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1、 ...wd... Answers to Textbook Questions and Problems CHAPTER3National Income: Where It Comes Fromand Where It Goes Questions for Review 1. The factors of production and the production technology determine the amount of output an economy can p

2、roduce. 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 im

3、provement 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 co

4、mpares 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 doll

5、ars 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 (

6、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

7、. 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 margina

8、l 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, an

9、d 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

10、 ′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

11、a= 0.25. Hence, F(K,L) = AK0.25L0.75. 5.Consumption depends positively on disposable income—i.e. the amount of income after alltaxes have been paid. Higher disposable income means higher consumption. The quantity of investment goods demanded depends negatively on the real interestrate. For an inve

12、stment 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 ratemakes 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

13、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

14、. 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 dete

15、rmine 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

16、 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

17、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.

18、 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 t

19、o 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 wo

20、rkers, 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 rise

21、s 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 p

22、roduction 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 dou

23、bles 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 = 1000.51000.5 = 100. b. According to the text, the f

24、ormulas 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 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 (l

25、and) 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

26、 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 t

27、he 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 fun

28、ction 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 t

29、o 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

30、 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

31、hire labor until its marginal productequals the real wage, and hire capital until its marginal product equals the realrental rate. Using these facts and the above marginal products for theCobb–Douglas production function, we find: W/P = MPL = (1 – α)Y/L. R/P = MPK = αY/K. Rewriting this: (W/P)L

32、= 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 capital,respectively. Given that the value of α = 0.3, then the above formulas indicatethat labor receives 70 percent of total output (or income) andcapital receives 30 percent of

33、 total output (or income). b.To determine what happens to total output when the labor force increases by 10percent, 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 thatα = 0.3. We also know tha

34、t labor L increases by 10 percent: Y1 = AK0.3L0.7. Y2 = AK0.3(1.1L)0.7. Note that we multiplied L by 1.1 to reflect the 10-percent increase in the laborforce. To calculate the percentage change in output, divide Y2 by Y1: That is, output increases by 6.9 percent. To determine how the increas

35、e 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 = 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 (

36、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 = 0.3AK–0.7L0.7. (R/P)2 = 0.3AK–0.7(1.1L)0.7. The rental price increases by the ratio So the rental pric

37、e 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)AKαL–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 wa

38、ge, and let (W/P)2 equal the final value of the real wage. To find (W/P)2, multiply L by 1.1 to reflect the 10-percent increase in the labor force: (W/P)1 = (1 – 0.3)AK0.3L–0.3. (W/P)2 = (1 – 0.3)AK0.3(1.1L)–0.3. To calculate the percentage change in the real wage, divide (W/P)2 by (W/P)1: Th

39、at is, the real wage falls by 2.8 percent. c.We can use the same logic as in part (b) to set Y1 = AK0.3L0.7. Y2 = A(1.1K)0.3L0.7. Therefore, we have: This equation shows that output increases by about 3 percent. Notice that α < 0.5 means that proportional increases to capital will increase out

40、put 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: The real rental price of capital falls by 6.5 percent because there are diminishingreturns to capital; that is, when capital increases, its margi

41、nal product falls. Finally, the change in the real wage is: Hence, real wages increase by 2.9 percent because the added capital increases themarginal productivity of the existing workers. (Notice that the wage and outputhave both increased by the same amount, leaving the labor share unchanged—afea

42、ture of Cobb–Douglas technologies.) d.Using the same formula, we find that the change in output is: This equation shows that output increases by 10 percent. Similarly, the rental price of capital and the real wage also increase by 10 percent: 5.Labor income is defined as Labor’s share of income

43、 is defined as 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 realwages. Otherwise, labor’s share

44、would deviate from 0.7. Thus, the first fact (a constantlabor share) implies the second fact (the trend in real wages closely tracks the trend inlabor 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 un

45、it 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 marginalproduct of farmers causes their real wage to rise. The real wage for farmers is measured as units of farm output

46、 per hour worked. Thereal wage is W/PF, and this is equal to ($/hour worked)/($/unit of farm output). c.If the marginal productivity of barbers is unchanged, then their real wage isunchanged.The real wage for barbers is measured as haircuts per hour worked. The real wage isW/PB, and this is equal t

47、o ($/hour worked)/($/haircut). d.If workers can move freely between being farmers and being barbers, then theymust 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 offarm goods is greater than the real wage in terms of haircuts

48、, then the price ofhaircuts must have risen relative to the price of farm goods. We know that W/P =MPL so that W = P′MPL. This means that PFMPLF = PHMPLB, given that thenominal wages are the same. Since the marginal product of labor for barbers hasnot changed and the marginal product of labor for fa

49、rmers has risen, the price of ahaircut must have risen relative to the price of the farm output. If we expressthis ingrowthrate terms, then the growth of the farm price + the growth of the marginalproduct of the farm labor = the growth of the haircut price. f.The farmers and the barbers are equally

50、 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 g

51、oods, 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 th

52、en equalized the nominal wage. 7.a.The marginal product of labor (MPL)is found by differentiating the productionfunction with respect to labor: An increase in human capital will increase the marginal product of labor because more human capital makes all the existing labor more productive. b. Th

53、e marginal product of human capital (MPH)is found by differentiating the production function with respect to human capital: An increase in human capital will decrease the marginal product of human capitalbecause there are diminishing returns. c.The labor share of output is the proportion of output

54、 that goes to labor. The totalamount 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. Thisquantity is divided by the total amount of output to compute the labor share: We can use the same logic to find

55、 the human capital share: so labor gets one-third of the output, and human capital gets one-third of the output.Since workers own their human capital (we hope!), it will appear that laborgets two-thirds of output. d. The ratio of the skilled wage to the unskilled wage is: Notice that the ratio is

56、 always greater than 1 because skilled workers get paidmore than unskilled workers. Also, when H increases this ratio falls because thediminishing returns to human capital lower its return, while at the same timeincreasing the marginal product of unskilled workers. e.If more colleges provide schola

57、rships, it will increase H, and it does lead to a more egalitariansociety. The policy lowers the returns to education, decreasing the gap betweenthe wages of more and less educated workers. More importantly, the policy evenraises the absolute wage of unskilled workers because their marginal productr

58、ises when the number of skilled workers rises. 8.The effect of a government tax increase of $100 billion on (a) public saving, (b) privatesaving, and (c) national saving can be analyzed by using the following relationships: National Saving= [Private Saving] + [Public Saving] = [Y – T – C(Y – T)]

59、 + [T – G] = Y – C(Y – T) – G. a.Public Saving—The tax increase causes a 1-for-1 increase in public saving. T increases by $100 billion and, therefore, public saving increases by $100 billion. b.Private Saving—The increase in taxes decreases disposable income, Y – T, by $100 billion. Since the m

60、arginal propensity to consume (MPC) is 0.6, consumption falls by 0.6 ′ $100 billion, or $60 billion. Hence, ΔPrivate Saving = –$100b – 0.6 (–$100b) = –$40b. Private saving falls $40 billion. c.National Saving—Because national saving is the sum of private and public saving, we can conclude that

61、 the $100 billion tax increase leads to a $60 billion increase in national saving. Another way to see this is by using the third equation for national saving expressed above, that national saving equals Y – C(Y – T) – G. The $100 billion tax increase reduces disposable income and causes consumption

62、 to fall by $60 billion. Since neither G nor Y changes, national saving thus rises by $60 billion. d.Investment—To determine the effect of the tax increase on investment, recall the national accounts identity: Y = C(Y – T) + I(r) + G. Rearranging, we find Y – C(Y – T) – G = I(r). The left s

63、ide of this equation is national saving, so the equation just says that national saving equals investment. Since national saving increases by $60 billion, investment must also increase by $60 billion. How does this increase in investment take place? We know that investment depends on the real inter

64、est rate. For investment to rise, the real interest rate must fall. Figure 3-1 illustrates saving and investment as a function of the real interest rate. The tax increase causes national saving to rise, so the supply curve for loanable funds shifts to the right. The equilibrium real interest rate

65、falls, and investment rises. 9. If consumers increase the amount that they consume today, then private saving and, therefore, national saving will fall. We know this from the definition of national saving: National Saving = [Private Saving] + [Public Saving] = [Y – T – C(Y – T)] + [T – G]. An

66、 increase in consumption decreases private saving, so national saving falls. Figure 3-2 illustrates saving and investment as a function of the real interest rate. If national saving decreases, the supply curve for loanable funds shifts to the left, thereby raising the real interest rate and reducing investment. 10. a. Private saving is the amount of disposable income, Y – T, that is not consumed: Sprivate = Y – T – C = 8,000 – 2,000 – [1,000 + (2/3)(8,000 – 2,000)] = 1,000. Public savin