# Center for Economic Research Discussion Papers

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This series of discussion papers arose from seminars and research conducted at the Center for Economic Research and includes work by three Nobel Prize in Economics laureates.

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### Browsing Center for Economic Research Discussion Papers by Subject "C63"

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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 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 Consumption-Based CAPM and Option Pricing under Jump-Diffusion Uncertainty(Center for Economic Research, Department of Economics, University of Minnesota, 2003-04) Kusuda, KojiIn Kusuda [45], we developed equilibrium analysis in security market economy with jump-Wiener information where no finite number of securities can complete markets. Assuming approximately complete markets (Bjork et al. [11] [12]) in which a continuum of bonds are traded and any contingent claim can be replicated with an arbitrary precision, we have shown sufficient conditions for the existence of approximate security market equilibrium, in which every agent is allowed to choose any consumption plan that can be supported with any prescribed precision. In this paper, we derive the Consumption-Based Capital Asset Pricing Model (CCAPM) using the framework in case of heterogeneous with additively separable utilities (ASUs) and of homogeneous agents with a common stochastic differential utility (SDU). The CCAPM says that the risk premium between a risky security and the nominal-risk-free security can be decomposed into two groups of terms. One is related to the price fluctuation of the risky security, and the other is related to that of commodity. Each group can be further decomposed into two terms related to consumption volatility and consumption jump in case of ASUs, and into three terms related to consumption volatility, continuation utility volatility, and jumps of consumption and continuation utility in case of SDU. Next, we present a general equilibrium framework of jump-diffusion option pricing models in each case of heterogeneous agents with CRRA utilities and of homogeneous agents with a common Kreps-Porteus utility. Finally, we construct a general equilibrium version of an affine jump-diffusion model with jump-diffusion volatility for option pricing using the framework.