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 out-
put 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 tech-
nology 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 ( PL) and the price of the good produced( ). 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 isP × MPL. The cost of hiring the additional
unit of labor is the wageW. 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 exam-
ple, 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 function has the form F(K,L) = AK L . The text
showed that the parameter α gives capital’s share of income. (Since income equals out-
put for the overall economy, it is also capital’s share of output.) So if capital earns one-
fourth of total income, then a = 0.25. Hence, F(K,L) = AK.2L0.7.
5. Consumption depends positively on disposable income—the amount of income after all
taxes have been paid. The higher disposable income is, the greater consumption is.
The quantity of investment goods demanded depends negatively on the real inter-
est 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.
11 12 Answers to Textbook Questions and Problems
6. Government purchases are a measure of the dollar value of goods and services pur-
chased 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 activi-
ties 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 econo-
my’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 multi-
plied by the marginal propensity to consumeM ( PC). The higher theMPC is, the greater is
the negative effect of the tax increase on consumption. Because output is fixed by the fac-
tors 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 mar-
ginal 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.
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
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.
2. 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, and output less than doubles, 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 inputs of capital and labor more than doubles output, then the pro-
duction 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 very fast at it.
α 1 – α
3. 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 = αY/K. Chapter 3 National Income: Where It Comes From and Where It Goes 13
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 = αY/K.
(W/P)L = MPL × L = (1 – α)Y.
(R/P)K = MPK × K = αY.
Note that the terms (W/P)L and (R/P)K are the wage bill and total return to capi-
tal, respectively. Given that the value of α = 0.3, then the above formulas indicate
that labor receives 70 percent of total output (or income), which is (1 – 0.3), 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:
α 1 – α
Y = AK L .
Let Y 1qual the initial value of output and Y equal 2inal output. We know that
α = 0.3. We also know that labor L increases by 10 percent:
Y1= AK L ..3 0.7
Y = AK (1.1L) . 0.7
Note that we multiplied L by 1.1 to reflect the 10-percent increase in the labor
To calculate the percentage change in output, divide Y by Y 2 1
Y 2 AK 0.(1.1L) 0.7
= 0.3 0.7
Y 1 AK L
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 = αAK α – L1 – .
We know that α = 0.3. We also know that labor ( L) increases by 10 percent. Let
(R/P) 1qual the initial value of the rental price of capital, and ( R/P) e2ual the
final rental price of capital after the labor force increases by 10 percent. To find
(R/P) , multiply L by 1.1 to reflect the 10-percent increase in the labor force:
(R/P) = 0.3AK –0.L .7
– 0.7 0.7
(R/P) 2 = 0.3AK (1.1L) .
The rental price increases by the ratio
(R/P) 0.3AK –0.(1.1L) 0.7
2 = –0.70.7
(R/P) 1 0.3AK L
So the rental price increases by 6.9 percent. 14 Answers to Textbook Questions and Problems
To determine how the increase in the labor force affects the real wage, con-
sider the formula for the real wage W/P:
α – α
W/P = MPL = (1 – α)AK L .
We know that α = 0.3. We also know that labor ( L) increases by 10 percent. Let
(W/P) e1ual the initial value of the real wage and ( W/P) equal th2 final value of
the real wage. To find (W/P) , mu2tiply L by 1.1 to reflect the 10-percent increase
in the labor force:
(W/P) =1(1 – 0.3)AK L .
(W/P) = (1 – 0.3)AK (1.1L).3 – 0.3
To calculate the percentage change in the real wage, divide ( W/P) b2
(W/P) :1 (W/P) )(.−( A1K 03 . L 03
2 = 03 .03
() /P 1 )(1−0 AK L
= (1.1) – 0.3
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 AK L ..30.7
Y 2 A(1.1K) L .
Therefore, we have:
Y 2 = A(1.1K) L
Y AK L.3 0.7
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) 0.3A(1.1K) –0.L 0.7
2 = –0.7 0.7
(R/P) 1 0.3AK L
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) 2 0.7A(1.1K) L
= 0.3 –0.3
(W/P) 1 0.7AK L
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 Chapter 3 National Income: Where It Comes From and Where It Goes 15
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:
Y 2 = (1.1A)K L
Y 0.3 0.7
1 AK L
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) 0.3(1.1A)K –0.L 0.7
2 = –0.70.7
(R/P)1 0.3AK L
(W/P) 0.3 –0.3
2 = 0.7(1.1A)K L
(W/P)1 0.7AK L.3 –0.3
4. Labor income is defined as
Labor’s share of income is defined as
ÊWL ˆ WL
Á ˜/Y. =
Ë P ¯ PY
If this ratio is about constant at, say, a value of 0.7, then it must be the case that 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
5. a. According to the neoclassical theory, technical progress that increases the margin-
al product of farmers causes their real wage to rise.
b. The real wage for farmers is measured as units of farm output per worker. The
real wage is W/P F and this is equal to ($/worker)/($/unit of farm output).
c. If the marginal productivity of barbers is unchanged, then their real wage is
d. The real wage for barbers is measured as haircuts per worker. The real wage is
W/P , and this is equal to ($/worker)/($/haircut).
e. If workers can move freely between being farmers and being barbers, then they
must be paid the same wage W in each sector.
f. 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 MPL = P MPL , given that the
F F H B
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 put it 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.
g. Both groups benefit from technological progress in farming. 16 Answers to Textbook Questions and Problems
6. a. The marginal product of labor MPL is found by differentiating the production
function with respect to labor:
MPL = dY
= 1 K H L1/3–2/.
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 pro-
duction function with respect to human capital: