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. I would like to thank Marzena Rostek for pointing out the connection.
3 proposed framework, markets do not discriminate among consumers and …rms, and both types of market participants are modeled as traders. Therefore, it is convenient to introduce the following notation: The set of all traders in the economy is denoted by , the number of traders is N I [ J N = I + J, and the typical trader is indexed by n = i; j.
Consumer i utility from consuming a non-numèraire is given by ui : R+ R. Array u = ui i ! f g 2I speci…es utility functions for all consumers. Another commodity is a numèraire or unit of account.
If a numèraire mi R is spent optimally outside of , it gives a positive constant marginal 2 E utility, which without loss of generality, is normalized to one. Next we present the assumptions on . Consumer i maximizes a quasi-linear utility function Ui(xi; mi) = ui(xi) + mi, satisfying E assumptions A1-A3 for all i : 2 I A1) Di¤erentiability: ui( ) is twice continuously di¤erentiable; A2) Strict monotonicity and strict concavity: u ( ) > 0, and u ( ) < 0; i0 i00 A3) Interiority: limx 0 ui0 (x) > ui0 (ei ) for all i; i0 ; ! 0 2 I Condition A3 gives a common lower bound on the marginal utility at zero and is strictly weaker than the standard Inada condition. For example, it allows for quadratic utility functions. Firms are producing non-numèraire commodity, using numèraire as an input. Cost functions cj( ) satisfy the assumptions A4-A7 for all j : 2 J A4) Di¤erentiability: cj( ) is twice continuously di¤erentiable; A5) Strict monotonicity and strict convexity: c ( ) > 0 and c ( ) > 0; j0 j00 A6) Zero inaction cost: cj(0) = 0;
A7) Interiority limy 0 cj0 (y) = 0 and 0 < limy 0 cj00(y) < . ! ! 1 Array c = cj j speci…es cost functions for all …rms. By assumption any economy = (u; e; c), f g 2J E has two or more traders, N 2, with at least one consumer, I 1. An economy that satis…es E assumptions A1-A7 is called smooth.
1.1 De…nition of a Noncompetitive Equilibrium
Suppose that trader n observes non-numèraire price p R+, while demanding tn R. It is 2 2 reasonable to assume then that any deviation from the observed trade in a thin market should a¤ect the price. A price change p = p p, triggered by a deviation, tn = tn tn, is called a trader’s price impact. More formally, price impact pn(tn) is a map R R. Provided that the ! optimization problem of each agent is convex, we can restrict attention to models of price impact that are linear. We can do so, because even if traders face a nonlinear demand, the only information they use is the …rst-order derivative of this function, evaluated at the equilibrium trade, which is, 4 of course, the slope of a linearized demand. Linear pn(tn) can be written as
pn(tn) = M ntn = M n(tn tn); (1) 4 In all the examples studied in this paper the equilibrium optimization problems of all agents are convex. In Section 4, in which we provide a game theoretic foundation for noncompetitive equilibrium, we also characterize settings for which the convexity condition is satis…ed.
4 where scalar Mn is a price impact, which (endogenously) will be positive. Price impact measures a change of the price of a commodity caused by an increase of the demand by one unit. When making decisions, traders take their price impacts Mn as given. With the observed values of p, tn and M n, trader n faces a linear inverse (residual) supply function
p (tn) =p + M n(tn tn): (2) p; tn;Mn
Given function p ( ), the total payment for non-numèraire trade tn is equal to p (tn)tn. p; tn;Mn p; tn;Mn The derivative of this product indicates how much the total cost of trade increases if the demand goes up marginally, or equivalently, what the bene…t is in terms of a numèraire from the marginal reduction of a non-numèraire demand. Consequently, the derivative can also be interpreted as a marginal revenue from selling the good
@(pp; tn;M n (tn)tn) MR(tn) = (3) @tn =p + M n(2tn tn):
In particular, at the observed trade tn and prices p, marginal revenue becomes MR(tn) =p +M ntn.
We now de…ne a noncompetitive equilibrium. Let M Mn n , and t tn n . Intuitively, f g 2N f g 2N a noncompetitive equilibrium is de…ned as a triple of prices, trades, and price impacts (p; t; M ) such that 1) all markets clear; 2) traders maximize their utility or pro…t functions, given market prices and assumed price impacts; and 3) assumed price impacts coincide with the true price impacts. We maintain two key principles of a competitive theory, namely, optimization and market clearing. The only departure from the competitive equilibrium model is that instead of a priori de…ning price impacts to be equal to zero, M = 0, we allow the traders to have nonzero estimates. The price impacts are not arbitrary though, but they do coincide with the price response of the market to a unilateral deviation in trade. For the sake of transparency, we …rst give a de…nition of equilibrium, assuming that the notion of consistency of price impacts is given, and then we present the de…nition of consistency.
De…nition 1. A vector (p; t; M ) is a noncompetitive equilibrium if: 1. Markets clear, n tn = 0; 2. For any n, trade tn maximizes a utility or pro…t, given residual supply function p ( ); P p; tn;Mn 3. Price impacts M are mutually consistent.
Formalization of the intuition that in equilibrium M n re‡ects a true price impact of trader n leads to a conceptual di¢ culty. A response of the economic system to a unilateral deviation is not de…ned in competitive theory. Consequently, to close the model, we specify an adjustment mechanism in an economy after a unilateral deviation. The proposed mechanism is based on the premise that all markets clear and all other agents optimize even if agent n is trading a suboptimal quantity tn + tn. This mechanism leads to the notion of a subequilibrium being triggered by a unilateral deviation tn.
5 De…nition 2. Given M n, a vector (~p; t~ n; M n) is a subequilibrium triggered by tn if: 1. Markets clear with a unilateral deviation, tn + tn + t~n = 0; n0=n 0 ~ 6 2. For any n0 = n, trade tn0 maximizes a utility or pro…t, given demand functions pp;~ t~ ;M ( ). 6 P n0 n0
If each (small) unilateral deviation tn triggers a unique subequilibrium, then trader n is e¤ectively facing an inverse residual supply function pn(tn) that assigns a price p~ from a subequilibrium triggered by tn. This fact allows us to close the de…nition of consistency of price impacts.
De…nition 3. M n is consistent with M n if there exists a smooth inverse supply function pn( ), assigning a subequilibrium price for any deviation tn, such that M n = @pn(0)=@tn. Vector M is mutually consistent if M n is consistent with M n for any n . 2 N
Intuitively if trader n sells tn above the equilibrium level, the market price goes down in order to encourage other traders to absorb the deviation and keep demand equal to supply. Our de…nition requires that a new price clears the market, and remaining traders respond optimally to a new price after unilateral deviation. The price concession needed to clear the market de…nes an inverse demand for trader n with slope M n. We show that with strictly decreasing marginal utilities and a …nite number of traders, price adjustment (and hence consistent price impact) is always non-zero, and the competitive assumption M = 0, is not consistent with our model. One of the advantages of a noncompetitive equilibrium framework over existing noncompetitive models such as Monopoly, Monopsony, Cournot or Monopolistic Competition is that the model does not a priori divide the traders into those who do and those who do not have market power because “they are large or small.”Here, price impact arises endogenously for each trader as a result of market clearing and optimization by all agents. This is why we call it endogenous market power. Also, unlike in a Walrasian framework, in a noncompetitive equilibrium, traders know they can a¤ect prices by changing their trades, and, therefore, they are not price takers. Notice that in De…nition 2, price impacts M are …xed in and out of equilibrium. Since agents assume they do not a¤ect the slopes by unilateral deviations, the traders are slope takers. In Section 4 (and in more detail in [24]) we characterize environments, given by games, in which slope-taking strategy is rational.
2 Optimal Choice and Market Power
Before characterizing a noncompetitive equilibrium, we …rst introduce a model of a consumer and a producer choice in thin markets.
2.1 Consumer Choice
We start with a geometry of a budget set, assuming that all three equilibrium elements (p; t; M ) are predetermined. A budget set B(p; ti; M i) of consumer i is a set of all a¤ordable trades (ti; mi), given a downward sloping demand function (2), or alternatively, given p; ti and M i. The set is
6 determined by a budget constraint
p (ti)ti + mi 0: (4) p; ti;Mi Inequality (4) implies that a non-numèraire demand is …nanced either by income from shares or a negative consumption of numèraire. Unlike in a competitive model, in (4) prices are not constant, but depend linearly on trade, and, therefore, the budget constraint is quadratic in ti and linear in mi. There are two important trades that satisfy (4) with equality, namely: the autarky trade, (ti; mi) = 0, and the equilibrium trade, (ti; m i) where, given strictly monotone preferences, m i = tip. Example 1. Let p = 1, ti = 1, and Mi = 1. The inverse residual supply of i, given by pp; ti;M i (ti) = 2 ti, de…nes a budget constraint (ti) +mi 0. Geometrically, a budget set has the shape of a parabola. The consumption numèraire is maximized at the trade for which marginal revenue is zero, ti = 0 and there exists no a¤ordable trade ti which gives mi that exceed bound 1.
The vector normal to the surface of a budget set, the budget set slope, is given by (MR(ti); 1), where the marginal revenue is de…ned as in (3). At the equilibrium trade (ti; m i), the normal vector becomes (p + M iti; 1) and is di¤erent from (p; 1), the actual prices at which the goods are traded. Intuitively, a marginal change of a trade, ti, raises the total spending directly by tip. In addition, ti changes prices by p = M iti, and, therefore, it increases the marginal spending on all goods that are traded by pti. At the autarky trade ti = 0, the slope is given by (p M iti; 1). In this case, the slope is di¤erent from (p; 1), as at ti = 0, market prices drop to p M iti. When the price impact converges to zero, budget set becomes a half-space and the normal vector turns into prices for all trades, as in a competitive framework. Apart from its shape, the other feature of a budget set that distinguishes it from a competitive budget set is that it depends on the equilibrium choice of trade ti. We now de…ne a stable trade, ti, given the observed values of M i and p. The notion of stability replaces the optimality condition in the Walrasian model. Intuitively, ti is stable at p and M i, if ti, together with the implied consumption of numèraire, m i = tip, maximizes utility among all trades that are a¤ordable, given the inverse demand function p ( ). p; ti;Mi
De…nition 4. Consumer’s trade ti is stable at p and M i if for all (ti; mi) B(p; ti; M i) 2
U(ti +e i; m i) U(ti + ei; mi):
The stable trade is not an optimal response to observed p and M i. Rather, it is an optimal response to the observed triple (p; ti; M i), as all three elements are needed to determine the a¤ordable alternatives completely. Mathematically, the stable trade is a …xed point of a mapping given by