Green technology advancement, energy input share and carbon emission trend studies

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Theoretical modelling

This section builds on Aghion et al.¹³ to analyse the impact of green technological progress on carbon emissions by constructing a three-sector production model incorporating capital, labour and energy. It is assumed that the firm’s production function takes the C-D form, denoted as:

$$Y_{t} = (A_{t} L_{t} )^{1 – \alpha – \beta } (B_{t} K_{t} )^{\beta } (\delta_{t} E_{t} )^{\alpha }$$

(1)

In the above equation, $Y_{t}$ represents the output of the final product, $K_{{\text{t}}}$ denotes capital inputs, $L_{t}$ denotes labour inputs, $E_{t}$ denotes energy inputs. The subscript t denotes the time, $A_{t}$ indicates labour-saving technology level, $B_{t}$ represents the technological level of capital factors, $A_{t}$ and $B_{t}$ are collectively referred to as the level of technology (without green technology), and $\delta_{t}$ denotes the level of green technology. Green technologies here denote improvements in energy efficiency. The parameters $\alpha$ and $\beta$ represent the shares of energy inputs and capital inputs in the production function, respectively, with $0 < \alpha < 1$ and $0 < \beta < 1$. The energy input share $\alpha$ represents the amount of energy invested in the production process and reflects the extent to which the economy consumes energy. A higher energy input share $\alpha$ indicates a higher degree of energy consumption.

Drawing on Shen et al.¹², it is assumed that the production of one unit of energy requires one unit of final product inputs and that the price of the final product is normalised to 1. The price of energy in the energy market is $p_{e}$, the price of labour in the labour market is $p_{l}$, and the price of capital in the capital market is $p_{k}$. Assuming that the final product market is a perfectly competitive market, according to the firm’s profit maximisation condition:

$$\mathop {{\text{max}}}\limits_{{K_{t} ,L_{t} ,E_{t} }} (A_{t} L_{t} )^{1 – \alpha – \beta } (B_{t} K_{t} )^{\beta } (\delta_{t} E_{t} )^{\alpha } – p_{kt} K_{t} – p_{lt} L_{t} – p_{et} E_{t}$$

(2)

The above equation takes a first-order derivative of $E_{t}$. Further collation gives the demand function for energy $E_{t}$ as:

$$E_{t} = \left[ {\frac{{\alpha (A_{t} L_{t} )^{1 – \alpha – \beta } (B_{t} K_{t} )^{\beta } \delta_{t}^{\alpha } }}{{P_{et} }}} \right]^{{\frac{1}{1 – \alpha }}}$$

(3)

From the above equation, the elasticity of demand for energy can be calculated as a constant $\frac{1}{1 – \alpha }$. The profit maximisation function of the energy producer is:

$$\mathop {MAX}\limits_{{P_{et} }} P_{et} E_{t} – E_{t}$$

(4)

Solving the first-order condition for maximisation of the above equation gives:

$$- \alpha P_{et}^{{\frac{ – 1}{{1 – \alpha }}}} + P_{et}^{{\frac{ – 2 + \alpha }{{1 – \alpha }}}} = 0 \, \Rightarrow \, P_{et} = \frac{1}{\alpha }$$

(5)

Therefore, the price of energy in the energy market is $\frac{1}{\alpha }$. Substituting into Eq. (3) gives the energy demand function as:

$$E_{t} = \alpha^{{\frac{2}{1 – \alpha }}} [(A_{t} L_{t} )^{1 – \alpha – \beta } (B_{t} K_{t} )^{\beta } ]^{{\frac{1}{1 – \alpha }}} \delta_{t}^{{\frac{\alpha }{1 – \alpha }}}$$

(6)

To facilitate the portrayal of the two-way relationship between energy inputs and economic output, as outlined in Kumbaroğlu et al.¹⁴, gross domestic product (GDP) is expressed as the difference between output and energy costs:

$$\begin{aligned} GDP_{t} = & Y_{t} – p_{et} E_{t} \\ { = } & (A_{t} L_{t} )^{1 – \alpha – \beta } (B_{t} K_{t} )^{\beta } (\delta_{t} E_{t} )^{\alpha } – \frac{1}{\alpha }E_{t} \\ { = } & \alpha^{{\frac{2\alpha }{{1 – \alpha }}}} (1 – \alpha )[(A_{t} L_{t} )^{1 – \alpha – \beta } (B_{t} K_{t} )^{\beta } ]^{{\frac{1}{1 – \alpha }}} \delta_{t}^{{^{{\frac{\alpha }{1 – \alpha }}} }} \\ \end{aligned}$$

(7)

Drawing on Lin and Li¹⁵, assuming that carbon emissions come only from the energy use process, the expression for carbon emissions is as follows:

$$D_{t} = \delta_{t}^{ – \lambda } E_{t}$$

(8)

In the above equation, $D_{t}$ represents carbon emissions, $\delta_{t}^{ – \lambda }$ measures the technological effect of the level of green technology on CO2 emissions, $\lambda$ denotes the elasticity of the impact of the level of green technology on carbon emissions, and $\lambda > 0$ indicating that the higher the carbon emissions of green technology, the smaller the direct carbon emissions, i.e., the technology effect of the level of green technology on CO2 emissions is negative.

Theoretical model analysis

Joining (6) and (8), the total carbon emissions can be obtained as follows:

$$D_{t} = \alpha^{{\frac{2}{1 – \alpha }}} \left[ {(A_{t} L_{t} )^{1 – \alpha – \beta } (B_{t} K_{t} )^{\beta } } \right]^{{\frac{1}{1 – \alpha }}} \delta_{t}^{{\frac{\alpha }{1 – \alpha } – \lambda }}$$

(9)

Joining Eqs. (7) and (9) gives:

$$D_{t} = GDP_{t} *\alpha^{2} (1 – \alpha )^{ – 1} \delta_{t}^{ – \lambda }$$

(10)

From the above equation, the total carbon emissions are determined by the total economy (GDP), the share of energy inputs ($\alpha$), the level of green technology ($\delta_{t}$) and the carbon elasticity of the level of green technology ($\lambda$).

Taking the natural logarithm of both sides of Eq. (7) and deriving the GDP growth rate $\left( {\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}} \right)$, the growth rate of the variable is equal to the rate of change of its natural logarithm) for both the left and right sides simultaneously, is:

$$\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}{ = }\frac{1 – \alpha – \beta }{{1 – \alpha }}\left( {\frac{{\dot{A}}}{A} + \frac{{\dot{L}}}{L}} \right) + \frac{\beta }{1 – \alpha }\left( {\frac{{\dot{B}}}{B} + \frac{{\dot{K}}}{K}} \right) + \frac{\alpha }{1 – \alpha }\frac{{\dot{\delta }}}{\delta }$$

(11)

Similarly, taking the natural logarithm on both sides of Eq. (10), further derivation leads to the growth rate of carbon emissions as follows:

$$\frac{{\dot{D}}}{D} = \frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP} – \lambda \frac{{\dot{\delta }}}{\delta }$$

(12)

Joining Eqs. (11) and (12), and organising, one obtains:

$$\frac{{\dot{D}}}{D} = \left( {\frac{\alpha }{1 – \alpha } – \lambda } \right)\frac{{\dot{\delta }}}{\delta } + \frac{1 – \alpha – \beta }{{1 – \alpha }}\left( {\frac{{\dot{A}}}{A} + \frac{{\dot{L}}}{L}} \right) + \frac{\beta }{1 – \alpha }\left( {\frac{{\dot{B}}}{B} + \frac{{\dot{K}}}{K}} \right)$$

(13)

In Eq. (13), $\frac{\alpha }{1 – \alpha } – \lambda$ is the coefficient of the impact of the rate of green technological progress on the growth rate of carbon emissions, which measures the comprehensive impact of the rate of green technological progress on the growth rate of carbon emissions.

In the following, the theoretical mechanism by which the rate of green technology progress affects the growth rate of carbon emissions is further analysed.

Equation (11) is obtained by taking the derivative of $\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}$ with respect to $\frac{{\dot{\delta }}}{\delta }$:

$${{\partial \left( {\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}} \right)} {\partial \left( {\frac{{\dot{\delta }}}{\delta }} \right)}}} \right. \kern-0pt} {\partial \left( {\frac{{\dot{\delta }}}{\delta }} \right)}}{ = }\frac{\alpha }{1 – \alpha } > 0$$

(14)

In the above equation, $\frac{\alpha }{1 – \alpha }$ represents the coefficient of influence of green technology progress rate on GDP growth rate. The larger the share of energy inputs $\alpha$, the larger the impact coefficient $\frac{\alpha }{1 – \alpha }$.

(12) in which $\frac{{\dot{D}}}{D}$ is derived with respect to $\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}$ avails the following:

$${{\partial \left( {\frac{{\dot{D}}}{D}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {\frac{{\dot{D}}}{D}} \right)} {\partial \left( {\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}} \right)}}} \right. \kern-0pt} {\partial \left( {\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}} \right)}}{ = 1}$$

(15)

The above equation indicates that the GDP growth rate has a coefficient of influence on the growth rate of carbon emissions of 1. By associating Eqs. (14) and (15), and by taking the derivative of $\frac{{\dot{D}}}{D}$ with respect to $\frac{{\dot{\delta }}}{\delta }$, the following is obtained:

$$\partial \left( {\frac{{\dot{D}}}{D}} \right){/}\partial \left( {\frac{{\dot{\delta }}}{\delta }} \right){ = }\partial \left( {\frac{{\dot{D}}}{D}} \right)/\partial \left( {\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}} \right) \cdot \left[ {{{\partial \left( {\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {\frac{{\mathop {GDP}\limits^{ \cdot } }}{GDP}} \right)} {\partial \left( {\frac{{\dot{\delta }}}{\delta }} \right)}}} \right. \kern-0pt} {\partial \left( {\frac{{\dot{\delta }}}{\delta }} \right)}}} \right] = 1 \cdot \frac{\alpha }{1 – \alpha } = \frac{\alpha }{1 – \alpha }$$

(16)

The above equation represents the impact coefficient of the green technology progress rate indirectly affecting the growth rate of carbon emissions by influencing the size of GDP, with the coefficient $\frac{\alpha }{1 – \alpha }$. Green technological advances lead the way in driving economic scale growth, thus driving further increases in the rate of growth of carbon emissions, a theoretical mechanism we call the scale effect. Since $\alpha > 0$, $\frac{\alpha }{1 – \alpha } > 0$, i.e., the scale effect is positive, the economic implication of the scale effect is that an increase in the rate of green technological progress results in an increasing rate of growth of the total economy ($GDP$), hence driving an increase in the rate of growth of carbon emissions.

Equation (16) shows that the size of the scale effect depends on the energy input share $\alpha$, thereby the larger $\alpha$ is, the larger the scale effect is. Therefore, the energy input share has a positive moderating effect on the scale effect, leading to the following theoretical proposition:

Proposition 1

Energy input share $\alpha$ positively moderates the scale effect.

The economic implication of Proposition 1 is that an increase in the share of energy inputs promotes an increase in the scale effect, which in turn drives up the growth rate of carbon emissions. In fact, the share of energy inputs $\alpha$ in the production process varies across economies due to differences in industrial structure, energy endowment, etc., and hence differences in scale effects. The higher the share of energy inputs $\alpha$, i.e., the more energy-input-dependent economic production is, the larger the scale effect of the rate of green technological progress and the higher the rate of growth of carbon emissions.

Equation (12) in which $\frac{{\dot{D}}}{D}$ is derived with respect to $\frac{{\dot{\delta }}}{\delta }$, thus obtaining:

$${{\partial \left( {\frac{{\dot{D}}}{D}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {\frac{{\dot{D}}}{D}} \right)} {\partial \left( {\frac{{\dot{\delta }}}{\delta }} \right)}}} \right. \kern-0pt} {\partial \left( {\frac{{\dot{\delta }}}{\delta }} \right)}} = – \lambda$$

(17)

In the above equation, $- \lambda$ represents the direct impact coefficient of the green technology progress rate on the carbon emission growth rate, which measures the technological impact of green technology progress on the carbon emission growth rate. In this case, the increase in the rate of progress of green technology directly contributes to the reduction of the growth rate of carbon emissions, and this theoretical mechanism is called the technology effect. Since $\lambda > 0$ and therefore $- \lambda < 0$, the technology effect is negative. The economic meaning of the technology effect is that the rate of green technological progress contributes to a decrease in the rate of growth of carbon emissions directly through technological progress.

The combined effect of the rate of green technological progress on the growth rate of carbon emissions depends on the combined effect of the scale and technology effects of the rate of green technological progress. Equation (13) by taking the derivative of $\frac{{\dot{D}}}{D}$ with respect to $\frac{{\dot{\delta }}}{\delta }$ in Eq, the following is obtained:

(18)

The above equation represents the combined effect of the rate of green technology progress on the growth rate of carbon emissions, which leads to theoretical proposition 2:

Proposition 2

As the rate of green technological progress continues to increase, green technological progress increases the growth rate of carbon emissions through the scale effect, with an impact coefficient of $\frac{\alpha }{1 – \alpha } > 0$. Green technological progress contributes to a reduction in the growth rate of carbon emissions through the technology effect, with an impact factor of $- \lambda < 0$. The combined effect coefficient of the rate of green technological progress on the growth rate of carbon emissions is $\frac{\alpha }{1 – \alpha } – \lambda$.

Proposition 2 gives the theoretical mechanism of the impact of the rate of progress of green technology on the growth rate of carbon emissions. The economic implication of Proposition 2 is that when the rate of progress of green technology increases, on the one hand, technological progress promotes the growth of economic scale (GDP), further promoting the increase in the growth rate of carbon emissions; on the other hand, green technology directly promotes the reduction in the growth rate of carbon emissions, and the combined effect depends on the relative size of the two.

Different scenarios of the relative magnitude of scale and technology effects are considered below:

When $\frac{\alpha }{1 – \alpha } – \lambda < 0$, the scale effect of green technological progress is smaller than the technological effect, and the growth rate of carbon emissions keeps decreasing when the rate of green technological progress increases, i.e. $\left( {\frac{\alpha }{1 – \alpha } – \lambda } \right)\frac{{\dot{\delta }}}{\delta } < 0$, which is consistent with the empirical facts. Since $\frac{1 – \alpha – \beta }{{1 – \alpha }}\left( {\frac{{\dot{A}}}{A} + \frac{{\dot{L}}}{L}} \right) + \frac{\beta }{1 – \alpha }\left( {\frac{{\dot{B}}}{B} + \frac{{\dot{K}}}{K}} \right) > 0$, the growth rate of carbon emissions, $\frac{{\dot{D}}}{D}$, decreases as the rate of green technological progress, $\frac{{\dot{\delta }}}{\delta }$, continues to increase. When the growth rate of carbon emissions $\frac{{\dot{D}}}{D} < 0$, it means that carbon emissions start to decrease continuously; when the growth rate of carbon emissions $\frac{{\dot{D}}}{D} = 0$, carbon emissions reach the maximum peak. Figure 3 shows the correlation between the growth rate of carbon emissions $\left( {\frac{{\dot{D}}}{D}} \right)$ and the rate of green technology progress $\left( {\frac{{\dot{\delta }}}{\delta }} \right)$ in this case.

As shown in Fig. 4, the carbon emissions at this point can be expressed as:

As shown in Fig. 4, carbon emissions first increase and then decrease with the progress of green technology. When the growth rate of carbon emissions x = 0, the carbon peak is achieved, and thereafter, as the level of green technology is further improved, the growth rate of carbon emissions drops to a negative value, and carbon emissions continue to decrease.

From Eq. (13), when carbon peaking is achieved, it needs to be satisfied:

$$\frac{{\dot{D}}}{D}{ = }\frac{1 – \alpha – \beta }{{1 – \alpha }}\left( {\frac{{\dot{A}}}{A} + \frac{{\dot{L}}}{L}} \right) + \frac{\beta }{1 – \alpha }\left( {\frac{{\dot{B}}}{B} + \frac{{\dot{K}}}{K}} \right) + \left( {\frac{\alpha }{1 – \alpha } – \lambda } \right)\frac{{\dot{\delta }}}{\delta } = 0$$

(19)

At this point, the direct effect of green technological progress, i.e., the reduction in the rate of growth of carbon emissions facilitated by green technological progress, is offset by the increase in the rate of growth of carbon emissions brought about by economic growth, and carbon peaking is achieved. At peak carbon, the level of green technology can be expressed as:

$$\frac{{\dot{\delta }}}{\delta } = {{\left[ {\frac{1 – \alpha – \beta }{{1 – \alpha }}\left( {\frac{{\dot{A}}}{A} + \frac{{\dot{L}}}{L}} \right) + \frac{\beta }{1 – \alpha }\left( {\frac{{\dot{B}}}{B} + \frac{{\dot{K}}}{K}} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {\frac{1 – \alpha – \beta }{{1 – \alpha }}\left( {\frac{{\dot{A}}}{A} + \frac{{\dot{L}}}{L}} \right) + \frac{\beta }{1 – \alpha }\left( {\frac{{\dot{B}}}{B} + \frac{{\dot{K}}}{K}} \right)} \right]} {\left( {\lambda – \frac{\alpha }{1 – \alpha }} \right)}}} \right. \kern-0pt} {\left( {\lambda – \frac{\alpha }{1 – \alpha }} \right)}}$$

(20)

The combined analysis of the two cases and the empirical facts show that $\frac{\alpha }{1 – \alpha } – \lambda < 0$ holds, thus leading to theoretical proposition 3:

Proposition 3

The technological effect of the rate of green technological progress on the growth rate of carbon emissions is greater than the scale effect, thus driving the growth rate of carbon emissions to decrease. When the growth rate of carbon emissions is 0, a carbon peak is achieved; when the growth rate of carbon emissions is reduced to negative, carbon emission reduction is achieved.

The economic meaning of proposition 3 is that the final impact of the green technology progress rate on the growth rate of carbon emissions is negative; that is, the green technology progress rate to promote the growth rate of carbon emissions continues to decline. It can be seen from formula (19) that when the rate of green technology progress drives the growth rate of carbon emissions to reduce to 0, a carbon peak is achieved; when the growth rate of carbon emissions is reduced to negative, carbon emissions begin to decrease, and carbon emission reduction is achieved. Proposition 3 explains the inverted “U”-shaped trend of carbon emissions in the empirical facts (as shown in Fig. 4), providing a theoretical basis for carbon peak and carbon neutrality from the perspective of the rate of progress of green technology.

2 days ago

0 2 11 minutes read