Download - SGPE Summer School: Macroeconomics Lecture 5
SGPE Summer School:
Macroeconomics
Lecture 5
Recap: The natural levels of production and
interest rate
Y n =C Y,Y e, r,A( )+ I r,Y e,K( )
where Y n = F(K,E(1-un )L)
Capital stock
was taken as
exogenous
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Recap: The natural rate of interest for given K
r
( ), , ,n eS Y Y r A
, ,eI r Y K
S,I
rn
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Introduction
Presentation Outline:
• Capital accumulation and growth
• Wage determination and unemployment
• Money and inflation in the long run
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Growth (Chapter 5)
Question:
• What factors determine the capital stock, production
and the real interest rate in the very long run?
Analysis of two cases:
• Constant population and technology
• Constant population growth and technical development
We also examine income differences between countries
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Growth: Given population & technology
Optimal capital stock:
… but what determines the real interest rate?
In a closed economy, long-run equilibrium, without
growth in population or technology, consumption must
be constant, so we must have
MPK
1+m-d = r
u '(Ct)
u '(Ct+1
) / (1+r)=1+ r
t
r = r
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Growth: Given population & technology
Thus we have, in long-run equilibrium without growth in
population or technology:
We call long-run equilibrium ‘steady state’ (this term will
be explained later)
MPK
1+m-d = r Þ K*
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Growth: Given population & technology
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Growth: Given population & technology
What happens if we start with K<K*?
• MPK is high so companies want to invest and the real
interest is high:
• With high real interest, consumers want to consume
less today than in the future; they save and accumulate
assets (= capital in a closed economy)
• The capital stock grows until it reaches equilibrium
• As the equilibrium level is reached, MPK falls and
growth returns to zero
If we start out with K>K* the opposite occurs
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Growth: Given population & technology
We can also illustrate the adjustment in the diagram with
savings and investment.
In the long-run equilibrium (steady state) :
If we start out with a lower capital K’ then investments
are higher and income and savings are lower for a given
real interest rate, so the real interest has to be higher
S(Y *,Y *,r,A) = I(r,Y *,K) where Y * = F(K*,EN n )
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Growth: Given population & technology
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Growth: Given population & technology
What happens if the consumer starts to care more about
the future (lower )?
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Growth: Given population & technology
What happens if the consumer starts to care more about
the future (lower )?
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Growth: Given population & technology
Convergence
Using CRS we can express the model in terms of capital
and income per effective worker
• CRS means that for any z
• We can choose
• Production function:
F zK, zEN( ) = zY
Y
EN= F
K
EN,1
æ
èç
ö
ø÷
z =1
EN
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Growth: Given population & technology
• Let and
• Production function:
• Production per effective worker depends only upon
capital per effective worker, regardless of the size of the
population. This is a consequence of CRS.
• Marginal product of capital:
• Condition for long-run equilibrium:
y ºY
EN, k º
K
ENf (k)= f
K
EN
æ
èç
ö
ø÷ º F
K
EN,1
æ
èç
ö
ø÷
y = f k( )
MPK = FK k,1( ) = f ' k( )
f '(k*)
1+m-d = r
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Growth: Given population & technology
K*= k*EN
Y*= f k*( )EN
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Growth: Given population & technology
For given population and technology:
• The capital stock and production reach their long-run
equilibrium levels over time
• In long-run equilibrium we have no growth
• If different countries have the same technology and
same subjective discount rate, they will in the long run
have the same real GDP per capita – convergence!
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Growth: Population growth and technical
development
• With given population and technology there is growth
only during the adjustment period if you start out from
low capital stock.
• To explain growth in the very long run we have to have
population growth and technical development.
Assume that the population is growing and technology is
improving at constant rates:
rDE
E= g
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Growth: Population growth and technical
development
Let us guess that real interest is constant on the long-term
growth path:
The optimal capital stock per effective worker is
determined by the same condition as before:
.r r=
f '(k*)
1+m-d = r
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Growth: Population growth and technical
development
But what factors determine the real interest rate? To see this we assume a logarithmic utility function:
Real interest is determined by the subjective discount rate ( ) and the pace of technological development (g).
Consumers must be ‘bribed’ not to consume more today.
r » r+g
r
u Ct( ) = ln C
t( )
u '(Ct)
u '(Ct+1
) / (1+r)=1+ r
t
Þ u ' Ct( ) =1/C
t
Þ 1+ rt= 1+r( )
Ct+1
Ct
= 1+r( ) 1+ g( )
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Growth: Population growth and technical
development
In steady state:
• K/EN and Y/EN are constant
• K and Y grow at the rate g+n
• K/N and Y/N grow at the rate g
K
EN= k * Þ K = k *EN Þ
DK
K=DE
E+DN
N= g +n
Y
EN= f k *( ) Þ Y = f k *( )EN Þ
DY
Y=DE
E+DN
N= g +n
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Growth: ‘The golden rule’
• Does a larger capital stock always mean more
consumption?
• Which capital stock maximises consumption in steady
state?
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Growth: ‘The golden rule’ (golden rule capital
stock)
• Investments in steady state where :
• Consumption per effective worker:
• To get the capital stock that maximises consumption,
take the derivative with respect to k:
I = DK +dK =DK
KK +dK = n+ g( )K +dK = n+ g +d( )K
DY
Y=DK
K= n+ g
Y
EN-I
EN= f k( )- n+ g +d( )k
f ' k( ) = n+ g +d
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Growth: ‘The golden rule’ (golden rule capital
stock)
Assume K=2Y and, for the sake of simplicity,
Marginal return on capital is higher than
Conclusion: the capital stock is lower than theconsumption-maximising capital stockIncreased saving would increase consumption in the longrun. The problem is that we are impatient ...
f ' k( ) = n+ g +d
f ' k( ) =MPK =aKa-1N1-a =aKaN1-a
K=a
Y
K»
1
3×1
2» 0.17
0 0.03 0.07 0.10n g
n+ g +d
m = 0
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Growth: The Solow model
Central assumptions:
• Constant savings rate (s) (no optimising)
• Closed economy: Investments = savings: I = sF(K,EN )
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Growth: The Solow model
• Change in capital stock: DK = sF(K,EN )-dK
sF(K) >dK Þ K grows
sF(K) <dK Þ K falls
sF(K*) =dK* steady-state
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Why are some countries richer than others?
GDP per capita in 2010 (PPP, 2005 constant prices, USD)
• USA 41 365
• Sweden 36 132
• China 7 130
• India 3 477
• Zimbabwe 319
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Why are some countries richer than others?
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Why are some countries richer than others?
Our theory: convergence
• Two countries with the same
should converge to the same income per capita in the
long run
• If one of the countries is poorer, growth should be
higher and over time the country should catch up with
the richer country
• Obviously, this is not happening – at least not very
quickly. Income differentials are large and persistent
E,a,m,un and d
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Why are some countries richer than others? Capital
stocks
What does ‘capital’ do in the model?
• Contributes to the production of goods and services• Is produced and increases through savings/investments• Decreases through capital consumption
Different kinds of capital are complements in production:
• Private capital: machinery, buildings• Public capital: infrastructure such as roads, airports,
telecommunication, schools, courts, admin, buildings• Human capital: education, experience
Differences in policies, taxes, corruption can lead to different capital stocks per worker
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Why are some countries richer than others?
Can differences in physical capital explain income
differences?
Compare two counties:
Differences in physical capital explain only part of the
differences in income
Y
N=K aE 1-aN 1-a
N= KaE 1-aN -a =
Ka
NaE1-a =
K
N
æ
èç
ö
ø÷
a
E 1-a
YIndia
/ NIndia
YUS
/ NUS
=KIndia
/ NIndia
KUS
/ NUS
æ
èçç
ö
ø÷÷
a
EIndia
EUS
æ
èçç
ö
ø÷÷
1-a
0.08
1.00=
0.07
1.8
æ
èç
ö
ø÷
1/3
EIndia
EUS
æ
èçç
ö
ø÷÷
2/3
= 0.34 ×EIndia
EUS
æ
èçç
ö
ø÷÷
2/3
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Why are some countries richer than others?
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Why are some countries richer than others?
Can differences in human capital explain incomedifferences?
• Hard to measure
• One measure is the average number of years the adultpopulation went to school
• To see how much it can explain we need to measurethe effect on productivity of one more year ofschooling
• Empirical estimates: Between 10 and 30% of thedifferences in income can be explained by differences inhuman capital
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Why are some countries richer than others?
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Why are some countries richer than others?
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Why are some countries richer than others?
Conclusion:
• At most 50 % of the income differences can be
explained by differences in physical capital and
schooling
• At least 50% of the differences in income must be
explained by other factors than the differences in
physical capital and schooling
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Why are some countries richer than others?
What other factors can explain differences in income?
• Inadequate access to technology – different E
• Overpopulation, lack of natural resources? No!
• Inadequate institutions (‘social infrastructure’)
– Corruption and lawlessness
– Lack of competition
– Poor public infrastructure
– Taxes and regulations
– Poorly functioning labour markets
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Why are some countries richer than others?
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Why are some countries richer than others?
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Why are some countries richer than others?
Does income per capita converge?
• Some convergence in Europe but no general convergence between rich and poor countries
• The growth funnel: – in rich countries, GDP per capita grows by about 2 per cent
per year– some poorer countries catch up, others fall behind
• Geographical differences: many Asian countries have had rapid growth while many African countries have fallen behind
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What determines technical development?
• In the long run it is technology that drives growth in
GDP per capita
• We have treated E as exogenous – our theory did not
explain what determines technical development
• Endogenous growth theory tries to explain what drives
technical development
• These models often contain an explicit sector for
research and development
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What determines technical development?
Production function for knowledge
where is the number of staff in R&D
Assume
– Growth depends on how many people work in R&D
– Growth depends on the amount of existing knowledge
that influences the development of new knowledge:
growth eventually slows down
growth goes on forever
‘We stand on the shoulders of giants’
DE = aNR
nEq
NR= lN Þ g =
DE
E= a(lN )n Eq-1
RN
q =1: g =DE
E= a(lN )n
q <1:
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