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Endogenous Market Power Endogenous Market Power Marek Weretkay August 5, 2010 Abstract In this paper we develop a framework to study thin markets, in which all traders, buyers, and sellers are large, in the sense that they all have market power (also known as bilateral oligopoly). Unlike many IO models, our framework does not assume a priori that some traders have or do not have market power because “they are large or small.” Here, market power arises endogenously for each trader from market clearing and optimization by all agents. This framework allows for multiple goods and heterogeneous traders. We de…ne an equilibrium and show that such equilibrium exists in economies with smooth utility and cost functions and is determinate. The model suggests that price impact depends positively on the convexity of preferences or cost functions of the trading partners. In addition, the market power of di¤erent traders reinforces that of others. We also characterize an equilibrium outcome: Compared to the competitive model, the volume of trade is reduced and hence is Pareto ine¢ cient. JEL classification: D43, D52, L13, L14 Keywords: Thin Markets, Bilateral Oligopoly, Walrasian Auction The problem of the exchange of goods among rational traders is at the heart of economics. The study of this problem, originating with the work of Walras and re…ned by Fisher, Hicks, Samuelson, Arrow, and Debreu, provides a well-established methodology for analyzing market interactions in an exchange economy. The central concept of this approach is competitive equilibrium, wherein it is assumed that individual traders cannot a¤ect prices. Price taking behavior is justi…ed by the informal argument that the economy is so large that each individual trader is negligible and hence has no impact on price. Such an argument, however, cannot be applied to thin markets, those markets in which the number of buyers and sellers is small and hence each buyer and seller does have market power. The two-sided or one-sided competitive models (e.g. Cournot) cannot be modi…ed in an obvious way to include only large traders, as the presence of competitive consumers that generate a downward sloping demand for an industry is essential for the Cournot game to be I am indebted to Truman Bewley, John Geanakoplos, for their guidance and support. I would also like to thank Rudiger Bachmann, Jinhui Bai, Andres Carvajal, Herakles Polemarchakis, Sergio Turner, Donald Brown and participants of the 1st Annual Caress-Cowels conference. I especially would like to thank Marzena Rostek for invaluable comments and suggestions. All errors are my own. yUniversity of Wisconsin-Madison, Department of Economics, 1180 Observatory Drive, Madison, WI 53706. E- mail: [email protected]. 1 well de…ned. The goal of this paper is to develop a framework that can be used to study markets with such bilateral oligopolies. There are many real-world examples of thin markets. A vast amount of empirical evidence shows that a signi…cant fraction of total trade in …nancial markets is actually accounted for by large institutional traders who do recognize their price impact.1 Other examples can be seen in many upstream markets, such as the automobile or pharmaceutical industries. It has been documented that in such markets, the market power of suppliers is countervailed by the monopsony power of a few buyers. Our approach has the following advantages over the existing models of market interactions. Unlike, for example, the competitive, monopoly, or Cournot model, ours does not impose a priori restrictions on price impacts of any trader (e.g. by assuming competitive consumers), but derives these impacts as part of an equilibrium based on the principles of market clearing and optimization. Neither does the model discriminate between consumers and producers or, more generally, buyers and sellers. Finally a noncompetitive equilibrium from this paper exists and is determinate (and hence is generically localy unique) under the general assumptions for utilities and cost functions. The model presented herein predicts certain speci…c e¤ects of thin trading: The price impact of each trader increases with the convexity of the preferences or cost functions of the trading partners and depends negatively on the number of traders. Price impacts mutually reinforce each other. Thin trading is associated with a smaller than competitive volume of trade, as buyers and sellers have incentives to reduce their trades. The e¤ects of noncompetitive trading on prices depend on the speci…cation of the economy. In a pure exchange economy with identical utility functions, the sign of the price bias is the same as the sign of the third derivative of the utility function. Typically, the welfare cost is not shared uniformly by traders, and in symmetric economies also depends on a sign of the third derivative. Therefore, the model gives testable predictions associated with thin trading. We should stress also that although the concept of a noncompetitive equilibrium is not de…ned as a Nash equilibrium in a game, the model does have a game theoretic foundation. In Section 4 (and in more detail in [24]) we show that the concept of noncompetitive equilibrium re…nes a set of Nash equilibria in a class of market games wherein players choose quantities contingent on prices (e.g. supply functions), and the games are known to have a continuum of equilibria.2 We show that the Nash equilibrium corresponding to a noncompetitive equilibrium is selected in the process of learning where traders estimate the slopes of their market demands, and act optimally, given their estimates. In addition, we provide a rationale for such selection: agents can learn their price impacts solely from the data from their past trades and market prices, and they do not need to have any information about the preferences, costs functions, or even the number of trading partners. Therefore, the assumptions behind the learning model …t many anonymous markets well. 1 For evidence on the robustness of price impacts of institutional traders in …nancial markets see [6], [7], [14], [15] and [17]. 2 The examples of such games include a Nash in Supply Functions game or a dynamic bargaining game in a Walrasian auction. 2 The proposed framework is related to the literature on Nash in Supply Functions games. Such games have been successfully applied to modeling oligopolistic industries (see [8], [13], and [16]) or …nancial markets (see [18] and [23]). It is well known that in deterministic games with Supply Functions equilibria are not determinate (see [8] and [16]). Therefore, the existing literature re…nes the set of Nash equilibria in various ways: By assuming symmetric producers and a market demand with unbounded noise (for example, in [16]), by focusing on quadratic models with some noise and by restricting strategy spaces to linear supply functions (in [18], [23]), or …nally, by requiring that traders report to the auctioneer their marginal cost functions (in [13]). The framework developed in this paper is more general than these models. It does not assume any symmetries among the traders and de…nes equilibrium for non-quadratic costs and utility functions. There is no exogenous, noisy demand and the model allows for many goods. The equilibrium selection results from endowing the agents with consistent beliefs about the slopes of their market demands and assuming that agents act optimally, given such beliefs.3 The model in this paper is also related to the literature on consistent conjectural equilibrium in a duopoly model (see [2], and [20]). A conjectural approach, however, does not have a game theoretic foundation and is less general, as the conjectural equilibrium exists only for symmetric producers with quadratic cost functions (see the debate [3] and [21]). Finally the model from this paper is, to some extent, related to the literature on market power in the general equilibrium framework that started in [19] and was followed by [1], [9], [10], [11] and [12]. These are highly complex models with equilibria that are not determinate. The remainder of the paper is organized as follows: In Section 1, we de…ne an economy and introduce the concept of a noncompetitive equilibrium. In Sections 2 we discuss how price impact a¤ects consumer and …rm choice. Section 3 is the heart of the paper, and therein, we characterize a noncompetitive equilibrium: Its existence, determinacy, ine¢ ciency, and convergence in deep markets to a competitive one. In Section 4 we give a game theoretic foundation for the proposed equilibrium and relate it to the existing literature. In the last section we study equilibrium with free entry. The proofs of all results are in the Appendix. 1 Economy and Equilibrium In this paper we develop a framework to study thin markets. We focus on an economy , de…ned E as follows. is a one-period economy with a numèraire and non-numèraire, and with two types E of traders, I consumers and J producers. is the set of all consumers, and is a set of …rms. I th J th Subscript i , for example xi, indicates that the variable refers to the i consumer, and j 2 I th subscript denotes variables of j …rm. In particular, xi R+ (ei R++) is a consumption th I I 2 2 (endowment) by the i the consumer, and x R+ (e R++) is an allocation (initial allocation) 2 2 of goods across consumers. Consumers’endowments of shares to …rms’pro…ts and endowments of numèraire are without loss of generality, normalized to zero. For consumer i, a trade is de…ned as a net demand ti xi ei, and for …rm j, a trade is a negative of a supply tj = yj. In the 3 To some extent, the theory of slope takers is related to the literature on auctions of shares.
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