## 3 The Models
### Optimal Decisions in Policy RE  
The profit structure of CLSC members and the SW of government remain similar to the previous subsection, but the demand function converts to \(D_{i}^{RE}=a-p_{i}^{RE}+\beta\theta_{i}^{RE}\). For example, in China, manufacturers and government organizations, such as China's National Development and Reform Commission in 2010, collaborated to promote the program \"Comments on Boosting the Re-manufacturing Development\" to encourage product reuse [52]. As discussed in Section 3.1, the profit functions for the retailer and manufacturer, and SW in Scenarios MRE and RRE are obtained as follows:  
\[\pi_{ri}^{RE}(p_{i}^{RE})=(p_{i}^{RE}-w_{i}^{RE})D_{i}^{RE}, \tag{4}\]\[\pi_{mi}^{RE}(w_{i}^{RE},\theta_{i}^{RE},\tau_{i}^{RE})=(w_{i}^{RE}-\lambda_{1} \theta_{i}^{RE}+X\tau_{i}^{RE})D_{i}^{RE}-(1-\eta_{i}^{RE})\kappa\tau_{i}^{RE2}- \lambda\theta_{i}^{RE2}, \tag{5}\]  
\[\pi_{gi}^{RE}(\eta_{i}^{RE})=\pi_{ri}^{RE}+\pi_{mi}^{RE}+\frac{D_{i}^{RE2}}{2}- \kappa\eta_{i}^{RE}\tau_{i}^{RE2}. \tag{6}\]  
Derivations of optimal decisions are similar to Policy C, hence omitted. The concavity of profit functions for CLSC members and SW for the government in Scenarios MRE and RRE is ensured by condition \(\Delta_{2m}>0\) and \(\Delta_{2r}>0\), respectively. Optimal decisions under the MS and RS games are presented in Lemma 3.4, respectively. **Lemma 3**. : _Optimal decision in Scenario MRE are obtained as follows: \(\eta_{m}^{RE}=\frac{6\lambda_{2}}{\Delta_{4}}\); \(w_{m}^{RE}=\frac{(a(4M_{3}x-M_{4}X^{2})+4c_{m}M_{3}x)\lambda_{2}}{\Delta_{2m}}\); \(p_{m}^{RE}=\frac{4(8(3a+c_{m})x-7 aX^{2})\lambda_{2}^{2}-2Z^{2}(a(6\kappa-X^{2 })+2m_{R}x)\lambda_{2}+2M_{3}YZx}{\Delta_{2m}}\); \(\theta_{m}^{RE}=\frac{(a-c_{m})M_{3}Zx}{\Delta_{2m}}\); \(\tau_{m}^{RE}=\frac{(a-c_{m})M_{4}X\lambda_{2}}{\Delta_{2m}}\); \(\pi_{mm}^{RE}=\frac{(a-c_{m})^{2}M_{3}x\lambda_{2}}{\Delta_{2m}}\); \(\pi_{rm}^{RE}=\frac{4(a-c_{m})^{2}M_{4}x\lambda_{2}}{\Delta_{2m}}\); \(\pi_{gm}^{RE}=\frac{(a-c_{m})^{2}M_{4}x\lambda_{2}}{\Delta_{2m}}\); \(Q_{m}^{RE}=\frac{2(a-c_{m})M_{4}x\lambda_{2}}{\Delta_{2m}}\), where \(\Delta_{2m}=M_{3}^{2}x-M_{4}X^{2}\lambda_{2}\)._  
**Lemma 4**. : _Optimal decision in Scenario RRE are obtained as follows: \(\eta_{r}^{RE}=\frac{N_{3}\lambda_{2}-Z^{2}\kappa}{2K(M_{2}+\lambda_{2})^{2}}\); \(w_{r}^{RE}=\frac{M_{2}(2(a+c_{m})(x-X^{2})\lambda_{2}+(c_{m}M_{3}+YZ)x)+X^{2}( c_{m}M_{2}+YZ)\lambda_{2}}{2\Delta_{2r}}\); \(p_{r}^{RE}=\frac{x2^{3}((a-c_{m})\beta-2Y)+Z((a-c_{m})(2 XZ+(4K+X^{2})\beta)-2 aN_{4}Z)\lambda_{2}+2(a(7N_{1}-2X^{2})+c_{m}(4K+X^{2})\lambda_{2}^{2}}\); \(\theta_{r}^{RE}=\frac{(a-c_{m})Z(M_{3}x+X^{2}\lambda_{2})}{2\Delta_{2r}}\); \(\tau_{r}^{RE}=\frac{(a-c_{m})X\lambda_{2}(M_{2}+\lambda_{2})}{\Delta_{2r}}\); \(\pi_{mr}^{RE}=\frac{(a-c_{m})^{2}\lambda_{2}(M_{3}x+X^{2}\lambda_{2})}{2\Delta _{2r}}\); \(\pi_{gr}^{RE}=\frac{(a-c_{m})^{2}\lambda_{2}((M_{1}+M_{3}+M_{3})x+X^{2}\lambda_{ 2})}{4\Delta_{2r}}\); \(Q_{r}^{RE}=\frac{(a-c_{m})\lambda_{2}(M_{3}x+X^{2}\lambda_{2})}{\Delta_{2r}}\), where \(\Delta_{2r}=M_{2}^{2}x-X^{2}(M_{2}+2\lambda_{2})\lambda_{2}\)._  
Recall that optimal subsidy rates in Policy C are directly proportional with market potential, and different results are obtained in Policy RE. Although the demand increases with market potential, it does not directly affect the subsidy rate. However, the government may have to spend more because the collection rates \(\tau_{m}^{RE}\) and \(\tau_{r}^{RE}\), or overall demand \(Q_{m}^{\text{C}}\) and \(Q_{r}^{\text{C}}\), increase with market potential. The following theorem highlights the characteristics of the optimal decision. **Theorem 2**. : _In Policy RE,_  
1. _The greening levels, collection rate, and social welfare are always higher under the RS game: however, the subsidy rate is higher under the MS game_
2. _The greening levels, collection rates, subsidy rates, and social welfare decrease with respect to_ \(\lambda_{1}\) _and_ \(\lambda_{2}\)_._  
We refer to Appendix D for the details of Theorem 2. The outcome of Theorem 2 differs from the previous one.