Optimal Supply of a Depletable Resource with a Backstop Technology: Heal's Theorem Revisited

Heal's theorem states that if the extraction cost of a depletable resource increases with cumulative extraction, and if a backstop technology exists, the user cost of the depletable resource declines to zero at the date of exhaustion. In this paper, we first present a simple method for proving this proposition, using a social planning model that determines the optimal rates both of extraction of the depletable resource and of production of the backstop technology. We then present two examples that show how this method can be used to solve more difficult problems in the theory of resource economics. The first example involves learning-by-doing in the backstop sector; that is, backstop costs decline with cumulative production. The second example involves uncertainty of backstop costs.