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Answers to Textbook Questions and Problems
CHAPTER 3 National Income: Where It Conies 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 x MPL. The cost of hiring the additional unit of labor is the wage W. Thus, this hiring decision has the following effect on profits:
AProfit = ^Revenue - ACost
= (Px MPL) 一 W.
If the additional revenue? P x \\4PL、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― hat is. until the MPL falls to the point where the change in profit is zero. In the equation above, the firm hires labor until AProfit = 0, which is when (P x 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 x L、and the return to capital. MPK x K? That is, under constant returns to scale, economic profit is zero. 4. A Cobb-Douglas production function has the form F(K.L) = AKaL{^? The text showed that the parameter a
gives capita?s share of income. So if capital earns one?fourth of total income, then a = 0.25. Hence, F(K.L) = A^25!075. 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.
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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 veterans1 benefits ? 7. Consumption, investment、and go\\-ermnent 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
l? 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:
55
Y= 100°100° = 100.
b. According to the text, the formulas for the marginal product of labor and the marginal product of capital
(land) are:
MPL = (l
MPK = aAK4'LZ?
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In this problem, a 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 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 = AKaL^? The text showed that the marginal
products for the Cobb-Douglas production function are:
MPL = (i -a)Y/L. MPK = aY/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 一 a)\? R/P = MPK = aY/K ?
Rewriting this:
(W/P)L = MPLxL = (\\ -a)r. (R/P)K = MPKxK = aY.
Note that the terms (\\V/P)L and (R/P)K are the wage bill and total return to capital, respectively? Given that the value of a = 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 = AKU EX
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