# Unfair and Risky? Profit Allocation in Closed-Loop Supply Chains by Cooperative Game Approaches
## Appendix B  
Proof of Proposition 2.: By substituting demand functions \(q_{n}^{L}\big{(}p_{n}^{L},p_{r}^{L}\big{)}\) and \(q_{r}^{L}\big{(}p_{n}^{L},p_{r}^{L}\big{)}\) into L's utility function \(u_{L}^{DC}\) and taking partial derivatives, we have \(\frac{\partial^{2}u_{L}^{DC}}{\partial p_{n}^{2}}=-\frac{2}{1-\delta}-\eta \sigma^{2}<0\), \(\frac{\partial^{2}u_{L}^{DC}}{\partial p_{r}^{2}}=-\frac{2}{1-\delta}-\frac{2}{ \beta}-\eta\sigma^{2}<0\) and \(\frac{\partial^{2}u_{L}^{DC}}{\partial p_{n}^{2}\partial p_{r}^{2}}=\frac{2}{1- \delta}-\eta\sigma^{2}\rho>0\). Then \(\frac{\partial^{2}u_{L}^{DC}}{\partial p_{n}^{2}}\frac{\partial^{2}u_{L}^{DC}}{ \partial p_{r}^{2}}-\left(\frac{\partial^{2}u_{L}^{DC}}{\partial p_{n}^{2} \partial p_{r}^{2}}\right)^{2}=\frac{4}{\delta(1-\delta)}+\frac{4\eta\sigma^{2 }(1+\eta)}{1-\delta}+\frac{2\eta\sigma^{2}}{\delta}+\eta^{2}\sigma^{4}(1-\rho )^{2}>0\) implies that L's utility function \(u_{L}^{DC}\) is strictly joint concave in \(p_{n}^{L}\) and \(p_{r}^{L}\), and \(u_{L}^{DC}\) has a unique optimal solution. By first-order conditions \(\frac{\partial u_{L}^{DC}}{\partial p_{n}^{L}}=0\) and \(\frac{\partial u_{L}^{DC}}{\partial p_{r}^{L}}=0\), L's optimal response functions are obtained as \(p_{n}^{L*}\big{(}w_{n}^{L},w_{r}^{L}\big{)}=(2w_{n}^{L}-\eta\sigma^{2}\delta^{2} \delta+2\eta\sigma^{2}\delta+2\eta\sigma^{2}w_{n}^{L}-\eta^{2}\sigma^{4}\delta^{2} w_{n}^{L}+\eta\sigma^{2}\rho w_{r}^{L}+\eta\sigma^{2}\delta w_{n}^{L}+\eta\sigma^{2} \delta w_{n}^{L}+\eta\sigma^{2}\delta w_{n}^{L}+\eta\sigma^{2}\delta w_{n}^{L}\)\(+\eta\sigma^{2}\delta w_{n}^{L}+\eta\sigma^{2}\delta w_{n}^{L}+\eta \sigma^{2}\delta w_{n}^{L}+\eta\sigma^{2}\delta^{2}w_{n}^{L}+2)/\)\(\big{(}\eta^{2}\sigma^{4}\rho^{2}\delta^{2}-\eta^{2}\sigma^{4}\rho^{2}\delta-\eta^{2} \sigma^{4}\delta^{2}+\eta^{2}\sigma^{4}\delta+4\eta\sigma^{2}\rho\delta+2\eta\sigma^{2} \delta+2\eta\sigma^{2}\delta+2\eta\sigma^{2}+4\eta\sigma^{2}+4\big{)}\) and \(p_{n}^{L}\big{(}w_{n}^{L},w_{r}^{L}\big{)}=(2\delta+2w_{r}^{L}+\eta\sigma^{2}w_{r}^{L}- \eta^{2}\sigma^{4}\delta^{2}w_{r}^{L}-\eta\sigma^{2}\rho\delta+\eta\sigma^{2} \delta w_{n}^{L}+\eta\sigma^{2}\delta w_{r}^{L}+\eta\sigma^{2}\rho\delta^{2}+\eta^{2} \sigma^{4}\delta w_{r}^{L}\)\(-\eta^{2}\sigma^{4}\rho^{2}\delta w_{r}^{L}+\eta\sigma^{2} \rho\delta w_{n}^{L}+\eta\sigma^{2}\rho\delta w_{n}^{L}+\eta\sigma^{\(\left(\eta^{2}\sigma^{4}\rho^{2}\delta^{2}-\eta^{2}\sigma^{4}\rho^{2}\delta-\eta^{ 2}\sigma^{4}\delta^{2}+\eta^{2}\sigma^{4}\delta+4\eta\sigma^{2}\rho\delta+2\eta \sigma^{2}\delta+2\eta\sigma^{2}+4\right)\). By substituting \(p_{n}^{L^{*}}\left(w_{n}^{L},w_{r}^{L}\right)\) and \(p_{r}^{L^{*}}\left(w_{n}^{L},w_{r}^{L}\right)\) into demand functions \(q_{n}^{L}\left(p_{n}^{L},p_{r}^{L}\right)\) and \(q_{r}^{L}\left(p_{n}^{L},p_{r}^{L}\right)\), the optimal quantities \(q_{n}^{L*}\left(w_{n}^{L},w_{r}^{L}\right)\) and \(q_{r}^{L*}\left(w_{n}^{L},w_{r}^{L}\right)\) for L are obtained as functions composed of wholesale prices \(w_{n}^{L}\) and \(w_{r}^{L}\). By substituting demand functions \(q_{n}^{F}\left(p_{n}^{F},p_{r}^{F}\right)\) and \(q_{r}^{F}\left(p_{n}^{F},p_{r}^{F}\right)\) into F's utility function \(u_{F}^{DC}\) and taking partial derivatives, we have \(\frac{\partial^{2}u_{F}^{DC}}{\partial p_{n}^{F}}=\frac{2\left(\lambda+1 \right)}{\partial-1}<0\), \(\frac{\partial^{2}u_{F}^{DC}}{\partial p_{n}^{F}}=\left(\lambda+1\right)\left( \frac{2}{\delta-1}-\frac{2}{\delta}\right)<0\), and \(\frac{\partial^{2}u_{F}^{DC}}{\partial p_{n}^{F}\partial p_{r}^{F}}=\frac{2 \left(\lambda+1\right)}{1-\delta}>0\). Then \(\frac{\partial^{2}u_{F}^{DC}}{\partial p_{n}^{F}}\frac{\partial^{2}u_{F}^{DC} }{\partial p_{r}^{F}}-\left(\frac{\partial^{2}u_{F}^{DC}}{\partial p_{n}^{F} \partial p_{r}^{F}}\right)^{2}=\frac{4\left(\lambda+1\right)^{2}\left(1-\delta \right)}{\delta\left(\delta-1\right)^{2}}>0\) implies that F's utility function \(u_{F}^{DC}\) is joint concave in \(p_{n}^{F}\) and \(p_{r}^{F}\), thus \(u_{F}^{DC}\) has a unique optimal solution.