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How Carbon Tax Policy Affects the Carbon Emissions of Manufacturers with Green Technology Spillovers?

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Abstract

This study examines the effects of carbon tax policies on carbon emissions in the context of green technology spillovers among manufacturers. We develop a differential game model that incorporates both horizontal and vertical green technology spillovers, analyzing scenarios with and without a carbon tax policy. Our equilibrium analysis demonstrates that a carbon tax policy effectively promotes carbon reduction in manufacturers experiencing green technology spillovers by increasing investment in green technology. The policy also mitigates the negative effect of free-riding among competing manufacturers (horizontal spillovers) while enhancing the positive effect of free-riding among non-competitors (vertical spillovers). Additionally, we find that raising carbon tax or fostering vertical spillovers can enhance the profitability of supply chain participants by expanding the market scale. Conversely, elevating consumer environment awareness yields greater benefits for those who innovate (the spiller) rather than those who adopt (the recipient). Our findings offer novel managerial insights for manufacturers navigating green technology spillovers in a landscape shaped by carbon tax policy.

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

We would like to express our deepest gratitude to the scholars who provided invaluable feedback and insights during the review of this paper. Their expertise and thoughtful comments significantly contributed to the quality and rigor of our work.

Funding

This work is supported by the Major Project of Philosophy and Social Science Research in Colleges and Universities of Jiangsu Province (Grant Number, 2021SJZDA131).

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  1. School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    Qiyao Liu & Xiaodong Zhu

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Qiyao Liu: investigation, writing original draft, software, and visualization. Xiaodong Zhu: conceptualization, supervision, methodology, writing, review, and editing. All authors reviewed the manuscript.

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Correspondence to Xiaodong Zhu.

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Appendix

Appendix

1.1 Proof of Theorem 1

According to formulas (6) to (7), the optimal profit value functions for the manufacturers and the supplier after time \(t\) are

$$\left\{\begin{array}{c}{{J}_{j}}^{N*}\left({e}_{j},{e}_{\overline{j} }\right)={e}^{-\rho t}\underset{{u}_{j}\left(t\right)\ge 0}{{\text{max}}}{\int }_{t}^{\infty }{e}^{-\rho \left(\xi -t\right)}\left[{\pi }_{j}{q}_{j}-\frac{{\mu }_{j}{{u}_{j}}^{2}\left(t\right)}{2}\right]dt,\\ {{J}_{S}}^{N*}\left({e}_{S}\right)={e}^{-\rho t}\underset{{u}_{S}\left(t\right)\ge 0}{{\text{max}}}{\int }_{t}^{\infty }{e}^{-\rho \left(\xi -t\right)}\left[{\pi }_{S}Q-\frac{{\mu }_{S}{{u}_{S}}^{2}\left(t\right)}{2}\right]dt,\end{array}\right.$$
(A.1)

where \(j\in \{L,F\}\).

Further, assume that

$$\left\{\begin{array}{c}{{V}_{j}}^{N}\left({e}_{j},{e}_{\overline{j} }\right)=\underset{{u}_{j}(t)\ge 0}{{\text{max}}}{\int }_{t}^{\infty }{e}^{-\rho (\xi -t)}[{\pi }_{j}{q}_{j}-\frac{{\mu }_{j}{{u}_{j}}^{2}(t)}{2}]d\xi ,\\ {{V}_{S}}^{N}\left({e}_{S}\right)=\underset{{u}_{S}\left(t\right)\ge 0}{{\text{max}}}{\int }_{t}^{\infty }{e}^{-\rho \left(\xi -t\right)}\left[{\pi }_{S}Q-\frac{{\mu }_{S}{{u}_{S}}^{2}\left(t\right)}{2}\right]d\xi ,\end{array}\right.$$
(A.2)

where \(j\in \{L,F\}\).

Then, the optimal profit value functions for the manufacturers and supplier after time \(t\) transforms into

$$\left\{\begin{array}{c}{{J}_{j}}^{N*}\left({e}_{j},{e}_{\overline{j} }\right)={e}^{-\rho t}{{V}_{j}}^{N}\left({e}_{j},{e}_{\overline{j} }\right),\\ {{J}_{S}}^{N*}\left({e}_{S}\right)={e}^{-\rho t}{{V}_{S}}^{N}\left({e}_{S}\right),\end{array}\right.$$
(A.3)

where \(j\in \{L,F\}\).

According to optimal control theory, the problem satisfies the following Hamiltonian-Jacobi-Bellman (HJB) equations:

$$\left\{\begin{array}{c}\rho {{V}_{j}}^{N}\left({e}_{j},{e}_{\overline{j} }\right)=\underset{{u}_{j}(t)>0}{{\text{max}}}[{\pi }_{j}{q}_{j}-\frac{{\mu }_{j}{{u}_{j}}^{2}(t)}{2}],\\ \rho {{V}_{S}}^{N}\left({e}_{S}\right)=\underset{{u}_{S}\left(t\right)>0}{{\text{max}}}\left[{\pi }_{S}Q-\frac{{\mu }_{S}{{u}_{S}}^{2}\left(t\right)}{2}\right],\end{array}\right.$$
(A.4)

where \(j\in \{L,F\}\).

It is easy to prove that formulas (A.4) are concave functions with respect to \({u}_{j}\left(t\right)\), \(j\in \{L,F,S\}\). The derivative of the solutions is equal to 0 to obtain

$$\left\{\begin{array}{c}{u}_{L}=\frac{{{V}_{LL}}^{N`}+{\alpha }_{1}{{V}_{LF}}^{N`}}{{\mu }_{L}},\\ {u}_{F}=\frac{{{V}_{FF}}^{N`}}{{\mu }_{F}},\\ {u}_{S}=\frac{{{V}_{SS}}^{N`}}{{\mu }_{S}},\end{array}\right.$$
(A.5)

where \({{V}_{ij}}^{N`}=\frac{\partial {V}_{i}}{\partial {e}_{j}},i,j\in \{L,F,S\}\).

Substitute formulas (A.5) into formulas (A.4) to obtain

$$\left\{\begin{array}{c}\rho {{V}_{L}}^{N}\left({e}_{L},{e}_{F}\right)={e}_{L}\left(s{\pi }_{L}-\delta {{V}_{LL}}^{N`}\right)-{e}_{F}\left(s{\pi }_{L}+\delta {{V}_{LF}}^{N`}\right)+\lambda Q{\pi }_{L}+\\ \frac{{({{V}_{LL}}^{N`}+{\alpha }_{1}{{V}_{LF}}^{N`})}^{2}}{2{\mu }_{L}}+\frac{{{V}_{FF}}^{N`}{{V}_{LF}}^{N`}}{{\mu }_{F}}+\frac{{{V}_{SS}}^{N`}({{V}_{LL}}^{N`}{\beta }_{1}+{{V}_{LF}}^{N`}{\beta }_{2})}{{\mu }_{S}},\\ \rho {{V}_{F}}^{N}\left({e}_{L},{e}_{F}\right)=[{e}_{F}\left(s{\pi }_{F}-\delta {{V}_{FF}}^{N`}\right)-{e}_{L}\left(s{\pi }_{F}+\delta {{V}_{FL}}^{N`}\right)+(1-\lambda )Q{\pi }_{F}+\\ \frac{({{V}_{FL}}^{N`}+{\alpha }_{1}{{V}_{FF}}^{N`})({V}_{LL}+{V}_{LF}{\alpha }_{1})}{{\mu }_{L}}+\frac{{{{V}_{FF}}^{N`}}^{2}}{2{\mu }_{F}}+\frac{{V}_{{\text{SS}}}({\beta }_{1}{{V}_{FL}}^{N`}+{\beta }_{2}{{V}_{FF}}^{N`})}{{\mu }_{S}},\\ \rho {{V}_{S}}^{N}\left({e}_{S}\right)=Q{\pi }_{S}+\frac{1}{2}{{V}_{SS}}^{N`}\left[\frac{2\left({{V}_{LL}}^{N`}+{\alpha }_{1}{{V}_{LF}}^{N`}\right){\alpha }_{2}}{{\mu }_{L}}+\frac{{{V}_{SS}}^{N`}}{{\mu }_{S}}-2\delta {e}_{S}\right].\end{array}\right.$$
(A.6)

Observe the structural features of formulas (A.6) and infer that the linear functions on \(e\) are the solutions to the HJB equations. Assume that

$$\left\{\begin{array}{c}{{V}_{L}}^{N}\left({e}_{L},{e}_{F}\right)={a}_{1}{e}_{L}+{a}_{2}{e}_{F}+{a}_{3},\\ {{V}_{F}}^{N}\left({e}_{L},{e}_{F}\right)={b}_{1}{e}_{L}+{b}_{2}{e}_{F}+{b}_{3},\\ {{V}_{S}}^{N}\left({e}_{S}\right)={c}_{1}{e}_{S}+{c}_{2}.\end{array}\right.$$
(A.7)

Substituting formula (A.7) and \({{V}_{ij}}^{`},i,j\in \{L,F,S\}\), into formula (A.6), we obtain

$$\left\{\begin{array}{c}{a}_{1}=\frac{s{\pi }_{L}}{\delta +\rho },{a}_{2}=-\frac{s{\pi }_{L}}{\delta +\rho },{a}_{3}=\frac{\lambda Q{\pi }_{L}}{\rho }+\frac{{s}^{2}{{\pi }_{L}}^{2}{\left(1-{\alpha }_{1}\right)}^{2}}{2\rho {\left(\delta +\rho \right)}^{2}{\mu }_{L}}-\frac{{s}^{2}{\pi }_{L}{\pi }_{F}}{\rho {\left(\delta +\rho \right)}^{2}{\mu }_{F}},\\ {b}_{1}=-\frac{s{\pi }_{F}}{\delta +\rho },{b}_{2}=\frac{s{\pi }_{F}}{\delta +\rho },{b}_{3}=\frac{Q\left(1-\lambda \right){\pi }_{F}}{\rho }+\frac{{s}^{2}{{\pi }_{F}}^{2}}{2\rho {\left(\delta +\rho \right)}^{2}{\mu }_{F}}-\frac{{s}^{2}{\pi }_{L}{\pi }_{F}{\left(1-{\alpha }_{1}\right)}^{2}}{\rho {\left(\delta +\rho \right)}^{2}{\mu }_{L}},\\ {c}_{1}=0,{c}_{2}=\frac{Q{\pi }_{S}}{\rho }.\end{array}\right.$$
(A.8)

Combine formulas (A.7) and (A.8) to obtain \({{V}_{ij}}^{`},i,j\in \{L,F,S\}\). Substitute this into formulas (A5) to obtain

$$\left\{\begin{array}{c}{{u}_{L}}^{N*}=\frac{s{\pi }_{L}(1-{\alpha }_{1})}{(\delta +\rho ){\mu }_{L}},\\ {{u}_{F}}^{N*}=\frac{s{\pi }_{F}}{(\delta +\rho ){\mu }_{F}},\\ {{u}_{S}}^{N*}=0.\end{array}\right.$$

Further manufacturers and supplier carbon reduction stabilization values can be obtained:

$$\left\{\begin{array}{c}{{e}_{L}}^{N*}=\frac{s{\pi }_{L}(1-{\alpha }_{1})}{\delta (\delta +\rho ){\mu }_{L}},\\ {{e}_{F}}^{N*}=\frac{s[{\pi }_{F}{\mu }_{L}+{\pi }_{L}{\alpha }_{1}{\mu }_{F}\left(1-{\alpha }_{1}\right)]}{\delta (\delta +\rho ){\mu }_{L}{\mu }_{F}},\\ {{e}_{S}}^{N*}=\frac{s{\pi }_{L}(1-{\alpha }_{1}){\alpha }_{2}}{\delta (\delta +\rho ){\mu }_{L}},\end{array}\right.$$

and carbon reduction trajectories as follows:

$$\left\{\begin{array}{c}{{e}_{L}}^{N*}\left(t\right)=\frac{s{\pi }_{L}(1-{\alpha }_{1})}{\delta (\delta +\rho ){\mu }_{L}}+{e}^{-\delta t}{e}_{0},\\ {{e}_{F}}^{N*}\left(t\right)=\frac{s[{\pi }_{F}{\mu }_{L}+{\pi }_{L}{\alpha }_{1}{\mu }_{F}\left(1-{\alpha }_{1}\right)]}{\delta (\delta +\rho ){\mu }_{L}{\mu }_{F}}+{e}^{-\delta t}{e}_{0},\\ {{e}_{S}}^{N*}\left(t\right)=\frac{s{\pi }_{L}(1-{\alpha }_{1}){\alpha }_{2}}{\delta (\delta +\rho ){\mu }_{L}}+{e}^{-\delta t}{e}_{0}.\end{array}\right.$$

The optimal profit trajectories are

$${{J}_{L}}^{N*}={e}^{-\rho t}\left({a}_{1}{{e}_{L}}^{N*}+{a}_{2}{{e}_{F}}^{N*}+{a}_{3}\right),$$
$${{J}_{F}}^{N*}={e}^{-\rho t}\left({b}_{1}{{e}_{L}}^{N*}+{b}_{2}{{e}_{F}}^{N*}+{b}_{3}\right),$$
$${{J}_{S}}^{N*}={e}^{-\rho t}\left({c}_{1}{{e}_{S}}^{N*}+{c}_{2}\right),$$

where \({a}_{1}=\frac{s{\pi }_{L}}{\delta +\rho },\;{a}_{2}=-\frac{s{\pi }_{L}}{\delta +\rho },\;{a}_{3}=\frac{\lambda Q{\pi }_{L}}{\rho }+\frac{{s}^{2}{{\pi }_{L}}^{2}{\left(1-{\alpha }_{1}\right)}^{2}}{2\rho {\left(\delta +\rho \right)}^{2}{\mu }_{L}}-\frac{{s}^{2}{\pi }_{L}{\pi }_{F}}{\rho {\left(\delta +\rho \right)}^{2}{\mu }_{F}}\), \({b}_{1}=-\frac{s{\pi }_{F}}{\delta +\rho },\;{b}_{2}=\frac{s{\pi }_{F}}{\delta +\rho },\;{b}_{3}=\frac{Q\left(1-\lambda \right){\pi }_{F}}{\rho }\), \(+\frac{{s}^{2}{{\pi }_{F}}^{2}}{2\rho {\left(\delta +\rho \right)}^{2}{\mu }_{F}}-\frac{{s}^{2}{\pi }_{L}{\pi }_{F}{\left(1-{\alpha }_{1}\right)}^{2}}{\rho {\left(\delta +\rho \right)}^{2}{\mu }_{L}}\), \({c}_{1}=0,{c}_{2}=\frac{Q{\pi }_{S}}{\rho }\).

1.2 Proof of Theorem 2

The solution process is similar to Theorem 1 and is omitted here. The parameters in the trajectories are as follows:

$$\begin{aligned}&{f}_{1}=\frac{\tau -s\tau +s{\pi }_{L}}{\delta +\rho },\\&{f}_{2}=\frac{s\left(\tau -{\pi }_{L}\right)}{\delta +\rho },\\&{f}_{3}=\frac{{[\tau -s\tau+s{\pi }_{L}+s\left(\tau -{\pi }_{L}\right){\alpha }_{1}]}^{2}}{2\rho {(\delta +\rho )}^{2}{\mu }_{L}}\\&\qquad\;+\,\frac{s\left(\tau -s\tau +s{\pi }_{F}\right)\left(\tau -{\pi }_{L}\right)}{\rho {(\delta +\rho )}^{2}{\mu }_{F}}\\& \qquad\;+\,\frac{\tau \left[\left(\tau -s\tau +s{\pi }_{L}\right){\beta }_{1}+s\left(\tau -{\pi }_{L}\right){\beta }_{2}\right]}{2\rho {(\delta +\rho )}^{2}{\mu }_{S}}\\& \qquad\;-\,\frac{\lambda Q(\tau -{\pi }_{L})}{\rho },\end{aligned}$$
$$\begin{aligned}&{g}_{1}=\frac{s\left(\tau -{\pi }_{F}\right)}{\delta +\rho },\\&{g}_{2}=\frac{\tau -s\tau +s{\pi }_{F}}{\delta +\rho },\\&{g}_{3}=\frac{\left[\tau -s(\tau -{\pi }_{L})(1-{\alpha }_{1})\right]\left[s(1-{\alpha }_{1})(\tau -{\pi }_{F})-\tau {\alpha }_{1}\right]}{\rho {(\delta +\rho )}^{2}{\mu }_{L}}\\&\qquad\;+\,\frac{{\left(\tau -s\tau +s{\pi }_{F}\right)}^{2}}{2\rho {(\delta +\rho )}^{2}{\mu }_{F}}\\&\qquad\;+\frac{\tau s({\beta }_{1}-{\beta }_{2})(\tau -{\pi }_{F})+{\tau }^{2}{\beta }_{2}}{\rho {(\delta +\rho )}^{2}{\mu }_{S}}\\&\qquad\;-\,\frac{\left(1-\lambda \right)Q\left(\tau -{\pi }_{F}\right)}{\rho },\\& {h}_{1}=\frac{s\tau }{\delta +\rho },\\&{h}_{2}=\frac{\tau {\alpha }_{2}\left[\tau -s\left(\tau -{\pi }_{L}\right)\left(1-{\alpha }_{1}\right)\right]}{\rho {\left(\delta +\rho \right)}^{2}{\mu }_{L}}\\&\qquad\;+\,\frac{{\tau }^{2}}{2\rho {\left(\delta +\rho \right)}^{2}{\mu }_{S}}-\frac{Q\left(\tau -{\pi }_{L}\right)}{\rho }.\end{aligned}$$

1.3 Proof of Proposition 1

  1. (a)

    \({{e}_{L}}^{T*}-{{e}_{L}}^{N*}=\frac{\tau \left[{\beta }_{1}{\mu }_{L}+\left(1-s+s{\alpha }_{1}\right){\mu }_{S}\right]}{\delta \left(\delta +\rho \right){\mu }_{L}{\mu }_{S}}>0,\) \({{e}_{F}}^{T*}-{{e}_{F}}^{N*}=\frac{\tau \left[{\beta }_{2}{\mu }_{L}{\mu }_{F}+\left(1-s\right){\mu }_{L}{\mu }_{S}+{\mu }_{F}{\mu }_{S}{\alpha }_{1}\left(1-s+s{\alpha }_{1}\right)\right]}{\delta \left(\delta +\rho \right){\mu }_{L}{\mu }_{F}{\mu }_{S}}>0,\) \({{e}_{S}}^{T*}-{{e}_{S}}^{N*}=\frac{\tau \left[{\mu }_{L}+\left(1-s+s{\alpha }_{1}\right){\alpha }_{2}{\mu }_{S}\right]}{\delta \left(\delta +\rho \right){\mu }_{L}{\mu }_{S}}>0\).

  2. (b)

    \({{u}_{L}}^{T*}-{{u}_{L}}^{N*}=\frac{\tau \left(1-s+s{\alpha }_{1}\right)}{\left(\delta +\rho \right){\mu }_{{\text{L}}}}>0,\) \({{u}_{F}}^{T*}-{{u}_{F}}^{N*}=\frac{\tau (1-s)}{\left(\delta +\rho \right){\mu }_{{\text{F}}}}>0,\) \({{u}_{S}}^{T*}-{{u}_{S}}^{N*}=\frac{\tau }{(\delta +\rho ){\mu }_{{\text{S}}}}\).

1.4 Proof of Proposition 2

  1. (a)

    \({{e}_{L}}^{N*}-{{e}_{F}}^{N*}=\frac{{(1-{\alpha }_{1})}^{2}{\mu }_{{\text{F}}}s{\pi }_{L}-{\mu }_{{\text{L}}}s{\pi }_{F}}{\delta (\delta +\rho ){\mu }_{{\text{L}}}{\mu }_{{\text{F}}}}\). As \(\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}{\mu }_{{\text{F}}}>0,\) \({(1-{\alpha }_{1})}^{2}{\mu }_{{\text{F}}}s{\pi }_{L}-{\mu }_{{\text{L}}}s{\pi }_{F}>0\) can be transformed into \(\frac{{\pi }_{L}}{{\pi }_{F}}>\frac{{\mu }_{{\text{L}}}}{{(1-{\alpha }_{1})}^{2}{\mu }_{{\text{F}}}}\).

  2. (b)

    \({{e}_{L}}^{T*}-{{e}_{F}}^{T*}=\frac{\tau {\mu }_{{\text{L}}}{\mu }_{{\text{F}}}\left({\beta }_{1}-{\beta }_{2}\right)-{\mu }_{{\text{L}}}{\mu }_{{\text{S}}}\left(s{\pi }_{F}+\tau -s\tau \right)+{\mu }_{{\text{F}}}{\mu }_{{\text{S}}}(1-{\alpha }_{1})[\tau -s\tau +s{\pi }_{L}+s\left(\tau -{\pi }_{L}\right){\alpha }_{1}]}{\delta (\delta +\rho ){\mu }_{{\text{L}}}{\mu }_{{\text{F}}}{\mu }_{{\text{S}}}}\). As \(\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}{\mu }_{{\text{F}}}{\mu }_{{\text{S}}}>0,\) \(\tau {\mu }_{{\text{L}}}{\mu }_{{\text{F}}}\left({\beta }_{1}-{\beta }_{2}\right)-{\mu }_{{\text{L}}}{\mu }_{{\text{S}}}\left(s{\pi }_{F}+\tau -s\tau \right)+{\mu }_{{\text{F}}}{\mu }_{{\text{S}}}\left(1-{\alpha }_{1}\right)\left[\tau -s\tau +s{\pi }_{L}+s\left(\tau -{\pi }_{L}\right){\alpha }_{1}\right]>0\) can be transformed into \({\beta }_{1}-{\beta }_{2}>{\Phi }_{2}(\tau )\), where \({\Phi }_{2}\left(\tau \right)=\frac{\left(\tau -s\tau +s{\pi }_{F}\right){\mu }_{{\text{L}}}{\mu }_{{\text{S}}}-\left[s{\pi }_{L}\left(1-{\alpha }_{1}\right)+\tau \left(1-s+s{\alpha }_{1}\right)\right]\left(1-{\alpha }_{1}\right){\mu }_{{\text{F}}}{\mu }_{{\text{S}}}}{\tau {\mu }_{{\text{L}}}{\mu }_{{\text{F}}}}\). If \({\beta }_{1}={\beta }_{2}=0,\) it can be transformed into \({\pi }_{L}>{\Phi }_{1}({\pi }_{F})\), where \({\Phi }_{1}\left({\pi }_{F}\right)=\frac{(\tau -s\tau +s{\pi }_{F}){\mu }_{1}+\tau (-1+{\alpha }_{1})(1-s+s{\alpha }_{1}){\mu }_{2}}{s{(1-{\alpha }_{1})}^{2}{\mu }_{2}}\).

1.5 Proof of Corollary 1

  1. (a)

    . \(\frac{\partial \Delta {e}^{N}}{\partial {\alpha }_{1}}=-\frac{2s{\pi }_{L}\left(1-{\alpha }_{1}\right)}{\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}}<0.\)

  2. (b)

    \(\frac{\partial \Delta {e}^{T}}{\partial {\alpha }_{1}}=-\frac{\tau +2s(1-{\alpha }_{1})({\pi }_{L}-\tau )}{\delta (\delta +\rho ){\mu }_{{\text{L}}}},\) As \(\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}>0,\) \(\tau +2s\left(1-{\alpha }_{1}\right)\left({\pi }_{L}-\tau \right)<0\) can be transformed into \(\tau +2s{\pi }_{L}-2s{\alpha }_{1}{\pi }_{L}-2s\tau \left(1-{\alpha }_{1}\right)<0\). If \(\tau >2{\pi }_{L},\) we have \(0<{\alpha }_{1}<\frac{\tau -2{\pi }_{L}}{2(\tau -{\pi }_{L})}\) and \(\frac{\tau }{2(\tau -{\pi }_{L})(1-{\alpha }_{1})}<s<1\).

  3. (c)

    \(\frac{\partial \Delta {e}^{T}}{\partial {\beta }_{1}}=\frac{\tau }{\delta (\delta +\rho ){\mu }_{{\text{S}}}}>0,\) \(\frac{\partial \Delta {e}^{T}}{\partial {\beta }_{2}}=-\frac{\tau }{\delta \left(\delta +\rho \right){\mu }_{{\text{S}}}}<0\).

1.6 Proof of Proposition 3

  1. (a)

    \({{e}_{L}}^{NH}={{e}_{L}}^{ND}=\frac{s{\pi }_{L}(1-{\alpha }_{1})}{\delta (\delta +\rho ){\mu }_{{\text{L}}}},\) \({{e}_{L}}^{NV}-{{e}_{L}}^{ND}=\frac{s{\pi }_{L}{\alpha }_{1}}{\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}}>0.\)

  2. (b)

    \({{e}_{L}}^{TH}-{{e}_{L}}^{TD}=-\frac{\tau {\beta }_{1}}{\delta \left(\delta +\rho \right){\mu }_{{\text{S}}}}<0,\) \({{e}_{L}}^{TV}-{{e}_{L}}^{TD}=\frac{s{\alpha }_{1}\left({\pi }_{L}-\tau \right)}{\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}}.\) As \(\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}>0,\) \(s{\alpha }_{1}\left({\pi }_{L}-\tau \right)>0\) can be transformed into \({\pi }_{L}>\tau >0\).

1.7 Proof of Proposition 4

  1. (a)

    \({{e}_{L}}^{NH}-{{e}_{L}}^{NV}=-\frac{s{\pi }_{L}{\alpha }_{1}}{\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}}<0\).

  2. (b)

    \({{e}_{L}}^{TH}-{{e}_{L}}^{TV}=\frac{s{\alpha }_{1}{\mu }_{{\text{S}}}(\tau -{\pi }_{L})-\tau {\beta }_{1}{\mu }_{{\text{L}}}}{\delta (\delta +\rho ){\mu }_{{\text{L}}}{\mu }_{{\text{S}}}}\). As \(\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}{\mu }_{{\text{S}}}>0,\) \(s{\alpha }_{1}{\mu }_{{\text{S}}}\left(\tau -{\pi }_{L}\right)-\tau {\beta }_{1}{\mu }_{{\text{L}}}>0\) can be transformed into \(s{\alpha }_{1}{\mu }_{{\text{S}}}\left(\tau -{\pi }_{L}\right)>\tau {\beta }_{1}{\mu }_{{\text{L}}}\). If \(0<\tau <{\pi }_{L},\) we have \(s>\frac{\tau {\beta }_{1}{\mu }_{{\text{L}}}}{(\tau -{\pi }_{L}){\alpha }_{1}{\mu }_{{\text{S}}}}\); else if \(\tau >{\pi }_{L},\) we have \(s<\frac{\tau {\beta }_{1}{\mu }_{{\text{L}}}}{(\tau -{\pi }_{L}){\alpha }_{1}{\mu }_{{\text{S}}}}\).

1.8 Proof of Proposition 5

  1. (a)

    . \({{e}_{F}}^{NH}-{{e}_{F}}^{NV}=\frac{s{\pi }_{L}\left(1-{\alpha }_{1}\right){\alpha }_{1}}{\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}}>0.\)

  2. (b)

    \({{e}_{F}}^{TH}-{{e}_{F}}^{TV}=\frac{\tau ({\alpha }_{1}{\mu }_{{\text{S}}}-{\beta }_{2}{\mu }_{{\text{L}}})-s(\tau -{\pi }_{L})(1-{\alpha }_{1}){\alpha }_{1}{\mu }_{{\text{S}}}}{\delta (\delta +\rho ){\mu }_{{\text{L}}}{\mu }_{{\text{S}}}}\). As \(\delta \left(\delta +\rho \right){\mu }_{{\text{L}}}{\mu }_{{\text{S}}}>0,\) if \(0<\tau <{\pi }_{L}\) and \({\alpha }_{1}{\mu }_{S}>{\beta }_{2}{\mu }_{L}\), we have \({{e}_{F}}^{TH}>{{e}_{F}}^{TV}\); else if \({\alpha }_{1}{\mu }_{S}<{\beta }_{2}{\mu }_{L}\), we have \(s>{\Phi }_{3}(\tau )\). If \(\tau >{\pi }_{L}\) and \({\alpha }_{1}{\mu }_{S}>{\beta }_{2}{\mu }_{L}\), we have \(s<{\Phi }_{3}(s)\), where \({\Phi }_{3}\left(\tau \right)=\frac{\tau ({\beta }_{2}{\mu }_{L}-{\alpha }_{1}{\mu }_{S})}{({\pi }_{L}-\tau )(1-{\alpha }_{1}){\alpha }_{1}{\mu }_{S}}\).

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Liu, Q., Zhu, X. How Carbon Tax Policy Affects the Carbon Emissions of Manufacturers with Green Technology Spillovers?. Environ Model Assess (2024). https://doi.org/10.1007/s10666-024-09965-x

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