  We study a class of deterministic mean field games on finite and infinite
time horizons arising in models of optimal exploitation of exhaustible
resources. The main characteristic of our game is an absorption constraint on
the players' state process. As a result of the state constraint the optimal
time of absorption becomes part of the equilibrium. This requires a novel
approach when applying Pontyagin's maximum principle. We prove the existence
and uniqueness of equilibria and solve the infinite horizon models in closed
form. As players may drop out of the game over time, equilibrium production
rates need not be monotone nor smooth.
