On the existence of equilibria in economies with infinitely many commodities and without ordered preferences

The present paper deals with the existence of equilibria in economies whose commodity space is L_∞(M, M, μ) and where the agents' preferences need not be complete or transitive. Applying a fixed point theorem of Browder, an equilibrium existence theorem for abstract economies (generalized qualitative games) is proven where each agent's choice set is contained in an arbitrary topological vector space. With the help of this theorem the existence of Walrasian general equilibrium for a suitably specified economic model is obtained. The final result is a generalization of T. F. Bewley's (J. Econ. Theory4 (1972), 514–540) equilibrium existence theorem to the case of non-ordered preferences.     

Edgeworth's conjecture with infinitely many commodities: commodity differentiation

Summary. Convergence of the cores of finite economies to the set of Walrasian allocations as the number of agents grows has long been taken as one of the basic tests of perfect competition. The present paper examines this test in the most natural model of commodity differentiation: the commodity space is the space of nonnegative measures, endowed with the topology of weak convergence. In Anderson and Zame [12], we gave counterexamples to core convergence in L ¹, a space in which core convergence holds for replica economies and core equivalence holds for continuum economies; in addition, we gave a core convergence theorem under the assumption that traders' utility functions exhibit uniformly vanishing marginal utility at infinity. In this paper, we provide two core convergence results for the commodity differentiation model. A key technical virtue of this space is that relatively large sets (in particular, closed norm-bounded sets) are compact. This permits us to invoke a version of the Shapley-Folkman Theorem for compact subsets of an infinite-dimensional space. We show that, for sufficiently large economies in which endowments come from a norm bounded set, preferences satisfy an equidesirability condition, and either (i) preferences exhibit uniformly bounded marginal rates of substitution or (ii) endowments come from an order-bounded set, core allocations can be approximately decentralized by prices. Received: July 29, 1996; revised version: January 14, 1997     

Equilibrium theory with asymmetric information and with infinitely many commodities

The traditional deterministic general equilibrium theory with infinitely many commodities cannot cover economies with private information constraints on the consumption sets. We bring the level of asymmetric information equilibrium theory at par with that of the deterministic one. In particular, we establish results on equilibrium existence for exchange economies with asymmetric (differential) information and with an infinite dimensional commodity space. Our new equilibrium existence theorems include, as a special case, classical results, e.g. Bewley [Existence of equilibria in economies with infinitely many commodities, J. Econ. Theory 4 (1972) 514-540] or Mas-Colell [The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986) 1039-1053].     

Decentralizing Edgeworth equilibria in economies with many commodities

Summary. This paper proves core-equivalence theorems for exchange economies without ordered preferences, defined on locally convex Riesz commodity spaces such that the price space is a lattice. Properness assumptions are borrowed from some recent equilibrium existence results. Received: January 15, 1998; revised version: August 19, 1998     

Determinacy of competitive equilibria in economies with many commodities

This paper provides a framework for establishing the determinacy of equilibria in general equilibrium models with infinitely many commodities and a finite number of consumers and producers. This paper defines a notion of regular economy for such models and gives sufficient conditions on the excess savings equations characterizing equilibria under which regular economies have a finite number of equilibria, each of which is locally stable with respect to perturbations in exogenous parameters, and under which regular economies are generic. This paper also defines two notions of concavity, called uniform concavity and weighted uniform concavity, which generalize standard finite-dimensional notions of differential concavity to an infinite-dimensional setting by prohibiting goods from becoming perfect substitutes asymptotically. For the case of economies in which there are countably many commodities, such as discrete time models or markets with countably many assets, results in this paper show that equilibria are generically determinate as long as utility functions and production sets are uniformly concave or weighted uniformly concave. Received: November 7, 1996; revised version: March 13, 1998     

Markets with infinitely many commodities and a continuum of agents with non-convex preferences

Summary Contrary to the finite dimensional set-up, the hypothesis of an atomless measure space of traders does not entail convexity of aggregate demand sets if there are infinitely many commodities. In this paper an assumption is introduced which sharpens the non-atomicity hypothesis by requiring that there are many agents of every type. When this condition holds, aggregate demand in an infinite dimensional setting becomes convex even if individual preferences are non-convex. This result is applied to prove the existence of competitive equilibria in such a context.Thanks to E. Dierker, K. Milford, M. Nermuth, C. Puppe, and N. Yannelis for comments and suggestions.     

Monopolistically competitive equilibria with differentiated commodities

Summary. An existence theorem of monopolistically competitive equilibrium of the economy in which commodities are subject to differentiation will be proved. We start with the existence theorem of Negishi (1961) and extend it to the commodity space of measures on a compact metric space. In so doing, we have to handle the price normalization carefully. Received: December 2, 1996; revised version: May 6, 1999     

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.     

On preferences with infinitely many subjective states

Models with subjective state spaces have been extremely useful in capturing novel psychological phenomena that consist of both a preference for flexibility and for commitment. Interpreting the utility representations of preferences as capturing these phenomena requires one to use the notion of a sign of a state. For linear preferences, we completely characterise the sign of a state in terms of its analytic representation as an integral with respect to a signed measure. In models with finitely many states, a state is either positive or negative, but never both. We show that in models with infinitely many states, a state can be both positive and negative. Thus, models with finitely many states may not capture all the behavioural features of an infinite model. Our methods are also useful in constructing utility functionals over menus with desired local properties.     

Competitive equilibria with limited enforcement

We show how to decentralize constrained efficient allocations that arise from enforcement constraints between sovereign nations. In a pure exchange economy these allocations can be decentralized with private agents acting competitively and taking as given government default decisions on foreign debt. In an economy with capital these allocations can be decentralized if the government can tax capital income as well as default on foreign debt. The tax on capital income is needed to make private agents internalize a subtle externality. The decisions of the government can arise as an equilibrium of a dynamic game between governments.     

Equilibrium theory with asymmetric information and infinitely many states

Radner (Econometrica 36, 31–58 1968) proved existence of a competitive equilibrium for differential information economies with finitely many states. We extend this result to economies with infinitely many states of nature. Each agent observes a public and a private signal. The publicly observed signal may take infinitely many values but, in order to get existence, we assume that private signals only take finitely many values. Actually, there is no hope to get a general existence result since Podczeck et al. (Vienna University Working Papers 2008) already proposed non-existence results. We would like to thank a referee and the Associate Editor for valuable suggestions and remarks. The actual version of the paper has benefited from their comments. This work commenced while V.F. Martins-da-Rocha and P.K. Monteiro were visiting RGEA. Carlos Hervés Beloso acknowledges support from BEC2003-09067-C04-01 (Ministerio de Ciencia y Tecnología and FEDER) and PGIDT04XIC30001PN (Xunta de Galicia). V.F. Martins-da-Rocha acknowledges partial financial support from PRONEX. P.K. Monteiro acknowledges the financial support of CNPq/Edital Universal 02/2006.     

Competitive equilibria in semi-algebraic economies

This paper develops a method to compute the equilibrium correspondence for exchange economies with semi-algebraic preferences. Given a class of semi-algebraic exchange economies parameterized by individual endowments and possibly other exogenous variables such as preference parameters or asset payoffs, there exists a semi-algebraic correspondence that maps parameters to positive numbers such that for generic parameters each competitive equilibrium can be associated with an element of the correspondence and each endogenous variable (i.e. prices and consumptions) is a rational function of that value of the correspondence and the parameters.This correspondence can be characterized as zeros of a univariate polynomial equation that satisfy additional polynomial inequalities. This polynomial as well as the rational functions that determine equilibrium can be computed using versions of Buchberger's algorithm which is part of most computer algebra systems. The computation is exact whenever the input data (i.e. preference parameters etc.) are rational. Therefore, the result provides theoretical foundations for a systematic analysis of multiplicity in applied general equilibrium.     

Competitive equilibria in overlapping generations Experiments

Summary The paper presents the results of four overlapping generations experiments performed at the California Institute of Technology. Overlapping generations markets were created in which each agent had a two period life span. With the exception of the first period, there were eight agents trading in each period; four buyers (two young and two old) and four sellers (two young and two old). Parameters were selected so that a small set of equilibria existed. The markets were open for twenty-nine periods with a demand shift occurring at the fifteenth and sixteenth periods.This work provides a method of computing all competitive equilibria for a class of environments—called the opposing shift environments. The main conclusion of the experiments is that the experimental price data converge to near the stationary portions of the competitive equilibria. Demographic dynamics are also explored as part of the price adjustment process. Dynamic features found in non-overlapping environments are also observed in the overlapping generations case.We thank E. J. Nanale, a Caltech undergraduate student, who helped design and conduct the experiments. Special thanks are due to Hsing Yang Lee for his help in adapting the Multiple Unit Double Auction Program to accomodate the overlapping generations environment and for his help with the data processing. We also thank David Grether who found a serious problem with the statistical analysis in an early draft. The research assistance of Kay-Yut Chen was also very valuable to the completion of this work. The comments and suggestions of P. Rangazas were very helpful. The research of the second author was supported in part by a National Science Foundation Grant and the California Institute of Technology Laboratory for Experimental Economics and Political Science.     

Competitive equilibria of economies with a continuum of consumers and aggregate shocks

This paper studies competitive equilibria of a production economy with aggregate productivity shocks. There is a continuum of consumers who face borrowing constraints and individual labor endowment shocks. The dynamic economy is described in terms of sequences of aggregate distributions. The existence of competitive equilibrium is proven and a recursive characterization is established. In particular, it is shown that for any competitive equilibrium, there is a payoff equivalent competitive equilibrium that is generated by a suitably defined recursive equilibrium.     

Walrasian equilibria as limits of noncooperative equilibria. Part II: Pure strategies

We examine the connection between Walrasian equilibria of a limit economy (with infinitesimal firms) and noncooperative (Cournot) equilibria of approximating finite economies (with significant firms). Nonconvex production sets, decreasing returns in the aggregate, and endogenous determination of the number of active firms are allowed. A Walrasian equilibrium is a limit of pure strategy noncooperative equilibria only if a condition (loosely analogous to downward sloping demand in the partial equilibrium constant returns to scale case) holds. The condition is also sufficient to guarantee the existence of a robust sequence of pure strategy noncooperative equilibria which converges to the Walrasian equilibrium.     

Undominated (and) perfect equilibria in Poisson games

In games with population uncertainty some perfect equilibria are in dominated strategies. We prove that every Poisson game has at least one perfect equilibrium in undominated strategies.