Fotini Simoglou

Fabio Amorim

Microeconomics Seminar - Odysseas Katsaitis

Theories of Economic Growth: Estimation of the Neoclassical Solow Model

Abstract

The objective of this paper is to demonstrate the theories of growth originated from

the Neoclassical Solow model and to estimate the Original Solow Model according to the

paper of Mankiw, Romer and Weil entitled “A Contribution to the Empirics of Economic

Growth”. A sample of data from 107 OECD countries will be estimated through the use of

the Solow Model equation

Introduction: The Original Solow Model, The Augmented Solow Model 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

of income by taking 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 Neoclassical Solow 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 (Y

L )=α + a1−a

ln ( s )− α1−α

ln ( n+g+δ )+∈

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) Y (t )=K (t )a ( A (t )⋅L (t ) )1−a

0 < a < 1

Y: Output

K: Capital

L: Labor

A: Level of technology

Then

,

(2 )⋯L( t )=L(0 )⋅ent

(3 )⋯A ( t )−A (0)⋅egt

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

K= KA⋅L

Y is the level of output per effective unit

{ L , A are assumed to grow ¿ ¿¿¿

y= YAL

=K (t )a

The evolution of K is driven by:

(4)

Κ̇ ( t )=sy ( t )−(n+g+δ )⋅Κ ( t ) Where δ is the rate of depreciation.

S⋅K (t )a−(n+g+δ )⋅Κ ( t )

The equation above implies that K converges to a steady-state value K* if in the above

equation K is zero then:

S⋅K ¿a=(n+g+δ )⋅Κ ¿

Or, by solving the equation for K(t):

Sn+g+δ

=Κ ¿ (t )Κ ¿ (t )a

Sn+g+δ

=Κ ¿ (t )1−a

(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

Υ (t )=Ka (t )⋅A (t ) where

K (t )=[ Sn+g+δ ]

11

And

Α ( t )=A (0 )⋅egt

Then the logs are taken in order to find the steady income per capita:

Y ( t )L( t )

=[ Sn+d+δ ]

a1−α

→ 11−α

⋅α= α1−α⋅Α (0)⋅egt

ln (Y (t )L (t ) )=ln [ S

n+g+δ ]1−α

+ ln A (0)+ ln egt

ln η λογαριθμική και η εκθετική είναι αντίστροφες συναρτήσεις.

Thus,

ln (Y ( t )L ( t ) )=ln A (0 )+gt +

a1−a

ln [ Sn+g+δ ]

ln (Y ( t )L ( t ) )=ln A (0 )+gt +

a1−a

(ln ( s )−ln ( n+g+δ ))

Therefore, the steady-state income per capita is the equation (6) :

ln (Y (t )L (t ) )=ln A (0 )+gt +

a1−a

ln( s )−α

1−αln(n+g+δ )

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 real income 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:

ln A( 0)=α+∈

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 (YL )=α + a

1−aln( s )− α

1−αln( n+g+δ )+∈

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

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 Real National

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 real GDP 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,

and in the sense 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 several ways. 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.

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

capital (OECD). Its effects 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

increasing productivity is the 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.

Mathematics behind the Augmented Solow Model:

Let the production function be

Υ (t )=K (t )a⋅H (t )b ( A( t )⋅L( t ))1−a−b

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,

y= YAL

k= KAL

h= HAL

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

Υ ( t )= YAL

=ka ( t )⋅hb ( t )

I have Sk∙Y(t)=(n+g+δ) ∙k(t) (1)

and Sh∙Y(t)=(n+g+δ) ∙h(t) (2)

Sk1−b⋅y ( t )1−b= (n+g+δ )1−b⋅k ( t )1−b

I put (1) to the power of 1-b

Shb⋅yb ( t )=(n+g+δ )b⋅hb( t ) I put (2) to the power of b

Then I multiply both:

Sk1−b⋅Shb⋅y ( t )=(n+g+δ )1⋅K ( t )1−b⋅hb( t )

Sk1−b⋅Shb⋅Ka ( t )⋅hb ( t )=(n+g+δ )⋅K1−b ( t )⋅hb ( t )

Then

S k1−b

⋅S hb

n+g+δ=

Κ ( t )1−b

K ( t )a⇔S k

1−b

⋅S hb

n+g+δ=Κ ( t )1−a−b

[Sk 1−b⋅Shb

n+g+δ ]11−a−b =K ¿( t )

Then: Sk∙y(t)=(n+g+δ)∙Κ(t)

(1)

Skα⋅y ( t )α= (n+g+δ )α⋅kα ( t ) I put (1) to the power of a

(2)

S1−α h⋅y1−α( t )=( n+g+δ )1−α⋅h1−α ( t ) I put (2) to the power of 1-a

Then I multiply them,

S ka

⋅S h1−a

⋅y ( t )=(n+g+δ )⋅Κα ( t )⋅h ( t )1−a

S ka

⋅S h1−a

⋅ka ( t )⋅hb ( t )=(n+g+δ )⋅Κ α( t )⋅h (t )(1−a)

Sk a⋅Sh1−a

n+g+δ=

h ( t )1−a

h( t )b⇔ S k

a

⋅S h1−a

n+g+δ=h ( t )

1−a−b

Then,

( S ka

⋅Sh1−a

n+g+δ )1

1−α−b=h¿( t )

I return to the production function:

Y ( t )=K a( t )⋅H b( t )⋅A ( t )⋅L( t )

Y ( t )L( t )

=Ka ( t )⋅Hb (t )⋅A ( t )

Where A(t)=A(0)∙egt I substitute:

Y ( t )L( t )

=(S k1−b

⋅S hb

n+g+δ )α1−α−β

⋅(Ska

⋅S h1−a

n+g+δ )b1−a−b

⋅A (0)⋅egt

Y ( t )L( t )

=Ska(1−b )1−a−b⋅Sh

ab1−a−b

(n+g+δ )a1−a−b

⋅Skab1−a−b⋅Sh

b(1−a )1−a−b

(n+g+δ )b1−a−b

⋅A (0 )⋅egt

Y ( t )L( t )

=Ska)1−a−b⋅Sh

b1−a−b

(n+g+δ )a+b1−a−b

⋅A (0 )⋅egt

I put logarithms:

lnY ( t )L( t )

=ln Ska1−a−b + ln Sh

b1−a−b −ln (n+g+δ )

a+b1−a−b + ln A (0 )+ ln egt

lnY ( t )L( t )

=a1−a−b

ln(Sk )+b1−a−b

ln ( Sh)−a+b1−a−b

ln (n+g+δ )+ ln A (0)+gt

This equation shows how 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)

h¿=( S ka

⋅S h1−a

n+g+δ )1

1−α−β

(11)

ln [Y ( t )L( t ) ]=ln A (0 )+gt−a+b

1−a−bln(n+g+δ )+

+a1−a−b

ln( Sk )+b1−a−b

ln( Sh )

From 10 I have:

h¿ 1−a−b=S ka

⋅S h1−a

n+g+δ

h¿1−a−b (n+g+δ )

S ka

=S h1−a

, so,

h¿1−a−b (n+g+δ )

S ka

=Sh

Now in 11 I have :

ln [Y ( t )L( t ) ]=ln A (0 )+gt−a+b

1−a−bln (n+g+δ )+

+a1−a−b

ln( Sk )+b1−a−b

ln(h¿1−a−b(n+g+δ )

S ka )

11−a

ln [Y ( t )L( t ) ]=ln A (0 )+gt−a+b

1−a−bln (n+g+δ )+

+a1−a−b

ln( Sk )+b1−a−b

⋅11−a

[ ln h¿1−a−b+ln(n+g+δ )−ln Ska ]

ln [Y ( t )L( t ) ]=ln A (0 )+gt−a+b

1−a−bln (n+g+δ )+

+a1−a−b

ln( Sk )+b1−a−b

⋅11−a

⋅(1−a−b) ln h¿+

+b1−a−b

⋅11−a

ln (n+g+δ )−b1−a−b

⋅11−a

ln S ka

⇔

ln [Y ( t )L( t ) ]=ln A (0 )+gt−[a+b

1−a

1−a−b−

b(1−a−b)(1−a ) ]⋅

¿ ln(n+g+δ )+[ a1−a

1−a−b−b

(1−a−b)⋅a

1−a ] ln( Sk )+

ln [Y ( t )L( t ) ]=ln A (0 )+gt−

[a−a2+b−ab−b ](1−a−b ) (1−a )

⋅

ln (n+g+δ )+[a−a2−ab(1−a−b ) (1−a ) ]⋅ln (Sk )+b

1−a(h¿ )⇔

ln [Y ( t )L( t ) ]=ln A (0 )+gt−

a (1−a−b )(1−a−b )(1−a )

ln(n+g+δ )+a(1−a−b)(1−a−b )(1−a(

ln(Sk )+b1−a

⋅ln (h¿ )

ln [Y ( t )L( t ) ]=ln A (0 )+gt−a+b

1−a−bln(n+g+δ )+

+a1−a−b

ln( Sk )+b1−a−b

ln(h¿1−a−b(n+g+δ )

S ka )

11−a

ln [Y ( t )L( t ) ]=ln A (0 )+gt−a+b

1−a−bln(n+g+δ )+

+a1−a−b

ln( Sk )+b1−a−b

⋅11−a

[ ln h¿1−a−b+ln(n+g+δ )−ln Ska ]

ln [Y ( t )L( t ) ]=ln A (0 )+gt−a+b

1−a−bln(n+g+δ )+

+a1−a−b

ln( Sk )+b1−a−b

⋅11−a

⋅(1−a−b) ln h¿+

+b1−a−b

⋅11−a

ln (n+g+δ )−b1−a−b

⋅11−a

ln S ka

⇔

ln [Y ( t )L( t ) ]=ln A (0 )+gt−[a+b

1−a

1−a−b−

b(1−a−b)(1−a ) ]⋅

¿ ln(n+g+δ )+[ a1−a

1−a−b−b

(1−a−b)⋅a

1−a ] ln( Sk )+

ln [Y ( t )L( t ) ]=ln A (0 )+gt−

[a−a2+b−ab−b ](1−a−b ) (1−a )

⋅

ln(n+g+δ )+[a−a2−ab(1−a−b ) (1−a ) ]⋅ln(Sk )+b

1−a(h¿ )⇔

ln [Y ( t )L( t ) ]=ln A (0 )+gt−

a (1−a−b )(1−a−b )(1−a )

ln(n+g+δ )+a(1−a−b)(1−a−b )(1−a(

ln(Sk )+b1−a

⋅ln (h¿ )

Mathematics Behind Theories of Endogenous Growth and Convergence:

Y* the steady state level of income given by the equation (11)

ln [Υ ( t )L( t ) ]=ln A (0)+gt− a+b

1−a−bln(n+g+δ )+ a

1−a−bln(Sk )+ b

1−a−bln(Sh )

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

a=b=13

and n+g+δ=0,06, then λ=0,02 that means that economy goes for

stabilization in 35 years.

d ln ( y ( t ))dt

+λ ln ( y ( t ))=λ ln ( y¿ )

I multiply by eλt

e λt d ln ( y ( t ))dt

+λeλt ln ( y ( t ))=λe λt ln ( y¿)

[e λt ln ( y ( t )) ]'=λeλt ln ( y¿ )f΄g+fg΄=(fg)΄

d ln ( y ( t ))dt

=λ [ln ( y¿ )−ln ( y ( t )) ]

e λt ln ( y ( t ))=eλt ln ( y¿ )+c I divide by eλt

ln ( y ( t ))=ln ( y¿)−eλt ln ( y¿)+e− λt ln ( y (0 ))

Where t=0 I find c which is income per effective worker at some initial date.

I subtract from the two sides the ln(y(0))

ln ( y ( t ))−ln ( y (0 ))=(1−e−λt ) ln ( y¿)+e− λt ln ( y (0 ))−ln ( y (0))

Equation 15:

ln ( y ( t ))−ln ( y (0 ))=(1−e−λt ) ln ( y¿)−(1−e− λt ) ln ( y (0 ))

I substitute the ln(y*) from equation 11 and I take the equation 16.

ln ( y ( t ))−ln ( y (0 ))=(1−e−λt ) a1−a−b

ln (Sk )+

+(1−e−λt ) b1−a−b

ln (Sh )−

−(1−e−λt ) a+b1−a−b

ln (n+g+δ )−(1−e−λt ) ln ( y (0))

Estimation of the original Solow Model by Mankiw, Romer and Weil:

Equation (7):

ln (YL )=α + a

1−aln( s )− α

1−αln( n+g+δ )+∈

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 Real National

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 real GDP 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).