Mathematical Modeling of Consumer's Preferences
Using Partial Differential Equations
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.