Effect of equity holding on a supply chain’s pricing and emission reduction decisions considering information sharing

Abstract

Against the backdrop of cap-and-trade regulations, we construct a low-carbon supply chain in which a seller holds the manufacturer’s equity. We investigate the impact of the seller’s equity holding and the cap-and-trade regulations on the supply chain members’ optimal decisions, profits and coordination in the cases of information symmetry and asymmetry. We find that, regardless of whether there is information asymmetry or symmetry, the manufacturer always benefits from an equity holding, while the seller can only earn higher profits through equity holding if the percentage of equity holding and the level of consumer low-carbon awareness are both sufficiently high. Both the seller and the manufacturer prefer the information asymmetry scenario when the manufacturer’s efficiency in emission reduction is below a particular threshold. In addition, equity holding can incentivize the manufacturer to share its information with the seller. Equity holding is beneficial for emission reduction and market demand but is not always beneficial for improving the supply chain’s total profit and reducing the supply chain’s total carbon emissions. Finally, with a transfer payment contract, a decentralized supply chain can obtain the same optimal profits as in the centralized supply chain scenario, and both the manufacturer and seller can earn higher profits than before.

Appendices

Appendix

Appendix 1: Proof of Proposition 1

Based on Eq. (1), we have: $H{ = }\left( {\begin{array}{*{20}c} {\frac{{\partial \Pi_{sc}^{{^{2} c}} }}{{\partial p^{2} }}} & {\frac{{\partial \Pi_{sc}^{{^{2} c}} }}{\partial p\partial e}} \\ {\frac{{\partial \Pi_{sc}^{{^{2} c}} }}{\partial e\partial p}} & {\frac{{\partial \Pi_{sc}^{{^{2} c}} }}{{\partial e^{2} }}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { - 2b} & {\theta - bp_{c} } \\ {\theta - bp_{c} } & { - (1 - \lambda p_{s} \delta )\eta_{m} + 2p_{c} \theta } \\ \end{array} } \right)$ and $\left| H \right|^{{2}} { = }2b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2}$. Since $- 2b < 0$ and $2b(1 - \lambda p_{s} \delta )\eta_{m} > (\theta + bp_{c} )^{2}$, $\Pi_{sc}^{C}$ is jointly concave in $e$ and $p$. Then, we obtain the optimal decisions of the supply chain, $e^{C}$ and $p^{C}$, under the centralized decision-making scenario according to the first-order conditions. By substituting $e^{C}$ and $p^{C}$ into $D = a - bp + \theta e$ and using Eq. (1), we obtain the market demand and the optimal profit for the supply chain under the centralized decision-making scenario. Thus, Proposition 1 is proven.□

Appendix 2: Proof of Corollary 1

We obtain $\frac{{\partial \Pi_{sc}^{C} }}{\partial \lambda } = p_{s} \{ \frac{{\delta \eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} }}{{2[2b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }} - 1\}$ based on Proposition 1. Suppose $F(\lambda ) = \delta \eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} - 2[2b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2}$; then, $\frac{{\partial \Pi_{sc}^{C} }}{\partial \lambda } > 0$ if $F(\lambda ) > 0$, and $\frac{{\partial \Pi_{sc}^{C} }}{\partial \lambda } \le 0$ if $F(\lambda ) \le 0$. It is easy to show that $F(\lambda )$ is concave in $\lambda$. Assuming $F(\lambda ) = 0$, we can obtain the equation roots, which can be written as $t_{1,2} = \frac{{4b\eta_{m} - 2(\theta + bp_{c} )^{2} \mp \sqrt {2\delta \eta_{m} } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)]}}{{4b\eta_{m} p_{s} \delta }}$. Since $2b(1 - \lambda p_{s} \delta )\eta_{m} > (\theta + bp_{c} )^{2}$, we obtain $\lambda < \frac{{2b\eta_{m} - (\theta + bp_{c} )^{2} }}{{2b\eta_{m} p_{s} \delta }}$. Thus, $t_{2}$ should be discarded since $t_{2} > \frac{{2b\eta_{m} - (\theta + bp_{c} )^{2} }}{{2b\eta_{m} p_{s} \delta }}$. Since $0 < \lambda < \frac{1}{2}$, we discuss the impact of $\lambda$ on the supply chain’s profits as follows.

• (a) If $t_{1} \le 0$, then $4b\eta_{m} - 2(\theta + bp_{c} )^{2} - \sqrt {2\delta \eta_{m} } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] \le 0$. Define $f(\sqrt {\eta_{m} } ) = 4b\eta_{m} - 2(\theta + bp_{c} )^{2} - \sqrt {2\delta \eta_{m} } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)]$. The roots of $f(\sqrt {\eta_{m} } ){ = }0$ are $\sqrt {\eta_{m} } { = }\frac{{\sqrt {2\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] \pm \sqrt {\Delta_{1} } }}{8b}$. Then, $F(\lambda ) > 0$ and $\frac{{\partial \Pi_{sc}^{C} }}{\partial \lambda } > 0$ for $\lambda \in (0,\min (\frac{1}{2},\frac{{2b\eta_{m} - (\theta + bp_{c} )^{2} }}{{2b\eta_{m} p_{s} \delta }}))$ when $0 < \eta_{m} \le \frac{{\{ \sqrt {2\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] + \sqrt {\Delta_{1} } \}^{2} }}{{64b^{2} }}$.

• (b) $\frac{{\{ \sqrt {2\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] + \sqrt {\Delta_{1} } \}^{2} }}{{64b^{2} }} < \eta_{m} < \frac{{\{ \sqrt {2\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] + \sqrt {\Delta_{2} } \}^{2} }}{{16b^{2} (2 - p_{s} \delta )^{2} }}$ if $0 < t_{1} < \frac{1}{2}$. Then, $F(\lambda ) \le 0$ and $\frac{{\partial \Pi_{sc}^{C} }}{\partial \lambda } \le 0$ for $\lambda \in (0,t_{1} ]$, $F(\lambda ) > 0$ and $\frac{{\partial \Pi_{sc}^{C} }}{\partial \lambda } > 0$ for $\lambda \in (t_{1} ,\min (\frac{1}{2},\frac{{2b\eta_{m} - (\theta + bp_{c} )^{2} }}{{2b\eta_{m} p_{s} \delta }}))$. This can be proven as above.

• (c) $\eta_{m} \ge \frac{{{{\{ }}\sqrt {2\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] + \sqrt {\Delta_{2} } {{\} }}^{2} }}{{16b^{2} (2 - p_{s} \delta )^{2} }}$ if $t_{1} \ge \frac{1}{2}$. Then, $F(\lambda ) < 0$ and $\frac{{\partial \Pi_{sc}^{C} }}{\partial \lambda } < 0$ for $\lambda \in (0,\frac{1}{2})$. Thus, Corollary 1 is proven.□

Appendix 3: Proof of Corollary 2

We obtain $\frac{{\partial \Pi_{sc}^{C} }}{{\partial p_{s} }} = \lambda \{ \frac{{\delta \eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} }}{{2[2b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }} - 1\}$ based on Proposition 1. Suppose $F(p_{s} ) = \delta \eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} - 2[2b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2}$. It is easy to prove that $\frac{{\partial \Pi_{sc}^{C} }}{{\partial p_{s} }}$ and $F(p_{s} )$ have the same sign and that $F(p_{s} )$ is concave in $p_{s}$. Assuming $F(p_{s} ) = 0$, we can obtain the equation roots, which can be written as:

$t_{3,4} = \frac{{4b\eta_{m} - 2(\theta + bp_{c} )^{2} \mp \sqrt {2\delta \eta_{m} } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)]}}{{4b\eta_{m} \lambda \delta }}$.

We have $p_{s} < \frac{{2b\eta_{m} - (\theta + bp_{c} )^{2} }}{{2b\eta_{m} \lambda \delta }}$ based on $2b(1 - \lambda p_{s} \delta )\eta_{m} > (\theta + bp_{c} )^{2}$. Thus, ${\text{t}}_{4}$ should be discarded since ${\text{t}}_{4} > \frac{{2b\eta_{m} - (\theta + bp_{c} )^{2} }}{{2b\eta_{m} \lambda \delta }}$. Then, we discuss the impact of $p_{s}$ on the supply chain’s profit as follows.

• (a) If $t_{3} \le 0$, then $4b\eta_{m} - 2(\theta + bp_{c} )^{2} - \sqrt {2\delta \eta_{m} } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] \le 0$. Define $f(\sqrt {\eta_{m} } ) = 4b\eta_{m} - 2(\theta + bp_{c} )^{2} - \sqrt {2\delta \eta_{m} } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)]$. Then, the discriminant of $f(\sqrt {\eta_{m} } ){ = }0$ is $\Delta_{1} { = }2(\theta + bp_{c} )^{2} {{\{ }}\delta [a - b(p_{c} e_{0} + c)]^{2} {{ + 16b\} }}$. It is easy to obtain the roots of $f(\sqrt {\eta_{m} } ){ = }0$ as $\frac{{\sqrt {2\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] \pm \sqrt {\Delta_{1} } }}{8b}$. Thus, $F(p_{s} ) > 0$ and $\frac{{\partial \Pi_{sc}^{C} }}{{\partial p_{s} }} > 0$ for $p_{s} \in (0,\frac{{2b\eta_{m} - (\theta + bp_{c} )^{2} }}{{2b\eta_{m} \lambda \delta }})$ when $0 < \eta_{m} \le \frac{{\{ \sqrt {2\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] + \sqrt {\Delta_{1} } \}^{2} }}{{64b^{2} }}$.

• (b) If $t_{3} > 0$, then $\eta_{m} > \frac{{\{ \sqrt {2\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] + \sqrt {\Delta_{1} } \}^{2} }}{{64b^{2} }}$. Thus, $F(p_{s} ) \le 0$ and $\frac{{\partial \Pi_{sc}^{C} }}{{\partial p_{s} }} \le 0$ in the case of $p_{s} \in (0,t_{3} ]$, and $F(p_{s} ) > 0$ and $\frac{{\partial \Pi_{sc}^{C} }}{{\partial p_{s} }} > 0$ in the case where $p_{s} \in (t_{3} ,\frac{{2b\eta_{m} - (\theta + bp_{c} )^{2} }}{{2b\eta_{m} \lambda \delta }})$. This can be proven in the same way as case (a).

  Thus, Corollary 2 is proven.□

Appendix 4: Proof of Proposition 2

Based on Eq. (5), we have $\frac{{\partial^{2} \Pi_{r}^{DS} }}{{\partial p^{2} }} = - 2b < 0$ and $\Pi_{r}^{DS}$ is concave in $p$. Thus, we obtain $p^{*} (e,w) = \frac{a + \theta e + b\lambda c + (1 - \lambda )bw}{{2b}}$ based on $\frac{{\partial \Pi_{r}^{DS} }}{\partial p} = 0$. By substituting $p^{*} (e,w)$ into Eq. (4), the Hessian matrix can be obtained as.

$\left( {\begin{array}{*{20}c} {\frac{{\partial \Pi_{m}^{{^{2} DS}} }}{{\partial e^{2} }}} & {\frac{{\partial \Pi_{m}^{{^{2} DS}} }}{\partial e\partial w}} \\ {\frac{{\partial \Pi_{m}^{{^{2} DS}} }}{\partial w\partial e}} & {\frac{{\partial \Pi_{m}^{{^{2} DS}} }}{{\partial w^{2} }}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { - (1 - \lambda p_{s} \delta )\eta_{m} + p_{c} \theta } & {\frac{{(1 - \lambda )(\theta - p_{c} b)}}{2}} \\ {\frac{{(1 - \lambda )(\theta - p_{c} b)}}{2}} & { - b(1 - \lambda )^{2} } \\ \end{array} } \right)$.

It is easy to prove that $\Pi_{m}^{DS}$ is jointly concave in $e$ and $w$ based on $4b(1 - \lambda p_{s} \delta )\eta_{m} > (\theta + bp_{c} )^{2}$. Then, we obtain the optimal decisions of the manufacturer, $e^{DS}$ and $w^{DS}$, under the information symmetry scenario according to the first order conditions. By substituting $e^{DS}$ and $w^{DS}$ into $p^{*} (e,w)$, we can obtain the optimal decision of the seller $p^{DS}$. Then, we obtain the market demand and the optimal profits of the seller and the manufacturer by substituting $e^{DS}$, $w^{DS}$ and $p^{DS}$ into $D = a - bp + \theta e$ and Eqs. (4) and (5). Thus, Proposition 2 is proven.□

Appendix 5: Proof of Corollary 3

Based on Proposition 2, we have $\frac{{\partial e^{DS} }}{\partial \lambda } = \frac{{4bp_{s} \delta \eta_{m} (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)]}}{{[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$, $\frac{{\partial e^{DS} }}{{\partial p_{s} }} = \frac{{4b\lambda \delta \eta_{m} (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)]}}{{[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$,$\frac{{\partial {\text{w}}^{DS} }}{{\partial p_{s} }} = \frac{{2\lambda \delta \eta_{m} (\theta + bp_{c} )(\theta - bp_{c} )[a - b(p_{c} e_{0} + c)]}}{{(1 - \lambda )[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$, $\frac{{\partial p^{DS} }}{\partial \lambda } = \frac{{p_{s} \delta \eta_{m} (\theta + bp_{c} )(3\theta - bp_{c} )[a - b(p_{c} e_{0} + c)]}}{{[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$, $\frac{{\partial {\text{D}}^{DS} }}{\partial \lambda } = \frac{{bp_{s} \delta \eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]}}{{[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$$\frac{{\partial p^{DS} }}{{\partial p_{s} }} = \frac{{\lambda \delta \eta_{m} (\theta + bp_{c} )(3\theta - bp_{c} )[a - b(p_{c} e_{0} + c)]}}{{[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$, $\frac{{\partial {\text{D}}^{DS} }}{{\partial p_{s} }} = \frac{{b\lambda \delta \eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]}}{{[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$, $\frac{{\partial \Pi_{m}^{DS} }}{\partial \lambda } = \frac{{p_{s} \delta \eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} }}{{2[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$ and $\frac{{\partial \Pi_{m}^{DS} }}{{\partial p_{s} }} = \frac{{\lambda \delta \eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} }}{{2[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$. Then, it is easy to prove Corollary 3 based on $[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} > 0$ and $a - b(p_{c} e_{0} + c) > 0$.□

Appendix 6: Proof of Corollary 4

We have $\frac{{\partial \Pi_{r}^{DS} }}{\partial \lambda } = p_{s} \{ \frac{{2b\delta \eta_{m}^{2} (1 - \lambda p_{s} \delta )(\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} }}{{[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{3} }} - 1\}$ based on Proposition 2. Let $G(\lambda ) = 2b\delta \eta_{m}^{2} (1 - \lambda p_{s} \delta )(\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} - [4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{3}$. Then, $\frac{{\partial \Pi_{r}^{DS} }}{\partial \lambda } > 0$ if $G(\lambda ) > 0$ and $\frac{{\partial \Pi_{r}^{DS} }}{\partial \lambda } \le 0$ if $G(\lambda ) \le 0$. $\frac{\partial G(\lambda )}{{\partial \lambda }}$ is convex in $\lambda$. Letting $\frac{\partial G(\lambda )}{{\partial \lambda }} = 0$, we can obtain the roots as $t_{5,6} = \frac{{24b\eta_{m} - 6(\theta + bp_{c} )^{2} \mp \sqrt {6\delta \eta_{m} } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)]}}{{24b\eta_{m} p_{s} \delta }}$. Based on $4b(1 - \lambda p_{s} \delta )\eta_{m} > (\theta + bp_{c} )^{2}$, we have $\lambda < \frac{{4b\eta_{m} - (\theta + bp_{c} )^{2} }}{{4b\eta_{m} p_{s} \delta }}$. Therefore,$t_{6}$ should be discarded since $t_{{6}} > \frac{{4b\eta_{m} - (\theta + bp_{c} )^{2} }}{{4b\eta_{m} p_{s} \delta }}$. Then, in the case where $0 < \lambda < \frac{1}{2}$ holds, we obtain $0 \le \eta_{m} \le \frac{{\{ \sqrt {6\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] + (\theta + bp_{c} )\sqrt {6{{\{ }}\delta [a - b(p_{c} e_{0} + c)]^{2} {{ + 96b\} }}} \}^{2} }}{{2304b^{2} }}$ if $t_{5} \le 0$. Then, $\frac{\partial G(\lambda )}{{\partial \lambda }} < 0$ if $\lambda \in (0,\min (\frac{1}{2},\frac{{4b\eta_{m} - (\theta + bp_{c} )^{2} }}{{4b\eta_{m} p_{s} \delta }}))$. Therefore, $G(\lambda ) > 0$ and $\frac{{\partial \Pi_{r}^{DS} }}{\partial \lambda } > 0$. Similarly, we can prove that in the case where $0 < \eta_{m} \le \frac{{\{ \sqrt {6\delta } (\theta + bp_{c} )[a - b(p_{c} e_{0} + c)] + (\theta + bp_{c} )\sqrt {6{{\{ }}\delta [a - b(p_{c} e_{0} + c)]^{2} {{ + 96b\} }}} \}^{2} }}{{2304b^{2} }}$ holds, $\frac{{\partial \Pi_{r}^{DS} }}{{\partial p_{s} }} > 0$ for $p_{s} \in (0,p_{s0} )$. Thus, Corollary 5 is proven.□

Appendix 7: Proof of Proposition 3

Based on Eq. (9), we have $\frac{{\partial^{2} \Pi_{r}^{DA} }}{{\partial p^{2} }} = - 2b < 0$. Thus, $\Pi_{r}^{DA}$ is concave in $p$. Then, we obtain $p^{*} (e,w) = \frac{a + \theta e + \lambda bc + (1 - \lambda )bw}{{2b}}$ based on $\frac{{\partial \Pi_{r}^{DA} }}{\partial p} = 0$. By substituting $p^{*} (e,w)$ into Eq. (8), the Hessian matrix can be obtained as.

$\left( {\begin{array}{*{20}c} {\frac{{\partial \Pi_{m}^{{^{2} DA}} }}{{\partial e^{2} }}} & {\frac{{\partial \Pi_{m}^{{^{2} DA}} }}{\partial e\partial w}} \\ {\frac{{\partial \Pi_{m}^{{^{2} DA}} }}{\partial w\partial e}} & {\frac{{\partial \Pi_{m}^{{^{2} DA}} }}{{\partial w^{2} }}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {p_{c} \theta - (1 - \lambda p_{s} \delta )\eta_{m} } & {\frac{{(1 - \lambda )(\theta - p_{c} b)}}{2}} \\ {\frac{{(1 - \lambda )(\theta - p_{c} b)}}{2}} & { - b(1 - \lambda )^{2} } \\ \end{array} } \right)$.

It is easy to prove that $\Pi_{m}^{DA}$ is jointly concave in $e$ and $w$ based on $4b(1 - \lambda p_{s} \delta )\eta_{m} > (\theta + bp_{c} )^{2}$. Then, we obtain the optimal decisions, $e^{DA}$ and $w^{DA}$, of the manufacturer under the information asymmetry scenario according to the first-order conditions. By substituting $e^{DA}$ and $w^{DA}$ into $p^{*} (e,w)$, we can obtain the optimal decision of the seller $p^{DA} = \int_{\mu - \varepsilon }^{\mu + \varepsilon } {\frac{{a + \theta e^{DA} + \lambda bc + (1 - \lambda )bw^{DA} }}{2b}} f(\eta_{m} )d_{{\eta_{m} }}$. Then, we have $p^{DA} = \frac{{8b\varepsilon (1 - \lambda p_{s} \delta )[3a + b(p_{c} e_{0} + c)] + (3\theta - bp_{c} )(\theta + bp_{c} )[a - b(p_{c} e_{0} + c)]\ln \left( {\frac{{4b(1 - \lambda p_{s} \delta )\left( {\mu + \varepsilon } \right) - (\theta + bp_{c} )^{2} }}{{4b(1 - \lambda p_{s} \delta )\left( {\mu - \varepsilon } \right) - (\theta + bp_{c} )^{2} }}} \right)}}{{32b^{2} \varepsilon (1 - \lambda p_{s} \delta )}}$. By substituting $e^{DA}$, $w^{DA}$ and $p^{DA}$ into $D = a - bp + \theta e$ and Eqs. (8) and (9), we obtain the market demand and the optimal profits of the seller and the manufacturer. Thus, Proposition 3 is proven.□

Appendix 8: Proof of Corollary 5

Based on $\frac{{\partial p^{DA} }}{\partial \lambda } = \frac{{(3\theta - bp_{c} )(\theta + bp_{c} )p_{s} \delta [a - b(p_{c} e_{0} + c)]}}{{32b^{2} \varepsilon (1 - \lambda p_{s} \delta )^{2} }}[\frac{{8b\varepsilon (1 - \lambda p_{s} \delta )(\theta + bp_{c} )^{2} }}{N} + M]$, we obtain $\frac{{\partial p^{DA} }}{\partial \lambda } > 0$ if $3\theta - bp_{c} > 0$ and $\frac{{\partial p^{DA} }}{\partial \lambda } \le 0$ if $3\theta - bp_{c} \le 0$. Similarly, $\frac{{\partial p^{DA} }}{{\partial p_{s} }} = \frac{{(3\theta - bp_{c} )(\theta + bp_{c} )\lambda \delta [a - b(p_{c} e_{0} + c)]}}{{32b^{2} \varepsilon (1 - \lambda p_{s} \delta )^{2} }}[\frac{{8b\varepsilon (1 - \lambda p_{s} \delta )(\theta + bp_{c} )^{2} }}{N} + M]$. Then,$\frac{{\partial p^{DA} }}{{\partial p_{s} }} > 0$ if $3\theta - bp_{c} > 0$, and $\frac{{\partial p^{DA} }}{{\partial p_{s} }} \le 0$ if $3\theta - bp_{c} \le 0$.

Similarly, we have $\frac{{\partial D^{DA} }}{\partial \lambda } = \frac{{p_{s} \delta (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]}}{{32b\varepsilon (1 - \lambda p_{s} \delta )^{2} }}[\frac{{8b\varepsilon (1 - \lambda p_{s} \delta )(\theta + bp_{c} )^{2} }}{N} + M] > 0$ and $\frac{{\partial D^{DA} }}{{\partial p_{s} }} = \frac{{\lambda \delta (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]}}{{32b\varepsilon (1 - \lambda p_{s} \delta )^{2} }}[\frac{{8b\varepsilon (1 - \lambda p_{s} \delta )(\theta + bp_{c} )^{2} }}{N} + M] > 0$. Thus, Corollary 5 is proven.□

Appendix 9: Proof of Proposition 4

It is easy to obtain $e^{DA} = e^{DS}$, $w^{DA} = w^{DS}$. Comparing the retail prices, we have $p^{DA} - p^{DS} = \frac{{(\theta + bp_{c} )[a - b(p_{c} e_{0} + c)](3\theta - p_{c} )\{ [4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]M - 8b\varepsilon (1 - \lambda p_{s} \delta )\} }}{{32b^{2} \varepsilon (1 - \lambda p_{s} \delta )[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]}}$. Then, $p^{DA} > p^{DS}$ if $(3\theta - p_{c} )\{ [4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]M - 8b\varepsilon (1 - \lambda p_{s} \delta )\} > 0$, which is equal to $\theta > \frac{{p_{c} }}{3}$ and $\eta_{m} > \frac{2\varepsilon }{M} + \frac{{(\theta + bp_{c} )^{2} }}{{4b(1 - \lambda p_{s} \delta )}}$(or $\theta < \frac{{p_{c} }}{3}$ and $0 < \eta_{m} < \frac{2\varepsilon }{M} + \frac{{(\theta + bp_{c} )^{2} }}{{4b(1 - \lambda p_{s} \delta )}}$), and $p^{DA} \le p^{DS}$ if $(3\theta - p_{c} )\{ [4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]M - 8b\varepsilon (1 - \lambda p_{s} \delta )\} \le 0$.

Comparing the market demand in the different scenarios, we have $D^{DA} - D^{DS} = \frac{{(\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]\{ [4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]M - 8b\varepsilon (1 - \lambda p_{s} \delta )\} }}{{32b\varepsilon (1 - \lambda p_{s} \delta )[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]}}$. Then, $D^{DA} > D^{DS}$ if $\eta_{m} > \frac{2\varepsilon }{M} + \frac{{(\theta + bp_{c} )^{2} }}{{4b(1 - \lambda p_{s} \delta )}}$, and $D^{DA} \le D^{DS}$ if $\eta_{m} \le \frac{2\varepsilon }{M} + \frac{{(\theta + bp_{c} )^{2} }}{{4b(1 - \lambda p_{s} \delta )}}$.Comparing the firms’ profits, we have.

$E(\Pi_{m}^{DA} ) - \Pi_{m}^{DS} = \frac{{\eta_{m} (\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} \{ [4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]M - 8b\varepsilon (1 - \lambda p_{s} \delta )\} }}{{16b\varepsilon (1 - \lambda p_{s} \delta )[4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$ and $E(\Pi_{r}^{DA} ) - \Pi_{r}^{DS} = \frac{{R(\theta + bp_{c} )^{2} [a - b(p_{c} e_{0} + c)]^{2} \{ [4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]M - 8b\varepsilon (1 - \lambda p_{s} \delta )\} }}{{b[32b\varepsilon (1 - \lambda p_{s} \delta )]^{2} [4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ]^{2} }}$, where $R = [8b\varepsilon (1 - \lambda p_{s} \delta ) + (\theta + bp_{c} )^{2} M][4b(1 - \lambda p_{s} \delta )\eta_{m} - (\theta + bp_{c} )^{2} ] + 32b^{2} \varepsilon (1 - \lambda p_{s} \delta )^{2} \eta_{m} > 0$. Then, $E(\Pi_{m}^{DA} ) > \Pi_{m}^{DS}$ and $E(\Pi_{r}^{DA} ) > \Pi_{r}^{DS}$ if $\eta_{m} > \frac{2\varepsilon }{M} + \frac{{(\theta + bp_{c} )^{2} }}{{4b(1 - \lambda p_{s} \delta )}}$, and $E(\Pi_{m}^{DA} ) \le \Pi_{m}^{DS}$ and $E(\Pi_{r}^{DA} ) \le \Pi_{r}^{DS}$ if $\eta_{m} \le \frac{2\varepsilon }{M} + \frac{{(\theta + bp_{c} )^{2} }}{{4b(1 - \lambda p_{s} \delta )}}$. Thus, Proposition 4 is proven.□

Appendix 10: Proof of Eq. (14)

Based on Eq. (13), we have $\frac{{\partial^{2} \Pi_{r}^{DSS} }}{{\partial p^{2} }} = - 2b < 0$. Thus, $\Pi_{r}^{DSS}$ is concave in $p$. According to $\frac{{\partial \Pi_{r}^{DSS} }}{\partial p} = 0$, we obtain $p^{*} (e,w) = \frac{a + \theta e + \lambda bc + (1 - \lambda )bw}{{2b}}$. By substituting $p^{*} (e,w)$ into Eq. (12), we have.

$\frac{{\partial \Pi_{m}^{DSS} }}{\partial e} = (1 - \lambda )(w - c)\frac{\theta }{2} - (1 - \lambda p_{s} \delta )\eta_{m} e + p_{c} (a - bp + \theta e + (e - e_{o} )\frac{\theta }{2}) + x_{1}$,

$\frac{{\partial \Pi_{m}^{DSS} }}{\partial w} = (1 - \lambda )(a - bp + \theta e - b\frac{1 - \lambda }{2}(w - c)) + p_{c} (e_{0} - e)b\frac{1 - \lambda }{2} + y_{1}$,

and the Hessian matrix can be expressed as:

$\left( {\begin{array}{*{20}c} {\frac{{\partial \Pi_{m}^{{^{2} DSS}} }}{{\partial e^{2} }}} & {\frac{{\partial \Pi_{m}^{{^{2} DSS}} }}{\partial e\partial w}} \\ {\frac{{\partial \Pi_{m}^{{^{2} DSS}} }}{\partial w\partial e}} & {\frac{{\partial \Pi_{m}^{{^{2} DSS}} }}{{\partial w^{2} }}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {p_{c} \theta - (1 - \lambda p_{s} \delta )\eta_{m} } & {\frac{{(1 - \lambda )(\theta - p_{c} b)}}{2}} \\ {\frac{{(1 - \lambda )(\theta - p_{c} b)}}{2}} & { - b(1 - \lambda )^{2} } \\ \end{array} } \right)$.

Thus, $\Pi_{m}^{DSS}$ is jointly concave in $e$ and $w$ when $4b(1 - \lambda p_{s} \delta )\eta_{m} > (\theta + bp_{c} )^{2}$. Then, we obtain the optimal decisions of the manufacturer, $e^{DSS}$ and $w^{DSS}$, under the information symmetry scenario with a transfer payment contract according to $\frac{{\partial \Pi_{m}^{DSS} }}{\partial e} = 0$ and $\frac{{\partial \Pi_{m}^{DSS} }}{\partial w} = 0$. By substituting $e^{DSS}$ and $w^{DSS}$, we obtain $p^{DSS}$. Thus, Eq. (14) is proven.□

Appendix 11: Proof of Eq. (21)

Based on Eq. (20), we have $\frac{{\partial^{2} \Pi_{r}^{DAS} }}{{\partial p^{2} }} = - 2b < 0$. Thus, $\Pi_{r}^{DAS}$ is concave in $p$. Then, we obtain $p^{*} (e,w) = \frac{a + \theta e + \lambda bc + (1 - \lambda )bw}{{2b}}$ based on $\frac{{\partial \Pi_{r}^{DAS} }}{\partial p} = 0$. By substituting $p^{*} (e,w)$ into Eq. (19), we have.

$\frac{{\partial \Pi_{m}^{DAS} }}{\partial w} = (1 - \lambda )(a - bp + \theta e - b\frac{1 - \lambda }{2}(w - c)) + p_{c} (e_{0} - e)b\frac{1 - \lambda }{2} + y_{1}$,

$\frac{{\partial \Pi_{m}^{DAS} }}{\partial e} = (1 - \lambda )(w - c)\frac{\theta }{2} - (1 - \lambda p_{s} \delta )\eta_{m} e + p_{c} (a - bp + \theta e + (e - e_{o} )\frac{\theta }{2}) + x_{1}$,

and the Hessian matrix can be obtained as.

$\left( {\begin{array}{*{20}c} {\frac{{\partial \Pi_{m}^{{^{2} DA}} }}{{\partial e^{2} }}} & {\frac{{\partial \Pi_{m}^{{^{2} DA}} }}{\partial e\partial w}} \\ {\frac{{\partial \Pi_{m}^{{^{2} DA}} }}{\partial w\partial e}} & {\frac{{\partial \Pi_{m}^{{^{2} DA}} }}{{\partial w^{2} }}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {p_{c} \theta - (1 - \lambda p_{s} \delta )\eta_{m} } & {\frac{{(1 - \lambda )(\theta - p_{c} b)}}{2}} \\ {\frac{{(1 - \lambda )(\theta - p_{c} b)}}{2}} & { - b(1 - \lambda )^{2} } \\ \end{array} } \right)$.

It is easy to prove that $\Pi_{r}^{DAS}$ is jointly concave in $e$ and $w$ based on $4b(1 - \lambda p_{s} \delta )\eta_{m} > (\theta + bp_{c} )^{2}$. Then, we obtain the optimal decisions, $e^{DAS}$ and $w^{DAS}$, of the manufacturer under the information asymmetry scenario with a transfer payment contract according to the first-order conditions. By substituting $e^{DAS}$ and $w^{DAS}$ into $p^{*} (e,w)$, we can obtain the optimal decision of the seller $p^{DAS} = \int_{\mu - \varepsilon }^{\mu + \varepsilon } {\frac{{a + \theta e^{DAS} + \lambda bc + (1 - \lambda )bw^{DAS} }}{2b}} f(\eta_{m} )d_{{\eta_{m} }}$.

Then, $p^{DAS} = p^{DA} + \frac{{2b(1 - \lambda )(3\theta - bp_{c} )Mx_{2} + [8b\varepsilon (1 - \lambda p_{s} \delta ) + (3\theta - bp_{c} )(\theta - bp_{c} )M]y_{2} }}{{16b^{2} \varepsilon (1 - \lambda )(1 - \lambda p_{s} \delta )}}$ where $M = \ln \left( {\frac{{4b(1 - \lambda p_{s} \delta )\left( {\mu + \varepsilon } \right) - (\theta + bp_{c} )^{2} }}{{4b(1 - \lambda p_{s} \delta )\left( {\mu - \varepsilon } \right) - (\theta + bp_{c} )^{2} }}} \right)$. Thus, Eq. (21) is proven.□