Energy is the important material foundation and support of economic and social development of every country and the main driving force of economic growth. It can affect the scale and speed of economic growth. On the other hand, economic growth accelerates the development of energy technology and promotes productivity growth. With the rapid development of economy, the mutual constraint between energy and economy is increasingly emerging.

Many studies have shown that energy plays an important role in economic growth. Luqman, et al.^[1] researched the asymmetric effect of renewable energy and nuclear energy on economic growth in Pakistan using an extended production function. They used the annual data from year 1990 to year 2016 with a nonlinear auto-regressive distributed lag model (NARDL). Analysis results showed that variables had an asymmetric co-integration relationship and the changes of renewable energy variable and nuclear energy variable had positive influences on economic growth. Režný and Bureš^[2] researched the influences of quantity and quality declines of extracTable fossil fuels and subsequent renewable energy technology quality decline on economic growth. They transformed the neoclassical growth model into the form of system dynamics and then extended the model using the important feedback mechanism to propose a new model named EENGM (energy extended neoclassical growth model). Research results showed that economic growth manifested in the form of real GDP (gross domestic product) was closely correlated to the aggravation of energy consumption and the increase in the usage amount of natural resources. Kouton^[3] tried to reveal the asymmetric heterogeneous relationship between energy use and economic growth in 19 African countries from year 1971 to year 2014, for which purpose, he used a panel NARDL and the asymmetric panel causality test. Empirical results showed that energy use and economic growth had an asymmetric relationship which depended on the stages of economic cycle. Kang, et al.^[4] used a trivariate VAR model with probability variability and the time-varying parameter method to determine the dynamic interactions of GDP growth, RES (renewable energy sources, such as wind energy, solar energy or hydropower) and NRES (non-renewable energy sources, such as electricity and coal) and carbon emission. Using a NARDL model, Toumi and Toumi^[5] applied the asymmetric causality into the research on the relationship of REC (renewable energy sources), CE (carbon emission) and real GDP of Kingdom of Saudi Arabia, and analyzed the nonlinear variation relationship of the variables using a NARDL model. Ayinde, et al.^[6] analyzed the relationship between national energy consumption and economic growth, industrial growth and urban growth in Nigeria. Using the data from year 1980 to year 2016, they analyzed the VEC model using the Granger causality, the impulse response function and the variance decomposition. They found a co-integration relationship using the Johansen Cointegration test and then built a VECM (vector error correction model). Ahmed, et al.^[7] studied the dynamic relationship of renewable energy sources, non-renewable energy resources, CO${}_2$ intensity and economic growth. Using the ARDL, the dynamic least square method, the FMOLS, and the Gregory-Hansen co-integration, they analyzed the time series data set in a specific range. Analysis results showed that the use of renewable energy sources could promote economic growth significantly. Gorus and Aydin^[8] chose eight Middle Eastern and North African countries possessing rich oil resources as the research objects and explored the causality of energy consumption, economic growth and carbon emission through single-country and muti-country Granger causality analyses. Comparing with the time-domain causality, through the panel frequency-domain analysis, they found that there were more causalities among variables of different frequencies. The panel causality test results showed that the energy saving policy had no significant short-term or long-term influence and had a negative long-term influence on economic growth. In the long term, energy consumption would promote economic growth. Can and Korkmaz^[9] explored the relationship between renewable energy sources and economic growth in Bulgaria. They used the Toda-Yamamoto analysis and the ARDL constraint test to analyze the relationship between renewable energy sources and economic growth from year 1990 to year 2016 on the basis of annual data. They studied the levels or differences of static variables using the augmented ADF (Dickey-Fuller), PP (Philips-Perron) and KPSS (Kwiatkowski-Philips-Schmidt-Shin) unit root tests. Finally, they found three different results from the research, indicating the variables had no long-term relationship. Sadraoui, et al.^[10] explored the co-integration relationship and Granger causality between economic growth and total energy consumption and the relationship between economic growth and financial development using the panel data analysis of Middle Eastern and Northern African areas from year 2000 to year 2018. Research results showed that energy consumption had a positive and significant influence on economic growth. In addition, they found there was a positive relationship between financial development and economic growth. Therefore, the study on the relationship between total energy usage and financial system was very important before the local government made specific energy and economic policies. Kahouli^[11] studied the relationship between economic growth (EG) and energy consumption (EC) of 34 OECD countries from year 1990 to year 2015. He analyzed the static relationship and, especially, the dynamic relationship, of EG and EC in the OECD by using three models to test the growth-energy relationship, the energy-growth relationship and their bidirectional connection. The empirical results supported the feedback effect between EG and EC. The result offered important suggestions on energy and economic policies to the decision makers of OECD. It ensured the sustainable development of economy and acted as an impulse to look for alternative energy sources to meet the energy demand increasing rapidly in the countries. Saad and Taleb^[12] analyzed and compared the short-term and long-term relationships of renewable energy sources and economic growth of 12 EU countries and drew inspiration for renewable energy policies. Therefore, they used the annual data of 12 EU countries from year 1990 to 2014 and the panel VEC model for an analysis, and then made a Granger causality test to test whether there was the causality between economic growth and renewable energy consumption. Research results showed that economic growth was the Granger cause for renewable energy sources unidirectionally in the short term. However, in the long term, the variables had a bidirectional causality. Tzeremes^[13] used the monthly data from January 1991 to May 2016 to study the relationship between energy consumption and economic growth of America at the level of industry. The researcher checked five sectors using the quantile causality, and results showed that there was the causality at the sectoral level in tails. The researcher first used the quantile causality to explain the relationship between energy consumption and economic growth. Bakirtas and Akpolat^[14] used the dumitrescui-hurlin panel Granger causality test to analyze the relationship of energy consumption, urbanization and economic growth in emerging market economics (Columbia, India, Indonesia, Kenya, Malaysia and Mexico) from year 1971 to year 2014. They used the bivariate and trivariate panel Granger causality analysis method to test the combined causality of two sequences. Research results showed that there was a panel Granger causality between economic growth and energy consumption. Stjepanovic^[15] studied the relationship between energy consumption and economic growth of 30 European countries. He used the panel data analysis method to study the relationship of variables based on fixed effect method. The data used coming from an European database Eurostat. According to the regression result of panel data, the researcher found that variables observed had strong correlations. DurĞun^[16] used a time sequence analysis method to test the causality between GDP and renewable energy sources per capita (including hydropower) of Turkey from year 1980 to year 2015. First, he made the ADF test and the Zivot-Andrews unit root test to the sequence and found the level value contained the unit root, and the progression was stationary in the first-order difference. The ARDL test result showed that the variables' con-integration level has the significance of 5%. He used the Toda-Yamamoto causality test to co-integration sequence and found that renewable energy sources were the unidirectional cause for economic growth. Gokmenoglu and Kaakeh^[17] made an empirical analysis on the relationship between nuclear energy consumption and economic growth of Spain. They analyzed the annual data from year 1968 to year 2014 using the unit root and stationary test, the Johansen co-integration test, the VECM and the Granger causality test. The empirical result showed that two variables had a long-term equilibrium relationship. The Granger causality test result proved that nuclear energy consumption could promote economic growth. Akinwale^[18] studied the short-term and long-term relationships of energy consumption, technological innovation and economic growth of Saudi Arabia. He used the Granger causality to determine the causal direction, and adopted the ARDL to analyze the data from year 1980 to year 2015. Research results indicated that variables had the co-integration relationship and there was a long-term relationship.

China's economy has been developing rapidly since the implementation of reform and opening-up policy, which has attracted a lot of foreign and domestic scholars. The scholars made a great number of analyses on the relationship between China's energy consumption and economic growth using various methods. Hao, et al.^[19] analyzed the dynamic relationship of rural GDP, rural energy consumption and rural investment based on the provincial panel data sets from year 1995 to year 2010. They tested the short-term and long-term causality of variables interested using the VECM and the FMOLS. Then, they analyzed the dynamic influence of process and the contribution of related factors using the impulse response function (IRF) and the variance analysis. The estimation result showed that rural energy consumption was the Granger cause for rural investment unidirectionally. In additional, the research result also indicated that rural GDP and rural energy consumption had the bidirectional causality in the short term. Rathnayaka, et al.^[20] studied the causality between China's energy consumption and economic growth from year 1980 to year 2013. First, they tested the stationarity of variables selected using the unit root test statistics. Then, they used the co-integration of Johansen and VECM to test the dynamic relationship of variables. The result showed that energy consumption and economic growth had the long-term bidirectional causality. Wang, et al.^[21] studied the co-integration, time dynamics and random relationship of China's economic growth, energy consumption and carbon emission using the data from year 1990 to year 2012. The co-integration test result showed that although there was the short-term dynamic adjustment mechanism, variables had a long-term co-integration relationship. In addition, they found through the pulse response analysis that carbon emission's influence on economic growth or energy consumption was slightly significant. Finally, economic growth and energy consumption had the bidirectional causality, and energy consumption was the unidirectional Granger cause for carbon emission. This finding had important significance for developing and implementing long-term energy and economic policies. To analyze the relationship between China's energy consumption and economic growth, Jiang and Bai^[22] adopted an adaptive time-frequency data analysis method for the multi-time-scale analysis on the characteristics of China's energy consumption growth rate. He used an integrated empirical mode decomposition method which was applicable to the analysis on nonlinear time sequence. The Granger causality test result showed that economic growth and energy consumption had the bidirectional causality in the short term, and the unidirectional causality in the long term. Hu, et al.^[23] explored the relationship between energy consumption and economic growth from the perspective of China's industrial sector. The research used the panel data of 37 industrial sectors of China from year 1998 to year 2010 and adopted not only the first generation of panel unit root test and panel co-integration test and also the second generation of test considering the correlation of sectional units. The empirical result showed that energy consumption and economic growth was co-integrated. The panel's FMOLS estimator indicated that the industrial sector's actual value added increased by 0.871% as energy consumption increased by 1%, and energy consumption increased by 1.103% as industrial sector's actual value added increased by 1%. They built a panel vector error correction model of causality test using the SGMM method. The research result showed that energy consumption was the unidirectional cause for economic growth in the long term.

Therefore, energy consumption has a great influence on economic growth. The paper researches the influence of energy consumption on economic growth of China's Yangtze River Economic Zone. The Yangtze River Economic Zone is composed by 11 provinces, cities and areas including Yunnan, Guizhou, Sichuan, Chongqing, Hubei, Hunan, Jiangxi, Anhui, Zhejiang, Jiangsu and Shanghai. The area has the widest hinterland and development space in China, and is developing into the most robust economic zone after coastal developed areas. From the perspective of industry, Yangtze River Economic Zone is a centralized place of most traditional sectors, such as steel and petrochemicals, and modern industrial sectors, such as automobile and electronics, of which the economic aggregate accounts for about 48% of national gross. However, from the perspective of resources, Yangtze River Economic Zone's upstream, midstream and downstream areas all lack energy sources considerably, of which the primary energy self-sufficiency rate is only 52%. In the middle of 1990s, the electricity shortage accounted for about 1/5 of total demand and the coal shortage was 0.17 billion tons or so each year in the Zone. In the future, with the further development, the energy demand will grow exponentially. Related research results show that Yangtze River Economic Zone's energy sources have become an important factor affecting the Zone's sustainable development^[24-28].

The paper makes an empirical analysis on the influence of energy consumption on economic growth in China's Yangtze River Economic Zone with a production function model using the econometrics analysis method. The production function model can reflect the relationship between input and output in economic growth. Generally, the input factors include capital and labor. Because the energy, as an important production factor input, has an increasingly greater influence on economic growth, the paper adds an energy input factor and decomposes the energy input factor into coal input, oil input, natural gas input and electricity input. There are many input factors, so the paper extends the conventional production function model to get an extended production function. Using the extended production function model built, the paper calculates the degrees of influences of four energy input factors on economic growth. The paper mainly uses two methods. The first is the output elasticity analysis. The paper calculates the output elasticity of four energy consumption and then compares the degrees of influences of energy consumption on economic growth from the aspect of output elasticity. The second is the analysis on contribution rate. The paper calculates the contribution rates of four energy consumption to economic growth and then compares the degrees of influences of the energy consumption on economic growth from the aspect of contribution rate^[29-33].

The general form of production function is

where $A$ is technological progress level, $K$ is capital input, $L$ is labor force input, and $Y$ is output.

If we introduce an energy factor, the production function can be expressed to be

where $A$ is technological progress level, $K$ is capital input, $L$ is labor force input, $E$ is energy input, and $Y$ is output.

The paper introduces the energy factor and divides the energy into coal, oil, natural gas and electricity. Because there are many input factors, the paper extends the conventional CES (constant elasticity of substitution) production function model and gets an extended CES production function

where $A(t)=A_0{\mathrm{e}}^{\lambda t}$ is technological progress level, $K$ is capital input, $L$ is labor force input, $(E_1,E_2,E_3,E_4)$ is energy input of coal, oil, natural gas and electricity, respectively, and $Y$ is economic output, which is the gross regional product in this paper.

Then, from the formula above, get the partial derivative

Similarly,

Therefore, the output elasticity of coal in the t${}^{\mathrm{t}\mathrm{h}}$ year is

The output elasticity of oil in the t${}^{\mathrm{t}\mathrm{h}}$ year is

The output elasticity of natural gas in the t${}^{\mathrm{t}\mathrm{h}}$ year is

The output elasticity of electricity in the t${}^{\mathrm{t}\mathrm{h}}$ year is

In this way, we can get the average output elasticity of coal, oil, natural gas and electricity from period 1 to period n, respectively.

The extended CES production function is

Suppose $L_i$ is the curve connecting $M_i(Y_i,A_i,K_i,L_i,E_{1i},E_{2i},E_{3i},E_{4i})$ to $M_{i+1}(Y_{i+1},A_{i+1}$, $K_{i+1},L_{i+1},E_{1,i+1},E_{2,i+1},E_{3,i+1},E_{4,i+1})$ $(i=1,2,\cdot \cdot \cdot ,n-1)$, then its parameter equation is

Therefore, factor A's value of influence on Y's changes in period i is

Factor K's value of influence on Y's changes in period i is

Similarly, factor L's value of influence on Y's changes in period i is

Factor $E_1$'s value of influence on Y's changes in period i is

Factor $E_2$'s value of influence on Y's changes in period i is

Factor $E_3$'s value of influence on Y's changes in period i is

Factor $E_4$'s value of influence on Y's changes in period i is

Write

then coal $E_1$'s contribution rate to economic growth in period i is

oil $E_2$'s contribution rate to economic growth in period i is

natural gas $E_3$'s contribution rate to economic growth in period i is

electricity $E_4$'s contribution rate to economic growth in period i is

And then, coal $E_1$'s contribution rate to economic growth from period 1 to period n is

oil $E_2$'s contribution rate to economic growth from period 1 to period n is

natural gas $E_3$'s contribution rate to economic growth from period 1 to period n is

electricity $E_4$'s contribution rate to economic growth from period 1 to period n is

The paper divides the energy consumption of Yangtze River Economic Zone into the coal consumption, the oil consumption, the natural gas consumption and the electricity consumption. To analyze the influence of energy consumption on economic growth in Yangtze River Economic Zone, the paper selects the gross regional product of Yangtze River Economic Zone (Y, of which the unit is 0.1 billion), fixed-asset investment (K, of which the unit is 0.1 billion), the number of employees (L, of which the unit is 10, 000 people), total coal consumption (E${}_1$, of which the unit is 10, 000 tons of standard coal), total oil consumption (E${}_2$, of which the unit is 10, 000 tons of standard coal), total natural gas consumption (E${}_3$, of which the unit is10, 000 tons of standard coal) and total electricity consumption (E${}_4$, of which the unit is 10,000 tons of standard coal) as indexes. The data came from the China Statistical Yearbook and the China Energy Statistics Yearbook. See Table 1 for the data. The paper uses Matlab 2016A as the data analysis and processing software.

Figure 1 shows the variation curves of gross regional product and energy consumption in Yangtze River Economic Zone.

First, build the following model

For the observed values $(K_i,L_i,E_{1i},E_{2i},E_{3i},E_{4i},Y_i)(i=1,2,\cdot \cdot \cdot ,n)$ known, write as

and let it have the minimum, and then get parameter estimate $\eta =(A_0,\lambda ,{\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4,{\delta }_5,{\delta }_6,\rho ,\mu )$. It is essentially a nonlinear optimization problem.

Calculate using the software to get

i.e.,

The model's coefficient of determination $R^2=1-\frac{\sum _{t=1}^n{(Y_t-{\hat{Y}}_t)}^2}{\sum _{t=1}^n{(Y_t-\overline{Y})}^2}=0.9859.$ It can be seen that the model has high fitting precision and small errors.

Table 2 and Figure 2 show the output elasticity of energy factors in these years calculated with the method given. We can see from the results that the output elasticity of four energy factors changed slightly from year 2000 to year 2016. Calculation results show that oil has the biggest output elasticity of which the average is 0.1945, followed by natural gas with the average of 0.1873, electricity with the average of 0.0819 and natural gas with the average of 0.0755.

Table 3 and Figure 3 show the contribution rates of energy factors to economic growth by years.

Table 3 and Figure 3 show that oil consumption has the biggest contribution rate to economic growth, followed by natural gas consumption, electricity consumption and coal consumption. From year 2000 to year 2016, the consumption of coal, oil, natural gas and electricity had the contribution rate of 2.29%, 15.60%, 8.84% and 6.36% to economic growth, respectively.

The paper mainly uses an extended production function model to analyze the influences of four energy consumption on economic growth in China's Yangtze River Economic Zone quantitatively from two perspectives.

1) The paper uses an extended production function model to calculate the output elasticity of four energy consumption to economic growth in Yangtze River Economic Zone and compares the influences of four energy consumption on economic growth in terms of output elasticity. Results show that coal consumption's output elasticity to economic growth is ${\tau }_{E_1}=0.0775$, oil consumption's output elasticity to economic growth is ${\tau }_{E_2}=0.1945$; natural gas consumption's output elasticity to economic growth is ${\tau }_{E_3}=0.1873$, and electricity consumption's output elasticity to economic growth is ${\tau }_{E_4}=0.0819$. Because ${\tau }_{E_2}>{\tau }_{E_3}>{\tau }_{E_4}>{\tau }_{E_1}$, oil has the greatest influence on economic growth, followed by natural gas, electricity and coal.

2) The paper uses an extended production function model to calculate the contribution rates of four energy consumption to economic growth in Yangtze River Economic Zone and compares the influences of energy consumption on economic growth in terms of contribution rate. Results show that coal consumption's contribution rate to economic growth is $\mathrm{C}\mathrm{R}{\mathrm{E}}_1=0.0229$, oil consumption's contribution rate to economic growth is $\mathrm{C}\mathrm{R}{\mathrm{E}}_1=0.1560$; natural gas consumption's contribution rate to economic growth is $\mathrm{C}\mathrm{R}{\mathrm{E}}_3=0.0884$, and electricity consumption's contribution rate to economic growth is $\mathrm{C}\mathrm{R}{\mathrm{E}}_4=0.0636$. Because $\mathrm{C}\mathrm{R}{\mathrm{E}}_2>\mathrm{C}\mathrm{R}{\mathrm{E}}_3>\mathrm{C}\mathrm{R}{\mathrm{E}}_4>\mathrm{C}\mathrm{R}{\mathrm{E}}_1$, oil has the greatest influence on economic growth, followed by natural gas, electricity and coal.

3) The gross output elasticity of four energy consumption to economic growth in Yangtze River Economic Zone is 0.5412, i.e., the gross regional product of Yangtze River Economic Zone increases by 0.5412% as four energy consumption increases by 1/5. The total contribution rate of four energy consumption to the Zone's economic growth is 33.09% indicating that about one third of economic growth of Yangtze River Economic Zone is brought by energy consumption. From the perspective of output elasticity and contribution rate, among four energy consumption, oil consumption has the greatest influence on the gross regional product of Yangtze River Economic Zone, followed by natural gas, electricity and coal. Overall, the energy consumption has a great influence on economic growth in China's Yangtze River Economic Zone.

1) The neoclassical economic growth theory believes that the economic growth of one country or area is decided by production factor input and technological progress at the same time, and the relationship between input and output reflecting economic growth can be expressed with the production function. Production factors generally include capital and labor while technological progress is exogenous. In an open economy, technological progress is endogenous, and energy, as an important production factor input, has an increasingly greater influence on economic growth. Therefore, the paper introduces the energy factor into the production function and divides the energy into coal, oil, natural gas and electricity. Because there are many input factors, the paper extends the production function and calculates the output elasticity and contribution rates of four energy consumption to economic growth with the extended production function. From the calculation results of output elasticity and contribution rates we can see that among the four energy consumption in China's Yangtze River Economic Zone, oil has the greatest influence on economic growth, followed by natural gas, electricity and coal. In addition, the gross contribution rate of four energy consumption to economic growth is nearly 1/3, indicating the great influences of four energy consumption on economic growth.

2) Different from the methods of other researchers, the paper uses an extended production function model to study the degrees of influences of energy consumption on economic growth from two aspects. The method proposed offers more theoretical reference for the studies on energy consumption's influence on economic growth. The empirical analysis results in the paper offer reference for decision makers to grasp economic growth's relationship with various energy consumption in current stage and for the country to make sustainable development strategies.

3) Research results show that in the future China first needs to improve the utilization efficiency of energy, especially the utilization of conventional energy such as oil, to realize the highly efficient utilization of conventional energy. Then, China needs to optimize the utilization structure of energy and improve the unreasonable energy consumption structure by reducing the proportion of conventional energy such as oil and coal gradually and increasing the proportion of clean energy such as electricity. In addition, China needs to make energy development strategies. In the new situation of reform and development, the energy shall have an increasingly greater influence on economic growth, so it's necessary to make a reasonable energy development plan meeting China's economic development requirements to ensure the good ecological environment and sustainable economic development.