1、Chapter 3 课后习题答案Solutions to the Problems in the TextbookConceptual Problems1. A 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-K.In this function, Y represents the l
2、evel of output, 1 - and are weights equal to the shares of labor (N) and capital (K) in production, while A measures all other contributions to output, in particular that of technology. It can be shown that labor and capital each contribute to economic growth by an amount that is equal to their indi
3、vidual growth rates multiplied by their respective shares in income. 2. The Solow growth 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
4、 eventually catch up to a richer one by saving at the same rate and making use of the same technological innovations. However, if these countries have different savings rates, they will reach different levels of income per capita, even though their long-term economic growth rates will be the same.3.
5、 A production function that omits the stock of natural resources 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 s
6、ignificant positive effect on a countrys level of GDP that could not be 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 o
7、nly captures the effect of technological progress but also the effect of changes in human capital (H), achieved through more education or training, 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) = AN
8、aKbHc with a + b + c = 1. In this case, the growth rate of output can be calculated as Y/Y = A/A + a(N/N) + b(K/K) + c(H/H).Clearly, here the magnitude of A/A is different from that derived from the simple Solow growth model using the production function Y = F(K, N) = ANaKb .5. The savings function
9、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 investment requirement,
10、 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 = (N/N).6. In the long run, (and in absence of technological progress) the growth rate
11、 of the steady-state output per capita is determined only the rate of population growth n = (N/N). However, in the short run, the savings rate and the rate of depreciation also can affect the economic growth rate of a nation, as can a one-time technological advance.7. Labor productivity is defined a
12、s Y/N, that is, the ratio of output (Y) to labor input (N). This implies that a surge in labor productivity occurs if output grows at a faster rate than labor input. The U.S. experienced such a surge in labor productivity in the mid and late 1990s due to large increases in GDP. This can be explained
13、 by the introduction of new technologies and more efficient use of existing technologies, for example increased investment in and use of computer technology. Furthermore, increased global competition has forced many firms to cut costs by reorganizing production and eliminating jobs. Therefore, with
14、large increases in output and a slower rate of job creation we should expect labor productivity to increase. (One should note, however, that a more highly skilled labor force also can contribute to an increase in labor productivity, since highly skilled workers can produce much more output than the
15、same number of less skilled workers.)Technical Problems1 a. According to Equation (2), the growth of output is equal to the growth in labor times labors share of income plus the growth of capital times capitals share of income plus the rate of technical progress, that is,Y/Y = (1 - )(N/N) + (K/K) +
16、A/A, where1 - is the share of labor (N) and is the share of capital (K). Therefore, if we assume that the rate of technological progress is zero (A/A = 0), output grows at an annual rate of 3.6 percent, sinceY/Y = (0.6)(2%) + (0.4)(6%) + 0% = 1.2% + 2.4% = + 3.6%,b. The so-called “Rule of 70“ sugges
17、ts that the length of time it takes for output 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%.c. Now that A/A = 2%, we can calculate economic growth as Y/Y = (0.6)(2
18、%) + (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%.2 a. According to Equation (2), the growth of output is equal to the growth in labor times the share of labor plus the growth of capital times the share of capit
19、al plus the growth rate of total factor productivity, that is,Y/Y = (1 - )(N/N) + (K/K) + A/A, where1 - is the share of labor (N) and is the share of capital (K). In this example = 0.3; therefore, if output grows at 3% and labor and capital grow at 1% each, we can calculate the change in total facto
20、r productivity in the following way 3% = (0.7)(1%) + (0.3)(1%) + A/A = A/A = 3% - 1% = 2%, that is, the growth rate of total factor productivity is 2%.b. If both labor and the capital stock are fixed, that is, N/N = K/K = 0, and output grows at 3%, then all the growth has to be attributed to the gro
21、wth in total factor productivity, that is, A/A = 3%.3 a. If the capital stock grows by K/K = 10%, the effect will be an additional growth rate in output ofY/Y = (0.3)(10%) = 3%.b. If labor grows by N/N = 10%, the effect will be an additional growth rate in output of Y/Y = (0.7)(10%) = 7%.c. If outpu
22、t grows at Y/Y = 7% due to an increase in labor by N/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 calculate the change in per capita income as follows:y/y = Y/Y - N/N = 7% - 10% = - 3%.
23、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 y/y = 7%. Therefore peoples welfare (or at least their living standard) will increase.4. Figure 3-4 shows output per head as a function of t
24、he 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 requirement, the economy is in a steady-state equilibrium. Now let us assume for simplicity
25、 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*.If the earthquake destroys one quarter of the capital stock but less than one quarter of the labor force, t
26、hen the capital-labor ratio will fall from k* to k1 and per-capita output will fall from y* to y1. Now saving is greater than the investment requirement, that is, sy1 (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,
27、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 will be less than the investment requirement and thus the capital-labor ratio and the level of output
28、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, the capital-labor ratio and the output per capita will not change. If the severity of the earthqua
29、ke 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 invested later in an effort to rebuild) or decreases (if people save less, since they decide that life i
30、s too short not to live it up). But in either case, a new steady-state equilibrium will be reached.5 a. An increase in the population growth rate (n) affects the investment requirement, that is, as n gets larger, the (n + d)k-line gets steeper. As the population grows, more needs to be saved and inv
31、ested 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 optimal capital-labor ratio will be determined by the intersection of the sy-curve and the new (n1
32、 + 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 determined by the new and higher rate of population growth n1 no. b. Starting from an initial steady-s
33、tate equilibrium at a level of per-capita output yo, the increase in the population growth rate (n) will cause the capital-labor ratio to decline from ko to k1. Output per capita will also decline, a process that will continue at a diminishing rate until a new steady-state level is reached at y1. Th
34、e growth rate of output will gradually adjust to the new and higher level n1.6 a. Assume the production function is of the form Y = F(K, N, Z) = AKaNbZc =Y/Y = A/A + a(K/K) + b(N/N) + c(Z/Z), with a + b + c = 1.Now assume that there is no technological progress (A/A = 0), and that capital and labor
35、grow at the same rate, that is, K/K = N/N = n. If we also assume that all available natural resources are fixed, such that Z/Z = 0, then the rate of output growth will be Y/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 befo
36、re, namelyY/Y = A/A + (a + b)n.If A/A cn, output will grow at a rate higher than n, in which case output per worker will increase.c. If the supply of natural resources 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 grow
37、th 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, even if we have a fixed supply of natural resources. 7. a. If the production function is of the form Y = K1/2(AN)1/2, and A is normalized to
38、1, we have Y = K1/2N1/2 . In this case, capitals and labors shares of income are both 50%.b. This is a Cobb-Douglas production function.c. A steady-state equilibrium 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 = k-1/2 = (n + d)/s = (0.07 + 0.03)/(
39、.2) = 1/2 = k1/2 = 2 = y = k = 4 . d. At the steady-state equilibrium, output per capita remains constant, since total output grows at the same rate as the population (7%), as long as there is no technological progress, that is, A/A = 0. But if total factor productivity grows at A/A = 2%, then total
40、 output will grow faster than population, that is, at 7% + 2% = 9%, so output per capita will grow at 2%.8 .If technological progress occurs, then the level of output per capita for any given capital-labor ratio increases. The function y = f(k) increases to y = g(k), and therefore the savings functi
41、on increases from sf(k) to sg(k). 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-state equilibrium will now be at a higher level of saving and output per capita, and
42、at a higher capital-labor ratio.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 investment ratio will increase in the transition period, since more investment will be required to reach the
43、 higher optimal capital-labor ratio.9. The Cobb-Douglas production function is defined asY = F(N, K) = AN1-K.The marginal product of labor can then be derived asMPN = (Y)/(N) = (1 - )AN-K = (1 - )AN1-K/N = = (1 - )(Y/N)= labors share of income = MPN*N/Y = (1 - )(Y/N)(N/Y) = (1 - ). 10 a. The product
44、ion function is of the formY = K1/2N1/2 = Y/N = (K/N)1/2 = y = k1/2.From k = sy/(n + d) = sk1/2/(n +d) = k1/2 = s/(n + d)= y* = s/(n + d) = (0.1)/(0.02 + 0.03) = 2= k* = sy*/(n + d) = (0.1)(2)/(0.02 + 0.03) = 4.b. Steady-state consumption equals steady-state income minus steady-state saving (or inve
45、stment), that is,c* = y sy = f(k*) - (n + d)k* .The golden-rule capital stock corresponds to the highest permanently sustainable level of consumption. Steady-state consumption is maximized when the marginal increase in capital produces just enough extra output to cover the increased investment requi
46、rement.From c = k1/2 - (n + d)k = (c/k) = (1/2)k-1/2 - (n + d) = 0= k-1/2 = 2(n + d) = 2(.02 + .03) = .1= k1/2 = 10 = k = 100.Since k* = 4 s = k1/2(n + d) = 10(0.05) = .5.d. If we have more capital than the golden rule suggests, we are saving too much and do not have the optimal amount of consumptio
47、n.Empirical Problems1. The average monthly growth rate of the U.S. population over the period 2000-2010 was 0.1 percent. Over the same period, total employment in educational services grew on average by 0.2 percent each month. This implies that there are more teachers per student at the end of this
48、ten-year period. Everything else being constant, one could infer that since class size was smaller, students got more attention and thus learned more and that therefore the average quality of U.S. workers improved. If this is true, then the prospects for future economic growth in the U.S. have impro
49、ved. 2. The graph below presents the change in employment in the information technology industry. The increase in employment in that industry in the 1990s can probably be explained by the increase in on-line business or other Internet activity and increased business investment 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
