Consumer Demand in the United States: Prices, Income, and Consumption Behavior, Third Edition

Consumer Demand in the United States

Lester D. Taylor · H.S. Houthakker

Consumer Demand in the United States Prices, Income, and Consumption Behavior Third Edition

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Lester D. Taylor Department of Economics University of Arizona Tucson AZ 85721 USA [email protected]

H.S. Houthakker (deceased)

ISBN 978-1-4419-0509-3 e-ISBN 978-1-4419-0510-9 DOI 10.1007/978-1-4419-0510-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009939326 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Dedicated to the memory of H. S. Houthakker (1924–2008)

In Memoriam

Hendrik Samuel Houthakker December 31, 1924–April 15, 2008 Memorial Church Harvard University September 25, 2008 It is a great honor and privilege to speak about Henk this afternoon. Apart from members of my own family, he was the most important person in my life. In the opening words of his article in the 1992 Festschrift for Henk, Paul Samuelson said in tribute: Originality, depth, and versatility—those are the marks of a master scholar. Hendrik Houthakker started off with a bang: a big bang. . . . this young and unknown Dutchman resolved a long-standing challenge before he reached his second quarter century.

Paul was referring, of course, to Henk’s proof in his famous 1950 Economica article that the strong axiom of revealed preference suffices for the integrability of preferences. At the time, this unsettled question was essentially of the same importance in economics as the Riemann Hypothesis was (and still is) in mathematics. To have solved it in the manner and ease with which he did was simply an initial display of the untrammeled power of Henk’s mind. Most of us are fortunate if we should come up with one really good idea in a career, but this was not the case with Henk. For there is virtually no area of economics that is absent something of importance from his fertile brain; the strong axiom is simply the best known. Let me note five other contributions that, in my opinion, are of nearly equal force: (1) His 1955 paper in the Review of Economic Studies that showed how an aggregate Cobb–Douglas production function can be obtained through aggregation over firms that are distributed according to the law of Pareto––I have always vii

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In Memoriam

felt this result in some ways to be even more amazing than his invention of the strong axiom. His 1956 paper in Kyklos that likens speciation in biology to the division of labor in economics. His 1960 paper in Econometrica on additive preferences in which he devised the direct and indirect addilog utility functions. His work over many years on futures markets that has done much to establish the importance of speculators to stability in commodity markets, as well as to dispel the confusion surrounding Keynes’s enigmatic concept of “backwardation.” His underlying ideas concerning consumption dynamics that informed our more than 40 years of collaboration on the study of consumer behavior.

The foregoing attests both to Henk’s originality and versatility, but there is more. For in addition to his formal research, Henk was extremely interested in economic policy and was a superb policy economist. As Dale [Jorgenson] mentioned, he served on two occasions at the Council of Economic Advisers in Washington, D.C., as a senior staff economist in 1966–1967 and then as one of the three Council Members in 1969–1971. As a policy economist, Henk had an amazing ability to see a question or problem, first in its totality, and, then (untainted by ideology) in its economic aspects separate from social and political considerations. A prime example of this is in an address that he gave in December 1964 in the Annual Meeting of the American Farm Bureau Federation. In this talk, Henk provides the most cogent and trenchant analysis of the social, political, and economic problems of agriculture in modern economies that I have ever seen and is an analysis that has as much relevance for world agricultural policy today as it did then. Despite English not being his first tongue, Henk (as noted by Bob [Russell]) was a craftsman with the language, and, as all who were acquainted with him know, he had a wonderfully subtle, and at times, piercing, sense of humor. A good example of this humor is given in some notes for a 1966 Treasury Consultants Meeting in Washington, D.C., in which he took to task ideas of price controls that were then currently being bandied about: Price controls on particular markets are all the more useless because they tend to be capricious in their coverage. For some reason, official attention has been concentrated on primary metals, which are no more important to the economy as a whole than any other sector of similar size. In line with this bias, the molybdenum industry was dissuaded from raising its prices by five percent, yet the whole molybdenum industry is no more important (in terms of value added) than Harvard University all by itself. When Harvard recently raised its tuition by about thirteen percent, we did not hear of President Pusey being summoned to Washington.

Let me now turn to things more personal. My first contact with Henk was in the fall of 1960 as a student in his courses in econometrics and mathematical economics (then Economics 221 and 214). He was a superb teacher and lecturer. He never used notes, but was always in total command of material, and spoke in beautifully clear and concise English. From the very start of classes, it was evident that here was a

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person with an extraordinary mind, but with a soul that was humble and unspoiled by fame. In line with the norm at the time, the door to his Littauer office was kept closed, but yet always open to a knock. I was in his office many, many times in the 1960s, and never once did I sense that he had more important things to do than to discuss economics with a student or colleague. In the summer of 1962, Henk had a contract with the U. S. Bureau of Labor Statistics to develop a model for forecasting consumption expenditures for use in an interagency growth study. Although that was the summer I was working on my dissertation on another topic, Henk nevertheless generously involved me in the project. This was the start of more than four decades of collaboration, and was the basis for the 1966 and 1970 editions of our book Consumer Demand in the United States. As a collaborator, Henk was very trusting, and was an absolute joy to work with. He was patient and generous in both time and thought and was never dogmatic or demanding with regard to his own ideas. He had a never-ending capacity to invent, and few problems, whether theoretical or econometric, were beyond his ken to solve. He was first and foremost a scientist, and, while cognizant that nothing can ever be perfect, was always insistent that finished work never contain a known error. Although we often discussed the possibility of a third edition of Consumer Demand, it was not until about 2000 that we actually set out working on one. By this time, both of us had logged a lot of additional work on demand and consumption, but mostly on different themes. Henk had focused principally on implementing the addilog models from his 1960 Econometrica paper and was testing the axioms of revealed preference, while I had pursued energy and telecommunications demand and done a lot of puzzling over how demand theory might be better integrated with psychology and the brain sciences. The result of our last combined effort is a totally new book that, besides including a complete redoing of the time-series analysis of the two earlier editions, adds an extensive cross-sectional investigation (reminiscent of what Henk and S. J. Prais did in their classic 1957 book) using data from the BLS quarterly consumer expenditure surveys. While work on this book was essentially completed a couple of years ago, it still awaits a publisher, and will now, it is sad to say, have to be dedicated to Henk’s memory. While Henk was the total package as an economist and an academic, research was clearly his first love. When he retired from Harvard in 1994, he did so because he wanted to finally be able to devote his full attention to research. Between then and about 2000, he completed what, in my view, are two of his finest pieces of research, papers that one simply could not expect of someone in their 70s. The first paper (as mentioned by Bob) represents an empirical test, using international data, of the weak and strong axioms of revealed preference, while the second involves the invention of a totally new functional form for describing the distribution of income. Unfortunately, Henk was never able to see these two papers through to publication, and they still await the light of day. Besides research, teaching, and public service, Henk was strongly committed to the general scholarship of economics and accordingly spent more than 20 years as a watchful steward of The Review of Economics & Statistics. However, while these years were clearly of benefit to the profession, they were very costly to Henk

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personally because of the way he approached his editorial duties. For his procedure was first to read every paper submitted to the Review as soon it was received and then to make a decision whether to pass the paper on to referees for formal review or to return it immediately to the author(s) as unsuitable for publication. Needless to say, this was a great boon to authors, especially to young ones. However, in doing this, Henk became concerned that, in sifting through the many ideas of others, he would not be able keep separate his own, and this, in turn, led him to put his own research and publishing on hold. The cost for Henk of this selfless act of intellectual integrity was, in my opinion, a Nobel Prize. As a final personal note: Harvard in the early 1960s was not particularly concerned with preparing its students for the classroom, and the only advice that I ever got was when Henk said to me as I was about to enter my first classroom: “Never be afraid of being wrong in front of a class.” Such is the sentiment of a modest and humble individual, and is a piece of wisdom that I have never forgotten. To Paul’s characterization of Henk as a master scholar—Originality, Depth, Versatility—two more can be added : Intellectual Integrity and Humility. Not only was Henk a master scholar, but he was a gentle and kind soul and a genuinely good person. Teresa, Louis, Johnny, and Isabella, the world is a lot richer because of your husband and father. Again, it has been a great privilege to be invited to make these remarks about Henk. Thank you. Lester D. Taylor

Preface

For 20 years or so, the authors thought of putting together a new edition of our 1970 book, Consumer Demand in The United States, but for a variety of reasons this never came to fruition. While both of us were continuing to work on demand and consumption, our interests in doing so were somewhat divergent. HSH was focusing more on implementing the addilog models first introduced in his 1960 Econometrica paper, testing the Axioms of Revealed Preference, and developing a new family of functions for analyzing the distribution of income, while LDT was pursuing energy and telecommunications demand, and puzzling over how the theory of demand might be better integrated into psychology, and even more, into how the brain is structured and functions. However, it is now clear that a mechanical updating of the 1970 book would have been of little utility, and that delay has in fact been useful. For an interval of 35 years allows, not only for the use of substantive new bodies of data, and taking advantage of notable advances in econometric techniques, but also for the exercise to be tempered by sober reflection (and hopefully wisdom!) arising from many added decades of experience. Not surprisingly, therefore, the present effort differs in major ways from the 1966 and 1970 books. While dynamics continues to be a main theme, the focus is away from development of a consistent system of equations for use in forecasting and projection (which was the primary motivation for the two earlier editions) to a more practical concentration on estimation of a broad array of price and total-expenditure elasticities. While the novelty in the 1966 and 1970 books was application of stateand flow-adjustment models, novelty in the present exercise takes three forms: (1) extensive analysis of cross-sectional data from the consumer-expenditure surveys conducted quarterly by the US Bureau of Labor Statistics (including merging into these surveys of price data collected quarterly by the American Chambers of Commerce Research Association), (2) attempts to graft the analyses onto recent advances in the neurosciences, and (3) extensive use of quantile regression in estimation. Thirty-five years is obviously a long time between editions of a book, and the once-young authors are now hardly so. As a consequence, there undoubtedly will be some readers who feel that the theoretical structures employed do not adequately reflect the current state of the literature. This may especially appear to be the case in the cross-sectional analysis, for the models estimated with the cross-sectional data generally predate 1990. There are four reasons for this. In the first place, when the xi

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focus is on the estimation of price and income elasticities, there is still no substitute, in our opinion, for the simple double-logarithmic function for ease of estimation, fit, and interpretation. Second, the current literature remains heavily conventional in its neoclassical orientation, which is at odds with a desire (though probably more on the part of LDT than HSH) to interpret results as much as possible in a neuroscience framework. Third, as a purely practical matter (and as will be seen), elasticities (viewed as mean tendencies) with the cross-sectional data are generally robust with respect to functional form and model sophistication. Finally, after years of estimating what probably numbers in the hundreds of thousands (maybe even millions) of demand functions, we have our own ideas of how to go about doing things, and truth be known - are probably pretty set in our ways! As in the first two editions, the book is targeted at a variety of readers. First, of course, is our fellow economists, who hopefully will find of interest our analyses of the BLS-CES quarterly surveys, application of the Bergstom-Chambers model (which allows for state- and flow-adjustment within the same framework) in the aggregate time-series analysis, and our attempt to graft demand analysis onto neuroscience roots. As before, most of the models estimated are derived (often in what might seem excruciating detail) from first principles, which teachers and students in applied econometrics classes (if not others) can find useful as illustration of how empirical research is done.1 Industry, marketing specialists, and government and other forecasters interested in price and income elasticities provide a second target audience. Finally, specialists in econometrics may find of interest our use of quantile regression, together with the frequent appearance of long-tailed, asymmetric distributions of residuals and our various methods of effecting non-linear estimation. Among other things, it is useful to contrast the preparation of the 1966 and 1970 books with that of the present exercise. Research for both the 1966 and the 1970 books was undertaken under contract for the US Bureau of Labor Statistics [for use in the (then) ongoing Interagency Growth Study], with some additional support provided by the Ford Foundation. The total amount involved was of the order of $75,000. Besides ourselves, upward of six research assistants, two computer programmers, and at least one secretary were involved. The 1966 edition took about 3 years to complete, and the 1970 edition about two. All work was done at Harvard Computer Center on an IBM 7094. Apart from several semesters of partial released time for LDT (together with the purchase of 4 years of survey data) provided by the Cardon Foundation for Economic Research in the Department of Agricultural & Resource Economics at the University of Arizona, no government, foundation, or private industry money has been used in the present effort. Work began on the cross-sectional analysis in the winter of 2000 and was pretty much completed by the fall of 2005. The time-series analysis was then started and was essentially finished in the summer of 2006. Thus, the present exercise has required about 7 years in total. While several of the graphs and charts have been prepared professionally, no research assistants, programmers, or secretarial assistance have

1 Also, writers of principles texts can now have a large array of up-to-date price and income elasticities at their disposal.

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been used. All statistical and econometric work has been done in SAS, and all on a personal computer, much of it during summers on a laptop. Among other things, equations, which often required a turnaround of 24 h to estimate in the 1966 and 1970 editions, could be estimated in a fraction of a second in the present effort. Most data in the first two editions were obtained in the form of worksheets from the Bureau of Labor Statistics and Department of Commerce or copied from the Survey of Current Business or Historical Statistics of The United States. In the present effort, the CES surveys were purchased on CDs from the Bureau of Labor Statistics, while all of the time-series data were downloaded from the Internet. To have participated in this much change in research technology over the course of our academic careers is obviously truly amazing. The present work is both less and more ambitious than its two predecessors. It is less ambitious in that the focus, within traditional applied demand analysis, is quite narrowly on the estimation of price and income elasticities. On the other hand, it is considerably more ambitious in the type of data analyzed and, especially, in our desire to couple consumption behavior with neuroscience and Darwinian considerations. Above all, we are cognizant of the present effort’s shortcomings. While our analysis of data from the BLS consumer expenditure surveys is the most extensive of which we are aware, we have hardly done justice to the information that these surveys contain. Socio-demographical information is employed simply as control variables, only crude efforts are made to take into account household composition effects, and aggregation information is ignored completely. Also, while we feel it incumbent for demand theory to take its cue from the brain sciences, the models and approaches that are pursued should be viewed as first attempts and our interpretations of empirical results as simply suggestive of what hopefully will be things to come. As with any large undertaking, we have benefited from the assistance, advice, and counsel of a number of individuals, colleagues, and organizations. We are particularly grateful to Sean McNamara of the Council for Community and Economic Research for making available to us the price data collected in the ACCRA quarterly surveys, to Donald Kridel, Gary Madden, Teodosio Perez-Amaral, Paul Rappoport, Barbara Sands and Dennis Weisman, to seminar participants at Kansas State and Curtin Universities, and to the Cardon Foundation for Economic Research in the Department of Agricultural & Resource Economics at the University of Arizona. We are grateful as well for the comments (both sympathetic and critical) of several anonymous readers. Also, we are especially indebted to Iris Geisler for invaluable help in penetrating the inner sanctums of SAS and to Nancy Bannister for her superb talents in preparing the manuscript for publication. Responsibility for omissions and errors remains, of course, with us. Finally, we should mention that we feel greatly honored by the impacts that the flow- and stock-adjustment models of the 1966 and 1970 editions of CDUS have had on applied demand and consumption analysis. Needless to say, the kudos of our peers in this regard has been exceedingly gratifying. Tucson, AZ, USA Cambridge, MA, USA

Lester D. Taylor H.S. Houthakker

Postscript

The economics profession lost one of its giants with the passing of Professor Houthakker on April 15, 2008. Memorial services for him were conducted in Woodstock, VT, on April 27, 2008, and later at Harvard University on September 25.1 While the manuscript of this book in its present form was essentially complete by late 2006, some work remained on Chapters 20 and 22, which represent Professor Houthakker’s last two major undertakings of research following his retirement from Harvard in 1994. Although drafts of two papers describing this research were written by Professor Houthakker in the 1990s, nothing was ever published. In the early summer of 2005, it was decided that these papers would be included as chapters in the present book. Unfortunately, their final editing has been without Professor Houthakker’s input. Chapter 20 on the distribution of income was in excellent shape, and required little more than careful proofreading and checking of equations. However, Chapter 22 on testing of the axioms of revealed preference had some missing elements. Fortunately, R. Robert Russell of the University of California at Riverside has authoritatively been able to fill these in. Finally, I grateful to Gillian Greenough of Springer-Verlag for her enthusiastic championing of the manuscript and its expert shepherding through publication. Tucson, AZ, USA

Lester D. Taylor

1

Obituaries appeared in both the Boston Globe and Washington Post on April 22, 2008. See www.boston.com.bostonglobe/obituaries/2008/0422/hendrik-houthakker; www.washingtonpost. com/wp-dyn/content/article/2008/04/22/AR2008042202871. See also an “Appreciation of His Life” by Thomas Blinkhorn in the December 23, 2008, issue of the Valley News, White River Junction, VT.

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Contents

1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . 1.1 Themes and Format of the Book . . . . . . . . . . . . . . . . 1.2 A Reader’s Guide to the Study . . . . . . . . . . . . . . . . . Part I

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Preliminaries

2 Demand Theory Under Review . . . . . . . . . . . . . . . . . 2.1 Conventional Theory of Consumer Choice . . . . . . . . . 2.2 Neoclassical Demand Theory as a 19th-Century Conservative Energy System . . . . . . . . . . . . . . . . 2.3 Dynamics: Some Preliminaries . . . . . . . . . . . . . . . 2.4 State- and Flow-Adjustment Models of Consumption . . . 2.5 A Neuroscience Approach to Consumer Behavior . . . . . 2.6 Brain Structure and Consumption Dynamics . . . . . . . 2.6.1 Assumptions and Terminologies . . . . . . . . . 2.6.2 Consumption Dynamics Associated with the Alpha and Beta Brains . . . . . . . . . . . . . . 2.6.3 Opponent Processes and Consumption Dynamics 2.6.4 Dynamics Associated with the Gamma Brain . . 2.6.5 From Consumption to Expenditure . . . . . . . . 2.6.6 Consumption/Income Relationships . . . . . . . 2.6.7 Rationality . . . . . . . . . . . . . . . . . . . . 2.7 The Maslovian Needs Hierarchy . . . . . . . . . . . . . . 2.7.1 Physiological Needs . . . . . . . . . . . . . . . 2.7.2 Security Needs . . . . . . . . . . . . . . . . . . 2.7.3 Community and Affection (Love) Needs . . . . . 2.7.4 Esteem Needs . . . . . . . . . . . . . . . . . . . 2.7.5 Self-Actualization Need . . . . . . . . . . . . . 2.8 Some Implications of a Hierarchy of Needs . . . . . . . . 2.9 Toward Empirical Application . . . . . . . . . . . . . . . 2.10 Emotions and Consumption Behavior . . . . . . . . . . . 2.11 Consumption Behavior and the Pursuit of Happiness . . . 2.12 Summary and Final Comments . . . . . . . . . . . . . . .

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3 Quantile Regression: A Robust Alternative to Least Squares 3.1 Some Background . . . . . . . . . . . . . . . . . . . . . 3.2 Quantile Regression . . . . . . . . . . . . . . . . . . . . 3.3 Illustrations and Comparison . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . Part II

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Analyses of Data from BLS Consumer Expenditure Surveys

4 Description of Data Used from the Ongoing BLS Consumer Expenditure Surveys . . . . . . . . . . 4.1 Some Background and History . . . . . . . . 4.2 The Current BLS Surveys . . . . . . . . . . . 4.3 Data Used in the Present Study . . . . . . . . 4.4 Control Variables . . . . . . . . . . . . . . . 4.5 Combining CES Surveys and ACCRA Prices 4.6 Levels of Aggregation . . . . . . . . . . . . .

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5 Stability of U.S. Consumption Expenditure Patterns: 1996–19991 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Principal Component Analyses of 14 CES Expenditure Categories . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Interpretation of Results . . . . . . . . . . . . . . . . . 5.3 Regression Models for PCs 1 and 2 . . . . . . . . . . . 5.4 Summary and Conclusions . . . . . . . . . . . . . . . 6 Price and Income Elasticities Estimated from BLS Consumer Expenditure Surveys and ACCRA Price Data: Some Preliminary Results . . . . . . . . . . . . . . . . . . 6.1 Background and Merging of Data Sets . . . . . . . . . 6.2 Models Estimated . . . . . . . . . . . . . . . . . . . . 6.3 Pooling Across Quarters and Years . . . . . . . . . . . 6.4 Effects of Other Variables . . . . . . . . . . . . . . . . 6.5 Equations for Total Consumption Expenditure as Function of After-Tax Income . . . . . . . . . . . . . 6.6 Tests for Heteroscedastic Error Terms . . . . . . . . . 6.7 Nonlinear Logarithmic Engel Curves . . . . . . . . . . 6.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 7 Estimation of Theoretically Plausible Demand Functions from U.S. Consumer Expenditure Survey Data . . . . . . 7.1 The Almost Ideal Demand System . . . . . . . . . . 7.2 The Linear Expenditure System . . . . . . . . . . . . 7.3 The Indirect Addilog Model . . . . . . . . . . . . . 7.4 The Direct Addilog Model . . . . . . . . . . . . . . 7.5 Some Technical Obiter Dicta Concerning Estimation 7.6 Discussion of Results . . . . . . . . . . . . . . . . . 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . .

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8 An Additive Double-Logarithmic Consumer Demand System 8.1 An Additive Double-Logarithmic Demand System . . . . . 8.2 Application to the CES-ACCRA Data Set for the Four Quarters of 1996 . . . . . . . . . . . . . . . . . . . . . . 8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 9 Quantile Regression Analysis of Asymmetrically Distributed Residuals . . . . . . . . . . . . . . . . . 9.1 Quantile Regression Estimation of the Additive Double-Logarithmic Model . . . . . . . . . . . 9.2 Price and Total-Expenditure Elasticities . . . . 9.3 Conclusions . . . . . . . . . . . . . . . . . . .

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10 CES Panel Dynamics: A Discrete-Time Flow-Adjustment Model 10.1 Double-Logarithmic Flow-Adjustment Model . . . . . . . . 10.2 Comparison with Static-Model Elasticities . . . . . . . . . . 10.3 State- vs. Flow-Adjustment Behavior . . . . . . . . . . . . . 10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Summary of Cross-Sectional Results . . . . . . . . . . . . . . . . 12.1 Stability of Expenditure Patterns . . . . . . . . . . . . . . . . 12.2 Joining of ACCRA Price Data with CES Expenditure Surveys 12.3 Summary of Price and Total-Expenditure Elasticities . . . . . 12.4 Estimation of Dynamical Cross-Sectional Models . . . . . . . 12.5 Effects of Other Variables . . . . . . . . . . . . . . . . . . . . 12.6 Asymmetrical Residuals and Quantile Regression . . . . . . . 12.7 Cross-Price Elasticities . . . . . . . . . . . . . . . . . . . . . 12.8 Evidence for Maslovian Hierarchical Preferences . . . . . . . 12.9 A First Look at the Relationship between Total Consumption and After-Tax Income . . . . . . . . . . . . . . 12.10 Epilogue to the CES Analysis: Update to 2005 . . . . . . . . . 12.11 Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Engel Curves for 29 Categories of CES Expenditure . 11.1 An Overview of the Results . . . . . . . . . . . . 11.2 Size of Estimated Total-Expenditure Elasticities . 11.3 Interpretation of Total-Expenditure Elasticities in Terms of Maslovian Hierarchy of Needs . . . . . 11.4 Summary and Conclusions . . . . . . . . . . . .

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Part III Analysis of Time-Series Data from National Income and Product Accounts 13 Analysis of Time-Series Data on Personal Consumption Expenditures from the U.S. National Income and Product Accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 NIPA PCE Categories . . . . . . . . . . . . . . . . . . . . . .

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13.2

Generalization of the Flow- and State-Adjustment Models . 13.2.1 Remarks . . . . . . . . . . . . . . . . . . . . . 13.2.2 Remarks . . . . . . . . . . . . . . . . . . . . . 13.3 Alternative Estimation Forms for the State-Adjustment and B-C Models . . . . . . . . . . . . . . . . . . . . . . . 13.4 Nonlinear Estimation . . . . . . . . . . . . . . . . . . . . 13.5 Models Estimated and Statistical Procedures . . . . . . . . 13.6 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 Annual PCE Models . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Food, Tobacco and Alcohol . . . . . . . . . . . . . . . . . . . 15.1.1 Food Purchased for Off-Premise Consumption . . . 15.1.2 Purchased Food and Beverages . . . . . . . . . . . . 15.1.3 Tobacco . . . . . . . . . . . . . . . . . . . . . . . . 15.1.4 Alcoholic Beverages . . . . . . . . . . . . . . . . . 15.2 Clothing, Accessories and Jewelry . . . . . . . . . . . . . . . 15.2.1 Shoes . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Clothing excluding Shoes . . . . . . . . . . . . . . 15.2.3 Cleaning, Storage and Repair of Clothing and Shoes 15.2.4 Jewelry and Watches . . . . . . . . . . . . . . . . . 15.2.5 Other clothing . . . . . . . . . . . . . . . . . . . . 15.3 Personal Care . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Toilet Articles and Preparations . . . . . . . . . . . 15.3.2 Barbershops, Beauty Salons and Health Clubs . . . . 15.4 Housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Owner-Occupied Housing . . . . . . . . . . . . . .

267 267 268 269 270 272 273 274 274 276 276 277 277 277 277 279 279

14 Quarterly PCE Models . . . . . . . . . . . . . . . . . . . . . 14.1 Quarterly Models . . . . . . . . . . . . . . . . . . . . . 14.1.1 Total Durable Goods . . . . . . . . . . . . . . 14.1.2 Motor Vehicles and Parts . . . . . . . . . . . . 14.1.3 Furniture and household equipment . . . . . . 14.1.4 Other Durable Goods . . . . . . . . . . . . . . 14.1.5 Total Non-durable Goods . . . . . . . . . . . . 14.1.6 Food . . . . . . . . . . . . . . . . . . . . . . 14.1.7 Clothing and Shoes . . . . . . . . . . . . . . . 14.1.8 Gasoline, Fuel Oil, and Other Energy . . . . . 14.1.9 Other Non-durable Goods . . . . . . . . . . . 14.1.10 Total Services . . . . . . . . . . . . . . . . . . 14.1.11 Housing . . . . . . . . . . . . . . . . . . . . . 14.1.12 Housing Operation . . . . . . . . . . . . . . . 14.1.13 Transportation . . . . . . . . . . . . . . . . . 14.1.14 Medical Care . . . . . . . . . . . . . . . . . . 14.1.15 Recreation . . . . . . . . . . . . . . . . . . . 14.1.16 Other Services . . . . . . . . . . . . . . . . . 14.2 Summary of Steady-State Elasticities and Budget Shares 14.3 A Dynamic Linear Expenditure System . . . . . . . . .

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Contents

15.4.2 Rental Housing . . . . . . . . . . . . . . . . . . 15.4.3 Rental Value of Farm Housing . . . . . . . . . . 15.4.4 Other Housing . . . . . . . . . . . . . . . . . . 15.5 Housing Operation . . . . . . . . . . . . . . . . . . . . . 15.5.1 Furniture, including Mattresses and Bedsprings . 15.5.2 Household Appliances . . . . . . . . . . . . . . 15.5.3 China, Glassware, Tableware and Utensils . . . . 15.5.4 Other Durable House Furnishings . . . . . . . . 15.5.5 Semi-Durable House Furnishings . . . . . . . . 15.5.6 Cleaning and Polishing Preparations and Miscellaneous Household, Supplies and Paper Products . . . . . . . . . . . . . . . . . . 15.5.7 Stationary and Writing Supplies . . . . . . . . . 15.5.8 Household Utilities . . . . . . . . . . . . . . . . 15.5.9 Domestic Services . . . . . . . . . . . . . . . . 15.5.10 Other Household Operation . . . . . . . . . . . 15.6 Medical Care . . . . . . . . . . . . . . . . . . . . . . . . 15.6.1 Drug Preparations and Sundries . . . . . . . . . 15.6.2 Ophthalmic Products and Orthopedic Appliances 15.6.3 Physicians . . . . . . . . . . . . . . . . . . . . 15.6.4 Dentists . . . . . . . . . . . . . . . . . . . . . . 15.6.5 Other Professional Services . . . . . . . . . . . 15.6.6 Hospitals and Nursing Homes . . . . . . . . . . 15.6.7 Health Insurance . . . . . . . . . . . . . . . . . 15.7 Personal Business . . . . . . . . . . . . . . . . . . . . . . 15.7.1 Brokerage Services . . . . . . . . . . . . . . . . 15.7.2 Bank Service Charges, Trust Services and Safe Deposit Box Rental . . . . . . . . . . . . . 15.7.3 Services Furnished Without Payment by Financial Intermediaries, Except Life Insurance Carriers . . . . . . . . . . . . . . . . 15.7.4 Expense of Handling Life Insurance and Pension Plans . . . . . . . . . . . . . . . . . . . 15.7.5 Legal Services . . . . . . . . . . . . . . . . . . 15.7.6 Funeral and Burial Expenses . . . . . . . . . . . 15.7.7 Other Personal Business Services . . . . . . . . 15.8 Transportation . . . . . . . . . . . . . . . . . . . . . . . . 15.8.1 User-Operated Transportation . . . . . . . . . . 15.8.2 New Autos . . . . . . . . . . . . . . . . . . . . 15.8.3 Net Purchases of Used Autos . . . . . . . . . . . 15.8.4 Other Motor Vehicles . . . . . . . . . . . . . . . 15.8.5 Tires, Tubes, Accessories and Other Parts . . . . 15.8.6 Repair, Greasing, Washing, Parking, Storage, Rental and Leasing . . . . . . . . . . . . . . . . 15.8.7 Gasoline and Oil . . . . . . . . . . . . . . . . .

xxi

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280 280 281 281 282 282 283 284 284

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285 285 286 290 290 291 291 292 292 293 294 295 298 300 300

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Contents

15.8.8 Bridge, Tunnel, Ferry and Road Tolls . . . . . . . 15.8.9 Transportation Insurance . . . . . . . . . . . . . . 15.8.10 Purchased Local Transportation . . . . . . . . . . 15.8.11 Intercity Transportation . . . . . . . . . . . . . . . 15.9 Recreation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.1 Books and Maps . . . . . . . . . . . . . . . . . . 15.9.2 Magazines, Newspapers and Sheet Music . . . . . 15.9.3 Non-Durable Toys and Sport Supplies . . . . . . . 15.9.4 Wheel Goods, Sports and Photographic Equipment, Boats and Pleasure Aircraft . . . . . . 15.9.5 Video and Audio Goods, including Musical Instruments and Computer Goods . . . . . . . . . 15.9.6 Radio and Television Repair . . . . . . . . . . . . 15.9.7 Flowers, Seeds and Potted Plants . . . . . . . . . . 15.9.8 Admissions to Specified Spectator Amusements . . 15.9.9 Clubs and Fraternal Organizations . . . . . . . . . 15.9.10 Commercial Participant Amusements: . . . . . . . 15.9.11 Pari-Mutuel Net Receipts . . . . . . . . . . . . . . 15.9.12 Other Recreation . . . . . . . . . . . . . . . . . . 15.10 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.10.1 Higher Education . . . . . . . . . . . . . . . . . . 15.10.2 Nursery, Elementary and Secondary Education . . 15.10.3 Other Education . . . . . . . . . . . . . . . . . . 15.11 Religious and Welfare Activities . . . . . . . . . . . . . . . 15.12 Foreign Travel and Other, Net . . . . . . . . . . . . . . . . 15.12.1 U. S. Foreign Travel . . . . . . . . . . . . . . . . 15.12.2 Expenditures Abroad by U.S. Residents . . . . . . 15.1.1 The Access/Usage Framework for Analyzing Telecommunications Demand . . . . . . . . . . . 15.1.2 A Generic Model of Usage Demand . . . . . . . . 15.1.3 A Framework for Estimating Market Demand Functions for New Products and Services 15.2.1 Demand Theory and Multi-Part Tariffs . . . . . . . 15.2.2 Empirical Representation of Rate Schedules . . . . 15.2.3 Electricity Demand in the 1970s: An Illustration . 15.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . 16 Discussion of the Time-Series Results . . . . . . . . . . 16.1 Tabulation of Annual Models . . . . . . . . . . . . 16.2 Elasticities of Current Study Compared with Those in the 1970 Edition of CDUS . . . . . . . . . . . . 16.3 Interpretation of Total-Expenditure Elasticities in Terms of Maslovian Hierarchy of Wants . . . . . . 16.4 Statistical Quality of the Time-Series Models . . . 16.5 Asymmetry in Residuals? . . . . . . . . . . . . . .

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310 310 311 312 315 315 316 316

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Contents

xxiii

17 Comparison of Time-Series and Cross-Sectional Elasticities 17.1 29-Category Level of Aggregation . . . . . . . . . . . . 17.2 CES-ACCRA 6-Category Level of Aggregation . . . . . 17.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . .

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389 389 392 393

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395 396 396 397 399 403

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407 409

19 The Dynamics of Personal Saving . . . . . . . . . . . . . . . . . . 19.1 B-C Model of Saving . . . . . . . . . . . . . . . . . . . . . . 19.2 Results for Personal Saving from the Flow of Funds Accounts 19.3 Extending Model to Include Capital Gains . . . . . . . . . . . 19.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

415 416 417 420 423

18 Overall Assessment of CES and PCE Elasticities . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Summary of Results . . . . . . . . . . . . . . . . 18.1.1 CES Data . . . . . . . . . . . . . . . . 18.1.2 NIPA Time-Series Data . . . . . . . . . 18.2 The Importance of Total-Expenditure Elasticities 18.3 Assessment of Price Elasticities . . . . . . . . . . 18.4 Total-Expenditure and Price Elasticities and Hierarchical Wants . . . . . . . . . . . . . . 18.5 Comparison of Annual and Quarterly Models . .

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Part IV Miscellaneous Studies of Income Distribution and Weak Axiom of Revealed Preference 20 The Stationarity of Consumer Preferences: Evidence from Twenty Countries . . . . . . . . . . . . . . . . . . . . . . . Hendrik S. Houthakker 20.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Latent and Revealed Preference in Finite Data Sets . . 20.2.1 What Does Revealed Preference Reveal? . . 20.2.2 The Weak Axiom . . . . . . . . . . . . . . . 20.2.3 Dominance . . . . . . . . . . . . . . . . . . 20.2.4 Matching and Connectedness . . . . . . . . 20.2.5 The Strong Axiom of Revealed Preference . 20.2.6 A Test Procedure for the Connected Case . . 20.2.7 Further Analysis of Matching . . . . . . . . 20.2.8 Test Procedure for the Matching Case . . . . 20.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Findings . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Concluding Remarks . . . . . . . . . . . . . . . . . . 21 Notes on Thick-Tailed Distributions of Wealth Lester D. Taylor 21.1 Introduction . . . . . . . . . . . . . . . . 21.2 Background . . . . . . . . . . . . . . . . 21.2.1 Scenario 1 . . . . . . . . . . . .

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21.3

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447 448 449 449 450 450 453 453 462

22 Conic Distributions of Earned Incomes . . . . . . . . . . . . 22.1 The Search for Functional Form . . . . . . . . . . . . . 22.2 Specification of the Conic Family of Distributions . . . . 22.2.1 The General Conic Distribution . . . . . . . . 22.2.2 The Conic-Quadratic Distribution . . . . . . . 22.2.3 The Conic-Linear Distribution . . . . . . . . . 22.3 Geometric Aspects . . . . . . . . . . . . . . . . . . . . 22.3.1 The Asymptotes . . . . . . . . . . . . . . . . 22.3.2 Modes and Inflections . . . . . . . . . . . . . 22.4 Descriptive Statistics . . . . . . . . . . . . . . . . . . . 22.4.1 The Median . . . . . . . . . . . . . . . . . . . 22.4.2 The Arithmetic Moments and Gini Coefficient 22.4.3 The Logarithmic Moments: Alternative Measures of Inequality . . . . . . . . . . . . . 22.4.4 The Lorenz Curve . . . . . . . . . . . . . . . 22.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 22.5.1 Medianization . . . . . . . . . . . . . . . . . 22.5.2 Comparison with Other Distributions . . . . . 22.5.3 Bias in Maximum Likelihood Estimation . . . 22.6 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.7 Empirical Results . . . . . . . . . . . . . . . . . . . . . 22.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .

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482 484 484 486 487 489 489 490 497

23 Final Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

505

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

509 517

21.4 21.5 21.6 21.7

Some Initial Simulations . . . . . . . . . 21.3.1 Scenario 2 . . . . . . . . . . . . 21.3.2 Comments . . . . . . . . . . . 21.3.3 Scenario 3 . . . . . . . . . . . . 21.3.4 Comments . . . . . . . . . . . Variations . . . . . . . . . . . . . . . . . Interpretation of Parameters and Scenarios Law of Pareto Tests . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . .

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Chapter 1

Introduction and Overview

At the time of our 1966 and 1970 books, the theoretical foundations for applied demand analysis were thought to be pretty much in place. An axiomatic basis for consumer choice had been established, problems with integrability had long been solved, as had the relationship between utility theory and revealed preference. The remaining questions were seen to be primarily practical, such as the functional structure of preferences, development of new functional forms for describing preferences and demand, estimation of systems of theoretically plausible demand functions, problems of aggregation, and a better understanding of the dynamics of consumer behavior. The literature on all of these topics is now large, and the review that we are now about to undertake does little more than identify currents and high points. On the theoretical front, tremendous strides have been made in investigating the internal structure of preferences (Blackorby et al., 1978; Blackorby and Diewert, 1979), aggregation and index-number theory (Afriat, 1977; Berndt et al., 1977; Diewert, 1976, 1977, 1978; Hildenbrand, 1994; Mantel, 1979; Sonnenschein, 1972; Stoker, 1984, 1985, 1986), and the estimation of systems of theoretically plausible demand functions (Barnett, 1979; Barten, 1968, 1969, 1977; Christensen et al., 1975; Deaton and Muellbauer, 1980a; Parks, 1969; Pollak and Wales, 1969, 1992; Powell, 1966). Diewert, Hildenbrand, Mantel, and Sonnenschein, inter alia, established the extent to which Slutsky restrictions apply to market demand, while the focus of Stoker was on the conditions under which aggregate demand and consumption functions can be meaningfully derived from their micro counterparts. In the work on equation systems, three different (though not necessarily inconsistent) paths have been followed. The first is to specify a particular utility function and then to derive the demand functions therefrom. Examples include the direct and indirect addilog utility functions of Houthakker (1960), as well as the dynamic quadratic utility function that was estimated in our 1970 book.1 A second approach has been to provide a second-order approximation to an arbitrary twice-differentiable direct or indirect utility function. Examples of this are the “Rotterdam” system of Barten, Theil, and others; the “translog” system of Jorgenson and his collaborators; and the “flexible” functional forms of Diewert and his associates. Finally, the third approach 1 See

also Taylor and Weiserbs (1972).

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_1, 

1

2

1 Introduction and Overview

is to specify an integrable demand system and then to estimate the system subject to the restrictions that integrability entails. Examples of this approach include the LES model of Stone (1954) and the almost ideal demand system (AIDS) of Deaton and Muellbauer(1980).2 While the research just described represents notable advances in consumption research, the incorporation of dynamics into formal demand theory has, as noted by Pollak (1990), not particularly kept pace. While an Internet search on habit formation and stock adjustment brings forth a large number of entries, most of these represent little more than mechanical applying of the original Houthakker–Taylor models.3 In our view, there are two reasons (not necessarily unrelated) as to why consumption dynamics has lagged other developments. The first is simply that the specification of dynamical structures is often addressed strictly empirically: What lag structures appear empirically to be the most appropriate? Put another way, most consumption theory continues to be formulated in static terms, and dynamics are added (if at all), often almost as an afterthought, at the estimation stage. The second reason relates to the fairly widespread view that dynamics are associated with taste change, which is a turn that conventional economic theory is reluctant to accept. Of course, this is not to say that, despite the influential paper of Stigler and Becker (1977), the assumption of immutable tastes and preferences has not found a great deal of scrutiny in the literature, for it has, and the same is true of the associated assumption of rationality.4 However, the place where this research is found is in behavioral economics, which mainstream economics, for the most part, has yet to embrace. 2 Interestingly,

the work that has probably had most impact on consumption analysis since 1970, both theoretical and applied, is the short four-page paper of Gorman (1961). In this paper, Gorman derived the explicit form that preferences must have in order for income-consumption curves (i.e., Engel curves) to be linear. For this to be the case, Gorman showed that the indirect utility function must have the form mi − f i (p) ui (p,m) = . g(p)

where f i (p) represents the expenditure necessary to reach a reference level of utility of zero for each individual (i), and g(p) is a price index (the same for every individual in an economy) that deflates the excess money income needed to attain a particular level of utility u¯ . Both fi and g are homogeneous of degree one in prices (p). The literature that this paper has given rise to is voluminous. [See, among others, Blackorby, Primont, and Russell (1978), Lewbel (1991), Banks, Blundell, and Lewbel (1997).] 3 However, this is not to say that there have not been advances on the Houthakker–Taylor models since 1970. See Phlips (1972), Spinnewyn (1969a, b), Weiserbs (1974), and (especially) Bergstrom and Chambers (1990). Also, it is worthy to note that an alternative approach to the dynamics of consumer behavior is that of Pollak (1970) and others, in which dynamics are associated with endogenous taste change. While, as will become clear in the next chapter, we do not (unlike Stigler and Becker (1977)) view tastes and preferences as immutable, the position in our 1966 and 1970 books was that the coefficients governing habit formation and stock adjustment are fundamental utility parameters. 4 See, among others, Kahneman (2003), Kahneman and Tversky (1979), Thaler (1991a, b), and Tversky and Kahneman (1981); see also Camerer et al. (2005).

1.1

Themes and Format of the Book

3

At this point, it is well to enquire as to the influence that the foregoing has had on the present exercise. The answer, in all candidness, is “not much.” As will be clear in the next chapter, while our efforts remain guided by conventional considerations (especially in the estimation of price and income elasticities), progress in understanding consumer behavior, in our view, does not lie in further refinements in neoclassical theory, but in locating demand analysis on a foundation in the neurosciences.

1.1 Themes and Format of the Book We now turn to the themes and format of the present effort, which is quite different from either of the two earlier editions of Consumer Demand in the United States. While dynamics continue to be the main theme, and much of the same ground is covered (especially in the time-series analysis), this is really a new undertaking, not only in scope and focus, but also in the way that the material is structured. In part because of our divergent interests and the way that our research has unfolded, the format of the book is largely one of interconnected essays and studies, with a focus on the following themes and motifs: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Estimation of a broad array of price and total-expenditure elasticities. Consumption behavior in relation to brain structure and functioning. Dynamics. Stability of tastes and preferences. Integration of price variation into cross-sectional survey data and estimation of complete systems of demand functions therefrom. Use of quantile regression. Shape of the distribution of residuals. Distribution of income and wealth. Test of axioms of revealed preference.

The book is structured into four parts. Part I consists of a review of the present state of demand theory and the various currents tugging at it (Chapter 2) and an overview of the econometric techniques that are employed in the empirical analyses (Chapter 3 and its appendix). Part II consists of nine chapters that focus on analysis of cross-sectional data from the consumer expenditures surveys conducted quarterly by the U. S. Bureau of Labor Statistics (BLS). Chapter 4 provides a description of the BLS surveys and discusses the procedures that have been employed in creating data sets used in estimation. Chapters 5 and 6 initiate the empirical analysis. Chapter 5 focuses on the stability of tastes and preferences using principal component analysis of 14 exhaustive categories of expenditure in the CES surveys for the 16 quarters between 1996 and 1999. The results suggest the existence of an underlying stability in preferences over these quarters that accounts for between 85 and 90% of the total variation in expenditure across households. Chapter 6, on the other

4

1 Introduction and Overview

hand, provides a look at price and income elasticities using data sets that combine expenditure data from the CES surveys with price data that are collected by the American Chambers of Commerce Research Association (ACCRA). The results from this exercise, which is based upon simple double-logarithmic functions, are useful to the sequel for two reasons: (a) they provide a benchmark set of elasticities for later comparison; and (b) they show that income (and total expenditure) elasticities are little effected by incorporation of prices into the Engel curves. Chapter 7 presents price and total-expenditure elasticities that are obtained from the estimation of complete systems of demand functions. Four systems are investigated: the almost ideal demand system of Deaton and Muellbauer, the linear expenditure system of Stone, and the indirect and direct addilog models of Houthakker. In Chapter 8, a fifth system is analyzed, namely, an additive doublelogarithmic system that represents an extension of Houthakker’s indirect addilog model. Chapter 9 is concerned with a rather intriguing result that emerged early in the analysis of the BLS surveys, namely, residuals from Engel curves and demand functions that display peaked and asymmetrical distributions with long tails. Regression error structures with these characteristics obviously have serious implications for least-squares estimators, and estimation based on minimizing the sum of absolute errors is suggested as a robust alternative. Since the CES surveys are partial panels (households in principle remain in the surveys for five quarters), the estimation of dynamical models is possible, and in Chapter 10 we take a look at such models using a double-logarithmic version of the flow-adjustment model. As price data from the ACCRA surveys can only meaningfully be applied at a level of six exhaustive CES categories of expenditure, in Chapter 11, we extend estimation of Engel curve (using only simple double-logarithmic functions) to a much more disaggregate CES level of 29 categories of expenditure. Finally, Chapter 12 (and its appendix) summarizes the results of the cross-sectional analyses and organizes the estimated price and total-expenditure elasticities into useful tables. Part III of the book consists of seven chapters (Chapters 13–19) that focus on aggregate time-series data from the National Income and Product Accounts (NIPA). Chapter 13 begins with listings of the time-series consumption categories analyzed (both annual and quarterly) and then turns to the development of the dynamical model that forms the workhorse of the analysis, namely, the model introduced by Bergstrom and Chambers in 1990 that encompasses both flow- and state-adjustment in the same framework. This model, which we refer to as the B-C model, is applied to quarterly data in Chapter 14 and to annual data in Chapter 15 and its appendices. Chapters 16, 17, and 18 provide comparisons of the quarterly and annual results (Chapter 16), time-series and cross-sectional results (Chapter 17), and timeseries results from the present effort to those obtained in the 1970 edition of CDUS (Chapter 18). Finally, Chapter 19 focuses on the relationship between aggregate saving and disposable personal income using data for saving from the U.S. Flow of Funds. The results from this chapter suggest that there is no simple explanation for the extremely low personal saving rates of recent years. Part IV of the study consists of three miscellaneous studies concerned with the distributions of income and wealth and tests of the axioms of revealed preference.

1.2

A Reader’s Guide to the Study

5

Finally, in Chapter 20, H. S. Houthakker provides an extensive battery of empirical tests of the weak and strong axioms of revealed preference using aggregate timeseries data from 20 countries. His conclusion, consistent with the results from the cross-sectional analyses of Chapters 5–12, is that there is a great deal of stationarity in consumer preferences. In Chapter 21, L. D. Taylor examines, in a sequence of Monte Carlo simulations, whether a Pareto upper tail for the distribution of wealth can arise in a Darwinian framework that emphasizes heritability, differential productive efficiency, and habit formation. The answer is in the affirmative. In Chapter 22, H. S. Houthakker introduces a new family of functions for describing the distribution of income that is defined by a quadratic equation in the logarithm of income and the logarithm of the upper tail of the distribution function of income. Such an equation defines a class of conic distributions that, unlike the lognormal and Pareto, can explain the entire distribution of earned incomes. The book concludes in Chapter 23.5

1.2 A Reader’s Guide to the Study As the study is rather wide-reaching in its purview, suggestions as to how readers with specific interests might navigate the material in it are obviously in order. To begin with, we recommend that everyone read Chapter 2 , even those whose interest might simply lie in seeing and comparing the many estimates of price and totalexpenditure elasticities that have been obtained in the study. While the material in this chapter is at times quite technical, mathematical details can be easily skipped over without loss of basic ideas. After Chapter 2, our suggestions are as follows: Readers interested merely in estimated values of price and total-expenditure elasticities: Chapters 12, 19, and 23. Teachers of graduate courses (and their students) in theoretical econometrics: Chapters 3, 9, and Appendix 3.1. Teachers of graduate and upper-division undergraduate courses (and their students) in applied econometrics: Chapters 3, 5–11, 13–15, 19, Appendices 15.1–15.3. Students of applied demand analysis: Chapters 5–19, Appendices 15.1–15.3. Economists interested in tests of the axioms of revealed preference: Chapter 20. Economists interested in the distribution of wealth and income: Chapters 21–22. Consultants and industry specialists interested in the structure of consumer expenditures: Chapters 5–6, 10–12, 14–19. Teachers of beginning economics courses interested in specific price and totalexpenditure elasticities: Chapters 12, 18, and 23. 5 Readers

immediately interested in summary and conclusions can jump directly to Chapters 12, 18, and 23.

Part I

Preliminaries

Chapter 2

Demand Theory Under Review

The conventional (i.e., neoclassical) theory of consumer choice is one of the great prides of economics, for, among other things, it provides a rigorous and elegant mathematical underpinning to the common-sense notion of the law of demand, that there is an inverse relationship between the price of a good and the amount of the good that a consumer is willing to buy. Macroeconomic theories over the years come and go, and to a lesser extent, the same is true of theories of production, but not the theory of consumer choice. For, although there periodically have been questions concerning the assumptions that underlie it, the theory has essentially retained its present form since the 1930s. It is, in short, one of the great invariants (along with the theory of least-squares estimation) in the core education of an economist. In the circumstances, accordingly, it might pretty much seem a waste of readers’ time to begin a demand study with a review of conventional theory, for one could easily begin, as is so often the case in demand studies: “Economic theory teaches us that quantity demanded is a function of price and income, that (in normal circumstances) demand functions slope downward with price and shift outward with income, and so on and so forth.” However, while the neoclassical theory (whether directly or in its equivalent alternative form in terms of revealed preference) will be a guiding framework within which the analyses of this book proceed, there will be a number of points at which results (and other considerations) emerge that tempt explanation or interpretation that go beyond that offered by this theory. For this reason, a brief review of conventional theory is in order.

2.1 Conventional Theory of Consumer Choice An economic agent, identified as an individual consumer, is assumed to allocate an income of y over n market goods qi , which can be purchased at unit prices of pi , in such a way that a “utility” function defined over the n goods, ϕ(q1 , . . . , qn ), is at a maximum. More formally, purchase decisions are assumed to follow as the solution to the following constrained maximization problem:

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_2, 

9

10

2 Demand Theory Under Review

Find the values of qi , i = 1, . . . , n, that maximize the function ϕ (q) = ϕ (q1 , . . . , qn ) ,

(2.1)

subject to the condition that 

pi qi = y.

(2.2)

To solve this problem, one first formulates the expression    pi qi ,  (q,λ) = ϕ (q) − λ y −

(2.3)

where λ is a Lagrangean multiplier representing the marginal utility of income, differentiates this expression with respect to qi and λ:   ∂ ∂qi =∂ϕ ∂qi − λpi , i = 1, . . . ,n,

(2.4)

  ∂ ∂λ =y − pi qi ,

(2.5)

equates the n + 1 derivatives to zero, and then solves the resulting first-order conditions for the n demand functions qi as functions of the n prices pi and income y1 : qi = qi (p1, . . . , pn, y), i = 1, . . . , n.

(2.6)

The explicit expressions for the demand functions in equation (2.6) obviously depends upon the analytic form of the utility function. For some utility functions, such as the Stone–Geary utility function (which yields the equations associated with the linear expenditure system), the demand functions are easily derived and often fairly easily estimated, while for other utility functions (such as the direct addilog utility function of Houthakker), the demand functions are both complicated and highly nonlinear. The demand functions corresponding to both of these utility functions, as well as those from several other utility functions, are estimated in Chapter 8.2

Solution of the first-order conditions requires that the utility function ϕ(q) satisfy a variety of regularity conditions. For present purposes, we will take these to include ϕ to be continuous in the n qi , with continuous first and second partial derivatives that are positive and negative, respectively. 2 Much of the elegance of this theory is that one can equally well take the demand functions in expression (2.6) as the starting point (the demand functions, after all, are what are in principle observable), and, under certain conditions, associate them with a utility function. These conditions (the so-called Slutsky conditions, which are both necessary and sufficient) are that the matrix, whose typical element is 1

∂qi ∂qi + qj , j = 1, . . . ,n, ∂pj ∂y

2.2

Neoclassical Demand Theory as a 19th-Century Conservative Energy System

11

2.2 Neoclassical Demand Theory as a 19th-Century Conservative Energy System3 The theory of consumer choice as just presented is widely viewed (and rightly so) as one of the real triumphs of economic theory, for it yields a conclusion (that income-compensated demand functions are downward-sloping in price) that is not only elegantly derived, but intuitively satisfying, and felt to be in accordance with real-world behavior. What is perhaps not appreciated is that this theory, in form, represents a 19th-century conservative physics energy system. The utility function is the counterpart to the potential energy function, while the income (or budget) constraint is the counterpart to kinetic energy. Marginal utilities form an invariant vector field of “forces,” and the “work” function has its counterpart in an expenditure line integral. The invariants in the choice problem are independent of the “path of motion,” while the conservative principle that is counterpart to the conservation of potential and kinetic energy is that the sum of utility and income is conserved. The structure that the vector field of marginal utilities has to satisfy (in vector-field terminology) is that the curl of the vector field must be equal to zero, which is to say that the vector field must be irrotational. In economics terminology, these restrictions represent the integrability conditions stated in Footnote 2.4 What we wish to do now is to discuss several possible applications of the neoclassical theory as illustrations of both appropriate and inappropriate uses of the theory. In our view, neoclassical theory is most appropriately used when the questions involved are of the “what if” variety. Such would be the case, for example, if the questions asked (with reference to a given choice set of market baskets of goods and services) were of the form: 1. Assuming the vector of prices to be held constant, which market basket would be chosen if income were A as opposed to the market basket chosen if income were B? 2. Assuming income to be held constant, which market basket would be chosen if the price vector were w as opposed to the market basket chosen if the price vector were z? The first question relates to the derivation of what are ordinarily referred to as Engel schedules, while the second question relates to the derivation of demand

be symmetric and of rank n–1. Demand functions that satisfy these conditions are said to be integrable (or, alternatively, theoretically plausible). 3 Much of the material in this section is taken from Appendix 1 of Taylor (2000). 4 Although economists are generally well-versed in linear algebra, this is usually not the case with respect to the concepts of vector fields. Mirowsky (1989) provides some discussion of the concepts in question, but not in sufficient detail for a full understanding of the relationship between neoclassical theory and 18th-century conservative physical systems. The needed concepts, which include gradient, vector cross (or outer) product, curl, and divergence, can be found in any standard advanced calculus book, such as Apostal (1957).

12

2 Demand Theory Under Review

schedules. In neither case is there any assumption that the behavior at issue is actually to be observed. Indeed, the presumption (although usually tacit) is that it is not. Contrast, now, the above questions with the following two questions: 1∗ . If an individual (with a given level of income A) were repeatedly (i.e., during a sequence of consumption periods) to be faced with the same price vector w, what would be the market basket of goods consumed? 2∗ . If in a subsequent sequence of periods, the individual continued to face the price vector w, but with income B rather than A, what would be the market basket consumed? In these two questions, the assumption is that consumption actually occurs within each of the consumption periods. For neoclassical theory, this should not matter, for the market baskets consumed in the sequence of consumption periods in (1∗ ) should be the same basket as would been chosen in (1), while the market baskets chosen in the sequence of consumption periods in (2∗ ) should be the same as would have been chosen in (2). Almost certainly, however, this is not what would be observed. With consumption actually taking place in (1∗ ) and (2∗ ), there would in all likelihood be an interaction between the consumption in one period and the marginal utilities operative during the next period. Because of the intrusion of real-time dynamics, the assumptions for a conservative vector field would accordingly be violated. For the questions in (1) and (2), in contrast, this would not be a problem, for one could reasonably expect that marginal utilities would remain invariant in the face of “what if” questions, since no consumption actually takes place. We should emphasize that the issue involved here is not, as a practical matter, the ignoring of some factors that impinge upon consumption and that behave randomly from one consumption period to the next. The issue, rather, is one involving realtime irreversible alterations of marginal utilities in a systematic, yet not necessarily predictable, manner. The question of whether the standard paradigm can be modified to accommodate such “endogenous taste changes” will be discussed later in this chapter. However, for now, we want to consider another example that illustrates the problem at issue. Suppose, as has frequently been the case in the telecommunications industry, there is a question before a regulatory commission regarding the amount of “repression” on telephone usage that might be expected to occur as a result of a proposed increase in tariffs. In assessing this question, the usual procedure is to use a price elasticity that has been estimated from an econometric model to calculate the expected amount of repression.5 The expected repression can be calculated either by using the price elasticity directly or (what is usually done) by simulating the model first under the status quo, then with the new tariff, and then by attributing

5 “Repression” in this situation refers to the revenue that would be lost because of the existence of a non-zero price elasticity.

2.3

Dynamics: Some Preliminaries

13

repression to the difference. The models in question are usually theoretically based, often with considerable dotting of i’s and crossing of t’s. Question: Is this procedure an appropriate application of the underlying neoclassical demand theory?6 The answer is clearly yes, in our opinion, provided the question is framed in terms of comparative statics (i.e., of the “what if” variety) as opposed to real-time dynamics. However, if the question is framed in terms of estimating what the actual impact of the tariff change would be on telecommunications usage, then the question would involve real-time dynamics and the answer would in general be no, for the invariances required of a conservative system would once again be unlikely to be fulfilled. The upshot, thus, would seem to be that the neoclassical theory of consumer choice can be meaningfully used in assessing alternative courses of action at a planning stage, but cannot be used in predicting the result from a course of action actually selected. As a general conclusion this seems to be a correct statement, for neoclassical theory imposes invariance assumptions that real-time consumption behavior almost certainly invalidates. Yet, this is not to say that conventional theory cannot be modified to accommodate the invalidations that arise. This can be done, but the modifications required have to be embedded in a theoretical structure that allows for an “arrow of time.” It is to this that we now turn.

2.3 Dynamics: Some Preliminaries That consumption behavior needs to be analyzed taking dynamics explicitly into account has been recognized for years and has been the subject of much empirical research, including much of our 1970 book. For the most part, however, the dynamics that have been postulated have been of a classical form. The notion of “earlier and later” is center stage, but the laws of motion ply in mechanical time, motion is reversible, and real time is important only for purposes of measurement and empirical application. Heraclitus’s dictive that “one cannot step in the same river twice” is not observed by most existing dynamical models. However, as we shall now discuss, it might seem that applied econometricians go to great lengths in attempts to assure that this is the case. Something that is often not recognized, at least in the terms being discussed in this section, is that the conditions that must be satisfied for the validity of the statistical inference in applied econometrics are in a sense isomorphic to the invariance conditions for a conservative vector field.7 For illustration, consider the estimation of a system of theoretically plausible demand functions that have been derived (say) 6

We leave till later the question of whether the parameters estimated in the econometric model can be identified with the parameters of the theoretical model. 7 That this is indeed the case was already implicit in Haavelmo’s classic monograph on the probability approach in econometrics (1944). It is also just below the surface in the reinterpretation of extension of the Haavelmo paradigm by Spanos (1989). LDT’s reinterpretation of the Haavelmo paradigm is given in Chapter 10 of Taylor (1994).

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2 Demand Theory Under Review

from the Stone–Geary–Samuelson utility function. Suppose that the demand functions are estimated, subject to all of the integrability conditions, from an appropriate body of data. In this situation, a conservative vector field of preferences is not only assumed, but is, in fact, imposed as part of estimation. Assuming the econometrican is well-schooled in the Haavelmo paradigm, however, this imposition is not as mechanical as it might appear. For a number of problems have to be dealt with before estimation actually occurs, most of them related to the fact that the data set being analyzed arises not out of controlled laboratory conditions, but from a natural experiment generated by history. Be that as it may, the sine qua non of the Haavelmo paradigm nevertheless is a presumption that the observations in the data set can be viewed as a sample drawn at random from a common underlying population. For this presumption to be justified, it becomes a major task of analysis to model not only the phenomenon of interest (namely, the relationship between consumption, prices, and income), but also the structure of the natural experiment that generated the data to begin with. Structural changes during the period of the sample have to be taken into account, as well as the fact that most behavior represented in the data set will reflect disequilibria. We put structural changes in italics because, in the natural experiment that generated the data, there are two different forms of structural change to contend with. The first refers to autonomous structural change arising from the “outside” that affects preferences, and therefore consumption, but which is not, itself, consumption caused. The other form of structural change is the endogenous change in the field of marginal utilities caused by real-time consumption. Of the two types of structural change, the second type is clearly the most problematic to take into account, for it requires an explicit representation of the feedback on preferences as consumption occurs. The usual way of accomplishing this empirically is to specify a model in which current consumption depends not only upon current income and prices, but upon past consumption as well.8 Endogenous preference changes are assumed to be reflected in the coefficient on past consumption.9 An alternative procedure, which relates to the state-adjustment model of the two editions of Consumer Demand in the United States, is to postulate a state-dependent vector field of marginal utilities in which the composition of the field at any point in time depends upon a set of state variables, which themselves evolve in response to real-time consumption together with the passage of mechanical time. If the econometrician is successful in specifying a model that appropriately takes into account the two types of structural change (and also appropriately reflects the probability structure of the random component of the implicit natural experiment), then the requirements of the Haavelmo paradigm will be satisfied. Moreover, it can also be argued that the invariance conditions for a conservative vector field will be 8 Care must be taken at this point to distinguish between mechanical dynamics that occur in mechanical time (and reflect disequilibrium behavior) and real-time dynamics that reflect the feedback of consumption on preferences. With the way that many models are specified, it is unlikely that these two forms of dynamics can be separately identified. The distinction between these two concepts of dynamics is discussed in much more detail in Section 2.6 below. 9 See, among others, Pollak (1970), Von Weizsaker (1971), and Yaari (1977).

2.4

State- and Flow-Adjustment Models of Consumption

15

satisfied as well. Estimation of the model can then proceed under the assumption that the parameters estimated in the econometric model are to be identified with the parameters of the underlying theory. The argument that the invariance conditions for a conservative vector field are satisfied when the Haavelmo assumptions are met can be stated as follows: If the real-time feedback of consumption on preferences is appropriately modeled, then the sample can be viewed as a sequence of state-dependent conservative vector fields in which the states are determined exogenously. Historical time is no longer reflected in the data, because its effects are held constant through the explicit modeling of how the states, themselves, evolve. Dynamics may still be present, but these would be of the mechanical variety and therefore invariant to historical time. Thus, each observation in the data set can be viewed as having arisen in answer to a “what if” type of question, namely: What quantities would be demanded if income were this and prices were thus and so? Though the conditions for a conservative vector field may be satisfied for purposes of estimation, it does not follow (as already discussed) that they would be satisfied for purposes of using the estimated model to predict the evolution of real-time consumption in response (say) to a change in prices. As just noted, the estimated model could be used to analyze “what if” types of questions in which there is no presumption of actual consumption ever occurring. But to predict realtime consumption would require also being able to predict real-time evolution of the state-dependent vector fields of preferences.10 The problem, clearly, is that predicting human consumption behavior is materially different from predicting the motion of an inanimate particle, for the motion of an inanimate particle does not interact with the forces defined in the field. With human behavior, on the other hand, motion does interact with these forces, for what one consumes today alters the marginal utilities that inform what one will consume tomorrow, perhaps in unpredictable ways.

2.4 State- and Flow-Adjustment Models of Consumption In the two editions of CSUS, two dynamical models were employed: a stateadjustment model, in which flows react to discrepancies between actual and desired states; and a flow-adjustment model, in which flows react to discrepancies between actual and desired flows. In applying these models to more than 80 categories of personal consumption expenditure from the National Income and Product Accounts for the years 1929–1964 (as well as higher levels of aggregation for a number of other countries), dynamical effects were found across a wide variety of durables,

10

This is obviously a tall order, but, as will be discussed later in this chapter, there is hope that, as we come to better understand how the brain is organized and functions, appropriate evolutionary laws (perhaps even of a mechanical variety) can be formulated.

16

2 Demand Theory Under Review

nondurables, and services. We begin in this section with brief descriptions of these models in their original form.11 The point of departure for the state-adjustment model is a demand function that relates the flow of expenditures for a good or service at time t, q(t), to the level of a state variable, s(t), the flow of income, x(t), and the level of price, p(t): q (t) = α + βs (t) + γ x (t) + λp (t) .

(2.7)

Time is measured continuously. Later in this chapter, we will discuss the inclusion of a state variable in genetic terms. However, for now, it is useful to return to our original motivation for its presence as presented in the 1970 edition of CDUS: A simple example will illustrate the principles involved. Let q(t) be an individual’s demand for clothing during a very short time interval around t, let x(t) be his income during that interval, and let s(t) be his inventory of clothes at time t. More exactly, let q(t) be the rate of demand at time t and x(t) be the rate [of flow] of income at that time. All other variables are ignored for the time being. Then the basic assumption is that q(t) = α + βs(t) + γ x(t),

(2.8)

so that the individual’s current demand for clothing depends not only on his current income, but also on his stock of clothing. We may expect that, for a person with given tastes and given income, the more clothes he has to begin with, the fewer he will be buy currently. In the case of a durable good such as clothing the stock coefficient β will be negative, but it will now be shown that equation (2.8) may also hold for other types of commodities if we allow a more general interpretation of s(t). In fact the equation can represent not only the stock-adjustment behavior just described, but also habit formation or inertia, which is apparently a more widespread phenomenon. Consider a commodity of which consumers do not normally hold physical inventories of any significance, say tobacco. By all accounts tobacco consumption is habit-forming, which means that it does not adjust immediately to changes in income (or in prices, for that matter) and that current consumption is positively influenced by consumption in the more or less recent past. In this case, we can say metaphorically that the consumer has built up psychological stock of smoking habits. His current consumption will be affected by that stock (or, if one prefers, “state variable”) just as it is for clothing, but the sign of β will now be positive: the more he has smoked in the past, the more he will smoke currently (tastes and income again being given). The question arises at once: How can we measure such a psychological stock? It will be shown in a moment that under certain reasonable assumptions there is no need to measure it, because s(t) can be eliminated from the regression equation. Yet it should be stressed first that this difficulty is not peculiar to habit-forming commodities, but arises almost as strongly for durable commodities such as clothing.12 In the latter case we cannot measure s(t) simply by the number of suits, shirts, and such, for some of these may worn out and due for replacement; moreover, their heterogeneity also makes direct measurement hard. Clearly some depreciated measure of inventories is needed, but the appropriate depreciation rates are usually not known a priori and would either have to be estimated from the data or 11

A generalization of these models suggested by Bergstrom and Chambers (1990) will be developed in Chapter 13. 12 In fact, there is often no a priori basis for deciding whether, in the demand for a commodity, habit formation or stock adjustment will predominate.

2.4

State- and Flow-Adjustment Models of Consumption

17

guessed. Hence even for durables, where the state variable has a concrete interpretation, it is desirable to eliminate it. This can be done in the following manner. First consider the accounting identity s˙(t) = q(t) − w(t),

(2.9)

where s˙(t)stands for the rate of change in the (physical or psychological) stock around time t and w(t) stands for the average “using up” of “depreciation” of that stock at the same time. From now on, moreover, we shall assume that w(t) = δs(t)

(2.10)

where δ is a constant depreciation rate. Hence the rate of depreciation at any time t is proportional to the stock at that time. The assumption of proportionality corresponds to the “declining balance” method of depreciation, which has been found to be realistic in many practical situations. Combining equations (2.9) and (2.10) we find that s˙(t) = q(t) − δs(t).

(2.11)

Integration of equation (2.11) shows, incidentally, that  s(t) =

t −∞

q(u)eδ(u−t) du

(2.12)

or, in words, the state variable at any time is equal to the sum of the discounted flows bought up to that time. This formula applies equally well to durable as well as to habit-forming commodities. Next eliminate s(t) from equation (2.11) by using equation (2.8): s(t) = q(t) −

 δ  q(t) − α − γ x(t) . β

(2.13)

Now differentiate equation (2.8) with respect to time and substitute equation (2.13) for s(t):

q˙ (t) = αδ + (β − δ)q(t) + γ x˙ (t) + γ δx(t),

(2.14)

which is a first-order differential equation involving only observable quantities q and x. [Houthakker and Taylor (1970, pp. 10–11).]

The long-run (or steady-state) solution for the dynamic system described by equations (2.7) and (2.11) is obtained by setting s˙(t) = 0, and then solving for q in terms of the exogenous quantities x and p: qˆ =

γδ λδ αδ + x+ p. δ−β δ−β δ−β

(2.15)

The dynamics of the system can then be easily characterized in terms of the derivatives for q with respect to x and p. Changes in income or price will have two types of effects on q: a short-run (or instantaneous) effect that arises before there is any feedback on the state variable and a long-run (or steady-state) effect that reflects full adjustment in the state variable. From equations (2.7) and (2.15), we see that

18

2 Demand Theory Under Review

the short- and long-run derivatives with respect to x and p are given by γ and λ and γ δ/(δ − β) and γ δ/(γ − β), respectively. As noted, the model can be applied to a full complement of consumption expenditures, whether they be durables, nondurables, or services. The only difference is in the interpretation of the state variable. For durables, the state variable can be seen as representing physical stocks (such as automobiles), while for nondurables and services, the state variable can be seen as representing psychological quantities (such as the stocks of smoking habits).13 The coefficient β is expected to be negative in the case of durables, but can be positive with nondurables and services. A positive β corresponds to habit formation, while a negative β corresponds to inventory (or stock) adjustment. The dynamics of the two cases are obviously different. Consider a change in income in conjunction with the short- and long-run derivatives for q, γ , and γ δ/(δ − β). The former is seen to differ from the latter by the factor δ/(δ − β). Since δ < 0, we will have δ(δ − β) less than 1 for β less than 0, and greater than 1 for β greater than 0.14 This implies that short-run derivatives will be greater than longrun derivatives for goods characterized by inventory adjustment, and the opposite for goods characterized by habit formation. Expenditure flow rates for goods subject to inventory adjustment respond quickly to changes in income or price, while flow rates for goods subject to habit formation respond sluggishly. Returning to equation (2.7) for a moment, we see that, in long-run equilibrium, qˆ = α + βˆs + γ x + λp.

(2.16)

Subtraction of (2.16) from (2.7) then yields. q(t) − qˆ = β[s(t) − sˆ],

(2.17)

which is the relationship that gives the state-adjustment model its name, for expenditures can be interpreted as adjusting so as to bring the state variable into line with its steady-state value. We now turn to the flow-adjustment model, whose structural equations (again in continuous time) are given by 13 An alternative (and physiologically appealing) interpretation of the state variable in the case of habit formation is as a measure of memory. Initial exposure to a consumption activity gives rise to new (or alters existing) synaptic connections, which can become strengthened with subsequent exposures. The coefficient β then measures the “pleasure” associated with the activity, while δ measures the rate at which memory of it fades. For an extraordinarily readable account of the biological processes involved, see Chapter 15 of Kandel (2006). Also, care must be taken not to equate habit formation described here with habituation in the psychological literature, which refers to “a simple non-associative form of learning in which a subject learns about the properties of a single, innocuous stimulus.” Again, see Kandel (2006). 14 Stability of the dynamical system described by equations (2.7) and (2.11) requires δ < β, which is to say that habit formation should never be so great as to overcome the tendency of the habit to wear off. The latter, should it occur, would characterize addiction.

2.4

State- and Flow-Adjustment Models of Consumption

19

q˙ (t) = θ [ˆq(t) − q(t)],

(2.18)

qˆ (t) = κ + μx(t) + ξ p(t).

(2.19)

Substituting for qˆ (t) in (2.18) from (2.19), we have q˙ (t) = θ κ + θ μx(t) + θ ξ p(t) − q(t).

(2.20)

The dynamics in this formulation can be interpreted in terms of an adjustment in the flow rate of expenditures so as to partially eliminate discrepancies between the actual flow rate of q and a desired rate qˆ as determined in equation (2.19). Consumers in this model are assumed to target the flow variable, whereas in the state-adjustment model, they are assumed to target a state variable. It is to be noted, however, that the decision variable in this model is qˆ , whereas the action variable is q˙ . Consequently, from equations (2.20) and (2.19) we see that the short-run derivatives of q with respect to income and price are θ μ and θ ξ , respectively, while the long-run derivatives are θ and ξ . As no state variable is present in the flow-adjustment model, dynamics arise because of delays and constraints in adjusting actual to targeted expenditure flows. In discrete time (about which more later), adjustment is of the standard geometric distributed-lag variety, in that a change in income or price this period will have some impact next period, a further (but smaller) impact next period, and so on. The dynamics are accordingly reminiscent of habit formation in the state-adjustment model. Inventory adjustment behavior cannot arise in this model.15 The dynamics in this case can be seen as representing “outside” varieties of dynamics, as discussed in the preceding section. This is in contrast to the state-adjustment model, where the feedback of consumption onto the state variable gives rise to dynamics of the “inside” variety, that is, of the type that allows current consumption to affect marginal utilities. A discrete time model for the state-adjustment model for empirical application is obtained by integrating equations (2.7) and (2.11) over finite intervals of time, and then by solving the resulting relationships for the reduced form for q. Specifically, we begin by integrating equations (2.7) and (2.11) over the interval from t to t + h:16 qt = αh + βst + γ xt + λpt ,

(2.21)

∗ s(t) = qt − δst ,

(2.22)

15 However, as will be noted in Section 10.3, inventory-adjustment behavior can be inferred in the flow-adjustment model for large values of θ. 16 In empirical application, h will be set to one, so that t will as usual refer to the current period and t–1 to the immediate past period. However, the use of an arbitrary period of h is useful in relating (at a theoretical level) the relative magnitudes of structural coefficients estimated from quarterly data (say) to those estimated from annual data.

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2 Demand Theory Under Review

where  qt =

t+h

 q(τ )dτ , st =

t

t+h

s(τ )dτ , etc.

(2.23)

t

Similarly, we will have for qt−h : qt−h = αh + βst−h + γ xt−h + λpt−h .

(2.24)

We now subtract equation (2.16) from equation (2.13): qt − qt−h = β(st − st−h ) + γ (xt − xt−h ) + λ(pt − pt−h ). (2.25)  ∗ 

We now approximate st − st−h by h/2 s (t + 1) + ∗ s (t) .17 Hence, from equation (2.22)  h qt + qt−h − δ(st + st−h ) , st − st−h ∼ = 2

(2.26)

so that substitution into equation (2.25) (and assuming that the approximation is good enough for practical purposes) gives qt − qt−h =

hβ [qt + qt−h − δ(st + st−h )] + γ (xt − xt−h ) + λ(pt − pt−h ). (2.27) 2

Returning now to equations (2.21) and (2.24), we have for st and st−h st = st−h =

1 (qt − αh − γ xt − λpt ), β

1 (qt−h − αh − γ xt−h − λpt−h ), β

(2.28) (2.29)

which, when substituted for st + st−h in equation (2.27), yields for qt − qt−h (after rearrangement and notational simplification): qt = A0 + A1 qt−h + A2 xt + A3 xt−h + A4 pt + A5 pt−h ,

(2.30)

where A0 = A1 = 17

αδh2 , A

1+

h(β−δ) 2

A

(2.31) ,

(2.32)

This involves application of the mean-value theorem applied to interval of length 2 h between t− h and t + h. If s(τ ) were to behave linearly within this interval, then the approximation would of course be exact.

2.4

State- and Flow-Adjustment Models of Consumption

A2 =

γ (1 + A

A3 = − A4 =

δh 2)

γ (1 − A

λ(1 + A

,

δh 2)

δh 2)

21

(2.33) ,

,

λ(1 − δh 2) , A h(β − δ) . A=1− 2 A5 = −

(2.34) (2.35) (2.36) (2.37)

For computational reasons, it is convenient to express xt as (xt − xt−h ) + xt−h ≡

xt + xt−h (and similarly for pt ), which transforms equation (2.30) into qt = A0 + A1 qt−h + A2 xt + A3 xt−h + A4 pt + A5 pt−h ,

(2.38)

where now A3 and A5 are equal to γ δh , A λδh . A5 = A

A3 =

(2.39) (2.40)

Equation (2.38) represents the finite approximation, as a difference equation, to the first-order differential equation in expression (2.14).18 The structural coefficients α, β, γ , λ, and δ are obtained from A0 , . . . , A5 in equation (2.38) as follows:19 A0 (2A2 − A3 ) , h2 A3 (A1 + 1) 2A3 2(A1 − 1) + , β= h(A1 − 1) h(2A2 − A3 ) 2A2 − A3 , γ = h(A1 + 1) 2A4 − A5 , λ= h(A1 + 1) α=

18

(2.41) (2.42) (2.43) (2.44)

The derivation of equation (2.38) is the original one from CDUS. An alternative not involving the calculus is given by Winder (1971). 19 The depreciation rate δ, it will be noted, is obtained from the coefficients A and A , as well as 4 5 from A2 and A3 , which means that δ is overidentified. The implications of this will be discussed in the next chapter.

22

2 Demand Theory Under Review

δ=

2A3 , h(2A2 − A3 )

=

2A5 . h(2A4 − A5 )

(2.45)

Turning now to a finite approximation for the flow-adjustment model, defined over the interval t to t + h, we have [from expressions (2.18) and (2.19)]

∗ qt = θ (ˆqt − qt ),

(2.46)

qˆ t = κh + μxt + ξ pt ,

(2.47)

where qt and the other quantities are as defined in expression (2.23). Using the meanvalue theorem as before, we approximate qt − qt−h by (h/2)( ∗ qt + ∗ qt−h ),20 which after rearrangement and notational simplification yields qt = A∗0 + A∗1 qt−h + A∗2 (xt + xt−h ) + A∗3 (pt + pt−h ),

(2.48)

where 2θ κh , 2 + θh 2 − θh , A∗1 = 2 + θh 2θ μh , A∗2 = 2 + θh 2θ ξ h , A∗3 = 2 + θh 2(1 − A∗1 ) θ= , h(1 + A∗1 ) A∗0 κ= , h(1 + A∗1 ) 2A∗2 , μ= h(1 + A∗2 ) 2A∗3 ξ= . h(1 + A∗1 ) A∗0 =

(2.49) (2.50) (2.51) (2.52) (2.53) (2.54) (2.55) (2.56)

20 The logic of this procedure is that we wish to obtain a difference equation in an approximation for q˙ (t), hence the focus on qt – qt−h .

2.5

A Neuroscience Approach to Consumer Behavior

23

2.5 A Neuroscience Approach to Consumer Behavior In the mid-1980s, a young colleague of one of the authors mentioned that real progress was not going to be made in understanding consumption behavior until we better understand how the brain is organized and functions. In the last 20 years, the neurosciences (together with evolutionary biology and psychology) have made great strides in this direction, and it seems to us that sufficient is now known about the brain for one to begin putting the theory of consumer behavior in a neurobiological framework. Such is the motivation for the rest of this chapter.21 We begin with the following list of postulates and principles: (1) (2) (3) (4)

All laws of physics, chemistry, and evolutionary biology are respected. Ditto for currently established structures and functioning of the brain. Human beings are social animals. There exists a hierarchy of five Maslovian needs that are genetic in nature, namely, physiological needs, security, love, self-esteem, and selfactualization.22

As a point of departure, we take the evolutionary phenomenon that the only goal of genes is to reproduce. Reproduction requires energy, or a need to make a living, which at the most basic level is simply being able to find enough food to survive. For most animals, time is proscribed in the search for food and other activities of physical survival. However, for human beings, at least for those fortunate enough to live in countries where income is in excess of the subsistence level, things are different, for there is time beyond that needed for basic survival. The question then becomes how to occupy this surplus time. In contrast to traditional demand theory, our view is that an individual’s main preoccupation is how to spend time rather than income. Unlike income, for which most people have to work, time appears each day as a gift. The amount is fixed, and its receipt cannot be escaped. Now, here is the key to how time will be utilized: The physiology and psychology of the human organism is such that unless a certain number of neurons are firing at any time, the individual is uncomfortable. In psychological terms, arousal is a too low a level. In general, stimulation in some form is required in order to maintain arousal, and much of this occurs through the consumption of market goods and services. Since goods and services are scarce, some of the individual’s time must be spent in acquiring the income needed for their purchase. When not asleep or at work, however, the basic question facing the individual is how to allocate time amongst consumption activities in order to maintain an acceptable level of arousal. Since the time that is available is fixed, people typically run out of it, so that time itself

21

Reference must obviously be made at this point to the flourishing subfield of behavioral economics that is increasingly combining psychology and economics with the neurosciences. Camerer et al. (2005) provide an excellent recent survey. 22 After the psychologist Abraham H. Maslow (1908–1970).

24

2 Demand Theory Under Review

becomes scarce. When this happens, time must be used more efficiently. Generally, this involves trading off time for market goods in consumption activities. For most people, the primary force in reducing the scarcity of time is increased efficiency at work (i.e., an increase in the real wage rate), but consumption activities generally become more time efficient as well.23 In general, an acceptable level of arousal will involve combining novelty with redundancy, where redundancy in this context refers to the amount of familiarity in a consumption activity and can be identified with what Stigler and Becker (1977) define as consumption capital.24 Too much redundancy leads to boredom because everything has an aura of deja vu (or “familiarity breeds contempt”), while too much novelty leads to confusion and immobility because one does not know what to expect. Some surprise is highly desirable, but too much causes discomfort because of limited capacity to process new information. Redundancy is created through exposure and arises through the internal dynamics of consumption. The quest for novelty, in contrast, is seen as inhering in the psyche. It functions independently of exposure (although exposure to a new activity can remind one how pleasant novelty can be) and provides the motivation for seeking new activities. The quest for novelty can thus be seen as leading to wants being endless, which in turn continually incites the appearance of new goods.25 We now turn to a detailed development of the ideas of these paragraphs, taking as a point of departure the assumption that tastes and preferences derive, at the most fundamental level, from motivations that are products of evolution, and are therefore genetically based. In one of the classic 20th-century treatises in psychology, Motivation and Personality, Abraham Maslow (1954) identified five fundamental genetically influenced motives driving human behavior, namely, physiological needs (air, food, water, and sex); security (both physical and psychological); love (desire for affection and a sense of belonging); self-esteem (desire for a stable, firmly based, high self-evaluation, self-respect, and respect for others); and self-actualization (desire for self-fulfillment).26 Although economists are traditionally hesitant to speak in terms of needs (as opposed to wants), physical minimums of food and shelter are obviously needed for survival, and to a lesser extent, the same can also be said to be true of security. Love, self-esteem, and self-actualization needs will be put to the side for the moment. The genes dictating survival behavior in terms of physiological and security needs should, because of evolutionary pressure, be fairly invariant across humans (just as they are within animal species), in which case one ought, accordingly, to be able to identify blocks of consumption that are reasonably invariant across individuals

23

How the latter is often accomplished is described in detail, with great insight and a highly entertaining manner, by Linder (1970). 24 See also Tversky and Sattath (1979). 25 In Chapter 18, it will be argued that, in giving rise to a continuing demand for new goods, the quest for novelty is a primary driver of economic growth in nonsubsistence economies. 26 These needs will be described and discussed in detail in Section 2.7 below.

2.6

Brain Structure and Consumption Dynamics

25

and time.27 At the subsistence level, there is thus little scope for choice. However, once subsistence is crossed, new factors come into play, which generate possibilities for choice. Included in these are cultural and social factors, but perhaps most importantly, the accumulation of consumption experience. At base, what we mean by consumption experience is the idea that every activity involves the firing of a network of neurons in the brain. The first time that a consumption activity is undertaken, a network of firing neurons has to be created, which is then in place (at least partially) the next time that the activity is contemplated. Once an individual has experienced an activity, the previous exposure forever alters the lens (i.e., the tastes and preferences) through which it is viewed. The feedback of consumption on tastes and preferences, which in turn affects the desire (or a lack thereof) for future consumption, clearly represents a form of consumption dynamics. A simple representation of this situation is with a single (generally unobservable) state variable as in the original state-adjustment model described in the preceding section. As has been noted, the state variable in the state-adjustment model allows for two types of dynamical behavior, depending upon the sign of the coefficient β, which attaches to it: β< 0 has been described as representing adjustment of current consumption (or more particularly expenditure) to the inventory of the good that is held, while β>0 has been described in terms of habit formation or inertia. However, while we ordinarily think of a negative β being associated with durable goods such as automobiles or appliances, this need not be the case, for stock-adjusting behavior can also be associated with rapid short-run satiation (such as occurs with repeated hearing of the same song or musical composition). Continual firing of the same neuronal network can, in this situation, be seen as leading to discomfort, in which case a negative β can be interpreted as counseling against short-run repetition of the activity. On the other hand, a positive β for a consumption activity, in addition to indicating the presence of habit formation, can also be interpreted as signifying that repetition of the activity leads to a strengthening of the associated neuronal network, in short, a form of learning.28 The problem with these interpretations of the state-adjustment model is a surfeit of riches, for the state variable can allow for many phenomena, none of which in ordinary circumstances can be separately identified. This being the case, it will be useful to discuss a brain-based framework that allows for a much richer dynamics than is possible with the simple state-adjustment model.

2.6 Brain Structure and Consumption Dynamics29 It is now generally accepted in the neurosciences that the human brain functions modularly, in that it consists of distinct, but interconnected, centers whose functionalities overlap (at least to some extent), but which in general practice a 27

Cf., Chapter 5. Cf., Cross (1983); see also Kandel (2006). 29 Much of the material in this section first appeared in Taylor (1992). 28

26

2 Demand Theory Under Review

division of labor.30 A related view is that the Homo sapiens brain really consists of three brains: a “lizard” brain that is located at the top of the spinal column that controls autonomic processes such as breathing and heartbeat; a “monkey” brain that lies at the base of the skull and that (among other things) is capable of learning from experience; and a third-level brain that occupies most of the brain cavity and what we ordinarily think of as the human brain.31 As noted earlier, a primary motivation in this chapter is a strong belief on our part that demand analysis can benefit materially from an approach that takes its cue from the organizational structure and functioning of the brain. The purpose of this section is to show a few of these benefits by examining some implications for consumption behavior of the three-level modular brain structure just described. We are especially interested in the consumption dynamics that such a structure can imply. One might think that simple brain structures might give rise to simple dynamics, but this is not the case, for we shall see that the dynamics generated by the lizard and monkey brains can be surprisingly complex. Indeed, most of the standard dynamical consumption models, including the original state- and flow-adjustment models, can be adduced from the functionalities of the lizard and monkey brains.

2.6.1 Assumptions and Terminologies Our point of departure will be a stylized human brain that is assumed to consist of three connected levels. These three levels will be referred to as the alpha, beta, and gamma brains, with the alpha brain at the lowest level and the gamma at the highest. The assumptions concerning the functionalities of the three levels are as follows:32 Alpha Brain: The alpha brain is assumed to control the autonomic process of the body, such as breathing, heartbeat, and the standard reflexes. This brain can be viewed as monitoring the basic physical life needs of the organism in terms of minimum consumption levels of food, clothing, and shelter. Beta Brain: The beta brain, unlike the alpha brain, is capable of learning from experience of figuring out the most efficient way of achieving a particular goal. It is capable of forming likes and dislikes and acting upon these in relationships with others, and in responding to changes in environment. Many emotions are seated in this brain, including fear, love, anger, anxiety, excitement, boredom, and regret. Habits can be formed and budget constraints recognized. 30

See Ellis and Young (1988), Fodor (1983), Gazzaniga (1985, 1988, 2008), and Shallice (1988). See Fisher (1982) and Franklin (1988). 32 Organizing the brain in this manner for analyzing consumption behavior is clearly highly stylized and is obviously not the only way that it might be done. In their masterly recent survey of behavioral economics, Camerer et al. (2005), for example, adopt a two-level theoretical structure that distinguishes between automatic and controlled processes, of which each in turn is decomposed into cognitive and affective processes. Our framework is consistent with theirs and, for present purposes, simpler. 31

2.6

Brain Structure and Consumption Dynamics

27

Gamma Brain: Unlike the alpha and beta brains, the gamma brain is capable of abstract reasoning and of thinking through the implications of a particular action. It can form expectations and act upon the basis of these expectations, and can embellish or dampen actions arising out of the alpha and beta brains. The gamma brain recognizes the passage of real time and can recognize that tastes and preferences can be real-time dependent, and is capable of initiating actions to counteract the destructive consequences of certain consumption activities. It is the overall monitor of consumption activity and searches for activities that contain sufficient novelty to relieve boredom, but with enough redundancy to forestall panic.33 The gamma brain is assumed to have a modular structure, consisting of a group of interconnected centers that have some functional overlap, but which in general practice a division of labor. Information is assumed to flow in both directions among these centers. Information flow in the vertical structure, however, is assumed to be one-way. The beta brain receives information from the alpha brain, the gamma brain receives information from both the alpha and the beta brains, but the alpha brain receives no information from the beta and gamma brains, and the beta brain receives no information from the gamma brain. The brain structure being assumed necessitates several breaks from conventional demand theory, beginning with the standard assumption that the individual optimizes with respect to a single utility indicator. While this assumption could probably be retained, it would require the presence of a “headquarters” center in the gamma brain of a type that we do not presently wish to postulate, the reason for which will shortly become clear. It is assumed instead that the three brains pursue individual goals, which in most circumstances are mutually compatible (or even reinforcing). The goal at each level will be taken to be “comfort.” Comfort at the alpha level is defined in terms of motivation to maintain the autonomic processes at proper levels. For processes, such as metabolism, that require goods as inputs, discomfort is dealt with by the sending of signals to the upper brains, which in turn decide upon the inputs needed and how they are to be provided. Comfort at the beta level will be defined in the next subsection. Comfort at the gamma level is defined in terms of an “acceptable” filling of time as discussed in Section 2.6 above. As described in Section 2.6, when not asleep or working the basic question facing an individual is how to occupy one’s time, which appears each day as a gift, so as to maintain neuronal activity at an acceptable level, which amounts to allocating time among consumption activities. Such decisions are assumed to lie in the province of the gamma brain. A second break with standard demand theory, at least at the level of principle, is the separation of consumption from expenditure. The view in the present context is that the individual has two roles in the theory of demand, as a consumer and as

33

For a discussion of novelty and redundancy in this context, see Scitovsky (1976), also Tversky and Sattath (1979).

28

2 Demand Theory Under Review

a purchasing agent. Time is the constraint on consumption behavior, while income is the constraint on expenditure. An individual, acting as a consumer, decides what consumption activities to pursue, and the goods required as input are assumed to be in inventory in amounts necessary for the activities to proceed. Income becomes a constraint when the individual, acting as a purchasing agent, receives orders for the goods to be stocked. A third break with standard demand theory relates to orderings of wants and preferences. Economists since Pareto have been reluctant to think in terms of hierarchical preferences, but that some needs are of higher order than others is a clear consequence not only of common sense but of the brain structure that is being assumed. The physiological needs of the autonomic processes controlled by the alpha brain are cases in point. Later, in Sections 2.8 and 2.9, we will find it useful to assume that needs associated with certain psychological processes have a hierarchical structure as well. Still another break with standard demand theory relates to rationality. Two concepts of rationality are involved: the psychologist’s concept, which focuses on the processes of decision-making; and the economist’s concept, which focuses on outcomes and is usually identified with transitivity of preferences. The individual here is assumed to be rational in the psychologist’s sense, but not necessarily in the economist’s sense. More will be said about this later in this section. We now turn to a discussion of factors that motivate consumption behavior of the beta and gamma brains. From evidence generated in experiments involving rats and pigeons “workers,” it is clear that the beta brain is capable of organizing much of the static demand and labor-supply behavior predicted by the Slutsky–Hicks–Allen theory of demand. Animal workers recognize budget constraints and respond in predictable ways to changes in prices, income, and wage rates.34 The beta brain is also clearly capable of learning from experience. The suggestion, accordingly, is that much of observed demand behavior can be viewed as arising in the beta brain—downward-sloping demand functions, negative incomecompensated substitution effects, positively sloping supply curves of effort, and a potentially rich dynamics arising from experience and state- and flow-adjustment processes. The assumption in this section is that the consumption behavior controlled by the beta brain is in response to signaled needs. These needs are psychological in origin as well as physiological. The physiological needs arise from the autonomic process controlled by the alpha brain, while the psychological needs are assumed to arise from psychological processes that reside in the beta brain itself. In both cases, it is postulated that the needs in question are defined in terms of desired values of certain flow or state variables. As actual values of these variables depart from the values desired, the beta brain initiates actions to eliminate the discrepancies. The dynamics involved, which involve a form of bang-bang control system, will be described in the chapter.

34

See Battalio et al. (1979) and Battalio and Kagel (1985).

2.6

Brain Structure and Consumption Dynamics

29

As noted, the primary motivation for the gamma brain is assumed to be to maintain a satisfactory overall state of physiological and psychological well-being.35 Doing this requires the processing of a great deal of information that flows not only from the external environment but also from the activities controlled by the alpha and beta brains, as well as those controlled by the gamma brain itself. The highest center in the gamma brain is viewed metaphorically as an executive processor whose basic function is to keep tabs on what is going on. Following Gazzaniga (1988), this executive processor is constantly engaged in “making sense” of information that arises from both within and without. It can acquire information both through interrogation (i.e., by directing questions such as why such and such is happening) and through the receiving of distress signals. Tastes and preferences are assumed to be described in terms of a set of state variables. As has been described, included in the state variables will be concrete quantities such as stocks of durable goods and stocks of financial assets, as well as (a possibly long list of) variables of purely psychological dimensions, together with neuronal networks that have been formed from past consumption activities. State variables will also include the core beliefs and myths that in an important sense can be said to guide an individual’s behavior, not just consumption behavior, but behavior in general. The set of beliefs will embody the individual’s moral code, or sense of right and wrong, and will reflect attitudes toward self-interest, altruism, and cooperation.36

2.6.2 Consumption Dynamics Associated with the Alpha and Beta Brains The consumption dynamics that arise from actions of the alpha and beta brains are essentially of the state- and flow-adjustment variety discussed in Section 2.4. For notation, let q, as before, denote the flow (measured instantaneously) of an input into an activity, and let s denote a state variable that is associated with the activity. Finally, let S ∗ denote a desired value of the state variable.37 Comfort, then, can be defined in terms of maintaining s equal to S∗ , in which case consumption can be defined in terms of manipulating q so as to achieve this relationship. 35

Comfort for the gamma brain could equally well be defined in terms of security in a broad sense—security in emotional and economic senses, as well as security in a physical sense. 36 Experimental evidence suggests that the executive processor (or what Gazzaniga refers to as the interpreter) is physically located in the left hemisphere of the gamma brain and is associated with the so-called language facility of this hemisphere. Gazzaniga sees core beliefs as arising out of the actions of this interpreter. See Gazzaniga (1985, Chapter 11). 37 There is no presumption of S∗ being determined by an optimization process. For activities involving autonomic processes, the S∗ s are physiologically determined; for activities involving psychological processes, they can be interpreted as representing minimum acceptable levels of comfort. On the other hand, this is not to say that elimination of a divergence of s from S∗ may not be subject to an optimization process.

30

2 Demand Theory Under Review

Assume s = S∗ and consider q˙ = γ [q(t) − α − βs(t)],

(2.57)

s˙ = q(t) − δs(t),

(2.58)

where q˙ = dq/dt, etc., and α, β, γ , and δ are parameters. Expression (2.57) can be interpreted as the consumption activity initiated by one of the lower-level brains in response to a divergence between s and S∗ . The parameters in this relationship represent both the behavioral response of the controlling brain and the production technology of the activity or process. Expression (2.58), on the other hand, represents the involuntary dynamic that describes the law of motion of s. The process described by expressions (2.57) and (2.58) will be in equilibrium, i.e., comfort will be achieved–– when both s˙ and q˙ are zero, in which case we can write q∗ = α + βS∗ ,

(2.59)

q∗ = δS∗.

(2.60)

Subtracting and adding q∗ to the term in brackets in expression (2.57) and then using expression (2.59) allows expression (2.57) to be written as q˙ = γ [q(t) − q∗ − β{s(t) − S∗ }],

(2.61)

which describes how q adjusts in response to a discrepancy between s and S∗ . For autonomic processes controlled by alpha brain, S∗ can be interpreted as a physiological determined constant. In the case of nutrition, for example, S∗ can represent a desired reserve of “nutritional well-being,” while δ represents the body’s physiological demands on this reserve, and q represents the rate at which the reserve is replenished. On the behavioral side, β can be interpreted as representing the beta brain’s desired response in q to a change in the actual value of s. This response, however, is tempered by the coefficient γ . The dynamical system described by expressions (2.57) and (2.58) embodies a number of well-known dynamic demand models, including both the flow- and stateadjustment models discussed in Section 2.5. The state-adjustment model is obtained as a limiting case by dividing expression (2.57) by γ and then letting γ become large. This corresponds to instantaneous adjustment in q. The flow-adjustment model, in contrast, corresponds to the state variable s being absent for the behavioral equation, which could arise either because the state variable is irrelevant (β = 0) or because the state variable depreciates instantaneously (s= 0). Other interpretations of this model are also possible. As it stands, the model is identical with the generalization of the original H-T state-adjustment model that has been suggested by

2.6

Brain Structure and Consumption Dynamics

31

Bergstrom and Chambers (1990) (which will be the focus of Chapter 13) and it can also support an error-correction interpretation.38

2.6.3 Opponent Processes and Consumption Dynamics39 While it is reasonable to define comfort in autonomic processes in terms of physiological constants, this is not the case for psychological processes, for allowances need to be made for desired values of the state variables to evolve in response to exposure to the activities involved. To this end, we now turn to a discussion of a framework based upon psychological opponent processes. The point of departure for an opponent-process model is the observation that many feelings of pleasure or pain seem to be followed by a contrary after-effect—pain by a feeling of relief and pleasure by a feeling of let down or emptiness.40 The model consists of three components, a primary process that is initiated by a stimulus, a secondary process that is triggered by the primary process, and an integrator that sums the hedonic effects of the primary and secondary processes. Solomon and Corbit (1974) refer to the primary process as an a-process and the secondary process as a b-process. The magnitude of the a-process is postulated to be closely correlated with the intensity, quality, and duration of the stimulus. The function of the b-process is to oppose or suppress the state generated by the a-process, and is postulated to be (i) of sluggish latency, (ii) inertial, or slow to build to its peak intensity, and (iii) slow to decay after the stimulus has terminated and the a-process has stopped. Finally, the b-process is hedonically opposite to that of the a-process. To illustrate the role that opponent processes can play in describing consumption dynamics, let us consider a consumption activity that has associated with it a pair of such processes, an a-process that is initiated by the stimulus that defines the activity and a b-process that is triggered by the onset of the a-process. For convenience, it will be assumed that the a-process is hedonically positive, so that the b-process is hedonically negative. The activity will have goods and time as inputs, and the stimulus will be identified with the physical consumption of the goods. The individual is assumed to influence the a-process by controlling the timing, intensity, and duration of the stimulus. The b-process, however, is governed by its own dynamics and can be controlled only through reapplying the stimulus (or by “redosing”). When 38 See Sargan (1964), Hendry et al. (1984), Salmon (1982), and Pagan and Wichens (1989). The error-correction interpretation is as follows: In equilibrium, q will be equal to α + βS, so that q(t) − α − βS(t) represents the disequilibrium error; q is then corrected in response to this error according to the adjustment parameter γ . 39 Much of the material from this section is taken from Taylor (1987, 1989a). 40 Opponent processes in psychology were introduced by Hurvich and Jameson (1957), but have been most extensively studied and developed by Richard Solomon and his associates. See, in particular, Solomon and Corbit (1974) and Solomon (1980). Opponent processes were brought to the attention of economists by Tibor Scitovsky in The Joyless Economy (1976). An edited version of the Solomon–Corbit paper was reprinted in the 1978 Handbook of the American Economic Association.

32

2 Demand Theory Under Review

redosing occurs in order to kill the effects of a painful b-process, the individual can be said to have reached a state of addiction.41 Following Solomon and Corbit (1974), let the hedonic function at time t that is associated with the consumption activity be given by ϕ(t) = ϕa (t) + ϕb (t)),

(2.62)

where ϕa ( > 0) + ϕb ( < 0) denote the hedonic values of the a- and b-processes, respectively. For ϕ a , it will be assumed that ϕa (t) = βd(t)q(t),

(2.63)

where β is a parameter, d(t) is a stimulus indicator variable, and q(t) represents the intensity of the stimulus. A stylized illustrative shape for ϕ is given in Fig. 2.1. The solid portions of these curves correspond to d(t) = 1,q(t) > 0, while the dashed portions correspond to d(t) = 0,q(t) = 0. Since ϕa = 0 for d = 0, the dashed portions accordingly represent ϕ b when d= 0. peak of primary affective reaction adaptation phase Intensity of primary affect

steady level

0 decay of after-reaction

Intensity of affective after-reaction

peak of affective after-reaction on off

off Stimulus/time

Fig. 2.1 Opponent processes

Redosing can be taken to occur when the discomfort associated with the bprocess reaches a point where it can no longer be tolerated. The timing of the redose for an addicting activity will therefore be determined by the shape of 41

An obvious example is redosing to escape the withdrawal pains from heroin. A less-destructive example is the so-called “salted-peanut syndrome,” in which short-run mini-addictions are set up following an initial succumbing to a first handful of peanuts, popcorn, potato chips, etc. In passing, we might note that much of the snack-food industry would appear to be premised on this syndrome.

2.6

Brain Structure and Consumption Dynamics

33

the b-process. The curve in Fig. 2.1 describes an activity that nonaddicting, as the dashed curve does not reach a threshold level of discomfort. For an addicting activity such as smoking, on the other hand, the curve ϕb does reach such a threshold, which leads to a redosing in order to eliminate the discomfort. The extreme, of course, is chain smoking, which corresponds to continuous redosing. Finally, we can also consider a physiologically addicting activity such as (normal) eating. In this case, ϕ b begins slowly, eventually reaches a threshold, and then keeps on falling, reflecting the fact that endless fasting ultimately results in death. For many consumption activities, the scope and duration of the activity are more or less predetermined. Movies and football games are of a certain length, apples of a certain size, etc., and, once begun, the activity is usually pursued to its natural conclusion. With eating an apple, the apples may be small, and one may not be enough. What determines how many will be eaten? The conventional answer to this question is that apples will be consumed to the point where the marginal utility from another apple (another bite?) divided by the price of apples is equal to the marginal utility of income. But this seems to be a poor description of behavior, for most people, if given the opportunity, will eat until the taste for apples is sated.42 This suggests a natural way of defining the duration of an activity: consumption ends at the point where the marginal utility from continuing the activity falls to zero. Taking consumption activities as the object of choice thus enables satiation to be brought into the analysis in a straightforward way. It seems reasonable to assume that each opponent process has associated with it a minimum of two state variables, one for the a-process and one for the b-process.43 The state variable for the a-process can be viewed as a generalized stock of consumption capital in the sense of Stigler and Becker (1977). Intangibles will be contained in this stock, and in many cases physical capital as well. Included in the intangibles will be knowledge of the activity that has been acquired through previous exposure, or what has been referred earlier as redundancy. This stock of consumption capital, which will be denoted by A, can be assumed to be subject to depreciation. The law of motion for the state variable for the a-process can accordingly be given the same form as for the state variable in the state-adjustment model of Section 2.5 [cf. expression (2.9)]: A˙ = q(t) − δA(t),

42

(2.64)

Or until, as has been suggested to one of the authors by Barbara Sands and Gordon Winston, the marginal utility of the activity falls below that of another activity. See Winston (1980, 1985, 1987). 43 One should in fact think in terms of a minimum of three state variables, rather than two, two of which represent quales (or feelings) and the third the neural networks that are brought into existence through exposure and experience.

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2 Demand Theory Under Review

in which case, the hedonic function ϕ a in expression (2.63) can more generally be written as ϕa (t) = βd(t)A(t).

(2.65)

The law of motion for the state variable for the b-process will be more complicated than expression (2.59), for, among other things, there is evidence from laboratory studies of opponent processes that when a stimulus of medium intensity is repeated many times within a short period of time, the reaction to the stimulus tends to diminish.44 This is a habituation (or “getting used to”) effect that can be interpreted as resulting from the strengthening of the b-process. Thus, the state variable for the b-process needs to allow for its hedonic effect to strengthen with repetition. Accordingly, let us define (as one possibility) a state variable B(t) for this process as ∗

B(t) = B0 +

n 

e−σ i q[i], σ > 0,

(2.66)

i=1

where B0 denotes the strength of the b-process at the time of the first exposure to the activity, q[i] denotes the intensity of the stimulus during the ith exposure, and n∗ denotes the number of distinct exposures. Finally, σ is a parameter that allows for B(t) to reach an eventual asymptote. The hedonic function ϕ b can then be defined as ψb (t) = −S(t,τ ),

(2.67)

= −e−k(t−τ ) B(t). It is important to note that two forms of time are represented in this expression: historical (or chronological) time, which refers to consumption decision points, and process (or activity) time, which refers to the time interval over which a consumption activity occurs. The former in indexed by t and the latter by τ . Note, too, that it is assumed that B(t) is independent of τ , i.e., the state variable for the b-process is affected only by the frequency and intensity of the stimulus associated with the activity. Finally, let B∗ denote a threshold level of discomfort such that B ≥ B∗ will trigger re-application of the stimulus. From the foregoing, it is clear that actions of the beta brain in regard to opponent processes and the nature of the consumption dynamics involved will depend upon whether or not the consumption activity is addictive, i.e., whether B(t) reaches the threshold level of discomfort B∗ . Activities for which this is the case can reasonably be assumed to be re-triggered automatically by the beta brain whenever B ≥ B∗ , hence the consumption dynamics for these activities should be similar to those already discussed in connection with the autonomic processes. B∗ serves the 44

See Solomon and Corbit (1974) and Solomon (1980).

2.6

Brain Structure and Consumption Dynamics

35

same function as the desired value (S∗ ) of the state variables. However, the situation is different for nonaddicting activities, for without the automatic trigger afforded by B ≥ B∗ , the beta brain lacks the capability of initiating an activity. “Redosing” in this case is an act of choice exercised by the gamma brain. In thinking about the dynamics associated with opponent processes, it is important not to confuse consumption dynamics with the internal dynamics of the processes themselves. Consumption dynamics are external to the processes and describe how the individual responds to signals to initiate the activities involved. Consumption dynamics operate in historical time and are indexed in terms of t. Process dynamics, in contrast, are internal to the processes, operate in process time, and are indexed in terms of τ . From the foregoing, it is clear that much of the consumption associated with activities that have negative b-processes can be viewed as being motivated by a desire to eliminate discomfort. The extreme, obviously, is an addicting activity for which ϕ is never positive. This would describe an activity in which ϕ a is “beaten down” by exposure and repetition. The individual may receive no stimulation from this activity ϕ b (t), but its absence causes discomfort.45

2.6.4 Dynamics Associated with the Gamma Brain We now turn to the activities of the gamma brain, which in contrast with the alpha and beta brains are not constrained to be reactive. The primary motivation of this brain, as has been noted, is assumed to be the maintenance of an acceptable level of physiological and psychological well-being. As with the alpha and beta brains, this is assumed to be accomplished through the monitoring of a set of state variables that, among other things, embodies the moral (or ethical) code of the individual. Well-being is assumed to be acceptable when actual values of these state variables are in proximity of certain desired values. The gamma brain has capacity to choose among activities (when real time permits this to be done), can initiate activities in response to distress signals from the alpha and beta brains, and in some cases can alter decisions of those two brains.46 In addition, the gamma brain can form expectations that become inputs into consumption plans that cover both near-term and long-term horizons. The gamma brain directs the acquisition of market goods and provides for their purchase through choosing the amount of time to be spent in the pursuit of income. 45 Cf. Scitovsky (1976, Chapter 6). This type of consumption is also related to what Hawtrey (1925, pp. 189–192) once referred to as defensive consumption, as opposed to creative consumption. Also, it should be noted that the focus in the discussion has been on opponent processes in which the a-process has been positive and the b-process negative. There are also activities in which this is reversed (i.e., where the a-process is negative and the b-process positive). Skydiving and bungee jumping provide instances. Solomon and Corbit (1974) discuss other examples. 46 For a discussion of the tempering of actions triggered by the emotions by reason, see, inter alia, Chapter 2 of Damasio (1999).

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2 Demand Theory Under Review

One of the capabilities of the gamma brain is to monitor the addictive processes of the beta brain and to assess whether an addiction is “good” or “bad.” If an addictive is deemed hostile, the gamma brain has the capability of taking corrective action (though not always successfully), that is, of initiating activities that can overcome or break the addictions. A “bad” addiction in this context can be defined in terms of the effect that the addicting activity has on the state variables that describe the individual’s overall feeling of well-being. Denote the desired values of these state variables by M∗ and the actual values by M.47 Assume, next, that the elements of M are functions of the threshold levels of pain that trigger redosing of the addicting activities, that is, let M(t) = G[B∗ (t),Z(t)],

(2.68)

where B∗ is as previously defined and Z represents all of the other factors affecting M. An addicting activity (with a negative b-process), j, can be defined as bad if, for some Mi ,∂Mi /∂B∗j < 0. Such an activity will cause Mi to depart negatively from M∗ i . Eventually, the shortfall can reach a value that causes the central monitor of the gamma brain to begin searching for ways to combat the addiction. As this occurs, conflict will almost certainly emerge, for the individual is not behaving in accord with his/her moral code. The result could be ambivalence or even mental pathology.48 Latter possibilities aside, the central monitor of the gamma brain can be viewed as taking action to combat addiction whenever Mi∗ − Mi reaches a particular threshold. Such actions can vary from establishment of “anti-markets” of the type discussed by Schelling (1978), participation in support groups (Alcoholics Anonymous, Weight-Watchers, etc.), “lashing to the mast” (a la Ulysses), to sheer acts of will (i.e., quitting the addicting activity “cold turkey”). Such “stop and go” consumption behavior can give rise to dynamics—specifically, to nonlinear dynamics 49 —that are more complicated than any that have been discussed in connection with the alpha and beta brains. We now turn to the gamma brain’s capacity to anticipate and to form expectations. As has been noted, the gamma brain can monitor both its own activities and those of the alpha and beta brains. When combined with a capacity to anticipate,

The elements of M∗ can be seen as representing the core beliefs of an individual [in the sense of Gazzaniga (1985)] and are assumed as invariant to consumption activities, so that in a sense they can be viewed as the preferences of conventional demand theory. However, they are more inclusive than the preferences of conventional demand theory, in that they will embody attitudes toward love and marriage, religion, interpersonal relations, self-interest, and altruism. 48 This would be reflective of situations in which “part of me wants to do this and part of me wants to do that,” but where one action forecloses the other. In neurobiological terms, there is conflict between the cognitive and affective spheres of the brain. See Camerer et al. (2005). 49 For an analysis of such dynamics in the context of opponent processes, see Lancry and Saldanha (1978). 47

2.6

Brain Structure and Consumption Dynamics

37

this means that the gamma brain has the potential for thinking through the consequences of not only its own actions, but also the actions of the two lower brains. If the consequences of a particular act are judged to be harmful, the decision to act to be revised or a consumption plan could be formulated that contains or undoes the harm.50 In formulating consumption plans, the gamma brain is assumed to have the capacity to form expectations regarding the future course of prices and income, and also to anticipate the wants and needs that relate to different stages of the life cycle. Expectations in these cases are with reference to calendar time rather than to process time and are more relevant to purchase decisions than to consumption decisions. A variety of expectations mechanisms could be considered at this point, including naive, extrapolative, adaptive, Bayesian, or rational. Of these, it would seem that rational expectation is the only one that requires the capability that has been ascribed to the gamma brain. Expectations for all of the other mechanisms can be formulated on the basis of current and past information, for which the potential for doing can be ascribed to the beta brain. Rational expectations, however, would seem to require abstract reasoning that only the gamma brain is capable of.

2.6.5 From Consumption to Expenditure The dynamics that have been considered to this point in this section relate to the consumption side of the consumption/expenditure dichotomy that was postulated earlier. This dichotomy has been characterized in terms of the individual wearing two “hats,” one as a consumer whose motivation is primarily to maintain an acceptable level of physiological and psychological well-being through time via consumption activities that use time and market goods as inputs, and the other as a purchasing agent whose motivation is primarily to obtain the market goods required in the most efficient way possible given the resources that are available. The next task is to examine the expenditure side of this dichotomy. The bridge between consumption and expenditure obviously has to occur in the gamma brain, where one can imagine that consumption needs are transmitted to the “purchasing agent” who then goes out into the markets to shop. Dynamics on the expenditure side arise from two different sources. The first source is the consumption dynamics that we have been discussing, that is, the dynamics that inhere in the consumption activities that are undertaken in efforts to maintain physiological and psychological well-being at acceptable levels. State variables change over time in response to exposure, experience, and depreciation, and this leads to timedependent changes in the amounts and types of market goods that are required by consumption activities as inputs. This is the type of dynamics associated with inertia, habit formation, and stock adjustment.

50 For a model endogenous taste change that allows for considerations such as this, see Manove (1973).

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2 Demand Theory Under Review

The second source of dynamics on the consumption side is described by how the individual, functioning as a purchasing agent, responds to changes in the external environment. The behavior reflected in these dynamics will include delays in adjusting changes in prices and wage rates, the appearance of new goods, and changes in the type and intensity of marketing efforts. Response delays can arise for a variety of reasons, including recognition lags and other imperfections in information flows, uncertainty, and certain intrinsic characteristics of the individual. Also, expectations can contribute to this category of dynamics if they are formulated on the basis of past values of income, prices, and interest rates. Complicated expectations aside, the dynamics represented in expressions (2.57) and (2.58) are appropriate for goods that involve the holding of physical inventories. In this case, equation (2.57) should be augmented by price and income, so that we would have q˙ (t) = [q(t) − α − βx(t) − γ x(t) − λp(t)]

(2.69)

s˙(t) = q(t) − δs(t),

(2.70)

where x and p denote the flow rate of income and level of price and s denotes the level of inventory. The coefficient δ can now be seen as representing the using up of the good in question as input into consumption activities. The coefficients α, β, γ ,and λ reflect intrinsic behavioral characteristics of the individual (as purchasing agent), while Φ can be viewed as incorporating delays and adjustments resulting from imperfections in information flows. The bridge between the consumption and expenditure sides in the gamma brain is represented in the using-up parameter δ. Strictly speaking, δ should be written as a function of time, for in substantial part it should be seen as psychologically based, and thus reflective of the “endogenous” taste changes occasioned by the internal dynamics of consumption activities, together with the gamma brain’s efforts to maintain an acceptable level of psychological well-being. Closing the system would accordingly require the specification of a law of motion for δ. Expressions (2.69) and (2.70) can also be used to describe the expenditure behavior of activities, such as concerts, theater, and sporting events, which are consumed at the time of purchase. Inventories will continue to be relevant for such activities, except that reference must now be to psychological quantities, rather than to physical stocks. For concerts, for example, one can imagine a stock of “music enjoyment” that periodically needs replenishing, and similarly for theater and sporting events. Moreover, many services, such as travel and telecommunications, are inputs into consumption activities that have conceptually well-defined state variables associated with them, which in some instances can be directly identified with the psychological quantities in question. Expectations clearly complicate. One way that they can be brought into the picture is rewrite equation (2.69) as: q˙ = [q(t) − α − βs(t) − γ xe (t) − λpe (t)],

(2.71)

2.6

Brain Structure and Consumption Dynamics

39

where xe and pe denote the purchasing agent’s expectations for income and price. To illustrate the complications that can be introduced, assume that the formation mechanisms for xe and pe are as follows: x˙ e (t) = ψ[x(t) − xe (t)]

(2.72)

t x (t) = e

w(τ )x(τ )dτ

(2.73)

p˙ (t) = θ [p(t) − pe (t)]

(2.74)

t− e

t p (t) = e

w(τ )p(τ )dτ ,

(2.75)

t−

where < 0,w(τ ) ≥ 0, and



w(τ )dτ = 1.

These expressions posit that expectations are formulated as a weighted average of past values of x and p, with a partial adjustment to new information. Differentiation of expression (2.71) with respect to time, and then using expressions (2.72) and (2.74) for˙xe and p˙ e yields: q¨ (t) = [˙q(t) − β˙s(t) − γ {x(t) − xe (t)} − λ{p(t) − pe (t)}],

(2.76)

which is a second-order differential equation in q.51 As the purchasing agent’s expenditure behavior at any point in time is obviously constrained by the income that is available, a final task is to incorporate the budget constraint into the analysis. For notation, let y denote income, q a vector of consumption activities, and x a vector of market goods and services that [per Becker (1965) and Lancaster (1971)] are inputs into these activities. Let p denote the vector of associated prices, and finally, for those goods for which physical inventories are relevant, let Xand X∗ denote vectors of actual and desired levels of inventories. Specifically, X∗ is assumed to represent a “safe” level of inventories (paper towels, milk, soap, etc.) over some relevant consumption horizon. In particular, let X∗ be defined by:

51 The point of the foregoing exercise is simply to show that the capacity to form expectations has material implications for expenditure dynamics. As simple adaptive expectations lead to a secondorder process, one can imagine (without getting caught up in details) the complexity that more sophisticated expectation mechanisms would entail. As has been noted, the dynamics represented in expressions (2.69) and (2.70) can be associated with decisions of the beta brain, as can also the formation of simple adaptive expectations. Dynamics arising out of more complex expectation mechanisms, however, must be attributed to the gamma brain.

40

2 Demand Theory Under Review

∗

t

X (t) ≥

x(τ )dτ ,

(2.77)

t−

where represents the consumption horizon, and x(τ ) is determined by the activity “production” functions, q(τ ) = f [x(τ )]. Next, let x(t) be defined by: X ∗ (t) − X(t), X ∗ (t) > X(t)) x(t) = 0 otherwise.

(2.78)

Finally, let it be assumed that the purchasing agent acquires x(t) at minimum cost subject to the income constraint:52 p (t)x(t) ≤ y(t).

(2.79)

2.6.6 Consumption/Income Relationships A clear implication of the three-brain structure of consumption behavior being discussed here is that activities controlled by the alpha and beta brains have first claim on income, followed by the needs of addicting psychological processes. It is only after the needs of these two categories have been satisfied that needs associated with boredom-relieving activities can assert claims. We can accordingly conclude that goods that are primary inputs into activities controlled by the alpha and beta brains will have smaller income elasticities than goods that are primarily inputs into activities controlled by the gamma brain. In terms of the traditional grouping of goods into luxuries and necessities, goods that inputs into activities associated with the alpha and beta brains will tend to be necessities, while those that are inputs into activities associated with the gamma brain will tend to be luxuries.53 An obvious empirical implication of the foregoing is that we should expect to observe greater variation in consumption patterns among individuals with high incomes than with low incomes. Consumption will be more proscribed the closer it is associated with activities that involve elimination of physiological and psychological discomfort, activities that in general will be controlled by the alpha and beta 52 Since saving is not being taken into account, the income constraint will normally be satisfied as an inequality. Because of possibilities to dissave or borrow, there is actually no necessity for any particular point in time that the income constraint be satisfied. Persistent deficits could not occur, however, not only because of the obvious threat of bankruptcy, but also because solvency for most individuals is an important part of their moral code (as defined by M∗ in the present section). Once the gamma brain recognizes that solvency is threatened, actions would be initiated to avert it. 53 Consumption decisions initiated by the alpha, beta, and gamma brains will be related to the Maslovian hierarchy of wants in Section 2.7 below.

2.6

Brain Structure and Consumption Dynamics

41

brains. One would expect these activities to have greater uniformity across individuals than activities controlled by the gamma brain, which allows for much greater expression of individual tastes. High income, in short, provides much greater scope for individual tastes to manifest themselves.54 Another implication of this analysis is that, in general, consumption behavior should be asymmetrical with respect to income increases and income decreases. Since goods that have first claim on income are those that are associated with eliminating physiological and psychological discomfort, expenditures for these goods will almost certainly not fall proportionately with any fall in income. For increases in income, on the other hand, novelty and new wants can be pursued, and expenditures will increase. While increases in expenditures may again be less than proportionate to the increase in income, the increases will almost certainly be larger (in absolute value) than any decreases would be for a likely fall in income. The likely result, accordingly, is a “rachet” in expenditures for changes from the current level of income.55

2.6.7 Rationality Earlier, it was noted that, in considering the framework of this section, it is important to distinguish between the economist’s concept of rationality and that of the psychologists. To repeat what was said earlier, the focus for psychologists is with the process (or processes) of decision-making, while for economists the focus is on the outcomes of decisions.56 In conventional demand theory, rationality is usually identified with the transitivity of preferences: If A is preferred to B, and B is preferred to C, then A is preferred to C. In general, the three-brain model that has been discussed in this section would appear to be consistent with the first concept of rationality, but not with the second. In its role as the ultimate arbiter of consumption decisions, the gamma brain takes into account both current information and consequences of past activities and forms expectations concerning the effects of current actions. Cognitive processes are many, and there is no reason to think that these

54

Moreover, such is consistent with the heteroscedasticity that is often observed in the residuals of Engel curves. 55 This paragraph assumes that the income level from which changes are measured is one to which the individual has more or less adjusted. The effects described accordingly refer to shortrun responses. Long-run responses may be more symmetrical to income increases and decreases, although it is doubtful because of experience and state-variable creation that consumption behavior is ever fully reversible, as is the case in conventional demand theory. Thus, at the level of the individual, the short-run dynamics of this section are consistent with the Relative- and Permanent-Income Hypotheses [Duesenberry (1949), Friedman (1957)] and also the Life-Cycle Model [Modigliani and Brumberg (1954, 1980)]. This is not necessarily the case for the long run, however, because of irreversibility. 56 See Simon (1986). For a discussion of how the concept of rationality has evolved in economics, see Arrow (1986).

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(in normal circumstances) are not coordinated in a compatible fashion by its central monitor to provide an acceptable level of comfort and well-being. Rationality in the second (or economists’) sense, however, is another matter. As a way of analyzing this question, let us imagine a conceptual experiment (as per Section 2.2) in which from a given set of initial conditions a short segment of real time is repeated. Rationality in the second sense would require that an individual make identical consumption decisions. However, unless all consumption activity is controlled by the alpha and beta brains, there is no reason for this to be the case. This is because, for consumption activities that do not involve the elimination of discomfort, the particular consumption activities that are engaged over any interval would seem to be inherently random as a consequence of the random firing of neurons. Put another way, there is no reason to think the moods (or dispositions) that drive the selection activities to stave off boredom would be repeated. Rationality in the second sense would therefore seem to be an event with probability zero. The foregoing is with reference to the individual as a consumer. A different conclusion can emerge with regard to expenditure behavior. The conceptual experiment to be imagined here is the usual one in which income is held constant and the individual is confronted with a sequence of vectors of relative prices that at some point begins to repeat itself. The question is then whether the expenditure pattern would also begin to repeat. If this were the case, the demand functions would be integrable, or equivalently the expenditure pattern would be consistent with the strong axiom of revealed preference.57 In either interpretation, rationality in the second sense would be present. Whether or not this would be the case turns on what besides income is assumed to be held constant. If everything on the consumption side were held constant, there would seem no reason why rationality in the second sense should not hold. The question, however, is whether everything on the consumption side can meaningfully be held constant. The problem is that the conceptual experiment envisioned implies a passage of time, and this could, among other things, affect expectations that are formed by the gamma brain, which in turn could lead to nonrepeating expenditure patterns, and hence to intransitivities.

2.7 The Maslovian Needs Hierarchy Over the course of a long and distinguished career, the psychologist Abraham Maslow put forth a compelling and influential argument that human behavior, at the most fundamental level, is motivated by five basic needs (listed in Section 2.5 above): physiological, security, love, self-esteem, and self-actualization. With the three-brain framework of the preceding as background, these will now be described.

57

See Houthakker (1950) and Chapter 22 below.

2.7

The Maslovian Needs Hierarchy

43

2.7.1 Physiological Needs Physiological needs are the needs that must be fulfilled in order for the body to survive, which, for our purposes will be taken to include, food, drink, basic shelter, and sex. Obviously, the most basic of these are the inputs into the autonomic processes controlled by the brain stem (i.e., the alpha brain) that are required to maintain the bodily functions in homeostasis, such as breathing, digestion, and appropriate chemical balances of the blood stream. As Maslow notes, the physiological needs are the most prepotent of all needs, in that for the human being who is missing everything in life in an extreme fashion, it is most likely that the major motivation would be the physiological needs rather than any others. A person lacking food, safety, love, and esteem will almost certainly hunger for food more strongly than for anything else. However, in non-third-world countries, chronic extreme hunger of the emergency type is rare. In most circumstances, people are experiencing appetite rather than hunger when they say, “I am hungry.” They are apt to experience sheer life-anddeath hunger only by accident, and then only a few times in their lives. And even when they say they are hungry, people may in fact be seeking more for comfort or dependence than for carbohydrates and protein.

2.7.2 Security Needs Once the physiological needs are satisfied, a new set of needs emerges—security; stability; dependency; protection; freedom from fear, anxiety, and chaos; need for structure, order, law, and limits; strength in the protector; and so on—which Maslow characterizes as safety needs. Once these needs come into play, all that has been said of the physiological needs is equally true of them, although in a lesser degree. When unsatisfied, these desires may serve as the almost exclusive organizer of behavior, and, as in the hungry human, the dominating goal is a strong determinant not only of a person’s current world outlook and philosophy, but also of the future and of values. Practically everything looks less important than safety and protection, and even the physiological needs, which being satisfied, may be underestimated. Indeed, as Maslow states: The safety needs can become very urgent on the social scene whenever there are real threats to law, to order, to the authority of society. The threat of chaos or of nihilism can be expected in most human beings to produce a regression from any higher needs to the more prepotent safety needs. A common, almost an expectable reaction, is the easier acceptance of dictatorship or of military rule. This tends to be true for all human beings, including healthy ones, since they too will tend to respond to danger with realistic regression to the safety need level and will prepare to defend themselves. But it seems to be most true of people who are living near the safety line. They are particularly disturbed by threats to authority, to legality, and to the representatives of the law. [Maslow (1970, p. 20; italics added.)]

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2 Demand Theory Under Review

2.7.3 Community and Affection (Love) Needs Once the physiological and safety needs are satisfied, the needs of human beings as social animals make an appearance, beginning with the needs for affection and community. In Maslow’s words:58 If both physiological and the safety needs are fairly well gratified, there will emerge the love and affection and belongingness needs, and the whole cycle already described will repeat itself with this new center. The love needs involve giving and receiving affection. When they are unsatisfied, a person will feel keenly the absence of friends, mate, or children. Such a person will hunger for relations with people in general—for a place in the group or family—and will strive with great intensity to achieve this goal. Attaining such a place will matter more than anything else in the world and he or she may even forget that once, when hunger was foremost, love seemed unreal, unnecessary, and unimportant. Now the pangs of loneliness, ostracism, rejection, friendlessness, and rootlessness are preeminent. We have very little scientific information about the belongingness need, although this is a common theme in novels, autobiographies, poems, and plays and also in the newer sociological literature.59 From these we know in a general way the destructive effects on children of moving too often; of disorientation; of the general over-mobility that is forced by industrialization; of being without roots, home and family, friends, and neighbors; of being a transient or a newcomer rather than a native. We still underplay the deep importance of the neighborhood, of one’s territory, of one’s clan, of one’s “kind,” one’s class, one’s gang, one’s familiar working colleagues. And we have largely forgotten our deep animal tendencies to herd, to flock, to join, to belong. [Maslow (1970, p. 20)]

2.7.4 Esteem Needs Upon gratification of the needs for affection and belonging, there arise a need or desire in people for a stable, firmly based, usually high evaluation of themselves, for self-respect and the esteem of others. For Maslow, these needs take two forms. 58

A cogent statement of the genetic basis of these needs is also supplied by Polyani (1974, pp. 209–210): The sentiments of trust and the persuasive passions by which the transmission of our articulate heritage is kept flowing, bring us back once more to the primitive sentiments of fellowship that exist previous to articulation among all groups of men and even among animals. Evidence of the primordial character of such conviviality and of the lively emotions engendered and gratified by its interplay is supplied by the experience both of animals and men. [. . .] Companionship among men is often sustained and enjoyed in silence. Mr. Utterson, in Stevenson’s Dr. Jekyll and Mr. Hyde, puts aside any business, however unimportant, to take his regular walk with his friend Mr. Richard Enfield, during which neither of them pronounces a single word. But conviviality is usually made effective by a more deliberate sharing experience, and most commonly by conversation. The exchange of greetings and conventional remarks is an articulation of companionship, and every articulate address of one person to another makes some contribution to their conviviality, in the sense of their reaching out to one another and sharing each other’s lives. Pure conviviality, that is, the cultivation of good fellowship, predominates in many acts of communication; indeed, the main reason for which people talk to one another is desire for company. The torment of solitary confinement is that it deprives one not of information but of conversation, however uninformative. 59 It should be kept in mind that Maslow first wrote these words in 1954.

2.8

Some Implications of a Hierarchy of Needs

45

The first is the desire for strength, achievement, adequacy, mastery and competence, positive self-image, confidence in the face of the world, and independence and freedom, while the second is the desire for reputation or prestige (defined as respect or esteem from other people), status, fame, and glory, dominance, recognition, attention, importance, dignity, or appreciation. As will be discussed below, much of consumption expenditure in high-income economies would seem to be motivated by this set of needs. Expenditures for “position goods” and on Veblenesque “conspicuous consumption” seem prime examples.

2.7.5 Self-Actualization Need Once physiological, security, belonging, and self-esteem needs are gratified, the fifth and final basic motivation in the Maslow hierarchy emerges, namely, selfactualization. For Maslow: Even if all these [i.e., the first four] needs are satisfied, we may still often (if not always) expect that a new discontent and restlessness will soon develop, unless the individual is doing what he or she, individually, is fitted for. Musicians must make music, artists must paint, poets must write if they are to be ultimately at peace with themselves. What can be, they must be. The must be true to their own nature. This need we may call self-actualization. [Maslow (1970, p. 22; italics in original)]

Self-actualization refers to people’s desire for self-fulfillment, that is, a tendency for them to become in fact what they are potentially, “to become everything that one is capable of becoming.” The specific form that these needs take can vary greatly from person to person. In one person, they may take the form of the desire to be an excellent parent, in another they may be expressed athletically, and in still another they may be expressed in painting pictures or invention. However, the common feature of the needs for self-actualization is that their emergence usually rests upon prior satisfaction of the physiological, safety, love, and esteem needs.60

2.8 Some Implications of a Hierarchy of Needs As just presented, the five Maslovian needs form a rigid hierarchy, in which higher needs do not come into play until lower ones are satisfied. As a framework for organizing real-world consumption behavior, such an interpretation of Maslow’s hierarchy is obviously too strict. Short of extreme physiological or security deprivation, an individual’s consumption behavior at any point in time is probably best seen

60

In the first edition of Motivation and Personality, Maslow (1954) discussed the fifth motivation, self-actualization, as being independent of age. However, subsequent empirical research has pretty much established its emergence only in individuals once they have reached their fifties, which (given the nature of self-actualization) of course makes intuitive sense.

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as being determined by whatever desire, at that moment, is most in need of gratification. Sometimes, this might be expenditures related to a physiological need, or it might be installation of a security device to thwart potential burglary, or it might be the purchase of artist supplies in pursuit of a long-term desire to paint. Indeed, for the very wealthy, it might be the purchase of a plot in a prestigious locale for the purpose of building a house that is larger than any of the individual’s friends or acquaintances. In short, for normal, well-adjusted individuals, whose basic physical and physiological needs are in “copacetic” equilibrium, the desire most in need of gratification at any point in time may fall into any of the higher-order needs, which is to say that probably the best that a forecaster can do is to make a probabilistic statement as to the basic need category that an about-to-be-undertaken consumption activity will fall into. However, over longer periods of time, the need hierarchy should be quite predictive of consumption behavior, specifically in determining the proportions of total expenditure that fall into each of the hierarchical categories. Related to the foregoing is the fact that a characteristic of the human organism when it is dominated by a certain need is that the whole philosophy of the future tends to change. For a chronically and extremely hungry person, for example, Utopia can be defined simply as a place where there is plenty of food. He or she may tend to think that, if only guaranteed food for the rest of life, they will be perfectly happy and will never want anything more (which of course dissipates once the hunger need is sated). Similar feelings can arise when there is a breakdown in security or (for the pathologically ill) when at the extremes of not belonging or loss of self-esteem. In these circumstances, tastes and preferences, as economists ordinarily think of them, can be extremely unstable, so ephemeral, in fact, that it is probably not meaningful even to speak of preference stability at the point in time. Stability in these circumstances is once again probably only meaningful over intervals of time.61 As with needs (as opposed to wants), mainstream economic theory has been reluctant to consider tastes and preferences as lexicographical.62 For with lexicographical preferences, there are no indifference classes, and accordingly no straightforward way, as in the Hicks–Allen framework, of deriving demand functions with income and substitution effects. However, a hierarchy of needs clearly implies a hierarchy of preferences. Consequently, developing a theory of consumer

61

Cf., Chapter 13 below, where a generalization of the dynamic models of Section 13.5 is presented. 62 Henceforth, needs and wants, for our purposes, will be used interchangeably. It may be of interest to note in passing that LDT once submitted a paper (co-authored with a graduate student) based upon lexicographical preferences to one of the leading economics journals. Rejection was almost immediate, with a statement (and stern admonishment!) from the editor (who had once been one of the author’s professors) that only a drunken sailor could conceivably display such preferences, and to never stray from the fold in this way again! Given the advances in evolutionary biology and the neurosciences, and emergence of journals in evolutionary and behavioral economics, such an attitude is hopefully no longer endemic.

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Some Implications of a Hierarchy of Needs

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behavior that allows for a Maslovian need hierarchy not only has to deal with preference instability, but hierarchical preferences as well.63 While Maslow is clear in stating that his five motivating needs are genetic in origin, he is equally clear that they are not the only determinants of behavior. Cultural and social factors are also important motivators of desires, and in normal circumstances, i.e., when individuals are in (what was referred above as) copacetic equilibrium with respect to physiological and security needs, are probably more important than their genetic counterparts. This being the case, one way that we might approach integration of Maslow’s hierarchy of needs into a theory of consumer behavior is to view preferences as consisting of the five Maslovian needs, each of which in turn consists of three functional components: genetic, cultural, and social. From the discussion to this point, it should be clear that the genetic component will be fundamental in all of the needs, but that its strength as a motivator of observed behavior might be expected to be inversely related to the order of the needs, i.e., strongest for the physiological and security needs and weakest for selfesteem and self-actualization. By fundamental in this context, we simply mean that the need, qua need, is genetic in origin.64 On the other hand, how the need is gratified can be culturally/socially determined, and the more so the higher is the order of the need. Also, at levels of income in which all base needs are satisfied, needs within needs can be created that are both hierarchically structured and culturally/socially based. By culture in this context, we have in mind two concepts, one broad and the other narrow. Culture in the broad sense refers to the institutions, broadly construed, that human beings have developed in order to survive and coexist as social animals. Genetics provide the “hardware” for this to occur, while culture provides the “software,” software that the culture itself transmits from one generation to the next. Language provides a cogent example. Genetics provide both the need and possibility for language, but the culture that an individual is born into determines the particular language that is learned. On the other hand, culture in the narrow sense is as we ordinarily think of the term, religion, education, civic, literature, theater, music, art, etc. Most of the time in this study, when we refer to needs that are culturally based, the narrow concept will be the one that is in mind, that is, needs that, as noted above, are really needs within needs. The actual form that these needs will take, however, will vary from culture (in the broad sense) to culture.

63

Early efforts to develop a theory of consumer choice based upon hierarchical preferences include Georgescu-Roegen (1954) and Ironmonger (1972). Both studies still reward reading, but especially that of Georgescu, who, among other things, presents an elegant treatment of probabilistic preferences. See also Encarnacion (1987). Another approach to probabilistic preferences, although in a discrete-choice context, is the random-utility models of McFadden and his associates [Domencich and McFadden (1975), Ben-Akiva and Lerman (1985), and Train (1986)]. 64 The genetic components arise from evolution, and thus are intrinsic to the organism, while the cultural and social components arise from without, and accordingly are extrinsic to the organism.

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Some conclusions and implications of the foregoing are as follows: (1) Tastes and preferences, as economists ordinarily think of them, have structure only when consumption behavior is measured over intervals of time. At points in time, consumption behavior is, at best, probabilistic. (2) Because of genetic variability and diversity in life experiences, no two individuals will ever have identical preference structures, even when consumption behavior is measured over intervals of time. Consequently, stable preferences can only meaningfully be spoken of only in connection with ensembles of individuals. (3) Stability in this context is to be taken in terms of stability of distributions across ensembles of consumers both in and across time. The order of the five Maslovian needs is genetically fixed across individuals, but their relative strengths are not. However, we should expect the distributions describing relative strengths to be stable across ensembles of individuals both at same points in time as well as over time. That is, the distributions of need strengths should, in general, be no different for individuals in a village in Papua New Guinea than for a similar-sized ensemble of individuals in the U.S., and the same for ensembles of individuals over time. In contrast, the distributions of cultural and social determinants of consumption will differ both across peoples as well as over time for people in the same culture, however in ways that, in general, should be (at least partially) measurable.65 (4) At any point in time, consumption decisions will be determined by the lowestorder want that is not already satisfied. (5) An empirical regularity that is taken to be one of the few genuine laws of economics is Engel’s Law for food expenditures,66 which states that the income elasticity of demand for food is positive, but less than 1. The needs hierarchy provides a rationalization for this phenomenon. For food is a physiological necessity literally at the very beginning of the need hierarchy, hence once its need is satisfied, it no longer figures in consumption behavior.67 As just noted, income elasticities with hierarchical preferences continue to have a straightforward interpretation. Price elasticities, however, appear to be another 65

See Chapter 5 below. After Ernst Engel (1857); see also Houthakker (1957). 67 The reference here is obviously to food at the survival level. However, much of food consumption is, as noted by Maslow, motivated by “appetites” (rather than by necessity), and is often combined with activities that serve needs other than physiological. A dinner party, for example, is an extraordinarily complicated consumption activity that is motivated by a wide variety of needs. As has been noted by Dennis Weisman in private correspondence, a slice of bread consumed at a dinner party will in general have a much different marginal utility than the same need for survival. Housing expenditures provide another complex example. As with food, expenditures for basic shelter ought to have an income elasticity that, again, is positive but less than 1. However, because housing expenditures frequently are motivated in terms of position goods [see Hirsch (1976) and Frank (1985)] and Veblenesque conspicuous consumption, which in general are associated with high-order needs, their income elasticity, depending upon how expenditures are measured, can (as we shall see) be well in excess of 1. 66

2.9

Toward Empirical Application

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matter, at least in terms of price effects as usually defined in terms of income and substitution effects. For, with indifference not in the picture, it is not clear how pure substitution effects might be measured. However, while indifference clearly has no meaning between need categories, it nevertheless might still be possible for indifference to arise within a given need category. Consider, for instance, an individual with a semi-chronic deficit in belonging and self-esteem needs who is able periodically to alleviate their symptoms through hosting of dinner parties for close friends and acquaintances. Now, dinner parties have many inputs—food, wine, candles, music, etc.—of which many combinations can lead to the same consumption end. In this situation, it would be perfectly reasonable for the individual, first, to specify a total budget for the party, and, then, to select the inputs in a way that gives rise to the maximum “satisfaction” for that expenditure.68 In this situation, different relative prices can lead to different combinations of purchases, in which case one should be able (in principle, anyway) to calculate pure price substitution effects.69

2.9 Toward Empirical Application Unfortunately, empirical implementation of a model incorporating a consumption hierarchy is hardly clean and straightforward. It would be one thing if consumption data currently available were to correspond to gratification of specific needs, but even in an ideal world of data collection, this would not be the case. For, as has been noted, most consumption activities serves a multiplicity of needs, some basic, some higher-order within a given need, and some multiple across needs. Accordingly, what our approach is going to be at this stage is to attempt to classify categories of expenditure from the BLS consumer expenditure surveys (and later from the National Income and Product Accounts) accordingly to the basic need categories that they serve. In this section, we will illustrate doing this with a six-category classification that has been devised for integrating price data collected by ACCRA with the CES expenditure data. The categories of consumption expenditure included in the six-category classification are: food consumed at home, shelter, utilities, transportation, health care,

68

This example obviously conflates a number of topics in demand theory, including household production functions (Becker, 1965; Lancaster, 1971), separability (Strotz, 1957), and two-stage budgeting (Blackorby et al.,1978); Pollak and Wales, 1992). A utility function appears to have been slipped in as well. However, none of these present problems for the framework that, bit-by-bit, is being developed here. First, given the postulate of a hierarchy of needs for motivating consumption behavior, the assumption of household production functions (i.e., consumption activities) that generate the means for gratifying desires seems almost mandatory. And the same is true for separability and two-stage budgeting. As for our slipping in a “satisfaction” function, a hierarchy of needs does not rule out utility functions, just the assumption of a single-valued function. Finally, it should be noted that, as mentioned earlier, a dinner party is a complex consumption activity that typically cuts across several basic need categories, in which case it has to be assumed that a non-varying mix of need categories is being tended to. 69 We will return to this point in several chapters below, wherein price elasticities using a number of different demand models are estimated.

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2 Demand Theory Under Review Maslovian Needs

Physiological

Food Consumed at Home

Security

Community and Affection

Shelter

Utilities

Self-Esteem

Transportation

Health Care

Self Actualization

Miscellaneous

Expenditures

Fig. 2.2 A cross-classification of six CES expenditure categories according to the five basic Maslovian needs

and miscellaneous expenditures. An initial effort to cross-classify expenditures from these categories with the five Maslovian needs is given in Fig. 2.2. Arrows in this figure run from needs to expenditure categories, indicating that expenditures are motivated in order to gratify needs. Width of the arrow shafts represents our (obviously subjective) attempt to specify the relative importance of individual needs for each category of expenditure. For physiological needs, arrows are shown as running to all categories of expenditure, but with food and shelter obviously being of most importance. Arrows from the security need run to shelter, utilities, health care, and miscellaneous expenditures. Out of ignorance on our part, the arrow shafts for this need are all made the same width. Arrows for the love/belonging need run to all expenditure categories except food, while for the self-esteem need, all categories are represented. The arrows for these needs are again depicted width the same thickness. Finally, the arrows for the self-actualization need, which again are depicted to run to all categories of expenditure, are assumed to be slightly thicker for shelter, transportation, and miscellaneous expenditures. Discussion and use of the schema depicted in this figure will be put off until Chapters 11 and 18.

2.10 Emotions and Consumption Behavior70 In this section, we shall consider the role that emotions might play in consumer behavior. The traditional economist’s view of the rational consumer is that they have no role at all, that choice arises out of purely rational utility calculations. Such a

70

While this section is brief, it is based upon extensive reading and study of the following body of research: Carter (1998, 2002), Churchland (2002), Damasio (1994, 1999, 2003), Dawkins (1976, 1986), Dennett (1991, 1995), Edelman (1989, 2004, 2006), Edelman and Toroni (2000), Franklin (1987), Gazzaniga (1985, 1988, 1992, 1998, 2008), Goldberg (2001), Goleman (1994), Greenfield (1997), Kandel (2006), Koch (2004), Le Doux (1996), Pinker (1994, 1997, 2002), Ridley (2001), and Rosenblum and Kuttner (2006), and Zeki (1999).

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Emotions and Consumption Behavior

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view, though, begs a number of important questions, such as, what informs and constitutes “utility,” and is also in conflict with an increasing body of neural research.71 Where traditional economics errors, in our opinion, is the tendency to view emotions in a humanist tradition as consisting purely of feelings, instead of a complex set of primal nervous systems whose evolutionary role has been to protect against harm in hostile environments. Take fear, for example. As put by Plutchik (1980): “Although a deer may run from danger, a bird may fly from it, and a fish may swim from it, there is a functional equivalence to all the different patterns of behavior; namely, they all have the common function of separating an organism from a threat to its survival.”72 What we associate with fear in human behavior is the feeling of being afraid. However, as posed by William James more than a hundred years ago: Do we run from a bear because we are afraid, or are we afraid because we run? James (1884) averred the former, but his answer, as it has turned out, was wrong. For the feeling of being afraid (at least in humans) is a conscious phenomena, and arises after a subconscious decision as to the appropriate survival response has already been made.73 For present purposes, our view regarding the role of emotions (which is now pretty standard in neural research) will be: (1) All emotions derive from primitive structures whose evolutionary roles have been to keep organisms away from harm. (2) Although, emotions are functions involved in survival, different emotions are involved with different survival functions: defending against danger, finding food and mates, caring for offspring, and so on, each of which may involve different brain systems. As a result, there is, in general, not a single emotional system in the brain, but many.74 (3) Signals triggering emotions enter the brain at the thalamus, and then divide along two pathways, one (via the amygdala) involving immediate response at the most primitive brain level (but which nevertheless may entail appraisal as to an appropriate response), and the second involving higher brain levels, which may ultimately give rise to feelings such as fear, joy, anger, etc.75 Since emotions, at the most base level, are responsible for survival, and since survival requires consumption, much of consumption behavior will accordingly be determined by the base emotions. This being the case, it seems straightforward to 71 See, in particular, the conclusions of Damasio (1994) arising from research on patients with damaged frontal lobes. Specifically, Damasio and his colleagues show that individuals with minimal cognitive, but major affective, deficits have difficulty making decisions, and often make poor decisions when they do. As noted by Camerar, Lowenstein, and Prelec (2005, p. 29): it is not enough to know what should be done; it is also necessary to feel it. Emotions accordingly can be viewed as what imbue tastes and preferences with both definition and “sharpness.” 72 As quoted by Le Doux (1996, p. 122). 73 See Chapters 3–5 of Le Doux (1996). 74 See Le Doux (1996, Chapter 4). 75 See Goleman (1994, Chapter 2).

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identify the subconscious structures associated with the base emotions with a good part of “tastes and preferences” as conventionally understood, and, more specifically, to identify the consumption involved with the most basic of the Maslovian wants. Higher-order wants, on the other hand, and the consumption that these give rise to, can then be seen as being associated with the “feelings” that are more traditionally associated with emotions such as love, joy, and happiness. Consumption of the former type will be governed by the lower-order (i.e., alpha and beta) brains, while the latter consumption will be associated primarily with the gamma brain.

2.11 Consumption Behavior and the Pursuit of Happiness Interestingly absent from Maslow’s hierarchy of wants is happiness. Clearly, everybody wants to be happy (as opposed to being unhappy), but is happiness itself a fundamental need? In our view, it is not. Genetically, the motivation of living organisms at the most basic level is simply to survive and reproduce. Happiness (or rather its possibility) is a state (or feeling) that comes later. Copacetic equilibrium, as we have defined it, can obviously be associated with happiness, but it is probably better identified in terms of an acceptable level of satisfaction or well-being. One might think that higher levels of satisfaction will be associated with higher levels of income and consumption, but this is not necessarily the case, for according to what are increasingly being accepted as objective measures of happiness, mean levels of such in the U.S. are no higher currently than they were 50 years ago.76 In many cases, consumption activity itself is thought to be the culprit.77 While the focus in this book is on what consumption behavior actually is, as opposed to what it might or ought to be, it nevertheless is of interest to take a few moments to speculate on how consumption patterns that give rise to unhappiness might be rationalized in terms of the neurobiological framework of this chapter. The extremes, of course, are consumption activities with negative b-processes that are physiologically or psychologically addicting. Drug dependence is an obvious example. Less extreme, but almost certainly a major motivator of much of consumption behavior connected to higher-order wants (especially those arising out of the self-esteem need), is consumption associated with position goods, a notion 76 See Di Tella and MacCulloch (2006), Frey and Stutzer (2002), Kahneman and Krueger (2006), and Layard (2005). 77 See Frank (1999), Frey and Stutzer (2002), Layard (2005), and Schor (1998). However, in our opinion, conclusions concerning happiness and income should be viewed with a lot of skepticism, for very little is currently known concerning the neurobiological mechanisms associated with happiness and how these might be affected by income and consumption activity [cf., Nettle (2005)]. For income levels above subsistence, happiness, as we ordinarily think of it, is likely more associated with continuing accomplishments and interpersonal relationships than with consumption activity (or its absence), with which each generation, no matter what its level of income, has to come to grips. Moreover, economists analyzing happiness pretty much overlook the fact that happiness has been a center of philosophical and religious controversy and debate since ancient times. Cf., McMahon (2006).

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Consumption Behavior and the Pursuit of Happiness

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that was first introduced by Fred Hirsch in his book, Social Limits to Growth, in 1976. In his categorization of such goods, Hirsch distinguishes between consumption goods that are physically in fixed supply and consumption goods for which the scarcity is socially induced. Examples of the former would be a Rembrandt painting or exclusive access to a natural landscape that is physically unique. In these cases, consumers derive at least part of their satisfaction from just the inherent characteristics of the goods, i.e., from the painting simply as a painting or acres simply as acres, rather than as objects that are scarce. However, with a second category of position goods: [. . .] consumer demand is concentrated on particular goods and facilities that are limited in absolute supply not by physical but by social factors, including the satisfaction engendered by scarcity as such. Such social limits exist in the sense that an increase in physical availability of these goods or facilities, either in absolute terms or in relation to dimensions such as population or physical space, changes their characteristics in such a way that a given amount of use yields less satisfaction. This is equivalent to a limitation absolute supply of a product or facility of a given “quality,” and it is in this sense that it is regarded here as a social limitation. This social limitation may be derived, most directly and most familiarly, from psychological motives of various kinds, notably envy, emulation, or pride. Satisfaction is derived from relative position alone, of being in front, or from others being behind. Command over particular goods and facilities in particular times and conditions becomes an indicator of such precedence its emergence as a status symbol. Where the sole or main source of satisfaction derives from the symbol rather than the substance, this can be regarded as pure social scarcity. Such satisfaction may also be associated with absolute physical scarcities. Thus to at least some people, part of the attraction of a Rembrandt, or of a particular natural landscape, is derived from its being the only one of its kind; as a result, physically scarce items such as these become the repository of pure social scarcity also . . . .[Hirsch (1976, pp. 20–21, italics added).]78

In recent years, entire volumes, notably Frank (1999) and Layard (2005), have been written about the negative consequences of position goods for happiness. For when consumption behavior is directed towards position goods, what one individual 78

As a contemporary instance of such a good, consider the “One-of-a-Kind Low Rider” described by Robert Nesmith on p. 18 of the February, 2007, issue of Art and Antiques: An exclusive Daimler 1931 Double Six 50 Sport Corsica Drophead Coupe still exists today—and that’s quite a feat considering that only one was made. A project begun in 1926 by the Coventry, England-based automaker that took roughly 8 years, this low-riding 12-cylinder was a direct response to arch-rival Rolls-Royce’s 1925 New Phantom. The Double Six 50 is amazingly quiet, thanks to its doublesleeve valve design. Originally a flex-head (hardtop) coupe, the car was later modified to the drophead, or convertible, style in 1933, with further coach modifications in ’34. It was thought that possibly two or even three versions of this automobile were made, but photographs confirm the same serial number: 30661. The car was pursued and bought by Ontario-based RM Auctions in 2004, when it was immediately sold to its present owner and meticulously restored, also by RM. The so-called Corsica Roadster won Best of Show at Pebble Beach Concours d’Elegance in August 2006. “The Pebble Beach Best of Show is the ‘Holy Grail’ of car awards,” says RM Auctions spokesperson Terrence Lobzun. “It’s as close to a time machine as you can get.” This car also serves as something else—a reminder to onlookers who “have everything” of one small detail: They don’t have this. [Italics added.] Now we know.

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gains in satisfaction, another loses, so that overall happiness (assuming that it can be summed across individuals) is at best not reduced.79 Since much of position-good consumption can almost certainly be taken to have its roots in the self-esteem need, which in turn can be seen as generating an endless sequence of wants within a need, it is easy to see why mean levels of satisfaction in high-income economies may be relatively independent of the level of income, yet nevertheless give rise to income (or total expenditure) elasticities that are in excess of 1.80 A final factor making for nonincreasing happiness in consumption behavior is simply the broad spectrum of consumption activities that are subject to (nonaddicting) habit formation. For such activities, a part of consumption expenditure (because of possible strengthening of negative opponent processes) will be focused on eliminating discomfort, rather than adding to satisfaction. Since, as will be shown in Chapter 17, about 65% of aggregate consumption expenditure in the U.S. is characterized by habit formation, this represents a potentially large drag on happiness.81

2.12 Summary and Final Comments Following a brief overview of conventional demand theory, the primary purpose of this chapter has been to develop a framework for the analysis of consumer behavior that is based on evolutionary and neurobiological foundations. While a theoretical 79

For a discussion of the welfare costs associated with consumption expenditures for position goods, see Frank (2007). 80 Some anecdotal insight into the power of the self-esteem/position-good motive in driving consumption expenditures of newly super-rich is provided in an article in The Wall Street Journal (December 16–17, 2006, pp. A1 and A6). The article notes that income of managers of the top 26 hedge-funds averaged $363 million in 2005, and includes a table showing mean expenditures for 10 luxuries for a group of hedge-fund managers (having an average net-worth of nearly $200 million) for the 12-month period ending in the first quarter of 2006. Heading the list are expenditures for fine art of nearly $4 million, followed by expenditures for yacht charters and jewelry of $430,000 and $376,000, respectively. Tenth on the list are expenditures for wine and spirits for the home of nearly $50,000. As is evident in the prices paid at auctions of over the last 20 years for paintings of Klimt, Monet, Picasso, Renoir, Van Gogh, and others, the importance of art as position goods to the ultra wealthy can hardly be overstated. [This paragraph, it should be noted, was written in early 2007 just as the financial meltdown that devastated most (if not all) of these hedge funds in the summer and fall of 2008 was getting started.] 81 Another basic need that can give rise to at-best zero-sum satisfaction consumption activities is the security need, specifically those expenditures that are caused by the need to protect self, home, and property from thievery and vandalism. Still another example of consumption expenditures that are of the ‘eliminate-discomfort’ variety are ones that are socially induced so as to be able “to appear in public without shame,” a notion, incidentally, that can be traced to the Talmud (Tamari, 1987). Related notions include Veblen’s Conspicuous Consumption (1899), “bandwagon” and “snob” effects of Leibenstein (1950), and “Keeping-up-with-the-Joneses” (Duesenberry, 1949). For an extensive discussion and analysis of the negative aspects of intertwined tastes and preferences, especially as they give rise to “competitive consumption,” see Schor (1998). Also, for a detailed account of the emergence of a consumerist society in the 17th and 18th centuries in France, see Roche (1997).

2.12

Summary and Final Comments

55

structure comparable to that of neoclassical theory (whether in Hicks–Allen–Slutsky or revealed preference formulations) is obviously absent, it is hoped that sufficient elements of this way of conceptualizing consumer behavior are present to provide—if nothing else as an augmentation to conventional theory—an interpretive framework for guiding the empirical analyses of this study and assessing results. In approaching this task, our basic premise has been that the framework must have a basis in evolutionary biology and be consistent with currently accepted findings regarding functioning of the human brain. As a point of departure, the “black box” of utility maximization has been replaced with the postulate that the behavior of human beings is motivated, following the psychologist Abraham Maslow, by five basic needs (or wants): physiological, security, love/belonging, self-esteem, and self-actualization. These wants are assumed to be hierarchical in the order presented. At the most basic level, physiological needs must be fulfilled before security needs emerge, security needs must be fulfilled before love/belonging needs emerge, and so on and so forth. However, the assumption of a hierarchy of wants conflicts with one of the basic premises of conventional demand theory, namely, that there is just a single want— utility—that can be described in terms of a single-valued function of the goods in a consumer’s choice set. With a hierarchy of wants, this premise can no longer be maintained. Although there may be equivalence classes (or indifference curves) within a need category, such cannot be the case between categories. Preferences of this form have obvious implications for defining income and price effects. While income effects can seemingly be defined in a straightforward way, this does not appear to be the case for price effects (at least as measured in terms of usual income and substitution terms) because of the absence of indifference between wants. A condition has been defined, copacetic equilibrium, intended to refer to a point in time, in which all of the basic needs of an individual are gratified. When this condition holds, as should be the case for normal individuals most of the time in high-income economies, the desires wanting to be satisfied will all be arising from higher-order wants within the higher basic needs, and, accordingly, the desires being satisfied at any point in time are almost certain (for neurological reasons) to be random.82 At best, therefore, consumption behavior at a point in time can only be modeled probabilistically. Consumption behavior over intervals of time, on the other hand, is another matter. For if desires waiting gratification at points in time are stable probabilistically, then the resulting probabilities when integrated over time can be seen as fashioning preferences that, in turn, give rise to expenditures that are stable proportions of

82

As mentioned at the start of Section 2.7, the “consumption” problem for most individuals, when not at work or asleep, is how to keep a sufficient number of neurons firing so as not to become bored. For individuals in copacetic equilibrium, the desire that, at any point in time, might be in play to motivate a consumption decision almost certainly emerges at random.

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2 Demand Theory Under Review

total expenditure.83 At a theoretical level, this allows for the estimation of demand functions, although, in general, not ones that can be rationalized in terms of utility maximization. An additional implication of hierarchical preferences (together with the notion that much of consumption activity is initiated by the lower-level alpha and beta brains) is that they provide a useful vehicle for rationalizing the size of income elasticities. In ordinary economics discourse, goods are classified according to whether they are inferior, normal, or luxury depending upon whether their estimated income elasticities are negative, positive but less than one, or greater than one, respectively. Just as conventional demand theory takes preferences as given (i.e., does not attempt an explanation as to where preferences come from), it also provides no explanation for why income elasticities are as they are, not even for a phenomenon as basic as Engel’s Law for food consumption. With hierarchical preferences, in contrast, Engel’s Law is an immediate consequence of satiation of the most basic physiological need. Other income elasticities may not be so easy to rationalize, but at least there is the potential for so doing. While the hierarchy of wants is genetic in origin, and therefore provides the foundation for an individual’s preferences, actual preference structures,i.e., the way that wants, especially the higher-order desires embedded arising within the basic needs (and controlled by the gamma brain), will be gratified, will come into being primarily through consumption experience. Thus, in an important sense, preference structures can be viewed as being self-assembled.84 In general, much of the selfassembly will be the result (per the discussion in Section 2.4 above) of dynamical formations of neural networks within the brain. Within a preference structure, some consumption activity will be triggered by the lowest-order brain (the brain stem, or alpha brain) in order simply to maintain homeostasis in the autonomic processes. Other consumption desires can emerge from unconscious slave processes (i.e., habit formation or even addictions), residing mostly in the mid-level brain centers (i.e., the beta brain), that have come into existence through previous consumption activity, while still other consumption behavior will be directed from the highest brain centers in conscious searches for new and novel gratifications. Finally, despite the assumption that there are only five basic needs, the number of wants themselves (when considered as higher-order wants that emerge from within the basic wants) is endless.85 Once one want is satisfied, there will always be another

83

By stable budget proportions in this context, we do not mean budget proportions that are totally invariant with respect to time and place, but rather budget proportions whose variation can be explained, at least to some extent, by variations in income, prices, and cultural-sociodemographical factors. Also, reference here, as noted in Section 2.7, is to ensembles of individuals. 84 By “self-assembled,” we mean that preferences are created on-the-fly within the organism in response to external exposure and experience. 85 This is hardly a novel precept, but is fundamental in the 18th-century Enlightenment thinking. Cf. J. Locke [in the Essay Concerning Human Understanding, as quoted in McMahon (2006, p. 320): “We are seldom at ease, and free enough from the solicitation of our natural or adopted desires, but a constant succession of uneasiness out of that stock, which natural wants, of acquired

2.12

Summary and Final Comments

57

to take its place. The basis for this assertion is the neurological need for a minimum level of arousal, which for most individuals will not occur with a fixed number of consumption activities. Repetition leads to satiation, which then leads to desires for new activities, and so on and so forth.86 On the other hand, this does not rule out that a desire for a new activity may in fact be for a minor variation on an existing activity. For while arousal may demand something new, the basic security need may demand something that is not too new. The optimal new activity, accordingly, will often be something that combines a small amount of novelty with a great deal of redundancy.87 But, even so, the quest for novelty would appear to be a real neurological phenomenon, and represents the magnet for a continual appearance of new goods.88

habits heaped up, take the will in their turns; and no sooner is one action dispatch’d, which by such a determination of the will we are set upon, but another uneasiness is ready to set us on work.” Cf. as well, de Tocqueville (also, as quoted in McMahon, pp. 333–334): In certain remote quarters of the Old World you may sometimes stumble upon little places which seem to have been forgotten among the general tumult and which have stayed still while all around them moves. The inhabitants are mostly ignorant and very poor; they take no part in affairs of government and often governments oppress them. But yet they seem serene and often have a jovial disposition. In America I have seen the freest and best educated of men in circumstances the happiest to be found in the world; yet it seemed that a cloud habitually hung on their brow, and they seem serious and almost sad even in their pleasures. The chief reason for this that the former do not give an moment’s thought to the toils they endure, whereas the latter never stop thinking the good things they have not got. 86 For normal individuals in copacetic equilibrium, the needs in question can often arise through what Polyani (1974, p. 173) refers to as intellectual passions, i.e., passions that reflect an individual’s need for discovery, which leads to new knowledge, which, once gratified, then in turn leads to a need for more new discovery, and so on and so forth. Intellectual passions thus perpetuate themselves by their fulfillment. 87 Symphony music directors are aware of this problem when they usually include no more than one contemporary composition in any single program. Old ears require the soothing security of familiar notes. See Tversky and Sattath (1979) for a discussion. 88 The higher-order intellectual passions, such as science, mathematics, religion, fiction, and fine arts (including music), are validated by becoming happy dwelling places of the human mind, and thus provide the outlet for an endless stream of new desires. [Cf. Polyani (1974, p. 280).] On the other hand, it should be noted that novelty is an individual experience, for what is novel to one may be redundant or of no interest to another.

Chapter 3

Quantile Regression: A Robust Alternative to Least Squares

As is usual in applied econometric exercises of the present type, the workhorse of the analyses is least-squares estimation, which is so well-known that it and its properties need no discussion. However, in the course of estimating Engel and demand functions from the BLS-CES surveys, a rather surprising phenomenon has steadily revealed itself, namely, a marked tendency toward asymmetry in the distribution of residuals. The graph in Fig. 3.1 is typical. Quite clearly, the distribution is not symmetrical, but rather is left-skewed (i.e., has a longer tail on the left than on the right) and has peak density well to the right of its OLS mean of 0.1 Taken at face value, error distributions of this type obviously have ominous implications for leastsquares estimation. Consequently, most of the cross-sectional models have also been estimated using quantile regression, which is a robust estimation procedure that is based upon minimizing the sum of absolute errors (as opposed to minimizing the sum of squared errors with least-squares estimation). While quantile regression is finding increasing empirical application, it remains relatively unknown in mainstream econometrics. The purpose of this chapter, accordingly, is to provide an overview of the technique and its properties.

3.1 Some Background Apart from examining for autocorrelation and heteroscedasticity, applied econometricians seldom give much attention to the properties of the stochastic terms of their regression models. Most of the time, least-squares estimation (in some form or another) is employed, based upon a belief (often no more than tacit) that the conditions needed for the validity of the Gauss–Markov and classical normal central-limit theorems are ever present. In fact, there are a lot of reasons as to why real-world 1

The graph in this figure is a kernel-smoothed density function of the residuals from a double-logarithmic equation fitted by ordinary least squares to 7,724 observations on health-care expenditures of individual households from the four quarterly BLS–CES surveys for 1996. The dependent variable in the regression is the logarithm of expenditures for health care, while the independent variables are the logarithms of total expenditure and the price of health care, plus a variety of socio-demographical variables.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_3, 

59

60

3 Quantile Regression: A Robust Alternative to Least Squares 0.5

Fig. 3.1 Kernel-smoothed density function OLS residuals for health care expenditures BLS-CES surveys 1996

0.45 0.4 0.35

f(r)

0.3 0.25 0.2

L

0.15 0.1 0.05 0 –6

–4

–2

0

2

4

6

OLS regression residuals

error terms may not behave in ways that these conditions require. If, for example, even one of the factors subsumed in an error term has an infinite variance, then neither of the theorems can be assumed to hold. Given the pervasiveness of phenomena (both natural and socio-economic, including the distributions of income and wealth and of short-term speculative price changes in economics) that are empirically described by scaling distributions with exponents greater than –2, that such phenomena may occasionally find presence in the error terms of econometric models is obviously to be expected.2 However, the concern of the present exercise is not with absence of second moments, per se, but rather with asymmetry in the distributions of residuals. Since the seminal articles of 1978 of Bassett and Koenker, quantile regression has emerged as a powerful robust alternative to least-squares procedures in situations in which the assumptions underlying least-squares estimation are questionable. As a point of departure, consider the standard regression model, which postulates a relationship between a dependent variable y and N independent variables x (viewed as a nonstochastic row vector), in terms of the conditional expectation of y, given x: E(y|x) = g(x),

(3.1)

for some function g. If g(x) is assumed to be linear in x, then we have the conventional linear regression model y = Xβ + u,

(3.2)

2 That the upper tail of the distribution of income is described by a scaling distribution with exponent greater than –2 (the law of Pareto) traces, of course, to Pareto (1897). However, developing the apparatus for applying scaling distributions to a wide range of economic, social, and natural phenomena is due to the polymath Benoit Mandelbrot. See, in particular, Mandelbrot (1983, 1997).

3.2

Quantile Regression

61

where u is assumed to be an unobservable stochastic term of mean zero with density function f(u). If f(u) has a finite variance, then, as is well-known, the Gauss–Markov for least squares states that the estimator of β that minimizes the quantity: ϕ(y, X, β) = (y − Xβ) (y − Xβ),

(3.3)

b = (X X)−1 X y,

(3.4)

with respect to β,

is the minimum-variance unbiased estimator of β in the class of all unbiased estimators linear in y. If, in addition, f(u) should be distributed normally (even if only asymptotically), then all of the apparatus of conventional statistical inference is available for hypothesis testing and confidence interval construction. However, as Koenker and Bassett note,3 one of the dark secrets of applied statistics and econometrics is that faith in the validity of Gauss–Markov and normality assumptions is often misplaced, in which case it is wise to consider nonlinear, or even, biased, estimators that may be superior to least squares. One such estimator, which in Gauss and Laplace has essentially the same progenitors as least squares, is the one associated with minimizing the sum of absolute errors (as opposed to the sum of squared errors), or LAE. Historically, LAE was never able to compete with least squares because of mathematical intractability and lack of a suitable sampling theory. Fortunately, this is no longer the case, for in the 1960s it was recognized that LAE estimators can be obtained as solutions to linear programming problems,4 and, thanks to Koenker and Bassett, a workable sampling theory is now available—not only for LAE, but also for its generalization to quantile regression.

3.2 Quantile Regression For a single random variable y, the LAE estimator is simply the median of y, that is, the point on the distribution of y for which half of the values of y lie above and half below. With the linear model, y = Xβ + u, the LAE estimator of β also corresponds to a median, though in this case to the “median” hyperplane, defined by Xb, for which half the values of y will lie “above” Xb and half will lie “below”.5 What “above” and “below” mean in this context is that half of the residuals will be positive and half will be negative. The median for a single random variable can 3

Koenker and Bassett (1978, p. 35). See Bassett and Koenker (1978, 1982), Koenker and Bassett (1978), Hahn (1995), Buchinsky (1994a, b, 1995, 1998), Horowitz (1998), Bassett and Chen (2000), Fitzenberger et al. (2003), and Mello and Perrelli (2003), and Koenker (2005). 5 However, in saying “half-and-half,” we are ignoring the fact that (degeneracy aside) N of the residuals (with N independent variables) will be exactly zero. 4

62

3 Quantile Regression: A Robust Alternative to Least Squares

accordingly be viewed as the LAE regression of the variable on a vector of 1s. Assuming that the linear model has an intercept (i.e., that one of the columns of X is a vector of 1s), the intercept with LAE can accordingly be interpreted as shifting the (N–1)-dimensional hyperplane defined by the N–1 non-constant variables so as to “split” the residuals half-and-half between positive and negative. Quantile regression (QR) is a generalization of the LAE estimator to quantiles other than the median. Specifically, the quantile regression for the qth quantile (0 ≤q ≤1), the QR vector, bq , for q will be given by minimizing with respect to β the expression:6 θ ( β| y,X,q) =



ρq |y − X β| ,

(3.5)

where: ρq =

q if y − Xβ ≤ 0 . 1 − q if y − Xβ > 0

(3.6)

The solution vector bq to this minimization problem is obtained straightforwardly as the solution to a linear programming problem. Just as the LAE regression, which corresponds to q = 0.5, has half of the residuals positive and half negative, the quantile regression for q (and T observations) will have qT residuals less than or equal to 0 and (1–q)T residuals greater than 0. This is true for all q. Under fairly relaxed regularity conditions, the QR estimators are distributed asymptotically normal with mean vector β q and covariance matrix ω (X X)−1 , where ω=

q (1 − q) f (q)2

.

(3.7)

and f(q) is the value of the density function of u at q.7 These asymptotic results hold even if u should lack a second moment. Just as ordinary least squares is the maximum-likelihood estimator when the error term in the regression model in expression (3.2) is distributed normally with covariance matrix σ 2 I, the LAE estimator can be obtained as the maximum-likelihood estimator when the error term follows the Laplace distribution,8 that is, when u is independently and identically distributed as: f (u) = Ae−a|u| , − ∞ < u < ∞. 6

(3.8)

See Bassett and Koenker (1978). Provided, of course, that f(q) is not 0. See Koenker and Bassett (1978). A number of finite sample approximations, including bootstrap procedures, to the asymptotic covariance matrix have been proposed and investigated in Monte Carlo studies. Buchinsky (1995) provides a good discussion and evaluation of these approximations. 8 See Taylor (1974, Chapter 17). 7

3.3

Illustrations and Comparison

63

To see this, we note that the logarithm of the likelihood function for the T×1 vector u will be given by LLF = A∗ − a



|ut |,

(3.9)

which, apart from the constants A∗ and a, is equivalent to minimizing |ut |. Interestingly, LAE can also be derived as the maximum-likelihood estimator when the distribution of the error term has a “bilateral” Pareto form:9 f (u) = A |u|−α , − ∞ < u < ∞, α > 1,

(3.10)

which is of interest because of the fact that, unlike for the Laplace distribution, this distribution does not possess a variance for α < 2. To see that LAE is maximum likelihood in this case, we first write the likelihood function as

LF = AT |ut |−α , (3.11) or, in logarithms as LLF = A∗ − α



ln |ut |.

(3.12)

Since α is a constant and the logarithmic function is an increasing monotonic trans|ut | is formation, the log-likelihood function will again be maximized when minimized. Moreover, the preceding can be extended to points on the distribution of u other than the median by writing the likelihood function as LF = AT

ρq |ut |−α ,

(3.13)

where ρ q is as defined in expression (3.6) (3.with u = y–Xb). The log-likelihood function is then  (3.14) ln ρq |ut |, LLF = A∗ − α which, for the same reasons as above, this function will be maximized when, as in expression (3.5), ρq |ut | is minimized.

3.3 Illustrations and Comparison The procedure that has been followed with the cross-sectional CES data has been to build models initially by least squares by, first, using a full complement of predictors consisting of income, price (where available), and a divers selection 9

A discussion of this distribution is given in the appendix to this chapter.

64

3 Quantile Regression: A Robust Alternative to Least Squares

of socio-demographical variables; and then eliminating all variables (other than income and price) having t-ratios less than 2 (in absolute value). The “final” models are then estimated as quantile regressions at two points on the distribution of residuals, first at the median (i.e., for q = 0.5), and then at the mode, where the modes are defined with respect to the kernel-smoothed distributions of the residuals from the median regressions.10 A detailed illustration of this procedure, applied to six exhaustive categories of CES expenditures is given in Chapter 9. For a present illustration, we will continue with the model whose OLS residuals are depicted in Fig. 3.1. The kernel-smoothed distributions of residuals from the quantile regressions at the median and the mode are given in Figs. 3.2 and 3.3. Three things are to be noticed about these graphs. The first is that the asymmetrical density function of Fig. 3.1 is definitely preserved in the quantile regressions. However, since the only material change is in the minimizing metric employed in estimation, this is probably to be expected. The second thing to notice is a lengthening of the lower tail of the density function for the quantile equations, especially for the residuals from the

0.5 0.45 0.4 0.35

f(r)

0.3 0.25 0.2 0.15

Fig. 3.2 Kernel-smoothed density function median regression residuals for health care expenditures BLS-CES surveys 1996

0.1 0.05 0 –5

–4

–3

–2

–1

0

1

2

3

4

5

median regres sion residuals

10

We unfortunately at this time are not able to provide a theoretical justification for use of the mode, but instead only a heuristic argument in that it seems reasonable for the regression to be anchored at the point on the error distribution with the largest concentration of mass. The kernelsmoothed density functions have been calculated using the unit normal density function as the kernel weighting function and a “support” of k = 1,000 intervals. Silverman’s rule-of-thumb: h= (0.9)min[std.dev., interquartile range/1.34](N −1/5 ), has been used for the smoothing parameter h. Two standard references for kernel density estimation are Silverman (1986) and Wand and Jones (1995). Ker and Goodwin (2000) provide an interesting practical application to the estimation of crop insurance rates.

3.3

Illustrations and Comparison

65 0.5

Fig. 3.3 Kernel-smoothed density function modal regression residuals for health care expenditures BLS-CES surveys 1996

0.45 0.4 0.35

f(r)

0.3 0.25 0.2 0.15 0.1 0.05

–5

–4

–3

–2

0 –1 0 1 2 modal regression residuals

3

4

5

modal regression.11 This latter, of course, is a reflection of the strong OLS sensitivity to outlier observations. Finally, the third thing to notice is the movement of the peak (or mode) of the densities to 0. The modal density, obviously, has the mode right at 0 and nearer to 0 for the median regression residuals than for the OLS residuals. All this, of course, is simply a reflection of the well-known relations to one another of the mean, median, and mode in skewed distributions. The estimated coefficients for total expenditure and price for the three models, together with R2 s and the value of the quantile at the mode of the median residuals, are given in Table 3.1. Table 3.1 Estimated regression coefficients health care expenditures CES surveys 1996 (t-ratios in parentheses) Regression Variable lntotexp lnphealth R2 mode

OLS 0.4703 (21.07) −1.2109 (−17.57) 0.2306

Median 0.4495 (18.65) −1.2170 (−16.35) 0.2287

Modal 0.4778 (20.65) −1.2312 (−17.23) 0.2277 0.5734

While it might be thought that the use of a robust method of estimation might lead to the disappearance of the asymmetrical OLS residuals in this context, this quite clearly is not the case. The quantile regressions appear only to lead to a sharpening of the asymmetry. Thus, the big question remains: Why such 11

Outliers affect quantile regressions only in terms of the number of observations above and below the regression hyperplane at the specified quantile, while OLS regressions are affected by both numbers and value.

66

3 Quantile Regression: A Robust Alternative to Least Squares

asymmetries? Are these a true implication of their nature? Or are they simply a consequence of misspecification? Unfortunately, at this point, we do not know the answer to this question, and accordingly will carry forth under the assumption that it does not matter.

3.4 Conclusion The purpose of this chapter has been to note a rather surprising phenomenon that has persistently emerged in the Engel curves and demand functions estimated with cross-sectional data from the BLS-CES surveys, namely, a marked tendency for residuals from the estimated models to be asymmetrically distributed. The kernelsmoothed distribution of residuals displayed in Fig. 3.1 is typical. Not only is the distribution asymmetrical, but its tails are thick and its peak is sharp. Error distributions with such characteristics are obviously at odds with the Gauss–Markov theorem. In view of this, most of the cross-sectional models are also estimated using quantile regression, which is an increasingly well-established robust alternative to least squares in circumstances in which the conditions for the validity of the Gauss–Markov and classical central limit theorems are questionable. The attraction of quantile regression in this situation is that, since the minimizing metric is absolute deviations rather than squared deviations, its parameter estimates are much less sensitive to outliers than are the parameter estimates of least squares. They should, as well, be much less sensitive to departures from symmetry. Quantile regression estimates are easily obtained as solutions to linear programming problems; and a now well-developed asymptotic distribution theory exists for quantile regression estimators, so that hypotheses of the usual type can be tested, confidence intervals constructed, etc., as for least squares. Also, and perhaps most importantly, the asymptotic properties of the estimators do not depend upon the underlying error distributions possessing a second moment.

Appendix: On a Bilateral Law of Pareto Lester D. Taylor Along with some form of a law of demand, the law of Pareto, which describes the upper tails of the distributions of income and wealth (as well as the distribution of firm size, together with any number of other social and natural phenomena), appears to represent one of the few genuine laws of economics. The expression for this probability law has the form: P (z > x) = 1 − F (x) = Ax−α , x ≥ x0 > 0, α > 0,

(3.15)

where F(x) represents the distribution function of x.12 As is well-known, the variance of this distribution is infinite for α < 2 for α ≤ 1 as is also mean. In this 12

The density function for the Pareto distribution is given by kx-(α+ 1) , where k = αx0α .

3.4

Conclusion

67

form, the law is obviously applicable only to positive phenomena, which is fitting for quantities like income, wealth, and firm size (and also to a wide variety of other social and physical phenomena for whose distributions the law has been found to describe). However, there is one circumstance for which it would be useful if there were a form of the law applicable to negative quantities as well, namely, error terms in regression models. The purpose of this note is to suggest a bilateral form of the law that can be used to this end.

1 A Bilateral Power Law Consider, first, the expression: h (x) = K |x|−(α+1) , for x < 0,

(3.16)

where K is a constant. One can then go on to define a function, f(x), as f (x) =

K |x|−(α+1) , for x < 0 , −(α+1) , for x > 0 1 − Kx

(3.17)

or equivalently as f (x) = K |x|−(α+1) , |x| > 0.

(3.18)

While f(x) could serve as a legitimate density function for error terms in a regression model, it has the drawback that it does not allow for an error value of zero. For a variety of reasons, it would be useful if the singularity at x = 0 could be made to go away. For the moment, let us assume that this can in fact be done, that is, let us assume that the expression: f (x) = K |x|−(α+1) , for − ∞ < x < ∞,

(3.19)

is in fact legitimate. For this to be so, it is necessary to be able to integrate f(x) through x = 0. Clearly, this is not possible by integrating over the real line, but does appear to be so by integrating over the complex plane. Accordingly, let us examine the integral:  H(x) =

∞

−∞

K|x|−(α+1) dx,

(3.20)

which can be broken into a sum of three integrals:  H(x) =

−1 −∞

K|x|−(α+1) dx +



1

−1

K|x|−(α+1) dx +



∞

K|x|−(α+1) dx,

1

or equivalently (because of symmetry of the first and third integrals) as

(3.21)

68

3 Quantile Regression: A Robust Alternative to Least Squares



∞

H(x) = 2

Kx−(α+1) dx +

1



1

−1

K|x|−(α+1) dx.

(3.22)

For the first integral on the right, we will have 

∞

2

 ∞ 2K (2 + α)x−(α+1) dx 2+α 1 2 K −α−2 ∞ x =− |1 2+α 2K (for α < 2). = 2+α

Kx−(α+1) dx = −

1

(3.23)

For the second integral:  J(x) = K

1

−1

|x|−(α+1) dx,

(3.24)

√ we make the substitution,13 x = ei θ (where i = −1), so that dx = i ei θ dθ and [using Euler’s identity (ei θ = cosθ + isinθ ) to obtain the limits for θ] therefore:  I(θ ) = −K 0

π

ieiθ dθ . |eiθ |(α+1)

(3.25)

However, since ei θ , as θ varies from 0 to 2π, represents points on the unit circle in the complex plane, |ei θ | accordingly measures the length of the radius vectors of this unit circle and is therefore equal to 1.14 Hence,  I (θ ) = −K =

π

ieiθ dθ

0  −Keiθ π0

 = −K (cos θ + i sin θ) π0 (using Euler’s identity) = −K [cos π + i sin π − (cos 0 + i sin 0)]

(3.26)

= 2 K. Consequently, for H(x), we will have 2K + 2K α+2 −2 K + 2 K(α + 2) = α+2 2 K(α + 1) . = α+2

H(x) = −

13 14

See Nahin (1998, p. 196). See Fisher (1990, p. 44).

(3.27)

3.4

Conclusion

69

Returning now to f(x) in expression (3.19) to be a density function, it is necessary that it integrate to 1 (i.e., H(x) must equal 1), from which, from the last line of expression (3.27), we find for the value of K: α+2 . 2(α + 1)

K=

(3.28)

Consequently, for f(x), we finally have: f (x) =

(α + 2) −(α+1) |x| , − ∞ < x < ∞, 1 < α < 2. 2(α + 1)

(3.29)

Symmetry of f(x) about the origin suggests that the mean of x ought to be zero. Formally:  E(x) = K =K

∞

−∞  −1

x|x|



−(α+1)

−∞  ∞

= −K =

x|x|−(α+1) dx

x|x|

 dx+K

−(α+1)

1 −1



dx + K

−1



∞

dx+K 1

1

−1

1 1

x|x|

−(α+1)

x|x|

−(α+1)

x|x|−(α+1) dx



dx + K

(3.30)

∞

x|x|

−(α+1)

dx

1

x|x|−(α+1) dx

Substituting once again ei θ for x in order to integrate over the complex plane, we have (dropping the constant K): 

1

−1

x |x|−(α+1) dx = −



π 0



= −K

 −(α+1) ie2iθ eiθ  dθ π

ie2iθ dθ

0

 1 π 2iθ 2e dθ 2 0 1 = − e2iθ |π0 2 1 = − [( cos θ + i sin θ )( cos θ + i sin θ )|π0 2 1 = [1 − 1] 2 = 0, =−

from which it follows that the mean of x is indeed zero.

(3.31)

70

3 Quantile Regression: A Robust Alternative to Least Squares

Turning now to the variance, it follows from the fact that the variance for the Pareto distribution for α less than 2 does not exist that the same will be true for this distribution. To see this, we need only examine E(x2 ) in the right-hand “tail” of f(x), namely:  ∞   2 |x >1 =K x2 |x|−(α+1) dx E x 1 ∞ x1−α dx =K 1  ∞ K (1 − α)x1−α dx = 1−α 1 K 2−α ∞ x |1 , = 1−α

(3.32)

which clearly does not exist for α < 2.

2 A Bilateral Pareto Regression Model The bi-lateral power (or Pareto) density function given in expression (3.29) is symmetrical about its mean like the normal distribution, but is much more peaked at the mean and has much fatter tails, indeed, so fat that, for α < 2, the distribution does not even possess a variance. We now turn to a practical application of this distribution, namely, as a density function underlying the distribution of the error term in the linear regression model: y = Xβ + u,

(3.33)

where specifically it is assumed that u has the density in expression (3.29): f (u) =

(α + 2) −(α+1) |u| , − ∞ < x < ∞, 1 < α < 2. 2(α + 1)

(3.34)

The task now is to find a feasible estimator of β under the assumption that u has the density function in expression (3.34) and that (as in the least squares model) the matrix X is nonstochastic, with rank N less than T. An obvious choice of estimator is the method of maximum likelihood. Assuming that u is identically and independently distributed with the density function in expression (3.34), we have for the logarithmic likelihood function: ln L(β) = C − (α + 1) ln |uj | = C − (α + 1) ln |yj − Xj β|,

(3.35)

where C is a constant independent of β. Since α is a constant and the logarithmic function is monotonic, maximization of expression (3.35) is equivalent to minimizing the sum of absolute errors (or LAE), that is, to minimizing:

3.4

Conclusion

71

φ(β) =



|yj − Xj β|

(3.36)

with respect to β. This is a happy result, for it means that estimation and statistical inference can be undertaken in the increasingly developed framework of quantile regression.15 Briefly put, the regression coefficients in quantile regression, which includes minimizing the sum of absolute errors as a special case, are obtained as solutions to linear programming problems. The resulting estimators are unbiased and under fairly mild regularity conditions are asymptotically normal (no matter what the distribution of the error term) and covariance matrix (as with least squares) that is proportional to (X’X)−1 .16 As noted in Section 3.3, for a single random variable x, the LAE estimator is simply the median of x, that is, the point on the distribution of x for which half of the values of x lie above and half below. With the linear model, y = Xβ + u, the LAE estimator of β also corresponds to a median, though in this case to the “median” hyperplane, defined by Xb, for which half the values of y will lie “above” Xb and half “below”.17 Quantile regression (QR) is a generalization of the LAE estimator to quantiles other than the median. Specifically, the quantile regression for the qth quantile (0 ≤q ≤1), the QR vector, bq , for q will be given by minimizing with respect to β the expression:18 φ(β|y, X, q) =



ρq |yj − Xj β|

(3.37)

where: ρq =

q if y − xβ≤0 1 − q if y − xβ>0

(3.38)

The solution vector bq to this minimization problem is obtained straightforwardly as the solution to a linear programming problem. 15 The seminal papers for quantile regression are Bassett and Koenker (1978), Koenker and Bassett (1978), and Bassett and Koenker (1982). See also, Buchinsky (1994a, 1994b, 1995, 1998), Buchinsky and Hahn (1998), Hahn (1995), Horowitz (1998), Bassett and Chen (2000), Fitzenberger, Koenker, and Machado (2002), and Mello and Perrelli (2003), as well as Koenker et al. (2003, 2005). 16 See, in particular, Bassett and Koenker (1982). For assessment of various methods of estimating the asymptotic covariance matrix, see Buchinsky (1994b). 17 What “above” and “below” mean in this context is that half of the residuals will be positive and half will be negative. In saying “half-and-half,” however, I am ignoring the fact that (degeneracy aside) N of the residuals (with N independent variables) will be exactly zero. The median for a single random variable can accordingly be viewed as the LAE regression of the variable on a vector of 1s. Assuming that the linear model has an intercept (i.e., that one of the columns of X is a vector of 1s), the intercept with LAE can accordingly be interpreted as shifting the (N–1)-dimensional hyperplane defined by the N–1 non-constant variables so as to “split” the residuals half-and-half between positive and negative. 18 See Bassett and Koenker (1978).

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3 Quantile Regression: A Robust Alternative to Least Squares

It has been known for years that estimators obtained by minimizing the sum of absolute errors are robust in the presence of fat-tailed distributions and other departures from the Gauss–Markov assumptions for optimal least-squares estimation. However, what has been lacking is a density function for the regression errors that allows for infinite variance. The only distribution that has previously been known to provide a maximum-likelihood basis for LAE is the Laplace (or bilateral exponential) distribution, but that distribution, while it has thicker tails than the normal, still has a finite variance. Thus, the bilateral Pareto distribution of this note appears to be a possibly important step in providing a statistical underpinning for robust regression estimation.

3 An Illustration We now turn to an exploration of the foregoing to a body of cross-sectional data from the Consumer Expenditure Surveys conducted quarterly by the U. S. Bureau of Labor Statistics. The graph in Fig. 3.4 is a kernel-smoothed density function of the residuals from a double-logarithmic equation fitted by ordinary least squares to 8,056 observations on health care expenditures of individual households from the four quarterly BLS-CES surveys for 1996.19 At a glance, it is clear that the distribution is not normal, for it has a sharp peak, a long tail, and is asymmetrical.20 0.5 0.45 0.4 0.35 f(r)

0.3 0.25 0.2 0.15 0.1 0.05 0

–6

–4

–2

0

2

4

6

residuals

Fig. 3.4 Kernel-smoothed OLS residuals health care expenditures BLS-CES surveys 1996

19 The model in question is a simple double-logarithmic function using a data set used in Chapter 6 that combines the BLS expenditure data with price data collected by the American Chambers of Commerce Researchers’ Association (ACCRA). 20 As will be discussed more below, these characteristics are the norm with data from the BLS–CES surveys.

3.4

Conclusion

73

Table 3.2 OLS and LAE regression coefficients health care expenditures BLS-CES surveys 1996 Variable

OLS

QR(Med.)

lnphealth

−1.2137 (−16.70) 0.4642 (20.71) 0.2247

−1.2164 (−15.71) 0.4408 (16.75) 0.2234

lntotexp R2

Needless to say, regression error terms with distributions like this are present problems for least-squares estimation. We see next the model estimated by LAE (that is, by quantile regression at the median). Table 3.2 gives the estimated coefficients for OLS and LAE, while Fig. 3.5 gives the distribution of the LAE residuals. On the surface, there appears to be little difference. The two estimates of the price elasticity are virtually identical, while the LAE elasticity for total expenditure is only modestly changed.21 Note, too, the small change in R2 . The distribution of the LAE residuals in Fig. 3.5 is very similar to the one in Fig. 3.4 for OLS, for it has long tails, a sharp peak, and is clearly not normal. 0.5 0.45 0.4 0.35

f(r)

0.3 0.25 0.2 0.15 0.1 0.05 0 –6

–4

–2

0

2

4

6

residuals

Fig. 3.5 Kernel-smoothed LAE residuals health care expenditures BLS-CES surveys 1996

Since the OLS and LAE equations are so similar, the temptation is probably strong to conclude that, at least with this data set, there is little reason not to use least squares. However, the residuals in Figs. 3.4 and 3.5 clearly raise a red flag, for peaked asymmetrical distributions with long tails are clearly not in keeping with the Gauss–Markov theorem. If the graphs in these figures were isolated cases, that 21 Other predictors in the model include a variety of socio-demographical variables. In general, the coefficients on these variables are also little changed between the two estimations.

74

3 Quantile Regression: A Robust Alternative to Least Squares

would be one thing, but they are not, at least with respect to the estimation of demand functions and Engel curves from the BLS-CES data. For, as noted in the text, we have estimated hundreds (maybe even in the thousands) of models with the CES data, and the residual distributions in Figs. 3.4 and 3.5 are typical. However, the question for now is the relationship that the distributions in Figs. 3.4 and 3.5 might bear (if any) to the distribution that has been discussed in this appendix. As is evident from expression (3.29), an important implication of the bilateral Pareto distribution is that the tails below and above the mean should be linear in logarithms, with slope equal to α. A straightforward way of testing for this is to separate the residuals according to positive and negative, and then examine the relationship between the logarithms of the densities (or alternatively the upper and lower tails of the distribution function) and the logarithms of their corresponding supports.22 Graphs of these relationships for the tails of the distribution function, derived from the density function for the LAE residuals in Fig. 3.5, are given in Figs. 3.6 and 3.7. 0 -8

–6

–4

–2

–1

0

2

4

–2

ln CDF

–3 –4 –5 –6 –7 –8 –9

ln |support|

Fig. 3.6 Graph of ln CDF vs. ln |support| lower tail of LAE regression residuals health care expenditures 1996

In Fig. 3.6, we see that the lower tail of the distribution is strongly logarithmically linear for residuals less than about −4.50.23 but this is probably very misleading for this represents a point on the cumulative distribution function of less than 0.0013. The rest of the graph in Fig. 3.6 shows some curvature, but it is fairly mild for points on the distribution function between 0.0013 and about 0.28, which corresponds to residuals between −4.5 and −0.60. For the upper tail of the distribution, we see in Fig. 3.7 that there is once again a strongly linear section (which again is probably misleading) between about 1.6 and 2.5, which corresponds to points on the cumulative distribution of 0.97 and 0.996. Again, the curvature is fairly mild for residuals 22

Obviously, absolute values of the support will be used for negative residuals. Keep in mind that the residuals are from an equation in which the dependent variable is in logarithms. 23

3.4

Conclusion

75 0

–8

–6

–4

–2

–1

0

2

4

–2

ln CDF

–3 –4 –5 –6 –7 –8 –9

ln |support|

Fig. 3.7 Graph of ln CDF vs. ln |support| upper tail of LAE regression residuals health care expenditures 1996

between 0.8 and 1.6, which corresponds to values of 0.83 and 0.97 on the cumulative distribution. As an additional (and more formal) exercise, a linear equation has been fitted (by least squares!) to the portions of the graphs in Figs. 3.6 and 3.7 that display only mild curvature, specifically, to points on the cumulative distribution that lie between the 5th and 40th percentiles on the lower tail and to points between the 60th and 95th percentiles on the upper tail. The coefficients in these regressions represent direct estimates of the parameter α. The results are as follows:24 Lower tail: 1.0463 ln CDF = −1.8831 (−161.99) − (−50.36) ln |support|

R2 = 0.9344

df = 178

(3.39)

Upper tail: 1.5918 ln (1 − CDF) = −3.1945 (−65.50) − (−38.67) ln |support|

R2 = 0.8085

df = 354 (3.40)

Both estimates of α are seen to be less than 2, as required with the bilateral Pareto density. However, the two estimates (1.04 and 1.59) are notably different, which is in conflict with the requirement that a single value of α applies to both tails. Nevertheless, it would seem that the use of LAE in this circumstance has more to be said for it than the use of least squares.

24

T-ratios are in parentheses.

76

3 Quantile Regression: A Robust Alternative to Least Squares

4 Conclusion The purpose of this appendix has been to generalize the Pareto distribution in a way that allows it to take on values over the entire real line. The generalization (referred to as the bilateral Pareto distribution) suggested is: f (u) =

(α + 2) −(α+1) , − ∞ < x < ∞, 1 < α < 2, |u| 2(α + 1)

(3.41)

which can be obtained by integrating the function f(x) = K|u|−(α + 1) in the complex plane through the trouble point u = 0. For values of α between 1 and 2, this distribution possesses a mean, but not a variance, and therefore may be appropriate as the probability underpinning in regression models in which error terms may be thought to have fat tails. The maximum-likelihood estimator for the distribution turns out to be minimizing the sum of absolute errors, which means the increasingly developed framework for quantile regression can be used for estimation and statistical inference. Use of the distribution has been explored in the context of the residuals from a double-logarithmic demand model for health care expenditures as estimated in the text. The distribution of residuals from the model has a sharp peak, and is asymmetrical, with long tail to the left, all of which is ominous for least-squares estimation. There is evidence of bilateral Pareto behavior in both tails, but the strong asymmetry means that a common value of α cannot be assumed. The suggestion in this situation is to move estimation to a different quantile. In any event, what the results of the illustration suggest is that applied econometricians, especially those working with large data sets, ought to begin examining the shapes of their residual distributions and to consider assuming the distribution of this appendix as a statistical underpinning (together with its attendant maximum-likelihood estimator of minimizing the sum of absolute errors) when appropriate.

Part II

Analyses of Data from BLS Consumer Expenditure Surveys

Chapter 4

Description of Data Used from the Ongoing BLS Consumer Expenditure Surveys

4.1 Some Background and History1 Household consumer expenditure surveys, which represent one of the oldest forms of acquiring economic data, had their origin in a concern with differences in consumption between the rich and the poor in England in the late 18th century. In the 1790s, two seminal budget surveys of workingmen were developed by David Davies (1795) and Frederick Eden (1797).2 In 1787, Davies, then a rector in the parish of Barkham, undertook a study of the working poor in his parish and began by collecting detailed budgets of six “typical” parish agricultural laborers. He circulated these budgets widely among friends and acquaintances throughout the kingdom and encouraged them to undertake the same. In 1795, Davies edited 127 of these budgets and used them as an empirical basis in a dispassionate plea for a minimum-wage law tied to the price of wheat. Both the budgets and plea were published in the Case of Labourers in Husbandry. Two years later, Frederick Eden, who was concerned with the effects on the working poor of the high price of wheat in 1794 and 1795, published the budgets (using essentially the same format as Davies) for 60 agricultural families and 26 non-agricultural families from various parts of England as an appendix in his three-volume work: The State of the Poor. Two developments in Europe ushered in the modern era of budget studies in the mid-19th century.3 The first was the political and social unrest that swept Europe in the 1840s, which caused concern for the economic welfare of the working classes, and thus to the collection of economic data, including budgetary statistics. The second development was an interest in the analysis of social data that, in substantial part, was tied to the contributions to probability theory of Laplace, Gauss, Poisson, the Bernoullis, and others. Budget studies were undertaken in Saxony and Prussia in 1848 and in Belgium in 1855 by Eduoard Ducpetiaux, who published detailed

1 The material in this section draws heavily on Stigler (1954, 1965), Koenker (1977), and Chapter 17 of the BLS Handbook of Methods (Bureau of Labor Statistics, U. S. Department of Labor, 1997). 2 See Stigler (1954) and Koenker (1977). 3 Stigler (1954).

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_4, 

79

80

4 Description of Data Used from the Ongoing BLS Consumer Expenditure Surveys

budget information on nearly 200 Belgian households. Engel’s 1857 study, which was the first statistical budget investigation (and probably still the most famous), in which Engel’s law of consumption for food was first proposed,4 was based on Ducpetiaux’s households survey. Engel’s study of consumption and his law for food expenditures gave rise to a number of subsequent efforts to adduce statistical laws of consumption from budgetary data, including one for rent by the director of the Berlin statistical bureau, Hermann Swabe, in 1868, and ones for subsistence, clothing, lodging, fuel and light, and sundries by the Massachusetts Commissioner of Labor Statistics, Carroll Wright. Carroll’s study, which followed Engel’s methodology, was the first comprehensive budget study undertaken in the U.S. The first nation-wide consumer expenditure survey in the U.S. was undertaken by the Bureau of Labor Statistics (BLS) in 1888–1891 for the purpose of analyzing workers’ expenditure patterns as elements of production costs. A similar survey was conducted in 1901, from which weights were obtained to construct an index of prices of food purchased by workers. This index was used as a deflator for all kinds of goods until World War I. A third survey, spanning 1917–1919, provided more comprehensive weights for computing the cost-of-living index now known as the consumer price index. A fourth survey, covering only urban wage earners and clerical workers, was undertaken in 1934–1936, primarily for the purpose of revising the 1917–1919 weights. Until the 1934–1936 survey, the principal purpose of the BLS surveys was to provide weights for consumption deflators and to study the welfare of selected groups. However, the economic depression of the 1930s pointed to the need for more comprehensive measures of economic activity, both macro and micro. At the macro level, this need was expressed in the construction by the Department of Commerce for the first time of aggregate national income and product accounts, while at the micro level, the BLS, in cooperation with four other federal agencies, undertook another national survey in 1935–1936 that extended the collection of consumer expenditure data to both rural and urban segments of the population.5 Sixth and seventh nation-wide surveys were conducted by the BLS in 1950 and 1960–1961, both primarily for the purpose of obtaining new weights for the consumer price index. The 1950 survey was an abbreviated version of the 1935–1936 survey and covered only urban consumers. The 1960–1961 survey, however, once again included both urban and rural households. The next major survey of consumer expenditures undertaken by the Bureau of Labor Statistics was in 1972–1973. This survey, while providing continuity in content with previous surveys, departed from previous BLS surveys in the way the 4

“The poorer a family, the greater the proportion of its total expenditure that must be devoted to the provision of food.” [Engel (1857, 1895); see also Houthakker (1957).] As noted by Stigler, Engel’s law of consumption for food was the first empirical generalization from budget data. 5 Studies using the data from the 1935–1936 survey were highly instrumental in the development of the Relative Income Hypothesis of Duesenberry (1949), the Life-Cycle Model of Modigliani and Brumberg (1954a, b), and the Permanent Income Hypothesis of Friedman (1957).

4.2

The Current BLS Surveys

81

information in it was collected. Unlike for prior surveys, which were conducted inhouse by BLS, sample selection and all field work was conducted, under contract with BLS, by the Bureau of the Census. A second significant change was the use of two independent surveys to collect the expenditure information––a diary survey and a panel interview survey. Finally, a third major change was a switch from an annual recall to a quarterly recall in the interview survey and a daily record-keeping of expenditures in the diary survey. However, as in all previous surveys, the data collected were again used to revise the weights in the consumer price index.

4.2 The Current BLS Surveys6 The rapidly changing conditions of the 1970s made it clear that more timely information was needed than could be supplied in consumer expenditure surveys conducted every 10–12 years. As a consequence, the Bureau of Labor Statistics initiated a continuing survey in 1979. In comparison with previous surveys, the key innovations of the current survey are that data are collected quarterly and it is semi-panel in nature. Once a household (or consumer unit in BLS terminology) is selected for inclusion, it remains in the survey for five consecutive calendar quarters. Entry and exit of households is staggered so that each quarter’s sample consists of a mixture of “new” and “old” households. As in the 1972–1973 survey, the present survey consists of two separate surveys, an interview survey and a diary survey, each with a different data collection technique and sample. The following description of the surveys is taken from Chapter 17 of the BLS Handbook of Methods (1997, pp. 161–164): The unit for which expenditure reports are collected is the set of eligible individuals constituting a consumer unit, which is defined as (1) all members of a particular housing unit who are related by blood, marriage, adoption, or some legal arrangement, such as foster children; (2) a person living alone or sharing a household with others, or living as a roomer in a private home, lodging house, or in a permanent living quarters in a hotel or motel, but who is financially independent. or (3) two or more unrelated persons living together who pool their income to make joint expenditure decisions. Students living in university-sponsored housing are also included in the sample as separate CUs. Survey participants record dollar amounts for goods and services purchased during the reporting period whether or not payment was made at the time of the expenditure. The expenditure amounts include all sales and excise taxes for all items purchased by the consumer unit for itself or for others. Excluded from both surveys are all business-related expenditures and expenditures for which the family is reimbursed. Interview Survey The Interview survey is designed to collect data on the types of expenditures which respondents can be expected to recall for a period of 3 months or longer. In general, expenses reported in the Interview survey are either relatively large, such as property, automobiles, or

6

This section provides a technical description of the BLS-CES surveys, mostly in BLS’s own language. While necessary for completeness, some readers may find the section useful only for reference, and can pass on to Section 4.3 without loss of continuity.

82

4 Description of Data Used from the Ongoing BLS Consumer Expenditure Surveys major appliances, or are expenses which occur on a fairly regular basis, such as rent, utility bills, or insurance premiums. Each occupied sample unit is interviewed once per quarter for five consecutive quarters. After the fifth interview, the sample unit is dropped from the survey and replaced by a new consumer unit. For the survey as a whole, 20 percent of the sample is dropped and a new group added each quarter. New families are introduced into the sample on a regular basis as other families complete their participation. Another feature of the current survey is that data collected in each quarter are considered independently, so that estimates are not dependent upon a family participating in the survey for a full five quarters. For the initial interview, information is collected on demographic and family characteristics and on the inventory of major durable goods of each consumer unit. Expenditure information is also collected in this interview, using a 1-month recall, but is used, along with the inventory information, solely for bounding purposes, i.e., to classify the unit for analysis and to prevent duplicate reporting of expenditures in subsequent interviews. The second through fifth interviews use uniform questionnaires to collect expenditure information in each quarter. Data collected in these questionnaires which are arranged by major expenditure component (e.g., housing, transportation, medical, and education) form the basis of the expenditure estimates derived from the Interview survey. Wage, salary, and other information on the employment of each CU member is also collected or updated in each of these interviews. The expenditure data are collected via two major types of questions asked. The first type of question asks for the purchase month directly for each reported expenditure. The second type of question asks for a quarterly amount of expenditures. The use of these two types of questions varies depending on the types of expenditures collected. Approximately 65 percent of the data were collected using the direct monthly method, whereas 35 percent were collected using the quarterly recall approach. In the fifth and final interview, an annual supplement is used to obtain a financial profile of the consumer unit. This profile consists of information on the income of the CU as a whole, including unemployment compensation; income from royalties, dividends, and estates; alimony and child; etc. A 12-month recall period is used to collect income and asset type data. Diary Survey The primary objective of the Diary survey is to obtain expenditure data on small, frequently purchased items which are normally difficult to recall. These items include detailed expenditures for food and beverages, both at home and in eating places; housekeeping supplies and services; nonprescription drugs; and personal care products and services. The Diary survey is not limited to these types of expenditures, but rather includes all expenses which the consumer unit incurs during the survey wee. Expenses incurred while away from home overnight and for credit and installment plan payments are excluded. Two separate questionnaires are used to collect Diary data: a Household Characteristics Questionnaire and a Record of Daily Expenses. The Household Characteristics Questionnaire is used to record information pertaining to age, sex, race, marital status, and family composition, as well as information on the work experience and earnings for each CU member. The socioeconomic information is used by BLS to classify the consumer unit for publication of statistical tables and economic analysis. Data on household characteristics also provide the link in the integration of Diary expenditure data with Interview expenditure data for publishing a full profile of consumer expenditures by demographic characteristics. The daily expense record is designed as a self-reporting product-oriented diary on which respondents record a detailed description of all expenses for two consecutive 1-week periods. Data collected each week are considered independently. The diary is divided by day of purchase and by broad classifications of goods and services—a breakdown designed to aid the respondent when recording daily purchases. The items reported are subsequently coded

4.2

The Current BLS Surveys

83

the Bureau of the Census so that BLS can aggregate individual purchases for representation in the Consumer Price Index and for presentation in statistical tables. Integrated Survey Data The integrated data from the BLS Diary and Interview surveys provide a complete accounting of consumer expenditures and income, which neither survey component alone is designed to do. Some expenditure items are collected only by either the Diary or the Interview survey. For example, the Diary collects data on detailed food expenditures, which are not collected in the Interview. The Interview collects data for expenditures on overnight travel and information on reimbursements, such as for medical care costs or automobile repairs, which are not collected in the Diary. For items unique to one or the other survey, the choice of which to use is obvious. However, there is considerable overlap in coverage between the surveys. Because of the overlap, the integration of the data presents the problem of determining the appropriate survey component from which to select the expenditure items. When data are available from both survey sources, the more reliable of the two is selected as determined by statistical methods. The selection of the survey source of items is evaluated periodically. Selection of Households The Consumer Expenditure Survey is a national probability sample of households designed to represent the total civilian noninstitutional population. The selection of households begins with the definition and selection of primary sampling units (PSUs), which consist of counties (or parts thereof), groups of counties, or independent cities. The set of sample PSUs used for the survey consists of 101 areas, of which 87 urban areas have also been selected by BLS for the consumer price index program. The sampling frame (i.e., the list from which housing units are chosen) for this survey is now generated from the 1990 census 100-percent detail file, which is augmented a sample drawn from new construction permits and coverage improvement techniques to eliminate recognized deficiencies in the census. In addition, the sample for the Diary survey is doubled during the last 6 weeks of the year to collect expenditure data during peak shopping period of the Christmas and New Year holidays. The population of interest is the total U. S. civilian population. Within this framework, the eligible population is noninstitutional persons (for example, those living in houses, condominiums, or apartments) and all people residing in group quarters such as housing facilities for students and workers. Military personnel living on base are not included. The Bureau of the Census selected a sample of approximately 8020 addresses to participate annually in the Diary survey. This results in an effective annual sample size of 5870 households, since many diaries are not completed due to refusals, vacancies, ineligibility, or the nonexistence of the household address. The actual workload of the diaries is spaced over 52 weeks of the year. The Interview survey is a rotating panel survey in which approximately 8910 addresses are contacted in each of the calendar quarters. Allowing for bounding interview, which are not included in the estimates, and nonresponse (including vacancies), the number of completed interviews per quarter is targeted 6160. Each month, one-fifth of the units are new to the survey. Each panel is interviewed for five quarters and then dropped from the survey.7

7 Quality

control for the Consumer Expenditure Survey is provided by a re-interview program, which provides a means of evaluating individual interviewer performance to determine how well the specified procedures are being carried out. Sub-samples of approximately 6% of households in the interview survey and 17% in the diary survey are re-interviewed on an ongoing basis (BLS Handbook of Methodology, 1997, Chapter 17, p. 161.). Non-response rates due to inability to contact or failure to participate for the two surveys for 1994 were 83% for the interview survey and 81% for the diary survey. (BLS Handbook of Methodology, 1997, Chapter 17, p. 164.)

84

4 Description of Data Used from the Ongoing BLS Consumer Expenditure Surveys

4.3 Data Used in the Present Study The Bureau of Labor Statistics makes available, at reasonable cost, information on CD-ROM’s for all of the quarterly surveys dating from 1982. At the time that work with the CES surveys was initiated in early 2002, the 1999 surveys were the most recent available. Surveys prior to 1996 are available in ASCII text format only; beginning in 1996, they are available in both ASCII and PC SAS data sets. Since the analysis was to be done using SAS, the 16 CES surveys for 1996Q1 through 1999Q4 were accordingly settled on as the ones to be analyzed. The BLS public use microdata CD-ROM’s contain a variety of data files from both the interview and diary surveys, organized under three headings: interview, EXPN, and diary.8 The interview survey data are organized into four major data files for each quarter: (1) FMLY: a family characteristics and income file, (2) MEMB: a member characteristics and income file, (3) MTAB: a detailed monthly expenditure file, and (4) ITAB: a monthly income file. The diary survey data are also organized in four major files: (1) FMLY: a family characteristics and income file, (2) MEMB: a member characteristics and income file, (3) EXPN: detailed weekly expenditures, and (4) DTAB: annual income. Finally, the EXPN files contain expenditure and non-expenditure information collected directly from all of the major sections of the quarterly interview survey and are more detailed than the other interview survey files alone. The empirical results reported in the next eight chapters are all based upon information taken from the interview FMLY data files. The first step in the preparation of the regression data sets was to take into account the fact that, since the diary data are collected over all 52 weeks of a year, data for 3-month periods may not respect calendar quarters. For example, for a consumer unit that reports data for November, December, and January 1996–1997, part of the data falls in 1996Q4 and part in 1997Q4. In this case, the information for the two calendar quarters is aggregated and assigned to 1997Q4. However, to take into account that two calendar quarters may be involved, “seasonal” dummy variables are constructed that can take three possible nonzero values: 1: if, for the current quarter, total expenditure for the consumer unit in the last quarter was 0; 2/3: if, for the current quarter, total expenditure for the consumer unit in the last quarter was nonzero, but less than the total expenditure in the current quarter; 1/3: if, for the current quarter, total expenditure for the consumer unit was greater in the last quarter than in the current quarter. The next step in preparing the regression data sets was to eliminate those consumer units with after-tax incomes of less than $5000, and then, finally, to eliminate, for each expenditure category, those consumer units for which their reported expenditures were 0 or negative.

8

See http://www.bls.gov/cex/csxmicro.htm

4.5

Combining CES Surveys and ACCRA Prices

85

4.4 Control Variables There will be frequent references in the next several chapters to a “full” of sociodemographico-regional control variables. The variables in this list include:

Mnemonic

Description

no_earnr fam_size age_ref dsinglehh drural dnochild dchild1 dchild4

number of income-earners in household size of household age of head of household dummy variable for single household dummy variable for rural area of residence dummy variable for no children in household dummy variable for children in household under age 4 dummy variable for oldest child in household between 12 and 17 and at least one child less than 12 dummy variable for education of head of household: grades 1 through 8 dummy variable for education of head of household: some high-school, but no diploma dummy variable for education of head of household: high-school diploma dummy variable for education of head of household: some college, but did not graduate dummy variable for education of head of household: bachelor’s degree dummy variable for education of head of household: post-graduate degree dummy variable for residence in northeast dummy variable for residence in midwest dummy variable for residence in south dummy variable for residence in west dummy variable for white head of household dummy variable for black head of household dummy variable for male head of household dummy variable for owned home dummy variable for household receiving food stamps dummy variable for if household includes members older than 64 seasonal quarterly dummy variables.

ded10 dedless12 ded12 dsomecoll ded15 dgradschool dnortheast dmidwest dsouth dwest dwhite dblack dmale down dfdstmps dpers64 D1, D2, D3, D4

4.5 Combining CES Surveys and ACCRA Prices Much of our cross-sectional analysis has involved the use of data sets that combine expenditure data from the CES surveys with price data that are collected in surveys conducted by the American Chambers of Commerce Research Association (ACCRA).9 In its ongoing price surveys, ACCRA collects price information in more than 300 cities in the U.S. for the 62 items of expenditure given in Table 4.1. From these 62 items, for each city in the survey, ACCRA constructs six price indices

9

See http://www.ACCRA.com

86

4 Description of Data Used from the Ongoing BLS Consumer Expenditure Surveys Table 4.1 Items included in ACCRA price surveys

Groceries

Housing

Utilities

Transportation Health care

Miscellaneous

t-bone stk.

Apt. rent

All electric

Bus fare

Hosp. room

Gd. beef Sausage Fry chicken Tuna Gal. Milk Dz. Eggs Margarine Parm. cheese Potatoes Bananas Lettuce Bread Cigarettes Coffee Sugar Cereal Sweet peas Tomatoes Peaches Kleenex Cascade Crisco Orange juice Frozen corn Baby food Coke

Home price Mort. rate Home P+I

Part electric Other energy Telephone

Tire bal. Gasoline Aspirin

Dr. appt. Dentist

Hamburger sandwich Pizza 2-pc. chicken Hair cut Beauty salon Tooth paste Shampoo Dry clean Men’s shirt Underwear Slacks Washer repair Newspaper Movie Bowling Tennis balls Monopoly set Liquor Beer Wine

(food, housing, etc.) and then from these an all-items index.10 In appending the ACCRA price data to the CES data sets, the ACCRA city indices in a state for each quarter are aggregated to the state level using city populations from the U.S. Census of 2000 as weights.11 The resulting ACCRA prices are then attached to CES consumer units in the CES data sets according to state of residence.12

4.6 Levels of Aggregation The cross-sectional analysis of the CES surveys has been pursued at three different levels of aggregation. The first of these, which is the focus of the 10

In principle, the ACCRA all-items indices are comparable, on a city basis, to BLS city CPIs. See http://www.census.gov/Press-Release/www/2003/SF4.html 12 In instances in which CES does not code state-of-residence for reasons of non-disclosure, the consumer units in question are dropped from the sample. 11

4.6

Levels of Aggregation

87

principal-component analysis in the next chapter, involves 14 CES categories of expenditure that exhaust total expenditure as follows:13

Food Alcoholic Bev. Housing Apparel Transportation Health Entertainment Personal Care Reading Education Tobacco Misc. Expend. Cash Contributions Pensions and Personal Insurance

The second level of aggregation is the one involving models in which price as well as total-expenditure elasticities are estimated. Because of the limited coverage of prices in the ACCRA surveys, all of these models are estimated at a level of six categories of expenditure that correspond to the six ACCRA price indices. These six, again exhaustive, categories are given in Table 4.2.14 Finally, the third level of aggregation of the CES data for which models are estimated is given in Table 4.3. This Table summarizes in one place all three levels of aggregation of the analysis. The categories listed at the far left under CES Data Sets represents the 14 categories of expenditure that are the object of the principal-component analysis in the next chapter. The categories listed at the left under CES-ACCRA Data Sets represent the six categories of expenditure for which Table 4.2 Six categories of expenditure CES-ACCRA data sets

13

CES category of expenditure

ACCRA price index

Food consumed at home Shelter Utilities

Groceries Housing Utilities

This is the highest level of aggregation (with one exception) that BLS provides in Public Use Microdata sets. The exception is that BLS does not aggregate the two food categories (food at home and food away from home) into a single food category, as is done here. 14 With the exception of “Misc. Expend.,” all of the CES categories are as defined in the CES interview data files. “Misc. Expend.” is defined as CES total expenditure minus expenditures in the first five categories.

88

4 Description of Data Used from the Ongoing BLS Consumer Expenditure Surveys Table 4.3 Expenditure categories BLS consumer expenditure surveys

CES data sets

CES-ACCRA data sets

Food:

Food consumed at home Shelter Utilities Transportation Health care Miscellaneous expenditures: Food away from home Apparel and services Household operation Household furnishings and equipment Entertainment Care products and services Reading Education Tobacco products and supplies Miscellaneous Cash contributions Personal insurance and pensions

Food consumed at home Food away from home Alcoholic beverages Housing: Shelter: Owned dwellings Rented dwellings Other lodging Utilities: Natural gas Electricity personal Telephone services Water and other public services Household operation Household furnishings and equipment Apparel and services: Men and boys Women and girls Transportation: Gasoline and motor oil Health care Entertainment Personal care products and services Reading Education Tobacco products and supplies Miscellaneous Cash contributions Personal insurance and pensions

both price and total-expenditure elasticities are estimated in Chapters 6–10. Finally, the 29 categories of expenditure, which provide the focus of analysis in Chapter 11, are all of those (with exception of the aggregated category for food) listed at the far left.

Chapter 5

Stability of U.S. Consumption Expenditure Patterns: 1996–19991

A cornerstone of macroeconomic analysis since the publication of Keynes’s General Theory in 1936 has been the strong belief in a stable aggregate consumption function. At the micro level, there has been an equally strong belief in invariant individual tastes and preferences. The usual approach in testing for structural stability is to examine consumption, expenditure, or demand functions estimated over different time periods for evidence of changes in marginal propensities to consume, price and income elasticities, and other parameters. The analysis in this chapter takes a different tack. Rather than analyzing stability (or its absence) in terms of invariance in behavioral parameters (i.e., the coefficients in consumption, demand, or Engel functions), the focus is on direct relationships among exhaustive categories of expenditure, using household expenditure information from the ongoing quarterly BLS consumer expenditure surveys. Sixteen quarters of data for 1996 through 1999 are analyzed.2 The results provide strong empirical evidence in support of structural stability in underlying consumption relationships that account for about 85% of the variation in U. S. consumer expenditure.

5.1 Principal Component Analyses of 14 CES Expenditure Categories The data analyzed in the analysis are taken from household expenditure information that is collected quarterly in diary and interview surveys by the U. S. Bureau of Labor Statistics. The analysis proceeds via a principal component analysis of 14 exhaustive categories of consumption expenditure for each of the 16 quarters in

1

Most of the material in this chapter is taken from Taylor (2007). All data in this analysis are taken from the public use microdata CD-ROMs of Consumer Expenditure, 1996–1999, obtained from the Bureau of Labor Statistics, U.S. Department of Labor. Since the BLS surveys collect expenditure data only, the information analyzed is necessarily in current prices.

2

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_5, 

89

90 Table 5.1 Consumption categories BLS-CES quarterly surveys

5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999 Category

Mnemonic

Food Alcoholic beverages Housing Apparel Transportation Health Entertainment Personal care Reading Education Tobacco Miscellaneous Cash Contributions Pensions and insurance

Food Alco. bev. Housing Apparel Trans. Health Entertain. Per. care Reading Educ. Tobacco Misc. Cash ctrb. Pens. Ins.

the data set, and then by examining the stability of the underlying eigenvectors.3 The 14 categories of expenditure that are the focus of the analysis are listed in Table 5.1. To fix the technical ideas underlying the analysis, let X denote an n-by-m matrix of n observations on m variables, and suppose that we want to find an m-by-m matrix K that transforms the variables represented by the columns of X into a set of new variables Z = (Z1 , Z2 , . . . , Zm ) that are orthogonal to one another, viz., Z = XK,

(5.1)

Z Z = [λ] ,

(5.2)

such that

where [λ] is an m-by-m diagonal matrix. From equation (5.1), we then have for Z Z: Z Z = K X XK,

(5.3)

so that

3 It is important to note that the motivation for this approach, as opposed to working with expenditure vectors directly, is not to reduce dimensions of variation, but rather as a vehicle for discovering non-apparent linear relationships. If relationships among expenditures (no matter how hidden) are invariant over time, then so too will the associated underlying eigenvectors. Moreover, as will be seen, invariance in specific eigenvectors allows for certain proportions of expenditure variation across various categories of expenditure to be identified as representing stable characteristics of households’ underlying tastes and preferences (whatever these might be).

5.1

Principal Component Analyses of 14 CES Expenditure Categories

K X XK = [λ] .

91

(5.4)

Since X X is real, symmetric, and positive definite, it follows that the columns of K will be the (normalized) eigenvectors associated with the k latent roots of X X, which in turn are given by the diagonal elements of [λ]. To find the columns of Z (which are called the principal components of X X), we proceed as follows. Since K is an orthonormal matrix, K K = I, which means that the trace of Z Z will be equal to the trace of X X. This being the case, we can find for the “first” principal component (PC) of X X the z that makes the maximum contribution to the trace of Z Z, subject to the condition that k1 k1 = 1, that is, we want to maximize:    (z1 κ) = z 1 z1 − κ z 1 X Xz1 − 1 ,

(5.5)

with respect to z1 and κ, where κ is a Lagrangian multiplier associated with the constraint k1 k1 = 1. From the first-order conditions: 2z1 − 2κX Xz1 = 0

(5.6)

z 1 X Xz1 − 1 = 0,

(5.7)

z 1 z1 = κ.

(5.8)

we eventually find that

Since the z1 that is desired is the one that makes the maximum contribution to the trace of Z Z, which in turn is equal to the sum of the latent roots of X X, it accordingly follows that the k that yields the “first” PC will be the k that is associated with the largest latent root of X X. The “second” PC of X X will then be obtained as the z that makes the second largest contribution to the trace of Z Z, but now subject not only to k2 k2 = 1, but also to k2 k1 = 0. Skipping details, the k2 that yields this PC will be the eigenvector associated with the second largest latent root of X X. The remaining m−2 principal components are obtained accordingly. For the situation at hand, each row of X will represent the expenditures for a particular household in each of the 14 expenditure categories listed in Table 5.1, as taken from a particular quarterly BLS-CES survey. The latent roots (normalized so that they sum to one) for the 16 quarterly surveys analyzed are tabulated in Table 5.2.4 From the table, we see that the largest principal component (i.e., the PC associated with the latent root) accounts for about 60% of the total variation in

4

All mathematical and statistical calculations have been undertaken using SAS.

92

5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999 Table 5.2 Latent roots for 14 CES expenditure categories 1996–1999

Latent 1996Q1 root

1996Q2

1996Q3

1996Q4

1997Q1

1997Q2

1997Q3

1997Q4

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.60385 0.25284 0.04442 0.02493 0.02011 0.01746 0.01371 0.01047 0.00796 0.00280 0.00064 0.00052 0.00018 0.00011

0.61274 0.23209 0.06506 0.02908 0.01971 0.01307 0.01076 0.00666 0.00516 0.00439 0.00058 0.00045 0.00018 0.00008

0.57089 0.20789 0.12853 0.02905 0.02073 0.01308 0.00898 0.00829 0.00618 0.00499 0.00068 0.00047 0.00017 0.00008

0.55001 0.21223 0.14030 0.02769 0.02085 0.01742 0.01142 0.01059 0.00495 0.00332 0.00059 0.00039 0.00015 0.00008

0.61914 0.24360 0.03369 0.02302 0.01793 0.01698 0.01464 0.01076 0.00970 0.00883 0.00074 0.00061 0.00018 0.00017

0.62163 0.24700 0.03809 0.03222 0.01982 0.01050 0.01018 0.00891 0.00587 0.00415 0.00082 0.00053 0.00019 0.00010

0.60412 0.23415 0.07533 0.03197 0.01971 0.01015 0.00825 0.00785 0.00379 0.00333 0.00070 0.00040 0.00017 0.00009

0.60375 0.24056 0.03773 0.03247 0.02554 0.01893 0.01404 0.01173 0.00801 0.00572 0.00074 0.00053 0.00017 0.00010

Latent 1998Q1 root

1998Q2

1998Q3

1998Q4

1999Q1

1999Q2

1999Q3

1999Q4

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.61481 0.24573 0.03584 0.02919 0.02085 0.01608 0.01575 0.01080 0.00616 0.00353 0.00057 0.00043 0.00015 0.00012

0.58659 0.23366 0.06613 0.05235 0.02091 0.01192 0.00934 0.00898 0.00498 0.00396 0.00054 0.00039 0.00015 0.00009

0.59920 0.22984 0.06721 0.03528 0.02068 0.01701 0.00989 0.00761 0.00629 0.00568 0.00058 0.00045 0.00017 0.00008

0.60645 0.26096 0.03868 0.03051 0.01685 0.01496 0.00985 0.00873 0.00589 0.00561 0.00068 0.00051 0.00020 0.00011

0.54761 0.24881 0.10108 0.03848 0.01799 0.01159 0.01008 0.00839 0.00781 0.00694 0.00053 0.00043 0.00016 0.00008

0.52546 0.22686 0.15219 0.03249 0.02081 0.01833 0.00826 0.00737 0.00433 0.00303 0.00049 0.00034 0.00013 0.00006

0.61113 0.25281 0.03938 0.02498 0.02046 0.01569 0.01073 0.00991 0.00863 0.00492 0.00063 0.00050 0.00015 0.00009

0.60950 0.24208 0.03486 0.03225 0.02293 0.01592 0.01415 0.01283 0.00849 0.00544 0.00074 0.00053 0.00017 0.00013

expenditure, while the second largest principal component accounts for about 25%.5 In contrast, the five smallest PCs (i.e., those associated with latent roots 10 through 14) account for less than 1% of the total variation in expenditure. Given the minuteness of these roots, the variation they measure might be thought to be meaningless noise. However, this is not the case, for, as will be seen below, all of the “small” PCs can in fact be identified with specific categories of expenditure; #14, for example, 5

It is important to keep in mind that the 60 and 25% refer to the total variation in consumption expenditure (where “total variation” is defined as the sum of squared expenditures over all of the households in a sample over all 14 categories of expenditure), and accordingly does not refer to the proportion of total consumption that, on the average, is accounted for by the principal components in question. With regard to the latter, the largest principal component typically accounts for about 40% of the total expenditure, while the second largest accounts for about 10%.

5.1

Principal Component Analyses of 14 CES Expenditure Categories

93

which typically accounts for less than one-hundredth of one percent of the total variation in expenditure, is highly correlated with the expenditures for reading materials. But this is getting ahead of the story. Since the transformation from nonorthogonal X to orthogonal Z involves a fullrank linear transformation, the (linear) relationships among the columns of X will now be represented in the eigenvectors forming the columns of K.6 From this, it follows that the questions involving stability of expenditure patterns can equivalently be investigated in terms of the stability of the columns of K. Our (exceedingly simple) approach to investigating this stability has proceeded via a sequence of regression equations, in which the eigenvectors for a quarter are regressed on their counterparts for other quarters. High R2 s in these regressions will obviously be in support of stability. A total of 35 regressions have been estimated, of which 15 entail contiguous quarters and 5 (arbitrary) noncontiguous quarters. The final regressions involve a pooled framework to be discussed below. The R2 s for the contiguous and noncontiguous regressions are tabulated in Table 5.3. Eigenvectors are represented in columns in the table, while quarters are represented in rows. The very first element of the table (0.997) thus represents the R2 in the regression of the eigenvector associated with largest latent root for 1996Q2 on the same for 1996Q1. Similarly, for the noncontiguous entries, the first element (0.984) represents the R2 in the regression of the eigenvector associated with largest latent root for 1999Q1 on the same for 1997Q3. Since our concern is with eigenvector stability across time, what we obviously are looking for are columns with uniformly high R2 s. This is clearly evident in columns 1, 2, 13, and 14, and to a lesser extent in columns 5, 11, and 12. The principal components associated with these eigenvectors typically account for about 90% of the total variation in consumer expenditure, hence a great deal of stability in consumption patterns (at least by this measure) appears to be present. Additional evidence in support of stability is offered by a final set of 14 “pooled” contiguous-quarter eigenvector regressions, in which the coefficients on the “lagged” quarter are constrained to be equal across the 15 quarters from 1996Q2 through 1999Q4, which is to say that the equations estimated are of the form: ki j t = α + βki j (t−1) + ui j t ,

(5.9)

for i, j = 1, . . . ,14 and t = 1996Q2, . . . ,1999Q4. The R2 s for these 14 equations are tabulated in Table 5.4. The R2 s are seen to be very high for eigenvectors 1, 2, 13, and 14 (0.9600 or higher) and moderately high for numbers 5 and 11 (0.4550 and 0.7556).7 6

Orthogonal and non-orthogonal in this context refers to the columns of Z and X. “Seasonal” effects are allowed for in the equations through inclusion of three quarterly dummy variables, both singly and interacted with the lagged eigenvector. The only equations displaying any seasonal effects at all are for eigenvectors 2, 4, 5, 7, 8, and 12. The strongest seasonal effects are in the equation for number 12. For eigenvector 2, the only seasonal variable with a t-ratio greater than 2 is the linear term for the second quarter. 7

1

0.997 0.971 0.998 0.978 0.985 0.996 0.997 0.997 0.973 0.987 0.987 0.978 0.963 0.971

0.984 0.947 0.996 0.973 0.974

0.909 0.975 0.994 0.985 0.987

0.899 0.975 0.984 0.991 0.993 0.998 0.997 0.998 0.989 0.982 0.984 0.907 0.871 0.963

2

0.003 0.997 0.064 0.721 0.862

0.954 0.993 0.983 0.066 0.328 0.993 0.029 0.640 0.617 0.534 0.186 0.030 0.000 0.024

3

0.834 0.995 0.064 0.244 0.867

0.051 0.974 0.970 0.000 0.120 0.974 0.002 0.016 0.002 0.591 0.256 0.153 0.732 0.002

4

0.824 0.807 0.499 0.062 0.954

0.234 0.946 0.769 0.174 0.019 0.946 0.932 0.860 0.694 0.813 0.855 0.637 0.702 0.527

5

0.000 0.070 0.027 0.067 0.439

0.028 0.765 0.008 0.292 0.128 0.765 0.010 0.000 0.022 0.080 0.898 0.006 0.009 0.985

6

0.013 0.083 0.062 0.055 0.048

0.070 0.886 0.076 0.178 0.338 0.886 0.065 0.062 0.454 0.003 0.936 0.802 0.023 0.038

7

0.679 0.949 0.017 0.044 0.012

0.891 0.684 0.012 0.153 0.003 0.684 0.462 0.023 0.003 0.015 0.865 0.800 0.485 0.082

8

0.015 0.008 0.035 0.874 0.739

0.931 0.107 0.005 0.061 0.078 0.107 0.955 0.972 0.142 0.888 0.059 0.000 0.026 0.001

9

0.878 0.012 0.075 0.923 0.819

0.836 0.144 0.001 0.015 0.031 0.144 0.980 0.989 0.211 0.915 0.094 0.549 0.878 0.014

10

0.767 0.682 0.876 0.963 0.552

0.856 0.958 0.998 0.895 0.637 0.958 0.456 0.990 0.740 0.582 0.592 0.942 0.827 0.868

11

0.811 0.748 0.905 0.972 0.633

0.992 0.968 0.999 0.919 0.705 0.968 0.453 0.992 0.793 0.667 0.671 0.956 0.863 0.898

12

0.998 0.990 0.981 0.713 0.996

0.994 0.998 0.999 0.752 0.746 0.998 0.998 0.973 0.972 0.998 0.999 0.996 0.998 0.999

13

0.999 0.990 0.983 0.748 0.996

0.999 0.997 0.999 0.783 0.778 0.997 0.999 0.975 0.974 0.999 0.999 0.997 0.999 0.999

14

a The

contiguous equations have the form, ki j t = α + βki j (t– 1) + ui , where ki j t represents the jth element of the ith eigenvector in quarter t, for i, j = 1, . . ., 14 and t = 1996Q2, . . ., 1999Q4.

1999Q2/1997Q3 1998Q4/1996Q2 1999Q4/1998Q1 1998Q3/1997Q1 1997Q2/1996Q2

Noncontiguous Quarters

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q4 1999Q1 1992Q2 1999Q3 1999Q4

Contiguous quartersa

Eigenvector Quarter

Table 5.3 R2 s for Eigenvector regressions 14 CES expenditure categories 1996–1999

94 5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999

5.1

Principal Component Analyses of 14 CES Expenditure Categories

Table 5.4 R2 s for Pooled Eigenvector regressions 14 CES expenditure categories 1996–1999

95

Eigenvector

R2

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.9774 0.9708 0.2415 0.2377 0.4550 0.1087 0.0750 0.0951 0.1693 0.2022 0.7556 0.0728 0.9600 0.9626

The conclusion from Tables 5.3 and 5.4 (especially Table 5.4) would seem to be that there is something pretty special about the principal components associated with both the two largest and two smallest latent roots. The components associated with latent roots 5 and 11 appear to be somewhat special as well. One way of examining what might be going on with the principal components is to obtain the “factor” loadings for the PCs by regressing each of them on the fourteen underlying categories of expenditure as predictors. For illustration, “loadings” for the first and last quarters of the study are tabulated in Table 5.5.8 “Key” loadings are highlighted in bold print. The following results emerge from this table: (1) The stability of eigenvectors 1, 2, 13, and 14 is immediately apparent. (2) PC 2 is virtually identical with PC 1, except for a negative loading on transportation. (3) PCs 6–10 and 13 and 14 are all associated with a single expenditure category (cf. PC 7 with education in 1996Q1, PC 10 with apparel in 1999Q4, etc.). (4) The expenditure categories with high single loadings vary between quarters for PCs 6–10 (cf. for example, the loadings on education and health for PC 7 for 1196Q1 with the same for 1999Q4). PCs 13 and 14, on the other hand, are obviously associated with expenditures for personal care and reading materials, respectively, a result, incidentally, that holds for all 16 quarters of data.9 8

Since the principal components are (by construction) exact linear combinations of the 14 underlying categories of expenditure, the R2 s of the regressions will obviously all be equal to 1. Equally obviously, the resulting vectors of “factor loadings” simply reproduce the corresponding eigenvectors. However, formulating the principal components in regression terms has always seemed to us to enlighten interpretation. 9 Since the principal components (by construction) are orthogonal, expenditure categories with high single loadings imply separability in an underlying utility function, a point to be returned to below.

Food Alco. bev. Housing Apparel Transp. Health Entertain. Pers. care Reading Education Tobacco Misc. Cash ctrb. Pens. Ins.

1996Q1

Expenditure

2

3

4

5

6

7

8

0.235 0.137 0.532 0.452 −0.443 −0.456 −0.023 −0.095 0.015 0.009 0.027 0.005 −0.020 0.002 0.004 0.006 0.686 0.614 −0.382 −0.053 0.049 0.003 −0.030 −0.003 0.095 0.058 0.170 0.044 −0.102 0.076 0.069 0.077 0.643 −0.763 −0.068 −0.012 0.009 0.002 0.001 0.002 0.076 0.044 0.229 0.571 0.771 0.140 −0.002 −0.020 0.104 0.062 0.252 0.116 −0.336 0.873 −0.077 −0.044 0.014 0.008 0.022 0.012 −0.007 −0.002 0.001 0.002 0.009 0.007 0.015 0.000 −0.001 0.004 0.001 0.008 0.040 0.032 0.056 −0.028 −0.008 0.049 0.991 0.028 0.010 0.005 0.018 0.016 −0.012 −0.014 −0.005 −0.001 0.025 0.017 0.067 0.040 −0.026 −0.011 −0.041 0.990 0.012 0.010 0.020 −0.009 0.006 0.009 0.003 0.034 0.181 0.109 0.642 −0.670 0.287 −0.036 −0.068 −0.026

1

Principal component

−0.149 0.003 −0.028 0.963 −0.004 −0.005 −0.160 0.027 0.010 −0.093 −0.008 −0.097 0.018 −0.067

9

0.003 −0.005 −0.007 −0.025 0.001 −0.006 −0.007 0.002 0.007 −0.005 −0.003 −0.033 0.999 −0.022

10

−0.041 0.870 −0.004 −0.010 −0.001 0.002 −0.015 0.004 0.002 −0.004 0.491 −0.009 0.004 −0.010

11

−0.015 −0.492 0.001 0.009 −0.001 −0.004 0.007 −0.016 0.003 0.007 0.870 0.002 0.001 0.004

12

Table 5.5 Loadings of the principal components on 14 CES expenditure categories 1996Q1 and 1999Q4

−0.022 −0.012 −0.005 −0.035 −0.001 −0.008 −0.005 0.988 0.144 −0.000 0.011 0.004 −0.004 −0.007

13

−0.004 0.001 −0.004 −0.010 0.001 −0.003 −0.006 −0.145 0.989 −0.002 −0.005 −0.008 −0.007 −0.009

14

96 5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999

Food Alco. Bev. Housing Apparel Transp. Health Entertain. Pers. Care Reading Education Tobacco Misc. Cash Ctrb. Pens. Ins.

1999Q4

Expenditure

2

3

4

0.227 0.157 0.306 0.483 0.015 0.010 0.018 0.010 0.653 0.642 −0.326 −0.143 0.069 0.051 0.069 0.068 0.682 −0.729 −0.052 −0.022 0.074 0.046 0.083 0.128 0.095 0.056 0.150 0.114 0.013 0.009 0.011 0.015 0.008 0.005 0.010 0.005 0.046 0.045 0.056 0.767 0.010 0.005 0.013 0.012 0.036 0.058 −0.167 −0.172 0.016 0.016 0.021 −0.006 0.184 0.135 0.856 −0.304

1

Principal component 6

7

8

9

10

11

−0.627 0.187 −0.365 −0.139 −0.013 −0.112 −0.041 −0.019 0.002 −0.012 0.014 0.000 −0.018 0.712 0.100 −0.151 −0.006 −0.032 0.004 −0.035 −0.006 −0.072 0.004 −0.014 0.140 0.022 0.978 −0.014 0.030 0.004 −0.006 −0.010 0.003 0.002 −0.002 −0.258 0.169 0.897 −0.302 −0.042 0.015 −0.003 −0.109 −0.021 0.239 0.921 −0.082 −0.163 −0.018 −0.015 −0.001 0.005 −0.003 0.002 0.026 0.002 −0.007 0.083 0.010 0.007 0.002 0.018 0.009 0.629 0.049 0.045 −0.040 −0.009 −0.007 0.007 −0.026 0.003 −0.000 0.002 −0.000 −0.010 0.705 0.081 0.961 0.004 0.069 −0.058 0.012 0.003 −0.004 0.063 0.054 0.059 0.993 −0.036 −0.002 0.328 0.056 −0.015 −0.096 −0.036 −0.021 −0.006

5

Table 5.5 (continued)

−0.003 −0.701 0.004 0.021 0.000 −0.011 0.003 0.004 −0.017 0.007 0.713 0.000 −0.000 0.004

12

−0.020 0.001 −0.007 −0.031 −0.001 −0.015 −0.002 0.994 0.103 −0.003 −0.001 0.007 −0.002 −0.004

13

−0.004 −0.020 −0.002 −0.017 −0.000 −0.010 −0.009 −0.104 0.994 −0.000 0.006 0.018 −0.003 −0.005

14

5.1 Principal Component Analyses of 14 CES Expenditure Categories 97

98

5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999

(5) PCs 11 and 12, it will be noted, are clearly associated with alcoholic beverages and tobacco, for these are the only components that have nontrivial loadings on those two categories of expenditure. This unique association holds for both PC 11 and PC 12 over all 16 quarters of data, as do the positive loadings for PC 11. On the other hand, while the signs for the loadings on alcoholic beverages and tobacco for PC 12 are always opposite to one another (as in the table), their order (i.e., whether +, −, or −, +) is not constant across quarters.10

5.2 Interpretation of Results The principal-component/eigenvector analyses of the preceding section are essentially simply exercises in linear algebra. We now turn to a discussion and (attempted!) interpretation of the results that have been obtained to this point, beginning with the strong structural stability (across all 16 quarters of data) in the four principal components that account for between 85 and 90% of the total variation in U.S. household consumption expenditure. The four principal components in question are the two “largest” (PCs 1 and 2) and the two “smallest” (PCs 13 and 14). Since the latter account for just a minor fraction of the total variation in expenditure, the former are obviously of most interest. As was noted above, two things stand out in connection with the “expenditure loadings” for PCs 1 and 2 in Table 5.5. The first is simply the congruence of the loadings, except for a switch in signs on transportation! If at an extreme (i.e., if, except for the signs on transportation, the loadings on the two components were in fact identical), this result would have the following implications: (1) the sum of the two principal components would be independent (mathematically) of expenditures for transportation; while (2) the difference of the two components would be exactly colinear with transportation expenditures.11 Although this congruency seems almost too bizarre to be fortuitous, thoughts as to what might be being reflected behaviorally will be postponed till later. The second thing that stands out in Table 5.5 for the first two principal components is the number of nontrivial loadings in each of the first two columns, as opposed to the at most two large loadings in most of the other columns. One way that the nontrivial loadings for PCs 1 and 2 can be viewed is by identifying a “basic” market basket of goods and services, consisting of expenditures for food, housing, apparel, transportation, health, entertainment, education, and personal insurance. Food, shelter, clothing, and health are, of course, intrinsic to survival, as indeed

10

This “flipping of signs,” in Table 5.4, accounts for the R2 of 0.0728 for eigenvector 12, compared with the R2 of 0.7556 for eigenvector 11. 11 For the data actually at hand, the regression of PC 1−PC 2 on transportation expenditures for 1996Q1 yields an R2 of 0.9960, while the regression of PC 1+PC 2 on transportation expenditures has an R2 of 0.0094. For 1999Q4, the comparable R2 s are 0.9993 and 0.0257.

5.2

Interpretation of Results

99

in our modern age (though perhaps at a higher level of “want”) are certain levels of expenditure for transportation, entertainment, education, and insurance and pensions. Accordingly, in view of their strong structural stability over the 16 quarters of data, what might be being captured in the first two principal components is a substantial part of consumption expenditure that reflects stable genetic, cultural, and demographic influences. Since this last statement is quite speculative, let us be clear as to what is being said. By genetic influences, we have in mind the discussion in Sections 8 and 9 of Chapter 2 in connection with the Maslovian needs hierarchy, which, at the most basic level, can be seen as biologically determined and common to all individuals. On the other hand, by cultural influences, we have in mind a slowly varying set of factors that drive various forms of social consumption. The consumption governed by these influences can be seen as determining the “social subsistence” component of consumption. Finally, a third “subsistence” component can be seen as arising from the influence of a variety of time-varying demographic factors, such as age, education, family size, place of residence, etc. The thrust, accordingly, of the statement at the end of the last paragraph is that these three sets of factors (genetic, cultural, and demographic) are sufficiently invariant so as to impart a basic structural stability to the 85-plus percent of the variation in consumption expenditure that is accounted for by the two largest principal components. An obvious next step is to see how much of the sample variation in these two principal components can be explained, in a conventional regression format, by variation in income and socio-demographic factors. However, before proceeding to this, it will be useful to examine briefly the loadings for the remaining principal components in Table 5.5. PCs 11 and 12, as was noted at the end of the preceding section, are of interest because of their unique association with alcoholic beverages and tobacco. Since the loadings for PC 11 are both positive and generally of the same magnitude over all 16 quarters of data, this component can clearly be “identified” as an “alcohol-tobacco” component. PC 12, on the other hand, is another matter. For, while the loadings on alcoholic beverages and tobacco for this component are always the only nontrivial ones, and are always of opposite signs, the order of the signs is not stable. If PC 11 is seen as representing the expenditures of households on alcoholic beverages and tobacco that both drink and smoke, then PC might be interpreted as representing expenditures for those households that do one or the other (but not both). However, the problem with this interpretation is the switching of signs. What does it mean for alcohol to have a negative loading in one survey, but positive in another, and vice-versa for tobacco? Some insight into this last question may be obtained from consideration of the instability apparent in the loadings for the principal components 3 through 10. Of these eight PCs, #’s 3, 4, and 5 typically have two or more nontrivial loadings over the 16 quarters of data, while #’s 6 through 10 invariably have just a single high loading, single high loadings, incidentally, that are always confined to one of apparel, entertainment, health, education, miscellaneous, or cash contributions. The thing that comes to mind in connection with expenditures in these categories is that they

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5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999

tend to be “lumpy” with respect to both time and households.12 One household, for example, might show a large apparel expenditure in a particular quarter because of a change in employment; while a second household might show a large health expenditure because of an accident; a third household might show a large education expenditure because of two children being in college; while a fourth household could show a large cash contribution because of warm feelings toward the nursing home that a parent had lived in; and so on and so forth. Lumpiness, combined with a certain amount of inherent randomness, of such expenditures accordingly means that the relative variation in expenditures across the six categories in question can shift from quarter to quarter, which in turn means (since the “sizes” of the principal components are determined according to relative contributions to total variation) that expenditure categories need not always identify with the same principal components. Entertainment, for example, might identify with PC 6 in one sample, but with PC 7 in another (as the case with 1996Q1 and 1999Q4 in Table 5.5). Such considerations would seem to apply as well to PCs 3, 4, and 5, and possibly might even account for the sign switches on alcoholic beverages and tobacco with PC 12.

5.3 Regression Models for PCs 1 and 2 The primary result to this point is the isolation of two stable consumption substructures that account for between 85 and 90% of the variation in U.S. household consumption expenditure. In this section, the principal components of consumption that define these two substructures are taken as dependent variables to be “explained,” in a traditional regression framework, as functions of income and a variety of socio-demographic variables. The results for 1996Q1 and 1999Q4 are presented in Tables 5.6, 5.7, 5.8, and 5.9. Both linear and logarithmic equations are estimated. The estimated regression coefficients (together with their associated t-ratios and p-values) are tabulated in Tables 5.6 and 5.7 for the linear equations and in Tables 5.8 and 5.9 for the logarithmic models.13 Of the 23 independent variables in the models, all are dummy variables, except for income, the number of earners in the household (no_earnr), the age of the reference person for the household (age_ref), and household size (fam_size).14 In view of the superior fit for the

12

Another possibility [as suggested by a referee of the paper version of this chapter (Taylor (2007)] is that, since the expenditure data are in current prices, the instability in these principal components may reflect changes in relative prices. 13 The principal components for 1996Q1 and 1999Q4 have been calculated from samples consisting of 3,670 and 7,704 households, respectively. However, households with incomes less than $5000 are eliminated from the regression analyses, as are also the negative observations in the logarithmic equations for PC 2 (all observations are positive with PC 1). The resulting sample sizes are consequently 2,750 and 5,637 for 1996Q1 and 1999Q4 for the linear equations, and 2,433 and 5,022 for the logarithmic equations. 14 The independent variables in these regressions are as defined in Section 4.4.

5.3

Regression Models for PCs 1 and 2

101

Table 5.6 Regression models for PC 1 1996Q1 and 1999Q4 linear models 1996Q1 Variable

1999Q4 Parameter

t-ratio

p-value

intercept 376.17 0.30 0.7672 income 0.1264 17.45 < 0.0001 no_earnr 254.97 3.30 0.0010 age_ref −2.787 −0.75 0.4530 fam_size 236.69 3.95 < 0.0001 dsinglehh −272.02 −1.61 0.1077 drural −241.10 −1.38 0.1678 dnochild −208.12 −1.17 0.2431 dchild1 −6.929 −0.03 0.9748 dchild4 −173.17 −0.72 0.4746 ded10 402.19 1.60 0.1102 dedless12 520.44 0.45 0.6556 ded12 523.84 0.45 0.6523 dsomecoll 961.31 0.83 0.4094 ded15 1733.83 1.48 0.1382 dgradschool 1355.00 1.15 0.2489 dnortheast −294.53 −1.70 0.0887 dmidwest −371.10 −2.33 0.0198 dsouth −315.05 −2.05 0.0406 dwhite 533.27 1.86 0.0636 dblack 549.61 1.67 0.0944 dmale 219.14 1.82 0.0684 dfdstmps −767.69 −3.45 0.0006 d1 −19.06 0.06 0.9530 R2 = 0.2751 d.f. = 2726 Mean of PC 1: $3395

Variable

Parameter

t-ratio

p-value

intercept 692.94 0.67 0.5009 income 0.1392 27.93 < 0.0001 no_earnr 184.83 2.84 0.0045 age_ref 1.653 0.52 0.6016 fam_size 118.32 2.16 0.0308 dsinglehh −614.57 −4.31 < 0.0001 drural −208.46 −1.28 0.2013 dnochild −109.08 −0.74 0.4579 dchild1 100.64 0.51 0.6112 dchild4 −12.73 −0.06 0.9537 ded10 −381.10 −1.59 0.1123 dedless12 989.88 1.02 0.3094 ded12 1069.92 1.11 0.2690 dsomecoll 1651.84 1.71 0.0881 ded15 2028.42 2.09 0.0370 dgradschool 2722.23 2.78 0.0055 dnortheast −334.89 −2.34 0.0194 dmidwest −544.34 −4.08 <0.0001 dsouth −591.42 −4.75 <0.0001 dwhite 364.93 1.70 0.0886 dblack 143.25 0.56 0.5757 dmale −21.91 −0.23 0.8205 dfdstmps −723.16 −2.90 0.0037 d1 304.35 −1.81 0.0708 R2 = 0.2641 d.f. = 5613 Mean of PC 1: $3933

logarithmic models, the discussion that follows will focus primarily on Tables 5.8 and 5.9. For PC 1 in Table 5.8, after income, which not surprisingly is the strongest predictor, we find the expenditures represented by this component to be positively related to the number of earners in a household, family size, and education, and negatively related to single-person households, living in a rural area, living in the northeast, midwest, or south (as opposed to living in the west), and the receipt of food stamps. Moreover, the R2 for this equation is a very respectable (for a cross-section sample) of about 0.50. For PC 2 in Table 5.9, we again find a very strong effect of income, and strong negative effects associated with single households, rural households, and living in the midwest or south (again relative to living in the west). The R2 for this component, however, is much lower than for PC 1. Since the loadings for PC 1 and PC 2 for the most part differ only in the sign of the loading on transportation expenditures, this difference obviously has to be manifested somewhere, and is seen to reside principally in the change in sign on the number of earners in the household (with little loss in statistical significance), emergence of strong positive effects of children in the household (countered by a decrease in the importance of the raw size

102

5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999 Table 5.7 Regression models for PC 2 1996Q1 and 1999Q4 linear models

1996Q1 Variable

1999Q4 Parameter

intercept 1027.65 income 0.00728 no_earnr −200.92 age_ref 5.304 fam_size −40.13 dsinglehh −208.30 drural −168.74 dnochild −305.09 dchild1 157.98 dchild4 74.76 ded10 −479.07 dedless12 −361.81 ded12 −146.77 dsomecoll −111.31 ded15 73.14 dgradschool 477.90 dnortheast 189.55 dmidwest −317.26 dsouth −425.33 dwhite −92.38 dblack −286.95 dmale −80.40 dfdstmps 119.65 d4 217.66 R2 = 0.0525 mean of PC 2: $775

t-ratio

p-value

0.81 0.4203 7.02 < 0.0001 −2.59 0.0097 1.42 0.1549 −0.67 0.5042 −1.23 0.2197 −0.96 0.3362 −1.70 0.0883 0.72 0.4729 0.31 0.7584 −1.90 0.0580 −0.31 0.7574 −0.13 0.8999 −0.10 0.9242 0.06 0.9503 0.41 0.6854 1.09 0.2750 −1.99 0.0472 −2.76 0.0059 −0.32 0.7488 −0.87 0.3842 −0.67 0.5053 0.54 0.5918 0.67 0.5027 d.f. = 2726

Variable

Parameter

intercept 847.70 income 0.0624 no_earnr −213.69 age_ref 2.162 fam_size 20.60 dsinglehh 110.81 drural −331.83 dnochild −205.48 dchild1 216.71 dchild4 421.44 ded10 239.27 dedless12 −489.89 ded12 −183.46 dsomecoll −379.46 ded15 257.19 dgradschool 291.65 dnortheast 129.17 dmidwest −155.69 dsouth −314.92 dwhite −54.45 dblack 107.72 dmale −24.33 dfdstmps −79.86 d4 214.13 R2 = 0.0560 mean of PC 2: $1054

t-ratio

p-value

0.85 0.3957 12.87 < 0.0001 −3.39 0.0007 0.70 0.4811 0.39 0.6982 0.80 0.4231 −2.10 0.0359 −1.44 0.1492 1.13 0.2588 1.98 0.0475 1.03 0.3038 −0.52 0.6038 −0.20 0.8450 −0.40 0.6861 0.27 0.7850 0.31 0.7588 0.93 0.3521 −1.20 0.2285 −2.61 0.0091 −0.26 0.7932 0.43 0.6642 −0.26 0.7950 −0.33 0.7410 1.31 0.1898 d.f. = 5613

of the household), and a greatly reduced negative effect of food stamps. However, the result that perhaps most leaps out of the columns in Tables 5.8 and 5.9, is the virtually identical income elasticities for the two principal components, at a value of about 0.45.15

5.4 Summary and Conclusions The analysis in this chapter has involved a detailed examination of the stability of U.S. household consumption patterns by employing a combined principalcomponent/regression analysis

15

The income elasticities for the two PCs over the 16 quarters of data vary from 0.39 to 0.46, and never differ in any quarter by more than 0.03.

5.4

Summary and Conclusions

103

Table 5.8 Regression models for PC 1 1996Q1 and 1999Q4 logarithmic models 1996Q1

1999Q4

Variable

Parameter

t-ratio

p-value

intercept income no_earnr age_ref fam_size dsinglehh drural dnochild dchild1 dchild4 ded10 dedless12 ded12 dsomecoll ded15 dgradschool dnortheast dmidwest dsouth dwhite dblack dmale dfdstmps d1 R2 = 0.5066

3.0146 0.4371 0.0563 −0.0018 0.0470 −0.1695 −0.0774 −0.0532 0.0250 −0.0098 0.0479 0.2015 0.2481 0.3521 0.4989 0.4461 −0.0589 −0.1164 −0.0686 0.0410 0.0475 0.0529 −0.1722 −0.0293

10.44 < 0.0001 24.68 < 0.0001 3.73 0.0002 −2.59 0.0096 4.12 < 0.0001 −5.19 < 0.0001 −2.33 0.0200 −1.57 0.1166 0.60 0.5487 −0.21 0.8306 0.20 0.8444 0.91 0.3635 1.12 0.2619 1.59 0.1123 2.24 0.0251 1.99 0.0464 −1.79 0.0738 −3.85 0.0001 −2.35 0.0191 0.88 0.3795 0.76 0.4465 2.30 0.0215 −4.01 < 0.0001 0.48 0.6340 d.f. = 2409

Variable

Parameter

t-ratio

p-value

intercept income no_earnr age_ref fam_size dsinglehh drural dnochild dchild1 dchild4 ded10 dedless12 ded12 dsomecoll ded15 dgradschool dnortheast dmidwest dsouth dwhite dblack dmale dfdstmps d4 R2 = 0.4823

2.7709 0.4491 0.0518 −0.0000 0.0343 −0.1718 −0.1034 −0.0412 0.0182 0.0128 −0.1324 0.3625 0.4193 0.5286 0.6570 0.7546 −0.0781 −0.1241 −0.1581 0.0659 0.0043 0.0075 −0.2071 −0.0406

13.39 < 0.0001 36.89 < 0.0001 4.68 < 0.0001 −0.02 0.9824 3.76 0.0002 −7.12 < 0.0001 −3.80 0.0001 −1.68 0.0927 0.55 0.5810 0.35 0.7258 −3.31 0.0009 2.23 0.0255 2.60 0.0094 3.27 0.0011 4.05 < 0.0001 4.62 < 0.0001 −3.27 0.0011 −5.58 < 0.0001 −7.62 < 0.0001 1.85 0.0650 0.10 0.9204 0.46 0.6446 −4.93 < 0.0001 −1.45 0.1478 d.f. = 4998

(1) Five stable consumption structures are isolated that regularly account for between 85 and 90% of the total variation in 14 (exhaustive) categories of consumption expenditure. (2) Four of the structures in question are associated with both the two “largest” and the two “smallest” principal components of consumption expenditure. The largest principal component typically accounts for about 60% of the total variation in expenditure, while the second largest component accounts for another 25%. At the other extreme, the two smallest components typically account for less than one-half of one percent of the total variation. (3) The two largest principal components are stable across several categories of expenditure, while the two smallest components each identify with just a single category of expenditure. The fifth stable principal component identifies with expenditures for alcoholic beverages and tobacco. (4) Except for opposing signs on transportation expenditures, the two largest components have virtually identical loadings on the 14 categories of expenditure.

104

5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999 Table 5.9 Regression models for PC 2 1996Q1 and 1999Q4 logarithmic models

1996Q1

1999Q4

Variable

Parameter

t-ratio

p-value

intercept income no_earnr age_ref fam_size dsinglehh drural dnochild dchild1 dchild4 ded10 dedless12 ded12 dsomecoll ded15 dgradschool dnortheast dmidwest dsouth dwhite dblack dmale dfdstmps d1 R2 = 0.2358

2.6381 0.4484 −0.0674 0.0005 0.0293 −0.0663 −0.2581 −0.1286 0.1249 0.1369 −0.0909 −0.0016 −0.0281 0.0357 0.3491 0.2292 0.0174 −0.2002 −0.1907 −0.0385 −0.0696 −0.0576 −0.0751 −0.2073

5.64 < 0.0001 15.13 < 0.0001 −2.63 0.0086 0.41 0.6784 1.48 0.1384 −1.22 0.2238 −4.54 < 0.0001 −2.25 0.0245 1.81 0.0709 1.79 0.0739 −1.14 0.2560 −0.00 0.9962 −0.08 0.9354 0.10 0.9182 0.65 0.5145 0.65 0.5141 0.32 0.7456 −3.99 < 0.0001 −3.91 < 0.0001 −0.43 0.6707 −0.67 0.5024 −1.50 0.1335 −1.07 0.2849 −2.03 0.0427 d.f. = 2409

Variable

Parameter

t-ratio

p-value

intercept income no_earnr age_ref fam_size dsinglehh drural dnochild dchild1 dchild4 ded10 dedless12 ded12 dsomecoll ded15 dgradschool dnortheast dmidwest dsouth dwhite dblack dmale dfdstmps d4 R2 = 0.2737

2.4847 0.4379 −0.0527 −0.0003 0.0419 0.0331 −0.2670 −0.0990 0.1871 0.1894 −0.0532 0.0296 0.0270 0.1252 0.3489 0.5086 0.0351 −0.0932 −0.1633 0.0546 0.0963 −0.0336 −0.0667 −0.0228

8.15 < 0.0001 24.06 < 0.0001 −3.15 0.0016 −0.38 0.7068 3.03 0.0024 0.92 0.3569 −6.47 < 0.0001 −2.69 0.0071 3.80 0.0001 3.48 0.0005 −0.90 0.3682 0.12 0.9010 0.11 0.9090 0.53 0.5963 1.47 0.1416 2.13 0.0335 0.99 0.3209 −2.81 0.0050 −5.28 < 0.0001 1.01 0.3133 1.50 0.1336 −1.39 0.1640 −1.08 0.2814 −0.54 0.5864 d.f. = 4998

(5) Virtually identical, as well, are the income elasticities of demand for the two largest principal components, with values that vary between 0.39 and 0.46 over the 16 quarters of CES data.

Let us now speculate a bit (and we want to emphasize that it is speculation) about what all this might mean. As was noted in Section 3, one interpretation of the stability of the two largest principal components of consumption is that the expenditure structures represented in these components derive from three (basically invariant) motivating bases (or substrates) of behavior:

(i) Biological (i.e., genetic) factors that define certain levels of expenditure for food, housing, clothing, transportation, health, education, and entertainment. (ii) Cultural factors that give rise to a variety of social patterns of consumption. (iii) Demographic factors such as age- and ethnic-distributions of the population, labor-force participation, place of residence, etc.

5.4

Summary and Conclusions

105

Of the expenditures associated with these factors, those of biological origin should obviously be more invariant (since they drive from a shared genetic basis) than those associated with cultural and demographic factors. Nevertheless, over moderate periods of time (such as the 4 years represented in the 16 quarters of data analyzed in this study), invariance ought to apply to cultural and demographic factors as well. On an average, about 50% of the total consumption expenditure is associated with the two largest principal components. As has just been suggested, these expenditures can be identified with tastes and preferences that are (1) common to individuals (i.e., genetically based), or (2) reflect stable cultural and demographic agglomerations. The remaining half of the total expenditure (under this interpretation) can therefore be attributed to those aspects of tastes and preferences that vary across individuals and households as represented in the structures of the principal components (specifically, PCs 3 through 10) that vary from quarter-to-quarter depending upon the idiosyncracies of particular surveys. A household’s consumption behavior, can accordingly be viewed as emanating from four distinct substrates: (1) a genetic substrate that is common to households; (2) a cultural substrate that is stable (over moderate periods of time) across households; (3) a demographic substrate that varies across households, but which is distributionally stable (again, over moderate periods of time); and (4) an idiosyncratic substrate that reflects genetic and experiential variation across households. For the 16 quarters of data that have been analyzed in this study, substrates (1), (2), and (3) can be seen as being represented in the principal components 1, 2, 11, 13, and 14, while substrate (4) would appear to be represented in the principal components 3–10 and 12. This interpretation of the results of this chapter, if valid, might accordingly be seen as having the following implications: (1) The genetic factors ingrained in the two largest principal components of consumption should be constant both over time and across cultures. However, this should not be the case for the cultural or demographic factors. Hence, because of slowly occurring changes in the latter factors, the relationships between the eigenvectors associated with the largest two principal components of consumption for any two points in time should be weaker the greater the temporal separation. Similarly, one should be expected to find weaker relationships between eigenvectors (at the same point in time) across countries than within countries. (2) The proportion of total variation in consumption expenditures accounted for by the two largest principal components ought, in general, to be a decreasing function of the level of income. There are two aspects to this implication. The first is simply the idea that, as income increases, genetically motivated consumption will probably be subject to satiation, implying a low income elasticity, which in turn would imply reduced relative variation in this expenditure across households. The second aspect is that, as the “core” expenditures associated with the two largest principal components of consumption decrease with income as a proportion of total expenditure, the individual idiosyncrasies of households

106

5 Stability of U.S. Consumption Expenditure Patterns: 1996–1999

should become increasingly important. Thus, not only will the “core” constituents of expenditure claim a decreasing proportion of total expenditure as a function of income for a given relative variation in consumption expenditures, but the variation in “non-core” expenditures will itself become a relatively more important part of the total.16 (3) Since the principal components are (by construction) orthogonal, their loadings on individual expenditure categories provide information concerning separability of consumers’ underlying utility functions (assuming that such in fact exist). Taking the high stable loadings for the two largest principal components at face value, what these loadings suggest is that functional structure in utility functions, rather than being pursued in terms of commodities (food, shelter, etc.), might better be approached in terms of underlying motivating substrates (biological, cultural, etc.).17 In closing, let us return to the question of stability in the aggregate consumption function that introduced this chapter. In approaching this question, we must be careful to distinguish between two different concepts of stability, one that refers to the structure of tastes and preferences and another that refers to the relationship between aggregate consumption and aggregate income. The most important finding of the paper would seem to be that, with reference to tastes and preferences, there exist two stable structures of consumption that, at a micro level, account for about 50% of total expenditure. In turn, these two structures are shown to have income elasticities, both of which are of the order of 0.45, that show little variation over the 16 quarters of data that are analyzed. The suggestion, accordingly, is that roughly 50% of total consumption expenditure can be said to have a simple stable relationship with income. Whether the micro-based results of this paper extend to the relationship between aggregate consumption and aggregate income is, of course, another matter. The thought that they might is rich with possibilities, but for now will be put to the side until Chapter 19.

16 However, the implication in this paragraph is not as clear-cut as it might seem because of the possibility of culturally based “habit formation” effects becoming increasingly more important with the increase in income. Interestingly, however, it will be noted in Chapter 16 that the proportion of the total consumption expenditure in the U.S. associated with habit formation was about the same, in 2006 as in 1964. 17 The utility function in the literature that might provide a point of departure for incorporating this notion is the Stone–Geary utility function with its “minimum required quantities.” See Stone (1954) and Chapter 8 below.

Chapter 6

Price and Income Elasticities Estimated from BLS Consumer Expenditure Surveys and ACCRA Price Data: Some Preliminary Results

This chapter represents our initial effort to estimate both price and income elasticities for several broad categories of consumer expenditure from cross-sectional data sets that combine price information that is collected quarterly for more than 300 cities and urban areas in the U.S. by American Chambers of Commerce Research Association (ACCRA)1 with individual household consumer expenditure data from quarterly consumer expenditure surveys that are conducted by the U.S. Bureau of Labor Statistics. Sixteen quarters of data are analyzed (1996–1999) for six categories of expenditure, namely, food consumed at home, housing, utilities, transportation, health care, and miscellaneous expenditures. Given that this is a first effort, the exercise is accordingly more concerned with feasibility than with theoretical or econometric elegance. Among other things, only simple double-logarithmic demand functions are estimated, and the econometrics do not extend beyond ordinary least squares. Nevertheless, the results obtained make intuitive sense, and suggest that integrating price information with household expenditure surveys is worthy of continuing research.

6.1 Background and Merging of Data Sets Household budget surveys have had a variety of uses in the long and venerable history, ranging from concern with the “state of the poor” in the late 18th- and mid-19th-century England and Continental Europe to a need for weights to be used in construction of consumer price indices.2 For economists, the principal use of data from household budget surveys has usually been the analysis of relationships between consumption expenditures and income [i.e., in the analysis of what, 1

Now Council for Community and Economic Research (C2LI). The standard reference for the history of early empirical studies of consumer behavior using data from household budget surveys is Stigler (1954); see also Houthakker (1957). Important 20thcentury studies with a family budget focus include Allen and Bowley (1935), Zimmerman (1935), Singer (1937), Schultz (1938), Prais and Houthakker (1955), Deaton and Muellbauer (1980a), and Pollak and Wales (1992), and Deaton (1997).

2

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_6, 

107

108

6 Price and Income Elasticities – Some Preliminary Results

since Engel (1857), have been known as Engel Curves]. Since most budget surveys collect only expenditure data, rather than both quantities and prices, it is generally not possible, absent heroic theoretical assumptions on the structure of consumer preferences, to estimate full-blown demand functions, and hence to obtain estimates of both income and price elasticities.3 In the absence of information on prices, estimation of demand functions using expenditure data obviously requires price data from some other source. For the BLS-CES surveys, the natural place to turn for such data is in the price surveys that the Bureau of Labor Statistics pursues monthly as input into the construction of consumer price indices. Prices for several hundred categories of expenditure for some 140 urban areas are collected in these surveys, so that cross-sectional price variation is in principle available. However, the problem is that indices reflecting areal variation in price levels at a point in time are not currently constructed by BLS, but rather only indices that measure price variation over time. Thus, the fact that the BLS all-items index for October, 2003, is 190.3 for Philadelphia and 196.3 for San Francisco cannot be interpreted as saying that the all-items CPI was 1.03% higher in San Francisco than in Philadelphia, but only that the all-items index in Philadelphia was 190.3% higher in October, 2003, than it was during the base years of 1982–1984, and similarly for San Francisco. Thus, the areal price indices that are currently constructed by BLS unfortunately cannot serve the need at hand. A second source of price information is in surveys that are conducted quarterly by ACCRA in 320 or so U.S. cities.4 Prices are collected by ACCRA for about 60 items of consumption expenditure, from which city-specific indices can be constructed that can be used to measure price differences both through time for a specific city and across cities at a point in time. In principle, this is precisely the form of price information that is required. From the 60 or so items for which price data are collected, ACCRA constructs indices for six broad categories of expenditure, namely, groceries, housing, utilities, transportation, health care, and miscellaneous. The items underlying the six ACCRA categories are those given in Table 4.1. In the analyses to follow, the six ACCRA categories are allied with comparable categories in the BLS-CES surveys. In particular, the ACCRA category “groceries” is identified with the CES category “food consumed at home,” while the other four specific ACCRA categories are identified with CES counterparts of the same name. Finally, the ACCRA miscellaneous category is identified with CES total expenditure

3 The reference here is to price elasticities estimated from conventional household budget surveys. Deaton (1990) provides an exception. In contrast, estimation of price elasticities for goods, such as telephone or utility services, in which the data used in estimation are collected from the records of vendors or from the actual bills of consumers are fairly commonplace. Cf. Taylor and Kridel (1993) and Rappoport and Taylor (1997). 4 See www.coli.org.

6.1

Background and Merging of Data Sets

109

minus the sum of expenditures for the first five categories. Since, to protect confidentiality, place of residence in the CES samples is specified only in terms of state and size of urban area, the ACCRA city price indices have had to be aggregated to a state level. Weights used in the aggregation are city population from the U.S. Census of 2000. The resulting state-level price indices are then attached to households in the CES samples according to states of residence. While attaching prices from ACCRA surveys to the CES samples in the manner described yields a cross-sectional consumption data set in which both price and income elasticities can be estimated, it is important to keep in mind that any attempt to extract price elasticities from household budget data, not just the effort in this and subsequent chapters, is laden with difficulties. The easiest case, of course, is where a good is both narrowly defined and homogeneous, and the price variation is due solely to price differences between regions. In this circumstance, the problem is simply one of obtaining an appropriate set of prices. With nonhomogeneous goods, on the other hand, the situation is much more complicated. For not only does price become ambiguous, but so too does the concept of quantity. Quality differences, which are almost always present in some degree in consumer expenditure data, are especially troublesome in this regard, as is also nonhomogeneity arising from broad categories of goods. Not surprisingly, both problems have attracted a great deal of attention in the literature.5 Finally, a third form of price variation that warrants consideration is that caused by regional differences in the cost of living. A haircut, for example, may be more expensive in New York City than in Wichita, in part because of scarcity, but in part also because of differences in the cost of living. As noted, the ideal circumstance (at least in principle) is where goods are narrowly defined and homogeneous (i.e., no grouping or quality gradations), and the price variation is due entirely to different prices for the same good (i.e., no cost-ofliving effects). The task in this situation is simply to match expenditures for each household with the prices that the households paid. Since expenditure is quantity times price, it obviously does not matter whether consumption is measured in terms of quantity or expenditure. Price and expenditure elasticities can be translated into one another through the addition or subtraction of 1. Unfortunately, however, the ideal circumstance just described is obviously not the one at hand. Consumption categories in the CES surveys are not narrowly defined, quality gradations are almost certainly present, and the same is true of regional differences in the cost of living. While efforts are made in the analyses to mitigate the problems that these lapses entail, notions that the price elasticities obtained are the clean, pristine ones of theory have to be put to the side.6

5 6

An early (but still relevant) discussion of these problems is given in Prais and Houthakker (1955). Possible dynamical effects in budget survey data will be addressed in Chapter 10.

110

6 Price and Income Elasticities – Some Preliminary Results

6.2 Models Estimated For convenience and ease of estimation and interpretation, the model employed in this chapter is simple double-logarithmic equations that relate expenditure to income, price, and a variety of socio-demographical variables (most of which are dummy variables) as follows: ln E = a + b ln y + c ln p + dz + u,

(6.1)

where E, y, p, and u denote expenditure for the category in question, total expenditure as a measure of income,7 ACCRA price, and a random error term, respectively, and z represents a set of socio-demographical variables (age, labor-force status, family size, education, etc.).8 The econometric procedure, using ordinary least squares, has been to apply the model in equation (6.1) to each of the six categories of CES expenditure (food consumed at home, shelter, utilities, transportation, health care, and miscellaneous expenditures) to the 16 quarterly CES surveys between 1996 and 1999. Since E denotes expenditure, and can thus be written as the product of quantity times price, an alternative representation of equation (6.1) is ln Q = a + b ln y + (c − 1) ln p + dz + u,

(6.2)

where Q denotes a pseudo quantity index, defined as Q = E/p. Since ln q in expression (6.2) is equal to ln E–ln p, models (6.1) and (6.2) are obviously equivalent, the only difference being that the coefficient attaching to ln p in equation (6.1) represents an expenditure elasticity, while c–1 in equation (6.2) represents the more conventional price elasticity. Equation (6.2) will accordingly be the focus of attention. OLS estimates of the price and total expenditure elasticities from equation (6.2) for the six categories and 16 quarters of expenditure are tabulated in Tables 6.1, 6.2, 6.3, 6.4, 6.5, and 6.6. Means and standard deviations for each of the 16 estimates are given at the bottom of the tables.9 Total expenditure elasticities from equations in which the price variables are excluded are presented as well. The latter equations are

7

Use of total expenditure, rather than disposable income, as the budget constraint (as was done in the two earlier editions of CDUS) is now standard practice in applied demand analysis, and has both practical and theoretical justifications. Its practical justification is that, at least over short periods of time, consumers have more control over their expenditures than over their receipts of income, so that total expenditure is the better measure of “true” income, which then be given either a permanent-income or a life-cycle interpretation. 8 A full listing of the socio-demographical variables that are included as predictors is given in Section 4.4. 9 The means are the simple means of the 16 quarterly estimates, while the standard deviations are the sample standard deviations of each “sample” of 16 estimates (rather than means of the estimated standard deviations).

6.2

Models Estimated

111

Table 6.1 Food consumed at home (t-ratios in parentheses) ln Q = α + βln y + γ ln p + . . .

ln E = α + βln y + . . .

Yr./Qtr.

Tot. exp.

Price

R2

Tot. exp.

R2

#Obs.

1996Q1

0.3158 (15.74) 0.2782 (16.47) 0.3019 (18.29) 0.3193 (18.97) 0.3417 (20.67) 0.3268 (18.70) 0.3081 (18.73) 0.3344 (20.65) 0.3559 (21.22) 0.3453 (20.36) 0.2954 (17.81) 0.2990 (17.54) 0.3236 (21.80) 0.3098 (21.96) 0.2882 (21.17) 0.3368 (24.80) 0.3175 0.0220

−0.2636 (−1.71) −0.2737 (−1.78) −0.4009 (−2.81) −0.2409 (−1.46) −0.2182 (−1.37) −0.2241 (−1.29) −0.0472 (−0.27) −0.4310 (−2.90) −0.7242 (−4.79) −0.3724 (−2.22) −0.5176 (−3.16) −0.5322 (−3.57) −0.3707 (−3.34) −0.4030 (−3.28) −0.5198 (−3.75) −0.57.35 (−5.45) −0.3821 0.1686

0.4859

0.3233 (16.08) 0.2826 (16.70) 0.3050 (18.43) 0.3244 (19.26) 0.3490 (21.12) 0.3333 (19.09) 0.3162 (19.22) 0.3399 (21.02) 0.3591 (21.52) 0.3493 (20.59) 0.2982 (17.99) 0.3037 (17.86) 0.3311 (22.32) 0.3154 (22.30) 0.2906 (21.36) 0.3407 (25.10)

0.4852

2189

0.4431

3054

0.4377

3312

0.4234

3373

0.4337

3408

0.4195

3400

0.4396

3415

0.4458

3478

0.4507

3502

0.4265

3432

0.4117

3419

0.4105

3332

0.4261

4257

0.4150

4673

0.4155

4549

0.4484

4538

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.4434 0.4436 0.4257 0.4383 0.4210 0.4391 0.4487 0.4568 0.4288 0.4159 0.4184 0.4292 0.4164 0.4169 0.4533

included as a check on bias that might arise in situations in which price information is not available. In assessing the results in these tables, reservations concerning the appropriateness of the ACCRA price indices for the tasks at hand will, for now, be put to the side. The key results in these tables are as follows: (i) Price effects are strong, both numerically and statistically, for all six categories of expenditure. With the exception of transportation and miscellaneous expenditures, and to a lesser extent for food consumed at home, the estimated price elasticities are generally stable (as measured by the sample standard deviations of each of the 16 elasticity estimates) over the 16 quarters of data. Estimated

112

6 Price and Income Elasticities – Some Preliminary Results Table 6.2 Shelter expenditures (t-ratios in parentheses) ln Q = α + βln y + γ ln p + . . .

ln E = α + βln y + . . .

Yr./Qtr.

Tot. exp.

Price

R2

Tot. exp.

R2

#Obs.

1996Q1

0.8205 (26.58) 0.8809 (34.08) 0.8380 (33.07) 0.8527 (34.25) 0.8501 (32.29) 0.8310 (31.96) 0.9076 (35.20) 0.8443 (33.45) 0.9065 (35.12) 0.8198 (33.37) 0.7952 (31.56) 0.8515 (34.00) 0.8716 (37.41) 0.8400 (38.72) 0.8400 (40.43) 0.8641 (41.55) 0.8509 0.0302

−0.4825 (−6.19) −0.3273 (−4.69) −0.5351 (−8.62) −0.7249 (−15.03) −0.5580 (−8.57) −0.4751 (−6.22) −0.4010 (−4.94) −0.4505 (−5.69) −0.5244 (−6.75) −0.5734 (−7.65) −0.5981 (−7.92) −0.6235 (−8.66) −0.7483 (−14.19) −0.6771 (−10.67) −0.6275 (−10.74) −0.7365 (−15.08) −0.5665 0.1221

0.4800

0.8383 (26.99) 0.8918 (34.05) 0.8440 (33.03) 0.8587 (34.35) 0.8596 (32.48) 0.8445 (32.35) 0.9221 (35.58) 0.8580 (33.86) 0.9191 (35.54) 0.8264 (33.52) 0.8026 (31.77) 0.8629 (34.45) 0.8806 (37.82) 0.8484 (39.11) 0.8469 (40.63) 0.8724 (41.93)

0.4522

2172

0.4410

3253

0.4507

3287

0.4555

3345

0.4395

3389

0.4527

3394

0.4720

3398

0.4533

3480

0.4662

3486

0.4562

3436

0.4214

3407

0.4494

3316

0.4276

4248

0.4294

4652

0.4425

4524

0.4562

4500

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.4681 0.4792 0.4945 0.4650 0.4581 0.4826 0.4583 0.4705 0.4663 0.4331 0.4552 0.4451 0.4453 0.4515 0.4823

price elasticities are, for the most part, of the order of –0.40 for food, −0.75 for shelter, −0.95 for utilities, and −1.10 for health care.10 (ii) Total expenditure effects are even stronger statistically, and highly stable through time. Estimated total expenditure elasticities are generally of the order of 0.30 for food, 0.85 for housing, 0.40 for utilities, 1.35 for transportation, 0.50 for health care, and 1.10 for miscellaneous expenditures.

10

The estimated price elasticities for transportation and miscellaneous expenditures, probably due to the fewness of items included in the corresponding ACCRA price indices, show too much variation to be explicit concerning a stable central tendency.

6.2

Models Estimated

113

Table 6.3 Utilities expenditures (t-ratios in parentheses) ln Q = α + βln y + γ ln p + . . .

ln E = α + βln y + . . .

Yr./Qtr.

Tot. exp.

Price

R2

Tot. exp.

R2

#Obs.

1996Q1

0.4166 (17.18) 0.3479 (16.87) 0.3558 (18.10) 0.3872 (18.49) 0.4098 (21.07) 0.4234 (19.56) 0.4009 (20.19) 0.4013 (20.44) 0.3907 (19.26) 0.3803 (18.94) 0.3371 (17.80) 0.3522 (18.49) 0.4078 (23.93) 0.3618 (22.00) 0.3293 (21.47) 0.3887 (24.28) 0.3807 0.0296

−1.0866 (−9.81) −0.8029 (−8.33) −0.8331 (−10.01) −1.0621 (12.96) −1.0626 (−12.72) −1.0344 (−11.19) −0.7322 (−9.25) −0.7661 (−8.50) −0.9592 (−11.84) −0.8534 (−9.87) −0.9908 (−11.30) −0.8420 (−11.14) −1.0625 (−14.56) −1.1279 (−15.84) −1.0643 (−15.51) −0.9846 (−13.63) −0.9540 0.1287

0.4619

0.4171 (17.20) 0.3478 (16.85) 0.3564 (18.13) 0.3864 (18.48) 0.4086 (21.08) 0.4231 (19.56) 0.4038 (20.32) 0.4040 (20.59) 0.3912 (19.31) 0.3822 (19.06) 0.3372 (17.82) 0.3545 (17.24) 0.4068 (23.93) 0.3601 (21.92) 0.3284 (21.45) 0.3890 (24.35)

0.4340

2174

0.3832

3258

0.3802

3284

0.3898

3353

0.4219

3370

0.3783

3385

0.3945

3374

0.3921

3465

0.4203

3478

0.3963

3426

0.4056

3398

0.3873

3308

0.4103

4224

0.3901

4651

0.3659

4499

0.4029

4488

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. dev.

0.3972 0.4302 0.4117 0.4609 0.4207 0.4379 0.4306 0.4496 0.4220 0.4430 0.4279 0.4374 0.4223 0.4174 0.4411

(iii) Comparison of columns 2 and 5 in the tables shows that, for these data sets, estimates of total expenditure elasticities are virtually unchanged by the presence or absence of price as a predictor. Two things, in our view, stand out about these results, namely, the statistical strength of the estimated elasticities and their stability over all of the 16 quarters of data. The magnitudes of the vast majority of t-ratios associated with the price elasticities are really pretty amazing (at least to us), for while high t-ratios are in general the norm with income (or total expenditure) elasticities, this is definitely not the case for price elasticities. Indeed, if our own experience in trying (for now more than 100 combined years!) to coax price elasticities out of time-series data is any

114

6 Price and Income Elasticities – Some Preliminary Results Table 6.4 Transportation expenditures (t-ratios in parentheses) ln Q = α + βln y + γ ln p + . . .

ln E = α + βln y + . . .

Yr./Qtr.

Tot. exp.

Price

R2

Tot. exp.

R2

#Obs.

1996Q1

1.2643 (35.02) 1.3874 (46.77) 1.3418 (47.77) 1.3482 (46.69) 1.3168 (43.60) 1.4231 (48.21) 1.3617 (47.45) 1.3625 (47.74) 1.2589 (43.00) 1.4084 (48.51) 1.3719 (48.45) 1.3423 (45.58) 1.3281 (49.53) 1.3535 (54.77) 1.3786 (56.49) 1.3405 (55.12) 1.3486 0.0445

−1.9735 (−5.87) −1.2211 (−3.89) −1.7018 (−6.60) −0.9323 (−9.96) −2.1704 (−5.18) −1.8912 (−5.13) −1.0356 (−2.74) −1.1157 (−2.99) −1.0742 (−3.31) −1.4878 (−4.04) −1.3145 (−4.22) −1.3306 (−3.91) −1.4461 (−4.64) −0.9164 (−3.95) −1.6981 (−5.74) −1.4638 (−4.84) −1.4233 0.3799

0.5473

1.2604 (34.87) 1.3864 (46.79) 1.3382 (47.65) 1.3493 (46.80) 1.3088 (43.48) 1.4187 (48.12) 1.3616 (47.49) 1.3618 (47.87) 1.2583 (43.14) 1.4054 (48.55) 1.3703 (48.48) 1.3399 (45.67) 1.3256 (49.54) 1.3540 (54.89) 1.3769 56.41) 1.3387 (55.10)

0.5382

2080

0.5546

3113

0.5837

3183

0.5571

3232

0.5249

3253

0.5502

3265

0.5610

3285

0.5561

3358

0.5350

3363

0.5864

3296

0.5699

3263

0.5545

3188

0.5099

4081

0.5368

4491

0.5589

4380

0.5449

4350

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. dev.

0.5564 0.5872 0.5745 0.5270 0.5533 0.5622 0.5570 0.5363 0.5882 0.5708 0.5549 0.5120 0.5381 0.5610 0.5461

guide, the usual situation is, first, to hope for estimated coefficients for price to be negative, and then to rejoice if associated t-ratios are greater than 2! Price elasticities with t-ratios of the magnitudes in Tables 6.1–6.5 are thus both gratifying and rare.11 Impressive, as well, is the general stability of the price-elasticity estimates over all 16 quarters of data. Interestingly, the category displaying the least stability (as measured by the standard deviations of the estimates over the 16 11 Of the 96 estimated price elasticities tabulated in Tables 6.1–6.6, not a single one is positive, and only one has a t-ratio less than 1 (in absolute value), this for food for the 1997 Q3.

6.3

Pooling Across Quarters and Years

115

Table 6.5 Health care expenditures (t-ratios in parentheses) ln Q = α + βln y + γ ln p + . . .

ln E = α + βln y + . . .

Yr./Qtr.

Tot. exp.

Price

R2

Tot. exp.

R2

#Obs.

1996Q1

0.4659 (9.44) 0.4309 (11.14) 0.4752 (12.17) 0.5018 (12.62) 0.5395 (13.97) 0.5374 (13.49) 0.5577 (14.43) 0.5330 (14.15) 0.5023 (12.60) 0.4959 (12.82) 0.5125 (13.19) 0.5028 (12.51) 0.4899 (14.00) 0.5379 (16.54) 0.4925 (15.77) 0.5302 (16.20) 0.5066 0.0326

−0.9920 (−4.25) −0.8805 (−4.79) −1.3450 (−7.62) −1.2315 (−11.35) −0.9872 (−5.54) −1.0873 (−5.88) −1.1957 (−6.58) −0.9234 (−5.01) −0.9894 (−5.25) −1.0128 (−5.58) −1.1604 (−6.18) −1.3255 (−6.78) −1.2036 (−7.38) −1.2165 (−8.01) −1.2328 (−8.03) −1.2021 (−7.70) −1.1242 0.1433

0.2340

0.4661 (9.51) 0.4326 (11.21) 0.4682 (12.04) 0.4964 (12.50) 0.5398 (14.07) 0.5353 (13.52) 0.5537 (14.39) 0.5346 (14.26) 0.5026 (12.71) 0.4957 (12.87) 0.5089 (13.17) 0.4935 (12.40) 0.4852 (13.95) 0.5327 (16.49) 0.4882 (15.69) 0.5257 (16.25)

0.2145

1797

0.2183

2689

0.2231

2771

0.2232

2790

0.2405

2802

0.2094

2854

0.2132

2877

0.2205

2935

0.2211

2934

0.2269

2880

0.2234

2850

0.2365

2768

0.2263

3590

0.2330

3947

0.2171

3797

0.2164

3840

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.2380 0.2585 0.2664 0.2669 0.2340 0.2404 0.2427 0.2409 0.2482 0.2473 0.2565 0.2476 0.2496 0.2395 0.2379

quarters)—transportation expenditures—is the one with the weakest coverage of prices, in that bus fares, tire balancing, and gasoline are the only prices represented in the ACCRA price index. In contrast, the two categories showing the most stability of price elasticity estimates—shelter and utilities—are the ones with the most extensive price coverage.

6.3 Pooling Across Quarters and Years The stability in income and price elasticities over the 16 quarters of data in Tables 6.1–6.6 clearly supports the estimation of models from data sets pooled over

116

6 Price and Income Elasticities – Some Preliminary Results Table 6.6 Miscellaneous expenditures (t-ratios in parentheses) ln Q = α + βln y + γ ln p + . . .

ln E = α + βln y + . . .

Yr./Qtr.

Tot. exp.

Price

R2

Tot. exp.

R2

#Obs.

1996Q1

1.2219 (24.62) 1.0562 (56.75) 1.1097 (59.17) 1.0709 (58.48) 1.1234 (41.24) 1.1555 (25.47) 1.1124 (33.17) 1.2046 (25.88) 1.2169 (31.69) 1.2099 (25.59) 1.1214 (26.26) 1.2817 (26.00) 1.0675 (46.42) 1.0368 (68.82) 1.0281 (68.87) 1.0516 (71.81) 1.1293 0.0777

−1.7384 (−2.72) −1.6996 (−−6.81) −1.2656 (−6.02) −0.9711 (−13.34) −1.2668 (−4.50) −1.0627 (−2.04) −1.5219 (−3.75) −1.6727 (−2.94) −1.5295 (−3.56) −1.7656 (−3.15) −1.2734 (−2.42) −1.2319 (−1.98) −1.2408 (−4.53) −1.0721 (−5.68) −1.2701 (−7.00) −1.2818 (−7.01) −1.3665 0.2540

0.4480

1.2178 (24.59) 1.0531 (56.63) 1.1087 (59.17) 1.0713 (58.60) 1.1212 (41.32) 1.1550 (25.55) 1.1071 (33.25) 1.1995 (25.88) 1.2101 (31.83) 1.2059 (25.25) 1.1200 (26.29) 1.2796 (26.13) 1.0657 (46.53) 1.0363 (69.02) 1.0268 (68.90) 1.0496 (71.94)

0.4454

2195

0.7444

3274

0.7529

3312

0.7464

3375

0.5966

3408

0.3740

3418

0.4986

3419

0.3562

3497

0.4569

3514

0.3545

3446

0.3872

3427

0.3511

3336

0.5953

4269

0.7407

4697

0.7441

4563

0.7502

4555

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.7469 0.7542 0.7482 0.5976 0.3748 0.5002 0.3583 0.4580 0.3567 0.3888 0.3511 0.5962 0.7414 0.7449 0.7511

quarters and years. The results are presented in Tables 6.7 and 6.8. The estimates in Table 6.6 refer to within-year quarters pooled by year, while the estimates in Table 6.8 are for a pooling of all 16 quarters. The models in Table 6.8, it should be noted, allow for price elasticities to vary by year.12 12 This is done through interacting logarithms of prices with yearly dummy variables (D97, D98, and D99). As D96 is the left-out year, the coefficients on the interaction terms accordingly represent deviations from the coefficient for D96·ln p. Although allowance could also be made for price elasticities to vary by quarter as well as by year, this has not been done. Neither have models been estimated that allow for income elasticities to vary across quarters and years.

6.3

Pooling Across Quarters and Years

117

Table 6.7 CES expenditures Quarters pooled by years (t-ratios in parentheses) ln Q = α + βln y + γ lnp + . . . Year

ln E = α + βln y + . . .

Tot. exp.

Price

R2

Tot. exp.

R2

#Obs.

0.3073 (34.43) 0.3315 (40.09) 0.3256 (38.94) 0.3158 (45.26)

−0.2920 (−3.74) −0.2528 (−3.13) −0.5424 (−6.91) −0.4717 (−8.06)

0.4419

0.3119 (34.89) 0.3381 (40.92) 0.3295 (39.47) 0.3203 (45.92)

0.4400

11256

0.4311

13701

0.4224

13685

0.4233

18017

0.8559 (64.91) 0.8675 (67.54) 0.8467 (67.74) 0.8563 (79.69)

−0.5726 (−18.79) −0.4915 (−13.26) −0.5829 (−15.65) −0.7061 (−25.81)

0.4754

0.8640 (65.06) 0.8799 (68.21) 0.8559 (68.32) 0.8645 (80.41)

0.4459

12057

0.4501

13661

0.4458

13646

0.4367

17924

0.3733 (35.32) 0.4103 (40.97) 0.3669 (36.94) 0.3704 (45.83)

−0.8960 (−23.10) −0.8785 (−20.58) −0.9012 (−22.07) −1.0620 (−29.79)

0.4159

0.3736 (35.35) 0.4118 (41.16) 0.3681 (37.10) 0.3695 (45.81)

0.3876

12069

0.3922

13594

0.3976

13610

0.3864

17862

1.3378 (88.88) 1.3591 (93.78) 1.2599 (92.96) 1.3454 (108.14)

−1.0844 (−16.42) −1.4933 (−8.04) −1.2599 (−8.11) −1.2178 (−9.40)

0.5646

1.3372 (88.88) 1.3566 (93.79) 1.3388 (93.07) 1.3444 (108.16)

0.5572

11608

0.5460

13161

0.5593

13109

0.5364

17302

0.4555 (22.24) 0.5220 (27.15)

−1.1542 (−15.10) −1.0323 (−11.40)

0.2392

0.4525 (22.15) 0.5213 (27.26)

0.2078

10047

0.2103

11468

Food 1996 1997 1998 1999

0.4332 0.4275 0.4260

Shelter 1996 1997 1998 1999

0.4616 0.4540 0.4540

Utilities 1996 1997 1998 1999

0.4323 0.4308 0.4239

Transportation 1996 1997 1998 1999

0.5476 0.5603 0.5379

Health care 1996 1997

0.2356

118

6 Price and Income Elasticities – Some Preliminary Results Table 6.7 (continued) ln Q = α + βln y + γ lnp + . . .

ln E = α + βln y + . . .

Year

Tot. exp.

Price

R2

Tot. exp.

R2

#Obs.

1998

0.4823 (24.54) 0.4948 (30.29)

−1.1242 (−11.97) −1.2167 (−15.61)

0.2357

0.4793 (24.55) 0.4900 (30.16)

0.2143

11432

0.2130

15174

−1.1263 (−14.05) −1.3688 (−6.14) −1.3756 (−5.19) −1.1854 (−11.45)

0.6465

1.1090 (88.39) 1.1543 (59.59) 1.2111 (54.78) 1.0505 (124.99)

0.6445

12157

0.4271

13742

0.3784

13723

0.6998

18084

1999

0.2336

Miscellaneous expenditures 1996 1997 1998 1999

1.1099 (88.38) 1.1574 (59.49) 1.2141 (54.67) 1.0518 (124.72)

0.4285 0.3797 0.7007

As is to be expected, the results with the pooled models are similar to those reported in Tables 6.1–6.6. T-ratios are increased, of course, as a consequence of substantial increases in sample sizes and degrees of freedom. Transportation and miscellaneous expenditures continue to have the greatest variation in estimated price elasticities; and shelter, utilities, and health care the least. From the estimated coefficients on the interaction terms, food consumed at home is seen to have the strongest year-to-year variability, for the elasticities for both 1997 and 1998 are both substantially larger than for 1996 and statistically significant. Shelter and utilities are the only other expenditure categories with statistically significant differences. Finally, as in Tables 6.1–6.6, total expenditure elasticities are seen both to be stable across years and little affected by the exclusion of the price variable.

6.4 Effects of Other Variables Since the focus in this exercise is on price and income elasticities, the other variables in the models are viewed as controls, their inclusion being necessitated in order to avoid bias problems associated with “omitted” variables. However, this is not to say that these other predictors are neither of importance nor of interest in their own right. In general, age, family size, and dummy variables denoting age and number of children, rural/urban, region of the country lived in, race, sex, homeownership, and whether the household is recipient of food stamps are usually significant in some form.13 The number of wage earners and education are occasionally important. Seasonal effects, on the other hand, are usually unimportant. The variables that

13

Since demographic factors related to family size and children are especially important in food consumption, the approach that has been employed here is simple-minded in comparison with what is available in the literature, such as, for example, the use of household production functions

0.3198 (79.71) 0.8574 (140.65) 0.3797 (79.93) 1.3452 (192.49) 0.4906 (52.49) 1.1262 (142.92)

Food

Misc.

Health care

Trans.

Utilities

Shelter

Income

Category

D97price −0.0811 (−1.17) 0.0625 (2.24) −0.0934 (−2.55) −0.2402 (−1.85) −0.0389 (−0.46) −0.0984 (−0.61)

Price

−0.2759 (−4.93) −0.6103 (−27.43) −0.8700 (−27.94) −1.1086 (−18.80) −1.0917 (−17.79) −1.1672 (−11.73)

−0.2918 (−4.28) 0.0272 (0.94) −0.0634 (−1.69) 0.0471 (0.40) −0.0119 (−0.14) −0.2548 (−1.55)

D98price −0.1584 (−2.57) −0.0400 (−1.57) −0.1468 (−3.99) −0.1662 (−1.56) −0.1205 (−1.51) 0.0525 (0.35)

D99price

Ln Q = a + b ln y + c ln p + dD97 ln p + eD98 ln p + fD99 ln p + . . . ln E = a + b ln y + . . .

0.4987

0.2349

0.5502

0.4245

0.4602

0.4303

R2 0.3245 (80.92) 0.8669 (141.78) 0.3800 (80.07) 1.3440 (192.61) 0.4873 (52.41) 1.1244 (143.15)

Income

Table 6.8 CES expenditures 1996−1999 pooled (t-ratios in parentheses)

0.4972

0.2104

0.5475

0.3894

0.4440

0.4273

R2

57706

48121

55180

57135

57288

56659

#Obs.

6.4 Effects of Other Variables 119

120

6 Price and Income Elasticities – Some Preliminary Results

are most sensitive to inclusion of ACCRA prices are the regional dummy variables (northeast, midwest, south, and west). Obviously, this is hardly a surprise, for, in the absence of price variables, these variables will pick up differences in regional price levels.

6.5 Equations for Total Consumption Expenditure as Function of After-Tax Income Since the budget constraint in the analysis is total expenditure, rather than after-tax income, no allowance is made for savings (except that implicit, by some definitions, in expenditures for durable goods and insurance and pension contributions). In this section, we take a brief look at the saving behavior that is implicit in the BLS survey data through estimation, again by using only OLS, of double-logarithmic regressions of total consumption expenditure on after-tax income. The resulting estimates for the 16 quarters for 1996 through 1999 are tabulated in Table 6.9. Three results are evident in this table. The first is the relatively small size of the elasticities, for all are less than 0.50. The second result is a remarkable stability in the elasticities; all lie in a range from 0.42 to 0.45. Finally, the third item of note is a second remarkable stability, this time in R2 s, which range from 0.5638 to 0.6103. An elasticity less than 1 is, of course, always to be expected in crosssectional relationships between total consumption and income, whether because it is simply always empirically the case or because of a theoretical explanation via variants of the permanent-income hypothesis. But even so, the small values that are obtained are troublesome, for they clearly imply saving behavior that seriously conflicts with the extremely low observed aggregate personal saving rates for the years in question. The use of total expenditure as the budget constraint accordingly seems a wise choice.14

6.6 Tests for Heteroscedastic Error Terms A common statistical problem with equations estimated with data from consumer surveys is heteroscedastic error terms, and it would be surprising if this problem should be absent in the present exercise. Accordingly, let us turn our attention to Table 6.10, which tabulates the variances by quintile of the OLS residuals from double-logarithmic demand equations for the six categories of expenditure for or adult-equivalent scales. Cf. Deaton and Paxson (1998), Lanjouw and Ravallion (1995), Prais and Houthakker (1955), and Rothbarth (1943). Browning (1992) provides a useful survey. 14 Reasons for the extremely low elasticities for total expenditure are not particularly evident. However, one factor that comes to mind may be improper implicit weighting of high incomes. Since the bulk of saving is done by households with high incomes, any “over-representation” of these households in the CES samples used in estimation would lead to a downward bias in the estimates of the elasticity for total expenditure, in which case the appropriate correction would be to weight households by their place in the distribution of income. The fact that households with after-income less than $5000 are excluded from the samples may be a factor as well.

6.6

Tests for Heteroscedastic Error Terms

121

Table 6.9 Elasticities for total expenditure (t-ratios in parentheses) lntotexp = α + βln income + . . . Yr./Qtr.

Income

R2

#Obs.

1996Q1

0.4484 (28.60) 0.4216 (33.69) 0.4412 (35.25) 0.4185 (33.67) 0.4527 (20.67) 0.4306 (35.42) 0.4194 (32.87) 0.4280 (34.23) 0.4195 (35.87) 0.4445 (36.58) 0.4497 (36.03) 0.4305 (33.80) 0.4208 (37.71) 0.4211 (39.54) 0.4266 (39.66) 0.4327 (41.43)

0.5955

2737

0.5726

4174

0.5976

4128

0.5765

4203

0.5958

4283

0.5822

4301

0.5693

4310

0.5759

4379

0.6103

4375

0.5952

4288

0.5948

4253

0.5725

4185

0.5746

5270

0.5639

5743

0.5638

5596

0.5738

5613

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4

Table 6.10 Variances of OLS equation residuals by Quintile BLS-ACCRA surveys 1999Q4 Quintile Category

First

Second

Third

Fourth

Fifth

Food at home Shelter Utilities Transportation Health care Misc. Exp.

0.1317 0.4601 0.4380 0.4149 0.6371 0.2582

0.0065 0.0172 0.0061 0.0193 0.0344 0.0072

0.0039 0.0075 0.0038 0.0131 0.0185 0.0035

0.0045 0.0071 0.0052 0.0159 0.0202 0.0039

0.0645 0.0492 0.0468 0.1098 0.1689 0.0303

122

6 Price and Income Elasticities – Some Preliminary Results

the fourth quarter of 1999. Since the variances in the first and fifth quintiles are uniformly many multiples of those in the middle three quintiles, it is evident simply by eye that heteroscedasticity is indeed present. As is well-known, the consequence of heteroscedasticity (assuming that the other Guass–Markov conditions are fulfilled) is a loss of efficiency. In principle, heteroscedasticity can be straightforwardly corrected through the use of generalized least-squares, but, given the magnitude of the present undertaking, the costs of the needed corrections are not trivial. It would be much preferable (again, on the assumption that the other Guass–Markov conditions are met) if we could conclude that the possible loss-of-efficiency consequences of the heteroscedasticity can be put to the side. As a test of this contingency, we have re-estimated the equations underlying the residuals in Table 6.10 by generalized least squares, in which, for the GLS estimator, the observations in each quintile are weighted by the inverse of the standard deviation of the OLS residuals in that quintile. The results are tabulated in Table 6.11. Table 6.11 Price and total-expenditure elasticities OLS and GLS equations BLS-CES surveys 1999Q4 (t-ratios in parentheses) OLS

GLS

Category

Tot. exp.

Price

Tot. exp.

Price

Food at home

0.3378 (27.40) 0.9370 (56.18) 0.4023 (28.09) 1.2819 (61.53) 0.4922 (16.48) 1.0548 (73.62)

−0.5261 (−6.31) −0.7542 (−15.37) −0.9955 (−13.85) −1.7526 (−8.77) −1.1395 (−7.50) −1.3825 (−8.96)

0.3284 (54.92) 0.9184 (122.06) 0.3949 (67.46) 1.2327 (103.92) 0.5065 (34.58) 1.1075 (166.74)

−0.5844 (−14.72) −0.7624 (−37.36) −1.0264 (−36.72) −1.6116 (−15.87) −1.2215 (−16.76) −1.3960 (−22.84)

Shelter Utilities Transportation Health care Misc. Exp.

Two things are readily apparent in this table, the first—and in the present circumstances, what we were hoping would in fact be the case—is the generally small differences in estimated elasticities between the OLS and GLS equations. From this, we conclude that little serious seems to be jeopardized (as far as least-squares estimation is concerned) by ignoring the heteroscedasticity. The second thing that stands out in the table is the uniform increase in t-ratios in the GLS equations. This latter, and extremely surprising, result is clearly not an implication of heteroscedasticity, and accordingly has to reflect something more fundamental at work. We have already had a glimpse into what this might be in Chapter 3, namely, a marked tendency toward residuals’ distributions that are asymmetrical and long-tailed. The kernel-smoothed distributions of residuals from the OLS equations for food and shelter in Fig. 6.1 illustrate the phenomenon in question. As displayed in these two panels, the central parts of residuals distributions (as represented, say, by the

–4

Nonlinear Logarithmic Engel Curves Food Consumed at Home 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 –2 0 2

123 Shelter 0.8 0.7 0.6 0.5 f(r)

f(r)

6.7

0.4 0.3 0.2 0.1 0

4

–4

–3

OLS residuals

–2

–1

0

1

2

3

4

OLS residuals

Fig. 6.1 Kernel-smoothed density functions OLS regression residuals BLS-CES surveys 1999Q4

middle three quintiles) tend to be sharply peaked, with modes that typically (though not invariably) lie to the right of the (OLS) mean of zero and tails that are both asymmetrical and thick. The thick tails (and hence apparent heteroscedasticity) are reflected in Table 6.10 by variances for the first and last quintiles that are multiples of those in the middle three, while asymmetry is reflected in one of the large variances (in these cases, the variance for the first quintile) being substantially larger than the other. What these figures suggest is that apparent heteroscedasticity may reflect a much larger problem for least-squares estimation, namely, the possibility that error variances are not finite. In Chapter 3, it was suggested that this latter contingency, in conjunction with asymmetry, counsels the use of a robust method of estimation, specifically, quantile regression. The possible need for this will be looked into in detail in Chapter 9. For now, our conclusion, taking the results in Table 6.11 at face value, will be that heteroscedasticity, while apparently present in the models that we will be estimating, can be safely ignored.15

6.7 Nonlinear Logarithmic Engel Curves In this section, we explore a contingency noted by one of the prepublication reviewers of the present work, namely, the possible importance of higher-order income terms in the Engel/demand functions to be estimated. While there are many ways that this question could be approached,16 in keeping with the intended simplicity of this chapter, we shall examine the contingency in terms of a set of doublelogarithmic models for the six CES-ACCRA categories in which the square of the 15

The thick tails illustrated in Fig. 6.1 for the OLS residuals may account for the anomalous increases in t-ratios in the GLS equations in Table 6.9. Since variances in the tails are so much greater than in the central parts of the distributions, the weighting involved for GLS (i.e., weighting each observation in a quintile by the inverse of the standard deviation of the OLS residuals for that quintile) will obviously greatly diminish the influence on the estimated elasticities of the observations whose residuals lie in the tails. 16 See, for example, Lewbel (1991), and Banks, Blundell, and Lewbel (1997).

124

6 Price and Income Elasticities – Some Preliminary Results

logarithm of total expenditure is included as an additional predictor. Addition of this second-order term obviously implies that the elasticity with respect to total expenditure is no longer a constant, but rather varies with its level. Specifically, the total-expenditure elasticity will now be given by ∂ ln Q = b + 2c ln y. ∂ ln y

(6.3)

Price elasticities, of course, are not affected, and remain constants as before. The results for 1999 (pooled over the year’s four quarters) are tabulated in Tables 6.12 and 6.13. Coefficients and t-ratios are given in Table 6.12 and elasticities in Table 6.13. Table 6.12 CES expenditures, 1999 nonlinear double-logarithmic Engel/Demand curves (t-ratios in parentheses) ln Q = a + blntotexp + c(lntotexp)2 + dlnp + . . . ln Q = a + blntotexp + dlnp + . . . Category

lntotexp

(lntotexp)2 lnp

R2

Lntotexp

lnp

R2

Food

1.4286 (15.36) 3.1280 (21.68) 2.2720 (20.77) −0.2489 (−1.42) 2.4402 (11.08) 3.7279 (33.57)

−0.0617 (−11.98) −0.1252 (−15.67) −0.1050 (−17.36) 0.0880 (9.11) −0.1058 (−8.75) −0.1486 (−24.15)

0.4280

0.3172 (48.87) 0.8615 (84.89) 0.3773 (51.93) 1.3437 (112.80) 0.5176 (34.16) 1.0535 (127.84)

−0.5923 (−15.50) −0.7067 (−25.88) −1.0671 (−30.01) −1.1539 (−11.21) −1.1897 (−15.84) −1.1100 (−16.35)

0.4235

Health

Misc.

0.7342 0.5398 0.2921 −1.1942

1.3549 1.0815 0.7266 −1.1086

0.5176 −1.1897

1.0535 −1.1100

Shelter Utilities Trans. Health Misc.

−0.5999 (−15.74) −0.7118 (−26.24) −1.0618 (−30.12) −1.1646 (−11.34) −1.1942 (−15.94) −1.1086 (−16.59)

0.4597 0.4332 0.5397 0.2413 0.7096

0.4523 0.4236 0.5375 0.2375 0.7002

Table 6.13 Elasticities for equations in Table 6.12 Elasticity

Food

Shelter

Utilities

Trans.

ln Q = a + blntotexp + c(lntotexp)2 + d ln p + . . . Total expenditure 10th Percentile 0.4423 1.1237 0.5915 1.1720 Mean 0.3286 0.8945 0.4003 1.3264 90th Percentile 0.1813 0.5968 0.1495 1.5318 Price −0.5991 −0.7118 −1.0618 −1.1646 ln Q = a + blntotexp + Dln p + . . . Total expenditure Price

0.3172 −0.5923

0.8615 −0.7066

0.3773 −1.0671

1.3467 −1.1539

6.8

Conclusions

125

When we look at the results in Table 6.12, we see: (1) The data strongly support addition of the squared term in lntotexp. T-ratios are large for both lntotexp and (lntotexp)2 for five of the six categories, and in all six for (lntotexp)2 . (2) The coefficients for lntotexp and (lntotexp)2 are positive and negative, respectively, in all categories, but transportation, interestingly, is the category for which the linear term has the small t-ratio (−1.42). (3) Inclusion of (lntotexp)2 has virtually no effect on estimates of the price elasticities. From expression (6.3), we see that a positive b and a negative c imply that the elasticity with respect to total expenditure will be larger in the lower tail of the distribution of total expenditure than in the upper tail, while the reverse will be true for negative b and positive c. To gauge the effect of these nonlinearities, totalexpenditure elasticities are calculated in Table 6.13 at three different points on the total-expenditure distributions, namely, at the mean and at the 10th and 90th percentiles. For food, we see that the estimated elasticities range from 0.44 at the 10th percentile to 0.18 at the 90th, while for shelter they vary from 1.12 to 0.60, and so on and so forth. In contrast, for transportation expenditure, which is the only one with a negative b and a positive c, the elasticities vary from 1.17 at the 10th percentile to 1.53 at the 90th percentile. However, at the mean levels of total expenditure the elasticities from the quadratic models are essentially the same as in the linear models (0.33 vs. 0.32 for food, 0.89 vs. 0.86 for shelter, etc.). While the results of this section obviously make a case for including a quadratic total-expenditure term in the Engel/demand functions to be estimated, there nevertheless seems, given the goals of the study, little statistical cost in failing to do so. For price elasticities are little affected by the presence or absence of the square of lntotexp, and the same is true for the total-expenditure elasticities in the central part of total-expenditure distribution.17 The caveat, of course, is that the total-expenditure elasticities that are estimated should not be employed in circumstances in which the distribution of income is of concern and importance, as, for example, in the evaluation of the welfare effects of income and commodity taxes.18

6.8 Conclusions As noted at the outset, the purpose of this chapter has been to explore the feasibility of estimating consumer demand functions from data sets that combine household level expenditure data from the ongoing quarterly consumer expenditure surveys 17

While, as is to be expected, inclusion of (lntotexp)2 leads to an increase in R2 , the increase is not large for any of the categories. Moreover, spot inclusion of a third-order term in lntotexp in several estimations [as is implicit in the Gorman rank-3 model of Banks, Blundell, and Lewbel (1997)] generally leads to all three total-expenditure terms becoming insignificant. 18 Cf. Muellbauer (1975) and Banks, Blundell, and Lewbel (1997).

126

6 Price and Income Elasticities – Some Preliminary Results

conducted by the U.S. Bureau of Labor Statistics with price data from the price surveys undertaken quarterly by the American Chambers of Commerce Research Association (ACCRA). Simple double-logarithmic demand functions have been estimated for six (exhaustive) categories of expenditure for 16 quarters for the years 1996–1999. In general, strong price effects are obtained that, for the most part, are stable over time. Among other things, the exercise confirms that commingling of expenditure and price data from disparate surveys is indeed feasible, and provides justification for estimation in the next two chapters of several systems of demand functions that satisfy the restrictions of conventional demand theory. The exercise also suggests that heteroscedasticity, while present in the models estimated, will not invalidate conclusions reached by OLS estimation. A second important finding is the virtually zero impact on estimates of total expenditure elasticities of a presence or absence of price in the estimating equations. This is a significant result, for it provides confidence that the total-expenditure elasticities that are estimated for expenditure categories for which price information is not available should be free of omitted-variable bias.19 A third finding is the empirical support provided for the use of total expenditure as the budget constraint as opposed to after-tax income. As noted, the small totalexpenditure elasticity (and therefore high implicit saving elasticity) that is obtained in the double-logarithmic regressions of total expenditure on after-tax income is clearly at odds with the low observed aggregate personal saving rates in the U.S. between 1996 and 1999. Use of total expenditure for the budget constraint thus seems justified. A fourth important conclusion concerns the noninclusion of a second-order term in lntotexp in the estimated Engel/demand functions. While the results in Section 7 clearly indicate the statistical importance of this term, its exclusion does not appear to affect either estimates of price elasticities or, more importantly, estimates of totalexpenditure elasticities in the central part of the income distribution. Finally, despite the nearly overwhelming interest of economists in price elasticities, the unfortunate truth is that, except for a few specific categories of consumption (such as food consumed at home, telecommunications, transportation, and household utilities), there is little empirical evidence as to their values. For this reason, it makes little sense at this stage of the analysis to get caught up in discussion of whether or not the price elasticities that have been obtained in this chapter are “plausible.” Such discussion will accordingly be put off until more results with the BLS data are obtained, as well as the results from the time-series analysis.

19

See Chapter 11.

Chapter 7

Estimation of Theoretically Plausible Demand Functions from U.S. Consumer Expenditure Survey Data

The estimation of demand systems in which the demand equations satisfy all of the restrictions of neoclassical theory of demand has long been considered a triumph of applied econometrics, for the resulting equations both honor the budget constraint and are consistent with an underlying (usually ordinal) utility function. Many such “theoretically plausible” systems are represented in the literature, ranging from simple linear expenditure equations to systems embodying exotic Gorman polar forms.1 The purpose of this chapter is to report results from applying four of the more popular of these systems to the data sets described in Chapter 6 that combine data for six exhaustive categories of household expenditure from the quarterly consumer expenditure surveys with price data obtained in quarterly surveys conducted by ACCRA. The four theoretically plausible systems investigated are: the almost ideal demand system of Deaton and Muellbauer (1980b), the linear expenditure system of Stone (1954), and the indirect and direct addilog systems of Houthakker (1960). For each system, the procedure has been to employ a common set of socio-demographical-regional variables (the same as used in the preceding chapter and always in the same form) to control for variation in sociodemographical and regional characteristics across households.2 Unlike in Chapter 6, the analysis of this chapter is confined to data sets that pool across the four quarters of 1996.

1

Besides the systems estimated below, included among these are the well-known Rotterdam and Translog systems [Barten (1969, 1977), Christensen et al. (1975)], as well as the “flexible functional forms” of Diewert (1971, 1973, 1974). “Gorman polar forms” are inspired by the famous 4-page paper of Gorman (1961), which derived the general expression for the indirect utility function in order for Engel curves to be linear in functions of income. [Besides Gorman, see, among others, Muellbauer (1975), Blackorby et al. (1978), Lewbel (1991), Banks et al. (1997), and Lewbel and Pendakur (2006).] 2 Since the focus continues to be on the estimation of price and total expenditure elasticities, these other variables will once again be ignored.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_7, 

127

128

7 Estimation of Theoretically Plausible Demand Functions

7.1 The Almost Ideal Demand System Since its introduction by Deaton and Muellbauer in 1980, the almost ideal demand System (AIDS) has been a workhorse of applied demand analysis, not least because of the ease with which its constituent demand equations can be estimated.3 The estimating equations of the Deaton–Muellbauer system have the form: wi = αi +



 γij ln pj + βi ln (y p), i, j = 1, . . . , n,

(7.1)

where wi denotes the budget share of the ith good, pj denotes the price of the jth good, y denotes the budget constraint, and p is the price index defined by4 ln p = α0 +



  αj ln pj + (1 2) γij ln pj pi .

(7.2)

The matrix of own- and cross-price and total-expenditure elasticities obtained from estimating this system of equations for the six BLS-ACCRA expenditure categories for the four quarters of 1996 are tabulated in Table 7.1.5 Discussion of the numbers in this and the tables that follow is postponed until Section 7.4.

7.2 The Linear Expenditure System The linear expenditure system (LES) has its theoretical basis in the Stone–Geary– Samuelson utility function, which has the form φ(q) =



βi ln (qj − αi ), i = 1, . . . , n,

(7.3)

where α i has a standard interpretation of “minimum required quantities,” and β i is subject to the constraint that they sum to 1. The demand functions corresponding to expression (7.6) are accordingly: 3

The AIDS model is a simple representation of a Gorman polar form. Because saving is not included as an “expenditure” category, the budget constraint in all of the analyses of this chapter is total expenditure (defined as the sum of expenditures for food consumed at home, shelter, utilities, transportation, and miscellaneous expenditures), rather than CES after-tax income. To simplify estimation, we have used the ACCRA all-items index in place of p as defined in expression (7.2). Others’ experience in estimating the Deaton–Muellbauer system suggests that any bias that this might cause should not be large. 5 The elasticities for the almost ideal demand system models are calculated (at sample mean values) according to the following formulae: 4

ηtot.exp. = 1 + β i /wi ηownprice = −1 + (γii /wi )−[[β i pi wi ∗ ]/pwi ] ηcross-price = (γij /wi )−[β i pj wj ∗ ]/pwi where wi ∗ is the weight of the ith expenditure category in the ACCRA all-items index.

7.2

The Linear Expenditure System

129

Table 7.1 Price and total expenditure elasticities almost ideal demand system CES-ACCRA surveys 1996 (calculated at sample mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total expenditure

−0.2981 −0.1105 −0.1071 −0.6134 0.7813

0.6644 −0.8285 0.1638 −0.2520 0.0023

0.0599 0.1909 −0.7222 −0.2471 0.4260

−0.0013 0.1902 0.0523 −1.3739 −0.0129

0.1400 0.2782 −0.0669 −0.7627 −0.9375

−0.5044 −0.5777 0.1783 1.5824 0.8318

0.4469 0.8876 0.4612 1.7250 0.6338

0.4395

−0.2179

−0.2267

−0.0154

0.0470

−1.1448

1.2150

qj = αj +

βj (y − pi αi ) , j = 1, . . . , n. pj

(7.4)

Multiplication by pj then yields the equations that give the linear expenditure system its name:  pj qj = pj αj +βj (y − pi αi ), j = 1, . . . , n. (7.5) Expenditures on the commodities in the linear expenditure system are thus seen to consist of amounts pj αj that are independent of income, plus proportions βj of the “uncommitted ” (or “supernumerary”) income, y − pi αi , that remains. However, since the minimum required expenditures are not observed, they have to be estimated. Estimation has proceeded via an iterative scheme, in which estimates of the αj s are obtained as the coefficients on pj in (homogeneous) regressions of pj qj on pj and y − pj αj , where the αj s are the estimated αj s from the preceding iteration. Iteration proceeds until stable estimates of the βj s are obtained.6 The results for the LES model are given in Tables 7.2 and 7.3, estimating equations in Table 7.2 and the calculated price and total-expenditure elasticities in Table 7.3.7 The estimates are 50 iterations on the αs. T-ratios for total expenditure are seen to be large in all six equations, but to be less than 2 for the coefficient on price in the equations for transportation and health care. Since, because of the strong separability structure, cross-price effects in the LES model arise only via 6

This estimation scheme differs from the one employed by Stone (1954), in that Stone iterated on βj rather than on αj . Iteration on αj not only appears to be simpler, but convergence to stable estimates of the βj s is quite rapid. 7 The LES elasticities in this table are calculated according to the following formulae:

ηtot. exp . = βj /wj    ηown−price = pj αj /pj qj 1 − βj − 1   ηcross−price = −βj pi αi /pj αj .

130

7 Estimation of Theoretically Plausible Demand Functions

Table 7.2 Estimating equations linear expenditure system CES-ACCRA surveys 1996 (t-ratios in parentheses) Coefficient Category

Price

Tot. Exp.

R2

Min. Req. Expenditures

Food

2.1476 (5.77) 3.9558 (11.03) 0.6394 (3.53) 0.6213 (0.31) −0.4149 (−1.07) 5.5404 (2.67) –

0.0166 (20.28) 0.1259 (50.72) 0.0138 (27.57) 0.3884 (85.66) 0.0212 (17.46) 0.4341 (107.65) –

0.8167 – 0.6861 – 0.8432 – 0.6026 – 0.4022 – 0.8160 – –

$232.33 – 531.28 – 71.97 – 68.79 – −48.79 – 583.37 – $1439.60

Shelter Utilities Transportation Health care Misc. exp. Total

Table 7.3 Price and total expenditure elasticities linear expenditure system CES-ACCRA surveys 1996 (calculated at sample mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total expenditure

−0.7576 −0.0380 −0.0052 −0.0049 0.0034

−0.0551 −0.7465 −0.0174 −0.0163 0.0114

−0.0446 −0.1019 −0.8902 −0.0132 0.0092

−1.3116 −2.9994 −0.4063 −0.9770 0.2718

0.1025 0.2344 0.0318 0.0304 −1.0895

−0.1729 −0.3953 −0.0536 −0.0512 0.0358

0.1356 0.6094 0.1541 2.5267 0.3021

−0.0418

−0.1382

−0.1119

−3.2935

0.2574

−0.9083

1.2294

income effects, we would normally expect all cross-price elasticities to be negative. However, all cross-price effects associated with the price of health care are seen to be positive, which seems to have a weird implication that all expenditure categories behave as inferior goods with respect to the price of health care. The (mechanical) reason for this is that, contrary to what we might again normally expect, the estimated minimum required expenditure for health care is negative (as implied by the negative sign for the price coefficient in the equation for health care), which (per the formula in footnote 7.5) implies positive values for all of the cross-elasticities with respect to its price.8

8

An analysis of negative minimum required quantities can be found in Solari (1971). However, in view of the small t-ratio for price in the equation for health care, the “true” value for health care (if not 0) could in fact be positive.

7.3

The Indirect Addilog Model

131

Perhaps the most interesting fact about the minimum required quantities in the present context is that they can be linked (at least in interpretation) with the expenditures identified with the first (i.e., largest) principal component that emerged in the principal component analysis of total expenditures in Chapter 5. The largest principal component, it will be recalled, was found to be extremely stable and to account, on the average, for about 40% of the total expenditure. Here, from the last column in Table 7.2, we see that the proportion of total expenditure accounted for by the total of the minimum required expenditures is $1440 a quarter, which is about 15% of quarterly total expenditure. While the latter proportion is much smaller than the former, both are consistent with the notion (discussed in Chapter 2) that consumption behavior in fact derives from an underlying Maslovian hierarchy of wants.

7.3 The Indirect Addilog Model9 In terms of convenience and goodness of fit, simple double-logarithmic demand functions are pretty much without peer in applied demand analysis. Nevertheless, problems abound at the theoretical level, for double-logarithmic demand functions are neither integrable nor additive. In recognition of this, Houthakker (1960) introduced two near-logarithmic demand systems that are both additive and consistent with conventional demand theory. The first of these (the indirect addilog model) is derivable from an indirect utility function, while the second (the direct addilog model) can be derived from a direct utility function. The indirect addilog model is the easier of the two to implement and will be presented first. The procedure employed by Houthakker is to transform double-logarithmic demand functions into an additive system via the fact that any nonadditive function θ i (y) can be made additive by the transformation yθi (y) , gi (y) = θk (y)

(7.6)

since gi (y) = y. The application of this transformation to the double-logarithmic function pi qi = Ai (y/pi )βi , i = 1, . . . , n,

(7.7)

gives an additive system of functions:10 9

In addition to Houthakker (1960), good discussions of the indirect and direct addilog models can be found in Phlips (1983). 10 This expression can be obtained more conventionally by applying Roy’s theorem to the indirect utility function:   β ϕ (y/p) = Aj y/pj j .

132

7 Estimation of Theoretically Plausible Demand Functions −βj

fj (y, p) =

Aj yβj pj

j = 1, . . . , n.

−βj

Aj βj yβj −1 pj

(7.8)

Division of fj (y, p) by fi (y, p) then yields −β

Aj yβj pj j pj qj = , −β pi qi Ai yβi pi j

(7.9)

which, upon taking logarithms, becomes ln qj −ln qi = aij + (βj +1)( ln y−ln pj )−(βi +1)( ln y − ln pi ), j = 1, . . . , n, j = i. (7.10) The price and total expenditure elasticities obtained from applying the system in expression (7.10) to the six BLS-ACCRA categories appear in Table 7.4.11 Table 7.4 Price and total expenditure elasticities indirect addilog model CES-ACCRA surveys 1996 (calculated at sample mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total expenditure

−0.2451 −0.0651 −0.0734 −0.0186 −0.0484

−0.1598 −0.7502 −0.0734 −0.0186 −0.0484

−0.1598 −0.0651 −0.2540 −0.0186 −0.0484

−0.1598 −0.0651 −0.0734 −1.1023 −0.0484

−0.1598 −0.0651 −0.0734 −0.0186 −0.3603

0.1598 −0.0651 −0.0734 −0.0186 −0.0484

0.3963 0.9458 0.4412 1.3816 0.5726

−0.0214

−0.0214

−0.0214

−0.0214

−0.0214

−1.0392

1.3213

7.4 The Direct Addilog Model The utility function corresponding to Houthakker’s direct addilog model is ϕ(q) =

11



β

αj qj j .

(7.11)

The total expenditure and price elasticities for the indirect addilog equations are calculated as follows:

ηtot.exp. = 1 + βj − βj wj ηown-price =−1−(1−w)βj ηcross-price = βi wi .

7.4

The Direct Addilog Model

133

From the first-order conditions (7.4) and (7.5), one obtains β −1

qj j

=

1 λpj , j = 1, . . . , n αj βj

(7.12)

which in turn yield the demand functions λ= β −1

qj j

=

β

αj βj qj j y

,

ypj , j = 1, . . . , n. β αj βj αj βj qj j

(7.13) (7.14)

In view of the severe nonlinearity of the equations in (7.14), the usual procedure (as with the indirect addilog model) preparatory to estimation of the βj s is to divide fj (y, p) by fi (y, p), and taking logarithms, to obtain (βj − 1) ln qj − (βi − 1) ln qi = aji + ln pj − ln pi , j = 1, . . . , n, j = I, (7.15) which upon rearrangement and division by (βj −1) yields ln qj = bji + b1j ln qi + b2j ( ln pj − ln pi ), j = 1, . . . , n, j = I,

(7.16)

where  b1j = (βi − 1) (βj − 1),

(7.17)

 b2j = 1 (βj − 1).

(7.18)

In view of equations (7.17) and (7.18), estimation of the n−1 equations in expression (7.16) accordingly requires that the equations be estimated subject to the n−2 constraints: b1j b2k = b2j b1k , j, k= 1, . . . , n, j = k = i.

(7.19)

As will be related in the next section, initial efforts to estimate the βj s on the basis of expression (7.16) ran into difficulties, and expression (7.14) has been resorted to instead. In logarithms, expression (7.14) becomes ln qj = aj +

1 ln λpj , j = 1, . . . , n. βj − 1

(7.20)

Estimation of the equations in (7.20) has proceeded via quasi-iteration on λ, with λ approximated by the function λ∼ = y−κ .

(7.21)

134

7 Estimation of Theoretically Plausible Demand Functions

Table 7.5 Price and total expenditure elasticities direct addilog model CES-ACCRA Surveys 1996 (calculated at sample mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

−1.0954 0.0099 0.0099 0.0099 0.0099

0.0261 −1.1378 0.0261 0.0261 0.0261

0.0075 0.0075 −1.1039 0.0075 0.0075

0.5031 0.5031 0.5031 −2.8917 0.5031

0.0309 0.0309 0.0309 0.0309 −1.4564

1.2940 1.2940 1.2940 1.2940 1.2940

0.4639 0.4885 0.4664 1.4248 0.6242

0.0099

0.0261

0.0075

0.5031

0.0309

−2.3213

1.5174

Misc.

Total expenditure

The price and total expenditure elasticities with a value for κ equal to 0.5 are presented in Table 7.5.12

7.5 Some Technical Obiter Dicta Concerning Estimation Before taking up a comparison and discussion of the results presented in Tables 7.1– 7.4 for the four demand systems, some remarks concerning their estimation are in order. (1) As noted, of the four demand systems analyzed, the AIDS model is by far the easiest to estimate, as the parameters of the system are obtained as the coefficients in the regressions of the budget shares wj on ln pi (i = 1, . . ., n) and ln (y/p). (2) For the linear expenditure system, on the other hand, things are not so straightforward, for the minimum required quantities, αj , are unobserved, and estimation accordingly has to proceed by some sort of iterative scheme. Stone’s original procedure was to iterate on the βs, but the procedure here has been to iterate on the αs in expression (7.5), whereby (beginning with α = 0) the αs estimated as the coefficients on pj in iteration k−1 are used to construct y − pj αj for iteration k. The elasticities in Table 7.3 are derived from 50 iterations. Convergence using this procedure is quite rapid, and results for 40 or more iterations are little changed from those for 10 iterations.13

12

The elasticities in this table are calculated from the following systems of simultaneous equations: (1−βj )∂ln qj /∂ln y+ wi βi ∂ln qi /∂ln y= 1, j, i= 1, . . . , n (1−βj )∂ln qj /∂ln pi + wi βi ∂ln qi /∂ln pi = 1, j, i= 1, . . . , n. In view of the small t-ratios for the αs (the coefficients on the price terms) in the equations for transportation and health care expenditures in Table 7.2, estimates for these parameters are the slowest to stabilize.

13

7.5

Some Technical Obiter Dicta Concerning Estimation

135

(3) Still another procedure for estimating the LES model would be to estimate α and β directly from the first-order conditions (7.4) by iterating on λ, in which case the estimating equations would be given by14 qj = αj + (λpj )−1 βj , j = 1, . . . , n,

(7.22)

from which estimates of αj and βj can be obtained as coefficients in the −1 regression of qj on a constant and (λ pj ) . Since λ for the LES is equal to 1/(y − pi αi ), a natural value with which to begin the iterations is λ = 1/y. (4) From expression (7.10), we see that the n−1 equations comprising the estimating equations for the indirect addilog model have to be estimated subject to the restriction that β i has the same value in each equation. This has been effected by estimating the equations in expression (7.10) by “stacking” the observations for the n−1 equations, and then estimating the equations in expression (7.10) in a seemingly unrelated-regressions framework, with a common coefficient on ln pi , but separate coefficients on the ln pj (as well as on all other variables). (5) Of the four equation systems analyzed, the direct addilog model, despite what would otherwise appear to be a fairly simple utility function, is by far the most problematic to estimate. The demand functions [cf. expression (7.11)] are intractable empirically and estimation (as with the indirect addilog model) is usually to proceed in terms of logarithmic deviations from a “left-out” category. In this case, however, complications in the form of nonlinear restrictions on the parameters across equations are brought into play. Three approaches have been pursued in estimation. The first, which is the one yielding the elasticities in Table 7.6, is (as noted) to estimate the parameters directly from the firstorder conditions through quasi-iteration on λ. The strength of this approach is that it avoids having to deal explicitly with nonlinearities. Its drawback is that, Table 7.6 Price and total expenditure elasticities direct addilog model CES-ACCRA surveys 1996 constrained nonlinear (calculated at sample mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total expenditure

−0.4573 −0.0661 −0.0661 −0.0661 −0.0661

−0.1969 −0.7280 −0.1969 −0.1969 −0.1969

0.2956 0.2956 −0.4633 0.2956 0.2956

−0.2021 −0.2021 −0.2021 −1.0729 −0.2021

−0.0332 −0.0332 −0.0332 −0.0332 −1.2384

−0.0846 −0.0846 −0.0846 −0.0846 −0.0846

0.6996 0.9498 1.1546 1.8827 2.1553

−0.0661

−0.1969

0.2956

−0.2021

−0.0332

−0.3451

0.4646

14 This is the procedure that was used in estimating the dynamic version of the LES in Chapter 8 of the 1970 edition of CDUS.

136

7 Estimation of Theoretically Plausible Demand Functions

at least in the way that we have implemented it, the procedure is not entirely objective.15 The second approach tried was to reformulate the equations in expression (7.16) as ln qj = bji + b2j [(βi − 1) ln qi + ln pj + ln Pi ], j = 1, . . . , n, j = I, (7.23) and then iterate on (β i −1). The results, however, made little sense. Finally, the third approach is simply a “brute force” one of estimating the parameters of the n−1 equations in expression (7.16) jointly (i.e., in a seemingly unrelated-regressions format) subject to the n−2 nonlinear restrictions given in expression (7.19). This procedure, which entails the estimation of a model with 38,621 observations, 135 variables, and 4 nonlinear constraints among the coefficients, has been estimated using a nonlinear programming algorithm found in SAS.16 The elasticities from this estimation are given in Table 7.6.

7.6 Discussion of Results The four demand systems analyzed in this exercise represent a broad spectrum of utility structures, ranging from the severe separability restrictions of the LES and indirect and direct addilog models to the only very minimal restrictions of the almost ideal demand system. The four systems have all been estimated from a common data set, namely, one consisting of 7,724 observations from the four quarterly CES surveys for 1996 augmented with price data from the four ACCRA price surveys for that year. To keep the discussion manageable, focus will be restricted to a comparison of price and total expenditure elasticities. Total-expenditure elasticities are collected in Table 7.7 and own-price elasticities in Table 7.8. Elasticities estimated from simple double-logarithmic equations are included in the tables as well. Turning now to magnitudes, we of course expect own-price elasticity for food to be small, which for the most part is the case. With regard to the general magnitude of own-price elasticities, it is to be kept in mind that their default values (their values in the absence of statistical significance) is minus 1 in the AIDS and in the two addilog models.17 Hence, although estimates of standard errors have not 15

Implementation of this procedure involves estimation of the n equations in expression (7.20) with λ defined as y−κ . An initial informal search over values of κ suggested values in the range of 0.15−0.50. R2 s of the estimated equations for values of κ in this range are relatively stable, as are the implied price and total expenditure elasticities. Interestingly, however, values for κ ≥ 0.25 yield several negative βs, which imply negative marginal utilities. Since the latter are theoretically implausible, a value for κ of 0.20 has been assumed. 16 The algorithm in question is proc nlp in the SAS OR software. Values of b and b from 1j 2j unrestricted OLS regressions were used as seeds in the constraints. However, three of the estimated βs turn out to be negative (which, as just noted, imply negative marginal utilities). 17 Likewise, for these models, the default value for the total-expenditure elasticities is 1. The system that would appear to give the most anomalous results is the direct addilog model in Table 7.5,

7.6

Discussion of Results

137

Table 7.7 Total expenditure elasticities AIDS, LES, indirect, and direct addilog models CES-ACCRA surveys 1996 Category

AIDS

LES

Indirect Addilog

Direct Addiloga

Double Log

Budget Share

Food Shelter Utilities Trans. Health care Misc.

0.4469 0.8876 0.4612 1.7250 0.6338 1.2150

0.1356 0.6094 0.1541 2.5267 0.3021 1.2294

0.3963 0.9458 0.4413 1.3817 0.5726 1.3213

0.4639 0.4885 0.4664 1.4248 0.6242 1.5174

0.2982 0.7826 0.3611 1.3718 0.4436 1.2091

0.1266 0.2066 0.0897 0.1537 0.0703 0.3531

a The

elasticities for the direct addilog model are from Table 7.6.

Table 7.8 Own-price elasticities AIDS, LES, indirect, and direct addilog models CES-ACCRA surveys 1996 Category

AIDS

LES

Indirect Addilog

Direct Addiloga

Double Log

Budget Share

Food Shelter Utilities Trans. Health care Misc.

−0.2981 −0.8285 −0.7222 −1.3739 −0.9375 −1.1448

−0.7576 −0.7465 −0.8902 −0.9770 −1.0895 −0.9083

−0.2450 −0.7502 −0.2540 −1.1023 −0.3603 −1.0392

−0.4573 −0.7280 −0.4633 −1.0729 −1.2384 −0.3451

−0.6663 −0.8198 −0.8806 −1.1990 −1.2029 −1.1142

0.1266 0.2066 0.0897 0.1537 0.0703 0.3531

a The

elasticities for the direct addilog model are from Table 7.6.

been calculated, the estimated elasticities that are substantially different than −1 are probably the only ones with any real statistical significance. What is perhaps most surprising is the price elasticities in the double-logarithmic equations, for which the statistical default values are 0 (rather than −1). All are substantial, with those for transportation, health care, and miscellaneous in excess of 1.18

for the fact that all of the own-price elasticities for this model (even the one for food!) are in excess of 1 (in absolute value) does not seem plausible. (The expenditure elasticities, on the other hand, are pretty much in line with those in the other models.) However, as noted, the direct addilog model is difficult to estimate, and this, rather than the underlying integrity of the model, may be the problem. The price elasticities from the constrained non-linear estimation in Table 7.6 seem much more plausible, but (as noted in footnote 14) three of the estimated βs in this model are negative. 18 The t-ratios for the own price in these equations range from −2.84 and −4.47 for miscellaneous and transportation expenditures to 16.84 and −23.80 for shelter and utilities. The R2 s for the double-log equations (with 7,724 observations) range from 0.1871 for health care to 0.5711 for miscellaneous expenditures. The double-log equations, it should be noted, are estimated with only own prices in the equations. Inclusion of all of the prices in the equations inserts sufficient multicollinearity into the models for the results not to be meaningful. Obviously, one of the benefits of estimating demand systems is that they provide restraints on multicollinearity.

138

7 Estimation of Theoretically Plausible Demand Functions

The AIDS model is the only one of the four demand systems that in principle allows for non-income-effect (i.e., Hicksian) substitution and complementarity, hence it is no surprise that the largest cross-price effects are to be found in Table 7.1.19 Since the numbers in the tables are both elasticities and based upon uncompensated derivatives, for which the Slutsky symmetry conditions do not apply, it is possible for a category (say A) to be a complement with respect to the price of another good (say B), but for B to be a substitute with respect to the price of A. Food and housing in Table 7.1 provide an example of this, for the cross-elasticity of food with respect to the price of housing is 0.11, but the cross-elasticity of housing with respect to the price of food is 0.66. There are many other instances of this switching of signs, so many, in fact, that asymmetry is pretty much the norm.

7.7 Conclusions The motivation for this chapter has been the application of four popular theoretically plausible consumer demand systems to a common cross-sectional data set that combines expenditure data from the quarterly BLS consumer expenditure surveys with price data that are collected quarterly in cost-of-living surveys conducted by ACCRA. Six broad categories of expenditure that exhaust total expenditure have been analyzed: food consumed at home, housing, utilities, transportation, health care, and miscellaneous expenditures. The four demand systems analyzed are the almost ideal demand system, the linear expenditure system, and the indirect and direct addilog models. The focus of the exercise has been on ease (or lack thereof) of estimation and comparison of price and total expenditure elasticities. The AIDS model is the most straightforward to estimate, while the direct addilog is the most difficult. Despite absolute differences in magnitudes that in some instances are rather large, there is substantial agreement in the rank-orderings of elasticities. In general, the largest elasticities (for both own price and total expenditure) are for transportation, miscellaneous, and shelter expenditures, while the smallest elasticities (again for both own price and total expenditure) are for food and utility expenditures. Engel’s law for food is confirmed in all instances. In general, elasticities are smallest with the indirect addilog model and largest, perhaps even implausibly so, with one of the versions of the direct addilog model.

19

From the formula for cross-price elasticities for LES model in footnote 5 above, we see that these depend upon ratios of minimum required expenditures. Its minimum required expenditure being small, together with a large β, accounts for the large cross-elasticities for transportation.

Chapter 8

An Additive Double-Logarithmic Consumer Demand System

Despite obvious theoretical shortcomings, the double-logarithmic function, because of ease of estimation and generally superior fit, is often the demand function of choice in applied demand analysis. However, the drawback to double-logarithmic demand functions is that they are not theoretically plausible, in that they are neither consistent with an underlying utility function nor additive (in the sense of satisfying the budget constraint). The purpose of this chapter is to develop and apply a double-logarithmic demand system that is additive. This is accomplished through an extension of the indirect addilog model of Houthakker that allows for all prices, not just the own price of a good, to be included in each of the demand functions. The system is applied to the six exhaustive categories of consumption expenditure, using the same combined CES-ACCRA data set for the four quarters of 1996, that was analyzed in the last chapter.

8.1 An Additive Double-Logarithmic Demand System As described in the preceding chapter, Houththakker, in his development of the indirect addilog model, employed a mathematical device that enables any nonadditive function θ i (y) to be made additive in terms of y by the transformation yθi (y) , gi (y) = θk (y)

(8.1)

since gi (y) = y. Application of this transformation to the double-logarithmic demand function n

qi = Ai yβi  pγj , i, j = 1, . . . , n,

(8.2)

j=1

then gives an additive system of functions: L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_8, 

139

140

8

An Additive Double-Logarithmic Consumer Demand System n

Aj yβj +1  pγk

fj (y, p) =

Aj

yβj

k=1 n

 pγk

, j = 1, . . . , n,

(8.3)

k=1

The denominator in this expression for fi (y, p) is obviously a very complicated function of prices (p) and income (y), indeed so much so that estimation of the functions directly is pretty much intractable. However, following Houthakker’s derivation of the indirect addilog model, the messy denominators can be eliminated through division of fi (y, p) by fi (y, p), so that Aj yβj +1 pγk qj = . qi Ai yβi +1 pγk

(8.4)

Upon taking logarithms, this expression then becomes ln qj − ln qi =aij +(βj − βi ) ln y+



(γjk − γik ) ln pk , i, j, k = 1, . . . , n, j = i, (8.5)

where aij = lnAj − lnAi . Expression (8.5) is thus seen to be consist of n–1 double-logarithmic equations, in which the “dependent” variables are logarithmic differences and the “independent” variables are the logarithms of income and the n prices. The coefficients that are estimated in these equations are not β j and γ jk , but rather (βj −βi ) and (γjk −γjk ), which would appear to leave the individual βs and γ s unidentified. However, it will be shown below how, by making use of the additivity constraints on income and price elasticities, unique estimates of these underlying parameters can be obtained.

8.2 Application to the CES-ACCRA Data Set for the Four Quarters of 1996 We now apply this model to the data set of the last chapter, that is, to the combined CES-ACCRA sample for six exhaustive categories of expenditure for the four quarters of 1996. With n equal to 6, expression (8.5) yields five equations to be estimated. Although the results are independent of the particular category to be “left out,” the dependent variables are defined as logarithmic deviations from miscellaneous expenditures (category 6). Each of the five estimating equations has the same set of independent variables, namely, the logarithm of real total expenditure (defined as the logarithm of nominal total expenditure minus the logarithm of the price), the logarithms of the prices of food consumed at home, housing, utilities, transportation, health care, and miscellaneous expenditures, together with the standard list of socio-demographical and regional variables. The estimated coefficients and t-ratios for total expenditure and price variables are tabulated in Table 8.1.

8.2

Application to the CES-ACCRA Data Set for the Four Quarters of 1996

141

Table 8.1 Estimating equations for additive double-log model (t-ratios in parentheses) Variable

Food

Shelter

Utilities

Transport.

Health care

lnpfood

−0.4126 (−1.30) 0.1816 (2.31) 0.0503 (0.48) −0.3366 (−2.87) 0.2005 (1.37) 0.3553 (1.13) −0.8184 (−43.58) 0.4127

0.0324 (0.08) −0.5929 (−5.85) 0.3162 (2.35) 0.0261 (0.17) 0.8588 (4.54) −0.7041 (−1.73) −0.2209 (−9.91) 0.1278

0.2317 (0.71) 0.0803 (0.90) −0.7500 (−7.04) −0.3024 (−2.52) −0.0730 (−0.49) 0.7053 (2.19) −0.8045 (−41.86) 0.4022

−0.3595 (−0.79) 0.0439 (0.39) 0.0155 (0.10) −1.3018 (−7.76) −0.3201 (−1.52) 2.0341 (4.50) 0.2051 (7.62) 0.0473

−1.2066 (−2.44) 0.0266 (0.22) 0.2976 (1.84) −0.0857 (−0.47) −0.4915 (−2.16) 1.0309 (2.10) −0.6763 (−23.15) 0.3256

lnphous lnputil lnptrans lnphealth lnpmisc lntotexp R2

In interpreting the coefficients (and their statistical significance) in this table, it needs to be kept in mind that the coefficients being estimated are deviations from the “left-out” category, miscellaneous expenditures. Consequently, the fact that all of the coefficients on lntotexp are negative, except for the one on transportation expenditures, implies that the total expenditure elasticities will be largest for miscellaneous and transportation expenditures. The large t-ratios associated with these coefficients in turn imply that differences in the total-expenditure elasticities for each of the goods and miscellaneous expenditures are not only large, but also highly statistically significant. Turning next to the individual βs and γ s, the six βs are easily obtained from the five coefficients on lny, plus a sixth equation representing the constraint that the budget-share weighted income elasticities sum to 1. Calculation of the γ s, on the other hand, is a bit more complicated. Thirty-six equations are required to solve for them, 30 of which are obviously the equations connecting the γ s to the coefficients on lnpj –lnpi in the five estimating equations. One would then think that the Hicks–Allen additivity conditions (i.e., that the income and own- and cross-price elasticities sum to 0 for each expenditure category) would provide the additional equations needed for identification. However, this is unfortunately not the case, for when linear relationships embodying these restrictions are included, five of the six price coefficients in the “left-out” equation turn out to be colinear with the 31 other coefficients. Consequently, in order to achieve identification, it has (arbitrarily) been assumed that the own-price elasticity for food is the negative of food’s total-expenditure elasticity. This identifies γ 11 (and therefore implicitly γ 61 ). For the remaining identifying restrictions, we have used the five apparently colinear relationships between γ6 k (for k = 2, . . . , 6),γ61 , and γji for j = 1, . . . ,5 and

142

8

An Additive Double-Logarithmic Consumer Demand System

Table 8.2 Estimated parameters additive double-logarithmic demand system CES-ACCRA surveys 1996 Category

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total expenditure

Food Shelter Utilities Trans. Health care Misc.

−0.3604 0.0073 −0.1716 −0.2368 −0.0622

0.0845 −0.7818 0.0943 0.1259 0.5961

0.2839 −0.1164 −0.9718 −0.2026 −0.3357

−0.3074 −0.1450 −0.2064 −1.2020 −0.5829

−1.1544 −0.1623 0.0757 0.0141 −0.7542

0.0522 −0.1889 −0.2219 0.0998 −0.2627

0.3604 0.9579 0.3743 1.3839 0.5025

−0.4484

−1.5079

−0.0985

1.2297

0.2272

−0.8037

1.1788

Table 8.3 Price and total expenditure elasticities additive double-logarithmic demand system CES-ACCRA Surveys 1996 (calculated at sample mean values) Food Food −0.3604 Shelter 0.2782 Utilities 0.3241 Trans. 0.3988 Health care 0.2197 Misc. 0.7916

Shelter

Utilities

0.0535 −0.5148 0.1295 0.1328 −0.1932 1.3208

−0.1397 0.3728 −0.7858 0.3842 0.4595 0.4868

Trans. 0.2924 0.3607 0.3127 −0.9925 0.5252 −0.3475

Health care 1.1297 0.4165 0.1198 0.2083 −0.5842 0.2003

Misc. 0.0929 0.3163 0.2385 0.1863 0.2311 −0.4019

Total expenditure 0.3604 0.9579 0.3743 1.3839 0.5025 1.1788

i = 1, . . . ,6.1 The resulting estimates of the βs, γ s and price and income elasticities are tabulated in Tables 8.2 and 8.3.2 When we look first at the price elasticities, we see that all own-price elasticities are negative, one of which (transportation) is virtually negative unity. Perhaps not surprisingly, the smallest own-price elasticity is the one that was imposed in estimation (as the negative of the total expenditure elasticity) for food. The largest cross-elasticities are for shelter and health care, specifically with respect to the prices of miscellaneous expenditures and food, both of which are positive. Food, utilities, transportation, and miscellaneous expenditures all have relatively small cross-price elasticities. However, since the obtaining of cross-price elasticities is relatively novel in applied demand analysis (especially with cross-section data), it is not clear at this point just how seriously the numbers in this table are to be taken, 1

An unfortunate implication of this procedure is that the resulting price elasticities do not satisfy the Hicks–Allen additivity conditions. 2 The elasticities in these tables are calculated as follows: ηtot.exp. = βj , j = 1, . . . ,6. ηownprice = γjj − wk γjk , k = 1, . . . ,6 ηcorss−price = wj γji − wk γjk , i, k = 1, . . . , 6, i = j, where wk is the budget weight of the kth expenditure category. The elasticities are aligned by column. Hence, the cross-elasticity for food with respect to the price of shelter is 0.0073.

8.2

Application to the CES-ACCRA Data Set for the Four Quarters of 1996

143

whether they might simply be artifacts of the data arising from the rather limited coverage of the ACCRA price data (particularly in the case of transportation and miscellaneous expenditures), or just what. The only conclusion that is perhaps really warranted at this point is that more empirical experience is needed. Elasticities with respect to total expenditure are seen to range from 0.36 for food to 1.38 for transportation. The next largest total-expenditure elasticities are for miscellaneous expenditures (1.18) and shelter (0.96). Those for health care (0.50) and utilities (0.37), in contrast, are much closer to the total-expenditure elasticity for food. Since the model under investigation in this exercise can be viewed as an extension of Houthakker’s indirect addilog model so as to include prices of all goods in each demand function, it accordingly is of interest to see how the two models compare. The price and total-expenditure elasticities for the indirect addilog model estimated in the last chapter are given in Table 8.4. Since the indirect addilog model entails rather strong separability assumptions, the only cross-price effects recorded are those arising from pure income effects, hence the rather small (and negative) “off-diagonal” terms in this table. Accordingly, the only comparisons of real interest are the own-price and total-expenditure elasticities from the two models. These are tabulated in Table 8.5. As an added comparison, elasticities from simple (i.e., nonadditive) double-logarithmic equations from Chapter 6 are included as well. Table 8.4 Price and total expenditure elasticities indirect addilog model CES-ACCRA surveys 1996 (calculated at sample mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total expenditure

−0.1084 −0.1656 −0.0729 −0.0126 −0.0524

−0.1204 −0.6647 −0.0729 −0.0126 −0.0524

−0.1204 −0.1656 −0.1530 −0.0126 −0.0524

−0.1204 −0.1656 −0.0729 −0.9338 −0.0524

−0.1204 −0.1656 −0.0729 −0.0126 −0.2284

0.1204 −0.1656 −0.0729 −0.0126 −0.0524

0.4406 0.9518 0.5327 1.3738 0.6287

−0.0289

−0.0289

−0.0289

−0.0289

−0.0289

−0.9124

1.3361

For the own-price elasticities, we see that the largest discrepancies across the three models are for food consumed at home, utilities, and health care. Otherwise, orders of magnitudes are pretty much the same. The elasticities for food and utilities are much smaller (in absolute value) in the indirect addilog model than in the double-log models. Because of the strong separability embodied in the indirect addilog model, together with the fact that the nonadditive double-log equations have been estimated with only the own-price (which obviously implies a crude informal separability), we should probably expect the elasticities for the indirect addilog model to be in closer agreement with the nonadditive model than for the additive and nonadditve double-log models. However, this is not the case, for the two double-log models in general yield comparable values. For the total-expenditure elasticities,

144

8

An Additive Double-Logarithmic Consumer Demand System

Table 8.5 Own-price and total expenditure elasticities addilog and double-log models CESACCRA surveys 1996 (calculated at sample mean values) Own-price elasticities Category Food Shelter Utilities Trans. Health care Misc.

Total-expenditure elasticities

Indirect addilog

Additive double-log

Nonadditive double-log

Indirect addilog

Additive double-log

Nonadditive double-log

−0.1084 −0.6647 −0.1530 −0.9338 −0.2284

−0.3604 −0.5148 −0.7858 −0.9925 −0.5842

−0.7020 −0.6153 −0.8937 −1.0777 −1.1617

0.4406 0.9518 0.5327 1.3738 0.6287

0.3604 0.9579 0.3743 1.3839 0.5025

0.3069 0.9088 0.3154 1.3224 0.4484

−0.9124

−0.4019

−1.1109

1.3361

1.1788

1.1217

agreement across the three models is in general quite close, particularly for the indirect addilog and additive double-log models.

8.3 Conclusions The focus in this chapter has been on the development of an additive system of double-logarithmic demand functions. The model takes its cue from the indirect addilog model of Houthakker (1960), using a mathematical device that allows for any nonadditive function to be transformed into an additive one. The resulting system of equations, in which demand for each good is a function of every price, satisfies the aggregation condition imposed by the budget constraint that the budgetshare income elasticities sum to 1. However, unlike Houthakker’s indirect addilog model, of which the demand system of this paper can be viewed as an extension, the demand equations of the system do not appear to be derivable from an underlying utility function, and thus are not integrable.3 The additive double-log equations have been applied to a cross-sectional data set consisting of six exhaustive categories of consumption expenditure from the four quarterly BLS consumer expenditure surveys for 1996 augmented with price data collected in quarterly cost-of-living surveys conducted by ACCRA. Own-price elasticities are all negative and range from −0.36 for food consumed at home to −0.99 for transportation expenditures. Total-expenditure elasticities, on the other hand, vary from 0.36 for food (in keeping with Engel’s law) to 1.38 for transportation. Cross-price elasticities suggest complex patterns of substitution and complementation, in that consumption of many goods appears to be complementary with respect 3

Because the Hicks−Allen aggregation conditions are consequences of the budget constraint, rather than of integrability, it would seem that they also ought to be satisfied. However, as has been noted, singularities among the identifying restrictions get in the way of these conditions being imposed.

8.3

Conclusions

145

to the price of some other good, but consumption of that other good appears to be a substitute with respect to the price of the first good. Indeed, such asymmetric substitution structures appear almost to be the norm. In view of the long-standing popularity of double-log functions in applied demand analysis, the demand system of this chapter, which is logarithmic and additive and allows for the demand for each good to be a function of all prices, would appear to be a useful addition to the toolkits of applied demand analysis. The system is reasonably straightforward to apply, and, at least in the application here, appears to give plausible results. While the system is not integrable, this, at least in our opinion, is a small price to pay to be able to work with a set of double-logarithmic demand functions that are additive.

Chapter 9

Quantile Regression Analysis of Asymmetrically Distributed Residuals

The purpose of this chapter is to investigate the phenomenon noted in Chapter 3 of a marked tendency toward asymmetrical distributions of residuals from equations estimated with data from the CES quarterly consumption surveys. Fig. 9.1 provides yet another display of this tendency, this time for the residuals from the model for shelter for 1996 Q1 in Table 6.2. Again, we see that the (kernel-smoothed) distribution of residuals is not symmetrical, but is left-skewed, and has peak density well to the right of the OLS mean of 0.1 As noted in Chapter 3, error distributions with such characteristics have ominous implications for least squares estimation. In this chapter, we shall illustrate, in the context of the additive double-logarithmic model of the last chapter, our procedure for dealing with this asymmetry using quantile regression.

9.1 Quantile Regression Estimation of the Additive Double-Logarithmic Model We now turn to the estimation by both OLS and quantile regression of the additive double-logarithmic model with the CES-ACCRA data set that underlies the analysis of the two preceding chapters. It will be recalled that the estimating equations of this model have the form: lnj qj − lni qi = aij + (βj − βi ) ln y +



(γjk − γjk ) ln ρk , i, j, k = 1, . . . , n, j = i, (9.1) which consists of n–1 double-logarithmic equations, in which the “dependent” variables are logarithmic differences; and the “independent” variables are the logarithms of total expenditure, the n prices, and the standard list of socio-demographicalregional variables. As before, the “left-out” category is miscellaneous expenditures.

1

The modal density is at the 60th quantile in this figure, which is close to the mean modal density of the 61st quantile for all 16 quarters. See Footnote 1 of Chapter 3 for a description of the kernelsmoothing procedure that has been employed.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_9, 

147

148

9 Quantile Regression Analysis of Asymmetrically Distributed Residuals 0.7 0.6 0.5 f(r)

0.4 0.3 0.2 0.1 0 -6

-4

-2

0 OLS residuals

2

4

6

Fig. 9.1 Kernel-smoothed density function OLS residuals for shelter 1996 Q1

In addition to OLS, two quantile regressions are estimated for each category following the procedure described in Chapter 3, the first at q = 0.5 (i.e., the median) and the second at q = q∗ , where q∗ is the modal quantile of the kernel-smoothed residuals from the median regression. The two quantile regressions will be denoted by QR(median) and QR(mode). The estimated regression coefficients and t-values (asymptotic for the quantile equations) for the five categories of expenditure are tabulated in Table 9.1, while the kernel-smoothed distributions of residuals are given in Figs. 9.2, 9.3, 9.4, 9.5, and 9.6.2 The coefficients in the quantile regressions are generally very similar, as are also the coefficients for total expenditure between OLS and the QR equations. R2 s are similar as well.3 Often, however, there are rather marked differences between the OLS and QR price coefficients.4 On the other hand, the real story (for now) is in Figs. 9.2–9.6, for in these we see that all of the residuals distributions (except for the one for health care) are skewed and all have a long tail (or tails). What is particularly interesting about the skewness and long tails in Figs. 9.2–9.6 is that, unlike for the residuals in Fig. 9.1, the error terms vj underlying the residuals in these figures are themselves transforms, specifically, differences either of the form: vj = uj − ui , j = i,

(9.2)

or 2 To save space, graphs for the QR median residuals are suppressed. The distributions of the QR median and mode residuals are very similar, the only real difference being movement of the mode to 0 in the modal regression. 3 The R2 s for the quantile regressions are calculated as the R2 in the OLS regression of predicted on actual values. 4 It must be kept in mind that the coefficients in Table 9.1 refer to elasticity differences, rather than elasticities themselves. Tables of the latter will be presented in the next section.

–1.2085 (−3.45) 0.2840 (3.19) 0.3134 (2.79) −0.0960 (−0.76) 0.0026 (0.02) 0.8421 (2.38) −0.9106 (−43.46) 0.3962

pfood

R2 mode

totexp

pmisc

phlthcare

ptrans

putil

phous

OLS

Variable

Food

QR(Mode)

−0.7668 (−2.26) 0.1074 (1.25) 0.2266 (2.08) −0.1435 (−1.17) 0.2198 (1.35) 0.6040 (1.76) −0.8998 (−44.43) 0.3931 0.4784

QR(Med.)

−0.7179 (−2.10) 0.1006 (1.16) 0.2227 (2.03) −0.1484 (−1.20) 0.2563 (1.56) 0.4984 (1.45) −0.8944 (−43.84) 0.3933

−0.9667 (−2.69) −0.6254 (−6.85) 0.5077 (4.40) 0.2314 (1.78) 0.2337 (1.35) 0.6745 (1.86) −0.4279 (−19.91) 0.2289

OLS

Housing

−0.8665 (−2.45) −0.6717 (−7.48) 0.3997 (3.52) 0.1815 (1.42) 0.1938 (1.14) 0.7991 (2.24) −0.3668 (−17.35) 0.2267

QR(Med.) −0.8494 (−2.40) −0.6665 (−7.43) 0.4046 (3.56) 0.1727 (1.35) 0.2031 (1.19) 0.7673 (2.15) −0.3685 (−17.44) 0.2264 0.4962

QR(Mode) −0.9727 (−2.51) 0.2548 (2.59) −0.4839 (−3.89) 0.0509 (0.36) −0.3291 (−1.77) 1.5005 (3.84) −0.8428 (−36.40) 0.3172

OLS

Utilities

−0.8703 (−2.44) 0.1471 (1.62) −0.6233 (−5.45) −0.0430 (−0.33) −0.0755 (−0.44) 1.4356 (3.99) −0.8381 (−39.34) 0.3144

QR(Med.) −0.7235 (−2.07) 0.1289 (1.45) −0.7446 (−6.64) −0.0605 (−0.48) −0.1435 (−0.85) 1.4604 (4.14) −0.8377 (−40.14) 0.3137 0.4559

QR(Mode) −1.4176 (−2.99) 0.1731 (1.44) 0.1650 (1.08) −1.2485 (−7.28) −0.5675 (−2.49) 3.0436 (6.35) 0.1704 (6.00) 0.0705

OLS

Transportation

−0.5409 (−1.17) 0.0125 (0.17) 0.0136 (0.09) −1.1576 (−6.96) −0.3620 (−1.63) 2.2411 (4.82) −0.1018 (−3.70) 0.0560

QR(Med.) −0.3084 (−0.68) −0.0040 (−0.03) 0.0389 (0.27) −1.2697 (−7.81) −0.2003 (−0.92) 1.9042 (4.20) −0.1318 (−4.90) 0.0531 0.4545

QR(Mode)

Table 9.1 Additive double-log regression equations CES surveys 1996 (t-ratios in parentheses) Health Care

−1.9658 (−3.86) 0.0525 (0.41) 0.4332 (2.65) 0.0494 (0.27) −0.6667 (−2.72) 1.7214 (3.35) −0.7661 (−25.16) 0.3064

OLS

−1.8155 (−3.42) 0.0635 (0.47) 0.1625 (0.95) −0.0228 (−0.12) −0.8908 (−3.49) 2.1671 (4.05) −0.7730 (−24.37) 0.3039

QR(Med.)

−1.7564 (−3.33) 0.0098 (0.07) 0.0882 (0.52) −0.0171 (−0.09) −0.9595 (−3.78) 2.3660 (4.44) −0.7765 (−24.59) 0.3039 0.5209

QR(Mode)

9.1 Quantile Regression Estimation of the Additive Double-Logarithmic Model 149

150

9 Quantile Regression Analysis of Asymmetrically Distributed Residuals OLS 0.6 0.5 f(r)

0.4 0.3 0.2 0.1 0 –6

–4

–2

0 residuals

2

4

6

2

4

6

MODE 0.6 0.5 f(r)

0.4 0.3 0.2 0.1 0 –6

–4

–2

0 residuals

Fig. 9.2 Kernel-smoothed density functions, OLS and QR(mode) residuals for food consumed at home

vj = ln uj − ln ui , j = i,

(9.3)

where uj denotes the error term attaching to the demand function for qj in expression (9.1).5 Accordingly, one might have thought that the skewness and fat tails manifested by the residuals in Fig. 9.1 would be undone by the differencing, but Figs. 9.2–9.6 show that this is obviously not the case.

9.2 Price and Total-Expenditure Elasticities The price and total-expenditure elasticities calculated from the coefficients in Table 9.1 are tabulated in Tables 9.2, 9.3, and 9.4.6 Three things stand out in these tables: 5

Expression (9.2) will hold if the error term appended to qj is exponential, while (9.3) will hold if the error term is multiplicative. At this point, since skewness and long tails are facts, it makes little difference for present purposes which of the two specifications is taken to be correct. 6 The QR elasticities, like the OLS elasticities, are calculated according to the formulae listed in Footnote 2 of the preceding chapter.

9.2

Price and Total-Expenditure Elasticities

151

OLS 0.6 0.5 f(r)

0.4 0.3 0.2 0.1 0 –6

–4

–2

0 residuals

2

4

6

2

4

6

MODE 0.6 0.5 f(r)

0.4 0.3 0.2 0.1 0 –6

–4

–2

0 residuals

Fig. 9.3 Kernel-smoothed density functions, OLS and QR(mode) residuals for shelter

(1) There is close agreement across the three tables in estimates of the total expenditure elasticities. (2) There is also fairly close agreement, although not as strong as for the expenditure elasticities, in the own-price elasticities. The biggest discrepancy is in the own-price elasticities for health care: −0.94 for the modal QR vs. −0.58 for OLS. (3) As noted, the own-price elasticities for the quantile median and modal models are close in magnitude, but this is less the case for the cross-elasticities. Not surprisingly, differences (for a given expenditure category) seem pretty much to be related to discrepancies between the median and the mode. For housing, the mode of distribution of the residuals (as calculated from the median QR regression) is virtually the same as the median (the mode is at quantile 0.4962), and there is in general little difference in the estimated elasticities. On the other hand, the differences are noticeably larger (again, as a general proposition) in the cross-price values for utilities and transportation, which would appear to be a consequence that, for both of these categories, the mode differs from the median by nearly five percentiles.7 7

This, of course, is obviously only an argument of prima facie plausibility.

152

9 Quantile Regression Analysis of Asymmetrically Distributed Residuals

OLS 0.6 0.5

f(r)

0.4 0.3 0.2 0.1 0 –6

–4

–2

0

2

4

6

2

4

6

residuals

MODE 0.6 0.5

f(r)

0.4 0.3 0.2 0.1 0

–6

–4

–2

0 residuals

Fig. 9.4 Kernel-smoothed density functions, OLS and QR(mode) residuals for utilities

9.3 Conclusions The focus in this chapter has been on what appears to be an unappreciated problem in applied econometrics, namely, a situation in which the distribution of regression residuals is asymmetrical with fat tails, a circumstance that clearly has ominous implications for least-squares estimation. The “corrective” proposed in this exercise has been the use of quantile regression, which is an increasingly used robust regression procedure that corresponds to estimation by minimizing the sum of absolute errors at particular quantiles on the distribution of a model’s residuals. Two quantile regressions have been estimated––the first at the quantile corresponding to the median of the regression residuals and the second at the mode of the median regression’s residuals. The procedure has been illustrated with a system of additive double-logarithmic demand functions applied to a cross-sectional data set consisting of 7,754 household observations for six exhaustive expenditure categories from the four quarterly BLS consumer expenditure surveys (augmented with price data collected quarterly by the American Chambers of Commerce Research Association) for 1996.

9.3

Conclusions

153 OLS

f(r)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

–6

–4

–2

0 residuals

2

4

6

MODE

f(r)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

–6

–4

–2

0 residuals

2

4

6

Fig. 9.5 Kernel-smoothed density functions, OLS and QR(mode) residuals for transportation

The residuals from the estimating equations (whether from least squares or quantile regressions) all display the asymmetries in question. Empirically, in the face of these asymmetries, OLS appears reasonably robust in the estimation of income (or total-expenditure) effects, but much less so in the estimation of price (especially cross-price) effects. Not surprisingly, differences appear particularly sensitive to low R2 s and strong asymmetry in the distribution of residuals. In terms of magnitude, the empirical estimates of total expenditure elasticities from the modal QR regressions (and even the OLS models, for that matter) make sense, the only mild surprise (at least for us) being the fairly high value (1.23) for transportation expenditures. The own-price elasticities also seem plausible.8 The cross-price elasticities, though, are another matter, for accumulated knowledge of cross-price effects is in general meager, hence remarks concerning their plausibility pretty much lack a basis.9 In closing this chapter, we would like to emphasize that the asymmetries that have been the focus of this chapter are a real phenomenon. We say this on the basis

8

However, it must be kept in mind that in order to identify the parameters for the “left-out” category (miscellaneous expenditures), it has been assumed that the own-price elasticity for food is the negative of the total expenditure elasticity. 9 The cross-price elasticity in Table 9.4 that most obviously stands out is the value of 3.29 for the elasticity of health care with respect to the price of miscellaneous expenditures. It will be interesting to see whether this strong substitution effect holds up in future research, or is simply an artifact of this particular data set.

154

9 Quantile Regression Analysis of Asymmetrically Distributed Residuals OLS

f(r)

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 –4

–2

f(r)

–6

–6

–4

–2

0 residuals MODE 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 residuals

2

4

6

2

4

6

Fig. 9.6 Kernel-smoothed density functions, OLS and QR(mode) residuals for health care

of thousands (!) of Engel curves and demand functions with the CES data for 1996 through 1999 that have been estimated. Virtually without exception, residuals that have been examined have an asymmetrical distribution. To use in this circumstance estimation that is more robust than least squares seems mandatory. Whether quantile regression at the mode of the distribution of the residuals is the best for the circumstances is, of course, another matter. However, as mentioned earlier, the procedure has intuitive appeal, and this seems sufficient reason, for now, for its use.

Table 9.2 Price and total expenditure elasticities, additive double–logarithmic model, BLS-ACCRA surveys, 1996 OLS regressions (calculated at mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total Expenditure

−0.4275 0.4058 0.0927 0.0308 0.0889 −0.3261

−0.0750 −0.5036 0.2870 0.3583 0.3100 −0.4678

−0.0811 0.3766 −0.7046 0.1778 −0.2428 0.3323

−0.5258 0.2949 −0.0558 −1.1216 −0.4912 1.8754

−1.0742 0.1743 0.2125 0.1763 −0.5806 2.4787

0.8916 0.1218 −0.2207 0.1269 0.0863 −1.1682

0.4275 0.9102 0.4953 1.5085 0.5721 1.3381

9.3

Conclusions

155

Table 9.3 Price and total expenditure elasticities, additive double-logarithmic model, BLSACCRA surveys, 1996 median QR regressions (calculated at mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total Expenditure

−0.4556 0.2929 0.1008 −0.0013 0.2821 −0.5663

−0.2239 −0.4793 0.2779 0.3286 0.2196 −0.2656

−0.2277 0.3395 −0.7452 0.1041 −0.0497 0.3709

0.1017 0.2049 −0.1082 −1.0105 −0.3362 1.1764

−1.1729 0.2559 0.0407 0.1243 −0.8651 3.1780

0.6426 0.1923 −0.1219 0.1471 0.0258 −1.0647

0.4556 1.0112 0.5119 1.2481 0.5770 1.3500

Table 9.4 Price and total expenditure elasticities, additive double-logarithmic model, BLSACCRA surveys, 1996 modal QR regressions (calculated at mean values)

Food Shelter Utilities Trans. Health care Misc.

Food

Shelter

Utilities

Trans.

Health care

Misc.

Total Expenditure

−0.4658 0.3047 0.1096 0.0647 0.1790 −0.4193

−0.2591 −0.4692 0.2910 0.3842 0.2141 −0.2509

−0.1332 0.3262 −0.8583 −0.3936 −0.1324 0.4402

0.2819 0.1994 −0.0747 1.0582 −0.1893 0.8860

−1.1661 0.2071 −0.0254 0.1932 −0.9484 3.2860

0.5903 0.1973 −0.1136 0.2115 0.0110 1.0202

0.4658 0.9971 0.5279 1.2339 0.5892 1.3657

Chapter 10

CES Panel Dynamics: A Discrete-Time Flow-Adjustment Model

In Chapter 4, it was noted that the design of the ongoing CES surveys incorporates a semi-panel feature, in that, following recruitment, households remain in the survey for four successive quarters. A household that was recruited, say, for the first quarter of a year will provide expenditure data for all four quarters of that year before exiting; a household that is recruited in the second quarter of the same year will provide expenditure data for the second, third, and fourth quarters of that year, plus the first quarter of the next year, before exiting; and so on and so forth. In each of the quarterly surveys, there will thus be some households for which data are available not only for the current quarter, but for as many as three preceding quarters. Using only such households, data sets can be constructed that allow for the estimation of dynamic models. This chapter reports the results of pursuing this task using a double-logarithmic version of the flow-adjustment model of Chapter 2.1

10.1 Double-Logarithmic Flow-Adjustment Model Specifically, the model that is estimated has the form2 ln qit = a + b ln qi(t−1) + c ln yit + d ln pit + z + uit ,

(10.1)

1

The state-adjustment model cannot be estimated in this context because of not being able to construct meaningful measures of quantity from the CES expenditure data and ACCRA price indices. The flow-adjustment model, since it can be formulated logarithmically, does allow for such a construction, and hence can be estimated. 2 To arrive at this model (which is a discrete-time version of the continuous-time version of the flow-adjustment model discussed in Section 2.4), let:

  (i) ln qit − ln qi(t−1) = θ ln qˆ it − ln qi(t−1)

where:

(ii) ln qˆ it = κ + μ ln yit + ξ ln pit . Inserting expression (ii) into (i) and combining terms yields the estimating equation in expression (10.1), with a = θκ, b = 1–θ, c = θμ, and d = θξ . L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_10, 

157

158

10

CES Panel Dynamics: A Discrete-Time Flow-Adjustment Model

where q represents pseudo-quantity, y total expenditure, p price, z the standard set of socio-demographico-regional variables, and i and t represent the expenditure category and quarter, respectively.3 This model has been applied (using both OLS and quantile regression) to 15 quarterly “legacy” samples for 1996Q2 through1996Q4. “Legacy” in this context denotes that the data sets used in the estimation include only those households for which expenditure information is available for both the current and immediately preceding quarters. Results are tabulated in Tables 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, and 10.7. For space considerations, coefficients and t-ratios for total expenditure and price in the estimating equations are tabulated for just the four quarters of 1999. These are given in Table 10.1. However, estimates of the short- and long-run total expenditure and price elasticities for all 15 quarters of the exercise are presented in Tables 10.2–10.7.4 In Table 10.1, we see that t-ratios are extremely large (by conventional standards) for both lagged and total expenditure. The coefficients for price are all negative, and, except for 1999Q4 for miscellaneous expenditures using OLS and 1999Q2 for transportation in the quantile regression, all have t-ratios in excess of 2. R2 s are all extremely respectable for cross-sectional data. Although OLS and quantile models give generally similar results, t-ratios in the quantile regressions are often larger for lagged expenditure and smaller for both total expenditure and for price. Residuals for most quarters have the usual asymmetry, particularly for transportation and miscellaneous expenditures, where the modes are seen to be well to the right of the median. Interestingly, however, the differences between OLS and quantile coefficients turn out to be least for these categories. Turning now to the short- and long-run elasticity estimates for the 15 quarters in Tables 10.2–10.7, the following results stand out: (1) As has been found throughout the analysis to this point, total-expenditure elasticities are much more stable across quarters than are price elasticities. That this is the case is evident both by eye and by the size of the respective two-standard-deviation confidence intervals. (2) In general, there are larger differences in the short-run elasticities between the OLS and quantile regressions than in the long-run elasticities. This is most pronounced for shelter, where the mean quantile short-run total-expenditure elasticity is less than half that for OLS (0.21 vs. 0.45). (3) Except for short-run total-expenditure elasticity for food, there is little evidence of trends in the estimated elasticities.

3

As in preceding chapters, the logarithm of the pseudo-quantity is once again defined as lnq =lnE–lnp, where lnE =ln(pq); u, of course represents an unobservable error term. 4 From Footnote 1, the short-run elasticities will be equal to c and d for total expenditure and price, respectively, while the long-run elasticities will be equal to c/(1–b) and d/(1–b). Unlike in preceding chapters, elasticities estimated are from equations in which socio-demographico-regional with t-ratios less than 2 are eliminated.

q(t–1)

1999Q1

Utilities

1999Q4

1999Q3

1999Q2

1999Q1

Shelter

1999Q4

1999Q3

1999Q2

1999Q1

0.4639 (32.13)

0.6235 (37.36) 0.5601 (40.02) 0.5691 (43.41) 0.5433 (4128)

0.4113 (21.03) 0.4757 (27.31) 0.4606 (28.46) 0.4302 (28.13)

Food consumed at home

Quarter

OLS

0.1697 (11.37)

0.4212 (16.58) 0.4817 (16.58) 0.5218 (25.74) 0.4522 (22.87)

0.1943 (11.85) 0.1867 (12.22) 0.1766 (13.30) 0.2173 (16.64)

Total expenditure

−0.6062 (−8.25)

−0.3876 (−9.63) −0.2156 (−5.16) −0.2430 (−6.24) −0.4697 (−10.60)

−0.2869 (−3.09) −0.2586 (−2.86) −0.3793 (−3.98) −0.4242 (−5.72)

Price

0.6301

0.6736

0.6717

0.6555

0.6556

0.5682

0.5350

0.5352

0.5220

R2

0.5735 (44.56)

0.8116 (73.20) 0.7755 (88.51) 0.7647 (89.10) 0.7330 (80.98)

0.4729 (25.22) 0.5082 (30.69) 0.5446 (35.15) 0.5421 (37.23)

q(t–1)

QR(Mode)

0.1377 (10.35)

0.2137 (12.66) 0.2423 (16.68) 0.2747 (20.69) 0.2822 (19.96)

0.1720 (10.94) 0.1679 (11.56) 0.1532 (12.05) 0.1613 (12.97)

Total expenditure

−0.4635 (−7.07)

−0.2928 (−10.95) −0.0416 (−1.59) −0.1719 (−6.75) −0.3565 (−15.91)

−0.3445 (−3.87) −0.2382 (−2.77) −0.3273 (−3.59) −0.4294 (−6.09)

Price

0.6230

0.6465

0.6717

0.6313

0.6361

0.5616

0.5302

0.5341

0.5191

R2

0.5068

0.5353

0.5269

0.5188

0.5367

0.5210

0.5405

0.5218

0.5027

Mode

2074

2899

2900

2608

2086

2928

2917

2630

2087

#Obs.

Table 10.1 Coefficients for total expenditure and price double-logarithmic flow-adjustment model CES-ACCRA samples four quarters 1999 (t-ratios in parentheses)

10.1 Double-Logarithmic Flow-Adjustment Model 159

0.5909 (43.14) 0.5497 (44.23) 0.6064 (47.73)

1999Q2

1999Q3

1999Q2

1999Q1

Health care

1999Q4

1999Q3

1999Q2

1999Q1

0.5773 (29.85) 0.5393 (31.71) 0.5222 (30.34)

0.1467 (8.17) 0.1593 (10.30) 0.1587 (10.96) 0.1203 (8.57)

Transportation

0.2393 (6.71) 0.3844 (12.00) 0.2735 (9.23)

1.1794 (34.13) 1.1561 (39.27) 1.2589 (45.83) 1.2053 (44.34)

0.1500 (12.13) 0.1551 (13.69) 0.1495 (12.79)

Total expenditure

−1.3845 (−10.09) −1.2127 (−10.08) −1.5725 (−13.11)

−1.5412 (−5.42) −1.2247 (−7.14) −1.4605 (−6.49) −1.5929 (−6.62)

0.7002

−0.5364 (−8.99) −0.4488 (−11.17) −0.4779 (−8.08)

0.4419

0.4673

0.4823

0.5615

0.5699

0.5486

0.5224

0.7024

0.6811

R2

Price

0.6769 (39.27) 0.6950 (47.04) 0.6516 (39.93)

0.1620 (8.61) 0.1748 (10.31) 0.1580 (10.45) 0.1438 (9.98)

0.7266 (63.11) 0.7126 (65.16) 0.7574 (70.46)

q(t–1)

QR(Mode)

0.1737 (5.46) 0.3269 (39.27) 0.2267 (8.06)

1.0296 (28.33) 0.9319 (28.89) 1.1594 (40.40) 1.0150 (36.36)

0.0921 (8.86) 0.0930 (9.32) 0.0939 (9.39)

Total expenditure

−1.2000 (−9.81) −1.2160 (−11.64) −1.4246 (12.52)

−1.3326 (−4.47) −0.9686 (−5.18) −1.0980 (−4.67) −1.3492 (−5.47)

−0.3591 (−7.16) −0.2986 (8.52) −0.3366 (−6.66)

Price

0.4348

0.4590

0.4779

0.5596

0.5692

0.5464

0.5206

0.6925

0.6682

0.6927

R2

0.5238

0.5240

0.5123

0.5257

0.5369

0.5400

0.5943

0.4790

0.5143

0.4879

Mode

2358

2122

1674

2782

2782

2506

1979

2910

2908

2604

#Obs.

10

1994Q4

1999Q3

q(t–1)

Quarter

OLS

Table 10.1 (continued)

160 CES Panel Dynamics: A Discrete-Time Flow-Adjustment Model

0.5427 (29.91)

1999Q4

1999Q4

1993Q3

1999Q2

1999Q1

0.3548 (21.00) 0.4137 (25.30) 0.4561 (30.24) 0.4620 (29.92)

Miscellaneous expenditures

q(t–1)

Quarter

OLS

0.8153 (25.57) 0.8521 (29.20) 0.7431 (28.69) 0.7396 (27.36)

0.2955 (9.54)

Total expenditure

0.6870

0.7123

0.6995

0.6634

0.4350

−1.4188 (−11.64)

−1.2876 (−5.83) −0.4271 (−2.01) −0.7065 (−3.37) −0.2616 (−1.39)

R2

Price

0.3614 (25.09) 0.3904 (28.08) 0.4492 (36.51) 0.4280 (33.54)

0.6960 (45.76)

q(t–1)

QR(Mode)

0.8407 (30.92) 0.8508 (34.29) 0.7093 (33.56) 0.7399 (33.54)

0.2284 (8.80)

Total expenditure

Table 10.1 (continued)

−1.2106 (−6.43) −0.6778 (−3.75) −0.8336 (−4.88) −0.5175 (−3.34)

−1.2055 (−11.80)

Price

0.6850

0.7104

0.6970

0.6614

0.4259

R2

0.5571

0.5777

0.5679

0.5780

0.4964

Mode

2561

2556

2297

1815

2351

#Obs.

10.1 Double-Logarithmic Flow-Adjustment Model 161

162

10

CES Panel Dynamics: A Discrete-Time Flow-Adjustment Model

Table 10.2 Total expenditure and price elasticities double-logarithmic flow-adjustment model food consumed at home OLS

QR(Mode) Price

Quarter

Short run

Long run

Short run

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.1618 0.2183 0.1957 0.1888 0.1797 0.1957 0.1814 0.2059 0.2236 0.2082 0.1688 0.1943 0.1867 0.1766 0.2173 0.1935 0.0185

0.2752 0.3919 0.3652 0.3642 0.3421 0.3573 0.3485 0.3681 0.4299 0.3582 0.3299 0.3300 0.3561 0.3274 0.3814 0.3550 0.0345

−0.2488 −0.1226 −0.2060 −0.4022 −0.2653 −0.1988 −0.3538 −0.6219 −0.3108 −0.4923 −0.6262 −0.2869 −0.2586 −0.3783 −0.4242 −0.3465 0.1478

elasticity

Total expenditure

Total expenditure

Price

Long run

Short run

Long run

Short run

Long run

−0.4232 −0.2201 −0.3845 −0.7759 −0.5050 −0.3630 −0.6797 −1.1117 −0.5976 −0.8469 −1.2238 −0.4873 −0.4932 −0.7032 −0.7445 −0.6373 0.2766

0.1387 0.1841 0.1636 0.1518 0.1394 0.1616 0.1381 0.1623 0.1743 0.1802 0.1416 0.1720 0.1679 0.1532 0.1613 0.1593 0.0152

0.2691 0.3688 0.3508 0.3270 0.3289 0.3485 0.3087 0.3240 0.3653 0.3404 0.3300 0.3263 0.3414 0.3365 0.3523 0.3311 0.0267

−0.2919 −0.0573 −0.2210 −0.4513 −0.3162 −0.3538 −0.3757 −0.3108 −0.4923 −0.3430 −0.2869 −0.3445 −0.2382 −0.3273 −0.4294 −0.3208 0.1297

−0.5663 −0.1148 −0.4738 −0.9720 −0.7461 −0.6797 −0.8399 −0.5976 −0.7718 −0.6479 −0.4873 −0.6536 −0.4843 −0.7189 −0.9378 −0.6698 0.2741

#Obs. 1247 1872 2109 2212 2168 2162 2194 2275 2209 2171 2147 2087 2630 2917 2928

Total Expenditure Elasticities

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

short-run long-run

0

5

10 quarter

15

20

Price Elasticities 0 elasticity

–0.2

0

2

4

6

8

10

12

14

–0.4

16

18 short-run long-run

–0.6 –0.8 –1 –1.2 quarter

Two-Standard Deviation Confidence Intervals [QR(mode)]: Total Expenditure: SR: 0.1289 < e < 0.1897; LR: 0.2777 < e < 0.3845 Price: SR: −0.0614 > e > −0.5802; LR: 0.0870 > e > –1.6618.

10.1

Double-Logarithmic Flow-Adjustment Model

163

Table 10.3 Total expenditure and price elasticities double-logarithmic flow-adjustment model shelter OLS

QR(Mode)

Total expenditure

Price

Quarter

Short run

Long run

Short run

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.4424 0.4733 0.4570 0.4256 0.4134 0.4654 0.4723 0.4872 0.4191 0.4182 0.4639 0.4212 0.4817 0.5218 0.4552 0.4545 0.0311

1.2816 1.0287 1.0441 1.0769 1.0791 1.2319 1.0657 1.1700 1.0886 1.1324 1.1087 1.1873 1.0950 1.2110 0.9967 1.1153 0.0787

−0.3050 −0.2951 −0.5093 −0.2552 −0.1377 −0.2069 −0.1842 −0.2080 −0.2529 −0.1735 −0.2560 −0.3876 −0.2156 −0.2430 −0.4697 −0.2733 0.1066

Total expenditure

Price

Long run

Short run

Long run

Short run

Long run

−0.8836 −0.6414 −1.1636 −0.6458 −0.3630 −0.5476 −0.4156 −0.4995 −0.6569 −0.4698 −0.6119 −1.0295 −0.4901 −0.5639 −1.0285 −0.6671 0.2450

0.1805 0.2208 0.2272 0.2020 0.1771 0.1886 0.2242 0.1825 0.1490 0.1756 0.2195 0.2137 0.2423 0.2427 0.2822 0.2107 0.0371

1.2765 1.0059 0.9136 1.0791 1.0327 1.2231 0.9960 1.1155 1.1153 1.0540 1.0257 1.1343 1.0793 1.1674 1.0569 1.0850 0.0920

−0.1520 −0.2111 −0.3649 −0.1280 0.0006 −0.0795 −0.1520 −0.0795 −0.0953 −0.1118 −0.1680 −0.2928 −0.0416 −0.1719 −0.3565 −0.1600 0.1076

−1.0750 −0.9617 −1.4672 −0.6838 0.0035 −0.4903 −0.6753 −0.4859 −0.7133 −0.6711 −0.7851 −1.5542 −0.1853 −0.7306 −1.3352 −0.7874 0.4372

#Obs. 1321 1998 2087 2192 2163 2158 2197 2277 2205 2174 2136 2085 2608 2900 2899

elasticity

Total Expenditure Elasticities 1.4 1.2 1 0.8 0.6 0.4 0.2 0

short-run long-run

0

5

10 quarter

15

20

Price Elasticities 0.5 elasticity

0 –0.5

0

5

10

15

–1 –1.5 –2 quarter

Two-Standard Deviation Confidence Intervals [QR(mode)]: Total Expenditure: SR: 0.1365 < e < 0.2849; LR: 0.9010 < e < 1.2690 Price: SR: 0.0552 > e > −0.3752; LR: 0.0870 > e > −1.6618.

20

short-run long-run

164

10

CES Panel Dynamics: A Discrete-Time Flow-Adjustment Model

Table 10.4 Total expenditure and price elasticities. Double-logarithmic flow-adjustment model utilities OLS

QR(Mode)

Total expenditure

Price

Quarter

Short run

Long run

Short run

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.1055 0.1426 0.1636 0.1985 0.1302 0.1578 0.1794 0.1314 0.1415 0.1219 0.1213 0.1697 0.1500 0.1551 0.1495 0.1479 0.0243

0.3119 0.3483 0.3607 0.3976 0.4362 0.3748 0.3964 0.3358 0.4100 0.3156 0.3401 0.3165 0.3667 0.3444 0.3798 0.3623 0.0370

−0.2355 −0.3901 −1.0069 −0.1377 −0.2669 −0.3441 −0.3284 −0.5515 −0.2347 −0.4170 −0.4516 −0.6062 −0.5364 −0.4448 −0.4779 −0.4902 0.2476

Total expenditure

Price

Long run

Short run

Long run

Short run

Long run

−0.6961 −0.9529 −2.2198 −2.1246 −0.8941 −0.8173 −0.7256 −1.4094 −0.6809 −1.0798 −1.2661 −1.1308 −1.3112 −0.9878 −1.2142 −1.1673 0.4671

0.0869 0.1219 0.0983 0.1565 0.0959 0.1018 0.1028 0.0894 0.0904 0.0894 0.0802 0.1377 0.0921 0.0930 0.0939 0.1020 0.0209

0.3035 0.3529 0.2852 0.3830 0.4086 0.3429 0.3581 0.3125 0.3867 0.3043 0.3056 0.3667 0.3369 0.3236 0.4003 0.3418 0.0387

−0.1875 −0.3172 −1.0204 −1.0679 −0.2411 −0.2465 −0.1801 −0.4572 −0.1323 −0.3414 −0.3914 −0.5364 −0.3591 −0.4779 −0.3366 −0.4027 0.2776

−0.6549 −0.9184 −2.0603 −2.6136 −1.0273 −0.8303 −0.6273 −1.5981 −0.5659 −1.1620 −1.4916 −1.0868 −1.3135 −1.0390 −1.4348 −1.2882 0.6868

Obs. 1321 1996 2097 2195 2160 2145 2176 2272 2204 2172 2134 2074 2604 2908 2910

Total Expenditure Elasticities

0.5 elasticity

0.4 0.3

short-run long-run

0.2 0.1 0 0

5

10 quarter

15

20

elasticity

Price Elasticities 0 0 –0.5 –1 –1.5 –2 –2.5 –3 –3.5

5

10

15

20 short-run long-run

quarter

Two-Standard Deviation Confidence Intervals [QR(mode)]: Total Expenditure: SR: 0.9329 < e < 1.1609; LR: 1.1210 < e < 1.4270 Price: SR: −0.4667 > e > −1.7551; LR: −0.5544 > e > −2.1576.

10.1

Double-Logarithmic Flow-Adjustment Model

165

Table 10.5 Total expenditure and price elasticities double-logarithmic flow-adjustment model transportation OLS

QR(Mode)

Total expenditure

Price

Quarter

Short run

Long run

Short run

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.1055 0.1426 0.1636 0.1985 0.1302 0.1578 0.1794 0.1314 0.1415 0.1219 0.1213 0.1697 0.1500 0.1551 0.1495 0.1479 0.0243

0.3119 0.3483 0.3607 0.3976 0.4362 0.3748 0.3964 0.3358 0.4100 0.3156 0.3401 0.3165 0.3667 0.3444 0.3798 0.3623 0.0370

−0.2355 −0.3901 −1.0069 −0.1377 −0.2669 −0.3441 −0.3284 −0.5515 −0.2347 −0.4170 −0.4516 −0.6062 −0.5364 −0.4448 −0.4779 −0.4902 0.2476

Total expenditure

Price

Long run

Short run

Long run

Short run

Long run

−0.6961 −0.9529 −2.2198 −2.1246 −0.8941 −0.8173 −0.7256 −1.4094 −0.6809 −1.0798 −1.2661 −1.1308 −1.3112 −0.9878 −1.2142 −1.1673 0.4671

0.0869 0.1219 0.0983 0.1565 0.0959 0.1018 0.1028 0.0894 0.0904 0.0894 0.0802 0.1377 0.0921 0.0930 0.0939 0.1020 0.0209

0.3035 0.3529 0.2852 0.3830 0.4086 0.3429 0.3581 0.3125 0.3867 0.3043 0.3056 0.3667 0.3369 0.3236 0.4003 0.3418 0.0387

−0.1875 −0.3172 −1.0204 −1.0679 −0.2411 −0.2465 −0.1801 −0.4572 −0.1323 −0.3414 −0.3914 −0.5364 −0.3591 −0.4779 −0.3366 −0.4027 0.2776

−0.6549 −0.9184 −2.0603 −2.6136 −1.0273 −0.8303 −0.6273 −1.5981 −0.5659 −1.1620 −1.4916 −1.0868 −1.3135 −1.0390 −1.4348 −1.2882 0.6868

Obs. 1321 1996 2097 2195 2160 2145 2176 2272 2204 2172 2134 2074 2604 2908 2910

elasticity

Total Expenditure Elasticities 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

short-run long-run

0

5

10 quarter

15

20

Price Elasticities 0 0

5

10

15

20

elasticity

–0.5 short-run long-run

–1 –1.5 –2 quarter

Two-Standard Deviation Confidence Intervals [QR(mode)]: Total Expenditure: SR: 0.9329 < e < 1.1609; LR: 1.1210 < e < 1.4270 Price: SR: −0.4667 > e > −1.7551; LR: −0.5544 > e > −2.1576.

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Table 10.6 Total expenditure and price elasticities double-logarithmic flow-adjustment model health care OLS

QR(Mode)

Total expenditure

Price

Quarter

Short run

Long run

Short run

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.1676 0.3277 0.3250 0.2793 0.3020 0.2641 0.2664 0.2224 0.3088 0.3111 0.3046 0.2393 0.3844 0.2735 0.2955 0.2848 0.0508

0.3872 0.6986 0.7279 0.5886 0.6267 0.5907 0.6137 0.4782 0.6894 0.6534 0.6734 0.5661 0.8344 0.5724 0.6462 0.6231 0.1051

−1.1083 −1.2081 −1.0707 −1.2369 −1.1425 −1.4274 −1.1275 −0.9983 −1.2490 −1.2567 −1.1907 −1.3845 −1.2127 −1.5725 −1.4188 −1.2403 0.1534

Total expenditure

Price

Long run

Short run

Long run

Short run

Long run

−2.5602 −2.5754 −2.3980 −2.6067 −2.3708 −3.1926 −2.5973 −2.1464 −2.7886 −2.6396 −2.6325 −3.2754 −2.6323 −3.2911 −3.1026 −2.7206 0.3443

0.1215 0.2920 0.2586 0.2183 0.3236 0.1750 0.1965 0.2245 0.2726 0.2146 0.2692 0.1737 0.3269 0.2267 0.2284 0.2348 0.0569

0.3306 0.7273 0.7606 0.6828 0.8149 0.6068 0.6415 0.6082 0.7973 0.6755 0.8345 0.5376 1.0718 0.6506 0.7513 0.6994 0.1632

−1.0737 −1.2493 −1.0495 −1.1654 −1.2474 −1.3514 −1.1668 −1.1261 −1.1924 −1.1284 −1.1345 −1.2000 −1.2160 −1.4246 −1.2055 −1.1952 0.0979

−2.9216 −3.1116 −3.0868 −3.6453 −3.1427 −4.6859 −3.8093 −3.0509 −3.4876 −3.5518 −3.5065 −3.7140 −3.9869 −4.0890 −3.9655 −3.5836 0.4821

#Obs. 1033 1591 1678 1775 1732 1741 1773 1832 1785 1743 1715 1674 2122 2358 2351

elasticity

Total Expenditure Elasticities 1.2 1 0.8 0.6 0.4 0.2 0

short-run long-run

0

5

10 quarter

15

20

Price Elasticities 0 0

5

10

15

20

elasticity

–1 –2

short-run long-run

–3 –4 –5

quarter

Two-Standard Deviation Confidence Intervals [QR(mode)]: Total Expenditure: SR: 0.1210 < e < 0.3486; LR: 0.3730 < e < 1.0258 Price: SR: −0.9994 > e > −1.3910; LR: −2.6194 > e > −4.5478.

10.2

Comparison with Static-Model Elasticities

167

Table 10.7 Total expenditure and price elasticities double-logarithmic flow-adjustment model miscellaneous expenditures OLS

QR(Mode)

Total expenditure

Price

Quarter

Short run

Long run

Short run

1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.8656 0.9440 0.9251 0.8476 0.7207 0.7841 0.8131 0.9116 0.8174 0.9041 0.8684 0.8153 0.8521 0.7431 0.7396 0.8368 0.0694

1.5269 1.4845 1.3632 1.3737 1.3398 1.3214 1.3935 1.4204 1.4943 1.5226 1.4385 1.2636 1.4534 1.3662 1.3747 1.4091 0.0770

−1.7700 −0.5837 −1.0585 −1.0318 −0.2407 −0.6956 −0.6281 −1.2380 −0.6183 −0.4834 −0.3943 −1.2876 −0.4271 −0.7065 −0.2616 −0.7617 0.4313

Total expenditure

Price

Long run

Short run

Long run

Short run

Long run

−3.1222 −0.9179 −1.5598 −1.6723 −0.4475 −1.1722 −1.0764 −1.9290 −1.1304 −0.8141 −0.6531 −1.9957 −0.7285 −1.2990 −0.4863 −1.2670 0.7072

0.8207 0.9115 0.8578 0.8492 0.7818 0.7488 0.8121 0.8647 0.8185 0.8573 0.7583 0.8407 0.8508 0.7093 0.7399 0.8148 0.0561

1.4846 1.4211 1.3025 1.3974 1.2939 1.3538 1.3753 1.3309 1.4734 1.4161 1.3405 1.3165 1.3957 1.2878 1.2935 1.3655 0.0645

−1.2888 −0.9843 −0.9837 −1.0365 −0.5159 −0.9054 −0.5984 −1.2308 −0.5661 −0.7833 −0.4602 −1.2106 −0.6778 −0.8336 −0.5175 −0.8395 0.2792

−2.3314 −1.5346 −1.4936 −1.7056 −0.8539 −1.6370 −1.0134 −1.8944 −1.0191 −1.2939 −0.8135 −1.8957 −1.1119 −1.5134 −0.9047 −1.4011 0.4491

Total Expenditure Basticities

0

5

10

15

20

–0.5

1.5 short-run long-run

1 0.5

elasticities

elasticity

1109 1687 1803 1909 1868 1841 1915 1987 1916 1881 1854 1815 2297 2556 2561

Price Elasticities

0

2

#Obs.

–1

short-run long-run

–1.5 –2

0 0

5

10 quarter

15

20

–2.5

quarter

Two-Standard Deviation Confidence Intervals [QR(mode)]: Total Expenditure: SR: 0.7026 < e < 0.9438; LR: 1.2365 < e < 1.4945 Price: SR: −0.2811 > e > −1.3979; LR: −0.5029 > e > −2.2993

10.2 Comparison with Static-Model Elasticities A perennial question in the analysis of budget surveys is the extent to which dynamics are reflected in expenditure data.5 If dynamics are absent, then the price and income elasticities that are estimated can be interpreted as measuring long-run (or steady-state) values, whereas if dynamics are present the estimates are neither fish 5 Dynamics, in the sense being considered here, appear to have been first discussed as a problem in the analysis of family budgets by Prais and Houthakker (1955).

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CES Panel Dynamics: A Discrete-Time Flow-Adjustment Model

nor fowl, in the sense of being neither short run nor long run. Following the debate in the 1950s concerning the efficacy of incorporating income elasticities that are estimated from budget surveys into time-series regressions for estimating price elasticities, the view has pretty much been that the situation with budget data is the former, that is, that short-term dynamics are largely absent, so that the estimates obtained (assuming that models are otherwise properly specified) represent steadystate values. The basis for this argument is that, whereas time-series estimates of price and income elasticities will reflect short-run adjustment to changes in income and prices, cross-section estimates will reflect long-run, steady-state adjustment.6 The latter is seen as being the case even if households, although they may be in temporal disequilibrium, are affected equally by cyclical and other time-varying factors. On the other hand, when dynamic phenomena other than inter-temporal variety are present, the argument needs modification. Reference here is to the fact that, for most commodities, purchases are influenced by existing stocks (including stocks of “habits”) as well as by income, price, family size, etc. Stocks (as noted in Chapter 2) inject a dynamic element into the consumption process in that a change in income, say, gives rise to a change in purchases before stocks have a chance to adjust. The new purchases affect stocks, which in turn feeds back on purchases, and so on, with long-run equilibrium being achieved when stocks cease adjusting. Since income and prices change through time, this type of dynamic adjustment will of course be reflected in time-series data, and both types of adjustment (that is, short run and long run) can be isolated if a proper model is employed.7 Less obvious, however, is the fact that this type of dynamic behavior can also be reflected in the budget data, even if all households are affected equally by inter-temporal phenomena. An extreme instance would be when a household has purchased a durable good (an automobile, say) in a time period just before information is to be provided to the budget survey. Current expenditure will obviously be affected by the recent purchase, and proper modeling requires that this influence be taken into account. Fortunately, the results with the dynamic model of this chapter allows for contingencies of this kind to be treated as an empirical question, for if they are absent, then the elasticities that are estimated in static models should be comparable in size to the long-run elasticities that have been estimated with the dynamic models of this chapter. In Table 10.8, which tabulates the mean estimates from the logarithmic flowadjustment models in this chapter and the same from the static logarithmic models in Chapter 6, we see that this is generally the case.8 For total-expenditure elasticities,

6 For detailed discussions of the differences between cross-section and time-series estimates of the same parameters, see Kuh and Meyer (1957) and Kuh (1959, 1963). 7 Besides Chapter 2, see Deaton (1975), Deaton and Muellbauer (1980a), Phlips (1983), and Pollak and Wales (1992). 8 Since quantile regression is not employed in Chapter 6, the comparison in this table is with means from the OLS regressions of that chapter.

10.2

Comparison with Static-Model Elasticities

169

Table 10.8 Comparison of total expenditure and price elasticities static and dynamic models CESACCRA data sets mean values 1996Q2−1999Q4 Logarithmic Flow-adjustment model Static model

Total expenditure

Price

Category

Total expenditure

Price

Short run

Long run

Short run

Long run

Food Shelter Utilities Transportation Health care Misc. Exp.

0.3175 0.8509 0.3807 1.3486 0.5066 1.1293

−0.3821 −0.5665 −0.9540 −1.4233 −1.1242 −1.3665

0.1935 0.4545 0.1479 1.1971 0.2848 0.8368

0.3550 1.1153 0.3623 1.4376 0.6231 1.4091

−0.3465 −0.2733 −0.4902 −1.3025 −1.2403 −0.7617

−0.6373 −0.6671 −1.1673 −1.5658 −2.7206 −1.2670

agreement is particularly close for food, utilities, and transportation, and reasonably so for shelter and health care, while for price elasticities, agreement is reasonably close for all but food and health care. Rank orderings by size is identical in both models for both elasticities (i.e., for total expenditure, the elasticity is smallest for food and largest for transportation, etc., and similarly for price). On the other hand, some large discrepancies in elasticities between the two models are evident as well. This is especially the case in the price elasticities for food and health care, for which the long-run elasticities in the flow-adjustment model are roughly double those of the static. For total expenditure, the largest differences are for shelter (0.85 in the static model vs. 1.12 for flow-adjustment) and miscellaneous expenditures (1.13 vs. 1.41). Also, it is to be noted that, with few exceptions, the short- and longrun elasticities from the flow-adjustment models bracket the estimates of the static models, the exceptions being food and utilities for total expenditure and health care and miscellaneous expenditures for price. In general, the results in Table 10.8 are very encouraging for the use of static models with budget-survey data when an absence of panel information does not allow estimation of dynamic models. While there is some ambiguity with regard to price elasticities (as evidenced in the large discrepancies in the static and dynamic estimates for food and health care), this is much less the case for total-expenditure elasticities. Accordingly, we are comfortable at this point in interpreting totalexpenditure elasticities estimated in static models from the CES surveys as long-run in nature.9

9

In passing, it might be useful to note that, were the dynamical analysis of this chapter to be formulated in a co-integration framework [Engle and Granger (1987)], the steady-state elasticities would in fact be identified with the elasticities estimated in the static equations.

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CES Panel Dynamics: A Discrete-Time Flow-Adjustment Model

10.3 State- vs. Flow-Adjustment Behavior In the discussion of the dynamics of state- and flow-adjustment models in Chapter 2, recall that the key distinguishing feature between the two models is that, in the stateadjustment model, expenditures (in relation to prices and income) are adjusted so as to achieve a stock equilibrium, while, in the flow-adjustment model, expenditures are adjusted so as to achieve a flow equilibrium. With the former, depending upon whether the commodity is stock-adjusting or habit-forming, short-run elasticities can be either greater or smaller (in absolute value) than the long-run elasticities, while with the latter, short-run elasticities will always be smaller than their counterparts for the long run. However, the fact (as noted earlier) that the state-adjustment model cannot be explicitly estimated with the CES data does not mean that some form of state-adjustment behavior cannot be indirectly inferred. To see how this might be done, let us rewrite here for convenience the two “structural” equations for the flow-adjustment model of this chapter from Footnote 1 above:   ln qit − ln qi(t−1) = θ ln qˆ it − ln qi(t−1) ,

(10.2)

ln qˆ it = κ + μ ln yit + ξ ln pit ,

(10.3)

where

which, upon combining, can be written as ln qit = θ κ + (1 − θ ) ln qi(t−1) + θ μ ln yit + θ ξ ln pit .

(10.4)

The short-run elasticities will be given by θ μ and θ ξ , respectively, while the longrun elasticities will be given by μ and ξ.10 In the state-adjustment model, goods subject to habit formation always have short-run elasticities smaller than in the long run, while the reverse is true for goods subject to stock adjustment. Among other things, this implies that goods subject to stock adjustment will have short-run elasticities that are large in relation to the shortrun elasticities for habit-formation goods. In the flow-adjustment model, short-run elasticities are necessarily smaller than their long-run counterparts, but can nevertheless be large in relation to these, depending upon the value of θ . If, for example, μ is large, then a large value for θ will imply a large value for θ μ.11 On the other hand, a large value of θ implies a small value for 1−θ , which in turn implies that the last period’s expenditure will have little effect on current expenditure. The extreme, of course, is a θ equal to 1, in which case current expenditure is a function only of The short-run elasticities are defined as ∂lnqit /∂lnyit and ∂lnqit /∂lnpit , holding lnqi (t–1) constant, while the long-run elasticities are obtained from expression (10.3) [or equivalently from expression (10.4), by setting lnqit equal to lnqi(t–1) ]. 11 Stability of the underlying consumption process of course requires θ≤ 1. 10

10.4

Conclusions

171

current income and price. We are thus back to a static model, but now to the one whose interpretation [via expression (10.3)] is that of being in the long-run steady state. However, for now, the point that we wish to make is that large short-run elasticities in flow-adjustment models can itself be interpreted as reflective of underlying stock-adjusting behavior (as opposed to small short-run elasticities, which would be reflective of underlying habit-formation behavior). In particular, such behavior should be expected whenever the expenditure category being analyzed contains sizeable amounts of expenditures for durable goods. In the present context, expenditures for durable goods are largely confined to transportation and miscellaneous expenditures. Expenditures for automobiles and tires are included in transportation, while expenditures for most other durable goods (furniture and household equipment, recreational vehicles, etc.) are included among miscellaneous expenditures. Accordingly, it is appropriate to find in Table 10.8 that the largest short-run elasticities (in relation to their long-run counterparts) are indeed for these two categories. Thus, while it is not possible, because of an absence of quantity data, to estimate state-adjustment behavior explicitly in the CES surveys, such can nevertheless be inferred.

10.4 Conclusions In this chapter, we have focused on the semi-panel nature of the CES surveys. Double-logarithmic dynamic demand models of the flow-adjustment type have been estimated using panel information for consecutive quarters for 15 of the CES-ACCRA data sets beginning with 1996Q2. The results are of interest both in themselves and as a check on the elasticities that have been estimated using static models. In general, the statistical results parallel those in previous chapters. Total expenditure and price continue to be statistically significant predictors, and models for the most part are stable over the quarters. As in the static models, totalexpenditure elasticities are much more stable than price elasticities. Dynamic effects are strong, with typically substantial differences between short- and long-run effects. In general, the results from comparing the long-run estimates with static-model estimates from Chapter 6 support the long-held conventional view that information represented in cross-sectional data is primarily steady state in nature. While somewhat mixed for price elasticities, the evidence in support of this conclusion is especially strong for total expenditure. This is an important result, for it provides strong justification in our opinion for interpreting conventionally estimated cross-sectional Engel curves (in circumstances in which panel information is not available) as reflecting long-run equilibrium behavior.

Chapter 11

Engel Curves for 29 Categories of CES Expenditure

In the chapters to this point, price information collected in quarterly surveys by the American Chambers of Commerce Research Association has been combined with the CES quarterly surveys to create data sets from which price as well as total-expenditure elasticities have been estimated. However, because of the limited coverage of the ACCRA data, estimation has been restricted to a fairly high level of aggregation. While the price and total-expenditure elasticities obtained at this level are obviously of interest in their own right, one of the main findings with the CES data is that the presence or absence of prices in demand functions makes little difference to the estimates of total-expenditure elasticities. This apparent absence of an omitted-variables bias is an important result, for it suggests that estimates of total-expenditure elasticities at lower levels of aggregation, for which expenditure data are available but price information is not, can reasonably be assumed to be free of bias as well. Anyway, such will be our assumption in this chapter as we extend estimation of total-expenditure elasticities to the 29 categories of expenditure that are listed in Table 4.3. The results for the 16 quarterly CES surveys for 1996Q1 through 1999Q4 are summarized in Tables 11.1–11.29. Only simple double-logarithmic models have been used in estimation. OLS models have been estimated, to begin with, employing the full set of socio-demographico-regional variables described in the preceding chapters as controls. “Final” models are arrived at (still using OLS) by the elimination of all of the control variables that have t-ratios less than 2. These “final” models are then re-estimated by quantile regression at both the median and the mode. The end result is a set of equations totally nearly 1,400. However, to keep the amount of material presented manageable, elasticities from only the OLS and QR(mode) equations are tabulated. This information (along with QR R2 s and modal values) is presented in a separate table for each category. In addition to the 16 elasticity estimates, means and standard deviations of the estimates are included, as are graphs of the 16 QR(mode) estimates for each category, along with two-standard-deviation confidence intervals constructed from the same. To conserve space, t-ratios for the estimated total expenditure elasticities are suppressed. In general, t-ratios in the

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_11, 

173

174

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.1 Total-expenditure elasticities, food consumed at home, CES surveys 1996Q1–1999Q4 OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.3238 0.2882 0.3144 0.3382 0.3726 0.3371 0.3237 0.3553 0.3465 0.3224 0.3033 0.2867 0.3261 0.3160 0.2914 0.3261 0.3232 0.0239

0.3062 0.2739 0.2952 0.3165 0.3513 0.3193 0.3114 0.3264 0.3144 0.3005 0.2824 0.3004 0.3071 0.3005 0.2839 0.3157 0.3072 0.0193

0.4741 0.4382 0.4351 0.4279 0.4359 0.4273 0.4361 0.4449 0.4437 0.4183 0.4140 0.4144 0.4330 0.4173 0.4048 0.4461

0.5285 0.5276 0.5770 0.5276 0.5222 0.5454 0.5436 0.5464 0.5634 0.5616 0.5095 0.5280 0.5553 0.5685 0.5261 0.5475 0.5424 0.0192

2737 4174 4128 4203 4283 4301 4310 4379 4375 4288 4253 4185 5270 5743 5596 5613

Food Consumed at Home Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

0.4 elasticity

Quarter

0.3 0.2 0.1 0 0

4

8 quarter

12

16

Two-Standard Deviation Confidence Interval [QR(mode)]: 0.2686 < e < 0.3498

equations are large, with values of 10 and more being standard and less than 5 unusual. The largest values are for housing (never less than 50) and smallest for tobacco (typically between 3 and 7).1

1 Of the 696 equations represented in Tables 11.1–11.29, only one (!) t-ratio is less than 2: 1.95 for tobacco in the OLS equation for 1996Q2.

11

Engel Curves for 29 Categories of CES Expenditure

175

Table 11.2 Total-expenditure elasticities, food consumed outside the home, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.8747 0.7893 0.8931 0.8626 0.8039 0.7794 0.8296 0.8195 0.8008 0.7960 0.8045 0.8035 0.8063 0.7415 0.7774 0.8033 0.8411 0.0472

0.9320 0.8643 0.9523 0.8971 0.8151 0.7881 0.8468 0.8543 0.8420 0.8369 0.8527 0.8272 0.8496 0.7783 0.8049 0.8192 0.8476 0.0354

0.3604 0.3552 0.3434 0.3336 0.3180 0.3251 0.3216 0.3508 0.3431 0.3241 0.3244 0.3297 0.3167 0.3073 0.3062 0.3282

0.5374 0.5514 0.5728 0.5849 0.5024 0.4369 0.5460 0.5496 0.5234 0.5283 0.5332 0.5555 0.5833 0.5251 0.5476 0.5642 0.5401 0.0354

2346 3594 3540 3657 3709 3743 3789 3808 3795 3730 3678 3653 4587 5004 4855 4847

Food Consumed Outside the Home Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

1 elasticity

0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.7532 < e < 0.9420

16

176

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.3 Total-expenditure elasticities, alcoholic beverages, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean

0.4462 0.4856 0.5109 0.4410 0.4963 0.4328 0.4587 0.5471 0.5080 0.5368 0.4994 0.4894 0.5107 0.4151 0.4393 0.5302 0.4842

0.3983 0.4912 0.4844 0.4480 0.4580 0.4094 0.4703 0.5538 0.4990 0.4855 0.4807 0.4797 0.5464 0.3802 0.4184 0.5236 0.4704

0.0816 0.0958 0.0931 0.0782 0.0924 0.0746 0.0976 0.1171 0.1020 0.1153 0.1066 0.1040 0.1112 0.0745 0.0788 0.0900

0.5591 0.5851 0.6080 0.6092 0.5420 0.5379 0.5445 0.6242 0.5598 0.6063 0.6529 0.5684 0.5802 0.5279 0.5741 0.5855 0.5746

1347 1984 2061 2033 2096 2100 2121 2066 2091 1992 2044 2042 2602 2797 2767 2733

Alcoholic Beverages Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4 0.6 elasticity

0.5 0.4 0.3 0.2 0.1 0

0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.3700 < e < 0.57

16

11

Engel Curves for 29 Categories of CES Expenditure

177

Table 11.4 Total-expenditure elasticities, housing, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.8763 0.7767 0.8190 0.8159 0.7976 0.7914 0.8580 0.8383 0.8969 0.7997 0.7848 0.7973 0.8368 0.8045 0.7869 0.8415 0.8204 0.0349

0.8748 0.8734 0.8928 0.8666 0.8373 0.8494 0.9173 0.8619 0.9321 0.8530 0.8516 0.8509 0.9120 0.8812 0.8362 0.8769 0.8730 0.0285

0.5936 0.5719 0.5818 0.5913 0.6049 0.5761 0.6109 0.5840 0.6235 0.5935 0.5681 0.6084 0.6118 0.5929 0.5673 0.6248

0.5928 0.5872 0.6021 0.5810 0.5336 0.5790 0.5527 0.5449 0.5890 0.6055 0.5903 0.5644 0.5979 0.6248 0.5669 0.5856 0.5811 0.0237

2748 4189 4137 4214 4293 4329 4314 4400 4392 4307 4266 4197 5284 5778 5617 5627

Housing Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

1 elasticity

0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.8160 < e < 0.9300

16

178

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.5 Total-expenditure elasticities, shelter, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.8616 0.9517 0.8899 0.8674 0.8607 0.8727 0.9041 0.8922 0.9961 0.9126 0.8926 0.9175 0.9265 0.9321 0.9223 0.9712 0.9120 0.0395

0.8611 0.9386 0.8853 0.8363 0.8373 0.8077 0.8474 0.8568 0.9266 0.9057 0.8484 0.8837 0.8899 0.9253 0.9253 0.9336 0.8810 0.0402

0.4378 0.4325 0.4372 0.4433 0.4306 0.4402 0.4621 0.4686 0.4643 0.4355 0.4240 0.4631 0.4305 0.4414 0.4390 0.4540

0.6001 0.6102 0.5970 0.6279 0.6143 0.6147 0.5938 0.5885 0.6281 0.5937 0.5979 0.6277 0.6268 0.6040 0.6138 0.6473 0.6116 0.0165

2714 4159 4098 4169 4253 4292 4284 4363 4349 4287 4234 4256 5245 5719 5550 5548

Shelter Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

elasticity

1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.8006 < e < 0.9614

16

11

Engel Curves for 29 Categories of CES Expenditure

179

Table 11.6 Total-expenditure elasticities, owned dwellings, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.9707 0.9893 0.8430 0.8142 0.9194 0.8638 0.9188 0.8320 0.9681 0.9178 0.8398 0.9574 0.9662 0.9598 0.9044 0.9855 0.9156 0.0597

0.9851 0.9920 0.8503 0.8208 0.8973 0.8508 0.9322 0.8175 0.9950 0.8934 0.8567 0.9616 0.9550 0.9562 0.9257 0.9722 0.9166 0.0618

0.4589 0.4524 0.4398 0.4538 0.4400 0.4357 0.4314 0.4387 0.4674 0.4205 0.4088 0.4605 0.4127 0.4379 0.4072 0.4311

0.6363 0.6053 0.5962 0.6114 0.6290 0.5973 0.6337 0.6052 0.6094 0.6153 0.5950 0.5804 0.6282 0.6352 0.6390 0.6111 0.6143 0.0176

1757 2726 2729 2735 2761 2832 2840 2875 2865 2800 2818 2777 3486 3890 3787 3751

Owned Dwellings Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

elasticity

1.2 1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.7930 < e < 1.0402

16

180

11

Engel Curves for 29 Categories of CES Expenditure

elasticity

Table 11.7 Total-expenditure elasticities, rented dwellings, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.5040 0.5123 0.4213 0.5423 0.5028 0.5475 0.6223 0.5838 0.6053 0.5314 0.5710 0.4814 0.5862 0.5309 0.5372 0.5262 0.5379 0.0499

0.5773 0.5897 0.5452 0.5512 0.6078 0.5688 0.6207 0.6230 0.6137 0.5998 0.5928 0.5444 0.6177 0.5704 0.5642 0.5542 0.5838 0.0279

0.2952 0.2808 0.2267 0.2931 0.2898 0.3547 0.3718 0.3584 0.3662 0.3175 0.2878 0.2820 0.3152 0.3016 0.3012 0.2811

0.5671 0.6083 0.5939 0.5985 0.5607 0.5284 0.5885 0.5899 0.5614 0.5820 0.5896 0.5902 0.6190 0.5669 0.5678 0.5687 0.5801 0.0219

977 1468 1411 1475 1537 1499 1484 1524 1508 1523 1459 1322 1819 1903 1835 1887

Rented Dwellings Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.5280 < e < 0.6396

16

11

Engel Curves for 29 Categories of CES Expenditure

181

Table 11.8 Total-expenditure elasticities, other lodging, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.7824 0.8820 0.9109 0.8004 0.8546 0.8384 0.8380 0.9141 0.8835 0.9056 0.7194 0.6942 0.7369 0.7734 0.7995 0.9274 0.8288 0.0738

0.6865 0.9651 1.0082 0.7306 0.8508 0.8304 0.9211 1.0115 0.8605 0.9496 0.7453 0.8162 0.7029 0.8200 0.8411 0.9213 0.8540 0.1035

0.1338 0.1902 0.1953 0.1865 0.1858 0.1570 0.1696 0.1930 0.1884 0.1716 0.1387 0.1238 0.1536 0.1587 0.1940 0.2118

0.4392 0.5511 0.5358 0.4726 0.4481 0.4360 0.6156 0.5328 0.4731 0.5099 0.5606 0.5973 0.4352 0.5375 0.6378 0.4670 0.5156 0.0660

509 922 1219 1129 849 931 1216 1172 854 931 1191 1125 1052 1234 1587 1505

Other Lodging Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

elasticity

1.2 1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.6470 < e < 1.0610

16

182

11

Engel Curves for 29 Categories of CES Expenditure

elasticity

Table 11.9 Total-expenditure elasticities, utilities, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.3798 0.3395 0.3590 0.4034 0.3904 0.4019 0.3991 0.4059 0.4096 0.3735 0.3504 0.3755 0.3995 0.3598 0.3357 0.3814 0.3790 0.0242

0.3091 0.2968 0.3166 0.3312 0.3334 0.3269 0.3347 0.3381 0.3421 0.3120 0.2829 0.3066 0.3392 0.3208 0.3020 0.3371 0.3206 0.0176

0.4079 0.3733 0.3670 0.3787 0.3964 0.3634 0.3837 0.3857 0.4034 0.3836 0.3785 0.3805 0.4138 0.3921 0.3612 0.3992

0.5408 0.5477 0.5243 0.4811 0.4959 0.5117 0.5267 0.5068 0.5652 0.5039 0.5430 0.5211 0.5634 0.5320 0.5149 0.5442 0.5266 0.0233

2720 4163 4094 4179 4237 4282 4258 4362 4347 4277 4221 4151 5228 5714 5518 5543

Utilities Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.2854 < e < 0.35

16

11

Engel Curves for 29 Categories of CES Expenditure

183

Table 11.10 Total-expenditure elasticities, natural gas, CES surveys1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.2599 0.2523 0.1882 0.1786 0.2101 0.2408 0.1478 0.1063 0.1813 0.2720 0.1380 0.1265 0.2355 0.2137 0.1723 0.1959 0.1950 0.0495

0.2342 0.2680 0.1789 0.1832 0.2262 0.2105 0.1388 0.1127 0.1787 0.2277 0.1531 0.1062 0.2305 0.1962 0.1422 0.1903 0.1861 0.0463

0.2675 0.3003 0.2078 0.1998 0.2490 0.2587 0.2613 0.2010 0.2192 0.2124 0.2234 0.2078 0.2380 0.2679 0.1982 0.2095

0.5525 0.5645 0.5597 0.4685 0.5160 0.5920 0.5071 0.5323 0.5432 0.5463 0.5258 0.4979 0.5239 0.5379 0.5020 0.4699 0.5275 0.0335

1414 2197 2196 2150 2159 2226 2212 2272 2260 2236 2225 2178 2764 3034 2963 2999

Natural Gas Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

0.3

elasticity

0.25 0.2 0.15 0.1 0.05 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.0935 < e < 0.2787

16

184

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.11 Total-expenditure elasticities, electricity, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.2332 0.2184 0.2298 0.2654 0.2323 0.2308 0.2501 0.2520 0.2357 0.2193 0.1949 0.2371 0.2367 0.2472 0.2109 0.2690 0.2352 0.0243

0.2375 0.1767 0.2124 0.2395 0.2278 0.2334 0.2370 0.2577 0.2235 0.1983 0.1894 0.2448 0.2378 0.2364 0.2041 0.2649 0.2263 0.0242

0.3160 0.2766 0.3393 0.3425 0.3074 0.2868 0.3294 0.3336 0.3021 0.2849 0.3551 0.3564 0.3092 0.2916 0.3183 0.3484

0.4812 0.4691 0.4677 0.4737 0.4350 0.4820 0.5300 0.4964 0.5169 0.4657 0.5124 0.5392 0.5280 0.5212 0.4783 0.5038 0.4935 0.0291

2552 3936 3856 3907 3963 4005 4000 4064 4065 4000 3957 3901 4915 5378 5211 5229

Electricity Total Expenditure Elasticities (QR) 1996Q1 – 1999Q2

0.3 elasticity

0.25 0.2 0.15 0.1 0.05 0

0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.1779 < e < 0.2747.

16

11

Engel Curves for 29 Categories of CES Expenditure

185

elasticity

Table 11.12 Total-expenditure elasticities, telephone services, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.4455 0.4333 0.4628 0.4578 0.4757 0.5270 0.4909 0.4707 0.4925 0.4357 0.4437 0.4832 0.4958 0.4548 0.4061 0.4170 0.4147 0.0380

0.3703 0.3712 0.4195 0.3972 0.4143 0.5075 0.4267 0.4176 0.4489 0.3984 0.3920 0.4047 0.4767 0.4248 0.3563 0.3930 0.4137 0.0390

0.1834 0.1834 0.1872 0.2031 0.1875 0.1833 0.2221 0.2237 0.2265 0.2089 0.2083 0.2180 0.2332 0.2145 0.2001 0.2164

0.4947 0.4395 0.4677 0.5190 0.4506 0.5033 0.4740 0.5163 0.4715 0.5848 0.4432 0.4643 0.5042 0.4974 0.4840 0.4959 0.4867 0.0370

2665 4066 4016 4089 4155 4209 4179 4265 4271 4198 4146 4076 5150 5599 5435 5450

Telep hone Services Total Expenditure Elasticities (QR) 1996Q1 – 1999Q2

0.6 0.5 0.4 0.3 0.2 0.1 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.3357 < e < 0.4917.

16

186

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.13 Total-expenditure elasticities, water and other public services, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.1750 0.1537 0.1701 0.1839 0.1585 0.1612 0.1506 0.2133 0.1761 0.1497 0.1535 0.1687 0.2303 0.1362 0.1574 0.2046 0.1714 0.0255

0.1765 0.1374 0.1399 0.1886 0.1534 0.1578 0.1402 0.2208 0.1928 0.1338 0.1622 0.1569 0.1920 0.1416 0.1586 0.1938 0.1654 0.0258

0.1160 0.1117 0.1574 0.1852 0.1398 0.1512 0.1584 0.1524 0.1373 0.1321 0.1419 0.1697 0.1603 0.1052 0.1151 0.1675

0.5543 0.5784 0.5592 0.5507 0.5502 0.5773 0.5657 0.5630 0.5895 0.5414 0.6086 0.5447 0.5360 0.5951 0.6063 0.6006 0.5701 0.0241

1651 2557 2564 2640 2652 2692 2728 2738 2694 2636 2681 2576 3287 3626 3550 3534

Water & Other Public Services Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

0.25 elasticity

0.2 0.15 0.1 0.05 0

0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.1138 < e < 0.2170

16

11

Engel Curves for 29 Categories of CES Expenditure

187

Table 11.14 Total-expenditure elasticities, housing operations, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.9066 0.9436 0.8303 0.9620 1.0161 0.9009 0.9361 0.7338 0.9380 0.8056 0.7338 0.7612 0.8761 0.7936 0.7457 0.9079 0.8620 0.0905

1.0488 1.0474 0.8652 1.0174 1.0854 0.9713 0.9352 0.7767 1.0567 0.9061 0.7733 0.7526 0.9040 0.8328 0.8060 0.9012 0.9175 0.1112

0.2910 0.2906 0.2619 0.2470 0.2792 0.2937 0.2250 0.2228 0.2814 0.2142 0.1807 0.2010 0.2446 0.2423 0.1976 0.2394

0.5505 0.5047 0.4985 0.5123 0.5374 0.5357 0.5629 0.4659 0.4950 0.4965 0.5029 0.4937 0.5041 0.4523 0.4740 0.4709 0.5035 0.0308

1150 1793 1881 1896 1811 1942 2034 2048 1980 2067 2151 2103 2511 3006 3023 3014

Housing Operations Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

1.2 elasticity

1 0.8 0.6 0.4 0.2 0

0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.6951 < e < 1.1399

16

188

11

Engel Curves for 29 Categories of CES Expenditure

elasticity

Table 11.15 Total-expenditure elasticities, house furnishings and equipment, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1997Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

1.1656 0.9459 1.0019 0.8529 1.0791 1.0662 1.0330 1.0710 1.0798 0.9646 0.9427 1.0203 1.0355 1.0430 0..8792 1.2716 1.0058 0.0842

1.2646 1.0366 1.0488 0.9288 1.1361 1.0879 1.0643 1.1086 1.1074 1.0023 0.9429 1.0886 1.0978 1.0994 0.9695 1.1571 1.0590 0.0852

0.2415 0.1608 0.1604 0.1557 0.2359 0.1623 0.1909 0.2056 0.2390 0.1694 0.1813 0.1723 0.2228 0.1914 0.1629 0.5349

0.5227 0.5143 0.4699 0.4857 0.5493 0.4123 0.5365 0.5142 0.5249 0.5844 0.3955 0.5185 0.5451 0.5185 0.5086 0.5424 0.4958 0.0496

1780 2683 2680 2648 2934 2788 2840 2761 2966 2737 2768 2630 3571 3651 3612 5366

House Furnishings & Equipment Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.8886 < e < 1.2294

16

11

Engel Curves for 29 Categories of CES Expenditure

189

elasticity

Table 11.16 Total-expenditure elasticities, apparel and services, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

1.0409 0.9255 0.8700 0.9178 1.0417 0.8985 0.9628 0.8777 1.0690 0.8953 0.8472 0.9046 0.8653 0.8624 0.8271 0.8356 0.9151 0.0842

1.0886 0.9082 0.8906 0.8980 1.0148 0.8955 0.9181 0.8947 1.0433 0.8560 0.8042 0.8794 0.8836 0.8638 0.8160 0.8145 0.9043 0.0805

0.3679 0.2913 0.2893 0.3255 0.3223 0.2612 0.3215 0.3057 0.3946 0.2705 0.2984 0.3102 0.3181 0.2963 0.2778 0.3023

0.6081 0.5598 0.6678 0.5968 0.5671 0.5778 0.6098 0.5533 0.6188 0.5889 0.6144 0.5985 0.5806 0.4948 0.6293 0.5316 0.5873 0.0410

2494 3666 3657 3763 3896 3792 3215 3863 3961 3728 3748 3675 4748 4963 4808 4876

Apparel & Services Total Expendtures Elasticities (QR) 1996Q1 – 1999Q4

1.2 1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.7433 < e < 1.0653.

16

190

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.17 Total-expenditure elasticities, women and girls’ apparel, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.8336 0.7962 0.7955 0.6916 0.7931 0.7700 0.7651 0.8381 0.8156 0.7209 0.6910 0.7907 0.7567 0.7069 0.6926 0.7031 0.7600 0.0596

0.8295 0.7797 0.7729 0.6745 0.7697 0.7469 0.7408 0.8863 0.8453 0.7152 0.7178 0.7805 0.7336 0.7143 0.6813 0.7001 0.7555 0.0594

0.2029 0.1609 0.1621 0.1577 0.1978 0.1425 0.1729 0.1907 0.1926 0.1555 0.1616 0.1855 0.1884 0.1940 0.1717 0.1642

0.5973 0.5245 0.5312 0.6106 0.5332 0.5077 0.6687 0.5735 0.6002 0.6147 0.6267 0.6048 0.6048 0.6093 0.5974 0.6522 0.5907 0.0457

1836 2513 2588 2658 2947 2639 2628 2653 2934 2529 2521 2470 3432 3349 3301 3300

Women and Girls' Apparel Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4 1 elasticity

0.8 0.6 0.4 0.2 0

0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.6367 < e < 0.8743.

16

11

Engel Curves for 29 Categories of CES Expenditure

191

Table 11.18 Total-expenditure elasticities, men and boys’ apparel, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.7844 0.6784 0.6796 0.6132 0.7882 0.6415 0.6297 0.6169 0.7182 0.6166 0.6240 0.6589 0.6604 0.6933 0.5424 0.6190 0.6602 0.0638

0.8461 0.6995 0.7042 0.5795 0.7991 0.6647 0.6523 0.6401 0.7508 0.6043 0.6591 0.6675 0.6809 0.6184 0.5756 0.6132 0.6722 0.0753

0.1945 0.1244 0.1344 0.1351 0.1814 0.1249 0.1370 0.1368 0.1872 0.1067 0.1526 0.1566 0.1599 0.1395 0.1322 0.1366

0.4769 0.5261 0.5193 0.4611 0.6156 0.5527 0.5484 0.5613 0.5855 0.4928 0.6185 0.6001 0.5716 0.5311 0.5566 0.5306 0.5468 0.0462

1587 1905 2034 2161 2628 2003 2074 2193 2508 1932 2028 2002 3031 2528 2596 2630

Men & Boys' Apparel Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

1

elasticity

0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.5216 < e < 0.8228.

16

192

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.19 Total-expenditure elasticities, transportation, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

1.2211 1.3278 1.3049 1.3411 1.2489 1.3491 1.3318 1.2848 1.2035 1.3057 1.3226 1.3643 1.3075 1.2885 1.3596 1.2716 1.3021 0.0472

1.1109 1.2200 1.2207 1.2315 1.1778 1.2341 1.2290 1.1885 1.1079 1.1979 1.2194 1.2530 1.1869 1.1388 1.2925 1.1571 1.1979 0.0505

0.5318 0.5405 0.5716 0.5556 0.5133 0.5462 0.5535 0.5520 0.5209 0.5575 0.5587 0.5520 0.5124 0.5253 0.5427 0.5349

0.5246 0.5551 0.5705 0.5255 0.5536 0.5316 0.5458 0.5432 0.5609 0.5338 0.5285 0.5328 0.5461 0.5090 0.5501 0.5424 0.5408 0.0156

2612 3982 3968 4035 4098 4138 4142 4218 4198 4110 4052 3999 5032 5517 5371 5366

Transportation Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

1.4 1.2 elasticity

1 0.8 0.6 0.4 0.2 0 0

5

10 quarter

15

Two-Standard Deviation Confidence Interval [QR(Mode)]: 1.0969 < e < 1.2989.

20

11

Engel Curves for 29 Categories of CES Expenditure

193

Table 11.20 Total-expenditure elasticities, gasoline and motor oil, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.3831 0.3763 0.4583 0.3967 0.3925 0.3590 0.3766 0.4773 0.4462 0.3846 0.3813 0.3582 0.3652 0.3511 0.3788 0.3731 0.3911 0.0370

0.3653 0.3660 0.4363 0.3977 0.3744 0.3785 0.3895 0.4699 0.4552 0.3843 0.3595 0.3825 0.3720 0.3453 0.3756 0.3642 0.3885 0.0352

0.3279 0.3310 0.3547 0.3453 0.3469 0.3241 0.3377 0.3457 0.3457 0.3113 0.3192 0.3111 0.3129 0.3334 0.3436 0.3333

0.4833 0.5673 0.5324 0.5180 0.4736 0.4664 0.4950 0.5137 0.5455 0.5380 0.5115 0.5147 0.5609 0.5476 0.4626 0.5206 0.5156 0.0326

2476 3790 3762 3831 3879 3935 3955 4024 3982 3885 3870 3790 4789 5218 5101 5093

Gasoline and Motor Oil Total Expenditure Elasticities (QR) 1996Q1 & 1999Q4

0.5 elasticity

0.4 0.3 0.2 0.1 0

0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.3181 < e < 0.4589.

16

194

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.21 Total-expenditure elasticities, health care, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.5037 0.4996 0.5050 0.5223 0.5831 0.5503 0.5478 0.5292 0.5295 0.5374 0.5261 0.5543 0.5304 0.5564 0.4870 0.5192 0.5306 0.0305

0.5196 0.4710 0.5148 0.5159 0.5658 0.5691 0.5351 0.5196 0.5683 0.5605 0.5068 0.5559 0.5441 0.5538 0.5144 0.5134 0.5329 0.0280

0.2083 0.2110 0.2224 0.2155 0.2377 0.2114 0.2216 0.2171 0.2252 0.2234 0.2144 0.2309 0.2120 0.2209 0.2176 0.2182

0.5578 0.5225 0.6079 0.5993 0.5510 0.6062 0.5413 0.5835 0.5326 0.5189 0.5714 0.5337 0.5426 0.5452 0.5905 0.5785 0.5614 0.0299

2248 3445 3457 3476 3515 3618 3640 3703 3689 3608 3566 3475 4429 4855 4689 4789

Health Care Total Expenditure Elasticities (QR) 1996Q1 – 1999Q4

0.6

elasticity

0.5 0.4 0.3 0.2 0.1 0

0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.4769 < e < 0.5889.

16

11

Engel Curves for 29 Categories of CES Expenditure

195

Table 11.22 Total-expenditure elasticities, entertainment, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

1.0015 0.9829 1.0479 1.0257 0.9789 0.9785 1.0064 0.9631 0.9547 0.9603 1.0091 0.9602 0.9529 0.9321 1.0017 0.9495 0.9783 0.0319

0.9985 0.9382 1.0624 1.0119 0.9972 0.9818 0.9802 0.9603 1.0112 0.9278 1.0145 0.9236 0.9778 0.9237 1.0128 0.9237 0.9732 0.0416

0.3702 0.3164 0.3657 0.3454 0.3788 0.3110 0.3429 0.3413 0.3977 0.3385 0.3589 0.3530 0.3442 0.3315 0.3689 0.3492

0.5865 0.5416 0.5488 0.5223 0.5673 0.5535 0.4994 0.5433 0.5919 0.4869 0.5195 0.5455 0.5603 0.4994 0.5681 0.5835 0.5404 0.0338

2500 3764 3726 3813 3963 3930 3918 4004 4069 3910 3899 3844 4894 5295 5147 5162

Entertainment Total Expenditure Elasticities (QR) 1996Q1–1999Q4

1.2 elasticity

1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.8947 < e < 1.0611.

16

196

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.23 Total-expenditure elasticities, personal care products and services, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std.Dev.

0.4116 0.4690 0.4765 0.4950 0.5187 0.4757 0.4377 0.5055 0.5167 0.5247 0.4928 0.4264 0.4481 0.4161 0.3907 0.4216 0.4642 0.0450

0.4095 0.4767 0.4724 0.4967 0.5352 0.5079 0.4188 0.5091 0.5153 0.5480 0.4815 0.4393 0.4727 0.4298 0.4957 0.4333 0.4776 0.0418

0.2402 0.2109 0.2352 0.2393 0.2498 0.2387 0.2405 0.2492 0.2582 0.2402 0.2486 0.2283 0.2306 0.2098 0.2064 0.2031

0.5143 0.5363 0.4789 0.5503 0.5783 0.5285 0.5542 0.5422 0.5397 0.6075 0.4692 0.5093 0.5682 0.4984 0.5696 0.5180 0.5352 0.0368

2164 3234 3231 3199 3309 3302 3347 3422 3419 3321 3330 3253 4072 4533 4405 4390

Personal Care Products & Services Total Expenditure Elasticities (QR) 1996Q1–1999Q4

0.6 elasticities

0.5 0.4 0.3 0.2 0.1 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.3940 < e < 0.5612.

16

11

Engel Curves for 29 Categories of CES Expenditure

197

elasticities

Table 11.24 Total-expenditure elasticities, reading materials, CES surveys 1996q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.5001 0.5507 0.5749 0.6048 0.6024 0.5787 0.4970 0.5962 0.5917 0.5929 0.5617 0.5457 0.4296 0.5261 0.4692 0.5625 0.5503 0.0488

0.4997 0.5408 0.5615 0.6264 0.6293 0.5788 0.4872 0.6080 0.6164 0.6219 0.5173 0.5426 0.4726 0.5146 0.4568 0.5467 0.5513 0.0575

0.1814 0.1644 0.1593 0.1711 0.1623 0.1620 0.1662 0.1831 0.1892 0.1594 0.1413 0.1509 0.1714 0.1460 0.1818 0.1574

0.5278 0.5309 0.4987 0.5165 0.5577 0.4983 0.5502 0.5272 0.5891 0.5772 0.5134 0.5423 0.5803 0.5341 0.5123 0.5582 0.5384 0.0283

1933 2951 2880 2933 3044 2949 2942 2996 3080 2897 2806 2816 3569 3762 3615 3671

Reading Materials Total Expenditure Elasticities (QR) 1996Q1–1999Q4

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.4363 < e < 0.6663.

16

198

11

Engel Curves for 29 Categories of CES Expenditure

elasticities

Table 11.25 Total-expenditure elasticities, education, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.9092 1.1619 0.7445 0.9720 0.9476 0.9024 0.9247 1.0398 1.1318 0.8469 1.0344 1.0553 0.9777 0.9900 0.6307 0.8448 0.9446 0.1359

0.9614 1.1969 0.6597 1.0084 1.1300 0.9878 0.9807 1.0154 1.0997 0.9206 1.0163 0.7862 1.0892 1.0706 0.6319 0.9662 0.9701 0.1573

0.1711 0.2049 0.1555 0.2060 0.1542 0.1631 0.1852 0.2277 0.2070 0.1894 0.2048 0.2177 0.2076 0.1860 0.1534 0.1860

0.6246 0.5507 0.5003 0.4533 0.6781 0.6242 0.5800 0.4902 0.5106 0.5914 0.4511 0.5423 0.5430 0.6371 0.4960 0.4714 0.5381 0.0782

475 661 677 1042 783 700 734 1031 747 678 714 1023 924 956 964 1339

Education Total Expenditure Elasticities (QR) 1996Q1–1999Q4

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.6555 < e < 1.2847.

16

11

Engel Curves for 29 Categories of CES Expenditure

199

elasticity

Table 11.26 Total-expenditure elasticities, tobacco products and smoking supplies, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.1202 0.0744 0.1416 0.1013 0.2247 0.2084 0.1685 0.1762 0.1856 0.1249 0.1213 0.1196 0.1628 0.1490 0.1138 0.1601 0.1470 0.0401

0.1351 0.0968 0.1970 0.1601 0.2407 0.1802 0.1662 0.1931 0.1973 0.0941 0.1262 0.1367 0.1567 0.1399 0.0742 0.2160 0.1568 0.0464

0.0339 0.0364 0.0136 0.0341 0.0372 0.0444 0.0348 0.0492 0.0377 0.0724 0.0516 0.0450 0.0318 0.0216 0.0523 0.0371

0.5363 0.5628 0.5536 0.5789 0.5767 0.6007 0.5876 0.5515 0.5909 0.5633 0.5602 0.5636 0.5994 0.6123 0.5916 0.6362 0.5791 0.0259

895 1325 1326 1310 1311 1348 1323 1367 1337 1299 1286 1216 1527 1665 1616 1572

Tobacco Products & Smoking Supplies Total Expenditure Elasticities (QR) 1996Q1–999Q4

0.3 0.25 0.2 0.15 0.1 0.05 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.0640 < e < 0.2495.

16

200

11

elasticity

Table 11.27 Total-expenditure 1996Q1–1999Q4

Engel Curves for 29 Categories of CES Expenditure

elasticities,

miscellaneous

expenditures,

CES

Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.7345 0.7299 0.7367 0.6183 0.6649 0.7036 0.5621 0.6620 0.8144 0.8349 0.7191 0.6927 0.5462 0.8259 0.6660 0.6563 0.6980 0.0841

0.4953 0.8432 0.4912 0.4309 0.5090 0.8452 0.3707 0.4419 0.4896 0.9477 0.4856 0.4215 0.3529 1.0142 0.4736 0.5096 0.5701 0.2125

0.0850 0.0982 0.0955 0.0763 0.0908 0.0971 0.0636 0.0843 0.0901 0.1269 0.0818 0.0728 0.0615 0.1185 0.0907 0.0872

0.3395 0.5225 0.3441 0.3571 0.3243 0.5892 0.3488 0.3656 0.3253 0.5891 0.3483 0.3389 0.3130 0.5677 0.3342 0.3912 0.3999 0.1022

1423 2339 2100 1980 2180 2420 2112 2054 2187 2343 2007 1862 2567 3109 2495 2433

surveys

Miscellaneous Expenditures Total Expenditure Elasticities (QR) 1996Q1–1999Q4

1.2 1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.1451 < e < 0.9951.

16

11

Engel Curves for 29 Categories of CES Expenditure

201

elasticity

Table 11.28 Total-expenditure elasticities, cash contributions, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

1.0741 1.2121 1.0268 0.9935 1.0782 1.0282 1.0447 1.1555 1.1399 1.0635 0.9706 1.0869 0.8696 1.1734 0.9930 1.0450 1.0597 0.0850

1.0000 1.1268 1.1169 1.1810 1.0749 1.0176 1.1090 1.1760 1.2357 1.1794 0.9969 1.0679 0.8873 1.1717 1.0524 1.0364 1.0894 0.0902

0.2102 0.1921 0.2396 0.1361 0.2067 0.2163 0.2238 0.2098 0.1776 0.2260 0.2634 0.2204 0.1572 0.2284 0.1917 0.1948

0.5984 0.5300 0.4953 0.6011 0.5139 0.5076 0.4923 0.4212 0.6053 0.5103 0.4902 0.5015 0.4968 0.5113 0.6049 0.5300 0.5256 0.0517

424 691 650 618 641 641 619 640 653 639 645 611 794 852 781 803

Cash Contributions Total Expenditure Elasticities (QR) 1996Q1–1999Q4

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.9090 < e < 1.2698.

16

202

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.29 Total-expenditure elasticities, personal insurance and pensions, CES surveys 1996Q1–1999Q4 Quarter

OLS

QR(Mode)

R2

Mode

#Obs.

1996Q1 1996Q2 1996Q3 1996Q4 1997Q1 1997Q2 1997Q3 1997Q4 1998Q1 1998Q2 1998Q3 1998Q4 1999Q1 1999Q2 1999Q3 1999Q4 Mean Std. Dev.

0.8588 0.8755 0.8909 0.8672 0.9807 0.9703 0.8843 0.8991 1.0115 0.9550 0.8753 0.9005 0.9611 0.9438 0.9167 0.9394 0.9206 0.0461

0.9031 0.9488 0.9296 0.9135 1.0279 1.0220 0.9389 0.9414 1.0644 1.0163 0.9262 0.9708 1.0449 1.0230 0.9691 0.9691 0.9749 0.0508

0.5515 0.5430 0.5530 0.5068 0.5475 0.5044 0.5179 0.5075 0.5586 0.5349 0.5519 0.5501 0.5192 0.5042 0.5086 0.5304

0.5182 0.5637 0.5491 0.5308 0.5797 0.5635 0.5654 0.5313 0.5593 0.5147 0.5259 0.5457 0.5403 0.5450 0.5413 0.5143 0.5430 0.0197

2350 3555 3510 3591 3649 3676 3669 3747 3745 3650 3645 3553 4525 4936 4790 4838

Personal Insurance and Pensions Total Expenditure Elasticities (QR) 1996Q1–1999Q4

1.2 elasticities

1 0.8 0.6 0.4 0.2 0 0

4

8 quarter

12

16

Two-Standard Deviation Confidence Interval [QR(Mode)]: 0.8733 < e < 1.0765.

11.1 An Overview of the Results The key results, statistically, in these tables are as follows: (1) As noted in Footnote 1, with only a single exception, all elasticities are highly significant statistically. (2) Estimated total-expenditure elasticities range from 1.20 for transportation to 0.16 for tobacco (both QR mean values).

11.1

An Overview of the Results

203

(3) Engel’s law for food is confirmed, not only for food consumed at home, but for off-premise food consumption as well.2 (4) In the graphs of the QR elasticities at the bottom of the tables, not even in a single instance do we see evidence of a trend in the estimated total-expenditure elasticities over the 16 quarters of estimation. (5) In keeping with the results in Chapters 6 and 10, we again see a marked stability in the estimated quarterly Engel curves. That this is the case is evident, not only by eye from the graphs, but also by the relatively small sizes of the standard deviations of the 16 quarterly estimates.3 Stability is evident as well in the relatively small variation in the R2 over the 16 quarters for each category. (6) Distributions of residuals continue to be asymmetrical. Most mean modal values are to the right of the median, as only four are to the left (electricity, telephone, housing operations and miscellaneous expenditures).4 The categories with the greatest “positive” asymmetry (i.e., modal value to the right of the median) are shelter and apparel (with mean values around 0.60), while the category with the greatest “negative” asymmetry is miscellaneous expenditures (with a mean value of 0.40). On the other hand, a number of categories display mean modal values that, statistically, are quite close to the median, including other lodging, electricity, telephone, housing operation, gasoline and motor oil, and cash contributions. (7) For the most part, the OLS and QR elasticity estimates are within 0.05 units of one another. Transportation and miscellaneous expenditures show the largest positive differences (both of the order of 0.10), while rented housing, housing operation, house furnishings and equipment, and cash contributions display the largest negative differences (all of the order of −0.05). Since the category with the largest “negative” asymmetry, miscellaneous expenditures, also has one of the largest positive differences in the OLS and QR elasticities, an obvious consideration is whether there is a relation between the differences and the degree of asymmetry. However, the scatter diagrams in Fig. 11.1a suggest that this is not the case. The first panel is in arithmetic units, while the second is in logarithms. Miscellaneous expenditures is clearly an outlier in each instance.

2 Obviously

expect Engel’s law to be confirmed for food consumed at home. Food consumed outside the home, on the other hand, is clearly another matter. We would expect the law to apply for employment-related outside-of-home food expenditures, but not necessarily for food expenditures associated with recreation and entertainment. 3 Interestingly, with few exceptions (specifically, for electricity, telephone, health care, entertainment, and miscellaneous expenditures), the variability of the QR estimates is less than for OLS. 4 Reminder: The modal value in this context refers to the mode of the distribution of residuals from the QR(median) regression, and represents the quantile at which the QR(mode) regression is estimated.

204

11

A

Arithmetric Units

0.15 OLS - QR(mode)

Engel Curves for 29 Categories of CES Expenditure

0.1 0.05 0 -0.05

0

0.1

0.2

0.3

-0.1

0.4

0.5

0.6

mode

B

Logarithms lnOLS - lnQR(mode)

0.7

0.25 0.2 0.15 0.1 0.05

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

0 -0.1 -0.05 0 -0.1

lnmode

Fig. 11.1 Relationship between mean OLS and QR(mode) elasticities and mean modal values

11.2 Size of Estimated Total-Expenditure Elasticities Let us now turn our attention to Table 11.30, which presents a summary tabulation of the 16-quarter means and standard deviations of the OLS and QR elasticities from Tables 11.1–11.29. As already noted, Engel’s law is confirmed, not only for food consumed at home, but also for food consumed outside of the home. Housing in the aggregate (which includes all housing-associated expenditures) is seen to have a mean total-expenditure elasticity of 0.87.5 For shelter as a whole, the elasticity is 0.88. The elasticity for owned dwellings (0.92) is larger than for rented dwellings (0.58), which in keeping with a generally held attitude that homeownership is a more desirable state than renting. Utilities, all with total-expenditure elasticities less than 0.5, are firmly in the necessity category, while housing operations and house furnishings and equipment border on luxury. Apparel, with elasticity as high as 0.9 may seem a bit of a surprise, though not that the value for women and girls is larger than for men and boys.6 5 Unless

otherwise noted, text references are to QR elasticities. the elasticity for aggregate apparel expenditure is larger than either of its constituents may seem an error. However, it must be kept in mind that the double-logarithmic equations of this chapter do not constitute an additive system; hence elasticities for aggregates are not budget-weighted averages of their components. Also, because some households have no expenditures for women

6 That

11.2

Size of Estimated Total-Expenditure Elasticities

205

Table 11.30 Summary of total-expenditure elasticities, CES surveys, 29 categories 1996Q–1999Q4 OLSQR

QR(mode)

Expenditure Category

Mean

Std. Dev.

Mean

Std. Dev.

Food consumed at home Food consumed outside of home Alcoholic beverages Housing Shelter Owned dwellings Rented dwellings Other lodging Utilities Natural gas Electricity Telephone services Water and other public services Housing operations House furnishings and equip. Apparel and Services Men and boys’ Women and girls’ Transportation Gasoline and motor oil Health care Entertainment Personal care and products Reading materials Education Tobacco products Miscellaneous expenditures Cash contributions Personal insurance and pensions

0.3232 0.8116 0.4842 0.8201 0.9120 0.9156 0.5379 0.8288 0.3790 0.1950 0.2352 0.4620 0.1714 0.8620 1.0058 0.9151 0.6603 0.7600 1.3021 0.3911 0.5306 0.9783 0.4642 0.5503 0.9446 0.1470 0.6980 1.0597 0.9206

0.0239 0.0382 0.0405 0.0349 0.0395 0.0597 0.0499 0.0738 0.0242 0.0495 0.0192 0.0318 0.0255 0.0905 0.0842 0.0758 0.0638 0.0522 0.0472 0.0370 0.0239 0.0319 0.0431 0.0488 0.1359 0.0401 0.0841 0.0850 0.0461

0.3072 0.8476 0.4704 0.8730 0.8810 0.9166 0.5838 0.8540 0.3206 0.1861 0.2263 0.4137 0.1654 0.9175 1.0590 0.9043 0.6722 0.7555 1.1979 0.3885 0.5329 0.9732 0.4776 0.5513 0.9701 0.1568 0.5701 1.0894 0.9747

0.0191 0.0354 0.0388 0.0237 0.0165 0.0176 0.0219 0.0660 0.0233 0.0335 0.0291 0.0370 0.0241 0.0308 0.0496 0.0410 0.0462 0.0458 0.0156 0.0326 0.0299 0.0338 0.0368 0.0283 0.0782 0.0258 0.1022 0.0518 0.0197

For the authors, perhaps the biggest surprise in this parade of total-expenditure elasticities is the fact that the estimated elasticity for transportation expenditures (1.20) is the largest of all the categories.7 A priori, our thoughts were that this elasticity would be generally of the order of 1, but smaller than the elasticities for expenditure categories associated with higher-order wants such as entertainment. However, what this view probably overlooks is that the motivation fueling the

and girls apparel, and similarly for men and boys, the data sets for the components are not exactly the same as for the aggregate. 7 Included in this category are expenditures for new and used vehicles, gasoline and oil, vehicle finance charges, maintenance and repairs, insurance, rental, leases, licenses and related charges, and public transportation.

206

11

Engel Curves for 29 Categories of CES Expenditure

marked switch to SUVs and light trucks in the 1990s almost certainly had a strong recreational component. Another factor is the fact that the public-transportation component of the category includes expenditures for air travel, which also will have a strong recreation component. Continuing down the list, we find that the elasticities for health and personal care, reading materials, and miscellaneous expenditures are all generally of the order of 0.5, while the ones for entertainment, education, cash contributions, and personal insurance and pensions are generally of the order of 1. All of these values make sense.8 What is perhaps most interesting of the latter group of categories is that the total-expenditure elasticity for cash contributions (1.09) is the largest, indeed, topped only by the value for transportation. However, given the many and complex motivations that drive eleemosynary activities, that expenditures associated with them should have an elasticity greater than 1 is hardly surprising.

11.3 Interpretation of Total-Expenditure Elasticities in Terms of Maslovian Hierarchy of Needs We now turn to a different way of interpreting the total-expenditure elasticities, namely, in terms of the Maslovian hierarchy of motivations that was discussed in Chapter 2. This hierarchy, to recall, consists of five basic needs, from lowest to highest, as follows: physiological, security, community (love), esteem, and self-actualization. Since these needs represent a hierarchy of wants in the order presented, the total-expenditure elasticities associated with the expenditures that satisfy the wants should form a hierarchy as well. Elasticities should be smallest for expenditures associated with basic physiological needs and highest for expenditures associated with self-actualization. In ideal circumstances, expenditure categories would be defined a one-to-one correspondence with wants, so that the want hierarchy would order total-expenditure elasticities as well. But this of course is not the case. The best we can do at present is to associate each of the expenditure categories of this chapter with the five Maslovian needs, from which we can hazard, at least tentatively, a theoretical ordering of elasticities. This theoretical ordering can then be compared with the empirical results that have been obtained. Since expenditure categories are generally broadly defined in terms of goods, and since most goods (even if narrowly defined) support multiple wants, each expenditure category will necessarily be multiply connected. The task facing us, then, is to devise, for each expenditure category, a qualitative set of weights over the five wants that can be used to assign the total-expenditure elasticity for the category to an appropriate ordinal location. To this end, we shall work with three ordinal elasticity sets as follows:

8 At

sufficiently high income, even cosmetic surgery probably becomes, for many, a necessity!

11.3

Interpretation of Total-Expenditure Elasticities

207

(i) A small elasticity set consisting of categories whose expenditures tend to be dominated by physiological or security needs; (ii) A medium elasticity set consisting of categories whose expenditures tend to be dominated by love or esteem needs; (iii) A large elasticity set consisting of categories whose expenditures tend to be dominated by love, esteem, or self-actualization needs. For “small,” “medium,” and “large,” it is assumed that:9 Small elasticity set (“necessities”): Total-expenditure elasticities that lie between 0 and 0.50. Medium elasticity set (“near-luxuries”): Total-expenditure elasticities that lie between 0.50 and 0.75. Large elasticity set (“luxuries”): Total Expenditure elasticities that are greater than 0.75. A first attempt at implementing this scheme for the 29 expenditure categories of this chapter is given in Table 11.31. The three important columns of the table are the second, third, and fifth. In the second column, expenditure categories are associated with Maslovian needs, the most important of which are highlighted in bold. In the third column, the weights implicit in column 2 are used to assign totalexpenditure elasticities to one of the three elasticity sets (small, medium, or large) described above. Finally, the fifth column provides an informal test of the scheme by comparing the mean QR(mode) total-expenditure elasticities from Table 11.30 (listed in column 4) with their theoretical elasticity sets in column 3. For example, expenditures for food consumed at home are identified in column 2 to be motivated by all of the Maslow needs except for security, with physiological needs assumed as the most important. Physiological needs being dominant, accordingly leads to assigning the total-expenditure elasticity for the category to the “small” elasticity set. In column 4, we see that the mean QR(mode) elasticity, 0.3072, falls into the “small” elasticity set, which, in turn, accounts for the Y (for “yes”) in the fifth column under “agreement.” As we are under no illusions concerning the subjectivity involved in this exercise, it is important that we convey as much as possible the thoughts that have guided us in arriving at the judgments expressed in columns 2 and 3. Since most of the 29 categories of expenditure are themselves aggregates, assignments have been made by focusing on expenditures at the lowest level of detail contained in the BLS microdata survey CDs.10 For most categories, consideration at this level often 9 The

designation of the elasticity sets in terms of “necessities,” “near-luxuries,” and “luxuries” is obviously at odds with the long-standing convention in economics of defining necessities as goods with income elasticities between 0 and 1 and luxuries as goods with income elasticities greater than 1. However, when one thinks about it, the present characterization is no less arbitrary, and for present purposes seems much more useful. 10 This detail is the 6-digit universal classification codes (UCCs).

208

11

Engel Curves for 29 Categories of CES Expenditure

Table 11.31 Determination of elasticity sets, 29 CES expenditure categories, 1996Q1–1999Q4 Expenditure Category Food consumed at home Food consumed outside of home Alcoholic beverages Housing Shelter Owned dwellings Rented dwellings Other lodging Utilities Natural gas Electricity Telephone services Water and other public service. Housing operations House furnishings and equip. Apparel and Services Men and boys’ Women and girls’ Transportation Gasoline and motor oil Health care Entertainment Personal care and products Reading materials Education Tobacco products Miscellaneous expenditures Cash contributions Personal insurance and pensions

Maslovian needs

Theoretical Estimated elasticity set elasticity

Agreement

P, L, E, SA P, L, E, SA P, L, E, SA P, S, L, E, SA P, S, L, E, SA P, S, L, E, SA P, S, L, E, SA P, S, L, E, SA P, S, L, E, SA P, S, L P, S, L, E, SA S, L, E, SA P, S, L, E

small large medium large large large medium large small small small medium small

0.3072 0.8476 0.4704 0.8730 0.8810 0.9166 0.5838 0.8540 0.3206 0.1861 0.2263 0.4137 0.1654

Y Y N Y Y Y Y Y Y Y Y N Y

P, S, L, E P, S, L, E, SA P, L, E, SA P, L, E, SA P, L, E, SA P, S, L, E, SA P, S, L P, S, L, E P, L, E, SA P, L, E, SA S, L, E, SA P, S, L, E, SA P, L P, S, L, E, SA S, L, E, SA S, L, E, SA

large large large medium large large medium medium large medium large large small medium large large

0.9175 1.0590 0.9043 0.6722 0.7555 1.1979 0.3885 0.5329 0.9732 0.4776 0.5513 0.9701 0.1568 0.5701 1.0894 0.9747

Y Y Y Y Y Y N Y Y N N Y Y Y Y Y

Codes: P: Physiological needs S: Security needs L: Love (community) needs E: Esteem needs SA: Self-actualization needs. Y: Estimated elasticity lies within limits defined by theoretical elasticity set. N: Estimated elasticity does not lie within limits defined by theoretical elasticity set. Bold: Indicates “dominant”; absence indicates no clear-cut dominance.

leads to all five of the Maslovian needs being flagged. As a first example, consider the two food categories. Now, food consumption is an extremely complex activity involving highly interactive motivations. Physiological needs are obviously most basic, but love (community), esteem, and self-actualization needs are present as well. Physiological needs are assumed to be dominant for food consumed at home, while love and esteem needs are assumed to be so for food consumed outside the

11.3

Interpretation of Total-Expenditure Elasticities

209

home. The result is assignment of food consumed at home to the low elasticity set and food consumed outside of the home to the high elasticity set. Perhaps an even more complicated category is housing (and its components), which encompasses not only shelter, but its operations (including utilities), maintenance, furnishings, and equipment. Like food, shelter is basic to survival, but also runs the gamut in serving other motivations, security (keep out saber-tooth tigers), love (watering hole for family and friends), esteem (look at my castle), and self-actualization (a sanctuary for realizing who I am). Accordingly, shelter is assumed to be motivated by all five Maslovian needs, with none dominant for owned dwellings, physiological and security dominant for rented dwellings, and love, esteem, and self-actualization dominant (because of heavy representation of vacation travel and entertainment) for other lodging. Except for telephone services, expenditures for utilities are viewed largely as providing “slave” services and are consequently assigned to the small elasticity set. However, telephone services are seen as serving both security and higher-order needs and accordingly assigned to the medium elasticity set. Housing operations might seem mis-assigned, but this category includes expenditures for baby-sitting, nanny, and housekeeping services and thus is motivated in substantial part by higher-order wants. Household furnishings and equipment includes expenditures for essentially all equipment used in (and outside) the home (small and large appliances, heating and cooling equipment, lawnmowers, telephones, etc.), except for entertainment-associated equipment and services (TVs, radios, cameras, etc.). Computers and computer-related expenditures are also included in this category. As with owned dwellings, expenditures in this category are seen as motivated by all five Maslovian needs, with none being dominant and are assigned to the large elasticity set. We have already touched on reasons for assignment of transportation and cash contributions to the large elasticity set. Other categories that are assigned to the large elasticity set include entertainment, education, reading materials, and personal insurance and pensions, for expenditures in these categories are almost certainly motivated in substantial part by higher-order wants. With health concerns related to tobacco use increasingly paramount, expenditures for tobacco products are obviously to be assigned to the small elasticity set.11 Finally, health care itself (as well as apparel and services and miscellaneous expenditures) is assigned to the medium elasticity set. Turning now to the “test” results in column 5 of Table 11.31, we see that 24 of the 29 expenditure categories appear to be correctly classified according to their theoretical assignment. The five categories that are incorrectly classified by this criteria are alcoholic beverages, telephone services, gasoline and motor oil, personal care and products, and reading materials, all of which have estimated total-expenditures that are in elasticity sets that are one set below the one to which they have been theoretically assigned. However, alcoholic beverages and personal care and products

11 In this case (and for alcoholic beverages as well), strong habit formation (including addiction) is

identified with physiological needs.

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Engel Curves for 29 Categories of CES Expenditure

have estimates near their theoretical lower limits (0.47 and 0.48, respectively), and gasoline and motor oil are not too far off. The largest discrepancy is clearly for reading materials, whose empirical elasticity is 0.55, as opposed to a theoretical lower limit of 0.75.12 An alternative way of assessing consistency of the estimated income elasticities of this chapter with the Maslovian needs hierarchy is in terms of the stability of the total-expenditure elasticities over the 16 quarters of estimation, for an implication of the Maslovian hierarchy would seem to be that estimated elasticities in the small elasticity set should have greater stability than estimated elasticities in the medium elasticity set, and similarly for estimated elasticities in the medium elasticity set vis-à-vis those in the large elasticity set. Table 11.32 provides a check of this implication. In this table, we have ordered the 29 expenditure categories from Table 11.31 according to the size (smallest to largest) of the standard deviations (taken from Table 11.30) of their respective mean total-expenditure elasticities. Standard deviations are given in column 2 of the table and theoretical elasticity sets in column 3. Although there is some obvious mis-classification according to the implication just described, the implication is nevertheless pretty much borne out, for, in general, categories with the smallest standard deviations tend to be the ones with elasticities assigned to the small elasticity set, while categories with the largest standard deviations tend to be the ones with elasticities assigned to the large elasticity set. Health care, housing, shelter, entertainment, and miscellaneous expenditures appear to be the most egregiously mis-classified categories.

11.4 Summary and Conclusions In this chapter, we have extended the analysis of the preceding chapters to a 29-category disaggregation of CES expenditure. Because of limited coverage in the ACCRA price surveys, price elasticities at this level of disaggregation cannot be estimated. Total-expenditure elasticities for each of the 29 categories for the 16 quarterly CES surveys from 1996 through 1999 have been estimated using simple double-logarithmic Engel functions. The socio-demographico-regional variables of previous chapters have been included as controls. Models have been estimated by both OLS and quantile regression. For the first time, an effort is made in this chapter to correlate values of total-expenditure elasticities with the five hierarchical Maslovian needs as discussed in Chapter 2. The needs identified as driving expenditures in a category are used to assign the total-expenditure elasticity for the category 12 Our thought in assigning this category to the large elasticity set was that expenditures within the

category would be heavily motivated by entertainment and self-actualization needs. However, with hindsight it is probably the case that the expenditures in this category (which are not large to begin with) are dominated by expenditures for newspapers and magazines. Unfortunately, sub-aggregates at the 6-digit UCC level are not included on the BLS microdata CDs, so that checking whether this is in fact the case is not possible.

11.4

Summary and Conclusions

211

Table 11.32 Standard deviations of mean estimated total-expenditure elasticities, and theoretical elasticity sets 29 CES categories, 1996Q1–1999Q4 Expenditure Category

Std. Dev. Of Mean Elasticity

Theoretical Elasticity Set

Utilities Food consumed at home Electricity Water and other public services Rented dwellings Health Housing Gasoline and motor oil Telephone Services Shelter Entertainment Personal care and products Natural gas Tobacco products Food consumed outside of home Alcoholic beverages Transportation Personal insurance and pensions Reading materials Women and girls’ apparel Owned dwellings Men and boys’ apparel Apparel and Services House furnishings and equipment Cash contributions Other lodging Housing operations Education Miscellaneous expenditures

0.0176 0.0195 0.0242 0.0258 0.0279 0.0280 0.0285 0.0352 0.0390 0.0402 0.0416 0.0418 0.0463 0.0464 0.0472 0.0502 0.0505 0.0508 0.0575 0.0594 0.0618 0.0753 0.0804 0.0852 0.0902 0.1035 0.1112 0.1573 0.2125

small small small small small medium large small small large large medium small small large medium large large large medium large medium medium large large large large large medium

to one of three theoretical elasticity sets, which in turn are quantified (based on break-points of 0, 0.5, and 1) in terms of necessities, near-luxuries, and luxuries. A comparison is then made between the empirical estimates of the elasticities and these theoretical elasticity sets. The principal results of the chapter are as follows: (1) Estimates of total-expenditure elasticities are almost invariably extremely highly significant statistically, with (as at the higher level of aggregation of the preceding chapters) generally marked stability over the 16 quarters of estimation. (2) As in the preceding chapters, the distributions of residuals from the estimated equations tend to be asymmetrical, with modes (for the QR equations) that are typically more than four percentiles from the median. (3) Engel’s law for food is confirmed in every instance.

212

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Engel Curves for 29 Categories of CES Expenditure

(4) Mean total-expenditure elasticities over the 16 quarters greater than 1 are found for transportation, housing operation, and cash contributions. Elasticities are close to 1 for entertainment, education, and personal insurance and pensions, and about 0.9 for housing. Elasticities are smallest for natural gas, electricity, water and other public services, and tobacco products. Finally, we are cautiously encouraged by the general consistency of the results of this chapter with the notion that consumption behavior is motivated by an underlying hierarchy of wants. Two informal tests of this notion have been undertaken. The first entails comparison of the empirical total-expenditure elasticity for an expenditure category with the elasticity set that it should theoretically be a member of as determined by the category’s underlying hierarchy of motivating wants. Of the 29 expenditure categories analyzed, the elasticities for 24 are correctly classified according to this criterion. Moreover, of the five categories that are incorrectly classified, only one appears to be seriously so. The second test that has been undertaken involves correlation of the stability of the estimated total-expenditure elasticity for an expenditure category (as measured by the 16-quarter standard deviation) with the category’s theoretical elasticity set, the idea being that categories whose expenditures are determined by higher-order wants should display greater variation in estimated elasticities over time than categories whose expenditures are determined by lower-order wants. At the extremes, categories whose expenditures are determined by the low-order wants should have small total-expenditure elasticities, while categories whose expenditures are determined by higher-order wants should have large elasticities.13 Again, the results are generally consistent with this indeed being the case.

13 Higher-order

wants correlate with high income, which in turn allows for individual idiosyncrasies to manifest themselves.

Chapter 12

Summary of Cross-Sectional Results

In this chapter, we bring to a close this part of the study by presenting a summary and overview of the results that have been obtained in our cross-sectional analysis of data contained in the BLS quarterly consumer expenditure surveys over the period between 1996 and 1999. Specifically, our goals in the cross-sectional analysis have been as follows: (1) To estimate total-expenditure elasticities for as many cross-sectional categories of consumption expenditure in the CES surveys as feasible. (2) To explore the feasibility of joining expenditure and socio-demographic information from the CES surveys with price data collected by the American Chambers of Commerce Research Association (ACCRA) for the purpose of estimating cross-sectional price elasticities. (3) To exploit the semi-panel nature of the CES surveys to estimate dynamic flowadjustment models. (4) To analyze the cross-sectional stability of expenditure patterns of U.S. households over time. (5) To explore the feasibility of a genetically based neurobiological model of expenditure motivation as a framework for organizing and interpreting results from conventionally estimated consumption and demand models. (6) To explore the use of quantile-regression estimation in circumstances in which certain of the conditions required for the validity of least-squares estimation may be questionable.

12.1 Stability of Expenditure Patterns One of the main results to emerge in the analysis of the CES surveys is the strong evidence in support of stable consumption patterns. A first look at this stability was provided in the principal-component analyses of 14 exhaustive expenditure categories in Chapter 5, where, it will be recalled, the focus was on analyzing the internal structures among consumption categories (as manifested in eigenvalues and eigenvectors), without regard to explicit control of income (or total L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_12, 

213

214

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Summary of Cross-Sectional Results

expenditure) and prices. The two largest principal components of expenditure were found to be extremely stable over the 16 quarterly surveys analyzed, and to invariably account for about 85% of the variation in total expenditure. Taken together, these results clearly attest to the stability of households’ underlying tastes and preferences. Stability in the internal structures of tastes and preferences (whatever their origin) should in turn translate into stable relationships between expenditures and prices and income. The results in numerous tables and graphs in the preceding chapters show that this is indeed the case. Estimates of total-expenditure elasticities are absent of trends and in general are remarkably stable over the 16 quarters of estimation. While estimates of price elasticities show greater variability, they, too, are largely free of trends.

12.2 Joining of ACCRA Price Data with CES Expenditure Surveys While a major purpose of the CES surveys is to provide weights for the BLS consumer price indices, the CES surveys collect only expenditure information, not quantities or prices. The price information used by BLS is collected in separate surveys. The ideal, of course, would be to merge expenditure and price information from the two BLS sources, but this would involve a massive effort that is yet to be undertaken for the purpose at hand. The only other readily accessible source of consumer price data is that collected in quarterly surveys by ACCRA. While the range of price information represented in the ACCRA surveys clearly falls far short of that collected in the BLS price surveys, it nevertheless has been possible to match CES expenditures and ACCRA prices at an exhaustive six-category level of expenditure. This has enabled, for the first time in our own experience, the estimation of conventional demand functions using household survey data, in which both income (or total-expenditure) and prices appear as predictors. The results are richly encouraging. Despite the limited coverage of prices for several of the aggregate categories, meaningful negative own-price elasticities, with substantial t-ratios, are obtained in virtually every demand equation estimated, whether the equation be separate or part of a theoretically plausible system. To this point, this is one of the most important results of the exercise. A related finding is that, statistically, the presence or absence of prices in the equations makes little difference to the estimates of the total-expenditure elasticities that are obtained, which is to say that total expenditure and prices are for the most part uncorrelated. This, too, is an important result, for it means that Engel curves can be estimated at levels of aggregation for which price information is not available, with confidence that absence of prices does not lead to biased estimates of the Engel elasticities. As far as we are aware, the effort here represents the first time that this contingency has been explored empirically.

12.3

Summary of Price and Total-Expenditure Elasticities

215

12.3 Summary of Price and Total-Expenditure Elasticities1 Price and total-expenditure elasticities from the various models estimated with the BLS-ACCRA data sets are summarized in Table 12.1. The estimates in this table refer to 1996, and in each case are estimated from data sets that are pooled across quarters.2 Of the seven models in the table, the AIDS, LES, indirect addilog, and direct addilog represent integrable demand systems (i.e., satisfy all of the restrictions of neoclassical demand theory). The additive double-log model is additive (i.e., satisfies the budget constraint), but is not integrable. The simple double-log and the logarithmic flow-adjustment models are obviously neither additive nor integrable. The conclusions that emerge from this table (and the tables that underlie it in Chapter 7 and 10) are as follows: Table 12.1 Own-price and total-expenditure elasticities, BLS-ACCRA data sets, 1996Q1– 1996Q4 (Pooled) Category

Simple Dbl-Log

AIDS

LES

Indirect addilog

Direct addilog

Additive dbl-log

Log Fl-Adjust.

−0.2981 −0.8285 −0.7222 −1.3739 −0.9375 −1.1448

−0.7576 −0.7465 −0.8902 −0.9770 −1.0895 −0.9083

−0.2450 −0.7502 −0.2540 −1.1023 −0.3603 −1.3092

−0.4573 −0.7280 −0.4633 −1.0729 −1.2384 −0.3451

−0.3604 −0.5148 −0.7858 −0.9925 −0.5842 −0.4019

−0.3372 −0.9791 −1.7547 −1.3830 −2.4748 −1.5674

0.1317 0.5902 0.1731 2.6826 0.2887 1.3278

0.3963 0.9458 0.4413 1.3817 0.5726 1.3213

0.4639 0.4885 0.4664 1.4248 0.6242 1.5174

0.3604 0.9088 0.3154 1.3224 0.4484 1.1217

0.3387 1.1258 0.3478 1.4000 0.6162 1.3814

Price elasticities: Foodathome Shelter Utilities Trans. Health care Misc. Exp.

−0.2919 −0.5726 −0.8960 −1.0843 −1.1542 −1.1263

Total-expenditure elasticities: Foodathome Shelter Utilities Trans. Health care Misc. Exp.

0.3073 0.8559 0.3733 1.3378 0.4555 1.1263

0.4469 0.8876 0.4612 1.7250 0.6338 1.2150

Source: Simple Dbl-Log: Table 8.5 [OLS]; AIDS: Table 7.1; LES: Table 7.2; Indirect Addilog: Table 7.3; Direct Addilog: Table 7.5; Log Fl. Adjust.: Tables 10.2–10.6 [OLS, long run].

1 The focus in this section is on just the price and total-expenditure elasticities estimated from the 6-category CES-ACCRA data sets. 2 The estimates are from OLS in all cases.

216

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Summary of Cross-Sectional Results

(1) Engel’s law for food is confirmed in every instance. In magnitude, elasticities for food are smallest in every model for price and in every model except for the direct addilog for total expenditure.3 (2) In general, elasticities are in closer agreement across models for totalexpenditure than for price. Magnitudes for both total-expenditure and price agree closely for food and transportation in all but the LES model. Disagreements for shelter, utilities, health care, and miscellaneous expenditures are most marked in the LES, indirect and direct addilog, and additive double-log models, with the LES and direct addilog models most at odds.4 (3) Except in the flow-adjustment model for utilities, demand is seen to be inelastic for food, shelter, and utilities; and elastic in all of the models for transportation and miscellaneous expenditures. For total expenditure, elasticities are greater than 1 in every model for transportation and miscellaneous expenditures. (4) As has been noted, the elasticities (of both price and total expenditure) in the table are all derived from parameter estimates that nearly invariably have the right sign and large (even huge!) t-ratios.

12.4 Estimation of Dynamical Cross-Sectional Models Once households are selected for inclusion in the CES surveys, diary data are collected from them for four consecutive quarters. Thus, for households not appearing for the first time in a survey, information is available that allows for dynamical models of the type discussed in Chapter 2 to be estimated. However, since the models in question require information on quantities, this can only be obtained implicitly in terms of logarithms through use of ACCRA price indices, which means that state-adjustment models, because of the depreciation relationship, cannot meaningfully be formulated non-linearly. The only dynamical model that accordingly can meaningfully be estimated is a logarithmic flow-adjustment model. The logarithmic flow-adjustment results, as summarized in Table 12.1 (and also in Table 10.8), are encouraging. Dynamical effects are obviously present in all six expenditure categories. Statistically, all signs are correct, and with only a couple of exceptions t-ratios are all at least 2.5 Comparison of the elasticity estimates in columns 1 and 7 of Table 12.1 above provides support for the conventionally held view that parameter estimates in static cross-sectional models are long run in nature. While the long-run price elasticities may appear on the high side—four of

3

The elasticities for the direct addilog model are from the constrained non-linear equations in Table 7.5. The elasticities for the logarithmic flow-adjustment model refer to the long run. 4 As was noted in Chapter 7, the direct addilog model is extremely problematic to estimate, and the disparity with the other models may be an artifact of estimation. The price elasticities in the LES model, on the other hand, may simply reflect strong attraction toward the default values of −1. 5 Cf. Table 10.1.

12.6

Asymmetrical Residuals and Quantile Regression

217

the categories are indicated to have elastic demands—the long-run total-expenditure elasticities are very similar in magnitude with their simple double-log counterparts. The latter avers well for interpreting the Engel elasticities for the 29 expenditure categories in Chapter 11 as representing steady-state behavior.

12.5 Effects of Other Variables Since the focus in the exercise is on the estimation of price and income elasticities, other variables in the models are viewed as controls, their inclusion being necessitated in order to avoid bias problems associated with “omitted” variables. In general, age, family size, and dummy variables denoting age and number of children, rural/urban, region of the country lived in, race, sex, homeownership, and whether the household is recipient of food stamps are usually significant in some form.6 The number of wage-earners and education are occasionally important. Seasonal effects, on the other hand, are usually unimportant. The variables that are most sensitive to inclusion of ACCRA prices are the regional dummy variables (northeast, midwest, south, and west). Obviously, this is hardly a surprise, for, in the absence of price variables, these variables will pick up differences in regional price levels.

12.6 Asymmetrical Residuals and Quantile Regression As described in detail in Chapter 9, an intriguing phenomenon that has emerged with the CES surveys is a characteristic occurrence of asymmetry in distributions of residuals. Such asymmetries, together with the presence of long tails, have negative implications for least-squares estimation, and for this reason many of the models have been estimated by quantile regression in addition to least squares. Quantile regression, as is now well-known, is much less sensitive to pathologies in underlying error terms (even absence of second moments!) than is least squares, and accordingly provides a meaningful robust estimation alternative. Our procedure has in general been to select models, to begin with, using least squares, and then to obtain final estimates of elasticities derived from quantile regressions applied at the modal values of the distributions of residuals from median quantile regressions. Movement of the regression quantiles to modes of the residuals from median regressions seems reasonable in light of the marked asymmetries that are usually present in the OLS and median estimations.

6

As noted in Footnote 11 of Chapter 6, since demographic factors related to family size and children are especially important in food consumption, the approach that has been employed here in taking these into account is simple-minded in comparison with what is available in the literature, such as, for example, the use of household production functions or adult-equivalent scales.

218

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Summary of Cross-Sectional Results

As has been noted, the modes of residual distributions are typically at least four percentiles away from the medians. As is probably to be expected, when modes are near medians, OLS and quantile regression usually give similar estimates. However, when asymmetry is marked (as is typically the case), the quantile estimates, especially those at the mode, can be quite different. Interestingly, coefficients on price (and on the lagged dependent in the dynamical models) are typically the most affected.

12.7 Cross-Price Elasticities Five of the seven models whose own-price and total-expenditure elasticities are tabulated in Table 12.1 also allow for the estimation of cross-price elasticities.7 Of the five, only two, the AIDS and additive double-log, allow for unrestricted estimation of both income and substitution effects.8 The LES model, because of its strong separability assumptions, allows for income effects only, while the two addilog models impose restrictions on the ratios of the cross-price derivatives.9 In view of these restrictions, it is accordingly not surprising that the cross-price effects in these three models are typically indicated to be small (cf. Tables 7.2, 7.4, and 7.6). With the AIDS and additive double-log models, on the other hand, a number of the crossprice effects are indicated to be quite large, for example, transportation and health care in the AIDS model and health care and miscellaneous expenditures in the additive double-log model. Unfortunately, however, there is little consistency between the two models, and at this point there is simply not enough empirical experience concerning cross-price elasticities to give either of the models (especially at this level of aggregation) an edge.

12.8 Evidence for Maslovian Hierarchical Preferences Although results from the time-series analysis are yet to be presented and discussed, results at this point from the cross-sectional analysis it would seem can, at the minimum, be viewed as being consistent with a genetically based hierarchical-ordering of preferences. The principal-component analysis in Chapter 7 of the internal structures of consumption expenditures over four years of quarterly surveys shows a marked stability in the two most important of these structures (i.e., as represented by 7

The full matrices of elasticities for the five models are given in Tables 7.1, 7.2, 7.4, and 7.6 (for the AIDS, LES, and two addilog models) and Table 8.3 (for the additive double-log model). 8 The AIDS model is integrable (at least in principle) and its elasticities therefore satisfy both the Cournot and Hicks–Allen–Slutsky aggregation conditions. The additive double-log model is not integrable, and satisfies only the Cournot condition. 9 For the indirect addilog model, the ratio of cross-price derivatives is equal to the ratio of totalexpenditure elasticities, while for the direct addilog model, the ratio of cross-price derivatives is equal to the ratio of quantities.

12.10

Epilogue to the CES Analysis: Update to 2005

219

the two largest principal components of total expenditure) that can be interpreted as representing both genetic and socially induced substrates of behavior, which in turn can be interpreted as arising out of a Maslovian hierarchy of wants.10 Further consistency is provided by a generally successful agreement between a theoretical ordering (as adduced from an assumed underlying Maslovian structure) of total-expenditure elasticities by size and variability with their empirically estimated counterparts. Agreement (cf. Tables 11.31 and 11.32 of the preceding chapter) is not exact, but is certainly suggestive.

12.9 A First Look at the Relationship between Total Consumption and After-Tax Income For reasons discussed in Chapter 6, the budget constraint in our analysis has been total expenditure, rather than after-tax income, which means that no allowance is made for saving (except for that implicit, by some definitions, in expenditures for durable goods and insurance and pension contributions). However, as a prelude to the detailed analysis of saving that will be the focus of Chapter 19, a set of simple double-logarithmic equations relating total-expenditure to after-tax income for the 16 quarters of CES surveys were presented in Table 6.9. While, like their expenditure-category counterparts, the total-expenditure equations are remarkably stable over the 16 quarters—both elasticities and R2 s all lie within the range of 0.05 of one another—the size of the elasticities of consumption elasticities seems implausibly small (all estimates are less than 0.50), even for cross-sectional data.

12.10 Epilogue to the CES Analysis: Update to 2005 At the time that analysis of the CES data was started in early 2002, information from the 1999 surveys was the latest available. At the time of this writing (April 2007), data are available for six additional years through 2005. In an appendix to this chapter, we bring the analysis of the CES surveys as close to the present as possible by reporting the results from estimating double-log Engel curves at a 15-expenditure category of aggregation for these 24 additional quarters of data. Our primary concern is whether the stability that has been found to characterize total-expenditure elasticities over the period 1996–1999 continues through 2005.

10

We take pains at this point to emphasize that the sense of this sentence is both speculative and interpretive; all that we are really meaning to imply is that the results of the cross-sectional analysis are consistent with an underlying hierarchy of wants.

220

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Summary of Cross-Sectional Results

12.11 Looking Ahead This brings, for now, our cross-sectional analysis to a close. In the next several chapters, we shall turn to what was the primary activity of original Houthakker and Taylor, namely, application of the state- and flow-adjustment models to the timeseries data for consumption from the National Income and Product Accounts. In Chapter 17, we shall return to the analyses of this part of the study in order to compare the CES-based price and total-expenditure elasticities with the time-series estimates.

Appendix: Addendum to the CES Analysis: Estimates of Total-Expenditure Elasticities for 24 Additional Quarters, 2000–2005 In this appendix, we bring the analysis of the CES surveys as close to the present as possible by reporting the results from estimating double-logarithmic Engel curves at a 15-expenditure category of aggregation for these 24 additional quarters of data. Our primary concern is whether the stability that has been found to characterize total-expenditure elasticities over the period 1996–1999 continues through 2005.11 As in the preceding chapter, double-logarithmic equations, using the same set of socio-demographical-regional variables as there, are estimated for the 15 categories of expenditure for each of the 24 quarters of additional data. Graphs of the estimated total-expenditure elasticities from these equations, together with the same for 1996q1–1999q4, are presented in Fig. 12.1, while in Table 12.2, we have tabulated mean values and standard deviations of the 40 total-expenditure elasticities for both 1996–2005 and the two sub-periods of interest. Two simple OLS equations are included at the base of the charts in Fig. 12.1, the first a regression of the 40 elasticities on a constant and dummy variable (d) separating the two periods (i.e., 1996q1–1999q4 and 2001q1–2005q4) and the second a regression of the elasticities on a 40-quarter time trend (T). The first of these regressions can obviously be employed to test a hypothesis that the means differ between the two periods, while the second can be used to test for a trend in the elasticities. When we look at the graphs, first we see apparent downward trends in the elasticities for food at home, apparel, entertainment, and pensions and personal insurance, and upward trends in the elasticities for health care and personal care. Of the remaining categories, there are suggestions of trends in the elasticities for food consumed outside the home, transportation, education, and cash contributions. All of these observations are confirmed statistically in the dummy variable and trend regressions,

11 This breakdown of expenditures is the same as analyzed in Chapter 5, except that food expenditures are disaggregated into food consumed at home and food consumed outside of the home. Time constraints have prevented integration of ACCRA price data into the analysis in order to investigate stability of price elasticities.

Appendix: Addendum to the CES Analysis

221

for the t-ratios on the trend term in all of the equations, but for alcoholic beverages, housing, reading, tobacco, and miscellaneous expenditures, are well in excess of 2 (in absolute value). On the other hand, except for apparel, entertainment, education, cash contributions, and pensions and personal insurance, the trends are not particularly strong quantitatively. For food consumed at home, transportation, and health care, for example, the trend values of the total-expenditure elasticities differ by only −0.04, −0.10, and 0.07 between 1996q1 and 2005q4, respectively. Indeed, as can be calculated from the information given in Table 12.2, even in the case of education expenditures, for which the trend is strongest—0.00629 per quarter—the difference

Estimated Total-Expenditure Elasticities 15 CES Categories of Expenditure 1996Q1 - 2005Q4 Food Consumed at Home

Food Consumed Outside of Home

0.4

0.95

0.35

elasticity

elasticity

0.9 0.85

0.3

0.8 0.75 0.7 0.65

0.25

0.6 0

4

8

12

16

20

24

28

32

36

40

0

4

8

12

16

e = 0.3232 – 0.0214d (65.41) (–3.35)

20

24

28

32

36

40

32

36

40

quarter

quarter 2

R = 0.2278

e = 0.3311 + 0.00101T R2 = 0.2824 (53.92) (–3.37)

Alcoholic Beverages

R = 0.2867

e = 0.8175 – 0.00157T (69.05) (-3.12)

R2 = 0.2043

Housing

1

0.6

2

e = 0.8116 – 0.0438d (93.33) (–3.12)

0.95 elasticity

elasticity

0.55 0.5 0.45

0.9 0.85 0.8

0.4

0.75 0.7

0.35 0

4

8

12

16

20

24

28

32

40

0

4

8

12

16

20

24

28

quarter

quarter

e = 0.4842 – 0.00893d (41.88) (–0.60)

36

2

R = 0.0093

2 e = 0.4673 – 0.000565T R = 0.0207 (0.90) (31.54)

2

e = 0.8201 + 0.00537d (115.62) (0.59)

R = 0.0090

e = 0.8243 – 0.000050T (69.05) (–0.13)

R2 = 0.0004

Fig. 12.1 Estimated total-expenditure elasticities 15 CES categories of expenditure 1996Q1–2005Q4

222

12

Summary of Cross-Sectional Results

Apparel

Transportation R2 = 0.3214

1.15

1.4

1.05

1.35

0.95 0.85 0.75

1.3 1.25 1.2

0.65

1.15 0

4

8

12

16

20

24

28

32

36

40

0

4

8

12

16

quarter

24

28

32

36

40

32

36

40

2 R = 0.2674

e = 1.3254 – 0.00251T (83.73) (–3.72)

Health Care

Entertainment R2 = 0.5702

e = 0.9783 – 0.0826d (69.35) (2.74)

2 R = 0.1646

e = 0.5306 + 0.0270d

1.15

0.65 0.6

1.05 elasticity

elasticity

20 quarter

2 R = 0.3923

e = 0.9442 – 0.00378T (52.58) (–4.95)

0.55 0.5

0.95 0.85

0.45

0.75

0.4 0

4

8

12

16

20

24

28

32

36

0

40

4

8

12

16

(108.50)

20

24

28

quarter

quarter (–7.10) R2 = 0.3771

e = 0.5112 + 0.00174T (60.03) (4.80)

e = 1.0213 – 0.00358T (88.81) (–7.47)

Personal Care

R2 = 0.5948

Reading 0.65

0.65 0.6

0.6 elasticity

0.55 elasticity

R2 = 0.1668

e = 1.3020 – 0.0466d (99.41) (–2.76)

elasticity

elasticity

e = 0.9151 – 0.00806d (62.16) (–4.24)

0.5 0.45 0.4

0.55 0.5 0.45

0.35

0.4

0.3

0.35 0

4

8

12

16

20

24

28

quarter

32

36

40

0

4

8

12

16

20

24

28

quarter

e = 0.4642 + 0.0565d (28.69) (2.71)

2 R = 0.1616

e = 0.5503 – 0.0404d (43.59)

R2 = 0.1389

e = 0.4201 + 0.00381T (23.95) (5.10)

2 R = 0.4067

e = 0.5534 – 0.00133T (32.97) (–1.86)

R2 = 0.0838

Fig. 12.1 (continued)

32

36

40

Appendix: Addendum to the CES Analysis

223

Education

Tobacco 0.35

1.25

0.3

1.15

0.25

1.05

elasticity

elasticity

1.35

0.95 0.85

0.2 0.15

0.75

0.1

0.65

0.05

0.55

0 0

4

8

12

16

20

24

28

32

36

40

0

4

8

12

16

quarter e = 0.9446 + 0.1305d (30.95) (3.31) e = 0.8940 + 0.00629T (23.74) (3.93)

24

28

R2 = 0.2887

e = 0.1470 + 0.0199d (12.94) (1.36)

R2 = 0.0463

e = 0.1456 + 0.000651T (1.04) (9.84)

2 R = 0.0275

Miscellaneous Expenditures

32

36

40

Cash Contributions

2 R = 0.0609

e = 0.6980 – 0.0381d (37.12) (–1.57)

20

quarter

R2 = 0.2261

e = 1.0601 – 0.1202d (37.95) (–3.33)

0.9

1.3 1.2

0.8 elasticity

elasticity

1.1 0.7 0.6

1 0.9 0.8

0.5

0.7

0.4

0.6 0

4

8

12

16

20

24

28

32

36

40

0

4

8

12

16

quarter

20

24

28

32

36

quarter

2 e = 0.6873 – 0.000595T R = 0.0083 (27.60) (–0.56)

e = 1.1070 – 0.0058T (32.17) (–3.97)

R2 = 0.2930

Pensions & Personal Insurance 1.15

elasticity

1.05 0.95 0.85 0.75 0.65 0

4

8

12

16

20

24

28

quarter (78.10)

(–7.52)

e = 0.9451 – 0.00451T (57.11) (–6.46)

Fig. 12.1 (continued)

R2 = 05236

32

36

40

e = 0.9206 – 0.1145d R2 = 05983

40

224

12

Summary of Cross-Sectional Results

Table 12.2 Total-expenditure elasticities, means and standard deviations, BLS-CES surveys, 40 quarters, 1996–2005 1996–1999

2000–2005

1996–2005

Category

Mean

Std. Dev.

Mean

Std. Dev.

Mean

Std. Dev.

Food home Food away Alcoholic beverages Housing Apparel Transportation Health care Entertainment Personal care Reading Education Tobacco Misc. Exp. Cash contributions Pensions and pers. ins.

0.3232 0.8116 0.4842 0.8201 0.9151 1.3020 0.5306 0.9783 0.4642 0.5503 0.9446 0.1470 0.6980 1.0601 0.9206

0.0239 0.0382 0.0405 0.0349 0.0758 0.0472 0.0239 0.0319 0.0431 0.0487 0.1359 0.0401 0.0841 0.0850 0.0461

0.3019 0.7677 0.4753 0.8255 0.8345 1.2554 0.5576 0.8956 0.5207 0.5100 1.0751 0.1669 0.6599 0.9399 0.8061

0.0166 0.0324 0.0496 0.0231 0.0446 0.0555 0.0343 0.0385 0.0756 0.0516 0.1122 0.0486 0.0688 0.1261 0.0478

0.3104 0.7853 0.4789 0.8233 0.8667 1.2741 0.5468 0.9287 0.4981 0.5261 1.0229 0.1590 0.6751 0.9880 0.8519

0.0222 0.0407 0.0459 0.0281 0.0706 0.0567 0.0331 0.0543 0.0698 0.0537 0.1368 0.0460 0.0766 0.1254 0.0734

between the trend values for the total-expenditure elasticity between 1996q1 and 2005 q4 (0.245) is not sufficiently large for the trend values to lie outside of a 95% confidence interval constructed with respect to the 40-quarter mean. Our conclusion, accordingly, is that, despite drifts in several total-expenditure elasticities over the 10 years since 1996, these do not seem of sufficient quantitative importance to revise any of the conclusions based on data for 1996 through 1999 that have been drawn to this point.12

12 As

the analysis in this appendix has come at a very late date in the study, our focus has been very narrowly on mean behavior of the estimated elasticities. Given time, a number of things warrant investigation, including apparent “heteroscedasticity” in the estimated elasticities for food consumed at home, alcoholic beverages, housing, apparel, and miscellaneous expenditures. Also, it would clearly be of interest whether the variation (including volatility) in the estimated elasticities can be explained in terms of changes in income and socio-demographical factors.

Part III

Analysis of Time-Series Data from National Income and Product Accounts

Chapter 13

Analysis of Time-Series Data on Personal Consumption Expenditures from the U.S. National Income and Product Accounts

We now turn to the aggregate time-series data on personal consumption expenditures from the U.S. National Income and Product Accounts (NIPA), the analysis of which was the primary focus in the 1966 and 1970 editions of CDUS. Both quarterly and annual data are analyzed using the dynamical models described in Chapter 2. The present chapter elaborates the models estimated, together with various methods of estimation. Chapter 14 reports the empirical results for the quarterly data, and Chapter 15 does the same for the annual data. Chapter 16 presents summary tables of the estimated price and total-expenditure elasticities from Chapter 14 and 15, while Chapter 17 provides a comparison of the same with the cross-sectional estimates for the CES data from earlier chapters. In CDUS, the ultimate goal of the analysis was to provide (under a contract with the Bureau of Labor Statistics) an exhaustive set of econometric equations that could be used for forecasting personal consumption expenditures in what was the then ongoing Inter-Agency Growth Study.1 The present effort does not have forecasting as a goal, but rather, as has already been noted, the estimation of price and totalexpenditure elasticities for as broad a range of expenditure categories as possible. This is not to say that the models that are presented in the next two chapters could not be used for forecasting, they could, but doing so would require a deal of care, for the models (indeed, the vast majority) have been estimated using nonlinear least squares. Why this is so will become clear in a moment.

1 The Inter-Agency Growth Study was a cooperative venture involving the Bureau of Labor Statistics, the Bureau of Economic Analysis, and the Council of Economic Advisers. Consumption equations were estimated for 84 categories of personal consumption expenditure appearing in Table 2.3 of the (then) National Income and Product Accounts. However, a number of the categories for which equations were estimated were based upon data that were not officially published. This was done in order to line up NIPA definitions of consumption with those employed by the Bureau of the Census in the input-output table that was at the core of the Study. All NIPA data in the present effort have been downloaded from the BEA website, and thus can be viewed as officially published.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_13, 

227

228

Analysis of Time-Series Data

13.1 NIPA PCE Categories The categories of personal consumption expenditure for which time-series data have been downloaded from the BEA website are listed in Tables 13.1 and 13.2, quarterly in Table 13.1 and annual in Table 13.2. Models have been estimated for all of the quarterly categories listed in Table 13.1 and for all but 1.3 (food furnished Table 13.1 PCE categories quarterly, 1947–2005 Durable goods Motor vehicles and parts Furniture and household equipment Other durable goods Nondurable goods Food Clothing and shoes Gasoline, fuel oil, and other energy goods Gasoline and oil Fuel oil and coal Other nondurable goods Services Housing Household operation Electricity and gas Other household operation Transportation Medical care Recreation Other Services

Table 13.2 PCE consumption categories annual, 1929–2004 1.0

2.0

Food and tobacco 1.1 Food purchased for off-premise consumption 1.2 Purchased meals and beverages 1.3 Food furnished to employees (including military) 1.4 Food produced and consumed on farms 1.5 Tobacco products 1.6 Food excluding alcoholic beverages 1.7 Alcoholic beverages purchased for off-premise consumption 1.8 Other alcoholic beverages Clothing, accessories, and jewelry 2.1 Shoes 2.2 Clothing and accessories except shoes 2.2a Women’s and children’s 2.2b Men’s and boys’ 2.3 Standard clothing issued to military personnel 2.4 Cleaning, storage, and repair of clothing and shoes 2.5 Jewelry and watches 2.6 Other

13.1

NIPA PCE Categories

229 Table 13.2 (continued)

3.0

4.0

5.0

6.0

7.0

8.0

Personal care 3.1 Toilet articles and preparations 3.2 Barbershops, beauty parlors, and health clubs Housing 4.1 Owner-occupied nonfarm dwellings—space rent 4.2 Tenant-occupied nonfarm dwellings—rent 4.3 Rental value of farm dwellings 4.4 Other Household operation 5.1 Furniture, including mattresses and bedsprings 5.2 Kitchen and other household appliances 5.3 China, glassware, tableware, and utensils 5.4 Other durable house furnishings 5.5 Semidurable house furnishings 5.6 Cleaning and polishing preparations and miscellaneous household supplies and paper products 5.7 Stationery and writing supplies 5.8 Household utilities 5.8a Electricity 5.8b Gas 5.8c Water and other sanitary services 5.8d Fuel oil and coal 5.8e Telephone and telegraph 5.9 Domestic service 5.10 Other Medical care 6.1 Drug preparations and sundries 6.2 Ophthalmic products and orthopedic appliances 6.3 Physicians 6.4 Dentists 6.5 Other professional services 6.6 Hospitals and nursing homes 6.6a Hospitals 6.6a1 Nonprofit 6.6a2 Proprietary 6.6a3 Government 6.6b Nursing homes 6.7 Health insurance 6.7a Medical care and hospitalization 6.7b Income loss 6.7c Workers’ compensation Personal business 7.1 Brokerage charges and investment counseling 7.2 Bank service charges, trust services, and safe deposit box rental 7.3 Services furnished without payment by financial intermediaries except life insurance carriers 7.4 Expense of handling life insurance and pension plans 7.5 Legal services 7.6 Funeral and burial expenses 7.7 Other Transportation 8.1 User-operated transportation

230

Analysis of Time-Series Data Table 13.2 (continued)

9.0

10.0

11.0 12.0

8.1a New autos 8.1b Net purchases of used autos 8.1c Other motor vehicles 8.2 Tires, tubes, accessories, and other parts 8.3 Repair, greasing, washing, parking, storage, rental, and leasing 8.4 Gasoline and oil 8.5 Bridge, tunnel, ferry, and road tolls 8.6 Insurance 8.7 Purchased local transportation 8.7a Mass transit systems 8.7b Taxicab 8.8 Purchased intercity transportation 8.8a Railway 8.8b Bus 8.8c Airline 8.8d Other Recreation 9.1 Books and Maps 9.2 Magazines, newspapers, and sheet music 9.3 Nondurable toys and sport supplies 9.4 Wheel goods, sports and photographic equipment, boats, and pleasure aircraft 9.5 Video and audio goods, including musical instruments, and computer goods 9.5a Video and audio goods, including musical instruments 9.5b Computers, peripherals, and software 9.6 Radio and television repair 9.7 Flowers, seeds, and potted plants 9.8 Admissions to specified spectator amusements 9.8a Motion picture theaters 9.8b Legitimate theaters and opera, and entertainments of nonprofit institutions (except athletics) 9.8c Spectator sports 9.9 Clubs and fraternal organizations 9.10 Commercial participant amusements 9.11 Pari-mutuel net receipts 9.12 Other Education and research 10.1 Higher education 10.2 Nursery, elementary, and secondary schools 10.3 Other Religious and welfare activities Foreign travel and other, net 12.1 Foreign travel by U.S. residents 12.2 Expenditures abroad by U.S. residents 12.3 Less: Expenditures in the U.S. by nonresidents 12.4 Less: Personal remittances in kind to nonresidents

to employees), 2.3 (clothing issued to military personnel), 12.3 (foreign travel in the U.S.), and 12.4 (remittances abroad) for the annual categories. The first two of these four excluded categories are imputed items little related to income or

13.2

Generalization of the Flow- and State-Adjustment Models

231

price, while the last two, both primarily balancing item in the NIPA accounts, are more related to exchange rates and income levels abroad than to U.S. income and prices. Unlike in the 1966 and 1970 editions of CDUS, plausible price elasticities have been obtained in every category estimated, both quarterly and annual. While a major reason for this rather amazing departure from conventional experience in teasing price elasticities from time-series data is the much greater relative price variation in the data brought about by the energy crises of the 1970s, technological change (especially in electronics and information technology), and trade globalization, there are a couple of procedural reasons as well. The first relates to the relaxation of statistical criteria for keeping a variable in a model, from proper sign and a t-ratio at least 1 (in absolute value) in CDUS to simply correct sign in the present study. The second reason, to be discussed in Section 4 below, is much more technical, in that it entails the fact that the nonlinear identification constraints that are imposed in estimating the state-adjustment models involve multiple solutions, and we have taken the liberty of seeking solutions that make economic sense.

13.2 Generalization of the Flow- and State-Adjustment Models In a paper published in 1990, A. R. Bergstrom and M. J. Chambers proposed a framework that generalized the HT models to encompass both state- and flowadjustment and reflected notable advances in the specification and estimation of continuous-time models. The model specified by Bergstrom and Chambers is as follows:   dq (t) = γ α (t) z (t) + βs (t) − q (t) dt + ε (t) for t ≥ 0,   ds (t) = q (t) − δs (t) dt for t ≥ 0,

(13.1) (13.2)

where: d: the differential operator t: continuous-time parameter q(t): rate of expenditure s(t): stock (or state) variable z(t): an mx1 vector of non-random variables whose elements include real disposable income, the real price of the good, and possibly other exogenous variables ε(t): a white noise innovation α: a 1xm vector of parameters β, γ , δ scalar parameters.

232

Analysis of Time-Series Data

13.2.1 Remarks 1. Equation (13.2) is simply the depreciation equation of the HT state-adjustment model written in differential form. 2. The generalization of the HT models is represented in expression (13.1). Dividing through by γ dt (and ignoring the white noise innovation), we see that the Bergstrom–Chambers (B-C) model reduces to the HT state-adjustment model in the limit as γ → ∞. On the other hand, the flow-adjustment model obtains when β = 0. As in the state-adjustment model, steady-state equilibrium is achieved when ds(t) and dq(t) are equal to zero, which then implies that qˆ = αz + βˆs,

(13.3)

where the “hats” once again denote steady-state values. The dynamics in the B-C model can accordingly be interpreted as state adjustment combined with error correction.2 State adjustment is reflected in the presence of the state variable s(t) in equation (13.1), while error correction is represented in the term: αz(t) + βs(t) − q(t),

(13.4)

in equation (13.1) (which reflects the short-run disequilibria in state adjustment) in conjunction with the partial-adjustment parameter γ . The B-C model thus allows for much richer dynamics than either of the HT models. A second important contribution of the Bergstrom–Chambers analysis is the attention given to the stochastic structure of the model. Unlike in CDUS, where the stochastic structure was treated rather lightly, Bergstrom and Chambers specify their model as a stochastic process and use great care in deriving the properties of the error term in the discrete-time estimation equation. These properties, which reflect the structure of the underlying stochastic process, are then imposed in estimation. We now turn to a reformulation of the original HT models that takes its cue from the Bergstrom–Chambers model, but which involves two extensions related to the assumption that there exists stable relationships governing the rate of expenditure (or its derivative) continuously at every point in time. This is certainly not descriptive of actual expenditure, as most of the time expenditure rates are zero.3 Moreover, at points where expenditure rates are positive, stable relationships seem unlikely. Stability seems to be plausible integrated over intervals of time, but not at points in time. We will deal with the first problem by introducing a random 2 The use of error-correction models, which are based primarily upon the ideas of classical control theory, stems largely from Sargan (1964). Cf. also, Hendry and Mizon (1978). 3 Non-zero expenditure rates with an interval of time might plausibly be expected to follow a Poisson probability law.

13.2

Generalization of the Flow- and State-Adjustment Models

233

variable that is zero most of the time, but occasionally equal to one, corresponding to the points in time that expenditure rates are positive. The second problem is dealt with by assuming that behavioral parameters are random variables in the time continuum and then by defining parameter stability in terms of convergent integral equations. Our point of departure is the model given in expressions (13.5), (13.6), and (13.7): qˆ (τ ) = θ (τ ) [α (τ ) z (τ ) + β (τ ) s (θ )]

(13.5)

  dq (τ ) = γ qˆ (τ ) − q (τ ) dτ + ε (dτ )

(13.6)

  ds (τ ) = q (τ ) − δ (τ ) s (τ ) dτ ,

(13.7)

where: d: the differential operator τ : the time parameter, τ ≥ 0 qˆ (τ ): desired rate of expenditure flow at time τ q(τ ): actual rate of expenditure flow at time τ z(τ ): vector of exogenous variables at time τ s(τ ): state variable at time τ θ (τ ): random variable that is zero most of the time, but is occasionally equal to 1 α(τ ): vector of random parameters β(τ ), γ (τ ), δ(τ ): scalar random parameters ε(τ ): a white noise innovation.

13.2.2 Remarks 1. Equations (13.5) and (13.7) are essentially the structural equations for the HT state-adjustment model. Note, however, that the dependent variable in equation (13.5) is a desired rate of flow rather than the actual rate. 2. Equation (13.6) corresponds to equation (13.1) of the model of Bergstrom and Chambers. 3. That expenditure is zero at most points in the time continuum is reflected in the dichotomous random variable θ (τ ) in expression (13.6). 4. That the relationships governing expenditure are unstable through the time continuum is reflected in the behavioral parameters being assumed to be random variables. 5. The white noise innovation that drives the stochastic process through time is included in equation (13.6). All of the parameters in the model are assumed

234

Analysis of Time-Series Data

to be independently and identically distributed (iid) with respect to the time continuum. The stochastic properties of the parameters will be described below. The circumstances described by expressions (13.5), (13.6), and (13.7) are, to put it mildly, fluid, for nearly everything is random. Stability is achieved in the model through integration over time. For notation, let (as in Chapter 2): 1 qt = h



t

q(τ )[μ( . . . )]dτ ,

(13.8)

t−h

where μ(. . .) is a joint probability measure over all of the random variables in the system. In particular, it is assumed that μ(. . .) has properties such that: qˆ = θ (αt zt + βst ),

(13.9)

q(t) − q(t − h) = γt (ˆqt − qt ) + εt ,

(13.10)

s(t) − s(t − h) = qt − δt st ,

(13.11)

where: αt zt =

⎧ ⎨ 1 t ⎩h t−h

α(τ )dτ

⎫⎧ ⎬ ⎨ 1 t ⎭ ⎩h

⎫ ⎬ z(τ )dτ

⎭

,

(13.12)

t−h

and similarly for β t , etc., and where θ is a variable that is equal to 1 if there is at least one τ in h for which θ (τ ) is 1. We will assume that h is sufficiently long that θ is in fact 1, and that the integral equations defining the parameters α t , β t , γ t , and δ t and converge to constants α, β, γ , and δ. Hence, we can write: qˆ = αzt + βt st ,

(13.13)

q(t) − q(t − h) = γ (ˆqt − qt ) + εt ,

(13.14)

s(t) − s(t − h) = qt − δst .

(13.15)

Before turning to matters of estimation, we perhaps should ask what has been gained by beginning from a state of randomness. To begin with, we think randomness as a point of departure is realistic. No existing demand theory can explain consumption well over short periods of time, and, in keeping with the discussion in Chapter 2, in our opinion none will be possible until the organization and operation of the brain is better understood. Stable preferences in our view emerge only with aggregation over time, and the model that we are proposing (which will henceforth be referred to as the B-C model) provides a descriptive framework for how this can occur. How large h must be for the integral equations defining the parameters in expression (13.12)

13.2

Generalization of the Flow- and State-Adjustment Models

235

to converge to constants is clearly an empirical question. A quarter is probably long enough, but a month may not be. Indeed, this may account for the unsatisfactory results that are often obtained in consumption studies using monthly data. To obtain the estimating equation for the B-C model, we return to expressions (13.5), (13.6), and (13.7), which we rewrite, dropping the time index τ and [in view of expressions (13.13), (13.14), and (13.15)] treating the parameters as non-random, in derivative form as q˙ = γ (αz + βs − q) + ε,

(13.16)

s˙ = q − βs.

(13.17)

Differentiating (13.16) with respect to time and then substituting for s˙ from (13.18), we have q¨ = γ [α˙z + β(q − βs) − q˙ ] + ε˙ .

(13.18)

From (13.16), we now solve for s: s=

1 β



 q˙ − αz + q − ε , γ

(13.19)

so that     q˙ q¨ = γ α˙z + βq − δ − αz + q − ε − q˙ + ε˙ , γ

(13.20)

which, after simplification, can be written as q¨ = a1 q˙ + a2 q + a3 z˙ + a4 z + b1 ε˙ + b2 ε,

(13.21)

a1 = −(γ + δ),

(13.22)

a2 = γ (β − δ),

(13.23)

a3 = αγ ,

(13.24)

a4 = αγ δ,

(13.25)

b1 = 1,

(13.26)

b2 = γ δ.

(13.27)

where:

236

Analysis of Time-Series Data

Since expression (13.21) is a second-order differential equation in q, finite approximation requires two integrations over the time interval h, whose length will now be assumed to be equal to 1. Accordingly, for the first integration, we will have

∗ q˙ t = a1 ∗ qt + a2 qt + a3 ∗ zt + a4 zt + b1 ∗ εt + b2 εt ,

(13.28)

where (as in Chapter 2) ∗ q˙ t = q˙ (t + 1) − q˙ (t), ∗ qt = q(t + 1) − q(t), etc. The second integration over the interval h = 1 then yields

∗ qt+1 − ∗ qt = a1 (qt+1 − qt ) + a2 qt + a3 (zt+1 − zt ) + a4 zt . +b1 (εt+1 − εt ) + b2 εt

(13.29)

Using now the approximation employed in Section 2.4 for qt −qt−1 , namely,  1 ∗

qt + ∗ qt−1 , 2

(13.30)

∗ qt+1 = 2(qt+1 − qt ) − ∗ qt ,

(13.31)

qt − qt−1 = we will have for ∗ qt+1 ,

so that for the left-hand side of (13.29):

∗ qt+1 − ∗ qt = 2(qt+1 − qt ) − 2 ∗ qt .

(13.32)

Consequently, 2(qt+1 − qt ) − 2 ∗ qt = a1 (qt+1 − qt ) + a2 qt + a3 (zt+1 − zt ) +a4 zt + b1 (εt+1 − εt ) + b2 εt ,

(13.33)

and 2(qt+2 − qt+1 ) − 2 ∗ qt+1 = a1 (qt+2 − qt+1 ) + a2 qt+1 + a3 (zt+2 − zt+1 ) +a4 zt+1 + b1 (εt+2 − εt+1 ) + b2 εt+1 . (13.34) Adding (13.33) and (13.34), we then have 2(qt+2 − qt ) − 2( ∗ qt+1 + ∗ qt ) = a1 (qt+2 − qt ) + a2 (qt+1 + qt ) +a3 (zt+2 − zt ) + a4 (zt+1 + zt ) + b1 (εt+2 − εt ) + b2 (εt+1 + εt ),

(13.35)

so that (after once again using the approximations for the s), we obtain: 2(qt+2 − qt ) − 4(qt+1 − qt ) = a1 (qt+2 − qt ) + a2 (qt+1 + qt ) + a3 (zt+2 − zt ) +a4 (zt+1 + zt ) + b1 (εt+2 − εt ) + b2 (εt+1 + εt ), (13.36) which, after collecting terms and simplification, can be written as

13.2

Generalization of the Flow- and State-Adjustment Models

qt = A1 qt−1 + A2 qt−2 + A3 (zt − zt−2 ) + A4 (zt−1 + zt−2 ) , +B1 (εt − εt−2 ) + B2 (εt−1 + εt−2 )

237

(13.37)

where: 4 + a2 4 + γ (β − δ) = , 2. − a1 2+γ +δ

(13.38)

γ (β − δ) + γ + δ − 2 (a2 − a1 − 2) , = 2 − a1 2+γ +δ

(13.39)

A1 = A2 =

A3 = a3 = αγ ,

(13.40)

A4 =

a4 αγ δ , = 2 − a1 2+γ +δ

(13.41)

B1 =

b1 1 , = 2 − a1 2+γ +δ

(13.42)

B2 =

b2 γδ . = 2 − a1 2+γ +δ

(13.43)

The structural parameters, α, β, γ , and δ, are most easily obtained sequentially, beginning with δ, as follows: A4 , A3 a4 α= , δ δ=

(13.44) (13.45)

Next, let A12 =

A1 . A2

Then γ =

(2 + δ)(A12 − A1 (1 − A12 )) − 8A12 , A1( (1 − A12 ) − A12

(13.46)

A2 (2 + γ + δ) − γ − δ − 2 + δ. γ

(13.47)

β=

Expression (13.37) represents the estimation equation for the B-C model. For comparison, it is useful to rewrite the estimation equation for the state-adjustment model as given in expression (13.41) of Chapter 2 in the notation of this chapter (with h equal to 1): qt = A1 qt−1 + A2 (zt − zt−1 ) + A3 zt−1 + ut .

(13.48)

238

Analysis of Time-Series Data

The differences between the two estimating equations are as follows: 1. qt−2 is included as an additional predictor in the B-C equation. 2. zt −zt−1 is replaced by zt −zt−2 . 3. zt−1 is replaced by zt−1 +zt−2 . Inclusion of the second lag of the dependent variable in the B-C equation reflects the fact that the underlying continuous-time differential equation is second order, rather than first order as in the state-adjustment model. Flow-adjustment in the B-C model, on the other hand, is reflected in the presence of zt−1 +zt−2 .4

13.3 Alternative Estimation Forms for the State-Adjustment and B-C Models The challenge for estimation in state-adjusting models is how to deal with the state variable. Although the variable is observable in principle for durable goods, this is not the case for non-durables and services, and even for durables direct measurement in most instances is infeasible. For this reason, the procedure in CDUS (and in most applications since) has been to treat the state variable as unobservable (or latent) and to eliminate it from the estimating equations via the depreciation equation. However, depreciation equations, beginning from an initial condition, can also be used to generate state variables, which in turn means that they can be included directly in the estimating equations as predictors in either of the state-adjustment or B-C models. To illustrate the procedure for the B-C model, we return to expressions (13.11), (13.12), and (13.13), rewriting them as

∗ qt = γ (αzt + βst − qt ) + εt ,

(13.49)

∗ st = qt − δst .

(13.50)

With our usual approximation of qt − qt−1 =

1 ∗ ( qt + ∗ qt−1 ), 2

(13.51)

we then have: qt =

2−γ γα γβ εt + εt−1 qt−1 + (zt + zt−1 ) + (st + st−1 ) + . 2+γ 2+γ 2+γ 2+γ

(13.52)

The equation for generating the state variable is obtained from expression (13.13), again using the approximation for st − st−1 , as per expression (13.51), 4

Cf. expression (2.48).

13.3

Alternative Estimation Forms for the State-Adjustment and B-C Models

st =

2−δ qt + qt−1 + st−1 . 2+δ 2+δ

239

(13.53)

Substitution of this expression for st on the right-hand side of expression (13.52) then yields (after solving for qt): qt = A1 qt−1 + A2 (zt + zt−1 ) + A3 st−1 + ut ,

(13.54)

where:

ut =

A1 =

(2 − γ )(2 + δ) + γ δ , (2 + γ )(2 + δ) − γ δ

(13.55)

A2 =

γ α(2 + δ) , (2 + γ )(2 + δ) − γ δ

(13.56)

A3 =

4γβ , (2 + γ )(2 + δ) − γ δ

(13.57)

2+δ (εt + εt−1 ). (2 + γ )(2 + δ) − γ δ

(13.58)

Parallel procedures applied to the state-adjustment model described by expressions (13.23) and (13.24) in Chapter 2 (again written in the notation of this chapter): qt = αzt + βst + εt ,

(13.59)

∗ st = qt − δst ,

(13.60)

qt = A1 qt−1 + A2 zt + A3 st−1 + ut ,

(13.61)

yield the estimating equation:

where: A1 =

β , 2+δ−β

(13.62)

A2 =

α(2 + δ) , 2+δ−β

(13.63)

A3 =

β(2 − δ) , 2+δ−β

(13.64)

ut =

1 . 2+δ−β

(13.65)

Estimates of the state variables have been constructed from equation (13.51) from annual data that begin in 1929 for specific values of δ and an initial condition of 0. Data for 1929 through 1946 have been used to this end, which allows for

240

Analysis of Time-Series Data

the estimates to be independent of the initial condition (at least for plausible values of δ). The estimates that are obtained, however, are obviously conditional upon δ. The value of δ that is ultimately used is the one that yields the smallest residual sum of squares in the estimating equation.5 While this approach has some obvious drawbacks, it also has some strengths. One of these is that it allows for the inclusion of more than a single state variable in a model, while another is that it allows for a nonlinear version of the state-adjustment model to be formulated. The former will be illustrated in an appendix to Chapter 15 with a 1990 application (using quarterly data) to the joint demands for new and used automobiles. For the latter, we return to expressions (13.59) and (13.60) above, with the behavioral equation (ignoring the error term) written as follows: ln qt = α ln zt + βst .

(13.66)

Expression (13.60) remains as its stands. Substitution of equation (13.53) for st in equation (13.66) then yields the estimating equation: ln qt − A1 qt = A1 qt−1 + A2 ln zt + A3 st−1 ,

(13.67)

where: A1 =

β , 2+δ

A2 = α, A3 =

β(2 − δ) . 2+δ

(13.68) (13.69) (13.70)

Since A1 appears on the left-hand side, as well as the right, estimation proceeds through iteration on this coefficient.6

13.4 Nonlinear Estimation In the notation of this chapter, prices and income (or rather total expenditure) has been subsumed in the vector z. With total expenditure (x) and price (p) treated separately, the estimating equation for the B-C model becomes

5 In the present effort, this approach has only been pursued with annual data. However, results from an earlier application with quarterly data that was undertaken in 1990 will be reported later. 6 Note that all of the structural parameters, including δ, are identified in expressions (13.67), (13.68), and (13.69) [and also in expressions (13.61), (13.62), and (13.63)]. Since an estimate of δ is present in the construction of the state variable, one could regenerate the state variable using this “new” value of δ, thereby setting up iteration on δ as well as β.

13.4

Nonlinear Estimation

241

qt = A0 + A1 qt−1 + A2 qt−2 + A3 (qt − qt−2 ) + A4 (qt−1 + qt−2 ) , +A5 (pt − pt−2 ) + A6 (pt−1 + pt−2 ) + ut

(13.71)

where now: γμ , 2+γ +δ γ μδ A4 = , 2+γ +δ γλ , A5 = 2+γ +δ γ λδ A6 = . 2+γ +δ

A3 =

(13.72) (13.73) (13.74) (13.75)

As is clear from comparison of equations (13.72) and (13.73) with equations (13.74) and (13.75), the depreciation rate δ is over-identified, for one estimate can be obtained from the ratio of A4 to A3 , while a second independent estimate can be derived from the ratio of A6 to A5 . To obtain a single estimate of δ accordingly requires that equation (13.71) be estimated subject to the restriction A4 A5 − A3 A6 = 0.

(13.76)

A similar type of constraint applies in the estimation of the state-adjustment model, whose estimating equation with total expenditure and price included is qt = A0 + A1 qt−1 + A2 (xt − xt−1 ) + A3 xt−1 + A4 (pt − pt−1 ) + A5 pt−1 + ut , (13.77) μ(2 + δ) , 2+δ−β 2μδ A3 = , 2+δ−β λ(2 + δ) A4 = , 2+δ−β 2λδ A5 = . 2+δ−β

A2 =

(13.78) (13.79) (13.80) (13.81)

Again, one estimate of δ can be obtained from A2 to A3 , and a second from A4 to A5 . In this case, the identifying restriction is A2 A5 − A3 A4 = 0,

(13.82)

which is the same form as for the B-C model in (13.76). Since the identifying restrictions (13.76) and (13.82) are non-linear, both models have been estimated

242

Analysis of Time-Series Data

by nonlinear least squares using the nonlinear programming procedure (proc nlp) in SAS.7 The interesting thing about the nonlinearity of the identifying restrictions is that they provide for a multiplicity of solutions, a fact that was first pointed out to the authors by Professor Paul Mathieu of the University of Namur (Belgium) following publication of the 1966 edition of CDUS.8 In experimenting with “seed” values for the coefficients to be estimated in the SAS proc nlp procedure, it was found that the solutions are extremely sensitive to the value specified for the constant term (A0 ). Indeed, a different solution appears to be associated with each and every value that might be specified. In this circumstance, normal estimation considerations would dictate that the solution settled on to be that led to the smallest value for the residual sum of squares. Often, however, solutions associated with relatively small minimization values give estimates of the parameters that are economic nonsense (positive price or negative total-expenditure elasticities). In these situations, experimentation with the constant term almost invariably leads to parameter estimates that indeed make economic sense. As has been noted, this is the liberty that we have taken in estimation of the B-C and state-adjustment models.9

13.5 Models Estimated and Statistical Procedures The analysis begins (with both annual and quarterly data), for each PCE category, with the estimation of a standard set of models: state-adjustment, flow-adjustment (both linear and logarithmic), and the B-C model. With the annual data, the stateadjustment model is estimated in both its original form (i.e., with the state variable eliminated) and with the state-variable not-eliminated (per the discussion in Section 13.3 above), both linear and logarithmic. With annual data, models are estimated for both the entire period for which data are available in the National Income Accounts, usually 1929–2004, and 1947–2004. “Final” models are selected on the basis of

7

Proc nlp includes a variety of computing algorithms, Newton, Newton–Raphson, etc. for a number of minimization criteria. Newton–Raphson is the default, and is the one that we have used. 8 A different estimation procedure was used in estimating the state-adjustment model in both editions of CDUS, one that iterated on the La Grange multiplier associated with the non-linear restriction. This multiplier appears at four off-diagonal places in the covariance matrix of the independent variables, and the point of the procedure that we employed was to find the value of the multiplier that led to the constraint in expression (13.79) being satisfied. In applying the model to Belgian data, Professor Mathieu found, as with the present non-linear estimation procedure, that there is an extremely large number of solutions that do so. 9 In doing so, we of course are making ourselves vulnerable to the charge that “a tenacious investigator can always find what he wants” [to quote the late Henri Theil (from somewhere in his great Economic Forecasts and Policy)]. However, what it seems to us that what the multiplicity of solutions implies is that the data (in the context of the underlying model) in fact contain solutions that make economic sense, and that what we are doing is finding one (or more) of these. In any event, it is an interesting phenomenon and clearly one that warrants further investigation.

13.6

Data

243

overall statistical quality, as reflected in signs, t-ratios, and the presence or absence of autocorrelation in the residuals. Preference, to begin with, is always given to the B-C model. If this model, following nonlinear estimation, is not acceptable, then attention turns to the flow- and state-adjustment models. Once a final model is tentatively selected, the Durbin-h (dh) and Dickey–Fuller (d-f) statistics are calculated (from the OLS versions of the state-adjusting models) as tests of first-order autocorrelation and possible existence of a unit root.10 If significant autocorrelation is detected, an appropriate covariance matrix of the error term is constructed, the models re-estimated in a generalized least-squares format (non-linearly for the models with state adjustment). The resulting coefficient estimates are then transformed into estimates of the appropriate underlying structural parameters and elasticities. In CDUS, variables were retained in a model (whether flow- or state-adjustment) if the sign were correct and the t-ratio were at least 1 (in absolute value). In the present effort, we have, in keeping with our goal of obtaining whenever possible plausible estimates of price and total-expenditure elasticities, relaxed the statistical criteria to simply having the correct sign. However, this does not mean that we end up with a large set of elasticities (especially price) that are flimsy statistically, for it will be seen that this is not the case. Indeed, t-ratios for price as well as total expenditure are often well in excess of 2, with those for price not infrequently being larger than the ones for total expenditure. The distributions of final PCE models according to type is tabulated in Table 13.3. Perhaps the most interesting feature is that all of the flow-adjustment models are logarithmic. Table 13.3 Distribution of PCE models by type annual and quarterly

Annual Quarterly

Flow adjustment

Log flow adjustment

State adjustment

Log state adjustment

B-C

Log B-C

0 0

17 4

28 0

0 0

62 16

1 0

13.6 Data As noted, all of the PCE data employed in these chapters has been downloaded from the Bureau of Economic Analysis website of the U.S. Department of Commerce, 10 The purpose of calculating the Dickey–Fuller statistic is to check whether the time-series structure of the data series used in estimation is consistent with the steady-state behavior that is postulated in the underlying theoretical models. The theoretical models imply that, in steady-state (or long-run) equilibrium, there is a linear relationship among expenditures, income, and price. Consistency between the theoretical models and the real world (as represented in the data series used to measure expenditures, income, and prices) requires that the coefficients in the steady-state relationships form co-integrating vectors. See Engle and Granger (1987).

244

Analysis of Time-Series Data

specifically from the Personal Income and Outlays tables.11 All dependent variables are measured as real personal consumption expenditures per capita, while income is measured as total PCE per capita. The relative price for a PCE category is defined as the (chained) price index for the category divided by the price index for total PCE (base: 2000 = 100). All data transformation and variable construction have been performed in SAS. For the annual models, the data set used in estimation is for 1929–2004, with the WW2 years (1942–1945) excluded. For the quarterly models, the data set used in estimation is for 1947Q1–2005Q4. Thus, the estimating data sets consist of 72 and 236 observations, respectively. With regard to exogenous variables other than prices and total expenditure, all of the annual models initially include a dummy variable separating pre- and post-WW2, while the proportion of the population that is of age 65 or greater has been included in several of the medical-care categories. The only other exogenous variable considered is the trend that started in 1962 (the year of the first Surgeon General’s report on smoking and cancer) in the models for tobacco.

11

http://www.bea.gov/bea/dn/home/personalincome.htm.

Chapter 14

Quarterly PCE Models

In this chapter, we present the results from the models that have been applied to the quarterly PCE data for 1947Q1 through 2005Q4 for the PCE categories listed in Table 13.1. The format in both this and the next chapter will be to tabulate the results for each category of expenditure in a separate table. The information presented will include the coefficients and their associated t-ratios, R2 , Durbin–Watson (dw), Durbin-h (h), and Dickey–Fuller (d-f) statistics for the estimating equation, structural coefficients for the underlying theoretical model, together with the totalexpenditure and price elasticities. For the B-C models, the estimated coefficients and t-ratios are from the final nonlinear least-squares equations, while the dw, dh, and d-f statistics are all calculated using the residuals from the OLS equation.1 The overall statistical quality of the estimated models is summarized by the following system of iconic symbols:2  : Fair   : Good    : Excellent  : Leaves a lot to be desired. Models are estimated for 20 categories of expenditure with the quarterly data. Of these, three are for the aggregate expenditures for durables, nondurables, and services, while four (gasoline and oil, fuel oil and coal, electricity and natural gas, and other household operation) are for sub-components of gasoline, fuel oil and other 1

The Dickey–Fuller test for the existence of a unit root in the residuals proceeds by estimating the homogeneous regression:

ˆut = aˆut−1 + vt , and then testing the hypotheses:

H0 : a = 0 H1 : a < 0.

The d-f statistic reported is the t-ratio for the estimate of a. [See Dickey and Fuller (1979).] The extremes are:   : at most one t-ratio under 2 (in absolute value), no apparent autocorrelation problems in residuals; : several t-ratios less than 2, apparent autocorrelation problems.

2

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_14, 

245

246

14

Quarterly PCE Models

energy and household operation. Of the 20 models, the B-C model is used in 16 and the logarithmic flow adjustment in four. The elasticities in the B-C models are calculated at the mean values for the sample period. The coefficients in the logarithmic flow-adjustment models, of course, already represent elasticities. In the B-C model, three elasticities are calculated for both total expenditure and price as follows: flelastpce(price) denotes the elasticities associated with the flow-adjustment parameter γ , srelastpce(price) denotes the elasticities associated with state adjustment, while lrelastpce(price) represents the long-run (or steady-state) elasticities. In the logarithmic flow-adjustment model, srelastpce(price) denotes the elasticities associated with the flow-adjustment parameter θ , while lrelastpce(price) again refers to the long run. The elasticities that are of most economic interest are the ones referring to the long run, for the long-run PCE elasticity (in its relation to 1) governs whether expenditure in the category in question accounts for an increasing or decreasing share of households’ total expenditure, while the long-run price elasticity describes households’ ultimate responsiveness to changes in relative prices. Of the structural parameters in the B-C model, the ones of most direct interest are β,δ, and μ. The sign of β determines whether expenditures for a category are stockadjusting (β < 0) or habit-forming (β > 0). The parameter δ is the depreciation rate of the underlying state variable, and determines how rapidly state effects dissipate. Finally, β and δ in relation to μ determine the steady-state budget share for the expenditure category in question.3

14.1 Quarterly Models 14.1.1 Total Durable Goods Table 14.1 Total durable goods Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 Θ κ

−0.0625 0.8053 0.2229 −0.1351 0.2157 −0.3157

−0.13 19.61 4.07 −2.90 srelastpce lrelastpce

μ: sr μ: lr λ: sr λ: lr 0.2470 1.1450

0.2470 1.1450 −0.1496 −0.6937 h d-f

srelastprice lrelastprice R2 dw 0.04 −15.23

−0.1496 −0.6937 0.9967 2.00

#obs. = 235 Logarithmic flow-adjustment model, quarterly, 1947Q2–2005Q4, model quality:   !

A priori, one would think that expenditures for durable goods would be characterized more by state-adjustment than by flow-adjustment. However, of the various

3 As noted in Section 2.4, stability of the underlying dynamical system requires that δ be greater than β. In the B-C model, this is equivalent to the sum of A1 and A2 being less than 1.

14.1

Quarterly Models

247

models estimated, the logarithmic flow-adjustment is decidedly the best. The price terms in the B-C and state-adjustment models are of little importance, and the estimated coefficients for qt-1 are very close to 1. The long-run price elasticity is estimated to be about −0.70, while the long-run total-expenditure elasticity is seen to be greater than 1 at 1.15. The statistical quality of the model is excellent.

14.1.2 Motor Vehicles and Parts Table 14.2 Motor vehicles and parts Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ –

−12.0296 0.4772 0.4902 0.1429 0.00141 −6.0074 −0.0594 6.1010 −

−1.11 5.99 6.70 9.43 0.47 −7.41 −0.48 − −

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−1.9717 −0.0167 0.0049 0.1899 1.1587 0.0434 −7.9815 −48.6995 −1.8231

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw −

22.9102 3.7551 0.8577 −8.1159 −1.3303 −0.3038 0.9924 1.71 −

h = 4.36 d-f = −13.06 #obs. = 234. B-C Model, quarterly, 1947Q3–2005Q4, model quality: .

We see that expenditures for motor vehicles and parts is characterized by both state and flow adjustment in this, the first of many instances of the B-C model. The stock coefficient β is negative, as is to be expected (which indicates that expenditures are stock-adjusting, as opposed to habit-forming). The depreciation rate at about 0.005, however, is clearly implausibly small. Flow adjustment, on the other hand, is rapid, as indicated by the fairly substantial value for γ . The long-run total-expenditure elasticity is estimated to be about 0.85, while the long-run price elasticity is −0.30. The steady-state value for μ of 0.043 indicates that the steady-state share of motor vehicles and parts is between four and four and four and a half percent of households’ budgets. While all of these results are plausible, the statistical quality of the model is unfortunately not very good. Only the changes in total PCE and relative price are statistically important, and the large value of the Durbin-h suggests that some autocorrelation remains in the residuals.4

4 An attempt to correct for the latter in a non-linear generalized-least-squares framework was unfortunately not successful. More will be said about how expenditures for motor vehicles and parts might more properly be modeled in the next chapter.

248

14

Quarterly PCE Models

14.1.3 Furniture and Household Equipment Table 14.3 Furniture and household equipment Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 Θ κ

0.9543 0.8197 0.1192 −0.1920 0.1982 5.2932

3.07 21.78 3.77 −4.78

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.1310 0.6613 −0.2110 −1.0648 0.1310 0.6613

srelastprice lrelastprice R2 dw h d-f

−0.2110 −1.0648 0.9983 2.45 −4.20 −19.17

#obs. = 235 Logarithmic flow-adjustment model, quarterly, 1947Q3–2005Q4, model quality:    !

The dominance of flow adjustment in total durable-goods expenditure apparently has its roots in this sub-component, again in a situation in which one would ordinarily think that stock adjustment would prevail. However, both the state-adjusting models flirt with dynamic instability. The logarithmic flow-adjustment model presented here is excellent statistically, and yields a steady-state total-expenditure elasticity of 0.66 and a steady-state price elasticity that is slightly elastic.

14.1.4 Other Durable Goods Table 14.4 Other durable goods Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−5.2769 0.8444 0.1056 0.0207 0.00152 −0.4436 −0.0326 2.5642

−2.11 8.04 1.01 7.58 17.23 −2.16 −2.57

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−2.0579 −0.0082 0.0367 0.0372 0.0953 0.0304 −0.7960 −2.0411 −0.6513

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

5.2890 2.0628 1.6878 −1.0788 −0.4207 −0.3552 0.9990 1.95

h = 1.71 d-f = −14.85 #obs. = 234 B-C model, quarterly, 1947Q3–2005Q4 A1 + A2 = 0.95, model quality:  .

Occasionally , B-C models are encountered with the time-series data in which the sum of A1 and A2 is either greater than or else extremely close to one, but which otherwise are strong statistically. In these situations, our procedure has been to impose a constraint that the sum of these two coefficients be equal to 0.95.5 Included in 5 This simply adds another constraint (in this case, linear) to the SAS non-linear programming procedure used in estimation. The value of 0.95 is arbitrary.

14.1

Quarterly Models

249

this catchall category of durable-goods expenditures are expenditures for durable recreational equipment, which, being related to leisure-time activities, we naturally expect should have a substantial long-run total-expenditure elasticity. This is seen to be the case. The long-run price elasticity (−0.36), on the other hand, is much more modest in magnitude.

14.1.5 Total Non-durable Goods Table 14.5 Total nondurable goods Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

100.5352 0.6149 0.3351 0.1749 0.01096 −2.8351 −0.1494 4.2194

5.35 9.51 5.18 11.27 29.43 −0.98 −0.97

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

23.7322 −0.0057 0.0313 0.2591 1.0931 0.2192 −4.1200 −17.7213 −3.5534

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

3.2249 0.7643 0.6466 −0.3905 −0.0926 −0.0783 0.9995 1.82

h = 4.97 d-f = −13.89 #obs. = 234 B-C model, quarterly, 1947Q2–2005Q4 A1 + A2 = 0.95, model quality: 

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 Θ κ

0.4944 0.8743 0.0763 −0.0323 0.1341 3.9331

3.87 26.30 3.67 −2.08

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.0814 0.6072 −0.0345 −0.2571 0.0814 0.6072

srelastprice lrelastprice R2 dw h d-f

−0.0345 −0.2571 0.9991 1.82 1.57 −13.98

#obs. = 235 Logarithmic flow-adjustment model, quarterly, 1947Q3–2005Q4, model quality:    !

As with other durables, the B-C model for this category has been estimated under the constraint A1 + A2 = 0.95. The results indicate that nondurable expenditures are characterized by mild stock adjustment. Both steady-state elasticities are inelastic (price decidedly so), and the steady-state budget share is estimated to account for about 26% of the total expenditure. Unfortunately, the statistical quality of this model is not high, for in addition to imposition of a stability constraint, the t-ratios for the price terms are both less than 1, and a value of nearly 5 for the Durbin-h statistic indicates the presence of positive autocorrelation in the residuals.

250

14

Quarterly PCE Models

However, try as we might, a plausible model using non-linear generalized least squares (NLGLS) could not be found. In light of the statistical defects with the B-C model, we have also tabulated the logarithmic flow-adjustment model for this category. The results in this model appear to be much better, for the estimate of A1 is comfortably below 1, and the t-ratio for the price term is greater than 2. Also, the value of the Durbin-h statistic (1.57) is now in an acceptable range. On the other hand, the estimates of the steadystate elasticities with this model pretty much agree with those of the B-C model.

14.1.6 Food Table 14.6 Food Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

99.9991 0.8590 0.0954 0.0598 0.00332 −4.9998 −0.2779 2.5642

2.61 15.61 1.71 6.64 2.78 −3.38 −2.05

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

39.8631 −0.0134 0.0278 0.1082 0.2713 0.0729 −9.0413 −22.6208 −6.0949

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 Dw

1.4632 0.5832 0.3932 −0.8465 −0.3375 −0.2275 0.9978 1.80

h = 7.17 d-f = −13.77 #obs. = 234 B-C model, quarterly, 1947Q3–2005Q4, model quality:  .

With this category, it is to be kept in mind that it includes expenditures for both food consumed in the home and food consumed outside of the home, the dynamics of which, since expenditures for food consumed outside the home would seem to have a larger recreational component, we should probably expect to be different. The former, we might expect to be stock-adjusting and the latter habit-forming.6 What the results show is that, on balance, expenditures for food, though stock-adjusting (β < 0), have small state-adjustment effects (both β and δ are small) and (since γ is greater than 1) fairly rapid flow adjustment. Both steady-state elasticities are decidedly inelastic. The only negative aspect of the model for this category is the large Durbin-h coefficient in the residuals.7 However, estimation in an NLGLS framework does not lead to credible results.

6 Models for the two sub-components are estimated separately with the annual data in the next chapter. Interestingly, both appear subject to stock adjustment. 7 The large negative Dickey–Fuller statistic, on the other hand, clearly indicates the absence of a unit root.

14.1

Quarterly Models

251

14.1.7 Clothing and Shoes Table 14.7 Clothing and shoes Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 Θ κ

0.3364 0.9150 0.0340 −0.0608 0.0888 3.9566

2.00 41.67 4.16 −3.20

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.0563 0.6345 −0.0635 −0.7150 0.0563 0.6345

srelastprice lrelastprice R2 dw h d-f

−0.0635 −0.7150 0.9990 2.05 −0.42 −15.67

#obs. = 235 Logarithmic flow-adjustment model, quarterly, 1947Q2–2005Q4, model quality:    !

The results for this category show expenditures for clothing and shoes to be characterized by sluggish flow adjustment. The statistical quality of the model is, in general, excellent, as all t-ratios are at least 2, and the residuals (as indicated by the dw, h, and d-f statistics) are well-behaved. Both steady-state elasticities are estimated to be inelastic.

14.1.8 Gasoline, Fuel Oil, and Other Energy Table 14.8 Gasoline, fuel oil, and other energy Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

29.9974 0.6540 0.2853 0.0240 0.00328 −0.6078 −0.00832 3.8383

4.62 8.27 3.72 3.85 1.29 −5.19 −1.04

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

7.8152 −0.0393 0.0068 0.0365 0.1400 0.0054 −0.9255 −3.5525 −0.1340

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

3.4686 0.9037 0.1340 −0.5550 −0.1446 −0.0214 0.9903 1.98

h = 0.63 d-f = −15.05 #obs = 234 B-C model, quarterly, 1947Q3−2005Q4, model quality:  .

Taken a face value, the results for this category are not terribly encouraging for households looking for increases in price to reduce the demand (though of course on the part of other households!) for energy, for the long-run price elasticity is estimated to be close to 0. On the other hand, the almost equally small steady-state total-expenditure elasticity suggests that economic growth will not lead to rampant increases in the total household energy demand. The structural coefficients

252

14

Quarterly PCE Models

indicate that energy expenditures are subject to mild stock adjustment, but fairly rapid short-run flow adjustment. Except for the small t-ratios for lagged total PCE and price, the statistical quality of the model is good. 14.1.8.1 Gasoline and Oil Table 14.9 Gasoline and oil Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

10.0001 0.6622 0.3109 0.0141 0.00268 −0.4524 −0.00859 3.9106

4.48 8.27 4.04 2.88 1.03 −4.51 −0.87

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

2.5571 −0.0109 0.0095 0.0214 0.0835 0.0099 −0.6848 −2.6781 −0.3189

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

2.5934 0.6632 0.3087 −0.5413 −0.1384 −0.0645 0.9970 2.05

h = −1.42 d-f = −15.62 #obs. = 234 B-C model, quarterly, 1947Q3–2005Q4, model quality:  

The results for expenditures for gasoline and oil parallel those for its total household energy. Expenditures are characterized by mild stock adjustment, fairly rapid shortrun flow adjustment, and small steady-state total-expenditure and price elasticities. Except for small t-ratios for the estimates of A4 and A6, the statistical quality of the model is good. The long-run estimate of μ indicates a steady-state budget share for gasoline and oil of about 1%. More will be said about gasoline demand in the next chapter. 14.1.8.2 Fuel Oil and Coal Expenditures for fuel oil and coal are one of the few instances of an inferior category in the NIPA data (negative total-expenditure elasticities) and the only one with the quarterly data. Both steady-state elasticities are about −0.40. The category is indicated to be subject to habit formation, which in this case implies that reduction in expenditure, following an increase in total expenditure, occurs with a delay. Apart from one of the t-ratios being less than 1, the only serious defect in the model is an unduly large Durbin-h statistic.

14.1.9 Other Non-durable Goods The model for this category is a B-C model that shows expenditures to be subject to mild habit formation, with fairly rapid short-run flow adjustment. Both steady-state

14.1

Quarterly Models

253 Table 14.10 Fuel oil and coal

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

19.0000 0.8413 0.0911 −0.00203 −0.000505 −0.1999 −0.0498 2.4463

1.29 11.82 1.04 −3.91 −0.95 −1.45 −2.44

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

8.1757 0.0615 0.1269 −0.0038 −0.0093 −0.0075 −0.3735 −0.7367 −0.3189

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

−0.9787 −0.4001 −0.7891 −0.5098 −0.2084 −0.4111 0.9921 1.96

h = 4.78 d-f = −14.93 #obs. = 234 B-C model, quarterly, 1947Q3–2005Q4, model quality:  . Table 14.11 Other nondurable goods Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

49.9983 0.8211 0.1387 0.0366 0.00353 −5.0078 −0.4827 2.2926

2.73 7.37 1.19 6.35 3.80 −5.22 −2.85

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

15.8689 0.0085 0.0482 0.0645 0.1727 0.0783 −8.8172 −23.7410 −10.7010

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

2.1830 0.8107 0.9840 −2.0040 −0.7443 −0.9033 0.9995 1.99

h = 0.60 d-f =−15.19 #obs. = 234 B-C model, quarterly, 1947Q3–2005Q4, model quality:    !

elasticities are nearly unit elastic. The statistical quality is excellent, as all but one t-ratio are greater than 2 and the residuals are well-behaved.

14.1.10 Total Services Because of an extremely strong upward trend in their expenditures, services are a difficult category to model. Price has the wrong sign in the flow-adjustment models, and the coefficient on qt−1 is nearly 1 in the state-adjustment model. In the model above, services expenditures are shown to be characterized by mild habit formation, but with a depreciation rate that is small, which means that the habit effect is slow to dissipate. The steady-state total expenditure is nearly 1.5, while the steady-state price elasticity is close to −1. The results also imply that, should

254

14

Quarterly PCE Models

Table 14.12 Total services Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

84.9699 0.9658 0.0245 0.1278 0.0109 −14.0200 −1.1960 2.0851

114.25 10.82 0.26 5.39 0.91 −1.54 −0.55

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

40.7518 0.0298 0.0426 0.2526 0.5275 0.8412 −27.7548 −57.2886 −92.2886

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.9246 0.4434 1.4745 −0.5839 −0.2800 −0.9312 0.9999 1.90

h=− d-f = −14.61 #obs. = 234. B-C model, quarterly, 1947Q3–2005Q4, model quality: .

present trends continue, services would eventually come to account for upward of 85% of total expenditure.8 In keeping with the generally poor results with the flowand state-adjustment models for this category, the B-C model is not of particularly high quality. Three t-ratios are less than 1, and (as noted in the preceding footnote) the model is close to being dynamically unstable. However, on the positive side, the steady-state elasticities seem reasonable, and the residuals appear to be well-behaved.9

14.1.11 Housing The model for housing expenditures is a B-C model of high statistical quality. The results show fairly substantial habit formation to be present in housing expenditures, with sluggish short-run flow adjustment.10 Both steady-state elasticities are quite elastic, and the steady-state μ indicates a long-run budget proportion for housing of about 18%. The statistical quality of the estimated model is high, as (with one exception) t-ratios are in excess of 2, and the residuals are well-behaved.

8

Mechanically, this large value arises from the fact that the sum of A1 and A2 is 0.99, which is close to causing the underlying model to be dynamically unstable. Accordingly, it might seem reasonable to impose a constraint on the sum of A1 and A2 (as for Other Durables above). Unfortunately, doing this does not lead to a plausible non-linear estimation solution. 9 This is the first of a number of instances in this and the next chapter in which the Durbin-h statistic (because of the presence of a negative number under a square root) cannot be calculated. 10 Something to be kept in mind in assessing the results for this category is that the owneroccupied component of housing is defined in terms of space-rental value, rather than as investment in new housing stock. Thus, what in principle is being measured is the flow of services from the housing stock. Strong flow-adjustment/habit-formation effects are accordingly to be expected.

14.1

Quarterly Models

255 Table 14.13 Housing

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

74.9870 1.4493 −0.4652 0.00714 0.00290 −2.0271 −0.8223 0.5421

5.53 21.03 −6.84 2.65 4.49 −3.27 −5.14

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

138.3262 0.1626 0.2028 0.0362 0.0196 0.1823 −10.2642 −5.4642 −51.7696

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.1219 0.2248 1.1337 −0.2134 −0.3936 −1.9892 0.9999 2.14

h = −2.14 d-f = −16.34 #obs. = 234. B-C model, quarterly, 1947Q3–2005Q4, model quality:    !

14.1.12 Housing Operation Table 14.14 Housing operation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

19.9595 0.6547 0.3069 0.0214 0.00205 −1.5384 −0.1476 3.8873

1.81 8.35 3.99 3.52 2.07 −2.64 −1.31

A B Δ M μ: fl μ: lr Λ λ: fl λ: lr

5.1345 0.0186 0.0480 0.0326 0.1268 0.0534 −2.3485 −9.1308 −3.8418

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

2.2528 0.5793 0.9474 −1.1663 −0.3000 −0.4907 0.9986 2.11

h = −3.36 d-f = −16.12 #obs. = 234. B-C model, quarterly, 1947Q3−2005Q4, model quality:    !

Housing operation comprises of expenditures for electricity and natural gas and other household operation. An excellent B-C model has been obtained for the category. The results show weak habit formation, rapid short-run flow adjustment, and steady-state elasticities that are both close to being elastic. All t-ratios but for the intercept and lagged price are in excess of 2, and the residuals are well-behaved. 14.1.12.1 Electricity and Natural Gas The quarterly series for the components of housing operation began in 1970, rather than 1947, hence the data sets used in estimation comprise only 186 quarters.

256

14

Quarterly PCE Models

Table 14.15 Electricity and natural gas Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

35.1049 0.6018 0.3206 0.00144 0.00205 −0.2410 −0.1775 3.8761

4.11 7.67 4.10 0.49 2.31 −0.63 −1.46

A B Δ M μ: fl μ: lr Λ λ: fl λ: lr

9.0567 0.3057 0.3682 0.0023 0.0090 0.0137 −0.3883 −1.5051 −2.2883

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.3530 0.0911 0.5367 −0.3663 −0.0945 −0.5569 0.9770 2.17

h = −4.44 d-f = −14.76 #obs. = 186. B-C Model, quarterly, 1970Q3–2005Q4, model quality: .

This B-C model for expenditures for electricity and natural gas shows strong habit formation and rapid short-run flow adjustment. The estimated steady-state elasticities are both inelastic, each being about 0.55 (in absolute value). Statistically, the model is not of high quality, as only one of the four total expenditure and price coefficients has a t-ratio greater than 2. The residuals, however, are well-behaved. More will be said about electricity and natural gas demands in the next chapter. 14.1.12.2 Other Household Operation Table 14.16 Other household operation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

7.9713 1.2481 −0.2709 0.00769 0.000782 −0.7095 −0.0722 1.1250

0.39 12.81 −2.35 1.20 0.87 −1.81 −0.41

α β δ μ μ: fl μ: lr λ γ : fl λ: lr

7.0857 0.0186 0.0509 0.0217 0.0244 0.0342 −2.0028 −2.2331 −3.1581

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.7763 0.6900 1.0881 −0.4919 −0.4373 −0.6895 0.9996 2.00

h=− d-f = −13.58 #obs. = 186. B-C Model, quarterly, 1970Q3–2005Q4, model quality: .

The B-C model for this category shows expenditures for other household operation, which, among other things, includes expenditures for domestic services and cleaning supplies, to be subject to mild habit formation that wears off slowly (β = 0.02,

14.1

Quarterly Models

257

δ = 0.05). In the steady state, demand is estimated to be slightly elastic with respect to total expenditure, but inelastic with respect to price. As with electricity and natural gas, the statistical quality of the estimated model is not high, as none of the t-ratios for total expenditure and price reach 2.

14.1.13 Transportation Table 14.17 Transportation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

19.9561 1.3502 −0.4075 0.00637 0.00252 −0.6008 −0.2372 0.7036

1.40 15.26 −4.34 1.77 2.42 −1.89 −1.05

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

28.3642 0.0792 0.1974 0.0263 0.0185 0.0439 −2.4774 −1.7430 −4.1389

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 Dw

0.4344 0.6175 1.0316 −0.2386 −0.3391 −0.5665 0.9988 2.00

h=− d-f = −15.30 #obs. = 234. B-C Model, quarterly, 1947Q3–2005Q4, model quality:  .

Included in this category are taxis and local mass transit, intercity rail, intercity bus, and airline travel, of which we should expect intercity rail travel (and possibly intercity bus as well) to be an inferior good and airline travel to be a luxury. As it turns out, the results (with the B-C model) show expenditures in the long run to be slightly elastic with respect to total expenditure and inelastic with respect to price. Expenditures for transportation services are characterized by habit formation (β = 0.08) and are estimated to account for about four and half percent of the total expenditure in the long run. The overall statistical quality of the model is good.

14.1.14 Medical Care Two B-C models have been tabulated for this category, one with the sum of A1 and A2 constrained to be equal to 0.95 and the other unconstrained. Both models show expenditures for medical care to be characterized by strong habit formation. In the unconstrained model, A1 + A2 = 0.99, which implies a steady value of 56% for the budget share for medical care, compares with about 20% in 2004. Moreover, even with the constraint on A1 and A2, the steady-state budget share is estimated to be 44%. However, whether the steady-state budget share is 56 or 44%, the important point is that, given present trends, the budget share for medical care is increasing

258

14

Quarterly PCE Models

Table 14.18 Medical care Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

7.0000 1.2690 −0.2809 0.0134 0.00668 −2.0000 −1.0001 0.8873

0.59 16.22 −3.50 1.87 4.29 −1.32 −2.52

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

7.8887 0.2289 0.2500 0.0472 0.0419 0.5613 −7.0714 −6.2748 −84.0208

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.2848 0.3209 3.8133 −0.1986 −0.2239 −2.6599 0.9999 1.96

h = −0.44 d-f = −15.14 #obs. = 234. B-C Model, quarterly, 1947Q3–2005Q4, model quality:  .

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

24.8361 1.0749 −0.1249 0.0128 0.0221 −1.9007 −3.2863 0.7730

2.71 11.87 −1.38 1.57 11.36 −1.53 −6.21

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

32.1303 0.7730 0.8637 0.0601 0.0465 0.4411 −8.9423 −6.9123 −65.6241

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.3156 0.4083 2.9966 −0.2188 −0.2831 −2.0775 0.9999 1.96

h = −0.44 d-f = −15.14 #obs. = 234. B-C Model, quarterly, 1947Q3–2005Q4 A1 + A2 = 0.95, model quality:  .

rapidly, as reflected in large estimates of the long-run total-expenditure elasticity. Interestingly, estimates of the long-run price elasticity are large as well. The statistical quality of both models is good. All t-ratios are greater than 1, and the residuals are problem free.

14.1.15 Recreation The B-C model for this category shows recreations expenditures to be subject to habit formation and fairly rapid short-run flow adjustment. The statistical quality of the estimated model is excellent. All t-ratios (except for the one for the estimate of A2) are greater than 2, and the residuals appear well-behaved. The long-run

14.1

Quarterly Models

259 Table 14.19 Recreation

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

7.9975 0.9984 −0.0178 0.00673 0.00130 −0.8376 −0.1619 1.8712

2.02 12.53 −0.23 3.70 3.80 −3.34 −3.29

α β δ μ μ: fl μ: lr λ γ : fl λ: lr

4.2740 0.0761 0.0966 0.0143 0.0267 0.0672 −1.7760 −3.3233 −8.3682

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.8559 0.4574 2.1551 −0.7012 −0.3747 −1.7656 0.9997 1.96

h=− d-f = −15.01 #obs = 234 B-C Model, quarterly, 1947Q3–2005Q4, model quality:    !

total-expenditure elasticity is large, which, given that recreation is tied to higher wants, is obviously to be expected. The long-run price elasticity is also highly elastic.

14.1.16 Other Services Table 14.20 Other services Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

49.9969 1.1074 −0.1815 0.0359 0.0126 −3.0241 −1.0579 1.3202

1.12 10.40 −1.61 2.00 2.54 −3.28 −1.16

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

37.8721 0.0768 0.1749 0.0952 0.1256 0.1697 −8.0061 −10.5693 −14.2770

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.9694 0.7343 1.3094 −0.4695 −0.3556 −0.6342 0.9994 1.92

h=− d-f = −14.78 #obs. = 234. B-C Model, quarterly, 1947Q3–2005Q4, model quality:    !

Expenditures for miscellaneous services, which, among other services, includes personal-business services and international travel and remittances, is seen to be subject to habit formation, and moderately rapid short-run flow adjustment. The estimates of the steady-state elasticities show the one for total expenditure to be elastic, but inelastic for price. The overall statistical quality of the estimated model is excellent.

260

14

Quarterly PCE Models

14.2 Summary of Steady-State Elasticities and Budget Shares The steady-state elasticities and budget shares for the quarterly models, together with the observed budget shares for 2004, are pulled together in one place in Table 14.1. For the aggregates of durables, nondurables, and services, we see that both durables and services have total-expenditure elasticities greater than 1. All three aggregate price elasticities, on the other hand, are less than 1 (in absolute value), and thus in the inelastic range. Of the durable components, the totalexpenditure elasticities range from 0.66 for furniture and household equipment to about 1.70 for other nondurables, while the price elasticities range from −0.30 for motor vehicles and parts to −1.06 for furniture and household equipment. Table 14.21 Steady-state price elasticities and budget shares. PCE data quarterly, 1947–2004 Elasticity Category

Tot. exp.

Price

Durable goods Motor vehicles and parts Furniture and household equipment Other durable goods Non-durable goods Food Clothing and shoes Gasoline, fuel oil, and other energy Gasoline and oil Fuel oil and coal Other nondurable goods Services Housing Household operation Electricity and gas Other household operation Transportation Medical care Recreation Other services

1.1492 0.8577 0.6615 1.6878 0.6466 0.3922 0.6375 0.1340 0.3088 −0.7891 0.9840 1.4745 1.1337 0.9474 0.5367 1.0881 1.0816 2.9966 2.1552 1.3094

−0.6934 −0.3038 −1.0648 −0.3552 −0.0783 −0.2275 −0.7150 −0.0214 −0.0645 −0.4111 −0.9033 −0.9312 −1.9892 −0.4907 −0.5569 −0.6855 −0.5665 −2.0775 −1.7656 −0.6362

∗

Steadystate

2004

Budget share∗

Budget share

–

0.1174 0.0511 0.0427 0.0236 0.2932 0.1393 0.0394 0.0355 0.0328 0.0027 0.0789 0.5894 0.1465 0.0551 0.0231 0.0320 0.0367 0.1726 0.0205 0.1378

0.0434 – 0.0304 0.2192 0.0725 – 0.0054 0.0099 −0.0075 0.0783 0.8412 0.1823 0.0137 0.0137 0.0342 0.0044 0.4411 0.0672 0.1697

“–” indicates that a logarithmic flow-adjustment model was estimated for the category.

For the nondurable components, total-expenditure elasticities range from about −0.80 for fuel oil and coal to nearly 1 for other nondurables, while price elasticities range from under −0.10 for gasoline and oil to −0.90 for other nondurables. Finally, for the components of services, except for electricity and natural gas (which is 0.54), total-expenditure elasticities are all greater than 1, ranging from 0.54 for electricity and natural gas to 3 for medical care. Price elasticities, on the other hand, range

14.3

A Dynamic Linear Expenditure System

261

from about −0.55 for electricity and natural gas and transportation to around −2 for housing and recreation. Although the steady-state budget shares (which are organic parameters for the B-C models) might appear as rather arcane theoretical quantities, they in fact are of thorough-going practical interest, for, given current trends, they represent “magnet” points toward which budget shares are being drawn. Note that, for categories of expenditure that are subject to stock adjustment (Motor Vehicles and Parts, Gasoline and Oil, etc.) the steady-state budget shares are below currently observed shares, while the opposite is true of categories that are characterized by habit formation. Although the steady-state share for medical care (and largely because of medical care, for services as a whole) is almost certainly implausibly large, it nevertheless probably accurately reflects present trends. More will be said about all this in the next chapter, and again in Chapters 16 and 17.

14.3 A Dynamic Linear Expenditure System We now turn to a dynamic linear expenditure system (LES), which, to recall, was applied (along with several other theoretically plausible equation systems) in Chapter 7 to a set of six exhaustive categories of expenditure from the BLS consumer expenditure surveys. In this section, we apply the dynamic version of the LES, which was first analyzed by Phlips (1972),11 to the 16 quarterly PCE categories of expenditure listed (in the first indentation) in Table 14.3. To derive the model that is estimated, we begin with the first-order conditions for the Stone–Geary–Samuelson utility function from Chapter 7, written in continuous time: βi = λ(t)pi (t), qi (t) − γi

(14.1)

where γ denotes the minimum required quantity and λ denotes the marginal utility of income. The other variables have their usual interpretation. Solving for qi (t), we then have: qi (t) = γi + βi /λ(t)pi (t).

(14.2)

In this case, λ(t) will be given by: λ(t) =

 i

11

See also Taylor and Weiserbs (1972).

xi (t) −

βi

j pi (t)γi (t)

.

(14.3)

262

14

Quarterly PCE Models

The minimum required quantity is assumed to be a function of a state variable, s(t), as follows: γi (t) = θi + αsi (t),

(14.4)

s˙i (t) = qi (t) − δsi (t).

(14.5)

where (as before):

Combining equations (14.2) and (14.4), we then have qi (t) = θi + αsi (t) + βi /λ(t)pi (t).

(14.6)

Proceeding to the finite approximation of expressions (14.5) and (14.6), we will have12 qit = θi + αsit + βi /(λpi )t ,

(14.7)

∗ sit = qit − δsit .

(14.8)

With our usual replacement of sit − sit−1 by 1/2( ∗ sit + ∗ sit−1 ), we then have after rearrangement: −1 qit = Ki0 + Ki1 qit−s + Ki2 (λpi )−1 t + Ki3 (λpi )t−1 , i = 1, . . . n; j = 1, . . . ,n, (14.9)

where: Ki0 = −

2θi δi , 2 − αi − δi

(14.10)

Ki1 =

2 + αi + δi , 2 − αi − δi

(14.11)

Ki2 =

βi (2 − δi ) , 2 − αi − δi

(14.12)

Ki3 = −

βi (2 + δi ) , 2 − αi − δi

(14.13)

and δi =

12

2(Ki2 + Ki3 ) , Ki2 − K13

(14.14)

In going from equations (14.6) to (14.7), a second approximation has been invoked involving the term 1/λ(t)pi (t) , namely, that the integral of a reciprocal is equal to the reciprocal of the integral. In this case, tractability unfortunately requires this additional approximation.

14.3

A Dynamic Linear Expenditure System

Ki2 − Ki3 , 1 + Ki1

(14.15)

2(Ki2 + Ki3 ) 2(1 − Ki1 ) − , Ki2 − Ki3 1 + Ki1

(14.16)

Ki0 (Ki2 − Ki3 ) . (1 + Ki1 )(Ki2 + Ki3 )

(14.17)

βi = αi =

263

θi =

The estimating equations for the dynamic state-adjustment LES are given in expression (14.9). Note that income does not appear explicitly in these equations, but rather indirectly through the marginal utility of income. Estimation proceeds through iteration on the vector λ, the goal being to secure an estimate of λ, that yields a set of coefficients, Kˆ ij (i = 1, . . ., m, j = 0, . . ., 3), such that the budget constraint, 

pit qˆ it = xt ,

(14.18)

i

where: −1 ˆ qˆ it = Kˆ i0 + Kˆ i1 qit−s + Kˆ i2 (λpi )−1 t + Ki3 (λpi )t−1 ,

(14.19)

is satisfied for each quarter.13 The results from applying this dynamic state-adjustment model, after 10 iterations on λ, to the 13 quarterly PCE categories are given in Table 14.2. In addition to the coefficients in the estimating equations and structural coefficients, the parameters tabulated include the long-run minimum required quantity (lrmrq), the steady-state β (lrbeta, which represents the steady-state budget share), and the steady-state elasticities for total expenditure and price (lrelastpce, lrelastprice). The R2 , dw, h, and d−f statistics are given as well. In general, the results leave quite a bit to be desired, for three of the estimated coefficients on the lagged dependent variable are greater than 1 (other nondurables, transportation, and other services), and the coefficients for four others (furniture and equipment, housing, medical care, and recreation) are close to 1, all of which are associated with strong habit formation. A value of greater K1 than 1 results inα being greater than δ, which implies that the category in question is addicting, and in turn leads to nonsense values for the steady-state elasticities, as is indeed

(1)

The algorithm for estimation begins with λt equal to 1/xi . The left-hand side of expression (14.19) is then calculated (denoted by xˆ t ), from which a new value of λt , obtained as:

13

(2)

(1)

(14.20)

κt = xt /ˆxt .

(14.21)

λt where κ t is defined as

= κt λ t ,

264

14

Quarterly PCE Models

Table 14.22 Dynamic LES model. 16 PCE categories quarterly, 1947–2005

K0 K1 K2 K3 A B Δ Θ Lrmrq Lrbeta Lrelastpce Lrelastprice R2 Dw H d-f #obs.

Motor vehicles and parts

Furniture and equipment

Coefficient

t-ratio

Coefficients

t-ratio

Coefficients

t-ratio

4.7285 0.8784 127.7339 −122.0688 −0.0841 144.8010 0.0454 5.3603 1.8785 0.0218 0.3851 −0.1424 0.9878 1.86 1.35 −14.17 235

0.77 21.99 7.44 −6.86 − − − − − − − − − − − − −

−1.3482 0.9965 39.1236 −38.8602 0.00327 39.2625 0.00679 1.3530 2.6115 0.0326 0.6915 −0.2557 0.9996 1.67 2.81 −12.90 235

−1.02 36.32 6.51 −5.92 − − − − − − − − − − − − −

−5.5480 0.8602 14.8391 −10.8862 0.1570 17.1543 0.3073 −6.4136 −13.1082 0.0158 0.6595 −0.2439 0.9989 1.73 2.39 −13.32 235

−4.63 26.42 4.86 −3.20 − − − − − − − − − − − − −

Food

K0 K1 K2 K3 A B Δ Θ lrmrq lrbeta lrelastpce lrelastprice R2 Dw H d-f #obs

Other durables

Clothing and shoes

Gasoline and other energy

Coefficient

t-ratio

Coefficients

t-ratio

Coefficients

t-ratio

107.9160 0.9310 56.8088 −51.0654 0.0350 60.9437 0.1065 115.7709 172.3933 0.0390 0.2368 −0.0876 0.9976 1.60 3.00 −12.60 235

3.96 57.38 6.61 −6.21 − − − − − − − − − − − − −

7.6786 0.9558 37.7529 −36.1567 −0.0098 39.4778 0.0432 8.0294 7.6769 0.0162 0.3070 −0.1135 0.9994 1.90 0.7 −14.52 235

2.55 67.00 9.53 −9.08 − − − − − − − − − − 7 − −

13.3882 0.9649 5.2905 −4.8253 0.0563 5.4810 0.0920 14.9063 38.4171 0.00607 0.1826 −0.0675 0.9891 2.19 −1.51 −16.82 235

3.11 91.49 7.19 −6.55 − − − − − − − − − − − − −

14.3

A Dynamic Linear Expenditure System

K0 K1 K2 K3 A B Δ Θ lrmrq lrbeta lrelastpce lrelastprice R2 dw H d-f #obs

265

Other nondurables

Housing

Coefficient

t-ratio

Coefficient

t-ratio

Coefficient

t-ratio

−1.0764 1.0001 12.1266 −11.6978 0.0279 10.8321 0.2955 −1.0763 553.4772 −2.6814 −34.8576 12.8894 0.9993 1.68 2.41 −13.09 235

−0.47 68.16 2.69 −2.48 − − − − − − − − − − − − −

−1.3482 0.9965 39.1236 −38.8602 0.00327 39.2625 0.00679 1.3530 2.6115 0.0326 0.6915 −0.2557 0.9996 0.90 8.46 −8.20 235

−1.02 36.32 6.51 −5.92 − − − − − − − − − − − − −

−5.5480 0.8602 14.8391 −10.8862 0.1570 17.1543 0.3073 −6.4136 −13.1082 0.0158 0.6595 −0.2439 0.9989 2.49 −3.86 −19.58 235

−4.63 26.42 4.86 −2.61 − − − − − − − − − − − − −

Transportation

Medical care

Housing operation

Recreation

Other services

Coefficient t-ratio Coefficient t-ratio Coefficient t-ratio Coefficient t-ratio K0 107.9160 K1 0.9310 K2 56.8088 K3 −51.0654 A 0.0350 B 60.9437 Δ 0.1065 Θ 115.7709 lrmrq 172.3933 lrbeta 0.0390 lrelastpce 0.2368 lrelastprice −0.0876 R2 0.9976 dw 1.09 H 7.12 d-f −9.36 #obs. 235

3.96 −77.179 −4.36 −7.1029 57.38 0.9964 523.26 0.9861 6.61 37.7529 3.51 6.5108 −6.21 −36.1567 −2.03 −5.3585 − 0.5057 − 0.1801 − 12.9602 − 6.6025 − 0.5063 − 0.1942 − −77.4545 − −7.2029 − 10995 − −99.7033 − 0.7911 − 0.0392 − 5.7047 − 1.1552 − −2.1094 − −0.4272 − 0.9998 − 0.9997 − 1.30 − 1.94 − 5.33 − 0.33 − −10.61 − −14.94 − 235 − 235

−3.78 −3.8785 −0.20 161.18 1.0040 67.77 3.67 29.2149 4.36 −2.76 −29.0757 −4.44 − 0.00875 − − 29.0989 − − 0.00477 − − −3.8631 − − 4.6434 − − −0.0150 − − −0.1213 − − 0.0449 − − 0.9993 − − 1.53 − − 3.64 − − −12.07 − − 235 −

266

14

Quarterly PCE Models

the case for other nondurables and other services.14 These two categories apart, we see that motor vehicles and parts and clothing and shoes are indicated to be subject to stock adjustment (α < 0), while the other categories are all indicated to be habit-forming (α > 0). The residuals for eight of the categories appear plagued by positive autocorrelation by the Durbin-h statistic, while the null hypothesis of a unit root cannot be rejected by the d-f statistic in two categories (Housing and Transportation).15 The steady-state elasticities from the model, together with their counterparts from the B-C and logarithmic flow-adjustment models from the preceding section, are tabulated in Table 14.3. As can be seen, the only categories for which there is reasonably close agreement in elasticities between the logarithmic flow-adjustment and B-C models and the LES model are for Food, Gasoline and Other Energy Goods, and Medical Care. Table 14.23 Comparison of steady-state elasticities Log flow adjustment

B-C model

PCE category

Tot. exp.

Price

Tot. exp.

Price

Tot. exp.

Price

Motor vehicles and parts Furniture and household equipment Other durable goods Food Clothing and shoes Gasoline and other energy goods Other non-durable goods Housing Household operation Transportation Medical care Recreation Other services

− 0.6615

− −1.0648

0.8577 −

−0.6934 −

0.3851 0.6915

−0.1424 −0.2557

− − 0.6345 −

− − −0.7150 −

1.6878 0.3932 − 0.1340

−0.3552 −0.2275 − −0.0214

0.6595 0.2368 0.3070 0.1826

−0.2439 −0.0876 −0.1135 −0.0675

0.9840 1.1337 0.9474 1.0316 3.8133 2.1552 1.3094

−0.9033 −1.9832 −0.4907 −0.5665 −2.6591 −1.7656 −0.6342

− 0.5568 0.5057 − 5.7047 1.1552 −

− −0.2059 −0.1870 − −2.1094 −0.4272 −

− − − − − − −

− − − − − − −

Dynamic LES

Log flow adjustment, B-C, and LES models 13 PCE categories quarterly, 1947–2005.

For the categories withKl close to 1, δ, though greater than α, is not much greater, and leads to steady-state elasticities that may appear unduly large. 15 In general, this model has not seemed sufficiently promising to us to pursue estimation by generalized least squares. 14

Chapter 15

Annual PCE Models

We now turn to the results that have been obtained with the annual personal consumption expenditure categories from the National Income Accounts. Models are estimated for all but five of the 112 categories of expenditure listed in Table 13.2. Categories for which models are not estimated are food furnished to employees (1.3), food produced and consumed on farms (1.4), standard clothing issued to military personnel (2.3), and the two balancing items in category 12.0 (12.3 and 12.4). The information tabulated is the same as for the quarterly models in Chapter 14. The presentation is organized in terms of the 12 major categories of expenditures (food, clothing and shoes, housing, etc.) listed at the far left in Table 13.2. With the exception of the five categories listed above, models are estimated for every level and sub-level of aggregation in the table. The most amazing thing about the results of this chapter is the fact that, with few exceptions, plausible price elasticities have been obtained for every category of expenditure! This is in sharp contrast to the experience in both editions of CDUS, where prices appeared in few more than half of the models estimated. For purely statistical reasons reflecting the variations injected into relative prices by deregulation, the energy crises of the 1970s, and the electronic revolutions of the 1980s and 1990s, we clearly expected to find prices to be of much greater importance this time around, but not to the extent that has turned out to be the case. Truth be told, some of this is obviously attributable to the relaxed statistical criteria that we are presently employing, together with the existence of multiple solutions in the nonlinear estimation algorithm. The conclusion, nevertheless, is that U. S. household consumption expenditures are strongly affected by variation in relative prices. That said, we now turn to results.

15.1 Food, Tobacco and Alcohol The ordinary nonlinear least squares state-adjustment model for this category has a small Durbin–Watson coefficient and a large Durbin-h statistic, thus indicating the

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_15, 

267

268

15

Annual PCE Models

Table 15.1 Food, tobacco, and alcohol Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

300.0098 0.8690 0.1328 0.00741 −5.9104 −0.3298

1.29 8.97 7.25 1.01 −2.18 −0.79

α β δ μ μ: lr λ

19532 −0.0827 0.0574 0.1381 0.0566 −6.1481

λ: r srelastpce lrelastpce srelastprice lrelastprice R2

−2.5186 0.6565 0.2689 −0.1949 −0.0799 0.9998

dw = 2.01 h = −0.04 d–f = −7.59 #obs. = 71 State-Adjustment Model, Annual, 1947–2004, GLS, Model Quality:  .

need for a nonlinear GLS model.1 Apart from the negative sign on β, which implies stock adjustment, the results generally make sense. Both the steady-state elasticities are small, which is what is obviously expected in a category dominated by food expenditures2 .

15.1.1 Food Purchased for Off-Premise Consumption The model for food consumed at home is a B-C model, which allows for both flow and stock adjustment. Flow adjustment in this case is seen to be rapid, while stock adjustment is weaker than in the state-adjustment model for the aggregate category. 1 The error term corresponding to the estimation equation for the state-adjustment model [from the argument leading to equation (2.30)] has the form:

vt =

2+δ 2−δ ut + ut−1 , 2−δ−β 2−δ−β

(15.1)

where ut is assumed to be well-behaved. The non-linear GLS estimator is then obtained as a weighted non-linear least squares equation:

where:

ˆ β)y ˆ = W(δ, ˆ β)X ˆ + W(δ, ˆ β)v, ˆ W(δ,

(15.2)

ˆ β) ˆ = [( ˆ δ, ˆ β)] ˆ −1 , ˆ β) ˆ  W(δ, W(δ,

(15.3)

ˆ δ, ˆ β) ˆ is the covariance matrix of v constructed from equation (15.1) and δˆ βˆ are the (ordinary) ( non-linear least-squares estimates ofδ and β. All matrix constructions are done in SAS. 2 The implausibly large α for this category is obviously not to be taken seriously. As has been noted, the non-linear constraints on the depreciation rate δ leads to solutions in the estimation algorithm that are sensitive to the starting value for A0 . Our procedure, accordingly, has been, in effect, to search over this starting value until an economically sensible solution is obtained (or until it becomes clear that one is probably not attainable). The αs are thus essentially arbitrary and consequently should be ignored.

15.1

Food, Tobacco and Alcohol

269

Table 15.2 Food purchased for off-premise consumption Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

100.0033 1.1249 −0.1941 0.0587 0.00164 −4.7604 −0.1325 1.4358

1.71 12.63 −2.33 5.85 1.17 −3.72 −1.05

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

69.6513 −0.0692 0.0139 0.1411 0.2026 0.0236 −11.4377 −16.4219 −1.9140

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.6845 1.1732 0.1963 −12.1477 −0.7136 −0.1194 0.9915 1.55

h = 2.43 d–f = −6.03 #obs. = 58 B-C Model, Annual, 1947–2004, Model Quality: .

In keeping with Engel’s law, the steady-state total-expenditure elasticity is small, as is also the steady-state elasticity for price. Because of its richer dynamical structure, positive autocorrelation in the residuals is generally not a problem with the B-C model. Nevertheless, the Durbin-h statistic of 2.43 indicates some may be present in this case. However, the results from applying non-linear generalized least squares are implausible.

15.1.2 Purchased Food and Beverages A much higher quality B-C model is obtained for food consumed outside the home than for food consumed at home. However, once again flow adjustment is rapid Table 15.3 Purchased food and beverages Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

94.8869 0.9381 0.0975 0.0257 0.00829 −2.6980 −0.8701 1.7687

2.00 10.44 −1.04 3.14 3.04 −2.09 −1.23

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

53.6474 −0.0159 0.1612 0.0571 0.0101 0.0520 −5.4571 −10.6031 −5.9948

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.6363 0.9231 0.8421 −1.0508 −0.5941 −0.5408 0.9950 1.79

h=− d–f = −7.19 #obs = 70 B-C Model, Annual, 1931–2004 (1942–1945 excluded), Model Quality:    !

270

15

Annual PCE Models

(as γ is greater than 1), and β is negative, again indicating stock adjustment, though the stock effect is milder than for home food consumption. Not surprisingly, the steady-state elasticities are considerably larger than for food consumed at home, yet both remain firmly inelastic. Perhaps the most interesting difference between the two categories is in the estimated steady-state budget shares (the μ: lr’s), which imply that the budget share of food consumed outside of the home will ultimately more than double that of food consumed in the home (0.0520 vs. 0.0236)3 .

15.1.3 Tobacco In the two editions of CDUS, tobacco was described as the quintessential (legal) good characterized by habit formation (indeed, if not addiction). However, with the health hazards known to be associated with its use, tastes for the population at large have obviously changed, and tobacco consumption is now a problem to model.4 The model finally arrived at is a B-C model estimated with data that began in 1962 (the year of the first Surgeon General’s Report on the health risks of smoking).5 Interestingly—and, indeed, as should be expected in the circumstances—tobacco consumption is now seen to be an inferior good. Flow adjustment is rapid, and the Table 15.4 Tobacco Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

74.9903 0.8072 0.0796 −0.00594 −0.00123 −0.9632 −0.1991 2.5279

0.83 4.82 0.54 −1.02 −0.57 −1.44 −3.00

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

29.6652 −0.000136 0.1033 −0.0109 −0.0275 −0.0109 −1.7647 −4.4609 −1.7624

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

−1.1223 −0.4440 −0.4434 −0.6028 −0.2385 −0.2382 0.9857 1.96

h=− d–f = −6.29 #obs. = 70 B-C Model, Annual, 1962–2004, Model Quality: . 3 These

compare with budget shares for these two categories in 2004 (see Table 15.1 below) of 0.0838 and 0.0530. 4 In principle, it would seem that tobacco consumption should be modeled in two stages, a discrete probit/logit first stage, in which the proportion of the population that consumes tobacco is explained, and then a continuous second-stage that models the actual amount consumed. 5 In our initial efforts to model this category using data over the entire period, a trend beginning in 1962 was included as an additional predictor in an (inadequate) attempt to account for structural changes triggered by the Surgeon General Reports.

15.1

Food, Tobacco and Alcohol

271

stock coefficient (β) is slightly negative, which, while anomalous for those who smoke, is consistent with tobacco being inferior for the population at large. 15.1.3.1 Food excluding Tobacco and Alcoholic Beverages This category is a variation on the item listed in the PCE table, in that it excludes tobacco as well as alcoholic beverages from aggregate food expenditures. Both B-C and state-adjustment models are tabulated. The state-adjustment model is estimated by nonlinear GLS, while the B-C model is not, thus providing an illustration of the point made earlier that, because of the B-C models richer dynamical structure, residuals from it are seldom plagued with positive autocorrelation. Although the two models show contradictory short-term dynamics—the state-adjustment model signals habit formation, the B-C model stock adjustment—their implicit steady-state budget shares are essentially the same. Because of the superior statistical quality of the state-adjustment model, it will be the one referred to subsequently. Table 15.5 Food excluding tobacco and alcoholic beverages Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

180.8744 0.7302 0.0901 0.0303 −4.9220 −1.6566

1.98 6.06 6.76 2.46 −2.84 −2.24

α β δ μ μ: lr λ λ: lr

1546.69 0.0928 0.4047 0.0866 0.1124 −4.7320 −6.1408

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.6413 0.8322 −0.1339 −0.1737 0.9996 1.98 0.09

d–f = −8.48 #obs. = 71. State-Adjustment Model, Annual, 1930–2004 (1942–1945 excluded), GLS, Model Quality: !

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

199.9677 1.0265 −0.3383 0.0330 0.0151 −1.1496 −0.5253 1.1545

2.83 8.67 −2.57 2.58 3.33 −0.64 −1.22

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

173.2093 −0.2284 0.2284 0.0968 0.1118 0.0484 −3.3688 −3.3892 −1.6847

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.8272 0.7165 0.3583 −0.1100 −0.0953 −0.0477 0.9954 1.92

h = 1.19 d–f = −7.93 #obs. = 70. B-C Model, Annual, 1931–2004 (1942–1945 excluded) Model Quality: .

272

15

Annual PCE Models

15.1.4 Alcoholic Beverages Table 15.6 Alcoholic beverages Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

80.9987 1.0926 −0.2418 0.00325 0.00964 −1.4872 −0.4415 1.4240

1.32 8.22 −1.81 1.33 3.32 −2.11 −1.10

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

63.3425 −0.0516 0.1484 0.00871 0.0111 0.00646 −3.9589 −5.0625 −2.9379

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.4870 0.3808 0.2826 −1.6465 −1.2876 −0.9555 0.9821 1.78

h = 2.22 d–f = −6.79 #obs = 66. B-C Model, Annual, 1947−2004, Model Quality: .

15.1.4.1 Alcoholic Beverages for Off-Premise Consumption Table 15.7 Alcoholic beverages for off-premise consumption Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

71.0000 1.0643 −0.2683 0.00132 0.000576 −0.7520 −0.3271 1.2121

2.08 5.68 −1.62 0.80 1.55 −1.44 −2.01

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

58.5777 −0.0710 0.2175 0.0037 0.0045 0.0028 −2.1277 −2.5790 −1.6041

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.3570 0.2946 0.2221 −1.8180 −1.4999 −1.1307 0.9913 1.95

h = 0.55 d–f = −7.44 #obs. = 58. B-C Model, Annual, 1947−2004, Model Quality: .

15.1.4.2 Other Alcoholic Consumption The three models for alcoholic beverages will be discussed as a group. Category 15.7 refers to alcoholic beverages purchased in liquor and wine establishments for home (or other off-premise) consumption, while category 15.8 refers to wine and spirits consumed in restaurants, clubs, and other venues. Item 15.6 obviously refers to the aggregate. The B-C model is seen to apply in all three. The model for 15.8 is strong statistically, but the ones for 15.6 and 15.7 are only fair. With regard to short-term dynamics, flow adjustment is rapid throughout, both the aggregate and

15.2

Clothing, Accessories and Jewelry

273

off-premise consumption are characterized by stock adjustment, while other alcohol consumption is characterized by habit formation. Steady-state demand is price elastic for both the sub-categories, but somewhat inexplicably not for the aggregate. The elasticity with respect to total expenditure is elastic for only 15.8. The latter is clearly in keeping with expenditures at restaurants, clubs, etc., being heavily entertainment oriented. Table 15.8 Other alcoholic beverages Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

104.9981 0.7749 −0.1403 0.00102 0.000901 −0.8528 −0.7559 2.4511

2.78 5.79 −1.06 2.08 2.37 −2.31 −2.49

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

42.8379 0.3583 0.4430 0.00203 0.00498 0.0106 −1.7027 −4.1734 −8.9035

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.5569 0.2001 1.0465 −2.4583 −1.0030 −5.2445 0.7796 1.88

h=− d–f = −7.10 #obs = 58. B-C Model, Annual, 1947−2004, Model Quality:    !

15.2 Clothing, Accessories and Jewelry The equation for the aggregate clothing category is a B-C model that is estimated with data from post-WW2. Statistically, the equation appears strong, for the estimated coefficients for both total expenditure and price are sizeable multiples of their standard errors, and the residuals appear free of autocorrelation. Unfortunately, however, the statistical strength is somewhat of an illusion. For, because of a strong Table 15.9 Clothing, accessories and jewelry Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

449.9906 1.0210 −0.1210 0.0253 0.0258 −1.6185 −1.6506 1.2248

4.60 4.30 −0.51 2.20 2.17 −5.06 −5.11

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

367.3993 0.3575 0.5099 0.0771 0.0944 0.2577 −4.9351 −6.0446 −16.5062

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.8526 1.5125 5.0589 −1.4349 −1.1715 −3.9183 0.9987 2.04

h=− d–f = −7.62 #obs = 58. B-C Model, Annual, 1947−2004, A1 + A2 = 0.90, Model Quality: .

274

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Annual PCE Models

trend in the dependent variable, it has been necessary to constrain the sum of A1 and A2 to be 0.90. Even so, the steady-state elasticities seem implausibly large. Probably all that should be concluded, accordingly, is that demand for aggregate clothing-related expenditures is elastic in the long run with respect to both price and total expenditure.

15.2.1 Shoes The equation for shoes is a logarithmic flow adjustment of excellent statistical quality. All estimated coefficients are multiples of their standard errors, autocorrelation in the residuals is absent, and the steady-state elasticities, both inelastic, are plausible. Table 15.10 Shoes Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3  κ

2.2551 0.6003 0.1538 −0.3774 0.4995 5.6421

4.66 7.66 5.40 −5.25

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.1922 0.3849 −0.4717 −0.9442 0.1922 0.3849

srelastprice lrelastprice R2 dw h d–f

−0.4717 −0.9442 0.9851 1.99 0.08 −8.23

#obs = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: !

15.2.2 Clothing excluding Shoes Take away expenditures for shoes from total clothing expenditures and what do you get? A B-C equation for clothing that is both strong statistically and has plausible steady-state elasticities! Clothing is seen to be subject to mild stock adjustment, with Table 15.11 Clothing excluding shoes Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

0.7761 0.7648 −0.1950 0.0366 0.00217 −1.1405 −0.0676 3.0666

0.07 7.80 2.18 5.82 2.33 −4.18 −1.55

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

0.2531 −0.0037 0.0296 0.0608 0.1866 0.0540 −1.8953 −5.8121 −1.6835

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

5.9009 1.9242 1.7092 −2.5420 −0.8289 −0.7363 0.9990 2.25

h = −1.46 d–f = −8.46 #obs = 58. B-C Model, Annual, 1947−2004, Model Quality:    !

15.2

Clothing, Accessories and Jewelry

275

a small depreciation rate and rapid flow adjustment. Steady-state clothing demand is impressively elastic with respect to total-expenditure, but inelastic with respect to price. 15.2.2.1 Mens and Boys Clothing 15.2.2.2 Women and Girls Clothing Not surprisingly, men and boys’ and women and girls’ clothing expenditures are seen to have different dynamical structures. The B-C model for women and girls clothing shows mild habit formation, while the same model for men and boys clothing displays stock adjustment. Flow adjustment, however, is rapid for both. Expenditures in the long run for both categories are slightly elastic with respect to total expenditure, but inelastic with respect to price. Statistically, the model is excellent for men and boys clothing and good for women and girls. Table 15.12 Mens and boys clothing Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

2.0997 0.9334 0.0111 0.0128 0.000733 −0.1242 −0.00711 2.1329

1.54 9.68 0.11 7.14 4.45 −2.64 −1.91

α β δ μ μ:fl μ:lr λ λ: fl λ:lr

0.9845 −0.0256 0.0286 0.0250 0.0533 0.0132 −0.2122 −0.5167 −0.1279

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

4.2635 1.9989 1.0557 −0.4880 −0.2288 −0.1208 0.9989 1.71

h = 1.60 d–f = −7.04 #obs = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    ! Table 15.13 Women and girls clothing Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

19.9972 0.7732 0.1270 0.0121 0.00121 −0.4942 −0.0900 2.7685

0.22 3.21 0.35 0.40 2.60 −3.52 −0.35

α β δ μ μ: fl μ:lr λ λ:fl λ:lr

7.2232 0.0035 0.0911 0.0213 0.0589 0.0221 −0.8675 −2.4017 −0.9019

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

3.0572 1.1073 1.1481 −1.8902 −0.6828 −0.7098 0.9981 1.75

h=− d–f = −8.23 #obs = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

276

15

Annual PCE Models

15.2.3 Cleaning, Storage and Repair of Clothing and Shoes This is the first of the three PCE categories for which total expenditure is excluded as a predictor, the other two being for the aggregate of intercity transportation expenditures (Table 15.78) and intercity bus travel (Table 15.80). The estimated equation is a B-C model, which displays moderate habit formation and substantial short- and long-run price elasticities. Apart from the absence of total expenditure as a predictor (which is probably not unreasonable), the statistical quality of the estimated equation is good. Table 15.14 Cleaning, storage and repair of clothing and shoes Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

19.0995 1.2597 −0.3283 − − −0.6785 −0.1547 0.9772

1.31 16.65 −5.05 − − −5.79 −1.17

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

19.5443 0.0056 0.1140

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

− − − −4.1949 −4.2926 −4.5137 0.9914 2.18

−2.1463 −2.0975 −2.2569

h = −0.97 d–f = −9.03 #obs = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), No PCE, Model Quality:  .

15.2.4 Jewelry and Watches Table 15.15 Jewelry and watches Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

5.0002 0.9342 0.000649 0.0124 0.000711 −0.1969 −0.0113 2.1089

−2.69 6.62 0.01 4.72 1.46 −2.77 −1.88

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

−2.3710 −0.0353 0.0286 0.0244 0.0515 0.0109 −0.3863 −0.8147 −0.1727

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

9.8739 4.6820 2.0936 −1.8715 −0.3774 −0.1688 0.9952 1.68

h=− d–f = −6.65 #obs = 58. B-C Model, Annual, 1947−2004, Model Quality:  .

15.3

Personal Care

277

The B-C model shows expenditures for jewelry and watches to be subject to rapid flow adjustment, with a mild stock effect that depreciates slowly. As is to be expected, the steady-state total-expenditure elasticity is large and the price elasticity small.

15.2.5 Other clothing This small category of expenditures, which consists of watch, clock, and jewelry repairs, costume and dress suit rental, and miscellaneous personal services, accounted for about one-half of one percent of the total expenditure in 2004. The estimated B-C model displays rapid flow adjustment, mild habit formation that dissipates slowly, and surprisingly large steady-state elasticities.

Table 15.16 Other clothing Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−0.00951 0.7141 0.2362 0.00437 0.000643 −0.4192 −0.0618 3.3397

−0.01 5.91 2.26 1.57 1.70 −1.77 −0.87

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

−0.0028 0.0334 0.0736 0.0071 0.0236 0.0129 −0.6795 −2.2693 −1.2420

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

7.2357 2.1666 3.9601 −4.2864 −1.2835 −2.3460 0.9925 1.92

h = 1.32 d–f = −7.91 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.3 Personal Care 15.3.1 Toilet Articles and Preparations 15.3.2 Barbershops, Beauty Salons and Health Clubs The personal care category consists of expenditures for toilet articles and preparations (Table 15.18) and barbers, beauticians, and health clubs (Table 15.19). B-C models apply to the components as well as to the aggregate. While all three of the models are weak statistically, the dynamics and steady-state parameters are all plausible. Habit formation prevails throughout the category, and spending is highly sensitive to price and on balance is a luxury.

278

15

Annual PCE Models

Table 15.17 Personal care Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

5.2157 1.1963 −0.2603 0.00562 0.000698 −0.2045 −0.0254 1.1944

1.28 8.90 −2.11 2.68 1.36 −0.89 −0.79

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

4.3667 −0.0253 0.0621 0.0153 0.0183 0.0109 −0.5576 −0.6660 −0.3964

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.2538 1.0497 0.7463 −1.0992 −0.9203 −0.6527 0.9956 1.91

h = 2.47 d–f = −7.86 #obs = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: . Table 15.18 Toilet articles and preparations Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

6.0000 1.2602 −0.3000 0.000425 0.000282 −0.0510 −0.0339 0.7930

0.71 9.86 −2.34 0.54 1.02 −1.51 −0.70

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

7.5660 0.2565 0.3318 0.0017 0.0013 0.0071 −0.2010 −0.2896 −0.8516

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.1647 0.2077 0.8800 −0.2297 −0.2896 −1.2269 0.9969 1.82

h=− d–f = −7.55 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: . Table 15.19 Barbershops, beauty salons and health clubs Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

14.9999 0.8361 0.0740 0.00124 0.000834 −0.3014 −0.2026 2.2041

0.86 4.96 0.51 0.76 1.12 −1.30 −0.59

α β δ μ μ:fl μ:lr λ λ:fl λ: lr

6.8056 0.2434 0.3361 0.0026 0.0056 0.0093 −0.6209 −1.3685 −2.2587

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.8189 0.3715 1.3481 −2.7369 −1.2418 −4.5054 0.9572 1.88

h=− d–f = −7.72 #obs = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.4

Housing

279

15.4 Housing Table 15.20 Housing GLS Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

0.4258 0.8631 0.1414 −0.1549 0.1470 3.1095

2.48 22.36 3.13 −5.91

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.1518 1.0326 −0.1662 −1.1309 0.1518 1.0326

srelastprice lrelastprice R2 dw h d–f

−0.1662 −1.1309 0.9996 2.05 −0.25 −8.52

#obs. = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: !

15.4.1 Owner-Occupied Housing Expenditures for owner-occupied housing is an imputed item referring to the spacerental value of owner-occupied housing, and is not to be confused with mortgage and interest payments. In 2004, the category accounted for about 11% of the total personal consumption expenditure. As for the aggregate for housing (which accounts for about 14% of the total expenditure), the model estimated is a logarithmic flowadjustment model. Both models are estimated by GLS and are of excellent statistical quality.6 Flow adjustment is sluggish, as evidenced in the small values of θ . Both steady-state elasticities are elastic7 . Table 15.21 Owner-occupied housing GLS Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3  κ

0.2252 0.8845 0.1459 −0.1662 0.1228 1.9495

1.06 25.34 2.97 −4.55

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.1548 1.2632 −0.1764 −1.4391 0.1548 1.2632

srelastprice lrelastprice R2 dw h d–f

−0.1764 −1.4391 0.9997 2.04 −0.18 −8.44

#obs. = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: ! 6 The

error term in the estimating equation of the flow-adjustment model [from the argument underlying equation (2.48)] is θ ut = (εt + εt−1 ). 2+θ 7 The total-expenditure elasticity for housing is clearly boosted by the demand for second homes, which is obviously highly income-elastic. [For a quantitative analysis of the effects of multiple homeownership on the income elasticity for housing, see Belsky et al. (2006)].

280

15

Annual PCE Models

15.4.2 Rental Housing The equation estimated for rental housing, which accounts for about 3% of the total consumption expenditure, is a B-C model with sluggish flow adjustment and a fairly strong habit formation. Steady-state demand is elastic with respect to price, but inelastic with respect to total expenditure. Since renting is generally considered as inferior to owning, neither of these results are surprising8 . Table 15.22 Rental housing Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

59.9962 1.3378 −0.3878 0.000628 0.000823 −0.3118 −0.4081 0.2807

4.54 8.90 −2.58 0.67 3.06 −0.82 −3.62

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

213.7094 0.3930 0.6544 0.0066 0.0018 0.0164 −3.2519 −0.9152 −8.1607

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.0450 0.1604 0.4015 −0.1414 −0.5037 −1.2608 0.9983 2.08

h = −0.99 d–f = −8.54 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), A1 + A2 = 0.95, Model Quality: .

15.4.3 Rental Value of Farm Housing Table 15.23 Rental value of farm housing Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

19.9996 0.7764 0.0261 −0.000105 −0.000290 −0.0250 −0.0692 1.1897

2.60 3.88 0.12 −1.08 −2.07 −0.97 −4.01

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

16.8113 1.0016 1.3810 −0.0004 −0.0005 −0.0015 −0.0960 −0.1142 −0.3494

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

−0.1569 −0.1319 −0.4802 −0.2776 −0.2334 −0.8497 0.9607 1.82

h=− d–f = −7.41 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  . The equation for this category is estimated with A1 + A2 = 0.95. Without this constraint, the sum of these two coefficients is very close to the dynamically unstable value of 1.

8

15.5

Housing Operation

281

Like owner-occupied housing, expenditure for farm housing is an imputed quantity referring to the space-rental value of farm housing. The B-C model indicates farm housing to be subject to extremely strong habit formation, with a small steady-state total-expenditure elasticity, and a much larger (but still inelastic) price elasticity.

15.4.4 Other Housing This miscellaneous housing category consists of expenditures for transient hotels, motels, other traveler accommodations, clubs, schools, and other group housing. The model estimated is a B-C model of excellent statistical quality that displays rapid flow adjustment, fairly strong stock adjustment, and steady-state elasticities both less than 1 (in absolute value).

Table 15.24 Other housing Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

5.9970 0.6982 0.1502 0.00654 0.00159 −0.5286 −0.1287 2.2064

1.57 5.24 1.26 3.38 2.49 −2.93 −1.65

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

2.7180 −0.3216 0.1218 0.0128 0.0283 0.0035 −1.0370 −2.2880 −0.2849

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

3.4232 1.5515 0.4261 −1.3844 −0.6274 −0.1723 0.9927 2.12

h=− d–f = −8.69 #obs. = 7. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    !

15.5 Housing Operation This eclectic category, which includes a wide range of expenditures related to housing operation, accounted for 10% of the total consumption expenditure in 2004. The model estimated is a B-C model, which includes a dummy variable for pre- and postWW2, and shows rapid flow adjustment, a negative state effect, and steady-state elasticities that are both inelastic.

282

15

Annual PCE Models

Table 15.25 Housing operation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

417.9798 0.4057 0.1703 0.0736 0.0367 −5.5286 −2.7559 27.0423 4.2260

1.33 3.82 1.41 4.85 3.10 −3.35 −1.26 1.89

A B  M μ: fl μ: lr λ: fl λ: lr

98.9064 −0.0755 0.2492 0.1128 0.4766 0.0866 −8.4711 −35.7991 −6.5010

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

4.4303 1.0483 0.8045 −2.7289 −0.6457 −0.4956 0.9995 1.81 0.96

d–f = −7.44 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

15.5.1 Furniture, including Mattresses and Bedsprings The B-C model for household furniture expenditures mirrors the one for all of household operation. Flow adjustment is rapid, β is negative (as is to be expected for a durable category), and the steady-state elasticities are again both less than 1. Table 15.26 Furniture, including mattresses and bedsprings Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

10.9999 0.4027 0.3027 0.0118 0.00231 −0.3369 −0.0659 5.1749

1.02 3.09 1.75 4.72 2.47 −2.26 −1.03

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

2.1256 −0.1093 0.0978 0.0166 0.0860 0.0079 −0.4734 −2.4500 −0.2236

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

9.5959 1.8543 0.8757 −2.7349 −0.5285 −0.2496 0.9958 1.48

h=− d–f = −6.23 #obs. = 7. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

15.5.2 Household Appliances Included in this category are refrigerators and freezers, cooking stoves, dishwashers, laundry equipment, room air conditioners, sewing machines, vacuum cleaners, and other appliances. This is the second category (tobacco being the first) in which the state variable for the B-C model appears explicitly in the estimating equation. The state variable is constructed, using equation (13.50) with data from 1929 through

15.5

Housing Operation

283

1945. A value of 0.10 for the depreciation rate δ has been used in the construction. Flow adjustment is again rapid, and, since appliances are durable, we naturally expect the coefficient on the state variable to be negative. Steady-state demand is slightly elastic with respect to total expenditure, but inelastic with respect to price. The statistical quality of the model, which is estimated by generalized least squares, is excellent.

Table 15.27 Household appliances Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 γ α β

25.1390 0.1237 0.00626 −0.0839 −0.0717 75.2959 23.5711 −0.0186

1.92 1.04 4.85 −2.42 −1.92

δ μ μ:fl μ:lr λ λ:fl λ:lr flelastpce

0.1000 0.0059 0.4420 0.0050 −0.0786 −5.9219 −0.0663 96.3526

srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

1.2797 1.0794 −20.4575 −0.2717 −0.2292 0.9935 2.13 −1.24

d–f = −8.08 #obs. = 59. B-C Model (with state variable), Annual, 1947−2004, GLS, Model Quality:    !

15.5.3 China, Glassware, Tableware and Utensils Again, we have an excellent B-C model. Flow adjustment is rapid, and, in line with china and kitchen and eating utensils being durable, state adjustment is negative. Steady-state demand is elastic for price but inelastic for total expenditure. Table 15.28 China, glassware, tableware and utensils Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

18.9985 0.8240 −0.0274 0.00307 0.000678 −0.5665 −0.1253 2.2107

2.81 6.40 −0.19 3.68 3.73 −3.74 −3.04

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

8.5938 −0.0882 0.1106 0.0060 0.0132 0.0033 −1.1074 −2.4482 −0.6159

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

3.0748 1.3909 0.7736 −5.0148 −2.2684 −1.2617 0.9856 2.55

h=− d–f = −10.21 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    !

284

15

Annual PCE Models

15.5.4 Other Durable House Furnishings Expenditures in this category are for such house furnishings as floor coverings; picture frames; mirrors; art products; portable lamps; window coverings and hardware; telephone equipment; writing equipment; and hand, power, and garden tools. Given this mixture of goods, some of which are either of limited durability or associated with decoration and recreation, it is perhaps not surprising that the model estimated is logarithmic flow adjustment. Flow adjustment is rapid, and the category is indicated to be a mild luxury in the long run but with a moderately small price elasticity. The only drawback to the estimated model is strong positive autocorrelation in the residuals, for the Durbin–Watson coefficient is low and the Durbin-h high. The d-f statistic of –5.78, however, indicates absence of a unit root. Unfortunately, application of generalized least squares to correct the positive autocorrelation leads to unacceptable results. Table 15.29 Other durable house furnishings Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

−2.6436 0.3856 0.6895 −0.2069 0.8869 −4.3026

−4.69 3.98 6.82 −2.82

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.9953 1.1222 −0.2986 −0.3367 0.9953 1.1222

srelastprice lrelastprice R2 dw h d–f

−0.2986 −0.3367 0.9903 1.22 5.54 −5.78

#obs. = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:  .

15.5.5 Semi-Durable House Furnishings The model for this category, which includes expenditures for textile house furnishings (including piece goods allocated to house furnishing use), lamp shades, and brooms and brushes, is one of the poorest of the time-series analysis, for neither total Table 15.30 Semi-durable house furnishings Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

0.5908 0.9281 0.0266 −0.1028 0.0746 8.2191

0.60 11.93 0.40 −1.03

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.0276 0.3701 −0.1066 −1.4296 0.0276 0.3701

srelastprice lrelastprice R2 dw h d–f

−0.1066 −1.4296 0.9812 2.08 − −10.24

#obs. = 71. Log Flow-Adjustment Model, Annual, 1947−2004, Model Quality: .

15.5

Housing Operation

285

expenditure nor price of much importance statistically.9 However, this unfortunate feature aside, demand is seen to be elastic with respect to price in the long run.

15.5.6 Cleaning and Polishing Preparations and Miscellaneous Household, Supplies and Paper Products This category represents expenditures related to maintenance of a variety of household durables and living activities. The associated state variable can be interpreted as a generalized stock of “living activities” that the expenditures of the category serve. A positive β is accordingly appropriate, as is an inelastic steady-state total-expenditure elasticity. On the other hand, since maintenance and restocking of household supplies can be postponed, an elastic steady-state price elasticity is appropriate as well.

Table 15.31 Cleaning and polishing preparations and miscellaneous household, supplies and paper products Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

30.0001 0.8997 0.00210 0.000826 −0.6005 −0.2362

2.12 21.24 1.39 2.25 −1.94 −2.06

α β δ μ μ: lr λ λ: lr

232.7996 0.3841 0.4896 0.0018 0.0082 −0.5079 −2.3561

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.1726 0.8010 −0.3603 −1.6715 0.9955 1.59 1.82

d–f = −6.75 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:  .

15.5.7 Stationary and Writing Supplies Despite the poor statistical quality of this model, the results generally make sense. One certainly does not see written communication as a luxury activity (steady-state demand is inelastic with respect to total expenditure), yet, once started, writing a project can be habit-forming (consistent with a positive β). The habit, however, is indicated to wear off rather quickly. On the other hand, given the telephone and computer as substitutes for written communication, a large steady-state price elasticity is to be expected.

9 The

B-C model yields even worse results.

286

15

Annual PCE Models

Table 15.32 Stationary and writing supplies Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

7.0006 1.0040 −0.0565 0.000186 0.000138 −0.0784 −0.0599 1.5116

0.76 6.98 −0.43 0.35 0.82 −1.12 −0.67

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

4.6311 0.3035 0.3710 0.0005 0.0007 0.0026 −0.2014 −0.3044 −1.1067

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.2362 0.1562 0.8586 −0.7126 −0.4714 −2.5906 0.9892 1.92

h=− d–f = −7.86 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.5.8 Household Utilities Primarily as a consequence of deregulation of the telephone and energy industries and the energy crises of the 1970s, a great deal of volatility has been injected into utility relative prices, with the consequence that, in statistical terms, price is now a more significant predictor of utility expenditures than total expenditure. This is aptly illustrated in the B-C model for total utility expenditures (which account for a little over 3% of the total consumption expenditure). As is to be expected, expenditure for the category as a whole is subject to a fairly strong habit formation, rapid flow adjustment, and has substantial steady-state elasticities.

Table 15.33 Household utilities Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

49.9997 0.8244 0.1413 0.00293 0.000893 −1.3075 −0.3982 2.4483

3.72 5.81 1.04 1.87 2.15 −4.58 −3.11

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

20.4219 0.2713 0.3046 0.0057 0.0139 0.0521 −2.5382 −6.2145 −23.2163

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.3276 0.1338 1.2240 −0.8834 −0.3608 −3.3004 0.9965 1.79

h=− d–f = −7.38 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    !

15.5

Housing Operation

287

15.5.8.1 Electricity The model for electricity (Table 15.34) is logarithmic flow adjustment. It, too, is an excellent model statistically, with sluggish flow adjustment (θ =0.11). Steady-state electricity demand is elastic with respect to price, but inelastic with respect to total expenditure.10 Table 15.34 Electricity GLS Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

0.3821 0.8945 0.0920 −0.1417 0.1113 3.6225

1.53 35.19 2.31 −4.77

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.0971 0.8723 −0.1496 −1.3435 0.0971 0.8723

srelastprice lrelastprice R2 dw h d–f

−0.1496 −1.3435 0.9984 2.52 −2.25 −10.50

#obs. = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: !

15.5.8.2 Natural Gas In the B-C model for natural gas (Table 15.35), we see that price is the main economic determinant. As natural gas is used primarily for heating, the state variable can be taken to represent the stock of gas-consuming appliances. Since natural gas Table 15.35 Natural gas Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

17.1001 0.8299 0.1291 0.000117 0.000112 −0.1495 −0.1432 1.7457

4.05 6.79 1.12 0.66 0.66 −2.29 −3.66

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

9.7953 0.9028 0.9579 0.0005 0.0008 0.0084 −0.4028 −0.7032 −6.7924

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.0198 0.0113 0.1970 −0.3805 −0.2180 −3.7863 0.9896 2.05

h=− d–f = −8.40 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  . 10 The

energy crises of the 1970s sparked a rash of electric demand studies, some of which are reviewed in Appendix 15.1 of this chapter.

288

15

Annual PCE Models

fuels services provided by durable goods, habit formation is to be expected. Flow adjustment is rapid, and steady-state demand is elastic with respect to price, but highly inelastic with respect to total expenditure. 15.5.8.3 Fuel Oil and Coal Completing the energy categories, we see that total expenditure is absent altogether from the state-adjustment model for fuel oil and coal (Table 15.36), which is hardly surprising, given that both the fuels either are or border on being inferior. The extremely large positive values for β and δ imply extremely sluggish habit formation and a substantial steady-state price elasticity. While the latter might seem unduly inflated, fuel oil and coal have electricity and natural gas as substitutes in the long run, hence a more-than-moderate price elasticity is to be expected. Table 15.36 Fuel oil and coal Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

18.8365 0.9478 − − −0.5830 −0.1764

1.90 32.44 − − −3.77 −2.14

α β δ μ μ: lr λ λ: lr

205.7468 1.1246 1.1782 − − −0.5081 −11.1602

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

− − −0.2778 −6.1020 0.9789 1.83 0.73

d–f = −7.54 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:  .

15.5.8.4 Water and Sanitary Services Turning now to expenditures for water (Table 15.37), the estimated state-adjustment model for the category shows moderate habit formation, with steady-state Table 15.37 Water and sanitary services Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

4.0021 0.9565 0.00454 0.000417 −0.5914 −0.0544

2.12 21.24 1.39 2.25 −1.94 −2.06

α β δ μ μ: lr λ λ: lr

57.7700 0.1335 0.2489 0.0035 0.0075 −0.6566 −1.4160

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.4151 0.8952 −0.3556 −0.7670 0.9918 1.89 0.72

d–f = −8.07 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:  .

15.5

Housing Operation

289

elasticities of 0.90 and −0.77. The value of 0.90 for the steady-state totalexpenditure elasticity is consistent with not all of water’s uses being a necessity. 15.5.8.5 Telephone and Telegraph Category (Table 15.38) includes expenditures for both local and long-distance telephone services.11 Like electricity demand, telecommunications demand has been much studied in the last 30 years, and the present model is certainly not one of the better efforts.12 The B-C model that is presented above is, in general, simply the best of the bad lot with the NIPA data. A logarithmic flow-adjustment model (which a priori would seem the clear choice) is best in terms of statistical significance of price and total expenditure, but has a coefficient on lagged consumption that is a bit larger than 1. This strong trend in the dependent variable is “controlled” in the B-C model by constraining the sum of A1 and A2 to be equal to 0.95. Since traditional telephony usage has been much affected by competition and by the rise of the Internet, it is perhaps not surprising to find price effects to be much more important statistically than total expenditure. However, strong stock adjustment makes little sense, as does the extremely low steady-state price elasticity13 . Table 15.38 Telephone and telegraph Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−5.0011 0.9996 0.0496 0.00682 0.000778 −0.0112 −0.00127 1.1495

4.05 6.79 1.12 0.66 0.66 −2.29 −3.66

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

4.3497 −0.6351 0.0567 0.0191 0.0220 0.0016 −0.0311 −0.0358 −0.0026

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.9539 1.6997 0.1393 −0.0422 −0.0367 −0.0030 0.9988 1.92

h=− d–f = −7.85 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), A1 + A2 = 0.95, Model Quality: .

11 The

last telegram was sent in early 2006. high points from this literature are included in Appendix 15.2 of this chapter. See also Taylor (1994). 13 The steady-state price elasticity is typical of the value found for access to the telephone network (as opposed to usage). Usage elasticities, on the other hand, have usually been found to be much larger, ranging from −0.20 or so for short-haul local calls, to −0.6 or so for long-haul long-distance calls, and to larger values for international calls. What the unsatisfactory results for this category really seem to point up is that a single-equation framework is not the proper one for modeling telecommunications demand. A more appropriate framework, in which access is separated from usage, is discussed in Appendix 15.2 to this chapter. 12 Some

290

15

Annual PCE Models

15.5.9 Domestic Services Expenditures for domestic services is modeled by a B-C model in which the state variable appears as an explicit predictor. As in the model for household appliances, the state variable has been constructed from the data from 1929 to 1945 using equation (13.52) with δ = 0.10. The result is an excellent model statistically, with slight stock adjustment, extremely rapid flow adjustment, and inelastic steady-state elasticities. Table 15.39 Domestic services Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 γ α β

63.2452 0.4982 0.00195 −1.0407 0.0367 76.1460 59.0097 −0.00946

3.14 3.17 2.67 −4.47 2.46

δ μ μ: sr μ: lr λ λ: sr λ: lr flelastpce

0.1000 0.00182 0.1385 0.00166 −0.9710 −73.9396 −0.8871 21.6681

srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

0.2846 0.2600 −69.1174 −0.9077 −0.8292 0.9866 1.68 −

d–f = −6.43 #obs. = 59. B-C Model (with state variable), Annual, 1946−2004, Model Quality:    !

15.5.10 Other Household Operation This miscellaneous category consists of maintenance and repair expenditures for appliances and house furnishings, moving and warehouse expenses, postage and parcel delivery charges, premiums plus premium supplements (less normal losses Table 15.40 Other household operation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−29.9918 0.7792 −0.00337 0.00945 0.00581 −0.8066 −0.4959 2.1804

3.47 7.47 −1.39 3.58 2.12 −1.64 −2.83

α β δ μ μ:fl μ:lr λ λ:fl λ:lr

13.7552 0.0767 0.3074 0.0195 0.0424 0.0259 −1.6603 −3.6204 −0.4146

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

5.0438 2.3133 3.0829 −2.4533 −1.1251 −1.4995 0.9988 1.92

h=− d–f = −7.85 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    !

15.6

Medical Care

291

and dividends paid to policyholders) for insurance on personal property (except motor vehicles), and other miscellaneous household operation services. The estimated B-C model shows rapid flow adjustment, mild habit formation, and highly elastic steady-state elasticities14 .

15.6 Medical Care Expenditures for medical care currently account for about 20% of the total consumption expenditure, and are the most rapidly growing of all the large PCE categories. The model for the aggregate of these expenditures is a B-C equation, of excellent statistical quality, in which habit formation is strong, but dissipates quickly (β = 0.60, δ = 0.84). Steady-state demand is highly elastic with respect to total expenditure, but slightly inelastic with respect to price. The implied steady-state budget share is 37%. Table 15.41 Medical care Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

40.9924 0.9800 −0.1063 0.0274 0.0463 −2.3121 −3.9012 −66.0014 0.9909

0.76 8.59 −1.00 3.28 4.14 −2.45 −2.99 −2.70

A B  M μ: fl μ: lr λ λ: fl λ: lr

41.3688 0.5992 0.8436 0.1061 0.1052 0.3665 −8.9473 −8.8659 −30.8881

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

0.6304 0.6362 2.1962 −0.2622 −0.2646 −0.9134 0.9997 2.20 −2.90

d–f = −9.00 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    !

15.6.1 Drug Preparations and Sundries Expenditures for this drug category exclude drug preparations and related products dispensed by physicians, hospitals, and other medical services. The GLS logarithmic flow-adjustment model shows fairly rapid short-run adjustment to changes in price and total expenditure, and steady-state elasticities that are elastic for total expenditure but inelastic for price15 .

14 For an analysis that focuses explicitly on the demand for first-class mail using data from the U.S.

Postal Service, see Taylor (1989b). has been one of the more interesting categories, statistically, to estimate. Not surprisingly, expenditures for the category display a strong trend, which leads to coefficients on the lagged

15 This

292

15

Annual PCE Models

Table 15.42 Drug preparations and sundries GLS Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

−5.7074 0.4650 1.0038 −0.2147 0.7303 −10.6683

−4.74 4.40 5.12 −2.68

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

1.3703 1.8763 −0.2932 −0.4014 1.3703 1.8763

srelastprice lrelastprice R2 dw h d–f

−0.2932 −0.4014 0.9987 2.04 −0.41 −8.43

#obs. = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: !

15.6.2 Ophthalmic Products and Orthopedic Appliances The equation for this category is a B-C model that is of only fair statistical quality. A dummy variable that separates pre- and post-WW2 years accounts for the extra coefficient in the equation. Expenditures in the category are indicated to be subject to very mild habit formation, and to be extremely insensitive to changes in both price and total expenditure in the long run. Table 15.43 Ophthalmic products and orthopedic appliances Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

5.0003 0.8659 0.0258 0.00190 0.000430 −0.1487 −0.0338 −3.9999 2.2345

1.08 4.75 0.18 1.77 2.21 −1.79 −1.21 −2.19

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

2.2379 0.00790 0.1132 0.00369 0.00826 0.00397 −0.2894 −0.6466 −0.3111

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

0.2017 0.0903 0.0970 −0.0999 −0.0447 −0.0481 0.9870 1.99 −

d–f = −8.16 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.6.3 Physicians This category consists of offices of physicians, HMO medical centers, and freestanding ambulatory surgical and emergency centers and accounts for about 4%

dependent variable in all of the OLS models (B-C and state adjustment, as well as log flow adjustment) close to 1 and a strong positive autocorrelation in the residuals. GLS applied to the B-C and state-adjustment models corrects the autocorrelation, but not the near instability.

15.6

Medical Care

293

of the current consumption expenditures. The estimated equation is a B-C model of good statistical quality. Flow adjustment is rapid, habit formation is strong, both steady-state elasticities are in excess of 2, and the long-run budget share is estimated to be about 11% of the total expenditure. Table 15.44 Physicians Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

34.9835 0.8150 0.0897 0.0107 0.0104 −1.8090 −1.7617 2.1499

3.45 5.89 0.49 2.50 1.81 −1.19 −4.90

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

16.2721 0.3841 0.4870 0.0231 0.0496 0.1093 −3.9015 −8.3879 −18.4733

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.0652 0.6152 2.9127 −1.0407 −0.4841 −2.2921 0.9971 2.01

h=− d–f = −8.23 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

15.6.4 Dentists We present three models for this category because it is one of the few (indeed perhaps the only category) in which all three of the dynamic models (B-C, state adjustment, and log flow adjustment) yield plausible equations. While the B-C model is the weakest of the three statistically (both the state- and flow-adjustment models get three smiles), because both flow- and stock-adjustment are strong, Table 15.45 Dentists Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

18.0001 0.6506 0.2109 0.00634 0.00272 −1.1998 −0.4984 3.3488

0.44 11.11 0.06 2.57 0.13 −1.99 −0.14

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

5.3751 0.0928 0.2077 0.0108 0.0363 0.0196 −1.9907 −6.6665 −3.5976

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

3.1663 0.9456 1.7089 −2.6262 −0.7842 −1.4172 0.9930 2.28

h=− d–f = −7.29 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

294

15

Annual PCE Models

Table 15.45 (continued) Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

21.9998 0.8222 0.00890 0.00335 −1.6007 −0.6032

3.02 14.53 4.11 3.10 −3.97 −2.87

α β δ μ μ: lr λ λ: lr

172.6851 0.2691 0.4643 0.00792 0.0189 −1.4259 −3.3919

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.6915 1.6448 −0.5617 −1.3362 0.9933 2.19 −0.94

d–f = −9.07 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:    !

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 K4 θ

−2.6951 0.4413 0.9020 −0.6951 −0.1597 0.7753

−6.13 5.63 6.46 −5.47 −3.68

κ μ: sr μ: lr λ: sr λ: lr srelastpce

−4.8240 1.2517 1.6145 −0.2216 −0.2858 1.2517

lrelastpce srelastprice lrelastprice R2 dw h

1.6145 −0.2216 −0.2858 0.9896 2.22 −1.24

d–f = −9.14 #obs. = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: !

it is clearly the model of choice. As for expenditures for physicians, there is strong habit formation, rapid flow-adjustment, and elastic steady-state price and total-expenditure elasticities.

15.6.5 Other Professional Services As with physicians and dentists, the equation for other professional medical services is a B-C model (although in this case of only middling statistical quality).16 However, while habit-formation is once again strong and the steady-state elasticities are both highly elastic, flow adjustment is very sluggish.

16 This

category consists of chiropractors; optometrists; mental health practitioners (except physicians); physical, occupational, and speech therapists; radiologists; podiatrists; all other miscellaneous health practitioners; ambulance services; kidney dialysis centers; family planning services; outpatient mental health and substance abuse centers; all other outpatient care centers; blood and organ banks; all other miscellaneous ambulatory health care services; home, health, furniture, and equipment rental; medical and diagnostic laboratories; and home health care.

15.6

Medical Care

295 Table 15.46 Other professional services

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

8.9990 1.4385 −0.4857 0.00337 0.00388 −0.4297 −0.4953 −10.9991 0.1645

0.64 13.52 −4.32 1.02 2.23 −0.76 −1.35 −1.63

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

54.7097 0.2270 0.5763 0.0561 0.00923 0.0926 −7.1599 −1.1777 −11.8137

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

0.5903 3.5890 5.9217 −0.4030 −2.4503 −4.0429 0.9980 2.29 −3.56

d–f = −9.48 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.6.6 Hospitals and Nursing Homes Table 15.47 Hospitals and nursing homes Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6

−36.2631 0.9808 0.00568 0.01136 −1.5034 −3.0068 11.4522

−2.02 58.38 3.33 3.33 −3.51 −3.51 4.08

α β δ μ μ: lr λ λ: lr

−71.8303 1.9806 2.0000 0.00287 0.2960 −0.7590 −78.3434

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.0412 4.2547 −0.0508 −5.3438 0.9992 1.91 0.37

d–f = −7.90 #obs. = 71. A3 = 2A2, A5 = 2A4 (δ = 2) State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), GLS, Model Quality: .

15.6.6.1 Hospitals Table 15.48 Hospitals Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3  κ

−1.2666 0.9256 0.3109 −0.2796 0.0773 −17.0171

−2.10 24.01 2.53 −2.51

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.3229 4.1768 −0.2904 −3.7563 0.3229 4.1768

srelastprice lrelastprice R2 dw h d–f

−0.2904 −3.7563 0.9980 2.37 −1.28 −7.91

#obs. = 45. Log Flow-Adjustment Model, Annual, 1960−2004, GLS, Model Quality:    !

296

15

Annual PCE Models

Non-Profit Hospitals Table 15.49 Non-profit hospitals Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

10.1253 0.9663 0.0104 0.00262 −0.8315 −0.2099

0.73 25.53 1.26 1.44 −0.73 −0.87

α β δ μ μ: lr λ λ: lr

137.8196 0.2547 0.2889 0.00923 0.0779 −0.7390 −3.2367

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

1.0248 8.6490 −0.5243 −4.4247 0.9983 1.93 0.23

d–f = −6.44 #obs. = 45. State-Adjustment Model, Annual, 1960−2004, GLS, Model Quality: .

Proprietary Hospitals Table 15.50 Proprietary hospitals Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3  κ

−2.9966 0.8403 0.4941 −0.2413 0.1736 −18.7640

−2.99 15.08 3.17 −3.02

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.5370 3.0939 −0.2624 −1.5118 0.5370 3.0939

srelastprice lrelastprice R2 dw h d–f

−0.2624 −1.5118 0.9899 1.70 1.09 −8.85

#obs. = 45. Log Flow-Adjustment Model, Annual, 1960−2004, Model Quality:    !

Government Hospitals Table 15.51 Government hospitals Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−1.8056 1.2272 −0.3383 0.00165 0.00348 −0.1778 −0.3744 0.0658

−0.37 8.25 −2.50 1.04 2.53 −0.97 −2.02

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−27.4400 −1.5800 1.0526 0.0784 0.00516 0.0313 −8.4273 −0.5545 −3.3695

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.3145 4.7790 1.9108 −0.1551 −2.3568 −0.9423 0.9919 2.16

h=− d–f = −7.03 #obs. = 45. B-C Model, Annual, 1960−2004, Model Quality:  .

15.6

Medical Care

297

15.6.6.2 Nursing Homes The equation for hospitals and nursing homes (Table 15.47) is a GLS stateadjustment model in which the depreciation rate δ is constrained to be equal to 2 (which involves constraining the estimates of A4 and A6 to be twice those of A3 and A5 ).17 While this restriction was imposed quite frequently in the two earlier editions of CDUS, this is the only instance in the present effort. In its absence, the estimates of A4 and A6 are about triple the estimates of A3 and A5 , which makes for a problem in the estimate of β. As it is, the model is still close to having problems, for β is also very close to 2, implying extraordinarily strong habit formation, which in turn leads to very large steady-state elasticities. Conservatively, all that should be concluded from the model is that habit formation is strong, short-run adjustment is very sluggish, and steady-state demand is elastic with respect to both price and total expenditure. Table 15.52 Nursing homes Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

−2.3739 0.7681 0.6524 −0.6179 0.2623 −10.2368

−2.60 21.64 2.56 −1.66

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.7380 2.8133 −0.6990 −2.6646 0.7380 2.8133

srelastprice lrelastprice R2 dw h d–f

−0.6990 −2.8133 0.9914 2.32 −3.06 −9.65

#obs. = 45. Log Flow-Adjustment Model, Annual, 1960−2004, Model Quality:  .

The models for nonprofit, proprietary, and government hospitals are state adjustment, log flow adjustment, and B-C, respectively. Statistical quality of the three models range from excellent for proprietary hospitals to pretty bad for nonprofits. Strong habit formation is explicit in the model for nonprofit hospitals, and implicit in the one for proprietary hospitals. In each case, short-run adjustment is slow. On the other hand, the B-C model for government hospitals shows very different dynamics. While flow-adjustment continues to be extremely sluggish, β has a large negative value, making for steady-state elasticities much smaller than their short-run counterparts. Even so, as in the other two hospital categories, the long-run total-expenditure elasticity is elastic and the price elasticity nearly so. Beginning in 1959, the Bureau of Economic Analysis began disaggregating hospitals and nursing homes (Tables 15.54 and 15.48) and further disaggregating hospitals into non-profit, proprietary, and government (Tables 15.54, 15.52, and 15.56, respectively). Logarithmic flow-adjustment models are estimated for both

17 This

category consists of (1) current expenditures (including consumption of fixed capital) of non-profit hospitals and nursing homes and (2) payments by patients to proprietary and government hospitals and nursing homes.

298

15

Annual PCE Models

hospitals and nursing homes (the former by GLS). In each case, short-term adjustment is slow and steady-state demand is elastic in respect to both price and total expenditure.

15.6.7 Health Insurance

Table 15.53 Health insurance Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

1.3299 1.0092 −0.1906 0.00587 0.00341 −0.3588 −0.2084 1.3463

0.50 6.47 −1.18 4.85 2.79 −4.11 −2.35

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

0.9878 0.0453 0.2903 0.0159 0.0213 0.0188 −0.9692 −1.3049 −1.1486

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.5833 1.1760 1.3966 −0.4170 −0.3098 −0.3671 0.9963 2.01

h=− d–f = −8.23 #obs. = 45. B-C Model, Annual, 1960−2004, Model Quality:    !

15.6.7.1 Medical and Hospitalization Insurance

Table 15.54 Medical and hospitalization insurance Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−9.0001 1.2137 −0.3899 0.00347 0.00303 −0.1579 −0.1378 0.6364

−2.16 5.43 −1.82 1.80 2.57 −1.39 −1.41

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−14.1426 0.0111 0.4363 0.0167 0.0107 0.0172 −0.7626 −0.4853 −0.7825

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.0305 1.6193 1.6615 −0.1656 −0.2602 −0.2670 0.9939 2.04

h=− d–f = −6.67 #obs. = 44. B-C Model, Annual, 1961−2004, Model Quality: .

15.6

Medical Care

299

15.6.7.2 Income Loss Insurance Table 15.55 Income loss insurance Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

0.4508 0.9772 −0.0272 0.00000184 0.00000127 −0.00128 −0.000863 1.6465

0.52 17.52 −0.49 0.05 0.05 −1.48 −0.88

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

0.2738 0.2841 0.3447 0.0000004 0.0000007 0.0000253 −0.00310 −0.00510 −0.0176

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.0001 0.0136 0.0766 −0.3649 −0.2216 −1.2606 0.9169 1.76

h=− d–f = −6.00 #obs. = 44. B-C Model, Annual, 1961−2004, A2 + A3 = 0.95, Model Quality: .

15.6.7.3 Workman’s Compensation Table 15.56 Workman’s compensation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

2.0003 1.2019 −0.2988 0.00149 0.000281 −0.1801 −0.0339 1.1050

1.60 7.58 −1.70 3.81 1.62 −3.00 −1.85 −

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

1.8103 0.0462 0.0942 0.00432 0.00477 0.00290 −0.5216 −0.5763 −0.3500

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.7818 1.6125 1.0820 −1.0216 −0.9246 −0.6204 0.9890 1.75

h=− d–f = −5.76 #obs. = 44. B-C Model, Annual, 1961−2004, Model Quality:  .

As with hospital expenditures, the Bureau of Economic Analysis began in 1959 disaggregating health insurance loss (Table 15.53) into medical and hospitalization (Table 15.54), income loss (Table 15.55), and workman’s compensation (Table 15.56).18 B-C models are estimated for all of the categories, with equations 18 Workman’s

compensation consists of premiums less benefits for health, hospitalization, accidental, and dismemberment insurance; Income loss insurance consists of premiums less benefits

300

15

Annual PCE Models

that range in statistical quality from poor to excellent, the equation for the aggregate being the one that is excellent. All three sub-categories are subject to habit formation, two (Tables 15.54 and 15.55) have rapid flow adjustment, all three have steady-state total-expenditure elasticities that are elastic, while two (again Tables 15.54 and 15.55) have elastic steady-state price elasticities.

15.7 Personal Business The equation for total personal business expenditures, which currently account for about 9% of the total expenditure (a significant proportion of which entails imputation), is a B-C model of only fair statistical quality. Expenditures in the category are subject to moderate habit formation, rapid flow adjustment, and are elastic in the long run with respect to total expenditure. On the other hand, they are very inelastic with respect to price. Table 15.57 Personal business Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

38.0006 0.9629 −0.1845 0.0266 0.0208 −0.9336 −0.7294 −52.9998 1.3347

0.68 4.32 −0.73 2.05 1.51 −1.06 −0.60 −1.81

A B  M μ: fl μ: lr Λ λ: fl λ: lr

28.4704 0.0815 0.3906 0.0741 0.0990 0.0937 −2.6058 −3.4780 −3.2930

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

1.4664 1.0986 1.3883 −0.2937 −0.2201 −0.2781 0.9958 1.87 −

d–f = −7.67 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.7.1 Brokerage Services The equation for brokerage services is a B-C model estimated from postwar data. Flow adjustment is reasonably rapid, habit formation prevails, and both steady-state elasticities are well in excess of 1 in the elastic range.

for income loss insurance; Workman’s compensation consists of premiums plus premium supplements less normal losses and dividends paid to policyholders for privately administered workers’ compensation.

15.7

Personal Business

301 Table 15.58 Brokerage services

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−10.9957 1.0165 −0.1794 0.00269 0.00306 −0.0873 −0.0995 1.0735

−2.86 2.22 −0.51 1.62 2.16 −1.57 −1.92

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

10.2425 0.2933 0.5696 0.00912 0.00979 0.0188 −0.2963 −0.3180 −0.6107

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

2.2001 2.0494 4.2243 −0.9839 −0.9166 −1.8893 0.9719 1.98

h=− d–f = −7.39 #obs. = 58. B-C Model, Annual, 1947−2004, Model Quality:  .

15.7.2 Bank Service Charges, Trust Services and Safe Deposit Box Rental The B-C model (which is only of fair quality statistically) for this category shows fairly strong habit formation, an elastic long-run elasticity with respect to total expenditure. Perhaps not surprisingly, however, the long-run elasticity with respect to price is highly inelastic.

Table 15.59 Bank service charges, trust services and safe deposit box rental Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

5.0000 1.1039 −0.2548 0.00235 0.00229 −0.0606 −0.0590 −5.0000 0.9048

−1.36 7.81 −1.77 1.91 3.08 −0.97 −0.96 −1.94

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−5.5262 0.2042 0.4870 0.00880 0.00796 0.0152 −0.2272 −0.2056 −0.3913

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

1.0324 1.1410 1.9649 −0.1186 −0.1310 −0.2256 0.9963 1.88 −

d–f = −7.69 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

302

15

Annual PCE Models

15.7.3 Services Furnished Without Payment by Financial Intermediaries, Except Life Insurance Carriers This entirely imputed category accounts for about two and a half percent of the total expenditure.19 The B-C model for it shows rapid flow adjustment, habit formation, an elastic steady-state total-expenditure elasticity, but inelastic price elasticity.

Table 15.60 Services furnished without payment by financial intermediaries, except life insurance carriers Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

6.5838 0.9030 −0.1053 0.00874 0.00736 −0.6353 −0.5346 1.5626

1.06 5.93 −0.50 1.92 1.99 −1.73 −1.71

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

4.2144 0.1629 0.4207 0.0223 0.0348 0.0364 −1.6492 −2.5306 −2.6428

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.6492 1.0554 1.7222 −0.7159 −0.4581 −0.7476 0.9905 2.04

h=− d–f = −8.34 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.7.4 Expense of Handling Life Insurance and Pension Plans In this the B-C model, there is habit formation, rapid flow adjustment, and highly elastic steady-state elasticities. The implied steady-state budget is indicated to be nearly 6%, which seems rather large for a category that currently accounts for less than 2% of the total expenditure, and most (if not all) of which is imputed.

19 The category consists of (1) operating expenses of commercial life insurance carriers and frater-

nal benefit life insurance and (2) administrative expenses of private non-insured pension plans and publicly administered government employee retirement plans. For commercial life insurance carriers, excludes expenses for accident and health insurance and includes profits of stock companies and services furnished without payment by banks, credit agencies, and investment companies.

15.7

Personal Business

303

Table 15.61 Expense of handling life insurance and pension plans Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

52.3000 0.6142 0.2947 0.00981 0.00526 −2.7999 −1.4999 3.7948

2.57 4.16 −1.87 1.82 1.86 −2.53 −2.17

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

13.7821 0.1951 0.2679 0.0157 0.0595 0.0577 −4.4733 −16.9750 −16.4583

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

3.2785 0.8640 3.1787 −4.2925 −1.3112 −4.1619 0.9728 2.13

h=− d–f = −8.02 #obs. = 58. B-C Model, Annual, 1947−2004, Model Quality:  .

15.7.5 Legal Services There are a couple of things about this B-C model for legal services that might cause the legal profession some heartburn. The first is that expenditures for legal services is seen to be stock-adjusting (rather than habit-forming), and the long-run elasticity is substantially less than 1, which implies that household legal services are not a growth industry. On the other hand, the profession can take some solace in the fact that the estimated steady-state price elasticity is small!

Table 15.62 Legal services Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

16.0297 0.7598 0.1161 0.00714 0.000896 −0.7756 −0.0973 2.8043

2.24 7.29 1.01 2.43 1.13 −1.63 −1.01

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

5.7160 −0.0450 0.0627 0.0124 0.0348 0.00722 −1.3460 −3.7747 −0.7840

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

2.6464 0.9435 0.5497 −1.1784 −0.4202 −0.2448 0.9598 1.90

h=− d–f = −7.78 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

304

15

Annual PCE Models

15.7.6 Funeral and Burial Expenses The B-C model for funeral and burial expenses displays rapid flow adjustment, and a fairly strong negative stock effect. The steady-state price elasticity is substantial, but, consistent with the fact that total expenditure is statistically of little importance, the steady-state elasticity with respect to total expenditure is minute. Table 15.63 Funeral and burial expenses Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

40.0001 0.3193 0.2675 0.000202 0.0000486 −0.6005 −0.1444 −3.9996 5.4859

2.36 2.16 2.41 0.52 0.53 −3.35 −2.03 −1.62

A B  M μ: fl μ: lr Λ λ: fl λ: lr

7.2915 −0.1660 0.1204 0.000280 0.00154 0.000118 −0.8327 −4.5679 −0.3501

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

0.3454 0.0630 0.0265 −4.8534 −0.8847 −0.3719 0.9135 2.01 −

d–f = −8.23 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.7.7 Other Personal Business Services Expenditures in this category include current expenditures (including consumption of fixed capital) of trade unions and professional, employment agency fees, moneyorder fees, spending for classified advertising, tax-return preparation services, and other personal business services, much of which again involves imputation. Yet, despite this, the B-C model for the category is of excellent statistical quality that Table 15.64 Other personal business services Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

28.0142 0.8155 −0.1558 0.00239 0.00279 −0.4106 −0.4792 −5.8233 1.4747

3.64 5.57 −1.03 3.35 4.76 −3.90 −3.58 −3.32

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

18.9971 0.1152 0.5836 0.00657 0.00968 0.00818 −1.1301 −1.6664 −1.4081

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

1.9351 1.3123 1.6351 −1.7822 −1.2085 −1.5059 0.9950 1.78 1.54

d–f = −7.35 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    !

15.8

Transportation

305

shows expenditures for the services in the category to be habit-forming and to have highly elastic steady-state elasticities.

15.8 Transportation In 2004, aggregate expenditures for transportation represented about 12% of the total consumption expenditure. This state-adjustment model shows that, on balance, transportation expenditures are subject to stock adjustment, with a moderately large, but nevertheless, inelastic steady-state total-expenditure elasticity and a very small steady-state price elasticity20 . Table 15.65 Transportation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6

4.8982 0.7195 0.2710 0.0308 −3.6407 −0.4143 11.4522

0.10 13.84 8.66 4.35 −1.03 −0.84 2.95

α β δ μ μ: lr λ λ: lr

139.5909 −0.2056 0.1207 0.2972 0.1099 −3.9937 −1.4770

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

2.3933 0.8851 −0.2302 −0.0851 0.9998 1.98 0.11

d–f = −8.15 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:  .

15.8.1 User-Operated Transportation The equation for user-operated transportation, which consists of expenditures for new autos, net purchases of used autos, and other motor vehicles (which includes trucks and recreational vehicles), is a state-adjustment model that displays stock adjustment, a nearly elastic steady-state total-expenditure elasticity and (as for transportation as a whole) a very small steady-state price elasticity. Since motor vehicles are the epitome of a durable good, we naturally expect a classic stock effect (i.e., a negative β) and elasticities that are much larger in the short run than in the long run. A nearly unitary steady-state total-expenditure elasticity is to be expected as well. On the other hand, the steady-state price elasticity of −0.13 seems implausibly small.

20 The

“extra” variable in this equation, as well as in the ones for categories 8.1 and 8.1a, is a dummy variable separating pre- and post-WW2 years.

306

15

Annual PCE Models

Table 15.66 User-operated transportation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6

4.8668 0.6991 0.2489 0.0319 −4.6817 −0.5995 49.9498

0.08 12.54 7.71 4.47 −1.23 −0.97 3.01

α β δ μ μ: lr λ λ: lr

120.8827 −0.2373 0.1368 0.2742 0.1059 −5.1578 −1.9927

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

2.2075 0.9310 −0.3277 −0.1266 0.9978 2.06 −0.30

d–f = −8.53 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:  .

15.8.2 New Autos This state-adjustment equation for expenditures for new automobiles is very sound statistically. The four total-expenditure and price coefficients are all several times their estimated standard errors, the residuals are free of autocorrelation, and the category is subject to strong stock adjustment with a depreciation rate that, although perhaps a bit low, is nevertheless plausible. While the steady-state elasticities may seem implausibly small, it is to be kept in mind that the vast majority of households do not participate in the new car market when their economic circumstances change, but rather in the used car market21 . Table 15.67 New autos Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6

65.4653 0.6265 0.0284 0.0214 −2.9040 −0.2193 39.9992

4.73 10.30 3.48 3.80 −5.04 −3.08 3.86

α β δ μ μ: lr λ λ: lr

2713.27 −0.3808 0.0785 0.0336 0.00574 −3.4361 −0.5871

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

1.5899 0.2717 −1.5711 −0.2685 0.9685 2.31 −1.39

d–f = −9.54 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:    !

21 It follows from this that, in principle, demands for new and used cars should be modeled jointly,

the idea being that, initially, an increase in income (say) will be reflected in the increase in demand for used cars, which (since their stock is fixed) will lead to an increase in the price of the used cars. Eventually, the result will be an increase in the price of late-model used cars relative to the price of new cars, which then triggers the purchase of new cars by the relatively small group of households who are new car buyers. A model of this type in which the demands for new and used cars are modeled jointly is developed in Appendix 15.3 to this chapter.

15.8

Transportation

307

15.8.3 Net Purchases of Used Autos Expenditures in this category consist primarily of used car dealers’ margins and non-dealer net transactions.22 The estimated equation is a B-C model, which unfortunately is of poor statistical quality. A value of 0.25 for δ is imposed in the estimation, and price is of little consequence. Also, the steady-state totalexpenditure elasticity of 0.26 seems implausibly low, as is the R2 of 0.81 with time-series data. Table 15.68 Net purchases of used autos Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

69.9924 0.7110 −0.2461 0.00307 0.00154 −0.1403 −0.0701 3.7948

6.03 3.81 −1.37 2.40 2.40 −0.47 −0.47

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

38.0872 −0.3452 0.2500 0.00683 0.0126 0.00287 −0.3120 −0.5734 −0.1310

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.1161 0.6073 0.2551 −0.2442 −0.1329 −0.0558 0.8065 2.07

h = −0.73 d–f = −7.77 #obs. = 58. B-C Model, Annual, 1947−2004 A4 = 0.5A3, A6 = 0.5A5 (δ = 0.25), Model Quality: .

15.8.4 Other Motor Vehicles Expenditures for this category, which encompasses trucks and recreational vehicles, is the fastest growing component of households’ motor-vehicle expenditures. In 2004, the category accounted for a little less than 3% of the total expenditure; according to the above B-C model, the share for the category will eventually approach 5%. The model shows sluggish flow adjustment, habit formation, and steady-state elasticities that are highly elastic with respect to total expenditure and just short of elastic with respect to price. While habit formation clearly seems an anomaly, this obviously reflects the model’s way of taking into account the recent extremely rapid increase in expenditures for the vehicles in this category.

22 Net

transactions are measured by net change in the stock of used cars owned by persons. Included, as well, is an item, “employee reimbursement,” which presumably is small. (See “Methodology Paper Series, MP6,” U.S. Department of Commerce, Bureau of Economic Analysis, U.S. Government Printing Office, June 1990, p. 46.)

308

15

Annual PCE Models

Table 15.69 Other motor vehicles Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−5.0193 1.4418 −0.4918 0.00399 0.00231 −0.1889 −0.1111 0.4331

−0.24 7.18 −2.45 1.04 3.16 −1.66 −0.85

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−11.5901 0.1365 0.2939 0.0152 0.0109 0.0470 −1.1895 −0.5151 −2.2210

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.8130 1.8774 3.5055 −0.2189 −0.5054 −0.9436 0.9940 1.59

h=− d–f = −6.29 #obs. = 65. B-C Model, Annual, 1936−2004, A1 + A2 = 0.95, Model Quality: .

15.8.5 Tires, Tubes, Accessories and Other Parts Since the components of this category are obviously related to the stock of motor vehicles, a positive β is reasonable. Flow adjustment is rapid. Also, since tires and purchase of accessories can usually be postponed, their purchase in the face of price increases can often be put off until income has increased. Hence, an inelastic steadystate price elasticity, but elastic total-expenditure elasticity is reasonable. Table 15.70 Tires, tubes, accessories and other parts Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

3.9998 0.9041 −0.0981 0.00356 0.00144 −0.0714 −0.0290 −4.0002 1.4747

0.41 6.98 −0.66 2.19 2.20 −1.93 −0.87 −1.04

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

2.7889 0.0133 0.2028 0.00792 0.0142 0.00744 −0.1592 −0.2855 −0.1494

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw H

2.6203 1.4616 1.3718 −0.6661 −0.3715 −0.3487 0.9970 1.87 −

d–f = −7.78 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

15.8.6 Repair, Greasing, Washing, Parking, Storage, Rental and Leasing The B-C model for this category shows sluggish flow adjustment, stock-adjustment, an elastic steady-state total-expenditure elasticity, and inelastic price elasticity. Essentially, the only difference between the equation for this category and the

15.8

Transportation

309

Table 15.71 Repair, greasing, washing, parking, storage, rental and leasing Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

3.0182 1.2610 −0.4575 0.0115 0.00550 −0.4658 −0.2236 0.7029

0.14 7.02 −2.60 2.07 1.83 −0.96 −0.60

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

4.2853 −0.1713 0.2400 0.0480 0.0337 0.0280 −1.9502 −1.3707 −1.1379

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.4805 2.1064 1.2290 −0.3636 −0.5173 −0.3018 0.9953 1.92

h=− d–f = −7.86 #obs. = 70. B-C Model, Annual, 1947−2004, Model Quality: .

preceding one for tires, etc. is that stock adjustment prevails in place of habit formation. Short-term stock adjustment thus makes sense, because once maintenance and repairs occur, they should be expected to last for a while.

15.8.7 Gasoline and Oil Like tires and accessories, gasoline and oil consumption is related to the stock of motor vehicles; hence, the positive β obtained in the state-adjustment model for this category is indeed in order. The statistical quality of the estimated equation is excellent, for three of the four total-expenditure and price coefficients have t-ratios in excess of 2, and the residuals are well-behaved. Interestingly, both steady-state elasticities are inelastic. Indeed, the value of about −0.30 for the steady-state price elasticity points to a major reason why gasoline prices require such large price changes to equate supply and demand. Table 15.72 Gasoline and oil Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6

31.9948 0.8919 0.0107 0.00179 −0.9782 −0.1633 13.0042

1.95 20.52 2.69 2.15 −6.05 −1.25 2.08

α β δ μ μ: lr λ λ: lr

664.4745 0.0679 0.1822 0.0104 0.0165 −0.9475 −1.5106

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.3193 0.5088 −0.1928 −0.3073 0.9976 2.42 −1.79

d–f = −9.88 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:    !

310

15

Annual PCE Models

15.8.8 Bridge, Tunnel, Ferry and Road Tolls The equation for various transportation tolls is another excellent B-C model. The equations shows rapid flow adjustment, a negative β, and, not surprisingly, inelastic steady-state elasticities. Table 15.73 Bridge, tunnel, ferry and road tolls Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

2.0014 0.7140 0.0504 0.00715 0.00146 −0.0851 −0.0174 2.7069

1.84 6.70 0.49 2.43 2.36 −2.02 −1.90

α β δ μ M: fl μ: lr Λ λ: fl λ: lr

0.7394 −0.1072 0.1021 0.00127 0.00343 0.000620 −0.1511 −0.4091 −0.0737

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

4.1593 1.5366 0.7496 −3.4429 −1.2719 −0.6205 0.9660 1.99

h=− d–f = −7.36 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−2945 excluded), Model Quality:    !

15.8.9 Transportation Insurance This category consists of premium plus premium supplements less normal losses and dividends paid to policyholders for motor vehicle insurance. The equation estimated is a B-C model showing rapid flow adjustment, strong state inertia, and steady-state elasticities that are mildly inelastic. Table 15.74 Transportation insurance Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

9.9913 1.0214 −0.0785 0.00103 0.000643 −0.2593 −0.1626 1.4151

2.72 8.83 −0.60 1.67 1.56 −2.28 −1.81

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

7.0624 0.4154 0.5840 0.00220 0.00312 0.00764 −0.4941 −0.6992 −1.7117

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.3649 0.2579 0.8933 −0.3809 −0.2691 −0.9324 0.9910 1.87

h=− d–f = −7.66 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.8

Transportation

311

15.8.10 Purchased Local Transportation 15.8.10.1 Local Mass Transportation 15.8.10.2 Taxicab Expenditures for local transportation (Table 15.75) include expenditures for local mass transportation (Table 15.76) and taxicabs (Table 15.77). In 2004, this category represented less than two-tenths of 1% of total expenditure. State-adjustment models are estimated for the total and taxicabs, while a B-C model is estimated for local mass transit. The stock coefficient β is negative for Table 15.76, positive for Table 15.77, and slightly negative for the total. While total expenditure is in the models for both local mass transit and taxicabs, it is absent in the equation for the total. The steady-state total expenditure elasticities are inelastic; the steady-state price elasticity is elastic for taxicabs, but inelastic for local transit and the total. All three models leave things to be desired statistically. Table 15.75 Purchased local transportation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

11.8403 0.9068 − − −1.2591 −0.0713

1.14 23.53 − − −3.96 −0.74

α β δ μ μ: lr λ λ: lr

775.2458 −0.0395 0.0582 − − −1.2832 −0.7647

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

− − −1.4640 −0.8723 0.9817 2.80 −3.56

d–f = −11.46 #obs. = 71. State-Adjustment Model, Annual, 1931−2004 (1942−1945 excluded), No PCE, Model Quality: . Table 15.76 Local mass transportation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

13.9991 0.7828 0.1013 0.00978 0.00134 −0.3896 −0.0535 −8.0025 2.6891

0.95 4.18 0.62 1.03 0.44 −2.56 −0.30 −2.67

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

5.2059 −0.0340 0.0685 0.00173 0.00465 0.00116 −0.6892 −1.8534 −0.4606

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

1.3663 0.5081 0.3395 −3.0639 −1.1394 −0.7613 0.9913 2.08 −0.76

d–f = −8.50 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

312

15

Annual PCE Models

Table 15.77 Taxicab Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

13.9814 0.7482 0.000220 0.000112 −0.2537 −0.1288

2.62 10.29 0.51 0.54 −1.16 −1.13

α β δ μ μ: lr λ λ: lr

71.5824 0.3949 0.6829 0.000188 0.0000445 −0.2165 −0.5134

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.1286 0.3048 −0.8661 −2.0535 0.9053 2.25 −1.70

d–f = −9.24 #obs. = 71. State-Adjustment Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.8.11 Intercity Transportation Four categories comprise purchased intercity transportation, namely, expenditures for intercity railway (Table 15.79), intercity bus (Table 15.80), airline (Table 15.81), and other (Table 15.82). Except for intercity railway, for which a state-adjustment model is estimated, the equations for all of the category are B-C models. The equation for the aggregate (Table 15.78) shows rapid flow adjustment, slight habit formation, and steady-state elasticities that are moderately elastic and inelastic with respect to total expenditure and price, respectively. Table 15.78 Intercity transportation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

4.9976 0.9610 −0.1096 0.00313 0.00107 −0.1853 −0.0631 1.6932

0.73 7.69 −0.97 1.56 3.03 −2.12 −1.26

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

2.9516 0.000814 0.1704 0.00713 0.0121 0.00716 −0.4229 −0.7160 −0.4249

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.9097 1.1279 1.1333 −0.7947 −0.4693 −0.4716 0.9928 1.85

h = 2.45 d–f = −7.58 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−2945 excluded), Model Quality:  .

15.8.11.1 Intercity Railway 15.8.11.2 Intercity Bus Except for travel in the northeast corridor between Washington, D. C., and Boston, intercity rail travel (Table 15.79) is now almost an anomaly, hence, it is hardly

15.8

Transportation

313 Table 15.79 Intercity railway

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A1 A2 A3 A4 A5

4.0022 0.8133 −0.00109 −0.0000860 −0.2042 −0.0161

2.97 18.12 −1.97 −2.11 −3.57 −1.81

α β δ μ μ: lr λ λ: lr

177.1226 −0.1240 0.0819 −0.00116 −0.000460 −0.2163 −0.0861

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

−1.8499 −0.7360 −1.7067 −0.6790 0.9724 2.04 −0.18

d–f = −8.41 #obs. = 71. State-Adjustment Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

Table 15.80 Intercity bus Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

6.1145 0.6282 0.1342 − − −0.0797 −0.0418 3.0926

2.70 5.35 1.25 − − −4.03 −2.40

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

1.9771 0.0566 0.2623 − − − −0.1381 −0.4269 −0.1760

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

− − − −2.9530 −0.9549 −1.2175 0.9119 2.16

h = −3.52 d–f = −8.86 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), No PCE, Model Quality:    !

surprising to find it to be an inferior good with negative total-expenditure elasticities. Somewhat similarly, total expenditure is absent altogether in the equation for intercity bus travel (Table 15.80), though for this category steady-state demand is moderately elastic with respect to price.

15.8.11.3 Airline In the B-C equation for Table 15.81 (which is of only fair quality statistically), airline travel is seen to be rapidly flow-adjusting and mildly stock-adjusting. The steady-state total-expenditure elasticity is moderately elastic, but the price elasticity (consistent with the price coefficients having small t-ratios) is highly inelastic.

314

15

Annual PCE Models

Table 15.81 Airline Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−3.0999 1.1384 −0.2312 0.00390 0.000601 −0.0280 −0.00431 1.2989

−1.56 10.55 −2.02 3.67 1.75 −0.81 −0.85

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−2.3865 −0.0435 0.0771 0.0101 0.0132 0.00648 −0.0730 −0.0944 −0.0465

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

3.0072 2.3151 1.4801 −0.1649 −0.1269 −0.0811 0.9940 1.77

h=− d–f = −7.33 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−2945 excluded), Model Quality: .

15.8.11.4 Other Intercity Transportation Finally, in the equation for other purchased intercity transport (most of whose components are ancillary to airline travel),23 expenditures are seen to flow adjust quickly and to be subject to mild habit formation. Moreover, steady-state demand is elastic with respect to both price and total expenditure.

Table 15.82 Other intercity transportation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

1.2498 0.9042 −0.0259 0.000427 0.000179 −0.0599 −0.0251 1.9353

0.39 6.00 −0.16 2.14 0.74 −0.49 −1.43

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

0.6458 0.0793 0.2096 0.000915 0.00177 0.00147 −0.1284 −0.2484 −0.2065

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

2.3152 1.1963 1.9242 −1.7617 −0.9103 −1.4642 0.9908 1.91

h = 0.81 d–f = −7.81 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

23 Expenditures

in the category consist of baggage charges, coastal and inland waterway fares, travel agents’ fees, airport bus fares, and limousine services.

15.9

Recreation

315

15.9 Recreation Expenditures for recreation accounted for about eight and half percent of the total consumption expenditure in 2004, and, consistent with being associated with higher-order wants, is one of the fastest growing major components of consumption expenditure. In general, we should expect expenditures for many of the components of this category to be characterized by substantial inertia and strong habit formation. The above equation for the aggregate is a log flow adjustment that displays rapid flow adjustment and steady-state elasticities that (in absolute value) are both in excess of 1. However, the equation is not particularly good statistically, for the large Durbin-h statistic indicates the presence of strong positive autocorrelation in the residuals. Application of GLS unfortunately leads to a nonsense estimate of K1 . Table 15.83 Recreation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

0.1366 0.2388 0.9689 −0.8839 1.2289 0.1795

0.30 2.51 8.36 −7.06

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

1.5642 1.2728 −1.4270 −1.1612 1.5642 1.2728

srelastprice lrelastprice R2 dw H d–f

−1.4270 −1.1612 0.9983 1.19 5.67 −6.27

#obs. = 71. Log Flow-Adjustment Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

15.9.1 Books and Maps The equation for books and maps is an excellent state-adjustment model that displays habit formation and elastic steady-state price and total-expenditure elasticities24 . Table 15.84 Books and maps Coefficient

Value

t-ratio

Coefficient

Value

Coefficient Value

A0 A1 A2 A3 A4 A5 A6

11.0033 0.8385 0.00480 0.00117 −0.6500 −0.1581 −2.9959

2.10 11.08 5.30 2.50 −4.94 −2.28 −1.97

α β δ μ μ: lr λ λ: lr

146.1101 0.1012 0.2769 0.00459 0.00723 −0.6211 −0.9790

srelastpce 1.0316 lrelastpce 1.6261 srelastprice −0.8637 lrelastprice −1.3614 R2 0.9901 dw 1.79 h 1.20

d–f = −7.41 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:    ! 24 A

dummy variable separating the pre- and post-WW2 years is included in this model.

316

15

Annual PCE Models

15.9.2 Magazines, Newspapers and Sheet Music A state-adjustment model is also estimated for magazines, newspapers, and sheet music,25 but with different short-term dynamics than for books and maps. Slight stock-adjustment is indicated, rather than habit formation, and the steady-state elasticities are both inelastic. Table 15.85 Magazines, newspapers and sheet music Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

12.9995 0.8823 0.00410 0.000435 −0.8689 −0.0917

1.95 16.87 3.40 1.46 −3.36 −1.09

α β δ μ μ: lr λ λ: lr

460.8783 −0.0166 0.1069 0.00430 0.00373 −0.8610 −0.7454

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.6012 0.5205 −0.6854 −0.5934 0.9599 1.91 0.43

d–f = −7.88 #obs. = 71. State-Adjustment Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

15.9.3 Non-Durable Toys and Sport Supplies The equation for nondurable toys and sports supplies is a log flow-adjustment model that is of excellent statistical quality. Not surprisingly, steady-state demand is indicated to be elastic with respect to total-expenditure, but is inelastic with respect to price. Table 15.86 Non-durable toys and sport supplies Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 K4 

−6.4747 0.3565 1.0954 −0.2790 0.1628 0.9489m

−6.22 5.03 8.66 −3.70 3.96

κ μ: sr μ: lr λ: sr λ: lr srelastpce

−10.0610 1.7021 1.6151 −0.4335 −0.4113 1.7021

lrelastpce srelastprice lrelastprice R2 dw h

1.6151 −0.4355 −0.4113 0.9976 2.02 −0.10

d–f = −8.37 #obs. = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: !

25 Inclusion

anomalous.

of sheet music in the same category as magazines and newspapers seems a bit

15.9

Recreation

317

15.9.4 Wheel Goods, Sports and Photographic Equipment, Boats and Pleasure Aircraft The goods in this category are all associated with higher-order wants that are strongly related to the growth in income. Hence, it is not surprising that, despite goods in the category being durable, expenditures for them are subject to habit formation, with steady-state elasticities that are both highly elastic. The equation is an excellent one statistically. Table 15.87 Wheel goods, sports and photographic equipment, boats and pleasure aircraft Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

4.9843 0.9359 0.00742 0.000644 −0.5803 −0.0504

0.49 15.43 2.05 2.21 −2.30 −0.95

α β δ μ μ: lr λ λ: lr

212.6121 0.0246 0.0908 0.00733 0.0101 −0.5734 −0.7864

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

1.2408 1.7015 −1.1064 −1.5172 0.9573 2.35 −1.45

d–f = −8.95 #obs. = 71. State-Adjustment Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    !

15.9.5 Video and Audio Goods, including Musical Instruments and Computer Goods 15.9.5.1 Video and Audio Goods, including Musical Instruments 15.9.5.2 Computers, Peripherals and Software As with the “big-boy toys” of the preceding category (Table 15.87), expenditures for this category are among the three or four fastest growing components of consumer expenditure.26 While the equation for the aggregate of the category (Table 15.88) is a logarithmic flow-adjustment model,27 the equations for the two sub-components (Tables 15.89 and 15.90) are both B-C models. Despite the durable nature of video, audio, and computer goods, expenditures in both sub-categories are subject to habit formation and exhibit substantial steady-state total-expenditure elasticities (Table 26 In

1950, expenditures of this category represented less than four one-hundredths of one percent of total expenditure; in 2004, they accounted for nearly three percent. 27 Finding a plausible equation for 9.0 is rather problematic. In view of the B-C equations for the two sub-categories, one would think that a B-C (or state-adjustment model) model would apply to the aggregate as well, but plausible non-linear solutions were not able to be found. And while the log flow-adjustment model that is tabulated gives plausible results, the apparent positive autocorrelation in the residuals (as indicated by the Durbin-h statistic of 2.51) unfortunately cannot be corrected by GLS.

318

15

Annual PCE Models

Table 15.88 Video and audio goods, including musical instruments and computer goods Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3  κ

−6.5913 0.4281 1.2086 −0.4590 0.8008 −11.5261

−4.79 6.02 6.99 −6.69

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

1.6926 2.1135 −0.6427 −0.8026 1.6926 2.1135

srelastprice lrelastprice R2 dw h d–f

−0.6427 −0.8026 0.9953 1.52 2.51 −7.45

#obs. = 45. Log Flow-Adjustment Model, Annual, 1960−2004, Model Quality:  .

Table 15.89 Video and audio goods, including musical instruments Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−10.0000 1.2005 −0.2996 0.00243 0.00158 −0.00175 −0.00114 0.8738

−2.53 4.44 −0.95 1.62 2.88 −0.64 −1.27

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−11.4446 0.1447 0.3262 0.00889 0.00777 0.0160 −0.0605 −0.0560 −0.1151

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

1.3974 1.5993 2.8748 −0.3666 −0.4195 −0.7541 0.9987 1.63

h=− d–f = −6.61 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−2945 excluded), Model Quality .

Table 15.90 Computers, peripherals and software Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−65.0000 1.3003 −0.4003 0.00764 0.00411 −0.00436 −0.00234 0.6932

−0.82 1.92 −0.59 0.71 1.03 −1.76 −1.24

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−93.7680 0.0553 0.2690 0.0327 0.0226 0.0411 −0.0186 −0.0129 −0.0234

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

6.7901 9.7953 12.3324 −0.5596 −0.8072 −1.0164 0.9989 1.94

h=− d–f = −4.64 #obs. = 25. B-C Model, Annual, 1981−2004 A1 + A2 = 0.90, Model Quality: .

15.9

Recreation

319

15.90, especially), while the steady-state price elasticities are of the order of −0.75 to −1.00.

15.9.6 Radio and Television Repair Since it is now usually less expensive to buy a new radio or TV than to repair existing sets (which are probably seriously technologically obsolete, anyway), it is not surprising to find expenditures in this category to be inferior (which, in view of the positive β, is itself indicated to be habit-forming!). The essentially unit steadystate price elasticity is, of course, consistent with price being the primary driver in decisions as to whether to repair or buy new. Table 15.91 Radio and television repair Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

5.4990 0.8645 −0.000581 −0.0000816 −0.1080 −0.0152

2.78 20.66 −4.37 −1.41 −3.63 −2.07

α β δ μ μ: lr λ λ: lr

135.6909 0.00579 0.1511 −0.000579 −0.000603 −0.1076 −0.1120

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

−0.5558 −0.5780 −0.9547 −0.9928 0.9661 1.76 0.93

d–f = −6.66 #obs. = 58. State-Adjustment Model, Annual, 1947−2004, Model Quality:    !

15.9.7 Flowers, Seeds and Potted Plants In this excellent state-adjustment equation, expenditures for flowers and potted plants are indicated to be habit-forming, with moderately elastic steady-state price and total-expenditure elasticities. Table 15.92 Flowers, seeds and potted plants Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

4.0000 0.9002 0.000688 0.000223 −0.0801 −0.0268

2.78 20.66 −4.37 −1.41 −3.63 −2.07

α β δ μ μ: lr λ λ: lr

37.9350 0.2957 0.4007 0.000586 0.00223 −0.0702 −0.2677

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.2719 1.0369 −0.3101 −1.1829 0.9925 1.63 1.83

d–f = −6.31 #obs. = 58. State-Adjustment Model, Annual, 1947−2004, Model Quality:    !

320

15

Annual PCE Models

15.9.8 Admissions to Specified Spectator Amusements 15.9.8.1 Motion Picture Theaters 15.9.8.2 Legitimate Theaters and Opera and Entertainments of Nonprofit Institutions (except Athletics) 15.9.8.3 Spectator Sports This broad category comprises admissions to motion pictures, legitimate theater and opera, and spectator sports.28 B-C equations are estimated for motion pictures (Table 15.94), and theater and opera (Table 15.95), as well as for the aggregate Table 15.93, while a state-adjustment equation is estimated for spectator sports (Table 15.96). Short-run flow adjustment is rapid in the B-C models. Theater and opera and spectator sports are subject to habit formation, while motion-picture expenditures are stock-adjusting. The steady-state total-expenditure elasticity is comfortably greater than 1 for both theater and opera and spectator sports, but less than 1 for motion pictures. The steady-state price elasticities are inelastic except for theater and opera. Given the attachment that most sports fans have to their college and professional teams, a small price elasticity for expenditures for spectator sports is obviously hardly surprising. Statistical quality of the models range from fair to good.

Table 15.93 Admissions to specified spectator amusements Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

9.9866 0.8010 0.0923 0.0122 0.000142 −0.4574 −0.0530 −6.0133 2.6762

2.10 7.23 0.98 2.09 0.55 −1.64 −0.47 −2.02

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

3.7317 −0.0876 0.00579 0.0215 0.0575 0.00133 −0.8004 −2.1419 −0.0497

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

9.7383 3.6389 0.2258 −2.0663 −0.7721 −0.0479 0.9272 2.66 −8.34

d–f = −10.67 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

28 This

category consists of admissions to professional and amateur athletic events and racetracks.

15.9

Recreation

321 Table 15.94 Motion picture theaters

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

12.4855 0.8803 0.0240 0.00146 0.000160 −0.3832 −0.0420 −9.0115 2.2549

3.04 4.35 0.14 1.76 1.15 −1.37 −0.89 −2.48

α β δ μ μ: fl μ:lr λ λ: fl λ: lr

5.5370 −0.0366 0.0548 0.00279 0.00628 0.00167 −0.7324 −1.6516 −0.4393

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

2.0092 0.8910 0.5344 −2.8627 −1.2695 −0.7614 0.9785 2.80 −35.51

d–f = −11.47 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: . Table 15.95 Legitimate theaters and opera and entertainments of nonprofit institutions (except athletics) Coefficient

Value

t−ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ −

8.9978 0.6643 −0.0913 0.000692 0.00111 −0.1217 −0.1956 1.7536 −

2.07 4.72 −1.12 1.63 2.23 −1.65 −2.17 − −

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

5.1312 0.2484 0.8032 0.00180 0.00315 0.00260 −0.3163 −0.5547 −0.4580

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw −

2.1606 1.2321 1.7837 −2.1133 −1.2051 −1.7446 0.9388 2.17 −

h=− d–f = −8.84 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  . Table 15.96 Spectator sports Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 −

−0.0790 0.8115 0.00164 0.000345 −0.0323 −0.00682 −

−0.04 5.24 2.41 1.10 −0.35 −0.54 −

α β δ μ μ: lr λ λ: lr

−1.2130 0.0277 0.2359 0.00161 0.00182 −0.0319 −0.0361

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

1.1064 1.2538 −0.1351 −0.1531 0.9855 1.99 0.03

d–f = −8.22 #obs. = 58. State-Adjustment Model, Annual, 1947−2004, Model Quality: .

322

15

Annual PCE Models

15.9.9 Clubs and Fraternal Organizations This category consists of current expenditures (including consumption of fixed capital) of non-profit clubs and fraternal organizations and dues and fees paid to proprietary clubs. The estimated B-C model shows fairly rapid flow adjustment, habit formation, and steady-state demand that is highly elastic with respect to both price and total expenditure.

Table 15.97 Clubs and fraternal organizations Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ −

8.9973 1.1573 −0.2759 0.000975 0.00513 −0.2277 −0.1198 1.0249 −

2.29 8.30 −2.26 1.29 2.18 −1.06 −2.32 − −

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

8.7788 0.0728 0.2630 0.00313 0.00321 0.00432 −0.7305 −0.7486 −1.0100

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw −

1.0727 1.0467 1.4471 −1.6192 −1.5799 −2.1844 0.9818 2.11 −

h=− d–f = −8.65 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

15.9.10 Commercial Participant Amusements: The logarithmic flow-adjustment model for this eclectic collection of recreational activities29 displays weak flow inertia, and steady-state demand that is highly elastic with respect to total expenditure, but inelastic with respect to price. Given that most of the expenditures in the category are leisure-time activities associated with higherorder wants, a substantial long-run total-expenditure elasticity is of course to be expected.

29 The

category consists of billiard parlors, bowling alleys, dancing, riding, shooting, skating and swimming places, amusement devices and parks, golf courses, skiing facilities, marinas, sightseeing, private flying operations, casino gambling, recreational equipment rental, and other commercial participant amusements.

15.9

Recreation

323 Table 15.98 Commercial participant amusements

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 K4 θ

−9.2769 0.3783 1.4185 −0.2465 −0.4342 0.9022

−5.62 3.41 5.27 −1.04 −3.95

κ μ: sr μ: lr λ: sr λ: lr srelastpce

−14.9207 2.0584 2.2815 −0.3965 −0.3577 2.0584

lrelastpce srelastprice lrelastprice R2 dw h

2.2815 −0.3965 −0.3577 0.9766 2.34 −4.03

d–f = −9.71 #obs. = 71. Log Flow-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: !

15.9.11 Pari-Mutuel Net Receipts The B-C model for this category shows extremely rapid flow adjustment, habit formation (that dissipates slowly relatively to its strength), and steady-state elasticities that are moderately elastic for total expenditure and extremely elastic with respect to price. That race-track betting is habit-forming (indeed, nearly addicting) is of course not surprising; however, the extreme large long-run price elasticity seems a bit high). Table 15.99 Pari-mutuel net receipts Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

5.4978 0.6993 0.2630 0.000178 0.0000566 −0.2008 −0.0638 3.4110

2.11 6.99 2.94 1.12 0.87 −2.25 −1.58

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

1.6118 0.1280 0.1588 0.000291 0.000991 0.00150 −0.3278 −1.1182 −1.6915

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

0.7132 0.2091 1.0790 −4.8083 −1.4096 −7.2733 0.9211 1.98

h = 0.1 d–f = −8.10 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:  .

15.9.12 Other Recreation The estimated equation for this category is a state-adjustment model of excellent statistical quality. Since the category includes a number of leisure-time activities

324

15

Annual PCE Models

Table 15.100 Other recreation Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6

31.9574 0.9090 0.0106 0.00287 −0.4669 −0.4132 −2.9959

3.48 36.78 4.11 4.16 −1.92 −4.82 −2.88

α β δ μ μ: lr λ λ: lr

179.8938 0.5930 0.6812 0.00430 0.0331 −0.6173 −4.7490

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

0.3187 2.4516 −0.3246 −2.4969 0.9992 2.05 −0.19

d–f = −8.43 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:    !

connected to income growth and higher-order wants,30 it is no surprise to find strong habit formation and steady-state elasticities that are highly elastic with respect to both price and total expenditure.

15.10 Education 15.10.1 Higher Education 15.10.2 Nursery, Elementary and Secondary Education 15.10.3 Other Education Expenditures in the education category (Table 15.101) consist of expenditures for higher education (Table 15.102),31 nursery, secondary, and elementary education (Table 15.103),32 and other education (Table 15.104).33 The estimated equation for the aggregate education, as well as for higher and secondary education, is a 30 The

category consists of lotteries, pet and pet services, cable TV, film processing, photographic studios, sporting and recreational camps, video rentals, internet access fees, and recreational services not elsewhere classified. 31 For private institutions, expenditures in this category consist of current expenditures (including consumption of fixed capital) less receipts (such as those from meals, rooms, and entertainment) accounted for elsewhere in consumer expenditures, and less expenditures for research and development financed under contracts or grants. For government institutions, expenditures equal student payment of tuition. 32 For this category, expenditures are of the same type as for higher education, except that expenditures for child day-care services are included in religious and welfare activities. 33 This category consists of (1) fees paid to business schools and computer and management training, technical and trade schools, other schools and instruction, and educational support services and (2) current expenditures (including consumption of fixed capital) by non-profit research organizations and by grant-making foundations for education and research.

15.10

Education

325 Table 15.101 Education

Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

−0.3251 0.9247 0.1704 −0.1956 0.7827 −4.3156

−1.59 26.19 3.81 −4.03

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.1771 2.2624 −0.2032 −2.5963 0.1771 2.2524

srelastprice lrelastprice R2 dw h d–f

−0.2032 −2.5963 0.9970 2.02 −0.10 −8.35

#obs. = 71. Log Flow-Adjustment Model, Annual, 1931−2004, Model Quality:    !

Table 15.102 Higher education Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

−0.9631 0.8148 0.2783 −0.1645 0.2041 −5.2004

−1.91 16.62 3.87 −2.97

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.3067 1.5026 −0.1812 −0.8880 0.3067 1.5026

srelastprice lrelastprice R2 dw h d–f

−0.1812 −0.8880 0.9913 2.08 −0.35 −8.57

#obs. = 71. Log Flow-Adjustment Model, Annual, 1931−2004, Model Quality:    !

Table 15.103 Nursery, elementary and secondary education Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

K0 K1 K2 K3 θ κ

−1.5498 0.7798 0.4535 −0.4096 0.2474 −7.0389

−3.22 10.71 3.63 −1.90

μ: sr μ: lr λ: sr λ: lr srelastpce lrelastpce

0.5085 2.0551 −0.4603 −1.8603 0.5085 2.0551

srelastprice lrelastprice R2 dw h d–f

−0.4601 −1.8603 0.9679 1.94 0.30 −8.02

#obs. = 71. Log Flow-Adjustment Model, Annual, 1931−2004, Model Quality:    !

326

15

Annual PCE Models

Table 15.104 Other education Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

17.9964 0.7882 −0.0801 0.00159 0.00275 −0.2001 −0.3473 −6.0036 1.4139

1.28 4.11 −0.43 1.16 3.19 −1.45 −1.58 −2.54

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

12.7278 0.4261 0.8680 0.00480 0.00679 0.00943 −0.6059 −0.8567 −1.1901

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

1.3754 0.9738 1.9107 −1.0135 −0.7168 −1.4080 0.9924 2.04 −

d–f = −8.36 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality: .

logarithmic flow-adjustment model, while a B-C model is estimated for other education.34 Short-run inertia is seen to mark expenditures for Tables 15.102 and 15.103. Steady-state elasticities for total expenditure are greater than 1 in all of the categories, while steady-state price elasticities are inelastic only for higher education. The estimated equations are of excellent statistical quality, except for other education, which is of only fair quality.

15.11 Religious and Welfare Activities The estimated equation for eleemosynary activities is a B-C model of excellent statistical quality.35 Expenditures in the category are strongly habit-forming, flow adjust rapidly, and have highly elastic steady-state price and total-expenditure elasticities. In light of present trends, the long-run equilibrium budget share is indicated to be about four and half percent of the total expenditure, which compares with a bit more than two and a half percent in 2004.

34 A

dummy variable separating pre- and post-WW2 years is included in the model for other education. 35 For non-profit institutions, expenditures equal current expenditures (including consumption of fixed capital) of religious organizations, child day-care services (excluding educational programs), social advocacy organizations, human rights organizations, civic and social organizations, residential mental health and substance abuse facilities, homes for the elderly, other residential care facilities, social assistance services, political organizations, museums, libraries, and grant-making and giving services. The expenditures are net of receipts (such as those from meals, rooms, and entertainment) that are accounted for elsewhere in consumer expenditures, and exclude relief payments with the U.S. and expenditures by grant-making foundations for education and research. For proprietary and government institutions, expenditures equal receipts from users.

15.12

Foreign Travel and Other, Net

327

Table 15.105 Religious and welfare activities Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 A7 γ

69.0625 0.5669 0.2103 0.00960 0.00905 −1.3116 −1.2368 −10.8223 3.4254

2.60 4.35 1.83 4.21 3.22 −1.94 −3.16 −1.73

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

20.1517 0.2797 0.4715 0.0165 0.0566 0.0406 −2.2579 −7.7342 −5.5522

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw h

2.5058 0.7315 1.7989 −2.0942 −0.6114 −1.5034 0.9984 2.04 −

d–f = −7.23 #obs. = 70. B-C Model, Annual, 1931−2004 (1942−1945 excluded), Model Quality:    !

15.12 Foreign Travel and Other, Net 15.12.1 U. S. Foreign Travel 15.12.2 Expenditures Abroad by U.S. Residents While a model has been estimated for categories 12.0, 12.1, and 12.2,36 equations have not been estimated for the two balancing PCE foreign categories, 12.3 and 12.4. The equation for the aggregate is a logarithmic flow-adjustment model, while the equations for categories 12.1 and 12.2 are state-adjustment and B-C models, respectively. Interestingly, both Tables 15.107 and 15.108 display a negative β, indicating that travel and expenditures abroad are viewed as durable goods. Steady-state demand is highly elastic with respect to total expenditure for both categories, but inelastic, especially for expenditures abroad, with respect to price. Table 15.106 Foreign travel and other, net Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5

0.2577 0.9637 0.00191 0.0000338 −0.5666 −0.0100

−0.04 21.35 0.43 0.28 −2.16 −0.18

α β δ μ μ: lr λ λ: lr

−56.7840 −1.0191 0.0178 0.00193 0.00931 −0.0572 −0.2759

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

1.8525 0.8936 −4.4234 −2.1338 0.9205 1.61 1.73

d–f = −7.29 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality: .

36 Beginning

in 1981, category expenditures abroad by U.S. residents includes U.S. students’ expenditures abroad.

328

15

Annual PCE Models

Table 15.107 U.S. foreign travel Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6

83.0001 0.5973 0.00259 0.00498 −0.3675 −0.7078 −20.0049

2.20 3.58 2.29 2.48 −2.52 −2.34 −2.01

α β δ μ μ: lr λ λ: lr

1012.48 −0.1838 0.2663 0.0195 0.0115 −0.7429 −0.4395

srelastpce lrelastpce srelastprice lrelastprice R2 dw h

2.3241 1.3691 −0.7122 −0.4113 0.9911 2.08 −0.47

d–f = −8.57 #obs. = 71. State-Adjustment Model, Annual, 1930−2004 (1942−1945 excluded), Model Quality:    !

Table 15.108 Expenditures abroad by U.S. residents Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4 A5 A6 γ

−9.9986 1.2780 −0.3180 0.00347 0.000702 −0.1311 −0.0265 1.0835

−2.41 4.96 −1.24 2.83 2.66 −1.56 −1.88

α β δ μ μ: fl μ: lr λ λ: fl λ: lr

−9.2278 −0.0612 0.0101 0.0989 0.1071 0.0140 −0.3744 −0.4057 −0.0531

flelastpce srelastpce lrelastpce flelastprice srelastprice lrelastprice R2 dw

44.0696 40.6721 5.7720 −0.9506 −0.8773 −0.1245 0.9550 2.09

h = −1.65 d–f = −8.01 #obs. = 70. B-C Model, Annual, 1947−2004, A1 + A2 = 0.95, Model Quality:  .

Appendix 15.1 Demand for Telecommunications The model for telephone expenditures is one of the least satisfactory of the entire study. However, telephone demand is much studied (including two books on the subject by one of the authors), and many excellent models now exist in the literature.37 In this appendix, we will describe the basic framework that guides the specification of many of these, as well as presenting an approach that can be used to estimate market demand functions for goods and services for which there are no historical data. 37 This

appendix scarcely scratches the surface of this literature. Taylor (1994) provides a comprehensive review. See also Taylor (2002, 2003).

Appendix 15.1 Demand for Telecommunications

329

15.1.1 The Access/Usage Framework for Analyzing Telecommunications Demand The thing that most distinguishes telecommunications demand from the demands for most goods and services is the fact that telecommunications services are not consumed in isolation, but involve a network. There must be access to the network before it can be used, which means that there is a clear-cut distinction between access and usage. Telephone companies charge for both, so the distinction is important for modeling telecommunications demand. The standard approach for modeling this distinction is a consumer-surplus framework, in which the consumer surplus from usage (conditional on access) determines whether access will in fact be demanded. The procedure, simply illustrated using a convenient static demand model for usage, is as follows. Specifically, let the demand function for usage be given by:38 q = Ae−αp yβ eu ,

(15.5)

where q denotes telephone usage, p the price per unit of usage, y income, and u a random error term. The consumer surplus from usage (conditional on access) will be given by: CS = =

∞

−αz yβ eu dz p Ae −αp β Ae y eu , α

(15.6)

or in logarithms: ln CS = a − αp + β ln y + u,

(15.7)

where a = lnA. Let π denote the cost of access to the network. Access will then be demanded whenever the consumer surplus from usage is greater than this cost, that is, if: CS >π ,

(15.8)

or, equivalently in logarithms, whenever: a − αp + β ln y + u > ln π .

(15.9)

Rearranging, we then find that access will be demanded if: 38 This

model is often referred to as the “Perl” model, as it was first applied in the context of telephone demand by Lewis Perl in an unpublished 1983 study [Perl (1983)]. Its attractive feature is that it allows for a usage price of 0 (which was very useful at the time because of the interest in implementing usage-sensitive pricing for local calls).

330

15

u > ln π − a + αp − β ln y.

Annual PCE Models

(15.10)

Since u is assumed to be random, Equation (15.10) relates the demand for access to usage and the price of access as a probability function, namely as: P(access) = 1 − ( ln π − a + αp − β ln y),

(15.11)

where Φ denotes the distribution function for u. In most circumstances, Equation (15.11) is estimated as either a probit or logit model, depending upon the assumption that is made concerning the distribution of u. If u is assumed to be normal, then equation (15.11) is estimated as a probit model, while if u is specified to have a logistic distribution, then the model is estimated straightforwardly as a logit model. The access/usage framework can be applied to either individual or aggregate data. If the data are individual, the model is usually interpreted as a random-utility model, in which case the error term u is seen as representing a draw from a universe of utility functions described by the distribution function Φ. On the other hand, if the data are aggregate, then Φ can be interpreted as describing the distribution of consumer surplus in a population of consumers. Both types of models are well represented in the literature.39 As an illustration, we will describe briefly a model that applies the foregoing to the demand for residential access to the telephone network that was developed at Southwestern Bell in the mid-1980s for the purpose of evaluating the impacts of higher local-service rates on its residential customers. The model was estimated from a data set consisting of census tracts (using data from the 1980 Census) in the five states (Arkansas, Kansas, Missouri, Oklahoma, and Texas) that were then served by Southwestern Bell.40 With census tracts as the unit of observation, equation (15.11) must be aggregated over the households in a tract. The procedure in doing this is over the joint distribution of u and y, where u and y are assumed to be independently distributed, u normal and y log-normal. Let v = u + β ln y.

(15.12)

It then follows that v will be N(μy , σ 2 + β 2 σy2 ). Let Pj denote the proportion of households in census tract j that have a telephone. From expression (15.11), this proportion will be given by Pj = P(vj > ln πj − a + αpj ),

39 See

(15.13)

Chapter 5 and 7 of Taylor (1994) and the references listed therein. Train, McFadden, and Ben-Akiva (1987) merit especial mention. 40 The presentation that follows is taken from Taylor and Kridel (1990) and Chapter 5 of Taylor (1994).

Appendix 15.1 Demand for Telecommunications

331

or equivalently by     Pj = P wj > w∗j = 1 −  w∗j ,

(15.14)

where v − βμyj wj =  1/2 s. σ 2 + β 2 σyj2

(15.15)

Let Fj denotes the probit for census tract j calculated from equation (15.14), and write: Fj =

ln πj − a + αpj − β ln yj .  1/2 2 2 2 σ + β σyj

(15.16)

Clearing the fraction, we have  1/2 Fj σ 2 + β 2 σyj2 = ln πj − a + αpj − βμyj + εj∗ ,

(15.17)

1/2  . εj∗ = σ 2 + β 2 σyj2

(15.18)

where

Equation (15.18) offers a number of challenges for estimation, but before discussing these the possibility that households in some areas might be able to choose between either flat-rate or measured service should be taken into account.41 Three possibilities are to be considered: 1. Only flat-rate service is available; 2. Only measured service is available; 3. Both flat-rate and measured service are available. In flat-rate only areas, access will be demanded if [from equation (15.11), since p = 0] u > ln πf − a − β ln y,

(15.19)

while in measured only areas, access will be demanded if u > ln πm − a + αp − β ln y,

(15.20)

where πf and πm denote the monthly fixed charges for flat-rate and measured service, respectively. In areas in which both flat-rate and measured services are options, 41 With

measured service, there is a usage charge for local calls, unlike for flat-service, for which the cost of local calls are included in the access fee.

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Annual PCE Models

access will be demanded if either expressions (15.19) or (15.20) hold. From these two inequalities, it follows that access will be demanded if   u > min ln πf , ln πm − a − β ln y .

(15.21)

Finally, given that access is demanded, flat-rate service will be selected if ln πf < ln πm + αp,

(15.22)

while measured service will be demanded if the inequality goes the other way. To incorporate these choices into the analysis, we define: δ1 = 1 if only flat-rate service is available, and 0 otherwise; δ2 = 1 if only measured service is available, and 0 otherwise; δ3 = 1 if both flat-rate and measured service is available, and 0 otherwise. With these dummy variables, we can rewrite equation (15.18) as  1/2 = δ1 ln πfj + δ2 ln δ2 ln πmj Fj σ 2 + β 2 σyj2 +δ3 min ( ln πfj , ln πmj + αpj ) − a + αpj − β ln μyj + εj∗ .

(15.23)

With addition of a number of socio-demographic variables as predictors, this is the equation to be estimated. Several things are to be noted regarding the estimation: (1) The equation is nonlinear in β. (2) σ , the variance of u is a parameter to be estimated. (3) min(lnπ f , lnπ m +α p) must be calculated for the areas in which both flat-rate and measured services are available, but doing this requires knowledge of α. Estimation begins by first rewriting expression (15.1.23) as:42 zj = −a + αpj − βμyj + εj∗ ss,

(15.24)

where  1/2 − δ1 ln πfj − δ3 min ( ln πfj , ln πmj + αpj ). zj = Fj σ 2 + β 2 σyj2

(15.25)

If σ 2 , β, and α were known, zj could be calculated (using observed values for Fj , σy2 , π f , and π m , and equation (15.25) could be estimated as a linear regression. none of the Southwestern Bell’s service territory had mandatory measured service, δ 2 is dropped.

42 As

Appendix 15.1 Demand for Telecommunications

333

However, these parameters are not known, so estimation proceeds by an iterative search procedure as follows: 1. The initial search is over values of σ 2 , β, and α. The value of σ 2 that is finally selected is the one that maximizes the correlation between the actual and predicted values of Fj . 2. The second step is to fix σ 2 at the value obtained in Step 1, keep β fixed and search over α. Again, the value of α selected is the one that maximizes the correlation between the actual and predicted values of Fj . 3. The final step involves interation on β, where an iteration consists of the estimation of equation (15.24), with zj [from equation (15.25)] calculated using the values of σ 2 and α obtained from Steps 1 and 2 and β from the immediately preceding iteration. Iteration stops when the estimate of β stabilizes.43 The model, with a number of socio-demographical variables included as predictors, was applied to a data set consisting of 8,423 census tracts from the 1980 census in the five states that were served at the time by Southwestern Bell (namely, Arkansas, Kansas, Missouri, Oklahoma, and Texas). The coefficients and t-ratios from the estimation are presented in Table 15.109. As expected, the sign for income is positive, while the signs of the price terms are negative, as are local-loop milage charges (mileage). With regard to the socio-demographical variables, penetration (access demand) is lower for black, Hispanic, and American Indian households, lower for renters than for homeowners, and lower in rural areas than in urban. Unemployment has a negative effect, while residence longevity (immob) has a positive effect, as do average age and size of household (avgage and avgsizehh), and the number of lines that can be reached in the local-calling area (lines). However, because of the way that the model is structured, care is required in interpreting the coefficients. The coefficient for income, 0.98, represents the elasticity for usage with respect to income. Likewise, 0.51 times the mean (0.52) of immob is the usage elasticity with respect to mobility and similarly for the other variables. The access elasticities, on the other hand, cannot be observed directly, but must be induced by changing a variable, and then calculating the resulting change in penetration. For example, consider a 10% increase in income. This yields a 0.42% increase in penetration, and hence an access elasticity of 0.042.44

43 The estimation procedure was evaluated before applying it to the Southwestern Bell data set by a

Monte Carlo study. The Monte Carlo results showed that the estimators have moderate biases that appear to diminish with sample size. The results also show that the variance of u has little effect on the accuracy of the estimated coefficients. For details, see Taylor and Kridel (1990) and Kridel (1987). 44 Two points should be kept in mind in assessing this calculation. First, since the variance of income is also an argument in the model, only small changes in income should be considered. Second, the implied elasticity income elasticity may seem small. However, since 93% of the households in the data set (on average) already had telephone service, an increase in income can have no effect on their access decision.

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Annual PCE Models

Table 15.109 Southwestern bell access demand model Variable

Coefficient

t-ratio

Constant Income Price of local minute Price of access Renter Rural Black Hispanic Amerindian Immob Avgage Mileage Employment Lines Avgsizehh Nuisance variance (σ 2 ) R2

0.43 0.98 −6.87 −1.00 −1.56 −0.82 −1.18 −2.78 −7.38 0.51 0.05 −0.45 2.88 0.37 0.55 5.60 0.42∗

1.2 19.0 − − −14.3 −19.0 −12.9 −24.9 −14.2 4.7 9.9 3.8 14.3 4.8 10.3 −

∗

The R2 is calculated as the square of the correlation between the actual and predicted values for Fj . Source: Taylor and Kridel (1990, Table 1).

As noted, the main purpose of the Southwestern Bell study was to quantify the impact on penetration in the company’s jurisdictions of higher access charges.45 Table 15.110 shows the predicted effects on penetration of a doubling of flat-rate access charges under two different scenarios, the first holding measured-services rates, where available, constant (as a lower-priced alternative), and the second a doubling of all rates. The table shows the (then) actual penetration rates, the predicted penetration rates under the new prices, and the implicit elasticities associated with each of the scenarios.46 While the calculated access-price elasticities are small in all cases, they are seen to vary inversely with the level of penetration, and to be moderated by the presence of a lower-priced alternative.47 45 Historically,

residential telephone service in the U.S. (as in most countries) was heavily cross-subsidized by revenues from long-distance service. However, competition in long-distance markets, together with complications caused by the breakup of the old Bell System in 1984, put considerable pressure on local telephone companies to substantially increase local rates, which in turn caused great concern as to the effects that these increases might have on low-income households. Hence, the motivation for the Southwestern Bell study [as well as the earlier one of Perl (1983)]. 46 The elasticities are calculated as arc elasticities between the existing and predicted penetration rates. 47 While access-price elasticities were (and still are) invariably found to be small, the problem with them is that tiny elasticities applied to huge customer bases give large numbers, and this caused great alarm at the time to those concerned with universal telephone service (i.e., the availability to low-cost telephone service to virtually every household). In Texas, for example, the predicted

Appendix 15.1 Demand for Telecommunications

335

Table 15.110 Estimated effects on telephone penetration, 100 percent increase in local rates southwestern bell study With lower-priced alternative∗

100% increase in all rates∗∗

State

Actual Penetration

Penetration

Elasticity

Penetration

Elasticity

Arkansas Kansas Missouri Oklahoma Texas All states

89.1 95.3 95.4 92.7 91.4 92.5

83.6 − 92.3 − 89.3 89.6

−0.0636 − −0.0330 − −0.0232 −0.0319

83.2 93.0 92.3 89.3 87.7 88.8

−0.0685 −0.0244 −0.0330 −0.0374 −0.0413 −0.0408

∗ Measured service, where available, is the lower-priced alternative. Measured service was not available in Kansas and Oklahoma. ∗∗ Both flat-rate and measured-rate, where available, are doubled. Source: Taylor and Kridel (1990, Table 2).

15.1.2 A Generic Model of Usage Demand In this section, we will discuss a model for analyzing usage demand that, in one form or another, has found widespread application to the analysis of the demand for toll calling, not only in the U.S., but in many other countries as well. To fix ideas, suppose that we have two telephone exchanges, A and B, which are not part of the same local-calling area, so that calls between the two exchanges are toll calls. Let the number of telephones in the two areas be T and R, respectively. The total number of possible connections between the two exchanges will therefore be T·R. Let M denote the number of calls that “sent-paid” from A to B during some time period (a quarter, say), and let θ denote the proportion of the potential connections (T·R) that are realized, so that M = θ (T · R).

(15.26)

Expression (15.26) might describe the relationship that would be observed in a stationary world, i.e., where income, price and all other factors that affect toll calling are constant. However, these other factors do not remain constant, and we can allow for them through the value of θ . In particular, let us suppose that income (y) and price (p) of a toll call from A to B affect θ according to θ = syβ pγ,

(15.27)

decrease in Table 15.110 of 3.7% points in penetration from a doubling of access prices translated into more than 200,000 households, which was a frighteningly large number. Numbers like this were grist to consumer groups, congressmen, state legislators, and regulators who were opposed to higher access charges, and attempts by telephone companies to increase local rates not surprisingly triggered large choruses of emotional response.

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Annual PCE Models

where a, β, and γ are constants. The relationship in expression (15.26) accordingly becomes M = ayβ pγ (T · R).

(15.28)

Finally, let us suppose that M is affected by a change in potential toll connections that may be either more or less than proportional, in which case the model becomes M = ayβ pγ (T · R)λ ,

(15.29)

where λ is a constant, presumably of the order of 1. Taking logarithms of both sides of expression (15.29), we then obtain: ln M = α + β ln y + γ ln p + λ ln (T · R)

(15.30)

Since most applications of this model has been with time-series data, dynamical considerations can easily be taken into account through postulation of a flow-adjustment mechanism as follows:48 ˆ = α + β ln y + γ ln p + λ ln (T · R) ln M

(15. 31)

  ˙ M ˆ , = ϕ ln M − ln M M

(15.32)

in which case one form of an estimating model would be: ln Mt = A0 + A1 ln Mt−1 + A2 yt + A3 pt + A4 ln (T · R) + ut .

(15.33)

An example of elasticities obtained with the model in expression (15.30), applied to quarterly times-series data of Bell Canada for Ontario and Quebec, is given in Table 15.111.49 The time period covered is 1974Q1 through 1983Q3. Four toll markets are analyzed in this application: two length of hauls in distance (100 miles or less and greater than 100 miles) and peak and off-peak tariff periods. The results for the four estimated models are presented in Table 15.111. The price elasticities are seen to lie in the range of −0.324 (sum of price 1 and price 2) for off-peak shorthaul to −0.370 for peak long-haul, while income elasticities vary between 0.469 for 48 In

some applications, the “market size” variable in expressions (15.30) and (15.33), ln(T⊕R), is decomposed into lnT + lnR, taking lnT to the left hand (with a coefficient of 1), and by expressing the dependent variable as the logarithm of calls per telephone. Income is then usually expressed as income per capita or income per household. “Market size” as a predictor is then measured as the number of telephones that can be reached. 49 The results from this model were used by Bell Canada in the Interexchange (IX) Hearings before the Canadian Radio/Television and Telecommunications Commission in 1986.

Appendix 15.1 Demand for Telecommunications

337

Table 15.111 Price, income and market-size elasticities for toll-calling in Ontario and Quebec Peak

Off-Peak

Elasticity

0−100 miles

101+ miles

0−100 miles

101+ miles

Income Price∗ Price 2∗∗ Market-size

0.640 −0.282 − 1.025

0.547 −0.370 − 0.829

0.635 −0.265 −0.059 1.158

0.469 −0.354 −0.036 1.043

∗ Both

income and price are included as polynomial distributed lags. Two price variables are included in the off-peak equations to take into account a change in the weekend discount structure that occurred in June 1977. Source: Taylor (1994, Table 6.1).

∗∗

off-peak long-haul to 0.640 for short-haul peak. Finally, the market size (i.e., TR) elasticities range from 0.829 for peak long-haul to 1.158 for off-peak short-haul.50 The price and income elasticities are typical of those found in the literature.51

15.1.3 A Framework for Estimating Market Demand Functions for New Products and Services52 For a variety of reasons, analyzing telecommunications demand is much more difficult currently than was the case 25 years ago. At that time, the old Bell System was still intact, which made for easy obtaining of comprehensive and high-quality data, and competition, deregulation, and technological change and did not blurr the distinction between telecommunications, computers, and television and made a plethora of new products and services available to consumers. As a consequence, many of the telecommunication and entertainment products and services that are now offered (or about to become offered) to households are simply not amenable to conventional forms of econometric analysis. Accordingly, in this section, we shall describe a procedure for obtaining estimates of market demand functions that are derived from willingness-to-pay information elicited in household surveys. The

50 The

coefficients attaching to the market-size variable were extremely controversial in the IX Hearings before the CRTC because, in principle, they provide information as existence and size of the network externality. For LDT’s analysis of this controversy, see Appendix 3 of Taylor (1994). 51 For an application of the model in expression (15.30) to FCC U.S. Interlata toll calling (using data collected by the Federal Communications Commission), see Taylor and Taylor (1993). The price and income elasticities obtained in that study are –0.63 and 0.97, respectively. 52 This analysis in this section is based upon Rappoport et al. (2004).

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15

Annual PCE Models

procedure will be illustrated with reference to VOIP (Voice Over Internet Protocol) telephone service.53 In Section 1, we saw that the standard procedure for estimating access demand involves obtaining information on the consumer surplus from usage by estimating a demand function, and then integrating beneath this demand function. In the present context, however, our procedure is essentially the reverse, for what we will now have as information are statements on the part of respondents in a survey as to the most that they would be willing to pay (WTP) for a particular type of VOIP service. This most that they are willing to pay accordingly represents (at least in principle) the maximum price at which the respondent would purchase that type of service. Thus, for any particular price of VOIP, VOIP will be demanded for WTPs that are this value or greater, while VOIP will not be demanded for WTPs that are less than this value. Hence, implicit in the distribution of WTPs is an aggregate demand function (or more specifically, penetration function) for VOIP service. In particular, this function will be given by: D (π ) = proportion of WTPs that are greater than or equal to π = P (WTP ≥ π)

(15.34)

= 1 − CDF (π ) , where CDF (π ) denotes the cumulative distribution function of the WTPs. Once CDFs of WTPs are constructed, price elasticities can be obtained (without intervention of the demand function) via the formula (or empirical approximations thereof):

ηπ =

∂D(π ) . ∂π

(15.35)

In the example before us, information on willingness to pay for VOIP service was collected from an omnibus national survey of about 8,000 households in April and May, 2004, by the Marketing Systems Group (MSG) of Philadelphia. The omnibus survey, Centris,54 is an ongoing random telephone survey of U.S. households. Each of the participants in the surveys utilized here was asked one (but not both) of the following two questions regarding their willingness to pay: (a) What is the most you would be willing to pay on a monthly basis for a service that provides unlimited local and long distance calling using your computer?

53 VOIP

is a common term that refers to the different protocols that are used to transport real-time voice and the necessary signaling by means of Internet Protocol (IP). Simply put, VOIP allows the user to place a call over IP networks. 54 http://www.Centris.com

Appendix 15.1 Demand for Telecommunications

339

(b) What is the most you would be willing to pay on a monthly basis for a service that provides unlimited local and long distance calling using your computer with Internet connection at a cost of $20 per month?55 The first question was asked to those households that currently have broadband access, while the second version was asked to only those households that did not have broadband access. We now turn to the calculation of price elasticities in line with expression (15.32) above. The most straightforward way of doing this would be to define the elasticities as simple arc elasticities between selected adjacent points on the empirical CDF’s. Unfortunately, however, because the survey-elicited WTP’s tend to bunch at intervals that are multiples of 5 dollars, the values that emerge from this procedure are highly unstable, and accordingly of little practical use. To avoid this problem, elasticities are calculated using a kernel-based non-parametric procedure in which the “pileups” at intervals of 5 dollars are “smoothed out.” Since kernel estimation may be seen as somewhat novel in this context, some background and motivation may be useful. The goal in kernel estimation is to develop a continuous approximation to an empirical frequency distribution that, among other things, can be used to assign density, in a statistical valid manner, in any small neighborhood of an observed frequency point. Since there is little reason to think that, in a large population, “pileups” of WTP’s at amounts divisible by $5 reflect anything other than the convenience of nice round numbers, there is also little reason to think that the “true” density at WTPs of $51 or $49 ought to be much different than the density at $50. The intuitive way of dealing with this contingency (i.e., “pileups” at particular discrete points) is to tabulate frequencies within intervals, and then to calculate “density” as frequency within an interval divided by the length of the interval (i.e., as averages within intervals). However, in doing this, the “density” within any particular interval is calculated using only the observations within that interval, which is to say that if an interval in question (say) is from $40 to $45, then a WTP of $46 (which is as “close” to $45 as is $44) will not be given weight in calculating the density for that interval. What kernel density estimation does is to allow every observation to have weight in the calculation of the density for every interval, but a weight that varies inversely with the “distance” that the observations lie from the center of the interval in question. For the analytics involved, let gˆ (x) represent the density function that is to be constructed for a random variable x (in our case, WTP) that varies from x1 to xn . For VOIP WTP, for example, the range x1 to xn would be 0 to $700.56 Next, divide this range (called the “support” in kernel estimation terminology) into k sub-intervals. The function gˆ (x) is then constructed as:

55 $20

was selected since dial-up prices were approximately $20. is for the households with broadband access. For those households without broadband access [i.e., for households responding to Question (b)], the range is from 0 to $220.

56 This

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15

Annual PCE Models

  N  K (xi − xj )/h , i = 1, . . . , k,. gˆ (x) = Nh

(15.36)

j=1

In this expression, K denotes the kernel-weighting function, h represents a smoothing parameter, and N denotes the number of observations. For the case at hand, the density function in expression (15.36) has been constructed for each interval using the unit normal density function as the kernel weighting function and a “support” of k = 1,000 intervals.57 From the kernel density functions, VoIP price elasticities can be estimated using numerical analogues to expression (15.35).58 The resulting calculations, undertaken at WTPs of $70, $60, $50, $40, $30, $20, and $10 per month, are presented in Table 15.112.59 The estimated elasticities are seen to range from about −3.0 for WTPs of $60−70 to about −0.6 for WTPs of $10. Interestingly, the values in column 1 (for households that already have broadband access) for the most part mirror those in column 2 (which refer to households that do not). Since this appears to have been the first effort to obtain estimates of price elasticities for VOIP, comparison of the numbers in Table 15.112 with existing estimates is obviously not possible. Nevertheless, it is of interest to note that the values that have been obtained are similar to existing econometric estimates for the demand for broadband access to the Internet.60 Table 15.112 VOIP elasticities based on WTP Kernel-smoothed CDF WTP

With Broadband

Without Broadband

$70 60 50 40 30 20 10

−2.8616 −3.0217 −2.7794 −1.7626 −1.0753 −0.7298 −0.5454

−2.9556 −2.4730 −3.0093 −1.5630 −1.0527 −0.7564 −0.6025

Source: Rappoport et al. (2006, Table 1).

57 Silverman’s

rule-of-thumb:  h = (0.9)min[std. dev., interquartile range/1.34](N − 1/5),

has been used for the smoothing parameter h. Two standard references for kernel density estimation are Silverman (1986) and Wand and Jones (1995). Ker and Goodwin (2000) provide an interesting practical application to the estimation of crop insurance rates. 58 The kernel-based elasticities are calculated as “arc” elasticities using points (at intervals of ± $5 around the value for which the elasticity is being calculated) on the kernel CDF’s via the formula: 59 Since Question (b) postulates an access cost of $20, the WTP’s for households without broadband access are assumed to be net of this $20. 60 See, e.g., Rappoport et al. (1998, 2002b, 2003), Kridel et al. (1999).

Appendix 15.1 Demand for Telecommunications

341

Since the elasticities of this procedure are constructed from information elicited directly from households, and thus entail the use of contingent-valuation (CV) data, the seriousness (in light of the longstanding controversy surrounding the use of such data) with which the elasticities that are constructed from willingness-to-pay data are to be taken might be open to question.61 However, in our view, the values in Table 15.112 are indeed plausible and warrant serious consideration.62 Added credence for the results, it seems to us, is provided by the fact that, with VOIP service, we are dealing with a product (voice telephony) with which respondents are familiar and already demand, unlike in circumstances (such as in the valuation of a unique natural resource or the absence of a horrific accident) for which no generally meaningful market-based valuation can be devised.

Appendix 15.2 Electricity Demand and Multi-Part Tariffs The usual assumption in applied demand analysis is that consumers face prices that are independent of the amounts consumed. However, in most electric-utility service areas, electricity has historically been supplied on some form of a multi-part tariff, which implies that the unit price that a consumer faces varies with the number of units that are purchased.63 The purpose of this appendix is to describe procedures that can be used when sufficient information exists to take such forms of pricing into account.64

15.2.1 Demand Theory and Multi-Part Tariffs Thanks to research that was undertaken in the late 1970s, the implications of multipart tariffs for the conventional theory of consumer choice are now fairly well understood, and, although the details are complex, the basic ideas and results are

61 The

critical literature on contingent valuation methods is large. See the NOAA Panel Report (1993), Smith (1993), Portnoy (1994), Hanneman (1994), Diamond and Hausman (1994), and McFadden (1994). On the other hand, particularly successful uses of CV data would seem to include Hammitt (1986) and Kridel (1988). 62 The estimated elasticities suggest that at a price somewhat less than $30 demand shifts from inelastic to elastic. Interestingly, in the afternoon following a luncheon presentation of this analysis, Vonage, largest of the VOIP providers, announced a price reduction to $24.99 per month for all new and existing customers a price that in the fall of 2009 continues in effect. 63 Electricity is only one such service. Other examples include telecommunications, water, and natural gas. 64 The discussion in this appendix is based upon the procedures that were developed by Taylor in collaboration with Gail R. Blattenberger in a series of contracts with the Electric Power Research Institute (EPRI) in the 1970s and early 1980s. See Taylor (1975), Taylor et al. (1977), Blattenberger et al. (1983), and Taylor et al. (1983). All of the results in these papers (and more) are available in an unpublished manuscript of Taylor and Blattenberger (1986).

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easily illustrated in a few simple diagrams. For simplicity, let us assume that a consumer purchases just two goods, electricity, q, and a composite good, z. Assume that electricity is supplied on a rate schedule that consists of a fixed charge and m “block” rates, while z can be purchased in unlimited quantities at a constant price p. Let r0 denote the fixed charge, and ri the unit price of electricity in the ith block, defined by ki − ki−1 . Obviously, k0 is zero and the last block is openended. Finally, let the consumer’s income be denoted by x. The budget constraint will accordingly take the form: r0 +

j−1 

ri (ki+1 − ki ) + rj (q − kj ) + pz = x,

(15.37)

i=1

where it is assumed that equilibrium occurs with the jth block of the rate schedule. The cost of electricity, c(q), is then: c(q) = r0 +

j−1 

ri (ki+1 − ki ) + rj (q − kj ),

(15.38)

(ri − rj )(ki+1 − ki ) + rj q.

(15.39)

i=1

or equivalently as c(q) = r0 +

j−1  i=1

With the cost of electricity written as in expression (15.39), we see that the last term, rj q, represents what the cost would be if all q units of electricity could be purchased at the rate rj . The rest of the expression then represents what we will call the infra-marginal premium for consumption in the jth block, namely, tj = r 0 +

j−1 

(ri − rj )(ki+1 − ki ).

(15.40)

i=1

With decreasing-block rates (ri > rj for i < j) and a positive fixed charge (r0 > 0), tj will be positive. The term marginal block will be used to denote the block containing the actual consumption level, extra-marginal will then refer to higher blocks and infra-marginal to lower. The problems involved with decreasing-block rates are easily illustrated graphically. A simple rate schedule, specifically designed for this purpose, is as follows: Fixed charge: 0 to k2 units: k2 to k3 units: k3 units or more:

ro r1 = 0 r2 /unit r3 /unit.

Appendix 15.2 Electricity Demand and Multi-Part Tariffs Fig. 15.1 Budget constraint with decreasing-block pricing

343

C(q)

k1

k2

k3

q

The graph of this budget constraint is shown in Fig. 15.1 The horizontal segment of the constraint corresponds to the price of 0 for consumption of the first k2 units of electricity. The linear segment between k2 and k3 has slope equal to −r2 /p and corresponds to the r2 part of the rate schedule, while the linear segment from k3 on, with slope equal to −r3 /p, corresponds to the r3 part of the schedule. Finally, the fixed-charge ro on would be represented as a point on the vertical axis. The non-linear, non-convex budget constraint represented in this figure has a number of consequences for the equilibrium of the consumer, demand functions, and Engel curves. These consequences are illustrated in Figs. 15.2, 15.3, 15.4, 15.5, 15.6, and 15.7, which show the budget sets and equilibrium indifference curves in a variety of circumstances.65 To begin with, Fig. 15.2 shows equilibria for two

z

Fig. 15.2 Consumer equilibria with decreasing-block pricing 65 These

diagrams were first presented in Taylor (1975); cf. also, Taylor (1994, Chapter 3).

q

344 Fig. 15.3 Effect of a change in intramarginal price

15

Annual PCE Models

z

q

Fig. 15.4 Effect of a change in both intramarginal and marginal price

z

q

Fig. 15.5 Price change that leads to a switch of blocks

z

q

Appendix 15.2 Electricity Demand and Multi-Part Tariffs Fig. 15.6 Effect of an income change

345

z

q

Fig. 15.7 Multiple equilibria z

q

different indifference maps. The indifference map with solid curves yields equilibrium on the facet of the budget constraint with slope equal to −r2 /p, while the indifference map with dashed curves shows equilibrium on the facet with slope equal to −r3 /p. Figure 15.3 describes an increase in r2 , but not r3 , while Fig. 15.4 displays an increase in both r2 and r3 . In these two figures, equilibrium following the price increases remains on the same facet of the budget constraint. However, that this need not always be the case is clear from Fig. 15.5, for the indifference curves in this figure show equilibrium switching from the facet with slope −r3 /p to the facet with slope −r2 /p. In this case, the increase in r3 causes the consumer to drop back to a higher marginal rate class. Switching into a different rate class can also be brought about by a change in income, as is evidenced in Fig. 15.6. Finally, Fig. 15.7 shows a case in which the budget constraint is tangent to the same indifference curve at two different points, resulting in multiple equilibria.

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15

Annual PCE Models

From Figs. 15.2, 15.3, 15.4, 15.5, 15.6, and 15.7, we are able to draw the following conclusions:66 (1) Because of the piece-wise linearity of the budget constraint, the equilibrium of the consumer cannot be derived, as is conventionally the case, using the differential calculus. While demand functions and Engel curves still exist, they cannot be derived analytically through solution of the first-order conditions for utility maximization. Moreover, this is true for all goods, not just for electricity. (2) In Fig. 15.5, it is evident that the demand functions are discontinuous, with gaps at the points where equilibrium switches from one facet of the budget constraint to another.67 (3) In Fig. 15.6, the same is seen to be true for Engel curves. (4) In Fig. 15.7, it is seen that there will be particular configurations of prices for which the demand functions are not single-valued. This follows from the nonconvexity of the budget constraint. Specifically, the demand functions will be multi-valued whenever there is a configuration of prices that yields multiple tangents of the budget constraint to the same indifference curve. (5) Similarly, there may exist levels of income for which the Engel curves are discontinuous and multi-valued. (6) Finally, in Figs. 15.3 and 15.6, we see that a change in an infra-marginal price that does not cause a block switch is equivalent to a change in income. The fact that demand functions with multi-part tariffs are discontinuous and multi-valued clearly creates problems for empirical analysis. Conventional econometric estimation requires analytic functions, but with multi-part tariffs the demand functions are correspondences, and thus cannot be represented analytically. While these are problems that would have to be dealt with if the data being analyzed refer to individual consuming units (as in the CES surveys),68 the discontinuities fortunately tend to disappear with aggregation. Specifically, it can be established that as one aggregates over a group of consumers, the mean demand function for the group becomes continuous in the limit as the group becomes large.69 The conditions for this result to hold are that tastes or income must vary across consumers in such a way that, at any point in the price set, the probability is zero that a discontinuity exists on the demand function for any given consumer, i.e., at each point 66 All

of these results, including conditions involving integrability, are derived mathematically in Blattenberger (1977). 67 Under general conditions, Blattenberger shows that the number of discontinuities of the demand function, with respect to the jth block rate, cannot exceed m − j + 1. 68 The only study of electricity demand using household-level data of which we are aware that deals specifically with the problems caused by a decreasing-block rate schedule is in the 1980 Ph.D. dissertation by Steven H. Wade at the University of Arizona. 69 This is another important result obtained by Blattenberger (1977). “Mean demand function” in this situation refers to mean quantities demanded at the points of discontinuity.

Appendix 15.2 Electricity Demand and Multi-Part Tariffs

347

in the price set, at most a few members of the population of consumers can have a discontinuity in their demand functions. Blattenberger (1977) shows that any reasonable distribution of tastes or income (including the Pareto) yields this result as the number of consumers becomes large.70 Consequently, in the discussion that follows, demand functions will be interpreted as referring a representative consumer and treated as though they were continuous.

15.2.2 Empirical Representation of Rate Schedules Let us now turn to the way that block-rate schedules can be represented in demand functions. Much of the time in any empirical model, multicollinearity and identification problems will obstruct inclusion of the entire rate schedule, so that a way is needed to summarize the information in the schedule. One way of doing this is by employing the concepts of marginal and infra-marginal rates. The budget constraint in expression (15.1), which can be rewritten using expressions (15.39) and (15.40) as tj + rj q + pz = x,

(15.41)

provides a vehicle for doing so. A price change that affects infra-marginal rates will show up as a change in the infra-marginal premium, tj , while a change that affects the marginal rate will obviously be reflected in rj (assuming that equilibrium continues to occur in the jth block). In particular, an increase in the infra-marginal rate will increase tj , while a decrease in the infra-marginal rate will lead to a decrease in tj . In neither case is the marginal rate rj affected (unless, of course, the change in the infra-marginal rate is so large as to cause a switch in blocks). In Fig. 15.8, the budget constraint in Fig. 15.1 is redrawn in terms of its representation in expression (15.41). The customer charge r0 is represented by distance ab on the vertical axis, while the infra-marginal premium tj is measured by the distance ac.71 It is clear from the figure that an increase in the infra-marginal premium, whether it arises from an increase in r0 or an increase in r2 , is indistinguishable from a decrease in income, and vice-versa, which is to say we have ∂q ∂q =− . ∂tj ∂x

70 Strictly

(15.42)

speaking, “large” in this context means infinity. However, experimental evidence obtained by Stephen Rassenti (1979) suggests that the mean demand function may be reasonably continuous for as few as 12 consumers. 71 Note that income, rather than the composite good z, appears on the vertical axis in this figure. This is a mere re-labeling since, with two goods and non-satiation, income not spent on electricity is necessarily spent on the second good.

348 Fig. 15.8 Two-part tariff representation of the budget constraint

15

Annual PCE Models

a

r0

{

b c

0

q

In view of this, one can define the budget constraint as x − tj . Changes in the infra-marginal premium that arise from a change in the customer charge or in the infra-marginal rate do not affect the marginal rate, but a change in the marginal rate does affect the infra-marginal premium. For, from the definition of tj in expression (15.41), we see that  ∂tj =− (ki+1 − ki ) = kj , ∂rj j−1

(15.43)

i=1

since k1 = 0. From the foregoing, we see that the impact of a change in the rate schedule on the amount of electricity consumed varies, depending upon whether the change affects the infra-marginal or marginal rate. (A change in the extra-marginal rate obviously has no effect.) Thus, a change in the rate schedule that affects infra-marginal rates (including the fixed charge) affects consumption only through the income effect, whereas a change in the marginal rate affects consumption through both the income effect and the substitution effect. As a consequence, the rate schedule in this situation can be represented in the demand function through two terms, the marginal rate and the infra-marginal premium.72

72 The procedure in these paragraphs is an illustration of the fact, first pointed out by Gabor (1955),

that any multi-part tariff can be replaced by an equivalent two-part tariff. In his 1975 survey article, Taylor had not grasped the full implications of Gabor’s theorem, and suggested that a rate schedule be represented by the marginal rate and the infra-marginal expenditure, as opposed to the inframarginal premium. Nordin (1976) set the record straight, as did Blattenberger (1977). Also, it should be mentioned that, in early 1974, Professor Aly Erceawn of Quaid-i-Azam University in Pakistan, brought to Taylor’s attention that many of the problems associated with decreasing-block pricing were discussed in his Ph.D. dissertation at Vanderbilt University (1974). Finally it should be noted that, while the analysis has been with respect to a decreasing-block rate schedule, the Gabor theorem applies as well to increasing-block pricing. The infra-marginal “premium” in this case could be interpreted as representing a subsidy, as opposed to a lump-sum tax.

Appendix 15.2 Electricity Demand and Multi-Part Tariffs

349

The representation just outlined suffices for consumers all of whom consume in the same block of the rate schedule. However, with aggregate data, because of taste and income differences (which are necessary for the aggregation theorem to hold), consumers will not all be consuming within the same block. Consequently, what is the marginal block for some consumers will be infra-marginal to other consumers, and extra-marginal to still others. In this situation, it is thus necessary to develop meaningful measures of a “fixed charge” (which the infra-marginal premium will now be referred to) and “marginal price” that can be seen as applying to a rate schedule as a whole. We will now describe a procedure for how this can be done. For concreteness, the unit of observation will be assumed to be a state. The procedure entails parameterizing a rate schedule for a state as a whole in terms of the total cost of consuming electricity on that schedule as a function of quantity, and then the marginal price as the slope of this function and the fixed charge as its intercept is defined. Construction of such a schedule proceeds as follows. First, let the total cost of consuming q units of electricity, c(q), on an individual rate schedule be as given in expression (15.2) above. Obviously, the position and shape of c(q) depends upon the rates and block demarcations of the particular schedule. The next step is to define the mean total-cost function for a state as a whole as ¯ C(q) =

T 

wi ci (q),

(15.44)

i=1

where ci (q) is the total-cost function in equation (15.3) for the ith rate schedule in the state, T denotes the total number of schedules, and wi denotes the proportion of ¯ customers in the state that purchase electricity on schedule i. The function C(q) is well-defined, but very complicated. Experience with U.S. data has shown, however, that the function is well-represented by a linear approximation:73 ¯ C(q) = a1 + a2 q + u,

(15.45)

where a1 and a2 are parameters and u is the approximation error. The parameters ¯ on q, using as a1 and a2 can then be estimated in a least-squares regression of C(q) 74 ¯ “observations” points on C(q). The marginal price is obtained from the total-cost function by differentiation with respect to q. With a linear approximation, this derivative is a constant, in which case the marginal price is independent of the amount of electricity that is consumed. This is an important property of the linear approximation, for it rules out

73 See

Taylor et al. (1977). ¯ In their application of this procedure, Taylor et al. used 291 points on C(q), calculated at 5 kilowatt hour intervals between 50 and 1,500 kilowatt hours. Forty-six of the estimated equations for the 48 contiguous U.S. states had an R2 greater than 0.99. 74

350

15

Annual PCE Models

any possibility of simultaneity bias.75 The marginal price is thus represented by the parameter a2 , while the infra-marginal premium is readily interpreted as the fixed charge a1 . Estimates of these coefficients then become the “observations” for the infra-marginal premium and marginal price in the electricity demand model.76

15.2.3 Electricity Demand in the 1970s: An Illustration The foregoing procedure for calculating marginal prices and infra-marginal premiums was used by Taylor, Blattenberger, and Verleger (TBV) in an extensive study of electricity demand that was undertaken for the Electric Power Research Institute in the mid-1970s. Although the results are dated, they are nevertheless of interest for purposes of illustration. As in the present effort, a logarithmic flowadjustment model was estimated using a combined time-series/cross-section data set consisting of annual observations on the 48 continental U.S. states for 1956 through 1974. Independent variables included fixed charges and marginal prices (as described above) for both electricity and natural gas, together with heating and cooling degree-days, the price of fuel oil, and a measure of the availability of pipeline delivered natural gas.77 The short- and long-run income and price elasticities from the estimated model [taken from Taylor and Blattenberger (1986)] are given in Table 15.113. The elasticities from the present study are included in this table as well. The numbers in this table pretty much speak for themselves. For despite differences in specifications, data sets, and definitions of variables, the short- and long-run income and price elasticities are pretty much the same—a steady-state income elasticity of the order of 0.8 and a steady-state own-price elasticity that is moderately 75 The

simultaneity bias in question is what would arise if the marginal price continued to depend on the amount of electricity that is consumed. 76 The drawback with this procedure is that in principle it requires information on each and every residential rate schedule in effect during the period of the data sample. At the time of the TBV study, this information was available in The National Electric Rate Book published by The Federal Energy Regulatory Commission. Unfortunately, the last year of publication of The National Electric Rate Book was 1975, and the nearest alternative source of information is that published by the Edison Electric Institute in Typical Electric Bills. Electricity expenditure data for residential customers in Typical Electric Bills are calculated for the purchase of 100, 250, 500, 750, and 1,000 kilowatt hours of electricity in urban communities of 2,500 or more, thus giving five points on a total-cost function. By aggregating across cities in a state, estimates of the fixed charge and marginal price (a set for each state) can then be obtained in the way just described (unfortunately, though, with only 3 degrees of freedom). A detailed evaluation of the two procedures, using a time-series/cross-section data set consisting of annual observations for the 48 contiguous states for 1956–1974 is given in Taylor et al. (1977). 77 The measure of natural gas availability was interacted with the income, price, and heating and cooling degree-day variables in order to allow for elasticities to differ depending on whether households had access to pipeline-delivered natural gas. The dependent variable in the model is the logarithm of kilowatt hours per electricity customer, while income is expressed as personal state income per customer. The model is estimated as a variance-components model by OLS.

Appendix 15.2 Electricity Demand and Multi-Part Tariffs

351

Table 15.113 Price and income elasticities, Taylor-Blattenberger-Verleger study of residential electricity demand in the U.S. Elasticity Variable Income: Short-run Long-run Fixed charge, electricity: Short-run Long-run Marginal price, electricity: Short-run Long-run Fixed charge, natural gas: Short-run Long-run Marginal price, natural gas: Short-run Long-run

TBV 0.077 0.802

Present Study 0.097 0.872

−0.041 −0.430 −0.101 −1.052

−0.150 −1.344

0.030 0.311 0.002 0.018

Sources: Taylor and Blattenberger (1986, Chapter 8, Table 3), Chapter 15 above.

elastic.78 Two comments are accordingly in order: (1) despite the variety of factors that have affected the electricity market over the last 30 years (energy crises, deregulation, etc.), the residential demand function for electricity appears to have great stability; (2) since the TBV specification is clearly superior to that of the present study, the fact that the income and own-price elasticities are similar greatly increases the confidence that can be placed in the NIPA model.79

15.2.4 Conclusion The purpose of this appendix has been to describe a procedure that can be used in situations in which the price of a good or service depends upon the quantity that is purchased. While the illustration is with reference to electricity purchased on a decreasing-block rate schedule, the procedure is clearly general, and is applicable to virtually any form of a nonlinear pricing structure. Key to the method is the theorem

78 The

TBV elasticities are calculated at the mean value of the measure of natural-gas availability. passing, it is interesting to note that, according to expression (15.42), the partial derivative with respect to the fixed charge should be the negative of the partial derivative with respect to income. However, when the elasticities in the table are converted to derivatives, this hypothesis is decidedly rejected. The negative effect of an increase in the infra-marginal premium is much larger than the positive effect of an increase in income. A similar result was found by Hill et al. (1983) using data for Arizona in a time-of-day electricity pricing experiment.

79 In

352

15

Annual PCE Models

of Andre Gabor (1955), which states that any multi-part tariff can be replaced by an equivalent two-part tariff. Use of this theorem allows almost any complicated pricing mechanism to be characterized in terms of two quantities: a marginal price and an infra-marginal premium, which measures the “up-front” charge (which may be negative) which, if paid, would enable the consumer to purchase as many units as desired at the marginal price.80

Appendix 15.3 Used Car Prices and the Demand for Automobiles In discussing the equations for automobile expenditures in the text, it was noted that the demands for new and used automobiles should more properly be modeled jointly in order to take into account the dependence of new car sales on the price of used cars. Specifically, it was noted that, since the vast majority of households purchase used rather than new cars, new car sales will be triggered when, because of an increase in the demand for used cars, the price of late-model used cars ultimately rise in relation to the price of new cars. In an unpublished paper presented at the 6th World Congress of the Econometric Society in Barcelona, Spain, in September 1990,81 a model was developed that takes into account this dependence. In this appendix, we present an updated version of that model, using annual data. The model, in continuous time, is as follows: πˆ (τ ) = α + βs(τ ) + γ x(τ ) + λp(τ ),

(15.46)



dπ (τ ) = θ πˆ (τ ) − π (τ ) dτ + ε(dτ ),

(15.47)

qˆ (τ ) = δˆs(τ )

(15.48)





q(τ ) = ϕ q(τ ) − qˆ (τ ) + ψ π(τ ˆ ) − κp(τ ) ,

(15.49)



dq(τ ) =  q(τ ) − qˆ (τ ) dτ + η(dτ ),

(15.50)

ds(τ ) = [q(τ ) − δs(τ )]dτ

(15.51)

80 As a historical note, it is useful to point out that both Gabor and Buchanan (1955) clearly under-

stood the theoretical implications of block tariffs and quantity discounts, and had their insights been integrated with those of Houthakker (1951), much of the sterile debate of the following 20 years as to which price—marginal or average—could have been avoided. Buchanan pointed out the ambiguities surrounding demand functions in the presence of quantity discounts, while Gabor’s key result was his theorem. Finally, it should be mentioned that the problems that are encountered with multi-part tariffs with electricity and natural gas are similar to the problems associated with overtime wages and a progressive income-tax structure in the context of labor-supply models. See Burtless (1978) and Hausman (1985). Riess and White (2005) provide a recent application of the Burtless-Hausman procedure to residential electricity demand in California. 81 see Taylor and Houthakker (1990).

Appendix 15.3 Used Car Prices and the Demand for Automobiles

353

where: πˆ : market-clearing price in used car market π : actual price in used car market s: stock of cars sˆ: steady-state demand for stock of cars x: income p: price of new cars q: demand for new cars qˆ : steady-state demand for new cars ε, η: white noise error terms. The basic logic of the model can be seen graphically in Fig. 1. The curve labeled D represents the willingness to pay to hold a particular stock of cars, while the vertical spike at S represents the stock that is available to be held. The market is cleared at a price of πˆ . The horizontal line at κp denotes the price at which the stock of cars can be enlarged through the purchase of new cars. New and used car prices are in equilibrium when πˆ and κp are equal. In the situation depicted, κp is less than πˆ , which results in new car purchases equal to q.82 Equation (15.46) represents the determination of the equilibrium price πˆ obtained from equating demand D (assumed to be a function of price and income) and supply S.83 Equation (15.47) allows a partial adjustment in used car prices to disequilibrium in the used car market. Equations (15.48)–( 15.50) describe the market for new car purchases. The steady-state demand for new cars is defined in equation (15.48) in terms of the replacement demand (i.e., depreciation) on the steady-state stock of cars. The latter, in turn, is obtained from equation (15.44), with πˆ replaced by κp. Equation (15.49) then represents the new (as opposed to the steady-state) demand for new cars, and consists of two components: an “inventory-adjustment effect represented by the deviation of the current flow of new car purchases from the desired (or steady-state) flow, as represented by the term β[q(τ ) − qˆ (τ )], and new demand arising from disequilibrium between new and used car prices, as represented by the term ϕ[πˆ (τ ) − κp(τ )]. New demand can clearly be negative, as well as positive.” Equation (15.50) allows for a partial adjustment in new demand. Finally, equation (15.51) is the familiar law of motion describing how car stocks evolve in response to new car purchases and depreciation. Estimation of the model proceeds in two stages, the first for used car prices obtained from equations (15.46), (15.47), and (15.51), and the second for new car

82 The

model is that of investment in a durable good in which existing examples are traded in a well-developed secondary market. It was first articulated by Keynes in Chapter 11 of The General Theory; cf. also Davidson (1978). The parameter κ is a number less than 1 that adjusts used car prices in such a way that, in equilibrium, new and used cars are perfect substitutes. 83 The price for new cars is included in equation (15.3.1) as a correction for inflation (when needed).

354

15

Annual PCE Models

purchases obtained from equations (15.48)–(15.50). The estimating equation for used car prices is given by πt = A0 + A1 πt−1 + A2 πt−2 + A3 (qt + qt−1 − δ(st + st−1 )) + A4 (xt − xt−2 ) + A5 (pt − pt−2 ) + ut ,

(15.52)

while the estimating equation for new car purchases is qt = B0 + B1 qt−1 + B2 zt + B3 (πt − κpt ) + vt ,

(15.53)

zt = (1/ϕ)[κ(pt + pt−1 ) − γ (xt + xt−1 )],

(15.54)

where:

using estimates of β, κ, and γ from equation (15.52). The structural coefficients are then obtained sequentially as A1 = A2 = A2 = A4 = A5 =

4 , 2+θ θ −2 , 2+θ θβ , 2+θ θγ , 2+θ θλ , 2+θ

(15.55) (15.56) (15.57) (15.58) (15.59)

so that:  2(1 + A2 ) 2(2 − A1 ) = , θ= A1 1 − A2 A3 (2 + θ ) β= , θ A4 (2 + θ ) , γ = θ A5 (2 + θ ) λ= θ

(15.61) (15.62) (15.63)

For the coefficients in new car estimating equation, we have α , ϕ 2 + ϕ , B1 = 2 − ϕ B0 = −

(15.64) (15.65)

Appendix 15.3 Used Car Prices and the Demand for Automobiles

δ(β + 1) , 2 − ϕ ψ B3 = , 2 − ϕ

B2 = −

355

(15.66) (15.67)

and: =

2(B1 − 1)(B2 − δ) − B2 (1 + B1 ) , δ(1 + B1 ) B2 + δ ϕ= , ϕ(B2 − δ) B0 ϕ α=− ,  B3 (2 − ϕ) . ψ= 

(15.68) (15.69) (15.70) (15.71)

The models have been estimated with annual PCE data over the period 1929–2004. The stock of cars, st , which appears in both the estimating equations, has been constructed by the equation st =

2−δ qt + qt−1 + st−1 , 2+δ 2+δ

(15.72)

beginning in 1929. A value of 0.15 has been used for the depreciation rate δ. The data set used in the estimation of both the models is 1947–2004, which allows for the value of st to be independent (at least for plausible values of δ) of the initial value of 0 employed in its construction. Although κ is in principle estimable in equation (15.53), it, too, has been specified a priori.84 The dependent variables in the estimating equations are the real price of used cars (from category 8.1b (Table 15.68) in the PCE tables) in equation (15.52) and real personal consumption expenditures for new motor vehicles [defined as the sum of real expenditures for new cars (Table 15.67) and other motor vehicles (Table 15.69)] in equation (15.53).85 Unlike in the text, expenditures are measured in aggregate, rather than per capita. Since the two estimating equations form a simultaneous system, estimation is by instrumental variables/two-stage least-squares.86 The results are presented in 84 However, the values for both δ and κ

(0.15 for δ and 0.25 for κ) have not been selected arbitrarily, but rather from informal searches within their parameter spaces. 85 Expenditures for light trucks and SUVs are included in category 8.1c. 86 In equation (15.52), q t−1 is the instrument for qt + qt−1 − δ(st + st−1 ), while xt and πt−1 are the instruments for zt and πt − κpt in equation (15.53). In expression (15.60), it is seen that θ is over-identified, which necessitates equation (15.52) being estimated subject to the constraint A1 + A2 = 1. Also, it should be noted that, since both expenditures and prices are expressed in real terms (and thus adjusted for inflation), there is no need to include the term pt − pt−2 in equation (15.52).

356

15

Annual PCE Models

Table 15.114 Results for used car prices and the demand for automobiles annual PCE data, 1947−2004 Equation (15.52) Used Car Prices, A1 + A2 = 1, Instrumental Variables/Two-Stage Least-Squares Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

A0 A1 A2 A3 A4

0.3147 0.8749 0.1251 −0.0234 0.0141

0.20 6.83 0.98 −1.62 1.26

α β γ δ κ

177.6114 −0.0415 0.0251 0.1500 0.2500

θ R2 dw h d–f

2.5719 0.5522 1.95 0.73 −7.28

Equation (15.53) Expenditures for New Vehicles, Instrumental Variables/Two-Stage Least-Squares Coefficient

Value

t-ratio

Coefficient

Value

Coefficient

Value

B0 B1 B2 B3

15.8176 0.7023 0.0145 0.2017

2.01 6.40 2.95 2.83

Φ φ ψ δ κ

0.2188 −2.0889 2.2650 0.1500 0.2500

R2 dw h d–f

0.9826 1.40 3.77 −5.49

Statistically, the estimated models are pretty much of the same quality. While tratios are larger in the equation for new vehicles, positive autocorrelation is evident in its residuals, but not in the residuals for used car prices.87 The short-term dynamics in the system are governed by θ in the equation for used car prices and by Φ, φ, and ψ in the equation for new vehicle purchases. θ and ψ are both estimated to be well in excess of 1, which implies that used car prices adjust rapidly to disequilibrium in the used car market, and similarly for the desired new car purchases in response to the disequilibrium between new and used car prices. On the other hand, the small value for Φ indicates that rapid adjustment does not extend to actual newcar purchases. Finally, the negative value for φ indicates (as we should expect) that new car purchases are inventory-adjusting. The steady-state properties of the system are obtained by reformulating equation (15.46) as a demand function for the stock of automobiles, as opposed to a price (or inverse-demand) function:88 sˆ(τ ) = α ∗ + β ∗ π (τ ) + γ ∗ x(τ ),

(15.73)

where sˆ can be interpreted as the desired stock of automobiles corresponding to π and x, and where: 87 The R2 for used car prices may seem low for time-series data. However, because of the constraint A1 + A2 = 1, the dependent variable in this equation is πt − πt−2 ; in levels, the R2 is about 0.95. 88 Since quantities and prices are measured in real terms, the inflation-correction term is dropped.

Appendix 15.3 Used Car Prices and the Demand for Automobiles

357

α α∗ = − , β

(15.74)

1 , β γ γ∗ = − . β β∗ =

(15.75) (15.76)

Table 15.115 Structural coefficients for automobile stock demand function Coefficient

Value

α∗ β∗ γ∗

4280 24.0964 0.6048

Price and Total-Expenditure Elasticities, New-Vehicle Expenditures Total Stock of Vehicles

New vehicles: Short-Run

Price

New Vehicles: Long-Run

Price

Price

Tot. Exp.

Used

New

Tot. Exp.

Used

New

Tot. Exp.

Used

New

4.8401

−3.9579

−0.9895

1.5798

1.6660

−0.4165

2.5411

2.0779

−0.5195

The values for these parameters implied by the estimates in Table 15.114 are given in Table 15.115. Since the stock of automobiles is fixed in the short run, so that at πˆ , sˆ = s, the negative coefficient for the state variable when the good involved is durable is simply a consequence of the fact that demand functions slope downward. An increase in the stock of cars leads [via equation (15.46)] to a decrease in the equilibrium price, which in turn leads [via equation (15.49)] to a decrease in the flow rate of demand for new cars. The coefficient γ ∗ in expression (15.76) is of interest because it represents the instantaneous marginal propensity to consume for automobiles. From Table (15.115), we see that the estimate of this parameter is 0.60. This value, in turn, contrasts with the value for the long-run marginal propensity to consume for new car purchases of 0.09, obtained by substituting qˆ /δ for sˆfrom equation (15.48). Also, not to be confused with γ∗ is an observed short-run marginal propensity to consume for new cars, that can be defined as the initial year impact on new vehicle purchases that would arise from a dollar increase in real total expenditure. This quantity can be calculated from equation (15.52) [using equation (15.53)] as

358

15

 B2 γ ∂q ∂q =− + B3 A3 + A5 ∂x ϕ ∂x B2 γ A5 B3 − ϕ = . 1 − A3 B3

Annual PCE Models

(15.77)

Using values of A3 , A5 , B2 , B3 , γ , and φ from Table 15.113, this formula yields a value of 0.0027 for this quantity. It is, of course, much smaller than γ∗ because of the substantial inertia in the adjustment of actual new car purchases to a change in desired purchases. The final quantities that we shall present for the model of this appendix are the short-and long-run elasticities with respect to price and total expenditure. These elasticities are tabulated at the bottom of Table 15.113. As in the text, the elasticities are calculated at the mean values of the annual data over the period 1947–2004. While the stock elasticities might seem large in relation to the ones for new vehicles, it must be kept in mind that they are, in effect, instantaneous elasticities, for they refer to the change in the desired stocks in response to a change in the marketclearing price. Interestingly, the long-run (new vehicle) elasticities that are estimated with this model are generally consistent with the single-equation steady-state elasticities estimated in the text, for they are larger than for purchases of new autos (Table 15.67), but smaller than for other motor vehicles (Table 16.69). More importantly, however, the extended model of this appendix allows for an increase in the demand for automobiles (arising, say, from an increase in income) to ultimately affect new car purchases through an increase in used car prices relative to the price of new cars, specifically, a rise in the price of late-model used cars. While the implicit PCE deflator for used cars is obviously an imperfect measure of the prices that in principle drive the model, it nevertheless is gratifying that the results are in fact as positive as they are.89

89 Specifically,

the used-car deflator will, in general, be a measure of the mean age of used cars, rather than of those of just late-model (say, of 8 years, instead of 1–2 years). While this is consistent with the value for κ of 0.25 that is used in estimation and the calculations, it obviously leads to estimates of new car elasticities that are almost certainly unduly large.

Chapter 16

Discussion of the Time-Series Results

16.1 Tabulation of Annual Models In this chapter, we provide a detailed summary and discussion of the time-series results just recorded, beginning with a summary tabulation of the steady-state price and total-expenditure elasticities. These are given in Table 16.1. In addition to the elasticities, the table includes the form of dynamic model estimated, the implied steady-state budget shares for the B–C and state-adjustment models, and actual budget shares for 2004. Of the 108 models listed in Table 16.1, 66 are B–C, 24 are state adjustment, and 18 are logarithmic flow adjustment. Since the B–C model is a generalization of the state- and (linear) flow-adjustment models (and accordingly allows for an additional degree of freedom in parameter space), that it is applied with much the greatest frequency should not be surprising. For the categories in which flowadjustment models are estimated, it is interesting that not once does a linear model gives better results statistically than the logarithmic version. Hence, once again in applied demand analysis, we find a general superiority of double-log over linear models. Since all of the models are dynamical, they obviously have short-term dynamics of some form. The B–C model has both flow and state adjustment, while the flowand state-adjustment models, of course, have the dynamics that their names imply. In all cases, the short-term adjustment can be rapid or sluggish depending upon the signs and size of particular parameters. In the B–C model, flow adjustment is rapid if γ is greater than 1, and sluggish if not, while in the flow-adjustment model, the flow adjustment is rapid if θ is close to 1 and sluggish if θ is close to 0. Finally, in the B–C and state-adjustment models, the short-term state adjustment is rapid if β is negative (stock adjustment) and sluggish if β is positive (habit formation). In general, what the results show is that flow adjustment is typically rapid in the B–C models, and about evenly split between rapid and sluggish in the log flow-adjustment models. In Table 16.2, we have we have tabulated the stock coefficients (β) and depreciation rates (δ) for the B–C and state-adjustment models. A count of signs of the

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_16, 

359

360

16

Discussion of the Time-Series Results

stock coefficients shows that 52 are positive, while only 24 are negative.1 The 52 categories subject to habit formation accounted for about 63% of the total expenditure in 2004, while the 24 subject to inventory adjustment accounted for about 28%.2 Interestingly, these are essentially the same proportions as in 1964. Table 16.1 Steady-state total-expenditure and price elasticities, disaggregated NIPA PCE categories, annual data, 1929–2004 Elasticity PCE

Budget share

Category

Model

Price

Steady_state

2004

1.0 Food and tobacco 1.1 Food purchased for off-premise consumption 1.2 Purchased meals and beverages 1.3 Food furnished to employees (including military) 1.4 Food produced and consumed on farms 1.5 Tobacco products 1.6a Food excluding tobacco and alcoholic beverages 1.7 Alcoholic beverages B-C 1.7a Alcoholic beverages purchased for off-premise consumption 1.8b Other alcoholic beverages 2.0 Clothing, accessories, and jewelry 2.1 Shoes 2.2 Clothing and accessories except shoes 2.2a Women’s and children’s

S. A. B-C

0.2689 0.1963

−0.0799 −0.1194

0.0566 0.0236

0.1488 0.0838

B-C

0.8421

−0.5408

0.0520

0.0530

−

not estimated

−

0.0013

−

not estimated

−

0.00006

B-C B-C

−0.4434 0.3583

−0.2382 −0.0477

−0.0109 0.0484

0.2826

−0.9555

0.0658

0.0160

−

B-C

0.2221

−1.1307

0.0028

0.0103

B-C

0.2001

−5.2445

0.0106

0.0057

B-C.

5.5089

−3.9183

−

0.0543

log F. A. B-C

0.3848 1.7092

−0.4717 −0.7363

− 0.0540

0.0066 0.0334

B-C

1.1481

−0.7098

0.0221

0.0209

0.0107 0.1114

1 The count refers to models at the lowest levels of disaggregation. For food, tobacco, and alcoholic beverages, for example, the signs are for categories 1.5, 1.6a, and 1.7a and 1.7b. 2 The five categories for which models are not estimated account for about a negative 1% of total expenditure. The remaining 10% is accordingly accounted for by categories estimated by logarithmic flow-adjustment models.

16.1

Tabulation of Annual Models

361 Table 16.1 (continued) Elasticity

Category

Model

PCE

2.2b Men’s and boys’ 2.3 Standard clothing issued to military personnel 2.4 Cleaning, storage and repair of clothing and shoes 2.5 Jewelry and watches 2.6 Other clothing and accessories 3.0 Personal care 3.1 Toilet articles and preparations 3.2 Barbershops, beauty parlors and health clubs 4.0 Housing 4.1 Owner-occupied non-farm dwellings: space rent 4.2 Tenant-occupied non-farm dwellings: rent 4.3 Rental value of farm dwellings 4.4 Other housing 5.0 Household operation 5.1 Furniture, including mattresses and bedsprings 5.2 Kitchen and other household appliances 5.3 China, glassware, tableware and utensils 5.4 Other durable house furnishings 5.5 Semi-durable house furnishings 5.6 Cleaning and polishing preparations and miscellaneous household supplies and paper products

B-C −

1.0557 not estimated

Budget share Price

Steady_state

2004

−0.1208

0.0132 −

0.0124 0.00004

B-C

−

−4.5137

−

0.0019

B-C

4.6820

−0.1688

0.0109

0.0070

B-C

3.9601

−2.3460

0.0129

0.0053

B-C B-C

0.7463 0.8800

−0.6543 −1.2269

0.0109 0.0071

0.0125 0.0068

B-C

1.3480

−4.5054

0.0093

0.0126

log F. A. log F. A.

1.0326 1.2632

−1.1309 −1.4391

− −

0.1486 0.1092

B-C

0.4015

−1.2608

0.0164

0.0302

B-C

−0.4802

−0.8497

−0.0015

0.0016

B-C B-C B-C

0.4261 0.8045 0.3565

−0.1723 −0.4556 −0.0349

0.0035 0.0866 0.0032

0.0077 0.1000 0.0092

B-C

1.0794

−0.2292

0.0050

0.0043

B-C

0.7736

−1.2617

0.0033

0.0043

log F. A.

1.1222

−0.3367

−

0.0094

log F. A.

0.3701

−0.1066

−

0.0049

S. A.

0.6206

−2.8667

0.0064

0.0089

362

16

Discussion of the Time-Series Results

Table 16.1 (continued) Elasticity PCE

Budget share

Category

Model

Price

Steady_state

2004

5.7 Stationery and writing supplies 5.8 Household utilities 5.8a Electricity 5.8b Natural gas 5.8c Water and other sanitary services 5.8d Fuel oil and coal 5.8e Telephone and telegraph 5.9 Domestic service 5.10 Other household operation 6.0 Medical care 6.1 Drug preparations and sundries 6.2 Ophthalmic products and orthopedic appliances 6.3 Physicians 6.4 Dentists 6.5 Other professional services 6.6 Hospitals and nursing homes 6.6a Hospitals 6.6a1 Nonprofit 6.6a2 Proprietary 6.6a3 Government 6.6b Nursing homes 6.7 Health insurance 6.7a Medical care and hospitalization 6.7b Income loss 6.7c Workers’ compensation 7.0 Personal business 7.1 Brokerage charges and investment counseling 7.2 Bank service charges, trust services and safe deposit box rental

B-C

0.8586

−2.5906

0.0026

0.0022

B-C log F. A. B-C S. A.

1.2240 0.8723 0.1970 0.8952

−3.3004 −1.3435 −3.7863 −0.7670

0.0521 − 0.0084 0.0075

0.0312 0.0147 0.0067 0.0074

S. A. B-C

− 0.1393

−6.1020 −0.0030

− 0.0016

0.0024 0.0160

B-C B-C

0.2600 0.4980

−0.8292 −0.2810

0.0017 0.0042

0.0024 0.0070

B-C log F. A.

2.1962 1.8763

−0.9134 −0.4014

0.3664 −

0.2029 0.0308

B-C

0.0970

−0.0481

0.0040

0.0028

S. A. B-C B-C

17.6060 1.7089 5.9217

−18.1884 −1.4172 −4.0429

0.6607 0.0196 0.0926

0.0394 0.0097 0.0268

S. A.

4.2547

−5.2438

0.2960

0.0800

log F. A. S. A. log F. A. B-C log F. A. B-C B-C

4.1768 8.6490 3.0939 1.9108 2.8133 1.3937 1.6615

−3.7563 −4.4247 −1.5118 −0.9423 −2.6646 −0.3671 −0.2670

− 0.0779 − 0.0313 − 0.0188 0.0172

0.0671 0.0431 0.0084 0.0156 0.0129 0.0146 0.0122

B-C B-C

0.0776 1.0820

−1.2606 −0.6204

0.00003 0.0029

0.00029 0.0021

B-C B-C

1.3883 4.2243

−0.2781 −1.8893

0.0937 0.0188

0.0745 0.0010

B-C

1.9649

−0.2256

0.0152

0.0110

16.1

Tabulation of Annual Models

363 Table 16.1 (continued) Elasticity PCE

Budget share

Category

Model

Price

Steady_state

2004

7.3 Services furnished without payment by financial intermediaries, except life insurance carriers 7.4 Expense of handling life insurance and pension plans 7.5 Legal services 7.6 Funeral and burial expenses 7.7 Other personal business 8.0 Transportation 8.1 User-operated transportation 8.1a New autos 8.1b Net purchases of used autos 8.1c Other motor vehicles 8.1d Tires, tubes, accessories and other parts 8.1e Repair, greasing, washing, parking, storage, rental and leasing 8.1f Gasoline and oil 8.1g Bridge, tunnel, ferry and road tolls 8.1h Insurance 8.2 Purchased local transportation 8.2a Mass transit systems 8.2b Taxicab 8.3 Purchased intercity transportation 8.3a Intercity railway 8.3b Intercity bus

B-C

1.7222

−0.7476

0.0364

0.0240

B-C

3.1787

−4.1619

0.0577

0.0120

B-C B-C

0.5497 0.0265

−0.2448 −0.3719

0.0072 0.00012

0.0099 0.0020

B-C

1.6351

−1.5059

0.0082

0.0052

S. A. S. A.

0.8851 0.9310

−0.0851 −0.1266

0.1099 0.1059

0.1192 0.1121

S. A. B-C

0.2718 0.2551

−0.2685 −0.0558

0.0057 0.0029

0.0119 0.0065

B-C

3.5055

−0.9436

0.0470

0.0288

B-C

1.3718

−0.3487

0.0074

0.0066

B-C

1.2290

−0.3018

0.0280

0.0230

S. A. B-C

0.5088 0.7496

−0.3073 −0.6205

0.0165 0.00062

0.0280 0.00007

B-C S. A.

0.8933 −

−0.9324 −0.8724

0.0076 −

0.0065 0.0017

B-C

0.3395

−0.7613

0.0012

0.0013

S. A. B-C

0.1286 1.1332

−2.0535 −0.4716

0.00044 0.0072

0.00046 0.0054

S. A. B-C

−0.7360 −

−0.6790 −1.2175

−0.00046 −

0.00007 0.00025

364

16

Discussion of the Time-Series Results

Table 16.1 (continued) Elasticity PCE

Budget share

Category

Model

Price

Steady_state

2004

8.3c Airline 8.3d Other intercity transportation 9.0 Recreation 9.1 Books and maps 9.2 Magazines, newspapers and sheet music 9.3 Nondurable toys and sport supplies 9.4 Wheel goods, sports and photographic equipment, boats, and pleasure aircraft 9.5 Video and audio goods, including musical instruments, and computer goods 9.5a Video and audio goods, including musical instruments 9.5b Computers, peripherals and software 9.6 Radio and television repair 9.7 Flowers, seeds and potted plants 9.8 Admissions to specified spectator amusements 9.8a Motion picture theaters 9.8b Legitimate theaters and opera and entertainments of nonprofit institutions (except athletics) 9.8c Spectator sports 9.9 Clubs and fraternal organizations

B-C B-C

1.4801 1.9242

−0.0811 −1.4642

0.0065 0.0015

0.0040 0.0011

log F. A. S. A. S. A.

1.2729 1.6261 0.6012

−1.4270 −1.3614 −0.5934

− 0.0072 0.0037

0.0855 0.0050 0.0047

log F. A.

1.6151

−0.4223

−

0.0078

S. A.

1.7015

−1.5172

0.0101

0.0085

log F. A.

2.1135

−0.6427

−

0.0160

B-C

2.8748

−0.7541

0.0160

0.0097

B-C

12.3324

−1.0164

0.0411

0.0062

S. A.

−0.5780

−0.9928

−0.00060

0.00055

S. A.

1.0369

−1.1829

0.0022

0.0022

B-C

0.2259

−0.0479

0.0013

0.0046

B-C

0.5344

−0.7614

0.0017

0.0012

B-C

1.7837

−1.7446

0.0026

0.0015

S. A. B-C

1.2538 1.4471

−0.1531 −2.1844

0.0018 0.0043

0.0018 0.0028

16.1

Tabulation of Annual Models

365 Table 16.1 (continued) Elasticity

Budget share

Category

Model

PCE

Price

Steady_state

2004

9.10 Commercial participant amusements 9.11 Pari-mutuel net receipts 9.12 Other recreation 10.0 Education and research 10.1 Higher education 10.2 Nursery, elementary and secondary schools 10.3 Other education 11.0 Religious and welfare activities 12.0 Foreign travel and other, net 12.1 Foreign travel by U.S. residents 12.2 Expenditures abroad by U.S. residents 12.3 Less: Expenditures in the United States by nonresidents 12.4 Less: Personal remittances in kind to nonresidents

log F. A.

2.2815

−0.3577

−

0.0121

B-C

1.0790

−7.2733

0.0015

0.00065

S. A. log F. A.

2.4516 2.2625

−2.4969 −2.5965

0.0331 −

0.0206 0.0257

log F. A. log F. A.

1.5026 2.0551

−0.8880 −1.8603

− −

0.0143 0.0063

B-C B-C

1.9107 1.7989

−1.4080 −1.5034

0.0094 0.0406

0.0063 0.0267

S. A.

0.8936

−2.1338

0.000932

0.00001

S. A.

1.3691

−0.4213

0.0115

0.0112

B-C

5.7720

−0.1245

0.0140

0.00085

−

not estimated

−

−

−

not estimated

−

−

Codes: B-C: Bergstrom–Chambers model; S. A.: state-adjustment model; log F. A.: logarithmic flow- adjustment model.

Table 16.2 Stock adjustment and habit formation in the B-C and state-adjustment models Category

Model

β

δ

2004 Budget share

1.0 Food and tobacco 1.1 Food purchased for off-premise consumption 1.2 Purchased meals and beverages 1.5 Tobacco products 1.6a Food excluding tobacco and alcoholic beverages

S. A. B-C

−0.0827 −0.0692

0.0574 0.0139

0.1488 0.0838

B-C B-C B-C

−0.0159 −0.0001 0.0928

0.1612 0.1033 0.4047

0.0530 0.0107 0.1114

366

16

Discussion of the Time-Series Results

Table 16.2 (continued) Category

Model

β

δ

2004 Budget share

1.7 Alcoholic beverages 1.7a Alcoholic beverages purchased for off-premise consumption 1.7b Other alcoholic beverages 2.0 Clothing, accessories, and jewelry 2.2 Clothing and accessories except shoes 2.2a Women’s and children’s 2.2b Men’s and boys’ 2.4 Cleaning, storage and repair of clothing and shoes 2.5 Jewelry and watches 2.6 Other clothing and accessories 3.0 Personal care 3.1 Toilet articles and preparations 3.2 Barbershops, beauty parlors and health clubs 4.2 Tenant-occupied non-farm dwellings: Rent 4.3 Rental value of farm dwellings 4.4 Other housing 5.0 Household operation 5.1 Furniture, including mattresses and bedsprings 5.2 Kitchen and other household appliances 5.3 China, glassware, tableware and utensils 5.6 Cleaning and polishing preparations and miscellaneous household supplies and paper products 5.7 Stationery and writing supplies 5.8 Household utilities 5.8b Natural Gas 5.8c Water and other sanitary services 5.8d Fuel oil and coal 5.8e Telephone and telegraph 5.9 Domestic service 5.10 Other household operation 6.0 Medical care 6.2 Ophthalmic products and orthopedic appliances 6.3 Physicians 6.4 Dentists 6.5 Other professional services

B-C B-C

−0.0516 −0.0710

0.1484 0.2175

0.0160 0.0103

B-C B-C. B-C

0.3583 0.5099 −0.0037

0.4430 0.3575 0.0296

0.0057 0.0543 0.0334

B-C B-C B-C

0.0035 −0.0236 0.0056

0.0911 0.0286 0.1124

0.0209 0.0124 0.0019

B-C B-C B-C B-C B-C

−0.0353 0.0334 −0.0253 0.2565 0.2434

0.0286 0.0736 0.0621 0.3318 0.3361

0.0070 0.0053 0.0125 0.0068 0.0126

B-C

0.3930

0.6944

0.0302

B-C B-C B-C B-C

1.0016 −0.3216 0.0755 −0.1093

1.3810 0.1218 0.2491 0.0978

0.0016 0.0077 0.1000 0.0092

B-C

−0.0186

0.1000

0.0043

B-C

−0.0882

0.1106

0.0043

S. A.

0.3841

0.4896

0.0089

B-C B-C B-C S. A.

0.3035 0.2713 0.9028 0.1325

0.3710 0.3046 0.9579 0.3489

0.0022 0.0312 0.0067 0.0074

S. A. B-C B-C B-C B-C B-C

1.1246 −0.6351 −0.0095 −0.0484 0.5992 0.0079

1.1782 0.0567 0.1000 0.0175 0.8436 0.1132

0.0024 0.0160 0.0024 0.0070 0.2029 0.0028

S. A. B-C B-C

0.3841 0.0928 0.2270

0.4870 0.2077 0.5763

0.0394 0.0097 0.0268

16.1

Tabulation of Annual Models

367 Table 16.2 (continued)

Category

Model

β

δ

2004 Budget share

6.6 Hospitals and nursing homes 6.6a1 Nonprofit 6.6a3 Government 6.7 Health insurance 6.7a Medical care and hospitalization 6.7b Income loss 6.7c Workers’ compensation 7.0 Personal business 7.1 Brokerage charges and investment counseling 7.2 Bank service charges, trust services and safe deposit box rental 7.3 Services furnished without payment by financial intermediaries, except life insurance carriers 7.4 Expense of handling life insurance and pension plans 7.5 Legal services 7.6 Funeral and burial expenses 7.7 Other personal business 8.0 Transportation 8.1 User-operated transportation 8.1a New autos 8.1b Net purchases of used autos 8.1c Other motor vehicles 8.1d Tires, tubes, accessories and other parts 8.1e Repair, greasing, washing, parking, storage, rental and leasing 8.1f Gasoline and oil 8.1g Bridge, tunnel, ferry and road tolls 8.1h Insurance 8.2 Purchased local transportation 8.2a Mass transit systems 8.2b Taxicab 8.3 Purchased intercity transportation 8.3a Intercity railway 8.3b Intercity bus 8.3c Airline 8.3d Other intercity transportation 9.1 Books and maps

S. A. S. A. B-C B-C B-C

1.9806 0.2547 −1.5800 0.0453 0.0111

2.0000 0.2889 1.0526 0.2903 0.4363

0.0800 0.0431 0.0156 0.0146 0.0122

B-C B-C B-C B-C

0.2841 0.0462 0.0815 0.2933

0.3447 0.0942 0.3906 0.5696

0.00029 0.0021 0.0745 0.0010

B-C

0.2042

0.4870

0.0110

B-C

0.1629

0.4207

0.0240

B-C

0.1951

0.2679

0.0120

−0.0450 −0.1660 0.1152 −0.1256 −0.2373 −0.3808 −0.3452 0.1365 0.0133

0.0670 0.1204 0.5836 0.1207 0.1378 0.0785 0.2500 0.2939 0.2028

0.0099 0.0020 0.0052 0.1192 0.1121 0.0119 0.0065 0.0288 0.0066

0.1713

0.2400

0.0230

S. A. B-C

0.0679 −0.1072

0.1822 0.1021

0.0280 0.00007

B-C S. A. B-C S. A. B-C

0.4154 −0.0395 −0.0340 0.3949 0.0008

0.5840 0.0582 0.0685 0.6829 0.1704

0.0065 0.0017 0.0013 0.00046 0.0054

S. A. B-C B-C B-C

−0.1240 0.0566 0.0435 0.0793

0.0819 0.2633 0.0771 0.2096

0.00007 0.00025 0.0040 0.0011

S. A.

0.1012

0.2769

0.0050

B-C B-C B-C S. A. S. A. S. A. B-C B-C B-C B-C

368

16

Discussion of the Time-Series Results

Table 16.2 (continued) Category

Model

β

δ

2004 Budget share

9.2 Magazines, newspapers and sheet music 9.4 Wheel goods, sports and photographic equipment, boats, and pleasure aircraft 9.5a Video and audio goods, including musical instruments 9.5b Computers, peripherals and software 9.6 Radio and television repair 9.7 Flowers, seeds and potted plants 9.8 Admissions to specified spectator amusements 9.8a Motion picture theaters 9.8b Legitimate theaters and opera and entertainments of nonprofit institutions (except athletics) 9.8c Spectator sports 9.9 Clubs and fraternal organizations 9.11 Pari-mutuel net receipts 9.12 Other recreation 10.3 Other education 11.0 Religious and welfare activities 12.0 Foreign travel and other, net 12.1 Foreign travel by U.S. residents 12.2 Expenditures abroad by U.S. residents

S. A.

−0.0166

0.1069

0.0047

S. A.

0.0246

0.0908

0.0085

B-C

0.1447

0.3262

0.0097

B-C

0.0553

0.2690

0.0062

S. A. S. A.

0.0058 0.2957

0.1511 0.4007

0.00055 0.0022

B-C

−0.0876

0.0058

0.0046

B-C B-C

−0.0366 0.2484

0.0548 0.8032

0.0012 0.0015

S. A. B-C

0.0277 0.0728

0.2359 0.2630

0.0018 0.0028

B-C S. A. B-C B-C S. A. S. A.

0.1280 0.5930 0.4261 0.2797 −1.0191 −0.1838

0.1588 0.6812 0.8680 0.4715 0.0178 0.2663

0.00065 0.0206 0.0063 0.0267 0.00001 0.0112

B-C

−0.0612

0.0101

0.00085

In general, inventory adjustment (negative βs) are found in the categories that one should expect, that is, for durable goods such as automobiles, furniture, kitchen appliances, and glass and flatware. Interestingly, as found in the 1970 edition of CDUS, expenditures for men and boys clothing is subject to inventory adjustment, while expenditures for women and girls clothing are characterized by habit formation. Habit formation (including that implicit in the log flow-adjustment models) likewise occurs where it is generally expected to be found: housing, personal care products and services, stationary and writing supplies, personal business and scattered throughout medical care, recreation, and education, as well as for categories that are “slave” to durable categories, such as household utilities and maintenance, tires, vehicle repair, gasoline and oil, transportation tolls, and car insurance.

16.2

Elasticities of Current Study Compared with Those in the 1970 Edition of CDUS

369

On the other hand, this is not to say that there are not some apparent anomalies, such as for telephone expenditures (which one would not expect to be inventoryadjusting) and expenditures for other vehicles (8.1c). However, as mentioned in the commentaries on the models in Chapter 15, the model for telephone expenditures in general leaves a lot to be desired, while category 8.3c includes expenditures for trucks and SUVs, which in recent years have the fastest growing segment of the automobile market. Finally, Tobacco (1.5), with its slight negative β, would also appear to be an anomaly. However, while habit formation is clearly in order for those who smoke, tobacco consumption for the population at large is indicated now to be an inferior good. Hence, a negative β is probably appropriate.

16.2 Elasticities of Current Study Compared with Those in the 1970 Edition of CDUS In Table 16.3, we provide a comparison of the steady-state elasticities from the present study with those that were estimated in the 1970 edition of CDUS. Obviously, the thing that most leaps out from this table is the much larger number of price elasticities (indeed, one for every category) for the present study. As has been noted, an important reason for this is the substantial statistical variation that was injected into the relative prices by the various crises and events of the last 35 years. Table 16.3 Price and total-expenditure elasticities in current study and 1970 edition of CDUS Current study Category 1.0 Food and tobacco 1.1 Food purchased for off-premise consumption 1.2 Purchased meals and beverages 1.3 Food furnished to employees (including military) 1.4 Food produced and consumed on farms 1.5 Tobacco products 1.6a Food excluding tobacco and alcoholic beverages 1.7 Alcoholic beverages 1.7a Alcoholic beverages purchased for off-premise consumption 1.8b Other alcoholic beverages 2.0 Clothing, accessories, and jewelry 2.1 Shoes

Total expenditure 0.2689 0.1963 0.8421 not estimated

1970 CDUS Price

Total expenditure

Price

−0.0799 −0.1194

not estimated 0.7115

−

−0.5408

δ=0 1.0342

−

not estimated

not estimated

9.7481 0.3583

−2.3459 −0.0477

0.8615 not estimated

0.2816 0.2221

−0.9555 −1.1307

0.6207 not estimated

0.2001 5.0589 0.3848

−5.2445 −3.9183 −0.4717

not estimated not estimated δ=0

−1.8919 −

370

16

Discussion of the Time-Series Results

Table 16.3 (continued) Current study Category 2.2 Clothing and accessories except shoes 2.2a Women’s and children’s 2.2b Men’s and boys’ 2.3 Standard clothing issued to military personnel 2.4 Cleaning, storage and repair of clothing and shoes 2.5 Jewelry and watches 2.6 Other clothing and accessories 3.0 Personal care 3.1 Toilet articles and preparations 3.2 Barbershops, beauty parlors and health clubs 4.0 Housing 4.1 Space rental value of owner-occupied non-farm dwellings 4.2 Space rental value of tenant-occupied non-farm dwellings 4.3 Rental value of farm dwellings 4.4 Other housing 5.0 Household operation 5.1 Furniture, including mattresses and bedsprings 5.2 Kitchen and other household appliances 5.3 China, glassware, tableware and utensils 5.4 Other durable house furnishings 5.5 Semi-durable house furnishings 5.6 Cleaning and polishing preparations and miscellaneous household supplies and paper products 5.7 Stationery and writing supplies 5.8 Household utilities 5.8a Electricity 5.8b Natural gas 5.8c Water and other sanitary services 5.8d Fuel oil and coal 5.8e Telephone and telegraph 5.9 Domestic service 5.10 Other household operation

Total expenditure

1970 CDUS Price

Total expenditure

Price

−0.7363

0.5151

−

−0.7098 −0.1208

0.9256 0.7141 not estimated

− −

−

−4.5137

β=δ

4.6820 3.9601 0.7463 0.8800 1.3480

−0.1688 −2.3460 −0.6543 −1.2269 −4.5054

1.6647 1.1657 not estimated 3.7406 1.3598

1.0326 1.2632

−1.1309 −1.4391

not estimated 2.6045

−1.2150

0.4015

−1.2608

1.5315

−0.1839

0.4802 0.4261 0.8045 0.3565

−0.8497 −0.1723 −0.4556 −0.0349

1.1283 1.2735 not estimated 0.5275

−0.6044 −

1.0794

−0.2292

1.4037

−0.3476

0.7736

−1.2617

0.7749

−2.5512

1.1222 0.3701 0.6206

−0.3367 −0.1066 −2.8667

1.1759 0.6466 1.6227

0.8586 1.2240 0.8723 0.1970 0.8952

−2.5906 −3.3004 −1.3435 −3.7863 −0.7670

1.8277 not estimated 1.9364 3.1087 0.5861

− 0.1393 0.2600 0.4980

−6.1020 −0.0030 −0.8292 −0.2810

δ=0 β=δ δ=0 1.2657

1.7092 1.1481 1.0557 not estimated

−0.6726 − −3.0391 −

−

− − −

−0.5638 −1.8926 − −0.1359

−0.2885

16.2

Elasticities of Current Study Compared with Those in the 1970 Edition of CDUS

371

Table 16.3 (continued) Current study Category 6.0 Medical care 6.1 Drug preparations and sundries 6.2 Ophthalmic products and orthopedic appliances 6.3 Physicians 6.4 Dentists 6.5 Other professional services 6.6 Hospitals and nursing homes 6.6a Hospitals 6.6a1 Nonprofit 6.6a2 Proprietary 6.6a3 Government 6.6b Nursing homes 6.7 Health insurance 6.7a Medical care and hospitalization 6.7b Income loss 6.7c Workers’ compensation 7.0 Personal business 7.1 Brokerage charges and investment counseling 7.2 Bank service charges, trust services and safe deposit box rental 7.3 Services furnished without payment by financial intermediaries, except life insurance carriers 7.4 Expense of handling life insurance and pension plans 7.5 Legal services 7.6 Funeral and burial expenses 7.7 Other personal business 8.0 Transportation 8.1 User-operated transportation 8.1a New autos 8.1b Net purchases of used autos 8.1c Other motor vehicles 8.1d Tires, tubes, accessories and other parts 8.1e Repair, greasing, washing, parking, storage, rental and leasing 8.1f Gasoline and oil 8.1g Bridge, tunnel, ferry and road tolls

Total expenditure

1970 CDUS Price

Total expenditure

Price − −0.3973

2.1962 1.8763 0.0970

−0.9134 −0.4014 −0.0481

not estimated 3.0422 1.3906

17.6060 1.7089 5.9217 4.2547 4.1768 8.6490 3.0939 1.9108 2.8133 1.3937 1.6615

−18.1884 −1.4172 −4.0429 −5.2438 −3.7563 −4.4247 −1.5118 −0.9423 −2.6646 −0.3671 −0.2670

1.1465 0.9976 1.3289 3.7114 not estimated not estimated not estimated not estimated not estimated 2.0162 not estimated

0.0776 1.0820 1.3883 4.2243

−1.2606 −0.6204 −0.2781 −1.8893

not estimated not estimated not estimated −2.9560

1.9649

−0.2256

β=δ=0

1.7222

−0.7476

1.0714

−

3.1787

−4.1619

1.1642

−

0.5497 0.0265 1.6351 0.8851 0.9310 0.2718 0.2551 3.5055 1.3718

−0.2448 −0.3719 −1.5059 −0.0851 −0.1266 −0.2685 −0.0558 −0.9436 −0.3487

β=δ=0 0.6462 β=δ=0 not estimated 1.0749 not estimated not estimated not estimated 1.9290

−1.1904

1.2290

−0.3018

0.8955

−0.3801

0.5088 0.7496

−0.3073 −0.6205

1.3572 4.4578

− − − −

−0.9162

−

− −

− −

372

16

Discussion of the Time-Series Results

Table 16.3 (continued) Current study Category 8.1h Insurance 8.2 Purchased local transportation 8.2a Mass transit systems 8.2b Taxicab 8.3 Purchased intercity transportation 8.3a Intercity railway 8.3b Intercity bus 8.3c Airline 8.8d Other intercity transportation 9.0 Recreation 9.1 Books and maps 9.2 Magazines, newspapers and sheet music 9.3 Nondurable toys and sport supplies 9.4 Wheel goods, sports and photographic equipment, boats, and pleasure aircraft 9.5 Video and audio goods, including musical instruments, and computer goods 9.5a Video and audio goods, including musical instruments 9.5b Computers, peripherals and software 9.6 Radio and television repair 9.7 Flowers, seeds and potted plants 9.8 Admissions to specified spectator amusements 9.8a Motion picture theaters 9.8b Legitimate theaters and opera and entertainments of nonprofit institutions (except athletics) 9.8c Spectator sports 9.9 Clubs and fraternal organizations 9.10 Commercial participant amusements 9.11 Pari-mutuel net receipts 9.12 Other recreation 10.0 Education and research

Total expenditure

1970 CDUS Price

Total expenditure

Price

0.8933 − 0.3395 0.1286 1.1332

−0.9324 −0.8724 −0.7613 −2.0535 −0.4716

1.2596 not estimated 1.3785 δ=0 not estimated

−

−0.7360 − 1.4801 1.9242

−0.6790 −1.2175 −0.0811 −1.4642

−3.2708 1.8944 5.8723 δ=0

1.2729 1.6261 0.6012

−1.4270 −1.3614 −0.5934

not estimated 1.4223 β=δ=0

1.6151

−0.4223

2.0107

−1.0186

1.7015

−1.5172

3.7162

−2.3889

2.1135

−0.6427

2.9950

2.8748

−0.7541

not estimated

12.3324

−1.0164

not estimated

−0.5780 1.0369

−0.9928 −1.1829

5.1978 3.3208

0.2259

−0.0479

not estimated

0.5344 1.7837

−0.7614 −1.7446

3.4075 1.2604

1.2538 1.4471

−0.1531 −2.1844

1.0697 0.0706

− −

2.2815

−0.3577

1.9143

−

1.0790 2.4516 2.2625

−7.2733 −2.4969 −2.5965

2.2770 2.1498 not estimated

−1.1967

−2.7785 −2.1657 −

−

−

−3.8427 −2.6514

−3.6685 −0.3109

− −1.0073

16.2

Elasticities of Current Study Compared with Those in the 1970 Edition of CDUS

373

Table 16.3 (continued) Current study Total expenditure

Category 10.1 Higher education 10.2 Nursery, elementary and secondary schools 10.3 Other education 11.0 Religious and welfare activities 12.0 Foreign travel and other, net 12.1 Foreign travel by U.S. residents 12.2 Expenditures abroad by U.S. residents 12.3 Less: Expenditures in the United States by nonresidents 12.4 Less: Personal remittances in kind to nonresidents

1970 CDUS Total expenditure

Price

Price

1.5026 2.0551

−0.8880 −1.8603

2.1512 2.7674

1.9107 1.7989 0.8936 1.3691

−1.4080 −1.5034 −2.1338 −0.4213

1.1251 1.8456 not estimated 3.0873

5.7720

−0.1245

1.7339

not estimated

δ=0

not estimated

1.1980

− − − −1.0156 −1.7707 −

−

Sources: Table 16 and Table 4.13 of CDUS (1970).

To make comparison of elasticities easier, Table 16.4 includes only those categories of expenditure for which steady-state estimates are available in both studies. These number 59 for total expenditure and 27 for price. Finally, scatter diagrams of the two sets of elasticity pairs in Table 16.4 are presented in Figs. 16.1 and 16.2. There are a number of reasons why elasticities of the two studies might differ, including taste change, data revision, use of different models, and changes in socio-demographical factors. Also, since elasticities with linear models (which all of the models estimated are, except for the logarithmic flow adjustment) depend upon budget shares and the point on the demand function at which they are calculated, increases in income and changes in relative prices can be factors as well. Whatever, in the discussion to follow, we shall take both sets of elasticities at face value, and draw conclusions accordingly.

Table 16.4 Total-expenditure and price elasticities, current study and 1970 CDUS Total-expenditure Category 1.1 1.5 1.7 2.2 2.2a 2.5

Current Study 0.1963 9.7481 0.2816 1.7092 1.1481 4.6820

Price 1970 CDUS 0.7115 0.8615 0.6207 0.5151 0.9256 0.1657

Category 1.5 2.5 3.1 4.1 4.2 4.3

Current Study

1970 CDUS

−2.3459 −0.1688 −1.2269 −1.4391 −1.2608 −0.8497

−1.8919 −0.6726 −3.0391 −1.2150 −0.1839 −0.6044

374

16

Discussion of the Time-Series Results

Table 16.4 (continued) Total-expenditure

Price

Category

Current Study

1970 CDUS

Category

Current Study

1970 CDUS

2.6 3.1 3.2 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8a 5.8b 6.1 6.2 6.3 6.4 6.5 6.6 6.7 7.3 7.4 7.6 8.1 8.1d 8.1e 8.1f 8.1 g 8.2a 8.3a 8.3d 9.1 9.3 9.4 9.5 9.6 9.7 9.8a 9.8b 9.8c 9.9 9.11 9.12 10.1 10.2

3.9601 0.8800 1.3480 1.2632 0.4015 0.4802 0.3565 1.0794 0.7736 1.1222 0.3701 0.6206 0.8586 0.8723 0.1970 1.8763 0.0970 17.6060 1.7089 5.9217 4.2547 1.3937 1.7722 3.1787 0.0265 0.9310 1.3718 1.2290 0.5088 0.7496 0.3395 −0.7360 1.4801 1.6261 1.6151 1.7015 2.1135 −0.5780 1.0369 0.5344 1.7837 1.2538 1.4471 2.2815 2.4516 1.5026 2.0551

1.1657 3.7406 1.3598 2.6045 1.5315 1.1283 0.5275 1.4037 0.7749 1.1759 0.6466 1.6227 1.8277 1.9364 3.1087 3.0422 1.3906 1.1465 0.9976 1.3289 3.7114 2.0162 1.0714 1.1642 0.6462 1.0749 1.9290 0.8955 1.3572 1.2596 1.3785 −3.7162 5.8723 1.4223 1.4223 3.7162 2.9950 5.1978 3.3208 3.4075 1.2604 1.0697 0.0706 1.9143 2.1498 2.1512 2.7674

5.8a 5.8c 6.1 6.2 6.7 8.1d 8.2 8.3a 8.3b 9.3 9.4 9.6 9.7 9.8a 9.8b 9.12 11.0 12.1

−1.3435 −0.7670 −0.2810 −0.0481 −0.3671 −0.3487 −0.8724 −0.6790 −1.2175 −0.4223 −1.5172 −0.9928 −1.1829 −0.7614 −1.7446 −2.4969 −1.5034 −0.4213

−1.8926 −0.1359 −0.2885 −0.3973 −0.9162 −1.1904 −1.1967 −2.7785 −2.1657 −1.1086 −2.3889 −3.8427 −2.6514 −3.6685 −0.3109 −1.0073 −1.0156 −1.7707

16.2

Elasticities of Current Study Compared with Those in the 1970 Edition of CDUS

375

Table 16.4 (continued) Total-expenditure Category 10.3 11.0 12.1 12.2

Price

Current Study

1970 CDUS

1.9107 1.7989 1.3691 5.7720

Category

Current Study

1970 CDUS

1.1251 1.8456 3.0873 1.7339

From the scatter diagrams in Figs. 16.1 and 16.2, the impression is one of disparity.3 However, for the total-expenditure elasticities, this impression is somewhat misleading, for, if outliers are ignored, about two-thirds of the elasticity pair-points lie within a reasonable distance of a 45◦ line.4 In terms of major categories, the agreement in total-expenditure elasticities is fairly good for household operation (category 5), transportation (category 8), and recreation (category 9), education (category 10), and eleemosynary activities (category 11), but not for clothing and shoes (category 2) or housing (category 4). Of the 43 categories with total-expenditure elasticities greater than 1 in the 1970 study, 29 of these show the same in the present effort. On the other hand, there are now 7 categories indicated to be luxuries, alcoholic beverages (1.7) and jewelry and watches (2.5) being notable instances.5

6

4

2

0 0

–1

1

2

3

4

5

6

–2

–4

current study

Fig. 16.1 Steady-state total-expenditure elasticities

3

The elasticity-pairs for 1.5 (tobacco) and 6.3 (physicians) are excluded from Figure 16.1. Given the scale of the axes, we should in fact say a 22.5-degree line. 5 In the 1970 study, there was only one instance of inferiority, namely, 8.3a (intercity railway). In the present study, not only does 8.3a continue to be inferior, but so, too, are 1.5 (tobacco) and 9.6 (radio and television repair). The latter is an interesting phenomenon, for, in the 1970 study, the 4

376

16

Discussion of the Time-Series Results 0

–2.5

–2

–1.5

–1

–0.5

0 –0.5 –1 –1.5 –2 –2.5 –3 –3.5 –4 –4.5

current study

Fig. 16.2 Comparison of steady-state price elasticities

With regard to estimates of the steady-state price elasticities, the first impression in Fig. 16.2 is essentially the correct one, for only a handful of elasticity-pairs lie with a reasonable distance of the line of equality. However, of the 16 categories indicated to be elastic in the 1970 study 14 continue to be so, while only 2 are now indicated to be elastic, these being rented housing (4.2) and theater and opera (9.8b).

16.3 Interpretation of Total-Expenditure Elasticities in Terms of Maslovian Hierarchy of Wants We now turn to an analysis of the steady-state total-expenditure elasticities in terms of the five Maslovian wants similar to the one that was undertaken in Chapter 11 for the cross-sectional total-expenditure elasticities from the CES surveys. Recall that the analysis entails the devising of a qualitative set of weights over the five wants that can be used to assign the total-expenditure elasticity for a category to an appropriate ordinal location. To this end, three ordinal elasticity sets are defined as follows: (i) A small elasticity set that consists of categories whose expenditures tend to be dominated by physiological or security needs;

steady-state total-expenditure elasticity for radio and television repair was indicated to be highly elastic.

16.3

Interpretation of Total-Expenditure Elasticities

377

(ii) A medium elasticity set that consists of categories whose expenditures tend to be dominated by love or esteem needs; (iii) A large elasticity set consists of categories whose expenditures tend to be dominated by love, esteem, or self-actualization needs. For small, medium, and large, it is once again assumed that: Small elasticity set (“necessities”): Steady-state total-expenditure elasticities that lie between 0 and 0.50. Medium elasticity set (“near luxuries”): Steady-state total-expenditure elasticities that lie between 0.50 and 0.75. Large elasticity set (“luxuries”): Steady-state total-expenditure elasticities that are greater than 0.75. The analysis is conducted in Table 16.5, and follows the same format as Table 11.31 for the CES total-expenditure elasticities. Again, to recall, the three important columns of the table are the second, third and fifth. In the second column, expenditure categories are associated with Maslovian needs, the most important of which are highlighted in bold. In the third column, the weights implicit in column 2 are used to assign total-expenditure elasticities to one of the three elasticity sets (small, medium, or large) described above. Finally, the fifth column provides an informal test of the scheme by comparing the steady-state total-expenditure elasticities from Table 16.1 (listed in column 4) with their theoretical elasticity sets in column 3. The judgments expressed in columns 2 and 3 of the table are based upon the same subjective considerations (and indeed employ the same assignments) as in Chapter 11. Table 16.5 Determination of elasticity sets, NIPA time-series expenditure categories Expenditure category 1.0 Food and tobacco 1.1 Food purchased for off-premise consumption 1.2 Purchased meals and beverages 1.5 Tobacco products 1.6a Food excluding tobacco and alcoholic beverages 1.7 Alcoholic beverages 1.7a Alcoholic beverages purchased for off-premise consumption 1.8b Other alcoholic beverages

Maslovian Needs

Theoretical Elasticity set

Estimated Elasticity

P,L,E,SA P,L,E,SA

Small Small

0.2689 0.1963

yes yes

P,L,E,SA

Medium

0.8421

no

P,L P,L,E,SA

Small Small

−0.4434 0.3583

P,L,E,SA P,L,E,SA

Large Large

0.2816 0.2221

no no

P,L,E,SA

Large

1.0465

yes

Agreement

yes yes

378

16

Discussion of the Time-Series Results

Table 16.5 (continued) Expenditure category 2.0 Clothing, accessories, and jewelry 2.1 Shoes 2.2 Clothing and accessories except shoes 2.2a Women’s and children’s 2.2b Men’s and boys’ 2.4 Cleaning, storage and repair of clothing and shoes 2.5 Jewelry and watches 2.6 Other clothing and accessories 3.0 Personal care 3.1 Toilet articles and preparations 3.2 Barbershops, beauty parlors and health clubs 4.0 Housing 4.1 Space rental value of owner-occupied non-farm dwellings 4.2 Space rental value of tenant-occupied non-farm dwellings 4.3 Rental value of farm dwellings 4.4 Other housing 5.0 Household operation 5.1 Furniture, including mattresses and bedsprings 5.2 Kitchen and other household appliances 5.3 China, glassware, tableware and utensils 5.4 Other durable house furnishings 5.5 Semi-durable house furnishings 5.6 Cleaning and polishing preparations and miscellaneous household supplies and paper products 5.7 Stationery and writing supplies 5.8 Household utilities 5.8a Electricity 5.8b Natural Gas

Maslovian Needs

Theoretical Elasticity set

Estimated Elasticity

P,L,E,SA

Large

5.0589

yes

P,L,E,SA, P,L,E,SA

Medium Large

0.3848 1.7092

no yes

P,L,E,SA P,L,E,SA P,L,E

Large medium small

1.1481 1.0557 −

yes no yes

L,E,SA L,E,SA

Large Large

4.6820 3.9601

yes yes

P,L,E,SA P,L,E,SA

medium medium

0.7463 0.8800

yes yes

P,L,E,SA

Large

1.3480

yes

P,S,L,E,SA P,S,L,E,SA

Large Large

1.0326 1.2632

yes yes

P,S,L,E,SA

medium

0.4015

no

P,S,L,E,SA

medium

0.4802

no

P,S,L,E,SA P,S,L,E P,S,L,E,SA

Large Large medium

0.4261 0.8045 0.3565

no yes no

P,S,L,E,SA

Large

1.0794

yes

P,L,E,SA

Large

0.7736

yes

P,L,E,SA

Large

1.1222

yes

P,S,L

small

0.3701

yes

P.S,L

small

0.6206

no

P,S,L,E,SA

Large

0.8586

yes

P,S,L,E P,S,L,E,SA P,S,L

small small small

1.2240 0.8723 0.1970

no no yes

Agreement

16.3

Interpretation of Total-Expenditure Elasticities

379

Table 16.5 (continued) Expenditure category 5.8c Water and other sanitary services 5.8d Fuel oil and coal 5.8e Telephone and telegraph 5.9 Domestic service 5.10 Other household operation 6.0 Medical care 6.1 Drug preparations and sundries 6.2 Ophthalmic products and orthopedic appliances 6.3 Physicians 6.4 Dentists 6.5 Other professional services 6.6 Hospitals and nursing homes 6.6a Hospitals 6.6a1 Nonprofit 6.6a2 Proprietary 6.6a3 Government 6.6b Nursing homes 6.7 Health insurance 6.7a Medical care and hospitalization 6.7b Income loss 6.7c Workers’ compensation 7.0 Personal business 7.1 Brokerage charges and investment counseling 7.2 Bank service charges, trust services and safe deposit box rental 7.3 Services furnished without payment by financial intermediaries, except life insurance carriers 7.4 Expense of handling life insurance and pension plans 7.5 Legal services 7.6 Funeral and burial expenses 7.7 Other personal business 8.0 Transportation 8.1 User-operated transportation 8.1a New autos 8.1b Net purchases of used autos 8.1c Other motor vehicles 8.1d Tires, tubes, accessories and other parts

Maslovian Needs

Theoretical Elasticity set

Estimated Elasticity

P,S,L,E

small

0.8952

no

P,S,L S,L,E,SA S,E, P,S,L P,S,L,E P,S,L,E

small medium small small Large Large

− 0.1393 0.2600 0.4980 2.1962 1.8763

yes no yes yes yes yes

P,S

small

0.0970

yes

P,S,L,E P,S,L,E P,S,L,E P,S,L,E P,S,L,E P,S,L,E P,S,L,E P,S,L,E P,S,L,E P,S,L,E P,S,L,E

Large Large Large Large Large Large Large Large Large Large Large

17.6060 1.7089 5.9217 4.2547 4.1768 8.6490 3.0939 1.9108 2.8133 1.3937 1.6615

yes yes yes yes yes yes yes yes yes yes

S S S,L,E,SA S,L,E,SA

small small Large Large

0.0776 1.0820 1.3883 4.2243

yes no yes yes

S,L,E,SA

Large

1.9649

yes

S,L,E,SA

Large

1.7222

yes

S,L,E,SA

Large

3.1787

yes

S,L,E L,E S,L,E,SA P,S,L,E,SA P,S,L,E,SA P,S,L,E,SA P,S,L,E,SA

medium small Large Large Large Large small

0.5497 0.0265 1.6351 0.8851 0.9310 0.2718 0.2551

yes yes yes yes yes yes yes

P,S,L,E,SA P,S,L,E,SA

Large Large

3.5055 1.3718

yes yes

Agreement

380

16

Discussion of the Time-Series Results

Table 16.5 (continued) Expenditure category 8.1e Repair, greasing, washing, parking, storage, rental and leasing 8.1f Gasoline and oil 8.1g Bridge, tunnel, ferry and road tolls 8.1h Insurance 8.2 Purchased local transportation 8.2a Mass transit systems 8.2b Taxicab 8.3 Purchased intercity transportation 8.3a Intercity railway 8.3b Intercity bus 8.3c Airline 8.8d Other intercity transportation 9.0 Recreation 9.1 Books and maps 9.2 Magazines, newspapers and sheet music 9.3 Nondurable toys and sport supplies 9.4 Wheel goods, sports and photographic equipment, boats, and pleasure aircraft 9.5 Video and audio goods, including musical instruments, and computer goods 9.5a Video and audio goods, including musical instruments 9.5b Computers, peripherals and software 9.6 Radio and television repair 9.7 Flowers, seeds and potted plants 9.8 Admissions to specified spectator amusements 9.8a Motion picture theaters 9.8b Legitimate theaters and opera and entertainments of nonprofit institutions (except athletics) 9.8c Spectator sports

Maslovian Needs

Theoretical Elasticity set

Estimated Elasticity

P,S,L,E,SA

Large

1.2290

yes

P,S,L P,S,L,E,SA

medium Large

0.5088 0.7496

yes yes

S,L,E,SA S,L

Large Small

0.8933 −

yes yes

S,L S,L S,L,E,SA

Small Small Large

0.3395 0.1286 1.1332

yes yes yes

S,L,E,SA S,L,E,SA S,L,E,SA S,L,E,SA

Small Small Large Large

−0.7360 − 1.4801 1.9242

yes yes yes yes

P,L,E,SA L,E,SA S,L,E,SA

Large Large medium

1.2729 1.6261 0.6012

yes yes yes

L,E,SA

Large

1.6151

yes

L,E,SA

Large

1.7015

yes

S,L,E,SA

Large

2.1135

yes

S,L,E,SA

Large

2.8748

yes

S,L,E,SA

Large

12.3324

yes

S,L,E,SA L,E,SA

Small Large

−0.5780 1.0369

yes yes

L,E,SA

Large

0.2259

no

L,E,SA L,E,SA

medium Large

0.5344 1.7837

yes yes

L,E,SA

Large

1.2538

yes

Agreement

16.3

Interpretation of Total-Expenditure Elasticities

381

Table 16.5 (continued) Expenditure category 9.9 Clubs and fraternal organizations 9.10 Commercial participant amusements 9.11 Pari-mutuel net receipts 9.12 Other recreation 10.0 Education and research 10.1 Higher education 10.2 Nursery, elementary and secondary schools 10.3 Other education 11.0 Religious and welfare activities 12.0 Foreign travel and other, net 12.1 Foreign travel by U.S. residents 12.2 Expenditures abroad by U.S. residents

Maslovian Needs

Theoretical Elasticity set

Estimated Elasticity

L,E,SA

Large

1.4471

yes

L,E,SA

Large

2.2815

yes

L,E,SA L,E,SA S,L,E,SA S,L,E,SA S,L,E,SA

Large Large Large Large Large

1.0790 2.4516 2.2625 1.5026 2.0551

yes yes yes yes yes

S,L,E,SA S,L,E,SA

Large Large

1.9107 1.7989

yes yes

L,E,SA L,E,SA

Large Large

0.8936 1.3691

yes yes

L,E,SA

Large

5.7720

yes

Agreement

Codes: P: Physiological needs S: Security needs L: Love (community) needs E: Esteem needs SA: Self-actualization needs Y: Estimated elasticity lies within limits defined by theoretical elasticity set. N: Estimated elasticity does not lie within limits defined by theoretical elasticity set. Bold: Indicates “dominant”; absence indicates no clear-cut dominance.

Of the 107 estimated steady-state elasticities listed in Table 16.3, 91 are seen to be correctly classified according to the Maslovian criteria, while 16 are not.6 Of the 16 mis-classifications, the aggregate and off-premise component of alcoholic beverages (1.7 and 1.7b) are perhaps the most anomalous. Spectator amusements (9.8) also shows a large discrepancy, yet, significantly, the elasticities for all of its component are correctly classified. In some cases, the elasticities of categories that are mis-classified, such as purchased meals (1.2), shoes (2.1), and rented and farm housing (4.2 and 4) are fairly close to their theoretical limits, while in other instances (as with spectator admissions), aggregation issues might be involved. Finally, the fact that three of the five household utility categories (as well as the aggregate) are mis-classified is particularly troubling.

6

Interestingly (coincidentally?), the percentage correctly classified here (84.11) is close to that found in Chapter 11 for the cross-sectional total-expenditure elasticities (2.4 of 29, or 82.76%). Unfortunately, however, neither percentage is sufficient to fail to reject (at conventional levels of significance) a null hypothesis that the classification scheme proffered is correct.

382

16

Discussion of the Time-Series Results

16.4 Statistical Quality of the Time-Series Models Overall, the statistical quality of the estimated NIPA equations is good. Of the 112 models tabulated in Chapter 15, 39 are excellent (   !), while only 6 get a dreaded frown (). Of the remaining 67, 37 are fair () and 30 are good ( ). In general, positive autocorrelation in the residuals has turned out not to be much of a problem in the least-squares equations (especially in the B–C models), and, in the instances where it is, usually disappears through application of an appropriate generalized-least-squares estimator. Also, only in an instance or two, does the Dickey–Fuller test even hint of a unit root in the residuals.7 Moreover, despite the fact that statistical significance (as measured by conventional standards) has, for the most part, not been a consideration in the selection of “final” models, nevertheless most of the time t-ratios for total expenditure are in excess of 2, and often for price as well. Finally, as is to be expected with aggregate time-series data, R2 s are typically greater than 0.99.

16.5 Asymmetry in Residuals? A curious result in the cross-sectional analysis of earlier chapters, it will be recalled, was an emergence of asymmetry and long tails in the residuals from the equations estimated with data from the BLS consumer expenditure surveys. Given the pervasiveness of the phenomenon with the cross-sectional data, one has to wonder whether something similar may be lurking in the residuals of the time-series models of the present chapters. While 70 residuals from the annual equations of Chapter 15 are probably too few in number to indicate anything definitive, the 200-plus residuals from the quarterly equations of Chapter 14 may at least be suggestive. Whatever, it will be interesting to take a brief look. The kernel-smoothed residuals for four of the quarterly models in Chapter 14 (food, housing, motor vehicles and parts, and medical care) are presented in Fig. 16.3. All are residuals from B–C models. Asymmetry, although not nearly as dramatic as in the cross-sectional equations, is nevertheless clearly present in the residuals for all but motor vehicles and parts. Also, long tails are evident in all four. Since the four categories in Fig. 16.3 are all estimated with the B–C model, residuals from a couple of categories (furniture and household equipment and clothing and shoes) estimated with logarithmic flow-adjustment models have been examined as well. Their graphs are given in Fig. 16.4. While the kernel-smoothed density functions for these categories appear symmetrical, long tails are again evident.

7 As noted in Chapter 13, the Dickey–Fuller test is used simply to investigate whether the orders of integration of the dependent and independent variables are the same, in short, as a specification test as to whether the integrative-order of the dynamic model that is specified is in accordance with that in the time-series used in estimation.

16.5

Asymmetry in Residuals?

383

So much for the quarterly models, what about the residuals from the annual models? Fig. 16.5 presents the kernel-smoothed residuals from the equations for two annual categories, food excluding tobacco and alcohol (1.6a) and clothing and medical care (6.0). Both are B–C models. The graphs are striking, for both asymmetry and long tails are present in spades! Indeed, it might be thought that we

A

0.025 0.02 density

0.015 0.01 0.005 0 –150

–100

–50

0

50

100

150

residual

Food

B

0.016 0.014 0.012 density

0.01 0.008 0.006 0.004 0.002 0 –200

–150

–100

–50

0

50

100

150

residual

Motor Vehicles & Parts

C

0.07 0.06

density

0.05 0.04 0.03 0.02 0.01 0 –40

–20

0 residual

Housing

Fig. 16.3 Kernel-smoothed residuals quarterly B–C models

20

40

200

384

16

D

Discussion of the Time-Series Results

0.035 0.03 0.025 density

0.02 0.015 0.01 0.005

–80

–60

–40

0 0

–20

20

40

60

80

residual

Medical Care

Fig. 16.3 (continued)

25

A

20 density

15 10 5 0

–0.18

–0.12

–0.06

0

0.06

0.12

0.18

residual

Furniture & Equipment 30

B

25

density

20 15 10 5 0 –0.08

–0.04

0

0.04

residual

Clothing & Shoes Fig. 16.4 Kernel-smoothed residuals quarterly log flow-adjustment models

0.08

16.5

Asymmetry in Residuals?

385

A

0.012 0.01 density

0.008 0.006 0.004 0.002 0

–125 –100

–75

–50

–25

0 25 residual

50

75

100

125

Food

density

B

–125

–75

0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 –25 25 residual

75

125

Medical Care

Fig. 16.5 Kernel-smoothed residuals annual B–C models

were once again looking at residuals from cross-sectional equations! However, as noted earlier, 70 observations are probably too few to establish anything definitive. Nevertheless, the evidence here is certainly consistent with the conclusion from the cross-sectional analysis that, in general, error terms attaching to consumption behavior appear to be at variance with the predicates of the Gauss–Markov theorem. In the cross-sectional analysis, our “solution” for asymmetry and long tails was to estimate models by quantile regression as well as by least squares. Because of the nonlinearity of the estimating equations for the B–C and state-adjustment models, quantile regression has not been pursued in the time-series analysis. However, to illustrate what might have been involved had we done so, quantile-regression estimates for two logarithmic flow-adjustment models (whose estimating equations are not nonlinear) are presented in Fig. 16.6. The categories for which this is done are 2.1 (shoes) and 4.1 (owner-occupied housing). Again, asymmetry and long tails are evident in both cases.

386

16

A

Discussion of the Time-Series Results

12 10 density

8 6 4 2 0 –0.3

–0.2

–0.1

0

0.1

0.2

0.3

residual

B

14 12

density

10 8 6 4 2 0 –0.3

–0.2

–0.1

0

0.1

0.2

0.3

residual

K0 K1 K2 K3 R2

Annual Data Shoes OLS equation Coefficient t-ratio 2.2551 4.66 0.6003 7.66 0.1538 5.40 –0.3774 –5.25 0.9851

coefficient 2.8322 0.5477 0.1588 –0.4556 0.9986

Q. R. equation t-ratio 5.62 6.72 5.36 –6.09

Fig. 16.6 Comparison of OLS (A,C) and quantile (B,D) regressions log flow-adjustment model

16.5

Asymmetry in Residuals?

387

C

35 30

density

25 20 15 10 5 0 –0.08

–0.06

–0.04

–0.02

0

0.02

0.04

0.06

0.08

residual

D

40 35 30

density

25 20 15 10 5 0 –0.08

–0.06

–0.04

–0.02

0

0.02

0.04

0.06

residual

K0 K1 K2 K3 K4 R2

Owner-Occupied Housing OLS equation Q. R. equation Coefficient t -ratio coefficient t-ratio –0.5345 -6.90 –0.5774 –6.32 0.9191 42.60 0.9279 36.48 1.2296 15.62 1.2970 13.96 –1.1047 –12.99 –1.1728 –11.69 –0.0470 –4.56 –0.0544 –4.48 0.9986 0.9986

Fig. 16.6 (continued)

0.08

Chapter 17

Comparison of Time-Series and Cross-Sectional Elasticities

In this chapter, we undertake a comparison of the elasticities estimated from the NIPA time-series data with their counterparts estimated in earlier chapters from the BLS consumer expenditure surveys. Two comparisons will be made. The first is of total-expenditure elasticities for the CES 29 categories of expenditure analyzed in Chapter 11, while the second is of both total-expenditure and price elasticities at the CES 6-category level analyzed in Chapter 6. In both cases, comparison is to steady-state NIPA elasticities.1

17.1 29-Category Level of Aggregation Despite the conceptual problems between the two data sources noted in Footnote 1, we will put these to the side for the moment, and proceed to regroup the NIPA categories to conform as closely as possible to the CES categories. The regrouping involved is described in Table 17.1. Six new NIPA categories are seen to be created, namely, for housing (consisting of NIPA categories 4.0 and 5.0), other lodging (consisting of 4.3 and 4.4), etc.2 New models are then estimated (as in Chapter 15)

1 Although the exercises in this chapter might seem in principle to be straightforward, this is unfortunately not the case, for there are many differences between the time series produced by the Department of Commerce and the household surveys undertaken by the Department of Labor. While both data-collection enterprises are intended to measure the same phenomenon, namely, the consumption expenditures of American households, they differ not only in methods, but also in concepts. The classification of expenditures is in general quite different. Some time-series items do not appear in the surveys at all, including several fictitious business services (such as “services rendered without payment by financial institutions”), food and clothing for the military, expenditures by non-profit organizations, and expenditures by foreign visitors to the U.S. The treatment of owner-occupied housing is markedly different, for the time series give an imputed value for space rental value, while the surveys give actual expenditures by households. 2 Aggregation proceeds by summing the real expenditures for the categories involved, and then obtaining the real price from the ratio of summed nominal expenditures to the summed real expenditures divided by the deflator for total PCE.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_17, 

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17 Comparison of Time-Series and Cross-Sectional Elasticities

Table 17.1 Correspondence between CES and NIPA categories of expenditure 29-category level of aggregation CES categories

NIPA categories

Food consumed at home Food consumed outside of home Alcoholic beverages Housing Shelter Owned dwellings Rented dwellings Other lodging Utilities Natural gas Electricity Telephone services Water and other public services Housing operations House furnishings and equip. Apparel and services Men and boys Women and girls Transportation Gasoline and motor oil Health care Entertainment Personal care and products Reading materials Education Tobacco products Miscellaneous expenditures Cash contributions Personal insurance and pensions

1.1 1.2 1.7 4.0 + 5.0 4.0 4.1 4.2 4.3 + 4.4 5.8 5.8b 5.8a 5.8e 5.8c 5.5 + 5.7 + 5.9 + 5.10 5.1 + 5.2 + 5.3 + 5.4 2.0 2.2b 2.2a 8.0 8.1f 6.0 9.0−9.1−9.2 3.0 9.1 + 9.2 10.0 1.5 not analyzed 11.0 7.4

Note: NIPA codes are from Table 13.2.

for each of these new categories.3 Table 17.2 contains the CES and NIPA totalexpenditure elasticities to be compared. The CES elasticities are the estimates (mean values for the 16 quarters, 1996–1999) from the OLS double-logarithmic equations in Table 11.30 of Chapter 11, while the NIPA estimates, except for the six new categories, are from Table 16.1.4 In general, what shows in Table 17.2 is as follows: (1) Although not invariably, the NIPA elasticities tend to be larger than their CES counterparts, indeed, in some instances, notably larger. 3

All six of the estimated equations are B-C models. The equations for total housing, other housing, and reading materials are excellent statistically, while the equations for the other three new categories are only fair to good. 4 A NIPA category corresponding to the CES miscellaneous category has not been constructed.

17.1

29-Category Level of Aggregation

391

Table 17.2 Comparison of CES and NIPA total-expenditure elasticities, 29 categories Expenditure category

CES

Food consumed at home Food consumed outside of home Alcoholic beverages Housing Shelter Owned dwellings Rented dwellings Other lodging Utilities Natural gas Electricity Telephone services Water and other public services. Housing operations House furnishings and equipment Apparel and services Men and boys Women and girls Transportation Gasoline and motor oil Health care Entertainment Personal care and products Reading materials Education Tobacco products Cash contributions Personal insurance and pensions

0.3232 0.8116 0.4842 0.8201 0.9120 0.9156 0.5379 0.8288 0.3790 0.1950 0.2352 0.4620 0.1714 0.8620 1.0058 0.9151 0.6603 0.7600 1.3021 0.3911 0.5306 0.9783 0.4642 0.5503 0.9446 0.1470 1.0597 0.9206

NIPA 0.1963 0.8421 0.2821 0.8800 1.0326 1.2632 0.4015 0.6849 1.2240 0.1970 0.8723 0.1393 0.8952 0.3685 0.8437 5.0589 1.9989 1.1481 0.8851 0.5088 2.1962 2.1641 0.7463 0.8800 2.2625 −0.4434 1.7989 3.1787

(2) However, putting a half a dozen or so NIPA outliers to the side (specifically, apparel, health care, entertainment, education, and personal insurance and pensions), agreement in fact is fairly reasonable, as is apparent from the scatter diagram of the elasticity pairs that is given in Fig. 17.1. This particularly is the case for the two food categories, alcoholic beverages, total housing, shelter and its components, and furniture and equipment. (3) Agreement is worst for household utilities and its components and apparel and its components. Of the two most pronounced NIPA outliers, apparel and services and personal insurance and pensions, the former is clearly the more disturbing, for the time series and surveys should in principle be pretty much measuring the same thing.5 On the 5 The problem with the NIPA clothing elasticity, as was noted in Chapter 15, is the extremely strong trend in the dependent variable, which appears to adversely affect the estimates of the structural coefficients relative to one another.

392

17 Comparison of Time-Series and Cross-Sectional Elasticities 2.5

NIPA Elasticities

2 1.5 1 0.5 0 0.5

1

1.5

2

2.5

–0.5 –1

CES Elasticities

Fig. 17.1 Scatter diagram of CES and NIPA total-expenditure elasticities 29 CES categories

other hand, personal insurance and pensions are measured as actual expenditures in the surveys, but are imputed values in the time series. While the CES and NIPA elasticities for natural gas are virtually identical, the differences are large for electricity and water. As with apparel, these disparities are disquieting, for one would again think that the expenditures being measured are essentially the same.

17.2 CES-ACCRA 6-Category Level of Aggregation We now turn to a comparison of CES and NIPA elasticities, for both total expenditure and price, that have been estimated at the 6-category CES-ACCRA level of aggregation. Interestingly, the amount of regrouping required to construct NIPA categories that conform to the CES categories (again, taking the NIPA categories at face value) is less than for the 29 categories, as is evident in Table 17.3. Indeed, the only new equations to be estimated are for food at home and miscellaneous expenditures. The relevant elasticities are presented in Table 17.4. The CES-ACCRA elasticities, for both the double-logarithmic and logarithmic flow-adjustment models, are taken from Table 10.8. The NIPA elasticities are from Table 16.1, except for the ones for food at home and miscellaneous expenditures, which are from the newly estimated equations.6 The picture in Table 17.4 between the CES-ACCRA and NIPA elasticities is hardly one of tightness. Food at home, housing, and transportation are the only categories for which the NIPA total-expenditure elasticities are reasonably close to their CES-ACCRA counterparts, while the price elasticities for health care are the only ones for which this is the case. The lack of agreement is seen graphically in 6

The equation for food at home is a B-C model of fair-to-middling statistical quality that shows rapid flow-adjustment and a negative β. The equation for miscellaneous expenditures is a stateadjustment model of excellent statistical quality that shows very mild inventory adjustment.

17.3

Summary

393

Table 17.3 Correspondence between CES and NIPA categories of expenditure, 6-category CESACCRA level of aggregation CES categories

NIPA categories

Food at home Shelter Utilities Transportation Health care Misc. exp.

1.6a−1.7a 4.0 5.8a 8.0 6.0 Total PCE − sum of above

Table 17.4 Comparison of CES and NIPA total-expenditure elasticities, 6 CES-ACCRA categories CES-ACCRA

NIPA

Double-log

Log Flow adjustment

Category

Total-exp.

Price

Total-exp.

Price

Total-exp.

Price

Food at home Shelter Utilities Transportation Health care Misc. exp.

0.3175 0.8509 0.3807 1.3486 0.5066 1.1293

−0.3821 −0.5665 −0.9540 −1.4233 −1.1242 −1.3665

0.3550 1.1153 0.3623 1.4376 0.6231 1.4091

−0.6373 −0.6671 −1.1673 −1.5658 −2.7206 −1.2670

0.2079 1.0326 1.2240 0.8851 2.1962 0.8771

−0.1255 −1.1309 −3.3004 −0.0851 −0.9134 −0.4395

Fig. 17.2, which shows scatter diagrams of the elasticity pairs for the CES-ACCRA and NIPA estimates.7 For not many elasticity pairs lie near straight lines running through the origin in the first and third quadrants. Not surprisingly, disparities are more pronounced for price elasticities than for total expenditure, and, interestingly, larger for the CES-ACCRA log flow-adjustment models than for the double-log models. Utility expenditures are seen to be a necessity for CES-ACCRA, but a luxury for NIPA. The former seems proper, but not the latter. The total-expenditure elasticities for health-care expenditures are difficult to understand as well.

17.3 Summary In this chapter, we have undertaken a comparison of the elasticities obtained from the time-series data on personal consumption expenditures from the National Income and Product Accounts with the same obtained from cross-sectional data from the quarterly consumer expenditure surveys conducted by the Bureau of Labor Statistics. Two comparisons have been made, the first at a 29-category level of disaggregation, for which only total-expenditure elasticities are available from the CES 7 Both total-expenditure and price elasticities are included in this diagram, total-expenditure obviously in the first quadrant and price in the third.

394

17 Comparison of Time-Series and Cross-Sectional Elasticities

A

3 2

NIPA

1 0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -2 -3 -4 CES-ACCRA (dbl log)

B

3 2

NIPA

1 0 -3

-2

-1

0

1

2

-1 -2 -3 -4 CES-ACCRA (log fl. adj.)

Fig. 17.2 Scatter diagram of 6-category elasticity estimates CES ACCA vs. NIPA

analysis, and the second for 6 aggregated categories of expenditure for which both price and total-expenditure elasticities have been estimated in CES equations. Not only is one source of data cross-sectional and the other time series, but they differ as well in that CES surveys are of expenditures of individual households, while the time-series data refer to national aggregates per capita. Also, it has to be kept in mind that different agencies, with differing missions and production procedures (both conceptual and methodological), are involved in the collection and construction of the two data sets. All considered, therefore, it should probably not be too surprising that the comparisons show as much disparity as they do. In the next chapter, we shall attempt to get to the bottom of some of the disparities.

Chapter 18

Overall Assessment of CES and PCE Elasticities

As has been noted at several points, the primary purpose of this exercise has been to estimate price and total-expenditure elasticities for as broad an array of consumption categories as possible from cross-sectional data from the BLS quarterly surveys of consumer expenditures and time-series data from the personal consumption expenditure tables of the National Income and Product Accounts. A substantial number of elasticities have been estimated, and at this point readers are probably weary of the prospect of seeing more. Accordingly, this chapter will bring this part of the study to a close by summarizing the results that have been obtained to this point and assessing their significance for future U.S. consumption behavior. However, to begin with, it will be useful to remind ourselves of the important differences between the present effort and the earlier editions of CDUS. In general, the most important difference is the extensive analysis of the crosssectional consumption data that are now available in the BLS quarterly consumer expenditure surveys.1 Although these surveys have been ongoing since 1982, the present study is, as far as we are aware, the first to exploit the consumption information that they contain on a wide-scale basis, and then to compare the elasticities obtained with their counterparts from the time-series analysis. Sixteen quarters of CES data, covering the years 1996 through 1999, have been analyzed from three different directions. The first analysis, as reported in Chapter 5, is a principalcomponent analysis of 14 exhaustive categories of consumption expenditure that has been undertaken for the purpose of assessing the stability of underlying tastes and preferences. The second analysis is estimation of traditional Engel curves. This has been done at various levels of aggregation; the most detailed being one at a 29-category level of disaggregation. Finally, the third undertaking has been to augment the expenditure data in the CES surveys with data collected in quarterly price surveys by the American Chambers of Commerce Research Association (ACCRA) in order to estimate both price and total-expenditure elasticities. While, due to the limited coverage of the ACCRA surveys, this has been feasible only at high levels

1

While the 1970 edition of CDUS included an analysis of the 1962–1963 survey of consumer expenditures, its purpose was primarily to check the consistency of consumption data in the survey with that in the National Income and Product Accounts.

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of aggregation, the results are impressive, and we accordingly view it as the most innovative part of the cross-sectional analysis. In the time-series analysis, the most important innovation has been the use of a new dynamical model, first proposed by Bergstrom and Chambers (1990), that generalizes the state- and flow-adjustment models by combining them in the same structure. As a consequence, of the more than 100 dynamical time-series models estimated, the B-C model appears with much the greatest frequency. Still another important innovation in the present study (at least in our eyes) has been the attempt to base the theory of consumer demand on a neurobiological foundation, by postulating that consumption behavior is motivated, in terms of a hierarchy of five Maslovian wants (after the late psychologist Abraham Maslow). The hierarchy is used to construct three theoretical classes of total-expenditure elasticities, against which the elasticities that have been estimated empirically from the cross-sectional and time-series analyses are compared.

18.1 Summary of Results 18.1.1 CES Data The key results from the analysis of the CES data are as follows: (1) The structure of consumer expenditures is highly stable over the 16 quarters of CES surveys analyzed, as revealed in the principal-component analysis of the 14-category disaggregation of expenditures. The two largest principal components invariably account for about 85% of the total variation in expenditures, quarter by quarter (which is equivalent to about 45% of the level of total expenditure), and are highly stable (as reflected in their underlying eigenvectors) over time. (2) Stability of underlying tastes and preferences is also reflected in small variation in the estimates of total-expenditure elasticities, quarter to quarter, for all 16 quarters in the analysis. (3) One of the most intriguing results to emerge in the estimation of CES equations is distribution of residuals that are asymmetrical and have long tails. These features, as shown in detail in Chapter 9, in principle have disturbing implications for least-squares estimation, as they are clearly inconsistent with the predicates underlying the Gauss–Markov theorem. Since the phenomena are so pervasive in the least-squares equations, quantile regression (which is an increasingly used robust estimator) has been employed in addition to least squares. Specifically, our procedure has been to estimate the models (as obtained by OLS to begin with) by quantile regression, first at the median of the distribution of the residuals and then at the mode. In general, it turns out that the QR estimates are similar to those of OLS, which means that, despite asymmetries and long tails, OLS does not appear to give misleading results.

18.1

Summary of Results

397

(4) As discussed in Section 12.2, the results with the combined CES/ACCRA data are richly encouraging. Despite the limited coverage of prices for several of the aggregate categories, meaningful negative own-price elasticities, with substantial t-ratios, are obtained in virtually every demand equation estimated, whether the equation be separate or part of a theoretically plausible system. This is clearly one of the most important results of the exercise. (5) In Chapter 8, a system of additive double-log equations are applied to the CES-ACCRA data sets. The own-price elasticities are all negative, and range from −0.36 for food consumed at home to −0.99 for transportation expenditures. Total-expenditure elasticities, on the other hand, vary from 0.36 for food (in keeping with Engel’s law) to 1.38 for transportation. Cross-price elasticities suggest complex patterns of substitution and complementation, in that consumption of many goods appears to be complementary with respect to the price of some other good, but consumption of that other good appears to be a substitute with respect to the price of the first good. Indeed, such asymmetric substitution structures appear almost to be the norm. In view of the long-standing popularity of double-log functions, this additive system would appear to be a useful addition to the toolkits of applied demand analysis. The system is reasonably straightforward to apply, and, at least in the application here, appears to give plausible results. While the system is not integrable, this, at least in our opinion, is a small price to pay to be able to work with a set of double-logarithmic demand functions that are additive. (6) A long-standing debate in econometrics concerns interpretation of the coefficients that are estimated from cross-sectional data. The results with the large variety of models (including the logarithmic flow-adjustment model, which exploits the semi-panel structure of the CES surveys) are all consistent with the view that cross-sectional estimates are essentially steady-state in nature, and hence (at least in principle) are commensurable with steady-state estimates obtained in dynamical time-series models. (7) Finally, experimentation with the CES-ACCRA data sets, in which prices are excluded from the models, shows that estimated coefficients on totalexpenditure are little affected. This is an important result, for it implies that the theoretical interpretation of Engel curves as measuring the relationship between quantity demanded and income (as represented here by total expenditure) in fact appears to hold empirically as well.

18.1.2 NIPA Time-Series Data Dynamical models have been estimated for more than 100 categories of expenditure (some of which are aggregates) from the annual personal consumption tables in the National Income and Product Accounts and for 20 categories of expenditure (again, some of which are aggregates) from the quarterly accounts. The key results that have emerged from the time-series analyses are as follows:

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18

Overall Assessment of CES and PCE Elasticities

(1) First, and foremost, is that price elasticities have been obtained in every timeseries category of expenditure for which a model has been estimated. This is in marked contrast with what was found in the two earlier editions of CDUS, where price elasticities were obtained in only about half of the expenditure categories analyzed. In part, this much greater showing of price effects is a statistical consequence of the greatly increased variation in relative prices that has occurred since 1970 as a result of a variety of energy crises, deregulation in several key industries, and substantially increased international trade. However, as has been emphasized at several points, correct signs is essentially the only statistical constraint that has been imposed on the predictors, and, in light of the multiple solutions in the nonlinear estimation of the state-adjustment and B-C models, we have taken the liberty of selecting the ones that are economically meaningful. Nonetheless, the price elasticities that are obtained are for the most part plausible, and their emergence (together with the price elasticities obtained with the CES-ACCRA data) is the most important contribution of the entire exercise in our view. (2) Relative prices determine resource allocation, but income elasticities (in our case total-expenditure elasticities), specifically, income elasticities that are greater than 1, determine the structure and pace of economic growth. Totalexpenditure elasticities appear in all but one of the time-series models, and are absent only for fuel oil and coal, and negative only for travel by intercity railway and television and radio repair. In only a few instances do the steady-state values appear unreasonable (usually because of strong trends in the dependent variable, which leads to heated habit formation). (3) Of the 108 time-series equations estimated, 66 are B-C models, 24 are state adjustment, and 18 are flow adjustment. Thus, the generalization of the stateand flow-adjustment models to allow for both forms of adjustment to be present in the same model has definitely been a step forward. Interestingly, in the 18 categories in which flow-adjustment models are estimated, all are logarithmic, again pointing up the general empirical superiority of a double-logarithmic function. (4) In general, positive autocorrelation in the residuals is much less of a problem in the present study than was the case in the two earlier editions of CDUS, which is almost certainly due, at least in substantial part, to the additional parameter that is estimated in the B-C model. The result is only a handful of equations that were needed to be estimated by generalized least squares, and of those, most are state- or flow-adjustment models. Moreover, in no instance does a unit root in the residuals appear to be present. (5) Statistically, the estimated equations are typically of good quality or better. Indeed, in only five cases do the estimated equations leave a lot to be desired. As is to be expected with aggregate time-series data, R2 s are high, more often than not greater than 0.99. (6) Because of the fewness of observations in relation to the cross-sectional data sets, estimation by quantile regression has not been pursued in the time-series analysis. Nevertheless, in the several instances in which OLS residuals are

18.2

The Importance of Total-Expenditure Elasticities

399

examined (in Chapter 16) in the manner of Chapter 9, asymmetry and long tails (especially the latter) are clearly present. Again, however, estimation by quantile regression yields coefficient estimates that differ little from their OLS counterparts. In the next two sections, we will undertake a systematic evaluation of the elasticities that have been estimated in this study.

18.2 The Importance of Total-Expenditure Elasticities Most of the time, the focus of economists, in keeping with the conventional definition of economics as “the study of the allocation of scarce resources amongst alternative ends”, is on price elasticities. As a consequence, income (or totalexpenditure) elasticities get little attention, and there is little, or even no, appreciation that, in a free-market economy, it is income elasticities that determine the structure and pace of economic growth. Indeed, a point that is generally not well understood is that economic growth requires the existence of goods with income elasticities of demand in excess of one. If this were not the case, i.e., if all income elasticities were less than one, the aggregate average propensity to consume would be falling (as was thought to be the case by Keynes in the General Theory), and growth would stagnate, since there would be no “animal spirits” to drive investment. A constant emergence of “luxury” goods therefore provides the sine qua non for sustained economic growth, as ever increasing investment is needed in order to provision their production.2 In Tables 18.1, 18.2, 18.3, and 18.4, we provide distributions of the estimated CES and NIPA total-expenditure elasticities according to whether they are elastic (greater than 1), inelastic (between 0.35 and 1), and “highly” inelastic (less than 0.35) for both CES and NIPA estimates at different levels of aggregation. Table 18.1 refers to CES categories, while Tables 18.2, 18.3, and 18.4 are for quarterly and annual NIPA categories. The elasticities are taken from Table 10.8, Table 11.30, Table 14.3, and Table 16.1. All CES estimates are from double-log OLS equations. What is most significant in these four tables is the agreement as to the categories with large total-expenditure elasticities, in particular, for transportation, shelter, recreation, education, and religion and welfare. Recreation and education (and even religious and welfare) expenditures should naturally be expected to have high total-expenditure elasticities, but, as transportation (specifically, automobiles and activities associated with them) and housing have fueled economic growth for

2 Our reasons as to why we believe that income elasticities greater than 1 emerge naturally through hierarchical preferences and the “quest for novelty” in individuals’ consumption activities have been set out in Section 2.5. For elaboration of the point of this paragraph, see Chapters 12 and 13 of Taylor (2000).

400

18

Overall Assessment of CES and PCE Elasticities

Table 18.1 Distribution of total-expenditure elasticities, CES ACCRA/CES categories, doublelog models Elastic

Inelastic

Highly inelastic

Category

Elasticity

Category

Elasticity

Category

Elasticity

CES-ACCRA 6 Trans. services Misc. exp. –

1.3486 1.1293 –

Shelter Utilities Health care

0.8509 0.3807 0.5066

Food at home – –

0.3175 – –

CES 29 Transportation Hld. furn. and equip. Cash contributions – – – – – – – – – – – – –

1.3021 1.0058 1.0597 – – – – – – – – – – – – –

Purchased meals Alcoholic bev. Owned dwellings Rented dwellings Other lodging Telephone Housing op. Apparel Health care Gasoline and oil Entertainment Personal care Reading materials Education Misc. exp. Per. ins. and pens.

0.8116 0.4842 0.9156 0.5379 0.8288 0.4620 0.8620 0.9043 0.5306 0.3911 0.9783 0.4642 0.5503 0.9446 0.6980 0.9206

Food at home Natural gas Electricity Water Tobacco – – – – – – – – – – –

0.3232 0.1950 0.2352 0.1714 0.1470 – – – – – – – – – – –

Sources: CES-ACCRA 6: Table 8, Chapter 10 (OLS, double-log); CES 29: Table 30, Chapter 12 (OLS). Table 18.2 Distribution of total-expenditure elasticities, NIPA categories, quarterly Elastic

Inelastic

Highly inelastic

Category

Elasticity Category

Elasticity Category

Elasticity

Oth. dur. goods Housing Trans. services Medical care Recreation Other services

1.6878 1.1337 1.0316 3.8133 2.1552 1.3094

0.8577 0.6615 0.3932 0.6345 0.9840 0.9574

0.1340 – – – – –

Motor veh. and parts Furn. and hld. equip. Food Clothing and shoes Oth. nondur. goods Household operation

Gas. and oth. energy. – – – – –

decades, one might think that they would be increasingly viewed as core necessities. Their estimated expenditure elasticities, however, suggest otherwise.3

3 Both housing and transportation clearly serve needs (particularly, community, esteem, and selfactualization) other than simply providing basic shelter and getting from point A to point B in the cheapest way possible. The implications of these higher-order needs for economic growth in the future will be discussed in Section 18.4 below.

18.2

The Importance of Total-Expenditure Elasticities

401

Table 18.3 Distribution of total-expenditure Elasticities, NIPA categories, annual 12 Elastic

Inelastic

Highly inelastic

Category

Elasticity Category

Elasticity Category

Elasticity

Clth., acc. and jewelry Housing Medical care Personal business Recreation Education and research Religious and welfare

5.5089 1.0326 2.1964 1.3383 1.2729 2.2635 1.7989

0.7463 0.8045 0.8851 0.8936 – – –

0.2689 – – – – – –

Personal care Household operation Transportation For. travel, net – – –

Food and tobacco – – – – – –

Table 18.4 Distribution of total-expenditure elasticities, NIPA categories, annual 80 Elastic Category

Inelastic Elasticity

Wom. and girls’ clth.

1.1481

Men and boys’ clth. Jewelry and watches

1.0557 4.6820

Oth. cloth. and acc.

3.9601

Barbershops, etc.

1.3480

Owner occ. housing Kitchen and hld. appl. Drugs

1.2632 1.0794

Physicians

1.8763 17.6060

Dentists

1.7089

Oth. prof. services

5.9217

Hospitals Nursing homes

4.1768 2.8133

Health insurance

1.3927

Broker charges

4.2243

Bank service fees Oth. financial fees

1.9649 1.7222

Oth. pers. Bus.

1.6351

Highly inelastic

Category

Elasticity

Category

Purchased food Shoes Toilet articles

0.8421

Food at home

0.1963

0.3848 0.8800

Tobacco Alc. bev. off-prem Oth. alc. bev.

−0.4434 0.2221

Rented housing Other housing Furniture China, etc. Semi-dur. furn. Hld. clean. sup., etc. Writing materials Electricity Water Oth. hld. operation Legal services Gasoline and oil Travel tolls Auto insurance Magazines, etc.

0.4015 0.4161

Elasticity

0.2001 −

0.3665 0.7736

Clean and repair of clth. Farm housing Natural gas

0.3701

Fuel oil and coal

−

0.6206

Telephone

0.1393

0.8586

0.2600

0.8952 0.4980

Domestic services Opthal. and Orthpd. Funeral New autos

0.0265 0.2718

0.5497

Used autos

0.2551

0.5088

Intercity railway

−0.7360

0.7496 0.8933

Intercity bus Radio and TV repair −

− −0.5780

0.8723

0.6012

−0.4802 0.1970

0.0970

−

402

18

Overall Assessment of CES and PCE Elasticities

Table 18.4 (continued) Elastic Category Oth. motor veh. Tires, parts, etc. Auto repair, etc. Airline Oth. intercity trans. Books and maps Nondur. toys Wheel goods Video and audio Computer and software Flowers Theater and opera Spectator sports Clubs and frat. Org. Comm. part. activities Pari-mutuel receipts Other recreation Higher education Secondary educ. Other education For. travel, US res. For. exp., US res.

Inelastic

Highly inelastic

Elasticity

Category

Elasticity

Category

Elasticity

3.5055

Motion pictures − − − − − − − − −

0.5344

−

−

− − − − − − − − −

− − − − − − − − −

− − − − − − − − −

1.0369 1.7837 1.2538 1.4471 2.2815

− − − − −

− − − − −

− − − − −

− − − − −

1.0970 2.4516 1.5026 2.0551 1.9107 1.7989 5.7720

− − − − − − −

− − − − − − −

− − − − − − −

− − − − − − −

1.3718 1.2290 1.4801 1.9242 1.6261 1.6151 1.7015 2.8748 12.3324

Source: Table 16.1.

On the other hand, the CES total-expenditure elasticities for medical care seem seriously out of line, not only in terms of common sense, but also with the estimates obtained in the time-series analysis.4 The time-series elasticities make sense, whereas the CES values (of the order of 0.50) do not. However, the large differences between the CES and NIPA estimates may be primarily reflecting differing definitions, for the CES surveys record only direct expenditures, and do not include contributions from Medicare and Medicaid. Whatever, it is clear that medical-care expenditures (which already represent about 20% of total expenditure) will continue (perhaps unceasingly) for an increasing proportion of households’ budgets, and thus will continue to be a major contributor to economic growth.5 4

However, as was noted in Chapter 15, because of the strong trend in physician expenditures, the extremely large value for the category’s total-expenditure elasticity, while clearly in excess of 1, is almost certainly greatly exaggerated. In contrast, the nearly equally large elasticity for computers and software is probably not exaggerated (although it is almost certainly on a declining trend). 5 Two caveats are necessary concerning the discussion in this section. The first is that, since the budget constraint in all of the analyses is total expenditure, rather than disposable income, saving is

18.3

Assessment of Price Elasticities

403

At the other end of the spectrum of expenditure elasticities (i.e., those with very small values), we find food consumed at home, tobacco, funeral expenses, farm housing, natural gas, fuel oil and coal, non-airline intercity travel, and radio and TV repair, all of which are expected to be hard-core necessities or even inferior (as with tobacco, farm housing, fuel oil and coal, intercity rail and bus travel, and radio and television repair).6 Obviously, these categories will hardly be the center-pieces of future growth.

18.3 Assessment of Price Elasticities Tables 18.5, 18.6, 18.7, and 18.8 present distributions of the cross-sectional and time-series price elasticities similar to the classification used with the totalexpenditure elasticities, in this case whether the estimated elasticities are elastic (less than −1), inelastic (between −1 and −0.35), or “highly” inelastic (greater than −0.35). As illustrated in Chapters 11 and 17, the Maslovian hierarchy of needs provides a framework for ordering total-expenditure elasticities, but (at least at this stage of insight) offers little guidance as to relative magnitudes of price elasticities.7 However, in more conventional terms, one should expect: (1) Large elasticities for goods for which there are close substitutes; (2) Small elasticities for normal goods (whether luxury or necessity) for which there are few or no close substitutes; (3) Moderate elasticities for luxuries (and near-luxuries), pretty much independently of the existence (or nonexistence) of substitutes;

ignored, which means that nothing is actually implied about the aggregate consumption function. The second caveat is that the total-expenditure elasticities in Table 18.1 are from models that are not additive, in that the budget-share weighted elasticities (in an exhaustive disaggregation of total expenditure) will not necessarily sum to 1. The CES ACCRA 6 elasticities in Table 18.1, for example, are from the logarithmic flow-adjustment model, which, being double-logarithmic, is not additive. However, the total-expenditure elasticities from the additive double-log system of Chapter 8, which is additive, are quite close to the ones in Table 18.1, the only difference of note being the elasticity for housing is 0.96, as opposed to 1.11 in the log flow-adjustment model. 6 As noted in Chapter 15 (and also in Appendix 15.3 to that chapter), in assessing the apparent small total-expenditure elasticity, one must keep in mind that only a small fraction of households are actual new-car buyers. 7 Nevertheless, one can deduce at least one important implication from the hierarchy of wants, namely, for goods that primarily serve low-order needs, for which there are no close substitutes. In such situations, income effects (of the traditional type) will be important at low levels of income. Own-price elasticities will be small, but cross-elasticities could be large because of the large income effects. Food, housing, and heating oil come to mind as examples. By contrast, some goods that serve higher-order needs (e.g., non-basic housing, restaurant meals, and jewelry) could have substantial price elasticities, even in the absence of close substitutes.

404

18

Overall Assessment of CES and PCE Elasticities

Table 18.5 Distribution of price elasticities, CES-ACCRA data double-log models Elastic

Inelastic

Category

Elasticity

Category

Elasticity

Trans. services Health care Misc. expenditures

−1.0777 −1.1617 −1.1109

Food Shelter Utilities

−0.7020 −0.6153 −0.8937

Source: Table 8.5.

Table 18.6 Distribution of steady-state price elasticities, NIPA quarterly Elastic

Inelastic

Highly inelastic

Category

Elasticity

Category

Elasticity

Category

Elasticity

Furniture, hld. equip. Housing

−1.0648

Motor veh. and parts Other dur. goods

−0.6934

Food

−0.2275

−0.3552

−0.0214

Medical care

−2.6591

−0.7150

–

Recreation

−1.7656

–

–

–

–

–

–

–

–

– –

– –

Clothing and shoes Household operation − 0.4907 Other nondur. goods−0.9033 Trans. Services Other services

Gas. and oth. engy. –

−0.5665 −0.6342

– –

– –

−1.9832

Source: Table 14.3.

Table 18.7 Distribution of steady-state price elasticities, NIPA 12 Elastic

Inelastic

Highly inelastic

Elasticity

Category

Elasticity

Category

Elasticity

Clothing Housing

−3.9183 −1.0326

−0.6543 −0.4556

−1.4270 −2.5965 −1.5034

−0.9134 – –

Food Personal business Transportation – –

−0.0799 −0.2781

Recreation Education Religious and welfare Foreign travel, net

Personal care Housing operation Medical care – –

−0.0851 – –

−2.1338

–

–

–

–

Source: Table 16.1.

18.3

Assessment of Price Elasticities

405

Table 18.8 Distribution of steady-state price elasticities, NIPA 80 Elastic Category

Inelastic Elasticity

Other alc. Bev. Clean and repr, clth. Other clothing Toilet articles

−5.2445 −4.5137

Barber, beauty Own-occ. housing Rental housing

−4.5054 −1.1309

China, etc. Hld. clean. Sup. Writing supplies

−1.2617 −2.8667 −2.5906

−2.3460 −1.2269

−1.2632

Highly inelastic

Category

Elasticity Category

Elasticity

Purchased food Alc.bev. off-prem Shoes Wom. and girls cloth. Farm housing Water

−0.5418 −0.9555

Food at home Tobacco

−0.1194 −0.2383

−0.4717 −0.7098

Men and boys’ cloth Jewelry and watches

−0.1208 −0.1688

−0.8497 −0.7670

Other housing Furniture

−0.1723 −0.0349

Domestic services Drugs Health ins. Fin. intermed.

−0.8292

Kitchen appliances

−0.2292

−0.4014 −0.3671 −0.7486

−0.3367 −0.0030 −0.2810

Funeral Oth. motor veh. Travel tolls Auto ins. Mass transit Intercity railway Mag. and newsp. Nondur. toys Video and audio Radio-TV repair Motion pictures Com. part. act. Higher educ.

−0.3719 −0.9436 −0.6205 −0.9324 −0.7613 −0.6790

Other dur. furn. Telephone Other household. op. Opthal., orthoped Banking services Legal services New autos Used autos Tires, tubes, parts

−0.0481 −0.2256 −0.2448 −0.2685 −0.0558 −0.3487

−0.5834

Auto repair, etc.

−0.3018

−0.4223 −0.7541

Gasoline and oil Airline

−0.3073 −0.0811

−0.9928

Spectator sports

−0.1531

−0.7614 −0.3577 −0.8880

For. exp.,U.S. res. − −

−0.1245 − −

−0.4213

−

−

−

−

−

Electricity Natural gas Fuel and coal Physicians Dentists Oth. prof. serv.

−1.3435 −3.7863 −6.1020 −18.1884 −1.4172 −4.0429

Hosp. and nurs. hom. Broker Life ins. and pens. Other pers. Bus.

−5.2438

Taxicab Intercity bus Oth. intercity trans. Books and maps

−2.0535 −1.2175 −1.4642

Wheel goods, etc. Computers Flowers Theater and opera Clubs and frat. org. Pari-mutuel recpts. Other recreation Secondary educ. Other educ.

−1.5172

For. trav., U.S. res. −

−1.0164 −1.1829 −1.7446

− − −

− − −

− − −

− − −

−2.1844

−

−

−

−

−7.2733

−

−

−

−

−2.4969 −1.8603 −1.4080

− − −

− − −

− − −

− − −

Source: Table 16.1.

−1.1893 −4.1619 −1.5059

−1.3614

406

18

Overall Assessment of CES and PCE Elasticities

(4) Finally, for goods that are “slave” to stock variables providing services— gasoline would seem a good example—elasticities might be expected to be small. Expenditure categories with large price elasticities include housing, health care, education, and eleemosynary activities. Categories with very small elasticities include food consumed at home, tobacco, jewelry and watches, transportation (except in the CES-ACCRA equation), furniture and kitchen appliances, telephone, gasoline and oil, ophthalmic and orthopedic devices, most components of personal business, airline travel, spectator sports, and foreign travel by U. S. residents. For the most part, these results are in keeping with the expectations expressed above (including the Maslovian implications noted in Footnote 7). Even so, there might appear to be some anomalies, the most obvious, perhaps, being the large elasticities for medical care and the small values for new and used automobiles. Concerning medical care, two contingencies come to mind. The first is that many medical treatments and procedures can be postponed (which represents a form of substitution), while the second is that, with person health insurance, Medicare, and Medicaid, the actual prices paid by consumers for medical care may be a great deal less than the “posted” prices represented in the medical-care price indices. On the other hand, the low elasticities for new and used cars do not resonate with the continual discounts and “cash backs” that the automobile manufacturers seem always to be offering. For such are clearly not consistent with the low price elasticity of demand. However, what really matters in the market for automobiles, whether new or used, is the relative price of new to used autos, for, as described in Appendix 15.3, an increase in income that increases the demand for automobiles represents an outward shift, initially, in the demand for used cars, which at some point translates into an increase in the price of late model used cars relative to the price of new cars. And it is this that ultimately triggers an increase in the demand for new cars. In this situation, while a small price elasticity for used cars may make sense, the equally small elasticity for new cars may simply be a reflection of a mis-specified model. The analysis in Appendix 15.3 suggests that this may in fact be the case.8 While it is a simple mathematical truism that, with a fixed budget, something somewhere has to give when prices change, the story that emerges from the elasticities in Tables 18.5, 18.6, 18.7, and 18.8 is one of considerable long-run price sensitivity. This is especially the case for categories such as housing and health care, where a priori it might be thought that consumers, because of the nature of the goods and services involved, pay little attention to price. The results here clearly suggest otherwise.9 On the other hand, the elasticities for telephone services and 8 It must also be kept in mind, when assessing the price elasticity for new autos, that expenditures for light trucks and sport-utility vehicles are included in other vehicles, whose elasticity of −0.94 seems much more plausible. 9 Also, because relatively so few price elasticities were obtained in the first two editions of CDUS, it should not be thought that the much greater number that have been found in the present study

18.4

Total-Expenditure and Price Elasticities and Hierarchical Wants

407

airline travel, to take a couple of examples at face value, might seem unduly low. However, what needs to be considered in both of these cases is that what anecdotally might appear to be a lot of price sensitivity on the part of consumers is really a reflection of price competition among telephone companies and airlines for customers. Following deregulation of these industries in the late 1970s and early 1980s, telephone calls and airline seats essentially became commodities. Demands became very sensitive to price differentials among vendors, but remained relatively insensitive to the overall levels of rates and fares.

18.4 Total-Expenditure and Price Elasticities and Hierarchical Wants We will complete our discussion of the price and total-expenditure elasticities that have been obtained in this study by returning to the notion that the tastes and preferences that underlie demand functions are characterized in terms of a hierarchy of wants. Two questions need to be asked at this point: (1) Are the results that have been obtained consistent with such a notion? (2) If so, does the notion add anything to the analysis? In Chapter 11, and again in Chapter 16, the steady-state total-expenditure elasticities were interpreted in terms of the five Maslovian wants. Expenditure categories were associated with the five wants, from which theoretical orderings of elasticities was constructed. Total-expenditure elasticities should be smallest for expenditures associated with basic physiological needs, and highest for expenditures associated with self-actualization. The theoretical orderings were then compared with the empirical estimates obtained in the cross-sectional and time-series analyses. In both instances, more than 80% of the estimated elasticities are correctly classified. While null hypotheses that the proposed classifications are “correct” would be rejected (at conventional levels of significance), the theoretical assignments are clearly much better than at random. As suggested in Footnote 7, the implications of hierarchical wants for price elasticities are much more subtle than for income (or total-expenditure) elasticities. However, as noted there, one implication that is straightforward to adduce concerns goods that primarily serve low-order needs and for which there are no close substitutes. In such situations, income effects (of the traditional type) will be important at low levels of income. Own-price elasticities will be small, but cross-price elasticities might be large because of the large income effects. Fuel oil consumption

simply reflects a much greater price awareness, post-1964 (the last year of data in the 1970 edition of CDUS), on the part of U.S. households. While this possibility cannot be completely discounted, a much more likely cause is the greatly increased variation in relative prices of the last 35 years that has enabled ever-present price effects to be detected statistically.

408

18

Overall Assessment of CES and PCE Elasticities

during the energy crises of the 1970s provides a good anecdotal example. As will be recalled, crude-oil prices were quadrupled in the fall of 1973, and the cost of fuel oil was affected accordingly. In New England, the average cost of heating an average home rose, in a short period of time, from about $500 per winter to as high as $1500. While the latter could be reduced to some extent by turning down thermostats and closing off rooms, there was essentially no scope for substitution (at least in the short run). As average income at the time was of the order of only $10,000 per household, there was a steep cut in real income (of upward of 10%) that obviously had to lead to reduced expenditures on other goods and services. If one were to analyze these cuts with income and expenditures measured in current prices, then the effects would be measured in terms of cross-price elasticities. On the other hand, if the cuts were analyzed in real income and expenditures, the effects would be measured in terms of income (or total-expenditure) elasticities. Either way, the predictions made by a want hierarchy would be that expenditures would be reduced least for goods associated with low-order wants, and most for goods associated with the higherorder wants, which is to say that what we ordinarily think of as necessities would be affected the least and luxuries the most. Might we have any empirical results to check the above? The answer is yes, with the equation systems that have been estimated with the CES-ACCRA data. The implication of the two preceding paragraphs is that the income component of cross-price elasticities should increase in the direction of higher-order wants. If, for example, A is associated primarily with physiological needs, B with community needs, C with both community and esteem needs, and D primarily with self-actualization needs, then the income component of the cross-price elasticities should be ordered, smallest to largest, from B to D. Table 18.9 reproduces the own- and cross-price elasticities for the linearexpenditure demand system from Table 7.3 that was estimated in the chapter with CES-ACCRA data for 1996. Of the several demand systems estimated in that chapter, we have selected the LES model because its cross-elasticities (since its underlying utility function is logarithmically additive) are pure income effects. Of the 6 categories in the CES-ACCRA data set, food consumed at home should be

Table 18.9 Price and total expenditure elasticities linear expenditure system CES-ACCRA surveys 1996 (calculated at sample mean values)

Food Shelter Utilities Trans. Health care Misc. exp.

Food

Shelter

Utilities

Trans.

Health care

Misc. exp.

−0.7576 −0.0380 −0.0052 −0.0049 0.0034 −0.0418

−0.0551 −0.7465 −0.0174 −0.0163 0.0114 −0.1382

−0.0446 −0.1019 −0.8902 −0.0132 0.0092 −0.1119

−1.3116 −2.9994 −0.4063 −0.9770 0.2718 −3.2935

0.1025 0.2344 0.0318 0.0304 −1.0895 0.2574

−0.1729 −0.3953 −0.0536 −0.0512 0.0358 −0.9083

Source: Table 7.3.

18.5

Comparison of Annual and Quarterly Models

409

associated with the lowest order wants, and probably transportation, health care, and miscellaneous expenditures with the highest. This being the case, we should find the smallest cross-elasticities for shelter and utilities and larger ones for transportation, health care, and miscellaneous expenditures. In the event, this is pretty much what turns out to be the case, for except for health care (which is an anomaly since its minimum required quantity is estimated as negative), the cross-elasticities with respect to food are indeed smallest for shelter and utilities, and largest for transportation and miscellaneous expenditures.10 Continuing the logic, we see that the cross-elasticities with respect to the price of shelter are larger for transportation and miscellaneous expenditures than for food and utilities; larger for transportation and miscellaneous expenditures with respect to the price of utilities than for food and shelter; larger for miscellaneous expenditures with respect to the price of transportation than for food, shelter, and utilities; and finally larger for transportation with respect to the price of miscellaneous expenditures than for food, shelter, and utilities. While all this is clearly in accord with common sense, and is hardly definitive in support of the hierarchical preferences, the latter nevertheless provides a coherent framework for interpreting and codifying the common sense.

18.5 Comparison of Annual and Quarterly Models One of the strengths of specifying a model in continuous time and then deriving the estimating forms thereof through integration is that it establishes a strict mathematical relationship between structural parameters estimated from models integrated over different periods of time. Specifically, from expressions (2.41)– (2.45) in Chapter 2 that, in the state-adjustment model, the values for β,δ,μ, and λ are inversely proportional to the length of the interval integrated over, while α is proportional to the square of the interval. Thus, if h [in expressions (2.41)–(2.45)] is taken to be 1 for an integrating interval of a year, then β and δ, for example, estimated over a quarter should be one-fourth of their values (sampling variation aside) estimated over 12 months. Similar relations will hold for the parameters in the B-C and flow-adjustment models. In Chapter 14, we presented models estimated with quarterly PCE data for 1947 through 2004 at the BEA 13-group level of aggregation, as well as for durables, nondurables, and services and two sub-components each of expenditures for gasoline and other energy products and household operation. We have also estimated models for this classification of expenditures with annual data over the same period. In this section, we present a comparison of the two sets of estimates. The details are given in Table 18.10 and in Figs. 18.1 and 18.2. Structural parameters and elasticities are tabulated in Table 18.10, while Fig. 18.1 summarizes the former and Fig. 18.2 the latter, in a scatter diagram. 10

Cross-elasticities with respect to a given price are listed in rows.

Model

log F. A. B-C log F. A. B-C log F. A. B-C log F. A. B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C

log F. A. B-C log F. A. B-C

Category

Quarterly models: Durable goods Mot. veh. and parts Furn. and hld. eq. Other dur. Nondurable goods Food Cloth. and shoes Gas. and oth. energy Gasoline and oil Fuel oil and coal Other nondur. Services Housing Household operation Electricity and nat. gas Other household op. Transportation Medical care Recreation Other services

Annual models: Durable goods Mot. veh. and parts Furn. and hld. eq. Other dur.

− 1.9002 − 1.8577

− 6.1010 − 2.5642 − 2.5086 − 3.8383 3.9106 2.4462 2.6926 2.0851 0.5421 3.8873 3.8761 2.1228 0.7036 0.8873 1.8712 1.3202 − −0.2202 − 0.0865

− −0.0167 − −0.0082 − −0.0134 − −0.0393 −0.0109 0.0615 0.0085 0.0298 0.1626 0.0186 0.3057 −0.0484 0.0792 0.2290 0.0761 0.0768 − 0.5563 − 0.2457

− 0.0049 − 0.0367 − 0.0279 − 0.0068 0.0095 0.1247 0.0482 0.0427 0.2028 0.0480 0.3682 0.0175 0.1974 0.2500 0.0966 0.1749 1.0688 2.0431 0.5999 2.1522

1.1449 0.8577 0.6615 1.6878 0.6072 0.3932 0.6345 0.1340 0.3088 −0.7891 0.9840 1.4745 1.1337 0.9474 0.5367 1.0881 1.0316 3.8133 2.1552 1.3094

−0.6243 −1.1461 −0.5366 −3.5472

−0.1496 −0.6934 −1.0648 −0.3552 −0.2571 −0.2275 −0.7150 −0.0214 −0.0645 −0.4110 −0.9033 −0.9312 −1.9832 −0.4907 −0.5569 −0.6855 −0.5665 −2.6591 −1.7656 −0.6342

Price

18

0.7820 − 0.4697 −

0.2157 − 0.1981 − 0.1341 − 0.0888 − − − − − − − − − − − − −

δ

Elasticities β Tot. exp.

γ

Parameters θ

Table 18.10 Comparison of annual and quarterly models, NIPA data, 1947 – 2004

410 Overall Assessment of CES and PCE Elasticities

Model

log F. A. B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C B-C

Category

Nondurable goods Food Cloth. and shoes Gas. and oth. energy Gasoline and oil Fuel oil and coal Other nondur. Services Housing Household operation Electricity and nat. gas Other household op. Transportation Medical care Recreation Other services

δ

1.0631 − − − − − − − − − − − − − − −

− 0.0348 0.7739 0.1693 −0.0331 0.6377 0.0528 0.1016 0.1627 0.0547 1.2205 0.1699 −0.3185 0.8535 0.6507 0.1272

− 1.5554 5.9795 3.5077 3.1164 1.9545 2.3639 4.3234 1.4676 2.2152 4.0359 0.5404 0.5625 0.4260 1.0655 2.2368

− 0.2365 0.8854 0.2523 0.0177 0.8387 0.1264 0.1389 0.2256 0.1035 1.2657 0.6396 0.3798 1.0000 0.7657 0.3879

0.6070 0.7035 0.4983 2.0616 0.1970 −1.2925 1.9515 2.5486 1.9104 1.5842 0.6946 2.2177 1.8077 5.1632 4.7885 2.3388

Tot. exp.

β

θ

γ

Elasticities

Parameters

Table 18.10 (continued)

−0.1488 −0.3069 −1.6982 −1.6912 −0.0550 −1.4222 −1.3126 −0.8962 −1.9864 −0.5987 −1.9941 −1.7943 −0.6289 −4.2491 −3.0347 −0.6774

Price

18.5 Comparison of Annual and Quarterly Models 411

412

18

Overall Assessment of CES and PCE Elasticities

7 6 5 annual

4 3 2 1 0 -1

0

1

2

-1

3

4

5

6

7

quarterly

Fig. 18.1 Scatter diagram of short-term dynamical parameters, annual and quarterly PCE models, 1947–2004 6 4

annual

2 0 –4

–3

–2

-1

0

1

2

3

4

-2 -4 -6 quarterly

Fig. 18.2 Scatter diagram of steady-state price and total-expenditure elasticities, annual and quarterly PCE models, 1947–2004

The overall results of the comparison are follows: (1) Although we made no effort to force the same form of model with the annual data as with the quarterly model, it is thus of interest that, with only one exception, the models are the same. The exception is for clothing and shoes, for which a B-C model is estimated with annual data, as opposed to a log flow-adjustment model with quarterly data. While this is what we should expect to find, assuming proper specification, it is reassuring when what we expect is in fact found.11

11 Moreover, the difference between models for clothing and shoes is not material, for with the quarterly data the log-flow-adjustment model is only marginally better than the B-C model, and vice-versa with the annual data.

18.5

Comparison of Annual and Quarterly Models

413

(2) The biggest difference between the quarterly and annual estimates is in shortterm dynamics.12 That this is the case is evident at a glance from a comparison of Figs. 18.1 and 18.2. Figure 18.1 shows the parameters describing the shortterm dynamics of the models (θ for the log flow-adjustment model, and γ , β, and δ for the B-C model) from the annual equations plotted against their counterparts from the quarterly equations, while Fig. 18.2 depicts the same for the steady-state price and total-expenditure elasticities. The extreme of agreement between the annual and quarterly estimates would, of course, be represented by all points in the figures lying on or near straight lines running through the origin from the third to first quadrants. While least-square lines (were they to be fitted) would be statistically close to homogeneous, the scatter in Fig. 18.1 is clearly visually greater than in Fig. 18.2. (3) Volatility in short-term dynamics is particularly evident in the estimates of θ in the log flow-adjustment models for nondurables (1.06 in the annual equation vs. 0.13 in the quarterly model) and in the estimates of γ in the B-C models for housing and other household operation. Probably most disturbing, however, are the changes in sign on β in the B-C equations for other durables, food, gasoline and other energy products, other household operation, and transportation. (4) Of differences in the total-expenditure elasticities, by far the most notable is for gasoline and other energy products, which is indicated to be a core necessity with the quarterly data to a roaring luxury in the annual model (0.13–2.06)! A comparable shift is seen in the steady-state price elasticity for this category (−0.02 to −1.69). (5) If we compare the first and third quadrants of Fig. 18.2, the steady-state price elasticities appear to be a bit more stable between the two estimations than the total-expenditure elasticities. Interestingly, for reasons that will be gone into in a moment, one might have thought the opposite to have been the case. The conclusion that emerges from this comparison is that, apart from the form of models, consistency between the quarterly and annual models is not great, especially with regard to short-term dynamics. Although nothing can be said definitively, several factors come to mind that might contribute to this. First, and perhaps foremost, is simply the data themselves, for while the quarterly measure the same activities, the information bases for the two differ, and is wider and more complete for the annual data. A second factor relates to seasonal adjustment of the quarterly data. Since the current- and constant-dollar data are seasonally adjusted individually, spurious covariances among the variables, both dependent and independent, may thereby be created. Finally, these two considerations aside, it may be that, per the

12 Since the quarterly PCE data are measured at annual rates, h is implicitly equal to 1 in both estimations, so that the parameters in the quarterly equations are in principle the same as in the annual equations.

414

18

Overall Assessment of CES and PCE Elasticities

discussion in Section 13.2, a quarter may not be sufficiently long for the integral equations describing the behavioral parameters to converge. While all of these considerations are basically speculative, our rather strong inclination, nevertheless, is to give greater credence to the annual estimates than to the quarterly.

Chapter 19

The Dynamics of Personal Saving

In the analyses to this point, total expenditure has been used for the budget constraint, which (among other things) leaves the relationship between total expenditure, saving, and income up in the air. In this chapter, we shall take a brief look at the results from applying the B-C model to personal saving. Doing this is relevant not only to the present effort, but also for any insight that it might provide into the current extremely low personal saving rate in the U.S. (during the dozen or so years preceding the financial crisis of 2007–09). However, a major problem in analyzing personal saving is deciding upon the particular measure to analyze. Three measures are available. The first measure is from the National Income and Product Accounts, the second is from the U.S. Flow of Fund Accounts constructed by the Federal Reserve, and, finally, the third is the one constructed by the Securities and Exchange Commission. Of the three measures of saving, the one that receives the most media attention (and also the one that tends to show the lowest rates of personal saving, indeed, about two percentage points lower on the average) is the NIPA definition, which is unfortunate, for it is the one that probably has the most flaws. Particularly questionable with this measure is the fact that, being constructed as the residual between disposable personal income and personal outlays,1 it depends indirectly on the imputed quantities represented in personal consumption expenditures, especially those related to owner-occupied housing. The imputed expenditures for this category are troublesome, for it would seem that, as a deduction from income, these should more properly reflect depreciation rather than space-rental value. As it turns out, whether for this or other reasons, the equations with the NIPA measure that have been estimated are sufficiently unreasonable that the only results reported are with the Flow of Funds measure.

1

Personal outlays differ from total consumption by inclusion of small allowances in the former for personal interest and transfers to government and the rest of the world.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_19, 

415

416

19

The Dynamics of Personal Saving

19.1 B-C Model of Saving For notation, let s denote saving, x income, and A a state variable that can be interpreted as representing the underlying stock of financial assets. The model (in continuous time) that is postulated is sˆ(t) = α + βA(t) + γ x(t),

(19.1)

ds(t) = θ [s(t) − sˆ(t)]dt + ε(dt)

(19.2)

dA(t) = [s(t) − δA(t)]dt.

(19.3)

Since this model is the B-C model with different notation (and absent price), its estimating equation will be st = B0 + B1 st−1 + B2 st−2 + B3 (xt − xt−2 ) + B4 (xt−1 + xt−2 ) + ut ,

(19.4)

with B0 , . . ., B4 functions of the structural parameters α, β, γ , and δ per expressions (13.38)–(13.43), and: ut =

1 γδ εt + εt−1 . 1+γ +δ 1+γ +δ

(19.5)

Remarks: 1. In application of state-adjustment models to consumption goods, δ is a positive number that measures the rate at which durable goods depreciate or, if β is positive, the rate at which habits wear off. With saving, however, the underlying state variable represents the stock of financial assets is an appreciating asset, which means that δ measures the mean rate of return on this stock and hence will be negative.2 2. Moreover, since A represents the stock of financial assets, it is the epitome of a durable good, which means that β should be expected to be negative. Stability of the underlying dynamical process then requires that β be even more negative than δ. 3. In addition, the fact that δ is negative (but greater than β) implies that the steady-state value of γ , the marginal propensity to save, will also be negative!

2

In the 1970 edition of CDUS, the most successful model of saving was one in which it is assumed that the depreciation rate on the stock of financial assets is zero, the argument for this being that interest and dividends on financial assets are already included in personal (and therefore disposable) income; assuming a non-zero depreciation rate would accordingly entail double-counting of interest and dividend income. However, despite its earlier success, application of this model to both NIPA and Flow of Funds measures of saving now yields uniformly unsatisfactory results and thus has not been pursued.

19.2

Results for Personal Saving from the Flow of Funds Accounts

417

The interpretation and implications of this rather counter-intuitive result will be discussed below.

19.2 Results for Personal Saving from the Flow of Funds Accounts In this section, we present the results from applying the model of the preceding section to data for personal saving from the U.S. Flow of Funds (FOF) Accounts. The FOF has two definitions of saving, gross saving, which refers to the increase in financial and tangible assets less the increase in liabilities, and net saving, which differs from gross saving by consumption of fixed capital.3 Equations are estimated both aggregate and per capita for both definitions of saving for both annual and quarterly data (annual 1947–2005, quarterly 1952–2005) in current dollars. The income variable is disposable personal income from the National Income and Product Accounts, and is also expressed in current dollars. The estimated equations are given in Table 19.1. The key features of the results are as follows:4 1. As required by the logic of the model, all estimates of δ are negative. 2. Similarly, all estimates of β are negative. 3. Because of δ being negative (but greater than β), all estimates of the steady-state marginal propensity to save (γ : ss) are negative as well. 4. All estimates of θ are well in excess of 1, which implies that actual saving adjusts rapidly to an increase in desired saving.5 5. Despite some apparent autocorrelation in the residuals, the estimated equations are generally of good statistical quality.6 Since income and saving are measured in current dollars, the estimates of δ can be interpreted as representing estimates of the implicit real rate of return on financial assets. In Table 19.2, we see that the estimates range from 0.0268 for aggregate gross saving to 0.0644 for aggregate net saving, and are larger for net saving than

3

The FOF saving data were downloaded from the Federal Reserve website: http://www.federalreserve.gov. The definitions of gross and net saving are taken from the Guide to the Flow of Funds Accounts, Volume 1, p. 150. 4 Except for the quarterly model for aggregate gross saving, the equations have been estimated with the restriction that B1 + B2 = 0.95. Otherwise, the models are either dynamically unstable (B1 + B2 ≥ 1) or else very close to being so. 5 Since saving is complement to consumption, this extremely rapid short-run adjustment to changes in income is, of course, simply the reverse side of the strong inertia in consumption expenditures. 6 While the Durbin-h statistics for the equations in Table 19.1 are all greater than 2, there is no evidence of unit roots in the residuals. Estimation by GLS unfortunately leads to estimates of θ that are implausibly negative.

418

19

The Dynamics of Personal Saving

Table 19.1 Flow of Funds personal saving, B-C model, quarterly and annual Quarterly

Annual

Aggregate Coefficient

Value

Per capita t-ratio

Value

Aggregate t-ratio

Value

Per capita t-ratio

Value

t-ratio

Gross FOF saving: B0 B1 B2 B3 B4 θ α β γ δ ϕ: sr ϕ: ss R2 dw

−1.2277 0.2218 0.7504 0.5071 0.00679 15.0280 −0.0817 −0.0216 0.5744 −0.00669 8.6317 −0.2574 0.9970 1.43

−0.45 3.99 13.56 13.76 −2.13 − − − − − − − − −

−6.3525 −0.66 0.3094 5.44 0.6406 6.67 0.3790 12.19 −0.00662 −7.98 9.9715 − 0.6371 − −0.0387 − 0.4547 − −0.00873 − 4.5341 − −0.2205 − 0.9962 − 1.58 −

6.0852 1.00 0.4050 2.86 0.5450 3.85 0.2180 4.38 −0.0164 −3.32 7.3405 − 0.8290 − −0.0692 − 0.2762 − −0.0377 − 2.0272 − −0.3274 − 0.9951 − 1.61 −

27.1702 1.16 0.3766 2.85 0.5734 4.34 0.2112 5.26 −0.0137 −4.21 7.9930 − 3.3993 − −0.0637 − 0.2633 − −0.0325 − 2.1040 − −0.2750 − 0.9946 − 1.67 −

−1.44 6.93 11.01 12.16 −11.54 − − − − − − − − −

−8.1715 −0.75 0.4226 7.46 0.5274 9.31 0.3268 9.88 −0.00773 −8.20 6.9480 − −1.1761 − −0.0440 − 0.4203 − −0.0118 − 2.9203 − −0.1546 − 0.9644 − 1.50 −

5.6796 0.78 0.5600 3.40 0.3900 2.37 0.1446 2.52 −0.0186 −3.11 4.9023 − 1.1586 − −0.0993 − 0.2017 − −0.0377 − 0.9886 − −0.3726 − 0.9425 − 1.56 −

24.7347 0.89 0.5308 3.44 0.4192 2.72 0.1333 2.93 −0.0152 −3.84 5.2540 − 4.7078 − −0.0913 − 0.1835 − −0.0571 − 0.9591 − −0.3042 − 0.9432 − 1.61 −

Net FOF saving: B0 B1 B2 B3 B4 θ α β γ δ ϕ: sr ϕ: ss R2 dw

−3.7306 0.3670 0.5830 0.4669 −0.0128 5.3764 −0.6939 −0.0448 0.5797 −0.0137 4.7648 −0.2566 0.9692 1.32

for gross saving.7 While values of 0.05–0.06 for the latter may seem high as a real return, they may be suggestive that households (and nonprofit entities), since gross and net saving in the Flow of Funds differ by consumption of fixed capital, do not view depreciation of fixed-capital assets as negative saving, in short, a sort of saving illusion. Assuming that the steady state is defined (per usual) as s˙ = 0, then (as noted) a negative δ together with dynamic stability implies that the steady-state marginal Since the quarterly data are measured at annual rates, the quarterly estimates of δ in Table 19.1 are multiplied by 4 in order to annualize them.

7

19.2

Results for Personal Saving from the Flow of Funds Accounts

419

Table 19.2 Annual real rates of return estimated from gross and net FOF saving models FOF gross saving

FOF net saving

Quarterly: Aggregate Per capita

0.0268 0.0349

0.0472 0.0548

Annual: Aggregate Per capita

0.0376 0.0325

0.0644 0.0571

Source: Table 19.1. Table 19.3 Steady-state marginal propensities to save estimated from gross and net FOF saving models FOF gross saving

FOF net saving

Quarterly: Aggregate Per capita

−0.2574 −0.1324

−0.2566 −0.2750

Annual: Aggregate Per capita

−0.3274 −0.2750

−0.3726 −0.3042

Source: Table 19.1.

propensity to save is negative. In Table 19.3, we see that estimates of this parameter range from −0.13 to −0.37. Since the steady-state is being defined in terms of s˙ = 0, what a value of −0.13 implies (to take gross saving per capita as an example) is that an interest rate of 3.25% on the gross stock of assets (physical as well as financial) could in principle support a total consumption expenditure that is 13% greater than the level of income.8 Perhaps a more interesting calculation than the foregoing is the implicit saving rates that are implied by the estimated models for a constant percentage increase in income. Given the form of the models—specifically, the fact that they are not logarithmic—such calculations must be obtained through simulation of equation (19.4). This is done in Table 19.4 for a variety of growth rates in income for the annual aggregate and per capita equations for gross saving from Table 19.1. Saving rates are calculated for 30 periods, beginning from 2003, for three assumed growth rates in income; 6, 8, and 10% for aggregate income and 4, 6, and 8% for income per capita. The results are not encouraging, for aggregate income would have to grow more than 6% per annum in order for the saving rate at the end of 30 periods to be positive, while per capita income has to grow more than 6% for this to be the case. Both growth rates (including an allowance for inflation of, say, 3%) are well above historical averages.

8

Note that nothing is implied about other income, which may be nothing more than social security.

420

19

The Dynamics of Personal Saving

Table 19.4 Steady-state saving rates, aggregate and Per-Capita FOF gross saving Period

Aggregate

Per-Capita

6% growth in income 4% growth in income: 21 22 23 24 25 26 27 28 29 30

−1.36 −1.63 −1.87 −2.09 −2.29 −2.48 −2.65 −2.80 −2.95 −3.08

−3.25 −3.64 −4.00 −4.34 −4.66 −4.96 −5.23 −5.49 −5.73 −5.96

8% growth in income: 6% growth in income: 21 22 23 24 25 26 27 28 29 30

2.68 2.53 2.40 2.29 2.18 2.08 2.00 1.92 1.85 1.79

1.26 1.04 0.84 0.66 0.49 0.33 0.19 0.06 −0.06 −0.18

10% growth in income: 8% growth in income: 21 22 23 24 25 26 27 28 29 30

5.85 5.78 5.71 5.65 5.60 5.56 5.51 5.48 5.45 5.42

4.76 4.64 4.54 4.44 4.35 4.27 4.20 4.14 4.08 4.03

19.3 Extending Model to Include Capital Gains In the view of many, a major reason for the extremely low personal saving rates of recent years is the substantial capital gains in the stock and housing market, the idea being that, with capital gains adding to wealth, households can reduce the amounts saved out of current income. In this section, we shall extend the model to take the possibility of this into account. This will be done by assuming that any capital-gains effect is transitory by rewriting equation (19.2) as ds(t) = θ [q(t) − qˆ (t)]dt + λz(t)dt + ε(dt),

(19.6)

19.3

Extending Model to Include Capital Gains

421

where z denotes capital gains. Equations (19.1) and (19.3) remain changed. In line with earlier derivations, the estimating equation now becomes qt = B0 + B1 qt−1 + B2 qt−2 + B3 (xt − xt−2 ) + B4 (xt−1 + xt−2 ) +B5 (zt − zt−2 ) + B6 (zt−1 + zt−2 ) + ut ,

(19.7)

with: λδ 2+θ +δ λ . B6 = 2+θ +δ B5 =

(19.8) (19.9)

The results from estimating equation (19.7) for quarterly data for both gross and net saving using a measure of capital gains taken from the Flow of Funds Accounts are tabulated in Table 19.5.9 From expressions (19.8) and (19.9), since λ is expected to be negative, B5 should be positive and B6 negative. We see that this is in fact the case. Since capital-gains effects are assumed to be transitory, we might expect their inclusion to affect model dynamics, but, in general, this is not the case. The estimates of θ , β, and δ are virtually identical in the equations for net saving, and only mildly affected in the equations for gross saving.10 The largest change in the structural parameter is in the steady-state marginal propensity to save (γ : ss) in the per capita model for gross saving, which increases (in absolute value) from −0.15 to −0.25. However, before concluding that the substantial capital gains of recent years explains the extremely low saving rates over the same period, it is well to consider just what it is that capital gains represent. To do this, we need to distinguish between capital gains that are realized and those that are not. The distinction is important because realized capital gains usually involve the sale of an asset for money, in which case whence did the money come? If the money is from current income, then it represents saving by the purchaser, and (assuming that the proceeds are spent on consumption) dissaving by the seller. The transaction is then a wash as far as aggregate saving is concerned. However, if the money is newly created (rather than coming out of current income), then the net effect of the transaction will be to reduce aggregate saving. Consequently, for realized capital gains, the effect on saving will, at best, be neutral. This will also be the case for unrealized capital gains, but for a different reason. In this case, rather than operating through a transfer of purchasing power, effects on saving must arise through households, because of the capital gains, feeling more wealthy and spending some of this wealth now, as opposed to later. Long-time estimates of this “wealth effect” indicate that about 4% of increases 9

Two measures of capital gains, accruing on assets owned by households and non-profit organizations, are available in the Flow of Funds, the first measured at market prices and the second measured at current cost. The first measure is the one used. Since δ is now over-identified, the models are estimated by non-linear least squares. Also, except in the equation for gross saving per-capita, equation, the sum of B1 and B2 are constrained to be equal to 0.95. 10 See Table 19.1.

422

19

The Dynamics of Personal Saving

Table 19.5 Flow of Funds personal saving, B-C model with capital gains, quarterly Aggregate Coefficient

Value

Per Capita t-ratio

Value

t-ratio

−1.91 2.16 5.65 6.99 −5.45 2.13 −1.99 − − − − − − − − − −

−2.7524 0.3221 0.6601 0.3861 −0.00401 0.00626 −0.000065 10.0900 −0.2728 −0.0158 0.4624 −0.00520 −0.000786 4.6656 −0.2256 0.9964 1.60

−0.54 2.53 5.43 4.64 −0.76 1.82 −0.65 − − − − − − − − − −

−2.17 4.04 5.88 6.16 −5.74 1.56 −1.59 − − − − − − − − − −

−8.6750 0.4390 0.5110 0.3297 −0.00777 0.00469 −0.000110 6.6317 −1.3081 −0.0443 0.4285 −0.0118 −0.000948 2.8418 −0.1554 0.9653 1.56

−1.13 4.23 4.78 3.69 −3.67 1.27 −1.30 − − − − − − − − − −

Gross FOF saving: B0 B1 B2 B3 B4 B5 B6 θ α β γ δ λ ϕ: sr ϕ: ss R2 dw

−3.3161 0.2625 0.6875 0.5061 −0.0110 0.00720 −0.000157 11.5292 −0.2781 −0.0401 0.5905 −0.0109 −0.00219 7.0418 −0.2205 0.9974 1.46

Net FOF saving: B0 B1 B2 B3 B4 B5 B6 θ α β γ δ λ γ : sr γ : ss R2 dw

−3.9078 0.3869 0.5631 0.4729 −0.0130 0.00560 −0.000153 7.7256 −0.5059 −0.0451 0.5945 −0.0137 −0.00149 4.5929 −0.2594 0.9712 1.44

in wealth (whether realized or not) are spent on consumption, which means that, cet. par., saving out of current income will be reduced by the same amount.11 11 See Poterba (2000) and Gramlich (2002). Related to the discussion of this paragraph is the slippery notion that realized capital gains represent income. While this is true for any individual, it is not the case, in a closed system, for the economy as a whole. A fallacy of composition is involved,

19.4

Conclusions

423

19.4 Conclusions Application of the B-C model to aggregate and per capita saving (as measured in the US. Flow of Funds Accounts) leads to the following results and conclusions: 1. Usually, rates of return, if accounted for at all, are included in the models of saving as predictors. With the present model, the rate of return, defined as the (negative) depreciation rate (δ) on the stock of financial assets (which is treated as an unobservable state variable), is estimated as a parameter. Since saving and income are measured in current dollars, inflation is thereby taken into account, so that the estimates of δ are in fact estimates of real rates of return. In general, all of the estimates of δ are in the ballpark of the ongoing estimates of the real interest rate. 2. A second innovation of the model of this chapter is that, in providing for an estimate of the steady-state marginal propensity to save (γ : ss), this parameter, combined with the estimated rate of return on the stock of financial assets, allows for the estimation of the steady-state dollar flows that are generated by this stock. The estimates indicate that these returns are generally of the order of 25−20% of the current income. The implication of this is that, for households whose other income is at or above the average level of income, this allows for a “retirement” level of consumption greater than what they enjoyed (at least on the average) during their working years. On the other hand, for households in retirement whose only other source of income is social security, then their retirement consumption is almost certainly quite diminished. This, in a nutshell, points up the great downside risks of low levels of saving during a household’s working years. 3. Looking to the future, projections (obtained through simulation of the estimated models) portend saving rates that, for historical growth rates in income, are small or even negative. To get projected saving rates that are not alarmingly small requires growth rates in income that are well in excess of historically plausible values. This, too, points up one of the really important policy concerns of current consumption and saving behavior. 4. Results when capital gains are included in the model indicate that these have expected negative effects on saving. However, these effects appear quantitatively small, and are probably not sufficient to explain the extremely low personal saving rates of recent years.

for, as noted, realizing a capital gain involves an exchange of existing assets, and the only income that is generated is brokerage and middlemen fees. Consequently, the effect on aggregate saving of realized capital gains is at best not to reduce it. On the other hand, a negative effect of unrealized capital gains on aggregate saving is best seen in terms of a money (or more properly, a wealth) illusion. For more detailed discussion of this and other types of fallacies of composition involving aggregate saving and consumption (including aggregate real-balance effects), see Chapter 7 of Taylor (2000).

Part IV

Miscellaneous Studies of Income Distribution and Weak Axiom of Revealed Preference

Chapter 20

The Stationarity of Consumer Preferences: Evidence from Twenty Countries Hendrik S. Houthakker

20.1 Motivation It is well-known that the basic theorems on individual demand, usually derived from the utility hypothesis, do not necessarily extend to market demand (Hicks, 1939). Some of these theorems may be lost by aggregation because the distribution of income intervenes. According to Hildenbrand (1983, 1994), however, market demand behaves much like individual demand if the distribution of quantities bought among consumers has greater dispersion at higher levels of income. This type of heteroscedasticity had been detected in cross-section data by Prais and Houthakker (1955). Hildenbrand (1994) provides more extensive evidence from household surveys that the distribution of demand is dispersed in accordance with his assumptions, but he does not analyze market demand as such.1 The purpose of this paper is to test whether market demand satisfies the axioms of revealed preference.2 If it does, the market demand functions are stationary, which means that tastes remain unchanged, so that quantities consumed depend only on income and prices. There are obvious reasons—such as advertising, demographic change, short-term dynamics (habit formation and inventory adjustment), and new products—for doubting the stationarity of market demand. It is all the more remarkable that, as shown in Section 20.3, violations of the weak axiom are extremely rare in annual time series of major expenditure categories from 20 OECD countries extending over as many as 35 years; no violations of the strong axiom were detected

I am indebted to Tomas Sjostrom and Atsushi Maki for valuable comments and other help, and Romeo Reyes for research assistance. 1 Hildenbrand’s assumption about dispersion is actually somewhat more complicated than stated here. His arguments for not analyzing market demand are criticized in a book review (Houthakker, 1995). That review also discusses the precise sense in which market demand functions are similar to individual demand functions. 2 The theory of revealed preference, which, of course, owes its genesis to Samuelson (1938), has spawned a huge literature—Varian (2006), for example, reports that a search in January 2005 of JSTOR business and economics journals for the phrase “revealed preference” yielded 997 articles!—and includes especially important contributions by Houthakker (1950), Afriat (1965, 1967), Diewert (1973), and others. Varian provides a useful and informative history and survey.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_20, 

427

428

20 The Stationarity of Consumer Preferences

at all. In most of these series, moreover, the weak axiom alone will rank the observations in terms of revealed preference, and the ranks so obtained may be interpreted as ordinal utilities.

20.2 Latent and Revealed Preference in Finite Data Sets The weak and strong axioms of revealed preference were designed to be testable from any set of data, finite or infinite. From a strictly mathematical point of view they are indeed testable, but this does necessarily mean that tests are computationally feasible.3 Although the weak axiom can be tested without difficulty, direct tests of the strong axiom present a formidable problem in large data sets. This section will present concepts that make tests of the strong axiom computationally feasible. A concise restatement of revealed preference theory (including some new material) provides the needed background.

20.2.1 What Does Revealed Preference Reveal? The weak axiom of revealed preference, as it came to be called later, was introduced by Samuelson (1938) as a testable alternative to the utility hypothesis, which at that time was thought to apply only to the individual consumer. Although Samuelson initially proposed the weak axiom (WARP) as a replacement for the older hypothesis, revealed preference theory and utility theory are now widely viewed as complementary approaches. Actually what matters is not utility as such, but a more basic concept which may be called “latent preference” to distinguish it from revealed preference. Latent preference is the answer to questions such as “would you (as a consumer) rather have one apple and two oranges, or two apples and one orange?”4 In other words, latent preference refers to unconstrained choice between two consumption bundles, whereas revealed preference refers to choice among many bundles under income and price constraints. The totality of all latent preferences expressed by a consumer is his or her “latent preference ordering.” Under certain conditions, it is possible to represent this ordering by a utility function that gives bundle 1 a higher value than bundle 2 if 1 is latently preferred to 2. The weak and strong axioms of revealed preference are theorems concerning latent preference derived from the theoretical properties just mentioned.

3 The question of testability has also been addressed by Varian (1982), whose definition of revealed preference is somewhat different from the one used here. See also Varian (2006). 4 The word “latent” does not imply that questions of this sort cannot be meaningfully answered, but there are obvious practical difficulties when commodities are numerous.

20.2

Latent and Revealed Preference in Finite Data Sets

429

20.2.2 The Weak Axiom WARP may be expressed—somewhat inaccurately as we shall see—by the implication: If pi = pj and

 h

j

pih qh ≤

 h

pih qih , then



j

h

ph qih >



j j p q , h h h

(20.1)

where the subscripts refer to commodities, the superscript identifies the prevailing budget constraint, the ps are prices, and qs are quantities; it is understood that situations i and j are not identical. More precisely, the two price-income situations are different if there is at least one good k for which 

pik i i h ph qh

j

= 

pk

j j h ph qh

.

(20.2)

Thus, the second premise of equation (20.1) says that the bundle qj (that is, the vector of quantities bought in situation j) costs no more at the prices pi of the situation i than the bundle qi , which was actually bought in that situation. This means that qj could have been bought but was not, so that qi is “revealed to be preferred to” qj . WARP infers from this that in situation j, when qj was actually bought, qi was too expensive, since otherwise qi would have been bought. The weak axiom defines the relation “revealed to be preferred to,” which pertains to pairs of quantity bundles in a certain domain. In the pure theory of consumption, the domain is usually the nonnegative orthant of a Euclidian space, but in this paper the domain consists of a finite set of n price-quantity observations. To say that no violations of WARP were found does not necessarily mean that WARP holds throughout the nonnegative orthant, most of which is outside the realm of experience. It should perhaps be made clear that a violation of WARP does not imply that revealed preference theory must be rejected. What it does imply is that one of the bundles involved in that violation does not fit into the same underlying preference ordering as the other bundles in the domain. As hinted earlier, the expression of WARP by equation (20.1) is incomplete. Strictly speaking, the premise of equation (20.1) should include a condition that all bundles in the domain come from the same preference ordering (generated, for instance, by a utility function). To avoid prolixity, this additional condition has been suppressed, but it should be borne in mind when interpreting the results. It is also important to note that, regardless of its original motivation, WARP says nothing about the nature of the quantities qh . These quantities may be bought by a household, by a group of households somehow defined, or by the market as a whole.5

5 See

Russell (1992) and the literature quoted therein.

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20 The Stationarity of Consumer Preferences

Returning to the formal aspects, the notation will be simplified by defining a “truth function” (that is, a function whose values are 0 for false and 1 for true) as follows:   j Definition 1: R(i, j) = 1 if and only if i = j and h pih qh ≤ h pih qih , from which one sees that R(i, j) equals zero in all other cases; in particular, R(i, i) = 0. With this definition, the standard statement of WARP in (1) can be replaced by R(i,j) · R(j,i) = 0

(20.1a)

In the subsequent analysis a matrix with R(i, j), where i and j both run from 1 to n, will be referred as an R matrix.

20.2.3 Dominance In a rapidly growing economy, all categories of consumption may be increasing at positive (but in general different) rates. This phenomenon, which affects the testing of WARP, can be analyzed by introducing the concept of “dominance”: j

Definition 2: qi dominates qj if, for all h = 1, 2,. . ., m, qih ≥ qh , and there is at least j one k such that qik > qk . Since prices and quantities are positive, it is clear that R(i, j) = 1 and R(j, i) = 0 when qi dominates qj , so that WARP is identically satisfied. In other words, pairs of observations that exhibit dominance tell us nothing about the truth or falsity of WARP. It is therefore important to know how much dominance is present in any set of data. For this purpose, let n be the number of observations in a data set, and np = n(n−1)/2 the number of observation pairs. Also, let t be the number of pairs in which one member dominates the other. The “dominance index” is then defined as D=

t np

which can vary from 0 to 1. If an “effective pair” is a pair of observations without dominance, then the number of effective pairs is ne = np−t. These concepts will be discussed more substantively in Section 20.3.

20.2.4 Matching and Connectedness Within the framework set by WARP, there are two patterns of revealed preference. It is possible that in a particular data set there are pairs of bundles in which neither bundle is revealed to be preferred to the other; in formula: R(i,j) = 0 and R(j,i) = 0

(20.3)

20.2

Latent and Revealed Preference in Finite Data Sets

431

Specifically, when qi was bought qj may have been too expensive, and when qj was bought qi may have been too expensive. When equation (20.3) holds, qi and qj will be said to “match” each other.6 On the other hand, a given set of data may be such that for all i and j (except if they are equal): Either R(i,j) = 1 or R (j,i) = 1,

(20.4)

which means that in every pair of bundles one bundle is revealed preferred to the other; hence matching never occurs. The revealed preference relation is then said to be “connected” in that data set. Most of the time series investigated here are characterized by equation (20.4) and some by equation (20.1a). Since equation (20.4) is simpler it will be pursued first (in Section 20.3). To do so, we need the strong axiom of revealed preference (SARP).

20.2.5 The Strong Axiom of Revealed Preference In the simplified notation introduced earlier, SARP can be written: If there exists bundles j, k,..., s such that: R (i,j) = 1 and R(j,k) = 1 and . . . R(s,t) = 1,

(20.5)

then R(t, i) = 0. SARP extends WARP to a chain of bundles of any length greater than one; for a chain of two, SARP is logically equivalent to WARP. It is convenient next to define a special relation for chains of bundles (or more precisely for the beginning and end of a chain): Definition 3: S(i, k) = 1 if and only if R(i, k) = 0 and (20.5) is satisfied. This definition covers only “descending” chains, in which each element is revealed to be preferred to the next. The case R(i, k) = 1 is excluded because the value of S(i, k) can only convey information if R(i, k) = 0. The relation S(i, k) may be described as “implicit revealed preference” to distinguish it from the “explicit” R(i, j). The first clause of this definition makes explicit and implicit revealed preference mutually exclusive.7 Using equation (20.5), we can now state the principal property of S(i, k): S(i,k) = 1 ⇒ R(k,i) = 0. 6 In

(20.6)

the literature “matching” has also been called “pseudo-indifference.” Actually, expression (20.2) is only a provisional definition, which will be restated in Section 20.2.7 below. 7 It also follows from the definition of S(i, k) that, unlike its explicit counterpart, implicit preference is transitive. Cf. Houthakker (1950) and Samuelson (1950).

432

20 The Stationarity of Consumer Preferences

It is illuminating to ask under what circumstances one bundle can be implicitly preferred to another. Suppose, for instance, that R(1, 2) = 1; when can this chain be extended to bundle 3 to yield new—and valid—information about the underlying preference ordering? If (explicit) revealed preference is connected we must have either R(1, 3) = 1 or R(3, 1) = 1. If it is further assumed that R(2, 3) = 1 we might be tempted to infer that bundle 1 is implicitly preferred to bundle 3, but this is not new information if R(1, 3) = 1, and a contradiction of equation (20.6) if R(3, 1) = 1. This argument obviously applies to chains of any length. It leads to the conclusion that, when WARP holds, implicit preference cannot arise in the connected case. In that case equation (20.6) is vacuous because its first clause is never satisfied, but equation (20.5), which does not involve the relation S, remains relevant. When there is matching, by contrast, valid new information can be obtained. In the above example it is now possible that R(1, 3) = R(3, 1) = 0. From R(1, 2) = 1 and R(2, 3) = 1 we could then infer that S(1, 3) = 1, which is new information and consistent with WARP. Thus implicit preference can sometimes fill in the blanks when matching is present. To do so, incidentally, it is not necessary to consider all chains of any length as the definition of S(i, k) might suggest; this could take a prohibitive length of time. It is enough to consider the final two elements of any sequence of ones.

20.2.6 A Test Procedure for the Connected Case Although expression (20.6) resembles expression (20.1a), there is practically an important difference between the two axioms. Even in large data sets it is easy to test WARP by checking where there are pairs of bundles for which R(i, j) = 1 or R(j, i) = 1. The number of needed comparisons is of the order n2 . A direct test of SARP, by contrast, would require looking at all possible chains of three or more elements, which is of the order n!8 The main purpose for the analysis below is to make direct tests of SARP unnecessary. The matter is relatively simple when the connectedness condition (20.4) holds; the validity or failure of condition (20.4) itself can be readily ascertained by looking for bundle pairs that satisfy R(i, j) = 0. Two new concepts are needed: Definitions 4 and 5: A(i) is the set of all bundles k for which R(i, k) = 1. a(i) is the number of bundles k for which R(i, k) = 1. We can now invoke the following: Theorem I: Suppose (a) WARP holds throughout a finite set of data H; (b) R is connected throughout H; and (c) for any pair of bundles such that R(i, j) = 1, A(j) 8 When there are 100 observations—fewer than in the largest sample investigated here—the number

of different chains is an integer with over 150 decimal digits. Examining all these chains could strain even the largest supercomputer.

20.2

Latent and Revealed Preference in Finite Data Sets

433

is a proper subset of A(i). Then the bundles qi can be totally ordered by R, in which case SARP is satisfied throughout H. Proof: A “total ordering” is an ordering without ties. If the data set contains n observations, there are n(n−1)/2 pairs of bundles, and consequently the same number of cases of revealed preference. This is only possible if there is a unique “maximum bundle” that is revealed to be preferred to all other bundles, of which there are n−1. There cannot be two or more maximum bundles, since they would have to match, contrary to assumption (b). One of these n−1 bundles must be revealed to be preferred to the remaining n−2 bundles, and so down to zero. A total ordering is obtained by assigning each bundle qi a rank a(i) as defined above, where a(i) is unique by virtue of condition (c). From this assignment it is clear that R(i,j) = 1 if and only a(i) > a(j).

(20.7)

To prove that violations of SARP are excluded by (6), suppose that in a chain of three, R(i, j) = R(j, k) = 1. Then expression (20.7) tells us that a(i) > a(j) > a(k), R(k, i) = 1 is impossible. This argument applies to chains of any length.9 Remark 1: The necessity of condition (c) can be seen from the following R-matrix (as defined in Section 20.2.2): R∗ = [(0 0 1 0) ( 1 0 0 0 )( 0 1 0 0 )( 1 1 1 0 )] WARP is obviously satisfied; so is the condition in expression (20.3) since there are 6 ones, as there should be for n = 4. There is also a maximum bundle, represented by the fourth row. The first three rows, however, do not obey condition (c), are not totally ordered, and constitute a violation of SARP. Remark 2: Condition (c) can be efficiently tested by observing that each of the sets A(h) can be described by a string of zeros and ones. Thus if R(i, j) = 1, and A(j) has a one in position k, then A(i) must also have a one in position k; in addition A(i) has a one in position j and possibly in other positions. Remark 3: Instead of considering a “maximum bundle” we could equally well have considered a “minimum bundle” and run the proof in reverse. Remark 4: The ranks assigned in the proof may be interpreted as values of a utility function, whose only arguments are the qh . For the finite data sets considered here, however, the mathematical form of this function cannot be inferred from revealed preference alone. 9

A more compact proof of this theorem is provided by R. Robert Russell in the appendix to this chapter.

434

20 The Stationarity of Consumer Preferences

20.2.7 Further Analysis of Matching To prepare for an alternative test procedure we have to return to the concept of matching introduced in Section 20.2.4. The provisional definition in expression (20.1a) must now be extended to incorporate the relation S(i, j) of Section 20.5. Definition 6: M(i, j) = 1 if and only if R(i, j), R(j, i), S(i, j), and S(j, i) all equal zero. Strictly speaking it is enough to say that the last four truth functions are all equal; if WARP holds they cannot all equal 1. Thus two bundles match each other if neither is explicitly or implicitly revealed to be preferred to the other. According to this definition, matching is reflexive [M(i, i) = 1] and symmetric [M(i, j) = M(j, i)]. Except in an important special case to be discussed shortly, matching is not transitive, for M(i, j) = M(j, k) = 1 neither implies nor excludes M(i, k) = 1. The concept of an R-matrix mentioned toward the end of Section 20.2.2 now has to be extended to include implicit preference: Definition 7: An RS-matrix is a matrix whose elements equal one whenever either R(i, j) = 1 or S(i, j) = 1, and zero otherwise. The matching relation is more complicated than the two revealed preference relations encountered so far. To begin with, we have to distinguish two types of matching, both present in the data analyzed here: Definition 8: Two bundles qi and qj match each other completely if (a) M(i, j) = 1 and (b) for all k = i = j both R(i, k) = R(j, k) and R(k, i) = R(k, j). If (a) is satisfied but (b) is not, the two bundles match each other incompletely. When matching is complete, which is the special case alluded to earlier, it is obviously transitive in addition to remaining reflexive and symmetric. A relation that has these three properties is called an equivalence relation. Thus if any and all matches in a data set are complete the bundles can be assigned to equivalence classes; all the bundles in such a class have the same rank as defined in Section 20.2.6.10 When matching is not complete it is not transitive, and bundles other than the matching pair may contain information relative to the ranks of two matching bundles. To illustrate this possibility, let an RS-matrix be RS∗ = [( 0 0 0 )( 1 0 0 )( 0 0 0 )] Bundles 1 and 3 match; so do 2 and 3. However, both matches are incomplete because R(2, 1) = 1. The ranking criterion based on the total number of ones in each row leads to a(1) = 0, a(2) = 1, and a(3) = 0. This information suggests that 10 In utility-function terms, equivalence classes obviously correspond to bundles lying on the same indifference curves.

20.2

Latent and Revealed Preference in Finite Data Sets

435

3 should somehow be ranked ahead of 1. The most convenient way involves the introduction of a second rank b(i) based on the column sums of the RS-matrix: Definition 9: b(i) is the number of bundles j for which R(j, i) = 1. In the connected case of Section 20.2.6, b(i) would be redundant since it would equal n−a(i)−1, but that identity does not hold when there are matching bundles. As is true of a(i), the largest possible value of b(i) is n−1 since there is at least one zero in each row and column of the RS-matrix. Thus, each bundle now has two ranks, which are of equal importance. However, it is important to note that, since a(i) represents the number of bundles j to which i is revealed to be preferred, while b(i) measures the number of bundles j that are revealed to be preferred to i, the ordering yielded by b(i) will be opposite that from a(i), that is, from smallest to largest, rather than from largest to smallest. The ranks add to n−1 if the bundle in question does not match any other bundles; otherwise the ranks will add to a smaller integer, namely n−1 less the number of matches in which that bundle is involved. The earlier interpretation of a(i) as a utility function (see remark 4 to Theorem I) does not carry over when incomplete matching is present.11 In the above example, we find by adding up the relevant columns that b(1) = 1, b(2) = 0, and b(3) = 0. Bundle 3 is now seen to indeed rank ahead of bundle 1.

20.2.8 Test Procedure for the Matching Case Matching evidently calls for modification in the test procedure described in Section 2.6. The basic idea of ranking the bundles will be retained, but there will now be two orderings, each of them partial rather than total. We saw that for ranking it is no longer possible to consider only the bundles j such that R(i, j) = 1 or S(i, j) = 1; those for which R(j, i) = 1 or S(j, i) = 1 also plays a part. We need one more definition: Definition 10: B(i) is the set of all bundles k for which R(k, i) = 1 or S(k, i) = 1. Theorem II: Suppose (a) WARP holds throughout a finite set of data; (b) for any pair of bundles such that R(i, j) = 1 or S(i, j) = 1, A(j) is proper subset of A(i); (c) for any pair of bundles such that R(j, i) = 1 or S(j, i) = 1, B(i) is a proper subset of B(j). Then the bundles qi can be partially ordered and there are no violations of SARP as expressed by (4). Proof: Theorem II generalizes Theorem I, which it includes as a special case, but is more intricate to prove.12 11

It can easily be seen that, when all matches are complete, no information is provided by b(i) since it is the same for all bundles in an equivalence class. 12 Unfortunately, a proof of this theorem was not included in the printed copy of this paper in the possession of LDT, nor could one be found in Professor Houthakker’s notes or computer files

436

20 The Stationarity of Consumer Preferences

20.3 Data The data used in this paper are from the OECD National Accounts (Vol. II, various issues, and diskettes covering all or part of the period 1960–1994).13 These are annual time series of household consumption at current and constant prices classified in various ways. The initial analyses used eight categories constituting a “minimum list,” for which most member countries provide estimates. The tables published before 1995 also show a more detailed breakdown, but for many countries—especially in the earlier years—there are no consistent entries or no entries at all. Because of changes in the base year for the constant-price data, there were actually two time series for several countries, one usually covering 1962– 1980 and the other 1987–1989; for various reasons not all available time series for all member nations could be used. The recent release of the 1960–1994 diskettes opened up new possibilities of analysis. These diskettes provide much longer time series for several countries than are available in print. They also contain a new breakdown of consumption into seventeen categories while retaining the old breakdown into eight. The two classifications are shown in Table 20.1.

Table 20.1 Classification of consumption by purpose 8 categories

17 categories

Food, beverages, and tobacco Clothing and footwear Gross rent, fuel, and power Furniture, furnishings, and household equipment and operation

Food Non-alcoholic beverages Alcoholic beverages Tobacco

Medical care and health expenses Transport and communications Recreation, entertainment, education and cultural services Miscellaneous goods and services

Rent Fuel and power Other household operation Personal transport equipment Education Personal care Expenditure in restaurants, cafes, and hotels Other goods and services

following his passing. A proof has been constructed by R. Robert Russell, and is included in the appendix to this chapter. 13 I also analyzed annual and quarterly time series from the US National Income and Product Accounts. The results from the annual series are similar to those from the OECD data reported in Section 20.4; those from the quarterly series are less clear-cut because of complicated patterns of matching. These additional analyses are not included here.

20.4

Findings

437

A third breakdown of consumption into four categories (durable goods, semidurable goods, nondurable goods, and services) sheds some light on dynamic effects, particularly inventory adjustment and habit formation. The analyses reported here cover 20 nations: the G7 countries (Canada, France, Germany, Italy, Japan, the UK, and the US) plus Australia, Austria, Belgium, Denmark, Finland, Greece, Ireland, Mexico, the Netherlands, Norway, Spain, Sweden, and Switzerland. One problem with the OECD data is that for some countries the published tables show an unexplained statistical discrepancy in the constant-price series (mostly in the early years). The ramifications of this discrepancy are not clear, but it probably does not affect the results of this paper. It should also be mentioned that cross-border consumption items (expenditures abroad by residents and expenditures in the country by nonresidents) are not taken into account. Prices in each category of consumption were calculated by dividing the constantprice figures into the current-price figures. In other words, the prices used are deflators.14 The consumption data are per capita at base-year prices; per capita figures were used to eliminate trends due mainly to population growth, thus reducing the incidence of dominance (see Section 20.2.3 above). Population data were taken from Volume I of the OECD National Accounts.

20.4 Findings As stated in the introduction, the initial purpose of this paper was to test whether the weak axiom holds in aggregate time series. My prior expectation was that there would be some contradictions, and when none appeared, I looked at more and more countries until finally a few violations of WARP came to light; by then all OECD countries with usable data had been covered.15 In the course of this search, it became apparent that violations of the strong axiom are also rare.16 The results are presented according to the number of consumption categories. See Table 20.1 and the text thereafter for a listing of these categories. Table 20.2 has the most detailed breakdown with 17 categories. It is especially illuminating because the series are mostly long and display little dominance; for three countries

14

One has to ask whether this widely used procedure might account for the nearly universal failure to reject WARP shown in Section 4. At one level the answer is clear: the use of deflators does not exclude all violations of WARP, since a few did appear. It is conceivable, however, that using deflators somehow biases the WARP test against rejection. This question needs further investigation. 15 When I completed my review of Hildenbrand’s work (Houthakker, 1995) no violations had yet been detected. 16 The testing procedure for a given data set has been as follows: (1) Check for dominance, eliminating all bundle pairs in which dominance exists; (2) Check each bundle pair for violations of WARP; (3) When no matching is present, check for the violation of SARP by searching for violations of condition (c) in Theorem I; (4) If matching is present, check for violations of SARP by searching for violations of either conditions (b) and (c) of Theorem II.

438

20 The Stationarity of Consumer Preferences Table 20.2 Summary of results using 17 categories of consumption

Country:

Austria Denmark France Germany Italy

Norway Sweden U.K. U.S.A.

Base year: First obsn.: Last obsn.: Da : Neb : Matches: WARP viol’s:

1990 1960 1994 0.313 409 0 0

1990 1962 1991 0.248 327 3 0

1980 1966 1994 0 406 0 0

1980 1970 1993 0.181 226 0 0

1970 1970 1994 0.057 283 2 0

1985 1970 1994 0.077 277 0 0

1990 1980 1994 0 105 0 0

1990 1970 1994 0 300 0 0

1985 1960 1993 0.012 554 1 0

Notes: a Dominance index (see Section 20.2.3); b Number of effective pairs of observations (see Section 20.2.3)

there is no dominance at all. The total number of effective pairs in this table is 2,887, none of which contradicts WARP. Matches occur in three countries, but there are no violations of SARP either. Table 20.3, with eight categories, has the largest number of data sets. The main reason, as explained in Section 3, is that in the early stages of this work, only those eight categories were analyzed. Many of the country series are relatively short, but a few (for Canada, Germany, and Switzerland) cover the entire period from 1960 to 1994. There is considerable overlap among the data sets for the same country (assuming there is more than one). Since this “data overlap” makes the total number of pairs less meaningful, an alternative overall measure of dominance for Table 20.3 is needed.17 The obvious candidate is the greatest lower bound, obtained by selecting from each country the data set with the largest number of effective pairs, and then summing these numbers over all countries. In any case it is clear that dominance is more pronounced here than in Table 20.2. Two violations of WARP were detected, one in Mexico and the other in Spain. As in the previous table, matches are infrequent, and they do not lead to contradictions of the strong axiom. As in Table 20.2, there is only one data set per country in Table 20.4, so the kind of overlap noted for Table 20.3 does not arise. However, dominance is more prevalent than in Table 20.3. The series for Italy, for example, have the same length in Tables 20.2 and 20.4, but here there are only 17 effective pairs compared with 277 in the earlier table. The total number of effective pairs in Table 20.4 is 879, of which one contradicts WARP.

17 Another kind of overlap may be called “category overlap.” The eight categories in Table 20.3 are combinations of the seventeen in Table 20.2, and to avoid duplication data sets eight categories are considered only when the seventeen were not available for the same base year and time span.

20.5

Discussion

439

Table 20.3 Summary of results using eight categories of consumption Country

Base year

First obsn.

Last obsn.

Da

neb

Matches

WARP viol’s

Australia Austria Austria Belgium Belgium Canada Canada Canada Denmark Finland Finland France France Germanyc Germanyc Germanyc Greece Ireland Italy Japan Japan Mexico Netherlands Norway Spain Spain Sweden Switzerland Switzerland Switzerland U.K. U.S.A. U.S.A. U.S.A.

1989 1975 1985 1975 1985 1971 1985 1986 1980 1975 1985 1970 1980 1975 1980 1991 1970 1985 1980 1985 1990 1980 1990 1985 1970 1986 1975 1970 1985 1990 1985 1975 1980 1985

1981 1964 1977 1963 1975 1963 1977 1960 1977 1963 1977 1963 1977 1963 1977 1960 1977 1977 1977 1977 1970 1980 1977 1977 1964 1980 1963 1963 1977 1960 1977 1963 1977 1982

1994 1980 1989 1980 1994 1980 1989 1994 1994 1980 1989 1980 1993 1980 1989 1994 1989 1989 1989 1989 1994 1993 1994 1989 1979 1994 1980 1980 1989 1994 1989 1979 1989 1993

0.220 0.853 0.487 0.922 0.642 0.712 0.179 0.466 0.653 0.621 0.808 0.915 0.588 0.882 0.667 0.862 0.103 0.073 0.769 0.885 0.810 0.121 0.536 0.333 0.867 0.486 0.209 0.425 0.346 0.395 0.641 0.728 0.628 0.348

71 20 40 12 68 44 64 318 141 58 15 13 56 18 26 82 70 51 18 9 57 80 71 52 16 54 121 88 51 360 28 37 29 43

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 1 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0

Notes: For a and b , see Table 20.2; c excluding East Germany.

20.5 Discussion At the outset, I referred to four well-known phenomena that cast doubt on the possibility of stationary consumer preferences. These will now be discussed in more detail. As regards advertising, a widespread view of consumption—more likely to be found among non-economists than among economists—is that consumers buy goods and services to satisfy their “needs,” and that at least some of these needs are created by suppliers through various kinds of persuasion. In this view, to take a topical example, people smoke cigarettes because tobacco companies advertise. One major difficulty, however, is that advertising is nearly always brand specific

440

20 The Stationarity of Consumer Preferences Table 20.4 Summary of results using four categories of consumptionc

Country

Base year

First obsn.

Last obsn.

Da

neb

Matches

WARP viol’s

Austria Canada Denmark Finland Greece France Italy Japan Norway U.K. U.S.A.

1990 1986 1980 1990 1970 1980 1985 1990 1990 1990 1985

1960 1960 1966 1960 1970 1970 1970 1970 1962 1970 1960

1994 1994 1994 1994 1994 1993 1994 1994 1991 1994 1993

0.877 0.739 0.653 0.872 0.477 0.848 0.943 0.927 0.832 0.837 0.877

78 155 141 76 157 42 17 22 73 49 69

0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 1 0 0

Notes: For services.

a

and b , see Table 20.2; c durable goods, semi-durable goods, nondurable goods, and

rather than generic; its purpose is to increase or defend the sponsor’s market share. The data analyzed here are not disaggregated to a level where branding may be important. A large part of advertising, moreover, is primarily informational; grocery stores, for example, publicize their prices and little else. It is also true that many products are hardly advertised at all; electricity is a case in point. Demographic changes are also prominent in popular discussions of consumption. In the media, we hear and read ad nauseam about baby boomers and Generation X, but the significance, if any, of these concepts appears to be more sociologic than economic. Although other demographic changes, particularly declining birth and death rates, are probably important in the long run, their effects may only show up in much longer consumption series than are available from OECD. Dynamic effects such as habit formation and inventory adjustment have figured prominently in time-series investigations of consumer’s expenditures, for instance in Houthakker and Taylor [1966, 1970 (and Chapters 14 and 15 of this book)]. If these effects are as pervasive as the econometric studies suggest, it might be thought that they should give rise to numerous violations of WARP. It is therefore surprising that they show up only to a limited extent in the present analysis, which is not of an econometric nature.18 An apparent case of inventory adjustment is provided by Norway in Table 20.4, one of the three WARP violations detected. In the 1980s a few years with strong sales of durables were followed by very weak sales during a recession that started about 1987. It may be noted that WARP was not contradicted by the Norwegian data in Table 20.2, where the expenditure breakdown is much more detailed. 18

However, care is required in interpreting this result, since the tests concern market (rather than individual) demand, dynamic effects of the habit-formation and inventory adjustment variety can be present in the underlying populations of consumers, but may not show up in the tests, because their distributions across commodities and socio-demographic characteristics of consumers are relatively stable through time.

20.6

Concluding Remarks

441

Finally, with respect to new goods, whose emergence is continual in consumer markets, two cases are to be considered. The first is the situation where the appearance of a new good leads to an enlargement of the consumer’s choice set through the fulfillment of a want that had not previously been attended, while the second case is the situation where the new good simply fulfills an already existing want. The former is probably inconsistent with conventional definitions of stationary preferences, while the latter is not. The results of this paper suggest the second case, that is, that most new goods simply substitute for existing goods in want fulfillment.

20.6 Concluding Remarks The main conclusion from this research is that contradictions of the weak axiom in market demand are rare indeed. In Tables 20.2–20.4, there are at least 5518 effective pairs of observations with data overlap (using the greatest lower bound for Table 20.2), in which there are only three contradictions. Thus Hildenbrand’s insight concerning the importance of heteroscedasticity among consumers for the properties of market demand is overwhelmingly confirmed. The results just summarized lead naturally to two further questions. One concerns parametric forms of the utility function that are consistent with the data for each country, assuming of course that these data satisfy the axioms of revealed preference. This question is of central importance for demand analysis, where a number of different specifications have been proposed, but it needs additional research. The second open question is whether the preference orderings of different countries are the same, which would mean that each country’s expenditures can be expressed by its income and prices in common units.

Appendix: Notes on Houthakker’s Paper on “Stationarity of Consumer Preferences” R. Robert Russell University of California, Riverside

Proofs of the Theorems Theorem I: Suppose (a) WARP holds throughout a finite data set H; (b) R is connected throughout H; and (c) for any pair of bundles such that R (i, j) = 1, A (j) is a proper subset of A (i). Then the bundles qi are totally ordered by R, in which case SARP is satisfied throughout H. Proof: First note that (a)–(c) imply that a(i) = a(j) for all pairs i, j. Hence, the relation RP , defined by

442

20 The Stationarity of Consumer Preferences

qi RP qj ⇔ a(i) > a(j) ∀i, j, totally orders the bundles and satisfies qi RP qj if and only if R(i, j) = 1. Transitivity and asymmetry of > (and hence RP) implies SARP. Theorem II: Suppose (a) WARP holds throughout a finite data set H; (b) for any pair of bundles such that R(i, j) = 1 or S(i, j) = 1, A(j) is a proper subset of A(i); (c) for any pair of bundles such that R(i, j) = 1 or S(i, j) = 1, B(i) is a proper subset of B(j). Then the bundles qi can be partially ordered, and there are no violations of SARP. Proof: To establish SARP, suppose on the contrary that, for some sequence of bundles, qi ,qj ,qk ,. . .qs ,qt , R(i, j) = R(j, k) = . . . = R(s, t) = R(t, i) = 1. Then (b) implies that a(i) > a(j) > a(k) > . . . > a(s) > a(t) > a(i), a contradiction. To construct the partial ordering, define the “pseudo-indifference” relation RI by  RI(i, j) =

1 if R(i, j) = R(j, i) = S(i, j) = S(j, i) = 0 ∧ a(i) = a(j) ∧ b(i) = b(j) 0 otherwise

and the preference relation RP by RP(i,j) = 1 if (i) R(i,j) = 1 or S(i,j) = 1 or (ii) R(i,j) = S(i,j) = 0 ∧ a(i) > a(j) ∧ b(i) ≤ b(j) or (iii) R(i,j) = S(i,j) = 0 ∧ a(i) ≥ a(j) ∧ b(i) < b(j); R(i,j) = 0 otherwise. It is obvious that RI satisfies reflexivity, symmetry, and transtivity (i.e., RI is an equivalence) and that RP is irreflexive and asymmetric. Using (b) and (c) of the theorem, we see that (i) implies a(i) > a(j) and b(i) < b(j), a special case of the second parts of (ii) and (iii). Transitivity of RP therefore follows immediately from transitivity of > and ≥. Thus, the union of RI and RP induces a partial ordering on the data set that respects the definitions of R and S: R(i, j) = 1 or S(i, j) = 1 implies RP (i, j) = 1.

Discussion of the Theorems The proof of Theorem II suggests that it can be strengthened in two ways. First, since the proof uses only the cardinality implications (e.g., a(i) > a(j)) of the subset conditions, these conditions can be replaced by the inequality conditions used in the proof. Second, because of transitivity of the binary relation > (equivalently, transitivity of the binary relation ⊂), conditions (b) and (c) of Theorem II can be replaced by the simpler (b) and (c) in the following (equivalent) restatement:

20.6

Concluding Remarks

443

Theorem II: Suppose (a) WARP holds throughout a finite data set H; (b) for any pair of bundles such that R(i, j) = 1, a(i)> a (j); (c) for any pair of bundles such that R(j, i) = 1, b(i) < b(j). Then the bundles qi can be partially ordered, and there are no violations of SARP. Thus, the requisite conditions can be checked without the computationally burdensome search for chains of revealed preference: one only needs to check for violations of WARP and for satisfaction of the counting rules in (b) and (c) of Theorem II. Of course, the conditions (a)–(c) (different for the two theorems) are only sufficient for SARP and the existence of a linear or partial ordering. Thus, if these conditions are not all satisfied, SARP itself must be tested. The burden, however, is lesser in the case where no matching takes place, as shown by the following result: Theorem III: If SARP and condition (b) of Theorem I hold (in which case there exists a total ordering), condition (c) of Theorem I holds. Proof: Suppose (c) is violated for some pair i, j: R(i, j) = 1 and a(i) ≤ a(j). Then there exists a commodity k such that R(j, k) = 1 and R(i, k) = 0. Since SARP holds, R(i, j) = R(j, k) = 1 implies R(k, i) = 0, a violation of condition (b). Thus, If (b) holds and (c) does not hold, SARP is necessarily violated; that is, if (b) holds, test (3) in footnote 16 of Houthakker’s paper is necessary and sufficient for satisfaction of SARP. The case where matching takes place is more difficult. The following matrix of R(i, j) values for four bundles shows that neither (b) nor (c) in Theorem II are implied by SARP: ⎡

0 ⎢0 ∗ RS = ⎢ ⎣0 0

1 0 0 0

0 1 0 0

⎤ 0 1⎥ ⎥. 0⎦ 0

In this example, SARP holds but R(1, 2) = 1, a(1) < a(2), and b(1) = b(2). As it turns out, Houthakker finds that, in those cases where matching is present, conditions (b) and (c) of Theorem II are met; so SARP is necessarily satisfied and no additional checks are necessary. But in other data sets, it may happen that (b) or (c) is not satisfied, in which case the satisfaction of SARP remains an open question. Nevertheless, Houthakker’s theorem substantially reduces the computational burden of testing for violations of SARP and for the existence of an ordering that rationalizes the data. Finally, note that, as stated, the conditions of Theorem II imply only a partial ordering. The following matrix satisfies the conditions of Theorem II, but the test procedure of footnote 16 yields only a partial ordering of bundles 1, 3, and 4 (with 4 at the top and 3 at the bottom):

444

20 The Stationarity of Consumer Preferences

⎡

0 ⎢ 0 RS∗ = ⎢ ⎣0 1

0 0 0 0

1 0 0 1

⎤ 0 0⎥ ⎥. 0⎦ 0

Bundle 2 is not ranked. Nevertheless, we know from Afriat’s (1967) theorem that (a weaker condition than) SARP implies the existence of a complete ordering—a utility function—that rationalizes the data. In this case, because of the extensive matching, an ordering with bundle 2 in any position would (trivially) rationalize the data.

Chapter 21

Notes on Thick-Tailed Distributions of Wealth Lester D. Taylor

21.1 Introduction One of the best-known empirical regularities in economics is the law of Pareto, according to which the upper tails of the distributions of income and wealth are described by the relationship: P (u > x) = A x−α ,

(21.1)

for income and wealth levels (x) greater than some value of x0 and α <2. However, despite its ubiquity across economies and time, the matter of why this law holds remains an open question. Is there, as Mandelbrot has suggested, simply a base “prime mover” that gives rise to scalable distributions, not only for income and wealth, but also to a large number of other economic, social, and natural phenomena? If so, then the question of “why” is, by definition, not answerable. On the other hand, if thick upper tails can somehow emerge through processes that do not assume a Pareto prime mover, then the question is not only potentially answerable, but is also of a great deal of interest. Examining the possibility of this is the purpose of the present chapter.

21.2 Background In a production/exchange/consumption economy, the distribution of wealth can be viewed as the outcome of a sequence of processes as follows: production, which generates income, which funds consumption and saving, which determines which producers prosper (and therefore go on to live another day) and the accumulation of wealth. At base, this sequence of processes involves the conflation of three factors, namely, talents, tastes, and heritability. Talents, in conjunction with the principle

I am grateful to Teodosio Perez-Amaral and Barbara Sands for comments on this chapter and to the Cardon Chair Endowment in the Department of Agricultural and Resource Economics at the University of Arizona for financial assistance.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_21, 

445

446

21

Notes on Thick-Tailed Distributions of Wealth

of remuneration, determine the distribution of income. Tastes, in turn, determine how income is allocated between consumption and saving, and how consumption is allocated across producers. Finally, for wealth to accumulate over time, it obviously must be heritable. Since heritability is a passive (but nevertheless necessary) enabler, it follows that the real drivers of the distributions of income and wealth are the distributions of talents and tastes. If, as is the usual assumption, talents are remunerated according to marginal-productivity principles, a thick-tailed distribution of talents will yield a thick-tailed distribution of income, and (with heritability and a saving rate that does not decrease with the level of income), ultimately to a thick-tailed distribution of wealth. However, as has been noted, the interesting question is whether the latter can emerge in the presence of only mild assumptions on the distribution of talents (and tastes). With the examination of this question as the goal, let us now turn to a highly stylized, artificial economy consisting of n agents producing and consuming n (possibly distinct) abstract consumption goods (whose prices are all 1) as follows:

21.2.1 Scenario 1 (1) Each agent is both a producer and a consumer. Agents (for survival) consume 1 unit of a good during a period, but are able to produce (at constant cost) up to n units of their own good. (2) During each period, agents “purchase” from each other (including themselves) at random according to draws from a uniform distribution. (3) If an agent makes more than one “sale” during a period, the excess “income” (where excess is defined as the number of sales minus one) is saved and carried over as “wealth” to the next period. (4) If an agent fails to make a sale during a period, consumption for the period can be financed by withdrawing 1 unit from the agent’s stock of wealth. However, if the stock should be empty, the agent is assumed to “die,” and is replaced by a new agent who begins from scratch. Talents are represented in (1), tastes in (2), and heritability in (3). Talents are fixed and assumed to be the same for everyone. Tastes, on the other hand, are random, but are also assumed to be the same across consumers in the sense that the goods purchased by consumers are determined by draws from a common uniform distribution. With n consumers each consuming 1 unit in a period, the randomness of tastes implies that there will be a nonzero probability that some producers will have multiple sales in a period, while other producers will have no sales. The result will be a nonuniform distribution of income, and hence, through heritability, a nonuniform distribution of wealth. An idea of the departure from nonuniformity under the conditions postulated in (1)–(3) is given in Table 21.1 for n equal to 2 through 5. From the numbers in this

21.3

Some Initial Simulations

447

Table 21.1 Wealth distributions, n = 2, . . . 5 N

P(0)

P(1)

P(2)

P(3)

P(4)

P(5)

2 3 4 5

0.33 0.40 0.43 0.45

0.33 0.30 0.29 0.27

0.33 0.20 0.17 0.16

0.10 0.09 0.08

0.03 0.03

0.01

table, it is evident that the most likely outcome for a producer is to have no sales at all in a period, with a probability that increases with n. On the other hand, it is also evident that the probability of any producer becoming “super rich” during a single period is small. However, whether this small probability carries over to the distribution of wealth remains to be seen.

21.3 Some Initial Simulations The exercise proceeds by simulating the economy described in rules (1) through (4) above over a number of periods and by examining the resulting behavior of the distribution of “wealth.” Table 21.2 tabulates simulation results for n equal to 100, 1,000, 5,000, and 10,000 for 10 periods. Two things are immediately apparent: (1) the distribution of wealth under the assumptions postulated is independent of the number of agents for n ≥ 1,000 and (2) there is no evidence of a really fat upper tail. Common talents and purely random preferences (combined with heritability) thus do not appear to be sufficient for a thick upper tail. The next steps, accordingly, will be to modify the assumptions governing the distributions of talents and tastes. This will be done in two stages. A second scenario

Table 21.2 Simulated wealth distributions—Scenario 1 Number of agents P(x)

100

1,000

5,000

10,000

P(0) P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(≥10)

0.333 0.258 0.111 0.061 0.091 0.091 0.015 0.015 0.000 0.000 0.000

0.308 0.235 0.165 0.111 0.083 0.051 0.020 0.010 0.010 0.003 0.001

0.285 0.229 0.177 0.132 0.075 0.048 0.025 0.015 0.006 0.003 0.004

0.296 0.229 0.179 0.111 0.082 0.046 0.028 0.013 0.010 0.003 0.002

448

21

Notes on Thick-Tailed Distributions of Wealth

will keep tastes random as in Section 21.2.1, but will allow for a non-common distribution of talents, while a third scenario will inject some stability into tastes. The assumptions for Section 21.3.1 are as follows:

21.3.1 Scenario 2 (5) Agents, as producers, are now assumed to be bifurcated into two groups, with one group having z times the “productivity” of the other (where z > 1). Assignment to the two groups is according to a binomial distribution with parameter pz. (6) Producers in the “high” productivity group (i.e., whose productivity is z times that in the “low” productivity group) are assumed to have “costs of production” that are 1/z of the costs of the producers in the low productivity group. Consequently, additions to “wealth” for these agents will equal to the number of sales in the period minus 1/z, rather than the number of sales minus 1. (7) Agents, as consumers, are (as in Section 21.2.1) assumed to consume 1 unit per period and to purchase randomly from the n producers with probability 1/n. (8) As before, agents with negative wealth at the end of a period are assumed to “die” and are replaced with new agents (having no wealth) with productivity z or 1 as described in (5).

The results from simulating this scenario for 10 periods, for z = 2, pz = 0.2, and again for n equal to 100, 1,000, 5,000, and 10,000 (a single simulation for each n), are tabulated in Table 21.3.

Table 21.3 Simulated wealth distributions—Scenario 2: z = 2, pz = 0.2 Number of agents P(x)

100

1,000

5,000

10,000

P(0) P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(≥10)

0.316 0.165 0.114 0.152 0.114 0.063 0.038 0.000 0.000 0.025 0.011

0.311 0.182 0.143 0.116 0.079 0.053 0.044 0.026 0.014 0.012 0.018

0.319 0.171 0.148 0.108 0.085 0.056 0.042 0.024 0.019 0.014 0.012

0.315 0.177 0.142 0.112 0.088 0.055 0.040 0.029 0.018 0.012 0.014

21.3

Some Initial Simulations

449

21.3.2 Comments (i) As with the first scenario, the distribution of wealth appears to stabilize for agents equal to 1,000 or more. (ii) The shape of the distribution, however, has changed. While the probability of no wealth is increased slightly (about 0.32 vs. about 0.30), the really interesting thing is the transfer of “mass” from wealth stocks of 1 and 2 units to stocks of 6 or more units. This would appear to be a reflection of an increased mean stock of wealth that is allowed for by the differential in talents. (iii) Variations in the values of z and pz will be examined below.

21.3.3 Scenario 3 In this scenario, the assumption of purely random tastes is replaced with an assumption that, once formed, tastes tend to persist from one period to the next (provided, of course, that the agent continues to live). Specifically, the assumptions of this scenario are as follows: (9) As in Section 21.3.1, agents, as producers, are assumed to be bifurcated into two groups, with one group having z times the “productivity” of the other (where z > 1). Assignment to the two groups is according to a binomial distribution with parameter pz. (10) Agents, as consumers, are still assumed to consume 1 unit per period, but now with some stability in tastes as follows: (i) In the initial period, agents purchase randomly from the n producers with probability 1/n. (ii) However, in subsequent periods, agents purchase from the producers acquired from in the preceding period (provided that the producers are still “alive”) with probability q and from the other n−1 producers with probability (1−q)/(n−1). (iii) If the producers “acquired from” in the preceding period are no longer “alive,” then agents purchase from all n agents with probability 1/n. (11) Producers in the “high” productivity group (i.e., whose productivity is z times that in the “low” productivity group) are assumed to have “costs of production” that are 1/z. Consequently, additions to “wealth” for these agents are the number of sales in the period minus 1/z, rather than the number of sales minus 1. (12) As before, agents with negative wealth at the end of a period are assumed to “die” and are replaced with new agents (having no wealth) with productivity z or 1 as described in (5).

450

21

Notes on Thick-Tailed Distributions of Wealth

Table 21.4 Simulated wealth distributions—Scenario 3: z = 2, pz = 0.2, q = 0.8 Number of agents P(x)

100

1,000

5,000

10,000

P(0) P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(≥10)

0.414 0.103 0.034 0.034 0.103 0.080 0.046 0.046 0.011 0.023 0.103

0.423 0.081 0.063 0.073 0.061 0.055 0.043 0.046 0.023 0.020 0.112

0.404 0.104 0.076 0.076 0.061 0.050 0.040 0.035 0.029 0.023 0.101

0.404 0.096 0.085 0.070 0.057 0.051 0.043 0.035 0.030 0.026 0.102

The simulation results (again for 10 periods) for this scenario for z = 2, pz = 0.2, and q = 0.8 are given in Table 21.4.

21.3.4 Comments (i) With this scenario, the wealth distribution appears more to stabilize at 5,000 agents, rather than at 1,000 as with the first two scenarios. (ii) Once again, the distribution is seen to change shape, with a shift in mass away from stocks of 1 to 3 units. This time, however, the shift is in both directions— back to zero, as well as into the tail. The distribution has become much more skewed to the right. In short, preference stability (or what in the discussion below will be referred to as “habit formation”) clearly appears to be a condition allowing for the emergence of “large” fortunes.

21.4 Variations We now turn to an examination of how sensitive Sections 21.3.1 and 21.3.3 are to the parameters governing the distribution of talents and tastes. Since the simulations involving 5,000 and 10,000 agents are fairly computer intensive, the examination will be confined to economies with 1,000 agents.1 Table 21.5 presents results for Section 21.3.1 for different combinations of z and pz, while Table 21.6 presents results for Section 21.3.3 for different combinations of z, pz, and q. “Anchors” for 1 Simulations (on a standard PC with Pentium III chip) for 10 periods for Section 21.3.1 take about

8 min for 5,000 agents and 40 min for 10,000 agents. For Section 21.3.3, the time required is about 10 min for 1,000 agents, an hour for 5,000 agents, and more than 10 h for 10,000 agents. All coding and simulations have been done in SAS.

21.4

Variations

451

Table 21.5 Simulated wealth distributions parameter variations—Scenario 1 Parameters P(x)

z pz 2 0.2

z pz 3 0.2

z pz 4 0.2

z pz 2 0.3

z pz 2 0.4

z pz 2 0.1

z pz 2 0.05

z pz 6 0.05

P(0) P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(≥10)

0.311 0.182 0.143 0.116 0.079 0.053 0.044 0.026 0.014 0.012 0.018 (1)

0.323 0.170 0.126 0.093 0.075 0.053 0.041 0.036 0.028 0.024 0.028 (2)

0.322 0.146 0.102 0.084 0.073 0.085 0.042 0.043 0.032 0.024 0.046 (3)

0.269 0.160 0.154 0.114 0.097 0.058 0.050 0.039 0.029 0.017 0.013 (4)

0.245 0.162 0.153 0.126 0.092 0.068 0.052 0.038 0.019 0.017 0.028 (5)

0.371 0.182 0.153 0.092 0.053 0.056 0.031 0.025 0.016 0.007 0.013 (6)

0.387 0.171 0.142 0.101 0.076 0.062 0.028 0.020 0.009 0.000 0.004 (7)

0.394 0.201 0.132 0.094 0.049 0.039 0.037 0.017 0.006 0.007 0.015 (8)

Table 21.6 Simulated wealth distributions parameter variations—Scenario 3 Parameters z pz q z pz q z pz q z pz q z pz q z pz q z pz q z pz q z pz q 2 0.2 0.8 2 0.2 1 2 0.2 0.5 4 0.2 1 6 0.2 1 2 0.4 0.8 2 0.6 0.8 4 0.6 1 6 0.2 0.1 P(0) P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(≥10)

0.423 0.081 0.063 0.073 0.061 0.055 0.043 0.046 0.023 0.020 0.112 (1)

0.480 0.079 0.063 0.050 0.051 0.037 0.030 0.028 0.019 0.023 0.141 (2)

0.353 0.135 0.115 0.090 0.068 0.045 0.041 0.035 0.032 0.028 0.059 (3)

0.412 0.092 0.082 0.050 0.044 0.047 0.030 0.023 0.025 0.033 0.162 (4)

0.395 0.101 0.086 0.045 0.050 0.035 0.021 0.026 0.020 0.034 0.188 (5)

0.350 0.099 0.075 0.073 0.054 0.056 0.050 0.047 0.037 0.024 0.135 (6)

0.261 0.095 0.091 0.069 0.064 0.068 0.049 0.065 0.039 0.047 0.151 (7)

0.277 0.097 0.068 0.039 0.028 0.035 0.027 0.038 0.046 0.057 0.289 (8)

0.250 0.121 0.137 0.106 0.073 0.072 0.052 0.043 0.034 0.038 0.084 (9)

comparison will be the columns for 1,000 agents from Tables 21.3 and 21.4 (that is z = 2 and pz = 0.2 for Section 21.3.1, and z = 2, pz = 0.2, and q = 0.8 for Section 21.3.3). For Section 21.3.1, in which only talents and their distribution across agents vary, the first interesting result appears to be the effect on the probability of zero wealth, especially with respect to the size of the binomial parameter governing the distribution of talents. In columns (1) to (3), in which the differential in productivity varies from 2 to 4, but pz remains constant at 0.2, the only noticeable effect is a mild flattening of the wealth distribution’s tail. Variation in pz, holding z constant at 2 (cf. columns (4)–(7)), however, presents a totally different picture. Here, as pz varies

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from 0.4 to 0.05, the probability of a stock of wealth of zero increases monotonically from about 0.25 to a value approaching 0.4. Especially interesting is the fact that a tripling of the productivity differential in the last column (i.e., an increase of z from 2 to 6, but holding pz constant at 0.05) appears to have little impact. The conclusion that emerges from Table 21.5, accordingly, is that it is how talents are distributed, rather than differences in talents, per se, that affects the distribution of wealth. Section 21.3.3, it will be recalled, allows not only for talents to vary across agents, but also for the emergence of a stability in tastes, in the sense that agents (assuming that they survive) tend, with a certain probability, to purchase from the same producers from one period to the next. In Table 21.4, we saw how this “habit formation” lead to a noticeable fattening of the tail of the wealth distribution. In Table 21.6, we see how sensitive the shape of the distribution, especially the tail, is to the value of q. A q of 1 implies that purchases of surviving agents are all from producers of their first purchase, while a q of 0 of course implies that preferences are purely random from period to period, as in Sections 21.2.1 and 21.3.1. In columns 1 and 2, we see that increasing q from 0.8 to 1 leads to an increase in the probability of zero wealth from 0.42 to 0.48, combined with an increase in the tail probability (i.e., the probability of a stock of wealth of 10 units or more) from 0.11 to 0.14. Columns (2) and (3) present this sensitivity in even sharper relief, in that a halving of q from 1 to 0.5 is seen to lead to a decrease in the probability of zero wealth from 0.48 to 0.35, together with a decrease in the tail probability from 0.14 to 0.06. In short, stable preferences appear to be a powerful force in making for skewed distributions of income and wealth. However, what is especially interesting about the simulations in Table 21.6 is the interaction between the parameters governing talents and the amount of preference stability. This is particularly evident in column (8), which combines a high differential in productivity (z = 4) with a fairly even distribution of talents (pz = 0.6) and extreme preference stability (q = 1). The contrast with the combination of parameters in this column with the combination in column (5), which has an even higher differential in productivity (z = 6), but a much smaller presence of this differential (pz = 0.2), and a relatively minute amount of preference stability (q = 0.1), is striking.2 The conclusion is thus the same as before: stable preferences are a potent enabler of the creation of heritable stocks of wealth.

2 So striking, in fact, that it is useful to report a variation on the parameters in column (8), in which z and q are kept at 4 and 1, respectively, but pz is reduced to 0.2. The result (for the simulation undertaken) is an increase in P(0) to 0.423, and a reduction in P(≥10) to 0.172. The point to note is that the latter remains high in relation to simulations in which q is less than 1.

21.6

Law of Pareto Tests

453

21.5 Interpretation of Parameters and Scenarios Despite the fact that the economies being simulated are highly stylized and artificial, the parameters governing production, consumption, and the distribution of talents have reasonably straightforward interpretations, and indeed not implausible connections to the real world. To begin with, Section 21.2.1 in effect represents a totally subsistence economy in which everyone has to consume 1 unit of “bread” in order to survive. Everyone has the same talent, so that there is no accumulation from differential productivity. The only way that wealth can accumulate is by agents being lucky enough through the randomness of preferences to sell multiple units of output. Experience in consumption does not affect choice in this scenario. If an agent is fortunate enough to survive to the next period, the agent purchased from in the next period is totally random. The numbers in Table 21.2 suggest that agents do not survive long in this scenario, and that the probability of becoming (and staying!) “super rich” is virtually nil. In short, there is little scope for a thick-tailed distribution of wealth ever arising in this scenario. Things are different in Section 21.3.1, for by positing variation in talents, this scenario allows for economic growth to occur. Growth will be faster, the larger is the z, and also the larger is the pz. This scenario accordingly allows for accumulation to take place, not only from luck, but also from differential productivity. The result is wealth distributions with decidedly thicker tails. Nevertheless, since preferences are still purely random in this scenario, being able to hang onto a fortune, once acquired, remains a chancy proposition. The introduction of preference stability through the parameter q in Section 21.3.3 posits, as has been noted at a couple of points, a probabilistic form of habit formation. At one extreme, a q of 1 implies that habit formation is complete, i.e., that subsequent consumption is completely determined by initial experience, while, at the other extreme, a q of 0 implies that preferences remain completely without structure, as in Sections 21.2.1 and 21.3.1. From the numbers in Table 21.6, it is clear that preference stability has a major impact on the shape of the wealth distribution, and indeed might even be sufficient (in conjunction with variation in talents) to give the distribution its real-world Pareto twist. We now turn to an examination of the extent to which this might be the case.

21.6 Law of Pareto Tests In his classic paper in the Journal of Political Economy that introduced the study of long-tailed distributions of the Pareto type into economics, Mandelbrot (1963) made use of a simple graph to make many of his points. Taking logarithms of both sides of equation (21.1): ln P (u > x) = ln A − α ln x,

(21.2)

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Notes on Thick-Tailed Distributions of Wealth

we see that a basic implication of the Pareto distribution is that the Pr (u > x) and x are linear in logarithms, with slope equal to minus α. In short, if the distribution of wealth is Pareto in its upper tail, then the graph of the upper tail should be a straight line when plotted on a double logarithmic grid. Figure 21.1 provides graphs for four of the Scenario 3 simulations from Table 21.6. The blue graph is for the simulation in Column (1) in Table 21.6 (i.e., z = 2, pz = 0.2, and q = 0.8), pink corresponds to Column (4), while yellow and green correspond to Columns (8) and (9).3 However, before drawing any conclusions from these graphs, several comments are in order: Scenario 3 npds = 10 (from left) Cols. 9, 1, & 8 of Table 21.6

0 0

1

2

3

4

-1

ln P(u > x)

-2 -3 -4 -5 -6 -7

ln x

Fig. 21.1 Scenario 3 npds = 10 (from left) Cols. 9, 1, & 8 of Table 21.6

(i) The law of Pareto is an abstract distribution that refers to a large number of agents involving birth, death, consumption, production, and heritability processes that are (in some sense) in equilibrium. Since real-world distributions are necessarily finite, the Law cannot explain the very upper reaches of the distributions of income and wealth (since these necessarily drop to zero, in a possibly precipitous fashion). What this means, accordingly, is that we should not expect to find linearity (or at least linearity with slope greater than minus 2) at the very

3

The graphs in these figures are for different realizations than those shown in Table 21.6. In the captions of the figures, npds refers to the number of periods in the simulations. Graphs have been constructed in Excel.

21.6

Law of Pareto Tests

455

extreme of the upper tail of the distribution of wealth, but rather, at best, only for an interval of values leading to the extreme. (ii) The distributions depicted in the figure are generated from simulations of economies that are extremely “small” in relation to the real world; “small” both in the sense of the number of agents (1,000) and the “age” of the economies (10). While the results in Tables 21.2–21.4 suggest that, for the scenarios being investigated, the wealth distribution stabilizes fairly quickly in terms of agents for a given number of periods, this may not be the case for the number of periods, for, if nothing else, the distribution will almost certainly “move to the right” as the number of periods increases. Looking first at the “anchor” case for Scenario 3 (i.e., the second graph from the left in Fig. 21.1), we find perhaps a hint of linearity for stocks of wealth between 9 and 19 units. However, the slope of the segment is clearly less than −2. In short, there is no evidence of a Pareto tail for this case. On the other hand, for the third graph from the left, for which the Scenario 3 parameters are z = 4, pz = 0.2, and q = 1, we see that there are essentially three connected regions of linearity. The first is for 2 through 7 units, the second for 8 though 18, and the third is for 19 on. Of the three segments, the second is clearly of most interest, especially if the slope is greater than −2. However, a regression of ln P(y > x) on ln x yields a slope of −2.15 (with a t-ratio of −35 and an R2 of 0.99). Turning now to the graph at the far right, a near-linear segment, somewhat similar to the second segment in the third graph from the left, is defined for x between 7 and 24. This time, a regression equation yields a slope of −2.07, with a t-ratio of −20 and an R2 of 0.96—in short, we are getting closer to possibly some form of the law of Pareto, but are not yet quite there! Finally, in the first graph on the left, we see that, in the face of extreme (but isolated) talent differences (z = 6, pz = 0.2), but little preference stability (q = 0.1), there is no evidence at all of a law of Pareto. If we were to stop at this juncture, the conclusion would pretty much have to be that, while strong habit formation (i.e., preference stability), combined with variations in talents and heritability, yield wealth distributions that are increasingly long-tailed, the processes stop short of the law of Pareto. However, as noted in comment (ii), an economy of only 10 periods is scarcely out of a “state of nature,” and it may be that the law of Pareto emerges only with the passage of many generations. As a partial test of this, a simulation entailing 100 periods has been undertaken, using what from the earlier simulations appear to be particularly sensitive values for the parameters, namely, z = 6, pz = 0.4, and q = 1. The resulting graph of ln P(y > x) against ln x is given in Fig. 21.2. Bingo! For the regression slope for the segment for x between 9 and 89 is −0.86 (with a t-ratio of −37 and an R2 of 0.94)! Not only is this consistent with the law of Pareto, but would appear to be a thick-tailed distribution with a vengeance—for a slope of −0.86 for such segment (being greater than −1)—would be consistent with the distribution that does not even have a mean! Whatever, further investigation of this scenario seems clearly to be in order. Figure 21.3 provides comparative graphs for z = 6, pz = 0.4, and q = 1 for periods of 100, 200, 300, and 400. First, we see that the log-log graphs, particularly the

456

21

Notes on Thick-Tailed Distributions of Wealth

Scenario 3 z = 6, pz = 0.4, q = 1 npds = 100 0 0

1

2

3

4

5

-0.5 -1

ln P(u > x)

-1.5 -2 -2.5 -3 -3.5 -4 -4.5

ln x

Fig. 21.2 Scenario 3 z = 6, pz = 0.4, q = 1 npds = 100

Scenario 3 z = 6, pz = 0.4, q = 1 (from left) npds = 100, 200, 300, 400 0 0

1

2

3

4

5

6

-1 -2

ln (u > x)

-3 -4 -5 -6 -7 -8

ln x

Fig. 21.3 Scenario 3 z = 6, pz = 0.4, q = 1 (from left) npds = 100, 200, 300, 400

7

21.6

Law of Pareto Tests

457

last three, are pretty much right-ward translations of one another. As in Fig. 21.2, the middle segments of the graphs are strikingly linear, and with what appears to be a common slope. The slope coefficients, t-ratios, and R2 s for linear least-squares regression lines fitted to these segments are given in Table 21.7. The slopes are seen to vary from −0.31 for the period of 100 to −0.51 for 200. The slopes for 300 and 400 are virtually the same as for 200.4 The t-ratios for the estimated slope coefficients are huge and R2 s are 0.9760 or higher. Table 21.7 Slopes of “middle” segments of graphs in Fig. 21.3 NPDS

Slope coefficient

t-ratio

R2

Interval for x

1−F(x)

100 200 300 400

−0.4091 −0.5120 −0.4945 −0.4861

−108 −82 −189 −174

0.9933 0.9760 0.9933 0.9896

1–92 35–191 32–277 49–370

0.256 0.189 0.196 0.190

Casual inspection of the slope coefficients for periods of 200, 300, and 400 leads to what would seem like an obvious conclusion that the slopes are simply random deviations from an underlying common value. However, a pooled regression shows that the slope coefficient for 200 periods differs from the slope coefficient for 300 periods by a t-ratio of −3.19, and that the slope for 200 periods differs from a common slope for 300 and 400 periods by a t-ratio of −4.49. (The t-ratio for the difference between 300 and 400 periods is 1.80.) However, in the present context, this difference, while highly significant statistically, seems of little practical importance. The results in Fig. 21.3 and Table 21.7 leave little question but that a distribution of wealth with a Pareto portion (i.e., a linear log-log segment, with slope greater than −0.2, in the upper tail) can emerge in the absence of a Pareto “prime mover.” However, whether one can conclude from this that the upper tail does in fact follow the law of Pareto is another matter. The column headed by “Interval for x” in Table 21.7 gives the “wealth” intervals over which the linear regressions are fitted, 12 through 92 units of wealth for the period of 100, 35 through 191 for 200, etc. The column headed by “1−F(x) gives the upper tails of the distribution of wealth that remains at the ends of the intervals—25.6% of agents have wealth greater than 92 units for the period of 100, and so on and so forth. Is 25% of the tail remaining (or about 20% in the case of periods of 200–400) sufficient to foreclose a Pareto law? The literature on thick-tailed distributions unfortunately does not provide much guidance to this question. Mandelbrot, who has pioneered statistical study of fattailed distributions of the Pareto-Levy type, has always seemed to be most concerned

4 If

a period is interpreted as a year, then periods of 2–400 should represent a plausible length of “history” for the distribution of wealth to reach an asymptotic form. The small size of the economy (1,000 agents) is not thought to be a problem because of the results presented in Table 21.4, which show the distribution to be reasonably independent of the number of agents.

458

21

Notes on Thick-Tailed Distributions of Wealth

about proper estimation of α. In a comment on the significance of the evidence provided by doubly logarithmic graphs, Mandelbrot had this to say: Limitations on the value of α lead to another quite different aspect of the general problem of observation. It concerns the practical significance of statements having only asymptotic validity. Indeed, to verify empirically the scaling distribution, the usual first step is to draw a doubly logarithmic graph: a plot of log10 [1−F(u)] as a function of log10 [u]. One should find that the graph is a straight line with the slope −α, or at least that this graph becomes straight as u increases. But, look closer at the sampling point of the largest u. Except for the distribution of incomes, one seldom has samples over 1000 or 2000 items; therefore, one seldom knows the value of u that is exceeded with the frequency 1−F(u) = 1000−1 or 2000−1 . That is, the “height” of the sampling doubly logarithmic graph will seldom exceed 3 units of the decimal logarithm of 1−F. The “width” of this graph will be at best equal to 3/α units of the decimal logarithm of u. However, if one wants to estimate reliably the value of the slope α, it is necessary that the width of the graph be close to 1 unit. In conclusion, one cannot trust any data that suggest that α is larger than 3. [Mandelbrot (1997, p. 87).]

Note that Mandelbrot does not say anything in this comment, nor anywhere else in his writings that I have been able to find, about how far into the tail of the distribution that one needs to be concerned with doubly logarithmic linearity. I say this, as an empirical matter, because of the fact that, since in any real-world circumstance we are necessarily dealing with distributions with a finite number of elements, there must eventually reach a point in the distribution where the density function must unceremoniously drop to zero (and stay there). Is this beyond 1 − F(x) = 1,000−1 or 2,000−1 , as (possibly) suggested by Mandelbrot, or might one take such a drop as beginning as early as 1 − F(x) = 0.20 (as in the graphs for periods of 200–400 in Fig. 21.3)? Before taking up a discussion of this question, it will be useful to consider a fourth scenario, in which the distribution of talents is assumed to be trinomial, rather than binomial. Specifically, this scenario allows for three categories of “productivity,” z, v, and 1, for z > v > 1, with probabilities pz, pv, and 1−pz−pv, respectively. Double-log graphs of ln P(u>x) and ln x corresponding to simulations for n = 1000 and periods of 100–400, for z = 5, v = 3, pz = 0.2, pv = 0.6, and q = 1, are given in Fig. 21.4. Comparison of the graphs in this figure with the graphs in Fig. 21.3 shows important similarities, but also an important difference. The similarities are: (1) the distributions for npds = 200–400 are essentially rightward translations of one another, and (2) there are again three reasonably well-defined linear segments. The important difference, however, is in the “middle” segment (for npds = 200–400), which occurs earlier in the distribution [between 1−F(x) equal to 0.40 and 0.68, as opposed to values of 0.55 and 0.80 for the distributions in Fig. 21.3], and now has a slope of about −0.33, as compared with slopes of about −0.5 for the comparable segments in Fig. 21.3. Thus, while the introduction of a “hump” in the distribution of talents retains a Pareto-like segment in the distribution of wealth, the point at which the segment stops, leaves a tail containing more than 30% of the mass of the distribution. This seems a bit too much to support any conclusion that the tails of the distributions in Fig. 21.4 are scaling.

21.6

Law of Pareto Tests

459 Scenario 4 z = 5, v = 3, pz = 0.2, pv = 0.6, q = 1 (from left) npds = 100, 200, 300, 400

0 0

1

2

3

4

5

6

7

-1 -2

ln P(u>x)

-3 -4 -5 -6 -7 -8

ln x

Fig. 21.4 Scenario 4 z = 5, v = 3, pz = 0.2, pv = 0.6, q = 1 (from left) npds = 100, 200, 300, 400

However, that they are thick-tailed is another matter, and in investigation of this, it is instructive to examine the behavior of the “spread” of the distributions represented in Fig. 21.4. This is done in Fig. 21.5, which presents graphs of the standard deviations for the distributions in Figs. 21.3 and 21.4. Of all the results presented to this point, this figure has to be the most dramatic,5 for, while a sequence of four elements hardly qualifies as determining asymptotic behavior, the two graphs are essentially perfectly linear, with slopes considerably greater than zero. In short, there is not even a hint of the standard deviation ever reaching an asymptote. An idea of how variances vary with the number of periods is given in Fig. 21.6, in which the variances divided by the number of periods for the distributions represented in Figs. 21.3 and 21.4 are plotted against the number of periods. From the upward slopes of these graphs—actually, there is even a hint of upward convexity!— we once again see no evidence of an eventual asymptote. The rightward translation of the distributions represented in Figs. 21.3 and 21.4 is obviously a reflection of increased mean holdings of wealth that occur with economic growth and heritability. This “aging” effect is quantified in Fig. 21.7, which shows the means of the wealth distributions from Figs. 21.3 and 21.4 plotted against the number of periods.

5 The point of this exercise is that, as discussed by Mandelbrot in his 1963 JPE paper, an implication of a distribution with an infinite variance is for sample variances to behave erratically as a function of increasing sample size, with no tendency to reach an asymptote.

460

21

Notes on Thick-Tailed Distributions of Wealth

Standard Deviations for Distributions in Figures 21.3 and 21.4

180 160

Standard Deviation

140 120 100 sd3 sd4

80 60 40 20 0 0

100

200 300 Number of Periods

400

500

Fig. 21.5 Standard deviations for distributions in Figs. 21.3 and 21.4

Variance/Number of Periods For Distributions in Figures 21.3 and 21.4 70 60

Variance/npds

50 40 var/npds3 var/npds4

30 20 10 0 0

100

200

300

400

500

Number of Periods

Fig. 21.6 Variance/number of periods for distributions in Figs. 21.3 and 21.4

21.7

Conclusions

461 Means for Distributions In Figures 21.3 and 21.4

200 180 160 140

Mean

120 100

m3 m4

80 60 40 20 0 0

100

200 300 Number of Periods

400

500

Fig. 21.7 Means for distributions in Figs. 21.3 and 21.4

21.7 Conclusions The purpose of this exercise has been to investigate whether a law of Pareto for the distributions of income and wealth can be obtained in the absence of assumptions concerning an initial Pareto “prime mover.” Three scenarios involving highly stylized, artificial economies (all of which can be given a Darwinian interpretation) have been simulated under varying assumptions regarding heritability, distribution of talents, and stability of tastes. Simulations with the first two scenarios make it pretty clear that talent differentials and pure randomness of tastes cannot suffice to produce wealth distributions with sufficient thickness to be interesting. However, things change in the third scenario, in which there is an allowance for preference stability, in the sense that once an agent experiences a good, that good is consumed with a nonzero probability in subsequent periods (so long, of course, as the agent remains “alive”). What the results with the third scenario show is that, with strong preference stability and substantial productivity differences among agents, thicktailed distributions of wealth can emerge that have certain Pareto features and are log-log translatable. How Pareto-like these distributions actually are, on the other hand, is another matter. Simulations with a fourth scenario, which allows for a more realistic distribution of talents across agents (but which retains extreme “habit formation”), yields distributions that remain log-log translatable and “segment” scalable. However, the “Pareto” segment of these distributions now leaves a tail that contains more than 30% of the mass of the distribution. Nevertheless, Mandelbrot-type exercises

462

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Notes on Thick-Tailed Distributions of Wealth

involving stability of variances suggest —indeed, convincingly in my view—that the distributions in question, whatever they might be, do not have a finite variance. Obviously, the results of these exercises cannot settle the question of whether income and wealth distributions with the law of Pareto upper tail can arise through normal birth, death, production, consumption, talent and taste differences, and heritability processes in the absence of an initial Pareto “prime mover.” Nevertheless, the results are clear in suggesting that presence of the latter may not be necessary. While the exercises reported here embody what are thought to be plausible mechanisms concerning the distribution of talents and the formation of preference stability, heritability in the scenarios has been limited to agents keeping their stocks of wealth from one period to the next (so long, that is, as they survive). Heritability involving talents and tastes within a “family” is a scenario waiting to be explored. Adding such assumptions would seem almost certainly to increase the probability of “super large” fortunes, and could lead to a softening of the sharp “kinks” in the log-log graphs in Figs. 21.3 and 21.4 that are evident at values of 1−F(x) of about 0.20 and 0.30, respectively.6 However, I feel that, when all is said and done, the strongest conclusion arising from the exercises of this paper is the uncovering of what would appear to be a powerful link between large fortunes and strong preference stability. Preference stability is a feature of the real world,7 as are large fortunes. A strong link between them would seem to open up an area of promising new research.

6

Also, as has been suggested to me by Barbara Sands, another interesting exercise would be to eliminate the restriction that agents with negative wealth are removed the instant that their wealth turns negative. Allowing them to continue living, at least for a few periods, might give rise to the “lower hump” that is characteristic of real-world income and wealth distributions. 7 See Chapters 5 and 20.

Chapter 22

Conic Distributions of Earned Incomes

22.1 The Search for Functional Form Since Pareto (1897) discovered the functional form of the distribution of high incomes efforts have been made to find a form that covers the entire income range. As was true of Pareto’s own work, most of these efforts were descriptive in the sense of looking for a distribution that would fit a set of data believed to be representative. Important steps in the search for appropriate distributions were the introduction of the generalized gamma for this specific purpose (Amoroso, 1925) and the application of the log normal (Gibrat, 1931) and stable distributions (Mandelbrot, 1960 Zolotarev, 1986). An empirical comparison of the large family of distributions by McDonald (1984) served to clarify their respective merits; thus, he reported poor performance of the log normal. McDonald also came up with a new contender, the generalized beta of the second kind, which will be discussed in Section 22.5.2. His comparison is not entirely conclusive, however, because he used a limited set of data and did not consider stable distributions. Mandelbrot (1962) has presented an interesting theoretical model of earnings based on stable distributions, and he has also made an important empirical observation to be discussed more fully in Section 22.2. In probability theory, stable distributions have attracted attention because they arise from general limit theorems concerning the addition of random variables. This is also the main reason why they have attracted interest in economics and many other disciplines.

I am indebted to Joshua Angrist, Richard Barakat, Gary Chamberlain, Herman Chernoff, Richard Freeman, Lawrence Katz, Benoit Mandelbrot, James Medoff, Donald Rubin, and Hal Stern for helpful conversations, and Ivor Frischknecht, Robert Plunkett, and Tycho Stahl for excellent research assistance. They are not responsible for any defects in the paper. The Harvard Institute for Economic Research provided financial support.

L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_22, 

463

464

22

Conic Distributions of Earned Incomes

There are two major obstacles to the empirical application of stable distributions (except for normal, which is not of interest in the present context). The main obstacle is that their characteristic coefficient (Pareto’s α) cannot exceed two. Although Pareto in his original data found this coefficient to be about 1.5, subsequent studies have generally arrived at larger values. There may be theoretical reasons for believing that α has a maximum of two, but contemporary income data suggest otherwise. This implies that the second and higher moments may exist. The other obstacle to the use of stable distributions is that some of their most basic properties are unknown. Except for a few special cases, neither the density nor the cumulative distribution function is available in closed form. The median and other order statistics are unknown in the asymmetric case. The conic distributions introduced in Section 22.2 resemble stable distributions to the extent that the logarithmic slope of the cumulative distribution function (cdf) approaches a negative constant asymptotically on the right, but this constant is not constrained by any particular minimum. In this respect they resemble the “fractal” or “hyperbolic” distributions emphasized by Mandelbrot (1977) in his more recent work. In that book, however, the term “hyperbolic” denotes any and all distributions with Pareto-type behavior in the upper tail, whereas the conic distributions can be defined unambiguously. The conic family of distributions presented in this paper belongs to the hyperbolic class and is apparently the only known member that can be expressed in closed form. While the conic distributions satisfy Pareto’s law for high incomes (except possibly for the size of the coefficient), this law tells us very little about the distribution of low and middle incomes. It is mostly by ability to fit the low and middle range, where the vast majority of income recipients are found, that alternative functional forms have to be judged. The empirical results in Sections 22.5.2 and 22.7 show that the two special cases of the conic family, called the “conic-linear” and the “conic-quadratic,” respectively, perform very well.1 Section 22.7 also shows that these distributions have another remarkable property: they account both for the overall distribution of earnings and for the distribution with narrowly defined cells obtained by cross-classification with respect to gender, age, education, and occupation. As an example, let F1 be a conic distribution describing the earnings of men, and F2 a distribution of the same type (with different parameters) describing the earnings of women. Furthermore, let F0 be the distribution of earnings among all persons, and w1 the proportion of males. Then, it appears that, to a high degree of approximation, F0 = w1 F1 + (1 − w1 ) F2 ,

(22.1)

1 As I hope to show elsewhere, these distributions are also useful in the analysis of daily price changes in financial markets.

22.2

Specification of the Conic Family of Distributions

465

where F0 has the same functional form as F1 and F2 . In other words, the all-persons distribution is not merely a finite mixture of the male and female distributions, but the former is of the same type as the latter. The conic distributions, therefore, appear to be (at least approximately) “self-mixable.”2 This is just an empirical observation, for which no theoretical explanation can be offered as yet. Since the distribution within narrow cells has had little attention previously, it is not clear whether the conic distributions are unique in this regard. The empirical advantages of the conic distributions come at quite moderate cost. Section 22.4 shows that the density and cdf can be expressed in terms of elementary functions, as can the median and other order statistics. The moments and the Gini coefficient are more difficult to derive, but can also be expressed in terms of known functions. Estimation of the parameters by maximum likelihood, discussed in Section 22.5, is fairly straightforward.

22.2 Specification of the Conic Family of Distributions The conic distributions implement, at least in part, a suggestion derived by Mandelbrot (in an unpublished sequel to his 1960 paper) from inspection of the worksheets underlying Houthakker (1959), which were in turn compiled from the 1950 census of population. Letvdenote income and F(v) the cumulative distribution function of income; also: V = ln v, U = ln {1 − F (v)} ≤ 0

(22.2)

Then Mandelbrot suggests that, when U is plotted as a function of V, it describes one branch of a hyperbola, one of whose asymptotes goes to the lower right with a slope of −α.3 The name of this parameter recalls the Pareto distribution, which the conic distributions resemble at their right-hand extremes, though not anywhere else. These distributions may be viewed as generalizations of the Pareto, where the relation between U and V is linear. Since there are other generalizations (see Johnson and Kotz, 1970, Chapter 19), it is better to avoid the term “generalized Pareto.” The conic distributions do not appear to be related to any standard distribution other than the Pareto.

2

This property recalls Mandelbrot’s concept of “self-similarity,” which plays a central role in his work. Self-similarity, however, requires that the micro and macro distributions agree in their parameters; this is not necessary for self-mixability. 3 In addition, Mandelbrot suggests that the plot of U against V has a point of inflection for some large value of V. Inflection is apparently required for the distribution to be stable, but in the data there is not much evidence that such a point exists. Since the possibility of inflection would complicate the analysis without improving the fit, it is not considered here. The conic distributions are in effect approximations to the (asymmetric) stable distribution that ignore the inflection.

466

0

22

+

+ ++ +

0 +

Conic Distributions of Earned Incomes

+

+ + ++ +

+ +

+ +

+ -1

+

-2 +

+ -2

+

-4

+

-3

+ + -6

+ -4 0 1 2 3 4 5 0 Note: Men are on the left, women on the right.

+

1

2

3

4

5

Fig. 22.1 Mandelbrot Plots for 1989 Earninigs of All Men and Women (18 and over)

Figure 22.2.1 presents two examples of “Mandelbrot Plots” using data from the March 1990 current population survey (see Section 6). The asymptotic behavior on the right-hand side is unmistakable for both sexes; the behavior on the left is also asymptotic, and will be discussed in Section 3.1.

22.2.1 The General Conic Distribution Mandelbrot’s suggestion is implemented mathematically by first writing the general equation of a conic section in U and V: c0 U 2 + 2c1 UV + c2 V 2 + 2c3 U + 2c4 V + c5 = 0.

(22.3)

From here on, it will be assumed that c0 = 0, which means that, without loss of generality, it can be equal to 1. With this slight modification, the coefficients in expression (22.3)—called the conic equation hereafter—must reflect four conditions ensuring that F(v) is a distribution function. Condition I requires that F (v0 ) = 0,

22.2

Specification of the Conic Family of Distributions

467

where v0 is the lower limit of v, which clearly must be positive.4 To translate this in terms of U and V, we note that, from equation (22.2), F(v) can be written as F (v) = 1 − exp {U (V)} ,

(22.4)

U (v0 ) = 0.

(22.5)

from which it follows that

Inserting this into equation (22.3) then gives c2 V02 + 2c4 V0 + c5 = 0,

(22.6)

which will have a real solution if c24 ≥ c2 c5 .

(22.7)

That solution is V0 =

−c4 ±

 c24 − c2 c5 c2

.

(22.8)

Condition II is that F(v) must exist for v ≥ v0 . From (22.4), this is clearly equivalent to U being real. The solution of equation (22.3) for U is U = −(c1 V + c3 ) ±



(c1 V + c3 )2 − (c2 V 2 + 2c4 V + c5 ).

(22.9)

For U to be real, the discriminant u = (c1 V + c3 )2 − (c2 V 2 + 2c4 V + c5 ) = (c21 − c2 )V 2 + 2(c1 c3 − c4 )V + c23 − c5

(22.10)

must be nonnegative for V ≥ V0 . Differentiation of u with respect to V shows that it attains an extremum when V=

c4 − c1 c3 c21 − c2

(22.11)

must be nonnegative for V ≥ V 0 . The second derivative of  u with respect to V is 2(c21 − c2 ),

(22.12)

4 If this condition is not satisfied U is a rational function of V, but the resulting distribution function does not appear to be capable of describing actual incomes or earnings and will not be considered further.

468

22

Conic Distributions of Earned Incomes

which must be positive for the extremum to be a minimum. Substituting equation (22.11) into equation (22.10), we find (after simplification and some rearrangement): ∗u = = =

−c24 + 2c1 c3 c4 − c2 c23 − c21 c5 + c2 c5 c2 c5 − c24

c21 − c2 − c1 (c1 c5 − c4 c3 ) + c3 (c1 c4 − c2 c3 )

(22.13)

c21 − c2

C , c21 −c2

where:    1 c1 c3    C =  c1 c2 c4  .  c3 c4 c5 

(22.14)

The minimum of the discriminant u (as a function of V) is thus seen to be proportional to the determinant C, which (since c21 − c2 must be positive) must be nonnegative in order for V0 to be real.5 The appearance of the determinant in (22.14) is not surprising, since the conic equation (22.3), with c0 = 1, can be written in matrix form as

⎡

⎤⎡ ⎤ 1 c1 c3 U U V 1 ⎣ c1 c2 c4 ⎦ ⎣ V ⎦ = 0. 1 c3 c4 c5

(22.16)

This further implies that the determinant in equation (22.14) must in fact be positive, not merely nonnegative. For, if it vanished, the left-hand side of equation (22.16) would be a product of two linear factors, and the distribution would be Pareto, rather than conic. Condition III is that F(v) must equal 1 at the upper limit of v, which is assumed to be infinite. Consequently, we must have [from (22.14)] lim U = ∞.

V→∞

(22.17)

It can be verified from (22.9) that this condition is satisfied if 5

It is possible, however, that the minimizing value of V given by equation (22.11) is less than V0 . In that case, it can be shown that equation (22.14) should be replaced by    1 c1 c3    ∗ C =  c1 c2 c4  , (22.15)  0 1 −V0  which again should be nonnegative.

22.2

Specification of the Conic Family of Distributions

− c1 ±



c21 − c2 < 0.

469

(22.18)

For the inequality to be meaningful, we accordingly must have c21 > c2 ,

(22.19)

which has already been found to be required in connection with equation (22.12). In the present context, it may be assumed that c1 > 0, which means that the ambiguous sign in equations (22.9) and 22.18) must be negative, and that equation (22.18) is satisfied. Condition IV requires the density f(v) to be nonnegative for v > v0 . This density function can be expressed in terms of U and V by noting [from the definitions of U and V in equation (22.2)] that uf (v) dU = dV 1 − F(v) = − exp{V − U(V)} f (v),

U  (V) =

(22.20)

hence, f (v) = −U  (V) exp{U(V) − V}.

(22.21)

Condition IV thus reduces to U  (V) < 0.

(22.22)

This derivative can be obtained from equation (22.3): U  (V) = −

c1 U + c2 V + c4 . U + c1 V + c3

(22.23)

The denominator (including the negative sign) in this expression is equal to the square-root term in equation (22.9) and thus is negative. Consequently, Condition IV accordingly requires the numerator to be negative as well, which, in turn, implies [using equation (22.5)] that c2 V0 + c4 ≤ 0.

(22.24)

Unfortunately, it does not appear that expression (22.23) lends itself easily to translating Condition IV into more general restrictions on the parameters. Instead, it is convenient to introduce an alternative to Condition IV, namely, Condition IVa: For F ≥ 0, F determines v, or equivalently, for U ≤ 0, U determines V. This condition is stronger than Condition IV because it excludes the possibility that the density vanishes for some v > vo .6 It may be noted in passing that Conditions 6 If the density did vanish at an interior point, the distribution would break up into two or more non-overlapping segments (the “rich” and the “poor,” if there were two segments). Conceivably, this pattern may have existed in feudal societies, but it is not of contemporary interest.

470

22

Conic Distributions of Earned Incomes

II and IVa set up a one-to-one correspondence between U and V. To implement Condition IVa, we note that the conic equation can be written c2 V 2 + 2(c1 U + c4 )V + U 2 + 2c3 U + c5 = 0.

(22.25)

The case where c2 = 0 leads to a linear equation in V; it will be disregarded until Section 22.2.3. When equation (22.25) is quadratic, the argument is similar to that applied to condition II, but there are some differences, so it will be restated. Here, the discriminant v is equal to v = 4[(c21 − c2 )U 2 + 2(c1 c4 − c2 c3 )U + c24 − c2 c5 ].

(22.26)

For U = 0, v must be nonnegative, since otherwise the lower limit V0 would not exist. Consequently, we must have: c24 − c2 c5 ≥ 0,

(22.27)

in agreement with equation (22.7). Since the derivative of  v with respect to U is proportional to (c21 − c2 )U + c1 c4 − c2 c3 ,

(22.28)

the discriminant accordingly reaches an extremum when U=−

c1 c4 − c2 c3 , c21 − c2

(22.29)

provided U ≤ 0. The case where this condition fails will be discussed below. Since the denominator is positive according to equation (22.19), it is sufficient that c1 c4 − c2 c3 ≥ 0.

(22.30)

It is clear from equation (2.19) that the second derivative of v [c21 − c2 ] is positive, so that the extremum is a minimum. For Condition IVa to hold, this minimum must be positive. By substituting equation (22.29) into equation (22.26), it can be shown, after some algebraic manipulations, that this minimum is equal to ∗v =

c2 |C| , c21 − c2

(22.31)

where |C| is the determinant of equation (22.14), which [in light of equation (22.16)] must be positive. Since the denominator is positive by virtue of equation (22.19), the sign of ∗v accordingly turns on the sign of c2 . Since it is assumed that c2 = 0, it would seem to follow that c2 > 0. However, this is not so, for it is quite possible that c2 < 0, in which case the proviso following (22.27) becomes operative. When c2 < 0, equation (22.27) still holds, so v is nonnegative for U = 0. For v to be nonnegative at all negative values of U, its derivative [which is proportional

22.2

Specification of the Conic Family of Distributions

471

to the expression in (22.28)] must remain negative when U decreases. When U is close to zero, the derivative is dominated by the constant term, which must therefore be negative, contrary to its sign in equation (22.30). Since the coefficient of U in equation (22.28) is positive according to equation (22.19), the derivative will remain negative for all U < 0. It is therefore convenient to rewrite equation (22.30) as c2 (c1 c4 − c2 c3 ) ≥ 0,

(22.32)

which covers all possibilities concerning the sign of c2 . These equations and inequalities, unless noted otherwise, hold for both variants of the conic distribution that will now be distinguished. The distinction between them hinges on the behavior of the density at the lower limit of V. For reasons that will become clear in a moment, the variants are called “conic-quadratic” (CQ) and “conic-linear” (CL) distributions, respectively. The general conic distribution discussed so far will be denoted by CG.

22.2.2 The Conic-Quadratic Distribution With the CQ distribution, it is assumed that the density vanishes at the lower end: f (vQ ) = 0,

(22.33)

where the subscript is changed from 0 to distinguish CQ from CL. This condition can be translated into U and V by using equation (22.21): U  (VQ ) = 0,

(22.34)

where according to equation (22.23):7 U  (VQ ) = −

c2 VQ + c4 = 0. c1 VQ + c3

(22.35)

Consequently, c4 = −c2 VQ ,

(22.36)

c1 VQ + c3 = 0.

(22.37)

provided that

Inserting equation (22.35) into equation (22.6) then yields c5 = c2 VQ2 . 7

Note that UQ = 0 from Condition I.

(22.38)

472

22

Conic Distributions of Earned Incomes

Also, substituting equation (22.36) for c2 VQ in equation (22.8) in turn implies that c2 c5 = c24 ,

(22.39)

in accordance with equation (22.27). The conic equation (22.3) then simplifies to: U 2 + 2c1 UV + c2 (V − VQ )2 + 2c3 U = 0.

(22.40)

This expression defines the conic-quadratic distribution; it is called “quadratic” because the highest power of V is two, which it will be if c2 = 0. Failure of this condition leads to the CL distribution, which will be discussed in Section 22.2.3. Expression (22.40) can be solved for U as follows: U(V) = −(c1 V + c3 ) −

 (c1 V + c3 )2 − c2 (V − VQ )2 ,

(22.41)

where an earlier conclusion concerning the sign of the square-root term has been invoked.8 For this solution to be meaningful, the expression under the square-root sign must be nonnegative. Considered as an equation in V, therefore, its discriminant must be positive. Inspection of this discriminant shows that it will be positive for all values of V > VQ only if c2 < 0.

(22.42)

To ensure that this restriction is satisfied, the CQ distribution will be estimated in a form that is slightly different from equation (22.40), by putting c2 = −b22 ,

(22.43)

where b2 is estimated in lieu of c2 . In view of equation (22.42), the expression under the square-root sign in equation (22.41) can be written as the sum of two squares. The square root of that expression will thus exceed the first term of equation (22.41) whenever V is above its lower limit.9 In the applications of interest here, V can range from a large negative number to a large positive. As shown earlier, the definition of U implies that10 U (V) < 0 and U (V) < 0 for V > VQ .

(22.44)

These inequalities ensure that 8

See the discussion following expression (22.19). The two squares collapse into a single one if and only if equation (22.39) is false, in which case the conic-quadratic distribution degenerates to the Pareto. 10 See equations (22.2) and (22.20). 9

22.2

Specification of the Conic Family of Distributions

f (v) > 0 for vQ < v < ∞.

473

(22.45)

Let us now look more closely at the behavior of F and f when v goes to infinity. From the definition of U, we have: F (∞) = 1 implies U (∞) = −∞.

(22.46)

Also, equation (22.41) entails that, for large values of V, U is proportional to V, with a proportionality factor equal to − c1 −



c21 − c2 ,

(22.47)

which is necessarily negative. Next, consider Mandelbrot’s suggestion that lim U (V) = α.

V→∞

(22.48)

By manipulation of equation (22.23), it can be shown that U (V) approaches equation (22.47) as V goes to infinity.11 Hence, we can write α = c1 +

 c21 − c2 ,

(22.49)

a relation that is useful in interpreting the empirical results. Yet another condition on the conic distribution results from postulating that the density vanishes at the extreme right, that is, f(∞) = 0, which in turn implies [from equation (22.21)] that:

lim −U (V) exp {U (V) + V} = 0.

V→∞

(22.50)

According to equations (22.48) and (22.49), U (V) remains finite for large V, so that it suffices to show that: lim [U (V) + V] = −∞.

V→∞

(22.51)

This will be the case if the proportionality factor equation (22.47) is less than −1, which is equivalent to α > 1.

(22.52)

In terms of the parameters in equation (22.49), this means that we must have 11 Divide the denominator into the numerator and then take the limit as V gets large, using l’Hospital’s rule.

474

22

Conic Distributions of Earned Incomes

2c1 − c2 > 1.

(22.53)

This condition will certainly hold if c1 ≥ 1/2, but it may also be satisfied by smaller values; in fact, it does not exclude values of c1 that are negative. This last possibility, however, gives rise to certain anomalies and is never encountered in applications to earned income. It will therefore be assumed that c1 > 0.

(22.54)

Finally, there is a condition that can be derived by solving equation (22.40) for V, rather than for U:  c1 U ± (c21 − c2 )U 2 − 2c2 (c1 VQ + c3 )U V(U) = VQ − . (22.55) c2 For the solution to be meaningful, the expression under the square-root sign must once again be positive for U < 0, which means (since c2 < 0) that we must have (c21 − c2 )U < (c1 VQ + c3 ).

(22.56)

For this inequality to hold, it is sufficient that c1 VQ + c3 > 0.

(22.57)

This condition, which supersedes equation (22.37), is also necessary, since U can be arbitrarily close to zero. Moreover, the ambiguous sign in equation (22.55) must in fact be positive in order to ensure that V everywhere has the correct sign.

22.2.3 The Conic-Linear Distribution As mentioned already, the CL distribution corresponds to the case c2 = 0,

(22.58)

in which case the conic equation (22.5) becomes: U 2 + 2c1 UV + 2c3 U + 2c4 V + c5 = 0,

(22.59)

where the terms involving U are the same as in equation (22.39). The CL equation, however, is linear in V provided c 1 = 0.12 If so, then 12 The cases where either condition does not hold are not of interest in the applications addressed in this paper, and will not be pursued. They could be relevant to other applications.

22.2

Specification of the Conic Family of Distributions

V=−

U 2 + 2c3 U + c5 . 2(c1 U + c4 )

475

(22.60)

There must again be a lower limit to V, which will be called VL since it need not be the same as for CQ. Because of equation (22.58), equation (22.6) leads to VL = −

c5 . 2c4

(22.61)

It is convenient to eliminate c 5 by writing c5 = −2c4 VL .

(22.62)

The solution of equation (22.59) is then U(V) = −(c1 V + c3 ) −



(c1 V + c3 )2 − 2c4 (V − VL ).

(22.63)

Important inequalities can be obtained by giving special values to V in equation (22.63): U(VL ) = −(c1 VL + c3 ) − |c1 VL + c3 | ,

(22.64)

 U(− c3 /c1 ) = − (2c3 c4/ /c1 ) + 2c4 VL .

(22.65)

and

The second of these equations is conditional on − c3 /c1 > VL ,

(22.66)

since otherwise U would not be real (and negative). Furthermore, the right-hand side of equation (22.64) should be zero, and the right-hand side of equation (22.65) should be negative. All of these conditions will be satisfied if: c1 > 0, c1 VL + c3 < 0, c4 < 0,

(22.67)

all of which will be assumed to hold. We also have U  (V) = −

c1 U + c4 , U + c1 V + c3

(22.68)

c4 , c1 VL + c3

(22.69)

which implies that U  (VL ) = −

476

22

Conic Distributions of Earned Incomes

which is negative, as it should be, by virtue of equation (22.67). For the CL distribution, therefore, the density at the lower limit of v does not vanish. At first sight, this may appear a blemish compared with the CQ distribution (where the density does vanish), but the results of Section 22.7 suggest that this apparent blemish is on balance an advantage of the CL distribution.13

22.3 Geometric Aspects In this section, the geometry of CQ and CL will be explored. First, the asymptotes of their distribution functions will be derived, and an alternative way of formulating CQ and CL will be presented. Then, the density functions will be analyzed in terms of their mode(s) and inflections.

22.3.1 The Asymptotes Let us first consider CQ. According to equation (22.19), α = c1 +

 c21 − c2 .

(22.70)

It is also convenient to put: β = c2 /α < 0,γ = −2(c1 VQ + c3 ) > 0,W = V − VQ > 0,

(22.71)

which implies: c1 = (α + β)/2,c2 = αβ,c3 = γ + (α + β)VQ /2.

(22.72)

After some manipulation, the conic equation (22.40) for CQ then simplifies to (U + αW)(U + βW) = γ U.

(22.73)

13 LDT note: At this point in the paper, HSH intended to include a figure showing the effect on the shape of the CQ density of different values of c1 . (The values used for the other parameters were approximately those estimated for all men over 18 who reported earnings in 1979.) Although he had obviously constructed the figure at some point, it (as with the figure noted in Footnote 4 above) was missing from the copy from which this chapter is taken. His discussion of the figure is as follows: For the middle and large values of c1 , most of the distribution has the expected shape, but in the lower tail there is a second mode. Since CQ is defined as having a density of zero at the lower limit of v, this second mode is internal [i.e., an artifact of the functional form]. For the smaller value of c1 , the only mode is at the lower end. The question of modality, which the Achilles heel of CQ (and also of CL), is discussed further in Section 22.5.3; it does not appear to affect the goodness of fit.

22.3 Geometric Aspects

477

This equation confirms that −α is the slope of the “Pareto asymptote” and also shows that the other asymptote (about which nothing has been said so far) has a slope of −β.14 For CQ, the latter slope is positive. Further insight into the geometry is provided by determining the intercepts in the equations for the two asymptotes: U = α0 − αW

(22.74)

U = β0 + βW.

(22.75)

To find the intercept in the first of these equations, for instance, we can insert it into equation (22.72), resulting in: {αγ − α0 (α − β)}W = α0 (γ − α0 ).

(22.76)

There is no finite solution to this equation. Differentiation with respect to W, however, yields αγ − α0 (α − β) = 0,

(22.77)

hence, α0 =

αγ . α−β

(22.77a)

Since α and γ are positive, and α is greater than β, it follows that α 0 is positive. A similar argument shows that β0 = −

βγ , α−β

(22.78)

which, since β is negative and γ positive, is also positive. The two asymptotes intersect where 2αβγ (α − β)2

(22.79)

(α + β)γ . (α − β)2

(22.80)

U0 = − and W0 =

14 Confirmation that the slope of the upper asymptote is −α is obtained by dividing expression (22.73) by (U + βW) and taking limits of the right-hand side as W gets large. That the slope of the lower asymptote (as W goes to zero) is −β is obtained similarly (though through the use of l’Hospital’s rule).

478

22

Conic Distributions of Earned Incomes

In the region above the horizontal axis, the function U(V) is of course, strictly speaking, not defined. The preceding argument is easily extended to the general conic CG, for which the definitions in equations (22.70) and (22.71) continue to apply, except that VQ is replaced by V0 . However, in addition, we need δ = −2(c2 V0 + c4 ),

(22.81)

(U + αW)(U + βW) = γ U + δV,

(22.82)

so that CG can be written as

which, unlike equation (22.73), is symmetric in U and V. The interested reader will have no difficulty in deriving the corresponding asymptotes. The alternative form for CL can be derived from equation (22.82) by observing that β = 0, so that we get U(U + αW) = γ U + δV,

(22.83)

where now, however, δ is reduced to −c4 , which is positive according to equation (22.67). In conjunction with β = 0, this means that the “second” asymptote, defined by equation (22.75) is above and parallel to the horizontal axis.

22.3.2 Modes and Inflections In the interval [v0 , ∞], the CQ and CL distributions can have zero, one, or more than one modes; accordingly, they can also have one or more “reverse” (or anti-) modes. In the case of CL one of the modes may be at the lower limit of earnings. Since the CQ and CL distributions are differentiable functions, their local maxima or minima are given by f  (v) = 0,v ≥ v0 ,

(22.84)

f  (v) = 0.

(22.85)

provided that

In terms of U and V, the first condition [from equation (22.14)] translates to f  (v) = − exp (U − 2 V){U  (U  − 1) + U  } = 0,

(22.86)

where U is given by equation (22.23) and the argument V has been suppressed. For the general conic, it can also be shown that U  = −

U  (U  + 2c1 ) + c2 . U + c1 V + c3

(22.87)

22.4 Descriptive Statistics

479

Since the exponential factor on the right-hand side of equation (22.86) does not vanish, the necessary condition for a mode or anti-mode is U  = −U  (U  − 1).

(22.88)

This will be called the “modal equation.” Expressing this equation in terms of U and V in order to determine the number and location of the modes and anti-modes calls for formidable algebra; in practice, graphical methods may be more convenient. As noted earlier, the possible multiplicity of modes is an undesirable feature of these distributions, but it appears that empirically any “excess” modes occur only for small values of V, where they do not affect the goodness of fit. Points of inflection in the density function also provide insight into the geometry of the conic distribution. At such points, the second derivative of the density function vanishes. From equation (22.86), this rather messy derivative is given by f  (v) = −e−U−3 V Z,

(22.89)

where: Z = (U  − 2)[U  (U  − 1) + U  ] + U  (2U  − 1) + U 

(22.90)

and U  = −3U 

U  + c1 . U + c1 V + c3

(22.91)

22.4 Descriptive Statistics In this section, formulae for the more important descriptive statistics of the conic distributions will be derived. The conditions under which these distributions possess higher moments will be investigated in the appendix. The Lorenz curve corresponding to the CL distribution will also be derived in this section.

22.4.1 The Median The definition of U(V) implies that the logarithm of the median (lmd) is given by U (lmd) = −ln (2) .

(22.92)

For CQ, substitution into equation (22.25) then yields:

lmd = VQ −

c1 ln (2) +



(c21 − c2 ) ln2 (2) + 2c2 (c1 VQ + c3 ) ln (2) c2

.

(22.93)

480

22

Conic Distributions of Earned Incomes

For CL, we get from equation (22.60): lmd =

ln2 (2) − 2c3 ln (2) + c5 . 2(c1 ln (2) − c4 )

(22.94)

The median (Md) itself can then be found by taking exponentials. As will be seen in Section 22.5, the existence of a relatively simple formula for the median is useful in estimation. Similar formulae hold for quartiles and other percentiles.

22.4.2 The Arithmetic Moments and Gini Coefficient The qth (arithmetic) moment around zero, called μq , can be written with the aid of equation (22.21) as: μq =



∞



∞

vq f (v)dv = −

v0

eU(V)+qV U  (V)dV =



V0

0

−∞

eU(V)+qV(U) dU (22.95)

provided the integral exists. The question of existence is discussed in the appendix. This integral is a special case of a more general one evaluated there. Since the Gini coefficient (G) is another special case, it will be derived first. It is convenient to express G in a formula derived by Dorfman (1979): G=

1 − μ−1 1



v2

[1 − F(v)]2 dv,

(22.96)

v1

where μ1 is the mean, while v1 and v2 are the lower and upper bounds of the domain of v, here equal to v1 and ∞, respectively. From the definition of U in equation (22.2), the integral in equation (22.96) can be written as 

v2

v1

 (1 − F(v)) dv = 2

∞

 2U(v)

e v0

dv =

∞

 2U(V)+V

e V0

dV = 2

0 −∞

e2U+V(U) dU. (22.97)

22.4.3 The Logarithmic Moments: Alternative Measures of Inequality Although less important than the arithmetic mean, the geometric mean is of interest for two reasons: (1) the conic distribution is expressed in terms of the logarithm of the variate; and (2) a measure of inequality proposed by Atkinson (1970), involving the geometric mean, is more closely related to economic theory than the Gini coefficient. Let us first look at the logarithmic mean λ1 , which can be defined, and written in standard notation, as

22.4 Descriptive Statistics

481



∞

λ1 =

 ln vdF(v) =

v0

0 −∞

VeV dU.

(22.98)

For CL, V is again given by equation (22.60), which can be more conveniently expressed as: 1 V=− 2 2c1



 c21 c5 − c4 B c1 U + B + , c1 U + c4

(22.99)

where B = 2c1 c3 − c4 . Consequently, equation (22.98) becomes λ1 = −

1 2c1



0 −∞

UeU dU −

B 2c21



0 −∞

eU dU −

c21 c5 − c4 B



2c21

0

−∞

eU dU . c1 U + c4 (22.100)

The first two integrals are equal to −1 and 1, respectively, while the last integral can be evaluated using Gradshteyn and Ryzhik (1965, 3.352). It is equal to: −c4 /c1 Ei(c4 /c1 ), c−1 1 e

(22.101)

where Ei(·) is the exponential integral function. Accordingly, it follows that   c21 c5 − c4 B −c4 /c1 1 λ1 = 2 B − c1 + e Ei(c4 /c1 ) . c1 2c1

(22.102)

The geometric mean (gm) is equal to exp(λ1 ). Atkinson’s measure of inequality is defined as one minus the ratio of the geometric mean to the arithmetic mean:15 A=1−

gm . μ1

(22.103)

The logarithmic variance λ2 is also occasionally used as an index of inequality. Its derivation for CL is similar to that of λ1 , but involves some additional integrals similar to equation (22.101) and available from the same source. The result is: λ2 =

15

c41 c4 + (c1 − c4 E∗ )(2c1 c4 C − C2 ) + 2c21 c4 CE∗ − c4 (CE∗ )2 4c61 c4

,

(22.104)

This is one of the three indices of inequality proposed by Atkinson (1970); the other two depend upon parameters that are difficult—perhaps impossible—to estimate. The index used here equals the fraction by which aggregate income could be reduced, while keeping aggregate utility constant, if that income were distributed equally among all recipients. From the viewpoint of economic theory, this is a more meaningful measure of income equality than the Gini coefficient. The validity of Atkinson’s index, however, depends upon three questionable assumptions: (1) that individual utilities can be aggregated; (2) that the utility function is Bernoullian; and (3) that output remains unchanged under an egalitarian redistribution of income.

482

22

Conic Distributions of Earned Incomes

where C is the determinant defined in expression (22.14) and E∗ = exp (−c4 /c1 )Ei(−c4 /c1 ).

(22.105)

22.4.4 The Lorenz Curve The three measures of inequality discussed above may give conflicting indications as to the movement of inequality over time. In fact, Section 22.8 will show that this happens in the CPS data for individual earnings. Better understanding is sometimes provided by the Lorenz curve (F), which represents the share in total income of those recipients whose income does not exceed a given limit. Accordingly, let F(v) be the cumulative distribution of income; then: 1 (F) = μ1



v

vf (v)dv,

(22.106)

v0

where μ1 is mean income as before. In the notation of equation (22.2), the integral in equation (22.106) can be written as  (F) =

0

eU+V dU,

(22.107)

Ul

where U l is some value of ln(1–F). The last integral can be evaluated by the method described in the appendix, but this will be left to another occasion. It follows from equation (22.106) that the curve (F) extends from 0 to 1, as does its argument, and that its first and second derivatives are both positive. If two such curves have no points in common (apart from 0 and 1), one of them unambiguously indicates greater inequality than the other. If they intersect, nothing can be said without introducing further assumptions. The Lorenz curve is also of interest because it permits estimation of the conic parameters when the data, as is not uncommon, take the form of income shares corresponding to certain quantiles, rather than of frequency distributions. This is not the case with the data described in Section 22.6, but there exist important data sets that are expressed only in terms of income shares, particularly those analyzed by Van Ginneken and Park (1984).

22.5 Estimation Maximum-likelihood estimation of the CQ and CL distributions is fairly straightforward since the interval frequencies are explicit functions of the four parameters. These functions are given, respectively, by equations (22.41) and (22.63), both in conjunction with equation (22.2). The log-likelihood function was maximized by Newton’s second-order method, with automatic derivative evaluation (Kalaba and

22.5 Estimation

483

Tischer, 1984) and built-in protection against overshooting.16 This function, whose maximum was typically in the neighborhood of −2, was determined with a precision of about eight decimal digits. The occasional presence of local maxima in addition to the global maximum was not a problem since non-global maxima were usually obvious. In any case, the main purpose was to verify whether CQ and CL fitted the data closely, so any failure to detect the global maximum would make the fit appear less close than it really was. In the course of the present research, conic distributions of various types were estimated for many thousands of samples, and multiple equilibria were detected in fewer than ten cases. Other cases may have gone undetected, but if so the corresponding parameters must have been close to those actually estimated. Goodness of fit may be measured by the difference (LR) between the highest log-likelihood attainable for a particular set of data and the log-likelihood corresponding to the parameters estimated for the conic distribution (CL or CQ). The highest log-likelihood LLsup would be attained by a hypothetical distribution that would fit the data perfectly. Such a distribution might have as many parameters as there are observations: LLsup =

T 

Ft ln Ft ,

(22.108)

Ft = F(vt ) − F(vt−1 ).

(22.109)

t=1

where T is the number of intervals, and

Thus, an LR of 0.01 means the maximum of the conic log-likelihood function is about 1% lower than the highest attainable. A comparison of functional forms using this criterion is provided in Section 22.5.2. An alternative measure of goodness of fit, χ 2 , permits the testing of hypotheses concerning functional form. These two measures are related, at least in a statistical sense, for it is easy to show that, if a distribution fits a set of data closely, then χ 2 is approximately equal to twice LR multiplied by the number of observations. However, when comparing the fit in samples of different size, the ranking produced by LR is quite different from the one produced by χ 2 . Moreover, it is well-known that a major problem with χ 2 in the context of estimation is its large value in large samples, for any specified distribution will be rejected by the χ 2 test if the sample is sufficiently large. Thus, all of the 12 functional forms considered by McDonald (1984) were decisively rejected in his three samples, each of which included between 35,000 and 40,000 observations. This is confirmed in Section 22.7, where acceptable values of χ 2 are found only in some of the smaller samples.

16

LDT note: The programs were written by HSH in True Basic.

484

22

Conic Distributions of Earned Incomes

One other feature of the estimation procedure should be mentioned at this point. Theoretically, the conic distributions extend to infinity at the right, but in computation some large value must be used instead. The number chosen was 1 million (that is, one billion dollars, since income is measured in thousands of dollars). As a check, the upper limit was put at 1000; the estimated parameters differ by at most 1 unit in the third decimal place. Accordingly, it can be concluded that the estimates are not sensitive to the magnitude of the proxy for infinity.

22.5.1 Medianization The principal difficulty encountered initially was the small size of the Hessian determinant [C in expression (22.14)]. Conceivably, the number of income intervals (22 of which is usually 13, including the open-ended top interval) is not large enough to estimate the four parameters with reasonable accuracy. There might also be a nearly exact relation—though not necessarily linear—between the parameters. In either case, there would be too many parameters. In order to reduce this number to three, income (or earnings) has been divided by its median, which is assumed to be known. This transformed variate accordingly has a median of 1, and allows (from the formulae in Section 22.4.1) for elimination of one of parameters (c3 in the case of CQ, and VL for CL). This expedient is clearly open to question, since the median in finite samples is a random variable, but comparison of parameter estimates obtained with and without “medianization” suggests that it is harmless. An example for CL is given in Table 22.1, which is based on the family income data from the current

Table 22.1 Estimated CL parameters without and with medianization Year

Np

c1

c3

c4

c5

LR

Det.

1970

4 3 3a 4 3 3a 4 3 3a

1.85975 1.86143

−4.72266 −0.46820 −4.72890 −5.86578 −0.66678 −5.86561 −5.83383 −0.47576 −5.83869

−0.26062 −0.25985

0.06074 −1.12952 0.06022 0.30994 −1.40480 0.30996 0.23329 −1.14000 0.23333

0.000704 0.000708

0.005253 −0.024548

0.000924 0.000924

0.004170 −0.026086

0.004358 0.004358

0.020183 −0.097195

1975

1980

1.98418 1.98413 1.75826 1.75940

−0.32720 −0.32722 −0.22526 −0.22527

Note: “np” refers to the number of estimated parameters. The “a” in “3a” indicates that some of the parameters have been adjusted to make them comparable to those in the first line, as per the text, while “Det” is the determinant of the Hessian matrix (the C of Section 22.2.1) corresponding to terminating estimates of the parameters. Source: Calculated from frequency distributions of family income in McDonald (1984, p. 655). The data for 1970 and 1975 were subsequently revised by the Census Bureau, but to maintain comparability with McDonald, these revisions were disregarded. The census definition of “family” was not the same in 1980 as in the earlier years.

22.5 Estimation

485

population survey used by McDonald (1984). For each year, the top line in the table gives the four parameters estimated without medianization and the second line the three parameters estimated with medianization as well as the implied value of the fourth parameter. The meaning of the third line will be explained in a moment. The goodness-of-fit measure LR and the determinant of the Hessian (C) are also included. It is clear from Table 22.1 that medianization changes the estimates of c1 and c4 very little; indeed, the only reason for printing them with five decimal digits is to show that they are not identical. The goodness of fit is also substantially unaffected. The main difference is in the estimates of c3 and c5 . These parameters do not remain invariant when the variate is transformed. However, it can easily be shown that c˜ 3 = c3 + c1 ln md, c˜ 5 = c5 + 2c4 ln md,

(22.110)

where ln md is the logarithm of the median and the tildes serve to distinguish the medianized parameters from the original. With these formulae, the two parameters on the second line can be adjusted to make them comparable with those on the first line. The result is shown in the line labeled “3a” for each year. Again, the effect of medianization is seen to be very small. Medianization is successful because the medians calculated from equations (22.93) and (22.94), using estimated parameters, are generally close to the sample medians.17 This implies, of course, that CQ and CL fit the data well, which can also be seen from the small values of LR in the table. As the last column of Table 22.1 shows, the Hessian determinants are about five times larger after medianization than before, nevertheless are still quite small. More precisely, the largest eigenvalue of the Hessians is usually much larger—by a factor of 100 or 100—than the smallest eigenvalue. This raises the question whether the number of estimated parameters can be further reduced by additional transformations of the data. However, experience to date suggests that the answer is negative. By using quartiles, for instance, it is theoretically possible to exclude c3 from estimation, but the empirical results are poor because the calculated quartiles are not sufficiently close to the actual ones. It appears, therefore, that the minimal number of estimated parameters is three.

22.5.2 Comparison with Other Distributions Since the medianized CL and CQ distributions—called CL3 and CQ3 hereafter— have only three parameters, they can be compared with the other three-parameter distributions that have been proposed. This is done in Table 22.2, which shows the LR for each distribution, and is again based upon the three samples used by McDonald (1984). The first distribution listed, a variant of Burr’s Type XII,

17 As in published census tables, the sample medians are calculated by linear interpolation within the interval in which the median is located.

486

22

Conic Distributions of Earned Incomes

Table 22.2 Goodness of fit for conic and other distributions with family income data Distribution

1970

1975

1980

Singh–Maddala Generalized gamma Conic-linear (CL3) Conic-quadratic (CQ3)

0.007974 0.016951 0.000704 0.000857

0.004081 0.007646 0.000924 0.000637

0.002501 0.002865 0.004358 0.001419

Source: See Table 22.1.

was introduced by Singh and Maddala (1976). It did well in McDonald’s comparison, but none of the 3 years does it fit as closely as CQ3. Yet, even so, it shows up better than the generalized gamma, which is included mainly because of its historical importance. The performance of the two conic distributions is quite similar for the 3 years, except for 1980, when CQ3 is markedly better. The relatively poor performance of CL3 for that year is something of an anomaly in light of its superiority to CQ3 in large samples shown below, particularly in Table 22.8. While the anomaly may be due to the estimation algorithm leading to a local, rather than a global, maximum, however this does appear to be the case. The goodness of fit was also calculated for generalized beta of the second kind (McDonald, 1984). However, as this distribution has four parameters, it was not estimated; the LR was calculated instead from the log-likelihood given by McDonald. Since the focus in this exercise is on individual earnings, rather than family income, a comparison based on the former concept is provided in Table 22.3. It is worthy of note that while the distribution of family incomes continues to be of policy interest, the data have become increasingly difficult to interpret in recent years because of a relative increase in the number of small families resulting from Table 22.3 Goodness of fit for Singh–Maddala and conic distributions for individual earnings data Distribution

Men

Women

All persons

All Occupations Singh–Maddala Conic-linear (CL3) Conic-quadratic (CQ3)

0.03319 0.03381 0.00433

0.01839 0.00155 0.00170

0.01479 0.00213 0.00230

Executives, Administrators, and Managers Singh–Maddala 0.03340 Conic-linear (CL3) 0.01028 Conic-quadratic (CQ3) 0.00122

0.03523 0.00234 0.00244

0.02010 0.00237 0.00203

Operators, Fabricators, and Laborers Singh–Maddala [a] Conic-linear (CL3) 0.00335 Conic-quadratic (CQ3) 0.00358

0.02372 0.00358 0.00359

0.01749 0.00424 0.00386

Note: [a] No estimate obtained. Source: PUMS data from 1980 census of population (see Section 22.6).

22.6 Data

487

higher divorce rates and lower rates of first marriage. Individual earnings data are fortunately not affected by this problem. It is evident that CL3 and CQ3 fit the individual earnings data much more closely than does the Singh–Maddala distribution. Indeed, the fit, as measured by LR, is often 10 or more times tighter for CL3 and CQ3, whose LR’s do not differ a great deal. Accordingly, it seems fair to conclude from Tables 22.2 and 22.3 that the conic distributions represent a marked improvement over their predecessors.

22.5.3 Bias in Maximum Likelihood Estimation Since maximum-likelihood estimates can be biased in finite samples, some indication of the size of bias is evidently useful. This is particularly important in the present context because the estimates of c1 obtained for CL and CQ are larger than past research might lead one to expect. For this purpose, Monte Carlo methods have been applied to the earnings of men and women for two alternative sets of parameters that are typical of the results in Section 22.7. Some of the findings (with reference to CL3), for 49 replications for each of the four sample sizes, are summarized in Table 22.4. Table 22.4 Means of estimated CL3 parameters in Monte Carlo studies Sample Size Parameter

True Value

500

1000

2000

4000

Set 1 c1 c3 c4

1.744 −0.542 −0.153

1.824 −0.621 −0.168

1.780 −0.573 −0.162

1.768 −0.560 −0.158

1.750 −0.540 −0.152

Set 2 c1 c3 c4

2.409 −1.617 −0.362

2.456 −1.742 −0.405

2.434 −1.672 −0.377

2.459 −1.669 −0.389

2.445 −1.674 −0.371

Note: The estimating equations were medianized (see Section 22.5.1).

By and large, the Monte Carlo means for the four samples sizes are not strikingly different from the true values. Nevertheless, a distinct pattern of bias appears to exist, for in the smaller classes, the estimates of c1 are too large and those of c3 and c4 too small. In the largest sample size 0.94 in set 2. This result suggests that one of the parameters in the definition of CL may be redundant. Similar phenomena are found for CQ.

22.6 Data One of the two principal sources of data used for this paper was the 1980 census of population and housing in the form of the combined A, B, and C 0.1% public use micro-samples (PUMS). According to the U. S. Bureau of the Census (1983,

488

22

Conic Distributions of Earned Incomes

p. 3): “There is negligible overlap among the [A, B, and C] samples.” The main advantages of the PUMS data are the large sample size (more than 500,000 person aged 18 and over) and the availability of published tabulations than can be used as a check. These advantages should be balanced against the relatively low accuracy of the census returns on individual earnings compared with the March current population survey (CPS), which was the other main source of data. A troublesome problem common to both the PUMS and CPS microdata is the use of “top-coding”—called “censoring” in the statistical literature—by the Census Bureau to safeguard the confidentiality of the underlying interviews. Thus, the PUMS data on earnings are top-coded at $75,000, while the top-code for the CPS data varies from $50,000 to $100,000 between 1976 and 1990, which are the years used in this study. This implies that for top-coded items the sample moments (and various measures of inequality) cannot be calculated.18 For the estimation of the distribution functions, this problem is not serious, because the top-code limit can be used as the lower bound of the open-ended interval at the right of the distribution. Indeed, one of the advantages of a close-fitting distribution is that it can be used to estimate the mean and other descriptive statistics despite the existence of top-coding.

22.7 Empirical Results The empirical results obtained with the various conic distributions are too extensive to be presented in full. By way of illustration, three analyses, all referring to annual earnings, are given in this section.19 The first analysis (Tables 22.5 and 22.6) classifies the PUMS data by sex, age, and education, while the second analysis (Tables 22.7 and 22.8) classifies the PUMS data by sex and occupation. Finally, the third analysis (also presented in Tables 22.7 and 22.8) uses CPS microdata classified by gender and year, and is supplemented by Figs. 22.1 and 22.2, which show inequality measures derived from the estimated functions. Only medianized estimates from CQ and CL are shown. In most cases, inclusion of the additional parameter of the CG does not appreciably improve the fit.20 Of the two tables presented for each analysis, one gives the estimated parameters and the

18 There is also “bottom-coding,” which is especially relevant to earnings from self-employment, since they can be negative. In PUMS, the largest recorded loss is $9995. The relatively few individuals with negative earnings are always included in the lowest earnings interval. It should also be mentioned that the interview forms do not have room for incomes exceeding $999,995, which is a limitation (although presumably minor) on the accuracy of the published tabulations. 19 Tables 22.1 and 22.2 presented some results for family income data derived from published current population reports. 20 CG is also more difficult to estimate because some of the four conditions derived in Section 22.2.1 are not automatically satisfied.

22.7 Empirical Results

489

Table 22.5a Estimates for men by age and education (1979 earnings, from 1980 census) CQ3

CL3

Age

Education

c1

b2

V0

c1

c3

c4

1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 6 6 7 7 7 7 7 7 7

2 3 4 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 7 1 7 1 2 3 4 5 6 7

3.411 3.190 5.092 3.674 2.046 2.215 2.339 2.286 1.942 1.651 2.060 1.821 1.904 2.149 1.841 1.470 1.117 1.547 1.949 2.199 1.990 1.644 1.343 1.081 1.467 1.961 2.079 1.958 1.490 1.476 1.725 [a] 2.073 2.460 2.321 2.128 1.471 1.228 1.753

0.583 0.430 1.335 0.606 0.228 0.228 0.195 0.173 0.169 0.143 0.180 0.106 0.152 0.131 0.107 0.112 0.055 0.108 0.183 0.160 0.113 0.087 0.087 0.058 0.107 0.184 0.176 0.112 0.077 0.114 1.385 [a] 0.224 0.235 0.213 0.185 0.145 0.093 0.166

−4.73 −4.59 −3.00 −4.25 −5.29 −5.37 −5.37 −5.71 −4.92 −5.87 −5.49 −9.21 −6.10 −6.05 −6.59 −5.32 −8.20 −6.59 −5.75 −6.08 −6.64 −7.42 −6.45 −7.83 −6.59 −6.23 −5.75 −7.92 −9.33 −7.22 −2.34 [a] −6.27 −6.95 −5.94 −7.20 −5.80 −7.50 −6.83

[a] 3.073 4.602 3.528 1.986 2.155 2.327 2.244 1.895 1.638 2.018 1.808 1.871 2.120 1.827 1.448 1.112 1.533 1.907 2.168 1.970 1.634 1.328 1.076 1.455 1.935 2.050 1.948 1.492 1.469 1.534 2.003 2.049 2.443 2.282 2.133 1.450 1.224 1.744

[a] −2.176 −8.531 −3.927 −0.590 −0.636 −0.361 −0.304 −0.094 −0.130 −0.299 −0.317 −0.227 −0.074 0.008 0.115 0.208 0.006 −0.383 −0.291 −0.038 0.044 0.138 0.205 0.008 −0.539 0.347 −0.201 −0.027 −0.119 −3.033 −5.587 −1.004 −1.519 −0.739 −0.912 −0.120 0.009 −0.542

[a] −0.630 −3.495 −0.194 −0.201 −0.201 −0.139 −0.124 −0.092 −0.091 −0.127 −0.086 −0.103 −0.074 −0.057 −0.044 −0.019 −0.056 −0.139 −0.116 −0.063 −0.045 −0.034 −0.020 −0.056 −0.162 −0.134 −0.082 −0.050 −0.075 −1.555 −2.340 −0.252 −0.324 −0.206 −0.215 −0.087 −0.052 −0.153

Note: [a] No estimates obtained (also in the following tables).

other the number of observations and goodness of fit. Because of their size, some of the tables have to be split. In view of the small-sample bias demonstrated in Section 22.5.3, only the estimates for sub-samples with at least 2000 observations are tabulated.

490

22

Conic Distributions of Earned Incomes

Table 22.5b Estimates for women by age and education (1979 earnings, from 1980 census) CQ3

CL3

Age

Education

c1

b2

V0

c1

c3

c4

1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 7 7 7 7 7 7 7

2 3 4 5 7 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 7 1 2 3 4 7 7 1 2 3 4 5 6 7

2.616 3.139 6.704 4.906 3.708 2.414 2.635 2.844 2.614 2.681 2.711 2.036 2.487 2.713 2.860 2.582 2.427 2.030 1.914 2.613 2.497 2.109 1.835 2.004 2.435 2.411 2.140 2.600 1.954 2.194 2.383 2.894 2.742 2.659 2.427

0.443 0.183 0.869 0.621 0.388 0.184 0.157 0.137 0.033 0.113 0.181 0.188 0.274 0.261 0.103 0.065 0.310 0.078 0.157 0.174 0.210 0.242 0.157 0.296 0.229 0.181 0.294 1.282 0.118 0.201 0.195 0.252 0.126 0.123 0.252

−5.53 −10.11 −4.61 −4.26 −6.48 −9.11 −9.43 −10.50 −33.32 −10.47 −8.56 −6.85 −5.57 −6.34 −15.97 −17.67 −5.31 −14.29 −7.40 −7.19 −6.88 −5.74 −7.35 −4.44 −6.23 −7.53 −5.01 −2.61 −11.04 −8.23 −7.63 −7.54 −11.46 −10.66 −6.67

2.652 3.150 6.421 4.838 3.712 2.412 2.630 2.847 2.616 2.684 2.700 2.021 2.453 2.684 2.875 2.589 2.365 2.029 1.909 2.148 2.466 2.069 1.840 1.990 2.424 2.412 2.109 2.259 1.956 2.195 2.377 2.892 2.751 2.664 2.409

−3.966 −2.135 −10.270 −4.418 −4.144 −1.651 −1.220 −1.143 −0.532 −0.664 −1.348 −0.811 −1.238 −1.554 −1.620 −0.613 −1.446 −0.551 −0.599 −0.747 −1.088 −0.946 −0.605 −0.829 −1.077 −0.983 −1.104 −4.518 −0.869 −1.601 −1.218 −2.225 −1.154 −0.893 −1.617

−1.099 −0.331 −3.030 −1.343 −0.919 −0.285 −0.213 −0.185 −0.037 −0.125 −0.249 −0.206 −0.337 −0.361 −0.170 −0.075 −0.390 −0.083 −0.158 −0.184 −0.248 −0.260 −0.166 −0.307 −0.277 −0.221 −0.335 −2.135 −0.145 −0.307 −0.258 −0.435 −0.174 −0.151 −0.362

Certain abbreviations for age, education, and occupational groups are used throughout the tables. Age groups are designated as follows: 1 = 18 through 24 2 = 25 through 34 3 = 35 through 44 4 = 45 through 54 5 = 55 through 64 6 = 65 and over 7 = all ages (18 and over).

22.7 Empirical Results

491

Table 22.5c Estimates for women by age and education (1979 earnings, from 1980 census) CQ3

CL3

Age

Education

c1

b2

V0

c1

c3

c4

1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 7 7 7 7 7 7 7

1 2 3 4 5 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 7 1 2 3 4 5 6 7

2.429 3.133 2.550 4.569 3.830 3.138 1.951 2.342 2.432 2.268 1.807 1.650 2.097 1.931 2.079 2.548 2.109 1.632 1.169 1.744 2.032 2.320 2.218 1.805 1.430 1.104 1.587 1.960 2.310 2.016 1.536 1.278 1.136 1.558 1.447 [a] 2.299 2.983 2.028 2.430 2.242 2.131 1.416 1.247 1.778

0.193 0.588 0.294 1.056 0.629 0.488 0.214 0.274 0.255 0.189 0.097 0.127 0.195 0.219 0.315 0.346 0.211 0.110 0.046 0.218 0.310 0.358 0.276 0.193 0.119 0.047 0.212 0.312 0.395 0.257 0.147 0.095 0.069 0.221 1.075 [a] 1.128 2.034 0.323 0.374 0.329 0.271 0.135 0.092 0.248

−8.88 −4.77 −5.86 −3.42 −3.79 −4.83 −6.16 −5.98 −5.79 −6.61 −8.96 −6.99 −6.48 −6.07 −4.76 −4.78 −6.18 −9.40 −12.35 −5.68 −4.67 −4.51 −5.05 −6.02 −7.65 −11.87 −5.43 −4.70 −4.13 −5.30 −6.88 −8.68 −10.05 −5.34 −2.37 [a] −2.87 −2.51 −5.08 −5.38 −5.01 −6.10 −7.22 −8.47 −5.80

2.402 3.055 2.516 4.270 3.719 3.060 1.909 2.286 2.345 2.228 1.794 1.641 2.060 1.888 1.993 2.435 2.074 1.620 1.168 1.708 1.936 2.212 2.044 1.765 1.413 1.104 1.549 1.905 2.207 1.959 1.529 1.270 1.136 1.531 1.389 2.372 1.888 1.615 1.973 2.362 2.146 2.109 1.403 1.245 1.749

−1.698 −4.063 −1.680 −7.491 −3.355 −3.347 −0.808 −1.441 −1.068 −0.715 −0.183 −0.202 −0.728 −0.824 −1.055 −1.377 −0.795 −0.392 0.114 −0.657 −0.937 −1.262 −0.875 −0.542 −0.213 0.122 −0.506 −1.034 −1.288 −0.858 −0.366 −0.118 0.003 −0.564 −2.219 −6.185 −3.702 −3.841 −1.406 −2.316 −1.368 −1.524 −0.302 −0.081 −1.029

−0.288 −1.403 −0.420 −2.842 −1.118 −0.933 −0.215 −0.348 −0.272 −0.185 −0.070 −0.092 −0.190 −0.220 −0.318 −0.038 −0.209 −0.094 −0.024 −0.195 −0.293 −0.380 −0.258 −0.162 −0.083 −0.024 −0169 −0.325 −0.419 −0.250 −0.121 −0.063 −0.042 −0.188 −1.174 −3.205 −1.525 −1.748 −0.393 −0.579 −0.378 −0.363 −0.104 −0.061 −0.272

492

22

Conic Distributions of Earned Incomes

Table 22.6a Goodness-of-fit Statistics for men by age and education (1979 earnings, from 1980 census) CQ3

CL3

Age

Education

# Obs.

LR

X2

LR

X2

1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 6 6 7 7 7 7 7 7 7

2 3 4 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 7 1 7 1 2 3 4 5 6 7

7241 16327 9027 36357 2471 4633 17486 12002 7244 6523 50359 3248 4258 12539 6096 3700 5136 34977 4761 4707 9824 4144 2803 3306 29545 4895 4227 7428 2886 23214 2709 8043 19495 26406 65427 35031 18003 18133 182495

0.00259 0.00292 0.01286 0.00394 0.00599 0.00375 0.00651 0.00325 0.00321 0.00392 0.00259 0.00568 0.00546 0.00552 0.00293 0.00196 0.00125 0.00334 0.00413 0.00693 0.00682 0.00148 0.00234 0.00114 0.00321 0.00842 0.00542 0.00586 0.00853 0.00404 0.00697 [a] 0.00422 0.00529 0.00485 0.01074 0.00170 0.00247 0.00433

55.872 163.970 582.172 602.022 28.639 37.689 241.670 85.841 46.034 51.146 277.971 37.411 49.705 146.647 36.696 14.417 12.984 235.510 40.026 74.466 141.743 12.112 13.290 7.648 189.314 96.812 48.716 93.596 51.015 190.064 34.723 [a] 185.560 332.569 735.841 822.930 61.287 90.262 1638.443

[a] 0.00257 0.00939 0.00306 0.00651 0.00407 0.00731 0.00345 0.00374 0.00340 0.00292 0.00574 0.00584 0.00605 0.00300 0.00239 0.00129 0.00357 0.00449 0.00704 0.00715 0.00153 0.00258 0.00112 0.00335 0.00845 0.00520 0.00571 0.00812 0.00381 0.00693 0.00935 0.00405 0.00478 0.00469 0.00948 0.00144 0.00201 0.00381

[a] 135.237 584.199 488.135 30.785 39.735 262.695 88.716 53.675 44.424 304.835 37.681 52.159 158.197 37.460 17.804 13.441 351.637 43.079 74.420 146.978 12.517 14.689 7.520 197.851 95.748 46.244 90.964 48.262 179.276 34.799 148.965 176.753 303.407 782.082 720.682 51.826 72.476 1439.630

Educations groups are defined as: 1 = no high school 2 = some high school, but less than 4 years 3 = high-school graduate, but no college

22.7 Empirical Results

493

Table 22.6b Goodness-of-fit statistics for women by age and education (1979 earnings, from 1980 census) CQ3

CL3

Age

Education

# obs.

LR

X2

LR

X2

1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 7 7 7 7 7 7 7

2 3 4 5 7 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 7 1 2 3 4 7 7 1 2 3 4 5 6 7

4656 16133 9971 2325 34234 3428 15833 9425 5682 3960 39642 3495 12035 4801 2347 2195 26453 2290 3607 9346 3235 21234 2523 2924 6507 2151 15830 5056 9805 19088 61316 30263 12949 9028 142449

0.00593 0.00572 0.00698 0.01280 0.00512 0.00255 0.00291 0.00408 0.00340 0.00583 0.00189 0.00361 0.00119 0.00230 0.00823 0.01231 0.00132 0.00367 0.00649 0.00242 0.00139 0.00248 0.00875 0.00398 0.00321 0.00794 0.00258 0.00456 0.99258 0.00310 0.00194 0.00453 0.00608 0.00828 0.00170

68.688 722.647 143.731 3038.283 1112.130 26.732 234.695 136.467 68.097 52.982 327.926 27.629 44.265 45.489 42.145 67.002 84.783 29.663 70.322 63.294 9.221 114.422 75.871 27.629 66.418 41.096 101.662 45.069 80.307 174.237 394.254 386.567 195.881 185.358 699.269

0.00582 0.00561 0.00615 0.01115 0.00482 0.00350 0.00283 0.00392 0.00338 0.00558 0.00178 0.00362 0.00108 0.00201 0.00804 0.01218 0.00135 0.00366 0.00647 0.00240 0.00134 0.00263 0.00864 0.00367 0.00292 0.00748 0.00238 0.00418 0.00255 0.00302 0.00183 0.00417 0.00584 0.00795 0.00156

68.895 735.766 127.195 2900.202 1114.660 26.333 230.148 133.800 68.134 50.976 313.728 68.134 39.513 40.988 41.168 66.634 82.831 29.584 69.544 61.720 8.869 118.391 75.871 25.469 61.713 39.005 92.822 41.047 80.080 171.641 376.287 364.722 189.685 179.551 643.675

4 5 6 7

= some college, but less than 4 years = college graduate, but no postgraduate study = some postgraduate study = all levels of education.

Occupation groups are abbreviated as follows: eam = executive, administrative, and managerial

494

22

Conic Distributions of Earned Incomes

Table 22.6c Goodness-of-fit statistics for all persons by age and education (1979 earnings, from 1980 census) CQ3

CL3

Age

Education

# Obs.

LR

X2

LR

X2

1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 7 7

1 2 3 4 5 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 3 7 1 2 3 4 5 6 7

2100 11897 32460 18998 4144 70591 3785 8061 33319 21427 12926 10483 90001 4828 7753 24564 10907 6047 7331 61430 7051 8314 19170 7379 4191 4674 50779 7430 7151 13935 5037 2767 2724 39044 4106 2318 3295 3295 13099 29300 45494 126743 65294 30952 27161 324944

0.00150 0.00234 0.00284 0.00763 0.00901 0.00273 0.00519 0.00216 0.00514 0.00317 0.00304 0.00255 0.00191 0.00457 0.00506 0.00589 0.00351 0.00292 0.00085 0.00328 0.00537 0.00715 0.00744 0.00229 0.00304 0.00195 0.00346 0.00794 0.00458 0.00474 0.00555 0.00154 0.00402 0.00326 0.00430 [a] 0.00791 0.00791 0.00573 0.00359 0.00304 0.00427 0.00452 0.00263 0.00264 0.00231

6.309 67.574 202.969 917.303 100.189 561.362 37.919 39.164 363.216 143.317 78.861 52.896 365.830 45.062 83.103 318.293 80.446 35.276 12.500 408.434 77.885 131.786 297.524 33.792 24.746 18.465 353.700 133.184 71.255 140.271 57.208 8.505 22.196 261.981 32.643 [a] 50.117 50.117 143.401 225.864 307.189 1132.890 622.812 163.223 142.958 1547.876

0.00154 0.00210 0.00271 0.00606 0.00717 0.00236 0.00544 0.00225 0.00555 0.00323 0.00309 0.00227 0.00201 0.00479 0.00564 0.00639 0.00332 0.00280 0.00077 0.00333 0.00616 0.00781 0.00822 0.00254 0.00293 0.00179 0.00376 0.00809 0.00448 0.00484 0.00514 0.00146 0.00376 0.00309 0.00446 0.00834 0.00746 0.00746 0.00550 0.00368 0.00286 0.00460 0.00388 0.00246 0.00231 0.00213

6.415 61.598 190.846 825.008 171.383 486.464 39.376 39.348 380.636 144.022 80.040 47.036 377.293 46.897 90.551 334.529 75.870 33.670 11.330 413.266 89.015 138.898 322.430 37.379 23.853 16.776 383.671 132.930 68.125 141.561 52.718 8.071 20.581 248.713 33.915 37.176 47.918 47.918 139.148 227.619 285.823 1197.835 534.145 152.463 124.155 1423.015

22.8 Discussion

495

fff = farming, forestry, and fisheries ofl = operators, fabricators, and laborers pcr = precision production, craft, and repair pso = professional specialties ser = service tsa = technical, sales, and administrative support.

22.8 Discussion A substantive discussion of the results presented in Section 22.7 would be out of place in this paper, which focuses on statistical rather than economic aspects. Instead, a few remarks on the estimates and goodness of fit will be presented. (1) The estimates show regular patterns with respect to the variables used for classification. Thus, in all three analyses, for both CL3 and CQ3, the estimates of c1 are generally higher for women than for men.21 In Table 22.5, similar regularities can be found involving age and education, except perhaps in the youngest and oldest age groups, where there are special factors such as school attendance and retirement. In Table 22.7, a downward time trend in the estimated c1 is apparent. Whether these patterns make economic sense is another question, but their existence suggests that the conic distributions do not produce random numbers. (2) The goodness of fit, as measured by LR, is generally satisfactory, in the sense that the likelihood function is within 1% of its theoretical maximum. Not surprisingly, LR tends to be lower in the larger aggregates, where it is often less than 0.001. Moreover, one may perhaps derive further encouragement from the declining tendency of LR in Table 22.8, which implies that the fit of CQ3 and CL3 improves with time. As mentioned earlier, the χ 2 statistics are nearly always significantly different from zero, but this is true for all known distributions with a small number of parameters, and therefore cannot be held against the conic family. It does suggest, however, that the family is not the last word on the subject. (3) The phenomenon of “self-mixability” discussed at the end of Section 22.1 can be observed throughout the tables, but is especially clear in Table 22.8. For CL in the years 1988–1990, for instance, LR is less than 0.001 for men, for women, and for all persons, indicating that CL fits the data very closely, not only for the two genders, but also for their union. 21

In this context, one anomaly has to be mentioned. In the early part of the period analyzed in Section 22.7, including the census year 1980, there were so few women with high earnings that the maximum-likelihood estimation effectively ignores them. As a result, the parameter estimates for women in those early years are based upon women with low and medium earnings only, and those with high earnings sometimes appear outside the “Pareto” asymptote defined in Section 22.3.1. This anomaly is not encountered in recent years.

496

22

Conic Distributions of Earned Incomes

Table 22.7 Estimates by gender and year (from current population surveys) CQ3 Year [b]

c1

CL3 b2

V0

c1

c3

c4

All men (18 and over) with earnings 1976 1.815 0.121 77 1.879 0.137 78 1.891 0.155 79 1.870 0.168 80 1.899 0.175 81 1.928 0.178 82 1.951 0.183 83 1.848 0.177 84 1.865 0.180 85 1.857 0.188 86 1.824 0.198 87 1.727 0.188 88 1.782 0.210 89 1.682 0.185 90 1.592 0.185

−9.08 −8.40 −7.61 −6.90 −6.74 −6.83 −7.05 −7.25 −7.23 −6.91 −6.49 −6.50 −5.98 −6.42 −6.20

1.847 1.872 1.882 1.853 1.878 1.900 1.924 1.821 1.832 1.822 1.779 1.688 1.728 1.641 1.550

−0.509 −0.591 −0.622 −0.583 −0.603 −0.657 −0.781 −0.775 −0.801 −0.779 −0.742 −0.640 −0.666 −0.576 −0.505

−0.117 −0.137 −0.153 −0.158 −0.164 −0.169 −0.184 −0.178 −0.182 −0.184 −0.185 −0.167 −0.182 −0.157 −0.148

All women (18 and over) with earnings 1976 3.021 0.232 77 2.820 0.214 78 2.982 0.274 79 2.718 0.221 80 2.674 0.195 81 2.786 0.261 82 2.989 0.309 83 2.747 0.292 84 2.684 0.286 85 2.627 0.275 86 2.649 0.291 87 2.547 0.289 88 2.452 0.296 89 2.388 0.288 90 2.316 0.246

−8.79 −9.15 −7.49 −8.45 −9.04 −7.21 −6.55 −6.63 −6.71 −6.85 −6.54 −6.42 −6.07 −6.13 −7.24

2.997 2.808 2.955 2.704 2.658 2.754 2.919 2.691 2.627 2.574 2.583 2.480 2.376 2.316 2.262

−2.596 −2.372 −2.600 −2.124 −1.854 −2.106 −2.427 −2.199 −2.150 −2.056 −2.084 −1.949 −1.777 −1.712 −1.782

−0.431 −0.385 −0.499 −0.373 −0.309 −0.421 −0.510 −0.461 −0.446 −0.419 −0.438 −0.417 −0.401 −0.384 −0.345

All persons (18 and over) with earnings 1976 1.966 0.193 77 1.968 0.208 78 1.993 0.235 79 1.935 0.229 80 1.926 0.229 81 1.968 0.239 82 1.986 0.251 83 1.872 0.233 84 1.892 0.232 85 1.870 0.233 86 1.856 0.236 87 1.784 0.228 88 1.822 0.249 89 1.749 0.227 90 1.682 0.200

−7.77 −7.38 −6.68 −6.63 −6.51 −6.35 −6.27 −6.51 −6.59 −6.51 −6.35 −6.34 −5.90 −6.19 −6.89

1.954 1.951 1.969 1.905 1.888 1.922 1.935 1.830 1.845 1.823 1.803 1.736 1.758 1.697 1.643

−1.221 −1.287 −1.341 −1.212 −1.141 −1.185 −1.292 −1.184 −1.202 −1.171 −1.123 −1.022 −1.026 −0.934 −0.900

−0.349 −0.269 −0.302 −0.277 −0.266 −0.278 −0.299 −0.270 −0.269 −0.264 −0.260 −0.241 −0.255 −0.228 −0.204

Note: [b] Year of CPS; earnings are of previous year.

22.8 Discussion

497

Table 22.8 Goodness-of-fit statistics by gender and year (from current population surveys) CQ3

CL3 χ2

LR

χ2

All men (18 and over) with earnings 1976 35325 0.00215 77 42863 0.00231 78 41977 0.00210 79 41876 0.00236 80 39470 0.00219 81 49155 0.00155 82 44236 0.00172 83 43800 0.00156 84 43174 0.00076 85 43555 0.00138 86 42852 0.00091 87 42301 0.00136 88 42451 0.00108 89 39866 0.00087 90 44634 0.00095

159.822 206.455 180.758 199.076 174.860 152.093 153.486 137.974 65.729 123.442 78.429 116.536 92.411 69.795 83.743

0.00202 0.00216 0.00186 0.00211 0.00189 0.00135 0.00148 0.00136 0.00057 0.00115 0.00078 0.00124 0.00090 0.00072 0.00079

150.783 193.828 159.878 177.816 149.832 132.462 131.390 119.648 49.815 103.023 67.263 106.433 77.448 57.357 69.496

All women (18 and over) with earnings 1976 25574 0.00037 77 31247 0.00091 78 31205 0.00064 79 32288 0.00114 80 30894 0.00117 81 39065 0.00108 82 35105 0.00095 83 34831 0.00130 84 35166 0.00071 85 36329 0.00089 86 35681 0.00108 87 35962 0.00149 88 36408 0.00076 89 34387 0.00106 90 41861 0.00083

20.181 88.566 50.687 94.125 76.522 90.706 84.995 107.176 52.430 88.188 87.342 111.722 64.665 86.649 75.970

0.00037 0.00089 0.00059 0.00108 0.00112 0.00094 0.00082 0.00114 0.00053 0.00072 0.00090 0.00131 0.00058 0.00086 0.00071

19.830 85.794 46.662 89.090 73.103 79.134 72.479 93.635 39.107 71.951 71.496 97.746 49.222 68.915 64.186

All persons (18 and over) with earnings 1976 60899 0.00150 77 74110 0.00148 78 73182 0.00112 79 74164 0.00146 80 70364 0.00125 81 88220 0.00111 82 79341 0.00115 83 78631 0.00109 84 78340 0.00051 85 79884 0.00077 86 78533 0.00061 87 78263 0.00101 88 78859 0.00061 89 74253 0.00074 90 86495 0.00078

192.789 230.685 168.056 217.854 175.603 195.323 182.659 173.870 80.293 126.344 98.813 158.998 96.111 110.306 135.247

0.00144 0.00143 0.00101 0.00139 0.00122 0.00105 0.00109 0.00101 0.00043 0.00070 0.00055 0.00093 0.00055 0.00062 0.00066

185.233 221.335 152.039 207.355 171.378 184.793 173.490 161.309 67.316 114.117 87.130 147.339 87.002 92.342 114.047

Year

# obs.

LR

498

22

Conic Distributions of Earned Incomes

(4) As regards the relative merits of CL3 and CQ3, it is clear that the former always fits better in the larger aggregates, though the difference in LR is generally small. In the more detailed breakdown of Tables 22.5 and 22.6, CQ3 is often superior––though again by a small margin––so it cannot be dismissed. Conceivably, there exists a member of the conic family that combines the respective advantages of CL3 and CQ3. In this connection, it is worth noting that the strong correlation between the estimates of different parameters, already noted in the Monte Carlo trials of Section 22.5.3, is also present in the empirical results of Section 22.7. However, no satisfactory two-parameter member of the conic family has so far been found. (5) As an illustration of the inequality measures discussed in Section 22.4, Figs. 22.2 and 22.3 show the Gini and Atkinson coefficients for CPS data on annual earnings by gender from 1976 to 1990.22 Both of these were calculated from CL3, which fits these aggregates slightly better than CQ3; using CQ3 would not have changed the picture perceptibly. When we look first at the Gini coefficient, we see that the overall inequality rose very little (from 0.4540 in 1976 to 0.4598 in 1990), but that it did rise markedly for men and fell less markedly for women. The approximate constancy of the overall Gini coefficient may be attributed to the rise in the proportion of women among earners over the period, and to a tendency toward convergence between male and female inequality. The Atkinson coefficients in Fig. 22.3 present a somewhat different pattern. As in Fig. 22.2, male inequality increased and female inequality decreased, but overall inequality decreased (from 0.3926 in 1976 to 0.3756 in 1990).23

Appendix: Integral Formulae for CL and CQ The integrals in equations (22.95) and (22.97) are special cases of  J(p,q) =

0 −∞

epU+qV(U) dU,

(22.111)

22 LDT note: As with the figures mentioned in Footnote 14, figures 20.2 and 20.3 are unfortunately not available. 23 Another measure of inequality, the logarithmic variance defined in equation (22.104), also declined markedly for all persons from 1.4346 to 1.2367 in 1990. Although not entirely clearcut, these findings cast serious doubt on the widespread belief that during the period under review “the rich got richer and the poor got poorer.” The conflict between the three inequality measures is attributable to the fact that the Lorenz curves (cf. Section 22.4.4) for 1976 and 1990 have interior points in common. As it happens, these curves are so close together as to be visually indistinguishable.

Appendix: Integral Formulae for CL and CQ

499

where p and q are positive constants.24 The integral on the right-hand side will first be evaluated for CL, for which V(U) is given by equation (22.63). The derivation is lengthy, and some intermediate steps will be suppressed. The exponent in equation (22.111) can be written as:   |C| 2c1 c3 − c4 2c1 p − q U−q − 2 , 2c1 2c21 2c1 (c1 U + c4 )

(22.112)

where|C| is the determinant defined in Section 22.2.1, except that c2 = 0 for CL. Putting c1 U + c4 = T, the exponent becomes: |C| 2c1 p − q c1 c4 p − c4 q + c1 c3 q T− +q 2 . 2 2 2c1 c1 2c1 T

(22.113)

Since the ratio in the middle, call it N for short, does not involve T, its exponential can be taken outside the integral, so that we have: J(p,q) =

−N c−1 1 e





 |C| 2c1 p − q exp T + q 2 dT. 2c21 2c1 T −∞ c4

(22.114)

The next substitutions: q |C| 2c1 p − q = κ 2, = η, 2 2c1 p − q c1

(22.115)

are of particular importance. The second substitution, where the numerator in the left-hand side is positive, is valid only if 2c1 p > q,

(22.116)

which means that q cannot be too large in relation to p. We know from expression (22.95) that, for moments, p = 1, and that q indicates the order of the moment. Thus, if c 1 < 1––a rare occurrence in the data analyzed here––no moments beyond the first exist.25 If expression (22.116) holds, η will also be positive. With these substitutions, the exponent beneath the integral sign in expression (22.114) can now be written: (1/2)ηκ[(T/κ) + (κ/T)].

24

(22.117)

In the present context they are natural numbers, but without this restriction, they can also be used to represent fractional moments. 25 Admittedly, expression (22.116) is only a sufficient condition, but if necessary can be safely conjectured.

500

22

Conic Distributions of Earned Incomes

The next transformation is −T/κ = et , which changes the integral in equation (22.114) into  κ

∞

exp ( − ηκ cosh t + t)dt,

(22.118)

w

where cosh t denotes the hyperbolic cosine of t and w = ln ( − c4 /κ). Since et = cosh t + sinh t, the integral in equation (22.118) can split into two parts, the first of which is:  ∞ e−ηκ cosh t cosh tdt = K1 (ηκ) − k1 (w,ηκ). (22.119) w

The functions K1 (z) and k1 (w,z) need some explanation. The first function is a member of Bessel family known by a variety of names, among which the “Macdonald function” is the shortest. The second function is called an “incomplete Macdonald function” by Agrest and Maksimov (1971). This function has the same relation to K1 (z) as the incomplete gamma function has to the ordinary gamma function.26 The incomplete Macdonald function can be expanded into a series that nearly always converges (although the conditions for convergence are not known): k1 (w,z) =

∞ 

( − 1)

j=0

jz

j

j!

γj+1 ,

(22.119a)

where 

w

γj =

cosh j tdt,

(22.120)

0

which satisfies the well-known recursion formula: γj =

1 sinh w cosh j

j−1

w+

j−1 γj−2 . j

(22.121)

Returning now to equation (22.118), the second integral in this expression is 

∞

w

e−ηκ cosht sinh tdt =

1 −ηκ cosh w e . ηκ

(22.122)

26 If the complete function is represented by an integral extending from 0 to infinity, then the incomplete function is the same integral extending from 0 to its second argument. The relevant integral representing the (complete) Macdonald function is given by Erdélyi [1953, Vol. II, p. 82, Equation (21)]. Agrest and Maksimov (1971) actually have two versions of the incomplete Macdonald function, of which only the “Bessel” form is need here.

Appendix: Integral Formulae for CL and CQ

501

Putting all these expressions together, we find that for CL: −N H, J(p,q) = c−1 1 e

(22.123)

where H = [κ{K1 (ηκ) − k1 (w,ηκ)} + η−1 exp ( − ηκ cosh w)].

(22.124)

For CQ, the original parameters are transformed as follows: A = c21 − c2 B = c1 c4 − c2 c3 N = c1 c3 − c4 H = (Nq + Bp)/A Q = q/c2 P = p − cQ R = B/A  E = P2 − AQ2 M = −ER S = a cosh (P/E). Some of these transformations may not be possible, in which case J(p, q) does not exist. In particular, the expression whose square root is taken in E should not be negative, and the argument of acosh (P/E) should not be less than one. In general, this means that q should not be too large in relation to p, which is to say that only lower moments exist. If the transformations are possible, it can then be shown that for CQ: J(p,q) = Re−H sinh (S)[k1 (M, − S) − K1 (M)] −

PeRP . EM

(22.125)

Chapter 23

Final Evaluation

A lot of ground has been covered to this point, and it is time to bring the investigation to a close. Nearly 40 years have passed since the second edition of CDUS, and obviously much has occurred during this time span, technical as well as empirical. Demand systems that embody all of the restrictions of standard neoclassical demand theory now abound, and there have been great strides in econometric sophistication. Especially important, in our view, have been attempts to place demand theory on a foundation informed by evolutionary psychology and evolutionary biology. On the empirical front, new sources of data, the energy crises of the 1970s, industry deregulation (especially of airlines, telecommunications, and electric utilities), the ongoing electronic revolution, trade globalization, and the strong economic growth since1990, have made, to the great delight of applied econometricians, for data sets with wonderfully enriched variation. All of these occurrences have allowed us, in our opinion, to present a much more detailed and satisfactory picture of the structure of consumer demand in the U.S. than was possible in the two earlier editions of CDUS. While the guiding framework throughout the exercise has been the conventional neoclassical theory of demand, our desire has been to interpret consumer behavior in a matrix molded by evolution. To this end, we have postulated that wants are endless, and that observed consumption behavior results from the interplay of a hierarchy of the five psychological needs identified by the late Abraham Maslow. While both our theoretical development of these notions and their empirical implementation are obviously very primitive and tentative at this stage, we feel strongly that the next big step in demand theory will be to ground the theory in the neural sciences. More prosaically, the major features of the present exercise includes:

1. Use of the Bergstrom–Chambers generalization of the flow- and state-adjustment models. 2. Analysis of four years of data from the consumer expenditure surveys conducted quarterly by the U.S. Bureau of Labor Statistics. 3. Estimation of price elasticities from cross-sectional data by joining the BLS consumer expenditure surveys with price information collected quarterly by the American Chambers of Commerce Research Association. L.D. Taylor, H.S. Houthakker, Consumer Demand in the United States, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-0510-9_23, 

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4. Estimation of a variety of complete systems of theoretically plausible demand functions, including the linear expenditure, almost ideal, and indirect- and directaddilog demand systems. 5. Development and estimation of an additive system of double-logarithmic demand functions. 6. Use of quantile regression in estimation. 7. Development of a new family of functions for describing the distribution of income. 8. Extensive tests of the weak and strong axioms of revealed preference using international data. Since results have been extensively summarized in the chapters in which they are obtained, we will emphasize in closing only those that, in our view, are of greatest overall importance. Heading the list is simply the many, many price elasticities that have been obtained. In the 1970 edition of CDUS, price elasticities were reported for only 44 of the 81 PCE categories for which models were estimated, and from this, it was fairly easy to conclude that relative prices are of only modest importance in explaining consumption expenditures. However, in the present exercise, price elasticities have been found in every time-series category that has been estimated! While two technical factors—relaxed statistical criteria and multiple solutions associated with the nonlinear estimation algorithm for the B-C and state-adjustment models—clearly contribute to this result, the factor that warrants most emphasis is the greatly increased relative price variation that has occurred since 1970. For it does not seem that relative prices suddenly awakened in 1970 as important predictors of consumption expenditures, but rather that the statistical variation needed to detect their significance was, as a general proposition, simply not available in the pre-1970 data. In short, the “natural” experiments induced by the 1970s energy crises, industry deregulation, etc., allow for a much truer picture of consumer behavior to be estimated. A second important contribution of the investigation has been the results obtained from the quarterly consumer expenditure surveys. Especially encouraging has been the results with the combined CES-ACCRA data sets that allow the estimation of both price and total-expenditure elasticities. While limited coverage of the ACCRA price surveys restricts analysis to high levels of aggregation, the price elasticities obtained are both plausible and statistically significant, and lend strong support for the investigation of data sets that combine the CES surveys with the BLS price collection efforts that underlay construction of the monthly consumer price index. Other important cross-sectional results include: (a). Strong evidence in favor of stable tastes and preferences. The results in support of this conclusion take two forms: (i) quarter-to-quarter stability of crosssectional total-expenditure (and, where estimated, price) elasticities, and (ii) quarter-to-quarter stability of the eigenvectors associated with the two largest principal components of 14 categories of CES expenditure.

23

Final Evaluation

505

(b). CES-based elasticities represent steady-state values. Evidence in support of this conclusion comes from the semi-panel nature of the CES surveys, which allows for the estimation of logarithmic flow-adjustment models. A third major result of the investigation has been the strong performance of the Bergstrom–Chambers model, which allows for the joint representation of flow and state adjustment. As in the two earlier editions of CDUS, dynamical adjustment is once again center stage, for of the more than 100 time-series models estimated, not one has a static form. Flow adjustment is almost invariably rapid in the B-C models, and habit formation clearly dominates inventory adjustment. Interestingly (but also probably somewhat surprisingly), the proportions of total-expenditure characterized by habit formation and inventory adjustment, about 63 and 27% in 2004, respectively, are little changed from their values calculated for 1964 in the 1970 edition of CDUS. A fourth result of the study that we feel is important to emphasize relates to the asymmetries that have appeared in the residuals of the cross-sectional equations, and which may be present in the time-series residuals as well. As noted in Chapter 3, asymmetry and fat tails have ominous implications for any form of least-squares estimation because of the fact that Gaussian measures of probability cannot deal with power-law events (i.e., events that, by conventional measures of variability, are hundreds, or even thousands, of standard deviations away from the mean). While we have no ready explanation for these curious phenomena, we caution researchers to be on the lookout for their presence, and perhaps to keep a robust method of estimation in mind as an alternative to least squares. Our final point relates to the total-expenditure elasticities that have been obtained, for it is these—specifically, ones that are in excess of 1—that ultimately drive economic growth. In the General Theory, Keynes felt that, because of a falling marginal propensity to consume in conjunction with a decreasing marginal efficiency of capital, economic growth would eventually come to an end. Events since WW2 have clearly shown Keynes to have been overly pessimistic. Where he was wrong, in our view, was his assumption that decreasing marginal utility of income would ultimately lead to a consumption bliss point. Empirically, there is no evidence of this, but actually the contrary, that wants, rather than being finite, are in fact endless, that once a want is satisfied, a new, higher-order want (often simply a variation on lowerorder themes) takes its place. And it is this phenomenon, expressed in income (and total-expenditure) elasticities that are greater than 1, that drives economic growth.1 With the foregoing in mind, it is useful to return to the tabulations of the CES and NIPA total-expenditure elasticities presented in Tables 18.1–18.4. As noted in that chapter, what is most significant is the broad agreement as to the consumption categories with total-expenditure elasticities that are greater than 1, namely, for 1 “Drives economic growth” in the sense that it is expected propensities to consume out of future income that are the critical determinants of investment in the new capacity to produce, hence the importance for investment of total-expenditure elasticities that are greater than 1. See Taylor (2000, Chapters 12 and 13).

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transportation, shelter, health care, recreation, education, and religion and welfare. Health care, recreation, and education (and even religious and welfare) expenditures should naturally be expected to be have high total-expenditure elasticities, but, as transportation (specifically, automobiles and activities associated with them) and housing have fueled economic growth for decades, one might think that they would be increasingly viewed as core necessities, and thus, if anything, becoming drags on economic growth. Their estimated elasticities, however, suggest otherwise. In turning our study of consumer demand in the U.S. over to readers for what is almost certainly a final time, we feel it important to emphasize that the study of consumer behavior remains an extremely fruitful area of research. While it might have seemed 50 years ago that the theory of consumer choice was essentially completed doctrine, this has not turned out to be the case. And while we are generally pleased with the large array of price and total-expenditure elasticities that we have been able to present, we wish we had a better explanation as to why their values are what they are. In reflecting on his genetic studies of bacteria, the great French Nobel molecular geneticist Francois Jacob distinguished between two categories of scientific investigation: day science and night science. Day science is rational, logical, and pragmatic, and is carried forward by precisely designed experiments. “Day science employs reasoning that meshes like gears, and achieves results with the force of certainty.” Night science, in contrast, “is a sort of workshop of the possible, where are elaborated what will be the building materials of science. Where hypotheses take the form of vague presentiments, of hazy sensations.”2 The present study, in its attempts to place the theory of consumer demand on a neurobiological foundation, has obviously been an exercise in night science. Explaining why price and total-expenditure elasticities have the values they do, in our opinion, awaits extension of these attempts to day science.

2

Jacob (1988, pp. 296–297), as quoted by Kandel (2006, p. 240).

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Index

A Access/usage framework, 328–334 ACCRA price data, 86, 107–126, 143, 214, 219, 220 Addiction, 18, 32, 36, 56, 209 Additive double-log demand system, 139–145, 403 Additivity, 140, 141, 142 Almost ideal demand system (AIDS), 2, 4, 128, 129, 136 Asymmetrical distributions of residuals, 60, 72, 73, 122, 123, 147–155, 158, 203, 211, 217–218, 382–387 B Behavioral economics, 2, 23, 26, 46 Bergstrom–Chambers (B-C) model, 238, 240, 241, 244–246, 252, 253, 287, 359, 360–368, 410, 411–412, 501, 503 Bilateral Pareto distribution, 72, 74 Bilateral Pareto regression model, 70–72 Brain structure alpha brain, 26 beta brain, 26 gamma brain, 27 Budget surveys, 79, 107–108, 109, 167, 168, 169 See also Consumer expenditure surveys (CES), BLS C Capital gains, 420–422 CES-ACCRA data set, 87, 88, 140–144, 147, 397, 398, 404, 408–409, 502 Co-integration framework, 169 Conic distributions, 461–499 Conic-linear distribution, 472–473 Conic-quadratic distribution, 468–472 Conservative energy system, 11–13

Consumer choice, conventional theory of, 9–10 Consumer expenditure categories, CES alcoholic beverages, 87, 88, 96, 176, 205, 208, 211, 390, 391 apparel and services men & boys, 88, 189, 205, 208, 211, 390, 391 women & girls, 88, 190, 205, 208, 211, 390, 391 cash contributions, 87, 88, 201, 205, 208, 211, 390, 391 education, 87, 88, 96–97, 198, 205, 208, 211, 390, 391, 400 entertainment, 87, 88, 195, 205, 208, 211, 390, 391, 400 food food away from home, 88 food consumed at home, 88, 123, 159, 174, 205, 208, 211, 221, 390, 391 health care, 88, 119, 194, 205, 208, 211, 390, 391, 393, 400, 409 housing electricity, 88, 184, 205, 208, 211, 390, 391, 400 household furnishings & equipment, 88, 188, 205, 208, 211, 390 household operation, 88, 187, 205, 208, 211, 390, 400 natural gas, 88, 183, 205, 208, 211, 291, 390, 391, 400 other lodging, 88, 181, 205, 208, 211, 390, 391, 400 owned dwellings, 88, 179, 205, 208, 211, 390, 391, 400 rented dwellings, 88, 180, 205, 208, 211, 390, 391, 400 shelter, 88, 96–97, 149, 177, 187, 205, 208, 211, 390, 391

517

518 Consumer expenditure categories, CES (cont.) telephone services, 88, 185, 205, 208, 211, 390, 391 utilities water & other public services, 88, 186, 205, 208, 211, 390, 391 miscellaneous, 88, 200, 205, 208, 211, 390 personal care products & services, 88, 196, 205, 208, 211, 390, 391 personal insurance & pensions, 87, 88, 202, 205, 208, 211, 390, 391 reading, 87, 88, 96–97, 197, 205, 208, 211, 390, 391, 400 tobacco products & supplies, 88, 199, 205, 208, 211, 390, 391 transportation gasoline & motor oil, 88, 193, 205, 208, 211, 390, 391 Consumer expenditure surveys (CES), BLS, 79–88, 90, 91, 107–126, 128, 132, 154, 213, 267, 382, 501 Consumption behavior and the pursuit of happiness, 52–54 Consumption capital, 24, 33 Consumption expenditure categories, NIPA annual admissions to specified spectator amusements, 236, 320, 364 airline, 236, 314, 364 alcoholic beverages purchased for off-premise consumption, 234, 278, 360 bank service charges, trust services, and safe deposit box rental, 235, 304, 362 barbershops, beauty parlors, and health clubs, 235, 284, 361 books and maps, 236, 316, 364 bridge, tunnel, ferry, and road tolls, 236, 311, 363 brokerage charges and investment counseling, 235, 303, 362 bus, 236, 314, 363 china, glassware, tableware, and utensils, 235, 288, 361 cleaning and polishing preparations, and miscellaneous household supplies and paper products, 235, 289, 361 cleaning, storage, and repair of clothing and shoes, 234, 281 clothing and accessories except shoes, 234, 280, 360

Index clothing, accessories, and jewelry, 234, 279, 360 clubs and fraternal organizations, 236, 322, 364 commercial participant amusements, 236, 322, 365 computers, peripherals, and software, 236, 318, 364 dentists, 235, 297, 362 domestic service, 235, 294, 362 drug preparations and sundries, 235, 295, 362 education and research, 236, 324, 365 electricity, 235, 291, 362 expenditures abroad by U.S. residents, 236, 327, 365 expense of handling life insurance and pension plans, 235, 305, 363 flowers, seeds, and potted plants, 236, 320, 364 food and tobacco, 234, 274, 360 food excluding alcoholic beverages, 234, 277, 360 food furnished to employees (including military), 234, 273, 360 food produced and consumed on farms, 234, 273, 360 food purchased for off-premise consumption, 234, 275, 360 foreign travel and other, net, 236, 326, 365 foreign travel by U.S. residents, 236, 326, 365 fuel oil and coal, 235, 292, 362 funeral and burial expenses, 235, 306, 363 furniture, including mattresses and bedsprings, 235, 287, 361 gas, 235, 291, 362 gasoline and oil, 236, 311, 363 government, 235, 300, 362 health insurance, 235, 300, 362 higher education, 236, 324, 365 hospitals and nursing homes, 235, 298, 362 hospitals, 235, 299, 362 household operation, 235, 286, 361 household utilities, 235, 290, 362 housing, 235, 284, 361 income loss, 235, 302, 362 insurance, 236, 312, 363 jewelry and watches, 234, 282

Index

519 kitchen and other household appliances, 235, 287, 361 legal services, 235, 305, 363 legitimate theaters and opera, and entertainments of nonprofitinstitutions (except athletics), 236, 321, 364 less: expenditures in the United States by nonresidents, 326, 365 less: personal remittances in kind to nonresidents, 326, 365 magazines, newspapers, and sheet music, 236, 317, 364 mass transit systems, 236, 312, 363 medical care and hospitalization, 235, 301, 362 medical care, 235, 295, 362 men’s and boys’, 234, 280, 361 motion picture theaters, 236, 320, 364 net purchases of used autos, 236, 308 new autos, 236, 308, 363 non-durable toys and sport supplies, 236, 317, 364 nonprofit, 235, 299, 362 nursery, elementary, and secondary schools, 236, 324, 365 nursing homes, 235, 300, 362 ophthalmic products and orthopedic appliances, 235, 296, 362 other alcoholic beverages, 234, 278, 360 other durable house furnishings, 235, 288, 361 other motor vehicles, 236, 309, 363 other professional services, 235, 298, 362 other, 234, 235, 236, 282, 286, 294, 306, 315, 323, 325, 361, 362, 363, 364, 365 owner-occupied nonfarm dwellings– space rent, 235, 284, 361 pari-mutuel net receipts, 236, 323, 365 personal business, 235, 303, 362 personal care, 235, 283, 361 physicians, 235, 296, 362 proprietary, 235, 299, 362 purchased intercity transportation, 236, 313, 363 purchased local transportation, 236, 312, 363 purchased meals and beverages, 234, 275, 360 radio and television repair, 236, 319, 364 railway, 236, 314, 363

recreation, 236, 316, 364 religious and welfare activities, 236, 326, 365 rental value of farm dwellings, 235, 285, 361 repair, greasing, washing, parking, storage, rental, and leasing, 236, 310, 363 semi-durable house furnishings, 235, 289, 361 services furnished without payment by financial intermediaries except lifeinsurance carriers, 235, 304, 363 shoes, 234, 279, 360 spectator sports, 236, 321, 364 standard clothing issued to military personnel, 234, 361 stationery and writing supplies, 235, 290, 362 taxicab, 236, 313, 363 telephone and telegraph, 235, 292, 362 tenant-occupied nonfarm dwellings—rent, 235, 361 tires, tubes, accessories, and other parts, 236, 310, 363 tobacco products, 234, 276, 360 toilet articles and preparations, 235, 283, 361 transportation, 235, 307, 363 user-operated transportation, 235, 308, 363 video and audio goods, including musical instruments, 236, 318, 364 video and audio goods, including musical instruments, and computer goods, 236, 318, 364 water and other sanitary services, 235, 291, 362 wheel goods, sports and photographic equipment, boats, and pleasure aircraft, 236, 317, 364 women’s and children’s, 234, 281, 360 workers’ compensation, 235, 302, 362 quarterly clothing and shoes, 257 durable goods, 252–253 electricity and gas, 261–262 food, 256 fuel oil and coal, 258 furniture and household equipment, 254 gasoline and oil, 258

520 Consumption expenditure categories, (cont.) gasoline, fuel oil, and other energy goods, 257 household operation, 261–263 housing, 260–261 medical care, 263–264 motor vehicles and parts, 253–254 non-durable goods, 255–256 other durable goods, 254–255 other household operation, 262–263 other non-durable goods, 258–259 other services, 265 recreation, 264–265 services, 259–260 transportation, 263 Control variables, 85, 173 Copacetic equilibrium, 46, 47, 52, 55, 57 Cross-sectional data consumer expenditures, 63–64, 72, 79, 85–88, 216–217 consumer prices, 108, 109, 158 D Depreciation rate, 17, 21, 247, 252, 253, 259, 274, 280, 287, 299, 308, 354, 359–360 Direct addilog model, 4, 131, 132–134, 138, 216, 218 Dynamics consumption, 29–35, 37–40 mechanical-time, 13, 14 real-time, 12, 13, 14, 15 E Elasticities income, 40, 48, 102, 107–126, 167–168, 210, 217, 350 long-run, 158, 168, 169, 170, 304, 305–306, 357 price, 48–49, 107–126, 150–152, 162, 163, 164, 165, 166, 215–216, 218, 350, 403–409 short-run, 158, 170–171 total-expenditure, 87, 128, 143–144, 150–152, 204–210, 211, 215–216, 220–229, 376–381, 399–403, 407–409 Emotions and consumer behavior, 50–52 Engel curves, 41, 74, 108, 123–125, 154, 171, 173–212, 214, 219, 220, 342, 397 Equation systems, 1, 135, 267, 408

Index F Flexible functional forms, 1, 127 Flow adjustment, 28, 29, 244, 252, 321, 503 Flow-adjustment models, 15–22, 157–171 Flow of Funds Accounts, 417–420, 421, 422 G Gabor’s theorem, 347, 351 Gauss–Markov theorem, 59, 61, 72, 73, 385, 396 Generalized least squares, 122, 249, 253, 256, 272, 274, 287, 289, 382, 398 Gorman polar form, 127, 128 rank-3 models, 125 H Haavelmo paradigm, 13, 14 Habit formation, 2, 5, 18, 25, 54, 56, 106, 170, 259, 260, 263, 267, 269, 278, 280, 282, 293, 296, 301, 309, 321, 365, 368, 440, 448, 450, 451, 459, 503 Heteroscedastic error terms, 120 Hicks–Allen aggregation, 144 Hierarchical needs, 42–49 See also Lexicographical preferences Household production functions, 49, 118, 120, 217 Houthakker–Taylor models, 2 I Indirect addilog model, 1, 4, 131–132, 133, 135, 139, 140, 143, 215, 218 Infra-marginal premium, 341, 346, 347, 348, 349, 350, 351 Integrable demand systems, 2, 215 See also Theoretically plausible demand functions Inter-Agency Growth Study, 233 K Kernel smoothing, 59, 60, 64, 65, 72 L Laplace distribution, 62, 63 Law of Pareto, 66–75 See also Pareto distribution Least absolute error (LAE) estimation, 61, 62, 71 Least squares estimation, 9, 59, 60, 72, 73, 76, 122, 123, 147, 213, 217, 382, 396, 503 Lexicographical preferences, 46 See also Hierarchical needs Life Cycle Model, 41, 80

Index Linear expenditure system (LES), 10, 128–131, 134, 267–272, 409 Long-tailed distributions, 451 M Marginal utilities, 11, 12, 14, 15, 19, 136 Maslovian needs, 23, 42–49, 50, 99, 206, 207, 208, 209, 210, 377–381 See also Wants; Hierarchical needs Mean demand functions, 345, 346 Median regression, 64, 65, 148, 203, 217 Modal regression, 65, 148 Multi-part tariffs, 340–351 N National Income and Product Accounts (NIPA), 15–16, 49, 80, 220, 233–250, 258, 293, 350, 360, 377, 382, 389, 390, 391, 392, 393, 394, 395, 397–399, 400, 401, 402, 404, 405, 411, 417, 436, 503 Needs (Maslovian) community and affection, 44 esteem, 44–45 physiological, 43 security, 43 self-actualization, 45 Neoclassical demand theory, 11–13, 215, 501 See also Consumer choice, conventional theory of Network externality, 336 Neurons (neural network), 23, 25, 27, 29, 42, 55 Neurosciences, 3, 23, 25–26, 46 Non-linear estimation, 137, 248, 260 O Opponent processes, 31–35, 36, 54 P Pareto distribution, 66, 70, 72, 74, 452, 463 See also Law of Pareto Penetration function, 337 Perl model, 328 Permanent income hypothesis, 80, 120 Position goods, 45, 48, 52–54 Principal component analysis, 87, 89–102, 131, 213–214, 218–219, 396, 502 Q Quantile regression, 59–76, 123, 147–155, 158, 168, 173, 210, 213, 217–218, 385, 396, 398, 399, 502 Quest for novelty, 24, 57, 399

521 R Rationality, 28, 41–42 Relative income hypothesis, 80 Repression (price), 12–13 Revealed preference (tests of weak and strong axioms), 427, 428–435, 502 Rotterdam system of demand functions, 1, 127 S Salted-peanut syndrome, 32 SAS, 84, 91, 136, 248, 250, 254, 274, 448 Satiation, 25, 33, 56, 57, 105, 346 Saving, 40, 120, 128, 415–423, 443, 444 Scaling (power) distributions, 60, 456 Slutsky conditions, 10 Stability dynamic, 48, 418–419 tastes and preferences, 48 State adjustment, 168, 248, 249, 252, 308, 319, 503 See also Stock adjustment State-adjustment models, 16, 18, 19, 25, 30–31, 33, 157, 170, 216, 237–246, 253, 269, 273, 274, 276, 277, 289, 291, 292, 293, 297, 298, 299, 307, 308, 311, 312, 313, 314, 316, 317, 320, 321, 323, 326, 365, 385, 416, 502 See also Stock-adjustment models State variables, 14, 16, 17, 18, 19, 25, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 238, 239, 244–246, 248, 268, 289, 293, 294, 356, 416 Stock adjustment, 2, 16, 18, 37, 170, 254, 256, 258, 272, 274, 280, 286, 297, 307, 308, 310, 365 See also State adjustment Stock adjustment models, see state-adjustment models Stone–Geary–Samuelson utility function, 14, 128, 267 See also Linear expenditure system (LES) T Theoretically plausible demand functions, 127–138 Translog demand systems, 1, 127 U Utility functions, 9, 10, 11, 14, 49, 106, 127, 128, 131, 132, 135, 144, 267, 329, 428, 429, 434, 479 V VOIP, 337, 338, 339, 340

522 W Wants, 24, 28, 37, 46, 52, 55, 56, 131, 212, 219, 376–381, 396, 407–409, 503

Index See also Maslovian needs; Hierarchical needs Willingness-to-pay, 336, 340