Competition, Collusion, and Game Theory ALDINE TREATISES IN MODERN ECONOMICS edited by Harry G. Johnson University of Chicago and London School Of Economics Competition, Collusion, and Game Theory

LESTER G. TELSER University of Chicago

P ALGRAVE MACMILLAN ABOUT THE AUTHOR

Lester G. Telser, Professor of Economics at the University of Chicago, received his Ph.D. from that university in 1956. One of the world's leading mathematical economists, he has been a Visiting Research Fellow, Cowles Foundation for Research in Economics, Yale University; Faculty Research Fellow, Ford Foundation; and Assistant Professor of Economics, Iowa State University.

ISBN 978-1-349-01540-5 ISBN 978-1-349-01538-2 (eBook) DOI 10.1007/978-1-349-01538-2

© Lester G. Telser 1971 Softcover reprint of the hardcover 1St edition 1971 978-0-333-13644-7

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First published in the U.S.A. 1972 by Aldine • Atherton, Inc. First published in the United Kingdom 1972

Published by THE MACMILLAN PRESS LTD London and Basingstoke Associated companies in New York Toronto Dublin Melbourne Johannesburg and Madras SBN 333 13644 6 TO JOSHUA AND TAMAR Foreword

For only a little over a decade, economic theorists have been working on a new and fundamental approach to the theory of competition and market structure, an approach inspired by appreciation of the earlier work of Edgeworth and Bohm-Bawerk and making use of the new tools of the theory of games as developed by von Neumann and Morgenstern. This new approach bases itself on the analysis of competitive behavior and its impli• cations for the characteristics of market equilibrium rather than on assump• tions about the characteristics of competitive and monopolistic markets. Its central concept is "the theory of the core of the market," and it is concerned, very broadly speaking, with the conditions under which markets will or will not achieve the characteristics of uniform prices and welfare optimality posited by traditional theory. This concern entails, among other things, a shift of emphasis from the prevailing concentration on oligopoly as a type of market structure and on advertising as a weapon of competition to the influence on the outcome of market processes of such factors as number of traders on the two sides of the market, transactions costs, brokerage, the way in which firms form their expectations of future demand, and the costs of collusion. Professor Lester' G. Telser has been at the forefront of the development of the new approach and is one of the few practitioners of it capable of com• municating to his fellow economists its theoretical techniques and, more important, its implications for the empirical analysis of market phenomena. This he does both by the construction and analysis of simple economic problems to illustrate the theory and by the presentation of empirical research of his own designed to formulate and test propositions suggested by the new "theory of the core" approach to market analysis. His own introduction provides a sufficiently concise summary of the scope of the book to make it vii Vlll Foreword unnecessary for me to attempt a still briefer summary here. Suffice it to say that this is the first book to present a comprehensive synthetic overview of an important new line of development in fundamental economic theory and that I am delighted to be associated with its publication as an Aldine Treatise.

HARRY G. JOHNSON Acknowledgments

The research presented in this book has received the generous support of the National Science Foundation, initiated with grant GS-365 in 1963 and continued with grant GS-1783 in 1966. This financial aid was indispensable in carrying out the work. It made possible my sojourn as a visiting research fellow at the Cowles Foundation for Research in Economics for the academic year 1964-65 where I first learned about the theory of the core. In 1969-70 I received a Ford Faculty Research Fellowship, which enabled me to finish this book. During this time I worked at the Center for Operations Research and Econometrics at the Catholic University of Louvain, Belgium. Much of the material in this book has been exposed to my students in the course "Theories of Competition," which I have taught in the economics department at the University of Chicago for the past five years. The com• ments of the students in this course have taught me a great deal; of particular helpfulness were those of Uri Ben-Zion, Y o ram Peles, and Donald Parsons. I have also been fortunate in having the aid of several very able research assistants. E. H. Thornber gave invaluable assistance in carrying out the empirical analysis reported in chapter VII. The work in chapter VIII was done with the assistance of Uri Ben-Zion, Harry Bloch, Josef May, and Stan Horowitz. Carl Berliner helped with appendix 4 of chapter VIII. I am also very grateful to many of my professional colleagues for their comments and criticisms of various parts of the book. In particular, Lloyd Shapley and Herbert Scarf saved me from some serious errors in chapter II when I presented this material at the Mathematical Social Science Board Seminar on Game Theory at the RAND Corporation, Santa Monica, California, in June 1969. The research in chapter VIII has benefited from the comments by H. Gregg Lewis, Walter Oi, George Stigler, and Leonard Weiss. Zvi Griliches has read and commented on both chapters VII and IX x Acknowledgements VIII. Yoram Barzel read with painstaking care chapters I, V, VII, and VIII. Much of the improvement in the exposition was inspired by the comments of Harry Johnson and Robert Wesner. Charles Cox and Howard Marvel eliminated errors in the final stages by reading the proofs and checking the index. I must assume, however, the sole responsibility for any errors and shortcomings in this book and must absolve all of those mentioned from any blame for these. One other person has been a constant source of encouragement as well as a sounding board for my ideas-my wife, Sylvia. The burden of editing and preparing the manuscript for publication, of endless checking, and of polishing the style was lightened thanks to her help. Contents

Foreword vii Acknowledgments xi Introduction xiii

I. APPLICATIONS OF CORE THEORY TO MARKET EXCHANGE 3 I. Introduction 3 2. Consumer Surplus and Tmnsferable Utility 4 3. Some Simple Tmding Situations 11 4. m Owners and n Nonowners 19 5. The Basic Core Constraints 28 6. Market Efficiency and Honest Brokers 31 7. Multiunit Trade 37 8. Increasing Returns and Public Goods from the Viewpoint of the Core 48 9. A Brief Historical Note 57 Appendix: Consumer Surplus 62

II. FURTHER ApPLICATIONS OF CORE THEORY TO MARKET EXCHANGE 68 1. Introduction 68 2. Balanced Collections of Coalitions 70 3. Empty Cores 78 4. The Feasibility of Trade 88 5. Group Rationality with Multiunit Trade 94 6. Competition and Numbers 104 7. The Number of Traders and the Emptiness of the Core 108 8. Conclusions 117

III. APPLICATIONS OF THE CoRE TO OLIGOPOLY 119 1. Introduction 119 2. Properties of the Core under Constant Returns 124 3. The Cournot-Nash Theory of Duopoly for Finite Horizons 131 4. The Coumot-Nash Theory of Duopoly for Infinite Horizons 142 xii Contents

IV. THEORIES OF EXPECTATIONS FOR N CoMPETING FIRMS 146 1. Introduction 146 2. Expectation Models with Quantity as the Policy Variable 149 3. Expectation Models for Price as the Policy Variable 164 4. Summary 172

V. COMPETITION OR COLLUSION? 175 l. Introduction 175 2. The Nature of Competition and Collusion 176 3. Equilibrium with Product Variety Illustrated for Spatial Competition 184 4. The Costs of Maintaining Collusion 192 5. Sharing the Collusive Return 206

VI. THE MONOPOLY AND CoURNOT-NASH EQUILffiRIA UNDER DYNAMIC CoNDITIONS 218 l. Introduction 218 2. Dynamic Demand Relations 221 3. Some Fundamentals on Optimal Policies 226 4. The Solvability of Certain Linear Equations 242 5. Properties of the Cooperative and Noncooperative Dynamic Equilibria 250 6. Conclusions 272

VII. EsTIMATES OF DEMAND, PRICE PoLICY, AND THE RATIO OF PRICE TO MARGINAL CosT BY BRAND FOR SELECTED CoNSUMER GooDS 274 l. Introduction 274 2. Estimates of the Demand Relation between Market Share and Prices 283 3. Estimates of the Relations among Competing Prices 294 4. Competition in a New Product 299 5. Conclusions 305

VIII. SOME DETERMINANTS OF THE RETURNS TO MANUFACTURING INDUSTRIES 312 l. Introduction 312 2. Description of the Data and Some Simple Summary Statistics 316 3. Multiple Regression Analysis of the Census Data 324 4. An Analysis of Employment Turnover in Selected Manufacturing Industries 339 5. A Brief Survey of Findings by Other Investigators 352 6. Conclusions 356

Appendix 1 : Estimation of Payrolls, Annual Earnings, and Employment of Nonproduction Workers 357 Appendix 2: Description of the Samples 359 Appendix 3: The Two-Digit Industry Effects 361 Appendix 4: The Relative Size Distribution of Firms 363

References 367 Name Index 371 Subject Index 372 Introduction

Although the nature of a market and what happens there is surely a proper subject of economic analysis, the student will search the literature in vain for more than passing mention of this fundamental topic prior to 1881 when Edgeworth published his profound analysis of markets. The next important contribution did not appear until a decade later in Bohm-Bawerk's cele• brated study of a horse market containing the first rigorous constructions of market supply and demand schedules. This paucity of early analysis is all the more surprising when we recall that in the 1870s economics embarked on its modem rigorous course with the contributions of Jevons, Menger, and Walras. After Bohm-Bawerk a half century passed before the next major contribution, the publication of von Neumann and Morgenstern's Theory of Games (1944). But the reviewers found in game theory little of relevance for economics, and it was not until 1959 that Martin Shubik pointed out that Edgeworth's theory could be married to game theory to produce a formidable new approach to the study of competition. This approach is now known as the theory of the core. Nor is this all. In 1838 Cournot deve• loped a mathematical theory of competition generally condemned and mis• understood in most textbooks, which turned out to be the forerunner of the minimax theorem of game theory as applied to nonzero-sum games. This became clear shortly after J. F. Nash in 1950 published his work on equili• brium points, and economists became aware of their connection to Cournot's theory. Rigorous research into competition has been growing lustily only since 1959. This curious tale may engage the attention of the historian of economic thought, but it is not our further concern. Why is it that prior to the recent studies of competition economists paid so little attention to the foundations of their discipline? One can find much attention given to questions of monopoly, cartel, and competition, but xiii xiv Introduction virtually all of this literature takes for granted some of the intrinsic properties of markets and competition without properly understanding them. For example, the common textbook description of competition runs in terms of given prices that the individual is powerless to change. Thus, a firm is said to operate in a competitive market if changes in its rate of sales exert no perceptible influence on the price of the product. Another common descrip• tion of competition asserts that price equals marginal cost in a competitive market. The former approach often goes on to observe that individuals are powerless to affect prices in a competitive market if there are many traders, so competition results from large numbers. The equality between price and marginal cost raises more complicated problems, as witness Marshall's need to appeal to theories of externalities to explain some phenomena obviously inconsistent with the posited equality. At best these discussions of competition are merely vague and uninformative, and at worst they are positively erroneous. We need a theory that defines competition and then deduces its implications. The theory of the core provides that long sought for theory of competition capable of meeting this need. It will be helpful to sketch how core theory looks at a competitive market for a single good. Each participant is assumed to start with a given stock of the good and to have preferences describing the terms at wh,ich he is willing to change his holdings. He is also assumed to act independently in finding a set of trades that will make him as well-off as possible. Since every trader is activated by the same motive, each is constrained by the conflicting desires of the others. Also, no one is forced to trade, and all are free to make con• tracts with anyone at mutually acceptable and, therefore, legally enforceable terms. A group of traders of whatever size may also make contracts embody• ing mutually acceptable terms of exchange. The basic tool of analysis is called "the characteristic function" in game theory. It describes the greatest gain obtainable by a coalition of traders under the most adverse conditions. We imagine that a number of traders combine and attempt to allocate their initial holdings among themselves so as to be as well-off as possible. In these cir• cumstances the group faces the most adverse conditions when the members are confined to trade among themselves. Therefore, the valuation of the optimal allocation of their goods among themselves by the given coalition determines the value of the characteristic function for the coalition. A trader's imputation is the difference between his valuation of his initial holdings and his valuation of his final holdings so that his imputation measures his gains from trade. Since individuals are free to join any coalition and since there is a value of the characteristic function for every coalition, it is assumed that the trader's imputation must be large enough so that no coalition can prevent a proposed allocation of goods by offering him better terms. In other words, the imputations are said to be in the core of the market if they satisfy a system of inequalities. A systematic study of the properties of these inequalities becomes a theory of competition. Introduction xv In the first two chapters I focus on a single market where one good is exchanged for money. This model is the analogue of partial equilibrium analysis according to Marshall. Consequently, we can learn much about the structure and performance of markets. Some of the questions considered are as follows: When does a competitive equilibrium exist? This is equivalent to finding when a market has a nonempty core. When is there a set of trades capable of implementing the core constraints? When will there be a common price per unit in the market? Under what conditions will there be Pareto optimality? How do transactions costs affect the equilibrium? What is the role of brokers in a market? How do changes in the number of traders affect the equilibrium? How efficient is random contact among the traders? Can there be an equilibrium if trade is confined to coalitions consisting of pairs of traders? In answering many of these questions we shall find that the concept of "a balanced game" is of primary importance. Core theory can also throw. light on problems involving public goods and increasing returns. For example, with the latter it is easy to show the possi• bility of an empty core, which implies that there is no competitive equili• brium. Therefore, core theory can determine when increasing returns lead to the situation known in the classical literature as a natural monopoly. The classical theory derives the properties of the market equilibrium in terms of supply and demand schedules, but, as is shown by core theory, this procedure is not always correct even for a competitive market. This leads to the subjects forming the contents of chapters III and IV. Chapter III applies core theory to oligopoly by developing a relation between the core and the Cournot-Nash equilibrium. One of the interesting questions dealt with in this chapter is, When will duopolists collude rather than compete? The answer is related to the famous problem in game theory known as the priso• ner's dilemma. Two prisoners suspected of a crime are captured and ques• tioned separately. Each is told that if they both confess then they will receive light sentences, while if one confesses but the other does not then the full penalty of the law will fall on the nonconfessor. If neither confesses, then both must go free because of there being no evidence upon which to base a conviction. The analogy with duopoly is clear. If the duopolists collude, they can divide the monopoly return. If they compete, then at best they can obtain the imputation implied by the Cournot-Nash equilibrium. If they agree to collude but one of them cheats, then the cheater can temporarily obtain a larger return than his share of the maximum, joint monopoly return. When does it pay to collude? Chapter IV extends the study of the Cournot-Nash equilibrium to the case of n competing firms on the assumption of static demand combined with uncertainty. The uncertainty requires consideration of how the firms choose their current actions in response to their forecasts offuture conditions. The problem divides into two parts. In the first the n firms are assumed to make identical goods with constant marginal and average cost. We consider XVI Introduction three kinds of forecasting equations. The first, originally proposed by Cournot, asserts that the current level of demand is the best forecast of next period's level. It is shown that the Cournot forecasting equation gives a stable equilibrium for two firms but not for three or more. The second forecasting equation, called "adaptive expectations," makes the forecast a geometrically weighted moving average of past observations. In this case it is shown that there is a stable equilibrium with three or less firms but that with four or more the weights attached to past observations must approach zero with sufficient rapidity, depending on the number of firms, for an equili• brium to exist. The third forecasting equation assumes that each firm uses the arithmetic mean of past observations as its prediction of future values. It is shown that in this case there is a stable equilibrium for any number of firms. The preceding results pertain when the n firms make perfect substitutes permitting them each an independent choice of quantity but not of price. The second part of the analysis assumes imperfect substitutability among firms' products. If their products are imperfect substitutes and if for any positive rates of sale there is a corresponding set of maximum nonnegative prices, then it is shown that a stable equilibrium exists for Cournot expecta• tions for any number of firms. A fortiori the same is true of adaptive expecta• tions and arithmetic mean expectations. A central question in this investigation is when competition or collusion will prevail. Since collusion can secure the maximum monopoly return, one might think that it would always prevail. But, then, why is there not a world of monopolies? Fortunately for consumer welfare, collusion has its costs, and it is not true that the participants to a collusive agreement can always obtain a higher net return then they would under competition. The main topic of chapter V is a study of the determinants of the costs of collusion and the conditions under which competition will prevail. This chapter also studies the nature of competition resulting from product variety, by using a spatial model which assumes a homogeneous product that can be made by an arbitrary number of identical firms whose average production cost is a decreasing function of the output rate. Customers are uniformly distributed along a straight road of finite length so that the demand function relating the rate of purchase to the delivered price is everywhere the same. Unit transport costs are constant with respect to quantity and distance. It is shown that, depending on the level of demand, competition can result in a single firm at the midpoint of the road segment that it serves, implying an efficient spacing of the firms. But there is no marginal cost pricing, and firms compete directly for patronage only at the boundary points of their market territories. The last topic of chapter V is the question of how the monopoly return for a group of colluding firms is shared among them. There is no fully satis• factory theory of this problem. One can apply core theory, but the results Introduction XVll are disappointing. Indeed, there is an embarrassingly large number of imputation theories based on the core which, unfortunately, either give trivial results or for which the core is empty. A trivial result is when the value of the characteristic function is zero for all coalitions except the one including all of the firms, and for this the value of the characteristic function equals the maximum, joint, net monopoly revenue. This chapter concludes with a modest conjecture about the sharing of the collusive return that seems pro• mising because some of its implications correspond with observable phenomena. Chapter VI continues the analysis of chapter IV by extending and general• izing the results to dynamic demand relations. N firms are assumed to make competing durable goods for which the demand depends on current and expected quasirents. The term quasirent refers to the buyer's evaluation of the cost of the services of one unit of the durable good for one time period. The development of the monopoly and the Cournot-Nash equilibria requires more advanced mathematical techniques in a dynamic setting; hence, this chapter includes 'Some discussion of topics in functional analysis pertinent to the economic application. It is shown that the existence of a monopoly equilibrium implies the existence of a Cournot-Nash equilibrium. Further, for a very general class of forecasting equations embracing all those studied in chapter IV, it is shown that a Cournot-Nash equilibrium exists if there is a monopoly. equilibrium. Thus, the existence of a monopoly equilibrium imposes restrictions on the nature of the dynamic demand relations that have several implications. One interesting result bears on the choice between rental and sale of a durable good if there is only one supplier. We often observe that monopolists prefer rental to sale. The analysis shows that sale of a durable can never yield a higher monopoly return than rental when nonnegativity conditions on stocks and rates of sale are taken into account. The final two chapters are mostly empirical. Chapter VII gives estimates of the demand relation between market shares and prices using monthly data for the leading brands of four advertised consumer products: Minute Maid, Snow Crop, Libby's, and Birds Eye in frozen orange juice concentrate; Chase and Sanborn and Maxwell House in instant coffee and in regular coffee; Folger's and Hills Brothers in regular coffee; and a leading brand in instant mashed potatoes. These results give the own and cross price elastici• ties from which one can infer limits for the ratios of price to marginal cost. These empirical studies also give evidence for judging the relevance of some of the theory in chapter VI. The first three products were well established during the sample period. It is shown that market shares in them quickly respond to price change and that price differentials among the leading brands tend to remain constant in the long run although there is considerable short• run variation. The persistent long-run price differentials apparently reflect brand cost differentials. A different picture emerges for instant mashed potatoes, a relatively new product during the sample period. The leading xv III Introduction brand studied was the innovator here and had a high but falling market share, not so much because it lost absolute sales as because later entrants were able to attract new customers. Consequently, the leading brand share fell as the total market grew. The price policy of the firm producing the leading brand is consistent with the estimates of the demand relation. These estimates show that market share did not respond as quickly to price change as was true for the leading brands of the established goods. Thus the firm producing the leading brand of the new product behaved as if it had a temporary monopoly. Combining the evidence for all of the products gives a single coherent explanation of the competitive relation among the brands. The last chapter studies the determinants of the returns to manufacturing industries using data for more than 400 four-digit manufacturing industries from the 1963 Census of Manufactures. This chapter contains the evidence bearing on some of the theory in chapter V. It is shown that rates of return are a nonlinearly increasing function of the four-firm concentration ratio. This is demonstrated by stratifying the sample of industries into three strata: 0 ~ C < 25, 25 ~ C < 50, and 50 ~ C ~ 100, where C equals the share of industry sales held by the four leading firms. There is about the same number of firms in each stratum. The coefficient relating the rate of return to the concentration ratio rises gently from the first to the second stratum and is sharply higher in the stratum C ?: 50. An important problem in this empirical work was to obtain a proper measure of the firm's capital including its intangible as well as its tangible capital. Accounting measures generally include such tangible items as the original cost of plant, equipment, inventory, and land; but there are other components of a firm's true capital contributing to its return although they do not appear on its balance sheet. These include the firm's outlays on research and development, on advertising, and on the training of its employees in firm-specific skills. The latter outlays comprise part of the firm• specific human capital. The empirical work stresses this capital's contribu• tion to the firm's return in two ways: first, by directly relating the firm's return to proxy measures of specific human capital and, second, by indirectly studying the implications of this theory on quit and layoff rates. The chapter contains a careful study of the determinants of labor force turnover using Bureau of Labor Statistics data. Although all eight chapters provide a unified treatment of the subject, they do fall into three somewhat autonomous groups that may be read independently. Chapters I to III emphasize the core, chapters IV, VI, and VII stress oligopoly, and chapters V and VIII relate industry structure to rates of return and other variables. Chapter V also draws on the analysis in chapter III, and chapter VI generalizes the theory in chapter IV. Chapter VII depends on the theory in chapter VI, and chapter VIII relies on the material in chapter V. Since each chapter gives the appropriate references to the material required from elsewhere in the book, the reader may follow his Introduction xix own interest in choosing the order in which to read the material and still be able to follow the reasoning. The mathematical level varies among the chapters. The most mathe• matically advanced is chapter VI followed by chapter II. Chapter I is the most elementary. Chapter II requires understanding of matrix algebra, the theory of linear inequalities, and calculus. Chapters IV and VI make much use of linear difference and differential equations. Some knowledge of advanced calculus and the theory of functions would also be helpful in understanding some parts of chapter VI. It is fair to describe the mathe• maticallevel in this book as less demanding than much of modern mathe• matical economics since little use is made of measure theory or topology. This level is adopted not because one should shun the use of powerful tools on principle but because it is clumsy to use them if less powerful ones are adequate. In fact the approach in chapters I and II is deliberately chosen to avoid the use of measure theory so that certain economic, rather than tech• nical, questions can be placed in the forefront. Similarly, the theory of dynamic monopoly and Cournot-Nash equilibria in chapter VI works in discrete time to avoid some purely technical complications arising in con• tinuous time that have no economic relevance. A full-scale analysis of advertising is not included in the book, but the interested reader is invited to consult several of my published articles on this subject, which are listed in the References at the end of the book. Perhaps the two most important of these are the ones published in 1962 and in 1964.

Finally, it is necessary to describe the reference system used. Theorems and lemmas are numbered sequentially in each chapter beginning with number one. Each number in cross-reference is preceded by a roman numeral which refers to the chapter in which the theorem or lemma appeared. Thus, theorem 11.3 means theorem 3 in chapter II. Equations are numbered sequentially beginning with one in each section and are always given in parentheses. For example, (11) means equation number 11 in the same section, while 1.5 (11) would mean equation (11) in chapter I, section 5. Tables and figures are numbered sequentially by chapter beginning with one. Citations in text follow the style commonly used in scientific literature; full details are given in the References. These references are by no means comprehensive, but they include the material of particular pertinence to the book. The fact is that symbolism is useful because it makes things difficult. What we wish to know is what can be proved from what. Now, in the beginnings, everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy of correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. Bertrand Russell, Mathematics and the Metaphysicians