-
Answers to Textbook Questions and
Problems
CHAPTER
3National
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
ΔP
rofit = 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
= .
Hence,
F
(
K,L
)
=
Consumption depends positively on
disposable income
—
. 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 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.
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.
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
–<
/p>
α
.
In
this problem,
α
is and
A
is 1. Substitute in the
given values for labor and land to find the
marginal product of labor is and
marginal product of capital (land) is . 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,
or
50.
d.
The new
level of output is .
e.
The new wage is . The new rental price
of land is .
f.
Labor now receives .
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
= , 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 = . We also
know that labor
L
increases by 10 percent:
Y
1
=
Y
2
= .
Note that we multiplied
L
by to reflect the 10-percent increase
in the labor force.
To
calculate the percentage change in output, divide
Y
2
by
Y
1
:
0.3
Y
2
AK<
/p>
(
1.1
L
)<
/p>
=
Y
1
AK
0.3
L
0.7
0
.7
=
(
1.1
)
0.7
=
1.069.
That is,
output increases by 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
= . 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 to reflect the 10-percent
increase in the labor force:
(
R/P
)
1
=
–
(
R/P
)
2
=
–
.
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 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
=
. 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 to
reflect the 10-percent increase in the labor
force:
(
W/P
)
1
= (1
–
–
.
(
W/P
)
2
= (1
–
–
.
To calculate the percentage change in
the real wage, divide
(
W/P
)
2
by (
W/P
)
1
< br>:
(
W
/
P
)
=
(
1
-
0.3
)
AK
(
1.1
L
)
(
W
/
P
)
(
1
-
0.3
)
AK
L
=
(
1.1
)
0.3
2
1
-
0.3
-
0.3
0.3
-
0.3
=
0.972
That is,
the real wage falls by percent.
c.
We can use the same logic
as in part (b) to set
Y
1
=
Y
2
=
A
Therefore, we have:
< br>0.7
Y
2
A
< br>(
1.1
K
)
< br>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
< 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
)<
/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 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>
/
P
)
2
1
=
0.7
A
(
1.1
K
)
L
-
0.3
0.7
A
K
0.3
L
-
0.3
0.3
0.3
=
(
1.1
)
=
1.029
Hence, real
wages increase by 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
=
Y
1
AK<
/p>
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
)
(
< br>R
/
P
)
2
1
=
0
.3
(
1.1
A
)
K
-
0.7
L
0.7
0.3
AK
-
0.7
L
0.7
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<
/p>
L
-
0.3
=
1.1
W
WL
?<
/p>
L
=
P
P
Labor’s share of income is
defined as
?
WL
?
WL
?
?
P
÷
÷
/
Y
=
PY
?
è
For
example, if this ratio is about constant at a
value of , then the value of
W
/
P
=
*
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. O
therwise,
labor’s share would deviate from . 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>
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
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
=
K
1/3
L
1/3
H
-
2/3
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
labor share:
(
1
K
1/3
H
1/3
L
-
2/3
)
L
3
Lab
or Share
=
1/3
1/3
1/3
K
H
L
1
=
3
We can use the
same logic to find the human capital
share:
(
< br>1
K
1/3
L
< br>1/3
H
-
2/3
)
H
3
Human
Capital Share
=
K
1/
3
H
1/3
L
1/3
1
=
3
-
-
-
-
-
-
-
-
-
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