Economic Modelling
Lecture 3
Steady State and Golden Rule of Saving in
Solow Model
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Solow Growth Model
Production function with capital and labour as its inputs:
(Close Economy without Government)
Firms’ Production Function
Market clearing:
Households’ Saving:
Investment requirement:
Closure rule:
Yt  Kt Lt
Yt  Ct  It
St  sYt
It  n   Kt
St  It
Dynamics: Capital accumulation: K  1  K
 It
t
t 1
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Per Capita Output and Per Capita Capital Stock in the Steady State
y  k
Y
y
L
i  n  k
SST
S  sy  sk
0.5ks
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ks
k
K
L
3
Capital Stock and output in the Steady State in
the Solow Model
K
Per Capita Capital Stock
k
L
Take log both sides
ln k   lnK  lnL
Differentiate it with respect to time to get growth rate of k :
dk  dK  dL
K L
k
dk  sY  K 

n
k  K 
dk   1 
 sk
   n

k 
Fundamental equation of economic growth:
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dk
 sk  1    n
k
4
Capital Stock and output in the Steady State in
the Solow Model
dk
Fundamental equation of economic growth:
 sk  1    n
k
dk
In steady state k  0 


1
sk
 n
Per Capita Capital Stock in the Steady State:
Per Capita Output in the Steady State:
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 s 1
ss
k 

  n 
y ss  


s 1

n
5
Solow Growth Model
Production function with Labour augmenting technology
(Close Economy without Government)
Firms’ Production Function
Yt  Kt AL t
Market clearing:
Yt  Ct  It
Households’ Saving:
Investment requirement:
Closure rule:
St  sYt
It  n   ga Kt
St  It
Dynamics: Capital accumulation: K  1  K
 It
t
t 1
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Capital Stock and output in the Steady State in
the Solow Model with technical progress
Y
~ K
~
y

Per Capita Effective Capital Stock k 
AL
AL
~

Take log both sides
ln  k   ln K  ln L  ln
 
Differentiate it with respect to time to get growth rate of k :
 
~
dk  dK  dL  dA
~
K
L A
k
~
dk  ~ 1 
    n  ga
~  sk

k 
 
A
~
dk  sY  K 
  n  ga
~ 
k  K 
~
dk ~ 1
   n  g 
Fundamental equation of economic growth: ~  sk
a
k
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Capital Stock and output in the Steady State in
the Solow Model with technical progress
Fundamental equation
of economic growth:
~
dk
In steady state ~  0 
k
~
dk
~ 1
~  sk    n  ga 
k
~ 1
sk
   n  ga
1
1

~ ss 

Per Capita Effective Capital Stock in the Steady State: k
s



n

g
a

Per Capita Effective Output in the Steady State:
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
~
ss
y  

1

s



n

g
a

8
Results from the steady state:
1.
Countries with higher saving rate have higher steady state level of
output and countries with lower saving rate have lower level of
output in the steady state.
2.
Countries with higher level of technology have higher level of output
and countries with lower level of technology have lower level of
output in the steady state.
3.
Countries with higher rate of population growth rate have lower level
of output in the steady state.
4.
Countries with higher capital share have higher output in the steady
state.
5.
Countries which differ in the initial capital stock eventually reach to
the same output level in the steady state.
6.
Growth of per capita income is zero in the steady state
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Calculation of Steady State: A Numerical Example
Let s = 0.32 and  = 0.08, and n =0  
y ss   k
Output in the steady state



 


s 1

 n
1
3

 1 


1
0.32  1  0.32  2
 1  
 2
 0.08  0  3  0.08 
 s
ss
y 
   n  
Consumption in the steady state:
ss 
1
3
C  Y  sY
ss
SS
SS
= 2-0.32(2) = 1.36.
1
Let s = 0.16 and  = 0.08, and n =0  
3

 1 


 s
y ss  
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3
1
0.16  1  0.16  2
 1  
  1.41
 0.08  0  3  0.08 
10
How does a higher saving rate affect the level of
output in the steady state?
y  k
y2
y1
S  sy  sk
2
2
2
i  n   k
2
2
High saving country
S  sy  sk
1
1
Low saving country
k1
k2
k
K
L
Note:
Saving rate affects level of income but not the growth rates.
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How does a higher rate of population growth affect
the level of output in the steady state?
y  k
y2
y1
i   n   k
1  1 
i   n   k
2  2 
S  sy  sk
High saving country
Higher
population
growth rate
means lower
output and
capital stock in
the steady state
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Low saving country
k1
k2
k
K
L
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Golden Rule for Saving and Capital Accumulation
y  k
y
Y
L
MPK  n  
k 1    n
i  n  k
C-max
S  sy  sk
Kg
Golden rule
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Kss
k
K
L
Steady State
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How High Should be the Saving Rate?
Saving Rate that Maximises Consumption
C-max = 1.25
y = 0.5*k0.5
y=2.5
k = 25
C
s1
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s2
s*=0.5
s4
s5
Saving rate
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Golden Rule of Saving
c  y i
c  y    n k
Max c  y    nk
c


1
 k
   n   0
dk


1


MPK



n
k
   n
Steady State
Golden Rule
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  
 1 
k  
   n 


1
 s 1
ss
k 

  n 
15
Reading and References
Text:
Blanchard Chapters 10, 11
Mankiw 7
Burda Wypslosz 3
Miles and Scot 5-6
Jones, Charles (CJ) Introduction to economic growth, 2002, 2nd Edition, Norton.
Articles:
•
Freeman Richard
•
Kaldor N. (1961) “ Capital Accumulation and Economic Growth” in F.A. Lutz and D.C.
Hague ed. The Theory of Capital, New York, St. Martin.
Lucas R.E. (1988) "On the Mechanics of Economic Development", Journal of Monetary
Economics, 22, 3-42.
Mankiw N.G., D. Romer and D. N. Weil (1992) “Contribution to the Empirics of Economic
Growth” Quarterly Journal of Economics, 107 407-437.
Parente S.L. and Prescott E. C. (1993) Changes in the Wealth of Nations, Federal Reserve
Bank of Minneapolis, Quarterly Review, Spring, pp. 3-16.
Romer, Paul (1989) Endogenous Technological Change, Journal of Political Economy, vol.
98, no. 5. Pt. 2, pp. S71-S102.
Temple, Jonathan R. W. and Voth, Hans-Joachim (1998). Human capital, equipment
investment, and industrialization. European Economic Review, 42(7), July, 1343-1362.
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•
•
•
•
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