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. https://arxiv.org/abs/2407.12754

[ joint work with Gianmarco Del Sarto, Marta Leocata, Giulia Livieri ]

Bio: Giulia was Assistant Professor at Scuola Normale Superiore (SNS) from February 2020 to November 2022. Previously she was a Post-doc researcher at SNS, where she obtained a Ph.D in Financial Mathematics in October 2017 with the score of 70/70 with laude. In 2013, Giulia did a post-graduate course in Mathematical Finance at the University of Bologna where she obtained a score of 30/30 with laude and an internship at Mediobanca a leading investment bank in Italy. Giulia graduated in 2012 in Mathematics at the University of Padova with the score of 100/100. Giulia's research focuses mainly on financial econometrics and stochastic analysis for the modelling of financial markets (both at high and low frequency) and Mean-Field Game (MFG). She is currently developing machine learning techniques for the standard memory, forecasting, and filtered problems that appear the parametric stochastic time series context. Also, Giulia aims to provide mathematical foundation based on the theory of MFGs to implement Deep Neural Network (DNN) models.