William R. Zame

Equilibrium theory in infinite dimensional spaces

Publisher Summary This chapter summarizes the account of the extension of the classical general equilibrium model to an infinite dimensional setting. The classical finite dimensional theory, the commodity space is the canonical finite dimensional linear space R n . By contrast, there is no canonical infinite dimensional linear space. Different economic applications require models involving different infinite dimensional linear spaces. The mathematical discipline of functional analysis has already been well developed as a tool for the abstract study of linear spaces. The chapter follows the methodology of functional analysis and attacks the existence problem. Advantage of this method is that it yields general results, capable of application in a wide variety of specific models. An important line of research in classical general equilibrium theory has been the relationship of the core to the set of competitive allocations. In the infinite dimensional setting, an extensive body of work has been developed, which centers around the infinite-dimensional version of the Debreu–Scarf core convergence theorem.

The Consumption-Based Capital Asset Pricing Model

This paper provides conditions on the primitives of a continuous-time economy under which there exist equilibria obeying the consumption-based capital asset pricing model. The paper also extends the equilibrium characterization of interest rates of Cox, Ingersoll, and Ross (1985) to multiagent economies. No Markovian state assumption is used. Copyright 1989 by The Econometric Society.

Clubs and the Market

This paper defines a general equilibrium model with exchange and club formation. Agents trade multiple private goods widely in the market, can belong to several clubs, and care about the characteristics of the other members of their clubs. The space of agents is a continuum, but clubs are finite. It is shown that (i) competitive equilibria exist, and (ii) the core coincides with the set of equilibrium states. The central subtlety is in modeling club memberships and expressing the notion that membership choices are consistent across the population.

Communication and equilibrium in discontinuous games of incomplete information

This paper offers a new approach to the study of economic problems usually modeled as games of incomplete information with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies (ties) in the underlying problem. The point of view taken here is that the tie-breaking rules that resolve these indeterminacies should be viewed as part of the solution rather than part of the description of the model. A solution is therefore a tie-breaking rule together with strategies satisfying the usual best-response criterion. When information is incomplete, solutions need not exist; that is, there may be no tie-breaking rule that is compatible with the existence of strategy profiles satisfying the usual best-response criteria. It is shown that the introduction of incentive compatible communication (cheap talk) restores existence.

Does Market Incompleteness Matter

This paper argues that incompleteness of intertemporal financial markets has little effect (on welfare, prices, or consumption) in an economy with a single consumption good, provided that traders are long-lived and patient, a riskless bond is traded, shocks are transitory, and there is no aggregate risk. In an economy with aggregate risk, a similar conclusion holds, provided traders share the same CRRA utility function and the right assets are traded. Examples demonstrate that these conclusions need not hold if the wrong assets are traded or if the economy has multiple consumption goods. Copyright The Econometric Society 2002.

Debt Constraints and Equilibrium in Infinite Horizon Economies with Incomplete Markets

Abstract This paper defines a notion of an equilibrium and a pseudo-equilibrium for infinite horizon economies with incomplete asset markets. This definition generalizes the usual ones for finite horizon economies with incomplete markets and for infinite horizon economies with complete markets. We establish the existence of a pseudo-equilibrium when assets are short-lived and denominated in general commodity bundles; we obtain a true equilibrium when assets are denominated solely in a single numeraire commodity, or in units of account. It seems to us that the notion of an equilibrium we define is a natural and compelling one; as evidence, we show that our notion actually coincides with several other - apparently quite distinct - notions of an equilibrium.

The nonatomic assignment model

SummaryWe formulate a model with a continuum of individuals to be assigned to a continuum of different positions which is an extension of the finite housing market version due to Shapley and Shubik. We show that optimal solutions to such a model exist and have properties similar to those established for finite models, namely, an equivalence among the following: (i) optimal solutions to the linear programming problem (and its dual) associated with the assignment model; (ii) the core of the associated market game; (iii) the Walrasian equilibria of the associated market economy.

Competitive Equilibria in Production Economies with an Infinite-Dimensional Commodity Space

The existence of competitive equilibrium is established for economies with a produc tion sector and an infinite dimensional space of commodities. The cru cial assumptions which are required (beyond those required in the fin ite dimensional setting) are bounds on marginal rates of substitution and marginal rates of transformation. Copyright 1987 by The Econometric Society.

Clubs and the Market: Large Finite Economies

We study large finite club economies in which agents can belong to several clubs, and care about the characteristics of the other club members. Club memberships must be integer consistent in aggregate. We show that states in the approximate core can approximately be decentralized by prices for private goods and for club memberships, that the approximate core is nonempty, and that approximate club equilibria exist. Our arguments use the convexification tools used for private goods economies, but we also develop a new tool to address the consistency requirement on memberships that are special to club economies. This tool allows us to overcome the integer consistency problems that are avoided in our (1999) paper by assuming a continuum of agents.

The Algebraic Geometry of Perfect and Sequential Equilibrium

Two of the most important refinements of the Nash equilibrium concept for extensive form games with perfect recall are Seltens (1975) {\it perfect equilibrium\/} and Kreps and Wilsons (1982) more inclusive {\it sequential equilibrium\/}. These two equilibrium refinements are motivated in very different ways. Nonetheless, as Kreps and Wilson (1982, Section 7) point out, the two concepts lead to similar prescriptions for equilibrium play. For each particular game form, every perfect equilibrium is sequential. Moreover, for almost all assignments of payoffs to outcomes, almost all sequential equilibrium strategy profiles are perfect equilibrium profiles, and all sequential equilibrium outcomes are perfect equilibrium outcomes. \par We establish a stronger result: For almost all assignments of payoffs to outcomes, the sets of sequential and perfect equilibrium strategy profiles are identical. In other words, for almost all games each strategy profile which can be supported by beliefs satisfying the rationality requirement of sequential equilibrium can actually be supported by beliefs satisfying the stronger rationality requirement of perfect equilibrium. \par We obtain this result by exploiting the algebraic/geometric structure of these equilibrium correspondences, following from the fact that they are {\em semi-algebraic sets\/}; i.e., they are defined by finite systems of polynomial inequalities. That the perfect and sequential