Cooperation and Efficiency in Markets

Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, 33615 Bielefeld, Germany Editorial Board: H. Dawid, D. Dimitrow, A. Gerber, C-J. Haake, C. Hofmann, T. Pfeiffer, R. Slowi´nski, W.H.M. Zijm

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Milan Horniaˇcek

Cooperation and Efficiency in Markets

123

Milan Horniaˇcek Comenius University Faculty of Social and Economic Sciences Institute of Public Policy and Economics Mlynské luhy 4 82005 Bratislava Slovakia [email protected]

ISSN 0075-8442 ISBN 978-3-642-19762-8 e-ISBN 978-3-642-19763-5 DOI 10.1007/978-3-642-19763-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011929346 c Springer-Verlag Berlin Heidelberg 2011  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book is a result of my research in game theory and oligopoly theory. It has also been influenced by my teaching experiences. Countable infinite repetition of strategic form noncooperative games with discounting of future payoffs has been the focus of my research interest for about 20 years. In most of my work, I have tried to analyze solution concepts that also take into account the deviations by the coalitions of players (if possible, by all coalitions). The emergence of cooperation between patient players, who are forced to be competitors in a static setting (or even in a dynamic setting with a finite horizon), is a fascinating phenomenon from both the intellectual and human points of view. From the intellectual point of view, the development of strategy profiles that are immune to the deviations by coalitions is a challenging and interesting work. From the human point of view, it is nice to know that – when there is a sufficiently long shadow of the future – selfish behavior, aimed at prospering at the expense of another individual, need not be rational. It is a good news that when all players agree on a cooperation scheme, no one can gain by distorting it unilaterally. If it turns out that no coalition can make any of its members better off without making another member worse off by distorting a cooperation scheme, it is a pleasant news. My sensitivity to the human side of results on infinite repeated games increased in last three years during which I was teaching the course “Models of competition and cooperation” at the Faculty of Social and Economic Sciences, Comenius University, Bratislava. This course includes the issues of the evolution and stability of cooperation. This motivated me to think more deeply about the eternal struggle between those who want to preserve mutually beneficial cooperation and those who want to distort or abuse it to achieve their selfish goals. My interest in cooperative behavior in infinite repeated games led me to think about an important economic phenomenon of our world at present – trading between (relatively small) farmers and other producers of foodstuffs (or producers of other consumer goods) and chain stores. The producers of foodstuffs are weaker partners in this relationship; chain stores have considerable market power. They take advantage of myopic price taking behavior of their suppliers. The resulting static supply functions narrow the space of attainable contracts. Obviously, an interaction with an infinite horizon enlarges this space. This motivated me to study a class of games that includes the models of infinite repeated interactions between chain stores and their suppliers as a special case – infinite countably repeated games between firms v

vi

Preface

on both sides of a market with discounting of future profits. This book deals with the analysis of such games that takes into account coordinated actions of all coalitions. A cooperation between firms on the same side of a market is vulnerable to a legal challenge on the basis of antitrust laws. In my model, some of the cooperating firms are on the same and some are on the different sides of a market. This raises a question as to whether or not the equilibrium behavior in it can be defended against the charges of the violation of antitrust laws. Therefore, I identify (some) conditions under which the equilibrium behavior in my model has socially desirable properties – it minimizes the cost on the levels of economic activity repeated along the equilibrium path and increases consumer welfare. The research underlying this book was supported by the grant VEGA 1/0138/09. The purchase of the Scientific WorkPlace software, which made the preparation of the book much easier, was financed from this grant. Bratislava, March 2011

Milan Horniaˇcek

Contents

1

Introduction . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 1.1 Choice of Solution Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 1.2 Choice of Non-collusive Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 1.3 Mathematical Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . Reference . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .

1 3 5 6 7

2

Model . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 9 2.1 Stage Game .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 9 2.2 Repeated Game. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 15 2.3 Solution Concepts .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 17 Reference . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 19

3

Existence of an SRPE and an SSPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 21 3.1 Existence of an SRPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 21 3.2 Existence of an SSPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 26

4

Efficiency of an SRPE and an SSPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 53 4.1 Natural Oligopoly .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 54 4.2 Natural Oligopsony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 58 4.3 Impact on Consumer Welfare.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 61 4.3.1 Comparison with a Monopsonist Choosing the Traded Quantities on the Demand Side and Price Taking Behavior on the Supply Side of the Analyzed Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 62 4.3.2 Comparison with Price Taking Behavior on the Demand Side and a Cournot Oligopoly on the Supply Side of the Analyzed Market .. . . . . . . . . . . . .. . . . . . . . 70 Reference . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 83

5

Afterword .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 85 Reference . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 90

Index . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 91 vii

Chapter 1

Introduction

This book analyzes countable infinite repetition of interaction between firms on both sides of a market, with discounting of future profits.1 Our first aim is to prove the sufficient conditions for the existence of equilibria, which take into account the deviations by the coalitions, in it. We explain the reasons for our choice of solution concepts in Sect. 1.1 of Chap. 1. Our second and third aims relate to the efficiency properties of the equilibria in our model. The second aim (covered in Sects. 4.1 and 4.2) is to show that – when the sum of the firms’ expected average discounted profits is maximized in the equilibrium – there is no waste of resources in the equilibria. The aggregate production cost of the equilibrium output of each type of good is minimized along the equilibrium path. If the firms on the demand side of the analyzed market are retailers (i.e., they resell the purchased goods in a retail market), then the aggregate selling cost of the equilibrium vector of sales in the retail market is minimized along the equilibrium path. The third aim (covered in Sect. 4.3) concerns the efficiency of the equilibria in our model from the point of view of welfare economics. We identify the conditions under which consumer welfare in each period along the equilibrium path in our model exceeds consumer welfare in a benchmark non-collusive equilibrium. In Sect. 1.2 of the Introduction, we explain our choice of benchmark non-collusive equilibria. In the last section of the Introduction, we explain the mathematical notations used in this book. The stage game in our model is a strategic form noncooperative game. It models the analyzed market with a finite number of producers and a finite number of buyers.2 All firms simultaneously make contract proposals to all firms on the other side of the analyzed market (i.e., each producer makes a contract proposal to each buyer and each buyer makes a contract proposal to each producer). A contract proposal

1

Throughout the book we use the well-established term “repeated game” for what [Mertens (1989)] refers to as the “supergame.” 2 Throughout the book, we use the term “analyzed market” for the market modeled by the stage game (of the repeated game that is the object of our study). Besides this market, the firms’ payoffs are influenced by the market in which the buyers in the analyzed market earn their revenue. The latter market is either a retail market for the goods traded in the analyzed market or, if buyers are themselves producers, a market for their products.

M. Horniaˇcek, Cooperation and Efficiency in Markets, Lecture Notes in Economics and Mathematical Systems 649, DOI 10.1007/978-3-642-19763-5_1, c Springer-Verlag Berlin Heidelberg 2011 

1

2

1 Introduction

contains a proposed traded quantity and a proposed price.3 A firm’s decision to not make a contract proposal to some firm on the other side of the analyzed market is modeled by the zero contract proposal, containing zero proposed quantity and zero proposed price. A pair of a producer and a buyer trades if and only if their contract proposals match and differ from the zero contract proposal. The quantities of the goods purchased by a buyer in the analyzed market determine his/her revenue in a market in which he/she is a seller. The firms can also withdraw from the analyzed market. A firm indicates its withdrawal from the analyzed market by making zero contract proposals to all firms on the other side of that market. In such a case, its payoff in the stage game is zero. The stage game payoff of a firm that did not withdraw from the analyzed market equals its profit. The described choice of the stage game has an underlying economic reason and an underlying game-theoretic reason. The contracts between the firms have to result from some process involving proposals and reactions to them.4 Such processes can be modeled by finite horizon extensive form noncooperative games. It is not possible to find one such process that would be superior from the point of view of realism or for some economic reasons. Therefore, instead of choosing one particular process of concluding contracts, we use a “shorthand” for all such processes. It does not specify the order of moves in the conclusion of contracts but merely expresses the basic fact that two firms can conclude a contract if and only if they agree on it. This “shorthand” can be modeled by a strategic form noncooperative game. This is important from the game-theoretic point of view. Suppose that the process of the conclusion of contracts was modeled by an extensive form game without simultaneous moves. Then, some unilateral single period deviations from the profile of equilibrium strategies in the repeated game would have to be punished by the rejection of a contract proposal. (Otherwise, a deviator can increase his/her payoff in the repeated game by proposing a contract increasing his/her stage game payoff at the expense of a trading partner.) Nevertheless, in some cases (for “moderate” deviations), the rejection would make both the proposer and the responder worse off than the acceptance of a deviator’s contract proposal. Therefore, the grand coalition can weakly Pareto improve the vector of payoffs in the repeated game5 by abandoning the punishment. Thus, the equilibrium strategy profile would not be renegotiation-proof.

3

Throughout the book, the term “price” always implies the unit price. This is certainly not unrealistic. Moreover, traded quantities and prices cannot be given by a supply function or a demand function in the analyzed market. These functions are derived under the assumption of price taking behavior on one side of the market. Such assumption would limit the possibilities for cooperation in the repeated game. 5 If there was at least one producer trading with all buyers along the equilibrium path and the stage game payoff functions were continuous, the grand coalition can also strictly Pareto improve the vector of payoffs in the repeated game. The method of doing so would be analogous to the one in Step 4 in Sect. 4.3.2. 4

1.1 Choice of Solution Concepts

3

1.1 Choice of Solution Concepts A subgame perfect equilibrium (first defined in [Selten (1965)] in German and in [Selten (1975)] in English) is a standard solution concept for countable infinite repeated games with discounting of future stage game payoffs. Nevertheless, it requires only the immunity of an equilibrium strategy profile to unilateral deviations. The continuation equilibrium payoff vectors need not be (even weakly) Pareto efficient.6 That is, it does not take into account the possibility that the grand coalition reaches an agreement to abandon the punishment of a previous unilateral deviation in order to increase the continuation payoff of each player. Moreover, a subgame perfect equilibrium does not require immunity to deviations by non-singleton proper coalitions (i.e., coalitions with at least two members that do not contain all players). In repeated games between firms it is plausible that players having closer business links or very intensive effects on one another will coordinate on a deviation. Thus far (to our best knowledge), there is only one solution concept for infinite horizon noncooperative games (of which the countable infinite repeated games with discounting of future stage game payoffs are a special case) that takes into account the deviations by all coalitions without imposing restrictions either on the strategies of individual players or on the strategy profiles to which a coalition can deviate.7 This solution is a strong perfect equilibrium developed by [Rubinstein (1980)]. It is an application of Aumann’s concept of a strong Nash equilibrium for strategic form noncooperative games ([Aumann (1959)])8 to infinite horizon noncooperative games, and requires that no coalition in no subgame can strictly Pareto improve the vector of the continuation payoffs of its members by a deviation. Since we want our main solution concept to be immune to all deviations by all coalitions, we have to base it on a strong perfect equilibrium. Nevertheless, we strengthen the latter concept by requiring that no coalition in no subgame can weakly Pareto improve the vector of the continuation payoffs of its members by a deviation. We call the resulting solution concept a strict strong perfect equilibrium (henceforth, SSPE – see Definition 2.2). If a deviation only weakly Pareto improves the vector of continuation payoffs of the members of a deviating coalition, then its members, whose continuation payoff does not change, are indifferent between participating and not participating in the deviation. Nevertheless, they can decide to deviate (perhaps, for some other reasons not captured by the model). Therefore, the replacement of a strong perfect equilibrium by an SSPE strengthens our results.

6

A continuation equilibrium in a subgame is a restriction of an equilibrium strategy profile to that subgame. The continuation equilibrium payoff vector is the payoff vector generated by the continuation equilibrium. 7 [Bernheim et al. (1987)] study only finite horizon games. Various concepts of renegotiationproofness for infinite horizon discrete time games with two players (mentioned below in the text) impose restrictions on the strategy profiles to which the grand coalition can deviate, or, in the case of [Maskin & Tirole (1988)], restrict attention to Markov strategies. 8 Despite the title of the paper a strong Nash equilibrium is a solution concept for noncooperative games. See also [Bernheim et al. (1987), p.2-3].

4

1 Introduction

In many infinite horizon games of economic interest a strong perfect equilibrium (and hence, also an SSPE) fails to exist. Namely, in countable infinite repeated games with discounting of future stage game payoffs (unless the stage game has a strong Nash equilibrium), a strong perfect equilibrium fails to exist when the weak Pareto efficient frontier of the set of individually rational stage game payoff vectors has no sufficiently large flat portion (i.e., a sufficiently large portion that is a subset of a hyperplane of the vector space with a dimension equal to the number of players). In our model, we do not face such a problem. The firms can coordinate (under the assumptions of Proposition 3.4) to play a stage game strategy profile that maximizes the sum of their stage game payoffs, and deviations can be punished by changing only the prices in the analyzed market. Then, the continuation payoffs during punishments have the same sum as the continuation payoffs in subgames where no firm is punished. Thus, they are strictly Pareto efficient. (In this case, the stage game strategy profiles, which lead to the same vector of traded quantities in the analyzed market and differ only in prices, generate a flat portion of the weak Pareto efficient frontier of the set of stage game payoff vectors.) A strong perfect equilibrium is sometimes criticized for being too strong. Such criticism is based on the criticism of a strong Nash equilibrium for strategic form noncooperative games (see [Bernheim et al. (1987), p. 3]). It is claimed that the deviations by coalitions, which are themselves not immune to a deviation by some strict subcoalition, should not be taken into account. Extending this claim to countable infinite repeated games, it can be argued that the deviations by coalitions, which are themselves not immune to a deviation by some strict subcoalition, need not be punished. Nevertheless, as [Horniaˇcek (1996), Sect. 6] shows, in countable infinite repeated games, this claim is not justified for deviations that last forever. The point here is that a deviation by a strict subcoalition of a deviating coalition can be punished by the remaining members of the deviating coalition. There are solution concepts for infinite horizon discrete time games that take into account (besides unilateral deviations) only the deviations by the grand coalition. There are various versions of renegotiation-proofness. With the exception of [Farrell (1993)], all of them are defined for games with two players. All of them impose restrictions on the strategy profiles to which the grand coalition can renegotiate (i.e., deviate). In a weakly renegotiation-proof equilibrium of [Farrell & Maskin (1989)] (which coincides with the concept of an internally consistent equilibrium of [Bernheim & Ray (1989)]), only renegotiation to another continuation equilibrium is allowed. [Farrell (1993)] imposes the same restriction. Moreover, he requires that all players, who did not deviate, block renegotiation – i.e., such players would not gain by renegotiation from a continuation equilibrium triggered by a deviation back to the initial continuation equilibrium. In a strong renegotiation-proof equilibrium of [Farrell & Maskin (1989)] (which coincides with the concept of a strongly consistent equilibrium of [Bernheim & Ray (1989)]), only renegotiation to a continuation equilibrium of some weakly renegotiation-proof equilibrium is allowed. [Maskin & Tirole (1988)] restrict attention to Markov strategies. Besides an SSPE, we also use a concept that we refer to as a strict renegotiationproof equilibrium (henceforth, SRPE). This equilibrium is a subgame perfect

1.2 Choice of Non-collusive Benchmarks

5

equilibrium with the property that the grand coalition cannot weakly Pareto improve the vector of continuation payoffs in any subgame (see Definition 2.1). Thus, unlike the solution concepts mentioned above, it does not impose any restriction on the strategy profiles to which the grand coalition can renegotiate. Clearly, the grand coalition cannot abandon a punishment of any unilateral deviation (because it would make at least one of its members strictly worse off). Of course, each SSPE is an SRPE but not vice versa. The results in Chap. 4 hold also for an SRPE. Thus, a collusive scheme, which can be sustained as an SRPE but not as an SSPE, has positive properties discussed there.

1.2 Choice of Non-collusive Benchmarks The economics literature on cartels and tacit collusion assumes that all participating firms act on the same side of a market. Usually all of them are on the supply side of a market and produce goods that are identical or close substitutes. This assumption is made – sometimes implicitly – also in official documents of antitrust (or, in the European Union, competition policy) authorities (e.g., [EU (2010b)]). Thus, it is not surprising that the attitude of these authorities to cartels and collusion is negative. There are some authors who recognize that – in certain circumstances – cartels and collusion can have a positive impact, or at least, criticize the approach of the antitrust policy to prices. Some economic historians study the role that cooperation between firms played in economic development of Germany prior to World War I, where cartels were legal. [Chandler (1990), part IV, especially Chapter 10] coined the term “cooperative managerial capitalism” for the institutional structure of the German economy at that time. Spatial economists point out that antitrust laws do not take into account the characteristics of spatial oligopolistic competition. For example, according to [Greenhut et al. (1987)], “the antimerger and price policies of the United States, as evidenced by the Celler Antimerger Act and the Robinson-Patman Act, are based on spaceless economic theory but are applied to spatial economic behavior” [Greenhut et al. (1987), p. 359]. They also point out that collusion between duopolists in the noncoincident-location case need not increase price discrimination [Greenhut et al. (1987), p. 149]. Moreover, some forms of spatial competition generate a higher price than that charged by a spatial monopolist. The Löschian competition, under which the firm presumes that its rivals will react identically to any proposed price change, is an example (see [Greenhut et al. (1987), p. 20]). We compare collusive schemes sustainable in an SRPE with two non-collusive benchmarks.9 Since in most real world markets, at least one side is characterized 9

Both benchmarks are non-spatial. Nevertheless, this does not reduce the value of our results because the comparison of various forms of competitive behavior with collusion is more favorable for the former in a non-spatial setting.

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1 Introduction

by imperfect competition (or even monopoly), our first benchmark model involves monopsony on the demand side and the second one, Cournot oligopoly on the supply side of the analyzed market. Nevertheless, in both, we assume price taking behavior on the other side of the market. The standard model of monopsony requires a supply or an inverse supply function. The model of Cournot oligopoly requires (an) inverse demand function(s). The former exists only in the case of price taking behavior on the supply side of the market. The latter exists only in the case of price taking behavior on the demand side of the market.10 The restriction to the monopsonistic market in the first benchmark model allows us to obtain results with fewer assumptions than we would need if we assumed oligopsony on the demand side. Moreover, this model is a plausible approximation of the relations between a chain-store and its suppliers. (As pointed out in the Preface, the problems in these relations constituted one of the motivations for the analysis in this book.) The purchased quantities are the monopsonist’s choice variables. This corresponds to the standard treatment of monopoly in the literature, in which quantity is the choice variable. If we replaced Cournot oligopoly by Bertrand oligopoly with differentiated products in the second benchmark model, we would obtain (under additional assumptions on demand curves) the same qualitative results. Working with Bertrand oligopoly, in which some groups of firms produce identical products, would lead to mathematical problems stemming from the discontinuity of the firms’ demand functions. Moreover, taking into account the result of [Kreps & Scheinkman (1983)], in the case of homogeneous goods, the Cournot oligopoly model is more general than the Bertrand oligopoly model.

1.3 Mathematical Notations Throughout the book, N denotes the set of positive integers and < the set of real numbers. For n 2 N; v implies that z  v but z ¤ v, and z  v implies that zj > vj for each j 2 f1; : : : ; ng. For n 2 N 4n is the simplex in
8 < :

z 2
n X j D1

9 =

zj D 1 : ;

For a set Z 
In the case of imperfect competition on both sides of a market, we would need a non-collusive model of bargaining in order to determine the prices and quantities. There is no generally known model of this type. Therefore, we prefer to work with a monopsony model and a Cournot oligopoly.

References

7

We endow each finite dimensional vector space with the Euclidean topology and each infinite dimensional Cartesian product of finite dimensional vector spaces with the product topology. Each subspace of any topological vector space Z inherits topology from Z. For a finite set K; # .K/ denotes its cardinality and 2K is the set of all subsets of K, including the empty set. For a twice differentiable function f W < ! <; the symbol f 0 .x/ denotes its first derivative at x and the symbol f 00 .x/ its second derivative at x.

References Aumann, R.J. “Acceptable Points in General Cooperative n-person Games,” Annals of Mathematical Studies Series 40 (1959), 287-324. Bernheim, B.D., B. Peleg, and M.D. Whinston: “Coalition-Proof Nash equilibria I. Concepts,” Journal of Economic Theory 42 (1987), 1-12. Bernheim, B.D. and D. Ray: “Collective Dynamic Consistency in Repeated Games,” Games and Economic Behavior 1 (1989), 295-326. Chandler, A.D. Jr.: Scale and Scope. The Dynamics of Industrial Capitalism. Cambridge, MA: The Belknap Press of the Harvard University Press, 1990. “Commission Action Against Cartels – Questions and Answers.” MEMO 10/290, 30 June 2010. http://europa.eu/rapid/pressReleasesAction.do?reference=MEMO/10/290&format=HTML& aged=0&language=EN&guiLanguage=en Farrell, J.: Renegotiation in Repeated Oligopoly Games. Mimeo. University of California at Berkeley, 1993. Farrell, J. and E. Maskin: “Renegotiation in Repeated Games,” Games and Economic Behavior 1 (1989), 327-360. Greenhut, M.L., G. Norman, and C. Hung: The Economics of Imperfect Competition. Cambridge: Cambridge University Press, 1987. Horniaˇcek, M.: “The Approximation of a Strong Perfect Equilibrium in a Discounted Supergame,” Journal of Mathematical Economics 25 (1996), 85-107. Kreps, D.M. and J.A. Scheinkman: “Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes,” Bell Journal of Economics 14 (1983), 326-337. Maskin, E. and J. Tirole: “A Theory of Dynamic Oligopoly II: Price Competition, Kinked Demand Curves, and Edgeworth Cycles,” Econometrica 56 (1988), 571-599. Mertens, J.-F.: “Supergames,” in The New Palgrave: Game Theory, ed. by J. Eatwell, M. Milgate, and P. Newman. New York: W.W. Norton, 1989, 238-241. Rubinstein, A: : “Strong Perfect Equilibrium in Supergames,” International Journal of Game Theory 9 (1980), 1-12. Selten, R.: “Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit,” Zeitschrift für die Gesamte Staatswissenschaft 121 (1965), 301-324 and 667-689. Selten, R.: “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games,” International Journal of Game Theory 4 (1975), 25-55.

Chapter 2

Model

In this chapter we describe our model of a market with strategically behaving agents on both sides. First, we characterize the stage game between the firms in the model. Then, we proceed to the formulation of the countable infinite repeated game with discounting of payoffs. In Sect. 2.3 we define the solution concepts that we apply to the repeated game: SRPE and SSPE.

2.1 Stage Game We denote the stage game by G. The stage game is a strategic form noncooperative game. There is a nonempty finite set J of producers and a nonempty finite set I of buyers. We have J D f1; : : : ; # .J /g and I D f# .J / C 1; : : : ; # .J / C # .I /g. Thus, J [ I is the set of players in G. (We use the terms “player(s)” and “firm(s)” interchangeably.) Buyers buy goods from producers. They either use the purchased goods to produce new good(s) (which they sell in the market(s) where customers are price takers) or sell them in the retail market to final consumers. Each producer produces one type of good, and can sell to any number of buyers in I: Therefore, we use the symbol J also for the set of goods in the model. We do not exclude the possibility that some or even all goods in J are identical. A coalition is a nonempty subset of J [ I . Thus, the set of all coalitions equals 2J [I n f¿g. A proper coalition is a nonempty strict subset of J [ I . In subscripts and superscripts we write C instead of .J [ I / nC for each coalition C  J [ I , and k instead of .J [ I / n fkg and k instead of fkg for each k 2 J [ I . For each j 2 J; there exists j > 0; which is the upper bound on the output of good j . These upper bounds can stem, for Q example, from capacity constraints. We   let Yj D 0; j for each j 2 J and Y D j 2J Yj . Thus, Yj is the set of feasible outputs of producer j . Producer j 2 J has cost function cj W Yj ! 0.

M. Horniaˇcek, Cooperation and Efficiency in Markets, Lecture Notes in Economics and Mathematical Systems 649, DOI 10.1007/978-3-642-19763-5_2, c Springer-Verlag Berlin Heidelberg 2011 

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2 Model

Let ( X D x D .xi /i 2I

   D xj i j 2J

i 2I

2Y

#.I /

j

X

) xj i 2 Yj 8j 2 J

:

(2.1)

i 2I

Here, xj i is the quantity of good j 2 J purchased by buyer i 2 I . We will call an element of X a “vector of traded quantities.” Of course, in each vector of traded quantities, the sum of the purchases of all buyers from each producer j 2 J has to be a feasible output of j . Clearly, X is a nonempty and compact subset of #.J /#.I / . (It contains, for example, the zero vector in R#.J /#.I / . It is a subset of
X i 2I

! xj i

:

(2.2)

j 2J

That is,  .x/ is the output vector, in which each producer’s output equals the sum of his/her deliveries to buyers specified by x. We set Xi D Y for each i 2 I . It is the set of feasible vectors of the purchases of buyer i from the producers in J . The cost function of buyer i 2 I , ci W Xi ! 0. For each i 2 I , Ui W X !
Only the quantities of the inputs purchased from the producers in J are the arguments of fk . We assume that the other inputs are fixed unless buyer k decides to leave the analyzed market.

1

2.1 Stage Game

11

inverse demand function for the m-th P output of buyer i . If all buyers are retaile j . .x// xj i , where for each j 2 J , ers, then Ui has the form Ui .x/ D j 2J P e j W Y !


    Ui x C  ci xiC >

i 2I WxiC >0

X

   cj j x C :

j 2J Wj .x C />0

Let X max D arg max

8 <X :

Ui .x/ 

i 2I

X

  cj j .x/ 

X

ci .xi / j x 2 X

i 2I Wxi >0

j 2J Wj .x/>0

9 = ;

(2.3) Thus, each x max 2 X max is a vector of traded quantities that maximizes the surplus from the trade in the analyzed market. That is, it maximizes the difference between the sum of the buyers’ revenue (from the sale of goods produced from the inputs purchased in the analyzed market or from the sale of purchased goods in the retail market) and the sum of the production costs of active producers (i.e., assuming that the producers who did not withdraw from the analyzed market have positive outputs) and the costs of active buyers (i.e., assuming that the buyers who did not withdraw from the analyzed market purchase a positive amount of at least one good). Since the sum of the producers’ revenue in the analyzed market equals the sum of buyers’ expenditure in it, x max also maximizes the sum of the profits in the analyzed market. X max is nonempty despite the discontinuity of the objective function in the maximization program in (2.3) caused by the fact that the firms who withdraw from the analyzed market do not incur any fixed cost. We can solve the latter maximization problem in two steps as follows. In the first step, we solve for each D 2 2J [I with D \ J ¤ ¿ and D \ I ¤ ¿, the maximization problem D D max

8 P < ŒUi .x/  ci .xi /  i 2I \D

P j 2J \D

  9 cj j .x/ j =

: x 2 X; x D 08 .j; i / … .D \ J /  .D \ I / ; ji

:

(2.4)

12

2 Model

We denote the set of its solutions by X .D/ max . Here, D is the maximal surplus from the trade in the analyzed market among the firms in D (i.e., when all firms outside D withdraw from the analyzed market). In this auxiliary problem, we assume that no firm in D withdraws from the analyzed market (i.e., each firm in D incurs a fixed cost). Since X is a nonempty and compact set and the objective function in the maximization problem in (2.4) is continuous, X .D/ max is nonempty and compact. The max .D/ max following statement holds for each x .D/ : if there exists a nonempty  .D/  2 X C  D such that for each j 2 J , j x > 0 if and only if j 2 C , and for each .D/ i 2 I , xi > 0 if and only if i 2 C , then C 

h

X i 2C \I

   i X Ui x .D/  ci xi.D/ 

j 2C \J

   cj j x .D/ > D ;

where the second inequality follows from the fact that each firm has a positive fixed cost (see part (iii) of Assumption 2.1 and part (ii) of Assumption 2.2).  There˚ fore, for each F 2 arg max D j D 2 2J [I ; D \ J ¤ ¿; D \ I ¤ ¿ , we have X .F / max  X max . As such, in the second step, we set X

.max/

D

[

˚ F 2 arg max D

X .F / max j  : j D 2 2J [I ; D \ J ¤ ¿; D \ I ¤ ¿ (2.5)

Since X .F / max is nonempty and compact for each o n F 2 arg max D j D 2 2J [I ; D \ J ¤ ¿; D \ I ¤ ¿ and the set of such coalitions F is nonempty and finite, X max is nonempty and compact. We use   to denote the maximal value of the objective function in the  maximization problem in (2.3). Assumption 2.4 ˚  implies that  > 0. For each x 2 X , we let EJ .x/ D j 2 J j j .x/ > 0 , EI .x/ D fi 2 I j xi > 0g, and E .x/ D EJ .x/ [ EI .x/. It is plausible to assume that the prices in the analyzed market cannot be arbitrarily high. We denote the upper bound on a price of good j 2 J by pjmax .2 We assume that pjmax > 0 for each j 2 J and max 2 X max ; 8j 2 EJ .x max / ; pjmax xjmax i  max fUi .x/ j x 2 X g ; 8x n o max 8i 2 k 2 I j xjk >0 : (2.6)

2

Of course, from the modeling point of view, we need the upper bounds on the prices in the analyzed market in order to ensure that the stage game payoffs are bounded and the repeated game (described in Sect. 2.2) is continuous at infinity.

2.1 Stage Game

13

Thus, (taking into account part (ii) of Assumption 2.2) for each x max 2 X max , every j 2 EJ .x max /, and each i 2 I with xjmax > 0, no purchase, in which the i max buyer i buys the quantity of good j equal to xjmax i at price pj , allows him/her h i#.I / Q max . P is the set of to earn a nonnegative profit. We let P D j 2J 0; pj feasible vectors of the prices in the analyzed market. Each of these vectors has the    . form p D pj i j 2J i 2I Now we describe the sets of pure strategies and payoff functions in stage game G D hJ [ I; .Ak /k2J [I ; .gk /k2J [I i. We have (

    #.I / X j qj i 2 Yj Aj D pj i ; qj i i 2I 2 0; pjmax  Yj

) ; 8j 2 J

(2.7)

i 2I

and Ai D



 #.J / 0; pjmax  Yj ; 8i 2 I:

(2.8)

A pure strategy of producer j 2 J in G is a collection of contracts (one for each buyer) that he/she proposes to the buyers. Each contract proposal specifies a proposed price and a proposed quantity.3 Of course, the sum of the proposed quantities cannot exceed a producer’s capacity. There is no need to treat a producer’s decision not to propose a contract to some buyer as a special case. We identify such decision with the contract proposal .0; 0/. For each j 2 J , we identify the producers’ pure strategy .0; 0/#.I / with his/her decision to withdraw from the analyzed market. For such a decision, the producer does not incur a fixed cost and his/her payoff in G equals zero.4 A pure strategy of buyer i 2 I in G is a collection of contracts (one for each producer)  that he/she proposes to the producers. Thus, each ai 2 Ai has the form ai D pij ; qij j 2J . Again, we identify the contract proposal .0; 0/ made to j 2 J with the decision not to trade with producer j . Further, for each i 2 I , we identify the buyer’s pure strategy .0; 0/#.J / with his/her decision to withdraw from the analyzed market. For such a decision, the buyer does not incur a fixed cost and his/her payoff in G equals Qzero. We let A D k2J [I Ak . For each a 2 A and every .j; i / 2 J  I trade between producer j and buyer i takes place if and only if j ’s contract proposal to i differs with i ’s contract proposal to j , i.e., if and  from .0; 0/  and coincides  only if pj i ; qj i D pij ; qij ¤ .0; 0/. For every a 2 A, x .a/ is the vector of

3

The producers can sell their product to different buyers at different prices. This enables us to construct a punishment for a deviation by a proper coalition of buyers, which does not harm the buyers who did not deviate, in the repeated game. 4 We assume here that a firm can leave the analyzed market in one period. In the repeated game (described in the following section), we assume that a firm can enter the analyzed market in one period and the entry requires only paying the fixed cost. Our qualitative results hold if the exit from and entry into the analyzed market took more than one period and entry required the incurring of a sunk cost (exceeding the single period fixed cost).

14

2 Model

by strategy profile a. We define   generated   it by xj i.a/ D qj i if traded quantities pj i ; qj i D pij ; qij and by xj i .a/ D 0 if pj i ; qj i ¤ pij ; qij . Further, for each a 2 A, y .a/ D  .x .a// is the output vector generated by strategy profile a. Similarly, for each a 2 A, e p .a/ 2 P is the vector of prices at which the quantities given by x .a/ are traded. We use the convention that a zero price is assigned to each zero traded quantity. Thus, e pj i .a/ D pj i D pij if xj i .a/ > 0 and e p j i .a/ D 0 if xj i .a/ D 0. For k 2 J [ I , player k’s payoff function in G, gk W A ! <, is defined by 8j 2 J W gj .a/ D

X

  e p j i .a/ xj i .a/  cj yj .a/ ;

i 2I

  if 9i 2 I with pj i ; qj i ¤ .0; 0/ ;   8j 2 J W gj .a/ D 0; if pj i ; qj i D .0; 0/ 8i 2 I; 8i 2 I W gi .a/ D Ui .x .a// 

X

(2.9) (2.10)

e p j i .a/ xj i .a/  ci .xi .a// ;

j 2J

  if 9j 2 J with pij ; qij ¤ .0; 0/ ;

  8i 2 I W gi .a/ D 0; if pij ; qij D .0; 0/ 8j 2 J:

(2.11) (2.12)

We define function g W A ! <#.J [I / by g .a/ D .gk .a//k2J [I and let n o V D v 2 <#.J [I / j 9a 2 A such that g .a/ D v :

(2.13)

Clearly, taking into account (2.9)–(2.12), we have

V D

[ D22J [I

9 8   P ˆ pj i xj i  cj j .x/ 8j 2 J nD; > v 2 <#.J [I / j vj D > ˆ > ˆ > ˆ i 2I > ˆ P > ˆ < vi D Ui .x/  ci .xi /  pj i xj i 8i 2 I nD; vk D 08k 2 D; = j 2J

ˆ ˆ x 2 X; xj i D 08 .j;h i / 2 Œ.Ji \ D/  I  [ ŒJ  .I \ D/ ; ˆ ˆ ˆ ˆ : pj i 2 0; pjmax 8 .j; i / 2 J  I

> > > > > > ;

:

(2.14) That is, V is the set of payoff vectors in G that can result from the pure strategy profiles. It follows from (2.13) that V is nonempty. Clearly, V is a compact subset of <#.J [I / . (For each D 2 2J [I the corresponding set in the union in (2.14) is compact because X is a compact set, function cj , j 2 J is continuous, and functions Ui and ci , i 2 I are continuous.) Therefore, V has a strict Pareto efficient frontier that we denote by } .V /. Since V contains payoff vectors generated by the pure strategy profiles in G, it need not be convex. Thus, a vector in } .V / can be weakly (or even strictly) Pareto dominated by a convex combination of other vectors in } .V /. We let V C D conV.

2.2 Repeated Game

15

In order to enable the firms to achieve payoff vectors in V C nV , we also allow objectively correlated strategies (see [Aumann (1974)]; henceforth, we use only the term “correlated strategies”) in G. We assume that all firms can observe the signals of a public randomizing device generating uniformly distributed signals from interval Œ0; 1. A correlated strategy of firm k 2 J [ I is the mapping k W Œ0; 1 ! Ak . It assigns to each signal ! 2 Œ0; 1, firm k’s pure strategy in G. We Q denote the set of correlated strategies of firm k 2 J [ I by „ and let „ D k k2J [I „k and Q „C D k2C „k for each C 2 2J [I n f¿g. Of course, a pure strategy is a special case of a correlated strategy. (That is, ak 2 Ak is – from the point of view of the outcome of G – identical to k 2 „k as defined by k .!/ D ak for each ! 2 Œ0; 1.) Taking into account the Carathéodory theorem (see [Hildenbrand (1974), p. 37]), in order to obtain any vector in V C , it is enough to use a correlated strategy profile that uses at most # .J [ I / C 1 pure strategy profiles with a positive probability. With a slight abuse of the notation, we will use the symbol gk , k 2 J [ I also for the payoff functions in G defined on „ and the symbol g for the function that assigns to each  2 „, the vector of expected payoffs in G. C Since  V is nonempty and compact, V is also nonempty and  compact.  We use } V C to denote its strict Pareto efficient frontier and let }  V C D } V C \   #.J [I / . Thus, }  V C is the set of individually rational strictly Pareto efficient
2.2 Repeated Game The repeated game is a countable infinite repetition of G with discounting of future profits (without discounting of current profit). That is, G is played in periods numbered by positive integers. All firms use the common discount factor ı 2 .0; 1/. We denote the repeated game with the discount factor ı by  .ı/ and its game form by . The actions in  are observable. That is, at the end of each period t 2 N ,

16

2 Model

every firm k 2 J [ I observes – for every firm r 2 .J [ I / n fkg – r’s pure strategy in G in period t. We denote the set of histories in  by H and the set of histories leading to period t 2 N by H .t / . Each element of H .t / contains decisions made by the players and signals of the public randomizing S device prior to period t. H .1/ contains only the empty history. We set Hf D t 2N H .t / . Thus, Hf is the set of nonterminal (i.e., finite) histories. We use H1 to denote the set of terminal (i.e., ˚ infinite)histories. Hence, H D Hf [ H1 . Each h 2 H1 has the form h D a.t / ; ! .t / t 2N , where a.t / 2 A and ! .t / 2 Œ0; 1 for each t 2 N . For each ˚   h 2 H1 , x .h/ D x a.t / t 2N is the sequence of vectors of traded quantities gen˚   erated by terminal history h, and y .h/ D y a.t / t 2N is the sequence of output vectors generated by h. The behavioral strategy of firm k 2 J [ I in  is the function sk W Hf ! „k . It assigns to each nonterminal history h 2 Hf , one of k’s correlated strategies in G. We denote Qthe set of the behavioral Q strategies of firm k 2 J [ I in  by Sk and let S D k2J [I Sk and SC D k2C Sk for each C 2 2J [I n f¿g. For s 2 S and C 2 2J [I n f¿g we let sC D .sk /k2C . We use the term “strategy profile” for s .C / 2 SC .5 For every k 2 J [ I , the terms “strategy of k” and “strategy  profile of  .C / 2 SC , fkg” are equivalent. For C 2 2J [I n f¿g with # .C /  2, s .C / D sk k2C   .C / .C / . and D 2 2C n f¿g, we let sD D sk k2D We express the firms’ payoffs in  .ı/ (computed before they implement their behavioral strategies and a signal of the public randomizing device in the first period is observed) as their expected average discounted profits (i.e., their expected average discounted stage game payoffs). For each k 2 J [ I , function k W S !n< isofirm / k’s payoff function in  .ı/.6 That is, when s 2 S generates a sequence v.t k of expected stage game payoffs of firm k 2 J [ I , we have k .s/ D .1  ı/

X t 2N

/ ı t 1 v.t : k

t 2N

(2.15)

We define function  W S ! <#.J [I / by  .s/ D .k .s//k2J [I . With this formulation of the payoff functions, the set of the vectors of the firms’ payoffs in  .ı/ equals V C . For each h 2 H .t / with t 2 N n f1g, we use h to denote the subhistory of h leading to period t 1. The subhistory contains all information contained in h except for the firms’ decisions and the signal of the public randomizing device in period

5 We use the symbol sC for a profile of the behavioral strategies of the members of a coalition C determined by a previously mentioned profile of the behavioral strategies (of all firms) s 2 S and the symbol s .C / for a profile of the behavioral strategies of the members of a coalition C that is not determined by any previously mentioned s 2 S. 6 Of course, the functional values of k depend on ı. Nevertheless, in order to avoid unnecessary notational complication, we use the symbol k instead of k;ı .

2.3 Solution Concepts

17

˚ ˚ t 1 t 2 t  1. (That is, if t > 2 and h D a.n/ ; ! .n/ nD1 , then h D a.n/ ; ! .n/ nD1 . If h 2 H .2/ , then h D ¿.) For each nonterminal history h 2 Hf , .h/ .ı/ is the subgame of  .ı/ following h. .h/ is its game form. Since  .ı/ is a game with observable actions, each of its subgames is a proper subgame. For any set B defined for  .ı/ and any h 2 Hf , the symbol B.h/ stands for the restriction of B to subgame .h/ .ı/ (e.g., Hf .h/ is the set of nonterminal histories in .h/ .ı//. Similarly, for any function f defined for  .ı/ and any h 2 Hf , the symbol f.h/ stands for the restriction of f to subgame .h/ .ı/ (e.g., k.h/ is firm k’s payoff function in .h/ .ı/). For h 2 Hf and h0 2 Hf .h/ , the subgame of (the subgame) .h/ .ı/ following history h0 in .h/ .ı/ is identical to the subgame .h;h0 / .ı/ of  .ı/. We use the latter symbol to denote this. For h 2 Hf , C 2 .C / 2J [I n f¿g, s .C / 2 SC .h/ , and h0 2 Hf .h/ , s.h 0 / is the restriction of strategy profile s .C / of coalition C in subgame .h/ .ı/ to subgame .h;h0/ .ı/. We assume that when a firm contemplates (either unilaterally or as a member of a coalition of deviating firms) a deviation from its behavioral strategy in a subgame .h/ .ı/ with h 2 H .t / ; it does so before the signal of the public randomizing device for period t is observed. This assumption corresponds to the excluding of a signal of the public randomizing device in period t from h 2 H .t / and defining a firm’s behavioral strategy as a mapping from the set of nonterminal histories to the set of its stage game correlated strategies (instead of as a mapping from the Cartesian product of the set of nonterminal histories with Œ0; 1 to the set of its stage game pure strategies). This allows the grand coalition to use any profile of stage game correlated strategies – and hence, (if V is not convex) to achieve a stage game payoff vector belonging to V C nV – in the first period of a subgame. (If the firms would have contemplated a deviation by the grand coalition only after observing the signal of the public randomizing device in the first period of a subgame, they would not have been able to use a profile of stage game correlated strategies with non-singleton support in the first period of a subgame.) The ability of the grand coalition to weakly Pareto improve the vector of current period stage game payoffs by a deviation contemplated after the current period signal of the public randomizing device is observed implies its ability to do so also by a deviation contemplated before the current period signal of the public randomizing device is observed, but not vice versa. An analogous comment holds for the deviations by any non-singleton coalition.

2.3 Solution Concepts As already stated in the Introduction, the SRPE and the SSPE are the solutions concepts that we apply to  .ı/. Definition 2.1. A strict renegotiation-proof equilibrium of  .ı/ is a strategy profile s  2 S with the following properties.

18

2 Model

(i) There do not exist a nonterminal history h 2 Hf , a firm k 2 J [ I , and its strategy sk 2 Sk.h/ such that k.h/



 sk ; sk.h/



   > k.h/ s.h/ :

(2.16)

(ii) There do not exist a nonterminal history h 2 Hf and a strategy profile s 2 S.h/ such that    k.h/ .s/  k.h/ s.h/ ; 8k 2 J [ I (2.17) and

   9 k 2 J [ I with k.h/ .s/ > k.h/ s.h/ :

(2.18)

It follows from part (i) of Definition 2.1 that an SRPE is a subgame perfect equilibrium. Part (ii) implies that all continuation equilibrium payoff vectors7 are strictly Pareto efficient. Definition 2.2. A strict strong perfect equilibrium of  .ı/ is a strategy profile s  2 S with the following properties. (i) There do not exist a nonterminal history h 2 Hf , a coalition C 2J [I n f¿; J [ I g, and its strategy profile s .C / 2 SC .h/ such that k.h/

      s .C / ; sC  k.h/ s.h/ ; 8k 2 C .h/

and 9 k 2 C with k.h/

      s .C / ; sC > k.h/ s.h/ : .h/

2

(2.19)

(2.20)

(ii) There do not exist a nonterminal history h 2 Hf and a strategy profile s 2 S.h/ such that    k.h/ .s/  k.h/ s.h/ ; 8k 2 J [ I (2.21) and

   : 9 k 2 J [ I with k.h/ .s/ > k.h/ s.h/

(2.22)

Thus, a strategy profile s  is an SSPE if no coalition in no subgame can increase the expected average discounted profit of at least one of its members without decreasing the expected average discounted profit of any other member by a deviation from the prescriptions of s  . That is, no coalition in no subgame has a deviation leading to a vector of expected average discounted profits of its members that weakly Pareto dominates the vector of their expected average discounted profits generated by s  . Part (ii) of Definition 2.2 implies that all continuation equilibrium payoff

7

Let s  2 S be an equilibrium strategy profile and let h 2 Hf . Then, the continuation  equilibrium

  , and the continuation equilibrium payoff vector is .h/ s.h/ . in subgame .h/ .ı/ is s.h/

References

19

vectors in an SSPE are strictly Pareto efficient. Part (i) also holds for singleton coalitions. Therefore, it implies that each SSPE is a subgame perfect equilibrium. Thus, each SSPE is an SRPE. Definition 2.2 does not exclude the possibility that a deviation by a coalition from its continuation equilibrium strategy profile in some subgame leaves the payoffs of all its members unchanged. Thus, taking into account the meaning of the term “strict Nash equilibrium” (which is a Nash equilibrium where each unilateral deviation by any player decreases his/her payoff), we should use the term “semi-strict strong perfect equilibrium” instead of SSPE. Nevertheless, in order to avoid terminological complexities, we use the latter term.

References Aumann, R.J.: “Subjectivity and Correlation in Randomized Strategies,” Journal of Mathematical Economics 1 (1974), 67-96. Hildenbrand, W.: Core and Equilibria of a Large Economy. Princeton, N.J.: Princeton University Press, 1974.

Chapter 3

Existence of an SRPE and an SSPE

In this chapter, we prove the sufficient conditions for the existence of an SRPE and an SSPE in the model described in Chap. 2. We let Assumptions 2.1–2.4 hold.

3.1 Existence of an SRPE Let vmax D max fUi .x/  ci .xi / j x 2 X g ; 8i 2 I; i    ˚ vmax D max pjmax yj  cj yj j yj 2 Yj ; 8j 2 J; j 8 9 < = X max vmin U ; 8i 2 I; D .x/  c .x /  p x j x 2 X i i ji i j : i ;

(3.1) (3.2) (3.3)

j 2J

  D cj j ; 8j 2 J: vmin j

(3.4)

Here, (taking into account that pjmax is the upper bound on the price of good j 2 J ) vmax (vmin ) is the highest (lowest) stage game profit of firm k 2 J [ I . k k   P Proposition 3.1. For each v 2 }  V C with k2J [I vk D   there exists ı 2 .0; 1/ such that for every ı 2 .ı; 1/,  .ı/ has an SRPE with equilibrium payoff vector equal to v :   P Proof. A. Preliminaries. Take (arbitrary) v 2 }  V C with k2J [I vk D   .   Using the definition of }  V C (see the second-last paragraph in Sect. 2.1), v is a strictly Pareto efficient payoff vector in  .ı/. Further, there exists   2 „, n 2 f1; : : : ; # .J [ I / C 1g, a.r/ 2 A, r 2 f1; : : : ; ng, and  D .r /r2f1;:::;ng 2 4n such that   0, g .  / D v , and for each r 2 f1; : : : ; ng,   leads to the occurrence of a.r/ . Moreover, for each j 2 J with  with  probability r .r/  pj i a D 0 for every i 2 I and each vj D 0, we have xj i a.r/ D 0 and e r 2 f1; : : : ; ng. Similarly, for each i 2 I with vi D 0, we have xj i .a.r/ / D 0 M. Horniaˇcek, Cooperation and Efficiency in Markets, Lecture Notes in Economics and Mathematical Systems 649, DOI 10.1007/978-3-642-19763-5_3, c Springer-Verlag Berlin Heidelberg 2011 

21

22

3 Existence of an SRPE and an SSPE

P and e p j i .a.r/ / D 0 for every j 2 J and each r 2 f1; : : : ; ng. As k2J [I vk D   ,   .r/ 2 X max for each r 2 f1; :::ng. Since the firms have a positive fixed cost, for x a each r 2 f1; : : : ; ng, a.r/ prescribes the withdrawal from the analyzed for  .r/market  every i 2 I with x a D 0. We every j 2 J with yj a.r/ D 0 as well as for    i  .r/ can assume without loss of generality that pj.r/ i ; qj i

pij.r/ ; qij.r/ D .0; 0/   for each .j; i / 2 J  I and every r 2 f1; : : : ; ng with xj i a.r/ D 0. Let       S S EJ D nrD1 EJ x a.r/ , EI D nrD1 EI x a.r/ , and E  D EJ [ EI . Thus, vk > 0 for each k 2 E  . From this and condition (2.6), it follows that for each     p j i a.r/ < pjmax i 2 EI , there exists r 2 f1; : : : ; ng such that xi a.r/ > 0 and e for each j 2 J . Further, for each j 2 Ej , there exist r 2 f1; : : : ; ng and i 2 I such          that xj i a.r/ > 0 and e p j i a.r/ > cj yj a.r/ =yj a.r/ .  ˚ We set  .r/ D ! 2 Œ0; 1 j   .!/ D a.r/ for each r 2 f1; : : : ; ng, and let function 1 W Œ0; 1 ! f1; : : : ; ng assign to every ! 2 Œ0; 1, r 2 f1; : : : ; ng that satisfies ! 2  .r/.  P     P Let ˛j D  nrD1 r i 2I e p j i a.r/ xj i a.r/ i for each j 2 EJ and h     P P p j i a.r/  pjmax xj i a.r/ for each i 2 EI . Clearly, ˛i D nrD1 r j 2J e ˛k < 0 for each k 2 E  . For k 2 E  and T 2 N , consider inequality D

     vk C ı 1  ı T ˛k  0: .1  ı/ vmax k

(3.5)

For the limit case ı D 1, weak inequality (3.5) holds as equality. Differentiating itsleft-hand side with respect to ı and evaluating the derivative at ı D 1, we get  1 max    vmax T ˛ ˛  v . This expression is positive if and only if T > v  v k k k k k k . We set  1   ˚ ˚  ˛k j k 2 E  : T D max min n 2 N j n > vk  vmax k

(3.6)

Then, for each k 2 E  , there exists ık1 2 .0; 1/ such that for every ı 2 Œık1 ; 1/, (3.5) holds for firm k. We set ı1 D max fık1 j k 2 E  g. Further, for each k 2 E  , there exists ık2 2 .0; 1/ such that for every ı 2 .ık2 ; 1/,    1  ı T vk C ˛k C ı T vk > 0:

(3.7)

(For the limit case ı D 1, (3.7) holds. Its left-hand side is continuous in ı.) We set ı2 D max fık2 j k 2 E  g and ı D max fı1 ; ı2 g.  For each D 2 2E satisfying the condition   i 2 I \ D & j 2 J \ D H) xj i a.r/ D 08r 2 f1; : : : ; ng

(3.8)

and every r 2 f1; : : : ; ng, let a.r/ j D 2 A differ from a.r/ in only two respects: e p j i a.r/ j D D 0 for each j 2 J \ D and every i 2 EI with

3.1 Existence of an SRPE

23

    p j i a.r/ j D D pjmax for each i 2 I \ D and every j 2 EI xj i a.r/ > 0, and e  .r/  > 0. (The other prices are equal to those prescribed by a.r/ and with xj i a    .r/  x a j D D x a.r/ .) Note that a.r/ j ¿ D a.r/ for each r 2 f1; : : : ; ng.  For each D 2 2E satisfying the condition (3.8), we define  j D 2 „ by .r/  j D .!/ D a j D, where r D 1 .!/, for every ! 2 Œ0; 1. B. Description of strategy profile s  . For each h 2 Hf n f¿g let L .h/ be the set of firms in E  that according to h, deviated from the prescriptions of s  in the .t / last period recorded in h. Thus, for each t 2 N n f1g n o and every h 2 H , we   .t 1/   have L .h/ D k 2 E  j ak ¤ sk .h / ! .t 1/ . (Here, sk .h / is firm k’s stage game correlated strategy prescribed by sk after the subhistory h of h, and   sk .h / ! .t 1/ is k’s stage game pure strategy prescribed by sk .h / for signal ! .t 1/ of the public randomizing device.) In the description of s  , we use for each k 2 E  , function k W Hf ! f0; 1; : : : ; T g ; which we define recursively below. For every t 2 N , each h 2 H .t / , and every k 2 E  , k .h/ is the number of remaining periods, including period t, for which the punishment of firm k (that started before period t or will start in period t) has to last. We set k .¿/ D 0 for each k 2 E  . Next, suppose that S there exists t 2 N n f1g t 1 such that we have already defined k .h0 / for each h0 2 mD1 H .m/ and every   .t / k 2 E . Consider h 2 H . For each j 2 EJ , we proceed as follows. If j … L .h/  .r/   > 0, then and there do not ˚ existi 2 L .h/  \ EI and r 2 f1; : : : ; ng with xj i a j .h/ D max j .h /  1; 0 . If j 2 L .h/ and there do not exist i 2 L .h/ \ EI   and r 2 f1; : : : ; ng with xj i a.r/ > 0, then j .h/ D T . If there exist i 2   L .h/ \ EI and r 2 f1; : : : ; ng with xj i a.r/ > 0, then j .h/ D 0. For each i 2 EI , we proceed as follows. If i … L .h/ and there do not exist j 2 L .h/ \ EJ   and r 2 f1; : : : ; ng with xj i a.r/ > 0, then i .h/ D max f i .h /  1; 0g. If i 2 L .h/, then i .h/ D T . If i … L .h/ and there exist j 2 L .h/ \ EJ and   r 2 f1; : : : ; ng with xj i a.r/ > 0, then i .h/ D 0. Now we describe the strategy profile s  . For each t 2 N and every h 2 H .t / , we let D .h/ D fk 2 E  j k .h/ > 0g and set s  .h/ D  j D .h/.1 Clearly, .s/ D v . For each r 2 f1; : : : ; ng, the sum of the firms’ stage game  profits given by a.r/ j D is the same for each D 2 2E satisfying the condition (3.8). The probabilities with which stage game pure strategy profiles a.r/ j D are used under s  depend only on r, and not on D. Therefore, X k2J [I

  X 

k.h/ s.h/ vk D   ; 8h 2 Hf : D

(3.9)

k2J [I

C. Strategy profile s  is an SRPE of  .ı/ for ı 2 .ı; 1/. Since  .ı/ is continuous at infinity,2 with regard to unilateral deviations, it suffices to show that there do not

1 2

Note that the definition of function k , k 2 E  ensures that D .h/ satisfies (3.8). The firms’ stage game payoffs are bounded, and the future stage game payoffs are discounted.

24

3 Existence of an SRPE and an SSPE

exist k 2 J [ I and h 2 Hf such that k can increase his/her expected average discounted profit in .h/ .ı/ by a unilateral single period deviation in its first period. C1. Unilateral deviations by producers. A unilateral deviation by j 2 J nEJ cannot lead to him/her trading with any buyer . Thus, for each ! 2 Œ0; 1, his/her unilateral deviation to any stage game pure strategy decreases his/her single period profit from zero to cj .0/ < 0, and does not affect future play. Therefore, the producer’s unilateral deviation to any stage game correlated strategy decreases his/her expected average discounted profit in the subgame of  .ı/ at the beginning of which the deviation takes place. Now, take (arbitrary) j 2 EJ and h 2 Hf . We distinguish between the two cases. First, j .h/ D 0. Without a deviation, j ’s expected average discounted profit in .h/ .ı/ is at least vj . A deviation gives him/her an expected stage game profit in the first period of .h/ .ı/ that is bounded from above by vmax j , and then he/she is punished for T periods and has the expected stage game profit vj C ˛j . Thus, a deviation gives him/her an expected average discounted profit in .h/ .ı/ that is not higher than    T vj C ˛j C ı T C1 vj  .1  ı/ vj C ıvj D vj ; (3.10) C ı 1  ı .1  ı/ vmax j where the inequality follows from (3.5). Therefore, a deviation does not increase j ’s expected average discounted profit in .h/ .ı/. Second, j .h/ > 0. Without a deviation, j ’s expected average discounted profit in .h/ .ı/ equals       1  ı j .h/ vj C ˛j C ı j .h/ vj  1  ı T vj C ˛j C ı T vj > 0: (3.11) (The latter payoff is positive by (3.7).) A deviation gives j a nonpositive expected stage game profit in the first period of .h/ .ı/. (A withdrawal from the analyzed market gives him/her zero stage game profit. His/Her failure to withdraw from the analyzed market at ! 2 Œ0; 1, for which sj.h/ prescribes a withdrawal, gives him/her stage game profit cj .0/ < 0. Any other pure strategy in G gives him/her a negative profit because the price of his/her output equals zero and he/she has a positive fixed cost.) Then, the punishment (giving him/her the stage game expected profit vj C ˛j for T periods) is resumed. Thus, a deviation gives him/her the expected average discounted profit in .h/ .ı/ that is not higher than       ı 1  ı T vj C ˛j C ı T C1 vj < 1  ı T vj C ˛j C ı T vj :

(3.12)

Therefore, a deviation decreases j ’s expected average discounted profit in .h/ .ı/. C2. Unilateral deviations by buyers. A unilateral deviation by i 2 I nEI cannot lead to him/her trading with any producer. Thus, for each ! 2 Œ0; 1, the buyer’s unilateral deviation to any stage game pure strategy decreases his/her single period

3.1 Existence of an SRPE

25

profit from zero to ci .0/ < 0 and does not affect future play. Therefore, his/her unilateral deviation to any stage game correlated strategy decreases his/her expected average discounted profit in the subgame of  .ı/ at the beginning of which the deviation takes place. Now, take (arbitrary) i 2 EI and h 2 Hf . We distinguish between the two cases. First, i .h/ D 0. Without a deviation, i ’s expected average discounted profit in .h/ .ı/ is at least vi . A deviation gives him/her an expected stage game profit in the first period of .h/ .ı/ that is bounded from above by vmax i , and then he/she is punished for T periods and provided the expected stage game profit vi C ˛i . Thus, a deviation gives him/her an expected average discounted profit in .h/ .ı/ that is not higher than    T C ı 1  ı .1  ı/ vmax vi C ˛i C ı T C1 vi  .1  ı/ vi C ıvi D vi ; (3.13) i where the inequality follows from (3.5). Therefore, a deviation does not increase i ’s expected average discounted profit in .h/ .ı/. Second, i .h/ > 0. Without a deviation, i ’s expected average discounted profit in .h/ .ı/ equals 

1  ı i .h/

     vi C ˛i C ı i .h/ vi  1  ı T vi C ˛i C ı T vi > 0:

(3.14)

(The latter payoff is positive by (3.7).) A deviation gives him/her a nonpositive expected stage game profit in the first period of .h/ .ı/. (A withdrawal from the analyzed market gives him/her zero stage game profit. His/Her failure to withdraw from the analyzed market at ! 2 Œ0; 1, for which si.h/ prescribes a withdrawal, gives him/her the stage game profit ci .0/ < 0. Any other pure strategy in G gives him/her a nonpositive profit because he/she pays price pjmax as defined in (2.6) for each good j 2 EJ that he/she buys, he/she has a positive fixed cost, and Ui .0/ D 0.) Then, the punishment (giving him/her the stage game expected profit vi C ˛i for T periods) is resumed. Thus, a deviation gives him/her an expected average discounted profit in .h/ .ı/ that is not higher than       ı 1  ı T vi C ˛i C ı T C1 vi < 1  ı T vi C ˛i C ı T vi :

(3.15)

Therefore, a deviation decreases i ’s expected average discounted profit in .h/ .ı/. C3. Payoff vector in each is strictly Pareto efficient. This claim follows  subgame  from the fact that .h/ s.h/ 2 V C for each h 2 Hf and from (3.9). t u The strategy profile described in the proof of 3.1 is based on the use   Proposition of the same vectors of the traded quantities (x a.r/ , r 2 f1; : : : ; ng) with the corresponding probabilities (r , r 2 f1; : : : ; ng) in each period, after any history. They are used even in the punishments for unilateral deviations by firms with a positive equilibrium payoff, and only the prices, at which a deviator trades in the analyzed

26

3 Existence of an SRPE and an SSPE

market, change in a way that harms him/her.3 (Firms with zero equilibrium payoff cannot increase their stage game payoff by any unilateral deviation.) These modified prices are in effect long enough to wipe out any single period gain from a deviation. Since the sum of the firms’ expected stage game profits is maximized in each period (even during punishments), the sum of their expected average discounted profits (which equals the average discounted sum of their expected stage game profits) is maximized in each subgame. Thus, the continuation equilibrium payoff vector is strictly Pareto efficient in each subgame.

3.2 Existence of an SSPE In this section, we first deal with a monopsonistic market, where I D f# .J / C 1g. In this case, under the conditions given below in Lemma 3.1, the stage game G has a strict strong equilibrium (henceforth, SSE). An infinite repetition of the latter is an SSPE of  .ı/ for each ı 2 .0; 1/. Definition 3.1. A correlated strategy profile   2 „ is an SSE of G if (i) there do not exist C 2 2J [I n fJ [ I; ¿g and  .C / 2 „C such that

and

      gk   ; 8k 2 C; gk  .C / ; C

(3.16)

     9k 2 C with gk  .C / ; C > gk   ;

(3.17)

(ii) there does not exist  2 „ such that

and

  gk ./  gk   ; 8k 2 J [ I;

(3.18)

  9k 2 J [ I with gk ./ > gk   :

(3.19)

Thus, a correlated strategy profile is an SSE of G if no coalition of players can, by a deviation, increase the expected payoff of at least one of its members without decreasing the expected payoff of another member. Part (i) of Definition 3.1 when applied to singleton coalitions implies that an SSE of G is a Nash equilibrium of G.

3

For simplicity, the strategy profile described in the proof of Proposition 3.1 prescribes for each punished producer trading at zero prices. When the assumptions of the latter proposition are satisfied, we can construct an SRPE in which all trading takes place at positive a   prices.  For  example,  punished producer j with v can sell his/her output at price cj yj a.r/ =yj a.r/ when j > 0   the vector of traded quantities x a.r/ , r 2 f1; : : : ; ng, is prescribed during his/her punishment. Nevertheless, this could lead to an increase in the duration of punishment, as well as to an increase in the lower bound on the values of the discount factor for which a strategy profile is an SRPE.

3.2 Existence of an SSPE

27

Part (ii) implies that the expected equilibrium payoff vector in an SSE is strictly Pareto efficient with respect to V C . An SSE satisfying Definition 3.1 would remain an SSE even if we allowed the firms’ mixed strategies in G (i.e., randomizations over their pure strategies based on the signals of their private randomizing devices, where the randomizations of the different firms are independent) and the firms’ strategies that assign their mixed strategies to the signals of the public randomizing device. A mixed strategy of firm k 2 J [ I with a finite support is a best response against ak 2 Ak if and only if each pure strategy in its support is a best response against ak . If a mixed strategy of firm k 2 J [ I with an infinite support is a best response against ak 2 Ak , then each subset of its support with a positive measure contains k’s pure strategy that is a best response against ak . Suppose that there exist an SSE   of G , k 2 J [ I , a 2 A, and a mixed strategy k of firm k such that a occurs with a positive probability when all firms follow   and k is a better response against ak than ak . Then, there exists ak‘ 2 Ak that (belongs to the support of k and) is a better response against ak than ak . Thus, a correlated strategy of firm k, which differs from k only in that it prescribes ak‘ for each ! 2 Œ0; 1 for which   prescribes a, is a  better response against k than k . This contradicts the assumption that   is an SSE of G. Next, suppose that there exists an SSE   of G , a coalition C with # .C / 2 Œ2; # .J [ I /  1 \ N , aC 2 AC , and a profile  .C / of the mixed strategies of the firms in C such that aC occurs with a positive probability when  the firms in J nC follow C and  .C / gives against aC a vector of the expected payoffs of the firms in C that weakly Pareto dominates the one generated by the prescriptions of C . Then, there exists a profile  .C / of correlated strategies of firms in C that assigns a positive probability to at most # .C / C 1 elements of AC and gives against aC the same vector of expected payoffs as  .C / . Thus, a profile of the correlated strategies of the firms in C , which differs from C only in that it prescribes  .C / instead of the prescription of C for the subset of Œ0; 1 for which  C prescribes aC , violates condition (i) of Definition 3.1. Of course, it can happen that  .C / does not weakly Pareto improve the vector of the expected payoffs of the firms in C against any aC that occurs with a positive probability when the firms in  , but it weakly Pareto improves the vector of the expected payoffs J nC follow C  . Then,  .C / changes the vector of the expected payoffs of the firms in C against C of the firms in C against at least two profiles of the pure strategies of the firms in  J nC that occur with a positive probability when the latter firms follow C . For each such aC , there exists a profile  .C / .aC / of the correlated strategies of the firms in C that gives the same vector of the expected payoffs of the firms in C against aC as  .C / (and it assigns a positive probability to at most # .C / C 1 elements of AC ). Thus, a profile of the correlated strategies of the firms in C , which differs from C only in that it prescribes  .C / .aC / instead of the prescription of  C for the subset of Œ0; 1 for which C prescribes aC , violates condition (i) of Definition 3.1. As far as the deviations by the grand coalition are concerned, any profile of mixed strategies gives a vector of the expected payoffs belonging to V C . Any element of V C can be generated by a profile of the correlated strategies.

28

3 Existence of an SRPE and an SSPE

Lemma 3.1. Assume that # .I / D 1, v 2 V C , and that there exist   2 „, n 2 f1; : : : ; # .J [ I / C 1g, a.r/ 2 A, r 2 f1; : : : ; ng , and  2 4n with   0 such that v D g .  /, and for each r 2 f1; : : : ; ng ; a.r/ occurs with probability r when the firms follow   ,   gk a.r/  0; 8k 2 J [ I; 8r 2 f1; : : : ; ng ;

(3.20)

and D 

X

  gk a.r/ ; 8D 2 2J [I with f# .J / C 1g  D; 8r 2 f1; : : : ; ng :

k2D

(3.21) Then, there exists an SSE of G with the equilibrium vector of expected payoffs equal to v . Proof. Consider (arbitrary) v 2 V C ,   2 „, n 2 f1; : : : ; # .J [ I / C 1g, a.r/ 2 A, r 2 f1; : : : ; ng, and  2 4n that satisfy the assumptions of Lemma 3.1. We can assume without loss of generality that for each r 2 f1; : : : ; ng, aj.r/ D .0; 0/  and a#.J prescribes the making of the contract proposal .0; 0/ to producer j if /C1  .r/  D 0. (If   does not have the latter property, we can replace it xj #.J /C1 a by another profile of correlated strategies that has the latter property and satisfies all assumptions of Lemma 3.1.4 This follows from two facts. First, prices proposed by the firms, which do not trade together, for their bilateral trade do not affect the payoffs. Second, if producer j and buyer i do not trade together, then replacing the traded quantity proposed by j to i and replacing the traded quantity proposed by i to j by zero does not change the vector of the traded quantities or payoffs.) We show that   is an SSE of G. Note that (3.20) implies v  0 and (3.21) implies D 

X

vk ; 8D 2 2J [I with # .J / C 1 2 D & # .D/ > 1:

(3.22)

k2D max Take (arbitrary) x max P 2 X max . Since P (3.21) also holds for D D E .x / and v  0, we have   D k2D vk D k2J [I vk . That is, v maximizes the sum ofthe firms’ payoffs over V C .This further implies that for each r 2   f1; :: : ; ng, .r/ x a 2 X max and gj a.r/ D 0 for each j 2 J with xj #.J /C1 a.r/ D 0. This also implies that the grand coalition cannot, by a deviation, increase the expected payoff of at least one of its members without decreasing the expected payoff of some other member.

4

The fact that we do not include the latter property among the assumptions of Lemma 3.1 is the only reason why we do not formulate the last sentence of the latter lemma as “Then,   is an SSE of G.”

3.2 Existence of an SSPE

29

˚  Consider (arbitrary) j 2 J , his/her unilateral deviation to j 2 „j n j , and ! 2 Œ0; 1 with j .!/ ¤ j .!/. Suppose that all other firms stick to the prescrip tions of j . Then, j .!/ either leads to j ’s withdrawal from the analyzed market (that gives him/her payoff 0  gj .  .!//) or to no trade between him/her and the monopsonist without his/her withdrawal from the analyzed  market  (thatgives  him/her payoff cj .0/ < 0  gj .  .!//). Therefore, gj j ; j  vj , and hence, j ’s unilateral deviation does not increase his/her expected payoff. The same conclusion holds when he/she deviates as a member of any coalition of producers, because deviations by other producers cannot affect his/her expected payoff from his/her deviation strategy. Next consider a monopsonist’s unilateral deviation to n o  #.J /C1 2 „#.J /C1 n #.J /C1  and consider (arbitrary) ! 2 Œ0; 1 with #.J /C1 .!/ ¤ #.J .!/ and /C1

g#.J /C1

     #.J .!/ ;  .!/ ¤ g#.J /C1   .!/ : #.J /C1 /C1

Take r 2 f1; : : : ; ng satisfying   .!/ D a.r/ . Suppose that all produc ers stick to the prescriptions of #.J . Then, #.J /jC1 .!/ either leads to /C1 the monopsonist’s withdrawal from the analyzed market (that gives him/her   payoff 0  g#.J /C1 a.r/ ) or to no trade between him/her and any producer without his/her withdrawal from the analyzed market (that gives him/her  payoff c#.J /C1 .0/ < 0  g#.J /C1 a.r/ ) or to trade with each pro   ducer in a nonempty set K  EJ x a.r/ . (He/She cannot trade with any   .r/  j 2 J nEJ x a because such j sticks to the prescription of j and makes the contract proposal .0;  .r/  0/ to the monopsonist.)  .r/ In the latter case, he/she buys quantity x a for price e p a from each j 2 K. Therefore, j #.J /C1 j #.J /C1    .r/   for each j 2 K. Thus, setting D gj a gj #.J /C1 .!/ ; #.J /C1 .!/ D D K [ f# .J / C 1g and using (3.21), we have   .!/ ;  .!/ #.J #.J /C1 /C1   X   D  gj #.J /C1 .!/ ; #.J /C1 .!/

g#.J /C1



j 2K

D D 

X

    gj a.r/  g#.J /C1 a.r/ :

j 2K

   ;   v#.J /C1 . Thus, a monopThis further implies that g#.J /C1 #.J /C1 #.J /C1 sonist’s unilateral deviation does not increase his/her expected payoff. Finally, consider a deviation by a ˚coalition C  J [ I with # .J / C 1 2 C  and C \ J ¤ ¿ to  .C / 2 „C n C . Suppose that the firms in J nC stick

30

3 Existence of an SRPE and an SSPE

 to the prescriptions of C . Take (arbitrary) r 2 f1; : : : ; ng and ! 2 Œ0; 1 with  .r/ .C / .!/ does not lead to the monopsonist’s trade with any  .!/ D a . If  producer, C the sum of the payoffs that is not higher than  the firms in P it gives .r/ .C / g .!/ leads to the monopsonist’s trade with each a . If  0  k k2C producer in a nonempty set M     J , we set K D M nC . (K can be empty. Clearly, K  EJ x a.r/ because the monopsonist cannot trade with any      j 2 J n EJ x a.r/ [ C – each such j sticks to the prescription of j and      makes the contract proposal .0; 0/.) Then, gj  .C / .!/ ; C .!/ D gj a.r/   .C /  for each j 2 K. Further, gj  .!/ ; C .!/  0 for every j 2 .C \ J / nM because such j does not trade. Thus, setting D D M [f# .J / C 1g and using (3.20) and (3.21) (and noting that C nD D .C \ J / nM and DnC D K), we have

X

gk

   .!/   .C / .!/ ; C

k2C

X

gk

k2C \D

 D 

   .!/  .C / .!/ ; C

X

j 2DnC

D D  

X

gj

    .C / .!/ ; C .!/

    X gj a.r/  gk a.r/

j 2DnC

k2C \D

X

  X   gk a.r/ D gk   .!/ ;

k2C

k2C

where weomit the  sum over if D  C . This further implies that P DnC P .C /     g ;  v . Thus, a deviation by C cannot increase k k2C k2C k C the expected payoff of at least one of its members without decreasing the expected payoff of some other member. t u The sufficient condition for the existence of an SSE in G given in Lemma 3.1 is easy to understand. Each pure strategy profile that occurs under a candidate equilibrium correlated strategy profile (i.e., under a correlated strategy profile which we want to be an SSE of G) with a positive probability has to be individually rational. Moreover, it has to give each coalition containing the monopsonist and at least one producer (i.e., each coalition that can trade without cooperation of any firm that does not belong to it) a sum of payoffs that is not lower than the sum of payoffs it can achieve without trading with any producer who does not belong to it. The latter condition ensures that no such coalition can increase the expected payoff of at least one of its members without decreasing the expected payoff of another member. This also ensures that (for any signal of the public randomizing device), the monopsonist cannot increase his/her payoff by a unilateral deviation that leads to his/her trade with at least one producer. (Each producer, with whom he/she would trade, would earn the same payoff as in the case when no deviation takes place. Thus, a deviation would have to increase the sum of the payoffs of the firms that participate in the trade.) Individual rationality implies that the monopsonist cannot increase his/her payoff by a unilateral withdrawal from the analyzed market (or any other unilateral

3.2 Existence of an SSPE

31

deviation that prevents him/her from trading with any producer) and no producer can increase his/her payoff by a unilateral deviation or as a member of a deviating coalition of producers. The following lemma shows that the sufficient condition in Lemma 3.1 is – with the exception of the requirement that the equilibrium profile of the correlated strategies uses with a positive probability at most # .J [ I / C 1 pure strategy profiles in G – also necessary for the existence of an SSE of G. Lemma 3.2. Assume that # .I / D 1 and v 2 V C is the vector of the equilibrium expected payoffs in an SSE of G. Then, v  0, (3.22) holds, and there exists   2 „ such that v D g .  / and each a 2 A that occurs with a positive probability when the firms follow   satisfies (3.20), (3.21), x .a / 2 X max , and P   max 2 X max . Further, k2E .x max / gk .a / D  for every x X

vk D   ; 8x max 2 X max :

(3.23)

k2E .x max /

Proof. Let v 2 V C be the equilibrium vector of the expected payoffs in an SSE of G. Since v 2 V C and the firms can use only pure or correlated strategies (and a pure strategy is a special case of a correlated strategy),5 there exists   2 „ with g .  / D v that is an SSE of G. Take (arbitrary) a 2 A that occurs with a positive probability when the firms follow   . If we had gk .a / < 0 for some k 2 J [I , firm k can increase its expected payoff by (unilateral deviation prescribing) withdrawal from the analyzed market when   prescribes a and by sticking to the prescriptions of k when   prescribes any other pure strategy profile. Therefore, g .a /  0. Since this holds for each a 2 A that occurs with a positive probability when the firms follow   , we have v  0. Suppose P that there exists a non-singleton coalition D such that # .J / C 1 2 D  .D/ and D > 2 <#.D/ satisfying v.D/  k2D gk .a /. Then, there exists v P .D/  .gk .a //k2D and k2D vk D D : Moreover, (using the definition of D in (2.4)) there exists x 2 X such that  each j 2 J nD and P xj #.J /C1 D 0 for D D U#.J /C1 .x/  c#.J /C1 .x/  j 2D\J cj xj #.J /C1 . Let h  i  .D/ pj D cj xj #.J /C1 C vj =xj #.J /C1

5

The arguments in the proof also apply to a profile of mixed strategies after the following modifications. First, we replace a firm’s payoff from a profile of pure strategies occurring with a positive probability in the equilibrium by its expected payoff from its pure strategy that belongs to the support of its equilibrium mixed strategy. Second, we replace the sum of the payoffs of the members of a non-singleton coalition containing the monopsonist from a profile of pure strategies occurring with a positive probability in equilibrium by the sum of their expected payoffs from a profile of their pure strategies belonging to the Cartesian product of the supports of their mixed strategies. These remarks hold under the assumption that members of a deviating coalition can communicate the signals of their private randomizing devices to other members.

32

3 Existence of an SRPE and an SSPE

  and aj D pj ; xj #.J  j 2 J \ D. Further, let a#.J /C1 prescribe the  /C1 for each contract proposal pj ; xj #.J /C1 to each j 2 J \ D and the contract proposal    for each k 2 D. Let .0; 0/ to every j 2 J nD. Then, gk .am /m2D ; aD D v.D/ k .D/ .D/  2 „D satisfies  .!/ D .am /m2D for each ! 2 Œ0; 1 with   .!/ D a , .D/  and  .!/ D  .!/ for every ! 2 Œ0; 1 with   .!/ ¤ a . Then, we have  .D/   D gk  ; D > vk for each k 2 D. This contradiction along with the fact that   is an SSE of G proves that (3.21) holds for each a 2 A that occurs with a positive probability when the firms follow   . Then, (3.22) follows from (3.21). For each x max 2 X max and every a 2 A that occurs with a positive probability when the firms follow   , applying (3.21) to D D E .x max / and using the fact that g .a /  0 yields X k2E .x max /

X     gk a D gk a D   : k2J [I

This implies (3.23), and also that x .a / 2 X max for each a 2 A that occurs with a positive probability when the firms follow   : t u The following proposition shows that the payoff vector satisfying the conditions stated in Lemma 3.1 is the equilibrium vector of the expected average discounted payoffs in an SSPE of  .ı/ for any discount factor. Proposition 3.2. Assume that #.I / D 1 and v 2 V C ; that there exists   2 „, n 2 f1; : : : ; # .J [ I / C 1g, a.r/ 2 A, r 2 f1; : : : ; ng, and  2 4n with   0 such that v D g .  /, and that for each r 2 f1; : : : ; ng, a.r/ occurs with probability r when the firms follow   and it satisfies (3.20) and (3.21). Then, for each ı 2 .0; 1/ ; there exists an SSPE of  .ı/ with the equilibrium vector of expected average discounted payoffs equal to v . Proof. Consider (arbitrary) v 2 V C ,   2 „, n 2 f1; : : : ; # .J [ I / C 1g, a.r/ 2 A, r 2 f1; : : : ; ng, and  2 4n satisfying the assumptions of Proposition 3.2. The latter assumptions coincide with the assumptions of Lemma 3.1. As already noted in the proof of Lemma 3.1, we can assume without loss of generality that for each  r 2 f1; : : : ; ng, aj.r/ D .0; 0/ and a#.J prescribes the making of the contract /C1 proposal .0; 0/ to producer j if xj #.J /C1 D 0. We have shown in the proof of Lemma 3.1 that   is an SSE of G. We define s  2 S by s  .h/ D   for each h 2 Hf . Thus, s  leads to the infinite repetition of   in each subgame of . Since   is a (correlated) Nash equilibrium of G, no firm can increase its expected average discounted profit in any subgame of  .ı/ by a unilateral deviation. No producer can increase his/her expected average discounted profit in any subgame of  .ı/ as a member of a deviating coalition of producers because the deviations by other producers cannot affect his/her expected average discounted profit from his/her deviation strategy.

3.2 Existence of an SSPE

33

As shown in the proof of Lemma 3.1, v maximizes the sum of the firms’ payoffs over V C . Therefore, in any subgame of  .ı/ and for any ı 2 .0; 1/, a deviation by the grand coalition (regardless of the number of periods it lasts) does not increase the sum of the firms’ expected average discounted profits. Thus, it cannot increase the expected average discounted profit of at least one firm without decreasing the expected average discounted profit of some other firm. Finally, consider (arbitrary) non-singleton coalition C  J [ I with # .J / C 1 2 C . As shown in the proof of Lemma 3.1, a deviation by such coalition C does not increase the sum of the stage game expected profits of its members. Therefore, in any subgame of  .ı/ and for any ı 2 .0; 1/, a deviation by such C (regardless of the number of periods it lasts) does not increase the sum of the expected average discounted profits of the members of C . Thus, it cannot increase the expected average discounted profit of at least one firm in C without decreasing the expected average discounted profit of some other firm in C . t u The following proposition shows that in each SSPE of  .ı/ and in every subgame .h/ of  .ı/, the vector of the expected average discounted profits is individually rational, maximizes the sum of the firms’ expected average discounted profits, and satisfies (3.23). Moreover, for each non-singleton coalition containing the monopsonist, the sum of the expected average discounted profits of its members is not lower than the maximal sum of their stage game profits that they can achieve without trading with the producers outside the coalition. The latter property implies that each profile of stage game pure strategies, which occurs with a positive probability in .h/ .ı/ when the firms follow the prescriptions of their equilibrium strategies in it, generates a vector of traded quantities that maximizes the sum of the firms’ stage game profits. All these properties would continue to hold (and the arguments in the proof of the following proposition would remain valid) if we allowed mixed strategies in the stage game (along with the assigning of stage game mixed strategies to nonterminal histories by the behavioral strategies in the repeated game). Therefore, the results in Chap. 4 that build on Proposition 3.3 would continue to hold even if we allowed mixed strategies in the stage game. Proposition 3.3. Assume that # .I s  2 S be an SSPE of  / D  1. Let ı 2 .0; 1/, P     .ı/, h 2 Hf , and v D .h/ s.h/ . Then, v  0, k2J [I vk D   , and v satisfies (3.22) and (3.23). Moreover, x .a / 2 X max for each a 2 A that occurs  with a positive probability in .h/ .ı/ when the firms follow the prescriptions of s.h/ .  Proof. Take (arbitrary)   ı 2 .0; 1/ ; s 2 S that is an SSPE of  .ı/, and h 2 Hf .

 Let v D .h/ s.h/ . If we have vk < 0 for some k 2 J [ I , firm k can increase its expected average discounted profit in .h/ .ı/ by withdrawing from the analyzed market with probability one in each period after every history in .h/ .ı/ leading to it. Therefore, v  0. If v does not satisfy (3.22) for some non-singleton  coalition D with # .J / C 1 2 D, there exists v.D/ 2 <#.D/ satisfying v.D/  vk k2D and P .D/ D D . Thus, D can increase the expected average discounted profit of k2D vk each of its members in .h/ .ı/ by using strategy ak , k 2 D described in the proof

34

3 Existence of an SRPE and an SSPE

of Lemma 3.2, with probability one in each period after every history in .h/ .ı/ leading to it. Thus, v satisfies (3.22). For each x max 2 X max , applying (3.22) to D D E .x max / and using the fact that v  0 yields X

vk D

k2E .x max /

X

vk D   :

(3.24)

k2J [I

The second equality in (3.24) implies that the claim in the last sentence of Proposition 3.3 holds. t u Now we give two examples of a stage game in which a vector of payoffs satisfying the assumptions of Proposition 3.2 exists. In the first, all firms have to participate in trade in order to maximize the sum of the stage game profits of all firms (that results from the unique vector of the traded quantities belonging to X max /. In the second example, X max has four elements and each producer can be replaced by another (that is identical to him) in the set of firms that participate in trade in the maximization of the sum of the stage game profits of all firms. Therefore, (the stage game payoff vector maximizing the sum of the profits of all firms) v assigns zero profit to each producer (i.e., the monopsonist captures the whole surplus from trade). Despite this, v is sustainable as the vector of the equilibrium expected average discounted profits in an SSPE of  .ı/ for any ı 2 .0; 1/. Since there exists only one buyer, a unilateral deviation by any producer, who is (according to the prescriptions of the equilibrium strategy profile) supposed to trade, can lead only to his/her nonparticipation in the trade. Thus, this cannot give him/her a positive stage game profit, and hence, he/she need not be punished. Example 3.1. J D f1; 2; 3; 4g and I D f5g. Producers 1 and 3 produce the same good, which we refer to as type one good; producers 2 and 4 produce  the same2 good, which we refer to as type two good. We have Y D Œ0; 5 and c j j yj D 10yj C 25 P for each j 2 J , c5 .x5 / D j 2J xj 5 C 15. The monopsonist is a retailer and U5 .x/ D P1 .x/ .x15 C x35 / C P2 .x/ .x25 C x45 / ;

(3.25)

where P1 W X ! Œ0; 121 defined by P1 .x/ D max f121  10 .x15 C x35 /  5 .x25 C x45 / ; 0g

(3.26)

is the inverse demand function for the type one good, and P2 W X ! Œ0; 121 defined by P2 .x/ D max f121  5 .x15 C x35 /  10 .x25 C x45 / ; 0g (3.27) is the inverse demand function for the type two good. We let pjmax D 326 for each j 2 J. In this example, the sum of the firms’ stage game profits is maximized when each producer supplies 1:5 unit of his/her output to the monopsonist. That is, X max D f.1:5; 1:5; 1:5; 1:5/g. The maximized sum of profits is 245 financial units. Thus,   D f1;2;3;4;5g D 245.

3.2 Existence of an SSPE

35

When each type of good is produced and supplied to the monopsonist by only one producer (and the remaining two producers withdraw from the analyzed market), the maximal sum of the firms’ stage game profits equals 223 financial units. Thus, f1;2;5g D f1;4;5g D f2;3;5g D f2;4;5g D 223. When one type of good is produced and supplied to the monopsonist by both of its producers and the other type is produced and supplied to the monopsonist by only one of its producers (and the other producer withdraws from the analyzed market), the maximal sum of the firms’ stage game profits equals (approximately) 237:27 financial units. Thus, f1;2;3;5g D .1;3;4;5/ D f1;2;4;5g D f2;3;4;5g D 237:27. When one type of good is produced and supplied to the monopsonist by both of its producers and the other type is not produced at all (with the withdrawal of both its producers from the analyzed market), the maximal sum of the firms’ stage game profits is 175 financial units. Thus, f1;3;5g D f2;4;5g D 175. When only one type of good is produced and supplied to the monopsonist by only one of its producers (and the remaining three producers withdraw from the analyzed market), the maximal sum of the firms’ stage game profits equals 140 financial units. Thus, fj;5g D 140 for each j 2 J . Consider the payoff vector v D .6:5; 6:5; 6:5; 6:5; 219/. It is easy to verify that it satisfies (3.22). We have v D g .a /, where aj D .36; 1:5/ for each j 2 J and a5 D .36; 1:5/j 2J . In this case, v is generated by a pure strategy profile in G, and as such, (3.22) implies (3.21). Thus, v and a satisfy all assumptions of Proposition 3.2. Example 3.2. J D f1; 2; 3; 4g and I D f5g. Producers 1 and 3 produce the same good, which we refer to as type one good; producers 2 and 4 produce the  same  good, which we refer to as type two good. WePhave Yj D Œ0; 100 and cj yj D 10yj C 100 for each j 2 J and c5 .x5 / D j 2J xj 5 C 300. The monopsonist uses the goods purchased from the producers in J to produce a good that he/she sells in a market, in which he/she is a monopolist. His/Her production function f W X ! Œ0; 200 has the form f .x/ D min fx15 C x35 ; x25 C x45 g. The inverse demand function for his/her product, P W Œ0; 200 ! Œ0; 102 has the form P .Q/ D max f102  Q; 0g, where Q is his/her output. This yields U5 .x/ D maxf102  minfx15 C x35 ; x25 C x45 g; 0g minfx15 C x35 ; x25 C x45 g: (3.28) Let pjmax D 66 for each j 2 J . Clearly, since both producers of each type of good have the same constant marginal cost and positive fixed cost, the maximization of the sum of the firms’ stage game profits requires that each type of good is produced by only one producer (and it does not matter which one). Further, given the form of the monopsonist’s production function, the outputs of both types of goods have to be equal. The maximization of the sum of the firms’ stage game payoffs under these constraints gives X max D f.40; 40; 0; 0/ ; .40; 0; 0; 40/ ; .0; 40; 40; 0/ ; .0; 0; 40; 40/g

(3.29)

36

3 Existence of an SRPE and an SSPE

and   D f1;2;5g D f1;4;5g D f2;3;5g D .3;4;5/ D 1100. When two producers produce one or both types of good, the optimal output of each type of good (that maximizes the sum of the firms’ stage game profits for a given set of producers who do not withdraw from the analyzed market) equals 40 but the sum of the firms’ stage game profits is reduced by the fixed costs of the redundant producer(s). Thus, D[f5g D 1000 for each D  J with # .D/ D 3, D \ f1; 3g ¤ ¿, and D \ f2; 4g ¤ ¿, as well as f1;2;3;4;5g D 900. Since the monopsonist needs a positive amount of each type of good in order to produce something and obtain a positive revenue, we have fj;5g D 400 for each j 2 J and f1;3;5g D f2;4;5g D 500. For each element of X max , there exists another with a different set of active producers. Therefore, there is only one payoff vector that satisfies (3.22), namely, v D .0; 0; 0; 0; 1100/. It can be generated by any profile of correlated strategies in G that assigns a positive probability only to the profiles of pure strategies that satisfy x .a/ 2 X max and g .a/ D v . Each such a prescribes the trading between the monopsonist and both active producers at price equal to 12:5 financial units. (An example of such a 2 A is given by a1 D a2 D .12:5; 40/, a3 D a4 D .0; 0/, and a5 D ..12:5; 40/ ; .12:5; 40/ ; .0; 0/ ; .0; 0//.) Clearly, each such profile of correlated strategies satisfies (3.20) and (3.21). Now we proceed to a sufficient condition for the existence of an SSPE of  .ı/ for ı 2 .0; 1/ close enough to one when there is more than one buyer. For each C 2 2J [I n fJ [ I g with C \ J ¤ ¿ and C \ I ¤ ¿, let

vmax C

#9 8 "  >  ˆ P < P ŒU .x/  c .x / C P max pj xjk  cj j .x/ = i i i D max i 2I \C j 2J \C k2I nC > ˆ ; : jx2X (3.30)

and 9 8 " # >  ˆ P < P U .x/  c .x /  P p max x cj j .x/ = i i i kj  k vmin ; D min C i 2I \C j 2J \C k2J nC > ˆ ; : jx2X (3.31) where we omit the sum over I nC if I  C and the sum over J nC if J  C . Since h i max max min the price of good j 2 J is from the interval 0; pj , vC (vC ) is the maximal (minimal) sum of the stage game payoffs of the firms in coalition C . Proposition 3.4. Assume that there exists v 2 V C with. v  0,   2 „, n 2 f1; : : : ; # .J [ I / C 1g, a.r/ 2 A, r 2 f1; : : : ; ng, and  2 4n with   0 such   .r/ that occurs with and  v D g . /, for each r 2 f1; : : : ; ng, a   r  S probability x a.r/ 2 X max . Further, assume that vi > 0 for each i 2 nrD1 EI x a.r/ ;      gj a.r/ > 0; 8r 2 f1; : : : ; ng ; 8j 2 EJ x a.r/ ;

(3.32)

3.2 Existence of an SSPE

37

      and for each r 2 f1; : : : ; ng and every .j; i / 2 EJ x a.r/  EI x a.r/ ,      such that pj.r/ 2 0; e p j i a.r/ if xj i a.r/ > 0 and there exists a price pj.r/ i i   D 0 if xj i a.r/ D 0, pj.r/ i X i 2EI .x .a.r/ //

X

     .r/ pj i xj i a.r/  cj yj a.r/ ;    8r 2 f1; : : : ; ng ; 8j 2 EJ x a.r/ ;

pj.r/ i xj i

i 2K





a

.r/

X i 2EI .x .a.r/ //



 cj

X

(3.33)

!   .r/ xj i a

i 2K .r/ pj i xj i

     a.r/  cj yj a.r/ ;

   8r 2 f1; : : : ; ng ; 8j 2 EJ x a.r/ ; n   o 8¿ ¤ K  i 2 I j xj i a.r/ > 0 ;

(3.34)

and

v.r/ C;M

" # 9 8 P P > ˆ max > ˆ > ˆ .x/  c .x /  p x U i i i ki > ˆ k > ˆ > ˆ i 2I \M k2J nC > ˆ " # > ˆ > ˆ > ˆ   P P > ˆ ˆ .r/ > ˆ C e p ki a Ui .x/  ci .xi /  xki > > ˆ > ˆ > ˆ i 2.C \I /nM Wxi >0 " k2J nC > ˆ # > ˆ > ˆ > ˆ   P P = < .r/ p x  c .x/ C j j jk jk D max j 2J \M k2I nC ˆ # > " > ˆ > ˆ > ˆ     P P > ˆ ˆ .r/ > ˆC e p jk a xjk  cj j .x/ j > > ˆ > ˆ > ˆ > ˆ j 2.C \J /nM Wj .x/>0 k2I nC ˚ > ˆ    > ˆ .r/ > ˆ x 2 X; x 2 x a ; 0 > ˆ ji ji > ˆ > ˆ > ˆ > ˆ 8 .j; i / 2 Œ..J \ C / nM /  I  [ ŒJ  ..I \ C / nM / ; > ˆ  .r/  ; : 8 .j; i / 2 .J nC /  .I nC / xj i D xj i a  X   gk a.r/ ; k2C

8r 2 f1; : : : ; ng ; 8C 2 2J [I n fJ [ I g with C \ J ¤ ¿ & C \ I ¤ ¿; 8M 2 2C ;

(3.35)

38

3 Existence of an SRPE and an SSPE

with strict if M ¤ ¿ and there exists either i 2 M \ I and j 2 J nC  inequality  with xj i a.r/ > 0 or j 2 J \ M and i 2 I nC with xj i a.r/ > 0.6 The sum over J nC (I nC , I \ M , .C \ I / nM , J \ M , .C \ J / nM ) is omitted here if J  C (I  C , I \ M D ¿, C \ I  M , J \ M D ¿, C \ J  M .) Then, there exists ı 2 .0; 1/such that for each ı 2 .ı; 1/,  .ı/ has an SSPE with the vector of the equilibrium expected average discounted payoffs equal to v . Proof. A. Preliminaries. Take (arbitrary) v 2 V C ,   2 „, n 2 f1; : : : ; #.J [ I / C 1g, a.r/ 2 A, r 2 f1; : : : ; ng, and  2 4n satisfying the assumptions of Proposition 3.4. We can assume without loss of generality that for each r 2 f1; : : : ; ng and   every .j; i / 2 J I with xj i a.r/ D 0, aj.r/ prescribes the making of the contract

the making of the contract proposal proposal .0; 0/ to buyer i and ai.r/ prescribes       S S .0; 0/ to producer j . Let EJ D nrD1 EJ x a.r/ , EI D nrD1 EI x a.r/ , E  D EJ [ EI ,

max vC j k 2 C 2 2J [I n fJ [ I g ; max ; D max v ; max wmax k k C \ J ¤ ¿; C \ I ¤ ¿

(3.36)

8k 2 E  ; wmin k

D min

vmin k ; min

J [I vmin n fJ [ I g ; C j k 2C 22 C \ J ¤ ¿; C \ I ¤ ¿



8k 2 E  ; ˛i D

n X

; (3.37)

2

3      X 4r e p j i a.r/  pjmax xj i a.r/ 5 ; 8i 2 EI ;

rD1

(3.38)

j 2J

9       r e p j i a.r/  pjmax xj i a.r/ j = ˇi D max ; 8i 2 EI ; (3.39)  .r/  ; : rD1 >0 j 2 EJ ; 9 r 2 f1; : : : ; ng with xj i a " # n      X X .r/ .r/ .r/ (3.40) r ; 8j 2 EJ ; e pj i a  pj i xj i a ˛j D  8 <

rD1

n P

i 2I

and 8 9   n  .r/   .r/  = .r/ < P pj i a xj i a  pj i j r e ˇj D max ; 8j 2 EJ : (3.41)  .r/  : rD1 ; i 2 EI ; 9 r 2 f1; : : : ; ng with xj i a >0 6

As in the case of the maximum in (2.3), the maximum in (3.35) is well defined despite the discontinuity of the objective function caused by the disregarding of the fixed costs of the buyers in .C \ I / nM with zero vector of purchases and the fixed costs of the producers in .C \ J / nM with zero output. We compute the maximum for each possible subset of C nM (including the empty set), assuming that all firms in it withdraw from the analyzed market, and then take the maximum of all such maxima.

3.2 Existence of an SSPE

39

It follows from the fact that vi > 0 for each i 2 EI , part (ii) of Assumption 2.3, and (2.6) that for each i 2 EI , there exists j 2 J and r 2 f1; : : : ; ng with e p j i .a.r/ / < pjmax and xj i .a.r/ / > 0. This implies that ˛i  ˇi < 0 for .r/

each i 2 EI . It follows from the properties of price pj i j 2 EJ : For k 2 E  and T 2 N consider inequality

that ˛j  ˇj < 0 for each

     wmin C ı 1  ı T ˇk  0: .1  ı/ wmax k k

(3.42)

For the limit case ı D 1, (3.42) holds as equality. Differentiatingits left hand side  with respect to ı and evaluating the derivative at ı D 1, we get  wmax  wmin  k k Tˇk . This expression is positive if and only if  1  max ˇk : T > wmin k  wk We set  1   ˚ ˚  max ˇk j k 2 E  : T D max min n 2 N j n > wmin k  wk

(3.43)

  Then, for each k 2 E  , there exists ık‘ 2 .0; 1/ such that for every ı 2 ık“ ; 1 , ˚  (3.42) holds for firm k. We set ı ‘ D max ık‘ j k 2 E  . Further, for each k 2 E  , there exists ık“ 2 .0; 1/ such that for every ı 2 ık“ ; 1 ,    1  ı T vk C ˛k C ı T vk > 0:

(3.44)

(For the limit case ı D  (3.44) holds. Its left-hand side is continuous in ˚ 1, inequality ı.) We set ı “ D max ık“ j k 2 E  . It follows from (3.35) that for each C 2 2J [I n fJ [ I g with C \ J ¤ ¿ and C \ I ¤ ¿ and every M 2 2C , we have vC;M D

n X rD1

r v.r/ C;M 

X

vk ;

(3.45)

k2C

with strict inequality if M  ¤ ¿ and there exist either i 2 M \ I , j 2 J nC , and r 2 f1; : : : ; ng with xj i a.r/ > 0, or j 2 J \ M; i 2 I nC , and r 2 f1; : : : ; ng   with xj i a.r/ > 0. For each C 2 2J [I n fJ [ I g with C \ J ¤ ¿ and  C \ I ¤ ¿ and every M 2 C 2 , which contains either i 2 M \ I such that xj i a.r/ > 0 for some j 2 J nC   and r 2 f1; : : : ; ng, or j 2 M \ J such that xj i a.r/ > 0 for some i 2 I nC and r 2 f1; : : : ; ng, there exists ıC;M 2 .0; 1/ such that for each ı 2 ŒıC;M ; 1/,

40

3 Existence of an SRPE and an SSPE

vC;M

  X  1  ıT vk C k2C

!

X

˛k

C ıT

k2M \E 

X

vk :

(3.46)

k2C

(Using (3.45) for M with the above described properties, for the limit case ı D 1, (3.46) holds as a strict inequality. Its right-hand side is continuous in ı.) We set 8 <

9 ıC;M j M 2 2C ; = ıC D max 9 .j; i / 2 Œ.J \ M /  .I nC / [ Œ.J nC /  .I \ M / ;   : ; & r 2 f1; : : : ; ng with xj i a.r/ > 0 8C 2 2J [I n fJ [ I g with C \ J ¤ ¿ & C \ I ¤ ¿: We set n n oo ı D max ı ‘ ; ı “ ; max ıC j C 2 2J [I n fJ [ I g ; C \ J ¤ ¿; C \ I ¤ ¿ : (3.47) As in the proof of Proposition 3.1, we set o n  .r/ D ! 2 Œ0; 1 j   .!/ D a.r/ for each r 2 f1; : : : ; ng, and let function 1 W Œ0; 1 ! f1; : : : ; ng assign to every ! 2 Œ0; 1, r 2 f1; : : : ; ng that satisfies ! 2  .r/. E For each every r 2 f1; : : : ; ng, let a.r/ j D differ from a.r/  .r/D 2 2 and  .r/  only in max > 0 and that e p j i a j D D pj for each .j; i / 2 J  .I \ D/ with xj i a   .r/  .r/  .r/ e p j i a j D D pj i for each .j; i / 2 .J \ D/  .I nD/ with xj i a > 0.     (The other prices are those prescribed by a.r/ and x a.r/ j D D x a.r/ .) Note that a.r/ j ¿ D a.r/ for each r 2 f1; : : : ; ng.  For each D 2 2E , define  j D by  j D .!/ D a.r/ j D, where r D 1 .!/. Clearly,  j ¿ D   . It follows from (3.33) (and the fact that for each r 2 f1; : : : ; ng and every j 2    J nEJ x a.r/ , aj.r/ prescribes the withdrawal from the analyzed market) that    gj a.r/ j D  0; 8D 2 2E ; 8j 2 J; 8r 2 f1; : : : ; ng and



gj . j D/  0; 8D 2 2E ; 8j 2 J:

(3.48)

(3.49)

Further, (3.34) implies that P i 2K

    e p j i a.r/ j D xj i a.r/  cj



P

i 2K



xj i a

.r/





   gj a.r/ j D

   ˚    8r 2 f1; : : : ; ng ; 8j 2 EJ x a.r/ ; 8¿ ¤ K  i 2 I j xj i a.r/ > 0 ; 

8D 2 2E :

(3.50)

3.2 Existence of an SSPE

41

B. Description of strategy profile s  . We say that a producer j and a buyer i deviated in a coordinated way in a single period from s  2 S (that will be defined – together with the function k , k 2 E  – recursively below) when (for an actually observed signal ! of the public randomizing device) j made to i the same contract proposal as i made to j and the latter contract proposal differed from the one prescribed by s  and proposed a positive traded quantity. For each h 2 Hf n f¿g, let LI .h/ be the set of buyers in EI who, according to h, deviated (unilaterally or as members of a deviating coalition) from the prescriptions of s  2 S in the last period recorded in h. Thus, for each t 2 N n f1g and every h 2 H .t / , we have n  o LI .h/ D i 2 EI j ai.t 1/ ¤ si .h / ! .t 1/ : (Here, si .h / is buyer i ’s stage game correlated strategy prescribed by si after   subhistory h of h and si .h / ! .t 1/ is i ’s stage game pure strategy prescribed by si .h / for signal ! .t 1/ of the public randomizing device.) Further, for each h 2 Hf n f¿g, let LJ .h/ be the set of producers in EJ who according to h, deviated in a coordinated way with at least one buyer from the prescriptions of s  in the last period recorded in h. We let L .h/ D LJ .h/ [ LI .h/. In the description of s  , we use for each k 2 E  , function k W Hf ! f0; 1; : : : ; T g that we define recursively below. For every t 2 N , each h 2 H .t / , and every k 2 E  , k .h/ is the number of remaining periods, including period t, for which the punishment of firm k (that started before period t or will start in period t) has to last. We set k .¿/ D 0 for each k 2 E  . Next,   suppose that there exists t 2 N n f1g such that we have already defined k h‘ for each k 2 E  and every S 1 H .m/ . Consider (arbitrary) h 2 H .t / . Take (arbitrary) k 2 E  . If h‘ 2 tmD1 k … L .h/ then k .h/ D max f k .h /  1; 0g. If k 2 L .h/ then k .h/ D T . Now, we describe the strategy profile s  . For each h 2 Hf , we let D .h/ D fk 2 E  j k .h/ > 0g and set s  .h/ D  j D .h/.   Clearly, .s  / D v . We have x a.r/ j D 2 X max for each r 2 f1; : : : ; ng and  every D 2 2E . Therefore, X k2J [I

  X 

k.h/ s.h/ vk D   ; 8h 2 Hf : D

(3.51)

k2J [I

C. Strategy profile s  is an SSPE of  .ı/ with ı 2 .ı; 1/ : C1. Deviations by coalitions of producers. When only producers deviate, a deviation by any k 2 J in any subgame cannot affect the expected average discounted profit of any j 2 J n fkg in any subgame. (A deviation by k cannot affect j ’s single period profit in the period in which it takes place and it has no impact on

42

3 Existence of an SRPE and an SSPE

the values of function m , m 2 E  .) Therefore, in order to show that a coalition of producers cannot weakly Pareto improve the vector of the expected average discounted profits of its members in any subgame, it is enough to show that no producer can increase his/her expected average discounted profit in any subgame by a unilateral deviation. As  .ı/ is a game continuous at infinity (see part C of the proof of Proposition 3.1), the single period deviation principle applies to unilateral deviations. For a unilateral deviation by j 2 J nEJ , the arguments in part C1 of the  proof of Proposition 3.1 also hold here. Thus, consider (arbitrary) D 2 2E and a unilateral single period deviation by a producer j 2 EJ in the first period of a subgame .h/ .ı/ with s  .h/ D  j D. Since this unilateral deviation does not change the functional value of function j , it suffices to show that it cannot increase j ’s expected profit in the first period of .h/ .ı/. In order to do so, it is enough to show that j cannot increase his/her stage game profit deviation   by a unilateral from a.r/ j D for any r 2 f1; : : : ; ng. If j … EJ x a.r/ j D , the argument is the same as that for the producers in J nEJ . Thus, take (arbitrary) r 2 f1; : : : ; ng    with j 2 EJ x a.r/ j D . Then, j ’s unilateral deviation from a.r/ j D (which changes the set of buyers with whom he/she trades – otherwise there would be no change in his/her stage game payoff) either leads to j ’s withdrawal from the analyzed market, to no trade between him/her and any buyer without his/her withdrawal from ˚ the analyzed  market,  or to  his/her trade with the buyers in a nonempty set K  i 2 I j xj i a.r/ j D > 0 . (He/She cannot trade with any i 2 I with   xj i a.r/ j D D 0 because such i makes the contract proposal .0; 0/ to him/her.) His/Her profit resulting from the deviation (when all other firms stick to the prescrip-  tions of a.r/ j D) is 0  gj a.r/ j D in the first case, cj .0/ < 0  gj a.r/ j D   in the second case,and using (3.50), it is not higher than gj a.r/ j D in the third case. C2. Deviations by coalitions of buyers. In order to show that a coalition of buyers cannot weakly Pareto improve the vector of expected average discounted profits of its members in any subgame, it is enough to show that no buyer can increase his/her expected average discounted profit in any subgame by a unilateral deviation. (For each subgame, if no buyer can increase his/her expected average discounted profit in it by a unilateral deviation, then no coalition of buyers can increase the sum of their expected average discounted profits. This follows from the summing of the inequalities in (3.52) and (3.54) over all buyers in the intersection of a deviating coalition and EI . The deviations by the buyers outside EI cannot affect the stage game payoffs of the buyers in EI . Further, the deviations by the buyers in EI cannot affect the stage game payoffs of the buyers outside EI .) As already noted, the single period deviation principle applies to unilateral deviations. (This principle applies also to the sum of the expected average discounted profits of a coalition because such a sum is a scalar.7 ) For a unilateral deviation by i 2 I nEI the arguments in 7

The argument for this claim is analogous to the one for unilateral deviations. Of course, in general the single period deviation principle does not apply to the deviations by coalitions. Nevertheless, it applies to the sum of the payoffs of the members of a deviating coalition because the latter sum is a scalar.

3.2 Existence of an SSPE

43

part C2 of the proof of Proposition 3.1 also hold here. Thus, consider a unilateral single period deviation by a buyer i 2 EI in the first period of a subgame .h/ .ı/ : We consider two distinguished cases. First, i .h/ D 0. Without a deviation, i ’s expected average discounted profit in .h/ .ı/ is at least vi > wmin i . A deviation gives him/her expected single period profit in the first period of .h/ .ı/ that is bounded from above by vmax  wmax and then i i he/she is punished for T periods by the single period expected profit vi C ˛i  vi C ˇi . Thus, a deviation gives him/her an expected average discounted profit in.h/ .ı/ that is not higher than    T C ı 1  ı .1  ı/ vmax vi C ˛i C ı T C1 vi i     .1  ı/ wmax C ı 1  ı T vi C ˇi C ı T C1 vi i   T  .1  ı/ wmin C ı 1  ı vi C ı T C1 vi < vi ; i

(3.52)

where the second weak inequality follows from (3.42). Therefore, a deviation decreases i ’s expected average discounted profit in .h/ .ı/. Second, i .h/ > 0. Without a deviation, i ’s expected average discounted profit in .h/ .ı/ equals 

1  ı i .h/

     vi C ˛i C ı i .h/ vi  1  ı T vi C ˛i C ı T vi > 0:

(3.53)

(The right-hand side of (3.53) is positive by (3.44).) A deviation gives him/her a nonpositive single period expected profit in the first period of .h/ .ı/. (His/Her withdrawal from the analyzed market leads to zero single period profit. When he/she does not withdraw from the analyzed market but does not trade with any producer, his/her single period profit is ci .0/ < 0. Taking into account (2.6), any stage game action that leads to him/her trading with at least one producer in EJ gives him/her a nonpositive single period profit because he/she pays the price pjmax for each good j 2 EJ that he/she buys.) Then, the punishment (giving him/her single period expected profit vi C ˛i for T periods) is resumed. Thus, a deviation gives him/her expected average discounted profit in .h/ .ı/ that is not higher than       ı 1  ı T vi C ˛i C ı T C1 vi < 1  ı T vi C ˛i C ı T vi :

(3.54)

Therefore, a deviation decreases i ’s expected average discounted profit in .h/ .ı/. C3. Deviations by a coalition C  J [I with C \J ¤ ¿ and C \I ¤ ¿. Consider (arbitrary) h 2 Hf and coalition C  J [ I with C \ J ¤ ¿ and C \ I ¤ ¿. It is enough to show that a deviation by C cannot increase the sum of the expected average discounted profits of the firms in C in .h/ .ı/.  .ı/ is a game continuous at infinity and the sum of the expected average discounted profits of the firms in C in .h/ .ı/ is a scalar. Therefore, it suffices to show that for any h 2 Hf , a single

44

3 Existence of an SRPE and an SSPE

period deviation by C in the first period of .h/ .ı/ cannot increase the sum of the expected average discounted profits of the members of C in .h/ .ı/.8 Of course, a change in prices, at which the producers in C trade with the buyers in C in the first period of .h/ .ı/, does not change the sum of the single period profits of the members of C . Moreover, a punishment that it triggers cannot increase the sum of the expected average discounted profits of the members of C in .h/ .ı/. Hence, it is enough to concentrate on the other changes brought about by the deviation by C . Let D D fk 2 E  j k .h/ > 0g. Then, when the members of C contemplate a single period deviation in the first period of .h/ .ı/, they know that without their deviation, a.r/ j D will occur with probability r . (Recall that they contemplate a deviation before they observe the signal of the public randomizing device in the first period .h/ .ı/.) Let Z be a nonempty subset of C whose members actually deviate in the first period of .h/ .ı/. That is, firm k 2 C belongs to Z if and only if for at least one r 2 f1; : : : ; ng and some interval ‘ .r/ of  .r/ with a positive length, it intends to use a stage game pure strategy different from ak.r/ j D. (A deviation at only one element of  .r/ has no affect on the single period profit in the current period or in the following periods because it takes place with zero probability.) If (for the given equilibrium strategies of the firms outside C and the deviation strategies of the firms in C n fi g) a deviation strategy of a buyer i 2 C \ I leads (for each ! 2 Œ0; 1) to the same quantities purchased by him/her and the same prices for all goods traded by him/her in positive quantities as the prescriptions of si , then we do not include i in Z. (Such deviation by i does not increase the sum of the expected single period profits of the members of C in the first period of .h/ .ı/. Thus, even without a punishment of i by the producers outside C , it cannot increase the sum of the expected average discounted profits of the members of C in .h/ .ı/.) We consider three distinguished cases. First, either there exists no coordinated deviation and Z \ I \ D .h/ D ¿, or for eachpair of  j 2 Z \ J and i 2 Z \ I who deviated in a coordinated  way,we have xjk a.r/ D 0 for each k 2 I nC and every r 2 f1; : : : ; ng and xki a.r/ D 0 for each k 2J nC and every r 2 f1; : : : ; ng; further, for each i 2 Z \ I \ D .h/, we have xki a.r/ D 0 for each k 2 J nC and every r 2 f1; : : : ; ng. This corresponds   to the situation in (3.45) with either M D ¿ or M ¤ ¿ but xjk a.r/ D 0 for   each .j; k/ 2 .M \ J /  .I nC / and every r 2 f1; : : : ; ng and xki a.r/ D 0 for each .k; i / 2 .J nC /  .M \ I / and every r 2 f1; : : : ; ng. It follows from (3.45)

When a deviation by a coalition C lasts for several periods or is infinite, in any single period, in which it takes place, it can happen that only the firms in a subcoalition Z of C intend to behave in a way that differs from the prescriptions of their equilibrium strategies. Nevertheless, in order to show that a deviation by C cannot increase the sum of the expected average discounted profits of the members of C (in a subgame in which it takes place), we have to consider all single period deviations by C (in the first period of the analyzed subgame), including those in which only the firms in the subcoalition Z of C intend to behave in a way that differs from the prescriptions of their equilibrium strategies. 8

3.2 Existence of an SSPE

45

that a deviation does not increase the sum of the expected single period profits of the members of C in the first period of .h/ .ı/. (If D .h/ ¤ ¿ and according to s  , the firms in C n .Z \ I \ D .h// are punishing the firms outside C or the firms in C n .Z \ I \ D .h// are punished by the firms outside C , then such punishments – which are triggered by the deviations contained in h – have the same effect on the sum of the expected profits of the members of C both with and without a deviation. The punishments of the producers in C by the buyers in C and the punishments of the buyers in C by the producers in C do not affect the sum of the expected profits of the members of C .) If a deviation triggers a punishment of a buyer in Z by the producers outside C in the following T periods, then such a punishment decreases the sum of the expected average discounted profits of the members of C in .h/ .ı/. Thus, a deviation cannot increase the sum of the expected average discounted profits of the members of C in .h/ .ı/. Second, there exists either j 2 .Z \J / nD .h/, who deviated in a coordinated way with some buyer and satisfies xjk a.r/ > 0 for some k 2 I nC and some   r 2 f1; : : : ; ng or i 2 .Z \ I / nD .h/ who satisfies xki a.r/ > 0 for some k 2 J nC and some r 2 f1; : : : ; ng. Let Z  be the set of all producers and buyers in Z who satisfy the condition described above. Take (arbitrary) k 2 Z  . A single period gain (in terms of the sum of their expected profits) of the members of C from a deviation cannot exceed wmax  wmin . It follows from (3.42) that this gain is wiped k k out by the punishment of k in the following T periods. (The punishment of any k 2 Z  by at least one firm outside C is sufficient for this. Note that we use ˇk , and not ˛k , in (3.42). If D .h/ ¤ ¿, the punishment of the firms in Z  affects neither the punishments of the firms outside C by the firms in C nZ nor the punishments of the firms in C nZ by the firms outside C triggered by the deviations contained in h. If a deviation by a firm in ZnZ  leads to the starting or resuming of its punishment by the firms outside C (as described in the analysis of the following case), then such a punishment further decreases the sum of the expected average discounted profits of the members of C in .h/ .ı/.) Thus, a deviation cannot increase the sum of the expected average discounted profits of the members of C in .h/ .ı/.  Third, there exists either j 2 Z \ J \ D .h/ with xjk a.r/ > 0 for some   k 2 I nC and some r 2 f1; : : : ; ng or i 2 Z \ I \ D .h/ with xki a.r/ > 0 for some k 2 J nC and some r 2 f1; : : : ; ng. Let Z C be the set of all producers and buyers in Z who satisfy the condition described above. Moreover, in this case, the set Z  ; described in the analysis of the second case, is empty. This case corresponds to the situation in (3.45) with M , which is equal to the union of Z C with the set of producers in ZnZ C who coordinated their deviation with some buyer in Z C and the set of buyers in ZnZ C who coordinated  their deviation with some producer in Z C . (Since Z  D ¿, we have xjk a.r/ D 0 for each j 2 M nZ C , every   k 2 I nC , and each r 2 f1; : : : ; ng, as well as xki a.r/ D 0 for each i 2 M nZ C , every k 2 J nC , and each r 2 f1; : : : ; ng.) Ignoring the punishments of the firms in C nZ C by the firms outside C and the punishments of the firms outside C by the

46

3 Existence of an SRPE and an SSPE

firms in C nZ C triggered by the deviations contained in h,9 the sum of the expected average discounted profits of the members of C without a deviation is X

  X  X X  vk C 1  ı k .h/ ˛k‘  1  ı T ˛k‘ ;

vk C

k2C

k2C

k2Z C

(3.55)

k2Z C

where ˛i‘ D

n X

3      X  e p j i a.r/  pjmax xj i a.r/ 5 ; 8i 2 I \ Z C (3.56) r 4 2

rD1

j 2J nC

and ˛j‘ D 

n X rD1

3      X  .r/ r 4 e p j i a.r/  pj i xj i a.r/ 5 ; 8j 2 J \ Z C : 2

i 2I nC

(3.57) We use zC to denote the sum of the expected stage game profits of the members of C in the first period of .h/ .ı/ resulting from a deviation. Using (3.46), zC is not higher than the right-hand side of (3.55). The punishment of the firms in Z C is resumed from the second period of .h/ .ı/. Thus, the sum of the expected average discounted profits of the firms in C resulting from a deviation (again, ignoring the punishments of the firms in C nZ C by the firms outside C and the punishments of the firms outside C by the firms in C nZ C triggered by the deviations contained in h, as well as new punishments of the buyers in ZnZ C by the firms outside C that further decrease the sum of the expected average discounted profits of the firms in C in .h/ .ı/) is 2 .1  ı/ zC C ı 4

X k2C

3    X  X X vk C 1  ı T ˛k‘ 5  vk C 1  ı T ˛k‘ : k2Z C

k2C

k2Z C

(3.58) It follows from (3.55) and (3.58) that a deviation cannot increase the sum of the expected average discounted profits of the firms in C in .h/ .ı/. C4. Payoff vector in each subgame is strictly  Pareto efficient. This argument follows  t u from (3.51) and the fact that k.h/ s.h/ 2 V C for each h 2 Hf . Let v be the equilibrium vector of the expected average discounted profits in an SSPE of  .ı/. Of course, the requirement of individual rationality implies that The punishments of the firms in C nZ C by the firms outside C and the punishments of the firms outside C by the firms in C nZ C triggered by the deviations contained in h have the same effect on the sum of the expected average discounted profits of the members of C both with and without a deviation.

9

3.2 Existence of an SSPE

47

all components of v are nonnegative. Since v 2 V C , this can be obtained from a profile of stage game correlated strategies   under which no more than # .J [ I /C1 profiles of stage game pure strategies occur with a positive probability. The other conditions in the statement of Proposition 3.4 are not necessary for the existence of an SSPE of  .ı/ with the equilibrium vector of the expected average discounted profits equal to v but (taken together) they are sufficient for it. They make possible the use of our method of proof of Proposition 3.4. Now, we give a brief intuitive outline of the latter method. Unilateral deviations by buyers with a positive equilibrium expected average discounted profit, as well as their deviations when they are members of a deviating coalition of buyers, are punished in the same way as in the proof of Proposition 3.1 – by increasing their purchasing prices for a finite number of periods. This number of periods is high enough to wipe out any gain from a deviation. Unilateral deviations by these buyers, as well as their deviations as members of a deviating coalition of buyers, during their punishment are punished by resuming the punishment. (Buyers with zero equilibrium expected average discounted profit withdraw from the analyzed market with probability one along the continuation equilibrium path in any subgame. No producer proposes a contract to them. Therefore, they cannot increase their expected stage game profit by a unilateral deviation or by taking part in a deviation by a coalition of buyers.) All producers, who along the equilibrium path trade a positive quantity of their product with some buyer, have a positive equilibrium expected average discounted profit. We denote the set of such producers by EJ . Similarly, all buyers, who along the equilibrium path purchase a positive quantity of at least one good, have a positive equilibrium expected average discounted profit. We denote the set of such buyers by EI . The equilibrium strategy profile s  described in the proof of Proposition 3.4 prescribes the same quantity traded between producer j 2 EJ and buyer i 2 EI in each period and after every history leading to it. It can prescribe one of the three prices for a trade between them. It prescribes the highest possible price when j is punishing i for the latter’s deviation. It prescribes a “normal” (intermediate) price when neither i nor j is being punished. It prescribes the lowest price when j is punished for a deviation coordinated with some buyer k 2 I n fi g. (A deviation by producer j with buyer k is coordinated if j made to k the same contract proposal as k made to j and the latter proposal was different from the one prescribed by s  and proposed a positive traded quantity.) Such coordinated deviation is the only reason for the punishment of a producer in EJ . As explained below, other types of a deviation by a producer in EJ cannot increase his/her expected stage game profit. (The producers with zero equilibrium expected average discounted profit – i.e., the producers outside EJ – withdraw from the analyzed market with probability one along the continuation equilibrium path in any subgame. No buyer proposes a contract to them. Therefore, they cannot increase their expected stage game profit by a unilateral deviation or by taking part in a deviation by a coalition of producers.) Conditions (3.32) and (3.33) imply that (when the firms follow s  ) each producer in EJ earns a nonnegative stage game profit in each period for every signal of the

48

3 Existence of an SRPE and an SSPE

public randomizing device. Therefore, he/she cannot increase his/her expected stage game profit by withdrawing from the analyzed market or by a deviation that leaves him/her with zero sales without withdrawing from the analyzed market. Condition (3.34) implies that he/she cannot increase his/her expected stage game profit by a deviation that (is unilateral or is a part of a deviation by a coalition of producers and) for some signals of the public randomizing device, reduces the set of buyers with whom he/she trades (while keeping unchanged the quantities traded with those buyers with whom he/she still trades and the prices at which he/she trades with them). Condition (3.34) is satisfied, for example, if the producers’ cost functions are differentiable and convex and for each producer in EJ , the lowest price of his/her product (that can be prescribed by s  ) is not lower than his/her marginal cost at his total output in each period (prescribed by s  /. The statement of Proposition 3.4 contains also the requirement that each profile of stage game pure strategies that occurs with a positive probability when the firms follow   generates a vector of traded quantities that maximizes the sum of the firms’ stage game profits. This requirement implies that v maximizes the sum of the firms’ payoffs over V C (i.e., over the set of all vectors of the expected average discounted profits). Therefore, v is strictly Pareto efficient. During the punishment only prices and not traded quantities are changed. Thus, the sum of the firms’ continuation equilibrium expected average discounted profits is the same in each subgame. Therefore, the continuation equilibrium vector of the expected average discounted profits in each subgame is strictly Pareto efficient. The condition (3.35) ensures that no coalition C (other than the grand one), which contains at least one producer and at least one buyer, can (in any subgame) increase the expected average discounted profit of at least one of its members without decreasing the expected average discounted profit of another member. (It also takes into account the possibility of withdrawal from the analyzed market by some firm(s) in C .) It contains – as special cases – three conditions. (See also (3.45) and (3.46). First, C cannot increase the sum of the expected stage game profits of its members by a deviation that only replaces some positive quantities traded by its members with zero. (This is the case of empty M .10 This is the first subcase of the first case considered in part C3 of the proof of Proposition 3.4.) Second, C cannot increase the sum of the expected stage game profits of its members by any deviation if it does not contain a buyer (a producer) who (according to the prescriptions of s  ) trades a positive quantity with a producer (a buyer) outside C with a positive probability. (This is the case when M is nonempty but no firm in it trades according to the prescriptions of s  with a firm outside C with a positive probability. It is the second subcase of the first case considered in part C3 of the proof of Proposition 3.4.) Third, a single period deviation cannot give the members of C the sum of the expected stage game profits equal or higher than the sum of their expected average discounted profits generated by s  if at least one buyer (producer) in C trades 10 In deviations considered in (3.35), only a pair of a producer and a buyer, who belong to M , can trade a positive quantity different from the one prescribed by s  . If either the producer or the buyer does not belong to M , a deviation can only replace a traded quantity prescribed by s  with zero.

3.2 Existence of an SSPE

49

(according to the prescriptions of s  ) a positive quantity with a producer (a buyer) outside C , and when a deviation takes place, each such buyer (producer) is supposed (according to s  ) to trade with the producers (buyers) outside C at the maximal possible prices (minimal possible prices) that can be prescribed by s  . (This is the case of nonempty M containing at least one firm that trades according to the prescriptions of s  with a firm outside C with a positive probability. It corresponds to the third case considered in part C3 of the proof of Proposition 3.4.) In this case, C also satisfies the following condition: no buyer (producer) k 2 C , who is supposed (according to s  ) to trade with at least one producer (buyer) outside C and is currently not being punished, deviates (in a way coordinated with some buyer). When the latter condition is not satisfied C may be able to increase the sum of the expected stage game profits of its members by a deviation. (This corresponds to the second case considered in part C3 of the proof of Proposition 3.4.) The strategy profile s  handles such deviation by meting a punishment to firm k that wipes out the increase in the sum of the expected stage game profits of the firms in C resulting from the deviation. Now, we give an example of a stage game that satisfies the requirements of Proposition 3.4.11 Example   3.3. We have J D f1; 2; 3g, I D f4; 5g ; Yj D Œ0; 20 for each j 2 J , cP j yj D 10yj C 50 for every j 2 f1; 2g, c3 .y3 / D 10y3 C 79, and ci .xi / D j 2J xj i C 20 for each i 2 I . Every i 2 I produces one type of good and its production function fi W Xi ! Œ0; 20 has the form fi .xi / D min fx1i C x3i ; x2i g. The inverse demand functions for the buyers’ outputs, P4 W <2C !
Example 4.3 in the following chapter also contains a stage game satisfying the assumptions of Proposition 3.4.

50

3 Existence of an SRPE and an SSPE

Thus, the maximal prices of the producers’ products given by p1max D p2max D     D 32:03 satisfy (2.6). We set p14 D p15 D p24 D p25 D 12:5. Since this price is equal to the average costs of producers 1 and 2 when their production is 20 units (and 1 .x  / D 2 .x  / D 20), (3.33) is satisfied. As this price exceeds the constant marginal cost of each producer, (3.34) holds. It remains to show that (3.35) holds. Consider first a deviating coalition containing one buyer and producers 1 and 2. Since the buyers are identical, we can take C D f1; 2; 4g without loss of generality. As 5 … C , each of the producers 1 and 2 can deliver to buyer 5 either 10 units or nothing. On the other hand, they can agree with buyer 4 on any feasible (i.e., not exceeding their production capacity) deliveries. If both of them agree with buyer 4 on the delivery of a positive quantity other than 10 units, we have M D f1; 2; 4g. If only one of them delivers 10 units to firm 5, the latter firm produces nothing and the maximum sum of the profits of the firms in C equals p3max

f1;2;4g C 12:5  10  10  10 D 70 C 25 D 45 < 58 D

X

  gk a : (3.59)

k2C

(In this case, each of the producers 1 and 2 delivers 5 units to buyer 4.) If both of them do so, firm 5 produces 10 units and optimal traded quantities within C result from the solution to the maximization program12 max .52  2x14 / x14  22x14  70 subject to 0  x14  10

(3.60)

(The sum of the fixed costs of the members of C is 120. The sum of the variable costs of producers 1 and 2 when their production is 10 units for buyer 5 equals 200. The revenue from the sale of these quantities to buyer 5 at price 12.5 equals 250. Subtracting the above given cost from this revenue gives 70.) The solution of this maximization programP is x14 D 7:5, giving the sum of the profits of the firms in C equal to 42:5 < 58 D k2C gk .a /. If none of the producers 1 and 2 trades with buyer 5 and no firm in C withdraws from the analyzed market, the maximal sum of the profits of the members of C is f1;2;4g D 70. If buyer 4 withdraws from the analyzed market (and hence, M D ¿), the maximal sum of the profits of the firms in C is 0. (This sum is obtained when producers 1 and 2 too withdraw from the analyzed market. Trading only 10 units with buyer 5 at price 13.9 gives each producer profit 11.) If one of the producers 1 and 2 withdraws from the analyzed market, the maximal sum of the profits of the firms in C equal 0. (Again, this sum is obtained by the withdrawal of all firms in C from the analyzed market. Without trading with both producer 1 and producer 2, firm 4 cannot produce anything.) The result for the deviating coalition f2; 3; 4g is worse than that for C D f1; 2; 4g. Producer 3 has a higher fixed cost than producer 1 by 29 financial units, whereas

Taking into account the form of production function f4 , the maximization of the sum of the profits of the members of C requires x14 D x24 .

12

3.2 Existence of an SSPE

51

g3 .a / D g1 .a /  28. He/She cannot trade with buyer 5 … C . The result for the deviating coalition f1; 2; 3; 4g is not better than that for C D f1; 2; 4g. Producer 3 cannot trade with buyer 5. Taking into account the form of production function f4 , producing the sum of the outputs of firms 1 and 3 exceeding 20 units cannot increase the sum of the profits of the members of coalition f1; 2; 3; 4g. The sum of the outputs of firms 1 and 3 not exceeding 20 units is produced with lower cost when only firm 1 produces it. Similarly, the best result for coalition f1:3:4g is not better than the best result for coalition f1; 4g given below. Next, consider a deviating coalition containing one buyer and one of the producers 1 and 2. We take C D f1; 4g without loss of generality.13 Firm 4 can buy either 10 units or nothing from producer 2. If it buys nothing, its output is zero and the maximal sum of the profits of the firms in C equals zero. (This sum is obtained by the withdrawal of both firms in C from the analyzed market. Selling 10 units of the output of firm 1 to buyer 5 at price 13.9 financial units gives profit 11.) If it buys 10 units from producer 2 and buys a positive amount other than 10 units from producer 1 (and hence, M D f1; 4g), it has to buy from producer 2 at price p2max D 32:03. Further, in this case, firm 1, if it sells 10 units to buyer 5, has to  sell them at price p15 D 12:5. Taking into account (2.6), the difference between the revenue and the costs of firm 4 – including the expenditure on the purchase of 10 units from firm 2 but excluding the expenditure on the purchase from firm 1 – is negative. The profit of firm 1 without the revenue from the sales to buyer 4 is lower than 12:5  10  10  10  50 D 25. Hence, the sum of the profits of the members of C is negative. If firm 4 buys 10 units from producer 2 and also 10 units from producer 1 (and hence, M D ¿), then a deviation can occur only in the refusal of producer 1 to trade with firm 5. Then, firm 5’s output equals zero and the sum of the profits of the members of C is 109. Now, consider a deviating coalition containing both buyers and one of the producers, 1 or 2. We take C D f1; 4; 5g without loss of generality.14 If C buys nothing from producer 2, the output of the firms 4 and 5 is zero and the maximal sum of the profits of the members of C equals zero (obtained when all of them withdraw from the analyzed market). If one of the firms 4 and 5 buys nothing from producer 2 and the other one buys 10 units from him – without loss of generality assuming that 5 buys nothing from 2 – firm 5’s withdrawal from the analyzed market is optimal for C . Then, the result for C is not better than that for coalition f1; 4g when producer 1 sells nothing to buyer 5 and buyer 4 buys 10 units from producer 2. As already shown, the sum of profits for f1; 4g is negative in this case. Hence, the sum of the profits of the members of C is negative. Next, suppose that both firms 4 and 5 buy 10 units each from producer 2. Then, both will buy a positive quantity from producer 1 The best result for a deviating coalition containing one buyer and producer 3 is worse than that for coalition f1; 4g : Producer 3 cannot trade with the buyer outside the deviating coalition. Further, his/her fixed cost exceeds the fixed cost of producer 1 by 29 > g1 .a /  g3 .a /. 14 The best result for C is better than that for coalition f3; 4; 5g. The fixed cost of producer 3 exceeds the fixed cost of producer 1 by 29 > g1 .a /  g3 .a /. Taking into account the form of production functions f4 and f5 , the best result for coalition f1; 3; 4; 5g is not better than the best result for C . 13

52

3 Existence of an SRPE and an SSPE

and at least one of them buys a quantity different from 10 units. (If both buy 10 units from producer 1, there would be no deviation. If any of them was buying nothing from producer 1, then its output would be zero and the best result for coalition C would be worse than the best result when such a buyer withdraws from the analyzed market. In this case, the requirement (3.35) cannot be violated because, as is already shown, it is not violated for coalition f1; 4g and gi .a / > 0 for each i 2 f4; 5g.) First, suppose (without loss of generality) that firm 5 buys a positive quantity other than 10 units from firm 1 and firm 4 buys 10 units from firm 1. As such, M D f1; 5g and firm 5 buys from producer 2 at price p2max . Taking into account (2.6), the difference between the revenue and the cost of firm 5 (excluding the expenditure on the purchase from firm 1 but including the expenditure on the purchase from firm 2) is negative. Thus, the sum of the profits of the members of coalition C is lower than that in the case of the withdrawal of firm 5 from the analyzed market. We have already shown that the sum of the profits of the members of C is negative in such a case. Second, suppose that both firms 4 and 5 buy a positive quantity other than 10 units from firm 1. Then, M D f1; 4; 5g and both firms 4 and 5 buy at price p2max from producer 2. Taking into account (2.6), for both firms 4 and 5, the difference between the revenue and the cost (excluding the expenditure on the purchase from firm 1 but including the expenditure on the purchase from firm 2) is negative. Thus, the sum of the profits of the members of C is negative. Finally, consider deviating coalition C D f2; 3; 4; 5g. If producer 3 produces nothing and C does not trade with producer 1, then firms 4 and 5 produce nothing and the maximal sum of the profits of the members of C is zero (obtained when all members of C withdraw from the analyzed market.) When firm 3 withdraws from the analyzed market, the result for coalition C is not better than the corresponding result15 for coalition f2; 4; 5g (which is the same as the corresponding result for coalition f1; 4; 5g analyzed above). Thus, suppose that producer 3 has a positive output and C does not trade with producer 1. (This gives a sum of the profits of the members of C higher than the positive output of producer 3 and the trade between C and producer 1. The reason for this is that the constant marginal cost of producer 3 is lower than price 13.9 – the minimum price at which buyers in C would have to buy from producer 1.) Taking into account the unique element of X max and the fact that producer 3 differs from producer 1 only in terms of the fixed cost, the maximal sum of the profits of the members of C in this case is    .c3 .0/  c1 .0// D 60  29 D 31 < 32 D

X

  gk a :

(3.61)

k2C

15

By “the corresponding result” we mean the result obtained from the same trades between the members of f2; 4; 5g and firms outside C .

Chapter 4

Efficiency of an SRPE and an SSPE

In this chapter, we deal with the efficiency and welfare properties of an SRPE. Since each SSPE is also an SRPE, the results also hold for the former. In Sects. 4.1 and 4.2, we analyze the cost (or transformational) efficiency of an SRPE and an SSPE. First, in Sect. 4.1, we partition the set of goods into groups of identical goods, which we call “types of goods.” We show that in an SRPE, in each subgame in which the sum of the firms’ expected average discounted profits is maximized, in every period along the equilibrium path, the active producers (i.e., those with a positive output) of each type of good can produce their total output prescribed by the equilibrium strategy profile with lower or the same total production cost as any other group of producers of the same type of good. Thus, in a SRPE, in each such subgame of  .ı/, in every period along the equilibrium path, for each type of good, the group of its active producers forms a natural oligopoly (as defined by [Baumol (1982)]) or (if there is only one active producer) a natural monopoly. Second, in Sect. 4.2, we concentrate on the case when all buyers are retailers. We show that in an SRPE, in each subgame in which the sum of the firms’ expected average discounted profits is maximized, in every period along the equilibrium path, the active buyers (i.e., those with a nonzero vector of purchased quantities) can sell the vector of their sales prescribed by the equilibrium strategy profile with lower or the same total selling cost as any other group of buyers. Thus, in an SRPE, in each such subgame of  .ı/, in every period along the equilibrium path, the group of active buyers forms a natural oligopsony or (if there is only one active buyer) a natural monopsony. In Sect. 4.3, we deal with the properties of an SRPE in our model with respect to consumer welfare. We compare them with the welfare properties of the static (i.e., single period) non-collusive equilibria under imperfect competition. Namely, we use two benchmarks: (1) a monopsony choosing the purchased quantities on the demand side of the analyzed market coupled with price taking behavior on its supply side, and (2) Cournot oligopoly on the supply side of the analyzed market coupled with price taking behavior on its demand side. We think that – as many real world industries are oligopolies or are formed by monopsonists with their own networks of suppliers and markets with perfect competition (or something “close” to it) on both sides are rare – this is a relevant comparison. We identify the conditions under

M. Horniaˇcek, Cooperation and Efficiency in Markets, Lecture Notes in Economics and Mathematical Systems 649, DOI 10.1007/978-3-642-19763-5_4, c Springer-Verlag Berlin Heidelberg 2011 

53

54

4 Efficiency of an SRPE and an SSPE

which a movement from the benchmark to a collusive outcome in an SRPE increases consumer welfare. We keep Assumptions 2.1–2.4 in this chapter.

4.1 Natural Oligopoly ˚  In this section, we assume that there exists a partition J .1/ ; : : : ; J .m/ of J with the property that for each k 2 f1; : : : ; mg, J .k/ contains goods that are identical and that differ from the goods contained in J .r/ for each r 2 f1; : : : ; mg n fkg. We call each k 2 f1; : : : ; mg a “type of good” and view the firms in J .k/ as the industry producing type k of good. It is natural to assume that the identity of the suppliers of any type of good does not affect the buyers’ revenue and cost functions. Therefore, in Sect. 4.1, we make, besides Assumptions 2.1–2.4, the following assumption.     Assumption 4.1. (i) For each iP2 I , we have ci .xi / D ci xi‘ for every xi ; xi‘ 2 P Xi2 satisfying j 2J .k/ xj i D j 2J .k/ xj‘ i for each k 2 f1; : : : ; mg. (ii) For each     P i 2 I , we have Ui .x/ D Ui x ‘ for every x; x ‘ 2 X 2 satisfying j 2J .k/ xjr D P ‘ j 2J .k/ xjr for each k 2 f1; : : : ; mg and every r 2 I . First, we give the definition of a natural oligopoly in a single product industry (tailored to our model).  P i Definition 4.1. Consider k 2 f1; : : : ; mgand output Qk 2 0; j 2J .k/ max . j   A set of firms J .k/  J .k/ with # J .k/  2 is a natural oligopoly in the industry producing type k of good for output level Qk if there exists an output vector Q y .k/ 2 j 2J .k/ Yj such that X

yj.k/ D Qk

(4.1)

j 2J .k/

and X j 2J .k/

  cj yj.k/ 

X j 2J .k/ Wyj >0

X Y   cj yj ; 8y 2 Yj with yj D Qk : j 2J .k/

j 2J .k/

(4.2) Thus, a group of firms in the industry  P producing i type k of good is a natural max oligopoly for output level Qk 2 0; j 2J .k/ j if it can produce output Qk with a lower or the same cost as any other group of firms (not excluding a singleton group) in the industry. The difference between our and the standard ([Baumol (1982)]) definition of a natural oligopoly lies in allowing the firms in an industry to have different cost functions, as well as in taking into account the capacity constraints and an upper bound on the number of firms capable of producing a given

4.1 Natural Oligopoly

55

type of good.1 Even if the firms in an industry are identical (i.e., if they have the same cost function and the same capacity), the capacity constraints can lead to the situation where the firms in a natural oligopoly do not have the same output (components of vector y .k/ are not equal). For example, suppose that all firms in the industry producing type k of good are identical and – without capacity constraints – one firm would produce output Qk with lower cost than any group of two or more firms because of the strictly concave cost function but the capacity of each firm in the industry is from the interval .0:5Qk ; Qk /. Then, there exists a natural duopoly for output Qk , in which one firm produces an output equal to its capacity and the other an output equal to the difference between Qk and its capacity. Next, we give the definition of a natural monopoly in a single product industry (tailored to our model). i  P . Definition 4.2. Consider k 2 f1; : : : ; mgand output Qk 2 0; j 2J .k/ max j A firm r 2 J .k/ is a natural monopoly in the industry producing type k of good for output level Q.k/ if Q.k/  r (4.3) and   cr Q.k/ 

X j 2J .k/ Wyj >0

X Y   cj yj ; 8y 2 Yj with yj D Qk : j 2J .k/

(4.4)

j 2J .k/

Thus, a firm in the producing  industry i type k of good is a natural monopoly for P max output level Qk 2 0; j 2J .k/ j if it can produce output Qk with a lower or the same cost as any other firm or group of firms in the industry. First, we show that the maximization of the sum of the firms’ stage game profits leads to the production of each type of good with a positive output by a natural oligopoly or a natural monopoly. Lemma 4.1. Let x max 2 X max . For each k 2 f1; : : : ; mg with   # J .k/ \ EJ .x max /  2 max / is a natural oligopoly in the industry producing type k of good for J .k/ \ EJ .xP output level j 2J .k/ j .x max /. For each k 2 f1; : : : ; mg with

  # J .k/ \ EJ .x max / D 1 the only firm in J .k/ \ EJ .x max P/ is a natural monopoly in the industry producing type k of good for output level j 2J .k/ j .x max /. 1

An analogous comment applies to the definition of a natural monopoly (Definition 4.2) below.

56

4 Efficiency of an SRPE and an SSPE

Proof. Take (arbitrary) x max 2 X max and k 2 f1; : : : ; mg with J .k/ \ EJ .x max / ¤ P ¿. Let Qk D j 2J .k/ j .x max /. Suppose that the claim of Lemma 4.1 does not .k/C  J .k/ with J .k/C ¤ J .k/ \ EJ .x max / hold. Then, Q there exists a nonempty P J and y 2 j 2J .k/C Yj such that j 2J .k/C yj D Qk and X

X

  cj yj <

j 2J .k/C

  cj j .x max / :

j 2J .k/ \EJ .x max /

Therefore, there exists x‘ 2 X satisfying xj‘ i D xjmax for each .j; i / 2     .k/ .k/C  i .k/ ‘  I , xj i D 0 for every .j; i / 2 J nJ  I , j x ‘ D yj for J nJ P P max each j 2 J .k/C , and j 2J .k/C xj‘ i D j 2J .k/ \EJ .x max / xj i for every i 2 I . ‘ max These properties of y and x imply that EI .x‘/ D EI .x / and 

X

  Ui x ‘ 

i 2EI .x ‘ /

>

X

i 2EI .x ‘ /

Ui .x max / 

i 2EI .x max /



X

X

  ci xi‘  X

X

   cj j x ‘

j 2EJ .x ‘ /

  ci ximax

i 2EI .x max /

  cj j .x max / :

(4.5)

j 2EJ .x max /

This contradiction with the definition of X max in (2.3) completes the proof.

t u

The basic idea of the proof of Lemma 4.1 is simple. Suppose that the production of type k of good takes place neither in a natural oligopoly nor by a natural monopoly. Then, some group of its producers, which differs from J .k/ \ EJ .x max /, can produce the same quantity with a lower cost. Thus, after a change in the set of the active producers of type k of good, it is possible to reduce the cost of its production while keeping unchanged its total output, outputs of all firms capable of producing other types of goods, and buyers’ purchases of each type of good. This increases the sum of the stage game profits of all firms (which equals the difference between the sum of the buyers’ revenues and the sum of the production cost and the buyers’ costs determined by their cost functions). Thus, we have a contradiction with the fact that the latter sum is maximized for each element of X max . Now, we show that an SRPE in  .ı/ leads – in each period along the equilibrium path in every subgame, in which the sum of the firms’ expected average discounted profits is maximized – to the production being carried out in a natural oligopoly or by a natural monopoly in each industry with a positive output. This result is the consequence of the preceding lemma. Proposition 4.1. Consider an SRPE s  of  .ı/, h 2 Hf , and the sequence x D fx .t/gt 2N 2 X 1 of the vectors of the actually traded quantities in .h/ .ı/

4.1 Natural Oligopoly

57

 generated by the prescriptions of s.h/ . Assume that

X

   r.h/ s.h/ D  :

(4.6)

r2J [I

Then, for each k 2 f1; : : : ; mg and every t 2 N with n o J .k/ .t/ D j 2 J .k/ j j .x .t// > 0 ¤ ¿; producing type k of good for outJ .k/ .t/ is  Pa natural oligopoly in the industry put level j 2J .k/ j .x .t// if # J .k/ .t/  2 or the only firm in J .k/ .t/ is a natural monopoly industry producing type k of good for output level  in the  P .k/  .x .t// if #J .t/ D 1. t u .k/ j j 2J Proof. Consider (arbitrary) SRPE s  of  .ı/, h 2 Hf , the sequence x 2 X 1 of the vectors of the actually traded quantities in .h/ .ı/ generated by the prescrip tions of s.h/ , and k 2 f1; : : : ; mg. Note that the sum of the firms’ expected average discounted profits (which equals the average discounted sum of their stage game expected profits) along the equilibrium path in .h/ .ı/ is maximized if and only if the sum of their stage game expected profits is maximized in each period. Thus, (4.6) implies that x .t/ 2 X max for each t 2 N . Therefore, the claim of Proposition 4.1 follows from Lemma 4.1. t u Propositions 4.1 and 3.3 have an obvious corollary. Corollary 4.1. Assume that # .I / D 1. Consider an SSPE s  of  .ı/, h 2 Hf , and the sequence x D fx .t/gt 2N 2 X 1 of the vectors of the actually traded  quantities in .h/ .ı/ generated by the prescriptions of s.h/ . Then, for each k 2 ˚  .k/ .k/ .t/ D j 2 J j j .x .t// > 0 ¤ ¿, f1; : : : ; mg and every t 2 N with J J .k/ .t/ is producing type k of good for out Pa natural oligopoly in the industry put level j 2J .k/ j .x .t// if # J .k/ .t/  2 or the only firm in J .k/ .t/ is a natural monopoly in the industry producing type k of good for output level  P .k/  .x .t// if # J .t/ D 1. j 2J .k/ j Proof. It follows from the assumptions of Corollary 4.1 and from Proposition 3.3 that (4.6) holds. Since s  is an SSPE of  .ı/, it is also an SRPE of  .ı/. Thus, the claim of Corollary 4.1 follows from Proposition 4.1. t u Thus, when the analyzed market is monopsonistic, in each SSPE of  .ı/, in every period along the equilibrium path in each subgame, each type of good with a positive output is produced in a natural oligopoly or by a natural monopoly.

58

4 Efficiency of an SRPE and an SSPE

4.2 Natural Oligopsony A natural oligopsony is an analogue of a natural oligopoly on the demand side of a market. Similarly, a natural monopsony is an analogue of a natural monopoly on the demand side of a market. We apply these concepts to the buyers in our model because they operate on the demand side of the analyzed market. We define them (in a way tailored to our model) for a given vector of quantities of the goods purchased from the producers in our model. In this section, we restrict our attention to the buyers in the analyzed market who are retailers. (In this case, the evaluation of the efficiency of a market structure of the retail market on the basis of the sum of selling costs, without taking into account the expenditure on the goods that are being sold, is justified.) Therefore, besides keeping Assumptions 2.1–2.4, in this section, we make the following assumption. Assumption 4.2. All buyers in I are retailers. For each i 2 I , function Ui has the form X Ui .x/ D Pj . .x// xj i ; j 2J

where Pj W Y !
4.2 Natural Oligopsony

59

Now, we give the definitions of a natural oligopsony and a natural monopsony in the analyzed market. Definition 4.3. A set of buyers I   I is a natural oligopsony in the analyzed market of the purchased quantities z 2 Y if # .I  /  2 and there exists   for a vector #.I  / such that xi i 2I  2 Y X xi D z (4.7) i 2I 

and X

  ci xi 

i 2I 

X

ci .xi / ;

i 2I Wxi >0

8x 2 X such that

X

xi D z:

(4.8)

i 2I

  It follows from (4.7) and (4.8) that xi i 2I  minimizes the cost of selling the vector of quantities z in the retail market by the buyers in the analyzed market. Nevertheless, a natural oligopsony is defined as a subset of the set of all buyers rather than a pair of a subset of the set of all buyers and a collection of vectors of the quantities purchased by them. Thus, when I  is the unique natural oligopsony in the analyzed market for a vector of purchased quantities z, there can still be more than one collection of vectors of the quantities purchased by the firms in I  that minimizes the cost of selling the vector of quantities z in the retail market. This approach to the definition of a natural oligopsony corresponds to both our and Baumol’s (see [Baumol (1982)]) approach to the definition of a natural oligopoly. Definition 4.4. A buyer i  2 I is a natural monopsony in the analyzed market for a vector of the purchased quantities z 2 Y if ci  .z/ 

X

ci .xi / ;

i 2I Wxi >0

8x 2 X such that

X

xi D z:

(4.9)

i 2I

It follows from (4.9) that the sale of the quantities of the goods specified by z in the retail market by buyer i  minimizes the cost of selling them. A natural monopsony and (in general) also a natural oligopsony are a result of both the economies of scale and the economies of scope.2 The former implies that it is not possible to decrease the sum of selling costs by dividing a vector of the quantities purchased by some buyer between two or more buyers3 in such a way 2 See [Baumol (1977)] for the characterization of the functions that exhibit both the economies of scale and the economies of scope. 3 The information about a natural oligopsony (natural monopsony) I  (i  ) in the analyzed market has the greatest value when the latter remains a natural oligopsony (natural monopsony) even after

60

4 Efficiency of an SRPE and an SSPE

that the composition of their sales (i.e., the ratios between the quantities of any two goods) would be the same. In the case of a natural oligopsony, the economies of scale are already exhausted. It is not possible to decrease the sum of selling costs by assigning the sum of the vectors of the quantities purchased by two or more buyers with the same composition to just one (or any lesser number) of the buyers. The economies of scope imply that it is not possible to decrease the sum of the selling costs by dividing a vector of the quantities purchased by some buyer between two or more buyers in such a way that each of them would specialize in sale of only some (or even only one) of the goods that their “predecessor” was selling. In the case of a natural oligopsony, the economies of scope are already exhausted. It is not possible to decrease the sum of selling costs by assigning the sum of the vectors of the quantities purchased by two or more buyers, which buy different goods, to just one (or any lesser number) of the buyers. First, we show that the maximization of the sum of the firms’ stage game profits leads to a natural oligopsony or a natural monopsony in the analyzed market. Lemma 4.2. Let x max 2 X max . If # .EI .x max //  2, then EI .x max / is a natural in the analyzed market for the vector of the purchased quantities P oligopsony max . If EI .x max / D fi  g, then buyer i  is a natural i 2I xi P monopsony in the analyzed market for the vector of the purchased quantities i 2I ximax . P Proof. Take (arbitrary) x max 2 X max and let z D i 2I ximax . Suppose that the claim of LemmaP4.2 does not hold. Then, we can find a nonempty I C  I and x ‘ 2 X such that i 2I C xi‘ D z, xi‘ D 0 for each i 2 I nI C , xi‘ > 0 for every i 2 I C , and X

  ci xi‘ <

X

  ci ximax :

i 2EI .x max /

i 2I C

  P P Note that i 2I C xi‘ D i 2EI .x max / ximax D z implies that j x ‘ D j .x max /  ‘ for each j 2 J and EJ x D EJ .x max /. Therefore, it follows from the properties of x ‘ that X X         X  ‘  Pj  x ‘ j x ‘  ci xi cj j x ‘  C ‘ j 2J i 2I j 2EJ .x / X X   max > Pj . .x // j .x max /  cj j .x max / j 2J



X

i 2EI



ci ximax



j 2EJ .x max /

:

(4.10)

.x max /

This contradiction with the definition of X max in (2.3) completes the proof.

t u

adding any finite number of replicas of all firms in I  (i  ) to I . In this case, the sum of the costs of selling the quantities of goods given by z cannot be decreased by any change in the number of buyers.

4.3 Impact on Consumer Welfare

61

The basic idea of the proof of Lemma 4.2 is simple. Suppose that either EI .x max / has at least two members but it is not a natural oligopsony in the analyzed market for the sum of the vectors of the quantities purchased by its members, or EI .x max / is a singleton but its only member is not a natural monopsony in the analyzed market for his/her vector of the purchased quantities. Then, for another group of buyers, it is possible to reduce the sum of selling costs while keeping the vector of total purchased quantities, as well as the output of each producer, unchanged. This increases the sum of the stage game profits of all firms (which equals the difference between the sum of the revenues from the sales in the retail market and the sum of the production costs of the active producers and the selling costs of the active buyers). Thus, we have a contradiction with the fact that the latter sum is maximized for each element of X max . Now we show that an SRPE in  .ı/ leads – in each period along the equilibrium path in every subgame, in which the sum of the firms’ expected average discounted profits is maximized – to a natural oligopsony or a natural monopsony in the analyzed market. This is a consequence of the preceding lemma. Proposition 4.2. Consider an SRPE s  of  .ı/, h 2 Hf , and the sequence x D fx .t/gt 2N 2 X 1 of the vectors of the actually traded quantities in .h/ .ı/ gen erated by the prescriptions of s.h/ . Assume that (4.6) holds. Then, for each t 2 N ,  I .t/ D fi 2 I j xi .t/ > 0g is a natural oligopsony if # .I  .t//  2 or the only buyer in it is a natural monopsonyP if # .I  .t// D 1 in the analyzed market for the vector of the purchased quantities i 2I xi .t/. t u Proof. Consider (arbitrary) SRPE s  of  .ı/, h 2 Hf , and the sequence x D fx .t/gt 2N 2 X 1 of the vectors of the actually traded quantities in .h/ .ı/ gen erated by the prescriptions of s.h/ . As noted in the proof of Proposition 4.1, (4.6) implies that x .t/ 2 X max for each t 2 N . Therefore, the claim of Proposition 4.2 follows from Lemma 4.2. t u

4.3 Impact on Consumer Welfare In this section, we describe the impact of a collusion in an SRPE (and hence, also of a collusion in an SSPE) on consumer welfare. In this comparison, we restrict our attention to the pure strategies in  .ı/. Such an approach is sufficient when V contains a payoff vector that is strictly Pareto efficient with respect to V C , it is the equilibrium payoff vector in an SRPE, and it weakly Pareto dominates a stage game payoff vector given by a benchmark used in the comparison. Besides Assumptions 2.1–2.4, we make the following assumption in Sect. 4.3. Assumption 4.3. (i) For each j 2 J , function cj is twice continuously differentiable. (ii) For every i 2 I , function Ui is twice continuously differentiable. (iii) For every i 2 I , function ci is twice continuously differentiable.

62

4 Efficiency of an SRPE and an SSPE

In Sect. 4.3.2, we relax part (ii) of Assumption 4.3. The results in this section are based on the fact that in an SRPE in each subgame, the vector of the firms’ equilibrium average discounted profits is strictly Pareto efficient with respect to V C . This implies that in each subgame along the equilibrium path, in every period, the vector of the firms’ stage game profits is strictly Pareto efficient with respect to V C .

4.3.1 Comparison with a Monopsonist Choosing the Traded Quantities on the Demand Side and Price Taking Behavior on the Supply Side of the Analyzed Market In this section, we compare the collusive outcome in an SRPE with the following benchmark: the producers in J are price takers in the analyzed market4 and there is only one buyer on its demand side. That is, I D f# .J / C 1g and x#.J /C1 D x for each x 2 X . We assume that the purchased quantities of the goods in J are the monopsonist’s decision variables and the unit prices he/she pays for them are determined by the inverse supply functions in the analyzed market. Of course, the monopsonist is also the in his/her product markets. i h monopolist max 1 be the inverse supply function for good j 2 J . It Let j W Yj ! 0; pj assigns to each quantity qj of good j purchased by the monopsonist, the unit price of good j (i.e., the unit price at which producer j wants to produce and bring to the analyzed market quantity qj ). The following assumption (which we use only in the current section) ensures – together with Assumption 4.3 – that the inverse supply functions can be defined in such a way that they are continuously differentiable on their domains. Assumption 4.4. For each j 2 J , function cj is strictly convex on Yj and pjmax  cj0 .j /. h i For each j 2 J , producer j ’s supply function j W 0; pjmax ! Yj is defined h      1    i by j pj D 0 if pj < cj0 .0/, j pj D cj0 pj if pj 2 cj0 .0/ ; cj0 j , 4

Following the microeconomics textbooks (e.g., [Varian (1992), p. 215]) that define a competitive firm as a price taking firm, we can use the term “perfect competition on the supply side of the analyzed market.” Nevertheless, according to Assumption 4.5 below, all goods in J are different. Hence, each type of good is sold in the analyzed market by only one firm (although it can be sold in other markets by other firms that are potential competitors of the firm in J selling it in the analyzed market). Since perfect competition is associated with a large number of firms producing a homogeneous good, we prefer the term “price taking behavior on the supply side of the analyzed market.” In the analysis of non-collusive behavior (similarly as in the game  .ı/), we allow the firms to leave the analyzed market. Nevertheless, we do not consider the entry of firms (that could drive the economic profits of the price taking firms down to zero). This is another reason why we do not use the term “perfect competition on the supply side of the analyzed market.”

4.3 Impact on Consumer Welfare

63

    and j pj D j if pj > cj0 j . (The third part of the definition of j follows     from the capacity constraint.) Then, it is straightforward that j1 qj D cj0 qj   for each qj 2 0; j . In order to ensure that j1 is well defined, continuous, and has a right-hand side continuous first derivative at zero and a left-hand side   contin uous first derivative at j , we set j1 .0/ D cj0 .0/ and j1 j D cj0 j . This choice of j1 .0/ is innocuous because it affects neither the revenue of producer j nor expenditure of the monopsonist. In order to obtain the results in this section, we also need the following assumption. Assumption 4.5. No two producers in J produce identical goods; further, we have

and

@2 U#.J /C1 .x/  0; 8j 2 J; 8k 2 J n fj g ; 8x 2 X @xj#.J /C1 @xk#.J /C1

(4.11)

@2 c#.J /C1 .x/  0; 8j 2 J; 8k 2 J n fj g ; 8x 2 X: @xj #.J /C1 @xk#.J /C1

(4.12)

Thus, we assume that the monopsonist’s marginal revenue from using any input j 2 J is nondecreasing in his/her use of any other input. Further, his/her marginal costs of using any input j 2 J is nonincreasing in his/her use of any other input. Assumption 4.5 is satisfied, for example, if the monopsonist is a retailer, all goods in J are complements, the inverse demand functions for them have nonnegative second partial derivatives, and the monopsonist’s cost function is additively separable.5 Example 4.1 contains a stage game satisfying Assumption 4.5. Let b x 2 X be the vector of the traded quantities in a long-run equilibrium in the benchmark considered in this section in which all firms are present in the analyzed market. (Since it is a long-run equilibrium, each firm earns a nonnegative profit in it. Since all firms have positive fixed costs, we have b x j#.J /C1 > 0 for each j 2 J .) Then, each producer j 2 J produces output b x j#.J /C1 and sells it for price   1 b x j#.J /C1 . In order to avoid an uninteresting case (in which our results in this j section would fail to hold because of the producers’ capacity constraints), we restrict our attention to the case when there is j 2 J with b x j#.J /C1 < j . Clearly, b x solves the maximization problem

Our results in this subchapter continue to hold when function U#.J /C1 is not differentiable on some subset of its domain but has right-hand side and left-hand side partial derivatives satisfying Assumption 4.5 on it. A stage game, in which the monopsonist is a retailer and each inverse demand function is linear on the subset of its domain on which it has a positive functional value, is an example of this. With a slight abuse of language, when subsets, at which Assumption 4.5 does not hold, have zero measure, we say that Assumption 4.5 is satisfied. 5

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4 Efficiency of an SRPE and an SSPE

max U#.J /C1 .x/  c#.J /C1 .x/ X   1 xj#.J /C1 xj#.J /C1  j j 2J

  subject to xj#.J /C1 2 0; j 8j 2 J:

(4.13)

Thus, b x satisfies the first-order conditions @U#.J /C1 .b x/ @xj#.J /C1



@c#.J /C1 .b x/ @xj#.J /C1

    x j#.J /C1  xj#.J /C1 cj00 b x j#.J /C1 D 0;  cj0 b

  8j 2 J with b x j#.J /C1 2 0; j ;

(4.14)

    x / @c#.J /C1 .b x/ @U#.J /C1 .b   cj0 b x j#.J /C1  xj#.J /C1 cj00 b x j#.J /C1  0; @xj#.J /C1 @xj#.J /C1 8j 2 J with b x j#.J /C1 D j ;

(4.15)

  @U#.J /C1 .b x / @c#.J /C1 .b x/ x j#.J /C1  0;   cj0 b @xj#.J /C1 @xj#.J /C1 8j 2 J with b x j#.J /C1 D 0:

(4.16)

It follows from (4.14), (4.15), and Assumption 4.4 that   @U#.J /C1 .b x / @c#.J /C1 .b x/ x j#.J /C1 > 0;   cj0 b @xj#.J /C1 @xj#.J /C1   8j 2 J with b x j#.J /C1 2 0; j : (4.17)     pj b x j#.J /C1  cj b Let b p D cj0 b x j#.J /C1 and b vj D b x j#.J /C1 for each j 2 J ,   P  j x #.J /C1  j 2J b b pD b p j j 2J , and b v#.J /C1 D U#.J /C1 .b x /  c#.J /C1 b pj b x j#.J /C1 . Consider the maximization problem x C z/  c#.J /C1 .b x C z/  max U#.J /C1 .b

X   b p j C rj b x j #.J /C1 C zj (4.18) j 2J

subject to      x j#.J /C1 C zj  cj b x j #.J /C1 C zj  b vj ; 8j 2 J; b p j C rj b

(4.19)

x j#.J /C1 ; 8j 2 J; zj  b

(4.20)

zj  j  b x j#.J /C1 ; 8j 2 J;

(4.21)

rj  b p j ; 8j 2 J;

(4.22)

pj ; 8j 2 J: rj  pjmax  b

(4.23)

4.3 Impact on Consumer Welfare

65

Thus, in the maximization problem (4.18)–(4.23), we seek a vector of the changes in the traded quantities given by vector b x and a vector of the changes in the prices given by vector b p that maximize the monopsonist’s profit without decreasing the profit of any producer j 2 J to below b vj . Taking into account the constraints (4.20)–(4.23), the set of feasible solutions of the latter maximization problem is bounded. It is clearly nonempty – it contains 0 2 <2#.J / . Since function cj , j 2 J is continuous and the constraints have the form of weak inequalities, the set of feasible solutions is also closed. Hence, it is compact. The objective function is continuous. Therefore, the maximization problem (4.18)–(4.23) has a solution. Let .b z;b r/ be a solution to the maximization problem (4.18)–(4.23). We have b p j Cb r j > 0 and b x j#.J /C1 Cb zj > 0 for each j 2 J . (A producer, for whom at least one of these inequalities fails, earns a negative profit.) Taking this into account, the constraint in (4.19) is binding for each j 2 J . (If this was not the case for some j 2 J , then a slight decrease in rj would increase the monopsonist’s profit without violating any constraint.) Suppose that b z D 0. Then, b r D 0. (If we had b r j < 0 for some j 2 J , pror j > 0 for some j 2 J , j ’s profit ducer j ’s profit would fall below b vj . If we had b would exceed b vj :) Take (arbitrary) j 2 J with b x j#.J /C1 < j . At .b z;b r / D .0; 0/ the monopsonist’s expenditure on good j equals the sum of producer j ’s cost of producing it and his/her profit b vj . Keeping this principle in mind, an infinitesimal increase in zj (while keeping the purchased quantities of and the expenditure on all other goods in J unchanged) increases the monopsonist’s expenditure on good j by cj0 b x j#.J /C1 and his/her profit by the left-hand side of (4.17), which is positive. Thus, it is possible to increase the monopsonist’s profit without violating any constraint. This contradicts the assumption that .b z;b r/ is a solution of the maximization problem (4.18)–(4.23) and proves that we haveb z ¤ 0: We cannot have b z < 0. Since all constraints in (4.19) are binding, this would   x j #.J /C1 Cb pj C b imply j1 b zj < b r j for each j 2 J with b zj < 0 and6 b rj D 0 z;b r / decreases for every j 2 J with b zj D 0. Thus, the replacement of .0; 0/ by .b the monopsonist’s profit. (If it does not thenthe replacement of b x by b x Cb z and the replacement of b p j by j1 b zj for each j 2 J with b x j #.J /C1 Cb zj < 0 would increase his/her profit. This would contradict the fact that b x solves (4.13)).

6 The profit of each j 2 J from its competitive supplyis strictly increasing in price on the set of  prices that are higher than cj0 .0/ and no higher than cj0 j . For

  cj0 .0/ < pj0 < pj  cj0 j we have pj

j

   pj  cj

j



  pj > pj

j

> pj0

j

 h pj0  cj   h pj0  cj

j j

 i pj0  i pj0 :

66

4 Efficiency of an SRPE and an SSPE

n o Suppose that b z D zC C z , zC > 0, z < 0, J C D j 2 J j zC > 0 , J D j n o C j 2 J j z \ J  D ¿. We already know that the replacement of b x j < 0 , and J

by b x C z (with the changes in prices that ensure that each producer’s profit remains unchanged) decreases the monopsonist’s profit. From this and Assumption 4.5, it follows that the replacement of b x CzC by b x CzC Cz (with the changes in prices that ensure that each producer’s profit remains unchanged) decreases the monopsonist’s z > 0. profit.7 Therefore, we haveb The replacement of .b x; b p / by .b x Cb z; b p Cb r / leads to a vector of the firms’ profits that is strictly Pareto efficient with respect to V and8 it weakly Pareto dominates b v. (There exists  > 0 such that the replacement of b r j by b r j C for each j 2 J gives a vector of the firms’ profits that strictly Pareto dominatesb v.) This is achieved without decreasing the output of any good while the output of at least one good rises. Assume that the firms are using pure strategies in game  .ı/ and that V contains a vector of the firms’ stage game profits v that weakly Pareto dominates b v, is strictly Pareto efficient with respect to V C , and is the vector of the firms’ equilibrium expected average discounted profits in an SRPE. Let .b x C z; b p C r/ 2 X  P generate v. Then, there exists r  r such that .z; r/ solves (4.18)–(4.23).9 This implies that z > 0. Thus, a movement from a non-collusive long run equilibrium to a collusive outcome in an SRPE does not lead to a decline in the output of any good 7 The changes in the monopsonist’s expenditure on goods in J  are the same in both replacements. Assumption 4.5 implies that      @ U#.J /C1 b x C zC  c#.J /C1 b x C zC @zj #.J /C1      x  c#.J /C1 b x @ U#.J /C1 b  ; 8j 2 J  : @zj #.J /C1

x C zC by b x C zC C z (as well as the replacement of b x by We can view the replacement of b b x C z ) as a sequence of “small” changes. Then, for each of them, we can approximate a change

in the functional value of the function U#.J /C1  c#.J /C1 by the scalar product of its gradient and a vector of the changes in its arguments.   8 This can be seen as follows. Let v be thevector of the firms’ profits generated by b x Cb z; b p Cb r . Suppose that there exists .e z;e r/ ¤ b z;b r such that b x Ce z; b p Ce r gives a vector of the firms’ v that weakly Pareto dominates v. Clearly, exists j 2 J with e vj > vj . (Otherwise, payoffs e  there  we would have a contradiction with the fact that b z;b r solves (4.18)–(4.23).) Thus, by decreasing e r j for each j 2 J withe vj > vj , we obtain a vector offirms’  profits v with vj D vj for each j 2 J and v#.J /C1 > v#.J /C1 . This contradicts the fact that b z;b r solves (4.18)–(4.23) and completes the argument. 9 Let r result from the changes in the prices that vj . Suppose that .r; z/  each j 2 J with profitb  leave z;b r of (4.18)–(4.23). Then, the sum of the compodoes not solve (4.18)–(4.23). Take solution b  nents of the vector of the profits generated by b x C b z; b p C b r exceeds the sum of the components   x C z; b p Cr (which equals the sum of the components of the vector of the profits v generated by  b of the vector of the profits generated by b x C z; b p C r ). Therefore, we can change the prices given p Cb r in such a way that we obtain a vector of the profits that weakly Pareto dominates v. This by b contradiction with the assumption that v is strictly Pareto efficient completes the argument.

4.3 Impact on Consumer Welfare

67

and yields an increase in the output of at least one good. This is a consequence of two facts. First, in a movement to a collusive outcome, an increase in the quantity of a good purchased by the monopsonist does not lead to a higher price given by the inverse supply curve, but only to a higher price that compensates for the increase in a producer’s costs. (Equation (4.17) used in the arguments above reflects this.) Second, (we assumed that) goods are complements. Thus, an increase in the quantity of one good purchased by the monopsonist does not make a decrease in the purchased quantity of another good profitable. First, suppose that the monopsonist uses the purchased goods to produce one type of new good. Then, z > 0 implies that his/her output rises. (An increase in the purchased quantities of the goods in J leads to higher expenditure for the monopsonist on them. Therefore, z > 0 can be optimal only if it leads to an increase in his/her output.) This leads (under the assumption that the good produced by the monopsonist is not a Giffen good) to lower price. Thus, the consumers of the good produced by the monopsonist are clearly better off. Next suppose that the monopsonist is a retailer. In order to assess the consequences of the movement from .b x; b p / to .b x C z; b p C r/ for consumer welfare, we have to also take into account prices. When the goods in J are complements (as we assumed in the justification of Assumption 4.5), the prices of some goods can rise. If none of them rises (as in Example 4.1), consumer welfare clearly increases. Now, we give the sufficient conditions under which consumer welfare increases even when some prices rise. Suppose that there is a finite set B of consumers who buy the goods sold by the monopsonist in the retail market but they can also buy other consumptions goods from other sellers, who also sell to other buyers. We assume that the prices of these goods do not change after collusion takes place. consumer   Each has strictly convex preferences. For consumer b 2 B, let b qb D b q bj j 2J be his/her vector of the consumption of the goods purchased from the monopsonist before collusion. Then, each consumer, whose consumption vector after collusion differs from his/her consumption vector before collusion, is better off after collusion if X j 2J

Pj . .b x Cb z//b q bj 

X

Pj . .b x //b q bj ;

j 2J

8b 2 B:

(4.24)

That is, for each consumer, his/her expenditure on the original quantities of the goods in J (purchased before collusion) at new prices (that prevail after collusion) is not higher than that at the original prices (that prevailed before collusion). If (4.24) holds, then each consumer can buy at new prices, the same quantities of all goods (including those that he/she does not buy from the monopsonist) as before collusion. Thus, as his/her preferences are strictly convex (which implies that his/her Marshallian demand vector is unique), his/her Marshallian demand vector at new prices gives him/her a higher utility than his/her Marshallian demand vector at original prices. Since collusion increases the sale of at least one good in J in the retail market, there exists at least one consumer whose consumption vector after collusion

68

4 Efficiency of an SRPE and an SSPE

differs from his/her consumption vector before collusion. If (4.24) holds for each such consumer, then collusion increases consumer welfare. Now, we give an example of a stage game satisfying Assumption 4.5. Example 4.1. Let J D f1; 2g, I D f3g, the monopsonist be a retailer, Y1 D Œ0; 5, Y2 D Œ0; 5, c1 .y1 / D y12 C 2, c2 .y2 / D y22 C 2, c3 .x13 ; x23 / D x13 C x23 C 1, p1max D 29, and p2max D 29. The monopsonist’s revenue function has the form U3 .x13 ; x23 / D P1 .x13 ; x23 / x13 C P2 .x13 ; x23 / x23 ;

(4.25)

where P1 W Y ! Œ0; 18 is the inverse demand function for good 1 in the retail market, P2 W Y ! Œ0; 18 is the inverse demand function for good 2 in the retail market, P1 .x13 ; x23 / D max f13  2x13 C x23 ; 0g ; (4.26) and P2 .x13 ; x23 / D max f13 C x13  2x23 ; 0g :

(4.27)

This example satisfies Assumptions 2.1–2.4, Assumption 4.3, and Assumptions 4.4–4.5 (U3 is not differentiable on the subset of its domain with the zero measure – see footnote 5). We have 1 .p1 / D 0:5p1 for p1 2 Œ0; 10 and 1 .p1 / D 5 for p1 2 Œ10; 29, 1 2 .p2 / D 0:5p2 for p2 2 Œ0; 10 and 2 .p2 / D 5 for p2 2 Œ10; 29, 1 .q1 / D 1 2q1 , and 2 .q2 / D 2q2 . In the long-run non-collusive equilibrium (obtained by solving the maximization problem (4.13)), b x 13 D 2 and b x 23 D 2. This gives b p1 D 4, b p 2 D 4, b v1 D 2, b v2 D 2, b v3 D 23, and the prices in the retail market P1 .2; 2/ D 11 and P2 .2; 2/ D 11. Consider a collusive outcome based on the maximization of the sum of the firms’ max profits. In this example X max D fx max g (i.e., X max is a singleton), where x13 D max  x 23 . This gives  D 31 and the prices in the retail market 3 >b x 13 and x23 D 3 > b P1 .3; 3/ D 10 < P1 .2; 2; / and P2 .3; 3/ D 10 < P2 .2; 2/. Letting (for example)   p13 D 4:9 and p23 D 4:9,10 profits are v1 D v2 D 3:7 and v3 D 23:6.11 Then, the firms can agree on moving from the long-run non-collusive equilibrium to the collusive outcome that we have just described. Such movement increases the profit of each firm. It also increases consumer welfare because the consumed quantity of each good increases and the price of every good decreases. The following example shows that a collusive outcome can increase the consumers’ welfare in comparison with the long-run non-collusive equilibrium in some cases even when Assumption 4.5 is not satisfied (namely, when the goods in J are substitutes or identical). 10

A reader may find this choice of prices strange. It is motivated by the desire to have prices that are not periodic numbers and have at most two non-zero decimal points (and hence, they can be paid by the banknotes and coins in circulation). 11 In this example, f1;3g D f2;3g D 9. Thus, the firms’ profits in the collusive outcome satisfy (3.22). Therefore, the described collusive outcome can be sustained in an SSPE of  .ı/ for any ı 2 .0; 1/. (See Proposition 3.2.)

4.3 Impact on Consumer Welfare

69

Example 4.2. Consider the stage game from Example 3.1 with one modification: producers’ fixed costs are 10 financial units instead of 25, i.e., cj .yj / D 10yj2 C 10 for each j 2 J .12 This leaves X max unchanged - it has the unique element x max D .1:5; 1:5; 1:5; 1:5/. This also increases the maximal sum of the profits to   D 305. (The new values of C for non-singleton coalitions containing the monopsonist can be computed from the original ones given in Example 3.1 by adding the product of 15 and the number of producers in a coalition.) We have P1 .x max / D P2 .x max / D 76. We will use the collusive arrangement based on x max with the vector of profits v D .5; 5; 5; 5; 285/.This results from price pj D 25 for each j 2 J . (It is easy to verify that v satisfies condition (3.22).) This example satisfies Assumptions 2.1–2.4, Assumption 4.3 (U3 is not differentiable on the subset of its domain with the zero measure – see footnote 5), and Assumption 4.4.     For each j 2 J , we have j pj D pj =20 for pj 2 Œ0; 100 and j pj D 5 for pj 2 Œ100; 326. The aggregate supply function for the type one good is ‰1 .p1 / D 0:1p1 for p1 2 Œ0; 100 and ‰1 .p1 / D 10 for p1 2 Œ100; 326. The aggregate supply function for the type two good is ‰2 .p2 / D 0:1p2 for p2 2 Œ0; 100 and ‰2 .p2 / D 10 for p2 2 Œ100; 326. We use Q1 to denote the aggregate output of the type one good (the sum of the outputs of producers 1 and 3) and Q2 to denote the aggregate output of the type two good (the sum of the outputs of producers 2 and 4). Then, the inverse supply function for type one good has the form ‰11 .Q1 / D 10Q1 , and the inverse supply function for the type two good has the form ‰21 .Q2 / D 10Q2. The monopsonist decides only on the aggregate amount of each type of good that he/she wants to buy. The price of each type of good is then determined by the inverse supply function for it defined P above.Thus, in the maximization problem (4.13) we have to replace the term  j 2J j1 xj#.J /C1 xj#.J /C1 by ‰11 .Q1 / Q1  ‰21 .Q2 / Q2 and the constraints by Q1 2 Œ0; 10 and Q2 2 Œ0; 10. Then, using the inverse supply functions for both types of goods and the producers’ supply functions, we have x D .0:5Q1 ; 0:5Q2 ; 0:5Q1 ; 0:5Q2 / : The solution of this modified maximization problem gives the long-run noncollusive equilibrium: b x j D 1:2 < xjmax , b p j D 24, b vj D 4:4 for each j 2 J , b v5 D 273, P1 .b x / D 85 > P1 .x max /, and P2 .b x / D 85 > P2 .x max /. Since v  b v, the firms can agree on moving from the long-run non-collusive equilibrium to the collusive outcome described above. Such movement increases the output and decreases the price of each type of good. Thus, it (irrespective of the distribution of

12

With the original fixed cost a long-run non-collusive equilibrium would not exist. The producers’ profits in the solution of the maximization program (4.13) would be negative. Nevertheless, the stage game in example 3.1 with original fixed cost can be used as an example that a collusion can make viable production that is not viable under a non-collusive arrangement.

70

4 Efficiency of an SRPE and an SSPE

the aggregate consumption vector between consumers) clearly increases consumer welfare.

4.3.2 Comparison with Price Taking Behavior on the Demand Side and a Cournot Oligopoly on the Supply Side of the Analyzed Market In this section, we compare the collusive outcome in an SRPE with the following benchmark: the producers compete in the Cournot manner in the analyzed market and the buyers are price takers in it, the buyers are themselves producers, each buyer can produce one type of good, and the buyers compete in the Cournot manner in the markets for their products.13 (We can formally model the buyers’ behavior in their output markets in this way even if some or all of these markets are separated, i.e., if the quantities brought to the market by some buyers do not affect the prices for which the other buyers sell their outputs.) We also identify I with the set of goods that are produced by the firms in I . For each i 2 I , there exists a production function fi W Y !
J .1/ ; : : : ; J .m/

o

of J with the property that for each k 2 f1; : : : ; mg, J .k/ contains goods that are identical and they differ from the goods contained in J .r/ for each r 2 f1; : : : ; mg n fkg. We also let Assumption 4.1 hold in this section. We assume (without loss of generality) that k 2 J .k/ for each k 2 f1; : : : ; mg. We also assume that the upper bounds on the prices of the goods of the same type are equal (i.e., 2  for each k 2 f1; : : : ; mg and every .j; n/ 2 J .k/ , we have pjmax D pnmax ). h i P Q Let Y† D m j 2J .k/ j . That is, Y† is the set of the feasible vectors kD1 0; of the outputs of the types of goods. For each y D .yk /k2f1;:::;mg 2 Y† and every k 2 f1; : : : ; mg, yk is the output of type k of good. We let Xi † D Y† for every i 2 I . For each i 2 I , there exist functions fi † W Xi † !
cost (expressed by the functional value of function ci ) depend only on the total used quantity of each type k 2 f1; : : : ; mg of input, and not on the used quantities of the individual inputs in J .k/ .

13

I am grateful to Martin Gregor for his comments on this section.

4.3 Impact on Consumer Welfare

71

We restrict our attention to the situation where an efficient use of inputs by each firm in I requires fixed proportions between the types of inputs (i.e., for each i 2 I , fi † is a Leontieff production function).14This is formally stated in the following assumption and it justifies the introduction of function Fi , i 2 I . Assumption 4.6. For each i 2 I , there exists Q.i / 2 Xi † such that Q.i /  0,       fi † Q.i / D 1, and for every Q.i / 2 Xi † , fi † Q .i / D fi † i Q.i / , where   / i D mink2f1;:::;mg Qk.i / =Q.i . k P  .i / and define D min  =Q For each i 2 I , let max .k/ j k2f1;:::;mg j 2J i k     function Fi W 0; max !
14

(4.28)

We can extend the analysis (at the expense of making it more cumbersome) to the situation where for each firm i 2 I and every k 2 f1; : : : ; mg, a positive output requires a positive input of at least one good in J .k/ but the goods in the latter group are close substitutes instead of being identical and their quantities are aggregated using the fixed coefficients (in general, different from one) that express the fixed (i.e., independent of their quantities) marginal rates of substitution between them.   15 , we have For i 2 I and i 2 0; max i   .i/ m @c X i† i Q  Q.i/  Ci0 . i / D k .i/ kD1 @ i Q k and, using part (iii) of Assumption 4.8,   1 @2 ci† i Q.i/ @Q.i/     Q.i/ A  0: Ci00 . i / D k n .i/ .i/ Q @ nD1 @ i Q n kD1 i k m X

0

m X

72

4 Efficiency of an SRPE and an SSPE

Assumption 4.9. (i) For each i 2 I , function Pi is twice continuously differentiable at every element of its domain at which it has a positive functional value, @Pi .q/  0; 8k 2 I n fi g ; @qk Y  ˚   0; max Fn . n / j n 2 0; max 8q 2 n with Pi .q/ > 0

n2I

(4.29)

and @Pi .q/ < 0; @qi Y 8q 2 with Pi .q/ > 0:

 n2I

 ˚  0; max Fn . n / j n 2 0; max n (4.30)

(ii) For each i 2 I , @2 Pi .q/  0; @2 qi Y 8q 2 with Pi .q/ > 0:

 n2I

 ˚  0; max Fn . n / j n 2 0; max n (4.31)

Assumption 4.10. For each j 2 J , function cj is convex. Part (i) of Assumption 4.9 is satisfied when (besides the twice continuous differentiability of inverse demand functions), the products of the firms in I are complements (i.e., not only the inputs used by the firms in I belonging to different elements of the partition of J are complements, but also the outputs of the firms in I are complements) and none of them is a Giffen good. Since the producers in J are Cournot competitors in this section, the convexity of their cost functions is sufficient (we do not need their strict convexity) for obtaining our results. In Example 4.3, we deal with a stage game that satisfies all assumptions made in this section and illustrates the results obtained in it.16 The following analysis is valid when a benchmark long-run non-collusive equilibrium exists. Such an equilibrium consists of two interdependent Cournot equi1 libria. Its  existence  requires the existence of an inverse demand function ˆn W max Y† ! 0; pn for each type n 2 f1; : : : ; mg of good: (Since the goods of the same type produced by different firms are identical, they have the same prices in a 16

The stage game in Example 3.3 satisfies all assumptions made in this section. Nevertheless, (due to the same form of buyers’ production functions and symmetry of inverse demand functions for 1 their products) inverse demand functions for the analyzed market (functions ˆ1 1 and ˆ2 in the notation used below in the text) do not exist in it.

4.3 Impact on Consumer Welfare

73

non-collusive benchmark equilibrium.) The direct functions in the analyzed  demand  Q max 0; p !
j 2J .n/

X

subject to xi  0 & xi C

e x k .p/ 2 Y:

(4.32)

k2I nfi g

In the remainder of this section, we restrict our attention to the case where for each n 2 f1; : : : ; mg, ˆ1 n is twice continuously differentiable with respect to all arguments and strictly decreasing and concave in the output of type n of good.17Then, the second Cournot equilibrium contained in the benchmark long-run non-collusive equilibrium – a Cournot equilibrium in the analyzed market e y 2 Y exists. For every 2 f1; : : : ; mg and each j 2 J . / , e y j solves the maximization problem 0 @yj C max ˆ1 

X

0 e yk ; @

k2J . / nfj g

X

1

1

e ykA

k2J .n/

A yj n2f1;:::;mgnf g

   cj yj subject to yj 2 Yj :

(4.33)

Thus, the benchmark long-run non-collusive equilibrium is defined by 00

00 1 X BB B e yj A e x @@ˆn @@ j 2J . /

17

 2f1;:::;mg

11

1

CC AA

C A

n2f1;:::;mg

This ensures that in the Cournot oligopoly in the analyzed market, each producer’s payoff function is continuous in the whole vector of arguments and concave (and hence, quasi-concave) in his/her own output.

74

4 Efficiency of an SRPE and an SSPE

00

0

and

B B@ @ˆn @

X j 2J . /

1

11

e yj A

CC AA

 2f1;:::;mg

; n2f1;:::;mg

i.e., by a vector of traded quantities and a vector of prices that have one component for each type of good. We carry out the analysis in this section without giving the sufficient conditions for the existence of the benchmark long-run non-collusive equilibrium. These conditions are quite complicated.18 If a benchmark long-run non-collusive equilibrium does not exist, then the non-collusive behavior tends to be erratic. This reinforces the case for allowing collusion between firms not limited to one side of a market. If a benchmark long-run non-collusive equilibrium exists, then the results in this section identify the conditions under which it is inferior to a collusive outcome from Q the pointof viewof consumer welfare. Let .b x; b p / 2 X  n2f1;:::;mg; 0; pnmax be the benchmark long-run non-collusive equilibrium, in which all firms are active (i.e., none of them withdraws from the analyzed market), andb v be the vector of the firms’ profits in this equilibrium. The following analysis is valid when the upper bounds on the prices in the analyzed market are high enough.19 It is enough to fix  ˚   2 0; min b x j i j .j; i / 2 J  I;b xj i > 0

(4.34)

and set





max fUi .x/ j x 2 X g .n/  max max ; j i 2 I;b xj i > 0 j j 2 J b xj i   8n 2 f1; : : : ; mg ; (4.35)

e p max n

and

18

.n/ pjmax D e p max : n ; 8n 2 f1; : : : ; mg ; 8j 2 J

(4.36)

Given the assumptions already made in this section, a sufficient condition for the existence of a Cournot equilibrium in the market for the buyers’ products is that Ui is concave in xi and ci is convex in the whole vector of arguments for each i 2 I . Nevertheless, in order to obtain continuous inverse demand functions for the types of goods in the analyzed market, we need for each vector of the prices of the types types of goods either Cournot equilibria in the market for the buyers’ products that give the same vector of aggregate demands for the types of goods or continuous selection from the equilibrium correspondence. Moreover, in order to apply the standard sufficient condition for the existence of a pure strategy Nash equilibrium of a strategic form noncooperative game (see, e.g., [Osborne & Rubinstein (1994), Proposition 20.3, p. 20]) to a Cournot oligopoly on the supply side of the analyzed market, the inverse demand functions for the analyzed market would have to be such that each producer’s profit is quasi-concave in his/her output. 19 In a non-collusive benchmark equilibrium, the upper bounds on the prices in the analyzed market are determined by the production capacities of the firms in J and the inverse demand functions for the types of goods. Nevertheless, these upper bonds need not be high enough for the validity of the following analysis (see Steps 1–4 below).

4.3 Impact on Consumer Welfare

75

This ensures that – when for .j; i / 2 J  I with b x j i > 0, the quantity of his/her product that producer j 2 J supplies to buyer i is not lower than b x j i  and all firms earn positive profits – the price of no type n 2 f1; : : : ; mg of good exceeds pnmax . Set  X   / b .i b .i / D Q D b x ; Q j i  . / j 2J

 2f1;:::;mg

 2f1;:::;mg

      / .i / b b D Q b .i b .i / i D min 2f1;::;mg Q ,b D b i , for each i 2 I , Q  =Q i 2I  i 2I P Q D j 2J . / j .b x / for every 2 f1; : : : ; mg, and Q D Q  2f1;:::;mg . That .i / b is, Q is the equilibrium input vector of firm i 2 I , with one component for each type of input, Q is the equilibrium aggregate output of type of input, and Q is the equilibrium vector of the aggregate outputs of the inputs used by firms in I , with one component for each type of input. Since .b x; b p / is a long run equilibrium, we have b v  0. This – together with the fact that all firms are active and they have positive fixed costs – further x / >0 and p n > 0for each   implies that j .b  b b b b > 0 for n 2 f1; : : : ; mg, as well as Fi i > 0, i > 0, and Pi Fk k k2I

every i 2 I . It follows from (4.32), Assumption 4.6, and part (ii) of Assumption 4.8 b .i / D b that Q i Q.i / for each i 2 I . In order to avoid an uninteresting case (in which our results in this section would fail to hold because of the producers’ capacity constraints), we restrict our attention to the case where j .b x / < j for each j 2 J .   max b Q Db p  for each This implies that i < i for each i 2 I . Of course, ˆ1  b

2 f1; : : : ; mg. Moreover, for every i 2 I , i solves the maximization problem     max Pi Fi . i / ; Fk b k



k2I nfi g

Ci . i /  i

m X  D1

Fi . i /

/ b p Q.i : 

(4.37)

Thus, b satisfies the following first-order conditions: 2 4Pi



Ci0

  Fk b k

k2I



C Fi . i /

@Pi

   Fk b k @qi

3 k2I

  5 Fi0 b i

m   X   .i / b ˆ1 Q Q ; i D   D1

8i 2 I: From (4.33), we get first-order conditions:

(4.38)

76

4 Efficiency of an SRPE and an SSPE

  Q @ˆ1 

  ˆ1 Q C j .b x/ 

@Q

   cj0 j .b x / D 0; 8 2 f1; : : : ; mg ; 8j 2 J . / : (4.39)

Combining (4.38) and (4.39), we obtain 2 4Pi



  Fk b k



k2I

D

m X  D1

C Fi . i /

@Pi



  Fk b k @qi

3

k2I

    5 Fi0 b i  Ci0 b i

" / Q.i 

 # 1 Q @ˆ    cj0 . / j . / .b ; 8i 2 I; x /  j . / .b x/ @Q

(4.40)

where, for each 2 f1; : : : ; mg, j . / can be any element of J . / . Recall that for every 2 f1; : : : ; mg, ˆ1  is strictly decreasing in Q . Thus, (4.40) implies that 2 4Pi



  Fk b k



k2I

C Fi . i /

@Pi

   Fk b k @qi

3 k2I

  5 Fi0 b i

m   X   / 0 i > Ci0 b j . / .b Q.i c x / ; 8i 2 I:  j . /

(4.41)

 D1

For each i 2 I , consider the maximization problem x C z/  ci .b xi C zi /  max Ui .b

m X X    b p  C rj i b x j i C zj i

(4.42)

 D1 j 2J . /

subject to x C z/ck .b x k C zk / Uk .b

m X X    b p  C rjk b x jk C zjk  b vk ; 8k 2 I n fi g ;  D1 j 2J . /

(4.43) X     x C z/  b vj ; 8 2 f1; : : : ; mg 8j 2 J . / ; b p  C rjk b x jk C zjk  cj j .b k2I

zjk  b x jk ; 8 .j; k/ 2 J  I; X X zjk  j  b x jk ; 8j 2 J; k2I

(4.44) (4.45) (4.46)

k2I

p  ; 8 2 f1; : : : ; mg ; 8 .j; k/ 2 J . /  I; rjk  b

(4.47)

rjk  pjmax  b p  ; 8 2 f1; : : : ; mg ; 8 .j; k/ 2 J . /  I

(4.48)

4.3 Impact on Consumer Welfare

77

Thus, in the maximization problem (4.42)–(4.48), we seek a vector of the changes in the traded quantities given by vector b x and a vector of the changes in the prices given by vector b p that maximize the profit of buyer i without decreasing the profit of any other firm k 2 .J [ I / n fi g below b vk . (Note that r can specify different changes in price b p  for different producers in J . / and for different buyers.) It maximizes a continuous function on a nonempty compact set, and hence has a solution. Let .b z;b r/ be a solution of (4.42)–(4.48) for firm i 2 I . We proceed in four steps. Step 1. The changes in the prices in the analyzed market without any change in the traded quantities cannot change the sum of the profits of the firms in I [ J . Thus, they cannot increase the profit of buyer i without decreasing the profit of any other firm. Suppose that (starting from b i ) buyer i infinitesimally increases i . Further, suppose that he/she does so by proportionally increasing the purchased quantity of each type 2 f1; : : : ; mg of good from every producer j 2 J . / with b x j i > 0 and increases his/her expenditure on the output of each j 2 J with b xj i > 0 by an amount equal to the increase in producer j ’s production cost (keeping the quantities traded by and the expenditures of all other buyers unchanged). Using (4.41), it increases i ’s profit without changing the profit of any firm belonging to J .20 Taking into account (4.29) and part (ii) of Assumption 4.7, it does not decrease the profit of any firm in I n fi g. Therefore,b z ¤ 0. b Step 2. Starting again from i and using (4.41), a decrease in i with a proportional decrease in the purchased quantity of each type 2 f1; : : : ; mg of good from every producer j 2 J . / with b x j i > 0, and a decrease in i ’s expenditure on the output of each j 2 J with b x j i > 0 by an amount equal to the decrease in producer j ’s production cost (again keeping the quantities traded by and the expenditure of all other buyers unchanged) decreases i ’s profit.21 (Decreasing i ’s expenditure on some

20

We have d



m P

P

D1

j 2J . / Wx j i >0

b

0

D



.i/ m X B Q

@ .i/ Q

D1 m X

D1

cj

  j b x b xj i C d i

X

b .i/ .i / i Q

xOj i





Q

1    C cj0 j b x b xj i A

b

j 2J . / Wx j i >0

n    o  max cj0 j b x j j 2 J . / ; b xj i > 0 : Q.i/

Recall that in (4.41), for each 2 f1; : : : ; mg, j . / can be any producer belonging to the set J . / . 21 This argument is based on an infinitesimal change in i . Nevertheless, taking into account (4.30), part (ii) of Assumption 4.9, parts (ii) and (iii) of Assumption 4.7, (4.28), and Assumption 4.10, the difference between the right- and the left-hand side of (4.41) is nonincreasing in i . Thus, an infinitesimal decrease in i does not decrease this difference. Therefore, we can view any decrease

78

4 Efficiency of an SRPE and an SSPE

good j by less than the decrease in firm j ’s production cost would further decrease {K’s profit. Decreasing the expenditure by more than the decrease in firm j ’s production cost would violate the constraint for firm j in (4.44). This conclusion does not depend on proportionality of decrease in purchased quantities.22 ) Taking into account (4.29) and part (ii) of Assumption 4.7, a decrease in i does not increase the profit of any firm k 2 I n fi g. Therefore, .b z;b r/ withb zi < 0 andb zk D 0 for each k 2 I n fi g decreases (in comparison with .0; 0/) the sum of the profits of all firms.23 This implies that it cannot increase or leave unchanged i ’s profit without decreasing the profit of any other firm. Thus, it cannot solve the maximization problem (4.42)–(4.48). Step 3. With respect to (4.29) and part (ii) of Assumption 4.7, a decrease in k for any k 2 I n fi g does not increase the profit of firm i . Thus, using the results in Step 2, a decrease in k for more than one k 2 I without an increase in n for any n 2 I decreases the sum of the profits of all firms. Therefore, .b z;b r/ with b z < 0 cannot increase or leave unchanged i ’s profit without decreasing the profit of any other firm. Thus, it cannot solve the maximization problem (4.42)–(4.48). ˚  zj i < 0 ¤ ¿ and BC D ˚Step 4. Suppose that B D .j; i / 2 J  I j xj i > 0&b .j; i / 2 J  I jb zj i > 0 ¤ ¿. Since (as already shown in Step 1) .0; 0/ cannot solve the maximization problem (4.42)–(4.48) and .b z;b r/, by assumption, solves it, these changes increase the sum of the profits of all firms in J [I in comparison with .b x; b p /. Suppose first that the changes in the inputs corresponding to B take place,

in i as a sequence of infinitesimal decreases, apply the argument in the text to each element of this sequence, and reach the conclusion stated in the text. An analogous comment applies to the other arguments in this section that are based on the signs of the derivatives. 22 Suppose that for 2 f1; : : : ; mg, producer j 2 J . / has share !j 2 .0; 1 in i ’s increased purchase of type of good. Let X   Z D j b x b xj i  b x ki k2J . / nfj g Then, we have     i Q.i/ C 1  !j Q .i/ dcj Z C !jb

d i and X j 2J . / W!j >0

   D cj0 j b x !j Q.i/

n    o    cj0 j b x !j Q.i/  Q.i/ max cj0 j b x j j 2 J . / ; !j > 0 :



Thus, we can still use (4.41). We have considered the specific changes in the prices in the analyzed market for which the change in i ’s expenditure on good j 2 J equals the change in firm j ’s cost. Nevertheless, the prices in the analyzed market do not affect the sum of the profits of all firms in J [I . An analogous comment applies also to the following arguments in this section concerning the sum of the profits of all firms in J [ I . 23

4.3 Impact on Consumer Welfare

79

and then the changes in the inputs corresponding to BC take place. Further, suppose that (for both groups), the changes in the quantities of the goods in J purchased by the firms in I are accompanied by the changes in their expenditure on these goods equal to the changes in the production cost of the firms belonging to J , and that in both groups, the changes take place sequentially. (Thus, in each step of the succession of changes, the change in the production cost of the affected producer in J is well defined. Since the argument here concerns a change in the sum of the profits of all firms, it does not depend on the order in which the changes within a group take place.) The conclusion in Step 3 implies that the first group of changes decreases the sum of the profits of all firms in comparison with .b x; b p /. Thus, the second group of changes increases the sum of the profits of all firms in comparison with the result of the first group of changes. Moreover, the increase in the sum of the profits of all firms in comparison with .b x; b p / is greater when only the second group of changes takes place than when both groups of changes take place. (Of course, when only the second group of changes takes place, the changes in the expenditures on the goods in J are equal to changes in the production costs of the firms in J C C and† be the sum of the profits stemming only from the latter changes.) Let † of all firms after only the second group of changes takes place and after both groups C C of changes take place, respectively. Thus, † > † . As the prices in the analyzed C market do not affect the sum of the profits, † is also the sum of the profits of all firms generated by .b x Cb z; b p Cb r/. For each k 2 J [ I , lete vk be the profit of firm k generated by .b x Cb z; b p Cb r/. Taking into account the constraints (4.43) and (4.44), we havee vk  b vk for each k 2 J [ I . We consider two distinguished cases. First, there exists ` 2 J such that b x `k Cmax fb z`k ; 0g > 0 for each k 2 I . (That is, after the second group of changes in the traded quantities alone takes place, producer ` trades with each buyer in I .) As such, we can find such changes in the prices in the analyzed market after which the profit vk and P of each firm k 2 .J [ I / n fi g equalsb C the profit of buyer i equals †  k2.J [I /nfi g b vk . (We can do this in two rounds. First, we change the prices in such a way that each j 2 J earns profit b vj .Then, if some buyer k 2 I n fi g earns a profit exceeding b vk , we change the prices in a way that increases his/her expenditure on good ` to the level that leaves him/her with profit b vk and decreases by the same amount the expenditure of buyer i on good `. We repeat the second step until we obtain the desired result.) Since C † 

X k2.J [I /nfi g

C b vk > † 

X

e vk D e vi ;

(4.49)

k2.J [I /nfi g

we have a contradiction with the assumption that .b z;b r/ solves the maximization problem (4.42)–(4.48). Second, there exists no such ` 2 J . Take (arbitrary) 2 f1; : : : ; mg, n 2 J . / , and d 2 I with b x nd > 0. (Since n .b x / > 0, such d exists.) For each k 2 I with b x nk Db znk D 0, take n o x jk > 0 : k D min j 2 J . / W b

80

4 Efficiency of an SRPE and an SSPE

Fix  2 .0; /. (See (4.34) for the fixing of .) Replace the zero delivery by producer n to buyer k by , reduce the delivery by producer k to buyer k by , reduce the delivery by producer n to buyer d by , and increase the delivery by producer k to buyer d by . These changes leave the outputs – and hence, the production costs – of the producers in J unchanged. Since Assumption 4.1 is satisfied in this section, these changes leave the sum of the revenues of the firms in I , and the sum of their costs expressed by the functional value of functions ck , k 2 I unchanged. Thus, they leave the sum of the profits of all firms unchanged. This transforms the second case to the first case, and we again obtain a contradiction with the assumption that .b z;b r / solves the maximization problem (4.42)–(4.48). Therefore (taking also into account Steps 1–3) we haveb z > 0. Of course, (with respect to Assumption 4.6 and part (ii) of Assumption 4.7)b z>0 implies that fk .b x k Cb zk /  fk .b x k / for each k 2 I . Moreover, the latter inequality is strict for at least one k 2 I . Otherwise, we would have Uk .b x Cb z/ D Uk .b x/ and (using part (ii) of Assumption 4.8) ck .b x k Cb zk /  ck .b x k / for each firm k 2 I (with strict inequalityfor P at least one  k 2 I), but (using part (ii) of Assumption 2.1) P c .b x Cb z/ > c x / . Thus, the sum of the profits of all firms  j j 2J j j 2J j j .b would decrease, contradicting the assumption that .b z;b r/ solves the maximization problem (4.42)–(4.48). Assume that the firms are using pure strategies in game  .ı/ and V contains a vector of the firms’ stage game profits v that weakly Pareto dominates b v, is strictly Pareto efficient with respect to V C , and is the vector of firms’ equilibrium expected average discounted profits in an SRPE. Let .b x C z; b p CP r/ 2 X P generate P v. From the fact that v weakly Pareto dominates b v (and hence, k2J [I vk > k2J [I b vk ) and the arguments in Steps 2 and 3 above, it follows that we cannot have z  0. Then, the argument analogous to the one in Step 4 above shows that z > 0.24 This, together with the fact that b x C z increases the sum of the profits of all firms in comparison with b x , implies that fk .b x k C zk /  fk .b x k / for each k 2 I with strict inequality for at least one k 2 I . That is, at least one firm in I increases its output and none decreases its output. If the goods produced by the firms in I are consumer goods, in order to evaluate the impact on consumer welfare, we have to also take into account the prices of the goods produced by the firms in I . If no firm in I raises its price (as in the following example), consumer welfare increases. If some firms in I raise their prices, we can proceed in an analogous way as in Sect. 4.3.1. The result in this section is driven by two differences in the firms’ motivation between a non-collusive benchmark equilibrium and a collusive outcome. First, in a Cournot equilibrium on the supply side of the analyzed market, the firms are discouraged from increasing their output by its consequence – a decrease in price. In a

24

In Step 4, we have shown that when only the increases in traded quantities take place, we can increase the profit of buyer i (without violating any constraint in (4.43)–(4.44)) by more than when also the decreases in traded quantities take place. Here, we show that when only the increases in traded quantities take place, we can give each firm a profit no lower than its profit when also the decreases in traded quantities take place and we can give at least one firm a higher profit.

4.3 Impact on Consumer Welfare

81

collusive outcome, the benefit of an increase in the output can be divided between a producer and a buyer (with a positive impact on other buyers because the buyers’ products are complements). Since both the firms in J and the firms in I have a market power in the market for their products (unlike in Sect. 4.3.1 where the firms in J are price takers), we have a case of double marginalization.25 Nevertheless, a remedy, which is analyzed here, is a collusion, and not a vertical integration. Second, in a Cournot oligopoly in the market for the buyers’ products (i.e., the products of the firms in I ), an individual buyer does not take into account the positive effect of his/her increased output on the revenues of other buyers. In a collusive outcome, these effects are taken into account. Now, we give an example of the situation analyzed above.   Example 4.3. We have J D f1; P2g, I D f3; 4g, Yj D Œ0; 50 and cj yj D 2yj C 15 for each j 2 J , ci .xi / D j 2J xj i C 1 for every i 2 I , and each buyer uses the inputs purchased from the producers in J to produce a single good. The buyers’ production functions have the form f3 .x3 / D min f0:5x13 ; x23 g and f4 .x4 / D min fx14 ; 0:5x24 g. The inverse demand functions for their products have the form P3 .q 3 ; q4 / D max f39  2q3 C q4 ; 0g

(4.50)

P4 .q3 ; q4 / D max f39 C q3  2q4 ; 0g :

(4.51)

and Since each type of good is produced only by one producer, J .1/ D f1g, J D f2g, and Y† D Y . The buyers’ production and cost functions implythat  .3/ max D 25, D 25, f ; Q2.3/ D Q Q.3/ D .2; 1/, Q.4/ D .1; 2/, max 3† 3 4 1 o   n o n .3/ .3/ .4/ .4/ .4/ .4/ min 0:5Q1 ; Q2 , f4† Q1 ; Q2 D min Q1 ; 0:5Q2 , and F i . i / D i and Ci . i / D 3 i C 1 for each i 2 I . For given prices p1 and p2 of the goods produced by firms 1 and 2, the Cournot equilibrium in the market for the outputs of firms 3 and 426 is .2/

and

25

3 2 3 .p1 ; p2 / D 12  p1  p2 5 5

(4.52)

2 3 4 .p1 ; p2 / D 12  p1  p2 : 5 5

(4.53)

This problem was first pointed out by Cournot ([Cournot (1897)]; the original French publication was in 1838). See [Spengler (1950)] for its analysis. 26 We compute the Cournot equilibrium using the buyers’ reaction functions obtained from the solutions to the optimization problem (4.37) for i D 3; 4. The formulae given in the text are valid for prices for which they give nonnegative outputs. This condition is satisfied in the non-collusive benchmark equilibrium. Therefore, here, we do not deal with the prices for which the above formulae are not valid. An analogous comment applies to the demand functions and the inverse demand functions for the analyzed market given below.

82

4 Efficiency of an SRPE and an SSPE

Using (4.52), (4.53), Q.3/ , and Q.4/ , we obtain the demand functions for the analyzed market 8 7 ˆ1 .p1 ; p2 / D 36  p1  p2 ; (4.54) 5 5 7 8 (4.55) ˆ2 .p1 ; p2 / D 36  p1  p2 ; 5 5 and from them, we have the inverse demand functions for the analyzed market 8 7 ˆ1 1 .Q1 ; Q2 / D 12  Q1 C Q2 ; 3 3

(4.56)

7 8 (4.57) ˆ1 2 .Q1 ; Q2 / D 12 C Q1  Q2 : 3 3 Using (4.56) and (4.57), we compute the Cournot equilibrium in the analyzed y 2 D 10=3. This gives the prices in the analyzed market market: b y 1 D 10=3 and b b p 1 D ˆ1 .10=3; 10=3/ D 98=9 and b p 2 D ˆ1 1 2 .10=3; 10=3/ D 98=9; the outputs of the buyers b 3 D 10=9 and b 4 D 10=9; the prices of the goods produced by them P3 .10=9; 10=9/ D P4 .10=9; 10=9/ D 341=9; the traded quantities in the analyzed market b x 13 D 20=9, b x 23 D 10=9, b x 14 D 10=9, b x 24 D 20=9; and the profitsb v1 D b v2 D 14:63 and b v3 D b v4 D 1:4691: The maximization of the sum of the profits of all firms gives the unique vector of the traded quantities in the analyzed market:  max max max max  ; x23 ; x14 ; x24 D .30; 15; 15; 30/ : x max D x13

(4.58)

Thus, the buyers’ outputs are 3 D 15 and 4 D 15, the prices of the goods produced by them are P3 .15; 15/ D P4 .15; 15/ D 24 < 341=9, the outputs of the firms in J are 1 .x max / D 45 and 2 .x max / D 45, and the maximized sum of the profits is   D 418. Note that the buyers’ outputs are higher and the prices of the goods produced by them are lower than in the non-collusive benchmark equilibrium. We choose the following upper bounds on the prices in the analyzed market: p1max D p2max D 671. These upper bounds satisfy both requirements, (2.6) and (4.35). Suppose that the firms agree on the following prices in the analyzed market: p1 D p2 D 5. This gives profits v1 D v2 D 120 and v3 D v4 D 89. Thus, v strictly Pareto dominates b v. Therefore, it is in the interest of all firms to move from the non-collusive benchmark equilibrium to the collusive outcome with prices p1 and p2 in the analyzed market. As already noted, such movement increases the buyers’ outputs and decreases the prices of the goods produced by them. Therefore, if the buyers’ products are consumer goods, it increases consumer welfare. It can be shown that this example satisfies all assumptions of Proposition 3.4. (Concerning requirements (3.33) and (3.34), we can take pji D 4 for each .j; i / 2 J  I .) Thus, the described collusive outcome can be sustained in an SSPE of  .ı/ for a ı sufficiently close to one.

References

83

References Baumol, W. J.: “On the Proper Cost Tests for Natural Monopoly in a Multiproduct Industry,” American Economic Review 67 (1977), 809-822. Baumol, W.J.: “Contestable Markets: An Uprising in the Theory of Industry Structure,” American Economic Review 72 (1982), 1-15. Cournot, A.: Research into the Mathematical Principles of the Theory Wealth. Edited by N. Bacon. New York: MacMillan, 1897. Osborne, M.J. and A. Rubinstein: A Course in Game Theory. Cambridge, MA: MIT Press, 1994. Spengler, J.J.: “Vertical Integration and Antitrust Policy,” Journal of Political Economy 58 (1950), 347-352. Varian, H.R.: Microeconomic Analysis. Third Edition. New York, W.W. Norton, 1992.

Chapter 5

Afterword

In this book, we have shown that (for a discount factor close enough to one and under certain conditions that we have identified in Chaps. 2–4), the firms on both sides of a market can organize and sustain a collusive scheme with the following properties. First, all continuation equilibrium payoff vectors are strictly Pareto efficient. Therefore, the grand coalition cannot abandon a punishment of a previous deviation by a smaller coalition (i.e., weakly Pareto improve the vector of the continuation payoffs of the members of a deviating coalition) without harming at least one firm that did not deviate.1 Thus, a collusive scheme is an SRPE. Moreover, if the requirements of Proposition 3.4 are satisfied or # .I / D 1 and the requirements of Proposition 3.2 are satisfied, then for each proper coalition C of the firms in every continuation equilibrium, for given continuation equilibrium strategies of the firms outside C , the vector of the payoffs of the members of C is strictly Pareto efficient. Thus, a collusive scheme is an SSPE. Second, when the sum of the firms’ continuation equilibrium payoffs is maximized in an SRPE in some subgame, then – in each period along the equilibrium path in such a subgame – the group of active producers of each type of good forms a natural oligopoly (or a natural monopoly if there is only one active producer) for the equilibrium output of that type of good. That is, the equilibrium output of each type of good is produced with the lowest possible production cost. This result also holds when the firms capable of producing a certain type of good have different cost functions. This shows that a collusive scheme sustainable in an SRPE with the

1

This shows why an SRPE, which is based on the strict Pareto efficiency of the continuation equilibrium payoff vectors, is preferable to various versions of renegotiation-proofness (e.g., [Farrell & Maskin (1989)] and [Maskin & Tirole (1988)]), which are based on the weak Pareto efficiency of the continuation equilibrium payoff vectors with respect to some set of “allowed” continuation payoff vectors (e.g., in a weakly renegotiation-proof equilibrium of [Farrell & Maskin (1989)] with respect to the set of all continuation equilibrium payoff vectors of an equilibrium that is weakly renegotiation-proof or as in [Maskin & Tirole (1988)], with respect to the set of all continuation payoff vectors that can be generated by Markov strategies.) In the latter case, an abandonment of the punishment of a previous deviation can increase the continuation payoff of each deviator without harming any player who did not deviate. In such a case, there need not be a player objecting to the abandonment of the punishment.

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maximization of the firms’ continuation equilibrium profits is superior to the noncollusive equilibria in the oligopolies containing firms with different cost functions from the point of view of production efficiency.2 When the buyers are retailers and the sum of the firms’ continuation equilibrium payoffs is maximized in an SRPE in some subgame, then – in each period along the equilibrium path in such a subgame – the group of active retailers forms a natural oligopsony (or a natural monopsony if there is only one active retailer) for the equilibrium vector of the sales in the retail market (which is also the equilibrium vector of the purchased quantities in the analyzed market). That is, the sum of the selling costs for the equilibrium vector of the sales in the retail market is minimized. This result also holds when the buyers have different cost functions (expressing their selling costs). We can combine this with the result on a natural oligopoly (or a natural monopoly) on the demand side of the analyzed market. Thus, the sum of the production and selling costs for the equilibrium vector of the sales in the retail market is minimized. For such an evaluation of the efficiency of a market structure on both sides of the analyzed market, the expenditure on the goods sold in it is irrelevant – the sum of the buyers’ expenditures equals the sum of the producers’ revenues. Third, we have identified two types of a market structure in the analyzed market for which a non-collusive equilibrium is (under some additional conditions) inferior from the point of view of consumer welfare to a collusive scheme sustainable in an SRPE that weakly Pareto improves the vector of the firms’ profits. (This result does not depend on the maximization of the sum of the firms’ continuation equilibrium payoffs in an SRPE.) A monopsonist on the demand side of the analyzed market, who chooses the purchased quantities, along with price taking behavior on the supply side is the first of them. In this case, we obtain the result that collusion increases consumer welfare if the goods sold in the analyzed market are complements (or independent) from the point of view of the monopsonist’s revenue and the marginal cost of using any of them is nonincreasing in the used quantity of any other good (see Assumption 4.5). Here, the result is driven by the fact that in a non-collusive equilibrium, the monopsonist is discouraged from increasing the purchased quantities of inputs (that would increase his/her sales to consumers) by the increases in their prices that would increase his/her expenditure on them by an amount exceeding the increases in the producers’ costs. Moreover, there is no “input cannibalism” – an increase in the use of one input does not decrease the marginal revenue from another input (and it does not increase the marginal cost of using another input).3 2

Consider, for example, a Cournot oligopoly producing a single homogeneous good, with differentiable inverse demand function at industry outputs giving a positive price and differentiable convex cost functions. Take a Cournot equilibrium in it. For each firm with a positive equilibrium output, its marginal cost equals the sum of the equilibrium price and the product of its output and the derivative of the inverse demand function. Thus, the firms with different outputs have different marginal costs. This implies that the sum of the costs of producing the equilibrium industry output is not minimized. 3 Collusion can also increase consumer welfare when an increase in one input decreases the difference in the marginal revenue from and the marginal cost of using another input. This happens if the latter effect is outweighed by the elimination of the increases in the prices of inputs (exceeding the

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A Cournot oligopoly on the supply side of the analyzed market coupled with price taking behavior on its demand side is another market structure that we have compared with a collusion. In this case, we obtain the result that collusion increases consumer welfare if (besides some standard and technical conditions that we do not repeat here) the buyers are themselves producers, each of them produces one type of good using Leontieff technology, their outputs are complements, and each buyer’s marginal cost of using any input is nondecreasing in the used quantity of any other input (see Assumptions 4.6–4.9). Here, the result is driven by the fact that in a non-collusive equilibrium, the producers of inputs are discouraged from increasing their outputs by the decreases in their prices and buyers do not take into account the positive impact of their increased outputs on the revenues of the other buyers. The possible positive impact of collusion on consumer welfare described above suggests that the antitrust policy in the US and the competition policy in the European Union (and the antitrust policies in other countries)4 should not view collusion that involves firms on both sides of a market as illegal. It should either view it as legal (for reasons described below) or in each case, examine its impact on consumer welfare. The legislators who included the prohibition of cartel agreements5 restricting competition into antitrust laws had obviously price cartels containing producers of either identical products or close substitutes in mind. Moreover, in the case of the Sherman Antitrust Act in the US, their thinking was influenced more by a fear of a concentration of political power enabled by a concentration of economic power than by a fear of the economic consequences of cartels.6 Nevertheless, currently antitrust laws prohibit all cartel agreements that directly or indirectly set prices. Thus, the prohibition applies also to cartels containing firms on both sides of a market and even if the products sold to final consumers are complements. Our analysis in Sect. 4.3 shows that under certain conditions, cartels containing firms on both sides of a market – besides being beneficial for participating firms – are beneficial for consumers. Such cartels should be legalized. Thus, the question is whether or not current antitrust laws (on both sides of the Atlantic) make this treatment possible. The collusive schemes analyzed in this book represent the agreements about the sequences of contracts (which specify prices and traded quantities) between trading

increases in the producers’ costs) caused by increased purchases of them. (Example 4.2 describes such a situation.) Collusion can also be beneficial for consumers when there is more than one buyer and the assumptions of Sect. 4.3.2 hold. 4 Unless we deal specifically with the competition policy or competition law of the European Union, we use the term “antitrust policy” or “antitrust law.” 5 We use the term “cartel agreement” also for cartel agreements that are not legally binding and for tacit collusion that has the same effects as a cartel agreement that is not legally binding. Antitrust laws prohibit each of these three types of coordinated behavior if it restricts competition. 6 In 1890, Congressmen (or their advisers) could hardly have the knowledge of economics needed for the evaluation of the economic consequences of cartels and other practices restricting competition. Alfred Marshall’s “Principles of Economics” was first published in 1890 – exactly the same year, in which the Sherman Antitrust Act was passed.

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partners. They are not legally binding. They make contracts in each period (except the first one) conditional on the history of contracts concluded up to that period. Only the firms participating in them are affected by the fixing of prices in them. These schemes do not fix the purchase or selling prices or any other trading conditions for any third party (i.e., for any firm that does not participate in them). It is true that an agreement on the quantities traded in the analyzed market affects the prices and quantities sold in the market for the products sold by the buyers (i.e., by the firms in I ; this market is the retail market if the firms in I are retailers and is the market for the goods produced by the firms in I if they are themselves producers). Nevertheless, it affects the prices and quantities sold in the latter market only through the fixing of the quantities of inputs used by the firms that sell in it. Any contract between a seller and a buyer, who is not a final consumer of the purchased goods, affects in the same way, the prices and quantities sold in the downstream market (i.e., in the market in which the buyer from the contract is a seller). Despite this, the contracts are not viewed as an instrument of price fixing or limiting production in downstream markets. Moreover, the collusive schemes analyzed in the present book can be beneficial for the buyers in the downstream market. Therefore, we think that they are not the cases of price fixing or limiting of production. If they are implemented in the US, they are not “a conspiracy in restraint of trade or commerce among the several States, or with foreign nations.” Thus, they do not violate Sect. 1 of the Sherman Antitrust Act. Since they do not involve acquisition by one corporation of the stock of another, they do not violate Sect. 7 of the Clayton Antitrust Act. If they are implemented in the European Union, they do not have as their “object or effect the prevention or restraint or distortion of competition within the Common Market.” Thus, they do not violate [EU (2010a), Article 101(1)]. Moreover, collusive schemes that we have analyzed increase the output of at least one good without decreasing the output of any other good. They are beneficial for consumers. Thus, they contribute to improving the production and distribution of goods.. Therefore, the European Commission should view them as compatible with the Common Market also on these grounds, according to [EU (2010a), Article 101(3)]. Allowing collusive schemes studied in the present book can yield an additional benefit. Suppose that all firms involved in the collusive schemes beneficial for consumers merge or through acquisitions create a business unit managed from one center (create a concentration in the terminology of the European Union’s competition law; henceforth we use only the term “merger”). Such a new business unit can achieve everything that can be achieved through analyzed collusive schemes. (It can achieve the same even more easily because the decisions of its top management are binding for the lower levels of management. Hence, there is no need to use punishment schemes that are needed to sustain a collusion in an SRPE and an SSPE.) Such a merger would have significant vertical components and if products of at least some participating firms (namely, those on the demand side of the market in question) are complements, it is likely that it would be cleared by the antitrust

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authorities on both sides of the Atlantic.7 If in the future, conditions change and for some part of the new business unit (created by the merger), continued participation in it is no longer useful (i.e., participation in the business unit created by the merger brings it lower profit than cooperation with some other firm), such a part cannot secede. In the case of a collusive scheme, a participating firm can withdraw from it whenever a withdrawal is useful for it.8 Thus, if a collusive scheme can be legally implemented, then the firms would prefer its implementation to a merger.9 Further, suppose that in the future, an antitrust authority finds that the operation of the business unit created by the merger violates the antitrust laws. The authorities in the European Union cannot order (or seek through the court) a divestiture. In the US, it is unlikely that the Antitrust Division of the Department of Justice will succeed in divestiture. In the case of a collusive scheme, if an antitrust authority finds that the participating firms violate antitrust laws, it can prohibit the continuation of their cooperation. Therefore, both from the point of view of participating firms and from the point of view of antitrust authorities, collusive schemes are preferable to mergers because the former can be undone much more easily. As already noted, collusion between firms on both sides of a market should not be viewed per se as a violation of antitrust laws. Antitrust authorities should adopt one of the following approaches to it. First, they can view it as per se legal (for reasons explained above) and intervene only when some specific action of participating firms is detrimental for consumers or violates the right of other firms to freely choose their business partners (e.g., by exclusionary vertical contracts that cannot be justified on efficiency grounds10). Second, they can introduce an approval process for cartel agreements between firms on both sides of a market, analogous to the current approval process for mergers, stipulating the right to revoke the approval if the conditions change in the future. In the latter case, the approved cartel agreements would be legally binding. Our results (especially those on monopsonistic markets – see Propositions 3.2 and 3.3 and Sect. 4.3.1) can shed light on an important real world economic problem: trading between (relatively small) farmers and other producers of foodstuffs (or producers of other consumer goods) and chain-stores. The producers of foodstuffs are weaker partners in this relationship. The chains-stores have considerable

7 The evaluation of the proposed mergers by the Antitrust Division of the US Department of Justice is quite accommodating, at least since the period of the first Reagan administration. The European Commission became more accommodating after critique of its handling of the proposed merger between General Electric and Honeywell (which it prohibited in July 2001). For an analysis of the impact of this case, see [Kolasky (2006)]. I am grateful to Tímea Antalicsová for drawing my attention to the latter paper. 8 Recall that in our models, punishments have the form of a modified contract. Thus, they are harmless for a firm that has found more attractive trading partners in another separated market. 9 [Chandler (1990), p. 424] points out that the legality of cartelization in Germany reduced the pressure for industry-wide mergers. 10 See [Whinston (2006), chapter 4] for the economic evaluation of the various forms of exclusionary vertical contracts.

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market power.11 Our results show that collusion can improve the situation of both the producers of foodstuffs and chain-stores and increase consumer welfare.

References Boel, M.F.: “Food Prices in Europe: What the CAP Can and Can’t Do,” Konrad Adenauer Stiftung Lecture, Brussels, 27 January 2009. http://europa.eu/rapid/pressReleasesAction.do? reference=SPEECH/09/25&format=HTML&aged=0&language=EN&guiLanguage=en Chandler, A.D. Jr.: Scale and Scope. The Dynamics of Industrial Capitalism. Cambridge, MA: The Belknap Press of the Harvard University Press, 1990. “Consolidated Version of the Treaty on the Functioning of The European Union,” Official Journal of the European Union C83/49, 30 March 2010. Farrell, J. and E. Maskin: “Renegotiation in Repeated Games,” Games and Economic Behavior 1 (1989), 327-360. Kolasky, W.: “GE/Honeywell: Narrowing, But Not Closing, the Gap,” Antitrust Magazine, Spring 2006, 69-76. Maskin, E. and J. Tirole: “A Theory of Dynamic Oligopoly II: Price Competition, Kinked Demand Curves, and Edgeworth Cycles,” Econometrica 56 (1988), 571-599. Whinston, M.D.: Lectures on Antitrust Economics. Cambridge, MA: MIT Press, 2006.

11

According to [Boel (2009)], “the retail sector has the strength of a giant.” Hence, in a model, its market power can be approximated by a monopsony power.

Index

Antitrust law, see antitrust policy Antitrust policy, vi, 5, 88, 89 cartel, see collusion Celler Antimerger Act, 5 Clayton Act, 88 collusion, 5, 67, 74, 81, 86–90 merger policy, 5, 88, 89 Robinson-Patman Act, 5 Sherman Act, 87, 88 Treaty on the Functioning of The European Union, 88 Aumann, R.J., 3, 15

Baumol, W.J., 53, 54, 59 Bernheim, B.D., 4 Boel, M.F., 90

Cartel, see collusion Chain-stores, v, 6, 89 Chandler, A.D. Jr., 5 Collusion, 5, 61, 67, 74, 81, 86–90 Competition policy, see antitrust policy Complements, 63, 67, 72, 81, 86–88 Consumer welfare, vi, 1, 53, 54, 61, 67, 68, 70, 74, 80, 82, 86, 87, 90 Cooperative managerial capitalism, 5 Cournot oligopoly, 6, 53, 81, 87 Cournot, A., 81

Double marginalization, 81

Farrell, J., 4

Greenhut, M.L., 5

Hung, C., 5 Kolasky, W., 89 Kreps, D.M., 6 Maskin, E., 4 Mertens, J.F., 1 Monopsonist, see monopsony Monopsonistic market, see monopsony Monopsony, 6, 26, 29, 30, 33–35, 53, 57, 62, 63, 65–69, 86, 89 natural monopsony, 53, 58–61, 86 Natural monopoly, 53, 55–58, 85, 86 Natural monopsony, 53, 58–61, 86 Natural oligopoly, 53–58, 85, 86 Natural oligopsony, 53, 58–61, 86 Norman, G., 5 Peleg, B., 4 Price taking, 6, 53, 86, 87 Ray, D., 4 Renegotiation-proof equilibrium, 4 internally consistent equilibrium, 4 strict renegotiation-proof equilibrium, see SRPE strong renegotiation-proof equilibrium, 4 strongly consistent eq., 4 weakly renegotiation-proof equilibrium, 4 Retail market, 1, 9–11, 58, 59, 61, 67, 68, 86 Rubinstein, A., 3 Scheinkman, J.A., 6 Selten, R., 3

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92 Spengler, J.J., 81 SRPE, 3–5, 17, 21, 53, 56, 61, 62, 66, 70, 80, 85, 86, 88 SSPE, 3, 5, 17–19, 21, 26, 32, 33, 36, 38, 53, 57, 61, 82, 85, 88 Strict renegotiation-proof equilibrium, see SRPE Strict strong perfect equilibrium, see SSPE

Index Strong Nash equilibrium, 3, 4 Strong perfect equilibrium, 3

Tirole, J., 4

Whinston, M.D., 4, 89