Optimal pricing,use, and exploration of uncertain natural resource stocks

Units of natural resources, to be called “mines,” are assumed distributed over an unexplored territory according to a Poisson process in space. At any moment, total reserves and unexplored land are given. Society can determine the rate of consumption and the rate of exploration. Reserves are drawn down by consumption and increased by discoveries made during exploration; the amount of unexplored land is decreased by exploration. At any moment, the payoff of society is a concave function of consumption less a linear function of exploration with future payoffs discounted. Optimal policies are considered and, in particular, it is shown for a large amount of unexplored land the shadow prices of reserves and of unexplored land move in random cycles but show only a slight upward tread, thereby casting some light on the failure of mineral prices to rise at the market rate of interest.