Extraction Costs in the Theory of Exhaustible Resources

This paper has two objectives. First, it reformulates the theory of optimal use of an exhaustible resource with more attention to the costs of extraction than has been customary in the literature. The output and shadow-price implications of optimal extraction are studied under these broader assumptions. Second, the paper provides some numerical solutions of a simple two-grade case, to give some feeling for the quantitative importance of changes in the supply of exhausti-ble resources. Our most striking result is, in fact, the suggestion that relatively large changes in resource availability generate very small changes in the sustainable level of finalconsumption. l The newer aggregative literature on the socially optimal manage-ment of a given pool of exhaustible resources has ignored one impor-tant characteristic of natural resources: the differential quality of mineral deposits. This note is intended to fill the gap by offering a treatment of differential extraction costs in the context of an aggrega-tive model. We embed the analysis of differential quality in one particular formulation of the macroeconomic problem of the optimal exploitation of exhaustible resources; but we think it would carry over straightforwardly to other ways of setting up the problem. l The basic economic model is that of Solow (1974), so it need only be sketched here. There is a single produced commodity, whose output (Q) can be either consumed directly (C) or accumulated as a stock of reproducible capital (K). Output is produced under constant returns to scale according to a well-behaved production function Q = F(K,R,L), whose inputs are the services of capital and labor and the using-up of resources. Population and employment (L) are con-stant, so-given constant returns to scale-we might as well nor-malize to L = 1, after which Q, K, C, and R can be thought of as output, capital, consumption, and resource use per head. R, as men-tioned, is the flow of the single natural resource into production. For reasons given in Solow (1974), we take F(K,R,L) to be the Cobb-