lecture 16: revisiting the solow model

Lecture 16: Revisiting the Solow model

José L. Torres

Universidad de Málaga

Advanced Macroeconomics

José L. Torres (Universidad de Málaga) Lecture 16: Revisiting the Solow model Advanced Macroeconomics 1 / 23

16. Revisiting the exogenous growth model

Important questions:

Why economies growth?Why some economies have larger income level than others?Why some economies growth at high rates and other economies at lowrates?What is a economic miracle?Is possible that today rich countries will be poor countries in thefuture? And the opposite?Is possible that some poor countries will remain poor forever?



PIB

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

Q 1/1970 Q 1/1975 Q 1/1980 Q 1/1985 Q 1/1990 Q 1/1995 Q 1/2000 Q 1/2005

Figure: GDP Spain 1970-2008José L. Torres (Universidad de Málaga) Lecture 16: Revisiting the Solow model Advanced Macroeconomics 3 / 23


Tasa de crecimiento del PIB

6

4

2

0

2

4

6

8

10

Q 1/1970 Q 1/1975 Q 1/1980 Q 1/1985 Q 1/1990 Q 1/1995 Q 1/2000 Q 1/2005

Figure: GDP growth rate SpainJosé L. Torres (Universidad de Málaga) Lecture 16: Revisiting the Solow model Advanced Macroeconomics 4 / 23


Richer countries across time:

Around 1600: China.

Around 1700: Italy.

XIX Century: Europe.

From 1900: United States


GDP (billions of dollars PPP) in 2010 (Source: IMF)

1 USA 14,526,0002 China 10,119,0003 Japan 4,323,0004 India 4,057,0005 Germany 2,944,0006 Russia 2,230,0007 UK 2,181,0008 Brazil 2,178,0009 France 2,135,00010 Italy 1,779,00011 Mexico 1,564,00012 South Korea 1,466,00013 Spain 1,372,000


GDP per capita (PPP dollars) in 2010 (Source: IMF)

1 Luxemburg 81,4662 Singapore 56,6943 Norway 51,9594 USA 46,8605 Switzerland 41,9506 Holland 40,9737 Australia 39,7648 Austria 39,7619 Ireland 39,49210 Taiwan 39,24511 Canada 39,17112 Sweden 38,204


Relative GDP per capita

1 USA 100.0 14 Holland 74.02 Luxembourg 93.6 16 Austria 72.03 Hong Kong 91.8 16 UK 70.94 Canada 91.2 17 Italy 70.95 Switzerland 88.5 18 Singapore 70.56 Norway 86.5 19 Iceland 70.37 Japan 84.2 20 Finland 66.98 Germany 82.0 21 New Zealand 63.39 Australia 80.6 22 Israel 54.910 Denmark 78.5 23 Spain 54.611 Sweden 77.9 24 Ireland 53.712 France 77.6 25 Chyprus 51.313 Belgium 75.1

Source: Penn World Tables


Relative GDP per capita

1 USA 100.037 Brazil 21.643 Colombia 18.855 Morocco 12.157 Peru 11.767 India 7.172 Nigeria 5.480 Mozambique 4.085 Uganda 3.090 Chad 2.3Source: Penn World Tables



Economic growth is an accumulative process.

Assuming an average annual growth rate of 1% from the year 0, thecurrent GDP would be of 485 millions:

(1+ 0, 01)2010 = 485.245.261 (1)

If a Roman could buy one computer in the year 0, today we can buy485 millions of computers



0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5x 104

Tiempo

Pro

ducc

ión

Figure: GDP with a annual growth rate of 1%José L. Torres (Universidad de Málaga) Lecture 16: Revisiting the Solow model Advanced Macroeconomics 11 / 23


The rise in GDP during a particular period can be calculated as:(1+ g)t

If the growth rate is g the level of GDP doubles approximately every70/g years:Spain. Average growth rate: 2%. Spain doubles GDP every 35 years.

South Korea. Average growth rate: 6%. South Korea doubles GDPevery 11.6 years.

China. Average growth rate: 16%. China doubles GDP every 4.6years.



Consider three economies with the same level of GDP in the initialperiod, but with different growth rates. After 110 years we obtain:

(1+ 0, 01)110 = 2, 98 (Country 1)

(1+ 0, 02)110 = 8, 83 (Country 2)

(1+ 0, 03)110 = 25, 82 (Country 3)



0 20 40 60 80 100 1200

5

10

15

20

25

30

Tiempo

Pro

ducc

ión

País 1País 2Pais 3



Aggregate production functions:

Yt = AtF (Kt , Lt ) (2)

Assumption: Constant Return to Scale.

Capital accumulation equation:

K̇t = It − δKt = sYt − δKt (3)

Assumption: Exogenous saving rate.



Population is increasing over time:

Lt = L0(1+ n)t ; n > 0 (4)

Assumption: Population growth is exogenous.



In per capita terms:

YtLt=AtK α

t L1−αt

Lt=AtK α

t L1t L−αt

Lt= AtK α

t L−αt =

AtK αt

Lαt= At

(KtLt

)α

(5)

yt = Atkαt (6)



Per capita production function:

yt = f (kt ) (7)

yt =YtLt

(8)

Capital stock per capita:

kt =KtLt

(9)



6

-•k

f (k)

.

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Capital accumulation is given by:

∆Kt = Yt − Ct − δKt (10)

where ∆Kt = Kt+1 −Kt . Operating:

Ct +Kt+1 − (1− δ)KtLt

=YtLt

(11)

CtLt+Kt+1Lt+1LtLt+1

− (1− δ)KtLt

=YtLt

(12)

CtLt+Kt+1Lt+1

Lt+1Lt

− (1− δ)KtLt

=YtLt

(13)



On the other hand:Lt+1Lt

= (1+ n) (14)

resulting

ct + kt+1(1+ n)− (1− δ)kt = yt (15)

Solving for capital stock per capita in t + 1:

kt+1 =(1− δ)kt + yt − ct

(1+ n)(16)

Using the definition of investment ( yt − ct = it = styt):

kt+1 =(1− δ)kt + styt

(1+ n)(17)

Operating:

kt+1 − kt + kt =(1− δ)kt + styt

(1+ n)(18)

or:

∆kt =(1− δ)kt + styt

(1+ n)− kt (19)

where ∆kt = kt+1 − kt . Finally, operating we arrive to:

∆kt =stAtkα

t − (n+ δ)kt(1+ n)

(20)



Solow-Swan modelProduction function Yt = AtF (Kt , Lt )Capital accumulation Kt+1 = (1− δ)Kt + ItInitial capital stock K0 > 0Feasibility condition Yt = Ct + ItInvestment It = stYtPopulation dynamics Lt = L0(1+ n)t



Steady State:sty t = (δ+ n)k t (21)

Assuming that technology is Cobb-Douglas:

stAtkαt = (δ+ n)k t (22)

kα−1t =

δ+ nstAt

(23)

k t =(

δ+ nstAt

) 1α−1

(24)

y t = Atkαt = At

(δ+ nstAt

) αα−1

(25)

c t = (1− st )y t = (1− st )At(

δ+ nstAt

) αα−1

(26)


lecture 16: revisiting the solow model

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