Optimal Exploration of an Exhaustible Resource with Stochastic Discoveries

The standard Hotelling model assumes that the stock of an exhaustible resource is known. We expand on the model by Arrow and Chang that introduced stochastic discoveries and for the first time completely solve such a model using impulse control. The model has two state variables: the "proven" reserves as well as a finite unexplored area available for exploration with constant marginal cost, resulting in a Poisson process of new discoveries. We prove that a frontier of critical levels of "proven" reserves exists, above which exploration is stopped, and below which it happens at infinite speed. This frontier is increasing in the explored area, and higher "proven" reserve levels along this critical threshold are indicative of more scarcity, not less. In this stochastic generalization of Hotelling's rule, the expected shadow price of reserves rises at the rate of interest across exploratory episodes. However, the actual trajectories of prices realized prior to exhaustion of the exploratory area may jump up or down upon exploration. Conditional on non-exhaustion, expected price arises at a rate bounded above by the rate of interest, consistent with most empirical tests based on observed price histories.