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Tutorial 4 Solutions
Economic Growth II: Population Growth, Technological Progress and Policy
Question 1
Draw a well-labelled graph that illustrates the steady state of the Solow model with population
growth. Use the graph to find what happens to steady-state capital per worker and income per worker
in response to each of the following exogenous changes.
(a) Better birth-control methods reduce the rate of population growth.
(b) A one-time, permanent improvement in technology increases the amount of output that can
be produced from any given amount of capital and labour.
Answer:
(a) A reduction in the population growth rate shifts the break-even investment curve downward,
as illustrated in the graph below. Since actual investment is now greater than break-even
investment, the level of capital per worker increases. In the new steady state, capital per
worker and output per worker are both higher.

(b) The once-off permanent technological improvement increases output per worker ሺሻ for any
given positive value of , so the investment curve shifts upward, as illustrated in the graph on
the next page. Since actual investment is now greater than break-even investment at the
original steady-state level, the level of capital per worker will increase. In the new steady
state, capital per worker and output per worker are both higher.
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Question 2
The economy of Beta can be described by the Solow growth model with population growth. The
following are some characteristics of the Beta economy:
saving rate () 0.15
depreciation rate () 0.06
steady-state capital per worker () 4.62
population growth rate () 0.005
steady-state output per worker 2
(a) What is the steady-state growth rate of output per worker in Beta?
(b) What is the steady-state growth rate of total output in Beta?
(c) What is the level of steady-state saving per worker in Beta?
Answer:
(a) In the steady state, capital per worker is constant, so output per worker is constant. Thus, the
growth rate of steady-state output per worker is zero.
(b) The population grows at a rate of 0.5 per cent; thus, capital must grow at a rate of 0.5 per cent
to maintain a constant capital per worker ratio in the steady state. Therefore, given the
constant returns to scale production function, the growth rate of total output is 0.5 per cent.
(c) In the steady state, saving per worker is 15 per cent of steady-state income (output) per
worker. Thus, steady-state saving per worker is 0.3.
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Question 3
An economy has a Cobb-Douglas production function: ൌ ఈሺሻଵିఈ . The economy has a capital
share of 1 3⁄ , a saving rate of 24 per cent, a depreciation rate of 3 per cent, a rate of population
growth of 2 per cent, and a rate of labour-augmenting technological change of 1 per cent. It is in
steady state.
(a) At what rates do total output, output per worker, and output per effective worker grow?
(b) Solve for capital per effective worker, output per effective worker, and the marginal product
of capital.
(c) Does the economy have more or less capital than at the Golden Rule steady state? How do
you know? Does the saving rate need to increase or decrease to achieve the Golden Rule
steady state?
(d) Suppose the change in the saving rate you described in part (c) occurs. During the transition
to the Golden Rule steady state, will the growth rate of output per worker be higher or lower
than the rate you derived in part (a)? After the economy reaches its new steady state, will the
growth rate of output per worker be higher or lower than the rate you derived in part (a)?
Explain your answers.
Answer:
(a) In the steady state, capital per effective worker is constant, leading to a constant level of
output per effective worker. Given that the growth rate of output per effective worker is zero,
the growth rate of total output is equal to the growth rate of effective workers. The labour
force grows at the rate of population growth rate , and the efficiency of labour grows at
a rate of . Therefore, total output grows at a rate of ൅ . Given that output grows at a rate
of ൅ and labour grows at a rate of , output per worker must grow at a rate of . This
follows from the rule that the growth rate of ⁄ is approximately equal to the growth rate of
minus the growth rate of . In the steady state in this question, total output grows at 3 per
cent, output per worker grows at 1 per cent, and output per effective worker is constant.
(b) First, find the expression for output per effective worker:
ൌ
ଵଷሺሻଶଷ
ൌ ൬
൰
ଵ
ଷ
ൌ ଵ/ଷ
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To solve for capital per effective worker, we start with the steady-state condition, substitute
in the expression for and the given parameter values, and solve for ∗:
∗ ൌ ሺ ൅ ൅ ሻ∗ 0.24ሺ∗ሻଵ/ଷ ൌ ሺ0.03 ൅ 0.02 ൅ 0.01ሻ∗
ሺ∗ሻଶ/ଷ ൌ 4
∗ ൌ 8
Then, output per effective worker is:
ൌ ሺ∗ሻଵ/ଷ ൌ 2
The marginal product of capital at the steady state is:
ൌ ൌ ሺ1/3ሻሺ∗ሻିଶ/ଷ ൌ ሺ1/3ሻሺ8ሻିଶ/ଷ ൌ 1/12 ൌ 0.083
(c) In the Golden Rule steady state, the marginal product of capital is ሺ ൅ ൅ ሻ ൌ 0.06. In the
current steady state, the marginal product of capital is 1/12, or 0.083. Since there are
diminishing marginal returns to capital, this economy has less capital per effective worker
than in the Golden Rule steady state. To increase capital per effective worker, there must be
an increase in the saving rate.
(d) In any steady state, output per worker grows at the rate . During the transition to the Golden
Rule steady state, the higher saving rate causes the growth rate of output per worker to rise
above until the economy converges to the new steady. That is, during the transition, the
growth rate of output per worker jumps up and then falls back to .
Question 4
Suppose a government is able to reduce its budget deficit permanently. Use the Solow growth model
with population growth and technological progress to graphically illustrate the impact of a permanent
government deficit reduction on the steady-state levels of capital and output per effective worker. Be
sure to label the i. axes, ii. curves, iii. initial steady-state levels, iv. terminal steady-state levels, and
v. direction the curves shift.
Then, answer the following questions:
(a) What is the per-effective-worker impact?
(b) Briefly interpret the predicted impact on output and capital per worker.
(c) How will the steady-state growth rates of income per worker and total income change?
(d) What is the growth rate in real wages?
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Answer:
Reducing the budget deficit will increase public saving and, consequently, the total saving rate or .
When increases, the investment function shifts upwards.

(a) The steady-state capital per effective worker will increase from ଵ∗ to ଶ∗ and the steady-state
income (output) per effective worker will increase from ଵ∗ to ଶ∗.
(b) Both steady-state capital per worker and steady-state output per worker will grow at the rate
of technological progress ().
(c) In the transition phase, the growth rate of capital per worker and output (income) per worker
will be higher than . However, once the new steady state is reached, the steady-state growth
rates of income per worker and total output will be as before. The steady-state growth rate of
income (output) per worker will remain at , and the steady-state growth rate of total income
(output) will remain at ሺ ൅ ሻ.
(d) The Solow model predicts that real wages increase at the same rate as ⁄ ; thus, at .
Question 5
Why might an economic policymaker choose the Golden Rule level of capital in an economy with
population growth but no technological change?
Answer:
It is reasonable to assume that the objective of an economic policymaker is to maximise the
economic wellbeing of the individual members of society. Since economic wellbeing depends on the
amount of consumption, the policymaker should choose the steady state with the highest level of
consumption. The Golden Rule level of capital represents the level that maximises consumption in
the steady state.
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If there is population growth but no technological change, output per worker increases by the
marginal product of capital if the steady-state capital stock per worker increases by one unit.
However, depreciation and the population rise by and n, respectively, so the net amount of extra
output per worker available for consumption is െ ሺ ൅ ሻ. The Golden Rule level of capital is
the level at which ൌ ൅ , where the marginal product of capital equals the summation of the
depreciation and population growth rates.
Question 6
One of the key distinctions made in the analysis of the Solow growth model is between changes in
levels and changes in growth rates. How does an increase in population growth rate change the
steady-state levels and growth rates of output and output per worker in the Solow model with no
technological change?
Answer:
This question concerns the levels and rates of total output () and output per worker (). The key to
the answer lies in the words “levels” and “rates”.
 The increase in the population growth rate will increase the steady-state level of output () and
the steady-state growth rate of (which will grow at a rate equal to the new higher population
growth rate) for the economy as a whole.
Suppose we have a country where the initial population growth rate is ଵ, and then it increases to
the higher growth rate of ଶ. Originally was growing at a rate of ଵ and then grows at a rate of
ଶ. Thus, the growth rate of is higher with the higher population growth rate. After the change
in the population growth rate, the level of will be higher than it would have been with the
original growth rate (ଵ) because it is growing faster.
 However, the increase in the population growth rate will not change the steady-state growth rate
of output per worker, . The growth rate remains zero in the long run.
In the Solow model without technological change, once the system reaches a steady state, and
are constant, i.e., we have ∗ and ∗. In steady state, the growth in output per worker and
capital per worker is zero regardless of the population growth rate.
 The steady-state levels of capital per worker and output per worker are both lower with the
higher population growth rate (ଶ). What is important to individual citizens is rather than .
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Question 7
An economy described by the Solow growth model has the following per-worker production
function: ൌ ଴.ହ.
(a) Solve for the steady-state value of as a function of , , and .
(b) A developed country has a saving rate of 28% and a population growth rate of 1% per year. A
less developed country has a saving rate of 10% and a population growth rate of 4% per year.
In both countries, ൌ 0.02 and ൌ 0.04. Find the steady-state value of for each country.
(c) What policies might the less-developed country pursue to raise its level of income?
Answer:
(a) To solve for the steady-state value of as a function of , , and , we begin with the
equation for the change in the capital stock in the steady state:
∆ ൌ ሺ∗ሻ െ ሺ ൅ ൅ ሻ∗ ൌ 0
Plugging the production function into the equation for the change in the capital stock, we find
that in the steady state:
∗ െ ሺ ൅ ൅ ሻሺ∗ሻଶ ൌ 0
(Note: If ൌ ଴.ହ then ଶ ൌ .)
Solving this, we find the steady-state value of :
∗ ൌ ሺ ൅ ൅ ሻ⁄
(b) The question provides us with the following information:

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Developed country Less-developed country
ൌ 0.28 ൌ 0.10
ൌ 0.01 ൌ 0.04
ൌ 0.02 ൌ 0.02
ൌ 0.04 ൌ 0.04
Using the equation for ∗ that we derived in part (a), we can calculate the steady-state values
of for each country.
Developed country: ∗ ൌ 0.28 ሺ0.04 ൅ 0.01 ൅ 0.02ሻ ൌ 4⁄
Less-developed country: ∗ ൌ 0.10 ሺ0.04 ൅ 0.04 ൅ 0.02ሻ ൌ 1⁄
(c) The equation for ∗ that we derived in part (a) shows that the less-developed country could
raise its level of income per capita by reducing its population growth rate or by increasing
its saving rate . Policies that reduce population growth include introducing birth control
methods and implementing disincentives for having children. Policies that increase the saving
rate include increasing public savings by reducing the budget deficit and introducing private
saving incentives like tax concessions that increase the return to savings.

Question 8
Prove each of the following statements about the steady state of the Solow model with population
growth and technological progress.
(a) The capital-output ratio is constant.
(b) Capital and labour each earn a constant share of an economy’s income. [Hint: By definition
ൌ ሺ ൅ 1ሻ െ ሺሻ].
(c) Both total capital income and total labour income grow at the rate of population growth plus
the rate of technological progress, ൅ .
(d) The real rental price of capital is constant, and the real wage grows at the rate of
technological progress . (Hint: The real rental price of capital equals total capital income
divided by the capital stock, and the real wage equals total labour income divided by the
labour force.)
Answer:
(a) The steady-state condition is ൌ ሺ ൅ ൅ ሻ. This implies that:
⁄ ൌ ሺ ൅ ൅ ሻ⁄
Since , , and are constant, the ratio ⁄ is also constant. We also know that ⁄ ൌ
ሾ ሺ ൈ ሻ⁄ ሿ ሾ ሺ ൈ ሻ⁄ ሿ⁄ ൌ ⁄ .
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Thus, we conclude that the capital-output ratio is constant in the steady state.
(b) Capital’s share of income () is ൈ ሺ ⁄ ሻ. We know from part (a) that the capital-
output ratio ⁄ is constant in the steady state. We know from the hint that is a
function of , which is constant in the steady state, so is constant. Thus, capital’s share
of income is constant in the steady state. Since labour’s share of income equals one minus
capital’s share, it is also constant.
(c) In the steady state, total income grows at a rate ൅ , which is the rate of population
growth plus the rate of technological change. In part (b), we showed the income shares for
capital and labour are constant. Since the income shares are constant and total income grows
at rate ൅ , total labour and capital income must each grow at rate ൅ .
(d) The real rental price of capital equals the marginal product of capital. In part (b), we saw that
is constant in the steady state because is constant in the steady state, so the real rental
price of capital is also constant in the steady state. The real wage equals the marginal product
of labour. We saw in part (c) that total labour income ൈ grows at rate ൅ in the
steady state, so we conclude that the real wage grows at rate :
∆ሺ ൈ ሻ
ൈ ൌ ൅
∆
൅
∆
ൌ ൅
൅ ∆ ൌ ൅
∆
ൌ

Question 9
Suppose a government can impose controls limiting the number of children people can have. Use the
Solow growth model with population growth and technological progress to graphically illustrate the
impact of the slower rate of population growth on the steady-state levels of capital and output per
effective worker. Be sure to label the i. axes, ii. curves, iii. initial steady-state levels, iv. terminal
steady-state levels, and v. direction the curves shift.
How will the steady-state growth rates of income per worker and total output change?
Answer:
Limiting the number of children will decrease the population growth rate, . The slope of the break-
even investment function will decrease.
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The steady-state capital per effective worker will increase from ଵ∗ to ଶ∗ and the steady-state income
per effective worker will increase from ଵ∗ to ଶ∗.
In the steady state, the growth rate of capital per worker will remain at .
The steady-state growth rate of income (output) per worker will remain at , and the steady-state
growth rate of total output will decrease from ሺଵ ൅ ሻ to ሺଶ ൅ ሻ.

Question 10
Assume that a country’s production function is ൌ ଴.ଷ଴.଻. The ratio of capital to output is 3, the
growth rate of output is 3 per cent, and the depreciation rate is 4 per cent. Capital is paid its marginal
product. (Note: You do not need to convert Y and to and for this question.)
(a) What is the marginal product of capital in this situation? (Hint: The marginal product of
capital may be computed using calculus by differentiating the production function and using
the capital-output ratio, or by using the fact that capital’s share equals multiplied by
divided by .)
(b) What must the saving rate be if the economy is in a steady state? (Hint: The saving rate
multiplied by must provide for gross growth of ሺ ൅ ൅ ሻ, where is the depreciation
rate.)
(c) If the economy decides to achieve the Golden Rule level of capital and actually reaches it,
what will be the marginal product of capital?
(d) What must the saving rate be to achieve the Golden Rule level of capital?

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Answer:
Given:
ൌ ଴.ଷ଴.଻

ൌ 3
൅ ൌ 0.03 (Growth rate of )
ൌ 0.04
(a) We want to express the in terms of our known values. To find the , we take the
first derivative of the production function with respect to .
ൌ
ൌ 0.3ି଴.଻଴.଻
ൌ 0.3ቆ଴.ଷ ቇ ଴.଻ since ି଴.଻ ൌ ଴.ଷିଵ ൌ ଴.ଷ ൈ ିଵ ൌ ଴.ଷ
ൌ 0.3 1 ሺ଴.ଷ଴.଻ሻ
ൌ 0.3
We now have the as a function of the capital-to-output ratio.
ൌ 0.3
ൌ 0.3 ൈ 13 since ൌ 1 ൬൰ൗ
ൌ 0.1
(b) At steady state, actual investment equals break-even investment. In this problem, we express
and in aggregate terms.
ൌ ሺ ൅ ൅ ሻ
ൌ ሺ ൅ ൅ ሻ ൌ ሺ0.04 ൅ 0.03ሻሺ3ሻ ൌ 0.21
Thus, the saving rate is 21 per cent.
(c) The Golden Rule level of capital stock is achieved when ൌ ൅ ൅ . Thus:
ൌ ൅ ൅ ൌ 0.04 ൅ 0.03 ൌ 0.07
(d) At the Golden Rule steady state:
∗ ൌ ሺ ൅ ൅ ሻ∗ீ
ൌ ሺ ൅ ൅ ሻ
∗ீ
∗
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We know that for a Cobb-Douglas production function, ൌ ሺ ⁄ ሻ. Solving this for the
capital-output ratio, we find:
⁄ ൌ ⁄
We can find the Golden Rule capital-output ratio using this equation:
∗ீ
∗ ൌ ⁄ ൌ 0.3 0.07⁄ ൌ 4.29
Thus, the saving rate must be 30% to achieve the Golden Rule level of capital:
ൌ ሺ0.04 ൅ 0.03ሻሺ4.29ሻ ൌ 0.3

Question 11
Suppose the government passes significant tax cuts on household income but does not reduce
spending, so the government budget deficit is larger. Use the Solow growth model with population
growth and technological progress to graphically illustrate the impact of the tax cut on the steady-
state levels of capital and output per effective worker. Be sure to label the i. axes, ii. curves, iii.
initial steady-state levels, iv. terminal steady-state levels, and v. direction the curves shift.
Briefly interpret the predicted impact, commenting on the income/output per worker. How will the
steady-state growth rates of income per worker and total output change?
Answer:
Decreasing while holding constant will decrease public savings (௣௨௕௟௜௖ ൌ െ ሻ and the total
saving rate, . The investment function will move downwards.

The steady-state capital per effective worker will fall from ଵ∗ to ଶ∗ and the steady-state income per
effective worker will decrease from ଵ∗ to ଶ∗.
In the transition phase, the growth rate in income per worker will be lower than in the steady state
and possibly negative. Once the new steady state is reached, the steady-state growth rates of capital
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and income per worker and total output will be as before. The steady-state growth rate of income
(i.e., output) per worker will remain at , and the steady-state growth rate of total output will remain
at ( ൅ ).

Question 12
What data would you need to determine whether an economy has more or less capital than in the
Golden Rule steady state?
Answer:
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 ( െ ) with the growth rate of total output
( ൅ ).
The growth rate of GDP is readily available. Estimating the net marginal product of capital requires a
little more work. Still, it can be calculated using available data on the capital stock relative to GDP,
the total amount of depreciation relative to GDP, and capital’s share in GDP.

Question 13
In the US, the capital share of GDP is about 30%, the average output growth is about 3% a year, the
depreciation rate is about 4% a year, and the capital-output ratio is about 2.5. Suppose that the
production function is Cobb-Douglas, so the capital share in output is constant, and the US has been
in a steady state.
(a) What must the saving rate be in the initial steady state? [Hint: Use the steady-state
relationship, ൌ ሺ ൅ ൅ ሻ.]
(b) What is the MPK in the initial steady state?
(c) Suppose public policy raises the saving rate so that the economy reaches the Golden Rule
level of capital. What will the MPK be at the Golden Rule steady state? Compare the MPK in
both steady states. Explain the difference.
(d) What will the capital-output ratio be at the Golden Rule steady state? [Hint: For the Cobb-
Douglas production function, the capital-output ratio is related to the MPK.]
(e) What must the saving rate be to reach the Golden Rule steady state?
Answer:
The US economy: A Cobb-Douglas production function in per-effective-worker terms has the form
ൌ ఈ where is capital’s share of income. The question tells us that ൌ 0.3, so we know the
production function is ൌ ଴.ଷ.
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In the steady state, we know that the growth rate of output equals 3 per cent, so we know that
ሺ ൅ ሻ ൌ 0.03.
The depreciation rate ൌ 0.04.
The capital-output ratio ⁄ ൌ 2.5. Because ⁄ ൌ ሾ ሺ ൈ ሻ⁄ ሿ ሾ ሺ ൈ ሻ⁄ ሿ⁄ ൌ ⁄ , we also
know that ⁄ ൌ 2.5. That is, the capital-output ratio is the same in terms of effective workers as in
levels.
(a) Begin with the steady-state condition, ൌ ሺ ൅ ൅ ሻ. Rewriting this equation leads to a
formula for the saving rate in the steady state:
ൌ ሺ ൅ ൅ ሻሺ ⁄ ሻ
Plugging in the provided values:
ൌ ሺ0.04 ൅ 0.03ሻሺ2.5ሻ ൌ 0.175
Thus, the initial steady-state saving rate is 17.5 per cent.
(b) We know that with a Cobb-Douglas production function, capital’s share of income is: ൌ ሺ ൈ ሻ ⁄ . Rewriting, we have:
ൌ ሺ ⁄ ሻ⁄
Plugging in the provided values, we find:
ൌ 0.3 2.5⁄ ൌ 0.12
(c) We know that at the Golden Rule steady state:
ൌ ሺ ൅ ൅ ሻ
Plugging in the provided values:
ൌ ሺ0.04 ൅ 0.03ሻ ൌ 0.07
At the Golden Rule steady state, the marginal product of capital is 7 per cent, whereas it is 12
per cent in the initial steady state. Hence, from the initial steady state, we need to increase the
saving rate to increase to achieve the Golden Rule steady state.
(d) We know that for a Cobb-Douglas production function, ൌ ሺ ⁄ ሻ. Solving this for the
capital-output ratio, we find:
⁄ ൌ ⁄
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
, we find:
∗ீ
∗ ൌ 0.3 0.07⁄ ൌ 4.29
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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 the formula for the saving rate in the steady state is:
ൌ ሺ ൅ ൅ ሻሺ ⁄ ሻ,
where ⁄ is the steady-state capital-output ratio. In the introduction to the answer, we
showed that ⁄ ൌ ⁄ , and in part (d), we found that the Golden Rule ⁄ ൌ 4.29.
Plugging in this value and the provided values:
ൌ ሺ0.04 ൅ 0.03ሻሺ4.29ሻ ൌ 0.30
The saving rate must rise from 17.5 to 30 per cent for the economy to reach the Golden Rule
steady state. 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.