Browsing by Author "Wong, Kam-Chau"

Now showing 1 - 5 of 5
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    Bounded Rationalities and Computable Economies
    (Center for Economic Research, Department of Economics, University of Minnesota, 1996-12) Richter, Marcel K.; Wong, Kam-Chau
    This paper studies economic equilibrium theory with a "uniformity principle" constraining the magnitudes (prices, quantities, etc.) and the operations (to perceive, evaluate, choose, communicate, etc.) that agents can use. For the special case of computability constraints, all prices, quantities, preference relations, utility functions, demand functions, etc. are required to be computable by finite algorithms. Then we obtain sharper versions of several traditional assertions on utility representation, existence of consumer demand functions, the fundamental welfare theorems, characterizations of market excess demands, and others. These positive results hold despite the fact that commodity and price spaces are no longer topologically complete. On the other hand, we give "computable counterexamples" to several traditional assertions, including the existence of a competitive equilibrium. The results can be interpreted as possibility and impossibility results in both computability-bounded rationality and in computational economics.
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    Bounded Rationalities and Definable Economies
    (Center for Economic Research, Department of Economics, University of Minnesota, 1996-12) Richter, Marcel K.; Wong, Kam-Chau
    Classical economic agents perform arbitrarily complex operations on arbitrarily complex magnitudes (real numbers). By contrast, real world agents have bounds on their abilities to perceive, think about, calculate with, and communicate magnitudes. There are many ways to model agents with bounded abilities, and here we mention two - one through bounds on computational abilities, and one through bounds on descriptive or definitional abilities. In both cases, we propose a "uniformity principle" constraining in a parallel fashion both the magnitudes (prices, quantities, etc.) and the operations (to perceive, evaluate, choose, communicate, etc.) that agents can use. We focus on the definitional bounds, deferring computational bounds to other papers (1996a,b). The languages allowed are those of ordered rings, and certain expansions; the structures are those of real closed ordered fields, and corresponding expansions. It is not obvious that a theory of definable economies is possible, since there may not be any definable structures that are reasonably close to the classical one. And even if such structures existed, it is not obvious that the classical theorems of economics would hold in them. Our two main conclusions are positive: In many interesting cases mathematical structures do exist with definability-bounded agents. Furthermore, many classical theorems of economic theory survive in a definable context: existence of demand and utility functions, existence of competitive equilibria, First and Second Welfare Theorems, characterization of aggregate excess demand, etc. Our proofs rely on theorems of mathematical logic (completeness (Tarski), model completeness (A. Robinson, Wilkie), o-minimality (van den Dries, Pillay and Steinhorn, Wilkie)) that allow us to establish existence of definable models and to transfer classical theorems to a definable framework. Although superficially different, the concepts underlying (Blume and Zame, 1992) are fundamentally close to the ones we use here.
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    Computability of Preference, Utility, and Demand
    (Center for Economic Research, Department of Economics, University of Minnesota, 1996-12) Richter, Marcel K.; Wong, Kam-Chau
    This paper studies consumer theory from the bounded rationality approach proposed in Richter and Wong (1996a), with a "uniformity principle" constraining the magnitudes (prices, quantities, etc.) and the operations (to perceive, evaluate, choose, communicate, etc.) that agents can use. In particular, we operate in a computability framework, where commodity quantities, prices, consumer preferences, utility functions, and demand functions are computable by finite algorithms (Richter and Wong (1996a)). We obtain a computable utility representation theorem. We prove an existence theorem for computable maximizers of quasiconcave computable utility functions (preferences), and prove the computability of the demand functions generated by such functions (preferences). We also provide a revealed preference characterization of computable rationality for the finite case. Beyond consumer theory, the results have applications in general equilibrium theory (Richter and Wong (1996a)).
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    Definable Utility in O-Minimal Structures
    (Center for Economic Research, Department of Economics, University of Minnesota, 1996-12) Richter, Marcel K.; Wong, Kam-Chau
    Representing binary ordering relations by numerical functions is a basic problem of the theory of measurement. It has applications in many fields of science, and arises in psychology and economics as the problem of establishing utility functions for preference relations. We obtain definable utility representations for (both continuous and upper semicontinuous) definable preferences in o-minimal expansions of real closed ordered fields. Such preferences have particular significance for modeling "bounded rationality" [18]. Our proofs are based on geometric theorems for definable sets, and provide new alternatives to the classical tools of separability (Debreu [5], Rader [16]) and metric-completeness (Arrow and Hahn [1]). The initial application of these ideas in economics was made by Blume and Zame (1992). Our results extend their Theorem 1 in several directions (see Remark la below).
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    Excess Demand Functions, Equilibrium Prices, and Existence of Equilibrium
    (Center for Economic Research, Department of Economics, University of Minnesota, 1995-10) Wong, Kam-Chau
    For continuous excess demand functions, the existing literature (e.g. Sonnenschein [1972, 1973], Mantel [1974], Debreu [1974], Mas-Colell [1977], etc.) achieves a complete characterization only when the functions are defined on special subsets of positive prices. In this paper, we allow the functions to be defined on a larger class of price sets, (allowing, for example the closed unit simplex, including its boundary). Besides characterizing excess demands for a larger class of economies, it is also a useful tool for proving other results. It allows us to characterize the equilibrium price set for a larger class of economies. It also permits extending Uzawa's observation [1962] by showing that Brouwer's Fixed-Point Theorem is implied by the Arrow-Debreu Equilibrium Existence Theorem ([1954], Thm. I.).