  A price-maker company extracts an exhaustible commodity from a reservoir, and
sells it instantaneously in the spot market. In absence of any actions of the
company, the commodity's spot price evolves either as a drifted Brownian motion
or as an Ornstein-Uhlenbeck process. While extracting, the company affects the
market price of the commodity, and its actions have an impact on the dynamics
of the commodity's spot price. The company aims at maximizing the total
expected profits from selling the commodity, net of the total expected
proportional costs of extraction. We model this problem as a two-dimensional
degenerate singular stochastic control problem with finite fuel. To determine
its solution, we construct an explicit solution to the associated
Hamilton-Jacobi-Bellman equation, and then verify its actual optimality through
a verification theorem. On the one hand, when the (uncontrolled) price is a
drifted Brownian motion, it is optimal to extract whenever the current price
level is larger or equal than an endogenously determined constant threshold. On
the other hand, when the (uncontrolled) price evolves as an Ornstein-Uhlenbeck
process, we show that the optimal extraction rule is triggered by a curve
depending on the current level of the reservoir. Such a curve is a strictly
decreasing $C^{\infty}$-function for which we are able to provide an explicit
expression. Finally, our study is complemented by a theoretical and numerical
analysis of the dependency of the optimal extraction strategy and value
function on the model's parameters.
