Titre du working paper : Optimal Exploration and Price Paths of a Non-renewable Commodity with Stochastic Discoveries
We address the long-standing challenge of adding optimal exploration to the classic Hotelling model of a non-renewable resource. The model we use, extending Arrow and Chang (1982), has two state variables: “proven” reserves and a finite area of resources available for exploration at constant marginal cost, with a Poisson process for new discoveries, and two controls, consumption (as opposed to storage) of the reserves already discovered, and exploration for more reserves. Exploration causes considerable mathematical difficulties; once started, the decision on whether to stop is made after the next discovery, which might occur after covering a large area of resources, rather than in the immediate neighbourhood of where exploration started. We prove that a frontier of critical levels of proven reserves exists, above which exploration ceases, and below which it proceeds at infinite speed. This frontier is increasing in explored area, and higher reserve levels along this critical threshold indicate more scarcity, not less. In a stochastic generalization of Hotelling’s rule, the path of the expected shadow price of reserves rises at the rate of interest as the horizon recedes, but it shifts after exploration. Conditional on non-exhaustion, expected price does not follow Hotelling’s rule, explaining its rejection in empirical tests based on price histories.
