Download Economics 154a, Spring 2005 Intermediate

Economics 154a, Spring 2005
Intermediate Macroeconomics
Problem Set 8: Answer Key
April 11, 2005
1. Do numerical problem #4 on p. 425 in Chapter 11 of the textbook:
An economy is described by the following equations:
Desired Consumption
C d = 300 + 0.5(Y − T ) − 300r ...
(a) Write the equation for the aggregate demand curve. ...
ANSWER: The IS curve is given by Y = C d + I d + G = 300 + 0.5(Y −
100) − 300r + 100 − 100r + 100 = 450 + 0.5Y − 400r. This can be rewritten as
0.5Y = 450 − 400r, or Y = 900 − 800r. The LM curve is M/P = L, or 6300/P =
0.5Y - 200r. To find the aggregate demand curve, substitute the LM curve into
the IS curve to eliminate r. To do this, multiply both sides of the LM curve by
4 to get 25, 200/P = 2Y − 800r, or 800r = 2Y − (25, 200/P ). Then substitute
this in the IS curve: Y = 900 − 800r = 900 − [2Y − (25, 200/P )]. This can be
rewritten as 3Y = 900 + (25, 200/P ), or Y = 300 + (8400/P ).
(b) Suppose that P=15. What are the short-run values of output, the real interest
rate, consumption, and investment?
ANSWER:With P = 15, the AD curve is Y = 300+(8400/15) = 860. From the
IS curve, 860 = 900−800r, which has the solution r = 0.05. Consumption is C =
300−0.5(860−100)−(300×0.05) = 665. Investment is I = 100−(100×0.05) = 95.
(c) What are the long-run equilibrium values of output, the real interest rate, consumption, investment, and the price level?
ANSWER:In the long run, Y = 700. From the IS equation, 700 = 900 − 800r,
which has the solution r = 0.25. The LM curve then is 6300/P = (0.5 × 700) −
(200 × 0.25) = 300, which has the solution P = 21. Consumption is C =
300−0.5(700−100)−(300×0.25) = 525. Investment is I = 100−(100×0.25) = 75.
In addition, do part (d) below:
(d) Suppose that the economy is the long-run equilibrium in part (c) and G increases
from 100 to 110 (T remains fixed at 100). Determine the new short-run equilibrium values of Y , r, C, and I (P remains fixed at its initial value). Calculate the
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multiplier on government spending (i.e., the change in Y divided by the change
in G). In addition, graph the Id, Sd, IS, and AD curves and show how they shift
when G increases from 100 to 110.
ANSWER: With P at 21 from part (c), an increase in G of 10 leads to a shift
out and up of the IS curve, and to a temporarily higher output level. The IS
curve is now Y = 400 + 0.5(Y − T ) − 400r + 110 or Y = 920 − 800r. The LM
curve does not move in the short run, so that it is still Y = 600 + 400r. The
equilibrium point is therefore found where 920 − 800r = 600 + 400r, so that
r = 320/1200 = 0.266. This implies output is Y = 600 + 400(320/1200) = 706.6.
Then C = 300+0.5(706.6−100)−300(0.266) = 523.33 and I = 100−100(0.266) =
73.3. Overall, output increased by 6.6 dollars for 10 dollars of extra government
expenditures which implies a ’multiplier’ of 6.6/10=66/Figures 1,2 and 3 depict
the movements of the S d , IS, and AD curves.
Figure 1: Shift of sd as G increases
Sd (G=110)
r
Sd (G=100)
Sd = 0.5Y+0.5T-300+300r-G
Id: 100-100r
Id,Sd
Figure 2: Shift of IS as G increases
∆G
r
IS’ (G=110)
IS (G=100)
LM: Y= 600+400r
B
A
IS: Y=700+2G-800r
Multiplier=∆Y/∆G
Y
∆Y
2
Figure 3: Shift of AD as G increases
LRAS: Y=700
P
B
SRAS: P=21
A
AD: 3Y=700+2G+25200/P
AD(G=110)
AD(G=100)
Y
(e) Optional bonus question: How do your answers in part (d) change if T increases
at the same time as G so as to keep the governments budget balanced?
ANSWER: In this case the government expenditure is offset by an increase in
taxes. This implies that the increase in aggregate demand is smaller than before.
The IS curve is now Y = 400 + 0.5(Y − 110) − 400r + 110 or Y = 910 − 800r.
Then, equilibrium r is r = 310/1200 = 0.258 and output is Y = 703.33. the
multiplier is now only 33.3%. Also, consumption decreases further, to C =
300 + (703.3 − 110) − 300(0.258) = 519.15 although investment does not decrease
as much, falling only to I = 100 − 100(0.258) = 74.2.
2. Do analytical problem #5 on p. 466 in Chapter 12 of the textbook.
To fight an ongoing 10% inflation the government makes raising wages or prices illegal.
However that government continues to increase the money supply...
(a) Using the Keynesian AD-AS framework, show the effects of the government’s
policies on the economy...
ANSWER: Figure 4 shows the effects of increasing the money supply while
holding the price level constant. Beginning at point A, the intersection of aggregate demand curve AD1 and short-run aggregate supply curve SRAS 1 , the
increase in the money supply shifts the aggregate demand curve to AD2 . Since
prices cannot rise, the short-run equilibrium is at point B, with output above its
full-employment level.
(b) After several years in which the controls have kept prices form rising, the government declares victory...
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Figure 4: Increasing money supply with constant price level
ANSWER: When the price controls are removed, the price level will jump
up, with the short-run aggregate supply curve shifting to SRAS 2 . The new
equilibrium is at point C, where there is full employment.
3. This problem studies the dynamic behavior of a macroeconomic model that consists
of three equations: Okuns law, an aggregate demand curve, and a Phillips curve...
(a) Suppose that the growth rate of the money supply is 12% per year...
ANSWER: We must show that if (gmT , uT , gy,T , πT ) = (12, 5, 2, 10) for T = t−1
and T = t, then it’s also true for all T > t. First, substitute ut = 5 in equation
1 to obtain ut+1 − 5 = −0.5(gyt+1 − 2). Then plug in the right hand of this last
e
equation into equation 3 to obtain πt+1 − πt+1
= −2(−0.5(gyt+1 − 2)) = gyt+1 − 2.
e
Now plug in equation 2 to obtain πt+1 − πt+1 = gmt+1 − πt+1 − 2 or πt+1 =
e
(gmt+1 + πt+1
− 2)/2. Finally, working backwards from the facts that gmt+1 = 12
e
and that πt+1
= πt = 10 we get πt+1 = (12 + 10 − 2)/2 = 10, gyt+1 = 12 − 10 = 2
and ut+1 = 5 − 0.5(2 − 2) = 5. Therefore this set of values is a steady state for
the economy.
(b) Suppose instead that gmt = 6 for all t. Show that the steady-state values of ut ,
gyt , and πt are 5, 2, and 4, respectively.
e
= πt = 4 and gmt+1 = 6 in the equations derived above
ANSWER: Setting πt+1
we get πt+1 = (6 + 4 − 2)/2 = 4, gyt+1 = 6 − 4 = 2 and ut+1 = 5 − 0.5(2 − 2) = 5.
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(c) Now suppose that the economy begins (in year 0) in the steady state discussed in
part (a) and that the monetary authority (the Fed) wants to move the economy
from this steady state to the steady state in part (b)...
ANSWER: Solve the above equations for general values:
Substitute ut from (1) into (3) and solve for π to obtain πt = πte + 8 − 2ut−1 + gyt .
Now substitute gyt from 2 to get πt = (πte + 8 − 2ut−1 + gmt )/2 which gives us
inflation as a result of ’exogenous’ variables. Solve for gmt = 2πt − πte − 8 + 2ut−1 .
Then, substitute into (3) again to obtain unemployment: ut = 3 + (ut−1 )/2 +
(πte − gmt )/4.
Date(t)
πte
desired π
gmt
ut
0 1
10 10
10 8
12 8
5 6
2
8
6
8
6
3
6
4
6
6
4
4
4
8
5
5
4
4
6
5
6
4
4
6
5
NOTE: This exercise is more difficult than it looks. Make sure you solve the
system of equations in general so that you are taking into account the changes
to ut as you go along.
(d) Calculate the sacrifice ratio associated with the transition path that you calculated in part (d). (Be sure to read Box 12.4 on p. 461 in Chapter 12 of the
textbook.)...
ANSWER: The sacrifice ratio is
(3 years*1 percentage point of unemployment)/(6 percentage points fall in inflation)=1/2.
(e) Suppose that inflation expectations are rational rather than adaptive. That is,
rather than set πte = πt−1 , assume instead that πte jumps immediately to its value
in the new steady state...
ANSWER: The following table presents the result of the rational expectations
assumption above, where we have assumed the monetary authority also sets
money at the steady state value immediately.
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Date(t)
πte
desired π
gmt
ut
0
10
10
12
5
1
4
4
6
5
2
4
4
6
5
3
4
4
6
5
4
4
4
6
5
5
4
4
6
5
6
4
4
6
5
Clearly the sacrifice ratio is 0 in this case.
NOTE: This exercise should not be interpreted as ’gullible’ expectations, with
π e = 4 no matter what, which will lead to a negative sacrifice ratio if the government sticks to its slow disinflation plan. Instead, the idea is that if agents
expect the government to set π at 4, the government could do it in one single
and painless step.
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