Low-carbon production or not? Co-opetition supply chain manufacturers’ production strategy under carbon cap-and-trade policy

Abstract

Given the background of supply chain globalization, whether supply chain manufacturers will actively choose low-carbon products for production under the restrictions of a low-carbon policy, is a subject of interest. Based on Stackelberg game theory, this study constructed a competition and cooperation game model of two supply chain manufacturers operating under the carbon cap-and-trade policy launched in China. The study also discussed the effects of the carbon cap-and-trade policy, carbon quota, and market competition intensity on the co-opetition supply chain manufacturers’ production options for low-carbon products. Interestingly, the results indicate that the low-carbon production selection strategies of downstream manufacturers are affected by the intensity of market competition when upstream manufacturers produce common products. However, when upstream manufacturers manufacture low-carbon products, the main factor affecting the low-carbon production selection strategy of downstream manufacturers becomes carbon policy factors, rather than the intensity of market competition.

Appendix

Proof of Theorem 1

In the range of $0 < p_{0} < p_{0}^{a} = {1 \mathord{\left/ {\vphantom {1 {e\left( {1 - \theta } \right)}}} \right. \kern-\nulldelimiterspace} {e\left( {1 - \theta } \right)}}$ and $p_{1}^{{CC^{ * } }} > 0,p_{2}^{{CC^{ * } }} > 0,w^{{CC^{ * } }} > 0,D_{1}^{{CC^{ * } }} > 0,D_{2}^{{CC^{ * } }} > 0$, we are able to find the partial derivatives with respect to $\theta$ separately:

$$\left\{ \begin{gathered} \frac{{\partial p_{1}^{{CC^{ * } }} }}{\partial \theta } = \frac{1}{{2\left( {1 - \theta } \right)^{2} }} > 0,\frac{{\partial p_{2}^{{CC^{ * } }} }}{\partial \theta } = \frac{{ep_{0} \theta \left( {\theta - 2} \right) + ep_{0} + 2}}{{4\left( {1 - \theta } \right)^{2} }} > 0,\frac{{\partial w^{{CC^{ * } }} }}{\partial \theta } = \frac{1}{{2\left( {1 - \theta } \right)^{2} }} > 0 \hfill \\ \frac{{\partial D_{1}^{{CC^{ * } }} }}{\partial \theta } = \frac{{ep_{0} \left( {1 + 2\theta } \right) + 1}}{4} > 0,\frac{{\partial D_{2}^{{CC^{ * } }} }}{\partial \theta } = \frac{{ep_{0} }}{4} > 0 \hfill \\ \end{gathered} \right.$$

Similarly, we seek the partial derivatives of $p_{0}$ separately:

$$\left\{ \begin{gathered} \frac{{\partial p_{1}^{{CC^{ * } }} }}{{\partial p_{0} }} = \frac{e}{2} > 0,\frac{{\partial p_{2}^{{CC^{ * } }} }}{{\partial p_{0} }} = \frac{e(\theta + 1)}{4} > 0,\frac{{\partial w^{{CC^{ * } }} }}{{\partial p_{0} }} = - \frac{e}{2} < 0 \hfill \\ \frac{{\partial D_{1}^{{CC^{ * } }} }}{{\partial p_{0} }} = \frac{{e\left( {2 + \theta } \right)(\theta - 1)}}{4} < 0,\frac{{\partial D_{2}^{{CC^{ * } }} }}{{\partial p_{0} }} = \frac{e(\theta - 1)}{4} < 0 \hfill \\ \end{gathered} \right.$$

Proof of Theorem 2

In the range of $\frac{c - 1}{{e\theta }} = p_{0}^{b} < p_{0} < p_{0}^{c} = \frac{2 + \theta (1 + c)}{{e(2 - \theta^{2} )}}$ and $c < \frac{1}{1 - \theta }$, we are able to find the partial derivatives with respect to $p_{0}$ and $\theta$ separately:

$$\left\{ \begin{gathered} \frac{{\partial p_{1}^{{CL^{ * } }} }}{{\partial p_{0} }} = \frac{e}{2} > 0,\frac{{\partial p_{2}^{{CL^{ * } }} }}{{\partial p_{0} }} = \frac{e(\theta + 1)}{4} > 0,\frac{{\partial w^{{CL^{ * } }} }}{{\partial p_{0} }} = 0 \hfill \\ \frac{{\partial D_{1}^{{CL^{ * } }} }}{{\partial p_{0} }} = \frac{{e(\theta^{2} - 2)}}{4} > 0,\frac{{\partial D_{2}^{{CL^{ * } }} }}{{\partial p_{0} }} = \frac{e\theta }{4} > 0 \hfill \\ \end{gathered} \right.$$

$$\left\{ \begin{gathered} \frac{{\partial p_{1}^{{CL^{ * } }} }}{\partial \theta } = \frac{1}{{2(\theta - 1)^{2} }} > 0,\frac{{\partial p_{2}^{{CL^{ * } }} }}{\partial \theta } = \frac{{2 + ep_{0} (\theta - 1)^{2} }}{{4(\theta - 1)^{2} }} > 0,\frac{{\partial w^{{CL^{ * } }} }}{\partial \theta } = \frac{1}{{2(\theta - 1)^{2} }} > 0 \hfill \\ \frac{{\partial D_{1}^{{CL^{ * } }} }}{\partial \theta } = \frac{{1 + c + ep_{0} \theta }}{4} > 0,\frac{{\partial D_{2}^{{CL^{ * } }} }}{\partial \theta } = \frac{{ep_{0} }}{4} > 0 \hfill \\ \end{gathered} \right.$$

Proof of Theorem 3

Because $p_{1}^{{LC^{ * } }} > 0,p_{2}^{{LC^{ * } }} > 0,w^{{LC^{ * } }} > 0,D_{1}^{{LC^{ * } }} > 0,D_{2}^{{LC^{ * } }} > 0$ in the range of $\frac{{2c - 2 - \theta - c\theta^{2} }}{e\theta } = p_{0}^{d} < p_{0} < p_{0}^{e} = \frac{1 + c\theta }{e}$, we can find the partial derivatives with respect to $p_{0}$ separately:

$$\left\{ \begin{gathered} \frac{{\partial p_{1}^{{LC^{ * } }} }}{{\partial p_{0} }} = 0,\frac{{\partial p_{2}^{{LC^{ * } }} }}{{\partial p_{0} }} = \frac{e}{4} > 0,\frac{{\partial w^{{LC^{ * } }} }}{{\partial p_{0} }} = - \frac{e}{2} < 0 \hfill \\ \frac{{\partial D_{1}^{{LC^{ * } }} }}{{\partial p_{0} }} = \frac{e\theta }{4} > 0,\frac{{\partial D_{2}^{{LC^{ * } }} }}{{\partial p_{0} }} = - \frac{e}{4} < 0 \hfill \\ \end{gathered} \right.$$

Similarly, we seek the partial derivatives of $\theta$ separately:

$$\left\{ \begin{gathered} \frac{{\partial p_{1}^{{LC^{ * } }} }}{\partial \theta } = \frac{1}{{2(\theta - 1)^{2} }} > 0,\frac{{\partial p_{2}^{{LC^{ * } }} }}{\partial \theta } = \frac{{2 + c(\theta - 1)^{2} }}{{4(\theta - 1)^{2} }} > 0,\frac{{\partial w^{{LC^{ * } }} }}{\partial \theta } = \frac{1}{{2(\theta - 1)^{2} }} > 0 \hfill \\ \frac{{\partial D_{1}^{{LC^{ * } }} }}{\partial \theta } = \frac{{1 + 2c\theta + ep_{0} }}{4} > 0,\frac{{\partial D_{2}^{{LC^{ * } }} }}{\partial \theta } = \frac{c}{4} > 0 \hfill \\ \end{gathered} \right.$$

Proof of Theorem 4

Based on Assumption 1, $p_{0}$ does not affect the operational strategies of manufacturers when they both produce low-carbon products. Thus, we seek partial derivatives of $\theta$ separately:

$$\left\{ \begin{gathered} \frac{{\partial p_{1}^{{LL^{ * } }} }}{\partial \theta } = \frac{1}{{2(\theta - 1)^{2} }} > 0,\frac{{\partial p_{2}^{{LL^{ * } }} }}{\partial \theta } = \frac{{2 + c(\theta - 1)^{2} }}{{4(\theta - 1)^{2} }} > 0,\frac{{\partial w^{{LL^{ * } }} }}{\partial \theta } = \frac{1}{{2(\theta - 1)^{2} }} > 0 \hfill \\ \frac{{\partial D_{1}^{{LL^{ * } }} }}{\partial \theta } = \frac{1 + c(1 + 2\theta )}{4} > 0,\frac{{\partial D_{2}^{{LL^{ * } }} }}{\partial \theta } = \frac{c}{4} > 0 \hfill \\ \end{gathered} \right.$$

Proof of Theorem 5

According to Eq. (9), we can first obtain $c^{2} - 2c - e^{2} p_{0}^{2} (1 - 2\theta ) + p_{0} (2e - 16q_{2} - 2ce\theta ) = 0$, and subsequently receive:

$$\left\{ \begin{gathered} p_{0}^{1} = \frac{{8q_{2} + ce\theta - e + \sqrt {64q_{2}^{2} + e^{2} (1 - c + c\theta )^{2} + 16eq_{2} (c\theta - 1)} }}{{e^{2} (2\theta - 1)}} \hfill \\ p_{0}^{2} = \frac{{8q_{2} + ce\theta - e - \sqrt {64q_{2}^{2} + e^{2} (1 - c + c\theta )^{2} + 16eq_{2} (c\theta - 1)} }}{{e^{2} (2\theta - 1)}} \hfill \\ \end{gathered} \right.$$

Under the conditions of $0 < c < \frac{1}{1 - \theta }$ and $p_{0}^{b} < p_{0} < p_{0}^{c}$, $p_{0}^{b} < p_{0}^{1} < p_{0}^{2} < p_{0}^{c}$ when ${0} < \theta < \frac{1}{2}$. Thereafter, we make $p_{0}^{1} { = 0}$ and $p_{0}^{2} = 0$

$$\left\{ \begin{gathered} q_{2a} = \frac{{e - ce\theta - \sqrt {ce^{2} (c - 2)(2\theta - 1)} }}{8} \hfill \\ q_{2b} = \frac{{e - ce\theta + \sqrt {ce^{2} (c - 2)(2\theta - 1)} }}{8} \hfill \\ \end{gathered} \right.$$

Proof of Theorem 6

According to Eq. (9), the function of $\Pi_{{M_{2} }}^{CL * } - \Pi_{{M_{2} }}^{CC * }$ becomes a linear equation of $p_{0}$ $\Pi_{{M_{2} }}^{{CL^{ * } }} - \Pi_{{M_{2} }}^{{CC^{ * } }} = \frac{{c^{2} - 2c + p_{0} (2e - 16q_{2} - ce)}}{16}$ when $\theta = \frac{1}{2}$. Subsequently, we obtain $p_{0}^{*} { = }\frac{{2e - 16q_{2} - ce}}{{16c^{2} - {2}c}}$ and $q^{*} = \frac{2e - ce}{{16}}$.

Proof of Theorem 7

Similar to the proof of Theorem 5

Proof of Theorem 8

According to Eq. (10), we can first obtain $\left( {ep_{0} - c} \right)\left( {2c\theta - ep_{0} - c + 2} \right) - p_{0} q_{2} = 0$ and then receive.

$$\left\{ \begin{gathered} p_{0}^{3} = \frac{{ce\theta + e - 8q_{2} + \sqrt {64q_{2}^{2} + e^{2} (1 - c + c\theta )^{2} - 16eq_{2} (c\theta + 1)} }}{{e^{2} }} \hfill \\ p_{0}^{4} = \frac{{ce\theta + e - 8q_{2} - \sqrt {64q_{2}^{2} + e^{2} (1 - c + c\theta )^{2} - 16eq_{2} (c\theta + 1)} }}{{e^{2} }} \hfill \\ \end{gathered} \right.$$

Under the conditions of $0 < c < {1 \mathord{\left/ {\vphantom {1 {\left( {1 - \theta } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \theta } \right)}}$ and $p_{0}^{d} < p_{0} < p_{0}^{e}$, $p_{0}^{4} < p_{0}^{3}$, regardless of the value of $\theta$. Thereafter, we make $p_{0}^{3} = 0$ and p₀⁴ = 0.

$$\left\{ \begin{gathered} q_{2c} = \frac{{e + ce\theta - \sqrt {ce^{2} (2 - c + 2c\theta )} }}{8} \hfill \\ q_{2d} = \frac{{e + ce\theta + \sqrt {ce^{2} (2 - c + 2c\theta )} }}{8} \hfill \\ \end{gathered} \right.$$