  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.
