1. INTRODUCTION
The overwhelming majority of the instruments of state regulation facilitating the economic growth are aimed at increasing the accumulation of private capital in physical and intangible forms. Traditionally, pursuing this goal, the government focuses on stimulating private savings and investments and uses such tools as reducing the tax burden and increasing current budget spending to boost effective demand (Slepov, 2017;Balynskaya and Ponomarev, 2018). Investments stimulate economic growth, activating demand in the short term, and in the long run – increasing production capabilities (Petrenko et al., 2018;Takhumova et al., 2018;Golovina and Golovin, 2013). Therefore, the level of investment directly reflects the economic situation of a country. However, in Russia, the share of budget investments against GDP has been constantly decreasing since 1993. Do such trends promote the growth of the Russian economy? To answer this question, it is necessary to analyze the practices of financing capital expenditures in developed countries. On the one hand, considering, for example, European countries, one can see that over the past 40 years, government investment in the largest countries of the European Union and on average in the European Union are declining against GDP. On the other hand, such a decrease occurs only in the most developed European countries, while in less developed one there is growth in or, at least, the same level of government investment against GDP (Välilä et al., 2005;Farzadnia et al., 2017).
It is impossible to increase the efficiency of budget regulation if the share of budget investments in GDP is reduced. According to international experts of the World Bank, sustainable economic growth at 7% of GDP per year over the period of 2025 years can be achieved only at the aggregate investment level of not less than 25% of GDP (more often 30% or higher), whereas in Russia the level of investment is 21 % of GDP (Aleksashenko, 2012). At the same time, the share of budget investment financed from the federal budget does not exceed 2% of GDP. In this paper the authors carry out a consistent analysis of GDP growth rates in selected countries, the dynamics of the extensive factors of economic growth – the growth of the capital stock and the size of labor force, the dynamics of intensive factors of economic growth – the level of R&D expenditures, the volume of foreign direct investment and R&D expenditures (Borensztein et al., 1998). In addition to this, the article considers in detail the impact of R&D expenditures on endogenous growth. Starting from Solow’s paper (Solow, 1957), technical progress has been generally seen as an exogenous factor. In reality, while some new technologies may occur spontaneously, others are the result of R&D expenditures. This predetermined a new line of research which included R&D expenditures in the production function, thus taking into account external factors related to technical progress (Schankerman, 1981;Novikova et al., 2016;Mazur et al., 2016). R&D expenditures can be estimated using the data on the costs of research, education and additional employee training (Kaldor, 1970). However, the method of cost estimate of the investments in knowledge and its depreciation rate are still to be developed. At the same time, there are publications attempting to link the growth of cost factors and technical progress (Bashkirova and Lessovaia, 2018;Muhammad, 2018). One of the main hypotheses states that technical progress, rising costs for research and development, and increased cost factors may be the results of increased knowledge (Schultz, 1962).
The new growth theory and the direction of the neoclassical theory – the theory of capital and investment emphasizes the growth of investment in human capital and knowledge (Fellner, 1969). Let us analyze the structural changes in the economy using the theories of Prebisch, Ali et al. (2016) and Jalalinezhad and Jenaabadi (2014). According to the results of Prebisch’s structuralist development approach and the theories of Kaldor and Thirwall, the goods and sectors of the national economy have different income elasticity of demand. That is why to ensure longterm economic growth, it is necessary to create an effective structure of a country’s trade specialization, in which the income elasticity of export demand grows at a faster rate than the import demand. This can be done by actively promoting a combination of selective (“vertical”) and horizontal instruments of industrial and technological policy. These tools should be aimed at changing the production structure of the developing economy and, thus, the structure of its trade specialization (that is, facilitating “dynamic comparative advantages”), which would enable faster economic development (BresserPereira, 2016;Fomina et al., 2018).
Among numerous works devoted to the comparative crosscountry analysis of the economic growth rates decomposition it is worth mentioning the following papers. The paper (Chen, 1977) analyzes the influence of production factors and total factor productivity (TFP) on the growth of the economies of Asian countries: Japan, Singapore, Taiwan, Korea, and Hong Kong from 1955 to 1970. The economic growth of China since 1952 with a forecast up to 2010 was considered in (Chow and Li, 2002). Determinants of economic growth with a discussion regarding the impact of TFP for the countries of Southeast Asia (Indonesia, Malaysia, Thailand, the Philippines and China) from 1950 to 1988 are considered in (Chen, 1997). The analysis of the main changes in TFP, which took into account the numerical estimates of the nonparametric Malmquist index, is presented in (Krüger, 2003;Gumel, 2017). The work (Chen and Yu, 2014) examines the growth of TFP over the period from 1991 to 2003 in the economies of 99 countries, both members of the OECD and the countries that do not belong to this union. The paper (Levenko et al., 2018) presented the results of the decomposition of the economic growth rates in 11 EU countries of Central and Eastern Europe for the period from 1996 to 2016, before and after the global financial crisis of 2007. This study demonstrated that before the crisis, the growth in TFP was the main driver of economic growth in Slovenia, Hungary and Slovakia, while capital growth was more significant in the Czech Republic, Croatia and Poland. During the global financial crisis, the contribution of TFP and capital growth varied greatly across the countries, reflecting the very diverse dynamics of the crisis. After the crisis, the contribution of TFP growth was insignificant in all countries of the sample, which coincided with the overall weak growth of their economies. A recent paper (Tamura et al., 2019) presents the results of the analysis of longterm trends, from 1800 to 2010, for a sample of 150 countries both regarding their economic growth and the dynamics of the main factors contributing to this growth, including the growth in human capital assets.
Besides, one should mention the results of a recent study (Yalçınkaya et al., 2017), in which the authors analyze the determinants of economic growth and make an attempt to determine whether the influence of the total factor productivity is connected with the development level of particular countries. The study examines the impact of gross fixed capital formation, labor factors and TFP of G7, G12 and G20 countries on real GDP per capita for the period from 1992 to 2014. The results show that TFP has a greater impact on economic growth than the total of fixed capital and labor for all groups of countries. In addition, it was established that the impact of TFP on economic growth in developed countries is stronger than in developing countries. The mutual influence of regulations, institutional development and productivity factors on OECD countries is discussed in (Égert, 2016). This study presents an attempt to explore the driving forces of multifactor productivity (MFP) at the country level, with a special focus on product market and labor market policies and the quality of institutions. The paper found that anticompetitive market rules in the market for products reduce the MFP level for OECD countries, while greater innovation and openness stimulate an increase in MFP. It was also established that the impact of product market regulation on MFP may depend on the level of labor market regulation. Improving institutions, creating more favorable conditions for business and reducing barriers to trade and investment reinforce the positive impact of R&D costs on MFP. It is also shown that differences between MFPs in different countries can be largely explained by differences in labor market regulation, barriers to trade, the volume of investments and the quality of institutions.
The rest of this study is organized in the following way: the second section considers the methodology for comparative crosscountry analysis of the decomposition of economic growth rates and factor productivity based on regression modeling, which was used to quantitatively measure the dependence of the considered variable on factors and to test the adequacy of the constructed model for a certain type of source data, to estimate the model parameters and to identify a set of regressors describing the dynamics of the resulting changes at an acceptable level of significance. The third section discusses sources of input data for regression modeling. The fourth section is devoted to the analysis and discussion of the results. The fifth and the sixth sections summarize the results obtained, and present the conclusion and the areas of further research.
2. METHODOLOGY
In this research, to perform the comparative analysis of the economic growth rate of several different countries for the period from 2000 to 2018, the authors used regression modeling, in which the aggregate production function is represented in the form of a multiplicative, power CobbDouglas function (Douglas, 1976;Felipe and Adams, 2005;Sharif and Butt, 2017). The latter, along with traditional extensive production factors (labor and capital), takes into account the intensive factor connected with the influence of the scientific and technological progress as an independent factor in the growth of the country’s national income (Turner et al., 2013;Ahmadi et al., 2018).
Regression analysis, unlike many other methods of economic growth modeling, allows one to: quantitatively measure the dependence of the considered variable on the factors; to test the accuracy of the constructed model for a certain type of source data; to estimate model parameters and to identify a set of regressors describing the dynamics of the resulting variable at an acceptable level of significance. It should be noted that in modern conditions with open national economies, the sum of elasticities of the gross output by production factors may differ significantly from 1 and indicate both increasing or decreasing aggregate returns from production factors. The elasticity coefficients of the gross output of the countries analyzed by production factors estimated using the regression modeling were applied for clustering the countries to determine the main actions that may stimulate their economic growth.
Let us decompose the economic growth rate into intensive and extensive components. To obtain an expression for the dependence of the economic growth rate on the factors growth rate, first, it is necessary to take the logarithm of both parts of the CobbDouglas production function
(${Y}_{t}={A}_{0}\cdot {K}_{t}^{\alpha}\cdot {L}_{t}^{\beta}$) and then differentiate the resulting expression:
where $\frac{{\dot{A}}_{0}}{{A}_{0}}$ represents the TFP growth rate, or total factor productivity (hereinafter – TFP), parameters α and β are the coefficients of output elasticity of capital and labor.
Thus, the growth rate of the economy is made up of the growth rates of capital, labor, and total factor productivity multiplied by elasticity coefficients of GDP for capital, labor, and TFP, respectively. In economic studies, the value of TFP traditionally characterizes the level of technological development and includes all changes that are not connected with the impact of extensive production factors. Most modern models of economic growth assume that total capital is the combination of physical and human capital: $K={K}_{{}_{I}}^{\gamma}\cdot {K}_{G}^{1\gamma}$, determined using with the equations of change in physical capital through investments in fixed assets – I, taking into account its retirement rate γ: ${\dot{K}}_{I}^{\gamma}=I\delta \cdot {K}_{I}^{\gamma}$ and the equations of change in human capital at the expense of state budget (on health care, education, science, R&D, etc.) – G and its retirement – ρ: ${\dot{K}}_{G}^{1\gamma}=G\rho \cdot {K}_{G}^{1\gamma}$.
The econometric equation linking the change in the rate of economic growth with the growth rate of the main factors of the model can be represented as follows:
where: ${g}_{Y}=\frac{\dot{Y}}{Y},{g}_{K}=\frac{\dot{K}}{K},{g}_{L}=\frac{\dot{L}}{L},{g}_{A}=\frac{{\dot{A}}_{0}}{{A}_{0}}$ are growth rates (of the country analyzed), respectively, of GDP, Capital, Labor and Total factor productivity.
3. RESULTS
3.1 Data Analysis
The data on the countries were collected and processed using World Bank Open Data.URL: https://data.worldbank.org/. This database contains the information on the meaning of all indicators for the chosen time horizon, which are required for the analysis.
To solve the regression equation for the selected countries the authors used the following indicators of the national economies for the period from 2000 to 2018.
The left part of the regression equation (3) presents the GDP growth rate, which is the result, which, in turn, depends on the utilized factors  the growth rates of capital, labor, and TFP. It means that the left part of the regression equation (3), the GDP growth rate, is the dependent variable of the considered model, which depends linearly on the independent variables  the growth rates of capital, labor, and TFP.
It should be noted that when using the CobbDouglas production function (${Y}_{t}={A}_{0}\cdot {K}_{t}^{\alpha}\cdot {L}_{t}^{\beta}$)it is assumed that the A0 factor has a fixed value (is a constant) for the analyzed period, and using the regression equation (3), it is assumed that the growth rate of TFP, gA, has a fixed value (is a constant) over the observed period. Thus, the considered model  the regression equation (3), shows that the GDP growth rate (% per annum) was chosen as the dependent variable, and the growth factors of capital and labor were chosen as factors that determine the GDP growth rate.
Such a choice of a dependent and independent variables for the numerical solution of the regression equation 3 is due to the articulated hypothesis and the task to assess the influence of changing Capital, Labor and Total factor productivity on GDP growth rate.
A standard approximation was used to calculate the annual rate of change in the total labor force, L, of the selected country. After taking the logarithm of the numerical values of L for the period 20002018, the authors carried out differentiation using the Differences [list] operation from the Wolfram Mathematica 11.0 package library.
Table 1 presents the data for the countries of the sample: the rate of annual economic growth (%), the rate annual capital growth (%), and the rate of annual labor force growth, estimated using the methodology considered above. Table 1 contains the values of indicators and factors obtained using descriptive statistics (average value, minimum and maximum values, and standard deviation). The calculated data for the annual growth rate of the labor force reflect the standard error.
3.2 Regression Analysis
The authors used a linear regression model (Linear Model Fit) of the Wolfram Mathematica 11.0 software for numerical calculation of the unknown values in the parameters of the regression equation for the following countries: Argentina, Australia, Brazil, Canada, China, France, Germany, India, Indonesia, Italy, Japan, Korea, Nigeria, Norway, Poland, Russia, Saudi Arabia, Turkey, UK, USA. The results of the calculations are presented in Table 2. The numerical values of all the calculated parameters reflect the standard error (in brackets). The numerical values of a0 parameter for all sample countries are given in %.
The data presented in Table 2 demonstrate that the most significant linear decomposition of the GDP change rate (economic growth) into the changes related to production factors (capital and labor) and TFP occurred in: the United States (R^{2}~0.91); Turkey (R^{2}~0.87); Brazil (R^{2}~0.87); Argentina (R^{2}~0.87); France (R^{2}~0.87); Turkey (R^{2}~0.83); Germany (R^{2}~0.79); Italy (R^{2}~0.87); Poland (R^{2}~0.75); Russia (R^{2}~0.72) and Japan (R^{2}~0.71). More than 70% of the changes in the GDP of these countries can be explained by the changes in the capital, labor and TFP factors. In the decomposition of the GDP change rate, a moderate impact (more than 0.22) of the capital change rate can be observed for half of the countries in the sample (USA – 0.23, Germany – 0.27, Turkey – 0.26, and Brazil – 0.22). For most countries in the sample, changes in the labor factor had a significant effect on the changes in the GDP rate: for the USA, this share is more than 0.89; for the UK – over 0.83; for China – over 0.72, for Brazil – over 0.74, and for Russia – over 1.2. One should also mention rather high annual growth rates of TFP from 2000 to 2018: more than 2% per year in such countries as Turkey, Russia, Poland, Indonesia, India, Nigeria, Australia, and China.
This study analyzes the structure of the economic growth factors for the countries with the most significant result (R^{2} > 70%) obtained in the considered method of decomposing GDP rates into shares related to the physical capital change rates (due to investment and retirement) and the change rates of labor and total factor productivity. It should be noted that in modern conditions with open national economies, the sum of elasticities of the gross output by production factors may differ significantly from 1 and indicate both increasing and decreasing aggregate returns on production factors. Table 2 shows that over the observation period of 20002018 only a few countries in the analyzed sample had an increasing return on production factors: USA, Russia, and South Korea. The rest of the countries in the sample had a decreasing return on traditional production factors, with increasing influence of the intensive factor (China, India, Indonesia, and Nigeria).
3.3 Cluster Analysis
The elasticity coefficients of the gross output of the countries analyzed by production factors estimated using regression modeling were used as the basis for clustering the countries to identify the actions that may best stimulate their economic growth. The authors used different distance functions and clustering methods. The paper contains a sample of 20 countries which was divided into separate clusters according to the estimated values of the parameters of the regression equation presented as a set of 3D points (in the parameter space). The study presents the results of the decomposition of economic growth rates from 2000 to 2018 (considering the change rates of extensive and intensive factors) for the sample of 20 countries which includes 11 countries with developed economies (USA, Great Britain, Canada, Australia, France, Germany, Italy, Japan, South Korea, Norway and Poland), and 9 countries with developing economies and emerging markets (Argentina, China, India, Brazil, Russia, Turkey, Indonesia, Nigeria, and Saudi Arabia). The findings were obtained using regression modeling and are presented in Table 2 (after performing the cluster analysis – with Find Clusters [data] in the Wolfram Mathematica 11.0 System). This enabled to form three groups of countries (see Figure 1) with comparable values of the obtained parameters {a0, α, β} (a0 – the TFP growth rate, α – GDP elasticity for capital, β – GDP elasticity for labor).
The cluster analysis enabled to determine three groups of countries with comparable values of the obtained parameters:

– Group 1 represented by the countries with the highest return on capital α if compared to other groups (Argentina, Canada, France, Germany, Italy, Japan, Turkey);

– Group 2 with the greatest return on labor factor β comparing to other groups (Australia, China, India, Indonesia, Nigeria, Poland);

– Group 3 with a high contribution of the TFP factor to the growth of the economies of the countries belonging to this group (Brazil, Korea, Norway, Russia, Saudi Arabia, UK, and USA).
It should be noted that: Group 1 is characterized by the highest return on capital α compared to other groups; Group 2 demonstrated the greatest return on labor factor β compared to other groups, and Group 3 showed the value a0 – the high contribution of the TFP factor to the economic growth for the countries in this group. As we can see from the list of the countries in each group, there are no specific differences related to the level of the national economy (a developed or developing one) at this stage of the development of these countries (20002018), according to the method used in this research that is based on the neoclassical approach with CobbDouglas production functions: all three groups include countries with both developed and developing economies. Nevertheless, we can see that the most representative countries with developed economies belong to Groups 1 and 3 with traditional extensive production factors: Canada, France, Italy, Germany, and Japan in Group 1, and USA, UK, and Norway in Group 3. Group 2, which includes countries actively using intensive production factors and/or with market specifics connected with the increase in prices on the international markets for the country's export products (for instance, oil for Nigeria), is represented by China, India, Indonesia, Nigeria, and Poland.
4. DISCUSSION
Classical models that describe natural and artificial systems, including economic systems, use formal mathematical methods and help researchers understand the phenomena occurring in them, highlight the main characteristics of the system, as well as the most important variables and parameters. For instance, the apparatus of differential calculus has been widely and continuously applied to reveal the dynamics and to forecast the evolution of variables in a dynamic system. To comprehend the complex behavior of real financial and economic systems requires nowadays it is necessary to improve the methods of their modeling all the time. However, traditional mathematical models do not reflect the most important specifics of economic systems since it is incredibly difficult to take into account numerous factors that these models have, such as political, social, environmental, and others. Consequently, this leads to giving descriptions that do not correspond to realworld observations (Machado and Lopes, 2015;Lima et al., 2018).
The most significant and frequently occurring factor in dynamic nonlinear systems is path dependence – the inertial evolution of a dynamic system or the effect of historical conditionality of development, which implies that the history leading to the assumed equilibrium can be neither unique nor predetermined. Such behavior of a dynamic system contrasts with the concept of equilibrium in the neoclassical economy, according to which the economic system reaches equilibrium regardless of its starting point or even if there is a shortterm crisis. In case of path dependence, initial conditions or some noise on the trajectory can have a significant impact on the final result of the economic system development and, probably, irreversible consequences in a nonlinear dynamic economic system.
When studying economic phenomena and processes, researchers traditionally use both limiting and average values of various indicators, represented as functions of the determining factors. For a single valued function, which describes the singlevalued dependence of the economic indicator on some factor, the average value of the indicator is determined as the ratio of the indicator to the corresponding value of factor, and the limiting value of indicator is the first order derivative for factor. In general, the dependence of the economic indicator on factor can be multivalued, that is, several different values of the economic indicator can correspond to the same value of factor. One of the reasons for the fact that the dependence of economic indicator on certain factor has many values is that economic subjects have memory. This means that the limit indicator at time point T may depend on all changes of the economic indicator and factor on a certain finite time interval (0,T) preceding the considered moment T.
The average and maximum values of indicators in traditional studies depend only on a given time point t=T and its infinitely small neighborhood t=T, that is, one assumes that economic agents have infinitely short memory. In terms of mathematics, this approximation is due to the use of derivatives of integer order. However, recently, economic research papers have actively used the concept of the derivative (integral differentiation) of noninteger order (Tarasova and Tarasov, 2016). There are various types of noninteger order derivatives in mathematics. Most studies use the Caputo fractional derivative (Tarasova and Tarasov, 2017;Etcuban and Pantinople, 2018). For instance, a recent paper (Rehman et al., 2018) gives a detailed interpretation of the Caputo fractional operator for economic processes and phenomena that take into account memory effects in economic systems. Another research (Luo et al., 2018) analyzes the possible causes of economic growth using fractional calculating techniques, while other researchers (Tejado et al., 2018;Akhmadeev et al., 2016;Nuriyev et al., 2018) present the results of analyzing the influence of various factors on economic growth in the Eurozone countries. This article provides models of economic growth for all states of the European Union (EU) for the period since 1970, built using the methods of integer and fractional calculation (differentiation and integration). In the models considered, gross domestic product (GDP) is a function of a country's area, gross capital formation (GCF), exports of goods and services, and other important social parameters of a country. The results obtained clearly demonstrate that the models of economic growth that take into account the memory effect and imply the fractional calculation methods of average and limiting values of economic system indicators more accurately correspond to the actual data, compared to the classical models in which similar indicators were calculated using integer calculus methods, omitting memory effects. In these works, fractionalorder derivatives are interpreted as economic characteristics (indicators), which take place between the average and marginal indicators, reflecting the effects with fading memory.
Considering all the above, the further research of possible factors of economic growth in the countries examined in this study should include the methods of fractional calculation and take into account the memory effect in the economic system.
5. CONCLUSION
The paper presents a comparative analysis of the decomposition of the growth rates in 20 countries (11 developed economies: USA, UK, Canada, Australia, France, Germany, Italy, Japan, South Korea, Norway and Poland; 9 countries with developing economies and emerging markets: Argentina, China, India, Brazil, Russia, Turkey, Indonesia, Nigeria, and Saudi Arabia), for the period from 2000 to 2018. In the research the authors used regression modeling based on the aggregate production function presented as multiplicative power CobbDouglas function. Having performed the regression analysis, the authors obtained numerical estimates of the model parameters for all analyzed countries: GDP elasticities of the considered countries by extensive factors – capital and labor, as well as the impact of the intensive factor – TFP growth rates that represent the dynamics of the resulting variable (GDP) at an acceptable level of significance.
By means of clusterization of the obtained results, the chosen countries have been divided into three groups, where each group included both developed and developing economies. The most representative countries with developed economies belong to Groups 1 and 3 and have traditional extensive production factors. These are Canada, France, Italy, Germany, and Japan in Group 1, and USA, UK, and Norway in Group 3. Group 2, which includes countries actively using intensive production factors and/or with market specifics connected with the increase in prices on the international markets for the country's export products (for instance, oil for Nigeria), is represented by China, India, Indonesia, Nigeria, and Poland.
The high values of the labor return coefficient observed in the economies of the United States, Russia and South Korea are directly connected with an increase in the total return of production factors. This fact is most significant for Russia since it indicates that an increase in human capital should be the most important factor in economic growth. Improving its quality will require reforms in the social sphere of the economy (education, health care, pension system, etc.). Economic growth in some countries (China, India, Indonesia, and Poland) over the analyzed period (20002018) occurred due to the growth of total factor productivity, that is, due to other factors rather than the weighted sum of labor and capital. This undoubtedly requires deeper structural reforms in these countries (transport, infrastructure, reforming some institutions such as property rights, judicial system, etc.) for active diversification of the economy.
Thus, the obtained numerical values of the elasticity coefficients of the gross output of the countries analyzed by production factors that were estimated using regression modeling can be used for clustering the countries to determine the main actions that may stimulate their economic growth.