Unformatted text preview: state values of y for each country.
Developed country:
Lessdeveloped country: y* = 0.28/(0.04 + 0.01 + 0.02) = 4.
y* = 0.10/(0.04 + 0.04 + 0.02) = 1. The equation for y* that we derived in part (a) shows that the lessdeveloped country could raise its level of income by reducing its population growth rate n or by
increasing its saving rate s. Policies that reduce population growth include introducing 8
Textbook Chapter methods of birth control and implementing disincentives for having children. Policies that increase the saving rate include increasing public saving by
Answer key for #2 and #4.
reducing the budget deficit and introducing private saving incentives such as
I.R.A.’s and other tax concessions that increase the return to saving.
c. 2. To solve this problem, it is useful to establish what we know about the U.S. economy:
α A Cobb–Douglas production function has the form y = k , where α is capital’s share of
income. The question tells us that α = 0.3, so we know that the production function is
0.3
y=k .
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/(L × E)]/[Y/(L × E)] = 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 steadystate condition, sy = (δ + n + g)k. Rewriting this equation
leads to a formula for saving in the steady state:
s = (δ + n + g)(k/y).
Plugging in the values established above: b. s = (0.04 + 0.03)(2.5) = 0.175.
The initial saving rate is 17.5 percent.
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). 70 Answers to Textbook Questions and Problems Plugging in the values established above, we find:
c. MPK = 0.3/2.5 = 0.12.
We know that at the Golden Rule steady state:
MPK = (n + g + δ).
Plugging in the values established above: d. 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.
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 steadystate 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 steadystate 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 result implies that if we set the saving rate equal to the share going
to capital (30%), we will achieve the Golden Rule steady state. 3. a. b. c. d. 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/(L × E)]/[Y/(L × E)] = K/Y, we can conclude that in the steady state,
the capital–output ratio is constant.
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.
We know that in the steady state, total income grows at n + g—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.
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. In terms of percentage changes, we can write this as
∆w/w + ∆L/L = ∆TLI/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.
4. a. The per worker production function is F(K,L)/L = AKα L1–α/L = A(K/L)α = Akα.
b. In the steady state, ∆k = sf(k) – (δ + n + g)k = 0. Hence, sAkα = (δ + n + g)k, or,
after rearranging:
1 s A 1− α *
k = δ + n+ g Plugging into the perworker production function from part (a) gives:
y =A
* 1 1 −α α 1−α s
δ + n + g Thus, the ratio of steadystate income per worker in Richland to Poorland is:
α (y *
Richland / y*
Poorland s R ich land 1− α δ+ n
) = R ich land + g s Poor land δ + n P oor land + g α 0.32 1− α = 0.05 + 0.01 + 0.02
0.10 0.05 + 0.03 + 0.02 α = [ 4 ] 1− α c.
d. If α equals 1/3, then Richland should be 41/2, or two times, richer than Poorland.
α α
4 1− α = 16, then it must be the case that = 2, which in turn requires that
If
1 − α α equals 2/3. Hence, If the CobbDouglas production function puts 2/3 of the
weight on capital and only 1/3 on labor, then we can explain a 16fold difference in
levels of income per worker. One way to justify this might be to think about capital
more broadly to include human capital—which must also be accumulated through
investment, much in the way one accumulates physical capital. ...
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 Fall '08
 SMITH
 Economics, Thermodynamics, Steady State, Ratio

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