中级宏观课后习题答案1-7.doc

Chapter 3 课后习题答案
Solutions to the Problems in the Textbook

Conceptual Problems

1. 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-qKq.

In this function, Y represents the level of output, 1 - q and q 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 individual 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 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. 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 significant positive effect on a country’s 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 only 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 that

Y = F(K, N, H) = ANaKbHc with a + b + c = 1.

In this case, the growth rate of output can be calculated as

DY/Y = DA/A + a(DN/N) + b(DK/K) + c(DH/H).

Clearly, here the magnitude of DA/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 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, 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, (and in absence of technological progress) the growth rate of the steady-state output per capita is determined only the rate of population growth n = (DN/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 as 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 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 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 same number of less skilled workers.)

Technical Problems

1 a. According to Equation (2), the growth of output is equal to the growth in labor times labor’s share of income plus the growth of capital times capital’s share of income plus the rate of technical progress, that is,

DY/Y = (1 - q)(DN/N) + q(DK/K) + DA/A,

where1 - q is the share of labor (N) and q is the share of capital (K). Therefore, if we assume that the rate of technological progress is zero (DA/A = 0), output grows at an annual rate of 3.6 percent, since

DY/Y = (0.6)(2%) + (0.4)(6%) + 0% = 1.2% + 2.4% = + 3.6%,

b. The so-called "Rule of 70" suggests 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 DA/A = 2%, we can calculate economic growth as

DY/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%.

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 capital plus the growth rate of total factor productivity, that is,

DY/Y = (1 - q)(DN/N) + q(DK/K) + DA/A,

where1 - 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, we can calculate the change in total factor productivity in the following way

3% = (0.7)(1%) + (0.3)(1%) + DA/A ==> DA/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, DN/N = DK/K = 0, and output grows at 3%, then all the growth has to be attributed to the growth in total factor productivity, that is, DA/A = 3%.

3 a. If the capital stock grows by DK/K = 10%, the effect will be an additional growth rate in output of

DY/Y = (0.3)(10%) = 3%.

b. If labor grows by DN/N = 10%, the effect will be an additional growth rate in output of

DY/Y = (0.7)(10%) = 7%.

c. If output grows at DY/Y = 7% due to an increase in labor by DN/N = 10% and this increase in labor is entirely due to population growth, then per capita income will decrease. Therefore, people’s welfare will decrease. We can calculate the change in per capita income as follows:

Dy/y = DY/Y - DN/N = 7% - 10% = - 3%.

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 Dy/y = 7%. Therefore people's welfare (or at least their living standard) will 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, 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 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, then 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, 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 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 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 invested later in an effort to rebuild) or decreases (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.

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 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 optimal capital-labor ratio will be determined by the intersection of the sy-curve and the new (n1 + 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-state 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. The 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 ==>

DY/Y = DA/A + a(DK/K) + b(DN/N) + c(DZ/Z), with a + b + c = 1.

Now assume that there is no technological progress (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 available natural resources 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, output will grow at a rate less than n since a + b < 1, and output per worker will fall.

b. If technological progress occurs (DA/A > 0), output will grow faster than before, namely

DY/Y = DA/A + (a + b)n.

If DA/A < cn, output will grow at a lower rate than n, in which case output per worker will still decrease.
If DA/A = cn, output will grow at the same rate as n, in which case output per worker will remain the same.
But if DA/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 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, 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 1, we have Y = K1/2N1/2 . In this case, capital's and labor's 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)/(.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, DA/A = 0. But if total factor productivity grows at DA/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
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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 function 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 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 higher optimal capital-labor ratio.

9. The Cobb-Douglas production function is defined as

Y F(N, K) = AN1-qKq.

The marginal product of labor can then be derived as

MPN = (DY)/(DN) = (1 - q)AN-qKq = (1 - q)AN1-qKq/N = = (1 - q)(Y/N)

== > labor's share of income = [MPN*N]/Y = [(1 - q)(Y/N)](N/Y) = (1 - q).

10 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)/(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 investment), 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 requirement

From c = k1/2 - (n + d)k ==> (Dc/Dk) = (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 < 100, we have less capital at the steady state than the golden rule suggests.

c. From k = sy/(n + d) = sk1/2/(n + d) ==> s = k1/2(n + d) = 100.05) = .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 consumption.

Empirical Problems

1. The average monthly growth rate of the U.S. population over the period 2000-2010 was 0.1 percent. Over the same period, t