Consumption in Incomplete Markets

Abstract

We develop a method to find approximate solutions, and their accuracy, to consumption-investment problems with isoelastic preferences and innite horizon, in incomplete markets where state variables follow a multivariate diffusion. We construct upper and lower contractions, fictitious complete markets in which state variables are fully hedgeable, but their dynamics is distorted. Such contractions yield pointwise upper and lower bounds for both the value function and the optimal consumption of the original incomplete market, and their optimal policies are explicit in typical models. Approximate consumption-investment policies coincide with the optimal one if the market is complete or utility logarithmic. An application to an incomplete market with stochastic excess returns, risk premia, and volatilities, shows that the welfare loss of the proposed approximations is equivalent to a capital loss of less than 3% for realistic parameter values.

Keywords: portfolio choice, consumption, incomplete markets, power utility

JEL Classification: G11, G12