Endogenous 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 ). 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 and heterogeneous traders. We de…ne an equilibrium and show that such equilibrium exists in economies with smooth and cost functions and is determinate. The model suggests that 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 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 . 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 . 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 . 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 power of a few buyers. Our approach has the following advantages over the existing models of market interactions. Unlike, for example, the competitive, , 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 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 . 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 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 spaces to linear supply functions (in [18], [23]), or …nally, by requiring that traders report to the auctioneer their 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 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, , 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 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 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

We start with a geometry of a , 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

(ti; m i) = arg max U(ti + ei; mi): (5) (ti;mi) B(p;ti;M i) 2

7 Because of the dependence of a budget set on an optimal choice of ti, neither the existence nor uniqueness of ti is guaranteed by the standard arguments that use the Weierstrass theorem and strict concavity of preferences. The next proposition establishes the two properties of a stable trade ti in a smooth economy.

Proposition 1. Under A1-A3, for any p; M i > 0, there exists unique ti stable at p; M i.

The necessary condition for interior ti to be optimal on B(p; ti; M i) is the tangency of a budget set with an upper contour set of the utility function at (ti; m i). Since the marginal utility of numèraire   and its price are always equal to one, the tangency condition reduces to ui0 (ti + ei) = MR(ti) = p + M iti. Proposition 1 implies that such an equation has a unique solution, and hence, there exists a stable demand function ti(p; M i) mapping nonnegative prices and price impacts into stable trades. Corollary 1 characterizes the properties of this function.

Corollary 1. Under A1-A3, a stable demand function ti(p; M i) is continuous for all p; M i > 0.

This function is also di¤erentiable for all p; M i 0 such that ti(p; M i) > ei.  For all interior trades a derivative of a stable demand function with respect to price is equal 1 to @ti(p; Mi)=@p = (Mi u ) . The slope of a non-numèraire stable demand function can be i00 decomposed into two parts: a substitution e¤ect associated with the inverse of the second derivative of a utility function and a market power e¤ect equal to a price impact of i on prices, Mi. Notice that the price impact makes the stable demand less responsive to changes in prices, and the demand function becomes steeper than the competitive demand. Gossen’sLaw asserts that when the choice of a competitive consumer is optimal, the marginal rate of substitution must coincide with the price ratio. The tangency condition extends this con- dition to a noncompetitive setting. The generalized Gossen’s Law says that the marginal rate of substitution is equal to a ratio of marginal revenues, where the marginal revenue of numèraire coincides with its price and is equal to one MRS = MR=1. Only when traders do not have price impacts, does this condition reduce to the standard Gossen’sLaw. Another implication of stability is that consumers with strictly monotone preferences choose trades for which marginal revenue is strictly positive (and equal to marginal utility) and hence they operate solely on the elastic part of the inverse supply.

2.2 Firm’sChoice

Modeling noncompetitive …rms in a general equilibrium framework is associated with the following complications. In a competitive equilibrium, price-taking behavior assures that in equilibrium the marginal cost of production is equal to the marginal utilities of the consumers and hence it does not matter in which of the two commodities the dividends are paid. When traders have market power, there is a wedge between the two values. If consumers are simultaneously shareholders, a gap creates natural incentives to transfer non-numèraire goods from a …rm directly to its owners and to bypass the markets. In addition, even if the dividends are paid solely in numèraire, there

8 does not exist obvious objective function for the …rm that all shareholders would agree on. In particular, the level of production-maximizing pro…t does not coincide with the optimal production plan from the perspective of each individual shareholder because, apart from receiving dividends, each consumer is tempted to use the …rm as leverage to pro…tably a¤ect market prices. The two considerations imply that in a noncompetitive setting a complete characterization of a …rm must explicitly specify which goods are transferred to owners as dividends and what the objective of a …rm is. In this paper we focus is on incorporated …rms, which by assumption maximize pro…ts in terms of numèraire and pay dividends exclusively in this good (money). Pro…t of the incorporated …rm j is given by

   (tj; mj) = (p   (tj)tj + mj): (6) p; tj ;Mj p; tj ;Mj

By convention, mj is money spent on inputs, and trade tj is a negative of the supply tj yj;  hence, the pro…t function has a minus sign. Given p, tj and M j, equation (6) de…nes a map of iso-pro…t curves. Analogously to a consumer’s budget set, with non-negligible price impact, any iso-pro…t curve has a parabolic shape, and the vector normal to it is equal to a negative of the marginal revenue (MR(tj); 1). Similarly to a consumer, …rms trade tj is stable at M j and p, if tj together with the required level of numèraire input to produce it, m j cj( tj), maximizes pro…t among all technologically  feasible trades, given the inverse demand function p   ( ). By identical arguments as in the case p; tj ;Mj  of consumers under A4-A7, there exists a function tj(p; M j) 0 that assigns unique stable trade  for any M j; p > 0, and such function is di¤erentiable for all prices and price impacts for which tj(M j; p) < 0. At the stable trade, the production set must be tangential to the iso-pro…t curve,     and hence the normal vectors are collinear cj0 ( tj) = MR(tj) =p  + Mjtj. The derivative of a 1 stable demand function with respect to price is given by @tj(p; Mj)=@p = (Mj + c ) . Thus the j00 greater the price impact, the less responsive the stable supply is to changes in prices.

2.3 Market Power

We now derive equilibrium price impacts and characterize the endogenously determined market power of the traders. Let us …rst …x the price impacts of other traders about their market power

M n = M n n =n. It follows from the de…nition of equilibrium that a small, unilateral deviation by f 0 g 06 trader n from the equilibrium trade to some tn triggers a subequilibrium with new prices and trades of other agents, so other trading partners can respond optimally to new prices and all markets clear with the deviation. Therefore, for any deviation tn = tn tn, prices adjust to a new level, assuring that markets clear with a unilateral deviation. Formally, such an o¤-equilibrium market clearing      condition can be written as tn + tn + n =n tn0 (p; Mn0 ) = 0, where tn0 (p; Mn0 ) is a stable trade 06 function of trader n n . Since in equilibrium markets clear, pair of p =p  and tn = 0 0 2 N nf g P satis…es the market clearing equation. When for all n = n, price impacts M n are non-negative, 0 6 0 by implicit function theorem market clearing condition de…nes a smooth inverse supply function

9 pn(tn) around tn = 0. The slope of the inverse supply function, the derivative of pn( ), is given  by

1 @p (0) @t (p; M ) n = n0 n0 = (7) @tn 2 @p 3 n =n X06 1 4 5 1 = M n + vn ; 2 0 0 3 n =n X06  4 5 where vn is a positive scalar de…ned as vi u (ti + ei) for a consumer and vj c ( tj) for a 0  i00  j00 producer, and is given by the convexity of the preferences or cost function. The consistency condition in De…nition 3 implies that in equilibrium the slope of inverse supply assumed by trader n should coincide with the price impact up to a level approximation of order one: M n = @pn(0)=@tn. This fact and equation (7) de…ne the consistent price impacts of trader n as a function of M n 1 M n = (M n + vn n0 = n); (8) N 1H 0 0 j 6 where ( ) is a harmonic average of N 1 positive scalars. The system of consistent price impacts H  M for all agents is determined as a solution of the system of N equations (8), each for one trader. We now characterize the system of consistent price impacts by inspecting the consistency condi- tion (8). The essential ingredient of a price impact for each trader is a convexity of utility and cost functions of their trading partners. The steeper the marginal utility or cost of the other traders, the greater the price concession needed to make others willing to absorb the o¤-equilibrium deviation, and hence, the greater the price impact. It follows that the trader with the highest convexity has the smallest price impact. More generally, trader ranking with respect to the strength of price impact is a reversed order of the convexities of their utility or cost functions. The dependence of the price impact (slope of the aggregate demand) on the convexity of preferences is standard and is shared by many noncompetitive models, for example a model in which the demand in the indus- try results from aggregation of the individual demands of I competitive consumers who maximize quasilinear utility functions. The individual demand of a consumer coincides with her marginal

(non-numèraire) utility, and hence, its slope is equal to the absolute value of ui00. The aggregate demand in the industry is obtained by summing horizontally the individual demands, and the slope of the inverse aggregated demand is given by (1=I) ( u i I). Our model modi…es this result in H i00j 2 the following way: A market demand faced by trader n is a horizontal sum of individual (stable) demands by all other traders in the economy, including producers, and therefore, we have N 1 elements in a harmonic mean. Unlike in a standard model, here all traders respond optimally to di¤erent (also o¤-equilibrium) prices. In addition, the slopes of consumers’individual demands are steeper, M i u , as price impacts make consumers more reluctant to adjust traders. i00 Price impacts are subject to a mutual reinforcement e¤ect. The price impacts of the other traders make their stable demand functions steeper, which reinforces the price impacts of a deviating

10 trader. This e¤ect, by the same argument, feeds back the price impacts of the absorbing traders. Such a positive mutual reinforcement e¤ect ampli…es the overall level of price impact in the economy. It does not matter for price impact whether the trader is a …rm maximizing pro…t or a consumer. In equilibrium, consumers and …rms with the same convexity have identical price impacts. More surprisingly, price impact does not depend on the trader’sshare in the total volume of trade. Even if one of the traders is a sole supplier of a commodity on the market, she has the same price impact as others, as long as the convexities are the same. This observation seems to go against the common economic intuition that only large traders have market power. Note that the same price impact a¤ects traders’ decisions in a di¤erent way, depending on the individual volume of trade. For example, with the price impact and the supply equal to one, a producer, by selling an additional unit of a good, is losing only one dollar on the original sales due to a reduced price. When the volume of trade is equal to one hundred units, the loss of pro…t then amounts to one hundred dollars. Consequently, the same price impacts will a¤ect large traders more severely, and hence traders’noncompetitive behavior is more pronounced. Finally, we would like to stress the role of a harmonic mean. Harmonic mean arises naturally in the consistency condition, as it puts higher weight on traders with smaller convexities. Observe that such traders absorb most of the unilateral deviation and hence are key in price impact deter- mination. The traders with almost linear utility or cost functions are willing to absorb a unilateral deviation almost without any price reward, making the price impacts of other traders negligible. This intuition is re‡ected in the property of the harmonic mean, namely, that whenever one of its elements converges to zero, the value of the mean also becomes zero.

3 Properties of Equilibrium

In this section, we state four theoretical results for a noncompetitive equilibrium in a smooth economy. (Recall that a smooth economy is one that satis…es assumptions A1-A7.) The …rst two results are relevant for the computation of an equilibrium. The third theorem is a version of the …rst welfare theorem, while the last …nding establishes the convergence to a unique competitive equilibrium in a large economy.

3.1 Existence

A good economic model gives predictions for any economy, satisfying general assumptions. The …rst theorem establishes the existence of such equilibria in a smooth economy.

Theorem 1. In a smooth economy, an equilibrium exists if and only if N > 2.

The assumptions of Theorem 1 are relatively weak, as they only require that there are more than two agents trading in a market. When the trade is restricted to two traders, as in the Edgeworth box economy or in the Robinson Crusoe economy, the mutual reinforcement of price impacts is without any discounting; price impacts explode to in…nity and equilibrium fails to exist. Non-existence of

11 equilibrium is contrary to a competitive model or traditional bargaining models. In Nash bargaining models, the outcome is Pareto e¢ cient, while in this and virtually any other noncompetitive model based on uniform price, (monopoly or Cournot), trade is associated with a dead-weight loss. In this context, the nonexistence of equilibrium can be interpreted as a manifestation of ine¢ ciency in which the rigidity of the uniform price contract combined with reinforcement of price impacts prevents trade.

3.2 Determinacy

The second desirable property of the model is that its predictions are sharp. Technically, the set of all equilibria should be small, possibly a singleton. It turns out that a equilibrium can be expressed as a solution to a system of nonlinear equations, where the number of equations is exactly equal to the number of endogenous unknowns. If no equation is (locally) redundant, the equilibria are locally unique. Unfortunately, local no-redundancy cannot be guaranteed for any arbitrary economy . E The next theorem formalizes intuition where economies for which the equilibria are not locally unique are not robust. For any n, let n0 ; n00 R be two scalars and …nally let  n0 ; n00 n . 2  f g 2N Fix economy . By , we denote an original economy with utilities perturbed by adding 2 E E 2 N 2 n0 xi + i00 (xi) to and cost functions by adding j0 yj + j00 (yj) . Let B(0; r) R  be an open P   ball, spanning a family of perturbed economies, with each member parameterized by  . We 2 P assume that the radius r > 0 is su¢ ciently small to preserve monotonicity and convexity of the preferences and cost functions on the relevant set of trades.5 Notice that has a nonzero Lebesque N 2 P measure in R  and it contains the original economy (as 0 ). 2 P Theorem 2. There exists a subset ^ with full Lebesque measure of , such that for any P  P P  ^, all equilibria of the economy  are locally unique. 2 P E Theorem 2 can be interpreted in the following way. For any economy , including the ones for E which the equilibria are not determinate, for almost all economies with small perturbations of preferences and technology, all equilibria are locally unique. Consequently, the non-determinacy of a noncompetitive equilibria is not a robust phenomenon. One of the implications of this theorem is that typically the number of equilibria is at most …nite. In the case of economies with quadratic utility and cost functions, the result can be strengthened as in such economies noncompetitive equilibria are globally unique. Theorem 2 is particularly important in the context of the existing noncompetitive literature. The problem of indeterminacy is shared by many noncompetitive models in di¤erent …elds: In a general equilibrium literature with market power (for a good survey see [12]), in Nash in Supply Functions in IO (see [8], [16] and Section 4.3.2).

3.3 Welfare

The …rst welfare theorem guarantees that when markets are perfectly competitive, the equilibrium allocation is Pareto e¢ cient. This result is a formalization of Adam Smith’s proposition about

5 Such su¢ ciently small radius exists by continuity of utility, cost functions, and set of trades.

12 the e¢ ciency of the invisible hand. Theorem 3 provides an analogous though opposite result for interactions in markets.

Theorem 3. In a smooth economy, in an equilibrium, allocation is Pareto e¢ cient if and only if J = 0 and the initial allocation is Pareto e¢ cient.

It follows that in economies with …rms, equilibrium is always Pareto ine¢ cient, and in a pure exchange economy, the allocation is ine¢ cient as long as initially there are some gains to trade. The negative message from Theorem 3 is partially counterbalanced by the next result, which asserts that although equilibrium allocation is not Pareto e¢ cient, in deep markets the ine¢ ciency is negligible, as all equilibria converge to a singleton, the unique competitive equilibrium. To state the result formally, we take advantage of a construct of a k replica economy. Theorem 4. For any smooth economy, and any " > 0, there exists k" 2; 3 ::: such that for 2 f g any k k" all equilibria in a k replica economy, satisfy for any n  " 1. M n(k ) "; k " k W alras 2. tn(k ) t "; k k  3. p(k") pW alras ". k k  Consequently, in deep markets, economic interactions are nearly competitive, and the competitive framework is a proper tool to use to study such economies. Can one empirically test for a noncompetitive behavior, given the information about trades, endowments, and prices from one trading period (t; e; p)? The relation between a competitive and noncompetitive equilibrium can be described in the following way. The triple (p; t; M ) is a noncompetitive equilibrium in = (u; e; c) only if (p; t) is a competitive equilibrium in the economy E ~ 1  2 with modi…ed preferences and technology, = (^u; e; c^), where u^i(xi) = ui(xi) 2 Mi(xi ei) , for 1 E 2 any consumer i , and c^j(yj) = cj + M j (yj) , for any …rm j . It follows that any allocation 2 I 2 2 J and price in a noncompetitive equilibrium can be rationalized as a competitive equilibrium by some other economy. As a result, noncompetitive equilibrium is empirically indistinguishable from a competitive equilibrium, given that the data is from one period only. Note that the preferences rationalizing a noncompetitive equilibrium as a Walrasian one exhibit the following property: Each consumer has preferences biased toward its own endowment. Therefore, if the time-series of initial endowments and trades are available, then given …xed preferences, one can identify the endowment e¤ects in the trades. Alternatively, one could test for noncompetitive equilibrium by assuming that in the economy, preferences are independent from endowments.

3.4 Model Predictions

Now we illustrate the e¤ects of noncompetitive trading on equilibrium allocation, prices, and welfare in a simple example of a pure exchange economy with quadratic utility functions. Quadratic functions are particularly convenient to work with because their second derivatives, and hence price impacts, do not depend on the level of equilibrium consumption. This property allows us to calculate equilibrium in a closed form.

13 Consider a pure exchange economy with I 3 consumers, each characterized by identical  quadratic utility function,

v 2 Ui(xi; mi) = ui(xi) + mi = xi (xi) + mi: (9) 2 where v is some scalar common to all consumers. The consumers di¤er in their initial endowments ei > 0. Theorem 1 implies that an equilibrium exists. In fact, in the case of quadratic utilities, such equilibrium is globally unique. Given identical quasilinear utility functions, in the competitive equilibrium, all traders consume the same amounts of non-numèraire good, equal to the average endowment in the whole economy W alras 1 x = I i ei =e , and the price is equal to the marginal utility at the per capita consumption pW alras = 1 ve. P We now compute a noncompetitive equilibrium. The consistency condition (8) gives a system of I equations and the same number of unknowns. With quadratic utilities, the consistent price 1 impacts do not depend on the level of consumption and are given by Mi = (Mi + v i = i). I 1 H 0 j 0 6   In an economy with two traders, consistency conditions become Mi = Mi0 + v for i = 1; 2, which is a system of two linear equations, and two unknowns, M 1 and M 2. In the …rst equation, the market power of consumer i = 1 is increasing in the price impact of agent i = 2, which in turn, by the second equation, reinforces back the price impact of trader i = 1. The e¤ect of mutual reinforcement of a price impact has already been discussed: The higher the price impact, the less elastic the demand with respect to the price and hence the greater the price change required to make the other consumers absorb the deviation. Geometrically, the two lines de…ned by the equations are parallel, and hence, they do not cross each other. It follows that the system has no solution, and equilibrium fails to exist (see Theorem 1).   For I > 2, the solution to the system of consistent price impacts is symmetric (Mi = Mi0 for any i; i I), and it is given by M i = v=(I 2). Price impact is inversely related to the number of 0 2 traders and positively to the convexity of preferences v. The remaining elements of the equilibrium, namely, the allocation and the price system, can be found by solving the system of the …rst order stability conditions and the equilibrium market clearing condition. The equilibrium consumption W alras is given by xi = x + (1 )ei where (I 2)=(I 1). Notice that the consumption of  agent i is a convex combination of the competitive consumption and the initial endowment. The allocation is closer to autarkic when the number of consumers in the economy goes down and a market becomes thinner. In an economy with symmetric quadratic preferences, equilibrium prices p are the same as the competitive ones, pW alras. As we show next, this …nding is not robust. To make the presentation more transparent, we …rst discuss the mechanism behind price bias in an economy with two types of traders, a buyer and a seller, and, to assure the existence of equilibrium, two traders of each type. The initial endowment of a buyer is eb and that of the seller is es > eb. Both types have quadratic utility functions as in (9) , except that their convexity coe¢ cients vi might be di¤erent now. The consistent price impacts can be found by solving (7), and the stable trades satisfy

14   1 viei p vi 1 viei p i  ti(p; Mi) =  =  =  ti(p; 0); (10) vi + Mi vi + Mi i i + Mi where ti(p; 0) corresponds to a Walrasian demand.

When utilities are identical, the price impacts of both types of traders also coincide, Mb = Ms, and therefore, the equilibrium market clearing condition, implicitly de…ning the equilibrium price, can be written as

v t (p; M ) = i t (p; 0) = t (p; 0) = 0: (11) i i v + M i i i=b;s i i i=b;s i=b;s X X X The unique price that solves the last equality in (11) is the competitive one pW alras and hence p =p W alras. The intuition behind this result is as follows: The identical convexity for the buyer and seller leads to the same market power for both types of consumers. For any …xed price, the buyer with price impact reduces its demand by the same proportion vi=(vi + M i) as the seller decreases its supply, and the market clears, and the price remains unchanged.

Now suppose that traders’convexities di¤er, for example vb > vs. Although in an asymmetric case, equilibrium does not have a closed form solution, it can be shown that vb > vs implies

M s > M b. Intuitively, when the buyers’ marginal utility is steeper than the sellers’, the seller’s deviation is associated with a higher price concession then that of the buyers. Consequently, the inequality vb=(vb + Mb) > vs=(vs + Ms) holds. This fact and the stable market clearing condition v 0 = t (p; M ) = i t (p; 0); (12) i i v + M i i i i i X2I X2I imply that at the noncompetitive equilibrium price, p, the Walrasian excess demand is strictly  negative i ti(p; 0) < 0. This is because in the last equality of (12), the constants for buyers are 2I above theP ones for sellers. Since the Walrasian aggregate demand function is strictly decreasing in price, in order to establish the market clearing in a competitive framework, the Walrasian price must go down. One can conclude that p > pW alras. By an analogous argument, the price bias is W alras negative when the seller’sconvexity is greater, vb < vb, that is p < p . This example illustrates that in a noncompetitive equilibrium, price can be greater, the same, or below the competitive one, depending on the values of the parameters of the considered economy. In general, when consumers have identical, quasi-linear, concave, but not necessarily, quadratic W alras utility functions, their Walrasian consumption is the same, xi =x  . In a noncompetitive equilibrium, the sellers are given incentives to reduce their supply, while the buyers cut down their demands. This leads to a disparity between the consumption of the buyers and the sellers, W alras xb < x < xs. If the second derivative of the utility function is increasing (u000 > 0), then the second derivative of the utility function of the buyer at the equilibrium consumption, vb = u (xb) , j 00 j exceeds the one for the seller vs = u (xs) . The quadratic example suggests that such endogenous j 000 j asymmetry in convexity may lead to a positive price bias over the price predicted by a competitive

15 model, p > pW alras. To make this observation formal, for any noncompetitive equilibrium (p; t; M ), s we partition the set of all traders into three groups: Buyers, b = i ti > 0 , sellers, = i i I f 2 Ij g I f 2 ti < 0 , and non-active traders, na = i t = 0 . Ij g I f 2 Ij g Proposition 2. Consider a smooth pure exchange economy with Pareto ine¢ cient initial allocation, in which utility functions are the same for all consumers. Then in any noncompetitive equilibrium

(p; t; M ), for any b b, s s and na na 2 I 2 I 2 I W alras    1. u000 > 0 implies upward price bias p > p and Ms > Mna > Mb, W alras    2. u000 = 0 implies no price bias p =p  and Ms = Mna = Mb, W alras    3. u000 < 0 implies downward price bias p < p and Ms < Mna < Mb.

There are two e¤ects generating the price bias. First, with positive third derivative u000 > 0, the sellers tend to reduce their supply more than the buyers, given any level of price impact M i, and hence the price of the good must go up to clear the market. This is a direct e¤ect. Second, the endogenous asymmetry in convexity implies that the sellers’price impacts exceed the ones of the buyers, further increasing the price. For example, with utility functions characterized by constant relative (CRRA) the third derivative is positive; therefore, in a pure exchange economy with such utility functions, one should expect upward price biases and a high relative market power of the sellers. We conclude this section by examining the e¤ects of noncompetitive interactions on welfare. In an economy with quasi-linear preferences, the utility is transferable, and one can measure welfare in terms of a numèraire. The aggregate gains to trade are de…ned as a total utility attained at the unique Pareto optimal allocation, less the utility in autarky. In the considered economy, the aggregate gains to trade are equal to W i ui(e) ui(ei). Straightforward calculation reveals  2I that W = Iv V ar(e ), where V ar(e ) is the variance of initial endowments in the economy. The 2 i i P aggregate gains to trade are positively correlated with the heterogeneity of the initial nonnumèraire endowments, the convexity of the utility function v, and the number of traders. Theorem 3 says that when the initial allocation is Pareto ine¢ cient, then a noncompetitive equilibrium allocation is not Pareto e¢ cient either. It follows that some part of the aggregate gains to trade is lost. The , DWL, is a fraction of the aggregate gains to trade W that vanishes,

ui(e) ui(xi) 1 DWL i = : (13)  W (I 1)2 P Equation (13) shows that the more traders in the economy, the more competitive interactions there are, and hence DWL is smaller. Since the inverse of DWL is quadratic in I, and W is linear, in absolute terms, welfare loss DWL W also disappears with I approaching in…nity, as  long as V ar(ei) is bounded. Interestingly, in the quadratic example the welfare loss is uniformly distributed among the traders – the individual welfare loss of each i, measured as a di¤erence between the utility in a competitive equilibrium and a noncompetitive one, as a fraction of the individual gains to trade, is also equal to 1=(I 1)2. Thus, no agent gains utility at the expense of other consumers thanks to his price impact. This prediction is not robust, and it critically relies on

16 the assumption of identical quadratic utility functions. The more realistic, convex marginal utility always leads to higher market power of the sellers, and the greater utility loss is on the buyers’side. In the extreme examples, sellers can have higher utility than in a competitive model, and buyers not only pay the DWL, but also a monopolistic rent.

3.5 Many Commodities

The noncompetitive analysis from this paper can be applied to economies with many non-numèraire commodities. Let l denote a particular commodity, L and is the number and the set of the L commodities respectively. The de…nitions of a noncompetitive equilibrium 1, 2, and 3 naturally L extend to the case of L > 1, where now prices and individual trades are vectors p; tn R and 2 price impacts Mn are L L, positive de…nite and symmetric matrices. With many goods budget  constraints and pro…t functions are quadratic forms, budget sets and iso-pro…t curves are elliptic paraboloids, and vectors orthogonal to these sets are given by the vectors of marginal revenues from selling each of the commodities. A simple example of an economy with many goods is a model with utility and cost functions that are additively separable, in which a component for each commodity satis…es A1-A7. An important example of such a model includes a two-period Arrow-Debreu economy with uncertainty and with Neumann-Morgenstern preferences. With separable utility functions, markets for di¤erent commodities are independent and L good economy is equivalent to L economies, each with one good. It follows that all the results for one-good economies discussed in previous sections apply to multi-good economies. In particular, by Theorem 1 noncompetitive equilibrium exists, in which the price impact matrices are positive de…nite and diagonal. The diagonal entries correspond to price impacts in particular markets and the cross-market price impacts, given by o¤-diagonal entries, are zero. Also, by Theorems 2, 3, and 4 such diagonal equilibria are determinate, Pareto ine¢ cient and converge to a competitive equilibrium when economy becomes large. With non-separable utility or cost functions the model is qualitatively di¤erent from a one- good economy and none of the results automatically extends to the multi-good setting. With non- separable utilities equilibrium price impacts inherit two properties of the Hessians of the utility function: positive de…niteness and symmetry. [22] characterized a unique equilibrium in a dynamic economy with L commodities (assets) and nonseparable utilities in an economy with quadratic preferences. More generally, under mild conditions of strict monotonicity and concavity (convexity) of preferences (of cost functions) the results from Theorems 2, 3, and 4 can be extended to all non- separable economies with many commodities. The existence of noncompetitive equilibrium requires additional assumptions that guarantee interiority of the solution and can be established using the same arguments as proof of Theorem 1.

17 4 Game Theoretic Foundation

To rationalize the concept of competitive equilibrium, Walras proposed a simultaneous auction. In this section we summarize results from [24] showing that the same Walrasian auction also provides a rationale for a noncompetitive equilibrium in markets with any number of traders. Thus, the noncompetitive framework from this paper naturally extends competitive theory to small markets.

4.1 Walrasian Auction

Consider a Walrasian auction, a static game in which traders, as their strategies, simultaneously choose demand schedules tn( ): R R, specifying a desired quantity for any price. The traders’  ! payo¤s are determined as follows: Given an arbitrary pro…le of submitted demand schedules t( ) =  tn( ) n N , a nonstrategic Walrasian auctioneer determines market clearing price p, (i.e. price for f  g 2    which n N tn(p) = 0) and agents’payo¤s are U(tn +e i; m n) where tn = tn(p) and m n =p tn. 2 CompetitiveP theory proposes the following solution to the problem of optimal bidding in a Walrasian auction. Since in large markets each trader is negligible, individual demand has no impact on a market clearing price p. In addition, since each trader makes trade choices contingent W alras on price, the optimal demand schedule for each n is given by t (p) arg maxt U(en +tn; tnp). n  n Given such competitive demands, the outcome of the Walrasian auction is (pW alras; tW alras), and hence the Walrasian auction provides a foundation for a competitive equilibrium. The competitive approach has a number of advantages that make it a natural tool to study large, anonymous markets. Optimality of tW alras( ) is robust to assumptions about the primitives n  of the economy. Also, unlike in standard game theoretic models, traders do not need to know the equilibrium strategies of trading partners as the only information a trader needs is own utility and endowment. A competitive solution is not suitable for anonymous markets with a small number of traders for the following reason: Competitive demand schedules of traders n = n de…ne a less than 0 6 perfectly elastic residual supply for trader n.6 Demand schedule tW alras( ) selects a trade on such n  a supply, for which marginal utility coincides with price, a suboptimal choice for a monopsonist. Trader n can increase utility by deviating from a competitive demand schedule. Thus tW alras( ) is  not a Nash equilibrium in a Walrasian auction when N < . The noncompetitive approach from 1 this paper corrects the price-taking “mistake,”by endowing all traders with correct beliefs about the slopes of their residual supplies, given by price impacts M n. If the traders’optimization problems are convex, statistic M n summarizes all the relevant information about the demand schedules of the trading partners, and it allows the traders to mutually best respond to their demands. This observation is formalized in Proposition 3. For any noncompetitive equilibrium (p; t; M ), let tn( ; M n) n N be stable demands. Suppose, for each n; stable demands of trading partners f  g 2 tn ( ; M n ) n =n de…ne a convex optimization problems. Then f 0  0 g 06

6 Given demand functions tn (p) one can …nd a residual supply for trader n by solving market-clearing 0 n0=n t + t (p) =f 0 g 6 condition n n =n n0 for price. 06 P

18 Proposition 3. (Nash Equilibrium) tn( ; M n) n constitutes a Nash equilibrium in a Walrasian f  g auction, with outcome given by (p; t; M ).

In general, the problem of a monopsony/monopsony may fail to be convex when the residual supply/demand is too concave, which in turn in the presented model may occur in economies with a small number of asymmetric traders, in which the marginal utility functions of some of them are very convex. When the utility functions are close to quadratic, in which the residual supplies are almost linear, the second order conditions are automatically satis…ed. In [24] we show that the result from Proposition 3 is quite general and holds in a number of market games, in which traders’ strategies are su¢ ciently reach to allow them to make choices contingent on prices. An important example includes a dynamic game de…ned by a tâtonnement process, in which an auctioneer adjusts the price in the direction of the excess aggregate demand, and the traders respond to the announced price by specifying the quantities they are willing to trade. The trade takes place only after the price attains the level that clears the market. In such a case, stable bidding constitutes a subgame-perfect Nash equilibrium. It is well known that the games based on demand function have a continuum of Nash equilibrium outcomes. As demonstrated in Proposition 4, outcomes in Nash equilibrium associated with stable demands tn( ; M n n are generically locally unique, and hence the noncompetitive approach re…nes f  g the set of Nash equilibria. In the next section we show that a selection of an equilibrium occurs naturally in markets in which traders have limited information about the residual economy.

4.2 Slope-Taking in Anonymous Markets

Suppose in the exchange economy described in Section 3.4 market interactions are repeated an in…nite number of times. The timeline is divided into periods indexed by T , and subperiods indexed by t. At the beginning of each subperiod, traders learn the realizations of their stochastic, t i.i.d. endowments ei, and choose demand schedules. Then the market clearing price and trades are determined in a Walrasian auction and consumption occurs. By assumption, agents trade in anonymous markets and hence do not know the primitives of the market. Thus, instead of being strategic players, in such markets traders often operate under the hypothesis that they face a residual supply T T pi (ti) = i + Mi ti + "i; (14) where "i is a random shock with unknown and Mi is a price impact of trader i. Note that by making choices contingent on prices, each slope-taker can select optimal trade for all possible realizations of residual supply by submitting stable demand schedules ti( ;Mi). (In the same way W alras  T as t ( ) maximizes preferences given a random perfectly elastic residual supply p (ti) = i +"i n  i for any realization of "i.) Consider the following adaptive learning process. All traders enter period T with some estimates T of their price impacts Mi and respond optimally to assumed supply (14) throughout the whole  T t t t period T , using tn( ;Mi ). At the end of period T , they use available time-series (p ; ti; ei)t T  2

19 collected within period T to re-estimate their price impacts and enter period T + 1 with new T +1 estimates Mi . T Since for i0 = i stable demands ti ( ;M ) = (1 vi ei p)=(Mi + v) are linear in p and ei 0 i0 0 0 0 6  T they de…ne a residual supply for i that has functional form as (14), even if estimates Mi i =i f 0 g 06 are incorrect. Its slope, a true price impact of trader i is given by 1=(I 1) (M T + v j = i). H j j 6 Stochastic component "t is a function of endowments of i = i and hence is independent from et, i 0 6 i which determines ti( ;Mi) and hence is correlated with equilibrium trade. Consequently, time-series  t   T ei t T provides a valid instrument for ti, and Mi is identi…able and can be consistently estimated f g 2 t t t from the available market level data (p ; ti; ei)t T . 2 The dynamics of the adaptive learning process described above can be easily seen in a simple version of the problem in which one assumes that within each period there are enough observations for the traders to perfectly estimate their true price impacts. In such a case system becomes deterministic M T +1 = 1= (I 1) M T + v j = i (15) i H j j 6 In addition it is convenient to assume that traders start with the same, e.g., competitive conjectures 0 T T (M = (0; 0; :::; 0)). Then, in each T solution is symmetric, (Mi = M ), and one can drop I 1 i0 equations and unknowns. Price impacts follow the same trajectory for all i, given by the linear di¤erence equation M T +1 = (M T + v)=(I 1). Since I > 2, the coe¢ cient multiplying M T is i i i less than one, and the traders’estimates monotonically approach the unique steady state, given by consistent price impacts M . Thus in the limit, traders submit stable demands with correct beliefs about their price impacts and hence play the Nash equilibrium associated with the noncompetitive equilibrium. In [24] we show that convergence to noncompetitive equilibrium occurs in a large class of environments. Even when (quadratic) preferences are heterogenous across traders, and hence the system of I di¤erence equations (15) cannot be reduced to a linear one, the consistency condition is a contraction (on relevant compact set) and hence it has a unique, globally stable, steady state M . The adaptive learning process leads to mutually consistent price impacts, regardless of initial, possibly heterogenous estimates M 0. Moreover learning is observed in the limit even when, within each given period, traders have …nite samples and hence estimates are imperfect, as long as a sample size monotonically increases to in…nity. We also study the dynamics of a learning process when preferences are non-quadratic.

4.3 Relation to the Literature

We now discuss how a noncompetitive framework complements the following strands of literature: 1) the Walrasian model with price takers; 2) the Klemperer and Meyer [16] model of Nash in Demands/Supplies; 3) the Bresnahan [2] consistent conjectural variation model.

20 4.3.1 Competitive Model

It is often claimed that the competitive equilibrium theory is a good predictor of the market interac- tions if the following three preconditions are satis…ed: 1) there are many traders; 2) if these traders freely and optimally adjust their traded quantities to prices; and 3) if they trade in centralized anonymous markets. The slope-taking approach from this paper allows one to formalize this claim using modern, game-theoretic language.7 Our equilibrium constitutes a Nash equilibrium in a game de…ned by a Walrasian auction, and it is a steady state of a learning process that is plausible in anonymous markets. When the number of traders is large, the consistent price impacts become negligible and the demands of the slope-taking traders, t( ; M ) (pointwise), converge to the Wal-  rasian demands tW alras( ). It follows that, in large markets, noncompetitive equilibrium becomes  equivalent to the competitive tâtonnement process proposed by Walras to rationalize a competi- tive equilibrium, and a noncompetitive equilibrium (p; t; M ) coincides with the equilibrium with price takers (pW alras; tW alras; 0). Thus, the competitive equilibrium approximates the outcomes in markets with many slope-taking traders. Our model allows one to study the directions and magnitudes of biases in the predicted prices and trades resulting from the assumption of in markets with …nitely many traders. For example, Carvajal and Weretka [5] extend the theory of complete markets to the anonymous markets.

4.3.2 Nash in Demands/Supplies

In their classic paper [16], Klemperer and Meyer study a model of oligopolistic industry with I identical producers who face aggregate demand and whose strategies are supply functions. To discipline submitted supply functions, they introduce uncertainty to a demand. For the case when uncertainty has unbounded support, Klemperer and Meyer establish the following properties of the equilibrium set: 1) Nash equilibrium exists; 2) it is symmetric; and 3) the set of all equilibria is connected (i.e., a supply schedule ti( ) that is bracketed by any two equilibrium demand schedules  tKM ( ) is also an equilibrium bid). They also show in the model with identical producers with i  quadratic cost functions and linear demand with additive unbounded noise that the equilibrium is unique. The slope-taking approach complements these results in the following ways: In the simple quadratic example with identical producers, the unique equilibrium coincides with the slope-taking equilibrium from this paper and the learning story based on re-estimation of the price impacts is applicable to equilibrium tKM ( ). The noncompetitive equilibrium approach pins down equilibrium  in the quadratic models without unbounded uncertainty. Such a unique equilibrium inherits prop- erties from the quadratic example considered by Klemperer and Meyer, as t( ; M ) is invariant to  adding a stochastic component to the model.

7 The Nash in demands argument cannot be directly applied to a Walrasian auction. For any arbitrarily large I < there is a continuum of Nash equilibria in demands, and the set of predicted equilibrium outcomes does not collapse1 to a singleton competitive equilibrium as I : ! 1

21 The additional advantage of the approach from this paper is tractability. In settings with iden- tical traders, tKM ( ) is found by solving a nonlinear di¤erential equation with an endogenously i  determined boundary condition, which makes the characterization of an equilibrium very chal- lenging. For example, even in the simple model with identical producers with non-quadratic cost functions, the uniqueness (or even determinacy), let alone characterization of an equilibrium, has yet to be established. The model complicates even more with heterogenous traders as an equilib- rium is given by the solution to I nonlinear di¤erential equations and there are no results that characterize equilibrium even with I 3 heterogenous traders. The noncompetitive equilibrium is  well de…ned and its properties are characterized in quite general settings. Finally, in non-quadratic settings, the Klemperer and Meyer equilibrium is demanding in terms of the level of rationality it requires from the traders. To play tKM ( ), the least the traders have to i  know is their true price impacts for any possible realizations of the residual supply, including the extreme ones. Since extreme realizations are infrequent, the price impacts for such realizations are not easily estimable from the data available in anonymous markets. Given the limited information, traders are likely to extrapolate their price impact from the typical realizations to the extreme ones, which the approach from this paper assumes. Such an approximation is without a signi…cant loss of utility for the trader as long as the preferences are close to quadratic and/or the trades are concentrated around typical values. This is why we view our framework as a less elegant but more realistic and tractable model of anonymous makers. In a quadratic model of an economy stable demands are linear and hence they constitute a lineaer Nash equilibrium. In fact, when restricted to quadratic models (with some noise), stable demands are equivalent to linear Nash equilibrium (see [24], Proposition 5), and hence the four theorems from Section 3.4 apply to outcomes of linear Nash equilibrium. The drawback of the symmetric linear Nash approach relative to a Klemperer and Meyer ap- proach and the model from this paper, is that it imposes ad hoc lineaerity restrictions on the strategies played in equilibrium, which re…nes the set of Nash equilibria. The linearity assump- tion is sometimes justi…ed by the tractability and the fact that the simplicity of linear equilibrium makes it a focal point of the game. The slope-taking from this paper approach o¤ers an alternative, behavioral justi…cation of a linear equilibrium based on the anonymity of the markets. The linear Nash equilibrium is the only steady state of the learning process in which traders independently estimate and re-estimate residual supplies and respond optimally given their estimates. In addition, the slope-taking approach allows one to study transitional dynamics of markets towards a linear Nash equilibrium. Another common criticism of the linear equilibrium is that the equilibrium is well de…ned only in a particular, quadratic, framework and it does not permit robustness analysis outside of this setting. The noncompetitive approach is more general as it de…nes equilibrium in nonquadratic models and, when viewed as an extension of linear equilibrium to nonquadratic settings, allows for such robustness analysis.

22 4.3.3 Conduct Parameter and Conjectural Variation

In the empirical literature, the competitiveness of an oligopolistic industry is often measured by the conduct parameter i estimated using the …rst-order condition

@P p +  q = MC(q ): (16) i i @Q i

Parameter i gives a markup of producer i over the marginal cost, and its estimated value typically varies (strictly) between zero and one. The conduct parameter is sometimes interpreted as a con- jecture of …rm i about the reaction of the industry to increased production of i, i.e., i tells how much total production in the industry goes up should trader i increase sales by one (See [4]). Such an interpretation is problematic from the theoretical point of view as no satisfactory game-theoretic model exists that explains how such di¤erent reactions could possibly arise in equilibrium. The ex- isting static games rationalize only two extreme values i = 0 (full reaction, ) and i = 1 (no reaction, Cournot). In the early Eighties, a conjectural variation literature emerged with an objective to rationalize the intermediate values of the conduct parameters ([2], [20]). This literature did not fully specify a strategic environment for traders, and hence the conjectural varia- tion approach could be only considered as a reduced form of some underlying game-theoretic model which has not been provided. The noncompetitive equilibrium has a game theoretic foundation and, when specialized to oligopolsitic industry, it results in a conduct parameter that ranges strictly between zero and one. The value of the conduct parameter decreases in the convexity of the cost functions and increases in the number of competitors. In the slope-taking model, positive “reaction” of the competing …rms arises because traders submit downward-sloping demands. By increasing quantity, trader i depresses the market clearing price, which crowds out the sales of the other producers. The behavioral interpretation of the conduct parameter as a conjectured reaction of the competing …rms does not apply to the model with slope-taking …rms. Here, agents trade in anonymous markets and have no information about other traders quantities, or identities. It is also instructive to contrast the consistency condition in the conjectural variation model of Bresnahan, [2] with the consistency of price impacts of noncompetitive equilibrium. In doing so, we restrict attention to a model in which conjectural equilibrium is well de…ned (exists and is unique), i.e., the duopoly with producers having identical quadratic cost functions who face a linear demand (see the debate between Bresnahan [2], [3], and Robson [21]). In both approaches, trader i has a conjecture about the reaction of the residual market to a unilateral deviation and acts optimally given such belief, both in and out of equilibrium. In the model of duopoly conjectures are constant in both models, and the two approaches are outcome-equivalent. The key modeling di¤erence that makes the slope-taking approach more general is as follows: In the conjectural variation model, trader i has a conjecture about the of his opponent. It is not clear how such a formulation in terms of best responses can be generalized to allow more than

23 two heterogenous producers.8 With many traders, after a unilateral deviation of i, it is reasonable to assume that agent j should not only best respond to the deviation of i but should also take into account the adjustments of the other traders (= i; j) and vice versa. In the noncompetitive 6 equilibrium, (by assumption) a unilateral deviation of i triggers a new (sub)equilibrium, in which all other traders simultaneously behave optimally given what the other traders are doing, and such an adjustment of the whole system de…nes the reaction of the residual market. In a subequilibrium, each trader j = i responds optimally not only to the deviation of trader i, but also to the optimal 6 adjustments of the remaining traders (= i; j). Consistent price impact is not a simple sum of the 6 best responses of other traders j = i to a unilateral deviation of i. Our approach allows one to study 6 markets with an arbitrary number of heterogenous traders, consumers, and producers. Another feature that distinguishes the noncompetitive equilibrium from Bresnahan’s conjectural variation is that it de…nes equilibrium in models with non-quadratic utility functions.

5 Equilibrium with Free Entry

In this section, we assume that an in…nite number of entrepreneurs has access to the technology they may potentially use to establish a …rm. They are willing to create a …rm as long as entering the industry is pro…table. The technology is given by cost function c. In the model with free entry, an economy is de…ned by two elements: A pure exchange economy, with consumers’utilities FE satisfying A1-A3 and technology = ((ui; ei)i I ; c). E 2 In the long-run equilibrium proposed by Marshall, free entry condition and price taking imply that the price is equal to a minimal , and …rms produce at the minimal e¢ cient scale. The number of active …rms is determined as a ratio of the aggregate demand at the equilibrium price, divided by the minimal e¢ cient scale. This model su¤ers from the following drawbacks: The equilibrium is well de…ned only for the technology characterized by U shaped average cost, and hence, the model does not give predictions for industries with, for example, increasing . In addition, the model assumes price-taking behavior even when the predicted number of …rms is small. We now rede…ne the concept of a noncompetitive equilibrium from this paper to allow for entry. By J we denote an exchange economy FE with a …xed number of J …rms using technology c. E J E The noncompetitive equilibrium in is symmetric if tj is the same for all j and …rm j is E 2 J active if tj = 0. The equilibrium with entry is de…ned iteratively in the following way: 6 De…nition 5. An equilibrium with entry in economy FE is a vector (p; t; M; J), such that: E 1. (p; t; M ) is a symmetric noncompetitive equilibrium in J with J active …rms; E 2. There does not exist a pro…table or pro…t reducing entry: for any J > J: a) if J > 0, there does not exist an noncompetitive equilibrium in J , or E b) if J = 0, there does not exist a noncompetitive equilibrium in J with J > 0. E 8 Perry [20] extends the conjectural variation approach to a model of oligopolistic industry with I > 2 producers with identical quadratic costs by assuming symmetry of the responses, invoking the symmetry of the model. The general consistency condition for the oligopolistic industry with many heterogenous traders has not been formulated.

24 The …rst condition assures that none of the active …rms has incentives to change the supplied quantity and all markets clear, while the second condition prevents non-active …rms from entering the industry. We assume that entry occurs if the pro…t without entry is strictly positive and the industry may accommodate additional …rms. Entry is also observed even when pro…t without entry is zero, provided that after entry the pro…t becomes positive. We assume that there are no incentives to enter the industry when the current observed pro…t and the pro…t after entry is zero. By Theorem 1, there exists a noncompetitive equilibrium in a pure exchange economy with J = 0, provided that I > 2. Such equilibrium is not an equilibrium with entry only if a noncompetitive equilibrium exists in economy J for some J > 0, with …rms making a positive pro…t. Such E equilibrium, in turn, is not a long-run equilibrium only if a noncompetitive equilibrium exists in the economy with J 0 > J …rms. This argument proves that long-run equilibrium exists or the economy can accommodate an arbitrarily large number of …rms, all making a strictly positive pro…t. With many …rms, the price impact is negligible; therefore, economies without long-run equilibrium are called competitive. Now, using the model of equilibrium with entry, we determine market structures in industries with di¤erent types of technologies. A) An active or inactive monopoly is the only compatible with the cost function given by c(yj) = F +c  yj, where F is a …xed and c a marginal cost. With two or more operating  …rms, constant marginal cost makes positive price impacts inconsistent with an o¤-equilibrium market-clearing condition.

Lemma 1. Consider a cost function that satis…es c ( ) = 0. The consistent price impacts satisfy 00  M j > 0 if and only if there is one …rm J = 1 in the industry .

Intuitively, with the price impact of …rm j equal to Mj > 0, and without any convexity added by the second derivative, its cost function c ( ), the response to a unilateral deviation by …rm j is strong 00  0  enough to make the price impact of j0 only a fraction of Mj. Because …rms are symmetric, the only possible consistent price impacts are M j = M j = 0. Since strictly positive price impacts M j > 0 0 are necessary for the existence of a stable supply function, given cost function, a noncompetitive equilibrium does not exist with J 2. Consequently, a monopoly is the richest market structure  compatible with such a cost function. Intuitively, given the technology, new …rms are not willing to challenge the incumbent. Their entry triggers perfectly competitive interactions, resulting in strictly negative pro…ts for all …rms.

B) Consider industries with cost function c(yj) =c  yj. For a technology that exhibits con-  stant returns to scale, the model with entry predicts an oligopolistic market structure (including a duopoly), in which …rms have no market power. The equilibrium price is equal to a marginal cost, p =c , and pro…ts are equal to zero, j = 0. With a constant marginal cost and J 2, Lemma 1 im-  plies that the positive price impact M j > 0 is inconsistent with a noncompetitive equilibrium. For

M j = 0, the stable supply of the oligopolistic …rm coincides with a competitive supply –each …rm is willing to produce an arbitrary amount at price c, and hence, any unilateral deviation of one of the traders is automatically absorbed by other active oligopolists without any e¤ect on price. Con- sequently, no price impact, M n = 0 for all traders n = i; j, is consistent with o¤-equilibrium market

25 clearing. The exact number of …rms and the individual level of production are not determined by the model. In fact, any J 2 is consistent with an equilibrium with entry. In addition, such  equilibrium is Pareto e¢ cient. Constant returns to scale are not compatible with a monopolistic industry because a monopolistic …rm trades with consumers characterized by decreasing marginal utility and hence, it faces a downward sloping demand. It follows that in equilibrium a monopoly has a strictly positive pro…t that encourages other …rms to enter the industry. Inactive …rm, J = 0, will also not be observed in the long run, as entering the industry is pro…table. C) We now examine a market structure in an industry with strictly convex cost functions. We start with the cost function with a positive …xed cost,

F + c (yj) if yj > 0 f(yj) = : (17) ( 0 otherwise ) where variable cost, c(yj), is strictly increasing, and strictly convex and marginal cost satis…es c0(0) = 0. With cost function (17), the average cost is either monotonically decreasing, and hence, the technology exhibits increasing returns to scale, or the average cost curve is U shaped and has a well-de…ned, minimal e¢ cient scale. In the equilibrium with entry, such a technology induces a monopolistic or oligopolistic market structure in which all …rms have positive price impacts and typically positive pro…ts. Industries with U shaped average costs have already been analyzed in a Marshalian framework, the most prominent model to determine a long-run market structure. For example, it is known that 2 with quadratic variable cost function c(yj) = vyj =2, in the competitive equilibrium with entry, each …rm produces at the minimal e¢ cient scale, 2F=v, and price is equal to a minimal average cost, p2F v. The number of consumers per each …rmp I=J can be found as a ratio of the individual production and the aggregate demand in the economy. A competitive equilibrium with entry is not internally consistent unless the economy is large. As with the predicted small number of active …rms, the assumption of price taking is not justi…ed. Now, using the framework from this paper, we calculate small scale biases in the predictions of competitive equilibrium with entry in an economy 2 with consumers’utility functions given by U(xi; mi) = 15xi 5xi +mi, and quadratic cost function with F = 10 and v = 10. The results are given in the Table 1.

Table 1. Small scale correction.

J 1 2 3 10 100 1 I 18-33 34-49 50-66 165-181 1649-1665 1- 1 I/J 18-33 17-24.5 16.66-22.0 16.5-18.1 16.48-16.65 16.48

y 1.34-1.41 1.38-1.41 1.38-1.41 1.406-1.413 1.413-14.14 1.414

p 14.21-14.56 14.17-14.41 14.15-14.35 14.134-14.215 14.142-14.15 14.142

 0.07-0.59 0.026-0.380 0.008-0.290 0.00-0.103 0.00-0.011 0

Notes: The …rst row, J, reports the number of …rms in the industry for which the simulation was conducted. I is the number of consumers compatible with J …rms; I=J is the number of consumers per …rm. y gives the range of production levels; p denotes

26 a price and  is pro…t. The …rst row de…nes the market structure in question. Column J = 1 represents a monopoly, J = 2 is a duopoly, and for J > 2 is an oligopoly with J …rms. In column “ ,”we report the predictions 1 of the competitive equilibrium with entry. Row I gives the numbers of consumers supporting the considered market structure, and row I=J is the number of consumers per each active …rm. The last three rows show the equilibrium level of individual production, price, and pro…t respectively. The pro…ts of oligopolistic …rms are positive, and the price is above the minimal average cost. In a small economy, the number of consumers per active …rm is signi…cantly above the value predicted by the Marshalian equilibrium. For example, a monopoly serves from 18 to 33 consumers and not predicted 16:48. One of the reasons is that indivisibility of a …xed cost prevents the other …rms from entering the industry, even if the incumbent has a relatively high pro…t. In addition, in a small economy, consumers have some market power, and hence, their aggregate demand is below the competitive demand. On the other hand, the simulation shows that the Marshalian equilibrium is a good approximation whenever the number of operating …rms is large. In an economy with more than one hundred active …rms, the gap in the predictions becomes negligible. D) When in the cost function …xed cost is zero, F = 0, the equilibrium with entry fails to exist. Theorem 1 asserts that for any value of J, one can …nd a noncompetitive equilibrium. It can also be shown that given identical cost function the equilibrium is symmetric, where all …rms make a positive pro…t. Hence, for any J there are always incentives to enter the industry. By de…nition, an economy without a long-run equilibrium is competitive.

References

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28 Appendix

Proof: Proposition 1. A trade that maximizes preferences Ui( ; ) on a budget set B(p; ti; M i),   (ti; mi) is a solution to the maxt ;m Ui(ti + ei; mi), subject to (ti; mi) B(p; ti; M i) and ti ei. i i 2  Ui( ; ) is strictly concave in ti, and constraints are convex, given that M i is positive. In addition,   by strict monotonicity of preferences in money, in an optimum budget constraint must be satis…ed with equality. The necessary Khun-Tucker optimality condition is given by

u p M i(2ti ti) 0; i0  pp; ti;M i (ti) ti + mi = 0; (18)  p i  i (ti) =p  + Mi (ti ti) ; (p; t ;M ) where the …rst inequality holds with equality for all commodities such that ti > ei. By the de…nition of a stable trade, ti is optimal on the budget set spanned by itself, and therefore the additional necessary stability condition is ti = ti. Thus (18) reduces to two conditions

u0 M iti p; (19) i  i m +p ti = 0:

For any positive M i, equation (19) has a unique solution ti. M i > 0, therefore the left hand side reservation is decreasing in ti. This function attains its maximal value, p , at ti = ei and is equal reservation to zero for some …nite value of ti. Therefore, for any p from interval [0; p ), there exists a reservation unique positive value of ti that solves inequality (19) with equality, and for p in [p ; ], the 1 unique value of ti satisfying (19) is zero. Given ti that solves (19), a unique stable m i is determined by the second equality (19). This shows that for any p 0 and positive M i, there exists a unique  ti that is stable.

Proof: Corollary 1. Proposition 1 de…nes stable trade function ti(p; M i). In addition, when ti ei, non-numéraire stable demand is implicitly de…ned by u (ti + ei) =p  + M iti. The  i0 derivative of the implicit function with respect to ti is given by M i u , which is positive. From the i00 implicit function theorem it follows that there exist neighborhoods of ti and p, and a di¤erentiable bijection, t (p; M i), solving (19). t ( ; ) coincides with a stable trade function. Since the argument i i   holds for any interior trade, ti(p; M i) is smooth. The implicit function theorem also implies that 2 2 1 the derivative of the stable trade function with respect to price is @ ti(p; Mi)=@p = (Mi u ) : i00 Proof: Theorem 1. We establish the existence of equilibrium for economies with more than two traders in the following steps. First we construct a truncated set of trades, that is non-empty, convex, and compact. Then we de…ne three continuous functions, and two non-empty, convex, and compact sets of the second derivatives, and the price impacts respectively. We show that a continuous function that consists of the three components de…ned in the previous steps has a …xed point. Next, we show that truncation of the trade set is without loss of generality, which …nally allows me to prove that the …xed point implies a noncompetitive equilibrium. In the last step we

29 argue that a noncompetitive equilibrium does not exist when there are only two traders.

Step 1. Truncated space of trades, . T

(Argument for Step 1): Given the upper bound on a price as u maxi ui0 (ei), for any consumer i 2I min  min a lower bound of the consumption in equilibrium xi is determined by ui(xi ) =u . By interiority, min min A3, and convexity of a utility function, xi is well de…ned and 0 < xi ei. Similarly for …rm max max  j we de…ne an upper bound on the production yj , as @fj0(yj ) =u . By monotonicity and convexity, production ymax > 0 exists and is unique. A truncated space of trades, denoted by , is j T a set of all trades such that markets clear and individual trades are within the following bounds

N 1 min max t R tn = 0; and ti ei + xi ; and 0 tj yj : (20) T  f 2 j  2   g n X2N Trade space is non-empty (for example, t = 0 is a member of ), convex (union of three convex T T sets). The set is also compact: it is closed as it is a pre-image of a closed set by a continuous function, and it is bounded as all trades are bounded from below; and they all sum up to zero, therefore they are also bounded from above.

 Step 2. Continuous function V ( ): .  T!V N (Argument for Step 2): De…ne V ( ): R++, as V ( ) (V1( );:::;VN ( )) where for each  T!     n = i; j; Vi(ti) u (ei + ti) and Vj(tj) = f ( tj). Function V ( ) is not well de…ned for trades in  i00 j00  which …rm’s supply of some good is zero. This is because the second derivative of cost functions are not de…ned on the boundary of R+. For tj = 0 function Vj(tj) is then de…ned as a limit

Vj(0) limtj 0 fj00( tj). By A7, such limit is well de…ned, …nite, and strictly greater than zero.  ! Utility and cost functions are twice continuously di¤erentiable; therefore, V (t) is continuous on .  N T We next show that function V ( ) maps into a subset R++ that is non-empty, convex,  T V  and compact. For any two non-negative scalars and , satisfying 0  < , de…ne a  N  N   1 non-empty, convex and compact box R++ as v R++ v 1  and v 1 , where N N V  V  f 2 j   g 1 R++ is a unit vector in R . The two scalars are de…ned as follows 2

 = sup max Vn(tn) and = inf min Vn(tn): (21) t n t n 2T 2N 2T 2N Since V (t) is continuous and is compact, both sup and inf are attained on , therefore 0 < T  T   < . By construction, the image of by function V ( ) is contained in . 1 T  V    Step 3. Continuous function H( ): .  M0  V !M0 N (Argument for Step 3): Given equilibrium convexity for all traders, v R++, the consistency 2 condition on M is equivalent to M being a …xed point of function H( ; ) (H1( ; );:::;HN ( ; )),        where each component Hn( ; ) is given by   1 Hn(M; v) = (M n + vn n0 = n); (22) N 1H 0 0 j 6 30 N N N   Note that H : R++ R++ R++. De…ne  = (N 2). By assumption N > 2, therefore   !    N  exists, 0 <  < : Let 0 M R+ v 1 be a non-empty convex and compact box. We 1 M   f 2  j  g  N N N now argue that H( ; ) maps 0 into a subset 0 . De…ne A( ; ): R++ R++ R++ as   V  M M    ! A( ; ) = (A1( ; );:::;AN ( ; )), where An( ; ) is given by        

1 1 An(M; v) = (Mn + vn ) : (23) N 1 2N 1 0 0 3 n =n X06 4 5    Since for any (M; v) , coe¢ cients satisfy Mn  and vn  2 M0  V 0  0  1 1 N 2 An(M; v)  +   +  = ; (24)  N 1  N 1 N 1  th where the last step holds by de…nition of . The n component of function H(M; v) (An(M; v)) is a harmonic (arithmetic) mean of elements Mn + vn for all n n, discounted by factor 0 0 0 2 N n 1=(N 1) < 1. Therefore, by the standard harmonic-arithmetic mean inequality Hn(M; v)     An(M; v) , and hence H : .  M0  V !M0 Step 4. Continuous function T ( ):  .  M0 !T (Argument for Step 4): Allocation function T ( ) is de…ned as follows 

1 2 1 2 T (M) arg max (ui(ti + ei) Mi (ti) ) (fj( tj) + Mj (tj) ) : (25)  2 2 2 3 t i j 2T X2I X2J 4 5 The function in (25) is continuous, strictly concave, and is compact, therefore, the allocation T function T ( ) is well de…ned. In addition, T ( ) is continuous by a maximum principle.   Step 5. Existence of a …xed point

    (Argument for Step 5): De…ne F : as F ( ; ; ) = (T ( );V ( );H( )). T  V  M0 ! T  V  M0       In Steps 2 - 4 we argued that all three components of the function are continuous, therefore F is   also continuous. In addition , and are non-empty, convex, and compact, therefore their T V M0 Cartesian product also has the three properties. Consequently, the Brouwer …xed point theorem applies and there exists a point (t ; v ;M ) satisfying (t ; v ;M ) = (t ; v ;M ).    F       Step 6. Trade t is in the interior of .  T (Argument for Step 6): We now show that t is in the interior of ,  T

1. tj 0 is not binding.  The partial derivative of (25) with respect to tj is positive for all tj < 0 and equal to zero

for tj = 0. If for some j, tj = 0, then t = 0 must hold for all j0 . Otherwise, one j0 2 J could increase the value of the function by reducing the supply of j0 and o¤set this change

31 by increased supply by …rm j. With all …rms producing zero, there must exist at least one

consumer with non-positive trade ti (as all trades sum up to zero) and the partial derivative of (25) with respect to ti evaluated at t , is u (t + ei) M t > 0 as M 0. Consequently, i i0 i i i i  marginally reducing tj (hence increasing the supply of j) and at the same time increasing ti by the same amount, increases the value of the objective function and does not violate the constraints de…ning . This contradicts the optimality of t on . T  T max 2. tj y is not binding.  j Suppose for some j; t = ymax. Given de…nition of ymax, partial derivative of the objective j j j function (25) with respect to tj, evaluated at t satis…es fj( t ) M t u. In addition, by j j j j  the fact that all trades sum up to zero, there must be at least one consumer i with strictly i i positive trade t > 0. For such a consumers partial derivative u (t +e ) M t < u (ei) u. i i0 l l i i i0  The two inequalities imply that marginally increasing tj (cutting on the supply) and reducing

the demand ti by the same amount increases value of (25) and such change does not violate the constraints. This contradicts optimality of t.

1 min 3. t ei + x is not binding. i  2 i 1 max max Suppose for some i, t = ei + x . By construction x < ei, therefore t < 0. In i 2 i i i addition u (t + ei) M t > ul:Since the trades sum up to zero, there must exist some other i0 i i i consumer i0 with t > 0 and for such consumer ui0 (t + ei ) M t < u. The two inequalities i0 0 i0 i0 imply that increasing the demand of i and o¤setting this change by the reduction of trade i0 does not violate the constraints and increases the value of (25). This contradicts optimality of t on .  T Step 7. Fixed point (t ; v ;M ) de…nes a noncompetitive equilibrium (p ; t ;M ) in .       E (Argument for Step 7): The objective function in (25) is strictly concave, which, together with the interiority demonstrated in Step 6, implies that t maximizes (25) on a larger set of trades de…ned by equality constraints n tn = 0. It follows that there exists a Lagrangian multiplier, 2N p, such that t solves the unconstraintP maximization problem, with the objective function (25) augmented by the additional term (constraints) p n tn. The …rst order optimality conditions 2N for this program are: for any consumer, u (t + ei) Mit = p , and for any …rm j the condition i0 i P i  becomes f ( t ) M t = p . For each i, de…ne m p t and for each …rm m fj( t ). j0 j j j  i   i j  J The two conditions and two de…nitions are necessary and su¢ cient for tn to be stable at prices p n and Mn (see (19).) Since t , the markets clear, n t = 0. Finally, by interiority of trades 2 T 2N v = u (t + ei) and v = f ( t ) therefore M = H(M ; v ), at t , which is su¢ cient for M i i00 i j j00 j  P     to be mutually consistent. This proves the existence of a noncompetitive equilibrium (p; t;M ) when N > 2. To see that p 0 observe that if there exists some …rm for which t < 0, then   j p = f ( t ) M t > 0. Otherwise, there must exist at least one consumer with t 0 and hence 0 j j j i  p = u (t + ei) M t > 0. i0 i i i Step 8. Nonexistence for N = 2.

32 (Argument for Claim 8): In proving the existence of equilibrium, when de…ning the upper bound on the price impacts, , we used the fact that N > 2. In this step we show that this condition is also necessary for the existence. To see it, observe that in the case of two traders the consistency conditions are M 1 = M 2 u , and M 2 = M 1 u . The two equations imply u = u , which is 200 100 100 200 impossible, since utility functions are strictly convex.

Proof: Theorem 2. For any n, let n0 ; n00 R++ be the positive scalars, n (n0 ; n00 ) 2  and  n n . Fix economy = (u; e; f). A vector  uniquely de…nes a perturbed economy  f ^ g 2N E  = (^u; e; f), the economy with preferences and technology modi…ed as follows E 2 u^i(xi) = u(xi) + i0 xi + n00 (xi) ; (26) and for …rm j ^ 2 fj(yj) = fj(yj) + j0 yj + j00 (yj) : (27)

Consider any open ball B(0; r) R2N with radius r > 0 su¢ ciently small to guarantee that P   for any perturbation  preserves monotonicity and concavity (convexity) of the utility (cost) 2 P functions on the set of relevant trades  (see (20)). By continuity of utility and cost functions, such T radius exists. Ball spans a family of perturbed economies around original economy . Observe P E that is a connected set with a positive Lebesgue measure in R2N . For any perturbed economy, P , with  the set of all noncompetitive equilibria is de…ned as a set of critical points of the E 2 P system of equations

(p; t; M ) = 0; (28) where u^ M iti p for all i i0 2 I ^    0 fj0( tj) Mjtj p for all j 1 (p; t; M ) 2 J ; (29)    ^ 1 1  Mn ( (Mn + Vn(tn)) ) for all n B n0=n 2 N C B 6 n C B n t C B P 2N C @ A and V^i(ti) u^ (ti +ei), and V^j(tj) = f^ ( tj). GivenP  …xed equation (28) constitutes a system of  i00 j00 2N + 1 equations and the same number of unknowns. In economy , all noncompetitive equilibria E are locally unique if ( ) is transverse to 0. Unfortunately, this is not necessarily true for arbitrary  . We will argue, however, that transversality holds for almost all economies in , and hence E P the lack of local uniqueness occurs only as a degenerate case. We …rst extend function ( ) to a  domain that includes perturbation parameters . Next we show that the extended function ( ; )   is transverse to zero, and, …nally, using a transversality theorem, we establish the transversality of

( ) for almost all ; and hence we demonstrate the generic determinacy of an equilibrium.  We prove the transversality of ( ; ) using a method of perturbation. First observe that any of   the …rst order conditions (N …rst equations) can be perturbed by varying corresponding n0 . Since

n0 enters exactly one equation in the whole system, such perturbation does not a¤ect any other equation. Each of the equations in the second group, de…ning consistent price impacts, can be

33  ~ perturbed by changing Mn to Mn and o¤setting this change by the adjustment of n00 so that the  ~ n sum of the two elements stays constant (Mn + Vn(tn) = Mn + Vn(t ) + n00 and simultaneously by changing n0 to compensate the e¤ects of the change in n00 in the …rst group of equations. Finally, the market clearing condition can be perturbed by varying tn for some n and compensating this change by adjusting n to keep the values of equations in the …rst two groups unchanged. This proves the transversality of ( p; t; M; ) to 0. Applying the transversality theorem, one can establish that there exists an open subset ^ with full Lebesgue measure of , such that for any  ^ P  P P 2 P restriction (p; t; M ) is transverse to zero. Consequently, in any perturbed economy  from ^, E P all equilibria are locally unique. Proof : Theorem 3. Let (p; t; M ) be a noncompetitive equilibrium. For any i , the 2 I      necessary and su¢ cient condition of stability of ti given p and Mi is ui0 (ti + ei) =p  + Miti and similarly for …rm j the condition becomes c ( tj) =p + M jtj:In a noncompetitive equilibrium 2 J j0 allocation is Pareto e¢ cient if for any n = i; j the marginal utilities and costs of all traders coincide u (ti +ei) = c ( tj). Also, since the consistency condition implies that price impacts are discounted i0 j0 harmonic averages of strictly positive scalars, the equilibrium price impacts are strictly positive.

Step 1. Pareto e¢ ciency of equilibrium implies J = 0 and e¢ ciency of endowments.

(Argument for Step 1): In equilibrium, trades are stable for all n , and hence marginal 2 N utilities and costs are equalized with marginal expenditure. This with Pareto e¢ ciency implies that M iti = M jtj = x for all i; j. Since price impacts are positive, for any n = i; j the trade    can be written as tn = x=Mn. Then by the market clearing condition, 0 = n x=Mn, and 2N hence x = ( (M ) 1) 10 = 0. Consequently, for any n equilibrium trades are given by n n P   2N tn = x=Mn =P 0. Consider an economy with J > 0. From the stability conditions for a …rm, it follows that p = f 0 (0) M j0 = 0 which in turn by the stability condition for a consumer implies j ui0 (ei) = 0 which contradicts strict monotonicity of preferences. It follows that only pure exchange economy, J = 0, might be consistent with Pareto e¢ ciency. In such economy, given ti = 0 for all i , the stability condition implies that u (ei) =p  and hence with no trade marginal utilities for 2 I i0 all consumers coincide. It follows that initial allocation is Pareto e¢ cient.

Step 2. J = 0 and e¢ cient endowments imply e¢ cient equilibrium allocation.

(Argument for Step 2): Fix an arbitrary equilibrium p; t; M . The …rst order conditions of  stability imply that t also solves 

1 2 t= arg max ui(ti + ei) M i (ti) p ti (30) t 2 i i X2I X2N The objective function is strictly concave, and hence there exists at most one t that is optimal.

But since u0 (ei) = u0 (ei ), no trade is such a solution and hence it is a unique equilibrium trade i i0 0 consistent with p; M . It follows that allocation coincides with endowments and hence equilibrium allocation is Pareto e¢ cient.

34 Proof: Theorem 4, In this proof we use notation from the proof of Theorem 1. By similar arguments as in Step 6 of this proof, the only trades consistent with noncompetitive equilibrium N are within set R+ . Given , the second derivatives of the utility and cost functions possibly T  T  observed in equilibrium are within a compact set , where the set and two scalars are de…ned in V Step 3. In the following arguments, I focus on a k replica of the economy, and therefore I assume that there are k traders of each type n.

  Step 1. In a k replica economy Mn N k 2 for any n.   (Argument for Step 1): In k replica each agent trades with k trades of each other type (there  are N 1, other types) and k 1 traders of its own type. Given v , price impacts M are th2 V consistent if, and only if, they are a …xed point of map H where the n element Hn( ), is given by  1 1 1 Hn(M; v) = k (M n + vn ) + (k 1)(M n + vn) : (31) 2 0 0 3 n =n X06 4 5 For each type n de…ne a map

1 1 An(M; v) = k (M n + vn ) + (k 1)(M n + vn) : (32) N k 1 0N k 1 0 0 0 11 n =n   X06 @ @ AA    maxn Mn+ Since in equilibrium v , equation (32) implies the following inequality An(M; v) N k 1 , 2 V     maxn Mn+  and by the harmonic-arithmetic mean inequality Hn(M; v) N k 1 . Since vector M is   consistent if, and only if, M = H(M; v) therefore the expression is also an upper bound on maxn M n+ maxn M n . Solving for maxn M n gives an upper bound on equilibrium price impacts  N k 1    Mn N k 2 for any n.   Step 2. Convergence to a competitive equilibrium.

(Argument for Step 2): For any " > 0 we de…ne k(")  + 2 1 (ignore the integer problem).  " N Given the result from the previous step, for such k(") in any noncompetitive equilibrium consistent   1  price impacts Mn satisfy Mn N k 2  = " for any n. In addition, for any M there exists a unique   t and p such that (p; t; M ) is a noncompetitive equilibrium. To see it, observe that stability for all i and j requires that

u0 (ti + ei) M iti = f 0( tj) M jtj: (33) i j   Since for all traders augmented marginal utility or cost is decreasing in tn and n tn = 0, there   2N is a unique t that solves (33) subject to market clearing, given …xed value M. TheP equilibrium price is then given by p = u (ti + ei) M iti for arbitrary i. This shows that there exist functions, tn(M ) i0 for any n and p(M ), mapping the vectors of price impacts into equilibrium trades and prices. 2 N Because (33) is continuous, by the standard argument t(M ) and p(M ) are also continuous in M .

Therefore, for any " there exists " such that for any M " , one obtains tn(M ) tn(0) " 0  0 k k 

35 and p(M ) p(0) ". But tn(0) and p(0) correspond to the competitive trade and price. To k k  conclude for any " de…ne k(") max  + 2 =N;  + 2 =N . It follows that for any k k("),  " "0  in a k replica economy in any noncompetitive equilibrium is in "-distance from a competitive    equilibrium. Proof : Proposition 2. The proof consists of the following steps: First, I prove two useful facts: In Step 1, I show that the convexity of buyers’utility function is always greater than those of the sellers, for any arbitrary level of price impact. Then in Step 2, I argue that in considered noncompetitive equilibrium, the price impacts of a seller exceed those of a buyer. The two results are used in the proof of the main result in Step 3. For any (…xed) equilibrium (p; t; M ), the set of all consumers is partitioned into three groups: buyers, b = i ti > 0 , sellers s = i ti < 0 , and nonactive traders na = i ti = 0 . I f 2 Ij g I f 2 Ij g I f 2 Ij g Since the initial allocation is not Pareto e¢ cient, the sets s and b are non-empty. For any p 0 I I  and Mi 0, I de…ne a stable convexity of trader i as a second derivative evaluated at the stable  trade ti(p; Mi),

vi(p; Mi) u00 (ti(p; Mi) + ei) : (34)  j i j In Step 1 we show that at equilibrium price p, and positive but otherwise arbitrary price impacts

Mb, Mna, Ms, buyers always have higher stable convexity, given u000 > 0.

Step 1. For any b b, s s, and na na, at equilibrium price p, and arbitrary positive Mb, 2 I 2 I 2 I Mna, Ms stable convexities satisfy

vb(p; Mb) > vna(p; Mna) > vs(p; Ms) > 0; (35)

(Argument for Step 1): Stable demand ti(p; Mi) is de…ned by the equality of marginal utility and the marginal revenue   ui0 (ti + ei) =p  + Miti: (36)   By de…nition tb(p; Mb) > 0, and it follows that ub0 (eb) > p. Therefore, from (36) at equilibrium price buyer’sstable trade is positive tb(p; Mb) > 0 for any Mb 0. Similarly, for a seller it is strictly  negative. The derivative of a stable trade function with respect to Mi is given by

@ti(p; Mi) ti(p; Mi) = i i : (37) @Mi M + v (p; Mi)

Since the denominator is always strictly positive, the stable trade is monotonically decreasing in

Mi for the buyers (on the whole domain), is increasing for the sellers, and is constant for nonactive traders. Consequently, for arbitrary price impact Mb > 0, the stable demand of a buyer is bracketed by a competitive and zero trade tb(p; 0) > tb(p; Mb) < 0. By analogous argument, for each seller the trade is bracketed by ts(p; 0) < ts(p; Ms) < 0. With identical quasilinear preferences the competitive consumption at p is the same for all traders (in fact the equality holds for all prices)

na tb(p; 0) + eb = e = ts(p; 0) + es; (38)

36 For any strictly positive (but otherwise arbitrary) price impacts Mb, Mna, Ms this equation and trade bracketing conditions imply the following inequality

tb(p; Mb) + eb < ena < ts(p; Ms) + es: (39)

With the second derivative of the utility function increasing in consumption, (u000 > 0), the relation between stable consumptions (39) implies the following ranking of convexities:

vb(p; Mb) > vna(p; Mna) > vs(p; Ms); (40) for arbitrary, strictly positive price impacts Mb, Mna, Ms.

Step 2. With u > 0, for any b b, s s, and na s, the equilibrium price impacts satisfy 000 2 I 2 I 2 I

M s > M na > M b: (41)

(Argument for Step 2): The system of consistent M for any i satis…es

1 M i = (M i + v(p; M i ) i0 = i): (42) I 1H 0 0 j 6

I such conditions imply that ranking of M i coincides with a reversed ranking of equilibrium con- vexities. Suppose for two traders i and i , vi(p; M i) > vi (p; M i ) and M i M i . Condition (42) 0 0 0  0 shows that M i is a harmonic average of I 2 elements that also de…ne M i and one element that 0       is strictly smaller as by assumption Mi0 + vi0 (p; Mi0 ) < Mi + vi(p; Mi) and hence Mi0 > Mi. Since   harmonic average is monotone, this implies Mi < Mi0 which is a contradiction. This, together with the result from Step 1 is su¢ cient for the result of Step 2.

W alras Step 3. u000 > 0 implies p > p .

(Argument for Step 3): Let M~ be any scalar satisfying M s > M~ > M s; for any b b, 2 I and s s. By the result from Step 2, such scalar exists. In stable market clearing condition 2 I    ~ i ti(p; Mi) = 0 replaces equilibrium price impacts, Mi, with the intermediate level M. The strict monotonicity of ti(p; ) in Mi shown in (37) and ranking of price impacts imply that aggregate P  excess demand evaluated at M~ is negative D(M~ ) ti(p; M~ ) < 0. In addition, for arbitrary  i common scalar M, a partial derivative of D( ) with respect to M is given by  P @D(M) t (p; M) = i : (43) @M M + v (p; M) i i X By v~ > 0, I denote a convexity of the utility function evaluated at the competitive consumption i given p; (p = u0(x )). By the result from Step 1, vs(p; M) < v~ < vb(p; M) for any M > 0. Replacing the convexities with a common value v~ in the derivative of D(M) gives a lower bound

37 of this derivative @D(M) t (M) D(M) > i i = : (44) @M M +v ~ M +v ~ P The inequality results from the fact that for the buyers (positive trades) replacing vi(p; M) with v~ decreases the denominator, while for sellers, (negative trade) the denominator goes up. Inequality (44) de…nes a lower bound on the slope of D(M). Observe that D(M) is upward sloping in M as long as M(M) < 0 and M 0. Since excess demand evaluated at the intermediate level M~ is  negative, D(M~ ) < 0, therefore D(M) is increasing in M around this price impact. Now we show that it is nondecreasing on the whole interval [0; M~ ]. To see it, consider a subset of all M from the interval, for which the excess demand is greater than zero D(M) 0. This subset is a pre-image  of a closed set by a continuous function, and hence is closed. Since it is also bounded, it is also compact. Therefore, it must contain an upper bound M^ . By construction, on the interval (M;^ M~ ], function D(M) < 0 and hence its derivative is strictly positive –function is increasing. This implies that for any M from this interval, D(M) < D(M~ ) < 0 and hence by continuity, at M^ , function D(M^ ) D(M~ ) < 0. But this contradicts the assumption that D(M^ ) 0. Consequently, for all   M (0; M~ ], excess demand D(M) is strictly negative and hence is increasing in M. It follows 2 that D(0) < D(M~ ) < 0. But this shows that at the noncompetitive equilibrium price we observe  excess supply D(0) = i ti(p; 0) < 0:The derivative of D(p; 0) with respect to p (for any price) is @D(p;0) = 1 < 0, hence it is decreasing in price. The unique competitive price solves @p i vi(p;0)P W alras W alras D(p ; 0) =P 0, therefore it satis…es p > p . Consequently, we observe a positive price bias in a noncompetitive equilibrium. The proofs for ui000 < 0 and ui000 = 0 are analogous the presented case, and hence we leave it to the reader.

Proof: Proposition 3. tn ; M n is the best response to tn ( ; M n ) n =n if its submission  f 0  0 g 06  results in an outcome of a Walrasian auction p; tn that maximizes the utility of trader n. Since t ; M gives rise to trade for which p; t equalizes the marginal utility of n with the n0 n0 n =n n  06 marginal revenue, given residual supply de…ned by tn ; M n and hence the …rst-order   0  0 n0=n  6  optimality condition is satis…ed at tn. Since the optimization problem is globally convex, tn is optimal and t ; M is a best response to t ; M . n n n0 n0 n =n   06 Proof : Lemma  1. Consider an economy with  J …rms with technology characterized by a constant marginal cost. The consistency condition for the price impacts and symmetry across j 1 1 1 1 …rms imply M = ((J 1)(M j) + (M i + vj) ) , which is equivalent to (M j) (2 J) = i I  1 2 i I (Mi + vi) . If J 2, then the left-hand side of the equation is less or equal to zero. But this 2  P Pis a contradiction since the right-hand side is strictly positive.

38