Consumption in incomplete markets

We develop a method to find approximate solutions, and their accuracy, to consumption–investment problems with isoelastic preferences and infinite horizon, in incomplete markets where state variables follow a multivariate diffusion. We construct upper and lower contractions; these are fictitious 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 is logarithmic.