The optimal product pricing and carbon emissions reduction profit allocation of CET-covered enterprises in the cooperative supply chain

Abstract

The carbon quota allocation rules of China’s pilot carbon emissions trading (CET) regions are various, which mainly include benchmarking, historical carbon intensity reduction and auctioning. When the allocation rules change, it is unresolved how to achieve the optimal product prices and effectively allocate the carbon emissions reduction profits of CET-covered enterprises in the cooperative supply chain. Thus, this paper uses the Stackelberg game, Nash equilibrium and the Shapley value based on cost modification to investigate these issues. The results indicate that: (1) The increasing carbon prices can always improve the retail prices only under the auctioning rule. Meanwhile, the growing low-carbon awareness of consumer cannot be always conducive to improving the wholesale and retail prices, and the similar product prices of non-CET-covered enterprises have greater impact on the wholesale prices than that on the retail prices. (2) Only under the free carbon quota allocation rules, can the optimal wholesale and retail prices under the Stackelberg game be always higher than those under the Nash equilibrium. Meanwhile, the auctioning rule can better reduce carbon emissions than the free allocation rules. (3) Improving carbon emissions reduction contribution and emission reduction costs can be conducive to increasing the carbon emission reduction profits of the supplier and retailer, while the impact of carbon emission reduction contribution on improving the carbon emission reduction profits is not always greater than that of the carbon emission reduction costs.

Appendices

Appendix A1

According to Eq. (7), the first-order partial derivatives of the payment utility function ($V$) about the product demands of the CET-covered and similar non-CET-covered enterprises ($q$ and $q_{0}$) can be obtained as Eqs. (A1-1) and (A1-2), respectively.

$$ q^{*} = \frac{{\alpha * \beta _{0} - \alpha _{0} * \gamma }}{{\beta * \beta _{0} - \gamma ^{2} }} - \frac{{\beta _{0} }}{{\beta * \beta _{0} - \gamma ^{2} }} * p_{r} + \frac{\gamma }{{\beta * \beta _{0} - \gamma ^{2} }} * p_{0} - \frac{{\beta _{0} }}{{\beta * \beta _{0} - \gamma ^{2} }} * k * \left( {\bar{e}_{{mr}} - \Delta e_{{mr}} } \right) + \frac{\gamma }{{\beta * \beta _{0} - \gamma ^{2} }} * k * e_{0} $$

(A1-1)

$$ q_{0}^{*} = \frac{{\alpha _{0} * \beta - \alpha * \gamma - \beta * p_{0} + \gamma * p_{r} - k * \beta * e_{0} + k * \gamma * \left( {\bar{e}_{{mr}} - \Delta e_{{mr}} } \right)}}{{\beta * \beta _{0} - \gamma ^{2} }} $$

(A1-2)

Then, we can obtain the following equations: $\frac{{\partial ^{2} V}}{{\partial q^{2} }} = - \beta$, $\frac{{\partial ^{2} V}}{{\partial q_{0}^{2} }} = - \beta _{0}$, $\frac{{\partial ^{2} V}}{{\partial q * q_{0} }} = - \gamma$ and $\frac{{\partial ^{2} V}}{{\partial q_{0} * q}} = - \gamma$, i.e., $\nabla V_{{(q,q_{0} )}} = \left| {\begin{array}{*{20}c} { - \beta } & { - \gamma } \\ { - \gamma } & { - \beta _{0} } \\ \end{array} } \right|$. Meanwhile, since $- \beta < 0$ and $\beta * \beta _{0} - \gamma ^{2} > 0$, thus the odd order sequential principal minor is less than zero, and the even order sequential principal minor is greater than zero, that is, $\nabla V_{{(q,q_{0} )}}$ is a negative definite matrix. Thus, $\left( {q^{ * } ,q_{0}^{ * } } \right)$ is the optimal solution.

Appendix A2

Under the Stackelberg game, we can first obtain the optimal marginal profit of the retailer ($\rho _{r}^{*}$) based on Eq. (A2-1).

$$ \frac{{\partial \Pi _{r} }}{{\partial \rho _{r} }} = 0 $$

(A2-1)

Then, the supplier determines its optimal marginal profit ($\rho _{m}^{*}$) based on the response function of the retailer ($\rho _{r}^{*}$) obtained from Eq. (A2-1), which can be obtained according to Eq. (A2-2).

$$ \frac{{\partial \Pi _{m} }}{{\partial \rho _{m} }} = 0 $$

(A2-2)

Based on the Eqs. (A2-1) and (A2-2), this paper obtains the optimal marginal profits of the supplier and retailer under three carbon quota allocation rules, and the results are shown in Table 3.

Table 3 The optimal marginal profits of the supplier and retailer under the Stackelberg game

Besides, Under the Stackelberg game and three carbon quota allocation rules, the second-order partial derivatives of the profit of the retailer ($\Pi _{r}$) about the marginal profit ($\rho _{r}$) are less than zero, as shown in Eqs. (A2-3)–(A2-8), respectively.

Under the benchmarking rule:

$$ \frac{{\partial \Pi _{r} }}{{\partial \rho _{r} }} = a - \rho _{m} - 2\rho _{r} - v_{{mr}} + c * p_{0} - k * \bar{e}_{{mr}} + k * \Delta e_{{mr}} + k * c * e_{0} + \left( {1 - t} \right) * h * \Delta e_{{mr}}^{2} + p_{c} * \left[ {\bar{e}_{r} - E_{{\sec tor}}^{r} + \left( {\lambda - 1} \right) * \Delta e_{{mr}} } \right] $$

(A2-3)

$$ \frac{{\partial ^{2} \Pi _{r} }}{{\partial \rho _{r}^{2} }} = - 2 < 0 $$

(A2-4)

Under the historical carbon intensity reduction rule:

$$ \frac{{\partial \Pi _{r} }}{{\partial \rho _{r} }} = a - \rho _{m} - 2\rho _{r} - v_{{mr}} + c * p_{0} - k * \bar{e}_{{mr}} + k * \Delta e_{{mr}} + k * c * e_{0} + \left( {1 - t} \right) * h * \Delta e_{{mr}}^{2} + p_{c} * \left[ {\bar{e}_{r} - l_{r} * E_{i}^{r} + \left( {\lambda - 1} \right) * \Delta e_{{mr}} } \right] $$

(A2-5)

$$ \frac{{\partial ^{2} \Pi _{r} }}{{\partial \rho _{r}^{2} }} = - 2 < 0 $$

(A2-6)

Under the auctioning rule:

$$ \frac{{\partial \Pi _{r} }}{{\partial \rho _{r} }} = a - \rho _{m} - 2\rho _{r} - v_{{mr}} + c * p_{0} - k * \bar{e}_{{mr}} + k * \Delta e_{{mr}} + k * c * e_{0} + \left( {1 - t} \right) * h * \Delta e_{{mr}}^{2} + p_{c} * \left[ {\bar{e}_{r} + \left( {\lambda - 1} \right) * \Delta e_{{mr}} } \right] $$

(A2-7)

$$ \frac{{\partial ^{2} \Pi _{r} }}{{\partial \rho _{r}^{2} }} = - 2 < 0 $$

(A2-8)

Thus, $\rho _{r}^{ * }$ is the optimal solution.

Under the Stackelberg game and three carbon quota allocation rules, the second-order partial derivatives of the profit of the supplier ($\Pi _{m}$) about the marginal profit ($\rho _{m}$) are also less than zero, as shown in Eqs. (A2-9)–(A2-14), respectively.

Under the benchmarking rule:

$$ \frac{{\partial \Pi _{m} }}{{\partial \rho _{m} }} = a - 1.5\rho _{m} - \rho _{r} - v_{{mr}} + c * p_{0} - k * \bar{e}_{{mr}} + k * \Delta e_{{mr}} + k * c * e_{0} + 0.5t * h * \Delta e_{{mr}}^{2} + 0.5p_{c} * \left[ {\bar{e}_{m} - E_{{\sec tor}}^{m} - \lambda * \Delta e_{{mr}} } \right] $$

(A2-9)

$$ \frac{{\partial ^{2} \Pi _{m} }}{{\partial \rho _{m}^{2} }} = - 1 < 0 $$

(A2-10)

Under the historical carbon intensity reduction rule:

$$ \frac{{\partial \Pi _{m} }}{{\partial \rho _{m} }} = a - 1.5\rho _{m} - \rho _{r} - v_{{mr}} + c * p_{0} - k * \bar{e}_{{mr}} + k * \Delta e_{{mr}} + k * c * e_{0} + 0.5t * h * \Delta e_{{mr}}^{2} + 0.5p_{c} * \left[ {\bar{e}_{m} - l_{m} * E_{i}^{m} - \lambda * \Delta e_{{mr}} } \right] $$

(A2-11)

$$ \frac{{\partial ^{2} \Pi _{m} }}{{\partial \rho _{m}^{2} }} = - 1 < 0 $$

(A2-12)

Under the auctioning rule:

$$ \frac{{\partial \Pi _{m} }}{{\partial \rho _{m} }} = a - 1.5\rho _{m} - \rho _{r} - v_{{mr}} + c * p_{0} - k * \bar{e}_{{mr}} + k * \Delta e_{{mr}} + k * c * e_{0} + 0.5t * h * \Delta e_{{mr}}^{2} + 0.5p_{c} * \left[ {\bar{e}_{m} - \lambda * \Delta e_{{mr}} } \right] $$

(A2-13)

$$ \frac{{\partial ^{2} \Pi _{m} }}{{\partial \rho _{m}^{2} }} = - 1 < 0 $$

(A2-14)

Thus, $\rho _{m}^{ * }$ is the optimal solution.

Appendix A3

Under the Nash equilibrium, the optimal marginal profits of the supplier and retailer can be jointly determined by Eqs. (A3-1) and (A3-2), and the results are shown in Table 4.

$$ \frac{{\partial \Pi _{m} }}{{\partial \rho _{m} }} = 0 $$

(A3-1)

$$ \frac{{\partial \Pi _{r} }}{{\partial \rho _{r} }} = 0 $$

(A3-2)

Table 4 The optimal marginal profits of the supplier and retailer under the Nash equilibrium

Under the Nash equilibrium and three carbon quota allocation rules, the second-order partial derivatives of the profits of the supplier and retailer ($\Pi _{m}$ and $\Pi _{r}$) about their marginal profits ($\rho _{m}$ and $\rho _{r}$) are less than zero, as shown in Eqs. (A3-3)–(A3-14), respectively.

Under the benchmarking rule:

$$ \frac{{\partial \Pi _{m} }}{{\partial \rho _{m} }} = q^{*} + \left[ {\rho _{m} - t * h * \Delta e_{{mr}}^{2} } \right] * \frac{{\partial q^{*} }}{{\partial \rho _{m} }} + p_{c} * \left[ {E_{{\sec tor}}^{m} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }} - \bar{e}_{m} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }}{\text{ + }}\lambda * \Delta e_{{mr}} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }}} \right] $$

(A3-3)

$$ \frac{{\partial \Pi _{r} }}{{\partial \rho _{r} }} = q^{*} + \left[ {\rho _{r} - \left( {1 - t} \right) * h * \Delta e_{{mr}}^{2} } \right] * \frac{{\partial q^{*} }}{{\partial \rho _{r} }} + p_{c} * \left[ {E_{{\sec tor}}^{r} * \frac{{\partial q^{*} }}{{\partial \rho _{r} }} - \bar{e}_{r} * \frac{{\partial q^{*} }}{{\partial \rho _{r} }}{\text{ + }}\left( {1 - \lambda } \right) * \Delta e_{{mr}} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }}} \right] $$

(A3-4)

$$ \frac{{\partial ^{2} \Pi _{m} }}{{\partial \rho _{m}^{2} }} = - 2 < 0 $$

(A3-5)

$$ \frac{{\partial ^{2} \Pi _{m} }}{{\partial \rho _{m}^{2} }} = - 2 < 0 $$

(A3-6)

Under the historical carbon intensity reduction rule:

$$ \frac{{\partial \Pi _{m} }}{{\partial \rho _{m} }} = q^{*} + \left[ {\rho _{m} - t * h * \Delta e_{{mr}}^{2} } \right] * \frac{{\partial q^{*} }}{{\partial \rho _{m} }} + p_{c} * \left[ {l_{m} * E_{i}^{m} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }} - \bar{e}_{m} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }}{\text{ + }}\lambda * \Delta e_{{mr}} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }}} \right] $$

(A3-7)

$$ \frac{{\partial \Pi _{r} }}{{\partial \rho _{r} }} = q^{*} + \left[ {\rho _{r} - \left( {1 - t} \right) * h * \Delta e_{{mr}}^{2} } \right] * \frac{{\partial q^{*} }}{{\partial \rho _{r} }} + p_{c} * \left[ {l_{r} * E_{i}^{r} * \frac{{\partial q^{*} }}{{\partial \rho _{r} }} - \bar{e}_{r} * \frac{{\partial q^{*} }}{{\partial \rho _{r} }}{\text{ + }}\left( {1 - \lambda } \right) * \Delta e_{{mr}} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }}} \right] $$

(A3-8)

$$ \frac{{\partial ^{2} \Pi _{m} }}{{\partial \rho _{m}^{2} }} = - 2 < 0 $$

(A3-9)

$$ \frac{{\partial ^{2} \Pi _{r} }}{{\partial \rho _{r}^{2} }} = - 2 < 0 $$

(A3-10)

Under the auctioning rule:

$$ \frac{{\partial \Pi _{m} }}{{\partial \rho _{m} }} = q^{*} + \left[ {\rho _{m} - t * h * \Delta e_{{mr}}^{2} } \right] * \frac{{\partial q^{*} }}{{\partial \rho _{m} }} + p_{c} * \left[ {\lambda * \Delta e_{{mr}} - \bar{e}_{m} } \right] * \frac{{\partial q^{*} }}{{\partial \rho _{m} }} $$

(A3-11)

$$ \frac{{\partial \Pi _{r} }}{{\partial \rho _{r} }} = q^{*} + \left[ {\rho _{r} - \left( {1 - t} \right) * h * \Delta e_{{mr}}^{2} } \right] * \frac{{\partial q^{*} }}{{\partial \rho _{r} }} + p_{c} * \left[ {l_{r} * E_{i}^{r} * \frac{{\partial q^{*} }}{{\partial \rho _{r} }} - \bar{e}_{r} * \frac{{\partial q^{*} }}{{\partial \rho _{r} }}{\text{ + }}\left( {1 - \lambda } \right) * \Delta e_{{mr}} * \frac{{\partial q^{*} }}{{\partial \rho _{m} }}} \right] $$

(A3-12)

$$ \frac{{\partial ^{2} \Pi _{m} }}{{\partial \rho _{m}^{2} }} = - 2 < 0 $$

(A3-13)

$$ \frac{{\partial ^{2} \Pi _{r} }}{{\partial \rho _{r}^{2} }} = - 2 < 0 $$

(A3-14)

Thus, $\rho _{m}^{ * }$ and $\rho _{r}^{ * }$ are the optimal solutions.

Appendix A4

No matter the cooperative supply chain chooses the Stackelberg game or Nash equilibrium, this paper can obtain the optimal carbon emissions reduction based on that the marginal product profits are equal to marginal costs, as shown in Eq. (A4-1).

$$ \rho _{m}^{*} + \rho _{m}^{*} = v_{{mr}} + h * \Delta e_{{mr}}^{2} $$

(A4-1)

Thus, when the cooperative supply chain chooses the Stackelberg game, we can further obtain the optimal carbon emissions reduction under the benchmarking rule, historical carbon intensity reduction rule, and auctioning rule, respectively, as shown Eqs. (A4-2)–(A4-4).

$$ \begin{aligned} \rho _{m}^{*} + \rho _{m}^{*} & = 0.75\left[ {a - v_{{mr}} + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)} \right] + 0.25 * h * \Delta e_{{mr}}^{2} - 0.25 * p_{c} * \Delta e_{{mr}} \\ & \quad + 0.25p_{c} * \left( {\bar{e}_{m} + \bar{e}_{r} - E_{{\sec otr}}^{m} - E_{{\sec otr}}^{r} } \right) = v_{{mr}} + h * \Delta e_{{mr}}^{2} \\ \end{aligned} $$

(A4-2)

$$ \begin{aligned} \rho _{m}^{*} + \rho _{r}^{*} & = 0.75\left[ {a - v_{{mr}} + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)} \right] + 0.25 * h * \Delta e_{{mr}}^{2} - 0.25 * p_{c} * \Delta e_{{mr}} \\ & \quad + 0.25 * p_{c} * \left( {\bar{e}_{m} + \bar{e}_{r} - l_{m} * E_{i}^{m} - l_{r} * E_{i}^{r} } \right) = v_{{mr}} + h * \Delta e_{{mr}}^{2} \\ \end{aligned} $$

(A4-3)

$$ \begin{aligned} \rho _{m}^{*} + \rho _{r}^{*} & = 0.75\left[ {a - v_{{mr}} + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)} \right] + 0.25p_{c} * \left( {\bar{e}_{m} - \bar{e}_{r} } \right) \\ & \quad + 0.25h * \Delta e_{{mr}}^{2} - 0.25 * p_{c} * \Delta e_{{mr}} = v_{{mr}} + h * \Delta e_{{mr}}^{2} \\ \end{aligned} $$

(A4-4)

The Eqs. (A4-2)–(A4-4) are quadratic function about the carbon emissions reduction (i.e., $\Delta e_{{mr}}$), which have two optimal solutions. In order to improve the carbon emissions reduction, this paper chooses the greater optimal solution as the carbon emissions reduction target, which are shown in Table 1.

Similarly, when the cooperative supply chain chooses the Nash equilibrium, we can further obtain the optimal carbon emissions reduction under the benchmarking rule, historical carbon intensity reduction rule, and auctioning rule, respectively, as shown Eqs. (A4-5)–(A4-7).

$$ \begin{aligned} \rho _{m}^{*} + \rho _{r}^{*} & = \frac{2}{3}\left[ {a - v_{{mr}} + c * p_{0} + k\left( {\Delta e_{{mr}} - \bar{e}_{{mr}} + c * e_{0} } \right)} \right] + \frac{{h * \Delta e_{{mr}}^{2} }}{3} - \frac{{p_{c} * \Delta e_{{mr}} }}{3} \\ & \quad + \frac{{p_{c} * \left( {\bar{e}_{m} + \bar{e}_{r} - E_{{\sec tor}}^{m} - E_{{\sec tor}}^{r} } \right)}}{3} = v_{{mr}} + h * \Delta e_{{mr}}^{2} \\ \end{aligned} $$

(A4-5)

$$ \begin{aligned} \rho _{m}^{*} + \rho _{r}^{*} & = \frac{2}{3}\left[ {a - v_{{mr}} + c * p_{0} + k\left( {\Delta e_{{mr}} - \bar{e}_{{mr}} + c * e_{0} } \right)} \right] + \frac{1}{3} * h * \Delta e_{{mr}}^{2} \\ & \quad - \,\frac{1}{3} * p_{c} * \Delta e_{{mr}} + \frac{{p_{c} * \left( {\bar{e}_{{mr}} - l_{m} * E_{i}^{m} - l_{r} * E_{i}^{r} } \right)}}{3} = v_{{mr}} + h * \Delta e_{{mr}}^{2} \\ \end{aligned} $$

(A4-6)

$$ \begin{aligned} \rho _{m}^{*} + \rho _{r}^{*} & = \frac{2}{3}\left[ {a - v_{{mr}} + c * p_{0} + k\left( {\Delta e_{{mr}} - \bar{e}_{{mr}} + c * e_{0} } \right)} \right] + \frac{1}{3} * h * \Delta e_{{mr}}^{2} \\ & \quad - \,\frac{1}{3} * p_{c} * \Delta e_{{mr}} + \frac{{p_{c} * \bar{e}_{{mr}} }}{3} = v_{{mr}} + h * \Delta e_{{mr}}^{2} \\ \end{aligned} $$

(A4-7)

The Eqs. (A4-5)–(A4-7) are quadratic function about the carbon emissions reduction (i.e., $\Delta e_{{mr}}$), which have two optimal solutions. In order to improve the carbon emissions reduction, this paper chooses the greater optimal solution as the carbon emissions reduction target, which are shown in Table 2.

Appendix A5

This paper further compares the optimal product prices between the Stackelberg game and Nash equilibrium. This paper lets the optimal wholesale prices under the Stackelberg minus the optimal wholesale prices under the Nash equilibrium. The results are shown as Eqs. (A5-1)–(A5-3), respectively, when the cooperative supply chain is under the benchmarking rule, historical carbon intensity reduction rule, and auctioning rule.

$$ \Delta _{1} = \frac{1}{6}\left[ {a - v_{{mr}} - h * \Delta e_{{mr}}^{2} + c * p_{0} + k * \left( {\Delta e_{{mr}} - \bar{e}_{{mr}} + c * e_{0} } \right)} \right] + \frac{{p_{c} * \left( {E_{{\sec otr}}^{m} + E_{{\sec otr}}^{r} - \bar{e}_{{mr}} + \Delta e_{{mr}} } \right)}}{6} $$

(A5-1)

$$ \Delta _{2} = \frac{1}{6}\left[ {a - v_{{mr}} - h * \Delta e_{{mr}}^{2} + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)} \right] + \frac{1}{6} * p_{c} * \left( {\Delta e_{{mr}} - \bar{e}_{{mr}} + l_{m} * E_{i}^{m} + l_{r} * E_{i}^{r} } \right) $$

(A5-2)

$$ \Delta _{3} = \frac{1}{6}\left[ {a - v_{{mr}} - h * \Delta e_{{mr}}^{2} + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)} \right] - \frac{{p_{c} * \left( {\bar{e}_{{mr}} - \Delta e_{{mr}} } \right)}}{6} $$

(A5-3)

Similarly, this paper further lets optimal retail prices under the Stackelberg minus the optimal retail prices under the Nash equilibrium. The results are shown as Eqs. (A5-4)–(A5-6), respectively, when the cooperative supply chain is under the benchmarking rule, historical carbon intensity reduction rule, and auctioning rule.

$$ \Delta _{{\text{4}}} = \frac{1}{{12}}\left[ {a - \left( {v_{{mr}} + h * \Delta e_{{mr}}^{2} } \right) + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)} \right]{\text{ + }}\frac{1}{{12}}p_{c} * \left( {E_{{\sec otr}}^{m} + E_{{\sec otr}}^{r} - \bar{e}_{{mr}} + \Delta e_{{mr}} } \right) $$

(A5-4)

$$ \Delta _{{\text{5}}} = \frac{1}{{12}}\left[ {a - \left( {v_{{mr}} + h * \Delta e_{{mr}}^{2} } \right) + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)} \right]{\text{ + }}\frac{1}{{12}}p_{c} * \left( {l_{m} * E_{i}^{m} + l_{r} * E_{i}^{r} - \bar{e}_{{mr}} + \Delta e_{{mr}} } \right) $$

(A5-5)

$$ \Delta _{{\text{6}}} = \frac{1}{{12}}\left[ {a - \left( {v_{{mr}} + h * \Delta e_{{mr}}^{2} } \right) + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)} \right] - \frac{{p_{c} * \left( {\bar{e}_{{mr}} - \Delta e_{{mr}} } \right)}}{{12}} $$

(A5-6)

Since there are $a - v_{{mr}} - h * \Delta e_{{mr}}^{2} + c * p_{0} + k * \left( {\Delta e_{{mr}} - \bar{e}_{{mr}} + c * e_{0} } \right) > 0$, $E_{{\sec otr}}^{m} + E_{{\sec otr}}^{r} - \bar{e}_{{mr}} + \Delta e_{{mr}} > 0$, and $l_{m} * E_{i}^{m} + l_{r} * E_{i}^{r} - \bar{e}_{{mr}} + \Delta e_{{mr}} > 0$, therefore there are $\Delta _{1} > 0$, $\Delta _{{\text{2}}} > 0$, $\Delta _{{\text{4}}} > 0$, and $\Delta _{{\text{5}}} > 0$. The result indicates that under the two free carbon quota allocation rules, the optimal wholesale prices and the retail prices when the cooperative supply chain chooses Stackelberg game are higher those when the cooperative supply chain chooses the Nash equilibrium. However, under the auctioning rule, when $\left( {\bar{e}_{{mr}} - \Delta e_{{mr}} } \right) < \frac{{a - \left( {v_{{mr}} + h * \Delta e_{{mr}}^{2} } \right) + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)}}{{p_{c} }}$, there are $\Delta _{{\text{3}}} > 0$ and $\Delta _{{\text{6}}} > 0$, which indicates that the optimal wholesale and retail prices when the cooperative supply chain chooses Stackelberg game are higher those when the cooperative supply chain chooses the Nash equilibrium. When $\left( {\bar{e}_{{mr}} - \Delta e_{{mr}} } \right) > \frac{{a - \left( {v_{{mr}} + h * \Delta e_{{mr}}^{2} } \right) + c * p_{0} + k * \left( {\Delta e_{{mr}} + c * e_{0} - \bar{e}_{{mr}} } \right)}}{{p_{c} }}$, there are $\Delta _{{\text{3}}} < 0$ and $\Delta _{{\text{6}}} < 0$, which indicates that the optimal wholesale and retail prices when the cooperative supply chain chooses Stackelberg game are lower those when the cooperative supply chain chooses the Nash equilibrium.