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Answers to
Textbook Questions and Problems
CHAPTER
3
National Income: Where It
Comes From and Where It Goes
Questions for Review
1.
The factors of production
and the production technology determine the amount
of
output an economy can produce. The
factors of production are the inputs used to
produce goods and services: the most
important factors are capital and labor. The
production technology determines how
much output can be produced from any given
amounts of these inputs. An increase in
one of the factors of production or an
improvement in technology leads to an
increase in the economy
’
s
output.
2.
When
a firm decides how much of a factor of production
to hire or demand, it
considers how
this decision affects profits. For example, hiring
an extra unit of
labor increases output
and therefore increases revenue; the firm compares
this
additional revenue to the
additional cost from the higher wage bill. The
additional revenue the firm receives
depends on the marginal product of labor
(
MPL
) and the
price of the good produced
(
P
). An additional unit of
labor produces
MPL
units of
additional output, which sells for
P
dollars per unit. Therefore, the
additional revenue to the firm is
P
?
MPL
. The cost of hiring the
additional unit
of labor is the wage
W
. Thus, this hiring
decision has the following effect on
profits:
Δ
Profit
=
Δ
Revenue
–
Δ
Cost
=
(
P
?
MPL
)
–
W
.
If the
additional revenue,
P
?
MPL
,
exceeds the cost (
W
) of
hiring the additional
unit of labor,
then profit increases. The firm will hire labor
until it is no
longer profitable to do
so
—
that is, until the
MPL
falls to the point where
the
change in profit is zero. In the
equation above, the firm hires labor until
Δ
Profit = 0, which is when
(
P
?
MPL
) =
W
.
This condition can be rewritten as:
MPL
=
W/P
.
Therefore, a
competitive profit-maximizing firm hires labor
until the marginal
product of labor
equals the real wage. The same logic applies to
the firm
’
s
decision regarding how much capital to
hire: the firm will hire capital until the
marginal product of capital equals the
real rental price.
3.
A production function has constant
returns to scale if an equal percentage
increase in all factors of production
causes an increase in output of the same
percentage. For example, if a firm
increases its use of capital and labor by 50
percent, and output increases by 50
percent, then the production function has
constant returns to scale.
If the production function
has constant returns to scale, then total income
(or
equivalently, total output) in an
economy of competitive profit-maximizing firms
is divided between the return to labor,
MPL
?
L
, and the return to
capital,
MPK
?
K
. That is, under constant
returns to scale, economic profit is zero.
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α
1
–
α
4.
A Cobb
–
Douglas
production function has the form
F
(
K,L
)
=
AK
L
. The text
showed
that the parameter
α
gives
capital
’
s share of income.
So if capital earns one-
0.25
0.75
fourth of total income, then
= 0.25. Hence,
F
(
K,L
)
=
AK
L
.
5.
Consumption
depends positively on disposable
income
—
i.e. the amount of
income
after all taxes have been paid.
Higher disposable income means higher consumption.
The quantity of
investment goods demanded depends negatively on
the real
interest rate. For an
investment to be profitable, its return must be
greater than
its cost. Because the real
interest rate measures the cost of funds, a higher
real
interest rate makes it more costly
to invest, so the demand for investment goods
falls.
6.
Government purchases are a measure of
the value of goods and services purchased
directly by the government. For
example, the government buys missiles and tanks,
builds roads, and provides services
such as air traffic control. All of these
activities are part of GDP. Transfer
payments are government payments to
individuals that are not in exchange
for goods or services. They are the opposite
of taxes: taxes reduce household
disposable income, whereas transfer payments
increase it. Examples of transfer
payments include Social Security payments to the
elderly, unemployment insurance, and
veterans
’
benefits.
ption, investment, and
government purchases determine demand for the
economy
’
s output, whereas
the factors of production and the production
function determine
the supply of
output. The real interest rate adjusts to ensure
that the demand for
the
economy
’
s goods equals the
supply. At the equilibrium interest rate, the
demand for goods and services equals
the supply.
8.
When the government increases taxes,
disposable income falls, and therefore
consumption falls as well. The decrease
in consumption equals the amount that
taxes increase multiplied by the
marginal propensity to consume
(
MPC
). The higher
the
MPC
is, the
greater is the negative effect of the tax increase
on consumption.
Because output is fixed
by the factors of production and the production
technology,
and government purchases
have not changed, the decrease in consumption must
be
offset by an increase in investment.
For investment to rise, the real interest
rate must fall. Therefore, a tax
increase leads to a decrease in consumption, an
increase in investment, and a fall in
the real interest rate.
Problems and Applications
1.
a.
According
to the neoclassical theory of distribution, the
real wage equals the
marginal product
of labor. Because of diminishing returns to labor,
an increase
in the labor force causes
the marginal product of labor to fall. Hence, the
real wage falls.
Given a
Cobb
–
Douglas production
function, the increase in the labor force
will increase the marginal product of
capital and will increase the real rental
price of capital. With more workers,
the capital will be used more intensively
and will be more productive.
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b.
The real
rental price equals the marginal product of
capital. If an earthquake
destroys some
of the capital stock (yet miraculously does not
kill anyone and
lower the labor force),
the marginal product of capital rises and, hence,
the
real rental price rises.
Given a
Cobb
–
Douglas production
function, the decrease in the capital
stock will decrease the marginal
product of labor and will decrease the real
wage. With less capital, each worker
becomes less productive.
c.
If a technological advance improves the
production function, this is likely to
increase the marginal products of both
capital and labor. Hence, the real wage
and the real rental price both
increase.
d.
High inflation
that doubles the nominal wage and the price level
will have no
impact on the real wage.
Similarly, high inflation that doubles the nominal
rental price of capital and the price
level will have no impact on the real
rental price of capital.
2.
a.
To find the
amount of output produced, substitute the given
values for labor
and land into the
production function:
Y
= 100
0.5
100
0.5
= 100.
b.
According to the text,
the formulas for the marginal product of labor and
the
marginal product of capital (land)
are:
MPL
= (1
–
α
)
AK
α
L
–α
.
MPK
=
α
AK
α–
1
L
1
–
α
.
In this
problem,
α
is 0.5 and
A
is 1. Substitute in the
given values for labor
and land to find
the marginal product of labor is 0.5 and marginal
product of
capital (land) is 0.5. We
know that the real wage equals the marginal
product
of labor and the real rental
price of land equals the marginal product of
capital (land).
c.
Labor
’
s share of
the output is given by the marginal product of
labor times
the quantity of labor, or
50.
d.
The new level of output is 70.71.
e.
The new wage is 0.71. The new rental
price of land is 0.35.
f.
Labor now receives 35.36.
3.
A production
function has decreasing returns to scale if an
equal percentage
increase in all
factors of production leads to a smaller
percentage increase in
output. For
example, if we double the amounts of capital and
labor output
increases by less than
double, then the production function has
decreasing returns
to scale. This may
happen if there is a fixed factor such as land in
the
production function, and this fixed
factor becomes scarce as the economy grows
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larger.
A
production function has increasing returns to
scale if an equal percentage
increase
in all factors of production leads to a larger
percentage increase in
output. For
example, if doubling the amount of capital and
labor increases the
output by more than
double, then the production function has
increasing returns to
scale. This may
happen if specialization of labor becomes greater
as the
population grows. For example,
if only one worker builds a car, then it takes him
a long time because he has to learn
many different skills, and he must constantly
change tasks and tools. But if many
workers build a car, then each one can
specialize in a particular task and
become more productive.
α
1
–
α
4.
a.
A
Cobb
–
Douglas production
function has the form
Y
=
AK
L
. The text
showed
that the marginal products for
the Cobb
–
Douglas production
function are:
MPL
= (1
–
α
)
Y/L
.
MPK
=
α
Y/K
.
Competitive profit-maximizing firms
hire labor until its marginal product
equals the real wage, and hire capital
until its marginal product equals the
real rental rate. Using these facts and
the above marginal products for the
Cobb
–
Douglas
production function, we find:
Rewriting this:
W/P
=
MPL
= (1
–
α
)
Y/L
.
R/P
=
MPK
=
α
Y/K
.
(
W/P
)
L
=
MPL
?
L
=
(1
–
α
)
Y
.
(
R/P
)
K
=
MPK
?
K
=
α
Y
.
Note that the terms
(
W/P
)
L
and
(
R/P
)
K
are the wage bill and total return to
capital, respectively. Given that the
value of
α
= 0.3, then the
above
formulas indicate that labor
receives 70 percent of total output (or income)
and capital receives 30 percent of
total output (or income).
b.
To determine
what happens to total output when the labor force
increases by 10
percent, consider the
formula for the Cobb
–
Douglas
production function:
Y
=
AK
α
L
1
–α
.
Let
Y
1
equal the initial value of output and
Y
2
equal final
output. We know that
α
=
0.3. We also know that labor
L
increases by 10 percent:
Y
1
=
AK
0.3
L
0.7
.
Y
2
=
A
K
0.3
(1.1
L
< br>)
0.7
.
Note that we
multiplied
L
by 1.1 to
reflect the 10-percent increase in the
labor force.
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To calculate the percentage
change in output, divide
Y
2
by
Y
1
:
0.
3
Y
2
AK
(
1.1
L
)
=
Y
1
AK
0.
3
L
0.7
0.7
=
(
1.1
)
0.7
=
1.069.
That is,
output increases by 6.9 percent.
To determine how the increase in the
labor force affects the rental price
of
capital, consider the formula for the real rental
price of capital
R/P
:
R/P
=
MPK
=
α
AK
α
–
1
L
1
–
α
.
We
know that
α
= 0.3. We also
know that labor (
L
)
increases by 10 percent. Let
(
R/P
)
1
equal the
initial value of the rental price of capital, and
let
(
R/P
)
2
equal the final rental price of capital
after the labor force increases by 10
percent. To find (
R/P
)
2
, multiply
L
by 1.1 to reflect the
10-percent increase
in the labor force:
(
R/P
)
1
= 0.3
AK
(
< br>R/P
)
2
=
0.3
AK
–
0.7
0.7
L
.
(1.1
L
)
.
0.7
–
0.7
The rental price
increases by the ratio
(
R
/
P
)
(
R
/
P
)
2
1
=
0.3
AK
-
0.7
(
1.1
L
)
0.3
AK
-
0.7
L
0.7
0.7
0.
7
=
(
1.1
)
=
1.069
So the rental
price increases by 6.9 percent. To determine how
the increase in
the labor force
affects the
real wage, consider the formula for the real wage
W/P
:
W/P
=
MPL
= (1
–
α
)
AK
α
L
–α
.
We know that
α
= 0.3. We also know that
labor (
L
) increases by 10
percent. Let
(
W/P
)
1
equal the initial value
of the real wage, and let
(
W/P
)
2
equal the final
value of the real wage.
To find (
W/P
)
2
, multiply
L
by 1.1
to reflect the 10-
percent increase in
the labor force:
0.3
–
0.3
(
W/P
)
1
= (1
–
0.3)
AK
L
.
(
W/P
)
2
= (1
–
0.3)
AK
(1.1
L
)
0.3
–
0.3
.
To calculate the percentage
change in the real wage, divide
(
W/P
)
2
by (
W/P
)
1
< br>:
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(
W
/
P
)
2<
/p>
(
W
/
P
)
1
=
(
1
-
0.3
)
AK
=
(
1.1
)
-
0.3
(
1
-
0.3
)
AK
0.3
(
1.1
L
)
0.3
-
0.3
-
0.3
L
=
0.972
That is,
the real wage falls by 2.8 percent.
c.
We can use the same logic as in part
(b) to set
Therefore, we
have:
Y
1
=
AK
0.3
L
0.7
.
Y
2
=
A
(1.1
K
)<
/p>
0.3
L
0.7
.
0.7
Y
2
A
(
1.1
K
)
L
=
Y
1
AK
0.3
L
0.7
0.3
=
(
1.1
)
0.3
=
1.029
This equation
shows that output increases by about 3 percent.
Notice that
α
<
0.5 means that proportional increases
to capital will increase output by less
than the same proportional increase to
labor.
Again using the same
logic as in part (b) for the change in the real
rental
price of capital:
(
R
/
P
)
(
R
/
P
)
< br>2
1
=
0.3
< br>A
(
1.1
K
< br>)
-
0.7
-
< br>0.7
L
0.7
0.3
AK
-
0.7
L
0.7
=
(
1.1
)
=
0.935
The real rental
price of capital falls by 6.5 percent because
there are
diminishing returns to
capital; that is, when capital increases, its
marginal
product falls.
Finally, the change in the real wage
is:
(
W
/
P
)
(
W
/
P
)
2
1
=
0.7
A
(
1.1
K
)
L
-
0.3
0.7
AK
0.3
L
-
0.3
0.3
0.3
=
(
1.1
)
=
1.029
Hence, real
wages increase by 2.9 percent because the added
capital increases
the marginal
productivity of the existing workers. (Notice that
the wage and
output have both increased
by the same amount, leaving the labor share
unchanged
—
a
feature of Cobb
–
Douglas
technologies.)
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d.
Using the
same formula, we find that the change in output
is:
0.3
0.7
Y
2
(
1.1
A
)
K
L
=
< br>Y
1
AK
0.3
L
0.7
=
1.1
This equation
shows that output increases by 10 percent.
Similarly, the rental
price of capital
and the real wage also increase by 10 percent:
(
R
/
P
)
(
R
/
P
)
2
1
=
0.3
(
1.1
A
)
K
-
0.7
L
0.7
0.3
AK
-
0.7
L
0.7
=
1.1
(
W
/
P
)
2
=
0.7
(
1.1
A
)
K
0.3
L
-
0.3
(
W
/
P
)
1
0.7
AK
0.3
L
-
0.3
5.
Labor income
is defined as
W
WL
<
/p>
?
L
=
P
P
Labor
’
s share of
income is defined as
?
WL
?
WL
?
÷
/
Y
=
?
P
÷
PY
è
?
For example,
if this ratio is about constant at a value of 0.7,
then the value of
W
/
P
=
0.7
*
Y
/
L
. This means that the real wage is
roughly proportional to labor
productivity. Hence, any trend in labor
productivity must be matched by an equal
trend in real wages. Otherwise,
labor
’
s share would deviate
from 0.7. Thus, the
first fact (a
constant labor share) implies the second fact (the
trend in real
wages closely tracks the
trend in labor productivity).
6.
a.
Nominal
wages are measured as dollars per hour worked.
Prices are measured as
dollars per unit
produced (either a haircut or a unit of farm
output). Marginal
productivity is
measured as units of output produced per hour
worked.
b.
According to the neoclassical theory,
technical progress that increases the
marginal product of farmers causes
their real wage to rise. The real wage for
farmers is measured as units of farm
output per hour worked. The real wage is
W
/
P
F
, and this is equal to ($$/hour worked)/($$/unit of
farm output).
c.
If the marginal
productivity of barbers is unchanged, then their
real wage is
unchanged. The real wage
for barbers is measured as haircuts per hour
worked.
The real wage is
W
/
P
B
, and
this is equal to ($$/hour worked)/($$/haircut).
d.
If workers can move freely between
being farmers and being barbers, then they
=
1.1
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must be paid the same wage
W
in each sector.
e.
If the nominal wage
W
is the same in both
sectors, but the real wage in terms
of
farm goods is greater than the real wage in terms
of haircuts, then the
price of haircuts
must have risen relative to the price of farm
goods. We know
that
W
/
P
=
MPL
so that
W
=
P
?
MPL
.
This means that
P
F
MPL
F
=
P
< br>H
MPL
B
, given
that
the nominal wages are the same.
Since the marginal product of labor for barbers
has not changed and the marginal
product of labor for farmers has risen, the
price of a haircut must have risen
relative to the price of the farm output. If
we express this in growth rate terms,
then the growth of the farm price + the
growth of the marginal product of the
farm labor = the growth of the haircut
price.
f.
The farmers and the
barbers are equally well off after the
technological
progress in farming,
given
the assumption that labor is
freely mobile between the two sectors and both
types of people consume the same basket
of goods. Given that the nominal wage
ends up equal for each type of worker
and that they pay the same prices for
final goods, they are equally well off
in terms of what they can buy with their
nominal income. The real wage is a
measure of how many units of output are
produced per worker. Technological
progress in farming increased the units of
farm output produced per hour worked.
Movement of labor between sectors then
equalized the nominal wage.
7.
a.
The
marginal product of labor
(
MPL
)
is found by differentiating the
production
function with respect to
labor:
dY
MPL
=
dL
1
1/3
1/3
-
2/3
=
K
H
L
3
An increase in
human capital will increase the marginal product
of labor
because more human capital
makes all the existing labor more productive.
b.
The marginal product of human capital
(
MPH)
is found by
differentiating the
production function
with respect to human capital:
dY
MPH
=
dH
1
1/3
1
/3
-
2/3
=
K
L
H
3
An
increase in human capital will decrease the
marginal product of human
capital
because there are diminishing returns.
c.
The labor
share of output is the proportion of output that
goes to labor. The
total amount of
output that goes to labor is the real wage (which,
under
perfect competition, equals the
marginal product of labor) times the quantity
of labor. This quantity is divided by
the total amount of output to compute the
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