An Optimal Growth Model with Time Lags

Abstract

The paper discusses the allocation of output among consumption and two types of capital with different gestation periods. Along an optimal path, we show that the imputed prices of capital goods, from the time they start production, do not exceed the prices of output, which are not less than the marginal instantaneous utility of consumption. A simple numerical example helps to illustrate some further implications of the model. RECENT PAPERS on optimal growth consider models of allocation of resources between consumption and investment. It is invariably assumed that investment results in an instantaneous increase in the stock of capital. Such assumptions obscure differences in the gestation periods among various capital goods. In [1] we discuss how a growth problem with time lags can be formulated and interpreted and explain the derivation of the necessary conditions for optimization. In this note we study the effects of differences in gestation periods on optimal investment plans for a growth problem including depreciation and population growth. Consider an economy where two capital goods and labor are used in the production of a single commodity. The per capita output at time t is given by the production function: f (kl, k2), where k, is the per capita stock of capital of type one, and k2 is the per capita stock of capital of type two. From now on, all variables will be per capita and we drop the designation. We assume the following: