Low Discounting and the Upper Long-Run Average Value in Dynamic Programming

Abstract

Consider a dynamic programming problem, where the discounted value functions converge to a limit function as the discount factor tends to 1. It is proved that the limit function must be the Markov upper long-run average value function, if the convergence holds in the weak topology on the space of all bounded measurable functions on the state space. Necessary and sufficient conditions for the existence of the weak limit are given. The results are applied to compact continuous dynamic programming problems used extensively in economics. Journal of Economic Literature Classification Numbers: C72, C73.