Firms’ strategy analysis under different retailing formats considering emission reduction efficiency and low-carbon preference

Abstract

In response to the cap-and-trade policy, many manufacturers begin to reduce emissions by various measures, such as investing in technologies and using clean materials, which will affect their operation strategies and further affect the whole supply chain. Based on this, we explore how the cap-and-trade policy affects firms’ operation decisions under different retailing formats. We study a supply chain with an e-commerce platform and one manufacturer, in which they can cooperate via reseller or marketplace formats. Meanwhile, we employ a new cost function of emission reduction by incorporating the manufacturer’s emission reduction efficiency (ERE). We find that in the absence of low-carbon preference, the manufacturer’s and the platform’s profits both increase as the manufacturer’s ERE improves in reseller format; but in marketplace format, the platform is worse off as the manufacturer’s ERE improves if the efficiency is larger than a threshold. Besides, under which format the platform or the manufacturer is better off is related to the product category. For some product categories, low-carbon preference could alter the retailing format under which the platform or the manufacturer gains a higher profit. Moreover, the platform and the manufacturer are both better off in the presence of low-carbon preference, regardless of retailing formats.

Appendices

Appendices

1.1 A.1 Proof of Theorem 1

In reseller format, the manufacturer’s objective function is $\pi _m= w(1-p_r)- \left( \left( e_m-\tau \right) (1-p_r)-F_m\right) p_c-\frac{(1-\eta ) \tau ^2}{2}$, and the platform’s profit function is $\pi _r=\left( p_r-w\right) (1-p_r)- \left( e_r (1-p_r)-F_r\right) p_c$. Here, we solve the models by backward induction. Given w, the platform decides the optimal retail price. Taking the first derivative of $\pi _r$ with respect to $p_r$ yields $\frac{\mathrm{d} \pi _r}{\mathrm{d} p_r}=1+w+p_c e_r-2 p_r$. Then, we can get the second derivative $\frac{\mathrm{d} \pi _r^2}{\mathrm{d} p_r^2}=-2<0$. Therefore, $\pi _r$ is concave in $p_r$.

Solving $\frac{\mathrm{d} \pi _r}{\mathrm{d} p_r}=1+w+p_c e_r-2 p_r=0$, we get

$$\begin{aligned}p_r^*=\frac{1}{2} \left( 1+w+p_c e_r\right) .\end{aligned}$$

Substituting $p_r^*$ into the manufacturer’s objective function, we have $\pi _m=p_c \left( \frac{1}{2} \left( e_m-\tau \right) \left( 1-w-p_c e_r\right) +F_m\right) -\frac{(1-\eta ) \tau ^2}{2}+\frac{1}{2} w \left( 1-w-p_c e_r\right)$. Taking the first partial derivatives of $\pi _m$ with respect to w and $\tau$ yields

$$\begin{aligned}\frac{\partial \pi _m}{\partial w}=\frac{1}{2} \left( 1-2 w-p_c \left( e_r-e_m+\tau \right) \right) , \\ \frac{\partial \pi _m}{\partial \tau }=\frac{1}{2} p_c \left( 1-w-p_c e_r\right) -(1-\eta ) \tau .\end{aligned}$$

Then, the second partial derivatives are obtained: $\frac{\partial \pi _m^2}{\partial w^2}=-1$, $\frac{\partial \pi _m^2}{\partial \tau ^2}=-(1-\eta )$, $\frac{\partial \pi _m^2}{\partial w\partial \tau }=\frac{\partial \pi _m^2}{\partial \tau \partial w}=-\frac{p_c}{2}$. Based on the assumption that $4 (1-\eta ) -p_c^2>0$, we get the Hessian matrix $H_1=\left( \begin{array}{cc} -1 &{} -\frac{p_c}{2} \\ -\frac{p_c}{2} &{} -(1-\eta ) \\ \end{array} \right)$ is negative. Therefore, $\pi _m$ is jointly concave in w and $\tau$.

Solving $\frac{\partial \pi _m}{\partial w}=0$ and $\frac{\partial \pi _m}{\partial \tau }=0$, we get

$w^{\mathrm{RN}}=\frac{2(1-\eta ) e_m p_c+\left( 2 (1-\eta ) -p_c^2\right) \left( 1-e_r p_c\right) }{4 (1-\eta ) -p_c^2}$, $\tau ^{\mathrm{RN}}=\frac{p_c \left( 1- \left( e_r+e_m\right) p_c\right) }{4 (1-\eta ) -p_c^2}$. Replacing w and $\tau$ with $w^{\mathrm{RN}}$ and $\tau ^{\mathrm{RN}}$ into $p_r^*$, we can obtain $p_r^{\mathrm{RN}}=1-\frac{(1-\eta ) \left( 1- \left( e_r+e_m\right) p_c\right) }{4 (1-\eta ) -p_c^2}$.

1.2 A.2 Proof of Corollary 1

Firstly, we recall the assumptions that $4 (1-\eta )-p_c^2>0$ and $1-(e_m+e_r)p_c>0$. Then, we take the first partial derivatives of $w^{\mathrm{RN}}$, $\tau ^{\mathrm{RN}}$, $p_r^{\mathrm{RN}}$, $\pi _m$ and $\pi _r$ with respect to some parameters.

• (i) $\frac{\partial w^{\mathrm{RN}}}{\partial e_m}=\frac{2 (1-\eta ) p_c}{4 (1-\eta )-p_c^2}>0$, $\frac{\partial \tau ^{\mathrm{RN}}}{\partial e_m}=-\frac{p_c^2}{4 (1-\eta )-p_c^2}<0$,

  $\frac{p_r^{\mathrm{RN}}}{\partial e_m}=\frac{(1-\eta ) p_c}{4 (1-\eta )-p_c^2}>0$, $\frac{\partial \pi _m^{\mathrm{RN}}}{\partial e_m}=-\frac{(1-\eta ) p_c \left( 1- \left( e_r+e_m\right) p_c\right) }{4 (1-\eta )-p_c^2}<0$,

  $\frac{\partial \pi _r^{\mathrm{RN}}}{\partial e_m}=-\frac{2 (1-\eta )^2 p_c \left( 1- \left( e_r+e_m\right) p_c\right) }{\left( 4 (1-\eta )-p_c^2\right) {}^2}<0$.

• (ii) From $\frac{\partial w^{\mathrm{RN}}}{\partial e_r}=-\frac{p_c \left( 2 (1-\eta )-p_c^2\right) }{4 (1-\eta )-p_c^2}$, we get when $\eta <1-\frac{1}{2}p_c^2$, $\frac{\partial w^{\mathrm{RN}}}{\partial e_r}<0$; and otherwise, $\frac{\partial w^{\mathrm{RN}}}{\partial e_r}>0$.

  $\frac{\partial \tau ^{\mathrm{RN}}}{\partial e_r}=-\frac{p_c^2}{4 (1-\eta )-p_c^2}<0$, $\frac{p_r^{\mathrm{RN}}}{\partial e_r}=\frac{(1-\eta ) p_c}{4 (1-\eta )-p_c^2}>0$,

  $\frac{\partial \pi _m^{\mathrm{RN}}}{\partial e_r}=-\frac{(1-\eta ) p_c \left( 1- \left( e_r+e_m\right) p_c\right) }{4 (1-\eta )-p_c^2}<0$,

  $\frac{\partial \pi _r^{\mathrm{RN}}}{\partial e_r}=-\frac{2 (1-\eta )^2 p_c \left( 1- \left( e_r+e_m\right) p_c\right) }{\left( 4 (1-\eta )-p_c^2\right) {}^2}<0$.

• (iii) $\frac{\partial w^{\mathrm{RN}}}{\partial \eta }=-\frac{2 p_c^2 \left( 1- \left( e_r+e_m\right) p_c\right) }{\left( -p_c^2-4 \eta +4\right) {}^2}<0$,

  $\frac{\partial \tau ^{\mathrm{RN}}}{\partial \eta }=\frac{4 p_c \left( 1- \left( e_r+e_m\right) p_c\right) }{\left( 4 (1-\eta )-p_c^2\right) {}^2}>0$,

  $\frac{p_r^{\mathrm{RN}}}{\partial \eta }=-\frac{p_c^2 \left( 1- \left( e_r+e_m\right) p_c\right) }{\left( -p_c^2-4 \eta +4\right) {}^2}<0$,

  $\frac{\partial \pi _m^{\mathrm{RN}}}{\partial \eta }=\frac{p_c^2 \left( 1- \left( e_r+e_m\right) p_c\right) {}^2}{2 \left( -p_c^2-4 \eta +4\right) {}^2}>0$,

  $\frac{\partial \pi _r^{\mathrm{RN}}}{\partial \eta }=\frac{2 (1-\eta ) p_c^2 \left( 1- \left( e_r+e_m\right) p_c\right) {}^2}{\left( -p_c^2-4 \eta +4\right) {}^3}>0$.

1.3 A.3 Proof of Theorem 2

In marketplace format, we only need to solve the manufacturer’s decisions. The manufacturer’s profit function is $\pi _m= (1-\phi ) p_r D- \left( \left( e_m+e_r-\tau \right) D-F_m\right) p_c-\frac{(1-\eta ) \tau ^2}{2}$. Here the manufacturer decides the retail price and emission reduction amount simultaneously.

Taking the first partial derivatives of $\pi _m$ with respect to $p_r$ and $\tau$ yields $\frac{\partial \pi _m}{\partial p_r}=p_c \left( e_r+e_m-\tau \right) +(1-\phi ) \left( 1-2 p_r\right)$, $\frac{\partial \pi _m}{\partial \tau }=p_c \left( 1-p_r\right) -(1-\eta ) \tau$. Then, the second partial derivatives can be obtained: $\frac{\partial \pi _m^2}{\partial p_r^2}=2 \phi -2$, $\frac{\partial \pi _m^2}{\partial \tau ^2}=-(1-\eta )$, $\frac{\partial \pi _m^2}{\partial \tau \partial p_r}=-p_c$. Based on the assumption that $2 (1-\eta ) (1-\phi )-p_c^2>0$, we get the Hessian matrix $H_2=\left( \begin{array}{cc} 2 \phi -2 &{} -p_c \\ -p_c &{} -(1-\eta ) \\ \end{array} \right)$ is negative. Therefore, $\pi _m$ is jointly concave in w and $\tau$. Solving $\frac{\partial \pi _m}{\partial p_r}=0$ and $\frac{\partial \pi _m}{\partial \tau }=0$, we obtain $p_r^{\mathrm{MN}}=\frac{(1-\eta ) \left( 1-\phi + \left( e_r+e_m\right) p_c\right) -p_c^2}{2 (1-\eta ) (1-\phi )-p_c^2}$, $\tau ^{\mathrm{MN}}=\frac{p_c \left( 1-\phi - \left( e_r+e_m\right) p_c\right) }{2 (1-\eta ) (1-\phi )-p_c^2}$.

1.4 A.4 Proof of Corollary 2

Similar to Corollary 1, Corollary 2 can be obtained easily by taking the first partial derivatives of the equilibrium solutions and profits with the parameters $e_m$, $e_r$, and $\eta$, and thus, we here omit the detailed solving process.

1.5 A.6 Proof of Proposition 1

According to equilibrium solutions obtained in Theorems 1 and 2, we can derive the platform’s profits in different retailing formats. Then, we get two thresholds $E_2$ and $E_3$ by solving $\pi ^{\mathrm{MN}}_r-\pi ^{\mathrm{RN}}_r=0$.

First, $\pi ^{\mathrm{MN}}_r-\pi ^{\mathrm{RN}}_r$

$$\begin{aligned} \begin{aligned}&=\frac{(1-\eta )\phi \left( 1-\phi -p_c \left( e_r+e_m\right) \right) \left( (1-\eta ) \left( 1-\phi +p_c \left( e_r+e_m\right) \right) -p_c^2\right) }{\left( 2 (1-\eta ) (1-\phi )-p_c^2\right) {}^2}\\&-\frac{(1-\eta ) ^2 \left( 1-p_c \left( e_r+e_m\right) \right) {}^2}{\left( 4 (1-\eta ) -p_c^2\right) {}^2}. \end{aligned} \end{aligned}$$

Then, we use $E=e_m+e_r$ and $A=\frac{\left( 2 (1-\eta ) (1-\phi )-p_c^2\right) {}^2}{\left( 4 (1-\eta ) -p_c^2\right) {}^2}$ to simplify the equation and obtain $(1-\eta )\phi \left( 1-\phi -p_c E\right) \left( (1-\eta ) \left( 1-\phi +p_c E\right) -p_c^2\right) -A (1-\eta ) ^2 \left( 1-p_c E\right) {}^2=0.$

Solving the equation, we can get two solutions $E^*=\frac{2 A (1-\eta ) +\phi p_c^2 \mp \phi \sqrt{4 (1-\eta ) ^2 \left( (1-\phi )^2-A (2-\phi )\right) -4 (1-\eta ) (1-A-\phi ) p_c^2+p_c^4}}{2 (1-\eta ) (A+\phi ) p_c}$.

1.6 A.7 Proof of Proposition 2

Similar to 1, the threshold $E_4$ can be obtained by solving $\pi _m^M-\pi _m^R=0$.

1.7 B.1 Proof of Theorem 3

The proof of this theorem is similar to that of Theorem 1; thus, we omit it.

1.8 B.2 Proof of Corollary 3

• (i) According to Theorem 3, we have

  $\frac{\partial w^{\mathrm{RP}}}{\partial e_m}=\frac{p_c \left( 2 (1-\eta )-\beta (\beta +p_c) \right) }{4 (1-\eta ) -\left( \beta +p_c\right) {}^2}$. In addition, based on the assumption $4 (1-\eta ) -\left( \beta +p_c\right) {}^2>0$, we can derive $2(1-\eta ) > \frac{1}{2}\left( \beta +p_c\right) {}^2$. If $\frac{1}{2}\left( \beta +p_c\right) {}^2>\beta (\beta +p_c)$ is satisfied, $2 (1-\eta )-\beta (\beta +p_c)>0$ always holds. From $\frac{1}{2}\left( \beta +p_c\right) {}^2>\beta (\beta +p_c)$, we can get $p_c>\beta$. Hence, when $p_c>\beta$, $\frac{\partial w^{\mathrm{RP}}}{\partial e_m}>0$. However, for the situation that $p_c\le \beta$, the $e_m$ has a positive or negative effect is related to the value of $2 (1-\eta )-\beta (\beta +p_c)$. When $2 (1-\eta )-\beta (\beta +p_c)>0$, namely, $\eta <1-\frac{1}{2}\beta (\beta +p_c)$, $\frac{\partial w^{\mathrm{RP}}}{\partial e_m}>0$; otherwise, $\frac{\partial w^{\mathrm{RP}}}{\partial e_m}<0$. Similarly, we get $\frac{\partial w^{\mathrm{RP}}}{\partial e_r}=-\frac{p_c \left( 2 (1-\eta ) -p_c \left( \beta +p_c\right) \right) }{4 (1-\eta ) -\left( \beta +p_c\right) {}^2}$. When $p_c<\beta$, $\frac{\partial w^{\mathrm{RP}}}{\partial e_r}>0$ always holds. For the condition that $p_c\ge \beta$, when $\eta <1-\frac{1}{2}p_c(\beta +p_c)$, $\frac{\partial w^{\mathrm{RP}}}{\partial e_r}>0$; otherwise, $\frac{\partial w^{\mathrm{RP}}}{\partial e_r}<0$. Next, we have $\frac{\partial w^{{\mathrm{RP}}}}{\partial \eta }=\frac{2 \left( \beta ^2-p_c^2\right) \left( 1-p_c \left( e_r+e_m\right) \right) }{\left( 4 (1-\eta )-\left( \beta +p_c\right) {}^2\right) {}^2}$, and thus, only when $\beta >p_c$, $\frac{\partial w^{{\mathrm{RP}}}}{\partial \eta }>0$ holds; otherwise, $\frac{\partial w^{{\mathrm{RP}}}}{\partial \eta }<0$. Then, $\frac{\partial w^{\mathrm{RP}}}{\partial \beta }=\frac{\left( 1-p_c \left( e_r+e_m\right) \right) \left( 4 \beta (1-\eta ) -p_c \left( \beta +p_c\right) {}^2\right) }{\left( 4 (1-\eta )-\left( \beta +p_c\right) {}^2\right) {}^2}$. According to the assumption $4(1-\eta )>(\beta +p_c)^2$, we have $4\beta (1-\eta )>\beta (\beta +p_c)^2$. Hence, we can derive that $4 \beta (1-\eta ) -p_c \left( \beta +p_c\right) {}^2>0$ if $p_c<\beta$. For the situation $p_c\ge \beta$, when $\eta < 1-\frac{p_c \left( \beta +p_c\right) {}^2}{4 \beta }$, $\frac{\partial w^{\mathrm{RP}}}{\partial \beta }>0$; otherwise, $\frac{\partial w^{\mathrm{RP}}}{\partial \beta }<0$.

• (ii) Taking the first partial derivatives of $\tau ^{\mathrm{RP}}$, we get

  $\frac{\partial \tau ^{\mathrm{RP}} }{\partial e_m}=-\frac{p_c \left( \beta +p_c\right) }{4 (1-\eta ) -\left( \beta +p_c\right) {}^2}<0$,

  $\frac{\partial \tau ^{\mathrm{RP}} }{\partial e_r}=-\frac{p_c \left( \beta +p_c\right) }{4 (1-\eta ) -\left( \beta +p_c\right) {}^2}<0$,

  $\frac{\partial \tau ^{\mathrm{RP}}}{\partial \eta }=\frac{4 \left( \beta +p_c\right) \left( 1-p_c \left( e_r+e_m\right) \right) }{\left( 4 (1-\eta )-\left( \beta +p_c\right) {}^2\right) {}^2}>0$,

  $\frac{\partial \tau ^{\mathrm{RP}} }{\partial \beta }=\frac{\left( 1-p_c \left( e_r+e_m\right) \right) \left( \left( \beta +p_c\right) {}^2+4 (1-\eta ) \right) }{\left( 4 (1-\eta ) -\left( \beta +p_c\right) {}^2\right) {}^2}>0$.

• (iii) Taking the first partial derivatives of $p^{\mathrm{RP}}_r$, we obtain $\frac{\partial p^{\mathrm{RP}}_r}{\partial e_m}=\frac{p_c \left( (1-\eta )-\beta (\beta +p_c)\right) }{4 (1-\eta ) -\left( \beta +p_c\right) {}^2}>0$. From the assumption $4 (1-\eta ) -\left( \beta +p_c\right) {}^2>0$, we have $(1-\eta )>\frac{1}{4}\left( \beta +p_c\right) {}^2$. Thus, when $\frac{1}{4}\left( \beta +p_c\right) {}^2>\beta (\beta +p_c)$, namely, $p_c>3\beta$, $(1-\eta )-\beta (\beta +p_c)>0$, and thus, $\frac{\partial p^{\mathrm{RP}}_r}{\partial e_m}>0$. For the situation $p_c\le 3\beta$, letting $(1-\eta )-\beta (\beta +p_c)>0$, we get $\eta <1-\beta (\beta +p_c)$. Hence, when $p_c\le 3\beta$, if $\eta <1-\beta (\beta +p_c)$, $\frac{\partial p^{\mathrm{RP}}_r}{\partial e_m}>0$; otherwise, $\frac{\partial p^{\mathrm{RP}}_r}{\partial e_m}<0$. Similarly, we have $\frac{\partial p^{\mathrm{RP}}_r}{\partial e_r}=\frac{p_c \left( (1-\eta )-\beta (\beta +p_c)\right) }{4 (1-\eta ) -\left( \beta +p_c\right) {}^2}=\frac{\partial p^{\mathrm{RP}}_r}{\partial e_m}$. Hence, the impact of $e_r$ on the retail price is the same with that of $e_m$ on it. Here, we omit the detailed process. Then, from $\frac{\partial p_r^{\mathrm{RP}}}{\partial \eta }=\frac{\left( 3 \beta -p_c\right) \left( \beta +p_c\right) \left( 1-p_c \left( e_r+e_m\right) \right) }{\left( 4 (1-\eta )-\left( \beta +p_c\right) {}^2\right) {}^2}$, we can get $\frac{\partial p_r^{\mathrm{RP}}}{\partial \eta }>0$ only when $p_c<3\beta$; otherwise, $\frac{\partial p_r^{\mathrm{RP}}}{\partial \eta }<0$.

1.9 B.3 Proof of Theorem 4

The proof is omitted because it is similar to that of Theorem 2.

1.10 B.4 Proof of Corollary 4

The proof is similar to that of Corollary 3; thus, we will not repeat the solving process.

1.11 B.5 Proofs of Propositions 3 and 5

The proofs of Propositions 3 and 5 are similar to those of Propositions 1, and 2, respectively. Here, we omit them.

1.12 C.1 Proof of Proposition 6

Based on equilibrium solutions obtained in Theorems 1 and 3, we can get $\tau ^{\text {RP}}-\tau ^{\mathrm{RN}}=\frac{\left( \beta +p_c\right) \left( 1-p_c \left( e_r+e_m\right) \right) }{4 (1-\eta ) -\left( \beta +p_c\right) {}^2}-\frac{p_c \left( 1-p_c \left( e_r+e_m\right) \right) }{4 (1-\eta ) -p_c^2}$. We can easily get

$\left( \beta +p_c\right) \left( 1-p_c \left( e_r+e_m\right) \right) >p_c \left( 1-p_c \left( e_r+e_m\right) \right)$ and $4 (1-\eta ) -\left( \beta +p_c\right) {}^2<4 (1-\eta ) -p_c^2$; thus, $\tau ^{\text {RP}}-\tau ^{\mathrm{RN}}>0$, namely, $\tau ^{\text {RP}}>\tau ^{\mathrm{RN}}$. Similarly, we can obtain

$\pi _r^{\text {RP}}-\pi _r^{\mathrm{RN}}=\frac{(1-\eta ) ^2 \left( 1-p_c \left( e_r+e_m\right) \right) {}^2}{\left( 4 (1-\eta ) -\left( \beta +p_c\right) {}^2\right) {}^2}-\frac{(1-\eta ) ^2 \left( 1-p_c \left( e_r+e_m\right) \right) {}^2}{\left( 4 (1-\eta ) -p_c^2\right) {}^2}>0$,

$\pi _m^{\text {RP}}-\pi _m^{\mathrm{RN}}=\frac{(1-\eta ) \left( 1-p_c \left( e_r+e_m\right) \right) {}^2}{2 \left( 4 (1-\eta ) -\left( \beta +p_c\right) {}^2\right) }-\frac{(1-\eta ) \left( 1-p_c \left( e_r+e_m\right) \right) {}^2}{2 \left( 4 (1-\eta ) -p_c^2\right) }>0$.

Proposition 6 is proved.

1.13 C.2 Threshold values

See Table 4

Table 4 Threshold values