Resource Extraction when a Future Substitute has an Uncertain Cost

In the last few years several economists have treated various aspects of uncertainty in the context of natural resource extraction. In particular, Dasgupta and Heal (1974) and Dasgupta and Stiglitz (1976) have studied cases where a perfect substitute for an exhaustible resource will become available at some future date. Both of these studies assume that this future date is uncertain, but that the characteristics of the substitute, represented by the costs of producing it, are known with certainty. The present paper analyses the opposite case: the date of availability is assumed to be known, but the costs of producing the substitute are uncertain. In Section 2 the model is presented, and Section 3 shows how uncertainty affects the socially optimal resource extraction under various assumptions about the society's risk aversion. In Section 4 the competitive solution is derived and compared with the social optimum, before some concluding comments are given in Section 5. 2. THE MODEL Let x(t) and y(t) be the rate of resource extraction and of production of a perfect substitute, respectively. The date when the substitute will become available, which is known at the time of planning (t = 0) is denoted by T. We shall disregard extraction costs, and denote the unit cost of producing the substitute by c. This cost is not known with certainty at t = 0, but we assume that there exists a probability distribution of c (consisting only of positive values) which all agents in the economy agree upon. No new information about the true value of c becomes available before T, but at T the true value of c is revealed. The social optimization problem is to maximize EW (= the expected value of W), where