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  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 & Management Optimization, doi: 10.3934/jimo.2020121
References:
[1]

Y. AchdouF. 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.  Google Scholar

[2]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477.  Google Scholar

[3]

Y. AchdouF. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[6]

A. BaranziniJ. Goldemberg and S. Speck, A future for carbon taxes, Ecol. Econ., 32 (2000), 395-412.  doi: 10.1016/S0921-8009(99)00122-6.  Google Scholar

[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.  Google Scholar

[8]

A. Bruvoll and B. M. Larsen, Greenhouse gas emissions in Norway: Do carbon taxes work?, Energ. Policy, 32 (2004), 493-505.   Google Scholar

[9]

K. ChangS. 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.   Google Scholar

[10]

S. H. ChangX. 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.  Google Scholar

[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.  Google Scholar

[12]

G. DaskalakisD. 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.   Google Scholar

[13]

M. Freidlin, Functional Internation and Partial Differential Equations, Annals of Mathematics Studies, 109. Princeton University Press, Princeton, NJ, 1985. doi: 10.1515/9781400881598.  Google Scholar

[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.  Google Scholar

[15]

O. Guéant, Mean Field Games and Applications to Economics, Ph.D Thesis, Université Paris-Dauphine, 2009. Google Scholar

[16]

O. GuéantJ.-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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[19]

Y. Y. HeL. 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.  Google Scholar

[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. Google Scholar

[21]

A. LachapelleJ.-M. LasryC.-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.  Google Scholar

[22]

A. LachapelleJ. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Mod. Meth. Appl. Sci., 20 (2010), 567-588.  doi: 10.1142/S0218202510004349.  Google Scholar

[23]

A. LapinS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. Mandell, Optimal mix of emissions taxes and cap-and-trade, J. Environ. Econ. Manage., 56 (2008), 131-140.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. SeifertM. Uhrig-Homburg and M. Wagner, Dynamic behavior of CO$_2$ spot prices, J. Environ. Econ. and Manage., 56 (2008), 180-194.   Google Scholar

[33]

M. ShiY. 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.   Google Scholar

[34]

S. Smith, The compatibility of tradable permits with other environmental policy instruments, Implementing Domestic Tradable Permits for Environmental Protection, (1999). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

G. Xu, Regulation comparative analysis and application of carbon tax and carbon trading in China, North. Econ., 6 (2011), 3-4.   Google Scholar

[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.  Google Scholar

[40]

H. B. YinP. G. MehtaS. 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.  Google Scholar

show all references

References:
[1]

Y. AchdouF. 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.  Google Scholar

[2]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477.  Google Scholar

[3]

Y. AchdouF. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[6]

A. BaranziniJ. Goldemberg and S. Speck, A future for carbon taxes, Ecol. Econ., 32 (2000), 395-412.  doi: 10.1016/S0921-8009(99)00122-6.  Google Scholar

[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.  Google Scholar

[8]

A. Bruvoll and B. M. Larsen, Greenhouse gas emissions in Norway: Do carbon taxes work?, Energ. Policy, 32 (2004), 493-505.   Google Scholar

[9]

K. ChangS. 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.   Google Scholar

[10]

S. H. ChangX. 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.  Google Scholar

[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.  Google Scholar

[12]

G. DaskalakisD. 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.   Google Scholar

[13]

M. Freidlin, Functional Internation and Partial Differential Equations, Annals of Mathematics Studies, 109. Princeton University Press, Princeton, NJ, 1985. doi: 10.1515/9781400881598.  Google Scholar

[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.  Google Scholar

[15]

O. Guéant, Mean Field Games and Applications to Economics, Ph.D Thesis, Université Paris-Dauphine, 2009. Google Scholar

[16]

O. GuéantJ.-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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[19]

Y. Y. HeL. 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.  Google Scholar

[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. Google Scholar

[21]

A. LachapelleJ.-M. LasryC.-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.  Google Scholar

[22]

A. LachapelleJ. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Mod. Meth. Appl. Sci., 20 (2010), 567-588.  doi: 10.1142/S0218202510004349.  Google Scholar

[23]

A. LapinS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. Mandell, Optimal mix of emissions taxes and cap-and-trade, J. Environ. Econ. Manage., 56 (2008), 131-140.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

J. SeifertM. Uhrig-Homburg and M. Wagner, Dynamic behavior of CO$_2$ spot prices, J. Environ. Econ. and Manage., 56 (2008), 180-194.   Google Scholar

[33]

M. ShiY. 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.   Google Scholar

[34]

S. Smith, The compatibility of tradable permits with other environmental policy instruments, Implementing Domestic Tradable Permits for Environmental Protection, (1999). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

G. Xu, Regulation comparative analysis and application of carbon tax and carbon trading in China, North. Econ., 6 (2011), 3-4.   Google Scholar

[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.  Google Scholar

[40]

H. B. YinP. G. MehtaS. 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.  Google Scholar

Figure 1.  Computed errors in the $ L^{\infty} $-norm at $ t = 0 $
Figure 2.  Evolution of normal distribution
Figure 3.  Evolution of the second distribution
Figure 4.  Evolution of uniform distribution
Figure 5.  Evolution of $ \bar{E}(x) $ and $ \bar{E}(E) $
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