Optimal trade-in delegation strategy considering third-party recycler intrusion and used products with different durability

Abstract

Considering that product replacement is very fast and consumers purchase products at different times, old consumers often have used products with different durability (different depreciation). Moreover, the third-party recycling industry is booming. To attract different old consumers, manufacturers always implement trade-in services to compete with third-party recyclers (3Ps). In practice, some manufacturers provide trade-in services themselves while others delegate retailers to provide trade-in services. To address the trade-in delegation decision-making problem of manufacturers, our paper constructs four theoretical models depending on whether the manufacturers delegate retailers and considers the 3Ps. We find that whether manufacturers delegate and whether retailers accept delegation mainly depend on the fixed costs of trade-in services, and the conditions for retailers to accept delegation are more stringent. Furthermore, 3Ps intrusion affects the optimal trade-in delegation strategy. In the extended case considering cap-and-trade policy, the optimal trade-in delegation strategy still holds and remains unchanged, and the trade-in delegation strategies don’t affect the total carbon emissions.

Appendices

Appendix A: Tables of the main models

See Tables 2

Table 2 Optimal decisions under models MT and RT

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Table 3 The equilibrium decisions in Model MY

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Table 4 Optimal decisions under model RY

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Table 5 Optimal decisions under models MTC and RTC

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Table 6 The equilibrium decisions in Model MYC

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Table 7 Optimal decisions under model RYC

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Appendix B: Proofs of the main models

2.1 Proof of optimal decisions under model MT

We derive the sub-game equilibrium results in Model MT by using backward inductions. From Eq. (10), the second-order derivative of the retailer’s profit $\Pi_{R}^{MT}$ with $p$ is ${{\partial^{2} \Pi_{R}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{MT} } {\partial p^{2} = {{\left( {2\alpha \left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right) - 2\left( {1 - \delta_{H} } \right)\left( {1 - N\delta_{L} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {2\alpha \left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right) - 2\left( {1 - \delta_{H} } \right)\left( {1 - N\delta_{L} } \right)} \right)} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}}}} \right. \kern-0pt} {\partial p^{2} = {{\left( {2\alpha \left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right) - 2\left( {1 - \delta_{H} } \right)\left( {1 - N\delta_{L} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {2\alpha \left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right) - 2\left( {1 - \delta_{H} } \right)\left( {1 - N\delta_{L} } \right)} \right)} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}}} < 0$, By solving first-order conditions ${{\partial \Pi_{R}^{MT} } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{MT} } {\partial p = 0}}} \right. \kern-0pt} {\partial p = 0}}$,we can get $p^{MT * } \left( {w,r_{H} ,r_{L} } \right) =$

$$\begin{gathered} \frac{1}{{2\left( {\alpha \left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right) - \left( {\delta_{H} - 1} \right)\left( {N\delta_{L} - 1} \right)} \right)}}\left( {\alpha \left( {N - 1} \right)\left( {\left( {w + r_{L} } \right)\delta_{H} - \left( {w + r_{H} } \right)\delta_{L} + r_{H} - r_{L} } \right)} \right. \hfill \\ \left. { + \left( {1 - \delta_{H} } \right)\left( {\left( {\delta_{L} w + r_{L} } \right)N - w - r_{L} + \delta_{L} - 1} \right)} \right) \hfill \\ \end{gathered}.$$

Then substituting $p^{MT * } \left( {w,r_{H} ,r_{L} } \right)$ into the manufacturer’s profit function Eq. (9), the Hessian matrix of the manufacturer’s profit $\Pi_{M}^{MT}$ with $w$, $r_{H}$,$r_{L}$ is $\left[ {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w\partial r_{H} }}} \right. \kern-0pt} {\partial w\partial r_{H} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w\partial r_{L} }}} \right. \kern-0pt} {\partial w\partial r_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{H} \partial w}}} \right. \kern-0pt} {\partial r_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{H} \partial r_{L} }}} \right. \kern-0pt} {\partial r_{H} \partial r_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{L} \partial w}}} \right. \kern-0pt} {\partial r_{L} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{L} \partial r_{H} }}} \right. \kern-0pt} {\partial r_{L} \partial r_{H} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\left( {\partial r_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{L} } \right)^{2} }}} \\ \end{array} } \right]$. We can have ${{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }} =$${{\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha + \left( {1 - N\delta_{L} } \right)\left( {\delta_{H} - 1} \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha + \left( {1 - N\delta_{L} } \right)\left( {\delta_{H} - 1} \right)} \right)} {\left( {1 - \delta_{L} } \right)\left( {1 - \delta_{H} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{L} } \right)\left( {1 - \delta_{H} } \right)}} < 0$,and $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w\partial r_{H} }}} \right. \kern-0pt} {\partial w\partial r_{H} }}} \\ {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{H} \partial w}}} \right. \kern-0pt} {\partial r_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} \\ \end{array} } \right| =$${{2\alpha \left( {1 - N} \right)\left( {1 - \left( {1 - N} \right)\alpha - N\delta_{L} } \right)} \mathord{\left/ {\vphantom {{2\alpha \left( {1 - N} \right)\left( {1 - \left( {1 - N} \right)\alpha - N\delta_{L} } \right)} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}} > 0$. Moreover, we also have $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w\partial r_{H} }}} \right. \kern-0pt} {\partial w\partial r_{H} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial w\partial r_{L} }}} \right. \kern-0pt} {\partial w\partial r_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{H} \partial w}}} \right. \kern-0pt} {\partial r_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{H} \partial r_{L} }}} \right. \kern-0pt} {\partial r_{H} \partial r_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{L} \partial w}}} \right. \kern-0pt} {\partial r_{L} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\partial r_{L} \partial r_{H} }}} \right. \kern-0pt} {\partial r_{L} \partial r_{H} }}} & {{{\partial^{2} \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MT} } {\left( {\partial r_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{L} } \right)^{2} }}} \\ \end{array} } \right| =$${{4\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } \mathord{\left/ {\vphantom {{4\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right) < 0}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right) < 0}}$. Therefore, we can easily observe that Hessian matrix is negative definite and $\Pi_{M}^{MT}$ is a jointly concave function in $w$, $r_{H}$,$r_{L}$. There is a unique optimal pricing solution for the manufacturer. By solving first-order conditions ${{\partial \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{MT} } {\partial w}}} \right. \kern-0pt} {\partial w}} = 0$,${{\partial \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{MT} } {\partial r_{H} }}} \right. \kern-0pt} {\partial r_{H} }} = 0$,${{\partial \Pi_{M}^{MT} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{MT} } {\partial r_{L} }}} \right. \kern-0pt} {\partial r_{L} }} = 0$,we can get $w^{MT * }$, $r_{H}^{MT * }$ and $r_{L}^{MT * }$. Substituting $w^{MT * }$, $r_{H}^{MT * }$ and $r_{L}^{MT * }$ into the price reaction function $p^{MT * } \left( {w,r_{H} ,r_{L} } \right)$, we have $p^{MT * }$. All the optimal decisions of firms are shown in Table 2.

The process of deriving equilibrium results in Model MTC is similar to that in Model MT, thus we omit it.

2.2 Proof of optimal decisions under model RT

We derive the sub-game equilibrium results in Model RT by using backward inductions. From Eq. (13), the Hessian matrix of the retailer’s profit $\Pi_{R}^{RT}$ with $p$,$r_{H}$ and $r_{L}$ is $\left[ {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p\partial r_{H} }}} \right. \kern-0pt} {\partial p\partial r_{H} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p\partial r_{L} }}} \right. \kern-0pt} {\partial p\partial r_{L} }}} \\ {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{H} \partial p}}} \right. \kern-0pt} {\partial r_{H} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{H} \partial r_{L} }}} \right. \kern-0pt} {\partial r_{H} \partial r_{L} }}} \\ {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{L} \partial p}}} \right. \kern-0pt} {\partial r_{L} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{L} \partial r_{H} }}} \right. \kern-0pt} {\partial r_{L} \partial r_{H} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\left( {\partial r_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{L} } \right)^{2} }}} \\ \end{array} } \right]$. We can have ${{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }} =$ ${{2\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha + \left( {1 - N\delta_{L} } \right)\left( {\delta_{H} - 1} \right)} \right)} \mathord{\left/ {\vphantom {{2\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha + \left( {1 - N\delta_{L} } \right)\left( {\delta_{H} - 1} \right)} \right)} {\left( {1 - \delta_{L} } \right)\left( {1 - \delta_{H} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{L} } \right)\left( {1 - \delta_{H} } \right)}} < 0$, and $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p\partial r_{H} }}} \right. \kern-0pt} {\partial p\partial r_{H} }}} \\ {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{H} \partial p}}} \right. \kern-0pt} {\partial r_{H} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} \\ \end{array} } \right| =$${{4\alpha \left( {1 - N} \right)\left( {1 - \left( {1 - N} \right)\alpha - N\delta_{L} } \right)} \mathord{\left/ {\vphantom {{4\alpha \left( {1 - N} \right)\left( {1 - \left( {1 - N} \right)\alpha - N\delta_{L} } \right)} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}} > 0$. Moreover, we also have $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p\partial r_{H} }}} \right. \kern-0pt} {\partial p\partial r_{H} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial p\partial r_{L} }}} \right. \kern-0pt} {\partial p\partial r_{L} }}} \\ {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{H} \partial p}}} \right. \kern-0pt} {\partial r_{H} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{H} \partial r_{L} }}} \right. \kern-0pt} {\partial r_{H} \partial r_{L} }}} \\ {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{L} \partial p}}} \right. \kern-0pt} {\partial r_{L} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\partial r_{L} \partial r_{H} }}} \right. \kern-0pt} {\partial r_{L} \partial r_{H} }}} & {{{\partial^{2} \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RT} } {\left( {\partial r_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{L} } \right)^{2} }}} \\ \end{array} } \right| =$${{8\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } \mathord{\left/ {\vphantom {{8\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right) < 0}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right) < 0}}$.Therefore, we can easily observe that Hessian matrix is negative definite and $\Pi_{R}^{RT}$ is a jointly concave function in $p$,$r_{H}$ and $r_{L}$. By solving first-order conditions ${{\partial \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{RT} } {\partial p}}} \right. \kern-0pt} {\partial p}} = 0$,${{\partial \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{RT} } {\partial r_{H} }}} \right. \kern-0pt} {\partial r_{H} }} = 0$,${{\partial \Pi_{R}^{RT} } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{RT} } {\partial r_{L} }}} \right. \kern-0pt} {\partial r_{L} }} = 0$, we can get $p^{RT * } \left( {w,b_{H} ,b_{L} } \right) = {{\left( {1 + w} \right)} \mathord{\left/ {\vphantom {{\left( {1 + w} \right)} 2}} \right. \kern-0pt} 2}$,$r_{H}^{RT * } \left( {w,b_{H} ,b_{L} } \right) = {{\left( {\delta_{H} + b_{H} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{H} + b_{H} } \right)} 2}} \right. \kern-0pt} 2}$ and $r_{L}^{RT * } \left( {w,b_{H} ,b_{L} } \right) = {{\left( {\delta_{L} + b_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{L} + b_{L} } \right)} 2}} \right. \kern-0pt} 2}$. Then substituting $p^{RT * } \left( {w,b_{H} ,b_{L} } \right)$, $r_{H}^{RT * } \left( {w,b_{H} ,b_{L} } \right)$ and $r_{L}^{RT * } \left( {w,b_{H} ,b_{L} } \right)$ into the manufacturer’s profit function Eq. (12), the Hessian matrix of the manufacturer’s profit $\Pi_{M}^{RT}$ with $w$,$b_{H}$,$b_{L}$ is $\left[ {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w\partial b_{H} }}} \right. \kern-0pt} {\partial w\partial b_{H} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w\partial b_{L} }}} \right. \kern-0pt} {\partial w\partial b_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{H} \partial w}}} \right. \kern-0pt} {\partial b_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\left( {\partial b_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{H} \partial b_{L} }}} \right. \kern-0pt} {\partial b_{H} \partial b_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{L} \partial w}}} \right. \kern-0pt} {\partial b_{L} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{L} \partial b_{H} }}} \right. \kern-0pt} {\partial b_{L} \partial b_{H} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\left( {\partial b_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{L} } \right)^{2} }}} \\ \end{array} } \right]$. We can have ${{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }} =$${{\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha + \left( {1 - N\delta_{L} } \right)\left( {\delta_{H} - 1} \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha + \left( {1 - N\delta_{L} } \right)\left( {\delta_{H} - 1} \right)} \right)} {\left( {1 - \delta_{L} } \right)\left( {1 - \delta_{H} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{L} } \right)\left( {1 - \delta_{H} } \right)}} < 0$, and $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w\partial b_{H} }}} \right. \kern-0pt} {\partial w\partial b_{H} }}} \\ {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{H} \partial w}}} \right. \kern-0pt} {\partial b_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\left( {\partial b_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{H} } \right)^{2} }}} \\ \end{array} } \right| =$${{2\alpha \left( {1 - N} \right)\left( {1 - \left( {1 - N} \right)\alpha - N\delta_{L} } \right)} \mathord{\left/ {\vphantom {{2\alpha \left( {1 - N} \right)\left( {1 - \left( {1 - N} \right)\alpha - N\delta_{L} } \right)} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}} > 0$. Moreover, we also have $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w\partial b_{H} }}} \right. \kern-0pt} {\partial w\partial b_{H} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial w\partial b_{L} }}} \right. \kern-0pt} {\partial w\partial b_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{H} \partial w}}} \right. \kern-0pt} {\partial b_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\left( {\partial b_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{H} \partial b_{L} }}} \right. \kern-0pt} {\partial b_{H} \partial b_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{L} \partial w}}} \right. \kern-0pt} {\partial b_{L} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\partial b_{L} \partial b_{H} }}} \right. \kern-0pt} {\partial b_{L} \partial b_{H} }}} & {{{\partial^{2} \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RT} } {\left( {\partial b_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{L} } \right)^{2} }}} \\ \end{array} } \right| =$${{\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } \mathord{\left/ {\vphantom {{\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right) < 0}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right) < 0}}$. Therefore, we can easily observe that Hessian matrix is negative definite and $\Pi_{M}^{RT}$ is a jointly concave function in $w$,$b_{H}$,$b_{L}$. There is a unique optimal pricing solution for the manufacturer. By solving first-order conditions ${{\partial \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{RT} } {\partial w}}} \right. \kern-0pt} {\partial w}} = 0$,${{\partial \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{RT} } {\partial b_{H} }}} \right. \kern-0pt} {\partial b_{H} }} = 0$,${{\partial \Pi_{M}^{RT} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{RT} } {\partial b_{L} }}} \right. \kern-0pt} {\partial b_{L} }} = 0$, we can get $w^{RT * }$,$b_{H}^{RT * }$ and $b_{L}^{RT * }$. Substituting $w^{RT * }$,$b_{H}^{RT * }$ and $b_{L}^{RT * }$ into the price reaction functions, we have $p^{RT * }$,$r_{H}^{RT * }$ and $r_{L}^{RT * }$. All the optimal decisions of firms are shown in Table 2.

The process of deriving equilibrium results in Model RTC is similar to that in Model RT, thus we omit it.

2.3 Proof of optimal decisions under model MYHH

We derive the sub-game equilibrium results in Model MYHH by using backward inductions. From Eq. (16),the Hessian matrix of the third-party’s profit $\Pi_{3P}^{MYHH}$ with $g_{H}$ and $g_{L}$ is $\left[ {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{3P}^{MYHH * } } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{3P}^{MYHH * } } {\left( {\partial g_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial g_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{3P}^{MYHH * } } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{3P}^{MYHH * } } {\partial g_{H} \partial g_{L} }}} \right. \kern-0pt} {\partial g_{H} \partial g_{L} }}} \\ {{{\partial^{2} \Pi_{3P}^{MYHH * } } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{3P}^{MYHH * } } {\partial g_{L} \partial g_{H} }}} \right. \kern-0pt} {\partial g_{L} \partial g_{H} }}} & {{{\partial^{2} \Pi_{3P}^{MYHH * } } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{3P}^{MYHH * } } {\left( {\partial g_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial g_{L} } \right)^{2} }}} \\ \end{array} } \right] =$$\left[ {\begin{array}{*{20}c} {{{ - 2\alpha \left( {1 - N} \right)} \mathord{\left/ {\vphantom {{ - 2\alpha \left( {1 - N} \right)} {\delta_{H} }}} \right. \kern-0pt} {\delta_{H} }}} & 0 \\ 0 & { - 2\left( {1 - N} \right)\left( {1 - \alpha } \right)} \\ \end{array} } \right]$. We can easily observe that Hessian matrix is negative definite and $\Pi_{3P}^{MYHH}$ is a jointly concave function in $g_{H}$ and $g_{L}$. There is a unique optimal pricing solution for the third-party. By solving first-order conditions ${{\partial \Pi_{3P}^{MYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{3P}^{MYHH} } {\partial g_{H} }}} \right. \kern-0pt} {\partial g_{H} }} = 0$, ${{\partial \Pi_{3P}^{MYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{3P}^{MYHH} } {\partial g_{L} }}} \right. \kern-0pt} {\partial g_{L} }} = 0$, we can get $g_{H}^{MYHH * } = {{\left( {s_{H} - c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{H} - c_{r} } \right)} 2}} \right. \kern-0pt} 2}$, $g_{L}^{MYHH * } = {{\left( {s_{L} - c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{L} - c_{r} } \right)} 2}} \right. \kern-0pt} 2}$.Then substituting $g_{H}^{MYHH * }$, $g_{L}^{MYHH * }$ into the retailer’s profit function Eq. (15), the second-order derivative of the retailer’s profit $\Pi_{R}^{MYHH * }$ with $p$ is ${{\partial^{2} \Pi_{R}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{MYHH} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }} = {{2\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha - \left( {1 - \delta_{H} } \right)\left( {1 - N\delta_{L} } \right)} \right)} \mathord{\left/ {\vphantom {{2\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha - \left( {1 - \delta_{H} } \right)\left( {1 - N\delta_{L} } \right)} \right)} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}} < 0$, By solving first-order conditions ${{\partial \Pi_{R}^{MYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{MYHH} } {\partial p}}} \right. \kern-0pt} {\partial p}} = 0$, we can get $p^{MYHH * } \left( {w,r_{H} ,r_{L} } \right) =$$\begin{gathered} \frac{1}{{2\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha - \left( {\delta_{H} - 1} \right)\left( {N\delta_{L} - 1} \right)} \right)}}\left( {\alpha \left( {N - 1} \right)\left( {\left( {w + r_{L} } \right)\delta_{H} - \left( {w + r_{H} } \right)\delta_{L} + r_{H} - r_{L} } \right)} \right. \hfill \\ \left. { - \left( {\delta_{H} - 1} \right)\left( {\left( {\delta_{L} w + r_{L} } \right)N - w - r_{L} + \delta_{L} - 1} \right)} \right) \hfill \\ \end{gathered}$. Then substituting $p^{MYHH * } \left( {w,r_{H} ,r_{L} } \right)$ into the manufacturer’s profit function Eq. (13), the Hessian matrix of the manufacturer’s profit $\Pi_{M}^{MYHH}$ with $w$, $r_{H}$,$r_{L}$ is $\left[ {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w\partial r_{H} }}} \right. \kern-0pt} {\partial w\partial r_{H} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w\partial r_{L} }}} \right. \kern-0pt} {\partial w\partial r_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{H} \partial w}}} \right. \kern-0pt} {\partial r_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{H} \partial r_{L} }}} \right. \kern-0pt} {\partial r_{H} \partial r_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{L} \partial w}}} \right. \kern-0pt} {\partial r_{L} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{L} \partial r_{H} }}} \right. \kern-0pt} {\partial r_{L} \partial r_{H} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\left( {\partial r_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{L} } \right)^{2} }}} \\ \end{array} } \right]$. We can have ${{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }} =$ ${{\left( {\left( {\alpha \left( {1 - N} \right) - N - 1} \right)\delta_{H} + \alpha \left( {1 - N} \right) + N + 1} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {\alpha \left( {1 - N} \right) - N - 1} \right)\delta_{H} + \alpha \left( {1 - N} \right) + N + 1} \right)} {2\left( {\delta_{H} - 1} \right)}}} \right. \kern-0pt} {2\left( {\delta_{H} - 1} \right)}} < 0$,and $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w\partial r_{H} }}} \right. \kern-0pt} {\partial w\partial r_{H} }}} \\ {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{H} \partial w}}} \right. \kern-0pt} {\partial r_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} \\ \end{array} } \right| =$${{\alpha \left( {N - 1} \right)\left( {\left( {N - 1} \right)\alpha + N + 1} \right)} \mathord{\left/ {\vphantom {{\alpha \left( {N - 1} \right)\left( {\left( {N - 1} \right)\alpha + N + 1} \right)} {\left( {\delta_{H} - 1} \right)}}} \right. \kern-0pt} {\left( {\delta_{H} - 1} \right)}} > 0$. Moreover, we also have $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w\partial r_{H} }}} \right. \kern-0pt} {\partial w\partial r_{H} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial w\partial r_{L} }}} \right. \kern-0pt} {\partial w\partial r_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{H} \partial w}}} \right. \kern-0pt} {\partial r_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{H} \partial r_{L} }}} \right. \kern-0pt} {\partial r_{H} \partial r_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{L} \partial w}}} \right. \kern-0pt} {\partial r_{L} \partial w}}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\partial r_{L} \partial r_{H} }}} \right. \kern-0pt} {\partial r_{L} \partial r_{H} }}} & {{{\partial^{2} \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{MYHH} } {\left( {\partial r_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{L} } \right)^{2} }}} \\ \end{array} } \right| =$${{2\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } \mathord{\left/ {\vphantom {{2\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } {\left( {1 - \delta_{H} } \right) < 0}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right) < 0}}$. Therefore, we can easily observe that Hessian matrix is negative definite and $\Pi_{M}^{MYHH}$ is a jointly concave function in $w$, $r_{H}$,$r_{L}$. There is a unique optimal pricing solution for the manufacturer. By solving first-order conditions ${{\partial \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{MYHH} } {\partial w}}} \right. \kern-0pt} {\partial w}} = 0$,${{\partial \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{MYHH} } {\partial r_{H} }}} \right. \kern-0pt} {\partial r_{H} }} = 0$,${{\partial \Pi_{M}^{MYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{MYHH} } {\partial r_{L} }}} \right. \kern-0pt} {\partial r_{L} }} = 0$,we can get $w^{MYHH * }$, $r_{H}^{MYHH * }$ and $r_{L}^{MYHH * }$. Substituting $w^{MYHH * }$, $r_{H}^{MYHH * }$ and $r_{L}^{MYHH * }$ into the price reaction function, we have $p^{MYHH * }$. All the optimal decisions of firms are shown in Table 3.

The processes of deriving equilibrium results in Model MYHL, MYLH, MYLL, MYCHH, MYCHL, MYCLH and MYCLL are similar to that in Model MYHH, thus we omit it.

2.4 Proof of optimal decisions under model RYHH

We derive the sub-game equilibrium results in model RYHH by using backward inductions. From Eq. (20),the Hessian matrix of the third-party’s profit $\Pi_{3P}^{RYHH}$ with $g_{H}$ and $g_{L}$ is $\left[ {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{3P}^{RYHH * } } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{3P}^{RYHH * } } {\left( {\partial g_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial g_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{3P}^{RYHH * } } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{3P}^{RYHH * } } {\partial g_{H} \partial g_{L} }}} \right. \kern-0pt} {\partial g_{H} \partial g_{L} }}} \\ {{{\partial^{2} \Pi_{3P}^{RYHH * } } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{3P}^{RYHH * } } {\partial g_{L} \partial g_{H} }}} \right. \kern-0pt} {\partial g_{L} \partial g_{H} }}} & {{{\partial^{2} \Pi_{3P}^{RYHH * } } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{3P}^{RYHH * } } {\left( {\partial g_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial g_{L} } \right)^{2} }}} \\ \end{array} } \right] =$$\left[ {\begin{array}{*{20}c} {{{ - 2\left( {1 - N} \right)\alpha } \mathord{\left/ {\vphantom {{ - 2\left( {1 - N} \right)\alpha } {\delta_{H} }}} \right. \kern-0pt} {\delta_{H} }}} & 0 \\ 0 & { - 2(1 - N)(1 - \alpha )} \\ \end{array} } \right]$. We can easily observe that Hessian matrix is negative definite and $\Pi_{3P}^{RYHH}$ is a jointly concave function in $g_{H}$ and $g_{L}$. There is a unique optimal pricing solution for the third-party. By solving first-order conditions ${{\partial \Pi_{3P}^{RYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{3P}^{RYHH} } {\partial g_{H} }}} \right. \kern-0pt} {\partial g_{H} }} = 0$,${{\partial \Pi_{3P}^{RYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{3P}^{RYHH} } {\partial g_{L} }}} \right. \kern-0pt} {\partial g_{L} }} = 0$,we can get $g_{H}^{RYHH * } = {{\left( {s_{H} - c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{H} - c_{r} } \right)} 2}} \right. \kern-0pt} 2}$, $g_{L}^{RYHH * } = {{\left( {s_{L} - c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{L} - c_{r} } \right)} 2}} \right. \kern-0pt} 2}$.Then substituting $g_{H}^{RYHH * }$ and $g_{L}^{RYHH * }$ into the retailer’s profit function Eq. (19), the Hessian matrix of the retailer’s profit $\Pi_{R}^{RYHH}$ with $p$, $r_{H}$ and $r_{L}$ is $\left[ {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p\partial r_{H} }}} \right. \kern-0pt} {\partial p\partial r_{H} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p\partial r_{L} }}} \right. \kern-0pt} {\partial p\partial r_{L} }}} \\ {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{H} \partial p}}} \right. \kern-0pt} {\partial r_{H} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{H} \partial r_{L} }}} \right. \kern-0pt} {\partial r_{H} \partial r_{L} }}} \\ {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{L} \partial p}}} \right. \kern-0pt} {\partial r_{L} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{L} \partial r_{H} }}} \right. \kern-0pt} {\partial r_{L} \partial r_{H} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\left( {\partial r_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{L} } \right)^{2} }}} \\ \end{array} } \right]$. We can have ${{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }} =$ ${{\left( {\left( {\alpha \left( {1 - N} \right) - N - 1} \right)\delta_{H} + \alpha \left( {1 - N} \right) + N + 1} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {\alpha \left( {1 - N} \right) - N - 1} \right)\delta_{H} + \alpha \left( {1 - N} \right) + N + 1} \right)} {\left( {\delta_{H} - 1} \right)}}} \right. \kern-0pt} {\left( {\delta_{H} - 1} \right)}} < 0$,and $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p\partial r_{H} }}} \right. \kern-0pt} {\partial p\partial r_{H} }}} \\ {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{H} \partial p}}} \right. \kern-0pt} {\partial r_{H} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} \\ \end{array} } \right| =$${{2\alpha \left( {1 - N} \right)\left( {1 + N - \alpha \left( {1 - N} \right)} \right)} \mathord{\left/ {\vphantom {{2\alpha \left( {1 - N} \right)\left( {1 + N - \alpha \left( {1 - N} \right)} \right)} {\left( {1 - \delta_{H} } \right)}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right)}} > 0$. Moreover, we also have $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p^{2} }}} \right. \kern-0pt} {\partial p^{2} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p\partial r_{H} }}} \right. \kern-0pt} {\partial p\partial r_{H} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial p\partial r_{L} }}} \right. \kern-0pt} {\partial p\partial r_{L} }}} \\ {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{H} \partial p}}} \right. \kern-0pt} {\partial r_{H} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\left( {\partial r_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{H} \partial r_{L} }}} \right. \kern-0pt} {\partial r_{H} \partial r_{L} }}} \\ {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{L} \partial p}}} \right. \kern-0pt} {\partial r_{L} \partial p}}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\partial r_{L} \partial r_{H} }}} \right. \kern-0pt} {\partial r_{L} \partial r_{H} }}} & {{{\partial^{2} \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{R}^{RYHH} } {\left( {\partial r_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial r_{L} } \right)^{2} }}} \\ \end{array} } \right| =$${{4\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } \mathord{\left/ {\vphantom {{4\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } {\left( {1 - \delta_{H} } \right) < 0}}} \right. \kern-0pt} {\left( {1 - \delta_{H} } \right) < 0}}$.Therefore, we can easily observe that Hessian matrix is negative definite and $\Pi_{R}^{RYHH}$ is a jointly concave function in $p$,$r_{H}$ and $r_{L}$. By solving first-order conditions ${{\partial \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{RYHH} } {\partial p}}} \right. \kern-0pt} {\partial p}} = 0$,${{\partial \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{RYHH} } {\partial r_{H} }}} \right. \kern-0pt} {\partial r_{H} }} = 0$,${{\partial \Pi_{R}^{RYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{RYHH} } {\partial r_{L} }}} \right. \kern-0pt} {\partial r_{L} }} = 0$, we can get $p^{RYHH * } \left( {w,b_{H} ,b_{L} } \right) = {{\left( {1 + w} \right)} \mathord{\left/ {\vphantom {{\left( {1 + w} \right)} 2}} \right. \kern-0pt} 2}$,$r_{H}^{RYHH * } \left( {w,b_{H} ,b_{L} } \right) =$ ${{\left( {\delta_{H} + b_{H} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{H} + b_{H} } \right)} 2}} \right. \kern-0pt} 2}$ and $r_{L}^{RYHH * } \left( {w,b_{H} ,b_{L} } \right) =$ ${{\left( {\delta_{L} + b_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{L} + b_{L} } \right)} 2}} \right. \kern-0pt} 2}$. Then substituting $p^{RYHH * } \left( {w,b_{H} ,b_{L} } \right)$,$r_{H}^{RYHH * } \left( {w,b_{H} ,b_{L} } \right)$ and $r_{L}^{RYHH * } (w,b_{H} ,b_{L} )$ into the manufacturer’s profit function Eq. (16), the Hessian matrix of the manufacturer’s profit $\Pi_{M}^{RYHH}$ with $w$,$b_{H}$,$b_{L}$ is $\left[ {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w\partial b_{H} }}} \right. \kern-0pt} {\partial w\partial b_{H} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w\partial b_{L} }}} \right. \kern-0pt} {\partial w\partial b_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{H} \partial w}}} \right. \kern-0pt} {\partial b_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\left( {\partial b_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{H} \partial b_{L} }}} \right. \kern-0pt} {\partial b_{H} \partial b_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{L} \partial w}}} \right. \kern-0pt} {\partial b_{L} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{L} \partial b_{H} }}} \right. \kern-0pt} {\partial b_{L} \partial b_{H} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\left( {\partial b_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{L} } \right)^{2} }}} \\ \end{array} } \right]$. We can have ${{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }} =$ ${{\left( {\left( {\alpha \left( {1 - N} \right) - N - 1} \right)\delta_{H} + \alpha \left( {1 - N} \right) + N + 1} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {\alpha \left( {1 - N} \right) - N - 1} \right)\delta_{H} + \alpha \left( {1 - N} \right) + N + 1} \right)} {2\left( {\delta_{H} - 1} \right)}}} \right. \kern-0pt} {2\left( {\delta_{H} - 1} \right)}} < 0$,and $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w\partial b_{H} }}} \right. \kern-0pt} {\partial w\partial b_{H} }}} \\ {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{H} \partial w}}} \right. \kern-0pt} {\partial b_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\left( {\partial b_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{H} } \right)^{2} }}} \\ \end{array} } \right| =$${{\alpha \left( {1 - N} \right)\left( {1 + N - \alpha \left( {1 - N} \right)} \right)} \mathord{\left/ {\vphantom {{\alpha \left( {1 - N} \right)\left( {1 + N - \alpha \left( {1 - N} \right)} \right)} {2\left( {1 - \delta_{H} } \right)}}} \right. \kern-0pt} {2\left( {1 - \delta_{H} } \right)}} > 0$. Moreover, we also have $\left| {\begin{array}{*{20}c} {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w\partial b_{H} }}} \right. \kern-0pt} {\partial w\partial b_{H} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial w\partial b_{L} }}} \right. \kern-0pt} {\partial w\partial b_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{H} \partial w}}} \right. \kern-0pt} {\partial b_{H} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\left( {\partial b_{H} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{H} } \right)^{2} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{H} \partial b_{L} }}} \right. \kern-0pt} {\partial b_{H} \partial b_{L} }}} \\ {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{L} \partial w}}} \right. \kern-0pt} {\partial b_{L} \partial w}}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\partial b_{L} \partial b_{H} }}} \right. \kern-0pt} {\partial b_{L} \partial b_{H} }}} & {{{\partial^{2} \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{M}^{RYHH} } {\left( {\partial b_{L} } \right)^{2} }}} \right. \kern-0pt} {\left( {\partial b_{L} } \right)^{2} }}} \\ \end{array} } \right| =$${{\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } \mathord{\left/ {\vphantom {{\left( {\alpha - 1} \right)\left( {N - 1} \right)^{2} N\alpha } {2\left( {1 - \delta_{H} } \right)}}} \right. \kern-0pt} {2\left( {1 - \delta_{H} } \right)}} < 0$. Therefore, we can easily observe that Hessian matrix is negative definite and $\Pi_{M}^{RYHH}$ is a jointly concave function in $w$,$b_{H}$,$b_{L}$. There is a unique optimal pricing solution for the manufacturer. By solving first-order conditions ${{\partial \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{RYHH} } {\partial w}}} \right. \kern-0pt} {\partial w}} = 0$,${{\partial \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{RYHH} } {\partial b_{H} }}} \right. \kern-0pt} {\partial b_{H} }} = 0$,${{\partial \Pi_{M}^{RYHH} } \mathord{\left/ {\vphantom {{\partial \Pi_{M}^{RYHH} } {\partial b_{L} }}} \right. \kern-0pt} {\partial b_{L} }} = 0$,we can get $w^{RYHH * }$,$b_{H}^{RYHH * }$ and $b_{L}^{RYHH * }$. Substituting $w^{RYHH * }$,$b_{H}^{RYHH * }$ and $b_{L}^{RYHH * }$ into the price reaction functions, we can have $p^{RYHH * }$,$r_{H}^{RYHH * }$ and $r_{L}^{RYHH * }$. All the optimal decisions of firms are shown in Table 4.

The processes of deriving equilibrium results in Model RYHL, RYLH, RYLL, RYCHH, RYCHL, RYCLH and RYCLL are similar to that in Model RYHH, thus we omit it.

Appendix C: proofs of the analysis

3.1 Proof of Theorem 1

For part (1), we can easily show that if $F_{M} \le T$, then $\Pi_{M}^{MT * } - \Pi_{M}^{RT * } \ge 0$, where

$$\begin{gathered} T = \frac{1}{{8\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha - \left( {\delta_{H} - 1} \right)\left( {N\delta_{L} - 1} \right)} \right)}}\left( {\left( {N - 1} \right)\left( {\left( {s_{H} - s_{L} - \delta_{H} + \delta_{L} } \right)^{2} \left( {N - 1} \right)\alpha^{2} } \right.} \right. + \left( {\left( {\left( {\delta_{H} - 2} \right)\delta_{L}^{2} } \right.} \right. + \hfill \\ \left( { - \delta_{H}^{2} + \left( {2s_{H} - 2s_{L} + 2} \right)\delta_{H} - s_{H}^{2} - 2s_{H} + 4s_{L} } \right)\delta_{L} + \left( {\left( {s_{L} - 2} \right)\delta_{H} + 2s_{H} } \right.\left. {\left. {\left. { - 2s_{L} } \right)s_{L} } \right)N + \left( {s_{H} - s_{L} - \delta_{H} + \delta_{L} } \right)^{2} } \right)\alpha \hfill \\ \left. {\left. { - N\left( {s_{L} - \delta_{L} } \right)^{2} \left( {\delta_{H} - 1} \right)} \right)} \right) \hfill \\ \end{gathered}$$

. For part (2), we can easily show that if $F_{R} \le {T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RT * } - \Pi_{R}^{MT * } \ge 0$.

3.2 Proof of Theorem 2

For part (1), we can easily show that if $F_{M} \le T_{HH}$, then $\Pi_{M}^{MYHH * } - \Pi_{M}^{RYHH * } \ge 0$, where

$$\begin{gathered} T_{HH} = \frac{1}{{8\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha - \left( {\delta_{H} - 1} \right)\left( {N\delta_{L} - 1} \right)} \right)}}\left( {\left( {N - 1} \right)\left( {\left( {s_{H} - s_{L} - \delta_{H} + \delta_{L} } \right)^{2} \left( {N - 1} \right)\alpha^{2} } \right.} \right. + \left( {\left( {\left( {\delta_{H} - 2} \right)\delta_{L}^{2} } \right.} \right. + \hfill \\ \left( { - \delta_{H}^{2} + \left( {2s_{H} - 2s_{L} + 2} \right)\delta_{H} - s_{H}^{2} - 2s_{H} + 4s_{L} } \right)\delta_{L} + \left( {\left( {s_{L} - 2} \right)\delta_{H} + 2s_{H} } \right.\left. {\left. {\left. { - 2s_{L} } \right)s_{L} } \right)N + \left( {s_{H} - s_{L} - \delta_{H} + \delta_{L} } \right)^{2} } \right)\alpha \hfill \\ \left. {\left. { - N\left( {s_{L} - \delta_{L} } \right)^{2} \left( {\delta_{H} - 1} \right)} \right)} \right) \hfill \\ \end{gathered};$$

we also can easily show that if $F_{M} \le T_{HL}$, then $\Pi_{M}^{MYHL * } - \Pi_{M}^{RYHL * } \ge 0$, where

$$\begin{gathered} T_{HL} = \frac{1}{{8\left( {\left( {\delta_{H} + 1} \right)\left( {N - 1} \right)\alpha + \left( {\delta_{H} - 1} \right)\left( {1 + N} \right)} \right)}}\left( {\left( {N - 1} \right)\left( {\left( {c_{r} - s_{H} + \delta_{H} + 1} \right)^{2} \left( {N - 1} \right)\alpha^{2} - N\left( {c_{r} + 1} \right)^{2} \left( {\delta_{H} - 1} \right)} \right.} \right. \hfill \\ + \left. {\left( {\left( {\delta_{H}^{2} + \left( {c_{r}^{2} - 2s_{H} - 1} \right)\delta_{H} - 2c_{r}^{2} + \left( {2s_{H} - 4} \right)c_{r} + s_{H}^{2} + 2s_{H} - 2} \right)N + \left( {c_{r} - s_{H} + \delta_{H} + 1} \right)^{2} } \right)\alpha } \right) \hfill \\ \end{gathered};$$

we can easily show that if $F_{M} \le T_{LH}$, then $\Pi_{M}^{MYLH * } - \Pi_{M}^{RYLH * } \ge 0$, where

$$\begin{gathered} T_{LH} = \frac{1}{{8\left( {1 + \delta_{L} } \right)\left( {N - 1} \right)\alpha - 16\left( {N\delta_{L} - 1} \right)}}\left( {\left( {1 - N} \right)\left( {\left( {c_{r} - s_{L} + \delta_{L} + 1} \right)^{2} \left( {N - 1} \right)\alpha } \right.} \right. + 2N\left( {s_{L} - \delta_{L} } \right)^{2} + \left( {\left( { - 3\delta_{L}^{2} + } \right.} \right. \hfill \\ \left. {\left. {\left. {\left( { - c_{r}^{2} - 4c_{r} + 6s_{L} - 3} \right)\delta_{L} + s_{L} \left( {2c_{r} - 3s_{L} + 2} \right)} \right)N + \left( {c_{r} - s_{L} + \delta_{L} + 1} \right)^{2} } \right)\alpha } \right) \hfill \\ \end{gathered};$$

we also can easily show that if $F_{M} \le T_{LL}$, then $\Pi_{M}^{MYLL * } - \Pi_{M}^{RYLL * } \ge 0$, where $T_{LL} = {{N\left( {c_{r} + 1} \right)^{2} \left( {1 - N} \right)} \mathord{\left/ {\vphantom {{N\left( {c_{r} + 1} \right)^{2} \left( {1 - N} \right)} {8\left( {N + 1} \right)}}} \right. \kern-0pt} {8\left( {N + 1} \right)}}$. For part (2), we can easily show that if $F_{R} \le {{T_{HH} } \mathord{\left/ {\vphantom {{T_{HH} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RYHH * } - \Pi_{R}^{MYHH * } \ge 0$; we also can easily show that if $F_{R} \le {{T_{HL} } \mathord{\left/ {\vphantom {{T_{HL} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RYHL * } - \Pi_{R}^{MYHL * } \ge 0$; we can easily show that if $F_{R} \le {{T_{LH} } \mathord{\left/ {\vphantom {{T_{LH} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RYLH * } - \Pi_{R}^{MYLH * } \ge 0$; we also can easily show that if $F_{R} \le {{T_{LL} } \mathord{\left/ {\vphantom {{T_{LL} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RYLL * } - \Pi_{R}^{MYLL * } \ge 0$. By analyzing the function image, we can show that $T$,$T_{HH}$,$T_{HL},$ $T_{LH}$ and $T_{LL}$ are always greater than zero.

3.3 Proof of Corollary 1

It is not difficult to check that $T = T_{HH}$.

3.4 Proof of Proposition 1

For part (1), it is not difficult to check that $w^{MT * } = w^{MYHH * } = w^{MYHL * } = w^{MYLH * } = w^{MYLL * }$ and $p^{MT * } = p^{MYHH * }$. For part (2), first, under Model MYHH, the condition that $0 < \delta_{L} < \delta_{L1}^{ * }$ needs to be met, where $\delta_{L1}^{ * } = {{g_{L}^{MYHL * } } \mathord{\left/ {\vphantom {{g_{L}^{MYHL * } } {\left( {p^{MYHL * } - r_{L}^{MYHL * } + g_{L}^{MYHL * } } \right)}}} \right. \kern-0pt} {\left( {p^{MYHL * } - r_{L}^{MYHL * } + g_{L}^{MYHL * } } \right)}} =$

$$\begin{gathered} \frac{1}{{\left( {\left( {c - c_{r} + 4} \right)\delta_{H} + c - 3c_{r} + 2s_{H} + 4} \right)\left( {N - 1} \right)\alpha + \left( {\left( {c - 3c_{r} + 4} \right)N + c - c_{r} + 6} \right)\left( {\delta_{H} - 1} \right)}}\left( {\left( {1 - N} \right)\alpha \left( {\left( {c + 7c_{r} + 4} \right.} \right.} \right. \hfill \\ \left. {\left. {\left. { - 8s_{L} } \right)\delta_{H} + c + 5c_{r} + 2s_{H} - 8s_{L} + 4} \right) + \left( {1 - \delta_{H} } \right)\left( {\left( {c + 5c_{r} - 8s_{L} + 4} \right)N + c + 7c_{r} - 8s_{L} + 6} \right)} \right) \hfill \\ \end{gathered}.$$

We can easily show that if $0 < \delta_{L} < \delta_{L1}$,then $p^{MT * } > p^{MYHL * }$, where $\delta_{L1} = {{\left( {\left( {N - 1} \right)\alpha \left( {\left( {c_{r} - s_{L} } \right)\delta_{H} + s_{H} - s_{L} } \right) + \left( {1 - \delta_{H} } \right)\left( {\left( {N + 1} \right)s_{L} - c_{r} - 1} \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {N - 1} \right)\alpha \left( {\left( {c_{r} - s_{L} } \right)\delta_{H} + s_{H} - s_{L} } \right) + \left( {1 - \delta_{H} } \right)\left( {\left( {N + 1} \right)s_{L} - c_{r} - 1} \right)} \right)} {\left( {\left( {s_{H} - c_{r} } \right)\left( {1 - N} \right)\alpha + \left( {1 - \delta_{H} } \right)\left( {1 - c_{r} N} \right)} \right)}}} \right. \kern-0pt} {\left( {\left( {s_{H} - c_{r} } \right)\left( {1 - N} \right)\alpha + \left( {1 - \delta_{H} } \right)\left( {1 - c_{r} N} \right)} \right)}}$.After proof, we found that $\delta_{L1} \le \delta_{L1}^{ * }$. Therefore, We can easily show that if $\delta_{L1} \le \delta_{L} < \delta_{L1}^{ * }$, then $p^{MT * } \le p^{{MY_{12} * }}$. Second, under Model MYLH, this condition that $0 < \delta_{H} < \delta_{H1}^{ * }$ needs to be met, where $\delta_{H1}^{ * } = {{g_{H}^{MYLH * } } \mathord{\left/ {\vphantom {{g_{H}^{MYLH * } } {\left( {p^{MYLH * } - r_{H}^{MYLH * } + g_{H}^{MYLH * } } \right)}}} \right. \kern-0pt} {\left( {p^{MYLH * } - r_{H}^{MYLH * } + g_{H}^{MYLH * } } \right)}} =$

$$\begin{gathered} \frac{1}{{\left( {\left( {c - c_{r} + 4} \right)\delta_{L} + c - 3c_{r} + 2s_{L} + 4} \right)(N - 1)\alpha + \left( {\left( {4c_{r} - 2c - 8} \right)\delta_{L} - 2s_{L} } \right)N + 2\left( {c - 2c_{r} + s_{L} - \delta_{L} + 5} \right)}}\left( {\alpha \left( {1 - } \right.} \right. \hfill \\ \left. N \right)\left( {\left( {c + 7c_{r} - 8s_{L} + 4} \right)\delta_{L} + c + 5c_{r} + 2s_{L} - 8s_{H} + 4} \right) + \left( {\left( {2c + 12c_{r} - 16s_{H} + 8} \right)\delta_{L} + 2s_{L} } \right)N - 2c - 12c_{r} - 10 \hfill \\ \left. { - 2s_{L} + 2\delta_{L} } \right) \hfill \\ \end{gathered}.$$

We can easily show that if $0 < \delta_{H} < \delta_{H1}$, then $p^{MT * } > p^{MYLH * }$, where

$$\begin{gathered} \delta_{H1} = \frac{1}{{\left( {s_{L} - c_{r} } \right)\left( {1 - N} \right)\alpha + \left( {s_{L} - c_{r} \delta_{L} } \right)N + \delta_{L} + c_{r} - s_{L} - 1}}\left( {\left( {N - 1} \right)\alpha \left( {\left( {c_{r} - s_{H} } \right)\delta_{L} + s_{L} - s_{H} } \right) + \left( {\left( { - c_{r} + 2s_{H} } \right)\delta_{L} } \right.} \right. \hfill \\ \left. {\left. { - s_{L} } \right)N + c_{r} - 2s_{H} + s_{L} - \delta_{L} + 1} \right) \hfill \\ \end{gathered}.$$

After proof, we found that $\delta_{H1} \le \delta_{H1}^{ * }$. Therefore, We can easily show that if $\delta_{H1} \le \delta_{H} < \delta_{H1}^{ * }$, then $p^{MT * } \le p^{MYLH * }$. For part (3), first, under Model MYLL, the conditions that $0 < \delta_{L} < \delta_{L2}^{ * }$ and $0 < \delta_{H} < \delta_{H2}^{ * }$ need to be met, where $\delta_{L2}^{ * } = {{g_{L}^{MYLL * } } \mathord{\left/ {\vphantom {{g_{L}^{MYLL * } } {\left( {p^{MYLL * } - r_{L}^{MYLL * } + g_{L}^{MYLL * } } \right)}}} \right. \kern-0pt} {\left( {p^{MYLL * } - r_{L}^{MYLL * } + g_{L}^{MYLL * } } \right)}} = {{\left( {\left( {8s_{L} - c - 5c_{r} - 4} \right)N - c - 7c_{r} + 8s_{L} - 6} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {8s_{L} - c - 5c_{r} - 4} \right)N - c - 7c_{r} + 8s_{L} - 6} \right)} {\left( {\left( {c - 3c_{r} + 4} \right)N + c - c_{r} + 6} \right)}}} \right. \kern-0pt} {\left( {\left( {c - 3c_{r} + 4} \right)N + c - c_{r} + 6} \right)}}$ and $\delta_{H2}^{ * } = {{g_{H}^{MYLL * } } \mathord{\left/ {\vphantom {{g_{H}^{MYLL * } } {\left( {p^{MYLL * } - r_{H}^{MYLL * } + g_{H}^{MYLL * } } \right)}}} \right. \kern-0pt} {\left( {p^{MYLL * } - r_{H}^{MYLL * } + g_{H}^{MYLL * } } \right)}} = {{\left( {\left( {8s_{H} - c - 5c_{r} - 4} \right)N - c - 7c_{r} + 8s_{H} - 6} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {8s_{H} - c - 5c_{r} - 4} \right)N - c - 7c_{r} + 8s_{H} - 6} \right)} {\left( {\left( {c - 3c_{r} + 4} \right)N + c - c_{r} + 6} \right)}}} \right. \kern-0pt} {\left( {\left( {c - 3c_{r} + 4} \right)N + c - c_{r} + 6} \right)}}$. We can easily show that if $0 < \delta_{L} < \delta_{L2}$ and $0 < \delta_{H} < \delta_{H2}$ satisfied at the same time, or if $\delta_{L2} \le \delta_{L} < \delta_{L2}^{ * }$ and $\delta_{H2} \le \delta_{H} < \delta_{H2}^{ * }$ satisfied at the same time, then $p^{MT * } > p^{MYLL * }$, otherwise,$p^{MT * } \le p^{MYLL * }$. Where $\delta_{L2} = {{\left( {\left( {\left( {N - 1} \right)c_{r} - 2 + \left( {N + 1} \right)s_{L} } \right)\alpha + c_{r} + 1 - \left( {N + 1} \right)s_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {\left( {\left( {N - 1} \right)c_{r} - 2 + \left( {N + 1} \right)s_{L} } \right)\alpha + c_{r} + 1 - \left( {N + 1} \right)s_{L} } \right)} {\left( {c_{r} N - 1} \right)}}} \right. \kern-0pt} {\left( {c_{r} N - 1} \right)}}$ and

$$\begin{gathered} \delta_{H2} = \frac{1}{{\left( {\left( {c_{r} + s_{L} } \right)N - c_{r} + s_{L} - 2} \right)\alpha - \left( {c_{r} \delta_{L} + s_{L} } \right)N + c_{r} - s_{L} + \delta_{L} + 1}}\left( {\left( {\left( {\left( {c_{r} + s_{H} } \right)\delta_{L} - s_{H} + s_{L} } \right)N + \left( {s_{H} - c_{r} } \right.} \right.} \right. \hfill \\ \left. {\left. {\left. { - 2} \right)\delta_{L} - s_{H} + s_{L} } \right)\alpha - \left( {c_{r} \delta_{L} + s_{L} } \right)N + c_{r} - s_{L} + \delta_{L} + 1} \right) \hfill \\ \end{gathered}.$$

After proof, we found that $\delta_{L2} \le \delta_{L2}^{ * }$ and $\delta_{H2} \le \delta_{H2}^{ * }$. For part (4), it is easy to prove that $r_{H}^{MT * } = r_{H}^{MYHH * } = r_{H}^{MYHL * }$ and $r_{H}^{MYLH * } = r_{H}^{MYLL * }$, it is not difficult to check that $r_{H}^{MYHL * } - r_{H}^{MYLH * } = {{\left( {\delta_{H} - s_{H} + 1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{H} - s_{H} + 1 + c_{r} } \right)} 2}} \right. \kern-0pt} 2} > 0$. It is also easy to prove that $r_{L}^{MT * } = r_{L}^{MYHH * } = r_{L}^{MYLH * }$ and $r_{L}^{MYHL * } = r_{L}^{MYLL * }$, it is also not difficult to check that $r_{L}^{MYLH * } - r_{L}^{MYHL * } = {{\left( {\delta_{L} - s_{L} + 1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{L} - s_{L} + 1 + c_{r} } \right)} 2}} \right. \kern-0pt} 2} > 0$.

3.5 Proof of Proposition 2

For part (1), it is easy to prove that $w^{RT * } = w^{RYHH * } = w^{RYHL * } = w^{RYLH * } = w^{RYLL * }$ and $p^{RT * } = p^{RYHH * } = p^{RYHL * } = p^{RYLH * } = p^{RYLL * }$. For part (2), first, it is easy to prove that $b_{H}^{RT * } = b_{H}^{RYHH * } = b_{H}^{RYHL * }$ and $b_{H}^{RYLH * } = b_{H}^{RYLL * }$, it is not difficult to check that $b_{H}^{RYHL * } - b_{H}^{RYLH * } = {{\left( {\delta_{H} - s_{H} + 1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{H} - s_{H} + 1 + c_{r} } \right)} 2}} \right. \kern-0pt} 2} > 0$. Similarly, it is easy to prove that $r_{H}^{RT * } = r_{H}^{RYHH * } = r_{H}^{RYHL * }$ and $r_{H}^{RYLH * } = r_{H}^{RYLL * }$, it is not difficult to check that $r_{H}^{RYHL * } - r_{H}^{RYLH * } = {{3\left( {\delta_{H} - s_{H} + 1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{3\left( {\delta_{H} - s_{H} + 1 + c_{r} } \right)} 4}} \right. \kern-0pt} 4} > 0$. Second, it is easy to prove that $b_{L}^{RT * } = b_{L}^{RYHH * } = b_{L}^{RYLH * }$ and $b_{L}^{RYHL * } = b_{L}^{RYLL * }$, it is not difficult to check that $b_{L}^{RYLH * } - b_{L}^{RYHL * } = {{\left( {\delta_{L} - s_{L} + 1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{L} - s_{L} + 1 + c_{r} } \right)} 2}} \right. \kern-0pt} 2} > 0$. Similarly, it is easy to prove that $r_{L}^{RT * } = r_{L}^{RYHH * } = r_{L}^{RYLH * }$ and $r_{L}^{RYHL * } = r_{L}^{RYLL * }$, it is not difficult to check that $r_{L}^{RYLH * } - r_{L}^{RYHL * } = {{3\left( {\delta_{L} - s_{L} + 1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{3\left( {\delta_{L} - s_{L} + 1 + c_{r} } \right)} 4}} \right. \kern-0pt} 4} > 0$.

3.6 Proof of Corollary 2

For part (1), since $s_{H} - r_{H}^{MYHj * } = {{\left( {s_{H} - \delta_{H} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{H} - \delta_{H} } \right)} 2}} \right. \kern-0pt} 2}$, it is clear that when $s_{H} > \delta_{H}$, we have $s_{H} > r_{H}^{MYHj * }$, otherwise, we have $s_{H} \le r_{H}^{MYHj * }$. Since $s_{L} - r_{L}^{MYiH * } = {{\left( {s_{L} - \delta_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{L} - \delta_{L} } \right)} 2}} \right. \kern-0pt} 2}$, it is clear that when $s_{L} > \delta_{L}$, we have $s_{L} > r_{L}^{MYiH * }$, otherwise, we have $s_{L} \le r_{L}^{MYiH * }$. For part (2), it is easy to show that $s_{H} - r_{H}^{MYLj * } = {{\left( {1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {1 + c_{r} } \right)} 2}} \right. \kern-0pt} 2} > 0$, therefore, $s_{H} > r_{H}^{MYLj * }$ always holds. Similarly, it is easy to show that $s_{L} - r_{L}^{MYiL * } = {{\left( {1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {1 + c_{r} } \right)} 2}} \right. \kern-0pt} 2} > 0$, and therefore $s_{L} > r_{L}^{MYiL * }$ also always holds.

3.7 Proof of Corollary 3

For part (1), since $s_{H} - b_{H}^{RYHj * } = {{\left( {s_{H} - \delta_{H} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{H} - \delta_{H} } \right)} 2}} \right. \kern-0pt} 2}$, $s_{H} - r_{H}^{RYHj * } = {{3\left( {s_{H} - \delta_{H} } \right)} \mathord{\left/ {\vphantom {{3\left( {s_{H} - \delta_{H} } \right)} 4}} \right. \kern-0pt} 4}$ and $b_{H}^{RYHj * } - r_{H}^{RYHj * } = {{\left( {s_{H} - \delta_{H} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{H} - \delta_{H} } \right)} 4}} \right. \kern-0pt} 4}$, obviously, we have $s_{H} > b_{H}^{RYHj * } > r_{H}^{RYHj * }$ if $\delta_{H} < s_{H}$ and $r_{H}^{RYHj * } \ge b_{H}^{RYHj * } \ge s_{H}$ if $\delta_{H} \ge s_{H}$. Similarly, since $s_{L} - b_{L}^{RYiH * } = {{\left( {s_{L} - \delta_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{L} - \delta_{L} } \right)} 2}} \right. \kern-0pt} 2}$, $s_{L} - r_{L}^{RYiH * } = {{3\left( {s_{L} - \delta_{L} } \right)} \mathord{\left/ {\vphantom {{3\left( {s_{L} - \delta_{L} } \right)} 4}} \right. \kern-0pt} 4}$ and $b_{L}^{RYiH * } - r_{L}^{RYiH * } = {{\left( {s_{L} - \delta_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {s_{L} - \delta_{L} } \right)} 4}} \right. \kern-0pt} 4}$, obviously, we have $s_{L} > b_{L}^{RYiH * } > r_{L}^{RYiH * }$ if $\delta_{L} < s_{L}$ and $r_{L}^{RYiH * } \ge b_{L}^{RYiH * } \ge s_{L}$ if $\delta_{L} \ge s_{L}$. For part (2), it is easy to show that $s_{H} - b_{H}^{RYLj * } = {{\left( {1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {1 + c_{r} } \right)} 2}} \right. \kern-0pt} 2} > 0$ and $b_{H}^{RYLj * } - r_{H}^{RYLj * } = {{\left( {1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {1 + c_{r} } \right)} 4}} \right. \kern-0pt} 4} > 0$, therefore, $s_{H} > b_{H}^{RYLj * } > r_{H}^{RYLj * }$ always holds. Similarly, it is easy to show that $s_{L} - b_{L}^{RYiL * } = {{\left( {1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {1 + c_{r} } \right)} 2}} \right. \kern-0pt} 2} > 0$ and $b_{L}^{RYiL * } - r_{L}^{RYiL * } = {{\left( {1 + c_{r} } \right)} \mathord{\left/ {\vphantom {{\left( {1 + c_{r} } \right)} 4}} \right. \kern-0pt} 4} > 0$, and therefore $s_{L} > b_{L}^{RYiL * } > r_{L}^{RYiL * }$ also always holds.

3.8 Proof of Proposition 3

For part (1), it is not difficult to check that $q_{n}^{MT * } = q_{n}^{MYHH * }$, $q_{rn}^{{{\rm I}MT * }} = q_{rn}^{{{\rm I}MYHH * }}$ and $q_{rn}^{{{\rm I}{\rm I}MT * }} = q_{rn}^{{{\rm I}{\rm I}MYHH * }}$. For part (2), first, under Model MYHH, the condition that $0 < \delta_{L} < \delta_{L1}^{ * }$ needs to be met. We can easily show that if $0 < \delta_{L} < \delta_{L1}$,then $q_{n}^{MT * } < q_{n}^{MYHL * }$ and $q_{rn}^{{{\rm I}MT * }} < q_{rn}^{{{\rm I}MYHL * }}$, and if $\delta_{L1} \le \delta_{L} < \delta_{L1}^{ * }$, then $q_{n}^{MT * } \ge q_{n}^{MYHL * }$ and $q_{rn}^{{{\rm I}MT * }} \ge q_{rn}^{{{\rm I}MYHL * }}$. Second, under Model MYLH, this condition that $0 < \delta_{H} < \delta_{H1}^{ * }$ needs to be met. We can easily show that if $0 < \delta_{H} < \delta_{H1}$, then $q_{n}^{MT * } < q_{n}^{MYLH * }$ and $q_{rn}^{{{\rm I}{\rm I}MT * }} < q_{rn}^{{{\rm I}{\rm I}MYLH * }}$, and if $\delta_{H1} \le \delta_{H} < \delta_{H1}^{ * }$, then $q_{n}^{MT * } \ge q_{n}^{MYLH * }$ and $q_{rn}^{{{\rm I}{\rm I}MT * }} \ge q_{rn}^{{{\rm I}{\rm I}MYLH * }}$. For part (3), first, under Model MYLL, the conditions that $0 < \delta_{L} < \delta_{L2}^{ * }$ and $0 < \delta_{H} < \delta_{H2}^{ * }$ need to be met. We can easily show that if $0 < \delta_{L} < \delta_{L2}$ and $0 < \delta_{H} < \delta_{H2}$ satisfied at the same time, or if $\delta_{L2} \le \delta_{L} < \delta_{L2}^{ * }$ and $\delta_{H2} \le \delta_{H} < \delta_{H2}^{ * }$ satisfied at the same time, then $q_{n}^{MT * } < q_{n}^{MYLL * }$,otherwise, $q_{n}^{MT * } \ge q_{n}^{MYLL * }$. For part (4), by analyzing the image of the functions $q_{rn}^{{{\rm I}MT * }} - q_{rn}^{{{\rm I}MYLH * }}$,$q_{rn}^{{{\rm I}MT * }} - q_{rn}^{{{\rm I}MYLL * }}$,$q_{rn}^{{{\rm I}{\rm I}MT * }} - q_{rn}^{{{\rm I}{\rm I}MYHL * }}$ and $q_{rn}^{{{\rm I}{\rm I}MT * }} - q_{rn}^{{{\rm I}{\rm I}MYLL * }}$, we can see that $q_{rn}^{{{\rm I}MT * }} > q_{rn}^{{{\rm I}MYLj * }}$ and $q_{rn}^{{{\rm I}{\rm I}MT * }} > q_{rn}^{{{\rm I}{\rm I}MYiL * }}$ always holds.

3.9 Proof of Proposition 4

For part (1), it is not difficult to check that $q_{n}^{RT * } = q_{n}^{RYHH * } = q_{n}^{RYHL * } = q_{n}^{RYLH * } = q_{n}^{RYLL * }$. For part (2), it is easy to prove that $q_{rn}^{{{\rm I}RT * }} = q_{rn}^{{{\rm I}RYHH * }} = q_{rn}^{{{\rm I}RYHL * }}$ and $q_{rn}^{{{\rm I}RYLH * }} = q_{rn}^{{{\rm I}RYLL * }}$, it is not difficult to check that $q_{rn}^{{{\rm I}RYHj * }} - q_{rn}^{{{\rm I}RYLj * }} = {{\alpha \left( {1 - N} \right)\left( {2s_{H} - c - c_{r} - \left( {c - c_{r} } \right)\delta_{H} } \right)} \mathord{\left/ {\vphantom {{\alpha \left( {1 - N} \right)\left( {2s_{H} - c - c_{r} - \left( {c - c_{r} } \right)\delta_{H} } \right)} {8\left( {1 - \delta_{H} } \right)}}} \right. \kern-0pt} {8\left( {1 - \delta_{H} } \right)}} > 0$. Similarly, it is easy to prove that $q_{rn}^{{{\rm I}{\rm I}RT * }} = q_{rn}^{{{\rm I}{\rm I}RYHH * }} = q_{rn}^{{{\rm I}{\rm I}RYLH * }}$ and $q_{rn}^{{{\rm I}{\rm I}RYHL * }} = q_{rn}^{{{\rm I}{\rm I}RYLL * }}$, it is not difficult to check that $q_{rn}^{{{\rm I}{\rm I}RYiH * }} - q_{rn}^{{{\rm I}RYiL * }} = {{\left( {1 - \alpha } \right)\left( {1 - N} \right)\left( {2s_{L} - c - c_{r} - \left( {c - c_{r} } \right)\delta_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \alpha } \right)\left( {1 - N} \right)\left( {2s_{L} - c - c_{r} - \left( {c - c_{r} } \right)\delta_{L} } \right)} {8\left( {1 - \delta_{L} } \right)}}} \right. \kern-0pt} {8\left( {1 - \delta_{L} } \right)}} > 0$.

3.10 Proof of Property 1

It is not difficult to check that $\Pi_{M}^{MT * } = \Pi_{M}^{MYHH * }$, $\Pi_{R}^{MT * } = \Pi_{R}^{MYHH * }$.

3.11 Proof of Theorem 3

For part (1), it is not difficult to check that $\Pi_{M}^{RT * } = \Pi_{M}^{RYHH * }$ and $\Pi_{R}^{RT * } = \Pi_{R}^{RYHH * }$. For part (2), by analyzing the image of the functions $\Pi_{M}^{RT * } - \Pi_{M}^{RYHL * }$, $\Pi_{M}^{RT * } - \Pi_{M}^{RYLH * }$ and $\Pi_{M}^{RT * } - \Pi_{M}^{RYLL * }$, we can see that $\Pi_{M}^{RT * } - \Pi_{M}^{RYLj * }$ and $\Pi_{M}^{RT * } - \Pi_{M}^{RYiL * }$ always holds. Similarly, by analyzing the image of the functions $\Pi_{R}^{RT * } - \Pi_{R}^{RYHL * }$, $\Pi_{R}^{RT * } - \Pi_{R}^{RYLH * }$ and $\Pi_{R}^{RT * } - \Pi_{R}^{RYLL * }$, we can see that $\Pi_{R}^{RT * } - \Pi_{R}^{RYLj * }$ and $\Pi_{R}^{RT * } - \Pi_{R}^{RYiL * }$ always holds.

3.12 Proof of Proposition 5

According to the optimal decisions in Table 2, $r_{H}^{MT * } = {{\left( {\delta_{H} + s_{H} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{H} + s_{H} } \right)} 2}} \right. \kern-0pt} 2}$, $r_{L}^{MT * } = {{\left( {\delta_{L} + s_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{L} + s_{L} } \right)} 2}} \right. \kern-0pt} 2}$, $r_{H}^{RT * } = {{\left( {3\delta_{H} + s_{H} } \right)} \mathord{\left/ {\vphantom {{\left( {3\delta_{H} + s_{H} } \right)} 4}} \right. \kern-0pt} 4}$, $r_{L}^{RT * } = {{\left( {3\delta_{L} + s_{L} } \right)} \mathord{\left/ {\vphantom {{\left( {3\delta_{L} + s_{L} } \right)} 4}} \right. \kern-0pt} 4}$, it is not difficult to find that both $r_{H}^{MT * }$ ($r_{L}^{MT * }$) and $r_{H}^{RT * }$ ($r_{L}^{RT * }$) increasing in $s_{H}$ ($s_{L}$) and $\delta_{H}$ ($\delta_{L}$).

3.13 Proof of Proposition 6

The proof of Proposition 6 is similar to that of Proposition 5, thus we omit it.

3.14 Proof of Theorem 4

For part (1), we can easily show that if $F_{M} \le T^{C}$, then $\Pi_{M}^{MTC * } - \Pi_{M}^{RTC * } \ge 0$, where

$$\begin{gathered} T^{C} = \frac{1}{{8\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha - \left( {\delta_{H} - 1} \right)\left( {N\delta_{L} - 1} \right)} \right)}}\left( {\left( {N - 1} \right)\left( {\left( {s_{H} - s_{L} - \delta_{H} + \delta_{L} } \right)^{2} \left( {N - 1} \right)\alpha^{2} } \right.} \right. + \left( {\left( {\left( {\delta_{H} - 2} \right)\delta_{L}^{2} } \right.} \right. + \hfill \\ \left( { - \delta_{H}^{2} + \left( {2s_{H} - 2s_{L} + 2} \right)\delta_{H} - s_{H}^{2} - 2s_{H} + 4s_{L} } \right)\delta_{L} + \left( {\left( {s_{L} - 2} \right)\delta_{H} + 2s_{H} } \right.\left. {\left. {\left. { - 2s_{L} } \right)s_{L} } \right)N + \left( {s_{H} - s_{L} - \delta_{H} + \delta_{L} } \right)^{2} } \right)\alpha \hfill \\ \left. {\left. { - N\left( {s_{L} - \delta_{L} } \right)^{2} \left( {\delta_{H} - 1} \right)} \right)} \right) \hfill \\ \end{gathered}.$$

For part (2), we can easily show that if $F_{R} \le {{T^{C} } \mathord{\left/ {\vphantom {{T^{C} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RTC * } - \Pi_{R}^{MTC * } \ge 0$.

3.15 Proof of Corollary 4

It is not difficult to check that $T = T^{C}$.

3.16 Proof of Theorem 5

For part (1), we can easily show that if $F_{M} \le T_{HH}^{C}$, then $\Pi_{M}^{MYCHH * } - \Pi_{M}^{RYCHH * } \ge 0$, where

$$\begin{gathered} T_{HH}^{C} = \frac{1}{{8\left( {\left( {\delta_{H} - \delta_{L} } \right)\left( {N - 1} \right)\alpha - \left( {\delta_{H} - 1} \right)\left( {N\delta_{L} - 1} \right)} \right)}}\left( {\left( {N - 1} \right)\left( {\left( {s_{H} - s_{L} - \delta_{H} + \delta_{L} } \right)^{2} \left( {N - 1} \right)\alpha^{2} } \right.} \right. + \left( {\left( {\left( {\delta_{H} - 2} \right)\delta_{L}^{2} } \right.} \right. + \hfill \\ \left( { - \delta_{H}^{2} + \left( {2s_{H} - 2s_{L} + 2} \right)\delta_{H} - s_{H}^{2} - 2s_{H} + 4s_{L} } \right)\delta_{L} + \left( {\left( {s_{L} - 2} \right)\delta_{H} + 2s_{H} } \right.\left. {\left. {\left. { - 2s_{L} } \right)s_{L} } \right)N + \left( {s_{H} - s_{L} - \delta_{H} + \delta_{L} } \right)^{2} } \right)\alpha \hfill \\ \left. {\left. { - N\left( {s_{L} - \delta_{L} } \right)^{2} \left( {\delta_{H} - 1} \right)} \right)} \right) \hfill \\ \end{gathered};$$

we also can easily show that if $F_{M} \le T_{HL}^{C}$, then $\Pi_{M}^{MYCHL * } - \Pi_{M}^{RYCHL * } \ge 0$, where

$$\begin{gathered} T_{HL}^{C} = \frac{1}{{8\left( {\left( {\delta_{H} + 1} \right)\left( {N - 1} \right)\alpha + \left( {\delta_{H} - 1} \right)\left( {1 + N} \right)} \right)}}\left( {\left( {N - 1} \right)\left( {\left( {c_{r} - s_{H} + \delta_{H} + 1} \right)^{2} \left( {N - 1} \right)\alpha^{2} - N\left( {c_{r} + 1} \right)^{2} \left( {\delta_{H} - 1} \right)} \right.} \right. \hfill \\ + \left. {\left( {\left( {\delta_{H}^{2} + \left( {c_{r}^{2} - 2s_{H} - 1} \right)\delta_{H} - 2c_{r}^{2} + \left( {2s_{H} - 4} \right)c_{r} + s_{H}^{2} + 2s_{H} - 2} \right)N + \left( {c_{r} - s_{H} + \delta_{H} + 1} \right)^{2} } \right)\alpha } \right) \hfill \\ \end{gathered};$$

we can easily show that if $F_{M} \le T_{LH}^{C}$, then $\Pi_{M}^{MYCLH * } - \Pi_{M}^{RYCLH * } \ge 0$,where

$$\begin{gathered} T_{LH}^{C} = \frac{1}{{8\left( {1 + \delta_{L} } \right)\left( {N - 1} \right)\alpha - 16\left( {N\delta_{L} - 1} \right)}}\left( {\left( {1 - N} \right)\left( {\left( {c_{r} - s_{L} + \delta_{L} + 1} \right)^{2} \left( {N - 1} \right)\alpha } \right.} \right. + 2N\left( {s_{L} - \delta_{L} } \right)^{2} + \left( {\left( { - 3\delta_{L}^{2} + } \right.} \right. \hfill \\ \left. {\left. {\left. {\left( { - c_{r}^{2} - 4c_{r} + 6s_{L} - 3} \right)\delta_{L} + s_{L} \left( {2c_{r} - 3s_{L} + 2} \right)} \right)N + \left( {c_{r} - s_{L} + \delta_{L} + 1} \right)^{2} } \right)\alpha } \right) \hfill \\ \end{gathered};$$

we also can easily show that if $F_{M} \le T_{LL}^{C}$, then $\Pi_{M}^{MYCLL * } - \Pi_{M}^{RYCLL * } \ge 0$,where $T_{LL}^{C} = {{N\left( {c_{r} + 1} \right)^{2} \left( {1 - N} \right)} \mathord{\left/ {\vphantom {{N\left( {c_{r} + 1} \right)^{2} \left( {1 - N} \right)} {8\left( {N + 1} \right)}}} \right. \kern-0pt} {8\left( {N + 1} \right)}}$. For part (2), we can easily show that if $F_{R} \le {{T_{HH}^{C} } \mathord{\left/ {\vphantom {{T_{HH}^{C} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RYCHH * } - \Pi_{R}^{MYCHH * } \ge 0$; we also can easily show that if $F_{R} \le {{T_{HL}^{C} } \mathord{\left/ {\vphantom {{T_{HL}^{C} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RYCHL * } - \Pi_{R}^{MYCHL * } \ge 0$; we can easily show that if $F_{R} \le {{T_{LH}^{C} } \mathord{\left/ {\vphantom {{T_{LH}^{C} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RYCLH * } - \Pi_{R}^{MYCLH * } \ge 0$;we also can easily show that if $F_{R} \le {{T_{LL}^{C} } \mathord{\left/ {\vphantom {{T_{LL}^{C} } 2}} \right. \kern-0pt} 2}$, then $\Pi_{R}^{RYCLL * } - \Pi_{R}^{MYCLL * } \ge 0$. By analyzing the function image, we can show that $T^{C}$,$T_{HH}^{C}$,$T_{HL}^{C}$,$T_{LH}^{C}$ and $T_{LL}^{C}$ are always greater than zero.

3.17 Proof of Corollary 5

It is not difficult to check that $T_{HH} = T_{HH}^{C}$, $T_{HL} = T_{HL}^{C}$, $T_{LH} = T_{LH}^{C}$ and $T_{LL} = T_{LL}^{C}$.

3.18 Proof of Corollary 6

For part (1), by calculation, we find that

$$\begin{gathered} C_{e}^{MTC} = \frac{1}{{4\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}\left( {e\left( {\alpha \left( {N - 1} \right)\left( {\left( {te + c - s_{L} } \right)\delta_{H} + \left( {s_{H} - c - te} \right)\delta_{L} - s_{H} + s_{L} } \right) + } \right.} \right.\left( {1 - \delta_{H} } \right)\left( {\left( {\left( {te + c} \right)\delta_{L} } \right.} \right. \hfill \\ \left. {\left. {\left. {\left. { - s_{L} } \right)N - te - c + s_{L} - \delta_{L} + 1} \right)} \right)} \right) \hfill \\ \end{gathered}$$

and

$$\begin{gathered} C_{e}^{RTC} = \frac{1}{{4\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}\left( {e\left( {\alpha \left( {N - 1} \right)\left( {\left( {te + c - s_{L} } \right)\delta_{H} + \left( {s_{H} - c - te} \right)\delta_{L} - s_{H} + s_{L} } \right) + } \right.} \right.\left( {1 - \delta_{H} } \right) \hfill \\ \left( {\left( {\left( {te + c} \right)\delta_{L} } \right.} \right.\left. {\left. {\left. {\left. { - s_{L} } \right)N - te - c + s_{L} - \delta_{L} + 1} \right)} \right)} \right) \hfill \\ \end{gathered}.$$

It is not difficult to check that $C_{e}^{MTC} = C_{e}^{RTC}$. For part (2), by calculation, we find that

$$\begin{gathered} C_{e}^{MTCHH} = \frac{1}{{4\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}\left( {e\left( {\alpha \left( {N - 1} \right)\left( {\left( {te + c - s_{L} } \right)\delta_{H} + \left( {s_{H} - c - te} \right)\delta_{L} - s_{H} + s_{L} } \right) + } \right.} \right.\left( {1 - \delta_{H} } \right)\left( {\left( {\left( {te + c} \right)\delta_{L} } \right.} \right. \hfill \\ \left. {\left. {\left. {\left. { - s_{L} } \right)N - te - c + s_{L} - \delta_{L} + 1} \right)} \right)} \right) \hfill \\ \end{gathered}$$

and

$$\begin{gathered} C_{e}^{RYCHH} = \frac{1}{{4\left( {1 - \delta_{H} } \right)\left( {1 - \delta_{L} } \right)}}\left( {e\left( {\alpha \left( {N - 1} \right)\left( {\left( {te + c - s_{L} } \right)\delta_{H} + \left( {s_{H} - c - te} \right)\delta_{L} - s_{H} + s_{L} } \right) + } \right.} \right.\left( {1 - \delta_{H} } \right) \hfill \\ \left( {\left( {\left( {te + c} \right)\delta_{L} } \right.} \right.\left. {\left. {\left. {\left. { - s_{L} } \right)N - te - c + s_{L} - \delta_{L} + 1} \right)} \right)} \right) \hfill \\ \end{gathered}.$$

It is not difficult to check that $C_{e}^{MYCHH} = C_{e}^{RYCHH}$. Similarly, by calculation, we find that

$$\begin{gathered} C_{e}^{MYCHL} = \frac{1}{{8\left( {\delta_{H} - 1} \right)}}\left( {e\left( {\left( {1 - N} \right)\left( {\left( {te + c - c_{r} } \right)\alpha - \left( {c_{r} + c + te} \right)N - te - c + c_{r} + 2} \right)\delta_{H} - \left( {c_{r} + c + te - 2s_{H} } \right)\left( {N - } \right.} \right.} \right. \hfill \\ \left. {\left. {\left. 1 \right)\alpha + \left( {c_{r} + c + te} \right)N - c_{r} + c + te - 2} \right)} \right) \hfill \\ \end{gathered}$$

and

$$\begin{gathered} C_{e}^{RYCHL} = \frac{1}{{8\left( {\delta_{H} - 1} \right)}}\left( {e\left( {\left( {1 - N} \right)\left( {\left( {te + c - c_{r} } \right)\alpha - \left( {c_{r} + c + te} \right)N - te - c + c_{r} + 2} \right)\delta_{H} - \left( {c_{r} + c + te} \right.} \right.} \right. \hfill \\ \left. {\left. {\left. { - 2s_{H} } \right)\left( {N - 1} \right)\alpha + \left( {c_{r} + c + te} \right)N - c_{r} + c + te - 2} \right)} \right) \hfill \\ \end{gathered}.$$

It is not difficult to check that $C_{e}^{MYCHL} = C_{e}^{RYCHL}$. By calculation, we find that $C_{e}^{MYCLH} =$$\frac{1}{{8\left( {\delta_{L} - 1} \right)}}\left( {e\left( {\left( {\left( {te + c - c_{r} } \right)\delta_{L} + te + c + c_{r} - 2s_{L} } \right)\left( {N - 1} \right)\alpha \left. {\left. { + \left( {2 - 2\left( {te + c} \right)N} \right)\delta_{L} + 2Ns_{L} + 2te + 2c - 2s_{L} - 2} \right)} \right)} \right.} \right.$ and

$$\begin{gathered} C_{e}^{RYCLH} = \frac{1}{{8\left( {\delta_{L} - 1} \right)}}\left( {e\left( {\left( {\left( {te + c - c_{r} } \right)\delta_{L} + te + c + c_{r} - 2s_{L} } \right)\left( {N - 1} \right)\alpha } \right.} \right. + \left( {2 - 2\left( {te + c} \right)N} \right)\delta_{L} + 2Ns_{L} + \hfill \\ 2te\left. {\left. { + 2c - 2s_{L} - 2} \right)} \right) \hfill \\ \end{gathered}.$$

It is not difficult to check that $C_{e}^{MYCLH} = C_{e}^{RYCLH}$. Similarly, by calculation, we find that $C_{e}^{MYCLL} = {{e\left( {2 + c_{r} - te - c - \left( {te + c + c_{r} } \right)N} \right)} \mathord{\left/ {\vphantom {{e\left( {2 + c_{r} - te - c - \left( {te + c + c_{r} } \right)N} \right)} 8}} \right. \kern-0pt} 8}$ and $C_{e}^{RYCLL} = {{e\left( {2 + c_{r} - te - c - \left( {te + c + c_{r} } \right)N} \right)} \mathord{\left/ {\vphantom {{e\left( {2 + c_{r} - te - c - \left( {te + c + c_{r} } \right)N} \right)} 8}} \right. \kern-0pt} 8}$. It is not difficult to check that $C_{e}^{MYCLL} = C_{e}^{RYCLL}$.

Appendix D: monotonicity analysis of profit

In Appendix D, this paper uses numerical examples to study the monotonicity of the optimal profits of the manufacturer and the retailer concerning parameters ($\delta_{H}$, $s_{H}$, $\delta_{L}$, $s_{L}$) under models MT, RT, MYHL, and RYHL.

First, in order to study the monotonicity of the optimal profits of the manufacturer and the retailer concerning $\delta_{H}$ under the models MT, RT, MYHL, and RYHL, we set $c = 0.95$, $c_{r} = 0.01$, $s_{H} = 0.92$, $s_{L} = 0.9$, $\alpha = 0.8$, $N = 0.6$, $\delta_{L} = 0.02$, $F_{M} = 0$, $F_{R} = 0$; among them, setting the fixed trade-in costs of the manufacturer and the retailer to zero does not affect the monotonicity of the optimal profits of the manufacturer and the retailer for any parameters. We increase $\delta_{H}$ from 0.65 to 0.95. The graph of the optimal profits of the manufacturer and the retailer concerning the parameter $\delta_{H}$ under the models MT, RT, MYHL, and RYHL is shown in Fig. 

Fig. 8

The impacts of $\delta_{H}$

8. From Fig. 8, we can find that the optimal profits of the manufacturer and the retailer are monotonically decreasing with respect to $\delta_{H}$. As Type I replacement consumers gradually increase their valuation of their used products, the manufacturer or the retailer has to increase the trade-in rebates for Type I used products to attract Type I replacement consumers to participate in the trade-in. The gradually increasing trade-in rebates reduce the profits of the manufacturer and the retailer.

Secondly, we study the monotonicity of the optimal profits of the manufacturer and the retailer concerning $s_{H}$ under the models MT, RT, MYHL, and RYHL. We set $c = 0.95$, $c_{r} = 0.01$, $\delta_{H} = 0.92$, $\delta_{L} = 0.02$, $\alpha = 0.8$, $N = 0.6$, $s_{L} = 0.9$, $F_{M} = 0$, $F_{R} = 0$; we increase $s_{H}$ from 0.9 to 0.95. The graph of the optimal profits of the manufacturer and the retailer with respect to the parameter $s_{H}$ under the models MT, RT, MYHL, and RYHL is shown in Fig. 

Fig. 9

The impacts of $s_{H}$

9. From Fig. 9, we can find that the optimal profits of the manufacturer and the retailer are monotonically increasing with respect to $s_{H}$. As $s_{H}$ increases, the manufacturer can obtain more profits from Type I used products, and the retailer can also indirectly obtain profits by requiring the manufacturer to increase the buyback price of Type I used products or increase the retail price of new products, thereby increasing the profits of both the manufacturer and the retailer.

Afterward, we also studied the monotonicity of the optimal profits of the manufacturer and the retailer with respect to $\delta_{L}$ under the models MT, RT, MYHL, and RYHL. We set $c = 0.95$, $c_{r} = 0.01$, $s_{H} = 0.92$, $s_{L} = 0.9$, $\alpha = 0.8$, $N = 0.6$, $\delta_{H} = 0.8$, $F_{M} = 0$, $F_{R} = 0$; we increase $\delta_{L}$ from 0 to 0.025. The graph of the optimal profits of the manufacturer and the retailer with respect to the parameter $\delta_{L}$ under the models MT, RT, MYHL, and RYHL is shown in Fig. 

Fig. 10

The impacts of $\delta_{L}$

10. It should be noted that the monotonicity of the retailer's optimal profit with respect to parameter $\delta_{L}$ is exactly the same as the monotonicity of the manufacturer's optimal profit with respect to parameter $\delta_{L}$. We omitted the graph of the retailer's optimal profit with respect to parameter $\delta_{L}$ under the MYHL (RYHL) model. This is because the retailer's optimal profit under the MYHL (RYHL) model does not change with the parameter $\delta_{L}$, and it is difficult to put it together with the graph of the retailer's optimal profit with respect to the parameter $\delta_{L}$ under the MT (RT) model, thus we omitted it. From Fig. 10, we see that the optimal profits of the manufacturer and the retailer under models MT and RT are monotonically decreasing with respect to $\delta_{L}$, but the profits of the manufacturer and the retailer under models MYHL and RYHL are not affected by changes in $\delta_{L}$. When third-party recyclers do not enter the market, as Type II replacement consumers gradually increase the valuations of their used products, the manufacturer or the retailer has to increase the trade-in rebates for Type II used products to attract Type II replacement consumers to participate in trade-ins. The gradually increasing trade-in rebates reduce the profits of the manufacturer and the retailer. When third-party recyclers enter the market, in order to preserve their own strength, when Type II replacement consumers increase the valuations of their used products, the manufacturer and the retailer will not provide them with higher trade-in rebates. Therefore, the profits of the manufacturer and the retailer do not change with changes in the durability of Type II used products.

Finally, we study the monotonicity of the optimal profits of the manufacturer and the retailer with respect to $s_{L}$ under the models MT, RT, MYHL, and RYHL. We set $c = 0.95$, $c_{r} = 0.01$, $\delta_{H} = 0.92$, $\delta_{L} = 0.02$, $\alpha = 0.8$, $N = 0.6$, $s_{H} = 0.92$, $F_{M} = 0$, $F_{R} = 0$; we increase $s_{L}$ from 0.9 to 0.92. The graph of the optimal profits of the manufacturer and the retailer with respect to the parameter $s_{L}$ under the models MT, RT, MYHL, and RYHL is shown in Fig. 

Fig. 11

The impacts of $s_{L}$

11. From Fig. 11, we see that the optimal profits of the manufacturer and the retailer under models MT and RT are monotonically increasing with respect to $s_{L}$, but the profits of the manufacturer and the retailer under models MYHL and RYHL are not affected by changes in $s_{L}$. When third-party recyclers do not enter the market, as $s_{L}$ increases, the manufacturer can obtain more benefits from Type II used products, and the retailer can also indirectly obtain benefits by requiring the manufacturer to increase the buyback price of Type II used products or increase the retail price of new products, increasing the profits of the manufacturer and the retailer. When third-party recyclers invade the market, due to the low durability of Type II used products, 3PC will compete with trade-in for the recycling of Type II used products. The losses caused by competition are offset by the profits generated by the increase in the residual value of Type II used products, resulting in the profits of the manufacturer and the retailer being unaffected by $s_{L}$.