OEM's optimal remanufacturing mode considering patent license under different commission contracts with platform

Abstract

While e-commerce platforms provide an online marketplace for manufacturer's new products and remanufactured products, they also participate in remanufacturing. Manufacturers can take advantage of patent licensing to make a profit from new entrants or still maintain remanufacturing internally. To better understand such a novel business competition and partnership, we develop a two-period game-theoretic model and analyze a market consisting of one OEM, one platform, and heterogeneous consumers in product valuation. Given that a platform may provide two forms of commission (proportional fee contract and per-unit fee contract), two remanufacturing modes of OEM (in-house remanufacturing and authorized remanufacturing) are investigated and it generates four business scenarios. Interestingly, different from previous research, our results reveal that remanufacturing mode selection not only depends on remanufacturing cost but is also affected by market parameters (that is the commission fee). In addition, we found that OEM's royalty fee and remanufacturing strategies show different characteristics to the change of commission under proportional fee contract because of the free-riding behavior of the platform. Finally, we compare the profits of OEM and platform under different commission contracts and gain insights that the proportional fee contract is always favorable for OEM, and commission contract choice of the platform depends on remanufacturing cost.

Appendices

Appendix 1

See Table

Table 7 Summary of thresholds

7.

Appendix 2

In each model, to get the sub-game Nash equilibrium, we first solve the optimization problem of OEM and platform in the second period through the backward induction method, then bring the optimal profit into the first period to get the total profit function, and finally solve for optimal decisions of the first period.

Model AI–Table 3

In the second period, OEM chooses to remanufacture internally and sells two types of products online on the platform. According to backward induction, we first solve the number of new and reman products ($q_{2n}$ and $q_{r}$) in the second period for any given $q_{1n}$. It’s easy to prove that the Hessian matrix of $\Pi_{M}^{AI}$ in ($q_{2n}$,$q_{r}$) is negative definite, so the profit function is jointly concave in ($q_{2n}$,$q_{r}$). We can express the Lagrange function of OEM’s profit as $L_{1}^{AI} = \left[ {\left( {1 - \phi } \right)p_{2n} - c_{n} } \right]q_{2n} + \left[ {\left( {1 - \phi } \right)p_{r} - c_{r} } \right]q_{r} + \lambda_{11} q_{r} + \lambda_{12} \left( {q_{1n} - q_{r} } \right)$, the KKT condition and first-order condition are used to solve the constrained optimization problem. The first-order conditions for $q_{2n}$ and $q_{r}$ are respectively: $\frac{{\partial L_{1}^{AI} }}{{\partial q_{2n} }}{ = }\left( {1 - \phi } \right)\left( {1 - 2q_{2n} - 2\delta q_{r} } \right) - c_{n} = 0$ and $\frac{{\partial L_{1}^{AI} }}{{\partial q_{r} }}{ = }\left( {1 - \phi } \right)\delta \left( {1 - 2q_{2n} - 2q_{r} } \right) - c_{r} + \lambda_{11} - \lambda_{12} { = }0$, the complementary relaxation conditions are $\lambda_{11} q_{r} { = }0$ and $\lambda_{12} \left( {q_{1n} - q_{r} } \right){ = }0$, and the feasible conditions are $\lambda_{11} \ge 0$, $\lambda_{12} \ge 0$, $q_{r} \ge 0$, $q_{1n} \ge q_{r}$,$q_{2n} > 0$. We can characterize the feasible region of equilibrium solutions by $c_{r}$. Given the number of new products $q_{1n}$ in the first period, solve the first-order condition, and we obtain the following:

NR ($q_{r} { = }0$): $q_{2n} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}}$, $q_{r} = 0$, $\lambda_{11} { = }c_{r} - \delta c_{n}$, $\lambda_{12} { = }0$. From $\lambda_{11} \ge 0$, we get the feasible region $c_{r} \ge c_{r1}^{AI}$.

PR ($0 < q_{r} < q_{1n}$): $q_{2n}^{{}} = \frac{1}{2}\left( {1 - \frac{{c_{n} - c_{r} }}{{\left( {1 - \delta } \right)\left( {1 - \phi } \right)}}} \right)$, $q_{r} = \frac{{\delta c_{n} - c_{r} }}{{2\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)}}{ = }K^{AI}$, $\lambda_{11} { = 0}$, $\lambda_{12} { = }0$. From $0 < q_{r} < q_{1n}$, we get the feasible region $c_{r} < c_{r1}^{AI}$ and $q_{1n} > K^{AI}$.

FR ($0 < q_{r} { = }q_{1n}$): $q_{2n} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}} - \delta q_{1n}$, $q_{r} { = }q_{1n}$, $\lambda_{11} { = 0}$, $\lambda_{12} { = }\delta c_{n} - c_{r} - 2\delta q_{1n} \left( {1 - \delta - \phi - \delta \phi } \right)$. From $\lambda_{12} \ge 0$, we get the feasible region $q_{1n} \le K^{AI}$.

Next, substituting the above response functions and constraints, we can express the OEM’s gross profit function in the first period as $\Pi_{M}^{AI} = \left[ {\left( {1 - \phi } \right)p_{1n} - c_{n} } \right]q_{1n} + \Pi_{M2}^{*}$, then we solve for $q_{1n}$. For the case of $q_{1n} > K^{AI}$, we can directly use the first-order condition; for the case of $q_{1n} \le K^{AI}$, we write the Lagrange function of $\Pi_{M}^{AI}$ as $L_{2}^{AI} = \left[ {\left( {1 - \phi } \right)p_{1n} - c_{n} } \right]q_{1n} + \Pi_{M2}^{*} + \lambda_{2} \left( {K^{AI} - q_{1n} } \right)$. The final results are as follows.

NR ($q_{r} { = }0$): $q_{1n}^{{}} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}}$, $q_{2n} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}}$, $q_{r} = 0$, and the feasible condition is $c_{r} \ge c_{r1}^{AI}$. The optimal profits of both members are $\Pi_{M}^{AI0} { = }\frac{{\left( {1 - c_{n} - \phi } \right)^{2} }}{{2\left( {1 - \phi } \right)}}$ and $\Pi_{E}^{AI0} { = }\frac{{\phi \left( {1 - \phi + c_{n} } \right)\left( {1 - \phi - c_{n} } \right)}}{{2\left( {1 - \phi } \right)^{2} }}$.

PR ($0 < q_{r} < q_{1n}$): $q_{1n}^{{}} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}}$, $q_{2n}^{{}} = \frac{1}{2}\left( {1 - \frac{{c_{n} - c_{r} }}{{\left( {1 - \delta } \right)\left( {1 - \phi } \right)}}} \right)$, $q_{r} = \frac{{\delta c_{n} - c_{r} }}{{2\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)}}{ = }K^{AI}$, and the feasible condition is $c_{r2}^{AI} \le c_{r} < c_{r1}^{AI}$. The optimal profits of both members are $\Pi_{M}^{AI + } { = }\frac{{\left( {1 - c_{n} - \phi } \right)^{2} }}{{4\left( {1 - \phi } \right)}} + \frac{1}{4}\left( {1 - \phi - 2c_{n} + \frac{{\left( {c_{n} \delta - c_{r} } \right)^{2} }}{{\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)}}} \right)$ and $\Pi_{E}^{AI + } { = }\frac{{\phi \left( {1 - \phi + c_{n} } \right)\left( {1 - \phi - c_{n} } \right)}}{{4\left( {1 - \phi } \right)^{2} }} + \frac{\phi }{4}\left( {1 - \frac{{\left( {\delta c_{n} - c_{r} } \right)^{2} }}{{\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)^{2} }}} \right)$.

FR ($0 < q_{r} { = }q_{1n}$): $q_{1n}^{{}} = \frac{{1 - \phi - c_{n} + \delta c_{n} - c_{r} }}{{2\left( {1 - \phi } \right)\left( {1 + \delta - \delta^{2} } \right)}}$, $q_{2n} = \frac{{\left( {1 - \phi } \right)\left( {1 - \delta^{2} } \right) + \delta c_{r} - c_{n} }}{{2\left( {1 - \phi } \right)\left( {1 + \delta - \delta^{2} } \right)}}$, $q_{r} { = }q_{1n}$, and the feasible condition is $c_{r} < c_{r2}^{AI}$. The optimal profits of both members are $\Pi_{M}^{{AI{ = }}} { = }\frac{{\left\{ {c_{r}^{2} + c_{n}^{2} \left( {2 - \delta } \right) + 2c_{n} c_{r} \left( {1 + \delta } \right) - 2c_{n} \left( {2 - \delta^{2} } \right)\left( {1 - \phi } \right) - 2c_{r} \left( {1 - \phi } \right) + \left( {2 - \delta } \right)\left( {1 + \delta } \right)\left( {1 - \phi } \right)^{2} } \right\}}}{{4\left[ {1 + \delta \left( {1 - \delta } \right)} \right]\left( {1 - \phi } \right)}}$ and $\Pi_{E}^{{AI{ = }}} { = }\frac{{\phi \left( {c_{r}^{2} + c_{n}^{2} \left( {2 - \delta } \right) + 2c_{n} c_{r} \left( {1 + \delta } \right) - \left( {2 - \delta } \right)\left( {1 + \delta } \right)\left( {1 - \phi } \right)^{2} } \right)}}{{4\left[ {1 + \delta \left( {1 - \delta } \right)} \right]\left( {1 - \phi } \right)}}$.

The equilibrium of model AI is obtained (See Table 3). By calculation, it is proved that the optimal profit is obtained under the PR strategy.

Model AT–Table 4

In the second period, OEM sells only new products, while the platform is authorized to produce and sell reman products. According to backward induction, there is a Cournot quantity competition game between the OEM and the platform in the second period and they make quantity decisions at the same time, then the OEM decides the royalty fee.

$\frac{{\partial \Pi_{M2}^{{}} }}{{\partial q_{2n} }}{ = }\left( {1 - \phi } \right)\left( {1 - 2q_{2n} - \delta q_{r} } \right) - c_{n}$, $\frac{{\partial^{2} \Pi_{M2}^{{}} }}{{\partial q_{2n}^{2} }}{ = } - 2\left( {1 - \phi } \right) < 0$, so $\Pi_{M2}^{{}}$ is concave in $q_{2n}$. From the first order condition $\frac{{\partial \Pi_{M2}^{{}} }}{{\partial q_{2n} }} = 0$, we have $q_{2n} = \frac{{\left( {1 - \phi } \right)\left( {1 - \delta q_{r} } \right) - c_{n} }}{{2\left( {1 - \phi } \right)}}$. We can write the Lagrange function of the platform profit as $L_{3}^{AT} = \phi p_{2n} q_{2n} + \left( {p_{r} - c_{r} - f} \right)q_{r} + \lambda_{31} q_{r} + \lambda_{32} \left( {q_{1n} - q_{r} } \right)$. The first-order condition of $q_{r}$ is $\frac{{\partial L_{3}^{AT} }}{{\partial q_{r} }}{ = } - \delta \phi q_{2n} + \delta \left( {1 - q_{2n} - 2q_{r} } \right) - c_{r} - f + \lambda_{31} - \lambda_{32} { = }0$, the complementary relaxation condition is $\lambda_{31} q_{r} { = }0$ and $\lambda_{32} \left( {q_{1n} - q_{r} } \right){ = }0$, and feasible conditions are $\lambda_{31} \ge 0$, $\lambda_{32} \ge 0$, $q_{r} \ge 0$, $q_{1n} \ge q_{r}$, $q_{2n} > 0$. Given the number of new products in the first period $q_{1n}$, we solve the first-order condition and get the following:

NR ($q_{r} { = }0$): $q_{2n} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}}$, $q_{r} = 0$, $\lambda_{31} { = }f + c_{r} - \frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}}$. From $\lambda_{31} \ge 0$, we get the feasible condition $f \ge \frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}} - c_{r}$.

PR ($0 < q_{r} < q_{1n}$):$q_{r} = \frac{{\delta \left( {1 + c_{n} } \right) - 2\left( {1 - \phi } \right)\left( {f + c_{r} } \right) - \delta \phi \left( {2 - c_{n} - \phi } \right)}}{{\delta \left( {1 - \phi } \right)\left( {4 - \delta - \delta \phi } \right)}}$, $q_{2n} = \frac{{\left( {1 - \phi } \right)\left( {2 + f + c_{r} - \delta } \right) - 2c_{n} }}{{\left( {1 - \phi } \right)\left( {4 - \delta - \delta \phi } \right)}}$. From $0 < q_{r} < q_{1n}$, we get the feasible condition $\frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}} - c_{r} - \frac{\delta }{2}q_{1n} \left( {4 - \delta - \delta \phi } \right) < f < \frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}} - c_{r}$.

FR ($0 < q_{r} { = }q_{1n}$): $q_{2n}^{{}} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}} - \frac{{\delta q_{1n} }}{2}$,$q_{r}^{{}} = q_{1n}^{{}}$,$\lambda_{32} { = } - f - c_{r} + \frac{\delta }{2}\left( {1 - \phi + \frac{{c_{n} \left( {1 + \phi } \right)}}{1 - \phi } - q_{1n} \left( {4 - \delta - \delta \phi } \right)} \right)$. From $\lambda_{32} \ge 0$, we get the feasible condition $f \le \frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}} - c_{r} - \frac{\delta }{2}q_{1n} \left( {4 - \delta - \delta \phi } \right)$.

Next, the equilibrium quantities of new and reman products ($q_{r}^{*}$, $q_{2n}^{*}$) are brought into the OEM’s profit function to solve the optimal royalty fee $f$.

NR ($q_{r} { = }0$): Write $L_{4}^{AT} = \left[ {\left( {1 - \phi } \right)p_{2n} - c_{n} } \right]q_{2n} + fq_{r} + \lambda_{4} \left( {f + c_{r} - \frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}}} \right)$, let $\frac{{\partial L_{4}^{AT} }}{\partial f} = \lambda_{4} = 0$, so $\lambda_{4} = 0$, there is no solution.

PR ($0 < q_{r} < q_{1n}$): From first-order condition we directly get $f = \frac{{\delta \left( {1 - \phi } \right)\left( {1 - \phi - c_{n} } \right) + c_{n} \delta }}{{2\left( {1 - \phi } \right)}} - \frac{1}{2} \cdot \frac{{4c_{r} \left( {2 - \delta } \right) + c_{n} \delta ^{2} }}{{8 - \delta \left( {3 + \phi } \right)}}$, then $q_{r}^{{}} = \frac{{2\left[ {\delta c_{n} - c_{r} \left( {1 - \phi } \right)} \right]}}{{\delta \left( {1 - \phi } \right)\left[ {8 - \delta \left( {3 + \phi } \right)} \right]}}{ = }K^{AT}$, $q_{2n}^{{}} = \frac{{\left( {1 - \phi } \right)\left[ {8 - \delta \left( {3 + \phi } \right) + 2c_{r} } \right] - c_{n} \left( {8 - \delta - \delta \phi } \right)}}{{2\left( {1 - \phi } \right)\left[ {8 - \delta \left( {3 + \phi } \right)} \right]}}$. From $0 < q_{r} < q_{1n}$, we get the feasible condition $c_{r} < c_{r1}^{AT}$ and $q_{1n} > K^{AT}$.

FR ($0 < q_{r} { = }q_{1n}$): We can write the Lagrange function as $L_{5}^{AT} = \left[ {\left( {1 - \phi } \right)p_{2n} - c_{n} } \right]q_{2n} + fq_{r} + \lambda_{5} \left( {\frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}} - c_{r} - \frac{\delta }{2}q_{1n} \left( {4 - \delta - \delta \phi } \right) - f} \right)$, let $\frac{{\partial L_{4}^{AT} }}{\partial f} = q_{r} - \lambda_{5} = 0$, so $\lambda_{5} { = }q_{r} > 0$ and $f{ = }\frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}} - c_{r} - \frac{\delta }{2}q_{1n} \left( {4 - \delta - \delta \phi } \right)$, the feasible condition is $0 < q_{1n} \le K^{AT}$.

Finally, substituting the above response functions of second period ($q_{2n}$, $f$, $q_{r}$) into the first period, OEM’s gross profit function is $\Pi_{M}^{AT} = \left[ {\left( {1 - \phi } \right)p_{1n} - c_{n} } \right]q_{1n} + \Pi_{M2}^{*}$, then we solve for the quantity decision ($q_{1n}$). For the case of $q_{1n} > K^{AT}$, we can directly derive $q_{1n}$ from the first-order condition; For the case of $q_{1n} \le K^{AT}$, we write the Lagrange function as $L_{6}^{AT} = \left[ {\left( {1 - \phi } \right)p_{1n} - c_{n} } \right]q_{1n} + \Pi_{M2}^{*} + \lambda_{2} \left( {K^{AT} - q_{1n} } \right)$. The final results are as follows:

NR ($q_{r} { = }0$): $q_{1n}^{{}} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}}$, $q_{2n} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}}$, $q_{r} = 0$, and the feasible condition is $c_{r} \ge c_{r1}^{AT}$. The optimal profits of both members are $\Pi_{M}^{AT0} { = }\frac{{\left( {1 - c_{n} - \phi } \right)^{2} }}{{2\left( {1 - \phi } \right)}}$, $\Pi_{E}^{AT0} { = }\frac{{\phi \left( {1 - \phi + c_{n} } \right)\left( {1 - \phi - c_{n} } \right)}}{{2\left( {1 - \phi } \right)^{2} }}$.

PR ($0 < q_{r} < q_{1n}$): $q_{1n}^{{}} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}}$, $f = \frac{{\delta \left( {1 - \phi } \right)\left( {1 - \phi - c_{n} } \right) + c_{n} \delta }}{{2\left( {1 - \phi } \right)}} - \frac{1}{2} \cdot \frac{{4c_{r} \left( {2 - \delta } \right) + c_{n} \delta^{2} }}{{8 - \delta \left( {3 + \phi } \right)}}$, $q_{r} = \frac{{\delta \left( {1 + c_{n} } \right) - 2\left( {1 - \phi } \right)\left( {f + c_{r} } \right) - \delta \phi \left( {2 - c_{n} - \phi } \right)}}{{\delta \left( {1 - \phi } \right)\left( {4 - \delta - \delta \phi } \right)}}$, $q_{2n} = \frac{{\left( {1 - \phi } \right)\left( {2 + f + c_{r} - \delta } \right) - 2c_{n} }}{{\left( {1 - \phi } \right)\left( {4 - \delta - \delta \phi } \right)}}$, and the feasible condition is $c_{r2}^{AT} \le c_{r} < c_{r1}^{AT}$. The optimal profits of both members are $\Pi_{M}^{AT + } { = }\frac{{\left\{ { - 4c_{n} c_{r} \delta \left( {1 - \phi } \right) + 2c_{r}^{2} \left( {1 - \phi } \right)^{2} + \delta \left( {\left( {\left( {1 - \phi } \right)^{3} - 2c_{n} \left( {1 - \phi } \right)^{2} } \right)\left( {8 - \delta \left( {3 + \phi } \right)} \right) + c_{n}^{2} \left( {8 - 8\phi + \delta \left( {\phi \left( {2 + \phi } \right) - 1} \right)} \right)} \right)} \right\}}}{{2\delta \left( {1 - \phi } \right)^{2} \left[ {8 - \delta \left( {3 + \phi } \right)} \right]}}$ and $\Pi_{E}^{AT + } { = }\frac{{\left\{ \begin{gathered} - 2c_{n} c_{r} \delta \left( {1 - \phi } \right)\left( {8 - \phi \left( {8 - \delta - \delta \phi } \right)} \right) + 2c_{r}^{2} \left( {1 - \phi } \right)^{2} \left( {4 - \delta \phi } \right) + \hfill \\ \delta \left[ {\phi \left( {1 - \phi } \right)^{2} \left[ {8 - \delta \left( {3 + \phi } \right)} \right]^{2} - c_{n}^{2} \left( {64\phi - \delta \left( {8 + 16\phi \left( {2 + \phi } \right) - \delta \phi \left( {5 + \phi \left( {4 + \phi } \right)} \right)} \right)} \right)} \right] \hfill \\ \end{gathered} \right\}}}{{2\delta \left( {1 - \phi } \right)^{2} \left[ {8 - \delta \left( {3 + \phi } \right)} \right]^{2} }}$.

FR ($0 < q_{r} { = }q_{1n}$):$q_{1n}^{{}} = \frac{{2\left( {1 - \phi } \right)\left( {1 - c_{r} - \phi } \right) - 2c_{n} \left( {1 - \delta - \phi } \right)}}{{\left( {1 - \phi } \right)\left[ {4\left( {1 - \phi } \right) + \delta \left[ {8 - \delta \left( {3 + \phi } \right)} \right]} \right]}}$, then $q_{2n}^{{}} = \frac{{1 - \phi - c_{n} }}{{2\left( {1 - \phi } \right)}} - \frac{{\delta q_{1n}^{{}} }}{2}$, $f{ = }\frac{{\delta \left[ {\left( {1 - \phi } \right)^{2} + c_{n} \left( {1 + \phi } \right)} \right]}}{{2\left( {1 - \phi } \right)}} - c_{r} - \frac{\delta }{2}q_{1n}^{{}} \left( {4 - \delta - \delta \phi } \right)$, $q_{r} { = }q_{1n}$, and the feasible condition is $c_{r} < c_{r2}^{AT}$. The optimal profits of both members are $\Pi_{M}^{{AT{ = }}} { = }\frac{{\left\{ \begin{gathered} c_{n}^{2} \left( {8\left( {1 - \phi } \right)^{2} + \delta^{2} \left( {1 + \phi } \right)^{2} } \right) + \left( {1 - \phi } \right)^{2} \left( {4c_{r}^{2} - 8c_{r} \left( {1 - \phi } \right) + \left( {1 - \phi } \right)\left( {8\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right)} \right) \hfill \\ + 2c_{n} \left( {1 - \phi } \right)\left( {4c_{r} \left( {1 - \delta - \phi } \right) - \left( {1 - \phi } \right)\left( {8\left( {1 - \phi } \right) - \delta \left( {4 - \delta \left( {3 + \phi } \right)} \right)} \right)} \right) \hfill \\ \end{gathered} \right\}}}{{4\left( {1 - \phi } \right)^{2} \left[ {4\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right]}}$, $\Pi_{E}^{AT = } { = }\frac{{\left\{ \begin{gathered} 4\delta \left( {4 - \delta \phi } \right)\left( {\left( {1 - \phi } \right)\left( {1 - c_{r} - \phi } \right) - c_{n} \left( {1 - \delta - \phi } \right)} \right)^{2} - 4c_{n} \delta \phi \left( {\left( {1 - \phi } \right)\left( {1 - c_{r} - \phi } \right) - c_{n} \left( {1 - \delta - \phi } \right)} \right) \hfill \\ \left( {4\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right) + \phi \left( {1 + c_{n} - \phi } \right)\left( {1 - c_{n} - \phi } \right)\left( {4\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right)^{2} + \hfill \\ 8\phi \left( {\left( {1 - \phi } \right)\left( {1 - c_{r} - \phi } \right) - c_{n} \left( {1 - \delta - \phi } \right)} \right)\left( {2c_{n} \left( {1 - \delta - \phi } \right) + \left( {1 - \phi } \right)\left( {2 + 2c_{r} - 2\phi + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right)} \right) \hfill \\ \end{gathered} \right\}}}{{4\left( {1 - \phi } \right)^{2} \left[ {4\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right]^{2} }}$.

The equilibrium of model AT is obtained (See Table 4). Through simple comparison, it is proved that the optimal profit is obtained under strategy PR.

Model FI–Table 5 The solution steps are similar to model AI and omitted here.

Model FT–Table 6 The solution steps are similar to model AT and omitted here.

Proof of Corollary 1

Based on initial assumptions, we examine the partial derivative of strategy thresholds of model AI to market parameters ($c_{n}$ and $\phi$), i.e. $\frac{{\partial c_{r1}^{AI} }}{{\partial c_{n} }}{ = }\delta > 0$, $\frac{{\partial c_{r2}^{AI} }}{\partial \phi }{ = }\delta \left( {1 - \delta } \right) > 0$,$\frac{{\partial c_{r2}^{AI} }}{{\partial c_{n} }}{ = }\delta \left( {2 - \delta } \right) > 0$. Therefore, Corollary 1 holds.

Proof of Corollary 2

Under the strategy PR of model AI, we take the partial derivative of decision variables to the proportional fee ($\phi$). Under the conditions $c_{r1}^{AI} > c_{r}$ and $1 - c_{n} - \phi > 0$($q_{1n} > 0$), it is clear that $\frac{{\partial q_{1n}^{AI + } }}{\partial \phi }{ = }\frac{ - 1}{{2\left( {1 - \phi } \right)}} + \frac{{1 - c_{n} - \phi }}{{2\left( {1 - \phi } \right)^{2} }} < 0$, $\frac{{\partial q_{2n}^{AI + } }}{\partial \phi }{ = }\frac{{c_{n} - c_{r} }}{{ - 2\left( {1 - \delta } \right)^{2} \left( {1 - \phi } \right)}} < 0$, $\frac{{\partial q_{r}^{AI + } }}{\partial \phi }{ = }\frac{{\delta c_{n} - c_{r} }}{{2\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)^{2} }} > 0$.

Under the strategy FR of model AI, we take the partial derivative of decision variables to the proportional fee ($\phi$). Given the feasible condition ($c_{r1}^{AI} > c_{r2}^{AI} > c_{r}$ and $1 - c_{n} - \phi > 0$), it is clear that:$\frac{{\partial q_{1n}^{{AI{ = }}} }}{\partial \phi }{ = } - \frac{1}{{2\left( {1 + \left( {1 - \delta } \right)\delta } \right)\left( {1 - \phi } \right)}} + \frac{{ - 1 + c_{n} + \phi - c_{n} \delta + c_{r} }}{{ - 2\left( {1 + \left( {1 - \delta } \right)\delta } \right)\left( {1 - \phi } \right)^{2} }} < 0$, $\frac{{\partial q_{2n}^{{AI{ = }}} }}{\partial \phi }{ = }\frac{{c_{n} - c_{r} \delta }}{{ - 2\left( {1 + \left( {1 - \delta } \right)\delta } \right)\left( {1 - \phi } \right)^{2} }} < 0$. Therefore, Corollary 2 holds.

Proof of Corollary 3

Based on initial assumptions, we examine the partial derivative of strategy thresholds of model AT to market parameters ($c_{n}$ and $\phi$), i.e. $\frac{{\partial c_{r1}^{AT} }}{\partial \phi }{ = }\frac{{c_{n} \delta }}{{\left( {1 - \phi } \right)^{2} }} > 0$,$\frac{{\partial c_{r1}^{AT} }}{{\partial c_{n} }}{ = }\frac{\delta }{1 - \phi } > 0$, $\frac{{\partial c_{r2}^{AT} }}{\partial \phi }{ = }\frac{1}{4}\delta \left( {\delta + \frac{{4c_{n} \left( {3 - \delta } \right)}}{{\left( {1 - \phi } \right)^{2} }}} \right) > 0$, $\frac{{\partial c_{r2}^{AT} }}{{\partial c_{n} }}{ = }\frac{{\delta \left( {12 - \delta \left( {3 + \phi } \right)} \right)}}{{4\left( {1 - \phi } \right)}} > 0$. Therefore, Corollary 3 holds.

Proof of Corollary 4

Under the strategy PR, take the partial derivative of decision variables to proportional fee ($\phi$) as:$\frac{{\partial q_{r}^{AT + } }}{\partial \phi }{ = }\frac{{2\left( {2c_{n} \left( {4 - \delta - \delta \phi } \right) - c_{r} \left( {1 - \phi } \right)^{2} } \right)}}{{\left( {1 - \phi } \right)^{2} \left( {8 - \delta \left( {3 + \phi } \right)} \right)^{2} }}$, $\frac{{\partial q_{2n}^{AT + } }}{\partial \phi }{ = }\frac{{2c_{r} \delta \left( {1 - \phi } \right)^{2} - c_{n} \left( {64 - 16\delta \left( {2 + \phi } \right) + \delta^{2} \left( {5 + \phi \left( {2 + \phi } \right)} \right)} \right)}}{{2\left( {1 - \phi } \right)^{2} \left( {8 - \delta \left( {3 + \phi } \right)} \right)^{2} }}$, $\frac{{\partial q_{1n}^{AT + } }}{\partial \phi }{ = }\frac{ - 1}{{2\left( {1 - \phi } \right)}} + \frac{{1 - c_{n} - \phi }}{{2\left( {1 - \phi } \right)^{2} }}$. Taking initial assumptions into consideration, we have $\frac{{\partial q_{r}^{AT + } }}{\partial \phi } > 0$, $\frac{{\partial q_{2n}^{AT + } }}{\partial \phi } < 0$, $\frac{{\partial q_{1n}^{AT + } }}{\partial \phi } < 0$.

Under the strategy FR, we take the partial derivative of decision variables to proportional fee ($\phi$) $\frac{{\partial q_{1n}^{{AT{ = }}} }}{\partial \phi }{ = }\frac{{ - 2\left( {1 - \phi } \right)^{2} \left( {4c_{r} + 8\delta - \delta^{2} \left( {4 - c_{r} } \right)} \right) - 2c_{n} \left( { - 8\delta \left( {1 - \phi } \right) + 4\left( {1 - \phi } \right)^{2} + 2\delta^{3} \left( {1 + \phi } \right) - \delta^{2} \left( {7 + \phi \left( {2 - \phi } \right)} \right)} \right)}}{{\left( {1 - \phi } \right)^{2} \left( {4\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right)^{2} }}$, the positive or negative sign of which depends on its numerator. Let the numerator be equal to 0, we have $c_{r} = c_{r1}^{A}$. So, given the feasible condition of strategy PR ($0 < c_{r2}^{AT} \le c_{r} < c_{r1}^{AT} < c_{n}$), we have $\frac{{\partial q_{1n}^{{AT{ = }}} }}{\partial \phi } < 0$ if $c_{r} > \max \{ c_{r1}^{A} ,0\}$; Otherwise $\frac{{\partial q_{1n}^{{AT{ = }}} }}{\partial \phi } > 0$.

Under the feasible conditions of $c_{r1}^{AT} > c_{r2}^{AT} > c_{r}$ and $\left( {1 - \phi } \right)\left( {1 - c_{r} - \phi } \right) - c_{n} \left( {1 - \delta - \phi } \right) > 0$($q_{1n} > 0$), it is straightforward to show that $\frac{{\partial q_{2n}^{{AT{ = }}} }}{\partial \phi }{ = }\frac{{2\delta \left( {4c_{r} + 8\delta - \delta^{2} \left( {4 - cr} \right)} \right)\left( {1 - \phi } \right)^{2} + c_{n} \left( \begin{gathered} - 16\left( {1 - \phi } \right)^{2} - 8\delta \left( {1 - \phi } \right)\left( {7 + \phi } \right) - 8\delta^{2} \left( {7 + \phi^{2} } \right) \hfill \\ - \delta^{4} \left( {5 + \phi \left( {2 + \phi } \right)} \right) + 2\delta^{3} \left( {17 + \phi \left( {6 + \phi } \right)} \right) \hfill \\ \end{gathered} \right)}}{{2\left( {1 - \phi } \right)^{2} \left( {4\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right)^{2} }} < 0$.

Therefore, Corollary 4 holds.

Proof of Corollary 5

Take the partial derivative of royalty fee ($f^{AT}$) to $\delta$ and $c_{r}$, we can easily derive:$\frac{{\partial f^{AT} }}{\partial \delta } = \frac{1}{2}\left( {1 - c_{n} - \phi + \frac{{c_{n} }}{1 - \phi } + \frac{{4c_{r} - 2c_{n} \delta }}{{8 - \delta \left( {3 + \phi } \right)}} - \frac{{\left( {3 + \phi } \right)\left( {4c_{r} \left( {2 - \delta } \right) + c_{n} \delta^{2} } \right)}}{{\left( {8 - \delta \left( {3 + \phi } \right)} \right)^{2} }}} \right) > 0$, $\frac{{\partial f^{AT} }}{{\partial c_{r} }} = - \frac{{2\left( {2 - \delta } \right)}}{{8 - \delta \left( {3 + \phi } \right)}} < 0$.$\frac{{\partial f^{AT} }}{{\partial c_{n} }} = \frac{1}{2}\delta \left( {\frac{\phi }{1 - \phi } - \frac{\delta }{{8 - \delta \left( {3 + \phi } \right)}}} \right)$, let $\frac{\partial f}{{\partial c_{n} }}{ = }0$ then $\delta { = }\frac{8\phi }{{\left( {1 + \phi } \right)^{2} }}$ is obtained. Considering the feasible condition of strategy PR, i.e. $0 < c_{r2}^{AT} \le c_{r} < c_{r1}^{AT} < c_{n}$, we find that when $\frac{{\partial f^{AT} }}{{\partial c_{n} }} < 0$, $\phi < \phi_{1}^{A}$ and $\delta > \frac{8\phi }{{\left( {1 + \phi } \right)^{2} }}$ should be satisfied; When $\frac{{\partial f^{AT} }}{{\partial c_{n} }} > 0$, we can derive the following two conditions: (1) $\phi < \phi_{1}^{A}$ and $\delta < \frac{8\phi }{{\left( {1 + \phi } \right)^{2} }}$, (2) $\phi > \phi_{1}^{A}$.

$\frac{{\partial f^{AT} }}{\partial \phi }{ = }\frac{1}{2}\delta \left( { - 1 + \frac{{c_{n} }}{{\left( {1 - \phi } \right)^{2} }} - \frac{{4c_{r} \left( {2 - \delta } \right) + c_{n} \delta^{2} }}{{\left( {8 - \delta \left( {3 + \phi } \right)} \right)^{2} }}} \right)$, let $\frac{\partial f}{{\partial \phi }}{ = 0}$ and $c_{r} { = }c_{r2}^{A}$ yields. Considering the feasible condition of strategy PR $0 < c_{r2}^{AT} \le c_{r} < c_{r1}^{AT} < c_{n}$ and the initial assumptions, we find that when $\frac{{\partial f^{AT} }}{\partial \phi } < 0$, $\phi < \phi_{2}^{A}$ and $\max \left\{ {c_{r2}^{A} ,c_{r2}^{AT} } \right\} < c_{r} < c_{r1}^{AT}$ should be satisfied; When $\frac{{\partial f^{AT} }}{\partial \phi } > 0$, we can derive the following two conditions: (1) $\phi < \phi_{2}^{A}$ and $c_{r2}^{AT} < c_{r} < \min \left\{ {c_{r2}^{A} ,c_{r1}^{AT} } \right\}$, (2) $\phi > \phi_{2}^{A}$ and $c_{r2}^{AT} \le c_{r} < c_{r1}^{AT}$.

Therefore, Corollary 5 holds.

Proof of Corollary 6

From the model FI, we examine the partial derivative of strategy thresholds to market parameters ($c_{n}$ and $\phi$) and get the following: $\frac{{\partial c_{r1}^{FI} }}{\partial r}{ = }\delta - 1 < 0$,$\frac{{\partial c_{r1}^{FI} }}{{\partial c_{n} }}{ = }\delta > 0$, $\frac{{\partial c_{r2}^{FI} }}{\partial r}{ = } - \left( {1 - \delta } \right)^{2} < 0$, $\frac{{\partial c_{r2}^{FI} }}{{\partial c_{n} }}{ = }\delta \left( {2 - \delta } \right) > 0$. Therefore, Corollary 6 holds.

Proof of Corollary 7

Under the strategy PR, we take the partial derivative of decision variables to per-unit fee ($r$), i.e.,$\frac{{\partial q_{1n}^{FI + } }}{\partial r}{ = } - \frac{1}{2} < 0$, $\frac{{\partial q_{r}^{FI + } }}{\partial r}{ = } - \frac{1}{2\delta } < 0$, $\frac{{\partial q_{2n}^{FI + } }}{\partial r}{ = }NA$.

Under the strategy FR, we take the partial derivative of decision variables to per-unit fee ($r$), i.e.,$\frac{{\partial q_{1n}^{{FI{ = }}} }}{\partial r}{ = } - \frac{2 - \delta }{{2 + 2\delta \left( {1 - \delta } \right)}} < 0$, $\frac{{\partial q_{2n}^{{\text{FI = }}} }}{\partial r}{ = } - \frac{1 - \delta }{{2 + 2\delta \left( {1 - \delta } \right)}} < 0$. Therefore, Corollary 7 holds.

Proof of Corollary 8

From the model FT, we examine the partial derivative of strategy thresholds to market parameters ($c_{n}$ and $\phi$), i.e., $\frac{{\partial c_{r1}^{FT} }}{\partial r}{ = }\frac{{\partial c_{r1}^{FT} }}{{\partial c_{n} }}{ = }\delta > 0$,$\frac{{\partial c_{r2}^{FT} }}{\partial r}{ = }\frac{{\partial c_{r2}^{FT} }}{{\partial c_{n} }} = \frac{3}{4}\delta \left( {4 - \delta } \right) > 0$. Therefore, Corollary 8 holds.

Proof of Corollary 9

Under the strategy PR, we take the partial derivative of decision variables with respect to per-unit fee ($r$), i.e.,$\frac{{\partial q_{1n}^{FT + } }}{\partial r}{ = } - \frac{1}{2} < 0$, $\frac{{\partial q_{r}^{FT + } }}{\partial r}{ = }\frac{2}{8 - 3\delta } > 0$, $\frac{{\partial q_{2n}^{FT + } }}{\partial r}{ = }\frac{ - 8 + \delta }{{16 - 6\delta }} < 0$.

Under the strategy FR, we take the partial derivative of decision variables with respect to per-unit fee ($r$), i.e.,$\frac{{\partial q_{1n}^{{FI{ = }}} }}{\partial r}{ = } - \frac{2 - 2\delta }{{4 + \delta \left( {8 - 3\delta } \right)}} < 0$, $\frac{{\partial q_{2n}^{{AT{ = }}} }}{\partial r}{ = } - \frac{{4 + \delta \left( {6 - \delta } \right)}}{{8 + 2\delta \left( {8 - 3\delta } \right)}} < 0$. Therefore, Corollary 9 holds.

Proof of Corollary 10

Take the partial derivative of royalty fee $f^{FT}$ with respect to market parameters ($c_{n}$ and $\phi$), we can easily derive: $\frac{{\partial f^{FT} }}{\partial r} = \frac{{\partial f^{FT} }}{{\partial c_{n} }} = \frac{{\delta^{2} }}{ - 16 + 6\delta } < 0$, $\frac{{\partial f^{FT} }}{\partial \delta } = \frac{{64 + 8c_{r} - \delta \left( {16 - 3\delta } \right)\left( {3 + c_{n} + r} \right)}}{{2\left( {8 - 3\delta } \right)^{2} }} > 0$, $\frac{{\partial f^{FT} }}{{\partial c_{r} }} = - \frac{{2\left( {2 - \delta } \right)}}{8 - 3\delta } < 0$. Therefore, Corollary 10 holds.

Proof of Proposition 1

Under the strategy PR, by taking the difference in $q_{r}$ of model AI and AT, we have $q_{r}^{AI + } - q_{r}^{AT + } { = }\frac{{ - c_{n} \delta \left( {4 + \delta \left( {1 - \phi } \right)} \right) + c_{r} \left( {4 + \delta + 4\phi - 5\delta \phi } \right)}}{{ - 2\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)\left( {8 - \delta \left( {3 + \phi } \right)} \right)}}$, it is not difficult to see that the denominator is less than 0. Let the numerator be equal to 0, $c_{r} { = }c_{r3}^{A}$ yields. Under the feasible conditions $0 < c_{r2}^{AT} < c_{r2}^{AI} < c_{r} < c_{r1}^{AI} < c_{r1}^{AT} < c_{n} < 1$, we have $q_{r}^{AI} < q_{r}^{AT}$ if $c_{r} > c_{r3}^{A}$, and $q_{r}^{AI} > q_{r}^{AT}$ if $c_{r} < c_{r3}^{A}$. For the same reason, $q_{2n}^{AI + } - q_{2n}^{AT + } { = }\frac{{c_{n} \delta \left( {6 - \delta - \delta \phi } \right) + c_{r} \left( {\delta + 3\delta \phi - 2\left( {3 + \phi } \right)} \right)}}{{ - 2\left( {1 - \delta } \right)\left( {1 - \phi } \right)\left( {8 - \delta \left( {3 + \phi } \right)} \right)}}$, let the numerator be equal to 0, we get $c_{r} { = }c_{r4}^{A}$. Therefore, Proposition 1 holds.

Similarly, Proposition 4 can also be proved.

Proof of Proposition 2

Under the strategy FR, by taking the difference in $q_{1n}$ of model AI and AT, we have $q_{1n}^{{AI{ = }}} - q_{1n}^{{AT{ = }}} { = }\frac{{\delta \left( {c_{n} \left( {4 + 8\phi - \delta \left( {3 + \delta } \right) - \delta \phi \left( {5 - \delta } \right)} \right) - \left( {1 - c_{r} - \phi } \right)\left( {4\left( {1 + \phi } \right) + \delta \left( {1 - 5\phi } \right)} \right)} \right)}}{{ - 2\left( {1 + \delta \left( {1 - \delta } \right)} \right)\left( {1 - \phi } \right)\left( {4\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right)}}$. Consequently, $q_{1n}^{{AI{ = }}} > q_{1n}^{{AT{ = }}}$ under the feasible condition $0 < c_{r} < c_{r2}^{AT} < c_{r2}^{AI} < c_{r1}^{AI} < c_{r1}^{AT} < c_{n} < 1$. For the same reason,$q_{2n}^{{AI{ = }}} - q_{2n}^{{AT{ = }}} { = }\frac{{\delta \left( {\left( {1 - c_{r} - \phi } \right)\left( {2\left( {1 - \phi } \right) + \delta \left( {2\left( {3 + \phi } \right) - \delta - 3\delta \phi } \right)} \right) - c_{n} \left( {2 - 2\phi + \delta \left( {4 + 6\phi - \delta \left( {7 - \delta + \phi \left( {3 - \delta } \right)} \right)} \right)} \right)} \right)}}{{ - 2\left( {1 + \delta \left( {1 - \delta } \right)} \right)\left( {1 - \phi } \right)\left( {4\left( {1 - \phi } \right) + \delta \left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right)}} < 0$.

Therefore, Proposition 2 holds. Similarly, Proposition 5 can also be proved.

Proof of Proposition 3

From the equilibrium of model AI and AT, the optimal profit of OEM in model AI is $\Pi_{M}^{{\text{AI*}}} { = }\frac{{\left( {1 - c_{n} - \phi } \right)^{2} }}{{4\left( {1 - \phi } \right)}} + \frac{1}{4}\left( {1 - \phi - 2c_{n} + \frac{{\left( {c_{n} \delta - c_{r} } \right)^{2} }}{{\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)}}} \right)$ and the optimal profit of OEM in model AT is $\Pi_{M}^{{\text{AT*}}} { = }\frac{{\left\{ { - 4c_{n} c_{r} \delta \left( {1 - \phi } \right) + 2c_{r}^{2} \left( {1 - \phi } \right)^{2} + \delta \left( {\left( {\left( {1 - \phi } \right)^{3} - 2c_{n} \left( {1 - \phi } \right)^{2} } \right)\left( {8 - \delta \left( {3 + \phi } \right)} \right) + c_{n}^{2} \left( {8 - 8\phi + \delta \left( {\phi \left( {2 + \phi } \right) - 1} \right)} \right)} \right)} \right\}}}{{2\delta \left( {1 - \phi } \right)^{2} \left[ {8 - \delta \left( {3 + \phi } \right)} \right]}}$. Taking the difference in optimal profits of OEM of these two models, we have $\Pi_{M}^{{\text{AI*}}} - \Pi_{M}^{{\text{AT*}}} { = }\frac{{ - 2c_{n} c_{r} \delta \left( {4 + \delta \left( {1 - \phi } \right)} \right)\left( {1 - \phi } \right) + c_{n}^{2} \delta^{2} \left( {4 - 8\phi + \delta \left( {1 + \phi } \right)^{2} } \right) + c_{r}^{2} \left( {1 - \phi } \right)\left( {4\left( {1 + \phi } \right) + \delta \left( {1 - 5\phi } \right)} \right)}}{{4\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)^{2} \left( {8 - \delta \left( {3 + \phi } \right)} \right)}}$. It is not difficult to see that the denominator is less than 0. Let the numerator be equal to 0, $c_{r} { = }c_{r5}^{A}$ yields. Under the condition $0 < c_{r2}^{AT} < c_{r2}^{AI} < c_{r} < c_{r1}^{AI} < c_{r1}^{AT} < c_{n} < 1$, we find that: when $\Pi_{M}^{AI + } > \Pi_{M}^{AT + }$, $\phi < \frac{1}{4}$ and $c_{r2}^{AI} < c_{r} < c_{r5}^{A}$ should be satisfied; When $\Pi_{M}^{AI*} < \Pi_{M}^{AT*}$, we can derive the following two conditions: (1) $\phi < \frac{1}{4}$ and $\max \left\{ {c_{r5}^{A} ,c_{r2}^{AI} } \right\} < c_{r} < c_{r1}^{AI}$; (2) $\phi > \frac{1}{4}$ and $c_{r2}^{AI} < c_{r} < c_{r1}^{AI}$.

Therefore, Proposition 3 holds. Similarly, Proposition 6 can also be proved.

Proof of Proposition 7

In strategy PR of model AI and FI, taking the difference in reman and new product quantities, $q_{r}^{AI} - q_{r}^{FI} = \frac{{r\left( {1 - \delta } \right)\left( {1 - \phi } \right) + \phi \left( {\delta c_{n} - c_{r} } \right)}}{{2\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)}} > 0$ and $q_{2n}^{AI} - q_{2n}^{FI} { = }\frac{{ - \left( {c_{n} - c_{r} } \right)\phi }}{{2\left( {1 - \delta } \right)\left( {1 - \phi } \right)}} < 0$ can be easily yield. Proportional fee $\phi$ and per-unit fee $r$ represent different sales cost forms for the OEM. In order to compare the difference between two pricing structures under the in-house remanufacturing scenario, we use $\phi$ to represent the commission fee in the two models, that is, let $r = \phi$. Therefore, given the condition $0 < c_{r2}^{FI} < c_{r2}^{AI} < c_{r} < c_{r1}^{FI} < c_{r1}^{AI} < c_{n} < 1$, we can derive the following:$\Pi_{M}^{{\text{AI*}}} - \Pi_{M}^{{\text{FI*}}} { = }\frac{{\phi \left( \begin{gathered} c_{r}^{2} - \phi \left( {1 - \phi } \right) - \delta^{2} \left( {2 - c_{n} \left( {2 - c_{n} } \right) - 3\phi + 2\phi c_{n} + \phi^{2} } \right) \hfill \\ + 2\delta \left( {1 - \phi - c_{n} \left( {1 - c_{n} - \phi } \right)} \right) - 2c_{r} \left( {1 - \phi - \delta \left( {1 - c_{n} - \phi } \right)} \right) \hfill \\ \end{gathered} \right)}}{{4\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)}} > 0$; When $\max \left\{ {c_{r2}^{AI} ,c_{r}^{I} } \right\} < c_{r}$, $\Pi_{E}^{{AI{*}}} - \Pi_{E}^{{FI{*}}} { = }\frac{{\phi \left( {2\left( {c_{r} - \delta \left( {1 - c_{n} } \right)} \right)\left( {1 - \delta } \right)\left( { - 1 + \phi } \right)^{2} - c_{r}^{2} + 2c_{n} c_{r} \delta - c_{n}^{2} \delta \left( {2 - \delta } \right) + 2\phi \left( {1 - \delta^{2} } \right)\left( {1 - \phi } \right)^{2} } \right)}}{{4\delta \left( {1 - \delta } \right)\left( {1 - \phi } \right)^{2} }} > 0$.

Therefore, Proposition 7 holds.

Proof of Proposition 8

Similar to proposition 7, we use $\phi$ to represent the commission fee in model AT and FT, let $r = \phi$. Guarantee that the feasible condition ($0 < c_{r2}^{AT} < c_{r2}^{FT} < c_{r} < c_{r1}^{AT} < c_{r1}^{FT} < c_{n} < 1$) are satisfied, it can be concluded that:$q_{r}^{AT} - q_{r}^{FT} = \frac{{2\phi \left( {1 - \phi } \right)\left( {8 - \delta \left( {3 + \phi } \right) + c_{r} } \right) - 2\phi c_{n} \left( {8 - \delta \left( {2 + \phi } \right)} \right)}}{{ - \left( {8 - 3\delta } \right)\left( {1 - \phi } \right)\left( {8 - 3\delta - \delta \phi } \right)}} < 0$; $q_{2n}^{AT} - q_{2n}^{FT} { = }\frac{{\phi \left( {c_{n} \left( {64 - 8\delta \left( {4 + \phi } \right) + \delta^{2} \left( {5 + \phi } \right)} \right) - \left( {1 - \phi } \right)\left( {64 + \delta \left( {2c_{r} + \delta \left( {3 + \phi } \right) - 8\left( {4 + \phi } \right)} \right)} \right)} \right)}}{{ - 2\left( {8 - 3\delta } \right)\left( {1 - \phi } \right)\left( {8 - 3\delta - \delta \phi } \right)}} > 0$.

$\Pi_{M}^{{\text{AT*}}} - \Pi_{M}^{{\text{FT*}}} { = }\frac{{ - \phi \left( \begin{gathered} 2c_{n} \left( {1 - \phi } \right)\left( {2c_{r} \left( {8 - \delta \left( {2 + \phi } \right)} \right) + \left( {8 - \delta } \right)\left( {1 - \phi } \right)\left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right) - \hfill \\ \left( {1 - \phi } \right)^{2} \left( {2cr^{2} + 4c_{r} \left( {8 - \delta \left( {3 + \phi } \right)} \right) + \left( {8 - 3\delta - \left( {8 - \delta } \right)\phi } \right)\left( {8 - \delta \left( {3 + \phi } \right)} \right)} \right) \hfill \\ + c_{n}^{2} \left( { - 64\left( {1 - \phi } \right) + \delta \left( {16 + \delta - 8\phi \left( {3 + \phi } \right) + \delta \phi \left( {4 + \phi } \right)} \right)} \right) \hfill \\ \end{gathered} \right)}}{{2\left( {1 - \phi } \right)^{2} \left( {8 - 3\delta } \right)\left( {8 - 3\delta - \delta \phi } \right)}} > 0$; $\Pi_{E}^{{AT{*}}} - \Pi_{E}^{{FT{*}}} { = }\frac{1}{2\delta }\left[ \begin{gathered} \frac{{ - 8c_{r}^{2} + 2c_{r} \delta \left( {8c_{n} + 3r\delta } \right) + 2\delta \left( \begin{gathered} - 64r\left( {1 - c_{n} - r} \right) + 48r\delta - 4\delta \left( {c_{n} + r} \right) \hfill \\ \left( {c_{n} + 11r} \right) + 3r\delta^{2} \left( { - 3 + 2c_{n} + 2r} \right) \hfill \\ \end{gathered} \right)}}{{\left( {8 - 3\delta } \right)^{2} }} \hfill \\ + \frac{\begin{gathered} 2c_{r}^{2} \left( {1 - \phi } \right)^{2} \left( {4 - \delta \phi } \right) - 2c_{n} c_{r} \delta \left( {1 - \phi } \right)\left( {8 + \phi \left( { - 8 + \delta + \delta \phi } \right)} \right) + \hfill \\ \delta \left( {\phi \left( {1 - \phi } \right)^{2} \left( {8 - \delta \left( {3 + \phi } \right)} \right)^{2} - c_{n}^{2} \left( {64\phi + \delta \left( { - 8 - 16\phi \left( {2 + \phi } \right) + \delta \phi \left( {5 + \phi \left( {4 + \phi } \right)} \right)} \right)} \right)} \right) \hfill \\ \end{gathered} }{{\left( { - 1 + \phi } \right)^{2} \left( { - 8 + \delta \left( {3 + \phi } \right)} \right)^{2} }} \hfill \\ \end{gathered} \right]$, Let $\Pi_{E}^{{AT{*}}} - \Pi_{E}^{{FT{*}}} { = }0$, we have $c_{r} = c_{r}^{T}$. So if $\max \left\{ {c_{r2}^{TF} ,c_{r}^{T} } \right\} < c_{r}$, then $\Pi_{E}^{AT} > \Pi_{E}^{FT}$. Therefore, Proposition 8 holds.