Reflection on the joint prevention and control of air pollution from the perspective of environmental justice—insights from a two-stage dynamic game model

Abstract

The practices of the joint prevention and control of air pollution (JPCAP) present two disadvantages: the low enthusiasm of governance subjects and an unsatisfactory governance effect. Revealing the existing problems and exploring their causes has been a key issue for promoting JPCAP. Given this, we especially establish a two-stage dynamic game model for air pollution control to explore the advantages and dilemmas of JPCAP by analyzing changes in environmental tax rate and social welfare. The results show that the unfair distribution of social welfare among cities is a key reason for the unsatisfactory effect of JPCAP. Therefore, we improve JPCAP by considering both production-oriented and consumption-oriented pollutions based on environmental justice. In the improved JPCAP mode, the social welfare of each city is higher than that of non-joint control of air pollution (NJCAP), in which the increased degree is positively related to the city’s negotiation ability. In addition, the consumption tax rate is negatively correlated with the negotiation ability of the central city and the trade transfer coefficient. This study not only provides a theoretical and methodological reference for formulating effective planning and compensation scheme for JPCAP but also can be extended to the practice and theoretical analysis of other cross-regional public issues.

Appendix

Proof of consumer surplus (${CS}_{a}$ and ${CS}_{b}$)

According to the established game model, consumer surplus equals the difference between consumers’ utility and costs. Given this, we obtain the consumer surplus of the two cities:

$$\begin{array}{l}{CS}_{a}={CS}_{b}={x}_{1a}+{x}_{2a}-{\left({x}_{1a}+{x}_{2a}\right)}^{2}-p\left({x}_{1a}+{x}_{2a}\right)\\ =\frac{1}{2}\left({q}_{1}+{q}_{2}\right)-\frac{1}{4}{\left({q}_{1}+{q}_{2}\right)}^{2}-\frac{1}{2}\left(1-{q}_{1}-{q}_{2}\right)\left({q}_{1}+{q}_{2}\right)\\ =\frac{1}{4}{\left({q}_{1}+{q}_{2}\right)}^{2}\end{array}$$

Proof of Formulas (22)–(24)

When the two cities participate in joint control, their optimizations can be expressed as:

$${\text{MaxW}}={CS}_{a}+{CS}_{b}+{\pi }_{1}+{\pi }_{2}+{tq}_{2}-\frac{1}{2}\left({k}^{2}+{m}^{2}{s}_{b}\right){q}_{2}^{2}$$

The first-order optimality condition (FOC) is:

$$\frac{\partial {W}^{\#}}{\partial {t}^{\#}}=\frac{\partial {CS}_{a}^{\#}}{\partial {t}^{\#}}+\frac{\partial {CS}_{b}^{\#}}{\partial {t}^{\#}}+\frac{\partial {\pi }_{a}^{\#}}{\partial {t}^{\#}}+\frac{\partial {\pi }_{b}^{\#}}{\partial {t}^{\#}}+\frac{\partial {\left({tq}_{2}\right)}^{\#}}{\partial {t}^{\#}}-\frac{1}{2}\frac{\partial {\left[\left({k}^{2}+{m}^{2}s\right){q}_{2}^{2}\right]}^{\#}}{\partial {t}^{\#}}=0$$

$$\iff {t}^{\#}=\frac{2\left(1+c\right)\left({k}^{2}+{m}^{2}s\right)-4c-1}{1+4{k}^{2}+4{m}^{2}s}$$

$$\because {q}_{1}=\frac{1-2c+mt}{3},{q}_{2}=\frac{1-2mt+c}{3}$$

$$\therefore {q}_{1}^{\#}=\frac{2\left[\left(1-c\right)\left({k}^{2}+{m}^{2}s\right)-c\right]}{1+4{k}^{2}+4{m}^{2}s}$$

$${q}_{2}^{\#}=\frac{1+3c}{1+4{k}^{2}+4{m}^{2}s}$$

Proof of Formulas (25)–(27)

To describe the social welfare accurately of the equilibrium state, we substitute equilibrium quantities into equations of social welfare. The results are as follows:

$$\because {W}^{\#}={CS}_{a}^{\#}+{CS}_{b}^{\#}+{\pi }_{1}^{\#}+{\pi }_{2}^{\#}+{g}^{\#}{q}_{2}^{\#}-\frac{1}{2}\left({k}^{2}+{m}^{2}s\right){\left({q}_{2}^{\#}\right)}^{2}$$

$$=\frac{1}{2}{\left({q}_{1}^{\#}+{q}_{2}^{\#}\right)}^{2}+\left(1-{q}_{1}^{\#}-{q}_{2}^{\#}-c\right){q}_{1}^{\#}+\left(1-{q}_{1}^{\#}-{q}_{2}^{\#}\right){q}_{2}^{\#}-\frac{1}{2}\left({k}^{2}+{m}^{2}s\right){\left({q}_{2}^{\#}\right)}^{2}$$

$${W}_{a}^{\#}=\frac{1}{4}{\left({q}_{1}^{\#}+{q}_{2}^{\#}\right)}^{2}+\left(1-{q}_{1}^{\#}-{q}_{2}^{\#}-c\right){q}_{1}^{\#}-\frac{1}{2}{k}^{2}{\left({q}_{2}^{\#}\right)}^{2}$$

$${W}_{b}^{\#}=\frac{1}{4}{\left({q}_{1}^{\#}+{q}_{2}^{\#}\right)}^{2}+\left(1-{q}_{1}^{\#}-{q}_{2}^{\#}-\right){q}_{2}^{\#}-\frac{1}{2}{m}^{2}s{\left({q}_{2}^{\#}\right)}^{2}$$

$${q}_{1}^{\#}=\frac{2\left[\left(1-c\right)\left({k}^{2}+{m}^{2}s\right)-c\right]}{1+4{k}^{2}+4{m}^{2}s},{q}_{2}^{\#}=\frac{1+3c}{1+4{k}^{2}+4{m}^{2}s}$$

$$\therefore {W}^{\#}=\frac{3{\left(1-c\right)}^{2}\left({k}^{2}+{m}^{2}s\right)+1+3{c}^{2}}{2\left(1+4{k}^{2}+4{m}^{2}s\right)}$$

$${W}_{a}^{\#}=\frac{20{\left(1-c\right)}^{2}{\left({k}^{2}+{m}^{2}s\right)}^{2}+2\left(1-22c+5{c}^{2}\right){k}^{2}+4\left(1-7c\right)\left(1-c\right){m}^{2}s+1+2c+17{c}^{2}}{4{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

$${W}_{b}^{\#}=\frac{4{\left(1-c\right)}^{2}{\left({k}^{2}+{m}^{2}s\right)}^{2}+4\left(3+8c+5{c}^{2}\right)\left({k}^{2}+{m}^{2}s\right)-2\left(1+3{c}^{2}\right){m}^{2}s+1-2-11{c}^{2}]}{4{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

Proof of Formulas (30)–(34 )

When the two cities control air pollution separately, their optimization can be expressed as:

$${\text{MaxW}}_{a}={CS}_{a}+{\pi }_{1}+{t}_{a}{kq}_{2}-\frac{{k}^{2}}{2}{q}_{2}^{2}$$

$${\text{MaxW}}_{b}={CS}_{b}+{\pi }_{2}+{t}_{b}{mq}_{2}-\frac{{m}^{2}{s}_{b}}{2}{q}_{2}^{2}$$

The first-order optimality conditions (FOCs) are:

$$\frac{\partial {W}_{a}^{*}}{\partial {t}_{a}^{*}}=\frac{\partial {CS}_{a}^{*}}{\partial {t}_{a}^{*}}++\frac{\partial {\pi }_{a}^{*}}{\partial {t}_{a}^{*}}+\frac{\partial {\left({t}_{a}{kq}_{2}\right)}^{*}}{\partial {t}_{a}^{*}}-\frac{1}{2}\frac{\partial {\left({k}^{2}{q}_{2}^{2}\right)}^{*}}{\partial {t}_{a}^{*}}=0\iff {t}_{a}^{*}\in \left[0,+\infty \right)$$

$$\frac{\partial {W}_{a}^{*}}{\partial {t}^{*}}=\frac{\partial {CS}_{b}^{*}}{\partial {t}_{b}^{*}}+\frac{\partial {\pi }_{b}^{\#}}{\partial {t}_{b}^{*}}+\frac{\partial {\left({t}_{b}{mq}_{2}\right)}^{*}}{\partial {t}_{b}^{*}}-\frac{1}{2}\frac{\partial {\left[{m}^{2}{sq}_{2}^{2}\right]}^{*}}{\partial {t}_{a}^{*}}=0\iff {t}_{b}^{*}=\frac{4{m}^{2}s\left(1+c\right)-4-c}{7+8{m}^{2}s}$$

$$\because {q}_{1}=\frac{1-2c+{mt}_{b}}{3},{q}_{2}=\frac{1-2{mt}_{b}+c}{3}$$

$$\therefore {q}_{1}^{*}=\frac{4{m}^{2}s\left(1-c\right)+1-5c}{7+8{m}^{2}s},{q}_{2}^{*}=\frac{5+3c}{7+8{m}^{2}s}$$

Proof of Formulas (34)–(36)

For the same reason as A.4., we get the following results:

$$\because {W}_{a}^{*}={CS}_{a}^{*}+{\pi }_{1}^{*}+{t}_{a}^{*}{kq}_{2}^{*}-\frac{1}{2}{\left({kq}_{2}^{*}\right)}^{2}$$

$$=\frac{1}{4}{\left({q}_{1}^{\#}+{q}_{2}^{\#}\right)}^{2}+\left(1-{q}_{1}^{\#}-{q}_{2}^{\#}-c\right){q}_{1}^{\#}-\frac{1}{2}{k}^{2}{\left({q}_{2}^{\#}\right)}^{2},$$

$${W}_{b}^{*}={CS}_{b}^{*}+{\pi }_{2}^{*}+{t}_{b}^{*}{mq}_{2}^{*}-\frac{1}{2}s{\left({mq}_{2}^{*}\right)}^{2}$$

$$=\frac{1}{4}{\left({q}_{1}^{\#}+{q}_{2}^{\#}\right)}^{2}+\left(1-{q}_{1}^{\#}-{q}_{2}^{\#}-\right){q}_{2}^{\#}-\frac{1}{2}{m}^{2}s{\left({q}_{2}^{\#}\right)}^{2},$$

$${W}^{*}={CS}_{a}^{*}+{CS}_{b}^{*}+{\pi }_{1}^{*}+{\pi }_{2}^{*}+{t}_{a}^{*}{kq}_{2}^{*}+{t}_{b}^{*}{mq}_{2}^{*}-\frac{1}{2}\left({k}^{2}+{m}^{2}s\right){\left({q}_{2}^{*}\right)}^{2}$$

$$=\frac{1}{2}{\left({q}_{1}^{\#}+{q}_{2}^{\#}\right)}^{2}+\left(1-{q}_{1}^{\#}-{q}_{2}^{\#}-c\right){q}_{1}^{\#}+\left(1-{q}_{1}^{\#}-{q}_{2}^{\#}\right){q}_{2}^{\#}-\frac{1}{2}\left({k}^{2}+{m}^{2}s\right){\left({q}_{2}^{\#}\right)}^{2},$$

$${q}_{1}^{*}=\frac{4{m}^{2}s\left(1-c\right)+1-5c}{7+8{m}^{2}s},{q}_{2}^{*}=\frac{5+3c}{7+8{m}^{2}s},{t}_{a}^{*}=0,{t}_{b}^{*}=\frac{4{m}^{2}s\left(1+c\right)-4-c}{\left(7+8{m}^{2}s\right)m}$$

$$\therefore {W}_{a}^{*}=\frac{40{m}^{4}{s}^{2}{\left(1-c\right)}^{2}+8{m}^{2}s\left(1-c\right)\left(5-11c\right)-{k}^{2}{\left(5+3c\right)}^{2}+20-32c+52{c}^{2}}{2{\left(7+8{m}^{2}s\right)}^{2}}$$

$${W}_{b}^{*}=\frac{{\left(1-c\right)}^{2}{m}^{2}{s}_{b}+4+2c+2{c}^{2}}{2\left(7+8{m}^{2}{s}_{b}\right)}$$

$${W}^{*}={W}_{a}^{*}+{W}_{b}^{*}$$

$$=\frac{48{m}^{4}{s}^{2}{\left(1-c\right)}^{2}+\left(79-126c+111{c}^{2}\right){m}^{2}s-{k}^{2}{\left(3c+5\right)}^{2}+48-18c+66{c}^{2}}{2{\left(7+8{m}^{2}s\right)}^{2}}$$

Proof of Formula (37)

To assess the impacts of pollution transmission on the environmental tax of air pollution and equilibrium quantities, we compute:

$$\frac{\partial {t}^{\#}}{\partial k}=\frac{12\left(1+3c\right)\left(k-ms\right)}{{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

$$\frac{\partial {q}_{1}^{\#}}{\partial k}=\frac{4\left(1+3c\right)\left(k-ms\right)}{{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

$$\frac{\partial {q}_{2}^{\#}}{\partial k}=\frac{-8\left(1+3c\right)\left(k-ms\right)}{{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

$$\frac{\partial \left({q}_{1}^{\#}+{q}_{2}^{\#}\right)}{\partial k}=\frac{-4\left(1+3c\right)\left(k-ms\right)}{{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

which leads to the following conclusion:

$$\frac{\partial {t}^{\#}}{\partial k}<0,\frac{\partial {q}_{1}^{\#}}{\partial k}<0,\frac{\partial {q}_{2}^{\#}}{\partial k}>0,\frac{\partial \left({q}_{1}^{\#}+{q}_{2}^{\#}\right)}{\partial k}>0$$

Proof of Formula (38)

To analyze the impacts of environmental injustice on social welfare, we calculate the partial derivatives of ${W}_{a}^{\#},{W}_{b}^{\#},{W}^{\#}$ with respect to the pollution transmission coefficient $k$ respectively:

$$\frac{\partial {W}_{a}^{\#}}{\partial k}=\frac{\partial {CS}_{a}^{\#}}{\partial k}+\frac{\partial {\pi }_{a}^{\#}}{\partial k}+\frac{\partial {\left({tkq}_{2}\right)}^{\#}}{\partial k}-\frac{1}{2}\frac{\partial {\left({k}^{2}{q}_{2}^{2}\right)}^{\#}}{\partial k}$$

$$=\frac{2\left(k-ms\right)\left(1+3c\right)\left[-1+10{k}^{2}+6{m}^{2}s-3c\left(3-2{k}^{2}+2{m}^{2}s\right)\right]}{{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

$$\frac{\partial {W}_{b}^{\#}}{\partial k}=\frac{\partial {CS}_{b}^{\#}}{\partial k}+\frac{\partial {\pi }_{a}^{\#}}{\partial k}+\frac{\partial {\left({tmq}_{2}\right)}^{\#}}{\partial k}-\frac{1}{2}\frac{\partial {\left({k}^{2}{q}_{2}^{2}\right)}^{\#}}{\partial k}$$

$$=\frac{\left(k-ms\right)\left(1+3c\right)\left[1-24{k}^{2}-16{m}^{2}s+\left(5-8{k}^{2}\right)c\right]}{{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{3}}$$

$$\frac{\partial {W}^{\#}}{\partial k}=\frac{\partial {W}_{a}^{\#}}{\partial k}+\frac{\partial {W}_{b}^{\#}}{\partial k}=\frac{-\left(k-ms\right){\left(1+3c\right)}^{2}}{{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

According to the above polynomials, we could easily obtain the following inequalities:

$$\frac{\partial {W}_{a}^{\#}}{\partial k}<0,\frac{\partial {W}_{b}^{\#}}{\partial k}>0,\frac{\partial {W}^{\#}}{\partial k}>0$$

Proof of Formula (39)

To assess the impacts of environmental injustice on the environmental tax of air pollution control and equilibrium quantities, we compute their first partial derivatives and get the following conclusions:

$$\frac{\partial {t}^{*}}{\partial k}=\frac{-24ms\left(5+3c\right)}{{\left(7+8{m}^{2}s\right)}^{2}}<0$$

$$\frac{\partial {q}_{1}^{*}}{\partial k}=\frac{-8ms\left(5+3c\right)}{{\left(7+8{m}^{2}s\right)}^{2}}<0$$

$$\frac{\partial {q}_{2}^{*}}{\partial k}=\frac{16ms\left(5+3c\right)}{{\left(7+8{m}^{2}s\right)}^{2}}>0$$

Proof of Formula (40)

Similar to A.7., we get the following results:

$$\frac{\partial {W}_{a}^{*}}{\partial k}=\frac{\partial {CS}_{a}^{*}}{\partial k}+\frac{\partial {\pi }_{a}^{*}}{\partial k}+\frac{\partial {\left({tkq}_{2}\right)}^{*}}{\partial k}-\frac{1}{2}\frac{\partial {\left({k}^{2}{q}_{2}^{2}\right)}^{*}}{\partial k}$$

$$=\frac{-8ms\left(5+3c\right)\left[\left(10+6c\right){k}^{2}+6\left(1-c\right){m}^{2}s-1-9c\right]}{{\left(7+8{m}^{2}s\right)}^{3}}<0$$

$$\frac{\partial {W}_{b}^{*}}{\partial k}=\frac{\partial {CS}_{b}^{*}}{\partial k}+\frac{\partial {\pi }_{a}^{*}}{\partial k}+\frac{\partial {\left({tmq}_{2}\right)}^{*}}{\partial k}-\frac{1}{2}\frac{\partial {\left({k}^{2}{q}_{2}^{2}\right)}^{*}}{\partial k}=\frac{{\left(5+3c\right)}^{2}ms}{{\left(7+8{m}^{2}s\right)}^{2}}>0$$

$$\frac{\partial {W}^{*}}{\partial k}=\frac{\partial {W}_{a}^{*}}{\partial k}+\frac{\partial {W}_{b}^{*}}{\partial k}=\frac{-ms\left(5+3c\right)\left[16\left(5+3c\right){k}^{2}+8\left(1-9c\right){m}^{2}s-43-93c\right]}{{\left(7+8{m}^{2}s\right)}^{3}}>0$$

Proof of Formula (41)

The differences between the environmental taxes, outputs of city a, city b and the sum of them are shown as follows:

$${t}^{\#}-{t}_{b}^{*}=\frac{2\left(1+c\right)\left({k}^{2}+{m}^{2}s\right)-4c-1}{1+4{k}^{2}+4{m}^{2}s}-\frac{4{m}^{2}s\left(1+c\right)-4-c}{7+8{m}^{2}s}$$

$$=\frac{2\left[\left({k}^{2}+{m}^{2}s\right)\left(1+c\right)-\left(4c+1\right)\right]\left(7+8{m}^{2}s\right)-\left[4{m}^{2}s\left(1+c\right)-4-c\right]\left(1+4{k}^{2}+4{m}^{2}s\right)}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$$=\frac{3\left[\left(10+6c\right){k}^{2}+6\left(1-c\right)s-1-9c\right]}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$${q}_{1}^{\#}-{q}_{1}^{*}=\frac{2\left[\left(1+c\right)\left({k}^{2}+{m}^{2}s\right)-c\right]}{1+4{k}^{2}+4{m}^{2}s}-\frac{4{m}^{2}s\left(1-c\right)+1-5c}{7+8{m}^{2}s}$$

$$=\frac{2\left[\left(1+c\right)\left({k}^{2}+{m}^{2}s\right)-c\right]\left(7+8{m}^{2}s\right)-\left[4{m}^{2}s\left(1-c\right)+1-5\mathrm{c}\right]\left(1+4{k}^{2}+4{m}^{2}s\right)}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$$=\frac{14\left(1-c\right)\left({k}^{2}+{m}^{2}s\right)-14c-16{m}^{2}s-4{m}^{2}s\left(1-c\right)-1-5c-4\left({k}^{2}+{m}^{2}s\right)+20\left({k}^{2}+{m}^{2}s\right)c}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$$=\frac{{k}^{2}\left(10+6c\right)+6{m}^{2}s\left(1-c\right)-9c-1}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$${q}_{2}^{\#}-{q}_{2}^{*}=\frac{1+3c}{1+4{k}^{2}+4{m}^{2}s}-\frac{5+3c}{7+8{m}^{2}s}=\frac{\left(1+3c\right)\left(7+8{m}^{2}s\right)-\left(5+3c\right)\left(1+4{k}^{2}+4{m}^{2}s\right)}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$$=\frac{7+21c+8{m}^{2}s+24{m}^{2}sc-\left(5+3c\right)-4\left(5+3c\right)\left({k}^{2}+{m}^{2}s\right)}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$$=\frac{-2{k}^{2}\left(10+6c\right)+12{m}^{2}s\left(c-1\right)+18c+2}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$$\left({q}_{1}^{\#}+{q}_{2}^{\#}\right)-\left({q}_{1}^{*}+{q}_{2}^{*}\right)=\left({q}_{1}^{\#}-{q}_{1}^{*}\right)+\left({q}_{2}^{\#}-{q}_{2}^{*}\right)$$

$$=\frac{{k}^{2}\left(10+6c\right)+6{m}^{2}s\left(1-c\right)-9c-1}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}+\frac{-2{k}^{2}\left(10+6c\right)+12{m}^{2}s\left(c-1\right)+18c+2}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$$=-\frac{{k}^{2}\left(10+6c\right)+6{m}^{2}s\left(1-c\right)-9c-1}{\left(1+4{k}^{2}+4{m}^{2}s\right)\left(7+8{m}^{2}s\right)}$$

$$\because {k}^{2}>\frac{1+9c-6{m}^{2}s\left(1-c\right)}{10+6c}$$

$$\therefore {t}^{\#}-{t}_{b}^{*}>0,{q}_{1}^{\#}-{q}_{1}^{*}>0,{q}_{2}^{\#}-{q}_{2}^{*}<0,\left({q}_{1}^{\#}+{q}_{2}^{\#}\right)-\left({q}_{1}^{*}+{q}_{2}^{*}\right)<0$$

Proof of Formula (42)

We calculate the differences of the social welfares of city a, city b, and the sum of them. The details are as follows:

$${W}^{\#}-{W}^{*}=\frac{3{\left(1-c\right)}^{2}\left({k}^{2}+{m}^{2}s\right)+1+3{c}^{2}}{2\left(1+4{k}^{2}+4{m}^{2}s\right)}$$

$$-\frac{48{m}^{4}{s}^{2}{\left(1-c\right)}^{2}+\left(79-126c+111{c}^{2}\right){m}^{2}s-{k}^{2}{\left(3c+5\right)}^{2}+48-18c+66{c}^{2}}{2{\left(7+8{m}^{2}s\right)}^{2}}$$

$$=\frac{{\left[\left(10+6c\right){k}^{2}+6\left(1-c\right){m}^{2}s-1-9c\right]}^{2}}{4\left(1+4{k}^{2}+4{m}^{2}s\right){\left(7+8{m}^{2}s\right)}^{2}}>0$$

$${W}_{a}^{\#}-{W}_{a}^{*}=\frac{20{\left(1-c\right)}^{2}{\left({k}^{2}+{m}^{2}s\right)}^{2}+2\left(1-22c+5{c}^{2}\right){k}^{2}+4\left(1-7c\right)\left(1-c\right){m}^{2}s+1+2c+17{c}^{2}}{4{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

$$-\frac{40{m}^{4}{s}^{2}{\left(1-c\right)}^{2}+8{m}^{2}s\left(1-c\right)\left(5-11c\right)-{k}^{2}{\left(5+3c\right)}^{2}+20-32c+52{c}^{2}}{2{\left(7+8{m}^{2}s\right)}^{2}}$$

$$=\frac{{\left[\left(10+6c\right){k}^{2}+6\left(1-c\right){m}^{2}s-1-9c\right]}^{2}\left(9+8{k}^{2}+16{m}^{2}s\right)}{4{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}{\left(7+8{m}^{2}s\right)}^{2}}>0$$

$${W}_{b}^{\#}-{W}_{b}^{*}=\frac{4{\left(1-c\right)}^{2}{\left({k}^{2}+{m}^{2}s\right)}^{2}+4\left(3+8c+5{c}^{2}\right)\left({k}^{2}+{m}^{2}s\right)-2\left(1+3{c}^{2}\right){m}^{2}s+1-2-11{c}^{2}]}{4{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}}$$

$$-\frac{{\left(1-c\right)}^{2}{m}^{2}s+4+2c+2{c}^{2}}{2\left(7+8{m}^{2}s\right)}=-\frac{{\left[\left(10+6c\right){k}^{2}+6\left(1-c\right){m}^{2}s-1-9c\right]}^{2}}{4{\left(1+4{k}^{2}+4{m}^{2}s\right)}^{2}\left(7+8{m}^{2}s\right)}<0$$

Proof of Formula (51)

The partial derivatives of the consumption environmental tax to environmental injustice and negotiation ability of central city are as follows:

$${t}^{c}=\frac{{\left[\left(10+6c\right){k}^{2}+6\left(1-c\right){m}^{2}s-1-9c\right]}^{2}\left[9+8{k}^{2}+16{m}^{2}s-2u\left(1+4{k}^{2}+4{m}^{2}s\right)\right]}{4e\left(1+3c\right)\left(1+4{k}^{2}+4{m}^{2}s\right){\left(7+8{m}^{2}s\right)}^{2}}$$

$$\frac{\partial {t}^{c}}{\partial e}=-\frac{{\left[\left(10+6c\right){k}^{2}+6\left(1-c\right){m}^{2}s-1-9c\right]}^{2}\left[9+8{k}^{2}+16{m}^{2}s-2u\left(1+4{k}^{2}+4{m}^{2}s\right)\right]}{4{e}^{2}\left(1+3c\right)\left(1+4{k}^{2}+4{m}^{2}s\right){\left(7+8{m}^{2}s\right)}^{2}}<0$$

$$\frac{\partial {t}^{c}}{\partial u}=-\frac{2\left(1+4{k}^{2}+4{m}^{2}s\right){\left[\left(10+6c\right){k}^{2}+6\left(1-c\right){m}^{2}s-1-9c\right]}^{2}}{4e\left(1+3c\right)\left(1+4{k}^{2}+4{m}^{2}s\right){\left(7+8{m}^{2}s\right)}^{2}}<0$$