Equilibrium control theory for Kihlstrom-Mirman preferences in continuous time

TL;DR

This paper develops an equilibrium control framework for continuous-time models with Kihlstrom-Mirman preferences, addressing dynamic inconsistency through a game-theoretic approach and solving associated PDEs.

Contribution

It introduces a novel equilibrium control theory for continuous-time Markov processes with Kihlstrom-Mirman preferences, linking utility theory with PDE-based solution methods.

Findings

Formulation of equilibrium strategies via Hamilton-Jacobi-Bellman equations

Verification of the PDE system as a sufficient condition for equilibrium

Application to a consumption-investment problem with CRRA-CES utility

Abstract

In intertemporal settings, the multiattribute utility theory of Kihlstrom and Mirman suggests the application of a concave transform of the lifetime utility index. This construction, while allowing time and risk attitudes to be separated, leads to dynamically inconsistent preferences. We address this issue in a game-theoretic sense by formalizing an equilibrium control theory for continuous-time Markov processes. In these terms, we describe the equilibrium strategy and value function as the solution of an extended Hamilton-Jacobi-Bellman system of partial differential equations. We verify that (the solution of) this system is a sufficient condition for an equilibrium and examine some of its novel features. A consumption-investment problem for an agent with CRRA-CES utility showcases our approach.