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Homework 2
Answer Key
March 18, 2005
1.
Compare two countries. Each country has a population growth rate of 2% (i.e. n =
.02) and a depreciation rate of 10% (i.e. d = .10). Both countries have the same CobbDouglas production function Yt = Kta(QLt)1-a with constant technology level Qt = 1.
Country A has an investment rate of s = .3 while Country B has an investment rate of .18.
Calculate the ratio of steady state labor productivity in Country A to steady state labor
productivity in Country B, when a = ⅓, ⅔, ½, and ⅞. If capital intensity is strong enough,
can differences in capital fundamentals explain differences in labor productivity seen in
different areas of the world?
Steady state occurs when gross investment per labor unit equals replacement
investment needs per labor unit
a
1 a
K a L1 a
s
(n  d )k SS  s  q  s 
 s k SS  k SS

L
nd
 
 
a
1
 s  1 a
 s  1 a
SS
k 

q




 nd 
nd 
When we want to compare the steady-state productivity across countries, keeping in mind
investment rates differ but depreciation and population growth are the same.
SS
q Ass
q
ss
B
 sA 


nd 
a
1 a
s 
 A 
 sB 
a
a
1 a
 .3 


 .18 
a
1 a
5
 
3
a
1 a
 sB  1 a


nd 
This is a function of the level of capital intensity. See the following table.
a
0.333333
0.5
0.666667
0.875
a
1 a
0.5
1
2
7
5
 
3
a
1 a
1.290994
1.666667
2.777778
35.72245
At a realistic capital intensity a = ⅓, the gap in capital fundamentals only explains a 30%
difference in productivity levels. But if capital is very important in production, capital
fundamentals will have a bigger impact on steady state productivity. In particular, the
higher is a, the more slowly capital returns will diminish as the capital stock is added.
When a is high, a country with a higher investment rate will be able to support a much
higher level of investment and have higher steady state capital stock. When a is high
enough, near a = ⅞, differences in capital fundamentals could explain differences in
productivity of a factor of more than 35.
2.
An economy is closed so that output is either used for household consumption, Ct
; capital investment, It; or government spending, Gt. The government collects a constant
share of output for its spending on output, Gt = τYt. The household consumes a constant
fraction of its after-tax income, Ct = c(1-τ)Yt, and invests the rest, It = (1-c)(1-τ)Yt.
Assume zero population growth, n = 0, and a realistic depreciation rate of 8%, d = .08.
The average rate of household consumption as a share of GDP not spent by government
over the last 10 years in Hong Kong is 66.4% , so set c = ⅔. The average of government
spending as a share of GDP in Hong Kong over the last 10 years is 9% .
a. Assume that production in country H is given by the Cobb-Douglas
function Yt = Kt⅓ (Lt) ⅔ with a = ⅓. Calculate steady state labor
productivity levels when government spending share is τ = .09. Calculate
steady state labor productivity levels when government spending share is τ
= .18.
With fixed technology, steady state is achieved when gross investment equals replacement
investment. When n = 0, all replacement investment goes toward replacing depreciation.
1
a
s
 s  1a
 s  1a
so that productivity is
dk  s k
 k
  k SS   
 q SS   
d
d 
d 
an increasing function of the investment rate.
In this economy the investment rate is a function of the taxes taken by government.
I
1 
s   (1   )(1  c) 
. We see that
Y
3
SS
 
SS
 
a
SS
a
1 a
1
1
.5
 1  
 1    1 3
1




SS
3
3


q 



 d 
 .08 
 .24 




The greater is the tax rate, the lower is long-term productivity
1 a
3

qSS
0.09 1.94722
0.18 1.848423
b. Now assume a production function with nondiminishing returns to capital,
Yt = Kt. Calculate the rate of growth of output when τ = .09 and is τ = .18.
The productivity of capital is constant Y  1 . The growth rate of output is equal to
K
the growth rate of capital. Capital accumulation is done through investment
K t 1  (1  d ) K t  I t  K t 1  K t  I t  dK t
 gtK1 
K t 1  K t
I
I Y
1
 t d  t t d  sd 
 .08
Kt
Kt
Yt K t
3
1   .24 1  (  .24)


3
3
3
The growth rate of output and capital is a negative function of the tax rate.
 gtK1 

gY
0.09 0.223333
0.18 0.193333
3. Examine an economy in which the production function is quadratic so
1
Yt  a  Lt  Lt 2 which implies a linear marginal product of labor curve,
2
Y
MPL  L  a  Lt . In this economy,
a. Assume that a = 100. What is the demand for labor in this economy if the
market real wage W = 10.
P
The demand for labor is W  a  LD  LD  a  W  100  W . With
P
P
P
W = 10, LD = 90.
P
b. Calculate the equilibrium real wage in this economy when a = 100 and the
labor supply is unaffected by the real wage and is fixed at LS = 100.if the
labor market is perfectly competitive and firms have no labor management
costs.
Labor supply can only equal labor demand when W = 0.
P
S
D
100  L  L  100  W  0   W  W  0
P
P
P
Now assume that firms must pay the labor management costs per labor unit so
E2
W
that are a linear function of the real wage c(
)
P W
P
c. The marginal cost of hiring an extra unit of labor is now,
2
W  c(W )  W  E . Calculate the demand for labor when E = 10,
P
P
P W
P
a = 100 and the real wage rate is W = 10. Calculate the demand for
P
labor when E = 10 and the real wage rate is W = 20.
P
102 100 100
= 10, the management costs are c(W ) 


 10 .
P
P W
W
10
P
P
Total labor costs per labor unit W  c(W )  20 . The firm will set the
P
P
marginal product of labor equal to the extra cost of labor
LD  100  W  c(W )  100  10  10  80 . When the wage rate is W =
P
P
P
20, wage costs are higher, but management costs are lower. The marginal
100
cost of labor is W  c(W )  W 
 25 . The demand for labor is
P
P
P W
P
D
L  100  W  c(W )  100  20  5  75 .
P
P
When W
Given the cost function, so that the marginal benefit of increasing real wages is
c
E2
per labor unit while the marginal cost per labor unit of raising


2
W
W
P
P
the real wage is 1.
 
d. If the efficiency wage minimizes total costs of hiring a worker by
equalizing the marginal benefit of increasing the wage level with the extra
cost, solve for the efficiency wage if E = 10. Calculate the demand for
labor at the efficiency wage. If the labor supply is LS = 100, what is the
level of unemployment (i.e. the difference between labor demand at the
efficiency wage and labor supply). What is the unemployment rate (i.e. the
level of unemployment divided by the labor supply)?
The cost minimizing wage sets the marginal benefit of raising wages to the margical cost,
E2
1.
 1  we  E  10 . We know the management costs when W =10 is 20 and the
2
P
e
w
 
demand for labor is 80. When the employment is 80 and the labor supply is 100, the
unemployment is 20. The unemployment rate is 20%; ur = 20/100 = .2.
e. Now assume that a negative technology shock reduces the productivity of
labor and decreases a from a = 100 to a = 90. If the efficiency wage rate
is unchanged, what is the new demand for labor and what is the new
unemployment rate?
The efficiency wage is unchanged at we = 10 implying total marginal labor
costs of 20. The demand for labor is
LD  90  W  c(W )  90  10  10  70 . Labor demand is 70,
P
P
unemployment is 30, and the unemployment rate is 30%.
4. Internet Question
Reported below are series for Current Dollar Gross Domestic Product and
Population for 10 East Asian Economies. Use the PPP and X-Rate conversion
factors for 2002 calculated by the World Bank (to be obtained from
http://siteresources.worldbank.org/ICPINT/Resources/Table5_7.pdf) to convert
the GDP of one of these into US dollars. Choose the country whose number
corresponds with the last digit of your student ID. For comparison, divide by the
population to get US dollar GDP per capita for both PPP converted GDP and XRate converted GDP. As an example, I have already completed the results for
China.
Last Digit
Student
Country
ID Number Name
Dave
China
0
Bangladesh
1
Hong Kong
2
India
3
Indonesia
4
Japan
5
Korea
6
Malaysia
7
Philippines
8
Singapore
9
Thailand
Currency
Unit
Millions of
Renminbi
Taka
HK Dollar
Rupee
Rupiah
Yen
Won
Ringgit
Peso
Sing Dollar
Baht
Current Price
GDP
10517230
3005801
1247381
24633240
1863274700
497897000
684263500
361624
3959648
158064
5446043
Population
(Millions)
1284.53
133.4
6.787
1055
211.06
127.377
47.615
24.526
79.503
4.185
63.437
Example
Dave
China
xp
1.8
e
8.28
US Dollars
PPP GDP
XRATE GDP
$5,842,905.56 $1,270,196.86
US Dollars
PPP GDP
XRATE GDP
per Capita
per Capita
$4,548.67
$988.84
US Dollar GDP per
US Dollar GDP
Capita
e
xp
xrate
PPP
xrate
PPP
0 Bangladesh
57.89
11.9
$51,922.63 $252,588.32
$389.23 $1,893.47
1 Hong Kong
7.8
6.9 $159,920.64 $180,779.86 $23,562.79 $26,636.19
2 India
48.61
8.8 $506,752.52 $2,799,231.82
$480.33 $2,653.30
3 Indonesia
9311.19
2357.7 $200,111.34 $790,293.38
$948.13 $3,744.40
4 Japan
125.39
146.2 $3,970,787.14 $3,405,588.24 $31,173.50 $26,736.29
5 Korea
738.7 1251.09 $926,307.70 $546,933.87 $19,454.12 $11,486.59
6 Malaysia
3.8
1.6
$95,164.21 $226,015.00 $3,880.14 $9,215.32
7 Philippines
51.6
12.1
$76,737.36 $327,243.64
$965.21 $4,116.12
8 Singapore
1.79
1.6
$88,303.91
$98,790.00 $21,100.10 $23,605.73
9 Thailand
42.96
12.6 $126,770.09 $432,225.63 $1,998.36 $6,813.46