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Chapter 21: The Simplest Short-Run Macro Model
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Answers to Study Exercises
Question 1
a) Ca + Ia ( + Ga + NXa in an open economy with government)
b) C + I ( + G + NX in an open economy with government)
c) actual; desired
d) autonomous; induced; a; b
e) autonomous
f) C + I; desired aggregate expenditure; actual national income (GDP)
g) $47 billion; 0.92; YD = Y (no taxes) and consumption is the only induced part of desired
expenditure
Question 2
a) accumulate; reduce; fall
b) fall; increase; rise
c) investment
d) upward; rise; 45-degree
e) more; simple multiplier (× $10 billion)
f) larger; 1/(1-z)
Question 3
a) The average propensity to consume (APC) is equal to consumption divided by disposable
income. See the completed table below.
YD
C
APC
MPC
0
100
200
300
400
500
600
700
800
150
225
300
375
450
525
600
675
750
--2.25
1.50
1.25
1.125
1.05
1.00
0.96
0.94
--75/100 = 0.75
75/100 = 0.75
75/100 = 0.75
75/100 = 0.75
75/100 = 0.75
75/100 = 0.75
75/100 = 0.75
75/100 = 0.75
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b) See the table above. The marginal propensity to consume (MPC) is equal to the change in
consumption divided by the change in disposable income that brought it about. The MPC is shown
as the change from one row in the table to the next.
c) See the figure below. The 45-degree line shows where consumption equals disposable income.
This occurs only once along the consumption function, at the break-even level of income, Y*,
which equals $600. The APC equals one at Y*. The slope of the consumption function is the MPC,
which in this case is 0.75.
Question 4
a) Desired saving is equal to disposable income minus desired consumption. We can compute
desired saving (S) from the table in Question 3. The plotted desired saving function is shown
below. The slope of the function is the marginal propensity to save, ∆S/∆YD, which equals one
minus the marginal propensity to consume. In this case the marginal propensity to save is equal to
0.25.
YD
C
S
0
100
200
300
400
500
600
700
800
150
225
300
375
450
525
600
675
750
–150
–125
–100
–75
–50
–25
0
25
50
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b) To show this to be true, begin by noting that disposable income must either be consumed or
saved. Thus
YD = C + S
Now divide both sides by YD to get
1 = C/YD + S/YD
This shows that the average propensity to consume plus the average propensity to save must equal
one.
Question 5
a) National income accounting is based on actual expenditures. Desired expenditures are not
observed whereas actual expenditures are. It would be impossible to base accounting on the
unobserved concept.
b) A sudden decrease in consumer expenditure would probably result in a sudden unanticipated
increase in inventories, as people are no longer buying as many goods as before. Inventories are
part of investment, and so measured (actual) investment would rise. This would be an increase in
inventory investment.
c) See the figure on next page. Suppose the economy begins at point A, in equilibrium with
national output and income equal to YE. The sudden reduction in consumer expenditure is a
downward shift in the consumption function and thus in the AE function. The immediate effect is
for desired expenditure to fall to point B, even though production remains at YE . The distance
between points A and B reflects the amount by which inventories are accumulating as a result of
the reduction in consumer expenditure. This accumulation of inventories is an unintended or
unplanned increase in investment. The reduction in desired consumption expenditure will
eventually leads to a reduction in equilibrium national income, and the economy will adjust
toward point C.
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Question 6
a) When actual national income is Y1, desired aggregate expenditure is b. Actual output is a and
so desired aggregate expenditure exceeds actual output. Inventories are being depleted — this is
unplanned negative inventory investment.
b) The depletion of inventories eventually leads firms to increase the level of output so they can
replenish their inventories. The rise in output generates a rise in income and this induces an
increase in desired aggregate expenditure. We move up and to the right along the AE curve. But
as long as AE exceeds actual output, the depletion of inventories leads firms to increase output.
The economy eventually settles down where AE cuts the 45-degree line. At this point actual
output is exactly equal to the level of desired aggregate expenditure, and the level of inventories
is constant.
c) When actual national income is Y2, desired aggregate expenditure is d. Actual output is e and
so desired aggregate expenditure is less than actual output. Inventories are being accumulated —
this is unplanned positive inventory investment.
d) The unintended accumulation of inventories eventually leads firms to reduce the level of
output. The reduction in output reduces income and this induces a decrease in desired aggregate
expenditure. We move down and to the left along the AE curve. But as long as AE is less than
actual output, the accumulation of inventories leads firms to reduce output. The economy
eventually settles down where AE cuts the 45-degree line. At this point actual output is exactly
equal to the level of desired aggregate expenditure, and inventories are neither being depleted
nor accumulated.
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Question 7
a) See the table below.
Y
0
2000
4000
6000
8000
10000
C = 500 + .9Y
I
500 + (.9)0 = 500
500 + (.9)2000 = 2300
500 + (.9)4000 = 4100
500 + (.9)6000 = 5900
500 + (.9)8000 = 7700
500 + (.9)10000 = 9500
100
100
100
100
100
100
AE = C + I
600
2400
4200
6000
7800
9600
b) Equilibrium national income is that level of national income where actual income, Y, equals
desired aggregate expenditure, AE. Thus the equilibrium level of national income is 6000.
c) Desired saving is equal to S = Y - C. When national income is 6000, desired saving is S = 6000 –
5900 = 100. Note that desired investment is also 100.
To see why S must equal I at the equilibrium level of income, note that the equilibrium
condition is that actual GDP, Y, equals desired aggregate expenditure, AE. Thus,
Î
Y = AE
Î
Y=C+I
Y–C=I
Î
S=I
Thus stating the equilibrium condition as Y = AE is equivalent to stating it as S = I.
Question 8
a) The (desired) aggregate expenditure function shows the total desired expenditure at each level of
national income.
AE = C + I ⇒ AE = 1400 + 0.8Y + 400 ⇒ AE = 1800 + 0.8Y
b) Equilibrium requires that Y = AE (this is the equation for the 45-degree line). The equilibrium
level of national income is therefore the value of Y that solves the following equation:
Y = 1800 + 0.8Y ⇒ Y(1 – 0.8) = 1800 ⇒ Y = 1800/(0.2) ⇒ Y = 9000
c) When Y = 9000, consumption equals 1400 + (0.8 × 9000) = 8600. Saving therefore equals Y – C
which is 9000 – 8600 = 400. Investment equals 400.
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Question 9
a) See the figure below. When wealth is 10000, the AE function is
AE = 500 + 0.75Y + (.05)(10000) + 150 ⇒ AE = 1150 + 0.75Y
Using the equilibrium condition, Y = AE, the equilibrium level of national income is the level of Y
that solves the following equation:
Y = 1150 + 0.75Y ⇒ Y(1 – 0.75) = 1150 ⇒ Y = 1150/.25 ⇒ Y = 4600
b) The value of the simple multiplier is 1/(1 – MPC). In this economy, MPC = 0.75 and so the
value of the simple multiplier is 1/(1 – 0.75) = 1/(0.25) = 4.
c) If desired investment increases from 150 to 250, this is an increase in autonomous expenditure of
100. Given the multiplier of 4, the change in equilibrium national income will be 400. This is
shown in the diagram above as the AE function shifts to AE′ and equilibrium national income rises
to 5000.
d) Let’s suppose that I has already increased to 250 as in part (c). If wealth now increases from
10000 to 15000, the level of autonomous consumption increases by 5000(.05) = 250. Thus the new
AE function becomes
AE = 1500 + (.75)Y
As households increase their autonomous consumption by 250, the AE function shifts up by 250
and the equilibrium level of national income increases by 250×4 = 1000. Equilibrium national
income rises to 6000.
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Question 10
The discussion is best worked out in terms of the figure below, which shows the equilibrium level
of national income in terms of desired saving and desired investment. The initial equilibrium is
point A. The rise in desired saving shifts up the saving function to S′ but does not affect the
investment function. The accompanying reduction in desired consumption leads to an unintended
accumulation of inventories and thus leads firms to reduce the level of output. After the multiplier
has worked itself out, the level of equilibrium income has fallen but the equilibrium level of saving
is unchanged. Thus, the attempt to increase saving leads to a reduction in national income but no
increase in overall saving — the “paradox of thrift”.