Transcript Slide 1

Introducing
Advanced
Macroeconomics:
Chapter 3 –
second lecture
Growth and business
cycles
CAPITAL
ACCUMULATION AND
GROWTH: THE BASIC
SOLOW MODEL
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The basic Solow model (repetition)
Yt  BKt L1t
 1
 Kt 
rt   B  
 Lt 

 Kt 
wt  1    B  
 Lt 
St  sYt
Kt 1  Kt  St   Kt
Lt 1  1 n Lt , n  1
• Parameters: B, ,s, and n . B constant: we focus on
capital accumulation
• Given K0 and L0 the model determines
Yt  , Kt  , Lt  , rt  , wt  and  St 
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Government sector
• Kt 1  Kt  St   Kt follows from the identity/definition
Kt 1  Kt  It   Kt and the national accounting identity
Yt  Ct  It  Yt  Ct  It  St  It
• Same construction with a public sector:
Yt  Ctp  Ctg  Itp  Itg 
Stp
Stg
Yt  Tt  Ctp  Tt  Ctg  I tp  I tg
• Insert into the identity Kt 1  Kt  Itp  Itg   Kt to get
St
Kt 1  Kt  Stp  Stg   Kt
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• We can use the model as it stands. We just have to
reinterpret Kt as the sum of private and government
capital stock and St as the sum of private and public
savings. Similarly the equation St  sYt should be
reinterpreted as:
St  Stp  Stg  sYt  Ctp  Ctg  1 s Yt
• The essential assumption underlying the Solow model
interpreted to include a government is thus that the
sum of private and public consumption as a
fraction of GDP is a constant, 1  s .
• This seems plausible empirically. And s  0.2 seems
plausible for many Western countries:
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The consumption share of GDP in several
Western countries
USA
1
United Kingdom
1
Ct / Yt
Ct / Yt
0.8
0.8
p
Ct / Yt
p
0.6
0.6
0.4
Ct / Yt
0.4
Cgt /
Cgt / Yt
Yt
0.2
0.2
0
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
Netherlands
1
1965
1970
1975
1980
1985
1990
1995
2000
1985
1990
1995
2000
Belgium
1
Ct / Yt
0.8
1960
Ct / Yt
0.8
p
0.6
p
Ct /
0.6
Yt
0.4
Ct / Yt
0.4
Cgt / Yt
g
Ct / Yt
0.2
0.2
0
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
1960
1965
1970
1975
Finland
1
1980
Denmark
1
0.8
Ct / Yt
0.8
Ct / Yt
0.6
Ct / Yt
p
0.6
p
Ct /
Yt
0.4
0.4
g
g
0.2
0
1960
Ct / Yt
Ct / Yt
1965
0.2
1970
1975
1980
1985
1990
1995
2000
0
1960
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1965
1970
1975
1980
1985
1990
1995
2000
The basic Solow model, short version
Yt  BKt L1t
Kt 1  Kt  sYt   Kt
Lt 1  1 n Lt
•
In previous lecture we saw that the Solow model
leads to the transition equation:
kt 1 
•
1
syt  1    kt 

1 n
and to the Solow equation:

1 

 sBkt   n    kt 
kt 1  kt 

1 n 


”technical term”
appearing because
of discrete time
savings per
capita  syt
Replacement investment to
compensate for depreciation
and growth of labour force
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The Solow diagram (repetition)
1

kt 1  kt 
sBk

t   n    kt 
1 n
(n +  )kt

sBkt
k*
kt
Why does growth in k t and yt have to stop?
Diminishing returns!
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Comparative analysis in the Solow diagram
1. The economy is initially in steady state with
parameters B, ,s, and n . What happens if the
savings rate increases permanently from s to a new
and higher level, s' ?
(n +  )kt

s’Bkt

sBkt
k0
k*
kt
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2. The economy is initially in steady state. No
parameters change, but an exogenous event, e.g., a
war or natural disaster, reduces the capital stock to
half size ”over night”. How is this analysed in the
Solow diagram?
(n +  )kt

sBkt
k* / 2
k*
kt
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Steady state
• Last time we saw that the Solow model implies
convergence to a unique steady state. From the
Solow equation
1
kt 1  kt 
sBkt   n    kt 

1 n
one easily computes
k B
*
and then
1
1
 s 


n




1
1

1
 s 
y  B k   B 

 n  
1


1
s


r*   B k*
 

 n  
*
* 
1
1
 
w*  1    B
1
1
 s 


 n  

1
 1    y*
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Some main lessons (repeated from previous lecture)
1

lny* 
B
lns  ln  n     .
1
1

1
 :
1 2
• The elasticity of y wrt. s is
an increase in the savings rate of 10%, e.g. from 20
to 22%, gives a long run increase in income per
worker of around 5% according to the basic Solow
model!
*
1
3
*
• The elasticity of y wrt. B is 1    2  1 ! Note that the
effect is stronger than one-to-one due to capital
accumulation.
• Another steady state prediction concerns the real
interest rate…
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The ”natural” interest rate
1
1
 s 
 s 
*
r*   






 
 n  
 n  
• The real rate of interest is determined by productivity
and thrift in the long run (Knut Wicksell):
– Higher   capital is more productive  demand of
capital per worker increases (ceteris paribus)  higher
equilibrium interest rate (the price of capital).
– Higher s /  n     supply of capital per worker increases 
the equilibrium interest rate decreases.
• Reasonable parameter values on an annual basis,
  1 / 3, s  0.22, n  0.005,   0.05, imply r*  8.3%
*
and *  3.3% . This value for  is very close to
empirical observations of the real interest rate!
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Structural policy
1. Crowding out:
•
•
•
•
Consider a permanent fall in s caused by a permanent
increase in government consumption as a percentage of
GDP.
What happens on impact? Yt is unaffected and still grows
at the rate of n, and yt is unaffected. But savings decrease
and consumption increases. There is full crowding out.
What happens in the longer run? During a transitional
period Yt grows more slowly than at the rate of n and yt
falls down to a new lower steady state level. There is more
than full crowding out. And the real interest rate
increases.
The government cannot increase GDP by raising
government expenditure in the long run. How about the
short run? (Book Two)
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2. Motives for tax-financed public services from a long
run perspective
•
Public investments (that would not be made by private
agents)
For government consumption:
• Public (non-rival and possibly non-excludable) goods
• Public consumption, e.g., on education and health care,
replacing private consumption, which means that s is not
affected
• Distributive reasons
• Externalities (education)
• General productivity effects of public consumption, e.g.,
judicial system, health care etc.
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3. Incentive policies:
– Policies that do not affect model parameters directly
through government expenditure/revenue, but indirectly
through the way they affect private behaviour. We cannot
analyze incentive policies explicitly because private
behaviour has not been derived from optimization.

– Golden rule:
1
s 1
*
1 
y B 
 
n




c*  B
1
1
 s 
1

s
 

 n  

1
The s that maximizes c* , which is s**   , is called the
golden rule savings rate.
– The model suggests structural policies that
•
promote technology
•
encourage savings (assuming that s is considerably
below s** ): institutions and incentive
•
reduce n
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Growth in the basic Solow model
• The long run prediction of the Solow model is its
steady state. What is the growth rate of GDP per
capita in steady state?
• Zero! Not in accordance with stylized facts.
• What is the growth rate of Yt in steady state? Since
Yt / Lt  yt and yt is constant, Yt and Lt must grow at
the same rate, n . GDP grows, but only at the same
rate as the labour force. Why is that?
• Assume that the economy initially is below steady

state implying that kt  k*. Then sBkt   n    kt ,
implying that capital per worker increases. But, once
again, because of diminishing returns the growth
in k t and yt will ultimately cease.
• However, there is transitory growth. How long-lasting
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is that?
Simulation
• Initially we are in steady state with the following
parameter values: B  1,   1 / 3,   0.05, n  0.03,
s  0.08 representing a developing country. This implies
that k*  y*  1.
• With effect first time in period 1, the savings rate increases
permanently to s'  0.22 corresponding to the savings rate
of a typical Western economy, implying
k* '  5.20 and y* '  1.73.
• Starting with k0  1 in period zero and one we now simulate
1
s' Bkt  1    kt 

1 n

over t  2 ,3,... . We also calculate yt  Bkt , ct  1  s'  yt and
gty  lnyt  lnyt 1and drawt the
 1, 2evolutions
,3,...
etc. for
of these
kt 1 
variables in the following figures.
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The evolution of yt , ct and gty after the increase
in s .
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The evolution of yt , ct and gty after the increase
in s . (continued)
The figures show that transitory growth is relatively
long-lasting.
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• In the Solow model, the transition towards steady
state is at least as important as the steady state
itself. And during this transition there is growth in k t
and yt . Hence, the basic Solow model is a
growth model!
• It is easy to find the growth rate of k t :
1
kt 1  kt 
sBkt   n    kt  

1 n
kt 1  kt
1

sBkt 1   n    

kt
1 n
This is called the modified Solow equation.
y
k
• The growth rate of yt follows from gt   gt .
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The modified Solow diagram
kt 1  kt
1
 1

sBk
  n   

t
kt
1 n
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• Growth in GDP per worker is higher the further below
steady state the economy is. This is in accordance
with conditional convergence.
• A permanent increase in s gives a jump upwards in
the growth rate of GDP per worker.
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Conclusions based on the basic
Solow model
• What can a (poor) country do to create a transitory
growth in GDP per worker resulting in a permanently
higher level of income and consumption per worker?
The basic Solow model provides the following
answers:
–
–
–
–
Increase the savings rate
Reduce the growth rate of the labour force
Reduce the rate of depreciation, i.e. invest better
Improve the level of technology
• How useful are these recommendations?
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