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) = AKαL1–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.
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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.
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b. According to the text, the formulas for the marginal product of labor and the marginal product of
capital (land) are:
MPL = (1 – α)AKαL–α.
MPK = αAKα–1L1–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).
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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 .
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 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 = AKαL1–α. The text showed that the marginal
products for the Cobb–Douglas production function are:
MPL = (1 – α)Y/L.
*
MPK = αY/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 = αY/K.
Rewriting this:
(W/P)L = MPL (R/P)K = MPK
L = (1 – α)Y. K = αY.
—
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 = AKαL1–α.
Let Y1 equal the initial value of output and Y2 equal final output. We know that α = . We also know that labor L increases by 10 percent:
Y1 =
Y2 = .
Note that we multiplied L by to reflect the 10-percent increase in the labor force. To calculate the percentage change in output, divide Y2 by Y1:
Y2=Y1 !
AK0.3(1.1L)AK0.3L0.70.70.7=(1.1)
=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 = αAKα–1L1–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)
$
21=0.3AK-0.7(1.1L)0.3AK-0.7L0.70.70.7=(1.1)
=1.069So 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)AKαL–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 (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:
(W/P)=(1-0.3)AK(1.1L)(W/P)(1-0.3)AKL=(1.1)0.321-0.3-0.30.3-0.3
=0.972That is, the real wage falls by percent.
c. We can use the same logic as in part (b) to set Y1 = Y2 = A
Therefore, we have:
0.7Y2A(1.1K)L=Y1AK0.3L0.70.3
=(1.1)
0.3
=1.029This 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)2(R/P)1
=0.3A(1.1K)-0.7-0.7L0.70.3AK-0.7L0.7
=(1.1)
=0.935The 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)
21=0.7A(1.1K)L-0.30.7AK0.3L-0.30.30.3=(1.1)
=1.029Hence, real wages increase by percent because the added capital increases the marginal
曼昆 宏观经济经济学第九版 英文原版答案3
