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Growth model with externalities for energetic transition via MFG with common external variable

Pierre Lavigne, Quentin Petit, Xavier Warin

TL;DR

This paper develops a novel mean-field game framework for multi-sector economic growth with a dynamically evolving externality driven by aggregate actions and exposed to common noise. The authors formulate the equilibrium via a stochastic maximum principle, recast it as a forward–backward SDE, and establish existence and uniqueness of a strong MFG equilibrium through a contraction argument; they also derive weak-equilibrium results and discuss a master-equation perspective. Computationally, they propose a fixed-point algorithm augmented with neural-network approximations to solve the resulting high-dimensional system, incorporating a fictitious-time scheme for stability. The numerical study on a two-sector brown/green model illustrates how externalities shape investment, capital dynamics, and a transition toward greener technologies under shared uncertainty. Overall, the work contributes to integrated assessment-type modeling by linking dynamic externalities, common noise, and MFG theory, with practical implications for policy design under environmental uncertainty.

Abstract

This article introduces a novel mean-field game model for multi-sector economic growth in which a dynamically evolving externality, influenced by the collective actions of agents, plays a central role. Building on classical growth theories and integrating environmental considerations, the framework incorporates common noise to capture shared uncertainties among agents about the externality variable. We demonstrate the existence and uniqueness of a strong mean-field game equilibrium by reformulating the equilibrium conditions as a Forward-Backward Stochastic Differential Equation under the stochastic maximum principle and establishing a contraction argument to ensure a unique solution. We provide a numerical resolution for a specified model using a fixed-point approach combined with neural network approximations.

Growth model with externalities for energetic transition via MFG with common external variable

TL;DR

This paper develops a novel mean-field game framework for multi-sector economic growth with a dynamically evolving externality driven by aggregate actions and exposed to common noise. The authors formulate the equilibrium via a stochastic maximum principle, recast it as a forward–backward SDE, and establish existence and uniqueness of a strong MFG equilibrium through a contraction argument; they also derive weak-equilibrium results and discuss a master-equation perspective. Computationally, they propose a fixed-point algorithm augmented with neural-network approximations to solve the resulting high-dimensional system, incorporating a fictitious-time scheme for stability. The numerical study on a two-sector brown/green model illustrates how externalities shape investment, capital dynamics, and a transition toward greener technologies under shared uncertainty. Overall, the work contributes to integrated assessment-type modeling by linking dynamic externalities, common noise, and MFG theory, with practical implications for policy design under environmental uncertainty.

Abstract

This article introduces a novel mean-field game model for multi-sector economic growth in which a dynamically evolving externality, influenced by the collective actions of agents, plays a central role. Building on classical growth theories and integrating environmental considerations, the framework incorporates common noise to capture shared uncertainties among agents about the externality variable. We demonstrate the existence and uniqueness of a strong mean-field game equilibrium by reformulating the equilibrium conditions as a Forward-Backward Stochastic Differential Equation under the stochastic maximum principle and establishing a contraction argument to ensure a unique solution. We provide a numerical resolution for a specified model using a fixed-point approach combined with neural network approximations.
Paper Structure (36 sections, 17 theorems, 157 equations, 4 figures, 1 table)

This paper contains 36 sections, 17 theorems, 157 equations, 4 figures, 1 table.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Contributions.
  4. Structure of the paper.
  5. Notations and Toolbox
  6. Notations
  7. Spaces of random variables and random processes.
  8. Toolbox
  9. SDE.
  10. BSDE.
  11. Technological investment model
  12. Stochastic context (S).
  13. Representative country
  14. Capital dynamics.
  15. Investment.
  16. ...and 21 more sections

Key Result

Theorem 2.1

There exists a unique solution $X \in S^2(\mathbb{F}, \mathbb{R}^d)$ to equation abstract-SDE.

Figures (4)

  • Figure 1: Two realisations of the pollution process.
  • Figure 2: Distributions of the investment $a_t$ (on the left) and the capital $k_t$ (on the right) for time 5, 10, and 20.
  • Figure 3: Distributions of the investment $a_t$ (on the left) and of the capital $k_t$ (on the right) for time 20, 30, and 45.
  • Figure 4: Quantiles 5%, 95% and mean of the pollution process

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Example 3.1
  • Remark 3.1
  • Example 3.2
  • Theorem 4.1
  • proof
  • Remark 4.1
  • Proposition 4.1
  • ...and 29 more