Exponential utility with non-negative consumption

TL;DR

This paper develops a mathematically tractable framework for exponential utility with non-negative consumption constraints, providing explicit equilibrium characterizations, multiple equilibrium examples, and asymptotic bond analysis.

Contribution

It introduces a novel approach to exponential utility with non-negative consumption constraints, including explicit solutions and equilibrium analysis in both complete and incomplete markets.

Findings

Explicit equilibrium characterizations in complete markets.

Existence of multiple and infinitely many equilibria.

Asymptotic analysis of zero coupon bonds using large deviations.

Abstract

We offer mathematical tractability and new insights for a framework of exponential utility with non-negative consumption, a constraint often omitted in the literature giving rise to economically unviable solutions. Specifically, using the Kuhn-Tucker theorem and the notion of aggregate state price density (Malamud and Trubowitz (2007)), we provide a solution to this problem in the setting of both complete and incomplete markets (with random endowments). Then, we exploit this result to provide an explicit characterization of complete market heterogeneous equilibria. Furthermore, we construct concrete examples of models admitting multiple (including infinitely many) equilibria. By using Cramer's large deviation theorem, we study the asymptotics of equilibrium zero coupon bonds. Lastly, we conduct a study of the precautionary savings motive in incomplete markets.