Browsing by Subject "Definability"

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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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    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).