Grandma's Dress, or What's New for Optimal Growth

Abstract

The recent revival of interest in optimal growth theory justifies the analysis of more flexible descriptions of preferences over time and their implications for optimal growth, in constrat with some of the newer investigations which attribute differing growth paths to tecnological factors. For the case of discrete time, using welfare functions for which the rate of time preference is variable, Beals and Koopmans showed in 1969 that the long run optimal capital path may depend on initial wealth, so that not all of the conclusions of optimal growth theory with a constant rate of time preference hold. Equivalent results for the case of continuous time have been reached by the present author. The analysis by the author of a particular case in which the resulting welfare function can be explicitly represented as an integral, illustrates the qualitative behavior of optimal growth paths. A more thorough analysis shows that a suitable limiting process allows one to define a utility function for continuous time with a variable rate of time preference which cannot be represented in closed form. The present investigation applies such preferences to optimal growth, with results similar to those obtained for the dscrete time case. In particular —when the rate of time preference is allowed to vary— a country may decide not to undertake the effort of economic development when its initial capital endowment is below some critical level. It is impossible to obtain such a result with a constant rate of time preference in the case of a simple neoclassical technology.