The Optimal Production of an Exhaustible Resource When Price is Exogenous and Stochastic

Abstract

This paper examines the optimal production of a resource such as oil when its price is determined exogenously (e.g. by a cartel such as OPEC), and is subject to stochastic fluctuations away from an expected growth path. We first examine the dependence of production on extraction cost, and show that the conventional exponential decline curve is indeed optimal if marginal cost is constant with respect to the rate of extraction but is a hyperbolic function of the reserve level. We next show that uncertainty about future price affects the optimal production rate in two ways. First, if marginal cost is a convex (concave) function of the rate of production, stochastic fluctuations in price raise (lower) average cost over time, so that there is an incentive to speed up (slow down) production. Second, the “option” value of the reserve, i.e. the ability to withhold production indefinitely and never incur the cost of extraction, provides an incentive to slow down the rate of production.

Research leading to this paper was supported by the National Science Foundation under Grant No. SES-8012667, and this support is gratefully acknowledged. The author also wishes to thank Robert Merton for helpful comments and suggestions.