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Solution key: Homework 5
Economics 101
Chapters 10
Multipliers: Assume the consumption function is C = 100 + .75(Y-T), G = 100, T = 100, I =100.
Suppose that the Republicans in Congress persuade Clinton to accept a tax cut to stimulate the
economy. But suppose they are also committed to a balanced budget, so that taxes and government
spending both are cut from 100 to 50. Do these two opposing fiscal policy actions completely cancel
each other out? Explain the economic intuition for why or why not?
You can plug numbers in and solve for output, to find that output decreases 50. But to see
the principle, let’s find a more general solution. The consumption function is of the form
C = a + b(Y-T), where b is the marginal propensity to consume. Plug this into the
equilibrium condition:
Y = C(Y- T ) + I + G :
Y = a + b(Y- T ) + I + G
Solve for Y: Y = (1/(1-b)) G - (b/(1-b)) T + (1/(1-b)) I + a/(1-b)
If the change in taxes and G are equal (T = G), then the combined multiplier is:
1/(1-b) - b/(1-b) = (1-b)/(1-b) = 1. This is known as the “balanced budget multiplier.”
The effect of the tax cut does not fully offset a cut in government purchases, because
taxes only work indirectly, raising consumption by the marginal propensity to consume,
whereas government purchases have an additional direct effect on expenditure.
Chapter 11
1) Deriving IS-LM: Assume the following model of the economy, with the price level fixed at 1.0:
C = 0.8(Y-T)
T = 1000
I = 800- 20r
(M/P)d = 0.4Y - 40r
G = 1000
(M/P)s= 1,200
a) Write a formula for the IS curve, expressing r as a function of Y. Graph it. Label intercept and slope .
r
a) Y = C + I + G
Y = 0.8(Y-1000) + 800 - 20r + 1000
IS: r = 50 - 0.01 Y.
50
Slope = -0.01
IS
Y
Interpretation of the slope:
On the IS curve higher interest rate is associated with lower income (because of it’s impact
on investment). For every point rise in interest rate income will fall by 100 units. This
happens because one point rise in r leads to 20 units fall in investment leading to a reduction
in equilibrium income of 100 (20*multiplier, where multiplier=1/(1-.8) = 5).
b) Write a formula for the LM curve expressing r as a function of Y. Graph it. Label intercept and slope.
r
(M/P)d = (M/P) s
0.4Y - 40r = 1200
LM: r = -30 + 0.01Y
LM
Slope = 0.01
-30
Y
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Interpretation of the slope:
On the LM curve higher income is associated with higher interest rate. This is because higher
income leads to higher demand for money which bids up the equilibrium interest rate. Here
every unit of increment in income will lead to 0.01 point rise in interest rate.
c)
What are the short-run equilibrium values of Y, r, Y-T, C, I, private saving, public saving and national
saving? Check by ensuring that C + I + G =Y and national saving equals investment.
Combining IS and LM: 50 - 0.01 Y = -30 + 0.01Y so Y = 4000
So: r = 50 - 0.01(4000) = 10. (using the IS equation again; you could also use the
LM equation).
Y-T = 4000 - 1000 = 3000. Private saving = Y - T - C = 4000 - 1000 - 2400 = 600
C = 0.8(3000) = 2400
Public saving = T - G = 1000 - 1000 = 0
I = 800 - 20(10) = 600
National saving = Private + Public = 600
d)
Assume that G increases by 200. By how much will Y increase in the short-run equilibrium?
Illustrate this with a graph of the IS-LM curves. Explain why the rise in government purchases does
not have the full multiplier effect on output here?
d) Y increases by 500.
r
r
LM1
LM
IS2
LM2
IS
IS1
4000 4500
4000 4250
Y
Y
The rise in r crowds out some investment
The fall in r raises investment.
so, expenditure rises less than the full multiplier amount.
Comment on (d)
An increase in government expenditure shifts out IS curve by the amount of multiplier. However
an increase in demand for money resulting from higher income will lead to an increase in interest
rate leading to a reduction in investment. Hence the final equilibrium income demanded in lower
than the figure suggested the multiplier effect alone.
e)
Assume that G is back at its original level of 1000, but (M/P)s increases by 200. By how much will Y
increase in the short-run equilibrium? Illustrate this with a graph of the IS-LM curves. What is
happening to investment here?
Y increases by 250.
2) True-False: Say if the following statements are True or False. If they are false write the
correct statement
a) A recession is defined as two periods of negative growth rate of the nominal GDP
False. Technically speaking a recession is defined as at least two consecutive
periods of negative growth rate in real GDP. In everyday usage of the term,
however, any sustained period of slow down in economic activity is called
recession. The period is measured from the peak of a business cycle to the
subsequent trough.
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b) Velocity of Money has increased much in the past thirty years because people use
more checks.
False. Checks are money themselves. The reason why people use less money and
the velocity of money has increased is the introduction of credit cards.
c) The IS curve is downward sloping because an increase in Taxes implies a
decrease in output.
False. The IS curve is downward sloping because an increase in the interest rate
implies an decrease in output.
d) An expansionary fiscal policy increases the interest rates.
True, an expansionary fiscal policy, shifts the IS right, and in the new equilibrium
both output and the interest rate increase.
e) The IS-LM model predicts that an increase in G and an increase in money supply
at the same time will certainly decrease income.
False. Such an increase shifts both curves to the right which increases income.
3)
a)
Explain in words why the IS-LM theory says the aggregate demand curve slopes downward.
A down sloping AD curve means a rise in price is associated with a fall in output. A rise in
price level affects the money market: the real money supply (M/P) falls for a given nominal
money supply, so the interest rate must rise to lower real money demand and maintain
equilibrium in the money market. This rise in the interest rate then affects the goods market:
investment falls, which lowers total planned expenditure, which lowers total production and
output.
b) What will happen to the slopes of the IS and AD curves if investment is less responsive to the
interest rate?
If investment is less responsive to the interest rate, the last step above will be smaller. So the
rise in price will lower output less: AD is steeper. The IS curve describes how much a rise in
the interest rate lowers investment demand and hence output. If investment is less responsive
to the interest rate, the fall in output will be smaller, so the IS curve will be steeper.
c)
What will happen to the slopes of the LM and AD curves if money demand is less responsive to
the interest rate?
If money demand is less responsive to the interest rate, then the story in part (a) will require a
bigger rise in the interest rate to lower money demand to equal money supply. So the price
rise will raise interest rates more, and lower investment and output more. So the AD curve is
flatter. The LM curve describes how much interest rates must rise when income rises, to keep
money demand constant. If money demand is less responsive to the interest rate, the interest
rate will have to rise more. So the LM curve is steeper.
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LM(steep)
M/p
r2
LM(flat)
r2
r1
L(r,Y2)
L(r,Y1)
L(r,Y1) L(r,Y2)
Y1
Y2
M/P1 M/P2
Y
LM(steep)
LM(flat)
r1
r2
r2
L(r,Y2)
L(r,Y1)
L(r,Y1) L(r,Y2)
IS
Y1
Y2flat
Y2steep
Y
P
P1
P2
Flat AD
associated with
steep LM
Y
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4) IS-LM Policy Analysis: Japan is considering how it might stimulate its currently sluggish
economy, but it is not sure whether it should use expansionary monetary policy or
expansionary fiscal policy. Consider two policy alternatives: (1) a rise in money supply and
(2) a cut in taxes -- which are scaled so that they raise equilibrium output the same amount.
Using the IS-LM curves, contrast these two alternative policies in terms of their implications
for the equilibrium interest rate and their likely effects on consumption and investment.
A rise in money supply raises output by lowering the interest rate. While a tax cut also
raises output, it raises the interest rate. The graph would look like the left panel below. So the
two policies have opposite effects on the interest rate. This means the expansionary monetary
policy would raise investment while the fiscal policy would crowd it out. Both policies would
raise consumption by raising overall income, but the fiscal policy would have a bigger effect
on consumption, since it gives an additional reason why disposable income is rising.
r
LM
r
LM1
LM2
IS2
IS
IS1
5) More on Short-run/Long run: Suppose that Congress is going to cut government purchases on a permanent basis.
Use the IS-LM/AS-AD tools to analyze the implications in the short run and in the long run. (Assume the following.
Prices are completely fixed in the short run and completely flexible in the long run. Taxes remain constant.
Consumption is a function of disposable income, with a constant marginal propensity to consume. Investment is a
function only of interest rate. Our starting point in full equilibrium output.)
a) Use the IS-LM and AS-AD graphs to show the short run and long run equilibria following this policy. Be
sure to label the axes, curves, use arrows to show shifts in curves, and mark the equilibrium points: 1 for
equilibrium, 2 for the short run equilibrium, and 3 for the long run equilibrium.
b) What happens to the following variables in the short run (rise, fall, no change, can’t tell): output, interest rate,
investment, consumption, and real money demand.
c) What happens in the long run? For each of the variables listed in (b) above, state if it returns in the long run
to its initial equilibrium value, if it is higher in the long run than its initial level, or if it is lower.
Ms/P1
Ms/P2
r
LM1
1
1
r1
r2
r3
2
3
LM2
2
3
L(r,Y1)
IS1
L(r,Y2)
IS2
M/P
Y2 Ybar
Y
1
P1
2
SRAS
P2
AD1
AD2
3
Y2 Ybar
5
Y
a) The cut in government purchases shifts the IS and AD curves left. In the long run, prices fall.
This raises the real money supply and shifts the LM curve right, enough to restore output to its
original level.
b) Short run: Y falls, r falls, I rises, C falls, real money demand falls (note that quantity money
demanded M/P does not changed)
c) Long run: Y returns to original level, r lower, I higher, C is back to its original level, real
money demand goes back to the original level (note again that quantity money demanded, M/P is
higher). This is exactly as in the Neoclassical Model.
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