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Midterm Exam
Economics 514
Macroeconomics Analysis
October 11, 2004
Short Questions (10 points each)
1. You measure the real wage rate in an economy as w = $25 per hour. The
consumption of an average household as being equal to C = 2400 per month. The
household is endowed with TIME = 24*30 = 720 hours in an average month and
is observed to spend L = 240 hours in labor. Assume that the household has a
utility function of U = ln Ct + ½ ln lst. Assume that the household chooses its
optimal, utility-maximizing combination of consumption and leisure. Calculate the
marginal rate of substitution between leisure and consumption.
The marginal rate of substitution can be derived as the ratio of the marginal utility of
1 1
1
MU ls
2 ls ls
C
2400
 MRS 
 12  12
leisure to the marginal utility of consumption
=
1
MU C
ls
480
C
2.5. Also at utility maximization, the slope of the indifference curves (-MRS) equals the
slope of the budget constraint (-w), so the MRS = w = 25. Whoops. Either answer was
accepted for full credit.
2.
You compare the price levels of several economies by measuring the price level in
each country of a common market basket of goods. To save time, you assume that
the typical market basket of goods includes only one good: Big Mac sandwiches
from McDonalds. Using the table below, convert the GDP per capita of China and
Japan into US dollars using the exchange rate method and the PPP method.
Calculate the real exchange rate with Japan and the US treating China as the home
country.
Country
USA
Japan
China
GDP per Capita
US$49735.14
¥3899026
3.
US$2.9
¥261.87
RMB10.411
RMB9073
XP
USA
Japan
China
Big Mac Price
X Rate
PPP
S DCU per $
111
8.28
s
2.306407
90.3 35126.36 43178.58
111 1.876293
3.59 1095.773 2527.298
8.28
Assume that money demand in a home and foreign economy is given by
M
 .2  Q i . Assume that at a particular point in time, the two economies have
P
equal supplies of money (i.e. M = MF) and equal levels of real output (Q = QF)
and in each economy the real interest rate is equal to the growth rate of output. In
the home economy, the money growth rate is 4% and in the foreign economy the
money growth rate is 16%. Calculate the spot/nominal exchange rate if absolute
PPP holds.
Absolute PPP implies SPF = P S 
and M = MF and Q = QF . S 
P

PF
gM F
g
M
5
5
M
Q i
MF

M QF
MF Q
iF
. With r = gQ , i=gM
i
QF iF
MF
 g
gM
 .16
.04
 4  2 Note, I also
accept .5 as an answer.
4.
We measure an economy in which the output per labor hour,
Q
= qt = $30, and
L
W
= wt = $20. We also measure that the ratio of GDP
P
to capital in the economy is equal to 1. Calculate TFP in this economy i.
GDP is given by Q  K  (ZL)1 Labor share of income is
WL WP
1 
 Q  2    13 . The level of TFP can be derived as
3
PQ
L
real wages are measured as
TFP  Z
5.
1
1
q

Q
2
1
2
 30 3 1 3  30 3  9.655
K
We find that the price level in a foreign economy in 1980 was equal to 1 and equal
to 1.75 in year 2000. We find that the price level in the domestic economy was
equal to 1 in year 1980 and equal to 1.5 in year 2000. The exchange rate between
the domestic economy equal to 1 in year 1980 and was equal to .75 in year 2000.
Calculate the average inflation rate in the domestic economy and the foreign
economy over the 20 year period 1981-2000. Estimate the average depreciation
rate of the domestic currency. Estimate the average real depreciation rate (i.e. the
growth rate of the real exchange rate). [Hint: Use the logarithm tables].
The average inflation rate over the period is the average growth of prices between 1981
1
and 2000 which can be derived as  81:00  [ln P2000  ln P1980 ]  0.020273255 . Similarly
20
1
F
F
F
 [ln P2000
 ln P1980
]  0.027980789 . The depreciation rate is
foreign inflation is  81:00
20
1
S
 [ln S2000  ln S1980 ]  -0.014384104 . The
the growth rate of the exchange rate is g81:00
20
s
S
F
g

g




real depreciation rate is 81:00
81:00
81:00
81:00  -0.00667657 .
X
0.25
0.5
0.75
1
1.25
1.5
1.75
2
ln(X)
-1.386294
-0.693147
-0.287682
0
0.223144
0.405465
0.559616
0.693147
-0.069315
-0.034657
-0.014384
0
0.011157
0.020273 -0.006677
0.027981 0.027981
0.034657
Long Answer Questions
6.
(20 points) Households in an economy get utility as a linear function from
consumption and leisure.
U  Ct  1 lst
2
Households in the economy are either unemployed in which case, they enjoy ls =
672 hours of leisure, or employed in which case they enjoy ls = 472 hours of
leisure and must work 200 hours. If they are unemployed they receive a benefit
level of C = b = 900. If they are employed, they get a consumption level of C = w
∙ 200.
a. Calculate the reservation wage (i.e. the wage offer that the household would
have to receive to give them a utility equal to the utility of an unemployed
person.
The reservation wage that sets utility of an unemployed person equal to the utility
of an employed person solves
b  1 2 672  900  336  1236  U u  U e  (200  w)  1 2 472  (200  w)  236
(200  w)  236  1236  w  5
Wage offers are uniformly distributed over a range of [0, 20]. This means that for
any reservation wage level, ŵ , the probability that a job applicant would get a
20  w
wage offer higher than ŵ is
.
20
b. What is the probability that a worker will receive a wage offer better than the
reservation wage from 6a.
20  5
 .75
The probability of finding a job that offers a wage greater that 5 is
20
c. If the probability that someone in the workforce has some event that would
cause them to leave their job if they have one is 5% (i.e. s = .05) and the
probability that someone searching for a job will receive a job offer is 100%
(i.e. p = 1), what is the steady-state unemployment rate.
s
.05
The steady state unemployment rate is ur SS 

 .05  .0625
s  pH ( w) .05  .75 .80
7. (20 points) In an economy, there is a fixed supply of labor (which we will
normalize to 1), Lt = 1, so the growth rate of the labor supply is n = 0. The
household can divide the labor supply between producing goods or to doing R & D.
The amount of time spent in producing goods is equal to u ∙ L = u. The amount of
time spent in R & D is (1-u) ∙ L = (1-u). The amount of goods produced are given
by a Cobb-Douglas production function where Kt is the capital stock and Zt is the
technology produced.
1
1
1
1
Qt  Kt 2 (Zt Lt ) 2  Kt 2 Zt 2
The amount of new technology produced is given by
Z  2  Z t (1  u ) Lt  2  Z t (1  u )
If the share of time spent in R & D is 2% (i.e. u = .98) what is the long-run growth
rate of labor productivity when the economy is on its balanced growth path.
Assume that the investment rate, s = .2. Calculate the capital deepening rate of
capital and the growth rate of labor productivity when capital productivity is 2.
What is the steady-state capital productivity level when the economy is on its
balanced growth path?
Z
The growth rate of technology is  2  .02  .04 . The growth rate of the capital labor
Z
ratio is equal to the growth rate of capital
K I
Q
Q
K  (1   ) K  I 
    s    .2   .08 . Under Cobb-Douglas, the
K
K
K
K
growth rate of output, is a weighted average of the growth rate of capital and
technology (when labor is fixed).
Q
gtQ  12 gtK  12 gtz  12 [.2  .08]  12 .04
K
When capital productivity is 2, the growth rate of output is .18. The steady state capital
productivity sets the growth rate of capital equal to the growth rate of technology so the
Q 
steady state capital productivity is    .6 .
K 
(10 points) Assume that the money demand is as described as by the BaumolTobin model. On a graph with P on the vertical axis and Q on the horizontal axis, draw
the relationship between P and Q taking M and i as given. Assume that Q is fixed (draw
this as vertical level of supply that is not affected by the price level). Where the two
curves cross is the equilibrium price level. Show in two graphs, how an increase in M or
an increase in I would affect the equilibrium price level.
P
M↑
PEQ
PEQ
Q
P
i↑
PEQ
PEQ
Q