1、 Solutions to the Problems in the Textbook Conceptual Problems 1. Endogenous or self-sustained growth supposedly can be achieved by policies designed to affect the proportion of GDP that goes towards investment. The neoclassical growth model of Chapter 3 predicted that the long-term growth rate is d
2、etermined only by population growth and technological progress and that changes in the savings rate have only transitory effects. The endogenous growth model, however, predicts that countries with a higher savings rate can achieve higher long-term growth and that a nations government can affect the
3、long-term growth rate by implementing policies that affect the savings rate. 2. A simple model with constant returns to scale for capital alone implies increasing returns to scale for all factors taken together, which could cause a single large firm to dominate the economy. However, such a model ign
4、ores the possibility that, in addition to these internal (private) returns, there are also external (social) returns to capital, that is, other firms can benefit from positive spillover effects of capital investment. In other words, more investment not only leads to a higher and more efficient capit
5、al stock but also to new ideas and new ways of doing things which can easily be copied by others. This implies that a single firm does not necessarily reap all of the benefits of increased capital investment and output, and therefore it becomes very difficult for that firm to dominate the market. 3.
6、 In the neoclassical growth model, an increase in the savings rate does not increase the long-term growth rate of output. However, because of the short-run adjustment process, there is some transitional gain that will lead to a higher level of output per capita. The endogenous growth model, on the o
7、ther hand, predicts that the government can influence the savings rate, which, in turn, will affect the long-term growth rate of output. 4.a. Chapter 4 suggests that the key to long-term economic growth is investment in human and physical capital with particular emphasis on research and development.
8、 4.b. (i) Investment tax credits may potentially affect economic growth in the long run by achieving a higher rate of technological progress. (ii) R&D subsidies and grants lead to technological advances that will have private and social returns. They are very effective in stimulating long-term econo
9、mic growth. (iii) According to the endogenous growth model, policies designed to increase saving will increase the long-term growth rate of output. However, empirical evidence does not lend much support to this notion. (iv) Increased funding for primary education has large private and social returns
10、 and is therefore an excellent means to stimulate long-term growth. However, a major drawback is that it may take a long time until these policies have their full effect. 5. The notion of absolute convergence states that economies with the same savings rate and rate of population growth will reach t
11、he same steady-state equilibrium if they have access to the same technology. The notion of conditional convergence states that economies that have access to the same technology and the same rate of population growth but different savings rates will reach steady-state equilibria at a different level
12、of output but have the same economic growth rate. Empirical evidence 1 tends to support the notion of conditional convergence across countries. 6. Endogenous growth theory assumes that the steady-state growth rate of output is affected by the rate at which the factors of production are accumulated.
13、Therefore, policies to stimulate the savings rate serve to increase the rate at which the capital stock is accumulated, which may, in turn, lead to a higher growth rate of output. While this notion may be important in explaining the growth rates of highly developed countries at the leading edge of t
14、echnology, it cannot explain the differences in growth rates across poorer countries. For these countries, the notion of conditional convergence seems to be more applicable. 7. Investing in physical capital will lead to a higher capital stock and to a higher level of output in the short run, but oft
15、en to the detriment of long-term growth unless there are significant external returns to capital. This would render investing in human capital a better strategy, since it has high returns and leads to an increase in long-term growth. Unfortunately, it may take a long time before a country can reap t
16、he benefits of investment in human capital. 8.a. A country that is able to choose its rate of population growth through population control policies can shift the investment requirement down, thereby increasing the level of steady-state output. With a lower rate of population growth it is possible to
17、 achieve a higher level of income per capita with a lower level of investment spending. Therefore, implementing population control policies may be an effective way to escape the poverty trap. 8.b. In an endogenous growth model, a lower population growth rate will increase a nations long-term growth
18、rate. The growth rate of per-capita output for an endogenous growth model was derived, as: y/y = sa - (n + d). Thus, we can see that, as population growth (n) decreases, the growth rate of output (y/y) increases. 9. The Asian Tigers (Hong Kong, Singapore, South Korea, and Taiwan) experienced a high
19、rate of growth between 1966 and 1990 by concentrating on improving the education of the population and increasing the savings rate, as suggested by the endogenous growth model. However, increases in the labor forces of these countries were also at work, as suggested by the neoclassical growth model.
20、 10. It is unclear whether countries can actually experience indefinite increases in their growth potential. However, if technological advances occur continuously and if intelligent resource management is practiced, it is conceivable that economic growth will continue for a very, very long time. As
21、Figure 4-2 in the text suggests, this is more likely to occur in highly industrialized countries. Technical Problems 1.a. A production function that displays both a diminishing and a constant marginal product of capital can be displayed by drawing a curved line (as in an exogenous growth model), fol
22、lowed by an upward-sloping line (as in an endogenous growth model). Such a graph is depicted below. 2 1.b. The first equilibrium (Point A in the graph below) is a stable low-income steady-state equilibrium. Any deviation from that point will cause the economy to eventually adjust again at the same s
23、teady-state income level (and capital-output ratio). The second equilibrium (Point B) is an unstable high-income steady-state equilibrium. Any deviation from that point will lead to either a lower income steady-state equilibrium back at Point A (if the capital-labor ratio declines) or ongoing growth
24、 (if the capital-labor ratio increases). In the latter case, not only total output but also output per capita continues to grow. y = f(k) y yB sy (n + d)k B yA A 0 kA kB k 1.c. A model like the one in this question can be used to explain how some countries find themselves in situations with no growt
25、h and low income while others have ongoing growth and a high level of income. In the first case, a country may have invested mostly in physical capital, leading to some short-term growth at the expense of long-term growth. In the second case, a country may have invested not only in physical capital
26、but also in human capital (education, skills, and training), reaping significant social returns. 2.a. If population growth is endogenous, that is, if a country can influence the rate of population growth through government policies, then the investment requirement is no longer a straight line. Inste
27、ad it is curved as depicted below. yC y = f(k) yB C n(y) + dk sf(k) yA B A 0 kA kB kC k 3 2.b. The first equilibrium (Point A) is a stable steady-state equilibrium. This is a situation of low income and high population growth, indicating that the country is in a poverty trap. The second equilibrium
28、(Point B) is an unstable steady-state equilibrium. This is a situation of medium income and low population growth. The third equilibrium (Point C) is a stable steady-state equilibrium. This is a situation of high income and low population growth. In none of these three cases do we have ongoing growt
29、h. For any capital stock k kb (even for k kc), the economy will adjust to kB. During the adjustment process to any of these two steady-state equilibria, the change in output per capita only will be transitory. 2.c. To escape the poverty trap (Point A), a country has several possibilities: First, it
30、can somehow find the means to increase the capital-labor ratio above a level consistent with Point B (perhaps by borrowing funds or seeking direct foreign investment). Second, it can increase the savings rate such that the savings function no longer intersects the investment requirement curve at eit
31、her Point A or Point B. Third, it can decrease the rate of population growth through specifically designed policies, such that the investment requirement shifts down and no longer intersects with the savings function at Points A or B. 3.a. If we incorporate endogenous population growth into a two-se
32、ctor model in Problem 2, we get a curved investment requirement line and a production function with first a diminishing and then a constant marginal product of capital as depicted below. (Note that the savings function has the same shape as the production function.) y y = f(k) yD sf(k) D n(y)+dk yC
33、yB yA C B A 0 kA kB kC kD k 3.b. Here we should have four intersections of the savings function sf(k) and the investment requirement n(y)+dk. The first equilibrium (at Point A) is a stable low-income steady-state equilibrium. Any deviation from that point will cause the economy to eventually adjust
34、again at the same steady-state income level (and capital-output ratio). The second equilibrium (at Point B) is an unstable low-income equilibrium. Any deviation from that point will lead to either a lower income steady-state equilibrium at Point A (if the capital-labor ratio declines) or a higher in
35、come steady-state equilibrium 4 at Point C (if the capital-labor ratio increases). Since Point C is a stable equilibrium, the economy will settle back at that point again whether the capital stock increases or declines. Point D is again an unstable equilibrium but at a high level of income. Any devi
36、ation from that point will lead to either a lower income steady-state equilibrium at Point C (if the capital-labor ratio declines) or ongoing growth (if the capital-labor ratio increases). 3.c. This model is more inclusive than either of the two models discussed previously and therefore has greater
37、explanatory power. While the graphical analysis is far more complicated, we can now see more clearly that a poor country cannot escape the poverty trap at Point A unless it somehow succeeds in increasing the capital-labor ratio (and thus per-capita income) beyond the level at Point B. 4.a. The produ
38、ction function is of the form Y = K1/2(AN)1/2 = K1/2(4K/NN)1/2 = K1/2(4K)1/2 = 2K. From this we can see that a = 2 since Y = Y/N = 2(K/N) = y = 2k. 4.b. Since a = y/k = 2, it follows that the growth rate of output and capital is y/y = k/k = g = sa - (n + d) = (0.1)2 - (0.02 + 0.03) = 0.15 = 15%. 4.c
39、. The term a in the equation above stands for the marginal product of capital. By assuming that the level of labor-augmenting technology (A) is proportional to the capital-labor ratio (k), it is implied that the level of technology depends on the amount of capital per worker that we have, which may
40、not be realistic. 4.d. In this model, we have a constant marginal product of capital and therefore we have an endogenous growth model. 5.a. The production function is of the form Y = 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)/
41、(0.02 + 0.03) = 2 = k* = sy*/(n + d) = (0.1)(2)/(0.02 + 0.03) = 4. 5.b. Steady-state consumption equals steady-state income minus steady-state saving (or investment), that is, c* = y sy = f(k*) - (n + d)k* . The golden-rule capital stock corresponds to the highest permanently sustainable level of 5
42、consumption. Steady-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 = (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*
43、= 4 s = k1/2(n + d) = 10(0.05) = 0.5. 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 consumption. Empirical Problems 1. As the graph below indicates, manufacturing output in the United Kingdom increased over the period 1950-20
44、01. In the same period manufacturing employment and the average number of hours worked by employees decreased significantly. This apparent contradiction can be explained by an increase in the productivity of employees, as illustrated by the increase in output per hour that employees produced. Additi
45、onal Problems 1. Increasing returns to scale imply that the level of output increases only if the level of all inputs are increased by the same amount. Comment on this statement. The level of output will increase as soon as the level of one input is increased, even if the level of other inputs remai
46、ns the same. This is always true, even under the assumption of decreasing or 6 constant returns to scale. Increasing returns to scale imply that the level of output increases at an increasing rate if the level of one factor input is increased while the levels of all other factor inputs remain the sa
47、me. Similarly, it follows that if there are increasing returns to scale and the level of all inputs are doubled, the level of output will more than double. 2. Assume the aggregate production function is of the following form: Y = aK. At what capital-labor ratio (k) can a steady-state equilibrium be
48、reached? From the production function: Y = aK, it follows that y = Y/N = a(K/N) = ak. Therefore, the savings function is sy = s(ak) and has a constant slope sa (n + d). Since the savings function and the investment requirement both have constant slopes, namely sa for the savings function and (n + d)
49、 for the investment requirement, these two lines will never intersect. Therefore, a steady-state equilibrium cannot be reached. 3. “The endogenous growth model predicts conditional convergence.” Comment. This statement is false. Conditional convergence is the notion that countries with different sav
50、ings rates but the same rate of population growth and access to the same technology will achieve the same long-term growth rate even though they may achieve a different standard of living. This is contrary to the endogenous growth model, which predicts that there is a positive correlation between sa
