GROWTH IN ADVANCED COUNTRIES: A COMPARISON OF ENDOGENOUS AND
EXOGENOUS MODELS
Nicolas Schrama (11881437)
Abstract
For many years, the wealthiest economies in the world have exhibited stagnant growth rates. This prompts the question of how wealthy economies can increase their growth rates. This research contributes to that literature by investigating how endogenous and exogenous growth models compare empirically in a long-run panel setting for the most advanced countries of the world. To that end, I use a fully endogenous R&D-based model, motivated by Romer (1990), to arrive at an estimable, empirical model. The main prediction of Romer’s model is that more labor employed in the research sector results in a permanent increase in the rate of economic growth. I contrast this model with an exogenous conditional convergence model in the spirit of Mankiw, Romer, and Weil (1992) and Islam (1995). The main prediction of the underlying theoretical model is that a higher savings leads to a higher level of GDP per capita in the long run. The analysis focuses on the fifty percent richest OECD members and conveys two main results. First, the Romer model does not accurately describe long-run growth in these countries. This finding coincides with the evidence in the literature. Second, the Solow model does not explain income level discrepancies between advanced countries. Previous research has found that it is effective in explaining income level discrepancies for a broad sample of countries.
Statement of Originality
This document is written by Student Nicolas Schrama who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
I: Introduction
For many years, the richest economies in the world have exhibited growth rates of approximately two percent per year. Meanwhile, other economies have started a process of catching up, displaying growth rates of ten percent or higher (World Bank, 2020a). This prompts the question of how wealthy economies can increase their growth rates. The neoclassical model (Solow, 1956, 1957; Swan 1956) is a milestone in the process of understanding what drives economic growth. However, it explains long-run economic growth only through an exogenous parameter. Starting with Romer (1986), endogenous growth theories incorporate mechanisms to explain long-run growth within the model. The collection of these models is also called new growth theory.
The main question of this research is how endogenous and exogenous growth models compare empirically in a long-run panel setting for the most advanced countries of the world. This paper defines advanced countries as the fifty percent richest OECD members. To answer the main question, I use a fully endogenous R&D-based model, motivated by Romer (1990), to arrive at an estimable, empirical model. Romer defines R&D as the process of creating knowledge in the form of designs for new capital goods, which combine to the total stock of knowledge. The main prediction of the underlying theoretical model is that more labor employed in the research sector results in a permanent increase in the rate of economic growth. I contrast this model with an exogenous conditional convergence model in the spirit of Mankiw, Romer, and Weil (1992) and Islam (1995). The main prediction of the underlying theoretical model is that a higher savings leads to a higher level of GDP per capita in the long run.
This paper will focus on the Romer model (1990) and the neoclassical model for the following reasons. First, economists agree that individuals and firms that undertake R&D foster technological change by creating new knowledge. Furthermore, they concur that technological change is the source of sustained increases in per capita income: many countries are richer than two-hundred years ago because of inventions made over this time (Parente, 2001). Meanwhile, the neoclassical model has remained one of the most empirically successful growth models: cross-sectional (Mankiw, Romer, and Weil; 1992) and panel (Islam; 1995) evidence corroborate its predictions.
The added value of this paper is the contribution of a panel approach to a literature characterized by time-series research, a focus on advanced countries, and incorporating the most recent years in the dataset.
In the next section, I summarize the literature on endogenous and exogenous growth models. Section three elaborates on the theoretical Romer and Solow models and the empirical models based on them. Section four discusses the employed data. In section five, the results are presented and analyzed, and section six concludes.
II: Literature review
Since the end of the Second World War, many economists have taken interest in the processes underlying economic growth. In this enterprise, they have considered five facts conventional wisdom: (a) markets contain many firms; (b) knowledge differs from other inputs due to their nonrival nature; (c) physical inputs such as labor and capital can be replicated; (d) technological progress arises from the actions of people; (e) many firms have market power and earn rents on inventions (Romer, 1994).
Economists have strived to incorporate these facts into their theories. Although the neoclassical model captured the first three, it does not explain long-run economic growth more deeply than through an exogenous parameter. Launched by Romer (1986), endogenous growth models attempted to integrate
the fourth fact. Work on models with similar macroeconomic features had started in the 1960s (Romer, 1994). Romer’s model was eventually enabled by industrial-organizational models worked out in the 1970s, which allow for aggregate models including many firms (a) with market power (e).
As Romer (1994, pp. 3) phrases it: “Endogenous growth distinguishes itself from neoclassical growth by emphasizing that economic growth is an endogenous outcome of an economic system, not the result of forces that impinge from outside”. Parente (2001) divides these models into two types, those with perfect competition and those with imperfect competition. Models of perfect competition revolve around the decision of agents to accumulate physical or human capital. These models generate growth endogenously due to the absence of diminishing returns to this input at the aggregate level. Seminal works in the category of perfect competition are Romer (1986), Lucas (1988), and Rebelo (1991). First, Romer, whose ideas go back to Arrow (1962), explains economic growth through the role of externalities in economic development. Second, Lucas, taking from Uzawa (1965), emphasizes the significance of human capital for economic growth. Finally, the AK model by Rebelo stresses the accumulation of physical capital.
Theories of imperfect competition commonly explicitly model the undertaking of R&D by private agents. Imperfect competition is introduced by giving successful researchers monopoly power. Without it, no self-interested agent engages in R&D. Pioneering papers in the category of imperfect competition are Romer (1990) and Grossman and Helpman (1991), who developed the R&D models of economic growth with the creation of knowledge at their root.
Models discussed thus far belong to the first generation. Time-series evidence by Jones (1995b) against scale effects in the creation of knowledge, which I elaborate on below, caused new growth theory to evolve into two second-generation theories (Ang & Madsen, 2011). First are the semi-endogenous models of, for example, Jones (1995a), Kortum (1997), and Segerstrom (1998), that abandon scale effects in knowledge creation by assuming diminishing returns to the stock of knowledge. Second are the Schumpeterian growth models of Peretto (1998), Dinopoulos and Thompson (1998), Young (1998), and Howitt (1999) among others. Although they preserve scale effects in knowledge production, they assume that an expanding amount of research varieties weakens the aggregate efficacy of research. As a result, research productivity per variety does not increase over time. Models that maintain the assumption of scale effects are called fully endogenous growth models. This includes those of the first generation.
Broadly, endogenous growth models have received three criticisms. First, Jones (1995b) advances a vital issue about models with constant or increasing returns to produced factors. Take for example R&D models. First-generation endogenous R&D models predict that growth in income per person is proportional to the number of workers engaged in R&D. However, Jones notes that, over the postwar period, the data refute this prediction for major industrialized countries: even though the number of researchers engaged in R&D and real R&D spending have approximately quintupled, economic growth has remained at a similar level.
Crucially, this issue does not only occur for R&D models. Other variables that first-generation endogenous growth models have plausibly identified as a determinant of long-run growth all have trended upwards strongly since the Second World War. Such variables include resources devoted to human capital accumulation, the number of highly educated workers, the extent of interactions among countries, and the world population (Romer, 2011).
These findings seem to support growth models of the second generation. However, R&D’s share of income and rates of investment in physical and human capital have also increased over time. Therefore, the lack of rising growth rates is puzzling to Schumpeterian models as well. Moreover, the parameter restrictions needed to eliminate scale effects on growth in these models are strong (Li, 2000). For semi-endogenous growth models, the absence of an increase in growth is unsurprising. They predict that a permanent increase in, for example, the number of workers in R&D leads to a limited period of increased growth, resulting in only a level effect on income in the long run (Romer, 2011).
A second criticism concerning endogenous growth models is their inability to explain the income divergence between developing and rich countries. According to Parente (2001), endogenous growth models do not account for why development miracles are a recent phenomenon and why they are limited to developing countries. Also, he argues that they do not explain why later entrants to modern economic growth have been able to double their income in shorter periods than earlier entrants.
Finally, Krugman (2013) argues that new growth theory “seems to have fizzled out”, since “too much of it involved making assumptions about how unmeasurable things affected other unmeasurable things”, asserting that some assumptions in endogenous growth models are hardly testable.
Though, some evidence favors endogenous growth models. Gong, Greiner, and Semmler (2004) test a semi-endogenous R&D-based model for the US and German economies and obtain reasonable parameter estimates. They conclude that the model they test is compatible with time-series evidence. Moreover, Huh and Kim (2013) test an AK model versus an exogenous growth model for each of the G7 countries. Their time series data support the endogenous growth model in four of the seven countries.
Meanwhile, economists have provided convincing evidence in favor of the Solow model. Mankiw, Romer, and Weil (1992) find that an augmented Solow model that includes human capital accumulation provides an excellent description of the cross-country data. They conclude that preserving the assumption of decreasing returns is unproblematic. Islam (1995) argues that the cross-sectional approach of Mankiw, Romer, and Weil (1992) leads to biased parameter estimates because they do not take country-fixed effects into account. However, the data in Islam’s panel approach also corroborate the Solow model. Finally, Parente (2001) adapts the Solow model to include costly technology adoption constrained by country policies and embeds this framework into a Malthusian model. He argues that this model can account for the evolution of the world income distribution after 1950 and explain the existence of growth miracles.
III: Methodology
In this section, I sketch a fully endogenous theoretical R&D-based model motivated by Romer (1990). Then, I derive an estimable, empirical model from that theoretical model. Following, a similar methodology is applied to the Solow model, allowing for an empirical comparison of the two models.
A: The Romer model
In the Romer model, profit-maximizing economic actors undertake R&D. This R&D fuels growth, which in turn affects the incentives for devoting resources to R&D (Romer, 2011). The model consists of three sectors. First, the research sector produces new knowledge and behaves competitively. Knowledge is a nonrival good, implying that it can be used simultaneously in multiple economic activities. Second, the intermediate goods sector uses the ideas produced by the research sector as inputs to manufacture intermediate capital goods. Firms in the intermediate goods sectors must have market power since they use a nonrival input. The model introduces market power through patents acquired from inventors, which are assumed to last forever. Third, the final goods sector produces final goods, using intermediate capital goods (K), labor (L), and human capital (H). Final goods may be either consumed or invested (Jones & Vollrath, 2013). In the original paper, Romer (1990) elaborates further on the microeconomic side of the model. However, that is beyond the scope of this research.
With respect to macroeconomics, the model describes only advanced countries since technological progress is driven by R&D in those regions. Precisely, the model explains growth for all advanced countries as a whole, rather than for each country individually (Jones & Vollrath, 2013). Less-developed countries, on the other hand, do not need to engage in R&D and can increase their per capita output more cheaply by adopting readily available technologies developed elsewhere (Parente, 2001).
The aggregate production function describes how technology, capital, labor, and human capital combine to produce output:
𝑌 = 𝐴(𝑡)𝐾!(𝑡)𝐿 " #$!(𝑡)𝐻
"#$!(𝑡)
The inputs behave as follows. Capital accumulates as individuals in the economy save and depreciates at the exogenous rate 𝛿:
Here, 𝑠 is the savings rate in the economy. Labor, analogous to the labor force, grows exponentially at the exogenous rate n. The labor force is divided between workers in the final goods sector, Ly(t), and
workers in the research sector, La(t), which are assumed to grow at the same rate. Human capital is also
used in both the final goods sector, Hy(t), and in the research sector, Ha(t), and is assumed to be equal in
these sectors, i.e. 𝐻"= 𝐻% = 𝐻. A(t) is the stock of knowledge or the number of ideas that has been created throughout history up until time t. It accumulates as follows:
𝐴̇(𝑡) = 𝐿&%(𝑡)𝐻%&(𝑡)𝐴'(𝑡)
The parameter 𝜑 represents how the accumulation of ideas depends on the existing stock of knowledge: if the creation of ideas in the past increases the productivity of researchers in the present, 𝜑 > 0; if past ideas decrease the productivity of researchers in the present, 𝜑 < 0; if past and current research sector productivity are unrelated, 𝜑 = 0. The sign of this relation depends on whether the most apparent discoveries are made first, or whether earlier discoveries make the creation of knowledge easier. The parameter 𝛽 represents the dependence of the average productivity of researchers on the number of agents in the research sector and their human capital. 𝛽 may vary from zero to one (Jones & Vollrath, 2013).
To generate long-run growth, 𝜑 must be equal to or larger than one. Otherwise, when knowledge accumulates, 𝐴̇ in a given period is a smaller share of A than it was in the previous period. This share decreases exponentially over time. Crucially, all output per capita growth results from the accumulation of knowledge along the balanced growth path. Therefore, when 𝜑 is smaller than one, stimulating the creation of ideas through La increases the long-run levels of technology and income but not their growth
rates. However, the time-series observations of Jones (1995b) show that larger values of 𝜑 are less likely. Hence, 𝜑 equals one is assumed to maintain fully endogenous growth. Thus, the described model indicates that La is the determinant of long-run growth. Based on these parameters and relations, the
derivation of the empirical model follows.
Consider the following production function: 𝑌 = 𝐴𝐾!𝐿 " #$!𝐻
"#$!
From here on, I will write 𝐻 instead of 𝐻" and 𝐻%, corresponding to the assumption that human capital is the same in the research and final goods sectors. Taking the derivative with respect to time of both sides: 𝑌̇ =𝜕𝑌 𝜕𝐴𝐴̇ + 𝜕𝑌 𝜕𝐾𝐾̇ + 𝜕𝑌 𝜕𝐿"𝐿"̇ + 𝜕𝑌 𝜕𝐻𝐻̇ Dividing both sides by 𝑌(𝑡):
𝑌̇ 𝑌= 𝐴𝜕𝑌 𝑌𝜕𝐴 𝐴 𝐴 ̇ +𝐾𝜕𝑌 𝑌𝜕𝐾 𝐾 𝐾 ̇ +𝐿"𝜕𝑌 𝑌𝜕𝐿" 𝐿" 𝐿" ̇ +𝐻𝜕𝑌 𝑌𝜕𝐻 𝐻 𝐻 ̇ Note that ()**)(= 𝛼, +*)+!)* != ,)* *),= 1 − 𝛼, and -)*
*)-= 1. Substituting these terms results in the growth rate of total GDP: 𝑌̇ 𝑌= 𝐴 𝐴 ̇ + 𝛼𝐾 𝐾 ̇ + (1 − 𝛼)𝐿" 𝐿" ̇ + (1 − 𝛼)𝐻 𝐻 ̇
The next step is to derive the growth rate of GDP per capita from the growth rate of total GDP. Define 𝑦 = 𝑌/𝐿 and 𝑘 = 𝐾/𝐿, then:
𝑦̇ 𝑦= 𝑌̇ 𝐿− 𝐿̇ 𝐿= 𝑌̇ 𝑌− 𝛼 𝐿̇ 𝐿− (1 − 𝛼) 𝐿̇ 𝐿= 𝐴 𝐴 ̇ + 𝛼𝑘 𝑘 ̇ + (1 − 𝛼)[𝐿" 𝐿" ̇ −𝐿̇ 𝐿] + (1 − 𝛼) 𝐻 𝐻 ̇ Also define 𝜇. =+" # +" so that 1 − 𝜇. = +" +"− +"# +"= +!"
+". Using that for small 𝑥, ln(1 − 𝑥) ≈ −𝑥: Δ ln(1 − 𝜇.) ≈𝐿.
"̇ 𝐿". −
𝐿.̇ 𝐿.
In the appendix, I show that in the long run, i.e. along the balanced growth path, GDP per capita and capital per capita grow at the same rate:
𝑦̇ 𝑦=
𝑘̇ 𝑘
This assumption holds only for major industrialized countries. It does not hold for emerging economies, as they are still accumulating capital.
Combining equations (1), (2), and (3) leads to: 𝑦̇ 𝑦= 1 1 − 𝛼 𝐴 𝐴 ̇ + (1 − 𝛼)Δln (1 − 𝜇.) + (1 − 𝛼)𝐻 𝐻 ̇
Recall from the description of the theoretical model that 𝐴̇ = 𝐿%&𝐻&𝐴 so that -̇-= 𝐿&%𝐻&. Substituting this equality into equation (4) yields:
𝑦̇ 𝑦= 1 1 − 𝛼𝐿% &𝐻&+ (1 − 𝛼)Δln (1 − 𝜇 .) + (1 − 𝛼) 𝐻 𝐻 ̇
This equation is not linear in 𝛽. Therefore, it cannot be estimated with OLS. The first-order Taylor approximation solves this problem: 𝑓(𝛽) ≈ 𝑓(𝑎) = 𝑓′(𝑎)(𝛽 − 𝑎) with 𝑓(𝛽) = 𝐿&%𝐻&. In the results of Gong, Greiner, and Semmler (2004), 𝛽 is close to zero. Therefore, I will use 𝑎 = 0. This yields 𝑓(𝛽) = 𝐿&%𝐻& ≈ 1 + 𝛽ln (𝐿
%𝐻). Combining this result with equation (5): 𝑦̇ 𝑦= 1 1 − 𝛼+ 1 1 − 𝛼𝛽ln (𝐿%𝐻) + (1 − 𝛼)Δln (1 − 𝜇.) + (1 − 𝛼) 𝐻 𝐻 ̇ which is linear in 𝛽.
Next, I rewrite the following terms from equation (6): 𝑦̇
𝑦≈ Δln (𝑦.) and
𝐻̇
𝐻≈ ∆ln (𝐻.) for small "̇" and ,̇,. Using equations (6), (7), and (8):
Δln (𝑦.) = 1 1 − 𝛼+
1
1 − 𝛼𝛽ln (𝐿%𝐻) + (1 − 𝛼)Δln (1 − 𝜇.) + (1 − 𝛼)∆ln (𝐻.)
Finally, adding coefficients (𝛾0), unobserved time- (𝜒.) and country- (𝜂1) fixed effects, and an error term (𝑢1,.) yields: Δ lnK𝑦1,.L = 𝛾3+ 𝛾#lnK𝐿%1,.𝐻1,.L + 𝛾4Δ lnK1 − 𝜇1,.L + 𝛾5∆ lnK𝐻1,.L + 𝜂1+ 𝜒.+ 𝑢1,. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
This completes the description of the model.
B: The Solow model
The theoretical Solow model is similar to the Romer model above. The production function is identical and labor, capital, and human capital accumulate identically. However, A grows at the exogenous rate
g, which is assumed to be the same for all countries. Therefore, the microeconomic foundations
necessary in the Romer model are no longer present. Also, because there is no research sector (and no intermediate goods sector), all labor is devoted to the final goods sector, i.e. 𝐿 = 𝐿". The main prediction of the theoretical model is that a higher savings leads to a higher level of GDP per capita. Furthermore, the exogenous rate g determines all long-run per capita growth. Therefore, I estimate the empirical Solow model using levels of GDP per capita. The derivation of the empirical model follows.
By adapting equation (14) in the appendix to reflect that 𝐿"= 𝐿 in the Solow model, the 𝑞 term simplifies to human capital, 𝐻. This yields the following expression for GDP per capita in the long run:
𝑦 = 𝐴#$!# 𝐻[ 𝑠 𝑛 + 𝛿]
! #$!
Taking the natural logarithm of both sides yields: ln(𝑦) = 1 1 − 𝛼ln(𝐴) + ln(𝐻) + 𝛼 1 − 𝛼ln(𝑠) − 𝛼 1 − 𝛼ln (𝑛 + 𝛿)
The ln (𝐴) term consists of two parts. First, ln (𝐴(0)) reflects technology, climate, institutions, and resource endowments (Mankiw, Romer, and Weil; 1992). Therefore, ln (𝐴(0)) differs across countries and is captured by country-fixed effects. Second, ln (𝑔𝑡) reflects accumulated technology at time 𝑡. Because 𝑔 is assumed to be the same for all countries, 𝑔𝑡 is a constant and captured by the constant term.
Adding coefficients (𝜌0), country- (𝜂1) and time- (𝜒.) fixed effects, and an error term (𝑒1,.) results in:
lnK𝑦1,.L = 𝜌3+ 𝜌#lnK𝑠1,.L + 𝜌4lnK𝑛1,.+ 𝛿1,.L + 𝜌5lnK𝐻1,.L + 𝜂1+ 𝜒.+ 𝑒1,. This completes the description of the model.
IV: Data
In the previous section, I sketched the theoretical Romer and the Solow models and transformed their parameters into estimable, empirical models. In this section, for each model, I discuss the construction of the variables in these empirical models and the data series they are based on. Finally, a description of the dataset follows.
A: The Romer model
The dependent variable in the empirical R&D model is ∆ln (𝑦1,.). It measures the growth of GDP per capita. The construction of this variable starts from the rdgpna series in the Penn World Tables 9.1 (Feenstra, Inklaar, & Timmer; 2015), which uses the national accounts to measure real GDP in millions of 2011 US dollars and is designed to be the dependent variable in growth regressions. Each observation is multiplied by one million and divided over the size of the labor force. Finally, I take the first difference of the natural logarithm. The size of the labor force uses the labor force, total series from the World Bank Data Indicators (World Bank, 2020b).
The first independent variable is ln (𝐿%1,.𝐻1,.), which is listed as idea_g. It reflects that more workers or a higher stock of human capital in the research sector increase the growth rate of the stock of ideas, which in turn increases GDP per capita growth. Therefore, its coefficient should be positive. Furthermore, idea_g is the main determinant of long-run growth in the model. It is the natural logarithm of the product of the number of workers in R&D and a measure of human capital.
(11)
(12)
The data for the number of workers in R&D originates from the researchers in R&D (per million people) series (World Bank, 2020d). Of all data series, this one has the lowest quantity of observations. The source of the human capital measure is the Human Capital Index (HCI) in the Penn World Tables. The HCI is constructed according to two methods: one based on the average years of schooling from Barro & Lee (2013) and an assumed rate of return to education based on Mincer equation estimates around the world (Psacharopoulos, 1994), and one based on an alternative dataset for average years of schooling constructed by Cohen & Leker (2014). Based on five heuristic rules, it is decided which is best for each country. A disadvantage of this measure of human capital is that the size of the effect on GDP growth is difficult to interpret.
The second independent variable is ∆ln (1 − 𝜇1,.), which is listed as ly_g. It measures the difference in the growth rates of the labor force in the final goods sector and the total labor force. A negative value indicates that the labor force in the research sector has grown more than the labor force in the final goods sector. Equation (9) reveals that a negative value of ly_g decreases the growth rate of GDP per capita. This may appear unintuitive since La is the determinant of long-run growth. However, ly_g reflects that when more workers are in R&D, fewer are producing output. In other words, idea_g
and ly_g portray the trade-off of increasing the share of the labor force in R&D: fewer workers produce final goods, but their future productivity is higher. If the Romer model describes long-run growth well, the increased productivity should outweigh the drag of the smaller labor force in final goods production in the long run. Thus, the expected sign of the coefficient of ly_g is positive. The construction of ly_g uses the number of workers in R&D and the total size of the labor force series. ly_g is the first difference of the natural logarithm of 1 − 𝜇1,..
The final independent variable in the model is ∆ln (𝐻1,.), which is listed as hc_g. It reflects that higher growth of human capital in the final goods sector increases the growth rate of output per capita. Therefore, its coefficient should have a positive sign. hc_g is the first difference of the natural logarithm of the Human Capital Index.
To use these data to answer the main question, further refinement is required. The long-run focus on the relation between R&D and economic growth in this research necessitates the separation of short-run cycles in the data from the trend. To this end, Stata offers four time-series filters. Two of them are high-pass filters – the Hodrick-Prescott (HP) and Butterworth (BW) filters – and two of them are band-pass filters – the Baxter-King (BK) and Christiano-Fitzgerald (CF) filters. Since the data are ideally filtered from short-run cycles, i.e. those with a high frequency, a high-pass filter is inappropriate. Moreover, the BK filter costs more observations than the CF filter, because it uses a moving average: applying the CF filter to a sample of advanced countries leaves 211 observations, whereas the BK filter leaves 137. Therefore, the CF filter is the most appropriate.
Furthermore, only cycles with a periodicity between eight and one-hundred years are passed through. This band filters out cycles shorter than eight years, which encompasses short-run effects and business cycles. The latter have a periodicity varying from one-and-a-half to eight years (Burns and Mitchell; 1946). The upper bound is one-hundred years to avoid filtering out low-frequency cycles.
Additionally, filtering requires specifying whether the time series is stationary. The available unit-root tests are the Im-Pesaran-Shin (IPS) and Fischer type tests because the sample is unbalanced. Both have a null hypothesis that all panels contain unit-roots. The alternative hypothesis of the IPS test is that some panels are stationary, and the alternative hypothesis of Fischer type tests is that at least one panel is stationary. Both tests, for differing assumptions with respect to asymptotics, reject the null hypothesis with p-values below 0.001, suggesting that the panels are stationary.
In the analysis, three samples are used. First, the full sample contains most countries in the world, i.e. 166. But eighty drop out due to a lack of research sector data. Second, the OECD sample includes the countries that are an OECD member at the start of 2020, which amounts to thirty-seven countries. However, Australia, New Zealand, and Switzerland drop out due to a lack of research sector data. Third, from the OECD sample, the advanced sample is constructed as follows. Examining the R&D expenditure of the OECD countries as a share of their GDP reveals large differences. Though the average R&D share over the last three years is 1.89%, countries in the bottom quarter in terms of income spend 1.13% or less on R&D, and countries in the top quarter spend 2.76% or more (World Bank, 2020c). Therefore, the advanced sample includes only the richest half of all OECD members. After applying the CF filter, Israel and Iceland drop out because time series filters forbid gaps in the data.
Countries are included in this sample based on income instead of R&D expenditure since this research aims to analyze whether the most advanced countries grow as a result of R&D. Selecting countries with the highest R&D expenditure in a sample of relatively rich countries would cause selection bias.
Figure 1 shows the spread of R&D expenditure as a percentage of GDP per capita for all OECD members.
As can be seen in the figure, the threshold of real GDP per capita for the advanced sample is forty thousand 2011 US dollars.
Summary statistics for the full sample are shown in Table 1.
The full sample for the Romer model contains a limited amount of observations on the number of researchers in R&D and the size of the labor force. The dataset contains observations from 1961 to 2017 of 166 countries.
Table 2 presents summary statistics of the advanced sample.
(1)
(2)
(3)
(4)
(5)
VARIABLES
N
mean
sd
min
max
HCI
7,495
2.097
0.728
1.007
3.974
Labor force
4,617
1.72e+07
6.77e+07
34,117
7.87e+08
La
1,396
88,637
222,271
11.55
1.71e+06
GDP_cap_g
4,451
0.016
0.0629
-0.672
0.941
(1)
(2)
(3)
(4)
(5)
VARIABLES
N
mean
sd
min
max
HCI
1,026
3.048
0.390
1.766
3.807
Labor force
504
1.61e+07
3.40e+07
142,030
1.64e+08
La
293
153634.8
290,760.6
1333,126
1,375,086
GDP_cap_g
486
0.036
0.091
-0.256
0.309
Figure 1: R&D expenditure as a percentage of GDP and GDP per capita for the OECD members
Table 2: Summary statistics of the full sample for the Romer model
The advanced sample contains observations of eighteen countries from 1961 to 2017. Thirteen countries remain after applying the CF filter. The remaining observations are from 1997 to 2017.
B: The Solow model
The dependent variable in the empirical Solow model is ln (𝑦1,.). It is the natural logarithm of GDP per capita and based on the same data as GDP per capita growth in the Romer model.
The first independent variable is ln (𝑠1,.), which is the natural logarithm of the savings rate. It is listed as savings. savings reflects that when a larger share of GDP is saved, capital per capita and thus output per capita accumulate faster when transitioning to the steady state. Also, they reach a higher level in the steady state. Therefore, savings’s coefficient should be positive. The savings rate is the share of gross capital formation at current PPPs in the Penn World Tables.
The second independent variable is ln (𝑛1,.+ 𝛿1,.), which is listed as pop_dep. It reflects that when the labor force grows faster or the depreciation rate is higher, capital per capita decreases more each period. As a result, a higher savings rate is required to achieve the same level of capital per capita. Hence, its coefficient should be negative. n is the growth rate of the labor force and is based on the same labor force data used in the Romer model. The construction of 𝛿 uses data from the Penn World Tables on the depreciation rate of the capital stock.
ln (𝐻1,.), listed as hc, is the final independent variable and reflects that a larger stock of human capital increases output. Therefore, it should have a positive coefficient. hc is the natural logarithm of the Human Capital Index.
To allow for a better comparison between the Romer model and the Solow model, I employ the
OECD and advanced samples again. For the Solow model, this should not improve empirical accuracy,
since it emphasizes differences in capital per capita instead of technology. Concerning time series filters, the CF filter is best suited for identical reasons as for the Romer model. However, since the Solow model deals with level variables, the non-stationary version is more appropriate. Unit-root tests support this decision.
Summary statistics of the full sample are presented in Table 3.
(1)
(2)
(3)
(4)
(5)
VARIABLES
N
mean
sd
min
max
HCI
7,495
2.097
0.728
1.007
3.974
Savings rate
8,960
0.219
0.168
-2.075
9.409
GDP_capita
4,617
34,465
37,049
595.2
238,553
n_plus_delta
4,451
0.0636
0.0270
-0.0578
0.329
Table 3: Summary statistics of the full sample for the Solow model
The full sample for the Solow model contains observations from 1961 to 2017 on 177 countries. The HCI statistics are the same as for the Romer model and are posted again for clarity.
Summary statistics of the advanced sample are in Table 4.
(1)
(2)
(3)
(4)
(5)
VARIABLES
N
mean
sd
min
max
HCI
1,026
3.048
0.390
1.766
3.807
Savings rate
1,026
0.284
0.0584
0.148
0.548
GDP_capita
504
80,318
22,752
44,724
160,344
n_plus_delta
486
0.0504
0.0142
0.00418
0.106
Table 4: Summary statistics of the advanced sample for the Solow model
The advanced sample contains observations from 1961 to 2017 of eighteen countries. After applying the CF filter, the sample includes data from 1991 to 2017.
V: Results and discussion
In the last two sections, I sketched two empirical models and described the construction of the variables within them. This section presents the empirical results and a discussion of them. For both the Romer and Solow models, the analysis starts from a fixed effects regression on the full sample. Then, the sample is restricted to the OECD and advanced samples. As discussed, the advanced sample is expected to be more accurate for the Romer model. Finally, I apply the time series filter to the advanced sample.
A: The Romer model
The results of the Romer model are in Table 5.
(1)
(2)
(3)
(4)
(5)
VARIABLES
Full sample
OECD
AdvancedAdvanced,
filtered
Savings rate
idea_g
0.002
0.000
0.030
0.001
0.007
(0.006)
(0.009)
(0.025)
(0.001)
(0.003)
ly_g
-1.753
-0.456
5.207
-2.263
-2.901
(3.832)
(4.256)
(6.688)
(4.879)
(4.899)
hc_g
-0.350
0.997
-0.030
-2.299*
0.461
(0.344)
(0.645)
(0.766)
(0.880)
(1.295)
savings
-0.343
(0.166)
Constant
0.010
0.026
-0.320
-0.014
-0.013
(0.067)
(0.097)
(0.287)
(0.009)
(0.009)
Observations
1,069
585
237
215
215
R-squared
0.261
0.453
0.538
0.509
0.551
Number of countries
86
34
15
13
13
Country FE
YES
YES
YES
YES
YES
Year FE
YES
YES
YES
YES
YES
Robust standard errors in parentheses
** p<0.01, * p<0.05
Column 1 contains the results for the full sample of countries, amounting to 1069 observations. No coefficients are significant and only the coefficient of idea_g has the anticipated sign. This is not surprising because the full sample includes many countries that the model is not intended to describe.
Column 2 contains the results for the OECD sample, which includes 585 observations. Compared with column 1, the coefficient of idea_g is zero, which opposes the prediction of the theoretical model. Further, the sign of the coefficient of hc_g is positive.
Column 3 contains the results of the advanced sample. As this sample includes only countries the model is intended to describe, the sample size is small at 237 observations. Relative to the results for the OECD sample, ly_g is positive and larger and hc_g is negative and smaller. Also, idea_g is larger than zero again.
Column 4 shows the results for the filtered advanced sample. The coefficient of idea_g is still close to zero and insignificant, the coefficient of ly_g is negative and insignificant, and the coefficient of hc_g is significant and negative, even though the model predicts that they are all positive.
Overall, the results are grim for the Romer model. The coefficients frequently do not have the expected sign and if they do, they are insignificant. The lack of significance makes interpreting the coefficients precarious. Three arguments may explain these results. First, the Romer model may not explain long-run growth. This conclusion is in line with the time-series evidence of Jones (1995b).
Second, once the analysis focuses on the right sample of countries, the sample size is small, making precise estimation difficult. However, the coefficient of the key variable, idea_g, is close to zero once the sample is limited to advanced countries. Therefore, it is unreasonable to attribute the poor results to sample size issues.
Third, two assumptions in the derivation of the empirical model may not hold: the assumption that capital per capita and output per capita have the same long-run growth rate and the assumption that 𝛽 in the equation describing the accumulation of knowledge is close to zero, i.e. 𝑎 = 0.
First, the assumption that the output-capital ratio is constant in the long run is verifiable using the capital stock at current PPPs in millions of 2011 US dollars and the rdgpna series from the Penn World Tables. Based on these series, Figure 2 shows the output-capital ratio over time for four major economies, the US, the UK, Germany, and the Netherlands.
As can be seen in Figure 2, the output-capital ratio has decreased over time. The output-capital ratio has decreased by approximately 1.3% per year from 1961 to 2017 for the advanced countries. After 1997, the decrease was approximately 3.1% on average. To relax the assumption of a constant output-capital ratio, the empirical model in column 5 of Table 5 includes the savings rate. The inclusion of the savings rate does not affect the significance, sign, or size of idea_g and ly_g considerably, but hc_g becomes positive and insignificant. Moreover, the savings rate itself is insignificant and has a negative sign. Thus, although the assumption of a constant output-capital ratio may be unrealistic, relaxing it does not alter the results.
Second, verifying whether 𝑎 = 0 is more complicated. Table 6 presents the results of testing the impact of this assumption by using diverging constants for 𝑎.
Figure 2: The output-capital ratio over time for the US, the UK, Germany, and the Netherlands.
(1)
(2)
(3)
(4)
(5)
(6)
VARIABLES
0
1/10
1/4
1/2
3/4
1
idea_g
0.001
(0.001)
ly_g
-2.263
-2.302
-2.368
-2.389
-2.369
-2.352
(4.879)
(4.849)
(4.824)
(4.821)
(4.829)
(4.837)
hc_g
-2.299*
-2.160*
-2.091
-2.156
-2.178
-2.160
(0.880)
(0.983)
(1.096)
(1.147)
(1.138)
(1.125)
idea_g_2
0.000
(0.000)
idea_g_3
-0.000
(-0.000)
idea_g_4
-0.000
(0.000)
idea_g_5
-0.000
(0.000)
idea_g_6
-0.000
(0.000)
Constant
-0.014
-0.015
-0.016
-0.016
-0.016
-0.016
(0.008)
(0.009)
(0.008)
(0.008)
(0.008)
(0.008)
Observations
215
215
215
215
215
215
R-squared
0.509
0.509
0.509
0.509
0.509
0.509
Number of
countries
13
13
13
13
13
13
Country FE
YES
YES
YES
YES
YES
YES
Year FE
YES
YES
YES
YES
YES
YES
Robust standard errors in parentheses
** p<0.01, * p<0.05
Column 1 of Table 6 is the same as column 4 in Table 5. The other columns in Table 6 diverge in the constant used for 𝑎 in the Taylor approximation. The results are shown for 𝑎 = 0.1, 𝑎 = 0.25, 𝑎 = 0.5, 𝑎 = 0.75, and 𝑎 = 1, as 0 ≤ 𝛽 ≤ 1. The key observation is that all idea_g coefficients are close to zero. Because all constants lead to the same result, uncertainty about which is correct is irrelevant. Thus, this assumption is not causing the poor results of the Romer model either.
To summarize, the results of the Romer model are poor. In column 4 of Table 5, which should theoretically contain the best estimates, idea_g has an effect close to zero and is insignificant, ly_g has the wrong sign and is insignificant, and hc_g has the wrong sign. The coefficient of idea_g being close to zero suggests that imprecise estimation is not the culprit. Also, testing and relaxing assumptions makes little difference. The evidence suggests that the Romer model does not explain long-run growth, even for advanced countries. This is a plausible conclusion, as it is in line with the time-series evidence of Jones (1995b).
B: The Solow model
The results for the Solow model are in Table 7.
(1)
(2)
(3)
(4)
VARIABLES
Full sample
OECD
Advanced
Advanced,
filtered
savings
0.100**
0.281**
0.084
0.029
(0.030)
(0.059)
(0.073)
(0.018)
pop_dep
0.073**
0.042*
0.006
-0.019**
(0.019)
(0.017)
(0.016)
(0.003)
hc
0.048
0.771
0.271
-0.371
(0.275)
(0.480)
(0.443)
(0.573)
Constant
10.08**
10.45**
10.86**
0.013*
(0.205)
(0.542)
(0.469)
(0.005)
Observations
3,844
992
486
468
R-squared
0.475
0.769
0.855
0.507
Number of countries
143
37
18
18
Country FE
YES
YES
YES
YES
Year FE
YES
YES
YES
YES
Robust standard errors in parentheses
** p<0.01, * p<0.05
Column 1 contains the results for the full sample with 3844 observations in total. The coefficient of savings is 0.100 (0.030). Thus, it is significant and has the expected sign. This implies that when the savings rate in a country increases by ten percent, e.g. from twenty to twenty-two percent, GDP per capita increases by one percent, e.g. from 30 000 to 30 300. The coefficient of pop_dep is 0.073 (0.019). Even though it is significant, it does not have the expected sign. According to this coefficient, when either the growth of the labor force or the depreciation rate increases by ten percent, GDP per capita increases by 0.73%. The coefficient of hc is 0.048 (0.275), meaning that it has the anticipated sign but is insignificant.
Column 2 contains the results for the OECD sample, which uses 992 observations. The results are similar to those in the first column, except that the coefficients of savings and hc are larger and the coefficient of pop_dep is smaller.
The results for the advanced sample are in column 3. The most notable change compared with column 2 is that the coefficients of savings and pop_dep have lost their significance. This could be related to the smaller sample size of 486 or be the result of little variation in savings rates for the wealthiest countries.
The results for the filtered series of this sample are in column 4. The coefficient of savings is smaller than in the third column, the coefficient of pop_dep has the anticipated negative sign and is significant, and the coefficient of hc_g is negative and insignificant.
Overall, the results suggest the following. First, the savings rate, the parameter at the heart of the Solow model, matters only when the sample is not too restricted. The theoretical model predicts that the savings rate influences the steady-state level of output per capita and should, therefore, be significant even in the advanced sample. Possibly, there is too little variation in savings rates in this sample. Second, human capital appears to be unimportant. Finally, the pop_dep parameter significantly correlates with income levels. However, the sign of the correlation differs per sample. This is surprising since the model predicts that it should be strictly negative.
Comparing the analyses of the Romer and Solow models, neither model provides an accurate description of advanced countries. For the Romer model, the coefficient of the most important parameter is close to zero and the others have the wrong sign. This is in line with the findings in the literature. The results for the conditional convergence model coincide, for a broad sample of countries, with the theoretical model’s prediction that a higher savings leads to a higher level of GDP per capita. This
finding is also in line with Mankiw, Romer, and Weil (1992) and Islam (1995). However, the results cease to corroborate the theoretical model for the advanced sample.
VI: Conclusion
This research aims to investigate how endogenous and exogenous growth models compare empirically in a long-run panel setting for the most advanced countries of the world. This is an essential research area since these models provide insight into how developed economies can further increase their wealth in the long run.
To answer the main question, I use a fully endogenous R&D-based model, motivated by Romer (1990), to arrive at an estimable, empirical model. The main prediction of Romer’s model is that more labor employed in the R&D sector results in a permanent increase in the rate of economic growth. This model is contrasted with a conditional convergence model in the spirit of Mankiw, Romer, and Weil (1992) and Islam (1995). The analysis focuses on the fifty percent wealthiest OECD members because R&D expenditure as a share of GDP differs significantly among the OECD members. Furthermore, the data are filtered from short-run effects and business cycles.
The results are grim for the Romer model, as the most important coefficient is close to zero and the others have the wrong sign. Relaxing assumptions made in the derivation of the empirical model also makes no difference. This suggests that the Romer model does not describe long-run growth, which is in line with Jones (1995b). The Solow model performs well for a broad sample of countries, which is in line with Mankiw, Romer, and Weil (1992) and Islam (1995). However, the results cease to corroborate the theoretical model for a sample of advanced countries. Thus, neither model accurately describes advanced countries in the long run.
All in all, this paper reinforces the conclusions conventional in the literature. At the same time, this means that the mists covering the mechanisms underlying long-run growth in advanced countries have yet to be cleared.
Appendix: the output-capital ratio in the long run
I start from the definition of 𝑘 and differentiate it with respect to time: 𝑘(𝑡) =((.)+(.) such that 𝑘(𝑡)̇ = )0(.)
). = +(̇
+$ −
(+̇
+$ = 𝑠𝑦 − 𝛿𝑘 − 𝑘𝑛. In the steady state, 𝑘̇ = 0 so that 𝑠𝑦 = 𝑘(𝑛 + 𝛿). Next, consider the production function the steady-state level of 𝑘 and 𝑦: 𝑦 =*
+ = 𝐴𝑘 ![+!,
+ ]
#$!. Since 𝑠𝑦 = 𝑘(𝑛 + 𝛿) in the steady state, 𝑘 = [ 8 9:;] % %&'+!, + 𝐴 % %&' and 𝑦 = [9:;8 ] ' %&'+!, + 𝐴 %
%&'. These simplify to:
𝑘 = 𝐴#$!# 𝑥𝑞 and 𝑦 = 𝐴#$!# 𝑥!𝑞 with 𝑥 = [9:;8 ] % %&' and 𝑞 =+!, + .
The last step makes the derivation of the growth of 𝑘 and 𝑦 in the steady state easier. Since 𝑘(𝑡) = 𝐴%&'% 𝑥𝑞(𝑡), 𝑘̇ =)0(.) ). = 𝑥 # #$!𝐴̇ ' %&'𝑞 + 𝑥𝑞̇𝑎 %
%&' such that 0̇
0= # #$![ -̇' -] % %&'+<̇ <. Similarly, "̇"=#$!# [-̇-'] % %&'+<̇ <. Thus, "̇ "= 0̇ 0= # #$![ -̇' -] % %&'+<̇ < with 𝑞 = +!,
+ , proving that 𝑘 and 𝑦 have the same long-run growth rate.
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