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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.**

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$\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