Answers to Questions for Chapter 3
1. Why is the demand for goods and services, denoted by Z in chapter 3,
always equal to Gross Domestic Product? Does this mean that supply and
demand are always equal?
In the National Income and Product Accounts (NIPA), the Flow of
demand Z is defined to include, as part of its second component
(Investment), any change in the aggregate Stock of inventories.
Aggregate Sales (the actual aggregate flow demand for goods and
services) is not always equal to the aggregate flow supply of goods and
services. The overall stock of inventories changes to absorb the
difference. In particular, the stock of inventories increases when
aggregate supply (Y) exceeds actual aggregate demand (Z) or sales; it
decreases when total final sales (Z) exceeds aggregate supply (Y). Only
if actual Investment is defined to include the actual change in
inventories (whether planned to occur or not) will total demand equal
GDP by definition.
2. Planned Investment is the sum of: (1) planned expenditure by firms on
currently produced capital goods and services (including software); (2)
intentional changes to inventories of finished and unfinished goods; (3)
expenditure by households on new houses and apartment buildings. The
purchase of an existing house is not included if it was produced a prior
period, because it is not part of this period’s GDP. The total planned
Investment is denoted I , where the over-bar indicates that this type of
spending is “exogenous” to the model. That just means it is not explained
within the model. What else should therefore have an over-bar?
The only part of expenditure that is explained in this model is
Consumption and not all of consumption is explained! Therefore,
almost everything is unexplained or “exogenous”. In particular, c0
should have an over-bar along with G , X , M . Likewise, the only
parameter of the model is the marginal propensity to consume, which is
also just given. To emphasize this, it too could have an over-bar, c1 . So
why did just I get the special designation? Because the unplanned
change in inventories, denoted inv , is included in actual investment
without the over-bar, the following definition is implied: I  I  inv .
All unplanned changes in inventories (whether they be unsold
consumption goods, unsold capital goods including unsold houses, or
goods produced in expectation of being sold to federal, state, and local
governments or to foreigners, but not sold) are tossed into one category
inv . For this reason the distinction between I and I has a special
significance and justifies the notation, even though other categories of
demand in addition to planned investment are exogenous.
3. The constant in the consumption function c0 is interpreted as the level of
consumption spending by people with zero incomes. Is a more reasonable
and more general interpretation implied?
Yes. The constant can be interpreted as that part of consumption
expenditure which is explained by wealth (a stock) rather than income
(a flow). When wealth increases, households at any level of income can
either liquidate assets in order to increase consumption or borrow
against wealth (using it as collateral for loans) in order to increase
consumption. Decreases in wealth can have the opposite effect,
resulting in lower consumption for people affected by the wealth
reduction, regardless of their income. Changes in wealth have effects on
consumption of people at all levels of income, not just those with no
income.
4. When the identity Z  C  I  G is replaced by equation (3.5), what is
the required interpretation of I  I ? It follows that the uses of Z prior to
(3.5) are potentially confusing. What symbol would have been a better
choice?
See above: the difference between actual investment and planned
investment is the unexpected change in inventories. A better choice
would have been the symbol for actual GDP. In short, Y  C  I  G
as opposed to Z  C  I  G .
5. The marginal propensity to consume is the simplest possible example of
“additional spending per dollar increase in total output” : z1  Z1 / Y .
Express the multiplier in terms of this ratio.
The multiplier for any ratio of the change in endogenous spending to the
change in total income is:
1
1
. In chapter 3,

1  z1 1  (Z1 / Y )
z1  c1 , although even this simplest case is made a bit more realistic in a
question at the end of the chapter where z1  c1 (1  t1 ) . A positive tax
rate reduces the effect of any change in income on total expenditure and
therefore reduces the multiplier. Subsequent models incorporate
further effects into z1  Z1 / Y by taking account of the effect of
changes in GDP on interest rates and therefore on those types of
expenditures which are sensitive to interest rates.
6. Rewrite equation (3.9) as Z  Z 0  z1Y . On the right side, where do we
put all types of “exogenous” demand and what are some examples? What is
a general name for anything in the second term?
All types of expenditure not explained by current income are included
in Z 0 . In this model, that includes consumption explained by wealth,
capital spending by firms (including purchases of new residences by
households), government spending, and expenditure by foreigners net of
our imports).
A general name for the second term is “endogenous” demand, which
here means demand explained by current income.
7. If output Y and expenditure Z are equal before and after some change in
any component of Z 0 , what is the relationship between Y and Z 0 .
What special case would yield the result Y  Z 0 ?
Subtract one equilibrium equation (for the change) from the other
(after the change):
Y new  Z 0new  z1Y new

 Y old  Z 0old  z1Y old

Y  Z 0  z1Y
In the simplest case, z1  c1 . Therefore:
Y (1  c1 )  Z 0
Y
1

 1 if
Z 0 1  c1
0  c1  1
Y  Z 0 if their ratio is unity, i.e. if z1  c1  0 .
8. Justify the geometric series form of the multiplier algebraically. As long
as the re-spending ratio z1 is a positive fraction, the multiplier is
approximated by a finite number of terms of a geometric series:
1
 1  z1  z12  z13  z14
1  z1
1  (1  z1 )(1  z1  z12  z13  z14 )
1  (1  z1 )  ( z1  z12 )  ( z12  z13 )  ( z13  z14 )  ( z14  z15 )
1  1  z15
9. In Figure 3-3, production and demand are both measured on the vertical
axis. Does that mean they are always equal? Explain.
There are two lines in the diagram. The 45º line identifies production
on the vertical axis with income on the horizontal axis, as in the national
income and product accounts where production includes any unplanned
change in inventories. The expenditure line as a function of income
includes only planned expenditures. Therefore, a vertical line cuts both
lines at distinct points when the aggregate level of inventories is
increasing or decreasing at an unexpected rate. When there is no such
unexpected change in overall inventories, aggregate supply and demand
are in balance.
The aggregate nature of macroeconomic models allows for offsetting
change in inventories across individual firms and industries when the
structure of the economy is changing.
10. Copy or redraw Figure 3-2 and place directly below it a new diagram
where savings- plus-taxes is measured on the vertical axis and total income
on the horizontal axis as in Figure 3-2. What is the intercept and what is the
slope of this new savings-plus-taxes line? In the new diagram, draw a
horizontal line cutting the vertical axis at I  G . Where does this line cross
the savings plus taxes line? What do you conclude about the verbal
description of equilibrium in this chapter’s model?
C, Z , Y
Expenditure
function
Consumption
function
Y
Savings
function
O
I G X M
Y
In the top half, both the consumption function and the parallel, higher
expenditure line are drawn. The intersection of the consumption
function with the 45º line occurs at a level of income on the horizontal
axis where saving is zero. Therefore, the saving line in the bottom part
cuts the horizontal axis at zero at that level of income. The total
expenditure line (parallel to the consumption function) in the top
diagram cuts the 45º line at the equilibrium level of income. In the
bottom diagram, the same point corresponds to an intersection between
the savings line and a horizontal line that measures all sources of
exogenous demand except c0 . At this level of income, savings is
positive.
The two diagrams show the same thing because of the way in which the
flow of savings is defined. The level of income where savings is zero is,
in the top diagram:
Y  C  c0  c1Y  Y 
c0
1  c1
In the bottom diagram:
S  Y  C  Y  c0  c1Y
S  0  Y  C  Y  c0  c1Y  0  Y 
c0
1  c1
The level of income where the aggregate level of inventories is not
changing unexpectedly is, in the top diagram:
Y C  I G X M
 c0  c1Y  I  G  X  M  Z 0  c1Y  Y 
Z0
1  c1
Z 0  c0  I  G  X  M
In the bottom diagram:
S  Y  C  Y  c0  c1Y  c0  (1  c1 )Y  c0  s1Y
S  I G X M
 c0  s1Y  I  G  X  M
 Y
c0  I  G  X  M Z 0
Z

 0
s1
s1 1  c1
The pictures and the equations tell the same story in different ways. In
this model, the multiplier is the reciprocal of the marginal propensity to
save, defined as one minus the marginal propensity to consume: two
ways of thinking about the multiplier. The condition for equilibrium
(no unexpected change in overall inventories) can also be restated as
equality between planned saving (unspent income overall) and the sum
of planned investment plus all other sources of exogenous demand,
namely, government spending and net exports. Note that because an
excess of imports over exports means that foreigners are lending us
purchasing, the trade deficit is just another name for foreign savings.
Therefore:
S  I G X M
S  (M  X )  I  G
S domestic  S foreign  I  G
In general, savings from all sources finances domestic exogenous
demand, here stated as the sum of private planned investment and
government spending.