A three-stage game model of plastic circular supply chain management in the context of cap-and-trade and plastic packaging regulations

Abstract

Circular supply chain management (CSCM) has become an emerging solution against plastic pollution and carbon emission associated with plastic production. This paper develops a three-stage game model to investigate decisions of three critical players in plastic CSCM, including a petrochemical company that produces virgin plastic and faces a cap-and-trade regulation, a multinational company (MNC) with a Scope 3’s emission reduction commitment that sells products in two markets with or without a plastic packaging regulation, and a plastic recycling company that collects and produces recycled plastic. Specifically, the petrochemical company first determines the optimal carbon reduction effort level. The plastic recycling company then determines the optimal price for recycled plastic. After observing the price of recycled plastic and the level of petrochemical company’s carbon reduction effort, the MNC finally determines the optimal proportion of recycled plastic to be used in product packaging. Our results show that the cap-and-trade regulation could encourage the petrochemical company to make greater effort to reduce carbon emission, but possibly lead to a reduction in the use of recycled plastic in packaging by the MNC. Higher plastic fines could incentivize the MNC to use more recycled plastic in the packaging, which is a disincentive for the petrochemical company to make carbon reduction effort. The MNC’s commitment to Scope 3’s carbon reduction contributes to an increased use of recycled plastics, but surprisingly, this results in more carbon emission from the petrochemical company. Interestingly, the total carbon emission of the plastics circular supply chain increases and then decreases as the Scope 3’s commitment by the MNC increases.

Appendices

Appendix

Proof of Lemma 1

In the scenario of the MNC violates the Market 1’s government-mandated proportion of recycled plastic ($0 \le r_{{}}^{{}} < \theta$), the profit maximization function is:

$$ \begin{gathered} \mathop {\max \Pi_{M}^{{}} }\limits_{(r)} = p_{{m_{1} }} \cdot D_{{m_{1} }} + p_{{m_{2} }} \cdot D_{{m_{2} }} - p_{r} (D_{{m_{1} }} + D_{{m_{2} }} )r - p_{v} (D_{{m_{1} }} + D_{{m_{2} }} )(1 - r) - kr^{2} \hfill \\ \, - \lambda \{ (D_{{m_{1} }} + D_{{m_{2} }} )[(e_{{m_{0} }} - e)(1 - r) + e_{r} r]\} - FD_{{m_{1} }} (\theta - r) \hfill \\ \end{gathered} $$

We solve the MNC’s optimization problem as follows. The first and second order derivative of $\Pi_{M}^{{}}$ in relation to $r$ are: $\frac{{\partial \Pi_{M}^{{}} }}{\partial r} = (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{{}} - e_{r} } \right) + p_{v} - p_{r}^{{}} ] + FD_{{m_{1} }} - 2kr$, $\frac{{\partial^{2} \Pi_{M}^{{}} }}{{\partial r^{2} }} = - 2k < 0$, which indicates the MNC’s profit is concave for $r$. Let $\frac{{\partial \Pi_{M}^{{}} }}{\partial r} = 0$, we have:

$r_{{}}^{*} = r_{f}^{*} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right) + p_{v} - p_{{r_{f} }}^{*} ] + FD_{{m_{1} }} }}{2k}$, where we have the prerequisite: $0 \le r_{{}}^{{}} < \theta$, namely $0 \le (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] + FD_{{m_{1} }} < 2k\theta$ must hold.

Proof of Lemma 2

In the scenario of the MNC complies with the Market 1’s government-mandated proportion of recycled plastic ($\theta \le r_{{}}^{{}} \le 1$), the profit maximization function is:

$$ \begin{gathered} \mathop {\max \Pi_{M}^{{}} }\limits_{(r)} = p_{{m_{1} }} \cdot D_{{m_{1} }} + p_{{m_{2} }} \cdot D_{{m_{2} }} - p_{r} (D_{{m_{1} }} + D_{{m_{2} }} )r - p_{v} (D_{{m_{1} }} + D_{{m_{2} }} )(1 - r) - kr^{2} \hfill \\ \, - \lambda \{ (D_{{m_{1} }} + D_{{m_{2} }} )[(e_{{m_{0} }} - e)(1 - r) + e_{r} r]\} \hfill \\ \end{gathered} $$

We solve the MNC’s optimization problem as follows. The first and second order derivative of $\Pi_{M}^{{}}$ in relation to $r$ are: $\frac{{\partial \Pi_{M}^{{}} }}{\partial r} = (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{{}} - e_{r} } \right) + p_{v} - p_{r}^{{}} ] - 2kr$, $\frac{{\partial^{2} \Pi_{M}^{{}} }}{{\partial r^{2} }} = - 2k < 0$, which indicates the MNC’s profit is concave for $r$. Let $\frac{{\partial \Pi_{M}^{{}} }}{\partial r} = 0$, we have:

$r_{{}}^{*} = r_{g}^{*} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right) + p_{v} - p_{{r_{f} }}^{*} ]}}{2k}$, where we have the prerequisite: $\theta < r_{{}}^{{}} < 1$, it is obviously that:

• if $0 < (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] < 2k\theta$, then $r_{{}}^{*} = r_{g}^{*} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right) + p_{v} - p_{{r_{f} }}^{*} ]}}{2k}$;

• if $2k\theta - FD_{{m_{1} }} < (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] < 2k\theta$, then $r_{{}}^{*} = r_{\theta }^{*} = \theta$;

• if $(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] \ge 2k$, then $r_{{}}^{*} = r_{1}^{*} = 1$.

Proof of Lemma 3

When the MNC violates the proportion of recycled plastic in packaging ($0 \le r_{{}}^{*} < \theta$), according to the Lemma 1, we have already obtain the MNC’s proportion of recycled plastic $r_{{}}^{*} = r_{f}^{*} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right) + p_{v} - p_{{r_{f} }}^{*} ] + FD_{{m_{1} }} }}{2k}$.

Substituting $r_{f}^{*}$ into the plastic recycling company’s profit function $\Pi_{R}^{{}}$. The second order derivative of $\Pi_{R}^{{}}$ in relation to $p_{r}$ is: $\frac{{\partial^{2} \Pi_{R}^{{}} }}{{\partial p_{r}^{2} }} = - \frac{{\eta (D_{{m_{1} }} + D_{{m_{2} }} )^{2} }}{{2k^{2} }} < 0$, which indicates the plastic recycling company’s profit is concave for $p_{r}$. Let $\frac{{\partial \Pi_{R}^{{}} }}{{\partial p_{r} }} = 0$, we have:

$$\begin{aligned} p_{{r_{f} }}^{*} (r_{{}}^{*} = r_{f}^{*} ) &= \frac{{[p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{f}^{*} - e_{r} } \right)](k + \eta )}}{2k + \eta } + \frac{{FD_{{m_{1} }} (k + \eta )}}{{(2k + \eta )(D_{{m_{1} }} + D_{{m_{2} }} )}}\\ & = p_{v} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right) - \frac{{k[p_{v} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right)]}}{2k + \eta } + \frac{{FD_{{m_{1} }} (k + \eta )}}{{(2k + \eta )(D_{{m_{1} }} + D_{{m_{2} }} )}}\end{aligned} $$

Substituting $r_{f}^{*}$ and $p_{{r_{f} }}^{*}$ into the petrochemical company’s profit function $\Pi_{P}^{{}}$. The second order derivative of $\Pi_{P}^{{}}$ in relation to $e$ is: $\frac{{\partial^{2} \Pi_{P}^{{}} }}{{\partial e^{2} }} = \frac{{\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} }}{{{\mkern 1mu} 2k + \eta }}{\mkern 1mu} - 2\gamma$.

Thus, to ensure the petrochemical company’s profit function is concave, we need the condition $2{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) \ge \lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2}$. Then, let $\frac{{\partial \Pi_{P}^{{}} }}{\partial e} = 0$, we have

$$ e_{{}}^{*} = e_{f}^{*} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} \{ 4{\mkern 1mu} k{\mkern 1mu} p_{c} - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c] + 2\eta p_{c} \} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} - \frac{{FD_{{m_{1} }} (D_{{m_{1} }} + D_{{m_{2} }} )p_{c} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} $$

Substituting $e_{f}^{*}$ into $p_{{r_{f} }}^{*}$, we could obtain the optimal recycled plastic price $p_{{}}^{*}$. Then, substituting $e_{f}^{*}$ and $p_{{r_{f} }}^{*}$ into $r_{{}}^{*}$, we could obtain the optimal proportion of recycled plastic.

Proof of Lemma 4–6

When the MNC complies with the proportion of recycled plastic in packaging ($\theta \le r_{{}}^{*} \le 1$), according to the Lemma 2, we have already obtain the MNC’s proportion of recycled plastic: when $2k\theta - FD_{{m_{1} }} < (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] < 2k\theta$, $r_{{}}^{*} = r_{\theta }^{*} = \theta$; when $2k\theta < (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] < 2k$,$r_{{}}^{*} = r_{g}^{*} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{g}^{*} - e_{r} } \right) + p_{v} - p_{{r_{g} }}^{*} ]}}{2k}$; when $(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] \ge 2k$, $r_{{}}^{*} = r_{1}^{*} = 1$.

• (1) $\theta < r_{{}}^{*} < 1$

  Substituting $r_{g}^{*}$ into the plastic recycling company’s profit function $\Pi_{R}^{{}}$. The second order derivative of $\Pi_{R}^{{}}$ in relation to $p_{r}$ is: $\frac{{\partial^{2} \Pi_{R}^{{}} }}{{\partial p_{r}^{2} }} = - \frac{{\eta (D_{{m_{1} }} + D_{{m_{2} }} )^{2} }}{{2k^{2} }} < 0$, which indicates the plastic recycling company’s profit is concave for $p_{r}$. Let $\frac{{\partial \Pi_{R}^{{}} }}{{\partial p_{r} }} = 0$, we have:

  $$ p_{{r_{g} }}^{*} (r_{{}}^{*} = r_{g}^{*} ) = \frac{{[p_{p} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right)](k + \eta )}}{2k + \eta } = p_{p} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) - \frac{{k[p_{p} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right)]}}{2k + \eta } $$

  Then, substituting $r_{g}^{*}$ and $p_{{r_{g} }}^{*}$ into the petrochemical company’s profit function $\Pi_{P}^{{}}$. The second order derivative of $\Pi_{P}^{{}}$ in relation to $e$ is: $\frac{{\partial^{2} \Pi_{P}^{{}} }}{{\partial e^{2} }} = \frac{{\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} }}{{{\mkern 1mu} 2k + \eta }}{\mkern 1mu} - 2\gamma$.

  Similar to the proof of Lemma 3, to ensure the petrochemical company’s profit function is concave, we also need the condition $2{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) \ge \lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2}$. Then, let $\frac{{\partial \Pi_{P}^{{}} }}{\partial e} = 0$, we have

  $$ e_{{}}^{*} = e_{g}^{*} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} \{ 4{\mkern 1mu} k{\mkern 1mu} p_{c} - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c] + 2\eta p_{c} \} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }}. $$

  Substituting $e_{g}^{*}$ into $p_{{r_{g} }}^{*}$, we could obtain the optimal recycled plastic price $p_{{}}^{*}$. Then, substituting $e_{g}^{*}$ and $p_{{r_{g} }}^{*}$ into $r_{{}}^{*}$, we could obtain the optimal proportion of recycled plastic.

• (2) $r^{*} = r_{\theta }^{*} = \theta$

  Substituting $r_{f}^{*}$ into the plastic recycling company’s profit function $\Pi_{R}^{{}}$, the first order derivative of $\Pi_{R}^{{}}$ in relation to $p_{r}$ is $\frac{{\partial \Pi_{R}^{{}} }}{{\partial p_{r} }} = \theta (D_{{m_{1} }} + D_{{m_{2} }} ) \ge 0$. Since we have $2k\theta - FD_{{m_{1} }} < (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] < 2k\theta$, that means $p_{v} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) - \frac{2k\theta }{{D_{{m_{1} }} + D_{{m_{2} }} }} \le p_{{r_{{}} }}^{*} \le p_{v} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) - \frac{2k\theta }{{D_{{m_{1} }} + D_{{m_{2} }} }} + \frac{{FD_{{m_{1} }} }}{{D_{{m_{1} }} + D_{{m_{2} }} }}$. Hence, the optimal recycled plastic price of the plastic recycling company is $p_{{r_{\theta } }}^{*} = p_{v} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{\theta }^{*} - e_{r} } \right) - \frac{2k\theta }{{D_{{m_{1} }} + D_{{m_{2} }} }} + \frac{{FD_{{m_{1} }} }}{{D_{{m_{1} }} + D_{{m_{2} }} }}$.

  Then, substituting $r_{\theta }^{*}$ and $p_{{r_{\theta } }}^{*}$ into the petrochemical company’s profit function $\Pi_{P}^{{}}$. The second order derivative of $\Pi_{P}^{{}}$ in relation to $e$ is: $\frac{{\partial^{2} \Pi_{P}^{{}} }}{{\partial e^{2} }} = - 2\gamma$, which indicates the petrochemical company’s profit is concave for $e$. Let $\frac{{\partial \Pi_{P}^{{}} }}{\partial e} = 0$, we have: $e_{\theta }^{*} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} )(1 - \theta )p_{c} }}{2\gamma }$.

  Substituting $e_{\theta }^{*}$ into $p_{{r_{\theta } }}^{*}$, we could obtain the optimal recycled plastic price $p_{{}}^{*}$. Then, substituting $e_{\theta }^{*}$ and $p_{{r_{\theta } }}^{*}$ into $r_{{}}^{*}$, we could obtain the optimal proportion of recycled plastic.

• (3) $r^{*} = r_{1}^{*} = 1$

  Substituting $r_{1}^{*} = 1$ into the plastic recycling company’s profit function $\Pi_{R}^{{}}$, the first order derivative of $\Pi_{R}^{{}}$ in relation to $p_{r}$ is $\frac{{\partial \Pi_{R}^{{}} }}{{\partial p_{r} }} = (D_{{m_{1} }} + D_{{m_{2} }} ) \ge 0$. Since we have $(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] \ge 2k$, that means $p_{{r_{{}} }}^{*} \le p_{v} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) - \frac{2k}{{D_{{m_{1} }} + D_{{m_{2} }} }}$. Hence, the optimal recycled plastic price of the plastic recycling company is $p_{{r_{1} }}^{*} = p_{v} + \lambda {\mkern 1mu} \left( {e_{{m_{0} }} - e_{1}^{*} - e_{r} } \right) - \frac{2k}{{D_{{m_{1} }} + D_{{m_{2} }} }}$.

  Then, substituting $r_{1}^{*}$ and $p_{{r_{1} }}^{*}$ into the petrochemical company’s profit function $\Pi_{P}^{{}}$. The second order derivative of $\Pi_{P}^{{}}$ in relation to $e$ is: $\frac{{\partial^{2} \Pi_{P}^{{}} }}{{\partial e^{2} }} = - 2\gamma$, which indicates the petrochemical company’s profit is concave for $e$. Let $\frac{{\partial \Pi_{P}^{{}} }}{\partial e} = 0$, we have: $e_{1}^{*} = 0$.

  Substituting $e_{1}^{*}$ into $p_{{r_{1} }}^{*}$, we could obtain the optimal recycled plastic price $p_{{}}^{*}$. Then, substituting $e_{1}^{*}$ and $p_{{r_{1} }}^{*}$ into $r_{{}}^{*}$, we could obtain the optimal proportion of recycled plastic.

Proof of Proposition 1

The proof of Proposition 1 could be obtained by the above Lemma 1–6.

Proof of Proposition 2

Since we have proved that when $0 \le (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right) + p_{v} - p_{r}^{*} ] + FD_{{m_{1} }} \ge 2k\theta$, the MNC will comply with government regulations on the proportion of recycled plastic. That means when the government unit fines $F \ge \frac{{2k\theta - (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right) + p_{v} - p_{{r_{f} }}^{*} ]}}{{D_{{m_{1} }} }}$, the MNC will not violates the regulation.

Let this threshold $\frac{{2k\theta - (D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} [\lambda \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right) + p_{v} - p_{{r_{f} }}^{*} ]}}{{D_{{m_{1} }} }} = R$. Taking the derivative of this threshold $T$ with respect to consumer demand in Market 1 $D_{{m_{1} }}^{{}}$, government-mandated minimum proportion of recycled plastic $\theta$ and commitment coefficient from Scope3 carbon reduction $\lambda$ respectively, we have $\frac{\partial R}{{\partial D_{{m_{1} }}^{{}} }} = - \frac{{2k\theta - D_{{m_{2} }} [\lambda \left( {e_{{m_{0} }} - e_{f}^{*} - e_{r} } \right) + p_{v} - p_{{r_{f} }}^{*} ]}}{{D_{{m_{1} }}^{2} }} \le 0$, $\frac{\partial R}{{\partial \theta }} = \frac{2k}{{D_{{m_{1} }} }} \ge 0$, $\frac{\partial R}{{\partial \lambda }} = - \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} \left( {e_{{m_{0} }} - e_{{}}^{*} - e_{r} } \right)}}{{D_{{m_{1} }} }} \le 0$.

Proof of Proposition 3

Since $e_{f}^{*} = e_{g}^{*} + \frac{{FD_{{m_{1} }} (D_{{m_{1} }} + D_{{m_{2} }} )p_{c} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }}$, and $\frac{{FD_{{m_{1} }} (D_{{m_{1} }} + D_{{m_{2} }} )p_{c} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} \ge 0$, we have $e_{g}^{*} \le e_{f}^{*}$. We have obtained that $e_{1}^{*} = 0$, hence, $e_{1}^{*} \le e_{g}^{*} \le e_{f}^{*}$ must hold.

Let $e_{\theta }^{*} - e_{f}^{*} \ge 0$, we have $(1 - \theta ) \ge \frac{{\gamma \{ 2p_{c} (2k + \eta ) - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c]\} }}{{2{\mkern 1mu} \gamma p_{c} {\mkern 1mu} (2k + \eta ) - \lambda p_{c}^{2} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }}$. Let $e_{\theta }^{*} - e_{g}^{*} \ge 0$ and $e_{\theta }^{*} - e_{f}^{*} \le 0$, we have $\frac{{\gamma \{ 2p_{c} (2k + \eta ) - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c]\} - FD_{{m_{1} }} p_{c} \gamma }}{{2{\mkern 1mu} \gamma p_{c} {\mkern 1mu} (2k + \eta ) - \lambda p_{c}^{2} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} \le (1 - \theta ) \le \frac{{\gamma \{ 2p_{c} (2k + \eta ) - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c]\} }}{{2{\mkern 1mu} \gamma p_{c} {\mkern 1mu} (2k + \eta ) - \lambda p_{c}^{2} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }}$. Let $e_{g}^{*} - e_{\theta }^{*} \ge 0$, we have $(1 - \theta ) \le \frac{{\gamma \{ 2p_{c} (2k + \eta ) - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c]\} - FD_{{m_{1} }} p_{c} \gamma }}{{2{\mkern 1mu} \gamma p_{c} {\mkern 1mu} (2k + \eta ) - \lambda p_{c}^{2} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }}$.

Proof of Proposition 4

Taking the derivative of the optimal recycled plastic proportion $r_{f}^{*}$ and $r_{g}^{*}$ with respect to the MNC’s Scope 3 commitment coefficient $\lambda$, we have:

$$ \frac{{\partial r_{f}^{*} }}{\partial \lambda } = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} )\left( {e_{{m_{0} }} - \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} \{ 4{\mkern 1mu} k{\mkern 1mu} p_{c} - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c] + 2\eta p_{c} \} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} + \frac{{FD_{{m_{1} }} (D_{{m_{1} }} + D_{{m_{2} }} )p_{c} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} - e_{r} } \right)}}{2k} $$

Since $\frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} \{ 4{\mkern 1mu} k{\mkern 1mu} p_{c} - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c] + 2\eta p_{c} \} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} - \frac{{FD_{{m_{1} }} (D_{{m_{1} }} + D_{{m_{2} }} )p_{c} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} = e_{f}^{*}$, and $e_{{m_{0} }} - e_{f}^{*} - e_{r} \ge 0$, hence $\frac{{\partial r_{f}^{*} }}{\partial \lambda } \ge 0$.

$$ \frac{{\partial r_{g}^{*} }}{\partial \lambda } = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} )\left( {e_{{m_{0} }} - \frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} \{ 4{\mkern 1mu} k{\mkern 1mu} p_{c} - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c] + 2\eta p_{c} \} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} - e_{r} } \right)}}{2k} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} )\left( {e_{{m_{0} }} - e_{g}^{*} - e_{r} } \right)}}{2k} \ge 0 $$

Since $\frac{{(D_{{m_{1} }} + D_{{m_{2} }} ){\mkern 1mu} \{ 4{\mkern 1mu} k{\mkern 1mu} p_{c} - (D_{{m_{1} }} + D_{{m_{2} }} )[\lambda p_{c} (\,2e_{{m_{0} }} - e_{r} ) + p_{v} (p_{c} - \lambda ) + \lambda c] + 2\eta p_{c} \} }}{{4{\mkern 1mu} \gamma {\mkern 1mu} (2k + \eta ) - 2\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} {\mkern 1mu} }} = e_{g}^{*}$, and $e_{{m_{0} }} - e_{g}^{*} - e_{r} \ge 0$, hence $\frac{{\partial r_{g}^{*} }}{\partial \lambda } \ge 0$.

Taking the derivative of the optimal recycled plastic proportion $r_{f}^{*}$ and $r_{g}^{*}$ with respect to the carbon trading price $p_{c}$, we have:

$\frac{{\partial r_{g}^{*} }}{{\partial p_{c} }} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} )^{5} {\mkern 1mu} \lambda^{3} (c - p_{v} ) + 2\gamma \lambda {\mkern 1mu} (2k + \eta )\left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} [\left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)(2e_{{m_{0} }} \lambda - e_{r} \lambda + p_{v} ) - 2{\mkern 1mu} (2k + \eta )]}}{{4{\mkern 1mu} {\mkern 1mu} (2k + \eta )[\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} - 2\gamma {\mkern 1mu} (2k + \eta )]^{2} {\mkern 1mu} }}$,

$\frac{{\partial r_{f}^{*} }}{{\partial p_{c} }} = \frac{{(D_{{m_{1} }} + D_{{m_{2} }} )^{5} {\mkern 1mu} \lambda^{3} (c - p_{v} ) + 2\gamma \lambda {\mkern 1mu} (2k + \eta )\left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} [\left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)(2e_{{m_{0} }} \lambda - e_{r} \lambda + p_{v} ) - 2{\mkern 1mu} (2k + \eta )]}}{{4{\mkern 1mu} {\mkern 1mu} (2k + \eta )[\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} - 2\gamma {\mkern 1mu} (2k + \eta )]^{2} {\mkern 1mu} }} + \frac{{FD_{{m_{1} }} \gamma \lambda (D_{{m_{1} }} + D_{{m_{2} }} )^{2} }}{{2[\lambda p_{c} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} - 2\gamma {\mkern 1mu} (2k + \eta )]^{2} {\mkern 1mu} }}$. Let $\frac{{\partial r_{g}^{*} }}{{\partial p_{c} }} \ge 0$, we have $\lambda \ge \frac{{\gamma (2e_{{m_{0} }} - e_{r} ){\mkern 1mu} (2k + \eta ) + \sqrt {K_{1} + \gamma {\mkern 1mu} (2k + \eta )\{ - 8kp_{v} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right) - 2c\left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)[p_{v} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right) - 2{\mkern 1mu} (2k + \eta )]\} } }}{{\left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} (p_{v} - c){\mkern 1mu} }}$. Let $\frac{{\partial r_{f}^{*} }}{{\partial p_{c} }} \ge 0$, we have:

$\lambda \ge \frac{{\gamma (2e_{{m_{0} }} - e_{r} ){\mkern 1mu} (2k + \eta ) + \sqrt {K_{1} + \gamma {\mkern 1mu} (2k + \eta )\{ 2p_{v} {\mkern 1mu} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)(FD_{{m_{1} }} - 4k) - 2c\left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)[p_{v} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right) - 2{\mkern 1mu} (2k + \eta ) + FD_{{m_{1} }} ]\} } }}{{\left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} (p_{v} - c){\mkern 1mu} }}$,where $K_{1} = \gamma {\mkern 1mu} (2k + \eta )\{ 2\gamma k( - 2e_{{m_{0} }} + e_{r} )^{2} + 2{\mkern 1mu} p_{v}^{2} \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)^{2} + \gamma \eta ( - 2e_{{m_{0} }} + e_{r} )^{2} - 4p_{v} \eta \left( {D_{{m_{1} }} + D_{{m_{2} }} } \right)\}$.

Proof of Proposition 5

Given the optimal petrochemical company’s carbon reduction effort level $e_{{}}^{*}$, the recycled plastic price under different scenarios is as follows:

$p_{{r_{f} }}^{*} = p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right) - \frac{{k[p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right)]}}{2k + \eta } + \frac{{FD_{{m_{1} }} (k + \eta )}}{{(2k + \eta )(D_{{m_{1} }} + D_{{m_{2} }} )}}$, $p_{{r_{\theta } }}^{*} = p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right) - \frac{2k\theta }{{D_{{m_{1} }} + D_{{m_{2} }} }} + \frac{{FD_{{m_{1} }} }}{{D_{{m_{1} }} + D_{{m_{2} }} }}$,

$p_{{r_{g} }}^{*} = p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right) - \frac{{k[p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{g}^{*} - e_{r} } \right)]}}{2k + \eta }$, $p_{{r_{1} }}^{*} = p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right) - \frac{2k}{{D_{{m_{1} }} + D_{{m_{2} }} }}$. If $p_{v} \le p_{r}^{*}$, then the price inversion between virgin and recycled plastics occurs.

Thus, let $p_{{r_{f} }}^{*} - p_{v} \ge 0$, $p_{{r_{\theta } }}^{*} - p_{v} \ge 0$,$p_{{r_{g} }}^{*} - p_{v} \ge 0$ and $p_{{r_{1} }}^{*} - p_{v} \ge 0$ respectively, we have $\lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right) + \frac{{FD_{{m_{1} }} (k + \eta )}}{{(2k + \eta )(D_{{m_{1} }} + D_{{m_{2} }} )}} \ge \frac{{k[p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right)]}}{2k + \eta }$, $\lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right) + \frac{{FD_{{m_{1} }} }}{{D_{{m_{1} }} + D_{{m_{2} }} }} \ge \frac{2k\theta }{{D_{{m_{1} }} + D_{{m_{2} }} }}$, $\lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right) \ge \frac{{k[p_{v} + \lambda {\mkern 1mu} \left( {e_{m} - e_{g}^{*} - e_{r} } \right)]}}{2k + \eta }$, $\lambda {\mkern 1mu} \left( {e_{m} - e_{{}}^{*} - e_{r} } \right) \ge \frac{2k}{{D_{{m_{1} }} + D_{{m_{2} }} }}$.