Suppose there are two types of sectors in a city: energy-intensive sectors and low-energy sectors. A city's carbon emissions are determined by both types. Enterprises in energy-intensive sectors are large in scale and monopolistic. The members of the low-energy sector are small-scale enterprises that participate in a perfectly competitive market, and in both sectors, the enterprises exhibit constant returns to scale. The production function for enterprises in the energy-intensive sectors is as follows:
\({Y_0}=AK_{0}^{{{\alpha _{01}}}}L_{0}^{n}E_{0}^{{{\alpha _{02}}}}\)
where \({Y_0}\)is the output of energy-intensive enterprises, is green total factor productivity, \({K_0}\)is investment in physical capital not related to energy,\({L_0}\) is the total labor measured as the total hours worked by employees of energy-intensive enterprises, and\({E_0}\)is the total energy input of the energy-intensive enterprises, measured by the carbon dioxide emissions of energy-intensive enterprises. \({\alpha _{01}}\)and \({\alpha _{02}}\)represent the output elasticity of the physical capital and energy input of the enterprise, respectively, is the labor output elasticity. Since both types of enterprises exhibit constant returns to scale,, which means\({\alpha _{01}}+{\alpha _{02}}+n=1\), and the cost constraints faced by energy-intensive enterprises are:
$${C_0}={r_0}{K_0}+\omega {L_0}+f{E_0}$$
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\({C_0}\) is the enterprise cost constraint, and \({r_0}\)is the unit cost of physical capital for the energy-intensive enterprises, that is, the bank's lending rate. \(\omega\)is the hourly wage, and is the cost of energy per unit of carbon dioxide emissions. The energy-intensive sector consists of large-scale enterprises with monopoly power. Therefore, the enterprises in this sector aim to minimize their costs subject to the given output condition. Solve the following Lagrangian equation:
$${\Omega _0}=\omega {L_0}+{r_0}{K_0}+f{E_0} - \mu \left[ {AK_{0}^{{{\alpha _{01}}}}L_{0}^{n}E_{0}^{{{\alpha _{02}}}}} \right]$$
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Taking the derivative with respect to physical capital, labor and energy consumption:
$$\left\{ \begin{gathered} \frac{{\partial {\Omega _0}}}{{\partial {K_0}}}={r_0} - \mu (A{\alpha _{01}}K_{0}^{{{\alpha _{01}} - 1}}L_{0}^{n}E_{0}^{{{\alpha _{02}}}}) \hfill \\ \frac{{\partial {\Omega _0}}}{{\partial {L_0}}}=\omega - \mu (AnK_{0}^{{{\alpha _{01}}}}L_{0}^{{n - 1}}E_{0}^{{{\alpha _{02}}}}) \hfill \\ \frac{{\partial {\Omega _0}}}{{\partial {E_0}}}=f - \mu (A{\alpha _{02}}K_{0}^{{{\alpha _{01}}}}L_{0}^{n}E_{0}^{{{\alpha _{02}} - 1}}) \hfill \\ \end{gathered} \right.$$
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The optimal input of physical capital for the energy-intensive enterprise is \({K_0}=\frac{{{Y_0}{\omega ^n}{f^{{\alpha _{02}}}}\alpha _{{01}}^{{n+{\alpha _{02}}}}}}{{A{n^n}\alpha _{{02}}^{{{\alpha _{02}}}}r_{0}^{{n+{\alpha _{02}}}}}}\), and the optimal amount of energy input is \({E_0}=\frac{{{Y_0}{\omega ^n}r_{0}^{{{\alpha _{01}}}}\alpha _{{02}}^{{n+{\alpha _{_{{01}}}}}}}}{{A{n^n}\alpha _{{01}}^{{{\alpha _{01}}}}{f^{n+{\alpha _{01}}}}}}\).
Similarly, the production function for the low-energy sector is
\({Y_1}=AK_{1}^{{{\alpha _{11}}}}L_{1}^{n}E_{1}^{{{\alpha _{12}}}}\)
Moreover, \({\alpha _{11}}+{\alpha _{12}}+n=1\), and the cost constraint faced by the enterprises in the low-energy sector is:
$${C_1}={r_1}{K_1}+\omega {L_1}+f{E_1}$$
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where \({r_1}\)is the unit cost of physical capital for the low-energy enterprises, and their cost of financing is constrained which is greater than the lending rate in the banking system due to crowding out in the traditional finance sector ; that is, \({r_1}\) is greater than \({r_0}\).
Under these cost constraints, the low-energy sector aims to to maximize its output; then, we can solve for its optimal factor input with the following Lagrangian equation:
$${\Omega _1}=AK_{1}^{{{\alpha _{11}}}}L_{1}^{n}E_{1}^{{{\alpha _{12}}}} - \lambda \left[ {\omega {L_1}+{r_1}{K_1}+f{E_1}} \right]$$
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Taking the derivative with respect to capital, labor and the energy input, we have:
$$\left\{ \begin{gathered} \frac{{\partial {\Omega _1}}}{{\partial {K_1}}}=A{\alpha _{11}}K_{1}^{{{\alpha _{11}} - 1}}L_{1}^{n}E_{1}^{{{\alpha _{12}}}} - \lambda {r_1} \hfill \\ \frac{{\partial {\Omega _1}}}{{\partial {L_1}}}=AnK_{1}^{{{\alpha _{11}}}}L_{1}^{{n - 1}}E_{1}^{{{\alpha _{12}}}} - \lambda \omega \hfill \\ \frac{{\partial {\Omega _1}}}{{\partial {E_1}}}=A{\alpha _{12}}K_{1}^{{{\alpha _{11}}}}L_{1}^{n}E_{1}^{{{\alpha _{12}} - 1}} - \lambda f \hfill \\ \end{gathered} \right.$$
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The optimal input of physical capital for the low-energy sector is \({K_1}=\frac{{{C_1}{\alpha _{11}}}}{{{r_1}}}\), and the optimal energy input is \({E_1}=\frac{{{C_1}{\alpha _{12}}}}{f}\). Therefore, the optimal output for low-energy enterprises is \({Y_1}={C_1}{r_1}^{{ - {a_{11}}}}\psi\), where \(\psi ={\alpha _{11}}^{{{\alpha _{11}}}}{n^n}{\alpha _{12}}^{{{\alpha _{12}}}}{\omega ^{ - n}}{f^{ - {\alpha _{12}}}}\).
Carbon intensity is the ratio of carbon emissions to output: \(\frac{E}{Y}=\frac{{{E_0}+{E_1}}}{{{Y_0}+{Y_1}}}=\frac{{\frac{{{Y_0}{\omega ^n}r_{0}^{{{\alpha _{01}}}}\alpha _{{02}}^{{n+{\alpha _{01}}}}}}{{A{n^n}\alpha _{{01}}^{{{\alpha _{01}}}}{f^{n+{\alpha _{01}}}}}}+\frac{{{C_1}{\alpha _{12}}}}{f}}}{{{Y_0}+{C_1}r_{1}^{{ - {\alpha _{11}}}}\psi }}\).
The development of digital finance has expanded the cost constraints on small and medium-sized enterprises by reducing the cost of borrowing for capital (Gu et al., 2021); that is, \(d{C_1}>0\), \(d{r_1}<0\). However, the enterprises in the energy-intensive sector obtain financial services mainly through traditional financial service providers such as banks, so they are not affected by the development of digital finance. As a result, the carbon intensity of the low energy sector is:
$$\frac{{{E_1}}}{{{Y_1}}}=\frac{{{C_1}{\alpha _{12}}}}{{{C_1}r_{{_{1}}}^{{ - {\alpha _{11}}}}\psi f}}=r_{{_{1}}}^{{{\alpha _{11}}}}\frac{{{\alpha _{12}}}}{{\psi f}}$$
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Therefore, the carbon intensity of the low-energy sector decreases with the decrease in \({r_1}\). The energy input \({E_1}={C_1}{\alpha _{12}}{f^{ - 1}}\)and the output \({Y_1}={C_1}r_{1}^{{ - {\alpha _{11}}}}\psi\) of the low-energy sector increase with the increase in \({C_1}\) and the decrease in \({r_1}\). Therefore:
$$\frac{{{E_0}+{E_1}}}{{{Y_0}+{Y_1}}}=\frac{{{E_0}}}{{{Y_0}+{Y_1}}}+\frac{{{E_1}}}{{{Y_0}+{Y_1}}}=\frac{{{E_0}}}{{{Y_0}+{Y_1}}}+\frac{1}{{{Y_0}/{E_1}+{Y_1}/{E_1}}}$$
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Since \({Y_1}\)increases with the increase in \({C_1}\) and the decrease in \({r_1}\), \(\frac{{{E_0}}}{{{Y_0}+{Y_1}}}\) decreases because of the increase in \({Y_1}\), and \(\frac{1}{{{Y_0}/{E_1}+{Y_1}/{E_1}}}\) decreases because of the lower carbon intensity in low-energy sector and the reduction of energy input (i.e., carbon emissions) in the low-energy sectors. Therefore, the total carbon emissions in the economy \(\frac{{{E_0}+{E_1}}}{{{Y_0}+{Y_1}}}\)are reduced due to the development of digital finance.
Factor distortions refer to the deviation of the market price of a kind of factor from its opportunity cost due to imperfect markets or government regulation. Following Hsieh and Klenow (2008), we construct the factor distortion index as follows:
$$\begin{gathered} DIS={(\frac{{\partial {Y_1}/\partial {K_1}}}{{{r_0}}})^{\frac{{{\alpha _{11}}}}{{{\alpha _{11}}+n}}}}{(\frac{{\partial {Y_1}/\partial {L_1}}}{\omega })^{\frac{n}{{{\alpha _{11}}+n}}}} \hfill \\ =A{(\frac{{{\alpha _{11}}K_{1}^{{{\alpha _{11}} - 1}}L_{1}^{n}E_{1}^{{{\alpha _{12}}}}}}{{{r_0}}})^{\frac{{{\alpha _{11}}}}{{{\alpha _{11}}+n}}}}{(\frac{{nK_{1}^{{{\alpha _{11}}}}L_{1}^{{n - 1}}E_{1}^{{{\alpha _{12}}}}}}{\omega })^{\frac{n}{{{\alpha _{11}}+n}}}} \hfill \\ =AK_{1}^{{{\alpha _{11}} - 1}}L_{1}^{{n - 1}}E_{1}^{{{\alpha _{12}}}}{({\alpha _{11}}\frac{{{L_1}}}{{{r_0}}})^{\frac{{{\alpha _{11}}}}{{{\alpha _{11}}+n}}}}{(n\frac{{{K_1}}}{\omega })^{\frac{n}{{{\alpha _{11}}+n}}}} \hfill \\ =A\psi r_{0}^{{\frac{{ - {\alpha _{11}}}}{{{\alpha _{11}}+n}}}}r_{1}^{{\frac{{ - n}}{{{\alpha _{11}}+n}}+1 - {\alpha _{11}}}} \hfill \\ \end{gathered}$$
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Since \(0<{\alpha _{11}}+n<1\), \(\frac{{ - n}}{{{\alpha _{11}}+n}}+1 - {\alpha _{11}}>\frac{{ - n}}{{{\alpha _{11}}+n}}+1 - \frac{{{\alpha _{11}}}}{{{\alpha _{11}}+n}}=0\). The factor distortion index \(DIS\)decreases with the reduction of \({r_1}\), then:
$$\frac{E}{Y}=\frac{{(\frac{{{Y_0}{\omega ^n}r_{0}^{{{\alpha _{01}}}}\alpha _{{02}}^{{n+{\alpha _{01}}}}}}{{A{n^n}\alpha _{{01}}^{{{\alpha _{01}}}}{f^{n+{\alpha _{01}}}}}})+\frac{{{C_1}{\alpha _{13}}}}{f}}}{{{Y_0}+{C_1}r_{1}^{{ - {\alpha _{11}}}}\psi }}=\frac{{(\frac{{{Y_0}{\omega ^n}r_{0}^{{{\alpha _{01}}}}\alpha _{{02}}^{{n+{\alpha _{01}}}}}}{{A{n^n}\alpha _{{01}}^{{{\alpha _{01}}}}{f^{n+{\alpha _{01}}}}}})+\frac{{{C_1}{\alpha _{13}}}}{f}}}{{{Y_0}+{C_1}{{(DIS \cdot r_{0}^{{\frac{{{\alpha _{11}}}}{{{\alpha _{11}}+n}}}}{A^{ - 1}})}^{\frac{{ - {\alpha _{11}} - n}}{{1 - {\alpha _{11}} - n}}}}{\psi ^{\frac{1}{{1 - {\alpha _{11}} - n}}}}}}$$
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\(\frac{{ - {\alpha _{11}} - n}}{{1 - {\alpha _{11}} - n}}<0\) , all else remaining unchanged, carbon intensity (\(E/Y\)) decreases as the factor distortion index \(DIS\) decreases. Therefore, the development of digital finance reduces carbon intensity by alleviating the distortion of factor prices.