Consumption Smoothing and Discounting in Infinite-Horizon, Discrete-Choice Problems

Suppose the consumption space is discrete. Our first contribution is a technical result showing that any continuous utility function of any stationary preference relation over infinite consumption streams has convex range, provided that the agent is sufficiently patient. Putting the result to use, we consider a model of endogenous discounting (a generalization of the standard model with geometric discounting) and show the uniqueness of the consumption-dependent discount factor as well as the cardinal uniqueness of utility. Comparative statics are then provided to substantiate the uniqueness. For instance, we show that, as in the more familiar case of an infinitely divisible good, the cardinal uniqueness of utility captures an agent’s desire to smooth consumption over time.