Reselling or marketplace mode for an online platform: the choice between cap-and-trade and carbon tax regulation

Abstract

This paper explores a manufacturer’s production decisions and a government’s choice between cap-and-trade and carbon tax regulation. The manufacturer sells its product through offline and online channels, and we consider the online-offline spillover effect (OOSE) to reflect the influence of online sales on offline sales. We further compare carbon emission, the manufacturer’s profit, and social welfare for reselling and marketplace modes under the two types of regulations. First, we find that under both policies, when the government allocates a tight cap (high tax rate), the manufacturer should decrease online sales if the OOSE increases, and vice versa. Interestingly, we also discover that the manufacturer’s profit may decrease with the cap regardless of the modes. Second, under a low (high) environmental coefficient, the social welfare using reselling mode is lower (higher) than using marketplace mode, and the two modes can generate the same social welfare under a moderate environmental coefficient. Third, using either mode, when the environmental damage coefficient is low (high), the government should implement cap-and-trade regulation (carbon tax regulation). Finally, based on the data of a real-world company, when the environmental damage coefficient is low (high), the company should use an online platform with marketplace (reselling) mode to achieve optimal profit.

Appendix

Proof of Theorem 1

The platform is the leader of the Stackelberg game, and we define the retail price is as follows: $p = \omega + \Delta .$ The manufacturer’s profit function and its derivative are:

$$ \pi_{m}^{cap} = p_{0} {\text{\{ }}Q + \gamma [\alpha - \beta (\omega { + }\Delta )]{\text{\} + }}\omega [\alpha - \beta (\omega { + }\Delta )] - e_{0} \{ Q + (1{ + }\gamma )[\alpha - \beta (\omega { + }\Delta )]\} + C(\lambda - bC), $$

$$ {{\partial \pi_{m}^{cap} } \mathord{\left/ {\vphantom {{\partial \pi_{m}^{cap} } {\partial \omega }}} \right. \kern-\nulldelimiterspace} {\partial \omega }} = \alpha - {2}\beta \omega - \beta [p_{0} \gamma + \Delta - e_{0} (\lambda - bC)(1 + \gamma )]. $$

We know that ${{\partial^{2} \pi_{m}^{cap} } \mathord{\left/ {\vphantom {{\partial^{2} \pi_{m}^{cap} } {\partial \omega^{2} }}} \right. \kern-\nulldelimiterspace} {\partial \omega^{2} }} = - 2\beta < 0,$ so set the first partial derivative of $\pi_{m}^{cap}$ with $\omega$ equal to zero. That is, ${{\partial \pi_{m}^{cap} } \mathord{\left/ {\vphantom {{\partial \pi_{m}^{cap} } {\partial \omega }}} \right. \kern-\nulldelimiterspace} {\partial \omega }} = \alpha - {2}\beta \omega - \beta [p_{0} \gamma + \Delta - e_{0} (\lambda - bC)(1 + \gamma )]{ = }0,$ so $\omega { = }{{\{ \alpha - \beta [p_{0} \gamma + \Delta - e_{0} (\lambda - bC)(1 + \gamma )]\} } \mathord{\left/ {\vphantom {{\{ \alpha - \beta [p_{0} \gamma + \Delta - e_{0} (\lambda - bC)(1 + \gamma )]\} } {(2\beta )}}} \right. \kern-\nulldelimiterspace} {(2\beta )}}.$

The platform’s profit function and its derivative are:

$$ \pi_{p}^{cap} = {{{{\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC) - \Delta ]\} } \mathord{\left/ {\vphantom {{\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC) - \Delta ]\} } 2}} \right. \kern-\nulldelimiterspace} 2} + \Delta } \mathord{\left/ {\vphantom {{{{\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC) - \Delta ]\} } \mathord{\left/ {\vphantom {{\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC) - \Delta ]\} } 2}} \right. \kern-\nulldelimiterspace} 2} + \Delta } 2}} \right. \kern-\nulldelimiterspace} 2}, $$

$$ {{\partial \pi_{p}^{cap} } \mathord{\left/ {\vphantom {{\partial \pi_{p}^{cap} } {\partial \Delta }}} \right. \kern-\nulldelimiterspace} {\partial \Delta }} = {{\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC) - 2\Delta ]\} } \mathord{\left/ {\vphantom {{\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC) - 2\Delta ]\} } 2}} \right. \kern-\nulldelimiterspace} 2}. $$

We know ${{\partial^{2} \pi_{p}^{cap} } \mathord{\left/ {\vphantom {{\partial^{2} \pi_{p}^{cap} } {\partial \Delta^{2} }}} \right. \kern-\nulldelimiterspace} {\partial \Delta^{2} }} = - \beta < 0,$ hence $\Delta { = }{{\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC)]\} } \mathord{\left/ {\vphantom {{\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC)]\} } {(2\beta )}}} \right. \kern-\nulldelimiterspace} {(2\beta )}}.$ Then we can derive the equilibrium outcomes presented in Theorem 1.

Proof of Corollary 1

${{\partial \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial \pi_{m}^{cap * } } {\partial C}}} \right. \kern-\nulldelimiterspace} {\partial C}} = be_{0} Q - 2bC + \lambda + ({1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8})be_{0} (1 + \gamma )\{ \alpha + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC)]\} ,$ ${{\partial^{2} \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial^{2} \pi_{m}^{cap * } } {\partial C^{2} }}} \right. \kern-\nulldelimiterspace} {\partial C^{2} }} = {{b[ - 16 + be_{0}^{2} \beta (1 + \gamma )^{2} ]} \mathord{\left/ {\vphantom {{b[ - 16 + be_{0}^{2} \beta (1 + \gamma )^{2} ]} 8}} \right. \kern-\nulldelimiterspace} 8}.$ These allow us to derive the equilibrium outcomes.

Proof of Theorem 2

Denote by $W_{R}^{cap}$ the social welfare under cap-and-trade regulation. Then we have $W_{R}^{cap} = \int_{0}^{{q^{ * } }} {(p - p^{ * } )dq + \pi_{R}^{s} } + \pi_{p}^{cap*} - v[e_{0} (q + Q + \gamma q)]^{2} .$ That is:

$$ W_{R}^{cap} = \int_{0}^{{q^{ * } }} {({\alpha \mathord{\left/ {\vphantom {\alpha \beta }} \right. \kern-\nulldelimiterspace} \beta } - {q \mathord{\left/ {\vphantom {q \beta }} \right. \kern-\nulldelimiterspace} \beta } - p^{ * } )dq + \pi_{m}^{cap*} } + \pi_{p}^{cap*} { + }\pi_{{\text{C}}}^{cap*} - v[e_{0} (q + Q + \gamma q)]^{2} , $$

$$ \begin{aligned} {{\partial W_{R}^{cap} } \mathord{\left/ {\vphantom {{\partial W_{R}^{cap} } {\partial C}}} \right. \kern-\nulldelimiterspace} {\partial C}} {=} & - {1 \mathord{\left/ {\vphantom {1 {16}}} \right. \kern-\nulldelimiterspace} {16}}b^{2} \beta Ce_{0}^{2} (1 {+} \gamma )^{2} [1 {+} 2e_{0}^{2} (1 {+} \gamma )^{2} v\beta ] {+} ({1 \mathord{\left/ {\vphantom {1 {16}}} \right. \kern-\nulldelimiterspace} {16}})be_{0} (1 {+} \gamma )\{ 3p_{0} \beta \gamma {-} \alpha [ - 3 {+} 2e_{0}^{2} (1 {+} \gamma )^{2} v\beta ] \\ & + e_{0} \beta (1 + \gamma )\lambda - 8Q\beta e_{0}^{2} v(1 + \gamma ) + 2e_{0}^{2} v\beta^{2} (1 + \gamma )^{2} [ - p_{0} \gamma + e_{0} (1 + \gamma )\lambda ]\} , \\ \end{aligned} $$

We know ${{\partial^{2} W_{R}^{cap} } \mathord{\left/ {\vphantom {{\partial^{2} W_{R}^{cap} } {\partial C^{2} }}} \right. \kern-\nulldelimiterspace} {\partial C^{2} }} = {{\{ e_{0}^{2} \beta b^{2} [ - 1 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )^{2} \} } \mathord{\left/ {\vphantom {{\{ e_{0}^{2} \beta b^{2} [ - 1 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )^{2} \} } {16}}} \right. \kern-\nulldelimiterspace} {16}} < 0$, so.

${{\partial^{2} W_{R}^{cap} } \mathord{\left/ {\vphantom {{\partial^{2} W_{R}^{cap} } {\partial C^{2} }}} \right. \kern-\nulldelimiterspace} {\partial C^{2} }} = {{\{ e_{0}^{2} \beta b^{2} [ - 1 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )^{2} \} } \mathord{\left/ {\vphantom {{\{ e_{0}^{2} \beta b^{2} [ - 1 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )^{2} \} } {16}}} \right. \kern-\nulldelimiterspace} {16}} < 0.$ Now, define

$$ A {=} {1 \mathord{\left/ {\vphantom {1 {16}}} \right. \kern-\nulldelimiterspace} {16}}be_{0} (1{+} \gamma )\{ 3p_{0} \beta \gamma {-} \alpha [ - 3 {+} 2e_{0}^{2} (1 {+} \gamma )^{2} v\beta ] {+} e_{0} \beta (1 {+} \gamma )\lambda {-} 8\beta Qe_{0}^{2} v(1 {+} \gamma ) {+} 2e_{0}^{2} v\beta^{2} (1 {+} \gamma )^{2} [ - p_{0} \gamma {+} e_{0} (1 {+} \gamma )\lambda ]\} . $$

If $A < 0,$ we get $v < \frac{{3(\alpha + p_{0} \beta \gamma ) + e_{0} \beta (1 + \gamma )\lambda }}{{2e_{0}^{2} \beta (1 + \gamma )\{ 4Q + (1 + \gamma )[\alpha + \beta p_{0} \gamma - e_{0} \beta (1 + \gamma )\lambda ]\} }},$ so the optimal total emission is $C^{ * } < 0.$ Therefore, we make $C^{ * } = 0$ without loss of generality.

If $A > 0,$ we denote that the optimal cap $C^{ * }$ exists in the interval of $\left[ {0,{\lambda \mathord{\left/ {\vphantom {\lambda b}} \right. \kern-\nulldelimiterspace} b}} \right]$ to maximize social welfare under cap-and-trade regulation.

• (1) If $C^{ * } > {\lambda \mathord{\left/ {\vphantom {\lambda b}} \right. \kern-\nulldelimiterspace} b},$ we have $v < \frac{{3(\alpha + p_{0} \beta \gamma )}}{{2e_{0}^{2} \beta (1 + \gamma )[4Q + (1 + \gamma )(\alpha + \beta p_{0} \gamma )]}},$ so we must have $C^{ * } = {\lambda \mathord{\left/ {\vphantom {\lambda b}} \right. \kern-\nulldelimiterspace} b}.$

• (2) If $C^{ * } < {\lambda \mathord{\left/ {\vphantom {\lambda b}} \right. \kern-\nulldelimiterspace} b},$ then

  $$ C^{ * } = \frac{{[3 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) - 8Qv\beta e_{0}^{2} (1 + \gamma ) + \beta e_{0} \lambda [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )}}{{e_{0} b\beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )}}. $$

We define $v_{2} ,v_{2}^{^{\prime}}$, with $v_{2}^{^{\prime}} > v_{2} ,$ as follows

$$ v_{2} = \frac{{3(\alpha + p_{0} \beta \gamma )}}{{2e_{0}^{2} \beta (1 + \gamma )[4Q + (1 + \gamma )(\alpha + \beta p_{0} \gamma )]}},v_{2}^{^{\prime}} = \frac{{3(\alpha + p_{0} \beta \gamma ) + e_{0} \beta (1 + \gamma )\lambda }}{{2e_{0}^{2} \beta (1 + \gamma )\{ 4Q + (1 + \gamma )[\alpha + \beta p_{0} \gamma - e_{0} \beta (1 + \gamma )\lambda ]\} }}. $$

Then we can easily solve for the optimal cap $C_{R}^{ * }$, total carbon emission $E_{R}^{cap}$ and social welfare $W_{R}^{cap}$ under different environmental damage conditions.

• (i) If $0 < v < v_{2} ,$ then $C_{R}^{ * } = {\lambda \mathord{\left/ {\vphantom {\lambda b}} \right. \kern-\nulldelimiterspace} b},$

  $$ W_{R}^{cap} {=} \frac{{(\alpha {+} p_{0} \beta \gamma )^{2} [7 {-} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ] {-} 16(\alpha {+} p_{0} \beta \gamma )e_{0}^{2} Qv\beta (1 {+} \gamma ) {-} 32e_{0}^{2} v\beta Q^{2} {+} 32p_{0} \beta Q}}{32\beta }, $$
  $$ E_{R}^{cap} = {{\{ e_{0} [4Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } \mathord{\left/ {\vphantom {{\{ e_{0} [4Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } 4}} \right. \kern-\nulldelimiterspace} 4},\quad \pi_{m}^{cap} { = }p_{0} Q + {{(\alpha + p_{0} \beta \gamma )^{2} } \mathord{\left/ {\vphantom {{(\alpha + p_{0} \beta \gamma )^{2} } {16\beta }}} \right. \kern-\nulldelimiterspace} {16\beta }}. $$

• (ii) If $v_{2} < v < v_{2}^{^{\prime}} ,$ then

  $$ C_{R}^{ * } = \frac{{[3 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) - 8Qv\beta e_{0}^{2} (1 + \gamma ) + \beta e_{0} \lambda [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )}}{{e_{0} b\beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )}}, $$
  $$ W_{R}^{cap} = \frac{{\alpha^{2} + 2Q(p_{0} - e_{0}^{2} Qv)\beta + 2\alpha \beta [p_{0} \gamma - 2e_{0}^{2} (1 + \gamma )vQ] + \beta^{2} p_{0} [p_{0} \gamma^{2} + 4e_{0}^{2} Qv(1 + \gamma )]}}{{2\beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ]}}, $$
  $$ E_{R}^{cap} = {{\{ e_{0} [Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } \mathord{\left/ {\vphantom {{\{ e_{0} [Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}, $$
  $$ \begin{aligned} \pi_{m}^{cap} { = } & p_{0} Q + \frac{{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\}^{2} }}{{\beta [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} }} \\ & - \frac{\begin{gathered} \{ \alpha [ - 3 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] - 3\beta p_{0} \gamma + 2\beta e_{0}^{2} v(1 + \gamma )[4Q + p_{0} \beta \gamma (1 + \gamma )]\} \hfill \\ \{ \alpha [ - 3 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] - 3\beta p_{0} \gamma + e_{0} \beta (1 + \gamma )[be_{0} Q + 2be_{0}^{3} Qv\beta (1 + \gamma )^{2} \hfill \\ - \lambda + 8e_{0} vQ + 2\beta e_{0} v(1 + \gamma )(p_{0} \gamma - e_{0} (1 + \gamma )\lambda )]\} \hfill \\ \end{gathered} }{{\{ be_{0}^{2} \beta^{2} (1 + \gamma )^{2} [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} \} }}, \\ \end{aligned} $$

• (iii) If $v > v_{2}^{^{\prime}} ,$ then $C_{R}^{ * } = 0,$

  $$ W_{R}^{cap} = \frac{\begin{gathered} 32Q(p_{0} - e_{0}^{2} Qv)\beta - [\alpha + p_{0} \beta \gamma - e_{0} \beta (1 + \gamma )\lambda ] \hfill \\ \{ \alpha [ - 7 + 2v\beta e_{0}^{2} (1 + \gamma )^{2} ] - 7\beta p_{0} \gamma + \beta e_{0} (1 + \gamma )[ - \lambda + 16Qe_{0} v + 2e_{0} v\beta (1 + \gamma )p_{0} \gamma - 2e_{0}^{2} v\beta (1 + \gamma )^{2} \lambda ]\} \hfill \\ \end{gathered} }{32\beta }, $$
  $$ E_{R}^{cap} = {{\{ e_{0} [4Q + (1 + \gamma )(a + \beta p_{0} \gamma - e_{0} \beta (1 + \gamma )\lambda )]\} } \mathord{\left/ {\vphantom {{\{ e_{0} [4Q + (1 + \gamma )(a + \beta p_{0} \gamma - e_{0} \beta (1 + \gamma )\lambda )]\} } 4}} \right. \kern-\nulldelimiterspace} 4},\pi_{m}^{cap} { = }p_{0} Q - e_{0} Q\lambda + {{[\alpha + p_{0} \beta \gamma - e_{0} \beta (1 + \gamma )\lambda ]^{2} } \mathord{\left/ {\vphantom {{[\alpha + p_{0} \beta \gamma - e_{0} \beta (1 + \gamma )\lambda ]^{2} } {(16\beta )}}} \right. \kern-\nulldelimiterspace} {(16\beta )}}. $$

Proof of Theorem 3

The proof follows the process used in proving Theorem 1. We omit the proof here.

Proof of Theorem 4

Denote by $W_{R}^{tax}$ the social welfare under carbon tax regulation. Then we have $W_{R}^{tax} = \int_{0}^{{q^{ * } }} {(p - p^{ * } )dq + \pi_{R}^{s} } + \pi_{p}^{tax*} - v[e_{0} (q + Q + \gamma q)]^{2} .$ That is:

$$ W_{R}^{tax} = \int_{0}^{{q^{ * } }} {({\alpha \mathord{\left/ {\vphantom {\alpha \beta }} \right. \kern-\nulldelimiterspace} \beta } - {q \mathord{\left/ {\vphantom {q \beta }} \right. \kern-\nulldelimiterspace} \beta } - p^{ * } )dq + \pi_{m}^{tax*} } + \pi_{p}^{tax*} { + }\pi_{{\text{C}}}^{tax*} - v[e_{0} (q + Q + \gamma q)]^{2} , $$

$$ \begin{aligned} {{\partial W_{R}^{tax} } \mathord{\left/ {\vphantom {{\partial W_{R}^{tax} } {\partial \tau }}} \right. \kern-\nulldelimiterspace} {\partial \tau }} = & {1 \mathord{\left/ {\vphantom {1 {16}}} \right. \kern-\nulldelimiterspace} {16}}\beta e_{0}^{2} (1{ + }\gamma )^{2} [ - 1 - 2v\beta e_{0}^{2} (1{ + }\gamma )^{2} ]\tau { + }{1 \mathord{\left/ {\vphantom {1 {16}}} \right. \kern-\nulldelimiterspace} {16}}e_{0} (1{ + }\gamma )\{ - 3\beta p_{0} \gamma \\ & + 8v\beta e_{0}^{2} (1{ + }\gamma )Q + 2e_{0}^{2} p_{0} v\beta^{2} \gamma (1 + \gamma )^{2} + \alpha [ - 3 + 2v\beta e_{0}^{2} (1{ + }\gamma )^{2} ]\} . \\ \end{aligned} $$

We know ${{\partial^{2} W_{R}^{tax} } \mathord{\left/ {\vphantom {{\partial^{2} W_{R}^{tax} } {\partial \tau^{2} }}} \right. \kern-\nulldelimiterspace} {\partial \tau^{2} }} = {1 \mathord{\left/ {\vphantom {1 {16}}} \right. \kern-\nulldelimiterspace} {16}}e_{0}^{2} \beta [ - 1 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )^{2} < 0,$ so we set the first derivative of $W_{R}^{tax}$ with respect to $\tau$ to zero. Hence $\tau^{ * } = {{\{ [ - 3 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) + 8Qv\beta e_{0}^{2} (1 + \gamma )\} } \mathord{\left/ {\vphantom {{\{ [ - 3 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) + 8Qv\beta e_{0}^{2} (1 + \gamma )\} } {\{ e_{0} \beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )\} }}} \right. \kern-\nulldelimiterspace} {\{ e_{0} \beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )\} }}.$

Define $B = {1 \mathord{\left/ {\vphantom {1 {16}}} \right. \kern-\nulldelimiterspace} {16}}e_{0} (1{ + }\gamma )\{ - 3\beta p_{0} \gamma + 8v\beta e_{0}^{2} (1{ + }\gamma )Q + 2e_{0}^{2} p_{0} v\beta^{2} \gamma (1 + \gamma )^{2} + \alpha [ - 3 + 2v\beta e_{0}^{2} (1{ + }\gamma )^{2} ]\} .$

If $B < 0,$ that is $v < {{[3(\alpha + p_{0} \beta \gamma )]} \mathord{\left/ {\vphantom {{[3(\alpha + p_{0} \beta \gamma )]} {\{ 2e_{0}^{2} \beta (1 + \gamma )[4Q + (1 + \gamma )(\alpha + \beta p_{0} \gamma )]\} }}} \right. \kern-\nulldelimiterspace} {\{ 2e_{0}^{2} \beta (1 + \gamma )[4Q + (1 + \gamma )(\alpha + \beta p_{0} \gamma )]\} }},$ then the optimal total emission is $\tau^{ * } < 0.$ Therefore, we can set $\tau^{ * } = 0$ without loss of generality.

If $B > 0,$ we denote that the optimal $\tau^{ * }$ exists in the interval of $\left[ {0,{{(\alpha + \beta p_{0} \gamma )} \mathord{\left/ {\vphantom {{(\alpha + \beta p_{0} \gamma )} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}}} \right]$ to maximize social welfare under carbon tax regulation.

• (1) If $\tau^{ * } < {{(\alpha + \beta p_{0} \gamma )} \mathord{\left/ {\vphantom {{(\alpha + \beta p_{0} \gamma )} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}},$ that is $v < {{[3(\alpha + p_{0} \beta \gamma )]} \mathord{\left/ {\vphantom {{[3(\alpha + p_{0} \beta \gamma )]} {\{ 2e_{0}^{2} \beta (1 + \gamma )[4Q + (1 + \gamma )(\alpha + \beta p_{0} \gamma )]\} }}} \right. \kern-\nulldelimiterspace} {\{ 2e_{0}^{2} \beta (1 + \gamma )[4Q + (1 + \gamma )(\alpha + \beta p_{0} \gamma )]\} }},$ then $\tau^{ * } { = }\frac{{[ - 3 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) + 8Qv\beta e_{0}^{2} (1 + \gamma )}}{{e_{0} \beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )}}.$

• (2) If $\tau^{ * } > {{(\alpha + \beta p_{0} \gamma )} \mathord{\left/ {\vphantom {{(\alpha + \beta p_{0} \gamma )} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}},$ then $\tau^{ * } { = }{{(\alpha + \beta p_{0} \gamma )} \mathord{\left/ {\vphantom {{(\alpha + \beta p_{0} \gamma )} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}}.$

We define $v_{3} ,v_{3}^{^{\prime}}$, and $v_{3}^{^{\prime}} = v_{2} < v_{3} ,$

$$ v_{3} {=} {{(\alpha {+} p_{0} \beta \gamma )} \mathord{\left/ {\vphantom {{(\alpha {+} p_{0} \beta \gamma )} {[2e{}_{0}^{2} Q\beta (1 {+} \gamma )]}}} \right. \kern-\nulldelimiterspace} {[2e{}_{0}^{2} Q\beta (1 {+} \gamma )]}},v_{3}^{^{\prime}} {=} v_{2} {=} {{[3(\alpha {+} p_{0} \beta \gamma )]} \mathord{\left/ {\vphantom {{[3(\alpha + p_{0} \beta \gamma )]} {\{ 2e_{0}^{2} \beta (1 + \gamma )[4Q + (1 + \gamma )(\alpha + \beta p_{0} \gamma )]\} }}} \right. \kern-\nulldelimiterspace} {\{ 2e_{0}^{2} \beta (1 + \gamma )[4Q + (1 + \gamma )(\alpha + \beta p_{0} \gamma )]\} }}. $$

Then we can easily solve for the optimal tax rate $\tau_{R}^{ * }$, total carbon emission $E_{R}^{tax}$, and social welfare $W_{R}^{tax}$ under the different environmental damage conditions as follows:

• (i) If $0 < v < v_{3}^{^{\prime}} { = }v_{2} ,$ then $\tau_{R}^{ * } { = }0,$

  $$ W_{\tau }^{ * } = \frac{{32Q(p_{0} {-} e_{0}^{2} Qv)\beta {-} (\alpha {+} p_{0} \beta \gamma )\{ - 7\alpha {+} 2\alpha e_{0}^{2} v\beta (1 {+} \gamma )^{2} {-} 7\beta p_{0} \gamma {+} 2e_{0}^{2} v\beta^{2} (1 {+} \gamma )[8Q {+} p_{0} \beta \gamma (1 + \gamma )]\} }}{32\beta }, $$
  $$ E_{R}^{tax} = {{\{ e_{0} [4Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } \mathord{\left/ {\vphantom {{\{ e_{0} [4Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } 4}} \right. \kern-\nulldelimiterspace} 4},\quad \pi_{m}^{tax} = {{\{ e_{0} [4Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } \mathord{\left/ {\vphantom {{\{ e_{0} [4Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } 4}} \right. \kern-\nulldelimiterspace} 4}. $$

• (ii) If $v_{2} < v < v_{3} ,$ then

  $$ \tau_{R}^{ * } { = }{{\{ [ - 3 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) + 8Qv\beta e_{0}^{2} (1 + \gamma )\} } \mathord{\left/ {\vphantom {{\{ [ - 3 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) + 8Qv\beta e_{0}^{2} (1 + \gamma )\} } {\{ e_{0} \beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )\} }}} \right. \kern-\nulldelimiterspace} {\{ e_{0} \beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )\} }}, $$
  $$ W_{R}^{tax} = \frac{{\alpha^{2} + 2Q(p_{0} - e_{0}^{2} Qv)\beta + 2\alpha \beta [p_{0} \gamma - 2e_{0}^{2} (1 + \gamma )vQ] + \beta^{2} p_{0} [p_{0} \gamma^{2} + 4e_{0}^{2} Qv(1 + \gamma )]}}{{2\beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ]}}, $$
  $$ E_{R}^{tax} = {{\{ e_{0} [Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } \mathord{\left/ {\vphantom {{\{ e_{0} [Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}, $$
  $$ \begin{aligned}\pi_{R}^{tax} & = \frac{{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\}^{2} }}{{\beta [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} }} \\ & \quad + Q\{ p_{0} - \frac{{\alpha [ - 3 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] - 3\beta p_{0} \gamma + 2e_{0}^{2} \beta v(1 + \gamma )[4Q + p_{0} \beta \gamma (1 + \gamma )]}}{{\beta (1 + \gamma )[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}\}.\end{aligned} $$

If $v > v_{3} ,$ then $\tau_{R}^{ * } { = }{{(\alpha + \beta p_{0} \gamma )} \mathord{\left/ {\vphantom {{(\alpha + \beta p_{0} \gamma )} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}},W_{R}^{tax} = p_{0} Q - ve_{0}^{2} Q^{2} ,E_{R}^{tax} = e_{0} Q,$$\pi_{R}^{tax} = {{[Q( - \alpha + p_{0} \beta )]} \mathord{\left/ {\vphantom {{[Q( - \alpha + p_{0} \beta )]} {[\beta (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta (1 + \gamma )]}}.$

Proof of Theorem 5

We have ${{\partial \pi_{m}^{cap} } \mathord{\left/ {\vphantom {{\partial \pi_{m}^{cap} } {\partial q}}} \right. \kern-\nulldelimiterspace} {\partial q}} = - {{[2(1 - \varphi )]} \mathord{\left/ {\vphantom {{[2(1 - \varphi )]} \beta }} \right. \kern-\nulldelimiterspace} \beta }q + \alpha {{(1 - \varphi )} \mathord{\left/ {\vphantom {{(1 - \varphi )} \beta }} \right. \kern-\nulldelimiterspace} \beta } + p_{0} \gamma - e_{0} (\lambda - bC)(1 + \gamma ).$

The optimization problem can be seen to involve only the decision of the production quantity. We know that ${{\partial^{2} \pi_{m}^{cap} } \mathord{\left/ {\vphantom {{\partial^{2} \pi_{m}^{cap} } {\partial q^{2} }}} \right. \kern-\nulldelimiterspace} {\partial q^{2} }} = {{[ - 2(1 - \varphi )]} \mathord{\left/ {\vphantom {{[ - 2(1 - \varphi )]} \beta }} \right. \kern-\nulldelimiterspace} \beta } < 0,$ so we can solve for the optimal quantity $q^{ * } = {{\{ \alpha + \beta [p_{0} \gamma - e_{0} (\lambda - bC)(1 + \gamma )]\} } \mathord{\left/ {\vphantom {{\{ \alpha + \beta [p_{0} \gamma - e_{0} (\lambda - bC)(1 + \gamma )]\} } {[2(1 - \varphi )]}}} \right. \kern-\nulldelimiterspace} {[2(1 - \varphi )]}},$ which gives us the optimal production decisions stated in Theorem 5.

Proof of Corollary 2

We have ${{\partial \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial \pi_{m}^{cap * } } {\partial C}}} \right. \kern-\nulldelimiterspace} {\partial C}} = be_{0} Q - 2bC + \lambda + ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2})be_{0} (1 + \gamma )\{ \alpha (1 - \varphi ) + \beta [p_{0} \gamma - e_{0} (1 + \gamma )(\lambda - bC)]\} ,$

${{\partial^{2} \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial^{2} \pi_{m}^{cap * } } {\partial C^{2} }}} \right. \kern-\nulldelimiterspace} {\partial C^{2} }} = {{[ - 4b(1 - \varphi ) + b^{2} e_{0}^{2} \beta (1 + \gamma )^{2} ]} \mathord{\left/ {\vphantom {{[ - 4b(1 - \varphi ) + b^{2} e_{0}^{2} \beta (1 + \gamma )^{2} ]} {[2(1 - \varphi )]}}} \right. \kern-\nulldelimiterspace} {[2(1 - \varphi )]}}.$ Set the first partial derivative of $\pi_{m}^{cap * }$ with respect to $C$ equal to zero, and define $C_{M}^{\gamma }$ as the stationary point. That is

$$ C_{M}^{\gamma } = \frac{{be_{0} \{ (1 + \gamma )[ - p_{0} \beta \gamma + e_{0} \beta (1 + \gamma )\lambda + \alpha ( - 1 + \varphi )] + 2Q( - 1 + \varphi )\} + 2\lambda ( - 1 + \varphi )}}{{b[ - 4(1 - \varphi ) + be_{0}^{2} \beta (1 + \gamma )^{2} ]}}. $$

• (i) When $0 \le \varphi {{ \le [4 - be_{0}^{2} \beta (1 + \gamma )^{2} ]} \mathord{\left/ {\vphantom {{ \le [4 - be_{0}^{2} \beta (1 + \gamma )^{2} ]} 4}} \right. \kern-\nulldelimiterspace} 4},$ we have ${{\partial^{2} \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial^{2} \pi_{m}^{cap * } } {\partial C^{2} }}} \right. \kern-\nulldelimiterspace} {\partial C^{2} }} < 0.$ Therefore, if $0 \le C \le C_{M}^{\gamma } ,$ then ${{\partial \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial \pi_{m}^{cap * } } {\partial C}}} \right. \kern-\nulldelimiterspace} {\partial C}} > 0;$ if $C_{M}^{\gamma } < C \le {\lambda \mathord{\left/ {\vphantom {\lambda b}} \right. \kern-\nulldelimiterspace} b},$ then ${{\partial \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial \pi_{m}^{cap * } } {\partial C}}} \right. \kern-\nulldelimiterspace} {\partial C}} < 0.$

• (ii) When ${{[4 - be_{0}^{2} \beta (1 + \gamma )^{2} ]} \mathord{\left/ {\vphantom {{[4 - be_{0}^{2} \beta (1 + \gamma )^{2} ]} 4}} \right. \kern-\nulldelimiterspace} 4} \le \varphi \le 1,$ we have ${{\partial^{2} \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial^{2} \pi_{m}^{cap * } } {\partial C^{2} }}} \right. \kern-\nulldelimiterspace} {\partial C^{2} }} > 0.$ Since we assume that the manufacturer has a positive demand and profit, we must have ${{\partial \pi_{m}^{cap * } } \mathord{\left/ {\vphantom {{\partial \pi_{m}^{cap * } } {\partial C}}} \right. \kern-\nulldelimiterspace} {\partial C}} > 0.$

Proof of Theorem 6

Following the process used in proving Theorem 2, we get the result shown in Theorem 6.

We define $v_{1} ,v_{1}^{^{\prime}}$, and $v_{1}^{^{\prime}} > v_{1} ,$

$$ \begin{aligned} v_{1} = & \frac{{\alpha (1 - \varphi ) + p_{0} \beta \gamma (1 - 2\varphi )}}{{2e_{0}^{2} \beta (1 + \gamma )\{ 2Q(1 - \varphi ) + (1 + \gamma )[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]\} }}, \\ v_{1}^{^{\prime}} = & \frac{{\alpha (1 - \varphi ) + p_{0} \beta \gamma (1 - 2\varphi ) + e_{0} \beta (1 + \gamma )\lambda }}{{2e_{0}^{2} \beta (1 + \gamma )\{ 2Q(1 - \varphi ) + (1 + \gamma )[\alpha (1 - \varphi ) + p_{0} \beta \gamma - e_{0} \beta (1 + \gamma )\lambda ]\} }}. \\ \end{aligned} $$

• (i) If $0 < v < v_{1} ,$ then $C_{M}^{ * } = {\lambda \mathord{\left/ {\vphantom {\lambda b}} \right. \kern-\nulldelimiterspace} b}.$

  $$ W_{M}^{cap} = \frac{\begin{gathered} [\alpha (1 {-} \varphi ) {+} p_{0} \beta \gamma ]^{2} [1 {-} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ] {+} 2[\alpha (1 {-} \varphi ) {+} p_{0} \beta \gamma ][\alpha (1 - \varphi ) + p_{0} \beta \gamma (1 - 2\varphi )] \hfill \\ - 8e_{0}^{2} v\beta \{ Q^{2} (1 - \varphi )^{2} + (1 - \varphi )Q(1 + \gamma )[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]\} + 8p_{0} \beta Q(1 - \varphi )^{2} \hfill \\ \end{gathered} }{{8\beta (1 - \varphi )^{2} }}, $$
  $$ \begin{aligned} E_{M}^{cap} & = {{[2e_{0} Q( - 1 + \varphi ) - e_{0} (1 + \gamma )(\alpha + p_{0} \beta \gamma - \alpha \varphi )]} \mathord{\left/ {\vphantom {{[2e_{0} Q( - 1 + \varphi ) - e_{0} (1 + \gamma )(\alpha + p_{0} \beta \gamma - \alpha \varphi )]} {[2( - 1 + \varphi )]}}} \right. \kern-\nulldelimiterspace} {[2( - 1 + \varphi )]}},\\ \pi_{m}^{cap} & = p_{0} Q + {{[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]^{2} } \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]^{2} } {[4\beta (1 - \varphi )]}}} \right. \kern-\nulldelimiterspace} {[4\beta (1 - \varphi )]}}.\end{aligned} $$

• (ii) If $v_{1} < v < v_{1}^{^{\prime}} ,$ then

  $$ C_{M}^{ * } = \frac{{\alpha (1 {-} \varphi )[1 {-} 2v\beta e_{0}^{2} (1 {+} \gamma )^{2} ] {+} p_{0} \beta \gamma [(1 {-} 2\varphi ) {-} 2v\beta e_{0}^{2} (1 {+} \gamma )^{2} ] {+} \beta \lambda e_{0} (1 {+} \gamma )[1 {+} 2e_{0}^{2} (1 {+} \gamma )^{2} v\beta ] {-} 4Qv\beta e_{0}^{2} (1 + \gamma )(1 - \varphi )}}{{e_{0} b\beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )}}, $$
  $$ W_{M}^{cap} = \frac{{\alpha^{2} + 2Q(p_{0} - e_{0}^{2} Qv)\beta + 2\alpha \beta [p_{0} \gamma - 2e_{0}^{2} (1 + \gamma )vQ] + \beta^{2} p_{0} [p_{0} \gamma^{2} + 4e_{0}^{2} Qv(1 + \gamma )]}}{{2\beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ]}}, $$
  $$ E_{M}^{cap} = {{\{ e_{0} [Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } \mathord{\left/ {\vphantom {{\{ e_{0} [Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}, $$
  $$ \begin{aligned} \pi_{m}^{cap} = & p_{0} Q - \frac{{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\}^{2} ( - 1 + \varphi )}}{{\beta [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} }} \\ & + \frac{\begin{gathered} \{ - \alpha [ - 1 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ]( - 1 {+} \varphi ) + 2\beta e_{0}^{2} v(1 + \gamma )[p_{0} \beta \gamma (1 + \gamma ) - 2Q( - 1 + \varphi )] + \beta p_{0} \gamma ( - 1 + 2\varphi )\} \hfill \\ \alpha [ - 1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]( - 1 + \varphi ) - \beta \{ be_{0}^{2} Q(1 + \gamma )[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] \hfill \\ - e_{0} (1 {+} \gamma )\lambda {-} 2e_{0}^{3} v\beta (1 {+} \gamma )^{3} \lambda {+} 2e_{0}^{2} v(1 {+} \gamma )[p_{0} \beta \gamma (1 {+} \gamma ) {-} 2Q( - 1 {+} \varphi )] {+} p_{0} \gamma ( - 1 {+} 2\varphi )\} \hfill \\ \end{gathered} }{{be_{0}^{2} \beta^{2} (1 + \gamma )^{2} [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} }}, \\ \end{aligned} $$

• (iii) If $v > v_{1}^{^{\prime}} ,$ then $C_{M}^{ * } = 0,$

  $$ \begin{aligned} W_{M}^{cap} & = p_{0} Q + \frac{1}{8\beta }\{ \alpha + \frac{\beta }{(1 - \varphi )}[p_{0} \gamma - e_{0} \lambda (1 + \gamma )]\}^{2}\\ & \quad + \frac{1}{4}\{ \alpha + \frac{\beta }{(1 - \varphi )}[p_{0} \gamma - e_{0} \lambda (1 + \gamma )]\} [{\alpha \mathord{\left/ {\vphantom {\alpha \beta }} \right. \kern-\nulldelimiterspace} \beta } + \frac{{(1 - 2\varphi )p_{0} \gamma }}{(1 - \varphi )} + \frac{{\lambda e_{0} (1 + \gamma )}}{(1 - \varphi )}] \hfill\\ & \quad - ve_{0}^{2} \{ Q + \frac{\alpha }{2}(1 + \gamma ) + (1 + \gamma )\frac{\beta }{2(1 - \varphi )}[p_{0} \gamma - e_{0} \lambda (1 + \gamma )]\}^{2} , \hfill \\ \end{aligned} $$
  $$ E_{M}^{cap} = {{\{ 2e_{0} Q( - 1 + \varphi ) + e_{0} (1 + \gamma )\left[ {\alpha ( - 1 + \varphi ) - p_{0} \beta \gamma + e_{0} (1 + \gamma )\lambda } \right]\} } \mathord{\left/ {\vphantom {{\{ 2e_{0} Q( - 1 + \varphi ) + e_{0} (1 + \gamma )\left[ {\alpha ( - 1 + \varphi ) - p_{0} \beta \gamma + e_{0} (1 + \gamma )\lambda } \right]\} } {[2( - 1 + \varphi )]}}} \right. \kern-\nulldelimiterspace} {[2( - 1 + \varphi )]}}, $$
  $$ \pi_{m}^{cap} { = }p_{0} Q - e_{0} Q\lambda - {{\{ a(1 - \varphi ) + \beta [p_{0} \gamma - e_{0} (1 + \gamma )\lambda ]\}^{2} } \mathord{\left/ {\vphantom {{\{ a(1 - \varphi ) + \beta [p_{0} \gamma - e_{0} (1 + \gamma )\lambda ]\}^{2} } {[4\beta ( - 1 + \varphi )]}}} \right. \kern-\nulldelimiterspace} {[4\beta ( - 1 + \varphi )]}}. $$

Proof of Theorem 7

Theorem 7 follows from the same process used in proving Theorem 6.

Proof of Theorem 8

Following the process used to prove Theorem 4, we have

$$ W_{R}^{tax} = \int_{0}^{{q^{ * } }} {({\alpha \mathord{\left/ {\vphantom {\alpha \beta }} \right. \kern-\nulldelimiterspace} \beta } - {q \mathord{\left/ {\vphantom {q \beta }} \right. \kern-\nulldelimiterspace} \beta } - p^{ * } )dq + \pi_{m}^{tax*} } + \pi_{p}^{tax*} { + }\pi_{{\text{C}}}^{tax*} - v[e_{0} (q + Q + \gamma q)]^{2} , $$

$$ \begin{aligned} {{\partial W_{R}^{tax} } \mathord{\left/ {\vphantom {{\partial W_{R}^{tax} } {\partial \tau }}} \right. \kern-\nulldelimiterspace} {\partial \tau }} = & - {1 \mathord{\left/ {\vphantom {1 {[4( - 1 + \varphi )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[4( - 1 + \varphi )^{2} ]}}e_{0} (1 + \gamma )[e_{0} \beta (1 + \gamma ) + 2e_{0}^{3} v\beta^{2} (1 + \gamma )^{3} ]\tau \\ & - {1 \mathord{\left/ {\vphantom {1 {[4( - 1 + \varphi )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[4( - 1 + \varphi )^{2} ]}}e_{0} \left( {1 + \gamma } \right)\{ - 2e_{0}^{2} v\beta (1 + \gamma )[p_{0} \beta \gamma (1 + \gamma ) - 2Q( - 1{ + }\varphi )] \\ & + p_{0} \beta \gamma (1 - 2\varphi ) + \alpha [ - 1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]( - 1 + \varphi )\} . \\ \end{aligned} $$

We know ${{\partial^{2} W_{R}^{tax} } \mathord{\left/ {\vphantom {{\partial^{2} W_{R}^{tax} } {\partial \tau^{2} }}} \right. \kern-\nulldelimiterspace} {\partial \tau^{2} }} = - {{\{ e_{0}^{2} \beta (1 + \gamma )^{2} [1 + 2v\beta e_{0}^{2} (1 + \gamma )]\} } \mathord{\left/ {\vphantom {{\{ e_{0}^{2} \beta (1 + \gamma )^{2} [1 + 2v\beta e_{0}^{2} (1 + \gamma )]\} } {[4(1 - \varphi )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[4(1 - \varphi )^{2} ]}} < 0,$ and we set the first derivative of $W_{R}^{tax}$ with respect to $\tau$ to zero. Hence

$$\begin{aligned}\tau^{* } & = \{ \alpha (1 - \varphi )[ - 1 + 2v\beta e_{0}^{2} (1 +\gamma )^{2} ] + p_{0} \beta \gamma [ - (1 - 2\varphi ) + 2v\beta e_{0}^{2} (1 + \gamma )^{2} ]\\ &\quad + 4Qv\beta e_{0}^{2} (1 +\gamma )(1 - \varphi )\}/ \{ e_{0} \beta [1 + 2e_{0}^{2} (1 + \gamma)^{2} v\beta ](1 + \gamma )\}.\end{aligned} $$

Define $\begin{aligned} B = & - {1 \mathord{\left/ {\vphantom {1 {[4( - 1 + \varphi )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[4( - 1 + \varphi )^{2} ]}}e_{0} \left( {1 + \gamma } \right)\{ - 2e_{0}^{2} v\beta (1 + \gamma )[p_{0} \beta \gamma (1 + \gamma ) - 2Q( - 1{ + }\varphi )] \\ & + p_{0} \beta \gamma (1 - 2\varphi ) + \alpha [ - 1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]( - 1 + \varphi )\} . \\ \end{aligned}$.

If $B < 0,$ we get $v < \frac{{\alpha (1 - \varphi ) + p_{0} \beta \gamma (1 - 2\varphi )}}{{2e_{0}^{2} \beta (1 + \gamma )\{ 2Q(1 - \varphi ) + (1 + \gamma )[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]\} }},$ and then the optimal total emission is $\tau^{ * } < 0.$ Therefore, we set $\tau^{ * } = 0$ without loss of generality.

If $B > 0,$ we denote that the optimal $\tau^{ * }$ exists in the interval of $\left[ {0,{{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}}} \right]$ to maximize social welfare under carbon tax regulation.

• (1) If $\tau^{ * } < {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}},$ that is $v < {{(\alpha + p_{0} \beta \gamma )} \mathord{\left/ {\vphantom {{(\alpha + p_{0} \beta \gamma )} {[2e{}_{0}^{2} Q\beta (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[2e{}_{0}^{2} Q\beta (1 + \gamma )]}},$ then

  $$\begin{aligned}\tau^{* } &= \{ \alpha (1 - \varphi )[ - 1 + 2v\beta e_{0}^{2} (1 +\gamma )^{2} ] + p_{0} \beta \gamma [ - (1 - 2\varphi ) + 2v\beta e_{0}^{2} (1 + \gamma )^{2} ] \\ &\quad + 4Qv\beta e_{0}^{2} (1 +\gamma )(1 - \varphi )\} / \{ e_{0} \beta [1 + 2e_{0}^{2} (1 +\gamma )^{2} v\beta ](1 + \gamma )\}.\end{aligned}$$

• (2) If $\tau^{ * } > {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}},$ then $\tau^{ * } { = }{{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}}.$

We define $v_{3} ,v_{3}^{^{\prime}}$, and $v_{3}^{^{\prime}} = v_{1} < v_{3} ,$

$$ v_{3} = {{(\alpha + p_{0} \beta \gamma )} \mathord{\left/ {\vphantom {{(\alpha + p_{0} \beta \gamma )} {[2e{}_{0}^{2} Q\beta (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[2e{}_{0}^{2} Q\beta (1 + \gamma )]}},v_{3}^{^{\prime}} = v_{1} = \frac{{\alpha (1 - \varphi ) + p_{0} \beta \gamma (1 - 2\varphi )}}{{2e_{0}^{2} \beta (1 + \gamma )\{ 2Q(1 - \varphi ) + (1 + \gamma )[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]\} }}. $$

• (i) If $0 < v < v_{1} ,$ then $\tau_{M}^{ * } = 0,$

  $$ \begin{gathered} W_{M}^{tax} = \hfill \\ - \frac{\begin{gathered} \alpha^{2} [ - 3{+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ]( - 1 {+} \varphi )^{2} {-} 2\alpha \beta ( - 1 {+} \varphi )\{ 2e_{0}^{2} v\beta (1 {+} \gamma )[p_{0} \beta \gamma (1 {+} \gamma ) {-} 2Q( - 1 + \varphi )] + p_{0} \gamma ( - 3 + 2\varphi )\} \hfill \\ + \beta \{ 8e_{0}^{2} Q^{2} v( - 1 {+} \varphi )^{2} {-} 8p_{0} Q( - 1 {+} \varphi )[ - 1 {+} e_{0}^{2} v\beta \gamma (1 {+} \gamma ) {+} \varphi ] {+} p_{0}^{2} \beta \gamma^{2} [ - 3 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} {+} 4\varphi ]\} \hfill \\ \end{gathered} }{{8\beta ( - 1 + \varphi )^{2} }}, \hfill \\ \end{gathered} $$
  $$ \begin{aligned}E_{R}^{tax} & = {{\{ 2e_{0} Q( - 1 + \varphi ) - e_{0} (1 + \gamma )[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]\} } \mathord{\left/ {\vphantom {{\{ 2e_{0} Q( - 1 + \varphi ) - e_{0} (1 + \gamma )[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]\} } {[2( - 1 + \varphi )]}}} \right. \kern-\nulldelimiterspace} {[2( - 1 + \varphi )]}},\\ & \pi_{m}^{tax} = p_{0} Q - {{[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]^{2} } \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + p_{0} \beta \gamma ]^{2} } {[4\beta ( - 1 + \varphi )]}}} \right. \kern-\nulldelimiterspace} {[4\beta ( - 1 + \varphi )]}}.\end{aligned} $$

• (ii) If $v_{1} < v < v_{3} ,$

  $$ \tau_{M}^{ * } \frac{{\alpha (1 - \varphi )[ - 1 + 2v\beta e_{0}^{2} (1 + \gamma )^{2} ] + p_{0} \beta \gamma [ - (1 - 2\varphi ) + 2v\beta e_{0}^{2} (1 + \gamma )^{2} ] + 4Qv\beta e_{0}^{2} (1 + \gamma )(1 - \varphi )}}{{e_{0} \beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )}}, $$
  $$ W_{M}^{tax} = \frac{{\alpha^{2} + 2Q(p_{0} - e_{0}^{2} Qv)\beta + 2\alpha \beta [p_{0} \gamma - 2e_{0}^{2} (1 + \gamma )vQ] + \beta^{2} p_{0} [p_{0} \gamma^{2} + 4e_{0}^{2} Qv(1 + \gamma )]}}{{2\beta [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ]}}, $$
  $$ E_{M}^{tax} = {{\{ e_{0} [Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } \mathord{\left/ {\vphantom {{\{ e_{0} [Q + (1 + \gamma )(\alpha + p_{0} \beta \gamma )]\} } {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}, $$
  $$ \begin{aligned}\pi_{m}^{tax} & = - \frac{{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\}^{2} ( - 1 + \varphi )}}{{\beta [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} }}{ + }p_{0} Q \\ &\quad - Q\frac{\begin{gathered} \{ - \alpha [ - 1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]( - 1 + \varphi ) \hfill \\ + 2\beta e_{0}^{2} v(1 + \gamma )[p_{0} \beta \gamma (1 + \gamma ) - 2Q( - 1 + \varphi )] + \beta p_{0} \gamma ( - 1 + 2\varphi )\} \hfill \\ \end{gathered} }{{\beta (1 + \gamma )[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}.\end{aligned} $$

• (iii) If $v > v_{3} ,$

  $$ \begin{aligned}\tau_{M}^{ * } & = {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma ]} {[\beta e_{0} (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta e_{0} (1 + \gamma )]}},W_{M}^{tax} = p_{0} Q - ve_{0}^{2} Q^{2},\\ & E_{M}^{tax} = e_{0} Q,\pi_{m}^{tax} = {{[p_{0} Q\beta + \alpha Q( - 1 + \varphi )]} \mathord{\left/ {\vphantom {{[p_{0} Q\beta + \alpha Q( - 1 + \varphi )]} {[\beta (1 + \gamma )]}}} \right. \kern-\nulldelimiterspace} {[\beta (1 + \gamma )]}}.\end{aligned} $$

Proof of Theorem 9

First, we explain the comparison of carbon emission and social welfare under cap-and-trade regulation.

$$ W = {1 \mathord{\left/ {\vphantom {1 {(2\beta )}}} \right. \kern-\nulldelimiterspace} {(2\beta )}}q^{2} + p_{0} \left( {Q + \gamma q} \right) + q({\alpha \mathord{\left/ {\vphantom {\alpha \beta }} \right. \kern-\nulldelimiterspace} \beta } - {q \mathord{\left/ {\vphantom {q \beta }} \right. \kern-\nulldelimiterspace} \beta }) - v[e_{0} (q + Q + \gamma q)]^{2} , $$

$$ {{\partial W} \mathord{\left/ {\vphantom {{\partial W} {\partial q}}} \right. \kern-\nulldelimiterspace} {\partial q}} = - 2e_{0}^{2} v(1 + \gamma )(q + Q + q\gamma ) + {{( - q + \alpha + p_{0} \beta \gamma )} \mathord{\left/ {\vphantom {{( - q + \alpha + p_{0} \beta \gamma )} \beta }} \right. \kern-\nulldelimiterspace} \beta }, $$

We know ${{\partial^{2} W} \mathord{\left/ {\vphantom {{\partial^{2} W} {\partial q^{2} }}} \right. \kern-\nulldelimiterspace} {\partial q^{2} }} = {{[ - 1 - 2\beta e_{0}^{2} v(1 + \gamma )^{2} ]} \mathord{\left/ {\vphantom {{[ - 1 - 2\beta e_{0}^{2} v(1 + \gamma )^{2} ]} \beta }} \right. \kern-\nulldelimiterspace} \beta } < 0,$ and we set the first derivative of $W$ with respect to $q$ to zero. Thus, we get $q = {{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\} } \mathord{\left/ {\vphantom {{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\} } {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}$ as the optimal decision for social welfare.

• (i) If $0 < v < v_{1} ,$ from Theorem 2 and Theorem 6, we know that

  $q_{M}^{ * } - q_{{\text{R}}}^{ * } = {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma (1 + \varphi )]} \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + \beta p_{0} \gamma (1 + \varphi )]} {4(1 - \varphi )}}} \right. \kern-\nulldelimiterspace} {4(1 - \varphi )}} > 0,$ and $q_{{\text{R}}}^{ * } < q_{M}^{ * } < q,$, so $W_{R}^{cap} < W_{M}^{cap} , \, and \, E_{R}^{cap} < E_{M}^{cap} .$

• (ii) If $v_{1} < v < v_{2} ,$ from Theorem 2 and Theorem 6, we know that $W_{M}^{cap} > W_{R}^{cap} .$

  $$ W_{M}^{cap} {-} W_{{\text{R}}}^{cap} {=} \frac{{\{ \alpha [ - 3 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ] {-} 3\beta p_{0} \gamma + 2\beta e_{0}^{2} v(1 {+} \gamma )[4Q {+} p_{0} \beta \gamma (1 {+} \gamma )]\}^{2} }}{{32\beta [1 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ]}} > 0. $$

• (iii) If $v_{2} < v < v_{1}^{^{\prime}} ,$ from Theorem 2 and Theorem 6, we know that $q_{{\text{R}}}^{ * } { = }q{ = }q_{M}^{ * } ,$ so $W_{R}^{cap} { = }W_{M}^{cap} , \, and \, E_{R}^{cap} = E_{M}^{cap} .$

• (iv) If $v_{1}^{^{\prime}} < v < v_{2}^{^{\prime}} ,$ from Theorem 2 and Theorem 6, we know that $W_{R}^{cap} > W_{M}^{cap} , \, and \, E_{R}^{cap} < E_{M}^{cap} .$

  $$ W_{{\text{R}}}^{cap} - W_{M}^{cap} { = }\frac{\begin{gathered} \{ \beta e_{0} (1 + \gamma )\lambda + 2e_{0}^{3} v\beta^{2} (1 + \gamma )^{3} \lambda - 2\beta e_{0}^{2} v(1 + \gamma )[p_{0} \beta \gamma (1 + \gamma ) - 2Q( - 1 + \varphi )] + \beta p_{0} \gamma (1 - 2\varphi ) \hfill \\ + \alpha [ - 1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]( - 1 + \varphi )\}^{2} \hfill \\ \end{gathered} }{{8\beta [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]( - 1 + \varphi )^{2} }} > 0, $$
  $$ q_{R}^{*} {-} q_{M}^{ * } \!=\! \frac{{\beta e_{0} (1 \!+\! \gamma )\lambda {+} 2e_{0}^{3} v\beta^{2} (1 {+} \gamma )^{3} \lambda {-} 2\beta e_{0}^{2} v(1 {+} \gamma )[p_{0} \beta \gamma (1{ +} \gamma ) {-} 2Q( - 1 {+} \varphi )] {+} \beta p_{0} \gamma (1 {-} 2\varphi ) {+} \alpha [ - 1 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ]( - 1 {+} \varphi )}}{{2[1 {+} 2e_{0}^{2} (1 {+} \gamma )^{2} v\beta ](1 {-} \varphi )}} \!<\! 0, $$
  $$ q_{R}^{*} < q_{M}^{ * } , \, and \, E_{R}^{cap} < E_{M}^{cap} . $$

• (v) If $v > v_{2}^{^{\prime}} ,$ from Theorem 2 and Theorem 6, we know that

$q_{R}^{ * } - q_{M}^{ * } = {{\{ \alpha (1 - \varphi ) + \beta [p_{0} \gamma - e_{0} \lambda (1 + \gamma )](1 - \varphi )\} } \mathord{\left/ {\vphantom {{\{ \alpha (1 - \varphi ) + \beta [p_{0} \gamma - e_{0} \lambda (1 + \gamma )](1 - \varphi )\} } {[4( - 1 + \varphi )]}}} \right. \kern-\nulldelimiterspace} {[4( - 1 + \varphi )]}} < 0,$ and $q < 0,$ so $q < q_{R}^{ * } < q_{M}^{ * } {\text{, W}}_{{\text{R}}}^{cap} > W_{M}^{cap} , \, and \, E_{R}^{cap} < E_{M}^{cap} .$

Next, we explain the comparison of carbon emission and social welfare under carbon tax regulation. We know that $q = {{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\} } \mathord{\left/ {\vphantom {{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\} } {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}$ is the optimal decision for welfare.

• (i) If $0 < v_{M} < v_{1} ,$ from Theorem 4 and Theorem 8 we know that $q_{R}^{tax} - q_{M}^{tax} = - {{[\alpha (1 - \varphi ) + p_{0} \beta \gamma (1 + \varphi )]} \mathord{\left/ {\vphantom {{[\alpha (1 - \varphi ) + p_{0} \beta \gamma (1 + \varphi )]} {[4(1 - \varphi )]}}} \right. \kern-\nulldelimiterspace} {[4(1 - \varphi )]}} < 0,$ and $q_{R}^{tax} < q_{M}^{tax} < q,$ so $W_{R}^{tax} < W_{M}^{tax} , \, and \, E_{M}^{tax} > E_{R}^{tax}$.

• (ii) If $v_{1} < v_{M} < v_{2} ,$ from Theorem 4 and Theorem 8, we know that $W_{R}^{tax} < W_{M}^{tax} ,$

  $$ q_{M}^{tax} {-} q_{{\text{R}}}^{tax} {=} - \frac{{\alpha [ - 3 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ] {+} \beta \{ - 3p_{0} \gamma {+} 2e_{0}^{2} v(1 + \gamma )[4Q + p_{0} \beta \gamma (1 + \gamma )]\} }}{{4 + 8e_{0}^{2} v\beta (1 + \gamma )^{2} }} > 0. $$

  So $q_{M}^{tax} > q_{{\text{R}}}^{tax} , \, and \, E_{R}^{tax} < E_{M}^{tax} .$

• (iii) If $v_{M} > v_{2} ,$ from Theorem 4 and Theorem 8, we know that $W_{R}^{tax} { = }W_{M}^{tax} , \, and \, E_{R}^{tax} = E_{M}^{tax} .$

Combining these parts, we get the result shown in Theorem 9.

Proof of Theorem 10

First, we explain the comparisons of carbon emission, manufacturer’s profit, and social welfare using reselling mode.

• (i) If $0 < v < v_{2} ,$ from Theorem 2 and Theorem 4, we know that $E_{R}^{tax} { = }E_{R}^{cap} , \, \pi_{m}^{tax} { = }\pi_{m}^{cap} , \, W_{R}^{tax} { = }W_{R}^{cap} .$

• (ii) If $v_{2} < v < v_{2}^{^{\prime}} ,$ from Theorem 2 and Theorem 4, we know that $E_{R}^{tax} { = }E_{R}^{cap} , \, \pi_{m}^{tax} < \pi_{m}^{cap} , \, W_{R}^{tax} = W_{R}^{cap} .$

  $$ \pi_{m}^{cap} {-} \pi_{m}^{tax} { = }\frac{\begin{gathered} - \{ \alpha [ - 3 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ] {-} 3\beta p_{0} \gamma {+} 2\beta e_{0}^{2} v(1 {+} \gamma )[4Q {+} p_{0} \beta \gamma (1 {+} \gamma )]\} \hfill \\ \{ \alpha [ - 3 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] - 3\beta p_{0} \gamma + \beta e_{0} (1 + \gamma )[ - \lambda + 8e_{0} vQ + 2\beta e_{0} v(1 + \gamma )p_{0} \gamma - 2\beta ve_{0}^{2} (1 + \gamma )^{2} \lambda ]\} \hfill \\ \end{gathered} }{{\{ be_{0}^{2} \beta^{2} (1 + \gamma )^{2} [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} \} }} > 0. $$

• (iii) If $v_{2}^{^{\prime}} < v < v_{3} ,$ from Theorem 2 and Theorem 4, we know that $E_{R}^{cap} > E_{R}^{tax} , \, \pi_{m}^{tax} < \pi_{m}^{cap} , \, W_{R}^{tax} > W_{R}^{cap} .$

  $$ E_{R}^{tax} {-} E_{R}^{cap} {=} \frac{{e_{0} (1 {+} \gamma )\{ - \alpha [ - 3 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ] {+} 3\beta p_{0} \gamma {+} \lambda \beta e_{0} (1 {+} \gamma ) {+} 2\beta (1 {+} \gamma )e_{0}^{2} v[ - 4Q {+} ( - p_{0} \gamma \beta (1 {+} \gamma ) {+} \beta e_{0} (1 {+} \gamma )^{2} \lambda )]\} }}{{4 + 8e_{0}^{2} v\beta (1 + \gamma )^{2} }}. $$

  Consider the formula:$- \alpha [ - 3 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] + 3\beta p_{0} \gamma + \beta e_{0} (1 + \gamma )\{ \lambda - 8e_{0} vQ + 2e_{0} v\beta (1 + \gamma )[ - p_{0} \gamma + e_{0} (1 + \gamma )\lambda ]\} .$ This formula is the molecular of $C_{R}^{ * }$, and $E_{R}^{tax} - E_{R}^{cap} < 0.$ Accordingly,

  $$ \pi_{R}^{cap} {-} \pi_{R}^{tax} {=} \frac{\begin{gathered} \{ 8Q[2 {+} 3e_{0}^{2} v\beta (1 {+} \gamma )^{2} ] {+} 5(\alpha {+} p_{0} \beta \gamma )(1 {+} \gamma ) {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{3} (\alpha {+} p_{0} \beta \gamma ) {-} e_{0} \beta (1 {+} \gamma )^{2} [1 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ]\lambda \} \hfill \\ \{ \alpha [ - 3 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ] {-} 3\beta p_{0} \gamma {-} \lambda \beta e_{0} (1 {+} \gamma ) {+} \beta (1 {+} \gamma )8e_{0}^{2} vQ {+} 2v\beta^{2} e_{0}^{2} (1 + \gamma )^{2} [p_{0} \gamma - e_{0} (1 + \gamma )\lambda ]\} \hfill \\ \end{gathered} }{{16\beta (1 + \gamma )[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} }} $$

  In the formula:$\alpha [ - 3 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] - 3\beta p_{0} \gamma + \beta e_{0} (1 + \gamma )\{ - \lambda + 8Qe_{0} v + 2e_{0} v\beta (1 + \gamma )[p_{0} \gamma - e_{0} (1 + \gamma )\lambda ]\} .$ is the molecular of $- C_{R}^{ * }$, and $C_{R}^{ * } < 0,$

  $$ \begin{aligned} & W_{R}^{cap} {-} W_{R}^{tax} =\\ & \quad \frac{{\{ \alpha [ - 3 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] - 3\beta p_{0} \gamma + \beta e_{0} (1 + \gamma )[\lambda - 8e_{0} vQ + 2e_{0} v\beta (1 + \gamma )( - p_{0} \gamma + e_{0} \lambda { + }e_{0} \gamma \lambda )]\}^{2} }}{{32\beta [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}} > 0.\end{aligned} $$

• (iv) If $v > v_{3}$,from Theorem 2 and Theorem 4, we know that $E_{R}^{tax} < E_{R}^{cap} , \, \pi_{m}^{tax} < \pi_{m}^{cap} , \, W_{R}^{tax} > W_{R}^{cap} .$

  $$ E_{R}^{tax} - E_{R}^{cap} = {{\{ e_{0} (1 + \gamma )[ - \alpha - p_{0} \beta \gamma + e_{0} \beta (1 + \gamma )\lambda ]\} } \mathord{\left/ {\vphantom {{\{ e_{0} (1 + \gamma )[ - \alpha - p_{0} \beta \gamma + e_{0} \beta (1 + \gamma )\lambda ]\} } 4}} \right. \kern-\nulldelimiterspace} 4} < 0, $$
  $$ \pi_{R}^{cap} - \pi_{R}^{tax} = \frac{{[\alpha + p_{0} \beta \gamma - e_{0} \beta (1 + \gamma )\lambda ]\{ 16Q + (1 + \gamma )[\alpha + p_{0} \beta \gamma - e_{0} \beta (1 + \gamma )\lambda ]\} }}{16\beta (1 + \gamma )} > 0, $$
  $$\begin{aligned}& W_{R}^{cap} - W_{R}^{tax} =\\ &\quad \frac{\begin{gathered} {[\alpha + p_{0} \beta \gamma - e_{0} \beta (1 + \gamma )\lambda ]} \hfill \\ \{ \alpha [ - 7 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ] - 7\beta p_{0} \gamma + \lambda \beta e_{0} (1 + \gamma ) - 16Qe_{0}^{2} v(1 + \gamma ) + 2e_{0}^{2} v\beta^{2} (1 + \gamma )^{2} [ - p_{0} \gamma + e_{0} (1 + \gamma )\lambda ]\} \hfill \\ \end{gathered} }{32\beta }.\end{aligned} $$

If $W_{R}^{cap} - W_{R}^{tax} > 0,$ we find $v > \frac{{7(\alpha + p_{0} \beta \gamma ) + e_{0} \beta (1 + \gamma )\lambda }}{{2e_{0}^{2} \beta (1 + \gamma )\{ 8Q + (1 + \gamma )[\alpha + \beta p_{0} \gamma - e_{0} \beta (1 + \gamma )\lambda ]\} }},$ and $v_{3} > \frac{{7(\alpha + p_{0} \beta \gamma ) + e_{0} \beta (1 + \gamma )\lambda }}{{2e_{0}^{2} \beta (1 + \gamma )\{ 8Q + (1 + \gamma )[\alpha + \beta p_{0} \gamma - e_{0} \beta (1 + \gamma )\lambda ]\} }},$ so $W_{R}^{tax} > W_{R}^{cap} .$

Next, we explain the comparisons of carbon emission, manufacturer’s profit, and social welfare using marketplace mode.

• (i) If $0 < v < v_{1} ,$ we know that $W = {1 \mathord{\left/ {\vphantom {1 {(2\beta )}}} \right. \kern-\nulldelimiterspace} {(2\beta )}}q^{2} + p_{0} \left( {Q + \gamma q} \right) + q({\alpha \mathord{\left/ {\vphantom {\alpha \beta }} \right. \kern-\nulldelimiterspace} \beta } - {q \mathord{\left/ {\vphantom {q \beta }} \right. \kern-\nulldelimiterspace} \beta }) - v[e_{0} (q + Q + \gamma q)]^{2} ,$

  $E_{M}^{tax} = E_{M}^{cap} = e_{0} (q + Q + q\gamma ).$ From Theorem 6 and Theorem 8, we know that $q^{tax*} = q^{cap*} = {{\{ \alpha (1 - \varphi ) + \beta [p_{0} \gamma - e_{0} \lambda (1 + \gamma )]\} } \mathord{\left/ {\vphantom {{\{ \alpha (1 - \varphi ) + \beta [p_{0} \gamma - e_{0} \lambda (1 + \gamma )]\} } {[2(1 - \varphi )]}}} \right. \kern-\nulldelimiterspace} {[2(1 - \varphi )]}}$. So $E_{M}^{tax} { = }E_{M}^{cap} , \, \pi_{m}^{tax} { = }\pi_{m}^{cap} , \, W_{M}^{tax} { = }W_{M}^{cap} .$

• (ii) If $v_{1} < v < v_{1}^{^{\prime}} ,$ from Theorem 6 and Theorem 8, we know that

  $q^{tax*} = q^{cap*} = {{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\} } \mathord{\left/ {\vphantom {{\{ \alpha + \beta [p_{0} \gamma - 2e_{0}^{2} Qv(1 + \gamma )]\} } {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}} \right. \kern-\nulldelimiterspace} {[1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]}}.$ So $E_{M}^{tax} { = }E_{M}^{cap} , \, \pi_{m}^{tax} < \pi_{m}^{cap} , \, W_{M}^{tax} = W_{M}^{cap} .$

  $$ \pi_{m}^{cap} {-} \pi_{m}^{tax} { = } - \frac{\begin{gathered} \{ \beta e_{0} (1 {+} \gamma )\lambda {+} 2e_{0}^{3} v\beta^{2} (1 {+} \gamma )^{3} \lambda {-} 2\beta e_{0}^{2} v(1 {+} \gamma )(p_{0} \beta \gamma (1 + \gamma ) - 2Q( - 1 + \varphi )) {+} \beta p_{0} \gamma (1 {-} 2\varphi ) \hfill \\ + \alpha [ - 1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]( - 1 + \varphi )\} \hfill \\ \{ \alpha [ - 1 {+} 2e_{0}^{2} v\beta (1 {+} \gamma )^{2} ]( - 1 {+} \varphi ) {-} 2\beta e_{0}^{2} v(1 {+} \gamma )[\beta p_{0} \gamma (1 {+} \gamma ) {-} 2Q( - 1 {+} \varphi )] {+} \beta p_{0} \gamma ( - 1 {+} 2\varphi )\} \hfill \\ \end{gathered} }{{be_{0}^{2} \beta^{2} (1 + \gamma )^{2} [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]^{2} }} > 0. $$

• (iii) If $v_{1}^{^{\prime}} < v < v_{3} ,$ from Theorem 6 and Theorem 8, we know that $E_{M}^{tax} < E_{M}^{cap} , \, \pi_{m}^{tax} < \pi_{m}^{cap} , \, W_{M}^{tax} > W_{M}^{cap} .$

  $$ \begin{aligned} & E_{M}^{tax} - E_{M}^{cap} =\\ &\quad \frac{{ - \left( {e_{0} \left( {1 {+} \gamma } \right)\left( {\left( {3 {-} 2e_{0}^{2} \left( {1 {+} \gamma } \right)^{2} v\beta } \right)\left( {\alpha {+} p_{0} \beta \gamma } \right) - 8Qv\beta e_{0}^{2} \left( {1 + \gamma } \right) + \beta e_{0} \lambda \left( {1 + 2e_{0}^{2} \left( {1 + \gamma } \right)^{2} v\beta } \right)\left( {1 + \gamma } \right)} \right)} \right)}}{{\left( {2\left( {1 + 2e_{0}^{2} v\beta \left( {1 + \gamma } \right)^{2} } \right)\left( { - 1 + \varphi } \right)} \right)}} < 0\end{aligned} $$

  In the formula:$[3 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) - 8Qv\beta e_{0}^{2} (1 + \gamma ) + \beta e_{0} \lambda [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )$ is the molecular of $C_{R}^{ * }$, and $C_{R}^{ * } < 0$

  $$ \begin{aligned}& \pi_{m}^{cap} - \pi_{m}^{tax} = \\ &\quad \frac{{\left( \begin{gathered} \left( \begin{gathered} \left( {1 {+} \gamma } \right)\left( {\beta \left( {e_{0} \left( {1 {+} \gamma } \right)\left( {1 {+} 2e_{0}^{2} v\beta \left( {1 {+} \gamma } \right)^{2} } \right)\lambda {-} p_{0} \gamma \left( {3 {+} 2e_{0}^{2} v\beta \left( {1 {+} \gamma } \right)^{2} - 2\varphi } \right)} \right) + \alpha \left( {3 + 2e_{0}^{2} v\beta \left( {1 + \gamma } \right)^{2} } \right)\left( { - 1 + \varphi } \right)} \right) \hfill \\ + 4Q\left( {1 + e_{0}^{2} v\beta \left( {1 + \gamma } \right)^{2} } \right)\left( { - 1 + \varphi } \right) \hfill \\ \end{gathered} \right) \hfill \\ \left( {\left( {3 - 2e_{0}^{2} \left( {1 + \gamma } \right)^{2} v\beta } \right)\left( {\alpha + p_{0} \beta \gamma } \right) - 8Qv\beta e_{0}^{2} \left( {1 + \gamma } \right) + \beta e_{0} \lambda \left( {1 + 2e_{0}^{2} \left( {1 + \gamma } \right)^{2} v\beta } \right)\left( {1 + \gamma } \right)} \right) \hfill \\ \end{gathered} \right)}}{{\left( {4\beta \left( {1{ + }\gamma } \right)\left( {1 + 2e_{0}^{2} v\beta \left( {1 + \gamma } \right)^{2} } \right)^{2} \left( { - 1 + \varphi } \right)} \right)}} > 0 \end{aligned}$$

  In the formula:$[3 - 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](\alpha + p_{0} \beta \gamma ) - 8Qv\beta e_{0}^{2} (1 + \gamma ) + \beta e_{0} \lambda [1 + 2e_{0}^{2} (1 + \gamma )^{2} v\beta ](1 + \gamma )$ is the molecular of $C_{R}^{ * }$, and $C_{R}^{ * } < 0$

  $$ W_{M}^{cap} - W_{M}^{tax} = \frac{{\left\{ \begin{gathered} \alpha [ - 1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ( - 1 + \varphi )] + \beta e_{0} (1 + \gamma )\lambda + 2e_{0}^{3} v\beta^{2} (1 + \gamma )^{3} \lambda - 2\beta e_{0}^{2} v(1 + \gamma ) \hfill \\ [p_{0} \beta \gamma (1 + \gamma ) - 2Q( - 1 + \varphi )] + \beta p_{0} \gamma (1 - 2\varphi ) \hfill \\ \end{gathered} \right\}^{2} }}{{8\beta [1 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ]( - 1 + \varphi )^{2} }} > 0, $$

• (iv) If $v > v_{3} ,$ from Theorem 6 and Theorem 8, we know that $E_{M}^{tax} < E_{M}^{cap} , \, \pi_{m}^{tax} < \pi_{m}^{cap} , \, W_{M}^{tax} > W_{M}^{cap} .$

  $$ E_{M}^{tax} - E_{M}^{cap} = \frac{{e_{0} (1 + \gamma )[\alpha (1 - \varphi ) + p_{0} \beta \gamma - e_{0} \beta (1 + \gamma )\lambda ]}}{2( - 1 + \varphi )} < 0, $$
  $$ \begin{aligned} & \pi_{m}^{cap} - \pi_{m}^{tax} =\\ & \quad \frac{{(1 + \gamma )[\alpha ( - 1 + \varphi ) - p_{0} \beta \gamma + e_{0} (1 + \gamma )\lambda ] + 4Q( - 1 + \varphi )[\alpha ( - 1 + \varphi ) - p_{0} \beta \gamma + e_{0} (1 + \gamma )\lambda ]}}{{4\beta (1{ + }\gamma )(1 - \varphi )}} > 0,\end{aligned} $$
  $$ \begin{aligned} & W_{M}^{cap} - W_{M}^{tax} = \\ &\quad \frac{\begin{gathered} {[\alpha ( - 1 + \varphi ) - p_{0} \beta \gamma + e_{0} \beta (1 + \gamma )\lambda ]} \hfill \\ \{ \alpha [ - 3 + 2e_{0}^{2} v\beta (1 + \gamma )^{2} ( - 1 + \varphi )] + \beta e_{0} (1 + \gamma )\lambda + 2e_{0}^{3} v\beta^{2} (1 + \gamma )^{3} \lambda - 2\beta e_{0}^{2} v(1 + \gamma )[p_{0} \beta \gamma (1 + \gamma ) - 2Q( - 1 + \varphi )]\} \hfill \\ \end{gathered} }{{8\beta ( - 1 + \varphi )^{2} }},\end{aligned} $$

If $W_{M}^{cap} - W_{M}^{tax} > 0,$ we find.

$v > \frac{{3\alpha ( - 1 + \varphi ) + \beta [ - e_{0} \beta (1 + \gamma )\lambda + p_{0} \beta \gamma ( - 3 + 4\varphi )]}}{{2e_{0}^{2} \beta (1 + \gamma )\{ 4Q( - 1 + \varphi ) + (1 + \gamma )[\alpha ( - 1 + \varphi ) - \beta p_{0} \gamma + e_{0} \beta (1 + \gamma )\lambda ]\} }},$ and $v_{3} > \frac{{3\alpha ( - 1 + \varphi ) + \beta [ - e_{0} \beta (1 + \gamma )\lambda + p_{0} \beta \gamma ( - 3 + 4\varphi )]}}{{2e_{0}^{2} \beta (1 + \gamma )\{ 4Q( - 1 + \varphi ) + (1 + \gamma )[\alpha ( - 1 + \varphi ) - \beta p_{0} \gamma + e_{0} \beta (1 + \gamma )\lambda ]\} }},$ so $W_{M}^{cap} > W_{M}^{tax} .$ Combining the above information, we get the result stated in Theorem 10.