Competitive Price Behavior of an Exhaustible Resource Where the Rate of Substitution is Constrained

The theory of price behavior of an exhaustible natural resource in a competitive market is now well known [1-5]. The approach taken by Nordhaus [3] is to relate the price of the resource to the cost of producing a perfect substitute from an alternative process, which he calls a However, he makes the simplifying assumption that capital in this process can be accumulated instantaneously when the natural resource is exhausted. In this paper the following question is addressed: To what extent would a more realistic assumption on capital accumulation affect the price behavior of the natural resource. One approach to this question is to defined an increasing, strictly convex investment cost function, Ct=O(lt), which includes adjustment costs. Then investment It is determined to equate the marginal cost 0'(1t) with the value of investment in the backstop technology. However, another approach is to introduce directly a constraint on the rate of investment, It<?M, where 1,, is a conlstant upper bound. This constraint implies that production capacity for investment ind the backstop technology is fixed; capacity does not adjust in the presence of abnormally high profits. These profits presumably accrue to the factors in short supply (e. g., skilled construction workers, specialized materials). The latter approach is taken here largely because it simplifies the analysis and allows more explicit results. It is shown that there is a qualitative difference in price behavior when the rate of investment in the backstop technology is constrained. The maximum price which the natural resource attains will no longer be bounded by the cost of production using the backstop technology.2 That is, the price of the natural resource must reach the cost of production from the backstop technology before the time of exhaustion in order to build capacity in advance. If demand is perfectly inelastic, price will make a discontinuous drop when the resource is exhausted. If demand is not perfectly inelastic, price will be continuous, but capacity in the