The Continuity of Optimal Dynamic Decision Rules

Abstract

In recent studies of the temporary competitive equilibrium, agents' current decision correspondences are derived using a standard recursion procedure, which is only applicable when the planning horizon is finite. This paper presents a general derivation of the current decision rule without restrictions on the time horizon or the number of states of the world in any period. It is shown that if utility is continuous in the product topology and if, in each period, expectations and the current constraint correspondence are continuous, then the current decision rule is upper semi-continuous. This result is obtained by associating with each current decision a set of feasible future plans. The expected utility of a current decision is then the expected utility of the best feasible future plan. The feasible future plan correspondence is shown to be continuous and the Maximum Theorem completes the proof.