PROBLEM SET 3
Problems for Chapter 3
1.
Suppose the consumption function in the U.S. is represented by the following
equation:
C = 200 + .5 YD, where YD = Y – T and T = 200.
a. What is the level of consumption in this economy if YD = 0? Briefly explain how
individuals “pay for” this consumption when YD = 0.
b. Given the above parameters, calculate the level of consumption if Y = 1200.
Suppose Y increases to 1300. What happens to the level of YD as Y increases to
1300 (i.e. calculate the change in YD)? What happens to the level of consumption
when Y rises to 1300 (i.e. calculate the change in consumption)? Using the
change in YD and the change in consumption you calculated, what is the marginal
propensity to consume (MPC) for this economy? Is this the same as the
parameter of the consumption function that represents the marginal propensity to
consume?
c. Write out the saving function for this economy. (HINT: S = YD – C. Substitute for
C from the information above and simplify the right side of the equation). What
is the level of saving when YD = 0? Explain how and why this occurs. What is
the marginal propensity to save for this economy? How do you know?
2.
Suppose the U.S. economy is represented by the following equations:
Z=C+I+G
C = 400 + .5 YD
YD = Y – T
G = 600
T = 400
I = 200
a. What is the marginal propensity to consume for this economy? What is the
marginal propensity to save?
b. Write out the equation that indicates how aggregate demand (Z) is a function of
income (Y) and the remaining autonomous expenditures. What will be the level
of aggregate demand if Y = 0? What does this level of demand represent?
Furthermore, given you equation, what will happen to the level of aggregate
demand (Z) as Y increases by $1? What does this number represent?
c. Based on your answer in part (b), calculate the level of demand (Z) for the
following levels of income: Y = 1600, Y = 1800, Y = 2200, and Y = 2400. Now
compare the level of demand that you calculated at each level of income with the
corresponding level of income. Is the economy in equilibrium at any of these
levels of income? Explain. Solve the model for equilibrium income
algebraically. Do you get the same answer for equilibrium?
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d. Use your calculations in part (c) to answer this question. When Y = 1600,
compare the levels of income and demand. How will firms respond to this
situation? Briefly explain. When Y = 2400, compare the levels of income and
demand. How will firms respond to this situation? Briefly explain.
3.
Suppose the U.S. economy is represented by the following equations.
a.
Z=C+I+G
C = 300 + .5 YD
YD = Y – T
G = 1000
T = 400
I = 200
Given the above variables, calculate the equilibrium level of output (Y). Using
the graph below, illustrate the equilibrium level of output for the economy.
Aggregate Demand (Z)
45o
4.
Y
b.
Now assume that consumer confidence increases causing an increase in
autonomous consumption from 300 to 400. What is the new equilibrium level of
output? How much does income change as a result of this event? What is the
multiplier for this economy?
c.
Graphically illustrate (in the above graph) the effects of the change in autonomous
consumption on the aggregate demand line and equilibrium Y. Clearly indicate in
your graph the initial and final equilibrium levels of output.
d.
Briefly explain why this increase in output is greater than the initial increase in
autonomous consumption.
Calculate the multiplier for each of the following values of the marginal propensity to
consume (MPC = c1): c1 = .4, c1 = .5, c1 = .6, c1 = .8, and c1 = .9. Briefly explain
what happens to the multiplier as the marginal propensity to consume increases.
Briefly explain the meaning of this intuitively.
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5.
For each of the following events, explain what effect the event has on autonomous
expenditures and equilibrium income/output:
a. Decrease in business confidence which results in a reduction in non-residential
investment.
b. Increase in consumer confidence which results in an increase in residential
investment.
c. Congress passes a budget which requires a reduction in government spending.
6.
7.
Suppose the U.S. economy is represented by the following equations.
Z=C+I+G
C = 20 + .8 YD
YD = Y – T
G = 60
T = 20
I = 20
a.
Calculate the equilibrium level of output/income.
b.
Suppose lump-sum taxes (T) decrease by 5. What is the new level of equilibrium
income?
c.
Ignore part (b). Suppose government spending (G) increases by 10. What is the
new level of equilibrium income?
d.
Ignore parts (b) and (c). Suppose that the government decides that the optimal
level of equilibrium income is 520. In what direction and by how much would the
government have to change government spending to achieve this level of income?
For each of the following, calculate the change in equilibrium output/income:
a. Government spending decreases by 100 and the marginal propensity to consume
is .5.
b. Investment spending increases by 20 and the marginal propensity to consume is
.9.
c. Taxes increase by 70 and the marginal propensity to consume is .6.
8.
Suppose equilibrium income decreases by 50 and the marginal propensity to consume
is .8. Assuming a change in government spending is responsible for the decrease in
equilibrium income, in what direction and by how much must government spending
have changed?
TEXTBOOK p. 60: #1, 2, 3
4
SELECTED ANSWERS
1.
a.
200. They draw down their savings or borrow.
b.
C = 800. C increases by 50 to 850. Disposable income increased by 100, and C
increases by 50; the marginal propensity to consume is .5.
c.
S = -200 + .5 YD. S = -200. This represents the dissaving that would occur to
pay for the consumption that occurs when disposable income is 0. The marginal propensity to
save is .5, which is 1 – MPC.
2.
a.
The MPC is .5. the marginal propensity to save is .5, which is 1 – MPC.
b.
Z = 1200 + .5(Y – 400) = 1000 + .5Y. Z is 1000 when Y = 0. This is the level of
autonomous expenditures. If Y increases by 1, Z rises by .5. This is the MPC.
c.
When Y is 1600, Z is 1800; when Y is 1800, Z is 1900; when Y is 2000, Z is
2000; when Y is 2200, Z is 2100; when Y is 2400, Z is 2200. Equilibrium occurs when Y is
2000 since Y = Z at that level of Y.
d.
When Y = 1600, Z > Y. Firms set production equal to demand. Since demand
exceeds Y, firms will increase Y over time. When Y is 2400, Y > Z. Production exceeds
demand, and firms will cut back production.
3.
a..
Y = 1500 + .5(Y – 400) = 1300 + .5Y. Solve for Y = 1300/.5 = 2600.
Z
ZZ
1400
1300
2600
2800
Y
b.
Plug in the numbers and you get: Y = 1400 + .5Y; Y = 2800. Output increases by
200. The multiplier is 2: every dollar change in autonomous C causes a 2 dollar
change in Y.
c.
See graph.
d.
As demand increases by 100, firms respond by increasing production. The higher
production represents higher income. As income increases, consumption rises
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again, causing firms to increase Y yet again. This is the multiplier process at
work.
4.
The multipliers will be (1/(1 – c1)): 1.7, 2, 2.5, 5, 10. the multiplier increases as the
MPC increases because a higher MPC means that more of any increase in income is
spent in each round of the multiplier.
5.
a. Autonomous expenditure and Y both fall.
b. Autonomous expenditure and Y both rise.
c. Autonomous expenditure and Y both fall.
6.
a.
Y = 420.
b.
∆Y = 5 x -.8 x -5 = +20.  new Y = 420 + 20 = 440.
c.
∆Y = 5 x +10 = +50.  new Y = 420 + 50 = 470.
d.
Desired change in Y (=∆Y) = new Y – old Y + 520 – 420 = +100
+100 = 5 x ∆G  ∆G = 100/5 = +20.
7.
a.
∆Y = (1/1-.5)) x (-100)  ∆Y = -200 (equilibrium Y decreases).
b.
∆Y = (1/1-.9)) x (+20)  ∆Y = +200 (equilibrium Y increases).
c.
∆Y = (1/1-.6)) x (-.6 x +70)  ∆Y = -105 (equilibrium Y decreases).
8.
-50 = (1/1-.5)) x ∆G  ∆G = -25 (government spending decreases by 25 if equilibrium
Y decreases by 50).
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ANSWERS TO TEXTBOOK PROBLEMS ON PAGE 60:
1.
a.
b.
c.
d.
e.
f.
g.
2.
a.
b.
c.
3.
a.
b.
c.
True.
False. Government spending including transfers was 31% of GDP in 2006
(see p. 45).
False. The propensity to consume is usually less than one because people
in general spend only part of additional income earned on consumption
goods (and save the rest).
True.
False. It states that aggregate demand equals output.
False. The increase in output is one times the multiplier.
False. An increase in the (marginal) propensity to consume leads to an
increase in output.
Y = 160 + 0.6(Y – 100) + 150 + 150
Y = 1000
YD = Y – T = 1000 – 100 = 900
C = 160 + 0.6(900) = 700
Y = 1000 (from 2a above).
Total Demand at Equilibrium = C (at equilibrium Y of 1000) + I + G
= 700 + 150 + 150 = 1000 = Y
Y = 160 + 0.6(Y – 100) + 150 + 110
Y = 900
Total Demand at Equilibrium = 160 + 0.6(900-100) + 150 + 110
= 900 = Y
Private (or household) S = (Y – T) – C
= (900–100) – [160+0.6(900-100)] = 160
Public S = T – G = 100 – 110 = -10
Private S + Public S = 160 – 10 = 150 = I.
The sum of public and private saving equal investment because this is the
alternate equilibrium condition which can be derived directly from
Y = C+I+G.