Alternative governmental carbon policies on populations of green and non-green supply chains in a competitive market

Abstract

As a severe challenge to governments, air pollution threatens many lives annually. This paper develops an evolutionary game theory model for supply chains (SCs) to encourage firms to reduce carbon emissions. A government dictates three carbon policies: carbon cap, carbon tax, and cap and trade. We construct six scenarios according to the carbon regulations and SC functions (i.e., centralized versus decentralized ones). First, the profit function of SCs that are the input of the game matrix is formulated, and optimal final prices and quantities of products are obtained. Then, an evolutionary game-theoretic model is formulated to analyze the behavior of populations by obtaining the evolutionary stable strategy under each scenario. In addition, three models of government intervention are presented, considering the cost associated with environmental pollution and government net revenue. Finally, a case study of motorcycle production dissects the results. The findings demonstrate that cap and trade can be introduced as an incentive policy since it encourages SCs to adopt green strategies and improves their profit. A carbon tax policy also succeeds in pressuring SCs to apply green strategies. However, a carbon cap policy fails in forcing SCs to adopt green strategies. Finally, it is concluded that decentralized supply chains (DSCs) handle high tax rates better than centralized supply chains (CSCs).

Appendix

Proof of Proposition 1

To obtain the optimal final price, wholesale price, and the quantity of products, the first division of manufacturers' profit function in GSCs and NGSCs must be calculated:

$$\frac{{{\text{d}}\pi_{{m_{i} }} }}{{{\text{d}}p_{i} }} = {\text{d}}p_{j} + \alpha_{i} + c_{i} - 2p_{i} + w_{i} = 0;\quad i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} ,$$

(56)

$$\frac{{{\text{d}}\pi_{{m_{j} }} }}{{{\text{d}}p_{j} }} = {\text{d}}p_{i} + \alpha_{j} + c_{j} - 2p_{j} + w_{j} = 0;\quad i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} .$$

(57)

Since ${d}^{2}{\pi }_{{m}_{i}}/d{p}_{i}^{2}<0$ and ${d}^{2}{\pi }_{{m}_{j}}/d{p}_{j}^{2}<0$, ${p}_{i}$ and ${p}_{j}$ can be obtained by solving Eqs. 56 and 57, respectively.

$$p_{{\begin{array}{*{20}c} {i } \\ \\ \end{array} }}^{\left( 1 \right)} = A_{i}^{\left( 1 \right)} + w_{i}^{\left( 1 \right)} + c_{i};\quad i = g_{1} ,ng_{1} ,$$

$$p_{{\begin{array}{*{20}c} {j } \\ \\ \end{array} }}^{\left( 1 \right)} = A_{j}^{\left( 1 \right)} + w_{j}^{\left( 1 \right)} + c_{j} ;\quad j = g_{2} ,ng_{2} .$$

Consequently, ${q}_{i}$ and ${q}_{j}$ are obtained by solving ${q}_{i}={\alpha }_{i}-{p}_{i}+d{p}_{j}$ and ${q}_{j}={\alpha }_{j}-{p}_{j}+d{p}_{i}$, respectively.

$$\begin{array}{*{20}c} {q_{i} = A_{i}^{{\left( 1 \right)}} ;} & {i = g_{1} ,ng_{1} ,} \\ {q_{j} = A_{j}^{{\left( 1 \right)}} ;} & {j = g_{2} ,ng_{2} } \\ \end{array}$$

Stackelberg follower’s best response is achieved. Therefore, this should be substituted in the supplier profit function to obtain the optimal wholesale price. The first division of the suppliers’ profit function is as follows:

$$\frac{{{\text{d}}\pi_{{s_{i} }} }}{{{\text{d}}w_{i} }} = \frac{{\left( {C_{i} - c_{i} - 2w_{i} } \right)d^{2} + \left( { - c_{j} - w_{j} - \alpha_{j} } \right)d - 2C_{i} + 2c_{i} + 4w_{i} - 2\alpha_{i} }}{{d^{2} - 4}} = 0,$$

(58)

$$\frac{{d\pi_{{s_{j} }} }}{{dw_{j} }} = \frac{{\left( {C_{j} - c_{j} - 2w_{j} } \right)d^{2} + \left( { - c_{i} - w_{i} - \alpha_{i} } \right)d - 2C_{j} + 2c_{j} + 4w_{j} - 2\alpha_{j} }}{{d^{2} - 4}} = 0,$$

(59)

where $i={g}_{1},{ng}_{1}, j={g}_{2},{ng}_{2}$.

Since ${d}^{2}{\pi }_{{s}_{i}}/d{w}_{i}^{2}<0$ and ${d}^{2}{\pi }_{{s}_{j}}/d{w}_{j}^{2}<0$, ${w}_{i}$ and ${w}_{j}$ can be obtained by solving Eqs. 58 and 59, respectively.

$$w_{i}^{\left( 1 \right)} = B_{i}^{\left( 1 \right)} + C_{i} ;\;\;i = g_{1} ,ng_{1} ,$$

$$w_{j}^{\left( 1 \right)} = B_{j}^{\left( 1 \right)} + C_{j} ;\;\;j = g_{2} ,ng_{2} .$$

Proof of Proposition 2

To obtain the optimal final price and quantity of products, the first division of GSC and NGSC profit functions must be calculated:

$$\frac{{{\text{d}}\pi_{i} }}{{{\text{d}}p_{i} }} = dp_{j} + C_{i} + \alpha_{i} + c_{i} - 2p_{i} = 0;\quad i = g_{1} .ng_{1} , j = g_{2} , ng_{2} ,$$

(60)

$$\frac{{{\text{d}}\pi_{j} }}{{{\text{d}}p_{j} }} = {\text{d}}p_{i} + C_{j} + \alpha_{j} + c_{j} - 2p_{j} = 0; \quad i = g_{1} ,ng_{1} , j = g_{2} , ng_{2} .$$

(61)

Since ${d}^{2}{\pi }_{i}/d{p}_{i}^{2}<0$ and ${d}^{2}{\pi }_{j}/d{p}_{j}^{2}<0$, ${p}_{i}$ and ${p}_{j}$ can be obtained by solving Eq. (60) and Eq. (61), respectively.

$$p_{i}^{\left( 2 \right)} = A_{i}^{\left( 2 \right)} + C_{i} + c_{i} ; \quad i = g_{1} ,ng_{1} ,$$

$$p_{j}^{\left( 2 \right)} = A_{j}^{\left( 2 \right)} + C_{j} + c_{j} ; \quad j = g_{2} , ng_{2} .$$

Consequently, ${q}_{i}$ and ${q}_{j}$ are obtained by solving ${q}_{i}={\alpha }_{i}-{p}_{i}+d{p}_{j}$ and ${q}_{j}={\alpha }_{j}-{p}_{j}+d{p}_{i}$, respectively.

$$q_{i} = A_{i}^{\left( 2 \right)} ; \quad i = g_{1} ,ng_{1} ,\quad j = g_{2} ,ng_{2} ,$$

$$q_{j} = A_{j}^{\left( 2 \right)} ; i = g_{1} ,ng_{1} ,\quad j = g_{2} ,ng_{2} .$$

Proof of Proposition 3

To obtain the optimal final price, wholesale price, and the quantity of products, the first division of manufacturers' profit functions in GSCs and NGSCs must be calculated:

$$\frac{{{\text{d}}\pi_{{m_{i} }} }}{{{\text{d}}p_{i} }} = \frac{1}{50} \left( {e_{i}^{{{\text{tsm}}}} + 50e_{i}^{{{\text{pro}}}} + \frac{1}{2}e_{i}^{{{\text{tmm}}}} } \right)\rho C_{c} + dp_{j} + \alpha_{i} + c_{i} - 2p_{i} + w_{i} = 0,$$

(62)

$$\frac{{{\text{d}}\pi_{{m_{j} }} }}{{{\text{d}}p_{j} }} = \frac{1}{50} \left( {e_{j}^{{{\text{tsm}}}} + 50e_{j}^{{{\text{pro}}}} + \frac{1}{2}e_{j}^{{{\text{tmm}}}} } \right)\rho C_{c} + dp_{i} + \alpha_{j} + c_{j} - 2p_{j} + w_{j} = 0,$$

(63)

where $i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} .$

Since ${\mathrm{d}}^{2}{\pi }_{{m}_{i}}/\mathrm{d}{p}_{i}^{2}<0$ as well as ${\mathrm{d}}^{2}{\pi }_{{m}_{j}}/\mathrm{d}{p}_{j}^{2}<0$, ${p}_{i}$ and ${p}_{j}$ can be obtained by solving Eqs. (62) and (63), respectively.

$$p_{i}^{\left( 3 \right)} = A_{i}^{\left( 3 \right)} + w_{i} + c_{i} + M_{{m_{i} }} ;\quad i = g_{1} ,ng_{1} ,$$

$$p_{j}^{\left( 3 \right)} = A_{j}^{\left( 3 \right)} + w_{j} + c_{j} + M_{{m_{j} }} ;\quad j = g_{2} ,ng_{2} .$$

Consequently, ${q}_{i}$ and ${q}_{j}$ are obtained by solving ${q}_{i}={\alpha }_{i}-{p}_{i}+d{p}_{j}$ and ${q}_{j}={\alpha }_{j}-{p}_{j}+d{p}_{i}$, respectively.

$$q_{i} = A_{i}^{\left( 3 \right)} \;\;\;\quad i = g_{1} ,ng_{1} ,$$

$$q_{j} = A_{j}^{\left( 3 \right)} \;\quad j = g_{2} ,ng_{2} .$$

Stackelberg follower’s best response is achieved. Therefore, this should be substituted in the suppliers’ profit function to obtain optimal wholesale prices. Since $\frac{{\mathrm{d}}^{2}{\pi }_{{s}_{i}}}{\mathrm{d}{w}_{i}^{2}}<0$ and ${\mathrm{d}}^{2}{\pi }_{{s}_{j}}/\mathrm{d}{w}_{j}^{2}<0$, ${w}_{i}$ and ${w}_{j}$ can be obtained by solving $\mathrm{d}{\pi }_{{s}_{i}}/\mathrm{d}{w}_{i}=$ 0 and $\mathrm{d}{\pi }_{{s}_{j}}/\mathrm{d}{w}_{j}=0$. The optimal wholesale prices are presented below:

$$w_{i}^{\left( 3 \right)} = B_{i}^{\left( 3 \right)} + C_{i} + M_{{s_{i} }} \quad\; i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} ,$$

$$w_{j}^{\left( 3 \right)} = B_{j}^{\left( 3 \right)} + C_{j} + M_{{s_{j} }} \quad \;\;\;i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} .$$

Proof of Proposition 4

To obtain the optimal final price and the quantity of products, the first division of GSCs' and NGSCs' profit functions must be calculated:

$$\frac{{{\text{d}}\pi_{i} }}{{{\text{d}}p_{i} }} = {\text{d}}p_{j} + C_{i} + \alpha_{i} - 2p_{i} + c_{i} - C_{c} \left( { - e_{i}^{{{\text{pro}}}} - \frac{1}{50}e_{i}^{{{\text{tsm}}}} - \frac{1}{100}e_{i}^{{{\text{tmm}}}} } \right) = 0,$$

(64)

$$\frac{{{\text{d}}\pi_{j} }}{{{\text{d}}p_{j} }} = {\text{d}}p_{i} + C_{j} + \alpha_{j} - 2p_{j} + c_{j} - C_{c} \left( { - e_{j}^{{{\text{pro}}}} - \frac{1}{50}e_{j}^{{{\text{tsm}}}} - \frac{1}{100}e_{j}^{{{\text{tmm}}}} } \right) = 0$$

(65)

where $i={g}_{1},{ng}_{1}$ and $j={g}_{2},{ng}_{2}$.

Since ${d}^{2}{\pi }_{i}/d{p}_{i}^{2}<0$ and ${d}^{2}{\pi }_{j}/d{p}_{j}^{2}<0$, ${p}_{i}$ and ${p}_{j}$ can be obtained by solving Eqs. (64) and (65), respectively.

$$p_{i}^{\left( 4 \right)} = A_{i}^{\left( 4 \right)} + C_{i} + M_{i} + c_{i} ;\;\;i = g_{1} ,ng_{1} ,$$

$$p_{j}^{\left( 4 \right)} = A_{j}^{\left( 4 \right)} + C_{j} + M_{j} + c_{j} ;\;\;j = g_{2} ,ng_{2} .$$

Consequently, ${q}_{i}$ and ${q}_{j}$ are obtained by solving ${q}_{i}={\alpha }_{i}-{p}_{i}+d{p}_{j}$ and ${q}_{j}={\alpha }_{j}-{p}_{j}+d{p}_{i}$, respectively.

$$q_{i} = A_{i}^{\left( 4 \right)} \; i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} ,$$

$$q_{j} = A_{j}^{\left( 4 \right)} \; i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} .$$

Proof of Proposition 5

To obtain the optimal final price, wholesale price, and the quantity of products, the first division of manufacturers’ profit functions in GSCs and NGSCs must be calculated:

$$\frac{{{\text{d}}\pi_{{m_{i} }} }}{{{\text{d}}p_{i} }} = {\text{d}}p_{j} + \alpha_{i} - 2p_{i} + w_{i} + c_{i} - \theta C_{T} \left( { - e_{i}^{{{\text{prp}}}} - \frac{1}{50}e_{i}^{{{\text{tsm}}}} - \frac{1}{100}e_{i}^{{{\text{tmm}}}} } \right),$$

(66)

$$\frac{{d\pi_{{m_{j} }} }}{dj} = dp_{i} + \alpha_{j} - 2p_{j} + w_{j} + c_{j} - \theta C_{T} \left( { - e_{j}^{{{\text{prp}}}} - \frac{1}{50}e_{j}^{{{\text{tsm}}}} - \frac{1}{100}e_{j}^{{{\text{tmm}}}} } \right),$$

(67)

where $i={g}_{1},{ng}_{1}$ and $j={g}_{2},{ng}_{2}$.

Since ${\mathrm{d}}^{2}{\pi }_{{m}_{i}}/\mathrm{d}{p}_{i}^{2}<0$ and ${\mathrm{d}}^{2}{\pi }_{{m}_{j}}/\mathrm{d}{p}_{j}^{2}<0$, ${p}_{i}$ and ${p}_{j}$ can be obtained by solving Eqs. (66) and (67), respectively.

$$p_{i}^{\left( 5 \right)} = A_{i}^{\left( 5 \right)} + w_{i}^{\left( 5 \right)} + N_{{m_{i} }} + c_{i}\quad \;\;\;i = g_{1} ,ng_{1} ,$$

$$p_{j}^{\left( 5 \right)} = A_{j}^{\left( 5 \right)} + w_{j}^{\left( 5 \right)} + N_{{m_{j} }} + c_{j}\quad\; j = g_{2} ,ng_{2} .$$

Stackelberg follower’s best response is achieved. Therefore, this should be substituted in the supplier profit function to obtain optimal wholesale prices. Since ${\mathrm{d}}^{2}{\pi }_{{s}_{i}}/\mathrm{d}{w}_{i}^{2}<0$ and ${\mathrm{d}}^{2}{\pi }_{{s}_{j}}/\mathrm{d}{w}_{j}^{2}<0$, ${w}_{i}$ and ${w}_{j}$ can be obtained by solving $\mathrm{d}{\pi }_{{s}_{i}}/\mathrm{d}{w}_{i}=$ 0 and $\mathrm{d}{\pi }_{{s}_{j}}/\mathrm{d}{w}_{j}=0$. The optimal wholesale prices are presented below:

$$w_{i}^{\left( 5 \right)} = B_{i}^{\left( 5 \right)} + C_{i} + N_{{s_{i} }} , i = g_{1} ,ng_{1} ,$$

$$w_{j}^{\left( 5 \right)} = B_{j}^{\left( 5 \right)} + C_{j} + N_{{s_{j} }} ,\;\;j = g_{2} ,ng_{2}$$

Proof of Proposition 6

To obtain the optimal final price and the quantity of products, the first division of GSCs' and NGSCs’ profit functions must be calculated:

$$\frac{{{\text{d}}\pi_{i} }}{{{\text{d}}p_{i} }} = {\text{d}}p_{j} + C_{i} + \alpha_{i} - 2p_{i} + c_{i} - C_{T} \left( { - e_{i}^{{{\text{pro}}}} - \frac{1}{50}e_{i}^{{{\text{tsm}}}} - \frac{1}{100}e_{i}^{{{\text{tmm}}}} } \right),$$

(68)

$$\frac{{{\text{d}}\pi_{j} }}{{{\text{d}}p_{j} }} = {\text{d}}p_{i} + C_{j} + \alpha_{j} - 2p_{j} + c_{j} - C_{T} \left( { - e_{j}^{{{\text{pro}}}} - \frac{1}{50}e_{j}^{{{\text{tsm}}}} - \frac{1}{100}e_{j}^{{{\text{tmm}}}} } \right),$$

(69)

where $i={g}_{1},{ng}_{1}$ and $j={g}_{2},{ng}_{2}.$

Since ${\mathrm{d}}^{2}{\pi }_{i}/\mathrm{d}{p}_{i}^{2}<0$ and ${\mathrm{d}}^{2}{\pi }_{j}/\mathrm{d}{p}_{j}^{2}<0$, ${p}_{i}$ and ${p}_{j}$ can be obtained by solving Eqs. (68) and (69), respectively.

$$\begin{gathered} p_{i}^{{\left( 6 \right)}} = A_{i}^{{\left( 6 \right)}} + C_{i} + N_{i} + c_{i} ;\;\;i~ = g_{1} ,ng_{1} , \hfill \\ p_{j}^{{\left( 6 \right)}} = A_{j}^{{\left( 6 \right)}} + C_{j} + N_{j} + c_{j} ;\;\;j~ = g_{2} ,ng_{2} . \hfill \\ \end{gathered}$$

Consequently, ${q}_{i}$ and ${q}_{j}$ are obtained by solving ${q}_{i}={\alpha }_{i}-{p}_{i}+d{p}_{j}$ and ${q}_{j}={\alpha }_{j}-{p}_{j}+d{p}_{i}$, respectively.

$$q_{i} = A_{i}^{\left( 6 \right)} ; i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} ,$$

$$q_{j} = A_{j}^{\left( 6 \right)} ; i = g_{1} ,ng_{1} , j = g_{2} ,ng_{2} .$$

See Tables 4, 5, 6, 7 and 8.