The value of secondary markets when consumers are socially conscious

Abstract

This study examines the effect of a secondary market with socially conscious consumers on a brand firm’s profitability. The existing literature on the introduction of a secondary market usually assumes a channel setting. In practice, however, brand firms begin introducing physical products to secondary markets with a consumer-to-consumer setting, which is not considered in existing theories. Therefore, we develop a two-period pricing model to investigate the effect of secondary markets with socially conscious consumers and conduct a sensitivity analysis to examine the impact of the main parameters on the equilibrium outcomes. Conventional wisdom suggests that the introduction of the consumer-to-consumer secondary market is always detrimental to suppliers of durable physical goods. However, we demonstrate that a brand firm may benefit from introducing the physical product secondary market if the consumers are socially conscious under certain conditions, which stands in contrast to the existing literature. Moreover, consumers always prefer the introduction of a secondary market; that is, a “win–win” outcome may be achieved. In summary, our findings provide useful implications regarding when managers should introduce a secondary market in case consumers are socially conscious.

Appendix

1.1 Equilibrium outcomes when not considering socially conscious consumers

Let $A_{1}=(4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))$, and $A_{2}=(\gamma + \lambda + \gamma \lambda )$.

$$\begin{aligned}{} & {} p_{1}^{bn*}=\frac{2 (\gamma + \lambda + 2 \gamma \lambda ) ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))}{A_{1}}\\{} & {} p_{2}^{bn*}=\frac{\gamma + \lambda - \lambda ^2 + \gamma \lambda (1 - \lambda +p_{1}^{bn*}}{2 A_2}\\{} & {} p_{s}^{bn*}=\frac{g \lambda (-8 \gamma ^2 - 2 \gamma (8 + 7 \gamma ) \lambda + (-8 + 3 (-2 + \gamma ) \gamma ) \lambda ^2 + 8 (1 + \gamma )^2 \lambda ^3)}{2 A_2 (A_{1})}\\{} & {} d_{1n}^{bn*}=\frac{-4 \gamma ^3 - 4 \gamma ^2 (3 + 2 \gamma ) \lambda -12 \gamma (1 + \gamma ) \lambda ^2 + (-4 + \gamma ^2 (8 + 5 \gamma )) \lambda ^3 + 4 (1 + \gamma )^2 \lambda ^4}{2 A_2 (A_{1})}\\{} & {} d_{2n}^{bn*}=\frac{4 (-1 + \lambda ) \lambda ^2 - \gamma ^2 (4 + \lambda (6 + \lambda )) + 2 \gamma \lambda (-4 + \lambda (-1 + 2 \lambda ))}{8 (-1 + \lambda ) \lambda ^2 + 2 \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 8 \gamma \lambda (-2 + \lambda ^2)}\\{} & {} d_{s}^{bn*}=\frac{4 \gamma ^3 + 4 \gamma ^2 (1 + 2 \gamma ) \lambda + 2 (-2 + \gamma ) \gamma (1 + \gamma ) \lambda ^2 - (1 + \gamma ) (4 + \gamma (2 + \gamma )) \lambda ^3 + 4 (1 + \gamma )^2 \lambda ^4}{2 A_2 (A_{1})}\\{} & {} \pi ^{bn*}=\frac{(8 \gamma ^2 + 4 \gamma (4 + 5 \gamma ) \lambda + (8 + \gamma (16 + 13 \gamma )) \lambda ^2 - 4 (1 + \gamma ) \lambda ^3) ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))}{4 A_2 A_1} \end{aligned}$$

1.2 Equilibrium outcomes when considering socially conscious consumers

$$\begin{aligned}{} & {} p_{1}^{bs*}=\frac{2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) (\gamma + \lambda + 2 \gamma \lambda + \gamma r \theta )}{A_{1}}\\{} & {} p_{2}^{bs*}=\frac{\gamma + \lambda - \lambda (\lambda + r \theta ) + \gamma \lambda (1 - \lambda - r \theta +p_{1}^{bs*})}{2 A_2}\\{} & {} p_{s}^{bs*}=\frac{ \left\{ \begin{array}{l} \gamma \lambda (-8 \gamma ^2 - 2 \gamma (8 + 7 \gamma ) \lambda + (-8 + 3 (-2 + \gamma ) \gamma ) \lambda ^2 + 8 (1 + \gamma )^2 \lambda ^3) ((-1 + \lambda ) \lambda \\ \quad +\gamma (-1 + (-1 + \lambda ) \lambda )) + \gamma (8 \gamma ^3 + 4 \gamma ^2 (6 + 5 \gamma ) \lambda + 4 \gamma (1 + \gamma ) (6 + \gamma ) \lambda ^2 - 4 (1 + \gamma ) (-2 \\\quad + \gamma (3 + 4 \gamma )) \lambda ^3 - (1 + \gamma ) (12 + \gamma (16 + 3 \gamma )) \lambda ^4 + 4 (1 + \gamma )^3 \lambda ^5) r \theta \end{array} \right\} }{2 A_2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) A_1}\\{} & {} d_{1n}^{bs*}=\frac{ \left\{ \begin{array}{l} 4 (-1 + \lambda )^2 \lambda ^4 + 4 \gamma (-1 + \lambda ) \lambda ^3 (-4 + \lambda (-1 + 3 \lambda ) - r \theta ) + \gamma ^4 (4 + 4 r \theta + \lambda (12 \\ \quad+ \lambda (4 + \lambda (-13 + 5 (-1 + \lambda ) \lambda )) + (8 + (-4 + \lambda ) \lambda ^2) r \theta )) + \gamma ^3 \lambda (4 (4 + 3 r \theta ) + \lambda (4 (7\\ \quad+ 3 r \theta ) + \lambda (\lambda (-29 + \lambda (9 + 4 \lambda ) - 3 r \theta ) - 8 (2 + r \theta )))) + 4 \gamma ^2 \lambda ^2 (6 + 3 r \theta + \lambda (3 \\ \quad+ \lambda (\lambda (-1 + 3 \lambda ) - 2 (5 + r \theta )))) \end{array} \right\} }{2 A_2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) A_1} \\{} & {} d_{2n}^{bs*}=\frac{\left\{ \begin{array}{l} 4 (-1 + \lambda ) \lambda ^3 (-1 + \lambda + r \theta ) + 2 \gamma \lambda ^2 (6 - 4 r \theta + \lambda (-3 - 2 r \theta + \lambda (-7 + 4 \lambda + 4 r \theta ))) \\ \quad+ \gamma ^3 (4 - \lambda (2 (-5 + r \theta ) + \lambda (-3 + 2 r \theta + \lambda (5 + \lambda - r \theta )))) + \gamma ^2 \lambda (12 \\ \quad- 4 r \theta + \lambda (12 - 10 r \theta + \lambda (-15 + r \theta + \lambda (-7 + 4 \lambda + 4 r \theta )))) \end{array} \right\} }{2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) A_1}\\{} & {} d_{s}^{bs*}=\frac{\left\{ \begin{array}{l} 4 (-1 + \lambda ) \lambda ^3 ((-1 + \lambda ) \lambda + (-2 + \lambda ) r \theta ) + \gamma ^2 \lambda (\lambda ^2 (16 + \lambda (-5 + 3 \lambda (-7 + 4 \lambda ))) + (-4 \\ \quad+ 3 (-2 + \lambda ) \lambda ) (-6 + \lambda (-1 + 4 \lambda )) r \theta ) + \gamma ^3 (\lambda (-8 + \lambda (-6 + 15 \lambda + 4 (-2 + \lambda ) \lambda ^3)) + 2 (-2 \\ \quad+ (-2 + \lambda ) \lambda ) (-2 + \lambda (-6 + \lambda + 2 \lambda ^2)) r \theta ) + \gamma ^4 (-4 + 4 r \theta + \lambda (-12 - \lambda (2 + \lambda ) (3 + (-5 \\ \quad+ \lambda ) \lambda ) + (8 + (-4 + \lambda ) \lambda ^2) r \theta )) + 2 \gamma \lambda ^2 (12 r \theta + \lambda (4 + \lambda (1 - 16 r \theta + \lambda (-11 + 6 \lambda + 6 r \theta )))) \end{array} \right\} }{2 A_2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) A_1}\\{} & {} \pi ^{bs*}=\frac{\left\{ \begin{array}{l} 4 \gamma \lambda ^3 (8 - (-3 + \lambda ) (-2 + \lambda ) \lambda (1 + 3 \lambda ) - 4 r \theta - 2 \lambda (1 + 3 (-2 + \lambda ) \lambda ) r \theta + (2 + (2 \\ \quad- 3 \lambda ) \lambda ) r^2 \theta ^2) + \gamma ^4 ((1 + \lambda - \lambda ^2)^2 (8 + \lambda (20 + 13 \lambda )) + 2 (-1 + (-1 + \lambda ) \lambda ) (-4 - 8 \lambda \\ \quad+ 5 \lambda ^3) r \theta + (-2 + (-2 + \lambda ) \lambda )^2 r^2 \theta ^2) + \gamma ^2 \lambda ^2 (48 + \lambda (48 - \lambda (107 + \lambda (50 + \lambda (-77 \\ \quad+ 12 \lambda )))) - 2 \lambda (20 + \lambda - 29 \lambda ^2 + 12 \lambda ^3) r \theta + (8 + \lambda (12 + (5 - 12 \lambda ) \lambda )) r^2 \theta ^2) + 2 \gamma ^3 \lambda (\lambda (44 \\ \quad+ \lambda (3 - (-2 + \lambda ) \lambda (-30 + \lambda (-21 + 2 \lambda )))) + \lambda (4 - \lambda (25 + 2 \lambda (3 + \lambda (-9 + 2 \lambda )))) r \theta + (4 \\ \quad +4 \lambda + \lambda ^3 - 2 \lambda ^4) r^2 \theta ^2 + 8 (2 + r \theta )) - 4 (-1 + \lambda ) \lambda ^4 (2 + \lambda ^2 + r \theta (-2 + r \theta ) + \lambda (-3 + 2 r \theta )) \end{array} \right\} }{2 A_2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) A_1} \end{aligned}$$

1.3 Proof of main results

Proof of proposition 1:

From the equilibrium outcomes in the above, regarding part(i), we can derive that $\frac{\partial p_{1}^{bn*}}{\partial \gamma }=2 \lambda ^2 (8 (-1 + \lambda )^2 \lambda ^2 + 2 \gamma (-1 + \lambda ) \lambda (-8 + \lambda (-7 + 8 \lambda )) + \gamma ^2 (8 + \lambda (14 + \lambda \ (-10 + \lambda (-13 + 8 \lambda )))))/A_1^2>0$; $\frac{\partial p_{2}^{bn*}}{\partial \gamma }=\lambda ^2 (24 \gamma ^4 + 16 \gamma ^3 (6 + 5 \gamma ) \lambda + 2 \gamma ^2 (72 + \gamma (96 + 31 \gamma )) \lambda ^2 + 6 \gamma (16 - \gamma (2 + \gamma ) (-8 + 7 \gamma )) \lambda ^3 - (-24 + \gamma (64 + \gamma \ (234 + \gamma (208 + 59 \gamma )))) \lambda ^4 + 2 (1 + \gamma ) (-24 + \gamma (-32 + (-11 + \gamma ) \ \gamma )) \lambda ^5 + 8 (1 + \gamma )^2 (3 + 2 \gamma (2 + \gamma )) \lambda ^6)/2A_1^2A_2^2>0$; $\frac{\partial p_{s}^{bn*}}{\partial \gamma }=\lambda ^2 (32 \gamma ^4 + 16 \gamma ^3 (8 + 7 \gamma ) \lambda + 4 \gamma ^2 (48 + \gamma (68 + 19 \gamma )) \lambda ^2 + 4 \gamma (32 + \gamma (36 - \gamma (22 + 23 \gamma ))) \lambda ^3 - (-32 + \gamma (80 + \ \gamma (372 + \gamma (352 + 97 \gamma )))) \lambda ^4 + 4 (-16 + \gamma (-44 - 33 \gamma + 5 \gamma ^3)) \lambda ^5 + 32 (1 + \gamma )^4 \lambda ^6)/2A_1^2A_2^2>0$

Regarding part(ii), we can derive that $\frac{\partial d_{1n}^{bn*}}{\partial \gamma }=\gamma \lambda ^3 (8 (-1 + \lambda ) \lambda ^3 + 8 \gamma ^2 \lambda (1 + 2 \lambda ) (-3 + 2 \lambda ^2) + 4 \gamma \lambda ^2 (-6 + \lambda (-3 + 8 \lambda )) + \gamma ^3 (-8 + \lambda (2 + \lambda ) \ (-14 + \lambda (-1 + 8 \lambda ))))/2A_1^2A_2^2<0$; $\frac{\partial d_{2n}^{bn*}}{\partial \gamma }=\lambda ^2 (4 \gamma ^2 + 4 \gamma (2 + 3 \gamma ) \lambda + (4 - (-8 + \gamma ) \gamma ) \ \lambda ^2 - 4 (1 + 2 \gamma (2 + \gamma )) \lambda ^3)/A_1^2>0$; $\frac{\partial d_{s}^{bn*}}{\partial \gamma }=\lambda (-32 \gamma ^4 - 8 \gamma ^3 (16 + 15 \gamma ) \lambda - 4 \gamma ^2 (48 + \gamma (76 + 29 \gamma )) \lambda ^2 + 2 \gamma (-64 + \gamma (-96 + \gamma (-12 + 19 \gamma ))) \lambda ^3 + (-32 + \gamma (48 + \ \gamma (276 + \gamma (292 + 93 \gamma )))) \lambda ^4 + 2 (1 + \gamma ) (28 + \gamma (52 + 7 \gamma (5 + \gamma ))) \lambda ^5 - 8 (1 + \gamma )^2 (3 + 2 \gamma (2 + \gamma )) \lambda ^6)/2A_1^2A_2^2<0$.

Regarding part(iii), we can derive that $\frac{\partial \pi _{1}^{bn*}}{\partial \gamma }=\lambda ^2 (-32 \gamma ^6 - 48 \gamma ^5 (4 + 3 \gamma ) \lambda - 4 \gamma ^4 (120 + \gamma (156 + 35 \gamma )) \lambda ^2 + 16 \gamma ^3 (1 + \gamma ) (-40 + \gamma (-20 + 13 \gamma )) \lambda ^3 + \gamma ^2 (-480 + \gamma \ (-480 + \gamma (856 + \gamma (1216 + 359 \gamma )))) \lambda ^4 + \gamma (-192 + \ \gamma (240 + \gamma (1744 + \gamma (1936 + (554 - 65 \gamma ) \gamma )))) \lambda ^5 + (-32 + \ \gamma (336 + \gamma (1076 - \gamma (-624 + \gamma (741 + 908 \gamma + 250 \gamma ^2))))) \lambda ^6 - (1 + \gamma ) \ (-96 + \gamma (32 + \gamma (672 + \gamma (876 + \gamma (331 + 3 \gamma ))))) \lambda ^7 + 4 (1 + \gamma )^2 (-24 + \gamma (-44 + \gamma (-1 + 7 \gamma (5 + 2 \gamma )))) \lambda ^8 + 32 (1 + \gamma )^5 \lambda ^9)/A_1^2A_2^2>0$; $\frac{\partial \pi _{2}^{bn*}}{\partial \gamma }=-\lambda ^2 (32 \gamma ^4 + 8 \gamma ^3 (16 + 15 \gamma ) \lambda + 4 \gamma ^2 (48 + \gamma (74 + 27 \gamma )) \lambda ^2 + 2 \gamma (64 + \gamma (84 - \gamma (16 + 35 \gamma ))) \lambda ^3 - (-32 + \gamma (72 + \ \gamma (356 + 5 \gamma (74 + 23 \gamma )))) \lambda ^4 + 4 (1 + \gamma )^2 (-16 + (-14 + \gamma ) \gamma ) \lambda ^5 + 32 (1 + \gamma )^4 \lambda ^6) (-4 (-1 + \lambda ) \lambda ^2 \ + 2 \gamma \lambda (4 + \lambda - 2 \lambda ^2) + \gamma ^2 (4 + \lambda (6 + \lambda )))/4A_1^3A_2^2>0$.$\Box$

Proof of proposition 2:

From the equilibrium outcomes, regarding part(i), by solving $\frac{\partial p_{1}^{bn*}}{\partial \lambda }=2 \gamma ^2 (8 \gamma ^2 + 2 \gamma (8 + 7 \gamma ) \lambda + (8 - \gamma (2 + 11 \gamma )) \lambda ^2 - 16 (1 + \gamma )^2 \lambda ^3 + (1 + \gamma ) (7 + 6 \gamma ) \lambda ^4)/A_1^2>0$, which reduces $0<\lambda <\lambda _{1}$, where $\lambda _{1}$ is third root of $\frac{\partial p_{1}^{bn*}}{\partial \lambda }=0$; by solving $\frac{\partial p_{2}^{bn*}}{\partial \lambda }=-8 \gamma ^6 - 16 \gamma ^6 \lambda + 2 \gamma ^4 (48 + \gamma (48 + 5 \gamma )) \lambda ^2 + 4 \gamma ^3 (1 + \gamma ) (56 + \gamma (40 + 9 \gamma )) \lambda ^3 + 2 \gamma ^2 (1 + \gamma ) (108 + \gamma (84 + \gamma + 10 \gamma ^2)) \lambda ^4 + 2 \gamma (1 + \gamma ) (48 + \gamma (-24 + \gamma (-128 + \ (-58 + \gamma ) \gamma ))) \lambda ^5 - (1 + \gamma )^2 (-16 + \ \gamma (128 + \gamma (128 + \gamma (14 + 3 \gamma )))) \lambda ^6 + 8 (1 + \gamma )^3 (2 + \gamma ) (-2 + 3 \gamma ) \lambda ^7 + 16 (1 + \gamma )^4 \lambda ^8/2A_1^2>0$, which reduces $0<\lambda <\lambda _{2}$, where $\lambda _{1}$ is second root of $\frac{\partial p_{2}^{bn*}}{\partial \lambda }=0$; by solving $\frac{\partial p_{s}^{bn*}}{\partial \lambda }=\gamma ^2 (32 \gamma ^4 + 16 \gamma ^3 (8 + 7 \gamma ) \lambda + 4 \gamma ^2 (48 + 17 \gamma (4 + \gamma )) \lambda ^2 - 8 \gamma (-16 + \gamma (-18 + \gamma (15 + 16 \gamma ))) \lambda ^3 - (-32 + \gamma (80 + \ \gamma (420 + \gamma (460 + 153 \gamma )))) \lambda ^4 - 16 (1 + \gamma )^3 (4 + \gamma ) \lambda ^5 + 4 (1 + \gamma )^2 (6 + \gamma (11 + 6 \gamma )) \lambda ^6)/2A_1^2A_2^2>0$, which reduces $0<\lambda <\lambda _{3}$, where $\lambda _{3}$ is the third root of $\frac{\partial p_{s}^{bn*}}{\partial \lambda }=0$.

Regarding part(ii), we can derive that $\frac{\partial d_{1n}^{bn*}}{\partial \lambda }=\gamma ^2 \lambda (-4 \lambda ^5 - 8 \gamma \lambda ^3 (1 + \lambda + 2 \lambda ^2) - \gamma ^4 (2 + 5 \lambda ) (4 + \lambda (6 + \lambda )) - 4 \gamma ^2 \lambda ^2 (6 + \lambda (9 + \lambda (4 + 5 \lambda ))) - 8 \gamma ^3 \lambda (3 + \lambda (8 + \lambda (4 + \ \lambda + \lambda ^2))))/2A_1^2A_2^2<0$; $\frac{\partial d_{2n}^{bn*}}{\partial \lambda }=\gamma (4 \lambda ^4 + 4 \gamma \lambda ^2 (1 + \lambda (2 + 3 \lambda )) + \gamma ^3 (4 + \lambda (16 + 11 \lambda )) + 4 \gamma ^2 \lambda (2 + \lambda (5 + 2 \lambda (1 + \lambda ))))/A_1^2>0$; by solving $\frac{\partial d_{s}^{bn*}}{\partial \lambda }=\gamma (32 \gamma ^4 - 8 \gamma ^3 (-16 + (-14 + \gamma ) \gamma ) \lambda + 4 \gamma ^2 (48 + \gamma (68 + (11 - 8 \gamma ) \gamma )) \lambda ^2 - 16 \gamma (1 + \gamma ) (-8 + \gamma (-1 + 2 \gamma (5 + \gamma ))) \lambda ^3 - (1 + \gamma ) \ (-32 + \gamma (112 + \gamma (316 + 5 \gamma (36 + \gamma )))) \lambda ^4 - 8 (1 + \gamma )^2 (2 + \gamma ) (4 + 3 \gamma ) \lambda ^5 + 8 (1 + \gamma )^3 (3 + 2 \gamma ) \lambda ^6)/2A_1^2A_2^2>0$, which reduces $0<\lambda <\lambda _{4}$, where $\lambda _{4}$ is the third root of 0$.\frac{\partial d_{s}^{bn*}}{\partial \lambda }=0$.

Regarding part(iii), by solving

$$\frac{\partial \pi _{1}^{bn*}}{\partial \lambda }=\gamma ^2 (-32 \gamma ^6 - 48 \gamma ^5 (4 + 3 \gamma ) \lambda - 12 \gamma ^4 (40 + \gamma (52 + 11 \gamma )) \lambda ^2 + 4 \gamma ^3 (-160 + \gamma (-240 + \gamma (-16 + 65 \gamma ))) \lambda ^3 + \gamma ^2 (-480 + \gamma \ (-480 + \gamma (976 + \gamma (1468 + 483 \gamma )))) \lambda ^4 + 2 \gamma (1 + \gamma ) (-96 + \gamma (216 + \gamma (736 + \ \gamma (472 + 27 \gamma )))) \lambda ^5 - (1 + \gamma ) (32 + \ \gamma (-368 + \gamma (-828 + \gamma (-236 + \gamma (441 + 242 \gamma ))))) \lambda ^6 - 2 (1 + \gamma ) (-48 + \gamma (-8 + \gamma (270 + \gamma \ (412 + \gamma (211 + 30 \gamma ))))) \lambda ^7 + (1 + \gamma )^2 \ (-88 + \gamma (-180 + \gamma (-88 + 5 \gamma (7 + 6 \gamma )))) \lambda ^8 + 4 (1 + \gamma )^3 (6 + \gamma (9 + 2 \gamma )) \lambda ^9)/A_1^2A_2^2>0,$$

which reduces $0<\lambda <\lambda _{5}$, where $\lambda _{5}$ is the fifth root of $\frac{\partial \pi _{1}^{bn*}}{\partial \lambda }=0$; by solving

$$\frac{\partial \pi _{2}^{bn*}}{\partial \lambda }=-(16 \gamma ^6 + 32 \gamma ^5 (1 + 2 \gamma ) \lambda + 4 \gamma ^4 (-12 + \gamma (12 + 19 \gamma )) \lambda ^2 - 8 \gamma ^3 (1 + \gamma ) (24 + 5 \gamma ) \lambda ^3 - 4 \gamma ^2 (1 + \gamma ) (52 + \gamma (26 + \gamma (-5 + 13 \gamma ))) \lambda ^4 - 2 \gamma (1 + \gamma ) (48 + \gamma (-36 + \gamma (-136 + \ \gamma (-43 + 12 \gamma )))) \lambda ^5 + (1 + \gamma )^2 (-16 + \ \gamma (136 + \gamma (128 + \gamma (-2 + 3 \gamma )))) \lambda ^6 - 8 (1 + \gamma )^3 (-4 + 5 \gamma (1 + \gamma )) \lambda ^7 - 16 (1 + \gamma )^4 \lambda ^8) (-4 (-1 + \lambda ) \lambda ^2 \ + 2 \gamma \lambda (4 + \lambda - 2 \lambda ^2) + \gamma ^2 (4 + \lambda (6 + \lambda )))/4A_1^3A_2^2>0,$$

which reduces $\max \{0,\lambda _{6}\}<\lambda <\min \{\lambda _{7},1\}$, where $\lambda _{6}$ and $\lambda _{7}$ are the second and third root of $\frac{\partial \pi _{1}^{bn*}}{\partial \lambda }=0$, respectively. $\Box$

Proof of proposition 3:

Regarding part(i), by comparing the optimal price of the new product in the first period and the second period, we can derive that $p_{1}^{bn*}-p_{2}^{bn*}=\lambda (6 \gamma ^2 + \gamma (10 + 7 \gamma ) \lambda + 4 (1 + \gamma ) \lambda ^2) ((-1 + \lambda ) \lambda + \ \gamma (-1 + (-1 + \lambda ) \lambda ))/A_1A_2>0$; by comparing the optimal price of the new product in the second period and the secondhand product, we can derive that $p_{2}^{bn*}-p_{s}^{bn*}=-(-1 + \lambda ) (-4 \gamma ^2 - 2 \gamma (4 + \gamma ) \lambda + (-4 + \gamma (2 + 7 \gamma )) \lambda ^2 + 4 (1 + \gamma ) \lambda ^3)/2A_1>0$.

Regarding part(ii), by comparing the optimal demand of the new product in the first period and the second period, we have $d_{1n}^{bn*}-d_{2n}^{bn*}=\gamma \lambda (\gamma +\lambda + 2 \gamma \lambda ) (2\lambda + \gamma (2 + 3\lambda ))/2A_1A_2<0$; by comparing the optimal price of the new product in the second period and the secondhand product, we have $d_{2n}^{bn*}-d_{s}^{bn*}=\gamma (4 (-2 + \lambda ) \lambda ^2 - \gamma ^2 (2 + 3 \lambda ) (4 + 3 \lambda ) + 2 \gamma \lambda (-8 + \lambda (-7 + 2 \lambda )))/2A_1A_2>0$. $\Box$

Proof of proposition 4:

We compare the profits and consumer surplus between the benchmark scenario and the introduction of the secondary market without socially conscious consumers. The results as are follows: $\pi ^{bn*}-\pi ^{b*}=4 \gamma ^4 + 8 \gamma ^4 \lambda + \gamma ^2 (-8 + (-11 + \gamma ) \gamma ) \ \lambda ^2 - 3 \gamma ^2 (1 + \gamma )^2 \lambda ^3 + (1 + \gamma ) (4 + \ \gamma (8 + 9 \gamma )) \lambda ^4 - 4 (1 + \gamma )^2 \lambda ^5/4A_1A_2<0$ and $cs^{bn*}-cs^{b*}=-4 \gamma ^3 (1 + 4 \gamma ) + 4 \gamma ^2 (3 + 6 \gamma + 8 \gamma ^2) \lambda + 4 \gamma (3 + \gamma (2 + (-2 + \gamma ) \gamma )) \lambda ^2 \ - 4 (1 + \gamma )^2 (-1 + 3 \gamma ^2) \lambda ^3 + (1 + \gamma ) (8 + 3 \gamma (4 + 5 \gamma )) \lambda ^4 - 12 (1 + \gamma )^2 \lambda ^5/4A_1A_2>0$. $\Box$

Proof of proposition 5:

In equilibrium, we can derive that

$$\begin{aligned}{} & {} \frac{\partial p_{1}^{bs*}}{\partial \gamma }=\frac{\left\{ \begin{array}{l} 2 \lambda (2 \gamma (-1 + \lambda ) \lambda ^2 (-8 + \lambda (-7 + 8 \lambda )) + 8 \gamma (\lambda - 2 \lambda ^3 + \lambda ^4) r \theta + 4 (-1 + \lambda )^2 \lambda ^2 (2 \lambda \\ \quad + r \theta ) + \gamma ^2 (\lambda (8 + \lambda (14 + \lambda (-10 + \lambda (-13 + 8 \lambda )))) +\\ (4 + (-1 + \lambda ) \lambda (-8 + \lambda (-3 + 4 \lambda ))) r \theta )) \end{array} \right\} }{(A_{1})^2}\\{} & {} \frac{\partial p_{2}^{bs*}}{\partial \gamma }=\frac{\left\{ \begin{array}{l} \lambda (8 (-1 + \lambda )^2 \lambda ^4 (3 \lambda + 2 r \theta ) + 16 \gamma (-1 + \lambda ) \lambda ^3 (\lambda (-6 + \lambda (-2 + 5 \lambda )) + (1 + \lambda ) (-4 + 3 \lambda ) r \theta )\\ \quad + 2 \gamma ^2 \lambda ^2 (\lambda (72 + \lambda (48 + \lambda (-117 + \lambda (-43 +\\ 52 \lambda )))) + 4 (1 + \lambda ) (12 + \lambda (-6 + \lambda (-12 + 7 \lambda ))) r \theta ) + 4 \gamma ^3 \lambda (\lambda (24 + \lambda (48 + \lambda (-9\\ \quad + \lambda (-52 + \lambda (-5 + 16 \lambda ))))) + 2 (8 + \lambda (14 + \lambda (-4 + (-2 + \\ \lambda ) \lambda (7 + 4 \lambda )))) r \theta ) + \gamma ^4 (16 r \theta + \lambda (24 + 48 r \theta + \lambda (80 + 32 r \theta + \lambda (62 - 24 r \theta + \lambda (-42 - 29 r \theta + \lambda (-59 + 2 \lambda (1 + 8 \lambda + 4 r \theta )))))))) \end{array} \right\} }{2 A_2^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial p_{s}^{bs*}}{\partial \gamma }=\frac{\left\{ \begin{array}{l} \lambda (16 (-1 + \lambda )^3 \lambda ^6 (2 (-1 + \lambda ) \lambda + (-2 + \lambda ) r \theta ) + 16 \gamma (-1 \\ \quad + \lambda )^2 \lambda ^5 ((-1 + \lambda ) \lambda (-4 + 3 \lambda ) (3 + 4 \lambda ) + 2 (-2 + \lambda ) (-3 + \lambda (-1 + 3 \lambda )) r \theta ) + 4 \gamma ^2 (-1 + \\ \lambda ) \lambda ^4 ((-1 + \lambda ) \lambda (120 + \lambda (140 + \lambda (-205 + \lambda (-137 + 120 \lambda )))) \\ \quad + 2 (-60 + \lambda (-10 + \lambda (133 + 5 \lambda (-3 + 2 \lambda (-8 + 3 \lambda ))))) r \theta ) + 8 \gamma ^3 (-1 + \\ \lambda ) \lambda ^3 (\lambda (-80 + \lambda (-140 + \lambda (170 + \lambda (265 \\ \quad + \lambda (-172 + 5 \lambda (-27 + 16 \lambda )))))) + (-80 + \lambda (-120 + \lambda (132 + \lambda (188 + \lambda (-100 + \lambda (-81 + 40 \lambda )))))) r \theta ) + \\ 2 \gamma ^5 \lambda (\lambda (-1 + (-1 + \lambda ) \lambda ) (-96 + \lambda (-280 + \lambda (8 + \lambda (524 \\ \quad + \lambda (137 + \lambda (-357 - 76 \lambda + 96 \lambda ^2)))))) + (96 + \lambda (304 + (-2 + \lambda ) \lambda (-32 + \lambda (288 + \\ \lambda (330 + \lambda (-73 + \lambda (-193 + 4 \lambda \\ \quad + 48 \lambda ^2))))))) r \theta ) + \gamma ^4 \lambda ^2 (\lambda (480 + \lambda (1120 + \lambda (-1080 + \lambda (-3072 + \lambda (1263 + \lambda (3014 + \lambda (-1109 + 16 \lambda (-67 +\\ 30 \lambda )))))))) + (480 + \lambda (880 + \lambda (-1024 + \lambda (-2128 + \lambda (956 \\ \quad + \lambda (1817 + \lambda (-621 + 8 \lambda (-73 + 30 \lambda )))))))) r \theta ) + \gamma ^6 (\lambda (1 + \lambda - \lambda ^2)^2 (32 + \lambda (112 + \\ \lambda (76 + \lambda (-92 + \lambda (-97 + 4 \lambda (5 + 8 \lambda )))))) + (32 +\lambda (144 + \lambda (168 + \lambda (-128 \\ \quad + \lambda (-348 + \lambda (-41 + \lambda (244 + \lambda (73 + \lambda (-85 + 8 \lambda (-3 + 2 \lambda )))))))))) r \theta )) \end{array} \right\} }{2 A_2^2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (A_{1})^2}\\{} & {} \frac{\partial d_{1n}^{bs*}}{\partial \gamma }=\frac{\left\{ \begin{array}{l} \gamma \lambda ^3 (\gamma + \lambda + \gamma \lambda - (1 + \gamma ) \lambda ^2)^2 (8 (-1 + \lambda ) \lambda ^3 + 8 \gamma ^2 \lambda (1 + 2 \lambda ) (-3 + 2 \lambda ^2) + 4 \gamma \lambda ^2 (-6 \\ \quad + \lambda (-3 + 8 \lambda )) + \gamma ^3 (-8 + \lambda (2 + \lambda ) (-14 + \lambda (-1 + \\ 8 \lambda )))) + \lambda (16 \gamma ^6 + 48 \gamma ^5 (2 + \gamma ) \lambda - 4 \gamma ^4 (-60 + (-48 + \gamma ) \gamma ) \lambda ^2 - 4 \gamma ^3 (-80 \\ \quad + \gamma (-60 + \gamma (44 + 29 \gamma ))) \lambda ^3 - 4 \gamma ^2 (-60 + \gamma ^2 (154 + \gamma (106 + \\ 11 \gamma ))) \lambda ^4 + \gamma (96 + \gamma (-240 + \gamma (-784 + \gamma (-400 + \gamma (144 + 95 \gamma ))))) \lambda ^5 \\ \quad + 4 (1 + \gamma )^2 (4 + \gamma (-56 + \gamma (19 + \gamma (60 + 7 \gamma )))) \lambda ^6 - (1 + \gamma )^2 (48 + \\ \gamma (-128 + \gamma (-196 + 3 \gamma (8 + 13 \gamma )))) \lambda ^7 - (1 + \gamma )^2 (-48 \\ \quad + \gamma (-32 + \gamma (76 + \gamma (56 + \gamma )))) \lambda ^8 + 4 (1 + \gamma )^3 (-4 + (-1 + \gamma ) \gamma (4 + \gamma )) \lambda ^9) r \theta \end{array} \right\} }{2 A_2^2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial d_{2n}^{bs*}}{\partial \gamma }=\frac{\left\{ \begin{array}{l} \lambda (8 (-1 + \lambda )^2 \lambda ^4 (\lambda - \lambda ^2 + 2 r \theta ) - 16 \gamma (-1 + \lambda ) \lambda ^3 (2 \lambda \\ \quad + \lambda ^3 (-5 + 3 \lambda ) + (4 + \lambda - 4 \lambda ^2 + \lambda ^3) r \theta ) + 2 \gamma ^2 \lambda ^2 (-(-1 + \lambda ) \lambda (24 + \lambda (48 + \lambda (-49 + \\ \lambda (-63 + 44 \lambda )))) + 4 (12 + \lambda (6 - \lambda (23 + \lambda (4 + \lambda (-16 + 5 \lambda ))))) r \theta ) + 4 \gamma ^3 \lambda (-(-1 + \lambda ) \lambda (-1 + (-1\\ \quad + \lambda ) \lambda ) (-8 + \lambda (-24 + \lambda (-3 + 16 \lambda ))) +\\ 2 (8 + \lambda (14 + \lambda (-8 + \lambda (-20 + \lambda (5 + 2 (5 - 2 \lambda ) \lambda ))))) r \theta ) + \gamma ^4 (-2 \lambda (1 + \lambda - \lambda ^2)^2 (-4 + \lambda (-12 + \lambda + 8 \lambda ^2))\\ \quad + (16 + \lambda (48 + \lambda (24 + \lambda (-48 +\\ \lambda (-39 - 8 (-3 + \lambda ) \lambda (1 + \lambda )))))) r \theta )) \end{array} \right\} }{2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial d_{s}^{bs*}}{\partial \gamma }=\frac{\left\{ \begin{array}{l} 8 (-1 + \lambda )^2 \lambda ^6 (-(-1 + \lambda ) \lambda (-4 + 3 \lambda ) - 2 (2 + (-2 + \lambda ) \lambda ) r \theta ) \\ \quad + 16 \gamma (-1 + \lambda ) \lambda ^5 (-(-1 + \lambda ) \lambda (12 + \lambda (-2 + \lambda (-17 + 8 \lambda ))) + (12 - \lambda (8 + \\ \lambda (11 + \lambda (-14 + 5 \lambda )))) r \theta ) + 2 \gamma ^2 \lambda ^4 (-(-1 + \lambda ) \lambda (-240 + \lambda (-100 + \lambda (610 + \lambda - 419 \lambda ^2 + 144 \lambda ^3))) - 4 (60 + \lambda (-20 + \lambda (-133 + \lambda (89 +\\ \lambda (61 + \lambda (-76 + 21 \lambda )))))) r \theta ) + 4 \gamma ^3 \lambda ^3 (-(-1 \\ \quad + \lambda ) \lambda (-160 + \lambda (-320 + \lambda (220 + \lambda (524 + \lambda (-167 + 4 \lambda (-59 + 22 \lambda )))))) - 2 (80 + (-2 + \\ \lambda ) \lambda (-40 + \lambda (76 + \lambda (94 + \lambda (-52 + \lambda (-35 + 24 \lambda )))))) r \theta ) + \gamma ^4 \lambda ^2 (\lambda (-480 + \lambda (-1240 + \lambda (640 \\ \quad + \lambda (2996 + \lambda (-283 + \lambda (-2604 + \lambda (329 + \\ 4 (213 - 62 \lambda ) \lambda ))))))) + (-480 + \lambda (-1120\\ \quad + \lambda (384 + \lambda (2000 + \lambda (-108 + \lambda (-1418 + \lambda (173 + 8 (53 - 16 \lambda ) \lambda ))))))) r \theta ) + \gamma ^6 (-\lambda (1 + \lambda - \\ \lambda ^2)^2 (32 + \lambda (120 + \lambda (116 + \lambda (-38 + \lambda (-93 + 2 \lambda (-7 + 8 \lambda )))))) \\ \quad - (-2 + (-2 + \lambda ) \lambda ) (-16 + \lambda (-64 + \lambda (-68 + \lambda (32 + \lambda (84 + \lambda (8 + \\ \lambda (-37 + 8 (-1 + \lambda ) \lambda ))))))) r \theta ) + 2 \gamma ^5 \lambda (-\lambda (-1 + (-1 + \lambda ) \lambda ) (-96 \\ \quad + \lambda (-304 + \lambda (-96 + \lambda (440 + \lambda (255 + \lambda (-215 + 8 \lambda (-13 + 6 \lambda ))))))) - \\ (96 + \lambda (352 + \lambda (248 + \lambda (-424 + \lambda (-484 \\ \quad + \lambda (217 + \lambda (298 + \lambda (-77 + 4 \lambda (-19 + 6 \lambda ))))))))) r \theta ) \end{array} \right\} }{2 A_2^2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial \pi _{1}^{bs*}}{\partial \gamma }=\frac{\left\{ \begin{array}{l} \lambda (16 (-1 + \lambda )^2 \lambda ^6 (2 (-1 + \lambda ) \lambda + (-2 + \lambda ) r \theta ) + 16 \gamma (-1 + \lambda ) \lambda ^5 ((-1 + \lambda ) \lambda (-12 + \lambda (-3 + 10 \lambda )) \\ \quad + (12 + \lambda (-2 + \lambda (-14 + 5 \lambda ))) r \theta - \\ 2 (-1 + \lambda ) r^2 \theta ^2) + 4 \gamma ^2 (-1 + \lambda ) \lambda ^4 (\lambda (120 + \lambda (60 + \lambda (-209 + \lambda (-33 + 80 \lambda )))) \\ \quad + 2 (60 + \lambda (70 + \lambda (-73 + 5 \lambda (-9 + 4 \lambda )))) r \theta + 4 (10 - \\ 7 \lambda ^2) r^2 \theta ^2) + 4 \gamma ^3 \lambda ^3 (\lambda (-160 + \lambda (-120 + \lambda (436 + \lambda (156 \\ \quad + \lambda (-387 + \lambda (-11 + 80 \lambda )))))) + (-160 + \lambda (-240 + \lambda (344 + \lambda (348 + \lambda (-264 + \\ \lambda (-79 + 40 \lambda )))))) r \theta - 2 (40 + \lambda ^2 (-61 + \lambda (7 + 16 \lambda ))) r^2 \theta ^2) + \gamma ^4 \lambda ^2 (\lambda (-480 + \lambda (-960 \\ \quad + \lambda (856 + \lambda (1936 + \lambda (-741 + \lambda (-1207 + 4 \lambda (83 + \\ 40 \lambda ))))))) + (-480 + \lambda (-1360 + \lambda (144 + \lambda (2240 + \lambda (176 \\ \quad + \lambda (-1091 + 76 \lambda + 80 \lambda ^2)))))) r \theta - 4 (80 + \lambda (80 + (-2 + \lambda ) \lambda (59 + 10 \lambda (7 + \\ \lambda )))) r^2 \theta ^2) + 2 \gamma ^5 \lambda (\lambda (-96 + \lambda (-312 + \lambda (-56 + \lambda (608 + \lambda (277 + \lambda (-454 \\ \quad + \lambda (-167 + 2 \lambda (63 + 8 \lambda )))))))) + (-96 + \lambda (-400 + \lambda (-304 + \\ \lambda (512 + \lambda (520 + \lambda (-273 + \lambda (-199 + 78 \lambda + 8 \lambda ^2))))))) r \theta \\ \quad + 2 (-40 + \lambda (-80 + \lambda (22 + \lambda (92 + \lambda (-4 + \lambda (-27 + 4 \lambda )))))) r^2 \theta ^2) + \gamma ^6 (\lambda (-32 + \\ \lambda (-144 + \lambda (-140 + \lambda (208 + \lambda (359 + \lambda (-65 + \lambda (-250 + \lambda (-3 + 56 \lambda )))))))) + (-32 + \lambda (-176 + \lambda (-264 + \lambda (80 + \lambda (408 + \lambda (69 + \\ \lambda (-216 + \lambda (-27 + 44 \lambda )))))))) r \theta \\ \quad + (-32 + \lambda (-96 + \lambda (-40 + \lambda (100 + \lambda (56 + \lambda (-43 + 4 \lambda (-3 + 2 \lambda ))))))) r^2 \theta ^2)) \end{array} \right\} }{A_2^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^3}\\{} & {} \frac{\partial \pi _{2}^{bs*}}{\partial \gamma }=\frac{\left\{ \begin{array}{l} \lambda (16 (-1 + \lambda )^2 \lambda ^5 (2 (-1 + \lambda ) \lambda + (-2 + \lambda ) r \theta ) + g^5 (\lambda (-1 + (-1 + \lambda ) \lambda ) (32 + \lambda (120 + \lambda (108 + \lambda (-70 \\ \quad + \lambda (-115 + 4 \lambda (1 + 8 \lambda )))))) + \\ (8 + 16 \lambda + \lambda ^3 (-12 + \lambda (-3 + 4 \lambda ))) (-4 + \lambda (-8 + \lambda (-1 + 4 \lambda ))) r \theta ) + 8 g (-1 + \lambda ) \lambda ^4 ((-1\\ \quad + \lambda ) \lambda (-20 + \lambda (-11 + 20 \lambda )) + 2 (10 + \lambda (-2 +\\ \lambda (-11 + 5 \lambda ))) r \theta ) + 4 g^2 \lambda ^3 ((-1 + \lambda ) \lambda (80 + \lambda (88 + \lambda (-139 + \lambda (-89 + 80 \lambda ))))\\ \quad + 2 (-40 + \lambda (-4 + \lambda (85 + \lambda (-13 + 10 \lambda (-5 + 2 \lambda ))))) r \theta ) + \\ 2 g^3 \lambda ^2 (\lambda (-160 + \lambda (-264 + \lambda (354 + \lambda (517 + \lambda (-349 + 10 \lambda (-27 + 16 \lambda )))))) + 4 (-40 + \lambda (-56 \\ \quad + \lambda (67 + 2 \lambda (43 + \lambda (-25 + \lambda (-19 + \\ 10 \lambda )))))) r \theta ) + g^4 \lambda (\lambda (-160 + \lambda (-512 + \lambda (-124 + \lambda (876 + \lambda (431 + \lambda (-569 + 4 \lambda (-51 + 40 \lambda ))))))) + (-160 \\ \quad + \lambda (-432 + \lambda (-72 + \lambda (584 + \\ \lambda (252 + \lambda (-309 + 16 \lambda (-7 + 5 \lambda ))))))) r \theta )) (4 (-1 + \lambda ) \lambda ^3 (-1 + \lambda + r \theta ) + 2 \gamma \lambda ^2 (6 - 4 r \theta + \lambda (-3 - 2 r \theta + \lambda (-7 + 4 \lambda + 4 r \theta ))) + \gamma ^3 (4 - \\ \lambda (2 (-5 + r \theta ) \\ \quad + \lambda (-3 + 2 r \theta + \lambda (5 + \lambda - r \theta )))) + \gamma ^2 \lambda (12 - 4 r \theta + \lambda (12 - 10 r \theta + \lambda (-15 + r \theta + \lambda (-7 + 4 \lambda + 4 r \theta ))))) \end{array} \right\} }{4 (g + \lambda + g \lambda )^2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + g^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^3}\\ \end{aligned}$$

Based on the above results, we can easily obtain the findings in Proposition 5. Note that $\gamma _{1}$ and $\gamma _{1}$ are the third and fourth root of $16 \lambda ^6 - 48 \lambda ^7 + 48 \lambda ^8 - 16 \lambda ^9 + (96 \lambda ^5 - 192 \lambda ^6 + 24 \lambda ^7 + 152 \lambda ^8 - 88 \lambda ^9 + 8 \lambda ^{10}) \gamma + (240 \lambda ^4 - 240 \lambda ^5 - 396 \lambda ^6 + 456 \lambda ^7 + 100 \lambda ^8 - 208 \lambda ^9 + 48 \lambda ^{10}) \gamma ^2 + (320 \lambda ^3 - 864 \lambda ^5 + 136 \lambda ^6 + 768 \lambda ^7 - 216 \lambda ^8 - 248 \lambda ^9 + 104 \lambda ^{10}) \gamma ^3 + (240 \lambda ^2 + 240 \lambda ^3 - 696 \lambda ^4 - 568 \lambda ^5 + 776 \lambda ^6 + 460 \lambda ^7 - 419 \lambda ^8 - 141 \lambda ^9 + 104 \lambda ^{10}) \gamma ^4 + (96 \lambda + 192 \lambda ^2 - 216 \lambda ^3 - 576 \lambda ^4 + 88 \lambda ^5 + 582 \lambda ^6 + 42 \lambda ^7 - 262 \lambda ^8 - 26 \lambda ^9 + 48 \lambda ^{10}) \gamma ^5 + (16 + 48 \lambda - 12 \lambda ^2 - 160 \lambda ^3 - 108 \lambda ^4 + 122 \lambda ^5 + 130 \lambda ^6 - 34 \lambda ^7 - 55 \lambda ^8 + 3 \lambda ^9 + 8 \lambda ^{10}) \gamma ^6=0$, respectively; $\theta _{1}=(8 \gamma ^6 \lambda ^2 + 40 \gamma ^5 \lambda ^3 + 44 \gamma ^6 \lambda ^3 + 80 \gamma ^4 \lambda ^4 + 152 \gamma ^5 \lambda ^4 + 64 \gamma ^6 \lambda ^4 + 80 \gamma ^3 \lambda ^5 + 168 \gamma ^4 \lambda ^5 + 56 \gamma ^5 \lambda ^5 - 27 \gamma ^6 \lambda ^5 + 40 \gamma ^2 \lambda ^6 + 32 \gamma ^3 \lambda ^6 - 192 \gamma ^4 \lambda ^6 - 286 \gamma ^5 \lambda ^6 - 102 \gamma ^6 \lambda ^6 + 8 \gamma \lambda ^7 - 52 \gamma ^2 \lambda ^7 - 272 \gamma ^3 \lambda ^7 - 351 \gamma ^4 \lambda ^7 - 144 \gamma ^5 \lambda ^7 - 5 \gamma ^6 \lambda ^7 - 24 \gamma \lambda ^8 - 64 \gamma ^2 \lambda ^8 + 40 \gamma ^3 \lambda ^8 + 230 \gamma ^4 \lambda ^8 + 204 \gamma ^5 \lambda ^8 + 54 \gamma ^6 \lambda ^8 + 24 \gamma \lambda ^9 + 124 \gamma ^2 \lambda ^9 + 224 \gamma ^3 \lambda ^9 + 173 \gamma ^4 \lambda ^9 + 50 \gamma ^5 \lambda ^9 + \gamma ^6 \lambda ^9 - 8 \gamma \lambda ^10 - 48 \gamma ^2 \lambda ^10 - 104 \gamma ^3 \lambda ^10 - 104 \gamma ^4 \lambda ^10 - 48 \gamma ^5 \lambda ^10 - 8 \gamma ^6 \lambda ^10)/(16 \gamma ^6 + 96 \gamma ^5 \lambda + 48 \gamma ^6 \lambda + 240 \gamma ^4 \lambda ^2 + 192 \gamma ^5 \lambda ^2 - 4 \gamma ^6 \lambda ^2 + 320 \gamma ^3 \lambda ^3 + 240 \gamma ^4 \lambda ^3 - 176 \gamma ^5 \lambda ^3 - 116 \gamma ^6 \lambda ^3 + 240 \gamma ^2 \lambda ^4 - 616 \gamma ^4 \lambda ^4 - 424 \gamma ^5 \lambda ^4 - 44 \gamma ^6 \lambda ^4 + 96 \gamma \lambda ^5 - 240 \gamma ^2 \lambda ^5 - 784 \gamma ^3 \lambda ^5 - 400 \gamma ^4 \lambda ^5 + 144 \gamma ^5 \lambda ^5 + 95 \gamma ^6 \lambda ^5 + 16 \lambda ^6 - 192 \gamma \lambda ^6 - 356 \gamma ^2 \lambda ^6 + 168 \gamma ^3 \lambda ^6 + 584 \gamma ^4 \lambda ^6 + 296 \gamma ^5 \lambda ^6 + 28 \gamma ^6 \lambda ^6 - 48 \lambda ^7 + 32 \gamma \lambda ^7 + 404 \gamma ^2 \lambda ^7 + 496 \gamma ^3 \lambda ^7 + 109 \gamma ^4 \lambda ^7 - 102 \gamma ^5 \lambda ^7 - 39 \gamma ^6 \lambda ^7 + 48 \lambda ^8 + 128 \gamma \lambda ^8 + 36 \gamma ^2 \lambda ^8 - 176 \gamma ^3 \lambda ^8 - 189 \gamma ^4 \lambda ^8 - 58 \gamma ^5 \lambda ^8 - \gamma ^6 \lambda ^8 - 16 \lambda ^9 - 64 \gamma \lambda ^9 - 84 \gamma ^2 \lambda ^9 - 24 \gamma ^3 \lambda ^9 + 32 \gamma ^4 \lambda ^9 + 24 \gamma ^5 \lambda ^9 + 4 \gamma ^6 \lambda ^9)$; $r_{1}=(8 \gamma ^6 \lambda ^2 + 40 \gamma ^5 \lambda ^3 + 44 \gamma ^6 \lambda ^3 + 80 \gamma ^4 \lambda ^4 + 152 \gamma ^5 \lambda ^4 + 64 \gamma ^6 \lambda ^4 + 80 \gamma ^3 \lambda ^5 + 168 \gamma ^4 \lambda ^5 + 56 \gamma ^5 \lambda ^5 - 27 \gamma ^6 \lambda ^5 + 40 \gamma ^2 \lambda ^6 + 32 \gamma ^3 \lambda ^6 - 192 \gamma ^4 \lambda ^6 - 286 \gamma ^5 \lambda ^6 - 102 \gamma ^6 \lambda ^6 + 8 \gamma \lambda ^7 - 52 \gamma ^2 \lambda ^7 - 272 \gamma ^3 \lambda ^7 - 351 \gamma ^4 \lambda ^7 - 144 \gamma ^5 \lambda ^7 - 5 \gamma ^6 \lambda ^7 - 24 \gamma \lambda ^8 - 64 \gamma ^2 \lambda ^8 + 40 \gamma ^3 \lambda ^8 + 230 \gamma ^4 \lambda ^8 + 204 \gamma ^5 \lambda ^8 + 54 \gamma ^6 \lambda ^8 + 24 \gamma \lambda ^9 + 124 \gamma ^2 \lambda ^9 + 224 \gamma ^3 \lambda ^9 + 173 \gamma ^4 \lambda ^9 + 50 \gamma ^5 \lambda ^9 + \gamma ^6 \lambda ^9 - 8 \gamma \lambda ^10 - 48 \gamma ^2 \lambda ^10 - 104 \gamma ^3 \lambda ^10 - 104 \gamma ^4 \lambda ^10 - 48 \gamma ^5 \lambda ^10 - 8 \gamma ^6 \lambda ^10)/(16 \gamma ^6 \theta + 96 \gamma ^5 \theta \lambda + 48 \gamma ^6 \theta \lambda + 240 \gamma ^4 \theta \lambda ^2 + 192 \gamma ^5 \theta \lambda ^2 - 4 \gamma ^6 \theta \lambda ^2 + 320 \gamma ^3 \theta \lambda ^3 + 240 \gamma ^4 \theta \lambda ^3 - 176 \gamma ^5 \theta \lambda ^3 - 116 \gamma ^6 \theta \lambda ^3 + 240 \gamma ^2 \theta \lambda ^4 - 616 \gamma ^4 \theta \lambda ^4 - 424 \gamma ^5 \theta \lambda ^4 - 44 \gamma ^6 \theta \lambda ^4 + 96 \gamma \theta \lambda ^5 - 240 \gamma ^2 \theta \lambda ^5 - 784 \gamma ^3 \theta \lambda ^5 - 400 \gamma ^4 \theta \lambda ^5 + 144 \gamma ^5 \theta \lambda ^5 + 95 \gamma ^6 \theta \lambda ^5 + 16 \theta \lambda ^6 - 192 \gamma \theta \lambda ^6 - 356 \gamma ^2 \theta \lambda ^6 + 168 \gamma ^3 \theta \lambda ^6 + 584 \gamma ^4 \theta \lambda ^6 + 296 \gamma ^5 \theta \lambda ^6 + 28 \gamma ^6 \theta \lambda ^6 - 48 \theta \lambda ^7 + 32 \gamma \theta \lambda ^7 + 404 \gamma ^2 \theta \lambda ^7 + 496 \gamma ^3 \theta \lambda ^7 + 109 \gamma ^4 \theta \lambda ^7 - 102 \gamma ^5 \theta \lambda ^7 - 39 \gamma ^6 \theta \lambda ^7 + 48 \theta \lambda ^8 + 128 \gamma \theta \lambda ^8 + 36 \gamma ^2 \theta \lambda ^8 - 176 \gamma ^3 \theta \lambda ^8 - 189 \gamma ^4 \theta \lambda ^8 - 58 \gamma ^5 \theta \lambda ^8 - \gamma ^6 \theta \lambda ^8 \ - 16 \theta \lambda ^9 - 64 \gamma \theta \lambda ^9 - 84 \gamma ^2 \theta \lambda ^9 - 24 \gamma ^3 \theta \lambda ^9 + 32 \gamma ^4 \theta \lambda ^9 + 24 \gamma ^5 \theta \lambda ^9 + 4 \gamma ^6 \theta \lambda ^9)$ $\Box$

Proposition 6:

In equilibrium, we have

$$\begin{aligned}{} & {} \frac{\partial p_{1}^{bs*}}{\partial \lambda }=\frac{\left\{ \begin{array}{l} 2 \gamma (\gamma (8 \gamma ^2 + 2 \gamma (8 + 7 \gamma ) \lambda + (8 - \gamma (2 + 11 \gamma )) \lambda ^2 - 16 (1 + \gamma )^2 \lambda ^3 \\ \quad + (1 + \gamma ) (7 + 6 \gamma ) \lambda ^4) - (4 \gamma ^2 + 2 \gamma (2 + \gamma )^2 \lambda + (-2 + \gamma )^2 (1 + \gamma ) \lambda ^2 - 8 (1 + \\ \gamma )^2 \lambda ^3 + 4 (1 + \gamma )^2 \lambda ^4) r \theta ) \end{array} \right\} }{(A_{1})^2}\\{} & {} \frac{\partial p_{2}^{bs*}}{\partial \lambda }=\frac{\left\{ \begin{array}{l} -16 (-1 + \lambda )^2 \lambda ^6 - 8 \gamma ^2 \lambda ^3 (\lambda (27 + \lambda (6 + \lambda (-46 + 3 \lambda (1 + 4 \lambda )))) \\ \quad + (-1 + \lambda ) (-8 + \lambda (-3 + 7 \lambda )) r \theta ) + 2 \gamma ^3 \lambda ^2 (-\lambda (112 + \lambda (192 + \lambda (-152 + \lambda (-199 + \\ 4 \lambda (17 + 8 \lambda ))))) - 4 (12 + \lambda (10 + (-2 + \lambda ) \lambda (11 + 9 \lambda ))) r \theta )\\ \quad + \gamma ^4 \lambda (-\lambda (96 + \lambda (384 + \lambda (170 + \lambda (-372 + \lambda (-159 + 8 \lambda (13 + 2 \lambda )))))) - 2 (32 + \\ \lambda (80 + \lambda (-16 - 83 \lambda + 20 \lambda ^3))) r \theta ) + \gamma ^5 (-2 \lambda ^2 (48 + \lambda (98 + \lambda (11 + \lambda (-57 \\ \quad + 2 \lambda (-5 + 6 \lambda ))))) - (16 + \lambda (80 + \lambda (64 + \lambda (-52 + \lambda (-41 + 8 \lambda (1 + \\ \lambda )))))) r \theta ) - 16 \gamma (-1 + \lambda ) \lambda ^4 (4 \lambda ^3 - r \theta + \lambda (-6 + r \theta )) + \gamma ^6 (8 - 8 r \theta + \lambda (16 - 16 r \theta \\ \quad + \lambda (-2 (5 + r \theta ) + \lambda (4 (-9 + r \theta ) + \lambda (\lambda (-2 + 3 \lambda ) - 5 (4 + r \theta )))))) \end{array} \right\} }{2 A_2^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial p_{s}^{bs*}}{\partial \lambda }=\frac{\left\{ \begin{array}{l} \gamma (-16 (-1 + \lambda ) ^2 \lambda ^6 (2 + (-4 + \lambda ) \lambda ) r \theta + 8 \gamma (-1 + \lambda ) \lambda ^5 ((-2 + \lambda ) (-1 + \lambda ) \lambda (-2 + 3 \lambda ) - 2 (-12 \\ \quad + \lambda (19 + \lambda (15 - 26 \lambda + 6 \lambda ^2))) r \theta ) + 4 \gamma ^2 \lambda ^4 (\lambda (48 - \\ 116 \lambda + \lambda ^3 (187 + 7 \lambda (-22 + 5 \lambda ))) - 2 (60 + \lambda (-68 + \lambda (-177 + \lambda (202 \\ \quad + \lambda (90 + \lambda (-139 + 30 \lambda )))))) r \theta ) + 4 \gamma ^3 \lambda ^3 (\lambda (120 + \lambda (-100 + \lambda (-395 + \\ \lambda (388 + \lambda (251 - 352 \lambda + 86 \lambda ^2))))) + (-160 + \lambda (-60 + \lambda (684 + \lambda (119 + \lambda (-822 \\ \quad + \lambda (37 + 8 (39 - 10 \lambda ) \lambda )))))) r \theta ) + \gamma ^4 \lambda ^2 (4 \lambda (160 + \lambda (120 + \\ \lambda (-660 + \lambda (-280 + \lambda (902 + \lambda (55 + 2 \lambda (-214 + 57 \lambda ))))))) + (-480 + \lambda (-960 + \lambda (1664 + \lambda (3120 \\ \quad + \lambda (-1652 + \lambda (-2712 + \lambda (837 + 16 (47 - \\ 15 \lambda ) \lambda ))))))) r \theta ) + \gamma ^7 ((1 + \lambda - \lambda ^2)^2 (32 + \lambda (112 + \lambda (68 + \lambda (-128 \\ \quad + \lambda (-153 + 8 \lambda (-2 + 3 \lambda )))))) - (16 + \lambda (48 + \lambda (4 + \lambda (-96 + \lambda (-52 + \lambda (46 + \\ \lambda (10 + \lambda (-16 + 7 \lambda )))))))) r \theta ) + \gamma ^6 (2 \lambda (-1 + (-1 + \lambda ) \lambda ) (-96 + \lambda (-280 \\ \quad + \lambda (32 + \lambda (622 + \lambda (251 + \lambda (-357 + 2 \lambda (-71 + 35 \lambda ))))))) - (32 + \\ \lambda (224 + \lambda (280 + \lambda (-416 + \lambda (-828 + \lambda (46 + \lambda (520 + \lambda (56 + \lambda (-111 + 8 \lambda (-3 + 2 \lambda )))))))))) r \theta )\\ \quad + \gamma ^5 \lambda (-192 r \theta + \lambda (480 - 752 r \theta + \lambda (1120 + \\ \lambda (200 (-6 + 11 r \theta ) + \lambda (8 (-439 + 140 r \theta ) + \lambda (951 - 1750 r \theta + \lambda (3442 - 920 r \theta \\ \quad + \lambda (-645 + 567 r \theta + 8 \lambda (-146 + 43 \lambda + 4 (7 - 3 \lambda ) r \theta )))))))))) \end{array} \right\} }{2 A_2^2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial d_{1n}^{bs*}}{\partial \lambda }=\frac{\left\{ \begin{array}{l} \gamma (16 (-1 + \lambda )^2 \lambda ^6 (-1 + 2 \lambda ) r \theta + 4 \gamma (-1 + \lambda ) \lambda ^5 (-(-1 + \lambda ) \lambda ^3 \\ \quad + 4 (1 + \lambda ) (-1 + 2 \lambda ) (-6 + 5 \lambda ) r \theta ) + 4 \gamma ^2 \lambda ^4 (-2 (\lambda ^2 + \lambda ^4 - 5 \lambda ^5 + 3 \lambda ^6) + \\ (-1 + 2 \lambda ) (60 + \lambda (24 + \lambda (-111 + \lambda (-8 + 39 \lambda )))) r \theta ) + \gamma ^7 (-\lambda (2 \\ \quad + 5 \lambda ) (-1 + (-1 + \lambda ) \lambda )^2 (4 + \lambda (6 + \lambda )) - (-2 + (-2 + \lambda ) \lambda ) (-8 + \\ \lambda (-12 + \lambda (24 + \lambda (42 + \lambda (-4 + 3 (-6 + \lambda ) \lambda ))))) r \theta ) + 4 \gamma ^3 \lambda ^3 (-\lambda ^2 (10 + \lambda (2 \\ \quad + \lambda (-6 + \lambda - 20 \lambda ^2 + 14 \lambda ^3))) + 2 (-40 + \lambda (10 + \lambda (193 + \\ \lambda (-23 + \lambda (-190 + \lambda (29 + 36 \lambda )))))) r \theta ) + \gamma ^4 \lambda ^2 (-8 \lambda ^2 (10 + \lambda (14 + \lambda (-14 + \lambda (-5 + \lambda \\ \quad - 10 \lambda ^2 + 8 \lambda ^3)))) + (-240 + \lambda (-320 + \lambda (1336 + \\ \lambda (1416 + \lambda (-1448 + \lambda (-1000 + \lambda (493 + 112 \lambda ))))))) r \theta ) + \gamma ^5 \lambda (-\lambda ^2 (80 + \lambda (228 + \lambda (8 \\ \quad + \lambda (-251 + \lambda (14 + \lambda (21 + 4 \lambda (-10 + 9 \lambda ))))))) + \\ (-96 + \lambda (-336 + \lambda (384 + \lambda (1544 + \lambda (128 + \lambda (-1282 + \lambda (-136 + 287 \lambda ))))))) r \theta ) + \gamma ^6 (-2 \lambda ^2 (-1 + (-1\\ \quad + \lambda ) \lambda ) (-20 + \lambda (-68 + \lambda (-36 + \\ \lambda (39 + 13 \lambda + 4 \lambda ^3)))) - (16 + \lambda (128 + \lambda (68 + \lambda (-528 + \lambda (-580 + \lambda (310 + \lambda (392 \\ \quad + \lambda (-96 + \lambda (-51 + 8 \lambda ))))))))) r \theta )) \end{array} \right\} }{2 A_2^2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial d_{2n}^{bs*}}{\partial \lambda }=\frac{\left\{ \begin{array}{l} 2 \gamma (\gamma + \lambda + \gamma \lambda - (1 + \gamma ) \lambda ^2)^2 (4 \lambda ^4 + 4 \gamma \lambda ^2 (1 + \lambda (2 + 3 \lambda )) + \gamma ^3 (4 + \lambda (16 + 11 \lambda )) + 4 \gamma ^2 \lambda (2\\ \quad + \lambda (5 + 2 \lambda (1 + \lambda )))) + (-8 \gamma ^5 (2 + \gamma ) - 16 \gamma ^4 (1 + \\ \gamma ) (4 + \gamma ) \lambda - 2 \gamma ^3 (48 + \gamma (88 + \gamma (44 + 5 \gamma ))) \lambda ^2 - 4 \gamma ^2 (1 \\ \quad + \gamma ) (16 + \gamma (20 + \gamma (4 + 3 \gamma ))) \lambda ^3 + \gamma (1 + \gamma ) (-16 - 40 \gamma + 22 \gamma ^3 + \gamma ^4) \lambda ^4 + 4 \gamma (1 + \\ \gamma )^2 (-8 + \gamma (2 + \gamma ) (4 + 3 \gamma )) \lambda ^5 - (1 + \gamma )^2 (16 + \gamma (-48 - 64 \gamma \\ \quad + 3 \gamma ^3)) \lambda ^6 - 8 (1 + \gamma )^3 (-4 + \gamma ^2) \lambda ^7 - 16 (1 + \gamma )^4 \lambda ^8) r \theta \end{array} \right\} }{2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial d_{s}^{bs*}}{\partial \lambda }=\frac{\left\{ \begin{array}{l} -16 (-1 + \lambda )^2 \lambda ^6 (2 + (-4 + \lambda ) \lambda ) r \theta + 8 \gamma \lambda ^5 ((-2 + \lambda ) (-1 + \lambda )^2 \lambda (-2 + 3 \lambda ) - 4 (6 - 15 \lambda + \lambda ^3 (23 \\ \quad + \lambda (-17 + 3 \lambda ))) r \theta ) + \gamma ^8 (-\lambda (2 + 5 \lambda ) (-1 + (-1 + \\ \lambda ) \lambda )^2 (4 + \lambda (6 + \lambda )) - (-2 + (-2 + \lambda ) \lambda ) (-8 + \lambda (-12 + \lambda (24 + \lambda (42 + \lambda (-4 + 3 (-6 + \lambda ) \lambda ))))) r \theta ) \\ \quad + 8 \gamma ^2 \lambda ^4 (\lambda (24 - 58 \lambda + \lambda ^3 (93 + \lambda (-76 + 17 \lambda ))) + \\ (-60 + \lambda (56 + \lambda (211 - \lambda (210 + \lambda (124 + 3 \lambda (-53 + 10 \lambda )))))) r \theta ) + 4 \gamma ^3 \lambda ^3 (\lambda (120 + \lambda (-100 + \lambda (-397 \\ \quad + \lambda (388 + \lambda (249 - 342 \lambda + 80 \lambda ^2))))) - \\ 2 (80 + \lambda (60 + \lambda (-390 + \lambda (-139 + \lambda (518 + \lambda (9 + 5 \lambda (-39 + 8 \lambda ))))))) r \theta ) + \gamma ^4 \lambda ^2 (8 \lambda (80 + \lambda (60 + \lambda (-335 \\ \quad + \lambda (-141 + \lambda (454 + \lambda (27 - 204 \lambda + \\ 50 \lambda ^2)))))) + (-480 + \lambda (-1280 + \lambda (1744 + \lambda (4664 + \lambda (-1836 + \lambda (-4232 + \lambda (1069 + 80 (13 - 3 \lambda ) \lambda ))))))) r \theta ) \\ \quad + \gamma ^5 \lambda (\lambda (1 + \lambda ) (480 + \lambda (640 + \\ \lambda (-1920 + \lambda (-1704 + \lambda (2767 + \lambda (715 + 8 \lambda (-171 + 35 \lambda ))))))) - 2 (96 \\ \quad + \lambda (496 + \lambda (160 + \lambda (-1768 + \lambda (-1268 + \lambda (1599 + 2 \lambda (480 + \lambda (-265 + \\ 12 \lambda (-7 + 2 \lambda ))))))))) r \theta ) + \gamma ^6 (\lambda (192 + \lambda (752 \\ \quad + \lambda (224 + \lambda (-2096 + \lambda (-1690 + \lambda (1707 + \lambda (1486 + \lambda (-591 + 8 \lambda (-48 + 13 \lambda ))))))))) - 2 (16 + \\ \lambda (160 + \lambda (308 + \lambda (-400 + \lambda (-1186 + \lambda (-41 + \lambda (901 + \lambda (96 + \lambda (-199 + 4 \lambda (-3 + 2 \lambda )))))))))) r \theta ) \\ \quad + \gamma ^7 (32 - 32 r \theta + \lambda (-176 (-1 + r \theta ) + \lambda (220 - \\ 72 r \theta + \lambda (8 (-43 + 78 r \theta ) + \lambda (-837 + 632 r \theta + \lambda (-4 (19 + 89 r \theta ) + \lambda (645 - 402 r \theta \\ \quad + \lambda (2 (95 + 56 r \theta ) + \lambda (-163 + 44 r \theta + 8 \lambda (-7 + 2 \lambda - r \theta )))))))))) \end{array} \right\} }{2 A_2^2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial \pi _{1}^{bs*}}{\partial \lambda }=\frac{\left\{ \begin{array}{l} \gamma (-32 \gamma ^7 - 192 \gamma ^6 \lambda - 144 \gamma ^7 \lambda - 480 \gamma ^5 \lambda ^2 - 624 \gamma ^6 \lambda ^2 - 132 \gamma ^7 \lambda ^2 - 640 \gamma ^4 \lambda ^3 - 960 \gamma ^5 \lambda ^3 \\ \quad - 64 \gamma ^6 \lambda ^3 + 260 \gamma ^7 \lambda ^3 - 480 \gamma ^3 \lambda ^4 - 480 \gamma ^4 \lambda ^4 + \\ 976 \gamma ^5 \lambda ^4 + 1468 \gamma ^6 \lambda ^4 + 483 \gamma ^7 \lambda ^4 - 192 \gamma ^2 \lambda ^5 + 240 \gamma ^3 \lambda ^5 + 1904 \gamma ^4 \lambda ^5 + 2416 \gamma ^5 \lambda ^5 \\ \quad + 998 \gamma ^6 \lambda ^5 + 54 \gamma ^7 \lambda ^5 - 32 \gamma \lambda ^6 + 336 \gamma ^2 \lambda ^6 + 1196 \gamma ^3 \lambda ^6 +\\ 1064 \gamma ^4 \lambda ^6 - 205 \gamma ^5 \lambda ^6 - 683 \gamma ^6 \lambda ^6 - 242 \gamma ^7 \lambda ^6 + 96 \gamma \lambda ^7 + 112 \gamma ^2 \lambda ^7 - 524 \gamma ^3 \lambda ^7 \\ \quad - 1364 \gamma ^4 \lambda ^7 - 1246 \gamma ^5 \lambda ^7 - 482 \gamma ^6 \lambda ^7 - 60 \gamma ^7 \lambda ^7 - 88 \gamma \lambda ^8 - \\ 356 \gamma ^2 \lambda ^8 - 536 \gamma ^3 \lambda ^8 - 321 \gamma ^4 \lambda ^8 + 12 \gamma ^5 \lambda ^8 + 95 \gamma ^6 \lambda ^8 + 30 \gamma ^7 \lambda ^8 \\ \quad + 24 \gamma \lambda ^9 + 108 \gamma ^2 \lambda ^9 + 188 \gamma ^3 \lambda ^9 + 156 \gamma ^4 \lambda ^9 + 60 \gamma ^5 \lambda ^9 + 8 \gamma ^6 \lambda ^9 + (-16 (-2 + \\ \gamma ) \gamma ^6 + 16 \gamma ^5 (12 + (4 - 3 \gamma ) \gamma ) \lambda - 8 \gamma ^4 (1 + \gamma ) (-60 \\ \quad + \gamma (2 + 3 \gamma )) \lambda ^2 + 4 \gamma ^3 (160 + \gamma (200 + \gamma (-44 + \gamma (-76 + 5 \gamma )))) \lambda ^3 - \gamma ^2 (-480 + \gamma (-400 + \\ \gamma (1184 + \gamma (1600 + \gamma (524 + 29 \gamma ))))) \lambda ^4 - 2 \gamma (1 + \gamma ) (-96 + \gamma (224 \\ \quad + \gamma (824 + \gamma (568 + 23 \gamma (4 + \gamma ))))) \lambda ^5 + (1 + \gamma ) (32 + \gamma (-368 + \gamma (-984 + \\ \gamma (-464 + \gamma (344 + \gamma (227 + \gamma )))))) \lambda ^6 + 4 (1 \\ \quad + \gamma )^2 (-24 + \gamma (4 + \gamma (120 + \gamma (117 + 23 \gamma )))) \lambda ^7 - 4 (1 + \gamma )^3 (-20 + \gamma (-36 + \gamma (-9 + 11 \gamma ))) \lambda ^8 - \\ 16 (1 + \gamma )^5 \lambda ^9) r \theta - \gamma (-2 \gamma ^2 (2 + \gamma ) - 2 \gamma (1 + \gamma ) (4 + \gamma ) \lambda \\ \quad + (-2 + \gamma ) (1 + \gamma ) (2 + \gamma ) \lambda ^2 + 4 (1 + \gamma )^2 \lambda ^3) (8 \gamma ^3 + 8 \gamma ^2 (3 + \gamma ) \lambda + 2 \gamma (12 + (2 - \\ 7 \gamma ) \gamma ) \lambda ^2 - 2 (1 \\ \quad + \gamma ) (2 + \gamma ) (-2 + 7 \gamma ) \lambda ^3 + 3 (1 + \gamma ) (-4 + (-4 + \gamma ) \gamma ) \lambda ^4) r^2 \theta ^2) \end{array} \right\} }{A_2^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^3}\\{} & {} \frac{\partial \pi _{2}^{bs*}}{\partial \lambda }=-\frac{\left\{ \begin{array}{l} (-4 (-1 + \lambda ) \lambda ^3 (-1 + \lambda + r \theta ) + 2 \gamma \lambda ^2 (-6 + 4 r \theta + \lambda (3 + 2 r \theta \\ \quad + \lambda (7 - 4 \lambda - 4 r \theta ))) + \gamma ^3 (-4 + \lambda (2 (-5 + r \theta ) + \lambda (-3 + 2 r \theta + \lambda (5 + \lambda - r \theta )))) - \\ \gamma ^2 \lambda (12 - 4 r \theta + \lambda (12 - 10 r \theta + \lambda (-15 + r \theta + \lambda (-7 + 4 \lambda + 4 r \theta ))))) (16 (-1 + \lambda )^2 \lambda ^7 (1 - \lambda + r \theta ) \\ \quad + 8 \gamma \lambda ^5 (-(-1 + \lambda )^2 \lambda (-14 + \lambda (-1 + 10 \lambda )) + 2 (2 +\\ \lambda ^2 - 8 \lambda ^3 + 5 \lambda ^4) r \theta ) + 8 \gamma ^2 \lambda ^4 (-(-1 + \lambda ) \lambda (38 + \lambda (14 + \lambda (-69 + \lambda + 20 \lambda ^2))) \\ \quad + (20 + \lambda (-2 + \lambda (-3 + \lambda (-3 + 2 \lambda ) (3 + 10 \lambda )))) r \theta ) + 2 \gamma ^3 \lambda ^3 (\lambda (200 - \\ \lambda (-176 + \lambda (612 + \lambda (195 + \lambda (-571 + 8 \lambda (7 + 10 \lambda )))))) + 4 (40 + \lambda (34 + \lambda (-39 \\ \quad + 2 \lambda (-3 + \lambda (-9 + 2 \lambda (-4 + 5 \lambda )))))) r \theta ) + \gamma ^4 \lambda ^2 (320 r \theta + \lambda (240 + \\ 688 r \theta - \lambda (8 (-86 + 13 r \theta ) + \lambda (372 + 500 r \theta + \lambda (2 (744 + 7 r \theta ) + \lambda (-393 + 77 r \theta \\ \quad + \lambda (-949 + 32 r \theta + 16 \lambda (18 + 5 \lambda - 5 r \theta )))))))) + \gamma ^6 (32 r \theta + \lambda (-48 +\\ 208 r \theta + \lambda (64 (-2 + 5 r \theta ) + \lambda (12 + 44 r \theta - \lambda (2 (-98 + 73 r \theta ) + \lambda (18 + 60 r \theta + \lambda (\lambda (-71 \\ \quad + \lambda (-47 + 40 \lambda ) - 17 r \theta ) + 18 (7 + r \theta )))))))) + \gamma ^7 (16 (-1 + \\ r \theta ) + \lambda (-80 + 48 r \theta + \lambda (4 (-31 + 9 r \theta ) + \lambda (4 (-3 + r \theta ) + \lambda (2 (64 + 5 r \theta ) + \lambda (76 - 4 r \theta \\ \quad + \lambda (-31 - 14 r \theta + 3 \lambda (-9 + \lambda + r \theta )))))))) + \gamma ^5 \lambda (160 r \theta + \\ \lambda (16 + 608 r \theta + \lambda (264 + 408 r \theta + \lambda (356 - 372 r \theta + \lambda (-514 \\ \quad - 292 r \theta + \lambda (-663 + 6 r \theta + \lambda (449 + 9 r \theta + \lambda (347 - 8 \lambda (23 + 2 \lambda - 2 r \theta )))))))))) \end{array} \right\} }{4 A_2^2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda ))^2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^3}\\ \end{aligned}$$

Based on the above results, we can easily obtain the findings in Proposition 6 by using Mathematica. Note that due to the long expression of the threshold, we have omitted it here. If necessary, we will provide the source file. $\Box$

Proposition 7:

In equilibrium, we have

$$\begin{aligned}{} & {} \frac{\partial p_{1}^{bs*}}{\partial r}=\frac{2 \gamma ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) \theta }{A_{1}}\\{} & {} \frac{\partial p_{2}^{bs*}}{\partial r}=\frac{\lambda (-1 + \gamma (-1 +\frac{\partial p_{1}^{bs*}}{\partial r}))\theta }{2 A_2}\\{} & {} \frac{\partial p_{s}^{bs*}}{\partial r}=\frac{ {\gamma (8 \gamma ^3 + 4 \gamma ^2 (6 + 5 \gamma ) \lambda + 4 \gamma (1 + \gamma ) (6 + \gamma ) \lambda ^2 - 4 (1 + \gamma ) (-2 + \gamma (3 + 4 \gamma )) \lambda ^3 - (1 + \gamma ) (12 + \gamma (16 + 3 \gamma )) \lambda ^4 + 4 (1 + \gamma )^3 \lambda ^5) \theta }}{2 A_2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))}\\{} & {} \frac{\partial d_{1n}^{bs*}}{\partial r}=\frac{{\gamma (4 \gamma ^3 + 4 \gamma ^2 (3 + 2 \gamma ) \lambda + 12 \gamma (1 + \gamma ) \lambda ^2 - 4 (1 + \gamma ) (-1 + \gamma + \gamma ^2) \lambda ^3 + (1 + \gamma ) (-4 + (-4 + \gamma ) \gamma ) \lambda ^4) \theta }}{2 A_2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))}\\{} & {} \frac{\partial d_{2n}^{bs*}}{\partial r}=\frac{\lambda (-2 \gamma ^2 (2 + \gamma ) - 2 \gamma (1 + \gamma ) (4 + \gamma ) \lambda + (-2 + \gamma ) (1 + \gamma ) (2 + \gamma ) \lambda ^2 + 4 (1 + \gamma )^2 \lambda ^3) \theta }{2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))}\\{} & {} \frac{\partial d_{s}^{bs*}}{\partial r}=\frac{{(-2 \gamma ^2 (2 + \gamma ) - 2 \gamma (1 + \gamma ) (4 + \gamma ) \lambda + (-2 + \gamma ) (1 + \gamma ) (2 + \gamma ) \lambda ^2 + 4 (1 + \gamma )^2 \lambda ^3) ((-2 + \lambda ) \lambda + \gamma (-2 + (-2 + \lambda ) \lambda )) \theta }}{2 A_2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))}\\{} & {} \frac{\partial \pi _{1}^{bs*}}{\partial r}= \frac{\left\{ \begin{array}{l} \gamma t (4 (-2 + \lambda ) (-1 + \lambda ) \lambda ^4 + 2 \gamma ^3 \lambda (2 (8 + 18 \lambda - 13 \lambda ^3 + \lambda ^4 + \lambda ^5) + (12 + \lambda (12 - \lambda (8 + 3 \lambda ))) r \theta ) + 4 \gamma \lambda ^3 (8 \\ \quad + 2 r \theta + \lambda (-2 - 8 \lambda + 3 \lambda ^2 - 2 r \theta )) +\\ \gamma ^4 (8 + 8 r \ + \lambda (28 + \lambda (20 + \lambda (-17 + \lambda (-12 + 7 \lambda ))) + 2 (8 + (-4 + \lambda ) \lambda ^2) r \theta )) \\ \quad + \gamma ^2 \lambda ^2 (24 (2 + r \theta ) + \lambda (48 + \lambda (\lambda (-23 + 12 \lambda ) - 8 (7 + 2 r \theta ))))) \end{array} \right\} }{A_2 (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2}\\{} & {} \frac{\partial \pi _{2}^{bs*}}{\partial r}=-\frac{\left\{ \begin{array}{l} \lambda (-2 \gamma ^2 (2 + \gamma ) - 2 \gamma (1 + \gamma ) (4 + \gamma ) \lambda + (-2 + \gamma ) (1 + \gamma ) (2 + \gamma ) \lambda ^2 + 4 (1 + \gamma )^2 \lambda ^3) t (4 (-1 + \lambda ) \lambda ^3 (-1 \\ \quad + \lambda + r \theta ) + 2 \gamma \lambda ^2 (6 - 4 r \theta + \lambda (-3 - 2 r \theta + \\ \lambda (-7 + 4 \lambda + 4 r \theta ))) + \gamma ^3 (4 - \lambda (2 (-5 + r \theta ) + \lambda (-3 + 2 r \theta + \lambda (5 \\ \quad + \lambda - r \theta )))) + \gamma ^2 \lambda (12 - 4 r \theta + \lambda (12 - 10 r \theta + \lambda (-15 + r \theta + \lambda (-7 + 4 \lambda + 4 r \theta ))))) \end{array} \right\} }{2 A_2 ((-1 + \lambda ) \lambda + \gamma (-1 + (-1 + \lambda ) \lambda )) (4 (-1 + \lambda ) \lambda ^2 + \gamma ^2 (-2 + \lambda ) (2 + 3 \lambda ) + 4 \gamma \lambda (-2 + \lambda ^2))^2} \end{aligned}$$

Based on the above results, we can easily obtain the findings in Proposition 7 using Mathematica. Note that due to the long expression of the threshold, we have omitted it here. If necessary, we will provide the source file. $\Box$

Proof of propositions 8 and 9:

We first compare the selling price and demand according to the equilibrium outcomes with socially conscious consumers. That is $p_{1}^{bs*}>p_{2}^{bs*}$ by solving $p_{2}^{bs*}>p_{s}^{bs*}$, which reduces $0<r<\min \{r_{13},1\}$; by solving $d_{1n}^{bs*}>d_{2n}^{bs*}$, which reduces $\theta _{7}<\theta <1$ and $r_{14}<r<1$; by solving $d_{2n}^{bs*}>d_{s}^{bs*}$, which reduces $0<r<\min \{r_{15},1\}$.

Next, we compare the profits and consumer surplus between the benchmark scenario and the introduction of the secondary market with socially conscious consumers. That is by solving $\pi ^{bs*}>\pi ^{b*}$, which reduces $\max \{0,\gamma _{12}\}<\gamma <1$, $\theta _{8}<\theta <1$, and $r_{16}<r<1$; $cs^{bs*}>cs^{b*}$. Note that due to the long expression of the threshold, we have omitted it here. If necessary, we will provide the source file. $\Box$

Proof of proposition 10:

We compare the selling price, demand, and profit according to the equilibrium outcomes without and with socially conscious consumers, and thus obtain the findings. That is $p_{1}^{bs*}>p_{1}^{bn*}$; $p_{2}^{bs*}<p_{2}^{bn*}$; $d_{1n}^{bs*}>d_{1n}^{bn*}$; $d_{2n}^{bs*}<d_{2n}^{bn*}$; $p_{s}^{bs*}>p_{s}^{bn*}$; $d_{s}^{bs*}>d_{s}^{bn*}$; by solving $\pi ^{bs*}>\pi ^{bn*}$, which reduces $\max \{0,\gamma _{13}\}<\gamma <1$, $\max \{0,\theta _{9}\}<\theta <1$, and $\max \{0,r_{17}\}<r<1$. Note that due to the long expression of the threshold, we have omitted it here. If necessary, we will provide the source file. $\Box$