Introduction

Energy, as the driving force of national economic development and an important material foundation of national security, its development and consumption issues have become the focus of academic and practical circles. So far, the mode of energy utilization has gone through the three stages: fuelwood age, coal age, and oil and gas age. Every change has had a dramatic influence on the development of economies and society. In the twenty first century, another mode or energy utilization — renewable energy — is becoming increasingly popular, which is determined by two factors. On the one hand, there is an increasingly prominent contradiction between economic development and the shortage of fossil energy sources. For instance, the 2008 International Monetary Fund (IMF) report (IMF, 2008) stated that the remain oil production capacity of Organization of Petroleum Exporting Countries (OPEC) was 5–7 million barrels during 2000–2002 and that this capacity was below one million barrels during 2004–2006. With the further increase of energy consumption demand, energy shortage has become the bottleneck of economic and social development all over the world. On the other hand, fossil energy consumption has caused serious environmental pollution problems. Fossil energy consumption emits a lot of pollutants, such as sulfur dioxide, carbon dioxide, oxynitrides, and dust, which has caused great damage and pollution to the ecological environment and adversely affected the long-term economic development (Zhang and Wu, 2011).

In view of the limitations of fossil energy and the pressure on the ecological environment, major countries in the world have put forward a series of policy measures to accelerate the development of renewable energy from the aspects of economy, politics, and technology, which have greatly improved the scale and technological level of the renewable energy industry. The International Renewable Energy Agency (IRENA) released the Global Energy Transformation: 2050 road map (2019 edition) in April 2019. It points out that electricity has become the main energy carrier. Meanwhile, renewable energy already accounts for more than half of newly installed electricity generation, and a big shift toward using renewable energy to generate electricity could lead to a 60% reduction in carbon dioxide emissions. When heating and transport renewables are taken into account, the figure is 75%. The developed countries and regions represented by Europe and the USA regard the development of renewable energy as an important measure to improve the ecological environment and get out of the weak economic situation. They have made great achievements. Europe’s renewable energy industry is growing at more than 20% per year. Moreover, the USA expects that renewable energy will provide more than 25% of its electricity by 2025.

To some extent, the exploitation of renewable energy, in addition to technological progress, is an important way to overcome the problems of economic collapse and environmental pollution caused by resource depletion. In this paper, both renewable energy consumption and fossil energy consumption are added to an endogenous economic growth model. Based on the theoretical analysis of the impact of renewable energy substitution on economic growth, this paper empirically examines the effect of the former on the latter. The marginal contribution comes in two aspects. First, different from the previous analyses of scholars based on the perspective of output or consumption, this paper introduces energy consumption into the Solow growth model based on the perspective of value volume, analyzes the impact of the proportion of renewable energy consumption on economic growth, and puts forward a new idea of theoretical research on the impact of energy consumption on economic growth. Second, based on the national level, this paper examines the effect of renewable energy consumption substitution on economic growth and puts forward more suitable and effective policy references, to promote the development of renewable energy and improve the quality and efficiency of energy conservation, emission reduction, and economic growth.

Our study aims to add renewable energy consumption and fossil energy consumption to an endogenous economic growth model. To the best of our knowledge, none of the existing studies has introduced renewable and non-renewable energy variables to Solow growth model. Second, based on the Solow growth model, we analyzed the influence of the proportion of renewable energy consumption on economic growth, and a new way to study the influence of energy consumption on economic growth was proposed. This is different from the viewpoint of other scholars based on output or consumption quantity. In addition, this is based on the national level, examined the impact of renewable energy consumption on economic growth, and proposed appropriate and effective policy references to promote the development of renewable energy and improve the efficiency of energy conservation and emission reduction. The rest of the article is structured as follows. The second section discusses literature review. The third section is the theoretical analysis to incorporate energy consumption into the Solow growth model. The fourth section introduces the data and economic models used and conducts regression analysis and discussion. Finally, the conclusion and policy implications are reported in the fifth section.

Literature review

Renewable energy has been studied earlier by some scholars. Clark et al. (1979) used a dynamic mathematical model to study the utilization of renewable resources. However, there were few research results over the next 20 years. At the beginning of the twenty first century, with the increasingly serious problem of energy shortage, the research on renewable energy gradually became a hot topic. Following, Grimaud and Rougé (2003) assumed that it is necessary to consider not only fuel energy, such as coal, oil, and gas, but also renewable energy when researching the economic growth effects of energy systems. Renewable energy mainly includes solar energy, wind energy, and water energy. Shi and Wang (2015) sorted out and summarized the historical track and characteristics of the energy revolution and believed that building a new energy system focusing on clean energy supply and changing energy consumption mode is the key to the energy revolution and can achieve sustainable economic development. The research on the economic impact of renewable energy in the academic circle has made some achievements in both theoretical discussion and empirical test.

Theoretical research

The theoretical research mainly includes two aspects: first, the renewable energy or the energy resources are directly defined as renewable, introduced them to the economic growth model, and discussed the relationship between the utilization of renewable energy, sustainable economic growth, and environmental pollution. The conclusions showed that the rational development of renewable energy can realize sustainable development of the economy (Kama, 2001; Koskela et al., 2002; Hartwick et al., 2003; Lúdvik and Turnovsky, 2004; Tao et al., 2008). Second, some studies introduced both renewable and non-renewable energy to the economic growth model, mathematically analyzed the impact on economic growth of renewable energy on the replacement of non-renewable energy, and believed that increasing renewable energy investment is an important way to achieve sustainable economic development (Vita, 2006; Bastianoni et al., 2009; Barreto, 2018). However, some scholars drew different conclusions. Zhang and Wei (2014) introduced non-renewable energy and renewable energy factors into Lucas endogenous growth model. The analysis showed that both renewable energy and non-renewable energy consumption can promote economic growth, but the impact of the former is less than that of the latter. Therefore, renewable energy substitution is not conducive to economic growth.

Empirical research

Different from the scholars in theoretical research who believe that the utilization of renewable energy and the substitution of renewable energy for non-renewable energy are conducive to the realization of sustainable economic development, the scholars in empirical research have not reached a consensus. According to the research conclusion, it can be divided into three parts:

First, most scholars support that the development of renewable energy is conducive to the realization of sustainable economic development and believe that the utilization of renewable energy can promote sustainable economic growth through energy conservation and emission reduction and optimizing the energy utilization structure (Apergis and Payne, 2010, 2012; Fang, 2011; Tugcu, et al., 2012; Ocal and Aslan, 2013; Ozturk and Bilgili, 2015; Cho et al., 2015; Inglesi-Lotz, 2016; Alper and Oguz, 2016; Yazdi and Shakouri, 2017; Saint Akadiri et al., 2019; Wang Y, et al., 2020; Rahman and Velayutham, 2020; Shahbaz et al., 2020; Usman et al., 2021).

Second, the impact of renewable energy development on economic growth has a threshold effect and conditions. Some scholars believe that the impact of renewable energy consumption on economic growth is not certain, which depends on how much renewable energy is used. Only when the specific threshold of renewable energy consumption is exceeded, the impact of renewable energy consumption on economic growth is positive and significant (Salim and Rafiq, 2012; Al-Mulali et al., 2013; Apergis and Salim, 2015; Zhang and Liu, 2015; Bilgili and Ozturk, 2015; Yao and Zhang, 2019; Ridzuan et al., 2020).

Third, there is an economic cost in applying renewable energy, and it is not benefit to economic growth. Ocal and Aslan (2013) used the autoregressive distribution lag (ARDL) method to study the causal relationship between renewable energy consumption and economic growth in Turkey. The result shows that renewable energy consumption has a negative impact on economic growth. Based on PVAR model, impulse response function, and variance decomposition analysis, Qi and Li (2017) empirically studied the impact of renewable energy consumption growth on economic growth in 18 EU member states from 1990 to 2014. The result shows that the growth of renewable energy consumption will have a certain negative impact on economic growth. Qi and Li (2018) further expanded the sample size to 28 EU member states and switched to the panel threshold model estimation method for research and found that the impact of renewable energy consumption substitution on the actual GDP is always negative, but the degree has changed. Aydoğan and Vardar (2020) estimates that there is a positive relationship between CO2 emissions and real GDP, non-renewable energy consumption, and agricultural value added in the long run, whereas a negative relationship is represented between CO2 emissions and square of real GDP and renewable energy consumption.

It can be seen that at this stage, domestic and foreign scholars have conducted in-depth research on renewable energy consumption and the economic effects when replacing the non-renewable energy with renewable energy and achieved a series of research results. On the whole, the theoretical research conclusions are relatively unified. Most scholars believe that renewable energy consumption is beneficial to the sustainable development of economy. In empirical research, different scholars have reached different conclusions due to different research samples and variable. At the same time, the domestic theoretical research on the economic effect of renewable energy substitution is relatively insufficient, and the existing research also draws lessons from foreign research for mathematical derivation, so the innovation is relatively insufficient. This paper introduces renewable energy consumption and non-renewable energy consumption into the economic growth model from the perspective of value quantity and mathematically analyzes the economic growth effect of renewable energy substitution and conducts an empirical study at the global national level and provides a policy reference for learning from the development experience of renewable energy in developed countries such as Europe and the USA and realizing high-quality economic growth in China.

Theoretical analysis

This section introduces energy consumption factors, including fossil energy and renewable energy, into Solow growth model (Solow, 1956) and draws on Barro (1990) and Davoodi and Zou (1998)’s analytical ideas and framework, to analyze the economic growth impact of renewable energy replacing traditional fossil energy.

Basic assumptions

First, the production function is expressed in the form of a C-D function. The output variables are economic output (\(Y\)), and the input factors include capital (\(K\)), labor (\(L\)), non-renewable energy (\(F\)), renewable energy (\(R\)), and exogenous technology (\(A\)). To simplify the analysis, economic output (\(Y\)), physical capital (\(K\)), non-renewable energy (\(F\)), and renewable energy (\(R\)) all use the per capita form table,Footnote 1 so the production function of per capita economic growth can be set as

$$y = Ak^{\alpha } f^{\beta } r^{\gamma }$$
(1)

In formula (1), \(y\) represents the output level per capita, and \(k\), \(f\), \(r\), and \(A\) represent the material capital per capita, the consumption of non-renewable energy per capita, the consumption of renewable energy per capita, and the rate of technological progress, respectively, further assuming that the technical-level will remain unchanged in the short term. The development of new energy is cost-constrained; \(\alpha\), \(\beta\), and \(\gamma\) represent the elastic coefficient, that is, \(0 < \alpha < 1\), \(0 < \beta < 1\), \(0 < \gamma < 1\), and \(\alpha + \beta + \gamma = 1\). And it is further assumed that each production factor satisfies the law of diminishing marginal output; that is, \(y_{k} > 0\), \(y_{kk} < 0\), \(y_{f} > 0\), \(y_{ff} < 0\), \(y_{r} > 0\), and \(y_{rr} < 0\).

Second, in the consumer utility function, consumers get utility from other consumption (\(c\)), non-renewable energy consumption (\(f\)), and renewable energy consumption (\(r\)); that is, their functional form is \(u = u(c,f,r)\). And it is further assumed that its specific form is

$$u(c,f,r) = \frac{{c^{1 - \sigma } - 1}}{1 - \sigma } + \frac{{f^{1 - \sigma } }}{1 - \sigma } + \frac{{r^{1 - \sigma } }}{1 - \sigma }$$
(2)

In formula (2), \(\sigma\) represents the risk aversion coefficient and \(\sigma > 0\). And it is further assumed that the consumption function satisfies \(u_{c} > 0\), \(u_{cc} < 0\), \(u_{f} > 0\), \(u_{ff} < 0\), \(u_{r} > 0\), and \(u_{rr} < 0\).

Third, in the energy consumption function, the existing literature on the energy consumption or production function is based on the perspective of either output or consumption. The shortcoming is that the strong assumption that energy production and consumption are the same does not match reality. Based on the perspective of value, this paper believes that the consumption of energy (non-renewable energy and renewable energy) depends on the price of energy (\(e\)) and purchase input (\(p_{e}\)) (\(e = y_{e} /p_{e}\)) and further assumes that \(y_{E} = \vartheta y\), where \(y\) is national income and \(\vartheta\) is the energy consumption proportion of national income. \(\phi = \vartheta /p_{e}\) is further assumed that, and then get the non-renewable energy consumption function, renewable energy consumption function and the total consumption function as follows:

$$e_{f} = \phi_{f} y$$
(3)
$$e_{r} = \phi_{r} y$$
(4)
$$e = e_{f} + e_{r} = (\phi_{f} + \phi_{r} )y$$
(5)

To analyze the proportion of renewable energy consumption, the existence of \(f = \theta_{f} e\), \(r = \theta_{r} e\), and \(\theta_{f} + \theta_{r} = 1\) are further reduced, where \(\theta_{r}\) can represent the proportion of renewable energy consumption.

Fourth, in the accumulation function of physical capital, it is assumed that all economic income is used for material capital accumulation, energy purchase, and consumption. Therefore, the dynamic equation for material capital accumulation is

$$\dot{k} = (1 - \phi_{f} - \phi_{r} )y - c$$
(6)

Model construction and analysis

According to the theory of dynamic optimization, the problem of dynamic optimization in this paper is how to achieve the goal of maximizing consumer utility under the three constraints of production function, energy consumption function, and material capital accumulation function. The dynamic optimal plan is

$$Max\int_{0}^{ + \infty } {u(c,f,r)e^{ - \rho t} dt}$$
(7)
$$u(c,f,r) = \frac{{c^{1 - \sigma } - 1}}{1 - \sigma } + \frac{{f^{1 - \sigma } }}{1 - \sigma } + \frac{{r^{1 - \sigma } }}{1 - \sigma }$$
$$s.t.y={Ak}^{a}{f}^{\beta }{r}^{\gamma }$$
$$e = (\phi_{f} + \phi_{r} )y$$
$$\dot{k} = (1 - \phi_{f} - \phi_{r} )y - c$$

Equation (7) represents the consumer utility maximization function, which represents the time discount rate.

The decision-making problem of representative consumers is an optimization problem. To solve this problem, the present value Hamiltonian function can be constructed in the form of

$$H = u(c,f,r) + \lambda [(1 - \phi_{f} - \phi_{r} )y - c]$$
(8)

In Eq. (8), \(\lambda\) represents the Hamiltonian multiplier, \(c\) represents the control variable, and \(k\) represents the state variable.

To solve the consumer utility maximization problem, for the Hamiltonian to find the first-order partial derivatives of other consumption and material capital, it can get

$$\left\{ \begin{gathered} \partial H/\partial c = c^{ - \sigma } - \lambda = 0 \hfill \\ \partial H/\partial k = \lambda (1 - \phi_{f} - \phi_{r} )A\alpha k^{\alpha - 1} f^{\beta } r^{\gamma } = \rho \lambda - \dot{\lambda } \hfill \\ \end{gathered} \right.$$
(9)

According to the optimality condition (9)

$$\left\{ \begin{gathered} c^{ - \sigma } = \lambda \hfill \\ (1 - \phi_{f} - \phi_{r} )A\alpha k^{\alpha - 1} f^{\beta } r^{\gamma } = \rho - \dot{\lambda }/\lambda \hfill \\ \end{gathered} \right.$$
(10)

Equation (11) can get from Eq. (10):

$$\frac{{\dot{\lambda }}}{\lambda } = \frac{d\ln \lambda }{{dt}} = \frac{{d\ln c^{ - \sigma } }}{dt} = \frac{ - \sigma d\ln c}{{dt}} = - \sigma \frac{{\dot{c}}}{c}$$
(11)

Under balanced economic growth, the economic growth rate is equal to the consumption growth rate:

$$g_{y} = \frac{1}{\sigma }[(1 - \phi_{f} - \phi_{r} )A\alpha k^{\alpha } f^{\beta } r^{\gamma } - \rho ]$$
(12)

Equation (13) is obtained by transforming the first term in the parentheses of Eq. (12):

$$g_{y} = \frac{1}{\sigma }[(1 - \phi_{f} - \phi_{r} )A\alpha k^{\alpha - 1} f^{\beta } r^{\gamma } (e^{1 - \alpha - \beta - \gamma } ) - \rho ]$$
$$= \frac{1}{\sigma }[(1 - \phi_{f} - \phi_{r} )A\alpha (\frac{k}{e})^{\alpha - 1} (\frac{f}{e})^{\beta } (\frac{r}{e})^{\gamma } - \rho ]$$
(13)

According to budget constraints

$$\frac{k}{e} = \frac{{[e/(\phi_{f} + \phi_{r} )]^{{\frac{1}{\alpha }}} /(Af^{\beta } r^{\gamma } )^{{\frac{1}{\alpha }}} }}{e} = e^{{\frac{1 - \alpha }{\alpha }}} (\phi_{f} + \phi_{r} )^{{ - \frac{1}{\alpha }}} A^{{ - \frac{1}{\alpha }}} f^{{ - \frac{\beta }{\alpha }}} r^{{ - \frac{\gamma }{\alpha }}}$$
(14)

Substituting \(\alpha + \beta + \gamma = 1\) into the above formula gives

$$\frac{k}{e} = (\phi_{f} + \phi_{r} )^{{ - \frac{1}{\alpha }}} A^{{ - \frac{1}{\alpha }}} (\frac{f}{e})^{{ - \frac{\beta }{\alpha }}} (\frac{r}{e})^{{ - \frac{\gamma }{\alpha }}}$$
(15)

Substituting Eq. (15) into the economic growth rate Eq. (13), the explicit relationship between the economic growth rate on the equilibrium path and renewable energy substitution can be obtained:

$$g_{y} = \frac{1}{\sigma }[(1 - \phi_{f} - \phi_{r} )\alpha (\phi_{f} + \phi_{r} )^{{\frac{1 - \alpha }{\alpha }}} A^{{\frac{1}{\alpha }}} (\frac{f}{e})_{{}}^{{\frac{\beta }{\alpha }}} (\frac{r}{e})_{{}}^{{\frac{\gamma }{\alpha }}} - \rho ]$$
(16)

Substituting \(f/e = 1 - r/e\) into the above formula, the partial derivative of (\(r/e\)) is found:

$$\frac{{\partial g}_{y}}{\partial \left(r/e\right)}=\frac{1}{\sigma }\left(1-{\varphi }_{f}-{\varphi }_{r}\right)\alpha {\left({\varphi }_{f}+{\varphi }_{r}\right)}^{\frac{1-\alpha }{\alpha } \frac{1}{{A}^{\alpha }}}{\left(\frac{f}{e}\right)}^{\frac{\beta }{\alpha }-1}{\left(\frac{r}{e}\right)}^{\frac{\gamma }{\alpha }-1}\frac{\gamma -\left(\beta +\gamma \right)\left(r/e\right)}{\alpha }$$
(17)

Inequality (18) is obtained by formula (17):

$$\begin{array}{cc}\left\{\begin{array}{c}{\partial g}_{y}/\partial \left(r/e\right)>0,\\ {\partial g}_{y}/\left(r/e\right)<0,\end{array}\right.& \begin{array}{c}r/e<\gamma /\left(\beta +\gamma \right)\\ r/e>\gamma /\left(\beta +\gamma \right)\end{array}\end{array}$$
(18)

The proposition to be tested in this article:

With the increase in the proportion of renewable energy, economic growth presents an inverse “U-shaped” curve.

The changing law of the inverted U-shaped curve obtained by our research is actually in line with reality, although all countries in the world are vigorously developing new energy at this stage.

Although all countries in the world are vigorously developing new energy at this stage, in the short term, traditional fossil energy is still the main energy consumption, and renewable energy is at a moderate development stage. It should be noted that the theoretical analysis also implies a strong assumption that the development technology of renewable energy remains unchanged in the short term and the development of renewable energy is subject to cost constraints. In fact, as the level of renewable energy development technology increases and costs decrease, the highest point of the inverted U-shaped curve will move further to the right until it disappears. This is also in line with the evolution of the energy industry.

Empirical analysis

Data selection and preliminary observation of variables

According to the ranking of the “Renewable Energy Country Attractiveness Index (RECAI)” announced in 2019, the top 40 countries were selected as the initial research samples. Selected according to the availability of relevant indicators, 34 countries were selected as the research samples, including Argentina, Egypt, Ireland, Australia, Pakistan, Brazil, Belgium, Poland, Denmark, Germany, France, the Philippines, Finland, South Korea, Netherlands, Canada, the USA, Peru, Mexico, South Africa, Norway, Portugal, Japan, Sweden, Thailand, Turkey, Spain, Greece, Italy, India, Indonesia, the UK, Chile, and China. During the study period, according to the data published in the “BP World Energy Statistics Report,” it can be calculated that the proportion of renewable energy consumption in 34 target countries (regions) in the world has been around 70%, which is quite representative (Fig. 1). In terms of indicator selection, taking hydro-power consumption and other renewable energy consumption (including wind, geothermal, solar, and garbage power generation) as the total consumption of renewable energy (unit, million tons of oil equivalent); taking the consumption of coal, oil, natural gas, and nuclear energy as non-renewable energy consumption (unit, million tons of oil equivalent); and taking the ratio of renewable energy consumption to total energy consumption (total renewable and non-renewable energy consumption) as the status of renewable energy alternatives of each country (unit, %), the data are derived from the BP Statistical Review of World Energy (2010–2018). The data on economic growth for each country use 2000 as the base period are expressed in actual gross domestic product (GDP) after eliminating inflation and are derived from the WDI database. At the same time, to ensure the stability and convergence of the data, the natural logarithm is taken for both renewable energy and GDP. Taking the data of 34 countries from the year of 2000 to 2017 as the study sample, taking the natural logarithm of renewable energy alternatives as the abscissa, and taking the natural logarithm of GDP as the ordinate, the resulting scatter diagram is shown in Fig. 2.

Fig. 1
figure 1

Distribution in 34 target countries

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Fig. 2
figure 2

Scatter diagram of the relationship between renewable energy substitution and economic growth

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The goodness-of-fit of the binomial trend line between renewable energy substitution and economic growth in Fig. 1 is 0.2134, which is higher than the linear trend direction (0.0159). Therefore, it is preliminarily judged that there is a U-shaped relationship between the replacement of renewable energy and economic growth. With the increase of the proportion of renewable energy substitution, economic growth shows a trend of first increasing and then decreasing. The proposition put forward by the theoretical analysis is preliminarily verified. This paper further uses metrological regression analysis method to conduct a rigorous empirical test.

Model building and regression analysis

Based on theoretical analysis and empirical observations, the regression model between renewable energy substitution and economic growth is set as follows:

$$\ln GDP_{it} = C + \alpha \ln R_{it} + \beta (\ln R_{it} )^{2} + \phi Z_{it} + \varepsilon_{it}$$

where \(\ln GDP_{it}\) and \(\ln R_{it}\) represent the GDP and the share of renewable energy, respectively, in year \(t\) of country \(i\) and are directly represented by the data calculated in the previous section. \(C\) represents the constant terms; \(\alpha\), \(\beta\), and \(\phi\) are the parameters to be estimated; and \(\varepsilon\) is the residual term. \(Z\) represents the control variables. Regarding the selection of control variables, capital and labor are the two core elements of economic growth, and their contribution to economic growth is self-evident. Therefore, they are introduced as necessary control variables into the regression model. The upgrading of industrial structure is also an important factor affecting economic growth, which provides new momentum for economic growth by optimizing the allocation of resource elements and eliminating outdated production capacity. Under the conditions of economic development, international trade is another important factor for economic growth. Various countries and regions exchange their presence through import and export trade to achieve economic development. The original data missing some years of each variable is obtained by interpolation. The selection and sources of variables are shown in Table 1.

Table 1 Variable selection and sources
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The statistical description of each variable is shown in Table 2.

Table 2 Statistical description of variables
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To avoid the problem of parameter estimation error and inconsistency of estimated variables within the group caused by the endogenous problem of independent variables, a systematic GMM regression method is used for regression analysis. The results are shown in Table 3.

Table 3 Regression results
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Table 3 shows that the \(p\) values of the regression equations \(AR(1)\) and \(AR(2)\) are less than 0.01 and greater than 0.1, respectively, under the conditions of control area and year, which indicates that the residual sequence of each equation has first-order auto-correlation and no second-order auto-correlation; the \(p\) value of the Sargan test is also greater than 0.1, which indicates that the equation is over-recognized effectively. At the same time, each variable also passes the significance test at a given level, so the set dynamic panel regression model is more ideal and basically conforms to the reality. From the influence of various variables, early economic growth is an important factor affecting current economic growth, and its influence coefficient is between 0.55 and 0.75. The proportion of renewable energy in total energy consumption \(\ln R\) increased by 1%, and economic growth increased by approximately 0.42–0.54%. This also provides empirical support for policy planning regarding developing renewable energy and replacing the traditional fossil energy of countries in the twenty first century, especially in the developed countries of Europe and the USA. The vigorous development of renewable energy can improve the quality and quantity of economic growth in many respects, such as stimulating investment, relieving the pressure of fossil energy demand, energy saving, and environmental protection. The square of the proportion of renewable energy consumption (\((\ln R)^{2}\)) has a significant negative impact on economic growth, which increases 1%, and the rate of economic growth decreases by approximately 0.052–0.066%. This shows that with the increase of the ratio of renewable energy to fossil energy substitution, there is a U-shaped relationship between this ratio and economic growth, which again verifies the proposition put forward by theoretical analysis. This paper argues that the reasons for the increase in the proportion of renewable energy and the decline in economic growth may be as follows: first, the marginal effect is diminishing. With the increase of the degree of new energy development, the input and cost of new energy development will be accelerated by the constraints of resource endowment, development technology, and cost and will exceed the marginal contribution to economic growth; second, the increase of the proportion of new energy consumption has, to some extent, squeezed out the positive effect of the traditional fossil energy consumption, especially for energy resources enrichment areas. However, with the improvement of the technology level, the decrease of marginal costs, and the strengthening of fossil energy constraints, this optimal substitution ratio may be further improved, which is also the focus of the author’s next stage of tracking research.

According to the method of finding the maximum value of quadratic function, the optimal ratio of renewable energy to total energy consumption can be calculated, and the result is approximately 58%. Based on the previous statistical analysis of 34 countries, the vast majority of countries have not yet reached that level; therefore, it is feasible for these countries to formulate corresponding renewable energy development plans, moderate the development of renewable energy, and carry out effective substitution of fossil energy consumption to make important contributions to energy saving, emissions’ reduction, and quality and efficiency improvement in economic development.

Our research conclusions on the relationship between the proportion of renewable energy consumption and economic growth are obviously different from those of other scholars in the literature review, the relationship between the proportion of renewable energy consumption and economic growth is an inverted “U” shape, which is neither linear nor “U” Shape. Renewable energy, as a new energy and new industrial activity, has a threshold in its economic value. In the initial development and application process, its economic value continues to increase with technological innovation. Once the threshold is exceeded, its economic value will show a downward trend as the proportion of renewable energy consumption increases. According to the calculation result of the optimal value in Table 1, the threshold is about 58%. According to the previous statistical analysis of 34 countries, most countries have not yet reached this level. Since the 2008 global financial crisis, major countries and regions in the world have taken the development of the renewable energy industry as an important measure to overcome economic weakness and global warming and have formulated a roadmap for renewable energy to replace fossil energy in the context of the energy transition. In general, most countries, such as the USA and China, hope that the proportion of new energy consumption will reach about 50% by 2050. Our conclusion shows that this strategic route is feasible. Appropriate development of renewable energy and effective replacement of fossil energy consumption can play an important role in energy saving, emission reduction, quality improvement, and efficiency improvement of economic development.

From the effect of control variables, the increase of total fixed capital, labor input, industrial level coefficient, and total import and export trade also has significant positive impact on economic growth. Therefore, countries around the world can also improve their level of economic growth by promoting the accumulation of fixed capital, expanding effective labor input, improving industrial structure, and expanding foreign trade.

Robustness test

To ensure the validity of the estimation results, the following three robustness tests are further carried out: the first one takes the sample countries which deleted the five with the largest share of renewable energy consumption, the second one takes the sample countries which deleted the five with the smallest share of renewable energy consumption, and the third test takes the sample countries which deleted the three with the largest share of renewable energy consumption and the three with the smallest share of renewable energy consumption. The system GMM method is again used for regression analysis of the three groups of data. The regression results of the three robustness tests are shown in Table 4.

Table 4 The robustness test
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It can be seen that the effect of the explanatory variables on economic growth obtained by the three sets of regression equations is consistent with the regression results in Table 1; the direction of influence has not changed, and the only difference is the degree of influence, which also indicates that the regression results in Table 1 are robust and reliable and can reflect the impact of the increase in the proportion of renewable energy replacing fossil energy on economic growth in 34 countries.

Conclusions and policy recommendations

This paper studies the effect on economic growth of replacing fossil energy with renewable energy. First, the energy factors that affect economic growth are divided into two types: fossil energy and renewable energy. Second, fossil energy and renewable energy are introduced into the Solow growth model to analyze the influence on economic growth with the various proportions of renewable energy in total energy consumption. Finally, the theoretical analysis results show that in the short term when the technological level, especially the technical-level of renewable energy development and utilization, remains unchanged, the economic growth presents an inverted U-shape, whereby the growth rate initially increases and then decreases with the increasing proportion of renewable energy in total energy consumption. In addition, based on the latest global ranking of the “Renewable Energy Country Attractiveness Index” (RECAI), published by Ernst and Young in 2019, select 34 countries as samples through screening to conduct empirical research on the impact of renewable energy substitution on economic growth. The results show that there is a significant inverted U-shaped relationship between the share of renewable energy in total energy consumption and economic growth. The optimal value of the share of renewable energy in total energy consumption is approximately 58%. Most countries are at the left-hand side of the inverted U-shaped curve, and moderate development of renewable energy to replace traditional fossil energy would play a significant role in driving national economic growth, especially for large energy-consumers.

According to theoretical and empirical analyses, as well as the plans for goals and experience of new energy development, vigorously developing renewable energy, replacing traditional fossil energy consumption with renewable energy consumption, and promoting the upgrading of energy consumption structure are very important driving force for economic growth. Policy recommendations for new energy development are proposed in terms of following aspects.

First, in-depth implementation of the innovation-driven development strategy improves renewable energy research and development technology. Renewable energy research and development technologies and equipment level directly determine the development of renewable energy. The long-term evolution of the energy industry indicates that technology and cost are the biggest constraints for a new energy to replace traditional energies. However, as the technical-level advances, the limits will gradually disappear. Therefore, the first prerequisite for improving the level of renewable energy development is to improve its technology research and development level. On the one hand, it is necessary to strengthen the top-level design of the renewable energy science and technology innovation system, improve the incentive mechanism for scientific and technological innovation, and promote research and development of basic, comprehensive, and strategic renewable energy technologies in a coordinated manner to enhance the overall competitiveness of energy science and technology, foster more technological advantages, and accelerate the transformation to economic advantage. On the other hand, integrate the renewable energy research strength from universities, enterprises, and governments, to build a range of renewable energy innovation centers and laboratories with deeper and broader cooperation. We should establish several technological innovation alliances to promote technological integrated innovation. Moreover, promote research and development of renewable energy equipment and break through bottlenecks in key technologies, materials, and parts in the manufacturing of renewable energy equipment to accelerate developing the ability to manufacture complete sets of major equipment independently. This can be achieved by pushing the upgrading of intelligent manufacturing in the upstream manufacturing industry of renewable energy and enhancing the development quality and efficiency of the entire industrial chain.

Second, adhere to the simultaneous development of centralized development and decentralized utilization, and adjust and optimize the development layout. Power generation of a new energy power station cannot be adjusted and dispatched artificially, and there is a strong connection between this power generation and the natural environment. On the one hand, the power varies with changes of temperature, humidity, day and night, seasons, and other factors, represented by the performance changing with time. On the other hand, indirectness, instability, and randomness are the internal factors that lead to bottlenecks of new energy development. The combination of internal and external factors leads to the inefficient utilization of new energy power, with the result that the structural surplus cannot even be absorbed. Therefore, to solve this problem, two paths can be taken. First, actively develop energy storage, change the dispatching operation mode, accelerate the breakthrough of operation control technologies such as grid balance and self-adaptation, and significantly improve the power system’s capacity for peak shaving and absorption of renewable energy. Second, accelerate the development of smart grids, build distributed energy systems, and convert energy on-site. Distributed power generation refers to the configuration of smaller generator sets (generally less than 30 MW) at the user site or close to the power site to meet the needs of specific users, support the economic operation of the existing distribution network, or meet both aspects at the same time requirements. Promote the application of distributed intelligent energy supply systems in industrial parks, urban communities, public buildings, and private houses and further improve the renewable energy dispatching operation and peak shaving cost compensation mechanism to meet the requirements of large-scale grid-connected renewable energy consumption and solve the problem of abandoning wind, water, and light in some areas.

The last thing to point out is that renewable energy consumption substitution is actually an advancement in energy consumption and utilization. In theory, the substitution of oil for coal and the substitution of natural gas for oil are essentially the same, and both are all advancements in energy utilization technology. Our article does not analyze the uniqueness of renewable energy substitution. On the other hand, our research does not consider the influence of technological progress, especially the technological innovation of the green utilization of fossil energy and the technological progress of the utilization of new energy. With the influence of technological progress, the phenomenon of “reverse substitution” may appear. These two aspects need to be further studied in the future.