Theories of Imperfectly Competitive Markets

Springer-Verlag Berlin Heidelberg GmbH

Luis C. Corch6n

Theories of Imperfectly Competitive Markets

2nd Revised and Enlarged Edition

, Springer

Professor Luis C. Corch6n University of Madrid "Carlos III" Department of Economics CI. Madrid, 126 28903 Getafe, Madrid Spain E-mail: lcorchon@eco.uc3m.es

Originally published as Volume 442 of the series Lecture Notes in Economics and Mathematical Systems

ISBN 978-3-642-07435-6

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Corch6n; Luis C.: Theories of imperfectIy competitive markets 1 Luis C. Corch6n. - 2., rev. and enl. ed. ISBN 978-3-642-07435-6 ISBN 978-3-662-04498-8 (eBook) DOI 10.1007/978-3-662-04498-8

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This Book is dedicated to my parents and relatives.

FOREWORD TO THE FIRST EDITION

One of the most fascinating debates of our times is the discussion over the merits and capabilities of market economies. Very often, one sees strong endorsements to the idea that markets provide an efficient way of allocating resources. Some years ago, opposite views on this issue used to be very pop­ular (at least in Europe) and were held by similarly qualified people. In my opinion, the contribution of economics to this question can not be dismissed on the grounds that economics still in its infancy and that this question is a "practical" one (whatever this means). Economics started with similar naive ideas, two hundred years ago. In particular it has taken a long time to real­ize that competition does not work in such a smooth way as many classical writers thought it did, and that many facts can not be explained by the the­ory of perfectly competitive markets. This issue is explored at depth in the Introduction to this book. In this sense the contribution of the Theory of in­dustrial Organization has been to make a convincing case for the view that monopoly and oligopoly can persist in the long run in a world populated by many rational entrepreneurs. Despite of the fact that we are far from having a satisfactory theory of how markets work, progress has been immense, and we certainly understand why current theories are still not completely satisfactory. This book is devoted to explaining the basics of such theory by focusing on market structure. The reader interested in the internal organization of the firm can read the excellent book by Milgrom and Roberts "Economics, Organiza­tion and Management" (Prentice Hall, 1992).

A customary complaint against Industrial Organization (1.0. in the sequel) is that it is seemingly capable to explain everything. I must admit that I have never fully understood this criticism. Take, for instance, the venerable IS-LM Keynesian model. By choosing conveniently both the liquidity preference and the expectations of investors (both unobservable variables), any pair (income, interest rate) can be an equilibrium. The usefulness of the IS-LM model relied on the fact that, by assuming certain properties on the shape of the relevant functions, is able to predict the consequences of, say, an increase in the mon­etary supply. These properties are derived either as sufficient conditions for local stability or simply as "reasonable" assumptions. Moreover the IS-LM model gives some idea of the kind of public policy that will improve market equilibrium. The fact that variations of the basic model yielded different pre­dictions was never regarded to be a basic flaw of these models or to be a signal that the endeavor of building a satisfactory model was completely hopeless. In line with these ideas, in this book we emphasize the comparative statics and welfare properties of the models here presented. With respect to other

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books on 1. 0., this book emphasize both the need for an unifying model and the variety of specific results that can be drawn from it. More specifically:

1) Considerable effort has been taken to present a unifying -but simple­framework in which applications can be discussed (a measure of this effort is that assumptions and results presented in Chapter 1 are used in Chapter 5). Therefore, the student is not asked to understand an array of different models each tailored to a single application. Important results are presented as Propo­sitions and formal proofs -which, in most cases involve elementary mathe­matics only- are always offered. This involves being selective and therefore several important topics (i.e. durable good monopoly, advertising, R&D, su­pergames, asymmetric information, macroeconomic models, vertical integra­tion, etc.) have been omitted. The approach of this book is to present a few basic models using elementary techniques, such that the interested reader can continue her study without the need of further advice.

2) This book stress the economic (rather than the game-theoretical) con­tent of the topic but at the same time do not present game theory and 1. O. as separate branches. Thus, on the one hand, equilibrium concepts are presented with their game-theoretical background. On the other hand, considerable at­tention is devoted to applications of the basic ideas, specially to the areas of welfare and comparative statics.

This book is the outgrow of courses given in the last year of a five-years Bsc. Degree (licenciatura) in Economics, in the Ph.D. program (Q.E.D.), both at the University of Alicante, and at a first year graduate course given at the Institute for Advanced Studies in Vienna. My colleagues Jose Al­calde, Pablo Amoros, Carmen Bevia, Carmen Herrero, Ramon Faull-Oller, Miguel Gonzalez-Maestre, Francisco Marhuenda, Antonio Molina, Bernardo Moreno, Ignacio Ortuno-Ortin, Jorge Padilla, Martin Peitz, Juan Pintos and Antonio Villar corrected many inadequacies and offered interesting comments. In this book I have made extensive use of papers jointly written with Isabel Fradera, Miguel Gonziilez-Maestre, Andreu Mas-Colell, Jose A. Silva, Am­paro Urbano and Simon Wilkie. My thanks to all of them for the pleasure that I derived from our joint work. I am greatly indebted with Walter Trockel for his encouragement to write this book and many helpful remarks. Becky Rip­ping corrected my English as much as she could. My secretaries Mercedes Mateo and Vera Emmen contributed to my productivity in this book with their customary efficiency. My thanks to all of them. Finally, this foreword would be incomplete without acknowledging that I have learn a lot from read­ing other books on Industrial Organization, specially those of Friedman (1977 and 1983), Tirole (1990), Basu (1993), Martin (1993), Krouse (1990), Carlton and Perloff (1990) and the Handbook of Industrial Economics (1989). Also,

IX

I found the survey of Schmalensee in Economic Journal (1988) very useful. Perhaps the most influential book in the approach taken here is the splendid "Oligopoly and the Theory of Games" (1977) by Jim Friedman, which I read when I was a graduate student. I want to believe that the style of my book owes a great deal to the magnificent and deeply influential Friedman's book,

FOREWORD TO THE SECOND EDITION

In this new edition I have introduced some changes with respect to the first edition: I have corrected several imprecisions and I have added new sections and exercises. The main change is a whole new chapter on games of incom­plete information. The appendix to Chapter 2 is also new. I any case, tried to keep the style of the book.

I want to express my gratitude to Dr. Werner A. Mueller the editor in charge of the series for his encouragement. My colleagues liiigo lturbe­Ormaetxe, Ramon Fauli-Oller, Felix Marcos and Bernardo Moreno have read the entire manuscript and have offered many suggestions. Miguel Gonzruez­Maestre helped me with the appendix to Chapter 3 and Diego Moreno offered many suggestion regarding Chapter 6. Finally, my friend Carlos Belando helped me with the problems associated with computers. My thanks to all of them.

TABLE OF CONTENTS

INTRODUCTION 1

CHAPTER 1: NASH EQUILIBRIUM 7

1. 1 Basic Concepts in Game Theory 7 1. 2 Aggregative Games I: The Main Assumptions. 9 1. 3 Aggregative Games II: Existence, Uniqueness and Inefficiency

of Nash Equilibria 13 1. 4 Aggregative Games III: Stability of Nash Equilibrium 15 1. 5 Price-Setting Games 19 1. 6 Additional References 23 1. 7 Appendix: Further Results on the Stability of Nash Equilibrium 23 1. 8 Exercises 27

CHAPTER 2: COMPARATIVE STATICS 35

2. 1 Introduction 35 2. 2 Aggregative Games I: Effects of an increase in the Number of Players 36 2.3 Aggregative Games II: Effects ofa Shift in Payoff Functions 40 2. 4 Inflation Transmission in Oligopoly and Perfect Competition 42 2. 5 Comparative Statics in Price-Setting Games 45 2. 6 Additional References 49 2. 7 Appendix: The Assumption of Differentiable Payoff Functions 49 2. 8 Exercises 51

CHAPTER 3: WELFARE AND COURNOT COMPETITION 59

3. 1 Introduction 59 3.2 Welfare and Cournot Equilibrium 61 3. 3 Welfare and Entry 62 3.4 Welfare and Free Entry Equilibrium 64 3. 5 Profitability and Free Entry 70 3. 6 Oligopolistic Competition and Constrained Efficiency 72 3. 7 Additional References 75 3. 8 Appendix: International Trade Policy in Oligopolistic markets 75 3. 9 Exercises 83

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CHAPTER 4: MONOPOLISTIC COMPETITION 91

4. 1 Introduction 91 4. 2 The representative consumer model 92 4. 3 The representative consumer model: General results 95 4. 4 A model of a large group 98 4. 5 A model with many consumers and price-setting firms 100 4. 6 The limit points of monopolistic competition 102 4. 7 Additional references 105 4. 8 Appendix: The existence of optimal and equilibrium allocations. 105 4. 9 Exercises 108

CHAPTER 5: TWO STAGE GAMES 113

5. 1 Introduction 113 5.2 A general model of two stage games 114 5. 3 Stackelberg equilibrium and entry prevention 116 5.4 Pricing of public firms in oligopolistic markets. 121 5. 5 Divisionalization 123 5.6 Revelation games 125 5. 7 Choice oftechnique 129 5. 8 Additional references 132 5. 9 Exercises 132

CHAPTER 6: GAMES OF INCOMPLETE INFORMATION 141

6. 1 Introduction. 141 6. 2 The main concepts 141 6. 3 Bayesian equilibrium and Nash equilibrium 144 6. 4 Oligopoly under incomplete information 145 6. 5 Resource allocation mechanisms 148 6. 6 Allocation of an indivisible good 151 6. 7 Additional references 154 6. 8 Exercises 154

REFERENCES 157

INTRODUCTION

The ambition of the theory of imperfectly competitive markets is to explain the working of markets in which the issue of strategic interaction among firms is central. Our analysis of this problem will be based on equilibrium concepts borrowed from Game Theory. This research program arises several questions on its feasibility like the empirical relevance of the results, the substantial theoretical insights obtained in this way, etc. Unfortunately, most of these questions can not be answered in the short run. This book is written in the hope that this research strategy is meaningful, but about its final success no­body can tell. Another important question is if simpler models could deliver the essential insights offered by the theory of imperfectly competitive mar­kets. This Introduction will be devoted to argue that, currently, there is no alternative to the approach presented in this book.

Consider the following fact: A square inch of soil in the Explanada of Ali­cante (located in front of the sea, right in the middle of downtown) cost several times more than a square inch of soil in San Vicente del Raspeig (located sev­eral miles toward the interior of the peninsula).l How can we explain such a thing? First notice that because of the large quantity of possible traders involved in this market, we can safely assume that any agent has to accept the market price, i.e. is a price-taker. Therefore we can use the venerable theory of perfectly competitive markets. Let us represent in the same diagram both the supply and demand of soil in the Explanada de Alicante and in San Vicente del Raspeig. First, the demand for land in the Explanada is, surely, above the demand of land in San Vicente, due to the fact that people value a house near to the sea, etc. Second the supply of land in San Vicente del Raspeig (which by simplicity we take to be completely inelastic) is greater than the corresponding supply in the Explanada de Alicante (also assumed to be completely inelastic). The equilibrium price in each market is found by in­tersecting the corresponding demand and the supply curves. We find that the equilibrium price of the soil in the Explanada de Alicante is larger than the equilibrium price of soil in San Vicente del Raspeig. As theorists, we leave the matter here since we have provided an explanation that accounts for the qualitative features displayed by these markets. It is left for the econometri­cians to find adequate functional forms and to test the hypothesis. It is clear that many facts can be accounted by this type of explanation (i.e. why fiats tend to be more expensive in Cambridge, Massachusetts -especially if they are close to Harvard Square- than in Rochester, New York, etc.).

1 The University of AJicante is located, alas, in San Vicente del Raspeig and not in the Explanada.

2

There are other facts that need a more sophisticated explanation. For in­stance a soft drink can costs 40 cents in a supermarket but almost 2$ in the bar of a train. This fact can be explained as follows: First suppose that any distributor of a certain brand can obtain as many cans as she wished at a fixed price, say p. If there are many supermarkets competing for customers of a particular brand the assumption that supermarkets act as price-takers in the market vis-a-vis with consumers can be regarded as a reasonable approxima­tion. Assuming that there are negligible overhead costs supermarkets sell this particular brand at the marginal cost p (and thus p equals 40 cents). However the bar in a moving train has a monopoly. This means that there are no perfect substitutes of cans sold by this bar. On the one hand, the cans that you could bring by yourself are not cold and thus, are an imperfect substitute. On the other hand you can buy a can in a supermarket at the end of the trip, but this delay has certain cost, so, again, this is an imperfect substitute. Depending on the particular problem to be analyzed the demand curve for cans of a particu­lar soft drink in a train may be more or less steep, reflecting the possibilities of substitution. In any case it will not be perfectly horizontal. The theory of monopoly predicts that the monopoly price will be above both the marginal cost and the price of this brand in a supermarket. Again a careful quantifica­tion would be needed before such explanation is found "correct". Notice that in order to explain this fact we had to invoke the theory of perfect competition and the theory of monopoly.

Consider now the following puzzling face A direct flight from Barcelona to Tokyo costs about 2.500$ and it takes about 11 hours. However if you first fly from Barcelona to Bern and then from Bern to Tokyo the cost falls to 1.400$ but it takes more than 15 hours. It is difficult to reconcile this fact with any theory that assumes that European airlines behave as perfectly competitive agents. Fortunately economic theorists invented the theory of the discriminatory monopolist. Let us assume that European airlines behave like a single monopolist (an assumption that some Europeans consider very real­istic). Suppose that there are two basic types of consumers. On the one hand we have rich consumers for whom time is very valuable (i.e business execu­tives) and on the other hand there are ordinary people who are neither very rich nor have a high valuation of their time. The tariff explained above can be used by a profit-maximizer perfectly discriminatory monopolist in order to discriminate between rich people and the rest of the world.3

2 A similar example was pointed out to me by Joaquim Silvestre. 3 A more difficult task would be to explain why such price discrimination arises also in USA where airlines are very competitive, specially if we do not want to support the view that the realism of the assumptions does not matter.

3

Until recently, a good number of economists thought that, like in the pre­vious cases, most facts in real markets could be explained by a combination of the theories of perfect competition and the monopoly, with the necessary amendments. Moreover, some of these economists would argue that the con­sideration of imperfectly competitive equilibria could only bring noise and tend to obscure the analysis. However the plain truth is that there is nothing mysterious about imperfect competition. The main models of oligopolistic competition were discovered in the XIX century by Antoine-Agustin Cournot (1838) and Joseph Bertrand (1883). In retrospect, they are nothing but straight­forward applications of the central concept in the theory of non-cooperative games, namely Nash equilibrium, (re)discovered by John Nash (1951). Prod­uct differentiation was considered by Harold Hotelling (1929) and Edward Chamberlin (1933). Finally Heinrich von Stackelberg (1934) was a forerun­ner of Subgame Perfection, a concept formalized by Reinhard Selten (1975). We will see in this book that the consideration of non competitive behavior enriches considerably the possibilities open to our analysis. The basic points are:

Firstly, some predictions and assumptions of the theory of perfect compe­tition are at odds with well-known facts.

Secondly, some models of imperfect competition predict facts that can not be predicted by the theory of perfect competition.

Thirdly the model of imperfect competition provides a deeper understand­ing of competition. because it shows how perfect competition mayor may not arise as a limit case. It also gives us valuable insights about why market equi­librium is not optimal and hints about how to correct this. Finally it broadens the traditional concept of competition. Let us take these points in turn.

1. The inadequacy of the perfectly competitive model to deal with increas­ing returns is well-known. In spite of the efforts of the disciples of Mar­shall, perfect competition and economies of scale are incompatible. Also, the perfectly competitive model ofInternational Trade " .. was hard pressed to explain why so much International Trade takes place between similarly en­dowed countries and why these countries import and export goods emanating from the same industry or why direct foreign investment occurs with firms re­siding in one country acquiring ownership and control of productive facilities located elsewhere" (G. M. Grossman (1992), p. 1). See also the Appendix to Chapter 3). Finally, most markets lack auctioneers and therefore, it is not clear how prices are set if all agents are supposed to be price-takers.4 The con-

4 The literature on market games have studied models in which agents are price-setters. The main conclusion of this literature is that the Walrasian equilibrium is sustained as a Nash

4

sideration of imperfectly competitive markets disposes of all these difficulties at once.

2. An alternative motivation to analyze imperfectly competitive markets comes from the fact that they make predictions about the direction of change of certain endogenous variables (i.e. prices, output, etc.) that are impossible under perfect competition.5 For instance in imperfectly competitive markets, the following facts are possible:

a) Under smooth dynamics, outputs may follow chaotic trajectories. b) The entry of a new firm in a market increases the output of the incumbent

firm. c) The entry of a new firm in a market decreases the total output and in­

creases the equilibrium price. d) The entry of a new firm in a market increases the profits of the incumbent

firm. e) A decrease in the costs of a firm may decrease the equilibrium output

and profits of this firm and the aggregate output too. t) An increase in the demand might decrease either the aggregate output or

the market price. We do not know the empirical relevance of points a) - I). What we know

is that there is a single assumption, which we will call A.2, under which none of the above "paradoxes" can occur. This assumption (plus some other mild assumptions) also implies the existence, uniqueness and stability of a Nash equilibrium. However, the list of estrange possibilities is not over. Even if A.2 is postulated, the following paradoxical facts may happen in imperfectly competitive markets:

I) A decrease in the costs of all firms can decrease the output of the most efficient firm in the market.

II) A merger of two or more firms can decrease the profits of all merged firms.

III) The entry of a new firm in the market might decrease social welfare. A policy devoted to foster small firms might decrease social welfare.

IV) Even if the entry of a firm would raise social welfare, this entry might not be profitable.

equilibrium of a market game under very weak assumptions. However, unless additional as­sumptions are postulated, there are Nash equilibria other than the Walrasian (these equilibria correspond, roughly speaking, to fix-price equilibria). Thus, this literature provides only a weak support for the view that only Walrasian outcomes can be supported as Nash equilibria of market games with price-setting agents, see J. P. Benassy (1986). 5 Here we mean perfect competition in partial equilibrium framework. We remark that most of the models presented in this book are in the spirit of partial equilibrium, even though they could be interpreted as general equilibrium models.

5

V) Under free entry an increase in the demand can decrease the number of

active firms. VI) Under free entry, profits can be arbitrarily large. Large rates of profits

are also compatible with free entry. VII) The optimal tariff is not necessarily decreasing on the degree of com­

petition. VIII) The optimal pricing of a good produced by a public firm may be

above or below marginal cost. IX) Even if firms are identical, those firms maximizing profits do not nec­

essarily obtain more profits than the others. Also, profit-maximizing owners will offer contracts to managers that will not induce profit maximization.

X) Entry deterrence might be superior, from the point of view of social welfare, to free competition.

XI) An incumbent firm may have incentives to increase fixed costs. Com­petition may also have the effect of increasing the total cost.

XII) Firms may wish to create a very large number of divisions, even if there are no cost advantages to do so.

XIII) Profit maximizing firms may switch to a new technique which de­creases the productivity of all firms.

This is by no means an exhaustive list. 6 We remark that we do not know the empirical relevance of points a-f and I-XIII above. In my opinion, some of them appear to be sufficiently interesting to merit some applied research.

3. Perhaps, the most important reason to justify the approach taken in this book is that we are able to obtain new and deep theoretical insights. Among them, the following three are worth to notice:

a) Market equilibrium is not optimal. Under homogeneous product, firms underproduce in relation to the optimum, aggregate output is less than the op­timal one and free entry yields overentry. These consequences are not gener­ally true under product differentiation. However a policy that increases output is generally welfare enhancing.

fJ) Under some circumstances, perfect competition is achieved as a limit­ing case: Thus, as pointed out by Bertrand, under price competition" product

6 We also remark that the models presented in this book yield other "paradoxes" worth to be noticed. Among them we single out the following ones: i) An identical increase in costs in a perfectly competitive and an oligopolistic market might increase less the price in the oligopolistic market than in the perfectly competitive market. ii) A monopolists facing potential competition might not be able to obtain positive profits. iii) To be a (Stackelberg) follower might be preferable to be a (Stackelberg) leader. iv) Equilibrium in the stock mar­ket makes impossible that all firms are owned by the same person. v) Even though entry deterrence may yield greater profits in the long run, non myopic, fully rational incumbents may accommodate entry in each period. vi) Rational consumers may elect unanimously a regulator who favors firms over consumers even if there is a pro-consumer candidate.

6

homogeneity, and constant returns to scale, if the game is played in one stage, perfect competition is the outcome of price competition. However if the game is played in two stages or there are decreasing returns to scale or product dif­ferentiation, the result no longer holds. Under quantity competition, product homogeneity and free entry, perfect competition arises when the minimum efficient scale of entry tends to zero, as first suggested by Cournot. However, increasing the number of firms in a market does not necessarily imply that perfect competition is achieved in the limit. Finally, under product differenti­ation, free entry and small efficient scale might not be enough to yield perfect competition.

'Y) The classical idea of competition is broaden, paying attention to ques­tions like timing (Le. stage games) and non-price competition (quality, loca­tion, design, etc.).

Bon Appetit!

CHAPTER 1. NASH EQUILIBRIUM

Abstract: Games and aggregative games. Characterization of games with strategic substitutes (Proposition 1.1). Existence (Proposition 1.2), unique­ness (Proposition 1.3), inefficiency (Proposition 1.4) and stability (Proposi­tions 1.5-6) of Nash Equilibrium in aggregative games with strategic sub­stitutes. Characterization of games with strategic complements (Proposition 1.7). Existence (Proposition 1.8), uniqueness (Proposition 1.9), inefficiency (Proposition 1.10) and stability (Proposition 1.11) of Nash Equilibrium in games with strategic complements. Appendix: Further results on the stability of Nash Equilibrium (Propositions 1.12-14).

1 BASIC CONCEPTS IN GAME THEORY

In this section we will review some important concepts from the theory of non-cooperative games. Those readers familiar with these notions may skip this section or might do a cursory reading.

A game in normal form, which, for simplicity, will be referred to as a game in the sequel, is a tuple (Ui( ),Si)iEI where I is the set of players (assumed to be finite), Si is the strategy set of player i E I which is the set of feasible actions for this player and Ui ( ) is the payoff (or utility) function of player i. Ui ( ) maps the strategies of all players into real numbers. Formally, Ui : X iE1

Si -+ lR. This function represents the preferences of player i over the actions taken by all players.

An important part of this book is devoted to the study of a particular class of games, in which the payoff function of each player can be written as a function of her own strategy (assumed to be one-dimensional) and the sum of the strategies of all players. This assumption has been called the Aggregation Axiom by Dubey, Mas-Colell and Shubik (1980), p. 346, and the correspond­ing games are called Aggregative Games. Clearly, the purpose of such an assumption is to give additional structure to the games under consideration; "Games with the above property clearly have much more of a structure than a game selected at random" (Shubik (1984) p. 325). Since the concept of an aggregative game will play an important role in this book, we will state it formally:

8

Definition 1.1 An Aggregative Game (UiO, Si)iEI consists of

a) A set of players (also called agents) I = {I, 2, ... , n}. b) A collection of strategy sets Si = 1R+, i = 1, ... , n. c) A collection of payoff functions Ui : Si x 1R+ --+ IR of the form

Ui(xi,x)where Xi E Siand X = LXj. jEI

In words, in an aggregative game, the Aggregation Axiom holds so that the (one dimensional) strategies of the players can be aggregated in an additive way. We remark that a more general definition of an aggregative game could be made assuming that X = f(Xl, ... , xn ), with f( ) strictly increasing but not necessarily linear. Most of our results on aggregative games are still valid under this more general definition by introducing suitable concavity assump­tions on f( ).

An aggregative game is a generalization of the well-known Cournot (1838) model. In this case, Ui = P(X)Xi - Ci(Xi), being Xi the output of firm i, X the total output, p( ) the inverse demand function and Ci ( ) the cost function of firm i7. This special case will be treated extensively in Chapter 3. We note that the concept of an aggregative game generalizes the Cournot model since it allows for the following:

a) Payoff functions different from profit (i.e. sales, Baumol (1959), per capita added value like in Labor-Managed firms (Vanek (1975)) or social wel­fare (Fershtrnan (1990)).

b) Uncertainty (Horowitz (1987) and Exercise 1.38). c) Taxes and subsidies (Dierickx, Matutes and Neven (1988)). d) Product heterogeneity (Yarrow (1985), p. 517 and Exercise 3.1).

Other examples of aggregative games are technological competition (Loury (1979)), competition under rationing schemes (Romano (1988)), the problem of the commons (Dasgupta and Heal (1979) pp. 55-78), contribution games, preference revelation, rent-seeking (Tullock (1980) and Exercises 1.40 and 1.41), pollution and wage-setting trade unions (Exercise 1.1).

A solution concept yields for each game a set of feasible strategies with some kind of internal stability property. By far the most popular solution con-

7 In our view, the assumption of profit maximization -which we will use from Chapter 3 onwards- lacks a convincing foundation in oligopolistic markets and has to be generalized to allow for the maximization of a more complex payoff function. Thus, in oligopolistic markets even if a firm is concerned only with profits, profit maximization is not, in general, the best policy to be pursued and moreover, it does not guarantee survival (see e.g. Vickers (1985) and Chapter 5). Also, it might be the case that firms may be interested in other objectives rather than just profits (see Baumol (1959) and Diercker and Grodal (1999».

9

cept is the so-called Nash Equilibrium (N.E. in the sequel). In a N.E. each player decides on a strategy which maximizes her payoff, when she has an­ticipated correctly the strategic choices made by other players. The correct forecast assumption can be motivated by assuming that all players share all the relevant information about the characteristics of all players, i.e. that infor­mation is complete. Thus, any reason underlying a particular strategic choice of a player is anticipated by any other player (this argument will be reconsid­ered in Section 3). More specifically, N.E. is spy-proof, i.e. if each player has a spy who can tell her the strategies chosen by the other players there is incentive to choose the N.E. strategies. Another interpretation ofN.E. is that of a self-enforcing agreement: players can talk but they can not make binding agreements, i.e. any player is free to reconsider her agreed upon choice of strategy. Therefore, any stable agreement must give incentives to any player to stick to her promised choices, if the rest of players keep their promised choice of strategies.

N.E. will be the main equilibrium concept used in this book. Before we state it formally let us introduce a new piece of notation. Let X-i =

(Xl, ... , Xi-l, Xi+l, ... ,xn ) be the list of strategies played by all players ex­cept i. In the case of aggregative games and when no ambiguity is pos­sible, X-i will also be used to denote the sum of all outputs minus i, i.e.

X-i = L#iXj.

Definition 1.2 Given a game (UiO, Si)iEI,a tuple (XniEI is a Nash Equilib­rium (N.E.) if Vi E I, Ui(xi, x:..i ) ~ Ui(Xi, x:..i) VXi E Si.

We now introduce a new concept: The best reply correspondence of a player (say i) picks up the set of optimal strategies for player i for given strate­gies of the remaining players. Letting S-i == X#iSj we have the following definition:

Definition 1.3 The best reply correspondence of player i is a correspondence R; : S-i ---+ Si such that Ri(x-i) = argmax aESi Ui(a, X-i).

With this definition in hand, the definition of a N.E. can be restated as follows: A tuple (XniEI is a N.E. if Vi E I, xi E Ri(xi). In other words, a N.E. is a tuple of strategies which are best reply to themselves.

2 AGGREGATIVE GAMES I: THE MAIN ASSUMPTIONS

Now we state and discuss the assumptions we will use in the case of ag-

lO

gregative games. We will always assume that the set of strategies of each player is compact (i.e. closed and bounded) and convex including zero. The latter assumption can be interpreted as the possibility of being inactive. The assumption that the strategy sets are bounded can be substituted by the as­sumption that payoff functions are such that players are never interested in choosing arbitrarily large strategies. The latter is the case in the Cournot model if average cost is bounded away from zero and for any output beyond certain limit, market price is zero or arbitrarily close to zero.

Our first assumption restricts the payoff function of each player to be twice continuously differentiable (C 2 ).

Notice that under Assumption 1 (A. 1 in what follows) if x; E interior of Si, the necessary condition of a N.E. in an aggregative game reads as follows

aUi(x;,x*) aUdx;,x*) 0 -..:.....:.......:...:...---.:..+ =

aXi ax Vi E I.

Let us define the marginal payoff of player i:

rr. _ rr. ( . ) = aU;(Xi, x) aUi (Xi, x) .1 t - .1 t x" X - ~ + ~ UXi uX

Vi E I . (I)

Our next assumption is a generalization of an assumption introduced by Frank Hahn (1962) in the context of the dynamic stability of Cournot equi­librium and subsequently much used in the literature on existence and com­parative statics ofCoumot equilibrium (see e.g. Friedman (1982) p. 496, As­sumption 3 and the references therein). In order to motivate our assumption we will first explain the original assumption. On the one hand this assump­tion requires the slope of the inverse demand function to be greater than the slope of the marginal revenue, i.e. that the inverse demand function be either concave or "not too" convex. On the other hand this assumption bounds the degree of economies of scale by the slope of the inverse demand function. Formally

) a2p( ) op( ) 0 db) ap() a2Ci ( ) 0 a ~Xi + ~ < an -~- -~ < . (2) uX uX uX uXi

In spite of the awkward interpretation, this assumption is useful because it has two important consequences. Firstly, it implies that profits are concave on Xi, given X-i. We will see that this is an important ingredient in the proof of the existence of a N.E. (see Proposition 1.2 below). Secondly it implies that the best reply function of any firm is decreasing on the sum of strategies of

11

the remaining players and that the slope of this function is greater than mi­nus one. These properties will be used to prove the stability of a N.E. (see Proposition 1.5 below). Thus, this assumption is a convenient way of encap­sulating both properties (these properties can be easily proved differentiating (2), see Exercise 1.2 part a). In any case there are perfectly reasonable exam­ples of demand and cost functions which do not fulfill equation (2) above (see Exercise 1.36).

In this book we will use a generalized version of (2) namely:

Assumption 2. 1'; (Xi, X) is strictly decreasing a) on X and b) on Xi, ViE I

It is left to the reader to verify that A. 2 a) and b) correspond to equations (2) a) and b) above ifpayoffs are profits (see exercise 1.3).

As we commented earlier, the second part of A. 2 (i.e. that 1'; ( ) is strictly decreasing on x) implies that the best reply function of each player is de­creasing. Games for which this property holds have been called by Bulow, Geanakoplos and Klemperer (1985) Games with Strategic Substitutes. Un­fortunately the property of strategic substitution is not strong enough to yield sufficient structure in order to tackle comparative statics questions. Our first result will be devoted to prove this point: We will show that given any strictly decreasing function mapping ffi. into ffi. we can find a concave and differen­tiable payoff function which satisfies the aggregation axiom such that it gen­erates the given function as a best reply mapping. Moreover, suppose that we require this payoff function to be the usual profit function in the hope that this extra assumption puts more restrictions on the properties of best reply func­tions. However, given any strictly decreasing function mapping ffi. into ffi. we can find a cost function for each agent and a linear demand function (common to all agents and such that the product is homogeneous, i.e. the market price depends on the sum of all outputs) such that the given function is the best re­ply function of this agent. The implication of these two results is surprising: The assumptions that payoff are profits, that the product is homogeneous and the demand function is linear do not have any extra implication once we have assumed payoff maximization! Formally:

Proposition 1.1. Let Xi = Ii(x-i), i = 1, ... , n be a collection of continuous functions defined on a compact set and such that Ii ( ) is strictly decreasing Vi. Then,

a) Vi, :lUi( ) E C1 , concave on Xi and decreasing on X and such that

Ii(x-i) == arg max yES, Ui(y, Y + X-i).

12

b) Vi, :3 Gi ( ) E C1 (interpreted as the cost function of player i) and a linear inverse demand function p = a - x such that

j;(X-i) == arg max ZESi (a - Z - X-i) Z - Gi(z) VX_i

Proof. a) Since Ii ( ) is strictly decreasing and continuous, it is invertible and

the inverse function, denoted by li-1 ( ), is continuous (see Bartle (1976) p. 156). Thus, li-1() is bounded and integrable (see Bartle (1976) p. 247).

Let qi ( ) be a primitive of li-1 ( ), i. e.

aqi(Xi) = f:-1( .) a ,x, . Xi

Define the payofffunction of player i asfollows: Ui(Xi, x) = qi(Xi)+X;-XiX. Notice that Ui ( ) is decreasing on x. Then, first order condition of payoff maximization reads

aUi aUi -1() _ -1( ) -a + -a = Ii Xi + 2Xi - Xi - X = Ii Xi - X-i = O. X Xi

Since li-1 ( ) is strictly decreasing, Ui ( ) is concave on Xi. Therefore the sec­ond order condition of payoff maximization is satisfied and thus, a) is proved.

b) Let p(x) == a - X and Gi(Xi) == aXi - x; - qi(Xi), where qi( ) is the function defined in part a) above. Since Xi is defined on a compact set, a can be taken to be large enough so that total and marginal costs are always non negative for each firm. Also, the integration constant can be taken so that qi(O) = 0 and thus Gi(O) = O. Then,

Ui(Xi,X) = P(X)Xi-G(Xi) = (a-x)xi-axi+x;+qi(Xi) = qi(Xi)+X;-XXi

which is identical to the utility function constructed in part a) above .•

The main consequence of Proposition 1.1 is that in games in which both the aggregation axiom and the strategic substitution assumption hold, the fact that best reply functions depend on the sum of strategies of the other players and that they are decreasing exhaust all the properties of best reply functions (see Exercise 1.4 for the case in which the aggregation axiom does not hold. Exercise 1.11 shows that the assumption that Ii ( ) is decreasing is essential for the proposition to hold). Thus, best reply functions are, to a large extent, arbitrary.

Proposition 1.1 may be regarded as analogous to the results obtained by Sonnenschein (1972), Mantel (1974) and Debreu (1974) on the lack ofstruc­tural properties of aggregate excess demand functions in General Equilib­rium (a good survey of this literature is provided by Shafer and Sonnenschein (1982)). However, notice that individual demand functions have some prop­erties, for instance the weak axiom of revealed preference. What the results

13

quoted above say is that these properties are lost when we sum individual demand functions and consider the aggregate demand function. We remark that in our case the root of the problem is not on the aggregation side, since the result holds for each agent. In fact, a closer result to Proposition 1.1 has been proved by Boldrin and Montrucchio (1986). They show that under cer­tain conditions, including a single consumer, dynamic decision rules obtained from utility maximization can follow any given dynamic path.

An implication of Proposition 1.1 is that under strategic substitution, the set of strategies in a N.E. is arbitrary and that comparative statics does not yield definitive answers (see Exercise 1.5). This situation has been called by Andreu Mas-Colell "Anything Goes Theorems". 8 Thus, we are led to conclude that we need stronger properties than strategic substitution in order to tackle comparative statics. As we will see A.2 will be sufficient to do just that.

Finally, we state our third assumption.

Assumption 3. Ui ( ) is strictly decreasing on x, Vi E 1

In the Cournot case, AJ just says that the inverse demand function is strictly decreasing. A.3 will be used only in some results in this and the following chapters. If Ui ( ) were strictly increasing in x, the relevant proofs can be easily modified and comparable results can be obtained.

3 AGGREGATIVE GAMES II: EXISTENCE, UNIQUE­NESS AND INEFFICIENCY OF NASH EQUILmRIA

In this Section we will state and prove some fundamental properties of N.E. in the case of aggregative games. The first thing to be checked is the existence of a N.E. or in other words the existence of a solution to the system of equations Xi E ~(X_i)' i = 1, ... , n. Existence of an equilibrium can be regarded as a minimal test on the coherence of the model. An incoherent model (or assertion) might produce strange results. For instance if we assume the existence of a greatest natural number it is possible to prove that this number is one. Moreover, the assumptions needed to show the existence of an equilibrium tell us the scope and the range of possible applications of the model.

8 In fact, "Anything Goes" is the title of a famous song written by Cole Porter.

14

Proposition 1.2. A NE. exists for any Aggregative Game satisfYing A.I-2.

Proof. We will check that any Aggregative Game in which A.1-2 hold, satisfies the classic conditions for the existence of a NE. (see e.g. Friedman (1977) p. 169). Payofffunctions are continuous in the strategy profile (Xi)iEI and concave on x (since by A.2 the derivative of the payofffunction with respect to its own strategy is decreasing on its own strategy). Strategy spaces are non empty, compact and convex, and this completes the proof.

From now on we will assume (when necessary) that the N.E. is interior. This implies that we disregard inactive firms. This assumption can be com­pletely dispensed with at the cost of some complications. Given the didactic nature of this book we have taken the simpler alternative.

The next important property is uniqueness. The importance of this prop­erty can be understood by assuming that there are several N.E .. In this case it is not clear how players can decide which strategy they should play even if they have complete information. In other words, ifN.E. is not unique, play­ers must coordinate on a particular N.E .. This can be done in several ways: sometimes one of the N.E. is a focal point, i.e. this particular N. E, has a special feature that makes clear that coordination can be achieved only in this N.E. (Schelling (1960)). Sometimes players feel that certain N.E. are partic­ularly risky and must be avoided (this is the criterion of risk dominance, see Harsanyi and Selten (1988)). However, at the present we do not have a theory of coordination which is universally regarded as satisfactory. Fortunately in our case uniqueness is obtained.

Proposition 1.3. There is a unique NE. for any Aggregative Game satisfying A.I-2.

Proof. Let us assume that the Proposition is false and let us denote with the superindexes 1 and 2 the strategies played by the agents in two different NE. At any NE. we have that T;(x~, x T ) = 0, r = 1,2. Thus if Xl = x 2 , A.2 and the previous equation imply that xI = x7 Vi = 1, ... , n, and this contradicts that both equilibria are different. Without loss of generality let us assume that Xl > x2• But again A.2 implies that xI < x7, Vi = 1, ... , n and this contradicts that xT = L~l xi, r = 1,2 .•

Let us remark that it is easy to find counterexamples to Proposition 1.3 when A.2 does not hold (see Exercises 1.7 and 1.40)

Proposition 1.4. Under A.l and A.3 given a NE., there is a tuple of strategies for which the payoff of each player is larger than the corresponding payoff in this NE., i.e. the NE. is inefficient.

Proof. By totally differentiating the utility jUnction of i we get that

dUi ~ T;(xi,x*)dxi + (8Ui(xi,x*)/8x) Ldxj jf.i

15

Evaluating this expression at a NE., where Ti(xi, x*) = 0, we see that if dXj < 0, Vj = 1, ... ,n, dUi > OW .•

Proposition 1.4 implies that there are incentives to collude (or merge) since by coordinating their strategies, firms can improve their payoffs (for refer­ences on mergers see Jacquemin and Slade (1989) and Salop et al. (1987)). However, once they have colluded, players have incentives to deviate from the collusive outcome (see Exercise 1.8).

4 AGGREGATIVE GAMES III: STABILITY OF NASH EQUILmRIUM

In Section 1 we argued that if all agents are perfectly rational (i. e. they are able to perform any required calculation) and they have complete information about the characteristics of all players we should expect that they would play only those strategies that constitute a N.E. (this is a variation of the so-called von Neumann-Morgenstern meta-argument). In fact, as we noticed in Section 3, for the argument to be totally convincing we require the N.E. to be unique. However, ifN.E. is not unique, or agents do not have complete information or they are not fully rational, the motivation for N.E. must be found elsewhere. One possibility is to axiomatize the equilibrium concept, i.e. to come up with a set of conditions that guarantee that, for any given game, any list of strategies played by agents coincide with the N.E. strategies for this game. Another possibility, that we will explore in this section, is to assume that the basic game is repeated over time and that agents are not fully rational. In particular, we will assume that agents use rules of thumb to decide which strategy to play in each period.

In dynamic models, time can be modeled either as a continuous or as a discrete variable. If time is assumed to be discrete, convergence becomes problematic, since the dynamic system might exhibit overshooting, i.e. move­ments in the "right" direction that are "excessive" and repeatedly pass over the N.E. without touching it (see Exercise 1.12). While overshooting might be important in some applications, it appears to be a sensible strategy to get rid of it, and to concentrate our attention on the case where such a complication does not arise. Thus, we will assume that time (denoted by t) is a continuous variable. If z is a generic variable, let dz / dt == i.

16

There are many ways to model the behavior of myopic agents. Here, we concentrate on two. On the one hand in the Best Reply Dynamics each agent changes her strategy in the direction of what would be her best reply (see Def­inition 3 above). On the other hand, in the Gradient Dynamics each agent in­creases the value of her strategy if and only ifher marginal payoff is positive. Of course both rules are too mechanical to be followed by smart agents. In particular agents do not learn anything from past mistakes or successes. Even worse, they are unable to take advantage of any information from the past be­yond yesterday. Moreover, agents never imitate other agents. Therefore these rules should be understood as preliminary examples of adaptive behavior and not as a full-fledged formalization of dynamic behavior. Formally:

Definition 1.4 Let R i ( ) be the best reply function of agent i. The Best Reply Dynamics system is given by Xi = Ri(x-i) - Xi, Vi E 1.

A motivation for Best Reply Dynamics is the following. Suppose that firms have theories on the particular value of the output set by other firms. Let us assume that the theories are common in the following sense: firms 1,2, ... ,i -1, i + 1, ... , n all expect the same output from firm i, denoted by Yi. Firms maximize their payoffs for given expected outputs of the competi­tors. Thus, Xi = Ri(Y-i), where Y-i is the expected output of the competitors of i. Expected outputs are revised as follows: if the actual output of firm i is greater (less) than the expected output, firms other than i increase (decrease) the output that they expect from i. A particularly simple formulation of this idea is given by Yi = Xi - Yi. Thus Yi = Ri(Y-i) - Yi. Notice that in this case the equation refers to expected values of the output.

We now tum to the definition of Gradient Dynamics.

Definition 1.5 Let T; ( ) be the marginal payoff of agent i. The Gradient Dy­namics system is given by Xi = T;(Xi' X-i), Vi E 1.

A motivation for Gradient Dynamics is that agents attempt to increase pay­offs by moving in the direction given by the gradient of their payoff function. Notice that any Best Reply Dynamics system R 1 ( ), ... , Rn ( ) can be trans­formed in Gradient Dynamics by letting Ui = XiRi(X-i) - x; /2, i = 1, .. , n. It is also easy to see that if payoffs are concave, the direction of change of the strategy implied by Best Reply Dynamics and Gradient Dynamics is the same (see Exercise 1.37).

Before we embark upon the proofs of our main results in this section, it may be convenient to explain some basic concepts of dynamic systems. The book by Hirsch and Smale (1974) is a highly recommendable introduction

17

to the theory of differential equations and the interested reader is urged to consult this book for fully rigorous definitions and results. Let li(Xi, X-i) = 0, i = I, ... n, be a system of equations with (Xi, X-i) E W, where W is a subset of an Euclidean space. Let xi, i = 1, ... , n be a solution of this system, i.e. 7i(xi, x~J = 0, i = I, ... n. Consider the following system of differential equations

Let Xi(t), i = I, ... n be a solution of the above system. This solution will be called a trajectory. If there exists a neigborhood of (xi, X~i) such that any trajectory converges to xi, i = I, ... n when time goes to infinity we say that the system is asymptotically stable. Let D((xi, X~i)' (Xi(t), X_i(t))) = D(t) be the distance between (xi,x~i) and (Xi(t),X_i(t)). It is clear that if D(t) goes to zero when t goes to infinity the given trayectory is asymptotically stable. This basic idea was generalized by the Russian engineer Liapunov in the following way: Let V : U --+ lR+ be a C2 function with U c W such that V(xi, x~i) = 0 and V(Xi, X-i) > 0 for any (Xi, X-i) 1= (xi, X~i)' If V(Xi(t),X-i(t)) < 0 for any (Xi, X-i) 1= (X:,X~i)' the solution (X:,X~i) is asymptotically stable. The function V is called a Liapunov function and it must be found in each particular case.

The next result is due to Frank Hahn (1962):

Proposition 1.5. The Best Reply Dynamics of any aggregative game fulfilling A. J and A. 2 is asymptotically stable to the unique N E ..

Proof. Let us find the relevant Liapunov function. Let V = ~ LiEI(Xi)2. Clearly, this function satisfies the conditions for a Liapunov function written above. Now, we have,

V· L(' )(L . a~(X_i) .) = Xi Xj - Xi = ax· iEI #i J

- LX; + LXi LXjaRi(x-i)/aXj' iEI iEI ji'i

The aggregation axiom implies that a~(X_i)/aXj = a~(X_i)/aXk' Tlj, k. Let qi == aRi(x_i)/aXj < o from A.2. Thus, the previous equation be­comes:

V= - LX; + LXiqi LXj. iEI iEI ji'i

If all qi multiply positive terms, we are done. So the worst case from our point of view occurs when all qi multiply negative terms. But since A. 2 implies that

18

qi > -1 we have that:

V < -(L x; + LXi LXj) = - ( I>i) 2 ::; 0 iEI iEI Ui iEI

and V o iff Xi = 0, Vi E I.

Thus, V is a Liapunov function and the proposition is proved. •

Exercise 1.13 shows that Proposition 1.5 fails when time is modeled as a discrete variable. Now let us tum our attention to the Gradient Dynamics.

Proposition 1.6. The Gradient Dynamics of any aggregative game satisfying A. 1 and A. 2 is asymptotically stable to the unique NE ..

Proof. The proof is very much like that of Proposition 1. 5, i. e. we will show that V = ~ 2:iEI(x;) is a Liapunov function. In our case we have that:

V· =""'('2(O~(Xi'X) O~(Xi'X)) .. ,,",.a~(Xi'X)) ~ x, a + a + x, ~ xJ a iEI x Xi jopi X

A.2 implies that all terms multiplying x; and Xi. 2:Ui Xj are negative. Thus if Xi 2:Ui Xj 2 0 Vi E I, we are done. The worst possible case from our point of view arises when all these terms are negative. But then Vi E I we have that:

. "",, oTi(Xi,X) . ,,",. (OT;(Xi'X) OT;(Xi'X)) Xi ~Xj a < Xi ~Xj a + a .

Ui X Ui X Xi

Th V· ,,",('2 '.,,", .. )(oTi(Xi,X) OT;(Xi'X)) us < ~ x, + X, ~ x, a + a

iEI jopi X Xi Again, the worst possible case arises when all these terms are negative. In

this case, By A. 2,

and V Oiff Xi = 0, Vi E I .•

Exercise 1.12 can be adapted to show that Proposition 1.6 does not hold under discrete time (see Exercise 1.14). See Exercise 1.16 for a simple proof of Proposition 1.6 under additional assumptions.

Finally, it must be remarked that a good understanding of Propositions 1.1-6 is obtained by picturing the case n = 2.

19

5 PRICE-SETTING GAMES

The framework of analysis developed in Sections 1.2-4 above is not ap­propriate when prices are assumed to be the relevant strategic variable. On the one hand, in general, the prices of all goods except i do not enter in an additive form in the demand function of i. Thus we should not expect that payoffs of i (profits, sales, ... , etc.) can be written as a function of the sum of all prices and the price of i. On the other hand, simple examples show that when goods are gross substitutes (i.e. when the demand of i is increasing with the price of j, 'r/j -=I- i) A.2 fails (see Exercise 1.18). Thus, in this section we provide a separate analysis of the case first considered by Bertrand (1883), in which agents are price-setters.

First, let us clarify the meaning of the assumption of price setting. Some­times it is claimed that, since firms usually post prices, this assumption cap­tures better actual competition than quantity setting. However the meaning of price setting is not that firms post pieces of papers with prices written on them but rather that given any change in the policy followed by a rival, the firm is committed to sell at a given price. This point is easily seen by considering that given a residual demand function it is immaterial if the firm sets a price or a quantity. What it matters is the reaction of competitors: are they going to keep their clients (quantity-setting)?, or are they going to maintain the price (price-setting)? Both reactions yield different residual demands and thus the corresponding N.E. are different (see Exercises 1.20-21 ).9

Let us elaborate this point in detail. Let us assume that there are no costs and that the aggregation axiom does not hold, so x stands for a n-list of outputs. Let Pi be the price of good i and Pi (x) be the inverse demand function of good i. In the case ofa Coumot equilibrium firm, say i, maximizes Xi,Pi(X) with X-i = ;Li, i.e. taking as given the output of the remaining firms. First order condition of profit maximization reads:

9 The Cournot model assumes that once quantities have been decided, the market price is such that the aggregate quantity is sold entirely. However it is silent about the specific mechanism in which market clearing occurs. We can think of an auction of the -already decided- aggregate output, as in the oil market. Alternatively, we may think of the output being sold to dealers, as it happens in the car market. The dealers are many, and so they behave like perfect competitors (see Chapter 2, Proposition 2.3). Finally, we can think that output is perisharable and once it has been produced, it must go, as in the fashion industry. In this case is clear that firms may decide to change prices in order to adequate output to demand, but firms are not price-setters because the output of the competitor will be sold entirely. A different interpretation of the Cournot model is that output stands for capacity (so capacity is assumed to be fully utilized). In this case, quantity-setting models attempts to capture long-run competition.

20

api(X) Pi(X) + Xi.-a- = O.

Xi In order to define a Bertrand equilibrium we need to invert the system of

inverse demand functions Pi = Pi ( X) i = 1, .... , n. Let us assume that this can be done (see Exercise 1.34) and write Xi = Xi(P), i = 1, .... , n, where P is a vector of prices. Now in a Bertrand equilibrium firm, say i, maximizes Xi (p) .Pi with P-i = [Li, i.e. taking as given the price set by the remaining firms. First order condition of profit maximization reads:

However,

aXi(p) 1 Xi + Pi'-a- = 0 or Pi + Xi· {) .( ) = O. Pi x, P

{)Pi

1 -# api(X) {)Xi(P) ----a;:'

{)Pi

This can be easily seen by obtaining aXil api by differentiating the system of equations Pi = Pi(X), i = 1, ... , n, taking P-i = P-i (see Exercise 1.34). From the system of equations resulting from differentiating the previous sys­tem it is also easy to see that if the aggregation axiom holds, any firm is a price-taker (Exercise 1.35). This was Bertrand's original result and the basis for his criticism of the Cournot model.

In this section, we will see how the results obtained in the previous sections can be adapted to the case of price-setting behavior.

Let Ui(Pi,P-i) be the payoff of i as a function of its own price, Pi, and the list of all prices minus i, P-i. Prices will be assumed to belong to some closed interval oflR+, denoted by Ki, i.e. Pi E Ki, Vi E I. Let K-i == X#iKj and K == X iE1 K i. As before T;(Pi, P-i) will denote the marginal payoff of i, i.e.

T,. = 'D (. .) = aU;(Pi,p-i) • • p" P-. - a Pi

The following assumptions parallel those made before:

Assumption I': UiO E C2, Vi E 1

Assumption 2': a) T;(Pi,P-i) is strictly decreasing on Pi, ViE 1

b) I au? (Pi,P-i) I > I:. i I au? (Pi,P-i) I, Vi E I a2pi 11 api Pj

) au? (Pi, P-i) 0 \-I . ..../.. . \-I' I C a > , v J r z, vZ E

Pi Pj

21

We found convenient to break Assumption 2' in three different parts. A. 2' a) simply states that UiO is strictly concave on Pi' A.2'c) implies that prices are Strategic Complements (see Exercise 1.2 part b). Clearly, Strategic Comple­mentarity is the polar case of Strategic Substitution considered before. Finally A.2' b) says that the marginal payoff of i is more sensitive to a change in its own price than to a change in all other prices. In other words, the Jacobian matrix of first order conditions of payoff maximization has a Dominant Di­agonal (for a definition of this property see next chapter, Definition 3. See Takayama (1974) pp. 380-390 for applications and basic results concern­ing this property). \0 Exercise 1.20 asks for an example of demand and cost functions fulfilling A.l '-2'. In the case of aggregative games and strategic substitutes, properties b) and c) can be encapsulated into a single property (i.e. that 7iO is strictly decreasing on x). The need of part b) is shown in the next Proposition, which is the logical counterpart of Proposition 1.1.

Proposition 1.7. Let r i : K-i ---t Ki be an arbitrary increasingfunction such that ri(p-i) > 8 > 0, Vp-i E K_i. Then there exists a demand function for good i which is linear on Pi and increasing on any other price (i.e. goods are gross substitutes) and a linear cost function such that riO is the best reply function of a profit maximizingfirm.

Proof. Let c be the marginal (and average) cost with c < 8. Consider the following demand function: Di(P) == ri(p-i) - p;J2 - c/2. Notice that Di(P) = (ri(P-i) - c)/2 ;::: 0, Vp E K, since ri(p-i) = Pi. Also, it is clear that it satisfies the assumptions of the Proposition. Taking Ui = Di(p)Pi -cDi (p) we readily see that the first order condition of profit maximization yields ri(p-i) = Pi. Finally it is easy to show that second order conditions of profit maximization also hold. •

We now present the counterparts of Propositions 1.2-3-4.

Proposition 1.8. A Nash Equilibrium exists for any game satisfying A.1' -2'a).

Proof. The proof follows closely that of Proposition 1.2 and therefore we will only sketch it.

Since payofffunctions are continuous in the strategy profile (Pi)iEI(by A.1') and concave on Pi(by A.2'), best reply functions exist and are continuous. Since they map a non empty, compact and convex set (K) into itself a fixed point exist. It is easy to see that this fixed point is a NE. •

10 The definition given in Arrow and Hahn (1971), pp. 233-35, requires that, in addition to the requirement above, the excess supply function of each good is increasing in its own price. This extra condition is needed in order to obtain uniqueness. See Proposition 1.9 below.

22

As in Section 1.3 we make the assumption that any N.E. equilibrium is interior, i.e. if (P:)iEI is a N.E., pi E interior K i, Vi E I. Then we have the following:

Proposition 1.9. There is a unique Nash Equilibriumfor any game satisfying A.l' - 2'a) and b).

Proof. Under our assumptions a vector of prices is a NE. if and only if:

1i (Pi , P-i) = 0, i = 1, ... , n

Notice that A.2'b) means that the Jacobian of the above system has a Domi­nant Diagonal. By A2'a), 1i( ) is decreasing on Pi. Then, the proofofthe re­sult follows from a standard uniqueness result (see, e.g. Arrow-Hahn (1971), pp. 234-5) .•

For the next Proposition we will need a new assumption:

Assumption 3': Ui( ) is strictly increasing on P-i, Vi E I

Proposition 1.10. Under A.l' and A. 3' given a NE., there is a tuple of strate­gies for which the payoff of each player is larger than the corresponding pay­offin this NE., i.e. the NE. is inefficient.

Proof. Identical to the proof of Proposition 1.3 .•

Finally we consider gradient dynamics (best reply dynamics is reviewed in exercise 1.19).

Proposition 1.11. The Gradient Dynamics under A. l' - 2' is asymptotically stable to the unique NE ..

Proof. Let Q(p) = maxiEI1i(P) and such that if Tr(P) = Ts(p) we choose the smallest index. We will show that V = Q(p)2 is a Liapunov jUnction. Let

us assume that at p, Q(p) = 1i(p), say. Thus we have that:

V = 2 Q = 21i 81i(p) + 1i l:Tj aT;,(p) 8Pi '.J.' 8pj

Jr'

The worst possible case from our point of view arises when all the terms multiplying aT;, (p) /8pj are positive. But then 1i. L#i Tj < 1:;2 and thus:

V < 1:;2(81i(p) + l:Tj 81i(p)) < 0 8Pi '.J.' 8pj

Jr'

where the last inequality follows from A.2. Since V = 0 iff Tj(p) = 0, V j E I, V is a Liapunov jUnction and the proof is complete .•

23

Further results on dynamics with price-setting firms are presented in the Appendix to this Lecture. In contrast with the case of strategic substitution, discrete dynamics leads to convergence to the N.E. in a large number of cases, see Milgrom and Roberts (1991). Again, a picture of the case n = 2 helps to understand Propositions 1.7-11.

Summing up, in this Chapter we have shown that there is a unique, stable and inefficient Nash Equilibrium in every aggregative game satis­fying A.1-2-3 and every game satisfying A.1'-2'-3'.

6 ADDITIONAL REFERENCES

Most of this Lecture is standard. A good reference is Friedman (1977) pp. 25-6 and 169-71. With respect to the existence of a N.E., A.2 can be dispensed of under symmetry (Mc Manus (1962, 1964». Continuity can also be dispensed of under strategic substitution (Bam6n and Fraysse (1985), Novshek (1985) and Kukushkin (1994» or under strategic complementarity (Vives (1990) and Milgromand Roberts (1990) and Exercise 1.23). See Amir (1996) for an application of supermodularity to the Cournot model. Strategic complementarity has been considered in a general set-up in which differen­tiability is not assumed by Topkis (1979), Bulow, Geanakoplos and Klem­perer (1985), Vives (1990) and Milgrom and Roberts (1990) Examples of non-existence ofN.E. are presented in Roberts and Sonnenschein (1977), and Friedman (1983) pp. 67-71. Caplin and Nalebuff(1991) and Dierker (1991) derive the quasi-concavity of the profit function from assumptions on the dis­tribution of consumers' tastes. Watts (1996) presents an uniqueness result. For Gradient Dynamics see Arrow, Hurwicz and Uzawa (1958).

7 APPENDIX: FURTHER RESULTS ON THE STABILITY OF NASH EQUILffiRIUMll

In this appendix we survey some additional literature on the stability of N.E. and prove some additional results.

The modem literature on the stability of N.E. has followed two different approaches to the study of best reply dynamics, namely discrete time and continuous time. Contributions in the former framework are found in Fried­man (1977), Moulin (1986), Lippman, Mamer and McCardle (1987), Vives

11 This appendix is based on Corch6n and Mas-Colell (1996).

24

(1990) and Milgrom and Roberts (1990), (1991). Their results say that un­der strategic complementarity a class of dynamic systems (that contains best reply dynamics) converge asymptotically to some equilibrium. A variation of best reply dynamics, initially proposed by Maskin and Tirole (1987), has been studied by Dana and Montrucchio (1986, 1987) for the case of two play­ers. They found that in general the possibility of chaotic behavior cannot be precluded.

In the continuous time framework, Negishi (1961) proved convergence of a similar dynamic system. Other convergence results have been obtained by Okuguchi (1964), Hirsch and Smale, (1974) pp. 265-273, Keenan and Rader (1985) and Gaunersdorfer and Hofbauer, (1994».

In this Appendix we will concern ourselves with the following question. What can be said about stability of oligopoly in continuous time that does not depend on special assumptions (beyond that strategy sets are one dimen­sional) about the underlying economy or the form of the process? Recall that the proofs of Propositions 1.5 and 1.6 made heavy use of A.2. In other words, can we say something about stability that is not based on ad hoc assumptions about the shape of best reply functions or the form of the dynamic process? In order to answer this question let us focus our attention on dynamic systems of the following form:

Si = Gi(s), i = 1,2, ... , n, or, in a more compact notation:

(3)

S = G(s) (4)

Where s E S = [a, b]n, a> O. The system is such that at a N.E., G(s*) = O.

Let DG(s) be the Jacobian of G(·), evaluated at s, and Tr(DG(s)) the trace of DG (s), i.e. the sum of all the diagonal elements. Let us consider the following properties:

(D) Gi (-) is a continuously differentiable function, for every i. (N) Tr(DG(s)) is negative for all s E S. (B) For every s E Sand i we have: if Si = a then Gi(s) > 0, and if Si = b

then Gi(s) < o. (T) The determinant of DG(s), denoted by IDG(s)l, is non-vanishing at

any N.E., s*.

(T) is a regularity condition. Two natural examples of dynamic systems satisfying properties (D), (N) and (B) above are Best Reply dynamics (see Definition 1.4) and Gradient dynamics (see Definition 1.5).

25

We first consider the case of two players. As we will see, in this case, con­vergence is obtained with considerable generality. Let ¢t(s) be the position of system (3) in time t as a function of s, an initial point in S. We will call ¢t(s) a trajectory. We are now ready to prove our next result:

Proposition 1.12. If n = 2 any dynamic process satisfying (D), (N), (B) and (T) generates a trajectory that converges to some equilibrium for any initial point.

Proof. Let Vol E denote the volume (area in our case) of any region E c S. By Liouville's theorem (see Arnold, 1973 p. 198. A nice and self-contained proof of this theorem can be found in Keenan and Rader (1985) pp. 467-8) the fact that Tr(DG(s)) < 0 implies that G contracts volume, that is, for any region E c S with Vol E > 0 andfor any t > 0 we have that Vol(¢t(E)) < Vol E.

Now suppose that from some initial point S, the trajectory ¢t (s) does not converge to an equilibrium. The situation must be that the trajectory spirals around a limit set which isformed by afinite number of M(possibly M = 0) of equilibria Sl, ... , S M, with Sl = S M, and of trajectories ILl' ... , IL M such that ILr connects Sr with Sr+l (if M = 0) the limit set is a limit cycle. See LeJschetz, 1946, pp.172). Under the above conditions the limit set contains a (topological) circle (it may contain more) for which its interior E is invariant and has Vol E > O. Hence for t > 0, Vol( ¢t(E)) = Vol E, contradicting the hypothesis .•

This result can not be extended to the case of n = 3, since in this case it is known that there are dynamic systems satisfying our conditions that yield chaotic behavior, i.e. Lorentz equations (see, e.g. Marsden and McCracken, 1976). Thus we are led to the task of finding suitable restrictions on the set of economies under consideration that help us to prove the convergence of dynamic systems satisfying properties (D), (N) (B) and (T). We have already encountered a similar question in the main text of this chapter. The answer there was negative (Propositions 1.1 and 1.7). Therefore, let us introduce a new restricted domain.

Definition 1.6 The set of Nice Economies (NIEC in the sequel) is composed by economies displaying the following characteristics:

1: Marginal costs are constant and identical. 2: The system of demandfunctions has a negative Dominant Diagonal, i.e. if Di : K ---+ 1R+ is the demand function of good i, we have that:

UDi(P) < o and 18Di (P) I > 2::I UDi(p)l,vp»O,ViEI UPi UPi up)"

#i

26

Notice that the dominant diagonal condition in Definition 6 refers to the first derivatives and the dominant diagonal condition in A.2' refers to the second derivatives. Therefore there is no logical implication between both conditions.

We remark that any economy belonging to the set NIEC has a unique and globally stable Walrasian equilibrium (see e.g. Arrow and Hahn, 1971). How­ever when oligopolistic competition is the relevant solution concept, the situ­ation is different, as is made clear by the following two propositions.

Proposition 1.13. Let 'A = Gi(Pl, .. Pi, .. Pn), with Pi E [a, b], i = 1, ... , n be a given differentiable dynamic system with 8Gd 8Pi < 0 for all i. Then, there exists an economy from the set NIEC which generates the above equations as a gradient system.

Proof. Assume costs to be zero (a similar construction to the one below would apply if costs were required to be positive).

Consider Gi(Pl, .. Pi, ··Pn). For given P-i thisfonction is integrable in Pi. Let Fi(Pl, ··Pi, ··Pn) == J:; Gi(S,P-i) ds. Then the demandfunctionfor firm i is Di = (Fi(Pl, .. Pi, .. Pn) + b)/Pi where b > O. Notice that by making b large enough, demand is positive for every price and the Dominant Diagonal property holds since L#i 8Di(p)/8pj is bounded and independent of b, but (8Di(p)/8Pi) depends monotonically on b and is as large as we wish. Finally, profitsfor firm i are 7ri == DiPi = Fi(Pl, ··Pi, ··Pn) + b. Clearly 87ri(P)/8Pi =

Gi(Pl, ··Pi, ··Pn) .•

Proposition 1.14. Let 'A = !i(P-i) - Pi, with Pi E [a, b]' i = 1, ... , n be a given differentiable dynamic system. Then, there exists an economy from the set NIEC which generates the above equations as best reply dynamics.

Proof. Using the construction given in the proof of Proposition 3 for a system of demand functions of the form, Di = !i(P-i) - pd2 + b/Piproduces the desired result .•

Proposition 1.13 shows that any list of n arbitrary functions can be rational­ized as gradient dynamics of n firms with zero costs (or, in general, identical and constant marginal costs) and a system of demand functions with a neg­ative Dominant Diagonal Jacobian. An identical result holds for the case of best reply dynamics (Proposition 1.14). Similar results can be proved when firms are quantity-setters. The reader is asked to compare Propositions 1.13 and 1.14 with Propositions 1.1 and 1.7 in the main text.

We end this Appendix by pointing out three important implication of Pro po­sitions 1.13-14: Firstly, we can construct nice economies with an arbitrarily

27

large number ofN. E .. Secondly, we may have N.E. with an arbitrarily low level of activity. Finally, there is a nice economy generating chaotic behavior as either Best Reply or Gradient dynamics. The reader is asked to provide specific constructions showing that these three implications are indeed true, see Exercise 1.17. We emphasize that Propositions 1.13 -14 and the three implications just mentioned, are obtained under restrictions on the class of allowable demand and cost junctions that for the perfectly competitive case, would imply that the equilibrium is unique and globally tatonement stable. This implies that oligopolistic models are more complex objects than per­fectly competitive models. Therefore, in order to obtain predictions, we have to make far stronger assumptions under oligopoly than under perfect compe­tition. This may help to make more palatable the assumptions used in this and subsequent chapters.

8 EXERCISES

1.1. Show that the examples mentioned below Definition 1.1 are special cases of an aggregative game (see CorchOn (1994».

1.2. Prove the following: a) Assuming A. 1 and A.2, show that the payoff function is strictly concave

on its own output and that the best reply correspondence of any player is a decreasing function of the sum of the strategies of the other players with slope less than one in absolute value.

b) Assuming A.I' and A.2' show that the best reply functions are increasing (hint: differentiate the first order condition of profit maximization).

1.3. Prove that A.2 generalizes equation (2) in the main text.

1.4. State and prove a result similar to Proposition 1.1a) in the case of an arbitrary function (hint: obtain the payoff function by integration of the best reply function). Comment on the implication of the hypothesis that players are maximizers. What about part b) of Proposition 1.1?

1.5. Show by means of a figure that under the assumptions of Proposition 1.1 and for the case of two players we have the following:

a) Given any set in JR.!, there is a pair of profit functions such that any point in this set is a Nash equilibrium of the game where these profit functions are payoff functions.

b) Comparative statics will not yield definitive answers (hint: construct best reply mappings which intersect at any given set of points, consider shifts of these curves and compare non adjacent equilibria).

c) Would dynamics be of any help in order to select equilibria?

28

1.6. Give a picture of two best reply correspondences for which there is no N.E .. How can you derive this correspondences from payoff maximization?

1.7. Give a counterexample to Proposition 1.3 when A.2 does not hold.

1.8. By using differential calculus show that if a tuple of strategies maximizes the sum of payoff functions, it is not, in general, a N.E ..

1.9. Assume that n = 2 and that payoff functions are C2•

a) Show by means of differential calculus that under A.2 the best reply function is a contraction.

b) Use a theorem on contraction mappings to show directly Propositions 1.2 and 1.3 (see Bartle (1976) p. 161-3 and Friedman (1977) pp. 170-1).

1.10. Derive necessary conditions of profit maximization for a monopolist in the following cases:

a) Advertising is possible (see Dorfman and Steiner (1954». b) There is price discrimination between two different markets (this is

called third degree price discrimination, see Carlton and Perloff (1990) pp. 445-447).

c) Compare the conditions obtained in a) and b) above with equation 1 in the main text.

1.11. Show that if p(O) > Ci(O) (where pO and C;( ) are the inverse demand and the marginal cost function of firm i respectively), the function rX_i(l -X-i) with r > 0, can not be a best reply function ofi ifi is a profit maximizer.

1.12. Suppose the following dynamic system, Xt -Xt-l = a- bXt-l, a, b > O. a) Show graphically for which values of b the system is asymptotically

stable. b) Consider now the continuous time analog of the above system, dx / dt =

a - bx and show graphically that it converges asymptotically for any positive b. Comment on the overshooting that occurs in the discrete case.

1.13. Let us assume that there are three identical firms with constant marginal

costs facing a linear inverse demand function. Show that, if time is modeled as a discrete variable, any trajectory starting from firms which are at an identical initial position is unstable.

1.14. Suppose that p = A - X and the cost function of a monopolist is C = (A - x)x + x2/2 + rx3/3 - rx2/2, r > 1.

a) By using graphical methods show that in continuous time the Gradient Dynamics converges to the profit-maximizing output except if the initial point

is x = O. b) Show that in discrete time the trajectories of Gradient Dynamics are

chaotic for r ~ 4 (see Holden (1986) p. 274).

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1.15. Show global stability of best reply dynamics and gradient systems when the speed of adjustment is different from one, i.e., Xi = ai(Ri(x-i) - Xi) and Xi = a/li(xi, X-i), ai > 0, Vi E 1

1.16. Show by means of a graphical argument that if payoffs are profits, Ci = C;Xi + x;d/2, Vi E I, A3 holds and UiO is concave on Xi for all X-i, the aggregate output of a Gradient Dynamics of any aggregative game is globally stable (hint: consider Xi = p(x) + XiP'(Xi) - C; - dXi, add up over i and show that the right hand side of this equation is strictly decreasing on x).

1.17. Construct excess demand functions for a nice economy such that: a) There is an arbitrarily large number ofN. E .. b) There is a unique N .E. with an arbitrarily low level of activity (hint: fix

a vector of outputs with each component arbitrarily close to zero, construct candidate reaction functions such that this level is the only point compatible with all these functions and show that there is an economy generating these functions as best reply functions)

c) Both Best Reply or Gradient dynamics of an economy with three goods, yield Lorentz equations (see, e.g. Marsden and McCracken, (1976) or Guck­enheimer and Holmes, (1983)).

1.18. Compute the best reply function of a price-setting firm facing a demand function linear in all prices in which goods are gross substitutes (assume that costs are zero).

1.19. Prove the convergence of best reply dynamics in the case of price setter agents.

1.20.- Let Pi = ai - f3iqi - 'Yqj, i I- jj i,j = 1,2 be the inverse demand functions for firms 1 and 2 wheref3i' ai > 0, andaif3j-aj'Y > 0,f31f32-'Y2 > O. Marginal costs are constant and identical for both firms.

a) Find the demand functions for both firms. b) Find both analytically and graphically the N .E. where firms are quantity­

setters. c) Find both analytically and graphically the N.E. where firms are price­

setters and show that Al ' and A2' hold. d) Compare and interpret the divergence between b) and c) above.

1.21.- This exercise generalizes the result obtained in Exercise 1.20. Let us assume that the product is differentiated with n varieties each offered by one

firm. Let Pi be the price of variety i and Xi = !i(Pl,"" .Pn) be the demand function of variety i,i = 1, ... , n. Assuming AI' - 2' - 3' and symmetry show that the output of each firm is larger in the N .E. in prices (Bertrand) than in the N.E. in quantities (Cournot) (hint: consider first order conditions

30

of payoff maximization). Explain this result first obtained by Vives (1985) and Okuguchi (1987).

1.22.- Show that if there are constant returns to scale and the demand function can be written as Xi = !i(P;).r(P_i), r(P_i) > 0, VP-i , the N.E. in prices is a dominant strategy equilibrium.

1.23.- Suppose that agents have identical payoff functions and that strategic complementarity and the aggregation axiom hold.

a) Show that there are no asymmetric N.E .. b) Show by means of a picture that there exists a symmetric N.E. even if

the best reply function is not continuous for any n 2 2.

1.24. Let us assume that in a market there are two identical firms competing in prices.

a) Define a N.E. in prices (i.e. a Bertrand equilibrium). b) Show that under constant returns to scale the perfectly competitive out­

put equals to the N.E. output. c) Show that under decreasing returns the N.E. in prices may fail to exists

(see Grossman (1981) pp. 1168-9).

Exercises 1.25-1.32 review some elementary aspects of N.E.. They pre­sume some previous knowledge of Game Theory.

1.25. Interpret the following situations as a prisoners dilemma game. a) The arms race. b) The fishing of whales.

1.26. Two persons have their savings (each one million $) deposited in a bank. This bank has lent 60% of its capital to a firm making long term investments and keeps the rest as reserves. Both players can either ask for their money now or wait until the firm pays back the credit. In the latter case each depositor receives an additional million each. If one person asks for her money now the bank becomes bankrupt but she gets the whole deposit. Ifboth persons ask for their money, they receive half of the deposits each. Analyze the N .E. of this game and compare it to those with a cooperative solution. Give an economic interpretation to each N.E .. Do expectations matter?

1.27. Interpret the Chicken game as a model of two firms with two strategies each: to cooperate (i.e. to price its product in order to maximize joint profits) or not to cooperate (i.e. to price non-cooperatively). Compare the two N.E. with those in the previous problem. Construct a similar game where payoffs give rise to a "Prisoners Dilemma-like" situation.

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1.28. Construct an example of a bipersonal game in which N.E. are efficient. Show graphically that if strategy spaces are one dimensional N.E. will not, in general, be efficient.

1.29. A group of n farmers have identical utility functions U = X - L~ where X is the quantity of a public good (freely enjoyed by any member of the community) and Li is the quantity of labor voluntarily offered by i and used in the production of the public good. The public good is produced under

constant returns to scale X = b 2::~=1 L i .

a) Calculate the value of X in a N.E. where the strategy of player i is L i .

b) Calculate X assuming that it is decided by maximizing a utilitarian social welfare function W = nU.

c) Analyze the differences between the value of X obtained in a) and b). Write this difference as a function of n and give an interpretation to this equa­tion.

d) Calculate a) and b) above if U = ..::.. - L~, a> O. Interpret a. n°

1.30. The utility function of driver i (i = 1, ... , n) depends on her speed (ai) and the speed of other drivers, represented by the n - 1 dimensional vector a-i. It is assumed to be decreasing on a_i(why?).

a) Show that if all drivers reduce infinitesimally their speed from the N.E. speed there is an increase in the payoff of each driver.

b) Show that the introduction of a speed limit that is binding at the Nash equilibrium, has an ambiguous effect on welfare.

c) Compare a) and b) above with Proposition 1.4.

1.31. Show by means of examples that a) the outcome of removing weakly dominated strategies depends on the

order in which they are removed. b) The N.E. may occur in weakly dominated strategies.

1.32. Consider a game with two players with payoff functions U1 = XIX2 -

xi/2 and U2 = X2 - x~xd2 where Xi E [0.5,2]' i = 1,2. a) Find both graphically and analytically the N.E .. b) Analyze the Best reply dynamics in discrete time.

1.33. Suppose that firms have subjective expectations about the form of pay­off functions. In particular, if 1I"i (x) is the real payoff function of firm i, this firm believes that payoffs are given by a (possibly different) function Ei(X). a) Define a Subjective N.E. (S.N.E.) when firms maximize subjective payoffs.

b) Show that under a concavity assumption, similar to A.2, if the slopes of 1I"i( ) and Ei( ) with respect to Xi are identical for all i, S.N.E. and N.E. coincide.

32

c) Make examples showing that this equivalence does not hold when the concavity assumption is removed.

1.34. Consider the system of equations Pi = Pi (X) i = 1, ... , n. a) Compute 8x;j8pi by differentiating taking P-i = P-i. b) Solve the system for the case n = 2. c) Give conditions under which the system of equations is solvable in gen­

eral.

1.35. Using the computations of the previous exercise, show that if the aggre­gation axiom holds, any firm is a price taker.

1.36. Give examples of demand and cost functions which do not satisfy equa­tion (2) in the main text.

1.37. Show that if payoffs are concave, x > 0 under Best Reply Dynamics if and only if x > 0 under Gradient Dynamics.

1.38. Suppose that the cost function of each firm depends on a parameter which may take two values with probabilities p and 1 - p. These probabilities are common knowledge among all firms. Suppose that the decision on how much to produced has to be taken before it is known the actual value of that parameter.

a) Define a Nash equilibrium of this game and show that the machinery developed in this chapter can be applied to this case.

b) Give an example in which the above scenario is relevant. c) Generalize to the case in which the parameter may take any finite num­

ber of values.

1.39. Show that the Cournot equilibrium is locally improvable (i.e. all firms can do a positive marginal profit) by a correlated strategy when marginal costs are decreasing. Show that the previous result is not true when marginal costs are strictly increasing (Gerard-Varet and Moulin (1978)).

1.40. Suppose that utility functions are Ui = Vi(2:7=1 Xj) - Xi, i = 1, ... , n, where Vi ( ) is differentiable and strictly concave. The interpretation is that Xi is i's contribution and 2:7=1 Xj is the quantity of a public good produced under constant returns to scale with unit marginal cost.

a) Find the N.E. when all utility functions are identical. Show that there are infinite N.E .. Compare this result with Proposition 1.3 (uniqueness ofN.E.). Which assumption is not satisfied in this case?

b) Find the N.E. when utility functions are all different (this is a non inte­rior N.E.). Show that N.E. is, in general, unique.

1.41. Suppose that an object with a value of V $ is going to be allocated by a committee to one of n competing agents. These agents can make expenses

33

that influence the decision of the committee. Let G i be the expense made by agent i. The probability that agent i obtains the object, denoted by Pi, is given by

Pi = ",n G.' Uj=l ]

Compute the N.E. of this game.

1.42. Suppose that in the problem described in the Exercise 1.41, there are some a priori probabilities ql, ... , qn thatthe object will be allocated to agents 1 to n. Now, the probability that the object is given to agent i is

qiGi Pi = ",n G'

Uj=l qj j

a) Compute the N.E. b) Show that ifn = 2, qi = Pi, i = 1,2. c) Show by means of a counterexample that the result obtained in b) above

does not hold for n > 2.

CHAPTER 2. COMPARATIVE STATICS

Abstract: The problem of comparative statics. Aggregative games with strategic substitution: The effect of entry of a new player on the strategies played in a NE. (Proposition 2.1) and on the payoff.s obtained in a NE. (Proposition 2.2). Cournot equilibrium and perfect competition (Proposi­tion 2.3). Idiosyncratic versus generalized shocks. Effects of an idiosyncratic shock on the strategies played in a NE. (Proposition 2.4) and on payoffs obtained in a NE. (Proposition 2.5). Effects of a generalized shock (Propo­sition 2.6). Effect of an increase in the marginal cost on equilibrium price in oligopoly and perfect competition (Propositions 2.7 and 2.8). Games with strategic complements: Effects of a shock on the strategies played in a NE. (Proposition 2.9) and on payoff.s obtained in a NE. (PropOSition 2.10). The effect of entry of a new player on the strategies played in a NE. (Proposition 2.11) and on the payoff.s obtained in a N E. (Proposition 2.12)

1 INTRODUCTION

In this chapter we continue the study of the two models of oligopolistic competition outlined in the previous chapter. Here we focus on the effects of changes either in the number of players or in the payoff functions (our exogenous variables) on the strategies played and the payoffs obtained in a Nash Equilibrium (our endogenous variables). In other words, we compare two Nash Equilibria each relative to a different set of exogenous variables. This kind of exercise is called Comparative Statics. It is generally accepted that a satisfactory model should yield unambiguous results concerning the direction of change of the endogenous variables.

Our results in this chapter point out that in the two models developed in Chapter I, comparative statics makes predictions about changes in the en­dogenous variables that agree with our intuition. In order to obtain these results we have to impose some assumptions that are familiar to us from the previous chapter, namely A.I-2 and A.I' -2'. An important caveat is that if A.2 or A.2' are not fulfilled, the models studied in this chapter may yield counterintuitive predictions. An Appendix discusses the case in which payoff functions are not differentiable everywhere.

36

Comparative statics is really meaningful when equilibrium is unique and stable. This is because if equilibrium is not unique, the direction of the change of endogenous variables may be affected by the particular equilibrium we start with. And if equilibrium is not stable we have no guarantee whatsoever that once the old equilibrium has been disrupted we will arrive to a new equilib­rium. Fortunately, as we have seen, Assumptions A.2 and A.2' imply that the N.E. is unique and stable so our comparative statics results are meaningful. The fact that assumptions used to prove uniqueness, stability and comparative statics are often very similar or identical, is well known in models of general equilibrium (see, e.g. Arrow and Hahn (1971».

2 AGGREGATIVE GAMES I: EFFECTS OF AN IN­CREASE IN THE NUMBER OF PLAYERS

There are two approaches to deal with an increase in the number of players. In the first approach, we look at the effect of the entry of a single player on strategies and payoffs. This approach provides testable implications of the model. In the second approach, we let the number of players to go to infinity and we look at the outcome in the limit of this process. The second approach is used to see what kind of competition we should expect in markets with a large number of participants and, in particular, if this outcome coincides with perfect competition. We will deal with both approaches in turn.

In the Cournot model, intuition suggests that the entry of a new firm in a market will have two effects. On the one hand, because competition is more intense after entry, profits and outputs of incumbent firms will decrease (the latter effect is called the business stealing effect). On the other hand, total output will increase reflecting the fall of the market price due to more com­petition. From the work of Mc Manus (1962), (1964), Frank (1965), Ruffin (1971), Okuguchi (1973), Seade (1980) and Szidarovsky and Yakowitz (1982) we know that these intuitive effects indeed occur under certain assumptions in the Cournot model. In this section we will show that these results carry through to the framework of an Aggregative Game. However we remark that our assumption A.2 is essential for these results to occur. In other words, if A.2 does not hold, the entry of a new firm may decrease total output or may increase the output or profits of incumbent firms. In this sense the results of this chapter show that our intuition about the comparative statics properties of oligopolistic markets is not necessarily correct since it depends heavily of the fulfillment of A.2.

Let us denote by x(n), xi(n) and Ui(n) the N.E. values of x, Xi and Ui in a

37

game with n players. The set I comprises players 1 to n. In order to simplify

notation, let y == Xn+l (n + 1). In the sequel, we will deal exclusively with interior N.E .. Thus, xi(n) > 0 for all n and y > 0, i.e. player n + 1 has actually entered into the market. This is an assumption that can be dispensed

with at the cost of complicating the presentation.

Proposition 2.1. Under A.l and A.2 we have the following:

a) x(n) < x(n + 1), and b) xi(n) > xi(n + 1), Vi E I.

Proof. The crucial fact used in this proof is that T; (x; , x*) = 0 in any N E. and T i ( ) strictly decreasing in Xi and x (i.e. A.2) imply that

sign{x(n) - x(n + I)} = -sign{xi(n) - xi(n + I)} ..

In words, x(n) and xi(n) move in opposite directions. First notice that x(n) = x(n + 1) can not hold since A.2, would imply that

xi(n) = xi(n + 1), Vi E I and since y > 0 this is impossible. Consider player i. We have four possible cases

1) x(n) > x(n + 1) and xi(n) > xi(n + 1).

2) x(n) > x(n + 1) and xi(n) < xi(n + 1)

3) x(n) < x(n + 1) and xi(n) > xi(n + 1) 4) x(n) < x(n + 1) and xi(n) < xi(n + 1).

However A.2 implies that cases 1) and 4) above are impossible because x and Xi move in the same direction and we have seen that this is impossible. Suppose that case 2) above holds. But if x(n) > x(n + 1), A.2 again implies that xj(n) < xj(n + 1), Vj -=1= i, i.e. case 2) above holds for all players. But this and y > 0 contradict the definition of x(n). Thus, case 3) holds for player i and thus, x(n) < x(n + 1). The last inequality and A.2 imply that xj(n) > xj(n + 1), Vj E 1..

If A.2 does not hold, Proposition 2.1 fails (see Exercises 2.1 and 2.2). The

intuition of why Proposition 2.1 may fail is clear: if the reaction function of the incumbent firm has a slope less than -1, the entry of a new firm might decrease aggregate output. Also, if reaction functions are upward-sloping the entry of a new firm makes incumbents to increase their outputs.

That aggregate output increases and individual output decreases with entry,

have additional implications in the symmetrical case where x; = xj for all i

andj: Ifindividual output falls with entry (i.e. x(n)ln > x(n+ 1)/(n+ 1)) it is easily computed that the rate of growth of aggregate output can not be larger than lin. Also, if aggregate output increases with entry (i.e. x(n) < x(n+ 1))

38

it is easily computed that the rate of growth of individual output must be larger than -l/(n + 1).

We now turn to study how payoffs change with entry.

Proposition 2.2. a) Under A.I, A.2 and A. 3, Ui (n) > Ui (n + 1), Vi E I. b) Under A.3 we have that U1 (1) > U1(2).

Proof. a) Let x_i(n) == x(n) - xi(n) be the sum of strategies ofal/players except i in a N.E. with n players. Under A.1-2, x_i(n) is increasing in n by Proposition 2.1. Let Vi(Xi' X_i) == Ui(Xi, X_i+Xi). By A.3, Vi( ) is decreasing on X -i. Then, if Proposition 2.2 were not true we had that

Vi(Xi(n + 1), x_i(n + 1)) 2': Vi(xi(n), x_i(n)) 2': Vi(xi(n + 1), x_i(n)),

where the last inequality follows from the definition of a N.E.. Therefore, x_i(n) 2': x_i(n + 1) which contradicts that x_i(n) is increasing in n.

b) Suppose it is not. Defining Vi( ) as before we have that

V1(Xl(2),X2(2)) 2': V1(Xl(1),0) 2': V1(Xl(2),0)

And since ViO is decreasing on X-i we get a contradiction .•

If A.2 does not hold, then Proposition 2.1 might not be satisfied (see ex­ercise 2.4). If A.2 holds but Ui ( ) is increasing on x we have the reverse conclusion (see exercise 2.3). Notice that in the case in which n = 1 A.1-2 are not required. If n = 1 but Ui ( ) is increasing in x, an identical reasoning to the one made in the proof above shows that entry increases the payoff of the incumbent player (see Exercise 2.3).

We now turn our attention to the second approach mentioned at the begin­ning of this section. In order to do that, we will assume that payoffs are profits and that all firms are identical (these two assumptions will be extensively used in Chapters 3 and 4). We first define what we mean by a Symmetric Walrasian Equilibrium (SWE). The notation is identical to the one introduced in Chapter 1: p is the market price, p( ) is the inverse demand function and c( ) is the cost function, assumed to be identical for all firms.

Definition 2.1 A Symmetric Walrasian Equilibrium (SWE) for a market with nfirms is a triple (pW(n) , xW(n), x;V(n)) such that

a) pW(n) = p(xW(n)). b) n x;V(n) = xW(n). c) x;V(n) maximizes pW(n)xi - C(Xi) for i = 1, ... , n.

Under the assumption that ~ 2': 0, i.e. non increasing returns to scale, the existence and uniqueness of a SWE can be established easily (a graphical

39

argwnent suffices). Notice that under constant returns to scale, Asymmetric Walrasian Equilibria (AWE) may exist: In particular, it is possible that some firms produce a large output and others do not produce at all. However, the aggregate output is identical in a SWE than in any of the AWE. Therefore, without loss of generality we will concentrate on SWE.

We will denote by x:;" the limit of the aggregate output in a N.E. when the nwnber of players tends to infinity. Similarly, x: is the limit of the aggregate output in a SWE when n tends to infinity. We are now prepared to establish our next result.

Proposition 2.3. a) Under A. J -2, x:;" exists and it is characterized by price equals to marginal cost.

b) Under A. J and ~ ~ 0, x: exists. d W _ * I

c/ Xoo - xoo'

Proof. a) Since all firms are identical, we can write the first order condition of payoff maximization as T(x(n), xi(n)) = 0 = T(x(n), xj(n)). From A.2 b) itfollows that xi(n) = xj(n)for all i and j, i.e. the NE. is symmetric. This allows us to write the first order conditions of profit maximization as

( ( )) x(n) 8p(x(n)) _ 8c(~) = 0 pxn + 8 8 . n x Xi

Now let n ~ 00 and consider the sequence x(n), n ~ 00. Notice that x( n) is well defined for each n (because under A. J -2 a NE. exists). Since we assumed in Chapter J that x belongs to a compact set, there is a convergent subsequence with a limit, which by continuity of p( ), a~~) and ~;}, is x:;". Now the first order condition of profit maximization reads as follows:

( * ) _ 8c(O) p Xoo - 8 .

Xi

Thus, part a) of the proposition is proved. Part b) is proved analogously.

Finally part c) is proved by noticing that p( XW (n)) = aC~r) for each nand

thus, p(x:) = a~~~) .•

Proposition 2.3 is due to Cournot ((1838), Chapter 8). It establishes condi­tions under which N.E. tends in the limit (i.e. when n ~ 00) to perfect com­petition. In other words, Proposition 2.3 provides a game-theoretical fO\.inda­tion of perfect competition. Exercises 2.17-18 ask to the reader to compute N.E. and SWE for given n and in the limit in two special cases.

A weakness of Proposition 2.3 is that in the limit, individual firms produce zero output but aggregate output is positive, a paradoxical situation indeed!. Moreover, if the assumption on returns to scale does not hold, the result fails

40

(see Exercise 2.35). This has motivated to some authors to replicate both the number of firms and the number of consumers. Exercises 2.17-8 ask to the reader to perform this exercise. However, under the assumption that out­puts are bounded, replication of demand produce strange results, see Exercise 2.36. A much better approach to the question of the limit ofN.E. is carried out in Chapter 3, see Proposition 3.5.

We end this section by noticing that the assumption that goods are per­fect substitutes is essential for the result to hold. Exercise 2.37 shows that Proposition 2.3 fails under complementarities. In Chapter 4 we will see un­der which condition this kind of result holds in markets with heterogeneous goods which are close substitutes.

3 AGGREGATIVE GAMES II: EFFECTS OF A SHIFT IN PAYOFF FUNCTIONS

In this Section we will study the effect on endogenous variables of an ex­ogenous shift in the payoff function. We will assume that the payoff func­tion of player i can be written as Ui = Ui (Xi, X, t i ) where ti is a one dimen­sional parameter which may be possibly different for different players. In the Cournot model ti represents either the factors behind the demand function -i.e. tastes, income, other prices, taxes- or behind the cost function -i.e. tech­nology, price of inputs or, as in Farrell and Shapiro (1990), the quantity of capital owned by firm i-. First order condition of payoff maximization now reads

Now we make the following assumption:

Assumption 4: T;.( ) is strictly increasing in t i .

This assumption allows us to interpret increases in ti as shifts to the right of the marginal payoff curve, i.e. ti can be regarded as a measure of the impact of a shock on the marginal payoff of player i. We will distinguish two types of shocks: idiosyncratic and generalized. In the first we consider the impact on the market of a variation in a single t i . This may correspond to a shift in the price of the factors used or the taxes paid only by player i or the quantity of capital held by i. In the second we consider a simultaneous variation in all t i , i = 1, ... ,n. This corresponds, for instance, to a shift in the common demand function or the price of a factor used (or a tax paid) by all

41

players in the industry. In this case any vector (t I , ... , tn ) can be represented by a single number t. In this case we will write the first order condition as 7i(Xi, x, t) = o.

In the case of an idiosyncratic shock, an increase in ti means that the marginal profitability of a given vector of strategies increases. Thus, intu­ition suggests that in this case, the strategy of player i will increase and the strategy of any other player will decrease. This intuition is formalized in the next Proposition:

Proposition 2.4. Under A. 1, A. 2 and A. 4 an increase in t i , will have the following effects:

a) Increases x b) Increases Xi. c) Decreases Xj for all j f. i.

Proof. Since the proof is fairly analogous to that of Proposition 2.1, we will indicate only the guidelines. Firstly it is proven that x can not be constant. Secondly, if x decreases, A.2 implies that the strategy of any player must increase in order to maintain first order conditions, and this contradicts the definition of x. Thus, x increases. Again, first order conditions of payoff maximizationfor all players except i plus A.2 imply that the strategy of any of these players must fall. Therefore, the definition of x implies that Xi must increase .•

This Proposition implies that an idiosyncratic shock affecting player i, af­fects the other players in the opposite direction (see Exercise 2.5). If Ti ( )

is strictly decreasing in t i , the inequality implied by A.4 is reversed so are the conclusions of Proposition 2.4. A.2 is needed for the result to hold (see Exercise 2.6).

For the next Proposition we will need an additional assumption. This as­sumption plus A.4 implies that a variation in ti affects both marginal and total payoff in the same direction.

Assumption 5: UiO is increasing and difftrentiable on ti.

Proposition 2.5. Under A. 1, A. 2, A.3, A.4 and A.5, an increase in t i , has the following effects:

a) Increases the payoff of i in a NE.. b) Decreases the payoffs of any other player in a NE..

42

Proof. First, it is easy -but tedious- to show that all variables are contin­uously difforentiable functions of ti in a neighborhood of equilibrium, since Assumption 2 implies that the Jacobian matrix of 1i( ) has a non-vanishing determinant. Then, taking into account the first order conditions for player j =f i, we have that

dUj _ aUj (dx dX j )

dti - ax dti - dti and Proposition 2.4 and A. 3 imply b) above.

In the case of player i we have that

dUi = aUi (dx _ dXi) + aui . dti ax dti dti ati

But by Proposition 2.4 part c), the strategy of any player but i has de­

creased and thus ;~ - ~ < O. Then, A.3 and A. 5 imply a) . •

Again, A.2 is needed for Proposition 2.5 to hold (see Exercise 2.7).

We will end this Section by studying the effects of a generalized shock. In this case, results are much less satisfactory: we can only predict changes in x.

Proposition 2.6. Under A. 1,2 and 4 an increase in t increases x.

Proof. First, by analogous reasoning to Proposition 2.1 it can be shown that x can not be constant. And if x decreases, all Xi must increase and this contradicts the definition of x .•

The effect of t on individual strategies and payoffs in equilibrium depends on how payoff functions are affected (see Exercise 2.8 and Dixit (1986) and Quirmbach (1988)). Finally without A. 2 Proposition 2.6 does not hold (see Exercise 2.9).

4 INFLATION TRANSMISSION: OLIGOPOLY VS. PERFECT COMPETITION

In this section we compare comparative statics properties of oligopolistic and perfectly competitive markets in a particular case.

Sometimes it is argued that inflation results from the combination of cost pushes (wages, raw material prices, etc.) and the oligopolistic structure of the markets. From this it follows that if the market could be forced to be more competitive, inflation would be reduced. This argument looks suspicious be­cause of the difference between high prices (in an oligopolistic market, ceteris

43

paribus, prices are higher than in a competitive market) and the sensitivity of prices to an exogenous cost push. In this section we consider the theoreti­cal plausibility of the above argument in the Cournot model (this section is entirely based on CorchOn (1992».

Let p be the market price of the good and p = p( x) be the inverse demand function. Throughout this Section we will assume that payoffs are profits and that firms are identical with constant marginal costs denoted by c.

We study the impact of an increase in marginal costs on equilibrium prices by means of two different measures. They are respectively, the derivative and the elasticity of equilibrium price with respect to marginal cost. Notice that in a perfectly competitive market price equals to marginal cost and thus, the value of each of the aforementioned measures is 1. In this section, we compare the value of each measure in an imperfectly competitive market with the corresponding value in a perfectly competitive market.

Definition 2.2 The inflationary sensitivity of a market is the derivative of the equilibrium price with respect to an exogenous change in marginal costs.

Thus, the inflationary sensitivity is ~ where p* denotes the Cournot equi­librium price. An alternative way of measuring the inflationary impact of an exogenous change in costs is the following:

Definition 2.3 The inflationary elasticity of a market is the elasticity of the equilibrium price with respect to the exogenously given marginal costs.

The inflationary elasticity, denoted by f.1, is f.1 == ~ :*' i.e. the elasticity of p* with respect to c. Both measures attempt to capture how rising costs trans­late into price increases, i.e. they measure how much inflation is transmitted by the market, given an exogenous cost push.

Notice that under imperfect competition price is greater than marginal cost and thus f.1 > 1 implies ~ > 1 and thus if the inflationary elasticity is larger in oligopoly than in perfect competition, the inflationary sensitivity of oligopoly is larger than the inflationary sensitivity under perfect competition (clearly, the converse is not true).

Let us define the key variables in our analysis. Let E == p",eex)).x be the elas-P x.

ticity of p' (x) with respect to x evaluated at the Cournot equilibrium. Roughly speaking E measures the degree of concavity (or convexity) of the inverse de­mand function. E ~ (resp. ::;) 0 iff p(x) is concave (resp. convex). Also

44

let f3 == P;'(1jx be the elasticity of the inverse demand function evaluated at Cournot equilibrium. It is easy to show that if f3 is constant, E = f3 - 1 and if f3 is increasing (resp. decreasing) on x, f3 > «) E + 1 (see Exercise 2.11).

Assuming interiority, if a list of outputs is a Cournot equilibrium, the first order condition of profit maximization hold, i.e. for all i

op(x) p(x) + (!f;"""Xi - e = o.

Adding up the previous equations over i, we obtain that

op(X) np(x) + (!f;"""x - en = O.

Then we have our next result:

(1)

Proposition 2.7. An oligopolistic market has a higher inflationary sensibility than a perfectly competitive one if and only if -1 - n < E < -1.

b) An oligopolistic market has identical inflationary sensibility than a per­fectly competitive one if and only if E = -1.

Proof. Differentiating (1) above we obtain that

dp* de = n+E+1

n (2)

Let us first prove part a). Suppose that ~ > 1. From (2) above we see that the denominator must be positive, i. e. n + E + 1 > 0 and ¥c > 1 implies

E + 1 < o. Conversely, If -1- n < E < -1, (2) above implies that ~ > o. Thus, if

¥c ::; 1, using (2) again, E + 1 2:: 0 and we reach a contradiction. Part b) follows from equation (2) directly .•

The necessary and sufficient condition in part a) of Proposition 2.7 is dis­cussed in Exercise 2.13. We just mention here that there are perfectly reason­able demand functions that fulfill this condition, for instance p = ;-r with o < 'Y < n. In this case, E = -'Y - 1 and 0 < 'Y < n imply the nec­essary and sufficient condition. However, if the inverse demand function is p = a - bx'\ a, b, a> 0, the condition is not fulfilled (see Exercise 2.12). In this case 1 + E = a > 0 and thus, one of the necessary conditions is violated. Finally, we remark under the assumptions made in this section, A.2 is equiv­alent to E > -no Thus, the necessary and sufficient condition above implies A.2.

With respect to the inflationary elasticity we have the following result:

45

Proposition 2.8. a) An oligopolistic market has a higher inflationary elastic­ity than a perfectly competitive one if and only if -n - 1 < E < (3 - 1.

b) The inflationary elasticity of oligopoly and perfectly competitive mar­kets are identical if and only if (3 = E + 1.

Proof. It is easily calculated that (3+n

J1= l+n+E

Since c and p* are positive from the definition of J1 we have that

(3)

. 1 . dp* . 1 ( 1) (4) szgn 0 J1 = szgn de = szgn 0 n + E + Let us prove part a) first. Suppose that J1 > 1. If n + E + 1 < 0 by

Proposition 2.7 we have that ~ :::::; 0 so (4) above implies that J1 :::::; 0 which is a contradiction. Therefore n + E + 1 > o. So if J1 > l,from (3) above we obtain that (3 > 1 + E.

Suppose that n + E + 1 > 0 and E < (3 - 1 but J1 :::::; 1. Then, the first two inequalities imply that (3 + n > o. Thus if J1 :::::; 1, we have that (3) implies 1 + E 2: (3 which contradicts E < (3 - 1.

Part b) follows trivially. •

The condition n + E + 1 > 0 has been mentioned before. The condition E < (3 - 1 is discussed in Exercise 2.11. Let us remark that from Propositions 2.7 and 2.8 it follows that there is no guarantee that an increase in n makes an oligopolistic market to be less inflationary.

Summing up, under our assumptions, inflation transmission in oligopoly depends on the shape of the inverse demand function. We have found per­fectly normal cases in which oligopolistic markets transmit less inflation (what­ever we measure it) than perfectly competitive ones. An important caveat is that the analysis made above assumes that input prices are constant. In the case in which wages (or the price of other input) are negotiated and depend on market conditions, the shape of the inverse demand function is no longer determinant of the question of inflation transmission, see Exercise 2.14.

5 COMPARATIVE STATICS IN PRICE-SETTING GAMES

In this section we focus our attention on the class of games studied in Section l.5. Under A.2', these games display strategic complementarity. The comparative statics properties of this class of games have been studied by

46

Lippman, Mamer and McCardle (1987), Vives (1990), Milgrom and Roberts (1990) and Milgrom and Shannon (1994). In this Section we will present a simplified version of their results.

It is interesting to remark that in the case considered in this Section, there is no need to distinguish between an idiosyncratic and a generalized shock because our results do not depend on the kind of shock. Thus, we will simply speak of a shock. As before, let 8U~~.t;) == Ti(p, t i) where ti is a parameter affecting the payoff function of i. We now assume the following:

Assumption 4'. 7';( ) is strictly increasing in ti.

This assumption is similar to that of A.4. Under A.4' an increase in ti implies a shift to the right of the marginal payoff curve of player i. However the interpretation is now different. Under quantity setting this shift could be interpreted as more aggressive behavior, since for a given strategy of the others, firm i increases output. Under price setting this shift implies less aggressive behavior, since for a given strategy of the others firm i charges a higher price. See Exercise 2.33 for further interpretation of this condition.

An important part in the proof the next result, will be played by the concept of a dominant diagonal matrix. defined below:

Definition 2.4 A n X n matrix with typical element aij has a Dominant Diag­onal if there are positive weights AI, A2, .... , An such that for all i = 1, ... , n

Ailaiil > L Ajlaijl· ifi

In this section, we first study the effect of a shock on prices and payoffs. The reason for this procedure is that results concerning entry will be derived from results concerning the effects of a shock.

Proposition 2.9. Let us assume that A. I', A. 2' andA. 4' hold. Let (dtl' ... , dtn)

be a infinitesimal variation of (tl' ... ,tn) with dti ;::: 0 Vi E 1. Then the fol­lowing happen:

a) Any price in the NE. increases or remains constant. b) If dti > 0 some i,:Jj such that Pj increases in the N.E.. c) If dti > 0 Vi E I. all prices increase.

Proof. Let us first prove part a). Diffirentiating the first order conditions of payoff maximization we obtain the following equations.

aTdp)d aTdp)d aTl (P)d - 0 a PI + ..... + a Pn + ~ tl -PI Pn utI

47

fJTn (p) ···················lit~·(pr·········aT~·(p)

a dPl + ...... + a dPn + a dtn = 0 PI Pn tn

The system above can be also written in matrix notation. Let T be a n x n

matrix with typical element -~, Dp be a n x 1 vector with typical element

dPi and tan x 1 vector with typical element dti%f:-. Then we have:

:r Dp=t

Notice that the matrix:r has the following properties:

I: All the elements in the main diagonal are positive and all the elements

off the main diagonal are negative.

2: It has a dominant diagonal (by A.2 'b, taking Ai = l,for all i) Since t is a non negative vector, by a classical result in Linear Algebra

(see, e.g. Takayama (1974) p. 382, Theorem 4. C. 3), there exists a unique Dp that solves the above system. Also Dp is non negative, so part a) is proved

If part b) of the Proposition were not true, Dp = 0 but this would contra­

dict that t is different from zero. Finally part c) is easily proved by contradic­tion .•

Notice that the Proposition above covers the case of both an idiosyncratic and a generalized shock. It is worth to remark that, under our conditions, the converse to Proposition 2.9 holds, i.e. if dt i ::::: 0 Vi E I, then the matrix :r has dominant diagonal (see Exercise 2.31). See Exercise 2.32 for an example showing that if A.4' is not fulfilled, Proposition 2.9 does not hold.

In order to prove the next result, let us make a new assumption:

Assumption 5'. UiO is increasing on t i .

Proposition 2.10. Let us assume that A. I', A. 2', A.3', A.4' and A.5', hold Then, an irifinitesimal variation of (t l , ... , tn) such that dt i > 0 Vi E I in­

creases the payoff of any player.

Proof. Differentiating the payoff at a N E. we obtain the following:

aUi (p*) aUi (p*) aUi (p*) dUi ~ a dPl + ..... + a dPn + at dtl

PI Pn 1

And since (j~(P') = 0, the resultfollows directly from the assumptions .• "Pi

See Exercise 2.34 for a similar result which depend on assumptions made on demand and cost functions.

We now tum our attention to the case of entry. We face the following difficulty: In the case of quantity-setter agents the output of the potential

48

entrant before entry is zero. However, when finns are price-setters, what is the price before entry of the product sold by a potential entrant finn n + 17. We take the view that before entry, the price of the product offered by the potential entrant is so high that it discourages conswners from buying this brand. In other words, we are asswning that the market rations conswners in the case of good n + 1 by means of high prices. Let us denote this price by poo. We will assume that Pn+ 1 < poo for all Pn+ 1 E K n+ 1. Then, interpreting poo and Pn+ 1 as parameters in the payoff functions of finns other than n + 1, strategic complementarity implies that marginal profits are increasing on p=. 12 In other words, A.2' c) implies that A.4' holds. This is the main insight behind the proof of the next two results.

Proposition 2.11. Let us assume that A. I', and A. 2' hold. Then the entry of a new firm decreases the prices charged in the NE. by all other firms.

Proof. Let us write the payoff function of firm i before and after the en­

try as Ui(Pi,P_i,pOO ) and Ui(Pi,P-i,Pn+I), i = 1, ... , n. Think of Pn+I as a generalized shock hitting firms 1, ... , n. In Proposition 2.9 we proved that (PI, ···,Pn) were increasing functions of the parameter representing the gen­eralized shock. Since Pn+l < p=, Proposition 2.9c) yields the result .•

We end this Section by studying the effect of entry on payoffs of incwn­bents.

Proposition 2.12. Let us assume that A. I', A. 2' and A.3' hold. Then, the entry of a new firm in the market decreases the payoffs of all incumbentfirms.

Proof. Let Pi (n) and Pi (n+ 1) be the price charged in the N E. by firm i before and after entry. Let p-i(n)) and p-i(n + 1)) the vector of prices charged by firms other than i before and after entry. Then we have the following:

Ui(Pi(n),p-i(n),p=) 2: Ui (Pi (n + l),p_i(n),pOO ) > Ui (Pi (n + 1),p_i(n + 1),p=) 2: Ui (Pi (n + 1),p_i(n + 1),Pn+l(n + 1))

where the first inequality follows from payoff maximization, the second in­equality from A.2' and Proposition 2.11 and the last inequality follows from the definition of pOO (notice that A. 4' and A. 5' are automatically fulfilled in this case) .•

Again, a good understanding of Propositions 2.1-6 and 2.9-12 can be ob­tained by making a picture in the case n = 2.

12 A more fonnal derivation ofpoo may be as follows: Let Dn+1(P,Pn+l) be the demand function of good n + 1 where p = (PI, ... , Pn). Let p~+ 1 (p) be the minimum price for which Dn+l (p, Pn+d = 0 as a function ofp. If Dn+l 0 is continuous, P~+I 0 is continuous as well. Take now maxp~+1 (p) over P E K. This maximum exists because P~+1 () is continuous and K is compact. Call this maximum P~+I' If P~+1 E Kn+l take poo = (max pK) + 1. If P~+1 rt Kn+l take poo = P~+I' It is clear that this poo has all the required properties.

49

Summing up, in this chapter we have studied how the entry of a new player and a shift in payoff functions affect the strategies played and the payoff obtained in a Nash Equilibrium. In the case of aggregative games with strategic substitution, under A.2, the effect of entry or an idiosyn­cratic shock, conform with our intuition. Without A.2 counterintuitive effects can occur. In the case of a generalized shock, even if A.2 is as­sumed, nothing can be said about payoffs and individual strategies. Also, we have studied conditions under which oligopolistic markets transmit more (or less) inflation than perfectly competitive ones. Under strategic complementarity and A.2' all effects of a change in an exogenous variable agree with our intuition and with the results obtained under strategic substitution.

6 ADDITIONAL REFERENCES

Applications of games with strategic complementarities to industrial or­ganization are reviewed in Vives (1993) and to macroeconomics in Silvestre (1993), see also Fudenberg and Tirole (1991) pp. 489-497. For some empir­ical evidence on the effects of entry in oligopolistic markets see Bresnahan and Reiss (1991) and the references therein. Zaleski (1992) reviews some empirical evidence on inflation and monopoly.

7 APPENDIX: THE ASSUMPTION OF DIFFEREN­TIABLE PAYOFF FUNCTIONS

In this Appendix we will argue that it is unlikely that payoff functions are differentiable everywhere. However, we will see that in some important cases, N .E. strategies are never located at points in which payoff functions are non differentiable.

Take the Coumot model spelled out in Chapter 1. For the time being let us work with a demand function instead of the inverse demand function used in the Coumot model. Suppose this function is linear, i.e. x = a - bp.

However, the assumption of linearity can not hold everywhere because for p> alb demand is negative. In order to avoid negative demand we may write the demand function as x = max{O, a - bp}. Thus, the demand function is not differentiable at p = o. However, as long as the N.E. occurs at an allocation yielding a positive price, the demand function is differentiable and the analysis carried out in this chapter still valid.

50

Suppose now that the above demand function corresponds to a given coun­try, say C, and this country enters into a free trade zone with demand function (before the entry ofC) equal to max{O, A - Bp} with A > a and A/ B > a/b. Thus, the demand function for the free trade zone, once country C has entered IS

x = 0 for p > A/B.

x = A - Bp for a/b ~ p ~ A/ B

x = a + A - (b + B)p for p < a/b.

Notice that the new demand function is not differentiable at p = a/b. Thus, if the N.E. occurs at an allocation yielding this price, payoff functions are not differentiable there and the analysis carried out in this chapter can not be applied (with the exception of Proposition 2.2, part b)). Fortunately, we will see that this is not the case and that if demand (or cost) are differentiable almost everywhere, N.E. occurs at differentiable points.

Let p = p( x) be the inverse demand function. Suppose this function is not differentiable at x = X, but it is differentiable in (x, x + E) and (x, x - E) for some E. If this lack of differentiability is caused by the aggregation of different demand curves it must be that

ap(x + 8) ap(x - 8) . ax > ax for all 8 suffiCiently small.

This just generalizes the fact that computing the slope of the inverse de­mand function in the linear model above,

1 1 --->--.

b+B B Suppose that there is a N.E. with aggregate output x and individual outputs

(Xl, .... , xn ). Consider now that firm i decreases infinitesimally its equilibrium output by dXi. Since we start at a N.E., this deviation can not be profitable,

dU - d (ap(x - dXi) A (A) aCi(Xi)) 0 i-Xi a Xi + P X - a ::;, or

x Xi

ap(X-dXi) A (A) aCi(Xi) 0 a Xi +p x - a ~. x Xi

Similarly, if firm i increases infinitesimally its output by dXi,

dU o -_ d .(ap(x + dXi) A. + (A) _ aCi(Xi)) < 0 1 Xl a Xl p X a _, or

x Xi

ap(X + dXi) A (A) aCi(Xi) < 0 Th a Xi + P x - a _. us, X Xi

51

ap(X - dXi) > ap(x + dXi) ax ax

contradicting our initial assumption. Thus, at least one of the deviations must be profitable and the N.E. can not occur at this allocation.

A similar analysis can be made in the case in which the cost function is not differentiable everywhere. Suppose again that there is a N.E. with aggregate output x and individual outputs (Xl, .... , xn). At output Xi, the cost function is not differentiable. In particular for outputs smaller than Xi, the cheaper production process is one suitable for small scale of operations with marginal cost denoted by 8~~:). For outputs larger than Xi, the cheaper production pro-

cess is one suitable for mass production with marginal cost denoted by a:;;) . We assume that

aK;(Xi) < aCi(Xi) aXi aXi'

i.e. the mass production process has smaller marginal costs at Xi (and possibly at any output larger than that). Suppose now that firm i decreases infinitesi­mally its equilibrium output by dXi. Since we start at a N.E., this deviation can not be profitable,

dU.=d .(ap(x),.+ (,)_aCi(Xi)) <0 , x, ~ x, p x ):l _, or uX UXi

ap( x) , . + ( ') _ aCi (Xi) > 0 ):l x, P x ):l _.

uX UXi Similarly, if firm i increases infinitesimally its output by dXi,

dU o = d .(ap(x) '. + (') _ aKi(Xi)) < 0 , x, ~ x, p x ):l _, or uX UXi

ap(X) '. + (') _ aKi(Xi) < 0 Th ):l x, p x ):l _. us, uX UXi

aKi(Xi) > aCi(Xi) aXi aXi

contradicting our initial assumption. Thus, at least one of the deviations must be profitable and the N.E. can not occur at this allocation.

8 EXERCISES

2.1. Show by means of an example that entry of a new firm might increase the output of the incumbent if A.2 does not hold (see Corch6n (1994), Example 1 ).

52

2.2. Show by means of an example that the entry of a new firm might decrease total output if A.2 does not hold (see Corch6n (1994), Example 2).

2.3. Assuming that payoffs are increasing in x, show that entry increases the payoff of incumbent players under A.2 or with n = 1.

2.4. Show by means of an example that entry of a new firm might increase the payoff of incumbents if A.2 does not hold (see Corchan (1994) Example 3).

2.5. Show that under the conditions of Proposition 2.4, dxldt i < dxddt i , i.e. that there are no multiplier effects. Compare with the results obtained in the case of price-setting firms (see Cooper and John (1988) and Fudenberg and Tirole (1991) p. 498).

2.6. Give an example in which A.2 is not satisfied and Proposition 2.4 does not hold (see Corchan (1994), Example 4).

2.7. Give an example in which A.2 is not satisfied and Proposition 2.5 does not hold (see Corch6n (1994), Example 5).

2.8. Give an example of a market in which a technological improvement in costs decreases the output and profits a/the most efficient firm (see Corch6n (1994) example 6).

2.9. Show by means of an example that without A. 2 Proposition 2.6 does not hold (see Corchan (1994) Example 7).

2.10.- Show that if payoff are profits, all firms are identical and there are non decreasing returns to scale, ui(l) ~ nUi(n).

2.11.- Show the following results: a) If j3 is constant, E = j3 - 1. If j3 is increasing (resp. decreasing) on x,

we have that j3 > (resp. <) E + 1. b) E «resp. » j3 - 1 implies that j3 is locally increasing. c) A necessary (but not sufficient) condition for j3 to be increasing on x is

the convexity of the inverse demand curve. d) Give an example of demand functions satisfying conditions in a)-c)

above (see Corch6n (1992)).

2.12.- Calculate directly the inflationary sensitivity and the inflationary elas­ticity in the following cases.

a) The inverse demand function is p = a - bx"', a, b, a > 0, b) The inverse demand function is p = AI x'Y with 0 < '"Y < n.

c) Show that if b < 0, the case considered in b) above is a special case of case a) above.

53

d) Show that the results in a) and b) above are in accordance with Proposi­tions 2.7 and 2.8.

2.13.- a) Show that 1+f+n > 0 if and only if dx/dc < 0, and that 1+f+n > o implies a unique Coumot equilibrium.

b) By differentiating (1) show that neither Strategic Substitution, nor Strate­gic Complementarity (see Bulow, Geanakoplos and Klemperer (1985)) is nec­essary or sufficient condition for an oligopolistic market to transmit more (or less) inflation than a perfectly competitive one.

c) Show that concavity of p( x), is a sufficient (but not a necessary) con­dition for the oligopolistic market to be less inflationary than the perfectly competitive one (see CorchOn (1992)).

2.14.- Calculate the inflationary elasticity and the inflationary sensitivity com­bining the following possibilities:

a) The bargaining outcome is either the one predicted by the Nash Bar­gaining solution or it is the outcome of trade unions maximizing the wage

bill. b) The inverse demand function is either linear or isoelastic (you have to

consider eight different cases). Does results in a) and b) above depend on the form of the inverse demand

function or on something else? (see CorchOn (1992) pp. 19-21).

2.15. Analyze a Coumot model with the following kind of linear taxes: ad valorem, on output and VAT. Assume that the market is characterized by a lin­ear inverse demand function and identical linear costs. Compute the Coumot equilibrium with each kind of taxes and indicate how comparative statics could be performed (see Dierickx, Matutes and Neven (1988)).

2.16.- Consider a market with a linear inverse demand function and n identical firms with cost functions C = x; /2ki where ki is the capital stock of the ith firm.

a) Compute the Coumot equilibrium for a given vector of capital stock. b) Show how outputs in the Coumot equilibrium depend on the vector of

capital stock (see Farrell and Shapiro, (1990) p. 277).

2.17.- Suppose that all firms are identical with quadratic cost functions and the inverse demand function is linear.

a) Find the Coumot and the Symmetric Walrasian equilibrium. b) Represent the Coumot and the Symmetric Walrasian equilibrium graph­

ically. c) Find the limit of both equilibria when the number of firms and/or con­

sumers tend to infinity.

54

2.18.- Suppose that all firms are identical with linear cost functions and the inverse demand function reads p = Ax"', 0 < a < 1.

a) Find the Cournot and the Symmetric Walrasian equilibrium. What would happen if a ~ I?

b) Represent the Cournot and the Symmetric Walrasian equilibrium graph­ically.

c) Find the limit of both equilibria when the number of firms and/or con­sumers tend to infinity.

2.19.- Show the effect of a merger on the Cournot equilibrium. Give an exam­ple in which profits of merged firms decrease as a consequence of the merger (see Salant, Switzer and Reynolds (1983)). Compare with the case of a verti­cal merger (see Salinger (1988)).

2.20.- Show that a Cournot equilibrium may fail to exist even ifthe inverse de­mand function is decreasing and continuous (see Novshek (1985). Compare with the arguments in Roberts and Sonnenschein (1977) pp. 107-109).

2.21.- Suppose that we have two identical firms producing costlessly in a market and the inverse demand function has a constant elasticity of -0'5.

a) Show that there is no Cournot equilibrium. b) Generalize to n firms and inverse demand function with constant elas­

ticity. Comment on the reason of the lack of existence of equilibrium. c) Compare with the results obtained in Exercise 2.18 above.

2.22.- Generalize the Cournot model to multiproduct firms. Find necessary and sufficient conditions of profit maximization. Try to apply the methods of proving Propositions 2.1-2.3 to this model.

2.23.- Consider a Cournot model with adjustment costs in which the inverse demand function is linear and cost functions are identical Ci = CXi + d( Xi -

Xi)2, where Xi is the status quo. a) Interpret d(Xi - Xi)2.

b) Find the Cournot equilibrium with adjustment costs assuming that Xl =

X2 = ...... = xn ·

c) Find the "long run" Cournot equilibrium in which for each firm the equilibrium output coincides with Xi' Interpret this kind of equilibrium.

d) In a dynamic set up where Xi is interpreted as last period output, starting from a symmetric allocation, find the sequence of Cournot equilibria and see if it converges to the long run Cournot equilibrium.

2.24.- Consider a market with oligopolistic and perfectly competitive firms with decreasing returns to scale. Show that such a market is qualitatively identical to a market with oligopolistic firms only. What about differentiabil­ity of the profit function in this case?

55

2.25.- Show that the effect ofa labor strike on profits depends on the number of struck finns (see Gaudet and Salant (1991), p. 662).

2.26.- Explain why a Stackelberg leader who takes over one finn in duopoly expands its output but a monopolist who takes over an industry of N finns contracts the output of each (see Gaudet and Salant (1991), p. 659).

2.27.- In the economy described in 2.18 above with n = 2, show that nei­ther strategic substitution nor strategic complementarity hold globally. Deter­mine the intervals in which strategic substitution and strategic complemen­tarity hold. Find a value of ex for which the Cournot equilibrium is in fact a Stackelberg equilibrium with either finn 1 or finn 2 as a leader.

2.28.- Describe a theory of slack-ridden oligopoly (see Selten (1986)). Show that entry of efficient firms may be blockaded by inefficient ones.

2.29.- Extend the original model of hierarchical control (Williamson (1967)) to the case of oligopoly (see Martin (1993) pp. 215-6).

2.30.- Let us assume two agents with payoff functions Ui(Sl, S2). Suppose that a N.E. exists. A social planner feels that the value of Sl in the N.E. is too low in relation with what she feels it would be optimal. She propose to subsidize agent 1, whose payoff function becomes U1(Sl, S2) + sl.l5, 8> O.

a) Give examples of this kind of situation. b) Give an example in which the value of Sl in the N.E. relative to the new

payoff function is less than in the old N.E .. Explain this result.

2.31.- Show that if dt i ~ 0 Vi E I, the matrix J has dominant diagonal (see Dierker and Dierker (1999), Theorem 1).

2.32.- Give an example showing that if A.4' is not fulfilled, Proposition 2.9 does not hold (see Dierker and Dierker (1999) Examples 2 and 3).

2.33.- Give a set of sufficient conditions for A.4' to hold (see Dierker and Dierker (1999), Proposition, p. 61).

2.34.- Give a set of assumptions on demand and cost functions yielding a result identical to Proposition 2.1 0 (see Dierker and Dierker (1999), Theorem 2, part c)).

2.35.- Assume that all finns are identical with a cost function C = aXT - bx; + CXi, with a, b, C > O. Market demand is linear.

a) Give conditions under which the average cost function is convex with a unique minimum.

b) Give conditions under which A.2 holds. c) Compute the Cournot equilibrium aggregate output. Show that when

the number of finns tend to infinity the Cournot equilibrium does not tend to the SWE.

56

d) Compare with Novshek (1980) pp. 477-8, example B.

2.36.- Let us assume that in a market we have n identical consumers yielding a linear demand function and a monopolist that can produce, at most, R units of a good at zero cost.

a) Show that the per capita welfare loss of the monopoly output tends to zero when n tends to infinity. What about total welfare loss and profits? (see Hart (1979), Sections 1-2).

b) Show that the monopoly equilibrium for large but finite n is perfectly competitive.

c) Analyze graphically the monopoly equilibrium in economies with a large number of consumers when average costs are increasing and decreas­ing. How is that the monopoly produces the perfectly competitive output?

2.37.- Let us assume that in an economy there are two products that are perfect complements, i.e. consumers need both goods in order to derive satisfaction from them.

a) Give examples of these goods. b) Which utility function may be used to represent the preferences of a

consumer over these two goods? c) If each good is offered by an independent seller, show that irrespectively

of the number of consumers and/or sellers, there is always aN .E. in quantities which involves zero output of the two goods (see Hart (1980) and Makowski (1980».

2.38.- Suppose that the inverse demand function is given by p = xl~'" , that all firms are identical and have constant returns to scale.

a) Compute the output of monopoly (i.e. all firms collude) and of perfect competition (i.e. all firms take the market price as given).

b) Let a ---. 1. Give an interpretation to a = 1. Compute the ratio of perfectly competitive and monopoly output. Show that this ratio tends to the number e when a ---. 1.

2.39.- Let us assume that the are n identical firms with constant marginal costs and that the demand function reads p = a - bOl..

a) Show that for some values of the parameters a, b and a, the above model generalizes both linear (Exercise 2.17) and isoelastic (Exercise 2.18) inverse demand functions (notice that in Exercise 2.17 the cost function is quadratic).

b) Find the Cournot equilibrium. c) Find the limit of both equilibria when the number of firms and/or con­

sumers tend to infinity. d) Suppose that firms collude. Find the equilibrium output and compare

with the one obtained in b) above.

57

2.40.- Suppose that an anti-trust authority attempts to prove that the n firms in a certain market collude. The antitrust authority knows that there are n identi­cal firms with constant marginal costs and that the inverse demand function is of the form indicated in Exercise 2.39. However the antitrust authority knows neither the parameters a, b,and a, nor the marginal cost. Show that the col­luding firms can construct false parameters and marginal costs such that if these parameters were true, prices and quantities were those corresponding to a Cournot equilibrium (see Phlips (1995), pp. 128-9).

2.41.- Suppose n identical firms producing an illegal drug. Demand and costs are linear, but the quantity marketed is only a fraction, say 0, of the quantity produced.

a) Compute the Cournot equilibrium for a given O. b) Show the effects of 0 on the endogenous variables.

2.42.- Applying methods similar to those used in Section 2.4 and Exercises 2-12 and 2.13 compute ~: in the case in which the inverse demand function is p = a - bxo. How this number depends on n?

CHAPTER 3. WELFARE AND COMPETITION

Abstract: Cournot and optimal outputs (Proposition 3.1). A Mechanism with efficient outcomes (Proposition 3.2). Entry and social welfare (Propo­sitions 3.3-4). Cournot Equilibrium with Free Entry (CEFE) and Walrasian Equilibrium (Proposition 3.5). Relationship between CEFE and optimal al­locations (Propositions 3.6-7). A reduction in the active number of.firms in CEFE increases social welfare (Propositions 3.8-9). Number of.firms and profits in CEFE (Propositions 3.10-11). Cournot equilibrium and constrained efficiency (Proposition 3.12). Appendix: Optimal trade policy (Propositions 3.13-17).

1 INTRODUCTION

In this chapter we will come across a much-debated question: Does the market mechanism allocate resources in an efficient way ? The classical an­swer to this question is the so-called First Theorem of Welfare Economics which says that perfect competition achieves an efficient allocation of re­sources. However this theorem can easily be adapted to show that a Cournot equilibrium is never efficient (see Proposition 3.1 below), casting a shadow of a doubt on the beneficial properties of the market. This calls for a deeper investigation into the welfare properties of imperfect competition and this is what this chapter is about. We will do this in two models: The quantity­setting model explained in Sections 1.2-1.4 and in a quantity-setting model with an endogenous number of firms that we present in Section 3.4 and study in Section 3.5.13

For reasons of tractability we will concentrate on a special case of the quantity-setter model which we spell now. There is an homogeneous good (however all results here hold for some kind of product differentiation, see Exercise 3.1). Players, referred to as firms from now on, have no objectives other than profits. Firm i has a cost function denoted by Ci ( ) such that Ci(O) = O. Let p be the market price of the product and p(x) be the inverse demand function mapping aggregate output into prices, i.e. p = p( x). This function is derived as follows. There is a representative consumer with a C3 ,

strictly increasing and quasi-linear utility function U = V(x) + l where l denotes the consumption of an outside good which will be called money and

13 The welfare properties of the price-setting model presented in Section 1.5 are largely unknown.

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v ( ) is assumed to be strictly concave. The budget constraint is px + l = M, where M is the exogenous income. Thus U = V(x) - px + M. Since M is constant it can be dropped. The inverse demand function comes from the first order condition of utility maximization p = a~ix) by setting a~ix) == p( x). We now explain how our previous assumptions look like in this framework. First, it is easily checked (recall Exercise 1.3) that A.2 reads:

a2p(x) ap(x) d ap(x) a2Ci (Xi) ~Xi+-~- < Oan -~-- a 2 < O. uX uX ux Xi

A.3 means that the inverse demand function p(x) is strictly decreasing which is equivalent to V ( ) be strictly concave. From now on, the N.E. in quantities will be called Cournot Equilibrium (C.E.), defined formally below:

Definition 3.1 A Cournot Equilibrium is a list of outputs (xi, ... , x~) such that Vi = 1, ... , n we have that:

n

j=l #i

In some propositions below we will assume that firms are identical with a cost function denoted by C( ). An important consequence of this assumption is that, under A2 b) all active firms produce the same output in any N.E., i.e. equilibrium is symmetric. This is easily seen by considering the first order conditions of profit maximization:

( *) ap(x*) * _ aC(xi) _ 0 _ (*) ap(x*) * _ aC(xi) p x + ~ Xi ~ - - p x + ~ Xj ~ or

uX UXi uX UXj

ap(x*) x* _ aC(xi) = ap(x*) x* _ aC(xi) (*) ax' aXi ax J aXj'

But A.2 b) implies that, given x' ,

ap(x*) _ aC(xr) ( _" ") ~ Xr ~ ,r - 2, J ux UXr

is strictly decreasing on Xn r = i, j. Then, equation (*) implies that x; = xi.

Preferences are such that utility is transferrable. This is because if we take one unit of profits and give it to the consumer her utility increases just in one unit. In other words, utility functions are quasi-linear. In this case, social welfare (denoted by W) is the sum of consumer and producers's utility functions, i.e. W == V(x) - 2:7=1 Ci(Xi) (see, e.g., Moulin (1988), pp. 170-1). Thus we can define an optimal allocation as follows:

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Definition 3.2 A list of outputs (x~, ... , x~) is socially optimal (or simply opti­mal) if it maximizes

n

V(x) - L Ci(Xi)' i=l

In the case of an interior maximum, first order conditions of the maximiza­tion above read,

aV(x) _ aCi(Xi) \-I' _ 1 J':l - J':l ,vZ - , ... , n. uX UXi

Since p = a~~x), the above equation is the well-known equality between price and marginal cost (see Exercise 3.21 for the case in which the cost func­tion is not differentiable).

2 WELFARE AND COURNOT EQUILIBRIUM

In this section we study the relationship between Coumot equilibrium out­puts and those which maximize social welfare. Then, denoting equilibrium (respectively optimal) variables with the superscript * (resp. 0), we have our first result in this chapter.

Proposition 3.1. Under A.i, A.2 and A.3, we have that XO > x*.

Proof. From the first order conditions of profit and welfare maximization, we have that Vi = 1, ... , n,

( *) ap(x*) * _ aCi(x;) _ 0 > ( 0) _ aCi(xf) p x + J':l Xi J':l - _ P X J':l'

uX UXi UXi (since we did not assume that the optimum is interior, the last inequality

may be strict). 1f x* ~ xo, A.3 implies that p(XO) ~ p(x*). Then, the previous inequality yields

ap(x*) * aCi(x;) aCi(xf) ap(x*) ° aCi(xf) --x· - > - > ---x· - -::-'--"-'-ax' aXi - aXi ax' aXi'

But, as we noticed before, A.2b} implies that

ap(x) aCi(Xi) --x· - ---::-'---'-ax' aXi

is decreasing in Xi given x. Therefore the previous inequality implies that xf > x;, Vi = 1, ... , n, which contradicts that x* ~ Xo .•

An implication of Proposition 3.1 is that Coumot equilibrium is not so­cially optimal. The reader is reminded that this is a different result from Proposition 1.4 which proved the inefficiency of Coumot equilibrium in terms of the payoffs offirms.

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We tum now our attention to the problem of how to drive equilibrium al­locations to the optimal ones by means of the appropriate policy. Under com­plete information of the regulatory agency it is easy to see that in most cases a carefully chosen subsidy per unit of output will achieve the desired end (see Exercise 3.3). However in this case a pure command economy can achieve optimal allocations by fiat. Therefore, the problem is meaningful when the regulatory body has incomplete information about either demand or costs. We will consider here the case in which only cost functions are private in­formation. In this case, there is a very simple mechanism, due to Loeb and Magath (1979) which achieves optimal allocations by means of a non-linear subsidy of the form V(x) - PXi for i = 1, ... , n. Under this scheme, profits for firm i are V(x) - Ci(Xi)' The N.E. is defined relative to this profit func­tion, i.e. Ui = V(x) - Ci(Xi)' This is just an special case ofa N.E. as defined in Chapter 1, Definition 2. Then, we have the following:

Proposition 3.2. Under A.l and A.2, 1) The optimal allocation can be sustained as HE. by the above subsidy scheme. 2) If social welfare is concave in Xl, ... , Xn , any HE. yields an optimal allo­

cation.

Proof. From the first order conditions of the optimal allocation we get that

8V(x) 8Ci(Xi) 0 d:f .. l't 'l 0 O· 1 -~-- - 8 :::; an ZJ stnct znequa Z y prevaz s Xi = , Z = , ... , n. ux Xi

It is easy to see that the equations that characterize the HE. are identi­cal to those above. Under A. 2, V ( ) - Ci ( ) is concave in Xi for given X -i. Therefore, the inequalities above characterize the HE. and the first part of the Proposition is proved The second part follows from the concavity of social welfare .•

The problem with the above mechanism is twofold. On the one hand it requires the controlling agency to know V(). On the other hand, the total surplus is allocated to firms. Both problems have been partially solved in subsequent contributions. A general problem is that mechanisms are seldom balanced, i.e. it is necessary to raise taxes in order to finance them. However taxes may produce an additional distortion on social welfare (see Exercises 3.4 and 3.9).

3 WELFARE AND ENTRY

We now focus our attention on the relationship between entry and changes

63

in welfare. A widely-held belief is that if entry barriers are removed (or if entry is subsidized) allowing new firms to enter into a market, social welfare will increase. However the next Proposition shows that this is true under very astringent assumptions.

Proposition 3.3. Under A.1, A.2, A.3 and 1) at;) is non decreasing on Xi and 2) firms are identical, social welfare increases with entry.

Proof. First we recall that, under the above assumptions, Cournot equilib­rium is symmetrical. Let ac(xi) be the average cost as afunction of output. Social welfare is W = V(x) - E~=l C(Xi). Under our assumptions it can be rewritten as W = V(x) - X ac(xi). Let us consider x and Xi as independent variables. Since ~~ = p - ac(xi) > 0 before entry, the same inequality will hold after entry since by continuity the entrant can find an output for which profits are positive. Thus, W is strictly increasing on x -given Xi- and de­creasing on Xi -given x- (from condition 1) above). Then, Proposition 3.3 follows from Proposition 2.1, where it was shown that entry of a new firm increases total output and decreases that of incumbent firms .•

Proposition 3.3 guarantees that under strong assumptions, in particular conditions 1) and 2) above, entry increases welfare. However it is easy to see that if these conditions do not hold, entry may decrease total welfare (see Exercises 3.5,3.6 and 3.7). The explanation of what happens in these cases is that the entry of a new firm into the market produces two effects. On the one hand the competitive effect, i.e. prices are driven downwards because of increased competition (see Proposition 2.1). On the other hand the technolog­ical effect, i.e. existing firms contract their output (by Proposition 2.1 again) and thus under either economies of scale or inefficient entrants, the economy as a whole produces less efficiently. Exercises 3.5-6-7 show that the second effect might prevail over the first one.

Now we will investigate whether some kind of converse to Proposition 3.3 may hold, i.e. whether potential welfare gains may imply entry. Let C(y) be the cost function of the entrant firm. This cost function is, possibly, different from the cost functions of incumbents. Let us define:

y ~ {Y I V(x(n) +y) - C(y) - t,c;(x;(n)) > W(n)}.

In words, Y is the set of outputs of the entrant which will improve welfare, under the assumption that the output of incumbent firms is given. Then we have the following:

64

Proposition 3.4. Let us assume Y f. 0 and that the entrant firm has a (Cl

convex cost function. Then, in any Cournot Equilibrium with n + 1 firms, the

entrant will produce a positive output.

Proof. We first show that 3y* such that av(x~~+y*) > aCa~*)' Let us maximize

V(x(n) + y) - C(y) with respect to y. Ifsuch a maximum exists, call it y', we

have that

8V(x(n)+y') _8C(y') . ifY-'-0 '>0 8y - 8y since I r, Y .

Since V(x(n) + y) - C(y) is strictly concave in y, by taking y* = y' - e,

for some e > 0, we get the desired result. Also if the maximum does not exist

it must be that 8V (x (n) + y) 8C (y') '>-I

8y > 8y vy > 0 since a reverse in sign for some y will imply the existence of a maximum. If

the reverse sign holds for all y, y = 0 is the only solution to the maximization

problem contradicting that Y f. 0. Therefore in both cases y* exists. Now if no entry is profitable we have

( () ) _ 8V (x (n) + y) C (y) '>-I P X n + y = :S -- vy.

8y Y Thus for y = y* these two inequalities contradict the existence of y* .•

Proposition 3.4 implies that under some conditions potential welfare im­provements are a good signal to entry. However this Proposition is weak in two respects. Firstly, it may be the case that the equilibrium output for the entrant does not belong to Y and thus entry decreases welfare. Secondly, the vector of equilibrium outputs with n + 1 firms might yield less welfare than the equilibrium outputs with n firms (see Exercises 3.5-6). Finally, notice that under economies of scale of the entrant firm Proposition 3.4 does not hold (see Exercise 3.8).

4 WELFARE AND FREE ENTRY EQUILIBRIUM

So far we have analyzed the relationship between Cournot equilibrium with a given number of firms and welfare. In this section we tum our atten­tion to the case where both the equilibrium and the optimal number of firms are endogenously determined. We will make the assumption that all firms are identical with cost functions denoted by C ( ). We are now ready to define an equilibrium in this context.

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Definition 3.3 A Cournot equilibrium with free entry (CEFE) is an integer m and a list of outputs (xi, ... , x;,,) such that Vi = 1, ... , m,

m

m

p(Lx: + z) z - C(z) ~ 0, Vz E JR.+ i=l

In other words, a number of firms and a list of outputs is a CEFE if these firms maximize profits taking the output of rivals as given, and no firm can enter into the market and make positive profits. The existence of such equilib­rium has been shown by Novshek (1980) when economies of scale are small (see also the Appendix to Chapter 4). The comparative statics of CEFE are analyzed by Corch6n and Fradera (1996) (see Exercises 3.26-27). Notice that in this kind of equilibrium entry and output setting are simultaneous deci­sions. See Exercise 5.41 for a model in which entry and output setting are sequential.

The main purpose of the model with free entry is to obtain a deeper un­derstanding of the relationship between N.E. and Walrasian equilibrium (or efficient allocations). In Chapter 2 we provided a first cut to this problem by letting the number of firms to go to infinity and focussing on the limit out­come of this process. There are two main drawbacks of this approach. First, if returns to scale are not non-increasing everywhere the result fails. Second, this approach does not say anything about what happens in finite but large economies. We will see how the consideration of CEFE solves these two problems.

We define the minimal efficient scale as the minimum output for which the average cost -denoted by ac( )- is minimized. We will denote the minimal efficient scale (assumed to exist) by 0:. If returns are non increasing 0: = O. We will denote by 'Y the aggregate output for which ac( 0:) = p( 'Y). Param­eter'Y could be interpreted as the Marshallian 10ng-TUTl perfectly competitive output when all firms are price-takers producing 0: and the number offirms is not necessarily an integer. Thus, 'Y is a measure of the market size. We will assume that 'Y > 0:, i.e. that economies of scale are small compared to the size of the market. Then, we have the following:

Proposition 3.5. Under A.3, x* E b - 0:, 'Yl. Proof. If x* > 'Y,p(x*) < ph) = ac(o:) ~ ac(z)Vz E JR.+> i.e. all active firms will have losses. But this implies that no firm will produce a positive output contradicting that x* > 'Y. If x* < 'Y - 0:, then a potential entrant can

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obtain positive profits by producing a quantity a since p( x * +a) > p( 'Y) and therefore a(p(x * +a) - ac(a)) > a(p("() - ac(a)) = 0 .•

Proposition 3.5 (the proof of which is due to Novshek (1980» implies that if the minimum efficient scale is small compared with the size of the mar­ket, equilibrium and optimal aggregate output will be close to each other. Therefore in this case, the perfectly competitive model is approximately cor­rect and welfare losses due to imperfect competition are small. Notice that firms can be made relatively small either by making a tend to zero (i.e. by reducing the non-convexity) or by making 'Y tend to infinity (i.e. by replicat­ing the consumer sector). The case of -unbounded- increasing returns (where 'Y < a) has been analyzed by Fraysse and Moreaux (1981), Dasgupta and Vshio (1981) and Guesnerie and Hart (1985), see exercises 3.10-13. Exer­cises 3.15-16 review other aspects of Proposition 3.5. Notice that Proposition 2.3 can be regarded as a limiting case of Proposition 3.5, valid only when returns to scale are non increasing, i.e. when a = O.

We now turn to the study of the relationship between optimal and free­entry equilibrium output. We first define an optimal allocation when the num­ber of active firms is variable.

Definition 3.4 The allocation {nO , (xl' ... , x~)} is optimal if nO r

V(XO) - L C(XO) :2: V(x) - L C(Xi), Vr E Nand V(Xl' ... , xr ) E 1R.+. i=l i=l

Notice that if the marginal (average) cost is always decreasing, there will only be one active firm in the optimum. If the marginal cost is always increas­ing, the optimum is symmetrical, i.e. all active firms produce the same output (see the discussion of A.6 below). Finally, under constant returns to scale, the number offirms is undetermined (see Exercise 3.17).

We turn our attention to the study of the relationship between CEFE and optimal allocations. In order to do this let us assume the following:

Assumption 6: a) Total and marginal costs are increasing. b) Average costs are V-shaped (strictly convex) with an unique minimum scale a.

As an example of a technology fulfilling A.6, think of a fixed cost and an increasing marginal cost14. For instance C(O) = 0 and C(Xi) = K + ex; for

14 The existence of a fixed cost implies that the cost function is discontinuous at O. Therefore A.I must be understood in the sequel as holding in a neighborhood of the (interior) optimal and eqUilibrium allocations.

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Xi > O. In this case, a = vi K / c. Two implications of A.6 are that the optimal allocation is symmetric and thus, Definition 3.4 reduces to the maximization of V(nxi) - nC(xi),with respect to (n, Xi) E lR~ and that if l' > a the optimum allocation, now denoted by the pair (nO, yO) is interior, i.e. (nO, yO) > > O. The reader is asked to prove these two implications in Exercise 3.19. 15

In the rest of this section, we will also assume that the number of firms can be treated as a continuous variable. This must be regarded as an approx­imation which simplifies the proofs and sharpens our results (which still are basically true if only integers are considered). An implication of this assump­tion is that active firms make zero profits in the CEFE (see Exercise 3.20 for further implications). Section 3.5 below examines how large profits can be at a CEFE when the number of firms is an integer. Under this additional assumption we have the following:

Lemma 1. Under A.l, A.3 and A.5, (n°, yO) is an interior optimum if and only if

aV a) ax (nO, yO) yO = C(yO) [1]

b) aV( ° 0) = aC( 0) [2] ax n , y ay y Moreover the optimal allocation is unique and yO = a.

Proof. Necessity of a) and b) follow from the necessary conditions for an interior optimum. Sufficiency follows from the fact that the solution to [1 J and [2 J is unique. Uniqueness of yO follows from the fact that [1 J and [2 J imply that

8~:1 (~Cy (a)) yO = a and n° = -----"---

a

where 8~:1(~~(a)) istheinverseof~~ evaluated at ~~(a) .•

Condition a) in Lemma 1 says that in the optimum, profits should be zero. Condition b) is the familiar marginal cost pricing condition (see Exercise 3 .21 for an example in which this requirement is neither a necessary nor a sufficient condition of optimality). Now we can prove our next result (hinted at by von Weizsaker( 1980)).

Proposition 3.6. Under A.l, A.3 and A.6 total and individual output are larger in the interior optimum than in the CEFE allocation.

15 Since all active finns produce the same output, we only need the output of a single finn to indicate the output in an optimum allocation.

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Proof. Consider the equation:

oV(x) G(Xi) oV-1 (C~:;}) _ -J'l- = --or x = 0 = F(Xi), (3)

uX Xi X which must hold in both optimal and equilibrium allocations. Since F( ) is increasing to the left of a (because average costs are decreasing there and a~~x) is decreasing in x) and decreasing to the right of a, F( ) is strictly quasi-concave with a maximum at Xi = a. By difforentiating [3 J we obtain

C(Xi) _ aC(Xi) Xi 8Xi

_x·a2V~x) • ax

which is positive when evaluated at equilibrium and is zero when evaluated at the optimum, i.e. at Xi = a. Therefore the conclusion of the proposition follows from a graphical argument .•

This result can be regarded as a excess capacity theorem in the sense that in equilibrium, firms underproduce in relation to the optimum16• The next Proposition (due to Perry (1984) and Suzumura and Kiyono (1987) and hinted at by von Weizsaker (1980» establishes a relationship between equilibrium and the optimal number of firms.

Proposition 3.7. Under A.i, A.2, A.3 andA.6, m > nO.

Proof. Notice that [3 J can be written as

X F(Xi) -= --=n, Xi Xi

where F( ) is the function defined in Proposition 3.6 above. Therefore,

d C(Xi) _ ac(x;} + a2V~x)x n _ Xi aXi ax

dx. - _ a2V~x) 2 ' • ax Xi

which evaluated at the optimum and the equilibrium is negative. Thus if we show that this derivative is negative in the interval (x;, a), the fact that a> x;, implies that nO < m.

Suppose that for some x; E (x;, a), :~ 2: O. Let x' = F(x;). Since

P = C(X;) and a2V~x) = ap(x) we have that: Xi ax ax '

( ') _ oG(x;) Op(X') I (') _ aG(x;) Op(X') I > 0 p X J'l + J'l Xi > P X J'l + a x - .

u~ ~ u~ X

oG(x~) op(x*) Since p(x*) - • + ---x~ = 0, we have that

OXi ox·

16 This result is not the same as the one in Proposition 3.1, since there it was assumed that the equilibrium and optimal number of firms was the same.

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( ') _ aC(x:) ap(x'), () _ aC(x:) ap(x*) ~ p x J;l + J;l Xi > p x* J;l + J;l X •.

UXi uX UXi uX But since from Proposition 3.6 x: > xi and x' > X*, the fact that A. 2

implies that both sides of the last inequality are strictly decreasing on x and Xi yields a contradiction .•

Proposition 3.7 asserts that market equilibrium yields overentry.17 Propo­sition 3.7 suggests that if some firms could be shut down, welfare would im­prove. Our next proposition (due to Suzumura-Kiyono (1987» shows that this conjecture is right. As in Chapter 2, xi(n) will denote a Cournot equilib­rium output given that n firms are active. This function can be extended to be differentiable. Moreover under A.l and A.2, Proposition 2.1 holds and this implies that this function is decreasing. Then we have:

Proposition 3.8. Under A.I, A.2, A.3 and A. 6 a (small) reduction in the equilibrium number of.firms in a CEFE increases welfare.

Proof. Let W(n) == V(nxi(n)) - nC(xi(n)). Then, we have that

dW aV(X*) ( dXi) _ C( )_ aC(Xi) dXi _ dn ax xi+n dn Xi n aXi dn-

dXi aC(Xi) = PXi-C(xi)+n-d (p a )

n Xi

By Proposition 2.1, t- < 0, and A. 3 implies that p > o~~:;}. Since profits

are zero in a CEFE, ~~ < o .•

Let us define a second best allocation as the one that maximizes social welfare with the restriction that outputs must be a Cournot equilibrium given that n firms are in the market (see Perry (1984». This implies that the number of firms in an industry can be controlled (i.e. by means of a licensing scheme) but that their oligopolistic behavior can not. Our next Proposition (due to Mankiw and Whinston (1986» shows that the over-entry result carries over a second-best scenario.

Proposition 3.9. Under A.I, A.2 and A.3, the number offirms in the second best optimum is less than the number of firms in a CEFE.

17 Keeping in mind that our results apply to some cases of product differentiation (see Ex­ercise 3.1) we reach the conclusion that in this case equilibrium with free entry generates excessive variety (since each firm produces a different product) and excess capacity. These two implications of monopolistic competition were first formulated by Chamberlin (1933). This topic is investigated further in the next chapter in Sections 4.3-4.

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Proof. We recall from Proposition 3.8 that

dW = px.-C + n dXi (p_ OC(Xi)). dn' dn OXi

In a second best optimum, ~~ = O. Again, by Proposition 2.1, ~ < 0, and A. 3 implies that in any Cournot equilibrium p > a~~:;). Thus, profits are

positive in the second best optimum. In a CEFE, profits are zero. Since by

Proposition 2.2, profits in any NE. are strictly decreasing on n, the result follows .•

5 PROFITABILITY AND FREE ENTRY

In the previous section we have made the simplifying assumption that the number of active firms is not necessarily an integer number. A consequence of this assumption is that profits are exactly zero at the CEFE. This may give the impression that, in general, in a CEFE profits are "small". This section (based on CorchOn and Fradera (1996» studies the question of how large profits can be in a CEFE. This question has important consequences on anti-trust policy since, sometimes, large profits are associated with monopoly or collusion. We will see that this is not necessarily the case. 18

In order to answer this question we have to be careful since by choosing units suitably, profits can be made as small as we wish. We take care of this problem by assuming that we can observe profits, number of firms and the market price, the latter acting as a kind of numeraire. This observation may come from a real market or from an experiment mimicking market forces. The question is: What triples {price, profits, number of active firms} can be generated as CEFE? In order to tie our hands as tight as possible we assume strong restrictions on the form of both demand, assumed to be of unit elastic­ity, and marginal cost, assumed to be constant. However, in despite of such strong restrictions, any triple {price, profits, number of active firms} can be generated as CEFE.

Proposition 3.10 Let us assume that it is known that: a) The demand curve is isoelastic with unit elasticity. b) Marginal costs are constant.

Let (n,p,ii") E NxJR.! be an observation of an umber of firms (n > l),aprice and a profit. Then, there is a demand function and a cost function fulfilling a) and b) above for which the CEFE is (n,p, ii").

18 In other words, a cartel is not always detectable by looking at real data. See Phlips (1995) for a thorough elaboration of this point.

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Proof. Let x be aggregate output, p = ~ be the candidate inverse demand function and Ci = eXi + k (if Xi > 0) and Ci = Of or Xi = 0 be the candidate cost function. Given these two functions a necessary and sufficient condition for (p, n, 7r) to be a CEFE is that

p=~, n=E( fA) and 7r=~-k, n -1 V k n 2

where E(z) == integer part of z. We will now construct an inverse demand and a cost function, i.e. we willfind A, k and e such that the CEFE of this economy is (n,p, 1i"). Indeed let

A (n+£)21i"n2 1i"n2 p(n-1) £2 + 2fn ,k = £2 + 2£n ' e = n

where 10 is an arbitrary number in (0,1). Let us now check that (n,p, 1i") are indeed a CEFEfor this economy.

p(n - 1) n _ p = n (n - 1) = P

(n + f)21i"n2 (n + f)21i" 1i"n2 _ 7r= - k = 7r

(£2 + 2fn)n 2 102 + 2fn 102 + 2fn

n = E( ~) = E(n + f) = n

Thus the proposition is proved. •

Next we assume that the fixed cost is also observable. In this case there is a loose relationship between the number of active firms and the rate of profits. In particular if 1i" are observed profits l9 , k is the observed fixed cost and n is the observed number of active firms, in any CEFE we have that I < ;&+*. Thus if there are 10 active firms, any (extraordinary) profit rate not above 21 % is compatible with free entry (see Exercise 3.14 for the case of a linear demand function). Profit rates reported in the magazine Fortune seldom rise beyond 7%, interest charges not considered.

Proposition 3.11.Let us assume that it is known that: a) The demand curve is isoelastic with unit elasticity. b) Marginal costs are constant.

Let (n, p, 1i", k) E N x lR~ be an observation of a number of firms, a price, a profit and afixed cost, such that i < ;&+*. Then:3 an economy for which

the CEFE equilibrium is (n,p, 1i") when the fixed costs is k. - --2 -

Proof. As before let k = i:.n2En. In this case, k. is given, so we solve this equation to find the value of 10,

19 Recall that profits here are extraordinary profits. In order to obtain these from accounting profits we must subtract from the latter the interest charges.

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_ _ 7r n Pf--2

E=-n+ n2 +T.

Thus similar calculations to the one performed in the proof of Proposition 3.10 show that the required parameters are:

A _(n+E)27rn2 _Nn-l) __ - J-2 ifn2 - 2 2 - ,c- - ,E- n+ n + - .

E + En n k

That E > 0 follows from the equation defining this variable. Also, if 1 2

E < 1 ...... -::;- < -+-k n2 n

Thus, the proposition is proved. •

Both results point out that CEFE is compatible with relatively high (rate of) profits. Thus, CEFE does not support the view that there is a well-defined inverse relationship between profitability and number of firms in an industry. On the one hand this has implications for the antitrust policy since it implies that a high rate of profits does not necessarily imply lack of competition. On the other hand it points out that the CEFE does not support the view, held by some applied researchers, that there is a positive relationship between concen­tration and profits across industries. In particular the fact that profits decrease with entry (Propositions 2.2-3) does not imply that there exist an inverse re­lationship between profits and concentration across industries. Finally, ac­cording to some empirical literature there are persistent differences between the rates of return of different industries (see e.g. Fraumeni and Jorgenson (1980». Standard explanations of this fact are the existence of differential risk among industries or slow entry. Our results point out that such finding is compatible with the industries being in a CEFE and facing no risk.

6 OLIGOPOLISTIC COMPETITION AND CON­STRAINED EFFICIENCY

So far the results obtained on the relationship between oligopolistic com­petition and social welfare are not encouraging. Except in large economies it appears to be no clear link between both concepts. In this section (based on Corch6n (1986» we develop an alternative approach in which Cournot equilibria will be shown to be constrained efficient.

Think of markets as places in which the welfare of agents is increased. More specifically we might regard firms as a device by means of which both

73

producers and consumers can increase their welfare levels. In an oligopolistic economy, firms are able to do that, subject to two restrictions. First there is imperfect coordination among firms and second the welfare of agents can be changed only if the new pair price-allocation is on the demand curve. This suggests that a fair comparison among markets and a planned economy should include additional restrictions on the allocations that a planner can achieve. Here we impose the restriction that the planner is constrained by the demand curve in the same sense than oligopoly is, but it is free from imperfect coor­dination problems. Our main result says that a planner with such a restriction can not do better than the market. This result suggests that imperfect coordi­nation is not harmful from the point of view of welfare.

A possible interpretation of why the planner is constrained by the demand curve is the following. Sometimes it is argued that the planner is a surrogate of the society, and that the allocations that she can achieve must be contained in those which can not be blocked by the grand coalition. Suppose that the status quo (i.e. the outcome corresponding to an oligopolistic equilibrium) is Pareto inefficient and that representatives of firms and consumers meet to­gether to discuss how to improve the situation. There is, however a restriction on any pair price-allocation which can be effectively achieved by this coali­tion. Consumers are many and if a particular consumer decides to break the agreement, she can hardly be discovered. Therefore if the agreed pair (price, allocation) is not on the demand curve consumers have incentives to betray the agreement since they will choose a bundle which maximizes their util­ity over the budget set. Therefore we can think of the demand curve as an constraint due to incentive compatibility problems and the fact that the final allocation has to be sold in a market. Formally:

Definition 3.5 An allocation (xi, ... , x;J is said to be Constrained Efficient (C.EF) if there exist no allocation (x~, ... , x'.,,) such that:

n n n n n n

i=l i=l i=l i=l i=l i=l n n

i=l i=l

In words, an allocation is C.EF. if there is no allocation such that the repre­sentative consumer and all firms are better off using the price system. Notice that if an allocation is C.EF. an intervention from the public sector will have, at least, one of the following effects.

a) Some firm is worse off or, b) the representative consumer is worse off or,

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c) the new allocation is not on the demand curve, so additional instruments (quantity constraints, taxes/subsidies etc.) must be used to "support" this al­location.

Our next result says that Cournot equilibrium is C.EF..

Proposition 3.12. Under A.3, any Cournot equilibrium is C.EF.

Proof. Suppose not. Thus there is a new allocation (x~, ... , x~) in which the representative consumer and all firms are better off. From the definition of a Cournot equilibrium we have,

p(xl)x: - Ci(x;) > p(x*)x; - Ci(x;) 2: p(x: +x:'.Jx: - Ci(x:), Vi = 1, ... , n,

or p(xl) > p(x; + x~J. Since p(x) is decreasing, n

~ I • I • w' 1 ~xi<x +xi-xi , v2= , ... ,n. i=1

Adding the previous inequalities we get Xl < x'. Differentiating the utility function of the consumer (as given in part a) in the definition above) we get

aV(x) _ p(x) _ ap(x). ax ax

Therefore, utility is strictly increasing on x since

aV(x) = p(x)and _ ap(x) > O. ax ax

Thus, welfare of the representative consumer did not improve and we obtain a contradiction .•

It must be noticed that Proposition 3.l2 does not carry through to other solution concepts or to some kinds of product differentiation. Some exten­sions and variations of this result are considered in Exercises 3.39-43 and in Corch6n-Urbano (1996).

Summing up, in this Chapter we have studied the relationship be­tween Cournot equilibrium and optimal allocations. A general conclu­sion is that there is a weak relationship between oligopolistic competition and welfare, except in large economies or in a constrained efficiency sce­nario. Most results obtained in this chapter depend on specific assump­tions (i.e. identical firms) and should therefore be regarded as less robust than those obtained in Chapter 2. The relationship between concentra­tion and profits in an industry is also found to be faint.

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7 ADDITIONAL REFERENCES

For a comprehensive survey of the literature on monopoly regulation see Baron (1989). See also the papers by Sappington and Sibley (1988) and Sib­ley (1989)). A thorough study of quantity and quality regulation in the case of monopoly can be found in Sheshinski (1976). Harris (1981) and Diercker, Diercker and Grodal (1999) present alternative concepts of second-best which differ from the one used here.

8 APPENDIX: INTERNATIONAL TRADE POLICY IN OLIGOPOLISTIC MARKETS

The classical theory of International Trade is based on the assumption that all agents are price-takers. This theory is just a reinterpretation of the General Equilibrium model formalized by Arrow, Debreu, McKenzie and others. In this model there exists a free trade competitive equilibrium which is Pareto efficient.

The classical approach did not fit very well with some facts. According to the classical theory, international trade should occur among countries with different initial endowments and technologies and it should involve different goods being traded. However a large part of the international trade occurs among similar countries (i.e. Germany and USA) and involves the export and import of goods which are roughly equivalent. Also, according to the classical theory domestic quotas are always harmful to the domestic country and tariffs are only advisable in the case in which the country is large (see Grossman (1992) pp. 1-2). Thus, many policies pursued by many governments around the world, appear to be unsound.

In the early eighties some economists began to take seriously the idea that international markets are not perfectly competitive and that an analysis based on oligopolistic or monopolistic competition was highly desirable. Specially influential was the paper by Brander and Krugman (1983) showing the possi­bility of an equilibrium in which each country was exporting and importing an homogeneous good. The countries involved were assumed to be identical. Later work showed that domestic quotas can increase the welfare of the do­mestic country and that tariffs can be profitably erected by small countries. Many other surprising insights followed. A survey of the huge literature on international trade with imperfectly competitive markets is beyond the scope

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of this Appendix. The interested reader may consult the books by Gross­man (1992), Helpman (1984), Helpman and Krugman (1989), Kierzkowski (1984), Krugman (1990) and the references there for further readings.

In this appendix, we present a simple model of international trade under oligopolistic competition and we derive the optimal trade policy. This model (based on Corch6n and Gonzalez-Maestre (1991)) is appropriate when the do­mestic country is small and her decisions do not influence the policy pursued by the foreign country. Other possible models suited for different scenarios are reviewed in Exercises 3.33-36 and 3.47-48. The description of the model is as follows:

There are n domestic firms with increasing cost functions denoted by Ci (

), i = 1, ... , n. Let Xi be the output of firm i. Let Xd = L:~=1 Xi be domestic output and trd = PXd - L:~1 Ci(Xi) be the aggregate domestic profit. We will assume that there is a unique foreign firm with an increasing cost function de­noted by C I (x I) where X I denotes its output. Thus aggregate output, denoted

by x, equals to Xd + XI'

The instruments in the hands of the government are tariffs and quotas. We will assume that the decision on them is irreversible and prior to the decision of firms on outputs, i.e., we assume the government acts as a Stackelberg leader.

Domestic social welfare (W), is the sum of producers' and consumer's surpluses plus the rents captured by the government via tariffs (R(xI))' The latter is assumed to be a tariff on output t, i.e., R( X I) = tx I' Thus,

8V(x) n

W = V(x) - px + trd + R(xI) = V(x) - ~xI - L Ci(Xi) + tXI' i=l

We will first consider the case where the Government has quotas as the only policy instrument (i.e. R( x I) = 0). We will assume that the output of the foreign firm and the quota are identical. Assuming A.l and interiority, first order conditions of profit maximization for domestic firms, given x I, read:

( ) 8p(xl + Xd) 8Ci(Xi) _ 0 p x I + Xd + Xi 8 - 8 - i = 1, ... , n.

x Xi A.2 implies that the above equations are also sufficient conditions of profit

maximization. It also implies the existence and uniqueness of a Cournot equi­librium for given quotas, see Propositions 1.2-3. Thus the output of any do­mestic firm and the aggregate output can be written as functions of the quota x I. Let W (x I) be the domestic welfare as a function of x I·

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Let us denote by x~ the minimum level of imports for which domestic output is zero. Let us denote the output of firm j in the Cournot equilibrium with zero quota by Xj, j = 1, ... , n, f. Our first result shows that the best quota compatible with oligopolistic competition is neither very small, nor a quota for which domestic production is almost zero, since domestic welfare can be raised by lowering (resp. increasing) the quota.

Proposition 3.13. If A.1-2-3 hold, ~~ (0) < 0 and if x~ :s xf. ~!'; (x~) > O.

Proof. Easy computations show that:

oV(X) (dXd + 1) _ op(x) (dXd + 1) xf _ p _ t oCi dXi ox dx f ox dx f i=l OXi dx f

dW

= oV(x) dXd _ op(x) xf (dXd + 1) _ t oCi dXi . ox dXf ox dXf i=l OXi dXf

Now, since for all firms for which the first order condition is folfilled with inequality .:::::1.ddx . = 0, we get that

XI

dW = oV(x) dXd _ op(x) Xf(dxd + 1) _ t(p + x/p(x)) dXi = dXf ox dXf ox dXf i=l ox dXf

= _ op(x) Xf(dxd + 1) _ op(x) t Xi dXi . (4) ox dXf ox i=l dXf

Finally, from Chapter 1 (see comments to equation (2) and Exercise 1.2 a)) we know that ~ddx > -1 and .:::::1.ddx . < 0, i = 1, ... , n and the resultfollows .•

XI XI

The intuition behind Proposition 3.13 is that the effect of an increase in quotas on social welfare can be decomposed into two elements. On the one hand we have a positive effect on welfare which comes from the increase in aggregate output caused by the increase in quotas. This increase in x de­creases both market price and expenditure on imports. On the other hand, production of domestic firms falls, which is socially inefficient since price is higher than marginal cost. If x f is zero the first effect vanishes so that only the second effect remains and if domestic output is zero only the first effect remains.

Let f = a;;,<;l 4~) == f(X). f was introduced in Section 2.4 where we

interpreted it as the degree of concavity of the inverse demand function.

Proposition 3.14. Let us assume A.1-2-3 and that domestic firms are identi­cal, the technology displays constant returns to scale and f( x) is non increas­ing on x. Then, W 0 is quasi-convex on x f.

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Proof. Let q == xflxd. Thenfrom equation (4) above,

dW _ op(x) ((dXd 1) dXi) -- ---Xdq-+ +-dXfOX dXf dXf

Also, from the first order condition of profit maximization of domestic firm i, we obtain the following:

8p(x) + x. a2pC;) ax • ax n(q + 1) + €

n(ap(x) + x.82pC;») + 8p(x) ax • ax ax

(n + l)(q + l)n + nf

Thus,

dW op(x) q2 - 1 - !. -=---Xd n dXfOX (n+l)(q+l)+€

By A.2, € > -n and thus (n + l)(q + 1) + € > O. So disregarding the case in which Xd = 0 (in which we know that ~:; > 0), ~:; = 0 if and only

if q =J(1 + t:ln). The last equation implies that the Xf for which ~:; = 0,

denoted by x~, is unique since q is increasing on x f and € is non increasing on

x (which in turn is increasing with xf). Finally if xf > x>, q increases and

€ decreases so ~:; > 0 and if xf < x~" by identical reasoning ~:; < 0 .•

Notice that A.2 and our assumption on € allows for inverse demand func­tions of the form p = a - bx"', with 0: ~ 0 or p = d-bx with b > 0, (in the latter case A.2 is satisfied if x has an upper bound small enough).

Proposition 3.14 suggests that if domestic firms have low marginal cost relative to the marginal cost of the foreign firm, i.e. if domestic firms are relatively efficient, the output of the foreign firm in a Cournot equilibrium with no quotas will be small and it is likely that it will be located in the decreasing part of W( ). Thus, autarky is the best policy. Conversely, if the domestic firm is relatively inefficient it is likely that the optimal policy is free trade. Thus, the optimal policy can be summarized in two rules: "Do not protect inefficient domestic firms" and "do not allow foreign mediocrities to enter".

An implication of Proposition 3.14 is that the optimal quota can be found by comparing social welfare levels at autarky and Cournot equilibrium with no quotas, i.e. free trade (see Exercise 3.29 for further implications of this Proposition). If ~:; (xf) ~ 0, the quasi-convexity of W( ) implies that do­mestic welfare is maximized in autarky. Therefore we turn to study sufficient conditions for ~:; (x f) ~ O.

Proposition 3.15. Suppose that A.1-2-3 hold and allfirms (including the for­eign) are identical with constant returns to scale. Then:

a) If n = 1, !,,~ (x I) :::; 0 if and only if p() is concave (i. e. € ~ 0)

b)Ifn> 1, ~!';(XI) < o. Proof. From the proof of Proposition 3.14 we have that:

dW ap( x) q2 - 1 - * -= - --Xd7""""-:-:-:-----:-7'---dXI ax (n+1)(q+1)+€

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Notice that _a~~)Xa. > o. Under A.2 and identical firms, the Cournot equi­

librium is symmetrical and thus q = lin. Multiplying the numerator and the

denominator of the above expression by n 2 we obtain that:

. dW _ . n€ + n 2 - 1 Szgn(-d (xI))=-Szgn (1)2 (5)

xI n€+ n+ where the denominator is positive (by A. 2). Thus if n = 1,

Sign (ddW (xI)) = - Sign € xI

If n > 1, A. 2 implies that the numerator of (5) is positive, so a) and b) above are proved •

Thus, under the conditions stated above, autarky is the best trade policy. See Exercise 3.30 for the case of many foreign firms.

Next we consider the case of a linear tariff on foreign output, R (x I) = tx I. In this case the first order condition of profit maximization for domestic firms looks like before, and the corresponding equation for the foreign firm is:

( ) ap(xI+xd) _ aCI(xI) _ -0 p x I + Xd + x I a a t - .

x XI

A corollary of Propositions 1.2 and 1.3 is that under A.1-2 there is a unique Cournot equilibrium relative to t, i.e. a unique vector of outputs satisfying the first order conditions. Let us define xi(t),xI(t) and x(t), respectively, as the Cournot equilibrium output corresponding to domestic firm i, the foreign firm, and aggregate output as a function oft. Let W(t) be the domestic wel­fare as a function of t. Let t* be the optimal tariff, that is, the tariff which maximizes the domestic welfare with the restriction that outputs will be de­termined as a Cournot equilibrium for a given tariff, i.e. W (t*) ~ W (t) for all t. We now concentrate on the characterization of the optimal tariff. We will assume constant marginal costs -denoted by c- and identical firms (including the foreign).

Now, let us define t. and f, respectively, by the following conditions:

p(X(t.)) - c = o. (1)

p(x(f)) - c - f = o. (II)

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In words, facing a tariff f the foreign firm chooses zero output, while 1;. is the tariff for which domestic firms will choose zero output. Notice that f and 1;. may be negative. Then we have the following auxiliary result:

Lemma. 2 Let us assume A.1-2 and that p(x) tends to 0 when x tends to infinity, then (i) 1;. and f exist, and (ii) Xi(t) is increasing, xf(t) is decreasing and x(t) decreasing. Moreover 1;. and f are unique.

Proof. To show existence, let z be such that p( z) = c. Now choose 1;. such that the first order condition of profit maximization for the foreign firm is satisfied when thisfirmproduces z and let z = x(1;.).

Let x* be the aggregate output corresponding to a Cournot equilibrium where theforeignfirm is inactive. Under A.1-2, x* exists. Let f = p(x*) - c. Facing tariff f the foreign firm will choose zero output.

That xf(t) is decreasing on t Xi(t) is increasing on t and x(t) decreasing on t,Jollows from a straightforward application of Proposition 2.4 to the case of identical firms by noticing that in our case the parameter representing the idiosyncratic shock is -t and thus, A.4 is satisfied Then it is obvious that 1;. and f are unique .•

The previous Lemma allows us to prove our next result:

Proposition 3.16. Under A.1-2-3, constant returns to scale and identical firms (including the foreign), d! (f) < 0 and d! (t.) > o. Proof.

dW dx dXf ap(x) dx dXd dXf -= p-- P--xf----c-+t-+xf= dt dt dt ax dt dt dt

dx ap(x) dXf = dt(P-xrfu-c)+ Tt(t+C-P)+Xf·

At the Cournot equilibrium p - c - t + a~:) x f = p - c + a~~) Xi = 0, so

that

dW = dx (t _ 2xf ap(x)) + dXf ap(x) + xf = dt dt ax dt ax

Since at t = f> 0, xf = 0, we have that: dW dx_ dx Tt(f) = dt t = dt (p - c)

which is negative, in view of part (ii) of Lemma 2. If t = 1;. then Xi = 0, and, from the Cournot equilibrium, p c and

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t = a~~)Xf' so that

dW ap( x) dXi ap( x) diW = -~xf(ndt(t - ~Xf) + xf·

The first order condition for a Cournot equilibrium implies that

ap(x) ap(x) p = c,and p + a;;-Xf - c - t. = 0, so ~xf = t.and

dW ap(x) dx dXf ap(x) dXi -(t) = ---xf( - - -) + xf = ---xf n- + xf > 0 dt - ax dt dt ax dt ' since ~ > O,from Lemma 2 .•

Proposition 3.16 part (i) implies that, under our assumptions, a prohibitive tariff (that is, a tariff which implies autarky) is never optimal. Part (ii) implies that it is not optimal to set a tariff yielding a very small (but positive) level of domestic production. In some sense, this result is equivalent to the one obtained in Proposition 3.13 for the case of quotas. However we will see in Proposition 3.17 below that, contrary to what happened with the quotas, the optimal tariff, is interior.

Proposition 3.17. Under A.I and A. 3, constant returns to scale and if all firms, including the foreign, are identical, the following properties hold: (1) Under A. 2, t* exists and t* E [t., ~. (2) Under A. 2, t* is positive. Moreover, if E( ) is non decreasing, W( ) is strictly quasi-concave and thus t* is unique. (3) If a~~) + alx\f)Xi > 0 in the optimum, then t* < 0 ifit is interior.

Proof. (1): Since p( x W ) - c = O,from the first order conditions of profit max­imization of the foreign firm we obtain that t. = a~~) x f. Thus, t. is negative. On the other hand, if t ~ t. then x = x f and

dW(t < t) = (t _ ap(x) x)dx + x. dt - - ax dt

From the profit maximization condition of the foreign firm it follows that

~~ = :~i~1«+2)' which substituted above gives

dW ap(x) 1 -(t < t) = [t+--(E+l)] > O,since A-2 implies E+l > O. dt - - ax a~~) (E + 2)

Thus, W(t) is strictly increasing if t ~ t.. On the other hand, Lemma 2 implies that there exists some t < t yielding higher welfare than any t 2: t. Therefore, no tariff outside the interval [t.,~ can yield larger social welfare.

By an argument identical to the one made at the beginning of Proposi­tion 2.5, all variables are continuously differentiable in a neigborhood of

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the Cournot equilibrium. Since this equilibrium is unique (apply Proposition 1.3), outputs in the Cournot equilibrium are continuous on t. Since social welfare is continuous on outputs, social welfare is continuous on t. Therefore, by the Weierestrass theorem, W(t) has a maximum in the interval [.t., ~. Since no tariff outside this interval can yield larger social welfare, t* exists.

(2): Since t* belongs to [i, ~,jrom Proposition 3.6 itfollows that.t. < t* < f. Therefore, the maximum is interior. Then the first order condition of social maximization equals zero, which yields

dW = dx (t _ 2xf ap(x)) + dXf op(x) + xf = O. dt dt ax dt ox

From the Cournot equilibrium conditions we obtain

d ap(x) + ap(x).!. + a2p<;,)x d 1 xfax ax n ax ' x dt = ap(x) A ; dt = nA;

ax

where A == a~~) (1 + ~) + az~x) (Xi + :;.) < 0 (from A. 2). Substitution of these derivatives in the previous equation gives,

dW = _1 [t _ 2xf op(x) + (op(x) + op(x)!:. + a2p(x) Xi)nxi + AXf] = O. dt nA ax ax ox n ox2

Using the definition of A it follows that

* op(x) a2p(x) op(x) 02p(x) t = -[2n(~ + ~Xi) + (~+ ~Xf)]Xf > 0,jromA.2

Now, it remains to show that if a~~) 2': 0, then W(t) is strictly quasi­concave, which ensures that t* is unique. From the first order condition of a Cournot equilibrium, itfollows that t = a~~) (xf - x;). Thus, we obtain

dW ap(x)x (2(ap(x) + a2p<;,)x) + (2 ap(x) + a2p<;,)x ).!.) _ =:i(ap(x))2 _ = ax fax ax' ax ax f n n ax dt ap(x) A

ax

By using the definitions of f(X) and q == Xd/X, we can rewrite the above as

dW p' dt = A {[2 + 2fq/n + 2/n + f(l- q)/n]xf - x;jn}

p' = Ax {(2 + 2/n)(1 - q) + f(l - q2)/n - q/n2}

Then sign {d~?)} = sign of H (t) where the latter is defined as

H(t) == (2 + 2/n)(1 - q) + f(l - q2)/n - q/n2.Thus,

ddH = -[2fq/n + 2 + 2/n + 1/n2] ddq + [(1 _ q2)/n] df . dx. t t dx dt

But A. 2 is equivalent to -f < n, which implies -fq < n + 1 + 1/2n. Therefore H(t) is strictly decreasing in t and WO is strictly quasiconcave. Thus t* is unique.

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(3): If the maximum is interior, dW(t*)/dt = 0, which, from a similar argu­ment than in part (2) implies a negative value of t* .•

Note that if p = a - x'>' with a > 0, or p = a - dbx with b > 0, then A.2 holds, :~ = ° in the first case, and :~ = b in the second. Thus, in these examples the optimal tariff exists, it is interior and unique.

Summing up, under no fixed costs, it is possible to characterize the shape of the optimal trade policy and the effect of certain trade policies on domestic welfare. However, most insights on the optimal trade policy obtained in the smooth case do not carry through to the case when fixed costs are positive, see Exercise 3.32.

9 EXERCISES

3.1.- Show that the results obtained here can be generalized to allow product differentiation if the utility function of the representative consumer can be

written as U = V (~ ¢ (Xi)) with ¢( ) strictly increasing and differentiable.

(Hint: make the following change of variable Yi == ¢(Xi), see Yarrow (1985)).

3.2.- Let us assume that the utility function of the representative consumer is U = ax - (b/2)x2 - pX. Calculate social welfare in a Cournot equilibrium with a given number of firms which have identical technologies, displaying constant returns to scale. Calculate the welfare loss induced by oligopolistic competition. Calculate the percentage of welfare loss with respect to maximal social welfare in the case of a monopoly.

3.3.- Show that under complete information if A.2 holds, any optimal allo­cation can be achieved as a Coumot equilibrium with subsidies. In the case of a monopoly, explain why it is necessary to subsidize it. Give an example in which the optimal allocation is not achievable with subsidies (see Gues­nerie and Laffont (1978), pp. 443-446). In the case of n > 1 and a linear inverse demand function show that there is a subsidy schedule that achieves the optimum and is balanced (see Gradstein (1995) p. 324).

3.4.- Let S be the subsidy needed to convince a monopoly to choose an output different from the profit maximizing one, i.e. S > II(x*) - (p(x)x - C(x)), where II(x*) are monopoly profits. Suppose that a completely informed plan­ner maximizes a social welfare function which is the sum of the consumer and the producer surpluses minus a term (denoted by E (S)) reflecting the distor­tions created by the taxes needed to finance S. Show that the value of S which maximizes social welfare is positive (see Romano (1988)).

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3.5.- Show by means of an example that under constant returns to scale the entry of an inefficient firm might decrease social welfare (see Schmalensee (1976». Give an intuitive explanation of what is going on in this example (see Bulow, Geanakoplos and Klemperer (1985), Section IV, Example E).

3.6.- Show that a public policy designed to help minor firms may reduce total welfare (see Lahiri and Ono, (1988». Relate this to Proposition 3.3.

3.7. - Show by means of an example that if firms are identical but there are economies of scale the entry of a firm might decrease social welfare. (Hint: consider a linear inverse demand function and a cost function ofthe following form C i = eXi + K if Xi > 0, C i = 0 if Xi = 0).

3.8. - Show by means of an example that under economies of scale, there might be potential welfare gains but no entry might be profitable. (Hint: consider an economy like the one in the previous example but with one firm).

3.9.- Suppose a firm with a single worker. The output is chosen by the planner in such a way that profits are zero. The planner knows the inverse demand function but she does not know the cost function. Suppose she asks the worker about the cost function. The worker maximizes the announced cost function Ai ( ) minus the true cost function Ci ( ) with the restriction that profits are zero. Show that the output corresponds to the one chosen by a monopolist with the true cost function and that the worker obtains a surplus identical to monopoly profits.

3.10.- Assuming that the inverse demand function is linear and that firms can produce either one unit of the good (at cost K) or zero (at cost 0), compute the CEFE and the welfare loss with respect to the optimum for a given K and for K ---> 0 (see A. Mas-Colell (1987), Section 8). How would the conclu­sions about the outcome in the limit change if the good were indivisible and consumers were identical and could only consume one or zero units of it?

3.11.- Assuming that the inverse demand function is linear and that firms are identical with the following cost function Ci = eXi + K if Xi > 0, C i = 0 if Xi = 0 calculate the CEFE and the welfare loss with respect to the optimum (see Dasgupta-Ushio, (1981».

3.12.- Show that under decreasing average costs, if the economy is large enough, at any CEFE, the price tends towards the infimum of the average cost (see Fraysse and Moreaux, (1981), Proposition 2 (i».

3.13.- Show by means of an example that under the conditions of Exercise 3.12, profits may not converge to zero (see Fraysse and Moreaux (1981) p.220).

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3.14. - Derive a bound on the rate of profits in the case in which demand func­tion is linear (see Corch6n and Fradera (1996)).

3.15.- Consider the case in which the technology is described by the following cost function C = sK, for (s-1)K :::; x :::; sK, s = 1, ... ,1, ... . Give exam­ples of firms for which this kind of cost function makes sense. Study optimal and equilibrium allocations assuming that the demand function is linear.

3.16.- Show by means of an example that if the inverse demand correspon­dence is not continuous, Cournot equilibrium does not necessarily tend to the Walrasian equilibrium (see Roberts (1980), pp. 268-269). However, using Proposition 3.5 show that the Cournot equilibrium output is arbitrarily close to the Walrasian equilibrium output.

3.17. - Let us assume that the market is characterized by quadratic utility and cost functions where the latter includes a fixed cost. Compute the output and the number of active firms in the CEFE and the optimal allocation according to Definition 4 (see Von Weizsiiker (1980)).

3.18.- Assuming a linear inverse demand function and identical firms with cost functions of the form C(Xi) = CXi + K if Xi > 0 and C(Xi) = 0 if Xi = 0, calculate the second best and compare it with the CEFE.

3.19.- Prove the following: a) Under A.3 and A.6a) both the CEFE and the optimum are symmetrical,

i.e. that active firms produce the same output. b) Under A.6, Definition 3.4 reduces to the maximization of V(nxi) -

nC(xi), for all (n, Xi) E lR~. c) If I' > a the optimum allocation (denoted by (n~ yO)) is interior, i.e. (n~

yO) »0.

3.20.- In a market like the one described in Exercise 3.11 above, compare the aggregate output for which the profits of incumbents in a CEFE are zero with the aggregate output for which no potential entrant can make positive profits. What are the reasons for the discrepancy between these outputs? In general, which output would be larger? What would happen if firms had to pay a fixed cost first and they were randomly selected to produce in a later stage?

3.21.- Show by means of an example that marginal cost pricing is not a nec­essary condition of optimality in a non-smooth world (see Beato-Mas-Colell (1985)).

3.22.- Analyze the divergence between private and social return to invest un­der quantity competition (see Farrell & Shapiro (1990) pp. 279-80).

3.23.- Interpret the models considered in this Chapter as models of General Equilibrium (hint: consider an additional good, labor, which is both the input

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and the numeraire). Show how the definitions of an optimal allocation in the text relate to the standard definition of Pareto efficiency.

3.24.- Consider a Coumot equilibrium with identical firms, constant marginal costs c and either a tax rate on output v or a tax rate on sales q. Show that if q = v / ( v + c) outputs in both equilibria are the same. What about the money raised by each tax? Which tax would be preferred by producers?

3.25.- Would it be possible to sustain a collusive outcome under free entry? (see Harrington 1991).

3.26.- Find graphically the CEFE. Show that, in general, is not unique (see Corch6n and Fradera (1996)).

3.27.- Show an example in which an increase in the demand decreases the number of active firms and the aggregate output in CEFE, even if A.2-3 hold (see Corch6n and Fradera (1996)). Compare with the Coumot model without entry.

3.28.- Characterize the form of the Best Reply correspondence in the case of free entry when A.2 holds (see Corch6n and Fradera (1996), Theorem 4).

3.29. Show that Proposition 3.13 implies that there is a quota for which the first order condition of social welfare maximization is satisfied but in which social welfare is minimized. Show that under the conditions of Proposition 3.14, first order conditions will never yield a maximum. Argue that quotas can never raise the welfare of domestic consumers.

3.30.- Show that Proposition 3.15 depends crucially on the assumption that there is only one foreign firm (see Corch6n and Gonzalez-Maestre (1991)) or on the fact that domestic firms have market power (see Helpman & Krugman (1989), p. 63).

3.31.- Show by means of an example that even if A.2 hold, the optimal tar­iff is not necessarily decreasing on the degree of competition. Conclude that there is no relationship between the optimal tariff and the imperfections asso­ciated with oligopolistic competition. Find the optimal tariff in the case of no domestic firms and give an example in which it is negative (see Corch6n and Gonzalez-Maestre (1991) and the references therein).

3.32.- Show that under fixed costs most of the conclusions stated in Proposi­tions 3.13-17 do not hold (see Corch6n and Gonzalez-Maestre (1991), Propo­sitions 3 and 4).

3.33.- Let us consider a model in which there is no domestic consumption. There are two firms, one domestic, one foreign. Give examples in which this kind of model is appropriate. Show that:

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a) When there is no foreign firm a monopoly is socially optimal. If there are several domestic firms they should be taxed.

b) If the domestic and the foreign firms are quantity setters a subsidy is socially optimal under N.E .. Discuss the case of Stackelberg leadership (see Helpman and Krugman (1989) pp. 88-91).

c) If the domestic and the foreign firms are price setters a tax is socially optimal.

d) Analyze the case in which both countries use taxes/subsidies (see Help­man and Krugman (1989) pp. 108-112). Compare with the Prisoners Dilemma.

3.34.- Consider the case in which there is domestic consumption but not do­mestic oligopolistic firms.

a) Discuss the Minimum Import Requirement (see Helpman and Krugman (1989) p. 56).

b) Show that under domestic perfectly competitive firms, it is possible that a positive tariff has no effect on the output of the monopolist (see Helpman and Krugman (1989) p. 60).

c) Show that an ad valorem tariff is always socially preferable to a tariff on the quantity (see Helpman and Krugman (1989) pp. 66-7).

3.35.- Consider two identical countries with an identical number of firms each.

a) Show that the N.E. in quantities implies reciprocal dumping (see Help­man and Krugman (1989) pp. 149-153).

b) Show that if a country imposes a tariff this might decrease the equilib­rium price in this country (see Helpman and Krugman (1989) pp. 152-3).

c) Show that under increasing returns, import protection might imply ex­port protection (see Krugman in Kierzkowsk (ed.) 1984).

3.36.- Consider two identical countries with a possibly different number of firms. Demand and costs are supposed to be linear. Give examples in which this kind of model is appropriate and compare it with the model in the Ap­pendix. Show that the following possibilities might arise.

a) Social Welfare might be greater in autarky than under free trade (see Martin (1993), pp. 386).

b) Study the optimal tariff when the number of domestic and foreign firms is the same. Show that it decreases with n (see Martin (1993), pp. 408).

c) Id. with the optimal subsidy (see Martin (1993), pp. 409).

3.37.- Calculate both the inflationary sensitivity and elasticity in the case of CEFE (see Definitions 2.1 and 2.2 in Chapter 2 and the reference therein).

3.38.- Let Si = xi/x. The Herfindahl concentration index, denoted by H, is defined as follows: H = E s;. Assuming constant returns to scale show

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that price-marginal cost margins and the elasticity of demand are functionally related with H (see Cowling and Waterson (1976)).

3.39.- Give a graphical proof of Proposition 3.12. Using the same picture, show that a Stackelberg equilibrium is not necessarily C.EF..

3.40.- Show that if the strict inequality in part b) of Definition 3.5 is replaced by a weak inequality, Proposition 3.12 does not necessarily hold (hint: con­sider an equilibrium with zero profits and increasing returns). Show that an analogous result to Proposition 3.12 holds if all firms in equilibrium have positive profits and n > l.

3.41. - Show that if the strict inequality in part b) of Definition 3.5 is replaced by a weak inequality, Proposition 3.12 does not necessarily hold in the case of n = 1 even if the monopoly makes positive profits. Show an analogous result to Proposition 3.12 in the case in which the output that maximizes the profits of the monopoly is unique,

3.42.- Prove that when n = 2, a N.E. with quantity (resp. price) -setting firms and strategic substitution (resp. complements) is C.EF.

3.43.- Show by means of an example that under product heterogeneity with more than 2 firms the N.E. (either in prices or quantities) is not C.EF..

3.45.- Consider a monopolist facing a subsidy schedule ¢(x). A perfectly informed regulator wants to maximize a weighted sum of consumer plus pro­ducer surpluses for given ¢( ).

a) Assuming that the consumer does not pay any taxes show that if ¢( ) is such that the firm, the consumer (assumed to be price-taker) and the regulator all would choose the same output, at this output, price equals marginal cost and, thus, is independent on the weights of the regulator. Show that in this point ¢ is decreasing.

b) Analyze the case in which the consumer faces a tax schedule of -¢(x). c) Give an interpretation of these results.

3.46.- Define domestic welfare as the sum of consumer surplus plus domestic profits plus money raised by the government. Suppose that a regulator awards a franchise monopoly among domestic and foreign firms. It is sometimes argued that domestic firms must be given priority even if they are less efficient than foreign firms since the profits they earn stay at home and thus, they contribute more to the domestic welfare than foreign firms. Suppose that the franchise, once awarded can not be resold. Show that if the franchise is allocated among all firms by a second price Vickrey auction (see Baron (1989) pp. 1489 and if.) domestic welfare is maximized.

3.4 7. This exercise applies results obtained in Exercises 2.17-18 -replication

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of demand and the number of firms- to the theory of International Trade. Suppose that we have k identical countries with n identical firms in each country. There are constant returns to scale. Inverse demand is either linear (as in Exercise 2.17) or isoelastic (as in Exercise 2.18).

a) Compute profits at the C.E. under autarky. b) Compute profits at the C.E. in the case in which there is free trade among

all countries. c) Compare the results obtained in a) and b) above.

3.48. In the framework of the previous exercise suppose that the solution concept is CEFE. Suppose that before free trade the n firms were at a CEFE in each country.

a) Compute the CEFE under free trade. b) Show that if demand is linear the equilibrium number of firms at the

CEFE is reduced if all countries open to free trade. c) Show that if demand is isoelastic the equilibrium number of firms at the

CEFE is identical under autarky and free trade.

CHAPTER 4: MONOPOLISTIC COMPETITION

Abstract: General features of Monopolistically Competitive markets. The model of a representative consumer: Are average costs decreasing in the op­timum? (Proposition 4.1). The effect on social welfare of an increase in the output or the number of.firms in equilibrium (Propositions 4.2-3). A model with a representative consumer and the Large Group assumption: The re­lationship between optimal and equilibrium qualities (Proposition 4.4) and between optimal and equilibrium output (Proposition 4.5). The horizontal differentiation model: The model of the circular city (Salop). Existence of an equilibrium (Proposition 4.6). The convergence of Monopolistic Competi­tion to Perfect Competition in large economies (Proposition 4.7). Appendix: Existence of optimal and equilibrium allocations (Propositions 4.8-9).

1 INTRODUCTION

So far, we have been concerned with a model in which the product was homogeneous (even though, as we remarked earlier, some kind of product heterogeneity could be introduced in that model) and its quality was given. In this chapter we will tum our attention to questions of product heterogeneity, quality and design. We will see how the basic ideas behind the model devel­oped in previous chapters can be accommodated to deal with such problems.

In this chapter we will consider markets in which products are substitutes, and each firm faces a large potential competition from similar brands. Ex­amples of this situation include restaurants, wine brands, candy bars, small shops, actors, etc. We will model this situation in a very stylized way assum­ing free entry of firms and that firms produce goods which are close substi­tutes. In other words, we will assume the existence of a countable infinite number of potential firms offering similar products. Therefore, the mod­els considered in this chapter have a mixed flavor. On the one hand they are monopolistic since each firm is the unique supplier of a product. On the other hand they are competitive since each entrepreneur faces a large potential competition from firms producing similar goods. Thus the situation captured in these models is usually termed (after Chamberlin (1933» Monopolistic Competition. Similar ideas were considered by Sraffa (1926) and Robinson (1931).

There have been several attempts to capture the previous situation by means of different models. These models differ only in the way in which consumers

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are modeled. The following possibilities have been considered in the litera­ture:

1) Models with a representative consumer in which the consumption sector is represented by a unique consumer with symmetric tastes over the set of differentiated commodities (see Spence (1976), Dixit and Stiglitz (1977) and (1979), Pettengill (1979), and Koenker-Perry (1980».

2) Models of horizontal product differentiation in which there is a large number of consumers (usually a continuum) each with a most preferred brand. The name of the model comes from the classical contribution by Hotelling (1929). See also Salop (1979) and d'Aspremont, Gabszewicz and Thisse (1979) and (1983). These models capture, for instance, product differenti­ation due to location.

3) Models of vertical product differentiation in which consumers have identical preferences but different incomes. Thus only rich consumers can afford high quality goods. The difference with 2) above is that if products are horizontally differentiated they have a positive demand when offered at the same price (in this sense the model with a representative consumer captures horizontal differentiation alone). If products are vertically differentiated and they are offered at the same price one brand captures the whole demand (see Gabszewicz and Thisse (1979) and Shaked and Sutton (1983». These models capture, for instance, product differentiation due to superior quality.

4) Models of characteristics in which consumers have preferences defined on the characteristics of the goods. This model originates in the work of Gorman (1956) and was popularized by Lancaster (1979).

5) Models in which preferences are distributed among consumers as a random variable (Sattinger (1984), Hart (1985 a, b), and Perloff and Salop (1985».

An important contribution by Anderson, de Palma and Thisse (1989) has shown that under certain assumptions, the models described in 1),4) and 5) all generate the same system of demand functions.

In this Chapter we will first concentrate on the model based on the repre­sentative consumer (Sections 2-4). Later on we will explain a simple model of horizontal differentiation due to Salop (1979) (Section 5). Finally we will study the limit points of Monopolistic Competition in the framework of a model of vertical differentiation (Section 6).

2 THE REPRESENTATIVE CONSUMER MODEL

In this section we will adapt the free entry model presented in Section 3.4

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to a world of product heterogeneity and quality design.

The set of potentially produced goods is the set of natural numbers. These goods are sometimes referred to as the differentiated commodity. Let us de­note by n the number of goods which are effectively produced. We remark that n is a variable of the model. Let us denote by Pi the price of good i. Each firm produces a good only. We will assume that firm i produces good i, i.e. no good is produced by two different firms. Thus, the set of potential firms is also the set of natural numbers and n is also the number of active firms.

In order to simplify the presentation, we will restrict ourselves to consid­ering only symmetrical allocations (see below and Exercise 4.2.). Therefore, there is no loss of generality if the analysis is carried out in terms of a rep­resentative firm producing x units of output at a quality vector k at a price Pi and with a technology represented by a cost function c = c(x, k). In our framework, costs depend on quality because different qualities of the same product are usually associated with different costs. We assume that c( ) is C1

except possibly at zero, i.e. we allow for the possibility of fixed costs. The cost function of any other firm is also c( ).20

There is a representative consumer with preferences which, for finite n, are representable by a C2 quasi-linear utility function, which is strictly increasing in the quantities of all goods. When all firms except i produce y units of output each at a common quality vector q and sell at a common price P this utility function can be written as U = V(k, q, n, x, y) - (n - l)py - PiX where V( ) is strictly concave. As in Chapter 3, the interpretation is that the true utility function is linear in an outside good (leisure, money, etc.). Thus, V ( ) represents tastes concerning the differentiated commodity and the two last terms of U come from the substitution of the consumption of the outside good in the budget constraint. V ( ) is assumed to be symmetrical such that

if X = Y and k = q then (n - 1) ~~ = ~~ and (n - 1) ~~ = ~~, where \7 denotes vector differentiation. An example of an utility function like this is V() = V(r(k).cj>(x) + (n - l)r(q).cj>(y)). This will be called in the sequel a generalized Spence-Dixit-Stiglitz (SDS) utility function. The properties of this function are studied in Exercise 4.1.

It may be useful to remark here that if two firms produce the same quality it does not imply that these goods are perfect substitutes since these products are intrinsically different. One may think of wine as the differentiated commodity

20 Notice that the representative consumer is the only consumer but the representative firm is not the only firm (there are n firms).

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and quality as years in storage. Different firms are located on different kinds of land and their products are different despite being of the same quality (i.e. the same vintage). Thus in our model, the specification of the product has two dimensions: one is fixed for each firm, but varies from firm to firm and the other is a decision variable for each firm. The latter can also be interpreted as advertising.

Let us now introduce two pieces of notation. If the variables x and y (resp. k and q) are bound to vary together so that x = y (resp. k = q) we will denote them by z (resp. a). In other words z (resp. a) is the common output (resp. quality). When no confusion can arise we will use z and a to denote symmetrical allocations, i.e. those in which x = y and k = q.

Social welfare, denoted by W, will be the sum of consumer and producer surpluses as in Chapter 3 (see Definition 2 there). We define a symmetrical optimum as the symmetrical allocation which maximizes social welfare, i.e. the sum of consumer and producers surpluses.

Definition 4.1 (aO, nO, ZO) is said to be a symmetrical optimum if (aO, aD, n°, ZO, ZO) maximizes W = V(a, a, n, z, z) - nc(z, a).

Notice that such an allocation is symmetrical because active firms produce the same quantity of output (inactive firms produce zero output). In some cases it can be shown that our symmetry assumptions imply that the full op­timum -i.e. the allocation which maximizes utility over the feasible set- is symmetric (see Exercise 4.2). In other cases it may be understood as a kind of restricted optimum, useful as long as it simplifies the analysis. In the Ap­pendix it is shown that a symmetrical optimum exists even if fixed costs yield a discontinuity in the cost function at zero output (Proposition 4.9).

Let us now turn to the definition of an equilibrium. The consumer chooses the quantities of goods 1, ... , n in order to maximize utility, taking as given qualities, prices and the set of available products. The latter means that if firm j is not active, the consumer is not allowed to demand this brand (in other words the price of j is infinity). The inverse demand function for the repre­sentative firm is derived from the first order conditions of utility maximiza­tion. Let aV(k~~n,x,y) be denoted by p(k, q, n, x, y). Thus Pi = p(k, q, n, x, y) is the inverse demand function for the representative firm which, because of the strict concavity ofV( ), is strictly decreasing on x. The following defini­tion generalizes the notion of a Cournot Equilibrium with Free Entry given in Definition 3.3.

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Definition 4.2 (a*, n*, z*) is an Exact Symmetrical Monopolistically Compet­itive Equilibrium (or in short an ESMC) if

a) (a*,z*) E arg max p(k,a*,n*,x,z*)x - c(k,x)and

b) p(q, a*, n* + 1, y, z*)y - c(y, q) :::; 0, Vy, q.

In the Appendix it is shown that the ESMC exists even if the cost function is discontinuous when the output is zero (Proposition 4.8). The above defini­tion has the disadvantage that the free entry condition b) is not easy to handle. Thus it is customary in the literature to redefine the equilibrium notion as follows.

Definition 4.3 (ae, ne, ze) is an Approximate Symmetrical Monopolistically Com­petitive Equilibrium (or in short an ASMC) if

a) (ae, ze) E arg max p(k, ae, ne, x, ze)x - c(k, x)and

b) p(ae, ae, ne, ze, ze)ze - c(ae, ze) = o. As we already noticed in Chapter 3, condition b) in the above definition

is different from condition b) in Definition 2 since the former assumes that profits of active firms are exactly zero. In general, for this condition to be fulfilled, the number of firms has to be assumed to be a continuous variable. This may be justified by assuming that optimal and equilibrium values of n are large. In the sequel we will use this assumption in the characterization of both optimum and equilibrium.

3 THE REPRESENTATIVE CONSUMER MODEL: GENERAL RESULTS

A recurrent criticism of the models of Monopolistic Competition is that they do not produce clear cut implications (e.g. see Stigler (1968), p. 320). However Chamberlin was, in fact, quite specific about this point. He pointed out several implications of the monopolistic competitive model. In particular:

1) The equilibrium output is located in the decreasing part of the average cost curve.

2) The optimal output is located in the minimum of the U -shaped average cost curve.

3) From 1) and 2) he concluded that the optimal output was larger than the equilibrium output. This is sometimes referred to as the excess capacity theorem.

4) Because firms produce too little in equilibrium, the equilibrium number of firms exceeds the optimal number of firms. This is sometimes referred to as the excess variety theorem.

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5) If the equilibrium number of firms could be reduced or its output ex­panded, social welfare will increase.

Was Chamberlin right or not? Point 1) can easily be seen to be correct (see Exercise 4.3). Unfortunately, point 2) does not necessarily hold if the product is differentiated (see Exercise 4.4). We will see that, under our assumptions, it is likely that the optimal output is located in the decreasing part of the av­erage cost curve (see Proposition 4.1 below). Then, even if average costs are assumed to be U-shaped, it is impossible to derive the excess capacity theorem. In the next Section we will see that, under additional assumptions on costs and preferences, point 3) mayor may not hold (see Proposition 4.5 below). Moreover, in general the excess variety theorem does not hold (see Exercise 4.6). Also, it is not possible to deduce from the excess capacity theorem the excess variety theorem (see Exercise 4.7), the reverse implica­tion being correct (see Exercise 4.8). Recall from the previous chapter that under some specification of the preferences and with homogeneous product, points 3) and 4) above hold (see Propositions 3.6-7). Lastly an increase in the number of firms or the equilibrium output, always increases welfare (see Propositions 4.2 and 4.3 below). Summing up, the model of monopolistic competition produces implications even though these implications do not al­ways coincide with those predicted by Chamberlin.

The rest ofthis Section will be devoted to show how to locate the optimum output in the average cost curve and to study the consequences of a change in the equilibrium output or the number of firms on social welfare.

Let us introduce some more notation. If y = f (x), f! will denote the elasticity of y with respect to x. Let f~ == (8V/8xt8V/8y)z be the elasticity of utility with respect to common output and f~ be the elasticity of utility with respect to the number of available brands. If f~ > f~ when elasticities are evaluated at some particular allocation we will say that people like variety (at this allocation) in the sense that utility increases faster with the number of brands, holding z as a constant, than with output, holding n as a constant (recall that tastes concerning the differentiated commodity are represented by V( ) and not by U( )). If the inequality is reversed we will say that people do not like variety (at this allocation). Finally,people are indifferent to variety (at some allocation) if the above equation holds with equality (see Exercise 4.9). Since the representative consumer is really a surrogate of consumers with different tastes, it seems natural to assume that people like variety. Indeed, at a ASMC, people must like variety (see the proof of Proposition 4.2 below). Now we have our first result.

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Proposition 4.1. Average costs are increasing, constant or decreasing in the optimum if and only if respectively people do not like, are indifferent or like variety at the optimum.

Proof. a(c(x, k)/x) = (ac _ ::..)~ = V _ 1)~.

ax ax x x x x2

From the first order conditions of an interior symmetrical optimum we have that f; = f~ f~ (see Exercise 4.10) and the result follows from that .•

Proposition 4.1 says that at the optimal allocation average costs must be declining at a rate that offsets the gains from additional brands. When n is bound to be an integer, Proposition 4.1 may fail (see Exercise 4.11).

Next, we investigate the effects of small variations of output or the number of firms at a ASMC on utility.

Proposition 4.2. An increase in the number offirms in a ASMC, holding ae

and ze as constant, will never decrease social welfare.

Proof. First notice that aw av av . an = an - c = an - PiX (by the zero profit conditwn).

Usingfirst order conditions of utility maximization and that

av = av + av = av + (n _ 1) av = n av . az ax ay ax ax ax

aw v we obtain that -a =- V(f~ - f;').

n ne

Now we will show that in equilibrium, the consumer must not dislike variety, i.e. f;; ~ f;. In order to see this, notice that the optimization performed by the consumer over goods implies that ~~ + ~~ = pn. Moreover since

the consumer can always reject an existent variety it must be that ~~ ~ pz. Dividing the last two equations and rearranging we obtain that f;; > f;. Thus aw > 0 .• an -

Proposition 4.3. An increase in the output in a ASMC, holding ae and n e

as constant, increases social welfare.

Proof. Computing aa~ we have that

oW = neav (1- ~ 8c). az ax pax

By the strict concavity of V ( ), p( ) is decreasing on x so the first order condition of profit maximization implies that Z~ < p and we are done .•

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Propositions 4.2 and 4.3 imply that the effect on welfare of a simultaneous increase of ze and a reduction of ne is ambiguous. However under additional assumptions the effect is not ambiguous, see Exercise 4.13.

4 A MODEL OF A LARGE GROUP

In this Section, we will study the relationship between optimal and equi­librium output. In order to do that, we will make two additional assumptions. Firstly we will assume that the utility function is a generalized SDS, i.e.

V( ) = V(r(k)¢(x) + (n - l)r(q)¢(y)) with V(O) = ¢(O) = O.

The function ¢( ) (resp. r(» measures the impact of the quantity con­sumed (resp. quality) on utility. Will call this utility function the generalized Spence-Dixit-Stiglitz (SDS) utility function. Also the cost function will be assumed to take the following form:

c(k, x) = F(k)f(x).

Some consequences of these assumptions on the elasticities used before are worked out in Exercise 4.15.

Secondly we introduce a Chamberlinian "large group" assumption. Let us define 8 == ¢(x)r(k) + (n -l)¢(y)r(q). The parameter 8 can be thought of as an weighted aggregate measure of the quantity consumed of the differentiated commodity. The inverse demand function for the representative firm reads

. _ OV(8) (k)O¢(x) P. - 08 r ax'

We now make the following assumption that will be maintained in the rest of this section.

Assumption L (Large Group Assumption). Allfirms regard 8 as a constant. In particular the representative firm regard 8 as independent of x and k.

Assumption L can be motivated on the grounds that the economy is large and thus a variation in x and k hardly changes ¢(x)r(k) + (n - l)¢(y)r(q), (see Spence (1976) p. 227 equation 52 and footnote 11, Dixit-Stiglitz (1977) p. 299 equations 8-9 and Tirole (1988) p.288. Costrell (1989) offers an alter­native motivation of this assumption).

We now restate the definition of an ASMC under our conditions. The reader is reminded that without the large group assumption, the model in this section would be a special case of the one presented in Section 2.

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Definition 4.4 (m*, x*, k*) is a Large Group Equilibrium (in short a LGE) if

) ( * k*) 8V(m*r(k*)¢(x*)) (k)8¢(x) (k) a x, E arg max J:) r -J:)-x - c x, k,x uS uX

b) 8V (m*r ~~*) ¢ (x*)) r (k*) 8¢8~') x* - c (x*, k*) = 0

We first study the relationship between optimal and equilibrium qualities.

Proposition 4.4. Under the above specification, if optimal and equilibrium qualities are unique, then aD = a*.

Proof. It can be shown that first order condition of welfare maximization and profit maximization with respect to qualities and the zero profit condition im­plies that optimal and equilibrium qualities must satisfo the same equation (see Exercise 4.18), so the Proposition follows from the uniqueness condi­tion. •

In other words, under the above assumptions, monopolistic competition does not introduce a distortion in the optimal qualities. This result was first noticed by Swan (1970) in the case of a monopoly (see also Tirole (1989) p.102 and Exercise 4.17). However we remark that under alternative spec­ifications of the cost function (see e.g. Yarrow (1985) and Ireland (1987», qualities are not optimal. For the rest of the Section, without loss of general­ity, we take r(ae ) = F(ae ) = 1.

We tum our attention to the study of the relationship between ZO and ze which was the focus of the papers by Spence (1976) and Dixit-Stiglitz (1977). We will assume the following.

Assumption S. The slope of E!(X) is greater than the slope of Et(X) for all x.

Assumption S can be motivated by noticing that it implies that second or­der condition of social welfare maximization are fulfilled (see Exercise 4.19). The intuition behind this result is that what A. S roughly says is that costs raise faster than utility.

Proposition 4.5. Under Assumptions Land S we have that: (a) If EtO is decreasing on x then ZO > z*. (b) If EtO is increasing on x then z* > ZOo (c) If EtO is constant on x then, ZO = Z*.

Proof. First order conditions of social welfare maximization imply that Et( ZO) = E!(ZO) (see Exercise 4.18). A. S implies that y > ZO <--+ Et(y) < E!(Y).

From Definition 4 we obtain that 1 + Et (ze) = E!(Ze) (see Exercise 4.18).

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Simple differentiation implies that

8€~(z) I ax > (resp. = or <) 0 ~ 1 + €~ (z) > (resp. = or <) €~(z) I {j2

where €t (z) = ~t. Then the proposition/ollows easily. •

5 A MODEL WITH MANY CONSUMERS AND PRICE­SETTING FIRMS

In this section we consider a model in which firms are price setters and the conswner sector is composed of conswners with different tastes. A par­ticularly useful and tractable example of such a model is due to Salop (1979) building on a similar model proposed by Hotelling (1929). We will call this model the horizontal differentiation or also the Salop model.

There is a continuwn of conswners. Each conswner has one most preferred brand, i.e. the type of soap, wine, etc. she most likes. In this presentation we will ignore the complications arising from the fact that conswners might not wish to buy at all. Therefore we will asswne that each consumer always buys one unit of the good. If a conswner buys one unit of a good which is located at a distance d from her most preferred specification at a price p she enjoys a utility of v - kd - p where v is the utility of buying one unit of her most preferred brand at zero price and k is a parameter indicating how important product differentiation is for the conswner, i.e. if k ~ 0 the product is almost homogeneous. If the good is spatially differentiated, k can be interpreted as the unit transportation cost paid by the conswner. Our asswnption that conswners always buy the good is equivalent to saying that v is large enough. Conswners differ in their most preferred good (see below) but share identical parameters v and k. In this model, k is fixed, but quality design could be easily incorporated to (see Riordan (1986)).

Conswners are supposed to be uniformly distributed in a circle of unit length. In the location interpretation of the model they live in a circular city (perhaps surrounding a lake). However any closed chord yields an equivalent model. The choice of a circle is made in order to avoid troublesome end­points (see Exercise 4.24). At each point of the circle there are L identical consumers. In this set up each firm competes only with its two neighbors (this was named a "chain market" by Chamberlin). In other words, the set of conswners buying from firm i is connected. Notice that in the SDS model a firm competes against all other firms.

The cost function of the representative firm -called firm i in the sequel-

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is e(O) = 0 and e(x) = ex + F if x > o. The subindex denoting a firm is ordered clockwise in the circle. Suppose that firm i decides to enter into the market between firms i -1 and i + 1 which are separated by a distance D (if n = 2, these two firms are the same and D = 1). Suppose that firm i chooses a location which is d units apart from i - 1. This firm charges a price Pi-I.

Firm i is D - d units apart from i + 1. The price set by this firm is PHI. Let us find a consumer who is indifferent about buying from firm i or from firm i - 1. If r is the distance of this consumer from firm i, r must solve

v - kr - Pi = V - k(d - r) - Pi-I

i.e. r = (kd - Pi + Pi-d/2k. It is clear that Lr is a measure of consumers located between i and i - 1. Let l be defined analogously to r but with respect to the consumers located between iand i + 1. An identical analysis to the previous one will show that

I = k(D - d) - Pi + PHI

2k Thus, total demand for i is

L LI L(kD + Pi-I + PHI - 2pi) x = r+ = 2k

We will assume that firms are located symmetrically, i.e. that d = lin, so D = 2d = 2/n. Notice that if prices of different firms are identical, the previous equation reduces to x = Lin. Now we define an equilibrium.

Definition 4.5 (pe, ne) is an Approximate Symmetrical Equilibrium in the Hor­izontal Differentiation model (or in short an ASEHD) if

L( k. + pe _ p) a) pe E arg m:x (p - e) n k

L( k + e e) b) (pe - e) ;;:e : - P - F = O.

The main difference with the equilibrium concepts used before is that here firms are assumed to be price (and not quantity) makers. We now have the following result:

Proposition 4.6. Under the above specification, there is a ASEHD and it is given by

pe = c + /¥- and ne = ~ Proof. Simple calculations show that if P = c + I¥- and n = #- con­

dition b) in Definition 4.5 is satisfied Also, no other price can yield higher profits if the firm has to remain active since pe fulfills the first and the second

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order conditions of profit maximization if F is sunk Since the firm is indif­ferent about being active and charging a price pe or shutting down, condition aJ in Definition 4.5 is also satisfied. •

Notice that in this model all consumers are served so the equilibrium output

ze = ~ = ,!if. See Exercises 4.25-30 for some additional results on the Salop modeY. '"

6 THE LIMIT POINTS OF MONOPOLISTIC COMPETI­TION

Right from its origins, the idea of Monopolistic Competition has been asso­ciated with a large economy. Even though is true that Monopolistic Competi­tion can be defined without any assumption regarding the size of the economy (as it was done in Definition 2 above) it is also true that in order to get ana­lytical results we had to assume that the number of firms can be treated as a continuous variable (see Definition 3 above and Propositions 4.1-3). More­over some models make a large group assumption like we did in Section 4.4. A recurrent criticism of the idea of Monopolistic Competition is that if the economy is large and therefore each firm faces an enormous number of sub­stitute products, a free entry equilibrium must be very close to the perfectly competitive equilibrium. Thus, Monopolistic Competition is just an embel­lishment of perfect competition. For instance in the Salop model when the economy is large, i.e. when ~ --+ 0, equilibrium price tend to marginal cost (see also Exercise 4.6).21

However, in the large group case if <l>(x) = x"', 0 < a < 1, equilibrium price do not tend to marginal costs when the economy is large (see Exer­cise 4.16). However, this is due to the fact that marginal utility (and thus the inverse demand function) is not differentiable at zero. Indeed, it is easily shown that in the model of Section 4.4 equilibrium prices must satisfy that p = (8c/8x)/(1 + €f). By an argument identical to the one used in Propo­sition 2.3, we will see that, in the limit, we obtain perfect competition. Let us assume that the commodity space, i.e. the set of all potential goods, is compact. When the fixed cost is small the number of active firms tends to infinite. Thus, given any product in the commodity space, there is a large number of firms producing goods which are very close substitutes, i.e. we are back, approximately, to the Cournot model and by Proposition 2.3 if n --+ 00,

x* --+ O. Now ifboth ¢' and ¢" are continuous on x (or at least the ratio ¢" / ¢'

21 Notice that K / L ---> 0 includes both the case in which the fixed cost converges to zero and the demand sector converges to infinity.

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is continuous), in a compact neighborhood of zero, say Z, by Weierestrass theorem ::Jr > ¢/'(x)/¢/(x)'r/x E Z. Therefore Ef == ¢"X/¢' must tend to zero when x tends to zero and thus, p -+ 8c/ ax. Summing up, if x* -+ 0 the first order condition of profit maximization becomes approximately that price equals marginal cost. The two basic requirements for this result to hold are that the commodity space must be compact and the inverse demand function be continuous (see Hart 1979). We remark that even if the commodity space is unbounded, convergence to perfect competition may occur (see Exercise 4.6). Thus, given continuity of demand, unboundedness of the commodity space is a necessary (but not sufficient) condition for the convergence to per­fect competition not to occur.

Given the above result there are basically two escape routes. Both assume that the commodity space is unbounded. One consists in building a model in which tastes of consumers are such that in large economies there are infinitely many operating firms each facing a decreasing demand (Hart (1985, a, b). This has the advantage that captures the spirit of monopolistic competition but requires considerable technical effort and makes assumptions on preferences that are strong. It also requires an additional assumption on the technology. Indeed let the cost function of the representative firm be e( x) = ex + K for x > O. The zero profit condition reads (p - e)x = K. Thus, if K -+ 0, either p -+ e or x -+ 0 but in the second case, the first order condition of profit maximization imply that p -+ e too. Therefore, if we want to achieve in the limit an outcome different from perfect competition, marginal costs can not be constant.

The second scape route considers a model in which firms retain monopoly power in large economies. This situation is called (after Shaked and Sutton (1983)) a Natural Oligopoly. This occurs when, potentially, there is room for a large number of firms (i.e. demand is large and/or fixed costs are small) but only a handful of firms are active in the market. The rest of this section will be devoted to explain how this might occur. However, let us remark that, strictly speaking, this kind of model does not capture monopolistic competi­tion. Rather, as the name suggests, is a model of oligopoly.

Let us denote total sales of the industry by t. In our framework, t is a proxy for the size of demand. In order to simplify the exposition we will assume that the cost function of any firm is e( k, x) = F (k) where k E 1R+. This can be interpreted by saying that variable cost is zero and F(k) is the fixed cost. Let q represent the highest quality offered by a competitor of a typical firm. We now present the following assumption:

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Assumption F: a) 3')' > 1 such that Vq if k = ')'q selling a product with quality k earns a fraction 10 of total sales. b) The elasticity of F( ) is bounded, i.e.

3(3 such that Vk, q, (F(k) - F(q))q < (3. (k - q)F(q) -

Assumption F has two parts. The first part asserts that by producing a good of sufficiently higher quality than the quality offered by the competitors, a typical firm may capture a (possibly small) fraction 10 of total industry sales. Notice that if the commodity space is understood as including the quality of goods, this assumption implies an unbounded commodity space. The second part requires that costs do not rise very fast with quality. See Shaked and Sutton (1983) for a model of vertical differentiation in which assumption F holds and there is a N.E. Under this assumption we have the following result (see Sutton (1991)).

Proposition 4.7. Under Assumption F the market share of any active firm in a ESMC is greater or equal than 10/(1 + (')' -1)(3) and n* ::; (1 + (')' -1)(3)/10, i.e. the number of active firms in a ESMC is bounded with a bound which is independent of the size of the economy t and the fixed cost F(k).

Proof. Let 0: be the market share of an active firm. By definition of an equi­librium we have that 0: t - F(a*) 2:: 10 t - F(k) where k = ')'a* and a* is the quality set by all other firms in a ESMC. Using part b) of assumption F we get that (3(')' - I)F(a*) 2:: HE - 0:). Since in equilibrium profits are not negative, 0: t 2:: F(a*) and therefore 0: 2:: 10/(1 + (')' - 1)(3). Since n* = 1/0: we obtain that n * ::; (1 + (')' - 1)(3) / E .•

Thus, no matter how large is the economy, i.e. t --t 00, or F(k) --t 022,

or both, firms retain monopoly power. In other words, limit theorems do not always hold when the commodity space is unbounded. In fact, it can be ar­gued that is not appropriate to consider a bounded commodity space when the economy becomes unbounded: If there is a relationship between the number of firms or the number of consumers and the number of products available in the market, it does not look very logical that the latter remains bounded if the former grows without any limit. Thus, an important conclusion of this section is that convergence to perfect competition in a world with product differenti­ation is a troublesome matter and may not occur. The book by Sutton (1991) offers ample empirical evidence of the occurrence of natural oligopolies.

22 For instance, let F(k) = a.kf3. In this case, we can make the fixed cost to be small by letting a --> O.

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Summing up, in this Lecture we have studied optimality and equi­librium with product differentiation. In the model with a representa­tive consumer our conclusions can be summarized as follows. 1) Opti­mal output is located on the decreasing part of the average cost curve if and only if the representative consumer likes variety, 2) welfare increases with respect to output and the number of firms in equilibrium. For some functional specifications and under a large group assumption 3) equilib­rium and optimal qualities coincide and 4) optimal output can be greater, equal or less than the equilibrium output. Finally we have seen that, un­der some circumstances, market power does not necessarily disappear in large economies.

7 ADDITIONAL REFERENCES

An enthusiastic defense of the basic insights of the Monopolistic Compe­tition can be found in Samuelson (1958). See Dos Santos Ferreira and Thisse (1992) and the references therein for a model with both horizontal and vertical differentiation. Sections 2-4 of this Lecture are taken from CorchOn (1991). The survey of Eaton and Lipsey (1989) presents a good exposition of the characteristics and the vertical differentiation models.

8 APPENDIX

In this Appendix we will show that equilibrium and optimal allocations exist despite the existence of fixed costs. In order to simplify the presentation we will assume that qualities are fixed. Let us introduce some additional notation. Let 7T(X, y, n) be the profit function of the representative firm when there are n - 1 competitors producing y units of output each. Let 7T'(y, n) == lim:z:-+o 7T(X, y, n) whenever defined. Also let ir(x, y, n) == 7T(X, y, n) if x> 0 and ir(O, y, n) == 7T'(y, n). Notice that if7T(x, y, n) is continuous except at x =

0, ir(x, y, n) is continuous everywhere. With this notation in hand definition 4.2 can be rewritten as follows:

(n· , z·) is an Exact Symmetrical Monopolistically Competitive Equilib­rium (or in short an ESMC) if

a) z· E arg max 7T(X, z*, nO) :z:

b) 7T(x,z*,n* + 1):::; OV'x. Now let us consider the following assumption:

Assumption E: a) (Regularity of payoff functions at x = 0) lirnx-+o 7T(X, y, n)

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exists. b) (Existence of a symmetrical NE. with given n) \:I(y, n), the range of x for which profits are non-negative is compact, 1l"(x, y, n) is CIon (x, y) and strictly concave on x. c) (Positive fixed costs) 3n' such that \:Iy, x> 0, \:In> n', 1l"(x, y, n) < O. d) (GrossSubstitution)1l"(x,y,n) > 1l"(x,y',n) ifandonlyify' > y. e) (Strategic Substitution) 81l"(x, y, n)j8x is decreasing on y and n.

Notice that parts b) and e) of the above correspond to assumptions A.l-2 in Lecture 1 and that part d) correspond to A.3. Parts a) and c) are new and they deal with the discontinuity of the fixed cost23 • Notice that if the inverse demand function is linear and marginal costs are constant and there is a fixed cost Assumption E holds. Now we can show the following:

Proposition 4.8./f Assumption E holds there is an ESMC.

Proof. First we will show that for given n, there is a z(n) satisfying a) in Definition 4.2' above when the payoff function for the representative firm is 1l"(x,y,n). Let R(y) be the best reply correspondence of the representative firm (see Definition 1.3) when its payofffunction is 1l"O. Because of Assump­tion E part b), R(y) is a continuous function defined on a compact and convex set. By Brower fixedpointtheorem 3z(n) such that z(n) = R(z(n)).

Let 7r(n) == 7r(z(n), z(n), n), i.e. 7r(n) is the true profit obtained by the representativefirm when there are nfirms in the market producing z(n) each. Suppose that 7r(n) ~ 0 \:In. Then z* = n* = 0 fulfills parts a) and b) in Definition 4.2' so an equilibrium has been found. Thus let us assume that 3m, such that 7r(m) > O. Then by part c) of Assumption E,3m' such that 7r(m') > 0 and 7r(m' + 1) ~ O. We claim that (m', z(m')) is a ESMC when the payofffunction of the representative firm is 7r(x, y, n). First notice that since profits are positive z( m') is also positive so 1l"O and 7rO coincide and z( n*) is also a best reply of the representative firm. Thus part a) in Definition 2' above is fulfilled.

Let us now prove part b). By definition of m' and since z(m' + 1) R(z(m' + 1)) we have that

02:: 7r(z(m' + 1), z(m' + 1), m' + 1) 2:: 7r(x, z(m' + 1), m' + 1) \:Ix

Thus, if part b) is not fulfilled

3x,7r(x,z(m'),m' + 1) > 0 2:: 7r(x,z(m' + l),m' + 1).

23 Part c) can be replaced by the following: 1) for each y > 0, 3n such that *(x, y, n') < ° tlx > 0, tin' > nand 2) *(x, y, n) < ° for any x small enough. I am indebted to Nikolai Kukushkin for pointing out this to me.

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But Assumption E, part d) implies that z(m' + 1) > z(m'). In order to finish the proof we will show that this leads to a contradiction. Indeed

o-rr(z(m'), z(m'), m')/ox = 0 = o-rr(z(m' + 1), z(m' + 1), m' + l)/ox ::;

::; o-rr(z(m' + 1), z(m' + 1), m')/ox < o-rr(z(m') , z(m'), m')/ox = 0

where the inequalities above follow from by Assumption E part e) and by the strict concavity of -rr0 on x .•

Notice that we have established the existence of a symmetrical equilibrium. However asymmetrical equilibria may also exist.

We will now consider the existence of a symmetrical optimum. Again for simplicity reasons we will assume that qualities are fixed. Thus Definition 4.1 can be rewritten as follows:

Definition 4.6 (n°, ZO) is said to be a symmetrical optimum if it maximizes W = V(n, z, z) - nc(z).

Then we present the following assumption:

Assumption 0: a) V 0 is a continuous jUnction. b) cO = K + I(z) if z > O(K > 0) and c(O) = 0 where 10 is continuous and strictly increasing. c) 3w > 0 such that n(K + I(z)) ::; w, '<Iz ;::: 0, '<In> o.

Parts a) and b) are self-evident. Part c) comes from the fact that the society has limited resources (endowments). An implication of c) is that only a finite number of firms can be established. Then, we have the following:

Proposition 4.9. Under Assumption 0, there exists a symmetrical opti­mum.

Proof. Fix n = m, say, and consider the following maximization problem:

MaxV(m,z,z) - mc(z)

By Assumption 0 part c) and since lOis strictly increasing z ::; 1-1 (w) Thus, if the above program has no solution it must be that

lim V(m, z, z) - mc(z) ;::: V(m, 0, 0) - mc(O) z-+O

and this implies that

0= c(O) ;::: lim c(z) = K > 0 z-+O

Thus a contradiction has been found and the above program has a solution. Call this solution z(n) and let W(n) == V(n,z(n),z(n)) - nc(z(n)). It is

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clear that the maximum number of active firms, denoted by n', is bounded since n ::::: wi K. Let n be such that W(n) ;::: W(j) j = 1, ... , n'. n exists because n' isfinite. By construction (n, z(n)) is an optimal allocation. •

9 EXERCISES

4.1.- Show that in the generalized SDS utility function, if x = y and k = q, we have that (n - 1) aV = aV and (n _ 1) \TV = \Tv. ax ay \Tk \Tq

4.2.- Show that under symmetry and convexity assumptions the full optimum -i.e. the allocation which maximize utility over the feasible set- is symmetric (see Dixit-Stiglitz (1977) pp. 300-1. See also Exercise 3.19).

4.3.- Show that conditions a) and b) in Definition 3 above, imply that ifp() is decreasing on X, x e will be located when average cost are decreasing.

4.4.- Lemma 1 in Chapter 3 shows that if the product is homogeneous, the average cost curve is convex and the number of firms can be taken as a con­tinuous variable, the optimal output occurs in the minimum of the average cost curve. Assuming now that there is product differentiation and the util­ity function of the representative consumer is V ( ) = V (L xi), 1 > a > 0, show that the optimal output does not occur in the minimum of the average cost curve.

4.5.- Give a General Equilibrium interpretation of the models reviewed in this Lecture (see Corch6n (1991) pp. 442-444).

4.6.- Let us assume that V( ) = a L~=l Xi - b L~=l x; - e L:n x LiN' Xj where Xi = output of firm i. The cost function of firm i reads Ci = eXi + %x; + K if Xi > ° and C(o) = 0.

a) Find the optimal allocation and show that the optimal output occurs in the decreasing part of the average cost curve. Relate this with Proposition 4.1.

b) Find the ASMC. c) Show that the optimal output always exceeds the equilibrium output. d) Show that if d = 0, for some values of band e, the optimum number of

firms may be larger than the equilibrium number of firms. Find the limit of the equilibrium price when K -t 0 and when b -t 0 and K -t O.

4.7.- Show by means of an example (different from Exercise 4.6 above) that excess capacity does not imply excess variety (hint: consider a market with increasing returns).

4.8.- Show that excess capacity implies excess variety (see CorchOn (1991), Propositions 4 and 6).

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4.9.- Show that if the utility function is a generalized SDS, people like variety if and only if E~ < 1.

4.10.- Show that the first order conditions of welfare maximization in Defi­nition 4.1 imply that E~ = Ek.E~ and that f; = E~.E~ (recall that n can be treated as a continuous variable).

4.11.- Give an example showing that when n is bound to be an integer, Propo­sition 4.1 fails.

4.12.- Show that the result in the Exercise 4.3 and Proposition 4.1 imply that if the average cost curve is U -shaped with a minimum which does not depend on k and that in the optimum people do not like variety, then Zo > ze.

4.13.- Compute the effect on social welfare of a simultaneous variation of ze

and n e (see Corch6n (1991), Proposition 10). Comment on the relationship between this effect and Propositions 4.4-5-6 in the main text.

4.14.- Show that an implication of Propositions 4.2-3 is that if the model of Monopolistic Competition is interpreted as a General Equilibrium model there is voluntary unemployment in the sense that welfare can be increased by increasing either the number of firms or output.

4.15.- Show that if the cost function can be written as e = F(k)f(x), then,

Ek = Ef and E~ = E~.

4.16.- Let us assume that qualities are fixed, the utility function is a gen­eralized SDS with ¢(x)= xc<, 0 < (Y < 1 and that the cost function reads C = ex + K if x> 0, C = 0 if x = o.

a) Compute the optimal and the equilibrium output. b) Find the equilibrium price. c) Compare the number of firms in equilibrium and in the optimum (see

Spence (1977) pp. 226-232).

4.17- Show that ifV( ) = v(x, y, n) +xr(k) + (n -l)yr(q) and e = xf(k) + t(x), optimal and equilibrium qualities coincide (see Riordan (1986)).

4.18.- If the utility function is a generalized SDS show that: a) The first order conditions of welfare maximization E~ = EkE~ and f; =

E~E~ reduce to Ek = Ef, and E~ = E~. b) The first order conditions of profit maximization and the zero profit

condition imply that Ek = Er and that 1 + E~' = E~.

4.19.- Show that if the slope of E~ (x) is greater than the slope of E~ (x) for all x the following implications hold

a) The optimum output is unique and

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b) the second order condition of welfare maximization with respect to out­put holds (hint: consider first order conditions of welfare maximization with respect to output and the number of firms).

4.20.- Let us assume that the basic economy is like the one in Exercise 4.16 above but the utility function is U = V(¢(x) + (n - l)¢(y)), i.e. the consumer does not like the outside good. Find the first order conditions of welfare and profit maximization. Compare with Exercise 4.16. Show that in this model excess capacity imply excess variety.

4.21. - Show that the CES model can be interpreted as a discrete choice model (see Anderson, De Palma, Thisse, (1987) pp. 139-140).

4.22.- Show that Propositions 4.4-5 can be proved with several consumers if each of them consumes a particular bundle of the differentiated commodity and if this bundle is consumed only by this consumer.

4.23.- Show that in the large group case a subsidy on output (or input) is Wel­fare improving irrespectively of the relationship between optimal and equilib­rium output (see Costrell (1990)). Relate this with Proposition 4.3.

4.24.- Consider a model like Salop's but with consumers located in an interval and two firms with zero costs (Hotelling (1929)).

a) Show that if firms are located sufficiently close each other, there is no N.E. in prices (see Friedman, 1983, pp. 56-62).

b) Analyze the case of quadratic transportation costs (see d' Aspremont, Gabszewicz and Thisse (1979) and (1983)).

c) What would happen if consumers and producers could bargain about locations? (see Hamilton, Mac Leod and Thisse (1991)).

4.25.- Find the demand function for a monopolist in the Salop model. Com­pare the elasticity of demand in this case and under oligopoly. Interpret the last result and find the monopoly equilibrium output (see Salop (1979)).

4.26.- Show that the ASEHD is unique and find the effects on the equilibrium values ofp, nand Z of variations in k, K, c, L and v in the Salop model.

4.27.- Compare optimal and ASEHD number of firms in the Salop model.

4.28.- Let us assume that in each point of the circle, there are (1 - u)L em­ployed consumers and uL unemployed consumers were u is given exoge­nously. Show that for any given u there exists an equilibrium according to Definition 5 above. What would happen if the existence of unemployment would introduce a tendency to reduce marginal costs? (see Weitzman (1982)).

4.29.- Analyze the kinked demand and the supercompetitive cases in the Salop model (see Salop, pp. 146-9). Would be sensible to assume quantity-setter firms in the Salop model?

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4.30.- Assume that there are two firms in the market. Show that if the firms are located too close to each other there is no Nash equilibrium in prices.

4.31.- In a model with two firms and locations show that an increase in the number of consumers in a location decreases the price and increases the mar­ket share of the firm located there (see Garella and Martinez-Giralt (1989)).

4.32.- Study the case in which the incumbent firms can blockade the entry of a more efficient firm in both the SDS and the Salop models. Compare with Exercise 2.26.

4.33.- Find the inflationary sensitivity and the inflationary elasticity of the equilibrium price (see Definitions 1-2 in Chapter 2) in the case ofSDS and the Salop models. Compare the result with those obtained in the case of Cournot equilibrium (see Propositions 2.7 and 2.8 and Corch6n (1992)).

4.34.- Show a case where the spatial model is equivalent to the characteristics model (see Peitz (1995), Theorem 1).

4.35.- Consider a market with identical firms. The cost function for firm i is Ci = eXi + K if Xi > 0 and Ci = 0 if Xi = O. The inverse demand function for firm i is Pi = a - bXi - 2:;=1 dXHi.8j - 2:;=1 dXj_i.8j. This kind of market is called a chain market (after Chamberlin).

a) Give an interpretation to the functional form of the inverse demand func­tion. Argue that when firms are located closer and closer 8 increases and tends to one in the limit. Thus, 8 can be taken as a measure of the density of firms in the market.

b) Does any example of a chain market come to the reader's mind? c) Calculate the Approximate Symmetrical Monopolistically Competitive

Equilibrium (ASMCE) where instead of a number of firms you find the den­sity 8. Analyze the ASMCE when K -+ O.

CHAPTER 5. TWO STAGE GAMES

Abstract: Games with Stages: Introduction and Examples. The Notion of Subgame Perfect Nash Equilibrium. Applications: Entry Deterrence and Stackelberg Equilibrium (Proposition 5.1). Contestable Markets: Existence, Uniqueness and Optimality of Sustainable Prices (Proposition 5.2). Optimal Rulesfor Public Firms (Proposition 5.3). Effects of Divis iona liz at ion (Propo­sition 5.4). Revelation Games: Effects of Manipulation (Propositions 5.5-6). Indeterminacy of Equilibria (Proposition 5.7). Choice of Technique (Propo­sition 5.8).

1 INTRODUCTION

So far we have studied models in which players choose their strategies si­multaneously. This assumption should not be taken as being literally true. All it matters is that when choosing their strategies players are not informed of the choices made by other players. This might be an adequate hypothesis in certain cases (i.e. price or quantity games) but in others is not appropri­ate. Thus, think of two firms choosing location and output. Suppose that each firm can discover the location of the competitor before any production is undertaken. Then it seems logical that each firm makes the output they plan to produce contingent on the actual location of both firms. Thus, in this framework a strategy for a firm is a location and afimction mapping possible locations of both firms into the set of feasible outputs for this firm. Therefore the consideration of time changes qualitatively the picture from the one stage case and requires a completely fresh analysis.

That this approach brings new insights can be seen by considering that in certain games a player can obtain a payoff larger than her Nash Equilibrium payoffby being committed to a strategy which yields to her strictly dominated payoffs (see Exercise 5.1). Thus, by making some of her strategies unfeasible a player can obtain a payoff larger than her Nash Equilibrium payoff.24 This

paradox can be explained by saying that the game possesses two stages and that in the first stage one player has the opportunity to alter the game in the second stage. This is precisely the situation in the famous leader-follower model due to Stackelberg which we will review later on.

In this chapter we will study games in which firms choose certain actions

24 This paradox is due to Schelling (1960). As Aumann (1985, 1986) has pointed out, a similar paradox arises when the solution concept is the core. The book by Dixit and Nalebuff (1991) is an excellent source for this kind of paradoxes.

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(e.g. quantity and quality of capital, location, design of the product, etc.) before others (e.g. price, output). The actions chosen in the first stage are irreversible, i.e. the firm becomes credibly committed to take certain actions. In Section 5.2 we will develop a general framework to analyze such situations. Sections 5.3-7 will present several important applications.

2 A GENERAL MODEL OF TWO STAGE GAMES

Let there be n > 1 players. Players have to make choices in two periods. Let us denote by ti the action taken by player i in the first period with ti E T;.. This choice can not be revised in the second period. Let Si be the action taken by i in the second period with Si E Si. Ti and Si are assumed to be subsets of an euclidean space. If player i has no choice in the first (resp. second) period, T;. (resp. Si) is a singleton. Let us introduce some notation. Let

S == (Sl' ... ' sn), t == (tl' ... ' tn), T == X~l Ti and S == Xf=lSi. The payoff function of player i is Ui : T X S ---t JR, i.e. Ui = Ui(t, s).

The information structure is as follows: First, there is complete informa­tion. Second, in each period players decide independently and simultane­ously. Third, in the second stage, any player knows the choices made by all players in the previous stage.

A strategy for player i is a pair (t i , :F;) -not a pair (ti, Si)- where :F; T ---t Si. The function Fi embodies a promise of the following sort: if in the first period t is played, in the second period player i will choose Si = Fi (t). This promise may be interpreted as a threat or as a compromise, depending on the context. The next example will try to convince the reader that not every function Fi should be admissible: Consider two firms that have to choose in the first stage if they enter or not in a market. In the second stage firms are quantity-setters. Suppose that firm 1 adopts the following strategy. It chooses to enter and a function Fi such that if firm 2 enters, firm 1 will choose an output for which the price of the product will be zero (assumed to exist). If firm 2 does not enter Fi selects the monopoly price. It is clear that the best reply of firm 2 is not to enter since otherwise it will trigger zero price in the second period. In this way, firm 1 can dictate to firm 2 what to do by threatening to do something nasty in period 2. However, this threat is scarcely credible since, if carried out, it will hurt the threatening agent. Therefore a rational agent will forecast that it is unlikely that such threats will be carried out. The notion of Subgame Perfect Nash Equilibrium (SPNE) introduced by Selten in 1965 (see Selten, 1975) deals with this problem by imposing a credibility requirement on the set of admissible threats. We will now explain

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how this can be done in our model. In order to do that, let us define a N.E. in the second stage, for given t E T, as follows:

Definition 5.1 A Nash Equilibrium in the Second Stage of the Gamefor a given t, (N ESSG) is an s' such that

Ui(S', t) ~ Ui(S;, Si, t) VSi E Si, Vi = 1, ... , n

Notice that, for given t, the definition above corresponds exactly to the notion of a N.E. used so far in this book. We will now assume two things: First, for any given t, there is a NESSG, i.e. there exists a correspondence V: T -+ S such that if S E V(t), then S is a NESSG for given t. Second, there is a selection of V, denoted by S( ), i.e. S(t) E D(t) Vt E T, such that all players believe unanimously that if t was chosen in the first period, then S(t) will be the NESSG (see Fudenberg and Tirole (1991) pp. 99-100 for a discussion of the second part of this assumption). These assumptions would hold if, for instance, for any given t the game in the second period satisfies the aggregation axiom and AI-2-3 (or AI' -2' -3 ') since in this case, for every t there is a unique NESSG (see Propositions 1.2-3 and 1.8-9). Now we are prepared for the definition of a Subgame Perfect Nash Equilibria.

Definition 5.2 (s*, to) is a Subgame Perfect Nash Equilibrium (SPNE) if s* = S (t*) ,and

Ui(S(t*), to) ~ Ui(S(t":.-i, ti), t":.-i, ti)' Vti E T;, Vi = 1, ... , n.

In words, (s*, to) is a SPNE if s* is a NESSG for given t* (and therefore the threat of playing s* if t* was chosen in the first period is believable) and t; maximizes the payoff of player i = 1, ... , n, taking into account the effect of ti on the NESSG. Notice that once t has been chosen in the first period, the choice of s is automatic, since s is the unique NESSG for given t. In this sense the definition of a SPNE of a two-stage game can be reduced to the definition of a N.E. of a suitably defined one-stage game. Indeed let us define the function Vi(t) == Ui( S(t), t). Then, (s*, to) is a SPNE of a two stage game with payoff functions Ui (s, t), i = 1, ... , n if and only if t* is aN .E. of a game with payoff functions Vi(t), i = 1, ... , n (see Exercise 5.2). However notice that the quasi-concavity of Vi (t) with respect to ti does not follow from the quasi-concavity of Ui( ) with respect to ti. This means that the existence of a SPNE cannot be guaranteed in general. Special cases are analyzed by Hellwig and Leininger (1987) and by Kalai, Fershtman and Judd (1988) (see Proposition 5.7 below).

The first order condition of payoff maximization in a SPNE reflects the fact that a variation in ti has two different effects. On the one hand it affects

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Ui( ) directly because ti has changed (this may be called the short-run effect, or borrowing from the military terminology, the tactical effect). On the other hand, it affects Ui( ) because the vector of strategies which are a N.E. for a given t changes (this may be called the long-run or strategic effect). Assum­

ing ~ = 8 i = lR+, i = 1, ... , n, differentiability and interiority, first order condition of payoff maximization in SPNE reads:

OUi(S (t) , t) + t OUi(S (t), t) oSj (t) = 0

Oti j=l OSj Oti

where the first (resp. second) term in equation represents the tactical (resp. strategical) effect. The reader is asked to compare the above equation with the necessary condition for a N.E. in a one-stage game (see Chapter 1, equation below A.l). Many paradoxical outcomes of SPNE can be explained by notic­ing that the strategic effect may be more important than the tactical effect. In other words, actions that look like poor (resp. good) choices in the short­run may be (resp. may be not) payoff maximizing because of their long-run

consequences. Notice too that from the definition of S( ) it follows that

OUi(S (t) , t) = O. OSi

There are no general properties of two-stage games. This is a possible reading of the result by Moore and Repullo (1988) that almost anything is implementable in SPNE, see also Proposition 5.7 and Section 7. Therefore the rest of this chapter will be devoted to present different examples of two­stage games.

3 STACKELBERG EQUILIBRIUM AND ENTRY PRE­VENTION

In order to grasp the fundamentals of two-stage games and SPNE, this section will be devoted to study the simplest possible case. We will assume that player 1 has to make a choice only in the first period. This player will be called the leader. Players 2, ... ,n have to make a choice only in the second period. These players will be called followers. Therefore, 8 1 and T i , i = 2, ... , n are singleton sets. Also, T1 = 8 i = lR+, i = 2, ... ,n. In the case of n = 2, the mapping V corresponds to the best reply correspondence of player 2. When n > 2, the mapping V maps the strategies chosen by 1 into the fixed point of best reply correspondences of players 2, ... ,n. In this section, S is such that it selects the most favorable equilibrium from the point of view of the leader. This can be motivated by assuming that the leader announces the vector of SPNE strategies and that followers only check that their announced

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strategies are a NESSG. In order to motivate the asymmetry of moves, we will think of player 1 as the incumbent, i.e. a firm that is already established in the market. Players 2, ... , n are the potential entrants. In the second period firms choose prices or quantities. The question we will address in the rest of this section is the possibility that the incumbent firm can successfully deter the entry of all potential entrants (a criticism of these models can be found in von Ungem-Stemberg (1987)).

We will first focus our attention on the case where firms are quantity­setters. Early work by Bain (1956) and Sylos-Labini (1962) (reviewed by Modigliani (1958)) suggest that incumbents can deter entry by committing themselves to sufficiently large outputs. This is the so called limit price the­ory (see Exercise 5.9). A criticism of this theory is that firms have preferences defined over profits and not over the number of competitors. In other words, the strategy of being committed to a large output may be less profitable than to allow entry. Osborne (1973) was the first to derive analytical conditions for entry to be deterred in an Stackelberg equilibrium by assuming smoothness of the relevant functions (see Exercise 5.10). This last assumption was critiqued by Dixit (1979) (see Exercise 5.11). The conclusion of both papers is that when there is a unique potential entrant, entry deterrence might, or might not, occur in a SPNE.

A more successful approach to the foundation of limit pricing consists in assuming that the number of potential entrants is very large. We will return to the model and the notation introduced in Chapter 3. Thus, let us denote by p(x) the inverse demand function, by x the aggregate output, by Xi the output of firm i and by c.;(Xi) the cost function of firm i = 1, ... ,n. Profits for firm i are P(X)Xi - Ci(Xi) == 7l"i(Xi, x). Now we assume the following:

AssumptionD: a) Cl(Xl)/Xl is decreasing on Xl.

b) Firms 2, ... , n have the same costfonction with c.;(0) = O. c.;( ) includes a fixed cost K. c) Total revenue is bounded, i.e. 3B such thatp(x)x ~ B, Vx E R d) n is large enough, in particular n - 1 > B / K.

Then we have:

Proposition 5.1. Under assumption D and if xi > 0, then xi = 0, i = 2, ... n, i.e. ifin a SPNE the leader is active, nofollower can be active.

Proof. Suppose it is not the case, so firms i, j, ... k, say, produce a positive output in a SPNE. Let x· denote aggregate output produced by 1, i, j, ... , k. By part d) of assumption D there is afirm, say m, such that it is inactive because

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not all firms can cover fixed costs. Since all firms are identical, 0 E R( x*) where R( ) is the best reply mapping offirm i = 2, ... , n .. Consider now a situation were firm 1 to produce x* . Now there is a NESSG such that firms i,j, ... k will be inactive since 0 E R(x*). The vector of outputs (x*, 0, ... ,0) yields more profits to firm 1 than those obtained in the allocation considered before because of part a) of assumption D (recall that profits for 1 are not negative since xi > 0). Since S( ) selects the most favorable NESSG from the point of view of agent 1, S(xi) must yield as much profits for 1 as those obtained by deterring the entry of 2, ... , n which, in turn, is larger than the SP NE profits for 1. Contradiction. •

Variants of Proposition 5.1 have been proved by Omori and Yarrow (1982), Gilbert (1986), Schwartz and Thomson (1986), Corch6n and Marcos (1988) and Vives (1988). See Gilbert and Vives (1986» for the case of several in­cumbent firms. The interpretation of this proposition is that if the number of potential competitors is sufficiently large the cost of entry prevention is less than the cost of allowing competition. Again what it matters here is the strate­gical versus the tactical effect. Proposition 5.1 can be regarded as a Domino theorem in the sense that the existence of an inactive firm causes all other fol­lowers to be inactive. See Exercise 5.13 for the welfare consequences of this equilibrium. The problem with this kind of models is how to make credible the commitment of firm 1 to the entry deterrence output. See Exercise 5.49 for an alternative interpretation of the model. See Exercise 5.12 for the con­nection between limit output and SPNE output. Exercises 5.4-8 review some aspects of the Stackelberg model when all firms are active.

We now tum our attention to models of entry prevention in which firms use prices as the relevant strategic variable. The most famous model in this field is the contestable market model (see Baumol, Panzar and Willig (1982) and Sharkey (1982». We will say that a market is contestable if it is subject to hit-and-run entry in a sense which will be made precise below.

In this presentation we will concentrate in the simplest possible case: All firms share the same cost function c( ) and the product is homogeneous. Let x = D(p) be the demand function, assumed to be decreasing. The central notion in the theory of contestable markets is that of a sustainable price. This price enable the incumbent firm to survive a hit-and-run attack by a potential competitor.

Definition 5.3 A price p is sustainable in a contestable market if a) pD(p) 2' c(D(p)). b) If p' < p, p'x' :::; c(x'), \Ix' :::; D(p').

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Condition a) says that at the sustainable price the incumbent is making non-negative profits. Condition b) says that no potential entrant can undercut the price set by the incumbent firm and to obtain positive profits. Notice that the potential entrant has to sell cheaper than the incumbent in order to attract any demand and that, unlike the incumbent, it can ration consumers. See Perry (1984) for a model in which the incumbent can use non-linear prices.

Sustainability can be regarded as a reduced form of a more elaborated model in which all the relevant game-theoretical aspects are conveniently summarized. In particular, it is easy to see that the notion of a sustainable price is a necessary condition of a SPNE of a two stage game in which strate­gies are prices and the leader (resp. follower) sets its price in the first (resp. second) stage.25

Letp = infp{pD(p) ~ c(D(p))}. In words, p is the lowest price at which costs are covered. We will assume that p exists and it is positive (these two assumptions can be easily derived from the fundamentals of the model, see Exercise 5.14). Now we have the following:

Proposition 5.2. a) If Assumption D part a) holds, p is sustainable. b) If c( ) and D( ) are Co and p is sustainable, then pD(p) = c(D(p)). c) If c( ) and D( ) are C1 and D(p) + fJ a~?L a~~) a~?) # 0 if P is sustain­able, then p = p.

Proof. a) If the Proposition were not true 3p' < p such that p'x' > c(x') with D(P') ~x'. Since x' # Othiswouldimplythatp' > c(x')jx'. Letx" = D(p'). Thus, since average costs are decreasing, p' > c( x') j x' ~ c( x" ) j x", and this contradicts the definition of p. b) Since p is sustainable it is only left to prove that pD(p) > c(D(p)) is impossible. Indeed, if this inequality holds, the continuity of c( ) and D( ) implies the existence of a p' < p but close enough to p such that p'D(p') > c(D(p')), and this implies that p is not sustainable. c) By Definition 5.3 part a), p < p is impossible. If p > P a potential entrant can charge p and sell D(fJ) (see part b) in Definition 5.3). Then we have two possible cases. If p D(p) > c(D(p)) p is not sustainable. If p D(p) =

c(D(p)) let us consider a slightly different price. The variation in profits for the entrant is:

d _(D(A) A 8D(p) 8c(x)8D(p))d 7r- p +p-------- p.

8p 8x 8p

25 However, sustainability can also be understood as a one-shot Nash equilibrium in prices. See Sharkey (1982) Chapter 8 for a game theoretical model close to the ideas behind the notion of contestability.

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Thus by choosing dp suitably, profits can be made positive .•

Part a) of Proposition 5.2 has to be understood as a result on the exis­tence of sustainable prices. Exercise 5.15 shows non existence of sustainable prices under different assumptions on the technology. Part b) of Proposition 5.2 implies that, if a market is a natural monopoly but contestable, monop­olistic rents are completely dissipated. Part c) of Proposition 5.2 says that the sustainable price in a market is unique. It also implies the so-called weak invisible hand theorem, i.e. that sustainable prices coincide with those prices that maximize social welfare under the restriction that costs must be cov­ered (these prices are called Ramsey-Boiteux prices). See Exercise 5.16 for a simple proof of this theorem. In other words, parts b) and c) of the above proposition imply that contestable markets self-regulate pretty well. We no­tice that the assumption that D{p) + P a~c:) - a~~) a~~) =f 0 is generic in the sense that if it is not fulfilled by a pair of demand and cost functions, a small perturbation of both functions will make this assumption true. This as­sumption is also essential for the conclusion to hold (see Exercise 5.17). See Exercise 5.15 for the consequences of replacing assumption D part a) by a weaker assumption.

The theory of contestable markets is appealing because it yields very defini­tive predictions on the equilibrium price from a very simple set of assump­tions, namely an individual rationality constraint for the incumbent firm -part a) in Definition 5.3- and a no-undercutting condition -part b) in Definition 5.3-. However, this model has received serious criticisms on two different fronts. On the one hand, from the empirical point of view, it is not clear as to what kind of markets the theory applies (see Graham, Kaplan and Sibley (1983)). After all, if a market is contestable, potential competition drives profits to zero and actual entry can not have any effect on profits. On the other hand, from the theoretical point of view the assumption that a potential entrant can undercut the price set by the incumbent firm and expect no retaliation from this firm seems very special. Thus the theory of contestable markets might be regarded as a yardstick model of competition with a purely normative value. See Exercise 5.3 for the paradoxical implications of free entry and price com­petition if there are sunk costs.

Other approaches to entry prevention use different strategic variables. Spe­cial cases include innovation and advertisement (see Salop (1979) and Exer­cise 5.18), the quantity of fixed costs or the wages paid by the firm (see Roger­son (1984) and Exercise 5.19) or the quantity of capital chosen by the incum­bent (see Spence (1977) and Dixit (1980) and Exercises 5.20-1). However, it is not always clear that the commitment of the incumbent firm is credible.

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For instance if the market for capital goods is perfect, the incumbent can vary the quantity of capital in the second period.

Finally we notice that the notion of a Cournot equilibrium with free entry (CEFE, see Chapter 3, Definition 3.3) can be adapted to the framework of a two stage game: In the first stage, firms decide to enter or not. In the second stage, the entrants compete in quantities. Under A.1-2 it can be shown that a two stage Cournot equilibrium is a CEFE (see Exercise 5.41).

4 PRICING OF PUBLIC FIRMS IN OLIGOPOLISTIC MARKETS

In this section, we will consider the optimal pricing for a public firm that competes with a private firm in the market for a certain good or service (e.g. transportation).

Let n = 2. Firm 1 is public. Its payoff is social welfare (W) defined as the sum of consumer and producer surpluses, W = V(XI,X2) - CI(XI)­C2(X2). V( ) will be assumed to be strictly concave in (Xl, X2) and three times continuously differentiable. If the product is homogeneous V(XI, X2) == V(XI + X2). The cost function of the public firm is assumed to be CI. Firm 2 is a profit-maximizing private firm. Let PI (resp. P2) be the market price of good 1 (resp. 2). We will assume that the consumer is a price taker and thus Pi = Z:', i = 1,2. Under our assumptions ~ < 0, i = 1,2, i.e. inverse demand functions are strictly decreasing. As in Definition 3.2, we say that a list of outputs is optimal if it maximizes total surplus. Both firms are assumed to be quantity-setters.

Since there is no a priori reason to assume that one particular firm moves first, we will consider explicitly the two possible scenarios, namely when the public firm moves in the first stage and the private in the second (i.e. when the public firm is leader) and viceversa (i.e. when the public firm is a follower). Let R2 ( ) be the best reply correspondence of firm 2, assumed to be CI . Recall from Chapter 1 (see the comments to A.2 and A.2') that

R 2 ( ) decreasing +-> good 1 is strategic substitute of good 2 and

R 2 ( ) increasing +-> good 1 is strategic complement of good 2.

When these relations hold at the equilibrium value of Xl, we will say that good 1 is locally a strategic substitute (resp. complement) of good 2. Then, we have the following:

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Proposition 5.3. a) If the product is homogeneous and both firms are iden­tical with constant returns to scale, any SPNE in which the public firm is the leader or the follower yields an optimal allocation. b) If the private firm is the leader, the optimal policy for the public firm (sup­posed to be active) is to set PI = ~aG .

Xl

c) If the public firm is the leader, in any SPNE, PI < fx;- if and only if good 1

is, locally, a strategic complement of good 2 and PI > fx;- if and only if good 1 is, locally, a strategic substitute of good 2.

Proof. a) Suppose that the public firm is leader. Then if XO is the optimal aggregate output (and, thus p(XO) = c), the pair Xl = XO,X2 = 0 constitutes the unique SPNE of this game because this allocation maximizes the payoff of firm 1 and X2 = 0 is the unique best reply of firm 2 in the second stage of the game. If the public firm is a follower, the threat of producing Xl = XO - X2 is credible and under the above conditions the resulting allocation is optimal. b) This follows from the first order condition of maximizing social welfare with respect to Xl for given X2.

c) Thefirst order condition of social welfare maximization with respect to Xl

reads

Or,

PI - {)CI = _ (P2 _ {)C2 ) {)R2 .

{)XI {)X2 aXI

From the first order condition of profit maximization of firm 2, (P2 - ~) > o. Thus, sign (PI - fx;-) = -sign ~ .•

Part a) of Proposition 5.3 is due to Harris and Wiens (1980). Part b) fol­lows from an observation in Beato and Mas-Colell (1984). Part c) is a sim­plification of a much more general result by Hagen (1979), whose paper is an excellent introduction to second-best theory.

There are two aspects of the above result which are worth noticing. Firstly, part b) asserts the optimality of price equals marginal cost in an economy with distortions in which the public firm is a follower. We remark that Beato and Mas-Colell (1984) pointed out that the rule price equals marginal cost can yield higher welfare that the pricing with the public firm as the leader, see Exercise 5.22. This is just an example of the fact that to be the leader does not guarantee higher payoffs than those obtained by being a follower. Secondly, the intuition behind the result in part c) is that since P2 > ~, the private firm underproduces with respect to the optimum. Thus, if goods are strategic substitutes the public firm tries to raise the output of the private firm

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by charging a more expensive price than in the first best. When goods are strategic complements the public firm tries to raise the output of the private firm by charging a cheaper price than in the first best. See Exercises 5.23-4 for additional results on this topic.

5 DIVISIONALIZATION

Some corporations create independent units, called in the sequel divisions, which compete in the same or related markets (see Milgrom and Roberts (1992) for empirical evidence of divisionalization). There are, at least, three different explanations of this policy: a) under decreasing returns production efficiency requires plant diversification, b) competition alleviates incentive problems inside the firm and c) divisionalization is a credible commitment to Stackelberg leadership of the group. In this section we study the impact on market equilibrium of this kind of decentralization by means of two exam­ples, focusing our attention on point c) above. In order to keep effects a) and b) away we will assume constant returns to scale and a given cost function. We will also assume that divisions' managers are profit maximizers (optimal incentives schemes for managers are considered in Exercise 5.45).

Suppose we have k corporations (subsequently called groups) in a market. Each group has access to an identical technology represented by a cost func­tion CXi where Xi is the output of a division and C is the (constant) marginal cost. Each group can build as many (identical) divisions as it likes. Each di­vision will be understood as a separate agent in the sense that it will behave independently of the rest of divisions in the group. Each group will attempt to maximize the overall profit received by all the divisions in the group. If a group, say j, builds m divisions and each of them produces an identical out­put Xj, profits for this group are trj = m(p(x )Xj - CXj) where p = p(x) is the inverse demand function and X is total output produced by all divisions of all groups.

In the first stage of the game each group decides (independently) the num­ber of divisions in this group. In the second stage divisions set (indepen­dently) the quantities to be produced. We now study the existence of a SPNE under additional assumptions on the inverse demand function.

Proposition 5.4. Under the following alternative conditions there is no SPNE.

a) p( ) is isoelastic with unit elasticity and k > 2. Or b) p() is linear.

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Proof. Suppose that the inverse demand function reads p = A/x, A > o. If in the second stage there are n divisions, profits per firm in a Cournot Equilibrium (CE) are A/n2. Therefore if a group sets up m divisions its profits are m.A/(m + t)2 where t is the number of divisions created by its competitors (therefore n = m + t). In a SPNE each group will attempt to maximize this expression for given t. Forgetting the integer problem, the first order condition of profit maximization is

A(t-m) =0. (m + t)3

It is clear that the second order condition is satisfied because the left hand side of the above equation is decreasing on m. Also the only solution to this equation is m = t, so we do not have to worry about the integer problem. Moreover since equilibrium is symmetrical t = (k - 1) m. Therefore if k = 2 equilibrium is completely undetermined since any number of divisions is a SPNE and if k > 2 SPNE implies an infinite number of divisions since the best reply of any group consists of building as many divisions as the total number of divisions set up by its competitors.

Suppose now thatthe inverse demandfunction reads p = a-x, a> c > o. If there are n divisions in the second stage, profits per firm in aCE are (a - c)2/(1 + n)2. Therefore if a group sets up m divisions its profits are m(a - c)2/(1 + m + t)2. Thefirst order condition of profit maximization is:

(a - c)2(1 + t - m) = 0 (1 + t + m)3 .

Since this condition is also sufficient we get that in any SPNE m = t + l. Therefore if k > 1 there is no SPNE since the best reply of each group is to set up one more firm than their competitors .•

The above result is robust to the consideration of more general forms of demand and product heterogeneity. This has been shown by GonzaIez­Maestre (1993) by using the Salop model and by CorchOn and GonzaIez­Maestre (2000) by using a model in which the inverse demand function is not restricted to be either linear or isoelastic and by considering product hetero­geneity a la Spence-Dixit-Stiglitz and uncertainty. In all these cases there is a number such that if the number of groups is greater than this number, there is no SPNE in pure strategies. Polasky (1992) has shown that there is no SPNE in mixed strategies. Existence of a (pure strategy) SPNE can be recovered by either imposing an exogenous bound on the number of divisions that can be created by each group or by making divisionalization sufficiently costly, see Baye, Crocker and Ju (1993) and CorchOn and GonzaIez-Maestre (2000). Exercises 5.46-49 consider further aspects of the theory of divisionalization.

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6 REVELATION GAMES

In this section we will concentrate on a special kind of two-stage games in which, by the choice ofti , player i can commit herself to playing in the second stage as if her utility function were Ui(S, t i ). Let ti be the true value of t i ,

which we will call the type of i. A type for a player can be regarded as a best reply function -or equivalently- a description of her relevant characteristics for the second stage of the game.26 Let Ui(S, ti ) be i's true payoff function. The name Revelation Game comes from the fact that in the first period players announce -and are committed to- a vector of types t. In order to make credible to play according to Ui( , ti ) and not to Ui( , t i ) players may delegate their choices in other players. Thus, in a revelation game, in the first period players announce a, possibly false, type and in the second period they behave as if this type were true. Special cases of revelation games include:

1) Incentives for Managers (see Vickers (1985), F ershtrnan and Judd (1987), Sklivas (1987», Macho-Stadler and Verdier (1991) and Salas (1992». In this case players are firm owners who are interested in profits alone. However, by means of the choice of managers' incentive schemes, they can alter the behavior of firms in the second period in which managers compete in prices or quantities according to these schemes. This papers assume that ownership and control are in the hands of different people and attempts to explain why firms do not maximize profits (see Exercise 5.25).

2) Pretension Games (see Alkan and Sertel (1981) and Koray and Sertel (1988». As an example, suppose that in order to built a motorway the govern­ment asks the potential builders to reveal their costs, and gives the contract to the firm with lower costs. In the first period firms announce a, possibly false, cost function. In the second period the awarded firm has to built the motorway with its true technology (another example is offered in Exercise 5.26).

3) Equilibrium in Supply Functions. In this case firms decide the kind of contract they offer to consumers. This might include a few simple options (see Singh and Vives (1984) and Exercise 5.27) or a menu of prices and quantities, i.e. a supply function (see Grossman (1981) and Exercise 5.28).

Under linear demand and costs, all games described in points 1) to 3) above have a SPNE (see, e.g. Exercise 5.25).

In the case of revelation games, the definition of a SPNE can be re-written as follows:

26 The notion of a type used here is a special case of the one used in Bayesian games (see Chapter 6) where a type describes all the information that is relevant in the description of a player. Here this information refers only to the characteristics of a player.

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Definition 5.4 (tj', ... , t~) is a Subgame Perfect Nash Equilibrium in a revela­tion game ijVi = 1, ... , n, Ui(S(t*), ti) ~ Ui(S(t~i' ti), ti), Vti E 7i

Thus in a revelation game the tactical effect is zero. We will concentrate on the study of two questions. The first is to compare the announced and the true type. The second is to compare outputs and prices in SPNE with those that would arise under truthful revelation.

Let f = '£,7=1 Sj. Think of Si as the output of finn i and f as the aggregate output. We postulate the aggregation axiom and thus the true payoff function of finn i can be written as Ui = Ui(Si, f, ti) where ti represents the true type offirm i. A possible payoff function for finn i in the second stage of the game is Ui = Ui(Si, f, ti) with ti E 7i and 7i ~ lR. We will assume that ti E 7i (i.e. the truth-telling strategy is possible). Now we make assumptions that are identical to those made in Chapters 1-2. Notice that notation is different: in this chapter since we substitute Xi and X by Si and f respectively. As we made in these chapters, let

r£1.( ) = OUi( ) OUi( ) 27 .L, - OSi + of .

Also, SPNE will be assumed to be interior. Under A.1-4 we can prove the following.

Lemma 1. a) Under A.1-3, there exist a unique NESSG. b) S (t) E C1 in a neighborhood of t. c) If, in addition, A.4 holds, an increase of ti increases Si and decreases Sj

Vj f:. i.

Proof. Part a) is identical to Propositions 1.2-3. Part b) follows from the fact that under our assumptions the Jacobian matrix of 'Ii ( ) has a non-vanishing determinant. Part c) is identical to Proposition 2.4 parts b) and c) .•

Proposition 5.5. Under A.1-4, t; > ti , Vi = 1, .. , n.

Proof. The first order condition of a SPNE is

T,.( ~ f* A.)OSi(t) ~Ui '" OSj(h) = 0 , s" , t. ot. + of ~ ot.

• j#i' From Lemma 1 and A.3 itfollows that 7i(si, /*, ~) < 0 = 7i(s;, /*, ti ),

which implies the result .•

Proposition 5.5 (which generalizes results by Vickers (1985), Fershtman and Judd (1987) and Sklivas (1987)) asserts that in the SPNE of a revelation

27 Notice that the mapping Ti ( ) refers to the first order condition of payoff maximization with respect to Si and it has no relationship with the set Ti (the set of types) .

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game with strategic substitution, players have incentives to overstate the value of i. This means that agents behave more aggressively than they would be under truthful behavior. However, in the case of strategic complementarity, agents behave less aggressively than under truthful behavior, see Exercise 5.25.

In order to understand the effect of manipulative behavior on resource al­location, let us define a sincere NESSG as a NESSG for given (i1 , ... , in), i.e. a NESSG in which payoff functions are the true ones. Let J' be the value of J in a sincere NESSG. Then we have the following:

Proposition 5.6. Under A.1-4, J' < j*.

Proof. From Proposition 5.5 we obtain that Ii ( si, j*, ii) < 0 = Ii (s:, J', ii), Vi = 1, ... ,n.

If J' 2:: j*, A.3 implies that

Ii(s:, j*, ii) < 0 = 1i(s:, J', ii)::; Ii(s:, j*, ii), Vi = 1, ... ,n. Thus, A.2 implies that s: < si, Vi = 1, .. , n and this contradicts the defini­

tion of f..

In the framework of strategic substitution and quantity-setter finns, Propo­sition 5.6 implies that when finns may lie about their type, social welfare improves as a consequence of the increase in total output (this is the basic intuition behind the "Pretend-but-Perfonn" mechanism proposed by Alkan, Sertel and Koray). However under price setting and strategic complementar­ity the effect is the opposite since prices increase as a consequence of strategic behavior (see Exercise 5.59).

The previous propositions depend crucially on the fact that Ii is one­dimensional. In the next proposition we will show that in a revelation game in which the space of types is rich enough, any payoff is supported by strategies which are a SPNE of this game, i.e. the equilibrium outcome is completely undetermined. This has been called by Kalai, Fershtman and Judd (1985) the "Folk Theorem of Game Theory" by analogy with a similar theorem in the framework of infinitely repeated games. The interpretation of this result is the following: Suppose that we are given a situation that violates our intuitive notions of how competition works. Can we construct a stage game such that the SPNE of this game exhibits the given counterintuitive result? The reader should try her hand to Exercises 5.3, 5.34, 5.40, 5.42 and 5.44 in order to be convinced that it is likely that the answer to this question is "yes". The result below presents a general construction of such outcomes in revelation games.

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Let us denote by 0 the minimum payoff that a player can guarantee to himself. Let Ri(s-i, ti ) be the best reply correspondence of player i in the second stage of the game if she had chosen type t i , i.e.

R;(B-i' ti ) = {Si E SdUi(Si, B-i, ti ) 2': Ui(S~, B-i, ti ) Vs~ E Si}

Recall from Exercise 1.4 that given an arbitrary function R;( ), there is a payoff function such that it generates R;( ) as the best reply function.

We will assume that each player, say i, has a strategy, denoted by Pi (p for penalty), such that, if used, it yields to anyone (including i) no more than the reservation utility O. This can be interpreted as an arbitrarily large output or as a price equal to zero. We will also assume that the type space is "rich" in the sense that it includes any conceivable payoff function. Formally:

Assumption R Let Vi : S --t 1R+ be an arbitrary function. Then, 3 ti E 1';, such that Vi(S, t i ) = Vi(s), 'Is E S.

Under these assumptions we have the following:

Proposition 5.7. Let us consider a revelation game with true payoff functions (Ul (, ti ), ... , un( , ti )). Let S = (SI,"" sn) be such that Ui(S, £) 2': 0, Vi = 1, ... , n. Then, under assumptions P and R above we have that: a) 3 f such S =S([) and b) Ui( S([), ti ) 2': Ui( S(Li' tD, ti ), 'It; E 1';" Vi = 1, ... , n, i.e. f is a SPNE of the revelation game.

Proof. Let us define the function R( ) = (Rl ( ), ... , Rn( )) , asfollows:

R;(S-i) = Si

Ri(p-i) = Si, where P-i is a vector with typical component Pj,j = 1, ... , n,j =1= i And

Ri(B-i) = piotherwise.

It is clear that S is the unique fixed point of R( ). Also from Exercise 1.4 we know that there is a payofffunction such that R;( ) is the best reply function corresponding to these payoffs, namely

k 2

V; = "'" - Sij + s .. o .. (s .) , ~ 2 'J ~ tiJ -"

j=1

where k is the dimensionality of Si. Assumption R implies that there exist a type such that this utility function belongs to the space of admissible utility functions This proves part a) above.

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In order to prove part b) let us assume that contrary to the Proposition, 3t: such that Ui( S(t), ti) < Ui( S(t-i, t:), i.;). Let s' = S(Li, t'J. It is clear that s: =I Si, and that s: =I Pi. But then the unique NESSG relative to (t-i, tD is one in which s~ = Pj, 'ifj =I i. By assumption P and Ui(S, i) ~ 0 itfollows that there are no profitable deviations for player i. Contradiction .•

An interpretation of the construction used in the proof of Proposition 5.7 is that each agent, say i, by means of the appropriate incentive scheme, construct a doomsday machine that is willing to play Pi in certain cases.28 In this sense, the situation is similar to the film Doctor Strangelove: Once the bombers have taken off there is no way to stop them. Of course the result obtained in Proposition 5.7 can be criticized because the reaction functions used in the proof are not very convincing since they are not continuous: Like in the film, a small mistake can unfold a catastrophe. However discontinuity has been used in order to simplify the proofs and it is not essential for the result to hold, see Exercise 5.30.

Proposition 5.7 says that SPNE ofa large class of revelation games is com­pletely undetermined. However, in some cases the introduction of uncertainty on payoff functions may restore uniqueness of SPNE. The idea is that reac­tion functions must be an optimal reply not only in equilibrium, but outside equilibrium just in case some exogenous shock takes us there (see Klemperer and Meyer (1986) and (1989». In terms of the construction used above, if some unforeseeable event takes us to B-i (=I L i , =I P-i), Pi is not an optimal choice since it is a dominated strategy.

7 CHOICE OF TECHNIQUE

In the previous section we saw that in the case of revelation games it is possible to support any outcome as SPNE. However, the construction used in the proof of Proposition 5.7 is not totally satisfactory since it is based on the fact that to achieve the "right" best reply function in the second stage is costless. Also the way by which this is achieved has some resemblance with warfare procedures but it is unnatural in market economies. In this section we will present an example of a counterintuitive result which is the outcome of the unique SPNE of a "natural" game in which different best reply functions are associated with different cost functions.

Suppose that n firms are competing in a homogeneous market with a given technology, say To. Suppose that a new technique, say Tl. is discovered. Is

28 Of course, the (unwanted) holocaust should not happen in equilibrium and that is why if all but one agent, say i, play P-i, i plays S;.

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it possible that profit maximizing firms switch from To to Tl and that as a consequence of this change aggregate productivity in the industry falls? In this section, we will see that this might be indeed the case.

We will assume that firms are quantity-setters, with the usual notation. The inverse demand function will be assumed to be linear p = a-x. Technology is ofthe following form. There is a fixed capital cost, F, to produce the output and a variable average cost denoted by c. The initial technology is the pair ( F o, co) = To. The newly available technology is the pair ( F l , Cl) = Tl , where Fl > Fo and Cl < co. We are interested in what happens to total factor productivity (when all firms adopt the same technology) defined as

EiENXi k = 0 1 EiEN(Fk + Ck Xi)' ,.

The extensive form game is as follows. At period one the new technology, Tl becomes available. The N firms each simultaneously choose to invest and adopt the new technology or remain with the old technology. The investment decisions are observable by all firms. At period two, each firm chooses its level of production, fully informed about its competitors' investment deci­sions. We will say that the "productivity paradox" holds, if in any SPNE all firms adopt the new technology, T l , and total factor productivity is lower than it would be at the equilibrium with only the old technology available. Let 8

== Fl - Fo.

Proposition S.S. When the demand curve is linear and C:-=-c;l ~ nn~l,for some values of 8, the productivity paradox occurs.

Proof. The Nash equilibrium output of each firm when all firms use technol­ogy To is xi = ~~cf and equilibrium profits are [~-:f] - Fo· If only one firm, say 1, chooses T l, then elementary calculations show that in this subgame the

equilibrium profits offirm 1 are [a+(n~~~o-nCl] 2 - Fl. Then afirm chooses

TI if and only if,

[a + (n - l)co - nCI] 2 _ [a - Co] > 8 (1) n+l n+l

Suppose now that all firms choose T I. Then, the SP NE output of each firm is xl = ~~ci and SPNE profits are [~~Ci] 2 - Fl. If only one firm, say 1, chooses

To the profits of this firm are [a+(n~~~l-nco] 2 - Fo. Thus there is a symmetric

SPNE where each firm invest in Tl if,

[a- CI]2 _ [a+(n-l)CI- nCo]2 >8 n+l n+l -

(2)

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lf the SPNE is symmetric, as each firm has the same output and the same technology, total factor productivity is the inverse of a firm's average cost. Comparing average costs for a firm before and after the innovation, a neces­sary and sufficient condition for productivity to fall is,

[a - Cl Fo] 8 > (co - Cl) --+--n+1 a-co

(3)

lf equations (1), (2) and (3) are compatible, the technology To is not adopted in equilibrium, all firms will adopt Tl in equilibrium, and total productivity in the industry will fall.

Wefirst show that (2) implies (1). That is, if there is an equilibrium where all firms adopt the new technology, then, it is profitable for at least one firm to adopt it. Thus the original technology cannot be used in a symmetric sub­game perfect equilibrium (in Exercise 5.51 you are asked to show that this is the onlySPNE of the game). Letw = a-co,y = a-Cl and z = Co-Cl. Then equation (1) can be written as (w + nz)2 - w2 ~ 8(n + 1)2, and equation (2) can be written as y2 - (y - nz)2 ~ 8(n + 1)2.

Suppose to the contrary that (2) holds and (1) does not, then we have that,

y2 _ (y _ nz? ~ 8(n + 1)2> (w + nz)2 _ w2

or

_n2w2 + 2zny > n2z2 + 2wnz

Dividing by nz and substituting in the original terms yields,

Co - Cl ~ n(co - cd a contradiction with the fact that co> Cl and n ~ 2. Thus when (2) holds (1) holds too.

lf (2) and (3) hold, the interval defined by

[(Co ~c,) (:~c; + a~oJ, (:~c;)' ~ (a+ (n :;~' ~=.)'] (4)

is non-empty for values of Fo sufficiently close to zero. Furthermore, if 8 lies in this interval, in equilibrium all firms adopt technology Tl and industry productivity falls.

Again substituting z and y, equation (3) becomes 8> z(yjn + 1)for Fo sufficiently close to zero. Thus in order for (2) and (3) to hold it must be that

y2 _ (y _ nz)2 ~ (n + 1)zy _ _ n2z2 + 2nzy ~ (n + 1)zy

which yields the condition,

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a - Cl n 2 -->-- (5) Co - Cl - n - 1

When equation (5) holds, then the interval (4) is non empty and so it is pos-sible to find a level of 8 such that in the new equilibrium, all firms adopt the new technology, and productivity falls .•

Notice that the condition c:-=-"d1 2': n~ 1 can not hold under monopoly (if n =

1 the right hand side of this condition is infinite) or under perfect competition (if n = 00 the right hand side of this condition is infinite too). In fact the expression nn~l has a minimum at n = 2 (duopoly). This is the value of n for which the condition is most likely to be satisfied (see Exercise 5.52).

Summing up, in this chapter we have studied models in which agents compete in two periods. This requires a refinement of the usual notion of equilibrium called Subgame Perfect Nash Equilibrium (SPNE). In gen­eral, nothing can be said about the properties of SPNE, but this approach can be justified by the number of insights obtained on several specific problems. However the limits of this approach are also clear: the ex­istence and the uniqueness of SPNE can not be guaranteed. Moreover some SPNE exhibit odd features that are hard to reconcile with our intu­ition about how markets work.

8 ADDITIONAL REFERENCES

Applications of the theory of two-stage games are discussed in Shapiro (1986) and Tirole (1988). They include learning by doing, R&D, licenses, advertisement, product selection and design, international trade, contract the­ory, switching costs and the internal organization of the finn. Section 5.5 is based on Corchan and Silva (1990). The book by Dixit and Nalebuff (1991) is a source of nice examples ofSPNE at work. The paper by Okuno-Fujiwara and Suzumura (1993) studies the welfare consequences of entry in a three­stage game.

9 EXERCISES

5.1.- Give an example of a bipersonal game in which, if player 1 plays the best reply to the strategy chosen by player 2, and this player plays a strictly dominated strategy, payoff for player 2 is greater than in a N.E.. Interpret this in terms of the strategic and the tactical effects as explained at the end of Section 5.2.

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5.2.- Prove that (8*, t*) are a SPNE ofa two stage game with payoff functions Ui (8, t), i = 1, ... , n if and only if t* is aN.E. of a game with payoff functions Vi(t) = Ui(S( t), t), i = 1, ... , n.

5.3.- Suppose that in the first stage firms make a decision over entry. Ifa firm enters it has to pay an entry cost E. In the second stage they compete in prices. Show that if the product is homogeneous and firms are identical with constant returns to scale, SPNE implies monopoly for any E. Comment on this result and the usual Bertrand model.

5.4.- Let us consider a market with n identical firms and homogeneous prod­ucts. The inverse demand function is decreasing and average costs are non decreasing. Let us assume that all the relevant functions are differentiable and that all equilibrium allocation are interior.

a) Find the first order conditions for a Stackelberg equilibrium with firm 1 as the leader and firms 2, ... ,n as followers. Assume that the behavior of followers is summarized by first order conditions of profit maximization (under which conditions is this procedure entirely justified?).

b) Assuming that the inverse demand function has constant elasticity, there are constant returns to scale and Stackelberg equilibrium is a Cournot equilib­rium as well, show that in any symmetric equilibrium the elasticity of demand is -n + l.

c) Under what conditions is the converse to b) true? d) Show that if demand and cost functions are linear, at a Stackelberg equi­

librium Xl = nxr , r = 2, ... ,n. e) Compare aggregate output in Stackelberg and Cournot equilibrium (as­

sume A.2).

5.5.- The Stackelberg disequilibrium point (for n = 2) is defined as the pair of outputs for which firm 1 (resp. 2) produces the output corresponding to a Stackelberg equilibrium with firm 1 (resp. 2) as a leader.

a) Can you give a game-theoretical interpretation of this concept? b) Find the Stackelberg disequilibrium if demand and cost functions are

linear.

5.6 - Let n = 2. Consider that both firms have the output as the relevant strategic variable. In this example output is interpreted as capacity or quantity of capital. Payoff functions are defined as follows:

Ui(Xl> X2) = xi(l - Xi - Xj), i i= j, i = 1,2.

a) Calculate the Stackelberg equilibrium (i.e. the SPNE) of this game when firm 1 is the leader.

b) Assuming that the firms choose capital simultaneously, calculate the Nash equilibrium (NE). Comment the differences with point a) above.

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c) Is entry deterrence possible in a SPNE? And in a NE? What would happen if there were a positive entry cost?

d) Compare payoffs of both firms. Is it true that payoffs of the leader in a SPNE are greater than payoffs of the follower in a SPNE? Is it true that the payoffs of the leader in a SPNE are greater than payoffs of this firm in a simultaneous move NE?

5.7 - Idem as exercise 5.6 but with the following payoff functions

Ui (PI, P2) = Pi (1 - Pi + P j) , i =I- j, i = 1, 2.

Give an interpretation of these payoff functions and compare results obtained here with those obtained in Exercise 5.6, especially point d).

5.8 - Let us assume that n = 2, all the relevant functions are linear and firms are price-setters. Find, both graphically and analytically, the Stackel­berg equilibrium of this game, compare it with the NE in prices and compare the payoffs of the leader and the follower,

5.9 - Let us define the limit quantity as the minimum output of the incumbent firm such that a potential firm can not enter into the market (the limit price is just the price corresponding to the limit quantity).

a) Calculate the limit quantity when all relevant functions are linear and there are fixed costs.

b) Compare the social welfare associated to the limit quantity with the one associated to a free entry Coumot equilibrium (see Definition 3 in Chapter 3) and with a second best allocation as defined in the paragraph previous to Proposition 3.9.

5.10.- Assuming that n = 2 and that all the relevant functions are smooth, de­rive, graphically and using calculus, necessary conditions of entry deterrence in a Stackelberg equilibrium (see Osborne (1973)).

5.11.- Assuming n = 2, that the inverse demand function is linear and there is a constant marginal cost with a positive fixed cost, derive, both graphically and analytically, conditions under which entry is deterred in a Stackelberg equilibrium. Relate this to the answer to Exercise 5.10 (see Dixit (1979)).

5.12.- Show that under some additional assumptions to those used in the proof of Proposition 5.1, the output in a SPNE is in fact the minimum output that prevents entry (see Corch6n and Marcos (1988)).

5.13.- Show that under economies of scale, entry deterrence is superior, from the point of view of social welfare, to Cournot competition (see Schwartz and Thomson (1986), Vives (1988) and CorchOn and Marcos (1988)). Explain in simple terms why this is so. What about per capita welfare loss in large economies? (see Dasgupta and Ushio (1981)).

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5.14.- State conditions under which p exists and is strictly positive.

5.15.-a) Show that if average costs are increasing, there are no sustainable prices.

b) Give an example in which sustainable prices do not exist even if the cost function is subadditive (see Sharkey (1982) p. 88).

5.16. - Show that if demand and costs are linear and there is a fixed cost, sus­tainable prices maximize social welfare subject to the constraint that produc­tion must break-even. Generalize this proof to the case where social welfare is an strictly concave function of the output (hint: Let x == D(p). If the unrestricted social welfare maximizing output is to right of x, we are done. Show that under mild additional assumptions the unrestricted social welfare maximizing output is never located to left of x).

5.17.- Show by means of an example that if D(p) + P a~r) - a~~) a~r) = 0 then part c) of Proposition 5.2 may not hold.

5.18.- Construct a model in which the expenditure in advertisement can deter entry (see Salop (1979».

5.19.- Show that under some circumstances an incumbent may have incen­tives to increase fixed costs (see Rogerson (1984». What would be the effect of this policy on social welfare?

5.20.- Let us assume that in order to deter entry, it is sufficient both that the incumbent firm has the option of producing the limit quantity, and that the unique condition for entry deterrence to be credible is that the incumbent has enough capacity to carry out this threat. Solve the profit maximizing quantities of output and capital for the incumbent firm with the restriction that entry is deterred (see Spence (1977». Give an scenario in the post-entry game where this approach can be justified as a SPNE.

5.21.- Suppose that the incumbent firm can affect the outcome of the second stage of the game by choosing capacity and thus altering its best reply func­tion of the second period in which firms act as quantity-setters.

a) Show by means of graphical examples that the SPNE of this game might or might not deter entry (see Dixit (1980».

b) Study the possible cases that might arise under entry deterrence or en­try accommodation depending on the sign of the strategic effect (see Tirole (1988) pp. 323-328). Relate your answer to the results obtained in Exercise 5.10.

c) Can you think of further applications of the same idea to oligopolistic competition? (see Tirole (1988) pp. 328-336).

5.22.- Show that with linear demand and costs, total surplus is higher when

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the public firm is a follower than when it is a leader (see Beato and Mas­Colell (1984)). Show by means of an example that under capacity constraints the previous result might be reversed (see Sheshinsky (1986) p. 1273).

5.23.- Suppose that the public firm is leader, the product is homogeneous and that the SPNE entails positive production of both private and public firms. Show that if A.l holds:

a) 8C1(xi)/8xl > 8C2(x;J/8x2 and thus under decreasing returns to scale and identical firms xi > X2'

b) If 8C1(O)/8xl > 8C2(O)/8x2 the SPNE entails positive production of the public firm.

5.24. Describe the optimal pricing policy of a public firm in an oligopolis­tic market in the case in which firms are price setters (see Ware and Winter (1986)).

5.25.- Consider a model with two firms offering differentiated products and in which demand and cost functions are linear. In the first stage firms decide the incentive schemes of managers, and in the second stage firms compete either in prices or quantities.

a) Find the SPNE, and show that profit-maximizing owners will offer con­tracts to managers that will not induce profit maximization. How to explain this paradox?

b) Compare outputs, prices and social welfare there with the one-shot Nash equilibrium (see Sklivas (1987)).

5.26.- Consider a duopoly with a linear demand function p = a - x and zero costs. In the first period firms decide what value of a they will use in their calculations in the second period, when they compete in quantities. Compute the SPNE of this pretension game.

5.27.- Consider a model in which in the first period firms decide to set either quantities or prices and in the second period firms behave as they decided before. Show that if goods are substitutes firms will choose to be quantity­setters in any SPNE (see Singh-Vives (1984)). Is this conclusion robust to the introduction of uncertainty? (see Klemperer and Meyer (1986)).

5.28.- Assume that firms can choose a reaction function in the first period. NESSG is determined by the fixed point of these reaction functions. Show that both Coumot and Bertrand equilibria are SPNE of this game (see Dixon (1986)).

5.29.- Let us assume that the product is differentiated and that firms are profit maximizers. In the first model, firms choose prices in the first stage and out­puts in the second stage. In the second model, outputs are selected in the first

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period and prices are chosen in the second period. Assume that for any given t, the NESSG entails no rationing in both models. Show that the SPNE of the first model is the Bertrand Equilibriwn and the SPNE of the second model is the Cournot Equilibrium (see Kreps and Scheikman (1983), Vives (1986) and Friedman (1988)). Argue that under some rationing rules these results do not hold (see Davidson and Deneckere (1986)).

5.30.- Asswning that n = 2, show by means of a graphical argument that Proposition 5.7 can be proved using continuous best reply mappings.

5.31.- Show that the best reply functions used in the proof of Proposition 5.7 have the following stability property: If strategies are chosen in the way prescribed by the Best Reply Dynamics (see Definition 1.4), convergence is assured in three periods for any initial condition.

5.32.- Consider a two stage game in which in the first stage firms may change hands and in the second stage they compete in quantities. Show that if the inverse demand function is p = 20 - x with zero costs, it is not possible in any SPNE for all firms to be owned by the same person (see Kamien and Zang, (1990) pp. 470-474).

5.33.- Construct a linear duopoly model in which firms can engage in cost re­ducing (or augmenting) activities in the first period (modeled as a continuous variable) and in the second stage they compete in quantities. Show that, under certain assumptions, firms will increase total cost in any SPNE (see Brander and Spencer (1983)).

5.34.- Show by means of an example that profit maximization does not imply survival (see Vickers (1985)).

5.35.- A monopoly has to decide its output in two periods. In the first period it is an unregulated monopoly and in the second, might be regulated with some probability that depends on the quantity of profits obtained in the first period. Regulation means that its output will be decided by the regulatory agency. Show that if monopoly profits and total expected profits are strictly concave on output and the firm is risk-neutral, it will produce the monopoly output in the first period.

5.36.- Suppose that in the first period firms choose capital and in the second they behave as price-takers. Show that there will be welfare losses (with respect to the first best) in a SPNE (see Dixon (1985)).

5.3 7. - Is it always the case that quantity (resp. price) competition yields strate­gic substitution (complementarity)? (see Tirole (1988) pp. 336-327).

5.38.- Show that if the best reply function of an incumbent firm is decreasing, to carry excess capacity is never a good strategy since the threat that if entry

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occurs incumbent's output will be expanded, is empty. What would happen under strategic complementarity?

5.39.- Suppose that in an homogeneous market there are n identical firms with quadratic cost functions. In the first stage of the game they decide if they joint a cartel or not. In the second stage those firms outside the cartel behave as price-takers and those firms inside the cartel choose the price in order to maximize cartel's profits over the market excess demand function. Show that there is a subgame perfect Nash equilibrium where the cartel exists (see d'Aspremont, Jacquemin, Gabszewicz and Weymark (1981)).

5.40.- Show how apparently innocent commitments are in fact designed to enhance monopoly power (see Salop (1986) and the references therein).

5.4l.-Define a two stage entry equilibrium as the SPNE of the following game: Firms decide to enter or not in the first period. If a firm decides to enter it pays a fixed cost. In the second period all firms that entered compete in quantities.

a) Show that under Al-2 any SPNE of the above game is a CEFE (see Chapter 3, definition 3.3).

b) Show that the reverse inclusion does not hold. c) Show that in some cases, CEFE does not exists, but a SPNE of the game

explained above does exist (see L6pez-Cufiat (1999)).

5.42.- Analyze the SPNE of a model where the incumbent firm is unable to commit to an entry-deterring output (this is the so-called Chain Store Paradox, see Selten (1978)).

5.43.- Analyze the Subgame Perfect Equilibrium of a model where firms de­cide in the first stage the amount of R&D that they undertake and in the second stage compete in quantities (see Spence (1984)).

5.44.- Consider the following model. In the first period, a representative of consumers choose a regulator. This regulator may be pro-consumer or pro­firm. In the first case she will enforce price equals marginal cost and in the second case she will enforce monopoly pricing. In the second stage the firm decides to invest or not to invest. The corresponding cost becomes sunk (think of the case of a public utility). In the third period the chosen regulator en­forces her policy. Assuming constant marginal costs of production show that in a SPNE consumers will choose a pro-firm regulator (see Blackmon and Zeckhauser (1990)). Analyze the case in which the order of stages one and two is switched over. What happens with the welfare of consumer if the pro consumer regulator dies?

5.45.- Analyze the subgame perfect Nash equilibrium of the following model. There are two firms with two divisions each. In the first stage owners decide

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the incentive scheme of the managers of the divisions controlled by the firm. In the second stage, managers compete in quantities. Give an interpretation to the result in terms of market versus technological interrelations (see Fauli­Oller and Giralt (1995) esp. p. 87).

5.46.- Work out Proposition 5.4 in the case ofa fixed cost (see Corch6n 1991)

5.47.- Compare Proposition 5.4 with the analysis of mergers in Exercise 2.19.

5.48.- Suppose that divisionalization is costly. Find the optimal number of divisions when the demand is linear (see Baye, Crocker and Ju (1993».

5.49.- Show that the Stackelberg leadership model used in Section 5.3 can be reinterpreted by considering that the leader can create divisions. Comment the credibility of this policy (see Schwartz and Thomson (1986).

5.50.- Compare Propositions 5.4 and 5.6. Why a SPNE exist in one case but not in the other?

5.51.- Show that there are no asymmetric SPNE in the game analyzed in Proposition 5.8. In particular, show that there are no SPNE in which a group of firms chooses a technology and the other firms choose the other technology (see CorchOn and Wilkie (1994), Appendix).

5.52.- In the case of the productivity paradox, discuss why monopoly or per­fect competition are different from oligopoly.

5.53.- Show that ifp = A/x the productivity paradox can occur (see CorchOn and Wilkie (1994».

5.54.- Analyze the SPNE of a game in which in the first stage firms can decide the quantity of R&D that they undertake (see Brander and Spencer (1983».

5.55.- Suppose a market with two firms and two periods. In the first period firms announce an output and in the second they produce and sell. Payoff functions are Ui = (a - 81 - 82)8i - f( ti - 8i)2, i = 1,2. The interpretation is that 8i is the output produced, ti is the output that they promised to produce, (a - 81 - 82)8i represents profits in the second period and f(ti - 8i)2 is a penalty for cheating. Compute the SPNE of this game and show that when f --+ 0 the SPNE tends to the Cournot equilibrium.

5.56.- Show that Walrasian equilibrium arises as the limit ofSPNE of a game in which each firm plays in a stage when the number of stages goes to infinity (see Boyer and Moreaux (1986».

5.57.- Suppose two stages and two firms. In the first stage firms decide R&D and advertisement. In the second stage they compete in quantities or prices. Analyze the SPNE of this game (see M. Antelo 1996).

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5.58.- Suppose that there are two firms in a market with linear demand, con­stant marginal costs and a positive fixed cost.

a) Calculate the entry deterring policy for firm 1. b) Calculate the Coumot equilibria in the case in which equilibrium is

unique and interior for both firms. c) Give conditions under which there is a Coumot equilibrium in which

firm 2 produces zero output. d) Show that if the antitrust authority does not know the cost function of

the incumbent, it can not distinguish between the situation in a) above and the situation in c) above (see Phlips (1995) p. 242).

5.59.- Adapt Propositions 5.5-6 to the case of strategic complementarity. Use A.l '-4'. Notice that in this case, the interpretation of t is the opposite to the one offered in the text in the case of strategic substitution.

5.60.- An owner ofa firm has a payoff function U = F(7r, e) where 7r stands for profit and e for effort. Profits are determined in the second stage by some kind of competition represented by the function 7r = 7r( e, 8) where 8 repre­sents the degree of competition in the product market (number of competitors, elasticity of demand, etc.). In the first period the owner has to decide about e. Write the first order condition of payoff maximization as W (7r( e, 8), e) = O. Show that the sign of * depends on ~!. In particular show that there are reasonable payoff functions for which an increase in competition yields less effort. Comment on the appropriateness of the market as an incentive scheme (see Hart (1980)).

5.61.- Two firms compete in an homogeneous market with linear demand. In the first stage they set prices. In the second stage they can choose between supply a high quality product with marginal cost c or a low quality product with marginal cost c. Consumers can not detect the quality of the product they are buying. In the third stage an independent agency inspect the good. If the good was of low quality it will found out with probability 7r and the firm will be penalized with a fine proportional to profits. Find the SPNE of this game.

5.62.- Suppose two firms compete in a market that lasts for T periods. Firm 2 is a quantity-setter. If firm 1 may be of two different types: If firm 1 is a quantity-setter, profits for this firm are IIc. If firm 1 is predatory, it will start a price war (lasting one period), and firm 2 will be out of the market. The price war costs L to firm 1. Before another firm enters into the market t periods are elapsed. By simplicity, assume that this new firm can not be driven out of the market. During these t periods, firm 1 is a monopolist and it earns IIrn. Show that if t is sufficiently large, a price war may arise as a part of a SPNE.

CHAPTER 6. INCOMPLETE INFORMATION

Abstract: Games with incomplete information. Bayesian Equilibrium and Nash equilibrium (Proposition 6.1). Payoffs of informed and uninformed players (Proposition 6.2). The revelation principle (Proposition 6.3). The revenue equivalence theorem (Proposition 6.4). The impossibility of efficient trading (Proposition 6.5).

1 INTRODUCTION

As we noticed in Chapter 1, the most important ingredient of the notion ofN.E. (and indeed of game theory) is that each player correctly anticipates the strategies chosen by other players. We offered there two interpretations of this assumption.

1) (Introspection) If information is complete (and N.E. unique), any reason underlying a particular strategic choice of a player can be discovered by any

other player. 2) (Learning) Suppose that the static game is played many times. At each

moment players expect other players to play the strategies that they played in the last period. If strategies converge, the divergence between expected and actual strategies must go to zero.

The problem with the first interpretation is that information is seldom com­plete. The problem with the second interpretation is that agents are not fully rational. In this chapter we present an approach suited to games in which in­formation is not complete and agents are fully rational. We will see that N .E. arises as an special case of this approach. In tune with most of this book, we will only consider static games.

2 THE MAIN CONCEPTS

A type for player i, denoted by ti, is anything that may affect the strategic choices made by player i. For instance ti may describe the payoff function of i, or it might describe the information of i about characteristics of players other than i, or it might describe the information of i about the information in the hands of other agents, etc. In other words, ti is the information in the hands of i. The set of possible types of player i is denoted by 1';. In most of this chapter, in order to keep the presentation simple we will assume

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that this set is finite. Also, in order to save notation, we will assume that all information sets have the same cardinality denoted by k., i.e. Ti = {tL t;, ... , tn, i = 1,2, ... , n. Let T = x~lT;. T will be called the type space or the set of states of the world. A typical element of T will be denoted by t = (tl' t 2 , ... , t n ). t will be called a profile of types or a state of the world. Let T-i = x#iTj and let Li be a typical element ofT_i'

Before players know their types, there is an a priori probability distribution of types. This distribution is shared by all players. This assumption can be motivated by noticing that before any two players know their types, their infonnation is, by definition, identica1.29 Let q( t) be the a priori probability oft. Let P(L;/ti) the probability that types of players 1,2, ... i - 1, i + 1, ... n are t l , t2 , ... , ti-l, ti+l , ... , tn given that player i is of type t;. In other words, P(L;/ti) is the conditional-or posterior- probability ofLi given t;. P(L;/ti) can be calculated from q(t) as follows:

( /) q(Li' ti) p Li ti =" ( ) .

utiETi q t-i, ti Types are correlated when priors and posteriors do not coincide. In this

case, if i knows that she is of type t i , this affects the conditional probabilities of other players to be of a certain type. Types are uncorrelated or independent when the information in the hands of i does not affect posterior probabilities of other players to be of a certain type. An interpretation of the case of com­plete infonnation is that the knowledge of one's type allows to know with certainty the types of all other players, i.e types are perfectly correlated.

The set of actions that can be taken by player i is denoted by Si. A typical element of Si is denoted by Si. Let S-i be a profile of actions for all players except i and S-i be the set of all possible S-i. Let S = (8i' Li) and S =

x 7=1 Si. If the profile of actions is S and the profile of types is t, the payoff of player i is given by Ui (s, t), where Ui : S x T --> R The case in which Ui (s, ) depends only on ti is called private values. This is the case when ti is interpreted as the infonnation of i on the payoffs of i. The case where Ui (s, ) depends on t is called common values. This is the case where the infonnation held by j affects the payoffs obtained by i. For instance, finn j may know the quality of an input used by finn i.

The next definition encapsulates the concepts explained above.

29 A posible objection against this argument is that in some problems, prior probabilities are not defined. For instance, it is hard to say the probability that oil prices in five years time will be 25% above current prices. Hence the distinction between risk (where prior probabilities are meaningful) and uncertainty where these priors do not exist. The approach presented in this chapter deals with risk.

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Definition 6.1 A Game of incomplete information consists of

a) A set of players I = {I, 2, ... , n}. b) A collection of action sets Si, i = 1, ... , n. c) A collection of type spaces T;, i = 1, ... n. d) Afonction q : T -+ JR~, representing prior probabilities. e) A collection ofpayofffonctions Ui : S x T -+ JR, i = 1, ... , n.

In the case of complete information, actions and strategies are equivalent concepts. Under incomplete information, the action taken by a player depends on the information held by this player. Formally,

Definition 6.2 A strategy for player i is afonction O'i : T; -+ Si. 30

One of the interpretations of the case of complete information is that all sets T;, i = 1,2, ... , n contain only one element. However, if the information set of a player, say i, contains only one element but the information set of another player, say j, contains more than one element, we may say that j is better informed than i since j can distinguish between things that i can not distinguish. In Section 4, we will work out the implications of differential information between two players.

An essential ingredient of the definition of a N .E. is that each player can anticipate the strategies used by the other players. This idea carries to the incomplete information framework. The definition of an equilibrium under incomplete information assumes that each player can -by introspection or learning- discover the strategies used by other players. However, in this case a strategy is a function, a much more complicated object than a real number and, therefore, much harder to calculate or learn. There is also an additional problem: if only actions are observable a player may be unable to check if the assumption she made about the strategies used by other players was right or not. Under complete information, if actions can be observed and only pure strategies are used, any player can check ifher assumption about the behavior of other players was right or not because actions and strategies coincide.3)

30 A strategy for player i is pooling when actions do not depend on the type of i. A strategy for player i is separating when different types are mapped into different actions. In dy­namic games of incomplete information this distinction is important since -if actions can be observed- separating strategies convey some information about the type of the player. Pooling strategies do not convey any information.

31 Under incomplete information this checking can be made when payoffs are delivered, because if i knows the actions, say s, and the payoff received, say U;, she can -in general­infer the state of the world, by solving the equation U;(s, t) = U;. However, the assumption

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In any case, the definition presented below is the only attempt made by eco­nomics to deal with the problem of incomplete information.

Definition 6.3 A list of strategies 0'* = (O'i, 0'2' ... , O'~) is a Bayesian Equilib­rium (B.E.) ijVi, Vti E T;, VS i E Si,

L p(L;jti)Ui(O'*(t), t):::: L p(L;jti)Ui(Si, O'''-i(Li), t).

An interpretation of the above definition is that strategies represent codes of conduct that are transmitted culturally. These codes might be expected to persist when no agent finds profitable to break the received code and behave differently. Another interpretation is that before the game is actually played agents consult a mediator. The mediator does not know who is who. She could only tell "if you were like this I would take this action, but if you were like that I would take this other action". These recommendations are public knowledge. A good list of recommendations -one for each type- is one that offers no incentive to deviate.32 In Section 5 we will use this interpretation to motivate one of the main results in this chapter, namely, the Revelation Principle.

3 BAYESIAN EQUILIBRIUM AND NASH EQUILIBRIUM

In this section we will see a reinterpretation of the definition of a B.E. that allows us to apply the machinery developed in Chapter 1 to the case ofN.E .. The basic trick consists in converting a game with incomplete information with n players in a complete information game with nk players where k is the cardinality of the information set of any player.

Consider a player i and a particular type for this player ti. An action chosen by this player with information ti will be denoted by Sir. Consider now the

that payoffs are actually delivered is not appropriate in all circumstances. For instance a planner may have a utility function which depend on her actions, the actions of all other players and the state of the economy. In this case the planner is unable to check if she is anticipating correctly the strategies used by other players because actual payoffs are never delivered to her.

32 Again, if the B.E. is not unique this is not a fully compelling argument, because agents may disregard the recommendations of the mediator in favor of a different behavior if each player expect the others to do the same. For instance, the mediator may reccornrnend a B.E. in which strategies are weakly dominated, or a B.E. in which the strategies are very risky (as noticed in games of complete information by Harsanyi and Selten (1988». In these cases, the recommendation ofthe mediator is not self-inforcing.

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first element ofLi , denoted by t~i' Let p(t~Jtn be the posterior probability that Li is precisely t~i' Let Lil be the list of actions taken by all players except i when their information is t~i' Expected payoffs for i are

p(t:'Jt~)Ui(SiT> S-i1, t~, t:'i) == Vir(SiT> S-il, 1).

The argument above is repeated for all elements ofT_i. For instance, for t'.:..i' we have that expected payoffs for i are

p(t~JtnUi(SiT> Liv, t~, t~i) == Vir(Sir, S-iv, v),

where p(t'.:..Jti) is the posterior probability of t'.:..i and S-iv is the list of actions chosen by players other than i when their information is t'.:..i' Now let

(n-1)k Vir(Sir, S-i1, ... , S-i(n-1)k) == L Vir(SiT> S-iv, v).

v=l The previous argument can be repeated for any information held by player

i. In this way we end up with k players with payoff functions

Vi1(Sil, S-il, ... , S-i(n-1)k), ...... ,Vik(Sik, S-ib ... , S-i(n-1)k),

where player i1 chooses an action Si1, etc. The previous argument is repeated for all other players. In this way we end up with a complete information game with nk players. It is a complete information game because any player now faces (n - l)k who can only be of one type. Now the following result is immediate:

Proposition 6.1. Consider a NE. of the complete information game with nk

players described above, s* = (sil, ... , S !k' s21' ... , s2k, ...... , S~l' ... , s~k)' Let a* = (ai, a2, ... , a~) be defined asfollows:

ai(t1) = (Si1,···,sik),a2(t2) = (S21'···'S2k)'······,a~(tn) = (S~l"",s~k)' Then a* is a B.E. of the incomplete information game. Moreover if a* (as defined above) is a B.E., s· (as defined above) is a NE. of the game with complete information.

Proof. See Exercise 6.1 .•

The main conclusion from the above result is that the machinery developed in Chapter 1 can be applied to show existence and uniqueness ofB.E .. We will dwell on this in the next section.

4 OLIGOPOLY UNDER INCOMPLETE INFORMATION

In this section, first we will present several models of oligopoly with in-

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complete information. After, we will attempt to answer the question, is infor­mation advantage rewarded?

A first example of incomplete information arises when costs are private information. For instance suppose that actions are outputs. Let the cost func­tion of firm i be C i (Si' t i) where Si is the output of firm i. Let S = L:~=l Si be aggregate output and p( ) be the inverse demand function. If payoffs are profits

Ui(S, t i ) = P(S)Si - Ci(Si,ti ), i = 1, ... ,n.

Suppose now that actions are prices. Let S be a vector of prices and Si be the price charged by i. Let D i ( ) be the demand function for firm i. Ifpayoffs are profits

Ui(S, t i ) = Di(s)Si - Ci(Di(s), t i ), i = 1, ... , n.

In the first case a strategy is a function mapping each possible cost function into a feasible output. In the second case, a strategy is a function mapping each possible cost function into a price. In Exercises 6.2 and 6.3 the reader is asked to compute the B.E., of a very special case of the examples above. In Exercises 6.4 and 6.5 the reader is asked to adapt assumptions A.I-3 and A.I'-3' to the framework of incomplete information with private values. Notice that in our new framework the aggregation axiom will not hold because the payoff function of i in t~ does not depend on the strategy chosen by i in any tf, m =I- r. Notice too that in the examples above we have private values.

A more complicated case arises when players have information about de­mand. For instance suppose that costs are zero, n = 2, inverse demand is linear P = a - b(Sl + S2), and actions are outputs. Firm 1 only knows the value of the parameter a and firm 2 only knows the value of b. Therefore, we may write a = tb b = t2 . Thus,

U1(s, t) = (tl - t 2(Sl + S2))Sl, U2(s, t) = (tl - t2(Sl + S2))S2.

Suppose now that the product is differentiated with demand functions Di =

1 - aisi + bisj , i,j = 1,2,i =I- j. and actions are prices. Firm i knows the impact of Si on the demand ofi and j =I- i, i.e. tl = (aI, b2) and t2 = (a2' b1). Let al = t ll , b2 = t12 and a2 = t21 , b1 = t22. Thus,

U1(s, t) = (1 - tllS l + t22S2)Sl, U2(s, t) = (1 - t2lS2 + t12S2)S2.

In this case we have common values. In Exercises 6.6 and 6.7 the reader is asked to solve a special case of the two examples above. Exercises 6.8 and 6.9 asked the reader to adapt assumptions A.I-3 and A.I '-3' to the framework of incomplete information with common values.

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We now want to study the impact of the differences in information on the payoffs obtained in a B.E .. Intuition suggests that a better informed player will obtain larger expected payoffs than a non-informed one. This is clearly true in environments in which agents are negligible because a well informed agent can take advantage of her superior information without affecting the payoffs obtained by other players. However in oligopolistic environments actions by one player affect payoffs of others. In fact we will see that under certain conditions, the conjecture stated above is true, i.e. a better informed player obtains larger payoffs than worse informed players. In order to do that we will assume the following:

Assumption I: a) Actions are outputs denoted by Xi. Let X = L~=l Xi.

b) There are two firms with payofffunctions p(x, t)Xi - C(t)Xi.

c) #TI = 2 and #T2 = 1. Firm 1 has complete information.

This assumption simplifies the problem as much as possible: There is an agent (firm 1), who always know the state of the world with certainty. The other agent (firm 2) can not distinguish between the two states of the world. Both firms are assumed to have the same technology because we want to focus on differences in payoffs due to differences in information only. However, the assumption of constant returns to scale is not made for convenience: without it, the result stated below, due to Einy, Moreno and Shitovitz (1999), would not hold (see Exercise 6.10). Let us introduce the following notation. Let R(x, t) == p(x, t) - c(t). Let Elli be expected profits -evaluated with the priors- for firm i in a B.E. Let t l and t 2 be two states of the world with prior probabilities q and (1 - q).

Proposition 6.2. Suppose Assumptions A.3 and I hold. Then EllI 2: Ell2· If XIW) -I- XI(t2), EllI > Elh

Proof. First order conditions of payoff maximization for firm 1 are

8R(XI(t; + X2, tl) XI(tl) + R(XI(tl) + X2, t l ) = O.And Xl

8R(XI(t2) + X2, t 2 ) (2) R( (2) 2) .<:l Xl t + Xl t + X2, t = o. UXI

From these two equations and A.3 we obtain that, in a B.E. R(x, ti) > 0, i = 1,2. First order conditions of expected payoff maximization for firm 2 are:

148

(1 - q)(OR(Xl(t;] + X2, t2) X2 + R(Xl(t2) + X2, t2)) = 0 X2

Plugging the first two equations into the third we obtain that:

qR(Xl(tl )+X2, tl)(l- x(\))+(1- q)R(Xl(t2)+X2' t2)(1- X(22)) = o. Xl t Xl t

Or,

1 1 Xl(tl ) - X2 2 2 Xl(t2) - X2 qR(Xl(t )+X2, t)( (1)) = (q-l)R(Xl(t )+X2, t)( (2))·

Xl t Xl t From the last equation and R(x, ti) > 0, i = 1,2, it/ollows that: i) If Xl (tl) = X2, then Xl(t2) = X2. ii) If Xl (t1) > X2, then Xl (t2) < X2· Thus, Xl (t1) > Xl (t2). iii) If Xl (t1) < X2, then Xl (t2) > X2· Thus, Xl (t1) < Xl(t2) Now we are prepared to compute ElIl - E1I2 =

qR(Xl (tl) + X2, t l )(Xl (tl) - X2) + (1 - q)(R(Xl (t2) + X2, t2))(Xl (t2) - X2).

Using the equation above we arrive at ElIl - E1I2 =

(l-q)( (R(XI (t2)+X2' t2)(Xl (t2)-X2)- (R(XI (t2)+X2' t2)(Xl (t2)-X2 Xl((t~)))) Xl t

2 2 2 Xl(t2) - Xl(t1) = (1 - q)R(Xl(t ) + X2, t )(Xl(t ) - X2)( ( 2) ).

Xl t

If case i) above arises, ElIl = E1I2. If cases ii) or iii) arise the signs 0/ Xl(t2) - X2 and Xl(t2) - Xl(t1) are identical and thus, ElIl > E1I2 .•

5 RESOURCE ALLOCATION MECHANISMS

At this point it is convenient to see markets from a broader perspective. Instead of focussing on particular market structures -oligopoly, monopolistic competition, etc.-, we will present a general framework to deal with resource allocation. This framework is called Mechanism Design or Implementation Theory. The interested reader can consult my book "Theory of Implementa­tion of Optimal Decisions in Economics" (1996) and the references therein for a more comprehensive treatment.

Under risk, an allocation is a mapping, say <I>, from the set of states of the world into the set of actions, i.e. <I> : T -+ S. Think of this allocation as state-contingent actions (or commodities). In order to compute an allocation the society needs a procedure. Think of the following: First, the members of the society are asked to submit certain messages. These messages encode

149

information about the state of the world. Once these messages have been sent, they are used to determine the actions to be taken. This procedure is precisely the definition of a mechanism.

Definition 6.4 A Mechanism oot consists in a collection of message spaces M 1 , ... , Mn, and an outcomefunction g : M ~ S, where M = Xi=lMi .

The models considered so far in this book can be regarded as special cases of the above definition when M = S and the outcome function is the identity mapping (see Exercise 6.11).

Notice that agents are no longer free to determine the actions. Instead they are free to submit a message. Once all messages have been sent, actions are determined.33 Since agents are supposed to understand the nature of the game, they will realize that payoffs and messages are associated. Thus, an strategy in this framework is a function mapping types into messages, i.e. ai : Ti ~ Mi. We will focus on those allocations which can be achieved when agents choose strategies that constitute a BE, i.e. when each player, say i, Vii E Ti, chooses mi(ti) such that, Vmi E Mi,

L p(Ldti)Ui(g(m(t)), t) ~ L p(Ldti)Ui(g(mi, m-i(t-i)), t).

This motivates the following definition

Definition 6.S The allocation <I> is implementable in B.E. by the mechanism {M1 , ... , Mn,g} if

a) 3m* E M which is a B.E. b) If m* is a B.E., g(m*(t)) = <I>(t), "It E T.

We will say that <I> is implementable when <I> is implementable in B.E. by some mechanism {M1 , ... , Mn, g}. An important particular class of mecha­nisms are those in which T;, = Mi for all i, i.e. in these mechanisms each player announces her own characteristic. These mechanisms are called reve­lation mechanisms or direct mechanisms as well. In a revelation mechanism a strategy is a mapping from and into T;" i.e. ai : Ti ~ Ti. A strategy in a revelation mechanism is truthful if it is the identity mapping. An allocation is truthfully implementable if the truthful strategy is a B.E. The next result,

33 In other words, there are no hidden actions. The problems associated with hidden actions are termed moral hazard In this presentation we deal with the problem of hidden information or adverse selection.

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known as the Revelation Principle (Myerson (1979), Harris and Townsend (1981)), provides a very useful link between implementation and truthful im­plementation.

Proposition 6.3. Suppose the allocation <I> is implementable in B.E.. Then <I> is truthfully implementable.

Proof. Suppose that <I> is implementable by the mechanism {MI , ... , Mn, g}. Let m be a strategy fulfilling the condition b) in Definition 5 above. Con­sider the revelation mechanism {TI , ... , Tn, <I> }.1fthe proposition is not true, 3i, ti , ti, such that

However if <I> was implementable, we should have that

And

<I>(ti' Li) = g(mi(ti), m_i(Li)).

Thus, the inequality above implies that

L p(L;jti)Ui(g(mi(ti ), m_i(Li)), t) > L p(L;jti)Ui(g(m(t)), t)

contradicting that m was a B.E., because agent i should have chosen mi(ti) instead of mi(ti) when she was of type t;..

The revelation principle can be interpreted in the following way. Each agent delegates her choice of a strategy in a mediator. The mediator ask to the player what her type is, and with this information chooses a strategy in order to maximize the player's payoff. The revelation principle states that no agent has incentives to fool the mediator telling a type different from the true one, because this will only result in the mediator being unable to pick up the best strategy.

The revelation principle states that truthful implementation is a necessary condition of implementation. Therefore, it is a very useful tool to check if an allocation is not implementable: It suffices to look at the revelation mecha­nism and to check that the truthful strategy is not a B.E. If this is the case, the allocation is not implementable in B.E. The revelation principle is useful too in order to discover properties of implementable allocations. In the next section we will provide examples of each of these uses in Propositions 6.4 and 6.5.

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However, truthful implementation is not a sufficient condition for imple­mentation: The revelation principle can not be used to check if an allocation is implementable because the truthful strategy may be not the only equilibrium strategy in the revelation mechanism. In other words, there may be several BE: In one of them all agents are using truthful strategies but in other BE some (or all) agents are using untruthful strategies (see Exercise 6.12). If such untruthful equilibria exist, unless there is a good reason to discard all but the truthful BE, truthful implementation does not imply implementation.

6 ALLOCATION OF AN INDIVISIBLE GOOD

In all the sections of this book we have considered the allocation of a pro­duced, infinitely divisible good in a market in which strategic interactions are important. It should be clear by now, that we have a pretty good under­standing of the case in which agents are completely informed. Unfortunately, our understanding of markets where agents are imperfectly informed is far from perfect. The strategy of economics in order to make progress in the understanding of markets with incomplete information has been to focus on the simplest possible case: The allocation of a non-produced indivisible good which is traded against an infinitely divisible good -called money. Important special cases of this class of situations are auctions and trading between two agents. In this section of this book we will explain two important results: The Revenue Equivalence Theorem (RET in the sequel), which refers to auc­tions and the Impossibility of Efficient Trade Theorem (IETT) which refers to bilateral trade.

The main difference with the framework presented in the previous sections is that we will assume here that the type space of each agent will be assumed here to be an interval of the real line, i.e. Ti = [t., ~ i = 1,2, ... , n. A type is interpreted as the monetary valuation of the good. An action is a pair (Pi, ei) where P; is the probability of obtaining the good and ei is the payment made (or received) by i. The payoff of i is Ui = Piti - ei. This implies that we are in a case of private values.

In this framework, the outcome function of a mechanism has two parts. One yields the probability of obtaining the good as a function of messages and the other gives the monetary payments to be made by agents as a func­tion of the messages they send. For instance if for the profile of messages m player i gets the good, and payments (el,e2, ... ,en) are made, g(m) ((1,0, .... , 0), (el' e2, ... , en)).

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It is straightforward to extend the notion of a BE to this framework (see Exercise 6.13). All our previous results, in particular the revelation principle, remain valid (see Exercise 6.14). Consider now the following assumption:

Assumption A: a) Types are un correlated b) Each ti is drawnfrom the same distributionfunction, denoted by FO. c) Agents are risk-neutral.

Parts a)-b) have the implication that the probability that ti is larger than any other type is (F(ti) )n-l. An implication of part c) and the revelation principle is that at any BE the expected payoff of i, denoted by Si, can be written as Fiti - Ei(ti) == Si(ti), where Ei(ti) is the expected payment.

A mechanism is allocation efficient when at all BE the type with the largest valuation obtains the object. In this case, Fi = (F(ti) )n-l. Notice that an al­location efficient mechanism is not necessarily efficient because it may waste money. A mechanism is strictly individually rational for the lowest type if at all BE and all i, Si(t) = Ui, with Ui = 0 if i is a buyer and Ui = 1 if i is a seller . All standard forms of auction satisfy allocation efficiency and strict individual rationality for the lowest type.

Suppose that we want to raise as much money as possible by auctioning an indivisible good (think of the indivisible as a licence to produce a good). The following result, due to Myerson (1981) and Riley and Samuelson (1981), says that a large class of mechanisms will raise the same amount of money. This is why this result is known as the Revenue Equivalence Theorem. The proof of this result offered here is due to Klemperer (1999).

Proposition 6.4. Suppose that Assumptions 1 and A hold. Consider two mechanisms which are allocation efficient and strictly individually rational for the lowest type. In each BE every bidder makes the same expected ex­penses in both mechanism and, therefore, the expected revenue raised by the two mechanisms is the same.

Proof. Since the mechanism is allocation efficient, expected payoffs for i when she is of type ti can be written as (F(ti) )n-1ti - Ei(ti). By the revelation principle the truthful strategy must be a BE, i.e. for all i and ti,

a(F(ti))n-l aE(ti) a t i - -a- =0.

ti ti Now, compute the derivative of payoffs with respect to the type:

aSi = a(F(ti)t-1 . (F( .))n-l _ aE(ti )

a a t, + t, a' ti ti ti

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Using the two equations above we obtain that

. {)Si (F( ))n-l 'VZ, 'Vti , -{) = ti . ti

This implies that the slope of Si is completely determined by the distribu­tion function F( ). This, together with Si CO = Ui , completely determines the function Si ( ). Thus, Si ( ) does not depend on the expected payments made by i. Thus, all individuals must make the same expected payments in two mech­anisms which are allocation efficient and strictly individually rational for the lowest type .•

The revenue equivalence theorem generalizes to common values and to the case of many indivisible goods provided agents want no more than one object, see Exercise 6.15. However the result does not generalize to correlated types, risk averse agents or different distribution functions, see Exercises 6.16-18. In other words, Assumption A is essential for the result to hold. Notice that it is straightforward to extend the result to any two mechanisms in which the function P is the same.

The concept of a mechanism allows us to investigate one of the oldest problems in economics, that of bilateral trade. An old tradition in economics asserts that any potential welfare gain between two agents can -and will- be realized by the appropriate contract.34 We will see that under incomplete information this view is not warranted.

Let us assume n = 2. Agent 2, called the seller, owns the indivisible good. Agent 1 is the potential buyer. An efficient mechanism is an allocation ef­ficient mechanism in which payments add up to zero, i.e. money is neither wasted nor asked to an external agent.35 In other words, an efficient mecha­nism is one that implements an efficient allocation of the indivisible good and the money.

We have the following impossibility result, due to Myerson and Satterth­waite (1983). The proof offered here is new but it was inspired by the paper ofUbeda (1999).

Proposition 6.5. Suppose n = 2 and that Assumptions 1 and A hold. Any mechanism strictly individually rational for the lowest type is not efficient.

34 This view is an important part of the so called "Coase Theorem". It is not clear at all that Coase agrees with the view that contracts work in such a smooth way, see Coase (1992).

35 The concept of efficiency used here corresponds to what is usually called ex-post effi­ciency. For other concepts of efficiency under risk or uncertainty the reader may consult the book by Mas-Colell, Whinston and Green (1995), pp.898-9.

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Proof. Consider the following direct mechanism. Mi = Ti i = 1,2, and g(m) = ((0,1),(0,0)) ifm2 2:: ml and g(m) = ((1,0), (-m2,ml)) if ml > m2. In words, if the valuation announced by the seller (m2) is larger than the valuation announced by the buyer (md, there is no trade and no payments. If the valuation announced by the buyer is larger than the valua­tion announced by the seller, there is trade, the buyer pays m2 and the seller receives mI. Thus, in this mechanism, we need money from outside because the sum of expected payments add up to a positive number. Also, the truth is a dominant strategy. Therefore the truth is a BE. A quick look at the proof of the revenue equivalence theorem shows that the theorem holds in the case in which allocation efficiency is required only at the truthful BE. Therefore, the sum of expected expenses made by all agents in any strictly individually rational and allocation efficient mechanism are positive. Thus no mechanism can be efficient .•

Proposition 6.5 says that under a (weak) individually rational constraint, no efficient allocation is implementable. It is important to remark that this result, like the revenue equivalence theorem, does not generalize to economies where Assumption A does not hold, see Exercise 6.l9.

Summing up, in this chapter, we have studied markets with incomplete information. In these markets we have found a new cause for inefficiency, not linked with the oligopolistic structure of the market but with incom­plete information.

7 ADDITIONAL REFERENCES

The first paper to deal with games with incomplete information was writ­ten by Vickrey (1961). The concept of equilibrium proposed there was a spe­cial case of Bayesian equilibrium. This concept was proposed by Harsanyi (1967-68). The impossibility theorem of Myerson and Satterthwaite has been studied by Makowski and Mezzetti (1993, 1994). They show that the theo­rem depends crucially on the assumption that types are uncorrelated. With correlated types, efficient and individually rational mechanisms exists, as first noticed by Myerson (1981) in the context of auctions.

8 EXERCISES

6.1. Show that a N .E. of the game with complete information is a B.E. of the game with incomplete information and viceversa.

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6.2. Suppose that inverse demand is linear, costs are linear and n = 2. Sup­pose further that #Tl = 1 and #T2 = 2. Compute the RE. of a game in which actions are outputs and the type of firm i is the marginal cost of i.

6.3. Suppose that the product is differentiated, demand and cost functions are linear and n = 2. Suppose further that #Tl = 1 and #T2 = 2. Compute the B.B. of a game in which actions are prices and the type of firm i is the marginal cost of i.

6.4. Adapt Assumptions A.1-3 to the framework of incomplete information with private values.

6.5. Adapt Assumptions Al '-3' to the framework of incomplete information with private values

6.6. Suppose that inverse demand reads p = a - b( Xl + X2) and costs are zero. Suppose further that #Tl = 1 and #T2 = 2. Compute the B.E. of a game in which actions are outputs, a = tl and b = t 2•

6.7. Suppose that the product is differentiated, demand functions read Di =

1 - aiPi + biPj, i, j = 1,2, i f j. Cost are zero. Suppose further that #Tl = 1 and #T2 = 2. Compute the RE. of a game in which actions are prices, tl =

(aI, b2) and t2 = (a2' bl ).

6.8. Adapt Assumptions A.1-3 to the framework of incomplete information with common values.

6.9. Adapt Assumptions Al '-3' to the framework of incomplete information with common values.

6.10. Give an example showing that, under decreasing returns to scale, Propo­sition 6.2 does not hold (see Einy, Moreno and Shitovitz (1999». Give the intuition behind this example.

6.11. Show formally that the Cournot and the Bertrand models are special cases of the definition of a mechanism.

6.12. Give an example in which there are two BE. In one of them all agents are using truthful strategies and in the other there are some agents using untruthful strategies. Choose the example in such a way that all agents with type spaces with more than one element are better off in the untruthful BE than in the truthful BE (see Postlewaite and Schmeidler (1986».

6.13. Write the definition of a BE in the case in which the type space is a subset of the real line.

6.14. Prove the revelation principle for a general type space.

6.15. Prove the revenue equivalence theorem for the case of many indivisible goods and for the case of common values (see Bulow and Klemperer (1996».

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6.16. Give an example in which types are correlated and the revenue equiv­alence theorem does not hold (see Klemperer (1999). The example is due to Myerson (1981)).

6.17.-Give an example in which agents are risk averse and the revenue equiv­alence theorem does not hold (see Klemperer (1999)).

6.18.-Give an example in which types are not drawn from a single distribution and the revenue equivalence theorem does not hold (see Klemperer (1999). The example is due to Myerson (1981)).

6. 19.-Give examples in which Proposition 6.4 does not hold (see Makowski and Mezzetti (1994)).

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