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Grupo de Estudos Monetários e Financeiros

Estudos do GEMF, N.º 15 de 2013


Mathematical Modeling of Consumer's Preferences 

Using Partial Differential Equations

Jorge Marques

Faculdade de Economia, Universidade de Coimbra


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.

JEL Classification:
C60, C62, D01.

smooth preferences, marginal rates of substitution, indifference map, ordinal utility, integrability conditions, differential equations.

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