On Optimal Depletion of Exhaustible Resources: Existence and Characterization Results

Abstract

IN RECENT YEARS, there has developed a significant literature on the economics of natural resources. A part of this literature is concerned with finding the characteristics of optimal programs for economies with exhaustible resource constraints; that is, of jointly determining the optimal depletion of such resources, and optimal investment in augmentable capital goods. It should be noted that the theory of optimal economic growth, in the form given it by Ramsey [17], and extensively developed by many others, had been primarily concerned with the latter problem. Natural resources were often assumed to be supplied exogenously in given amounts in each period, an approach clearly unsuitable for capturing the essence of problems associated with the optimal use of exhaustible resources. The more recent literature derives its inspiration from Hotelling's classic paper [8], stressing the increasing significance of such resources in production, as they are irrevocably run down. I shall consider, in this paper, a model of intertemporal allocation in which there is a produced good (which can be used for consumption or for further production), and an exhaustible resource (which is essential for production), the total initial stock of which is given. The use of the resource over the (infinite) planning horizon must not exceed this available stock. A planner is assumed to evaluate consumption in each period, in terms of a utility function, and to "maximize" the undiscounted sum of these one-period utilities, to obtain, simultaneously, the optimal depletion of the exhaustible resource, and the optimal investment 2 pattern. This model resembles, in some aspect or other, the frameworks examined by Dasgupta [3], Dasgupta and Heal [4, 5], Solow [18], and Stiglitz [19], to mention only a few. I will address three sets of issues in this framework. First, an interesting problem in the theory of optimal economic growth is to find suitable conditions under which a competitive program is optimal. I will show that a feasible program is optimal if and only if (a) it is competitive,3 and (b) it satisfies the transversality condition, that the value of the capital and resource stocks, at the competitive prices, converges to zero (Theorem 3.1). It follows from this result, that a competitive program is optimal if and only if it satisfies the above-mentioned transversality condition. It should be noted that in traditional models of optimal growth (in which exhaustible resources do not appear as essential factors of production), a feasible program, under certain technological curvature conditions, is shown to be optimal if and only if it is competitive, and the value of input