  In this paper, we present a mean field game to model the production behaviors
of a very large number of producers, whose carbon emissions are regulated by
government. Especially, an emission permits trading scheme is considered in our
model, in which each enterprise can trade its own permits flexibly. By means of
the mean field equilibrium, we obtain a Hamilton-Jacobi-Bellman (HJB) equation
coupled with a Kolmogorov equation, which are satisfied by the adjoint state
and the density of producers (agents), respectively. Then, we propose a
so-called fitted finite volume method to solve the HJB equation and the
Kolmogorov equation. The efficiency and the usefulness of this method are
illustrated by the numerical experiments. Under different conditions, the
equilibrium states as well as the effects of the emission permits price are
examined, which demonstrates that the emission permits trading scheme
influences the producers' behaviors, that is, more populations would like to
choose a lower rather than a higher emission level when the emission permits
are expensive.
