How Can Governments Choose Optimal Carbon Tax Policies?

By Gökçe Dayanıklı

The economic and social impacts of events like wildfires and floods are motivating more people to recognize the effects of a changing climate. The increase in carbon emissions levels fuels many such climate-related changes. At the national level, some governments plan to implement policies such as cap-and-trade markets or carbon taxes to incentivize industries to decrease their carbon footprints. However, it is challenging for organizations to achieve this objective when individualistic goals obstruct improvements to the social good. The economic sector with the highest global carbon footprint is electricity production [8]. Because producers commonly use nonrenewable energy resources—including oil and natural gas—to generate electricity, the creation of incentives for them to invest in renewable energy resources is crucial for an ultimate decrease in carbon emissions.

Yet when regulators set policies, it is difficult to anticipate and control the population’s reaction. For example, many people acted with their individual goals in mind during the current COVID-19 pandemic. Though governments implemented social distancing policies, some individuals opted to socialize in unsafe ways while others behaved even more cautiously than the policies recommended. Similarly, governments cannot control the producers in free markets because all producers try to optimize their own outcomes. Understanding a population’s possible reaction to a carbon tax level is thus the first step towards a well-informed decision-making process for policy implementation. 

For this reason, we propose a stochastic and dynamic model for electricity producers and the government [2]. In the model, producers choose the amount of renewable or nonrenewable energy that they will use in production—in accordance with a specific government-sanctioned carbon tax level—in order to optimize their goals, which include revenue maximization, demand¹ matching, carbon tax minimization, and energy resource cost minimization. The regulator simultaneously chooses the carbon tax level, aiming to optimize its own objectives by accounting for the producers’ reactions.

Modeling producers who behave individualistically to optimize their own outcomes creates a game setting
in the population. Therefore, we need to identify a Nash equilibrium behavior of the producers; in other words, we must find a behavior for which no one has a profitable unilateral deviation for a carbon tax level that is set by the government. We can employ game theory tools to approach this problem, though it is difficult to compute the Nash equilibrium when there are many players. In this case, we can use the mean field game (MFG) methodology [4, 6, 7] with some assumptions. This approach lets us focus on one representative producer and their interactions with the other producers through the distribution of all producers’ states. The idea is that each producer is insignificant; one producer’s deviation does not affect the law of the states (such as the law of electricity production) of all producers. One can therefore take the law exogenously while solving the optimal control problem of the representative producer. But at the end, the law of the state of this producer should correspond to the law that is fixed at the beginning. This requirement creates a fixed-point problem.

On the other hand, the model setup becomes a stochastic control problem if a social planner² manages the population; the model includes the law of the players’ states through the cost function, the state dynamics, or both. This is called the mean field control (MFC) problem. In this case, we cannot assume that the distribution is fixed at the beginning. Because the producers are identical and prescribed the same behavior by a social planner, we assume that all of the producers will deviate if one chooses to do so.

We then analyze two different settings for the producers [2]: 

1. Competitive producers, wherein each individual player tries to optimize their own outcomes (this setup allows one to model a free market)
2. Cooperative producers, wherein players behave like the production facilities of a monopolistic firm. 

Mathematically, the first setting corresponds to a game problem and the second setting corresponds to a control problem. We find forward-backward stochastic differential equation systems for both models that characterize the solutions and show their existence and uniqueness. Furthermore, we propose a numerical algorithm and use real-world data from the energy sector to analyze experiments.

We first investigate the Price of Anarchy (PoA) — the inefficiency of the Nash equilibrium against the social optimum. The cost of a representative player in a game setting to their cost in the control setting yields the PoA. This value is always larger than one; if the value increases, the competition becomes more inefficient for the producers. Figure 1 illustrates how the advantage of cooperation diminishes as the carbon tax increases, since this cost restricts the producers’ behavior.

Furthermore, we analyze the optimal carbon tax level that minimizes the total cost of the government. This cost is different from that of the producers and depends on elements like the need to match the population’s electricity demand and the reputation cost of imposing high taxes on the producers. We can hence find the optimal carbon tax policy that a government should implement. Figure 2 illustrates the cost accrued by the government for implementing different carbon tax levels and the penalties for failing to match the population’s electricity demand in both settings.

In conclusion, MFG and MFC models that include a regulator enable us to compute optimal policies by accounting for the reactions of many players. Other application areas, such as epidemic containment, have also been a point of interest for this methodology in recent years [1, 3, 5].

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¹ Electricity consumers are not included in the model and the demand is exogenous.

² The social planner is not the regulator (i.e., the government). Here, the social planner tries to optimize the producers’ outcomes, while the regulator optimizes its own outcome.

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Gökçe Dayanıklı presented this research during a minisymposium at the 2021 SIAM Conference on Control and Its Applications, which took place virtually in July 2021 in conjunction with the 2021 SIAM Annual Meeting.

Acknowledgments: I thank Mathieu Laurière for his valuable comments on this manuscript.

References
[1] Aurell, A., Carmona, R., Dayanıklı, G., & Laurière, M. (2021). Optimal incentives to mitigate epidemics: A Stackelberg mean field game approach. Preprint, arXiv:2011.03105.
[2] Carmona, R., Dayanıklı, G., & Laurière, M. (2021). Mean field models to regulate carbon emissions in electricity production. Preprint, arXiv:2102.09434.
[3] Carmona, R., & Wang, P. (2020). Finite-state contract theory with a principal and a field of agents. Manag. Sci., 67(8).
[4] Huang, M., Malhamé, R.P., & Caines, P.E. (2006). Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst., 6(3), 221-252.
[5] Hubert, E., Mastrolia, T., Possamaï, D., & Warin, X. (2020). Incentives, lockdown, and testing: From Thucydides’s analysis to the COVID-19 pandemic. Preprint, arXiv:2009.00484.
[6] Lasry, J.-M., & Lions, P.-L. (2006). Jeux à champ moyen. I – Le cas stationnaire. Comptes Rendus Mathématique, 343(9), 619-625.
[7] Lasry, J.-M., & Lions, P.-L. (2006). Jeux à champ moyen. II – Horizon fini et contrôle optimal. Comptes Rendus Mathématique, 343(10), 679-684. 
[8] United States Environmental Protection Agency. (2021, October 26). Global greenhouse gas emissions data. Retrieved from https://www.epa.gov/ghgemissions/global-greenhouse-gas-emissions-data.

Gökçe Dayanıklı is a Ph.D. candidate in the Department of Operations Research and Financial Engineering at Princeton University. She works on applications of mean field games and control and graphon games.