Undiscounted optimal growth with consumable capital and capital-intensive consumption goods

Abstract

This paper presents a complete characterization of the optimal policy in a two sector undiscounted growth model. The model is an extension of the Leontief two sector model, which analyzes the optimal allocation of capital and labor to a consumption good sector and an investment good sector. The paper extends this framework to include consumable capital. Thus, the planner has preferences over the consumption good and the consumable capital. Future welfare levels are treated equally as current ones. Geometric techniques are applied to characterize the optimal policy if the consumption good is capital-intensive. The results suggest that if the initial capital stock is below a threshold level, that depends upon the consumption of capital, there is no convergence to the golden rule stock. Otherwise, an economy with a low level of capital produces only investment goods and an economy with a high level of capital produces only consumption goods. Depending on the value of the marginal rate of transformation of capital between today and tomorrow, the transition dynamics of middle range countries emerge: a country can take an infinite sequence of monotonic adjustment to the golden rule stock and attains a full utilization of factors each period, or an optimal program exhibits an infinite sequence of nonmonotonic fluctuations.