Ken Urai

A Generalization of Continuity and Convexity Conditions for Correspondences in Economic Equilibrium Theory

Fixed-point theorems for multi-valued mappings and economic equilibrium existence theorems are generalized from the viewpoint that the continuity and/or convexity assumptions on a mapping may be replaced by weaker local conditions on the vector field defined by the mapping. The generalization gives us natural conditions on individual (possibly non-ordered) preferences or aggregated demand behaviours so that we may obtain several extensions of recent results in social and game-theoretic equilibrium theories. JEL Classification Numbers: C60, C62, C72, D50, D51.

Fixed point theorems and the existence of economic equilibria based on conditions for local directions of mappings

Fixed point theorems for set valued mappings are reexamined from a unified viewpoint on local directions of the values of a mapping on a subset of a Hausdorff topological vector space to itself. Some basic fixed point theorems, such as Kakutani’s and Browder’s, are generalized so that we could apply them to game theoretic and economic equilibrium existence problems with non-ordered preferences having neither global continuity nor convexity conditions. Relations of our main results to other mathematical theorems such as Fan-Browder’s theorem, maximal element existence theorem for ℒ-majorized mappings, Eaves’ theorem, KKM and KKMS theorem, are also studied.

Fixed point theorems in Hausdorff topological vector spaces and economic equilibrium theory

The aim of this paper is to develop fixed point theorems in Hausdorff topological vector spaces that are suitable for the purpose of economic equilibrium theory. The special concept we have used here is the “direction structure” that characterizes mappings in the economic theory, (preferences, excess demands, and the like,) adequately, and enables us to modify problems on mappings into those on a structure of the base set. Especially, since our mathematical generalization may directly be related to the continuity and/or convexity of individual preferences, we may obtain existence theorems of maximal points, Pareto optimal allocations, and price equilibria for Gale-Nikaido-Debreu abstract economies under quite natural conditions.

General Equilibrium Model for an Asymmetric Information Economy without Delivery Upper Bounds

Abstract In this paper, we introduce production into the standard general equilibrium model with asymmetric information, which was proposed by Dubey et al. (Cowles Foundation Discussion Paper 2000; Econometrica 2005). In such an economy, there is no rational explanation for producers’ delivery upper bounds while the endowments naturally limit consumers’ deliveries. However, we show that the typical equilibrium allocation of the asymmetric information economy necessarily and substantially depends on such exogenous upper bounds (Example 1 and Theorem 1). In other words, an equilibrium existence theorem without such upper bounds, even if such exists, will typically fail to treat the asymmetric information problem, e.g., the adverse selection problem. Hence, to treat the equilibrium existence problem under the informational asymmetry appropriately, we have to extend the standard model so that the delivery upper bounds need not to be specified explicitly. For this purpose, we propose a quite natural and realistic assumption with respect to the technological condition related to the market delivery, i.e., the existence of some small standardization, commoditization, and/or transaction costs of market deliveries is shown to be sufficient (Theorem 3).

General Equilibrium Model and Set Theoretic Finiteness

The axiom of infinity in the ZF set theory is usually considered essential for Brouwer’s fixed point theorem and the proof of the existence of economic equilibria. In this paper, the meanings of the infinity axiom are reconsidered from the view point of the minimal requirements for the general equilibrium theory. We provide several tools and conditions under which a settheoretically finite setting is sufficient, at least for an elementary equilibrium existence argument on the static general equilibrium model. In view of the social science, the purpose of general equilibrium theory is to describe the whole structure of the human society based on the price and the market mechanism. However, there exists a classical problematic feature relating to the objectivity, i.e., the introspection problem for us to describe the world including ourselves, which also brings about important questions about the true sense of our individual rationality. In order to consider about the introspective feature of the social science and the human rationality under an economic model, there may exist two approaches. One is to attempt to describe our true rationality into the model, so that we may treat directly the whole society as the totality of such rational individuals. The other approach is giving up to define the true sense of rationality (a general intelligence of human) and, instead, to seek a minimal requirement for our rationality to construct and recognize the world described by a certain economic model. In this paper, we base our argument on the second approach, and show that the static general equilibrium theory such as Debreu (1959), could essentially be treated under a finitistic mathematical argument. JEL Classification: C62, D51