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Answers to Textbook Questions and Problems
CHAPTER 9 Economic Growth II: Technology, Empirics, and Policy
Questions for Review
1. In the Solow model, we find that only technological progress can affect the steady-state rate of growth
in income per worker. Growth in the capital stock (through high saving) has no effect on the steady-state growth rate of income per worker; neither does population growth. But technological progress can lead to sustained growth.
2. In the steady state, output per person in the Solow model grows at the rate of technological progress g.
Capital per person also grows at rate g. Note that this implies that output and capital per effective
worker are constant in steady state. In the U.S. data, output and capital per worker have both grown at about 2 percent per year for the past half-century.
3. To decide whether an economy has more or less capital than the Golden Rule, we need to compare the
marginal product of capital net of depreciation (MPK – δ) with the growth rate of total output (n + g). The growth rate of GDP is readily available. Estimating the net marginal product of capital requires a little more work but, as shown in the text, can be backed out of available data on the capital stock relative to GDP, the total amount of depreciation relative to GDP, and capital’s share in GDP.
4. Economic policy can influence the saving rate by either increasing public saving or providing
incentives to stimulate private saving. Public saving is the difference between government revenue and government spending. If spending exceeds revenue, the government runs a budget deficit, which is negative saving. Policies that decrease the deficit (such as reductions in government purchases or
increases in taxes) increase public saving, whereas policies that increase the deficit decrease saving. A variety of government policies affect private saving. The decision by a household to save may depend on the rate of return; the greater the return to saving, the more attractive saving becomes. Tax incentives such as tax-exempt retirement accounts for individuals and investment tax credits for corporations increase the rate of return and encourage private saving.
5. The legal system is an example of an institutional difference between countries that might explain
differences in income per person. Countries that have adopted the English style common law system tend to have better developed capital markets, and this leads to more rapid growth because it is easier for businesses to obtain financing. The quality of government is also important. Countries with more government corruption tend to have lower levels of income per person.
6. Endogenous growth theories attempt to explain the rate of technological progress by explaining the
decisions that determine the creation of knowledge through research and development. By contrast, the Solow model simply took this rate as exogenous. In the Solow model, the saving rate affects growth temporarily, but diminishing returns to capital eventually force the economy to approach a steady state in which growth depends only on exogenous technological progress. By contrast, many endogenous growth models in essence assume that there are constant (rather than diminishing) returns to capital, interpreted to include knowledge. Hence, changes in the saving rate can lead to persistent growth.
Problems and Applications
1. a. In the Solow model with technological progress, y is defined as output per effective worker, and k
is defined as capital per effective worker. The number of effective workers is defined as L ? E (or LE), where L is the number of workers, and E measures the efficiency of each worker. To find output per effective worker y, divide total output by the number of effective workers:
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YK2(LE)2=LELEYK2L2E2=LELE
11111YK2=11LEL2E21Y?K?2÷=??LEèLE÷?2 y=k11
b. To solve for the steady-state value of y as a function of s, n, g, and δ, we begin with the equation
for the change in the capital stock in the steady state:
Δk = sf(k) – (δ + n + g)k = 0.
The production function y=k can also be rewritten as y2 = k. Plugging this production function
into the equation for the change in the capital stock, we find that in the steady state:
sy – (δ + n + g)y2 = 0.
Solving this, we find the steady-state value of y:
y* = s/(δ + n + g).
c. The question provides us with the following information about each country:
Atlantis:
s = 0.28 n = 0.01 g = 0.02 δ = 0.04
Xanadu:
s = 0.10 n = 0.04 g = 0.02 δ = 0.04
Using the equation for y* that we derived in part (a), we can calculate the steady-state values of y for each country.
Developed country: y* = 0.28/(0.04 + 0.01 + 0.02) = 4 Less-developed country: y* = 0.10/(0.04 + 0.04 + 0.02) = 1
2. a. In the steady state, capital per effective worker is constant, and this leads to a constant level of
output per effective worker. Given that the growth rate of output per effective worker is zero, this means the growth rate of output is equal to the growth rate of effective workers (LE). We know labor grows at the rate of population growth n and the efficiency of labor (E) grows at rate g. Therefore, output grows at rate n+g. Given output grows at rate n+g and labor grows at rate n, output per worker must grow at rate g. This follows from the rule that the growth rate of Y/L is equal to the growth rate of Y minus the growth rate of L.
b. First find the output per effective worker production function by dividing both sides of the
production function by the number of effective workers LE:
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YK3(LE)3=LELEYK3L3E3=LELE
12212YK3=11LEL3E31Y?K?3=?÷LEèLE?y=k311
To solve for capital per effective worker, we start with the steady state condition:
Δk = sf(k) – (δ + n + g)k = 0.
Now substitute in the given parameter values and solve for capital per effective worker (k):
Substitute the value for k back into the per effective worker production function to find output per effective worker is equal to 2. The marginal product of capital is given by
Substitute the value for capital per effective worker to find the marginal product of capital is equal to 1/12.
c. According to the Golden Rule, the marginal product of capital is equal to (δ + n + g) or 0.06. In the current steady state, the marginal product of capital is equal to 1/12 or 0.083. Therefore, we have less capital per effective worker in comparison to the Golden Rule. As the level of capital per effective worker rises, the marginal product of capital will fall until it is equal to 0.06. To increase capital per effective worker, there must be an increase in the saving rate. d. During the transition to the Golden Rule steady state, the growth rate of output per worker will increase. In the steady state, output per worker grows at rate g. The increase in the saving rate will increase output per effective worker, and this will increase output per effective worker. In the new steady state, output per effective worker is constant at a new higher level, and output per worker is growing at rate g. During the transition, the growth rate of output per worker jumps up, and then transitions back down to rate g.
3. To solve this problem, it is useful to establish what we know about the U.S. economy: ? A Cobb–Douglas production function has the form y = kα, where α is capital’s share of income.
The question tells us that α = 0.3, so we know that the production function is y = k0.3.
? In the steady state, we know that the growth rate of output equals 3 percent, so we know that (n +
g) = 0.03.
? The depreciation rate δ = 0.04. ? The capital–output ratio K/Y = 2.5. Because k/y = [K/(LE)]/[Y/(LE)] = K/Y, we also know that k/y =
2.5. (That is, the capital–output ratio is the same in terms of effective workers as it is in levels.)
a. Begin with the steady-state condition, sy = (δ + n + g)k. Rewriting this equation leads to a formula
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for saving in the steady state:
s = (δ + n + g)(k/y).
Plugging in the values established above:
s = (0.04 + 0.03)(2.5) = 0.175.
The initial saving rate is 17.5 percent.
b. We know from Chapter 3 that with a Cobb–Douglas production function, capital’s share of
income α = MPK(K/Y). Rewriting, we have
MPK = α/(K/Y).
Plugging in the values established above, we find
MPK = 0.3/2.5 = 0.12.
c. We know that at the Golden Rule steady state:
MPK = (n + g + δ).
Plugging in the values established above:
MPK = (0.03 + 0.04) = 0.07.
At the Golden Rule steady state, the marginal product of capital is 7 percent, whereas it is 12 percent in the initial steady state. Hence, from the initial steady state we need to increase k to achieve the Golden Rule steady state.
d. We know from Chapter 3 that for a Cobb–Douglas production function, MPK = α (Y/K). Solving
this for the capital–output ratio, we find
K/Y = α/MPK.
We can solve for the Golden Rule capital–output ratio using this equation. If we plug in the value 0.07 for the Golden Rule steady-state marginal product of capital, and the value 0.3 for α, we find
K/Y = 0.3/0.07 = 4.29.
In the Golden Rule steady state, the capital–output ratio equals 4.29, compared to the current capital–output ratio of 2.5.
e. We know from part (a) that in the steady state
s = (δ + n + g)(k/y),
where k/y is the steady-state capital–output ratio. In the introduction to this answer, we showed that k/y = K/Y, and in part (d) we found that the Golden Rule K/Y = 4.29. Plugging in this value and those established above:
s = (0.04 + 0.03)(4.29) = 0.30.
To reach the Golden Rule steady state, the saving rate must rise from 17.5 to 30 percent. This
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result implies that if we set the saving rate equal to the share going to capital (30 percent), we will achieve the Golden Rule steady state.
4. a. In the steady state, we know that sy = (δ + n + g)k. This implies that k/y = s/(δ + n + g). Since s, δ, n, and g are constant, this means that the ratio k/y is also constant. Since k/y =
[K/(LE)]/[Y/(LE)] = K/Y, we can conclude that in the steady state, the capital–output ratio is constant.
b. We know that capital’s share of income = MPK ? (K/Y). In the steady state, we know from part (a)
that the capital–output ratio K/Y is constant. We also know from the hint that the MPK is a function of k, which is constant in the steady state; therefore the MPK itself must be constant. Thus, capital’s share of income is constant. Labor’s share of income is 1 – [Capital’s Share]. Hence, if capital’s share is constant, we see that labor’s share of income is also constant.
c. We know that in the steady state, total income grows at n + g, defined as the rate of population
growth plus the rate of technological change. In part (b) we showed that labor’s and capital’s share of income is constant. If the shares are constant, and total income grows at the rate n + g, then labor income and capital income must also grow at the rate n + g.
d. Define the real rental price of capital R as R = Total Capital Income/Capital Stock = (MPK ? K)/K = MPK. We know that in the steady state, the MPK is constant because capital per effective worker k is
constant. Therefore, we can conclude that the real rental price of capital is constant in the steady state.
To show that the real wage w grows at the rate of technological progress g, define TLI = Total Labor Income L = Labor Force Using the hint that the real wage equals total labor income divided by the labor force: w = TLI/L. Equivalently, wL = TLI. In terms of percentage changes, we can write this as Δw/w + ΔL/L = ΔTLI/TLI. This equation says that the growth rate of the real wage plus the growth rate of the labor force
equals the growth rate of total labor income. We know that the labor force grows at rate n, and, from part (c), we know that total labor income grows at rate n + g. We, therefore, conclude that the real wage grows at rate g.
5. a. The per worker production function is F(K, L)/L = AKα L1–α/L = A(K/L)α = Akα
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曼昆-宏观经济经济学第九版-英文原版答案9
