MICRO SEMINAR PAPER-SOLOW

MICRO SEMINAR PAPER-SOLOW

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  Fotini Simoglou FabioAmorim MicroeconomicsSeminar- OdysseasKatsaitis Theoriesof Economic Growth: Estimation of the Neoclassical Solow Model
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  Abstract The objective ofthispaperis to demonstrate the theoriesof growthoriginatedfrom the Neoclassical Solow model andto estimate the Original Solow Model accordingtothe paperof Mankiw,RomerandWeil entitled “A Contributiontothe Empiricsof Economic Growth”. A sample of data from107 OECD countrieswill be estimated throughthe use of the SolowModel equation
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  Introduction: The OriginalSolowModel,The AugmentedSolowModel and Endogenous Growth Theories. With his neoclassical growth model, Robert Solow made a substantial contribution to Economics by influencing further studies on growth as well as many fields in economics (P. Edward, 11). As Robert Solow describe: The "neoclassical model of economic growth started a small industry. It stimulated hundreds of theoretical and empirical articles by other economists”. His model was published in 1956 in his paper named “A Contribution to the Theory of Economic Growth” and he introduced the neoclassical factor substitution in production to the Harrod-Domar model (Deardorff Alan, 1). As Robert Solow explains in his “Growth Theory and after” paper,Harrod and Domar concluded that in order for an economy to achieve steady growth at a constant rate, the saving rate has to equalize the “product of the capital output ratio” and the rate of labor force growth assuming that those elements are a given constant (R. Solow, 307). Solow was not satisfied with this assumption and tried to develop a more realistic model comprised of, as he described, “a reasonable degree of technological flexibility” (R.Solow, 308). The results achieved by Robert Solow suggested that there is a range of steady growth states and that the permanent rate of output growth per unit of labor depends on technology and not on savings (R. Solow, 309). Gregory Mankiw, David Romer and David N. Weil in their paper “A contribution to the empirics of economic growth”, explains that the original Solow model examines economic growth starting from a neoclassical production function with decreasing returns to capital (G. Mankiw, 407). They stated that the main argument of the Solow model is to demonstrate how real income is affected by saving and population growth (Mankiw, 410). According to their paper,Solow demonstrated that each country achieves a different “steady-state” level
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  of income bytaking the rates of saving and population growth as exogenous (G. Mankiw, 407). From this assumption and accounting that population growth and savings differ across countries, it can be concluded that a country with a higher rate of savings tend to be richer and a country with a higher rate of population growth tend to be poorer (G. Mankiw, 407). According to Mankiw, Romer and Weil, the model is consistent with its predictions in the sense that a substantial part of the cross-country income per capita variation can be assigned to the rate of savings and rate of population growth (Mankiw, 407). However,they observe that the Solow model do not correctly predict the size of this variation once it is found to be “too large” (Mankiw, 408). Therefore,as they conclude, the Solow is not a complete model but should not be ignored observing that it can correctly predict a great part of the cross country variation in income even by assuming decreasing returns to scale and by treating saving, population growth and technological advance as exogenous (Mankiw, 409). According to A. Bassemini and S. Scarpetta,the NeoclassicalSolow Model by treating those variables as exogenous and assuming decreasing returns to scale would indicate that the rate of growth of richer countries is slower than the rate of poorer countries. As a criticism however, they argue that this convergence indicated by the model has recently becoming weak among OECD countries. Moreover, it would indicate that long- term economic growth would not be affected by policies (A. Bassemini and S. Scarpetta,11). Mankiw, Romer and Weil derived the Solow Model and estimated the constant α by using the following equation: (7)           )ln( 1 )ln( 1 ln     gns a a L Y Their derivation started by assuming a Cobb-Douglas production function taking the rates of saving, population growth and technological progress as exogenous. So, production at time t is: (1)          aa tLtAtKtY   1
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     gn,ratesoutyexogenousl growtoassumedare, AL 0 <a < 1 Y: Output K: Capital L: Labor A: Level of technology Then,   gt nt eAtA eLtL   )0()()3( )0()(2   Effective units of labor A(t) L(t), grows at rate n+g The model implies that a constant fraction of output, s, is invested. K is the stock of capital per affective unit LA K K   Y is the level of output per effective unit  a tK AL Y y  The evolution of K is driven by: (4) )()()()( tgntsyt   Where δ is the rate of depreciation. )()()( tgntKS a   The equation above implies that K converges to a steady-state value K* if in the above equation K is zero then:   **  gnKS a Or, by solving the equation for K(t):
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     a t t gn S * *       a t gn S    1*  (5) (This is the steady state value) The steady-state capital labor ratio positively related to the rate saving and negatively related to the rate of population growth. The main prediction of the Solow model refers to the impact of saving and population growth on real income. We substitute the equation (5) into the production function and we obtain the      tAtKt a  where   1 1         gn S tK And     gt eAt  0 Then the logs are taken in order to find the steady income per capita: gt a e dn S tL tY            )0( )( )( 11 1 1          gt eA gn S tL tY ln)0(lnlnln 1                ln η λογαριθμική και η εκθετική είναι αντίστροφες συναρτήσεις. Thus,                  gn S a a gtA tL tY ln 1 )0(lnln      )ln()ln( 1 )0(lnln         gns a a gtA tL tY Therefore,the steady-state income per capita is the equation (6) :
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      )ln( 1 )ln( 1 )0(lnln             gns a a gtA tL tY According to Mankiw, Romer and Weil, the signs and the magnitudes of the saving and population growth coefficients are predicted in the model. An elasticity of income per capita related to saving rate of 0.5 and an elasticity related to n+g+δ of -0.5 is exhibited because capital share in income α is one third (Mankiw, 410). They wanted to test the predictions of the Solow Model that realincome is higher in countries with a high rate of savings and lower in countries with higher values of population growth rates (n+g+δ) assuming that n and δ are constant across countries (Mankiw, 410). Depreciation rates (δ) and the advancement of knowledge (g) were not expected to vary largely across countries. However,the term A(0) which is attributed to technology, resource endowments, climate, institutions etc..., can be different in each country ( Mankiw, 411). Thus, it is assumed that:  )0(ln A Being α a constant and ε a “country-specific shock”, then log of income per capita at time 0 is given by the equation (7):           )ln( 1 )ln( 1 ln     gns a a L Y This is the equation that Mankiw, Romer and Wiel used as their empirical specification for testing the original Solow Model. They assumed that s and n are independent of ε meaning that country specific factor that shifts the production function are not dependent on the rate of savings and population growth. Therefore, based on this fact they estimated the equation (7) with ordinary least squares (OLS) (Mankiw, 411). By doing so, they could find if there are any biases in the model as it predicts the signs and magnitudes of savings and
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  population growth. Theystate that if the elasticities of Y/L in respect to s is 0.5 and in respect to n+g+δ is -0.5, the model is correct (Mankiw, 412). Therefore, they examined and compared the value of α with the value found by factor shares. The data used by Mankiw, Romer and Wiel were taken from the RealNational Accounts from Summers and Heston and includes real income, government and private consumption, investment and population growth ranging from 1960 to 1985. The working age population (aged 15 to 64) was measured by n, the average share of real investment was measured by s in realGDP divided by the working age population Y/L(Mankiw, 412). They used three data samples of countries. The first one included 98 countries excluding the oil producers. The second was comprised of 75 countries and excluded countries in which population was less than 1 million in 1960 and the ones that got a D grade from Summers and Heston. The third was comprised from the 22 OECD countries that had population over 1 million. This third set of data was considered of high quality and uniform with small variation “omitted country-specific factors” but had the disadvantage that much of the variation of the important variables were not taken in consideration and the sample is small (Mankiw, 413). By running the equation (7), Mankiw, Romer and Weil, found three results that were consistent with the Solow Model. The most significant, as they argue, is that large amount of the cross-country variation in income per capita were accounted by differences in saving and in population growth (Mankiw, 414). This implies that saving and population growth account for most of the variation in income per capita. In the regression for the second sample, R2 (R squared) was found 0.59 (Mankiw, 414). The other consistent results are that the predicted signs for coefficients of saving and population growth are “high significant” and that there was no rejection on the assumed restriction that ln(s) coefficients and (n+g+δ) are of the same magnitude and different sign (Mankiw, 414). In order to obtain better results, Mankiw, Romer and Weil augmented the neoclassical Solow Model by introducing the accumulation of human capital. The literature identifies two ways in which educational investment can contribute to growth. First, human capital can participate in production as a productor factor,
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  and in thesense the accumulation of human capital could directly create growth of output. Second, human capital can contribute to raising technological progress, since education eases the innovation and adoption of new technologies affecting positively growth (Freire-Seren, p585).In the neoclassical Solow approach the exclusion of human capital explains why the effects of saving and population growth on income are too large. Firstly, because for a given level of human –capital accumulation, higher saving or lower population growth leads to higher level of income, when accumulation of human capital taken into account. Secondly, human –capital accumulation are correlated with saving and population growth rates (Mankiw, p407-408). So, in order to expand further and avoid large magnitudes of saving and population growth effects on income, we include a proxy for human-capital accumulation at the neoclassical model and find that accumulation of human capital is in fact correlated with saving and population and solving the problem of large magnitudes(Mankiw, p408). With the introduction of the human capital variable, the nature of growth process changes as the augmented Solow model differs from the neoclassical in severalways. First, due to the fact that in the augmented the elasticity of income with respect to stock of physical capital is not from capital’s share in income, this suggests that “capital receives its social return and thus there are no externalities to the accumulation of physical capital” (Mankiw, p432).Secondly, the accumulation of physical capital has a larger impact on income per capita than the original Solow model. A higher saving rate leads to higher income and higher level of human capital. Third, at the augmented, population growth has larger impact on income per capita than the original model. In the neoclassical model higher population growth lowers income as capital must be spread more thinly at working population. At the augmented, human capital has also be spread thinly and that means that higher population growth lowers total factor productivity. Fourth, another difference is that at the augmented convergence occurs more slowly than in the original model. Finally, the augmented model suggests that differences in saving, education and population growth should explain cross country differences in income per capita (Mankiw, p432). Furthermore, with the augmented model we see that diminishing returns to broad capital is less severe than to the physical capital in the original Solow model.
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  Convergence, in addition,to steady state is slower and transition effects last longer. So, regarding R.Gordon, Solow introduced an added element into the model to be consistent with history: growth in technology including better schooling (human capital) leading to improved organization and all fruits of innovation and research(Gordon, p284). Further than the neutral technological progress which leads to higher level of output achieved with the same quantity and combination of factor inputs, further than the fact that technological progress can lead in savings on either capital or labor, technological progress may also be labor-augmenting. “Labor augmenting technological progress occurs when quality of skills of labor force are upgraded and therefore human capital leads to further growth”(Todaro & Smith, p145). Lucas (1988) further improved the analysis also by adding the human capital factor distinguished from physical capital. “A poor country with little human capital cannot become rich just by accumulating physical capital and in fact its rate of return on investment in physical capital may be lower than in rich countries”(Gordon, p291). He also assumes that although there are decreasing returns to physical capital accumulation, when human capital is held constant, there is also constant returns to reproducible human and physical capital (Gordon, p291). This approach makes improved education the key to achieving economic growth. Further studies in economic growth resulted in endogenous growth models that are contrasted to the neoclassical Solow Model. While the Solow Model argues that exogenous technological progress mainly affect long run growth, endogenous models argue that growth rates can be modified by endogenous variables that are defined by government policies (C. Jones, 495). According to Nick Crafts,the main idea behind endogenous growth models is that investment decisions do affect long run growth. If that is the case,as he explains, government’s policy could favorably affect investments decisions (N. Craft,30). Thus, direct policies regarding taxations and subsidies as well as indirect policies regarding institutional reforms and intervention could raise investments and affect growth (N. Craft, 30). One of the main factors in determining the level of real output per capita is the physical accumulation of
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  capital (OECD). Itseffects could be more or less permanent depending on the extent to which technological innovation is embodied in new capital(OECD, p11-18). Based on previous theories, Gavin Cameron discuss about the effects of innovation to the rate of growth in his paper entitled “Innovation and Growth: a survey of the Empirical Evidence”. He pointed out that the original Solow model predicted that growth was related to technological changes but it didn’t explain the causes of those changes. Cameron then questioned if the lack of innovations wouldn’t reduce the rate of growth: “Is it likely that economic growth would continue in the absence of increased workforce skill levels, investment in R&D and public infrastructure, the installation of capital equipment embodying new technologies, or changes in types and varieties of goods?” (G.Cameron, 4). He then defined innovations as being research and development spending, patenting, innovation counts and technological spillovers between firms, industries and countries (G.Cameron, 1). For this measurement, he accounted for externalities that can influence the effects of innovations. The “standing on shoulders” implies that rival firms can take advantage on spill over’s caused by leaks in knowledge, mistakes in patenting and skilled labor movements. The “surplus appropriability” appears when the entire social gains related to the spill over’s do not return to the innovator. Another externality, the “creative destruction” occurs because new innovations can make obsolete previous productions. Finally, “stepping on toes” occurs when “congestion or network externalities” appear when the implementation of innovations are “substitutes or complements” (G.Cameron, 2) . The results based on Griliches study, demonstrate that there is a strong correlation between investments in R&D and output and that spillovers from technological innovation are very important and wide. In numbers, it is estimated that output is increased by 0.05% to 0.1% when the capital stock for R&D is raised by 1% and that the rate of social return stands between 20 and 50% (Cameron, 21). Cameron also stresses that,in contrast to the neoclassical model predictions, convergence do not happen when more complexes endogenous models are applied and that the issue is still not a common sense among economists (Cameron, 22). As a conclusion, Cameron suggests that the most important for
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  increasing productivity isthe innovation and consequently spillovers from domestic firms and organizations rather than international spillovers (Cameron, 22). According to him, foreign spillovers do not flow easily to the internal economy because of “secrecy, geographic and cultural barriers and for it to happen it would be necessary a heavy investment. He stresses that human capital formation is mainly obtained from investment in higher education (Cameron, 22). Besides the variables discussed above, other main determinants of growth are observed. Especially through reduction of uncertainty, growth may be positively affected by a stable macroeconomic environment. On the contrary, macroeconomic instability may have a negative impact on growth as it impairs productivity and investment (OECD,p11-18). Inflation and growth are variables that are connected with the macroeconomic environment. A positive and low risk economic environment for investment is created when inflation rates are stable. This stability reduces uncertainty in the economy and consequently enhance efficiency of the price mechanism (OECD, p11-18). Another determinant of growth is related to international trade. An open economy is more susceptible to growth because it benefits from comparative advantage,from transfer and diffusion of knowledge and technology, from increasing economies of scale and from the exposure from competition (Petrakos,p8-11). Institutional frameworks not only directly influence economic growth but also affect other determinants of growth such as physical capital, human capital and investment. Problems such as corruption, property rights and bureaucratic issues can be solved or minimized if a trustworthy institutional environment is part of the economy (Petrakos,p8-11). Political instability also affects growth since it may increase uncertainty, and therefore discourage investment and eventually impair economic growth.
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  Mathematics behind theAugmented SolowModel: Let the production function be         baba tLtAtHtKt   1 )()( Where H the stock of human capital and all other variables are defined as before A(t)=A(0)∙egt L(t)=L(0)∙ent (1) The evolution of K is k(t)=Sk∙y(t)-(n+g+δ) ∙k(t) (2) The evolution of H is h(t)=Sh∙y(t)-(n+g+δ) ∙h(t) Where: Sk is the fraction of income invested in physical capital and Sh the fraction invested in human capital Also, AL Y y  AL K k  AL H h  are quantities per effective unit of labor. We assume that a+b<1 which implies that there are decreasing returns to all capital (if a+b=1, then there are constant returns to scale and in this case, there is no steady state fro this model). If I go to (1), (2) and assume K(t)=0 and h(t)=0 then above equations imply that the economy converges to a steady state defined by )()()( thtk AL Y t ba  I have Sk∙Y(t)=(n+g+δ) ∙k(t) (1) and Sh∙Y(t)=(n+g+δ) ∙h(t) (2)
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    bbbb tkgntySk  1111 )()(  I put (1) to the power of 1-b   )()( thgntySh bbbb   I put (2) to the power of b Then I multiply both:   )()()( 111 thtKgntyShSk bbbb      )()()()( 11 thtKgnthtKShSk bbbabb    Then )( )( )( )( * 1 1 1 1 1 1 1 tK gn ShSk t gn hSkS tK t gn hSkS babb ba bb a b bb                        Then: Sk∙y(t)=(n+g+δ)∙Κ(t) (1)   )()( tkgntySk    I put (1) to the power of a (2)   )()( 1111 thgntyhS     I put (2) to the power of 1-a Then I multiply them, )1(1 11 )()()()()( )()()()( a ba aa aaa thtgnthtkhSkS thtgntyhSkS         ba aa b a aa th gn hSkS th th gn ShSk          1 11 1 )( )( )( 
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  Then, )(* 1 1 1 th gn ShkS ba a               I returnto the production function: )()()( )( )( )()()()()( tAtHtK tL tY tLtAtHtKtY ba ba   Where A(t)=A(0)∙egt I substitute: gt ba ba ba b ba a gt ba b ba ab ba ab ba a ba ab ba ba gt ba b aabb eA gn ShSk tL tY eA gn ShSk gn ShSk tL tY eA gn hSkS gn hSkS tL tY                                              )0( )( )( )( )0( )()( )( )( )0( )( )( 1 11 ) 1 1 )1( 1 1 11 )1( 1111      I put logarithms: gtAgn ba ba Sh ba b Sk ba a tL tY eAgnShSk tL tY gtba ba ba b ba a             )0(ln)ln( 1 )ln( 1 )ln( 1)( )( ln ln)0(ln)ln(lnln )( )( ln 111  
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  This equation showshow income per capita depends on population growth and accuracy of physical and human capital. Combining (11) with the equation for the steady-state level of human capital given in (10) yields an equation for income as a function of the rate of investment in physical capital, the rate of population growth and the level of human capital. (10)               1 1 1 * gn hSkS h aa (11) )ln( 1 )ln( 1 )ln( 1 )0(ln )( )( ln Sh ba b Sk ba a gn ba ba gtA tL tY               From 10 I have:      gn hSkS h aa ba 1 1* a a ba hS kS gnh    11* )(  , so, Sh kS gnh a ba   )(1* 
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  Now in 11I have :           )ln( 1 )ln( (1)(1( )1( )ln( )1)(1( )1( )0(ln )( )( ln )( 1 )ln( 11 )ln( 11 )0(ln )( )( ln )ln( 1)1(1 )ln( )1)(1(1 )0(ln )( )( ln ln 1 1 1 )ln( 1 1 1 ln)1( 1 1 1 )ln( 1 )ln( 1 )0(ln )( )( ln ln)ln(ln 1 1 1 )ln( 1 )ln( 1 )0(ln )( )( ln )( ln 1 )ln( 1 )ln( 1 )0(ln )( )( ln * * 2 2 1 1 * 1* 1 1 1* h a b Sk aba baa gn aba baa gtA tL tY h a b Sk aba abaa gn aba babbaa gtA tL tY Sk a a ba b ba a gn aba b ba ba gtA tL tY kS aba b gn aba b hba aba b Sk ba a gn ba ba gtA tL tY Skgnh aba b Sk ba a gn ba ba gtA tL tY kS gnh ba b Sk ba a gn ba ba gtA tL tY a a a aba a a ba                                                                                                                                                
• 18.
           )ln( 1 )ln( (1)(1( )1( )ln( )1)(1( )1( )0(ln )( )( ln )( 1 )ln( 11 )ln( 11 )0(ln )( )( ln )ln( 1)1(1 )ln( )1)(1(1 )0(ln )( )( ln ln 1 1 1 )ln( 1 1 1 ln)1( 1 1 1 )ln( 1 )ln( 1 )0(ln )( )( ln ln)ln(ln 1 1 1 )ln( 1 )ln( 1 )0(ln )( )( ln )( ln 1 )ln( 1 )ln( 1 )0(ln )( )( ln * * 2 2 1 1 * 1* 1 1 1* h a b Sk aba baa gn aba baa gtA tL tY h a b Sk aba abaa gn aba babbaa gtA tL tY Sk a a ba b ba a gn aba b ba ba gtA tL tY kS aba b gn aba b hba aba b Sk ba a gn ba ba gtA tL tY Skgnh aba b Sk ba a gn ba ba gtA tL tY kS gnh ba b Sk ba a gn ba ba gtA tL tY a a a aba a a ba                                                                                                                                                
• 19.
  Mathematics Behind TheoriesofEndogenous Growth and Convergence: Y* the steady state level of income given by the equation (11) )ln( 1 )ln( 1 )ln( 1 )0(ln )( )( ln Sh ba b Sk ba a gn ba ba gtA tL t              And Y(t) be the actual value of time and the speed of convergence is given by: Where λ=(n+g+δ)(1-a-b): Convergence rate. Here if we put 3 1  ba and n+g+δ=0,06, then λ=0,02 that means that economy goes for stabilization in 35 years.     )ln()(ln )(ln * yty dt tyd   I multiply by eλt      * ln)(ln )(ln yetye dt tyd e ttt        *' ln)(ln yetye tt   f΄g+fg΄=(fg)΄     cyetye tt  * ln)(ln  I divide by eλt        )0(lnlnln)(ln ** yeyeyty tt          )(lnln )(ln * tyy dt tyd  
• 20.
  Where t=0 Ifind c which is income per effective worker at some initial date. I subtract from the two sides the ln(y(0))            )0(ln)0(lnln1)0(ln)(ln * yyeyeyty tt    Equation 15:            )0(ln1ln1)0(ln)(ln * yeyeyty tt    I substitute the ln(y* ) from equation 11 and I take the equation 16.                )0(ln1ln 1 1 )ln( 1 1 )ln( 1 1)0(ln)(ln yegn ba ba e Sh ba b e Sk ba a eyty tt t t                 
• 21.
  Estimation ofthe originalSolowModel by Mankiw, Romer and Weil: Equation (7):           )ln( 1 )ln( 1 ln     gns a a L Y Mankiw, Romer and Wiel used the equation (7) as their empirical specification for testing the original Solow Model. They assumed that s and n are independent of ε meaning that country specific factor that shifts the production function are not dependent on the rate of savings and population growth. Therefore,based on this fact they estimated the equation (7) with ordinary least squares (OLS) (Mankiw, 411). By doing so, they could find if there are any biases in the model as it predicts the signs and magnitudes of savings and population growth. They state that if the elasticities of Y/L in respect to s is 0.5 and in respect to n+g+δ is -0.5, the model is correct (Mankiw, 412). Therefore,they examined and compared the value of α with the value found by factor shares. The data used by Mankiw, Romer and Wiel were taken from the RealNational Accounts from Summers and Heston and includes real income, government and private consumption, investment and population growth ranging from 1960 to 1985. The working age population (aged 15 to 64) was measured by n, the average share of real investment was measured by s in realGDP divided by the working age population Y/L(Mankiw, 412). They used three data samples of countries. The first one included 98 countries excluding the oil producers. The second was comprised of 75 countries and excluded countries in which population was less than 1 million in 1960 and the ones that got a D grade from Summers and Heston. The third was comprised from the 22 OECD countries that had population over 1 million. This third set of data was considered of high quality and uniform with small variation “omitted country-specific factors” but had the disadvantage that much of the variation of the important variables were not taken in consideration and the sample is small (Mankiw, 413). By running the equation (7), Mankiw, Romer and Weil, found three results that were consistent with the Solow Model. The most significant, as they argue, is that large amount of
• 22.
  the cross-country variationin income per capita were accounted by differences in saving and in population growth (Mankiw, 414). This implies that saving and population growth account for most of the variation in income per capita. In the regression for the second sample, R2 (R squared) was found 0.59 (Mankiw, 414). The other consistent results are that the predicted signs for coefficients of saving and population growth are “high significant” and that there was no rejection on the assumed restriction that ln(s) coefficients and (n+g+δ) are of the same magnitude and different sign (Mankiw, 414).