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I. Hakan Yetkiner
http://www.hakanyetkiner.com/
Izmir University of Economics
Department of Economics
ECON 405
ECONOMIC GROWTH AND DEVELOPMENT
Dr. Yetkiner
26 March 2013
KEY TO EXERCISE 03
Simple Arithmetic in Economic Growth
1. Let the law of motion for A be given by
A
L

A
A
a. In the steady state, the growth rate of A is constant. Show that the law of motion for
A implies that the steady state growth rate of A is the population growth rate n . Briefly
interpret this result.
By definition, a variable grows at a constant rate in the SS. If we apply the definition on
the differential equation,
 L A  L A 
L
A
d  A SS 

  0    SS SS 2 SS SS   SS  n  SS . We may check our results by


dt  ASS 
LSS
ASS
ASS


A
L
solving the differential equation. In particular,
implies A  L . Hence,

A
A
L
dA  L0 e nt dt . Integrating both sides yields A(t )  0 e nt  const . We may find const
n
L
L 
L

by imposing t  0 : A(0)  0  const . Hence, A(t )  0 e nt   A(0)  0  . It is easy
n
n 
n

A
A
to see that
approaches n at SS (to see this, first find
and next take its limit).
A
A
b. Identify the steady state value of L / A .
We may determine the SS value of L / A in two ways. First, using
see that
A
L
  , it is easy to
A
A
L
L
A SS
L
  SS  n   SS  SS  n /  .Secondly, using the information
ASS
ASS
ASS
ASS
1
I. Hakan Yetkiner
http://www.hakanyetkiner.com/
Izmir University of Economics
Department of Economics
L 
L0
L

.
e nt   A(0)  0  and L  L0 e nt , we may find 
L0  nt
n
n 
A L0 

  A(0) 
e
n
n 

L
Taking limit, where t   , one may show that  n /  .
A
A(t ) 
L0
2. In 2004, Turkey experienced about 10 percent real growth in GDP but employment has
not changed at all. Some have criticized this statistics by saying that “how could an
economy grow without an increase in employment?”. How would you answer this
question were you given a conventional neoclassical production technology (e.g.,
Y  F ( K , L)  AK  L1 )?
For the Cobb-Douglas case, Yˆ  Aˆ  Kˆ  (1   ) Lˆ . As long as, Aˆ  0 and/or Kˆ  0 ,
output grows. Labor (employment) growth is not necessary.
3. Suppose that a country’s production function is defined as Y  AK  L1 . From 1999 to
2000 a country's output rose from 4000 to 4500, its capital stock rose from 10,000 to
12,000, and its labor force declined from 2000 to 1750. Suppose that the elasticity of
output with respect to capital is α = 0.3, and with respect to labor (1-α) = 0.7.
Note that Yˆ  0.125 , Kˆ  0.20 , Lˆ  0.125 .
a. How much did capital contribute to economic growth over the year?
Capital’s contribution is Kˆ  0.06 .
b. How much did labor contribute to economic growth over the year?
Labor’s contribution is (1   ) Lˆ  0.0875 .
c. How much did productivity contribute to economic growth over the year?
Aˆ  Yˆ  Kˆ  (1   ) Lˆ  Aˆ  0.1525
4. Suppose the economy is characterized by a production function of the form
Y  F ( K , G, H )  K  G  H 1 
2
I. Hakan Yetkiner
http://www.hakanyetkiner.com/
Izmir University of Economics
Department of Economics
where G is government expenditures, H  hL human capital, and L is population
(workforce).
a. Suppose that G  G(0)e zt , h  h(0)e mt , and L  L(0)e nt . What is the ultimate form of
the production function?
The ultimate shape of the production function becomes:
Y  G(0)  ( L(0)h(0))1  K  e[ z ( mn)(1  )]t


b. Write the production function in intensive form (per capita).
Y  K  G  (hL)1   y  k  g  h1  , where y  Y / L , k  K / L , etc.
c. Derive the Solovian fundamental equation of growth by using the production

1  
technology. Show that capital per capita grows at rate
z
m at the steady1
1
state.
K  sK  G  (hL)1   K 
k  sk  g  h1   (n   )k 
k / k  sk  1 g  h1   (n   ) .
d (kSS / k SS )
 0  (  1)(kSS / k SS )   ( g SS / g SS )  (1     )(hSS / hSS ) .
dt

(1     )
Hence, (kSS / k SS ) 
z
m.
(1   )
(1   )
d. Derive the growth rate of output at the steady-state.
y  k  g  h1   yˆ SS  kˆSS  gˆ SS  (1     )hˆSS . If we substitute respective values
of k̂ SS , ĝ SS , and ĥSS , we find that:
 
(1     ) 
yˆ SS   
z
m   z  (1     )m
(1   )
 (1   )

1
1
yˆ SS 
z 
(1     )m
1
1
This is the SS growth rate of the system.
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