PRACTICE Assignment 1
Economics 514
2010
Algebra Problems
1. Immigration and Real Wages
The economy of Taiwan has a labor force of 10 million workers. The average worker
works 1000 hours per year meaning Lt = 10 billion. The average real wage rate for
workers is NT$100 per hour while the rental price of capital is R = .12. Assume a
P
Cobb-Douglas production function and price-taking firms are maximizing profits of
Yt  Kt ( At Lt )1   1 3
a. Calculate the capital and labor productivity level. Calculate the level of capital
and technology level.
Under Cobb-Douglass, capital productivity and labor productivity are proportional
to the capital and labor rental rate respectively
Y 1 R
W

  3*.12  .36
y 1
  1.5*100  150
 P
(1   ) P
K
Assuming that labor is 10 billion, then Y  y  L  150 10 billion  1.5 trillion . We can
Y
1.5 trillion

 4.167 trillion . We can use
Y
.36
K
the data on labor productivity and capital productivity to solve for A. Calculate the
use this to calculate ca;ital stock K 

y 
.5
productivity function yt  kt ( At )1  At  yt   t   150  .36   90
 kt 
b. In the 1990’s, Taiwan instituted a guest worker program which imported 300,000
workers from South East Asia. Assume that in the short-run the capital stock and
technology are fixed at the level solved for in part a. If Taiwan imports 300,000
additional workers and each guest worker also works 1000 hours, what will be the
new level of wages, capital rental prices and output?
Assuming fixed capital and technology, then we have a labor demand curve and a
vertical labor supply curve. An increase in labor supply will drive down real wages.
The increase in labor supply will also increase the productivity of capital increasing
capital demand. This will push up the capital rental rate.
1
The labor supply goes up 3% from 10 billion to 10.3 billion. We can solve for the new
capital labor ratio. The capital stock is 4.167 trillion.
In Trillions In Billions
K
L
Before Guest Workers 4.166667
10
After Guest Workers
4.166667
10.3
k
416.6667
404.5307
1
2
We can write labor productivity as yt  kt ( At )1  404.5307443 3 (90) 3  148.53 .
The new capital productivity is Y  y  148.53
 0.367 Calculate the level
K
k
404.53
of wages as proportional to labor productivity and the new capital rental rate as
R
Y
W
 (1   )  y  99.02 .
proportional to capital productivity    0.122
P
K
P
We can solve for output Y  y  L  148.54 10.3 billion  1.53 trillion
Note a 3% rise in the labor rate results in a 2% rise in output. According to the
production function the elasticity of output with respect to labor is 2/3.
c. Assume instead that Taiwan can rent capital at a constant rate of R = .12.
P
Calculate the wage rate, the capital stock and output if Taiwan imports these
workers.
If Taiwan can rent as much capital as it wants at a constant rental rate, the profit
Y
R
mazimizing capital productivity level remains constant  1   3*.12  .36 .
K
P
Holding technology constant, this means a constant labor

1
productivity yt  kt ( At )
y 
 yt  At   t 
 kt 

1
 90  .36 
.5
 150 . If the labor
W
 (1   )  y  23 150  100 .
P
As labor increases by 3%, capital and output also increase by 3% to keep capital and
labor productivity constant.
productivity is constant, the real wage rate is constant
In Trillions In Billions In Trillions
K
L
Y
Before Guest Workers 4.166667
10
1.5
After Guest Workers
4.291667
10.3
1.545
Now assume that we can divide workers into two categories: skilled and unskilled
workers. Denote the working hours of the unskilled workforce as Ut; the working
hours of the skilled workforce is St. Output is a Cobb-Douglas function of capital,
uskilled labor, and skilled labor so that
Yt  Kt  ( BSt ) (U t )1      13
Under this production function, the marginal product of skilled labor is proportional
to the average product of skilled labor; the marginal product of unskilled labor is
proportional to its average product
Yt
Y
MPU  Y
 1       t
S
U
St
Ut
Under perfect competition, the real wage for the skilled worker is equal to the
marginal product of skilled labor and the marginal product of unskilled labor.
MPS  Y
 
You identify skilled laborers as those with education beyond secondary school. There
are 2.5 million skilled workers in the Taiwan workforce and 7.5 million unskilled
workers. The wages of unskilled workers are $80. The rental price of capital is R =
P
.12.
d. Calculate the productivity of unskilled labor and capital. Use your knowledge of
the quantity of unskilled labor hours and unskilled labor productivity to calculate
output. Use your knowledge of output and capital productivity to calculate the
quantity of capital. Use your knowledge of output and the quantity of skilled labor
to calculate the average productivity of skilled labor and the real wage paid to
skilled labor. Calculate the skill premium of skilled labor, B.
The productivity of unskilled labor is propotional to the wages of unskilled
Y
Y
labor W P  80   1 3   t  t  240 With this unskilled labor productivity and a
Ut
Ut
quantity of unskilled labor, output is Yt = 240*Ut = 240*7.5 billion = 1.8 trillion.
W SK 
Yt 1
The productivity of skilled labor is 
    3 600  200 ; the real wage to
St
 P 
skilled labor is equal to marginal product Capital productivity is
Y 1 R
Y
1.8 trillion

  3*.12  .36 . Capital is K 

 5 trillion . We can

Y
K
P
.36
K
solve for B by
Yt
1.8
Yt  K t  ( BSt ) (U t )1  B  

 62208

1 
2.5  7.5 1 

K t ( St ) (U t )
5
(
)
(
)
 
1000 1000
e. Taiwan increases the unskilled labor force by 300,000. Calculate the effect on
wages of skilled and unskilled labor if:
i.
The capital stock is fixed at the level calculated in d.
With fixed capital and skilled workers, an increase in the unskilled labor force
will push down the productivity of the unskilled labor force and push up the
productivity of other factors.
First we can calculate output as
Before Guest Workers
After Guest Workers
In Trillions In Billions In Billions
K
S
U
B
5
2.5
7.5
5
2.5
7.8
In Trillions
Y
62208
1.8
62208 1.823687
We can then calculate productivity of skilled and unskilled workers
Y
Before Guest Workers
After Guest Workers
S
U
1.8
1.823687
2.5
2.5
Y/S
Skilled
Unskilled
Wages
Wages
240
240
80
233.806 243.1583 77.93534
Y/U
7.5
720
7.8 729.4748
ii.
Taiwan firms can hire as much capital as they like at a capital rental
rate of .12.
How much capital would be demanded at a fixed capital rental rate. Firms
will hire capital until the marginal product of capital drops to the level of the
rental rate. Under Cobb-Douglass, this is proportional to the average product
of capital.
Y
R
 .12   t   Kt  1 ( BSt ) (U t )1  
P
Kt
  ( BSt ) (U t )1  
Kt  

.12


With a fixed capital productivity
1
1
 ( BSt U t ) 12 

  5.099 trillion
2
3
 .36

Y 1 R

  P  3*.12  .36 , then we solve for Y
K
Y  .36* K  1.836 trillion . We can then solve for the productivity paid to skilled and
unskilled labor.
Y
Before Guest Workers
After Guest Workers
1.8
1.835647
S
U
2.5
2.5
Skilled
Unskilled
Y/S
Y/U
Wages
Wages
7.5
720
240
240
80
7.8 734.2588 235.3394 244.7529 78.44645
Skilled wages still rise, now by even more as skilled workers have more capital to work
with. Unskilled wages still fall as the number of skilled workers is fixed. However, the
additional capital mitigates this.
2. Long-term Growth
There are two economies, designated j = A and B. Each economy has the same
technology level, At, which grows at an annual rate of 2% (gA = .02). The annual
depreciation rate in both economies is 8% (δ = .08).
1
2
Yt  Kt 3 ( At Lt ) 3
Country A has an investment rate, s = .21and a population growth rate of .04. Country
B has an investment rate of s = .36 and a population growth rate of .02.
a.
Assume that both countries have an average capital productivity of 2
units of output per capital. Calculate the growth rate of labor
productivity in each country.
g y   g k  (1   ) g A
 Y

g y    s  (n   )   (1   ) g A
 K

s
A
B
n
0.21
0.36
b.
Productivity
Growth
0.04 0.113333
0.02
0.22
Calculate the ratio of labor productivity in country A to the labor
productivity in country B when both countries are on their balanced
growth path. What would be the ratio of labor productivity if the capital
intensity parameter in the Cobb-Douglas production function was equal
to α = ⅔ instead of α = ⅓ (i.e. if capital were more important in the
production process).
SS
 n gA
Y 

 K 
s
Y 
ytBGP    
K 

s
A
B
n
0.21
0.36
Ratio
Alpha
1/3
2/3
SS




1
 At 
y ABGP
yBBGP
   nA  g A

sA

   nB  g A

sB








1
Steady
State
Capital
Productivity
0.04 0.666667
0.02 0.333333
0.707107
0.25
3. Calculate Technology Levels along the Balanced Growth Path
An economy produces with a Cobb-Douglas production function.
Yt  Kt ( At Lt )1
yt  kt ( At )1
In this economy, you observe there is consistently $3.00 in GDP for every $2.00 in
labor compensation. The economy is assumed to be on its balanced growth path. Year
in and year out, the rate of growth of labor productivity is 2% (i.e. y  .02 ) ratio).
y
Assume a depreciation rate of 8% (δ=.08) and a population growth rate of country 1%
(n = .01). Country A has an investment rate of 30%. Calculate the level of capital
productivity. The current level of labor productivity yt = 35. Calculate the current
level of technology, At.
There is a bit of extraneous information here.


1
t
yt  kt A
 yt
1
k 
  t  At1 
 yt 
1

y 
At1   t  yt1
 kt 
 y 
 1
1
t
 At    yt  
 kt 

3
 11  3 2  2
  35 3   21.193
 30 

1
4. Endogenous Growth
New inventions are developed through research and development. Time spent in
research and development is given by LRt. The creation of new techniques for
producing goods is
A  At
LR
At 1  At  B  LRt  t 1
 B t
At
At
. Assume that a constant proportion of the population works in research in
development and that population grows at rate n = .01.
a. Calculate the long-term growth rate of technology.
Define the share of labor put into the R&D sector to be sRD. Then we can
write LRt = sRD Lt. We can write the growth rate of technology as a function of the
At 1  At
L
 B  s RD t . If the growth rate is constant, then the numerator must
At
At 
be growing at the same rate as the denominator. The growth rate of the numerator is
n so the growth rate of the denominator must also be n. The growth rate of At is
ratio
g A so g A  n
b. Technology is growing at its long-term rate. Suddenly, the effectiveness of the
R& D sector, B, increases dramatically. Draw a picture that graphs the path of
technology, A, over time.
Holding the ratio of labor to technology constant, an increase in the effectiveness of
the R & D sector. But if the growth rate of technology is greater than n we would see
the numerator grow more slowly than the denominator. Over time, technology growth
would slowdown until it returned to the long run level. However, after the initial
acceleration of growth, the technology level will be on a higher long-term path.
gA
n
time
A
n
time
5. Consumption-Leisure Trade-off
A worker receives a real wage, w, for each hour worked which can be spent on
consumption. Ct  wt Lt . The worker has T hours worth of time. The workers utility
function is a Cobb-Douglas function of consumption and time spent not working.
U  Ct (T  Lt )1
a. Calculate the optimal level of labor for such a worker as a function of the real
wage rate. Does the substitution effect of an increase in the real wage outweigh
the income effect (i.e. does an increase in the real wage increase optimal labor)?
We can maximize utility with respect to consumption and leisure subject to the budget
constraint by substituting the budget constraint into the utility function so utility is a
function of only of labor.
max U  ( wt Lt ) (TIME  Lt )1
L
U
  ( Lt ) 1 wt (TIME  Lt )1  1    (TIME  Lt )  ( wt Lt )  0
L
  (TIME  Lt )  1    Lt

 Lt   TIME
The worker devotes a constant share of time to work regardless of the real wage. The
income effects exactly cancel out the substitution effects and the labor supply curve is
vertical.
b. Now assume that the worker receives a lump-sum payment in every period, Rt:
Ct  wt Lt  Rt . Solve for the optimal labor supply as a function of wt and Rt. What
is the effect of Rt on the supply of labor.
Change the budget constraint
max U  ( wt Lt  Rt ) (TIME  Lt )1
L
U
  wt ( wt Lt  Rt ) 1 (TIME  Lt )1  1    (TIME  Lt )  ( wt Lt  Rt )  0
L
  wt (TIME  Lt )  1    wt Lt  Rt 

1     L
Rt 
 t 
 
wt 
R
 Lt   TIME  1    t
wt
Here, an increase in the lump-sum payment will decrease optimal labor through the
income effect. Notice that with a constant (positive) lump sum payment, the optimal
labor supply is an increasing function of the real wage.
 (TIME  Lt ) 
6. Frictional Employment
In an economy, there are a range of firms with projects of different levels of
profitability. These projects are uniformly distributed over the range [0, w ]. Workers
who accept a job get welfare of V ( wt )  wt while unemployed workers get a benefit
level of b and get welfare: U (b)  b . Workers have a separation rate of s = .02 and
every unemployed worker receives 1 offer per period (i.e. p = 1).
a. Assume that unemployment benefits equals 25% of the average wage
rate of firms (i.e. b = 1/8 w ). Calculate the reservation wage. What
fraction of wage offers will be accepted?
Workers will work as long as they receive a wage higher than the unemployment
benefit, so the reservation wage will be 1/8 w . Since wages are distributed
uniformly, 87.5% of wage offers will be above 1/8 w .
b. Calculate the long-term unemployment rate in this economy.
The job finding rate is f = .875. Thus the unemployment rate would be equal
to s/(s+f) = .02/.895 = 0.0223.