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Some Results on the Control of Polluting Firms According to Dynamic Nash and Stackelberg Patterns^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Model

- $x\ge 0$ the state variable (the number of polluting firms or the size of PO)
- $u\ge 0$ the control variable of the home country, i.e., the intensity of the home country’s counter pollution effort;
- $\upsilon \ge 0$ emissions’ rate (control variable of the polluting firms acting as organization);
- $g\ge 0$ endogenous growth rate of the group of polluters (of PO);
- $\varphi \ge 0$ the rate at which the counter pollution measures would reduces the size of PO;
- $\frac{a}{2}\ge 0$ the cost factor which faces the home country due to the unsuccessful discrimination among the overall firms during the abatement (or taxation);
- $\beta \ge 0$ percentage losses of the polluters per emission;
- $\gamma \ge 0$ average number of polluting firms which are not able to face the compliance costs.

## 3. Nash Equilibrium

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

- (a)
- A rising volume of emissions;
- (b)
- An increasing percentage lost of polluting firms (PO) per emissions;
- (c)
- An increasing rate at which pollution-control reduce the polluting firms (PO) $\left(\gamma \right)$

**Proposition**

**3.**

- (a)
- An increasing average number of the polluting firms abandonment$\left(\gamma \right)$
- (b)
- An increasing percentage looses per emission $\left(\beta u\right)$
- (c)
- An increasing value of the shadow price$u$(the shadow price of the polluting firms (PO)).

**Proposition**

**4.**

- i.
- The stationary values of the strategies in Nash equilibrium are the following:$$\begin{array}{l}{\widehat{u}}_{N}=\frac{\beta \left({b}_{2}-\widehat{\mu}\gamma \right)+{c}_{4}\left(\varphi +{c}_{3}/\widehat{\lambda}\right)}{{c}_{4}a+\widehat{\mu}{\beta}^{2}}\\ {\widehat{\upsilon}}_{N}=\frac{a\left({b}_{2}-\widehat{\mu}\gamma \right)-\widehat{\mu}\beta \left(\varphi +{c}_{3}/\widehat{\lambda}\right)}{{c}_{4}a+\widehat{\mu}{\beta}^{2}}\end{array}$$$\alpha $are given by (8a) and (8b), while the subscript N in (11) means the Nash solution.
- ii.
- the Nash equilibrium value for the number of polluting firms is given by$${\widehat{x}}_{N}=\frac{1}{g}\left[\left(\varphi -\frac{a}{2}{\widehat{u}}_{N}\right){\widehat{u}}_{N}+\left(\gamma +\beta {\widehat{u}}_{N}\right){\widehat{\upsilon}}_{N}\right]$$${b}_{2}$as in (11).

**Proof.**

_{1}and at the endogenous growth rate g. Taking into account (11), the stationary value of the polluting firms ${\widehat{x}}_{N}$ decreases with increasing endogenous rate $g$ (as the control factor ${c}_{3}$ equals to zero).

## 4. The Leader–Follower Game (Polluting Firms (the PO) as a Leader)

**Step 1:**The polluting firms, as group (i.e., the PO), announce their common strategy, $\upsilon $.

**Step 2:**For the given strategy $\upsilon $, the social planner of country (the follower) solves the same Nash optimal control problem. As it is mentioned in the Nash case (see (9)), the home’s optimal response to the strategy $\upsilon $ of the polluting firms (the PO), will be

**Step 3:**Now, in the last step, the leader has to solve the same as in the Nash case optimal control problem, but for the known reaction function (13) of the follower:

**Proposition**

**5.**

- i.
- The optimal strategies for the social planner and the polluting firms (the PO) of the hierarchical game are given, respectively by the following expressions$$\begin{array}{l}{\widehat{u}}_{S}=\frac{\beta \left({b}_{2}-\widehat{\mu}\gamma \right)+{c}_{4}\varphi +{b}_{3}\left({\beta}^{2}/a\right)}{{c}_{4}a+\widehat{\mu}{\beta}^{2}}\\ {\widehat{\upsilon}}_{S}=\frac{a\left({b}_{2}-\widehat{\mu}\gamma \right)-\beta \left(\widehat{\mu}\varphi -{b}_{3}\right)}{{c}_{4}a+\widehat{\mu}{\beta}^{2}}\end{array}$$
- ii.
- The number of polluting firms (the size of PO) is given by the expression$${\widehat{x}}_{S}=\frac{1}{g}\left(\left(\varphi -\frac{a}{2}{\widehat{u}}_{S}\right){\widehat{u}}_{S}+\left(\gamma +\beta {\widehat{u}}_{S}\right){\widehat{\upsilon}}_{S}\right)$$

**Proof.**

## 5. Comparison of the Two Solutions

- (i)
- The fewer the polluting firm’s losses per emission $\left(\beta \right)$, the smaller the difference $\Delta $. If the loss rate $\beta $ vanishes $(\beta =0)$, the Nash and Stackelberg equilibrium solutions become equal.
- (ii)
- If the polluting firms have no objective related to the unsuccessful discrimination of the social planner (${b}_{3}=0$), the Nash and Stackelberg equilibrium solutions are equal. If the same factor ${b}_{3}$ is positive, the group of polluting firms (the PO) announces a volume of emissions, υ
_{S}, such that the home country reacts with a higher counter-pollution effort, υ_{S}. As a result, the number of polluting firms (the size of the PO), $x$, increases which, in turn, increases the volume of emissions.As follows from the comparison of (11) and (17),$${\widehat{u}}_{S}>{\widehat{u}}_{N}and{\widehat{\upsilon}}_{S}{\widehat{\upsilon}}_{N}$$

**Proposition 6.**

**Proposition 7.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Proposition 4.**

**Proof of Proposition 5.**

## References

- Başar, Tamer, and Geert Jan Olsder. 1999. Dynamic Noncooperative Game Theory, 2nd ed. New York: Academic Press. [Google Scholar]
- Chen, Wanting, and Zhi-Hua Hu. 2018. Using evolutionary game theory to study governments and manufacturers’ behavioral strategies under various carbon taxes and subsidies. Journal of Cleaner Production 201: 123–41. [Google Scholar] [CrossRef]
- Dockner, Engelbert, Gustav Feichtinger, and Steffen Jørgensen. 1985. Tractable classes of Non Zero Sum open Loop Nash Differential Games: Theory and Examples. Journal of Optimization Theory and Applications 45: 179–97. [Google Scholar] [CrossRef]
- Dockner, Engelbert J., Steffen Jorgensen, Ngo Van Long, and Gerhard Sorger. 2000. Differential Games in Economics and Management Science. Cambridge: University Press. [Google Scholar]
- Grass, Dieter, Jonathan P. Caulkins, Gustav Feichtinger, Gernot Tragler, and Doris A. Behrens. 2008. Optimal Control of Nonlinear Processes. With Applications in Drugs, Corruption and Terror. Berlin and Heidelberg: Springer. [Google Scholar]
- Gromova, Ekaterina Viktorovna, Anna Viktorovna Tur, and Lidiya Ivanovna Balandina. 2016. A game-theoretic model of pollution control with asymmetric time horizons. Contributions to Game Theory and Management 9: 170–79. [Google Scholar]
- Halkos, George E. 1996. Incomplete information in the acid rain game. Empirica 23: 129–48. [Google Scholar] [CrossRef]
- Halkos, George, and George Papageorgiou. 2014. Controlling Polluting Firms: Nash and Stackelberg Strategies. MPRA Discussion Paper 58947. Munich: University Library of Munich. [Google Scholar]
- Madani, Kaveh. 2010. Game theory and water resources. Journal of Hydrology 381: 225–38. [Google Scholar] [CrossRef]
- Schüller, Katharina, Kateřina Staňková, and Frank Thuijsman. 2017. Game theory of pollution: National policies and their international effects. Games 8: 30. [Google Scholar] [CrossRef]
- Shi, Guang-Ming, Jin-Nan Wang, Bing Zhang, Zhe Zhang, and Yong-Liang Zhang. 2016. Pollution control costs of a transboundary river basin: Empirical tests of the fairness and stability of cost allocation mechanisms using game theory. Journal of Environmental Management 177: 145–52. [Google Scholar] [CrossRef]
- Wang, Hongwei, Lingru Cai, and Wei Zeng. 2011. Research on the evolutionary game of environmental pollution in system dynamics model. Journal of Experimental & Theoretical Artificial Intelligence 23: 39–50. [Google Scholar]
- Wood, Peter John. 2010. Climate Change and Game Theory. Environmental Economics Research Hub Research Reports. Canberra: The Crawford School of Economics and Government. Australian National University. [Google Scholar]
- Zhang, Ming, Hao Li, Linzhao Xue, and Wenwen Wang. 2019. Using three-sided dynamic game model to study regional cooperative governance of haze pollution in China from a government heterogeneity perspective. Science of the Total Environment 694: 133559. [Google Scholar] [CrossRef]
- Zhao, Liuwei, and Shuai Jin. 2021. Research on the Impact of Ecological Civilization Construction on Environmental Pollution Control in China—Based on Differential Game Theory. Discrete Dynamics in Nature and Society. [Google Scholar] [CrossRef]

$\varphi $ | $\alpha $ | $\beta $ | $\gamma $ | ${c}_{1}$ | ${c}_{2}$ | ${b}_{1}$ | ${b}_{2}$ | ${c}_{4}$ | ${\rho}_{1}$ | ${\rho}_{2}$ | |

${\widehat{u}}_{N}$ | + | – | ? | - | 0 | 0 | 0 | + | 0 | 0 | 0 |

${\widehat{\upsilon}}_{N}$ | - | ? | ? | 0 | 0 | 0 | 0 | + | + | 0 | 0 |

${\widehat{x}}_{N}$ | + | - | + | + | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Halkos, G.E.; Papageorgiou, G.J.
Some Results on the Control of Polluting Firms According to Dynamic Nash and Stackelberg Patterns. *Economies* **2021**, *9*, 77.
https://doi.org/10.3390/economies9020077

**AMA Style**

Halkos GE, Papageorgiou GJ.
Some Results on the Control of Polluting Firms According to Dynamic Nash and Stackelberg Patterns. *Economies*. 2021; 9(2):77.
https://doi.org/10.3390/economies9020077

**Chicago/Turabian Style**

Halkos, George E., and George J. Papageorgiou.
2021. "Some Results on the Control of Polluting Firms According to Dynamic Nash and Stackelberg Patterns" *Economies* 9, no. 2: 77.
https://doi.org/10.3390/economies9020077