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PRACTICE Assignment 1
Economics 514
2013
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  K t ( Lt )1   1 3
a. Calculate the capital and labor productivity level. Calculate the level of capitall.
Under Cobb-Douglass, capital productivity and labor productivity are proportional
to the capital and labor rental rate respectively
K  ( L )1
Y 1 R

  3*.12  .36  t t
 kt 1

K
P
Kt
kt  .36
Kt 
1
 1
1
 .36
1
3 1
3
 .36 2  .63 
53
3

125
27
125
10
27
2 13 2  5 3  13 2 5 10
kt 
 3
3
3 3 
3
9
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 is 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?
There will now be 10.3 billion workers. Given a fixed capital stock, the new capital
125 10
R
2
kt 

 4.4948;   kt 1  13  4.4948 3  .1224  .12
27 10.3
P
labor ratio would be
1
1
10
w  23 kt3  23  4.49483 =.1002<
9
Reducing the capital labor ratio raises capital rental rates and reducing wages.
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.
125
10
Kt 
10.3; w  So in the long run, the population size does not affect the
27
9
exchange rate.
w  MPL  (1   ) K t ( Lt )   (1   )kt 
2. 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 ) 
3. Household Problem
A household lives for two periods, 0 and 1. The household begins life with zero financial
wealth, earns Y0 = 100 in period 0 and has zero income in period 1. The utility function
of the household is U(c0,c1) = u(c0)+.75u(c1).
a.
Assume that the permanent income hypothesis holds, so that the real interest rate is
1+r = 4/3. Calculate consumption and savings in period 0.
The household budget constraint is to set present value of lifetime
consumption equal to present value of lifetime income.
c
c0  1  Y0
1 r
Maximize utility subject to the constraint:
c


max u (c0 )  .75  u (c1 )   c0  1  Y0  
1 r


c
c0  1  Y0
1 r
u '(c0 )  
.75  u '(c1 ) 

1 r
 u '(c0 )  1  r  .75  u '(c1 )
When 1+r = 4/3, Y0 = 100
1  r   4 3  u '(c0 )  u '(c1 )  c0  c1
c
c0  0  Y  c0  4 7  Y0  57.14285714
4/3
S0  Y0  c0  42.85714286
b. Now assume that the interest rate rises to 2. Calculate how this rise in the interest
rate affects savings under three assumptions about the intertemporal elasticity of
substitution.
We can write the constant intertemporal elasticity of substitution utility
1 1
c  1
1
function as u (c) 
.
The
marginal
utility
is
where the
u
'(
c
)

c
1  1
intertermporal elasticity of substitution is ψ.
i. Assume that the felicity function is the natural logarithm, u(c) = ln(c). What is the
new savings level after the rise in the interest rate.
The marginal utility is u’(c) = c-1. This is the same as the intertemporal elasticity when ψ
= 1. Given the intertemporal equation, u '(c0 )  1  r  .75  u '(c1 ) we would write this
 2  .75  1  c1  1.5  c0 . Given the budget constraint, we would
c1
1.5  c0
c0 
 Y0  c0  4 7  Y0  57.14285714
write
. The household was initially a saver.
2
S0  Y0  c0  42.85714286
as 1
c0
So a rise in the interest rate increases their possibility of consuming in the future without
cutting back current consumption (i.e. there is a positive income effect). When there is a
unit elasticity of substitution, the substitution effect is just strong enough to outweigh the
income effect, so interest rate has no effect on savings.
ii. Assume that the elasticity of substitution is weaker than unity so the utility
function.u(c) = -c-1. What is the intertemporal elasticity of substitution? Calculate the
level of savings when the interest rate is 4/3 and when the interest rate is 2. Does the
income effect outweigh the substitution effect, so savings increases?
When the utility function is u(c) = -c-1 then the marginal utility is u '(c)  c 2 which is the
same as the constant intertemporal elasticity of substitution is ψ = .5. Using the
intertemporal first order condition c0 2  2  .75  c12 . Raise both sides of this equation to
the power -.5. c0   2  .75   c1  c1  1.5  c0 . Substituting this into the intertemporal
.5
1.5  c0 2  1.5
2

c0  Y0  c0 
 Y0  62.02041029
budget constraint.
2
2
2  1.5
S0  Y0  c0  37.97958971
When the elasticity of substitution is less than 1, the substitution effect is not strong
enough to outweigh the income effect. A rise in the interest rate reduces savings.
c0 
iii. Assume that the elasticity of substitution is stronger than unity so the utility function
u(c) = 2×c.5.
When the utlity function is u(c) = 2×c.5then the marginal utility is u '(c)  c .5 which is
the same as the constant intertemporal elasticity of substitution is ψ = 2. Using the
intertemporal first order condition c0 .5  2  .75  c1.5 . Raise both sides of this equation to
the power -2 c0   2  .75   c1  c1  1.52  c0 . Substituting this into the intertemporal
2
1.52  c0 2  1.52
2

c0  Y0  c0 
 Y0  47.05882353
budget constraint.
2
2
2  1.52
S0  Y0  c0  52.94117647
When the elasticity of substitution is greater than 1, the substitution effect is strong
enough to outweigh the income effect. A rise in the interest rate increases savings.
c0 
c.
Assume that Y0 = 100 and Y1 = 100 and 1+r = 4/3. Calculate consumption in
period 0.
When 1+r = 4/3, c0  c1 , so if Y0 = Y1, then
c
Y
c0  1  Y0  1  c0  Y0  100 and savings is zero.
1 r
1 r
4. 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  SS 
ytBGP     
K  


s
A
B
Ratio
n
0.21
0.36
Alpha
1/3
2/3

1
 At 
Steady
State
Capital
Productivity
0.04 0.666667
0.02 0.333333
0.707107
0.25
y ABGP
yBBGP
   nA  g A

sA

   nB  g A

sB








1
5. 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 
3
 y 
 1
1
t
 At    yt  
 kt 

 11  3 2  2
  35 3   21.193

 30 
1
6. 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