Mathematical Modeling of Consumer's Preferences Using Partial Differential Equations

Jorge Marques
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Jorge Marques: Faculty of Economics, University of Coimbra, Portugal

No 2013-15, GEMF Working Papers from GEMF, Faculty of Economics, University of Coimbra

Abstract: The aim of this paper is to define consumer's preferences from the differentiable viewpoint in the sense of Debreu. In this framework given the marginal rates of substitution we can consider a vector field to represent consumer's preferences in the microeconomic theory. By definition the marginal rates of substitution satisfy a system of first order partial differential equations. For a continuously differentiable vector field that holds the integrability conditions we provide a general method to solve the system. In the special case of integrable preferences these conditions impose symmetry properties in the underlying preferences. Our results allow to characterize consumer's preferences in terms of the indifference map for the following classes: linear, quasi-linear, separable, homothetic, homothetic and separable. We show that this alternative approach is connected with the traditional formulation concerning the representability of preferences by utility functions. Moreover, we deduce even the general expression of utility functions that satisfy the integrability conditions in the context of ordinal utility.

Keywords: smooth preferences; marginal rates of substitution; indifference map; ordinal utility; integrability conditions; differential equations. (search for similar items in EconPapers)
JEL-codes: C60 C62 D01 (search for similar items in EconPapers)
Pages: 27 pages
Date: 2013-05
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