Optimal growth with depletable resources

Abstract

Optimal growth trajectories for Cobb-Douglas economies facing a depletable resource constraint and possessing a utilitarian objective function are examined. It is shown that for a large enough discount factor, there exists a locus of initial ‘balanced endowments’ from which the ‘balanced endowments trajectory’ is characterized by linear closed-loop optimal controls and exponentially growing or decaying economic activity. The rate of technical progress requisite to assure continual per capita economic growth is derived. Optimal solutions from arbitrary endowments are characterized. If a balanced endowments trajectory exists, then the optimal path from arbitrary initial capital and resource stocks converges monotonically to a balanced endowments trajectory. Initial endowments of capital and the resource both influence the long-run levels of economic activity, although not the long-run growth rates. The comparative magnitude of the elasticity of marginal utility and capital's share of output are critical in determining whether a larger initial capital stock leads to greater or smaller levels of long-run economic activity, but increases in the resource stock always lead to long-run increases in economic activity.