The Optimal Consumption Stream of an Agent Exposed to Uninsurable Idiosyncratic Risk

We directly construct (no hidden approximations!) the optimal consumption stream of an agent exposed to an arbitrary, uninsurable idiosyncratic risk process in the context of an incomplete market that is a generalization of the classic Constantinides and Duffie (1996). We exploit our construction to derive a number of important properties of the optimal stream. In particular, we show that the expected future consumption is a convex function of the idiosyncratic risk and is monotone increasing in the strength of idiosyncratic risk. For the special case of CRRA utility functions, the growth rate of consumption is also (sufficiently near zero) a monotone increasing function of idiosyncratic risk. Finally, an effective risk aversion and an effective discount factor are introduced and shown to be always larger than the conventional ones