Abstract
Along with the significant improvement of environmental consciousness, consumers not only consider the price and quality level of products, but also pay more attention to their green level. In order to strengthen the competitive advantage, the manufacturers should consider the green level of products in addition to their price and the quality level. In this paper, we investigate the green investment of two competing manufacturers in a supply chain based on price and quality competition and analyze the effect of green investment on the quality level of the product. The research shows that the manufacturer is willing to make a green investment with a relatively low value of green sensitivity regardless of whether the manufacturer’s rival makes a green investment. Further, we find that the profit of the manufacturer who makes a green investment is greater than the profit of the manufacturer who does not invest regardless of the market size. When both competing manufacturers make green investments, the profit of the manufacturer who is in a large potential market is higher than that of the manufacturer who is in a small potential market. While in a same potential market, the profits of the two competing manufacturers are the same. Finally, we conclude that the green investment counterintuitively will not always improve the quality level of the products.
Similar content being viewed by others
References
Bititci U, Garengo P, Drfler V, Nudurupati S (2012) Performance measurement: challenges for tomorrow. Int J Manag Rev 14(3):305–327
Chen J, Liang L, Yao D, Sun S (2017) Price and quality decisions in dual-channel supply chains. Eur J Oper Res 259(3):935–948
Chen Z, Lan Y, Zhao R (2018) Impacts of risk attitude and outside option on compensation contracts under different information structures. Fuzzy Optim Decis Mak 17:13–47
Choi S (2003) Expanding to direct channel: market coverages as entry barrier. J Interact Market 17(1):25–40
Choudhary V, Ghose A, Mukhopadhyay T, Rajan U (2005) Personalized pricing and quality differentiation. Manag Sci 51(7):1120–1130
Desai PS, Kekre S, Radhakrishnan S, Srinivasan K (2001) Product differentiation and commonality in design: balancing revenues and cost drivers. Manag Sci 47(1):37–51
Feng J, Lan Y, Zhao R (2017) Impact of price cap regulation on supply chain contracting between two monopolists. J Ind Manag Optim 13(1):347–371
Ghosh D, Shah J (2015) Supply chain analysis under green sensitive consumer demand and cost sharing contract. Int J Prod Econ 164:319–329
Hong Z, Guo X (2018) Green product supply chain contracts considering environmental responsibilities. Omega. https://doi.org/10.1016/j.omega.2018.02.010
Karaer O, Erhun F (2015) Quality and entry deterrence. Eur J Oper Res 240(1):292–303
Lan Y, Zhao R, Tang W (2015) An inspection-based price rebate and effort contract model with incomplete information. Comput Ind Eng 83:264–272
Lan Y, Liu Z, Niu B (2017) Pricing and design of after-sales service contract: the value of mining asymmetric sales cost information. Asia Pac J Oper Res 34(1):1740002
Lan Y, Peng J, Wang F, Gao C (2018) Quality disclosure with information value under competition. Int J Mach Learn Cybern 9(9):1489–1503
Li B, Zhu M, Jiang Y, Li Z (2016) Pricing policies of a competitive dual-channel green supply chain. J Clean Prod 112:2029–2042
Li W, Chen J (2018a) Backward integration strategy in a retailer Stackelberg supply chain. Omega 75(2):118–130
Li W, Chen J (2018b) Pricing and quality competition in a brand-differentiated supply chain. Int J Prod Econ. https://doi.org/10.1016/j.ijpe.2018.04.026
Matsubayashi N (2007) Price and quality competition: the effect of differentiation and vertical integration. Eur J Oper Res 180(2):907–921
Matsubayashi N, Yamada Y (2008) A note on price and quality competition between asymmetric firms. Eur J Oper Res 187(2):571–581
Wang S, Hu Q, Liu W (2017a) Price and quality-based competition and channel structure with consumer loyalty. Eur J Oper Res 262(2):563–574
Wang X, Lan Y, Tang W (2017b) An uncertain wage contract model for risk-averse worker under bilateral moral hazard. J Ind Manag Optim 13(4):1815–1840
Xu X, He P, Xu H, Zhang Q (2017) Supply chain coordination with green technology under cap-and-trade regulation. Int J Prod Econ 183:433–442
Yan Y, Zhao R, Lan Y (2017) Asymmetric retailers with different moving sequences: group buying versus individual purchasing. Eur J Oper Res 261(3):903–917
Yan Y, Zhao R, Liu Z (2018a) Strategic introduction of the marketplace channel under spillovers from online to offline sales. Eur J Oper Res 267(1):65–77
Yan Y, Zhao R, Chen H (2018b) Prisoner’s dilemma on competing retailers’ investment in green supply chain management. J Clean Prod 184:65–81
Yang K, Zhao R, Lan Y (2016) Impacts of uncertain project duration and asymmetric risk sensitivity information in project management. Int Trans Oper Res 23(4):749–774
Yang K, Lan Y, Zhao R (2017) Monitoring mechanisms in new product development with risk-averse project manager. J Intell Manuf 28(3):667–681
Yu Y, Han X, Hu G (2016) Optimal production for manufacturers considering consumer environmental awareness and green subsidies. Int J Prod Econ 182:397–408
Zahra B, Jafar H (2017) A mathematical model for green supply chain coordination with substitutable products. J Clean Prod 145:232–249
Zhang L, Wang J, You J (2015) Consumer environmental awareness and channel coordination with two substitutable products. Eur J Oper Res 241(1):63–73
Acknowledgements
The study of Dr. Wu was funded by the National Natural Science Foundation of China (Grant Number: 71601141) and the study of Yang was funded by Yanta Scholars Foundation of Xian University of Finance and Economics.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
Ethical approval
This study does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Communicated by Y. Ni.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Green investment in a price and quality-based supply chain
Appendix: Green investment in a price and quality-based supply chain
Proof of Proposition 1
For given \(w_i\) and \(q_i\), \(i=1,2,\) we conclude that \(\pi ^\mathrm{NN}_{r}\) is jointly concave in \(p_1\) and \(p_2\) because the Hessian matrix of \(\pi ^\mathrm{NN}_{r}\), which are \(\frac{\partial ^2 \pi ^{\mathrm{N N}}_{r}}{\partial p_{1}^2}=-\,2<0\), \(\frac{\partial ^2 \pi ^{\mathrm{N N}}_{r}}{\partial p_{1}^2} \frac{\partial ^2 \pi ^{\mathrm{N N}}_{r}}{\partial p_{2}^2} -\frac{\partial ^2 \pi ^{\mathrm{N N}}_{r}}{\partial p_{1}\partial p_2} \frac{\partial ^2 \pi ^{\mathrm{N N}}_{r}}{\partial p_{2}\partial p_1}=4(1-\alpha ^2)>0\), is negative definite. By solving the following first-order condition, \(\frac{\partial \pi ^\mathrm{NN}_{r}}{\partial p_1}=0\) and \(\frac{\partial \pi ^\mathrm{NN}_{r}}{\partial p_2}=0\), we obtain the unique optimal retail price (4).
Thus, substituting Eq. (4) into the manufacturers’ profit functions we obtain \(\pi ^\mathrm{NN}_{m_{i}}\). To ensure that \(\pi ^\mathrm{NN}_{m_{i}}\) is jointly concave in \(q_i\) and \(w_i\), we require that \(\frac{\partial ^2\pi ^{\mathrm{NN}}_{m_{i}}}{\partial q_{i}^2}=-\,\tau <0\), \(\frac{\partial ^2\pi ^{\mathrm{NN}}_{m_{i}}}{\partial q_{i}^2} \frac{\partial ^2\pi ^{\mathrm{NN}}_{m_{i}}}{\partial w_{i}^2} -\frac{\partial ^2\pi ^{\mathrm{NN}}_{m_{i}}}{\partial q_{i}\partial w_{i}} \frac{\partial ^2\pi ^{\mathrm{NN}}_{m_{i}}}{\partial w_{i}\partial q_{i}}=\tau -\frac{1}{4}>0\), which implies the Hessian matrix of \(\pi ^\mathrm{NN}_{m_{i}}\) is negative definite. By solving the first-order condition, that is \(\frac{\pi ^\mathrm{NN}_{m_{i}}}{\partial q_{i}}=0\) and \(\frac{\pi ^\mathrm{NN}_{m_{i}}}{\partial w_i}=0\), we have the unique optimal quality levels \(q_{i}^{\mathrm{NN}}\) and wholesale price \(w_{i}^{\mathrm{NN}}\) (6).
Thus substituting Eq. (6) into Eq. (4) and demands function (2), the optimal retail price \(p_{i}^{\mathrm{NN}}\) and the demand \(D_{i}^{\mathrm{NN}}\) (7) are obtained.
Substituting Eqs. (6) and (7) into \(\pi ^\mathrm{NN}_{r}\) and \(\pi ^\mathrm{NN}_{m_{i}}\), we obtain the retailer’s and the manufacturers’ optimal profits \(\pi ^\mathrm{NN}_{m_{i}}\) and \(\pi ^\mathrm{NN}_{r}\) (8).
The proof is completed. \(\square \)
Proof of Corollaries 1
From the concavity of the retailer’s profit and the manufacturers’ profit, we derive \(4\tau -1>0\). In order to make decision variables to be positive, we require
or
For \( f(2\alpha \tau -\beta )=a_{j} \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ a_{j}\left( 4\,\tau -1 \right) ^{2} +2\,a_{i} \left( 4\,\tau -1 \right) \left( 2\,\alpha \,\tau -\beta \right) \), we conclude that \(f(2\alpha \tau -\beta )>0\) when \(a_i<a_j\), and \(f(2\alpha \tau -\beta )\) has two different roots \(\frac{-\,a_i\pm \sqrt{a_{i}^{2}-a_{j}^{2}}}{a_{j}}(4\tau -1)\) when \(a_i>a_j\). Therefore, we have \(f(2\alpha \tau -\beta )<0\) when
and \(f(2\alpha \tau -\beta )>0\) when
From (24) (25) and (26), we can derive \(f(2\alpha \tau -\beta )<0\) when
or
From (24) (25) and (27), we can derive \(f(2\alpha \tau -\beta )>0\) when
or
- (i)
For any given \(\alpha \), we have
$$\begin{aligned} \frac{\partial q_i^{\mathrm{NN}}}{\partial \alpha }= & {} {\frac{2\,\tau \, \left( a_{j} \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ a_{j} \left( 4\,\tau -1 \right) ^{2} +2\,a_{i} \left( 4\,\tau -1 \right) \left( 2\,\alpha \,\tau -\beta \right) \right) }{ \left( 4\,{\alpha }^{2}{\tau }^{2}-4\,\alpha \,\beta \,\tau +{\beta }^{2}- 16\,{\tau }^{2}+8\,\tau -1 \right) ^{2}}},\\ \frac{\partial w_i^{\mathrm{NN}}}{\partial \alpha }= & {} {\frac{4\,\tau ^2\, \left( a_{j} \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ a_{j} \left( 4\,\tau -1 \right) ^{2} +2\,a_{i} \left( 4\,\tau -1 \right) \left( 2\,\alpha \,\tau -\beta \right) \right) }{ \left( 4\,{\alpha }^{2}{\tau }^{2}-4\,\alpha \,\beta \,\tau +{\beta }^{2}- 16\,{\tau }^{2}+8\,\tau -1 \right) ^{2}}},\\ \frac{\partial D_i^{\mathrm{NN}}}{\partial \alpha }= & {} {\frac{2\,\tau ^2\, \left( a_{j} \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ a_{j}\left( 4\,\tau -1 \right) ^{2} +2\,a_{i} \left( 4\,\tau -1 \right) \left( 2\,\alpha \,\tau -\beta \right) \right) }{ \left( 4\,{\alpha }^{2}{\tau }^{2}-4\,\alpha \,\beta \,\tau +{\beta }^{2}- 16\,{\tau }^{2}+8\,\tau -1 \right) ^{2}}},\\ \frac{\partial \pi _{m_{i}}^{\mathrm{NN}}}{\partial \alpha }= & {} {\frac{-2\,{\tau }^{2} \left( 4\,\tau -1 \right) \left( a_{j}\, \left( 2\,\tau \,\alpha -\beta \right) +a_{i}\, \left( 4\,\tau -1 \right) \right) \left( a_{j}\, \left( 2\,\tau \,\alpha -\beta \right) ^{2}+a_{j}\, \left( 4\,\tau -1 \right) ^{2}+2\,a_{i}\, \left( 4\,\tau -1 \right) \left( 2\,\tau \,\alpha -\beta \right) \right) }{ \left( 4\,{\alpha }^{2}{\tau }^{2}-4\,\alpha \,\beta \,\tau +{\beta }^{2}- 16\,{\tau }^{2}+8\,\tau -1 \right) ^{3}}}. \end{aligned}$$When \(a_i<a_j\), we have \( f(2\alpha \tau -\beta )>0\). Therefore, we conclude \(\frac{\partial q_i^{\mathrm{NN}}}{\partial \alpha }>0\), \(\frac{\partial w_i^{\mathrm{NN}}}{\partial \alpha }>0\), \(\frac{\partial D_i^{\mathrm{NN}}}{\partial \alpha }>0\), \(\frac{\partial \pi _{m_{i}}^{\mathrm{NN}}}{\partial \alpha }>0\). When \(a_i>a_j\), we have \( f(2\alpha \tau -\beta )<0\) from (28) and (29). Therefore, we obtain \(\frac{\partial q_i^{\mathrm{NN}}}{\partial \alpha }<0\), \(\frac{\partial w_i^{\mathrm{NN}}}{\partial \alpha }<0\), \(\frac{\partial D_i^{\mathrm{NN}}}{\partial \alpha }<0\), \(\frac{\partial \pi _{m_{i}}^{\mathrm{NN}}}{\partial \alpha }<0\). Similarly, we obtain \( f(2\alpha \tau -\beta )>0\) from (30) and (31), which make \(\frac{\partial q_i^{\mathrm{NN}}}{\partial \alpha }>0\), \(\frac{\partial w_i^{\mathrm{NN}}}{\partial \alpha }>0\), \(\frac{\partial D_i^{\mathrm{NN}}}{\partial \alpha }>0\), \(\frac{\partial \pi _{m_{i}}^{\mathrm{NN}}}{\partial \alpha }>0\) to be hold.
- (ii)
For any given \(\beta \), we have
$$\begin{aligned} \frac{\partial q_i^{\mathrm{NN}}}{\partial \beta }= & {} -\,{\frac{ a_{j} \left( 2\,\tau \,\alpha -\beta \right) ^{2}+ a_{j} \left( 4\,\tau -1 \right) ^{2}+2\,a_{i}\left( 4\,\tau -1 \right) \left( 2\,\tau \,\alpha -\beta \right) }{\left( 4\,{\alpha }^{2}{\tau }^{2}-4\,\alpha \,\beta \,\tau +{\beta }^{2}- 16\,{\tau }^{2}+8\,\tau -1 \right) ^{2}}},\\ \frac{\partial w_i^{\mathrm{NN}}}{\partial \beta }= & {} -\,{\frac{2\,\tau \, \left( a_{j} \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ a_{j} \left( 4\,\tau -1 \right) ^{2} +2\,a_{i} \left( 4\,\tau -1 \right) \left( 2\,\alpha \,\tau -\beta \right) \right) }{\left( 4\,{\alpha }^{2}{\tau }^{2} -4\,\alpha \,\beta \,\tau +{\beta }^{2}-16\,{\tau }^{2}+8\,\tau -1\right) ^{2}}},\\ \frac{\partial D_i^{\mathrm{NN}}}{\partial \beta }= & {} -\,{\frac{\tau \, \left( a_{j} \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ a_{j}\left( 4\,\tau -1 \right) ^{2} +2\,a_{i} \left( 4\,\tau -1 \right) \left( 2\,\alpha \,\tau -\beta \right) \right) }{\left( 4\,{\alpha }^{2}{\tau }^{2}-4\,\alpha \,\beta \,\tau +{\beta }^{2}-16\,{\tau }^{2}+8\,\tau -1 \right) ^{2}}},\\ \frac{\partial \pi _{m_{i}}^{\mathrm{NN}}}{\partial \beta }= & {} {\frac{\tau \left( 4\,\tau -1 \right) \left( a_{j}\, \left( 2\,\tau \,\alpha -\beta \right) +a_{i}\, \left( 4\,\tau -1 \right) \right) \left( a_{j}\, \left( 2\,\tau \,\alpha -\beta \right) ^{2}+a_{j}\, \left( 4\,\tau -1 \right) ^{2}+2\,a_{i}\, \left( 4\,\tau -1 \right) \left( 2\,\tau \,\alpha -\beta \right) \right) }{\left( 4\,{\alpha }^{2}{\tau }^{2}-4\,\alpha \,\beta \,\tau +{\beta }^{2} -16\,{\tau }^{2}+8\,\tau -1 \right) ^{3}}}. \end{aligned}$$When \(a_i<a_j\), we have \( f(2\alpha \tau -\beta )>0\). Therefore, we conclude \(\frac{\partial q_i^{\mathrm{NN}}}{\partial \beta }<0\), \(\frac{\partial w_i^{\mathrm{NN}}}{\partial \beta }<0\), \(\frac{\partial D_i^{\mathrm{NN}}}{\partial \beta }<0\), \(\frac{\partial \pi _{m_{i}}^{\mathrm{NN}}}{\partial \beta }<0\). When \(a_i>a_j\), we have \( f(2\alpha \tau -\beta )<0\) from (28) and (29). Therefore, we obtain \(\frac{\partial q_i^{\mathrm{NN}}}{\partial \beta }>0\), \(\frac{\partial w_i^{\mathrm{NN}}}{\partial \beta }>0\), \(\frac{\partial D_i^{\mathrm{NN}}}{\partial \beta }>0\), \(\frac{\partial \pi _{m_{i}}^{\mathrm{NN}}}{\partial \beta }>0\). Similarly, we obtain \( f(2\alpha \tau -\beta )>0\) from (30) and (31), which make \(\frac{\partial q_i^{\mathrm{NN}}}{\partial \beta }<0\), \(\frac{\partial w_i^{\mathrm{NN}}}{\partial \beta }<0\), \(\frac{\partial D_i^{\mathrm{NN}}}{\partial \beta }<0\), \(\frac{\partial \pi _{m_{i}}^{\mathrm{NN}}}{\partial \beta }<0\) to be hold. \(\square \)
Proof of Proposition 2
By the same reason in the proof of Proposition 1, we conclude that \(\pi ^\mathrm{GN}_{r}\) is jointly concave in \(p_1\) and \(p_2\). By solving \(\frac{\partial \pi ^\mathrm{GN}_{r}}{\partial p_1}=0\) and \(\frac{\partial \pi ^\mathrm{GN}_{r}}{\partial p_2}=0\), we obtain the unique optimal retail price (13).
Thus substituting Eq. (13) into the manufacturer \(M_1\)’s profit functions \(\pi ^\mathrm{GN}_{m_{1}}\), we obtain that the Hessian matrix of \(\pi ^\mathrm{GN}_{m_{1}}\) is as follows:
To ensure \(\pi ^\mathrm{GN}_{m_{1}}\) is jointly concave in \(q_1\), \(w_1\) and \(\theta _1\), we need that the leading principal minors \(S_1=-\,\tau <0\), \(S_2=\tau -\frac{1}{4}>0\), \(S_3=-\,\tau \mu +\frac{1}{4}\mu +\frac{1}{4}\gamma ^2\tau <0\), which implies the Hessian matrix is negative definite.
Similarly, substituting Eq. (13) into the manufacturer \(M_2\)’s profit functions \(\pi ^\mathrm{GN}_{m_{2}}\), we obtain that the Hessian matrix of \(\pi ^\mathrm{GN}_{m_{2}}\) is as follows:
To ensure \(\pi ^\mathrm{GN}_{m_{2}}\) is jointly concave in \(q_2\) and \(w_2\), we need that the leading principal minor \(S_1=-\,\tau <0\), \(S_2=\tau -\frac{1}{4}>0\). In order to maintain the concavity of the profit function \(\pi ^\mathrm{GN}_{m_{1}}\) and \(\pi ^\mathrm{GN}_{m_{2}}\) , the condition \(\tau >\frac{1}{4}\) and \(\mu (4\tau -1)>\gamma ^2 \tau \) must be hold simultaneously.
Solving the first-order condition, that is \(\frac{\pi ^\mathrm{GN}_{m_{i}}}{\partial q_{i}}=0\), \(\frac{\pi ^\mathrm{GN}_{m_{i}}}{\partial w_i}=0\), and \(\frac{\pi ^\mathrm{GN}_{m_{1}}}{\partial \theta _{1}}=0\), the unique optimal quality levels \(q_{i}^\mathrm{GN}\), optimal wholesale price \(w_{i}^\mathrm{GN}\) and optimal the green level \(\theta _1\) (15) are derived.
Thus substituting Eq. (15) into Eq. (13) and demands function (11), we get the optimal retail price \(p_{i}^\mathrm{GN}\) and the demand \(D_{i}^\mathrm{GN}\) (16).
Substituting Eqs. (15) and (16) into \(\pi ^\mathrm{GN}_{m_{i}}\) and \(\pi ^\mathrm{GN}_{r}\), we conclude the retailer’s optimal profit \(\pi ^\mathrm{GN}_{r}\) and the manufacturers’ optimal profits \(\pi ^\mathrm{GN}_{m_{i}}\) (17).
The proof is completed. \(\square \)
Proof of Corollaries 2
From the concavity of the retailer’s profit and the manufacturers’ profit, we derive \(4\tau -1>0\) and \(\gamma ^2 \tau -\mu (4\tau -1)<0\). In order to make decision variables to be positive, we require \(\frac{\gamma ^{2}\tau }{\mu }< 2\alpha \tau -\beta <4\tau -1 \).
We assume
- (i)
For any given \(\alpha \), we have
$$\begin{aligned}&\frac{\partial q_1^\mathrm{GN}}{\partial \alpha }= {\frac{\tau \,\mu \, \left( 4\,a_{1}\, \left( 4\,\tau -1 \right) \left( -\,\frac{1}{2}\,{\gamma }^{2}\tau +\mu \, \left( 2\,\alpha \,\tau -\beta \right) \right) +2\,a_{2}\, \left( \mu \, \left( 2\,\alpha \,\tau - \beta \right) ^{2}+ \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\, \tau -1 \right) -{\gamma }^{2}\tau \right) \right) \right) }{ \left( 2 \,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \, \tau -{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial q_2^\mathrm{GN}}{\partial \alpha }= -\,{\frac{2\,\tau \, \left( a_{1}\, \left( \left( \mu \, \left( 2\, \alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) ^{2}+\mu \, \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{ \gamma }^{2}\tau \right) \right) +a_{2}\, \left( 2\,\mu \, \left( 2\, \alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial w_1^\mathrm{GN}}{\partial \alpha }= {\frac{4\,{\tau }^{2}\mu \, \left( a_{1}\, \left( 4\,\tau -1 \right) \left( 2\,\mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2} \tau \right) +a_{2}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) \right) }{ \left( 2\, \alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau - {\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial w_2^\mathrm{GN}}{\partial \alpha }= -\,{\frac{4\,{\tau }^{2} \left( a_{1}\, \left( \left( \mu \, \left( 2\, \alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) ^{2}+\mu \, \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{ \gamma }^{2}\tau \right) \right) +a_{2}\, \left( 2\,\mu \, \left( 2\, \alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial \theta _1^\mathrm{GN}}{\partial \alpha }= {\frac{2\,\gamma \,{\tau }^{2} \left( a_{1}\, \left( 4\,\tau -1 \right) \left( 2\,\mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2} \tau \right) +a_{2}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) \right) }{ \left( 2\, \alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau - {\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial D_1^\mathrm{GN}}{\partial \alpha }= {\frac{2\,{\tau }^{2}\mu \, \left( a_{1}\, \left( 4\,\tau -1 \right) \left( 2\,\mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2} \tau \right) +a_{2}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) \right) }{ \left( 2\, \alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau - {\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial D_2^\mathrm{GN}}{\partial \alpha }= -{\frac{ \left( a_{1}\, \left( \left( \mu \, \left( 2\,\alpha \,\tau - \beta \right) -{\gamma }^{2}\tau \right) ^{2}+\mu \, \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) +a_{2}\, \left( 2\,\mu \, \left( 2\,\alpha \,\tau - \beta \right) -{\gamma }^{2}\tau \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) {\tau }^{2}}{ \left( 2\, \alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau - {\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}}, \nonumber \\&\frac{\partial \pi _{m_{1}}^\mathrm{GN}}{\partial \alpha } = -{\frac{2\,{\tau }^{2}\left( 2\,a_{1}\,L_{1}+a_{2}\, L_{2}\right) \left( \mu \,\left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \left( a_{1}\,\left( 4\,\tau -1 \right) +a_{2}\, \left( 2\,\alpha \,\tau -\beta \right) \right) }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{3} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2} \tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{3}}},\nonumber \\&\frac{\partial \pi _{m_{2}}^\mathrm{GN}}{\partial \alpha }= {\frac{8\,{\tau }^{2} \left( \tau -1/4 \right) \left( a_{1}\, L_{3} +a_{2}\, L_{4} \right) \left( a_{1}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) +a_{2}\, \left( \mu \, \left( 4\, \tau -1 \right) -{\gamma }^{2}\tau \right) \right) }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{3} \left( 2\,\alpha \,\mu \,\tau -{ \gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{3}}}, \end{aligned}$$(32)When \(\frac{\gamma ^{2}\tau }{\mu }< 2\alpha \tau -\beta <4\tau -1 \), from (32), we can conclude that \(\frac{\partial q_1^\mathrm{GN}}{\partial \alpha }>0\), \(\frac{\partial q_2^\mathrm{GN}}{\partial \alpha }<0\), \(\frac{\partial w_1^\mathrm{GN}}{\partial \alpha }>0\), \(\frac{\partial w_2^\mathrm{GN}}{\partial \alpha }<0\), \(\frac{\partial \theta _1^\mathrm{GN}}{\partial \alpha }>0\), \(\frac{\partial D_1^\mathrm{GN}}{\partial \alpha }>0\), \(\frac{\partial D_2^\mathrm{GN}}{\partial \alpha }<0\), \(\frac{\partial \pi _{m_{1}}^\mathrm{GN}}{\partial \alpha }>0\), \(\frac{\partial \pi _{m_{2}}^\mathrm{GN}}{\partial \alpha }<0\).
- (ii)
For any given \(\beta \), we have
$$\begin{aligned}&\frac{\partial q_1^\mathrm{GN}}{\partial \beta }= -\,{\frac{\mu \, \left( a_{1}\, \left( 4\,\tau -1 \right) \left( 2\,\mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) +a_{ 2}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ \left( 4 \,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2} \tau \right) \right) \right) }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2}\tau -\beta \, \mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial q_2^\mathrm{GN}}{\partial \beta }= {\frac{a_{1}\, \left( \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) ^{2}+\mu \, \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) +a_{2}\, \left( 2\,\mu \, \left( 2\,\alpha \,\tau - \beta \right) -{\gamma }^{2}\tau \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) }{ \left( 2\,\alpha \,\tau -\beta -4\, \tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial w_1^\mathrm{GN}}{\partial \beta }= -\,{\frac{2\,\tau \,\mu \, \left( a_{1}\, \left( 4\,\tau -1 \right) \left( 2\,\mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2} \tau \right) +a_{2}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) \right) }{ \left( 2\, \alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau - {\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial w_2^\mathrm{GN}}{\partial \beta }= {\frac{2\,\tau \, \left( a_{1}\, \left( \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) ^{2}+\mu \, \left( 4\, \tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2} \tau \right) \right) +a_{2}\, \left( 2\,\mu \, \left( 2\,\alpha \,\tau - \beta \right) -{\gamma }^{2}\tau \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) }{ \left( 2\,\alpha \,\tau - \beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2} \tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial \theta _1^\mathrm{GN}}{\partial \beta }= -\,{\frac{\gamma \,\tau \, \left( a_{1}\, \left( 4\,\tau -1 \right) \left( 2\,\mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2} \tau \right) +a_{2}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) \right) }{ \left( 2\, \alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau - {\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial D_1^\mathrm{GN}}{\partial \beta }= -\,{\frac{\tau \,\mu \, \left( a_{1}\, \left( 4\,\tau -1 \right) \left( 2\mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) + a_{2}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) ^{2}+ \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{ \gamma }^{2}\tau \right) \right) \right) }{ \left( 2\,\alpha \,\tau - \beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2} \tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial D_2^\mathrm{GN}}{\partial \beta }= {\frac{\tau \, \left( a_{1}\, \left( \left( \mu \, \left( 2\,\alpha \, \tau -\beta \right) -{\gamma }^{2}\tau \right) ^{2}+\mu \, \left( 4\,\tau -1 \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) +a_{2}\, \left( 2\,\mu \, \left( 2\,\alpha \,\tau - \beta \right) -{\gamma }^{2}\tau \right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) }{ \left( 2\,\alpha \,\tau - \beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2} \tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial \pi _{m_{1}}^\mathrm{GN}}{\partial \beta }= {\frac{\tau \, \left( 2\,a_{1}\,L_{1}+a_{2}\, L_{2}\right) \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \left( a_{1}\, \left( 4\,\tau -1 \right) +a_{2}\, \left( 2\,\alpha \,\tau -\beta \right) \right) }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{3} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{3}}},\nonumber \\&\frac{\partial \pi _{m_{2}}^\mathrm{GN}}{\partial \beta }= -{\frac{8\,\tau \, \left( \tau -1/4 \right) \left( a_{1}\, L_{3}+a_{2}\,L_{4}\right) \left( a_{1}\, \left( \mu \, \left( 62\,\alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) +a_{2}\, \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{3} \left( 2\,\alpha \,\mu \,\tau -{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{3}}}, \end{aligned}$$(33)When \(\frac{\gamma ^{2}\tau }{\mu }< 2\alpha \tau -\beta <4\tau -1 \), from (33), we can derive that \(\frac{\partial q_1^\mathrm{GN}}{\partial \beta }<0\), \(\frac{\partial q_2^\mathrm{GN}}{\partial \beta }>0\), \(\frac{\partial w_1^\mathrm{GN}}{\partial \beta }<0\), \(\frac{\partial w_2^\mathrm{GN}}{\partial \beta }>0\), \(\frac{\partial \theta _1^\mathrm{GN}}{\partial \beta }<0\), \(\frac{\partial D_1^\mathrm{GN}}{\partial \beta }<0\), \(\frac{\partial D_2^\mathrm{GN}}{\partial \beta }>0\), \(\frac{\partial \pi _{m_{1}}^\mathrm{GN}}{\partial \beta }<0\), \(\frac{\partial \pi _{m_{2}}^\mathrm{GN}}{\partial \beta }>0\).
- (iii)
For any given \(\gamma \), we have
$$\begin{aligned} \frac{\partial q_1^\mathrm{GN}}{\partial \gamma }= & {} -{\frac{2\,\mu \, \left( a_{2}\, \left( 2\,\alpha \,\tau -\beta \right) +a_{1}\, \left( 4\,\tau -1 \right) \right) \gamma \,\tau }{ \left( 2\, \alpha \,\tau -\beta -4\,\tau +1 \right) \left( \mu \, \left( 2\,\alpha \, \tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial q_2^\mathrm{GN}}{\partial \gamma }= & {} {\frac{2\,\mu \, \left( a_{2}\, \left( 2\,\alpha \,\tau -\beta \right) + a_{1}\, \left( 4\,\tau -1 \right) \right) \gamma \,\tau }{ \left( 2\, \alpha \,\tau -\beta -4\,\tau +1 \right) \left( \mu \, \left( 2\,\alpha \, \tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial w_1^\mathrm{GN}}{\partial \gamma }= & {} -{\frac{4\,{\tau }^{2}\mu \, \left( a_{2}\, \left( 2\,\alpha \,\tau - \beta \right) +a_{1}\, \left( 4\,\tau -1 \right) \right) \gamma }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) \left( \mu \, \left( 2 \,\alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -{\gamma } ^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial w_2^\mathrm{GN}}{\partial \gamma }= & {} {\frac{4\,{\tau }^{2}\mu \, \left( a_{2}\, \left( 2\,\alpha \,\tau -\beta \right) +a_{1}\, \left( 4\,\tau -1 \right) \right) \gamma }{ \left( 2 \,\alpha \,\tau -\beta -4\,\tau +1 \right) \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial \theta _1^\mathrm{GN}}{\partial \gamma }= & {} -{\frac{\tau \, \left( a_{2}\, \left( 2\,\alpha \,\tau -\beta \right) +a _{1}\, \left( 4\,\tau -1 \right) \right) \left( \mu \, \left( 2\, \alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) +{\gamma }^{ 2}\tau \right) }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) +\mu \, \left( 4\, \tau -1 \right) -{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial D_1^\mathrm{GN}}{\partial \gamma }= & {} -{\frac{2\,{\tau }^{2}\mu \, \left( a_{2}\, \left( 2\,\alpha \,\tau - \beta \right) +a_{1}\, \left( 4\,\tau -1 \right) \right) \gamma }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) \left( \mu \, \left( 2 \,\alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -{\gamma } ^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial D_2^\mathrm{GN}}{\partial \gamma }= & {} {\frac{2\,{\tau }^{2}\mu \, \left( a_{2}\, \left( 2\,\alpha \,\tau -\beta \right) +a_{1}\, \left( 4\,\tau -1 \right) \right) \gamma }{ \left( 2 \,\alpha \,\tau -\beta -4\,\tau +1 \right) \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial \pi _{m_{1}}^\mathrm{GN}}{\partial \gamma }= & {} -{\frac{\mu \,{\tau }^{2}\gamma \, \left( a_{2}\, \left( 2\,\alpha \,\tau -\beta \right) +a_{1}\, \left( 4\,\tau -1 \right) \right) ^{2} \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) -\mu \, \left( 4\,\tau -1 \right) +{\gamma }^{2}\tau \right) }{ \left( \mu \, \left( 2\,\alpha \, \tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) ^{3} \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2}}},\nonumber \\ \frac{\partial \pi _{m_{2}}^\mathrm{GN}}{\partial \gamma }= & {} -{\frac{8\,\gamma \,\mu \,{\tau }^{2} \left( a_{1} \left( \mu \left( 2\, \alpha \tau -\beta \right) -{\gamma }^{2}\tau \right) +a_{2} \left( \mu \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) \right) \left( \tau -1/4 \right) \left( a_{1} \left( 4\,\tau -1 \right) +a_{2} \left( 2\,\alpha \tau -\beta \right) \right) }{ \left( \mu \, \left( 2\, \alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -{\gamma }^{ 2}\tau \right) ^{3} \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2}}}.\nonumber \\ \end{aligned}$$(34)When \(\frac{\gamma ^{2}\tau }{\mu }< 2\alpha \tau -\beta <4\tau -1 \), from (33), we can derive that \(\frac{\partial q_1^\mathrm{GN}}{\partial \gamma }>0\), \(\frac{\partial q_2^\mathrm{GN}}{\partial \gamma }<0\), \(\frac{\partial w_1^\mathrm{GN}}{\partial \gamma }>0\), \(\frac{\partial w_2^\mathrm{GN}}{\partial \gamma }<0\), \(\frac{\partial \theta _1^\mathrm{GN}}{\partial \gamma }>0\), \(\frac{\partial D_1^\mathrm{GN}}{\partial \gamma }>0\), \(\frac{\partial D_2^\mathrm{GN}}{\partial \gamma }<0\), \(\frac{\partial \pi _{m_{2}}^\mathrm{GN}}{\partial \gamma }<0\).
For the profit of manufacturer \(\mathrm{M_1}\) with a green investment, we have \(\frac{\partial \pi _{m_{1}}^\mathrm{GN}}{\partial \gamma }>0\) if \( \frac{\gamma ^{2}\tau }{\mu }< 2\alpha \tau -\beta < \frac{\mu (4\tau -1)-\gamma ^{2}\tau }{\mu }\), we have \(\frac{\partial \pi _{m_{1}}^\mathrm{GN}}{\partial \gamma }<0\) if \(\max \left\{ \frac{\gamma ^{2}\tau }{\mu },\frac{\mu (4\tau -1)-\gamma ^{2}\tau }{\mu } \right\}< 2\alpha \tau -\beta <4\tau -1\). \(\square \)
Proof of Proposition 3
By the same reason in the proof of Proposition 1, we conclude that \(\pi ^\mathrm{GG}_{r}\) is jointly concave in \(p_1\) and \(p_2\). By solving \(\frac{\partial \pi ^\mathrm{GG}_{r}}{\partial p_1}=0\) and \(\frac{\partial \pi ^\mathrm{GG}_{r}}{\partial p_2}=0\), we obtain the unique optimal retail price (19).
Thus substituting Eq. (19) into the manufacturers’ profit functions \(\pi ^\mathrm{GG}_{m_{i}}\), we obtain that the Hessian matrix of \(\pi ^\mathrm{GG}_{m_{i}}\) is as follows:
To ensure \(\pi ^\mathrm{GG}_{m_{i}}\) is jointly concave in \(q_i\), \(w_i\) and \(\theta _i\), we need that the leading principal minors \(S_1=-\,\tau <0\), \(S_2=\tau -\frac{1}{4}>0\), \(S_3=-\,\tau \mu +\frac{1}{4}\mu +\frac{1}{4}\gamma ^2\tau <0\), which implies the Hessian matrix is negative definite.
By solving the first-order condition, that is \(\frac{\pi ^\mathrm{GG}_{m_{i}}}{\partial q_{i}}=0\), \(\frac{\pi ^\mathrm{GG}_{m_{i}}}{\partial w_i}=0\), and \(\frac{\pi ^\mathrm{GG}_{m_{i}}}{\partial \theta _{i}}=0\), the unique optimal quality levels \(q_{i}^\mathrm{GG}\), optimal wholesale price \(w_{i}^\mathrm{GG}\) and optimal the green level \(\theta _{i}^\mathrm{GG}\) (21) are derived.
By substituting Eq. (21) into Eq. (19) and demands function (1), the optimal retail price \(p_{i}^\mathrm{GG}\) and the demand \(D_{i}^\mathrm{GG}\) (22) are obtained.
Substituting Eqs. (21) and (22) into \(\pi ^\mathrm{GG}_{m_{i}}\) and \(\pi ^\mathrm{GG}_{r}\), we get the retailer’s optimal profit \(\pi ^\mathrm{GG}_{r}\) and the manufacturers’ optimal profits \(\pi ^\mathrm{GG}_{m_{i}}\) (23).
The proof is completed. \(\square \)
Proof of Corollaries 3
From the concavity of the retailer’s profit and the manufacturers’ profit, we have \(4\tau -1>0\) and \(\gamma ^2 \tau -\mu (4\tau -1)<0\). In order to make decision variables to be positive, we need \(\frac{\gamma ^{2}\tau }{\mu }< 2\alpha \tau -\beta <4\tau -1 \). We assume
We can conclude \(T_1<0\).
- (i)
For any given \(\alpha \), we have
$$\begin{aligned}&\frac{\partial q_i^\mathrm{GG}}{\partial \alpha }= -{\frac{2\,\tau \, T_1 }{ \left( 2\,\alpha \, \tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -2\,{ \gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial w_i^\mathrm{GG}}{\partial \alpha }= -{\frac{4\,{\tau }^{2} T_1 }{ \left( 2\,\alpha \, \tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -2\,{ \gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial \theta _i^\mathrm{GG}}{\partial \alpha }= -{\frac{2\,{\tau }^{2}\gamma \, T_1 }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -2\,{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{ 2}\mu }},\nonumber \\&\frac{\partial D_i^\mathrm{GG}}{\partial \alpha }= -{\frac{2\,{\tau }^{2} T_1 }{ \left( 2\,\alpha \, \tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -2\,{ \gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\&\frac{\partial \pi _{m_{i}}^\mathrm{GG}}{\partial \alpha }= {\frac{2\,{\tau }^{2} \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^ {2}\tau \right) \left( a_{i}\, \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) +a_{j}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) \right) { T_1}}{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{3} \left( 2\,\alpha \,\mu \, \tau -2\,{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{3}\mu }}. \end{aligned}$$(35)When \(\frac{\gamma ^{2}\tau }{\mu }< 2\alpha \tau -\beta <4\tau -1 \), from (35), we can conclude that \(\frac{\partial q_i^\mathrm{GG}}{\partial \alpha }>0\), \(\frac{\partial w_i^\mathrm{GG}}{\partial \alpha }>0\), \(\frac{\partial \theta _i^\mathrm{GG}}{\partial \alpha }>0\), \(\frac{\partial D_i^\mathrm{GG}}{\partial \alpha }>0\), \(\frac{\partial \pi _{m_{i}}^\mathrm{GG}}{\partial \alpha }>0\).
- (ii)
For any given \(\beta \), we have
$$\begin{aligned} \frac{\partial q_i^\mathrm{GG}}{\partial \beta }= & {} {\frac{T_1 }{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -2\,{\gamma }^{2}\tau -\beta \, \mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\ \frac{\partial w_i^\mathrm{GG}}{\partial \beta }= & {} {\frac{2\,\tau \, T_1 }{ \left( 2\,\alpha \, \tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -2\,{ \gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\ \frac{\partial \theta _i^\mathrm{GG}}{\partial \beta }= & {} {\frac{\tau \,\gamma \, T_1 }{ \left( 2\,\alpha \, \tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -2\,{ \gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}\mu }},\nonumber \\ \frac{\partial D_i^\mathrm{GG}}{\partial \beta }= & {} -{\frac{\tau \, T_1}{ \left( 2\,\alpha \, \tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,\alpha \,\mu \,\tau -2\,{ \gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{2}}},\nonumber \\ \frac{\partial \pi _{m_{i}}^\mathrm{GG}}{\partial \beta }= & {} -{\frac{{\tau } \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^ {2}\tau \right) \left( a_{i}\, \left( \mu \, \left( 4\,\tau -1 \right) -{\gamma }^{2}\tau \right) +a_{j}\, \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) -{\gamma }^{2}\tau \right) \right) { T_1}}{ \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{3} \left( 2\,\alpha \,\mu \, \tau -2\,{\gamma }^{2}\tau -\beta \,\mu +4\,\mu \,\tau -\mu \right) ^{3}\mu }}. \end{aligned}$$(36)When \(\frac{\gamma ^{2}\tau }{\mu }< 2\alpha \tau -\beta <4\tau -1 \), we can conclude from (36) that \(\frac{\partial q_i^\mathrm{GG}}{\partial \beta }<0\), \(\frac{\partial w_i^\mathrm{GG}}{\partial \beta }<0\), \(\frac{\partial \theta _i^\mathrm{GG}}{\partial \beta }<0\), \(\frac{\partial D_i^\mathrm{GG}}{\partial \beta }>0\), \(\frac{\partial \pi _{m_{i}}^\mathrm{GG}}{\partial \beta }<0\).
- (iii)
For any given \(\gamma \), we have
$$\begin{aligned} \frac{\partial q_i^\mathrm{GG}}{\partial \gamma }= & {} {\frac{ 2\,\left( a_{i}-a_{j} \right) \mu \,\tau \,\gamma }{ \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -2\,{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial w_i^\mathrm{GG}}{\partial \gamma }= & {} {\frac{4\, \left( a_{i}-a_{j} \right) \mu \,{\tau }^{2}\gamma }{ \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -2\,{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial D_i^\mathrm{GG}}{\partial \gamma }= & {} {\frac{2\, \left( a_{i}-a_{j} \right) \mu \,{\tau }^{2}\gamma }{ \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) +\mu \, \left( 4\,\tau -1 \right) -2\,{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial \theta _i^\mathrm{GG}}{\partial \gamma }= & {} {\frac{2a_{i}\, L_{5}+2a_{j}\, L_{6}}{\mu \, \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) +\mu \, \left( 4\, \tau -1 \right) -2\,{\gamma }^{2}\tau \right) ^{2}}},\nonumber \\ \frac{\partial \pi _{m_{i}}^\mathrm{GG}}{\partial \gamma }= & {} {\frac{{\tau }^{2}\gamma \, L_{7}L_{8}}{\mu \, \left( 2\,\alpha \,\tau -\beta -4\,\tau +1 \right) ^{2} \left( 2\,{\gamma }^{2}\tau -\mu \, \left( 2\,\alpha \,\tau -\beta \right) -\mu \, \left( 4\,\tau -1 \right) \right) ^{3}}},\nonumber \\ \end{aligned}$$(37)where
$$\begin{aligned} L_{5}= & {} \left( \mu \,\tau \, \left( 2\,\alpha \,\tau -\beta +4\, \tau -1 \right) \left( 3\,{\gamma }^{2}\tau -\mu \, \left( 4\,\tau -1 \right) \right) \right. \\&\left. -2\,{\gamma }^{2}{\tau }^{2} \left( {\gamma }^{2}\tau + \mu \, \left( 4\,\tau -1 \right) \right) \right) ,\\ L_{6}= & {} \mu \,\tau \, \left( 2\,\alpha \left( 3\,{\gamma }^{2}\tau -\mu \, \left( 2\, \alpha \,\tau -\beta \right) \right) \right. \\&\left. \tau \left( 3\,{\gamma }^{2}\tau - \mu \, \left( 2\,\alpha \,\tau -\beta \right) \right) \right. \\&\left. -\beta \left( 3\,{\gamma }^{2} \tau -\mu \, \left( 2\,\alpha \,\tau -\beta \right) \right) -1 \right) \\&+4\,\mu \,\tau ^{2} \left( 3\,{\gamma }^{2}\tau -\mu \, \left( 2\,\alpha \,\tau -\beta \right) \right) \\&-2\,{\gamma }^{2}{\tau }^{2} \left( {\gamma }^{2} \tau +\mu \, \left( 2\,\alpha \,\tau -\beta \right) \right) ,\\ L_{7}= & {} \left( a_{i}\, \left( {\gamma }^{2}\tau -\mu \, \left( 4\,\tau -1 \right) \right) \right. \\&\left. +a_{j}\, \left( {\gamma }^{2}\tau - \mu \, \left( 2\,\alpha \,\tau -\beta \right) \right) \right) ,\\ L_{8}= & {} 2\, \left( a_{i}-a_{j} \right) \left( {\gamma }^{2}\tau -\mu \, \left( 4\,\tau -1 \right) \right) \\&\times \left( \mu \, \left( 2\,\alpha \,\tau -\beta \right) -\mu \, \left( 4\,\tau -1 \right) \right) \\&-\, \left( a_{i}\,\left( {\gamma }^{2}\tau -\mu \, \left( 4\,\tau -1 \right) \right) \right. \\&\left. +a_{j}\, \left( {\gamma }^{2}\tau -\mu \, \left( 2\,\alpha \, \tau -\beta \right) \right) \right) \left( 2\,{\gamma }^{2}\tau -\mu \, \right. \\&\times \left. \left( 2\,\alpha \,\tau -\beta \right) -\mu \, \left( 4\,\tau -1 \right) \right) . \end{aligned}$$
From (37), when \(a_i<a_j \), we can conclude that \(\frac{\partial q_i^\mathrm{GG}}{\partial \gamma }<0\), \(\frac{\partial w_i^\mathrm{GG}}{\partial \gamma }<0\), \(\frac{\partial D_i^\mathrm{GG}}{\partial \gamma }<0\), \(\frac{\partial \pi _{m_{i}}^\mathrm{GG}}{\partial \gamma }<0\) and \(\frac{\partial \theta _i^\mathrm{GG}}{\partial \gamma }>0\), if \(\frac{3\gamma ^2 \tau }{\mu }< 2\alpha \tau -\beta <4\tau -1 \).
When \(a_i>a_j \), we can conclude that \(\frac{\partial q_i^\mathrm{GG}}{\partial \gamma }>0\), \(\frac{\partial w_i^\mathrm{GG}}{\partial \gamma }>0\), \(\frac{\partial D_i^\mathrm{GG}}{\partial \gamma }>0\). \(\square \)
Proof of Proposition 4
When manufacturer \(M_2\) does not make a green investment, from
we obtain that the manufacturer \(M_1\) will invest if \(\gamma \leqslant \sqrt{{\frac{\mu \, \left( \left( 4\,\tau -1 \right) ^{2}-\left( 2\,\alpha \,\tau -\beta \right) ^{2} \right) }{\tau \, \left( 4\,\tau -1 \right) }}}\). Otherwise, manufacturer \(M_1\) will not invest. By the same reason, when manufacturer \(M_1\) does not make a green investment, manufacturer \(M_2\) will invest if \(\gamma \leqslant \sqrt{{\frac{\mu \, \left( \left( 4\,\tau -1 \right) ^{2}-\left( 2\,\alpha \,\tau -\beta \right) ^{2} \right) }{\tau \, \left( 4\,\tau -1 \right) }}}\). Otherwise, manufacturer \(M_2\) will not invest.
When manufacturer \(M_2\) makes a green investment, from
we conclude that the manufacturer \(M_1\) will invest if
Otherwise, the manufacturer \(M_1\) will not invest.
Since manufacturers \(M_1\) and \(M_2\) are symmetrical, manufacturer \(M_2\) has the same green investment strategy as manufacturer \(M_1\). \(\square \)
Rights and permissions
About this article
Cite this article
Yang, S., Ding, P., Wang, G. et al. Green investment in a supply chain based on price and quality competition. Soft Comput 24, 2589–2608 (2020). https://doi.org/10.1007/s00500-019-03777-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-019-03777-y