Pricing and coordination strategies of dual-channel green supply chain considering products green degree and channel environment sustainability

ABSTRACT

To study the pricing and coordination problem of dual-channel green supply chain (DGSC) considering products green degree and the distinctive of channel environment sustainability, a DGSC with one manufacturer and one retailer is chosen as the study object, here, the manufacturer sells products through the direct channel and the retailer sells products by the traditional channel. Then, the market demand is revised considering products green degree and the channel environment sustainability. Based on these, effects of products green degree and the level of channel environment sustainability on supply chain members’ pricing strategies are analysed in both decentralised and centralised models, meanwhile, the effects of the cross-environmental-sustainability sensitivity coefficient on pricing strategies also are analysed. A new two-part compensation contract is proposed to coordinate the DGSC. Results shows that effects of the products green degree on pricing strategies in both decentralised and centralised models are same; however, its effects on supply members’ benefits are different. Influences of the level of channel environment sustainability on supply chain members’ pricing strategies and benefits in both decentralised and centralised models are different. Meanwhile, when the compensation fee meet a certain range, the two-part compensation contract can coordinate this supply chain.

KEYWORDS:

• Pricing strategies
• dual-channel supply chain
• environment sustainability
• product green degree
• coordination

1. Introduction

Green supply chain is thought to be future direction. Thus, establishing and implementing green supply chain has become a strategic task for every country industrial development (O’Rourke 2014). Facing government regulations and international green barriers, if enterprises want to win a sustainable competitive advantage, a long-term choice for enterprises is to take social and environmental responsibility and to provide environment-friendly green products (Lash and Wellington 2007; Golden, Subramanian, and Zimmerman 2011; Pan, Xi, and Wang 2015). Therefore, a growing number of manufacturers begin to implement their green product plans (Dangelico and Pujari 2010), such as, a clothing manufacturer called Patagonia (Ghosh and Shah 2012), Adidas, Haier, Appliance World Expo (Li et al. 2016) etc. In reality, many manufacturers have gain benefits from green actions (Rao and Holt 2005; Jayaram and Avittathur 2015).

According to the above descriptions, many retailers and manufacturers are selling or outputting green products to protect the modern environment. In this study, pricing strategies of the green products will be discussed in a supply chain. In reality, following the rapid development and arrival of e-commerce and growing consumer recognition on green products, selling green products online are being adopted by manufacturers. This model is called a dual-channel green supply chain (DGSC). From the views of improving enterprise market competitiveness and customer demand, implementing dual-channel strategy will be a good choice for manufacturers. A DGSC contains a direct sale channel of the manufacturer and a traditional retail channel of the retailer. For consumers, the DGSC offers a replaceable and convenient way to get green products and assists the manufacturer draw more consumers. However, the direct channel’s implementation also exacerbates completion between the traditional retailer and the manufacturer. Meanwhile, consumers’ choice buying green products from direct channel or traditional channel is influenced by the environmental protection awareness of consumers (Chen et al. 2017). Thus, considering products’ green natures, the environmental sustainability of channels and the channel conflicts, pricing and coordination strategies will be studied in a DGSC.

This study takes into account the environmental sustainability of channel in analysing pricing and coordination strategies of DGSC. According to analysis, it attempt to reply the following questions: How do the green degree of products and the level of environmental sustainability of channel influence the pricing strategies and revenues of members in the DGSC? How do to achieve the DGSC coordination? To solve these problems, Stackelberg game model is adopted to explore the DGSC. Here, assume that the manufacturer outputs one kind of green product and can determine the green degree of the products. In addition, to improve the green degree of the products, the manufacturer should invest in extra money for green technology. The hypothesis is similar to that in document (Li et al. 2016). The differences of our jobs are that the environmental sustainability of channel is considered in our study. Meanwhile, assume that supply chain members need extra money to improve the environmental sustainability of channels. The hypothesis is similar to that in document (Chen et al. 2017). The differences of our jobs are that a green supply chain in the situation of dual channels is investigated. Firstly, we add the environmental sustainability into the pricing question of a DGSC using Stackelberg game model in both decentralised and centralised models. Furthermore, the effects of the channel environment sustainability and the green degree of products on pricing strategies of a DGSC are analysed. Then, a two-part compensation contract is designed to achieve the DGSC coordination.

This article is organised as follows: Chapter 1 presents the introduction; Chapter 2 presents the literature review; Chapter 3 presents the model establishment and hypothesis; Chapter 4 presents pricing strategy of DGSC in the decentralised and centralised models; Chapter 5 presents the numerical simulation; and Chapter 6 presents the conclusions

2. Literature review

2.1. Environment benefits of channel

Many efforts had been done to explore the effects of online and offline retailing (Gould and Golob 1997; Cairns 2004). In the earlier study, Hanne et al. (2002) had discussed that implementing e-commerce in the food consumption and production system could help companies decrease greenhouse gas emissions directly or indirectly. Lean et al. thought that in particular circumstances, online channel could reduce the environmental influence of shopping (Loon et al. 2014). Meanwhile, decreasing accessional shopping routes and maximising the number of products per transaction could make the online channel more environmental sustainable. Hesse (2002), Matthews et al. (2002), and Rizet et al. (2010) thought that ‘if the environmental benefits of e-commerce could be achieved’, consumer trip by car should be reduced. In the conventional shopping situation, Brown and Guiffrida (2014) compared the carbon emissions of two ways in bring items home: one is traditional shopping involving ‘customer pickup with trip chaining’, another is online shopping involving last mile delivery. Results indicated that last mile delivery using minimum trucks would reduce more carbon emissions in specific conditions. Ji, Zhang, and Yang (2017) thought that implementing online shopping channel was profitable for manufacturers. However. The low-carbon sensitivity degree of consumers should meet certain conditions. Similarly Nabot (2016) discussed the influences of online and conventional retailing on the environment from the ‘last mile’ view. Results showed that online retailing had an important effect in reducing carbon emissions.

Moreover, some studies had confirmed that implementing environmental sustainability actions would bring benefits for online business. For the online retailing, if consumers’ low-carbon preference was high, the manufacturers would implement low-carbon sell, which could help them gain more social and economic benefits (Ji, Zhang, and Yang 2017). Though stating that shopping by online was well to the environment, the online retailer could gain revenues (Edwards, Mckinnon, and Cullinane 2011). Meanwhile, with the raise of consumer environmental awareness, manufacturers and retailers with better eco-friendly manipulates would profit (Liu, Anderson, and Cruz 2012). If consumers with green and environmental awareness noticed the environment sustainability differences between online and offline channels, these differences would influence their decisions in shopping online or offline.

Therefore, in a dual-channel supply chain, its supply chain members are encouraged to implement green actions, here, optimising the environmental sustainability of channels and improving the green degree of products are included.

2.2. Green products and its pricing problems

The current studies about green products focus on two aspect: one is about products’ recyclability, another is no considering the products’ natures. In the aspect of products recyclability, Chen and Sheu (2009) discussed how to enhance the recyclability of their products. Sheu (2011) considered the environmental benefits brought by recycling and discussed what time the revenues of supply chain members would be maximised. Based on this study, he found that outputting green products could help manufactures gain green benefits in a certain condition (Sheu and Chen 2012).

Many researches like to study the green products questions without considering the products’ natures, for instance recyclability. To obtain the maximised benefits polices of the green innovation would be maximised, Ghosh et al. chose a green supply chain with one manufacture and one retailer (Ghosh and Shah 2012). The green degree of products were decided by the manufacturer and were thought to have positive effects on the market demand and the manufacturer’s benefit. Zhang and Liu (2013) discussed the influences of green products on the market demand based on a three-stage supply chain. In this study, the green degree of products was a assume coefficient. Cao and Zhang (2013) also believed that the green degree of products could add the market demand. Based on a green supply with one supplier and one manufacturer, the pricing strategy problems of the supply chain members were discussed. Zhang, Wang, and Ren (2014) compared the differences of pricing strategy between manufactures outputting green products and non-green products. Li et al. (2016) discussed the pricing strategy of a DGSC with one manufacturer and one retailer using Stackelberg game model. However, in this study, the environmental sustainability of channels were not considered. Based on this study, Chen et al. considered the environmental sustainability of channels and discussed the pricing strategies of a dual-channel supply chain (Chen et al. 2017). However, this study did not consider the green degree of products.

Different from the above researches, this study explored pricing and coordination strategies in a DGSC considering the green degree of products under decentralised and centralised decisions using the Stackelberg game model.

2.3. Coordination strategies of DGSC

Opening the online channel brings some new channel conflicts problems. In the traditional supply chain environment, to mitigate and eliminate ‘double marginalisation’ (Posner 2005) problem of supply chain members, related coordination mechanisms had been proposed. However, in the dual-channel supply chain, the manufacturer was either the retailer’s supplier or the retailer’s competitor, it caused that the ‘double marginalisation’ problem and the channel conflict came to together. The general supply chain contracts could not coordinate the dual-channel supply chain (Boyaci 2005), such as wholesale price contracts, repurchase contracts, income sharing contracts etc.; therefore, many new contracts were proposed to coordinate the dual-channel supply chain. Park and Keh (2003) proposed that in the dual-channel supply chain, manufacturers could use quantitative discounts to encourage retailers to continue to work with them and to achieve the dual-channel supply chain coordination. Chiang (2010) designed sharing the costs from inventory and sharing the benefits from online to coordinate the dual-channel supply. Dan, Guang-Ye, and Zhang (2012) proposed a compensation mechanism to achieve the dual-channel supply chain coordination. In this mechanism, the manufacturer would provide some order forms from the direct channel, then the retailer provided some ‘franchise fee’ to compensate the manufacturer. This compensation mechanism was prove to be effective in achieving the dual-channel supply chain coordination. However, these methods are not easy to implement in the practical application process. Moreover, these contracts were not used to solve the DGSC coordination problem. To make up this gap, a two-part compensation contract was used by Li et al. (2016) and its effectiveness in coordinating the DGSC was proved. In this paper, the new two-part compensation contract was adopted from the coordination perspective of the two channels.

3. Model establishment

In this paper, a DGSC with one manufacturer and one retailer was chosen, here, the manufacturer only produces one type of green product and sells them through the direct channel, and the retailer sells the green products through the traditional channel. The manufacturer and the retailer are risk neutral and completely rational. The green manufacturer has enough production capacity. Meanwhile, the manufacturer is the leader of Stackelberg game and the retailer is the follower. In the traditional channel, the manufacturer provides the green products with a wholesale price $w$ to the retailer, and the production cost is $c$. Meanwhile, to open the direct channel, the manufacturer sells the green products to consumers with price $p_d$. The retailer decides its retail price $p_r$ based on the manufacturer’s decision.

Assume that consumers are price sensitive and environment preference. Therefore, if the products’ price is same, consumers will choose environmentally friendly products. In this paper, the green degree of products is considered and has positive effect on increasing consumer demand. Meanwhile, assume that consumers will turn from the low environment sustainability channel to the high environment sustainability channel. In this study, the linear demand model is chosen. The channel demand has linear relationships with retail price, green degree of products, and the environmental sustainability of channels. These assumption are similar to Ghosh and Shah (2012) and Chen et al. (2017). Therefore, the demand functions in the direct channel and the traditional channel are as follows:

(1) $D_r=\rho a-{\alpha }_r{p}_r+\beta p_d+{\kappa }_{r}g+\mu {\theta }_r-\lambda {\theta }_d$(1)

(2) $D_d=(1-\rho )a-{\alpha }_r{p}_d+\beta p_r+{\kappa }_{d}g+\mu {\theta }_d-\lambda {\theta }_r$(2)

In formulas (1) and (2), $D_r$ and $D_d$ stand for the traditional channel demand and the direct channel demand, respectively. $a$ is the total demand of the market and $\rho$ is the initial ratio of consumers buying products through the traditional channel. ${\alpha }_r$ and ${\alpha }_d$ presents the price sensitivity coefficient in the offline channel and the direct channel, respectively. $\beta$ shows the cross-price sensitivity coefficient, here, $\alpha >\beta >0$. $\mathrm{g}$ is the green degree of products. ${\kappa }_d$ and ${\kappa }_r$ are the expansion effectiveness coefficients of the green degree on the direct channel demand and the traditional channel demand. However, for the same green degree, the influences on consumers in the direct channel are lower than it in the traditional channel because seeing the actual products can help consumes examine the green products. Thus, ${\kappa }_r>{\kappa }_d$. ${\theta }_r$ and ${\theta }_d$ present the environmental sustainability level of the traditional channel and the direct channel, respectively. $\mu$ stands for the channel sensitivity coefficients on environmental sustainability. $\lambda$ shows the cross-environmental-sustainability sensitivity coefficients of channel, here, $\mu >\lambda >0$. To reduce calculation, assume that the retail price and the environmental sustainability in a channel are same Chen et al. (2017). To simplify expressions, let $\mu =1$ and $\beta =0$. Meanwhile, let $1>\lambda >0$. To reduce the channel conflicts, this study uses the same pricing strategy (Fruchter and Gapiero 2011; Webb 2002), there is $p_r=p_d=p$. Thus, the simplified demand functions in direct channel and offline channel are as follows:

(3) $D_r=\rho a-{\alpha }_{r}p+{\kappa }_{r}g+{\theta }_r-\lambda {\theta }_d$(3)

(4) $D_d=(1-\rho )a-{\alpha }_{d}p+{\kappa }_{d}g+{\theta }_d-\lambda {\theta }_r$(4)

To improve and keep the environmental sustainability of channels, the retailer and the manufacturer have to pay for extra money. Assume that the extra money $C_d$ paid by the manufacturer equals to ${\tau }_d{{\theta }_d}^2/2$, here, ${\tau }_d$ stand for the cost coefficient of the environmental sustainability of the direct channel. Similarly, assume that the extra money $C_r$ paid by the retailer equals to ${\tau }_r{{\theta }_r}^2/2$, here, ${\tau }_r$ stand for the cost coefficient of the environmental sustainability of the traditional channel. However, the inconsistency of ${\tau }_r$ and ${\tau }_d$ will harm the enthusiasm of the manufacturer and the retailer. Thus, in this paper, assume that ${\tau }_r$ and ${\tau }_d$ are equal to $\tau$.

In addition, to improve and keep the green degree of the products, the manufacturer has to pay for extra money for green technology, and assume that the extra money $C_g$ equals to $\delta {\mathrm{g}}^2/2$, here, $\delta$ is the cost coefficient of the green technology.

4. Basic decision model of DGSC

4.1. Decentralized model

In the decentralised model, the manufacturer and the retailer make decisions independently. The manufacturer decides the wholesale price, the green degree of the products, and the direct channel’s environment sustainability. The retailer decide the retail price and the traditional channel’s environment sustainability. The benefits of the manufacturer and the retailer are as follows:

(5) ${\pi }_d=(w)D_r+(p_d)D_d-\frac{\delta g^2}{2}-\frac{\tau {{\theta }_d}^2}{2}$(5)

(6) ${\pi }_r=(p-w)D_r-\frac{\tau {{\theta }_r}^2}{2}$(6)

Proposition 1:

(1) When $(1-\beta )\tau >1/2$, ${\pi }_r$ is a jointly concave function to $p$ and ${\theta }_r$.

(2) when $\lambda <0.556$, ${\pi }_d$ is an jointly concave function to $w$ and ${\theta }_d$ and is not an jointly concave function to $w$, $\mathrm{g}$ and ${\theta }_d$ (Appendix 1).

Base on the backward method, proposition 1, and formula (6), the optimal decisions of the retailer are as follows:

(7) $p_r(p_d,w,g,{\theta }_d)=\frac{\tau w-w+a\rho \tau +g{\kappa }_r\tau -\lambda {\theta }_d\tau }{2\tau -1}$(7)

(8) ${\theta }_r(p_d,w,g,{\theta }_d)=\frac{a\rho -w+g{\kappa }_r-\lambda {\theta }_d}{2\tau -1}$(8)

Based on proposition 1, if we want to obtain the optimal decisions of the manufacturer, the two-stage optimisation method should be used. Firstly, formulas (7) and (8) should be substituted formula (5), and $w$ and ${\theta }_d$ with respect to $g$ can be got.

(9) $w(g)=\frac{\tau (V_1+V_2+V_3)}{{\tau }^2(3{\lambda }^2+4\lambda -7){(\beta -1)}^2+2\beta {\lambda }^2\tau (1-\lambda )+{\lambda }^3(2\tau -\lambda )+2{\lambda }^2(1-\tau )+6{\tau }^3(1-{\beta }^3)+4\tau (1-\beta )-1}$(9)

(10) ${\theta }_d(g)=\frac{B_1+B_2+B_3+B_4+B_5}{{\tau }^2(3{\lambda }^2+4\lambda -7){(\beta -1)}^2+2\beta {\lambda }^2\tau (1-\lambda )+{\lambda }^3(2\tau -\lambda )+2{\lambda }^2(1-\tau )+6{\tau }^3(1-{\beta }^3)+4\tau (1-\beta )-1}$(10)

$\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{e},B_1=a(1-\rho )+\mathrm{g}{\kappa }_d-2a\tau ++{\tau }^2(g{\kappa }_d+2a\rho )(1-2\beta )+a{\beta }^2{\tau }^2+a\rho {\lambda }^2(1-\lambda )$;$B_3=6a\beta \lambda {\tau }^2-$

$a\beta {\lambda }^2{\tau }^2+6a\rho \lambda {\tau }^2+a\rho {\lambda }^2\tau +2a{\beta }^2\rho {\tau }^{2}a{\tau }^2-3a\lambda {\tau }^2;$

$B_2=(g{\kappa }_r\lambda -a{\lambda }^2+2a\beta \tau )(1-\tau )-(g{\kappa }_r\tau +2g{\kappa }_d\tau )(1-\beta );$

$B_4=-3a{\beta }^2\lambda {\tau }^2+{\beta }^2\mathrm{g}{\kappa }_d{\tau }^2+g\lambda (\beta \tau -\lambda -\beta \lambda \tau )({\kappa }_r-{\kappa }_d);$

$B_5=g\tau ({\kappa }_d\lambda +3{\beta }^2{\kappa }_r\tau -6\beta {\kappa }_r\tau )(\lambda +1).$

$V_1=a{\lambda }^3+g{\kappa }_r{\lambda }^2+g{\kappa }_d{\lambda }^3+2a{\tau }^2(1+{\beta }^2)(1-\rho );$

$V_2=a\tau (5\rho -3)(1-\beta )+a\lambda \tau (\lambda +1);V_3=g\lambda \tau (1-\beta )({\kappa }_r+{\kappa }_d).$

Based on formulas (9) and (10), $g$ can be got.

(11) $\mathrm{g}=-\frac{a\tau (N_1+N_2+N_3)}{(M_1+M_2+M_3+M_4+M_5)}$(11)

$\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{e},M_1=(6\beta {\kappa }_r{\kappa }_d{\tau }^3+7\delta \beta {\tau }^2-4\delta \beta \lambda {\tau }^2-3\delta \beta {\lambda }^2{\tau }^2)(2-\beta );$$M_2=(2\delta \beta {\lambda }^2\tau -2\beta {\kappa }_r{\kappa }_d{\tau }^2-2{\kappa }_r^2{\tau }^2)(\lambda -1)$;

$M_3=2\beta {\kappa }_d^2{\tau }^2(1-\tau )+4\delta \beta \tau +{\kappa }_r^2{\lambda }^2\tau +{\beta }^2{\kappa }_d^2{\tau }^3-3{\kappa }_r^2{\tau }^3;$

$M_4=2{\kappa }_r{\kappa }_d\lambda \tau (1+\tau )+2{\kappa }_r{\kappa }_d{\tau }^2(3\tau -1);$

$M_5={\kappa }_d^2\tau +$

$\delta {\lambda }^4-3\delta {\lambda }^2{\tau }^2+2\delta {\lambda }^2\tau (1-\lambda );$

$N_3=-\beta {\kappa }_d\rho \tau (1+\lambda )+6{\kappa }_r\rho {\tau }^2(2\beta -1);$

$N_1={\kappa }_d(1-\rho )+({\kappa }_d\lambda \rho +{\kappa }_r\lambda )(1+\tau );$

$N_2=2{\kappa }_d\tau {\lambda }^2(1-\beta )(1-\rho )+3{\beta }^2{\kappa }_r{\tau }^2(1-2\rho ).$

$w$ and ${\theta }_d$ can be got through putting $g$ into formulas (9) and (10). In addition, putting $g$, $w$ and ${\theta }_d$ into formulas (7) and (8), $p$ and ${\theta }_r$ can be got. Then, ${\pi }_r$ and ${\pi }_d$ can be got by putting $p$, ${\theta }_r$, $w$,$g$ and ${\theta }_d$ into formulas (5) and (6).

Proposition 2:

(1) $\frac{\mathrm{\partial }p}{\mathrm{\partial }{\theta }_r}>0$;

(2) $\frac{\mathrm{\partial }w}{\mathrm{\partial }{\theta }_d}>0$; $\frac{\mathrm{\partial }w}{\mathrm{\partial }g}>0$

(3) $\frac{\mathrm{\partial }p}{\mathrm{\partial }g}>0$; when $0<\lambda <\frac{1}{3}$, $\frac{\mathrm{\partial }p}{\mathrm{\partial }{\theta }_d}>0$; otherwise, $\frac{\mathrm{\partial }p}{\mathrm{\partial }{\theta }_d}<0$;$\frac{\mathrm{\partial }w}{\mathrm{\partial }{\theta }_r}<0$ (Appendix 2).

Proposition 2 shows that the retail price will increase with the rise of the environmental sustainability level of the traditional channel. Maybe it is because the increase of the environmental sustainability level of the traditional channel will add the extra cost of the retailer, and the retailer has to set a high retail price to obtain more benefits.

In addition, with the increase of the products green degree and the environmental sustainability level of the direct channel, the wholesale price will add. Maybe it is because the increases of the environmental sustainability level of the direct channel and the green degree of the products will add the extra cost of the manufacturer, and the manufacturer has to set a high wholesale price to obtain more benefits.

Finally, proposition 2 also shows the cross-influence among the environmental sustainability level of channels, the green degree and the price. The retail price will increase with the rise of the products green degree. However, the change circumstances of the retail price with the increase of the environmental sustainability level of the direct channel are decided by the cross-environmental-sustainability sensitivity coefficients of channels. In addition, with the increase of the environmental sustainability level of the traditional channel, the wholesale price will decrease.

4.2. Centralized model

In the centralised model, the profit maximisation of the entire supply chain are the aim of the retailer and the manufacturer. The central decision makers decide the retail prices in the two channels, the products’ green degree, and the environmental sustainability level of the channels. $p_j$ stands for the retail price in the centralised model. The environmental sustainability level of the traditional and the direct channels is ${\theta }_{rj}$ and ${\theta }_{dj}$, respectively. $g_j$ stands for the green degree of the products. Thus, the total benefit of the DGSC is as follows:

(13) ${\pi }_j=(p_{\mathrm{r}})D_r+(p_d)D_d-\frac{\delta {g_j}^2}{2}-\frac{\tau {{\theta }_{dj}}^2}{2}-\frac{\tau {{\theta }_{rj}}^2}{2}$(13)

Proposition 3:

${\pi }_j$ is an jointly concave function to $g_j$ and ${\theta }_{dj}$, and ${\theta }_{rj}$ and ${\theta }_{dj}$, respectively. Meanwhile, ${\pi }_j$ is an jointly concave function to$p_{\mathrm{r}}$ and $p_d$. But it is not an jointly concave function to $p_{\mathrm{r}}$, $p_{\mathrm{d}}$, $g_j$, ${\theta }_{rj}$ and ${\theta }_{dj}$ (Appendix 3).

Proposition 3 shows that to obtain the optimal values of $p_j$, $g_j$, ${\theta }_{rj}$ and ${\theta }_{dj}$. The two-stage method should be adopted. In the first stage, based on formula (13), we get ${\theta }_{rj}$ and ${\theta }_{dj}$ with regard to $p_j$ and $g_j$.

(14) ${\theta }_{rj}(p_j,g_j)={\theta }_{dj}(p_j,g_j)=\frac{p_j(1-\lambda )}{\tau }$(14)

In the second stage, putting ${\theta }_{rj}$ and ${\theta }_{dj}$ into formula (13), the optimal values of $p_j$ and $g_j$ can be got.

(15) $g_j=-\frac{a\tau ({\kappa }_r+{\kappa }_d)}{\tau {({\kappa }_r+{\kappa }_d)}^2+2\delta {(1-\lambda )}^2-4\tau \delta (1-\beta )}$(15)

(16) $p_j=-\frac{a\tau \delta }{\tau {({\kappa }_r+{\kappa }_d)}^2+2\delta {(1-\lambda )}^2-4\tau \delta (1-\beta )}$(16)

Putting $p_j$ and $g_j$ into formula (14), the optimal values of ${\theta }_{rj}$ and ${\theta }_{dj}$ can be got, meanwhile, the optimal benefits also can be got.

Proposition 4:

(1) $\frac{\mathrm{\partial }{p}_j}{\mathrm{\partial }{\theta }_{rj}}=\frac{\mathrm{\partial }{p}_j}{\mathrm{\partial }{\theta }_{dj}}>0$;

(2) ${\frac{\mathrm{\partial }{p}_j}{\mathrm{\partial }{g}_j}}_{\hspace{0.17em}}>0$ (Appendix 4).

Proposition 4 indicates that the retail price in the centralised model will increase following the increase of the channel environment sustainability level. Maybe it is because the increase of the channel environment sustainability level will add the demand, the decision maker can set a high retail price to gain more benefits. Meanwhile, with the increase of the products’ green degree, the retail price in the centralised model will grow. Maybe it is because the increase of the green degree of the products will add the extra cost of the decision maker, and the decision maker has to set a high retail price to obtain more benefits.

4.3. Coordinating DGSC

Clearly, the retail price in the decentralised model is higher compared with it in the centralised model. Meanwhile, the DGSC’s benefits is higher in the centralised model. Thus, in this section, a two-part compensation contract will be used to coordinate the decentralised DGSC. In reality, many researches have designed contracts to coordinate dual-channel supply chain (Cai, Zhang, and Zhang 2009; Cai 2010; Chen, Zhang, and Sun 2012). In this study, a simple two-part compensation contract will be used to coordinate the decentralised DGSC.

In this contract, the manufacturer will allocate a certain percentage orders ($\eta$) from the direct channel to the retailer, and then charges a certain fee $F$ from the retailer. Based on these, the benefits formulas of the retailer and the manufacturer are as follows.

(17) ${\pi }_r^C=(p^C-w^C)D_r+\eta (p^C-c)D_d-\frac{\tau {{\theta }_r}^2}{2}-F$(17)

(18) ${\pi }_d^C=(w^C-c)D_r+(1-\eta )(p^C-c)D_d-\frac{\delta g^2}{2}-\frac{\tau {{\theta }_d}^2}{2}+F$(18)

To achieve the DGSC coordination, we set $p_r^C=p_{rj}$ and $p_d^C=p_{dj}$, based on formulas (17) and (18), formulas (19) and (20) can be got.

(19) $w^C=\frac{a-({\theta }_r-{\theta }_d)(\lambda +1)-2a\rho -g({\kappa }_r-{\kappa }_d)}{2(1-\beta )}$(19)

(20) $p^C=\frac{(1-\lambda )({\theta }_r+{\theta }_d)+g({\kappa }_r+{\kappa }_d)+a}{4(1-\beta )}$(20)

Putting $w^c$ and $p^C$ into formulas (17) and (18), ${\pi }_r^C$ and ${\pi }_d^C$ can be got. In addition, ${\pi }_r^C+{\pi }_d^C={\pi }_j$. To coordinate the supply chain, ${\pi }_r^C>{\pi }_r$ and ${\pi }_d^C>{\pi }_d$ should also be met. From ${\pi }_r^C>{\pi }_r$ and ${\pi }_d^C>{\pi }_d$, we get $F<5T^2/144(1-\beta )$ and $T^2/48(1-\beta )<F$. Here, $T=a-4a\rho -3{\theta }_r+{\theta }_d-3g{\kappa }_r+g{\kappa }_d-\lambda {\theta }_r+3\lambda {\theta }_d$.

Thus, proposition 5 can be got.

Proposition 5:

When $w^C=(a-({\theta }_r-{\theta }_d)(\lambda +1)-2a\rho -g({\kappa }_r-{\kappa }_d))/2(1-\beta )$ and $T^2/48(1-\beta )<F<5T^2/144(1-\beta )$, the two-part compensation contract can achieve the DGSC coordination.

5. Numerical simulation

In this article, a numerical example will be presented to show the results’ effectiveness. The influences of the cross-environmental-sustainability sensitivity coefficient, the green degree, and the environmental sustainability level of the channels on the players’ benefits and pricing strategies in the two model above will be analysed. Then, the two-part compensation contract also is analysed. Assume that the related parameters are as follows: $\mathrm{a}=200,\delta =0.8,\tau =2,\rho =0.4,\beta =0.5,{\kappa }_r=0.05,{\kappa }_d=\mathrm{0.04.}$

(1) Effects of the cross-environmental-sustainability sensitivity coefficient on the related decision variables in both decentralised and centralised models are shown in Figure 1. From Figure 1, with the increase of the cross-environmental-sustainability sensitivity coefficient, the environmental sustainability levels of channels and the green degree of the products in both centralised and decentralised model will decrease. Meanwhile, to the changes of the cross-environmental-sustainability sensitivity coefficient, the environmental sustainability levels of channels are more sensitive than the products’ green degree in both centralised and decentralised model.

Figure 1. The plates of the influence of $\lambda$ on the results in both decentralised and centralised DGSC. (a) indicates the influence of $\lambda$ on $g,g_j,{\theta }_r,{\theta }_{rj},{\theta }_d$, and ${\theta }_{dj}$; (b) shows the influence of $\lambda$ on $p,w$ and $p_j$.

In addition, with the increase of the cross-environmental- sustainability sensitivity coefficient, the optimal wholesale price will grow, and the optimal retail price in both centralised and decentralised model will decrease. Meanwhile, when the cross-environmental-sustainability sensitivity coefficient is low 0.556, these results above are effective. At this time, the retail price is bigger than the wholesale price.

(2) Effects of the environmental sustainability levels of channels and the products’ green degree on the players’ pricing strategies in both decentralised and centralised models are shown in Figure 2. From Figure 2, with the increase of the environmental sustainability level of channels, the retail price in both decentralised and centralised models will increase. However, changes of the wholesale price are different with the changes of the environmental sustainability levels in different channels. With the increase of the environmental sustainability level of the traditional channel, the wholesale price will decrease. With the growth of the environmental sustainability level of the direct channel, the wholesale price will increase. In addition, with the rise of the products’ green degree, the retail price in both decentralised and centralised models and the wholesale price will increase. From Figure 2, we also can get that the retail price in the centralised model is lower compared with the retail price in the decentralised model.

Figure 2. The plates of the influence of $g(g_j),{\theta }_r$ and ${\theta }_d$on the results in both decentralised and centralised DGSC. (a) indicates the influence of ${\theta }_r$ on $p,w$, and $p_j$; (b) shows the influence of ${\theta }_d$ on $p,w$ and $p_j$; (c) presents the effect of $g(g_j)$ on $p,w$, and $p_j$.

(3) Effects of the environmental sustainability levels of channel and the products’ green degree on the players’ benefits in both decentralised and centralised models are shown in Figures 3 and 4. From Figure 3, with the growth of the products’ green degree, the retailer’s benefits in the decentralised model will increase. When, the value of the green degree can meet a certain range, with the growth of the products’ green degree, the manufacturer’s benefits in the decentralised model will increase. However, when the value of the green degree exceeds the certain range, the manufacturer’s benefits in the decentralised model will decrease. In centralised model, when the value of the green degree meet a certain range, the benefits of the supply chain will grow, otherwise it will decrease.

Figure 3. The plates of the influence of $g(g_j)$ on the benefits of the decentralised and the centralised DGSC. (a) indicates the influence of $g$ on ${\pi }_r$; (b) shows the influence of $g_j$ on ${\pi }_j$; (c) presents the effect of $g$ on ${\pi }_d$.

Figure 4. The plates of the influence of ${\theta }_r$ and ${\theta }_d$ on the benefits of the decentralised and the centralised DGSC. (a) indicates the influence of ${\theta }_r$ on ${\pi }_r,{\pi }_d$and${\pi }_j$; (b) shows the influence of ${\theta }_d$ on ${\pi }_r,{\pi }_d$and${\pi }_j$.

Therefore, for the decision makers, the green degree of the products should keep in a certain range because the excessive green degree will damage the decision makers’ benefits.

From Figure 4, with the increase of the environmental sustainability level of channels, the total benefits of DGSC in centralised model will grow. However, in the decentralised model, with the growth of the environmental sustainability level of the traditional channel, the retailer’s benefit will add and the manufacturer’s revenue will decrease slowly. With the growth of the environmental sustainability level of the direct channel, the retailer’s benefit will decrease slowly and the manufacturer’s revenue will increase.

These analytical results indicates that the improvements of the environmental sustainability levels of the traditional channel will damage the benefits of the manufacturer. The improvements of the environmental sustainability levels of the direct channel will damage the benefits of the retailer. Moreover, the benefits of supply chain in the centralised model is higher.

Therefore, for the decision makers, if they want to gain more benefits, improving their channel environment sustainability is an effective method.

(4) The two-part compensation contract can achieve the DGSC coordination, as shown in Figure 5. Based on the aforementioned analysis, in the centralised model, the retail price is lower and the benefits of DGSC is higher. Thus, there is ‘double marginalisation effect’, the supply chain should be coordinated. In this study, the two-part compensation contract is used. From Figure 5, when the value of F is in R, the retailer and the manufacturer can get more benefits, namely, the two-part compensation contract can achieve the DGSC coordination.

Figure 5. Coordination principle figure.

6. Conclusions

This study adds the channel environment sustainability into DGSC and analyses the effects of the channel environment sustainability and the products green degree on the pricing strategies in both decentralised and centralised models. Meanwhile, a two-part compensation contract is used to coordinate the DGSC. Then, the effects of the cross-environmental-sustainability sensitive coefficient on the retailer’s and the manufacturer’s decision variables are discussed in both decentralised and centralised models. The effects of products’ green degree and the channel environment sustainability levels on the retailer’s and the manufacturer’s pricing strategies and benefits are analysed in both decentralised and centralised models.

Based on the aforementioned analysis, some meaningful results are obtained. Firstly, the products’ green degree has important influences on pricing strategies and benefits of the decision makers. our study indicate that there is a positive relationship between the green degree and the retail and the wholesale prices. Moreover, when the value of the green degree is in a certain range, the manufacturer can get more benefits. Secondly, the channel environment sustainability has a significant effect on pricing strategies and benefits of the DGSC. Our research shows that there is a negative relationship between the environmental sustainability level of the traditional channel and the wholesale price, and there are positive relationships between the environmental sustainability level of the traditional channel and the retail price. However, the environmental sustainability level of the direct channel has positive effects on decision makers’ pricing strategies. In addition, the cross-environmental- sustainability sensitivity coefficient has a positive effect on the wholesale price and has negative effects on the retail prices in both decentralised and centralised model. The channel environmental sustainability level will affect the benefits of the centralised DGSC positively. However, the increase of the environmental sustainability level of the traditional channel will damage the manufacturer’s benefits and the improvement of the environmental sustainability level of the direct channel will damage the retailer’s benefits. Finally, when the value of the compensation fee F can met a certain range, the contract adopted can coordinate the DGSC.

This paper only studies a two-stage green supply chain system with on green manufacturer and one retailer. In fact, green supply chain system is very complex, there are double channel and multi-channel supply chain. Moreover, in a green supply chain system, it contains green supplier, green manufacturer, retailer and green third party logistics, however, in this article, only the relationships between green manufacturer and retailer are discussed. In addition, supply chain members are assumed to be risk-neutral.

In this study, a two-stage green supply chain was considered. In reality, multi-channel green supply chain and multi-stage supply chain are very common. In the future, the pricing policies of green supply chain should be studied with a multi-channel green supply chain or a multi-stage supply chain. In addition, next, a two-stage green supply chain with green supplier and green manufacturer should be discussed. We also can explore the related questions in different competitive environments.

Disclosure statement

The authors declare that there are no conflict interest.

Additional information

Notes on contributors

Pan Liu

Pan Liu conceived and designed the experiments and performed the experiments; Pan Liu analysed the data and wrote the article.

Appendix 1.

Based on formula (6), we can get the Hessian matrix.

$H_1=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_r)}^2}{{\mathrm{\partial }}^2{p}_{{ }_r}}& \frac{\mathrm{\partial }{({\pi }_r)}^2}{{\mathrm{\partial }}^2{p}_r{\theta }_r}\\ \frac{\mathrm{\partial }{({\pi }_r)}^2}{{\mathrm{\partial }}^2{\theta }_r{p}_r}& \frac{\mathrm{\partial }{({\pi }_r)}^2}{{\mathrm{\partial }}^2{\theta }_r}\end{array}\right)=\left(\begin{array}{cc}-2& 1\\ 1& -\tau \end{array}\right)=2\tau -1>0$

Clearly, when $\tau >1/2$, ${\pi }_r$ is an jointly concave function to $p_r$ and ${\theta }_r$.

Putting formulas (7) and (8) into formula (5), we can get the Hessian matrix.

$H_2=\left(\begin{array}{ccc}\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_d}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}w{p}_d}\end{array}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}g{p}_d}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_d{p}_d}\\ \begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_{d}w}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}w}\end{array}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}wg}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}w{\theta }_d}\\ \begin{array}{cc}\begin{array}{c}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_{d}g}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_d{\theta }_d}\end{array}& \begin{array}{c}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}gw}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_{d}w}\end{array}\end{array}& \begin{array}{c}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}g}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_{d}g}\end{array}& \begin{array}{c}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}g{\theta }_d}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_d}\end{array}\end{array}\right)=\left(\begin{array}{cccc}-2& \frac{\lambda }{2\tau -1}& \frac{2{\kappa }_d\tau -\lambda {\kappa }_r-{\kappa }_d}{2\tau -1}& \frac{\lambda +2\tau -1}{2\tau -1}\\ \frac{\lambda }{2\tau -1}& \frac{-2\tau }{2\tau -1}& \frac{{\kappa }_d\tau }{2\tau -1}& \frac{-\lambda \tau }{2\tau -1}\\ \frac{2{\kappa }_d\tau -\lambda {\kappa }_r-{\kappa }_d}{2\tau -1}& \frac{{\kappa }_d\tau }{2\tau -1}& -\delta & 0\\ \frac{\lambda +2\tau -1}{2\tau -1}& \frac{-\lambda \tau }{2\tau -1}& 0& -\tau \end{array}\right)$

As $H_2^1=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}w}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}wg}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}gw}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}g}\end{array}\right)=\left(\begin{array}{cc}\frac{-2\tau }{2\tau -1}& \frac{{\kappa }_d\tau }{2\tau -1}\\ \frac{{\kappa }_d\tau }{2\tau -1}& -\delta \end{array}\right)=\frac{-2\tau \delta (2\tau -1)-{({\kappa }_d\tau )}^2}{{(2\tau -1)}^2}<0$, therefore, ${\pi }_d$ is not an jointly concave function to $w$ and $\mathrm{g}$. $H_2^2=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}w}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}w{\theta }_d}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_{d}w}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_d}\end{array}\right)=\left(\begin{array}{cc}\frac{-2\tau }{2\tau -1}& \frac{-\lambda \tau }{2\tau -1}\\ \frac{-\lambda \tau }{2\tau -1}& -\tau \end{array}\right)=\frac{2{\tau }^2(2\tau -1)-{(\lambda \tau )}^2}{{(2\tau -1)}^2}$, because the positive and negative of $2{\tau }^2(2\tau -1)-(\lambda \tau {)}^2$ cannot be sure. Therefore, ${\pi }_d$ is not an jointly concave function to $w$ and ${\theta }_d$. $H_2^3=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_{\mathrm{d}}}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_{\mathrm{d}}{\theta }_d}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_d{p}_{\mathrm{d}}}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_d}\end{array}\right)=\left(\begin{array}{cc}-2& \frac{\lambda }{2\tau -1}\\ \frac{\lambda }{2\tau -1}& \frac{-2\tau }{2\tau -1}\end{array}\right)=\frac{4\tau (2\tau -1)-{(\lambda )}^2}{{(2\tau -1)}^2}$, when $4\tau (2\tau -1)>(\lambda {)}^2$, ${\pi }_d$ is an jointly concave function to $p_d$ and $w$. $H_2^4=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_d}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_{d}g}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}g{p}_d}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^{2}g}\end{array}\right)=\left(\begin{array}{cc}-2& \frac{2{\kappa }_d\tau -\lambda {\kappa }_r-{\kappa }_d}{2\tau -1}\\ \frac{2{\kappa }_d\tau -\lambda {\kappa }_r-{\kappa }_d}{2\tau -1}& -\delta \end{array}\right)=2\delta -\frac{{(2{\kappa }_d\tau -\lambda {\kappa }_r-{\kappa }_d)}^2}{{(2\tau -1)}^2}$, because the positive and negative of $2\delta -\frac{{(2{\kappa }_d\tau -\lambda {\kappa }_r-{\kappa }_d)}^2}{{(2\tau -1)}^2}$ cannot be sure, thus, ${\pi }_d$ is not an jointly concave function to $p_d$ and $\mathrm{g}$. $H_2^5=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_d}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{p}_d{\theta }_{\mathrm{d}}}\\ \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_{\mathrm{d}}{p}_d}& \frac{\mathrm{\partial }{({\pi }_{\mathrm{d}})}^2}{{\mathrm{\partial }}^2{\theta }_{\mathrm{d}}}\end{array}\right)=\left(\begin{array}{cc}-2& \frac{\lambda +2\tau -1}{2\tau -1}\\ \frac{\lambda +2\tau -1}{2\tau -1}& -\tau \end{array}\right)=2\tau -\frac{{(\lambda +2\tau -1)}^2}{{(2\tau -1)}^2}$, because the positive and negative of it cannot be sure, thus, ${\pi }_d$ is not an jointly concave function to $p_d$ and ${\theta }_{\mathrm{d}}$.

Based on the analysis above, we obtain formula (A1), (A2) and (A3)

(A1) $p_r=\frac{3{\theta }_r+3a\rho +3g{\kappa }_r-3\lambda {\theta }_d}{4}$(A1)

(A2) $p_d=\frac{{\theta }_d+a(1-\rho )+g{\kappa }_d-\lambda {\theta }_r}{2}$(A2)

(A3) $w=\frac{{\theta }_r-\lambda {\theta }_d+a\rho +g{\kappa }_r}{2}$(A3)

Appendix 2.

Based on formulas (A1), (A2) and (A3), we can get:

$\frac{\mathrm{\partial }{p}_r}{\mathrm{\partial }{\theta }_r}=\frac{3}{4}>0;\frac{\mathrm{\partial }{p}_r}{\mathrm{\partial }{\theta }_d}=-\frac{3}{4}<0;\frac{\mathrm{\partial }{p}_r}{\mathrm{\partial }g}=\frac{3}{4}>0;\frac{\mathrm{\partial }w}{\mathrm{\partial }{\theta }_r}=\frac{1}{2}>0;\frac{\mathrm{\partial }w}{\mathrm{\partial }{\theta }_d}=-\frac{\lambda }{2}<0;\frac{\mathrm{\partial }w}{\mathrm{\partial }g}=\frac{{\kappa }_r}{2}>0;\frac{\mathrm{\partial }{p}_d}{\mathrm{\partial }g}=\frac{{\kappa }_d}{2}>0$

$\frac{\mathrm{\partial }{p}_d}{\mathrm{\partial }{\theta }_d}=\frac{1}{2};\frac{\mathrm{\partial }{p}_d}{\mathrm{\partial }{\theta }_{\mathrm{r}}}=-\frac{\lambda }{2}.$

Appendix 3.

Based on formula (13), we can get the Hessian matrix

$H_3=\left(\begin{array}{ccccc}\frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_r}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_r{p}_d}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_{\mathrm{r}}{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_r{\theta }_{rj}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_r{\theta }_{d_j}}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_{\mathrm{d}}{p}_r}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_d}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_d{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_d{\theta }_{rj}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_d{\theta }_{d_j}}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j{p}_{\mathrm{r}}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j{p}_{\mathrm{d}}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j{\theta }_{rj}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j{\theta }_{dj}}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}{p}_{\mathrm{r}}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}{p}_{\mathrm{d}}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}{\theta }_{\mathrm{d}\mathrm{j}}}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{\mathrm{d}j}{p}_{\mathrm{r}}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}{p}_d}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}{\theta }_{\mathrm{r}\mathrm{j}}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}}\end{array}\right)=\left(\begin{array}{ccccc}-2& 0& {\kappa }_r& 1& -\lambda \\ 0& -2& {\kappa }_d& -\lambda & 1\\ {\kappa }_r& {\kappa }_d& -\delta & 0& 0\\ 1& -\lambda & 0& -\tau & 0\\ -\lambda & 1& 0& 0& -\tau \end{array}\right)$

As $H_3^1=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}{\theta }_{dj}}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}{\theta }_{rj}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}}\end{array}\right)=\left(\begin{array}{cc}-\tau & 0\\ 0& -\tau \end{array}\right)={\tau }^2>0$, ${\pi }_j$ is an jointly concave function to ${\theta }_{rj}$ and ${\theta }_{dj}$.

As $H_3^2=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j{\theta }_{d_j}}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}}\end{array}\right)=\left(\begin{array}{cc}-\delta & 0\\ 0& -\tau \end{array}\right)=\delta \tau >0$, ${\pi }_j$ is an jointly concave function to $g_j$ and ${\theta }_{dj}$,

As $H_3^3=\left(\begin{array}{cc}\frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_{\mathrm{r}}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_d{p}_r}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_r{p}_d}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{p}_d}\end{array}\right)=\left(\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right)=4$. Thus, ${\pi }_j$ is an jointly concave function to$p_{\mathrm{r}}$ and $p_d$.

As $H_3^4=\left(\begin{array}{ccc}\frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j{\theta }_{r_j}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{g}_j{\theta }_{dj}}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{rj}{\theta }_{dj}}\\ \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}{g}_j}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}{\theta }_{rj}}& \frac{\mathrm{\partial }{({\pi }_j)}^2}{{\mathrm{\partial }}^2{\theta }_{dj}}\end{array}\right)=\left(\begin{array}{ccc}-\delta & 0& 0\\ 0& -\tau & 0\\ 0& 0& -\tau \end{array}\right)=-{\tau }^2\delta <0$. ${\pi }_j$ is not an jointly concave function to $p_{\mathrm{r}}$, $p_{\mathrm{d}}$, $g_j$, ${\theta }_{rj}$ and ${\theta }_{dj}$.

Based on the aforementioned analysis, we obtain formula (A4) and (A5).

(A4) $p_{\mathrm{r}}=\frac{{\theta }_r-{\theta }_d\lambda +g{\kappa }_r+a\rho }{2}$(A4)

(A5) $p_{\mathrm{d}}=\frac{{\theta }_d-{\theta }_r\lambda +g{\kappa }_d+a(1-\rho )}{2}$(A5)

Appendix 4.

Based on formulas (A4) and (A5), we can get:

$\frac{\mathrm{\partial }{p}_r}{\mathrm{\partial }{\theta }_{rj}}=\frac{\mathrm{\partial }{p}_d}{\mathrm{\partial }{\theta }_{dj}}=\frac{1}{2}>0;\frac{\mathrm{\partial }{p}_r}{\mathrm{\partial }{\theta }_{\mathrm{d}j}}=\frac{\mathrm{\partial }{p}_d}{\mathrm{\partial }{\theta }_{rj}}=-\frac{\lambda }{2}<0$

${\frac{\mathrm{\partial }{p}_{\mathrm{r}}}{\mathrm{\partial }{g}_j}}_{\hspace{0.17em}}=\frac{\mathrm{\partial }{p}_d}{\mathrm{\partial }{g}_j}=\frac{{\kappa }_r}{2}>0.$