Paper: arXiv 2407.12754

Authors: Gianmarco Del Sarto, Marta Leocata, Giulia Livieri

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly rigorous mathematical framework applying Mean Field Games and McKean-Vlasov FBSDEs with common noise to cap-and-trade pollution regulation. However, it is primarily a theoretical and numerical simulation study lacking empirical backtesting or real-world dataset validation.

Research Flowchart

  flowchart TD
    A[Goal: Model Pollution Regulation in Competitive Firms via MFG] --> B[Model Formulation: AK Capital Model with Pollution & Cap-and-Trade]
    B --> C[Problem Transformation: Potential Game Equivalent to Mean-Field Control]
    C --> D[Computational Analysis: Solve Riccati ODEs & McKean-Vlasov FBSDE with Common Noise]
    D --> E[Key Finding: Explicit MFG Solutions & Approximate Emission Equilibrium Price]
    D --> F[Key Finding: Degree of Competition Critically Impacts Economic Outcomes]