Switching from complete to incomplete information

We construct an elementary mechanism [Dutta, B., Sen, A., Vohra, R., 1995. Nash implementation through elementary mechanisms in economic environments. Review of Economic Design 1, 173–203] that Nash implements the constrained Walrasian correspondence. We extend it to incomplete and non-exclusive information economies by enlarging the message space of agents. In addition, measurability restrictions on allocations with respect to prices proper to constrained rational expectations equilibria are imposed in the outcome function. We show that by imposing such restrictions, the mechanism Bayesian implements the constrained rational expectations equilibrium correspondence. This result shows game-theoretic connections between these two market equilibrium concepts. However, these connections are obtained at the price of strong restrictions on the behavior of agents.     

Unbounded exchange economies with satiation: How far can we go?

We unify and generalize the existence results in Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55 (6), 1403–1418], Dana et al. [Dana, R.-A., Le Van, C., Magnien, F., 1999. On the different notions of arbitrage and existence of equilibrium. Journal of Economic Theory 87 (1), 169–193], Allouch et al. [Allouch, N., Le Van, C., Page Jr., F.H., 2006. Arbitrage and equilibrium in unbounded exchange economies with satiation. Journal of Mathematical Economics 42 (6), 661–674], Allouch and Le Van [Allouch, N., Le Van, C., 2008. Erratum to “Walras and dividends equilibrium with possibly satiated consumers”. Journal of Mathematical Economics 45 (3–4), 320–328]. We also show that, in terms of weakening the set of assumptions, we cannot go too far.     

Impossibility of Nash implementation in two-person economies

In this note, we prove two impossibility results of Nash implementation in two-person economies. First, we will show the non-existence of continuous and balanced mechanisms which implement the Walrasian correspondence. Second, by adding a convexity assumption of mechanisms, we present the impossibility of continuous implementation of Pareto optimal and individually rational allocations in balanced way. Received: 16 April 1996 / Accepted: 25 April 1997     

Equilibrium theory with a measure space of possibly satiated consumers

Walras equilibria may not exist when consumers’ preferences are possibly satiated. To overcome this difficulty, several extended notions of equilibria have been proposed and all reduce to Walras equilibria under nonsatiation and free disposal. This includes the notions of equilibria with slack (also called dividend equilibria) as by Drèze and Müller [J. Economic Theory 23 (1980) 131], Makarov [J. Mathematical Economics 8 (1981) 87], Aumann and Drèze [Econometrica 54 (1986) 1271], Mas-Colell [Equilibrium theory with possibly satiated preferences, in: Majumdar, M. (Ed.), Proceedings of the Essays in Honour of David Gale on Equilibrium and Dynamics, Macmillan, London, pp. 201–213], monetary equilibria as by Kajii [J. Mathematical Economics 25 (1996) 75], or weak equilibria as by Polemarchakis and Siconolfi [J. Mathematical Economics 22 (1993) 85], which are all defined when there are finitely many consumers. This includes also the notion of free disposal equilibrium, when markets clear in a weak sense, allowing free disposal. Our paper considers an economy with a measure space of consumers and provides a general existence result of equilibria for the various existing notions. This result extends in particular the result by Hildenbrand [Econometrica 38 (1970) 608] on the existence of Walras equilibria.     

Some consequences of the unknottedness of the Walras correspondence

Two basic properties concerning the dynamic behavior of competitive equilibria of exchange economies with complete markets are derived essentially from the fact that the Walras correspondence has no knots.     

Nash implementation with an infinite-dimensional trade space

This paper deals with the problem of implementing the Walras correspondence via Nash equilibria, in exchange economies with infinitely many commodities and finitely many households with possibly non-ordered preferences. We explicitly construct a feasible mechanism enjoying some features, which have natural economic meanings. Under a fairly weak boundary condition, this game fully implements the Walras equilibria. If this condition is not fulfilled, our mechanism nevertheless implements the constrained Walras equilibria. Received: 11 December 2003, Accepted: 29 July 2005 JEL Classification: D41, D43, D51 We thank (without implicating) Prof. Jean-Marc Bonnisseau and Cuong Le Van for helpful comments. The views expressed in this paper reflect those of the authors and not necessarily those of Calyon.     

An axiomatization of the inner core using appropriate reduced games

I adapt a reduction process introduced by Serrano and Volij [Serrano, R., Volij, O., 1998. Axiomatization of neoclassical concepts for Economies. Journal of Mathematical Economics 30, 87–108] so that the reduced games of convex-valued games are convex-valued. I use the corresponding consistency property and its converse to axiomatize the inner core for games that are convex-valued, non-level and smooth.     

Theorems on the core of an economy with infinitely many commodities and consumers

It is known that the classical theorems of Grodal [Grodal, B., 1972. A second remark on the core of an atomless economy. Econometrica 40, 581–583] and Schmeidler [Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometrica 40, 579–580] on the veto power of small coalitions in finite dimensional, atomless economies can be extended (with some minor modifications) to include the case of countably many commodities. This paper presents a further extension of these results to include the case of uncountably many commodities. We also extend Vind’s [Vind, K., 1972. A third remark on the core of an atomless economy. Econometrica 40, 585–586] classical theorem on the veto power of big coalitions in finite dimensional, atomless economies to include the case of an arbitrary number of commodities. In another result, we show that in the coalitional economy defined by an atomless individualistic model, core–Walras equivalence holds even if the commodity space is non-separable. The above-mentioned results are also valid for a differential information economy with a finite state space. We also extend Kannai’s [Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815] theorem on the continuity of the core of a finite dimensional, large economy to include the case of an arbitrary number of commodities. All of our results are applications of a lemma, that we prove here, about the set of aggregate alternatives available to a coalition. Throughout the paper, the commodity space is assumed to be an ordered Banach space which has an interior point in its positive cone.     

Uniform allocation and reallocation revisited

Thomson (1995a) proved that the uniform allocation rule is the only allocation rule for allocation economies with single-peaked preferences that satisfies Pareto efficiency, no-envy,one-sided population-monotonicity, and replication-invariance on a restricted domain of single-peaked preferences. We prove that this result also holds on the unrestricted domain of single-peaked preferences. Next, replacing one-sided population-monotonicity by one-sided replacement-domination yields another characterization of the uniform allocation rule, Thomson (1997a). We show how this result can be extended to the more general framework of reallocation economies with individual endowments and single-peaked preferences. Following Thomson (1995b) we present allocation and reallocation economies in a unified framework of open economies. Received: 20 February 1999 / Accepted: 15 February 2000     

INFORMATION THEORETIC APPROACHES IN ECONOMICS

Economics has seen a recent rise in interest in information theory as an alternative framework to the conventional notion of equilibrium as a fixed state, such as Walrasian market‐clearing general equilibrium. The information theoretic approach is predicated on the notion of statistical equilibrium (SE) that takes a distribution over all possible states as an equilibrium, and therefore predicts the endogenous fluctuations of the system along with its central tendency simultaneously. For this reason, SE approaches can explain the observed data without relying on arbitrary assumptions about random noise and provide useful insights for many interesting economic problems that conventional methods have not been able to satisfactorily deal with. In this paper, we review the key elements of information theory focusing on the notions and applications of entropy and SE in economics, particularly paying attention to how entropy concepts open up a new frontline of economic research.     

Manipulation of the Walrasian mechanism in production economies with unbounded short-selling

Hurwicz (1979) and Otani and Sicilian (1982, 1990) characterized the Nash equilibrium allocations of the Walrasian demand manipulation game in successively more general exchange environments. In this paper, I extend the analysis to production economies with short-selling. First, I generalize Hurwicz’s and Otani and Sicilian’s theorem that any allocation at which each agent’s consumption bundle lies above her true offer curve can be supported in Nash equilibrium. I then show that for finite economies of any size the set of such allocations is often topologically large.Received: 17 January 2003, Accepted: 4 April 2005, JEL Classification: D51, D82For comments on this and earlier versions of the paper, I wish to thank Rick Bond, Bhaskar Chakravorti, Tom Gresik, Costas Syropoulos and William Thomson. I would especially like to thank Mike Jerison for helping to overcome a difficulty with a previous version. Also, the comments of the anonymous referees are gratefully acknowledged.     

The coordinate-wise core for multiple-type housing markets is second-best incentive compatible

We consider the generalization of Shapley and Scarf’s (1974) [Shapley, L., Scarf’s, H., 1974. On cores and indivisibility. Journal of Mathematical Economics 1, 23–37.] model of trading indivisible objects (houses) to so-called multiple-type housing markets. We show that the prominent solution for these markets, the coordinate-wise core rule, is second-best incentive compatible.     

No unbounded arbitrage,weak no market arbitrage and no arbitrage price system conditions; Equivalent conditions

Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162] prove that the no unbounded arbitrage (NUBA), a special case of a condition in Page [Page, F.H., 1987. On equilibrium in Hart’s securities exchange model. Journal of Economic Theory 41, 392–404], is equivalent to the existence of a no arbitrage price system (NAPS) when no agent has non-null useless vectors. Allouch et al. [Allouch, N., Le Van, C., Page F.H., 2002. The geometry of arbitrage and the existence of competitive equilibrium. Journal of Mathematical Economics 38, 373–391] extend the NAPS introduced by Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418] and show that this condition is equivalent to the weak no market arbitrage (WNMA) of Hart [Hart, O., 1974. On the existence of an equilibrium in a securities model. Journal of Economic Theory 9, 293–311]. They mention that this result implies the one given by Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162]. In this note, we show that all these conditions are equivalent.     

Representation of transferable utility games by coalition production economies

We prove that, by the method of construction of a coalition production economy due to Sun et al. [Sun, N., Trockel, W., Yang, Z., 2008. Competitive outcomes and endogenous coalition formation in an n$n$-person game. Journal of Mathematical Economics 44, 853–860], every transferable utility (TU) game can be generated by a coalition production economy. Namely, for every TU game, we can construct a coalition production economy that generates the given game. We briefly discuss the relationship between the core of a given TU game and the set of Walrasian payoff vectors for the induced coalition production economy.     

The equilibrium set of infinite dimensional Walrasian economies and the natural projection

The natural projection plays a fundamental role to understand the behavior of the Walrasian economies. In this paper, we extend this method to analyze the behavior of infinite dimensional economies. We introduce the definition of the social equilibrium set, and we show that there exists a bijection between this set and the Walrasian equilibrium set of an infinite dimensional economy. In order to describe the main topological characteristics of both sets, we analyze the main differential characteristics of the excess utility function and then, we extend the method of the natural projection as suggested by Y. Balasko.     

Implementation of the Walrasian correspondence by market games

Abstract. In this paper we present a set of axioms guaranteeing that, in exchange economies with or without indivisible goods, the set of Nash, Strong and active Walrasian Equilibria all coincide in the framework of market games. Received: 25 December 1998 / Accepted: 8 April 2002 We are very grateful to William Thomson for many suggestions. Also we would like to thank comments from J. Alcalde, R. Bhattacharya, C. Herrero, P. Marhuenda, D. Pérez-Castrillo, A. Romero-Medina, T. Shinotsuka, T. Sj?str?m, and A. Villar, an associated editor and an anonymous referee. The first author thanks financial support from DGCYT and Direcció General de Reserca under projects PB98-0870, and SGR2000-00054. The second author wishes to acknowledge financial help from DGCYT under project PB98-0940.     

The veto mechanism in atomic differential information economies

We establish new characterizations of Walrasian expectations equilibria based on the veto mechanism in the framework of differential information economies with a complete finite measure space of agents. We show that it is enough to consider the veto power of a single coalition, consisting of the entire set of agents, to obtain the Aubin private core. Moreover, we investigate on the veto power of arbitrarily small and big coalitions, providing an extension to mixed markets of well known Schmeidler (1972) and Vind’s (1972) results in terms of Aubin private core allocations.     

Existence of equilibria with infinitely many commodities: Banach lattices reconsidered

The purpose of this paper is twofold. The first aim is to present an extension of the results on the existence of Walrasian equilibrium to the infinite dimensional setting. The result depends on two crucial assumptions. These are the compactness of the collection of feasible allocations and the non-emptiness of the interior of the production set. The proof is a direct generalization of Bewley's (1972) proof for the L_∞ case. The second purpose of this paper is to show that the recent result of Mas-Colell (1986) on the existence of equilibrium for exchange economies on Banach lattices can be obtained through an argument based on the result outlined above. That is, exchange economies on Banach lattices with ‘uniformly proper’ preferences behave as though they were production economies in which the production sets have non-empty interior.