Transcript Sorensen-Chapter 3 (Part 1)

Introducing
Advanced
Macroeconomics:
Chapter 3 – first
lecture
Growth and business
cycles
CAPITAL
ACCUMULATION AND
GROWTH: THE BASIC
SOLOW MODEL
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The basic Solow model
•
How can a nation become rich, i.e., initiate a growth
process leading to higher GDP/consumption per capita
in the long run.
The basic Solow model provides some first answers:
It predicts how the evolution and the long run levels
of GDP and consumption per capita depend on
structural parameters such as the rate of investment
and the growth rate of the labour force.
Key elements of the Solow model:
•
•
–
–
–
–
In each period output is determined by the supplies of capital
and labour through the production function
Exogenous savings/investment rate, s, exogenous growth
rate of labour force, n, and exogenous depreciation rate, δ.
Explicit description of capital accumulation: Kt 1  Kt  It   Kt
Accumulation of capital is the main driving force for wealth.
“Basic” model: No technological progress.
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The ”micro world” of the Solow
model
•
•
•
•
Object: Closed economy.
Time: A sequence of periods/years, t  0,1, 2,...
Agents: Households and firms (and government).
Commodities and markets: Output, capital
services and labour services (one asset = physical
capital).
• The market for output: Supply = firms’ output, Yt.
Demand from households for consumption and
investment = Ct  It . Relative price = 1.
– One-sector model: Output can be used either for
consumption or for investment.
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• The market for capital services: Consumers own
the capital stock, K t , and rent its services to firms.
Supply of capital services = K ts . Firms’ demand = K td
– Relative price (in units of output) for renting one unit of
capital for one period: rt = real rental rate for capital.
– Real interest rate: t  rt   , where  is the rate of
depreciation. Or: rt  t   .
(Alternative interpretation: The firms own the capital,
borrow for the purchase of capital at an interest rate of  t
and bear the cost of depreciation themselfes).
– User cost  rt  t   .
• Labour market: Households supply = Lt (= the
labour force). Demand from firms = Ldt . Relative
price: wt = the real wage rate.
• Competitive markets: rt and wt adjust to equate
supply and demand in all markets  full (or natural)
utilization of ressources.
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The production side
•
… is modelled as if all production (all of GDP) comes
from one profit maximizing firm that produces value
added, Yt , from of capital services, K td (machineyears), and labour services, Ldt (man-years),
according to the production function:
Yt  F  K td ,Ldt 
1. Constant returns to scale: F   Ktd , Ldt    F  Ktd ,Ldt  .
The replication argument!
2. Positive marginal products:
F ' K  K d ,Ld   0, F ' L  K d ,Ld   0.
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3. Marginal products are decreasing in the amount of
the factor used:
''
''
FKK
 0, FLL
0
(diminishing returns) and growing in the amount of
the other factor used:
''
''
FLK
 0, FLK
0
•
Profit maximization: Given rt and wt , the firm
chooses Yt ,Ktd and Ldt to:
max Yt  rt Ktd  wt Ldt , s.t . Yt  F  Ktd ,Ldt 
The ususal necessary conditions for an optimum:
FK'  K td ,Ldt   rt , FL'  K td ,Ldt   wt
(These two equations do not determine K td and Lt
from given rt and wt , they only determine K td / Ldt )
d
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• Competitive market clearing  Ktd  Kt and Ldt  Lt ,
where K t and Lt are the supplies in period t:
FK'  Kt ,Lt   rt , FL'  K t ,Lt   wt
Since K t and Lt are predetermined in any given
period, rt and wt are determined this way.
K t and Lt predetermined: what does that mean?
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The income distribution
rt  FK'  K t ,Lt   rt K t  FK'  K t ,Lt  K t
wt  FL'  K t ,Lt   wt Lt  FL'  K t ,Lt  Lt
1. No pure profits. Euler’s rule 
F  Kt ,Lt   FK'  K t ,Lt  K t  FL'  K t ,Lt  Lt  0
2. The functional income distribution
'
'
F
K
,L
K
F


rt Kt
wt Kt
K
t
t
t
L  K t ,Lt  Lt

and

Yt
Yt
F  Kt ,Lt 
F  Kt ,Lt 
The income share of each factor is the elasticity of
the production function with respect to the factor in
question.
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According to empirics, labour’s share is relatively constant around 2/3 over
long periods except for short run fluctuations:
Labour’s share of domestic factor incomes
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Is there a production function that fulfills all of our
assumptions and has fixed output elasticities
independently of K t and Lt ? Yes, the Cobb-Douglas
production function:
F  Kt ,Lt   Bt K t L1t , Bt  0, 0    1,
where Bt is total-factor-productivity (TFP).
Check: It follows that
 1

 Kt 
 Kt 
'
rt  FK   Bt  
and wt  FL  1    Bt   
 Lt 
 Lt 
rt K t
wt Lt
  and
 1
Yt
Yt
'
The Cobb-Douglas function seems as a realistic long run
assumption. We even have reason to believe that   1 / 3.
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Households
•
The number of households in period t is Lt , which is
predetermined. Household behaviour:
1. Each supplies one unit of labour inelastically. Total supply  Lt .
2. Own the capital stock, K t , which is predetermined in period t .
Supply = K t (as long as rt  0 ).
3. The representative household decides Ct given Yt , and hence
St  Yt  Ct . The intertemporal budget constraint:
Kt 1  Kt  St   Kt , 0    1
We assume that the result of the consumer’s considerations is:
4. ”Biology”:
St  sYt .
Lt 1  1  n  Lt , n  1
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The complete model
Yt  BK t L1t 
 1
 Kt 
rt   B  
 Lt 

 Kt 
wt  1    B  
 Lt 
St  sYt
Kt 1  Kt  St   Kt
Lt 1  1  n  Lt ,
• Parameters: B, ,s, and n . NB: No subscript t on B:
”Basic” Solow model.
• Endogenous variables: Yt  , Kt  , Lt  , rt  , wt  and  St 
of which  Kt  and  Lt  are state variables:
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• Given K0 and L0 the model determines Yt  , Kt  , Lt  ...
• Government in the model? Yes, simply interpret St
as private plus government savings.
• We viewed the capital accumulation equation:
Kt 1  Kt  St   Kt
as the household’s budget constraint.
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Alternatively: By definition we have that
Kt 1  Kt  It   Kt
()
The condition for equilibrium in the output market, or
the national accounting identity, is: Yt  Ct  I t , and by
St  Yt .CHence
definition:
t
I t  St
( )
Combining () and ( ) gives the capital accumulation
equation again: s is the savings rate and the
investment rate.
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Analysing the basic Solow model
1. Define: yt  Yt / Lt and kt  Kt / Lt .
2. From Yt  BK t L1t we get the per capita production
function:
y  Bk  , 0    1
t
t
y
k
ln
y

ln
y


ln
k

ln
k

g


g


Note:
t
t 1
t
t 1
t
t
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3. Insert St  sYt into Kt 1  Kt  St   Kt to get:
Kt 1  Kt  sYt  1    Kt
4. Divide by Lt 1  1  n  Lt on both sides to find that:
kt 1 
1
syt  1    kt 

1 n

5. Insert yt  Bkt to get the transition equation:
1

kt 1 
sBk

t  1    kt 
1 n
6. Subtracting kt from both sides of the transition
equation gives 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
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and growth of labour
force Companies, 2005
The transition diagram

sBk
dk
1
t  1    kt

t 1
kt 1 
sBkt  1    kt  

.

1 n
dkt
1 n
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• About the slope, dkt 1 / dkt :
– It is everywhere strictly positive
– It goes to infinity as kt goes to zero
– It goes to 1    / 1  n  as kt   , and 1    / 1  n   1 
n    0 . Indeed we assume that n   (realistic!)
• The figure shows
that kt  k in the long run. Hence  1

yt  y*  B  k *  , ct  c*  1  s  y* and rt  r*   B  k * 
etc.
*
*
• The values k , y etc. define the ”steady state”.
*
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The Solow diagram
1
kt 1  kt 
sBkt   n    kt 

1 n
(n +  )kt

sBkt
k*
kt
Why does growth in kt and yt have to stop?
Diminishing returns!
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Steady state
• The long run levels k * , y* etc. depend on parameters.
How? What makes a nation rich?
• Look at the Solow equation:
kt 1  kt 
1
sBkt   n    kt 

1 n
In steady state kt 1  kt  0. Insert this to find
sB  k
  n    k
* 
k B
*
1
1
y  B k
*
 s 


n





* 
B
*
1
1
1
1


 s 


n





1
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• Some sharp predictions of the Solow model:
1

lny* 
B
lns  ln  n     .
1
1

1

1 2
The elasticity of y wrt. s is
(since we believe
1
that   3 ): an increase in s of 10%, e.g. from 20 to
*
y
22%, should give an increase in
of 5%!
*

1

1
2
The elasticity of y wrt. n   and wrt. B are
1
3

and 1   2 , respectively. Why is the latter not one?
Capital accumulation!
*

• We have reached empirically testable hypotheses!
Empirics:
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Real GDP per worker against the average
investment share across 85 countries
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Real GDP per worker against the average annual
labour force growth rate across 85 countries
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