Optimum Accumulation Under Uncertainty: the Case of Stationary Returns to Investment

Abstract

In the theory of optimum growth it has been found that models with discrete time are easier to treat rigorously than models with continuous time. But continuous-time models often have the advantage of providing simpler results. I shall illustrate this tension in the present paper by discussing the model for optimum growth under uncertainty that has received most attention in the literature (Phelps [6], Levhari and Srinivasan [4], Hahn [2], Hakansson [3], Brock and Mirman [1]). An existence theorem will be proved for the discrete-time case. By a heuristic argument, I obtain an equation for the optimum under continuous-time which makes possible results about the effects of uncertainty on the optimum policy more general than are available in discrete time. These latter results are somewhat surprising. By way of prelude I outline the reasons for research into optimum growth under uncertainty, and offer a classification of models. The model discussed in this paper is less appealing than some others; but it seems to be the easiest one.

This is a revised version of the first part of ‘Optimum Growth and Uncertainty’, May 1972 version. The first three sections formed part of a paper prepared for the Bergen workshop. Section IV was developed during the workshop.