Determinacy of competitive equilibria in economies with many commodities

This paper provides a framework for establishing the determinacy of equilibria in general equilibrium models with infinitely many commodities and a finite number of consumers and producers. This paper defines a notion of regular economy for such models and gives sufficient conditions on the excess savings equations characterizing equilibria under which regular economies have a finite number of equilibria, each of which is locally stable with respect to perturbations in exogenous parameters, and under which regular economies are generic. This paper also defines two notions of concavity, called uniform concavity and weighted uniform concavity, which generalize standard finite-dimensional notions of differential concavity to an infinite-dimensional setting by prohibiting goods from becoming perfect substitutes asymptotically. For the case of economies in which there are countably many commodities, such as discrete time models or markets with countably many assets, results in this paper show that equilibria are generically determinate as long as utility functions and production sets are uniformly concave or weighted uniformly concave. Received: November 7, 1996; revised version: March 13, 1998     

Decentralizing Edgeworth equilibria in economies with many commodities

Summary. This paper proves core-equivalence theorems for exchange economies without ordered preferences, defined on locally convex Riesz commodity spaces such that the price space is a lattice. Properness assumptions are borrowed from some recent equilibrium existence results. Received: January 15, 1998; revised version: August 19, 1998     

Equilibria in large economies with differentiated commodities and non-ordered preferences

Summary. The purpose of this paper is to provide an equilibrium existence result for economies with a measure space of agents, a finite set of producers and infinitely many differentiated commodities. The approach proposed in this paper, based on the discretization of measurable correspondences, allows us to extend the existence results in Ostroy and Zame (1994) and Podczeck (1997) to economies with a non-trivial production sector and with possibly non-ordered preferences. Moreover, our approach allows for more general consumption sets than the positive cone and following the direction introduced by Podczeck (1998), the uniform substitutability assumptions of Mas-Colell (1975), Jones (1983), and Ostroy and Zame (1994), are replaced by the weaker assumptions of uniform properness.Received: 11 June 2001, Revised: 6 March 2003, JEL Classification Numbers: C62, D51.Thanks to Charalambos D. Aliprantis, Erik J. Balder, Jean-Marc Bonnisseau, Bernard Cornet, Monique Florenzano, Konrad Podczeck, Rabee Tourky, Nicholas C. Yannelis and two anonymous referee for helpful discussions and suggestions.     

Non-cooperative games with a continuum of players whose payoffs depend on summary statistics

Summary. We consider static non-cooperative games with a continuum of small players whose payoffs depend on their own actions and finitely many summary statistics of the aggregate strategy profile. We prove the existence of an equilibrium in pure strategies without any convexity restrictions on payoffs or the common action space. We show that this result applies to a broad class of monopolistic competition models. Received: April 13, 2001; revised version: December 18, 2001 RID="*" ID="*" The result in this paper generalizes a result in my PhD dissertation supervised by M. Ali Khan and Joe Harrington. I thank them for support and encouragement. I also thank Sung Kim, Bruce Nanney, Ashvin Rajan, Kali Rath, and an anonymous referee for comments. The usual disclaimer applies.     

Equilibria in infinite-dimensional production economies with convexified coalitions

Summary. This paper deals with a private ownership production economy assuming that the commodity space is infinite-dimensional. It is first showed that the fuzzy core allocations, a concept that goes back to J.-P. Aubin, are in a one-to-one correspondence with certain core allocations of a continuum economy suitably defined. This result is obtained under convexity of preferences and production sets and separability of the commodity space. In the case of nonconvex preferences and production sets, the set of fuzzy coalitions can be enlarged in order to obtain that every allocation of the core accordingly defined is supported by a non zero price. The proof of the equivalence result when the positive cone of the commodity space has the empty interior, is obtained under assumptions of properness for preferences relations and production sets. Received: July 9, 1998; revised version: December 6, 1999     

Incentive compatible core and competitive equilibria in differential information economies

Summary. If the allocations of a differential information economy are defined as incentive compatible state-contingent lotteries over consumption goods, competitive equilibrium allocations exist and belong to the (ex ante incentive) core. Furthermore, any competitive equilibrium allocation can be viewed as an element of the core of the n-fold replicated economy, for every n. The converse holds under the further assumption of independent private values but not in general, as shown by a counter-example. Received: August 9, 1999; revised version: September 12, 1999     

The option of joint purchase in vertically differentiated markets

Summary. Within the framework proposed by Mussa and Rosen (1978) for modelling quality differentiation, consumers are assumed to make mutually exclusive purchases. A unique pure strategy equilibrium exists in this case. In this note, we allow consumers to buy simultaneously different variants of the differentiated good. We call this the “joint purchase option”. The paper proposes a detailed analysis of price competition when this option is opened: first, we show that either uniqueness, or multiplicity, or absence of price equilibrium arise, depending on the utility derived from joint purchase relative to exclusive purchase. Second, we characterize these equilibria, whenever they exist. Received: July 25, 2001; revised version: October 21, 2002 RID="*" ID="*" The second author gratefully acknowledges the financial support from Interuniversity Attraction Pole Program- Belgian State- Federal Office for Scientific, Technical and Cultural Affairs under contract PAI 5/26. Correspondence to: X.Y. Wauthy     

Indeterminacy of intertemporal equilibria under imperfect competition

Summary. It is known that overlapping generations models with imperfectly competitive firms may exhibit a continuum of stationary equilibria. The reason of this indeterminacy is that different price expectation functions of consumers lead to different objective demand functions against which firms maximize. All these expectation functions fulfill perfect foresight in the equilibrium, but they can be arbitrary off the equilibrium. In this paper it is shown that it is not this arbitrariness which is responsible for the indeterminacy, but that the continuum of stationary equilibria emerges even if expectation functions are rational. Received: March 25, 1999; revised version: February 16, 2000     

Edgeworth's conjecture with infinitely many commodities: commodity differentiation

Summary. Convergence of the cores of finite economies to the set of Walrasian allocations as the number of agents grows has long been taken as one of the basic tests of perfect competition. The present paper examines this test in the most natural model of commodity differentiation: the commodity space is the space of nonnegative measures, endowed with the topology of weak convergence. In Anderson and Zame [12], we gave counterexamples to core convergence in L ¹, a space in which core convergence holds for replica economies and core equivalence holds for continuum economies; in addition, we gave a core convergence theorem under the assumption that traders' utility functions exhibit uniformly vanishing marginal utility at infinity. In this paper, we provide two core convergence results for the commodity differentiation model. A key technical virtue of this space is that relatively large sets (in particular, closed norm-bounded sets) are compact. This permits us to invoke a version of the Shapley-Folkman Theorem for compact subsets of an infinite-dimensional space. We show that, for sufficiently large economies in which endowments come from a norm bounded set, preferences satisfy an equidesirability condition, and either (i) preferences exhibit uniformly bounded marginal rates of substitution or (ii) endowments come from an order-bounded set, core allocations can be approximately decentralized by prices. Received: July 29, 1996; revised version: January 14, 1997     

Preemptive entry in differentiated product markets

Summary. Models of spatial competition are typically static, and exhibit multiple free-entry equilibria. Incumbent firms can earn rents in equilibrium because any potential entrant expects a significantly lower market share (since it must fit into a niche between incumbent firms) along with fiercer price competition. Previous research has usually concentrated on the zero-profit equilibrium, at which there is normally excessive entry, and so an entry tax would improve the allocation of resources. At the other extreme, the equilibrium with the greatest rent per firm normally entails insufficient entry, so an entry subsidy should be prescribed. A model of sequential firm entry (with an exogenous order of moves) resolves the multiplicity problem but raises a new difficulty: firms that enter earlier can expect higher spatial rents, and so firms prefer to be earlier in the entry order. This tension disappears when firms can compete for entry positions. We therefore suppose that firms can commit capital early to the market in order to lay claim to a particular location. This temporal competition dissipates spatial rents in equilibrium and justifies the sequential move structure. However, the policy implications are quite different once time is introduced. An atemporal analysis of the sequential entry process would prescribe an entry subsidy, but once proper account is taken of the entry dynamics, a tax may be preferable. Received: April 26, 1999; revised version: September 22, 1999     

Core and Walrasian equilibria when agents' characteristics are extremely dispersed

Summary. It is shown that core-Walras equivalence fails whenever the commodity space is a non-separable Banach space. The interpretation is that a large number of agents guarantees core-Walras equivalence only if there is actually a large number of agents relative to the size of the commodity space. Otherwise a large number of agents means that agents' characteristics may be extremely dispersed, so that the standard theory of perfect competition fails. Supplementing the core-Walras non-equivalence result, it is shown that in the framework of economies with weakly compact consumption sets – as developed by Khan and Yannelis (1991) – the core is always non-empty, even if consumption sets are non-separable. December 12, 2001; revised version: December 6, 2002 RID="*" ID="*" Thanks to E. Dierker, M. Nermuth, R. Tourky, and N. C. Yannelis for helpful discussions and suggestions, and thanks to a referee for comments which helped to improve the final version.     

Arrow-Debreu equilibria with asymptotically heterogeneous expectations exist

Summary. In infinite horizon economies only local equivalence of beliefs is needed to ensure the existence of an Arrow–Debreu equilibrium. In fact, agents can even disagree completely in the long run in the sense that asymptotically, their beliefs are singular. Received: November 3, 2000; revised version: February 13, 2002     

Existence and optimality of oligopoly equilibria in linear exchange economies

Summary. We consider a linear exchange economy and its successive replicas. We study the notion of Cournot-Walras equilibrium in which the consumers use the quantities of commodities put on the market as strategic variables. We prove that, generically, if the number of replications is large enough but finite, the competitive behaviour is an oligopoly equilibrium. Then, under a mild condition, which may be interpreted in terms of market regulation and/or market activity, we show that any sequence of oligopoly equilibria of successive replica economies converges to the Walrasian outcome and furthermore that every oligopoly equilibrium of large, but finite, replica is Pareto optimal. Consequently, under the same assumptions on the fundamentals of the economy, one has an asymptotic result on the convergence of oligopoly equilibria to the Walras equilibrium together with a generic existence result for the Cournot-Walras. Received: June 20, 2002; revised version: November 20, 2002 RID="*" ID="*" Part of this paper was written while the second author was visiting the Universidad de Vigo. The support of the department of mathematics is gratefully acknowledged. Correspondence to: J.M. Bonnisseau     

Markov perfect equilibria in industries with complementarities

Summary.   This paper considers the existence and computation of Markov perfect equilibria in games with a “monotone” structure. Specifically, it provides a constructive proof of the existence of Markov perfect equilibria for a class of games in which a) there is a continuum of players, b) each player has the same per period payoff function and c) these per period payoff functions are supermodular in the player's current and past action and have increasing differences in the player's current action and the entire distribution of actions chosen by other players. The Markov perfect equilibria that are analyzed are symmetric, not in the sense that each player adopts the same action in any period, but rather in the sense that each player uses the same policy function. Since agents are typically distributed across many states they will typically take different actions. The formal environment considered has particular application to models of industries (or economies) in which firms face costs of price adjustment. It is in this context that the results are developed. Received: November 9, 1999; revised version: February 10, 2000     

Valuation in infinite-horizon sequential markets with portfolio constraints

Summary. We develop a theory of valuation of assets in sequential markets over an infinite horizon and discuss implications of this theory for equilibrium under various portfolio constraints. We characterize a class of constraints under which sublinear valuation and a modified present value rule hold on the set of non-negative payoff streams in the absence of feasible arbitrage. We provide an example in which valuation is non-linear and the standard present value rule fails in incomplete markets. We show that linearity and countable additivity of valuation hold when markets are complete. We present a transversality constraint under which valuation is linear and countably additive on the set of all payoff streams regardless of whether markets are complete or incomplete. Received: March 9, 2000; revised version: February 13, 2001     

Effect of credible quality investment with Bertrand and Cournot competition

We show how credible revelation and ability to commit to quality choice effect equilibrium qualities and welfare when product market is either Bertrand or Cournot competition. We show that results depend on the type of competition but not generally on the cost of quality function. We show that with Bertrand competition, the equilibrium qualities are lower with credible commitment. Competition is moderated and producer surplus is higher and consumer surplus lower. With Cournot competition, higher quality will be better but lower quality will be worse with credible commitment. Consumer surplus is always greater with credible commitment and if cost does not increase too quickly with quality, producer surplus will also increase. Thus credible commitment is a collusive device with Bertrand competition but it can improve social welfare with Cournot competition. Received: February 8, 2000; revised version: February 14, 2002 RID="*" ID="*" The idea of this paper originated in the weekly workshops of Mordecai Kurz at Stanford. I am forever in debted to Mordecai and fellow students – Luis Cabral, Peter DeMarzo, John Hillas, Michihiro Kandori, Steve Langois, Patrick McAllister, Steve Sharpe, Peter Streufert, Steve Turnbull and Gyu-Ho Wang – for their criticism and encouragement. I also benefited from comments from Yi-Heng Chen, Jin-Li Hu, Kala Krishna, Jinji Naoto, Thomas J. Prusa, and Shyh-Fang Ueng at various later stages of this work. Last but not least, I am grateful for the detailed comments of the referee.     

Recursive structure and equilibria in games with private monitoring

Summary. In each stage of a repeated game with private monitoring, the players receive payoffs and privately observe signals which depend on the players' actions and the state of world. I show that, contrary to a widely held belief, such games admit a recursive structure. More precisely, I construct a representation of the original sequential problem as a sequence of static games with incomplete information. This establishes the ground for a characterization of strategies and, hence, of behavior in interactive-decision settings where private information is present. Finally, the representation is used to give a recursive characterization of the equilibrium payoff set, by means of a multi-player generalization of dynamic programming. Received: February 11, 2002; revised version: July 22, 2002 RID="*" ID="*" I am very grateful to In-Koo Cho, Larry Epstein, Denis Gromb, Stephen Morris, Paolo Siconolfi, Lones Smith and Max Stinchcombe for several insights and suggestions. A referee's comments helped improving the exposition. Finally, I wish to thank the participants to the seminars at MEDS, NYU, Columbia University, Caltech, UCLA, University of Rochester, University of Texas-Austin, Northwestern Summer Microeconomics Conference 98, Summer in Tel Aviv 98, and NASM98.     

The set of sunspot equilibria in economies with incomplete financial markets: variable asset prices

Summary. I consider the set of equilibria of two-period economies with S extrinsic states of nature in the second period and I assets with linearly independent nominal payoffs. Asset prices are variable. If the number of agents is greater than (S-I), the payoff matrix is in general position and S 2I, the set of equilibrium allocations generically (in utility function space) contains a smooth manifold of dimension (S-1). Moreover, the map from states o f nature to equilibrium allocations (restricted to this manifold) is one-to-one at each equilibrium. Received: February 23, 1998; revised version: June 1, 2000     

On the efficiency of indirect taxes in differentiated oligopolies with asymmetric costs

This paper shows that unit taxation can be welfare superior to ad valorem taxation in asymmetric and differentiated oligopolies if the goods are sufficiently differentiated, the cost variance is sufficiently large and the ad valorem tax rate is sufficiently high. Moreover, this result holds under either Cournot competition or Bertrand competition.      

Existence of equilibria for economies with externalities and a measure space of consumers

Summary. This paper considers an exchange economy with a measure space of agents and consumption externalities, which take into account two possible external effects on consumers preferences: dependence upon prices and dependence upon other agents consumption. We first consider a model with a general externality mapping and we then treat the particular case of reference coalition externalities, in which the preferences of each agent a are influenced by prices and by the global or the mean consumption of the agents in finitely many (exogenously given) reference coalitions associated with agent a. Our paper provides existence results of equilibria in both models when consumers have transitive preferences. It extends in exchange economies the standard results by Aumann [2], Schmeidler [16], Hildenbrand [12], and previous results by Greenberg et al. [11] for price dependent preferences, Schmeidler [17] for fixed reference coalitions and Noguchi [15] for a more particular concept of reference coalitions. We also mention related results obtained independently by Balder [4].Received: 25 May 2004, Revised: 19 October 2004, JEL Classification Numbers: D62, D51, H23. Correspondence to: Bernard CornetThis paper has benefited from comments and valuable discussions with Erik Balder, Stefan Balint, Jean-Marc Bonnisseau, Alessandro Citanna, Gael Giraud, Filipe Martins-da-Rocha, Jean-Philippe Médecin, Jean-François Mertens, Nicholas Yannelis and an anonymous referee.