Decarbonization investment in a supply chain with a retail platform based on blockchain technology

Abstract

This study considers a manufacturer that invests in decarbonization and sells a product to consumers through a direct channel and an online retail platform. The platform decides to operate in either an agency selling or a reselling mode, and apply blockchain technology to improve its performance. The selected selling mode determines the market power wielded by the platform and the manufacturer. A five-stage game is constructed to examine the interplay between the manufacturer’s decarbonized investment decision and the platform’s decisions on blockchain application and selling mode selection. Results imply that the retail platform benefits from the manufacturer’s investment while the latter gains from the former’s blockchain application hinge on its selling mode selection. Specifically, the manufacturer benefits from blockchain application in an agency mode but is disadvantaged under a reselling mode. Besides, the blockchain application and the manufacturer’s investment have diametrically opposing influences on the platform’s selling mode selection. As the efficiency of the manufacturer’s investment increases, blockchain application becomes more favorable to the platform, particularly in a reselling mode. Furthermore, applying blockchain is not always conducive to social welfare, but the platform can increase the utility of blockchain application to consumers or reduce the commission rate to improve social welfare. Finally, switching the order of decarbonization investment and blockchain application does not affect their optimal operational decisions.

Appendix: Proofs

Proof of Proposition 1

Under an agency mode, we compare the optimal solutions of cases $A - \overline{E}\,\overline{B}$ and $A - \overline{E}B$, and those of the optimal solutions of cases $A - EB$ and $A - E\overline{B}$. We obtain $p_{A}^{{\overline{E}B}} - p_{A}^{{\overline{E}\,\overline{B}}} = \frac{\alpha \gamma (1 - \lambda )}{{2(1 - \alpha \lambda )}} > 0,$ $\Pi_{A - M}^{{\overline{E}B}} - \Pi_{A - M}^{{\overline{E}\,\overline{B}}} = \frac{{(2 - \lambda + \gamma (1 - \lambda ))^{2} - (2 - \lambda )^{2} }}{8(2 - \lambda )} > 0,$ $\Pi_{A - P}^{{\overline{E}B}} - \Pi_{A - P}^{{\overline{E}\,\overline{B}}} = \frac{\lambda }{{8(2 - \lambda )^{2} }}((2 - \lambda + \gamma (1 - \lambda ))(2 - \lambda + (3 - \lambda )\gamma ) - (2 - \lambda )^{2} ) > 0$. Moreover, we have $p_{A}^{EB} - p_{A}^{{E\overline{B}}} = \frac{4k(1 - \lambda )\gamma }{{(2 - \lambda )(8k - (2 - \lambda )\tau^{2} )}} > 0$, $e_{A}^{EB} - e_{A}^{{E\overline{B}}} = \frac{\tau (1 - \lambda )\gamma }{{8k - (2 - \lambda )\tau^{2} }} > 0$, $\Pi_{A - M}^{EB} - \Pi_{A - M}^{{E\overline{B}}} = \frac{{k(2 - \lambda + (1 - \lambda )\gamma )^{2} - k(2 - \lambda )^{2} }}{{(2 - \lambda )(8k - (2 - \lambda )\tau^{2} )}} > 0$, and $\Pi_{A - P}^{EB} - \Pi_{A - P}^{{E\overline{B}}} = \frac{8k\lambda }{{(2 - \lambda )^{2} (8k - (2 - \lambda )\tau^{2} )^{2} }}\left( {(2 - \lambda + (1 - \lambda )\gamma )(k(2 - \lambda + (3 - \lambda )\gamma ) - \frac{{(2 - \lambda )\gamma \tau^{2} }}{4}) - k(2 - \lambda )^{2} } \right)$$> \frac{8k\lambda }{{(8k - (2 - \lambda )\tau^{2} )^{2} }}((k(1 + \gamma ) - \frac{{\gamma \tau^{2} }}{4}) - k) > 0$, where the second inequality follows from $k > \frac{{\tau^{2} }}{4}$.

Similarly, under a reselling mode, we compare the optimal solutions of cases $R - \overline{E}\,\overline{B}$ and $R - \overline{E}B$, $R - E\overline{B}$ and $R - EB$, we have $p_{R}^{{\overline{E}B}} - p_{R}^{{\overline{E}\,\overline{B}}} = \frac{\gamma }{2} > 0,w_{R}^{{\overline{E}B}} = w_{R}^{{\overline{E}\,\overline{B}}} ,\Pi_{R - P}^{{\overline{E}B}} - \Pi_{R - P}^{{\overline{E}\,\overline{B}}} = \frac{4 + 3\gamma }{{24}} > 0,\Pi_{R - M}^{{\overline{E}B}} - \Pi_{R - M}^{{\overline{E}\,\overline{B}}} = - \frac{{\gamma^{2} }}{8} < 0$. Besides, we have $p_{R}^{EB} - p_{R}^{{E\overline{B}}} = \frac{\gamma }{2} > 0,w_{R}^{EB} = w_{R}^{{E\overline{B}}} ,e_{R}^{EB} = e_{R}^{{E\overline{B}}} ,$ $\Pi_{R - M}^{EB} - \Pi_{R - M}^{{E\overline{B}}} = - \frac{{\gamma^{2} }}{8} < 0,$ and $\Pi_{R - P}^{EB} - \Pi_{R - P}^{{E\overline{B}}} = \frac{\gamma }{8}(\gamma + \frac{8k}{{6k - \tau^{2} }}) > 0$.

Proof of Proposition 2

Under an agency mode, we compare the optimal solutions of cases $A - \overline{E}\,\overline{B}$ and $A - E\overline{B}$, cases $A - EB$ and $A - \overline{E}B$. We have $p_{A}^{{E\overline{B}}} - p_{A}^{{\overline{E}\,\overline{B}}} = \frac{{(2 - \lambda )\tau^{2} }}{{2(8k - (2 - \lambda )\tau^{2} )}} > 0$, $\Pi_{A - M}^{{E\overline{B}}} - \Pi_{A - M}^{{\overline{E}\,\overline{B}}} = \frac{{(2 - \lambda )^{2} \tau^{2} }}{{8(8k - (2 - \lambda )\tau^{2} )}} > 0$, $\Pi_{A - P}^{{E\overline{B}}} - \Pi_{A - P}^{{\overline{E}\,\overline{B}}} = \frac{{(2 - \lambda )\lambda \tau^{2} (16k - (2 - \lambda )\tau^{2} )}}{{8(8k - (2 - \lambda )\tau^{2} )^{2} }} > 0$ where the third inequality follows from $k > \frac{{\tau^{2} }}{4}$. Moreover, $p_{A}^{EB} - p_{A}^{{\overline{E}B}} = \frac{{(2 - \lambda + (1 - \lambda )\gamma )\tau^{2} }}{{2(8k - (2 - \lambda )\tau^{2} )}} > 0$, $\Pi_{A - M}^{EB} - \Pi_{A - M}^{{\overline{E}B}} = \frac{{(2 - \lambda + (1 - \lambda )\gamma )^{2} \tau^{2} }}{{8(8k - (2 - \lambda )\tau^{2} )}} > 0$, and $\Pi_{A - P}^{EB} - \Pi_{A - P}^{{\overline{E}B}} = \frac{{\lambda (2 + \gamma (1 - \lambda ) - \lambda )(16k(1 + \, \gamma )\tau^{2} - (2 + \gamma (3 - \lambda ) - \lambda ) \, \tau^{4} )}}{{8(8k - (2 - \lambda )\tau^{2} )^{2} }} > 0$.

Similarly, under a reselling mode, we compare the optimal solutions of cases R-$\overline{E}\,\overline{B}$ and R-$E\overline{B}$, R-EB and R-$\overline{E}B$. We have $p_{R}^{{E\overline{B}}} - p_{R}^{{\overline{E}\,\overline{B}}} > 0,w_{R}^{{E\overline{B}}} - w_{R}^{{\overline{E}\,\overline{B}}} = \frac{2k}{{6k - \tau^{2} }} - \frac{1}{3} > 0$, $\Pi_{R - P}^{{E\overline{B}}} - \Pi_{R - P}^{{\overline{E}\,\overline{B}}} = \frac{{12k\tau^{2} - \tau^{4} }}{{18(6k - \tau^{2} )^{2} }} > 0$, and $\Pi_{R - M}^{{E\overline{B}}} - \Pi_{R - M}^{{\overline{E}\,\overline{B}}} = \frac{{\tau^{2} }}{{6(6k - \tau^{2} )^{2} }} > 0$. Moreover, $p_{R}^{EB} - p_{R}^{{\overline{E}B}} = \frac{{2\tau^{2} }}{{3(6k - \tau^{2} )}} > 0,w_{R}^{EB} - w_{R}^{{\overline{E}B}} = \frac{{\tau^{2} }}{{3(6k - \tau^{2} )}} > 0$, $\Pi_{R - M}^{EB} - \Pi_{R - M}^{{\overline{E}B}} = \frac{{\tau^{2} }}{{6(6k - \tau^{2} )}} > 0$, and $\Pi_{R - P}^{EB} - \Pi_{R - P}^{{\overline{E}B}} = \frac{{6k(2 + 3\gamma )\tau^{2} - (1 + 3\gamma )\tau^{4} }}{{18(6k - \tau^{2} )^{2} }} > \frac{{(1 + 3\gamma )\tau^{2} }}{{18(6k - \tau^{2} )^{2} }} > 0$.

Proof of Proposition 3

The results are immediately obtained by respectively comparing the optimal profits of the manufacturer and the platform in Sects. 4.1.1 and 4.2.1.

Proof of Proposition 4

Define $f_{0} (\lambda ) = \Pi_{A - P}^{{E\overline{B}}} - \Pi_{R - P}^{{E\overline{B}}} = \frac{{2k^{2} }}{{(8k - (2 - \lambda )\tau^{2} )^{2} (6k - \tau^{2} )^{2} }}(4\lambda (6k - \tau^{2} )^{2} - (8k - (2 - \lambda )\tau^{2} )^{2} ),$ for any $\tau \ge 0$ and $k \ge \frac{{\tau^{2} }}{4}$, we have $f_{0} (0) = - (8k - 2\tau^{2} )^{2} < 0$ and $f_{0} (1) = (4k - \tau^{2} )^{2} (20k - 3\tau^{2} ) > 0$, which implies that there exists a threshold $\lambda_{3} \in (0,1)$ such that if $0 < \lambda \le \lambda_{3}$, $\Pi_{A - P}^{{E\overline{B}}} \le \Pi_{R - P}^{{E\overline{B}}}$, and if $\lambda_{3} < \lambda \le 1$, $\Pi_{A - P}^{{E\overline{B}}} \ge \Pi_{R - P}^{{E\overline{B}}}$, where $\lambda_{3} = \max \{ \lambda |f_{0} (\lambda ) = 0\}$.

In addition, we immediately obtain that if $0 < \lambda < \lambda_{2} = \frac{2}{3}$, $f_{1} (\lambda ) = \Pi_{A - M}^{{E\overline{B}}} - \Pi_{R - M}^{{E\overline{B}}} = \frac{k(2 - \lambda )}{{8k - (2 - \lambda )\tau^{2} }} - \frac{k}{{6k - \tau^{2} }} > 0$ and $e_{A}^{{E\overline{B}}} = \frac{(2 - \lambda )\tau }{{8k - (2 - \lambda )\tau^{2} }} > e_{R}^{{E\overline{B}}} = \frac{\tau }{{6k - \tau^{2} }}$. Moreover, as $f_{0} (\lambda_{2} ) = 8(2k - \frac{{\tau^{2} }}{3})^{2} > 0$ and $f_{0} (\lambda = \lambda_{1} ) = \frac{4}{9}(8k - \frac{11}{9}\tau^{2} ) > 0$, we obtain $\lambda_{2} > \lambda_{1} > \lambda_{3}$.

Proof of Proposition 5

Define

$f_{2} (\lambda ) \, = \Pi_{A - M}^{{\overline{E}B}} - \Pi_{R - M}^{{\overline{E}B}} = \frac{1}{24}(2 - 3\lambda + 3\gamma ((2 + \gamma )(1 - \lambda ) + \frac{\gamma }{2 - \lambda })),$ then

$f_{3} (\lambda ) \, = \Pi_{A - P}^{{\overline{E}B}} - \Pi_{R - P}^{{\overline{E}B}} = \frac{1}{72}( - 4 + 9\lambda + 6\gamma ( - 2 + 3\lambda ) + \frac{{9\gamma^{2} ( - 4 + \lambda (7 + ( - 5 + \lambda )\lambda ))}}{{(2 - \lambda )^{2} }}$.

Clearly, given $3\gamma ((2 + \gamma )(1 - \lambda ) + \frac{\gamma }{2 - \lambda }) > 0$ for any $\lambda \in (0,1)$, we have $\{ \lambda |f_{2} (\lambda ) = 0\} > \lambda_{2}$ and if $3\gamma ((2 + \gamma )(1 - \lambda ) + \frac{\gamma }{2 - \lambda }) > 1$, we have $\{ \lambda |f_{2} (\lambda ) = 0\} > 1$. The above results imply that there exists $\lambda_{5} = \min \{ 1,\lambda_{{5^{\prime}}} \}$, if $\lambda > \lambda_{5}$ then $f_{2} (\lambda ) > 0$; otherwise, $f_{2} (\lambda ) \le 0$, where $\lambda_{5} = \min \{ \lambda |f_{2} (\lambda ) = 0\}$.

Similarly, $\lambda_{4} > \lambda_{1}$ exists such that if $\lambda > \lambda_{4}$ then $f_{3} (\lambda ) > 0$, otherwise, $f_{3} (\lambda ) \le 0$.

Proof of Proposition 6

Define $f_{4} (\lambda ) = \Pi_{A - M}^{EB} - \Pi_{R - M}^{EB} = \frac{{\gamma^{2} }}{8} - \frac{k}{{6k - \tau^{2} }} + \frac{{k(2 - \lambda + (1 - \lambda )\gamma )^{2} }}{{(2 - \lambda )(8k - (2 - \lambda )\tau^{2} )}} = f_{1} (\lambda ) + g_{1} (\gamma ,\lambda ,k,\tau )$, where $g_{1} (\gamma ,\lambda ,k,\tau ) = \frac{{\gamma^{2} }}{8} + \frac{{k(2(2 - \lambda )((1 - \lambda )\gamma )) + (1 - \lambda )^{2} \gamma^{2} )}}{{(2 - \lambda )(8k - (2 - \lambda )\tau^{2} )}} > 0$ for any $0 \le \lambda \le 1$, $\gamma ,\tau > 0$, $k > \frac{{\tau^{2} }}{4}$. From the proof of Proposition 4, $f_{1} (\lambda_{2} = \frac{2}{3}) = 0$ and for any $\frac{2}{3} \le \lambda \le 1$, $\Pi_{A - M}^{EB} \le \Pi_{R - M}^{EB}$. In addition, we have $f^{\prime}_{1} (\lambda ) < 0$, that is, $f_{1} (\lambda )$ is decreasing in $\lambda$. Thus, we determine a threshold $\lambda_{6} > \lambda_{2} = \frac{2}{3}$, such that $f_{4} (\lambda_{6} ) = 0$, where $\lambda_{6} = \min \{ \lambda |f_{4} (\lambda ) = 0\}$.

Similarly, we define $f_{5} (\lambda ) = \Pi_{A - P}^{EB} - \Pi_{R - P}^{EB} = \frac{{2k\lambda (2 - \lambda + (1 - \lambda )\gamma )(4k(2 - \lambda + (3 - \lambda )\gamma ) - (2 - \lambda )\gamma \tau^{2} )}}{{(2 - \lambda )^{2} (8k - (2 - \lambda )\tau^{2} )^{2} }} - \frac{{(k(4 + 6\gamma ) - \gamma \tau^{2} )^{2} }}{{8(6k - \tau^{2} )^{2} }}$$= f_{0} (\lambda ) + g_{2} (\lambda ,\gamma ,k,\tau )$. From Proof of Proposition 4, $f_{0} (\lambda_{3} ) = 0$ and for any $\lambda_{3} \le \lambda \le 1$ $\Pi_{A - P}^{EB} \ge \Pi_{R - P}^{EB}$. For any $0 \le \lambda \le 1$, $g_{2} (\lambda ,\gamma ,k,\tau ) \ge 0( < 0)$ strongly depends on $k,\gamma ,\tau$. Define $\lambda_{7} = \min \{ \lambda |f_{5} (\lambda ) = 0\}$, then we have $\lambda_{7} \ge \lambda_{3} ( < \lambda_{3} )$ is also strongly depend on $\gamma$,$\tau$, and $k$.

Proof of Proposition 7 and 8

Under the reselling mode, when the manufacturer invests and does not invest in green technology, from Eqs. (4)–(7) we respectively obtain $CS_{R}^{EB} - CS_{R}^{{E\overline{B}}} = CS_{R}^{{\overline{E}B}} - CS_{R}^{{\overline{E}\,\overline{B}}} = \frac{{\gamma^{2} }}{8} > 0$.

From the equilibrium profits of the manufacturer and the platform in Sect. 4.2, we obtain that $\Pi_{R}^{EB} - \Pi_{R}^{{E\overline{B}}} = \Pi_{R - M}^{EB} - \Pi_{R - M}^{{E\overline{B}}} + \Pi_{R - P}^{EB} - \Pi_{R - P}^{{R\overline{B}}}$ $= \frac{k\gamma }{{6k - \tau^{2} }} > 0$ and $\Pi_{R}^{{\overline{E}B}} - \Pi_{R}^{{\overline{E}\,\overline{B}}} = \Pi_{R - M}^{{\overline{E}B}} - \Pi_{R - M}^{{\overline{E}\,\overline{B}}} + \Pi_{R - P}^{{\overline{E}B}} - \Pi_{R - P}^{{\overline{E}\,\overline{B}}} = \frac{\gamma }{6} > 0$. Besides, $EI_{R}^{{\overline{E}B}} - EI_{R}^{{\overline{E}\,\overline{B}}} = \mu$$(D_{d}^{{\overline{E}B}} + D_{r}^{{\overline{E}B}} - D_{d}^{{\overline{E}\,\overline{B}}} - D_{r}^{{\overline{E}\,\overline{B}}} ) = \mu (\frac{1}{2}(1 - p_{R}^{{\overline{E}B}} + \gamma ) + \frac{1}{2}(1 - p_{R}^{{\overline{E}B}} ) - (1 - p_{R}^{{\overline{E}\,\overline{B}}} )) = 0$.

Then, from Eq. (3), we obtain that for any $\mu > 0$$SW_{R}^{EB} - SW_{R}^{{E\overline{B}}} \, = CS_{R}^{EB} - CS_{R}^{{E\overline{B}}} + \Pi_{R}^{EB} - \Pi_{R}^{{E\overline{B}}} = \frac{{\gamma^{2} }}{8} > 0$, $SW_{R}^{{\overline{E}B}} - SW_{R}^{{\overline{E}\,\overline{B}}} = CS_{R}^{{\overline{E}B}} - CS_{R}^{{\overline{E}\,\overline{B}}} + \Pi_{R}^{{\overline{E}B}} - \Pi_{R}^{{\overline{E}\,\overline{B}}} - (EI_{R}^{{\overline{E}B}} - EI_{R}^{{\overline{E}\,\overline{B}}} ) > 0$.

Under the agency selling mode, when the manufacturer invests and does not invest in green technology, from Eqs. (4)–(7) we respectively obtain $CS_{A}^{EB} - CS_{A}^{{E\overline{B}}} = \frac{1}{4}\left( {2(p_{A}^{EB} )^{2} + \gamma^{2} + 2\gamma (1 + e^{EB} \tau ) + 2(1 + e_{A}^{EB} \tau )^{2} - 2p_{A}^{EB} (2 + \gamma + 2e_{A}^{EB} \tau )} \right) - \frac{1}{2}(1 + e_{A}^{{E\overline{B}}} \tau - p_{A}^{{E\overline{B}}} )^{2}$ $= \frac{{\gamma (32k^{2} (4 + 5\gamma - \lambda (2 + (4 - \lambda )\gamma )) - 8k(2 - \lambda )(\lambda (2 - \lambda (1 - \gamma ) - 3\gamma ) + 4\gamma )\tau^{2} }}{{4(2 - \lambda )^{2} (8k - (2 - \lambda )\tau^{2} )^{2} }} + \frac{{(2 - \lambda )^{2} (2 - (2 - \lambda )\lambda )\gamma \tau^{4} )}}{{4(2 - \lambda )^{2} (8k - (2 - \lambda )\tau^{2} )^{2} }}$ $> \frac{{8k + (2 - \lambda )(2 - (2 - \lambda )\lambda )\gamma \tau^{4} )}}{{4(2 - \lambda )(8k - (2 - \lambda )\tau^{2} )^{2} }} > 0,CS_{A}^{{\overline{E}B}} - CS_{A}^{{\overline{E}\,\overline{B}}} \, = \frac{\gamma (4 - 2\gamma + \gamma (5 - (4 - \lambda )\lambda ))}{{8(2 - \lambda )^{2} }} > 0,$ where the inequalities follows from $k > \frac{{\tau^{2} }}{4}$ and $\lambda \in (0,1)$.

The proof of Proposition 1 shows that $\Pi_{A - M}^{EB} > \Pi_{A - M}^{{E\overline{B}}}$, $\Pi_{A - P}^{EB} > \Pi_{A - P}^{{E\overline{B}}}$, $\Pi_{A - M}^{{\overline{E}B}} > \pi_{A - M}^{{\overline{E}\,\overline{B}}}$ and $\Pi_{A - P}^{{\overline{E}B}} > \Pi_{A - P}^{{\overline{E}\,\overline{B}}}$, i.e., $\Pi_{A}^{EB} > \Pi_{A}^{{E\overline{B}}}$ and $\Pi_{A}^{{\overline{E}B}} > \Pi_{A}^{{\overline{E}\,\overline{B}}}$.

In addition, given that $EI_{A}^{{\overline{E}B}} - EI_{A}^{{\overline{E}\,\overline{B}}} = \mu (D_{d}^{{\overline{E}B}} + D_{r}^{{\overline{E}B}} - D_{d}^{{\overline{E}\,\overline{B}}} - D_{r}^{{\overline{E}\,\overline{B}}} ) = \frac{\mu \gamma }{{4 - 2\lambda }} > 0$. Then, from Eq. (3) we obtain that $SW_{A}^{EB} - SW_{A}^{{E\overline{B}}} \, = CS_{A}^{EB} - CS_{A}^{{E\overline{B}}} + \Pi_{A}^{EB} - \Pi_{A}^{{E\overline{B}}} = CS_{A}^{EB} - CS_{A}^{{E\overline{B}}} > 0$, $SW_{A}^{{\overline{E}B}} - SW_{A}^{{\overline{E}\,\overline{B}}} = CS_{A}^{{\overline{E}B}} - CS_{A}^{{\overline{E}\,\overline{B}}} + \Pi_{A}^{{\overline{E}B}} - \Pi_{A}^{{\overline{E}\,\overline{B}}} - (EI_{A}^{{\overline{E}B}} - EI_{A}^{{\overline{E}\,\overline{B}}} )$$= \frac{\gamma (4u( - 2 + \lambda ) + 2(3 - \lambda )(2 - \lambda ) + \gamma (7 - (6 - \lambda )\lambda ))}{{8(2 - \lambda )^{2} }}.$ Define $SW_{A}^{{\overline{E}B}} - SW_{A}^{{\overline{E}B}} = 0$, we obtain that $\mu_{0} = \frac{1}{4}(6 + \gamma (4 - \frac{1}{2 - \lambda } - \lambda ) - 2\lambda )$. Clearly, if $\mu < \mu_{0}$, then $SW_{A}^{{\overline{E}B}} \ge SW_{A}^{{\overline{E}\,\overline{B}}}$, otherwise $SW_{A}^{{\overline{E}B}} < SW_{A}^{{\overline{E}\,\overline{B}}}$. Moreover, take derivatives of $\mu_{0}$ with respect to $\gamma$ and $\lambda$, and we have $\frac{{\partial \mu_{0} }}{\partial \gamma } = \frac{1}{4}(4 - \frac{1}{2 - \lambda } - \lambda ) > 0,\;\;\frac{{\partial \mu_{0} }}{\partial \lambda } = \frac{1}{4}( - 2 - \gamma (1 + \frac{1}{{(2 - \lambda )^{2} }})) < 0$.

Proof of Proposition 9

Under an agency selling mode, when the manufacturer does not invest in green technology, we compare the optimal profits if the platform applies BCT to analyze its effect, that is, $\Pi_{A - M}$ in Sects. 4.1.1 and 4.1.2. Taking the first derivative of $\frac{{\partial \Pi_{A - M}^{{\overline{E}B}} }}{\partial \alpha }$ with respect to (w.r.t) $\alpha$, we have $\frac{{\partial \Pi_{A - M}^{{\overline{E}B}} }}{\partial \alpha } = \frac{1}{4}\lambda (1 + 2\gamma + \frac{{\alpha \gamma^{2} (1 - \lambda )(4 - \alpha (1 + \lambda )(3 - \alpha \lambda ))}}{{(1 - \alpha \lambda )^{3} }})$, and $\max_{\lambda ,\alpha } \{ (4 - \alpha (1 + \lambda )(3 - \alpha \lambda ))\} > 0$. Combining that when $\alpha = 0$, $\Pi_{A - P}^{{\overline{E}B}} - \Pi_{A - P}^{{\overline{E}\,\overline{B}}} = - \Pi_{A - P}^{{\overline{E}\,\overline{B}}} < 0$ and when $\alpha = \frac{1}{2}$, $\Pi_{A - P}^{{\overline{E}B}} - \Pi_{A - P}^{{\overline{E}\,\overline{B}}} = \frac{\lambda }{{8(2 - \lambda )^{2} }}((2 - \lambda + \gamma (1 - \lambda ))(2 - \lambda + (3 - \lambda )\gamma ) - (2 - \lambda )^{2} ) > 0$, we immediately obtain that there exists a threshold $0 < \alpha_{1} < \frac{1}{2}$, such that $\Pi_{A - P}^{{\overline{E}B}} \ge \Pi_{A - P}^{{\overline{E}\,\overline{B}}}$ for any $\alpha_{1} < \alpha < 1$.

Using a similar method, we analyze the case that the manufacturer invests in green technology. Given $\frac{{\partial \Pi_{A - M}^{EB} }}{\partial \alpha } > 0$, $\Pi_{A - P}^{EB} - \Pi_{A - P}^{{E\overline{B}}} < 0$ when $\alpha = 0$, and $\Pi_{A - P}^{EB} - \Pi_{A - P}^{{E\overline{B}}} = \frac{8k\lambda }{{(2 - \lambda )^{2} (8k - (2 - \lambda )\tau^{2} )^{2} }}$$\left( {(2 - \lambda + (1 - \lambda )\gamma )(k(2 - \lambda + (3 - \lambda )\gamma ) - \frac{{(2 - \lambda )\gamma \tau^{2} }}{4}) - k(2 - \lambda )^{2} } \right) > \frac{8k\lambda }{{(8k - (2 - \lambda )\tau^{2} )^{2} }}((k(1 + \gamma ) - \frac{{\gamma \tau^{2} }}{4}) - k) > 0$.when $\alpha = \frac{1}{2}$, then we derive a threshold $0 < \alpha_{2} < \frac{1}{2}$, such that $\Pi_{A - P}^{EB} \ge \Pi_{A - P}^{{E\overline{B}}}$ for any $\alpha_{2} < \alpha < 1$.

Proof of Proposition 10

Under a reselling mode, when the manufacturer does not invest in green technology, we compare the profits if the platform applies and does not apply BCT to explore its effect, that is, $\Pi_{R - P}^{{\overline{E}B}}$ and $\Pi_{R - P}^{{\overline{E}\,\overline{B}}}$. From Sects. 4.2.1 and 4.2.2. We immediately obtain that when $\alpha = 1$, $\Pi_{R - P}^{{\overline{E}B}} - \Pi_{R - P}^{{\overline{E}\,\overline{B}}} = \frac{{\alpha (1 + \gamma )^{2} }}{16} - \frac{1}{18} > 0$, when $\alpha = \frac{1}{2}$, $\Pi_{R - P}^{{\overline{E}B}} - \Pi_{R - P}^{{\overline{E}\,\overline{B}}} = \frac{4 + 3\gamma }{{24}} > 0$, and when $\alpha = 0$, $\Pi_{R - P}^{{\overline{E}B}} - \Pi_{R - P}^{{\overline{E}\,\overline{B}}} = - \frac{1}{18} < 0$. Taking the derivative of $\Pi_{R - P}^{{\overline{E}B}}$ w.r.t $\alpha$, we obtain that $\frac{{\partial \Pi_{R - P}^{{\overline{E}B}} }}{\partial \alpha } = \frac{(1 + (2 - \alpha )\gamma )(1 - \alpha + (2 - \alpha (5 + \alpha ))\gamma )}{{4(1 + \alpha )^{3} }}$. Define $g(\alpha ) = 1 - \alpha + (2 - \alpha (5 + \alpha ))\gamma$. Given that $g(\alpha )$ is decreasing in $\alpha$, and $g(0) > 0$, $g(1) = - 4\gamma < 0$, which implies that $\Pi_{R - P}^{{\overline{E}B}}$ is increasing in $\alpha$ first and then decreasing in $\alpha$ in $(0,1)$. Then, combining the above results, we obtain a threshold $\alpha_{3}$ such that $\Pi_{R - P}^{{\overline{E}B}} \le \Pi_{R - P}^{{\overline{E}\,\overline{B}}}$ if $0 < \alpha \le \alpha_{3}$, and $\Pi_{R - P}^{{\overline{E}B}} > \Pi_{R - P}^{{\overline{E}\,\overline{B}}}$ if $1 > \alpha > \alpha_{3}$.

Using a similar method, we explore the effect of $\alpha$ on the platform's BCT application when the manufacturer invests in green technology by comparing the profits of the platform in Sects. 4.2.3 and 4.2.4. Taking the derivative of $\Pi_{R - P}^{EB}$ w. r. t $\alpha$, we obtain:

$\frac{{\partial \Pi_{R - P}^{EB} }}{\partial \alpha } = \frac{{(2k( - 1 + ( - 2 + \alpha )\gamma ) - ( - 1 + \alpha )\gamma \tau^{2} )(8k^{2} ( - 1 + \alpha + ( - 2 + \alpha (5 + \alpha ))\gamma ) + 2k(1 + (4 - \alpha (11 + 2\alpha ))\gamma )\tau^{2} + ( - 1 + 3\alpha )\gamma \tau^{4} )}}{{(4k(1 + \alpha ) - \tau^{2} )^{3} }}$.

Apparently, $\frac{{\partial \Pi_{R - P}^{EB} }}{\partial \alpha }$ is decreasing in $\alpha$ in $(0,1)$, combining $\frac{{\partial \Pi_{R - P}^{EB} }}{\partial \alpha }(\alpha = 0) = \frac{{(k(2 + 4\gamma ) - \gamma \tau^{2} )^{2} }}{{(4k - \tau^{2} )^{2} }} > 0$ and $\frac{{\partial \Pi_{R - P}^{EB} }}{\partial \alpha }(\alpha = 1) = - \frac{{4k(1 + \gamma )(16k^{2} \gamma + k(1 - 9\gamma )\tau^{2} + \gamma \tau^{4} )}}{{(8k - \tau^{2} )^{3} }} < 0$, where the inequality follows from $k > \frac{{\tau^{2} }}{4}$. The results imply that $\Pi_{R - P}^{EB}$ is increasing in $\alpha$ first and then decreasing in $\alpha$ in $(0,1)$. In addition, from Proof of Proposition 2, we obtain that $\Pi_{R - P}^{EB} - \Pi_{R - P}^{{E\overline{B}}} < 0$ when $\alpha = 0$, $\Pi_{R - P}^{EB} - \Pi_{R - P}^{{E\overline{B}}} = \frac{\gamma }{8}(\gamma + \frac{8k}{{6k - \tau^{2} }}) > 0$ when $\alpha = \frac{1}{2}$, and $\Pi_{R - P}^{EB} - \Pi_{R - P}^{{E\overline{B}}} = k^{2} (\frac{{4(1 + \gamma )^{2} }}{{(8k - \tau^{2} )^{2} }} - \frac{2}{{(6k - \tau^{2} )^{2} }}) > 0$ when $\alpha = 1$. Then, we obtain a threshold $\alpha_{3}$, $\Pi_{R - P}^{EB} - \Pi_{R - P}^{{E\overline{B}}} \le 0$ if $0 < \alpha \le \alpha_{4}$, and $\Pi_{R - P}^{EB} - \Pi_{R - P}^{{E\overline{B}}} > 0$ if $\alpha_{4} < \alpha \le 1$.

Proof of Proposition 11

From the Proof of Proposition 2, we immediately obtain that:

• (i) Under the agency selling mode, $p_{A}^{{E\hat{B}}} > p_{A}^{{\overline{E}\hat{B}}}$, $\Pi_{A - M}^{{E\hat{B}}} > \Pi_{A - M}^{{\overline{E}\hat{B}}}$, $\Pi_{A - P}^{{E\hat{B}}} > \Pi_{A - P}^{{\overline{E}\hat{B}}}$.

• (ii) Under the reselling mode, $p_{R}^{{E\hat{B}}} > p_{R}^{{\overline{E}\hat{B}}}$, $w_{R}^{{E\hat{B}}} > w_{R}^{{\overline{E}\hat{B}}}$, $\Pi_{R - M}^{{E\hat{B}}} > \Pi_{R - M}^{{\overline{E}\hat{B}}}$, $\Pi_{R - P}^{{E\hat{B}}} > \Pi_{R - P}^{{\overline{E}\hat{B}}}$,

where $\hat{B} \in \{ B,\overline{B}\}$. The results suggest that under either selling mode, the manufacturer decides to invest in green technology in stage 3′.

From the Proof of Proposition 1, we immediately determine that given the manufacturer’s optimal decision in stage 3', the following results hold:

• (i) $(i)$ Under the agency selling mode, $p_{A}^{EB} > p_{A}^{{E\overline{B}}}$, $\Pi_{A - M}^{EB} > \Pi_{A - M}^{{E\overline{B}}}$, $\Pi_{A - P}^{EB} > \Pi_{A - P}^{{E\overline{B}}}$, $e_{A}^{EB} > e_{A}^{{E\overline{B}}}$.

• (ii) Under the reselling mode, $p_{R}^{EB} > p_{R}^{{E\overline{B}}}$, $w_{R}^{EB} = w_{R}^{{E\overline{B}}}$, $e_{R}^{EB} = e_{R}^{{E\overline{B}}}$, $\Pi_{R - M}^{EB} < \Pi_{R - M}^{{E\overline{B}}}$, $\Pi_{R - P}^{EB} > \Pi_{R - P}^{{E\overline{B}}}$. The results suggest that under either selling mode, the platform chooses to apply BCT in stage 2′ while the manufacturer prefers the platform to select an agency selling mode.