Differentiability Properties of a Parametric Consumer Problem

TL;DR

This paper analyzes the mathematical properties of the budget map and indirect utility function in a consumer problem within a Banach space, using advanced variational analysis tools to establish differentiability and subdifferential formulas.

Contribution

It provides new differentiability and subdifferential formulas for the budget map and indirect utility function in a Banach space setting, extending previous variational analysis methods.

Findings

Established Lipschitz-likeness of the budget map

Derived formulas for subdifferentials of the nuisance function

Connected subdifferential estimates to economic interpretations

Abstract

We study the budget map and the indirect utility function of a parametric consumer problem in a Banach space setting by some advanced tools from set-valued and variational analysis. The Lipschitz-likeness and differentiability properties of the budget map, as well as formulas for finding subdifferentials of the infimal nuisance function, which is obtained from the indirect utility function by changing its sign, are established. Our investigation is mainly based on the paper by Mordukhovich [J. Global Optim. 28 (2004), 347--362] on coderivative analysis of variational systems and the paper of Mordukhovich, Nam, and Yen [Math. Program. 116 (2009), 369--396] on subgradients of marginal functions. Economic meanings of the obtained subdifferential estimates are explained in details.