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Optimal Consumption and Portfolio Rules with Durability and Local Substitution

Ayman Hindy
Chi-fu Huang

Abstract

We study a model of optimal consumption and portfolio choice which captures, in two different interpretations, the notions of local substitution and irreversible purchases of durable goods. The class of preferences we consider excludes all nonlinear time-additive and nearly all the non-time-additive utility functions used in the literature. We discuss heuristically necessary conditions and provide sufficient conditions for a consumption and portfolio policy to be optimal. Furthermore, we demonstrate our general theory by solving in a closed form the optimal consumption and portfolio policy for a particular felicity function when the prices of the assets follow a geometric Brownian motion process. The optimal consumption policy in our solution consists of a possible initial "gulp" of consumption, or a period of no consumption, followed by a process of cumulative consumption with singular sample paths. In almost all states of nature, the agent consumes periodically and invests more in the risky assets than an agent with time-additive utility whose felicity function has the same curvature and the same time-discount parameter. We compute the equilibrium risk premium in a representative investor economy with a single physical production technology whose rate of return follows a Brownian motion. In addition, we provide some simulation results that demonstrate the properties of the purchase series for durable goods with different half-lives.