宏观经济学 教案Chapter03.docx

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1、CHAPTER 3GROWTH AND ACCUMULATIONChapter Outline Growth accounting The Cobb-Douglas production function The marginal product of labor and capital Convergence Total factor productivity The importance of human capital The neoclassical growth model The steady-state equilibrium The golden-rule capital st

2、ock Investment and saving Technological advances and population growthChanges from the Previous EditionOther than updates of actual data, no significant changes have been made in Chapter 3.Introduction to the MaterialTo be able to explain why per-capita GDP grows at different rates in different coun

3、tries, students need to understand what determines growth in the factors of production and what causes improvements in the state of technology. It is clear that while short-run fluctuations in inputs can at times be important, the rate of (physical and human) capital accumulation and technological p

4、rogress are most crucial for long-term growth. Since even small differences in the economic growth rate can lead to large differences in a populations standard of living in the long run, it is extremely important to understand which policy measures are most successful in enhancing a nations economic

5、 growth rate.Chapter 3 introduces the neoclassical growth model, a framework that uses an aggregate Cobb-Douglas production function with constant returns to scale. For example, if we assume thatY = F(N, K) = ANW,then the rate of output growth can be written asAY/Y = AA/A + (1 - 0)(AN/N) + 0(AK/K),w

6、here (1-0) and 0 are weights equal to the shares of labor (N) and capital (K) in production. The above equation implies that labor and capital each contribute to output growth by an amount that is equal to their individual growth rates multiplied by their respective share in income. The1 .c. Now tha

7、t AA/A = 2%, we can calculate economic growth asAY/Y =(0.6)(2%) + (0.4)(6%) + 2% =1.2% + 2.4% + 2% = + 5.6%.Thus it will take 70/5.6 = 12.5 years for output to double at this new growth rate of 5.6%.2a According to Equation (2), the growth of output is equal to the growth in labor times the share of

8、 labor plus the growth of capital times the share of capital plus the growth rate of total factor productivity, that is,AY/Y = (1 - 0)(AN/N) + 0(AK/K) + AA/A, where1 - 0 is the share of labor (N) and 0 is the share of capital (K). In this example 0 = 0.3; therefore, if output grows at 3% and labor a

9、nd capital grow at 1% each, we can calculate the change in total factor productivity in the following way3% =(0.7)( 1 %) + (03)(1%) + AA/A= AA/A = 3% - 1% = 2%,that is, the growth rate of total factor productivity is 2%.2b If both labor and the capital stock are fixed, that is, AN/N = AK/K = 0, and

10、output grows at 3%, then all the growth has to be attributed to the growth in total factor productivity, that is, AA/A = 3%.3 .a. If the capital stock grows by AK/K = 10%, the effect will be an additional growth rate in output ofAY/Y = (03)(10%) = 3%.4 .b. If labor grows by AN/N = 10%, the effect wi

11、ll be an additional growth rate in output ofAY/Y = (0.7)(10%) = 7%.5 .c. If output grows at AY/Y = 7% due to an increase in labor by AN/N = 10% and this increase in labor is entirely due to population growth, then per capita income will decrease. Therefore, peoples welfare will decrease. We can calc

12、ulate the change in per capita income as follows:Ay/y = AY/Y - AN/N = 7%- 10% = -3%.6 .d. If the increase in labor is not due to population growth but instead due to an influx of women into the labor force, then income per capita will increase by Ay/y = 7%. Therefore peoples welfare (or at least the

13、ir living standard) will increase.7 . Figure 3-4 shows output per head as a function of the capital-labor ratio, that is, y = f(k), the savings function, that is sy = sf(k), and the investment requirement, that is, the (n + d)k-line. At the intersection of the savings function with the investment re

14、quirement, the economy is in a steady-state equilibrium. Now let us assume for simplicity that the earthquake does not affect peoples savings behavior and that the economy is in a steady-state equilibrium before *the earthquake hits, that is, the capital-labor ratio is currently k .yki k* k2 kIf the

15、 earthquake destroys one quarter of the capital stock but less than one quarter of the labor force, then the capital-labor ratio will fall from k to ki and per-capita output will fall from y! to yi. Now saving is greater than the investment requirement, that is, syi (d + n)ki, and the capital stock

16、and the level of output per capita will grow until the steady state at k： is reached again.However, if the earthquake destroys one quarter of the capital stock but more than one quarter of the labor force, then the capital-labor ratio will increase from k to k2. Saving (and gross investment) now wil

17、l be less than the investment requirement and thus the capital-labor ratio and the level of output per capita will fall until the steady state at k is reached again.If exactly one quarter of both the capital stock and the labor stock are destroyed, then the steady state will be maintained, that is,

18、the capital-labor ratio and the output per capita will not change.If the severity of the earthquake has an effect on peoples savings behavior, the savings function sy = sf(k) will move either up or down, depending on whether the savings rate (s) increases (if people save more, so more can be investe

19、d later in an effort to rebuild) ordecreases (if people save less, since they decide that life is too short not to live it up). But in either case, a new steady-state equilibrium will be reached.8 .a. An increase in the population growth rate (n) affects the investment requirement, that is, as n get

20、s larger, the (n + d)k-line gets steeper. As the population grows, more needs to be saved and invested to equip new workers with the same amount of capital that existing workers already have. Since the population will now be growing faster than output, income per capita (y) will decrease and a new o

21、ptimal capital-labor ratio will be determined by the intersection of the sy-curve and the new (m + d)k-line. Since per-capita output will fall, we will have a negative growth rate in the short run. However, the steady-state growth rate of output will increase in the long run, since it will be determ

22、ined by the new and higher rate of population growth m n0.yy。yioyy。yiokikok(ni + d)k5b Starting from an initial steady-state equilibrium at a level of per-capita output y0, the increase in the population growth rate (n) will cause the capital-labor ratio to decline from ko to ki. Output per capita w

23、ill also decline, a process that will continue at a diminishing rate until a new steady-state level is reached at yi. The growth rate of output will gradually adjust to the new and higher level ni.kokito tlt.a. Assume the production function is of the formY = F(K, N, Z)= AKaNbZc =AY/Y = AA/A + a(AK/

24、K) + b(AN/N) + c(AZ/Z), with a + b + c= 1.Now assume that there is no technological progress (AA/A = 0), and that capital and labor grow at the same rate, that is, AK/K = AN/N = n. If we also assume that all available natural resources are fixed, such that AZ/Z = 0, then the rate of output growth wi

25、ll beAY/Y = an + bn = (a + b)n.In other words, output will grow at a rate less than n since a + b 0), output will grow faster than before, namelyAY/Y = AA/A + (a + b)n.If AA/A cn, output will grow at a rate higher than n, in which case output per worker will increase.6 .c. If the supply of natural r

26、esources is fixed, then output can only grow at a rate that is smaller than the rate of population growth and we should expect limits to growth as we run out of natural resources. But if the rate of technological progress is sufficiently large, then output can grow at a rate faster than population,

27、even if we have a fixed supply of natural resources.7 .a. If the production function is of the form Y = KI/2(AN)1/2, and A is normalized to 1, we have Y = K1/2N,/2.In this case, capitafs and labors shares of income are both 50%.7b This is a Cobb-Douglas production function.8 .c. A steady-state equil

28、ibrium is reached when sy = (n + d)k.From Y = K1/2N1/2 = Y/N = K1/2N-1/2 = y = k1/2 = sk1/2 = (n + d)k= (n + d)/s =(0.07 + 0.03)/(.2)= 1/2 = k1/2 = 2 = y = k = 4 .9 .d. At the steady-state equilibrium, output per capita remains constant, since total output grows at the same rate as the population (7

29、%), as long as there is no technological progress, that is, AA/A = 0. But if total factor productivity grows at AA/A = 2%, then total output will grow faster than population, that is, at 7% + 2% = 9%, so output per capita will grow at 2%.8.a.If technological progress occurs, then the level of output

30、 per capita for any given capital- labor ratio increases. The function y = f(k) increases to y = g(k), and therefore the savings function increases from sf(k) to sg(k).10 b. Since g(k) f(k), it follows that sg(k) sf(k) for each level of k. Therefore, the intersection of the sg(k)-curve with the (n +

31、 d)k-line is at a higher level of k. The new steady-state equilibrium will now be at a higher level of saving and output per capita, and at a higher capital-labor ratio.11 c. After the technological progress occurs, the level of saving and investment will increase until a new and higher optimal capi

32、tal-labor ratio is reached. The investment ratio will increase in the transition period, since more investment will be required to reach the higher optimal capital-labor ratio.The Cobb-Douglas production function is defined asY = F(N, K)= ANeK0.The marginal product of labor can then be derived asMPn

33、 = (AY)/(AN) = (1 - 0)AN-eKe = (1 - 0)AN,-0Ke/N = = (1 - 0)(Y/N)= labors share of income = MPn*N/Y = (1 - 0)(Y/N)(N/Y) = (1 - 0).12 .a. The production function is of the formY = K1/2N1/2 = Y/N =(K/N),/2 二二y = k,/2.From k = sy/(n + d) = sk1/2/(n +d) = k1/2 = s/(n + d)= y* = s/(n + d) = (0.1)/(0.02 +

34、0.03) = 2二二k* 二 sy7(n + d)=(0.1)(2)/(0.02 + 0.03)= 4.13 .b. Steady-state consumption equals steady-state income minus steady-state saving (or investment), that is,= y - sy = f(k*) - (n + d)k.The golden-rule capital stock corresponds to the highest permanently sustainable level of consumption. Steady

35、-state consumption is maximized when the marginal increase in capital produces just enough extra output to cover the increased investment requirement.From c = k1/2 - (n + d)k = (Ac/Ak) = (l/2)k1/2 - (n + d) = 0= k1/2 = 2(n + d) = 2(.02 + .03) = . 1 = k1/2 = 10 = k = 100.Since k = 4 s = k1/2(n + d) =

36、 10(0.05) = .5.15 .d. If we have more capital than the golden rule suggests, we are saving too much and do notEmpirical Problems1. The average growth rate of the U.S. population over the period from 2002-2012 was about 10 percent. Over the same period, total employment in educational services grew b

37、y an average of 30 percent. This implies that there are more teachers per student at the end of this ten-year period. If one assumes that everything else is held constant, one could infer that the average class size was smaller, so individual students got more attention and therefore learned more. I

38、n other words, the average quality of students and, subsequently, U.S. workers improved. If this is true, then the prospects for future economic growth in the U.S. have improved.2. employment in information services400035003000thousands of employees 15OO1OOO500 OmVI0.1Riolaiz1Mo,60mS1RssizSS7S71M300

39、710,60.6661-Milmlss士 61-NilmsThe graph above presents the change in employment in the information technology industry. The increase in employment in that industry from 1993 to 2013 can probably be explained by the increase in on-line business or other Internet activity and increased business investm

40、ent in computer technology. But to some degree it may also be a result of the long economic upswing that occurred during the 1990s.As we can also see from this graph, employment in the information technology industry decreased again after 2000 and leveled off after 2010 to about the same level as in

41、 the early 1990s. This is most likely due to the fact that businesses had already invested sufficiently in new computer technology. As software became easier to use and people became more computer savvy, there was also less need to hire IT personnel.Additional Problems1. Assume an economy in which t

42、he level of technology and the stock of capital are fixed. If there is a sudden increase in the labor force due to immigration, how would the standard of living be affected? Would it make a difference if the increase in the labor force were instead caused by a higher labor force participation rate o

43、f women?If the increase in the labor force were entirely due to an influx of foreigners, then the population would increase and living standards would decrease, since per-capita income would decrease. However, if the increase in the labor force were due to an increase in the labor force participatio

44、n of women, then income per capita would increase, since the overall population would not increase. In this case, we would see an increase in living standards.3. Assume that labors share of income is three times as large as capitaFs share, and that capital, labor, and output all grow at the same rat

45、e (n). Use a Cobb-Douglas aggregate production function that has constant returns to scale to show that labor will be a larger source of output growth than capital if there is no technological progress.With the share of labor being (1 - 0) = 0.75 and the share of capital being 0 = 0.25, we get the f

46、ollowing aggregate production function:Y = AN1- AN3/4K,/4We can get the growth accounting equation by taking the total derivative of this equation and then dividing both sides of the equation by Y, that isAY/Y = AA/A + (1 - 0)(AN/N) + 0(AK/K) = 0 + (0.75)n + (0.25)n = n.Therefore we see that labor c

47、ontributes three times as much to growth as capital in the absence of technological progress, that is, if AA/A = 0.4. Assume a Cobb-Douglas aggregate production function in which labors share of income is 80% and capitaFs share of income is 20%. By how much will real GDP grow if labor grows at 3%, the capital stock grows at 2%, and total factor productivity increases by 2.4%?The growth of output can be calculated as the sum of the growth in labor times the l