The fuzzy core and the (\Pi, \beta)-balanced core

Summary. This note provides a new proof of the non-emptiness of the fuzzy core in a pure exchange economy with finitely many agents. The proof is based on the concept of -balanced core for games without side payments due to Bonnisseau and Iehlé (2003).Received: 8 May 2003, Revised: 8 August 2004, JEL Classification Numbers: D51, C71.Helpful comments of Jean-Jacques Herings and Hans Peters are gratefully acknowledged.     

Mistaken self-perception and equilibrium

Summary. Motivated by real-world information economics problems and by experimental findings on overconfidence, this paper introduces a general epistemic construction to model strategic interaction with incomplete information, where the players self-perception may be mistaken. This allows us to rigorously describe equilibrium play, by formulating appropriate equilibrium concepts. We show that there always exist objective equilibria, where the players correctly anticipate each others strategies without attempting to make sense of them, and that these outcomes coincide with the equilibria of an associated Bayesian game with subjective priors. In population games, these equilibria can also be always introspectively rationalized by the players, despite their possibly mistaken self-perception.Received: Received: May, 12, 2003; revised version: 3 December 2004, Revised: 3 December 2004, JEL Classification Numbers: J31, D82, D83.I thank Massimiliano Amarante, Eddie Dekel, Massimo Marinacci, Jean Francois Mertens, Giuseppe Moscarini, Paolo Siconolfi, Marciano Siniscalchi, Joel Sobel, and an anonymous referee.     

Infinite inequality systems and cardinal revelations

Summary. Many economics problems are maximization or minimization problems, and can be formalized as problems of solving linear difference systems of the form and r _k -r _l > c _kl , for r-unknowns, with given c-constants. They typically involve strict as well as weak inequalities, with infinitely many inequalities and unknowns. Since strict inequalities are not preserved under passage to the limit, infinite systems with strict inequalities are notoriously hard to solve. We introduce a unifying tool for solving them. Our main result (Theorem 1 for the countable case, Theorem [2] for the not-necessarily-countable case) introduces a uniform solvability criterion (the -Axiom), and our proof yields a method for solving those that are solvable. The axioms economic intuition extends the traditional ordinal notion of revealed preference to a cardinal notion. We give applications in producer theory, consumer theory, implementation theory, and constrained maximization theory.Received: 21 May 2003, Revised: 29 October 2004, JEL Classification Numbers: C69, D21, D11, D82. Correspondence to: Marcel K. RichterWe thank Professor Leonid Hurwicz for helpful discussions, and a referee for helpful comments.     

Stationary Markov equilibria for overlapping generations

Summary.  At a stationary Markov equilibrium of a Markovian economy of overlapping generations, prices at a date-event are determined by the realization of the shock, the distribution of wealth and, with production, the stock of capital. Stationary Markov equilibria may not exist; this is the case with intra-generational heterogeneity and multiple commodities or long life spans. Generalized Markov equilibria exist if prices are allowed to vary also with the realization of the shock, prices and the allocation of consumption and production at the predecessor date-event. (Stationary) Markov -equilibria always exist; as allocations and prices converge to equilibrium prices and allocations that, however, need not be stationary.Received: 2 March 2004, Revised: 2 April 2004, JEL Classification Numbers:   D50, D52, D60, D80, D90.Correspondence to: Felix KublerWe thank participants in seminars in Athens and Lund, at Penn, at IMPA and at Stanford, the 2002 CEME (NBER) General Equilibrium Conference and the 2002 SED meetings, and especially Martin Hellwig, George Mailath and an anonymous referee for very helpful comments.     

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.     

On behavioral heterogeneity

Summary. An index of behavioral heterogeneity for every finite population of households is defined. It is shown that the higher the index of behavioral heterogeneity the less sensitive depends the aggregate consumption expenditure ratio upon prices. As a consequence, a high index implies a tendency for the Jacobian of aggregate demand to have a dominant negative diagonal.Received: 12 May 2003, Revised: 13 February 2004, JEL Classification Numbers: D 11, E 10.Financial support by Deutsche Forschungsgemeinschaft, grants HI 232/2-3 and KN 567/1-1, is gratefully acknowledged.     

On the continuity of equilibrium and core correspondences in economies with differential information

Summary. We study upper semi-continuity of the private and coarse core and the Walrasian expectations equilibrium correspondences for economies with differential information, with Boylan (1971) topology on agents information fields.Received: 16 January 2004, Revised: 28 October 2004, JEL Classification Numbers: D50, D82, C70. Correspondence to: Ezra EinyWe wish to thank Carlos Herves, Nicholas Yannelis, and an anonymous referee for their helpful comments.     

Pooling and endogenous market incompleteness

Summary. We study a financial market economy with a continuum of borrowers and pooling of borrowers promises. Under these conditions and in the absence of designing costs, utility-maximizing decisions of price-taking borrowers may lead to financial market incompleteness. Parametrizing equilibria through the borrowers no-arbitrage beliefs, we link expectations to the financial market structure. Markets are complete if and only if borrowers beliefs are homogeneous. Price-taking behavior causes a coordination problem which in turn yields indeterminacy and inefficiency of equilibrium allocations.Received: 29 May 2003, Revised: 13 February 2004, JEL Classification Numbers: D50, D52.Correspondence to: Alessandro CitannaWe would like to thank David Cass, John Geanakoplos, Thorsten Hens, Atsushi Kajii, and an anonymous referee for their comments. The first author also thanks CERMSEM (Paris I) and Columbia University Graduate School of Business for the hospitality. A first version of this paper has appeared as GSIA Working Paper #1997-E137, Carnegie Mellon University, which itself revised Citanna and Villanacci (1995).     

Continuity of the equilibrium price density and its uses in peak-load pricing

Summary. With as the commodity space, the equilibrium price density is shown to be a continuous function of the commodity characteristics. The result is based on symmetry ideas from the Hardy-Littlewood-Pólya theory of rearrangements. It includes, but is not limited to, the case of symmetric (rearrangement-invariant) production costs and additively separable consumer utility. Examples arise in continuous-time utility pricing, e.g., electricity pricing. In this context, a continuously varying price has two uses. First, it precludes demand jumps that would arise from discontinuous switches from one price rate to another. Second, in the problems of operating and valuing hydroelectric and pumped-storage plants (studied elsewhere), price continuity guarantees that their capacities (viz., the reservoir and the converter), the energy stocks, and in the case of hydro also the river flows, have well-defined marginal values.Received: 9 May 2001, Revised: 8 July 2004, JEL Classification Numbers: C62, D51, D58, L94. Correspondence to: Anthony HorsleyPart of this work (CentER DP 9014) was completed at the Center for Economic Research, Tilburg University, whose financial support is gratefully acknowledged. The extension to storage was supported by ESRC grant R000232822. We also thank the referee and the editor for their comments.     

Asymmetric contests with general technologies

Summary. We investigate the pure-strategy Nash equilibria of asymmetric, winner-take-all, imperfectly discriminating contests, focussing on existence, uniqueness and rent dissipation. When the contest success function is determined by a production function with decreasing returns for each contestant, there is a unique pure-strategy equilibrium. If marginal product is also bounded, limiting total expenditure is equal to the value of the prize in large contests even if contestants differ. Partial dissipation occurs only when infinite marginal products are permitted. Our analysis relies heavily on the use of share functions and we discuss their theory and application.Received: 28 May 2003, Revised: 26 April 2004, JEL Classification Numbers: C72, D72. Correspondence to: Richard CornesMuch of the research in this paper was undertaken while the first author was a Visiting Scholar at the Centre for Economic Studies, University of Munich. The support of the Centre is gratefully acknowledged.     

On the characterization of excess demand functions

Summary. In this paper, we give the necessary and sufficient conditions that characterize the individual excess demand function when it depends smoothly on prices and endowments. A given function is an excess demand function if and only if it satisfies, in addition to Walras law and zero homogeneity in prices, a set of first order partial differential equations, its substitution matrix is symmetric and negative semidefinite. Moreover, we show that these conditions are equivalent to the symmetry and negative semidefiniteness of Slutsky matrix, Walras law and zero homogeneity of Marshallian demand functions.Received: 25 November 2002, Revised: 11 March 2004, JEL Classification Numbers: D11.Marwan Aloqeili: I would like to thank an anonymous referee for helpful comments.     

The Zilcha criterion for dynamic inefficiency

Summary. This paper reexamines the condition (1 + n), which Zilcha (1991) presents as a necessary and sufficient condition for dynamic inefficiency of stationary allocations in overlapping generation models with stochastic production. We show that this condition is necessary but not sufficient for a stationary allocation to be dynamically inefficient by Zilchas definition. We also show that there is a narrow but widely studied class of specifications in which the Zilcha test is both necessary and sufficient for dynamic inefficiency of stationary competitive equilibrium allocations. Outside this class, however, counterexamples can be constructed relatively easily.Received: 30 September 2002, Revised: 13 August 2004, JEL Classification Numbers: D51, D90, E13, E22. Correspondence to: Steven RussellWe thank Jon Burke, Subir Chakrabarti, Itzhak Zilcha and an anonymous referee for helpful conversations and/or comments.     

Identification of consumers’ preferences when their choices are unobservable

Summary. We provide conditions under which the heterogenous, deterministic preferences of consumers in a pure exchange economy can be identified from the equilibrium manifold of the economy. We extend those conditions to consider exchange economies, with two commodities, where consumers preferences are random. For the latter, we provide conditions under which consumers heterogenous random preferences can be identified from the joint distribution of equilibrium prices and endowments. The results can be applied to infer consumers preferences when their demands are unobservable.Received: 8 May 2003, Revised: 14 September 2004, JEL Classification Numbers: D12, D51.I am very grateful to an anonymous referee, Donald Brown, and Daniel McFadden for their detailed comments and insightful suggestions. Section 2 of this paper is joint work with Donald J. Brown; it is included here for publication with his permission. Those results were presented at the 1990 Workshop on Mathematical Economics at the University of Bonn, the 1992 SITE Workshop on Empirical Implications of General Equilibrium Models at Stanford University, and, more recently, at the June 2000 Conference in Honor of Rolf Mantel, in Buenos Aires, Argentina. The comments of the participants at those conferences and workshops are much appreciated. The research presented in this paper was supported by NSF Grants SES-8900291, SBR-9410182, and SES-0241858. This paper is dedicated to Marcel K. Richter, who has inspired much of my research.     

The equilibrium manifold keeps the memory of individual demand functions

Summary. It is shown that the property that the equilibrium manifold keeps the memory of the individual demand functions holds true if every individual demand function satisfies the following three properties: 1) It is a function of commodity prices and of consumers income; 2) Consumption belongs to the nonnegative orthant of the commodity space; 3) Walras law. Neither differentiability nor continuity are necessary. In addition, the demand functions do not have to be utility maximizing subject to budget constraints.Received: 2 September 2003, Revised: 26 January 2004, JEL Classification Numbers: D50, D51.A preliminary version of this paper was released in October 1999 under the title Deriving individual demand functions from the equilibrium manifold. I wish to thank an anonymous referee for thoughtful comments.     

Adaptive contracting: the trial-and-error approach to outsourcing

Summary. Adaptive contracting occurs when a principal experiments with the delegation of authority through leaving contracts incomplete. We highlight two potential benefits of adaptive contracting: First, the delegation of authority can be advantageous even if the agent acts opportunistically, since expected private benefits will be shared between the parties through price negotiation. Second, the principal extracts information from experimenting with delegation of authority and we identify a positive option value embodied in the principals ability to extend or withdraw the delegated authority in future contracting periods.Received: 14 April 2003, Revised: 26 January 2004, JEL Classification Numbers: D72, L33, L97.We thank John Sørensen and Henrik Severin Hansen for introducing us to the contractual and economical issues in local bus outsourcing in Denmark, Oliver Hart and Antonio Rangel for early discussions on adaptive contracting and Christian Aastrup for research assistance. We are grateful to Danish Transportation Research Institute (www.dtf.dk) for comments and financial support for this project.     

Satisficing behavior, Brouwer’s Fixed Point Theorem and Nash Equilibrium

Summary. We show that Nash Equilibrium points can be obtained by using response maps or reply functions that simply use better responses rather than best responses. We demonstrate the existence of a Nash Equilibrium as the fixed point of a better response map and since the better response map is continuous the fixed point can be established by simply using Brouwers fixed point theorem. The proof applies to games with a finite number of strategies as well as to games with a continuum of strategies. In case the games have a continuum of strategies the payoff functions have to be continuous on the action sets and quasi concave on the players action set.Received: 22 September 2003, Revised: 31 March 2004, JEL Classification Numbers: C72, D00, D40. Correspondence to: Robert A. BeckerWe have benefited from comments on an earlier draft made by participants at Indiana Universitys Microeconomics workshop (October 2002) and the Midwest Economic Theory Conference held at the University of Pittsburgh (May 2003). We also thank Roy Gardner for comments on earlier versions. We thank the Associate Editor, Mark Machina, for his detailed comments and suggestions. This project began when Subir Chakrabarti was a visitor in the Department of Economics, Indiana University, Bloomington in the Spring of 2002. He thanks that department for its support.     

Voting in assemblies of shareholders and incomplete markets

Summary. An economy with two dates is considered, one state at the first date and a finite number of states at the last date. Shareholders determine production plans by voting - one share, one vote - and at -majority stable stock market equilibria, alternative production plans are supported by at most percent of the shareholders. It is shown that a -majority stable stock market equilibrium exists if where S is the number of states at the last date and J is the number of firms. Moreover, an example shows that -majority stable stock market equilibria need not exist for smaller s.Received: 23 December 2002, Revised: 14 June 2004, JEL Classification Numbers: D21, D52, D71, G39. Correspondence to: Hervé CrésThe authors are grateful to an anonymous referee for helpful comments and suggestions. Financial support from the Danish Research Councils and hospitality of HEC is gratefully acknowledged by Mich Tvede and support from Fondation HEC is gratefully acknowledged by Hervé Crés.     

When and why not to auction

Summary. Standard auctions are known to be a revenue-maximizing way to sell an object under broad conditions when buyers are symmetric and have independent private valuations. We show that when buyers have interdependent valuations, auctions may lose their advantage, even if symmetry and independence of information are maintained. In particular, simple alternative selling mechanisms that sometimes allow a buyer who does not have the highest valuation to win the object will in general increase all buyers willingness to pay, possibly enough to offset the loss to the seller of not always selling to the buyer with the greatest willingness to pay.Received: 18 June 2003, Revised: 7 September 2004, JEL Classification Numbers: D44, D82. Correspondence to: Colin M. CampbellWe thank Richard McLean, Dan Kovenock, two referees, and seminar participants at the University of Pittsburgh, Johns Hopkins University, Washington University, the Federal Communications Commission, the Institute for Advanced Studies at Princeton, UC-Santa Barbara, UC-Santa Cruz, and the Ohio State University Department of Finance for helpful suggestions. We thank the NSF for funding.     

Remarks concerning concave utility functions on finite sets

Summary. A direct construction of concave utility functions representing convex preferences on finite sets is presented. An alternative construction in which at first directions of supergradients (prices) are found, and then utility levels and lengths of those supergradients are computed, is exhibited as well. The concept of a least concave utility function is problematic in this context.Received: 28 November 2002, Revised: 28 June 2004, JEL Classification Numbers: D11, C60.I am indebted to an anonymous referee, Marcel K. Richter and Kam-Chau Wong, for many valuable remarks and suggestions.     

An axiomatic approach to \boldsymbol{\varepsilon}-contamination

Summary.  Suppose that an economic agent is 100% certain that uncertainty she faces is characterized by a particular probability measure, but that she has a fear that, with 100% chance, her conviction is completely wrong and she is left perfectly ignorant about the true measure in the present as well as in the future. This situation is often called -contamination of confidence. The purpose of this paper is to provide a simple set of behavioral axioms under which the decision-makers preference is represented by the Choquet expected utility with the -contamination of confidence.Received: 25 November 2002, Revised: 15 November 2004, JEL Classification Numbers:   D81. Correspondence to: Hiroyuki OzakiWe are grateful to an anonymous referee. The referees comments greatly improved the exposition of the paper. The work reported here is partially supported by a grant from the Economic and Social Research Insitute, the Cabinet Office, the Government of Japan.