Optimal pricing and remanufacturing entry strategies of manufacturers in the presence of online reviews

Abstract

In recent years, online reviews are increasingly getting more concerns by firms and consumers because they can help mitigate consumers’ uncertainty on product quality and experienced attributes and significantly affect firms’ operational decisions. We in this study attempt to examine the remanufacturing entry and pricing strategies of manufacturers in the presence of online reviews. We consider an original equipment manufacturer (OEM) selling a new product over two periods in online retailing and determine whether and when to adopt a remanufacturing entry strategy in the market. We uncover online reviews’ quality-dimensional effect and experienced dimensional effect that jointly determine the OEM’s pricing and remanufacturing strategies. We show that in the presence of online reviews, the OEM will cautiously determine whether to adopt the first-period remanufacturing entry strategy and may also adopt the second-period remanufacturing entry strategy under certain conditions. Interestingly, the OEM will adopt the penetration pricing strategy for the new product and the remanufactured product (if available) when the actual product quality is sufficiently high, but the skimming pricing strategy otherwise, which is different from the uniform pricing strategy in the absence of online reviews. Our results also show that online reviews significantly affect the OEM’s profit and consumer surplus. In particular, when the actual product quality is high enough, the OEM and consumers will attain the “win–win” situation.

Appendices

Appendix: Proofs

Proof of Proposition 1

To prove Proposition 1, we first derive the optimal pricing decisions and profits of the OEM under the three different remanufacturing entry strategies in the absence of online reviews, which are presented in as follows.

(1) In the NN scenario.

Following Eq. (2), the OEM’s profit in the NN scenario in period 1 can be formulated as $\widetilde{\Pi }_{1}^{NN} = (1 - \tilde{p}_{1n}^{NN} )(\tilde{p}_{1n}^{NN} - c_{n} )$. By taking the first-order and second-order derivatives of $\widetilde{\Pi }_{1}^{NN}$ regarding $\tilde{p}_{1n}^{NN}$, we have $\frac{{\partial \widetilde{\Pi }_{1}^{NN} }}{{\partial \tilde{p}_{1n}^{NN} }} = 1 + c_{n} - 2\tilde{p}_{1n}^{NN}$ and $\frac{{\partial^{2} \widetilde{\Pi }_{1}^{NN} }}{{\partial (\tilde{p}_{1n}^{NN} )^{2} }} = - 2 < 0$. Thus, $\widetilde{\Pi }_{1}^{NN}$ is concave in $\tilde{p}_{1n}^{NN}$. By setting $\frac{{\partial \widetilde{\Pi }_{1}^{NN} }}{{\partial \tilde{p}_{1n}^{NN} }} = 0$, we have $\tilde{p}_{1n}^{{NN{*}}} = \frac{{1 + c_{n} }}{2}$, $\tilde{D}_{1n}^{{NN{*}}} = \frac{{1 - c_{n} }}{2}$ and $\widetilde{\Pi }_{1}^{{NN{*}}} { = }\frac{{\left( {c_{n} - 1} \right)^{2} }}{4}$. Moving to the second period, the OEM’s second-period profit is the same as that in the first period and thus we have $\tilde{p}_{2n}^{{NN{*}}} = \frac{{1 + c_{n} }}{2}$, $\tilde{D}_{2n}^{{NN{*}}} = \frac{{1 - c_{n} }}{2}$ and $\widetilde{\Pi }_{2}^{{NN{*}}} { = }\frac{{\left( {c_{n} - 1} \right)^{2} }}{4}$. Over the two periods, in equilibrium, the OEM’s total profit is $\widetilde{\Pi }_{{}}^{{NN{*}}} { = }\frac{{\left( {c_{n} - 1} \right)^{2} }}{2}$.

(2) In the NE scenario.

If the OEM introduces the remanufactured product in the second period, the OEM’s first-period is the same as that in the NN scenario and thus we have $\tilde{p}_{1n}^{{NE{*}}} = \frac{{1 + c_{n} }}{2}$ and $\widetilde{\Pi }_{1}^{{NE{*}}} { = }\frac{{\left( {c_{n} - 1} \right)^{2} }}{4}$. In the second period, the OEM’s profit can be formulated as $\widetilde{\Pi }_{2}^{NE} = (1 - \frac{{\tilde{p}_{2n}^{NE} - \tilde{p}_{2r}^{NE} }}{1 - \theta })(\tilde{p}_{2n}^{NE} - c_{n} ) + (\frac{{\tilde{p}_{2n}^{NE} - \tilde{p}_{2r}^{NE} }}{1 - \theta } - \frac{{\tilde{p}_{2r}^{NE} }}{\theta })(\tilde{p}_{2r}^{NE} - c_{r} )$. By taking the first-order and second-order derivatives of $\widetilde{\Pi }_{2}^{NE}$ regarding $\tilde{p}_{2n}^{NE}$ and $\tilde{p}_{2r}^{NE}$, respectively, we have $\frac{{\partial \widetilde{\Pi }_{2}^{NE} }}{{\partial \tilde{p}_{2n}^{NE} }} = 1 + \frac{{c_{r} - c_{n} + 2\tilde{p}_{2n}^{NE} - 2\tilde{p}_{2r}^{NE} }}{\theta - 1}$, $\frac{{\partial \widetilde{\Pi }_{2}^{NE} }}{{\partial \tilde{p}_{2r}^{NE} }} = \frac{{2\theta \tilde{p}_{2n}^{NE} - 2\tilde{p}_{2r}^{NE} + c_{r} - \theta c_{n} }}{{\theta - \theta^{2} }}$, $\frac{{\partial^{2} \widetilde{\Pi }_{2}^{NE} }}{{\partial (\tilde{p}_{2n}^{NE} )^{2} }} = \frac{2}{\theta - 1} < 0$ and $\frac{{\partial^{2} \widetilde{\Pi }_{2}^{NE} }}{{\partial (\tilde{p}_{2r}^{NE} )^{2} }} = \frac{2}{\theta (\theta - 1)} < 0$. It can easily obtain that $\widetilde{\Pi }_{2}^{NE}$ is concave in $\tilde{p}_{2n}^{NE}$ and $\tilde{p}_{2r}^{NE}$. In equilibrium, the optimal prices of the new and remanufactured products are $\tilde{p}_{2n}^{{NE{*}}} = \frac{{1 + c_{n} }}{2}$ and $\tilde{p}_{2r}^{{NE{*}}} = \frac{{\theta + c_{r} }}{2}$, respectively; and thus the demands are $\tilde{D}_{2n}^{{NE{*}}} = \frac{{1 - \theta + c_{r} - c_{n} }}{{2\left( {1 - \theta } \right)}}$ and $\tilde{D}_{2r}^{{NE{*}}} = \frac{{\theta c_{n} - c_{r} }}{{2\theta \left( {1 - \theta } \right)}}$, respectively. Then, we can obtain the second-period profit, i.e., $\widetilde{\Pi }_{2}^{{NE{*}}} { = }\frac{{\theta \left( {c_{n} - 1} \right)^{2} + \left( {c_{r} - \theta } \right)\left( {c_{r} + \theta - 2\theta c_{n} } \right)}}{{4\theta \left( {1 - \theta } \right)}}$. Over the two periods, in equilibrium, the OEM’s total profit is $\widetilde{\Pi }_{2}^{{NE{*}}} { = }\frac{1}{4}\left( {2 - 4c_{n} + c_{n}^{2} + \frac{{\left( {c_{n} - c_{r} } \right)^{2} }}{1 - \theta } + \frac{{c_{r}^{2} }}{\theta }} \right)$. To meet the conditions $\tilde{D}_{2n}^{{NE{*}}} \ge 0$ and $\tilde{D}_{2r}^{{NE{*}}} > 0$, the consumers’ remanufactured product receptivity $\theta$ should satisfy $\frac{{c_{r} }}{{c_{n} }} < \theta \le 1 + c_{r} - c_{n} < 1$.

(3) In the EE scenario.

In the EE scenario, the OEM’s profit in each period is the same as that in the NE scenario when the remanufactured product is introduced in the second period and thus we have $\tilde{p}_{1n}^{{EE{*}}} = \tilde{p}_{2n}^{{EE{*}}} = \frac{{1 + c_{n} }}{2}$ and $\tilde{p}_{1r}^{{EE{*}}} = \tilde{p}_{2r}^{{EE{*}}} = \frac{{\theta + c_{r} }}{2}$. The OEM’s total payoff is $\widetilde{\Pi }_{{}}^{{EE{*}}} = \frac{{\theta \left( {c_{n} - 1} \right)^{2} + \left( {c_{r} - \theta } \right)\left( {c_{r} + \theta - 2\theta c_{n} } \right)}}{{2\theta \left( {1 - \theta } \right)}}$ accordingly.

Comparing the OEM’s total profit under the three scenarios, we have.

$\widetilde{\Pi }_{{}}^{{EE{*}}} - \widetilde{\Pi }_{{}}^{{NE{*}}} = \widetilde{\Pi }_{{}}^{{NE{*}}} - \widetilde{\Pi }_{{}}^{{NN{*}}} = \frac{{\left( {c_{r} - \theta c_{n} } \right)^{2} }}{{4\theta \left( {1 - \theta } \right)}} > 0$ and $\widetilde{\Pi }_{{}}^{{EE{*}}} - \widetilde{\Pi }_{{}}^{{NN{*}}} = \frac{{\left( {c_{r} - \theta c_{n} } \right)^{2} }}{{2\theta \left( {1 - \theta } \right)}} > 0$. Then, under the condition that the new product’s demand is greater than zero, it is obviously that $\widetilde{\Pi }_{{}}^{{EE{*}}} > \widetilde{\Pi }_{{}}^{{NE{*}}} > \widetilde{\Pi }_{{}}^{{NN{*}}}$. Thus, when $\frac{{c_{r} }}{{c_{n} }} < \theta \le 1 + c_{r} - c_{n} < 1$, the OEM is profitable to introduce the remanufactured product in period 1 (EE scenario); otherwise, the OEM never sells the remanufactured product in the market (NN scenario).

Proof of Table 2

To prove Table 2, by considering consumer reviews, we apply the backward induction technique to derive the optimal decisions in the NN, NE and EE scenarios, respectively.

(1) In the NN scenario.

Following Eq. (3), the OEM’s profit in the second period can be formulated as $\Pi_{2}^{NN} = \frac{{\left( {p_{2n}^{NN} - c_{n} } \right)\left( {\frac{1}{2}\left( {p_{1n}^{NN} - 2p_{2n}^{NN} - \mu p_{2n}^{NN} + q + \mu + \mu q - 1} \right) + \varepsilon } \right)}}{2\varepsilon }$. It is derived that $\frac{{\partial \Pi_{2}^{NN} }}{{\partial p_{2n}^{NN} }} = \frac{{p_{1n}^{NN} - \mu p_{1n}^{NN} - 4p_{2n}^{NN} + q + \mu + \mu q + 2c_{n} + 2\varepsilon - 1}}{4\varepsilon }$ and $\frac{{\partial^{2} \Pi_{2}^{NN} }}{{\partial (p_{2n}^{NN} )^{2} }} = - \frac{1}{\varepsilon } < 0$. Thus, $\Pi_{2}^{NN}$ is concave in $p_{2n}^{NN}$ and let $\frac{{\partial \Pi_{2}^{NN} }}{{\partial p_{2n}^{NN} }} = 0$, then we have $p_{2n}^{{NN{*}}} (p_{1n}^{{NN{*}}} ) = \frac{{p_{1n}^{NN} - \mu p_{1n}^{NN} + q + \mu + \mu q + 2\varepsilon + 2c_{n} - 1}}{4}$.

By substituting $p_{2n}^{{NN{*}}} (p_{1n}^{{NN{*}}} )$ into $\Pi_{{}}^{NN}$ and taking the first-order and second-order derivatives of $\Pi_{{}}^{NN}$ regarding $p_{1n}^{NN}$, we have $\frac{{\partial \Pi_{{}}^{NN} }}{{\partial p_{1n}^{NN} }} = \frac{{p_{1n}^{NN} + q + 18\varepsilon - 32p_{1n}^{NN} \varepsilon - 2\mu \left( {p_{1n}^{NN} + \varepsilon - 1} \right) + \left( {p_{1n}^{NN} - q - 1} \right)\mu^{2} + 2c_{n} \left( {\mu - 1 + 8\varepsilon } \right) - 1}}{16\varepsilon }$ and $\frac{{\partial^{2} \Pi_{{}}^{NN} }}{{\partial (p_{1n}^{NN} )^{2} }} = \frac{{\left( {\mu - 1} \right)^{2} }}{16\varepsilon } - 2 < 0$. Thus, $\Pi_{{}}^{NN}$ is concave in $p_{1n}^{NN}$. Let $\frac{{\partial \Pi_{{}}^{NN} }}{{\partial p_{1n}^{NN} }} = 0$ and we have $p_{1n}^{{NN{*}}} = \frac{{q - 1 + 18\varepsilon + 2\mu + 2c_{n} \left( {\mu - 1 + 8\varepsilon } \right) - \mu \left( {2\varepsilon + \mu + q\mu } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$ and then we can derive $p_{2n}^{{NN{*}}} = \frac{{4\varepsilon \left( {4\varepsilon - 1 + \mu + 2q\left( {1 + \mu } \right)} \right) - c_{n} \left( {4\varepsilon \left( {\mu - 5} \right) + \left( {\mu - 1} \right)^{2} } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$.

Finally, at optimality, when considering the consumer reviews, the OEM’s total profit in the NN scenario is.

$\begin{gathered} \Pi_{{}}^{{NN{*}}} = \frac{{q^{2} \left( {1 + \mu } \right)^{2} + 2\varepsilon \left( {3 + 2\varepsilon + \mu } \right) + q\left( {1 + \mu } \right)\left( {\mu - 1 + 4\varepsilon } \right){ + }2c_{n}^{2} \left( {\mu + 1 + 4\varepsilon } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }} \\ { + }\frac{{ - c_{n} \left( {q\left( {1 + \mu } \right)\left( {3 + \mu } \right) + 2\left( {\mu - 1 + \varepsilon \left( {11 + \mu } \right)} \right)} \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }} \\ \end{gathered}$.

The demands of the new product in each period are.

$D_{1n}^{NN*} = \frac{{2\varepsilon \left( {7 + \mu } \right) - 2c_{n} \left( {8\varepsilon + \mu - 1} \right) + q\left( {\mu^{2} - 1} \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$;

$D_{2n}^{NN*} = \frac{{2\left( {4\varepsilon + \mu - 1 + 2q\left( {1 + \mu } \right) - c_{n} \left( {3 + \mu } \right)} \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$, respectively.

To avoid trivial cases and make sure that the new product’s demand is always greater than zero after introducing the remanufactured product, we assume that $q > \frac{2 - 2\varepsilon }{{1 + u}}$ and $\varepsilon \le \frac{1}{2}$ and thus $32\varepsilon - \left( {\mu - 1} \right)^{2} > 16 - \left( {\mu - 1} \right)^{2} > 0$ and $D_{2n}^{NN*} > \frac{{2\left( {1 - c_{n} } \right)\left( {3 + \mu } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }} > 0$. As for the first period, to make sure $D_{1n}^{NN*} > 0$, the actual quality $q$ should satisfy $q < \frac{{ - 2c_{n} - 14\varepsilon + 16\varepsilon c_{n} + 2\mu c_{n} - 2\varepsilon \mu }}{{\mu^{2} - 1}}$.

(2) In the NE scenario.

Following the Eq. (4), if the OEM introduces only the remanufactured product in the second period, the second-period profit can be formulated as $\Pi_{2}^{NE} = \frac{{\left( {p_{2n}^{NE} - c_{n} } \right)\left( {p_{2n}^{NE} - p_{2r}^{NE} + \left( {\theta - 1} \right)\left( {q + \varepsilon + E[\eta_{n} |R_{n} ]} \right)} \right)}}{{2\varepsilon \left( {\theta - 1} \right)}} + \frac{{\left( {p_{2r}^{NE} - c_{r} } \right)\left( {p_{2r}^{NE} - \theta p_{2n}^{NE} } \right)}}{{2\theta \varepsilon \left( {\theta - 1} \right)}}$, wherein $E[\eta_{n} |R_{n} ] = \frac{{(p_{1n}^{NE} - q - 1)(1 - \mu )}}{2}$. By taking the first-order and second-order derivatives of $\Pi_{2}^{NE}$ regarding $p_{2n}^{NE}$ and $p_{2r}^{NE}$, respectively, we have $\frac{{\partial \Pi_{2}^{NE} }}{{\partial p_{2n}^{NE} }} = \frac{{2p_{2n}^{NE} - 2p_{2r}^{NE} + c_{r} - c_{n} + \left( {\theta - 1} \right)\left( {q + \varepsilon + E[\eta_{n} |R_{n} ]} \right)}}{{2\varepsilon \left( {\theta - 1} \right)}}$,$\frac{{\partial^{2} \Pi_{2}^{NE} }}{{\partial (p_{2n}^{NE} )^{2} }} = \frac{1}{{\varepsilon \left( {\theta - 1} \right)}} < 0$, $\frac{{\partial \Pi_{2}^{NE} }}{{\partial p_{2r}^{NE} }} = \frac{{2p_{2r}^{NE} - 2\theta p_{2n}^{NE} + \theta c_{n} - c_{r} }}{2\theta \varepsilon (\theta - 1)}$ and $\frac{{\partial^{2} \Pi_{2}^{NE} }}{{\partial (p_{2r}^{NE} )^{2} }} = \frac{1}{{\theta \varepsilon \left( {\theta - 1} \right)}} < 0$. Let $p_{2n}^{{NE{*}}}$ and $p_{2r}^{NE*}$ denote the roots for $\frac{{\partial \Pi_{2}^{NE} }}{{\partial p_{2n}^{NE} }} = \frac{{\partial \Pi_{2}^{NE} }}{{\partial p_{2r}^{NE} }} = 0$ and thus ($p_{2n}^{{NE{*}}}$, $p_{2r}^{{NE{*}}}$) is the unique set of optimal pricing decisions for the new and remanufactured product, respectively, i.e.,

$p_{2n}^{{NE{*}}} (p_{1n}^{NE} ) = \frac{1}{2}\left( {c_{n} + q + \varepsilon + E[\eta_{n} |R_{n} ]} \right)$ and $p_{2r}^{{NE{*}}} (p_{1n}^{NE} ) = \frac{1}{2}\left( {c_{r} + \theta \left( {q + \varepsilon + E[\eta_{n} |R_{n} ]} \right)} \right)$.

Accordingly, by substituting ($p_{2n}^{{NE{*}}}$, $p_{2r}^{{NE{*}}}$) into the Eq. (4), we can get that $\Pi_{{}}^{NE} = \left( {c_{n} - p_{1n}^{NE} } \right)\left( {p_{1n}^{NE} - 1} \right) + \frac{{\left( {X_{1} - 2c_{n} } \right)\left( {2c_{n} - 2c_{r} + \left( {\theta - 1} \right)X_{1} } \right)}}{{32\varepsilon \left( {\theta - 1} \right)}} - \frac{{\left( {c_{r} - \theta c_{n} } \right)\left( {2c_{r} - \theta X_{1} } \right)}}{{16\theta \varepsilon \left( {\theta - 1} \right)}}$, wherein $X_{1} = p_{1n}^{NE} - \mu p_{1n}^{NE} + q + \mu q + 2\varepsilon + \mu - 1$. Since $\frac{{\partial^{2} \Pi_{{}}^{NE} }}{{\partial (p_{1n}^{NE} )^{2} }} = \frac{{\left( {\mu - 1} \right)^{2} }}{16\varepsilon } - 2 < 0$, let $\frac{{\partial \Pi_{{}}^{NE} }}{{\partial p_{1n}^{NE} }} = \frac{{p_{1n}^{NE} - 32\varepsilon p_{1n}^{NE} + q + 18\varepsilon - 1 - 2\mu \left( {p_{1n}^{NE} + \varepsilon - 1} \right) + \mu^{2} \left( {p_{1n}^{NE} - q - 1} \right) + 2c_{n} \left( {8\varepsilon + \mu - 1} \right)}}{16\varepsilon } = 0$, we can get the optimal first-period price of new product, $p_{1n}^{{NE{*}}} = \frac{{q - 1 + 18\varepsilon + 2\mu + 2c_{n} \left( {\mu - 1 + 8\varepsilon } \right) - \mu \left( {2\varepsilon + \mu + q\mu } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$ and then derive $p_{2n}^{{NE{*}}} = \frac{{4\varepsilon \left( {4\varepsilon - 1 + \mu + 2q\left( {1 + \mu } \right)} \right) - c_{n} \left( {4\varepsilon \left( {\mu - 5} \right) + \left( {\mu - 1} \right)^{2} } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$ and $p_{2r}^{{NE{*}}} = \frac{{c_{r} \left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right) - \theta c_{n} \left( {\mu - 1} \right)\left( {\mu - 1 + 8\varepsilon } \right) + 8\theta \varepsilon \left( {\mu - 1 + 4\varepsilon + 2q\left( {1 + \mu } \right)} \right)}}{{64\varepsilon - 2\left( {\mu - 1} \right)^{2} }}$.

Accordingly, the OEM’s total profit in the NE scenario is

$$ \begin{gathered} \Pi_{{}}^{{NE{*}}} = \frac{1}{{8\theta \varepsilon \left( {\theta - 1} \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}}(2\theta c_{n} c_{r} \left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right) + c_{r}^{2} \left( {\left( {\mu - 1} \right)^{2} - 32\varepsilon } \right) \\ + \theta \varepsilon (8\left( {\theta - 1} \right)\left( {q^{2} \left( {1 + \mu } \right)^{2} + 2\varepsilon \left( {3 + 2\varepsilon + \mu } \right){ + }8\left( {\theta - 1} \right)\left( {q\left( {1 + \mu } \right)\left( {\mu - 1 + 4\varepsilon } \right)} \right)} \right) + \\ c_{n}^{2} \left( {\theta \left( {8\varepsilon + \mu - 1} \right)^{2} - 16\varepsilon \left( {1 + 4\varepsilon + \mu } \right)} \right) - 8\varepsilon c_{n} \left( {\theta - 1} \right)\left( {q\left( {1 + \mu } \right)\left( {3 + \mu } \right) + 2\left( {\mu - 1 + \varepsilon \left( {11 + \mu } \right)} \right)} \right))) \\ \end{gathered} $$

and the demands of the new and remanufactured products in each period are:

$D_{1n}^{NE*} = \frac{{2\varepsilon \left( {7 + \mu } \right) - 2c_{n} \left( {8\varepsilon + \mu - 1} \right) + q\left( {\mu^{2} - 1} \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$;

$D_{2n}^{NE*} = \frac{{c_{r} - \theta c_{n} }}{4\varepsilon (1 - \theta )} + \frac{{2\left( { - 1 + 4\varepsilon + \mu + 2q\left( {1 + \mu } \right) - c_{n} \left( {3 + \mu } \right)} \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }} = D_{2n}^{NN*} + \frac{{c_{r} - \theta c_{n} }}{4\varepsilon (1 - \theta )}$;

$D_{2r}^{NE*} = \frac{{\theta c_{n} - c_{r} }}{4\theta \varepsilon (1 - \theta )}$, respectively.

Hence, we have $q < \frac{{ - 2c_{n} - 14\varepsilon + 16\varepsilon c_{n} + 2\mu c_{n} - 2\varepsilon \mu }}{{\mu^{2} - 1}}$ to make sure $D_{1n}^{NE*} > 0$; $\theta < \frac{{ - c_{r} - 8\varepsilon - 24c_{n} \varepsilon + 32c_{r} \varepsilon + 16q\varepsilon + 32\varepsilon^{2} + 2c_{r} \mu + 8\varepsilon \mu - 8c_{n} \varepsilon \mu + 16q\varepsilon \mu - c_{r} \mu^{2} }}{{ - c_{n} - 8\varepsilon + 8c_{n} \varepsilon + 16q\varepsilon + 32\varepsilon^{2} + 2c_{n} \mu + 8\varepsilon \mu - 8c_{n} \varepsilon \mu + 16q\varepsilon \mu - c_{n} \mu^{2} }}$ to make sure $D_{2n}^{NE*} > 0$ and $\theta > \frac{{c_{r} }}{{c_{n} }}$ to make sure $D_{2r}^{NE*} > 0$.

Since $q > \frac{2 - 2\varepsilon }{{1 + u}}$ and $\varepsilon < \frac{1}{2}$, we obtain that $1 - \mu - 8\varepsilon < - 3 - \mu < 0$ and $1 - 4\varepsilon - \mu - 2q\left( {1 + \mu } \right) < - 3 - \mu < 0$ and accordingly, we have $\begin{gathered} \frac{{ - c_{r} - 8\varepsilon - 24c_{n} \varepsilon + 32c_{r} \varepsilon + 16q\varepsilon + 32\varepsilon^{2} + 2c_{r} \mu + 8\varepsilon \mu - 8c_{n} \varepsilon \mu + 16q\varepsilon \mu - c_{r} \mu^{2} }}{{ - c_{n} - 8\varepsilon + 8c_{n} \varepsilon + 16q\varepsilon + 32\varepsilon^{2} + 2c_{n} \mu + 8\varepsilon \mu - 8c_{n} \varepsilon \mu + 16q\varepsilon \mu - c_{n} \mu^{2} }} - \frac{{c_{r} }}{{c_{n} }} \\ = \frac{{8\varepsilon \left( {c_{n} - c_{r} } \right)\left( {1 - 4\varepsilon - \mu - 2q\left( {1 + \mu } \right) + c_{n} \left( {3 + \mu } \right)} \right)}}{{c_{n} \left( {c_{n} \left( {1 - \mu } \right)\left( {1 - \mu - 8\varepsilon } \right) + 8\varepsilon \left( {1 - 4\varepsilon - \mu - 2q\left( {1 + \mu } \right)} \right)} \right)}} \\ > \frac{{8\varepsilon \left( {c_{n} - c_{r} } \right)\left( {(c_{n} - 1)\left( {3 + \mu } \right)} \right)}}{{c_{n} \left( {c_{n} \left( {1 - \mu } \right)\left( {1 - \mu - 8\varepsilon } \right) + 8\varepsilon \left( {1 - 4\varepsilon - \mu - 2q\left( {1 + \mu } \right)} \right)} \right)}} > 0 \\ \end{gathered}$.

Hence, in the NE scenario, the actual quality $q$ should satisfy $q < \frac{{ - 2c_{n} - 14\varepsilon + 16\varepsilon c_{n} + 2\mu c_{n} - 2\varepsilon \mu }}{{\mu^{2} - 1}}$ and $\frac{{c_{r} }}{{c_{n} }} < \theta < \frac{{ - c_{r} - 8\varepsilon - 24c_{n} \varepsilon + 32c_{r} \varepsilon + 16q\varepsilon + 32\varepsilon^{2} + 2c_{r} \mu + 8\varepsilon \mu - 8c_{n} \varepsilon \mu + 16q\varepsilon \mu - c_{r} \mu^{2} }}{{ - c_{n} - 8\varepsilon + 8c_{n} \varepsilon + 16q\varepsilon + 32\varepsilon^{2} + 2c_{n} \mu + 8\varepsilon \mu - 8c_{n} \varepsilon \mu + 16q\varepsilon \mu - c_{n} \mu^{2} }}$.

(3) In the EE scenario.

Following Eq. (5), if the OEM launches the remanufactured product in the first period, the OEM’s second period profit can be formulated as.

$\begin{gathered} \Pi_{2}^{EE} = \left( {p_{2n}^{EE} - c_{n} } \right)\left( {\frac{1}{2} - \frac{{p_{2r}^{EE} - p_{2n}^{EE} + q - \theta q + E[\eta_{n} |R_{n} ] - E[\eta_{r} |R_{r} ]}}{{2\varepsilon \left( {\theta - 1} \right)}}} \right) \\ + \frac{{\left( {p_{2r}^{EE} - c_{r} } \right)\left( {p_{2r}^{EE} - \theta p_{2n}^{EE} + \theta E[\eta_{n} |R_{n} ] - E[\eta_{r} |R_{r} ]} \right)}}{{2\theta \varepsilon \left( {\theta - 1} \right)}} \\ \end{gathered}$,

wherein $E[\eta_{n} |R_{n} ] = \frac{{(p_{1n}^{NE} - q - 1)(1 - \mu )}}{2}$ and $E[\eta_{r} |R_{r} ] = \frac{{(p_{1r}^{EE} - \theta q - 1)(1 - \mu )}}{2}$.

Using the same method above, we can get the optimal prices of the new and remanufactured products in the second period, respectively, i.e., $p_{2n}^{EE*} (p_{1n}^{EE} ,p_{1r}^{EE} ) = \frac{1}{4}\left( { - 1 + 2c_{n} + p_{1n}^{EE} + \mu - \mu p_{1n}^{EE} + q + 2\varepsilon { + }\mu q} \right)$ and $p_{2r}^{EE*} (p_{1n}^{EE} ,p_{1r}^{EE} ) = \frac{1}{4}\left( { - 1 + 2c_{r} + p_{1r}^{EE} + \mu - \mu p_{1r}^{EE} + \theta (q + 2\varepsilon + \mu q)} \right)$.

Accordingly, by substituting ($p_{2n}^{{EE{*}}}$, $p_{2r}^{{EE{*}}}$) into Eq. (5) and using the same method above, we can easily obtain that $p_{1n}^{{EE{*}}} = \frac{{q - 1 + 18\varepsilon + 2\mu + 2c_{n} \left( {\mu - 1 + 8\varepsilon } \right) - \mu \left( {2\varepsilon + \mu + q\mu } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$ and $p_{1r}^{{EE{*}}} = \frac{{\theta \left( {q + 18\varepsilon } \right) - \left( {1 + \theta q} \right)\mu^{2} + 2c_{r} \left( {8\varepsilon + \mu - 1} \right) + 2\mu - 2\theta \varepsilon \mu - 1}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$. In this case, the optimal $p_{2n}^{{EE{*}}}$ and $p_{2r}^{{EE{*}}}$ can be derived as $p_{2n}^{{EE{*}}} = \frac{{4\varepsilon \left( {4\varepsilon - 1 + \mu + 2q\left( {1 + \mu } \right)} \right) - c_{n} \left( {4\varepsilon \left( {\mu - 5} \right) + \left( {\mu - 1} \right)^{2} } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$ and $p_{2r}^{{EE{*}}} = \frac{{4\varepsilon \left( {2\left( {\mu - 1} \right) + \theta \left( {1 + 4\varepsilon - \mu + 2q\left( {1 + \mu } \right)} \right)} \right) - c_{r} \left( {4\varepsilon \left( {\mu - 5} \right) + \left( {\mu - 1} \right)^{2} } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$. Thus, in the EE scenario, the OEM’s optimal profit over the two periods is

$$ \begin{gathered} \Pi^{EE*} = \frac{1}{{\theta \left( {\theta - 1} \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}}( - \left( {\mu - 1} \right)^{2} - 2c_{r}^{2} \left( {1 + 4\varepsilon + \mu } \right) + c_{r} \left( { - 3 + 2\mu + u^{2} } \right) \\ + \theta c_{r} \left( {3 - 2\mu - \mu^{2} + 4c_{n} \left( {1 + 4\varepsilon + \mu } \right)} \right) + \theta^{2} X_{2} - \theta (X_{2} - \left( {\mu - 1} \right)^{2} + 2c_{n}^{2} \left( {1 + 4\varepsilon + \mu } \right)) \\ \end{gathered} $$

wherein.

$\begin{gathered} X_{2} = 2\varepsilon (2\varepsilon + \mu + 3) - (\mu - 1)^{2} + q^{2} \left( {1 + \mu } \right)^{2} + q\left( {1 + \mu } \right)\left( {\mu - 1 + 4\varepsilon } \right) \\ - c_{n} \left( {1 - \mu^{2} + 2\varepsilon \left( {11 + \mu } \right) + q\left( {3 + 4\mu + \mu^{2} } \right)} \right) \\ \end{gathered}$.

Thus, the demands of the new and remanufactured products in each period are obtained, i.e.,

$D_{1n}^{EE*} = \frac{{2(c_{n} - c_{r} )\left( {8\varepsilon + \mu - 1} \right) + \left( {\theta - 1} \right)\left( { - \left( {\mu - 1} \right)^{2} + 2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right)} \right)}}{{\left( {\theta - 1} \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}}$;

$D_{1r}^{EE*} = \frac{{2c_{r} \left( {8\varepsilon + \mu - 1} \right) - \left( {\mu - 1} \right)^{2} + \theta \left( {\left( {\mu - 1} \right)^{2} - 2c_{n} \left( {8\varepsilon + \mu - 1} \right)} \right)}}{{\theta \left( {\theta - 1} \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}}$;

$D_{2n}^{EE*} = \frac{{2\left( {(c_{n} - c_{r} )\left( {3 + \mu } \right) + \left( {\theta - 1} \right)\left( {1 + 4\varepsilon - \mu + 2q\left( {1 + \mu } \right)} \right)} \right)}}{{\left( {\theta - 1} \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}}$;

$D_{2r}^{EE*} = \frac{{4 - 4\mu + 2c_{r} \left( {3 + \mu } \right) - 2\theta \left( {2 - 2\mu + c_{n} \left( {3 + \mu } \right)} \right)}}{{\theta \left( {\theta - 1} \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}}$.

Hence, we have $\theta < 1 - \frac{{2\left( {c_{n} - c_{r} } \right)\left( {8\varepsilon + \mu - 1} \right)}}{{2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right) - \left( {\mu - 1} \right)^{2} }}$ to make sure $D_{1n}^{EE*} > 0$; $\theta > \frac{{\left( {\mu - 1} \right)^{2} - 2c_{r} \left( {8\varepsilon + \mu - 1} \right)}}{{\left( {\mu - 1} \right)^{2} - 2c_{n} \left( {8\varepsilon + \mu - 1} \right)}}$ and $c_{n} > \frac{{\left( {\mu - 1} \right)^{2} }}{{2\left( {8\varepsilon + \mu - 1} \right)}}$ to make sure $D_{1r}^{EE*} > 0$; $\theta < \frac{{1 + 2q + 4\varepsilon - \mu + 2q\mu - 3c_{n} + 3c_{r} - c_{n} \mu + c_{r} \mu }}{1 + 2q + 4\varepsilon - \mu + 2q\mu }$ to make sure $D_{2n}^{EE*} > 0$ and $\theta > \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }}$ to make sure $D_{2r}^{EE*} > 0$.

Comparing the boundaries of $\theta$,

$$ \begin{gathered} \frac{{1 + 2q + 4\varepsilon - \mu + 2q\mu - 3c_{n} + 3c_{r} - c_{n} \mu + c_{r} \mu }}{1 + 2q + 4\varepsilon - \mu + 2q\mu } - \left( {1 - \frac{{2\left( {c_{n} - c_{r} } \right)\left( {8\varepsilon + \mu - 1} \right)}}{{2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right) - \left( {\mu - 1} \right)^{2} }}} \right) \hfill \\ = \frac{{\left( {c_{n} - c_{r} } \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)\left( {2\varepsilon + (\mu + 1)\left( {q - 1} \right)} \right)}}{{\left( {1 + 4\varepsilon - \mu + 2q\left( {1 + \mu } \right)} \right)\left( { - \left( { - 1 + \mu } \right)^{2} + 2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right)} \right)}} > 0 \hfill \\ \end{gathered} $$

and $1 - \frac{{2\left( {c_{n} - c_{r} } \right)\left( {8\varepsilon + \mu - 1} \right)}}{{2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right) - \left( {\mu - 1} \right)^{2} }} < 1$ because of $32\varepsilon - \left( {\mu - 1} \right)^{2} > 16 - \left( {\mu - 1} \right)^{2} > 0$,$c_{n} > c_{r}$, $2\varepsilon + (\mu + 1)\left( {q - 1} \right) > 1 - \mu > 0$, $1 + 4\varepsilon - \mu + 2q\left( {1 + \mu } \right) > 5 - \mu > 0$ and $- \left( { - 1 + \mu } \right)^{2} + 2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right) > 2\left( {\mu - 1 + \varepsilon \left( {7 + \mu } \right)} \right) > 0$. Thus the upper boundary of $\theta$ is $1 - \frac{{2\left( {c_{n} - c_{r} } \right)\left( {8\varepsilon + \mu - 1} \right)}}{{2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right) - \left( {\mu - 1} \right)^{2} }}$.

$\frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }} - \frac{{\left( {\mu - 1} \right)^{2} - 2c_{r} \left( {8\varepsilon + \mu - 1} \right)}}{{\left( {\mu - 1} \right)^{2} - 2c_{n} \left( {8\varepsilon + \mu - 1} \right)}} = \frac{{\left( {c_{n} - c_{r} } \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)\left( {1 - \mu } \right)}}{{ - \left( {2 - 2\mu + c_{n} \left( {3 + \mu } \right)} \right)\left( {\left( {\mu - 1} \right)^{2} - 2c_{n} \left( {8\varepsilon + \mu - 1} \right)} \right)}} > 0$ and $0 < \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }} < 1$ because of $c_{n} > c_{r}$, $32\varepsilon - \left( {\mu - 1} \right)^{2} > 0$ and $c_{n} > \frac{{\left( {\mu - 1} \right)^{2} }}{{2\left( {8\varepsilon + \mu - 1} \right)}}$. Thus, the lower boundary of $\theta$ is $\frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }}$;

Furthermore, $1 - \frac{{2\left( {c_{n} - c_{r} } \right)\left( {8\varepsilon + \mu - 1} \right)}}{{2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right) - \left( {\mu - 1} \right)^{2} }} > \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }}$ only if $q < \frac{{\left( {\mu - 1} \right)^{3} - 2\varepsilon \left( {5 + \mu \left( {26 + \mu } \right)} \right)}}{{\left( {3 + \mu } \right)\left( {\mu^{2} - 1} \right)}}$.

Finally, in the scenario of EE, the actual quality $q$ and consumer receptivity to the remanufactured product $\theta$ should satisfy $q < \frac{{\left( {\mu - 1} \right)^{3} - 2\varepsilon \left( {5 + \mu \left( {26 + \mu } \right)} \right)}}{{\left( {3 + \mu } \right)\left( {\mu^{2} - 1} \right)}}$ and $\frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }} < \theta < 1 - \frac{{2\left( {c_{n} - c_{r} } \right)\left( {8\varepsilon + \mu - 1} \right)}}{{2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right) - \left( {\mu - 1} \right)^{2} }}$.

Proof of Proposition 2

According to Table 2, we have $p_{2r}^{{EE{*}}} - p_{2r}^{{NE{*}}} = \frac{{\left( {\mu - 1} \right)\left( {16\varepsilon \left( {1 - \theta } \right) + (\theta c_{n} - c_{r} )\left( {8\varepsilon + \mu - 1} \right)} \right)}}{{64\varepsilon - 2\left( {\mu - 1} \right)^{2} }}$. Then, we have $64\varepsilon - 2\left( {\mu - 1} \right)^{2} > 32 - 2\left( {\mu - 1} \right)^{2} > 0$ and $8\varepsilon + \mu - 1 > 3 - \mu > 0$ because $q > \frac{2 - 2\varepsilon }{{1 + u}}$ and $\varepsilon \le \frac{1}{2}$. Since $\theta c_{n} - c_{r} > 0$ always holds in the NE scenario, we can easily obtain that $\left( {16\varepsilon \left( {1 - \theta } \right) + (\theta c_{n} - c_{r} )\left( {8\varepsilon + \mu - 1} \right)} \right) > 0$ and $\frac{{\left( {\mu - 1} \right)\left( {16\varepsilon \left( {1 - \theta } \right) + (\theta c_{n} - c_{r} )\left( {8\varepsilon + \mu - 1} \right)} \right)}}{{64\varepsilon - 2\left( {\mu - 1} \right)^{2} }} < 0$ because of $\mu < 1$. Thus, $p_{2r}^{{NE{*}}} > p_{2r}^{{EE{*}}}$ always holds.

Proof of Proposition 3

5.1 Proof of Proposition 3(1)

According to the optimal decisions as reported in Table 2, we have $p^{NN*}_{1n} = p^{NE*}_{1n} = p^{EE*}_{1n}$ and $p^{NN*}_{2n} = p^{NE*}_{2n} = p^{EE*}_{2n}$. Without loss of generality, we take the NN scenario as an example. By taking the first-order and second-order derivatives of $p^{NN*}_{1n}$, $p^{NN*}_{2n}$ regarding $\mu$, respectively, we have.

$\frac{{\partial p^{NN*}_{1n} }}{\partial \mu } = \frac{{2c_{n} \left( {\left( {\mu - 1} \right)^{2} + 16\varepsilon \left( {1 + \mu } \right)} \right) - 2\left( {q + q\mu \left( {32\varepsilon + \mu - 2} \right) + \varepsilon \left( {32\varepsilon + \left( {\mu - 1} \right)\left( {15 + \mu } \right)} \right)} \right)}}{{\left( { - 32\varepsilon + \left( {\mu - 1} \right)^{2} } \right)^{2} }}$;

$\begin{gathered} \frac{{\partial^{2} p^{NN*}_{1n} }}{{\partial \mu^{2} }} = \frac{{4c_{n} X_{3} - 4q\left( {X_{3} + 8\varepsilon \left( {32\varepsilon + 3\left( {\mu - 1} \right)^{2} } \right)} \right) - 4\varepsilon \left( {X_{3} + 8\left( {1 - \varepsilon } \right)\left( {32\varepsilon + 3\left( {\mu - 1} \right)^{2} } \right)} \right)}}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{3} }} \\ < \frac{{4(c_{n} - q - \varepsilon )X_{3} }}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{3} }} \\ \end{gathered}$,

wherein $X_{3} = 256\varepsilon^{2} + \left( {\mu - 1} \right)^{3} + 24\varepsilon \left( {\mu - 1} \right)\left( {3 + \mu } \right)$. Since $c_{n} - q - \varepsilon < \frac{{c_{n} \left( {3 + \mu } \right) - 4}}{4} < 0$ and $\frac{{\partial X_{3} }}{\partial \varepsilon } = 512\varepsilon + 24\left( {\mu - 1} \right)\left( {3 + \mu } \right) > 0$, if $\varepsilon > \frac{1 - \mu }{2}$, we have $X_{3} > \left. {X_{3} } \right|_{{\varepsilon = \frac{1 - \mu }{2}}} = \left( {\mu - 1} \right)^{2} \left( {27 - 11\mu } \right) > 0$. Thus, we have $\frac{{\partial^{2} p^{NN*}_{1n} }}{{\partial \mu^{2} }} < \frac{{4(c_{n} - q - \varepsilon )X_{3} }}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{3} }} < 0$. Then, since $q > \frac{2 - 2\varepsilon }{{1 + \mu }} > \frac{1}{2} > \frac{{c_{n} }}{2}$ and $32\varepsilon + (\mu - 1)(\mu + 7) > 0$, we can derive that $\begin{gathered} \frac{{\partial p^{NN*}_{2n} }}{\partial \mu } = \frac{{4\varepsilon \left( {\left( {\mu - 1} \right)^{2} + 24\varepsilon + 8\varepsilon \mu + 2q\left( {32\varepsilon + (\mu - 1)(\mu + 3)} \right) - c_{n} (32\varepsilon + (\mu - 1)(\mu + 7)} \right)}}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{2} }} \\ > \frac{{4\varepsilon \left( {\left( {\mu - 1} \right)^{2} + 24\varepsilon + 8\varepsilon \mu + (2q - c_{n} )(32\varepsilon + (\mu - 1)(\mu + 7)} \right)}}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{2} }} > 0 \\ \end{gathered}$.

Proof of Proposition 3(2)

By taking the first-order derivatives of $p^{NN*}_{1n}$, $p^{NN*}_{2n}$ regarding $\varepsilon$, respectively, we have $\frac{{\partial p^{NN*}_{1n} }}{\partial \varepsilon } = \frac{{2\left( {\mu - 1} \right)\left( {16q\left( {\mu + 1} \right) - 8c_{n} \left( {3 + \mu } \right) + \left( {\mu - 1} \right)\left( {7 + \mu } \right)} \right)}}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{2} }}$ and $\frac{{\partial p^{NN*}_{1n} }}{\partial \varepsilon } \ge 0$ if $16q\left( {\mu + 1} \right) - 8c_{n} \left( {3 + \mu } \right) + \left( {\mu - 1} \right)\left( {7 + \mu } \right) \le 0$, i.e., $q \le \frac{{7 + 24c_{n} - 6\mu + 8c_{n} \mu - \mu^{2} }}{16 + 16\mu }$; and

$\begin{gathered} \frac{{\partial p^{NN*}_{2n} }}{\partial \varepsilon } = \frac{{4\left( {1 + 8\varepsilon \left( {16\varepsilon - 1} \right) - 3\mu - 2q\left( {\mu - 1} \right)^{2} \left( {1 + \mu } \right) + c_{n} \left( {\mu - 1} \right)^{2} \left( {3 + \mu } \right) + \mu \left( {8\varepsilon \left( {2 - \mu } \right) + \mu \left( {3 - \mu } \right)} \right)} \right)}}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{2} }} \\ > \frac{{4\left( {128\varepsilon^{2} + \left( {\mu - 1} \right)^{2} \left( {c_{n} \left( {3 + \mu } \right) - 8\varepsilon - \left( {1 + 3\mu } \right)} \right)} \right)}}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{2} }} \\ \end{gathered}$.

Let $X_{4} { = }128\varepsilon^{2} + \left( {\mu - 1} \right)^{2} \left( {c_{n} \left( {3 + \mu } \right) - 8\varepsilon - \left( {1 + 3\mu } \right)} \right)$ and it can be derived that $\frac{{\partial X_{4} }}{\partial \varepsilon } = 256\varepsilon - 8\left( {1 - \mu } \right)^{2}$. Since $\frac{2 - 2\varepsilon }{{1 + \mu }} < q < 1$, i.e., $\varepsilon > \frac{1 - \mu }{2}$, we can easily obtain that $\frac{{\partial X_{4} }}{\partial \varepsilon } = 256\varepsilon - 8\left( {1 - \mu } \right)^{2} > \left( {1 - \mu } \right)(248 + 8\mu ) > 0$. Thus, we have $X_{4} > \left. {X_{4} } \right|_{{\varepsilon = \frac{1 - \mu }{2}}} = \left( {\mu - 1} \right)^{2} \left( {27 + \mu + c_{n} \left( {3 + \mu } \right)} \right) > 0$ and $\frac{{\partial p^{NN*}_{2n} }}{\partial \varepsilon } > 0$.

Proof of Proposition 3(3)

By taking the first-order derivatives of $p^{NE*}_{2r}$, $p^{EE*}_{2r}$ regarding $\mu$ and $\varepsilon$, respectively, we have $\begin{gathered} \frac{{\partial p^{NE*}_{2r} }}{\partial \mu } = \frac{{4\theta \varepsilon \left( {\left( {\mu - 1} \right)^{2} + 24\varepsilon + 8\varepsilon \mu + 2q\left( {32\varepsilon + (\mu - 1)(\mu + 3)} \right) - c_{n} (32\varepsilon + (\mu - 1)(\mu + 7)} \right)}}{{\left( { - 32T + \left( { - 1 + u} \right)^{2} } \right)^{2} }} \\ = \theta \frac{{\partial p^{NN*}_{2n} }}{\partial \mu } > 0 \\ \end{gathered}$;

$\begin{gathered} \frac{{\partial p^{NE*}_{2r} }}{\partial \varepsilon } = \frac{{4\theta \left( {1 + 8\varepsilon \left( {16\varepsilon - 1} \right) - 3\mu - 2q\left( {\mu - 1} \right)^{2} \left( {1 + \mu } \right) + c_{n} \left( {\mu - 1} \right)^{2} \left( {3 + \mu } \right)} \right)}}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{2} }} \hfill \\ \, + \frac{{4\theta \left( {\mu \left( {8\varepsilon \left( {2 - \mu } \right) + \mu \left( {3 - \mu } \right)} \right)} \right)}}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)^{2} }} = \theta \frac{{\partial p^{NN*}_{2n} }}{\partial \varepsilon } > 0 \hfill \\ \end{gathered}$; and $\begin{gathered} \frac{{\partial p^{EE*}_{2r} }}{\partial \mu } = \frac{{4\varepsilon \left( {2\left( {32\varepsilon + \left( {\mu - 1} \right)^{2} } \right) - c_{r} \left( {32\varepsilon + \left( {\mu - 1} \right)\left( {7 + \mu } \right)} \right)} \right)}}{{\left( {32\varepsilon + \left( {1 - \mu } \right)^{2} } \right)^{2} }} \\ + \frac{{4\varepsilon \theta \left( {8\varepsilon \left( {\mu - 5} \right) - \left( {\mu - 1} \right)^{2} + 2q\left( {32\varepsilon + \left( {\mu - 1} \right)\left( {3 + \mu } \right)} \right)} \right)}}{{\left( {32\varepsilon + \left( {1 - \mu } \right)^{2} } \right)^{2} }} \\ \end{gathered}$.

Since $2\left( {32\varepsilon + \left( {\mu - 1} \right)^{2} } \right) - c_{r} \left( {32\varepsilon + \left( {\mu - 1} \right)\left( {7 + \mu } \right)} \right) > (2 - c_{r} )\left( {32\varepsilon + \left( {\mu - 1} \right)^{2} } \right) > 0$ and let $X_{5} = 8\varepsilon \left( {\mu - 5} \right) - \left( {\mu - 1} \right)^{2} + 2q\left( {32\varepsilon + \left( {\mu - 1} \right)\left( {3 + \mu } \right)} \right)$, we have $\frac{{\partial X_{5} }}{\partial q}{ = }64\varepsilon + 2\left( {\mu - 1} \right)\left( {3 + \mu } \right) > 0$ and $X_{5} > \left. {X_{5} } \right|_{{q = \frac{2 - 2\varepsilon }{{1 + \mu }}}} = \frac{{ - 13 - 128\varepsilon^{2} + 4\varepsilon \left( {\mu - 5} \right)^{2} + 9\mu - \left( {\mu - 5} \right)\mu^{2} }}{1 + \mu } > \left( {1 - \mu } \right)\left( {5 + 3\mu } \right) > 0$. Thus, it is easy to obtain that $\frac{{\partial p^{EE*}_{2r} }}{\partial \mu } > 0$. Then, we have $\frac{{\partial p^{EE*}_{2r} }}{\partial \varepsilon } = \frac{{4\left( {2\left( {1 - \mu } \right)^{3} + c_{r} \left( {1 - \mu } \right)^{2} \left( {3 + \mu } \right) + \theta \left( {128\varepsilon^{2} - 8\varepsilon \left( {1 - \mu } \right)^{2} - \left( {1 - \mu } \right)^{3} - 2q\left( {1 - \mu } \right)^{2} \left( {1 + \mu } \right)} \right)} \right)}}{{\left( {32\varepsilon + \left( {1 - \mu } \right)^{2} } \right)^{2} }}$.Since $2\left( {1 - \mu } \right)^{3} + c_{r} \left( {1 - \mu } \right)^{2} \left( {3 + \mu } \right) > (2 + c_{r} )\left( {1 - \mu } \right)^{3} > 0$ and let $X_{6} { = }128\varepsilon^{2} - 8\varepsilon \left( {1 - \mu } \right)^{2} - \left( {1 - \mu } \right)^{3} - 2q\left( {1 - \mu } \right)^{2} \left( {1 + \mu } \right)$, Similar to $X_{4}$, we have $\frac{{\partial X_{6} }}{\partial \varepsilon } > 0$ when $\varepsilon > \frac{1 - \mu }{2}$. Then, we have $X_{6} > \left. {X_{6} } \right|_{{\varepsilon = \frac{1 - \mu }{2}}} = \left( {\mu - 1} \right)^{2} \left( {27 + 5\mu - 2q\left( {1 + \mu } \right)} \right) > 0$ and $\frac{{\partial p^{EE*}_{2r} }}{\partial \varepsilon } > 0$ accordingly.

Proof of Proposition 4

According to Table 2, denoted by $\Delta p_{n}^{NN*} = p_{1n}^{{NN{*}}} - p_{2n}^{{NN{*}}} = \frac{{ - 1 + q + 22\varepsilon - 8\varepsilon \left( {q + 2\varepsilon } \right) + 2\mu - 2\left( {3 + 4q} \right)\varepsilon \mu - \left( {1 + q} \right)\mu^{2} + c_{n} \left( {\mu - 1} \right)\left( {1 + 4\varepsilon + \mu } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}$. It can be easily obtained that, when $q \le \frac{{\varepsilon \mu (4c_{n} - 6) - (\mu - 1)^{2} + c_{n} (\mu^{2} - 1 - 4\varepsilon ) + 22\varepsilon - 16\varepsilon^{2} }}{{8\varepsilon (1 + \mu ) + \mu^{2} - 1}}$, $\Delta p_{n}^{NN*} > 0$. In addition, considering the condition regarding NN scenario (i.e., $q < \frac{{ - 2c_{n} - 14\varepsilon + 16\varepsilon c_{n} + 2\mu c_{n} - 2\varepsilon \mu }}{{\mu^{2} - 1}}$), we have.

$\begin{gathered} \frac{{ - 2c_{n} - 14\varepsilon + 16\varepsilon c_{n} + 2\mu c_{n} - 2\varepsilon \mu }}{{\mu^{2} - 1}} - \frac{{\varepsilon \mu (4c_{n} - 6) - (\mu - 1)^{2} + c_{n} (\mu^{2} - 1 - 4\varepsilon ) + 22\varepsilon - 16\varepsilon^{2} }}{{8\varepsilon (1 + \mu ) + \mu^{2} - 1}} \hfill \\ = \frac{{\left( {1 - c_{n} } \right)\left( {32\varepsilon - \left( {1 - \mu } \right)^{2} } \right)\left( {4\varepsilon + \mu - 1} \right)}}{{\left( {8\varepsilon + \mu - 1} \right)\left( {1 - \mu^{2} } \right)}} > 0 \hfill \\ \end{gathered}$.

Thus, when $q \le \frac{{\varepsilon \mu (4c_{n} - 6) - (\mu - 1)^{2} + c_{n} (\mu^{2} - 1 - 4\varepsilon ) + 22\varepsilon - 16\varepsilon^{2} }}{{8\varepsilon (1 + \mu ) + \mu^{2} - 1}}$, we have $p_{1n}^{{NN{*}}} \ge p_{2n}^{{NN{*}}}$; otherwise, $p_{1n}^{{NN{*}}} < p_{2n}^{{NN{*}}}$.

Proof of Proposition 5

For simplicity, we set $\Delta \Pi_{1}^{*} = \Pi^{NE*} - \Pi^{NN*}$, $\Delta \Pi_{2}^{*} = \Pi^{EE*} - \Pi^{NN*}$ and $\Delta \Pi_{3}^{*} = \Pi^{EE*} - \Pi^{NE*}$ and thus we have $\Delta \Pi_{1}^{*} = \frac{{\left( {c_{r} - \theta c_{n} } \right)^{2} }}{{8\theta \varepsilon \left( {1 - \theta } \right)}} > 0$ directly.

By comparing the scenarios of EE and NN, we have $\begin{gathered} \Delta \Pi_{2}^{*} = \frac{{\left( {\mu - 1} \right)^{2} + 2c_{r}^{2} \left( {1 + 4\varepsilon + \mu } \right) + \theta \left( {\mu - 1} \right)\left( {2 - 2\mu + c_{n} \left( {3 + \mu } \right)} \right)}}{{\theta \left( {1 - \theta } \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}} \\ + \frac{{\theta^{2} \left( {\left( {\mu - 1} \right)^{2} - c_{n} \left( {\mu - 1} \right)\left( {3 + \mu } \right) + 2c_{n}^{2} \left( {1 + 4\varepsilon + \mu } \right)} \right)}}{{\theta \left( {1 - \theta } \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}} \\ - \frac{{c_{r} \left( {\left( {\mu - 1} \right)\left( {3 + \mu } \right) - \theta \left( {\mu - 1} \right)\left( {3 + \mu } \right) + 4c_{n} \theta \left( {1 + 4\varepsilon + \mu } \right)} \right)}}{{\theta \left( {1 - \theta } \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}} \\ \end{gathered}$.

By taking the first-order derivatives of $\Delta \Pi_{2}^{*}$ regarding $\varepsilon$, we have $\frac{{\partial \Delta \Pi_{2}^{*} }}{\partial \varepsilon } = 8\left( {c_{r} - \theta c_{n} } \right)^{2} > 0$, thus $\Delta \Pi_{2}^{*} > \left. {\Delta \Pi_{2}^{*} } \right|_{{\varepsilon = \frac{1 - \mu }{2}}}$. Next, we denote $X_{7} = \left. {\Delta \Pi_{2}^{*} } \right|_{{\varepsilon = \frac{1 - \mu }{2}}}$ and obtain $\overline{\mu }_{0} = \frac{{ - 1 - c_{r}^{2} + c_{r} \left( { - 1 + \theta + 2\theta c_{n} } \right) + \theta \left( {2 + c_{n} - \theta \left( {1 + c_{n} + c_{n}^{2} } \right)} \right)}}{{\left( {1 - \theta } \right)\left( {\theta - 1 + c_{r} - \theta c_{n} } \right)}}$ when $\frac{{\partial X_{7} }}{\partial \mu } = 0$ and $\frac{{\partial^{2} X_{7} }}{{\partial \mu^{2} }} = 2\left( {\theta - 1} \right)\left( {\theta - 1 + c_{r} - \theta c_{n} } \right) < 0$ respectively.

(1) when $\frac{{c_{r} }}{{c_{n} }} < \theta \le \frac{{2 + c_{r} }}{{2 + c_{n} }}$, $\overline{\mu }_{0}$ satisfies the conditions $q > \frac{2 - 2\varepsilon }{{1 + \mu }}$ and $0 < \mu < 1$. In this case, $X_{7} \ge \left. {X_{7} } \right|_{{\mu = \mu_{0} }} = \frac{{\left( {\theta c_{n} - c_{r} } \right)^{3} \left( {8 + c_{r} - \theta \left( {8 + c_{n} } \right)} \right)}}{{\left( {1 - \theta } \right)\left( {1 - c_{r} + \theta \left( { - 1 + c_{n} } \right)} \right)}} > 0$.

(2) When $\theta \le \frac{{c_{r} }}{{c_{n} }}$ or $\theta > \frac{{2 + c_{r} }}{{2 + c_{n} }}$, we have $\overline{\mu }_{0} > 1$ and $X_{7} \ge \left. {X_{7} } \right|_{\mu = 1} = 4\left( {c_{r} - \theta c_{n} } \right)^{2} > 0$.

Thus, we can obtain that regardless of the value of $\theta$, $\Delta \Pi_{2}^{*} > 0$ always holds.

By comparing the scenario of EE and NE, we have $\Delta \Pi_{3}^{*} = \frac{{8\varepsilon \left( {\theta - 1} \right)^{2} \left( {\mu - 1} \right)^{2} + 8\varepsilon (c_{r} - \theta c_{n} )\left( {\theta - 1} \right)\left( {\mu - 1} \right)\left( {3 + \mu } \right) + \left( {8\varepsilon + \mu - 1} \right)^{2} \left( {c_{r} - c_{n} \theta } \right)^{2} }}{{8\theta \varepsilon \left( {1 - \theta } \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}}$.

It can be derived that $\Delta \Pi_{3}^{*} = 0$ when $\overline{\theta }_{0} = \frac{{ - 2\sqrt 2 X_{8} + 4\varepsilon \left( {1 - \mu } \right)\left( {2 + 3c_{r} + \mu \left( {c_{r} - 2} \right)} \right) + c_{n} X_{9} }}{{8\varepsilon \left( {1 - \mu } \right)^{2} + 8c_{n} \varepsilon \left( {1 - \mu } \right)\left( {3 + \mu } \right) + c_{n}^{2} \left( { - 1 + 8\varepsilon + \mu } \right)^{2} }}$ or $\overline{\theta }_{1} = \frac{{2\sqrt 2 X_{8} + 4\varepsilon \left( {1 - \mu } \right)\left( {2 + 3c_{r} + \mu \left( {c_{r} - 2} \right)} \right) + c_{n} X_{9} }}{{8\varepsilon \left( {1 - \mu } \right)^{2} + 8c_{n} \varepsilon \left( {1 - \mu } \right)\left( {3 + \mu } \right) + c_{n}^{2} \left( { - 1 + 8\varepsilon + \mu } \right)^{2} }}$, wherein $X_{8} = \sqrt {\varepsilon \left( {c_{n} - c_{r} } \right)^{2} \left( {\mu - 1} \right)^{2} \left( {1 - 2\varepsilon } \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}$ and $X_{9} = 4\varepsilon \left( {1 - \mu } \right)\left( {3 + \mu } \right) + c_{r} \left( {8\varepsilon + \mu - 1} \right)^{2}$.

Next, combining the condition of each scenario as noted above, we denote the condition of NN as $q < \tilde{q}_{3}$, the condition of NE as $q < \tilde{q}_{3}$ and $\frac{{c_{r} }}{{c_{n} }} < \theta < \overline{\theta }_{2}$ and the condition of EE as $q < \tilde{q}_{2}$ and $\frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }} < \theta < \overline{\theta }_{3}$, wherein $\tilde{q}_{2} { = }\frac{{\left( {\mu - 1} \right)^{3} - 2\varepsilon \left( {5 + \mu \left( {26 + \mu } \right)} \right)}}{{\left( {3 + \mu } \right)\left( {\mu^{2} - 1} \right)}}$, $\tilde{q}_{3} = \frac{{ - 2c_{n} - 14\varepsilon + 16\varepsilon c_{n} + 2\mu c_{n} - 2\varepsilon \mu }}{{\mu^{2} - 1}}$, $\overline{\theta }_{2} = \frac{{ - c_{r} - 8\varepsilon - 24c_{n} \varepsilon + 32c_{r} \varepsilon + 16q\varepsilon + 32\varepsilon^{2} + 2c_{r} \mu + 8\varepsilon \mu - 8c_{n} \varepsilon \mu + 16q\varepsilon \mu - c_{r} \mu^{2} }}{{ - c_{n} - 8\varepsilon + 8c_{n} \varepsilon + 16q\varepsilon + 32\varepsilon^{2} + 2c_{n} \mu + 8\varepsilon \mu - 8c_{n} \varepsilon \mu + 16q\varepsilon \mu - c_{n} \mu^{2} }}$ and $\overline{\theta }_{3} = 1 - \frac{{2\left( {c_{n} - c_{r} } \right)\left( {8\varepsilon + \mu - 1} \right)}}{{2\varepsilon \left( {7 + \mu } \right) + q\left( {\mu^{2} - 1} \right) - \left( {\mu - 1} \right)^{2} }}$. Comparing the boundaries above, we have the following findings:

(a) $\frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }} > \frac{{3c_{r} + c_{r} \mu }}{{3c_{n} + c_{n} \mu }} = \frac{{c_{r} }}{{c_{n} }}$;

(b)$\overline{\theta }_{0} - \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }}$ increases with $\varepsilon$ and $\overline{\theta }_{0} - \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }} < \left. {\left( {\overline{\theta }_{0} - \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }}} \right)} \right|_{{\varepsilon = \frac{1}{2}}} = 0$;

(c)$\overline{\theta }_{1} - \frac{{c_{r} }}{{c_{n} }}$ increases with $\varepsilon$ and $\left. {\overline{\theta }_{1} - \frac{{c_{r} }}{{c_{n} }} > (\overline{\theta }_{1} - \frac{{c_{r} }}{{c_{n} }})} \right|_{{\varepsilon = \frac{1 - \mu }{2}}} = \frac{{2\left( {c_{n} - c_{r} } \right)\left( {2\left( {1 - \mu } \right) + c_{n} \left( {3 + \mu - \sqrt {\mu \left( {15 + \mu } \right)} } \right)} \right)}}{{c_{n} \left( {\left( {2 + 3c_{n} } \right)^{2} + 4\mu \left( {c_{n} - 1} \right)} \right)}} > 0$.

Similarly, we can obtain that $\overline{\theta }_{1} - \overline{\theta }_{3}$ increases with $\varepsilon$ and $\overline{\theta }_{1} - \overline{\theta }_{3} < \left. {(\overline{\theta }_{1} - \overline{\theta }_{3} )} \right|_{{\varepsilon = \frac{1}{2}}} = \frac{{\left( {c_{n} - c_{r} } \right)\left( {3 + \mu } \right)\left( { - 2 + q + \mu \left( {\mu - 7} \right) - q\mu^{2} + 2c_{n} \left( {3 + \mu } \right)} \right)}}{{\left( {6 - q + 3\mu + \left( { - 1 + q} \right)\mu^{2} } \right)\left( {2 - 2\mu + c_{n} \left( {3 + \mu } \right)} \right)}} < 0$; and.

$\overline{\theta }_{2} - \overline{\theta }_{3}$ increases with $q$ and $\overline{\theta }_{2} - \overline{\theta }_{3} \le 0$ when $q \le \tilde{q}_{1}$ wherein $\tilde{q}_{1} = \frac{{\left( {\mu - 1} \right)^{4} - 2\varepsilon \left( {\mu - 1} \right)^{2} \left( {31 + \mu } \right) + 2c_{n} \left( {\mu - 1} \right)\left( {8\varepsilon + \mu - 1} \right)^{2} - 512\varepsilon^{3} - 128\varepsilon^{2} \left( {\mu - 5} \right)}}{{\left( {256\varepsilon^{2} + \left( {\mu - 1} \right)^{3} } \right)\left( {1 + \mu } \right)}}$.

Thus, we can obtain the following findings:

(1) when $q \le \tilde{q}_{1}$, we have $\overline{\theta }_{2} \le \overline{\theta }_{3}$. If ${{c_{r} } \mathord{\left/ {\vphantom {{c_{r} } {c_{n} }}} \right. \kern-\nulldelimiterspace} {c_{n} }} < \theta \le \min \{ \overline{\theta }_{1} ,\overline{\theta }_{2} \}$, the OEM chooses the NE scenario; if $\min \{ \overline{\theta }_{1} ,\overline{\theta }_{2} \} < \theta \le \overline{\theta }_{3}$, the OEM chooses the EE scenario; otherwise, the NN scenario occurs because it doesn’t satisfy the conditions of both NE and EE.

(2) when $\tilde{q}_{1} < q \le \tilde{q}_{2}$, we have $\overline{\theta }_{2} > \overline{\theta }_{3}$. In this case, if ${{c_{r} } \mathord{\left/ {\vphantom {{c_{r} } {c_{n} }}} \right. \kern-\nulldelimiterspace} {c_{n} }} < \theta \le \overline{\theta }_{1}$ or $\overline{\theta }_{3} < \theta \le \overline{\theta }_{2}$, the OEM chooses the NE scenario; if $\overline{\theta }_{1} < \theta \le \overline{\theta }_{3}$, the OEM chooses the EE scenario; otherwise, the OEM will never sell the remanufactured product in the market, i.e., NN scenario.

(3) when $\tilde{q}_{2} < q \le \tilde{q}_{3}$, the EE scenario never occurs in the market and the OEM always sells the remanufactured product in the second period (i.e., NE scenario) if $\frac{{c_{r} }}{{c_{n} }} < \theta < \overline{\theta }_{2}$. The reason is that $\Delta \Pi_{1}^{*} = \Pi^{NE*} - \Pi^{NN*} > 0$ always holds when the conditions of NE are satisfied.

Proof of Proposition 6

10.1 Proof of Proposition 6(1)

Because of $p_{1n}^{NN*} = p_{1n}^{NE*} = p_{1n}^{EE*}$ and $\tilde{p}_{1n}^{NN*} = \tilde{p}_{1n}^{NE*} = \tilde{p}_{1n}^{EE*}$, we now take the NN scenario as an example. According to the optimal pricing decisions, we have $p_{1n}^{NN*} - \tilde{p}_{1n}^{NN*} = \frac{{\left( {\mu - 1} \right)\left( {1 - 4\varepsilon - \mu - 2q\left( {1 + \mu } \right) + c_{n} \left( {3 + \mu } \right)} \right)}}{{64\varepsilon - 2\left( {\mu - 1} \right)^{2} }}$.

Since $\mu - 1 < 0$, $p_{1n}^{NN*} - \tilde{p}_{1n}^{NN*} > 0$ if $1 - 4\varepsilon - \mu - 2q\left( {1 + \mu } \right) + c_{n} \left( {3 + \mu } \right) < 0$, i.e., $q > \frac{{1 + 3c_{n} + c_{n} \mu - \mu - 4\varepsilon }}{2 + 2\mu }$. In addition, we have $\frac{{1 + 3c_{n} + c_{n} \mu - \mu - 4\varepsilon }}{2 + 2\mu } < \frac{4 - 4\varepsilon }{{2 + 2\mu }} = \frac{2 - 2\varepsilon }{{1 + \mu }}$. Thus, $p_{1n}^{NN*} - \tilde{p}_{1n}^{NN*} > 0$ always holds when $q > \frac{2 - 2\varepsilon }{{1 + \mu }}$.

Similarly, we have $p_{1r}^{EE*} - \tilde{p}_{1r}^{EE*} = \frac{{\left( {\mu - 1} \right)\left( {2 - 2\mu + c_{r} \left( {3 + \mu } \right) + \theta \left( { - 1 - 4\varepsilon + \mu - 2q\left( {1 + \mu } \right)} \right)} \right)}}{{64\varepsilon - 2\left( {\mu - 1} \right)^{2} }}$.

Thus, we have $p_{1r}^{EE*} - \tilde{p}_{1r}^{EE*} > 0$ if $2 - 2\mu + c_{r} \left( {3 + \mu } \right) + \theta \left( { - 1 - 4\varepsilon + \mu - 2q\left( {1 + \mu } \right)} \right) < 0$, i.e., $\theta > \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{1 + 2q + 4\varepsilon - \mu + 2q\mu }$. Following the Proof of Table 2, we have $\theta > \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }}$ in the EE scenario. Since $1 + 2q + 4\varepsilon - \mu + 2q\mu > 5 - \mu > 2 - 2\mu + 3c_{n} + c_{n} \mu > 0$, we derive that $\frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{1 + 2q + 4\varepsilon - \mu + 2q\mu } < \frac{{2 - 2\mu + 3c_{r} + c_{r} \mu }}{{2 - 2\mu + 3c_{n} + c_{n} \mu }}$ always holds and thus $p_{1r}^{EE*} > \tilde{p}_{1r}^{EE*}$.

Proof of Proposition 6(2)

Regarding the second-period pricing decision of the new product, we also take the NN scenario as an example and we have $p_{2n}^{NN*} - \tilde{p}_{2n}^{NN*} = \frac{{32\varepsilon^{2} + \left( {1 - \mu } \right)^{2} + c_{n} \left( {1 - \mu } \right)\left( {8\varepsilon + \mu - 1} \right) + 8\varepsilon \left( { - 5 + \mu + 2q\left( {1 + \mu } \right)} \right)}}{{64\varepsilon - 2\left( {\mu - 1} \right)^{2} }}$.

We can easily find that $p_{2n}^{NN*} - \tilde{p}_{2n}^{NN*}$ is increasing in $q$ and $p_{2n}^{NN*} - \tilde{p}_{2n}^{NN*} > 0$ when $32\varepsilon^{2} + \left( {1 - \mu } \right)^{2} + c_{n} \left( {1 - \mu } \right)\left( {8\varepsilon + \mu - 1} \right) + 8\varepsilon \left( { - 5 + \mu + 2q\left( {1 + \mu } \right)} \right) > 0$, i.e., $q > \overline{q}_{1}$, wherein $\overline{q}_{1} = \frac{{8\varepsilon \left( {5 - \mu } \right) - \left( {\mu - 1} \right)^{2} - c_{n} \left( {1 - \mu } \right)\left( {8\varepsilon + \mu - 1} \right) - 32\varepsilon^{2} }}{{16\varepsilon \left( {1 + \mu } \right)}}$.

Similarly, by comparing the second-period pricing decisions of the remanufactured product in the NE and EE scenarios, respectively, we have $\begin{gathered} p_{2r}^{NE*} - \tilde{p}_{2r}^{NE*} = \frac{{\theta \left( {32\varepsilon^{2} + \left( {1 - \mu } \right)^{2} + c_{n} \left( {1 - \mu } \right)\left( {8\varepsilon + \mu - 1} \right) + 8\varepsilon \left( { - 5 + \mu + 2q\left( {1 + \mu } \right)} \right)} \right)}}{{64\varepsilon - 2\left( {\mu - 1} \right)^{2} }} \\ = \theta (p_{2n}^{NN*} - \tilde{p}_{2n}^{NN*} ) \\ \end{gathered}$.

We can easily verify that $p_{2r}^{NE*} - \tilde{p}_{2r}^{NE*} > 0$ if $q > \overline{q}_{1}$. Similarly, for the EE scenario, we have $p_{2r}^{EE*} - \tilde{p}_{2r}^{EE*} = \frac{{16\varepsilon \left( {\mu - 1} \right) - c_{r} \left( {\mu - 1} \right)\left( {8\varepsilon + \mu - 1} \right) + \theta \left( {32\varepsilon^{2} + \left( {\mu - 1} \right)^{2} + 8\varepsilon \left( {2q\left( {1 + \mu } \right) - 3 - \mu } \right)} \right)}}{{64\varepsilon - 2\left( {\mu - 1} \right)^{2} }}$.

Since $p_{2r}^{EE*} - \tilde{p}_{2r}^{EE*}$ is monotonically increasing in $q$ and $p_{2r}^{EE*} - \tilde{p}_{2r}^{EE*} > 0$ when $16\varepsilon \left( {\mu - 1} \right) - c_{r} \left( {\mu - 1} \right)\left( {8\varepsilon + \mu - 1} \right) + \theta \left( {32\varepsilon^{2} + \left( {\mu - 1} \right)^{2} + 8\varepsilon \left( {2q\left( {1 + \mu } \right) - 3 - \mu } \right)} \right) > 0$. That is, when $q > \overline{q}_{2} = \frac{{16\varepsilon \left( {1 - \mu } \right) + c_{r} \left( {\mu - 1} \right)\left( {8\varepsilon + \mu - 1} \right) - \theta \left( {32\varepsilon^{2} + \left( {\mu - 1} \right)^{2} - 8\varepsilon \left( {3 + \mu } \right)} \right)}}{{16\theta \varepsilon \left( {1 + \mu } \right)}}$, we have $p_{2r}^{EE*} > \tilde{p}_{2r}^{EE*}$.

Proof of Proposition 7

Denote by $\begin{gathered} \Delta \Pi^{NN*} = \Pi^{NN*} - \tilde{\Pi }^{NN*} \\ = \frac{{q^{2} \left( {1 + \mu } \right)^{2} + 2\varepsilon \left( {3 + 2\varepsilon + \mu } \right) + q\left( {1 + \mu } \right)\left( {\mu - 1 + 4\varepsilon } \right){ + }2c_{n}^{2} \left( {\mu + 1 + 4\varepsilon } \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }} \\ + \frac{{ - c_{n} \left( {q\left( {1 + \mu } \right)\left( {3 + \mu } \right) + 2\left( {\mu - 1 + \varepsilon \left( {11 + \mu } \right)} \right)} \right)}}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }} - \frac{1}{2}\left( {1 - c_{n} } \right)^{2} \\ \end{gathered}$.

By taking the first-order derivatives of $\Delta \Pi^{NN*}$ regarding $q$, we have $\frac{{\partial \Delta \Pi^{NN*} }}{\partial q} = \frac{{\left( {1 + \mu } \right)\left( { - 1 + 4\varepsilon + \mu + 2q\left( {1 + \mu } \right) - c_{n} \left( {3 + \mu } \right)} \right)}}{{32\mu - \left( {\mu - 1} \right)^{2} }} > \frac{{\left( {1 - c_{n} } \right)\left( {1 + \mu } \right)\left( {3 + \mu } \right)}}{{32\mu - \left( {\mu - 1} \right)^{2} }} > 0$. It can be derived that $\frac{{\partial \Delta \Pi^{NN*} }}{\partial q} = 0$ when $\overline{q}_{3} = \frac{{c_{n} \left( {1 + \mu } \right)\left( {3 + \mu } \right) + 4\varepsilon \left( {8Y_{1} - \mu - 1} \right) - \left( {\mu - 1} \right)\left( {1 + \mu - Y_{1} + \mu Y_{1} } \right)}}{{2\left( {1 + \mu } \right)^{2} }}$ or $\overline{q}_{30} = \frac{{c_{n} \left( {1 + \mu } \right)\left( {3 + \mu } \right) - 4\varepsilon \left( {8Y_{1} + \mu + 1} \right) - \left( {\mu - 1} \right)\left( {1 + \mu + Y_{1} - \mu Y_{1} } \right)}}{{2\left( {1 + \mu } \right)^{2} }}$, wherein $Y_{1} = \sqrt {\frac{{\left( {c_{n} - 1} \right)^{2} \left( {1 + \mu } \right)^{2} }}{{32\varepsilon - \left( {\mu - 1} \right)^{2} }}}$. In addition, we obtain that $\overline{q}_{30} - \frac{2 - 2\varepsilon }{{1 + \mu }} = - \frac{{\left( {1 - c_{n} } \right)\left( {1 + \mu } \right)\left( {3 + \mu } \right) + Y\left( {32\varepsilon + \left( {\mu - 1} \right)^{2} } \right)}}{{2\left( {1 + \mu } \right)^{2} }} < 0$. Thus, we have $\Pi^{NN*} > \tilde{\Pi }^{NN*}$ if $q > \overline{q}_{3}$.

The proof of NE and EE scenarios are similar to the proof above and thus we omit it here. Finally, we have $\Pi^{NE*} > \tilde{\Pi }^{NE*}$, if $q > \overline{q}_{4} = \frac{1}{{4\left( {1 + \mu } \right)^{2} }}\left( {2c_{n} \left( {1 + \mu } \right)\left( {3 + \mu } \right) + 8\varepsilon \left( { - 1 - \mu + 4\sqrt 2 Y_{2} } \right) - \left( {\mu - 1} \right)\left( {2 + 2\mu - \sqrt 2 Y_{2} + \sqrt 2 \mu Y_{2} } \right)} \right)$, wherein $Y_{2} = \sqrt {\frac{{\left( { - \left( {c_{r} - \theta c_{n} } \right)^{2} + 2\left( {\theta \left( { - 1 + c_{n} } \right)^{2} + \left( {c_{r} - \theta } \right)\left( {c_{r} + \theta - 2\theta c_{n} } \right)} \right)\varepsilon } \right)\left( {1 + \mu } \right)^{2} }}{{\theta \varepsilon \left( {1 - \theta } \right)\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)}}}$.

Similarly, we have $\Pi^{EE*} > \tilde{\Pi }^{EE*}$, if $q > \overline{q}_{5} = \frac{{ - 1 + 4\varepsilon + 4\varepsilon \mu + \mu^{2} + c_{n} \left( {\theta - 1} \right)\left( {1 + \mu } \right)\left( {3 + \mu } \right) - \theta \left( {1 + \mu } \right)\left( { - 1 + 4\varepsilon + \mu } \right) - 2\mu Y_{3} + Y_{3} - 32\varepsilon Y_{3} + \mu^{2} Y_{3} }}{{2\left( {\theta - 1} \right)\left( {1 + \mu } \right)^{2} }}$, wherein $Y_{3} = \frac{{\left( {1 + \mu } \right)\sqrt {\left( {\theta - 1} \right)\left[ \begin{gathered} 2c_{r}^{2} \left( {5 - 16\varepsilon + 2\mu + \mu^{2} } \right){ + }\theta^{2} \left( {32\varepsilon + 3\left( {\mu - 1} \right)^{2} + c_{n}^{2} \left( {3 + \mu } \right)^{2} - 2c_{n} \left( { - 7 + 32\varepsilon + 6\mu + \mu^{2} } \right)} \right) \hfill \\ + \theta \left( { - 32\varepsilon + c_{n}^{2} \left( { - 32\varepsilon + \left( {\mu - 1} \right)^{2} } \right) - 7\left( {\mu - 1} \right)^{2} + 2c_{n} \left( { - 7 + 32\varepsilon + 6\mu + \mu^{2} } \right)} \right) \hfill \\ + 4c_{r} \left( {3 - 2\mu - \mu^{2} + \theta \left( { - 3 + 2\mu + \mu^{2} + c_{n} \left( { - 5 + 16\varepsilon - 2\mu - \mu^{2} } \right)} \right)} \right) + 4\left( {\mu - 1} \right)^{2} \hfill \\ \end{gathered} \right]} }}{{\left( {32\varepsilon - \left( {\mu - 1} \right)^{2} } \right)\sqrt \theta }}$.

Proof of Proposition 8

In the NN scenario, denoted by $\Delta CS^{NN*} = CS^{NN*} - \widetilde{CS}^{NN*}$, it can be derived that $\frac{{\partial^{2} \Delta CS^{NN*} }}{{\partial q^{2} }} > 0$ and $\Delta CS^{NN*} = 0$ when $\overline{q}_{60} = \frac{\begin{gathered} - 256\varepsilon^{3} \left( {\mu - 3} \right) - 4c_{n} \left( {\mu - 1} \right)^{2} \left( {1 + \mu } \right) + 64c_{n} \varepsilon^{2} \left( {3\mu^{2} - 11} \right) + 8c_{n} \varepsilon \left( {9 - 7\mu - 5\mu^{2} + 3\mu^{3} } \right) + \hfill \\ \sqrt 2 \left( {\mu - 1} \right)^{4} Y_{4} + 16\varepsilon^{2} \left( { - 7 + 21\mu - 13\mu^{2} - \mu^{3} + 64\sqrt 2 Y_{4} } \right) - 4T\left( { - 1 + \mu } \right)\left( { - 5 - 12\mu + \mu^{2} - 16\sqrt 2 Y_{4} + 16\sqrt 2 \mu Y_{4} } \right)) \hfill \\ \end{gathered} }{{2\left( {1 + \mu } \right)\left( {16\varepsilon \left( {\mu - 1} \right)^{2} - \left( {\mu - 1} \right)^{2} \left( {1 + \mu } \right) + 64\varepsilon^{2} \left( {3\mu - 5} \right)} \right)}}$, or.

$\overline{q}_{6} = \frac{\begin{gathered} - 256\varepsilon^{3} \left( {\mu - 3} \right) - 4c_{n} \left( {\mu - 1} \right)^{2} \left( {1 + \mu } \right) + 64c_{n} \varepsilon^{2} \left( {3\mu^{2} - 11} \right) + 8c_{n} \varepsilon \left( {9 - 7\mu - 5\mu^{2} + 3\mu^{3} } \right) - \hfill \\ \sqrt 2 \left( {\mu - 1} \right)^{4} Y_{4} - 16\varepsilon^{2} \left( { - 7 + 21\mu - 13\mu^{2} - \mu^{3} + 64\sqrt 2 Y_{4} } \right) - 4T\left( { - 1 + \mu } \right)\left( { - 5 - 12\mu + \mu^{2} - 16\sqrt 2 Y_{4} + 16\sqrt 2 \mu Y_{4} } \right)) \hfill \\ \end{gathered} }{{2\left( {1 + \mu } \right)\left( {16\varepsilon \left( {\mu - 1} \right)^{2} - \left( {\mu - 1} \right)^{2} \left( {1 + \mu } \right) + 64\varepsilon^{2} \left( {3\mu - 5} \right)} \right)}}$,

wherein $Y_{4} = \frac{\begin{gathered} - 128\varepsilon^{3} \left( {\mu - 1} \right)^{2} + 128\varepsilon^{4} \left( {\mu - 1} \right)^{2} - 16\varepsilon \left( {\mu - 1} \right)^{2} \left( {1 + \mu } \right) + \left( {\mu^{2} - 1} \right)^{2} + 16\varepsilon^{2} \left( {13 - \mu - 5\mu^{2} + \mu^{3} } \right) \hfill \\ + c_{n}^{2} \left( { - 64\varepsilon^{2} \left( {\mu - 3^{2} } \right) + \left( {\mu^{2} - 1} \right)^{2} } \right) - 2c_{n} \left( {64\varepsilon^{3} \left( {\mu - 1} \right)^{2} - 8\varepsilon \left( {\mu - 1} \right)^{2} \left( {1 + \mu } \right) + \left( {\mu^{2} - 1} \right)^{2} + 8\varepsilon^{2} \left( {17 + 15\mu - 17\mu^{2} + \mu^{3} } \right)} \right) \hfill \\ \end{gathered} }{{\left( { - 32\varepsilon + \left( { - 1 + \mu } \right)^{2} } \right)^{2} }}$.Accordingly, we have $\overline{q}_{60} < \left. {\overline{q}_{60} } \right|_{{\varepsilon { = }\frac{1 - u}{2}}} = \frac{{58 + 4c_{n} \left( { - 36 + \mu + 9\mu^{2} } \right) - 2\mu \left( {34 + \mu \left( {9 + 2\mu } \right)} \right) + \sqrt 2 \left( {15 + \mu } \right)^{2} Y}}{{2\left( {1 + \mu } \right)\left( { - 73 + 39\mu } \right)}} < \frac{2 - 2\varepsilon }{{1 + \mu }}$. In addition, to satisfy the condition of NN (i.e., $q \le \overline{q}_{1}$), we can derive that $\overline{q}_{1} - \overline{q}_{6}$ increases with $\mu$ and $\left. {\overline{q}_{1} - \overline{q}_{6} } \right|_{{\mu = \frac{2 - q - 2\varepsilon }{q}}} < 0$, $\left. {\overline{q}_{1} - \overline{q}_{6} } \right|_{\mu = 1} > 0$. Thus, there exists a $\overline{\mu }_{1} \in (\frac{2 - q - 2\varepsilon }{q},1)$ so that $\overline{q}_{6} \le \overline{q}_{1}$ when $\mu \ge \overline{\mu }_{1}$. Thus, when $q \le \overline{q}_{6}$ and $\mu \ge \overline{\mu }_{1}$, we have $CS^{NN*} - \widetilde{CS}^{NN*} < 0$.

In the NE scenario, denoted by $\Delta CS^{NE*} = CS^{NE*} - \widetilde{CS}^{NE*}$, it can be verified that $\frac{{\partial \Delta CS^{NE*} }}{\partial q} > 0$ and $\Delta CS^{NE*} > \left. {\Delta CS^{NE*} } \right|_{{q = \frac{2 - 2\varepsilon }{{1 + \mu }}}} > 0$ accordingly. Thus, $CS^{NE*} > \widetilde{CS}^{NE*}$ always holds.