variable (assumings1 > s2 ) Outcome Given a l N Ra s1 s2 γ ν
N1 [1]
N2 [2]
Lt [3]
v1 [4]
v2 [5]
P[0] [6]
U [7]
+ 0 + − + + − −
− 0 + + − − + +
+ + + − − − − −
+ + + − − − − −
− + + + − − + +
+ − − − − − − −
+ + − − − − − −
Note: See also Table 11.11; +, Effect on outcome of change in given is positive; −, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome.
a
When a is increased, each farm produces more wheat from the same amounts of land and labor. With wheat more plentiful, soap becomes relatively more costly: i.e., P[0] increases. Utility rises everywhere. With enhanced productivity, farms are able to supply the market from further away: i.e., X increases. In
20 For
proofs here, see Nerlove and Sadka (1991).
330
l
N
Ra
s1
s2
γ
ν
21 For
11
The City and Its Hinterland
the regional economy, less soap is demanded at the higher exchange rate, and the amount of labor allocated to wheat production increases. When l is increased, each soap worker produces more. With soap now more plentiful, wheat becomes relatively more costly: i.e., P[0] drops. Utility rises everywhere. The income and substitution effects of the price change for wheat cancel each other out, so the demand for wheat overall and the allocation of labor in the regional economy remain unchanged When N is increased, there are more workers available to produce both commodities. Given that the income elasticity of demand21 for each product is 1.0, we would expect demand for each product to rise proportionally. However, to get sufficient additional wheat, the area under cultivation, radius X, must increase, and nonzero shipping cost means that wheat becomes relatively more costly: i.e., P[0] drops. Labor allocated to wheat production increases as a proportion of total regional labor, N. When Ra is increased, farms must pay more for each unit of land rented. This makes wheat more costly to produce. Wheat becomes relatively more expensive (P[0] drops), and the area under cultivation (radius X) drops. In the regional economy, more labor is now allocated to soap production. When s1 is increased, it costs more to ship wheat to soap workers. Less wheat reaches the market and the exchange rate, P[0], therefore drops. The demand for wheat decreases, and the radius of production (X) drops. In the regional economy, this is compensated in part by a shift in labor from soap to wheat production.22 In spatial equilibrium, utility is now everywhere lower. When s2 is increased, it costs more to ship soap to wheat workers. Put differently, a wheat worker at distance x pays only a smaller amount of wheat on net (i.e., after shipping cost) for each unit of soap purchased. The exchange rate at the city, P[0], drops. At remote distances, farm workers are no longer able to achieve the same utility: hence, X drops. In spatial equilibrium, utility is now everywhere lower. The amount of labor in soap production rises on net. The principal spatial effect is that the exchange rate now increases more rapidly with distance from the city. When γ is increased, wheat production becomes more reliant on labor relative to land. Access to land is less important in the regional economy, and the radius (X) of wheat production therefore shrinks. This in turn makes wheat relatively less costly to produce, and thus P[0] drops. When ν is increased, consumers have a stronger preference for soap compared to wheat. In the regional economy, labor is reallocated from wheat to soap production. The area under cultivation (radius X) decreases. Because wheat is now marginally less costly to supply to the city, soap becomes relatively less expensive: i.e., P[0] declines.
a given good, income elasticity of demand (i) is the percentage change in quantity of the good demanded (q) given a one percent change in the income (y) of the consumer: i = (y/q)dq/dy. 22 Equivalently, Nerlove and Sadka (1991, p. 97) conclude that a drop in transport costs is sufficient to induce labor to shift from rural to urban areas.
11.7
Final Comments
331
What about the regional economy in this model? In column [2] of Table 11.10, I show the regional economy for an example comparable to that in Model 11D except that s1 = s2 = 0.01. In column [3], I show a second example where s1 = s2 = 0.02. Contrasting these with the outcomes for Model 11D in column [1] gives us an idea of the role played by shipping cost. First, consider the market for labor. Suppose we were able to decrease shipping costs, sweeping from column [3] to column [1] of Table 11.10, the wage (paid in wheat) to a wheat worker adjacent to the city increases. In contrast, the wage (paid in soap) to a city worker remains fixed regardless of shipping cost. As we decrease shipping costs, labor is redeployed from wheat production to soap production. Finally, as we decrease shipping costs, the wage bills paid increase: slightly for farms and more for soap firms. Second, consider the market for land. As we decrease shipping costs, the rent (paid in wheat) per unit of land adjacent to the city decreases. As we decrease shipping costs, the total amount of land used in wheat production increases. Finally, as we decrease shipping costs, the total land rents paid increase slightly. Finally, consider the market for the two commodities: wheat and soap. As we decrease shipping costs, soap becomes more valuable: it costs more wheat to purchase one unit of soap. Total wheat production decreases and total soap production rises. In so doing, utility goes up. I explain these effects as follows. When shipping costs are high, the regional economy has to devote considerable labor to production of commodities that require land. Commodities that do not require land must therefore be shipped at great cost to remote consumers. The regional economy has a focus on wheat production and consumption. When shipping costs are decreased, the regional economy condenses around the city, the price of the city good falls, and consumers everywhere purchase relatively more soap and less wheat and benefit overall from the reduced cost of shipping. From the perspective of this book, it is both a strength and a shortcoming of Model 11E that it is a general equilibrium model. The strength is that we have now been able to link, for a simple regional economy, the prices of land, labor, and commodities. We can now see how a change in any of the givens in this model translates into changes in the equilibrium outcomes in all three sets of markets. This is a powerful analytical tool. The weakness is that, in a sense, I began this book with a much simpler question: how does the location of one firm affect the prices paid or received by other firms in the vicinity. Unfortunately, we can’t look at such questions directly with a general equilibrium model since all prices are now endogenous. However, there is an important linkage here; the competition that leads firms to choose a location is pushing firms toward the kind of equilibrium that Model 11E describes.
11.7 Final Comments In this chapter, the principal model has been 11E. I included Models 11A and 11D to help readers better understand aspects of Model 11E. In Table 11.13, I summarize
332
11
The City and Its Hinterland
Table 11.13 Assumptions in Models 11A through 11E
(a) Assumptions in common A1 Closed regional market economy B1 Exchange of crop (or crops) for money B3 All firms in the industry are identical I2 Cobb–Douglas production function: CRS J2 Nonzero opportunity cost (rent) for land J5 Land is input to production J6 Competitive market for land (b) Assumptions specific to particular models C2 Fixed local customers C4 Identical customers C5 Identical linear demand D7 Horizontal marginal cost curve J1 Zero opportunity cost (rent) for land E1 Zero shipping cost everywhere A4 Rectangular plane E2 Fixed unit shipping rate A2 Barter market C7 Maximize same utility function C10 Two kinds of customers L1 Endogenous market in labor L2 Fixed total amount of labor in region L3 Allocation between industries is endogenous
11A [1]
11B [2]
11C [3]
11D [4]
11E [5]
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x
x x x x x
x x x x x
x x
x x
x
x x x x x x
x x x x x x x x
the assumptions that underlie Model 11A through 11E. Many assumptions are in common to all these models: see the list in panel (a) of Table 11.13. The models differ in that (1) Models 11A through 11C assume given demand curves for agricultural products in the city and an infinitely elastic supply of labor to farms at a given wage, (2) Models 11D and 11E assume a fixed amount of labor in the region to be allocated between city and farm production, (3) Models 11A and 11D assume shipping costs are zero, (4) Models 11B and 11E assume that farms produce only one crop, and (5) Model 11C assumes farms produce two crops. What about Walrasian equilibrium and localization? • Model 11A has no shipping costs. It models three markets: wheat, labor, and land. The supply of labor and the supply of land are each infinitely elastic at the going wage (w) or rent (Ra ), respectively. The amounts of labor and land used by a farm are determined in the market for wheat. There is no Walrasian simultaneity here. Prices are not jointly determined here: instead, Ra , w, and other givens determine P (the price of wheat). • Model 11D also has no shipping costs. It too identifies three markets: wheat/soap, labor, and land. The supply of land is infinitely elastic at the going rent (Ra ). What
11.7
Final Comments
333
makes it different from 11A is that (1) the total supply of labor in the region is fixed and is allocated to either city or farm production and (2) firms in the city produce soap that is exchanged for wheat. There is Walrasian simultaneity here. The exchange rate of wheat for soap (P) and the wage (w) are jointly determined by Ra and other givens. • Models 11B and 11C incorporate shipping costs. Model 11B has three markets: wheat, labor, and land. Model 11C adds a fourth market (crop 2). The supply of labor is infinitely elastic at the going wage (w). Land may also be obtained but now subject to differentials in shipping cost and land rent. There is Walrasian simultaneity here. The locational rent (the amount by which rent exceeds Ra at a given location) and the price of wheat and the price of crop 2 in Model 11C are now jointly determined by Ra , w, and other givens. • Model 11E also incorporates shipping costs. Land may also be obtained, subject to differentials in shipping cost and land rent. There is Walrasian simultaneity here. The locational rent, the price of wheat, and the wage rate are now jointly determined by Ra and other givens. What about the regional economy? Model 11E has brought us a long way toward a comprehensive depiction of the regional economy. From Table 11.10, we can measure regional output and regional income. We are also able to divvy up regional income into a wage bill and a rent bill. In Model 11E, there is no profit to either soap or wheat production. The only profit is the advantage that arises to farms nearer the city, and this is dissipated in locational rents. In terms of the objectives of this book, this chapter is important for the ideas raised in the last two paragraphs. At the same time, there remain some troubling issues. One is that capital has no role to play in the models in this chapter; we cannot use these models to look at how firm receipts are distributed to labor, land, and capital. Second, I am troubled by curve EF in Fig. 11.9: the wage paid to wheat labor. As drawn, EF rises steadily as we consider farms further away from the city. The wage is highest at the outer edge of wheat production. I recognize that this is a nominal wage only. Since utility is the same for laborers throughout the region, soap is becoming more costly with distance from the city at an even faster rate than the wage. However, what bothers me is that wage reaches a precipice at the outer radius. Thünen envisaged it as a frontier wage. The outer boundary of wheat production is akin to a “big bang”: wage rises ever faster as we move away from the factory, but a step beyond that boundary there is no further demand for labor. Why the abrupt change? Third, the models in this chapter are concerned solely with the consequences of perfect competition. In this regard, this chapter is similar to Chapters 4 and 5. Throughout this book however, I have emphasized the implications of geography and shipping cost for the existence of local monopolies. There is no monopoly in the market for either soap or wheat in this chapter. Indeed, the only monopoly element evident is the locational rent generated at sites near the city. At this stage, we need a model in which local monopolies can arise.
Chapter 12
Local Production and Consumption Substitutability, Saturation, and the Regional Economy (Thünen–Miron Problem)
A factory, using labor and other inputs, produces a commodity for sale to farms that replaces local production. Because of transportation costs (including shipping costs and commuting cost), the factory purchases inputs (including labor) and sells its product within a market area. In so doing, the factory leads to a rearrangement of local production and consumption. As a monopolist, the firm sets price knowing this affects the locations of consumers as well as their demands. In Model 12A, the behavior of the farm in autarky is examined. In Model 12B, I consider a farm within the market area of the factory. In Model 12C, I introduce a monopolist with constant marginal cost. In Model 12D, the monopolist, also a monopsonist in its market for labor, sets an f.o.b. wage to attract labor to work in the factory. In Models 12B through 12D, I explore the significance of having a saturated market area wherein farm demand for the commodity is price inelastic. The models in this chapter build on Chapters 2 and 11 in that they introduce substitution between a locally produced good and the factory good. This allows us to better understand the role of market saturation and its impact on decisions by the monopolist regarding wage and price as the regional economy grows. Unlike Chapter 11, this chapter allows us to see how prices and localization get jointly determined in a regional economy when the firm is able to exploit the monopoly advantage created by a geography.
12.1 The Thünen–Miron Problem To me, Model 11E is intriguing because it links—for the first time in this book— prices in the commodity, land, and labor markets. At the same time, there are important respects in which it is also wanting. Model 11E assumes soap production occurs solely in the city. Why is this? After all, this might seem surprising given that model assumes constant returns to scale in soap production. If no economies of scale, why not have each farm produce its own soap as well as wheat? That way, the farm could save on shipping costs. Presumably the answer is, although the city has constant returns to scale in soap, it is not possible to replicate that low unit cost J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2_12, C Springer Science+Business Media, LLC 2010
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in production on the farm.1 In other words, there is an implicit assumption here about the relative efficiency of soap production in the city. Since farms need both wheat and soap to have a nonzero utility, farms cannot exist outside the city’s output market area in Model 11E. Allowing for a local economy where farms could exist in the absence of the soap factory—even if farm soap production is relatively inefficient—gives added insight into the impact of the factory. Model 11E assumed a fixed amount of labor. While this is not necessarily problematic, it differs from the treatment in earlier chapters where we typically have assumed an unlimited wellspring of labor and other inputs, each unit of an input available at the same going price. To what extent, therefore, might the outcomes of Model 11E be a result of this particular assumption. Model 11E is silent on the determination of capital investment and returns. Model 11E is based on a competitive market for the exchange of soap and wheat, whereas the nature of a spatial economy is that prices can vary locally and in so doing create opportunities for monopolistic behavior. The purpose of this chapter is to address these concerns in a sequence of models that help us think further about prices and localization. That these explorations lead to important and surprising ideas make them all the more fitting in this culminating chapter of the book. Marshall (1907, p. 278) defines manufacturing as any business that works up material for sale in distant markets since the ability to exploit economies of scale rests on the ability to choose freely the locality in which they will do their work. To Marshall, manufacturing is contrasted on the one hand with agriculture and other extractive industries that are geographically bound and on the other hand with industries that make or repair things to suit the needs of individual customers, from whom they cannot be far removed. Implicit in this argument is a substitution by consumers between a commodity formerly made locally and a commodity now produced in a factory elsewhere. In this chapter, I consider models that explicitly allow for local production of soap by individual farms. Following Marshall, the factory is then the remote producer who earns a profit by replacing local production. In this chapter, I present four models. The first two models have their origins in Miron (1975); the other two in Miron (1978b). All these models are concerned with a farm economy in equilibrium. Model 12A describes an equilibrium in which each farm is in autarky: producing to meet its own needs. Model 12B describes an equilibrium in the presence of a soap factory where the price of factory soap is given. Model 12C makes the price of factory soap endogenous. Model 12D makes endogenous both the price of soap and the wage paid to factory workers. These models represent a progression of thought—from simpler to more sophisticated models— which I then use to draw conclusions about Walrasian price setting in a regional economy. At the same time, these models provide important insights into the role of geography in economic thought as well as to the distribution of income.
1 This
is the same argument made in Nerlove and Sadka (1991, p. 100) that I quoted in Chapter 1.
12.2
Local Production in the Farm Economy
337
12.2 Local Production in the Farm Economy To set the stage for these models, let me describe the farm economy in common to them. To begin, let me comment on the treatment of farms (and firms) to this point in the book. In Chapters 10 and 11, a farm is implicitly an entrepreneurial unit. The farm decides what (which crop), where, how (mix of factor inputs), and how much to produce to maximize profit. In those respects, the farm is just like a firm in Chapters 2, 3, 4, 5, 6, 7, and 8. Subsequent to Chapter 2, I generally assume an unlimited wellspring of firms or farms; the number of firms or farms is neither constrained nor determined by the models presented in this book except insofar as the competitive market assumption presumes that the number of firms is sufficient to make each firm a price taker. The notable exception here is Chapter 9 wherein number of farmers is important in determining risk-return tradeoff. In this chapter, I treat the farm also as a labor unit. I assume the farm has a fixed amount of labor (h units of labor per time period) that it allocates so as to maximize utility.2 Each farm has a log-linear utility function (u) defining preferences over consumption of two commodities: wheat in amount q1 and soap in amount q2 3 See (12.1.1) in Table 12.1. I assume an unlimited wellspring of farms at some given level of utility. The farm allocates labor to wheat production (in amount h1 , remunerated in wheat) and farm soap production (in amount h2 , remunerated in soap): see (12.1.6). The farm bids a rent R per m2 of land for a unit of time. Relocation is costless. The farm chooses the amount of land (L) it will occupy. The farm produces a gross output of wheat (Q)—using h1 units of labor and L units of land as inputs—in a Cobb–Douglas production function with decreasing returns to scale: see (12.1.2). This gross wheat output can be divided into rental payments, RL, and a residual termed net wheat output: see (12.1.3). Because there are diminishing returns, net wheat output can be thought to blend (1) return to labor and (2) return to the farm enterprise. The farm also produces an amount of soap, v2 , using a labor input, with constant returns, to scale: see (12.1.4). Let z be the amount of soap that the farm purchases from the factory. The farm pays an effective price P[x] for each unit of soap purchased from the factory. If the factory uses an f.o.b. price, effective price also includes unit shipping cost: see (12.1.5). In Model 12C where I also consider the case of discriminatory pricing, P[x] would be price set locally by the factory. The amount of soap consumed by the farm can now be calculated: see (12.1.7).4 So too can the amount of wheat consumed: see (12.1.8).
2 In that sense, I treat the farm as equivalent to the region in Model 11E: a fixed amount of labor. An alternative model is developed in Curry (1984) wherein it is assumed that farms may each have a given, but different, amount of labor. Such differences introduce the possibility of trade among farms—an outcome that I ignore here by assuming all farms have the same amount of labor. 3 I ignore here problems that arise when decision-making unit (the farm here) is composed of more than one individual. Where individuals differ in their preferences, it is no longer necessarily true that their preferences can be well-ordered in the sense required in Models 12A or 12B. 4 The algebraically inclined reader might ask whether it is possible that z might be negative; that is, the farm sells soap rather than purchases it. I rule out that possibility here: implicitly I assume that
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Table 12.1 The farm economy Utility of farm u = qν2 q11 − ν
0 < ν < 1
(12.1.1)
Gross production of wheat on farm β Q1 = bh1 Lγ b, β, γ > 0; β + γ < 1
(12.1.2)
Production of wheat net of land rent (net wheat output) R[x] ≥ Ra v1 = Q1 − R[x]L
(12.1.3)
Production of soap on farm c > 0 v2 = ch2
(12.1.4)
Effective price of soap purchased f.o.b. from factory s, x ≥ 0 P[x] = Pb + sx
(12.1.5)
Labor budget h = h1 + h2 + h3
(12.1.6)
Consumption of soap q2 = v2 + z
h1 , h2 , h3 ≥ 0 z ≥ 0
Consumption of wheat (assuming factory labor paid in wheat) q1 = v1 + (wb − rx)h3 − (Pb + sx)z
(12.1.7) (12.1.8)
Notes: Rationale for localization (see Appendix A): Z3—Implicit unit cost advantage at some locales; Z8—Limitations of shipping cost. Givens (parameter or exogenous): b—Scalar in wheat production; c—Marginal product of labor in on-farm soap production; h—Total labor available to the farm; r—Marginal cost of commuting; s—Unit shipping rate; x—Distance from factory; β— Exponent of labor in wheat production; γ —Exponent in land in wheat production; ν—Preference for soap. Outcomes (endogenous): h1 —Labor allocated by farm to wheat production; h2 —Labor allocated by farm to soap production; h3 —Labor allocated by farm to factory work; L—Land used by farm in wheat production; P[x]—Effective price of factory soap at x; Pb —F.o.b. price of soap (exogenous in Model 12B); Q1 —Amount of wheat produced by farm; q1 —Consumption of wheat by farm; q2 —Consumption of soap by farm; R[x]—Market rent per unit land at x; u— Utility of farm; ν 1 —Production of wheat by farm net of land rent; ν 2 —Production of soap by farm; wb —Factory wage (Model 12D only); —Amount of soap purchased from factory.
In allocating labor among activities, the farm finds that each additional unit of labor allocated to wheat production earns a smaller additional return: a consequence of the diminishing returns assumed in (12.1.2). The implication of this argument, in reverse, is that the farm would never allow h1 to drop to zero; for a sufficiently small, but nonzero, h1 , we can make the marginal productivity of labor in wheat production as high as is needed by altering the amount of land used in production. Therefore, at least a small h1 is always more productive than labor allocated to soap production on the farm. In contrast, each additional unit of labor allocated to farm soap production earns the same marginal return, c: see (12.1.4). Since I treat wheat as numéraire here, the marginal value product of labor in soap production is cP[x]. The farm at distance x therefore allocates labor to wheat production until the marginal product there falls to cP[x]. whatever price at which the factory sells its soap, it is less costly than producing soap on the farm. If this were not true, why have a factory at all?
12.3
Model 12A: Farm in Autarky
339
12.3 Model 12A: Farm in Autarky Now consider a farm in autarky. In this section, I use an asterisk (∗ ) to indicate an autarky solution. This farm in autarky is so far away from the factory that it is not worthwhile either to purchase factory-made soap (therefore, z∗ = 0) or to sell farm labor to the factory (h∗3 = 0). See (12.2.1) and (12.2.3) in Table 12.2. Because there is nothing anywhere that makes one site more valuable than another, market rent everywhere is the same: see (12.2.2). The farm can be thought to maximize utility through its choice of labor for wheat production (h∗1 ) and its use of land (L∗ ). Table 12.2 Model 12A: farm in autarky Autarky z∗ = 0 R∗ = Ra h∗3 = 0
(12.2.1) (12.2.2) (12.2.3)
Land use (from maximizing farm’s net production of wheat) β / (1 − γ ) L = (γ b / Ra )1 / (1 − γ ) h1 β / (1 − γ ) v1 = (1 − γ )b1 / (1 − γ ) (γ / Ra )γ / (1 − γ ) h1 − 1 / β γ /β − (1 − γ ) / β Rγ / β v(1 − γ ) / β v2 = ch − cb γ (1 − γ ) a 1
(12.2.4) (12.2.5) (12.2.6)
Labor allocation (from maximizing farm’s utility) h∗1 = k1 h h∗2 = (1 − k1 )h
(12.2.7) (12.2.8)
where k1 = {β(1 − ν)} / {β(1 − ν) + ν(1 − γ )}
(12.2.9)
Other model outcomes (solved by back substitution) L∗ = (γ b / Ra )1 / (1 − γ ) (k1 h)β / (1 − γ ) Q∗1 = b(k1 h)β / (1 − γ ) (γ b / R)1 / (1 − γ ) v∗1 = (1 − γ )b(k1 h)β / (1 − γ ) (γ b / R)1 / (1 − γ ) v∗2 = c(1 − k1 )h u∗ = {c(1 − k1 )h}ν {(1 − γ )b(k1 h)β / (1 − γ ) (γ b / Ra )1 / (1 − γ ) }1 − ν
(12.2.10) (12.2.11) (12.2.12) (12.2.13) (12.2.14)
p∗ = b1 / (1 − γ ) (β / c)(γ / Ra )γ / (1 − γ ) (k1 h) − (1 − β − γ ) / (1 − γ ) X ∗ = (p∗ − Pb ) / s
(12.2.15) (12.2.16)
Notes: See also Table 12.1. Rationale for localization (see Appendix A): NA—No localization for farm in autarky. Givens (parameter or exogenous): b—Scalar in wheat production; c—Marginal product of labor in on-farm soap production; h—Total labor available to the farm; Ra —Opportunity cost of land; s—Unit shipping rate; β—Exponent of labor in wheat production; γ —Exponent in land in wheat production; ν—Preference for soap. Outcomes (endogenous): h1 ∗ —Labor allocated by farm to wheat production; h2 ∗ —Labor allocated by farm to soap production; h3 ∗ —Labor allocated by farm to factory work; k1 —A constant; L∗ —Land used by farm in wheat production; p∗—Effective price of soap (units of wheat per unit soap); Pb —F.o.b. price set by factory; Q1 ∗ — Amount of wheat produced by farm; q1 ∗ —Consumption of wheat by farm; q2 ∗ —Consumption of soap by farm; R∗ —Rent paid per unit land; u∗ —Utility of farm; v1 ∗ —Production of wheat by farm net of land rent: also amount of wheat consumed by farm; v2 ∗ —Production of soap by farm: also amount of soap consumed by farm; X∗ —Distance from factory beyond which farm is in autarky; z∗ —Amount of soap farm purchases from factory.
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First, consider the use of land. Given an amount of labor in wheat production, there is a quantity of land input that maximizes production of wheat net of land rent in (12.1.3); the partial solution for L (“partial” because we have not yet solved for the amount of labor to be employed in wheat production) is shown in (12.2.4) and for v1 in (12.2.5). Because of diminishing returns in wheat production (β < 1 − γ ), the farm’s net production of wheat rises less than proportionally with the allocation of labor (h1 ). By substituting (12.2.4), and (12.1.4) into (12.1.6), we get the farm’s production possibility frontier in autarky, which is shown in (12.2.6). It is convex with respect to the origin in (v1 , v2 ) space: see curve AB in Fig. 12.1. Model 12A: Production and consumption in autarky AB CD E HI OF OG
H A
Farm’s production possibility frontier Indifference curve Utility-maximizing production and consumption combination Implicit price ratio Wheat produced (net of rent); also wheat consumed Soap produced; also soap consumed
C
Wheat (net of land rent)
Fig. 12.1 Model 12A: production and consumption: farm in autarky. Notes: ν = 0.4, b = 1.5, β = 0.6, γ = 0.05, c = 0.001, h = 4,000, Ra = 100. Here, q1 = 116.6 and q2 = 2.05. Outcomes: h1 ∗ = 1,946, h2 ∗ = 2,054, L∗ = 0.06, Q1 ∗ = 122.73, v1 ∗ = 116.6, ν2 ∗ = 2.0, u∗ = 23.18, p∗ = 37.84. Horizontal axis scaled from 0 to 6; vertical from 0 to 250
F
E
D
G
Soap
B
I
Now, consider the allocation of labor. How does the farm allocate labor in autarky in order to reach point E in Fig. 12.1? Log-linear utility functions instance homothetic preferences; as its income is varied, the farm varies its consumption of each of the two commodities proportionally. Put differently, the log-linear utility function implies that each commodity has an income elasticity of demand of 1. The farm’s income corresponds to the total amount of labor (h) at its disposal. In autarky, it turns this labor into (1) soap at constant returns to scale or (2) wheat at diminishing returns to scale. In the absence of diminishing returns, the farm would therefore allocate its labor to wheat and soap production in the proportions ν and 1 – ν, respectively. However, with diminishing returns to wheat production, a utility-maximizing farm instead allocates a proportion, k1 , of h (its income here) for wheat with the remainder going to farm soap production. See (12.2.7) and (12.2.8). The proportion k1 varies directly with the returns to scale in wheat production (β and γ ) and inversely with ν. The closer to constant returns to scale, the more attractive it is to put labor in wheat production; the upper limit on k1 —as we approach constant returns to scale in wheat production—is 1 – ν.
12.3
Model 12A: Farm in Autarky
341
By setting h∗1 and h∗2 in this way, and then setting L∗ according to (12.2.4), the farm then gets net productions, ν1∗ and ν2∗ , that maximize utility. This happens at the point on the farm’s highest indifference curve: the one just tangent to the production possibility frontier. See indifference curve CD in Fig. 12.1. We can solve for all the other variables by back substitution starting from (12.2.7), (12.2.8), and (12.2.9). The level of utility, u∗ , that corresponds to curve CD is given in (12.2.13). The farm then produces the combination (ν1∗ , ν2∗ ) at point E along AB—given by (12.2.12) and (12.2.14), respectively—and then consumes those same quantities: q∗1 = ν1∗ and q∗2 = ν2∗ . These amounts are OF and OG, respectively in Fig. 12.1. Even though the farm in autarky is not exchanging wheat for soap with the factory, it still behaves as though there were an exchange rate (i.e., a ratio of prices) of wheat for soap. The negative of the slope of the line passing through E, and tangent to both curve AB and curve CD, is the implicit exchange rate, p∗ : see (12.2.15). The implicit exchange rate—which corresponds to line HI in Fig. 12.1—shows us the tradeoff at the margin (point E in Fig. 12.1): how much wheat (the numéraire good) the farm would have to give up to get one more unit of soap. From this, we can now calculate how far the farm must be from the factory to make the purchase of factory soap unattractive.5 At the boundary of the factory’s market area, the effective price of soap, (12.1.5), must just be equal to the implicit exchange rate. This implies that the maximum distance a farm can be from the factory is given by (12.2.16). Beyond X∗ , the farm is in autarky in that z∗ = 0. What about comparative statics in Model 12A? See Table 12.3. b
When b is increased, the farm can produce more wheat from the same combination of labor and land. Now relatively scarce, soap becomes more costly. The amount of land used for wheat production increases as does net output. Overall, the farm becomes better off. Table 12.3 Model 12A: comparative statics of farm in autarky
b c h Ra β γ ν
h∗1 [1]
h∗2 [2]
p∗ [3]
L∗ [4]
Q∗1 [5]
v∗1 [6]
v∗2 [7]
u∗ [8]
0 0 + 0 + + –
0 0 + 0 – – +
+ – – – ? ? +
+ 0 + – ? ? –
+ 0 + – ? ? –
+ 0 + – ? ? –
0 + + 0 – – +
+ + + – ? ? ?
Notes: +, Effect on outcome of change in given is positive; −, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome; ?, Effect on outcome of change in given is unknown.
5 For the moment, I leave aside complications that arise when the farm is part of the factory’s labor market area. These are discussed below in Model 12D.
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c
When c is increased, the farm can produce more soap from the same amount of labor. Now being relatively scarce, wheat becomes more costly. More soap is produced in total. Overall, the farm becomes better off. h When h is increased, the farm can produce more wheat and more soap. The farm allocates more labor to each. As land now becomes relatively scarce, wheat becomes more valuable. The amount of land used for wheat production increases as does net output. More soap is produced in total. Overall, the farm becomes better off. When Ra is increased, land becomes a relatively more costly input. As land Ra is now relatively more expensive, wheat becomes more costly. The amount of land used for wheat production declines as less wheat is produced on net. β When β is increased, labor becomes more important in the production of wheat. The farm reallocates labor from wheat to soap production. γ When γ is increased, land becomes more important in the production of wheat. The farm reallocates labor from wheat to soap production. ν When ν is increased, consumers have a stronger preference for soap compared to wheat. The farm reallocates labor from wheat to soap production. Soap becomes more valuable, and its output increases. The amount of land used for wheat production declines as less wheat is produced on net. Being more preferred, soap is now more valuable. In the models presented in Chapter 11, I had assumed constant returns to scale in wheat production. There, we saw (in Fig. 11.8 for example) a linear production possibility frontier. In contrast, Model 12A assumes diminishing returns to scale, and this generates a production possibility frontier in Fig. 12.1 that is nonlinear.6 Is the assumption of diminishing returns important in Chapter 12? Specifically, what happens to Model 12A as we let β + γ approach 1: i.e., constant returns to scale? The answer is that Curve AB in Fig. 12.1 gets pulled into the shape of a straight line, k1 converges to 1 – ν, and curve AB and the implicit price ratio line (curve HI in Fig. 12.1) converge. However, even in the absence of diminishing returns, it is still possible to find a combination of wheat and soap production that leaves the farm best off in autarky. The significance of diminishing returns becomes clearer when we consider—as we will do in Model 12B—farms inside the market area of the soap factory. What about the regional economy here? In Table 12.4, I summarize the economic characteristics of a utility-maximizing farm in autarky based on the same numerical example used to generate Fig. 12.1.7 Using p∗ to convert amounts of soap into equivalent amounts of wheat, the farm in autarky has a total output equivalent to about 200 units of wheat. As part of its output, about 123 units of wheat are produced (using 1,946 units of labor and 0.06 units of land) of which γ (here 5%) is spent on 6 More
specifically, curve AB in Fig. 12.1 is concave with respect to the origin. To solve Model 12A, I first use (12.2.13), (12.2.6), and (12.2.9) to find the labor allocated by the farm to wheat (h∗1 ) and soap (h∗2 ), and (12.2.16) to find the implicit price ratio (p∗ ). Next, I use (12.2.11) to solve for the land rented (L∗ ). Finally, I find total wheat output (Q∗1 ), net wheat produced (ν1∗ ), own production of soap (ν2∗ ), and utility (u∗ ). 7
12.3
Model 12A: Farm in Autarky
343
Table 12.4 Model 12A: example of the farm economy in autarky Wheat [1] Farm’s output Farm’s output (valued in wheat) Farm’s disbursements Land rent As percentage of wheat output Shipping cost Farm income (valued in wheat) Implicit wage bill Implicit profit of farm Farm’s consumption Farm’s utility Implicit price ratio Farm’s expenditure (valued in wheat) As percentage of total expenditure Land used by a farm Labor used by a farm
Soap [2]
Total [3]
122.73 122.73
2.05 77.73
200.47
6.14 5%
0.00
6.14
116.60 73.64 42.96 116.60
77.73 77.73 0.00 2.05
116.60 60% 0.06 1,946
77.73 40% 0.00 2,054
0.00 194.33 151.37 42.96 23.18 37.84 194.33 0.06 4,000
Notes: ν = 0.4, b = 1.5, β = 0.6, γ = 0.05, c = 0.001, h = 4,000, Ra = 100. Calculations by author.
renting land. The remaining 95% is the amount of wheat retained as income by the farm. The farm uses its remaining labor to produce soap. Its total income—(1) net wheat production plus (2) soap production valued in wheat—is 194 and it splits this with share ν spent on soap consumption and share 1 – ν spent on wheat consumption. The expression for k1 in (12.2.13) guarantees that these amounts consumed correspond exactly to the net amounts produced, hence autarky. Because here I have assumed diminishing returns to scale in wheat production, I can derive one further result about the regional economy. From the wheat production function, we know that the farm will allocate a proportion γ of its output of wheat to land rents. If it were purchasing labor in a competitive market, it would also allocate a proportion β to its wage bill; that would leave a proportion 1 − β − γ which is the profit of the farm enterprise. Of course, the farm here is not purchasing its labor from a market; the amount of labor on the farm is given. Nonetheless, we can divvy farm income from wheat production into an implicit wage bill and an implicit profit proportionally using β and 1 − β − γ . I show these amounts in Table 12.4. In the case of soap production, there are constant returns to scale, so the implicit wage bill would equal farm income from soap production; the implicit profit associated with soap production is zero. I add one final comment in light of Thünen’s concern with the distribution of income. In Model 12A, rent (Ra ) is given. Effectively, Ra and the other givens of this model determine the outcomes of the model, including farm consumption and utility. If Ra were to increase, farm utility would decline. So, to rephrase Thünen’s question, what determines whether Ra is low (farms pay low rents and have a high
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Local Production and Consumption
utility) or Ra is high (farms pay a high rent and have a low utility)? As with earlier models in this book, Model 12A is silent on this question; it tells us how much land the farm will consume and hence how much is spent on the rent bill. However, it says nothing about how Ra , the rent per unit land, is determined.
12.4 Model 12B: Farm Purchasing Soap from the Factory Now, consider the farm within the output market area of the soap factory. I leave aside until Model 12D the possibility of the farm being in the factory’s labor market. Within the factory’s output market area, the farm exchanges wheat for factory soap. In autarky, the farm produced for its own needs: in Fig. 12.1, E is the both the combination of output levels and the combination of consumption levels that maximize utility. Once inside the market area of the firm, the combination of wheat and soap that a farm produces (say V2 in Fig. 12.2) may now differ from the combination that it consumes (say Q2 in Fig. 12.2). The farm then trades away wheat to get the combination of wheat and soap that maximizes its utility. Let us assume here a vast population of farms, many of them in autarky, and wherein the market area for the soap factory is relatively small. In that way, we can think of the supply of farms, given free entry, as infinitely elastic at the level of utility (u∗ ) achievable in autarky. Given the f.o.b. price set by the soap factory and the unit shipping rate, farms freely relocate until the level of utility achievable anywhere in the market area is just equal to the utility achievable in autarky. Where an advantage might otherwise exist at a particular location, I assume that farms bid up the rent on land locally until that utility advantage disappears. In other words, spatial equilibrium means that rent rises everywhere within the factory’s market area by just enough to leave the farm there at the same level of utility as in autarky. The implication of this is that almost every measure of interest—the amounts of wheat and soap produced by the farm, the amounts of wheat and soap consumed, the allocation of farm labor, the amount of land used in wheat production, and the market rent for land will now vary by location within the factory’s market area.8 Because of that variation, I refer to the utility-maximizing solution for each variable as a schedule by location (a gradient): e.g., R[x]. To begin, consider a farm just inside the boundary: i.e., where x is only slightly smaller than X∗ . There are two differences here compared to the farm in autarky. For one, the effective price of factory soap now affects the farm’s consumption bundle: the mix of soap and wheat that maximizes its utility. For another, the lower price for factory soap encourages the farm to reallocate labor from soap to wheat production. Put differently, the farm finds it attractive to purchase factory soap, substitute some farm-made soap in favor of factory soap, shift some labor from soap production
8 Another implication is that we are using utility as though it were ratio or interval-scaled rather than ordinal.
12.4
Model 12B: Farm Purchasing Soap from the Factory
345
Model 12B: Production and consumption in factory's market area
A A2 A1
C V2
A0
Locus of production combinations within factory market area Terms of trade at x = 0 Farm’s production possibi lity frontier at x = X Farm’s production possibility frontier within X11 area Terms of trade at x = X 1 Farm’s production possibility frontier at x = X* Farm’s production possibility frontier at x = 0 Indifference curve Production and consumption at x = X* Locus of consumption combinations within factory market area Terms of trade at x within X1 area
E Q2
Q1 Q0
Wheat (net of land rent)
F F2 F1 F0
A 0A1V2E A 0Q0 A1B A 2B A1Q1 AB A oBo CD E EQ2Q1Q0 V2Q2
D
0
Soap G G2 G1 G0
B
Fig. 12.2 Model 12B: farm labor allocated to wheat production. Notes: ν = 0.4, b = 1.50, β = 0.6, γ = 0.05, c = 0.001, h = 4,000,Ra = 100, Pb = 20, s = 0.10. At x = X1 , x = 66.80, P = 26.68, R = 494, v1 = 168.97, v2 = 0, q1 = 101.38, q2 = 2.53. At x = 0, P = 20, R = 4,415, v1 = 150.58.97, v2 = 0, q1 = 90.35, q2 = 3.01. Horizontal axis scaled from 0 to 4.5; vertical from 0 to 200
to wheat production and thereby increase the amount of land that it uses to produce wheat. The extent of substitution in both production and consumption depends on (1) the effective price of factory soap, (2) the extent of diminishing returns to wheat production, and (3) the relative preference for soap as opposed to wheat. If, for example, the farm had constant returns to scale in wheat production, then all farms within the factory’s market area would give up all soap production and specialize entirely in wheat production. In the presence of diminishing returns, the farm may well find it advantageous to continue producing soap on the farm even when it purchases soap from the factory. Now, consider farms still closer to the factory. Starting from X∗ and moving closer to the factory, we observe the following. Because soap is now relatively cheaper, farms closer to the factory consume relatively more of it. Nonetheless, an efficient farm will produce somewhere along its production possibility frontier. As we consider farms still closer to the factory, the effective price of factory soap drops further and the farm is inclined to give up still more soap production despite the diminishing returns to wheat production. In fact, we may reach a distance (I call it X1 ) at which the farm has entirely given up its own soap production: there is
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Local Production and Consumption
Table 12.5 Model 12B: farm in X1 area (purchasing from factory, but still producing some soap of its own) Farm’s objective β Max U = (bh1 Lγ − RL − Pz)1 − ν (c(h − h1 ) + z)ν
(12.5.1)
Land input (from ∂U /∂L = 0) L[x] = (γ b / R[x])1 / (1 − γ ) h1 [x]β / (1 − γ )
(12.5.2)
Output Q[x] = b1 / (1 − γ ) h1 [x]β / .(1 − γ )) (γ / R[x])γ / (1 − γ ) V1 [x] = (1 − γ )Q[x]
(12.5.3) (12.5.4)
Soap purchases (from ∂U / ∂z = 0 ) P[x](z[x] + c(h − h1 [x])) = ν(V1 [x] + P[x]c(h − h1 [x]))
(12.5.5)
Utility (after back substitution to eliminate L and z) U[x] = (ν ν (1 − ν)1 − ν / P[x]ν ) ((1 − γ )b1 / (1 − γ ) h1 [x]β / (1 − γ ) (γ / R[x])γ / (1 − γ ) + P[x]c(h − h1 [x]))
(12.5.6)
Labor input (from ∂U / ∂h1 = 0) h1 [x] = b1 / (1 − β − γ ) (γ / R[x])γ / (1 − β − γ ) (β / (cP[x]))(1 − γ ) / (1 − β − γ )
(12.5.7)
Utility (after back substitution to eliminate h1 ) U[x] = (ν ν (1 − ν)1 − ν / P[x]ν ) (P[x]ch + (1 − b − γ )b1 / (1 − β − γ ) (γ / R[x])γ / (1 − β − γ ) (β / (cP[x]))β / (1 − β − γ ) )
(12.5.8)
u∗
Land rent that preserves U[x] = everywhere R[x] = γ ((1 − β − γ )ν ν (1 − ν)1 − ν / (u∗ P[x]ν − P[x]chν ν (1 − ν)1 − ν ))(1 − β − γ )/γ b1 / γ (β / (cP[x]))β / γ Optimized labor input (after back substitution to eliminate R) h1 [x] = (β / (cP[x]))(u∗ P[x]ν − P[x]chν ν (1 − ν)1 − ν ) / ((1 − β − γ )ν ν (1 − ν)1 − ν )
(12.5.10)
Effective price at which h1 [x] = h P[x] = ((βu∗ ) / (chν ν (1 − ν)1 − ν ))1 / (1 − ν)
(12.5.11)
Inner radius of X1 area (if X2 area exists) under f.o.b. pricing X1 = (((βu∗ ) / (chν ν (1 − ν)1 − ν ))1 / (1 − ν) − Pb ) / s
(12.5.9)
(12.5.11)
Notes: See also Table 12.1. Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitations of shipping cost. Givens (parameter or exogenous): b—Scalar in wheat production; c—Marginal product of labor in on-farm soap production; h—Total labor available to the farm; Pb —F.o.b. price for soap set by factory; Ra —Opportunity cost of land; s—Unit shipping rate; x—Distance from factory; β—Exponent of labor in wheat production; γ — Exponent in land in wheat production; ν—Preference for soap. Outcomes (endogenous): g1 [x]—A constant at x; h1 [x]—Farm labor in wheat production at x; h2 [x]—Farm labor in soap production at x; h3 [x]—Farm labor in factory work at x; k2 —A constant; L[x]—Land used by farm in wheat production at x; P[x]—Effective price of factory soap at x; q1 [x]—Consumption of wheat on farm at x; Q1 [x]—Production of wheat on farm at x; q2 [x]—Consumption of soap on farm at x; R[x]— Bid rent per unit land at x; u[x]—Utility of farm at x; v1 [x]—Production of wheat (net of rent) on farm at x; v2 [x]—Production of soap on farm at x; X∗—Radius beyond which farm is in autarky; X1 —Radius within which farm produces only wheat; z[x]—Quantity of factory soap purchased by farm at x.
12.4
Model 12B: Farm Purchasing Soap from the Factory
347
no longer any further possibility of substituting farm-made soap in favor of factory soap. If such a distance exists, farms at distance x where X1 < x < X ∗ constitute what I label an X1 market area. In the X1 area, farms both produce some soap and purchase some from the factory. See Table 12.5. All farms at distances x < X1 constitute a second part of the market area, that I label X2, wherein the factory has no opportunity to encourage farms to further substitute their farm soap for factory soap. In the X2 area, farms specialize in wheat production, and all soap consumed is purchased from the factory. See Table 12.6. It is not necessarily the case that X1 exists in every market. In other words, it is possible that the factory’s market area is entirely an X1 area. If an X2 market area exists—and this will depend entirely on the combination of the f.o.b. price for factory soap, the unit shipping cost, and the returns to scale in wheat production—the factory’s market area will be made up of two distinct rings. Put differently, the closer wheat production is to constant returns to scale, the larger will be the X2 area and the smaller the X1 area. Table 12.6 Model 12B: farm in X2 area (producing only wheat; all soap purchased from factory) Farm’s objective β Max U = (bh1 Lγ − RL − Pz)1 − ν (c(h − h1 ) + z)ν
(12.6.1)
Land input (from ∂U / ∂L = 0) L[x] = (γ b / R[x])1/(1 − γ ) hβ / (1 − γ )
(12.6.2)
Output Q[x] = b1 / (1 − γ hβ / (1 − γ )) (γ / R[x])γ / (1 − γ ) V1 [x] = (1 − γ )Q[x]
(12.6.3) (12.6.4)
Soap purchases (from ∂U / ∂z = 0) P[x]z[x] = νV1 [x]
(12.6.5)
Utility (after back substitution to eliminate L and z) U[x] = (ν ν (1 − ν)1 − ν / P[x]ν )(1 − γ )b1/(1 − γ ) hβ / (1 − γ ) (γ / R[x])γ / (1 − γ )
(12.6.6)
u∗
Land rent that preserves U[x] = everywhere R[x] = g((1 − γ )ν ν (1 − ν)1 − ν / (u∗ P[x]ν ))(1 − γ ) / γ b1 / γ hβ / γ
(12.6.7)
Note: See also Table 12.5.
Figure 12.1 was useful to us in thinking about the farm in autarky; a similar diagram is helpful in thinking also about a farm within the factory’s market area. Consider a farm’s production possibility frontier: curve AB from Fig. 12.1 now reproduced in Fig. 12.2. The quantity OB is the amount of soap (ν2 − ch) produced when all farm labor is used to produce soap alone. OB is not affected by land rent since soap production is assumed to require no land. Put differently, if we consider a site for the farm closer to the factory where rent is higher, point B on the curve will not move. The quantity OA is the amount of wheat (ν1 = b1/(1 − γ ) (yγ /(1 − γ ) − y1/(1 − γ ) )R − γ /(1 − γ ) hβ/(1 − γ ) ) produced when all farm labor is used to produce wheat alone. OA is net of land rent; OA will be smaller where land rent is higher. Put differently, as we consider farm sites within the factory’s market area, rent per square kilometer will be higher the closer the farm
348
12
Local Production and Consumption
is to the market in a way that offsets the advantage of being able to purchase soap for a lower unit shipping cost. The effect of competition among farms—in pushing up rents in the market for land—is to twist the production possibility frontier counterclockwise about point B on Fig. 12.2. The resulting production possibility frontiers, e.g., AB, A2 B, A1 B, and A0 B in Fig. 12.2, must be such as to leave farms at the respective sites indifferent among them. Consider a location at distance x somewhere in the X1 market area. Since the rent here is higher than Ra , the production possibility frontier twists counterclockwise around point B on Fig. 12.2. Suppose that it forms curve A2 B. The exchange rate is the effective price: Pb + sx. Imagine a straight line with a slope of − (Pb + sx); a so-called terms of trade line. To maximize utility for a farm, the terms of trade line must be tangent both to (1) the production possibility frontier (A2 B) at the production combination (point V2 in Fig. 12.2) and (2) the indifference curve CD at the consumption combination (point Q2 in Fig. 12.2). Put differently, if we put a terms of trade line at its point of tangency to the indifference curve, the rent per square kilometer must be such as to make the production possibility frontier reach the terms of trade line at just one point: either as a tangent (as in at point V2 on A2 B) or as a corner solution (as at A1 on A1 B or at A0 on A0 B). In each case, the terms of trade line guarantees that (1) the amount of wheat given up is just the amount needed to purchase the factory soap and (2) no other pair of production and consumption combinations allows the farm to reach a higher level of utility. As an example, suppose the farm at x = 0 in Fig. 12.2 produces 0A0 units of wheat net of rent and no soap (i.e., is in the X2 area), trades F0 A0 units of wheat (including shipping cost) to get F0 Q0 units of soap, and thereby consumes 0F0 units of wheat and 0G0 units of soap. In general, as we consider farms closer to the market, the locus of consumption combinations is traced out by the part of the indifference curve below and to the right of point E. Similarly, the locus of net production combinations is traced out initially by the curve EV2 A1 as we pass through the X1 market area and then curve A1 A0 as we pass through the X2 area. To solve Model 12B for a farm in the X1 market area, I first find the amount of land used by a farm at distance x from the factory. The partial solution for L—“partial” again because we have not yet solved for the amount of labor to be employed in wheat production—is shown in (12.5.2) and for v1 in (12.5.4). Next, I consider the farm’s allocation of labor. For the farm at distance x, each unit of labor allocated to soap production has a marginal value product of cP[x]. The farm therefore allocates labor to wheat production—with its diminishing returns—until the marginal value product there drops to cP[x]; any remaining labor is allocated to soap production. In terms of wheat production, the labor allocation is (12.5.7). Finally, I consider the amount of factory soap purchased by the farm. Because I have assumed a log-linear utility function, the farm allocates shares ν and 1 − ν of its total net income, ν1 + P[x]c(h − h1 ), to consumption of wheat and soap, respectively: see (12.5.5). I can then determine the utility of a farm at distance x: see (12.5.8). Under the assumption of spatial equilibrium, I can then reverse this to derive an expression for R[x] at each distance from the factory that gives the same level of utility. See (12.5.9).
12.4
Model 12B: Farm Purchasing Soap from the Factory
349
Now, consider what happens as β + γ approaches 1: i.e., we get closer to constant returns to scale in wheat production. The curvature disappears and AB approaches a straight line. The X1 market area also disappears. This is interesting because it implies the firm will have only an X2 market area: i.e., a corner solution wherein each farm sets h2 = 0 everywhere inside the factory’s output market area. Inside the factory’s market area, the net production locus then becomes simply an enlargement of the vertical segment A0 A1 . To solve Model 12B for a farm in the X2 market area, I first calculate the effective price at which the farm would allocate all labor to wheat production: see (12.5.10). I then determine the radius at which this effective price occurs: see (12.5.11). I then find the amount of land used by a farm at distance x from the factory. Since this is the X2 area, the farm allocates all labor to wheat production: h1 [x] = h. The partial solution for L is shown in (12.6.2) and for v1 in (12.6.4). The farm continues to allocate shares ν and 1 − ν of its total net income, now simply v1 , to consumption of wheat and soap, respectively: see (12.6.5). I can then determine the utility of a farm at distance x: see (12.6.6). Under the assumption of spatial equilibrium, I once again reverse this to derive an expression for R[x] at each distance from the factory that gives the same level of utility. See (12.6.7). It is helpful here to compare this set of areas (X1 and X2) with the solution to Model 11E which had no distinct subareas. In model 11E, farms existed only within a single area: the market area of the city’s soap producers. Here in Chapter 12, we have autarky plus up to two distinct market areas. Model 11E in this respect is most like the X2 area: farms there do not produce soap themselves. It is by taking into account the X1 area that Model 12B differs substantively from Model 11E. To help us think further about the X1 and X2 market areas, I have sketched a solution for land rent, R[x]. As shown in Fig. 12.3, rent is equal to Ra (amount OF in Fig. 12.3) at the autarky boundary, X∗ (amount OG in Fig. 12.3). As we move closer to the factory (from right to left in Fig. 12.3), we see rent rising steadily as we pass through the X1 area on the basis on substitution by the farm in both production and consumption. Rent continues to rise as we move through the X2 area on our way toward the factory, though now only on the basis of substitution of consumption. That rent should rise is not surprising. Rent has to be higher to offset the advantage of being closer to the factory: i.e., the lower effective price for purchased soap. What is perhaps surprising is the way in which rent rises consistently as we approach the factory through both the X1 and X2 market areas. One might have thought, for example, since the X1 area has two kinds of substitution (production and consumption), whereas the X2 market has only one (production), the rent schedule might have appeared less continuous—that is, kinked—as we passed through distance X1 . Apparently, as we approach the boundary X1 heading for the factory, the farm is setting its own soap production ever closer to zero so that at the boundary (X1 ) there is no discontinuity in the rate at which rent changes.9 9 This may seem surprising to readers who are algebraically inclined. After all, there is a potentially important difference between the rent gradients in the X2 and X1 areas: i.e., between (12.6.7) and (12.5.9). In (12.6.7), the denominator of R[x] is u∗ P[x]ν. In (12.5.9), the denominator includes the factor u∗ P[x]ν − P[x]ch ν ν (1 − ν)1 − ν . Is it possible in (12.5.9), unlike in (12.6.7), for the
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Local Production and Consumption
Model 12B: Land rent and location
Rent (per Square Kilometer)
A
AB BC EG OA OD OE OF OG
Schedule of rent (per square kilometer of land) paid by farms in M2 market area Schedule of rent (per square kilometer of land) paid by farms in M1 market area Width of the X1 market area Rent (per square kilometer of land) paid by farm at distance 0 from factory Rent (per square kilometer of land) paid by farm at distance (X1) Width of the X2 market area Opportunity cost of land (Ra) Distance (X*) beyond which farm is in autarky
B
D F 0
C E
Distance from factory G
Fig. 12.3 Model 12B: land rent and location. Notes: ν = 0.4, b = 1.50, β = 0.6, γ = 0.05, c = 0.001, h = 4,000, Ra = 100, Pb = 20, s = 0.10. Outcomes are X∗ = 178.43, X1 = 66.80, R[X ∗ ] = 100; R[X1 ] = 494. R[0] = 4,415. Horizontal axis scaled from 0 to 200; vertical from 0 to 5,000
What happens to the intensity of land use in wheat production (h1 /L)? Because it is more costly to rent land nearer the factory, we might expect that the farm finds it profitable to use land more intensively there. The production function (12.1.2) allows for substitution between land and labor in production of wheat. As noted above, we see the rent for a unit of land higher for a farm nearer the factory compared to a farm near the boundary X∗ . Therefore, we might expect substitution. This is exactly what happens in the X1 and X2 market as illustrated by the schedules for h1 /L in Fig. 12.4. In the X1 area, h1 /L rises as we move closer to the factory: the farm has labor being freed up from soap production that it can now apply to wheat production. In the X2 area, h1 /L continues to rise. What is perhaps surprising, again here, is the way in which h1 /L rises consistently as we cross over from the X1 to the X2 market areas. In the X2 area, farms here do not have any on-farm denominator factor to go to zero? If so, it would create a discontinuity in R[x] that would make it different from (12.5.9). The answer here is that, the denominator factor in (12.5.9) always drops as we move further away from the factory: albeit more slowly than does the denominator in (12.6.7). Put differently, farms that substitute (i.e., where h1 [x] < h) are able to outbid a farm that opts not to substitute (i.e., h1 [x] = h) everywhere in the X1 area. Nonetheless, rents decline with increasing distance everywhere in the X1 area, and the market boundary is reached at R[X ∗ ] = Ra .
Model 12B: Farm Purchasing Soap from the Factory
Fig. 12.4 Model 12B: labor intensity and location. Notes: ν = 0.4, b = 1.50, β = 0.6, γ = 0.05, c = 0.001, h = 4,000, Ra = 100, Pb = 20, s = 0.10. Outcomes are X ∗ = 178.43, X1 = 66.80, h1 /L = 31,710 at x = X ∗ ; h1 /L = 222,180 at x = X1· h1 /L = 2,228,206 at x = 0. Horizontal axis scaled from 0 to 200; vertical from 0 to 2,500,000
351
Model 12B; Labor intensity and locaton
Labour intensity in wheat production (h1/L)
12.4
A
D F 0
AB BC EG OD OE OF
Schedule of labor intensity (h1/L) in wheat production in M2 market area Schedule of labor intensity (h1/L) in wheat production in M1 market area Width of the X1 market area Labor intensity at x = X1 Width of the X2 market area Labor intensity at x = X*
B C G
E
Distance from factory
soap production from which to find additional labor. The argument is the same as in Fig. 12.3. As we approach the boundary X1 heading for the factory, the farm is setting its own soap production ever closer to zero so that at the boundary (X1 ), there is no discontinuity in the rate at which h1 /L changes. Finally, what happens to the density of farms (or, its reciprocal, the land area occupied by a farm) across the factory’s market area? Fig. 12.5 illustrates an outcome possible here. Again, consider first a farm at X∗ and then examine farms that are closer to the factory. Just inside the X1 area, as we move closer to the factory, lot size L initially increases. This might seem strange because we argued above that labor intensity, h1 /L, was rising with the increase in land rent as we move toward the factory. However, because h1 increases quickly as the farm gets out of soap production just inside X∗ , the farm is best off there to increase L. Once we move from further into the X1 area and then into the X2 area, we see the more conventional result, that lot size shrinks as we move toward the factory because of higher rent. This is not surprising since the possibility of further substitution among uses of labor is decreasing (in the X1 area) or zero (in the X2 area) as we consider farms still closer to the factory. What about comparative statics here? See Table 12.7. A change in given typically affects the boundaries of the X1 and X2 areas: i.e., X1 and X∗ . It may also affect the value of an outcome at x = 0, x = X1 , or x = X ∗ . b
When b is increased, the farm produces more wheat from the same combination of labor and land. Now relatively scarce, soap becomes more valuable. The outer boundary (X∗ ) and inner boundary (X1 ) both increase. The amount of land used by a farm for wheat production increases as does net output. Land rents rise but only in the X2 area.
352
12
Model 12B: land area and location AB BC EG OD OE OF
A 0
h
Pb Ra
s β
C
D
F
c
Schedule of land area (L) occupied by farm in X2 market area Schedule of land area (L) occupied by farm in X1 market area Width of the M1 market area Land area of farm at x = X1 Width of the M2 market area Land area of farm at x = X*
Land used per form (L)
Fig. 12.5 Model 12B: land area occupied by a farm. Notes: ν = 0.4, b = 1.50, β = 0.6, γ = 0.05, c = 0.001, h = 4,000, Ra = 100, Pb = 20, s = 0.10. Outcomes are X ∗ = 178.43, X1 = 66.80, L[X ∗ ] = 0.0614; L[X1 ] = 0.0180. L[0] = 0.0018. Horizontal axis scaled from 0 to 200; vertical from 0 to 0.08
Local Production and Consumption
B
E
Distance from factory G
When c is increased, the farm produces more soap from the same amount of labor. Factory soap is now less competitive. The outer boundary (X∗ ) and inner boundary (X1 ) both decrease. More soap is produced on-farm in total. The attractiveness of proximity to the factory is reduced, so land rent declines. When h is increased, the farm produces more wheat and more soap. The outer boundary (X∗ ) and inner boundary (X1 ) both decrease. The farm allocates more labor to each activity. The amount of land used for wheat production increases as does net output. The attractiveness of proximity to the factory is reduced, so land rent declines. When Pb is increased, soap purchases become more costly to the farm. The outer boundary (X∗ ) and inner boundary (X1 ) both decrease. When Ra is increased, land becomes a relatively more costly input. The outer boundary (X∗ ) and inner boundary (X1 ) both decrease. As land is now relatively more expensive, wheat becomes more costly. The amount of land used for wheat production declines and less wheat is produced on net. Land rents rise everywhere in the factory’s market area. When s is increased, soap purchases become more costly to the remote farm. The outer boundary (X∗ ) and inner boundary (X1 ) both decrease. Rent at both distances remains unchanged. When β is increased, labor is more important in the production of wheat. Farms reallocate labor into wheat production because of its improved yield.
. . + . . . . . 0
. . + . . . . . 0
X1 [2]
0 0 + 0 0 0 + + −
[3]
X∗
X1 [5]
+ 0 + 0 − 0 + + −
0 [4]
− + + + − 0 − + −
L[x]
+ 0 + 0 − 0 + + −
[6]
X∗ + + + + − 0 + − −
0 [7]
Q1 [x]
+ 0 + 0 − 0 + − −
X1 [8] + 0 + 0 − 0 + − −
[9]
X∗ + + + + − 0 + − −
0 [10]
v1 [x]
+ 0 + 0 − 0 + − −
X1 [11] + 0 + 0 − 0 + − −
[12]
X∗ . . . . . . . . .
0 [13]
v2 [x]
. . . . . . . . .
X1 [14] 0 + + 0 0 0 − 0 +
[15]
X∗ + − − − + 0 + − +
0 [16]
R[x]
0 0 0 0 + 0 − − +
X1 [17]
. . . . + 0 . . .
X∗ [18]
+ − − − − − + − −
X1 [19]
+ − − − − − + − +
X∗ [20]
Note: Effects estimated numerically by increasing each given in turn compared to values for givens shown in Table 12.8; +, Effect on outcome of change in given is positive; –, Effect on outcome of change in given is negative; 0, Change in given has no effect on outcome; ., Not applicable.
b c h Pb Ra s β γ ν
0 [1]
h1 [x]
Table 12.7 Model 12B: comparative statics of farm in autarky
12.4 Model 12B: Farm Purchasing Soap from the Factory 353
354
γ
ν
12
Local Production and Consumption
Now relatively scarce, soap becomes more valuable. The outer boundary (X∗ ) and inner boundary (X1 ) both increase. When γ is increased, land is more important in the production of wheat. As the farm is less able to substitute labor for land, the outer boundary (X∗ ) and inner boundary (X1 ) both decrease. The farm produces less wheat. The attractiveness of proximity to the factory is reduced, so land rent declines. When ν is increased, consumers have a stronger preference for soap compared to wheat. As soap is now more valuable, farms are willing to purchase factory soap from further away: the outer boundary (X∗ ) increases. At the same time, they are reallocating farm labor in favor of soap production, so farm soap production increases, and the inner boundary (X1 ) decreases. Less land is used
Table 12.8 Model 12B: example of farm economy (a) Farm by distance from factory
Distance (x) Effective price of factory soap (P) Land rent (R) Labor allocated by farm to wheat production (h1 ) Land used by farm to produce wheat (L) Gross output of wheat by farm (Q) Wheat output net of land rent (v1 ) Consumption of wheat by farm (q1 ) Output of soap on farm (v2 ) Amount of soap purchased from factory (z)
0 [1] 0.00 20.00 4415 4,000 0.0018 158.50 150.58 90.35 0.00 3.01
X1 [2] 66.80 26.68 494 4,000 0.0180 177.87 168.97 101.38 0.00 2.53
X∗ [3] 178.43 37.84 100 1,946 0.0614 122.73 116.60 116.60 2.05 0.00
Wheat [1]
Soap [2]
Total [3] 3,751,341
161 161
0.54 18
8
0
153 97 56 5% 103
18 18 0
103 60% 0.03 3,460
68 40%
(b) Average farm
Total farms in X1 and X2 areas Average farm output Average farm output (valued in wheat) Average farm disbursements Land rent Shipping cost Farm income (valued in wheat) Implicit return on labor Implicit profit of farm Land rent as % of wheat output Average farm consumption Utility Expenditure (valued in wheat) Implicit expenditure share Land used Labor used
179 8 11 171 115 56
2.51
540
23.18 171
4,000
Note: Calculations by author. ν = 0.4, b = 1.5, β = 0.6, γ = 0.05, c = 0.001, h = 4,000, Ra = 100, s = 0.10, Pb = 20.
12.5
Comments on Model 12B
355
for wheat production, less wheat is produced both gross and net of rent. Land rent rises because proximity to the factory is now more valuable. What about the regional economy here? Panel (b) of Table 12.8 summarizes examples of a typical farm within the factory’s market area. This example uses the same numerical values for the givens as we did for Model 12A in Table 12.4: plus values for the two new givens: Pb and s. The summary values in panel (b) include farms in both the X1 and X2 market areas. To help us better understand these summary data, I present in panel (a) a breakdown of key measures by location within the factory’s market area. Because of the assumptions in common to Models 12A and 12B, there are similarities in outcome between farms in autarky and farms in the factory’s trading area evidenced by a comparison of Tables 12.4 and 12.8. By assumption, the utility of a farm is the same in both. Both models also assume that the farm (1) splits its income 60–40 between wheat and soap consumption and (2) expends 5% of its wheat production on land rent. What then are the differences? Compared to the farm in autarky, the average farm within the factory’s market area produces more wheat (161 vs. 123) and less soap (0.54 vs. 2.05), consumes less wheat (103 vs. 116) and more soap (2.51 vs. 2.05), occupies less land (0.03 vs. 0.06), pays a higher rent bill (8 vs. 6), and puts more labor into wheat production (3,460 vs. 1,946). In addition, the two models differ in terms of leakages from the regional economy. In Model 12A, the only leakage is the rent bill; the farm in autarky— from Table 12.4—pays 6 units of wheat to an absentee landlord. In Table 12.8, the typical farm within the factory’s market area pays 8 in rent plus another 11 on shipping.
12.5 Comments on Model 12B In Chapter 2, we considered a monopolist with a factory at Place 1 who also sells that commodity to remote customers at Place 2. We assumed that each customer at Place 2 had the same fixed downward-sloped demand curve and that the total number of consumers at Place 2 was fixed. This was a relatively simple model that helped us better understand how the monopolist sets price geographically in the presence of a unit shipping cost. In Model 12B, we see how customers (farms) spread themselves out around a supplier (soap factory) in spatial equilibrium. This has two immediate implications for the model presented in Chapter 2. One is that the number of consumers at any place may well be responsive to the price charged by the monopolist. The second is that the demand by any one customer is not fixed; the consumer substitutes between wheat and soap, as well as between soap produced on-farm (locally) and soap purchased from the factory. A fundamental difference between Model 12B and Model 11E is the treatment of farm activity outside the market area of the factory. In Chapter 11, as in Thünen’s original work, the regional economy is seen as a kind of colonial plantation set against a backdrop of unrelenting wilderness. In the models of Chapter 11, no one would live outside the factory’s market area because the factory good is essential to well-being. Through the idea of a farm-made commodity that substitutes for the
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factory commodity, Model 12B, allows for economic life outside the factory’s market area. Although Model 12B does have farm incomes that rise steadily out to the boundary of the factory’s market area as in Model 11E, Model 12B envisages a transition from a farm just inside the factory’s market area to a farm in autarky. Let me digress for a moment on a related thought. As a student in Economics, I remember professors explaining the notion of a compensated demand curve. The idea was simple enough. Since price varies (by definition) along an ordinary demand curve, the real income (well-being) of a consumer is also varying. Imagine instead a demand curve in which, at every price, the income of the consumer is adjusted to leave them equally well off at that price compared to another price. As a student, I thought this an interesting theoretical concept that illustrated the important distinction between income and price substitution effects but without a counterpart in the real world. However, let us take a sample of farms at various distances and record data on the quantity of factory soap purchased and the effective price paid. We observe farms further from the factory paying a higher effective price. A graph of this schedule of data as a curve constitutes a compensated demand curve. At every distance from the factory, land rent adjusts just enough to ensure that the farm achieves the same level of utility. In spatial equilibrium, what we observe across the landscape is therefore a compensated demand curve for factory soap.
12.6 The Soap Factory as Profit Maximizer The purpose of this book is to link prices and localization. Model 12B is an important step in that direction. It shows how local economic activity (farms) come to cluster around a supplier (soap factory) in response to the cost of purchasing soap from the factory instead of own production. The factory creates a shadow in the geography of local soap production. No soap is produced on farms within radius X1 , and soap production is reduced as far away as X∗ . By the f.o.b. price it sets, the factory also comes to reshape the density of farms (and hence the localization of farms) out to X∗ . At the same time, Model 12B does not explain how the factory sets its f.o.b. price. In the remainder of this chapter, I examine implications of profit maximization by the factory. In this, I assume that the soap factory is a monopolist in that it faces a downward-sloped demand curve for its product. However, I retain the assumption from Model 12B that each farm can produce its own soap if advantageous. In the presence of shipping costs, the factory survives only if it is more efficient at making soap than are the farms. Model 12C bridges to Chapter 2 by similarly incorporating a constant marginal cost. Model 12D assumes that the firm’s variable production cost is a labor cost only and that labor has to be attracted from farms. By assuming only one input, I can keep 12D relatively simple as well as consistent with Model 11E. In this chapter, Model 12C is in part a foil to help us better understand the implications of Model 12D.
12.6
The Soap Factory as Profit Maximizer
357
Finally, note that the firm in 12D is a bilateral monopolist. It faces a downwardsloped demand curve for its output and an upward-sloped supply curve for its input. Where a monopolist engages in mill pricing of its output and/or labor input, it quickly becomes interested in the price elasticity of demand of any consumer or the price elasticity of its labor supply. These are important in determining the most profitable price and wage. In Models 12C and 12D, I will show that these two elasticities vary with distance from the firm. This makes the aggregate elasticities facing the firm dependent on the nature and extent of its labor supply and output demand markets: i.e., on the localization of farms. Let me put the same argument differently. Implicit in Model 11E is the idea that a wage, wb , is set to attract the farm labor needed to produce soap in the city. This is an idea explored further in Model 12D. The implication of this is that the firm faces an upward-sloped supply curve for labor; it can get more workers in its labor supply area only if it pays a higher wage. Congestion arises because, to attract still more labor, the firm must pay more to (1) attract workers from further away and (2) induce farms that already supply labor to provide still more. Put differently, congestion here is a lower price elasticity of labor supply. In Models 12C and 12D, I continue to assume a rectangular plane wherein distance can be measured using a straight line. This plane is taken to be boundless; we never have to worry about edge effects. Land on this plane is everywhere identical in terms of yield. Suppose that this land is initially occupied by identical farms as in Model 12A, each producing and consuming in autarky. There is no other use for this land. I continue to assume that the supply of farms is infinitely elastic at the autarky utility level, u∗ ; if anything were to happen that might cause utility to rise even momentarily, the concept of free entry implies new farms would flood into the region, pushing up land rents, until utility drops back to u∗ . Using comparative statics, Models 12C and 12D also allow us to think about how a spatial economy might change over time as a result of capital investment or improvements in technology. This is an underdeveloped area of competitive location theory. There have been efforts made, for example, to estimate the effects of transportation investments—by reducing the unit cost of shipping—on spatial patterns of agricultural production.10 In Models 12C and 12D, I focus on how the firm might respond to an improvement in productivity. I do this as an exercise in comparative statics. I make the assumption that the farm’s ability to produce wheat and soap remain the same over time but that the factory produces soap more and more efficiently (imagining here that productivity improves as time goes on). In part, this might be because the factory is accumulating capital that it reinvests in the production process. In part, it might also be because the firm’s ability to apply or innovate technology is improving. However, consistent with my comparative statics 10 Day
and Tinney (1969) extend a Thünen model by introducing the need for working capital which the farmer can accumulate over time. Also in regard to a Thünen model. Okabe and Kume (1983) employ a kind of cobweb cycle model to study how market prices for crops in the current season affect the farm’s decision of what crop to plant the following year. Fujita and Krugman (1995) look at how to model the growth and form of urban and rural areas jointly over time.
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approach, I make the assumption that the firm cannot or does not anticipate these changes in productivity.
12.7 Model 12C: The Factory as Monopolist Using f.o.b. Pricing In Model 12C, I assume that a factory using f.o.b. pricing can be described by the profit equation (12.9.3): see Table 12.9. The factory has an annual fixed cost, C0 , as well as a constant marginal cost, C, per unit output. The factory in this sense looks like the factory in Chapter 2. The fixed cost, C0 , is like an opportunity cost on capital invested in the factory; having incurred that cost, the factory can get any output it needs (at the constant marginal cost C) from this wellspring. I ignore here the role of farm labor in factory production; that will be incorporated in Model
Table 12.9 Model 12C: factory as spatial monopolist Demand by farm in X1 area z[P] = ((u∗ P − (1 − ν) − chν ν (1 − ν)(1 − ν) ) /((1 − β − γ )ν ν (1 − ν)(1 − ν) ))(ν(1 − γ ) + (1 − ν)β) − (1 − ν)ch (12.9.1) Demand by farm in X2 area z[P] = u∗ P − (1 − ν) (ν / (1 − ν))1 − ν
(12.9.2)
(a) Using f.o.b. pricing Factory’s profit Z = Pb Q − C0 − CQ
(12.9.3)
Demand for area) factory soap (assuming X2 Q = 2π x∈X2 (xz[x]/L[x])dx + 2π x∈X1 (xz[x]/L[x])dx
(12.9.4)
Demand for factory soap (if no X2 area) Q = 2π x∈X1 (xz[x]/L[x])dx
(12.9.5)
(b) Using discriminatory pricing Factory’sprofit (assuming X2 area) Z = 2π x∈X2 (xz[x]/L[x])(P[x] − C − sx)dx + 2π x∈X1 (xz[x]/L[x])(P(x) − C − sx)dx − C0
(12.9.6)
Factory’s profit (if no X2 area) Q = 2π x∈X1 (xz[x]/L[x])(P[x] − C − sx)dx − C0
(12.9.7)
Profit-maximizing price everywhere in X2 area (1 /(1 − ν)) P[x] = ((bu∗ ) / ((1 − γ )chν ν (1 − ν)1 − ν )
(12.9.8)
Notes: For inner boundary to X1 area when h1 [X1 ] = h, see (12.5.11). See Table 12.5 and Table 12.6. Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitations of shipping cost. Givens (parameter or exogenous): C—Marginal cost of factory soap production; Co —Fixed cost of factory soap production; x—Distance from factory. Outcomes (endogenous): L[x]—Land used per farm in wheat production at x; Pb —F.o.b. price for soap set by factory; P[x]—For factory, price for soap at farm at x that maximizes profit; Q—Factory output of soap; Z—Profit of factory; z[x]—Quantity of factory soap purchased by a farm at x.
12.7
Model 12C: The Factory as Monopolist Using f.o.b. Pricing
359
12D. Implicitly, Model 12C assumes all inputs to factory soap production, including labor, have horizontal supply curves. In effect, the factory can purchase as much or as little of each input, and the unit cost will always stay the same. Again, this is as in Chapter 2. As in Model 12B, I allow for the possibility of concentric ring areas around the factory. As in Model 12B, an X2 area where farms allocate labor only to wheat production and an outer X1 area where farms allocate labor to wheat and farm soap production are possible. The X2 area may not exist in all cases; I include it here for comprehensiveness. Demand for factory soap is therefore given by either (12.9.4) or (12.9.5) depending on the presence of an X2 area. The amount demanded by any one farm is given by (12.9.1) or (12.9.2); see the kinked curve ABC in Fig. 12.6 of which segments AB and BC are the demand schedules for factory soap by a farm in the X1 area and X2 area, respectively. Note that (12.9.1) and (12.9.2) are compensated demand curves; they already incorporate the adjustment in land rent that makes consumption of factory soap at each effective price give the same level of utility. Now consider the introduction of the soap factory into this landscape. For the soap factory to be profitable, several conditions must be met. First, in this barter economy, the factory’s f.o.b. price (Pb ) must be below p∗ ; otherwise it would not
Models 12B and 12C: individual demand for factory soap
A
Demand for factory soap (Z) by farm in X1 market area Demand for factory soap (Z) by farm in X2 market area Opportunity cost (p*) of producing soap on farm in autarky Effective price at which farm no longer produces any soap of its own
B
Effective price for soap to farm
D
AB BC OA OD
0
C
E
Soap purchased (z)
Fig. 12.6 Models 12B and 12C: demand by farm (z) for factory soap. Notes: ν = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, h = 4,000, Ra = 9,600, s = 0.02, C0 = 0, and C = 0.03. Outcomes are p∗ = 0.2906 X ∗ = 9.64 in 12B and 13.03 in 12C. Horizontal axis scaled from 0 to 600; vertical from 0 to 0.35
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get any customers. Second, the factory’s average cost of production (C + C0 /Q) must be sufficiently low to make production profitable. Assume the following hypothetical situation without a factory. Each farm is in autarky. The period of time is a year. The preference—budget share—for soap (ν) is 0.20. The scale parameter for wheat production (b) is 900. The returns on labor and land in wheat production (β and γ ) are 0.2 and 0.6, respectively. The marginal productivity of labor in farm soap production (c) is 0.16 blocks of soap per personhour of labor. The unit shipping rate (s) is 0.02 per unit of wheat transported 1 km. The total amount of labor available to the farm is 4,000 person-hours per year, and the annual opportunity cost of land (Ra ) is 9,600 m3 of wheat per square kilometer for the year. In this case, we can calculate the outcome for the farm in autarky from Table 12.2. From v, we know that the farm would like to allocate its 4,000 personhours annually into 3,200 person-hours for wheat production and 800 person-hours for farm soap production. However, because of diminishing marginal productivity in wheat production, the farm instead allocates only 2,667 person-hours to wheat production (h∗1 ) and the rest (1,333 person-hours) to soap (h∗2 ). This allows the farm to produce 213 blocks of soap (v∗2 ); in autarky, this is also the amount of soap consumed (q∗2 ) by the farm. The farm rents 0.0388 km2 of land and is thereby able to produce 620 units of wheat annually (Q∗1 ). The land rent bill (Ra L∗ ) totals 372 units of wheat, leaving a net output (v∗1 ) of at most 248 units of wheat; in autarky, this is also the amount of wheat consumed by the farm. The utility level achieved by the farm (u∗ ) is 241. For farms in autarky, the implicit exchange rate (p∗ ) is 0.2906. At the utility maximum, the farm is giving up 0.2906 m3 in potential wheat consumption for the last block of soap consumed. Assume now that a soap factory is placed somewhere on this rectangular plane. This factory occupies no space. This factory sells (exchanges) its product to farms (in return for wheat). Shipments arise because of the flows of wheat sold to the factory and soap purchased there. These movements incur shipping costs, but I assume these movements require no geographic space. In other words, even in the presence of the factory, all land on the rectangular plane is used by farms and only for wheat production. I assume initially here that the factory sells soap at an f.o.b. price (exchange rate), Pb . Then, I consider the case where a monopolist can improve profit using discriminatory pricing. Suppose the factory has C = 0.27, which is modestly smaller than the autarky-implicit exchange rate (p∗ ) in Table 12.4. Suppose we also assume C0 = 0 so that we make soap production in the factory much like soap production on the farm. Both are constant returns to scale; since C < p∗ here, factory production is more efficient. In this case, what Pb will earn the factory its maximum profit? More directly, how do we solve here for the exchange rate (Pb ) that maximizes factory profit? Our problem here is that neither equation (12.9.4) nor equation (12.9.5) can be reduced algebraically to gives an explicit solution for Pb . We must therefore rely on numerical approximations to the integral expressions in them (once again, I use the midpoint rule version of a Riemann sum11 ). We also have to use numerical 11 For
calculations reported here, I used N = 200 panels.
12.7
Model 12C: The Factory as Monopolist Using f.o.b. Pricing
361
methods to find the solution; one approach is to iterate to find the Pb that makes the profit earned by the factory as large as possible Doing this, I find the profit-maximizing factory will set an f.o.b. exchange rate (Pb ) of 0.2752 units of wheat per unit of soap. Appropriately, this is below the implicit exchange rate (0.2906) for farms in autarky. Under profit maximization, the radius of the factory’s output market is a mere 0.77 km and, there is no X2 area here. The total amount demanded from the factory is 881 units of soap. In terms of comparative statics, what would happen were C in fact smaller? In panel (a) of Table 12.10, I show the profit-maximizing exchange rate for a factory with f.o.b. pricing for values of C ranging from 0.27 down to 0.03. In each case, befitting a comparative statics analysis, I assume the firm treats C as a fixed parameter. Not surprisingly, when C is lower, so too is the profit-maximizing Pb . Where it is more efficient (i.e., has a smaller C), the firm sets a lower f.o.b. price; as it does so, radius, output, and profit all increase. If the profit-maximizing price becomes low enough, it becomes possible for farms sufficiently close to the factory to specialize entirely in wheat production: hence the emergence of an X2 area. As we know from a first course in Economics, the price set by a monopolist depends on the price elasticity of demand for soap; the more elastic the demand, Table 12.10 Model 12C: factory behavior as function of marginal cost (C) (a) Firm using f.o.b. pricing Unit cost Price
Radiuses
C
Pb
X1
X∗
[1]
[2]
[3]
0.27 0.23 0.18 0.13 0.08 0.03
0.275 0.245 0.208 0.173 0.136 0.098
– – 1.2 2.9 4.8 6.7
Output
Profit
Additional
[4]
X2 area (000) [5]
X1 area (000) [6]
Z (000) [7]
rent (000) [8]
0.8 2.3 4.1 5.9 7.7 9.6
– – 24 171 480 1051
0.9 23 109 198 294 386
0.0 0.3 4 16 44 97
0.0 0.3 3 11 32 73
(b) Firm using discriminatory pricing Unit cost Radiuses X∗
Prices
Output
C [1]
X1 [2]
[3]
P[0] [4]
P[X1 ] [5]
0.27 0.23 0.18 0.13 0.08 0.03
– – – 2.2 5.0 7.3
1.0 3.0 5.5 8.0 10.5 13.0
0.280 − 0.260 − 0.233 − 0.231 0.231 0.231 0.231 0.231 0.231
P[X∗ ]
Profit
Additional
[6]
X2 area (000) [7]
X1 area (000) [8]
Z (000) [9]
rent (000) [10]
0.291 0.291 0.291 0.291 0.291 0.291
– – – 35 203 798
1 26 164 469 884 1043
0.0 0.4 4 20 58 130
0.0 0.2 2 10 24 44
Note: ν = 0.2, b = 900, β = 0.2, γ = 0.60, c = 0.16, s = 0.02, h = 4,000, Ra = 9,600, C0 = 0. Calculations by author.
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the greater the revenue associated with a lower price. Here, elasticity varies from one consumer to the next for two principal reasons. One is that because the factory prices f.o.b. consumers pay the freight and face a higher effective price. As far as the factory is concerned, consumers closer to the factory appear to have demands that are more sensitive to f.o.b. price than do consumers who are further away. In Table 12.10 we see that as C is lowered, the radius of the market (X∗ ) increases which implies that the factory has more customers further away; hence the aggregate demand facing the factory becomes less price elastic from the factory’s perspective. The second is that to the extent that consumers are in the X2 area, their demand will be less sensitive to price because they no longer substitute farm-made soap for factory-made. In Table 12.10, I show that as C drops, eventually Pb becomes low enough for an X2 area to emerge and that as C drops further, the demand from the X2 area becomes an ever-more important share of its total demand. This too implies that, as C drops still further, the factory’s demand will become ever less price elastic. Panel (a) of Table 12.10 also shows “Additional rent”: i.e., the aggregate rent paid by farms over and above (π X ∗2 )Ra . Because we have assumed that landlords are absentee, the factory sees rent paid as a leakage from the local economy in that this takes away from the wheat being offered in exchange for soap. After all, in the absence of the factory, additional rent everywhere would be zero. At C = 0.03, the factory’s profit and the additional rent are 97 and 73,000 units of wheat, respectively. Presumably, if they were not paying this added rent, farms would have spent a share of this on factory soap. At the same time, note what happens to the price elasticity of demand over the range of C shown in Table 12.10. When C is large, there is no X2 area; all farmers are in the X1 area where there is substitution between farm and factory soap. As we make C smaller, an X2 area begins to emerge. When C = 0.03, demand is dominated by customers from the X2 area where there is no possibility of substitution. Add to that the idea that the consumer at X∗ is increasingly further away as C becomes smaller, and the overall price elasticity of demand is decreasing. Now, let me put the ideas in the last two paragraphs together. Despite earning larger profits, the factory owner for whom C drops has reason to be unhappy. First, the owner sees the leakage of profit in the form of higher rents paid by farms to absentee landlords. Second, f.o.b. pricing is proving to be an increasingly blunt tool for the job at hand. To maximize profit, the factory owner would want to decrease prices in the part of the market where demand is more price elastic. However, under f.o.b. pricing, the factory raises or lowers effective price by the same amount at every location.
12.8 Model 12C: The Factory Using Discriminatory Pricing With discriminatory pricing, the factory sets potentially a different price for each customer. As in Chapter 2, I assume here that nothing prevents or inhibits the factory from selling at different prices to different customers. Panel (b) of Table 12.9
12.8
Model 12C: The Factory Using Discriminatory Pricing
363
outlines Model 12C in the case of the factory pursuing discriminatory pricing. Table 12.5 and Table 12.6 still describe outcomes as long as we remember that P[x] is now the delivered price rather than effective price (f.o.b. price plus unit shipping cost). From those two tables, I can derive the farm’s demand for factory soap. See (12.9.1) and (12.9.2). The factory now sets a discriminatory price (P) for farms at each distance x so as to maximize overall profit. See (12.9.7). Because of mathematical complexity, I once again rely here on numerical methods to solve Model 12C for the price-discriminating monopolist. In panel (b) of Table 12.10, I show discriminatory pricing solutions for the same values of C examined above in respect to a factory pricing f.o.b. There are some obvious differences between f.o.b. and discriminatory pricing schemes. First, not unexpectedly, profit is higher when the firm uses discriminatory instead of f.o.b. pricing: e.g., 13,000 units of wheat vs. 97,000 at C = 0.03. Second, the price-discriminating factory has a larger market radius than does the f.o.b. pricer: e.g., 13.0 km vs. 9.6 km at C = 0.03. Third, as we move down Table 12.10, for the values of C listed, an X2 area first appears with f.o.b. pricing (at C = 0.18) and with discriminatory pricing only at C = 0.13. Further, at low values of C, the boundary radius X1 for discriminatory pricing is not the same as for f.o.b. pricing. However, these are differences that the reader might well have expected. There are two other differences that I think might be more surprising. First, in the case of discriminatory pricing, the factory sets the same delivered price across its X2 area even while it sets a delivered price that increases with distance across the X1 area. See Fig. 12.7 where I compare the schedule of effective prices under f.o.b. pricing (curve AJB there) with the schedule of delivered prices under discriminatory pricing (curve CDE). Earlier in Fig. 8.2, I showed another example where delivered prices (under discriminatory pricing) increase less quickly with distance than do effective prices (under f.o.b. pricing). Still earlier, in Chapter 2, I had labeled this behavior by the monopolist “partial freight absorption”. Partial freight absorption is still evident here in the X1 market area: compare the slopes of curves JB and DE in Fig. 12.7. What is novel about Fig. 12.7 is the horizontal segment (curve CD) of the schedule of delivered prices in Model 12C that spans the X2 market area there. To explain why this happens, I go back to the equation (12.9.2) for the farm’s demand for factory soap in the X2 area. This is not the linear demand curve that I had used earlier as in Chapter 2. Instead, it is a negative exponential demand curve for which the price elasticity (ignoring the negative sign) is the exponent on price: namely 1 − ν. Since 0 < ν < 1, the price elasticity must lie between 0 and 1. Put differently, the farm’s demand for soap in the X2 area is price inelastic. As such, it is most profitable for the firm to set the delivered price as high as possible there (consistent with being in the X2 area). The factory therefore sets delivered price everywhere in its X2 market area just low enough to keep farms from producing soap of their own: see (12.9.8).12
12 This
is the same kind of argument that I made in Model 7D about price setting by a repair contractor.
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Models 12B and 12C: Price by location AJ CD DE JB OF OG OH OI
Model 12B schedule of effective prices in X2 market area (Pb + x) Model 12C schedule of delivered price in X2 market area (P[x]) Model 12C schedule of delivered price in X1 market area (P[x]) Model 12B schedule of effective prices in X1 market area (Pb + x) Model 12C outer radius of X2 market area (X1) Model 12B outer radius of X1 market area (X*) Model 12C outer radius of X1 market area (X*) Model 12B outer radius of X2 market area (X1)
Effective price (P[x])
B
C
E
J D
A
0
I F Distance from factory (x)
G
H
Fig. 12.7 Schedule of price by location in Models 12B and 12C compared. Notes: ν = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, h = 4,000, Ra = 9,600, s = 0.02, C0 = 0, and C = 0.03. Outcomes are Pb = 0.0978, X ∗ = 9.64 in 12B and 13.03 in 12C. Horizontal axis scaled from 0 to 14; vertical from 0 to 0.4
Second, from the perspective of the factory, there is less leakage of rent with discriminatory pricing. For example, at C = 0.03, total additional rent paid by farms (rent paid by farms over and above (πX ∗2 )Ra ) is only 44,000 units of wheat: compared to 73,000 under f.o.b. pricing). Put differently, the factory is better able to benefit from the gains to be made from its efficient production. Much of the reduction in additional land rent comes in the X2 market area. See Fig. 12.8 which again illustrates the case of C = 0.03. There, the faint curve (ABC) is the schedule of rents under f.o.b. pricing while the kinked solid curve (DEF) is the schedule of rents under discriminatory pricing. Remember here that farms are in spatial equilibrium and that I have assumed they achieve the same level of utility under either pricing scheme. What the consumer saves in land rent in one pricing scheme over the other is offset by a higher price for soap. Now, let me put the ideas in the last two paragraphs together. In my view, the factory finds itself constrained by f.o.b. pricing in the sense of a leakage of potential
12.8
Model 12C: The Factory Using Discriminatory Pricing
365
Market rent per unit land (R [x])
Models 12B and 12C: Land rent by location AB BC DE EF OG OH OI OJ OK
D
Model 12B schedule of land rent in X2 market area Model 12B schedule of land rent in X2 market area Model 12C schedule of land rent in X2 market area Model 12C schedule of land rent in X1 market area Model 12B outer radius of X2 market area (X1) Model 12C outer radius of X2 market area (X1) Model 12B outer radius of X1 market area (X*) Model 12C outer radius of X1 market area (X*) Opportunity cost of land (Ra)
B
E
K
0
F
C
G H Distance from factory (x)
I
J
Fig. 12.8 Schedule of land rent (R[x]) by location in Models 12B and 12C compared. Notes: ν = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, h = 4,000, Ra = 9,600, s = 0.02, C0 = 0, and C = 0.03. Outcomes are Pb = 0.0978, X ∗ = 9.64 in 12B and 13.03 in 12C. Horizontal axis scaled from 0 to 14; vertical from 9,000 to 11,500
profit in the form of additional land rent (i.e., over and above Ra ). As the efficiency of the factory improves (i.e., C drops), f.o.b. pricing means that the factory lowers its effective price for everyone: even though customers in the adjacent X2 market area have a relatively inelastic demand. Using price discrimination, the firm would drop its price only for the more remote (and more elastic) customers still in the X1 market area. Some students, particularly those from outside Economics, may see price discrimination as something undesirable, reprehensible, or even illegal. They think that the pricing of commodities should somehow be more closely tied to the cost of manufacturing and distribution. I think that many economists would argue that, in fact, it can be difficult to discern whether firms are indeed using price discrimination. Suppose, for example, that our soap manufacturer announces that he will sell his product at a uniform price out to distance X1 and that beyond that the firm will pass along only a fraction of any additional unit shipping cost to the consumer. This can be cast in a way that alludes to (1) an efficient regional infrastructure for shipping so that the difference in the marginal cost of serving a customer here rather than there
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within the region is negligible and (2) the cooperative spirit of the firm in helping out remote customers. These assertions may or may not be true; what is clear is that the pricing scheme of the firm here is indistinguishable from price discrimination. This brings us to an important idea in this book. I began this book by arguing that geography is important to economic thought. However, we have now come to a point where, at least within a radius X1 , the factory maximizes its profit by a pricing scheme (uniform pricing) in which geography appears not to matter. In fact, to avoid the appearance that it is indeed engaged in price discrimination, the firm has the incentive to say things like “an efficient infrastructure for shipping services means that we can serve a large market area without differentiating price by location”. At the same time, just such a pricing policy—viewed as price discrimination—allows the factory to extract profit that would otherwise accrue to landlords. And, such profit arises specifically because of the role of unit shipping costs. Two caveats are in order here. First, is it always the case that the factory will have an X2 area around it? No. As argued above, the existence of an X2 area is dependent on the extent of diminishing returns to scale in wheat production. Only if wheat production is sufficiently close to constant returns to scale will we see the emergence of an X2 area. Second, is it always the case that farms everywhere in the X2 area will have a price-inelastic demand for soap? Again no. We get this result in Model 12C because we have assumed a log-linear utility function for each farm combined with a competitive market for land. What is clear, however, is that the absence of farm soap production in the X2 area makes farms less able to substitute when the price of factory soap is reduced and that this makes demand less price elastic.
12.9 Model 12D: The Factory as Bilateral Monopolist To this point, I have assumed that the farm does not allocate labor to factory work: i.e., h3 = 0 everywhere. However, if the factory indeed needs labor to produce its soap and farms are the only source of labor, this cannot be true. Presumably, the factory has to offer a wage high enough to make it attractive to farms nearby to divert labor to factory production. Consider first a farm immediately adjacent to the factory. If the factory wage (wb ) is high enough, the farm will allocate labor only to wheat production and to factory work: in other words, h2 = 0. Because the marginal returns to factory work and to on-farm soap production are each fixed, the farm would generally never engage in both of them simultaneously (except in the knife-edge case where returns are identical: i.e., where w[x] = cP[x]). We will typically then have a corner solution wherein either h2 = 0 or h3 = 0. If we now consider a farm a little further away from the factory, the effective factory wage will drop due to the cost of commuting.13 The allocation of labor to wheat
13 Note
that my treatment of commuting cost here is smooth; for each additional unit of labor allocated to factory work, the cost of commuting rises proportionally. In reality, commuting cost
12.9
Model 12D: The Factory as Bilateral Monopolist
367
production rises and to factory work declines. The critical distance from the factory is X2 : h3 > 0 and h2 = 0 within radius X2 ;h3 = 0 and h2 > 0 outside it. I label the geographic area inside radius X2 as the X3 area (the firm’s labor market area). There are two distinct ways of thinking about the factory here. One is that the factory is essentially equivalent to a cooperative at which farms gather to produce their needed soap more efficiently. In this case, the output area served by the factory and the area from which it draws its labor would be the same. The other is to think that the factory uses labor from a small area to produce a good sold over a larger area. The latter is more akin to the concept of a manufacturing plant used by Marshall. Therefore, in what follows, I assume X2 < X ∗ : i.e., the factory’s labor market area is smaller than its output market area. Model 12D examines how a bilateral monopolist with labor as its sole input adjusts wage and price (exchange rate) so as to maximize profit. Whether such adjustments are upward or downward will depend on the elasticities of demand for the factory’s output and of its supply of labor. In the non-spatial spatial, a usual tendency would be to assume fixed elasticities so that successive increments of economic growth or technical change would consistently give similar kinds of wage and price changes. Does the same result hold in a world incorporating geographic space? If not, why not? I return to this question shortly. Model 12D is laid out in Table 12.11. In contrast to Model 12C, assume the soap factory has only one variable input, labor. The production function (12.11.1) assumes that there are returns to scale: diminishing if 0 < δ < 1, constant if δ = 1, or increasing if δ > 1. In contrast, soap production on the farm, as we saw in Chapter 12, had constant returns to scale. At the same time, output is determined by a scalar k0 that I assume is fixed in any one time period but that changes over the years; however, assume the firm is unaware of this and treats k0 as fixed for each time period. Assume here that the factory is a relatively efficient producer of soap compared to farms. The demand for factory soap is given by (12.11.3) which totals the demands in the X1, X2, and X3 areas. As in Model 12C, the X2 area may or may not exist; once again, I include it here for comprehensiveness. The supply of factory labor is given in (12.11.5). It hires all workers at the same wage, wb .14 The net profit of the firm is given by (12.11.2). Assume the factory has a fixed cost, C0 , each time period. This is its only cost other than the wage bill. Now consider the introduction of a soap factory into this landscape. For the soap factory to be profitable, several conditions must be met. First, in this barter economy, the factory’s f.o.b. exchange rate (Pb ) must be below p∗ ; otherwise it would not get any customers. Second, its wage (wb ) must be high enough to attract the required workers. In the X1 area, a farm that produces soap of its own, and the marginal value product of labor is cP[x]; therefore, to have labor attracted to factory work is usually more lumpy than this. You might choose for example to work 8 hours a day, rather than 7, without incurring any added commuting cost. The models in this chapter do not take such considerations into account. 14 This too may seem strange. After all, why wouldn’t the factory use discriminatory pricing in its labor market. I use a constant wage here once again to simplify calculation.
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Table 12.11 Model 12D: factory as bilateral monopolist Factory’s production function Q = k0 Nδ (0 < δ < 1 ; k0 > 0)
(12.11.1)
Factory’ profit Z = P b Q − C0 − w b N
(12.11.2)
Demand for factory soap (assuming X2 area) Q = 2π x∈X3 (xz[x]/L[x])dx + 2π x∈X2 (xz[x]/L[x])dx + 2π x∈X1 (xz[x]/L[x])dx
(12.11.3)
Demand for factory soap (assuming no X2 area) Q = 2π x∈X3 (xz[x]/L[x])dx + 2π x∈X1 (xz[x]/L[x])dx
(12.11.4)
Supply offactory labor N = 2π x∈X3 (xh3 [x]/L[x])dx
(12.11.5)
Notes: In the absence of X2 area, outer boundary (X2 ) of X3 area occurs where the marginal return on factory work equals the marginal return on on-farm soap production, i.e., wb -rX2 = c(Pb + sX2 ). In the presence of X2 area, outer boundary (X2 ) of X3 area occurs where the marginal return on factory work equals the marginal return on wheat production, i.e., wb -rX2 = (βk2 /(ch(1-γ )))(Pb +tX2 )ν . See Table 12.5, Table 12.6, and Table 12.12. Rationale for localization (see Appendix A): Z1—Presence of fixed cost for factory; Z2—Implicit unit cost advantage at some locales; Z8—Limitations of shipping cost. Givens (parameter or exogenous): Co —Fixed cost of factory soap production; ko —Scale of production; δ—Returns to scale on labor in factory soap production. Outcomes (endogenous): h3 [x]—Schedule of labor allocated by farm to factory work; L[x]—Schedule of and used per farm in wheat production; N—Total supply of labor to factory; Pb —F.o.b. price for soap set by factory; Q—Total quantity of factory soap demanded; wb —Wage rate paid by factory; Z—Factory s profit; z[x]—Schedule of quantity of factory soap purchased by a farm.
implies wb > cP[0]. In fact, wb must be even higher than this if there is an X2 area present. Third, the factory’s scale of production (k0 ) must be sufficiently large to make production profitable. How do farms behave in this model? Farms that fall into the X1 or X2 areas will continue to have the behavior described in Tables 12.5 and 12.6. What is new here are the farms in the labor market area (X3). Here, farms allocate no labor to on-farm soap production: see (12.12.1) and (12.12.2) in Table 12.12. For a given allocation of labor to wheat production, we can then find the optimal size of farm (12.12.3) and the gross and net outputs, (12.12.4) and (12.12.5). For a given allocation of labor to wheat production, we can determine z from the familiar budget share equations (12.12.6) and (12.12.7) and reexpress utility as a function of h1 [x] in (12.12.8). Finally, we can then find the h1 [x] that maximizes utility in (12.12.9), reexpress utility in (12.12.10), and then invert it to get an expression for R[x] at each distance (12.12.11) that makes U[x] = u∗ everywhere. Let us now try a simple example. Suppose the factory has δ = 1.0, that is, constant returns to scale, just like farm soap production. Suppose we also assume C0 = 0 so that we make soap production in the factory using labor only, much like soap production on the farm. In this case, what k0 at the minimum is required for the factory to earn a profit? The answer to that question is fairly easy; k0 would
12.9
Model 12D: The Factory as Bilateral Monopolist
369
Table 12.12 Model 12D: farm in X3 area (farm engaged only in wheat production and factory work) Factory work and on-farm soap labor h2 [x] = 0 h3 [x] = h − h1 [x]
(12.12.1) (12.12.2)
Land use and production (from ∂U / ∂L = 0) L[x] = (γ b / R[x])1 / (1 − γ ) h1 [x]β / (1 − γ ) Q[x] = b1 / (1 − γ ) (γ / R[x])γ / (1 − γ ) h1 [x]β / (1 − γ ) V1 [x] =(1–γ )Q[x]
(12.12.3) (12.12.4) (12.12.5)
Goods consumption (from ∂U / ∂z = 0 ) P[x]z[x] = ν(V1 [x] + w[x](h − h1 [x])) V1 [x] − P[x]z[x] = (1 − ν)(V1 [x] + w[x](h − h1 [x])) U[x] = (ν ν (1 − ν)1 − ν / P[x]ν)(V1 [x] + w[x](h − h1 [x]))
(12.12.6) (12.12.7) (12.12.8)
Labor allocation (from ∂U / ∂h1 = 0) h1 [x] = (β / w[x])(1 − γ ) / (1 − β − γ ) b1 / (1 − β − γ ) (γ / R[x])γ / (1 − β − γ ) Utility U[x] = (ν ν (1 − ν)1 − ν / P[x]ν )(w[x]h + (1 − β − γ )b1 / (1 − β − γ ) (γ / R[x])γ / (1 − β − γ ) (β / cP[x])β / (1 − β − γ ) )
(12.12.9) (12.12.10)
Rent in spatial equilibrium (from U[x] = u∗ ) (12.12.11) R[x] = γ (((1 − β − γ )ν ν (1 − ν)1 − ν ) / (.u∗ P[x]ν − w[x]hν ν (1 − ν)1 − ν ))(1 − β − γ ) b1 / γ (b / w[x])β / γ Notes: Rationale for localization (see Appendix A): Z1—Presence of fixed cost for factory; Z2— Implicit unit cost advantage at some locales; Z8—Limitations of shipping cost. Givens (parameter or exogenous): b—Scalar in wheat production; c—Marginal product of labor in on-farm soap production; h—Total labor available to the farm; Ra —Opportunity cost of land; s—Unit shipping rate; β—Exponent of labor in wheat production; γ —Exponent in land in wheat production; ν— Preference for soap. Outcomes (endogenous): h1 [x]—Farm labor in wheat production at x; h2 [x]— Farm labor in soap production at x; h2 [x]—Farm labor in factory work at x; k2 —A constant (see Table 12.5); L[x]—Land used per farm in wheat production at x; q1 [x]—Consumption of wheat on farm at x; Q1 [x]—Production of wheat on farm at x; q2 [x]—Consumption of soap on farm at x; R[x]—Bid rent per unit land at x; u[x]—Utility at x; v1 [x]—Production of wheat (net of rent) on farm at x; v2 [x]—Production of soap on farm at x; z[x]—Quantity of factory soap purchased by farm at x.
have to be larger than c. Since we have assumed c = 0.16 above, let us therefore assume k0 = 0.18 as an illustrative case: put differently, assume factory workers are modestly more productive than are farms in producing soap per unit of labor allocated to this activity. How do we solve here for the exchange rate and wage (Pb , wb ) combination that maximizes factory profit. Our problem here is twofold. First, equations (12.11.3) and (12.11.5) cannot be reduced algebraically. We must therefore rely on numerical approximations (midpoint rule version of a Riemann sum15 ) to the integral
15 For
calculations reported here, I used N = 200 panels.
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expressions in them. The second problem is that we have to use numerical methods to find the solution. One approach is to scan values of Pb ; for each chosen Pb , we can then iterate to find the wb that provides sufficient labor to meet the demand for soap forthcoming at that price and thereby calculate the profit earned by the factory. Doing this, I find the profit-maximizing factory will set an f.o.b. exchange rate (Pb ) of 0.2800 m3 of wheat per block of soap. See Table 12.13. As expected, this is below the implicit exchange rate (0.2906) for farms in autarky. To sustain the demand this generates, the factory will offer a wage (wb ) of 0.0482 per person-hour. For a farm in autarky, the marginal value product of labor is cp∗ which is 0.0465 per person-hour; the factory is, as is necessary, offering a relatively better wage. Under profit maximization, the radius of the factory’s output market is a mere 0.53 km; the radius of its labor market is 0.26 km; and, there is no X2 area here. From the small sizes of the X1 and X3 areas, the transport rates, s and r, must be relatively high. The total amount demanded from the factory is 1,330 blocks of soap. For the factory, total revenue is 372, total cost is 356, and net profit is 16. Table 12.13 Model 12D: factory behavior under f.o.b. pricing as function of scalar (k0 ) Scalar Price Pb k0 [1] [2]
Radiuses Wage wb X2 X1 [3] [4] [5]
X∗ [6]
0.18 0.20 0.40 0.60 0.80
0.048 0.050 0.058 0.059 0.058
0.53 7 1.08 31 4.93 640 6.88 1,084 7.93 1,141
0.280 0.269 0.192 0.153 0.132
0.26 0.52 2.09 2.34 2.33
– – – 3.92 4.97
Output Labor N All X3 [7] [8] [9]
Profit X2 Add’l X2 X1 Z area rent [10] [11] [12] [13] [14]
1 1 – 5 5 – 256 114 – 629 190 189 913 222 387
0 1 142 250 303
0.0 0.1 12 34 54
No No No Yes Yes
0.0 0.1 9 27 43
Note: α = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, s = 0.02, r = 0.01, h = 4,000, Ra = 9,600, δ = 1.0, C0 = 0. Calculations by author.
Finally, let us look at the price elasticity of demand facing the factory here. Suppose the factory were to reduce its f.o.b. exchange rate by 10%: from 0.2800 to 0.2520. To meet the increased demand, the factory would then have to set wb = 0.0544. The total amount demanded from the factory would then be 29,351 blocks of soap. For the factory, total revenue would now be 7,396, total cost would be 8,876, and net profit would drop to –1,479. The price elasticity of demand here is ((8,876 − 1,330)/1,330)/((0.2520 − 0.2800)/0.2800) = −210.7. At first glance, this may seem surprising. After all, if the price elasticity is so large (in absolute value), why wouldn’t the factory further lower its price? However, the trick here is to be mindful also of what is happening to cost. In our example, cost is simply the wage bill. To meet the increase in demand when it drops price, the factory has to raise its wage to attract sufficient labor. Total cost is now 8,876: up from 356 previously. Put differently, cost rises almost 24-fold, whereas revenue goes up only 19-fold; that is why profit drops when Pb is set 10% below 0.2800. Model 12D allows us to think about economic development. Important here is the role of k0 . As k0 is made larger, the firm is able to produce more output from a
12.9
Model 12D: The Factory as Bilateral Monopolist
371
given quantity of labor. In practice, k0 might increase because the firm is using more capital to produce its output or because the technology of the firm has improved.16 In what follows, I treat changes in k0 as comparative statics. I do not ask what the firm might do if it knew that its k0 was going to increase over time. Instead, I treat ko as exogenous and simply look at the setting of Pb and wb . What happens when k0 is larger? Table 12.13 shows the profit-maximizing combinations of exchange rate and wage as k0 is varied systematically from 0.18 to 0.80. Not surprisingly, as k0 is increased, the profit-maximizing Pb decreases. Perhaps more surprising is that wb at first increases and then decreases as k0 is increased. What causes Pb to decline so quickly at first and then almost level out? See Fig. 12.9. What does this have to do with the first up then down trajectory for wb ? Model 12D: F.o.b. price and wage set by factory AB Wage set by factory to maximize profit CD F.o.b. price set by firm to maximize profit
C Factory’s wage rate price (Wb)
B Factory’s f,o.b. price (Pb)
Fig. 12.9 Model 12D: f.o.b. price and wage setting by a profit-maximizing factory. Notes: ν = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, h = 4,000, Ra = 9,600, s = 0.02, C0 = 0, and C = 0.03. Horizontal axis scaled from 0.0 to 0.9; price axis scaled from 0.1 to 0.3; wage axis scaled from 0.045 to 0.065
A
0
D
Scale of factory production (k0)
How a factory sets price depends on both the price elasticity of demand for soap and elasticity of supply of labor. Here, demand elasticity varies from one consumer to the next for two principal reasons. As we saw in Model 12C, because the factory prices f.o.b., consumers pay freight, and hence consumer demand appears to be more sensitive to f.o.b. price for consumers closer to the factory than for consumers further away. In Table 12.13 we see that as k0 rises, the radius of the market (X∗ ) 16 An
extension to the models considered in this chapter would treat capital and technology, by both firm and farms, as endogenous. See, for example, Zhang (1993).
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increases which implies that we have more customers further away from the factory, and the aggregate demand facing the factory will become less price elastic from the factory’s perspective. The second reason is that to the extent that consumers are in the X2 or X3 area their demand will be less sensitive to price because they no longer have the ability to substitute farm-made soap for factory-made. In Table 12.13, I show what happens to Pb and wb as k0 rises from 0.18 to 0.80. First, when k0 is greater than about 0.4, Pb becomes low enough for an X3 area to emerge; as k0 rises further, the demand from the X2 and X3 areas becomes an evermore important share of its total demand. This implies that above k0 , about 0.4, the factory’s demand becomes less price elastic as k0 rises. Second, when k0 is smaller than about 0.4, the effect is opposite. There, as k0 is increased, the factory finds that demand in the X1 area (more elastic demand) is growing faster than demand in the X2 area (less elastic demand). The final column of Table 12.13 shows “Additional rent”: i.e., the aggregate rent paid by farms over and above (π X∗2 )Ra . Because we have assumed that landlords are absentee, the factory sees rent paid as a leakage from the local economy in that this takes away from the wheat being offered by farmers in exchange for soap. The total additional rent is negligible for k0 below about 0.20. However, by k0 = 0.80, the total additional rent is 43,000 units of wheat. As in Model 12C, additional rent is a substantial relative to the total profit earned by the factory. Once again here, there is an incentive for the factory to use discriminatory pricing to capture some of the additional rent. What about the comparative statics of Model 12D? Suppose a factory finds itself in the situation described in the bottom row of Table 12.13 and correspondingly sets its f.o.b. price at 0.132 and its wage a 0.058. Now, suppose this factory were to increase either its Pb or wb by 10%. What effects would such changes have on outcomes in its market area. In Fig. 12.10, I show the effect of each change on the wheat labor gradient. Under profit maximization, h1 [x] is given by ABCD in Fig. 12.10: kink B happens at the boundary between the X3 and X2 areas and Kink C happens at the boundary between X2 and X1. If wb is increased by 10%, the gradient shifts to EFCD. The shift in kink from B to F is because of the expansion of the X3 area, since factory work is now more attractive. Within the X3 area, farms everywhere reduce the amount of labor, h1 [x], devoted to wheat labor in order to supply more factory labor. Beyond the new X3 area boundary, the higher factory wage has no effect on the wheat labor gradient. If Pb is increased by 10%, the gradient shifts to GHIJ. The shift in kinks from B to H and C to I and in X∗ from D to J is because all three areas (X3, X2, and X1) shrink because factory soap is now less attractive. In the X3 area, farms reallocate labor in favor of wheat production; in the X1 area, farms put more labor into farm soap production to offset factory soap that is now less attractive. A similar story can be told about the impacts of a change in wb or Pb on the rents (see Fig. 12.11), land use (see Fig. 12.12), and net wheat production (see Fig. 12.13). What about the regional economy in Model 12D. Continuing with the same numerical example, let me start with column [1] of Table 12.14 where ko is just 0.20. The factory’s market area is small: just 1.1 km in radius or about 100 farms. The factory earns a profit here of just 0.1. The average farm pays a rent of 374.
12.10
Final Comments About This Chapter
373
Model 12D: Comparative statics of wheat labor (h1) ABCD Labor allocated by a farm to wheat at profit-maximizing solution EFCD Labor allocated by a farm to wheat if factory wage (wb) increased by 10% GHIJ Labor allocated by a farm to wheat if factory f.o.b. price (Pb) increased by 10%
H
F
I
C
Wheat labor (h1[x ] )
B
G A E 0
J
D
Distance from factory (x )
Fig. 12.10 Model 12D: comparative statics of wheat labor. Notes: ν = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, h = 4,000, Ra = 9,600, s = 0.02, C0 = 0, k0 = 0.80, δ = 1.0, r = 0.01, Pb = 0.1445, and wb = 0.0585. Horizontal axis scaled from 0 to 9; vertical axis scaled from 0 to 5,000. Viewed from left to right, kinks occur at boundaries of X3, X2, and X1 areas, respectively
Moving to the right in Table 12.14 gives basic information about the factory and the farms in its market area. In addition to transportation costs, there are two leakages from the regional economy here that depend on ko : one is factory profit and the other is the additional (scarcity) rents paid to landlords. Across these results, the scarcity rents in total (after multiplying the per farm additional rent by the number of farms) is on the order of 3/4 of the landlord profit. The two leakages become more important the larger is ko . Taken together as a proportion of aggregate wheat production in the factory’s market area, they rise from 0.2% of wheat production when ko = 0.18 to 3.0% when ko = 0.80.
12.10 Final Comments About This Chapter In this chapter, the principal model has been Model 12D. I included Models 12A through 12C to help readers better understand aspects of Model 12D. In Table 12.15, I summarize the assumptions that underlie Model 12A through 12D. Many assumptions are in common to all these models: see panel (a) of Table 12.15. The models differ in that (i) a farm is assumed to be within the factory’s market
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Model 12D: Comparative statics of rent (R)
D
ABC Rent per unit land by a farm at profit-maximizing solution DBC Rent per unit land by a farm if factory wage (wb) increased by 10% EF Rent per unit land by a farm if factory f.o.b. price (Pb) increased by 10%
Rent per unit land (R[x])
Fig. 12.11 Model 12D: comparative statics of land rent. Notes: ν = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, h = 4,000, Ra = 9,600, s = 0.02, C0 = 0, k0 = 0.80, δ = 1.0, r = 0.01, Pb = 0.1445, and wb = 0.0585. Horizontal axis scaled from 0 to 9; vertical axis scaled from 9,600 to 23,600. Viewed from left to right, kinks occur at boundaries of X3, X2, and X1 areas, respectively
12
A E
B 0 Distance from factory (x) F C
Model 12D: Comparative statics of land (L) ABCD Amount of land rented by a farm at profit-maximizing solution EFCD Amount of land rented by a farm if factory wage (wb) increased by 10% GHIJ Amount of land rented by a farm if factory f.o.b. price (Pb) increased by 10%
I C Land used by farm (L [x ])
Fig. 12.12 Model 12D: comparative statics of land use. Notes: ν = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, h = 4,000, Ra = 9,600, s = 0.02, C0 = 0, k0 = 0.80, δ = 1.0, r = 0.01, Pb = 0.1445, and wb = 0.0585. Horizontal axis scaled from 0 to 9; vertical axis scaled from 0.00 to 0.06. Viewed from left to right, kinks occur at boundaries of X3, X2, and X1 areas, respectively
H B F J
D
G A E 0
Distance from factory (x)
area in Models 12B through 12D, (ii) the monopolist’s f.o.b. price gets set endogenously in Model 12C, and (iii) the monopsonist gets to set both price and wage in Models 12D. What about Walrasian equilibrium and localization? • Model 12A has no shipping costs. It models one exchange albeit implicitly: the implicit exchange of wheat for soap. The supply of land is infinitely elastic at
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Net wheat produced by a farm (V1[x])
Model 12D: Comparative statics of net wheat production (V1) ABCD Net wheat production by a farm at profit-maximizing solution EFCD Net wheat production by a farm if factory wage (wb ) increased by 10% GHIJ Net wheat production by a farm if factory f.o.b. price (Pb ) increased by 10%
H B F
I C
J D
G A E 0
Distance from factory [x]
Fig. 12.13 Model 12D: comparative statics of net wheat production. Notes: ν = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, h = 4,000, Ra = 9,600, s = 0.02, C0 = 0, k0 = 0.80, δ = 1.0, r = 0.01, Pb = 0.1445, and wb = 0.0585. Horizontal axis scaled from 0 to 9; vertical axis scaled from 0 to 350. Viewed from left to right, kinks occur at boundaries of X3, X2, and X1 areas, respectively
the rent (Ra ). The amounts of labor and land used by a farm in autarky in wheat production are implicitly traded off against the alternative use of labor in soap production. There is no Walrasian simultaneity here. Prices are not jointly determined here: instead, Ra and other givens determine p∗ (the implicit exchange rate). • Model 12B has shipping costs. It models exchange in three markets: wheat/soap, labor (implicitly), and land. Overall, the supply of land is infinitely elastic at the opportunity cost (Ra ); however, shipping costs now mean that, in Models 12D through 12D, there will be rent differentials by location that preserve a utility equilibrium. What makes 12B different from 12A is that farms in the market area now exchange some wheat for factory soap. There is still no Walrasian simultaneity here. The rent gradient (R[x]) is now determined by the factory’s f.o.b. price (Pb ), the opportunity cost of land (Ra ), and other givens. • Model 12C endogenizes f.o.b. price (Pb ). Model 12C has three markets: wheat, labor (implicitly), and land. There is now a Walrasian simultaneity here. The locational rent (the amount by which rent exceeds Ra at a given location) and factory’s f.o.b. price (Pb ) are now jointly determined by Ra and other givens. • Model 12D endogenizes f.o.b. price (Pb ) and wage (wb ). Model 12D has three markets—wheat, labor, and land—all now explicit. Land may also be obtained,
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Table 12.14 Model 12D: farm outcomes as scalar (k0 ) is varied ko
0.20 [1]
0.30 [2]
0.40 [3]
0.50 [4]
0.60 [5]
0.70 [6]
0.80 [7]
Pb wb Revenue (000s) Cost (000s) Profit (000s) X∗ Total market area Total farms
0.2684 0.0500 1.8 1.7 0.1 1.1 4 100
0.2232 0.0557 21 17 3.5 3.4 36 909
0.1920 0.0583 49 37 12 4.9 76 1,924
0.1684 0.0594 77 55 23 6.1 117 2,938
0.1529 0.0594 96 62 34 6.9 149 3,706
0.1420 0.0589 109 65 44 7.4 174 4,299
0.1314 0.0585 121 67 54 8.0 199 4,920
Per Farm Land area Farm soap produced Wheat produced Factory wages Land rent Opportunity cost (Ra L) Additional rent Wheat spent on factory soap Wheat consumption Soap consumption Factory soap purchased
0.0389 150 624 16 374 374 1 19 247 218 68
0.0393 126 634 16 380 377 3 26 244 229 103
0.0397 105 643 16 386 381 5 32 242 238 133
0.0400 90 650 15 390 384 6 35 240 246 156
0.0402 82 655 14 393 386 7 37 239 252 170
0.0404 77 659 12 396 388 8 38 238 256 179
0.0405 73 662 11 397 389 9 39 237 261 188
Note: α = 0.2, b = 900, β = 0.2, γ = 0.6, c = 0.16, s = 0.02, r = 0.01, h = 4,000, Ra = 9,600, δ = 1.0, C0 = 0. Calculations by author.
subject to differentials in shipping cost and land rent. There is Walrasian simultaneity here. The locational rent, the factory’s f.o.b. price (Pb ), and the factory wage (wb ) are now jointly determined by Ra and other givens.
What about the regional economy? Model 12D is the final step in this book toward a comprehensive depiction of the regional economy. We can measure regional output and regional income. We are also able to divvy up regional income into a wage bill, a rent bill, and profits. In Model 12D, there is the possibility of a profit to factory soap production. For a farm, the only profit is the advantage that arises when it locates nearer the factory, and this is dissipated in locational rents. Who gains and who loses from improvement in productivity at the factory? The factory gains. So do the absentee landlords. However, farms are no better off after the improvement in factory productivity. The reason here is our assumption of free entry. In this chapter, the utility of a farm never changes. If the price of factory soap drops, the size of the market and the intensity of land use shift until utility is same as before. In effect, we are assuming an infinitely elastic supply of farms at a given level of utility.
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Table 12.15 Assumptions in Models 12A through 12D
(a) Assumptions in common A1 Closed regional market economy A4 Rectangular plane B3 All farms identical B4 Local (farmsite) demand for own wheat and soap C6 Identical individuals at each farm C7 Maximize same utility function I1 Leontief technology far in soap production: one input: labor I3 Cobb-Douglas production function for wheat: DRS J2 Nonzero opportunity cost (rent) for land J4 Amount of land used is endogenous J5 Land is input to production of wheat J6 Competitive market for land L4 Farm is a labor unit (b) Assumptions specific to particular models E3 Unit shipping cost is prohibitive A2 Barter market: wheat for soap E2 Fixed unit shipping cost H1 Factory supplies soap at given f.o.b. price H2 Location of factory is given M1 Factory sells soap at f.o.b. price B1 Endogenous market for soap D1 Factory is monopolist/monopsonist D6 Factory has a fixed cost L1 Endogenous market for factory labor L3 Allocation of labor between farm and factory work is endogenous
12A [1]
12B [2]
12C [3]
12D [4]
x x x x x x x x x x x x x
x x x x x x x x x x x x x
x x x x x x x x x x x x x
x x x x x x x x x x x x x
x x x x x
x x x x x x x x
x x x x x x x x x x
x
There is an interesting relationship here between land rent and factory behavior in Model 12D. As k0 increases, the factory becomes more productive, and this translates directly into higher profit for the factory. However, it also translates into higher land rents as farms bid to be closer to the factory. What about impact of transportation costs? Model 12D assumes both commuting and shipping costs. What would happen if both s and r were to approach zero? If s is zero, farms everywhere face the same effective price for factory soap. If r is zero, farms everywhere have a choice between two activities with constant returns to scale; factory labor and on-farm soap production. It would engage only in the activity with the higher marginal value product of labor. Land rent everywhere would be Ra , and no farm would produce soap itself. Each farm would purchase the same amount of factory soap (z), and the factory wage would be set so as to draw the appropriate amount of labor from the farm. The factory would then have an infinitely elastic supply of labor. In such a case, the factory becomes like a cooperative where workers gather to produce more efficiently than they could, each alone, at the farm.
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In developing Model 12D, I seek an argument here applicable to any local production: not just on farms. Put differently, I use the farm as a metaphor for the local economy in general. To me, farm and factory are key elements in a story about the effects of globalization on the space economy. I view a farm as a local production unit; it might consist for example of the farms in a district along with ancillary local services and commodities: e.g., construction, education, finance, health, insurance, real estate, religion, and retail. In turn, I view the soap factory as any globalizing economic activity that is seeking to establish itself in local economies by replacing a commodity or service that had been locally produced with a commodity or service now produced elsewhere. In the models of this chapter, I want to explore how the introduction of an efficient large-scale manufacturer reshapes prices locally and thereby the local economy within its reach. At the same time, there are limits to this analogy. First, if farm is indeed a community or region it is not clear how one might determine a utility function for such a unit. I am referring here to the wellknown problem that divergent individual preferences can mean to the existence of a social welfare function for the community as a whole; a problem sometimes labeled Arrow’s Impossibility Theorem: see Arrow (1965). Second, if the farm is an urban area, it presumably includes people who do no farming at all. What is the nature of land consumption in that case?
12.11 The Connecting Topics The objective of this book has been to reinterpret competitive location theory by focusing on the relationship between Walrasian price setting and the localization of firms. This area of inquiry is rich in insights and leads to important arguments about the nature of the Social Sciences in general, and of Economics and Geography in particular, as areas of intellectual inquiry. I have written each chapter to illustrate both the potential of the models and their limitations. In so doing, I hope that this provides readers with starting points for thinking about where and how these sets of models might be extended. In Chapter 1, I outlined eight sets of topics that connect models across the chapters. I now return to this list of topics to draw overall conclusions.
Localization, clustering, or geographic concentration Is there a correct or best way of thinking about localization? Alternatively, do differences in conceptualization across chapters suggest different dimensions to localization that make it useful to have a variety of models?
In Chapters 2, 3, 4, and 5, 11, and 12, localization is a condition in which a local market comes to be supplied by a firm or establishment at a remote location. To
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me, this is a Marshallian notion of localization. In Chapters 6 and 7, localization is akin to vertical integration: the location of a firm beside a supplier or customer firm. In Chapter 9, localization is the clustering of farms in markets to reduce risk. In my view, Chapters 6, 7, and 9 are aspects of a Coasian perspective: i.e., having to do with risk and organization. In Chapters 11 and 12, localization is two processes happening at once: a concentration of production at the city or factory seen above as a Marshallian perspective and a geographical rearrangement of customers, including their production, into homogeneous concentric rings brought about by competition among farms in the land market that is a distinctly Thünen perspective. Localization in Chapter 10, also looks like the latter of these. To me, the three perspectives— Marshall, Coase, and Thünen—are distinct and complementary ways of thinking about localization.
Shipping cost, congestion, and risk When we argue that risk is important in one model but not in another, what does this actually mean? Is there a way of thinking about these three concepts that enables us to better distinguish among the roles played by them?
Models in Chapters 2 through 6, 8, and 10 through 12 in general envisage that unit shipping costs include things like the differential cost of brokerage, insurance, and warranty service. To the extent that some of these involve risk, we cannot disentangle risk from shipping cost. In Chapter 7, unit shipping costs are replaced by travel delay or more properly the risk of delay. Here, we can see the effects of risk directly. Elsewhere in the book, risk is also incorporated in ways that are identifiable (i.e., not directly related to shipping cost): e.g., Model 2E and the models in Chapters 7 and 9. In general, it should be possible to separate risk from other aspects of shipping cost throughout competitive location theory. In my view, this is a task waiting to be completed. Separating out the effect of congestion from unit shipping cost is equally desirable, but more difficult. Let me explain why. The connection between congestion— which I define as a condition wherein the firm’s marginal cost curve or industry supply curve is upward sloped—and unit shipping cost is best exemplified in Chapters 3, 4, 5, and 7, Model 8D, Model 9C, and Chapters 10, 11, and 12. In Chapter 3, where the relationship is perhaps clearest, the firm trades off use of less efficient factories against shipping costs in fulfilling its customers’ demands: in Model 8D, this is recast in terms of the market area served by a factory. In Chapters 4 and 5, arbitrageurs exploit opportunities for trade by purchasing where costs are relatively low: that often means buying in more efficient places where there is less congestion in production. However, in practice, local supply curves mix congestion and unit shipping cost. Congestion appears in Chapter 7 in the form of diminishing returns to labor in production. In Model 9C, there is a curious kind of congestion
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that arises as farms cluster to participate efficiently in risky markets. In Chapters 10, 11, and 12, land rents get driven up because of congestion; farmers want to get closer to the geographic place at which agricultural products are demanded to minimize shipping costs. With the exception of Chapter 3, I have been unable to find a simple way in which to distinguish the effects of congestion from unit shipping cost in thinking about local supply curves.
Organization of firm and market In this book, several location models give us insights into the organization of the firm and market. What are these insights? How do they build on the work of Coase and others on the economics of organization?
In Chapters 2 and 3, the only aspect of firm organization considered here is whether it is cost-minimizing to build or utilize an additional factory. Indirectly, Chapter 6 looks at firm organization in the sense that localization (the clustering of a firm with a client or supplier) looks like vertical integration. Chapter 7 is more directly related to firm organization; it looks at the choice between in-house production and outsourcing. Chapter 12 looks at firm organization insofar as the farm, much as in Chapter 7, is trying to decide what to do in-house. Much work remains to be done here. Competitive location theory needs better and more general ways of modeling firm organization. In Chapters 2, 3, 4, and 5, market endogeneity arises in the sense that a change in shipments can imply a change in the areal extent of a market. Chapter 7 looks at market endogeneity in terms of multiple firms outsourcing. Chapter 8 introduces the idea of range and the notion of a market area. In Model 9B, the organization of farms into cooperatives (markets) can be seen as an extension of the farm that arises because the farm alone cannot be large enough to meet its own needs. Chapters 9 (Model 9C) and 10 introduce the idea that markets for commodities and land are simultaneous. Chapters 11 and 12 add simultaneity with a market for labor to this. Because of its emphasis on the land market as the vehicle through which alternative economic activities compete for production sites, Thünen models are a promising avenue for getting at the idea of Walrasian equilibrium across a range of markets.
Division of labor, indivisibilities, economies of scale, urbanization economies, agglomeration economies, and local cost advantage Are these concepts in fact descriptions of the same thing? If not, when, how, and why are the differences among them important in the models in this book?
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The models in this book envisage that cost savings originate in different ways. In Chapter 2, the firm has a fixed cost that can be spread over whatever quantity the firm produces. The firm trades that cost off against the savings from the elimination of shipping costs. In Chapter 3, the firm weights shipping cost against local variations in unit production costs in deciding how to allocate the limited capacities of its factories. Similarly, in Chapters 4 and 5, arbitrageurs take advantage of lower cost advantages locally where unit shipping cost is not prohibitive. In Chapter 6, the firm finds local cost advantages only because of the shipping costs of inputs or products that are not ubiquitous. Chapter 7 (risk reduction by outsourcing) and Chapter 9 (risk reduction by market formation) explore the urbanization economies made possible by application of an insurance principle. In Chapter 10, the only local cost advantage is a smaller shipping cost. While these origins are different, the effects that they have on the location of firms are similar; where price of a product is given, firms are attracted to sites with low costs of production and shipment. At the same time, the significance of these different origins lies in what they tell us about model predictions.
Commodity prices and Walrasian equilibrium What is the nature of Walrasian equilibrium in the presence of geography?
The early chapters in this book tell us nothing about the simultaneity of prices; at best, they tell us something about how other prices affect the price at which a firm or industry sells. In Chapters 2, 4, and 5, the only prices that are jointly determined are the prices of the commodity at different geographic places. Chapters 3 and 6 do not solve for market prices. However, the use of shadow prices (particularly, the opportunity costs of using the limited production capacities of its factories) in Chapter 3 has parallels with market prices. Chapter 7 explores the link between the price for a firm’s good and the price it is willing to pay for outsourcing. Chapter 8 looks at the definition of market areas and the setting of price by a firm in response to the prices of competitors within its range or trade area. Chapter 9 explores price setting across markets of various sizes based on risk aversion. However, the later chapters are different. Chapter 10 introduces a simultaneity between commodity price(s) and land rent. The Samuelson models in Chapter 11 extend this simultaneity to also include wages. Chapter 12 extends this still further by allowing us to think about the simultaneous determination of profits, land rents, and commodity prices in a regional economy. In these models, it is the sharing of factor markets—the fact that firms producing one commodity have to compete with other firms for labor, land, and capital—that ensures a Walrasian equilibrium across markets.
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Regional economy, social welfare, factor incomes, and autarky What does each of our models tell us about the impacts of shipments on the regional economy?
The early chapters of this book do not tell us much about the regional economy as a whole. Chapters 2 through 7 each (1) raise the possibility that we might usefully think of a regional economy as an endogenously determined set of places linked by shipments or other flows and (2) make it possible to think about autarky as a state where customers either supply for their own needs or are supplied only by adjacent factories but (3) provide no way of measuring factor incomes. Models in this chapters at best give us an idea of producer surplus and consumer surplus for one industry in the region. The later chapters are more helpful. Although Chapter 9 has the potential for us to sketch out a regional economy, the model starts with farm outputs given. It is only starting in Chapter 10 that we are able to begin thinking about farm product, land rents, and farm income. In Chapter 11, our regional income and product accounts can be extended to include wages. Finally, in Chapter 12, we are also able to incorporate profits and show a breakdown of regional income into wages, land rents, and profits.
Monopoly and space The diverse links between geographical space and a monopoly that is at least local in extent make it difficult to know when or how to apply basic economic ideas about the consequences of competition. How, and to what extent, do the models in this book allow us to do this?
Over the course of the book, we dip back and forth between monopolistic and competitive perspectives. Chapter 2 introduced the idea of a monopolist engaged in discriminatory pricing. To a monopolist, the difference in price between two customers may never be as large as the difference in unit shipping cost between them. Chapter 8 helps us think about the nature of a monopoly when there are competitors within the firm’s range or trade area. Chapter 3, being silent on price, tells us little about how a monopolist behaves in a spatial economy. The same is true of Chapters 6 and 7 in terms of the price of the commodity sold by the firm. However, Chapter 7 also gives us ideas about the monopolist contractor providing inputs to firms. Chapters 4 and 5 introduced the idea that, in competitive equilibrium, the difference in price between any two geographic places would not exceed the unit shipping cost
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between them. Chapter 9 is at odds with this topic because of a confounding of producers and consumers. Chapters 10 and 11 do not get us much further along on this topic because they assume competitive markets. It is not until Chapter 12 that we are able to reincorporate the impact of a local monopolist (the soap factory).
Geography How important is the specific geography chosen to the outcomes or conclusions of the model? What might have been the consequences of using other geographical representations? Why represent geography in the ways we do? In some models in this book, the modeling of geography was relatively unimportant. In Chapters 2 through 5 and 7, for example, it is possible to generate almost all results using unit shipping costs rather than unit shipping rates; in that sense, the results of the model depend only on the cost of shipment, not on distance directly, and therefore the models are independent of how distance is measured. Chapter 6 is the first point in the book where the way that distances get measured substantially affects model outcomes. However, even here, the conditions for localization depend only on map angles and not on map distances. The models in Chapters 8, 9, 10, 11, and 12 all rest on the idea that distance is measured as the crow flies and that shipping cost is proportional to distance. Conclusions in the second half of the book are more sensitive to the way in which distances are measured.
Appendix A Assumptions and Rationale for Localization
Assumption Block A: The Regional Economy A1
Closed regional market economy. Unless otherwise stated, no shipments of goods or payments into or out of the region with exception of all rent payments; where present, these are treated as payments to absentee landlords and therefore are leakages from the regional economy. Precludes arbitrage from outside the region that might otherwise supplement demand or supply within the region. In general, markets exist for the exchange of any pair of commodities, crops, or services. Unless otherwise specified, I assume a fiat money economy in which a commodity, crop, or service is exchanged for money. The regional economy includes a market each for labor, capital, and land. Unless otherwise indicated, all models assume that the firm is a price-taker in all three markets and can obtain the quantity (if any) it desires of each at the going price. Firms (including farms) produce and/or supply commodities/crops to markets. Unless otherwise stated, the organizational design and objective of firms is to maximize profit. All firms are assumed to be efficient. The cost of shipping is zero for shipments to local customers. For other customers, unless otherwise specified, the cost of shipping is unit shipping rate times distance times quantity. No economies of the large haul. Unless otherwise stated, inputs used by a firm are assumed to be ubiquitously available: i.e., at the same effective price everywhere. Households demand commodities and so too may firms. In so doing, the behavior of households conforms to either (1) an assumed demand curve, or (2) utility maximization. In some models, households may also supply labor. In some models, a firm may be able to affect the price received for its output (monopoly) or the price paid for an input (monopsony). Otherwise, all markets are assumed to be competitive. In a competitive market, each firm supplies
J.R. Miron, The Geography of Competition, DOI 10.1007/978-1-4419-5626-2, C Springer Science+Business Media, LLC 2010
385
386
Appendix A
the same commodity and is a price taker in that market. This industry is also characterized by free entry. Firms keep entering market until excess profit for marginal new firm entering industry is driven to zero. Relocation of firms is therefore assumed to be costless. Each market clears: an exchange rate is established such that for each commodity or crop the quantity demanded at that rate is equal to the quantity that suppliers have the incentive to supply. There is sufficient production capacity overall to meet overall demand at the going price. Commodities are perishable and cannot be carried over from one time period to the next. No one carries over money (e.g., through savings or debt) from one period to the next. Unless otherwise assumed, there is no uncertainty. A2 Barter market. There is a barter market in the exchange of two commodities. A3 Punctiform landscape. A simplifying assumption in which each producer and customer for a commodity can be thought to be a point—occupying no space— on the landscape. Regardless of the number of customers at one point and the number of producers at another, the distance between the two points—and hence unit shipping cost—is invariant. A4 Rectangular plane. Location of any point can be described by Cartesian coordinates. Travel is possible in any direction on this plane; distances are therefore Euclidean (as the crow flies). Where appropriate, land is available throughout the region and has the same yield when used in the same way. A5 Linear landscape. Economic activity occurs only as points along a line in geographic space: i.e., a one-dimensional spatial economy. A6 Ribbon landscape. A regional landscape in which farms are limited to a strip of land of unit (say 1 km) width. There is a cost of shipping along the length of the ribbon; any cost of shipping across the width of the ribbon is ignored (assumed zero). A7 Transportation network. Region modeled as (1) set of vertexes; (2) some pairs of vertexes are joined by a transportation link; (3) a route—combination of links—joins every possible pair of vertexes; (4) every customer place is a vertex; (5) every input supply place is a vertex; (6) factory may be sited at any vertex or along any link joining two vertexes.
Assumption Block B: Market for Commodity or Crop B1 Endogenous market. A market for the exchange of any pair of commodities, crops, or services wherein equilibrium outcomes (e.g., quantity transacted and price or exchange rate) are determined in the model.
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387
B2 Upwardly sloped local supply curve. A higher price is required to attract local producers to supply more of the commodity to the local market. Implies congestion. B3
All firms in industry identical.
B4
Local demand. There is only a local demand for the commodity.
Assumption Block C: Demand C1
Fixed demand locations. The firm faces fixed (price-insensitive) demands for its products from customers at one or more locations.
C2
Fixed local customers. The number of consumers in the local market is fixed.
C3
Fixed remote customers. The number of consumers in each remote market is fixed.
C4
Identical customers. Consumers are identical in terms of preferences and incomes. However, effective price may vary from among customers.
C5
Identical linear demand. For each industry being modeled, each customer has the same linear inverse demand curve.
C6
Identical individuals.
C7
Maximize same utility function. Customers maximize the same log-linear utility function defined over the consumption of two goods.
C8
Other demand curve. Each customer has the same individual inverse demand curve, but this is not linear as in Assumption C5.
C9
Customers uniformly distributed. The density of customers (customers per square kilometer) is the same in every part of the geographic region being modeled.
C10 Two kinds of customers. All customers are either of kind A or kind B. All customers of kind A are identical. All customers of kind B are identical.
Assumption Block D: The Enterprise D1 Monopolist and/or monopolist. Firm exploits a downward-sloping demand curve or an upward-sloped supply curve. D2 Firm minimizes cost of production and shipping. D3 No capacity exceeded. Firm does not produce any more at a factory than stated capacity.
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D4 Choice of factory location(s). Factory is free to choose how many factories or where to put them up. D5 I factories. Firm has a factory at each of I different sites, each producing the same commodity but some factories more efficient than others. D6 Fixed cost. Firm has a fixed cost. D7 Horizontal marginal cost curve. The firm’s marginal cost of production— inclusive of normal profit but before any reduction for shipping cost and land rent—is constant. Put differently, the firm’s marginal cost curve is a horizontal line. This implies that the firm has a wellspring of production from its fixed capital investment—that is, any amount of production without congestion. D8 Reverse-L marginal cost curve. The firm’s marginal cost curve is horizontal up to the stated factory capacity, then becomes vertical. Put differently, the firm is unable to produce any more than the stated capacity of the factory regardless of cost. See panel (b) in Fig 3.2.
Assumption Block E: Shipping Cost E1 Zero shipping cost everywhere. All shipping is costless regardless of the distance shipped. E2 Fixed unit shipping rate. The cost of shipping a unit from a given origin to a given destination is the same for every unit per kilometer shipped. No economies of the large haul; no economies of the long haul; no modal transfer costs; no fixed costs unrelated to length of shipment (e.g., brokerage or loading/unloading fees). E3 Unit shipping cost is prohibitive. Unit shipping cost is too high to permit any shipment of the commodity from producer to a remote customer. E4 Unit shipping costs symmetric. Unit shipping cost from point i to point j is same as unit shipping cost from j to i. E5 Firm bears shipping cost to market. Firm bears cost of shipping to all customers served.
Assumption Block F: Competition F1
Competitor sells same product. Customers cannot distinguish between product of the firm and that of its competitor.
F2
Competitor sells a different product.
F3
Competitor sells at same f.o.b. price. Firm and competitor sell at same price f.o.b.
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389
F4
Competitor sells at price that ignores competition. Competitor’s price is either exogenous to model or determined within model in a way that does not take into account firm’s price.
F5
Competitor sells at a different f.o.b. price.
F6
Firm is myopic. Firm chooses location and/or price on the assumption that competitor(s) will not respond by changing either their price or location.
F7
Location of competitor is given.
F8
Cooperative shares shipping costs among members.
F9
Peer competitor in remote market. Entrant sells identical product in remote market, from factory built there, with same costs as firm.
F10 Customers shared. Firm and competitor split the market if both have the same effective price there: otherwise, firm with lower effective price gets all customers at that location.
Assumption Block G: Uncertainty G1
Uncertainty implicit in model.
G2
Uncertainty explicit in model.
G3
Risk incorporated into utility function.
Assumption Block H: Location H1 Some suppliers price f.o.b. Where input is available only from one or selected points, suppliers use f.o.b. pricing and firm pays shipping to acquire input. H2
Location(s) of firm given.
Assumption Block I: Economies of Scale I1
Leontief technology: CRS. Cobb–Douglas production function exhibiting constant returns to scale.
I2
Cobb–Douglas production function: IRS. Cobb–Douglas production function exhibiting increasing returns to scale.
I3
Cobb–Douglas production function: DRS. Cobb–Douglas production function exhibiting decreasing returns to scale.
I4
Diminishing returns to labor (congestion).
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Appendix A
I5
Delay for equipment repair.
I6
Delay awaiting repair contractor.
I7
Choose in-house or outsourced repairs.
Assumption Block J: Land Use J1
Zero opportunity cost (rent) for land. As in a frontier setting, no other use for land other than those specified in model.
J2
Nonzero opportunity cost (rent) for land.
J3
Farm occupies fixed amount of land.
J4
Amount of land used endogenous.
J5
Land is input to production.
J6
Competitive market for land. All participants are price-takers. Land allocated to highest bidder.
Assumption Block K: Capital K1 Fixed capital requirements. Firm has fixed capital requirements. In this book, capital refers to the plant and equipment required by the firm to produce a commodity. Capital is durable; it is typically used to produce goods repeatedly over many periods of time. In principle, capital is a physical measure; however, we typically assume for simplicity that it can be represented in diverse ways as a money value; e.g., the purchase cost, the resale value, the replacement value, or the stream of future profits that can be earned from it. At the same time, we need be mindful that in competitive markets for capital goods, the money value associated with a unit of capital is itself a price determined by supply and demand factors.
Assumption Block L: Labor L1
Endogenous market for labor.
L2
Fixed total amount of labor in region.
L3
Allocation of labor between industries is endogenous.
L4
Enterprise is a unit of labor.
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Assumption Block M: Firm’s Pricing Policy M1 Firm uses f.o.b. pricing. Firm sets price at factory gate. Customer pays shipping cost. M2 Price discrimination. Firm sets delivered price in local and remote markets.
Rationale for Localization Z1 Presence of fixed cost. With larger output, firm spreads the fixed cost and lowers its unit costs. Z2 Implicit unit cost advantage at some locales. Unit (variable) production costs and/or unit shipping costs are lower at some locales than others. There are many possible reasons for such differences: e.g., differences in effective prices for inputs (e.g., labor, land, capital, or materials), the firm’s division of labor across establishments, or local regulation. Z3
Implicit unit price advantage at some locales.
Z4
Risk spreading and insurance.
Z5
Capacity constraints and congestion.
Z6
Differences among consumers.
Z7
Variations in goods.
Z8
Limitations of shipping cost or travel delay.
Z9
Shipping cost on non-ubiquitous inputs.
Appendix B Glossary
Alonso William Alonso (born 1933), an American regional scientist, published Location and Land Use: Toward a General Theory of Land Rent in 1964. Algorithm An iteration is a numerical procedure that improves upon an initial estimate of the solution to a problem. An algorithm is a repetition of iterations that converges on the solution. Arbitrage The purchase of commodities in markets where price is low for the purpose of resale in markets where price is high. As the crow flies
The Euclidean distance between two places on a plane.
Autarky A condition wherein regions do not engage in trade. Local supply in one geographic market is not available to meet local demand in another geographic market for the same product. Autarky arises when shipping costs are too high to permit arbitrage. Average cost produced.
For a firm given its level of output, total cost divided by the amount
Basing point pricing A pricing strategy wherein the monopolist chooses some customer places (so-called “basing points”), and sets a delivered price at each of them. Customers elsewhere then pay the unit shipping cost to ship home the commodity from the basing point. Barter market I use barter here in the economic sense of an exchange—a trade of some amount of one good in return solely for an amount of another good with no money involved—that takes place in the context of a market. This is seen here strictly as a matter of business; I exclude here any exchange (e.g., an exchange of gifts) where the motivation is, at least in part, something else. See also “numéraire.” Bernoulli process A Bernoulli trial is a statistical experiment, which can result in only one of two possible outcomes. An experiment consisting of a series of independent Bernoulli trials is called a Bernoulli process. Beta analysis Beta is the increase in mean (return) required to offset a unit increase in variance if two alternatives are to be thought to be equally preferable
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Capacity The amount of output that can be produced by a given factory over a stated period of time. In a simple case, we imagine that the marginal cost of production is fixed per unit for any quantity up to capacity (marginal cost of production is a horizontal line up to capacity). To allow for the concept that no quantity greater than capacity is possible, we typically assume the marginal cost curve then becomes vertical. In other words, no matter how much the firm might want to increase quantity above capacity, it simply cannot produce a quantity in excess of capacity. Cartesian coordinates Representation of a point in two-dimensional geographic space by a pair of distance coordinates: (x, y). An example of such a planar representation is given by Universal Transverse Mercator (UTM) coordinates which take the form of easting and northing coordinates. Cartesian coordinates are an approximation to location in that they ignore both elevation and the fact that the earth is a sphere. The advantage of Cartesian coordinates is that distance calculations are simplified compared to spherical coordinates. . Euclidean distance Cobb–Douglas production function A firm using two inputs to produce an output is said to have a Cobb–Douglas production function if the maximum level of output, Q, obtainable from q1 units of input 1 and q2 units of input 2 can be g expressed as Q = aqb1 q2 where a, b, and g are parameters. This firm has returns to scale that are constant if b + g = 1, decreasing If b + g < 1, and increasing if b + g > 1. Assuming competitive markets for inputs and output and constant or diminishing returns to scale, a profit-maximizing firm will spend the proportion b of its revenue on purchases of input 1 and g on purchases of input 2, the cross price elasticity will be zero, and the firm’s expansion path will be linear. Comparative cost analysis A technique for finding the least cost location for a firm by enumerating possible locations and then calculating the unit cost of production at each. Comparative statics A comparison of outcomes (endogenous values) predicted by a model when a given (exogenous variable or parameter) is changed by a small amount. Some models describe market equilibrium; here comparative statics details the changes in equilibrium when a given is changed by a small amount. In other cases, models describe optimal outcomes; here comparative statics details changes in optimal outcome when a given is changed. Competitive location theory Area of scholarship that looks at how competition among firms and households leads to geographic patterns in market equilibrium (otherwise known as spatial equilibrium). Complementary slackness theorem In a Linear Programming solution, each inequality constraint has both a slack (or surplus) and a shadow price. The Complementary Slackness Theorem says that at least one of the slack (or surplus) and shadow price must be zero for every constraint. Congestion A consequence arising in an economic landscape whereby the firm finds it relatively more costly or less profitable to marginally increase the quantity of
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a commodity that it supplies. Congestion may arise, for example, because of (1) limitations in the firm’s ability to manage a larger output, (2) limitations at the factory or in the supply chain (supplier, warehouse, mode transfer station, or transportation network), or (3) a deterioration in the firm’s profit from the response of competitors. Usually, congestion is measured over the short run: i.e., before the firm has the opportunity to adjust its investment in land, plant, and equipment. Conjectural variation In game theory, a player’s perception (assumption) about the reaction of another player to the first player’s choice of action. Consumer benefit For a market, the amount that consumers might have been willing to pay rather than go without the commodity; alternatively, the area under the demand curve to the left of the quantity demanded. Consumer surplus The area below the demand curve and above the market price to the left of market quantity. Put differently, Consumer Surplus is the amount (in dollars) that consumers would have been willing to pay over and above the market price rather than go without the product. Core theory As this book is focused on microeconomic applications, I take this to include neoclassical theory of consumer demand, theory of the firm, and welfare economics. Corner solution A condition in which the solution to an optimization problem Y = Max x [f(x)] occurs at a limiting value of x. Cournot Antoine Augustin Cournot (born 1801), a French economist and mathematician, published the original version of his Recherches sur les principes mathématiques de la théorie des richesses in 1883. Cross-price elasticity For any two commodities, labeled 1 and 2, respectively, cross price elasticity (c) is the percent change in quantity of commodity 1 demanded (q1 ) given a one percent change in the price of commodity 2(p2 ):c = (p2 /q1 )(dq1 /dp2 ). Delaunay triangulation
See Thiessen polygon
Delivered price Firm sets price for commodity delivered to customer; consumer does not pay a separate shipping charge. Where every consumer pays the same price, the firm engages in uniform pricing. Where each consumer pays potentially a different price, the profit-maximizing firm engages in discriminatory pricing. Demand cone A term characterizing the tendency for individual quantity demanded to fall off with increasing distance from a supplier pricing f.o.b. because of the rise in effective price. Diminishing marginal utility The idea that utility increases at a decreasing rate as a person consumes more of the commodity. In the twentieth century, economists in general moved away the notion of diminishing marginal utility of commodities
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because it was seen to be based on a cardinal measurement of utility. Instead, the ordinal assumption of a diminishing marginal rate of substitution between commodities was invoked. In this book, I retain the use of diminishing marginal utility because students find it helpful. .utility function. Discrete choice model A statistical model used to predict consumer choice from among a set of discrete and denumerable alternatives. This includes the Multinomial Logit Model and Nested Logit Model. Discriminatory pricing A pricing scheme used by a monopolist to enhance profit that results in different prices for different markets or submarkets, it is sometimes called “third-degree price discrimination.” Diseconomies of scale
See Economies of scale.
Division of labor First proposed by Adam Smith, this is the idea that the greater the output (scale) of a firm the better able it is to take advantage of the added productivity from having specialized labor. Domestic market
See Local Market.
Dual See Duality Theorem. Duality Theorem In Linear Programming, the Duality Theorem states that for every Primal linear program there exists another linear program called a Dual whose solution yields the shadow prices to the Primal. Dupuit Arsène Jules Etienne Dupuit (born 1804), a French engineer and economist, published an important set of articles on the economics of transportation between 1842 and 1865. Several of these papers are republished in De Bernardi (1933).). Ekelund & Hébert (1999 Economies of scale An attribute of a production function whereby the firm can produce more efficiently at a higher level of output. In the usual conceptualization, indivisibilities in production technology are thought to make possible a lower unit production cost possible when the firm achieves a particular scale of output. In some cases, a firm is thought to experience economies of scale through all relevant levels of output. In other cases, the firm is thought to have a most efficient scale of output (lowest possible unit production cost) above which the firm begins to experience diseconomies of scale. Economies and diseconomies of scale are generally measured over the long run: i.e., giving the firm sufficient time to adjust its investment in land, plant, and equipment. Marshall (pp. 278–279) saw three kinds of economies of scale: economy of skill (division of labor enabled by a larger scale of production), economy of machinery (indivisibilities enabled by a larger scale of production), and economy of materials (less wastage in a larger operation). Effective price Price paid by purchaser per unit of the commodity. It includes transaction costs paid by the purchaser related to search, negotiation, and acquisition (including shipping) with respect to the product.
Appendix B
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Efficient firm An efficient firm (1) incurs the least possible cost in achieving its desired output and (2) seeks the maximum revenue possible from that output. It adopts an organizational structure than enables it to be efficient. It has a production function which shows, for each combination of inputs, the maximum possible output that can be produced with those inputs. The firm also has a cost function which shows, for each level of output (Q), the minimum possible cost of achieving that Q. Finally, the firm knows the demand for its product and exploits that information along with the knowledge of its cost and production functions to maximize its own profit. See also “firm.” Endogenous In a model, an endogenous value is an outcome; a value predicted by the model based on other (endogenous) values. Endogenous variables may have a stochastic component. Establishment As commonly used in Censuses, the whole or part of a firm that is based, or carries on business, at a particular physical site (i.e., a contiguous set of facilities). Each firm can be partitioned into one or more establishments. Euclidean distance In two-dimensional space, the straight-line distance √ between two points with Cartesian coordinates (x1 , y1 ) and (x2 , y2 ) is given by ((x1 −x2 )2 + (y1 − y2 )2 ). See shortest path Excess demand In a market at a given price P, excess demand is the amount if any by which local demand exceeds local supply. Local here excludes demand or supply by arbitrageurs. Excess profit Any profit in excess of normal profit, that is, profit in excess of the return attributable to an unpriced factor of production such as entrepreneurial skill or owner equity. See also Ricardian rent and monopoly excess profit. Excess supply In a market at a given price P, excess supply is the amount if any by which local supply exceeds local demand. Local here excludes demand or supply by arbitrageurs. Exclusion Theorem In the Weber–Launhardt model, where input places and market places lie along a straight line, no location can be more efficient than one of these places. As a result of this theorem, we may use comparative cost analysis at each place to find the best location. See also Hakimi Theorem. Exogenous In a model, an exogenous value is a given; a value used in the model to predict other (endogenous) values. Usually, exogenous values include parameters, independent (or predictor) variables, and lagged or nearby values of endogenous variables. Usually, exogenous variables are not thought to be stochastic. Expansion path
See linear expansion path.
Expected value The expected value, E[X], for a discrete stochastic variable X is defined as E[X] = x xP[X = x] where x is a realization of X (i.e., a value that X
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can take on), P[X = x] is the probability of that realization, and the summation is over all possible realizations. Expendable An attribute of consumer demand for a product such that there is a price above which the consumer demands none of it. The opposite of an expendable commodity is an indispensable commodity: i.e., a commodity, which must be consumed in some quantity, however small, even when its price is high. Externality A consequence on one economic actor (e.g., a firm) arising from the behavior of another that is not priced. Also known as a spillover. Fetter Frank A. Fetter (born 1863), an American economist, authored “The economic law of market areas” published in the Quarterly Journal of Economics in 1924. Fiat money economy Any national economy in which the medium of exchange is a paper currency issued on behalf of the national government that does not obligate it to convert that currency into another store of value (such as gold) on demand. I assume here that the paper money supply is maintained by an authority (central bank) to ensure that the currency is a good store of value (or, equivalently, that the level of inflation is low). Firm A group of persons engaged—that is, organized by contract or fiat rather than market price mechanisms—in the production of a commodity (be it a commodity or service) for the purpose of earning a profit. In a competitive economy, profit arises because the firm incurs lower cost than is possible relying on market price mechanisms alone. Fixed coefficients technology
See Leontief technology.
Fixed cost A cost incurred by the firm for a period of operation that does not depend on the quantity of output produced. F.o.b. price Abbreviation for “free on board.” Firm sets price at factory, warehouse, or store; customer pays freight from that place. Also known as mill pricing. According to Marshall (1907, p. 325), the label “f.o.b.” arises from the practice of merchants to quote a price for their commodity on board a vessel in port, each purchaser then incurring any shipping cost from there. Free entry A market condition in which nothing prevents new firms from entering an industry other than considerations of profitability. If demand shifts so as to cause a rise in market price, new firms enter the industry until the price is driven down to the point where potential new firms find it is no longer profitable to get into that industry. In competitive location theory, the concept of free entry is often associated with the work of Lösch on imperfect competition in space. However, free entry is also integral to perfect competition in general and to the supply curve over the long run in particular.
Appendix B
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Freight In this book, freight refers to the cost of shipping a quantity of commodity from one place to another. For the most part, this means fuel, labor, maintenance, and other costs associated with the shipment as though the firm was undertaking its own shipping activity. Elsewhere, the book refers to shipping rates which presumably are the prices charged by a firm for the service of picking up a shipment at one place and delivering it to another. In this book, we do not distinguish between costs and prices associated with shipping. General equilibrium model A representation of an economy in which supply and demand curves are endogenous. Global net social welfare Net Social Welfare (See definition below) summed over all regions and net of shipping costs. Hakimi Theorem In the Weber–Launhardt model, on a transportation network defined as consisting of M vertexes (each vertex at least one of a customer location, a supplier location, or a place where two or more network segments intersect) with accompanying network segments, no location can be more efficient than one or more of the vertexes. As a result of this theorem, we may use comparative cost analysis at each vertex to find the best location. See also Exclusion Theorem. Half-freight A market outcome in which the prices at which a monopolist sells the same commodity in two markets differ by half the difference in shipping costs of shipping to the two markets. In general, this arises when consumers in the two markets are identical and have linear demand curves. Hitchcock Frank Lauren Hitchcock (born 1875), an American mathematician, published “The distribution of a product from several sources to numerous localities” in Journal of Mathematics and Physics 20 (1941), pp. 224–230. Hitchcock–Koopmans problem A Linear Program wherein the firm seeks to minimize the sum of the costs of production at a set of factories, subject to capacity constraints (one for each factory), and the shipments from these factories to meet the demand requirements of customers at each distinct place. Also known as the Transportation Linear Program, it was originally solved using a Stepping Stone Algorithm. In general, linear programs are solved using the Simplex Algorithm. Among others who also contributed in important ways to the Transportation Linear Program and its solution were Boldyre, Dantzig, Ford and Fulkerson, Kantorovich, Robinson, and Tolstoi. Home market See Local Market. Homotheticity See Linear Expansion Path. Hoover Edgar Malone Hoover (born 1907), an American economist, published Location Theory and the Shoe Leather Industries in 1937. Hotelling Harold Hotelling (born 1895), an American economist and statistician, published a seminal article on locational competition entitled “Stability in Competition” in the Economic Journal in 1929. See also Samuelson (1960).
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Income elasticity of demand For a given commodity, income elasticity of demand (i) is the percentage change in quantity of the commodity demanded (q) given a one percent change in the income (y) of the consumer: i = (y/q)dq/dy. Indispensable See expendable. Indivisibility An attribute of a production process such that production cannot be replicated at a smaller scale with the same efficiency. Industry marginal cost curve A schedule showing the marginal cost to an industry (usually over the longer run where capital invested adjusts as needed) as a function of the quantity to be supplied. The market supply curve is thought to be the same as the industry marginal cost curve. We can measure the industry marginal cost curve over either the short run (no additional competitors or factories) or the long run (additional competitors and/or factories possible). Instrument variable In a model to be solved as an optimization (i.e., which specifies an objective function to be optimized), an instrument variable is a given whose value is chosen for the purpose of that optimization. For example, given the problem Maxx Y = f (X), Y is the objective function and X is the instrument variable. Insurance principle With insurance, a consumer or firm incurs a small upfront cost (the premium) now to protect themselves against the possibility of a substantial, though unlikely, loss during some future period. What enables a market in insurance is the prospect of a profit by the insurer—that total payouts to the insured plus other costs of business do not exceed the revenue earned from premiums paid by consumers willingly insured. This is the insurance principle. In general, it requires that the risk of loss for each customer be small and that the occurrences of loss be statistically independent. Integrated market A condition of two places arising when the cost of shipping product from one market to the other is zero. Inverse demand curve A demand function is generally expressed as a schedule of quantity demanded (Q) at various prices (P): i.e., Q = f [P]. An inverse demand function rearranges this as the price consumers are willing to pay as a function of the quantity supplied to the market: i.e., P = f −1 [Q]. Inverse supply curve A supply function is generally expressed as a schedule of quantity supplied (Q) at various prices (P): i.e., Q = g[P]. An inverse demand function rearranges this as the price suppliers at the margin in order ensure that a given quantity is supplied to the market: i.e., P = g−1 [Q]. Insurance principle Any economic activity that has the effect of pooling risk across a group of economic actors so that Isolated market For this commodity, local producers and demanders transact in this and only this market. No one else transacts products there: e.g., no one purchases there for the purpose of reselling elsewhere or sells there a commodity purchased elsewhere.
Appendix B
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Kinked A condition wherein a function (e.g., demand schedule, supply schedule, excess supply, excess demand, or Price Difference Curve) takes the form of a polyline or piecewise-linear spline function: i.e., forms a continuous function with a discontinuous derivative. Koopmans Tjalling Charles Koopmans (born 1910), a Dutch-born American economist and Nobel Laureate in 1975, published “Optimum utilization of the transportation system” in D.H. Leavens (ed.), The Econometric Society Meeting (Washington, DC, September 6(18), 1947; Proceedings of the International Statistical Conferences, Volume V, 1948, pp. 136–146. Kuhn–Kuenne algorithm A numerical algorithm to solve the Weber–Launhardt factory location problem in two-dimensional space for any number of input and output places. See Algorithm Launhardt Wilhelm Launhardt (born 1832), a German engineer and economist, was an early advocate of mathematics in economics. His Mathematische Begründung der Volkswirtschaftslehre was published in 1885. An English translation of this book, Mathematical Foundations of Economics, was first published in 1993. His main economic contribution lies in founding location theory. See Backhaus (2000, 2002). Leontief technology A production technology (also known as a fixed coefficients technology) characterized by the fact that each unit of output produced requires exactly the same amount of an input regardless of the relative prices of inputs. In other words, there is no substitutability among inputs in production. It is named after Wassily Leontief, an American Economist and Nobel Laureate (1973), whose main area of research was input–output analysis. Linear expansion path 1. A condition of a production function wherein, if as firm expenditure (output) is increased by a fixed proportion holding prices of inputs constant, the efficient firm purchases the same proportion more of each input. 2. A condition of a utility function wherein, if as income is increased by a fixed proportion holding prices of commodities constant, the rational consumer purchases the same proportion more of each commodity. Put differently, each commodity has an income elasticity of +1.0. Such a utility function is also said to exhibit homotheticity. Linear programming The solving of optimization problems in which a linear objective function is maximized or minimized subject to linear inequality and nonnegativity constraints. Liquidity risk
See price risk.
Local demand At a place or submarket, the demand for a product by resident consumers for local consumption or by resident firms for local production. Local demand does not include demand by anyone in their capacity as an arbitrageur for resale to others.
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Local market A local market is a set of agents (suppliers and demanders) engaged in the sale and purchase of a commodity wherein market price—not a single price set in a global market—varies from one local market to the next because of some impediment (characterized by a unit shipping cost between markets). Where a mechanism links participants so that market price varies systematically from local market to local market (i.e., local markets are not in autarky), local markets can be termed submarkets. A domestic market or home market each instance a local market. Local supply At a place or submarket, the supply of a product by firms from local production. Local supply does not include supply offered by anyone in their capacity as an arbitrageur importing for resale. Localization economies Localization economies are reductions in unit production cost that arise when several firms in the same industry locate in close proximity. For some reason, having other firms in the same business in close proximity allows your firm to be more efficient. Log-linear utility function A log-linear utility function in two commodities takes where q1 the form ln [U] = aln[q1 ] + (1 − a) ln [q2 ] or equivalently U = qa1 q1−a 2 is the quantity of commodity 1 and q2 is the quantity of commodity 2. This utility function has the desirable properties that marginal utility is positive for each commodity and that there is diminishing marginal utility. At the same time, a log-linear utility function has the special properties that each commodity is indispensable, that the proportion of the consumer budget spent on each commodity is fixed (a for commodity 1, 1 – a for commodity 2), and that the cross- price elasticity of demand is zero (in other words, the demand for one commodity is independent of the price of the other commodity). See also utility function. Logistics The management of production, inventory, and shipments so as to enable a firm to achieve its objectives. Lösch August Lösch (born 1906), a German economist, worked on equilibrium in a spatial economy. His main contribution was Die raumliche Ordnung der Wirtschaft, published in German in 1939. An English translation of this book, The Economics of Location, was first published in 1954. Löschian Equilibrium In an industry where firms produce the same commodity And compete by choosing geographic locations, a long-run equilibrium wherein firms no longer have the incentive to enter or leave the industry. Every firm earns normal profit only. Marginal cost curve A schedule showing the marginal (additional) cost incurred by the firm (usually over the short run wherein capital invested is held constant) as a function of the quantity to be supplied. Marginal cost of production The increment to the firm’s total cost incurred by production of the last unit produced.
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Marginal factory For a firm with multiple factories or an industry composed of multiple firms each with its own factory, the marginal factory is the factory, which produces the last (most costly) unit to the market. Marginal shipping cost In the I = 2, J = 1 Weber–Launhardt location problem, there are four places on a two-dimensional map; the output place, the production place, and two input places. The marginal shipping cost is the extra profit per unit output that could be earned if only the production place were one kilometer closer to another place. Marginal shipping costs are w1 ,w2 , and wm . Marginal revenue The addition to the firm’s total revenue created by the last unit supplied by the firm to the market. If the firm’s demand curve is horizontal, it is a price-taker (i.e., in a perfectly competitive market), and therefore its marginal revenue is simply the price. On the other hand, if the firm’s demand curve is downward sloping, marginal revenue is the price of the last unit sold minus the revenue lost on all other units sold because the market price has now been reduced because of the marginal unit of product supplied. Market A market can be described as succinctly as a locus where buyers and sellers intersect. The purchase or sale of a commodity typically involves search activity on the part of both buyers and sellers; activity that is costly and time consuming. The classical notion of a market is a combination of institutions and mechanisms operating at a site at which (or portal through which) offerings of a commodity are on view and information about prices is readily available. In this way, markets are seen to facilitate the efficient exchange of the commodity. : isolated market; submarket. Market area The geographic area containing most of the customers of a firm (usually a store) and wherein its market share is highest. The market areas of two competing stores may overlap; the union of the market areas of all stores in a market is the trade area. See trade area. Marshall Alfred Marshall (born 1842), an English economist (and a founder of the neoclassical school in which economists study wealth and human behavior to understand why we make the choices we do), published the first edition of his Principles of Economics in 1890. Revised extensively over the years in a succession of eight editions, Principles of Economics postulated important arguments about implications of geography for the nature and pattern of economic activity. Maximum price In a linear inverse demand curve, maximum price is the Y-intercept: the price at and above which the quantity demanded is zero. Midpoint rule A method for numerically estimating the area under a curve, Y = f [x], between two values of x: x = a and x = b. The interval from a to b is divided into N panels of equal size, and a height, f [xi ], is calculated at midpoint of each panel i: i.e., xi = a + (i − 0.5)(b − a)/N for i = 1, 2, . . . , N. The area under the curve is then approximated by the sum of the areas of N rectangles, each of width
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(a − b)/N and height f [xi ] for that panel. This is known as the midpoint rule version of a Riemann sum. Mill pricing
See f.o.b. pricing.
Minimum price The lowest price at which a profit-maximizing firm would participate in the market. Revenue just covers the variable cost of production. Minimum price is not sustainable over the longer run because fixed costs of production are not covered. Model A simplified representation of the real world whose purpose is to describe, simulate, explain, predict, or control a phenomenon of interest. Although models can take a variety of forms from physical to chart to mathematical, I use the term in this book to mean only something that can be expressed mathematically. In this situation, a model includes values that are exogenously given (typically parameters and exogenous or lagged variables) and equations that link these to the values of endogenous variables. Monopolist Firm can affect the price it receives for its product by varying quantity it supplies to the market. In popular usage, a monopolist is sometimes thought to be the only producer of a commodity. That is not the case here. Instead, I envisage the firm to be (1) one of a small number of firms supplying a market and/or (2) producing a commodity differentiated from, but substitutable for, commodities produced by rival firms. Monopoly excess profit For the marginal firm, any profit in excess of normal profit arising from the firm’s ability to exploit a downward-sloping demand curve. See also Ricardian rent and excess profit. Myopic Any behavior of a firm such that it does not foresee any reaction by its competitors to its choices: e.g., with respect to price, quality of commodities sold, or geographic location. Net revenue
See Profit.
Net social welfare For a region, the economic well-being to society as a whole: measured either as (1) consumer surplus plus producer surplus, or (2) consumer benefit minus producer cost. See global net social welfare. No money illusion In a demand model, no money illusion means that price is relative to the units in which other money variables are measured. Put differently, if all money variables were to double, quantity demanded would be unchanged. Non-spatial A feature of a model wherein geography plays no role. Typically, shipping cost and/or commuting costs are assumed zero or are otherwise ignored. Non-tariff barrier A limitation, other than a tariff, such as a quota or other regulation that makes importing less attractive or feasible. Normal profit The return attributable to an unpriced factor of production such as entrepreneurial skill or owner equity.
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Numéraire In an exchange of two commodities, the numéraire is the commodity in whose units the value of the other commodity is expressed. In a fiat money economy, the numéraire is generally currency. In the bartering of, say, wheat for soap, if a unit of wheat is worth four units of soap, then soap is numéraire. The choice of numéraire is generally arbitrary; we could say that a unit of soap is worth 0.25 units of wheat, making wheat the numéraire. Oligopoly A market condition in which there are only a few vendors. No vendor has a monopoly, but none of the vendors is strictly a price-taker. Opportunity cost of capital The return on the best alternative investment opportunity available to the firm at a similar level of risk. Palander Tord Palander (born 1902), a Swedish economist, completed his PhD thesis, entitled Beiträge zur Standortstheorie (Contributions to Location Theory) at Stockholm University in 1935. Perfect competition An attribute of a market wherein each supplier and each demander is a price-taker. Perfectly elastic demand A condition of the demand curve facing a firm or an industry whereby consumers willingly demand any quantity at a given price but are unwilling to pay any more. Also known as a horizontal demand curve. Pickup or delivery pricing A pricing strategy wherein the firm sets two prices— an f.o.b. price and a delivered price—and allows the customer to choose between them. Place A geographic region sufficiently small in area that we can ignore shipping costs on shipments of the commodity within that region. Polyline A piecewise-linear spline function composed of two or more segments. Each segment (piece) is itself a linear function. The notion of a spline is that adjacent pieces share an endpoint: i.e., forms a continuous function with a discontinuous derivative. Predohl Andreas Predohl, born 1893, published “The theory of location and its relation to general economics’” in the Journal of Political Economy in 1928. Price Difference Curve In a two-region model of trade, the Price Difference Curve shows the amount of the commodity shipped from the lower priced to the higher priced region that would result in a given difference in prices between the two regions. Calculated as the vertical difference between the excess supply curve in the lower price region and the excess demand curve in the higher price region. In this text, I refer to this derivation as the Samuelson Model. Price discrimination
See discriminatory pricing.
Price elasticity of demand Price elasticity of demand (e) is the percentage change in quantity demanded (q) for associated with a one percent change in price (p):e = (p/q)(dq/dp). Since dq/dp is typically negative, most treatments take the absolute
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value of e. However, for the benefit of noneconomists, I leave the negative sign to remind the reader that when price goes up, we expect quantity demanded to drop. Price risk A loss (or increase in cost) arising because of an unforeseen change in market conditions that causes price to change over the short term; price risk is associated with price volatility. In a search-theoretic perspective. sellers hold an asset until the price bid by a potential purchaser exceeds the vendor’s reservation price. Here, a distinction can be drawn between price risk and liquidity risk. Liquidity risk is the loss arising because of the delay in obtaining a bid at or above the reservation price. In practice, it is difficult to distinguish between price risk and liquidity risk. The approach in this book is to treat liquidity risk as simply an element of price risk. Price taker A condition under which a market participant (supplier or demander) is unable to affect the price they receive or pay for a unit of the product by varying the quantity that they supply or demand. The supplier (demander) sees the demand (supply) for its product as horizontal: i.e., infinitely elastic at the given market price. Primal In Linear Programming, a problem in general cast as a maximization subject to less-than-or-equal-to and nonnegativity constraints. See Dual. Producer cost For a market, the aggregate cost incurred by suppliers; alternatively, the area under the supply curve to the left of the quantity supplied. Producer surplus The area under the market price and above the supply curve to the left of the market equilibrium. In common parlance, producer surplus is the amount (in dollars) received by efficient producers over and above what is needed to secure the participation of the marginal producer in the market. Profit The amount by which a firm’s revenue for a period exceeds its costs inclusive of a normal return on any unpriced factors such as owner equity or management skill. Also known as net revenue or excess profit. Punctiform A commonly used abstraction of geographic space wherein economic activity is clustered at distinct geographic points (places). Put differently, economic activities themselves do not use land or otherwise occupy space. In this abstraction, shipping between activities located at the same place incurs a negligible (zero) cost; shipping between activities at two different places incurs a nonzero cost that is invariant with respect to the number or volume of economic activities there. Pythagorean formula The square of the length of the hypotenuse of a right angle triangle in two-dimensional space is the sum of the squares of the other two sides. Quadratic programming The solving of optimization problems in which a quadratic objective function is to be maximized or minimized subject to linear inequality and nonnegativity constraints. Range For an expendable commodity sold at an f.o.b. price, the distance at which shipping cost is sufficiently high to cause demand to drop to zero.
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Rational By rational, I would like to mean simply that an economic actor makes choices consistently:). Becker (1962). A consumer is deemed to be rational if, in choosing between alternatives, he or she exhibits preferences among these alternatives that are (1) well-ordered and (2) stable over time. Here well-ordered means that if the consumer prefers alternative A to B and also prefers B to C, then that consumer will prefer A to C. The phrase “stable over time” is to suggest that if the consumer preferred A to B yesterday, then other things being equal, he or she will still prefer A to B today. However, economists typically impose additional constraints on rationality, including diminishing marginal utility, limitations on separability, and notions of expected utility. For an interesting discussion of the nature and paradox of rational choice, see Sugden (1991). Rectangular plane In this book, a representation of the world as though it were simply a two-dimensional surface. Ignored here, for simplicity of exposition, are (1) spherical properties of a globe and (2) differences in elevation. On a rectangular plane, a place can be represented simply by a pair of Cartesian coordinates. Reductionism An approach to understanding which focuses on an aspect of the process under study so as to better analyze the problem. In that way, we hope to better our understanding of the process under study: i.e., our ability to describe, simulate, explain, predict, or control it. Relativist A relativist sees explanations simply as bettering current understanding. Ricardian Rent An excess profit that arises because of an asset or market situation unique to a firm that prevents competitors from entering the market and/or earning the same profit. Samuelson Paul Anthony Samuelson (born 1915), an American economist and Nobel Laureate in 1970, published “Spatial price equilibrium and linear programming” in the American Economic Review in 1952. Samuelson conjecture Samuelson conjectured that if trade was to be desirable, it had to improve well-being across regions. He proposed that global net social welfare (GNSW) be measured as the sum of net social welfare for all regions less the shipping costs (treated here as a deadweight loss. Semi-net revenue For a firm, revenue minus variable cost. In this book, variable cost includes both production and shipment. The firm’s net revenue (profit) is seminet revenue minus fixed cost. Shadow price A standard Linear Programming problem maximizes an objective function linear in endogenous variables subject to linear constraints and nonnegativity constraints on the endogenous variables. A shadow price, one for each linear constraint, is the amount by which the value of the objective function could be increased if only the constraint were relaxed by one unit (made one unit less binding).
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Shortest path a. On a two-dimensional plane, the Euclidean distance between two points. b. On a network, the path from one vertex to another that is the shortest. Slack In Linear Programming, the amount by which the left-hand side of a lessthan-or-equal-to inequality is less than the right-hand side. See Surplus. Social Welfare A measure of the benefit arising from a competitive market calculated as either (1) the sum of consumer surplus and producer surplus or (2) the amount by which Consumer Benefit exceeds Producer Cost. Spatial equilibrium An equilibrium in which market participants at different locations have no incentive to change the way they participate in a market: e.g., their location, type of commodity produced, price, or quantity produced. Spatial price equilibrium In the case of multiple places, a condition in which arbitrageurs have no further incentive to purchase in a low-price market for resale in a high-price market. Spillover
See Externality.
Submarket A market for identical, or similar, commodities is said to be formed of submarkets when prices in the submarkets differ but are linked in some respect. In a strong version of submarkets, the price in one submarket is a fixed premium on the price in another submarket. In a weak version, a rise in price in one submarket causes the price in the other submarket to change, but there is no fixed premium. Supply step function A supply schedule that takes the form of ascending horizontal steps. Arises when, for example, the firm has a number of factories of differing efficiency each of which has a constant unit production cost. Surplus In Linear Programming, the amount by which the left-hand side of a greater-than-or-equal-to inequality is larger than the right-hand side. See Slack. Symmetric shipping cost A feature of shipping rates such that the cost of shipping a unit of product from Place i to Place j is the same as the cost of shipping a unit from j to i. Tariff A tax charged on an imported commodity: often ad-valorem (i.e., a percentage of the value of the commodity). Tariff jumping A business strategy in which a firm constructs a factory at a foreign site (also known as foreign direct investment) to avoid tariffs, quotas or other restrictions that would otherwise apply on the importation of its commodities into that country. Theory-model dissonance A mathematical model typically consists of givens (exogenous values and parameters), relationships (behavioral and identity), and outcomes (endogenous). Implicit here is a causality; givens determine outcomes (and not vice versa). Implicit in both the givens and the relationships is an underlying theory. The theory-model conundrum arises because there are two ways to think about the theory here. One is that the theory identifies parameters and exogenous
Appendix B
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values and provides the basis for the relationships used to solve for the endogenous variables. The other is that the theory is a set of deductions arising from a set of assumptions. The conundrum is that it is not easy to move back and forth between these two ways of viewing theory. It would be nice, for example, if assumptions corresponded to givens and deductions to relationships. However, there is no particular reason to expect this. This makes for some interpretation of models; we cannot just look at a mathematical structure. Thick A market condition in which there are many buyers and sellers. From a search-theoretic perspective on markets a seller in a thick market does not have to wait long to get a fair price for their commodity. Thiessen Alfred Henry Thiessen (born 1872), an American climatologist, defined polygons around individual rainfall stations to estimate total rainfall across a region. Thiessen polygon A map polygon formed on a rectangular plane by constructing perpendicular bisectors to the straight lines joining a place (typically a store) to similar places nearby. The partitioning of a map in this way is called a Voronoi Diagram. The map of the lines for which perpendicular bisectors are drawn is called the Delaunay triangulation. Thin A market condition in which there are few buyers and sellers. From a searchtheoretic perspective on markets a seller in a thin market typically must wait longer to get a fair price for their commodity. Thünen Johann Heinrich von Thünen (born 1783), a German economist, published the original version of The Isolated State in 1826. An English translation of this book, Isolated State: An English Edition of Der Isolierte Staat, was first published in 1966. Total cost For a firm, the sum of variable and fixed costs of production inclusive of any unpriced resources such as entrepreneurial talent. Trade area The geographic area covered by a market in which a firm (often, a store) participates. The trade area is thought to be composed of market areas for each store in the market. See market area. Transportation Linear Program
See Hitchcock–Koopmans problem.
Ubiquitous A feature of an economic landscape wherein some input (factor of production) is available everywhere at the same price. Uniform pricing The practice of a monopolist by which the same price is set for different customer places even though the marginal cost of serving a customer differs from one place to the next. Unit shipping cost
See unit transaction cost.
Unit shipping rate
Unit transaction cost per kilometer shipped.
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Unit transaction cost Cost incurred by buyer related to search and information gathering, negotiation, and acquisition (including freight and transfer, storage and inventory, agency and brokerage fees, credit, cost of insurance and other loss risks, installation and removal, warranty and service, and taxes and tariffs) per unit shipped (purchased) inclusive of normal profit (the profit attributable to an unpriced factor of production such as entrepreneurial skill or owner equity). Unit transaction cost may vary seasonally. Throughout this book, this is termed unit shipping cost. Urbanization economies Urbanization economies are reductions in unit production cost made possible when firms in different industries locate in close proximity. For some reason, having a diversity of firms in other businesses nearby makes your firm more efficient. Utility function An ordered (i.e., ordinal) scoring of choices (bundles of commodities whose consumption is desirable) that reflect consumer preferences and that are transitive and evidence diminishing marginal utility. As a ranking, a given utility function is said to be unique up to a monotonic transformation. For example, the utility functions f [x, y] = xb y1−b where 1 < a < 0 and g[x, y] = axb yc where b > 0, c > 0, and b + c < 1, calculated at consumption of x units of commodity 1 and y units of commodity 2, generate the same rank ordering: i.e., g[x, y] is a monotonic transformation of f [x, y]. The two utility functions above exhibit diminishing marginal utility. Also see “Rational.” Variable cost A cost incurred by the firm for a period of operation that varies with the quantity of output produced. Von Thünen
See Thünen.
Voronoi Georgy Fedoseevich Voronoi (born 1868), a Russian mathematician worked on polygonal partitioning of a two-dimensional plane. This follows on earlier work by Rene Descartes (1644) and Johann Peter Gustav Lejeune Dirchlet (1850). Voronoi diagram
See Thiessen polygon.
Walras Marie-Ésprit Léon Walras (born 1834), a French (Swiss) economist, first published his Élements d’économie politique pure in 1874. Weber Alfred Weber (born 1868), a German economist, published über den Standort der Industrie in 1909. An English translation of this book, Theory of the Location of Industries, was first published in 1929. Weber–Launhardt map triangle In the I = 2, J = 1 version of the Weber– Launhardt Model, the triangle formed when three map points on a rectangular plane—each with a pair of Cartesian coordinates—are drawn on a map. Weber–Launhardt weight triangle In the I = 2, J = 1 version of the Weber– Launhardt Model, the “equilibrium of forces” triangle created using the weights w1 , w2 , and wm . See Marginal Shipping Cost.
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First Author Index
A Adams, R. M., 142 Aguirre, I., 62 Aitken, S. C., 27 Aji, M. A., 2 Alao, N., 13, 20 Alba, R. D., 27 Alchian, A. A., 21, 37 Alderighi, M., 208 Alexandersson, G., 3 Alonso, W., 13, 260 Alperovich, G., 23, 166 Anas, A., 17 Anderson, S. P., 15, 53, 108, 203 Appa, G. M., 70 Aranya, R., 17 Arnott, R., 13, 17 Arrow, K. J., 9, 11, 128, 378 Arsham, H., 82 Asami, Y., 23, 154 Atwood, R. S., 3 Azar, O. H., 44 B Backhaus, U., 12, 60, 282 Bacon, R. W., 201, 207 Baker, C. B., 127 Baldwin, R., 16 Banerjee, A. V., 301 Barnes, F. A., 3 Barnes, T. J., xi Basu, K., 203 Batty, M., 13, 202 Baulch, B., 143–144 Baumol, W. J., 1, 11, 62, 100, 182 Becker, G. S., 23 Beckmann, M. J., 13, 18, 71, 301–302, 306 Belderbos, R., 60 Belleflamme, P., 3
Benassi, C., 203 Bennett, R. J., 3 Benson, B. L., 52 Berliant, M., 13 Bernoulli, D., 15, 238 Berry, B. J. L., 13, 227 Bester, H., 203 Bhagwati, J. N., 301 Bigman, D., 6 Birkett, M. S., 3 Bishop, R. L., 18, 238 Black, D., 178 Blackorby, C., 129 Blaug, M., 13–14 Blomqvist, A. G., 301 Bobst, B. W., 23 Bodington, C. E., 18 Boots, B. N., 13, 202, 214 Borts, G. H., 13, 26 Boventer, E. von, 86 Bowles, S., 1, 11 Bowring, J., 301 Boyd, R., 127, 144, 146–147 Boyer, M., 203 Bradfield, M., 166 Braid, R. M., 203 Brand, C. J., 3 Brander, J. A., 60 Breedlove, L. B., 3 Bressler, R. G., 13 Brown, S. E., 227 Brueckner, J. K., 301 Buckley, P. J., 27 Bukenya, J. O., 22, 144 Bullard, C., 70 Bunge, W., 13 Buongiorno, J., 144 Burns, M. E., 129 Byers, J. A., 214
445
446 C Cai, D. H., 301 Calvo, G. A., 301 Camacho, A., 233 Capozza, D. R., 13, 15, 203 Carlson, A. S., 3 Carrothers, G. A. P., 227 Casella, A., 233 Caves, R. E., 27 Chamberlin, E. H., 17 Charnes, A., 17, 82–83 Chaudhuri, S., 301 Chipman, J. S., 129 Chisholm, M., 13, 15, 21, 73 Clark, C., 270–271 Clark, J. B., 16 Clark, W. A. V., 27 Coase, R. H., 11, 16, 31 Coelho, P. R. P., 23 Combes, P. P., 9, 13 Cook, G. A. S., 3 Cooley, C.H., 16 Couclelis, H., 27 Cournot, A. A., 12, 14–16, 28, 97–124 Cox, K. R., 27 Cromley, R. G., 23, 72, 154, 275 Cuddington, J. T., 301 Cunha, C. B., 70 Currie, J. M., 135 Curry, L., 2, 4, 13–14, 27, 321, 337 D Daitoh, I., 301 Daly, M. T., 13, 202 Damania, D., 203 Dana, J. D. Jr., 197 Daniel, T. E., 127 Dantzig, G. B., 17–18, 82 Davis, D. R., 15 Davis, P., 227 Day, R. H., 18 Dean, R. D., 23, 154 De Bernardi, M., 12 DeCanio, S. J., 15, 52 De Felice, F., 6 De Fraja, G., 48 de Frutos, M. A., 203 De Geer, S., 12 Dempsey, B. W., 271 Deneckere, R., 153 Dennis, K., xiv de Palma, A., 203, 227 De Smith, M. J., 6
First Author Index Deutsch, A., 15, 52 De Vany, A., 144 De Vroey, M., 19 Devletoglou, N. E., 203 Dicken, P., 7, 13 Dickinson, H. D., 271–272 Dorfman, R., 17 Dorta-González, P., 203 Downward, P., 21 Dudey, M., 227 Dunn, E. S. Jr., 13, 272 Dunning, J. H., 27 Dupuit, 12 Duranton, G., 9, 13, 27 E Eaton, B. C., 13, 20, 62, 203 Economides, N., 13, 203, 231–266 Edgeworth, F. Y., xiv Edwards, W., 301 Eiselt, H. A., 6 Ekelund, R. B. Jr., 12, 20 Ellison, J., 161 Ellsberg, D., xvi, 246 Emerson, D. L., 166 England, K. V. L., 21 Enke, S., xi, 11, 97–124, 126, 142 Epping, G. M., 14, 202 Erickson, R. A., 14, 202 Eswaran, M., 166, 201 F Faminow, M. D., 52 Fan, W., 16 Fehr, E., 19 Feldman, M. P., 10 Feldman, S. L., 157 Fernandez Lopez, M., 18 Fetter, F. A., 13, 201 Fields, G. S., 301 Fik, T. J., 14 Fisher, I., xi Fjeldsted, B. L., 70 Flaam, S. D., 19 Florian, M., 126, 136 Ford, L. R., 70, 82 Fotheringham, A. S., 14, 202 Fox, K. A., 127 Francis, R. L., 6, 27 Friedman, M., xii Friesz, T. L., 126 Fujita, M., 13, 16, 20, 357 Furlong, W. J., 15, 52 Furtran, W. H., 142
First Author Index
447
G Gabriel, S. A., 142, 239 Gabszewicz, J. J., 13, 203 Gale, D., 83 Gale, S., 27 Gannon, C. A., 203 Garrison, W. L., 7, 14, 18, 20, 273 Gass, S. I., 82 Gee, J. M. A., 53 Gehrig, T., 233 Georgescu-Roegen, N., 239 Ghatak, S., 301 Ghosh, A., 6, 13–14, 27, 202, 205 Giarratani, F., 3 Giersch, H., 17 Gillen, W. J., 202 Gilligan, T. W., 15, 52, 144 Glaeser, E. L., 9 Glazer, A., 233 Goettler, R., 203 Gold, J. R., 27 Golledge, R. G., 14, 27, 202 Gorman, W. M., 23 Gould, P. R., 27 Graham, R., 214 Gras, N. S. B., 17 Greenhut, M. L., 13, 15, 18, 24, 29–67, 201 Green, M. B., 70 Grether, E. T., 26 Griffith, D. A., 14 Güder, F., 126, 136 Guise, J. W. B., 127 Gupta, B., 203, 217
Hart, A. B., 17 Hart, P. W. E., 27 Hartshorne, R., 14 Hartwick, J. M., 201, 318 Harvey, D. W., 27 Hay, D. A., 203 Head, K., 3, 15 Heckscher, E., 3 Henderson, J. M., 18, 70 Henderson, J. V., 13 Henkel, J., 233 Herbert, J. D., 13 Hicks, J. R., 129 Hitchcock, F. L., 13, 69–95 Hoare, A. G., 301 Hohenbalken, B., 62, 227 Hoover, E. M., 13 Hornby, J. M., 301 Horn, H., 60 Hotelling, H., 203 Houthakker, H. S., 82 Hsu, S. K., 23, 43, 154, 166, 296 Huang, T., 203 Huck, S., 203 Huff, D. L., 14, 202, 227 Hughes, J. W., 15, 52 Hummels, D., 296 Huntingdon, E., 4 Huriot, J. M., 13 Hurter, A. P., 6, 166, 203 Hurwicz, L., 11, 18, 238 Hwang, H., 62, 155, 169, 170, 219 Hyson, C. D., 201–202 Hyson, W. P., 201–202
H Haddock, D. D., 15, 50, 52 Hadley, G., 18 Haggett, P., 27 Haig, R. M., 17 Hall, R. W., 82, 220 Hamacher, H. W., 6 Hamilton, J. H., 203 Hammond, S., 3 Hamoudi, H., 203 Handler, G. Y., 174 Hanemann, W. M., 129 Hanson, G. H., 15, 17 Harker, P. T., 126, 136 Harley, C. K., 296 Harris, C. C. Jr., 13, 17, 70 Harris, J. R., 301 Harris, R. G., 296 Harsanyi, J. C., 18, 238–239
I Inaba, F. S., 147 Irmen, A., 203 Isard, W., 11, 13, 18, 23, 154, 157, 203 Ishikawa, T., 28 Islam, N., 301 Iyer, G., 203, 227 J Jacobs, J., 9 Javorcik, B. S., 9 Jayet, H., 249 Jevons, H. S., 11, 14 Johansson, B., 10, 13 John, F., 18 Johnson, A. S., 272 Jones, D. W., 14 Jones, E. D., 17 Jorgenson, D. W., 301
448 Judge, G. G., 18, 114, 117, 125–149 Jung, C., 144 Justman, M., 8, 37 K Kahn, A. B., 82 Kahn, J. A., 182 Kakutani, S., 70, 126 Kantorovich, L., 70 Keir, M., 10, 12 Kellerman, A., 14 Keller, W., 11 Kelley, A. C., 301 Kennedy, M., 142 Khalili, A., 166 Khan, M. A., 301 Kilkenny, M., 166 Kim, S., 3 Koo, A. V. C., 18, 238 Koopmans, T. C., 12–13, 70 Koropeckyj, I. S., 157 Kroll, Y., 239 Krugman, P., 13, 357 Krugman, P. R., 7, 16 Krumme, G., 27 Krzyzanowski, W., 13 Kuhn, H. W., 18, 83, 126 Kusumoto, S. I., 166 Kuznar, L. A., 81 L Labys, W. C., 13, 22, 144 Lahr, M. L., 18 Lai, F. C., 203 Lambertini. L., 203 Lampard, E. E., 17 Landon, C., 3 Landon-Lane, J. S., 301 Larch, M., 37 Launhardt, W., 12, 15, 151–176, 282 Lawrence, O. L., 3 Lederer, P. J., 203 Lee, C. I., 301 Lee, D. B., 13 Lee, M. L., 301 Lefeber, L., 18 Leigh, A. H., 271 Lemke, C. E., 126 Lenstra, J. K., 18 Lentnek, B., 14, 177–199, 202 Leontief, W. W., 17 Lerner, A. P., 203 Le, T., 301 Levy, D. T., 52
First Author Index Levy, H., 239 Lewis, W. A., 301 Lindberg, O., 157 Lo, L., 14, 202, 224, 227 Long, W. H., 207 Loomes, G., 18, 238 Lord, R. A., 52 Lösch, A., 13, 203 Louveaux, F., 15, 173–174 Luna, H. P. L., 135 M Macdonell, J., 16, 296 Machlup, F., 52 MacKinnon, J. G., 126 Magnanti, T. L., 94 Mai, C. C., 23, 154–155, 166, 170, 219 Maki, U., 20 Manne, A. S., 29–67 Marchionatti, R., 10 Marcotte, P., 126–127 Mark, D. M., 27 Markowitz, H. M., 239 Markusen, J. R., 13, 27, 29 Marschak, J., 18, 238 Marschak, T., 11 Marshall, A., 272, 336 Martins, A. P., 301 Martin, W., 130 Massey, D., 17 Mathur, V. K., 13, 23, 154, 166 Matsumura, T., 203 Mayer, T., 13 McCann, P., 14–15, 20, 27, 153, 178 McCann, R. J., 71 McChesney, F. S., 50 McCool, T., 301 McGuire, M., 249 McLafferty, S. L., 6, 14, 202, 205 McLaren, J., 193 Meardon, S. J., 13, 16 Megdal, S. B., 37 Miles, R. E., 214 Miller, H. J., 202, 227 Miller, S. M., 166 Mills, E. S., 13, 203 Minta, S. C., 27 Mirchandani, P., 94 Mirchandani, P. B., 6 Miron, J. R., 1, 14, 202, 227, 336 Mirowski, P., 10 Moene, K. O., 301 Mogus, A., 144
First Author Index Montgomery, C. A., 142 Morehouse, N. F., xi Moses, L. N., 13, 153, 166 Mosteller, F., x, xi Motta, M., 60 Mu, L., 14, 202 Mulligan, G. F., 14–15, 202 Murata, K., 269 Murphy, J. A., 135 Myrdal, G., 13 N Nachum, L., 3 Neary, J. P., 16, 60, 301 Needham, D., 15, 52 Nerlove. M. L., 20, 301, 327, 329–330, 336 Netz, J. S., 6 Neven, D., 60 Nickel, S., 6, 13 Nijkamp, P., 14, 18 Norback, P. J., 60 North, D., 296 North, D. C., 13 Nourse, H. O., 26 O Ohlin, B., 3, 13 Ohta, H., 201 Ó hUallacháin, B., 178 Okabe, A., 275, 357 Orden, A., 70 Osborne, M. J., 203 Oster, G., 301 Ottaviano, G., 7, 16 P Pang, J. S., 126 Papageorgiou, G. J., 14 Papageorgiou, Y. Y., 14 Parr, J. B., 13–14, 202, 215, 227 Pate, J., 202 Peet, J. R., 14, 21 Peneder, M., 13 Penfold, R., 14, 62 Perry, M., 3 Phillips, D., 14 Phipps, M., 28 Pinch, S., 3 Pipkin, J. S., 27 Pires, C. P., 203 Pitts, T. C., 14, 202 Plummer, P., 202 Plummer, P. S., 205 Ponsard, C., 13
449 Pred, A., 14, 27, 157, 202 Predohl, A., 12 Prescott, E. C., 203 Puu, T., 13 Q Quigley, J. M., 10, 17 R Raina, J. L., 27 Ranis, G., 301 Rautman, C. A., 70 Reggiani, A., 27 Reilly, W. J., 227 ReVelle, C., 6, 62 Ricardo, D., 13 Richardson, H. W., 18, 26 Romer, P. M., 9 Rossi-Hansberg, E., 13 Roth, T. P., xii Rowse, J., 136 Rugman, A. M., 3 Rushton, G., 14, 27, 202 S Sakashita, N., 13 Salop, S. C., 203 Sarker, S., 301 Sasaki, K., 13 Satchi, M., 301 Sauer, C. O., 21 Scarf, H., 126 Scherer, F. M., 13 Schmalensee, R., 129 Schmenner, R. W., 28 Schmitt, N., 62 Schneider, E., 272 Schulz, N., 227 Schumpeter, J. A., 268, 272 Scotchmer, S., 11 Scott, A. J., 7, 14, 27 Seale, A. D., 127 Seim, K., 203 Selten, R., 203 Serra, D., 62 Shaked, A., 203 Sharpe, W. F., 239 Sheppard, E., 14, 202, 227 Sibson, R., 214 Siebert, H., 13, 18, 26, 114 Simon, H. A., 18, 238 Skott, P., 301 Slade, M. E., 227 Slichter, S. H., 11
450 Smart, M., 28 Smith, D., 157 Smithies, A., 13, 52, 203 Smith, M. J., 6, 126 Smith, R. H. T., 13 Smith, T., 203 Smith, V. L., 135 Solomon, B. D., 14, 202 Solow, R. M., 16 Soper, J. B., 15, 52 Soubeyran, A., 249 Spengler, J., 16 Spivey, W. A., 18 Stahl, K., 13, 227, 233 Steininger, K. W., 13 Stern, N. H., 13, 202, 301 Stevens, B. H., 13 Stigler, G. J., 1, 10, 52, 272 Stiglitz, J., 11 Stiglitz, J. E., 13 Stimson, R. J., 14, 27, 202 Storper, M., 11 Stuart, H. W., 203 Sugden, R., 18, 238 T Tabuchi, T., 7, 203 Takayama, T., 13, 18, 114, 117, 127 Tarrow, S., 21 Teitz, M. B., 6, 14 Tellier, L. N., 166 Thisse, J., 3, 7, 11, 13, 15–16, 52, 157, 166, 203 Thomas, I., 6, 203 Thore, S., 137 Thrall, G. I., 14 Tramel, T. E., 127 Tucker, A. W., 18, 83, 136 Turk, M. H., 10 Turnovsky, S. J., 129 U Ullman, E., 273 Ulph, A., 23
First Author Index V Vaux, H. J., 142 Velupillai, K. V., 10 Venables, A. J., 13 Vickrey, W. S., 22 Vidale, M. L., 82 Vining, R., 26 Vollrath, D., 301 W Wagener, U. A., 82 Wagner, H. M., 83 Wagner, W. B., 14 Walker, R., 28 Walmsley, D. J., 27 Webber, M. J., 13–14, 202–203 Weber, S., 249 Weinschenck, G., 13 Wendell, R. E., 14 West, D. S., 62 Wheaton, W. C., 13 Whitaker, J. K., 2 White, A. G., 11 White, H. C., 11 Willig, R. D., 129 Wilson, M. G. A., 14 Wilson, R. H., 184 Wingo, L. Jr., 260 Wolpert, J., 27 Woodward, R. S., 166 Wu, J. H., 126 Y Yeh, C. N., 23 Z Zarembka, P., 301 Zenou, Y., 301 Zeuthen, F., 239 Zhang, W. B., 14, 371 Zhu, T., 203 Ziss, S., 62
Subject Index
A Advertising, 15, 32, 42 Agency, xi, xv, 15, 35, 44, 180, 236 Agglomeration economies, 2–3, 10, 25, 154, 177–180, 187, 193–197, 248, 380 Aggregation, 38, 108 Algorithm, 70, 82, 136, 171, 214, 293, 327 Arbitrage/Arbitrageur(s), 37, 43, 45, 48, 71, 91, 97–98, 100, 107–111, 113–115, 120, 125–149, 258, 273, 381, 401 Assumptions, 5, 20–21, 24–25, 31, 33–36, 43, 47, 52–53, , 56, 65–66, 71, 74, 93, 100, 103–106, 108, 111, 115, 122–123, 131, 138, 144, 151–155, 170, 172, 175, 180, 184, 188, 194, 197, 202, 205, 212, 219, 221–223, 225, 227–228, 232, 240, 242, 260, 264, 270, 272–273, 283, 293, 302, 312, 332, 342, 349, 357, 390 As the crow flies/Rectangular distance, 75, 155, 172, 205, 383, 393 Autarky, 1, 9, 25, 91, 97, 100–111, 113–114, 117, , 122–126, 131–135, 147–148, 248, 255, 322, 335–336, 339–347, 349–350, 353, 355, 360–361, 370, 375, 382, 393 Average cost, 33–35, 197, 251, 360 B Barter, 36, 233–235, 249, 264, 318, 332, 359, 367, 377, 386, 393, 405 Basing point pricing, 15, 52 Behavioral geography, 4, 26–27 Bernoulli process, 242, 256 Beta analysis, 239, 247, 255–256 Bid rent, 272, 283, 286–289, 294, 306, 308–312, 314–315, 325–326, 346, 369 Boundary/Boundaries/Range, 4, 10, 17, 25, 27, 32, 46, 48–55, 56, 63, 73, 81, 89,
142, 160, 188, 202–203, 205–227, 250, 258–260, 262, 267, 275, 306, 314–315, 327, 329, 333, 336, 341, 344, 349–352, 354, 356–358, 372–375 Bounded, 26, 204, 267 Broker/Brokerage, 15, 35, 44, 236, 379, 388 C Capacitated network, 94 Capacity, 8, 32, 69–81, 84–94, 99, 103, 180, 187, 201, 204, 212, 220–221, 227–228, 386–388, 391 Cartesian coordinates, 153–154, 159, 212, 386 Case study, xiii–xiv, 4, 73, 144–147 Clustering, see Localization/Clustering/ Co-location Cobb–Douglas production function, 152, 166–169, 175, 177, 302, 312, 320, 332, 337, 377, 389, 394 Co-location, see Localization/Clustering/ Co-location Comparative cost analysis, 175 Comparative statics, x, 41–42, 49, 55, 61–62, 81, 83, 88, 106–107, 112, 119–120, 159, 165, 172, 175, 185, 190, 196, 256, 263, 277, 283–285, 288, 295, 305, 309, 311, 315, 317, 322, 329, 341, 351, 353, 357, 361 Competitive location theory, vii, x–xi, xv–xvi, 1, 6–8, 12–14, 18–20, 22, 24, 26–28, 30, 99, 201, 357, 378, 380 Competitive market, 1, 4, 15, 32, 37–38, 71, 75, 86–87, 93, 97–99, 104, 124, 130–132, 149, 152, 218, 262, 264, 267, 269–271, 274, 286, 297, 299, 302, 315, 332, 336–337, 343, 366, 383, 385, 390, 394, 403, 408 Complementary slackness theorem, 83, 85
451
452 Congested/Congestion, 1–3, 8, 25, 28, 32, 44, 69–70, 72, 74, 77, 94, 97, 103–104, 111, 113, 133, 138, 142, 154–155, 172, 184, 194–195, 197, 276, 357, 379–380, 387–389, 391 Constant returns to scale, 20, 166–169, 301–302, 318, 320–322, 335, 337, 340, 342–343, 345, 347, 349, 360, 366, 389 Consumer benefit, 33–34, 46, 54, 101–102, 105, 109, 125, 129–131, 133, 140, 206, 210, 274, 276, 280, 291, 294, 303–305, 308–310, 312–313 Consumer surplus, 33–34, 46, 54, 67, 98, 101–102, 105, 109, 114, 124, 126, 129–135, 137–138, 140, 206, 210, 274, 276, 280–281, 304, 310, 316, 382 Contestable market, 1, 62 Contractor, 17, 177, 188–189, 193–199, 363, 382, 390 Cooperative, 231, 249–250, 252, 255, 258, 262, 264, 272, 366–367, 377, 380, 389 Core theory, viii, x, xii, xiv–xv, 395 Corner solution, 97, 117, 163, 185–186, 189, 191, 195, 287–288, 312, 348, 366 Cross-price elasticity, 225, 267, 285 Cumulative causation, 17 D Deductive approach, xiii–xiv, 6 Delaunay triangulation, 214 Delivered price, 45, 48, 52, 63, 71, 87, 209–210, 363–364, 391 Demand cone, 207 Demand curve, 4, 33–43, 45–50, 54, 56, 63, 71, 87, 97–98, 100–102, 105–110, 112–116, 118, 120–122, 125–126, 129–136, 149, 154, 170–171, 198, 202, 206–208, 210, 216–217, 219, 223–224, 273, 277–278, 280, 283–286, 288–291, 293–294, 303–306, 308–310, 313, 315–316, 322, 332, 355–357, 359, 363, 385, 387 Demand curve, free spatial, 208 [Demand/Supply] by arbitrageurs, 37, 97, 109–110, 114, 120 Diminishing marginal utility, xii, 23–24, 36, 130, 223, 238 Diminishing returns, 152, 198, 270, 301, 337–338, 340, 342–345, 348, 366, 379, 389 Discipline of difference, 21 Discrete choice model, 205, 227
Subject Index Discriminatory pricing, 15, 22, 45, 52, 71, 202, 205, 337, 358, 360–365, 367, 372, 382, 395 Diseconomies of scale, 3, 106 Distribution of income, 7, 67, 94, 124, 130, 149, 176, 199, 268, 300, 318, 336, 343 Division of labor, 1–3, 9, 25, 43, 67, 69, 95, 99, 124, 179, 187, 391, 391 Domestic market, 9 Dual, 83–88, 220, 301–303 Dual economy, 301–302 Duality Theorem, 83–84 Dynamics, 28 E Economies of scale, 1, 3, 9, 25, 32, 70, 72–73, 99, 124, 151, 166, 176, 179, 196, 335–336, 380, 389 Effective price, 15, 25, 35, 38, 45, 52–53, 138, 151–153, 195, 203–207, 209, 211, 213, 215–218, 221–222, 231, 232, 264, 337–339, 341, 344–346, 348–349, 354, 356, 359, 362–365, 385, 387, 389, 391 Efficient firm, 26, 32, 35, 69, 103, 105, 111, 121, 130–131, 133, 153, 182, 318, 320 Endogenous, x, xv–xvi, 11, 19, 25, 28, 41, 44, 46, 54, 56, 63, 74, 78, 82, 112, 119, 139–140, 160, 167, 171–172, 176, 189, 213, 258–259, 286, 303, 322, 336, 339, 382, 386, 390 Endowment[s], 14, 234–235, 240–245, 248–253, 259–263, 265–267, 300–301 Establishment, 11, 27, 30, 42, 53, 69, 178, 198, 202, 204, 220, 258, 378, 391 Euclidean distance, 26, 154, 156, 159, 172, 324 Excess demand, 5, 76, 91, 110, 114–120, 126, 135 Excess profit, 30, 33–34, 39–40, 46–48, 63, 75, 78, 101, 104, 113, 129–134, 219, 269, 274–275, 286, 294, 299, 304, 306–307, 312, 320, 323, 328 Excess supply, 76, 109, 114–120 Exclusion Theorem, 158, 174 Exogenous, 11, 28, 41–42, 44, 46, 54–56, 61, 63, 74, 83, 101, 107, 112, 116, 157, 171–172, 186, 189, 191, 194, 198, 206, 213, 216, 224, 294, 317, 319, 323, 325–327, 346, 369, 371 Expansion path, 32, 152, 166, 168–169, 253 Expected utility, xii, 233, 239 Expected value, 239, 242–243, 245 Expendable, 37, 40, 48, 231–232, 262 Explanation, xii–xiii, xv–xvi, 1, 3–4, 7, 21–22, 76, 117–118, 124
Subject Index Explanatory, x–xii, xiv, xvi, 203 Externality, 19, 179, 250, 259, 398 F Fiat money economy, 31, 69, 100, 126, 143, 153, 182, 233–234, 273, 385 Fixed coefficients technology, 318 Fixed cost, 31, 33–34, 37, 39–40, 46, 54, 56, 62–63, 66–67, 72, 93, 103, 166–167, 171–172, 180–182, 191, 195, 206–207, 209–210, 217–218, 228, 263, 273, 318, 320, 358, 367–369, 377, 381, 388, 391 F.o.b. price, 15, 48, 52, 64, 154, 157, 167, 171–172, 175, 201–209, 211–218, 221–224, 227–228, 337–339, 344, 346–347, 356, 358–365, 367, 368, 370, 372–377, 388–389 F.o.b. pricing, 208–209, 211, 214, 217–218, 221–223, 228, 346, 361–365, 370 Free entry, 17, 99, 169, 219, 299, 303, 312, 323, 326, 344, 357, 386 Free spatial demand curve, see Demand curve, free spatial Freight, 15, 21, 29, 35, 44, 47, 51–53, 116, 142, 143, 145, 155–158, 165, 209, 236, 268, 362–363, 371 Functionalist, x, xi, xii, xvi G General equilibrium model, 318, 322, 325, 331 Geographic angle[s], 163 Global net social welfare, 114, 125, 127, 134, 136 Gradient, 18, 60, 97, 123, 297–298, 309, 344, 349, 372, 375 H Hakimi Theorem, 174–175 Half-freight, 47, 51–52, 60, 66, 209 Hitchcock-Koopmans problem, 69–95, 127, 220–221 Home market, 9, 14–16, 30, 35, 40, 45, 48, 57, 60, 66–67 Homothetic, 23, 340 Homotheticity, 253 I Income elasticity of demand, 330, 340 Indivisibility/Indivisibilities, 1, 3, 9, 25, 31, 70, 166, 179, 188, 198, 380 Industrial restructuring, 17, 193 Industry marginal cost curve, 99, 103–104
453 Information, 2, 4, 11, 15, 22, 35, 37, 44, 53, 67, 147, 202, 205, 227, 233, 236, 264–265, 321, 372–373 Installation and removal, 15, 35, 44–45, 236 Instrumentalist, x, xii, xvi Instrument variable, 82–84 Insurance/Insurance principle, 9, 15, 35, 44, 177–199, 236, 240–241, 250, 258, 265, 378–379, 381, 391 Integrated market, 107–113, 117, 120, 135, 141 Inventory/Inventories, 15, 23, 33, 35, 42–44, 51, 53, 70, 94, 127, 152–153, 177, 180–193, 205, 207, 236–237 Inverse demand curve, 33–43, 45–46, 50, 54, 56, 63, 100–101, 107, 110, 113, 122, 130, 132–133, 170, 202, 206–208, 219, 224, 273–274, 280, 286, 294, 303, 308, 313 Inverse supply curve, 100–101, 110, 116, 132–133, 171 Investment, 3, 6, 9, 16, 21, 27–28, 31–32, 35, 38, 44, 54–55, 57–58, 70, 74–75, 93, 128, 142, 147, 179, 197, 239–240, 259, 270–271, 336, 357 Isolation/Isolated market, 30, 93, 97, 133, 194, 248, 273 K Kink[s]/Kinked, 76–77, 108, 111, 116–118, 125, 127, 148, 207, 216–217, 290–291, 295, 297, 349, 359, 364, 372–375 Kuhn-Kuenne algorithm, 171 L Labor market/Market for labor, 15–16, 66, 299–302, 318–319, 321–322, 326–327, 331, 335, 341, 344, 367–368, 370, 380 Land market/Market for land, 232, 264, 266, 270, 274, 277–278, 282–283, 286–287, 311, 322, 326, 331, 348, 366, 379–380 Leontief [production] technology, 32, 152–153, 156, 165–166, 167, 171, 175, 251, 265, 377 Linear expansion path, 32, 166, 168, 253 Linear program[ming], 18, 69–70, 78, 81–86, 125, 136, 220 Local cost advantage, 1, 3, 25, 95, 176, 380–381 Local demand, 8, 37, 71, 76, 87, 91, 97, 102, 105, 107–111, 113–115, 117–118, 121–123, 125, 127, 132–142, 148, 232 Local economy, 2, 25, 275, 336, 356, 362, 378
454 Localization/Clustering/Co-location, 24–25, 378 Localization economies, 9, 25, 27, 154, 178 Local market, 2, 9, 14, 22, 58, 98, 103–104, 107, 122, 133, 137, 147, 231–266, 273, 378 Local supply, 41, 76, 91, 94, 97–111, 113–115, 117–118, 121–125, 132–134, 137–142, 148–149, 151, 175, 177, 232, 267, 273, 379–380 Locational analysis, 4, 26–27 Locational behavior, 2, 6, 27 Location decisions, vii Logistics, 69–95, 125 Log-linear utility function, 23–24, 224–226, 240, 242, 253, 322, 337, 340, 348, 366 Long[er] [run/term], 3, 75, 99, 103–104, 237 Löschian Equilibrium, 219 M Map triangle, 159–163, 165 Marginal cost curve, 32–33, 40, 53, 72–73, 75, 94, 98–99, 101–104, 132, 141, 148, 175, 221, 228, 273, 297, 332, 379 cost of production, 32, 35, 50, 55, 70, 72, 94, 103–104, 123, 130–131, 273–275, 329 factory, 75–80, 90–92 revenue, 33–34, 38–39, 46–47, 71, 98, 101–102, 206–207, 209–210, 215, 273 shipping cost, 22, 161, 178, 252 supplier, 104, 113, 133, 139, 278 Market area, 201–229, 232, 251, 255, 260–262, 335–336, 341–342, 344–345, 347–352, 355–356, 359, 368, 372–373, 375–376, 379–381 economy, 2, 6, 66, 94, 99, 123, 132, 135, 175, 198, 205, 221–223, 228, 264, 268, 297, 332, 377 equilibrium, 16, 19, 24, 28, 34, 41, 46, 54, 102, 105–108, 112–113, 120–122, 130, 132, 217, 276, 283, 288, 293, 299, 303, 305, 308, 312–314, 321, 326 Market for labor, see Labor market/Market for labor Market for land, see Land market Maximum price, 34, 37, 39–40, 43, 45–47, 54, 56–57, 63, 132, 207, 212, 231, 274–275, 280, 286, 294 Midpoint rule, 308–309, 312, 360, 369–370
Subject Index Mill pricing, 15, 357 Minimum price, 37, 39, 56, 207, 216, 275 Monopolist, 17, 22, 25, 29–50, 52–54, 59–60, 66, 71, 75, 87, 97–99, 101–102, 104–105, 129–132, 144, 170, 175, 194–197, 201–203, 205–208, 210, 220, 228, 273, 335–336, 355–363, 366–374, 377, 382–383 Monopoly, 1, 5, 16, 25–26, 29–67, 101, 124, 129–133, 144, 203, 209, 277 Monopoly excess profit, 34, 46, 54, 101, 129–133 Multimarket equilibrium, 24 Multinational, 15, 27 Myopic, 63, 217–219, 228, 248, 264 N National income, 128–129 Neoclassical, viii, xii, xvi, 1, 8, 14, 23, 166, 217, 238, 240, 268–270, 272 Net revenue, 39–43, 47–49, 54–56, 63–64, 98, 129, 156, 194, 275, 280 Net social welfare, 114, 125–127, 134–136, 291–292, 294–295, 303–305, 308–309, 312–313 Network/Networks, 3, 8, 10–11, 14, 17, 26, 28, 44, 48, 74, 94, 113, 142–143, 151, 172–176 No money illusion, 40 Non-spatial, 30–43, 76–87, 90, 91, 100, 240–249, 256, 276, 284, 299, 301, 318–323, 367 Nontariff barrier, 128, 143–145 Normal profit, 15, 32, 38–39, 105, 108, 133, 155, 219, 264, 272, 275, 279, 305–306 Numéraire, 242, 318, 338, 341 O Oligopoly, 17, 144 Opportunity cost, 6, 31–35, 38, 40–42, 44, 46, 49–51, 53–61, 63–65, 67, 69, 72, 75, 77–78, 84–87, 93, 101, 103, 127, 151, 165, 220, 241, 250, 259, 263–264, 297, 302, 304, 309, 311, 319, 325, 325, 332, 339, 346, 350, 358–360, 365, 369, 375–377, 381 Optimal location theory, 6, 30, 75, 201 Optimization, 17–18, 24, 70, 75, 82, 126, 136, 171 Outsource/Outsourced/Outsourcing, 30, 177–179, 188–193, 197–198, 258, 262, 380–381
Subject Index P Palander, 12 Perfect competition, 97–124, 125–149, 217, 333 Perfectly elastic demand, 154 Pickup or delivery pricing, 52 Polyline, 76–78, 114–117, 207, 255, 291 Predohl, 12 Price difference curve, 76, 100, 114–120, 125 elasticity of demand, 24, 34, 40, 46–47, 54, 101, 110, 145, 170, 185, 189, 219, 357, 361–362, 370–371 gradient, 60, 97, 123 risk, 237–238 taker, 4–5, 26, 32, 37–38, 44, 99–101, 103, 113, 133, 144, 155, 217, 232, 269–270, 337 Primal, 82–86, 88, 220 Producer cost, 33–34, 46, 54, 98, 101–102, 105, 109, 125, 129–131, 133–134, 206, 210, 274, 276, 280–281, 292, 303–305, 308–310, 312–313, 316 Producer surplus, 34, 46, 54, 98, 101–102, 105, 109, 114, 126, 129–135, 137–140, 206, 210, 274, 280, 310, 316, 382 Production possibility frontier, 319, 321–322, 340–342, 345–348 Profit, 33–34, 40–60, 170–176, 177, 180–183, 194–197, 201–202, 205–210, 213–214, 217–219, 238, 249, 258, 264–265, 268–269, 271–272, 274–275 Punctiform, 26, 66, 70, 94, 100, 111, 123, 132, 175, 198, 221–222, 223, 228 Pythagorean formula, 154, 212–213 Q Quadratic programming, 18, 136, 292, 309 Quasi supply curve, 40, 290–291, 294, 296, 314 R Range, see Boundary/Boundaries/Range Rate[s] of return, 56–61, 63–64, 179, 238 Rational, xii, xv, 1, 8, 33–34, 46, 54, 56, 63, 67, 74, 77, 100–101, 110, 116, 128, 131–132, 148, 155, 157, 160, 167, 171–172, 174, 176, 178, 180, 189, 191, 194, 206, 210, 212–213, 216, 218, 221–224, 236–241, 249–251, 253, 259, 272–274, 280, 286, 294, 302–303, 308, 303, 319, 325, 338–339, 346, 358, 368–369
455 Reasoning, 1–28, 126–127 Rectangular distance, see As the crow flies/Rectangular distance Rectangular plane, 26, 75, 151, 153, 159, 175, 204–205, 207, 211–216, 221–223, 225, 228, 264, 267, 279–285, 293–297, 302, 308, 327, 332, 357, 360, 377 Reductionism/Reductionist, 20–21 Regional economy, 1, 25, 44, 62, 67, 231–232, 262, 266–268, 278–279, 282–283, 288, 293, 295, 298, 299–302, 306–307 Regionalization, 17, 25 Relativist, 22 Rent, 13, 78, 105, 135, 182, 196, 231, 241, 249, 253, 259–265, 267–284, 286–291, 293–295, 297–300, 302–304, 306–315, 318–326, 328, 331–333, 337–340, 343–352 Retail/Retailing, 42–44, 50–51, 55, 153, 182, 202, 211–212 Ribbon, 26, 267, 274–277, 283–284, 286, 290, 293–297 Ricardian rent, 39, 75, 78, 105, 269, 277, 299, 311, 315 Risk-return, 233, 239–240, 246–248, 254–257, 261–263, 337 Risk[s], 1–2, 4, 6, 8–9, 11, 15, 62, 64–65, 179–180, 182–183, 187, 189, 191–195, 197–198, 231, 233, 236–241, 245, 247–248, 250–251, 253–259 S Samuelson conjecture, 126 Scale of production, 9, 67, 70, 72–74, 154–155, 165–167, 179, 368, 396 Semi-net revenue, 39, 43, 47–48, 54–56, 63–64, 129, 156 Sensitivity analysis, 81, 141 Shadow price, 69–70, 75, 78–81, 83–93, 220, 381 Shipper, 28, 44, 94, 135, 147, 177, 182–183, 198 Shipping cost/Unit shipping cost, 9, 15–16, 21–22, 25, 28–29, 35, 43–48, 50, 52, 53–67, 69, 72–77, 81, 83–87, 90–94, 97, 99–101, 107–113, 107, 132–136, 140–141, 188, 198, 202–206, 209–210, 213, 216, 218, 221–224, 231–232, 243, 245, 249–255, 258–266, 379–383 Shortest path, 26, 159, 172, 205 Short [run/term], 3, 32, 41, 50, 70, 72–75, 99, 102, 180, 197, 237 Slack, 76, 81, 83, 85, 88–91
456 Social Welfare, 25, 34, 46, 54, 101–102, 105, 114, 125–136, 274, 276, 280–281, 291–292, 294–295, 303–305, 308–309, 312–313, 378, 382 Spatial equilibrium, 7, 12, 19–20, 100, 127, 151, 204, 231, 277, 326–329, 330, 344, 348–349, 355–356, 364 Spatial price equilibrium, 13–14, 93, 99–100, 113–122, 126–127, 135–136, 142–145, 147–148, 205, 291–292, 300 Specialization, 10 Store of value, 31, 35–36, 234 Submarket, 9, 25, 45, 51, 93, 97, 123, 148–149, 202, 209 Substitutability/Substitution/Substitutable, 5, 15, 30, 32, 36–37, 43, 51, 86, 129, 145, 151–153, 165–169, 176–177, 184, 188, 191, 201–202, 204, 207–208, 221–228, 232, 240, 260, 265, 275–276, 299–300, 302–306 Supply step function, 77 Surplus, 33–34, 46, 54, 67, 81, 88, 89, 98, 101–102, 105, 109, 114, 129–135, 137–140, 210, 274, 280–281, 301, 304, 310, 382 Symmetric shipping cost, 56 T Takayama, 13, 18, 114, 125–149 Tariff jumping, 60, 128 Tariff[s], 10, 15, 35, 44–45, 60, 128, 142–146, 236 Taxes, 15, 35, 44–45, 236, 270 Theory-model dissonance, 408 Thick, 236–237 Thiessen polygon, 204, 214–215 Thin, 236–237 Total cost, 32–34, 43, 45–46, 53–54, 70, 74, 79, 82, 84, 87–88, 131, 166–167, 175, 181, 184, 186, 190, 192, 210, 263, 370 Trade area, 211–212, 214, 232, 381–382 Transaction cost, 15–16, 36, 133, 236, 264–265 Transfer, 3, 8, 15, 35, 44, 94, 236 Transportation linear program, 70 Transportation network, 3, 8, 26, 74, 94, 133, 154, 172–175 U Ubiquitous, 5, 75, 111, 152, 154–155, 157–158, 166–167, 171–172, 176, 178, 270, 302, 318, 381
Subject Index Unbounded, 26, 204, 267, 329 Uncertainty, 11, 23, 53, 62, 64, 66, 154, 179, 183, 187, 191–192, 198, 205, 226–227, 231, 236–240, 264, 275, 297–298 Uniform pricing, 22, 52 Unit production cost, 3, 69, 73–76, 78, 80, 87, 90–92, 276, 381 Unit shipping cost, 9, 15–16, 21–22, 25, 28–29, 35, 43–48, 50, 52, 53–67, 69, 72–77, 81, 83–87, 90–94, 97, 99–101, 107–113, 107, 132–136, 140–141, 188, 198, 202–206, 209–210, 213, 216, 218, 221–224, 231–232, 243, 245, 249–255, 258–266, 379–383 See also Shipping cost/Unit shipping cost Unit shipping rate, 44, 55, 83, 100, 152, 167, 175, 206, 210, 212–213, 216, 218, 221–224, 250, 252, 254, 270–271, 274–275, 277, 279–280, 283–284, 286, 292, 294, 297, 299, 306, 308, 313, 325, 332, 338–339, 344, 346, 360, 369, 383 Unit transaction cost, 231, 264 Unpriced, 15, 30, 32–33, 39, 101, 103, 128, 133, 250, 279, 305 Urbanization [economies/economy], 9, 25, 154, 178–179, 198, 380–381 Utility function, 24, 223–226, 228, 232, 238, 240–242, 250, 253, 264, 322, 332, 337, 340, 348, 366, 378 V Variable cost, 32, 37, 39–40, 74, 77, 98, 273 Vertical integration, 11, 30, 193, 379–380 Voronoi, 214 Voronoi diagram, 214 W Walrasian [multimarket] equilibrium/ process/linkage/perspective, 1, 5–6, 24–25, 123, 233, 235, 237, 265–266, 268–269, 288, 297–298, 332–333, 374–376, 380–381 Warranty, 15, 35, 42, 44–45, 205, 236, 379 Weber–Launhardt map triangle, 159–160, 163 Weber-Launhardt problem, 12, 151–155 Weber-Launhardt weight triangle, 161–162 Wellspring, 31–32, 69, 103, 179, 336–337, 358
