Dynamic Programming of Economic Growth

Abstract

In the present paper a class of problems of optimal economic growth is formulated in terms of the ‘functional equation’ approach of dynamic programming (Bellman, 1957).² A study is made of the continuity and concavity properties of the state valuation function, i.e. the function indicating the maximum total discounted welfare (utility) that can be achieved starting from a given initial state of the economy. Under suitable conditions this function is characterized by a certain functional equation. Both the cases of a finite and an infinite planning horizon are treated, the latter case being discussed under the assumption of constant technology and tastes. Here iteration of a certain transformation associated with the functional equation is shown to provide convergence to the state valuation function. Exact solutions are given for the case of linear-logarithmic production and welfare functions.

This research was supported in part by the Office of Naval Research under Contract ONR 222(77) with the University of California. I am indebted to D. Blackwell and W. Gorman for helpful comments on earlier drafts of this paper. In particular, D. Blackwell provided an alternative, and simpler, proof of Theorem 4.3.