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3. Is Growth Good?
Students like to discuss the benefits versus the costs of growth. While it’s easy for the government to
calculate output (GDP), it’s much harder to account for the quality of life. And everyone is aware of the
tradeoffs between growth and quality of life, as seen in such side effects of growth as pollution, traffic, and
suburban sprawl.
1. The three sources of economic growth are capital growth, labor growth, and productivity growth. The
growth accounting approach is derived from the production function.
2. A decline in productivity growth is the primary reason for the slowdown in output growth in the
United States since 1973. Productivity growth may have declined because of deterioration in the legal
3. The rise in productivity growth in the 1990s occurred because of the revolution in information and
communications technologies (ICT). Not only were there improvements in ICT, but also government
4. A steady state is a situation in which the economy’s output per worker, consumption per worker,
and capital stock per worker are constant.
5. If there is no productivity growth, then output per worker, consumption per worker, and capital per
worker will all be constant in the long run. This represents a steady state for the economy.
6. The statement is false. Increases in the capital-labor ratio increase consumption per worker in the
steady state only up to a point. If the capital-labor ratio is too high, then consumption per worker may
7. (a) An increase in the saving rate increases long-run living standards, as higher saving allows for
more investment and a larger capital stock.
8. Endogenous growth theory suggests that the main sources of productivity growth are accumulation
of human capital (the knowledge, skills, and training of individuals) and technological innovation
9. Government policies to promote economic growth include policies to raise the saving rate and
policies to increase productivity. One way to increase the saving rate is to increase the real return
to saving by providing a tax break, as Individual Retirement Accounts did in the United States.
Unfortunately, the response of saving to increases in the real rate of return is small. Another way to
increase the saving rate is to reduce the government budget deficit. However, the theory of Ricardian
1. Hare: $5000 (1.03)70 $39,589
Tortoise: $5000 (1.01)70 $10,034
2.
20 Years Ago Today Percent Change
Y1000 1300 30%
K2500 3250 30%
N 500 575 15%
(a) A/A Y/Y aK K/K aN N/N
30% (0.3 30%) 0.7 15%
30% 9% 10.5%
10.5%
3. (a)
Year K N Y K/N Y/N
1 200 1000 617 0.20 0.617
2 250 1000 660 0.25 0.660
3 250 1250 771 0.20 0.617
4 300 1200 792 0.25 0.660
This production function can be written in per-worker form since Y/N K.3N.7/N K.3/N.3
and 4, and so is Y/N.
(b)
1 200 1000 1231 0.20 1.231
2 250 1000 1316 0.25 1.316
3 250 1250 1574 0.20 1.259
4 300 1200 1609 0.25 1.341
This production function can’t be written in per-worker form since Y/N K.3N.8/N K.3/N.2. Note
4. To answer this problem, an approximate solution can be found by finding the ratio GDP (2008)/GDP
(1950), taking the natural logarithm of that ratio and dividing by 58. This is the answer given in the
1950 2008 Ratio Rate
Australia 7,412 25,301 3.41 2.1%
Canada 7,291 25,267 3.47 2.1%
France 5,186 22,223 4.29 2.5%
5. (a) sf(k) (n d)k
0.3 3k.5 (0.05 0.1)k
0.9k.5 0.15k
0.9/0.15 k/k.5
6 k.5
k 62 36
0.4 3k.5 (0.05 0.1)k
1.2k.5 0.15k
1.2/0.15 k/k.5
8 k.5
k 82 64
0.3 3k.5 (0.08 0.1)k
0.9k.5 0.18k
0.9/0.18 k/k.5
5 k.5
k 52 25
0.3 4k.5 (0.05 0.1)k
1.2k.5 0.15k
1.2/0.15 k/k.5
8 k.5
k 82 64
6. (a) In steady state, sf(k) (n d)k
0.1 6k.5 (0.01 0.14)k
0.6k.5 0.15k
0.6/0.15 k/k.5
4 k.5
k 42 16 capital per worker
y 6k.5 6 4 24 output per worker
c .9 y .9 24 21.6 consumption per worker
7. First, derive saving per worker as sy y c g [1 0.5(1 t) t] 8k.5 0.5(1 t)8k.5 4 (1
t)k.5
1600, y 8 1600.5 320, c 0.5(1 0)320 160, and (n d)k 0.1 1600 160
investment
per worker.
(b) When t 0.5, sy 4 (1 0.5)k.5 2k.5 national saving per worker
1. (a) The destruction of some of a country’s capital stock in a war would have no effect on the steady
state, because there has been no change in s, f, n, or d. Instead, k is reduced temporarily, but
equilibrium forces eventually drive k to the same steady-state value as before.
(b) Immigration raises n from n1 to n2 in Figure 6.3. The rise in n lowers steady-state k, leading to a
lower steady-state consumption per worker.
2. (a) Solow model
The rise in capital depreciation shifts up the (n d)k line from (n d1)k to (n d2)k, as shown in
Figure 6.5. The equilibrium steady-state capital-labor ratio declines. With a lower capital-labor
ratio, output per worker is lower, so consumption per worker is lower (using the assumption that
the capital-labor ratio is not so high that an increase in k will reduce consumption per worker).
3. (a) With a balanced budget T/N g. National saving is S s(Y – T) sN[(Y/N)(T/N)]
sN(y – g). Setting saving equal to investment gives
S I,
sN(y – g) (n d)K,
s(y – g) (n d)k,
4. St sYt – hKt Nt(syt – hkt). Setting St It yields Nt(syt – hkt) (n d)Kt. Dividing through by Nt and
eliminating time subscripts for steady-state variables gives sy – hk (n d)k. Rearranging and using
the expression y f(k) gives sf(k) (n d h)k.
The steady-state value of capital per worker, k*, is given by the intersection of the (n d h)k line
5. The initial level of the capital-labor ratio is irrelevant for the steady state. Two economies that are
identical except for their initial capital-labor ratios will have exactly the same steady state.
Since the two economies must have the same growth rate at the steady state, and since the economy
with the higher current capital-labor ratio has higher current output per worker, then the country with
the lower current capital-labor ratio must grow faster.
6. The growth accounting equation is
Y/Y A/A (aK K/K) (aN N/N).
We are just increasing the amount of capital and labor, and there is no change in productivity, so
A/A 0. If the production function can be written in per-worker terms, then total output must
7. Assume there are a constant number of workers, N, so that Ny Y and Nk K. Since y Akah1–a and
h Bk, then y Aka(Bk)1–a (AB1–a)k. Then Y Ny (AB1–a)K XK, where X equals AB1–a. This puts
3. The rise in productivity growth in the 1990s occurred because of the revolution in information and
communications technologies (ICT). Not only were there improvements in ICT, but also government
4. A steady state is a situation in which the economy’s output per worker, consumption per worker,
and capital stock per worker are constant.
5. If there is no productivity growth, then output per worker, consumption per worker, and capital per
worker will all be constant in the long run. This represents a steady state for the economy.
6. The statement is false. Increases in the capital-labor ratio increase consumption per worker in the
steady state only up to a point. If the capital-labor ratio is too high, then consumption per worker may
7. (a) An increase in the saving rate increases long-run living standards, as higher saving allows for
more investment and a larger capital stock.
8. Endogenous growth theory suggests that the main sources of productivity growth are accumulation
of human capital (the knowledge, skills, and training of individuals) and technological innovation
9. Government policies to promote economic growth include policies to raise the saving rate and
policies to increase productivity. One way to increase the saving rate is to increase the real return
to saving by providing a tax break, as Individual Retirement Accounts did in the United States.
Unfortunately, the response of saving to increases in the real rate of return is small. Another way to
increase the saving rate is to reduce the government budget deficit. However, the theory of Ricardian
1. Hare: $5000 (1.03)70 $39,589
Tortoise: $5000 (1.01)70 $10,034
2.
20 Years Ago Today Percent Change
Y1000 1300 30%
K2500 3250 30%
N 500 575 15%
(a) A/A Y/Y aK K/K aN N/N
30% (0.3 30%) 0.7 15%
30% 9% 10.5%
10.5%
3. (a)
Year K N Y K/N Y/N
1 200 1000 617 0.20 0.617
2 250 1000 660 0.25 0.660
3 250 1250 771 0.20 0.617
4 300 1200 792 0.25 0.660
This production function can be written in per-worker form since Y/N K.3N.7/N K.3/N.3
and 4, and so is Y/N.
(b)
1 200 1000 1231 0.20 1.231
2 250 1000 1316 0.25 1.316
3 250 1250 1574 0.20 1.259
4 300 1200 1609 0.25 1.341
This production function can’t be written in per-worker form since Y/N K.3N.8/N K.3/N.2. Note
4. To answer this problem, an approximate solution can be found by finding the ratio GDP (2008)/GDP
(1950), taking the natural logarithm of that ratio and dividing by 58. This is the answer given in the
1950 2008 Ratio Rate
Australia 7,412 25,301 3.41 2.1%
Canada 7,291 25,267 3.47 2.1%
France 5,186 22,223 4.29 2.5%
5. (a) sf(k) (n d)k
0.3 3k.5 (0.05 0.1)k
0.9k.5 0.15k
0.9/0.15 k/k.5
6 k.5
k 62 36
0.4 3k.5 (0.05 0.1)k
1.2k.5 0.15k
1.2/0.15 k/k.5
8 k.5
k 82 64
0.3 3k.5 (0.08 0.1)k
0.9k.5 0.18k
0.9/0.18 k/k.5
5 k.5
k 52 25
0.3 4k.5 (0.05 0.1)k
1.2k.5 0.15k
1.2/0.15 k/k.5
8 k.5
k 82 64
6. (a) In steady state, sf(k) (n d)k
0.1 6k.5 (0.01 0.14)k
0.6k.5 0.15k
0.6/0.15 k/k.5
4 k.5
k 42 16 capital per worker
y 6k.5 6 4 24 output per worker
c .9 y .9 24 21.6 consumption per worker
7. First, derive saving per worker as sy y c g [1 0.5(1 t) t] 8k.5 0.5(1 t)8k.5 4 (1
t)k.5
1600, y 8 1600.5 320, c 0.5(1 0)320 160, and (n d)k 0.1 1600 160
investment
per worker.
(b) When t 0.5, sy 4 (1 0.5)k.5 2k.5 national saving per worker
1. (a) The destruction of some of a country’s capital stock in a war would have no effect on the steady
state, because there has been no change in s, f, n, or d. Instead, k is reduced temporarily, but
equilibrium forces eventually drive k to the same steady-state value as before.
(b) Immigration raises n from n1 to n2 in Figure 6.3. The rise in n lowers steady-state k, leading to a
lower steady-state consumption per worker.
2. (a) Solow model
The rise in capital depreciation shifts up the (n d)k line from (n d1)k to (n d2)k, as shown in
Figure 6.5. The equilibrium steady-state capital-labor ratio declines. With a lower capital-labor
ratio, output per worker is lower, so consumption per worker is lower (using the assumption that
the capital-labor ratio is not so high that an increase in k will reduce consumption per worker).
3. (a) With a balanced budget T/N g. National saving is S s(Y – T) sN[(Y/N)(T/N)]
sN(y – g). Setting saving equal to investment gives
S I,
sN(y – g) (n d)K,
s(y – g) (n d)k,
4. St sYt – hKt Nt(syt – hkt). Setting St It yields Nt(syt – hkt) (n d)Kt. Dividing through by Nt and
eliminating time subscripts for steady-state variables gives sy – hk (n d)k. Rearranging and using
the expression y f(k) gives sf(k) (n d h)k.
The steady-state value of capital per worker, k*, is given by the intersection of the (n d h)k line
5. The initial level of the capital-labor ratio is irrelevant for the steady state. Two economies that are
identical except for their initial capital-labor ratios will have exactly the same steady state.
Since the two economies must have the same growth rate at the steady state, and since the economy
with the higher current capital-labor ratio has higher current output per worker, then the country with
the lower current capital-labor ratio must grow faster.
6. The growth accounting equation is
Y/Y A/A (aK K/K) (aN N/N).
We are just increasing the amount of capital and labor, and there is no change in productivity, so
A/A 0. If the production function can be written in per-worker terms, then total output must
7. Assume there are a constant number of workers, N, so that Ny Y and Nk K. Since y Akah1–a and
h Bk, then y Aka(Bk)1–a (AB1–a)k. Then Y Ny (AB1–a)K XK, where X equals AB1–a. This puts
