## Appendix A Optimal Decision in Scenario MC  
To obtain optimal response for the retailer in Equation (1), one needs to solve \(\frac{d\pi_{\text{em}}^{C}}{dp_{\text{em}}^{C}}=0\). On simplification, \(p_{m}^{C}(w_{m}^{C},\theta_{m}^{C},\tau_{m}^{C},\rho_{m}^{C})=\frac{a+u_{m}^{ C}+\theta_{m}^{C}\beta+\rho_{m}^{C}}{2}\). Because \(\frac{d^{2}\pi_{\text{em}}^{C}}{dp_{m}^{C}}=-2<0\), the profit function for the retailer is concave. Therefore, substituting \(p_{m}^{C}\) in Equation (2), the profit function for the manufacturer is obtained as:  
\[\pi_{mm}^{C}(w_{m}^{C},\theta_{m}^{C},\tau_{m}^{C},\rho_{m}^{C})=\frac{(w_{m}^{ C}-\theta_{m}^{C}\lambda_{1}+X\tau_{m}^{C}-c_{m})(a-w_{m}^{C}+\beta\theta_{m}^{C}+ \rho_{m}^{C})-2\kappa{\tau_{m}^{C}}^{2}-2\lambda_{2}{\theta_{m}^{C}}^{2}}{2}\]  
To obtain optimal response for the manufacturer on wholesale price and investment efforts, we need to solve \(\frac{d\pi_{\text{em}}^{C}}{dw_{m}^{C}}=0\), \(\frac{d\pi_{\text{em}}^{C}}{d\pi_{\text{em}}^{C}}=0\), and \(\frac{d\pi_{\text{em}}^{C}}{d\theta_{m}^{C}}=0\), simultaneously. After simplification, the following response is obtained:  
\[{w_{m}}^{C}=\frac{\kappa(a+c_{m}+\rho_{m}^{C})(4\lambda_{2}-\beta(\beta- \lambda_{1}))+(\kappa(\beta^{2}-\lambda_{1}^{2})-X^{2}\lambda_{2})(a+\rho_{m}^ {C})}{\kappa(8\lambda_{2}-Z^{2})-X^{2}\lambda_{2}}\]  
\[\tau_{m}^{C}=\frac{(a-c_{m}+\rho_{m}^{C})X\lambda_{2}}{\kappa(8\lambda_{2}-Z^{ 2})-X^{2}\lambda_{2}}\text{and }\theta_{m}^{C}=\frac{\kappa Z(a-c_{m}+\rho_{m}^{C})}{\kappa(8\lambda_{2}-Z^{ 2})-X^{2}\lambda_{2}}\]  
Because, the manufacturer's profit function is a function of three variables, we compute the Hessian matrix(\(H_{m}^{C}\)) to verify concavity as follows:  
\[H_{m}^{C}=\left[ \begin{array}{cc}\frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}& \frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}&\frac{\beta_{m}^{2}c}{\lambda_{m}^{C}} \\ \frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}&\frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}& \frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}\\ \frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}&\frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}& \frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}\\ \frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}&\frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}& \frac{\beta_{m}^{2}c}{\lambda_{m}^{C}}\\ \end{array} \right]=\left[ \begin{array}{cc}-1&\frac{-X}{2}&\frac{\beta+ \lambda_{1}}{2}\\ -X&-2\kappa&\frac{\beta}{2}\\ \frac{\beta+\lambda_{1}}{2}&\frac{\beta+\lambda_{1}}{2}\\ \frac{\beta+\lambda_{1}}{2}&\frac{\beta+\lambda_{1}}{2}&-2\lambda_{2}-\lambda_{1 }\beta\\ \end{array} \right]\]The values of first, second, and third principal minors are obtained as \(H_{m1}^{C}=-1<0\); \(H_{m2}^{C}=\frac{8\kappa-X^{2}}{4}\); and \(H_{m3}^{C}=-\frac{(8\kappa-X^{2})\lambda_{2}-\kappa(\beta-\lambda_{1})^{2}}{2}\), respectively. Therefore, profit function for the manufacturer is concave if \(8\kappa>X^{2}\) and \((8\kappa-X^{2})\lambda_{2}-\kappa(\beta-\lambda_{1})^{2}>0\). Substituting optimal responses in Equation (3), the SW function for the government organization is obtained as \(\pi_{gm}^{C}(\rho_{m}^{C})=\frac{\kappa\lambda_{2}(a-c_{m}+\rho_{r}^{C})((a-c _{m})((14\kappa-X^{2})\lambda_{2}-XZ^{2})+xZ^{2}\rho_{m}^{C}-(2\kappa-X^{2}) \lambda_{2}\rho_{r}^{C})}{((8\kappa-X^{2})\lambda_{2}-\kappa(\beta-\lambda_{1} )^{2})^{2}}\). Therefore, one can obtain optimal subsidy rate by solving \(\frac{d\pi_{gr}^{C}}{d\rho_{m}^{C}}=0\). On simplification, optimal subsidy rate is obtained as \(\rho_{m}^{C}=\frac{6(a-c_{m})\kappa\lambda_{2}}{\kappa(2\lambda_{2}-Z^{2})-X^ {2}\lambda_{2}}\). Note that \(\frac{d^{2}\pi_{gr}^{C}}{d\rho_{m}^{C}}=-\frac{2\kappa\lambda_{2}\Delta_{1}}{( \kappa(8\lambda_{2}-(\beta-\lambda_{1})^{2})-X^{2}\lambda_{2})^{2}}<0\), i.e., \(\pi_{gm}^{C}\) is concave with respect to subsidy rate if \(\Delta_{1}=\kappa(2\lambda_{2}-(\beta-\lambda_{1})^{2})-X^{2}\lambda_{2}\). By using back substitution, we obtain optimal decision as presented in Lemma 1.