Optimal Consumption and Portfolio Rules with Local Substitution

We study the problem of optimal consumption and portfolio choice for a class of utility functions that capture the notion that consumptions at nearby dates are almost perfect substitutes. The class we consider excludes all time additive and almost all the non time additive utility functions used in the literature. We provide necessary and 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 accumulated 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.