Browsing by Author "Richter, Marcel K."
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Item Bounded Rationalities and Computable Economies(Center for Economic Research, Department of Economics, University of Minnesota, 1996-12) Richter, Marcel K.; Wong, Kam-ChauThis 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.Item Bounded Rationalities and Definable Economies(Center for Economic Research, Department of Economics, University of Minnesota, 1996-12) Richter, Marcel K.; Wong, Kam-ChauClassical 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.Item Coalitions, Core, and Competition(Center for Economic Research, Department of Economics, University of Minnesota, 1971-06) Richter, Marcel K.Item Computability of Preference, Utility, and Demand(Center for Economic Research, Department of Economics, University of Minnesota, 1996-12) Richter, Marcel K.; Wong, Kam-ChauThis 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)).Item Definable Utility in O-Minimal Structures(Center for Economic Research, Department of Economics, University of Minnesota, 1996-12) Richter, Marcel K.; Wong, Kam-ChauRepresenting 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).Item Duality and Rationality(Center for Economic Research, Department of Economics, University of Minnesota, 1976-09) Richter, Marcel K.Item Implicit Functions and Diffeomorphisms without C1(Center for Economic Research, Department of Economics, University of Minnesota, 1995-03) Hurwicz, Leonid; Richter, Marcel K.We prove implicit and inverse function theorems for non-C1 functions, and characterize non-C1 diffeomorphisms.Item An Integrability Condition, with Applications to Utility Theory and Thermodynamics(Center for Economic Research, Department of Economics, University of Minnesota, 1977-02) Hurwicz, Leonid; Richter, Marcel K.Item Nontransitive-Nontotal Consumer Theory(Center for Economic Research, Department of Economics, University of Minnesota, 1985-11) Kim, Taesung; Richter, Marcel K.We show the parallel nature of two approaches to nontransitive or nontotal consumers: through "weak" preferences and through "strict" preferences. This yields specific results (e.g., a new equilibrium existence theorem with weak preferences), general results (a Metatheorem translating between the two approaches), and general concepts (a new notion of "rationality"). We give revealed preference axioms to characterize preferences, and prove equivalences among several axiom systems, showing that the apparent weakness of some axiom systems is illusory. We resolve a Weak Axiom conjecture, and we introduce a new axiom. We prove that our nonclassical consumers generalize classical equilibrium theory.Item Optimization and Lagrange Multipliers: Non-C1 Constraints and "Minimal" Constraint Qualifications(Center for Economic Research, Department of Economics, University of Minnesota, 1995-03) Hurwicz, Leonid; Richter, Marcel K.When do Lagrange multipliers exist at constrained maxima? In this paper we establish: a) Existence of multipliers, replacing C1 smoothness of equality constraint functions by differentiability (for Jacobian constraint qualifications) or, for both equalities and inequalities, by the existence of partial derivatives (for path-type constraint qualifications). This unifies the treatment of equality and inequality constraints. b) A notion of "minimal" Jacobian constraint qualifications. We give new Jacobian qualifications and prove they are minimal over certain classes of constraint functions. c) A path-type constraint qualification, weaker than previous constraint qualifications, that is necessary and sufficient for existence of multipliers. (It only assumes existence of partial derivatives.) A survey of earlier results, beginning with Lagrange's own multipliers for equality constraints is contained in the last section. Among others, it notes contributions and formulations by Weierstrass; Bolza; Bliss; Caratheodory; Karush; Kuhn and Tucker; Arrow, Hurwicz, and Uzawa; Mangasarian and Fromovitz; and Gould and Tolle.Item Revelations of a Gambler(Center for Economic Research, Department of Economics, University of Minnesota, 1977-02) Richter, Marcel K.; Shapiro, LeonardItem The Second Welfare Theorem of Classical Welfare Economics(Center for Economic Research, Department of Economics, University of Minnesota, 2001-08) Hurwicz, Leonid; Richter, Marcel K.We extend the Second Fundamental Theorem of Welfare Economics in several directions. For pure exchange economies, we drop all insatiability requirements on preferences. For economies with production, we use a concept of directional optimality to provide necessary and sufficient conditions for a given allocation to be competitive. This enables us to show, for example, that not all consumers need to be locally nonsatiated, if the economy is "connected." (An example due to Stanley Reiter shows that such extra conditions are unavoidable.) We use weak assumptions on feasibility sets, allowing, but not requiring, short sales and a very general form of disposability. We do not require that preferences be reflexive, transitive, total, or negatively transitive; and we replace full continuity of preferences by a semicontinuity condition for strict preferences. This provides decentralization results extending some of Arrow's original results [1], as well as those in Arrow and Hahn [2, Theorem 4, pp. 93-94] Debreu [6, Theorem 6.4, p. 95], [4, p. 281], and elsewhere.Item Testing Strictly Concave Rationality(Center for Economic Research, Department of Economics, University of Minnesota, 1987-07) Matzkin, Rosa L.; Richter, Marcel K.We prove that the Strong Axiom of Revealed Preference tests the existence of a strictly quasiconcave (in fact, continuous, generically Coo, strictly concave, and strictly monotone) utility function generating finitely many demand observations. This sharpens earlier results of Afriat, Diewert, and Varian that tested ("nonparametrically") the existence of a piecewise linear utility function that could only weakly generate those demand observations. When observed demand is also invertible, we show that the rationalizing can be done in a Coo way, thus extending a result of Chiappori and Rochet from compact sets to all of Rn. For finite data sets, one implication of our result is that even some weak types of rational behavior - maximization of pseudotransitive or semitransitive preferences -- are observationally equivalent to maximization of continuous, strictly concave, and strictly monotone utility functions.Item Ville Axioms and Consumer Theory(Center for Economic Research, Department of Economics, University of Minnesota, 1977-02) Hurwicz, Leonid; Richter, Marcel K.