Download Midterm 1 / Questions and Answers

ECON 202 / MACROECONOMICS / SPRING 2005
MIDTERM 1 / Questions and Answers
QUESTION (1)
Assume that there are two factors of production, K and L, and that they are both fully employed at K =
K and L = L. Furthermore, assume that the economy is described by the following set of equations:
Y = Y = F(K, L) = 1,200
Y = C+I+G
C = 125 + 0.75 (Y – T)
I = I(r) = 200 – 10 r
G = G = 150, and
T = T = 100.
(a)
(b)
(c)
(d)
(e)
Solve for the level of consumption.
Solve for the equilibrium real interest rate.
Draw a graph to show the saving, investment, and the equilibrium real interest rate.
Calculate private saving and public saving.
Suppose government purchases increase. What would happen to investment and to the
equilibrium real interest rate? Why?
ANSWER
(a) C = 125 + 0.75 (Y – T) = 125 + 0.75 (1200 – 100) = 125 + 0.75 x 1100 = 950.
(b) S = Y – C – G = 1200 – 950 – 150 = 100, and since S = I in the equilibrium of the goods
market, I = 100 also, which means that 100 = 200 – 10 r, solving for which we find r = 10%.
(c) Draw a graph with real rate of interest measured along the vertical axis and saving and
investment measured along the horizontal axis. S is represented by a vertical line at 100, and I
is represented by a downward sloping line which intersects the vertical axis at r = 20% and
the horizontal axis at I = 200. The point of intersection between these two lines corresponds
to I = S = 100 and r = 10%.
(d) Private saving = Disposable Income – Consumption = Y – T – C = 1200 – 100 – 950 = 150,
Public saving = T – G = 100 – 150 = – 50.
(e) As G increases, private saving does not change (because of Y being constant) but public
saving decreases (because public saving = T – G), therefore total saving decreases. This
means that investment must decrease also, which, everything else being the same, can happen
only if real interest rate increases in the new equilibrium of the goods market.
QUESTION (2)
Consider the following (Cobb-Douglas type) production function:
Y = 60 K1/3 L2/3
(a) Calculate Y for (i) K = 64 and L = 8, (ii) K = 128 and L = 16, and (iii) K = 192 and L = 24.
Explain how your results illustrate the property of constant returns to scale.
(b) Derive the algebraic expression for the marginal product of labor.
(c) Calculate the MPL at K = 64 and L = 8.
(d) What would the real wage rate have to be in order that the firms in this economy decide to
hire a total of 8 units of labor, if the amount of capital employed in the economy is already K
= 64 units? What would the total real payments to labor be? What would labor’s share of
total output be?
ANSWER
(a) (i) Y = 60 K1/3 L2/3 = 60 (64)1/3 (8)2/3 = 960,
(ii) Y = 60 K1/3 L2/3 = 60 (128)1/3 (16)2/3 = 960 x 2 = 1920,
(iii) Y = 60 K1/3 L2/3 = 60 (192)1/3 (24)2/3 = 960 x 3 = 2880.
As we go from (i) to (ii), L increases by 16 / 8 = 2 times and K increases by 128 / 64 = 2
times, and Y also increases 1920 / 960 = 2 times. As we go from (i) to (iii), L increases by 24
/ 8 = 3 times and K increases by 192 / 64 = 3 times, and Y also increases 2880 / 960 = 2 times.
Therefore in both cases, Y increases by the same factor by which K and L increase at the same
time. These results indicate that there are constant returns to scale both from (i) to (ii) and
from (i) to (iii) (or, from (ii) to (iii)).
(b) Marginal product of labor is mathematically the derivative of Y with respect to L, and it is
equal to (2/3) 60 K1/3 L2/3 - 1 = 40 K1/3 L-1/3 = 40 (K / L)1/3. Alternatively, we can use the
definition MPL ≈ F(K, L+1) – F(K, L) = 60 K1/3 (L+1)2/3 – 60 K1/3 L2/3.
(c) MPL = 40 (K / L)1/3 = 40 (64 / 8)1/3 = 40 x 2 = 80. Or, using the approximation,
MPL ≈ 60 K1/3 (L+1)2/3 – 60 K1/3 L2/3. = 60 (64)1/3 (9)2/3 – 60 (64)1/3 (8)2/3 = 1038,42 – 960
= 78,42, which is approximately equal to 80.
(d) Real wage rate = W / P = MPL = 80,
Total real payments to labor = (W / P) x L = 80 x 8 = 640,
Labor’s share of total output = total real payments to labor / total output = 640 / 960 = 2 /3.
QUESTION (3)
Suppose that money demand is represented by the equation (M/P)d = 0.25 Y.
(a) Use the quantity equation to calculate the income velocity of money.
(b) How would you expect velocity to change in an environment of hyperinflation?
(c) How would you expect velocity to change in an environment of higher financial
sophistication?
ANSWER
(a) The quantity equation is as follows: M V = P Y and it can be rewritten as M / P = Y / V.
Here, M denotes the quantity of nominal money circulating in the economy, and, because of
the money market equilibrium, it is equal to the nominal quantity of money demanded. In
other words, M / P = (M/P)d . Therefore, we can compare the quantity equation M/P =
Y/V with the money demand (M/P)d = 0.25 Y , and find that Y/V = 0.25Y, or 1/V =
0.25, which implies V = 1 / 0.25 = 4.
(b) During hyperinflations, any nominal amount of money very quickly loses its real value (the
rate of decrease in the real value of any nominal quantity of money is equal to the inflation
rate which is very high during hyperinflations). Therefore, the opportunity cost of holding
money becomes very high and by far exceeds the utility gained from holding money. As a
result, people (or firms) try to hold as little money as possible, and get rid of it as quickly as
possible by spending it. But this means that money changes hands very quickly and this is
nothing but the velocity being high. So, during hyperinflations, velocity increases.
(c) Higher financial sophistication enables people to do their payments by holding smaller
amounts of money (as in the case of the introduction of ATMs or credit cards). This means
that, with the same volume of transactions, or with the same price level and real GDP (in
symbols, with the same P and Y), the amount of money held is lower, implying that P Y / M is
higher. But since PY / M is nothing but the velocity V, higher financial sophistication means
a higher V. So, the velocity increases
QUESTION (4)
Recall that the exact formula for calculating the real interest rate is:
1 + r = (1 + i) / (1 + π)
Suppose you start with $100 and earn an interest rate of 300 percent per year.
(a) How much money would you have at the end of the year?
(b) If inflation were 150 percent per year, and the initial price level were 1.25, what would the
price level be at the end of the year?
(c) Calculate the real value of your wealth at the beginning and at the end of the year.
(d) Use these real values to calculate the real interest rate.
(e) Use the equation at the beginning of this problem to verify your result.
ANSWER
(a) An interest rate of 300 percent per year means that the interest you earn at the end of the year
is 300 percent of, or 3 times, the principle. So, interest = $100 x 3 = $300. The money you
would have at the end of the year = principle + interest = $100 + $300 = $400. You can
calculate this directly by $100 x (1 + 3) = $400.
(b) An inflation rate of 150 percent means that the increase in the price level is 150 percent of, or
1.5 times, the price level at the beginning of the year. So, the increase in the price level = 1.25
x 1.5 = 1.875. Therefore, the price level at the end of the year would have grown to 1.25 +
1.875 = 3.125. You can calculate this directly by 1.25 x (1 + 1.5) = 3.125.
(c) Recall that the real value of the wealth = the nominal value / the price level. At the beginning
of the year, real value = $100 / 1.25 = 80. At the end of the year, real value = $400 / 3.125
= 128.
(d) The real interest rate = the change in the real value / the real value at the beginning, so r =
(128 – 80) / 80 = 0.6 = 60%.
(e) 1 + r = (1 + i) / (1 + π) = (1 + 3) / (1 + 1.5) = 4 / 2.5 = 1.6, so r = 0.6.
QUESTION (5)
The two parts of the question are totally unrelated. Please take note of it.
(a) Futilia, a typical small open economy, has been imposing quotas on imports with the intention
of keeping her trade balance under control. The newly elected liberal government decides to
lift the quotas, but is facing severe resistance from the opposition who claims that the lifting of
quotas will severely hurt the trade balance. Show the impact of the lifting of the quotas on the
real exchange rate, the overall trade balance, and on exports and imports separately.
b) In Agresiland, the Central Bank decides to aggressively manage the exchange rate such that
the rate of nominal depreciation is kept equal to the inflation difference between that of
Agresiland and that of its trade partners, which let us assume is equal to zero (π* = 0). What
would be the implication of that on the real exchange rate if exchange rates were determined
in line with the PPP principle?
ANSWER
(a) As the quotas are lifted, imports will increase at any value of the real exchange rate (this is an
outward shift in the imports demand curve). As a result of this, net exports will decrease at
any value of the real exchange rate (similarly, this is an inward shift in the net exports demand
curve). In the long run, the output is given (unaffected by the real exchange rate), and the
difference between saving and investment is constant, so saving – investment is represented
by a vertical line in the graph with real exchange rate along the vertical axis and net exports
along the horizontal axis. As the net exports demand curve shifts inward due to the lifting of
quotas and the saving – investment schedule does not change its position, the real exchange
rate will decrease. The net exports (hence, the overall trade balance) does not change because
the difference saving – investment does not change. However, exports and imports do change
separately. Since the real exchange rate decreases, exports will increase, and as exports
increase with net exports remaining constant, imports will have to increase, too. But this net
increase in imports can be separated into two parts: Imports will increase due to lifting of
quotas (an outward shift in the imports demand curve) and they will have to decrease due to
the decrease in the real exchange rate (a movement along the imports demand curve), with the
increase effect outweighing the decrease effect.
(b) The underlying assumption of the PPP principle is that the real exchange rate does not change.
Alternatively, if the nominal exchange rate is determined according to the PPP principle, then
%Δe = %ΔP* – %ΔP, where e is the nominal exchange rate, P* is the foreign price level, P is
the domestic price level, and %Δ indicates “percentage change in…” But the percentage
change in the real exchange rate is given by %Δε = %Δe + %ΔP – %ΔP*, and therefore,
using the PPP principle here gives %Δε = %ΔP* – %ΔP + %ΔP – %ΔP*, which simplifies
to %Δε = 0. This means nothing but that the real exchange rate does not change.
QUESTION (6)
The three parts of the question are totally unrelated. Please take note of it.
(a) Suppose that the natural rate of unemployment uN = 12.5%, the periodicity is one month, and
the fraction of workers who find jobs (the rate of job finding) is 56%. What must be the
fraction of workers who lose jobs every month (rate of job separation)?
(b) Suppose that a country experiences reduction in productivity (due to an economic crisis, for
example). How would this affect the demand for labor? If the real wage were rigid (for
structural, legislative, or whatever reasons), show graphically the impact of this change on the
unemployment rate.
(c) Assume that apples cost $0.50 in 2002 and $1 in 2007, whereas oranges cost $1 in 2002 and
$0.50 in 2007. If 10 apples and 5 oranges were produced in 2002, and 5 apples and 10 oranges
in 2007, find the nominal GDP for both years and the real GDP for 2007 using 2002 as the
base year in this two-good economy.
ANSWER
(a) Recall that the natural rate of unemployment is given by u = s / (s + f), where u is the
unemployment rate, s is the rate of job separation, and f is the rate of job finding. If the
natural rate of unemployment is 12.5%, or 0.125, and the rate of job finding is 56%, or 0.56,
we have 0.125 = s / (s + 0.56). This can be rewritten as 0.125 s + 0.07 = s, or as 0.07 =
0.875 s, yielding s = 0.08. So, 8% of the unemployed workers lose jobs (or, more generally,
are separated from their jobs) every month.
(b) The demand-for-labor curve is identical with the marginal-product-of-labor curve. As the
productivity of the labor declines, marginal product of labor decreases at any employment
level, so the marginal-product-of-labor curve shifts down (or, leftward), implying that the
demand-for-labor curve shifts leftward. The supply-of-labor curve is unaffected. Now, if the
real wage rate is rigid, the quantity demanded of labor decreases, but the quantity supplied of
labor stays constant. This means that the rate of unemployment increases.
(c) the nominal GDP in 2002 = 10 apples x $0.50 / apple + 5 oranges x $1 / orange = $10,
the nominal GDP in 2007 = 5 apples x $1 / apple + 10 oranges x $0.50 / orange = $10, and
the real GDP in 2007 = 5 apples x $0.50 / apple + 10 oranges x $1 / orange = $12.50.