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Mathematical Modeling of Consumer's Preferences Using Partial Differential Equations


  • Jorge Marques

    (Faculty of Economics, University of Coimbra, Portugal)


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.

Suggested Citation

  • Jorge Marques, 2013. "Mathematical Modeling of Consumer's Preferences Using Partial Differential Equations," GEMF Working Papers 2013-15, GEMF, Faculty of Economics, University of Coimbra.
  • Handle: RePEc:gmf:wpaper:2013-15.

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    References listed on IDEAS

    1. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, December.
    2. Debreu, Gerard, 1972. "Smooth Preferences," Econometrica, Econometric Society, vol. 40(4), pages 603-615, July.
    3. Birendra K. Rai & Chiu Ki So & Aaron Nicholas, 2012. "A Primer On Mathematical Modelling In Economics," Journal of Economic Surveys, Wiley Blackwell, vol. 26(4), pages 594-615, September.
    4. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
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    More about this item


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

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D01 - Microeconomics - - General - - - Microeconomic Behavior: Underlying Principles

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