Modeling and computation of mean field game with compound carbon abatement mechanisms

Shuhua Zhang; Junying Zhao; Ming Yan; Xinyu Wang

doi:10.3934/jimo.2020121

Article Contents

2021, Volume 17, Issue 6: 3333-3347. Doi: 10.3934/jimo.2020121

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Modeling and computation of mean field game with compound carbon abatement mechanisms

1.
Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
2.
Department of Mathematics, Tianjin University of Commerce, Tianjin 300134, China

^* Corresponding author: Junying Zhao
^* Corresponding author: Junying Zhao

Received: October 31, 2019

Revised: February 29, 2020

Early access: June 2020

Published: November 2021

This project was supported in part by the National Basic Research Program (2012CB955804), the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322), the Major Program of Tianjin University of Finance and Economics (ZD1302), Tianjin Philosophy and Social Science Planning Project (TJGLQN18-005), and Tianjin Science and Technology Development Strategic Research Planning Project (18ZLZXZF00130)

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Abstract

In this paper, we present a mean field game to model the impact of the coexistence mechanism of carbon tax and carbon trading (we call it compound carbon abatement mechanism) on the production behaviors for a large number of producers. The game's equilibrium can be presented by a system which is composed of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman (HJB) partial differential equation. An implicit and fractional step finite difference method is proposed to discretize the resulting partial differential equations, and a strictly positive solution is obtained for a non-negative initial data. The efficiency and the usefulness of this method are illustrated through the numerical experiments. The sensitivity analysis of the parameters is also carried out. The results show that an agent under concentrated carbon emissions tends to choose emission levels different from other agents, and the choices of agents with uniformly distributed emission level will be similar to their initial distribution. Finally, we find that for the compound carbon abatement mechanism carbon tax has a greater impact on the permitted emission rights than carbon trading price does, while carbon trading price has a greater impact on carbon emissions than carbon tax.

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Mathematics Subject Classification: Primary: 35Q91, 49L20, 65L60.

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Figure 1. Computed errors in the $ L^{\infty} $-norm at $ t = 0 $

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Figure 2. Evolution of normal distribution

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Figure 3. Evolution of the second distribution

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Figure 4. Evolution of uniform distribution

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Figure 5. Evolution of $ \bar{E}(x) $ and $ \bar{E}(E) $

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