  We establish the existence and uniqueness of a solution to the master
equation for a mean field game of controls with absorption. The mean field game
arises as a continuum limit of a dynamic game of exhaustible resources modeling
Cournot competition between producers. The proof relies on an analysis of a
forward-backward system of nonlocal Hamilton-Jacobi/Fokker-Planck equations
with Dirichlet boundary conditions. In particular, we establish new a priori
estimates to prove that solutions are differentiable with respect to the
initial measure.
