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.     

Social equilibrium and cycles on compact sets

One proof of existence of general equilibrium assumes convexity and continuity of a preference correspondence on a compact convex feasible set W. Here the existence of a local equilibrium for a preference field which satisfies, not convexity, but the weaker local acyclicity is shown. The theorem is then applied to a voting game, σ, without veto players. It is shown that if the dimension of the policy space is no greater than ν(σ) ? 2, where ν(σ) is the Nakamura number of the game, then no local cycles may occur and a local equilibrium must exist. With convex preferences, then, there will exist a choice of the game from W.     

An Elementary Approach to Approximate Equilibria with Infinitely Many Commodities

We prove the existence of approximate equilibria in a finite exchange economy with a countably infinite number of commodities and nonconvex preferences, when every trader has an excess demand set that is finitely spannable, i.e., that could be covered by a union of its convex subsets in finitely many steps. We show that the bound on the norm of the per capita aggregate excess demand is reciprocally related to the square root of the population. Extensions are also made to the case where countably many commodities are indivisible. The proofs are elementary. Journal of Economic Literature Classification Numbers: D50, C62, D52.     

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.     

Private Value Allocations in Large Economies with Differential Information

We consider a Radner-type (e.g., Radner, 1968, Econometrica36, 31–58) pure exchange economy with differential information and a continuum of agents. We show that under appropriate assumptions the set of Aumann–Shapley private value allocations in such an economy coincides with the set of Radner competitive equilibrium allocations. Journal of Economic Literature Classification Numbers: C71, D51, D82.     

Pareto optima and equilibria when preferences are incompletely known

An exchange economy in which agents have convex incomplete preferences defined by families of concave utility functions is considered. Sufficient conditions for the set of efficient allocations and equilibria to coincide with the set of efficient allocations and equilibria that result when each agent has a utility in her family are provided. Welfare theorems in an incomplete preferences framework therefore hold under these conditions and efficient allocations and equilibria are characterized by first order conditions.     

Efficiency, monotonicity and rationality in public goods economies

Summary. In economies with public goods, we identify a necessary and sufficient condition for the existence of cost monotonic, Pareto optimal and individually rational mechanisms. These exist if and only if the preferences of the agents satisfy what we call the equal ordering property. We also show that when this condition holds the egalitarian equivalent correspondence is the only cost monotonic selection from the core of the economy. Furthermore, it is unambiguous in the sense that the agents are indifferent among all the allocations in it. Received: February 26, 1996; revised version: January 31, 1997     

On equilibrium in Hart's securities exchange model

In this paper, we use a no unbounded arbitrage condition to give a very direct proof of the existence of equilibrium in Hart's unbounded securities exchange model (J. Econ. Theory, 9 (1974), 293–311). We also examine the relationship between the no unbounded arbitrage condition and the sufficiency conditions of Hart, ibid. and Hammond (J, Econ. Theory, 31 (1983), 170–175). We present an example to show that if traders are not sufficiently risk averse, then Hammond's overlapping expectations condition is not, in general, equivalent to the no unbounded arbitrage condition or Hart's sufficiency conditions, and therefore, is not sufficient to guarantee the existence of equilibrium. We also present an example to show that it is possible for the no unbounded arbitrage condition to hold without overlapping expectations, and therefore, it is possible for equilibrium to exist without overlapping expectations.     

When sunspots don't matter

Summary We show that a finite, competitive economy isimmune to sunspots if (i) preferences are strictly convex, (ii) the set of feasible allocations is convex, and (iii) the contingent-claims market is perfect. The conditions (i)–(ii) cover some, but not all, economies with nonconvex technologies. Based on an indivisible-good example, we show that even economies with strictly convex preferences and full insurance arenot in general immune from sunspots. We also show that (1) the sufficient conditions (i)–(iii) are not necessary for sunspot immunity and (2)ex-ante efficiency is not necessary for immunity from sunspots.This paper is based on an earlier paper, Indivisibilities in Production, and Sunspot Equilibrium, presented at the 1990 S.E.D.C. Meetings, Minneapolis-St. Paul, June 1990. The research support of NSF Grant SES-9012780, the Center for Analytic Economics, and the Thorne Fund is gratefully acknowledged.     

Lotteries, Sunspots, and Incentive Constraints

We study a prototypical class of exchange economies with private information and indivisibilities. We establish an equivalence between lottery equilibria and sunspot equilibria and show that the welfare and existence theorems hold. To establish these results, we introduce the concept of the stand-in consumer economy, which is a standard, convex, finite consumer, finite good, pure exchange economy. With decreasing absolute risk aversion and no indivisibilities, we prove that no lotteries are actually used in equilibrium. We provide a simple numerical example with increasing absolute risk aversion in which lotteries are necessarily used in equilibrium. We also show how the equilibrium allocation in this example can be implemented in a sunspot equilibrium. Journal of Economic Literature Classification Numbers: D11, D50, D82.     

Optimal growth with many consumers

A method is described for constructing all Pareto-optimal allocations for a dynamic economy with many heterogeneous consumers, under certainty, in which both the technology and consumer preferences are recursive but preferences need not be additively separable over time. Optimal (perfect foresight competitive equilibrium) allocations are obtained through the study of a dynamic program. For an economy with one consumption good, sufficient conditions are given for the existence of a unique interior stationary distribution of consumption and wealth. For a two-person exchange economy, sufficient conditions are given for the global asymptotic stability of the unique interior stationary point.     

A revised concept of distributional equity

Recent studies have shown that in a private goods economy with production, allocations which are considered to be distributionally fair may not exist. As a solution for such cases, just allocations are introduced and examined in some detail. Such allocations are shown, by rigorous proof, to exist in general.     

Efficiency and equilibrium when preferences are time-inconsistent

We consider an exchange economy with time-inconsistent consumers whose preferences are additively separable. If consumers have identical discount factors, then allocations that are Pareto efficient at the initial date are also renegotiation-proof. In an economy with a sequence of markets, competitive equilibria are Pareto efficient in this sense, and for generic endowments, only if preferences are locally homothetic.     

Competitive bargaining equilibrium

In a simple exchange economy we propose a bargaining procedure that leads to a Walrasian outcome as the agents become increasingly patient. The competitive outcome therefore obtains even if agents have market power and are not price-takers. Moreover, where in other bargaining protocols the final outcome depends on bargaining power or relative impatience, the outcome here is determinate and depends only on preferences and endowments. Our bargaining procedure involves bargaining over prices and maximum quantity constraints, and it guarantees convergence to a Walrasian outcome for any standard exchange economy. In contrast, without quantity constraints we show that equilibrium is generically inefficient.     

Non-existence of competitive equilibria with dynamically inconsistent preferences

This paper shows the robust non-existence of competitive equilibria even in a simple three period representative agent economy with dynamically inconsistent preferences. We distinguish between a sophisticated and naive representative agent. Even when underlying preferences are monotone and convex, at given prices, we show by example that the induced preference of the sophisticated representative agent over choices in first-period markets is both non-convex and satiated. Even allowing for negative prices, the market-clearing allocation is not contained in the convex hull of demand. Finally, with a naive representative agent, we show that perfect foresight is incompatible with market clearing and individual optimization at given prices.     

Pareto-optimal Nash equilibria are competitive in a repeated economy

Consider a finite exchange economy first as a static, 1 period, economy and then as a repeated economy over T periods when the utility of each agent is the mean utility over T. A family of strategic games is defined via a set of six general properties the most distinct of which is the ability of agents to move commodities forward in time. Now consider Pareto optimal allocations in the T period economy which are also Nash equilibria in this family of strategic games. We prove that as T becomes large this set converges to the set of competitive utility allocations in the one period economy. The key idea is that a repetition of the economy when agents can move commodities forward in the time acts as a convexification of the set of individually feasible outcomes for player i holding all other strategies fixed.     

Strategy-Proof Probabilistic Mechanisms in Economies with Pure Public Goods

Public good economies where agents are endowed with strictly convex continuous single-peaked preferences on a convex subset of Euclidean space are considered. Such an economy arises for instance in the classical problem of allocating a given budget to finance the provision of several public goods where the agents have monotonically increasing strictly convex continuous preferences. A probabilistic mechanism assigns a probability distribution over the feasible alternatives to any profile of reported preferences. The main result of the paper establishes that any strategy-proof (in the sense of A. Gibbard, Econometrica45 (1977), 665-681) and unanimous mechanism must be a random dictatorship. Journal of Economic Literature Classification Numbers: D70, D71, H40, C60.     

Experts against adverse selection: A note on the existence of equilibrium with costly appropriable information

The existence of competitive equilibrium in Laffont's (J. Econ. Theory10 (1975)) model of adverse selection with costly information is studied. The existence of an equilibrium with finite prices is demonstrated without unusual restrictions on preferences or the technology of information production. This is made possible by changing the way in which the behaviour of information producing agents is modelled, and allowing for some public information.     

On the Existence of Undominated Nash Equilibria in Normal Form Games

In this paper the existence problem of undominated Nash equilibrium in normal form games is analyzed. It is shown that an undominated Nash equilibrium exists, if (a) strategy sets are convex polytopes inRⁿand (b) utility functions are affine with respect to each player's own strategy. It is shown by counterexamples that, first, it is not sufficient to have concave utility functions instead of affine under condition (b) even when condition (a) is satisfied, and, second, it is not sufficient to have just compact and convex strategy sets instead of polytopes in condition (a) even when condition (b) is satisfied.     

On exchange rates and efficiency

Summary This paper analyzes an exchange economy in which several assets serve as stores of value and where agents have completely heterogeneous preferences and endowments. It describes the set of perfect foresight equilibria in which all assets have positive prices. There are international policies with determinate exchange rates if the world economy satisfies a strong efficiency criterion. Also, the corresponding equilibrium allocations are in the core of the world economy for certain international policies. Hence, a system of fixed exchange rates can support efficient allocations to the extent that countries agree on a division of seigniorage in the creation of international reserves.I would like to thank an annonymous referee, Suchan Chae, Mike Woodford, Ken Kasa, Helen Popper, Kathryn Marshall, and seminar participants at the University of Kansas, the Federal Reserve board, the 1990 meetings of the Western Economic Association, the University of Western Ontario, the University of Chicago, the University of Pennsylvania, Purdue University, and Victoria University of Wellington for comments on earlier drafts.