the'basic'solow'model'' - sweetsweet.ua.pt/afreitas/growthbook/part...

Miguel'Lebre'de'Freitas' Introduc4on'to'economic'Growth'

The'basic'Solow'model''

“A'thri=y'society'will,'in'the'long'run,'be'wealthier'than'an'impa4ent'one,'but'it'will'not'grow'faster”'[Robert'Lucas'Jr.]''

.''

'


Introduc4on'

•  The'main'innova4on'of'the'NeoJClassical'model'in'respect'to'the'

Malthusian'Model'is'the'replacement'of'land'by'“capital”'in'the'

produc4on'func4on.''

•  Contrary'to'land,'capital'can'be'produced'and'accumulated'

•  This'opens'an'avenue'to'overcome'the'diminishing'returns'to'

labour:'by'allowing'the'capital'stock'to'expand,'the'Solow'model'

avoids'the'nega4ve'rela4onship'between'produc4vity'and'the'size'

of'popula4on'that'plagues'the'Malthus'model.'''

•  S4ll,'because'capital'itself'faces'diminishing'returns,'capital'

accumula4on'alone'cannot'generate'long'term'growth.'In'the'

Solow'model,'a'highet'investment'rate'leads'to'a'higher'level'of'per'

capita'income,'but'it'does'not'generate'long'term'growth.''


Diminishing(returns(versus(constant(returns(to(scale('

NYy =

N0N

β

"#

$%&

'=NKAy 0

0y

1N

A

B

C

1y

β

"#

$%&

'=NKAy 1


The'basic'model'

•  Produc4on'func4on'with'CRS'on'capital'(K)'and'labour'(N)''

•  TFP'(A)'constant'''•  Perfect'compe44on''

•  Keynesian'consump4on'func4on'with'

exogenous'savings'rate''

•  Popula4on'expanding'at'a'constant'rate'•  Constant'deprecia4on'rate''


The'flow'income'chart''

Households

F.Markets

Firms

sY

KKI δ+= !

( )YsC −= 1( )KrwNY δ++=


Factor'income'shares'

•  Perfect'compe44on'and'CRS'imply'constant'

factor'income'shares'

•  The'share'of'labour'income'in'na4onal'income'

is'equal'to'1�β'the'elas4city'of'labour'in'the'produc4on'func4on''

•  The'share'of'capital'is'β.''


Dynamics'and'Equilibrium''

*k NKk /=0k

0y

y = Akβ

syQ

P

1k

R*y

(n+δ)k

S

O


The'Solow'model'and'the'Kaldor’'facts'

1.  Output'per'worker'grows'over'4me'at'a'sustained'rate'J'Not'OK'

2.  The'capital'stock'per'worker'grows'over'4me'at'a'sustained'rate'–'Not'OK'

3.  The'capitalJoutput'ra4o'exhibits'no'clear'trend'over'4me'–'OK''

4.  The'real'return'to'capital'is'rela4vely'constant'over'4me'–'OK''

5.  The'shares'of'labour'and'of'capital'on'na4onal'income'are'roughly'constant'over'4me'J'OK'

6.  There'are'wide'differences'in'the'growth'rate'of'produc4vity'across'countries'–'Not'OK.''


Saving'rate'and'per'capita'GDP'

5

6

7

8

9

10

11

0 5 10 15 20 25 30 35

Investment as a percentage of GDP (average 1950-2000)

Rea

l GD

P p

er c

apita

in 2

000

(logs

, 199

6 U

S d

olar

s)


Popula4on'growth'and'per'capita'GDP'

5.5

6.5

7.5

8.5

9.5

10.5

-1% 0% 1% 1% 2% 2% 3% 3% 4% 4% 5%

Average population growth (1950-2000)

Log

of G

DP

per

cap

ita in

200

0 (1

996

US

dol

ars)


Demographic'transi4on'and'poverty'

trap''

NKk /=

βAky =

sy

( )[ ]kyn δ+

L A H

y


What'happens'when's'increase?'

NYy /=

*1k NKk /=

y = Akβ

(n+δ)k

s1 ys0 y

*0k

*1y*0y

(Y/K)0(Y/K)1

0

1

The'produc4vity'of'capital'(and'r)'

declines'


What'happens'when'A'increases?'

NYy /=

*1k NKk /=

(n+δ)k sysy

*0k

*1y

*0y

(Y/K)0βkAy 1=

βkAy 0=

0

1

!

The'produc4vity'of'capital'(and'r)'is'

constant'


The'golden'rule'

NYy /=

NKk /=

y = Akβ

(n+δ)k

sG y

Gk

GG ys )1( −

GG ys


How'efficient'is'a'move'towards'the'

golden'rule?'''

•  If's>sG'a'free'lunch'is'available'(dynamic'

inefficiency)''

•  But'if's<sG,'the'move'towards'the'golden'rule'

obeys'to'a'trade'off.'Without'a'social'welfare'

func4on,'one'cannot'evaluate'this'move.'

•  Why'would'a'decentralized'economy'deliver'

too'much'savings?'Distor4ons?''


Endogenous'savings'

•  Two'period'model'

''''Max'''C1 +

C21+ r

=Q1 +Q21+ r

U = lnC1 +lnC2i + r

S.T.''

1+ r = 1+ ρ( ) c2c1= 1+ ρ( ) 1+γ( )

Implies'

where' 1+γ =c2c1

γ ≈ r − ρCon4nuous'4me'equivalent:''


What'happens'when'ρ'declines?''

KY∂

∂

NKk /=*0k

δρ +0

ββ−

=∂

∂1kA

KY

δρ +1

*1k

A'

B' C'

s = β n+δρ +δ

< β


The'Solow'residual'

•  Measures'the'contribu4on'of'factors'other'than'labour'and'physical'capital'to'the'change'in'per'capita'GDP'

•  The'contribu4on'of'capital'and'labour'is'assessed'taking'into'account'the'corresponding'income'shares'''

•  TFP'change'is'computed'as'the'difference'between'the'actual'growth'of'output'and'the'growth'implied'by'factor'accumula4on.''

•  Note'that'this'corresponds'to'the'Solow'neutral'rate'of'technological'progress'''


Discussion''

•  The Solow model accounts for the fact that historical ratios of capital to output and real interest rates tend to be relatively stable in the long run.

•  It also offers the credible suggestion that countries with high savings rates and low population growth rates should enjoy higher per capita incomes.

•  In its current formulation the model fails, however, to explain the most basic fact of modern economic growth: that per capita income tends to increase over time.

•  Continuous growth of per capita income could be obtained in the context of the Solow model if saving rates rose continuously over time. But then, interest rates should exhibit declining trends, and this does not happen in reality.

•  The obvious solution is to allow technology to expand over time: as we just saw, with technological progress, per capita output expansion does not imply a declining interest rate. '

the'basic'solow'model'' - sweetsweet.ua.pt/afreitas/growthbook/part...

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