Download Intermediate Macroeconomics，Assignment 5

Intermediate Macroeconomics，Assignment 5
Due May 20th (Friday), in-class
1. Consider the impact of an increase in thriftiness in the Keynesian cross. Suppose the
consumption function is
C = C� + c(Y − T)
where C� is a parameter called autonomous consumption and c is the marginal
propensity to consume.
a). What happens to equilibrium income when the society becomes more thrifty, as
represented by a decline in C� ?
b). What happens to equilibrium saving?
c). Why do you suppose this result is called the paradox of thrift?
d). Does this paradox arise in the classical model of Chapter 3? Why or why not?
Answer:
a) By equalizing the actual expenditure and the planned expenditure we have
Y = C� + c(Y − T) + I(r) + G,
which solves the equilibrium income as
C� − cT + I(r) + G
.
Y=
1−c
Therefore a decline in C� reduces the equilibrium income.
(We can also use the Keynesian cross to graphically show this result)
b) The equilibrium saving is
S= Y−C−G
= Y − C� − c(Y − T) − G
= (1 − c)Y − C� + cT − G
= I(r)
which is irrelevant with respect to C� , and therefore the equilibrium saving will NOT
change after a decline in C� . (We can also state that saving must equal investment as we
are looking at a closed economy. Hence saving does not change with C� . )
c) Because a reduction in consumption normally increases saving, given government
spending as exogenous. However, according to a Keynesian cross model, a reduction
in autonomous consumption does not affect equilibrium saving, which is a paradox.
d) This paradox does not arise in the classical model in Chapter 3. This is because in
that model, a reduction in autonomous consumption does not affect equilibrium
income and hence does not affect total consumption. As a result, a reduction in
autonomous consumption does increases the equilibrium saving, which is consistent
with our intuition.
2. This problem asks you to analyze the IS–LM model algebraically. Suppose
consumption is a linear function of disposable income:
C(Y – T) = a + b(Y −T),
where a > 0 and 0 < b < 1. Suppose also that investment is a linear function of the
interest rate:
I(r) = c − dr,
where c > 0 and d > 0.
a). Solve for Y as a function of r, the exogenous variables G and T, and the model’s
parameters a, b, c, and d.
b). How does the slope of the IS curve depend on the parameter d, the interest rate
sensitivity of investment? Refer to your answer to part (a), and explain the intuition.
c). Which will cause a bigger horizontal shift in the IS curve, a $100 tax cut or a $100
increase in government spending? Refer to your answer to part (a), and explain the
intuition.
Now suppose demand for real money balances is a linear function of income and the
interest rate:
L(r, Y) = eY − fr,
where e > 0 and f > 0.
d). Solve for r as a function of Y, M, and P and the parameters e and f.
e). Using your answer to part (d), determine whether the LM curve is steeper for large or
small values of f, and explain the intuition.
f). How does the size of the shift in the LM curve resulting from a $100 increase in M
depend on
i. the value of the parameter e, the income sensitivity of money demand?
ii. the value of the parameter f, the interest sensitivity of money demand?
g). Use your answers to parts (a) and (d) to derive an expression for the aggregate
demand curve. Your expression should show Y as a function of P; of exogenous policy
variables M, G, and T; and of the model’s parameters. This expression should not
contain r.
h). Use your answer to part (g) to prove that the aggregate demand curve has a negative
slope.
i). Use your answer to part (g) to prove that increases in G and M, and decreases in T,
shift the aggregate demand curve to the right. How does this result change if the
parameter f, the interest sensitivity of money demand, equals zero?
Answer:
a).
Y=
a−bT+c−dr+G
1−b
.
b) According to a), the slope of the IS curve is
−d
1−b
, which is negatively proportional
to the interest rate sensitivity of investment. The intuition is that when investment is
more sensitive to the interest rate, income, which includes investment as one of its
components, becomes more sensitive to the interest rate as well.
c) According to a), a $100 tax cut causes $
100b
1−b
increase in equilibrium income while a
100
$100 increase in government spending causes $ 1−b increase in equilibrium income. The
latter one is larger the then former one as 0<b < 1. The intuition is that tax cut’s effect
on income has to be through the change of consumption while the effect of increasing
government spending is directly on equilibrium income.
e
d) r = f Y −
1M
f P
.
e) The LM curve is steeper for smaller f. The intuition is that when f is small, the
money demand is not very sensitive to the change of interest rate, and therefore the
income is not sensitive to the change of interest rate.
f) The size of the shift in the LM curve resulting from a $100 increase in M is
negatively related to f and is not relevant with respect to e.
g) By combining these two equation and eliminating r we have
Y=
dM[(1−b)f+de] 1
1−b
P
(a−bT+c+G)f
+ (1−b)dM[(1−b)f+de].
h) Obviously, the AD has a negative slope.
i) Obvious. When f=0, increases in M still shift the aggregate demand curve to the
right, but changes in T or G will not affect the AD curve.
3. Suppose that the price level relevant for money demand includes the price of imported goods and
that the price of imported goods depends on the exchange rate. That is, the money market is
described by
M/P = L(r, Y ),
where P = l*Pd + (1 – l)*Pf /e. The parameter l is the share of domestic goods in the price index P.
Assume that the price of domestic goods Pd and the price of foreign goods measured in foreign
currency Pf are fixed.
a. Suppose that we graph the LM* curve for given values of Pd and Pf (instead of the usual P). Is this
LM* curve still vertical? Explain.
b. What is the effect of expansionary fiscal policy under floating exchange rates in this model?
Explain. Contrast with the standard Mundell–Fleming model.
c. Suppose that political instability increases the country risk premium and, thereby, the interest rate.
What is the effect on the exchange rate, the price level, and aggregate income in this model? Contrast
with the standard Mundell–Fleming model.
Answer:
a. No, it is not vertical anymore. Instead it is upward sloping. This is because a rising exchange rate e
lowers the price of foreign goods Pf and hence lowers the aggregate price P. With a lower P, the real
money supply is increased, which induces a higher income Y.
b. With an upward sloping LM* curve, an expansionary fiscal policy under floating exchange rates
raises the exchange rates and the equilibrium income (In a standard Mundell–Fleming model with
vertical LM* curve, fiscal policy does not change equilibrium income under floating exchange rates).
c. With an increased interest rate, the IS* curve is shifted to the left due a reduced investment.
Therefore the exchange rate is lowered and the income is lowered as well. In a standard
Mundell–Fleming model with vertical LM* curve, only the exchange rate is lowered while the
aggregate income is not affected).
4. Use the Mundell–Fleming model to answer the following questions about the state of California (a
small open economy).
a. What kind of exchange-rate system does California have with its major trading partners (Alabama,
Alaska, Arizona, . . .)?
b. If California suffers from a recession, should the state government use monetary or fiscal policy to
stimulate employment? Explain. (Note: For this question, assume that the state government can print
dollar bills.)
c. If California prohibited the import of wines from the state of Washington, what would happen to
income, the exchange rate, and the trade balance? Consider both the short-run and the long-run
impacts.
Answer:
a. California has a fixed exchange-rate regime with other states.
b. Since California has a fixed exchange-rate regime, it should use fiscal policy to stimulate
employment. As a matter of fact, it cannot use monetary policy along at all as it has to maintain a
fixed exchange rate.
c. Imposing import restriction essentially shifts the net export schedule to the right and hence shift
the IS* curve to the right. Consequently both the income and the trade balance are increased while
the exchange rate is not affected at the short-run. In the long run, things are more complicated. This
is because with a rising demand of domestic goods, their price is raised, which reduces the real money
supply and hence shifts the LM* curve to the left. As a result the real exchange rate is raised (recall
that in the short-run the (real) exchange rate will not be affected), and both aggregate income and
trade balance is unaffected.