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 rate of growth of output per person slowed worldwide after 1972. This slowdown
appears to reflect a slowdown in productivity growth—the rate at which the production
function is improving over time. Various explanations have been proposed, but the
slowdown remains a mystery. In the second half of the 1990s, productivity grew more
quickly again in the United States and, it appears, a few other countries. Many commentators
attribute the productivity revival to the effects of information technology.
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.
68
C H A P T E R 8 Economic Growth II
Problems and Applications
1. a. 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).
b. The question provides us with the following information about each country:
Developed country: s = 0.28 Less-developed country: s = 0.10
n = 0.01 n = 0.04
g = 0.02 g = 0.02
δ = 0.04 δ = 0.04
Using the equation for y* that we derived in part (a), we can calculate the steadystate
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.
c. The equation for y* that we derived in part (a) shows that the less-developed 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
methods of birth control and implementing disincentives for having children.
Policies that increase the saving rate include increasing public saving by
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.
2. 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 = k
α
, 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 = k
0.3
.
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/(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 steady-state 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:
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).
√
Chapter 8 Economic Growth II 69
70 Answers to Textbook Questions and Problems
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 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. 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.
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—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.
Chapter 8 Economic Growth II 71
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.
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, Δk = sf(k) – (δ + n + g)k = 0. Hence, sAkα
= (δ + n + g)k, or,
after rearranging:
Plugging into the per-worker production function from part (a) gives:
Thus, the ratio of steady-state income per worker in Richland to Poorland is:
c. If α equals 1/3, then Richland should be 41/2
, or two times, richer than Poorland.
d. If = 16, then it must be the case that , which in turn requires that
α equals 2/3. Hence, If the Cobb-Douglas production function puts 2/3 of the
weight on capital and only 1/3 on labor, then we can explain a 16-fold 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.
k
sA
n+g
*
=
+
⎡
⎣
⎢
⎤
⎦
⎥
1−
⎛
⎝⎜
⎞
⎠⎟
δ
1
α
y * 1
=
+ +
⎡
⎣
⎢
⎤
⎦
⎥
−
⎛
⎝⎜
⎞
⎠⎟ −
⎛
⎝⎜
⎞
⎠⎟
A
s
1
1
α
α
α
δ n g
y y
s
g
s
n
Richland Poorland
R ichland
Poorland
* *
/( ) = + +
+
δ
δ
nR ichland
PPoorland g+
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1−
α
α
=
0.32
0.05 + 0.01 + 0.02
0.10
0.05 + 0.03 + 0.02
4
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
= [ ]
1−
1−
⎛
α
α
α
α⎝⎝⎜
⎞
⎠⎟
4 1
α
α−
⎛
⎝⎜
⎞
⎠⎟ α
α1 −
⎛
⎝⎜
⎞
⎠⎟ = 2
72 Answers to Textbook Questions and Problems
5. How do differences in education across countries affect the Solow model? Education is
one factor affecting the efficiency of labor, which we denoted by E. (Other factors affecting
the efficiency of labor include levels of health, skill, and knowledge.) Since country
1 has a more highly educated labor force than country 2, each worker in country 1 is
more efficient. That is, E1 > E2. We will assume that both countries are in steady state.
a. In the Solow growth model, the rate of growth of total income is equal to n + g,
which is independent of the work force’s level of education. The two countries will,
thus, have the same rate of growth of total income because they have the same
rate of population growth and the same rate of technological progress.
b. Because both countries have the same saving rate, the same population growth
rate, and the same rate of technological progress, we know that the two countries
will converge to the same steady-state level of capital per effective worker k*. This
is shown in Figure 8–1.
Hence, output per effective worker in the steady state, which is y* = f(k*), is the
same in both countries. But y* = Y/(L × E) or Y/L = y* E. We know that y* will be
the same in both countries, but that E1 > E2. Therefore, y*E1 > y*E2. This implies
that (Y/L)1 > (Y/L)2. Thus, the level of income per worker will be higher in the
country with the more educated labor force.
c. We know that the real rental price of capital R equals the marginal product of capital
(MPK). But the MPK depends on the capital stock per efficiency unit of labor.
In the steady state, both countries have k*1 = k*2 = k* because both countries have
the same saving rate, the same population growth rate, and the same rate of technological
progress. Therefore, it must be true that R1 = R2 = MPK. Thus, the real
rental price of capital is identical in both countries.
d. Output is divided between capital income and labor income. Therefore, the wage
per effective worker can be expressed as:
w = f(k) – MPK • k.
As discussed in parts (b) and (c), both countries have the same steady-state capital
stock k and the same MPK. Therefore, the wage per effective worker in the two
countries is equal.
Workers, however, care about the wage per unit of labor, not the wage per
effective worker. Also, we can observe the wage per unit of labor but not the wage
per effective worker. The wage per unit of labor is related to the wage per effective
worker by the equation
Wage per Unit of L = wE.
Investment,break-eveninvestment
k
Capital per effective worker
k*
(δ + n + g) k
sf (k)
Figure 8–1
Thus, the wage per unit of labor is higher in the country with the more educated
labor force.
6. a. In the two-sector endogenous growth model in the text, the production function for
manufactured goods is
Y = F(K,(1 – u) EL).
We assumed in this model that this function has constant returns to scale. As in
Section 3-1, constant returns means that for any positive number z,
zY = F(zK, z(1 – u) EL). Setting z = 1/EL, we obtain:
Using our standard definitions of y as output per effective worker and k as capital
per effective worker, we can write this as
y = F(k,(1 – u)).
b. To begin, note that from the production function in research universities, the
growth rate of labor efficiency, ΔE / E, equals g(u). We can now follow the logic of
Section 8-1, substituting the function g(u) for the constant growth rate g. In order
to keep capital per effective worker (K/EL) constant, break-even investment
includes three terms: δk is needed to replace depreciating capital, nk is needed to
provide capital for new workers, and g(u) is needed to provide capital for the
greater stock of knowledge E created by research universities. That is, break-even
investment is (δ + n + g(u))k.
c. Again following the logic of Section 8-1, the growth of capital per effective worker
is the difference between saving per effective worker and break-even investment
per effective worker. We now substitute the per-effective-worker production function
from part (a), and the function g(u) for the constant growth rate g, to obtain:
Δk = sF(k,(1 – u)) – (δ + n + g(u))k.
In the steady state, Δk = 0, so we can rewrite the equation above as:
sF(k,(1 – u)) = (δ + n + g(u))k
As in our analysis of the Solow model, for a given value of u we can plot the leftand
right-hand sides of this equation:
The steady state is given by the intersection of the two curves.
Chapter 8 Economic Growth II 73
Y
EL
F
K
EL
u= −
⎛
⎝⎜
⎞
⎠⎟,( ) .1
Investment,break-eveninvestment
Capital per effective worker
[δ + n + g(u)]k
sF (k, 1 – u)
Figure 8–2
d. The steady state has constant capital per effective worker k as given by Figure
8–2 above. We also assume that in the steady state, there is a constant share of
time spent in research universities, so u is constant. (After all, if u were not constant,
it wouldn’t be a “steady” state!). Hence, output per effective worker y is also
constant. Output per worker equals yE, and E grows at rate g(u). Therefore, output
per worker grows at rate g(u). The saving rate does not affect this growth
rate. However, the amount of time spent in research universities does affect this
rate: as more time is spent in research universities, the steady-state growth rate
rises.
e. An increase in u shifts both lines in our figure. Output per effective worker falls
for any given level of capital per effective worker, since less of each worker’s time
is spent producing manufactured goods. This is the immediate effect of the
change, since at the time u rises, the capital stock K and the efficiency of each
worker E are constant. Since output per effective worker falls, the curve showing
saving per effective worker shifts down.
At the same time, the increase in time spent in research universities increases
the growth rate of labor efficiency g(u). Hence, break-even investment [which
we found above in part (b)] rises at any given level of k, so the line showing breakeven
investment also shifts up.
Figure 8–3 below shows these shifts:
In the new steady state, capital per effective worker falls from k1 to k2. Output per
effective worker also falls.
f. In the short run, the increase in u unambiguously decreases consumption. After
all, we argued in part (e) that the immediate effect is to decrease output, since
workers spend less time producing manufacturing goods and more time in
research universities expanding the stock of knowledge. For a given saving rate,
the decrease in output implies a decrease in consumption.
The long-run steady-state effect is more subtle. We found in part (e) that output
per effective worker falls in the steady state. But welfare depends on output
(and consumption) per worker, not per effective worker. The increase in time
spent in research universities implies that E grows faster. That is, output per
worker equals yE. Although steady-state y falls, in the long run the faster growth
rate of E necessarily dominates. That is, in the long run, consumption unambiguously
rises.
Nevertheless, because of the initial decline in consumption, the increase in u
is not unambiguously a good thing. That is, a policymaker who cares more about
74 Answers to Textbook Questions and Problems
Investment,break-eveninvestment
Capital per effective worker
[δ + n + g(u2
)]k
sF (k, 1 – u1
)
B
A
[δ + n + g(u1
)]k
sF (k, 1 – u2
)
k2
k1
Figure 8–3
current generations than about future generations may decide not to pursue a policy
of increasing u. (This is analogous to the question considered in Chapter 7 of
whether a policymaker should try to reach the Golden Rule level of capital per
effective worker if k is currently below the Golden Rule level.)
More Problems and Applications to Chapter 8
1. a. The growth in total output (Y) depends on the growth rates of labor (L), capital
(K), and total factor productivity (A), as summarized by the equation:
ΔY/Y = αΔK/K + (1 – α)ΔL/L + ΔA/A,
where α is capital’s share of output. We can look at the effect on output of a 5-percent
increase in labor by setting ΔK/K = ΔA/A = 0. Since α = 2/3, this gives us
ΔY/Y = (1/3) (5%)
= 1.67%.
A 5-percent increase in labor input increases output by 1.67 percent.
Labor productivity is Y/L. We can write the growth rate in labor productivity
as
= – .
Substituting for the growth in output and the growth in labor, we find
Δ(Y/L)/(Y/L) = 1.67% – 5.0%
= –3.34%.
Labor productivity falls by 3.34 percent.
To find the change in total factor productivity, we use the equation
ΔA/A = ΔY/Y – αΔK/K – (1 – α)ΔL/L.
For this problem, we find
ΔA/A = 1.67% – 0 – (1/3) (5%)
= 0.
Total factor productivity is the amount of output growth that remains after we
have accounted for the determinants of growth that we can measure. In this case,
there is no change in technology, so all of the output growth is attributable to measured
input growth. That is, total factor productivity growth is zero, as expected.
b. Between years 1 and 2, the capital stock grows by 1/6, labor input grows by 1/3,
and output grows by 1/6. We know that the growth in total factor productivity is
given by
ΔA/A = ΔY/Y – αΔK/K – (1 – α)ΔL/L.
Substituting the numbers above, and setting α = 2/3, we find
ΔA/A = (1/6) – (2/3)(1/6) – (1/3)(1/3)
= 3/18 – 2/18 – 2/18
= – 1/18
= – .056.
Total factor productivity falls by 1/18, or approximately 5.6 percent.
2. By definition, output Y equals labor productivity Y/L multiplied by the labor force L:
Y = (Y/L)L.
Chapter 8 Economic Growth II 75
Δ(Y/L)
Y/L
ΔL
L
ΔY
Y
Using the mathematical trick in the hint, we can rewrite this as
= + .
We can rearrange this as
= – .
Substituting for ΔY/Y from the text, we find
Using the same trick we used above, we can express the term in brackets as
ΔK/K – ΔL/L = Δ(K/L)/(K/L).
Making this substitution in the equation for labor productivity growth, we conclude
that
3. We know the following:
ΔY/Y = n + g = 3.6%
ΔK/K = n + g = 3.6%
ΔL/L = n = 1.8%
Capital’s share = α = 1/3
Labor’s share = 1 – α = 2/3.
Using these facts, we can easily find the contributions of each of the factors, and
then find the contribution of total factor productivity growth, using the following equa-
tions:
Output
=
Capital’s
+
Labor’s
+
Total Factor
Growth Contribution Contribution Productivity
ΔY αΔK (1 – α)ΔL ΔA
Y K L A
3.6% = (1/3)(3.6%) + (2/3)(1.8%) + ΔA/A
We can easily solve this for ΔA/A, to find that
3.6% = 1.2% + 1.2% + 1.2%.
We conclude that the contribution of capital is 1.2% per year, the contribution of labor
is 1.2% per year, and the contribution of total factor productivity growth is 1.2% per
year. These numbers match the ones in Table 8–3 in the text for the United States from
1948–2002.
76 Answers to Textbook Questions and Problems
Δ(Y/L)
=
ΔA
+
αΔK
+ (1 – α)
ΔL
–
ΔL
Y/L A K L L
Δ(Y/L)
=
ΔA
+
αΔ(K/L)
.
Y/L A K/L
=
ΔA
+
αΔK
–
αΔL
A K L
= + −
⎡
⎣⎢
⎤
⎦⎥
Δ
α
Δ ΔA
A
K
K
L
L
.
Δ(Y/L)
Y/L
ΔL
L
ΔY
Y
ΔY
Y
ΔL
L
Δ(Y/L)
Y/L