On optimal growth under uncertainty

In a recent paper Brock and Mirman showed that in a one-sector model of economic growth under uncertainty the long-run behavior of the optimal capital stock is governed by the basic properties of an acyclic ergodic Markov process. This paper considers a similar model and has two purposes. First, necessary and sufficient conditions for optimal policy functions are derived in a regime in which future utilities are discounted. These conditions lead, in an example, to an explicit optimal policy function, which is used to display the steady-state solution for the capital stock under an optimal policy. Secondly, in the Brock and Mirman paper it was assumed that the production functions are ordered. We show that all the properties proved by Brock and Mirman are satisfied even when the production functions are not ordered.