Optimal pricing and green decisions in a dual-channel supply chain with cap-and-trade regulation

Abstract

The issue of low carbon emission reduction is getting more and more attention. This paper focuses on analyzing the equilibriums and digging deep into the impacts of cap-and-trade regulation, joint emission abatement scheme, and online direct channel on the performances of supply chain. By constructing two decentralized models in the single/joint emission abatement schemes under cap-and-trade regulation in a dual-channel supply chain, we find that consumer’s environmental preference effectually motivates both the manufacturer and retailer to reduce emissions with joint emission abatement scheme. The analysis results show that introducing an online channel is always good for improving enterprises’ profits, as well as protecting the environment in the joint emission abatement. Besides, the retailer should provide consumers with better service in the retail channel, which can further promote purchase power and drive the sustainable development of the supply chain. The manufacturer, when consumers have strong environmental preferences, should actively participate in cap-and-trade mechanism no matter single or joint emission abatement scheme is enforced.

Appendices

Appendix 1

Firstly, we derive the best response retail price for the retailer in Step-2. Then, the manufacturer makes the optimal decisions based on this retail price. With Eq. (4), ${\Pi }_{R}^{COD}$ is concave in ${p}_{r}^{COD}$ given that $\frac{{\partial }^{2}{\Pi }_{R}^{COD}}{{\partial {p}_{r}^{COD}}^{2}}=-2<0$. Let $\frac{\partial {\Pi }_{R}^{COD}}{\partial {p}_{r}^{COD}}=0$, we can obtain the best response retail price:

$${p}_{r}^{COD}=(a+{ w}^{COD}+e {\theta }^{COD}+\gamma {p}_{d}^{COD})/2$$

(A-1)

Substituting Eq. (A-1) to Eq. (3) and express manufacturer’s profit by ${p}_{d}^{COD},{ w}^{COD}$, and ${\theta }^{COD}$. Assuming $e<\frac{2\sqrt{k(3-2\gamma -{\gamma }^{2})}}{3+\gamma }+(\gamma -1){p}_{e}$ to ensure the equilibrium solutions in single emission abatement strategy are positive, we can derive the Hessian matrix of ${\Pi }_{M}^{COD}$ with respect ${p}_{d}^{COD},{ w}^{COD},$ and ${\theta }^{COD}$ is negative definite. Thus, a unique optimal solution for Eq. (3) exists. Then the optimal solution of the manufacturer can be derived by the Karush–Kuhn–Tucker (KKT) conditions. The Lagrangian function is:

$$L={\Pi }_{M}^{COD}+{\omega }_{1}\left(E-\left(\tau -{\theta }^{COD}\right)\left({D}_{r}^{COD}+{D}_{d}^{COD}\right)-X\right)+{\omega }_{2}{ p}_{d}^{COD}+{\omega }_{3}{ w}^{COD}+{\omega }_{4} {\theta }^{COD},$$

where ${\omega }_{1}$, ${\omega }_{2}$,${\omega }_{3}$, and ${\omega }_{4}$ are Lagrangian multipliers. The KKT conditions are:

$$\left\{\begin{array}{c}\partial L/\partial {p}_{d}^{COD}=0,\\ \genfrac{}{}{0pt}{}{\partial L/\partial { w}^{COD}=0,}{\partial L/\partial {\theta }^{COD}=0,}\\ \begin{array}{c}{\omega }_{1}\left(E-\left(\tau -{\theta }^{COD}\right)\left({D}_{r}^{COD}+{D}_{d}^{COD}\right)-X\right)=0,\\ {\omega }_{2}{ p}_{d}^{COD}=0,\\ \begin{array}{c}{\omega }_{3}{ w}^{COD}=0,\\ \begin{array}{c}{\omega }_{4} {\theta }^{COD}=0,\\ {\omega }_{1}, {\omega }_{2}, {\omega }_{3} , {\omega }_{4 }\ge 0.\end{array}\end{array}\end{array}\end{array}\right.$$

We derive that when ${\omega }_{1}={\omega }_{2}={\omega }_{3}= {\omega }_{4 }=0$, the optimal solution exists in case A. The optimal solution is illustrated in Theorem 1.

Appendix 2

Proof of Proposition 1

According to Theorem 1, we can derive the following first-order conditions:

$$\frac{\partial {\theta }^{{COD}^{*}}}{\partial e}=\frac{(3+\gamma )(a-(1-\gamma )\tau {p}_{e})(4k(1-\gamma )+{e}^{2}(3+\gamma )+2e(1-\gamma )(3+\gamma ){p}_{e}+{(\gamma -1)}^{2}(3+\gamma ){p}_{e}^{2})}{{(4k(\gamma -1)+{e}^{2}(3+\gamma )-2x(2\gamma +{\gamma }^{2}-3){p}_{e}+{(\gamma -1)}^{2}(3+\gamma ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {D}_{r}^{{COD}^{*}}}{\partial e}=\frac{2k(1-\gamma )(3+\gamma )(e+(1-\gamma ){p}_{e})(a-(1-\gamma )\tau {p}_{e})}{{(4k(\gamma -1)+{e}^{2}(3+\gamma )-2x(2\gamma +{\gamma }^{2}-3){p}_{e}+{(\gamma -1)}^{2}(3+\gamma ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {D}_{d}^{{COD}^{*}}}{\partial e}=\frac{2k(3+\gamma )(1-\gamma )(2+\gamma )(e+(1-\gamma ){p}_{e})(a-(1-\gamma )\tau {p}_{e})}{{(4k(\gamma -1)+{e}^{2}(3+\gamma )-2x(2\gamma +{\gamma }^{2}-3){p}_{e}+{(\gamma -1)}^{2}(3+\gamma ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {\Pi }_{M}^{{COD}^{*}}}{de}=\frac{k{(3+\gamma )}^{2}(e+(1-\gamma ){p}_{e}){(a-(1-\gamma )\tau {p}_{e})}^{2}}{{(4k(\gamma -1)+{e}^{2}(3+\gamma )-2x(2\gamma +{\gamma }^{2}-3){p}_{e}+{(\gamma -1)}^{2}(3+\gamma ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {\Pi }_{R}^{{COD}^{*}}}{\partial e}=\frac{-(4{k}^{2}{(\gamma -1)}^{2}(3+\gamma )(e+(1-\gamma ){p}_{e}){(a-(1-\gamma )\tau {p}_{e})}^{2})}{{(4k(\gamma -1)+{e}^{2}(3+\gamma )-2e(2\gamma +{\gamma }^{2}-3){p}_{e}+{(\gamma -1)}^{2}(3+\gamma ){p}_{e}^{2})}^{3}}$$

Based on the above assumption $e<\frac{2\sqrt{k(3-2\gamma -{\gamma }^{2})}}{3+\gamma }+(\gamma -1){p}_{e}$, we can derive $\partial {\theta }^{{COD}^{*}}/\partial e>0$, $\partial {D}_{r}^{{COD}^{*}}/\partial e>0$, $\partial {D}_{d}^{{COD}^{*}}/\partial e>0$, $\partial {\Pi }_{M}^{{COD}^{*}}/\partial e>0$,$\partial {\Pi }_{R}^{{COD}^{*}}/\partial e>0$.

Proof of Proposition 2

The proof process is similar to that of Proposition 1, thus substituting ${p}_{e}=0$ into the relevant first-order conditional expressions of Proposition 1.

Appendix 3

On the one hand, we have concluded that the wholesale price equals to direct price in the “Case A: single emission abatement strategy in the green dual-channel supply chain” section. To be consistent with previous conclusions, we are going to solve the model in case B based on ${p}_{d}^{CTD}={w}^{CTD}$. One the other hand, a rational retailer will compare wholesale price and direct price, and then, choose the lower one to wholesale the products. Thus, prices in a dual-channel supply chain must meet the conditions ${p}_{d}^{CTD}\ge {w}^{CTD}$. Therefore, the equality ${p}_{d}^{CTD}={w}^{CTD}$ is realistic. Firstly, we derive best response retail price and service level. With Eq. (8), given that ${\partial }^{2}{\Pi }_{R}^{CTD}/{\partial {p}_{r}^{CTD}}^{2}=-2<0$ and $Det\left(\left\{\frac{{\partial }^{2}{\Pi }_{R}^{CTD}}{{\partial {p}_{r}^{CTD}}^{2}},\frac{{\partial }^{2}{\Pi }_{R}^{CTD}}{\partial {p}_{r}^{CTD}\partial {s}^{CTD}}\right\},\left\{\frac{{\partial }^{2}{\Pi }_{R}^{CTD}}{\partial {p}_{r}^{CTD}\partial {s}^{CTD}},\frac{{\partial }^{2}{\Pi }_{R}^{CTD}}{{\partial {s}^{CTD}}^{2}}\right\}\right)= Det(\left\{-2,\beta \right\},\left\{\beta ,-\eta \right\})=2\eta -{\beta }^{2}>0$, we can derive that ${\Pi }_{R}^{CTD}$ is jointly concave in ${p}_{r}^{CTD}$ and ${s}^{CTD}$. Therefore, let $\frac{\partial {\Pi }_{R}^{CTD}}{\partial {p}_{r}^{CTD}}= \frac{\partial {\Pi }_{R}^{CTD}}{\partial {s}^{CTD}}=0$, we obtain the retailer’s best response function:

$${p}_{r}^{CTD}=(-{ w}^{CTD}{\beta }^{2}+a\eta +{ w}^{CTD}\eta +{ w}^{CTD}\gamma \eta +e\eta {\theta }^{CTD})/(2\eta -{\beta }^{2})$$

(C-1)

$${s}^{CTD}=\beta (a-{ w}^{CTD}+{ w}^{CTD}\gamma +e{\theta }^{CTD})/(2\eta -{\beta }^{2})$$

(C-2)

We substitute Eq. (C-1) and Eq. (C-2) to Eq. (7) and get a new simplified manufacturer’s profit expressed by ${p}_{d}^{CTD},{ w}^{CTD}$, and ${\theta }^{CTD}$. Assuming ${\beta }^{2}<2\eta$ and $({\beta }^{2}-(3+\gamma )\eta ){\left(e+\left(\gamma -1\right){p}_{e}\right)}^{2}+2(\gamma -1)(k({\beta }^{2}-2\eta )+2e(-{\beta }^{2}+(3+\gamma )\eta ){p}_{e})>0$ to ensure the equilibrium solutions in joint emission abatement strategy are positive, we can derive the Hessian matrix of ${\Pi }_{M}^{CTD}$ with respect ${w}^{CTD}$ and ${\theta }^{CTD}$ is negative definite. Thus, the unique optimal solutions for Eq. (7) exist. Then, the optimal solutions of the manufacturer can be derived by KKT conditions. We show the equilibrium solutions of case B in Theorem 2.

Appendix 4

Proof of Proposition 3

According to Theorem 2, we have the following first-order conditions:

$$\frac{\partial {\theta }^{{CTD}^{*}}}{\partial e}=\frac{({\beta }^{2}-(3+\gamma )\eta )(a-(1-\gamma )\tau {p}_{e})(2k(1-\gamma )({\beta }^{2}-2\eta )+{e}^{2}({\beta }^{2}-(3+\gamma )\eta )+2e(1-\gamma )({\beta }^{2}-(3+\gamma )\eta ){p}_{e}+{(\gamma -1)}^{2}({\beta }^{2}-(3+\gamma )\eta ){p}_{e}^{2})}{{(2k(\gamma -1)({\beta }^{2}-2\eta )+{e}^{2}({\beta }^{2}-(3+\gamma )\eta )+2e(\gamma -1)(-{\beta }^{2}+(3+\gamma )\eta ){p}_{e}+{(\gamma -1)}^{2}({\beta }^{2}-(3+\gamma )\eta ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {s}^{{CTD}^{*}}}{\partial e}=\frac{2k\beta (\gamma -1)({\beta }^{2}-(3+\gamma )\eta )(e-(\gamma -1){p}_{e})(a+(\gamma -1)\tau {p}_{e})}{{(2k(\gamma -1)({\beta }^{2}-2\eta )+{e}^{2}({\beta }^{2}-(3+\gamma )\eta )+2e(1-\gamma )({\beta }^{2}-(3+\gamma )\eta ){p}_{e}+{(\gamma -1)}^{2}({\beta }^{2}-(3+\gamma )\eta ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {D}_{r}^{{CTD}^{*}}}{\partial e}=\frac{2k(\gamma -1)\eta ({\beta }^{2}-(3+\gamma )\eta )(e-(\gamma -1){p}_{e})(a+(\gamma -1)\tau {p}_{e})}{{(2k(\gamma -1)({\beta }^{2}-2\eta )+{e}^{2}({\beta }^{2}-(3+\gamma )\eta )+2e(\gamma -1)(-{\beta }^{2}+(3+\gamma )\eta ){p}_{e}+{(\gamma -1)}^{2}({\beta }^{2}-(3+\gamma )\eta ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {D}_{d}^{{CTD}^{*}}}{\partial e}=\frac{2k(\gamma -1)(-{\beta }^{2}+(2+\gamma )\eta )({\beta }^{2}-(3+\gamma )\eta )(e-(\gamma -1){p}_{e})(a+(\gamma -1)\tau {p}_{e})}{{(2k(\gamma -1)({\beta }^{2}-2\eta )+{e}^{2}({\beta }^{2}-(3+\gamma )\eta )+2e(\gamma -1)(-{\beta }^{2}+(3+\gamma )\eta ){p}_{e}+{(\gamma -1)}^{2}({\beta }^{2}-(3+\gamma )\eta ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {\Pi }_{M}^{{CTD}^{*}}}{de}=\frac{-(k{({\beta }^{2}-(3+\gamma )\eta )}^{2}(-e+(\gamma -1){p}_{e}){(a+(\gamma -1)\tau {p}_{e})}^{2})}{{(2k(\gamma -1)({\beta }^{2}-2\eta )+{e}^{2}({\beta }^{2}-(3+\gamma )\eta )+2e(\gamma -1)(-{\beta }^{2}+(3+\gamma )\eta ){p}_{e}+{(\gamma -1)}^{2}({\beta }^{2}-(3+\gamma )\eta ){p}_{e}^{2})}^{2}},$$

$$\frac{\partial {\Pi }_{R}^{{CTD}^{*}}}{\partial e}=\frac{2{k}^{2}{(\gamma -1)}^{2}({\beta }^{2}-2\eta )\eta ({\beta }^{2}-(3+\gamma )\eta )(e-(\gamma -1){p}_{e}){(a+(\gamma -1)\tau {p}_{e})}^{2}}{{(2k(\gamma -1)({\beta }^{2}-2\eta )+{e}^{2}({\beta }^{2}-(3+\gamma )\eta )+2e(\gamma -1)(-{\beta }^{2}+(3+\gamma )\eta ){p}_{e}+{(\gamma -1)}^{2}({\beta }^{2}-(3+\gamma )\eta ){p}_{e}^{2})}^{3}}$$

According to ${\beta }^{2}<2\eta$ and $({\beta }^{2}-(3+\gamma )\eta ){\left(e+\left(\gamma -1\right){p}_{e}\right)}^{2}+2(\gamma -1)(k({\beta }^{2}-2\eta )+2e(-{\beta }^{2}+(3+\gamma )\eta ){p}_{e})>0$, we can derive $\partial {\theta }^{{CTD}^{*}}/\partial e>0$, $\partial {s}^{{CTD}^{*}}/\partial e>0$, $\partial {D}_{r}^{{CTD}^{*}}/\partial e>0$, $\partial {D}_{d}^{{CTD}^{*}}/\partial e>0$, $\partial {\Pi }_{M}^{{CTD}^{*}}/\partial e>0$,$\partial {\Pi }_{R}^{{CTD}^{*}}/\partial e>0$.

Proof of Proposition 4

The proof process is similar to that of Proposition 3, thus substituting ${p}_{e}=0$ into the relevant first-order conditional expressions of Proposition 3.