  We develop a model based on mean-field games of competitive firms producing
similar goods according to a standard AK model with a depreciation rate of
capital generating pollution as a byproduct. Our analysis focuses on the
widely-used cap-and-trade pollution regulation. Under this regulation, firms
have the flexibility to respond by implementing pollution abatement, reducing
output, and participating in emission trading, while a regulator dynamically
allocates emission allowances to each firm. The resulting mean-field game is of
linear quadratic type and equivalent to a mean-field type control problem,
i.e., it is a potential game. We find explicit solutions to this problem
through the solutions to differential equations of Riccati type. Further, we
investigate the carbon emission equilibrium price that satisfies the market
clearing condition and find a specific form of FBSDE of McKean-Vlasov type with
common noise. The solution to this equation provides an approximate equilibrium
price. Additionally, we demonstrate that the degree of competition is vital in
determining the economic consequences of pollution regulation.
