Pricing new and remanufactured products under patent protection and government intervention

Abstract

This study proposes a three-period closed-loop supply chain model framework with remanufacturing, in which an original equipment manufacturer (OEM) sells the new products through the entire selling season, and a remanufacturer collects the used products and sells the remanufactured products in the third period to a group of strategic consumers with green preferences. In the basic model, firms determine their respective prices to pursue profit maximization, considering consumers’ strategic behaviour and green preference. Then, we extend the model by incorporating patent protection and government intervention and investigate their impacts on firms’ operational strategies, profitability, and consumer surplus. Our results show that all the optimal prices increase in the costs of the new and remanufactured products, while the optimal licensing fee strictly decreases in both marginal costs. We further uncover that consumers’ strategic behaviour can alleviate but consumers’ green preference will exacerbate the effects of marginal costs on product prices. Compared with the basic model, under patent protection and government intervention, one can see that government subsidies help more consumers find an ideal product rather than leave the market with nothing. In particular, it boosts sales of the remanufactured product and improves the performance of the remanufacturer’s profitability, although the OEM charges a higher patent-licensing fee. Interestingly, the profit of the OEM also increases in the per-unit government subsidy because the OEM could benefit indirectly from government subsidies through its patent-licensing revenue. Based on the combined effects of patent protection and government intervention, we therefore propose an efficient and easy-to-implement subsidy/tax scheme, which can assist the social planner in encouraging firms to set prices at the level of reaching social welfare maximization.

Appendix

1.1 Proofs

1.1.1 Proof of Proposition 1

This model is a Stackelberg game model. In the Stackelberg game, the OEM is the leader, determining the new product prices in different periods first, and the remanufacturer is the follower, setting the remanufactured product price based on the leader’s decision. Then, using backward induction, we first consider the remanufacturer’s price optimization problem.

The objective function of the remanufacturer can be expressed as:

$$ \mathop {\max }\limits_{{p_{r} }} \Pi_{R}^{B} (p_{r} ) = (p_{r} - c_{r} )D_{R} = (p_{r} - c_{r} )\left( {\frac{{\alpha \beta p_{1} - p_{r} }}{1 - \theta } - \frac{{p_{r} }}{\theta }} \right) = \frac{{\alpha \beta \theta p_{1} p_{r} - p_{r}^{2} - \alpha \beta \theta p_{1} c_{r} + p_{r} c_{r} }}{\theta (1 - \theta )}. $$

To show that the objective function is concave, we consider the first- and second-order derivatives with respect to $p_{r}$ and obtain $\frac{{\partial \Pi_{R}^{B} }}{{\partial p_{r} }} = \frac{{\alpha \beta \theta p_{1} - 2p_{r} + c_{r} }}{\theta (1 - \theta )}$, $\frac{{\partial^{2} \Pi_{R}^{B} }}{{\partial p_{r}^{2} }} = \frac{ - 2}{{\theta (1 - \theta )}} < 0$. Then, let $\frac{{\partial \Pi_{R}^{B} }}{{\partial p_{r} }} = 0$, we have $p_{r} = \frac{{\alpha \beta \theta p_{1} + c_{r} }}{2}$ at optimality.

Substituting $p_{r} = \frac{{\alpha \beta \theta p_{1} + c_{r} }}{2}$ into Eq. (1), the objective function of the OEM is given by

$$ \begin{aligned} \mathop {\max }\limits_{{p_{1} }} \Pi_{M}^{B} (p_{1} ) & = (p_{1} - c_{n} )D_{M1} + (\alpha p_{1} - c_{n} )D_{M2} + (\alpha \beta p_{1} - c_{n} )D_{M3} \\ & = (p_{1} - c_{n} )\left( {1 - \frac{{p_{1} - \alpha \delta p_{1} }}{1 - \delta }} \right) + (\alpha p_{1} - c_{n} )\left( {\frac{{p_{1} - \alpha \delta p_{1} }}{1 - \delta } - \frac{{\alpha p_{1} - \alpha \beta \delta p_{1} }}{1 - \delta }} \right) \\ & \quad + (\alpha \beta p_{1} - c_{n} )\left( {\frac{{\alpha p_{1} - \alpha \beta \delta p_{1} }}{1 - \delta } - \frac{{\alpha \beta p_{1} - p_{r} }}{1 - \theta }} \right). \\ \end{aligned} $$

Considering the second-order derivative of the objective function with respect to the new product price $p_{1}$, we have $\frac{{\partial^{2} \Pi_{M}^{B} }}{{\partial p_{1}^{2} }} = - 2\frac{{(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )}}{1 - \delta } - \frac{{\alpha^{2} \beta^{2} (2 - \theta )}}{1 - \theta } < 0$. Immediately, the optimal regular price of the new product can be shown as follows by letting $\frac{{\partial \Pi_{M}^{B} }}{{\partial p_{1} }} = 0$:

$$ p_{1}^{B*} = \frac{{(1 - \delta )[2(1 - \theta ) + \alpha \beta (2 - \theta )c_{n} + \alpha \beta c_{r} ]}}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}}. $$

Finally, $p_{1}^{B*}$, $p_{2}^{B*}$ and $p_{3}^{B*}$ are achieved in the basic model:$p_{2}^{B*} = \alpha p_{1}^{B*}$ and $p_{3}^{B*} = \alpha \beta p_{1}^{B*}$.

Then, plugging $p_{1}^{B*}$ into $p_{r} = \frac{{\alpha \beta \theta p_{1} + c_{r} }}{2}$, the optimal price of the remanufactured product $p_{r}^{B*} = \frac{{\alpha (1 + \delta )(1 - \alpha ) + \alpha (2 - \alpha )c_{n} + (3 + \delta - \alpha \delta )c_{r} }}{2(3 - \alpha + \delta - \alpha \delta )}$ can be obtained. Note that the demand for the remanufactured product should satisfy the constraint $\frac{{\alpha \beta p_{1}^{B*} - p_{r}^{B*} }}{1 - \theta } - \frac{{p_{r}^{B*} }}{\theta } \le \gamma (1 - \frac{{p_{1}^{B*} - \alpha \delta p_{1}^{B*} }}{1 - \delta })$, which equals the condition $\gamma \ge \frac{{(1 - \delta )(\alpha \beta \theta p_{1}^{B*} - c_{r} )}}{{2\theta (1 - \theta )[1 - \delta - (1 - \alpha \delta )p_{1}^{B*} ]}}$. Moreover, to ensure nonzero demand for each product type in Table 2, the condition $\phi_{2} \le p_{r}^{B*} < \phi_{3}$ should be satisfied in the derived optimal prices, which is equivalent to the condition for cost $c_{r}$, i.e., $\frac{ - 2\alpha (1 - \theta - \beta ) - \alpha \beta \theta (1 + \delta )}{{1 - \delta }}\Omega_{B} \le c_{r} < \alpha \beta \theta \Omega_{B}$, where $\Omega_{B} = \frac{{(1 - \delta )[2(1 - \theta ) + \alpha \beta (2 - \theta )c_{n} + \alpha \beta c_{r} ]}}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}}$. Now, we complete the proof for Proposition 1. Q.E.D.

1.1.2 Proof of Proposition 2

• (a) Consider the derivatives of the optimal prices with respect to the cost of the new product $c_{n}$, and we have

  $$ \frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}{ = }\frac{\alpha \beta (1 - \delta )(2 - \theta )}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}} > 0, $$
  $$ \frac{{\partial p_{2}^{B*} }}{{\partial c_{n} }}{ = }\frac{{\alpha^{2} \beta (1 - \delta )(2 - \theta )}}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}} > 0, $$
  $$ \frac{{\partial p_{3}^{B*} }}{{\partial c_{n} }}{ = }\frac{{\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}} > 0, $$
  $$ \frac{{\partial p_{r}^{B*} }}{{\partial c_{n} }} = \frac{\alpha \beta \theta }{2} \cdot \frac{\alpha \beta (1 - \delta )(2 - \theta )}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}} > 0. $$

By observing the above derivatives, we easily obtain $\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }} > \frac{{\partial p_{2}^{B*} }}{{\partial c_{n} }} > \frac{{\partial p_{3}^{B*} }}{{\partial c_{n} }} > \frac{{\partial p_{r}^{B*} }}{{\partial c_{n} }}$.

Similarly, we take the derivatives of the optimal prices with respect to the cost of the remanufactured product $c_{r}$ and have

$$ \frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }} = \frac{\alpha \beta (1 - \delta )}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}} > 0, $$

$$ \frac{{\partial p_{2}^{B*} }}{{\partial c_{r} }} = \frac{{\alpha^{2} \beta (1 - \delta )}}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}} > 0, $$

$$ \frac{{\partial p_{3}^{B*} }}{{\partial c_{r} }} = \frac{{\alpha^{2} \beta^{2} (1 - \delta )}}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}} > 0, $$

$$ \frac{{\partial p_{r}^{B*} }}{{\partial c_{r} }} = \frac{\alpha \beta \theta }{2} \cdot \frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }} + \frac{1}{2} = \frac{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + \alpha^{2} \beta^{2} (1 - \delta )(4 - \theta )}}{{8(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 4\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}} > 0. $$

By further comparing the magnitudes of $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$ and $\frac{{\partial p_{r}^{B*} }}{{\partial c_{r} }}$, we have

$\frac{{\partial p_{r}^{B*} }}{{\partial c_{r} }} - \frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }} = \frac{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + \alpha^{2} \beta^{2} (1 - \delta )(4 - \theta ) - 2\alpha \beta (1 - \delta )}}{{8(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 4\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}}$.

We find the sufficient condition to guarantee $\frac{{\partial p_{r}^{B*} }}{{\partial c_{r} }} > \frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$ is $\alpha \beta (4 - \theta ) > 2$. Immediately, if $\alpha \beta (4 - \theta ) > 2$, we have $\frac{{\partial p_{r}^{B*} }}{{\partial c_{r} }} > \frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }} > \frac{{\partial p_{2}^{B*} }}{{\partial c_{r} }} > \frac{{\partial p_{3}^{B*} }}{{\partial c_{r} }} > 0$.

Therefore, all the optimal prices increase in the costs $c_{n}$ and $c_{r}$.

• (b) Then, we consider the effects of consumers’ green degree on the above derivatives. First, we take the derivatives of $\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}$, $\frac{{\partial p_{2}^{B*} }}{{\partial c_{n} }}$, $\frac{{\partial p_{3}^{B*} }}{{\partial c_{n} }}$, and $\frac{{\partial p_{r}^{B*} }}{{\partial c_{n} }}$ with respect to $\theta$, having

  $$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}} \right) = \frac{{4\alpha \beta (1 - \delta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} > 0, $$
  $$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{2}^{B*} }}{{\partial c_{n} }}} \right) = \frac{{4\alpha^{2} \beta (1 - \delta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} > 0, $$
  $$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{3}^{B*} }}{{\partial c_{n} }}} \right) = \frac{{4\alpha^{2} \beta^{2} (1 - \delta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} > 0, $$
  $$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{r}^{B*} }}{{\partial c_{n} }}} \right) = \frac{\alpha \beta }{2} \cdot \frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }} + \frac{\alpha \beta \theta }{2} \cdot \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}} \right) > 0. $$

Similarly, we consider the derivatives of $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$, $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$, $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$, and $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$ with respect to $\theta$, having

$$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}} \right) = \frac{{2\alpha \beta (1 - \delta )\{ 2[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + \alpha^{2} \beta^{2} (1 - \delta )\} }}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} > 0, $$

$$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{2}^{B*} }}{{\partial c_{r} }}} \right) = \frac{{2\alpha^{2} \beta (1 - \delta )\{ 2[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + \alpha^{2} \beta^{2} (1 - \delta )\} }}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} > 0, $$

$$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{3}^{B*} }}{{\partial c_{r} }}} \right) = \frac{{2\alpha^{2} \beta^{2} (1 - \delta )\{ 2[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + \alpha^{2} \beta^{2} (1 - \delta )\} }}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} > 0, $$

$$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{r}^{B*} }}{{\partial c_{r} }}} \right) = \frac{\alpha \beta }{2} \cdot \frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }} + \frac{\alpha \beta \theta }{2} \cdot \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}} \right) > 0. $$

Thus, as $\theta$ increases, the effects of marginal costs on the optimal prices get stronger.

In a similar fashion, we next consider the effects of consumers’ strategic degree on the derivatives $\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}$, $\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}$, $\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}$, $\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}$ and $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$, $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$, $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$, $\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}$. We have

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}} \right) = \frac{{ - 4\alpha \beta (2 - \theta )(1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0, $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{2}^{B*} }}{{\partial c_{n} }}} \right) = \frac{{ - 4\alpha^{2} \beta (2 - \theta )(1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0, $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{3}^{B*} }}{{\partial c_{n} }}} \right) = \frac{{ - 4\alpha^{2} \beta^{2} (2 - \theta )(1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0, $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{r}^{B*} }}{{\partial c_{n} }}} \right) = \frac{\alpha \beta \theta }{2} \cdot \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{1}^{B*} }}{{\partial c_{n} }}} \right) < 0; $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}} \right) = \frac{{ - 4\alpha \beta (1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0, $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{2}^{B*} }}{{\partial c_{r} }}} \right) = \frac{{ - 4\alpha^{2} \beta (1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0, $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{3}^{B*} }}{{\partial c_{r} }}} \right) = \frac{{ - 4\alpha^{2} \beta^{2} (1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0, $$

$$ \frac{{\partial p_{r}^{B*} }}{{\partial c_{r} }}{/}\partial \delta = \frac{\alpha \beta \theta }{2} \cdot \frac{{\partial p_{1}^{B*} }}{{\partial c_{r} }}{/}\partial \delta < 0. $$

Therefore, the effects of marginal costs on the optimal prices get weaker as $\delta$ increases. Q.E.D.

1.1.3 Proof of Proposition 3

To show the monotonic property of the optimal prices over consumers’ purchase behaviour, we first take the first-order derivatives of prices with respect to $\delta$ as follows:

$$ \frac{{\partial p_{1}^{B*} }}{\partial \delta } = \frac{{ - [2(1 - \theta ) + \alpha \beta (2 - \theta )c_{n} + \alpha \beta c_{r} ]4(1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0. $$

$$ \frac{{\partial p_{2}^{B*} }}{\partial \delta } = \frac{{ - \alpha [2(1 - \theta ) + \alpha \beta (2 - \theta )c_{n} + \alpha \beta c_{r} ]4(1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0, $$

$$ \frac{{\partial p_{3}^{B*} }}{\partial \delta } = \frac{{ - \alpha \beta [2(1 - \theta ) + \alpha \beta (2 - \theta )c_{n} + \alpha \beta c_{r} ]4(1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0, $$

$$ \frac{{\partial p_{r}^{B*} }}{\partial \delta } = \frac{\alpha \beta \theta }{2} \cdot \frac{{ - [2(1 - \theta ) + \alpha \beta (2 - \theta )c_{n} + \alpha \beta c_{r} ]4(1 - \theta )[(1 - \alpha )^{2} + \alpha^{2} (1 - \beta )^{2} ]}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }} < 0. $$

Therefore, all the optimal prices decrease in consumers’ strategic degree $\delta$, and we immediately have $\left| {\frac{{\partial p_{1}^{B*} }}{\partial \delta }} \right| > \left| {\frac{{\partial p_{2}^{B*} }}{\partial \delta }} \right| > \left| {\frac{{\partial p_{3}^{B*} }}{\partial \delta }} \right| > \left| {\frac{{\partial p_{r}^{B*} }}{\partial \delta }} \right|$.

Similarly, we consider the derivative of $p_{1}^{B*}$ regarding $\theta$ yielding

$$ \frac{{\partial p_{1}^{B*} }}{\partial \theta } = 2\alpha \beta (1 - \delta )\frac{{Ac_{n} + Bc_{r} - 2\alpha \beta (1 - \delta )}}{{\{ 4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )\}^{2} }}, $$

where $A = 2[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )]$ and $B = 2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta ) + \alpha^{2} \beta^{2} (1 + \delta )$. Then, we have $\frac{{\partial p_{1}^{B*} }}{\partial \theta } > \frac{{\partial p_{2}^{B*} }}{\partial \theta } > \frac{{\partial p_{3}^{B*} }}{\partial \theta } > 0$ if the condition $Ac_{n} + Bc_{r} > 2\alpha \beta (1 - \delta )$ is satisfied; otherwise, $\frac{{\partial p_{1}^{B*} }}{\partial \theta } < \frac{{\partial p_{2}^{B*} }}{\partial \theta } < \frac{{\partial p_{3}^{B*} }}{\partial \theta } < 0$.

Note that $\frac{{\partial p_{r}^{B*} }}{\partial \theta } = \frac{\alpha \beta \theta }{2} \cdot \frac{{\partial p_{1}^{B*} }}{\partial \theta } + \frac{{\alpha \beta p_{1}^{B*} }}{2} = \frac{\alpha \beta }{2}(\theta \frac{{\partial p_{1}^{B*} }}{\partial \theta } + p_{1}^{B*} )$; thus, we obviously have $\frac{{\partial p_{r}^{B*} }}{\partial \theta } > 0$ if $\frac{{\partial p_{1}^{B*} }}{\partial \theta } > 0$. However, if $\frac{{\partial p_{1}^{B*} }}{\partial \theta } < 0$, we would like to consider the derivative of $\theta \frac{{\partial p_{1}^{B*} }}{\partial \theta } + p_{1}^{B*}$ and have $\frac{\partial }{\partial \theta }\left( {\theta \frac{{\partial p_{1}^{B*} }}{\partial \theta } + p_{1}^{B*} } \right) = 2\frac{{\partial p_{1}^{B*} }}{\partial \theta } < 0$. Immediately, we can obtain the minimum of $\theta \frac{{\partial p_{1}^{B*} }}{\partial \theta } + p_{1}^{B*}$ at $\theta = 1$, that is,

$$ \min \left[ {\theta \frac{{\partial p_{1}^{B*} }}{\partial \theta } + p_{1}^{B*} } \right] = \left. {\theta \frac{{\partial p_{1}^{B*} }}{\partial \theta } + p_{1}^{B*} } \right|_{\theta = 1} = \frac{{[A + \alpha^{2} \beta^{2} (1 - \delta )]c_{n} + c_{r} [B + \alpha^{2} \beta^{2} (1 - \delta )] - 2\alpha \beta (1 - \delta )}}{{2\alpha^{3} \beta^{3} (1 - \delta )}}. $$

Therefore, if $[A + \alpha^{2} \beta^{2} (1 - \delta )]c_{n} + c_{r} [B + \alpha^{2} \beta^{2} (1 - \delta )] > 2\alpha \beta (1 - \delta )$, we have $\frac{{\partial p_{r}^{B*} }}{\partial \theta } > 0$; otherwise, we have $\frac{{\partial p_{r}^{B*} }}{\partial \theta } \le 0$. Q.E.D.

1.1.4 Proof of Proposition 4

Similar to the proof of Proposition 1, we first consider the derivative of the profit function $\Pi_{R}^{P} = (p_{r} - c_{r} - f)D_{R} = (p_{r} - c_{r} - f)(\frac{{\alpha \beta p_{1} - p_{r} }}{1 - \theta } - \frac{{p_{r} }}{\theta }) = \frac{{(\alpha \beta \theta p_{1} - p_{r} )(p_{r} - c_{r} - f)}}{\theta (1 - \theta )}$ and have $\frac{{\partial \Pi_{R}^{P} }}{{\partial p_{r} }} = \frac{{\alpha \beta \theta p_{1} - 2p_{r} + c_{r} + f}}{\theta (1 - \theta )}$, $\frac{{\partial^{2} \Pi_{R}^{P} }}{{\partial p_{r}^{2} }} = \frac{ - 2}{{\theta (1 - \theta )}} < 0$. Then, we let $\frac{{\partial \Pi_{R}^{P} }}{{\partial p_{r} }} = 0$ and obtain the structure of the optimal price for the remanufactured product $p_{r} = \frac{{\alpha \beta \theta p_{1} + c_{r} + f}}{2}$.

Next, we plug $p_{r} = \frac{{\alpha \beta \theta p_{1} + c_{r} + f}}{2}$ into the profit function of the OEM under patent protection and take the second-order derivatives with respect to $p_{1}$ and $f$. We have the first- and second-order leading principal minors below:

$$ \left| {H_{1} } \right| = \frac{{\partial^{2} \Pi_{M}^{P} }}{{\partial p_{1}^{2} }} = - \frac{2(1 - \alpha )(1 - \alpha \delta )}{{1 - \delta }} - \frac{{2\alpha^{2} (1 - \beta )(1 - \beta \delta )}}{1 - \delta } - \frac{{\alpha^{2} \beta^{2} (2 - \theta )}}{1 - \theta } < 0, $$

$$ \left| {H_{2} } \right| = \left| {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{M}^{P} }}{{\partial p_{1}^{2} }}} & {\frac{{\partial^{2} \Pi_{M}^{P} }}{{\partial p_{1} \partial f}}} \\ {\frac{{\partial^{2} \Pi_{M}^{P} }}{{\partial f\partial p_{1} }}} & {\frac{{\partial^{2} \Pi_{M}^{P} }}{{\partial f^{2} }}} \\ \end{array} } \right| = \left| {\begin{array}{*{20}c} { - \frac{2(1 - \alpha )(1 - \alpha \delta )}{{1 - \delta }} - \frac{{2\alpha^{2} (1 - \beta )(1 - \beta \delta )}}{1 - \delta } - \frac{{\alpha^{2} \beta^{2} (2 - \theta )}}{1 - \theta }} & {\frac{\alpha \beta }{{1 - \theta }}} \\ {\frac{\alpha \beta }{{1 - \theta }}} & { - \frac{1}{\theta (1 - \theta )}} \\ \end{array} } \right| > 0, $$

which show that the Hessian Matrix is negative definite, so there is an optimal solution to the maximum of the OEM’s profit. Let $\frac{{\partial \Pi_{M}^{P} }}{\partial f} = \frac{\alpha \beta }{{1 - \theta }}p_{1} - \frac{{\theta c_{n} + c_{r} }}{2\theta (1 - \theta )} - \frac{1}{\theta (1 - \theta )}f = 0$, we have $f = \frac{{2\alpha \beta \theta p_{1} - \theta c_{n} - c_{r} }}{2}$, and substitute it into the function $\frac{{\partial \Pi_{M}^{P} }}{{\partial p_{1} }} = 0$. Then, we have the optimal price for the new product in the first period $p_{1}^{P*} = \frac{{(1 - \delta )(1 + \alpha \beta c_{n} )}}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}}$. Immediately, $p_{2}^{P*} = \alpha p_{1}^{P*}$, $p_{3}^{P*} = \alpha \beta p_{1}^{P*}$, $f^{P*} = \frac{{2\alpha \beta \theta p_{1}^{P*} - \theta c_{n} - c_{r} }}{2}$, and $p_{r}^{P*} = \frac{{\alpha \beta \theta p_{1}^{P*} + c_{r} + f^{P*} }}{2}$. By a similar argument for the proof of Proposition 1, we can derive the condition shown in Proposition 4 immediately but omit it here for brevity. Q.E.D.

1.1.5 Proof of Corollary 1

• (a) Similar to the proof of Proposition 2, considering the derivatives of $p_{1}^{P*}$, $p_{2}^{P*}$, $p_{3}^{P*}$, $f^{P*}$ and $p_{r}^{P*}$ with respect to $c_{n}$, we have

$\frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }} = \frac{(1 - \delta )\alpha \beta }{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}} > 0$, $\frac{{\partial p_{2}^{P*} }}{{\partial c_{n} }} = \frac{{(1 - \delta )\alpha^{2} \beta }}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}} > 0$, $\frac{{\partial p_{3}^{P*} }}{{\partial c_{n} }} = \frac{{(1 - \delta )\alpha^{2} \beta^{2} }}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}} > 0$, $\frac{{\partial f^{P*} }}{{\partial c_{n} }} = \alpha \beta \theta \frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }} - \frac{\theta }{2} < 0$, and $\frac{{\partial p_{r}^{P*} }}{{\partial c_{n} }} = \frac{\alpha \beta \theta }{2}\frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }} + \frac{1}{2}(\alpha \beta \theta \frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }} - \frac{\theta }{2}) = \alpha \beta \theta \frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }} - \frac{\theta }{4}$.

Moreover, we have $\frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }} > \frac{{\partial p_{2}^{P*} }}{{\partial c_{n} }} > \frac{{\partial p_{3}^{P*} }}{{\partial c_{n} }} > \frac{{\partial p_{r}^{P*} }}{{\partial c_{n} }} > \frac{{\partial f^{P*} }}{{\partial c_{n} }}$.

Because the optimal prices for the new product are independent of the cost of the remanufactured product, it is straightforward to have $\frac{{\partial p_{1}^{P*} }}{{\partial c_{r} }} = \frac{{\partial p_{2}^{P*} }}{{\partial c_{r} }} = \frac{{\partial p_{3}^{P*} }}{{\partial c_{r} }} = 0$. Then, we consider the derivatives of $f^{P*}$ and $p_{r}^{P*}$ with respect to $c_{r}$, leading to $\frac{{\partial f^{*} }}{{\partial c_{r} }} = - \frac{1}{2} < 0$ and $\frac{{\partial p_{r}^{P*} }}{{\partial c_{r} }} = \frac{1}{4} > 0$. Therefore, $f^{P*}$ is decreasing in $c_{r}$, but $p_{r}^{P*}$ increases in $c_{r}$.

• (b) Then, we investigate the effects of consumers’ green degree on the above derivatives. Clearly, the effects of $c_{r}$ and $c_{n}$ on all the optimal prices for the new product are independent of $\theta$. Regarding other derivatives, we have

  $$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial f^{*} }}{{\partial c_{n} }}} \right) = \alpha \beta \frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }} - \frac{1}{2} = \frac{{ - (1 - \alpha )(1 - \alpha \delta ) - \alpha^{2} (1 - \beta )(1 - \beta \delta )}}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}} < 0, $$
  $$ \frac{\partial }{\partial \theta }\left( {\frac{{\partial p_{r}^{P*} }}{{\partial c_{n} }}} \right) = \alpha \beta \frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }} - \frac{1}{4} = \frac{{ - (1 - \alpha )(1 - \alpha \delta ) - \alpha^{2} (1 - \beta - \beta \delta + 2\delta \beta^{2} - \beta^{2} )}}{{4(1 - \alpha )(1 - \alpha \delta ) + 4\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}}. $$

As $\theta$ increases, the effect of $c_{n}$ on the licensing fee becomes weaker, while the effect of $c_{n}$ on the price for the remanufactured product depends on the sign of $- (1 - \alpha )(1 - \alpha \delta ) - \alpha^{2} (1 - \beta - \beta \delta + 2\delta \beta^{2} - \beta^{2} )$.

Since the derivatives $\frac{{\partial f^{*} }}{{\partial c_{r} }}$ and $\frac{{\partial p_{r}^{P*} }}{{\partial c_{r} }}$ are constant, they remain unchanged as $\theta$ or $\delta$ varies. However, as $\delta$ increases, we have

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }}} \right) = \frac{{ - \alpha \beta [2(1 - \alpha )^{2} + 2\alpha^{2} (1 - \beta )^{2} ]}}{{[2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )]^{2} }} < 0, $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{2}^{P*} }}{{\partial c_{n} }}} \right) = \frac{{ - \alpha^{2} \beta [2(1 - \alpha )^{2} + 2\alpha^{2} (1 - \beta )^{2} ]}}{{[2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )]^{2} }} < 0, $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{3}^{P*} }}{{\partial c_{n} }}} \right) = \frac{{ - \alpha^{2} \beta^{2} [2(1 - \alpha )^{2} + 2\alpha^{2} (1 - \beta )^{2} ]}}{{[2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )]^{2} }} < 0, $$

$$ \frac{\partial }{\partial \delta }\left( {\frac{{\partial f^{P*} }}{{\partial c_{n} }}} \right) = \alpha \beta \theta \cdot \frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }}} \right) < 0,\frac{\partial }{\partial \delta }\left( {\frac{{\partial p_{r}^{P*} }}{{\partial c_{n} }}} \right) = \alpha \beta \theta \cdot \partial \frac{{\partial p_{1}^{P*} }}{{\partial c_{n} }}/\partial \delta < 0. $$

Therefore, as $\delta$ increases, the effects of $c_{n}$ on all optimal prices and the licensing fee become weaker. Q.E.D.

1.1.6 Proof of Corollary 2

• (a) Similar to the proof of Proposition 3, we consider the derivatives of the optimal prices with respect to consumer’s strategic degree $\delta$, obtaining

  $$ \frac{{\partial p_{1}^{P*} }}{\partial \delta } = \frac{{ - (1 + \alpha \beta c_{n} )[2(1 - \alpha )^{2} + 2\alpha^{2} (1 - \beta )^{2} ]}}{{[2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )]^{2} }} < 0, $$
  $$ \frac{{\partial p_{2}^{P*} }}{\partial \delta } = \frac{{ - \alpha (1 + \alpha \beta c_{n} )[2(1 - \alpha )^{2} + 2\alpha^{2} (1 - \beta )^{2} ]}}{{[2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )]^{2} }} < 0, $$
  $$ \frac{{\partial p_{3}^{P*} }}{\partial \delta } = \frac{{ - \alpha \beta (1 + \alpha \beta c_{n} )[2(1 - \alpha )^{2} + 2\alpha^{2} (1 - \beta )^{2} ]}}{{[2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )]^{2} }} < 0, $$
  $$ \frac{{\partial f^{*} }}{\partial \delta } = \alpha \beta \theta \frac{{\partial p_{1}^{P*} }}{\partial \delta } < 0,\frac{{\partial p_{r}^{P*} }}{\partial \delta } = \alpha \beta \theta \frac{{\partial p_{1}^{P*} }}{\partial \delta } < 0. $$

Therefore, all the optimal prices still strictly decrease in consumers’ strategic degree under the patent protection effect. Moreover, the optimal licensing fee also decreases as consumers become more strategic.

• (b) Recall that the optimal prices $p_{1}^{P*}$, $p_{2}^{P*}$ and $p_{3}^{P*}$ are not related to consumers’ green degree, so we only consider the derivatives of $f^{P*}$ and $p_{r}^{P*}$ over $\theta$, yielding

  $$ \frac{{\partial f^{P*} }}{\partial \theta } = \frac{{2\alpha \beta p_{1}^{P*} - c_{n} }}{2} = \frac{{C_{1} - c_{n} }}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}}, $$
  $$ \frac{{\partial p_{r}^{P*} }}{\partial \theta } = \frac{\alpha \beta }{2}p_{1}^{P*} + \frac{1}{2}\frac{{\partial f^{P*} }}{\partial \theta } = \alpha \beta p_{1}^{P*} - \frac{{c_{n} }}{4} = \frac{{C_{2} - c_{n} }}{{4(1 - \alpha )(1 - \alpha \delta ) + 4\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}}, $$

  where $C_{1} = \frac{\alpha \beta (1 - \delta )}{{(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )}}$, $C_{2} = \frac{2\alpha \beta (1 - \delta )}{{(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta ) - \alpha^{2} \beta^{2} (1 - \delta )}}$, and $C_{1} < C_{2}$. Therefore, we have $\frac{{\partial f^{P*} }}{\partial \theta } > 0$ and $\frac{{\partial p_{r}^{P*} }}{\partial \theta } > 0$ if $c_{n} < C_{1}$; we have $\frac{{\partial f^{P*} }}{\partial \theta } < 0$ and $\frac{{\partial p_{r}^{P*} }}{\partial \theta } > 0$ if $C_{1} < c_{n} < C_{2}$; otherwise, we have $\frac{{\partial f^{P*} }}{\partial \theta } < 0$ and $\frac{{\partial p_{r}^{P*} }}{\partial \theta } < 0$ if $c_{n} > C_{2}$. Q.E.D.

Proof of Propositions 5 is similar to proof of Propositions 1 and 4, so we omit it here. Q.E.D.

1.1.7 Proof of Proposition 6

Recall that $p_{1}^{B*} = \frac{{(1 - \delta )[2(1 - \theta ) + \alpha \beta (2 - \theta )c_{n} + \alpha \beta c_{r} ]}}{{4(1 - \theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(2 - \theta )}}$, and $p_{1}^{G*} = p_{1}^{P*} = \frac{{(1 - \delta )(1 + \alpha \beta c_{n} )}}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}}$. To find the sufficient condition for $p_{1}^{B*} > p_{1}^{P*} = p_{1}^{G*}$, we let the numerator of $p_{1}^{B*}$ be larger than that of $p_{1}^{P*}$($p_{1}^{G*}$) and the denominator of $p_{1}^{B*}$ be smaller than that of $p_{1}^{P*}$($p_{1}^{G*}$), leading to

$$ \left\{ {\begin{array}{*{20}l} {1 - 2\theta + \alpha \beta (1 - \theta )c_{n} + \alpha \beta c_{r} \ge 0} \hfill \\ {2(1 - 2\theta )[(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta )(1 - \beta \delta )] + 2\alpha^{2} \beta^{2} (1 - \delta )(1 - \theta ) \le 0} \hfill \\ \end{array} } \right. $$

$$ \Rightarrow \frac{{(1 - \alpha )(1 - \alpha \delta ) + \alpha^{2} (1 - \beta - \beta \delta + \beta^{2} )}}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta )(1 - \beta \delta ) + \alpha^{2} \beta^{2} (1 - \delta )}} \le \theta \le \frac{{1 + \alpha \beta (c_{n} + c_{r} )}}{{2 + \alpha \beta c_{n} }}. $$

Then, we immediately have $p_{2}^{G*} = p_{2}^{P*} \le p_{2}^{B*}$, $p_{3}^{G*} = p_{3}^{P*} \le p_{3}^{B*}$ under the above condition. Moreover, because $p_{1}^{G*} = p_{1}^{P*}$, we clearly have $\frac{{2\alpha \beta \theta p_{1}^{G*} - \theta c_{n} - c_{r} + m}}{2} = f^{G*} > f^{P*} = \frac{{2\alpha \beta \theta p_{1}^{P*} - \theta c_{n} - c_{r} }}{2}$ and $\alpha \beta \theta p_{1}^{G*} - \frac{{\theta c_{n} }}{4} + \frac{{c_{r} }}{4} - \frac{m}{4} = p_{r}^{G*} \le p_{r}^{P*} = \alpha \beta \theta p_{1}^{P*} - \frac{{\theta c_{n} }}{4} + \frac{{c_{r} }}{4}$. Recall that $p_{r}^{B*} = \frac{{\alpha \beta \theta p_{1}^{B*} + c_{r} }}{2}$, and we can obtain $p_{r}^{G*} \le p_{r}^{P*} \le p_{r}^{B*}$ if $p_{1}^{P*} \le \frac{{p_{1}^{B*} }}{2} + \frac{{c_{r} + \theta c_{n} }}{4\alpha \beta \theta }$. Till now, the proof for Proposition 6 is complete. Q.E.D.

1.1.8 Proof of Proposition 7

By comparing the optimal profits of the OEM (respectively the remanufacturer) under the patent protection effect and the combined effects, we have the following result:

$\Pi_{M}^{G*} - \Pi_{M}^{P*} = \frac{{mc_{n} }}{4(1 - \theta )} + \frac{{m^{2} }}{8\theta (1 - \theta )} > 0$ and $\Pi_{R}^{G*} - \Pi_{R}^{P*} = \frac{{4m(\alpha \beta \theta p_{1}^{G*} - p_{r}^{G*} ) + m(\theta c_{n} - c_{r} )}}{16\theta (1 - \theta )} > 0$. Moreover, the difference increases as the per-unit subsidy $m$ increases. Q.E.D.

1.1.9 Proof of Proposition 8

Considering the second-order derivatives of the social welfare function with respect to $p_{1}$ and $p_{r}$, we have the first- and second-order leading principal minors: $\left| {H_{1} } \right| = \frac{{\partial^{2} \Pi_{SW} }}{{\partial p_{1}^{2} }} = \frac{{(1 - \alpha \delta )^{2} + \alpha^{2} \delta (1 - \beta \delta )^{2} - 2(1 - \alpha \delta )(1 - \alpha ) - 2\alpha^{2} (1 - \beta \delta )(1 - \beta )}}{1 - \delta }{ + }\frac{{\alpha^{2} \beta^{2} (\delta^{2} - 2)}}{1 - \theta } < 0$,$\left| {H_{2} } \right| = \left| {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{SW} }}{{\partial p_{1}^{2} }}} & {\frac{{\partial^{2} \Pi_{SW} }}{{\partial p_{1} \partial p_{r} }}} \\ {\frac{{\partial^{2} \Pi_{SW} }}{{\partial p_{r} \partial p_{1} }}} & {\frac{{\partial^{2} \Pi_{SW} }}{{\partial p_{r}^{2} }}} \\ \end{array} } \right| = \left| {\begin{array}{*{20}c} {\frac{{(1 - \alpha \delta )^{2} + \alpha^{2} \delta (1 - \beta \delta )^{2} - 2(1 - \alpha \delta )(1 - \alpha ) - 2\alpha^{2} (1 - \beta \delta )(1 - \beta )}}{1 - \delta }{ + }\frac{{\alpha^{2} \beta^{2} (\delta^{2} - 2)}}{1 - \theta }} & {\frac{{ - \alpha \beta (\delta^{2} - 2)}}{1 - \theta }} \\ {\frac{{ - \alpha \beta (\delta^{2} - 2)}}{1 - \theta }} & {\frac{{\delta^{2} - 2}}{\theta (1 - \theta )}} \\ \end{array} } \right| > 0$Note that the Hessian Matrix is negative definite, so there is an optimal solution to the maximum of the social welfare. Let $\frac{{\partial \Pi_{SW}^{{}} }}{{\partial p_{1} }} = 0$ and $\frac{{\partial \Pi_{SW}^{{}} }}{{\partial p_{r} }} = 0$, and we have

$\left\{ {\begin{array}{*{20}l} { - \frac{{(\delta^{2} - 2)(\alpha \beta p_{1} - p_{r} )}}{1 - \theta }{ + }\frac{{(\delta^{2} - 2)p_{r} }}{\theta } - \frac{{c_{n} }}{1 - \theta } + \frac{{c_{r} }}{\theta (1 - \theta )} = 0} \hfill \\ {\frac{{2\alpha - 1 - \alpha^{2} \delta + 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) - \alpha^{2} (2 - \delta^{2} )(1 + \beta^{2} )}}{1 - \delta }p_{1} + \alpha \beta c_{n} = 0} \hfill \\ \end{array} } \right.$.

Through calculation, we obtain $p_{1}^{SW*} = \frac{{\alpha \beta c_{n} (1 - \delta )}}{{1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(1 + \beta^{2} )}}$, and immediately, $p_{2}^{SW*} = \alpha p_{1}^{SW*}$, $p_{3}^{SW*} = \alpha \beta p_{1}^{SW*}$, and $p_{r}^{SW*} = \frac{{(2 - \delta^{2} )\alpha \beta \theta p_{1}^{SW*} - c_{n} \theta + c_{r} }}{{2 - \delta^{2} }}$. Q.E.D.

The proof of Corollary 3 is similar to the proofs of Proposition 2 and Corollary 1 , so we omit it here. Q.E.D.

1.1.10 Proof of Corollary 4

Considering the derivatives of the optimal prices with respect to consumers’ strategic degree $\delta$, we have

$$ \frac{{\partial p_{1}^{SW*} }}{\partial \delta } = \frac{{ - \alpha \beta c_{n} [(1 - \alpha )^{2} + \alpha^{2} (2 + \delta^{2} - 2\delta )(1 - \beta )^{2} ]}}{{[1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(1 + \beta^{2} )]^{2} }} < 0, $$

$$ \frac{{\partial p_{2}^{SW*} }}{\partial \delta } = \frac{{ - \alpha^{2} \beta c_{n} [(1 - \alpha )^{2} + \alpha^{2} (2 + \delta^{2} - 2\delta )(1 - \beta )^{2} ]}}{{[1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(1 + \beta^{2} )]^{2} }} < 0, $$

$$ \frac{{\partial p_{3}^{SW*} }}{\partial \delta } = \frac{{ - \alpha^{2} \beta^{2} c_{n} [(1 - \alpha )^{2} + \alpha^{2} (2 + \delta^{2} - 2\delta )(1 - \beta )^{2} ]}}{{[1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(1 + \beta^{2} )]^{2} }} < 0, $$

$$ \frac{{\partial p_{r}^{SW*} }}{\partial \delta } = \alpha \beta \theta \frac{{\partial p_{1}^{SW*} }}{\partial \delta } + \frac{{( - c_{n} \theta + c_{r} )2\delta }}{{(2 - \delta^{2} )^{2} }} < 0. $$

Therefore, all the optimal prices decrease in $\delta$. Moreover, we immediately have $\frac{{\partial p_{r}^{SW*} }}{\partial \delta } < \frac{{\partial p_{1}^{SW*} }}{\partial \delta } < \frac{{\partial p_{2}^{SW*} }}{\partial \delta } < \frac{{\partial p_{3}^{SW*} }}{\partial \delta } < 0$.

Observing the optimal prices for the new product, we find that they are all independent of consumers’ green degree $\theta$, but we have

$$ \frac{{\partial p_{r}^{SW*} }}{\partial \theta } = - \frac{{[1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(\delta \beta^{2} + 1)]c_{n} }}{{(2 - \delta^{2} )[1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(1 + \beta^{2} )]}}. $$

Therefore, if $1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(\delta \beta^{2} + 1) \le 0$, we have $\frac{{\partial p_{r}^{SW*} }}{\partial \theta } \ge 0$; otherwise, we have $\frac{{\partial p_{r}^{SW*} }}{\partial \theta } < 0$. Q.E.D.

1.1.11 Proof of Proposition 9

Similar to the proofs of Propositions 1, 4 and 8, we first consider the derivatives of profit function $\Pi_{R}^{o}$ with respect to $p_{r}$: $\frac{{\partial \Pi_{R}^{o} }}{{\partial p_{r} }} = \frac{{\alpha \beta \theta p_{1} - 2p_{r} + c_{r} - m_{r} + f}}{\theta (1 - \theta )}$, $\frac{{\partial^{2} \Pi_{R}^{o} }}{{\partial p_{r}^{2} }} = \frac{ - 2}{{\theta (1 - \theta )}} < 0$, and obtain the structure of the optimal price for remanufactured product $p_{r} = \frac{{\alpha \beta \theta p_{1} + c_{r} - m_{r} + f}}{2}$.

Then, we plug $p_{r} = \frac{{\alpha \beta \theta p_{1} + c_{r} - m_{r} + f}}{2}$ into the profit function of the OEM $\Pi_{M}^{o}$ and calculate its derivatives with respect to $p_{1}$ and $f$. The corresponding Hessian Matrix is negative definite, so there is an optimal solution to the maximum of the social welfare. Let $\frac{{\partial \Pi_{M}^{o} }}{{\partial p_{1} }}$ and $\frac{{\partial \Pi_{M}^{o} }}{\partial f}$, and we have

$$ \begin{aligned}& \frac{{\partial \Pi_{M}^{o} }}{\partial f} = \frac{\alpha \beta }{{1 - \theta }}p_{1} - \frac{{\theta (c_{n} - m_{n} ) + (c_{r} - m_{r} )}}{2\theta (1 - \theta )} - \frac{1}{\theta (1 - \theta )}f \Rightarrow f \\&\quad= \frac{{2\alpha \beta \theta p_{1} - \theta (c_{n} - m_{n} ) - (c_{r} - m_{r} )}}{2}, \end{aligned}$$

$$ \begin{aligned} \frac{{\partial \Pi_{M}^{o} }}{{\partial p_{1} }} & = 1 - \frac{2(1 - \alpha )(1 - \alpha \delta )}{{1 - \delta }}p_{1} - \frac{{2\alpha^{2} (1 - \beta )(1 - \beta \delta )}}{1 - \delta }p_{1} - \frac{{2\alpha^{2} \beta^{2} (1 - \theta )}}{1 - \theta }p_{1} + \frac{\alpha \beta (1 - \theta )}{{1 - \theta }}(c_{n} - m_{n} ) = 0 \\ \Rightarrow p_{1} & = \frac{{1 - \delta + \alpha \beta (1 - \delta )(c_{n} - m_{n} )}}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta )(1 - \beta \delta ) + 2\alpha^{2} \beta^{2} (1 - \delta )}}. \\ \end{aligned} $$

Therefore, the optimal prices and licensing fee are given by

$$ p_{1}^{o*} = \frac{{1 - \delta + \alpha \beta (1 - \delta )(c_{n} - m_{n} )}}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta )(1 - \beta \delta ) + 2\alpha^{2} \beta^{2} (1 - \delta )}}, $$

$$ p_{2}^{o*} = \alpha p_{1}^{o*} ,p_{3}^{o*} = \alpha \beta p_{1}^{o*} , $$

$$ f^{o*} = \frac{{2\alpha \beta \theta p_{1} - \theta (c_{n} - m_{n} ) - (c_{r} - m_{r} )}}{2},p_{r}^{o*} = \frac{{\alpha \beta \theta p_{1}^{o*} + c_{r} - m_{r} + f^{o*} }}{2} Q.E.D. $$

1.1.12 Proof of Proposition 10

Let $p_{1}^{o*} = p_{1}^{SW*}$, $p_{r}^{o*} = p_{r}^{SW*}$, and we have

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\alpha \beta c_{n} (1 - \delta )}}{{1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(1 + \beta^{2} )}} = \frac{{1 - \delta + \alpha \beta (1 - \delta )(c_{n} - m_{n} )}}{{2(1 - \alpha )(1 - \alpha \delta ) + 2\alpha^{2} (1 - \beta )(1 - \beta \delta ) + 2\alpha^{2} \beta^{2} (1 - \delta )}}} \hfill \\ {\frac{{c_{n} \theta - c_{r} }}{{2 - \delta^{2} }} = \frac{{\theta (c_{n} - m_{n} ) - (c_{r} - m_{r} )}}{4}} \hfill \\ \end{array} .} \right. $$

Then, using algebraic manipulation, we obtain the unique results below:

$$ m_{n}^{*} = \frac{1}{\alpha \beta } - \frac{{[1 + \alpha^{2} \delta - 2\alpha \delta + \alpha^{2} \delta^{2} (1 - \beta )^{2} ]c_{n} }}{{1 - 2\alpha + \alpha^{2} \delta - 2\alpha^{2} \beta (1 + \delta - \delta^{2} ) + \alpha^{2} (2 - \delta^{2} )(1 + \beta^{2} )}},m_{r}^{*} = \frac{{(c_{n} \theta - c_{r} )(2 + \delta^{2} )}}{{2 - \delta^{2} }} + \theta m_{n}^{*} ,Q.E.D. $$