Emission reduction cooperation in a dynamic supply chain with competitive retailers

Abstract

The increasing carbon emissions cause severe environmental issues that threaten the survival of human beings. Such a situation promotes supply chain firms and governments to take various measures to achieve the sustainable goal. This paper investigates the cost-sharing cooperation strategy of emission reductions in a dynamic supply chain consisting of a manufacturer and two distinct retailers with different scales. In the presence of carbon tax and green reputation, we model a Stackelberg differential game to derive and compare the optimal equilibrium decisions and further conduct a numerical study to analyze the chain members’ profits between the cases of no cost sharing and cost sharing. The manufacturer can actualize this sharing, respectively, with either of the competitive retailers. The results show that cost sharing is a better choice for supply chain members than no sharing. In the cost-sharing scenario, the manufacturer and two retailers prefer the case in which the emission reduction cost that a retailer undertakes is relatively high, because the manufacturer can increase emission reduction level to build its green reputation, and thereby the retailers can improve profits. When the proportion of emission reduction cost that a retailer undertakes locates in a certain interval, all the supply chain participants can be better off in the scenario where small-scale retailer shares the emission reduction cost. In addition, the increase in carbon tax improves the flexibility of cost-sharing ratio for the members achieving this consensual scenario on the cost-sharing cooperation.

Appendices

Appendix A

1.1 Proof of Proposition 1

Letting ${V}_{{r}_{1}}^{N}$, ${V}_{{r}_{2}}^{N}$, and ${V}_{m}^{N}$ represent the value functions of the large-scale retailer, the small-scale retailer, and the manufacturer in no-cost-sharing strategy, the HJB equations are given by

$$\rho V_{{r_{i} }}^{N} = \mathop {{\text{max}}}\limits_{{p_{1} }} \left\{ {\left( {p_{i} - w_{i} } \right) \cdot \left( {\alpha_{i} G - \beta p_{i} + \gamma p_{3 - i} - \gamma p_{i} } \right) + \frac{{\partial V_{{r_{i} }}^{N} }}{\partial G}\left( {\theta \tau - \delta G} \right)} \right\},i = 1,2$$

(23)

$$\rho V_{m}^{N} = \mathop {{\text{max}}}\limits_{\tau } \left\{ \begin{gathered} w_{1} \cdot \left( {\alpha_{1} G - \beta p_{1} + \gamma p_{2} - \gamma p_{1} } \right) + w_{2} \cdot \left( {\alpha_{2} G - \beta p_{2} + \gamma p_{1} - \gamma p_{2} } \right) \hfill \\ - s\left( {1 - \tau } \right)E_{0} \cdot \left( {\alpha_{1} G + \alpha_{2} G - \beta p_{1} - \beta p_{2} } \right) - \frac{k}{2}\tau^{2} + \frac{{\partial V_{m}^{N} }}{\partial G}\left( {\theta \tau - \delta G} \right) \hfill \\ \end{gathered} \right\}$$

(24)

Using the first-order condition to maximize the right-hand side of Eq. (23), we can obtain

$$p_{{r_{i} }}^{N} = \frac{{\alpha_{i} G + \gamma p_{3 - i} + \left( {\beta + \gamma } \right)w_{i} }}{{2\left( {\beta + \gamma } \right)}}, i = 1,2$$

(25)

Simultaneous Eqs. (25) yield Eq. (11).

Substituting Eq. (11) into the right-hand side of Eq. (24) yields

$$\rho {V}_{m}^{N}={w}_{1}\left(\beta +\gamma \right)\left[\frac{2\left(\beta +\gamma \right){\alpha }_{1}+\gamma {\alpha }_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}G+\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{1}+\gamma \left(\beta +\gamma \right){w}_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]+{w}_{2}\left(\beta +\gamma \right)\left[\frac{2\left(\beta +\gamma \right){\alpha }_{2}+\gamma {\alpha }_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}G+\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{2}+\gamma \left(\beta +\gamma \right){w}_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]-s{E}_{0}\left[\frac{\left(\beta +\gamma \right)\left({\alpha }_{1}+{\alpha }_{2}\right)}{2\beta +\gamma }G-\frac{\beta \left(\beta +\gamma \right)\left({w}_{1}+{w}_{2}\right)}{2\beta +\gamma }\right]+s{E}_{0}\tau \left[\frac{\left(\beta +\gamma \right)\left({\alpha }_{1}+{\alpha }_{2}\right)}{2\beta +\gamma }G-\frac{\beta \left(\beta +\gamma \right)\left({w}_{1}+{w}_{2}\right)}{2\beta +\gamma }\right]-\frac{k}{2}{\tau }^{2}+\frac{\partial {V}_{m}^{N}}{\partial G}\left(\theta \tau -\delta G\right)$$

(26)

Using the first-order condition to solve the right-hand side of Equation (26) with respect to $\tau$, we can obtain Eq. (12).

Conjecture the manufacturer’s value function as a quadratic form, which is expressed as follows:

$$V_{m}^{N} \left( G \right) = I_{1} G^{2} + I_{2} G + I_{3}$$

(27)

where ${I}_{1}$, ${I}_{2}$, and ${I}_{3}$ are the coefficients to be determined. From Equation (27), we have

$$\frac{{\partial V_{m}^{N} }}{\partial G} = 2I_{1} G + I_{2}$$

(28)

By Substituting Eqs. (27) and (28) into (26), and letting the corresponding coefficients of ${G}^{2}$ on both sides of equation be equal, we can obtain

$$- 2\theta^{2} \left( {2\beta + \gamma } \right) \cdot I_{1}^{2} + \left[ {k\left( {2\beta + \gamma } \right)\left( {\rho + 2\delta } \right) - 2\theta sE_{0} \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right)} \right] \cdot I_{1} - \frac{{\left( {sE_{0} } \right)^{2} \left( {\beta + \gamma } \right)^{2} }}{{2\left( {2\beta + \gamma } \right)}}\left( {\alpha_{1} + \alpha_{2} } \right)^{2} = 0$$

(29)

Solving Eq. (29) yields

$$I_{1} = \frac{{2\theta sE_{0} \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) - k\left( {2\beta + \gamma } \right)\left( {\rho + 2\delta } \right) + \Delta_{1} }}{{ - 4\theta^{2} \left( {2\beta + \gamma } \right)}}$$

(30)

where ${\Delta }_{1}\ge 0$ is required to guarantee the existence of the solution. Note that when ${I}_{1}$ takes a smaller root, the green reputation level will not converge to a steady-state value. Thus, the smaller root is abandoned.

Similarly, ${I}_{2}$ and ${I}_{3}$ are obtained as the following:

$$I_{2} = \frac{{2A_{1} }}{{k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} }}$$

(31)

$${I}_{3}=\frac{{w}_{1}}{\rho }\left[\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right)(\beta +\gamma ){w}_{1}+\gamma {\left(\beta +\gamma \right)}^{2}{w}_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]+\frac{{w}_{2}}{\rho }\left[\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right)(\beta +\gamma ){w}_{2}+\gamma {\left(\beta +\gamma \right)}^{2}{w}_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]+\frac{s{E}_{0}\beta \left(\beta +\gamma \right)}{\left(2\beta +\upgamma \right)\uprho }\left({w}_{1}+{w}_{2}\right)+\frac{{\beta }^{2}{(s{E}_{0})}^{2}}{2k\rho }{\left[\frac{(\beta +\gamma )}{(2\beta +\gamma )}\left({w}_{1}+{w}_{2}\right)\right]}^{2}+\frac{2{\theta }^{2}{A}_{1}^{2}}{k\rho {\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{1}\right]}^{2}}-\frac{2\theta s{E}_{0}\beta {A}_{1}}{k\rho \left[k\left(2\beta +\gamma \right)\rho +\Delta \right]}\frac{(\beta +\gamma )}{(2\beta +\gamma )}\left({w}_{1}+{w}_{2}\right)$$

(32)

Substituting Eqs. (11), and (12) into the right-hand side of Eq. (23) yields

$$\rho {V}_{{r}_{1}}^{N}=\left(\beta +\gamma \right){\left[\frac{2\left(\beta +\gamma \right){\alpha }_{1}+\gamma {\alpha }_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}G+\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{1}+\gamma \left(\beta +\gamma \right){w}_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]}^{2}+\frac{\partial {V}_{{r}_{1}}^{N}}{\partial G}\left[\frac{s{E}_{0}\theta \left(\beta +\gamma \right)}{\left(2\beta +\gamma \right)k}\left({\alpha }_{1}+{\alpha }_{2}\right)G-\frac{s{E}_{0}\beta \left(\beta +\gamma \right)\theta }{\left(2\beta +\gamma \right)k}\left({w}_{1}+{w}_{2}\right)+\frac{{\theta }^{2}}{k}\frac{\partial {V}_{m}^{N}}{\partial G}-\delta G\right]$$

(33)

Conjecture the large-scale retailer’s value function as a quadratic form, which is expressed as follows:

$${V}_{{r}_{1}}^{N}\left(G\right)={J}_{1}{G}^{2}+{J}_{2}G+{J}_{3}$$

(34)

where ${J}_{1}$, ${J}_{2}$, and ${J}_{3}$ are the coefficients to be determined. From Equation (34), we have

$$\frac{\partial {V}_{{r}_{1}}^{N}}{\partial G}=2{J}_{1}G+{J}_{2}$$

(35)

By Substituting Equations (28), (34), and (35) into (33), and letting the corresponding coefficients of ${G}^{2}$ on both sides of equation be equal, we can obtain

$$J_{1} = \frac{{k\left( {2\beta + \gamma } \right)\left( {\beta + \gamma } \right)\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{1} + \gamma \alpha_{2} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]^{2} }}{{\rho k\left( {2\beta + \gamma } \right) - 2sE_{0} \theta \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) - 4\theta^{2} \left( {2\beta + \gamma } \right)I_{1} + 2k\left( {2\beta + \gamma } \right)\delta }}$$

(36)

Substituting Equation (30) into Equation (36) yields

$$J_{1} = \frac{{k\left( {2\beta + \gamma } \right)\left( {\beta + \gamma } \right)}}{{\Delta_{1} }}\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{1} + \gamma \alpha_{2} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]^{2}$$

(37)

Similarly, ${J}_{2}$ and ${J}_{3}$ are obtained as the following:

$$J_{2} = \frac{{2A_{2} }}{{k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} }}$$

(38)

$$J_{3} = \frac{{\left( {\beta + \gamma } \right)}}{\rho }\left[ {\frac{{\left( { - 2\beta^{2} - 4\beta \gamma - \gamma^{2} } \right)w_{1} + \gamma \left( {\beta + \gamma } \right)w_{2} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]^{2} + \frac{{4A_{1} A_{2} \theta^{2} }}{{k\rho \left[ {k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} } \right]^{2} }} - \frac{{2\beta \theta sE_{0} \left( {\beta + \gamma } \right)A_{2} }}{{k\rho \left( {2\beta + \gamma } \right)\left[ {k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} } \right]}}\left( {w_{1} + w_{2} } \right)$$

(39)

Substituting Eqs. (11), and (12) into the right-hand side of Eq. (23) yields

$$\rho {V}_{{r}_{2}}^{N}=\left(\beta +\gamma \right){\left[\frac{2\left(\beta +\gamma \right){\alpha }_{2}+\gamma {\alpha }_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}G+\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{2}+\gamma \left(\beta +\gamma \right){w}_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]}^{2}+\frac{\partial {V}_{{r}_{2}}^{N}}{\partial G}\left[\frac{s{E}_{0}\theta \left(\beta +\gamma \right)}{\left(2\beta +\gamma \right)k}\left({\alpha }_{1}+{\alpha }_{2}\right)G-\frac{s{E}_{0}\beta \left(\beta +\gamma \right)\theta }{\left(2\beta +\gamma \right)k}\left({w}_{1}+{w}_{2}\right)+\frac{{\theta }^{2}}{k}\frac{\partial {V}_{m}^{N}}{\partial G}-\delta G\right]$$

(40)

Conjecture the small-scale retailer’s value function as a quadratic form, which is expressed as follows:

$${V}_{{r}_{2}}^{N}\left(G\right)={K}_{1}{G}^{2}+{K}_{2}G+{K}_{3}$$

(41)

where ${K}_{1}$, ${K}_{2}$, and ${K}_{3}$ are the coefficients to be determined. From Eq. (41), we have

$$\frac{\partial {V}_{{r}_{2}}^{N}}{\partial G}=2{K}_{1}G+{K}_{2}$$

(42)

By Substituting Equations (28), (41), and (42) into (40), and letting the corresponding coefficients of ${G}^{2}$ on both sides of equation be equal, we can obtain

$$K_{1} = \frac{{k\left( {2\beta + \gamma } \right)\left( {\beta + \gamma } \right)\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{2} + \gamma \alpha_{1} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]^{2} }}{{\rho k\left( {2\beta + \gamma } \right) - 2sE_{0} \theta \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) - 4\theta^{2} \left( {2\beta + \gamma } \right)I_{1} + 2k\left( {2\beta + \gamma } \right)\delta }}$$

(43)

Substituting Equation (30) into Equation (43) yields

$$K_{1} = \frac{{k\left( {2\beta + \gamma } \right)\left( {\beta + \gamma } \right)}}{{\Delta_{1} }}\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{2} + \gamma \alpha_{1} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]^{2}$$

(44)

Similarly, $K_{2}$ and $K_{3}$ are obtained as the following:

$$K_{2} = \frac{{2A_{3} }}{{k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} }}$$

(45)

$$\begin{aligned} K_{3} = & \frac{{\left( {\beta + \gamma } \right)}}{\rho }\left[ {\frac{{\left( { - 2\beta^{2} - 4\beta \gamma - \gamma^{2} } \right)w_{2} + \gamma \left( {\beta + \gamma } \right)w_{1} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]^{2} \\ & + \frac{{4A_{1} A_{3} \theta^{2} }}{{k\rho \left[ {k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} } \right]^{2} }} - \frac{{2\beta \theta sE_{0} \left( {\beta + \gamma } \right)A_{3} }}{{k\rho \left( {2\beta + \gamma } \right)\left[ {k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} } \right]}}\left( {w_{1} + w_{2} } \right) \\ \end{aligned}$$

(46)

1.2 Proof of Proposition 2

By substituting Eq. (12) into (1), we obtain the following differential equation:

$$\dot{G}\left(t\right)-\frac{k\left(2\beta +\gamma \right)\rho -{\Delta }_{1}}{2k\left(2\beta +\gamma \right)}G=\frac{2\left(2\beta +\gamma \right){\theta }^{2}{A}_{1}-\beta \theta s{E}_{0}\left(\beta +\gamma \right)\left({w}_{1}+{w}_{2}\right)\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{1}\right]}{k(2\beta +\gamma )\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{1}\right]}$$

(47)

Given that $G\left(0\right)={G}_{0}$, solving Equation (47) yields Eq. (15), where ${G}_{\infty }^{N}$ can also be showed in Eq. (93).

By substituting Eqs. (15) and (93) into (12), we can obtain Eqs. (13) and (94).

By substituting Eqs. (15) and (93) into (11), we can obtain Eqs. (14) and (95).

By substituting Eqs. (15) and (93) into (2), we can obtain Eqs. (16) and (96).

1.3 Proof of Proposition 3

Proposition 3 shows that the monotonicity of retail price relates to ${G}_{0}$ and ${G}_{\infty }^{N}$ for two retailers. When ${G}_{0}>{G}_{\infty }^{N}$, the retail price decreases with time. Therefore, the skimming pricing strategy should be adopted. When ${G}_{0}<{G}_{\infty }^{N}$, the retail price increases with time, thereby requiring the adoption of the penetration pricing strategy.

Proposition 3 also shows that the monotonicity of emission reduction level relates to ${G}_{0}$, ${G}_{\infty }^{N}$, and $\frac{k\left(2\beta +\gamma \right)\left(\rho +2\delta \right)-{\Delta }_{1}}{2k\theta \left(2\beta +\gamma \right)}$ for the manufacturer. When $\frac{k\left(2\beta +\gamma \right)\left(\rho +2\delta \right)-{\Delta }_{1}}{2k\theta \left(2\beta +\gamma \right)}\left({G}_{0}-{G}_{\infty }^{N}\right)>0$, the emission reduction level decreases with time. Therefore, the skimming emission reduction strategy should be adopted. When $\frac{k\left(2\beta +\gamma \right)\left(\rho +2\delta \right)-{\Delta }_{1}}{2k\theta \left(2\beta +\gamma \right)}\left({G}_{0}-{G}_{\infty }^{N}\right)<0$, the emission reduction level increases with time, thereby meaning the adoption of the penetration emission reduction strategy.

1.4 Proof of Corollary 4

According to Eq. (95), we can get

$${p}_{{r}_{1\infty }}^{N}-{p}_{{r}_{2\infty }}^{N}=\frac{\left({\alpha }_{1}-{\alpha }_{2}\right){G}_{\infty }^{N}-(\beta +\gamma )({w}_{2}-{w}_{1})}{2\left(\beta +\gamma \right)+\gamma }$$

(48)

Thus, we can get the relationship between ${p}_{{r}_{1\infty }}^{N}$ and ${p}_{{r}_{2\infty }}^{N}$, as shown in Corollary 4.

According to Equation (96), we can get

$${D}_{{r}_{1\infty }}^{N}-{D}_{{r}_{2\infty }}^{N}=\frac{(\beta +\gamma )}{2\left(\beta +\gamma \right)+\gamma }\left[\left({\alpha }_{1}-{\alpha }_{2}\right){G}_{\infty }^{N}-(\beta +2\gamma )({w}_{1}-{w}_{2})\right]$$

(49)

Thus, we can get the relationship between ${D}_{{r}_{1\infty }}^{N}$ and ${D}_{{r}_{2\infty }}^{N}$, as shown in Corollary 4.

1.5 Proof of Proposition 5

Letting ${V}_{m}^{C}$, ${V}_{{r}_{1}}^{C},$ and ${V}_{{r}_{2}}^{C}$ represent the value functions of the manufacturer, the large-scale retailer, and the small-scale retailer in cost-sharing strategy, the HJB equations are given by

$$\rho {V}_{m}^{C}=\,\underset{\tau }{\mathrm{max}}\left\{{w}_{1}\cdot \left({\alpha }_{1}G-\beta {p}_{1}+\gamma {p}_{2}-\gamma {p}_{1}\right)+{w}_{2}\cdot \left({\alpha }_{2}G-\beta {p}_{2}+\gamma {p}_{1}-\gamma {p}_{2}\right)-s\left(1-\tau \right){E}_{0}\cdot \left({\alpha }_{1}G+{\alpha }_{2}G-\beta {p}_{1}-\beta {p}_{2}\right)-(1-\varphi )\frac{k}{2}{\tau }^{2}+\frac{\partial {V}_{m}^{C}}{\partial G}\left(\theta \tau -\delta G\right)\right\}$$

(50)

$$\rho {V}_{{r}_{1}}^{C}=\underset{{p}_{1}}{\mathrm{max}}\left\{\left({p}_{1}-{w}_{1}\right)\cdot \left({\alpha }_{1}G-\beta {p}_{1}+\gamma {p}_{2}-\gamma {p}_{1}\right)-\varphi \frac{k}{2}{\tau }^{2}+\frac{\partial {V}_{{r}_{1}}^{C}}{\partial G}\left(\theta \tau -\delta G\right)\right\}$$

(51)

$$\rho {V}_{{r}_{2}}^{C}=\underset{{p}_{2}}{\mathrm{max}}\left\{\left({p}_{2}-{w}_{2}\right)\cdot \left({\alpha }_{2}G-\beta {p}_{2}+\gamma {p}_{2}-\gamma {p}_{2}\right)+\frac{\partial {V}_{{r}_{2}}^{C}}{\partial G}\left(\theta \tau -\delta G\right)\right\}$$

(52)

Using the first-order condition to maximize the right-hand side of Eqs. (51) and (52), we can obtain

$${p}_{{r}_{i}}^{C}=\frac{{\alpha }_{i}G+\gamma {p}_{3-i}+(\beta +\gamma ){w}_{i}}{2(\beta +\gamma )},i=\mathrm{1,2}$$

(53)

Simultaneous Eq. (53) yield Eq. (17).

Substituting Eqs. (17) into the right-hand side of Equation (50), and using the first-order condition to solve the right-hand side with respect to $\tau$, we can obtain Eq. (18).

Substituting Eqs. (17), and (18) into the right-hand side of Eq. (50) yields

$$\rho {V}_{m}^{C}={w}_{1}\left(\beta +\gamma \right)\left[\frac{2\left(\beta +\gamma \right){\alpha }_{1}+\gamma {\alpha }_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}G+\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{1}+\gamma \left(\beta +\gamma \right){w}_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]+{w}_{2}\left(\beta +\gamma \right)\left[\frac{2\left(\beta +\gamma \right){\alpha }_{2}+\gamma {\alpha }_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}G+\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{2}+\gamma \left(\beta +\gamma \right){w}_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]-s{E}_{0}\left[\frac{\left(\beta +\gamma \right)\left({\alpha }_{1}+{\alpha }_{2}\right)}{2\beta +\gamma }G-\frac{\beta \left(\beta +\gamma \right)\left({w}_{1}+{w}_{2}\right)}{2\beta +\gamma }\right]+s{E}_{0}\left[\frac{s{E}_{0}\left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({\alpha }_{1}+{\alpha }_{2}\right)G-\frac{s{E}_{0}\beta \left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)+\frac{\theta }{k\left(1-\varphi \right)}\frac{\partial {V}_{m}^{C}}{\partial G}\right]\left[\frac{\left(\beta +\gamma \right)\left({\alpha }_{1}+{\alpha }_{2}\right)}{2\beta +\gamma }G-\frac{\beta \left(\beta +\gamma \right)\left({w}_{1}+{w}_{2}\right)}{2\beta +\gamma }\right]-(1-\varphi )\frac{k}{2}{\left[\frac{s{E}_{0}\left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({\alpha }_{1}+{\alpha }_{2}\right)G-\frac{s{E}_{0}\beta \left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)+\frac{\theta }{k(1-\varphi )}\frac{\partial {V}_{m}^{C}}{\partial G}\right]}^{2}+\frac{\partial {V}_{m}^{C}}{\partial G}\left(\theta \left[\frac{s{E}_{0}\left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({\alpha }_{1}+{\alpha }_{2}\right)G-\frac{s{E}_{0}\beta \left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)+\frac{\theta }{k\left(1-\varphi \right)}\frac{\partial {V}_{m}^{C}}{\partial G}\right]-\delta G\right)$$

(54)

Conjecture the manufacturer’s value function as a quadratic form, which is expressed as follows:

$${V}_{m}^{C}\left(G\right)={X}_{1}{G}^{2}+{X}_{2}G+{X}_{3}$$

(55)

where ${X}_{1}$, ${X}_{2}$, and ${X}_{3}$ are the coefficients to be determined. From Eq. (55), we have

$$\frac{\partial {V}_{m}^{C}}{\partial G}=2{X}_{1}G+{X}_{2}$$

(56)

By Substituting Eqs. (55) and (56) into (54), and letting the corresponding coefficients of ${G}^{2}$ on both sides of equation be equal, we can obtain

$$-2{(2\beta +\gamma )\theta }^{2}\cdot {{X}_{1}}^{2}+\left[k\left(1-\varphi \right)\left(2\beta +\gamma \right)\left(\rho +2\delta \right)-2\theta s{E}_{0}(\beta +\gamma )({\alpha }_{1}+{\alpha }_{2})\right]\cdot {X}_{1}-\frac{{(s{E}_{0})}^{2}{(\beta +\gamma )}^{2}}{2(2\beta +\gamma )}{({\alpha }_{1}+{\alpha }_{2})}^{2}=0$$

(57)

Solving Eq. (57) yields

$$X_{1} = \frac{{2\theta sE_{0} \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) + \left( {1 - \varphi } \right)\left[ {\Delta_{2} - k\left( {2\beta + \gamma } \right)\left( {\rho + 2\delta } \right)} \right]}}{{ - 4\left( {2\beta + \gamma } \right)\theta^{2} }}$$

(58)

where ${\Delta }_{2}\ge 0$ is required to guarantee the existence of the solution. Note that when ${X}_{1}$ takes a smaller root, the green reputation level will not converge to a steady-state value. Thus, the smaller root is abandoned.

Similarly, ${X}_{2}$ and ${X}_{3}$ are obtained as the following:

$${X}_{2}=\frac{2{A}_{4}}{k(2\beta +\gamma )\rho +{\Delta }_{2}}$$

(59)

$${X}_{3}=\frac{{w}_{1}}{\rho }\left[\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right)(\beta +\gamma ){w}_{1}+\gamma {\left(\beta +\gamma \right)}^{2}{w}_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]+\frac{{w}_{2}}{\rho }\left[\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right)(\beta +\gamma ){w}_{2}+\gamma {\left(\beta +\gamma \right)}^{2}{w}_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]+\frac{s{E}_{0}\beta \left(\beta +\gamma \right)}{\rho \left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)+\frac{{\beta }^{2}{(s{E}_{0})}^{2}}{2k\rho (1-\varphi )}{\left[\frac{\left(\beta +\gamma \right)}{\left(2\beta +\gamma \right)}({w}_{1}+{w}_{2})\right]}^{2}+\frac{2{\theta }^{2}{A}_{4}^{2}}{k\rho \left(1-\varphi \right){\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{2}\right]}^{2}}-\frac{2\beta \theta s{E}_{0}{A}_{4}}{k\rho \left(1-\varphi \right)\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{2}\right]}\frac{\left(\beta +\gamma \right)}{\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)$$

(60)

Substituting Eqs. (17), and (18) into the right-hand side of Equation (51) yields

$$\rho {V}_{{r}_{1}}^{C}=\left(\beta +\gamma \right){\left[\frac{2\left(\beta +\gamma \right){\alpha }_{1}+\gamma {\alpha }_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}G+\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{2}+\gamma \left(\beta +\gamma \right){w}_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]}^{2}-\frac{k\varphi }{2}{\left[\frac{s{E}_{0}\left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({\alpha }_{1}+{\alpha }_{2}\right)G-\frac{s{E}_{0}\beta \left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)+\frac{\theta }{k\left(1-\varphi \right)}\frac{\partial {V}_{m}^{C}}{\partial G}\right]}^{2}+\frac{\partial {V}_{{r}_{1}}^{C}}{\partial G}\left(\theta \left[\frac{s{E}_{0}\left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({\alpha }_{1}+{\alpha }_{2}\right)G-\frac{s{E}_{0}\beta \left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)+\frac{\theta }{k\left(1-\varphi \right)}\frac{\partial {V}_{m}^{C}}{\partial G}\right]-\delta G\right)$$

(61)

Conjecture the large-scale retailer’s value function as a quadratic form, which is expressed as follows:

$${V}_{{r}_{1}}^{C}\left(G\right)={Y}_{1}{G}^{2}+{Y}_{2}G+{Y}_{3}$$

(62)

where ${Y}_{1}$, ${Y}_{2}$, and ${Y}_{3}$ are the coefficients to be determined. From Eq. (62), we have

$$\frac{\partial {V}_{{r}_{1}}^{C}}{\partial G}=2{Y}_{1}G+{Y}_{2}$$

(63)

By Substituting Eqs. (56), (62), and (63) into (61), and letting the corresponding coefficients of ${G}^{2}$ on both sides of equation be equal, we can obtain

$$\left[k\left(2\beta +\gamma \right){\left(1-\varphi \right)}^{2}\left(\rho +2\delta \right)-2s{E}_{0}\theta \left(1-\varphi \right)\left(\beta +\gamma \right)\left({\alpha }_{1}+{\alpha }_{2}\right)-4(1-\varphi )(2\beta +\gamma ){\theta }^{2}{X}_{1}\right]\cdot {Y}_{1}=\frac{k\left(\beta +\gamma \right){\left(1-\varphi \right)}^{2}}{2\left(\beta +\gamma \right)+\gamma }{\left[2\left(\beta +\gamma \right){\alpha }_{1}+\gamma {\alpha }_{2}\right]}^{2}-\frac{\varphi {\left(s{E}_{0}\right)}^{2}{\left(\beta +\gamma \right)}^{2}}{2\left(2\beta +\gamma \right)}\left({\alpha }_{1}+{\alpha }_{2}\right)-2\left(2\beta +\gamma \right)\varphi {\theta }^{2}{X}_{1}^{2}-2\varphi \theta s{E}_{0}(\beta +\gamma )\left({\alpha }_{1}+{\alpha }_{2}\right){X}_{1}$$

(64)

Solving Eq. (64) yields

$$Y_{1} = \frac{\begin{gathered} \frac{{k\left( {\beta + \gamma } \right)\left( {1 - \varphi } \right)^{2} }}{{2\left( {\beta + \gamma } \right) + \gamma }}\left[ {2\left( {\beta + \gamma } \right)\alpha_{1} + \gamma \alpha_{2} } \right]^{2} - \frac{{\varphi \left( {sE_{0} } \right)^{2} \left( {\beta + \gamma } \right)^{2} }}{{2\left( {2\beta + \gamma } \right)}}\left( {\alpha_{1} + \alpha_{2} } \right) \hfill \\ - 2\left( {2\beta + \gamma } \right)\varphi \theta^{2} X_{1}^{2} - 2\varphi \theta sE_{0} \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right)X_{1} \hfill \\ \end{gathered} }{\begin{gathered} k\left( {2\beta + \gamma } \right)\left( {1 - \varphi } \right)^{2} \left( {\rho + 2\delta } \right) \hfill \\ - 2sE_{0} \theta \left( {1 - \varphi } \right)\left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) - 4\left( {1 - \varphi } \right)\left( {2\beta + \gamma } \right)\theta^{2} X_{1} \hfill \\ \end{gathered} }$$

(65)

Substituting Eq. (58) into Eq. (65) yields

$$Y_{1} = \frac{{A_{5} }}{{\left( {1 - \varphi } \right)^{2} \Delta_{2} }}$$

(66)

Similarly, ${Y}_{2}$ and ${Y}_{3}$ are obtained as the following:

$$Y_{2} = \frac{{2A_{6} }}{{k\left( {2\beta + \gamma } \right)\left( {1 - \varphi } \right)^{2} \rho + \left( {1 - \varphi } \right)^{2} \Delta_{2} }}$$

(67)

$${Y}_{3}=\frac{(\beta +\gamma )}{\rho }{\left[\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{1}+\gamma \left(\beta +\gamma \right){w}_{2}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]}^{2}-\frac{\varphi {\beta }^{2}{\left(s{E}_{0}\right)}^{2}}{2k\rho {\left(1-\varphi \right)}^{2}}{\left[\frac{\left(\beta +\gamma \right)}{\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)\right]}^{2}-\frac{\varphi {\theta }^{2}{X}_{2}^{2}}{2k\rho {\left(1-\varphi \right)}^{2}}+\frac{{X}_{2}\varphi \theta s{E}_{0}\beta }{k{\left(1-\varphi \right)}^{2}\rho }\frac{\left(\beta +\gamma \right)}{\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)-\frac{{Y}_{2}\theta s{E}_{0}\beta }{k\rho \left(1-\varphi \right)}\frac{\left(\beta +\gamma \right)}{\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)+\frac{{X}_{2}{Y}_{2}{\theta }^{2}}{k\rho (1-\varphi )}$$

(68)

Substituting Eqs. (17), and (18) into the right-hand side of Eq. (52) yields

$$\rho {V}_{{r}_{2}}^{C}=\left(\beta +\gamma \right){\left[\frac{2\left(\beta +\gamma \right){\alpha }_{2}+\gamma {\alpha }_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}G+\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{2}+\gamma \left(\beta +\gamma \right){w}_{1}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]}^{2}+\frac{\partial {V}_{{r}_{2}}^{C}}{\partial G}\left(\theta \left[\frac{s{E}_{0}\left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({\alpha }_{1}+{\alpha }_{2}\right)G-\frac{s{E}_{0}\beta \left(\beta +\gamma \right)}{k\left(1-\varphi \right)\left(2\beta +\gamma \right)}\left({w}_{1}+{w}_{2}\right)+\frac{\theta }{k\left(1-\varphi \right)}\frac{\partial {V}_{m}^{C}}{\partial G}\right]-\delta G\right)$$

(69)

Conjecture the small-scale retailer’s value function as a quadratic form, which is expressed as follows:

$${V}_{{r}_{2}}^{C}\left(G\right)={Z}_{1}{G}^{2}+{Z}_{2}G+{Z}_{3}$$

(70)

where ${Z}_{1}$, ${Z}_{2}$, and ${Z}_{3}$ are the coefficients to be determined. From Eq. (70), we have

$$\frac{\partial {V}_{{r}_{2}}^{C}}{\partial G}=2{Z}_{1}G+{Z}_{2}$$

(71)

By Substituting Eqs. (56), (70), and (71) into (69), and letting the corresponding coefficients of ${G}^{2}$ on both sides of equation be equal, we can obtain

$$\left[k\left(2\beta +\gamma \right)(1-\varphi )\left(\rho +2\delta \right)-2s{E}_{0}\theta \left(\beta +\gamma \right)\left({\alpha }_{1}+{\alpha }_{2}\right)-4(2\beta +\gamma ){\theta }^{2}{X}_{1}\right]\cdot {Z}_{1}=\frac{k(1-\varphi )\left(\beta +\gamma \right)}{2\left(\beta +\gamma \right)+\gamma }{\left[2\left(\beta +\gamma \right){\alpha }_{2}+\gamma {\alpha }_{1}\right]}^{2}$$

(72)

Solving Eq. (72) yields

$$Z_{1} = \frac{{\frac{{k\left( {1 - \varphi } \right)\left( {\beta + \gamma } \right)}}{{2\left( {\beta + \gamma } \right) + \gamma }}\left[ {2\left( {\beta + \gamma } \right)\alpha_{2} + \gamma \alpha_{1} } \right]^{2} }}{{k\left( {2\beta + \gamma } \right)\left( {1 - \varphi } \right)\left( {\rho + 2\delta } \right) - 2sE_{0} \theta \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) - 4\left( {2\beta + \gamma } \right)\theta^{2} X_{1} }}$$

(73)

Substituting Eq. (58) into Eq.(73) yields

$$Z_{1} = \frac{{k\left( {\beta + \gamma } \right)\left[ {2\left( {\beta + \gamma } \right)\alpha_{2} + \gamma \alpha_{1} } \right]^{2} }}{{\left[ {2\left( {\beta + \gamma } \right) + \gamma } \right]\Delta_{2} }}$$

(74)

Similarly, ${Z}_{2}$ and ${Z}_{3}$ are obtained as the following:

$$Z_{2} = \frac{{2A_{7} }}{{k\left( {2\beta + \gamma } \right)\left( {1 - \varphi } \right)\rho + \left( {1 - \varphi } \right)\Delta_{2} }}$$

(75)

$$\begin{aligned} Z_{3} = & \frac{{\left( {\beta + \gamma } \right)}}{\rho }\left[ {\frac{{\left( { - 2\beta^{2} - 4\beta \gamma - \gamma^{2} } \right)w_{2} + \gamma \left( {\beta + \gamma } \right)w_{1} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right] - \frac{{2\beta \theta sE_{0} A_{7} }}{{k\rho \left( {1 - \varphi } \right)^{2} \left[ {k\left( {2\beta + \gamma } \right)\rho + \left( {1 - \varphi } \right)\Delta_{2} } \right]}}\frac{{\left( {\beta + \gamma } \right)}}{{\left( {2\beta + \gamma } \right)}}\left( {w_{1} + w_{2} } \right) \\ & + \frac{{4\theta^{2} A_{4} A_{7} }}{{k\rho \left[ {k\left( {1 - \varphi } \right)\left( {2\beta + \gamma } \right)\rho + \left( {1 - \varphi } \right)\Delta_{2} } \right]^{2} }} \\ \end{aligned}$$

(76)

1.6 Proof of Corollary 6

Based on Eq. (11) in Proposition 1, and Eq. (17) in Proposition 5, Corollary 6 can easily be obtained.

1.7 Proof of Proposition 7

By substituting Eq. (18) into (1), we obtain the following differential equation:

$$\dot{G}\left(t\right)-\frac{k\left(2\beta +\gamma \right)\rho -{\Delta }_{2}}{2k\left(2\beta +\gamma \right)}G=\frac{2\left(2\beta +\gamma \right){\theta }^{2}{A}_{4}-\beta \theta s{E}_{0}\left(\beta +\gamma \right)\left({w}_{1}+{w}_{2}\right)\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{2}\right]}{k(1-\varphi )(2\beta +\gamma )\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{2}\right]}$$

(77)

Given that $G\left(0\right)={G}_{0}$, solving Eq. (77) yields Eq. (21), where ${G}_{\infty }^{C}$ can also be showed in Eq. (97).

By substituting Eqs. (21) and (97) into (18), we can obtain Eqs. (19) and (98).

By substituting Eqs. (21) and (97) into (17), we can obtain Eqs. (20) and (99).

By substituting Eqs. (21) and (97) into (2), we can obtain Eqs. (22) and (100).

1.8 Proof of Proposition 8

Proposition 8 shows that the monotonicity of retail price relates to ${G}_{0}$ and ${G}_{\infty }^{C}$ for the two retailers. When ${G}_{0}>{G}_{\infty }^{C}$, the retail price decreases with time. Therefore, the skimming pricing strategy should be adopted. When ${G}_{0}<{G}_{\infty }^{C}$, the retail price increases with time, thereby requiring the adoption of the penetration pricing strategy.

Proposition 3 also shows that the monotonicity of emission reduction level relates to ${G}_{0}$, ${G}_{\infty }^{C}$, and $\frac{k(2\beta +\gamma )\left(\rho +2\delta \right)-{\Delta }_{2}}{2k\theta (2\beta +\gamma )}$ for the manufacturer. When $\frac{k(2\beta +\gamma )\left(\rho +2\delta \right)-{\Delta }_{2}}{2k\theta (2\beta +\gamma )}\left({G}_{0}-{G}_{\infty }^{C}\right)>0$, the emission reduction level decreases with time. Therefore, the skimming emission reduction strategy should be adopted. When $\frac{k(2\beta +\gamma )\left(\rho +2\delta \right)-{\Delta }_{2}}{2k\theta (2\beta +\gamma )}\left({G}_{0}-{G}_{\infty }^{C}\right)<0$, the emission reduction level increases with time, thereby meaning the adoption of the penetration emission reduction strategy.

1.9 Proof of Corollary 9

According to Eq. (99), we can obtain

$${p}_{{r}_{1\infty }}^{C}-{p}_{{r}_{2\infty }}^{C}=\frac{\left({\alpha }_{1}-{\alpha }_{2}\right){G}_{\infty }^{C}-(\beta +\gamma )({w}_{2}-{w}_{1})}{2\left(\beta +\gamma \right)+\gamma }$$

(78)

Thus, we can get the relationship between ${p}_{{r}_{1\infty }}^{II}$ and ${p}_{{r}_{3\infty }}^{II}$, as shown in Corollary 9.

According to Eq. (100), we can obtain

$${D}_{{r}_{1\infty }}^{II}-{D}_{{r}_{2\infty }}^{II}=\frac{(\beta +\gamma )}{2\left(\beta +\gamma \right)+\gamma }\left[\left({\alpha }_{1}-{\alpha }_{2}\right){G}_{\infty }^{I}-(\beta +2\gamma )({w}_{1}-{w}_{2})\right]$$

(79)

Thus, we can get the relationship between ${D}_{{r}_{1\infty }}^{C}$ and ${D}_{{r}_{1\infty }}^{C}$, as shown in Corollary 9.

1.10 Proof of Corollary 10

Based on Eqs. (93) and (97), we can easily obtain ${{G}_{\infty }^{N}<G}_{\infty }^{C}$. Meanwhile, based on Eqs. (94) and (98), we can also obtain ${\tau }_{\infty }^{N}<{\tau }_{\infty }^{C}$.

Thus, we get the relationship of the steady-state retail prices, demands in two strategies according to ${{G}_{\infty }^{N}<G}_{\infty }^{C}$ as shown in Corollary 10.

Appendix B

The variable substitutions ${\Delta }_{1},{\Delta }_{2},{\mathrm{A}}_{1}-{\mathrm{A}}_{7},{X}_{1},{X}_{2},{Y}_{1}$, and ${Z}_{1}$ are shown as follows:

$${\Delta }_{1}\equiv \sqrt{k(2\beta +\gamma )(\rho +2\delta )\left[k\left(2\beta +\gamma \right)\left(\rho +2\delta \right)-4\theta s{E}_{0}\left(\beta +\gamma \right)\left({\alpha }_{1}+{\alpha }_{2}\right)\right]}$$

(80)

$${\Delta }_{2}\equiv \sqrt{k(2\beta +\gamma )(\rho +2\delta )\left[k\left(2\beta +\gamma \right)\left(\rho +2\delta \right)-\frac{4\theta s{E}_{0}(\beta +\gamma )({\alpha }_{1}+{\alpha }_{2})}{\left(1-\varphi \right)}\right]}$$

(81)

$$\begin{aligned} A_{1} \equiv\, & kw_{1} \left[ {\frac{{2\left( {\beta + \gamma } \right)^{2} \alpha_{1} + \gamma \left( {\beta + \gamma } \right)\alpha_{2} }}{{2\left( {\beta + \gamma } \right) + \gamma }}} \right] + kw_{{r_{2} }} \left[ {\frac{{2\left( {\beta + \gamma } \right)^{2} \alpha_{2} + \gamma \left( {\beta + \gamma } \right)\alpha_{1} }}{{2\left( {\beta + \gamma } \right) + \gamma }}} \right] \\ & - ksE_{0} \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) + sE_{0} \beta \left( {\beta + \gamma } \right)\left( {w_{{r_{1} }} + w_{{r_{2} }} } \right) \cdot \frac{{\Delta_{1} - k\left( {2\beta + \gamma } \right)\left( {\rho + 2\delta } \right)}}{{2\theta \left( {2\beta + \gamma } \right)}} \\ \end{aligned}$$

(82)

$$\begin{aligned} A_{2} \equiv\, & 2k\left( {2\beta + \gamma } \right)\left( {\beta + \gamma } \right)\left[ {\frac{{\left( { - 2\beta^{2} - 4\beta \gamma - \gamma^{2} } \right)w_{{r_{1} }} + \gamma \left( {\beta + \gamma } \right)w_{{r_{2} }} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{1} + \gamma \alpha_{2} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right] \\ & + \frac{{k\left( {2\beta + \gamma } \right)\left( {\beta + \gamma } \right)}}{{\Delta_{1} }}\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{1} + \gamma \alpha_{2} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]^{2} \left[ {\frac{{4\theta^{2} \left( {2\beta + \gamma } \right)A_{1} }}{{k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} }} - 2sE_{0} \theta \beta \left( {\beta + \gamma } \right)\left( {w_{{r_{1} }} + w_{{r_{2} }} } \right)} \right] \\ \end{aligned}$$

(83)

$$\begin{aligned} A_{3} \equiv\, & 2k\left( {2\beta + \gamma } \right)\left( {\beta + \gamma } \right)\left[ {\frac{{\left( { - 2\beta^{2} - 4\beta \gamma - \gamma^{2} } \right)w_{{r_{2} }} + \gamma \left( {\beta + \gamma } \right)w_{{r_{1} }} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{2} + \gamma \alpha_{1} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right] \\ & + \frac{{k\left( {2\beta + \gamma } \right)\left( {\beta + \gamma } \right)}}{{\Delta_{1} }}\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{2} + \gamma \alpha_{1} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]^{2} \left[ {\frac{{4\theta^{2} \left( {2\beta + \gamma } \right)A_{1} }}{{k\left( {2\beta + \gamma } \right)\rho + \Delta_{1} }} - 2sE_{0} \theta \beta \left( {\beta + \gamma } \right)\left( {w_{{r_{1} }} + w_{{r_{2} }} } \right)} \right] \\ \end{aligned}$$

(84)

$$\begin{aligned} A_{4} \equiv\, & kw_{{r_{1} }} \left[ {\frac{{2\left( {\beta + \gamma } \right)^{2} \alpha_{1} + \gamma \left( {\beta + \gamma } \right)\alpha_{2} }}{{2\left( {\beta + \gamma } \right) + \gamma }}} \right] + kw_{{r_{2} }} \left[ {\frac{{2\left( {\beta + \gamma } \right)^{2} \alpha_{2} + \gamma \left( {\beta + \gamma } \right)\alpha_{1} }}{{2\left( {\beta + \gamma } \right) + \gamma }}} \right] \\ & - ksE_{0} \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) + sE_{0} \beta \left( {\beta + \gamma } \right)\left( {w_{{r_{1} }} + w_{{r_{2} }} } \right) \cdot \frac{{\Delta_{2} - k\left( {2\beta + \gamma } \right)\left( {\rho + 2\delta } \right)}}{{2\theta \left( {2\beta + \gamma } \right)}} \\ \end{aligned}$$

(85)

$$\begin{aligned} A_{5} \equiv\, & \frac{{k\left( {1 - \varphi } \right)^{2} \left( {\beta + \gamma } \right)}}{{2\left( {\beta + \gamma } \right) + \gamma }}\left[ {2\left( {\beta + \gamma } \right)\alpha_{1} + \gamma \alpha_{2} } \right]^{2} - \frac{{\varphi \left( {sE_{0} } \right)^{2} \left( {\beta + \gamma } \right)^{2} }}{{2\left( {2\beta + \gamma } \right)}}\left( {\alpha_{1} + \alpha_{2} } \right)^{3} \\ & - 2\left( {2\beta + \gamma } \right)\varphi \theta^{2} X_{1}^{2} - 2\varphi \theta sE_{0} \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right)X_{1} \\ \end{aligned}$$

(86)

$${A}_{6}\equiv\, 2k{\left(1-\varphi \right)}^{2}\left(\beta +\gamma \right)\left[\frac{2\left(\beta +\gamma \right){\alpha }_{1}+\gamma {\alpha }_{2}}{2\left(\beta +\gamma \right)+\gamma }\right]\left[\frac{\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{{r}_{1}}+\gamma \left(\beta +\gamma \right){w}_{{r}_{2}}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}\right]-2\left(2\beta + \gamma \right)\varphi {\theta }^{2}{X}_{1}{X}_{2}+\frac{\varphi \beta s{E}_{0}{\left(\beta +\gamma \right)}^{2}}{\left(2\beta +\gamma \right)}\left({\alpha }_{1}+{\alpha }_{2}\right)\left({w}_{{r}_{1}}+{w}_{{r}_{2}}\right)-\varphi s{E}_{0}\theta \left(\beta +\gamma \right)\left({\alpha }_{1}+{\alpha }_{2}\right){X}_{2}+ 2\beta s{E}_{0}\theta \varphi \left(\beta +\gamma \right)\left({w}_{{r}_{1}}+{w}_{{r}_{2}}\right){X}_{1}-2\beta s{E}_{0}\theta \left(\beta +\gamma \right)\left(1-\varphi \right)\left({w}_{{r}_{1}}+{w}_{{r}_{2}}\right){Y}_{1}+ 2{\theta }^{2}(1-\varphi )(2\beta +\gamma ){X}_{2}{Y}_{1}$$

(87)

$$\begin{aligned} A_{7} \equiv\, & 2k\left( {1 - \varphi } \right)\left( {\beta + \gamma } \right)\left[ {\frac{{2\left( {\beta + \gamma } \right)\alpha_{2} + \gamma \alpha_{1} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right]\left[ {\frac{{\left( { - 2\beta^{2} - 4\beta \gamma - \gamma^{2} } \right)w_{{r_{2} }} + \gamma \left( {\beta + \gamma } \right)w_{{r_{1} }} }}{{4\left( {\beta + \gamma } \right)^{2} - \gamma^{2} }}} \right] \\ & + 2\left( {2\beta + \gamma } \right)\theta^{2} X_{2} Z_{1} - 2sE_{0} \beta \theta \left( {\beta + \gamma } \right)\left( {w_{{r_{1} }} + w_{{r_{2} }} } \right)Z_{1} \\ \end{aligned}$$

(88)

$$X_{1} { = }\frac{{2\theta sE_{0} \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) + \left( {1 - \varphi } \right)\left[ {\Delta_{2} - k\left( {2\beta + \gamma } \right)\left( {\rho + 2\delta } \right)} \right]}}{{ - 4\left( {2\beta + \gamma } \right)\theta^{2} }}$$

(89)

$$X_{2} = \frac{{2A_{4} }}{{k\left( {2\beta + \gamma } \right)\rho + \Delta_{2} }}$$

(90)

$$Y_{1} = \frac{{A_{5} }}{{\left( {1 - \varphi } \right)^{2} \Delta_{2} }}$$

(91)

$$Z_{1} = \frac{{\frac{{k\left( {1 - \varphi } \right)\left( {\beta + \gamma } \right)}}{{2\left( {\beta + \gamma } \right) + \gamma }}\left[ {2\left( {\beta + \gamma } \right)\alpha_{2} + \gamma \alpha_{1} } \right]^{2} }}{{k\left( {2\beta + \gamma } \right)\left( {1 - \varphi } \right)\left( {\rho + 2\delta } \right) - 2sE_{0} \theta \left( {\beta + \gamma } \right)\left( {\alpha_{1} + \alpha_{2} } \right) - 4\left( {2\beta + \gamma } \right)\theta^{2} X_{1} }}$$

(92)

$${G}_{\infty }^{N}=\frac{4\left(2\beta +\gamma \right){\theta }^{2}{A}_{1}-2\beta \theta s{E}_{0}\left(\beta +\gamma \right)\left({w}_{{r}_{1}}+{w}_{{r}_{2}}\right)\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{1}\right]}{{\Delta }_{1}^{2}-{\left[k(2\beta +\gamma )\rho \right]}^{2}}$$

(93)

$${\tau }_{\infty }^{N}=\frac{k\left(2\beta +\gamma \right)\left(\rho +2\delta \right)-{\Delta }_{1}}{2k\theta (2\beta +\gamma )}{G}_{\infty }^{N}+\frac{2{A}_{1}\theta \left(2\beta +\gamma \right)-s{E}_{0}\beta (\beta +\gamma )({w}_{{r}_{1}}+{w}_{{r}_{2}})\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{1}\right]}{k(2\beta +\gamma )\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{1}\right]}$$

(94)

$${p}_{{r}_{i\infty }}^{N}=\frac{2\left(\beta +\gamma \right){\alpha }_{i}+\gamma {\alpha }_{3-i}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}{G}_{\infty }^{N}+\frac{2{(\beta +\gamma )}^{2}{w}_{{r}_{i}}+\gamma (\beta +\gamma ){w}_{{r}_{3-i}}}{4{(\beta +\gamma )}^{2}-{\gamma }^{2}}, i=\mathrm{1,2}$$

(95)

$${D}_{{r}_{i\infty }}^{N}=\frac{2{\left(\beta +\gamma \right)}^{2}{\alpha }_{i}+\gamma (\beta +\gamma ){\alpha }_{3-i}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}{G}_{\infty }^{I}+\frac{(\beta +\gamma )\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{{r}_{i}}+\gamma {\left(\beta +\gamma \right)}^{2}{w}_{{r}_{3-i}}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}, i=\mathrm{1,2}$$

(96)

$${G}_{\infty }^{C}=\frac{4\left(2\beta +\gamma \right){\theta }^{2}{A}_{4}-2\beta \theta s{E}_{0}\left(\beta +\gamma \right)\left({w}_{{r}_{1}}+{w}_{{r}_{2}}\right)\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{2}\right]}{(1-\varphi )\left[{\Delta }_{2}^{2}-{\left[k(2\beta +\gamma )\rho \right]}^{2}\right]}$$

(97)

$${\tau }_{\infty }^{C}=\frac{k\left(2\beta +\gamma \right)\left(\rho +2\delta \right)-{\Delta }_{2}}{2k\theta (2\beta +\gamma )}{G}_{\infty }^{C}+\frac{2{A}_{4}\theta \left(2\beta +\gamma \right)-s{E}_{0}\beta (\beta +\gamma )({w}_{{r}_{1}}+{w}_{{r}_{2}})\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{2}\right]}{k(1-\varphi )(2\beta +\gamma )\left[k\left(2\beta +\gamma \right)\rho +{\Delta }_{2}\right]}$$

(98)

$${p}_{{r}_{i\infty }}^{C}=\frac{2\left(\beta +\gamma \right){\alpha }_{i}+\gamma {\alpha }_{3-i}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}{G}_{\infty }^{C}+\frac{2{(\beta +\gamma )}^{2}{w}_{{r}_{i}}+\gamma (\beta +\gamma ){w}_{{r}_{3-i}}}{4{(\beta +\gamma )}^{2}-{\gamma }^{2}},i=\mathrm{1,2}$$

(99)

$${D}_{{r}_{i\infty }}^{C }=\frac{2{\left(\beta +\gamma \right)}^{2}{\alpha }_{i}+\gamma (\beta +\gamma ){\alpha }_{3-i}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}{G}_{\infty }^{C}+\frac{(\beta +\gamma )\left(-2{\beta }^{2}-4\beta \gamma -{\gamma }^{2}\right){w}_{{r}_{i}}+\gamma {\left(\beta +\gamma \right)}^{2}{w}_{{r}_{3-i}}}{4{\left(\beta +\gamma \right)}^{2}-{\gamma }^{2}}, i=\mathrm{1,2}$$

(100)