Research on single/cooperative emission reduction strategy under different power structures

Abstract

This paper develops low-carbon decisions in a two-echelon supply chain considering consumers’ low-carbon preference and cap-and-trade (C&T) regulation. Two different power structures are considered, including manufacturer-dominated (MD) and retailer-dominated (RD) cases. The single emission reduction (SER) mode where only the manufacturer invests in low-carbon technology and the cooperative emission reduction (CER) mode where the manufacturer invests in low-carbon technology and the retailer invest in low-carbon promotion are investigated respectively. It is found that a relatively loose C&T regulation helps to promote the cooperation of supply chain enterprises. Under both MD and RD cases, CER mode is always a rational choice for supply chain enterprises. Under SER mode, the manufacturer’s profit will not always decrease when he loses the dominant position. However, the RD case is profitable for the retailer and the supply chain. Under CER mode, the dominant role is important for both the manufacturer and the retailer. However, the profit of the supply chain under RD case may be lower than that under MD case. Through numerical analysis, we found that the fluctuation of carbon price has a more significant impact on the manufacturer’s emission reduction decision under CER mode than that under SER mode. In addition, with the increase of unit carbon price, the RD case performs better than the MD case in promoting supply chain’s low-carbon level and profit.

Appendix

Proof of Lemma 1

In order to obtain the Stackelberg equilibrium, the best response of the follower and in the second stage should be determined at first. The leader’s decision problem is solved based on the follower’s response. Thus, we firstly solve the retailer’s response. With $\frac{{\partial^{2} \pi_{r}^{SMS} }}{{\partial \left( p \right)^{2} }} = - 2\beta < 0$, we can know that the retailer’s profit is concave in the retail price. Thus, let $\frac{{\partial \pi_{r}^{SMS} }}{\partial p} = 0$, we can get the best response of the retailer on the manufacturer’s emission reduction rate and wholesale price: $p^{SMS} = \frac{1 + \eta \tau + \beta w}{{2\beta }}$. Substituting the price into the manufacturer’s profit function, we can get $\frac{{\partial^{2} \pi_{m}^{SMS} }}{{\partial \left( w \right)^{2} }} = - \beta < 0$, $\frac{{\partial^{2} \pi_{m}^{SMS} }}{\partial w\partial \tau } = \frac{{\partial^{2} \pi_{m}^{SMS} }}{\partial \tau \partial w} = \frac{1}{2}\left( {\eta - \beta p_{e} e} \right)$, and $\frac{{\partial^{2} \pi_{m}^{SMS} }}{{\partial \left( \tau \right)^{2} }} = \eta p_{e} e - k_{m}$. Thus, the Hessian of $\pi_{m}^{SMS}$ is $H\left( {\pi_{m}^{SMS} } \right) = \left[ \begin{gathered} - \beta \quad \quad \;{{\quad \quad \quad \;\left( {\eta - \beta p_{e} e} \right)} \mathord{\left/ {\vphantom {{\quad \quad \quad \;\left( {\eta - \beta p_{e} e} \right)} 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ {{\left( {\eta - \beta p_{e} e} \right)} \mathord{\left/ {\vphantom {{\left( {\eta - \beta p_{e} e} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}\quad \quad \;\eta p_{e} e - k_{m} \hfill \\ \end{gathered} \right]$. In this model, $k_{m}$ is significant large. Thus, $H_{11} < 0$ and $\det \left( H \right) = \left( {k_{m} - \eta p_{e} e} \right)\beta - \frac{{\left( {\eta - \beta p_{e} e} \right)^{2} }}{4} > 0$. Hence, $\pi_{m}^{SMS}$ is concave in $w$ and $\tau$. Let $\frac{{\partial \pi_{m}^{SMS} }}{\partial w} = 0$ and $\frac{{\partial \pi_{m}^{SMS} }}{\partial \tau } = 0$, we can get $\tau^{SMS*} { = }\frac{{\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}$, $w^{SMS*} { = }\frac{{2k_{m} \left[ {1 + \beta c + \beta p_{e} \left( {e - \alpha } \right)} \right] - \left( {\eta + \beta p_{e} e} \right)\left[ {\eta p_{e} \left( {e - \alpha } \right) + p_{e} e + \eta c} \right]}}{{4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}$, and $p^{SMS*} { = }\frac{{k_{m} \left[ {3 + \beta c + \beta p_{e} \left( {e - \alpha } \right)} \right] - \left( {\eta + \beta p_{e} e} \right)\left[ {\eta p_{e} \left( {e - \alpha } \right) + p_{e} e + \eta c} \right]}}{{4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}$.

Proof of Proposition 1

$\frac{{\partial \tau^{SMS*} }}{\partial \eta }{ = }\frac{{\left[ {4\beta k_{m} + \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} > 0$,

$\frac{{\partial w^{SMS*} }}{\partial \eta }{ = }\frac{{\left[ {4\eta k_{m} - p_{e} e\left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} > 0$,

$\frac{{\partial p^{SMS*} }}{\partial \eta }{ = }\frac{{\left[ {2k_{m} \left( {3\eta + \beta p_{e} e} \right) - p_{e} e\left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} > 0$.

Therefore, $\frac{{\partial p^{SMS*} }}{\partial \eta } > \frac{{\partial w^{SMS*} }}{\partial \eta }$. In addition, with $\pi_{m}^{SMS*} { = }\frac{{k_{m} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} }}{{2\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}$ and $\pi_{r}^{SMS*} { = }\frac{{\beta k_{m}^{2} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} }}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }}$, we can know that both $\pi_{m}^{SMS*}$ and $\pi_{r}^{SMS*}$ increase as $\eta$ increases.

Proof of Proposition 2

(1) $\frac{{\partial \tau^{SMS*} }}{{\partial k_{m} }}{ = } - \frac{{4\beta \left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} < 0$;

$\frac{{\partial \pi_{m}^{SMS*} }}{{\partial k_{m} }}{ = } - \frac{1}{2}\left[ {\frac{{\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}} \right]^{2} < 0$;

$\frac{{\partial \pi_{r}^{SMS*} }}{{\partial k_{m} }}{ = } - 2k_{m} \beta \frac{{\left( {\eta + \beta p_{e} e} \right)^{2} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} }}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{3} }} < 0$.

Thus, $\tau^{SMS*}$, $\pi_{m}^{SMS*}$, and $\pi_{r}^{SMS*}$ increase as $k_{m}$ decreases.

(2) $\frac{{\partial w^{SMS*} }}{{\partial k_{m} }}{ = } - \frac{{2\left[ {\eta^{2} - \left( {\beta p_{e} e} \right)^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }}$,

$\frac{{\partial p^{SMS*} }}{{\partial k_{m} }}{ = } - \frac{{\left( {3\eta - \beta p_{e} e} \right)\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }}$.

Thus, when $0 \le \eta \le {{\beta p_{e} e} \mathord{\left/ {\vphantom {{\beta p_{e} e} 3}} \right. \kern-\nulldelimiterspace} 3}$, then ${{\partial w^{SMS*} } \mathord{\left/ {\vphantom {{\partial w^{SMS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} \ge 0$ and ${{\partial p^{SMS*} } \mathord{\left/ {\vphantom {{\partial p^{SMS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} \ge 0$; when ${{\beta p_{e} e} \mathord{\left/ {\vphantom {{\beta p_{e} e} 3}} \right. \kern-\nulldelimiterspace} 3} < \eta \le \beta p_{e} e$, then ${{\partial w^{SMS*} } \mathord{\left/ {\vphantom {{\partial w^{SMS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} \ge 0$ and ${{\partial p^{SMS*} } \mathord{\left/ {\vphantom {{\partial p^{SMS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} < 0$; when $\beta p_{e} e < \eta \le 1$, then ${{\partial w^{SMS*} } \mathord{\left/ {\vphantom {{\partial w^{SMS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} < 0$ and ${{\partial p^{SMS*} } \mathord{\left/ {\vphantom {{\partial p^{SMS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} < 0$.

Proof of Lemma 2

According to the reverse solution method, we first solve the manufacturer’s wholesale price and emission reduction rate decision. By substituting $p = w + \Delta p$ into Eq. (2), we can get $\frac{{\partial^{2} \pi_{m}^{SRS} }}{{\partial \left( w \right)^{2} }} = - 2\beta < 0$, $\frac{{\partial^{2} \pi_{m}^{SRS} }}{\partial w\partial \tau } = \frac{{\partial^{2} \pi_{m}^{SRS} }}{\partial \tau \partial w} = \eta - \beta p_{e} e$, $\frac{{\partial^{2} \pi_{m}^{SRS} }}{{\partial \left( \tau \right)^{2} }} = 2\eta p_{e} e - k_{m}$. The Hessian of $\pi_{m}^{SRS}$ is $H\left( {\pi_{m}^{SRS} } \right) = \left[ \begin{gathered} - 2\beta \quad \quad \;\quad \quad \eta - \beta p_{e} e \hfill \\ \eta - \beta p_{e} e\quad \quad \;2\eta p_{e} e - k_{m} \hfill \\ \end{gathered} \right]$. In the model, $k_{m}$ is significantly large, and thus $H_{11} < 0$ and $\det \left( H \right) = 2\beta \left( {k_{m} - 2\eta p_{e} e} \right) - \left( {\eta - \beta p_{e} e} \right)^{2} > 0$. Hence, $\pi_{m}^{SRS}$ is concave in $w$ and $\tau$. Let $\frac{{\partial \pi_{m}^{SRS} }}{\partial w} = 0$ and $\frac{{\partial \pi_{m}^{SRS} }}{\partial \tau } = 0$, we can get $w^{SRS} { = }\frac{{k_{m} \left[ {1 + \beta c - \beta \Delta p + \beta p_{e} \left( {e - \alpha } \right)} \right] - \left( {\eta + \beta p_{e} e} \right)\left[ {\eta p_{e} \left( {e - \alpha } \right) + p_{e} e\left( {1 - \beta \Delta p} \right) + \eta c} \right]}}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}$ and $\tau^{SRS} { = }\frac{{\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta \Delta p - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}$.

Next, we solve the retailer’s optimal decision problem. Substituting $w^{SRS}$ and $\tau^{SRS}$ into the profit function of the retailer, we can get $\frac{{\partial^{2} \pi_{r}^{SRS} }}{{\partial \left( {\Delta p} \right)^{2} }} = - \frac{{2\beta^{2} k_{m} }}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }} < 0$. Hence, $\pi_{r}^{SRS}$ is concave in $\Delta p$. Let $\frac{{\partial \pi_{r}^{SRS} }}{\partial \Delta p} = 0$, we can get $\Delta p = \frac{{1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)}}{2\beta }$. By substituting $\Delta p$ into $w^{SRS}$ and $\tau^{SRS}$, we can get the optimal decisions of the manufacturer.

Proof of Proposition 4

(1) $\frac{{\partial \tau^{SRS*} }}{{\partial k_{m} }}{ = } - \frac{{\beta \left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} < 0$; $\frac{{\partial \pi_{m}^{SRS*} }}{{\partial k_{m} }}{ = } - \frac{1}{8}\left[ {\frac{{\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}} \right]^{2} < 0$;

$\frac{{\partial \pi_{r}^{SRS*} }}{{\partial k_{m} }}{ = } - \frac{{\left( {\eta + \beta p_{e} e} \right)^{2} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} }}{{4\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} < 0$.

(2) $\frac{{\partial w^{SRS*} }}{{\partial k_{m} }}{ = } - \frac{{\left[ {\eta^{2} - \left( {\beta p_{e} e} \right)^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{2\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }}$,

$\frac{{\partial p^{SRS*} }}{{\partial k_{m} }}{ = } - \frac{{\left( {\eta - \beta p_{e} e} \right)\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{2\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }}$.

Thus, we can conclude that when $\eta > \beta p_{e} e$, ${{\partial w^{SRS*} } \mathord{\left/ {\vphantom {{\partial w^{SRS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} < 0$, ${{\partial p^{SRS*} } \mathord{\left/ {\vphantom {{\partial p^{SRS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} < 0$; when $\eta \le \beta p_{e} e$, ${{\partial w^{SRS*} } \mathord{\left/ {\vphantom {{\partial w^{SRS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} \ge 0$, ${{\partial p^{SRS*} } \mathord{\left/ {\vphantom {{\partial p^{SRS*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} \ge 0$.

Proof of Lemma 3

We first solve the retailer’s decisions, including pricing and low-carbon promotion level. With $\frac{{\partial^{2} \pi_{r}^{SMC} }}{{\partial \left( p \right)^{2} }} = - 2\beta < 0$, $\frac{{\partial^{2} \pi_{r}^{SMC} }}{\partial p\partial s} = \frac{{\partial^{2} \pi_{r}^{SMC} }}{\partial s\partial p} = \gamma$, and $\frac{{\partial^{2} \pi_{r}^{SMC} }}{{\partial \left( s \right)^{2} }} = - k_{r}$, we can get the Hessian of $\pi_{r}^{SMC}$ which is $H\left( {\pi_{r}^{SMC} } \right) = \left[ \begin{gathered} - 2\beta \quad \quad \;\gamma \hfill \\ \gamma \quad \quad \;\quad - k_{r} \hfill \\ \end{gathered} \right]$. In this model, $k_{r}$ is significantly large, and thus $H_{11} < 0$ and $\det \left( H \right) = 2\beta k_{r} - \gamma^{2} > 0$. It can be concluded that the retailer’s profit function is a joint concave function of retail price and low-carbon promotion level. Thus, let $\frac{{\partial \pi_{r}^{SMC} }}{\partial p} = 0$ and $\frac{{\partial \pi_{r}^{SMC} }}{\partial s} = 0$, we can get $s^{SMC} = \frac{{\gamma \left( {1 + \eta \tau - \beta w} \right)}}{{2\beta k_{r} - \gamma^{2} }}$ and $p^{SMC} = \frac{{k_{r} \left( {1 + \eta \tau + \beta w} \right) - \gamma^{2} w}}{{2\beta k_{r} - \gamma^{2} }}$. Substituting them into the manufacturer’s profit function, we can get $\frac{{\partial^{2} \pi_{m}^{SMC} }}{{\partial \left( w \right)^{2} }} = - \frac{{2\beta^{2} k_{r} }}{{2\beta k_{r} - \gamma^{2} }} < 0$, $\frac{{\partial^{2} \pi_{m}^{SMC} }}{\partial w\partial \tau } = \frac{{\partial^{2} \pi_{m}^{SMC} }}{\partial \tau \partial w} = \frac{{\beta k_{r} \left( {\eta - \beta p_{e} e} \right)}}{{2\beta k_{r} - \gamma^{2} }}$, and $\frac{{\partial^{2} \pi_{m}^{SMC} }}{{\partial \left( \tau \right)^{2} }} = \frac{{2\beta k_{r} \eta p_{e} e - 2\beta k_{m} k_{r} + \gamma^{2} k_{m} }}{{2\beta k_{r} - \gamma^{2} }}$. Thus, we can get the Hessian of $\pi_{m}^{SMC}$: $H\left( {\pi_{m}^{SMC} } \right) = \left[ \begin{gathered} - \frac{{2\beta^{2} k_{r} }}{{2\beta k_{r} - \gamma^{2} }}\quad \quad \;\quad \quad \quad \quad \quad \;\;\frac{{\beta k_{r} \left( {\eta - \beta p_{e} e} \right)}}{{2\beta k_{r} - \gamma^{2} }} \hfill \\ \frac{{\beta k_{r} \left( {\eta - \beta p_{e} e} \right)}}{{2\beta k_{r} - \gamma^{2} }}\quad \quad \;\frac{{2\beta k_{r} \eta p_{e} e - 2\beta k_{m} k_{r} + \gamma^{2} k_{m} }}{{2\beta k_{r} - \gamma^{2} }} \hfill \\ \end{gathered} \right]$. Suppose that $2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} > 0$, then $H_{11} < 0$ and $\det \left( H \right) = \frac{{\beta^{2} k_{r} \left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}{{\left( {2\beta k_{r} - \gamma^{2} } \right)^{2} }} > 0$. Thus, $\pi_{m}^{SMS}$ is jointly concave in $w$ and $\tau$. Let $\frac{{\partial \pi_{m}^{SMC} }}{\partial w} = 0$ and $\frac{{\partial \pi_{m}^{SMC} }}{\partial \tau } = 0$, we can gain the optimal decision of the manufacturer and the retailer in Lemma 3.

Proof of Proposition 5

$\frac{{\partial \tau^{SMC*} }}{\partial \eta }{ = }\frac{{k_{r} \left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) + k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} > 0$,

$\frac{{\partial w^{SMC*} }}{\partial \eta }{ = }\frac{{k_{r} \left[ {2\eta k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - p_{e} e\beta k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} > 0$,

$\frac{{\partial p^{SMC*} }}{\partial \eta }{ = }\frac{{k_{r} \left[ {2\beta k_{m} k_{r} \left( {3\eta + \beta p_{e} e} \right) - 2k_{m} \eta \gamma^{2} - p_{e} e\beta k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} > 0$.

Thus, $\frac{{\partial p^{SMC*} }}{\partial \eta } > \frac{{\partial w^{SMC*} }}{\partial \eta }$. In addition, with Lemma 1 and Eq. (5), we can conclude that $s^{SMC*}$, $\pi_{m}^{SMC*}$, and $\pi_{r}^{SMC*}$ increase with the increase of $\eta$.

Proof of Lemma 4

We first solve the manufacturer’s wholesale price and emission reduction rate. With $\frac{{\partial^{2} \pi_{m}^{SRC} }}{{\partial \left( w \right)^{2} }} = - 2\beta < 0$, $\frac{{\partial^{2} \pi_{m}^{SRC} }}{\partial w\partial \tau } = \frac{{\partial^{2} \pi_{m}^{SRC} }}{\partial \tau \partial w} = \eta - \beta p_{e} e$, and $\frac{{\partial^{2} \pi_{m}^{SRC} }}{{\partial \left( \tau \right)^{2} }} = 2\eta p_{e} e - k_{m}$, we can get the Hessian of $\pi_{m}^{SRC}$: $H\left( {\pi_{m}^{SRC} } \right) = \left[ \begin{gathered} - 2\beta \quad \quad \;\quad \quad \eta - \beta p_{e} e \hfill \\ \eta - \beta p_{e} e\quad \quad \;2\eta p_{e} e - k_{m} \hfill \\ \end{gathered} \right]$. In this model, $k_{m}$ is significantly large, and thus $H_{11} < 0$ and $\det \left( H \right) = 2\beta \left( {k_{m} - 2\eta p_{e} e} \right) - \left( {\eta - \beta p_{e} e} \right)^{2} > 0$. Hence, $\pi_{m}^{SRC}$ is concave in $w$ and $\tau$. Let $\frac{{\partial \pi_{m}^{SRC} }}{\partial w} = 0$ and $\frac{{\partial \pi_{m}^{SRC} }}{\partial \tau } = 0$, we can get $w^{SRC} { = }\frac{\begin{gathered} k_{m} \left[ {1 + \beta c - \beta \Delta p + \beta p_{e} \left( {e - \alpha } \right) + \gamma s} \right] \hfill \\ - \left( {\eta + \beta p_{e} e} \right)\left[ {\eta p_{e} \left( {e - \alpha } \right) + p_{e} e\left( {1 - \beta \Delta p + \gamma s} \right) + \eta c} \right] \hfill \\ \end{gathered} }{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}$ and $\tau^{SRC} { = }\frac{{\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta \Delta p - \beta p_{e} \left( {e - \alpha } \right){ + }\gamma s} \right]}}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}$.

Next, we solve the retailer’s optimal decision problem. By substituting $w^{SRC}$ and $\tau^{SRC}$ into the profit function of the retailer, we can get $\frac{{\partial^{2} \pi_{r}^{SRC} }}{{\partial \left( {\Delta p} \right)^{2} }} = - \frac{{2\beta^{2} k_{m} }}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }} < 0$, $\frac{{\partial^{2} \pi_{r}^{SMC} }}{\partial p\partial s} = \frac{{\partial^{2} \pi_{r}^{SMC} }}{\partial s\partial p} = \frac{{\beta \gamma k_{m} }}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}$, and $\frac{{\partial^{2} \pi_{r}^{SRC} }}{{\partial \left( s \right)^{2} }} = - k_{r}$. We can get the Hessian of $\pi_{r}^{SRC}$: $H\left( {\pi_{r}^{SRC} } \right) = \left[ \begin{gathered} - \frac{{2\beta^{2} k_{m} }}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}\quad \quad \;\;\frac{{\beta \gamma k_{m} }}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }} \hfill \\ \frac{{\beta \gamma k_{m} }}{{2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} }}\quad \quad \quad \quad \;\quad \quad - k_{r} \hfill \\ \end{gathered} \right]$. Suppose that $k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} > 0$, we can get $H_{11} < 0$ and $\det \left( H \right) = \frac{{\beta^{2} k_{m} \left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}{{\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} > 0$. Thus, $\pi_{r}^{SRC}$ is concave in $p$ and $s$. Let $\frac{{\partial \pi_{r}^{SRC} }}{\partial p} = 0$ and $\frac{{\partial \pi_{r}^{SRC} }}{\partial s} = 0$, we can get the optimal decisions of the manufacturer.

Proof of Proposition 7

(1) $\frac{{\partial \tau^{SRC*} }}{{\partial k_{m} }}{ = } - \frac{{k_{r} \left( {4\beta k_{r} - \gamma^{2} } \right)\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} < 0$;

$\frac{{\partial s^{SRC*} }}{{\partial k_{m} }}{ = } - \frac{{2\gamma k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} < 0$;

$\frac{{\partial \pi_{m}^{SRC*} }}{{\partial k_{m} }}{ = } - \frac{1}{2}\frac{{\left\{ {k_{r} \left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]} \right\}^{2} \left[ \begin{gathered} k_{m} \left( {4\beta k_{r} + \gamma^{2} } \right) - \hfill \\ 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} \hfill \\ \end{gathered} \right]}}{{\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{3} }} < 0$;

$\frac{{\partial \pi_{r}^{SRC*} }}{{\partial k_{m} }}{ = } - \frac{{k_{r}^{2} \left( {\eta + \beta p_{e} e} \right)^{2} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} }}{{\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} < 0$.

(2) $\frac{{\partial w^{SRC*} }}{{\partial k_{m} }}{ = } - \frac{{k_{r} \left( {\eta + \beta p_{e} e} \right)\left[ {2k_{r} \left( {\eta - \beta p_{e} e} \right) + p_{e} e\gamma^{2} } \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }}$,

$\frac{{\partial p^{SRC*} }}{{\partial k_{m} }}{ = } - \frac{{k_{r} \left( {\eta + \beta p_{e} e} \right)\left[ {2\beta k_{r} \left( {\eta - \beta p_{e} e} \right) + \gamma^{2} \left( {\eta + 2\beta p_{e} e} \right)} \right]\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\beta \left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }}$.

Thus, we can conclude that when $\eta > \beta p_{e} e$, then ${{\partial w^{SRC*} } \mathord{\left/ {\vphantom {{\partial w^{SRC*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} < 0$, ${{\partial p^{SRC*} } \mathord{\left/ {\vphantom {{\partial p^{SRC*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} < 0$; when $\eta \le \beta p_{e} e$, then ${{\partial w^{SRC*} } \mathord{\left/ {\vphantom {{\partial w^{SRC*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} \ge 0$, ${{\partial p^{SRC*} } \mathord{\left/ {\vphantom {{\partial p^{SRC*} } {\partial k_{m} }}} \right. \kern-\nulldelimiterspace} {\partial k_{m} }} \ge 0$.

Proof of Theorem 1

(1) With Lemma 1 and Lemma 3, we can get.

$\tau^{SMS*} - \tau^{SMC*} { = } - \frac{{2k_{m} \gamma^{2} \left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} \le 0$.

(2) $w^{SMS*} - w^{SMC*} { = } - \frac{{k_{m} \gamma^{2} \left( {\eta - \beta p_{e} e} \right)\left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{\beta \left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}$.

Thus, when $\eta > \beta p_{e} e$, then $w^{SMS*} < w^{SMC*}$; when $\eta \le \beta p_{e} e$, then $w^{SMS*} \ge w^{SMC*}$.

(3) $p^{SMS*} - p^{SMC*} { = } - \frac{{k_{m} \gamma^{2} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]\left[ {2\beta k_{m} + \left( {\eta - \beta p_{e} e} \right)\left( {\eta + \beta p_{e} e} \right)} \right]}}{{\beta \left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} \le 0$. Thus, $p^{SMS*} \le p^{SMC*}$.

Proof of Theorem 2

By comparing the profits of the manufacturer and the retailer in the two models, we can get the following conclusions:

$\pi_{m}^{SMS*} - \pi_{m}^{SMC*} { = } - \frac{{\left\{ {k_{m} \gamma \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]} \right\}^{2} }}{{\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} \le 0$;

$\pi_{r}^{SMS*} - \pi_{r}^{SMC*} { = } - \frac{{\left\{ {k_{m} \gamma \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]} \right\}^{2} \left[ {8\beta k_{m}^{2} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{4} } \right]}}{{2\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} \left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} \le 0$.

Thus, $\pi_{m}^{SMS*} \le \pi_{m}^{SMC*}$, $\pi_{r}^{SMS*} \le \pi_{r}^{SMC*}$, $\pi_{{}}^{SMS*} \le \pi_{{}}^{SMC*}$.

Proof of Theorem 3

$\tau^{SRS*} - \tau^{SRC*} { = } - \frac{{k_{m} \gamma^{2} \left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{2\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} \le 0$.

$w^{SRS*} - w^{SRC*} { = } - \frac{{k_{m} \gamma^{2} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]\left[ {k_{m} - p_{e} e\left( {\eta + \beta p_{e} e} \right)} \right]}}{{2\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} \le 0$.

$p^{SRS*} - p^{SRC*} { = } - \frac{{k_{m} \gamma^{2} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]\left[ {3\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)\left( {\eta + 2\beta p_{e} e} \right)} \right]}}{{2\beta \left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} \le 0$.

Thus, we can get $\tau^{SRS*} \le \tau^{SRC*}$; $w^{SRS*} \le w^{SRC*}$; $p^{SRS*} \le p^{SRC*}$.

Proof of Theorem 4

$\pi_{m}^{SRS*} - \pi_{m}^{SRC*} { = } - \frac{{\left\{ {k_{m} \gamma \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]} \right\}^{2} \left[ {k_{m} \left( {8\beta k_{r} - \gamma^{2} } \right) - 4k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}{{8\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} \le 0$;

$\pi_{r}^{SRS*} - \pi_{r}^{SRC*} { = } - \frac{{\left\{ {k_{m} \gamma \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]} \right\}^{2} }}{{4\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }} \le 0$.

Thus, we can get $\pi_{m}^{SRS*} \le \pi_{m}^{SRC*}$, $\pi_{r}^{SRS*} \le \pi_{r}^{SRC*}$, $\pi_{{}}^{SRS*} \le \pi_{{}}^{SRC*}$.

Proof of Theorem 5

(1) $\tau^{SMS*} - \tau^{SRS*} { = } - \frac{{\left( {\eta + \beta p_{e} e} \right)^{3} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]}}{{2\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} < 0$.

(2) $\pi_{m}^{SMS*} - \pi_{m}^{SRS*} { = }\frac{{k_{m} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} \left[ {4\beta k_{m} - 3\left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}{{8\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}$.

Thus, when $4\beta k_{m} - 3\left( {\eta + \beta p_{e} e} \right)^{2} \ge 0$, then $\pi_{m}^{SMS*} - \pi_{m}^{SRS*} \ge 0$; when $4\beta k_{m} - 3\left( {\eta + \beta p_{e} e} \right)^{2} < 0$, then $\pi_{m}^{SMS*} - \pi_{m}^{SRS*} < 0$. With $2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} > 0$, we can conclude that when $\frac{{\left( {\eta + \beta p_{e} e} \right)^{2} }}{2\beta } < k_{m} < \frac{{3\left( {\eta + \beta p_{e} e} \right)^{2} }}{4\beta }$, then $\pi_{m}^{SMS*} < \pi_{m}^{SRS*}$; when $k_{m} \ge \frac{{3\left( {\eta + \beta p_{e} e} \right)^{2} }}{4\beta }$, then $\pi_{m}^{SMS*} \ge \pi_{m}^{SRS*}$.

(3) $\pi_{r}^{SMS*} - \pi_{r}^{SRS*} { = } - \frac{{k_{m} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} \left\{ {4\beta k_{m} \left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right] + \left( {\eta + \beta p_{e} e} \right)^{4} } \right\}}}{{4\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} \left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} < 0$.

(4) $\pi_{{}}^{SMS*} - \pi_{{}}^{SRS*} { = } - \frac{{k_{m} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} \left( {\eta + \beta p_{e} e} \right)^{2} \left[ {8\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}{{8\left[ {4\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} \left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}} < 0$.

Proof of Theorem 6

(1) $s^{SMC*} - s^{SRC*} { = }\frac{{\gamma k_{m} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]\left[ {k_{m} \gamma^{2} - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}{{\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}$,

$\tau^{SMC*} - \tau^{SRC*} { = }\frac{{k_{r} \left( {\eta + \beta p_{e} e} \right)\left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]\left[ {k_{m} \gamma^{2} - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}{{\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}$.

Thus, we can conclude that when $k_{m} \gamma^{2} - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} > 0$, then $s^{SMC*} > s^{SRC*}$, $\tau^{SMC*} > \tau^{SRC*}$; when $k_{m} \gamma^{2} - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} \le 0$, then $s^{SMC*} \le s^{SRC*}$, $\tau^{SMC*} \le \tau^{SRC*}$.

(2) $\pi_{m}^{SMC*} - \pi_{m}^{SRC*} { = }\frac{{k_{m} k_{r} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} \left\{ \begin{gathered} \left[ {k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} + \hfill \\ 2k_{r}^{2} \left[ {\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {2\beta k_{m} - \left( {\eta + \beta p_{e} e} \right)^{2} } \right] \hfill \\ \end{gathered} \right\}}}{{2\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]\left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} }}$,

$\pi_{r}^{SMC*} - \pi_{r}^{SRC*} { = } - \frac{{k_{m} k_{r} \left[ {1 - \beta c - \beta p_{e} \left( {e - \alpha } \right)} \right]^{2} \left\{ {\left[ {k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} + 2k_{m}^{2} \left( {\beta k_{r} - \gamma^{2} } \right)\left( {2\beta k_{r} - \gamma^{2} } \right)} \right\}}}{{2\left[ {2k_{m} \left( {2\beta k_{r} - \gamma^{2} } \right) - k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]^{2} \left[ {k_{m} \left( {4\beta k_{r} - \gamma^{2} } \right) - 2k_{r} \left( {\eta + \beta p_{e} e} \right)^{2} } \right]}}$.

Thus, we can get $\pi_{m}^{SMC*} - \pi_{m}^{SRC*} > 0$ and $\pi_{r}^{SMC*} - \pi_{r}^{SRC*} < 0$.