-
*-
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:
ΔProfit
= ΔRevenue –
ΔCost
=
(
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
ΔProfit = 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.
4.
A
Cobb
–
Douglas production
function has the form
F
(
K,L
)
=
AK
α
L
1
–
α
. The text
showed that the
parameter
α
gives capital’s
share of income. So if capital earns
one
-fourth of total income, then
?
=
0.25. Hence,
F
(
K,L
)
=
AK
0.25
L
< br>0.75
.
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.
7.
Consumption, 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 t
he 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.
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).
3.
c.
Labor’s share of the output is given by
the marginal product of labor times the quantity
of labor, o
r
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.
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 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.
4.
a.
A
Cobb
–
Douglas production
function has the form
Y
=
AK
α
L
1
–
α
. 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:
W/P
=
MPL
= (1
–
α
)
Y/L
.
R/P
=
MPK
=
α
Y/K
.
Rewriting this:
(
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
=
A
K
0.3
L
0.7
.
Y
2
=
< br>AK
0.3
(1.1
L
)
0.7
.
Note that we multiplied
L
by 1.1 to reflect the
10-percent increase in the labor force.
To calculate the percentage change in
output, divide
Y
2
by
Y
1
:
Y
2
=
Y
1
AK
0.3
(
1.
1
L
)
AK
0
.3
L
0.7
0.7
< br>0.7
=
(
1.1
)
=
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
–
0.7
L
0.7
.
(
R/P
)
2
= 0
.3
AK
–
0.7
(1.1
L
)
0.7
.
The rental price increases by the ratio
(
R
/
P
)
(
R
/
P
)
2
1
=
0.3
AK
-
0.7
(<
/p>
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 (<
/p>
W/P
)
2
,
multiply
L
by 1.1
to reflect the 10-percent increase in the labor
force:
(
W/P
)
1
= (1
–
0.
3)
AK
0.3
L
–
0.3
.
(
< br>W/P
)
2
= (1
–
0.3)
AK
0.3
(1.1
L
)
–
0.3
.
To calculate the percentage change in
the real wage, divide
(
W/P
)
2
by (
W/P
)
1
< br>:
*-
(
W
/
P
)
2
(
W
/
P
)
1
=
(
1
-
0.3
)
AK
=
(
1.1
)
-
0.3<
/p>
(
1
-
0.3<
/p>
)
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
Y
1
= <
/p>
AK
0.3
L
0
.7
.
Y
2
=
A
(1.1
K
)
0.3
L
0.7
.
Therefore, we have: <
/p>
0.7
Y
2
A<
/p>
(
1.1
K
)<
/p>
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
< br>/
P
)
(
R
/
P
)
2
1
=
0.3
A
(
1.1
K
)
-
0.7
-
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.)
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
=
< br>0.3
(
1.1
A
)
K
-
0.7
L
0.7
0.3
AK
-
0.7
L
0.7<
/p>
5.
Labor income is defined as
=
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<
/p>
-
0.3
<
/p>
=
1.1
W
WL
?
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
< br>/
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>
P
B
, and this is
equal
to ($$/hour worked)/($$/haircut).
d.
If workers can move freely between
being farmers and being barbers, then they 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