The Optimal Exploitation of an Unknown Reserve

The general problem of the optimal utilization of scarce natural resources has received considerable recent attention in the literature. Most study has been of the case where the level of the resource available to the economy is known with certainty. In those instances where uncertainty has been explicitly introduced, the focus has been on analysing the effect of the existence of a technologically advanced substitute for the scarce resource which will become available at some unknown future date (Dasgupta and Heal (1974)). The more difficult problem of optimal planning when premature exhaustion is a real possibility due to a lack of precise knowledge about the total supply of the resource has received little attention. (Gilbert (1976) is a notable exception.) This paper begins an investigation of that problem. Any treatment of resource utilization with unknown reserves must confront two rather different issues. The first of these is the possibility of exhaustion. In models where the level of the resource is known with certainty, the exhaustion of an " essential" natural resource occurs only asymptotically. When the resource base is of unknown size, however, the date on which it will have been completely exploited is also a random variable. The choice of the rate at which to consume the resource must necessarily be influenced by the effect which the consumption rate has on the probability of exhaustion at subsequent points in time. The second issue is that of learning about the distribution of reserves over time. Even though the level of total reserves may be unknown, it must certainly be the case that the activities of exploration and extraction provide information about the distribution of remaining reserves. Moreover, the nature of this information will be influenced by the exploration and extraction decisions which are made along the way. Thus, an optimal programme of extraction should properly balance both consumption and information benefits against extraction and opportunity costs at each point in time. This paper, while treating the first issue with some generality, will take a rather limited view of the way in which information about the distribution of total reserves is assimilated. We shall assume that the planner begins with a given probability distribution of possible endowments of the natural resource, and that he updates this distribution over time by conditioning on the knowledge of his cumulative consumption at each instant. Thus we will abstract from the activity of exploration. This is a regrettable omission which we hope to correct in a later paper. The treatment of this initial problem seems essential to further progress in any event. The plan of the paper is as follows. We begin with a review of the problem of optimal depletion without production in a certain environment. Variational techniques are then employed to deduce necessary conditions for a solution to the problem of optimal exploitation under uncertainty. It is seen that the possibility of premature exhaustion considerably alters the requirements of an optimal path. We then determine a more complete characterization of an optimum and prove a number of qualitative and comparative static results,