曼昆宏观经济经济学第九版英文原版答案9.

1、Answers to Textbook Questions and Problems CHAPTER 9 Economic Growth II: Technology, Empirics, and Policy Questions for Review 1. In the Solow model, we find that only tech no logical progress can affect the steady-state rate of growth in in come per worker. Growth in the capital stock (through high

2、 sav ing) has no effect on the steadystate growth rate of in come per worker; n either does populati on growth. But tech no logical progress can lead to susta ined growth. 2. In the steady state, output per pers on in the Solow model grows at the rate of tech no logical progressg. Capital per person

3、 also grows at rate g. Note that this implies that output and capital per effective worker are con sta nt in steady state. In the U.S. data, output and capital per worker have both grow n at about 2 perce nt per year for the past half-ce ntury. 3. To decide whether an economy has more or less capita

4、l than the Golden Rule, we need to compare the marginal product of capital net of depreciation ( MPK - S ) with thegrowth rate of total output ( n + g). The growth rate of GDP is readily available. Estimat ing the n et marg inal product of capital requires a little more work but, as shown in the tex

5、t, can be backed out of available data on the capital stock relative to GDP, the total amount o f depreciation relative to GDP, and capital s share in GDP. 4. Econo mic policy can in flue nce the sav ing rate by either in creas ing public sav ing or provid ing incen tives to stimulate private sav in

6、g. Public sav ing is the differe nce betwee n gover nment reve nue and government spending. If spending exceeds revenue, the government runs a budget deficit, which is negative saving. Policies that decrease the deficit (such as reductions in government purchases or in creases in taxes) in crease pu

7、blic sav ing, whereas policies that in crease the deficit decrease sav ing. A variety of gover nment policies affect private sav ing. The decisi on by a household to save may depe nd on the rate of retur n; the greater the return to sav ing, the more attractive sav ing becomes. Tax incen tives such

8、as tax-exempt retireme nt accou nts for in dividuals and in vestme nt tax credits for corporati ons in crease the rate of retur n and en courage private sav ing. 5. The legal system is an example of an institutional difference between countries that might explain differe nces in in come per pers on.

9、 Coun tries that have adopted the En glish style com mon law system tend to have better developed capital markets, and this leads to more rapid growth because it is easier for bus in esses to obta in financing. The quality of gover nment is also importa nt. Coun tries with more gover nment corrupti

10、on tend to have lower levels of in come per pers on. 6. En doge nous growth theories attempt to expla in the rate of tech no logical progress by explai ning the decisi ons that determ ine the creati on of kno wledge through research and developme nt. By con trast, the Solow model simply took this ra

11、te as exoge no us. In the Solow model, the sav ing rate affects growth temporarily, but diminishing returns to capital eventually force the economy to approach a steady state in which growth depe nds only on exoge nous tech no logical progress. By con trast, many en doge nous growth models in esse n

12、ce assume that there are con sta nt (rather tha n diminishing) retur ns to capital, interpreted to include knowledge. Hence, changes in the saving rate can lead to persistent growth. Problems and Applications 1. a. In the Solow model with tech no logical progress, y is defi ned as output per effecti

13、ve worker, and k is defi ned as capital per effective worker. The n umber of effective workers is defi ned as L E (or LE), where L is the number of workers, and E measures the efficiency of each worker. To find output per effective worker y, divide total output by the number of effective workers: 1

14、1 Y = K2(LE)2 LE - LE Y 1 1 1 K2L2E2 LE - LE Y = K2 LE - - 1 1 L2E2 1 Y ? K ?2 = LE 1LE7 1 y = k2 b. To solve for the steady-state value of y as a fun cti on of s, n, g, and S , we beg in wtthe equati on for the cha nge in the capital stock in the steady state: Ak = sf(k) S + + g)k = 0. The producti

15、on function y = . k can also be rewritten as y2 = k. Plugging this production function into the equation for the change in the capital stock, we find that in the steady state: 2 sy S + + g)y = 0. Solv ing this, we find the steady-state value of y: y* = s/( S r+ g). c. The questi on provides us with

16、the follow ing in formati on about each cou ntry: Atla ntis: s = 0.28Xa nadu: s = 0.10 n = 0.01n = 0.04 g = 0.02g = 0.02 S = 0.04S = 0.04 * Using the equation for y that we derived in part (a), we can calculate the steady-state values of y for each cou ntry. Developed cou ntry:y* = 0.28/(0.04 + 0.01

17、 + 0.02) = 4 Less-developed cou ntry:y = 0.10/(0.04 + 0.04 + 0.02) = 1 2. a. In the steady state, capital per effective worker is constant, and this leads to a constant level of output per effective worker. Give n that the growth rate of output per effective worker is zero, this means the growth rat

18、e of output is equal to the growth rate of effective workers ( LE). We know labor grows at the rate of population growth n and the efficiency of labor ( E) grows at rate g. Therefore, output grows at rate n+g . Given output grows at rate n+g and labor grows at rate n, output per worker must grow at

19、rate g. This follows from the rule that the growth rate of Y/L is equal to the growth rate of Y minus the growth rate of L. b. First find the output per effective worker production function by dividing both sides of the producti on fun cti on by the n umber of effective workers LE: 2 Y LE =K(LE) =LE

20、 1 2 2 Y K3L3E3 LE LE Y K LE 1 1 L3E3 Y ?K 7 ?十 LE 出E? y= k3 To solve for capital per effective worker, we start with the steady state con diti on: Ak = sf(k) 8 +=)+ g)k = 0. Now substitute in the give n parameter values and solve for capital per effective worker (k): 0.24fc = (0.03 4- D.02 4- OMk A

21、5 = 4 = 3. Substitute the value for k back into the per effective worker production function to find output per effective worker is equal to 2. The marg inal product of capital is give n by Substitute the value for capital per effective worker to find the marginal product of capital is equal to 1/12

22、. c. According to the Golden Rule, the marginal product of capital is equal to (8+ n + g) or 0.06. In the curre nt steady state, the marg inal product of capital is equal to 1/12 or 0.083. Therefore, we have less capital per effective worker in comparison to the Golden Rule. As the level of capital

23、per effective worker rises, the marginal product of capital will fall un til it is equal to 0.06. To in crease capital per effective worker, there must be an in crease in the sav ing rate. d. During the transition to the Golden Rule steady state, the growth rate of output per worker will in crease.

24、In the steady state, output per worker grows at rate g. The in crease in the sav ing rate will in crease output per effective worker, and this will in crease output per effective worker. I n the new steady state, output per effective worker is con sta nt at a new higher level, and output per worker

25、is growing at rate g. During the transition, the growth rate of output per worker jumps up, and then transitions back down to rate g. 3. To solve this problem, it is useful to establish what we know about the U.S. economy: a. ? A Cobb -Douglas production function has the form y = k , where a is capi

26、taliscrihaes of 0 3 The questi on tells us thata = 0.3, so we know that the producti on fiynctik n is ? In the steady state, we know that the growth rate of output equals 3 perce nt, so we know that ( n + g) = 0.03. ? The depreciati on rate 8 = 0.04. ? The capital -output ratio K/Y = 2.5. Because k/

27、y = K/(LE)/Y/(LE) = K/Y, we also know that k/y = 2.5. (That is, the capital utput ratio is the same in terms of effective workers as it is in levels.) Chapter 9 Econo mic Growth II: Tech no logy, Empirics, and Policy73 a. Beg in with the steady-state con diti on, sy = ( S + + g)k. Rewrit ing this eq

28、uati on leads to a formula for saving in the steady state: s = ( S n+ g)(k/y). Plugging in the values established above: s = (0.04 + 0.03)(2.5) = 0.175. The initial saving rate is 17.5 percent. b. We know from Chapter 3 that with a Cobb Douglas production function, capital share of s in come a MPK (

29、K/Y). Rewrit ing, we have MPK = a K/Y). Plugging in the values established above, we find MPK = 0.3/2.5 = 0.12. c. We know that at the Golden Rule steady state: MPK = (n + g + S.) Plugging in the values established above: MPK = (0.03 + 0.04) = 0.07. At the Golden Rule steady state, the marginal prod

30、uct of capital is 7 percent, whereas it is 12 percent in the initial steady state. Hence, from the initial steady state we need to increase k to achieve the Golden Rule steady state. d. We know from Chapter 3 that for a Cobb -Douglas production function, MPK = a Y/K). Solving this for the capital -o

31、utput ratio, we find K/Y = aM/PK. We can solve for the Golden Rule capital -output ratio using this equation. If we plug in the value 0.07 for the Golden Rule steady-state marginal product of capital, and the value 0.3 fora , we find K/Y = 0.3/0.07 = 4.29. In the Golden Rule steady state, the capita

32、l -output ratio equals 4.29, compared to the current capital -output ratio of 2.5. e. We know from part (a) that in the steady state s = ( S n+ g)(k/y), where k/y is the steady-state capital -output ratio. In the introduction to this answer, we showed that k/y = K/Y, and in part (d) we found that th

33、e Golden Rule K/Y = 4.29. Plugging in this value and those established above: s = (0.04 + 0.03)(4.29) = 0.30. To reach the Golden Rule steady state, the saving rate must rise from 17.5 to 30 percent. This result implies that if we set the saving rate equal to the share going to capital (30 percent),

34、 we will achieve the Golde n Rule steady state. 4. a. In the steady state, we know that sy = ( S r+ g)k. This implies that k/y = s/( S r+ g). Since s, S, and g are constant, this means that the ratio k/y is also constant. Since k/y = K/( LE)/ Y/( LE) = K/Y, we can conclude that in the steady state,

35、the capital utput ratio is con sta nt. b. We know that capital s share of income= (K/Y). In the steady state, we know from part (a) that the capital utput ratio K/Y is constant. We also know from the hint that the MPK is a function of k, which is constant in the steady state; therefore the MPK itsel

36、f must be constant. Thus, capital s share of in cocensta nt. Labor s share of in come iQapitalShare. Hen ce, if capital is cbasa nt, we see that labor s share of in come is also con sta nt. c. We know that in the steady state, total in come grows at n + g, defi ned as the rate of populati on growth

37、plus the rate of tech no logical cha nge. In part (b) we showed that labor s and capital s share of in come is con sta nt. If the shares are con sta nt, and total in come grows at the rate n + g, the n labor in come and capital in come must also grow at the rate n + g. d. Define the real ren tal pri

38、ce of capital R as R = Total Capital In come/Capital Stock =(MPK K)/K =MPK. We know that in the steady state, the MPK is constant because capital per effective worker k is con sta nt. Therefore, we can con clude that the real ren tal price of capital is con sta nt in the steady state. To show that t

39、he real wage w grows at the rate of tech no logical progress g, defi ne TLI = Total Labor In come L = Labor Force Using the hint that the real wage equals total labor in come divided by the labor force: w = TLI/L. Equivale ntly, wL = TLI. In terms of perce ntage cha nges, we can write this as Aw/w +

40、 zk/L = zTLI/TLI. This equation says that the growth rate of the real wage plus the growth rate of the labor force equals the growth rate of total labor in come. We know that the labor force grows at rate n, and, from part (c), we know that total labor in come grows at rate n + g. We, therefore, con

41、 clude that the real wage grows at rate g. 5. a. The per worker producti on fun cti on is F(K, L)/L = AK “L1% = A(K/L)“ = Ak a b. In the steady state, k = sf(kR-( S + + g)k = 0. Hence, sAk = ( S r+ g)k, or, after rearranging: e k* = e sA U ed + n+ g U Chapter 9 Econo mic Growth II: Tech no logy, Emp

42、irics, and Policy82 Pluggi ng into the per-worker producti on fun cti on from part (a) gives ?oe.、e Thus, the ratio of steady-state in come per worker in Richla nd to Poorla nd is (y* Richla nd /y* Poorla nd )= SRichla nd 挈d + nRichland Spoorla nd U1-a U d + n Poorland +g U a e 0.32 z 0.10u1-a =e-/u

43、 企0.05+0.01 + 0.02 0.05+0.03+0.02U c. If a equals 1/3, then Richland should be14, or two times, richer than Poorland. ? a L ?Va L d. If 4e = 16, then it must be the case that a a - s彳.、e which in turn requires that a equals 2/3. Hence, if the Cobb -Douglas production function puts 2/3 of the weight

44、on capital and only 1/3 on labor, then we can explain a 16-fold difference in levels of income per worker. One way to justify this might be to think about capital more broadly to in clude huma n capital which must also be accumulated through in vestme nt, much in the way one accumulates physical cap

45、ital. 6. How do differe nces in educati on across cou ntries affect the Solow model? Educati on is one factor affecting the efficiency of labor , which we denoted by E. (Other factors affecting the efficiency of labor in clude levels of health, skill, and kno wledge.) Since cou ntry 1 has a more hig

46、hly educated labor force than country 2, each worker in country 1 is more efficient. That is, E1 E2. We will assume that both cou ntries are in steady state. a. In the Solow growth model, the rate of growth of total in come is equal to n + g, which is in depe ndent of the work forcelevel dfesJucati

47、on. The two cou ntries will, thus, have the same rate of growth of total in come because they have the same rate of populati on growth and the same rate of tech no logical progress. b. Because both cou ntries have the same sav ing rate, the same populati on growth rate, and the same rate of tech no

48、logical progress, we know that the two cou ntries will con verge to the same steady * state level of capital per effective worker k . This is shown in Figure 9-1. Capital per efled ive wofker SI LK VLrmlEfl里uutl电 jq JEunq谄uiul Hence, output per effective worker in the steady state, which is y = f(k

49、), is the same in both countries. But y = Y/(L * E) or 丫/L = y E. We know that y will be the same in both countries, but that E1 E2. Therefore, y E1 y E2. This implies that ( Y/L)1 (Y/L)2. Thus, the level of income per worker will be higher in the cou ntry with the more educated labor force. c. We k

50、now that the real rental price of capital R equals the marginal product of capital ( MPK). But the MPK depenqs on the capital stock per efficiency unit of labor. In the steady state, both countries have k 1 = k 2 = k because both countries have the same saving rate, the same population growth rate,

51、and the same rate of tech no logical progress. Therefore, it must be true that R1 = R2 = MPK . Thus, the real rental price of capital is identical in both countries. d. Output is divided betwee n capital in come and labor in come. Therefore, the wage per effective worker can be expressed as w = f(k)

52、 -MPK ? k. As discussed in parts (b) and (c), both countries have the same steady-state capital stock k and the same MPK . Therefore, the wage per effective worker in the two countries is equal. Workers, however, care about the wage per un it of labor, not the wage per effective worker. Also, we can

53、 observe the wage per un it of labor but not the wage per effective worker. The wage per un it of labor is related to the wage per effective worker by the equati on Wage per Un it of L = wE. Thus, the wage per un it of labor is higher in the cou ntry with the more educated labor force. 7. a. In the

54、two-sector en doge nous growth model in the text, the producti on fun cti on for man ufactured goods is Y = F K,(1 -u) EL. We assumed in this model that this fun cti on has con sta nt retur ns to scale. As in Secti on 3-1, constant returns means that for any positive number z, zY = F (zK, z(1 -u) EL

55、). Setting z = 1/EL, we obtai n El=F?(1-u)2 EL EL? Using our standard definitions of y as output per effective worker and k as capital per effective worker, we can write this as b. y = Fk,(1 -u) To beg in, n ote that from the producti on fun cti on in research uni versities, the growth rate of labor

56、 efficiency,E/Eequals g(u). We can now follow the logic of Section 9-1, substituting the c. function g(u) for the constant growth rate g. In order to keep capital per effective worker ( K/EL) constant, break-even investment includes three terms: k is deeded to replace depreciating capital, nk is n e

57、eded to provide capital for new workers, and g(u) is n eeded to provide capital for the greater stock of knowledge E created by research universities. That is, break-even investment is S + n + g (u) k. Again following the logic of Section 9-1, the growth of capital per effective worker is the differ

58、e nee betwee n sav ing per effective worker and break-eve n in vestme nt per effective worker. We now substitute the per-effective-worker producti on fun cti on from part (a) and the fun cti on g(u) for the constant growth rate g, to obtain Ak = sF k,(1 -u) - S + n + g (u)k In the steady state, k =

59、0, sAwe can rewrite the equation above as sF k,(1 -u) = S + + g(u)k. As in our analysis of the Solow model, for a given value of u, we can plot the left and right sides of this equati on -Haul毎MH (5-n+g(uX Figure 9-2 CapKal per ctkx；U e worker d. The steady state is give n by the in tersecti on of t

60、he two curves. The steady state has constant capital per effective worker k as given by Figure 9-2 above. We also assume that in the steady state, there is a constant share of time spent in research universities, so u is constant. (After all, if u were not constant, it wouldn t be a “ steady ” state