Equilibrium in a Market Game

This paper studies the equilibrium of an exchange economy with the same number of agents and commodities. It is shown that under rather mild conditions on demand the market have a price equilibrium—independently of the divisibility of the commodities. The result extends Gale’s equilibrium theorem for indivisible commodities to the case where some commodities are perfectly divisible. The proof is based on a topological lemma contained in previous paper of one of the authors.     

Approximate equilibria in markets with indivisible commodities

This paper presents results on the existence of approximate equilibria in large but finite exchange economies in which all the traders have non-convex consumption sets. As such, the paper gives a natural extension, to the finite economy, of Mas-Colell's results on existence with indivisible commodities and a continuum of agents. It also presents an approximate equilibrium theorem without any assumption of compactness or bounded non-convexity of preferences.     

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 complete removal of individual uncertainty: multiple optimal choices and random exchange economies

Summary. The aim of this paper is to develop some measure-theoretic methods for the study of large economic systems with individual-specific randomness and multiple optimal actions. In particular, for a suitably formulated continuum of correspondences, an exact version of the law of large numbers in distribution is characterized in terms of almost independence, which leads to several other versions of the law of large numbers in terms of integration of correspondences. Widespread correlation due to multiple optimal actions is also shown to be removable via a redistribution. These results allow the complete removal of individual risks or uncertainty in economic models where non-unique best choices are inevitable. Applications are illustrated through establishing stochastic consistency in general equilibrium models with idiosyncratic shocks in endowments and preferences. In particular, the existence of “global” solutions preserving microscopic independence structure is shown in terms of competitive equilibria for the cases of divisible and indivisible goods as well as in terms of core for a case with indivisible goods where a competitive equilibrium may not exist. An important feature of the idealized equilibrium models considered here is that standard results on measure-theoretic economies are now directly applicable to the case of random economies. Some asymptotic interpretation of the results are also discussed. It is also pointed out that the usual unit interval [0,1] can be used as an index set in our setting, provided that it is endowed together with some sample space a suitable larger measure structure. Received: September 14, 1998; revised version: January 6, 1999     

Bargaining, coalitions and competition

We study a Gale-like matching model in a large exchange economy, in which trade takes place through non-cooperative bargaining in coalitions of finite size. Under essentially the same conditions of core equivalence, we show that the strategic equilibrium outcomes of our model coincide with the Walrasian allocations of the economy. Our method of proof makes use of the theory of the core. With respect to previous work, our positive implementation result applies to a substantially larger class of economies: the model relaxes differentiability and convexity of preferences, and also admits an arbitrary number of divisible and indivisible goods.     

Cournot-Nash and Lindahl equilibria in pure public “bad” economies

Summary. Using a general equilibrium framework, this paper analyzes the equilibrium provision of a pure public bad commodity (for example pollution). Considering a finite economy with one desired private good and one pure public “bad” we explicitly introduce the concept of Lindahl equilibrium and the Lindahl prices into a pure public bad economy. Then, the Lindahl provision is analyzed and compared with the Cournot-Nash provision. The main results for economies with heterogeneous agents state that the asymptotic Lindahl allocation of the pure public bad is the null allocation. In contrast, the asymptotic Cournot-Nash provision of the public bad might approach infinity. Other results were obtained in concert with the broad analysis of the large finite economies with pure public bad commodities. Received: July 26, 2001; revised version: March 12, 2002 RID="*" ID="*" We are indebt to Nicholas Yannelis and anonymous referee for their valuable comments and suggestions. Correspondence to: B. Shitovitz     

Exchange economies with asymmetric information: competitive equilibrium and core

Summary. A replica theorem is shown to hold for exchange economies with asymmetric information. In a replicated exchange economy with asymmetric information the set of all core elements with equal treatment is nonempty, but it is in general only a subset of the core. Nevertheless, the replica theorem and the presence of at least one core element with equal treatment suffice to show existence of a competitive quasi-equilibrium. Conditions on the initial endowments and the communication system are given to ensure that every competitive quasi-equilibrium is a competitive equilibrium.Received: 24 February 2003, Revised: 3 July 2003JEL Classification Numbers: C70, D50, D82.I thank an anonymous referee whose comments led to an improvement of the paper.     

Existence of an equilibrium with arbitrary tax schemes for financing local public goods

The existence of a public competitive equilibrium for any arbitrary (nondistortive) tax schemes (in particular, proportional taxes) for economies with local public goods, is proved. Allowing preferences to be both nontransitive and noncomplete enables an explicit introduction of the government as an (additional) agent in the economy. Moreover, we allow for “spillovers” of the public goods among localities, and for the production sets to depend on the amount of public goods produced in the economy. The only restriction on the tax system is that every individual is able to afford it and that the government's budget never runs a surplus. Since every equilibrium allocation is a Pareto optimum, every tax scheme is optimal (in the sense that its equilibrium allocation is a Pareto optimum).     

A Model of Commodity Differentiation with Indivisibilities and Production

This paper presents an existence theorem in a general equilibrium model of a production economy with commodity differentiation and indivisibilities. The model is motivated by the existence of markets with indivisible commodities, such as the markets for automobiles and computers. As is standard in the literature, the space of commodity characteristics is described by a compact metric space and a commodity vector is described by an integer-valued Borel measure on the space of commodity characteristics. An atomless measure space of producers and consumers is assumed to overcome the problem of non-convexity of the production and consumption sets induced by indivisibilities. This paper is based on a chapter from James Marton’s dissertation. We would like to thank Marcus Berliant, Wilhelm Neuefeind, and an anonymous referee for their valuable comments and suggestions.     

WELFARE ECONOMICS AND ECONOMIES OF SCALE*

For timeless economies where technology provides economies of scale, it is shown that, under a variety of circumstances, an allocation is optimal if and only if it is a free-entry monopsonistic equilibrium, in which producers take account of labour costs rising with the level of employment. The results refine Fujita's one-commodity theory, and extend it to many commodities and a heterogeneous population. Production takes place in towns. When transport costs for commodities are zero, one good is produced in each town. Land use and worker travel costs are allowed. For the equivalence theorem to hold, all production and property in a town should have a single owner. Models with transport costs are also discussed. For them, it is shown that in general there is no equilibrium.     

EFFICIENT MIXED CLUBS: NONLINEAR-PRICING EQUILIBRIA WITH ENTREPRENEURIAL MANAGERS

Scotchmer and Wooders show that efficient clubs are homogeneous when consumers are divisible in Berglas's anonymous crowding model. However, if consumers are not divisible or if clubs have multiple facilities with economies of scope, mixed clubs are efficient. In such a model, we consider clubs with multiple membership policies for different types of consumers, and show the existence and efficiency of equilibrium with nonlinear policies. We employ entrepreneurial equilibrium, all equilibrium concept with profit-seeking entrepreneurs. Our theorem can be regarded as showing the existence of a core allocation that satisfies envy-free property in the absence of non-anonymous crowding effects.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

A competitive equilibrium for a warm-glow economy

The warm-glow model (Andreoni in J Political Econ 97:1447–1458, 1989; Econ J 100:464–477, 1990) of public goods provision has received widespread interest, yet surprisingly most attention has focused on the voluntary contribution equilibrium of the model, and only very little attention has been devoted to the competitive equilibrium. In this paper, we introduce the concept of competitive equilibrium for a warm-glow economy (henceforth, warm-glow equilibrium) and establish both existence and welfare properties. The warm-glow equilibrium concept may prove to be very useful to the normative and positive theory of public goods provision. First, it is a price-based mechanism achieving efficient outcomes. Second, not only could the warm-glow equilibria outcomes serve as a point of reference to measure free-riding and welfare loss but also, as suggested by Bernheim and Rangel (Behavioral Economics and Its Applications, 2007), in large economies they may be approximated by Walrasian equilibria outcomes.     

Characterization and incentive compatibility of Walrasian expectations equilibrium in infinite dimensional commodity spaces

Summary. We consider a differential information economy with infinitely many commodities and analyze the veto power of the grand coalition with respect the ability of blocking non-Walrasian expectations equilibrium allocations. We provide two different Walrasian expectations equilibrium equivalence results. First by perturbing the initial endowments in a precise direction we show that an allocation is a Walrasian expectations equilibrium if and only if it is not privately dominated by the grand coalition. The second characterization deals with the fuzzy veto in the sense of Aubin but within a differential information setting. This second equivalence result provides a different characterization for the Walrasian expectations equilibrium and shows that the grand coalition privately blocks in the sense of Aubin any non Walrasian expectations equilibrium allocation with endowment participation rate arbitrarily close to the total initial endowment participation for every individual. Finally, we show that any no free disposal Walrasian expectations equilibria is coalitional Bayesian incentive compatible. Since the deterministic Arrow-Debreu-McKenzie model is a special case of the differential information economy model, one derives new characterizations of the Walrasian equilibria in economies with infinitely many commodities.Received: 29 October 2003, Revised: 24 February 2004, JEL Classification Numbers: D51, D82, D11. Correspondence to: Emma Moreno-GarcíaThe authors are grateful to an anonymous referee for his/her careful reading and helpful comments and suggestions.C. Hervés and E. Moreno acknowledge support by Research Grant BEC2000-1388-C04-01 (Ministerio de Ciencia y Tecnología and FEDER); and support by the Research Grant SA091/02 (Junta de Castilla y León).     

The core of an economy with satiation

Summary. It has long been known that in economies with satiation, the set of competitive equilibria does not coincide with the limiting core of an economy. In dividend equilibria, introduced independently by a number of authors, the budget excess is allowed to be divided among consumers as dividends and equilibrium existence is restored. In this paper a new notion of blocking which leads to core - dividend equilibrium equivalence is introduced. Specifically, it is shown that an allocation is a dividend equilibrium if and only if a corresponding equal treatment allocation of a large economy is in the core.Received: 21 June 2003, Revised: 3 December 2003, JEL Classification Numbers: D50.Alexander Konovalov: The author would like to thank M. Florig, P. J. J. Herings, V. Marakulin, D. Talman, V. Vasilev, and anonymous referees for helpful comments and suggestions.     

On the second optimality theorem of welfare economics

To what extent does the second optimality theorem of welfare economics (every Pareto optimal allocation can be repesented as a Walras equilibrium allocation) remain valid when preferences are allowed to be locally satiated? It is always valid for an exchange economy, and is valid for a production economy if there is a consumer who is not locally satiated, but not in general for a production economy where all consumers are locally satiated. A generalized equilibrium is defined, which includes the Walras equilibrium as a special case. Every Pareto optimum can be represented as a generalized equilibrium allocation. Furthermore, every Pareto optimal utility distribution can be realized by a Walras equilibrium allocation.     

Search cost,trading strategies and optimal market structure

The theory of Walrasian equilibrium yields a set of prices at which the aggregate competitive demand for each commodity equals its aggregate competitive supply. However, even at equilibrium prices the theory of competitive equilibrium does not explicitly offer explanation regarding the manner in which trades are actually executed. This paper considers a model where trade takes place in a decentralized fashion and examines in a dynamic game-theoretic framework, the role of social institution of money and markets in facilitating exchange. The steady state Nash equilibrium derived in the paper demonstrates how, depending on the level of transaction costs associated with a market setup (synonymously, trading posts to exchange possible pairs of goods) appropriate monetary trade emerges, which like a hub and spoke network (Starr and Stinchcombe, 1999) makes some markets non-functioning and in equilibrium only the markets having trade through the medium of exchange continue to exist. However, despite the obvious advantages of a market setup in reducing search costs, pure random search for a complementary trading partner (as considered by Ostroy and Starr, 1974; Kiyotaki and Wright , 1989; and others) prevails in many economies, especially, in many developing economies. This paper models this feature of developing economies by introducing differences in transaction costs across agents and shows why sustainable equilibria might exist exhibiting random search for certain commodities even in the presence of established markets.     

On Nash implementation of Walrasian equilibria in a market for pure indivisibles without money

This paper deals with the Walrasian property of Nash and strong equilibria of a specific strategic market game which refers to a pure exchange economy involving purely indivisible commodities and no money. The game is of sealed-bid auction type and it is shown that any Nash equilibrium at which no agent is in status quo is a strong equilibrium and implements a Walrasian equilibrium. Moreover, it appears that two modifications of the game's rules ensure that any strong equilibrium outcome is Walrasian. These results are identical to those obtained by Svensson for markets involving purely indivisible goods and money.     

Correlated equilibrium and sunspot equilibrium

Summary We show by an example that the sunspot equilibria of a competitive economy are not equivalent to the correlated equilibria if sunspots generate transfers between (extrinsic) states of nature (through a contingent commodities market). Nevertheless, we prove that the sunspot equilibrium allocations of a standard overlapping generations economy coincide with the (strategic form) correlated equilibrium allocations of a natural market game mimicking the economy.     

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

In this paper, we provide a set of sufficient conditions under which recursive competitive equilibrium exists and is unique for a large class of distorted dynamic equilibrium models with capital and elastic labor supply. We develop a lattice based approach to the problem. The class of economies for which we are able to obtain our existence and uniqueness result is considerably larger than those considered in previous work. We conclude by applying the new results to some important examples of monetary economies often used in applied work. Journal of Economic Literature Classification Numbers: C62, D51, D90, E10.