# Mid-Term Exam # 1 ```Name____________________
Student ID____________________
Mid-Term Exam # 1
Economics 514
Macroeconomic Analysis
Thursday, October 12th , 2006
the blue book! Do not hand in your blue book!
1. Marginal Rate of Substitution A worker has no capital income and, thus, has a
budget line given by Ct  wt Lt where w is real wages, C is consumption, and L is
labor. The worker can use their time for labor or leisure, ls: TIME  lst  Lt . The
workers preferences are given by the utility function of
1
1
U t  2  Ct 2  2lst 2 
Normalize the time period to be TIME = 1 and set the real wage w = 25.
Calculate the marginal rate of substitution between consumption and leisure when
the worker chooses his optimal consumption and labor. [Hint marginal utility of
1
1
consumption is MUC = Ct 2 and marginal utility of leisure, MUls = 2lst 2 ].
MRS = w = 25
2. Forecasts of economic growth. Country A and Country B have the same level of
Y
labor productivity, y0 = 36, and capital productivity, 0
= 1. ), a population
K0
growth rate of 2% (n = .02), and a depreciation rate of 6% (δ = .06). Country A
has an investment rate of 20% (s = .2). Country B has an investment rate of 10%
x
x^20
x
x^20
1.01
1.22
1.07
3.87
1.02
1.49
1.08
4.66
1.03
1.81
1.09
5.60
1
1.04
2.19
1.1
6.73
1.05
2.65
1.11
8.06
1.06
3.21
1.12
9.65
a. Country A and B have Cobb-Douglas production functions of the form.
Yt  K t ( At Lt )1   1 3
where both share the same technology level, At. Assume a technology growth of
2% (i.e. gA = .02). Solve for the level of technology at time 0, A0. Assume that 20
years from now, both countries are on their balanced growth path. Calculate the
level of labor productivity in both countries. What is the ratio of labor
productivity in country A relative to country B?


1
 Y  Y 
 Y  1  Y 
At1   t   t   At   t   t   A0  36
 Kt   Lt 
 Kt   Lt 
Technology level is A20  1.49*36  53.64

1
 Yt 
 Yt  1
 Yt  2
   yt    At    At
 Lt 
 Kt 
 Kt 
Y 
 K 
SS

n  g A   .1

s
s
For country A, this is &frac12; so labor productivity at the balanced growth path is
Y 
ytBGP   t 
 Kt 
1
2
At  2 At
For country B, steady state capital productivity is 1, so
y20  2  53.64  75.86
on the balanced growth path, the level of productivity is 53.64. The ratio is the
square root of 2.
2
Assume Country A and B both have an AK production function:
Yt  AK t
where both share the same constant technology level, A. Calculate the level of
technology, A. Calculate the rate of growth of labor productivity for country A
and country B. What is the level of labor productivity in these two economies in
20 years? What is the ratio of labor productivity in country A relative to country
B?
Yt  AKt  A 
Yt
 1,  yt  kt
Kt
gtk  gty  s  A  (n   )  s  .08
For country A, the growth rate of labor productivity is constant at 12%, after 20 years
it will increase to 9.65*36 = 347.4. . For country B, the growth rate is 2%, and so
labor productivity is 53.64. The ratio is 6.48.
3
Changing Demographics Country F has a production function of the form:
Yt  K t ( At Lt )1   1 3
And accumulates capital according to the following function Kt+1 = (1-δ) Kt + It , the
I
investment rate is constant s  t while the labor force growth rate and technology
Yt
growth rate are constant n0 and gA.. The economy reaches a steady state capital
productivity level. Then, at time T, due to a change in fertility rates, population
growth drops permanently to a new level n1 &lt; n0.
a. Draw a picture that shows labor productivity growth as a function of capital
productivity with the old and the new population growth rate. Draw arrows on the
graph to indicate the transition to the new steady state
NEW
 Y

g y     s  (n   )   (1   ) g A
 K

OLD
gA
[Y/K]SS
Y/K
[Y/K]SS
b. Explain, in one paragraph or less what happens to labor productivity growth after
population growth changes.
As population and labor force grows, less investment needs to be devoted to equipping
new workers and more can be devoted toward giving each worker more capital to work
with. Capital will now grow more quickly than the previous rate which was the growth
rate of technology. As capital deepens, diminishing returns will insure that output growth
is slower than capital growth, reducing capital productivity over time and decelerating
output growth. Soon, the level of capital productivity will reach a new lower steady state
level where capital growth and technology growth are equal.
4
Draw a picture of the path of capital productivity over time.
Y/K
time
c. Draw a picture of the path of labor productivity over time
y
time
5
3. Use the first order conditions of the profit maximization problems of firms to
derive a labor demand curve and the first order conditions of the utility
maximization problems to derive a labor supply curve. You will then put these
together to get an equilibrium.
a. The production function of an economy is Cobb-Douglas.
1
1
Yt  Kt 2 Lt 2
Real profits are given by
1
1
2
2
PK
t t Lt  Wt Lt  Rt Kt
W
R
Taking the real wage, w = , t
and the real capital rental price, rt, t
as
Pt
Pt
given solve for the profit maximizing demand for capital and labor. If the supply
of capital is elastic at a rate of r = .15, what is the profit-maximizing capital labor
ratio. According to the first order condition for optimal labor demand, what would
be the wage rate (defined as w*) at that capital-labor ratio.
The marginal product of capital is
1
2
MPKt  1 2 kt 2  .15  kt  1/ .3  100 9
MPLt  1 2 kt 2  5 / 3  w *
1
6
b. Workers preferences are given by a log/log utility function.
ln Ct  2ln lst
where Ct is consumption and lst is leisure. Assume that worker’s have 1 day
available for work or leisure, Lt + lst = 1. Assume that non-labor income is equal
to 1, so the workers budget constraint is
Ct  wt (1  lst )  1
3
Solve for the first order condition that shows optimal labor supply as a function of
the real wage. Solve for the optimal amount of labor (defined as L*) that you
would have at the real wage that you solved for in part a, w*. How much
consumption could be purchased when workers work L* hours at wage rate w*?
What would be the optimal amount of capital hired by firms if labor was L*?
Ct  5 3 (1  lst )  1 3
5
3
1
Ct
 2
lst
5
 Ct  lst
6
5
lst  5 3 (1  lst )  1  2  5 3 lst
6
12
 5 10 
    lst  2   lst  .8
6
6 6 
L  .2
k  K  100  K  100 * L  100 *.2  20
L
9
9
9
9
7
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