# Optimal Environmental Policy in a Dynamic Transboundary Pollution Game: Emission Standards, Taxes, and Permit Trading

^{1}

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## Abstract

**:**

## 1. Introduction

_{2}), and ozone, which cause global warming and environmental degradation [7]. Transboundary pollution not only destroys the ecological environment, but also endangers human health. Outdoor air pollution caused by PM2.5, SO

_{2}and ozone leads to premature death for more than 3 million people worldwide every year [8]. Therefore, in view of the impact of neighboring regions, when the government regulates the environmental policies of local enterprises, it should not only consider the impact of local environmental pollution, but also consider the impact of the behavior of the government and enterprises in adjacent areas, which increases the difficulty of decision-making and may cause new changes in the optimal choice of environmental policies [9]. Most related studies conducted previously were limited to the optimal environmental policy of either local or domestic governments, and some scholars compared the emission taxes, standards, and permit trading policies under the conditions of different market situations and implementation environments [10,11,12]. However, these studies did not compare the environmental and social welfare effects of the three environmental policies, particularly the conditions of the optimal policy choice for the transboundary pollution problem, which is the main objective of the present study. The consideration of transboundary pollution is unique; for two or more adjacent regions, each region suffers not only from the damage caused by the pollution stock generated by local firms, but also from the pollution stock generated by firms in the neighboring region [13,14]. Therefore, the government of each region will also be affected by the neighboring region when implementing environmental policies.

## 2. Literature Review

## 3. Parameter Description, Assumptions and Methods

#### 3.1. Parameter Description and Assumptions

#### 3.2. Methods

#### 3.2.1. Differential Game

#### 3.2.2. Stackelberg Game

## 4. Dynamic Games of Environmental Policies

#### 4.1. Emission Standards (Scenario S)

**Proposition**

**1.**

#### 4.2. Emission Taxes (Scenario T)

**Proposition**

**2.**

#### 4.3. Tradable Emission Permits (Scenario P)

**Proposition**

**3.**

#### 4.4. Comparing Emission Standards, Taxes, and Permit Trading

- (1)
- Difference in the trajectory and steady state of the pollution stock

**Proposition**

**4.**

- (2)
- Difference in social welfare under three policies

**Proposition**

**5.**

## 5. Analysis of the Equilibrium Results

#### 5.1. Equilibrium Trajectories

#### 5.2. Steady-State Equilibrium Results

## 6. Conclusions and Applications

#### 6.1. Conclusions

#### 6.2. Limitations and Prospects

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. For Proposition 1

## Appendix B. For Proposition 2

## Appendix C. For Proposition 3

## References

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**Figure 1.**The equilibrium trajectories of the pollution stock in the three scenarios under the initial. pollution stock ${x}_{0}=800$ (

**left**), and ${x}_{0}=100$ (

**right**).

**Figure 2.**The equilibrium trajectories of the revenues in the three scenarios under the initial pollution. stock ${x}_{0}=800$ (

**left**), and ${x}_{0}=100$ (

**right**).

**Figure 3.**Steady-state equilibrium and damage parameter ${d}_{i}$; (

**a**) Steady-state equilibrium pollution stock; (

**b**) Steady-state net revenue of region $i$; (

**c**) Steady-state net revenue of region $j$.

**Figure 4.**Steady-state equilibrium and the market size ${a}_{i}$; (

**a**) Steady-state equilibrium pollution stock; (

**b**) Steady-state net revenue of region $i$; (

**c**) Steady-state net revenue of region $j$.

**Figure 5.**Steady-state equilibrium and the natural decay rate $\delta $; (

**a**) Steady-state equilibrium pollution stock; (

**b**) Steady-state net revenue of each region.

Notation | Description |
---|---|

${\alpha}_{m}$ | The positive constant parameter measuring the reservation price, $m=i,j$ |

$a$ | The market size of each region |

${\chi}_{m}$ | The quantity of the product purchased by consumers in each region, $m=i,j$ |

${\gamma}_{m}$ | The cost coefficient of emission reduction, $m=i,j$ |

$\delta $ | The natural decay rate |

$\rho $ | The discount rate |

${d}_{m}$ | The damage parameter, $m=i,j$ |

$x\left(0\right)$ | The initial level of the pollution stock, $x\left(0\right)={x}_{0}>0$ |

$p\left(t\right)$ | The product price at time $t$ |

${q}_{m}\left(t\right)$ | The output of the firm at time $t$$,\text{}m=i,j$ |

${r}_{m}\left(t\right)$ | The pollution abatement level at time $t$$,\text{}m=i,j$ |

$x\left(t\right)$ | The pollution stock in the two regions at time $t$ |

${\pi}_{m}\left(t\right)$ | The instantaneous profit of the firm at time $t$$,\text{}m=i,j$ |

${\theta}_{m}\left(t\right)$ | The emission standard set by the government in each region at time $t$$,\text{}m=i,j$ |

${\tau}_{m}\left(t\right)$ | The emission tax rate at time $t$$,\text{}m=i,j$ |

${\overline{E}}_{m}\left(t\right)$ | The emission quota of the firm assigned by the government in each region at time $t$$,\text{}m=i,j$ |

${V}_{m}$ | The value function of each region, $m=i,j$ |

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**MDPI and ACS Style**

Xu, H.; Luo, M.
Optimal Environmental Policy in a Dynamic Transboundary Pollution Game: Emission Standards, Taxes, and Permit Trading. *Sustainability* **2022**, *14*, 9028.
https://doi.org/10.3390/su14159028

**AMA Style**

Xu H, Luo M.
Optimal Environmental Policy in a Dynamic Transboundary Pollution Game: Emission Standards, Taxes, and Permit Trading. *Sustainability*. 2022; 14(15):9028.
https://doi.org/10.3390/su14159028

**Chicago/Turabian Style**

Xu, Hao, and Ming Luo.
2022. "Optimal Environmental Policy in a Dynamic Transboundary Pollution Game: Emission Standards, Taxes, and Permit Trading" *Sustainability* 14, no. 15: 9028.
https://doi.org/10.3390/su14159028