Manufacturer’s R &D cooperation contract: linear fee or revenue-sharing payment in a low-carbon supply chain

Abstract

This paper studies one manufacturer’s optimal co-development payment under different power structures in a low-carbon supply chain with one manufacturer, one technology firm, and one retailer. The manufacturer could collaborate with the technology firm on developing low-carbon technology through linear fee or revenue-sharing payment. Our results demonstrate that the manufacturer’s optimal payment depends on the technology firm’s low-carbon level. If the manufacturer collaborates with a relatively low-level (high-level) technology firm, she should choose linear fee (revenue-sharing) payment, regardless of power structures. If the technology firm’s low-carbon level is intermediate, the optimal payment varies with power structures. Meanwhile, the manufacturer’s optimal payment is also beneficial to the environment due to more emission reductions. Besides, no matter under which payment type, as the manufacturer’s market power becomes strong, both the manufacturer’s and the technology firm’s profits increase.

Appendices

Appendices

A. Linear fee payment

When the manufacture collaborates with the technology firm through linear fee payment, the three members’ profit functions are as follows:

$$\begin{aligned}&\varPi _m^L(f,e_m,w)=w \left( a-p+r \left( e_m+e_h+s e_h\right) \right) -\frac{e_m^2}{2}-f e_h, \end{aligned}$$

(A.1)

$$\begin{aligned}&\varPi _h^L(e_h)=f e_h-\frac{e_h^2}{2 k_h},\end{aligned}$$

(A.2)

$$\begin{aligned}&\varPi _t^L(p)=(p-w) \left( a-p+r \left( e_m+e_h+s e_h\right) \right) . \end{aligned}$$

(A.3)

1.1 A-1 Proof of MS game

Here, we use backward induction to solve the problem. In stage 3, the retailer decides the retail price after the manufacturer sets the wholesale price. From Eq. (A.3), we obtain $\frac{\partial {\varPi ^L_t}^2}{\partial p^2}=-2<0$, so $\varPi _t^L(p)$ is concave in p. Let $\frac{\partial \varPi _t^L}{\partial p}=a+r \left( s e_h+e_h+e_m\right) -2 p+w=0$, we obtain $p^*=\frac{1}{2} \left( a+r s e_h+r e_h+r e_m+w\right)$.

Replace $p^*$ in $\varPi _m^L(f,e_m,w)$, we obtain $\frac{\partial {\varPi ^L_m}^2}{\partial w^2}=-1 <0$, so $\varPi _m^L(f,e_m,w)$ is a concave function of w. Let $\frac{\partial \varPi ^L _m}{\partial w}=\frac{1}{2} \left( a+r (s+1) e_h+r e_m-2 w\right) =0$, we have $w^*=\frac{1}{2} \left( a+r s e_h+r e_h+r e_m\right)$.

Replace $w^*$ and $p^*$ in $\varPi _m^L(f,e_m,w)$, we obtain $\frac{\partial {\varPi ^L_m}^2}{\partial e_m^2}=\frac{1}{4} \left( r^2-4\right) <0$, so $\varPi _m^L(e_m,w)$ is concave in $e_m$. From Eq. (A.2), we obtain $\frac{\partial {\varPi ^L_h}^2}{\partial e_h^2}=-\frac{1}{k_h}<0$, so $\varPi ^L _h(e_h)$ is concave in $e_h$. Let $\frac{\partial \varPi ^L _h}{\partial e_h}=f-\frac{e_h}{k_h}=0$ and $\frac{\partial \varPi ^L _m}{\partial e_m}=\frac{1}{8} \left( 2 r \left( a+r (s+1) e_h\right) +2 \left( r^2-4\right) e_m\right) =0$, we obtain $e_m^{*}=\frac{r \left( a+f r (s+1) k_h\right) }{4-r^2}$ and $e_h^{*}=f k_h$.

Replace $e_m^{*}$ and $e_h^{*}$ in $\varPi _m^L(f,e_m,w)$, we have $\varPi _m^L(f)=\frac{a^2+f k_h \left( 2 \left( a r (s+1)+f \left( r^2-4\right) \right) +f r^2 (s+1)^2 k_h\right) }{2 \left( 4-r^2\right) }$. Then we get $\frac{\partial \varPi _m^2}{\partial f^2}=\frac{k_h \left( r^2 (s+1)^2 k_h+2 \left( r^2-4\right) \right) }{4-r^2}$. When $k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$, $\frac{\partial \varPi _m^2}{\partial f^2}<0$ holds, and thus $\varPi _m^L(f)$ is concave in f. Let $\frac{\partial \varPi _m}{\partial f}=\frac{k_h \left( a r (s+1)+f r^2 (s+1)^2 k_h+2 f \left( r^2-4\right) \right) }{4-r^2}=0$, we obtain $f^{LM}=\frac{a r (s+1)}{2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h}$. Replace $f^{LM}$ with f in $e_m^*$, $e^*_h$, $w^*$ and $p^*$, we can obtain the optimal solutions and the profits.

1.2 A-2 Proof of VN game

In the VN game, the manufacturer and the retailer decide the wholesale and retail prices simultaneously in the third stage. Here, we assume $p=m_1+w$. Replace $p=m_1+w$ in Eqs. (A.1) and (A.3), we obtain $\frac{\partial {\varPi ^L_m}^2}{\partial w^2}=-2<0$ and $\frac{\partial {\varPi ^L_t}^2}{\partial m_1^2}=-2<0$, so $\varPi _m^L(f,e_m,w)$ is concave in w, and $\varPi _t^L(p)$ is concave in $m_1$. Let $\frac{\partial \varPi ^L_m}{\partial w}=a+r \left( s e_h+e_h+e_m\right) -m_1-2 w=0$ and $\frac{\partial \varPi ^L_t}{\partial m_1}=a+r \left( s e_h+e_h+e_m\right) -2 m_1-w=0$, we obtain $w^*=\frac{1}{3} \left( a+r s e_h+r e_h+r e_m\right)$ and $m^*_1=\frac{a}{3}+\frac{1}{3} r s e_h+\frac{r e_h}{3}+\frac{r e_m}{3}$.

Then, we back to the second stage and solve the manufacturer’s and the technology firm’s emission reduction decisions. Replace $w^*$ and $m_1^*$ in $\varPi _m^L(f,e_m,w)$, we obtain $\frac{\partial {\varPi ^L_m}^2}{\partial e_m^2}=\frac{1}{9} \left( 2 r^2-9\right) <0$ and $\frac{\partial {\varPi ^L _h}^2}{\partial e_h^2}=-\frac{1}{k_h}<0$, so $\varPi ^L_m(f,e_m,w)$ is concave in $e_m$, and $\varPi ^L_h(e_h)$ is concave in $e_h$. Let $\frac{\partial \varPi ^L_m}{\partial e_m}=\frac{1}{18} \left( 4 r \left( a+r (s+1) e_h\right) +2 \left( 2 r^2-9\right) e_m\right) =0$ and $\frac{\partial \varPi ^L_h}{\partial e_h}=f-\frac{e_h}{k_h}=0$, we obtain $e_m^{*}=\frac{2 \left( a r+f r^2 s k_h+f r^2 k_h\right) }{9-2 r^2}$ and $e_h^{*}=f k_h$.

Replace $e_m^{*}$ and $e_h^{*}$ in $\varPi ^L_m$, we have $\varPi ^L_m(f)=\frac{a^2+f k_h \left( 2 a r (s+1)+f r^2 (s+1)^2 k_h+f \left( 2 r^2-9\right) \right) }{9-2 r^2}$. Then we get $\frac{\partial \varPi ^L_m}{\partial f}=\frac{2 k_h \left( a r (s+1)+f r^2 (s+1)^2 k_h+f \left( 2 r^2-9\right) \right) }{9-2 r^2}$ and $\frac{\partial {\varPi ^L_m}^2}{\partial f^2}=\frac{2 k_h \left( r^2 (s+1)^2 k_h+2 r^2-9\right) }{9-2 r^2}$. When $k_h<\frac{9-2 r^2}{r^2 (s+1)^2}$, $\frac{\partial {\varPi ^L_m}^2}{\partial f^2}<0$, and thus $\varPi ^L_m(f)$ is concave in f. Let $\frac{\partial \varPi ^L_m}{\partial f}=0$, we obtain $f^{LN}=\frac{a r (s+1)}{9-2 r^2-r^2(s+1)^2 k_h}$. Replace $f^{LN}$ with f in $e_m^*$, $e^*_h$, $w^*$ and $p^*$, we can obtain the optimal solutions and the profits.

1.3 A-3 Proof of RS game

In the RS game, the manufacturer decides the wholesale prices after the retailer sets the retail price in the third stage. Here, we still assume $p=m_1+w$. Replace $p=m_1+w$ in Eq. (A.1), we obtain $\frac{\partial {\varPi ^L_m}^2}{\partial w^2}=-2<0$, so $\varPi _m^L(f,e_m,w)$ is concave in w. Let $\frac{\partial \varPi ^L_m}{\partial w}=a+r \left( s e_h+e_h+e_m\right) -m_1-2 w=0$, we obtain $w^*=\frac{1}{2} \left( a+r s e_h+r e_h+r e_m-m_1\right)$. Replace $w^*$ in Eq. (A.3), we obtain $\frac{\partial {\varPi ^L_t}^2}{\partial m_1^2}=-1 <0$, so $\varPi ^L_t$ is concave in $m_1$. Let $\frac{\partial \varPi ^L_t}{\partial m_1}=\frac{1}{2} \left( a+r (s+1) e_h+r e_m-m_1\right) -\frac{m_1}{2}=0$, we obtain $m^*_1=\frac{1}{2} \left( a+r s e_h+r e_h+r e_m\right)$.

Then, we back to the second stage and solve the manufacturer’s and the technology firm’s emission reduction decisions. Replace $w^*$ and $m1^*$ in $\varPi _m^L(f,e_m,w)$, we obtain $\frac{\partial {\varPi ^L_m}^2}{\partial e_m^2}=\frac{1}{8} \left( r^2-8\right) <0$ and $\frac{\partial {\varPi ^L _h}^2}{\partial e_h^2}=-\frac{1}{k_h}<0$, so $\varPi ^L _m(f,e_m,w)$ is concave in $e_m$, and $\varPi ^L_h(e_h)$ is concave in $e_h$. Let $\frac{\partial \varPi ^L_m}{\partial e_m}=\frac{1}{16} \left( 2 r \left( a+r (s+1) e_h\right) +2 \left( r^2-8\right) e_m\right) =0$ and $\frac{\partial \varPi ^L_h}{\partial e_h}=f-\frac{e_h}{k_h}=0$, we obtain $e_m^{*}=\frac{r \left( a+f r (s+1) k_h\right) }{8-r^2}$ and $e_h^{*}=f k_h$.

Replace $e_m^{*}$ and $e_h^{*}$ in $\varPi ^L _m$, we obtain $\varPi ^L _m(f)=\frac{a^2+f k_h \left( 2 \left( a r (s+1)+f \left( r^2-8\right) \right) +f r^2 (s+1)^2 k_h\right) }{2 \left( 8-r^2\right) }$. Then we can get $\frac{\partial \varPi _m}{\partial f}=\frac{k_h \left( a r (s+1)+f r^2 (s+1)^2 k_h+2 f \left( r^2-8\right) \right) }{8-r^2}$ and $\frac{\partial {\varPi ^L _m}^2}{\partial f^2}=\frac{k_h \left( r^2 (s+1)^2 k_h+2 \left( r^2-8\right) \right) }{8-r^2}$. When $k_h<\frac{2 \left( 8-r^2\right) }{r^2 (s+1)^2}$, $\frac{\partial \varPi _m^2}{\partial f^2}<0$, and thus $\varPi ^L _m$ is concave in f. Let $\frac{\partial \varPi ^L_m}{\partial f}=0$, we obtain $f^{LR}=\frac{a r (s+1)}{2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h}$. Replace $f^{LR}$ with f in $e_m^*$, $e^*_h$, $w^*$ and $p^*$, we can obtain the optimal solutions and the profits.

1.4 A-4 Proof of Lemma 1

We derive the effect of power structures on decisions in this lemma. Note that we compare these decisions when the three power structures all exist, namely, $k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$.

• (a) We first compare the optimal linear fee f under different power structures. Taking the difference between $f^{LM}$ and $f^{LN}$, we have $f^{LM}-f^{LN}=\frac{a r (s+1)}{\left( 9-2 r^2-r^2(s+1)^2 k_h\right) \left( 2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h\right) }>0$.

  Taking the difference between $f^{LR}$ and $f^{LN}$, we have $f^{LN}-f^{LR}=\frac{7 a r (s+1)}{\left( 9-2 r^2-r^2(s+1)^2 k_h\right) \left( 2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h\right) }>0$. Thus $f^{LM}>f^{LN}>f^{LR}$.

• (b) We then compare the emission reductions under different power structures. Taking the difference between $e^{LM}_m$ and $e^{LN}_m$, we have $e^{LM}_m-e^{LN}_m=\frac{2 a r}{\left( 9-2 r^2-r^2(s+1)^2 k_h\right) \left( 2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h\right) }>0$.

  Taking the difference between $e^{LN}_m$ and $e^{LR}_m$, we have $e_m^{LN}-e_m^{LR}=\frac{14 a r}{\left( 9-2 r^2-r^2(s+1)^2 k_h\right) \left( 2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h\right) }>0$. Thus $e^{LM}_m>e^{LN}_m>e_m^{LR}$.

  Taking the difference between $e^{LM}_h$ and $e^{LN}_h$, we have $e^{LM}_h-e^{LN}_h=\frac{a r (s+1) k_h}{\left( 9-2 r^2-r^2(s+1)^2 k_h\right) \left( 2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h\right) }>0$.

  Taking the difference between $e^{LN}_h$ and $e^{LR}_h$, we have $e_h^{LN}-e_h^{LR}=\frac{7 a r (s+1) k_h}{\left( 9-2 r^2-r^2(s+1)^2 k_h\right) \left( 2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h\right) }>0$. Thus $e^{LM}_h>e^{LN}_h>e_h^{LR}$.

1.5 A-5 Proof of Proposition 1

• (a) Taking the difference between $\varPi _m^{LM}$ and $\varPi _m^{LN}$, we obtain

  $$\begin{aligned} \varPi _m^{LM}-\varPi _m^{LN}=\frac{a^2}{\left( 9-2 r^2-r^2(s+1)^2 k_h\right) \left( 2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h\right) }>0. \end{aligned}$$

  Taking the difference between $\varPi _m^{LN}$ and $\varPi _m^{LR}$, we obtain

  $$\begin{aligned} \varPi _m^{LN}-\varPi _m^{LR}=\frac{7 a^2}{\left( 9-2 r^2-r^2(s+1)^2 k_h\right) \left( 2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h\right) }>0. \end{aligned}$$

  Thus $\varPi _m^{LM}>\varPi _m^{LN}>\varPi _m^{LR}$.

  Taking the difference between $\varPi _h^{LM}$ and $\varPi _h^{LN}$, we obtain

  $$\begin{aligned} \varPi _h^{LM}-\varPi _h^{LN}=\frac{a^2 r^2 (s+1)^2 k_h \left( 17-2 r^2 (s+1)^2 k_h-4 r^2\right) }{2 \left( 9-2 r^2-r^2(s+1)^2 k_h\right) {}^2 \left( 2 \left( r^2+4\right) -r^2 (s+1)^2 k_h\right) {}^2}. \end{aligned}$$

  When $\varPi _h^{LM}-\varPi _h^{LN}>0$, we have $k_h<\frac{17-4 r^2}{2 r^2 (s+1)^2}$ and $\frac{17-4 r^2}{2 r^2 (s+1)^2}>\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$. Hence, when $k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$, $\varPi _h^{LM}>\varPi _h^{LN}$ holds.

  Taking the difference between $\varPi _h^{LN}$ and $\varPi _h^{LR}$, we obtain

  $$\begin{aligned} \varPi _h^{LN}-\varPi _h^{LR}=\frac{7 a^2 r^2 (s+1)^2 k_h \left( 2 r^2 (s+1)^2 k_h+4 r^2-25\right) }{2 \left( 9-2 r^2-r^2(s+1)^2 k_h\right) {}^2 \left( 2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h\right) {}^2}. \end{aligned}$$

  When $\varPi _h^{LN}-\varPi _h^{LR}>0$, we have $k_h<\frac{25-4 r^2}{2 r^2 (s+1)^2}$ and $\frac{25-4 r^2}{2 r^2 (s+1)^2}>\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$. Hence, when $k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$, $\varPi _h^{LN}>\varPi _h^{LR}$ holds. From what has been discussed above, we obtain $\varPi _h^{LM}>\varPi _h^{LN}>\varPi _h^{LR}$.

• (b) Taking the ratio of $\varPi _t^{LM}$ to $\varPi _t^{LN}$, we obtain $\varPi _t^{LM}/\varPi _t^{LN}=\frac{4 \left( 9-2 r^2-r^2(s+1)^2 k_h\right) {}^2}{9 \left( 2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h\right) {}^2}$.

  Set $\varPi _t^{LM}/\varPi _t^{LN}>1$, we obtain $4 \left( 9-2 r^2-r^2(s+1)^2 k_h\right) {}^2-9 \left( 2 \left( 4-r^2\right) \right. \left. -r^2 (s+1)^2 k_h\right) {}^2>0$. Then we derive $\frac{2 \left( 3-r^2\right) }{r^2 (s+1)^2}<k_h<\frac{2 \left( 21-5 r^2\right) }{5 r^2 (s+1)^2}$. According to the constraint $k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$, we can get $\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}>\frac{2 \left( 3-r^2\right) }{r^2 (s+1)^2}$ and $\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}<\frac{2 \left( 21-5 r^2\right) }{5 r^2 (s+1)^2}$. Thus when $\frac{2 \left( 3-r^2\right) }{r^2 (s+1)^2}<k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$, we obtain $\varPi _t^{LM}>\varPi _t^{LN}$; when $1<k_h<\frac{2 \left( 3-r^2\right) }{r^2 (s+1)^2}$, we obtain $\varPi _t^{LM}<\varPi _t^{LN}$.

  Taking the ratio of $\varPi _t^{LR}$ to $\varPi _t^{LN}$, we obtain $\varPi _t^{LR}/\varPi _t^{LN}=\frac{32 \left( 9-2 r^2-r^2(s+1)^2 k_h\right) {}^2}{9 \left( 2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h\right) {}^2}$.

  Set $\varPi _t^{LR}/\varPi _t^{LN}>1$, we obtain $32 \left( 9-2 r^2-r^2(s+1)^2 k_h\right) {}^2-9 \left( 2 \left( 8-r^2\right) \right. \left. -r^2 (s+1)^2 k_h\right) {}^2>0$.

  Then we derive $k_h<\frac{2 \left( 72-23 r^2-42 \sqrt{2}\right) }{23 r^2 (s+1)^2}$ or $k_h>\frac{2 \left( 72-23 r^2+42 \sqrt{2}\right) }{23 r^2 (s+1)^2}$. In addition, $\frac{2 \left( 72-23 r^2-42 \sqrt{2}\right) }{23 r^2 (s+1)^2}<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$ and $\frac{2 \left( 72-23 r^2+42 \sqrt{2}\right) }{23 r^2 (s+1)^2}>\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$. Hence, when $k_h<\frac{2 \left( 72-23 r^2-42 \sqrt{2}\right) }{23 r^2 (s+1)^2}$, we obtain $\varPi _t^{LR}>\varPi _t^{LN}$; when $\frac{2 \left( 72-23 r^2-42 \sqrt{2}\right) }{23 r^2 (s+1)^2}<k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$, we obtain $\varPi _t^{LR}<\varPi _t^{LN}$.

  Taking the ratio of $\varPi _t^{LM}$ to $\varPi _t^{LR}$, we obtain $\varPi _t^{LM}/\varPi _t^{LR}=\frac{\left( 2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h\right) {}^2}{8 \left( 2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h\right) {}^2}$.

  Set $\varPi _t^{LM}/\varPi _t^{LR}>1$, we obtain $\left( 2 \left( 8-r^2\right) -r^2 (s+1)^2 k_h\right) {}^2-8 \left( 2 \left( 4-r^2\right) \right. \left. -r^2 (s+1)^2 k_h\right) {}^2>0$. Then we derive $\frac{2 \left( 8 \left( 3-\sqrt{2}\right) -7 r^2\right) }{7 r^2 (s+1)^2}<k_h<\frac{2 \left( 8 \left( 3+\sqrt{2}\right) -7 r^2\right) }{7 r^2 (s+1)^2}$. In addition, $\frac{2 \left( 8 \left( 3-\sqrt{2}\right) -7 r^2\right) }{7 r^2 (s+1)^2}-\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}=-\frac{8 \left( 1+2 \sqrt{2}\right) }{7 r^2 (s+1)^2}<0$ and $\frac{2 \left( 8 \left( 3+\sqrt{2}\right) -7 r^2\right) }{7 r^2 (s+1)^2}-\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}=\frac{8 \left( 2 \sqrt{2}-1\right) }{7 r^2 (s+1)^2}>0$. Thus when $\frac{2 \left( 8 \left( 3-\sqrt{2}\right) -7 r^2\right) }{7 r^2 (s+1)^2}<k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$, $\varPi _t^{LM}>\varPi _t^{LR}$; when $1<k_h<\frac{2 \left( 8 \left( 3-\sqrt{2}\right) -7 r^2\right) }{7 r^2 (s+1)^2}$, $\varPi _t^{LM}<\varPi _t^{LR}$.

  Therefore, if $k_h<k^{L}_1=\frac{2 \left( 72-23 r^2-42 \sqrt{2}\right) }{23 r^2 (s+1)^2}$, we have $\varPi _t^{LR}>\varPi _t^{LN}>\varPi _t^{LM}$; if $k^{L}_1<k_h<k^{L}_2=\frac{2 \left( 8 \left( 3-\sqrt{2}\right) -7 r^2\right) }{7 r^2 (s+1)^2}$, then $\varPi _t^{LN}>\varPi _t^{LR}>\varPi _t^{LM}$; if $k^{L}_2<k_h<k^{L}_3=\frac{2 \left( 3-r^2\right) }{r^2 (s+1)^2}$, then $\varPi _t^{LN}>\varPi _t^{LM}>\varPi _t^{LR}$; if $k^{L}_3<k_h<\frac{2 \left( 4-r^2\right) }{r^2 (s+1)^2}$, then $\varPi _t^{LM}>\varPi _t^{LN}>\varPi _t^{LR}$.

B. Revenue-sharing payment

When the manufacture collaborates with the technology firm through revenue-sharing payment, the three members’ profit functions are as follows:

$$\begin{aligned}&\varPi _m^S(\phi ,e_m,w)=(1-\phi )w \left( a-p+r \left( e_m+e_h+s e_h\right) \right) -\frac{e_m^2}{2}, \end{aligned}$$

(B.1)

$$\begin{aligned}&\varPi _h^S(e_h)=w \phi \left( a-p+r \left( e_m+e_h+s e_h\right) \right) -\frac{e_h^2}{2 k_h},\end{aligned}$$

(B.2)

$$\begin{aligned}&\varPi _t^S(p)=(p-w)\left( a-p+r \left( e_m+e_h+s e_h\right) \right) . \end{aligned}$$

(B.3)

1.1 B-1 Proof of MS game

The decision sequence is similar to that of “Appendix A-1”. In stage 4, the retailer decides the retail price after the manufacturer sets the wholesale price in stage 3. From Eq. (B.3), we obtain $\frac{\partial {\varPi ^S_t}^2}{\partial p^2}=-2<0$, so $\varPi _t^S(p)$ is concave in p. Let $\frac{\partial \varPi _t^S}{\partial p}=a+r \left( s e_h+e_h+e_m\right) -2 p+w=0$, we obtain $p^*=\frac{1}{2} \left( a+r s e_h+r e_h+r e_m+w\right)$. Replace $p^*$ in $\varPi _m^S(\phi ,e_m,w)$, we obtain $\frac{\partial {\varPi ^S_m}^2}{\partial w^2}=-(1-\phi ) <0$, so $\varPi _m^S(\phi ,e_m,w)$ is a concave function of w. Let $\frac{\partial \varPi ^S_m}{\partial w}=-\frac{1}{2} (\phi -1) \left( a+r (s+1) e_h+r e_m-2 w\right) =0$, we have $w^*=\frac{1}{2} \left( a+r s e_h+r e_h+r e_m\right)$.

Replace $w^*$ and $p^*$ in $\varPi _m^S(\phi ,e_m,w)$, we obtain $\frac{\partial {\varPi ^S_m}^2}{\partial e_m^2}=\frac{1}{4} \left( r^2 (1-\phi )-4\right) <0$, so $\varPi _m^S(\phi ,e_m,w)$ is concave in $e_m$. From Eq. (B.2), we obtain $\frac{\partial {\varPi ^S_h}^2}{\partial e_h^2}=\frac{1}{8} \left( 2 r^2 (s+1)^2 \phi -\frac{8}{k_h}\right)$. Set $\frac{1}{8} \left( 2 r^2 (s+1)^2 \phi -\frac{8}{k_h}\right) <0$, we obtain $\phi <\frac{4}{r^2 (s+1)^2 k_h}$, so $\varPi ^S_h(e_h)$ is concave in $e_h$. Let $\frac{\partial \varPi ^S _m}{\partial e_m}=\frac{1}{2} \left( -\frac{1}{2} r (\phi -1) \left( a+r (s+1) e_h+r e_m\right) -2 e_m\right) =0$ and $\frac{\partial \varPi ^S _h}{\partial e_h}=\frac{1}{8} \left( 2 r (s+1) \phi \left( a+r (s+1) e_h+r e_m\right) -\frac{8 e_h}{k_h}\right) =0$, we obtain $e_m^{*}=\frac{a r (1-\phi )}{4-r^2 (s+1)^2 \phi k_h-r^2(1-\phi )}$ and $e_h^{*}=\frac{a r (s+1) \phi k_h}{4-r^2 (s+1)^2 \phi k_h-r^2(1-\phi )}$.

Replace $e_m^{*}$ and $e_h^{*}$ in $\varPi _m^S$, we obtain $\varPi _m^S(\phi )=\frac{a^2 (1-\phi ) \left( 4-r^2 (1-\phi )\right) }{2\left( 4-r^2 (s+1)^2 \phi k_h+r^2(1-\phi )\right) {}^2}$. Then we derive $\frac{\partial \varPi ^S_m}{\partial \phi }=\frac{a^2 \left( 8-r^2 (s+1)^2 k_h \left( r^2 (\phi -1)-2 \phi +4\right) +2 r^2 (\phi -1)\right) }{\left( r^2 (s+1)^2 \phi k_h-r^2(\phi -1)-4\right) {}^3}$, $\frac{\partial {\varPi ^S_m}^2}{\partial \phi ^2}$$=\frac{a^2 r^2 \left( (s+1)^2 k_h \left( r^2 (s+1)^2 k_h \left( r^2 (2 \phi -3)-4 (\phi -3)\right) -2 r^4 (\phi -1)-32\right) +4 \left( r^2 (\phi -1)+4\right) \right) }{\left( -r^2 (s+1)^2 \phi k_h+r^2 (\phi -1)+4\right) {}^4}$.

When $\frac{\partial {\varPi ^S_m}^2}{\partial \phi ^2}<0$, we derive $\phi >\phi _1=\frac{(s+1)^2 k_h \left( 3 \left( r^2-4\right) r^2 (s+1)^2 k_h-2 r^4+32\right) +4 \left( r^2-4\right) }{2 r^2 \left( (s+1)^2 k_h-1\right) \left( \left( r^2-2\right) (s+1)^2 k_h-2\right) }$. Because $0<\phi <1$, we restrict $\phi _1<1$ to ensure the second derivative is negative. From this, we obtain $\frac{4}{\left( 8-r^2\right) (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$. Set $\frac{\partial \varPi ^S_m}{\partial \phi }=0$, we obtain $\phi ^{SM}=\frac{\left( 4-r^2\right) \left( r^2 (s+1)^2 k_h-2\right) }{r^2 \left( \left( 2-r^2\right) (s+1)^2 k_h+2\right) }$. To ensure the optimal solution exists, we restrict $\phi ^{SM}<\frac{4}{r^2 (s+1)^2 k_h}$, and then derive $\frac{2}{\left( r^2-4\right) (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$. Then, we restrict $\phi ^{SM}<1$ and derive $\frac{2}{r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$. Here, $\frac{2}{r^2 (s+1)^2}-\frac{4}{\left( 8-r^2\right) (s+1)^2}=\frac{2 \left( 8-3 r^2\right) }{r^2 \left( 8-r^2\right) (s+1)^2}>0$. Thus, from what has been discussed above, the constraint condition is $\frac{2}{r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$. Replace $\phi ^{SM}$ with $\phi$ in $e_m^*$, $e^*_h$, $w^*$ and $p^*$, we can obtain the optimal solutions and the profits.

1.2 B-2 Proof of VN game

In the VN game, the manufacturer and the retailer decide the wholesale and retail prices simultaneously in the third stage. Here, we also assume $p=m_1+w$. Replace $p=m_1+w$ in Eqs. (B.1) and (B.3), we obtain $\frac{\partial {\varPi ^S_m}^2}{\partial w^2}=2 (\phi -1)<0$ and $\frac{\partial {\varPi ^S_t}^2}{\partial m_1^2}=-2<0$, so $\varPi _m^S(\phi ,e_m,w)$ is concave in w, and $\varPi _t^S(p)$ is concave in $m_1$. Let $\frac{\partial \varPi ^S_m}{\partial w}=(1-\phi ) \left( a+r (s+1) e_h+r e_m-m_1-2 w\right) =0$ and $\frac{\partial \varPi ^S_t}{\partial m_1}=a+r \left( s e_h+e_h+e_m\right) -2 m_1-w=0$, we obtain $w^*=\frac{1}{3} \left( a+r s e_h+r e_h+r e_m\right)$ and $m^*_1=\frac{1}{3} \left( a+r s e_h+r e_h+r e_m\right)$.

Replace $w^*$ and $m_1^*$ in $\varPi _m^S(\phi ,e_m,w)$, we obtain $\frac{\partial {\varPi ^S_m}^2}{\partial e_m^2}=\frac{2}{9} r^2 (1-\phi )-1<0$ and $\frac{\partial {\varPi ^S _h}^2}{\partial e_h^2}=\frac{2}{9} r^2 (s+1)^2 \phi -\frac{1}{k_h}$. Thus $\varPi ^S _m(\phi ,e_m,w)$ is concave in $e_m$. Set $\frac{2}{9} r^2 (s+1)^2 \phi -\frac{1}{k_h}<0$, we have $\phi <\frac{9}{2 r^2 (s+1)^2 k_h}$, and thus $\varPi ^S _h(e_h)$ is concave in $e_h$ under this constraint. Let $\frac{\partial \varPi ^S_m}{\partial e_m}=-\frac{2}{9} r (\phi -1) \left( a+r (s+1) e_h+r e_m\right) -e_m$ and $\frac{\partial \varPi ^S_h}{\partial e_h}=\frac{2}{9} r (s+1) \phi \left( a+r (s+1) e_h+r e_m\right) -\frac{e_h}{k_h}=0$, we obtain $e_m^{*}=\frac{2 a r (\phi -1)}{2 r^2 (s+1)^2 \phi k_h-2 r^2 (\phi -1)-9}$ and $e_h^{*}=-\frac{2 a r (s+1) \phi k_h}{2 r^2 (s+1)^2 \phi k_h-2 r^2 (\phi -1)-9}$.

Replace $e_m^{*}$ and $e_h^{*}$ in $\varPi ^S_m$, we obtain $\varPi ^S_m=-\frac{a^2 (\phi -1) \left( 2 r^2 (\phi -1)+9\right) }{\left( -2 r^2 (s+1)^2 \phi k_h+2 r^2 (\phi -1)+9\right) {}^2}$. Then we derive $\frac{\partial \varPi ^S_m}{\partial \phi }=\frac{a^2 \left( 9 \left( 2 r^2 (\phi -1)+9\right) -2 r^2 (s+1)^2 k_h \left( 4 r^2 (\phi -1)-9 (\phi -2)\right) \right) }{\left( 2 r^2 (s+1)^2 \phi k_h-2 r^2 (\phi -1)-9\right) {}^3}$, $\frac{\partial {\varPi ^S_m}^2}{\partial \phi ^2}$$=\frac{2 a^2 r^2 \left( 4 (s+1)^2 k_h \left( r^2 (s+1)^2 k_h \left( r^2 (4 \phi -6)-9 (\phi -3)\right) -4 r^4 (\phi -1)-81\right) +18 \left( 2 r^2 (\phi -1)+9\right) \right) }{\left( -2 r^2 (s+1)^2 \phi k_h+2 r^2 (\phi -1)+9\right) {}^4}$.

Set $\frac{\partial {\varPi ^S_m}^2}{\partial \phi ^2}<0$, we have $\phi >\phi _2=\frac{2 (s+1)^2 k_h \left( 3 \left( 2 r^2-9\right) r^2 (s+1)^2 k_h-4 r^4+81\right) +9 \left( 2 r^2-9\right) }{2 r^2 \left( (s+1)^2 k_h-1\right) \left( \left( 4 r^2-9\right) (s+1)^2 k_h-9\right) }$. Because $0<\phi <1$, we restrict $\phi _2<1$ and derive that $\frac{9}{2 \left( 9-r^2\right) (s+1)^2}<k_h<\frac{9}{2 r^2 (s+1)^2}$. Let $\frac{\partial \varPi ^S_m}{\partial \phi }=0$, we obtain $\phi ^{SN}=\frac{\left( 9-2 r^2\right) \left( 4 r^2 (s+1)^2 k_h-9\right) }{2 r^2 \left( \left( 9-4 r^2\right) (s+1)^2 k_h+9\right) }$. To ensure the optimal solution exists, we restrict $0<\phi ^{SN}<1$ and derive $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{9}{2 r^2 (s+1)^2}$; then we restrict $\phi ^{SN}<\frac{9}{2 r^2 (s+1)^2 k_h}$ and derive $\frac{9}{2 \left( 2 r^2-9\right) (s+1)^2}<k_h<\frac{9}{2 r^2 (s+1)^2}$. Thus, from what has been discussed above, the constraint condition is $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{9}{2 r^2 (s+1)^2}$. Replace $\phi ^{SN}$ with $\phi$ in $e_m^*$, $e^*_h$, $w^*$ and $p^*$, we can obtain the optimal solutions and the profits.

1.3 B-3 Proof of RS game

Similarly, we still assume $p=m_1+w$. Replace $p=m_1+w$ in Eq. (B.1), we obtain $\frac{\partial {\varPi ^S_m}^2}{\partial w^2}=-2(1-\phi ) <0$, so $\varPi _m^S(\phi ,e_m,w)$ is concave in w. Let $\frac{\partial \varPi ^S_m}{\partial w}=(1-\phi ) \left( a+r (s+1) e_h+r e_m-m_1-2 w\right) =0$, we obtain $w^*=(1-\phi ) \left( a+r (s+1) e_h+r e_m \right. \left. -m_1-2 w\right)$. Replace $w^*$ in Eq. (B.3), we obtain $\frac{\partial {\varPi ^S_t}^2}{\partial m_1^2}=-1<0$, so $\varPi ^S_t$ is concave in $m_1$. Let $\frac{\partial \varPi ^S_t}{\partial m_1}=\frac{1}{2} \left( a+r (s+1) e_h+r e_m-m_1\right) -\frac{m_1}{2}=0$, we obtain $m^*_1=\frac{1}{2} \left( a+r s e_h+r e_h+r e_m\right)$.

Then, we back to the first stage and solve the manufacturer’s and the technology firm’s emission reduction decisions. Replace $w^*$ and $m_1^*$ in $\varPi _m^S(\phi ,e_m,w)$, we obtain $\frac{\partial {\varPi ^S_m}^2}{\partial e_m^2}=\frac{1}{8} \left( r^2 (1-\phi )-8\right) <0$ and $\frac{\partial {\varPi ^S _h}^2}{\partial e_h^2}=\frac{1}{16} \left( 2 r^2 (s+1)^2 \phi -\frac{16}{k_h}\right)$, so $\varPi ^S _m(\phi ,e_m,w)$ is concave in $e_m$. Set $\frac{1}{16} \left( 2 r^2 (s+1)^2 \phi -\frac{16}{k_h}\right) <0$, we have $\phi <\frac{8}{r^2 (s+1)^2 k_h}$, and $\varPi ^S_h(e_h)$ is concave in $e_h$ under this condition. Let $\frac{\partial \varPi ^S_m}{\partial e_m}=\frac{1}{16} \left( 2 e_m \left( r^2(1-\phi )-8\right) -2 r (\phi -1) \left( a+r (s+1) e_h\right) \right) =0$, $\frac{\partial \varPi ^S_h}{\partial e_h}=\frac{1}{16} \left( 2 r (s+1) \phi \left( a+r (s+1) e_h+r e_m\right) -\frac{16 e_h}{k_h}\right) =0$. We obtain $e_m^{*}=\frac{a r (\phi -1)}{r^2 (s+1)^2 \phi k_h+r^2(1-\phi ))-8}$ and $e_h^{*}=-\frac{a r (s+1) \phi k_h}{r^2 (s+1)^2 \phi k_h+r^2 (1-\phi )-8}$.

Replace $e_m^{*}$ and $e_h^{*}$ in $\varPi ^S_m$, we obtain $\varPi ^S_m(\phi )=\frac{a^2 (1-\phi ) \left( r^2 (\phi -1)+8\right) }{2 \left( 8-r^2 (s+1)^2 \phi k_h+r^2 (\phi -1)\right) {}^2}$. Then we derive $\frac{\partial \varPi ^S_m}{\partial \phi }=\frac{a^2 \left( 4 \left( r^2 (\phi -1)+8\right) -r^2 (s+1)^2 k_h \left( r^2 (\phi -1)-4 \phi +8\right) \right) }{\left( r^2 (s+1)^2 \phi k_h+r^2 (1-\phi )-8\right) {}^3}$, $\frac{\partial {\varPi ^S_m}^2}{\partial \phi ^2}$$=\frac{a^2 r^2 \left( -2 (s+1)^2 k_h \left( r^4 (\phi -1)+64\right) +r^2 (s+1)^4 k_h^2 \left( r^2 (2 \phi -3)-8 (\phi -3)\right) +8 \left( r^2 (\phi -1)+8\right) \right) }{\left( 8-r^2 (s+1)^2 \phi k_h+r^2 (\phi -1)\right) {}^4}$. Set $\frac{\partial {\varPi ^S_m}^2}{\partial \phi ^2}<0$, we derive $\phi >\phi _3=\frac{-2 \left( 64-r^4\right) (s+1)^2 k_h+3 r^2 \left( 8-r^2\right) (s+1)^4 k_h^2+8 \left( 8-r^2\right) }{2 r^2 \left( (s+1)^2 k_h-1\right) \left( \left( 4-r^2\right) (s+1)^2 k_h+4\right) }$. Because $0<\phi <1$, we restrict $\phi _3<1$ and derive $\frac{8}{\left( 16-r^2\right) (s+1)^2}<k_h<\frac{8}{r^2 (s+1)^2}$. Let $\frac{\partial \varPi ^S_m}{\partial \phi }=0$, we obtain $\phi ^{SR}=\frac{\left( 8-r^2\right) \left( r^2 (s+1)^2 k_h-4\right) }{r^2 \left( \left( 4-r^2\right) (s+1)^2 k_h+4\right) }$. To ensure the optimal solution exists, we restrict $0<\phi ^{SR}<1$ and derive $\frac{4}{r^2 (s+1)^2}<k_h<\frac{8}{r^2 (s+1)^2}$, and under this condition, $\phi >\phi _3$ and $\phi <\frac{8}{r^2 (s+1)^2 k_h}$ both hold. Thus, the constraint condition is $\frac{4}{r^2 (s+1)^2}<k_h<\frac{8}{r^2 (s+1)^2}$. Replace $\phi ^{SR}$ with $\phi$ in $e_m^*$, $e^*_h$, $w^*$ and $p^*$, we can obtain the optimal solutions and the profits.

1.4 B-4 Proof of Lemma 2

Here, we compare these decisions when the MS and VN games exist, namely, $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$.

• (a) Taking the difference between $\phi ^{SM}$ and $\phi ^{SN}$, we obtain $\phi ^{SM}-\phi ^{SN}=\frac{(s+1)^2 k_h \left( 2 r^4 (s+1)^2 k_h-17 r^2+18\right) +18}{2 r^2 \left( \left( r^2-2\right) (s+1)^2 k_h-2\right) \left( \left( 4 r^2-9\right) (s+1)^2 k_h-9\right) }$. When $\phi ^{SM}-\phi ^{SN}>0$, we derive $(s+1)^2 k_h \left( 2 r^4 (s+1)^2 k_h-17 r^2+18\right) +18>0$. Then we obtain $k_h<\frac{17 r^2-\sqrt{145 r^4-612 r^2+324}-18}{4 r^4 (s+1)^2}$ and $k_h>\frac{17 r^2+\sqrt{145 r^4-612 r^2+324}-18}{4 r^4 (s+1)^2}$. Due to $\frac{17 r^2-\sqrt{145 r^4-612 r^2+324}-18}{4 r^4 (s+1)^2}<0$ and $\frac{9}{4 r^2 (s+1)^2}-\frac{17 r^2+\sqrt{145 r^4-612 r^2+324}-18}{4 r^4 (s+1)^2}=\frac{18-8 r^2-\sqrt{145 r^4-612 r^2+324}}{4 r^4 (s+1)^2}>0$, so $\phi ^{SM}-\phi ^{SN}>0$ always holds under the above constraint.

• (b) Taking the difference between $e_m^{SM}$ and $e_m^{SN}$, we obtain $e_m^{SM}-e_m^{SR}=\frac{a r}{2 \left( 2 r^4-17 r^2+36\right) (s+1)^2 k_h}>0$.

  Taking the difference between $e_h^{SM}$ and $e_h^{SN}$, we obtain $e_h^{SM}-e_h^{SN}=\frac{a r (s+1) k_h}{4 r^4 (s+1)^4 k_h^2-34 r^2 (s+1)^2 k_h+72}>0$ always holds when $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$.

1.5 B-5 Proof of Proposition 2

Here, we compare the three members’ profits when the MS and VN games exist, namely, $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$.

• (a) Taking the difference between $\varPi _m^{SM}$ and $\varPi _m^{SN}$, we obtain $\varPi _m^{SM}-\varPi _m^{SN}=\frac{a^2 \left( \left( 36-17 r^2\right) (s+1)^2 k_h+36\right) }{8 \left( 4-r^2\right) \left( 9-2 r^2\right) (s+1)^2 k_h \left( 4-r^2 (s+1)^2 k_h\right) \left( 9-2 r^2 (s+1)^2 k_h\right) }>0$.

  Taking the difference between $\varPi _h^{SM}$ and $\varPi _h^{SN}$, we obtain $\varPi _h^{SM}-\varPi _h^{SN}=\frac{a^2 \left( (s+1)^2 k_h \left( 2 \left( 36-17 r^2\right) (s+1)^2 k_h+17 r^2+36\right) -36\right) }{8 \left( 4-r^2\right) \left( 9-2 r^2\right) (s+1)^4 k_h^2 \left( 4-r^2 (s+1)^2 k_h\right) \left( 9-2 r^2 (s+1)^2 k_h\right) }>0$.

• (b) Taking the ratio of $\varPi _t^{SM}$ to $\varPi _t^{SN}$, we obtain $\frac{\varPi _t^{SM}}{\varPi _t^{SN}}=\frac{4 \left( 9-2 r^2\right) ^2 \left( 9-2 r^2 (s+1)^2 k_h\right) {}^2 \left( \left( 2-r^2\right) (s+1)^2 k_h+2\right) {}^2}{9 \left( 4-r^2\right) ^2 \left( 4-r^2 (s+1)^2 k_h\right) {}^2 \left( \left( 9-4 r^2\right) (s+1)^2 k_h+9\right) {}^2}$.

  When $\frac{\varPi _t^{SM}}{\varPi _t^{SN}}>1$, we obtain that only $k_h>k^{S}_1=\frac{72 \left( r^2-3\right) }{\left( 23 r^4-102 r^2-\sqrt{-47 r^8+348 r^6+252 r^4-6480 r^2+11664}+108\right) (s+1)^2}$ is in the feasible region $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$. Hence, when $k^{S}_1<k_h<\frac{4}{r^2 (s+1)^2}$, we have $\varPi _t^{SM}>\varPi _t^{SN}$; when $\frac{9}{4 r^2 (s+1)^2}<k_h<k^{S}_1$, we have $\varPi _t^{SM}<\varPi _t^{SN}$.

1.6 B-6 Proof of Lemma 3

Here, we compare these decisions when the VN and RS games exist, namely, $\frac{4}{r^2 (s+1)^2}<k_h<\frac{9}{2 r^2 (s+1)^2}$.

• (a) Taking the difference between $\phi ^{SN}$ and $\phi ^{SR}$, we obtain $\phi ^{SN}-\phi ^{SR}=\frac{7 \left( (s+1)^2 k_h \left( 2 r^4 (s+1)^2 k_h-25 r^2+36\right) +36\right) }{2 r^2 \left( \left( r^2-4\right) (s+1)^2 k_h-4\right) \left( \left( 4 r^2-9\right) (s+1)^2 k_h-9\right) }>0$.

• (b) Taking the difference between $e_m^{SN}$ and $e_m^{SR}$, we obtain $e_m^{SN}-e_m^{SR}=\frac{7 a r}{2 \left( 2 r^4-25 r^2+72\right) (s+1)^2 k_h}>0$.

  Taking the difference between $e_h^{SN}$ and $e_h^{SR}$, we obtain $e_h^{SN}-e_h^{SR}=\frac{7 a r (s+1) k_h}{4 r^4 (s+1)^4 k_h^2-50 r^2 (s+1)^2 k_h+144}>0$ always hold when $\frac{4}{r^2 (s+1)^2}<k_h<\frac{9}{2 r^2 (s+1)^2}$.

1.7 B-7 Proof of Proposition 3

Here, we compare the three members’ profits when the VN and RS games exist, namely, $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$.

Taking the difference between $\varPi _m^{SN}$ and $\varPi _m^{SR}$, we obtain

$$\begin{aligned} \varPi _m^{SN}-\varPi _m^{SR}= & {} \frac{7 a^2 \left( \left( 72-25 r^2\right) (s+1)^2 k_h+72\right) }{8 \left( 8-r^2\right) \left( 9-2 r^2\right) (s+1)^2 k_h \left( 8-r^2 (s+1)^2 k_h\right) \left( 9-2 r^2 (s+1)^2 k_h\right) }\\&>0. \end{aligned}$$

Taking the difference between $\varPi _h^{SN}$ and $\varPi _h^{SR}$, we obtain

$$\begin{aligned} \varPi _h^{SN}-\varPi _h^{SR}=\frac{7 a^2 \left( (s+1)^2 k_h \left( 2 \left( 72-25 r^2\right) (s+1)^2 k_h+25 r^2+72\right) -72\right) }{8 \left( 8-r^2\right) \left( 9-2 r^2\right) (s+1)^4 k_h^2 \left( 8-r^2 (s+1)^2 k_h\right) \left( 9-2 r^2 (s+1)^2 k_h\right) }. \end{aligned}$$

When $\varPi _h^{SN}-\varPi _h^{SR}>0$, we derive $k_h>\frac{1}{2 (s+1)^2}$ or $k_h<\frac{72}{\left( 25 r^2-72\right) (s+1)^2}$. Due to $\frac{4}{r^2 (s+1)^2}-\frac{1}{2 (s+1)^2}=\frac{8-r^2}{2 r^2 (s+1)^2}>0$, so $\varPi _h^{SN}>\varPi _h^{SR}$ always holds when $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$.

Taking the ratio of $\varPi _t^{SN}$ to $\varPi _t^{SR}$, we obtain $\frac{\varPi _t^{SM}}{\varPi _t^{SN}}=\frac{9 \left( r^2-8\right) ^2 \left( r^2 (s+1)^2 k_h-8\right) {}^2 \left( \left( 4 r^2-9\right) (s+1)^2 k_h-9\right) {}^2}{32 \left( 9-2 r^2\right) ^2 \left( 9-2 r^2 (s+1)^2 k_h\right) {}^2 \left( \left( r^2-4\right) (s+1)^2 k_h-4\right) {}^2}$.

When $\frac{\varPi _t^{SM}}{\varPi _t^{SN}}>1$, we found that all thresholds $k_h$ are not in the feasible region $\frac{9}{4 r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$. Hence, $\varPi _t^{SN}>\varPi _t^{SR}$ always holds under the above constraint.

C. The manufacturer’s optimal payment

Here, we explore the manufacturer’s optimal payment by comparing the manufacturer’s profit under different payments. Meanwhile, the impact of payment type on the environment is also analyzed.

1.1 C-1 Proof of Lemma 4

We analyze the effect of payment type on the environment in the MS game. Meanwhile, the condition $\frac{2}{r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$ holds to ensure the cases LM and SM exist.

• (a) Taking the difference between $e_m^{LM}$ and $e_m^{SM}$, we obtain $e_m^{LM}-e_m^{SM}=\frac{2 a r}{2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h}-\frac{2 a}{\left( 4 r-r^3\right) (s+1)^2 k_h}$. This is an increasing function with respect to $k_h$, thus when $k_h=\frac{2}{r^2 (s+1)^2}$, the minimum of $e_m^{LM}-e_m^{SM}$ is $\frac{a r}{r^4-7 r^2+12}>0$. Hence $e_m^{LM}>e_m^{SM}$ always holds.

• (b) Taking the difference between $e_h^{LM}$ and $e_h^{SM}$, we obtain $e_h^{LM}-e_h^{SM}=\frac{2 a \left( -\left( 3-r^2\right) r^2 (s+1)^2 k_h-2 r^2+8\right) }{r (s+1) \left( 4-r^2 (s+1)^2 k_h\right) \left( 2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h\right) }$. Set $e_h^{LM}-e_h^{SM}>0$, we derive $k_h<\frac{2 \left( 4-r^2\right) }{r^2 \left( 3-r^2\right) (s+1)^2}$. Hence, when $\frac{2}{r^2 (s+1)^2}<k_h<\frac{2 \left( 4-r^2\right) }{r^2 \left( 3-r^2\right) (s+1)^2}$, $e_h^{LM}>e_h^{SM}$; when $\frac{2 \left( 4-r^2\right) }{r^2 \left( 3-r^2\right) (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$, $e_h^{LM}<e_h^{SM}$.

1.2 C-2 Proof of Proposition 4

We analyze the effect of payment type on the profits in the MS game. Meanwhile, the condition $\frac{2}{r^2 (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$ holds to ensure the cases LM and SM exist.

• (a) Taking the difference between $e_m^{LM}$ and $e_m^{SM}$, we obtain $\varPi _m^{LM}-\varPi _m^{SM}=\frac{a^2}{2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h}-\frac{2 a^2}{r^2 \left( 4-r^2\right) (s+1)^2 k_h \left( 4-r^2 (s+1)^2 k_h\right) }$. Let $\varPi _m^{LM}-\varPi _m^{SM}=0$, we derive two roots are $k_h=\frac{9-2 r^2+\sqrt{17-4 r^2}}{r^2 \left( 4-r^2\right) (s+1)^2}$ and $k_h=\frac{9-2 r^2-\sqrt{17-4 r^2}}{r^2 \left( 4-r^2\right) (s+1)^2}$. However, only $k_h=\frac{9-2 r^2+\sqrt{17-4 r^2}}{r^2 \left( 4-r^2\right) (s+1)^2}$ is in the feasible region. Hence, when $\frac{2}{r^2 (s+1)^2}<k_h<\frac{9-2 r^2+\sqrt{17-4 r^2}}{r^2 \left( 4-r^2\right) (s+1)^2}$, $\varPi _m^{LM}-\varPi _m^{SM}>0$; when $\frac{9-2 r^2+\sqrt{17-4 r^2}}{r^2 \left( 4-r^2\right) (s+1)^2}<k_h<\frac{4}{r^2 (s+1)^2}$, $\varPi _m^{LM}-\varPi _m^{SM}<0$.

• (b) Taking the difference between $\varPi _t^{LM}$ and $\varPi _t^{SM}$, we obtain $\varPi _t^{LM}-\varPi _t^{SM}=\frac{4 a^2}{\left( 2 \left( 4-r^2\right) -r^2 (s+1)^2 k_h\right) {}^2}-\frac{a^2 \left( \left( 2-r^2\right) (s+1)^2 k_h+2\right) {}^2}{\left( 4-r^2\right) ^2 (s+1)^4 k_h^2 \left( 4-r^2 (s+1)^2 k_h\right) {}^2}$.

  Set $\varPi _t^{LM}-\varPi _t^{SM}=0$, we derive four roots, but only $k_{h1}=\frac{8-r^4+3 r^2+\sqrt{r^8-10 r^6+33 r^4-48 r^2+64}}{r^2 \left( 6-r^2\right) (s+1)^2}$ is in the feasible region. Hence, when $\frac{2}{r^2 (s+1)^2}<k_h<k_{h1}$, $\varPi _t^{LM}-\varPi _t^{SM}>0$; when $k_{h1}<k_h<\frac{4}{r^2 (s+1)^2}$, $\varPi _t^{LM}-\varPi _t^{SM}<0$, where $k_{h1}=k^{M}_3$.

1.3 C-3 Proof of Lemmas 5 and 6

The proof is similar to that of Lemma 4, so here we omit it.

1.4 C-4 Proof of Propositions 5 and 6

The proof is similar to that of Proposition 4, so here we omit it.