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Mathematical Modeling of Consumer's Preferences Using Partial Differential Equations

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  • Jorge Marques

    (Faculty of Economics, University of Coimbra, Portugal)

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    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.

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    Bibliographic Info

    Paper provided by GEMF - Faculdade de Economia, Universidade de Coimbra in its series GEMF Working Papers with number 2013-15.

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    Length: 27 pages
    Date of creation: May 2013
    Date of revision:
    Handle: RePEc:gmf:wpaper:2013-15.

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    Keywords: smooth preferences; marginal rates of substitution; indifference map; ordinal utility; integrability conditions; differential equations.;

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    1. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, Oxford University Press, number 9780195102680, October.
    2. Debreu, Gerard, 1972. "Smooth Preferences," Econometrica, Econometric Society, Econometric Society, vol. 40(4), pages 603-15, July.
    3. Birendra K. Rai & Chiu Ki So & Aaron Nicholas, 2012. "A Primer On Mathematical Modelling In Economics," Journal of Economic Surveys, Wiley Blackwell, Wiley Blackwell, vol. 26(4), pages 594-615, 09.
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