On optimal cycles in dynamic programming models with convex return function

Abstract

In this paper we study the behavior of optimal paths in dynamic programming models with a strictly convex return function. Such a model has been investigated in Dawid and Kopel (1997) who assume that the growth of a renewable resource is governed by a piecewise linear function. We prove that in their model the optimal cycles undergo the following qualitative changes or bifurcations: a cycle of period n "bifurcates" into a cycle of period n+1 for increasing elasticity of the return function. We also show that under the assumption of a concave differentiable growth function the qualitative properties of the optimal policy remain valid: oscillating behavior is optimal. Furthermore, we demonstrate numerically that the period of a cyclic optimal path increases if the convexity of the return function (measured by the elasticity) increases.