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1、CHAPTER 3Solutions to the Problems in the TextbookConceptual Problems:1. The production function provides a quantitative link between inputs and output. For example, the Cobb-Douglas production function mentioned in the text is of the form:Y = F(N,K) = AN1-qKq, where Y represents the level of output
2、. (1 - q) and q are weights equal to the shares of labor (N) and capital (K) in production, while A is often used as a measure for the level of technology. It can be easily shown that labor and capital each contribute to economic growth by an amount that is equal to their individual growth rates mul
3、tiplied by their respective share in income. 2.The Solow model predicts convergence, that is, countries with the same production function, savings rate, and population growth will eventually reach the same level of income per capita. In other words, a poor country may eventually catch up to a richer
4、 one by saving at the same rate and making technological innovations. However, if these countries have different savings rates, they will reach different levels of income per capita, even though their long-term growth rates will be the same.3.A production function that omits the stock of natural res
5、ources cannot adequately predict the impact of a significant change in the existing stock of natural resources on the economic performance of a country. For example, the discovery of new oil reserves or an entirely new resource would have a significant effect on the level of output that could not be
6、 predicted by such a production function.4.Interpreting the Solow residual purely as technological progress would ignore, for example, the impact that human capital has on the level of output. In other words, this residual not only captures the effect of technological progress but also the effect of
7、 changes in human capital (H) on the growth rate of output. To eliminate this problem we can explicitly include human capital in the production function, such thatY = F(K,N,H) = ANaKbHc with a + b + c = 1. Then the growth rate of output can be calculated as DY/Y = DA/A + a(DN/N) + b(DK/K) + c(DH/H).
8、5.The savings function sy = sf(k) assumes that a constant fraction of output is saved. The investment requirement, that is, the (n + d)k-line, represents the amount of investment needed to maintain a constant capital-labor ratio (k). A steady-state equilibrium is reached when saving is equal to the
9、investment requirement, that is, when sy = (n + d)k. At this point the capital-labor ratio k = K/N is not changing, so capital (K), labor (N), and output (Y) all must be growing at the same rate, that is, the rate of population growth n = (DN/N).6.In the long run, the rate of population growth n = (
10、DN/N) determines the growth rate of the steady-state output per capita. In the short run, however, the savings rate, technological progress, and the rate of depreciation can all affect the growth rate. 7.Labor productivity is defined as Y/N, that is, the ratio of output (Y) to labor input (N). A sur
11、ge in labor productivity therefore occurs if output grows at a faster rate than labor input. In the U.S. we have experienced such a surge in labor productivity since the mid-1990s due to the enormous growth in GDP. This surge can be explained from the introduction of new technologies and more effici
12、ent use of existing technologies. Many claim that the increased investment in and use of computer technology has stimulated economic growth. Furthermore, increased global competition has forced many firms to cut costs by reorganizing production and eliminating some jobs. Thus, with large increases i
13、n output and a slower rate of job creation we should expect labor productivity to increase. (One should also note that a higher-skilled labor force also can contribute to an increase in labor productivity, since the same number of workers can produce more output if workers are more highly skilled.)
14、Technical Problems:1.a.According to Equation (2), the growth of output is equal to the growth in labor times the labor share plus the growth of capital times the capital share plus the rate of technical progress, that is,DY/Y = (1 - q)(DN/N) + q(DK/K) + DA/A, where 1 - q is the share of labor (N) an
15、d q is the share of capital (K). Thus if we assume that the rate of technological progress (DA/A) is zero, then output grows at an annual rate of 3.6 percent, sinceDY/Y = (0.6)(2%) + (0.4)(6%) + 0% = 1.2% + 2.4% = + 3.6%,1.b.The so-called Rule of 70 suggests that the length of time it takes for outp
16、ut to double can be calculated by dividing 70 by the growth rate of output. Since 70/3.6 = 19.44, it will take just under 20 years for output to double at an annual growth rate of 3.6%,1.c.Now that DA/A = 2%, we can calculate economic growth as DY/Y = (0.6)(2%) + (0.4)(6%) + 2% = 1.2% + 2.4% + 2% =
17、+ 5.6%.Thus it will take 70/5.6 = 12.5 years for output to double at this new growth rate of 5.6%.2.a.According to Equation (2), the growth of output is equal to the growth in labor times the labor share plus the growth of capital times the capital share plus the growth rate of total factor producti
18、vity (TFP), that is,DY/Y = (1 - q)(DN/N) + q(DK/K) + DA/A, where 1 - q is the share of labor (N) and q is the share of capital (K). In this example q = 0.3; therefore, if output grows at 3% and labor and capital grow at 1% each, then we can calculate the change in TFP in the following way 3% = (0.3)
19、(1%) + (0.7)(1%) + DA/A = DA/A = 3% - 1% = 2%, that is, the growth rate of total factor productivity is 2%.2.b.If both labor and the capital stock are fixed and output grows at 3%, then all this growth has to be contributed to the growth in factor productivity, that is, DA/A = 3%.3.a.If the capital
20、stock grows by DK/K = 10%, the effect on output would be an additional growth rate of DY/Y = (.3)(10%) = 3%.3.b.If labor grows by DN/N = 10%, the effect on output would be an additional growth rate of DY/Y = (.7)(10%) = 7%.3.c.If output grows at DY/Y = 7% due to an increase in labor by DN/N = 10%, a
21、nd this increase in labor is entirely due to population growth, then per capita income would decrease and peoples welfare would decrease, sinceDy/y = DY/Y - DN/N = 7% - 10% = - 3%.3.d.If this increase in labor is due to an influx of women into the labor force, the overall population does not increas
22、e and income per capita would increase by Dy/y = 7%. Therefore peoples welfare would increase.4. Figure 3-4 shows output per head as a function of the capital-labor ratio, that is, y = f(k). The savings function is sy = sf(k), and it intersects the straight (n + d)k-line, representing the investment
23、 requirement. At this intersection, the economy is in a steady-state equilibrium. Now let us assume that the economy is in a steady-state equilibrium before the earthquake hits, that is, the steady-state capital-labor ratio is currently k*. Assume further, for simplicity, that the earthquake does no
24、t affect peoples savings behavior. If the earthquake destroys one quarter of the capital stock but less than one quarter of the labor force, then the capital-labor ratio falls from k* to k1 and per-capita output falls from y* to y1. Now saving is greater than the investment requirement, that is, sy1
25、 (d + n)k1, and the capital stock 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 increases from k* to k2. Saving
26、 now will 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 is maintained, that
27、is, the capital-labor ratio and the output per capita do not change. If the severity of the earthquake has an effect on peoples savings behavior, then 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
28、 invested in an effort to rebuild) or decreases (if people save less, since they decide that life is too short not to live it up). y y = f(k) y2 y* (n+d)k y1sy 0 k1 k* k2 k5.a. An increase in the population growth rate (n) affects the investment requirement, and the (n + d)k-line gets steeper. As th
29、e population grows, more saving must be used to equip new workers with the same amount of capital that the existing workers already have. Therefore output per capita (y) will decrease as will the new optimal capital-labor ratio, which is determined by the intersection of the sy-curve and the (n1 + d
30、)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 determined by the new and higher rate of population growth. y(n1 + d)k y = f(k) yo (no + d)k y1 sy 0k1 ko
31、k5.b. Starting from an initial steady-state equilibrium at a level of per-capita output y*, the increase in the population growth rate (n) will cause the capital-labor ratio to decline from k* to k1. Output per capita will also decline, a process that will continue at a diminishing rate until a new
32、steady-state level is reached at y1. The growth rate of output will gradually adjust to the new and higher level n1. y y* y1 to t1 t k k* k1 to t1 t6.a.Assume the production function is of the form Y = F(K, N, Z) = AKaNbZc =DY/Y = DA/A + a(DK/K) + b(DN/N) + c(DZ/Z), with a + b + c = 1. Now assume th
33、at there is no technological progress, that is, DA/A = 0, and that capital and labor grow at the same rate, that is, DK/K = DN/N = n. If we also assume that all natural resources available are fixed, such that DZ/Z = 0, then the rate of output growth will be DY/Y = an + bn = (a + b)n.In other words,
34、 output will grow at a rate less than n since a + b 0, then output will grow faster than before, namelyDY/Y = DA/A + (a + b)n.If DA/A c, then output will grow at a rate larger than n, in which case output per worker will increase.6.c.If the supply of natural resources is fixed, then output can only
35、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. However, if the rate of technological progress is sufficiently large, then output can grow at a rate faster than population, even if we have a fixed supply of n
36、atural resources. 7.a. If the production function is of the form Y = K1/2(AN)1/2, and A is normalized to 1, then we have Y = K1/2N1/2 . In this case capitals and labors shares of income are both 50%.7.b. This is a Cobb-Douglas production function.7.c. A steady-state equilibrium is reached when sy =
37、(n + d)k. From Y = K1/2N1/2 = Y/N = K1/2N-1/2 = y = k1/2 =sk1/2 = (n + d)k = k-1/2 = (n + d)/s = (0.07 + 0.03)/(.2) = 1/2 = k1/2 = 2 = y = k = 4 . 8.a. If technological progress occurs, then the level of output per capita for any given capital-labor ratio increases. The function y = f(k) increases t
38、o y = g(k), and thus the savings function increases from sf(k) to sg(k). y g(k) y2 f(k) (n +d)k sg(k) y1sf(k) 0 k1 k2k 8.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 + d)k-line is at a higher level of k. The new steady
39、-state equilibrium will now be at a higher level of saving and output per capita, and at a higher capital-labor ratio.8.c. After the technological progress occurs, the level of saving and investment will increase until a new and higher optimal capital-labor ratio is reached. The ratio of investment
40、to capital will also increase in the transition period, since more has to be invested to reach the higher optimal capital-labor ratio. k k2 k1 0 t1 t2t9.The Cobb-Douglas production function is defined asY = F(N,K) = AN1-qKq.The marginal product of labor can then be derived asMPN = (DY)/(DN) = (1 - q)AN-qKq = (1 - q)AN1-qKq/N = = (1 - q)(Y/N)= labors share of income = MPN*(N)/Y = (1 - q)(Y/N)*(N)/(Y) = (1 - q)