Increasing Returns and Inframarginai Economics - Vol. 3 <*"*
Inframarginai Contributions to
Development Economics
edited by
Christis Tombazos Xiaokai Yang
Inframarginal Contributions to
Development Economics
Increasing Returns and Inframarginal Economics Series Editors: James Buchanan, Yew-Kwang Ng, (Xiaokai Yang) Associate Editor: Guang-Zhen Sun
Published Vol. 1 An Inframarginal Approach to Trade Theory Edited by Xiaokai Yang, Wenli Cheng, Heling Shi & Christis G. Tombazos Vol. 2
Readings in the Economics of the Division of Labor: The Classical Tradition Edited by Guang-Zhen Sun
Vol. 3
Inframarginal Contributions to Development Economics Edited by Christis Tombazos & Xiaokai Yang
Increasing Returns and Inframarginal Economics - Vol. 3 f ^ f a j
Inframarginal Contributions to
Development Economics
edited by
Christis Tombazos Xiaokai Yang Monash University, Australia
YJ? World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI
• HONG KONG • TAIPEI • CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
INFRAMARGINAL CONTRIBUTIONS TO DEVELOPMENT ECONOMICS Increasing Returns and Inframarginal Economics — Vol. 3 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-256-658-9
Printed in Singapore by World Scientific Printers (S) Pte Ltd
ToXiaofpi
This page is intentionally left blank
Preface
As in the case of any anthology of reprinted contributions, primary credit for this volume goes to the authors of the various chapters. We are also grateful to the original publishers for their permission to reprint articles that appear in this volume. Unlike the remaining contributions, Chapter 1 is written specifically for this volume. It discusses the origins of inframarginal analysis, and provides an introduction to the implementation of the inframarginal methodology in the context of a simple example. The production of the camera-ready copy of this manuscript was a difficult and labourious task that was managed, almost single-handedly, by our in-house copy editor Aiko Yoshibayashi. We thank Aiko for her extraordinary attention to detail, and her hard work. We also gratefully acknowledge the generous financial support of the Department of Economics and the Faculty of Business and Economics of Monash University. Finally, we thank the Research Centre for Increasing Returns and Economic Organisation of Monash University for providing the hardware and office space without which this project would not have been possible.
Christis G. Tombazos Managing Editor
vn
This page is intentionally left blank
Contents
Preface
vii
Part 1. Introduction 1. Returning to the Classical Tradition: The Relevance and Application of Inframarginal Analysis to Development Economics Christis G. Tombazos and Stephen Matteo Miller
3
Part 2. The Origins of Inframarginal Applications to the Study of Economic Development 2. The Marginal Cost Controversy Ronald H. Coase 3. Economics and Biology: Specialization and Speciation Hendrik S. Houthakker 4. Externality James M. Buchanan and Wm. Craig Stubblebine
29
47 55
Part 3. Development Strategies, Income Distribution, and Dual Structures 5. Economic Development, International Trade, and Income Distribution Xiaokai Yang and Dingsheng Zhang IX
77
Contents
X
6. Pursuit of Relative Utility and Division of Labor Jianguo Wang andXiaokai Yang
111
Part 4. Urbanization 7. Development, Structural Changes and Urbanization 137 Xiaokai Yang 8. An Equilibrium Model Endogenizing the Emergence of a Dual Structure between the Urban and Rural Sectors 167 Xiaokai Yang and Robert Rice 9. Agglomeration Economies, Division of Labour and the Urban Land-Rent Escalation: A General Equilibrium Analysis of Urbanisation 195 Guang-Zhen Sun andXiaokai Yang Part 5. Entrepreneurship and the Firm 10. Theory of the Firm and Structure of Residual Rights Xiaokai Yang and Yew-Kwang Ng 11. The Theory of Irrelevance of the Size of the Firm Pak-Wai Liu andXiaokai Yang 12. Specialization, Product Development, Evolution of the Institution of the Firm, and Economic Growth Jeff Borland and Xiaokai Yang
231 259
291
Part 6. Endogenous Transaction Costs and Property Rights 13. Endogenous Specialisation and Endogenous Principal-Agent Relationship Xiaokai Yang and Yeong-Nan Yeh 14. A Model Formalizing the Theory of Property Rights Xiaokai Yang and Ian Wills
329
363
Contents
15. Economy of Specialization and Diseconomy of Externalities C. Y. Cyrus Chu and C. Wang
XI
391
Part 7. Investment, Endogenous Growth, and Social Experiments 16. The Division of Labor, Investment and Capital Xiaokai Yang 17. A New Theory of Industrialization Heling Shi and Xiaokai Yang
409 437
Part 8. Infrastructure, Labor Surplus, Insurance, and the Trade-off Between Leisure and Income 18. Population Density and Infrastructure Development 463 C. Y. Cyrus Chu 19. Uncertainty, Insurance, and Division of Labor 479 Monchi Lio 20. Push or Pull? The Relationship between Development, Trade and Primary Resource Endowment 497 Mei Wen and Stephen P. King Index
531
Parti
Introduction
This page is intentionally left blank
CHAPTER 1 RETURNING TO THE CLASSICAL TRADITION: THE RELEVANCE AND APPLICATION OF INFRAMARGINAL ANALYSIS TO DEVELOPMENT ECONOMICS
Christis G. Tombazos and Stephen Matteo Miller* Monash University 1. Why Consider the Inframarginal Approach to Economic Development? Economic development is an observable phenomenon, improvement in the well-being of people, while development economics is the study of that phenomenon. A relatively new sub-discipline of economics known alternatively as "inframarginal economics", "new classical economics" or "the economics of the division of labour," may offer its deepest insights in the study of economic development. This reason alone justifies compiling a collection of essays relating to economic development from the perspective of inframarginal economics, which differs in important ways from more traditional frameworks. We do not expect that inframarginal economics will be readily embraced by all people interested in the field of development. Arndt1 provides a fascinating account of the history of the idea of economic development, though it concentrates especially on the 20th century. It shows just how much recent debate there has been about how to define development. For the most part, until the collapse of the Soviet Union, economic development as practiced has been part of a political response from the West to counter the spread of communism to developing countries. Yet, as convincing and detailed as Arndt's story sounds, it does not seem likely that the idea of development only emerged within the last one hundred years. For instance, Arndt makes little mention of Adam Smith. * We thank Xiaokai Yang for useful discussions. 3
4
C. G. Tombazos, S. M. Miller
As Yang and Ng2 note in the introductory chapter of their well known book, neoclassical economics has focused primarily on the problem of resource allocation, while inframarginal economics, or the economics of the division of labour, gives primary attention, just like Adam Smith, to issues of economic organisation. Economic development is much less about a simple allocation of resources, and much more about explaining first the type of economic organisation that emerges, and only then analysing the resource allocation problem. Put another way, economic organisation puts boundaries on how resources are allocated, and therefore, the underlying reason for variations in economic development are ultimately related to organisational issues about economic activity. This line of reasoning is also present in the new institutional economics (NIE) view of economic development, including Bates3 and North4, among other efforts prior to and since then (see http://coase.org for more references). The motivations are different though, as work in the NIE tradition frequently begins with observations on economic history, while the new classical economics begins from the history of economic thought. Both approaches share common criticisms of standard neoclassical economics. Yet, work among practitioners of the former often make minor adjustments of neoclassical models to fit the historical fact that transaction costs matter for economic performance. Practitioners of the latter typically begin by attempting to mathematically formalise certain aspects of Smith's framework, and then search for evidence to verify inframarginal, and hence, Smithean views of economic performance. So while the issues and explanations of economic performance are often the same, the approaches are different, but it is important to bear in mind that neither approach fully embodies what Smith actually said about economic development. 2. Smith's View of Economics and Economic Development Many economists today would certainly point to Adam Smith as an early contributor to, if not the founder of, economics. Smith was also preoccupied with where improvements in economic performance, or growth, come from. Hence, he might also be viewed as the first development economist. Economic ideas had, of course, been circulating
Returning to the Classical Tradition
5
before him. Levy5 even claims that the anti-economic content of Plato's Republic was aimed at the economic ideas found in Homer's Illiad. There, the Gods are perfect rational agents, and understand the benefits of positive sum trade, as they make deals with each other, while humans are primitive, and understand only negative sum trade, or war. What Plato's Republic did was to establish an elitist view, that the division of labour for all occupations must be planned, and the agents needed to do so were Philosopher Kings. This view was dominant for roughly the next two thousand years. In The Wealth of Nations6, Smith challenged this view by appealing to our egalitarian sensibilities. In short, he suggested that the division of labour, which was constrained by market size and population, could arise if each person were left to choose their own occupation. If transaction costs are not prohibitive, a larger market and population makes it possible for people to reap the returns to specialisation. However, he was a great synthesizer of economic ideas, and to appreciate his contribution, it is necessary to consider his three works, The Theory of Moral Sentiments (TMS)7, Lectures on Jurisprudence (LJ)8, and An Enquiry into the Nature and Consequences of the Wealth of Nations (TWN)6. Smith's interest in development has its origins in his work on moral philosophy. As Levy9'10 puts it: It seems to me that Smith is relying on a belief that modestly informed spectators would reach a common judgment about the level of wellbeing of the median member of different societies to defend economic growth. Smith has told us three things. First, he tells us that the as-if well-being of children is easy for everyone to impute. Second, he has told us that the children of the median member of society will do best in a rapidly growing society. Third, he has told us that one and two suffice for us to conclude that rapid growth is good (see page 313). So, herein lies the justification for growth. In order to understand where growth comes from, it is first necessary to point out that for Smith, it is man's inherent desire to trade that drives the division of labour, and volume of trade that determines the extent of the division of labour. Furthermore, trade is possible because people share language; whether it's Chinese, French, Jamaican, English, or Swahili. As long as people
6
C. G. Tombazos, S. M. Miller
can communicate, it is language (see Smith TWN I. ii par. 26, Levy1' for an extensive interpretation of Smith's views). Thus, a larger market size with lots of commercial activity means a greater division of labour (see Smith TWN I. iii par. I6). The division of labour increases worker productivity because of improvements to human capital ("dexterity") and physical capital ("the invention of a great number of machines"), as each factor becomes more specialised, and because specialisation itself implies time is no longer lost in switching from one task to another (see Smith TWN I. i par. 56). Rising productivity is of course an indicator of a growing economy. Lectures on Jurisprudence* includes an early draft of the Wealth of Nations that discusses the role of specialisation in economic growth. It also includes a noteworthy discussion of the impediments to growth. Specifically, any factor that drives a wedge between the natural (distortion-free) price, and the observed market price, slows down commercial activity, and hence, the potential for economic growth. These wedges might be caused by monopoly, by taxes or tariffs, or by prices or risk due to various forms of uncertainty including property rights (see Juris-Prudence Part II. Of Police in Smith8 where he discusses among other things, the relationship between the division of labour and growth, as well as retardants to growth.). 3. Where has the Division of Labour Gone? Stigler12 observes that specialisation is the key facet of Smith's explanation of economic growth and development that is surprisingly missing from all of neoclassical economics. For instance, it is notably absent from the neoclassical framework of Marshall and Jevons. Reasons for this anomaly have been offered by a number of authors. Buchanan and Yoon13 speculate that the neoclassical thinkers sidestepped the issue of the division of labour because they considered it irreconcilable with their theory of distribution. This theory depends critically on constant returns. Yet, division of labour, and the benefits of specialisation, suggest increasing returns to scale. Others, including Houthakker (p.62)14 and Yang (p.8)15, add technical hurdles to what may have discouraged the founders of marginalist economics from addressing issues relevant to the division of labour.
Returning to the Classical Tradition
7
These hurdles originate from the fact that division of labour ultimately boils down to a binary decision of whether or not to produce a particular good. Mathematically, such outcomes are represented by corner solutions and require models that can accommodate such discontinuities. Yet, the marginal revolution relied on calculus that, in turn, requires smooth and continuous functions. Such functions can investigate interior outcomes, but are not suitable to study the discontinuities that characterise the division of labour. Clearly, nineteenth century mathematical formalism represented a genuine hurdle in reconciling the theory of distribution with classical questions of economic organisation. However, Buchanan and Yoon13 have recently argued that Marshall's decision to banish division of labour from the core preoccupations of the marginal revolution on the basis that it would undermine the requirement of constant returns was unnecessary. As these authors explain, improvements in the division of labour are in fact consistent with constant returns to scale. This is so because enhancements in the division of labour are generally realised when agents switch from self-production to market production. Hence, as the division of labour intensifies, the economy-wide value of the final output relative to the total value of all inputs increases. This leads to what Buchanan and Yoon refer to as "generalised increasing returns". Still, given any particular pattern of the division of labour, as the quantity of inputs employed by any one firm expands, output increases proportionally. It is the reorganisation of the economy that redefines what is produced and exchanged in the market that generates generalised increasing (economy-wide) returns, not the increase in inputs used by any given firm in the absence of any change in the economy-wide division of labour. In other words, generalised increasing returns and constant returns to scale can happily coexist. Applications of corner solutions and discontinuities to economic models can be traced to Coase16 and Koopmans17. Following Stigler12, a number of papers from faculty and students at the University of Chicago, including Rosen18' 19, Becker and Murphy20, and Tamura21' 22, among others, began to explore the economics of specialisation. Just as this research was beginning at the University of Chicago, Xiaokai Yang, imprisoned in China as a dissenter to the ideas that defined the Cultural
8
C. G. Tombazos, S. M. Miller
Revolution, began a remarkable journey to recreate economic theory using a framework that establishes a bridge between modern economics and Smith (see Yang15 for an excellent summary of relevant contributions). Unlike Smith, inframarginal economics makes no claims about the relationship between language and trade, nor does it ask what the theoretical justifications for growth are. Instead, it begins with the idea that there are gains from specialisation, that are possible in market economic activity, but there are also transaction costs from using the market. People are consumer-producers. They choose how much of their waking hours to allocate to a specific activity. As the division of labour evolves, individuals move away from self-producing all of their consumption towards producing only one good, which earns income that is used to purchase a wide variety of consumption goods. If transaction costs are high, people are unlikely to find it worthwhile to engage in market activity, and will instead self-produce their consumption in autarky. If, on the other hand, transaction efficiency is high enough, people who otherwise start off identical, will become increasingly specialised in particular occupations, thereby capturing the benefits of specialisation. Table 1 below compares some aspects of inframarginal and neoclassical models with Smith's theory of economic growth. Clearly, the inframarginal literature is closer than neoclassical theory in capturing Smith's thoughts on development, notwithstanding the absence of any moral content. The aim of this volume therefore is to bring together a number of recent articles on this topic in an effort to summarise the inframarginal explanation of the origins of economic development. As illustrated by the articles included in this anthology, inframarginal analysis has profound implications on the study of economic development. To assist the non-specialist reader with these contributions, the remaining of this article serves as an introduction to the practical implementation of inframarginal analysis. In particular, the next section outlines the key "steps" of inframarginal analysis, and section 5 implements these steps in the context of a simple model.
Returning to the Classical Tradition
9
Table 1: Classical, Inframarginal, and Neoclassical Perspectives Classical
Inframarginal
(Adam Smith)
(New Classical)
Neoclassical
Not explicitly
Yes
Yes
Moral justifications for growth
Yes
Not Yet
Not Yet
Firms are collections of people
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No?
Yes
Yes
No
No
No
Yes?
Use of models is necessary
Assume people start off identical and choose to be different in their occupation Growth occurs by realising gains from specialisation Transaction costs determine the level of specialisation and hence growth Trade, and hence growth can occur even without exogenous differences Population
growth
adversely
affects economic performance
4. The "Steps" of Inframarginal Analysis Commonly, inframarginal analysis of economic models involves the following steps: i.
Design the model [important features of any model include items such as: types of agents, dimensions, production and consumption characteristics, and institutional structure (in particular: rules of exchange — e.g. presence of transaction costs, tariffs, etc.)]
ii.
Identify the prime configurations and market structures {note: prime configurations are patterns of production and consumption that may characterise individual agents within countries, and which may be consistent with meaningful frameworks of
C. G. Tombazos, S. M. Miller
exchange — or market structures). Autarky is viewed as meaningful "pattern of exchange" — i.e. a pattern of exchange with zero flows For each configuration, within each structure, employ marginal analysis to optimise for each agent's objective function Using demand and supply from previous step, solve for the market clearing conditions, which determine the price that would prevail if the corner equilibrium that is represented in each structure is in fact the general equilibrium. Of course, depending on the production function, prices may be determined prior to evaluation of market clearing conditions (as per the nonsubstitution theorem) Still, step (iv) may facilitate the determination of necessary conditions for the structure at hand to prevail Undertake a demarcation of parameter space in parameter value subsets within each of which the local corner equilibrium is the general equilibrium. The general equilibrium requires that under the relative price that prevails in any given corner or interior equilibrium, all individuals in all countries prefer the configurations consistent with the structure at hand, than any other possible shadow configurations. (Note that shadow configurations include all possible configurations other than the configuration prevalent in the structure and for the agent under examination) The final step entails examination of the resulting inframarginal comparative statics
Returning to the Classical Tradition
11
5. Even a Simple Inframarginal Model has Much to Say About Development The two good inframarginal model has been summarised in a number of contributions (Yang and Ng2, Yang15), but typically the purpose is to provide an intuition for more complicated models. Here, the two good inframarginal model is presented to summarise its implications for economics development. The context in which it is used here is to highlight the differences between an economy based on subsistence agriculture, and one that is more specialised, and hence developed. Just as Adam Smith pointed out that the difference between the philosopher and the common porter is really much smaller than the philosopher would lead us to believe, the model begins with Smith's inherent assumption that each person starts of the same. Differences arise from individual choices about how to specialise in productive activities. What follows represents a simple implementation of the steps outlined in section 4. Step i — Model design: o o
There is a continuum of identical consumers-producers that has massM Agents ex ante identical, and have the following utility function for food (denoted/) and non-food («) commodities: max ui = (f + kfd)(n
o
+ knd)
f, n: goods that are self provided fd, nd: goods bought (demanded) from the market 0 < k < 1 is the trading efficiency coefficient (the lower k is, the higher the transaction cost) Each consumer-producer has an allocation of labour-time (/) and land (/) that is exhausted across the two goods:
tf+t„=l
12
C. G. Tombazos, S. M. Miller
o
The production functions for food and non-food for a consumerproducer are variants of the Cobb-Douglas production function, in terms of labour (/,-) and land (?,): f + fs =max 0,(lf-ajtbf
\(ln-*)'t]
n + ns = max 0.
o
where the term a is a fixed learning cost. Finally there is a budget constraint such that in general:
Pffs+Pnns=[pffd+Pnnd] The idea behind inframarginal models is typically that the efficiency of transactions, often denoted by k, determines the extent of the division of labour, which in turn, determines income levels. The lower the efficiency of transactions (the higher the costs of transacting), the more expensive it is to rely on the market, and hence, people find selfproduction makes them better off. This is depicted below: k is low
k is medium range
i
i
Autarky
Partial specialisation
k is high
i Complete specialisation
The inframarginal approach typically calls for backing out the transaction efficiency parameter, to find the point at which the economy switches between configurations. These range from autarky, when each individual self-produces its consumption, to extreme specialisation, when each individual produces one-good, from which the proceeds are used to finance consumption of all other goods. To back out those transaction efficiency thresholds (the points at which the model switches behaviour), the first required step is to find the optimal supply and demand for each, initially identical, consumer-producer under autarky, under partial
Returning to the Classical
specialisation (for specialisation.
more
complicated
13
Tradition
models),
and
complete
Autarky
Partial specialisation
Complete specialisation
i
LIL
i Optimal
Optimal
Optimal S&D
S&D
S&D
Hence, there is an optimal supply and demand for each person in each configuration, which corresponds to a different extent of the division of labour. These optimal supply and demand conditions are then used to get an indirect or real utility/welfare function for each individual in each configuration: Autarky
Partial specialisation
Complete specialisation
i
1 Welfare
Welfare
Welfare
Once the real utility is known for each individual, it is then possible to determine the thresholds of transaction efficiency at which people in the economy find themselves in autarky, partial specialisation and complete specialisation. With this overview in mind, the model can be analysed in different configurations (in this case two). Steps ii — Configurations and market structures
(a) Structure A, autarky
(b) Structure D, division of labour
Figure 1: Configurations and Structures
C. G. Tombazos, S. M. Miller
14
Step Hi — Marginal solutions From the general problem, it is easy to see that autarky is a much simpler world because there are no markets, and hence, there are no prices and there is no budget constraint. Hence, human behaviour is not governed by relative price changes. People don't respond to what is not observable. Still, they have preferences so that opportunity cost is still very much in their minds. Structure A (Autarky): gen<;ral problem simplifies to: max u, = f -n if1-
f = (lf-ajtbf n = {ln-a)°tbn lf+ln=\ tf + tn=\ This constrained maximisation problem can be simplified to an unconstrained optimisation problem, in this case solved for the non-food crop by substituting in all the constraints: a maxu tbn x uAA=({l-l = 111n)-a)" — <„| — a |(l-tj i i ~ / „ {ln-a)
The four first order necessary Kuhn-Tucker conditions are: | ^ = -fl.((l-/i,)-a)"-1 (!-/„)*•(/„-«)"<+
a.({l-ln)-a)a{l-tn)b.{ln-artbn<* dl
Returning to the Classical Tradition
15
^L = ^ . ( ( l - / . ) - a H l - / . r •(/.-«)•< + b.((l-ln)-a)a(l-tn)b.(ln-a)aCl<0
"K In autarky, because of the Cobb-Douglas utility specification, both goods must be consumed, and hence, both goods must be produced, as there is no market for goods that the household decides not to produce. Looked at another way, if either /„ or tn is either zero or one, then the first order condition is not satisfied. Hence, an interior solution must exist in autarky, such that the household produces all goods that it consumes. Rearranging and simplifying the first order necessary condition for labour yields:
|^((iH)-«r'-a-«r=((i-'.)-«H'»-<*r' From the labour constraint, this implies that:
Since the consumption shares are equal in this specification, the model implies that labour is therefore allocated equally to both food and non-food goods. Hence, half of each person's labour is devoted to each good, but less than half of that labour is actually used in producing the goods, because there are learning costs, which are incurred twice, once for each good. Turning now to the land first order necessary condition:
£ ^ L : ( l - , ) 6 - V =(!_,)>.,*-i
From the land constraint, this implies that:
16
C. G. Tombazos, S. M. Miller
uA=({l-ln)-a)\x-tn)b.{ln-a)atbn={*.5-af What is interesting to note about this real utility function is that, in addition to the absence of prices (since markets are not used), well-being is, marginally, independent of transaction efficiencies. This condition will be recalled later so that the threshold at which the economy switches from autarky to the complete division of labour can be determined. Structure D (division of labour): Food producers Under division of labour, everything changes. Now households face a budget constraint, and they face transaction costs in purchasing consumption goods. Now households specialise in producing one crop (food or non-food), such that all labour is allocated to a single activity, the proceeds from which are used to purchase all other goods (in multiple good models), in this case one. Now, labour shares are no longer the choice variable, instead the choice is how much of the good produced to supply to the market, and how much to keep for domestic consumption. The model in this configuration also makes room for Adam Smith's brilliant insight, that you can still have trade (within a country) between people who start off identical. The driving force behind this is that people now choose to be different in their occupations, because there are gains from specialising, a feature that is not inherent to the typical neoclassical model. Under specialisation there are two cases, the food crop and the nonfood crop producers. The mathematics for multiple good models can quickly become cumbersome (for the model with m goods there are 2m possible zero/non-zero combinations to consider). Fortunately, there is a so-called Wen theorem, proven by Wen (1998), which holds that no individual buys and self-provides the same good. This simplifies the mathematics of the model significantly. Applying this theorem to the
Returning to the Classical Tradition
17
food crop specialist, since a food producer would not buy that same food crop, but would instead purchase all other goods, the general problem simplifies to: maxwD = ( / + (0))((0) + knd) = f -knd (lf-a)atbf-f
f =
/ / + (0) = 1
f/+(0) = l
Pfr+{0) = {0) +
d
Pnn
The next step is to substitute the second and third constraints for labour and land, into the production function. So now / = (l — or) - fs, since all labour and land is allocated to food production. This can be substituted into the utility function. Additionally, the budget constraint can be solved for non-food purchased from the market n = —— so that it too can be substituted into the utility function. After making these substitutions, the problem is now to maximise the following utility function. max uDJ nf =
,((!-„)"-/•).*.
f
V Pn
This problem calls for maximising utility in terms of food supplied. Hence, du
D,f
= -k-
df
r
Pff Pn
du
DJ
df
^.((l-a)a-r).k<0
+
Pn
=o
In autarky, the marginal cost of producing more of one good entails consuming less of the other. Here the marginal cost of supplying more to the market requires having less of that good to consume at home (since the Wen theorem implies that no person buys and self-provides the same good). Applying the Kuhn-Tucker theorem to these two first order
18
C. G. Tombazos, S. M. Miller
necessary conditions reveals that food supplied to market must be positive. If it were zero,' then — • ((1 - a)")- k<0, which is infeasible if prices, transaction efficiency, and the learning costs are all positive. Hence, the individual's optimal food supply to the market is:
duD,f dfs 'J
.fs=il-aT 2
The greater the returns to specialisation (a), or the lower the learning cost of producing food, the greater amount of food is supplied. When combined with the budget constraint, this implies that the amount of food demanded is: n'
=
l a a =Pf( ~ )
Pff' Pn
Pn
2
As in neoclassical economics, the first law of demand, that generally requires the quantity of the good demanded to be negatively related to its relative price, is satisfied. Also, as under autarky, the point here is to find the real utility function, in terms of prices, for food crop producers. Inserting the optimal supply and demand into the utility function yields, uDf=f-knd
=k-^-± '— 4 Pn Points to observe about this measure of well-being is that, ceteris paribus, food producers are better off as the relative price of food rises, as the transaction efficiency rises, as the returns to specialisation {a) rises, or as the learning cost for producing food crops falls. Structure D (division of labour): Non-food producers Following the logic from above, after simplifying the general problem, the problem for non-food producers is symmetric:
maxuD,n=((0) + fd)(n + (0)) = kfd-n n = {ln-a)"tbn-ns
(0) + / „ = l
Returning to the Classical Tradition
19
(<>) + *„ = 1 {0)
s
=pffd+(0)
Pnn
+
which, after substituting in the constraints, reduces to: max ur
Pf
By similar reasoning, the symmetry of the model implies that for nonfood crop producers, their supply function is characterised by: duD,n_^s dns
=
(l-a)a
'
2
and their demand for the food crop is: fd
=
P„ns Pf
=
Pn C 1 -")" 2
Pf
Substituting in the optimal supply and demand functions into the utility function yields: U
D,n
-
*
Pn {l-afa .
4 Pf As with the real utility function for food producers, ceteris paribus, non-food crop specialists are better off as the relative price of the crop they produce rises, as the transaction efficiency rises, as the returns to specialisation (a) rises, or as the learning cost for producing non-food crops falls.
Step iv — Market demand and supply Compared to the neoclassical view, market demand and supply are defined in a slightly different, but consistent way. First, note that markets in this model only exist under complete specialisation. In the two good case, the market demand for one good actually equals the individual demands for that good multiplied by the mass of suppliers of the other
20
C. G. Tombazos, S. M. Miller
good, since no person buys and self-produces the same good. Hence for food, market demand is given by:
Pn (l~aT
Fd=Mn-fd=Mn
l
Pf
where again, the first law of demand holds, and also, the quantity of food demanded is increasing in the population of consumers, the number of non-food producers. Market supply is defined by:
From this, it is evident that market supply of food is increasing in the number of food producers, a reduction in fixed learning costs of producing food, and an increase in the returns to specialisation. While the first is due to the fact that more producers means more supply. The second is because a reduction in learning costs means more time per person is actually devoted to producing the crop. Finally, an increase in the returns to specialisation results from the fact that for each unit of time allocated, a higher amount of output is achieved. Equating demand and supply results in:
= Mf Pf
(1-a)-
= FS
2
which implies that:
K M
f
_Pf Pn
Since Walras's Law holds, in the two-good case presented here, it is sufficient to consider only the market clearing conditions for food to determine equilibrium. However, just to be sure, with properties that are symmetric to food crops, market demand for non-food crops is given by: Nd =Mf-nd
=Mf-^-^ Pn
'— 2
Returning to the Classical Tradition
21
where again the first law of demand for non-food crops holds, and demand is once more increasing in the population of consumers, in this case food producers. Non-food crop supplies are defined by:
(l-a)a Ns =M
-ns =M-±
'— 2 which is likewise increasing in the number of producers, declining in the costs of learning to produce the crop, and increasing in the returns to specialisation. Market equilibrium for non-food crops is defined by:
pf (i-a)a
, f
Pn
2
(i-«r " 2
which yields an identical condition: MIL=PL M
f
Pn
From this, it is possible to determine the relative price between food and non-food crops. Consider rearranging that condition as follows: M
nPn =
M
fPf
Since everyone is identical, and no two people are charged different prices, then it must be the case that this condition holds between any pair of food and non-food specialists, hence, the two prices are equal: Pn=Pf With market equilibrium defined, it is now possible to address the final step, which is to identify the point in transaction efficiency space at which the world shifts from autarky to the division of labour. Step v-vi — Autarky vs specialisation Recall that it is the indirect utility function that helps to determine in which configuration the individuals find themselves. The indirect utility functions for each subsistence farmers, food crop specialists, and nonfood crop specialists are, respectively:
22
C. G. Tombazos, S. M. Miller
= (0.5-a)2 (1 -a)
uDf
4
Pn
\2a U
D,n
( I " -a) 4
=
Pf The logic concerning how to interpret these results is simple. If the individual is "happier", that is, has a higher real income, in one configuration than in any other, she will choose that configuration. Also, within a particular configuration (the division of labour in this case), it must be the case that the utility of a particular occupation equals that of all others. Hence, for two goods and two configurations, there are only two conditions to check. Within the division of labour configuration it must be the case that utility across occupations is identical:
. (I-af
, (I-af
4
4
The more important case to consider entails a comparison between real utility under autarky with that under the division of labour. If the real income under autarky is greater than real income under specialisation, then autarky will be chosen: v
(0 S
IT
1
^ /r.
V
'
~
U
D,f
~
U
D,n
Alternatively, division of labour chosen is: ^-(0.5
a)
'
= UDj
=
UDn
Isolating both cases in terms oik, suggests that autarky is chosen over the division of labour if the transaction efficiency parameter k falls below:
Returning to the Classical Tradition
23
, fas-«r (l-af and division of labour is chosen if:
{l-af Hence, the model implies that a fundamental problem in generating higher real income is to reduce transaction inefficiencies, which illustrates the similarity between the inframarginal and transaction cost approaches to economic development. So again, even this simple model can provide a more or less formal presentation of some of Smith's ideas about development: 1. Transaction inefficiencies are a key impediment to realising higher levels of well-being. 2. Higher levels of well-being are attainable if people make choices about their professional specialisation, as opposed to having that decision made for them by someone else. 3. Trade occurs even between ex ante identical individuals. 6. Concluding Remarks As illustrated in the context of the simple model discussed in the previous sections, the inframarginal perspective facilitates a reconciliation between division of labour considerations and neoclassical frameworks of distribution. The resulting analysis sheds light on a host of topics relevant to development economics that have been largely neglected by marginalist orthodoxy. Examples include the benefits from specialisation when trading partners are ex ante identical, the interaction of technological comparative advantage, factor endowments, and economies of specialisation, the role of transaction costs in the organisation of the economy, and so forth. Naturally, this field of research has generated considerable interest in recent years. However, little has been done by way of organising the
24
C. G. Tombazos, S. M. Miller
accumulated knowledge in a single volume. This anthology fills this gap by collecting key inframarginal contributions to development economics. As such, this volume serves both as an introduction of the field to the new researcher as well as a useful source of reference to those actively involved in this area.
References 1.
H. W. Arndt, Economic Development: The History of an Idea (University of Chicago Press, Chicago, 1987).
2.
X. Yang and Y. K. Ng, Specialization and Economic Organization: A New Classical Economic Framework (North-Holland, Amsterdam, 1993).
3.
R. Bates, States and Markets in Africa (University of California Press, Berkeley, 1981).
4.
D. C. North, Institutions, Institutional Change and Economic Performance (Cambridge University Press, Cambridge, 1990).
5.
D. M. Levy, Economic Ideas of Ordinary People: From Preferences to Trade (Routledge, London, 1992).
6.
A. Smith, An Inquiry into the Nature and Causes of the Wealth of Nations (Liberty Fund, Indianapolis, 1981).
7.
A. Smith, The Theory of Moral Sentiments (Liberty Fund, Indianapolis, 1982).
8.
A. Smith, Lectures on Jurisprudence (Liberty Fund, Indianapolis, IN, 1982).
9.
D. M. Levy, The European Journal of the History of Economic Thought, 2 (1995).
10.
D. M. Levy, How the Dismal Science Got Its Name: Classical Economics and the Ur-Text of Racial Politics (The University of Michigan Press, Ann Arbor, 2001).
11.
D. M. Levy, Economic Inquiry, 35 (1997).
Returning to the Classical Tradition
25
12.
G. Stigler, Journal of Political Economy, 84 (1976).
13.
J. M. Buchanan and Y. J. Yoon, History of Political Economy, 31 (1999).
14.
H. S. Houthakker, Kyklos, 9 (1956).
15.
X. Yang, Economics: New Classical Versus Neoclassical Frameworks (Blackwell Publishers, Maiden and Oxford, 2001).
16.
R. Coase, Economica, 13 (1946).
17.
T. C. Koopmans, Three Essays on the State of Economic Science (McGraw-Hill, New York, 1957).
18.
S. Rosen, Economica, 45(1978).
19.
S. Rosen, Journal of Labor Economics, 1 (1983).
20.
G. S. Becker and K. M. Murphy, The Quarterly Journal of Economics, 107 (1992).
21.
R. Tamura, Journal of Political Economy\ 99 (1991).
22.
R. Tamura, Journal of Economic Theory, 58 (1992).
This page is intentionally left blank
Part 2
The Origins of Inframarginal Applications to the Study of Economic Development
This page is intentionally left blank
CHAPTER 2 THE MARGINAL COST CONTROVERSY*
Ronald H. Coase University of Chicago Law School 1.
The State of the Debate
I wish to discuss in this article the question of how prices ought to be determined in conditions of decreasing average costs. In particular, I wish to discuss one answer to this question which is by now familiar to most economists and which may be summarised as follows: (a) The amount paid for each unit of the product (the price) should be made equal to marginal cost. (b) Since, when average costs are decreasing, marginal costs are less than average costs, the total amount paid for the product will fall short of total costs. (c) The amount by which total costs exceed total receipts (the loss, as it is sometimes termed) should be a charge on the Government and should be borne out of taxation. This view has been supported by Professor H. Hotelling,1 Professor A. P. Lerner,2 Mr. J. E. Meade and Mr. J. M. Fleming.3 It has aroused * Reprinted from Economica, 13, Ronald H. Coase, "The Marginal Cost Controversy," 169-182, 1946, with permission from Blackwell. ' "The General Welfare in Relation to Problems of Taxation and of Railway and Utility Rates," Econometrica, July, 1938. 2 In his book, The Economics of Control (1944). Professor Lerner had earlier set out this view in articles in the Review of Economic Studies and in the Economic Journal. 3 "Price and Output Policy of State Enterprise," Economic Journal, December, 1944. See also J. E. Meade, Economic Analysis and Policy, 1936, pp. 182-186, American edition (1938), pp. 195-199.
29
30
R. H. Coase
considerable interest and has already found its way into some textbooks on public utility economics.4 But despite the importance of its practical implications, its paradoxical character, and the fact that there are many economists who consider it fallacious, it has so far received little written criticism.5 It may have been the sheer quantity of literature in favour of this solution, and the relatively small amount of written adverse criticism which led Mr. J. M. Fleming to claim that it "is not, I think, open to serious criticism" and to lament the fact that it was not more widely understood and accepted "outside the narrow ranks of the economists." But a different solution, which I believe in essentials to be the correct one, had already been suggested by Mr. C. L. Paine in 19376 and by Professor E. W. Clemens in 1941. 7 I wrote in 1945 a short note
See C. Woody Thompson and Wendell R. Smith, Public Utility Economics, pp. 271-273, and Irston R. Barnes, The Economics of Public Utility Regulation, pp. 586-588. See also Professor Emery Troxel, "I Incremental Cost Determination of Utility Prices," "II Limitations of the Incremental Cost Patterns of Pricing," "III Incremental Cost Control under Public Ownership," Journal of Land and Public Utility Economics, November, 1942, February, 1943, and August, 1943; and Professor James C. Bonbright, "Major Controversies as to the Criteria of Reasonable Public Utility Rates," Papers and Proceedings, American Economic Association, December, 1940. Professor Bonbright points out that the "extreme social conservatism of most public utility and railroad specialists had prevented" this solution "from gaining wide acceptance, or even from receiving any considerable notice, in the literature of rate theory." However, he thought that it might become a live issue in the next few years (after 1940) as a result of Professor Hotelling's article, which Professor Bonbright considered to be "one of the most distinguished contributions to rate-making theory in the entire literature of economics." 5 It is true that Professor Ragnar Frisch criticized Professor Hotelling's article shortly after it appeared. But, though much of interest emerged in Professor Frisch's note and the subsequent discussion with Professor Hotelling, it appears, at least to the non-mathematical reader, that Professor Frisch's attack was not directed at the foundations of Professor Hotelling's argument but rather to what seemed to him to be defects in its formulation. See Ragnar Frisch, "The Dupuit Taxation Theorem" and "A Further Note on the Dupuit Taxation Theorem" and H. Hotelling, "The Relation of Prices to Marginal Costs in an Optimum System" and "A Final Note", Econometrica, April, 1939. 6 See C. L. Paine, "Some Aspects of Discrimination by Public Utilities," Economica, November, 1937. 7 See E. W. Clemens, "Price Discrimination in Decreasing Cost Industries," American Economic Review, December, 1941.
The Marginal Cost
Controversy
31
criticising the solution as set out by Mr. Meade and Mr. Fleming, and a further note by Dr. T. Wilson9 underlined the fact that agreement among economists had not yet been reached. I now propose to examine the Hotelling-Lerner solution, as I shall call it, in greater detail and to point out the fundamental defects which I believe it contains. 2.
Isolation of the Problem
Any actual economic situation is complex and a single economic problem does not exist in isolation. Consequently, confusion is liable to result because economists dealing with an actual situation are attempting to solve several problems at once. I believe this is true of the question I am discussing in this article. The central problem relates to a divergence between average and marginal costs. But, in any actual case, two other problems usually arise. First, some of the costs are common to numbers of consumers and any consideration of the view that total costs ought to be borne by consumers raises the question of whether there is any rational method by which these common costs can be allocated between consumers. Secondly, many of the so-called fixed costs are in fact outlays which were made in the past for factors, the return to which in the present is a quasi-rent, and a consideration of what the return to such factors ought to be (in order to discover what total costs are) raises additional problems of great intricacy.10 These are, I think, the other two problems which usually exist simultaneously with a divergence between average and marginal costs. They are, however, separate or at least separable questions. Thus, the example used by Professor Hotelling, the problem of pricing in the case of a bridge,11 is in fact an extremely complex case rather than the simple one it appears to be on the surface.
"Price and Output Policy of State Enterprise: A Comment," Economic Journal, April, 1945. 9 "Price and Output Policy of State Enterprise," Economic Journal, December, 1945. 10 See F. A. Hayek, Collectivist Economic Planning, the section entitled "The Criterion of Marginal Costs," pp. 226-231. 11 This example was originally used by Dupuit in an article in the Annates des Ponts et Chaussees (1844) which was reprinted in De Vutilite et de sa mesure (1933).
32
R. H. Coase
I propose to isolate the question at issue by examining an example in which, although there is a divergence between marginal and average costs, all costs are attributable to individual consumers; in which all costs are currently incurred; and in which, to avoid a further complication which might trouble some readers concerning the meaning of marginal cost, all factors are in perfectly elastic supply. Assume that consumers are situated around a central market in which a certain product is available at constant prices. Assume that roads run out from the central market but that each road passes only one consumer of the product. Assume also that a carrier can carry on each journey additional units of the product at no additional cost (at least to a point beyond the limit of consumption of any individual consumer).12 Assume further that the product is sold at the point of consumption. It is clear that the cost of supplying each individual consumer would be the cost of the carrier plus the cost at the central market of the number of units consumed by that particular consumer of the product. The marginal cost would be equal to the cost of a unit of the product at the central market. The average cost would be higher than the marginal cost and would decline as the cost of the carrier was spread over an increasing number of units.13 The Hotelling-Lerner solution would presumably be that the amount which consumers should pay for each unit of the product should be equal only to marginal cost. The effect would be for consumers to pay for the cost of the product at the central market and for the Government, or rather the taxpayer, to bear the costs of carriage. It is the validity of this solution that I wish to examine. But first it is necessary to turn to a consideration of fundamentals.
12
An indivisibility must be present in all cases of decreasing average costs. Although I assume that it is not possible to employ less than a carrier, his services may be assumed to be in perfectly elastic supply in that payment will vary proportionately with the time he is employed and that the additional employment of carriers will not raise their price. 13 The assumption that the total costs consist of two distinct kinds, one of which enters into marginal cost while the other does not, is not essential. We could have assumed that the costs of carriage increased as additional units were carried but that the marginal costs of carriage were below the average. It will, however, aid in exposition if we keep to the original assumption.
The Marginal Cost
3.
Controversy
33
What is Optimum Pricing?
I take a pricing system to be one in which individual consumers have command over various sums of money which they use to obtain goods and services by spending this money in accordance with a system of prices. It is, of course, not the only method of allocating goods and services, or more properly, the use of factors of production between consumers. It would be possible for the Government to decide what to produce and to allocate goods and services directly to consumers. But this would have disadvantages as compared with the use of a pricing system. No Government could distinguish in any detail between the varying tastes of individual consumers (which is, of course, why a "points" system of rationing in wartime is adopted for many items);14 without a pricing system, a most useful guide to what consumers' preferences really are would be lacking; furthermore, although a pricing system puts additional marketing costs on to consumers and firms, these may in fact be less than the organising costs which would otherwise have to be incurred by the Government.15 These are the reasons which would lead an enlightened Government to adopt a pricing system — and we shall see later that they are very relevant to the problem we are considering. If it is decided to use a pricing system, there are two main problems that have to be solved. The first is how much money each individual consumer shall have — the problem of the optimum distribution of income and wealth. The second is, what is to be the system of prices in accordance with which goods and services are to be made available to consumers — the problem of the optimum system of prices. It is with the second of these problems that I am concerned in this article. The first is partly, though not entirely, a question of ethics. But it is important to realise that there are these two problems and that both have to be solved if a pricing system is to produce satisfactory results. As I am in this section dealing with the second only of these problems, I shall assume that the distribution of income and wealth can be taken to be the optimum. For an individual consumer, the system of prices represents the terms on which he can obtain various goods and services. According to what 15
Compare Lerner, op. cit., p. 53. See R. H. Coase, "The Nature of the Firm," Economica, November, 1937.
34
R. H. Coase
principles should prices be determined? The first would appear to be that for each individual consumer the same factor should have the same price in whatever use it is employed, since otherwise consumers would not be able to choose rationally, on the basis of price, the use in which they prefer a factor to be employed. The second would appear to be that the price of a factor should be the same for all consumers since otherwise one consumer would be obtaining more for the same amount of money than another consumer. If the optimum distribution of income and wealth had been obtained, the effect of charging different prices for the same factor to different people would be to upset that distribution. It is a more subtle application of this second rule that the price fixed should be such as to allow factors to go to the highest bidders. That is, the price should be one which equates supply and demand and it should be the same for all consumers and in all uses.16 This implies that the amount paid for a product should be equal to the value of the factors used in its production in another use or to another user. But the value of the factors used in the production of a product in another use or to another user is the cost of the product. We thus arrive at the familiar but important conclusion that the amount paid for a product should be equal to its cost. It will be this principle which will enable us to discuss the problems of individual pricing without tracing throughout the economic system all the changes consequent upon the alteration of a single price. 4.
The Argument for Multi-Part Pricing
How does this general argument for basing prices on costs apply to the case we are considering — the case of decreasing average costs? The writers whose views I am considering seem to assume that the alternatives with which one is faced are to charge a price equal to marginal cost (in which case a loss is made) or to charge a price equal to average cost (in which case no loss is made). There is, however, a third possibility — multi-part pricing. In this section, I set out the argument for multi-part pricing when there are conditions of decreasing average costs.
Compare also Lerner, op. cit., pp. 45-50.
The Marginal Cost
Controversy
35
It is clear that if the consumer is not allowed to obtain at the marginal cost additional units of products, produced under conditions of decreasing average costs, he is not being allowed to choose in a rational manner between spending his money on consuming additional units of the product and spending his money in some other way, since the amount which he would be called upon to spend to obtain additional units of the product would not reflect the value of the factors in another use or to another user. But for the same reason it can be argued that the consumer should pay the total cost of the product. A consumer does not only have to decide whether to consume additional units of a product; he has also to decide whether it is worth his while to consume the product at all rather than spend his money in some other direction. This can be discovered if the consumer is asked to pay an amount equal to the total costs of supplying him, that is, an amount equal to the total value of the factors used in providing him with the product. To apply this argument to our example, the consumer should not only pay the costs of obtaining additional units of the product at the central market, he should also pay the cost of carriage. How can this be brought about? The obvious answer is that the consumer should be charged one sum to cover the cost of carriage while for additional units he should be charged the cost of the goods at the central market. We thus arrive at the conclusion that the form of pricing which is appropriate is a multi-part pricing system (in the particular case considered, a two-part pricing system), a type of pricing well known to students of public utilities and which has often been advocated for just the reasons which I have set out in this article.17
See H. F. Havlik, Service Charges in Gas and Electric Rates, and references therein. See also Barnes, op. cit., p. 588. Havlik himself appears to support the view that costs which are attributable to individual consumers should be charged to those consumers. He does, however, use a variant of the Hotelling-Lerner solution when dealing with the case in which what he terms marginal customer costs, "the additional costs of taking on a customer and maintaining the connection, without actually supplying him with electricity," are less than average customer costs. In this case, "revenues from a customer charge would be less than total customer costs" and it would be "justifiable" for the Government "to give a subsidy" (pp. 92-93). Havlik does not discuss how the subsidy ought to be raised. In this article I am, however, concerned simply with the case in which all costs are attributable to individual consumers and to this case Havlik's variant of the Hotelling-Lerner solution, which is concerned with common costs, does not apply.
36
R. H. Coase
Now it is, I think, extremely significant that none of the advocates of the Hotelling-Lerner solution should have examined the possibilities of multi-part pricing as a solution of the problem they are considering. They write as though the only possible method of pricing is to charge a single price per unit and the problem they have to solve is what that price should be. It may be that their reason for not examining multi-part systems of pricing was that they were sure they had in fact found the optimum system of pricing. We must therefore compare the results of adopting the Hotelling-Lerner solution with those of using multi-part pricing. 5.
Multi-Part Pricing Compared with the Hotelling-Lerner Solution
The Hotelling-Lerner solution, if adopted in the case of my example, would mean that the cost of the goods at the central market would be paid for by consumers but that the cost of carriage would be borne out of taxation. My objections to this solution as compared with adopting a two-part system of pricing fall under three heads: first, that it leads to a maldistribution of the factors of production between different uses; second, that it leads to a redistribution of income; and third, that the additional taxation imposed will tend to produce other harmful effects. First, the Hotelling-Lerner solution would appear to remove the means whereby consumers make a rational choice between the use as carriers and the use for some other purpose of the factors which enter into the cost of carriage. In this use, the factor would be free; in another use (provided that it entered into marginal cost) it would have to be paid for. Similarly, this solution would mean that consumers would choose between different locations without taking into account that the costs of carriage vary as between one location and another. The answer which the supporters of the Hotelling-Lerner solution would make to this objection would appear to be that the Government should estimate for each individual consumer in my example whether he would buy the product and also what location he would prefer, if he had
The Marginal Cost
Controversy
37
to pay the total cost.18 Only if the consumer would thus have been prepared to pay the total cost of supplying the product to a given location will provision for supplying it to that location be made under the Hotelling-Lerner scheme. Professor Hotelling points out that to decide whether the demand was sufficient to warrant the costs of building a bridge "would be a matter of estimation of vehicular and pedestrian traffic originating and terminating in particular zones, with a comparison of distances by alternative routes in each case, and an evaluation of the saving in each class of movement".19 If it were possible to make such estimates, at low cost and with considerable accuracy and without knowledge of what had happened in the past when consumers had been required to pay the total cost, this would be likely to lead, in my opinion, not to a modification of the pricing system but rather to its abolition. The pricing system, as I pointed out earlier, is a particular method of allocating the use of factors of production between consumers and the arguments for its adoption derive their main force from the view that such estimates of individual demand by a Government would be very inaccurate. It should be noted here that neither Professor Lerner nor Mr. Meade in fact make any considerable claim for the accuracy of these estimates. Indeed, Professor Lerner in an earlier section of his book argues for a pricing system on precisely the grounds that it is impossible for a Government to make such estimates.20 Neither Professor Hotelling, Professor Lerner nor Mr. Meade give in my view sufficient weight to the stimulus to correct forecasting which comes from having a subsequent market test of whether consumers are willing to pay the total cost of the product. Nor do they recognize the importance of the aid which the results of this market test give in enabling more accurate forecasts to be made in the future. Professor Hotelling says: "Defenders of the current theory that the overhead costs of an industry must be met out of the sale of its products or services hold that this is necessary in order to find out whether the creation of the See Lerner, op. cit., pp. 186-199 and Meade, loc. cit., pp. 324-325. And it would seem that Professor Hotelling's mathematical formulation comes to much the same thing, see loc. cit., p. 262 and p. 268. 19 Loc. cit., pp. 247-248. 20 Op. cit., pp. 61-64.
38
R. H. Coase
industry was a wise social policy. Nothing could be more absurd." This, he says, "is an interesting historical question".21 And he adds later: "When the question arises of building new railroads or new major industries of any kind or of scrapping the old, we shall face, not a historical, but a mathematical and economic problem".22 Nowhere in Professor Hotelling's article does one find recognition of the fact that it will be more difficult to discover whether to build new railroads or new industries if one does not know whether the creation of past railroads or industries was wise social policy. And it is certainly not absurd to take into account the fact that decisions are likely to be better made if afterwards there is some test of whether such decisions were wise social policy than if such an enquiry is never made. I do not myself believe that a Government could make accurate estimates of individual demand in a regime in which all prices were based on marginal costs. But it may be well to consider what would be likely to be done if a Government attempted to carry out the Hotelling-Lerner policy. Consider the example I have been discussing. Certain consumers would have to be designated as able to buy the product. The Government would then undertake to pay whatever costs for carriage were incurred on behalf of these consumers. A Government would have a difficult task in deciding where to draw the line. If it adopted a narrow view of the qualifications required of those allowed to consume this product, consumers who really preferred to use the factor employed in the carriage of the product in this way, would be prevented from doing so. If on the other hand it was liberal in its view, many would find that they were no longer deterred from consuming the product or living at a greater distance from the central market by the cost of the factor used in carriage, that is, by its value in alternative uses or to an alternative user. It would, of course, be possible for the Government to follow a liberal policy to one class of consumers and a narrow policy to others at the same time. It is not easy to guess what policy a Government would be likely to follow. But in Great Britain I suspect that it would
21 22
hoc. cit., p. 268. hoc. cit., p. 269.
The Marginal Cost
Controversy
39
tend to err on the liberal side and that there would consequently be too great an employment of the factor used in the carriage of the product. But even if the Government were able to estimate individual demands accurately, the Hotelling-Lerner solution would be subject to another objection. The Government is supposed to estimate which consumers would be willing to pay the cost of carriage (and we shall assume for the moment that it estimates correctly). But it does not in fact ask these consumers to pay this sum. This money is then available for these consumers to spend on some other commodity. Consumers who buy products which are produced under conditions of decreasing average costs will therefore obtain products for any given expenditure embodying a greater value of factors than those who do not. There is a redistribution of income in favour of consumers of goods produced under conditions of decreasing average costs.24 There would not, I think, be any dispute that what is equivalent to a redistribution of income does in these circumstances take place. Professor Hotelling is, however, the only one of the writers whose views I am examining who deals explicitly with this point. I shall therefore examine his reasons for thinking that this objection is of little substance. First of all, I believe that Professor Hotelling considers this objection to be largely irrelevant because the initial distribution of income, at least in the United States, is not in fact the optimum. He does not directly say this but it is evident from his whole approach to the question.25 When he argues that the loss resulting from an application of the marginal cost rule should be borne out of income taxes, inheritance taxes and taxes on the site value of land, he is, I think, partly doing so because he believes that the wealthy and the landlords already have too large a share of the total wealth and income. But why should consumers of goods produced under conditions of decreasing average costs be the only ones to benefit from this All the essentials of this argument have been set out in another connection by Cannan, in his History of Local Rates in England (Second Edition). See Chapter VIII, "The Economy of Local Rates" and especially his remarks on p. 187. 24 This assumes that the taxes from which the loss is made good do not fall entirely on consumers of goods produced under conditions of decreasing average costs. This is, of course, so because it is proposed that the taxes to be used should be income and similar taxes. 25 See, for example, his remarks, loc. cit., p. 259.
40
R. H. Coase
redistribution? The reason why Professor Hotelling sees little harm in using pricing policy partly as a means of redistributing income is, I think, that he does not consider the distinction between consumers of products produced under conditions of decreasing average costs and consumers of products produced under conditions of constant or increasing average costs to be of great importance. He argues that a Government carrying out his policy would undertake a great variety of public works. "A rough randomness in distribution would be ample to ensure such a distribution of benefits that most persons in every part of the country would be better off by reason of the programme as a whole".26 This comes to saying that in a regime of marginal cost pricing, all consumers will buy goods produced under conditions of decreasing average costs; that what is lost by any particular consumer in the redistribution involved in one scheme will be offset as a result of the redistribution following on another scheme; and that as a consequence, the significant redistribution would be from the wealthy and landlords to all others. It would be indeed pedantic to object to the achievement of a desirable aim merely because it is done in an unusual way. But this argument stands or falls by the assumption that there will be no significant redistribution as between consumers of different kinds of products. There is no reason to assume that this will be so. The gain which individual consumers would derive from the Hotelling-Lerner policy would depend on the extent to which they were willing to pay the total cost for products produced under conditions of decreasing average costs (given their initial income); and on the absolute divergence between marginal and average costs in the case of these goods; and on the extent to which the additional income derived as a result of the Hotelling-Lerner policy was spent on goods produced under conditions of decreasing average costs; and on the absolute divergence between marginal and average costs in these cases. It would be possible to appraise the character of the redistribution only after a detailed factual enquiry. There seems, however, no reason to suppose that it would be a negligible redistribution. The public utility industries provide some of the most striking instances of products supplied under conditions of decreasing average costs. Let us assume that they are the only industries in which these Loc. cit., p. 259.
The Marginal Cost
Controversy
41
conditions are found. Consumers who live in regions of low density of population would probably not be willing to pay the total costs of supply of public utility services which in their case would be very high, and would consequently gain nothing as a result of the Hotelling-Lerner policy because they would not be given the services. Consumers who live in cities would find their gains limited because, equipment there being relatively intensively used, the divergence between marginal and average cost would probably be much less than elsewhere; while since they probably already use all the public utility services, the additional income would be likely to be spent on other than public utility services. It would be those living in small towns, which have some but not all the public utility services and where the divergence between marginal and average cost was great, who would, I think, tend to gain most from the Hotelling-Lerner policy. I see no reason to suppose that there would not be some redistribution, possibly very considerable, as a result of this policy if it were generally applied. Professor Hotelling admits this possibility but claims that by a subsequent redistribution a situation could be produced in which everyone was better off than before.27 He does not describe how this redistribution would be effected. But it would obviously be an inferior arrangement to adopting a multi-part system of pricing which makes it unnecessary to have subsequent redistributions of income at all. I am, however, at a loss to understand how ordinary taxation procedures could be used to redistribute income from consumers of goods produced under conditions of decreasing average costs to all other consumers. An attempt to do this might be made by means of a tax on the consumption of goods produced under conditions of decreasing average costs. But this would either be equivalent to introducing multi-part pricing (if a lump sum tax was levied on consumers) or, if a tax per unit of consumption is imposed, it would bring about a divergence between the amount paid for additional units and marginal cost, a result which it is the object of the Hotelling-Lerner solution to avoid. I now turn to the third objection to the Hotelling-Lerner solution. The loss incurred is, it is said, to be made good by increased taxation. The taxes which Professor Hotelling and the others who support this solution have in Loc. cit., pp. 257-258.
42
R. H. Coase
mind are income taxes, inheritance taxes and taxes on the site value of land. Let us assume for the time being that the form of tax used to make good the loss is an income tax. But income taxes are usually so framed that marginal units of income are taxed, and therefore an income tax will have the same unfortunate effect on consumers' choice as a tax on goods and will produce results similar in character to those which follow from charging an amount for additional units of output greater than marginal cost. After the appearance of Professor Hotelling's first article, he seems to have had his attention drawn to this point by Professor Lerner. Professor Hotelling says in the discussion with Professor Frisch which followed his original article that "an income tax of the usual kind is a sort of excise tax on effort and on waiting, as well as on other less defensible ways of getting an income. An income tax is to some extent objectionable because it affects the choice between effort and leisure and the choice between immediate and postponed consumption. Thus some of the same kind of loss attaches to an income tax as to excise taxes proper. How serious this effect may be is a question for factual research; but there is some reason to suppose an income tax superior to excise taxes on individual commodities in this respect....".28 Professor Hotelling does not give any reasons why he thinks income taxes will tend to be less harmful in this respect than excise taxes. It may be so but it is obviously desirable to know what the circumstances are in which income taxes are less harmful and when they are likely to be found before applying the Hotelling-Lerner solution — if, that is, this policy would lead to increases in income taxes.29 Professor Hotelling attempts to avoid this difficulty by suggesting that "the public revenues, including those required to operate industries with sales at marginal cost, should be derived primarily from rents of land and other scarce goods, inheritance and windfall taxes, and taxes designed to reduce Econometrica, April, 1939, pp. 154-155.1 would add that income taxes also affect the choice between doing a job for oneself and employing some one to do it for one and in consequence an income tax dissipates some of the advantages of specialisation. See F. W. Paish, "Economic Incentive in Wartime," Economica, August, 1941, p. 244. 29 This problem seems to have been overlooked in the theory of public finance. The usual discussion of the burden of indirect taxation assumes that the alternative is a lump sum payment. See for example, M. F. W. Joseph, "The Excess Burden of Indirect Taxation," Review of Economic Studies, June, 1939. Compare also J. R. Hicks, Value and Capital, p. 41.
The Marginal Cost
Controversy
43
socially harmful consumption." This is not a very satisfactory solution. First of all, it assumes that such taxes will be sufficient to raise the sum required. Secondly, it assumes that the disturbance to the distribution of income and wealth due to the additional taxation on those who derive their incomes in these ways is better than the loss which would occur if the additional taxation was spread more evenly over people in the country. Alternatively, Professor Hotelling's suggestion involves the assumption that the optimum distribution of income and wealth has not already been achieved and that those who derive their incomes in these ways have not been taxed enough in the past. But, of course, if this is so, this further taxation is desirable quite apart from questions of pricing policy and there is little need to link it to the problem of pricing under conditions of decreasing average costs. Furthermore, the question would still remain of how the pricing problem should be solved when the optimum distribution of income and wealth was achieved. Professor Hotelling's suggestion for avoiding the loss which would result from increased income taxes is one of limited validity. In this section, I have compared the results of using a multi-part pricing system with those which would follow from the Hotelling-Lerner policy. I have shown that the Hotelling-Lerner solution would bring about a maldistribution of the factors of production, a maldistribution of income and probably a loss similar to that which the scheme was designed to avoid, but arising out of the effects of increased income taxes. These results would be avoided by the use of a multi-part system of pricing. 6.
Average Cost Pricing Compared with the Hotelling-Lerner Solution
Professor Hotelling, Professor Lerner, Mr. Meade and Mr. Fleming do not seem to have realised that many of the problems which they were trying to solve could have been dealt with by means of multi-part pricing and that this system of pricing would in fact have produced results not open to the objections which could be brought against the Hotelling-Lerner solution. But in fairness to them it must be pointed out that their attack was directed Econometrica, April, 1939, p. 155.
44
R. H. Coase
against charging a single price which was based on average cost and not against multi-part pricing. Is the argument valid in this case? If multi-part pricing is not possible, is it not preferable to adopt the Hotelling-Lerner solution rather than to adopt pricing based on average costs? In this case, the argument for the Hotelling-Lerner solution is considerably strengthened — and this in two respects. First of all, it is clear that if consumers are not allowed to buy additional units at marginal cost, there is a maldistribution of the factors of production. The nature of the gain which would accrue in this respect through the adoption of the Hotelling-Lerner solution has already been discussed in earlier sections.31 The second respect in which the argument for the Hotelling-Lerner solution is strengthened concerns the effectiveness of average cost pricing in providing a market test of the willingness of consumers to pay the total costs. In the previous section, I pointed out that multi-part pricing furnished such a test. How does this apply to the case of average cost pricing? The fact that consumers are willing to buy at a price which covers average costs certainly shows that they prefer to obtain that value of factors in that form rather than in any other which is open to them.32 The difficulty is, as Professor Hotelling points out, that the reverse is not true. It has long been known to economists that in cases in which the demand curve lies at all points below the average cost curve, it may be possible, by means of price discrimination, to raise the average revenue sufficiently to bring it up to average cost. If therefore pricing is on an average cost basis, there will be certain cases in which consumers would have been willing to pay the total cost but in which, owing to the limitations of this particular method of pricing, this would not be possible. Production could be undertaken in such cases if the Hotelling-Lerner policy was followed. These are the advantages of the Hotelling-Lerner solution as compared with average cost pricing. But the disadvantages which were examined in the previous section still remain. These have to be balanced 31
It might be thought that if all goods were priced on an average cost basis, since all prices would be raised above the marginal cost level, the choice of the consumers would be unaffected. But this would be true only if the rise in price were proportionate to marginal cost and this is most unlikely to be true. See the discussion between Professor Frisch and Professor Hotelling in Econometrica, April, 1939. 32 Compare Wicksteed, The Common Sense of Political Economy, pp. 675-676.
The Marginal Cost Controversy
45
one against the other. The first advantage which the Hotelling-Lerner solution possesses as compared with average cost pricing is that it allows a better choice at the margin in consumption. But this advantage would be reduced and might be offset by the loss which would result if the Hotelling-Lerner solution involved increased income taxes. The second advantage is that a Government could undertake production in cases in which consumers would be willing to pay the total cost but which could not be undertaken with average cost pricing. But it has to be remembered that this policy is one in which the Government estimates individual demands and is therefore subject to the limitations which we discussed in the previous section. Not all cases in which production would not be undertaken with average cost pricing ought to be undertaken. A Government which made many errors in its estimates of individual demands could easily offset any good such a policy might produce. Average cost pricing may prevent some things from being done which perhaps ought to be done but it is also a means of avoiding certain errors in production, some of which would inevitably be made if the Hotelling-Lerner policy were followed. As I indicated earlier, I do not myself believe that it is reasonable to assume that the Government could make accurate estimates of individual demands if all prices were based on marginal cost. Finally, there is the redistribution of income and wealth which the Hotelling-Lerner solution would involve and which, as I pointed out in the previous section, would appear to be difficult to rectify in the absence of multi-part pricing, without reintroducing the kind of tax which would prevent that rational choice at the margin which the Hotelling-Lerner solution aims to achieve. It will be seen from the discussion in this section that the question of average cost pricing as against the Hotelling-Lerner solution does not present any clear-cut case. The claim which is made for the Hotelling-Lerner solution as inevitably superior to average cost pricing must therefore be rejected. 7.
The Problems that Remain
In this article, I have been examining the problem of pricing under conditions of decreasing average costs. I have, however, confined myself
46
R. H. Coase
to one particular case, that in which all costs are attributable to individual consumers and in which all costs are currently incurred. Given these assumptions, I showed that the Hotelling-Lerner solution was inferior to a multi-part system of prices and that as compared with average cost pricing the balance of advantage was not clear. The next steps would appear to be to examine the problem of pricing when there are common costs. If there are costs which cannot be attributed to individual consumers, does the Hotelling-Lerner solution then come into its own, as Mr. H. F. Havlik has suggested?33 Should such common costs be borne out of taxation? Or is the right approach to discover some basis in accordance with which these costs should be allocated among consumers? Finally, there is the question of expenditures which have already been incurred for factors. Are these costs to be borne out of taxation? Or should they be borne by consumers? If the analysis in this article is accepted, these would seem to be the next questions to be examined.
See footnote 17.
CHAPTER 3 ECONOMICS AND BIOLOGY: SPECIALIZATION AND SPECIATION*
Hendrik S. Houthakker* Stanford
University
It is well known that Charles Darwin's work on evolution, according to his own statement, was partly inspired by Malthus' theory of population. To this extent economics may therefore count itself among the sources of modern biology. Apart from this initial link, however, economics has had less contact with biology than with almost any other major science. With the physical sciences, particularly classical mechanics and thermodynamics, economics at least has some conscious affinity of method, and with the social sciences it shares the subject matter (though little else), but with biology it appears to have nothing in common. It would be presumptuous for an economist to argue that closer relations between economics and biology would benefit the other field. All I want to point out here is that economists may derive some useful insight from observation of the non-human living world. This is particularly true for that much-neglected but centrally important chapter of economics: the division of labor, or specialization as it may be more appropriately called in the present context. Specialization, as we shall see, is closely, connected with what biologists call speciation, or the formation of species.
Reprinted from Kyklos, 9 (2), Hendrik S. Houthakker, "Economics and Biology: Specialization and Speciation," 181-189, 1956, with permission from Blackwell. * I am indebted to my colleagues, Melvin Reder and Tibor Scitovsky for useful comments.
47
48
H. S. Houthakker
Adam Smith, in his unsurpassed discussion, declared the division of labor to be peculiar to human societies. He viewed it as the result of a mysterious "propensity to truck," from which animals are somehow immune. We do not, he observed, see two dogs make a fair and reasonable exchange of one bone for another. Related arguments to this effect are also supported by canine examples. It seems that Smith's interest in the animal world did not go beyond dogs; otherwise he might have thought of ants and bees. Even in his day something like division of labor was known to exist in insect societies, and Linnaeus had already described aphids as ants' cows. Smith was carefully ambiguous about the question whether the propensity to truck can itself be reduced to more immediately rational considerations, but he did point out the advantages of following that propensity. These advantages are the increase in skill, the saving of time otherwise necessary to switch from one job to another, and the enhanced possibilities of using tools and machinery. Without much explanation he also indicated how far specialization can go in the famous statement that "the division of labor is limited by the extent of the market." It is not to the credit of economists that in the 180 years following the publication of the Wealth of Nations so little should have been done to clarify this statement, the simplicity of which is quite deceptive. Most economists have probably regarded the division of labor, in Schumpeter's words, as an "eternal commonplace," yet there is hardly any part of economics that would not be advanced by a further analysis of specialization and related phenomena. It should be added, however, that such an analysis involves the use of methods that are rather unlike those by which the classical questions of economics are discussed. These classical questions are treated with the aid of traditional calculus methods (often disguised in literary form) but the latter are not suited to deal with indivisibilities. It is in fact from indivisibilities that the division of labor takes its start, and the basic indivisibility is that of the individual, whether human or animal. This may seem like a play on words, or what is almost the same thing, bad metaphysics, but it is more serious than that. For our purpose we may regard an individual as a coordinated complex of activities. The indivisibility of the individual consists in the fact that, although it may be capable of a great many different activities, it can perform only few activities simultaneously because most activities utilize the same resources and more particularly that coordinating resource which is
Economics and Biology: Specialization and Speciation
49
known as the brain. The larger the number of simultaneous activities, the greater the difficulty of coordinating them and of carrying out each one properly, and the smaller therefore the output from each activity. This applies not only to simultaneous activities, but also to activities that are spread out over time. In the first place some shorter or longer interval is usually needed to switch from one activity to another; in the second place it is usually easier to perform activities that are known from previous experience than to perform them for the first time. All this, the economist will note at once, can be put in terms of increasing returns. We have increasing returns to the extent that, if several activities are replaced by a single one, there is less need for coordination and switching time and more scope for acquiring experience. The output of the single activity may thus be raised above the combined outputs of the several activities. If we have two activities, for instance, the simplest shape of the production possibility curve exhibiting these features will be as in Figure 1. If the individual produces only the first commodity he can obtain AP units, and if only the second AQ units, but if he produces both simultaneously he is bounded by the straight line P'Q' The segments PP or QQ' therefore represent the loss of production due to the need for coordination, etc. If the individual cannot trade with others he will normally be unable to specialize, at least if both commodities are necessary to him. He will then be at a point such as R, where an indifference curve touches the production possibility curve.
Figure 1
50
H. S. Houthakker
As soon as another individual appears on the scene specialization may arise (Figure 2). The individual may then get to a point such as V, whose exact location depends on the respective offer curves (cases of incomplete specialization are also possible). The post-trade point V is better than either of the pretrade points R and U, and both parties therefore gain, but it is possible that only one party gains and the other remains where he was. The case here described differs from the Ricardian case of constant cost in that there will be a tendency to specialization even if there are no differences in relative efficiency (that is, even if P'Q' and S'T are parallel). Specialization may even go against the comparative advantage (that is,P'g'may have a flatter slope than S T ' ) . Under the Ricardian assumptions the benefits of specialization consist wholly in the utilization of comparative advantages, but in the present case, which is essentially that of Adam Smith, the benefit comes mainly from the avoidance of coordination costs (in the widest sense) by the two individuals. It may be noted that Smith went out of his way to deny the existence of innate differences between individuals; in his theory such differences were unnecessary. Like the Declaration of Independence, which dates from the same year 1776, the Wealth of Nations is based on the premise that all men are born equal.
Figure 2
Although specialization will lead to the avoidance of individual coordination costs, it may in turn call for coordination between the two individuals. The simplest example is transportation cost. These external
Economics and Biology: Specialization and Speciation
51
coordination costs may be heavy enough to outweigh the saving in internal coordination costs. In a box diagram such as Figure 2, transportation costs may be represented by shifting all points referring to the second individual to the left, thus narrowing the area between P'Q' and S'T' within which specialization is advantageous. The division of labor is thus limited by the extent of the market. Let us consider a slightly different example of specialization, which has attracted the attention of students of spatial competition such as Hotelling1 and Loesch.2 Imagine a population which is spread out, not necessarily evenly, along a road and which consumes a single commodity. This good is sold by firms who deliver it at the customer's door. All firms have the same cost function, consisting of a part proportional to their sales, a part proportional to the aggregate distance over which they have to deliver, and a part that is constant. There is free entry, so that no firm can make a profit over its total cost. Consumers buy in the cheapest market, but otherwise their demand is inelastic. A complete solution of this problem would have to specify the location of each firm and the prices it charges to its various customers. For our purpose we need only mention some general characteristics of the solution. The area served by each firm has to be just large enough to provide a margin of revenue over variable costs equal to the fixed costs. At the boundary between the territories of two firms both must charge the same price, since otherwise customers near the boundary would switch to the cheaper firm. This price at the boundary has to be equal to variable cost (the cost of the commodity plus transportation cost from the firm's location), without any allowance for fixed costs. The latter have to be recouped from the customers inside the territory, and the prices charged to them will be the highest that will prevent new firms from establishing themselves at the boundaries between existing firms. In the case of a uniform distribution of demand, for instance, the price charged will be the same everywhere. What matters to us at the moment is not the level of prices, but rather the fact that the above conditions determine the number of firms, though no explicit general formula for that number has as yet been derived. It is clear, however, that the number of firms depends critically on the ratio 1
H. Hotelling, "Stability in Competition," Economic Journal, 1929, p. 41. A. Loesch, Die raumliche Ordnung der Wirtschaft, Jena 1940 (English transl. The Economics of Location, New Haven, Conn., 1954). 2
52
H. S. Houthakker
between the fixed costs and the transportation costs, or more generally, between the external and the internal coordination costs. Another example may serve to show that the above case is of wider importance than may appear at first sight. People's feet are of different sizes, and shoes are the more comfortable the more their size corresponds to the size of the foot. On the other hand, the unit cost of a pair of shoes of a given size decreases with the number of pairs of that size that is produced, because it is expensive to switch from one size to another. Hence people may find it preferable to wear cheap shoes that do not fit their feet exactly rather than expensive shoes made to their measure. Under free entry (or even under monopoly) there will again be a definite number of sizes that is actually produced. In this case the transportation cost from the previous example has its parallel in the reduction of price consumers are willing to pay for badly fitting shoes. By now the listener may well wonder what all this has to do with biology. The connection is not really very remote. One of the most striking phenomena of the living world is its organization into species, of which there are hundreds of thousands, or even millions, each of them with a greatly varying number of members. Though vastly more complicated, this pattern has some analogy to that of the firms along a road, or of the discretely varying shoe sizes. Each species may be compared to one size of shoe, and its members to the shoes produced of that size. The advantages and disadvantages of the biological pattern can therefore not be too dissimilar from those of the economic pattern. We may, for instance, consider each species as adapted to one particular range of foods (though this is by no means the whole story). The wider the range of foods, the more complicated the anatomical and physiological arrangements necessary to obtain and digest each type of food efficiently; or in the above terminology, the higher the internal coordination cost. On the other hand, a wider range of foods makes possible a larger number of members of the species, because the danger of starvation is smaller. For each species there will consequently be an optimum range and an optimum number of individuals, as there was in the case of shoes. In reality this balance between range and number is only one element, though probably a basic one, in the structure of advantages and disadvantages. It may of course be asked: advantages and disadvantages to whom? When it comes to the explanation of the characteristics of separate species biologists tend to use the word "advantage" quite freely, and
Economics and Biology: Specialization and Speciation
53
usually in the sense of something which promotes survival. This may easily lead to paradoxical results; thus R. A. Fisher in his Genetical Theory of Natural Selection3 maintains that evolution leads to an increase in fitness, which he defines as the chance of survival, from which it follows, oddly enough, that a species has the best chance of survival just before it becomes extinct. If we consider not separate species, but the whole animal world, the notion of advantage becomes even more tenuous. Since the purpose of nature is unknown, and it may indeed be meaningless to ask if there is a purpose, teleological explanations may appear to be ruled out. This, it is often held, contributes the dividing line between the social and the biological sciences. It appears, however, that this view overestimates the purposefulness of human behavior. It is true that the rational man has long been a favorite with economists, but recently his standing has somewhat declined. The development of the theory of choice, for instance, has cast doubt on many of the qualities with which the rational man was traditionally endowed. The realization of the importance of uncertainty has further undermined the concept of rationality, for which no satisfactory definition appropriate to uncertainty has yet been found. Moreover, it is sometimes argued that economic activities are primarily distinguishable by the manner in, rather than by the motive with which they are carried out. Although nature may have no discernible motives, it may nevertheless operate in the same way as if it had a motive. Since there are therefore no major objections, what is the advantage of extending economic analysis to biological phenomena? There is, I think, one important advantage. In economics we also observe species, such as different commodities, occupations, etc., but the boundaries between them are often vague for lack of a precise criterion of classification. Consequently any attempt to count the number of "individuals" in a "species" would meet with considerable conceptual difficulties. In biology, however, there is a more definite criterion, even though it is not absolutely precise. Species are defined by the criterion of interfertility, the ability to produce fertile offspring. The species of biology are therefore much more clearly defined than those of economics, and this is their principal advantage as an object of research.
3
Oxford, 1930.
54
H. S. Houthakker
In this lecture I have indulged in rather wild speculation, but I hope I need not apologize for it. Provided it is administered in small doses and counterbalanced by large quantities of more solid inquiry, speculation is healthy for any science even though most of its results turn out to be wrong.
CHAPTER 4 EXTERNALITY*
James M. Buchanan 3 and Wm. Craig Stubblebine b "George Mason University
Claremont McKenna
College
Externality has been, and is, central to the neo-classical critique of market organisation. In its various forms — external economies and diseconomies, divergencies between marginal social and marginal private cost or product, spillover and neighbourhood effects, collective or public goods — externality dominates theoretical welfare economics, and, in one sense, the theory of economic policy generally. Despite this importance and emphasis, rigorous definitions of the concept itself are not readily available in the literature. As Scitovosky has noted, "definitions of external economies are few and unsatisfactory".1 The following seems typical: External effects exist in consumption whenever the shape or position of a man's indifference curve depends on the consumption of other men. [External effects] are present whenever a firm's production function depends in some way on the amounts of the inputs or outputs of another firm.2 It seems clear that operational and usable definitions are required.
Reprinted from Economica, 29, James M. Buchanan and Wm. Craig Stubblebine, "Externality," 371-384, 1962, with permission from Blackwell. 1 Tibor Scitovosky, "Two Concepts of External Economies", Journal of Political Economy, vol. LXII (1954), p. 143. 2 J. de V. Graaf, Theoretical Welfare Economics, Cambridge, 1957, p. 43 and p. 18.
55
56
J. M. Buchanan, W. C. Stubblebine
In this paper, we propose to clarify the notion of externality by defining it rigorously and precisely. When this is done, several important, and often overlooked, conceptual distinctions follow more or less automatically. Specifically, we shall distinguish marginal and inframarginal externalities, potentially relevant and irrelevant externalities, and Pareto-relevant and Pareto-irrelevant externalities. These distinctions are formally developed in section 1. As we shall demonstrate, the term, "externality", as generally used by economists, corresponds only to our definition of Pareto-relevant externality. There follows, in section 2, an illustration of the basic points described in terms of a simple descriptive example. In section 3, some of the implications of our approach are discussed. It is useful to limit the scope of the analysis at the outset. Much of the discussion in the literature has been concerned with the distinction between technological and pecuniary external effects. We do not propose to enter this discussion since it is not relevant for our purposes. We note only that, if desired, the whole analysis can be taken to apply only to technological externalities. Secondly, we shall find no cause for discussing production and consumption externalities separately. Essentially the same analysis applies in either case. In what follows, "firms" may be substituted for "individuals" and "production functions" for "utility functions" without modifying the central conclusions. For expositional simplicity only, we limit the explicit discussion to consumption externalities. 1. We define an external effect, an externality, to be present when, u*=u*(XxtX2,...,Xm,YJ.
(1)
This states that the utility of an individual, A, is dependent upon the "activities", (Xl,X2,....,Xm), that are exclusively under his own control or authority, but also upon another single activity, Yx , which is, by definition, under the control of a second individual, B, who is presumed to be a member of the same social group. We define an activity here as any distinguishable human action that may be measured, such as eating bread,
Externality
57
drinking milk, spewing smoke into the air, dumping litter on the highways, giving to the poor, etc. Note that A's utility may, and will in the normal case, depend on other activities of B in addition to Y\, and also upon the activities of other parties. That is, A's utility function may, in more general terms, include such variables as (Y2,Y3,....,Ym; Z,,Z 2 ,....,Z m ). For analytical simplicity, however, we shall confine our attention to the effects of one particular activity, Yu as it affects the utility ofA. We assume that A will behave so as to maximise utility in the ordinary way, subject to the externally determined values for Yh and that he will modify the values for the Xs, as Yx changes, so as to maintain a state of "equilibrium". A marginal externality exists when, «jj*0.
(2)
Here, small it's are employed to represent the "partial derivatives" of the utility function of the individual designated by the super-script with respect to the variables designated by the subscript. Hence, u§ = duA/8Y] , assuming that the variation in Y\ is evaluated with respect to a set of "equilibrium" values for the Xs, adjusted to the given value for Yx. An infra-marginal externality holds at those points where, «$=0,
(3)
and(l) holds. These classifications can be broken down into economies and diseconomies: a marginal external economy existing when, u* > 0 ,
(2A)
that is, a small change in the activity undertaken by B will change the utility of A in the same direction; a marginal external diseconomy existing when, u*<0.
(2B)
An infra-marginal external economy exists when for any given set of values for (Xl,X2,....,XJ,say,(Cl,C2,...,Cm),
58
J. M. Buchanan, W. C. Stubblebine
u*=0, and \u§dY^ >0. (3A) o This condition states that, while incremental changes in the extent of 5's activity, Yh have no affect on A's utility, the total effect of 5's action has increased A's utility. An infra-marginal diseconomy exists when (1) holds, and, for any given set of values for (Xx,X2,...,Xm), say, (C,,C2,...,Cm), then, u* = 0 , a n d \u*dYi < 0 .
(3B)
o
Thus, small changes inS's activity do not changer's level of satisfaction, but the total effect of 5's undertaking the activity in question is harmful to A. We are able to classify the effects of B's action, or potential action, on A's utility by evaluating the "partial derivative" ofA's utility function with respect to Y\ over all possible values for Y\. In order to introduce the further distinctions between relevant and irrelevant externalities, however, it is necessary to go beyond consideration of A's utility function. Whether or not a relevant externality exists depends upon the extent to which the activity involving the externality is carried out by the person empowered to take action, to make decisions. Since we wish to consider a single externality in isolation, we shall assume that B's utility function includes only variables (activities) that are within his control, including Y\. Hence, B's utility function takes the form, u»=u*(Xx,Y2,...,Ym).
(4)
Necessary conditions for utility maximisation by B are,
KK=f*lfrBt>
&
where 1} is used to designate the activity of B in consuming or utilising some numeraire commodity or service which is, by hypothesis, available on equal terms to A. The right-hand term represents the marginal rate of substitution in "production" or "exchange" confronted by B, the party taking action on Yu his production function being defined as,
/*=/*(^,y2,...,yj,
(6)
Externality
59
where inputs are included as activities along with outputs. In other words, the right-hand term represents the marginal cost of the activity, Yu to B. The equilibrium values for the Yi 's will be designated as Yt 's. An externality is defined as potentially relevant when the activity, to the extent that it is actually performed, generates any desire on the part of the externally benefited (damaged) party (A) to modify the behaviour of the party empowered to take action (B) through trade, persuasion, compromise, agreement, convention, collective action, etc. An externality which, to the extent that it is performed, exerts no such influence is defined as irrelevant. Note that, so long as (1) holds, an externality remains; utility functions remain interdependent. A potentially relevant marginal externality exists when,
*0.
uA
(7)
This is a potentially relevant marginal external economy when (7) is greater than zero, a diseconomy when (7) is less than zero. In either case, A is motivated, by 5's performance of the activity, to make some effort to modify this performance, to increase the resources devoted to the activity when (7) is positive, to decrease the quantity of resources devoted to the activity when (7) is negative. Infra-marginal externalities are, by definition, irrelevant for small changes in the scope of B's activity, Y\. However, when large or discrete changes are considered, A is motivated to change S's behaviour with respect to Y\ in all cases except that for which, u*\ _ = 0 , a n d
(8) K
M\y-V
7
u*(C1,C2,...,Cmjl)Zu*(Cl,C2,...,Cm,Yl)JoT&\l Y^%. When (8) holds, A has achieved an absolute maximum of utility with respect to changes over Y\, given any set of values for the X's. In more prosaic terms, A is satiated with respect to Y\3. In all other cases, where infra-marginal external economies or diseconomies exist, A will have 3
Note that, ui\
_ = 0 , is a necessary, but not a sufficient, condition for irrelevance.
60
J. M. Buchanan, W. C. Stubblebine
some desire to modify 5's performance; the externality is potentially relevant. Whether or not this motivation will lead A to seek an expansion or contraction in the extent of 5's performance of the activity will depend on the location of the infra-marginal region relative to the absolute maximum for any given values of the JTs.4 Pareto relevance and irrelevance may now be introduced. The existence of a simple desire to modify the behaviour of another, defined as potential relevance, need not imply the ability to implement this desire. An externality is defined to be Pareto-relevant when the extent of the activity may be modified in such a way that the externally affected party, A, can be made better off without the acting party, B, being made worse off. That is to say, "gains from trade" characterise the Pareto-relevant externality, trade that takes the form of some change in the activity of B as his part of the bargain. A marginal externality is Pareto-relevant when5 (-)«£ /uXj > [ « * / « * - f*m
KK
>(r)[KK
]
~frJfrB\=?^
_ and when u{ /uXj < 0, and
(9)
u*/uXj >0.
In (9), Xj and Yj are used to designate, respectively, the activities of A and B in consuming or in utilising some numeraire commodity or service that, by hypothesis, is available on identical terms to each of them. As is indicated by the transposition of signs in (9), the conditions for Pareto relevance differ as between external diseconomies and economies. This is because the "direction" of change desired by A on the part of B is different in the two cases. In stating the conditions for Pareto relevance under ordinary two-person trade, this point is of no significance since trade in one good flows only in one direction. Hence, absolute values can be used.
4
In this analysis of the relevance of externalities, we have assumed that B will act in such a manner as to maximise his own utility subject to the constraints within which he must operate. If, for any reason, B does not attain the equilibrium position defined in (5) above, the classification of his activity for A may, of course, be modified. A potentially relevant externality may become irrelevant and vice versa. 5 We are indebted to Mr. M. McManus of the University of Birmingham for pointing out to us an error in an earlier formulation of this and the following similar conditions.
61
Externality
The condition, (9), states that^'s marginal rate of substitution between the activity, Yu and the numeraire activity must be greater than the "net" marginal rate of substitution between the activity and the numeraire activity for B. Otherwise, "gains from trade" would not exist between A and B. Note, however, that when B has achieved utility-maximising equilibrium, U»/!*%=/*/f°.
(10)
That is to say, the marginal rate of substitution in consumption or utilisation is equated to the marginal rate of substitution in production or exchange, i.e., to marginal cost. When (10) holds, the terms in the brackets in (9) mutually cancel. Thus, potentially relevant marginal externalities are also Pareto-relevant when B is in utility-maximising equilibrium. Some trade is possible. Pareto equilibrium is defined to be present when, (-)u$Juj
= W*/H£ - fyj fr. \> and when u^luj, < 0 , and (11)
KK =(-)[%1/M" -frjfy^\
when U
ZK
>0
-
Condition (11) demonstrates that marginal externalities may continue to exist, even in Pareto equilibrium, as here defined. This point may be shown by reference to the special case in which the activity in question may be undertaken at zero costs. Here Pareto equilibrium is attained when the marginal rates of substitution in consumption or utilisation for the two persons are precisely offsetting, that is, where their interests are strictly opposed, and not where the left-hand term vanishes. What vanishes in Pareto equilibrium are the Pareto-relevant externalities. It seems clear that, normally, economists have been referring only to what we have here called Pareto-relevant externalities when they have, implicitly or explicitly, stated that external effects are not present when a position on the Pareto optimality surface is attained.6
6
This applies to the authors of this paper. For recent discussion of external effects when we have clearly intended only what we here designate as Pareto-relevant, see James M. Buchanan, "Politics, Policy, and the Pigovian Margins", Economica, vol. XXVIX (1962),
62
J. M. Buchanan, W. C. Stubblebine
For completeness, we must also consider those potentially relevant infra-marginal externalities. Refer to the discussion of these as summarised in (8) above. The question is now to determine whether or not, A, the externally affected party, can reach some mutually satisfactory agreement with B, the acting party, that will involve some discrete (non-marginal) change in the scope of the activity, Y\. If, over some range, any range, of the activity, which we shall designate by AYX, the rate of substitution between Y\ and Xj for A exceeds the "net" rate of substitution for B, the externality is Pareto-relevant. The associated changes in the utilisation of the numeraire commodity must be equal for the two parties. Thus, for external economies, we have AuA
I AuA /
AY, J AXj
, N AuB lAuB > (-) AY / AYj X
AfB lAfB AY, / AYj
(12)
and the same with the sign in parenthesis transposed for external diseconomies. The difference to be noted between (12) and (9) is that, with infra-marginal externalities, potential relevance need not imply Pareto relevance. The bracketed terms in (12) need not sum to zero when B is in his private utility-maximising equilibrium. We have remained in a two-person world, with one person affected by the single activity of a second. However, the analysis can readily be modified to incorporate the effects of this activity on a multi-person group. That is to say, 5's activity, Yu may be allowed to affect several parties simultaneously, several's, so to speak. In each case, the activity can then be evaluated in terms of its effects on the utility of each person. Nothing in the construction need be changed. The only stage in the analysis requiring modification explicitly to take account of the possibilities of multi-person groups being externally affected is that which involves the condition for Pareto relevance and Pareto equilibrium. For a multi-person group i
pp. 17-28, and, also, James M. Buchanan and Gordon Tullock, The Calculus of Consent, Ann Arbor, 1962.
Externality
( - ) Z < / " £ >[ M */ W * - / y f / / ^ l 1=1
63
- when w | / w £ < 0, and, (9A) 1_
Z < K >(-)[«?/"? -fi/tf]r=y
]
when
0 -
That is, the summed marginal rates of substitution over the members of the externally affected group exceed the offsetting "net" marginal evaluation of the activity by B. Again, in private equilibrium for B, marginal externalities are Pareto-relevant, provided that we neglect the important element involved in the costs of organising group decisions. In the real world, these costs of organising group decisions (together with uncertainty and ignorance) will prevent realisation of some "gains from trade" — just as they do in organised markets. This is as true for two-person groups as it is for larger groups. But this does not invalidate the point that potential "gains from trade" are available. The condition for Pareto equilibrium and for the infra-marginal case summarised in (11) and (12) for the two-person model can readily be modified to allow for the externally affected multi-person group.
The distinctions developed formally in section 1 may be illustrated diagrammatically and discussed in terms of a simple descriptive example. Consider two persons, A and B, who own adjoining units of residential property. Within limits to be noted, each person values privacy, which may be measured quantitatively in terms of a single criterion, the height of a fence that can be constructed along the common boundary line. We shall assume that 5's desire for privacy holds over rather wide limits. His utility increases with the height of the fence up to a reasonably high level. Up to a certain minimum height, A's utility also is increased as the fence is made higher. Once this minimum height is attained, however, A's desire for privacy is assumed to be fully satiated. Thus, over a second range, ^4's total utility does not change with a change in the height of the fence. However, beyond a certain limit, A's view of a mountain behind B's property is progressively obscured as the fence goes higher. Over this third range, therefore, A's utility is reduced as the fence is constructed to
64
J. M. Buchanan, W. C. Stubblebine
higher levels. Finally, A will once again become wholly indifferent to marginal changes in the fence's height when his view is totally blocked out. We specify that B possesses the sole authority, the only legal right, to construct the fence between the two properties. The preference patterns for .4 and for B are shown in Figure 1, which is drawn in the form of an Edgeworth-like box diagram. Note, however, that the origin for B is shown at the upper left rather than the upper right corner of the diagram as in the more normal usage. This modification is necessary here because only the numeraire good, measured along the ordinate, is strictly divisible between A and B. Both must adjust to the same height of fence, that is, to the same level of the activity creating the externality. As described above, the indifference contours for A take the general shape shown by the curves aa, a'a', while those for B assume the shapes, bb, b'b'. Note that these contours reflect the relative evaluations, for A and B, between money and the activity,Yx. Since the costs of undertaking the activity, for B, are not incorporated in the diagram, the "contract locus" that might be derived from tangency points will have little relevance except in the special case where the activity can be undertaken at zero costs. Figure 2 depicts the marginal evaluation curves for A and B, as derived from the preference fields shown in Figure 1, along with some incorporation of costs. These curves are derived as follows: Assume an initial distribution of "money" between A and B, say, that shown at Mon Figure 1. The marginal evaluation of the activity for A is then derived by plotting the negatives (i.e., the mirror image) of the slopes of successive indifference curves attained by A as B is assumed to increase the height of the fence from zero. These values remain positive for a range, become zero over a second range, become negative for a third, and, finally, return to zero again.7
7
For an early use of marginal evaluation curves, see J. R. Hicks, "The Four Consumer's Surpluses", Review of Economic Studies, vol. XI (1943), pp. 31-41.
65
Externality
V M*
H,
H,
H,
H<
Height of Fence (B' s Activity)
Figure 1
5's curves of marginal evaluation are measured downward from the upper horizontal axis or base line, for reasons that will become apparent. The derivation of 2?'s marginal evaluation curve is somewhat more complex than that for A. This is because B, who is the person authorised to undertake the action, in this case the building of the fence, must also bear the full costs. Thus, as B increases the scope of the activity, his real income, measured in terms of his remaining goods and services, is reduced. This change in the amount of remaining goods and services will, of course, affect his marginal evaluation of the activity in question. Thus, the marginal cost of building the fence will determine, to some degree, the marginal evaluation of the fence. This necessary interdependence between marginal evaluation and marginal cost complicates the use of simple diagrammatic models in finding or locating a solution. It need not,
66
J. M. Buchanan, W. C. Stubblebine
ME,
\
ME, NMEA=MSCB NMEB= MSC„ MC Figure 2
however, deter us from presenting the solution diagrammatically, if we postulate that the marginal evaluation curve, as drawn, is based on a single presumed cost relationship. This done, we may plot 5's marginal evaluation of the activity from the negatives of the slopes of his indifference contours attained as he constructs the fence to higher and higher levels. 5's marginal evaluation, shown in Figure 2, remains positive throughout the range to the point H5, where it becomes zero. The distinctions noted in section 1 are easily related to the construction in Figure 2. To A, the party externally affected, B's potential activity in constructing the fence can be assessed independently of any prediction of 5's actual behaviour. Thus, the activity of B would, (1) exert marginal external economies which are potentially relevant over the range OHi,
Externality
67
(2) exert infra-marginal external economies over the range HiH2, which are clearly irrelevant since no change in 5's behaviour with respect to the extent of the activity would increase A's utility; (3) exert marginal external diseconomies over the range H2Ha, which are potentially relevant to A; and, (4) exert infra-marginal external economies or diseconomies beyond H4, the direction of the effect being dependent on the ratio between the total utility derived from privacy and the total reduction in utility derived from the obstructed view. In any case, the externality is potentially relevant. To determine Pareto relevance, the extent of S's predicted performance must be determined. The necessary condition for 5's attainment of "private" utility-maximising equilibrium is that marginal costs, which he must incur, be equal to his own marginal evaluation. For simplicity in Figure 2, we assume that marginal costs are constant, as shown by the curve, MC. Thus, S's position of equilibrium is shown at// B , within the range of marginal external diseconomies for A. Here the externality imposed by B's behaviour is clearly Pareto-relevant: A can surely work out some means of compensating B in exchange for B's agreement to reduce the scope of the activity—in this example, to reduce the height of the fence between the two properties. Diagrammatically, the position of Pareto equilibrium is shown at H^ where the marginal evaluation of A is equal in absolute value, but negatively, to the "net" marginal evaluation of B, drawn as the curve NMEB. Only in this position are the conditions specified in (11), above, satisfied.8
This diagrammatic analysis is necessarily oversimplified in the sense that the Pareto equilibrium position is represented as a unique point. Over the range between the "private" equilibrium for B and the point of Pareto equilibrium, the sort of bargains struck between A and B will affect the marginal evaluation curves of both individuals within this range. Thus, the more accurate analysis would suggest a "contract locus" of equilibrium points. At Pareto equilibrium, however, the condition shown in the diagrammatic presentation holds, and the demonstration of this fact rather than the location of the solution is the aim of this diagrammatics.
68
J. M. Buchanan, W. C. Stubblebine
3. Aside from the general classification of externalities that is developed, the approach here allows certain implications to be drawn, implications that have not, perhaps, been sufficiently recognised by some welfare economists. The analysis makes it quite clear that externalities, external effects, may remain even in full Pareto equilibrium. That is to say, a position may be classified as Pareto-optimal or efficient despite the fact that, at the marginal, the activity of one individual externally affects the utility of another individual. Figure 2 demonstrates this point clearly. Pareto equilibrium is attained at H3, yet B is imposing marginal external diseconomies on A. This point has significant policy implications for it suggests that the observation of external effects, taken alone, cannot provide a basis for judgment concerning the desirability of some modification in an existing state of affairs. There is not a prima facie case for intervention in all cases where an externality is observed to exist.9 The internal benefits from carrying out the activity, net of costs, may be greater than the external damage that is imposed on other parties. In full Pareto equilibrium, of course, these internal benefits, measured in terms of some numeraire good, net of costs, must be just equal, at the margin, to the external damage that is imposed on other parties. This equilibrium will always be characterised by the strict opposition of interests of the two parties, one of which may be a multi-person group. In the general case, we may say that, at full Pareto equilibrium, the presence of a marginal external diseconomy implies an offsetting marginal internal economy, whereas the presence of a marginal external economy implies an offsetting marginal internal diseconomy. In "private" equilibrium, as opposed to Pareto equilibrium, these net internal economies and diseconomies would, of course, be eliminated by the utility-maximising acting party. In Pareto equilibrium, these remain because the acting party is being compensated for "suffering" internal
9 Cf. Paul A. Samuelson, Foundations of Economic Analysis, Cambridge, Mass., 1948, p. 208, for a discussion of the views of various writers.
Externality
69
economies and diseconomies, that is, divergencies between "private" marginal costs and benefits, measured in the absence of compensation. As a second point, it is useful to relate the whole analysis here to the more familiar Pigovian discussion concerning the divergence between marginal social cost (product) and marginal private cost (product). By saying that such a divergence exists, we are, in the terms of this paper, saying that a marginal externality exists. The Pigovian terminology tends to be misleading, however, in that it deals with the acting party to the exclusion of the externally affected party. It fails to take into account the fact that there are always two parties involved in a single externality relationship. 10 As we have suggested, a marginal externality is Pareto-relevant except in the position of Pareto equilibrium; gains from trade can arise. But there must be two parties to any trading arrangement. The externally affected party must compensate the acting party for modifying his behaviour. The Pigovian terminology, through its concentration on the decision-making of the acting party alone, tends to obscure the two-sidedness of the bargain that must be made. To illustrate this point, assume that A, the externally affected party in our model, successfully secures, through the auspices of the "state", the levy of a marginal tax on S's performance of the activity, Yx. Assume further that A is able to secure this change without cost to himself. The tax will increase the marginal cost of performing the activity for B, and, hence, will reduce the extent of the activity attained in B's "private" equilibrium. Let us now presume that this marginal tax is levied "correctly" on the basis of a Pigovian calculus; the rate of tax at the margin is made equal to the negative marginal evaluation of the activity to A. Under these modified conditions, the effective marginal cost, as confronted by B, may be shown by the curve designated as MSCB in Figure 2. A new "private" equilibrium for B is shown at the quantity, H3, the same level designated as Pareto equilibrium in our earlier discussion, if we neglect the disturbing interdependence between marginal evaluation and marginal costs. Attention solely to the decision calculus of B here would suggest, perhaps,
This criticism of the Pigovian analysis has recently been developed by R. H. Coase; see his "The Problem of Social Cost", Journal of Law and Economics, vol. Ill (1960), pp. 1-44.
70
J. M. Buchanan, W. C. Stubblebine
that this position remains Pareto-optimal under these revised circumstances, and that it continues to qualify as a position of Pareto equilibrium. There is no divergence between marginal private cost and marginal social cost in the usual sense. However, the position, if attained in this manner, is clearly neither one of Pareto optimality, nor one that may be classified as Pareto equilibrium. In this new "private" equilibrium for B,
u
?Jui=ffJfrBru$K>
<13)
where uYi ux represents the marginal tax imposed on B as he performs the activity, Y\. Recall the necessary condition for Pareto relevance defined in (9) above, which can now be modified to read, (-)u$ juXj > [uBYJuBYj - f»/f*
+ u$JuXj ]
and uY> juAXj > ( - ) [ « * / « * - f*j'f^
+ u^luXj
, when u{ juXj < 0, (9B) _ , when u* lux > 0.
In (9B), Y1 represents the "private" equilibrium value for Yu determined by B, after the ideal Pigovian tax is imposed. As before, the bracketed terms represent the "net" marginal evaluation of the activity for the acting party, B, and these sum to zero when equilibrium is reached. So long as the left-hand term in the inequality remains non-zero, a Pareto-relevant marginal externality remains, despite the fact that the full "Pigovian solution" is attained. The apparent paradox here is not difficult to explain. Since, as postulated, A is not incurring any cost in securing the change in B's behaviour, and, since there remains, by hypothesis, a marginal diseconomy, further "trade" can be worked out between the two parties. Specifically, Pareto equilibrium is reached when,
(r)u$lux
= u§lul - ffl'fl,+uYlJux~\
< / < =(-)[
whenw^/wf <0, and (HA) -f'/f'+ufJu^whenu'/u^X).
Diagrammatically, this point may be made with reference to Figure 2. If a unilaterally imposed tax, corresponding to the marginal evaluation of A, is placed on 5's performance of the activity, the new position of Pareto
Externality
71
equilibrium may be shown by first subtracting the new marginal cost curve, drawn as MSCB, from 5's marginal evaluation curve. Where this new "net" marginal evaluation curve, shown as the dotted curve between points Hi and K, cuts the marginal evaluation curve for A, a new position of Pareto equilibrium falling between H2 and H3 is located, neglecting the qualifying point discussed in Footnote 8. The important implication to be drawn is that full Pareto equilibrium can never be attained via the imposition of unilaterally imposed taxes and subsidies until all marginal externalities are eliminated. If a taxsubsidy method, rather than "trade", is to be introduced, it should involve bi-lateral taxes (subsidies). Not only must B\ behaviour be modified so as to insure that he will take the costs externally imposed on A into account, but A's behaviour must be modified so as to insure that he will take the costs "internally" imposed on B into account. In such a double tax-subsidy scheme, the necessary Pareto conditions would be readily satisfied.11 In summary, Pareto equilibrium in the case of marginal externalities cannot be attained so long as marginal externalities remain, until and unless those benefiting from changes are required to pay some "price" for securing the benefits. A third point worthy of brief note is that our analysis allows the whole treatment of externalities to encompass the consideration of purely collective goods. As students of public finance theory will have recognised, the Pareto equilibrium solution discussed in this paper is similar, indeed is identical, with that which was presented by Paul Samuelson in his theory of public expenditures.12 The summed marginal rates of substitution (marginal evaluation) must be equal to marginal costs. Note, however, that marginal costs may include the negative marginal evaluation of other parties, if viewed in one way. Note, also, that there is nothing in the analysis which suggests its limitations to purely collective goods or even to goods that are characterised by significant externalities in their use. 11
Although developed in rather different terminology, this seems to be closely in accord with Coase's analysis. Cf. R. H. Coase, loc. cit. 12 Paul A. Samuelson, "The Pure Theory of Public Expenditure", Review of Economics and Statistics, vol. XXXVI (1954), pp. 386-9.
72
J. M. Buchanan, W. C. Stubblebine
Our analysis also lends itself to the more explicit point developed in Coase's recent paper.13 He argues that the same "solution" will tend to emerge out of any externality relationship, regardless of the structure of property rights, provided only that the market process works smoothly. Strictly speaking, Coase's analysis is applicable only to inter-firm externality relationships, and the identical solution emerges only because firms adjust to prices that are competitively determined. In our terms of reference, this identity of solution cannot apply because of the incomparability of utility functions. It remains true, however, that the basic characteristics of the Pareto equilibrium position remain unchanged regardless of the authority undertaking the action. This point can be readily demonstrated, again with reference to Figure 2. Let us assume that Figure 2 is now redrawn on the basis of a different legal relationship in which A now possesses full authority to construct the fence, whereas B can no longer take any action in this respect. A will, under these conditions, "privately" construct a fence only to the height H0, where the activity clearly exerts a Pareto-relevant marginal external economy on B. Pareto equilibrium will be reached, as before, at H3 determined, in this case, by the intersection of the "net" marginal evaluation curve for A (which is identical to the previously defined marginal social cost curve, MSC, when B is the acting party) and the marginal evaluation curve for B.u Note that, in this model, A will allow himself to suffer an internal marginal diseconomy, at equilibrium, provided that he is compensated by B, who continues, in Pareto equilibrium, to enjoy a marginal external economy. Throughout this paper, we have deliberately chosen to introduce and to discuss only a single externality. Much of the confusion in the literature seems to have arisen because two or more externalities have been handled simultaneously. The standard example is that in which the output of one firm affects the production function of the second firm while, at the same time, the output of the second firm affects the production function of the first. Clearly, there are two externalities to be analysed in such cases. In R. H. Coase, loc. cit. The H3 position, in this presumably redrawn figure, should not be precisely compared with the same position in the other model. We are using here the same diagram for two models, and, especially over wide ranges, the dependence of the marginal evaluation curves on income effects cannot be left out of account. 14
Externality
73
many instances, these can be treated as separate and handled independently. In other situations, this step cannot be taken and additional tools become necessary.15
15 For a treatment of the dual externality problem that clearly shows the important difference between the separable and the non-separable cases, see Otto Davis and Andrew Whinston, "Externalities, Welfare, and the Theory of Games", Journal of Political Economy, vol. LXX (1962), pp. 241-62. As the title suggests, Davis and Whinston utilise the tools of game theory for the inseparable case.
This page is intentionally left blank
Part 3
Development Strategies, Income Distribution, and Dual Structures
This page is intentionally left blank
CHAPTER 5 ECONOMIC DEVELOPMENT, INTERNATIONAL TRADE, AND INCOME DISTRIBUTION*
Xiaokai Yang" and Dingsheng Zhang b * "Monash University
b
Monash and Wuhan
University
1. Introduction The purpose of this paper is twofold. First, we develop a general equilibrium model with endogenous specialization to show that the relationship between economic development and inequality of income distribution is neither monotonic nor an inverted U-curve. Second, we use the general equilibrium model to investigate the interdependence between the extent of the market, the level of division of labor, productivity and the degree of inequality of income distribution. Let us motivate the two tasks. Krugman (1995a, b; 1996) and Krugman and Venables (1995) vividly document the fact that the relationship between economic development and income distribution came to the focus of public concern in the 1970s and 1990s. However, this was always a controversial issue in economic literature. Some models and theories were developed, with empirical evidence sometimes, to show that there is a positive correlation between economic
* Reprinted from Journal of Economics, 78 (2), Xiaokai Yang and Dingsheng Zhang, "Economic Development, International Trade, and Income Distribution," 163-190, 2003, with permission from Springer-Verlag. * We are grateful to Guangzhen Sun, Lin Zhou and Meng-Chun Liu for helpful discussion. We also gratefully acknowledge the Australian Research Council for financial support (Large Grant A79602713). We are solely responsible for the remaining errors.
77
78
X. Yang, D. Zhang
development and inequality of income distribution (Banerjee and Newman, 1993; Lewis, 1955; Palma, 1978; Li and Zou, 1998; Chang and Ram, 2000, for instance). Other models were developed, with empirical evidence sometimes, to show that there is a negative correlation between inequality and international trade, which relates to economic development (Aghion, Caroli and Garcia-Pena Losa, 1999; Alesina and Rodrik, 1994; Galor and Zeira, 1993; Thompson, 1995, and Fei, Ranis and Kuo, 1979; Frank, 1977; Balassa, 1986, for instance). Not only data from developing and newly industrialized countries are contradictory, but data from developed countries also generate controversy. Kuznetz (1955) proposed the hypothesis of an inverted Ucurve of the relationship between inequality and per capita income and provided some support for it from US data. Krugman and Venables (1995) use a general equilibrium model with global economies of scale to predict such an inverted U-curve of inequality. Greenwood and Jovanovic (1990) use a dynamic equilibrium model to predict the inverted U-curve. Some theories and data show that there is a negative or insignificant correlation between trade or development and inequality in the developed country (see, for instance, Krugman and Lawrence, 1994, and Katz and Murphy, 1992). Others show a positive correlation between trade and inequality in the developed country (see, for instance, Grossman, 1998; Murphy and Welch, 1991; Borjas, Freeman and Katz, 1992; Karoly and Klerman, 1994; Sachs and Shatz, 1995).1 However, Ram (1997) provides new empirical evidence for an uninverted-U pattern in the developed countries, taking into account data gathered since the Second World War. Jones (1998, p. 65) also shows that the ratio of GDP per worker in the 5th-richest country to GDP per worker in the 5th-poorest country fluctuated from 1960 to 1990. The controversy and new evidence call for new models and theories that can resolve it. In this paper, we develop a general equilibrium model to show that the relationship between inequality and trade that relates to economic development is neither monotonic nor an inverted U-curve.
1
Recent surveys on the relationship between trade and income distribution in developed countries can be found in Cline (1997), Williamson (1998), and Burtless (1995).
Economic Development, International Trade, and Income Distribution
79
In our general equilibrium model of endogenous specialization, economic development is described as an evolutionary process of division of labor that is driven by improvements in trading efficiency. Because of differences in trading efficiencies between countries and between different groups of residents in the same country, those individuals with better trading efficiencies will be involved in the division of labor and related trade before others are. Inequality increases because of the dual structure. As latecomers catch up, the dual structure disappears, so that inequality declines. As the leading group goes to an even higher level of specialization, leaving others behind, dual structure emerges and inequality increases again. As latecomers catch up, the dual structure disappears and inequality decreases again. This ratcheting process of inequality and equality generates fluctuation of the degree of inequality of income distribution. This implies that the relationship between inequality and economic development is neither monotonically positive nor monotonically negative. It may not be a simple inverted Ucurve. The intuition behind the model is as follows. If the relationship between inequality and economic development is monotonic, then in the very developed country, income distribution must be either extremely equal or extremely unequal. This is true too for an inverted U-curve between inequality and per capital income. However, we do not see either of these two extreme cases. Hence, inequality of income distribution must fluctuate as the equilibrium level of division of labor evolves. This evolution puts a group of individuals in a higher level of division of labor earlier than others. Hence, a dual structure in which some individuals are more specialized (or more commercialized) than others keeps emerging and disappearing as the equilibrium level of division of labor increases. A new data set in Deininger and Squire (1996) supports our hypothesis. It indicates that there is no systematical link between growth and changes in aggregate inequality. Recent regressions of Barro (1999), and Banerjee and Duflo (1999) also suggest that no monotonic correlation between inequality and growth performance can be supported by data.
80
X. Yang, D. Zhang
The second purpose of the current paper is to explore the intrinsic relationship between the development of division of labor and changes in inequality of income distribution. Kuznetz (1955) explained inequality of income distribution by per capita income. This is certainly not a general equilibrium view since per capita income is endogenous in a general equilibrium model, which itself should be explained by parameters. Murphy, Shleifer, and Vishny (1989) show that unequal income distribution restricts the extent of the market, so that economies of scale cannot be fully exploited and economic development is retarded. We put this idea together with Allyn Young's conjecture that "not only division of labor is dependent on the extent of the market, but also, the extent of the market is determined by the level of division of labor", to develop a general equilibrium mechanism that simultaneously determines the extent of the market, the level of division of labor, productivity and the degree of inequality of income distribution. We will examine effects of the coexistence of exogenous and endogenous comparative advantages on inequality of income distribution. Yang (1994), and Yang and Borland (1991) have drawn the distinction between David Ricardo's exogenous comparative advantage (Ricardo, 1817) and Adam Smith's endogenous comparative advantage (Smith, 1776).2 There is an extensive literature on exogenous comparative advantage in trade theory (see, for instance, Dixit and Norman, 1980). Separately, there are many models of endogenous comparative advantage in the growing literature on endogenous specialization (see Yang and Ng, 1998 for a recent survey on this literature and references there). This paper develops a general equilibrium model with the trade off between transaction costs and endogenous and exogenous comparative advantages. We shall show that as trading efficiencies are improved, the scope for efficiently balancing this trade off is enlarged. Hence, the equilibrium level of division of labor increases. The existence of exogenous comparative advantage implies that different
2
Endogenous comparative advantage is associated with economies of specialization and referred to by Grossman and Helpman (1991) as acquired comparative advantage, whereas exogenous comparative advantage is associated with constant returns to scale in production, referred to by them as natural comparative advantage.
Economic Development, International
Trade, and Income Distribution
81
types of individuals are sequentially involved in the division of labor. Part of the population is first involved in a low level of division of labor, the rest being left behind. The emergence of this dual structure increases inequality of income distribution. Then the rest of the population is involved in this low level of division of labor, catching up to rich fellows. This catch up process reduces duality and related inequality. Then, part of the population goes to an even higher level of division of labor, duality and related inequality increases. Then the rest who are left behind catch up; duality and inequality reduces again. This ratcheting process goes on, generating fluctuation of the degree of duality of economic structure and of the degree of inequality of income distribution. In this development process, the most important determinant of equilibrium aggregate productivity and welfare is trading efficiency rather than inequality of income distribution. This is called irrelevance of inequality (or equality) of income distribution to economic development. Early studies of structural changes and dual structure relied on the assumption of disequilibrium in some markets to predict dual structure and structural changes. For instance, Lewis (1955) tried to explain the dual structure between commercialized and self-sufficient sectors by the evolution of the division of labor and the related productivity progress (see also Ranis, 1988). Due to lack of appropriate analytical tools, he ended up with a model based on disequilibrium in the labor market caused by institutional wage. Chenery (1979) used market disequilibrium to explain structural changes. Recently, general equilibrium models have been used to study dual structure. In some of these models, such as in Khandker and Rashid's equilibrium model (1995), dual structure is exogenously assumed.3 They cannot predict the emergence and evolution of dual structure. In a recent literature on formal equilibrium models of high development economics, evolution of dual structure between the manufacturing sector with economies of scale in production and the agricultural sector with constant returns to scale can be predicted (see Krugman and Vanables, 1995, 1996, and Fujita 3
Also, in Dixit's dynamic planning model (1968), the existence of labor surplus is exogenously assumed.
82
X. Yang, D, Zhang
and Krugman, 1995). The equilibrium models with endogenous geographical location of economic activities of Krugman and Venables (1995), and Baldwin and Venables (1995) attribute the emergence of dual structure to the geographical concentration of economic activities in economic development that marginalizes peripheral areas. Kelly (1997), based on Murphy, Shleifer, and Vishny (1989), develops a dynamic general equilibrium model that predicts spontaneous evolution of a dual structure between the modern sector with economies of scale and the traditional sector with constant returns technology. As trading efficiencies are sufficiently improved, the level of division of labor increases and dual structure disappears. Our model in this paper is complementary to these general equilibrium models that predict the emergence and evolution of dual structure. We pay more attention to the effects that the evolution of individuals' levels of specialization and the coexistence of exogenous and endogenous comparative advantages exert on the emergence and evolution of dual structure.4 The rest of this paper is organized as follows, section 2 presents the 2 x 2 Ricardian model with transaction costs and endogenous and exogenous comparative advantages. Section 3 solves for general equilibrium and its inframarginal comparative statics. Section 4 extends the analysis to consider not only income distribution in the developing country but also that in the developed country. The concluding section summarizes the findings of the paper and suggests possible extensions.
4
Mokyr (1993, pp. 65-66) documents the evolution of individuals' level of specialization during the Industrial Revolution in Britain. This evolution is sometimes referred to as the "industrious revolution" which implies that self-provided home production is replaced with commercialized production. Yang, Wang and Wills (1992) find empirical evidence for this evolution from Chinese data. A difference in empirical implications generated by the Krugman and Venables model and our model is that the former yields scale effects (productivity of the manufacturing firm increases if and only if its size increases), but the latter does not.
Economic Development, International Trade, and Income Distribution
83
2. A Ricardian Model with Endogenous and Exogenous Comparative Advantages Consider a world consisting of country 1 and country 2, each with Mt (i= 1, 2) consumer-producers. The set of individuals is a continuum and the mass of population is 1. There are two types of individuals in country 1. The mass of type a persons is Ma and that of type b persons is Mb, where Ma + Mb + M2 = 1. All individuals in country 2 are assumed to be identical, but different from individuals in country 1. The utility function for individuals of type i is Ut = (*, + ktfYiy,
+ klyttP,
»= a, b, 2,
(1)
where xh yt are quantities of goods x and y produced for selfconsumption, xt , yt are quantities of the two goods bought from the market and kt is the trading efficiency coefficient for a type i individual. The transaction cost is assumed to take the iceberg form: for each unit of good bought, a fraction 1 - kt is lost in transit, the remaining fraction kt is received by the buyer. Type a persons and type b persons are distinguished by their trading efficiencies. We assume that ka = k,kb = t, and k > t. In other words, each type a persons has higher trading efficiency than each type b person. Also, we assume that k2 = k. The assumptions can be interpreted as follows. Country 2 is a developed country that has greater trading efficiencies. Country 1 is a less developed country where some residents (who might be urban or coastal residents) have the same trading efficiency as in the developed country, but the rest of the population has a lower trading efficiency. The production conditions are the same for all individuals in the same country, but different between the countries. Hence, the production functions for a type i consumer-producer are X +X
J J=L%'
yj+ysj=Ljy'
x2+xs2=If2x,
y2+ys2=rL2y,
j=*>b'
(2a)
(2b)
where x., yst are respective quantities of the two goods sold by a type i person; Lis is the amount of labor allocated to the production of good s (= x, y) by a type / person, and Lix + Liy = B > 1. For simplicity, we assume that B = 2. It is assumed that r, c > 1. This system of production
84
X. Yang, D. Zhang
functions and endowment constraint displays economies of specialization in producing good x and constant returns to specialization in producing good y. But an individual in country 2 has a higher productivity in producing good y than an individual in country 1. Economies of specialization are individual specific and activity specific, that is, they are localized increasing returns, which are compatible with the Walrasian regime.5 If all individuals allocate the same amount of labor to the production of each good, then an individual in country 1 has the same average labor productivity of good x as an individual in country 2. But the average and marginal labor productivity of good y for an individual in country 2 is higher. This is similar to the situation in a Ricardian model with exogenous comparative advantage. Country l's productivities are not higher than country 2 in producing all goods, but may have exogenous comparative advantage in producing good x. But if an individual in country 2 allocates much more labor to the production of x than an individual in country 1, her productivity is higher than that of the latter. Similarly, if an individual in country 1 allocates more labor to the production of good x than an individual in country 2, her productivity of good x will be higher. This is referred to as endogenous comparative advantage, since individuals' decisions on labor allocation determine 5
Marshall was aware of the incompatibility between his marvellous marginal analysis of demand and supply and classical economic thinking about the implications of specialization and division of labor. As a result he drew the distinction between internal economies of scale and external economies of scale. The latter is supposed to relate to economies of division of labor in the market place. However, as Allyn Young (1928) argued, "the view of the nature of the processes of industrial progress which is implied in the distinction between internal and external economies is necessarily a partial view. Certain aspects of those processes are illuminated, while, for that very reason, certain other aspects, important in relation to other problems, are obscured." As Young and Yang (2001) have pointed out, economies of specialization are diseconomies of a person's scope of production activities and economies of division of labor consist of economies of specialization of all individuals and economies of diversity of different occupations. They are different from (internal or external) economies of scale, though related to local economies of scale of an individual labor in producing a good. Hence, it seemed to Young that the concept of external economies of scale is a misrepresentation of the classical concept of economies of specialization and division of labor.
Economic Development, International Trade, and Income Distribution 85 difference in productivity between them. But an individual in country 1 has no endogenous comparative advantage in producing good y since her marginal and average productivity of y is always 1, lower than r, independent of her labor allocation. However, country 1 has exogenous comparative advantage in producing x and country 2 has exogenous comparative advantage in producing y. The decision problem for a type i individual involves deciding on what and how much to produce for self-consumption, to sell and to buy from the market. In other words, the individual chooses six variables xnx° ,xf ,yt,y° ,yf > 0 . Hence, there are 26 = 64 possible corner and interior solutions. As shown by Wen (1998), for such a model, an individual never simultaneously sells and buys the same good, never simultaneously produces and buys the same good, and never sells more than one good. We refer to each individual's choice of what to produce, buy and sell that is consistent with the Wen theorem as a configuration. There are three configurations from which the individuals can choose: (1) self sufficiency. Configuration A, where an individual produces both goods for self-consumption. This configuration is defined by x,,yt >0,
x' = x," = yt' = yf =0,i
= a,b,2.
(2) specialization in producing good x. Configuration (x/y), where an individual produces only x, sells x in exchange for y, is defined by Xi,x;,yf>o,
x?=yi=y;=o.
(3) specialization in producing good y. Configuration (y/x), where an individual produces only y, sells y in exchange for x, is defined by
yny:,x?>Q,
yf=Xt=x,'=Q.
The combination of all individual's configurations constitutes a market structure, or structure for short. Given the configurations listed above, fourteen structures may occur in equilibrium. Structure AAA, as shown in panel (1) of Figure 1, is an autarky structure where individuals in both countries choose self-sufficiency (configuration A). Here, the first two letters denote the configurations chosen by type a and type b persons, respectively in country 1 and the
86
X. Yang, D. Zhang
00- g)fc^ fcw0' ^ 'g) g) g) country 1 country 2 (1) Structure AAA
country 1 country 2 (7) Structure XXP
country 1 country 2 (2) Structure PAY
country 1 country 2 (8) Structure XXD
Figure 1: Configurations and Structures
country 1 country 2 (3) Structure XAP
country 1 country 2 (9) Structure XXY
Economic Development, International Trade, and Income Distribution
87
third letter denotes the configuration chosen by individuals in country 2. In each panel upper-left circles represent configurations chosen by type a individuals, lower-left circles represent configurations chosen by type b individuals, and circles on the right hand side represent configurations chosen by individuals in country 2. Structure PAY, shown in panel (2) of Figure 1, means that some type a persons choose configuration (x/y) and others choose autarky, all type b persons choose autarky, and individuals in country 2 choose configuration (y/x). Letter P stands for a type of person partially involved in the division of labor: some of them completely specialize and others choose autarky. This structure involves three types of dual structure. In a type I dual structure ex ante identical individuals are divided between specialization and autarky. In a type II dual structure, different types of individuals in the same country are divided between specialization and autarky. In a type III dual structure, one country is completely involved in the division of labor, while some residents in the other country are self-sufficient. Structure XAP, shown in panel (3) of Figure 1, means that all type a persons choose configuration (x/y), all type b persons choose autarky, some individuals in country 2 choose (y/x) and others choose autarky. Structure DAY, shown in panel (4) of Figure 1, means that some type a persons choose (x/y) and others choose (y/x), all type b persons choose autarky, and individuals in country 2 choose (y/x). Letter D stands for division of individuals of a certain type between (x/y) and (y/x). Structure XAY, panel (5) of Figure 1, means that all type a persons choose (x/y), all type b persons choose autarky, and individuals in country 2 choose configuration (y/x). Structure XAD implies that all type a persons choose (x/y), all type b persons choose autarky, some individuals in country 2 choose (y/x) and others choose (x/y). Structure XPY, panel (6), means that all type a persons choose (x/y), some type b persons choose autarky and others choose (x/y), and individuals in country 2 choose configuration (y/x).
88
X. Yang, D. Zhang
Structure XXP, panel (7), means that all type a and type b persons choose (x/y), some individuals in country 2 choose configuration (y/x), and others choose autarky. Structure XXD, panel (8), means that all type a and type b persons choose (x/y), some individuals in country 2 choose configuration (y/x), and others choose (x/y). Structure XXY, panel (9), implies all type a and type b persons choose (x/y) and all individuals in country 2 choose configuration (y/x). Structure XDY means that all type a persons choose (x/y), some type b individuals choose (x/y) and others choose (y/x), all individuals in country 2 choose configuration (y/x). Structure YDY means that all type a persons choose (y/x), some type b individuals choose (x/y) and others choose (y/x), all individuals in country 2 choose configuration (y/x). Structure DXY means that some type a persons and all type b persons choose (x/y), other type a persons choose (y/x), all individuals in country 2 choose configuration (y/x). Structure DYY means that some type a persons and all type b persons choose (y/x), other type a persons choose (x/y), all individuals in country 2 choose configuration (y/x). It can be shown that other structures cannot occur in equilibrium. 3. General Equilibrium and its Inframarginal Comparative Statics According to Sun, Yang and Zhou (1998), a general equilibrium exists and is Pareto optimal for the kind of models in this paper under the assumptions that the set of individuals is a continuum and preferences are strictly increasing and rational; both local increasing returns and constant returns are allowed in production and transactions. Also, the set of equilibrium allocations is equivalent to the set of core allocations. An equilibrium in this model is defined as a relative price of the two goods and all individuals' labor allocations and trade plans, such that (i) Each individual maximizes her utility, that is, the consumption bundle generated by her labor allocation and trade plan maximizes utility function (1) for given p.
Economic Development, International Trade, and Income Distribution
89
(ii) All markets clear. For simplicity, we assume that J3= 0.5. Let the number (measure) of type i individuals choosing configuration (x/y) be Mix, choosing (y/x) be Miy, and choosing A be MA. Since the interior solution is never optimal in this model of endogenous specialization and there are many structures based on corner solutions, we cannot use standard marginal analysis to solve for a general equilibrium. We adopt a two step approach in solving for a general equilibrium. In the first step, we consider a structure. Each individual's utility maximizing decision is solved for the given structure. Utility equalization condition between individuals choosing different configurations in the same country and market clearing condition are used to solve for the relative price of traded goods and numbers (measure) of individuals choosing different configurations. The relative price and numbers and associated resource allocation are referred to as a corner equilibrium for this structure. According to the definition, a general equilibrium is a corner equilibrium in which all individuals have no incentive to deviate, under the corner equilibrium relative price, from their chosen configurations. Hence, in the second step, we can plug the corner equilibrium relative price into the indirect utility function for each configuration, then compare corner equilibrium values of utility across all configurations. The comparisons are called a total cost-benefit analysis which yields the conditions under which the corner equilibrium utility in each constituent configuration of this structure is not smaller than any alternative configuration. This system of inequalities can thus be used to identify a subspace of parameter space within which this corner equilibrium is a general equilibrium. With the existence theorem of general equilibrium proved by Zhou, Sun and Yang (1998), we can completely partition the parameter space into subspaces, within each of which the corner equilibrium in a structure is a general equilibrium. As parameter values shift between the subspaces, the general equilibrium will discontinuously jump between structures. The discontinuous jumps of structure and all endogenous
90
X. Yang, D. Zhang
variables are called inframarginal comparative statics of general equilibrium. We now take the first step of the inframarginal analysis. As an example, we consider structure XAP. Assume that in this structure, M2y individuals choose configuration (y/x) and M2A individuals choose autarky in country 2, where M2y + M2A = M2. Ma individuals choose configuration (x/y) and Mb individuals choose configuration A in country 1. Since all individuals in country 2 are ex ante identical in all aspects, the maximum utilities in configurations A and (y/x) must be the same in country 2 in equilibrium. Marginal analysis of the decision problem for an individual in country 2 choosing autarky yields her maximum utility in configuration A: U2A = (r2c+1y)0'5, where y- ccl(c + l) e+1 . Marginal analysis of the decision problem for an individual in country 2 choosing configuration (y/x) yields the demand function xd2 = rip, the supply function^ = r, and indirect utility function: U2y = r(k/p)05. The utility equalization condition U2y = U2A yields p = pjpy = kr/2°ny. Similarly, the marginal analysis of the decision problem of a type a individual choosing configuration (x/y) yields the demand function, yd _ jc-\ ip ^ m e SUppiy function xsa = 2C_1 , and indirect utility function: Uax =2c~l(kp)05 . The marginal analysis of the decision problem for a type b individual choosing configuration A in country 1 yields her maximum utility in autarky UbA = (2c+lyf5. Inserting the corner equilibrium relative price into the market clearing condition for good x, Msx x" - M2y %d2 , yields the number of individuals selling good y, M2y = kMJAy, Mm = Ma, and MbA = Mh. Indirect utility functions for individuals choosing various configurations in the two countries are listed in Table 1.
Configurations Type a person Type b person Country 2
Table 1: Indirect Utility Functions Indirect utility functions (x/y) (y/x) t/„ = 2c-\kpf5 U!Ly = (k/pf5 $ UbK = ¥-\tpf Uby = (t/pf5 c 5 U2x = 2 ~\kpf U2y = r(k/pf5
A U^ = (2c+lrf5 UbA = (T+lrf5 U2A = (l^ryf5
Economic Development, International Trade, and Income Distribution
91
Following this procedure, we can solve for corner equilibrium in each structure. The solutions of all corner equilibria are summarized in Table 2. We can then take the second step to carry out total cost-benefit analysis for each corner equilibrium and to identify the parameter subspace within which the corner equilibrium is a general equilibrium. Consider the corner equilibrium in structure XAP as an example again. In this structure Ma individuals choose configuration (x/y) and Mh individuals choose autarky in country 1, and M2y individuals choose configuration (y/x) and M2A individuals choose autarky in country 2. For a type a person in country 1, equilibrium requires that her utility in configuration (x/y) is not smaller than in configurations (y/x) and A under the corner equilibrium relative price in structure XAY. Also equilibrium requires that all type b persons in country 1 prefer autarky to configurations (x/y) and (y/x) and that all individuals in country 2 are indifferent between configurations (y/x) and A and receive a utility level that is not lower than in configuration (x/y). In addition, this structure occurs in equilibrium only if M2ye (0, M2). All the conditions imply Um > Uay, Uax > UaA, UhA > UbM UbA > Uby, U2A=U2y>U2x,
M2ye(0,M2),
where indirect utility functions in different configurations and corner equilibrium relative price are given in Tables 1 and 2. The conditions define a parameter subspace: kt < 16/Vr, k e (4yVr, Min{4f, 4M2y/Ma}), where y = cc/(c + l) c+1 . Within this parameter subspace, the corner equilibrium in structure XAP is the general equilibrium. Following this procedure, we can carry out total cost-benefit analysis for each structure. The total cost-benefit analysis in the second step and marginal analysis of each corner equilibrium in the first step yields inframarginal comparative statics of general equilibrium, summarized in Table 3. From this Table, we can see that the parameter subspaces for structure AAD, XAA, and all structures in which country 1 exports y and country 2 exports x to occur in general equilibrium are empty.
92
Structure AAA PAY XAP DAY XAY XAD XPY XXP XXD XXY XDY YDY DXY DYY
X. Yang, D. Zhang Table 2: Comer Equilibria Numbers of individuals choosing various configurations A4A = Ma, MbA = Mb, M2A = M2 r23~c/k Mix = M2kr/4y. M^ = M a - M2kr/4y, MbA = Afb, M2A = M2 krlyl^1 MK = Ma, MbA = Mb, M2y = MJdAy, M2A= M2- (MJdAy) 21M ax = (Ma+ rM2)/2, May= (M a - rM2)/2, MbA = Mb,M2v = M2 •Mx = Ma, MbA = Mb, M2v = M2 M2r2'^/Afa ^ = M , MbA = Mb, M2y = (Ma + M2)/2 r2'" c Mx = Ma, Mbx = (foM2/4x) - Ma, M2v = M2 yl^lk krly2M M« = Ma, M,x = Mb, M2y = *(Ma + Mb)/4r, M 2 A =M 2 -WM a +M b )/4^ Max = M,, Mbx = Mb, M2y = (Ma + Mb + M2)I2, a1-* M2x = (M 2 - M - M)/2 Mx = Ma, Mbx = Mb, M2v = M2 M2r2i-c/(M!i+ Mb) 21Mx = Ma, Mbx = (Mb - Ma + rM2)/2, Mbv = (Ma + Mb - rM2)/2, M2v = M2 May = Ma, Mbx = (Ma + Mb + rM2)/2, 21Mbv = (Mb - Ma - rM2)/2, M2v = M2 2l-c Mbx = Mb, Max = ( M - Mb + rM2)/2, Mav = (Ma + Mh - rM2)/2, M2v = M2 2l-c Mbx = Mb, M« = ( M + Mb + rM2)/2, Mav = (Ma - Mb - rM2)/2, M2v = M2
Relative price of xto y
When a dual structure occurs in equilibrium, individuals in autarky appear as underemployed or underdeveloped in the sense that they receive none of the gains from trade and income differential between them and other individuals who are involved in the division of labor increases as a result of a shift of equilibrium from AAA to a dual structure. These self-sufficient individuals cannot find a job to work for the market. Figure 1 illustrates the equilibrium structures.6 We say the 6
The coexistence of two equilibria distinguishes the model of endogenous specialization from a standard Arrow-Debreu equilibrium model in which the number of multiple equilibria is always odd.
w
i« -^ £
A
x •*
s.
ft
13
^
^
11
T
^J->
I*
V
us
-
S3
Economic Development, International Trade, and Income Distribution
V V
*
5tv
•
^
V
fs
Si
X
"V.
5?*!'
v 1
x
V -?
ST
93
Table 3: {Continued) Mi/M2 >r
MJ M2< 1
XAP
MJM2 < Vr
AAA
MJ M2 e
ihr)
MJM2 > Vr
k< 4yM2IMa XAP
; < AyMJrMi
AyMJrMi
k> 4rM2/Ma XAY
XAY
XPY
4yMJrM2 PAY
4j-Ma/f-M2 XAY
MJ Mi >r
X
XPY
PAY
X
Economic Development, International Trade, and Income Distribution
95
level of division of labor increases if occurrence of letter A or P decreases and/or the occurrence of letter D, X, and/or Y increases in the equilibrium structure. We can identify five levels of division of labor. The lowest one is associated with structure AAA where all three types of individuals are self-sufficient. The second level is associated with structures PAY and XAP in each of which two types of individuals are self-sufficient. The third level is associated with structures DAY, XAY, XAD in each of which all of individuals of a certain type are selfsufficient. The fourth level is associated with structures XPY and XXP in each of which only part of the population of a certain type are selfsufficient. The fifth level is associated with structures XXD, XXY, XDY, YDY, DXY, DYY, in each of which all individuals are specialized. In this table y= ccl(c + l) c+1 . X stands for configuration (x/y) chosen by individuals of a certain type, Y stands for (y/x) chosen by individuals of a certain type, A stands for autarky, P stands for the partial division of labor where individuals of a certain type are divided between autarky and specialization, D stands for the division of individuals of a certain type between (x/y) and (y/x). Structure XAP involves type I and type II dual structures, structure XXP involves type I and type III dual structures. Structures DAY, XAY, XAD involve type II and type III dual structures. Structures PAY and XPY involve all three types of dual structure. With the definition of division of labor in mind, we can now closely examine Table 3. As trading efficiencies are improved, the equilibrium level of division of labor increases. If the equilibrium level of division of labor increases from level 1 to level 2, then dual structure emerges, or duality of economic structure increases. If the equilibrium level of division of labor increases from 2 to 3, duality decreases as those individuals who are left behind catch up. If the equilibrium level of division of labor increases from 3 to 4, duality increases again as some of individuals in a group that was in autarky before choose specialization. As the equilibrium level of division of labor increases from 4 to 5, duality decreases again as those who were left behind catch up. But it is also possible for improvements in trading efficiencies to generate a jump of equilibrium from level 2 to level 4 in the division of labor. Such a
96
X. Yang, D. Zhang
jump will reduce duality. A jump from level 3 in the division of labor to level 5 will reduce duality as well. This analysis implies that as different groups of the same type or different types of individuals gradually choose a higher level of specialization one by one, the degree of duality fluctuates. If we use the difference between the highest and the lowest per capita real income i.e. the difference in per capital real income between type a and type b individuals in country 1 as a measure of income inequality, its increase will raise inequality of income distribution and its decrease will reduce inequality. Hence, as different individuals gradually become involved in the network of the division of labor as a result of improved trading efficiencies, the equilibrium degree of inequality fluctuates. It does not monotonically increase or decrease. The relationship between inequality and per capita real income may not be an inverted U-curve. Inserting the corner equilibrium relative price in Table 2 into indirect utility functions in Table 1, we can compare the difference of per capita real incomes (equilibrium levels of utility) of two types of individuals in country 1, considering the parameter subspaces that demarcate the structures. The comparisons confirm our analysis. We first consider the path of evolution in the division of labor: AAA-»XAP->XAY-»XPY —»XXY. The difference in per capita real income between type a and type b individuals in country 1 is 0 in structure AAA. It is positive in structure XAP. It decreases as the equilibrium jumps from XAP to XAY. It then increases as the equilibrium jumps from XAY to XPY. It decreases again as the equilibrium jumps from XPY to XXY. We can find other evolutionary paths of the division of labor with a fluctuating degree of inequality of income distribution. Also, comparisons of ratios of the highest to the lowest real income between corner equilibria generate similar results. All of the results on the evolution of the division of labor, dual structure and trade pattern are summarized in the following proposition, illustrated in Figure 1. Proposition 1: As trading efficiency increases from a very low to a very high level, the equilibrium level of the domestic and international
Economic Development, International Trade, and Income Distribution
97
division of labor increases from complete autarky in both countries to complete division of labor in both countries. As different individuals are gradually involved in the division of labor, the duality of economic structure fluctuates, generating fluctuation of the equilibrium degree of inequality of income distribution in the less developed country. This degree is neither monotonically increasing nor monotonically decreasing. The relationship between the inequality and per capita real income might not be of an inverted U-curve. The inframarginal comparative statics of general equilibrium in Table 3 can be used to establish two corollaries. The first is that evolution in the division of labor generated by improvements in trading efficiencies will raise equilibrium aggregate productivity. In order to establish the above statement, we consider the aggregate PPF for individual 1 (from country 1) and individual 2 (from country 2). As shown in Figure 2 where c = 2, the PPF for individual 1 is curve BC, that for individual 2 is curve AC. In autarky, the two persons' optimum decisions for taste parameter j6 G (0, 1) are functions of /?. Let f3 change from 0 to 1; we can calculate equilibrium values of Y = yx + y2 and X=X\ + x2 as functions of J3. The values of X and Y for different values of /? constitute curve EGD in Figure 2. The equilibrium aggregate production schedule in structure AAA is a point on the curve, dependent on the value of j5. But the aggregate PPF for the two individuals is the curve EFD. Since in a structure with complete division of labor the equilibrium production schedule is point F which is on the aggregate PPF, the aggregate productivity in a structure with the complete division of labor is higher than in structure AAA. The difference between EFD and EGD can be considered as economies of the division of labor. Following the same reasoning, we can prove that the equilibrium aggregate productivity in a dual structure is lower than the PPF. Hence, Proposition 1 implies that as trading efficiencies are improved, the equilibrium level of division of labor and equilibrium aggregate productivity increase side by side.
98
X. Yang, D. Zhang y/
2+2a
2&A
C 2
0
B 4
8
^x
Figure 2: Economies of Division of Labor Based on Endogenous and Exogenous Comparative Advantage
The second corollary is that deterioration of a country's terms of trade and increase of gains received by this country from trade may concur. Suppose that the initial values of parameters satisfy (Ma + Mh)/M2 e (Vr, r), MJM2 e (1, r), kt > I6f/r. Within this parameter subspace, suppose that the initial value of/ satisfies /'< 4){M!i+ Mb)/M2r. Table 3 indicates that structure XPY occurs in equilibrium where country l's terms of trade are p' = y23~c/t'. Now assume that / increases to the value /", so that t"> 4XM a + Mb)/M2. According to Table 3, the equilibrium jumps to structure XXY where country l's terms of trade arep" = M2r21_7(Ma+ Mb). It can then be shown that country l's terms of trade deteriorate as a result of the change in / within the parameter subspace. But this shift of the equilibrium from XPY to XXY increases utility of each individual in country 1 if k'lt' < k"M2r/4f{Ma+ Mh). This has established the claim that there exists some parameter subspace within which the deterioration of a country's terms of trade may concur with an increase of gains that this country receives from trade. There are other parameter subspaces within which changes in parameters may
Economic Development, International
Trade, and Income Distribution
99
generate concurrence of the deterioration of one country's terms of trade and an increase in its gains from trade.7 This corollary generates the following policy implications. In the transitional stage of economic development and globalization, the terms of trade are against the less developed country that has relatively low trading efficiency: some residents in the less developed country receive autarky utility and most gains from trade go to the developed country. There are two policies to change this inferior position. One is to impose a tariff to improve the terms of trade and the other is to improve trading efficiency to expand the network of trade. The former is to increase the share of gains received by the less developed country from a shrunk pie because of the deadweight caused by the tariff. The latter is to provide a greater share of gains from trade by enlarging the pie. The expanded network of division of labor can generate productivity gains. As long as productivity improvements outpace the deterioration of the terms of trade, the less developed country can receive more gains from trade not only because of productivity gains, but also because of more equal division of gains from trade between the countries as all individuals are involved in the international and domestic division of labor. 4. Effects of Trade on Income Distribution in the Developed Country Since we assume that there is only one type of individuals in the developed country that has greater trading efficiency, the model in the previous section cannot explore effects of economic development and international trade on income distribution in the developed country. In this section we extend the model in the previous section by specifying two types of individuals in the developed country. The new model is exactly the same as the previous one except that there are two types of
7
Empirical and theoretical research on economic development and terms of trade can be found, for instance in Morgan (1970), Kohli and Werner (1998), Sen (1998).
Table 4: Equilibrium Structure
0
kt < 16 fir
AAAA
XAPA
0
ks> 16/Vr kt> 16 f/r
16AXPYA
16^/r XXYP
st < 16 fir
AADA
f< r
^ XAYA
s> 4y1r st> 16f
?>4
^ XPYA
XX
Economic Development, International Trade, and Income Distribution 101
individuals in country 2 as well. Types la and lb of individuals are in country 1 and types 2a and 2b of individuals are in country 2. For type 2a individuals, trading efficiency is k, for type lb individuals, trading efficiency is t, and for type 2b and la individuals, trading efficiency is s, where 1 > k > s > t > 0. For simplicity, we assume that M]a = Mu, = M2a = M2\, = 1/4. Following the method in the previous section, the inframarginal comparative statics of this new model can be solved as in Table 4, where y = ccl (1 + c)c+1, X denotes configuration (x/y), Y denotes (y/x), P denotes a division of individuals of a certain type between specialization and autarky, D denotes a division of individuals of a certain type between (x/y) and (y/x). The first of four letters denoting a structure represents the configuration chosen by type la persons, the second represents that chosen by type lb persons, the third represents that chosen by type 2a persons and the fourth represents that chosen by type 2b persons. Figure 3 gives an intuitive illumination of the evolution of the division of labor and economic development as a result of improvements in trading efficiencies. In this figure, an arrow -» denotes the direction of the evolution of the division of labor, the numbers 1 and 2 next to the arrows denote the two countries and the signs + and - denote increase and decrease, respectively, of the inequality of income distribution in a country. A question mark implies that either an increase or decrease of inequality is possible. From the results we can see that although evolutionary paths of division of labor AAAA—»AADA—»XAYA-» XAYD—»XXYY generates an inverted U-curve of inequality in the less developed country, path AAAA->XAPA->XPYA->XAYP->XXYY generates fluctuation of inequality. Other paths with ? may generate such fluctuation, too. Again, inframarginal comparative statics of general equilibrium confirm Proposition 1 even if income distribution in the developed country is considered. This yields Proposition 2.
102
X. Yang, D. Zhang
1:0 2: AAAA 1 2:0
AADA
-XAYD
1:.2:XXYY
XAPA
1:2-A
-XPYA
1:+ 2:1
-XXYP
1:2:-
Figure 3: Evolution in Division of Labor
Proposition 2: As trading efficiency is improved, the equilibrium level of division of labor and aggregate productivity increase. The degree of inequality of income distribution in the less developed country and the developed country fluctuates as duality of economic structure fluctuates. The degree of inequality is nonmonotonic and it might not be an inverted U-curve. An interesting feature of our inframarginal analysis of general equilibrium is that economic development and structural changes can take place in the absence of changes of tastes and production functions. All development and trade phenomena can be generated by changes in trading efficiencies that are neither on the demand side nor on the supply side. All recent general equilibrium models of "high development economics" share this feature (see Fujita and Krugman 1995; Krugman and Venables, 1995; Kelly, 1997; Shi and Yang, 1995; Yang, 1991, and Yang and Rice, 1994, for instance). Hence, they explore the limitation of the analysis that explains trade and development phenomena by changes in demand and/or supply sides. 5. Conclusion and Policy Implication In this paper, we have proposed a theory of ratcheting inequality and equality of income distribution using inframarginal analysis of a model of endogenous specialization. According to our view, there are two resources concerning the source of the income inequality. One is the
Economic Development, International Trade, and Income Distribution 103
difference in transaction condition between groups; the other is the relative number of people of different types. This theory accommodates the empirical evidence for the coexistence of positive and negative correlation between inequality and economic development which relates to trade. Because of the trade off between economies of the division of labor and transaction costs, the equilibrium level of the division of labor increases as trading efficiencies are improved. If trading efficiencies are different between countries or between different groups of individuals in the same country, different individuals will be involved in the division of labor sequentially. If some individuals have a higher level of specialization (or commercialization) than other individuals, a dual structure will increase inequality of income distribution. As those poor individuals increase their levels of specialization, the dual structure disappears and inequality decreases. But as the level of division of labor increases further, while some individuals' levels of specialization increase before others' do, a new dual structure will occur, which will disappear as latecomers catch up. This ratcheting process generates fluctuation of the equilibrium degree of inequality as the division of labor evolves. Therefore, there is neither a monotonic relationship between inequality of income distribution and economic development nor an inverted U-curve. In other words, inequality (equality) of income distribution is irrelevant to economic development. We should pay more attention to improvements in trading efficiencies that will enhance aggregate productivity and all individuals' welfare than to the relationship between inequality and economic development (or trade). Our model has policy implications. The government can reduce inequality of income distribution by implementing policies that affect trading efficient parameters. The government may increase trading effciency of a poor population by removing institutional obstacles of free migration from inland or rural areas to coast or urban areas, or by constructing a transportation infrastructure in poor areas. A logical extension of this paper is to add more goods and/or more countries and introduce a tariff into the Ricardian model. More structures may occur in equilibrium and the ratcheting process will be more pronounced. If political economics of income distribution and rent
104
X. Yang, D. Zhang
seeking is considered, negative effects of unfair income distribution on economic development may be analyzed.
References Aghion, P., Caroli, E., and Garcia-Pena Losa, C. (1999): "Inequality and Economic Growth: the Perspective of the New Growth Theories." Journal of Economic Literature 37: 1615-1660. Alesina, A., and Rodrik, D. (1994): "Distributive Politics and Economic Growth." Quarterly Journal of Economics 109: 465-490. Baldwin, R. E., and Venables, A. J. (1995): "Regional Economic Integration." In Handbook of International Economics, edited by Grossman, G. M. and Rogoff, K., Vol. 3. Amsterdam New York Oxford: Elsevier North-Holland. Balassa, B. (1986): "The Employment Effects of Trade in Manufactured Products between Developed and Developing Countries." Journal of Policy Modeling 8: 371-390. Banerjee, A., and Duflo, E. (1999): "Inequality and Growth: What Can the Data Say?" Working Paper, Department of Economics, MIT-Press. Banerjee, A., and Newman, A. (1993): "Occupational Choice and the Process of Development." Journal of Political Economy 101: 274-298. Barro, R. (1997): Determinants of Economic Growth. Cambridge, Mass., MIT-Press. Barro, R. (1999): "Inequality, Growth, and Investment." Working Paper 7038, National Bureau of Economic Research. Becker, G. (1981): A Treatise on the Family. Cambridge, Mass., Harvard University Press. Borjas, G., Freeman, R., and Katz, L. (1992): "How Much Do Immigration and Trade Affect Labor Market Outcomes?" Brookings Papers on Economic Activity 1: 1-67. Buchanan, J. M., and Stubblebine, W. C. (1962): "Externality." Economica 29: 371-384. Burtless, G. (1995): "International Trade and the Rise in Earnings Inequality." Journal of Economic Literature 33: 800-816.
Economic Development,
International
Trade, and Income Distribution
105
Chang, J., and Ram, R. (2000): "Level of Development, Rate of Economic Growth, and Income Inequality." Economic Development and Cultural Change 48: 787-800. Chenery, M. (1979): Structural Change and Development Policy. New York: Oxford University Press. Cline, W. (1997): Trade and Income Distribution. Washington, DC, Institute for International Economics. Coase, R. (1946): "The Marginal Cost Controversy." Economica 13: 169-182. Coase, R. (1960): "The Problem of Social Cost." Journal of Law and Economics 3: 1-44. Cypher, J., and Dietz, J. (1998): "Static and Dynamic Comparative Advantage: A Multiperiod Analysis with Declining Terms of Trade." Journal of Economic Issues 32: 305-314. Deininger, K., and Squire, L. (1996): "A New Data Set Measuring Income Inequality." World Bank Economic Review 10: 565-591. Dixit, A. (1987): "Trade and Insurance with Moral Hazard." Journal of International Economics 23: 201-220. Dixit, A. (1989): "Trade and Insurance with Adverse Selection." Review of Economic Studies 56: 235-248. Dixit, A., and Norman, V. (1980): Theory of International Trade. Cambridge: Cambridge University Press. Din, M. (1996): "International Capital Mobility and Development Policy in a Dual Economy." Review of International Economics 4: 185-201. Easton, S., and Michael, W. (1997): "Income, Growth, and Economic Freedom." American Economic Review, Papers and Proceedings 87: 328-332. Feenstra (1998): "Integration of Trade and Disintegration of Production in the Global Economy." Journal of Economic Perspectives 12: 31-50. Fei, J., Ranis, G., and Kuo S. (1979): Growth with Equity: The Taiwan Case. New York, published for the World Bank by Oxford University Press. Frank, C. (1977): Foreign Trade and Domestic Aid. Washington, DC, Brookings Institute. Frye, T., and Shleifer, A. (1997): "The Invisible Hand and the Grabbing Hand." American Economic Review, Papers and Proceedings 87: 354-358.
106
X. Yang, D. Zhang
Fujita, M., and Krugman, P. (1995): "When is the Economy Monocentric: von Thunen and Chamberlin Unified." Regional Science and Urban Economics 25: 505-528. Gallup, J., and Sachs, J. (1998): "Geography and Economic Development." Working Paper, Harvard Institute for International Development. Galor, O., and Zeira, J. (1993): "Income Distribution and Macroeconomics." Review of Economic Studies 87: 205-210. Gomory, R. E. (1994): "A Ricardo Model with Economies of Scale." Journal of Economic Theory 62: 394-419. Greewood, J., and Jovanovic, B. (1990): "Financial Development, Growth, and the Distribution of Income." Journal of Political Economy 98: 1076-1107. Grossman, G. (1998): "Imperfect Labor Contracts and International Trade." mimeo., Department of Economics, Princeton University. Grossman, S., and Hart, O. (1986): "The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Integration." Journal of Political Economy 94: 691-719. Grossman, G., and Helpman, E. (1990): "Comparative Advantage and Long-Run Growth." American Economic Review 80: 796-815. Jones, C. I. (1998): Introduction to Economic Growth. New York: Norton. Karoly, L., and Klerman, J. (1994): "Using Regional Data to Reexamine the Contribution of Demographic and Sectoral Changes to Increasing U.S. Wage Inequality." In The Changing Distribution of Income in an Open US Economy, edited by J. Bergstrand et al. Amsterdam: North-Holland. Katz, L., and Murphy, K. (1992): "Changes in Relative Wages, 1963-1987: Supply and Demand Factors." Quarterly Journal of Economics 107: 35-78. Kelly, M. (1997): "The Dynamics of Smithian Growth." Quarterly Journal of Economics 112:939-964. Khandker, A., and Rashid, S. (1995): "Wage Subsidy and Full Employment in a Dual Economy with Open Unemployment and Surplus Labor." Journal of Development Economics 48: 205-223. Kohli, U., and Werner, A. (1998): "Accounting for South Korean GDP Growth: Indexnumber and Econometric Estimates." Pacific Economic Review 3: 133-152. Krugman, P. (1996): "Does Third World Growth Hurt First World Prosperity?" Harvard Business Review 72: 113-121.
Economic Development, International
Trade, and Income Distribution
107
Krugman, P. (1995a): "Growing World Trade: Causes and Consequences." Brookings Paper on Economic Activity 1: 327-362. Krugman, P. (1995b): "Technology, Trade, and Factor Prices." NBER Working Paper No. 5355, National Bureau of Economic Research. Krugman, P., and Lawrence, R. (1994): "Trade, Jobs and Wages." Scientific American 270: 44-49. Krugman, P., and Venables, A. J. (1995): "Globalization and the Inequality of Nations." Quarterly Journal of Economics 110: 857-880. Kuznets, S. (1955): "Economic Growth and Income Inequality." American Economic Review 45: 1-28. Lewis, W. (1955): The Theory of Economic Growth. London: Allen and Unwin. Li, H., and Zou, H. (1998): "Income Inequality is not Harmful for Growth: Theory and Evidence." Review of Development Economics 2: 318-334. Mokyr, J. (1993): "The New Economic History and the Industrial Revolution." In The British Industrial Revolution: An Economic Perspective, edited by J. Mokyr. Boulder and Oxford: Westview Press. Murphy, K., Schleifer, A., and Vishny, R. (1989): "Income Distribution, Market Size, and Industrialization." The Quarterly Journal of Economics 104: 537-564. Murphy, K., and Welch, F. (1991): "The Role of International Trade in Wage Differentials." In Workers and Their Wages: Changing Patterns in the United States, edited by M. Kosters. Washington, American Enterprise Institute. Palma, B. (1978): "Dependency: A Formal Theory of Underdevelopment or a Methodology for the Analysis of Concrete Situations of Underdevelopment?" World Development 6: 899-902. Puga, D., and Venables, A. (1998): "Agglomeration and Economic Development: Import Substitution vs. Trade Liberalisation." Centre for Economic Performance, Discussion Paper No. 377. Ram, R. (1997): "Level of Economic Development and Income Inequality: Evidence from the Postwar Developed World." Southern Economic Journal 64: 576-583. Ranis, G. (1988): "Analytics of Development: Dualism." In Handbook of Development Economics, edited H. Chenery and T. Srinivasan, vol. 1, 4.
108
X. Yang, D. Zhang
Ricardo, D. (1817): The Principles of Political Economy and Taxation. London: Dent J.M., 1965. Rosen, S. (1978): "Substitution and the Division of Labor." Economica 45: 235-250. Rosen, S. (1983): "Specialization and Human Capital." Journal of Labor Economics 1: 43-49. Sachs, J. (1996): "Globalization and the U.S. Labor Market." The American Economic Review: Papers and Proceedings, May. Sachs, J., and Shatz, H. (1994): "Trade and Jobs in US Manufactures." Brookings Papers on Economic Activity 1: 1-84. Sachs, J., and Warner, A. (1995): "Economic Reform and the Process of Global Integration." Brookings Papers on Economic Activity 1. Sachs, J., and Warner, A. (1997): "Fundamental Sources of Long-Run Growth." American Economic Review, Papers and Proceedings 87: 184—188. Sen, P. (1998): "Terms of Trade and Welfare for a Developing Economy with an Imperfectly Competitive Sector." Review of Development Economics 2: 87-93. Smith, A. (1776): An Inquiry into the Nature and Causes of the Wealth of Nations. Reprint, Oxford: Clarendon Press 1976. Sun, G., Yang, X., and Zhou, L. (1998): "General Equilibria in Large Economies with Endogenous Structure of Division of Labor." Working Paper, Department of Economics, Monash University. Thompson, H. (1995): "Free Trade and Income Redistribution in Some Developing and Newly Industrialized Countries." Open Economies Review 6: 265-280. Wen, M. (1998): "An Analytical Framework of Consumer-Producers, Economies of Specialisation and Transaction Costs." In Increasing Returns and Economic Analysis, edited by K. Arrow, Y-K. Ng, and X. Yang. London: Macmillan. Williamson, J. (1998): "Globalization, Labor Markets and Policy Backlash in the Past." Journal of Economic Perspectives 12: 51-72. Yang, X. (1991): "Development, Structure Change, and Urbanization." Journal of Development Economics 34: 199-222. Yang, X. (1994): "Endogenous vs. Exogenous Comparative Advantages and Economies of Specialization vs. Economies of Scale." Journal of Economics/ Zeitschrift fur Nationalokonomie 60: 29-54.
Economic Development, International
Trade, and Income Distribution
109
Yang, X. (2001): Economics: New Classical versus Neoclassical Frameworks. London: Blackwell Publishers. Yang, X., and Borland, J. (1991): "A Microeconomic Mechanism for Economic Growth." Journal of Political Economy 99: 460-482. Yang, X., and Ng, S. (1998): "Specialization and Division of Labor: A Survey." In Increasing Returns and Economic Analysis, edited by K. Arrow, Y-K. Ng, and X. Yang. London: Macmillan. Yang, X., Wang, J., and Wills, I. (1992): "Economic Growth, Commercialization, and Institutional Changes in Rural China, 1979-1987." China Economic Review 3: 1-37. Young, A. (1928): "Increasing Returns and Economic Progress." The Economic Journal 38: 527-542.
This page is intentionally left blank
CHAPTER 6 PURSUIT OF RELATIVE UTILITY AND DIVISION OF LABOR*
Jianguo Wang a and Xiaokai Yang b * ^National University of Singapore
Monash
University
1. Introduction Adam Smith (1776) argued that the development of the division of labor is due to humankind's inclination for exchange. By contrast, Emile Durkheim (1933) argued that the division of labor may be the result of individuals' pursuit of relative economic standing. According to Durkheim, the division of labor is a result of the struggle for existence; different occupations can coexist without being obliged mutually to destroy one another because they pursue different objects. Frank (1984, 1985a,b) also argues that people's pursuit of relative economic standing generates a positive effect on specialization because individuals intend to open new fields to avoid competition and to rank themselves at higher positions. Henri Saint-Simon, Joseph Fourier, Robert Owen, and Karl Marx (e.g., 1844) and many modern management scientists believe that individuals normally dislike specialization but prefer diverse activities. However, the existing theories in the literature of this field have not explored the relationship between changes in preference for relative utility and the division of labor even in a descriptive stage, not to mention in the * Reprinted from Journal of Comparative Economics, 23 (1), Jianguo Wang and Xiaokai Yang, "Pursuit of Relative Utility and Division of Labor," 20-37, 1996, with permission from Elsevier. * We are grateful to two anonymous referees and to Yew-Kwang Ng, Robert Frank, Jeff Borland, Ben Heijdra, and Murray Kemp for helpful comments. Any remaining errors are ours. Ill
112
J. Wang, X. Yang
context of general equilibrium. In this paper, we shall develop two general equilibrium models to explore the relationship between the degree of individuals' desire for relative utility and the development of the division of labor. Unfortunately, the neoclassical microeconomic framework cannot be used to achieve this for the following reason. In the neoclassical microeconomic framework in which a production function or technology, either with constant returns to scale or with increasing returns to scale, is employed to describe the relationship between output and inputs, total factor productivity is independent of the level of specialization and of the internal organization of a firm. This paper shall combine Yang's (1990) framework with consumerproducers, economies of specialization, and transaction costs with Wang's (1993) approach to modeling individuals' behavior pursuing relative utility to formalize classical thoughts about the relationship between pursuit of relative position and the division of labor. Yang's framework can be used to investigate the relationship between total factor productivity, the level of specialization, and trade dependence. Wang's (1993) approach, which is built on the literature of pursuit of relative economic standing (see, for example, Basmann (1988), Frank (1984, 1985a,b, 1989), Mezias (1988), Miner (1990), Ng and Wang (1993), Stadt et al. (1985), Schoeck(1969), Veblen (1899), and Veenhoven (1991)), can be used to explore equilibrium implications of pursuit of relative utility. The blend of the two analytical vehicles will enable us to inquire into the relationship between individuals' pursuit of relative utility and the development of the division of labor. Two key concepts in this paper, division of labor and relative utility, should be given here first. Roughly speaking, an economic organizational pattern is said to involve the division of labor if it allocates the labor of different individuals to different activities. Hence, the levels of specialization of individuals and the number of professional activities are two sides of the level of division of labor. Relative utility is the utility ratio between individuals, e.g., the ratio of person i's utility to person/s utility. Note that relativity of utility and relative utility are two distinct concepts. Relativity of utility means that formation of preferences is relative. We have the relativity of utility so long as individuals pursue relative
Pursuit of Relative Utility
113
economic standing, which includes relative utility but may include something else such as relative income. The main finding of our analysis is that, given the existing structure of individuals' preferences and the degree of increasing returns to specialization, (i) a stronger desire for relative economic standing, e.g., envy, will promote the level of division of labor and productivity if it reduces the differential between preferences for goods or if it weakens an individual's preference for variety of her productive activities; and (ii) the consumption of relative economic standing may or may not generate positive effects on the division of labor, depending on specific conditions. In the first model, with two goods and M consumer-producers (section 2), each person as a consumer prefers diverse consumption and as a producer prefers specialized production. An increase in the division of labor increases, of course, the number of transactions. Since there is a cost occurring from each transaction, i.e., transaction cost, a higher level of the division of labor must generate a higher level of transaction costs. The trade-off between economies of specialization and transaction costs implies that the level of division of labor is determined by transaction efficiency. If transaction efficiency is low, then individuals choose autarky because transaction costs outweigh economies of specialization generated by the division of labor. If transaction efficiency is sufficiently improved, then individuals choose the division of labor because transaction costs are outweighed by economies of specialization. Each individual's utility is a function of the quantities of goods consumed as well as a ratio of her utility level to others' utility levels. The behavior pursuing relative utility is equivalent to behavior with envy. This implies that a person's utility can be expressed as a function of the quantities of all goods consumed and others' utility levels, which have negative effects on her utility level if she pursues relative utility. Each individual maximizes her utility function given others' utility levels under a well-specified and enforced private property system so that stealing is impossible. The equilibrium relative prices of traded goods, relative numbers of different specialists, and equilibrium level of division of labor are then functions of the parameters that characterize pursuit of relative utility. The comparative statics of general equilibrium yield the
114
J. Wang, X. Yang
equilibrium implications of individuals' pursuit of relative utility. Since, in the context of preference interdependence, an increase in the desire for relative utility implies a relative decrease in the desire for direct consumption, the increase in the desire for relative position will affect the structure of preferences for goods. The first equilibrium model with two goods is developed in section 2 to show that if an increase in the desire for relative position increases the differential between preferences for various goods, then the equilibrium is more plausibly to be associated with a lower level of division of labor. On the other hand, if an increase in the desire for relative position reduces the differential between preferences for various goods, then a higher level of division of labor is more plausibly the equilibrium. In the second model, section 3, the number of goods is assumed to be N instead of 2, and each individual is assumed to prefer a variety of her productive activities in addition to direct consumption and relative position. An increase in the desire for relative position will change the structure of preferences. If this increase weakens the preference for a variety of activities, then the equilibrium level of division of labor will be increased by the increased desire for relative position. 2. A Model with Two Consumer Goods and M Consumer-Producers 2.1. Specification of the model Consider an economy with M ex ante identical consumer-producers and two consumer goods. The self-provided amounts of the two goods are x and y, respectively. The amounts of the goods sold at the market are xs and ys, respectively. The amounts of the goods purchased in the market are xd and yd, respectively. The fraction 1 - k of a unit of goods purchased disappears in transit because of transaction costs. Hence, a person receives k when she buys one unit of goods. Each person is equipped with a system of production functions and an endowment constraint. xt + xf = Laxi,
yt + yf - Layi
(production function)
Pursuit of Relative
Lxj + Lyi = 1
Utility
115
(endowment constraint), (1)
where Lri is person i's labor share in producing good r. We define the share as person z's level of specialization in producing good r. The parameter a represents the degree of economies of specialization. This system of production functions displays economies of specialization if a > 1. This implies that, for a > 1, labor productivity of a good increases as a person's level of specialization in producing this good increases. Note that economies of specialization are individual specific or increasing returns are localized. rt + r/ is the quantity of good r(r = x, y) that is produced by person /. Due to transaction costs, krf is the quantity that is received by person i from the purchase of rf, and rt + ty is the quantity of good r consumed by person i. Hence, each consumer-producer's utility function is specified as1 f Y U^ix^kxfYiy^kyfY n^T >
(2a)
where J is a set of all individuals except person i, a and/? are respective preference parameters for the two goods, and y is the parameter that represents the preference for relative utility. We assume a + f3 + y -1 for simplicity or for normalization. However, this constant-returns- to-scale assumption is not essential. What is essential is that these preference parameters are interdependent. Even if we assume a + jB + y = g where g may be larger or smaller than 1, our main results will remain the same. 1
The strategic interactions are not taken into account in this model mainly because they do not make much difference since no uncertainty is considered and with a large population the strategic effect is negligible in the models. It may be argued that the utility function in (2a) involves circularity and information problems since, in a model of relative utility, Uj itself depends on Ut. Moreover, it might also be argued that individual;' cannot observe Uj. However, as shown in Wang (1993), the existence of envy is equivalent to the existence of pursuit of relative utility. If we start from a model allowing for real incomes as the source of envy, it can be converted into a model with utilities as the source of envy. Moreover, ex ante identical individuals who can only take others' utility levels as given when they make their optimum decision have no information problems for their decision making. Thus, individual / may know Uj via individual j's real income. Also, (2a) is not an acceptable formation if Mis a variable. However, since Mis a constant here, (2a) is acceptable. See Wang (1993) for the case of a variable M. In fact, relative utility may be more observable than relative income. For example, if we use one's smile frequency as one of her happiness indicators, obviously, her smile frequency is easier to be seen than her income.
116
J. Wang, X. Yang
Obviously, our utility function is an application of the Cobb-Douglas log-linear hypothesis, and its functional form is very similar to what Khinchin (1957) discusses. However, we do not use the logarithm of the objective function mainly because its algebra is nearly the same, and the results will be the same. The utility function characterizing the pursuit of relative utility can be rearranged as N -r-|V(l-(M-l)/)
U,=
a
{xi+kx?) {yi+ky?y
(2b)
where M is the population size. This utility function characterizes an individual's envy, which can be derived from behavior pursuing relative position, characterized by the utility function in (2a). Let/? be the price of good y in terms of x, then person i's decision problem is - i/(i-(w-i)r)
a
{xi+kxt) {yi+kyfY\YlU U, = lx,+kxfYlv,+kyfY \\U..j xt+xf -If.,xi' L„.+L = 1
ys=La.
v. + J i
(utilityfunction)
Si
xf + pyf = xf + pyf
(production function)
yi
(endowment constraint) (budget constraint). (3)
2.2. Individuals' decision problems and corner solutions The model is similar to Yang's except that the pursuit of relative utility is allowed. Therefore, our analysis focuses mainly on the relationship between the pursuit of relative utility and the development of division of labor and skips those relationships analyzed by Yang. Due to the great number of ex ante identical individuals and individual-specific economies of specialization or localized increasing returns, no individual has monopoly power to manipulate market supply and demand and, therefore, to influence prices of traded goods. Thus, the regime here is a special Walrasian one: the numbers of individuals selling different goods play the same role as prices do in the conventional Walrasian regime.
Pursuit ofRelative Utility
117
According to Yang (1990), an individual's optimum decision is always a corner solution in the models of this kind. Facing many corner solutions, she must decide which one is optimal. Since the utility levels are discontinuous from one corner solution to another, the conventional marginal analysis works only within a corner solution but not for choosing the optimum decision out of many corner solutions. Yang (1990) has solved this problem by applying the Kuhn-Tucker theorem. He first identifies a set of candidates for the optimum solution, rules out the interior solution and many corner solutions from this set, and then identifies an optimum decision by comparing utility levels across many corner solutions. This total benefit-cost analysis across corner solutions in addition to the marginal analysis of each corner solution is referred to by Buchanan and Stubblebine (1962) as inframarginal analysis. To follow Yang's approach, it is necessary to adopt some important concepts from him. A pattern of economic organization has a higher level of division of labor than another pattern if all individuals' levels of specialization in the former pattern are higher than in the latter. A pattern of economic organization is said to be autarky if every individual produces her own consumption. A combination of zero and nonzero variables that is compatible with the Kuhn-Tucker conditions and the budget constraint is called a configuration. A combination of configurations that is compatible with the market-clearing condition is called a market structure or simply a structure. If n goods are traded, n - 1 relative prices and n - 1 relative numbers of individuals choosing different configurations are determined by n - 1 independent market clearing conditions and n - 1 utility equalization conditions across n configurations. The n - 1 relative number and the n - 1 relative prices define a corner equilibrium in a given structure. All the corner equilibria constitute a set of candidates for the general equilibrium. General equilibrium is defined as a fixed point that satisfies the following conditions, (i) Each individual uses inframarginal analysis, which means total benefit-cost across corner solutions in addition to marginal analysis for each corner, to maximize her utility with respect to configurations and quantities of each good produced, consumed, and traded for a given set of relative prices of traded goods and a given set of the number of individuals selling different goods; and (ii) the
118
J. Wang, X. Yang
set of relative prices of traded goods and the set of number of individuals selling different goods clear the markets for traded goods and equalize utility for all individuals. According to this definition, the general equilibrium is the corner equilibrium that yields the highest per-capita real income because it satisfies the two conditions for general equilibrium and other corner equilibria do not satisfy condition (i). We will see in the following sections that computation of the equilibria is rather complicated and computational cost is quite high. In reality, equilibria are approached through some iterative improvements. However, this dynamic process must be described by a dynamic model (see Yang and Ng (forthcoming)) who show in a dynamic model that individuals can search for the efficient pattern of division of labor through the price system over time if pricing efficiency is not too low). In our static model, this dynamic iterative process is not described, and we can see only the outcome of the process. For a person's optimum decision, Yang (1990, p. 206) has proven the following lemma. Lemma: According to the necessary condition for an optimum decision, an individual does not produce and purchase a good at the same time, and sells at most one good. According to this lemma, there are only three configurations in the present model. Following an approach for solving for the equilibrium based on corner solutions developed by Yang (1990), we can solve for the equilibrium in two steps. First, we can solve for a corner solution for each configuration and a corner equilibrium for each structure. Then the general equilibrium is identified from the set of corner equilibria. A. Autarkic configuration Configuration autarky or simply A is a profile of zero and nonzero variables defined by xs=ys - xd=yd = 0, x,y, Lx, Ly > 0. In configuration A, each consumer-producer self-provides all goods she needs. Let xs = ys - xd = yd = 0 in (3), and person /'s decision problem for configuration A becomes
Pursuit ofRelative Utility
P maxC/, = L"*{\-L.T XI
XI
V
)
(
m
V JeJ
Yr'
119
i/(i-(Af-i)r)
(4a) J
The first-order condition dUldLxi = 0 yields the corner solution for structure A as
P
a a+p
n
' x =
L* =1-L* =x a + p' v
a
a + ft
(
y
v
P
(4b)
« + /?
where subscript / is dropped when no confusion is caused. Inserting (4b) into (4a) and using the factt/. = [/. for all i andy yields the per-capita real income in configuration A, UA, given by a a +p
U,
P
\"P
(4c)
a+p
B. Configuration(x\y) Configuration(x|y\ is defined by x, xs,yd> 0, Lx -1, xd - y = ys - Ly = 0 . A person choosing this configuration sells good x and buys good y. Inserting x, xs, yd > 0, Lx = 1 , xd = y = ys = Ly = 0 into (3), person f s decision problem for(x| jA is ,
maxC/,= (\-x!)a(kx;/pY
v y
-|i/(HM~1)r)
YlUj eJ
vJ
(5a)
• )
The first-order condition dUxi jdx\ = 0 yields the optimum decisions xs
- •
P
a + j3
y
d
—.
P
P{cc + PY
r
u
a ^af kp p(a + p) a+p
where Ux(p) is the indirect utility function for(x|_y).
,
(5b)
120
J. Wang, X. Yang
C. Configuration (y\x) Configuration (y\ x) is defined by y, ys, xd > 0, Ly = 1, yd = x = xs = Lx = 0. A person choosing (y\x) sells good y and buys good x. Inserting y, ys, xd >0 , Ly = 1, yd -x = xs = LX = 0 into (3), person z's decision problem for (y\x\ is max £/,
(1-j/fteO 0 \Uuj)
•
( 6a )
The first-order conditionrfC/^./<s^f = 0 yields the optimum decisions a a + /3
r=
,
pa pa ' a + (3
„ ( U =
fi Y kpa a+p {a + P)
(6b)
whereUy(p)is the indirect utility function for(j|x). 2.3. Corner equilibrium and general equilibrium and its comparative statics A. Market structures and corner equilibria There are two structures in the model: autarky denoted by A and the division of labor denoted by D. Due to the absence of trade, the comer solution for autarky is a comer equilibrium. The comer equilibrium for structure D is determined by two conditions: the market clearing condition, which is established by equilibrating market demand and supply, and the utility equalization condition, which comes from the assumption of free entry and utility-maximization behavior. We can solve for the relative price p and the relative number of individuals choosing (x| y) and yy\ x) from the market-clearing and utility-equalization conditions. The utility-equalization condition Ux(p) = Uy(p) yields p* = £(^-«)/(1->') . The market-clearing condition Mxxs - Myxd yields M^ - ap/p ,2 where Mx and My are the respective numbers of 2
Mmust be an integer. Yang and Ng (1993) have shown that if the integer condition is not satisfied, the equilibrium may not exist.
121
Pursuit of Relative Utility
individuals selling good x and good y and M^ = Mx/My . The market-clearing condition for goody is not independent of that forx due to Walras' law. Inserting the corner equilibrium level ofp into indirect utility function Ux(p) or U (p) yields the corner equilibrium utility level for structure D as UD =
\a
P a+P) \a+P a
V
fc2aft/(a+fi)
(V)
B. General equilibrium and its comparative statics Since the general equilibrium is the corner equilibrium that generates maximum per-capita real income, which can be identified by comparing the utility levels across structures, hence, having compared UD with UA, we have
uD>uA
iff
r
\
(a-\)(a+p)l7P
a k>k° = a + (3
P
\
a +p
(a-\)(a+p)l2a
(8)
Expression (8) yields the following proposition. Proposition 1: Equilibrium is the division of labor if the coefficient of transaction efficiency k> k° and is autarky ifk < k°. Figure 1 provides an intuitive illustration of Proposition 1. Two individuals in Figure la are used to illustrate autarky. In fact there are more individuals than shown in Figure 1. Circles represent configurations. The lines represent flows of goods self-provided. In autarky there is no market demand and supply. Productivity is low because of a low level of specialization. However, transaction costs do not exist because people do not do business with each other. Figure lb represents the division of labor, structure D. The lines represent the flows of goods. For structure D, there is a local business community composed of two persons. There are two markets, one for x and the other for j / .
122
J. Wang, X. Yang
a
b
A yC 3 33f Xs
>V )
f market V. for x .
(yx
1 market
v
for
y >
Figure 1: (a) Autarky and (b) the Division of Labor
There are two distinct professional sectors. The per-capita output of each good is higher than in autarky because of a higher level of specialization for each person. But transaction costs are higher than in autarky too, because each person must undertake two transactions to obtain the necessary consumption. Moreover, the degree of interdependence between individuals, the degree of production concentration of each good, and the degree of integration of society are higher for structure D than in autarky. The most important implication of the inframarginal analysis and framework with consumer-producers, economies specialization, and transaction costs is to predict many macroeconomic phenomena on the basis of a sound microfoundation; see Yang and Ng (1993, Chaps. 17 and 18). For the static simple model in this paper, it is predicted that aggregate demand and supply, the extent of the market will shrink as transaction efficiency k is reduced due to, say, oil crisis. According to our theory, demand and supply and the division of labor, and the division of labor and the extent of market are two sides of the same coin. Hence, macroeconomic phenomena that relate to aggregate demand can be explained by a change in the parameter of pursuing relative position through its effect on the equilibrium level of division of labor. Unfortunately, due to the limitation of space, we need another paper to investigate the macroeconomic implications of the model.
Pursuit of Relative
123
Utility
The effects of the degree of individuals' desire for relative utility, y, on the development of the division of labor can be seen from the effects of a change in y on k° via its effect on a and fl . If the effect of an increase in y on k° is negative (positive), the requirement in transaction efficiency for the division of labor to be the equilibrium is lower (higher). A total differentiation of k° with respect to y yields dink0 da
d\nk°
d\nk° d/3 —, (9) dy da dy dfi dy wheredfijdy = -(dajdy)-1 due toa + fi + y-l. The division of labor is more plausibly the equilibrium for a stronger desire for relative utility3 if d In k°/dy < 0 and less plausibly to be equilibrium if J In k°jdy > 0. A differentiation of k° in (8) yields 51nA:° a-\ '.ta-S^-Zb. » \ >0 2 da fS a + j3 a a + fi =
+
dink0
and^-^-O
dfi
if£a>0,y
V(10)
'
which implies that (i) dlnk°jdy <0 , if a < fi , da/dy>0 , and/or d0/dy / ? , da/dy<0 , and/or dfijdy >0 ; (ii) dlnk° Idy>0 , if a0 , or if a> fi , da/dy>0, and/or dfijdy < 0 ; (iii) dlnk°jdy-0 ifa = fl, no matter what sign dajdy and dfijdy are; and (iv) for the case where both dajdy<0 and dfijdy <0 , d\nk°jdy may be positive or negative, depending on the degree of dajdy < 0 anddfijdy < 0 . It is also easy to see that91nA:°/5a<0. Hence, a greater degree of increasing returns to specialization promotes the division of labor. This deduction is summarized in the following proposition. Proposition 2: A stronger desire for relative utility promotes the development of division of labor if it reduces the differential between preferences for different goods. It has no effects on the development of division of labor if this differential is zero. It has negative effects on the 3
A stronger desire for relative utility means that the parameter for relative utility ( y ) is increased relative to other preference parameters (e.g., a and /J).
124
J. Wang, X. Yang
development of division of labor if it increases the differential between preferences for different goods. If both da/dy < 0 and d/5/dy < 0, d In k^/dy may be positive or negative; i.e., a stronger desire for relative utility may or may not promote the development of division of labor, depending on the relative magnitude of da/dy < 0 to dfi/dy < 0. A greater degree of increasing returns to specialization promotes the development of division of labor. As we have seen, whether individuals choose autarky or the division of labor depends on the utility differenceU D -U A . UD-UA = 0 determines a critical value of transaction efficiency &° to generate the division of labor in a general equilibrium. If a stronger desire for relative utility lowers k°, it must promote the development of division of labor. But why does an increase in y lower k° only if it reduces the differential between preferences for goods? The reason is as follows. Because there exists a trade-off between a productivity gain from a higher level of division of labor due to increasing returns to specialization and a cost from more transactions, a differential between preferences for different goods, through its effects on the trade-off, will generate a tension, i.e., a changeable scope or room for the trade-off between allocational efficiency and organizational efficiency4 that restrains the development of division of labor. Because the smaller the differential between the preferences for different goods, the larger the scope for trading off allocational efficiency against organizational efficiency, a higher level of division of labor can be achieved by a change in y that reduces the differential. A change in y that increases this differential will slow down the development of the division of labor. When this differential vanishes, the division of labor can no longer be promoted by an increase in y, but it can still be made more favorable by a larger degree of increasing returns to specialization. However, if we allow a trade-off between work and leisure
4
In Yang (1990), a competitive equilibrium is defined as organization-efficient if the equilibrium level of the division of labor is the same as the Pareto-efficient level, while it is allocation-efficient if the equilibrium relative prices are the same as the price in the Pareto-optimum corner equilibrium.
Pursuit of Relative Utility
125
in the model, the development of division of labor can be generated by a change in y even if this differential equals zero. According to Proposition 2, some of the intuitions and conclusions of Marx, Durkheim, Veblen, and other economists are theoretically defensible, and others are indefensible. For example, Veblen (1899) and Frank (1985b) argue that the pursuit of relative economic standing, in general, generates social welfare losses, and Durkheim (1933) and Frank (1984, 1985a,b) argue that the pursuit of relative position, in general, promotes the level of specialization. Our model shows that these conclusions are not correct without qualification. Therefore, it becomes a matter for empirical economists to decide on the applicability of various conjectures to the historical record. 3. A Model with N Consumer Goods and M Consumer-Producers An increase in the desire for relative utility may generate implications for the level of division of labor via its effect on preferences for a variety of each person's activities. This section is devoted to the investigation of the effects in the context of an equilibrium model with N goods. Let us consider an economy with M consumer-producers and N consumer goods. The self-provided amount of good r by person i is xir. The amount sold in the market of good r by person i is x"ir .The amount purchased in the market of good r by person i is xdir. There are n traded goods and N- n nontraded goods. The transaction cost coefficient is 1 - k. Thus, kxir is the amount person i obtains when she purchases xfr . According to Lemma 1, each individual buys « - 1 goods, self-provides N - (w - 1 ) goods, and produces and sells 1 good. The amount consumed of good r by person i is xir +kxfr. The utility function is identical for all individuals (11) where a, a, and y e (0, 1) and aN + <J + y = 1; a and y are her preference parameters for consumer goods and for relative utility, respectively, and a is her preference parameter for the variety of her productive activities
126
J. Wang, X. Yang
because N- (n - 1) is the number of self-produced goods, where n is the number of traded goods and n - 1 is the number of goods purchased by an individual. Here, there are two assumptions different from those for the previous model: (i) a consumer's preferences for all consumer goods are the same, and (ii) an individual has a preference for the variety of productive activities. Thus, the effects of the changes in preference differentials between different consumer goods from a stronger desire for relative utility on the development of division of labor cannot be investigated in this model. The reason for this assumption is that, in this general model, our analysis is to focus on the effects of the pursuit of relative utility on the relationship between preferences for consumer goods as a whole and preference for diversely productive activities, and in turn, on the development of division of labor. According to Karl Marx (1844) and other economists, individuals, indeed, have preferences for the variety of productive activities. Then an individual as a consumer-producer must find a balance between preference for diverse activities and preferences for consumer goods as a whole. Moreover, if an individual's tastes differ across goods, this general model cannot be solved. The production functions are similar to that in the preceding section Xir+K=L%> r=l,...,N,a>l.
(12)
The transaction cost coefficient 1 - k can be seen as a variable transaction cost since 1 - k is proportional to the quantity purchased. For the model with N goods, a fixed transaction cost is needed to ensure that the second-order condition for an interior value of the number of traded goods is satisfied in equilibrium. If the second-order condition is not satisfied, then the equilibrium is either autarky or complete specialization and gradual evolution of the division of labor is impossible. A fixed labor transaction cost coefficient for each purchase is assumed to be c, which can be interpreted as, for example, a fixed communication cost for each transaction between the Walrasian auctioneer and a person. Assuming that an individual spends a fixed fraction c of her available hours on each purchase, the labor share an individual spends in n - 1 purchases isLc = c(n - 1 ) . The endowment constraint of labor is given by
127
Pursuit of Relative Utility
L c +JTz, f r =l,
Z.fr,Z.c e[0,l].
(13)
The budget constraint for a consumer-producer selling good r is
where g e G , G is a set of all goods purchased, and pr and pg are the respective prices of good r and good g. In our symmetric model, we have assumed that goods 1,2,..., n are traded by individuals selling good r, and goods n + 1, n + 2,..., N are not traded (« e [1, AT)). Let / / be a set of all nontraded goods. Applying Lemma 1 cited in section 2.2 to the model, for a person selling good r, we have xir>0,
xl >0,
x
ig=xig=Lig=°>
x?r=0, 4
Lir>0,
> 0
%,A*>0, x>Ol=x$l=0
V
£
G G
'
V/zei/.
(15)
Hence, the decision problem for an individual selling good r becomes
m a X :c/ / =x-nK)°n^( gcG heH
jv w+i ( uY
- r riTT
\jeJUjJ
s.t. xir + xsir - Lair, xih =Laih, \/h&H Lc = c(n -1)
(production function)
(total fixed cost in trade)
Lir + ~Y^Lih+Lc=\->
(endowment constraint)
heH
prx\r = ]T Pgx?g
(budget constraint),
(16)
where pr is the price of good r, and the decision variables are xir,xl,xfg,Ljr,Lih,xih,and n. Having inserted all constraints into [/and differentiating[/, with respect toLir,xsir,xfg, individual optimal decisions are given by
Lir=n{\-Lc)/N, X* =Kl(n-\)
= Llln,
x°r=(n-l)Li/n xih
=[(l-Lc)/Nj
128
J. Wang, X. Yang 4 =(Pr/Pg)xl/(n-l)
= {Pr/Pg)Ll/",
VgeG,
(17)
where Lc = c(n -1) . Inserting these optimal solutions into Uj , and noticing pr = pg in a symmetric model, Ui can be expressed as a function of n \an
U,=
ka[n-\)
'l-c(n-l)'
N
aa(N-n)
N
x(N-n + iy.
(18)
Maximizing Ui with respect to n yields the optimal number of traded goods, n , for an individual selling good r dhxU,L = -a(]nn*+\) dn
aaNc — -+ • l-c(«*-l)
N~n'+\ +a\nk = F^0.
(19)
It is not difficult to identify the sign of dn*/dy from (19). Since the larger optimal number of traded goods, n , means the higher level of division of labor, we can use the sign of dn*jdy to investigate the effect of a stronger desire for relative utility on the level of division of labor. Because n cannot be solved explicitly, we must apply the implicit function theorem to (19) so that dn* __ dF/dy dy dF/dn'
(20)
where y = \- aN - a . The second-order condition for interior optimal solution n is dF/dn* = d2 In Ul /dn2 < 0 . Since we assume a, a, and^ are interdependent and the sum of them is a constant, we have F = F[a(y), o(y)]. Umce,dF/dy = (dF/da)(da/dy) + (dF/da) (da/dy). Differentiating F and using F = 0 in (19) yields dF/da = a/a (N - n +1) > 0. Also dF/da < 0 can be derived from (19). Therefore, dF — <0, dy dF n — >0, dy
da if
da <0<
(21a)
dy dy ..da da n if >0> . dy dy
(21b)
Pursuit of Relative
129
Utility
Expressions (21a) and (21b) together with the second-order condition for the interior n , implies that dn*
n
.. da
< 0 if
dy This leads us to
n
da
<0<
dy
dn* ;
dy
da > 0 if —
dy
dy
_. da >0>
.
(22)
dy
Proposition 3:5 A stronger desire for relative utility has positive effects on the level of division of labor n {or the number of traded goods) if it weakens the preference for variety of each person's productive activities and increases the preference for direct consumption of goods. Proposition 3 is intuitive. If a stronger desire for relative utility weakens the preference for the variety of activities, a decrease in a, individuals will prefer more specialization so that the equilibrium level of division of labor is promoted. Figure 2, where it is assumed that m = N=A, provides an intuitive illustration of the evolution of division of labor when an increase in the degree of the desire for relative utility reduces the degree of the desire for variety of a person's productive activities and increases the desire for direct consumption of goods. The lines signify goods flows. The small arrows indicate direction of goods flows. The numbers beside the lines signify goods involved. A circle with number / signifies a person selling good i. Figure 2a denotes autarky, where each person self-provides four goods, i.e., n - 1 = 0, because of a low degree of the desire for relative utility that leads to a high degree of the desire 5 Compared to Proposition 2, no difference in preferences across consumer goods is assumed for the general model from which Proposition 3 is drawn. That is to say, an individual has the same preference for all consumer goods. Hence, the effects of the size of preference differentials between different consumer goods on the development of division of labor cannot be investigated in this general model. Otherwise, the conclusions from this more general model must be more ambiguous. This assumption is made since our main purpose is to focus on the effects of the size of preference differential between all consumer goods as a whole and the variety of her productive activities on the development of division of labor (in fact, here our suggestion is based on, as many economists have argued, that "individuals could also have preferences for variety in their activities"). Also if we assume preferences are different across N goods, the model is not tractable. Of course, in reality, the preferences are not symmetrical and solving for the equilibria is a dynamic process of iterative improvement.
130
J. Wang, X. Yang
'$>,'$>.
'$'$
Figure 2: The evolution of division of labor, (a) autarky, n = 1, four communities; (b) partial specialization, n = 2, two communities; (c) complete specialization, n = 4, the integrated market.
for variety of productive activities. Figure 2b denotes partial specialization, where each person sells one good, buys one good, trades two goods, and self-provides three goods, i.e., n = 2, because of a higher degree of the desire for relative utility that leads to a lower degree of the desire for variety of activities. Figure 2c denotes extreme specialization, where each person sells and self-provides one good, buys three goods, and trades four goods, i.e., n = 4, because of a very high degree of the desire for relative utility that leads to a very low degree of the desire for variety of activities. Note that here there are multiple equilibria but all of them yield the same utility and symmetric comparative statics. Since our model is symmetrical, only the number of traded goods is relevant, and patterns of
Pursuit of Relative
Utility
131
trade do not make any differences. For instance, for an equilibrium structure in Figure 2b, it is indeterminate if goods 1 and 2 or goods 2 and 3 are traded. Also, it is indeterminate who is specialized in which profession. An infinitesimal differential in parameters of tastes, production, and transactions across goods and individuals will rule out the multiplicity of equilibria. Since there exists a positive relationship between the level of division of labor and productivity and trade dependence as shown in Yang (1990), Yang and Borland (1991), and Borland and Yang (1992), the following corollary can be derived from Proposition 3. Corollary: A stronger desire for relative utility has positive effects on labor productivity and trade dependence if it decreases the desire for a variety of each person's productive activities and increases the desire for quantities of consumption goods. It can be shown that as long as economies of specialization are individual specific, that is, increasing returns to specialization are localized, and individuals pursue their relative utility under perfect competition, the equilibrium is Pareto optimal. For detailed proof of this proposition, see Yang and Ng (1993) and Wang (1993).6 4. Concluding Remarks We have explored the relationship between the pursuit of relative utility and the development of division of labor in two equilibrium models in this paper. It is shown that the equilibrium level of division of labor depends on some critical level of transaction efficiency, which is in turn determined by preference parameters for consumption, for a variety of activities, and for relative utility. An increase in the degree of the desire for relative utility will change the structure of preferences for consumption 6
According to Debreu's theory of value, the existence of global increasing returns and/or externalities distorts market. However, Yang and Ng (1993) and Wang (1993) show that (i) if increasing returns to specialization instead of to scale and are localized to each of many individuals who have no monopoly power (this is just the case in this paper), a Walrasian regime prevails; and (ii) relative utility effects do not distort market under perfect competition but relative consumption effects and relative income effects do.
132
J. Wang, X. Yang
and for a variety of activities. This increase will generate positive effects on the development of division of labor if it reduces the differential between preferences for different goods or if it reduces the desire for a variety of each person's productive activities. Our finding also means that the consumption of relative position may or may not generate positive or negative effects, depending on specific conditions. Thus, some of the intuitions and conclusions of Marx, Durkheim, Veblen, and other economists on the relationship between pursuit of relative position and division of labor are not correct without qualification.
References Basmann, Robert L., Molina, David J., and Slottje, Daniel J., "A Note on Measuring Veblen's Theory of Conspicuous Consumption." Rev. Econom. Statist. 70, 3:531-535, Aug. 1988. Borland, Jeff, and Yang Xiaokai, "A New Approach to Economic Organization and Growth." Amer. Econom. Rev. 82, 2:460-482, May 1992. Buchanan, James, and Stubblebine, Wm G., "Externality." Economica 29, 116:371-384, Nov. 1962. Durkheim, Emile, The Division of Labour in Society. New York: The Free Press, 1933. Frank, Robert H., "Are Workers Paid Their Marginal Products?" Amer. Econom. Rev. 74, 4:549-571, Sept. 1984. Frank, Robert H., "The Demand for Unobservable and Other Non-Positional Goods." Amer. Econom. Rev. 75, 1:101-116, Mar. 1985a. Frank, Robert H., Choosing The Right Pond: Human Behaviour and The Quest for Status. New York: Oxford Univ. Press, 1985b. Frank, Robert H., "Frames of Reference and The Quality of'Life." Amer. Econom. Rev. 79, 2:80-85, May 1989. Khinchin, Aleksandr I., Mathematical Foundations of Information Theory. New York: Dover, 1957.
Pursuit of Relative
Utility
133
Marx, Karl, In Dirk J. Struik, Ed.; Martin Milligan, Trans., Economic and Philosophic Manuscripts of 1844. New York: International Publishers, 1971. Mezias, Stephen J., "Aspiration Level Effects: An Empirical Investigation." J. Econom. Behav. Organiz. 10, 4:389^100, Dec. 1988. Miner, Frederick C , Jr., "Jealousy on the Job." Personnel. J. 69, 4:88-95, Apr. 1990. Ng, Yew-Kwang, and Wang, Jianguo, "Relative Income, Aspiration, Environment Quality, Individual and Political Myopia: Why May the Rat-Race for Material Growth Be Welfare-Reducing." Math. SocialSci. 26, 1:3-23, July 1993. Schoeck, Helmut, Envy: A Theory of Social Behaviour. New York: Harcourt, Brace & World, 1969. Smith, Adam, Reprint, In E. Cannan, Ed. An Inquiry into The Nature and Causes of The Wealth of Nations. Chicago: Univ. of Chicago Press, 1976. Stadt, Huib van de, Kapteyn, Arie, and Geer, Sara van de, "The Relativity of Utility: Evidence from Panel Data." Rev. Econom. Statist. 67, 2:179-187, May 1985. Veblen, Thorstein, The Theory of The Leisure Class. New York: Macmillan Co., 1899. Veenhoven, Ruut, "Is Happiness Relative?" Social Indicators Res. 24, 1:1-34, February 1991. Wang, Jianguo, "Pursuit of Relative Economic Standing," Ph.D. Dissertation, Department of Economics, Monash University, 1993. Yang, Xiaokai, "Development, Structure Change, and Urbanisation." J. Develop. Econom. 34, 1-2:199-222, Nov. 1990. Yang, Xiaokai, and Borland, Jeff, "A Microeconomic Mechanism for Economic Growth," J. Polit. Econom. 99, 3:460^182, July 1991. Yang, Xiaokai, and Ng, Yew-Kwang, Specialization and Economic Organisation: A New Classical Microeconomic Framework, Contributions to Economic Analysis Series. Amsterdam: North Holland, 1993. Yang, Xiaokai, and Ng, Yew-Kwang, "Specialization, Information, and Growth: A Sequential Equilibrium Analysis."/. Econom. Behav. Organiz., forthcoming.
This page is intentionally left blank
Part 4
Urbanization
This page is intentionally left blank
CHAPTER 7 DEVELOPMENT, STRUCTURAL CHANGES AND URBANIZATION*
Xiaokai Yang* Monash
1.
University
Introduction
As far as the problem of production was concerned, Adam Smith (1776) and Allyn Young (1928) emphasized the productivity implications of economic organization (the division of labor). Neoclassical microeconomics cannot explore such implications for the following reason. For production functions with constant returns to scale, an agent's productivity of a good is not greater when he produces only this good than when he produces many goods. On the other hand, production functions with increasing returns to scale cannot be used to characterize the level of specialization within a firm. The concept of economies of scale presupposes a complete separation of pure consumers from pure producers. 'Scale' relates to a firm which is a pure producer, but is irrelevant to a pure consumer. The separation of pure producers from pure consumers is a basis of Debreu's theoretical framework and neoclassical microeconomics. This artificial separation has perhaps misled economic theory. In autarky, there is neither a pure consumer nor a pure producer; each individual is a
Reprinted from Journal of Development Economics, 34 (1-2), Xiaokai Yang, "Development, Structural Changes and Urbanization," 199-222, 1990, with permission from Elsevier. * The author is grateful to Gene Grossman, Barry Nelebuff, Edwin Mills. Yew-Kwang Ng, Roslyn Anstie, and two referees for their comments. Thanks also go to the Ford Foundation and the Open Society Fund for financial support. The author is responsible for the remaining errors. 137
138
X. Yang
producer/consumer. The division of labor will increase the portion of a person's production that is not consumed by himself (i.e. the portion sold to other people) and increase the portion of a person's consumption that is not produced by himself (i.e. the portion purchased from other people). We can view this change as an increase in the degree of separation between production and consumption though each person is a producer/consumer even in the division of labor. The degree of such separation depends on the level of division of labor (or inversely on the degree of self-sufficiency). As to endogenizing the level of division of labor and thereby the degree of such separation, Debreu's framework is irrelevant since in his framework, pure consumers are completely separated from pure producers and the degree of the separation of consumption from production cannot be defined. In order to capture the ideas of Smith and Young, this paper specifies production functions for each producer/consumer such that an individual's productivity increases with the level of specialization and the aggregate transformation curve for the whole economy depends positively on the level of division of labor. There are several implications of this method of specifying production functions for the theory of equilibrium. First, each individual is a producer/consumer. He must decide how many goods are self-provided, i.e. what is the level of specialization. Hence, the model in this paper can be used to endogenize the level of division of labor. According to conventional microeconomics, pure consumers cannot choose the level of specialization since they must buy all goods from firms. Second, a Cobb-Douglas utility function is specified for each producer/consumer. Consequently each individual as a consumer prefers diverse consumption and as a producer prefers specialized production. This implies that the division of labor will incur great transaction costs. Therefore, there is a trade-off between economies of specialization and transaction costs. In other words, our method of specifying production functions makes the level of division of labor crucial for productivity; while transaction efficiency is critical for the determination of the level of division of labor. Because of increasing returns to specialization, the production possibility frontier (PPF) is associated with extreme
Development,
Structural Changes and Urbanization
139
specialization. Extreme specialization will, however, incur prohibitively great transaction costs since people prefer diverse consumption. Hence, the welfare frontier may differ from the PPF.1 A natural conjecture is that a competitive equilibrium will balance the trade-off between economies of specialization and transaction costs, and that improvement of transaction efficiency will move the equilibrium closer to the PPF, resulting in an increase in the division of labor. The major purpose of this paper is to prove this conjecture. In other words, Smith's conjecture of the 'invisible hand' and his insights into increasing returns to specialization will be reconciled in this paper. A crucial assumption leading to this result is that labor is specific for each person who is able to produce all goods. This assumption combined with the assumption of free entry ensures the first welfare theorem even if there exist increasing returns to specialization. Intuitively, we can see that if increasing returns to specialization are specific for each individual, economies of scale are limited. Since production functions are specified for each producer/consumer, prices are determined by the numbers of individuals selling different goods. This number cannot be manipulated by any individual because of the assumption of free entry and the assumption that each person is able to produce all goods. Hence, nobody is able to manipulate prices. Nevertheless, if labor can be divided in fine detail and the population size is very large, economies of division of labor may be very large. Hence, a competitive market may be compatible with the substantial economies of the division of labor. Therefore, we can use this method to develop the concepts of equilibrium and Pareto optimum in relation not only to the resource allocation for a given level of division of labor, but also to the determination of the level of division of labor and productivity. The implications of this method for other fields of economic analysis are important. The level of division of labor based on increasing returns to specialization is intimately related to the extent of market, trade dependence, trade pattern, market structure, and economic structure, so these can be made endogenous in our model. Productivity is related to 1 According to conventional microeconomics, the welfare frontier coincides with the PPF.
140
X. Yang
the level of division of labor which depends on transaction efficiency, which is in turn affected by urbanization, government policies, and institutional arrangements. Hence, our model can be used to investigate the impacts of urbanization, government policies, and institutional arrangements on the equilibrium level of division of labor (related to the market structure, trade dependence, and so on) and productivity. This paper shows that the equilibrium trade volume depends positively on the absolute degree of increasing returns to specialization in production and transaction, and negatively on the average distance between a pair of neighbors. The trade pattern is determined by the relative degree of increasing returns to specialization in producing different goods and relative preference for different goods. In addition, it is shown that increases in diversification of the economic structure, concentration of production, integration of the economy, specialization, and the output share of roundabout productive activities are different versions of the evolution of division of labor resulting from improvements of transaction efficiency which are in turn caused by urbanization, liberalization policies, or changes in institutional arrangements. Many economists have proposed similar ideas. Nevertheless few among them have been successful in formalizing them. On the other hand, the formal models proposed by mainstream economists are often inconsistent with these ideas. This may be due to the difficulty of formalizing the ideas of Smith and Young. For example, many economists [see, e.g. Helpman and Krugman (1985) and Herberg and Tawada (1982)] point to problems which are considered to make an equilibrium model with increasing returns to specialization and transaction costs unmanageable. Such problems include the issue of corner solutions based on increasing returns to specialization, the problem of infinite combinations of individual corner solutions in solving for equilibrium, notorious complications in dealing with indexes of variables in models with transaction costs, and the problem of existence of equilibrium. Formalizing the notion of increasing returns to scale is much easier than formalizing the notion of increasing returns to specialization. This might explain why it is hard to find microeconomic equilibrium models that formalize the theory of production proposed by Smith and Young.
Development, Structural Changes and Urbanization
141
In this paper, we try to formalize the essence of the ideas of Smith and Young as well as to keep an equilibrium model tractable. Our techniques for achieving these two goals are to specify a specific transaction technology and to devise a multiple-step approach to handling the issue of combinations of corner solutions. In order to get around the problem of the existence of equilibrium, we propose a specific model. Using this model, we can prove the existence of equilibrium although it is impossible to reach a general conclusion on this issue for a model with increasing returns to specialization. Fortunately, these measures are not only effective in keeping the model tractable, but also useful in working out the meaningful comparative statics of equilibrium. Many interesting economic phenomena which are not addressed in the conventional theory can be explained by the kind of model in this paper. This paper is organized as follows. Section 2 sets out a model with three goods. Sections 3-7 develop a multiple-step approach to handling the model with increasing returns to specialization. Section 3 solves for the corner solutions for the individual decision problem. Section 4 solves for all candidates for equilibrium in various market structures. Section 5 solves for the restricted Pareto optimum in each market structure and the full Pareto optimum. Section 6 investigates the relationship between equilibria and the Pareto optima. Section 7 solves for an equilibrium and investigates its comparative statics. Some simple conclusions are summarized in the final section. 2.
A Model with Three Goods
Let us first consider an economy with M consumers/producers and three consumer goods. The self-provided amounts of these goods are x, y, and z, respectively. By self-provided we shall mean that quantity of a good produced by an individual for his own consumption. The amounts of these goods sold at the market are xs, /, and z\ respectively. The amounts of these goods purchased in the market are xd, y\ and z , respectively. An 'iceberg' type of transaction technology is characterized by the coefficient k. Fraction i of a shipment disappears in
142
X. Yang
transportation. Thus, (1 - k)xd, (1 - k)yd , and (1 - k)zd are the amounts a person receives from the purchases of three goods, respectively. Furthermore, we assume that (1 - k) depends on the quantity of labor used in transaction. (1 - k) can be viewed as transaction service. Such services are categorized into self-provided ones and traded ones. Let 1 - k = T + Td where T is the self-provided quantity of transaction service and T1 is the quantity purchased of transaction service. The more transaction service T + Td, the greater portion of a purchase is received by its buyer. Here, T + Td < 1. Signifying the quantity sold of transaction service by Ts , transaction technology and production functions are thus given by x + xs=Lax,
y + ys=Lby,
z + z"=Lcz,
T + TS=L'T,
(2.1a)
where x + x5, y + ys, z + zs, and T + TS are the output levels of four goods and service, respectively. Ls is the amount of labor used in producing good (or service) s where s - x,y,z,T . (2.1) is assumed to be identical for all individuals. In such iso-elasticity production functions parameters a, b, c, and t characterize the returns to specialization. If a, b, c,t>\, then there are increasing returns to specialization. Adopting the concept of localized technology proposed by Sah and Stiglitz (1986), we assume that the total quantity of labor available for an individual is specific for him. Let this quantity be one; there is an endowment constraint of the specific labor for an individual: Lx+Ly+Lz+LT=l,
0
i = x,y,z,T.
(2.1b)
This method of specifying production functions is substantially different from the conventional one. This system of production functions differs from the production functions associated with the U-shaped average cost curve and those with global increasing returns to scale. The production possibility frontier (PPF) of this system of production functions is associated with extreme specialization. Therefore, if there is no transaction cost, equilibrium is the extreme division of labor (each individual has extreme specialization and different individuals specialize in producing different goods). In our model the reason that people prefer some division of labor is that internal economies of scale are very limited because labor is specific
Development, Structural Changes and Urbanization
143
for each individual. This point distinguishes increasing returns to specialization from increasing returns to scale. Assume, further, that the transaction service T (or 7t/) is related only to the quantity traded and there are transaction costs related to the distance between a pair of trade partners; thus there is a location problem. Suppose that all people are evenly located and the geographic distance between a pair of neighbors is a constant. The distance between a pair of trade partners may differ from the distance between a pair of neighbors. If all trade partners of an individual are located in a circle with radius R and with his location as the origin, it can be shown that the average distance between this individual and his trade partners is proportional to R and the number of these trade partners N is proportional to R2. Hence, the average distance between this individual and his trade partners is proportional to yJN. If all the trade partners supply different goods to this individual, the number of traded goods for him is n = N + 1. For simplicity, we assume that the number of trade partners of an individual, N=n-\, where n is the number of traded goods for him. Assume that the transaction cost coefficient K characterizes the transaction cost related to the average distance between a pair of trade partners; then the K fraction of (T + Td)xd or (T + Td)yd , or (T + Td)zd disappears on the way from a seller to a buyer. The relation between K and the number of trade partners of an individual, N, is given by K = s4N,
(2.2)
where s is a constant depending on the distance between a pair of neighbors and n. Taking (2.2) into account, the amounts consumed of the three goods are x + (l-K) (T + Td)xd , y + (\-K) (T + Td)yd , and z + (l-K) (T + Td)zd , respectively. The utility function is identical for all individuals: U = [x + (l-K)(T x[z + (l-K)(T
+ Td)xdJ[y + Td)zdJ,
+ (l-K)(T
+
Td)ydJ (2.3)
where0
144
X. Yang
We assume free entry for all individuals into any sector and that M is large. These assumptions imply that individuals treat prices parametrically. 3.
The Individual Optimal Decision
This section and the four sections to follow are devoted to devising a multiple-step approach to handling the model with increasing returns to specialization. In our model, the individual decision problem implicitly includes three problems: to choose the optimum level of specialization (in other words, the optimal number of traded goods is a decision variable), to choose the optimal composition of traded goods, and to choose the optimal quantities of consumption, production, and trade. This problem is unmanageable in a step. If there are increasing returns, some variables will take zero values in the individual optimal decision and in equilibrium. Hence, an individual needs to enumerate all possible combinations of zero and non-zero variables before identifying his optimal decision. Also we need to enumerate all possible combinations of these individual combinations of zero and non-zero variables before identifying the Pareto optimum and equilibrium although we can exclude some of these combinations from the list of candidates for equilibrium by application of the Kuhn-Tucker theorem (as shown in Appendix A). There are five steps in solving for equilibrium. This section will solve for individually rational decisions for given prices and for each combination of zero and non-zero variables. Section 4 solves for the candidate for equilibrium (or corner equilibrium) for each combination of the above individual combinations. Section 5 solves for the Pareto optimum candidate for equilibrium. Section 6 investigates the relationship between equilibrium and the Pareto optimum. Finally, equilibrium and its comparative statics are solved. The individual decision problem is maxU = [x + (\-K)(T
+ Td)xdJ[y
x[z + (\-K)(T + Td)zdJ,
+ (l-K)(T
+ Td)y<
(3.1)
Development,
Structural Changes and Urbanization
s.t. x + xs = Lax, y + ys =Lby, z + zs =LCZ S
T + T =L'T
145
(production function), (transaction technology),
Lx + L +Lz+LT=\
(endowment constraint), d
pTT> + Pxx° + pyy + pzz° = pTT + pxx" + pyyd + pzzd (trade balance), where/,/*,/'', and Lt (i = x,y,z,T) are decision variables, which may take on zero or positive values. pt is the price of good (or service) /. If a, b, c, t > 1, the optimal decision is certainly a corner solution. By combination of zero and non-zero values of the variables, there are several possible corner solutions. We shall call such a combination a 'structure'. An individual needs to enumerate and compare utilities in all structures before choosing a structure. Therefore, an individual must solve for the corner solution for each structure and his decision making process consists of two stages. In the first stage all structures are enumerated. An individual solves for the efficient allocation (how much should be produced, consumed, and traded of each good) for given prices and for each structure. In the second stage, he decides what should be produced, and what should be sold and purchased, i.e. which structure should be chosen. Section 4 will discuss this problem.2 There are 29 structures of four general types: (1) Autarky (x,y,z), i.e. an individual self-provides three goods. For this structure xs = y* = zs = xd = yd = zd = T = Td = Ts = LT = 0. In other words, the amounts sold and purchased of the three goods are zero, as are the amounts sold, purchased, and self-provided of transaction services. (2) Structure (i/j), i.e. an individual sells good (or service) / and purchases good (or service)/, i,j = x,y,z,T. For such structures ks =kd =id =j = j s =Lj=0
2
for
k*i,j,
This two-stage set up is just for convenience of exposition. In fact all these decisions are determined simultaneously.
146
X. Yang
where index k denotes the goods other than goods / and j . By two permutations of four factors, we obtain 12 structures of this type: (xly), (y/x), (x/z), (z/x), (y/z), (z/y), (T/x), (x/T), (T/y), (y/T), (T/z), and (zIT). (3) Structure (iljk), i.e. an individual sells good (or service) i and purchases goods (or services) j and k. i, j , k = x, y, z, T. For such structures rs =rd =id =j = k = j s =ks =Lj=Lk=0
for
r*i,j,k,
where index r denotes the goods other than goods i,j, and k. There are 12 structures of this type: (xlyz), (x/yT), (x/zT), iylxz), iylxT), iylzT), (zlxy), (zIxT), (zlyT), (Tlxy), (Tlxz), and (T/yz). (4) Structure (i/jkr), i.e. an individual sells good (or service) i and purchases goods (or service) j , k, and r, i, j , k, r = x, y, z, T. For such structures ^ = j = k = r = j s = ks = rs = Lj =Lk=Lr=0
for j,k, r ± i.
There are four structures of this type: (xlyzT), (y/xzT), (z/xyT), (T/xyz). Appendix A has proven Proposition 1: According to the necessary condition for an optimum decision, an individual does not produce and purchase a good at the same time, and sells only one good {if any). This proposition implies that the remaining structures, e.g., (ijk/ijk), (ij/jk), (ij/k), etc. do not satisfy the necessary condition for maximizing individual utility. Hence, these structures will not be concerned. Letting the relevant variables in problem (3.1) take on a zero value, an individual can solve for his optimal decisions for each structure. These individual decisions include individual supply, which is a constant depending on a, b, c, a, P, y, t, and s, and individual demand, depending positively on the relative prices of the goods he sells to the goods he purchases, and his supply. Inserting the individual optimal decision for each structure and for given prices into the utility function gives an indirect utility function for each structure. The indirect utility functions differ from structure to structure although the original utility function is identical for all individuals. In an indirect utility function, the number of relative prices is
Development,
Structural Changes and Urbanization
147
one less than the number of traded goods. Here there are 29 alternative structures for each individual that might be rational. 4. The Markets and Candidates for Equilibrium (Corner Equilibria) As in solving for the individual decisions, if there are increasing returns, we need to enumerate all combinations of structures before identifying the equilibrium and the Pareto optimum. This section first investigates how structures are combined to constitute markets. We will then enumerate all combinations of structures and solve for the 'candidates' for equilibrium (corner equilibrium). The corner equilibrium is an analogue to the corner solution in the optimization problem for an individual. It is a concept used to solve for the equilibrium in a calculation of several steps. Of course, a corner equilibrium would never come into being if it was not a full equilibrium3. Defining a combination of several mutually consistent structures as a market, there are many market configurations, such as those shown in Figure 1. Figure 1(a) is a market that combines structure (x/y) (an individual sells x and buys y, and self-provides z and T) and structure (y/x) (an individual sells y and buys x, and self-provides z and 7). We refer to this market as P. Figure 1(b) is a market that combines structure (x/yz) (an individual sells x and buys y and z, and self-provides 7), structure (y/x) (an individual sells y and buys x, and self-provides z and 7) and structure (z/x) (an individual sells z and buys x, and self-provides y and T). We refer to this market as B. Note that some structures are not mutually consistent, e.g., structure (x/y) and (x/z) are not mutually consistent, i.e. there is no demand for x and no supply of y and z although there are supply of x and demand for y and z for a combination of these two structures. Therefore, a combination of (x/y) and (x/z) cannot constitute a market.
3
Many papers on international trade, e.g. Ethier (1986), show that in models with increasing returns to scale, there are multiple equilibria and some of them are unstable. In our model all corner equilibria except the Pareto optimum one are not full equilibria. This point distinguishes our model from the models with increasing returns to scale.
148
X. Yang
(a) Market P
(b) Market B Figure 1
In the analysis of competitive equilibrium, we will use the concept of a basic market. A basic market is defined as a market for which one cannot obtain another market by dropping any of its structure. In Figure 1, the two markets are basic markets. However, a combination of structures (x/yz), iy/xz), (y/x) and (z/x) is not a basic market because we can obtain market B in Figure 1(b) by dropping structure (y/xz). Many structures can be combined with a basic market to obtain a new, non-basic market. For example, by combining market B with any of the other six structures which do not involve trade in T and are not in market B we can obtain ^
C{ - 26 markets, where C( isj combination of
six factors. For a basic market, the number of traded goods equals the number of structures. For a non-basic market, the number of traded goods is smaller than the number of structures. We will enumerate all possible markets, and solve for all candidate equilibria (corner equilibria), then find the Pareto optimum corner equilibrium and the full equilibrium.
Development, Structural Changes and Urbanization
149
From the individual optimal decisions, the individual supply and demand functions for each structure can be derived. Let Mt be the number of individuals selling good i; market clearing conditions can be specified for each basic market. For example, if the market is P, a combination of structure (x/y) and structure (y/x), shown in Figure 1(a), there are the market clearing conditions in market P: Mxxs = Myxd,
Mxyd =Myys,
(4.1 a)
where Mx and My are the numbers of the individuals selling x and y, respectively. The individual budget constraints for the two component structures are Pvx'=y,
Pvx*=y,
(4.1b)
where pxy=pxi'py. One of these two equations implies the other due to Walras' law. From (4.1a) and (4.1b), it can be derived that pxy=ydlx°=ylrxs,
(4.2)
where r = Mx I My is the ratio of the number of individuals choosing structure (x/y) to the number of individuals choosing structure (y/x). We need solve only for the relative numbers of individuals choosing different structures because we can derive Mx = rM/(l + r) and M - Mj (l + r) from Mx+ My = M. Hence, Mx and My can be found if r is known. According to the individual optimal decisions, xs and ys are constants depending upon a, b, c, a, /?, y, s, and /. Hence, (4.1b) implies p^ depends inversely on r. Inserting (4.2) into indirect utility functions in structures (x/y) and (y/x), the utilities may be expressed as functions of r. Setting U(x/y) = U(y/x), we have U(x/y) = r-f>G(xly) = r"G(ylx) = U(y/x),
(4.3)
where G's depend on a, b, c, a, ft, y, t, and s. The intersection of U(x/y) and U(y/x) determines a corner equilibrium value of r. Inserting the value of r back into the utility function, (4.3) gives a corner equilibrium utility U*, which is real income as well as the real returns to labor in this market because of the assumption that each person has one unit of labor.
150
X. Yang
For the category of non-basic markets, there is the following proposition: Proposition 2: There does not exist a corner equilibrium for any non-basic market. For a non-basic market, there are m structures and n traded goods, and m > n. According to Appendix A, in any optimal structure an individual sells at most one type of good. This implies that the types of structures selling the same type of good exceed one if m > n. The necessary conditions for a corner equilibrium lead to m - 1 conditions of utility equalization. These m - \ equations contain n - 1 relative prices. Noting the fact that these m - 1 equations are log-linear, nonhomogeneous, and independent of one another, this system has no consistent solution since m > n. Therefore, Proposition 2 has been established. Noting the following two points, we can prove the existence of a corner equilibrium for a given basic market. (A) For a given basic market with n traded goods and n component structures, the individual supplies of all traded goods are constants and the individual demands for all traded goods are functions of relative prices. The indirect utility functions are determined only by relative prices of the traded goods. Letting the n indirect utility functions equal one another, we obtain (n - 1) log-linear equations containing (n - 1) relative prices. Again, noting that these equations are non-homogeneous and independent of one another, we can solve for a vector of log-relative prices. The relative prices are positive. (B) Given the relative prices solved in (A), (n - 1) independent market clearing conditions can be transformed into a system of equations containing (n - 1) relative numbers of individuals selling the different goods. This system looks like a system of linear equations associated with an input-output system. For example, in the market with three traded goods, such a system is
Development, Structural Changes and Urbanization
-filia + -y(a + r)
1
b^M^y s
V
151
(4.4a)
Mzxpzxz
If the number of goods traded by individuals selling good z is less than that traded by the whole market, e.g. structure (z/x) trades two goods and structures (x/y, z) and (y/x, z) trade three goods, then (4.4a) becomes MyxPyxys
1
fx^
(4.4b)
where My= MJMj is the relative number of individuals selling good i to those selling good j . Myx and Mzx are unknown. It is easy to see that the solution vector (Myx, Mzx)' for this kind of system of linear equations is positive. Considering the positiveness of equilibrium relative prices shown in (A), the positiveness of the solution of (4.4) guarantees the existence of corner equilibrium in a basic market. This leads us to Proposition 3:
There exists a corner equilibrium for any basic market.
There are multiple corner equilibria. All these corner equilibria comprise a set of the candidates for equilibrium. These candidates satisfy the following conditions (i) All excess demands for goods are zero for uniform positive relative prices of goods and a uniform positive price of labor (the real returns to labor). (ii) Individuals maximize their utilities for given prices and for a given basic market. Note that we have not yet imposed full maximization by individuals at the moment since this can be done only when all candidates for equilibrium are enumerated. By enumerating all corner equilibria, we will find the Pareto optimum corner equilibrium and prove that it is a full equilibrium in the next two sections.
152 5.
X. Yang
The Pareto Optimum Corner Equilibrium
This section first proves that a corner equilibrium is Pareto efficient for a given basic market. Then, we prove that all non-basic markets are neither equilibria nor Pareto optimal. Finally the relationship between the real return to labor in a corner equilibrium and the preference and technology parameters will be investigated. If the market is P as shown in Figure 1(a), i.e., some individuals choose structure (x/y) and other individuals choose structure (y/x), then we can derive the necessary conditions for a restricted Pareto efficient allocation from the problem in Appendix B. By 'restricted', we shall mean that the market is given. From the necessary conditions of that problem, it can be shown that the corner equilibrium found in sections 3 and 4 is the restricted Pareto optimum for the market. Following this procedure, we can show that each corner equilibrium is the restricted Pareto optimum for a given basic market. Moreover, Appendix B has proven Proposition 4: Each corner equilibrium is Pareto efficient for a given basic market and all non-basic markets cannot satisfy the necessary conditions for the Pareto optimum. Combining this with Proposition 2, we conclude that neither equilibrium, nor the Pareto optimum is associated with a non-basic market. This leads us to Proposition 5: Both candidates for an equilibrium and for the Pareto optimum market are associated with some basic market. By comparing real returns to labor in different basic markets, we can find the Pareto optimum corner equilibrium: it is the corner equilibrium with the maximum real return to labor. This Pareto optimum corner equilibrium depends upon t, s and the differences among (a,a), (b, /?), (c, y). Given the number of traded goods, it can be shown that (i) the corner equilibrium including as traded goods those with larger preference
Development, Structural Changes and Urbanization
153
parameters is Pareto superior to that including as traded goods those with smaller preference parameters; and (ii) the corner equilibrium including as traded goods those with a high return to specialization is Pareto superior to that including as traded goods those with a small return to specialization, (i) and (ii) lead us to Proposition 6: The corner equilibrium including as traded goods those with a large preference parameter has a greater real return to labor, and the corner equilibrium including as traded goods those with higher returns to specialization has a greater real return to labor. Proof: See Appendix C. Assume that a = b = c = t and a = /? = y, then differences in the real returns to labor in various markets depends only upon the number of traded goods and service. The composition of traded goods has no effect on the real returns to labor. With this assumption, there are five possible market configurations: (1) Autarky. We refer to it as market A. This market consists of structure (x, y, z). (2) Partial division of labor in production. We refer to it as market P. This market consists of structure (x/y) and (y/x). (3) Complete division of labor in production. We refer to it as market C. This market consists of structure (x/yz), (y/xz) and (z/xy). (4) Partial division of labor in production and transaction. We refer to it as market PT. This market consists of structure (x/yT), (y/xT), and (Tlxy). (5) Complete division of labor in production and transaction. We refer to it as market CT. This market consists of structure (xlyzT), (ylxzT), (z/xyT), and (T/xyz). Having solved the corner equilibria in these five markets and compared the real returns to labor, we obtain
154
X. Yang
Proposition 7: (1) For s (the average distance between a pair of neighbors) < 0.54, the real returns to labor in markets P and PT cannot be the maximum. (i) Market A has the maximum real return to labor if a < 0.41 0.251og(l-1.414s). (ii) Market C has the maximum real return to labor z/0.41 - 0.251og (1 - 1.4145) < a < 2.77 - 1.091og (1 - 1.732*). (Hi) Market CT has the maximum real return to labor if 2.11 - 1.091og (1 - 1.7325) < a. (2) For 0.54 < s < 0.58, the real return to labor in market P cannot be the maximum. (i) Market A has the maximum real return to labor if a < 0.41 0.25 log (1-1.4145). (ii) Market C has the maximum real return to labor ifOAl - 0.251og (1 - 1.414s) 0.71, market A has the maximum real return to labor for any value of a.
Development, Structural Changes and Urbanization
155
Note, the Pareto optimum corner equilibrium has the maximum real return to labor. The next section will prove that all non-Pareto optimum corner equilibria are not full equilibria and that the Pareto optimum corner equilibrium is a full equilibrium. 6.
The Relationship between the Equilibrium and the Pareto Optimum
This section will establish Proposition 8: For an increasing returns economy, the Pareto optimum corner equilibrium is an equilibrium and equilibrium is Pareto optimal. Moreover, the Pareto optimum allocation with equal utilities for all individuals is an equilibrium. Because all non-basic markets are incompatible with equilibrium and the Pareto optimum, only basic markets are concerned in proving this proposition. To prove this proposition, it suffices to establish that (1)
if an equilibrium exists, it is an element of a set that consists of the corner equilibria in all basic markets; (2) the Pareto optimum corner equilibrium exists and is an equilibrium; (3) all non-Pareto optimum corner equilibria are not equilibria. On the basis of these three statements, it is trivial to show the final part of Proposition 8 because the only difference between the Pareto optimum and equilibrium is that the former may have the utilities different from individual to individual, but for the latter all people have equal utilities. It is easy to show (1) because the set of all markets includes all possible combinations of corner solutions for individual optimal decisions following the definition of a market. The equilibrium is certainly associated with an element of the market set since the definition of equilibrium requires optimization of individual decisions. This,
156
X. Yang
combined with Proposition 5, yields that the equilibrium, if it exists, is an element of a set that consists of the comer equilibria in all basic markets. To justify (2), we need to prove the existence of a Pareto optimum comer equilibrium and prove that this comer equilibrium satisfies the full maximization of individual utilities. Such full maximization will ensure that this comer equilibrium is a full equilibrium because a comer equilibrium satisfies all conditions for equilibrium except ensuring optimal choice of structure by an individual. Propositions 3 and 5 can be used to justify the existence of a Pareto optimum comer equilibrium. The definition of a Pareto optimum comer equilibrium combined with Proposition 4 implies that this comer equilibrium ensures the full maximization of individual utilities. Therefore, (2) can be proven. That is, there exists a Pareto optimum comer equilibrium and this comer equilibrium is an equilibrium. (3) is easy to prove. Since the notion of equilibrium in this paper is extremely neoclassical, equilibrium has to ensure that each person chooses the comer solution that maximizes his utility. Non-Pareto optimal comer equilibria are not equilibria because in each of these equilibria every person's utility is not maximized with respect to his comer solutions. Proposition 8 can thus be established. This section has developed an approach to analyzing a model with increasing returns to specialization. Such an approach has been used to show that the equilibrium achieves a Pareto optimum. By solving for all comer equilibria, we can analyze under what conditions the equilibrium will shift from one comer equilibrium to another. Therefore, this approach is useful for analyzing the development of market structure and equilibrium. The next section applies this approach to investigate the comparative statics of this model. 7.
Equilibrium and its Comparative Statics
Propositions 5, 7, and 8 lead us to Corollary 1: For sufficiently small s, increasing economies of specialization will cause the equilibrium to evolve from autarky first to
Development, Structural Changes and Urbanization
157
complete division of labor in production {market Q , then jump to complete division in production and transaction {market CT). For sufficiently large s, equilibrium is autarky for any degree of increasing returns to specialization. For the values of s in between, increasing economies of specialization will make equilibrium gradually evolve from autarky first to market C, then to market PT, finally to market CT. In this evolution, the goods with relatively large parameters of preference and degree of increasing returns to specialization will be traded before other goods are involved in the market. Here, s is the average distance between a pair of neighbors. Moreover, Propositions 4, 5, and 6 lead us to Corollary 2: For sufficiently large economies of specialization, decreasing s will involve more goods in the division of labor and the market. The two corollaries tell us that the number of traded goods is determined by the absolute degree of increasing returns to specialization and the average distance between a pair of neighbors, while the composition of traded goods is determined by the relative degree of increasing returns to specialization and the relative preference for different goods. Also, the corollaries say that a sufficiently large degree of increasing returns to specialization in production as well as in transaction is a necessary but not sufficient condition for the division of labor, while a sufficiently small s is a necessary but not sufficient condition for the division of labor. Sufficient increases in a as well as in Ms will produce a 'take-off of the division of labor and productivity. Moreover, these two corollaries mean that there is some substitution between a and lis. For example, if the degree of increasing returns to specialization is not large, then urbanization can promote the division of labor and productivity by increasing lis. If a is large, then we may still 4
Yang (1988) shows that introducing intermediate goods and the market for labor into the model in this paper, we can justify Coase's theory of the firm; while all results in this paper still hold.
158
X. Yang
have the developed division of labor even if population is dispersed (large s). If the degree of increasing returns to specialization in production as well as in transaction is a sufficiently large constant, then urbanization can decrease s, thereby producing a 'take-off of the equilibrium level of division of labor. This take-off will enhance trade dependence (the ratio of trade volume to income), the extent of market (demand for traded goods), and per capita real income (the real returns to labor, or real productivity). We have worked out the formula for these three variables in different markets. They are increasing when the market evolves to the complete division of labor from autarky as s decreases. Moreover, our model can be used to show that the following variables also change with the evolution. (1) Self-sufficiency decreases as specialization develops. (2) Diversification of the economy increases as the number of traded goods and professional sectors increase. (3) Degree of concentration of production increases as the total output share of a traded good produced by one producer rises. (4) Degree of integration of the economy develops as the number of trade partners of each individual increases. (5) Transaction efficiency increases and the ratio of the value of transaction service (or roundabout production, or intermediate production) to the value of consumer goods increases as the division of labor evolves. All these phenomena, some of them apparently contradictory, are different versions of the evolution of division of labor. This evolution is caused by the increases in lis and a. In order to explore the difference between the theory of structural change proposed here from the conventional one, two examples are discussed. First, we compare our theory of structural change with the theory of Lewis and Fei and Ranis. According to Lewis (1955) and Fei and Ranis (1964), development is a process in which surplus labor in the agricultural sector is transferred to the industrial sector. Sufficient surplus of agricultural output is a necessary condition for starting this
Development, Structural Changes and Urbanization
159
process. In order to compare their models with our theory, let us assume that in our model x is food, y is clothing, and z is house. If a is very small and s is very great, then our equilibrium is autarky. All individuals self-provide three goods and live scattered throughout the rural area. We can call this a traditional agricultural sector. In this economy each individual produces all goods he needs. A decrease in s generated by urbanization or an increase in a generated by a technical innovation will shift our equilibrium to the division of labor, e.g. to market CT. In this market the previous 'natural agents' are changed to professional farmers, workers specializing in the production of clothing or housing, and professional traders producing T, respectively. In this transition, we see that the professional farmers are less than the previous 'natural agents' (producing all goods) and population is concentrated in urban areas. This looks like a shift of labor from the agricultural sector to the industrial sector. But this actually is a process of the evolution of division of labor. Each individual switches to professional activity from self-sufficient activity. In autarky, low productivity is not because of 'labor surplus', but because of the low level of division of labor. Hence, 'labor surplus' as well as 'surplus of agricultural products' are not necessary conditions for this shift of economic structure. From our theory proposed in this paper, the necessary conditions for this transition of economic structure are the sufficiently high degree of increasing returns to specialization (a, b, c, t) and sufficiently small s. Therefore, the key issue for economic development and transition of economic structure is the initiation and speeding up of the evolution of the division of labor rather than the existence of a labor surplus. If s is interpreted as the transaction cost related to government interference with the market system, e.g., a trade tax and restrictions on market exchange, our model will tell us that a liberalization policy will substantially decrease s and raise the equilibrium level of division of labor thereby generating economic development. Indeed, such transaction costs imposed by a government in a less developed country are large and the success of liberalization policies in the development of 'four small tigers' (Taiwan, Hong Kong, South Korea, and Singapore) is the best support for this theory.
160
X. Yang
The second example is the theories of Kuznets (1966) and Chenery (1979). According to their theory of structural changes, the transition of economic structure is based on an increase in per capita income. According to our theory, however, the increase in per capita income and all other phenomena listed above are different versions of the evolution of the division of labor. It does not make sense to explain one version of this evolution by another. This paper explains the evolution by the improvements of transaction efficiency resulting from changes in the level of urbanization, policy, or institutional arrangements and explains structural changes by this evolution. Yang (1988) explores a mechanism behind the evolution of the division of labor in the context of a dynamic equilibrium model without exogenous changes in a and s. 8. Conclusions The major accomplishment in this paper is to distinguish increasing returns to specialization from increasing returns to scale and to develop a multiple-step approach to handling the model with increasing returns to specialization. Also, this paper shows that a competitive market can efficiently integrate economies of specialization (which is endogenous to individuals) into economies of division of labor in the whole society (which is exogenous to individuals). A free market endogenously determines an efficient level of division of labor by balancing a trade-off between increasing returns to specialization and transaction costs. In addition, the comparative statics of our model shed new light on urban economics and the issue of structural change.
Appendix A: The Choice of Structures This appendix proves Proposition 1 in section 3. First, the first part of this proposition is established. Assume xd> 0; xd can be solved from the budget constraint. Inserting this expression of xd and the rearranged production function of x into the utility function yields
Development, Structural Changes and Urbanization U = {LX -x> +(l-K)(l-k)[x° + pz(zs-zd)/px]}a[y
+pT(T> -T")/px
+ (l-K){l-k)y"Y[z
+ py{y>
+ (l-K)(l-k)^y.
161
-yd)/px (A.1)
s
Differentiating (A. 1) with respect to x yields dU/dxs = - A [1 - (1 - K)(\ - k)] < 0,
(A.2a)
where 0 < ( l - ^ ) ( l - & ) < 1, and A is a positive magnitude independent ofXs. Canceling yd or zd, orTdby using the budget constraint, it can be shown that dU/di' < 0 if i" > 0, i = x, y,z,T.
(A.2b)
Eq. (A.2) implies that the optimum amount sold of a good is zero if an individual buys this good. In other words, an individual will not buy and sell a good at the same time. Assume xd > 0 [this implies Xs = 0 due to (A.1-A.2)]; then the optimum quantity sold of at least another good has to be positive because of the budget constraint. Without loss of generality, we supposey"> 0.8U/dys = Ogives the necessary condition for the optimum ys . Inserting this condition into 8U/dLx and differentiating the resulting first-order derivative with respect to Lx again yields d2U/8L2x>0
if 8U/dLx=0.
(A.3a)
This implies that the optimum value of Lxis either zero or one if xd > 0. Lx = 1 conflicts the assumptions^ 0, implied by the assumption xd > 0, and requiring Ly > 0. Hence, (A. 3 a) means the optimum value of Lx is zero if xd> 0. (A.3b) In other words, an individual will not produce and purchase a good at the same time. This is just the first part of the above proposition. Next, the second part of the proposition is proven. Without loss of generality, we assume ys, zs > 0 . Because / cannot be negative for i = x, y, z, T and because i" = 0 if id > 0 due to (A.2), it can be shown that yd = zd = 0 if ys , zs > 0 . dU/dys = dll/dzs = 0 gives the necessary conditions for the optimum ys and zs. Inserting these conditions into dU/dLy and differentiating the resulting first-order derivatives with respect to Ly again yields d2U/dL2y>0
if dU/dLy=0.
(A.4a)
162
X. Yang
This implies that the optimum value of Ly is either zero or one. Ly = 0 conflicts the assumption ys > 0, and Ly = 1 conflicts the assumption z > 0. Hence, (A.4a) means ys and zs cannot be positive at the same time.
(A.4b)
In other words, an individual will not sell two goods at the same time. This is just the second part of the above proposition. Appendix B: Corner Equilibrium and the Restricted Pareto Optimum If market is P in Figure 1(a), i.e., some individuals choose structure (x/y), signified by subscript 1 and other individuals choose structure (y/x), signified by subscript 2, then we can obtain the necessary conditions for the Pareto efficient allocation from the problem below: d
max. z
i
Ux=x?\T(\-K)y?Yzl, i
(B.l)
Z
JV*;? > 2>>'2. >.. --• T S.t.
A | ~T J l | — Ltx,
Z | — ^1
Ljj
l^x J
y2+y°2=L»y,z2=(l-LT-Lyy,
,
T = LT,
MjXf = M2x( or rx{ = x(, M2y\=Mxydx
or ys2=ryf,
U2 = yp2 U(l - K)xd2 \a z\>u. where u is a constant. The first-order conditions are MRSI=MRSIJ\ MRSlx = MRTz\, MRSl=MRT$, MRS\y=MRFfy, MRSyx=MRTyx, 2 MRTL=MRTIT , yx yx
(B.2) MRSl=MRT>, MRS^MRT^,
(B.3) (B.4)
where MRS* =(dUk/dxJ)/(dUk/dxi) is the marginal rate of substitution between good i and j for individuals choosing structure k; MRT* = (dxJdL)/(dXj/dL) is the marginal rate of transformation between good i and j for individuals choosing structure k.
Development, Structural Changes and Urbanization
163
These are just the necessary conditions for corner equilibrium. From the corner equilibrium for market P, solved in subsection LB andI.C., we can derive that MRS^/T = Py/px=
MRS%T or MRS\X =
MRS^.
This is just (B.2). From this equilibrium we can obtain (B.3) and (B.4) too. Actually, the relative number of individuals, r, is also a decision variable in the problem of Pareto efficient allocation. However, from the problem (B.l), we can solve for the unique xf, x{ , ys2, and yf; they determine a unique r through the market clearing condition since the number of traded goods equals the number of structures in the basic market. For a non-basic market, e.g., that consists of structures (z/xy), (x/yz), (z/xy), and (x/y), we have a similar problem of Pareto optimal allocation as (B.l) which maximizes U(z/xy) subject to all individual production functions, transaction technologies, and balance between consumption and production giving that other utilities are not smaller than some constants. However, the relative numbers of individuals in structures (x/yz), and (x/y) are flexible. Name these two numbers as M\ and M2, respectively. For the relevant Lagrange function associated with this restricted optimization problem, LA, we can show dLA/dMt \M _o < 0 if AdU (z/xy)/dx + BdU (z/xy)/by < 0, 8LA/8M2 \M=0<0
(B.5a)
if AdU (z/xy)/dx + BdU (z/xy)/By > 0, (B.5b)
where we use the facts thatM = ^TM(. , M is the total number of individuals, and Mt is the number of individuals in structure /. LA is the Lagrange function. (B.5) implies that for any value ofAdU(z/xy)/dx + B8U(z/xy)/dy, the Pareto optimum requires either Mi = 0 or M2 = 0. In other words, this non-basic market cannot satisfy the necessary conditions for the Pareto optimum. Applying the above procedure to each market, we can show Proposition 4 in section 5.
164
X. Yang
Appendix C: A Proof to Proposition 6 Assume, for example, that the market is P, as shown in Figure 1(a); it is not difficult to show (i) the comer equilibrium including as traded goods those with larger preference parameters is Pareto superior to that including as traded goods those with smaller preference parameters; and (ii) the comer equilibrium including as traded goods those with great return to specialization is Pareto superior to that including as traded goods those with small return to specialization. (i) Assuming a = (5 and a = b = c = /, we have utilities for structure (x/y) and (y/x): log[/(x/j) = a(21ogx + l o g r - l o g r ) + ^logz + «log(l-icr), (C.la) logU(y/x) = a(2logx + logT + logr) + ylogz + a l o g ( l - K ) , (C.lb) where x = [2a/( 3 a + j ' ) ] / 2 , z = y/(3a + y)
andT = [a/(3a + y)J.
The condition of utility equalization becomes E = logU(x/y) - logU(y/x)
= 2alogr = 0.
(C.2)
(i) will be established, if it can be shown that d\ogU*/da\a=y
=dlogU*/da\a=r+(d\ogU*/dr*)(dr*/da)\a__r>0,(C3)
where U* is real income and r* = M*xJM*y is relative number of the individuals choosing different structures in the comer equilibrium. From (C.2) it can be derived that dr'lda\a__f = -(dE/da)/(8E/dr%__r where 8E/da\
=
dlogU'/da\
= 0,
(C.4a)
= 2 log r = 0 because r = 1 if a = y, and >0.
(C.4b)
(C.4) ensures that (C.3) holds. (C.3) implies that if individuals prefer x and y to z, the comer equilibrium with x and y as traded goods will have a greater real return to labor than that with x and z or with y and z as traded goods.
Development, Structural Changes and Urbanization
165
(ii) Assuming a = p = y and a = b = t, (C.l) holds if x = [2a/(3a + c)]72, z = [cl{3a + c)]c, and 71 = [2a/(3a + c)]°. Also, there is the condition of utility equalization (C.2). (ii) will be established, if it can be shown that d]ogU*/da\aae=dlogU'ma=c+(d\ogU'/drtXdr'/da^g=c>0.
(C.5)
From (C.2) with x = [2a/(3a + c)]72, z = [c/(3a + c)]c, and T = \2aLI (3a + c)]a, it can be derived that. dlogU*/da\a=c >0, and dr*jda\a=c =-(dE/da)/(dE/dr*)\a=c where 8E/da\ '
=0, (C.6)
=0.
\a=c
(C.6) ensures (C.5) to hold. (C.5) implies that if the returns to specialization in producing x and y are greater than that in producing z, then the corner equilibrium with x and y as traded goods will have a greater real return to labor than that with x and z or with y and z as traded goods. Proposition 6 can be established by using (i) and (ii).
References Chenery, M., 1979, Structural change and development policy (Oxford University Press, Oxford). Ethier, W.J., 1986, The theory of international trade, Discussion paper no. 1 (International Economics Research Center, University of Pennsylvania, Philadel
