Non-Smooth Integrability Theory

TL;DR

This paper develops a new method for deriving utility functions from non-differentiable demand functions, providing necessary and sufficient conditions, and exploring implications for econometric theory.

Contribution

It introduces novel conditions involving the Slutsky matrix and a PDE for demand functions, and establishes the uniqueness and continuity properties of the utility function.

Findings

Demand functions with Lipschitz continuity can be characterized by new conditions.

The space of such demand functions is compact under certain metrics.

The utility function mapping is continuous in relevant topologies.

Abstract

We study a method for calculating the utility function from a candidate of a demand function that is not differentiable, but is locally Lipschitz. Using this method, we obtain two new necessary and sufficient conditions for a candidate of a demand function to be a demand function. The first concerns the Slutsky matrix, and the second is the existence of a concave solution to a partial differential equation. Moreover, we show that the upper semi-continuous weak order that corresponds to the demand function is unique, and that this weak order is represented by our calculated utility function. We provide applications of these results to econometric theory. First, we show that, under several requirements, if a sequence of demand functions converges to some function with respect to the metric of compact convergence, then the limit is also a demand function. Second, the space of demand functions that have uniform Lipschitz constants on any compact set is compact under the above metric. Third, the mapping from a demand function to the calculated utility function becomes continuous. We also show a similar result on the topology of pointwise convergence.