Download C:\Documents and Settings\Ivan

Econ 154b
Spring 2005
Suggested Solutions to Problem Set 9: Questions 1-3
Question 1
(a) If the Fed targets the real interest rate, then money demand shocks are offset by changes in
the money supply, so the LM curve does not move. To see this, look at the figure below, which
depicts the result in a Keynesian model. Initially, real money supply is given by the MS 1 line, while
money demand is given by the curve MD 1 ŸY 1 , where Y 1 is the current level of output. Suppose a
money demand shock increases money demand for a given level of output; then the money demand
curve shifts to MD 2 ŸY 1 . This tends to increase the real interest rate. When the Fed sees the rise in
the real interest rate, it increases the money supply in response to reduce the real interest rate back to
its targeted level of 3%. It is successful in doing so if it increases the real money supply to MS 2 .
In terms of the LM curve, the result of targeting the real interest rate is shown in the figure
below. Beginning at LM 1 , the shock to money demand shifts the LM curve up and to the left to LM 2 .
If the Fed does not respond, this raises the real interest rate and reduces output. But when the Fed
responds by increasing the money supply by just the right amount, the LM curve shifts back to LM 1 .
Since the shock causes the money supply to change, but does not affect output, the money supply is
acyclical. By following the interest-rate-targeting rule, the AD curve is unaffected by money-demand
shocks (since they are offset by money-supply changes), so it is more stable than if the Fed did not
respond at all.
Finally, since the AD curve does not move, the results do not change for the misperceptions
version of the classical model.
(b) When there are IS shocks, the rule does not work very well. Suppose a shock shifts the IS
curve from IS 1 to IS 2 , as shown in the figure below. Targeting the real interest rate requires the Fed
to increase the money supply to shift the LM curve from LM 1 to LM 2 . While this maintains the real
interest rate at its initial level, output is above full-employment output. The money supply is
procyclical, since the shift in the IS curve caused output to rise, and the increase in the money supply
caused output to rise further. This response to IS shocks makes the aggregate demand curve less
stable, as it shifts the AD curve farther to the right in response to an IS shock than it would have if
the LM curve did not respond. Also, the Fed cannot maintain this policy in the long run, as the
economy must return to full-employment output with a higher real interest rate on the new IS curve.
If the Fed were to try to keep increasing the money supply in an attempt to shift the LM curve down
and to the right and reduce the real interest rate, the result would be higher inflation.
In the misperceptions version of the classical model, the shift in the AD curve will tend to raise
the price level (as the SRAS curve is upward-sloping in that model), which will put some pressure on
the LM curve to shift to the left. This will make it even more difficult for the Fed to stay on LM 2 and
keep the interest rate fixed (the Fed will need to increase the money supply even more). The
conclusions, however, are the same.
(c) When there are aggregate supply shocks, the rule also does not work very well. Suppose a
supply shock shifts the FE line from FE 1 to FE 2 , as shown below. In the short run, there is no effect
on output or the real interest rate, since the equilibrium occurs where the IS curve intersects the LM
curve. However, at this point output is below full-employment output. In the long run, the price level
will fall, shifting the LM curve from LM 1 to LM 2 to reach the new long-run equilibrium. If the
central bank tries to target the real interest rate at its initial level, it will have to reduce the money
supply as the price level falls to keep the LM curve from shifting and to prevent the real interest rate
from declining. Such a policy cannot be successful in the long run, as the economy must eventually
return to full-employment output with a lower real interest rate.
In the misperceptions version of the classical model, the shift in FE corresponds to an identical
shift in the LRAS curve. Once the LRAS curve has shifted to the right, there is a new
full-employment level, and the economy will tend to move to that level of output. If the Fed keeps
on decreasing the money supply, this will keep on shifting the AD curve to the left, trying to keep the
economy at the old (initial) level of output. Clearly, this is not sustainable, and the conslusions are
the same as with the Keynesian model.
Question 2
Suppose that the Fed assigns 0 points, rather than 1 point, to an unemployment rate of 3%.
Then, the appropriate payoffs to the Fed and the Firm in the new game are given in the following
table:
FIRM’S STRATEGY
Raise P
Don’t raise P
Raise M
A
(0,2)
B
(1,1)
Don’t raise M
C
(-1,0)
D
(1,3)
FED’S STRATEGY
The outcome will depend on whether we view this game as a simultaneous game or whether we
allow the Firm to choose first.
If we view this game as a simultaneous game (where the Fed and the Firm choose their actions
simultaneously), the game has two Nash equilibria: (Raise M, Raise P) and (Don’t raise M, Don’t
raise P). (Recall that a pair of actions, (s1,s2) is a Nash equilibrium if s1 is a best response to s2 and
s2 is a best response to s1. Here, for instance, (Raise M, Raise P) is a Nash equilibrium because if
the Fed raises M, then the best thing the Firm can do is to raise P; and if the Firm raises P, the best
thing the Fed can do is to raise M. We can check for the other Nash equilibrium similarly.) The
concept of a Nash equilibrium doesn’t tell us which of the two outcomes (A or D) will actually
occur. We would need to assume extra rules for choosing the equilibrium outcome of the game.
If, however, we let the Firm choose its action first, then we will have a unique outcome. The
reason is as follows: Firms think this way: (1) If we raise P, then the Fed will choose to raise M, so
the outcome is A. We get 2 points. (2) If we do not raise P, then the Fed will not raise M, so the
outcome is D. We get 3 points. So we are better off not raising P, because then the Fed will choose
not to raise M, and we will get 3 points.
The difference between this result and the outcome of the example in the text is that in this
model the Fed has no incentive to fool firms (and hence there is no credibility problem). In the text
example, the Fed gained by raising M when firms didn’t raise P, so outcome D was not chosen. But
now outcome D is the preferred outcome by both firms and the Fed.
Question 3
If changes in monetary policy are reflected primarily by changes in the short-term interest rate,
then we would expect a negative relationship between the unemployment rate and the interest rate.
The reason is that when unemployment is high, we would expect the Fed to lower the interest rate in
order to stimulate aggregate demand. This expectation is supported by the data, as the graph below
shows:
Unemployment Rate and 3-Month Treasury Bill
18
16
14
12
10
8
6
4
2
0
Unemployment Rate
Time (Month)
Jan-05
Jan-03
Jan-01
Jan-99
Jan-97
Jan-95
Jan-93
Jan-91
Jan-89
Jan-87
Jan-85
Jan-83
Jan-81
Jan-79
Jan-77
Jan-75
Jan-73
Jan-71
Jan-69
Jan-67
Jan-65
Jan-63
Jan-61
3MTB
Economics 154, Intermediate Macroeconomics
Answer to Problem 4, Problem set 9
ANSWER: The first step to solve the problem is defining L as a function of r and π:
L = 0.2Y − 500i = 0.2Y − 500r − 500π
With Y = 1000,
L = 200 − 500r − 500π
.
1. When r = 0.04, equating real money supply to money demand gives: M/P = L = 200 −
(500 ∗ 0.04) − 500π = 180 − 500π. Real seignorage revenue R = π × M/P = π180 − 500π 2 .
The following table shows seignorage revenue (R) for inflation rates between 0 and 0.30.
These values are plotted in Figure 1.
Figure 1: Seigniorage Revenue as a Function of Inflation
π
0.00
0.02
0.04
0.06
0.08
0.10
R
0.0
3.4
6.4
9.0
11.2
13.0
π
R
π
R
0.12
0.14
0.16
0.18
0.20
14.4
15.4
16.0
16.2
16.0
0.22
0.24
0.26
0.28
0.30
15.4
14.4
13.0
11.2
9.0
1
2. Seignorage is maximized at π = 0.18.
The calculus solution to the above problem is found by setting dR/dπ = 0. First find the
revenue equation above (R = π180 − 500π 2 ). Then take the derivative with respect to π
to find where it is flat with respect to this variable: dR/dπ = 180 − 2 × 500π. Now find π
such that dR/dπ = 0 : π = 180/1000 = 0.18. Since the graph above shows the equation is
concave down, R is maximized with respect to π where dR/dπ = 0.
3. The maximum amount of seignorage revenue is 16.2.
4. When r = 0.08, equating real money supply to money demand gives: M/P = L = 200 −
(500 × 0.08) − 500π = 160 − 500π . Real seignorage revenue R = πM/P = 160π − 500π 2 .
The following table shows seignorage revenue (R) for inflation rates between 0 and 0.30.
These values are plotted in Figure 2.
π
0.00
0.02
0.04
0.06
0.08
0.10
R
0.0
3.0
5.6
7.8
9.6
11.0
π
R
π
R
0.12
0.14
0.16
0.18
0.20
12.0
12.6
12.8
12.6
12.0
0.22
0.24
0.26
0.28
0.30
11.0
9.6
7.8
5.6
3.0
The maximum seignorage of 12.8 is attained when π = 0.16. This can be solved by calculus
just as above by noting that dR/dπ = 160 − 2 × 500π.
Figure 2: Seigniorage Revenue given r=0.08
To find the inflation rate that maximizes seignorage, use calculus rather than drawing a
graph. That is, express real seignorage revenue as a function of the inflation rate π, take
the derivative of seignorage with respect to π, set the derivative equal to 0, and solve for
the optimal value of π.
2