A limit theorem for an approximate core of a large but finite economy

We present a formulation of an approximate core that can be sustained as an approximate equilibrium for a ‘large enough’ finite exchange economy whose traders need not have transitive, continuous or convex preferences.     

An inner core equivalence theorem

Summary We extend the notion of the inner core of a finite economy to a large economy. We prove that competitive allocations and the core coincide with the inner core.This is based on Chapter 5 of my Ph.D thesis. I am indebted to my thesis advisor, Lloyd S. Shapley, for his guidance. Discussions with Michael Balch are gratefully acknowledged. I also thank Charles Stuart, Nicholas C. Yannelis and an anonymous referee for comments and suggestions that greatly improved the paper.     

A refinement of the core equivalence theorem

It is known that if blocking coalitions are restricted to be small relative to the size of the economy then the approximation of core allocations by competitive allocations can still be assured. This paper proves that the same type of approximation result is valid if the absolute size of the blocking coalition is bounded. The methods of Anderson (1978) are applied.     

When is the core equivalence theorem valid?

Summary An answer to the question of the title is provided. It is shown that under some conditions the core equivalence theorem holds true (if and) only if the price space is a vector lattice.Research of both authors was supported in part by a Chrysler Corporation grant to IUPUI.     

Short run core equivalence in an overlapping generations model

This essay establishes the equality between the set of competitive equilibrium allocations and the set of allocations which belong to the cores of a sequence of short run economies for an overlapping generations model. A short run economy is a finite period economy obtained by truncating the original economy so that the truncated economy is irreducible.     

An equivalence theorem in the theory of resource extraction

There exists a formal equivalence between a class of resource extraction problems with a stock-dependent net benefit function and Gale's canonical cake-eating problem.     

A Lagrangean approach to the limit theorem on the core of an economy

Conclusion It is shown that an allocation which remains in the core under replication gives rise to an utility allocation with the following property: The utility level of no class can be increased without decreasing the utility level of at least one other class even if the number of consumers is allowed to vary continuously. An examination of the first order conditions the corresponding maximum problem with constraints leads to the conclusion that the allocation can be sustained by a price vector as a competitive equilibrium. This is the limit theorem on the core, proved in this paper with a simple Lagrangean technique.The notion of Pareto efficiency with a variable number of consumers proves useful for other purposes as well. For an extensive discussion of that point the reader is referred to Schweizer (1981).Ours, of course, is not the first calculus approach to the limit theorem [see e. g. L. Johansen (1978)]. But, as far as I know, the method of differentiating the Lagrangean simply with respect to the number of consumers is novel and leads to a straightforward proof of the theorem which can be reproduced in class-rooms even at the undergraduate level.The author wants to thank Joseph Greenberg and Werner Hildenbrand as well as an anonymous referee for helpful comments.     

Costly enforcement of property rights and the Coase theorem

We examine a setting in which property rights are initially ambiguously defined. Whether the parties go to court to remove the ambiguity or bargain and settle before or after trial, they incur enforcement costs. When the parties bargain, a version of the Coase theorem holds. However, despite the additional costs of going to court, other ex-post inefficiencies, and the absence of incomplete information, going to court may ex-ante Pareto dominate settling out of court. This is especially true in dynamic settings, where obtaining a court decision today saves on future enforcement costs. When the parties do not negotiate and go to court, a simple rule for the initial ambiguous assignment of property rights maximizes net surplus. A paper circulated under the same title and dated 6 March 2000 contained the basic structure examined in this paper, but did not develop many of the results reported here. For comments, we would like to thank participants at the WZB-CEPR contests conference and seminar audiences at the University of Southern California, the University of British Columbia, UC Davis, and UC Riverside.     

On Kakutani's fixed point theorem,the K-K-M-S theorem and the core of a balanced game

Summary We provide elementary proofs of Scarf's theorem on the non-emptiness of the core and of the K-K-M-S thoerem, based on Kakutani's fixed point theorem. We also show how these proofs can be modified to apply a coincidence theorem of Fan instead of Kakutani's fixed point theorem, for some additional simplicity.The results presented here were first reported in Shapley (1987) and Vohra (1987). A version of our proof of Theorem 1 has also been presented in a recent book by C.D. Aliprantis, D.J. Brown and O. Burkinshaw,Existence and Optimality of Competitive Equilibria (1989) Springer-Verlag. We are grateful to Ky Fan, Wanda Gorgol, Tatsuro Ichiishi and Ali Khan for comments on earlier drafts. Vohra's research has been supported in part by NSF grant SES-8605630.     

The equilibrium of the bargaining game and core convergence theorem on an exchange economy with limited traders

Debreu and Scarf (1963), Hildenbrand (1974), Aumann (1964), Dierker (1975), Bewley (1973), and others have shown that the core of an exchange economy with infinitely many or finitely many traders converges. However, an exchange economy does not always consist of infinitely many or finitely many traders. This note provides proof of the core convergence theorem on an exchange economy with limited traders by a bargaining game methodology. The main contribution of this note is to innovate the equilibrium solution to the bargaining game in the exchange economy. In this note, the concept of common payoff is introduced; in the bargaining game of a coalition on its common payoff, all coalition members will get the same distribution, thus the distribution scheme of the cooperation surplus of the exchange economy is determined. This note shows that the bargaining game among the traders on the distribution of the cooperation surplus will make the pure exchange economy with limited traders converge to the Walrasian equilibrium, all the allocations other than the Walrasian equilibrium will be eliminated from the core of this economy.     

A theorem on open coverings of a simplex and Scarf's core existence theorem through Brouwer's fixed point theorem

Summary This paper provides simple proofs of a theorem on open coverings of a simplex and Scarf's core existence theorem through Brouwer's fixed point theorem.I want to thank Daniel Abrams, Yakar Kannai, Herbert Scarf, Nicholas Yannelis, Jingang Zhao, and an anonymous referee for their helpful comments. I also gratefully acknowledge the financial support from the Alfred P. Sloan Foundation.     

The Nash-threats folk theorem with communication and approximate common knowledge in two player games

We show that the use of communications to coordinate equilibria generates a Nash-threats folk theorem in two-player games with “almost public” information. The results generalize to the n-person case. However, the two-person case is more difficult because it is not possible to sustain equilibria by comparing the reports of different players, and using these “third parties” to effectively enforce contracts.     

Competitive and core allocations in large economies with differential information

Summary. We study the core and competitive allocations in exchange economies with a continuum of traders and differential information. We show that if the economy is “irreducible”, then a competitive equilibrium, in the sense of Radner (1968, 1982), exists. Moreover, the set of competitive equilibrium allocations coincides with the “private core” (Yannelis, 1991). We also show that the “weak fine core” of an economy coincides with the set of competitive allocations of an associated symmetric information economy in which the traders information is the joint information of all the traders in the original economy. Received March 22, 2000; revised version: May 1, 2000     

The least core, kernel and bargaining sets of large games

Summary. We study the least core, the kernel and bargaining sets of coalitional games with a countable set of players. We show that the least core of a continuous superadditive game with a countable set of players is a non-empty (norm-compact) subset of the space of all countably additive measures. Then we show that in such games the intersection of the prekernel and the least core is non-empty. Finally, we show that the Aumann-Maschler and the Mas-Colell bargaining sets contain the set of all countably additive payoff measures in the prekernel. Received: June 6, 1996; revised version: March 1, 1997     

Adult equivalence scales and food expenditures: an application to Mexican beef and pork purchases

In the analysis of food expenditures, the use of a simple count of household members as an estimate of household size implicitly assumes each household member has the same marginal impact. In this analysis of Mexican food expenditures, endogenously determined adult equivalence scales are estimated in such a way that these marginal impacts are allowed to vary by household member age and gender. The results of a series of hypothesis tests indicate a rejection of the implied null hypothesis of equal marginal expenditure impacts associated with the use of the traditional count based household size variable. This study also rejects the null hypothesis of the equality of adult equivalent scales across the commodities included in this analysis.