Download Macroeconomic Analysis ECON 6022A Fall 2011 Problem Set 4

Macroeconomic Analysis
ECON 6022A
Fall 2011
Problem Set 4
November 2, 2011
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The price level and money demand
Suppose the price level in the economy is P . Real money demand L(Y, i) is the same as we’ve discussed in
Lecture 6. So nominal money demand is as follows,
M d = P · L(Y, i)
where Y is the real income and i nominal interest rate.
1. Given P is the price level, interpret the economic meaning of a new variable pM =
1
.
P
2. Given Y and i, plot the relation between M d and pM , where pM is on the vertical axis.
3. Adding the money supply curve, M s to the graph above. And show the determination of equilibrium
p∗M .
4. Explain how p∗M responds to an increase in income Y , with the aid of the graph in 3.
5. (Optional) Plot the relation between M d and interest rate, i, where interest rate is on the vertical
axis.
6. (Optional) Adding money supply curve to the above graph. And show the determination of equilibrium i∗ .
7. (Optional) Given Y and we assume that price, P , is fixed in the short run. Suppose the nominal
money supply increases. Show the determination of new equilibrium interest rate, i∗∗ . Is it lower than
i∗ ? Explain why.
Solution:
1. Price of money. The number of baskets needed to purchase one unit of money. It is a real variable.
2. See Figure 2.
1
3. See Figure 3.
4. When income Y increases, people’s demand for money would increase. So the M d curve shifts to the
right. The price of money increases.
2
5. See Figure 5.
6. See Figure 6.
7. When the nominal money supply increases, without changes in prices in the short-run, the new equilibrium interest rate i∗∗ would be lower than i∗ . The return on non-monetary asset has to be lower,
so that households would hold more money in their portfolio. The mechanism is that households try
to get rid of the excess money in their portfolio. Therefore, the price for non-monetary asset (bonds)
increases and the return to non-monetary asset decreases.
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2
Money growth and inflation
Suppose the money demand function takes the following form
M d /P = L(Y, i) = Y ηY · ψ(i)
where, ψ 0 (i) < 0.
1. Show the income elasticity of money demand is ηY .
2. Suppose that the growth rate of nominal money supply is µ, and inflation rate π and income growth
rate g. The inflation rate is fully anticipated (π e = π). The real interest rate is r and we assume that
it is constant. Show the following:
π = µ − ηY · g
[hint: In a growing economy, what’s the relation between inflation and money supply growth? You
may find the following helpful:
ln xt+1 − ln xt = ln(
∆xt
∆xt
xt+1
) = ln(1 +
)≈
xt
xt
xt
Solution:
1. The income elasticity of money demand is
dM d /M d
dM d Y
Y
=
· d = P · ηY · Y ηY −1 · ψ(i) ·
= ηY .
η
dY /Y
dY
M
P · Y Y · ψ(i)
2.
Solution: Asset market equilibrium gives
Mts = Mtd
and therefore,
Mts /Pt = YtηY · ψ(it )
4
Given it = rt + πte and rt is constant, we have
it = r + πte
Given πt = πte , we have,
it = r + πt = r + π
And therefore we know that it = i = r + π is a constant. Given the money demand equation and the
equilibrium condition, we have the following,
ln(Mts ) − ln(Pt ) = ηy · ln(Yt ) + ln(ψ(it ))
and
s
ln(Mt+1
) − ln(Pt+1 ) = ηy · ln(Yt+1 ) + ln(ψ(it+1 ))
since it = it+1 = r + π,
s
[ln(Mt+1
) − ln(Mts )] − [ln(Pt+1 ) − ln(Pt )] = ηy · [ln(Yt+1 ) − ln(Yt )]
or,
∆ ln(M s ) − ∆ ln(P ) = ηy · ∆ ln(Y )
Using the approximation method provided in the hint,
∆P
∆Y
∆M s
−
= ηy ·
Ms
P
Y
therefore,
π = µ − ηy · g
3
Growth rate in money supply
ABC, 7th edition: Numerical Problem 6, Page 272.
Solutions:
(a) π e = ∆M/M = 10%. i = r + π e = 15%. M/P = L = 0.01 × 150/0.15 = 10. P = 300/10 = 30.
(b) π e = ∆M/M = 5%. i = r + π e = 10%. M/P = L = 0.01 × 150/0.10 = 15. P = 300/15 = 20. The
slowdown in money growth reduces expected inflation, increasing real money demand, thus lowering the
price level.
4
Nominal and real interest rates (optional)
Suppose the nominal interest rate at Period t is it and real interest rate is rt . Let πt be the inflation rate.
The inflation rate from year t to year t + 1, πt , is the ratio of the change in the price level (Pt+1 − Pt ) to the
initial price level, Pt . We assume that πt · rt ≈ 0. Show that the following holds:
it = rt + πt
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Solution: The nominal value of asset in year t is xt and in year t + 1 is xt+1 . The following relation is
satisfied:
xt+1 = xt · (1 + it )
Since pt+1 = pt · (1 + π), we have
xt+1
xt
=
· (1 + it )
pt+1
pt · (1 + πt )
Rearrange it, we have the following,
xt+1
xt (1 + it )
=
·
pt+1
pt (1 + πt )
xt+1
xt
and
are real value of the asset in year t + 1 and t, respectively. Therefore,
pt+1
pt
1 + rt =
(1 + it )
(1 + πt )
Using the assumption that πt · rt ≈ 0, the result is obtained. Or alternatively, we can approximate the
equation by using,
ln(1 + rt ) = ln(1 + it ) − ln(1 + πt )
and
rt ≈ it − πt
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