Notes on optimal transportation

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

The existence of fully rational expectations approximate equilibria with noisy price observations

Nonrevealing fully rational expectations approximate equilibria exist in microeconomic pure exchange economies in which uninformed agents have suitably dispersed noisy price observations. Such traders maximize a state-dependent expected utility conditional on the price vector they observe, the distributions of noisy price observations, and the correct equilibrium relationship between states of the world and prices. In equilibrium, aggregate excess demand is small with high probability in every state of the world (and its expectation is also small); this magnitude diminishes as the noisy price observations become more accurate. Equilibria are obtained by applying a fixed point argument to state-dependent excess demand functions which are smooth because of the noisy price observations.     

Selecting equilibria in common agency games

We characterize equilibrium payoffs of a delegated common agency game in a public good context where principals use smooth contribution schedules. We prove that under complete information, payoff vectors of equilibria with truthful schedules coincide with the set of smooth equilibrium payoffs, including non-truthful schedules. We next consider whether the presence of arbitrarily small amounts of asymmetric information is enough to refine this payoff set. Providing that the extensions of the equilibrium schedules beyond the equilibrium point are flatter than truthful schedules, the set of equilibrium payoffs is strictly smaller than the set of smooth (equivalently, truthful) equilibrium payoffs. Interestingly, some forms of asymmetric information do not sufficiently constrain the slopes of the extensions and fail to refine the payoff set. In the case of a uniform distribution of types and arbitrary out-of-equilibrium contributions, the refinement has no bite. If, however, one restricts out-of-equilibrium behavior in a natural way, the refinement is effective. Alternatively, we may consider an exponential distribution with unbounded support (and hence no out-of-equilibrium choices) and we find that the refinement selects a unique equilibrium payoff vector equal to Lindahl prices.As a separate contribution, equilibria with forcing contracts are also considered both under complete and asymmetric information.     

Essential equilibria of large generalized games

We characterize the essential stability of games with a continuum of players, where strategy profiles may affect objective functions and admissible strategies. Taking into account the perturbations defined by a continuous mapping from a complete metric space of parameters to the space of continuous games, we prove that essential stability is a generic property and every game has a stable subset of equilibria. These results are extended to discontinuous large generalized games assuming that only payoff functions are subject to perturbations. We apply our results in an electoral game with a continuum of Cournot-Nash equilibria, where the unique essential equilibrium is that only politically engaged players participate in the electoral process. In addition, employing our results for discontinuous games, we determine the stability properties of competitive prices in large economies.     

The price normalization problem in imperfect competition and the objective of the firm

Summary. General equilibrium models of oligopolistic competition give rise to relative prices only without determining the price level. It is well known that the choice of a numéraire or, more generally, of a normalization rule converting relative prices into absolute prices entails drastic consequences for the resulting set of Nash equilibria when firms are assumed to maximize profits. This is due to the fact that changing the price normalization amounts to altering the objective functions of the firms. Clearly, the objective of a firm must not be based on price normalization rules void of any economic content. In this paper we propose a definition of the objective of a firm, called maximization of shareholders' real wealth, which takes shareholders' demand explicitly into account. This objective depends on relative prices only. Real wealth maxima are shown to exist under certain conditions. Moreover, we consider an oligopolistic market and prove the existence of a Nash equilibrium in which each firm maximizes the real wealth of its shareholders. Received: July 10, 1997; revised version: July 27, 1998     

Coalition-proof Bertrand equilibria

Summary. This paper examines the coalition-proof Nash equilibria of a Bertrand model of price competition where firms supply all demand. When firms are asymmetric we prove existence and provide a sufficient condition for uniqueness. For symmetric firms, we show that an equilibrium is necessarily unique. We also examine whether this unique equilibrium outcome is implementable through a sequential move game where the firms take turns at announcing prices. Finally we examine the limiting property of such equilibria as the number of firms go to infinity.Received: 20 March 2002, Revised: 5 August 2003JEL Classification Numbers: D43, D41, L13.Correspondence to: Kunal SenguptaWe are deeply indebted to an anonymous referee for very helpful and incisive comments that led to substantial improvements in the paper. We also gratefully acknowledge the hospitality of the Department of Finance, Hong Kong University of Science and Technology where much of the work on this paper was carried out.     

Multiunit Pay‐Your‐Bid Auction with One‐Dimensional Multiunit Demands*

An arbitrary number of units of a good is sold to two bidders through a discriminatory auction. The bidders are homogeneous ex ante and their demand functions are two‐step functions that depend on a single parameter. We characterize the symmetric Bayesian equilibrium and prove its existence and uniqueness. We compare this equilibrium with the equilibrium of the multiunit Vickrey auction and with the equilibria of the single‐unit first price and second price auctions. We examine the consequences of bundling all units into one package. We study the impacts that variations of the “relative” supply have on the equilibrium, on the bidders' average payoffs per unit, and on the efficiency of the equilibrium allocation.     

Indeterminacy of equilibrium in stochastic OLG models

Summary. This paper studies the equilibria of a stochastic OLG exchange economies consisting of identical agents living for two periods, and having the opportunity to trade a single infinitely-lived asset in constant supply. The agents have uncertain endowments and the stochastic process determining the endowments is Markovian. For such economies, the literature has focused on studying strongly stationary equilibria in which quantities and prices are functions of the exogenous states of nature which describe the uncertainty: such equilibria are generalizations of deterministic steady states, and this paper investigates if they have the same special status as asymptotic limits of other equilibrium paths. The difficulty in extending the analysis of equilibria beyond the class of strongly stationary equilibria comes from the presence of indeterminacy: we propose a procedure for overcoming this difficulty which can be decomposed into two steps. First backward induction arguments are used to restrict the domain of possible prices; then if some indeterminacy is left, expectation functions are introduced to make the forward equilibrium equations determinate. The properties of the resulting trajectories, in particular their asymptotic properties, can then be studied. For the class of models that we study this procedure provides a justification for focusing on strongly stationary equilibria. For the model with positive dividends (equity or land) the justification is complete, since we show that the strongly stationary equilibrium is the unique equilibrium. For the model with zero dividends (money) there is a continuum of self-fulfilling expectation functions resulting in a continuum of equilibrium paths starting from any admissible initial condition: under conditions given in the paper, these equilibrium paths converge almost surely to one of the strongly stationary equilibria-either autarchy or the stochastic analogue of the Golden Rule. Received: November 19, 2001; revised version: March 22, 2002 RID="*" ID="*" We are grateful for the stimulating environment and research support provided by the Cowles Foundation at Yale University during the Fall 2000 when this paper was first conceived. We are also grateful to the participants of the SITE Workshop at Stanford University and the Incomplete Markets Workshop at SUNY Stony Brook during the summer 2001 for helpful discussions. Correspondence to: M. Magill     

Price competition in a differentiated products duopoly under network effects

We examine price competition under product-specific network effects, in a duopoly where the products are differentiated both horizontally and vertically. We emphasize the role of consumers’ expectations formation. When expectations are not influenced by prices, the market may be shared but shares must be equal unless product qualities differ or one firm, possibly even the low-quality one, may capture the entire market. When expectations are influenced by prices, which would be the case when there is commitment, competition becomes more intense and the high-quality firm tends to capture a larger market share. Under strong network effects there is a continuum of equilibria and the higher the prices, the smaller the difference between those prices can be. Requiring continuity of expectations, however, delivers a unique equilibrium where one firm captures the entire market.     

On Rothschild–Stiglitz as Competitive Pooling

Dubey and Geanakoplos (Q J Econ 117:1529–1570, 2002) have developed a theory of competitive pooling, which incorporates adverse selection and signaling into general equilibrium. By recasting the Rothschild–Stiglitz model of insurance in this framework, they find that a separating equilibrium always exists and is unique. We prove that their uniqueness result is not a consequence of the framework, but rather of their definition of refined equilibria. When other types of perturbations are used, the model allows for many pooling allocations to be supported as such: in particular, this is the case for pooling allocations that Pareto dominate the separating equilibrium.     

Public Sector Rationing and Private Sector Selection

We study the interaction between nonprice public rationing and prices in the private market. Under a limited budget, the public supplier uses a rationing policy. A private firm may supply the good to those consumers who are rationed by the public system. Consumers have different amounts of wealth, and costs of providing the good to them vary. We consider two regimes. First, the public supplier observes consumers’ wealth information; second, the public supplier observes both wealth and cost information. The public supplier chooses a rationing policy, and, simultaneously, the private firm, observing only cost but not wealth information, chooses a pricing policy. In the first regime, there is a continuum of equilibria. The Pareto dominant equilibrium is a means‐test equilibrium: poor consumers are supplied while rich consumers are rationed. Prices in the private market increase with the budget. In the second regime, there is a unique equilibrium. This exhibits a cost‐effectiveness rationing rule; consumers are supplied if and only if their cost–benefit ratios are low. Prices in the private market do not change with the budget. Equilibrium consumer utility is higher in the cost‐effectiveness equilibrium than the means‐test equilibrium.     

Price competition,free entry,and welfare in congested markets

In this paper we study the problem of price competition and free entry in congested markets. In particular, we consider a network with multiple origins and a common destination node, where each link is owned by a firm that sets prices in order to maximize profits, whereas users want to minimize the total cost they face, which is given by the congestion cost plus the prices set by firms. In this environment, we introduce the notion of Markovian Traffic Equilibrium to establish the existence and uniqueness of a pure strategy price equilibrium, without assuming that the demand functions are concave nor imposing particular functional forms for the latency functions. We derive explicit conditions to guarantee existence and uniqueness of equilibria. Given this existence and uniqueness result, we apply our framework to study entry decisions and welfare, and establish that in congested markets with free entry, the number of firms exceeds the social optimum.     

Bargaining with revoking costs

A simple two stage bilateral bargaining game is analyzed. The players simultaneously demand shares of a unit size pie. If the demands add up to more than one, the players simultaneously choose whether to stick to their demand or accept the other?s offer. While both parties sticking to their offers leads to an impasse, accepting a lower share than the original demand is costly for each party. The set of pure strategy subgame perfect equilibria of the game is characterized for continuously differentiable payoff and cost functions, strictly increasing in the pie share and the amount conceded, respectively. Higher cost functions are shown to improve bargaining power. The limit equilibrium prediction of the model, as the cost functions are made arbitrarily high, selects a unique equilibrium in the Nash Demand Game that corresponds to a Proportional Bargaining Solution of Kalai (1977).     

Coalition-proof supply function equilibria in oligopoly

In an industry where firms compete via supply functions, the set of equilibrium outcomes is large. If decreasing supply functions are ruled out, this set is reduced significantly, but remains large. Specifically, the set of prices that can be sustained by supply function equilibria is the interval between the competitive price and the Cournot price. In sharp contrast, when the number of firms is above a threshold we identify (e.g., three if demand is linear), only the Cournot outcome can be sustained by a coalition-proof supply function equilibrium.     

Equilibria in first price auctions with participation costs

This paper characterizes the equilibria of first price auctions with participation costs in the independent private values environment. Bidders use cutoff strategies to decide whether they will participate in the auction. It is shown that, when bidders are homogeneous, there always exists a unique symmetric equilibrium, and further, there is no other equilibrium when valuation distribution functions are inelastic. When distribution functions are elastic at the symmetric equilibrium, there exists an asymmetric equilibrium. Inelasticity/elasticity includes concavity/convexity of distribution functions as a special case. We find similar results when bidders are heterogeneous.     

Markov equilibria in dynamic matching and bargaining games

Rubinstein and Wolinsky [Rev. Econ. Stud. 57 (1990) 63] show that a simple homogeneous market with exogenous matching has a continuum of (non-competitive) perfect equilibria; however, the unique Markov-perfect equilibrium of this model is competitive. By contrast, in the more general case of heterogeneous markets, even the Markov property is not enough to guarantee the perfectly competitive outcome. We define a market game that allows for heterogeneous values on both sides of the market and exhibit a number of examples of (non-competitive) Markov-perfect equilibria, with and without discounting. Unlike the homogeneous case, these equilibria allow for inefficient trades and for trade at non-uniform prices. The non-competitive equilibrium may be unique.     

On the existence of rational expectations equilibrium

We prove a theorem on the existence of rational expectations general economic equilibrium when agents condition on prices as well as on private information, and maximize the expectation of a state-dependent utility function. The key to the result is a new idealization of what it means for a set of empirical distributions to support agents' expectations. This idealization depends on the notion that agents compare their expectations with continuous versions of the random empirical distributions that are generated by the workings of the economy. The existence theorem covers all strictly concave utility functions, arbitrary distributions of the state variable, and situations in which the dimension of the state variable is large relative to the number of commodities.     

Intertemporal pricing in markets with differential information

Summary This paper provides a theory of intertemporal pricing in a small market with differential information about the realizations of a stochastic process which determines demand. We study the sequential equilibria in stationary strategies of the stochastic game between a seller and buyer. The seller has zero cost of producing one unit of a non-durable good in all market periods. The buyer's value for the good is a random variable governed by a simple Markov process. At the beginning of each period the unit's value is determined by nature and is privately revealed to the buyer. The seller posts a single price offer each period, which the buyer either accepts or rejects. Only two types of price paths emerge in equilibrium: either prices are constant, or they have persistent cycles between a low and a high value. In both cases, however, prices are sticky in the sense that changes in price are less frequent than changes in the economy's fundamentals.We thank John Rust and Asher Wolinsky for helpful comments. We also gratefully acknowledge financial support from NSF grant SES 89-09242.     

Multiple equilibria and hysteresis in simple exchange models

This paper presents a simple two-good general equilibrium model of exchange in which demands are linear functions of relative prices. The solution of the model is represented by a cubic equation which can have either a single root which corresponds to a unique equilibrium price or three roots corresponding to multiple equilibria. The major properties of the model are that the relative price can make large discrete jumps in response to small parameter changes, and can display hysteresis. The paper shows how recent developments in non-linear methods can be used to examine the equilibrium properties of the model and establish conditions under which multiple equilibria can arise. The model is compared with that used by Shapley and Shubik.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.