# American Institute of Mathematical Sciences

November  2021, 17(6): 3333-3347. doi: 10.3934/jimo.2020121

## 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

Received  November 2019 Revised  March 2020 Published  November 2021 Early access  June 2020

Fund Project: 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)

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.

Citation: Shuhua Zhang, Junying Zhao, Ming Yan, Xinyu Wang. Modeling and computation of mean field game with compound carbon abatement mechanisms. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3333-3347. doi: 10.3934/jimo.2020121
##### References:
 [1] Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control. Optim., 50 (2012), 77-109.  doi: 10.1137/100790069. [2] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477. [3] Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.  doi: 10.1137/120882421. [4] R. S. AVi-Yonah and D. M. Uhlmann, Combating global climate change: Why a carbon tax is a better response to global warming than cap and trade, Stanford Environ. Law J., 28 (2009), 3-50. [5] F. Bagagiolo and D. Bauso, Mean-field games and dynamic demand management in power grids, Dyn. Games Appl., 4 (2014), 155-176.  doi: 10.1007/s13235-013-0097-4. [6] A. Baranzini, J. Goldemberg and S. Speck, A future for carbon taxes, Ecol. Econ., 32 (2000), 395-412.  doi: 10.1016/S0921-8009(99)00122-6. [7] A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7. [8] A. Bruvoll and B. M. Larsen, Greenhouse gas emissions in Norway: Do carbon taxes work?, Energ. Policy, 32 (2004), 493-505. [9] K. Chang, S. Wang and K. Peng, Mean reversion of stochastic convenience yields for CO$_2$ emissions allowances: Empirical evidence from the EU ETS, Spanish Rev. Financ. Econ., 11 (2013), 39-45. [10] S. H. Chang, X. Y. Wang and A. Shanain, Modeling and computation of mean field equilibria in producers' game with emission permits trading, Commun Nonlinear Sci. Numer. Simulat., 37 (2016), 238-248.  doi: 10.1016/j.cnsns.2016.01.020. [11] S. H. Chang and X. Y. Wang, Modelling and computation in the valuation of carbon derivatives with stochastic convenience yields, Plos One, 10 (2015), e0125679. doi: 10.1371/journal.pone.0125679. [12] G. Daskalakis, D. Psychoyios and R. Markellos, Modeling CO$_2$ emission allowance prices and derivatives: Evidence from the European trading scheme, J. Bank. Financ., 33 (2009), 1230-1241. [13] M. Freidlin, Functional Internation and Partial Differential Equations, Annals of Mathematics Studies, 109. Princeton University Press, Princeton, NJ, 1985. doi: 10.1515/9781400881598. [14] D. Gomes, R. M. Velho and M.-T. Wolfram, Socio-economic applications of finite state mean field games, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130405, 18 pp. doi: 10.1098/rsta.2013.0405. [15] O. Guéant, Mean Field Games and Applications to Economics, Ph.D Thesis, Université Paris-Dauphine, 2009. [16] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., Springer, Berlin, 2003 (2011), 205-266.  doi: 10.1007/978-3-642-14660-2_3. [17] O. Guéant, Mean field games with a quadratic Hamiltonian: A constructive scheme, Advances in Dynamic Games, Ann. Internat. Soc. Dynam. Games, Birkhäuser/Springer, New York, 12 (2012), 229-241. [18] O. Guéant, A reference case for mean field games models, J. Math. Pure Appl., 92 (2009), 276-294.  doi: 10.1016/j.matpur.2009.04.008. [19] Y. Y. He, L. Z. Wang and J. H. Wang, Cap-and-trade vs. carbon taxes: A quantitative comparison from a generation expansion planing perspective, Compu. Ind. Eng., 63 (2012), 708-716.  doi: 10.1016/j.cie.2011.10.005. [20] S. Hitzemann and M. Uhrig-Homburg, Empirical performance of reduced form models for emission permit prices, Working Paper, (2013), Available at SSRN: http://dx.doi.org/10.2139/ssrn.2297121. [21] A. Lachapelle, J.-M. Lasry, C.-A. Lehalle and P.-L. Lions, Efficiency of the price formation process in presence of high frequency participants: A mean field game analysis, Math. Financ. Econ., 10 (2016), 223-262.  doi: 10.1007/s11579-015-0157-1. [22] A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Mod. Meth. Appl. Sci., 20 (2010), 567-588.  doi: 10.1142/S0218202510004349. [23] A. Lapin, S. H. Zhang and S. Lapin, Numerical solution of a parabolic optimal control problem arising in economics and management, Appl. Math. Computa., 361 (2019), 715-729.  doi: 10.1016/j.amc.2019.06.011. [24] J.-M. Larsy and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire. (French) [Mean field games. I. The stationary case], C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019. [25] J.-M. Larsy and P. L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal. (French) [Mean field games. II. Finite horizon and optimal control], C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018. [26] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8. [27] J. Li, R. Bhattacharyya, S. Paul, S. Shakkottai and V. Subramanian, Incentivizing sharing in realtime D2D streaming networks: A mean field game perspective, 2015 IEEE Conference on Computer Communications, (2015), 15385329. doi: 10.1109/INFOCOM.2015.7218597. [28] S. Mandell, Optimal mix of emissions taxes and cap-and-trade, J. Environ. Econ. Manage., 56 (2008), 131-140. [29] G. E. Metcalf, Designing a carbon tax to reduce U. S. greenhouse gas emissions, Rev. Enviro. Econ. Policy, 3 (2009), 63-83.  doi: 10.3386/w14375. [30] W. A. Pizer, Combining price and quantity controls to mitigate global climate change, J. Pub. Econ., 85 (2002), 409-434.  doi: 10.1016/S0047-2727(01)00118-9. [31] M. J. Roberts and M. Spence., Effluent charges and licenses under uncertainty, J. Pub. Econ., 5 (1976), 193-208.  doi: 10.1016/0047-2727(76)90014-1. [32] J. Seifert, M. Uhrig-Homburg and M. Wagner, Dynamic behavior of CO$_2$ spot prices, J. Environ. Econ. and Manage., 56 (2008), 180-194. [33] M. Shi, Y. Yuan and S. Zhou, Carbon tax, cap-and-trade or mixed policy: Which is better for carbon mitigation?, J. Manage. Sci. Chin., 16 (2013), 9-13. [34] S. Smith, The compatibility of tradable permits with other environmental policy instruments, Implementing Domestic Tradable Permits for Environmental Protection, (1999). [35] S. Sorrell and J. Sijm, Carbon trading in the policy mix, Oxford Rev. Econ. Pol., 19 (2003), 420-437.  doi: 10.1093/oxrep/19.3.420. [36] R. N. Stavins, Addressing climate change with a comprehensive US cap-and-trade system, Oxford Rev. Econ. Policy, 24 (2008), 298-321.  doi: 10.1093/acprof:osobl/9780199573288.003.0010. [37] B. B. F. Wittneben, Exxon is right: Let us re-examine our choice for a cap-and-trade system over a carbon tax, Energ. Policy, 37 (2009), 2462-2464.  doi: 10.1016/j.enpol.2009.01.029. [38] G. Xu, Regulation comparative analysis and application of carbon tax and carbon trading in China, North. Econ., 6 (2011), 3-4. [39] A. Yanas, Dynamic games of environmental policy in a global economy: Tax versus quotas, Int. Econ., 15 (2007), 592-611.  doi: 10.1111/j.1467-9396.2007.00690.x. [40] H. B. Yin, P. G. Mehta, S. P. Meyn and U. V. Shanbhag, Synchronization of coupled oscillators is a game, IEEE Trans. Automat. Contr., 57 (2012), 920-935.  doi: 10.1109/TAC.2011.2168082.

show all references

##### References:
 [1] Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control. Optim., 50 (2012), 77-109.  doi: 10.1137/100790069. [2] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477. [3] Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.  doi: 10.1137/120882421. [4] R. S. AVi-Yonah and D. M. Uhlmann, Combating global climate change: Why a carbon tax is a better response to global warming than cap and trade, Stanford Environ. Law J., 28 (2009), 3-50. [5] F. Bagagiolo and D. Bauso, Mean-field games and dynamic demand management in power grids, Dyn. Games Appl., 4 (2014), 155-176.  doi: 10.1007/s13235-013-0097-4. [6] A. Baranzini, J. Goldemberg and S. Speck, A future for carbon taxes, Ecol. Econ., 32 (2000), 395-412.  doi: 10.1016/S0921-8009(99)00122-6. [7] A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7. [8] A. Bruvoll and B. M. Larsen, Greenhouse gas emissions in Norway: Do carbon taxes work?, Energ. Policy, 32 (2004), 493-505. [9] K. Chang, S. Wang and K. Peng, Mean reversion of stochastic convenience yields for CO$_2$ emissions allowances: Empirical evidence from the EU ETS, Spanish Rev. Financ. Econ., 11 (2013), 39-45. [10] S. H. Chang, X. Y. Wang and A. Shanain, Modeling and computation of mean field equilibria in producers' game with emission permits trading, Commun Nonlinear Sci. Numer. Simulat., 37 (2016), 238-248.  doi: 10.1016/j.cnsns.2016.01.020. [11] S. H. Chang and X. Y. Wang, Modelling and computation in the valuation of carbon derivatives with stochastic convenience yields, Plos One, 10 (2015), e0125679. doi: 10.1371/journal.pone.0125679. [12] G. Daskalakis, D. Psychoyios and R. Markellos, Modeling CO$_2$ emission allowance prices and derivatives: Evidence from the European trading scheme, J. Bank. Financ., 33 (2009), 1230-1241. [13] M. Freidlin, Functional Internation and Partial Differential Equations, Annals of Mathematics Studies, 109. Princeton University Press, Princeton, NJ, 1985. doi: 10.1515/9781400881598. [14] D. Gomes, R. M. Velho and M.-T. Wolfram, Socio-economic applications of finite state mean field games, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130405, 18 pp. doi: 10.1098/rsta.2013.0405. [15] O. Guéant, Mean Field Games and Applications to Economics, Ph.D Thesis, Université Paris-Dauphine, 2009. [16] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., Springer, Berlin, 2003 (2011), 205-266.  doi: 10.1007/978-3-642-14660-2_3. [17] O. Guéant, Mean field games with a quadratic Hamiltonian: A constructive scheme, Advances in Dynamic Games, Ann. Internat. Soc. Dynam. Games, Birkhäuser/Springer, New York, 12 (2012), 229-241. [18] O. Guéant, A reference case for mean field games models, J. Math. Pure Appl., 92 (2009), 276-294.  doi: 10.1016/j.matpur.2009.04.008. [19] Y. Y. He, L. Z. Wang and J. H. Wang, Cap-and-trade vs. carbon taxes: A quantitative comparison from a generation expansion planing perspective, Compu. Ind. Eng., 63 (2012), 708-716.  doi: 10.1016/j.cie.2011.10.005. [20] S. Hitzemann and M. Uhrig-Homburg, Empirical performance of reduced form models for emission permit prices, Working Paper, (2013), Available at SSRN: http://dx.doi.org/10.2139/ssrn.2297121. [21] A. Lachapelle, J.-M. Lasry, C.-A. Lehalle and P.-L. Lions, Efficiency of the price formation process in presence of high frequency participants: A mean field game analysis, Math. Financ. Econ., 10 (2016), 223-262.  doi: 10.1007/s11579-015-0157-1. [22] A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Mod. Meth. Appl. Sci., 20 (2010), 567-588.  doi: 10.1142/S0218202510004349. [23] A. Lapin, S. H. Zhang and S. Lapin, Numerical solution of a parabolic optimal control problem arising in economics and management, Appl. Math. Computa., 361 (2019), 715-729.  doi: 10.1016/j.amc.2019.06.011. [24] J.-M. Larsy and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire. (French) [Mean field games. I. The stationary case], C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019. [25] J.-M. Larsy and P. L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal. (French) [Mean field games. II. Finite horizon and optimal control], C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018. [26] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8. [27] J. Li, R. Bhattacharyya, S. Paul, S. Shakkottai and V. Subramanian, Incentivizing sharing in realtime D2D streaming networks: A mean field game perspective, 2015 IEEE Conference on Computer Communications, (2015), 15385329. doi: 10.1109/INFOCOM.2015.7218597. [28] S. Mandell, Optimal mix of emissions taxes and cap-and-trade, J. Environ. Econ. Manage., 56 (2008), 131-140. [29] G. E. Metcalf, Designing a carbon tax to reduce U. S. greenhouse gas emissions, Rev. Enviro. Econ. Policy, 3 (2009), 63-83.  doi: 10.3386/w14375. [30] W. A. Pizer, Combining price and quantity controls to mitigate global climate change, J. Pub. Econ., 85 (2002), 409-434.  doi: 10.1016/S0047-2727(01)00118-9. [31] M. J. Roberts and M. Spence., Effluent charges and licenses under uncertainty, J. Pub. Econ., 5 (1976), 193-208.  doi: 10.1016/0047-2727(76)90014-1. [32] J. Seifert, M. Uhrig-Homburg and M. Wagner, Dynamic behavior of CO$_2$ spot prices, J. Environ. Econ. and Manage., 56 (2008), 180-194. [33] M. Shi, Y. Yuan and S. Zhou, Carbon tax, cap-and-trade or mixed policy: Which is better for carbon mitigation?, J. Manage. Sci. Chin., 16 (2013), 9-13. [34] S. Smith, The compatibility of tradable permits with other environmental policy instruments, Implementing Domestic Tradable Permits for Environmental Protection, (1999). [35] S. Sorrell and J. Sijm, Carbon trading in the policy mix, Oxford Rev. Econ. Pol., 19 (2003), 420-437.  doi: 10.1093/oxrep/19.3.420. [36] R. N. Stavins, Addressing climate change with a comprehensive US cap-and-trade system, Oxford Rev. Econ. Policy, 24 (2008), 298-321.  doi: 10.1093/acprof:osobl/9780199573288.003.0010. [37] B. B. F. Wittneben, Exxon is right: Let us re-examine our choice for a cap-and-trade system over a carbon tax, Energ. Policy, 37 (2009), 2462-2464.  doi: 10.1016/j.enpol.2009.01.029. [38] G. Xu, Regulation comparative analysis and application of carbon tax and carbon trading in China, North. Econ., 6 (2011), 3-4. [39] A. Yanas, Dynamic games of environmental policy in a global economy: Tax versus quotas, Int. Econ., 15 (2007), 592-611.  doi: 10.1111/j.1467-9396.2007.00690.x. [40] H. B. Yin, P. G. Mehta, S. P. Meyn and U. V. Shanbhag, Synchronization of coupled oscillators is a game, IEEE Trans. Automat. Contr., 57 (2012), 920-935.  doi: 10.1109/TAC.2011.2168082.
Computed errors in the $L^{\infty}$-norm at $t = 0$
Evolution of normal distribution
Evolution of the second distribution
Evolution of uniform distribution
Evolution of $\bar{E}(x)$ and $\bar{E}(E)$
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