A Mean Field Game approach for pollution regulation of competitive firms

TL;DR

This paper models pollution regulation for competitive firms using mean-field game theory, deriving explicit solutions and analyzing the impact of competition on economic outcomes under cap-and-trade systems.

Contribution

It introduces a linear quadratic mean-field game model for pollution regulation, providing explicit solutions and analyzing equilibrium prices and competition effects.

Findings

Explicit solutions via Riccati differential equations.

Derived a McKean-Vlasov FBSDE for equilibrium pricing.

Showed competition level influences pollution regulation outcomes.

Abstract

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.