Department of Economics, Faculty of Economic and Management
Sciences, University of Pretoria
EKN 325 - Introductory Growth and Development Economics,
2019
Memorandum
Answer all four questions below
1a) Making use of the basic Solow model model and capital accumulation
equation k˙ = sy − (n + δ)k, compute the steady state output per worker,
y ∗ , and explain why the Chinese government may have implemented the one-child
policy in the late 1970s.
[15 marks]
I expect the student to compute the steady state, interpret the steady state and link to the
one-child policy in China.
𝒀 = 𝑲𝜶 𝑳𝟏−𝜶 (÷ 𝑳)
𝒚 = 𝒌𝜶
[2 marks]
In steady sate: 𝑘̇ = 0:
0 = 𝑠𝑦 − (𝑛 + 𝛿)𝑘
0 = 𝑠𝑘 𝛼 − (𝑛 + 𝛿)𝑘
(𝑛 + 𝛿)𝑘 = 𝑠𝑘 𝛼
𝑠
𝑘 1−𝛼 =
(𝑛 + 𝛿)
𝟏
𝒔
𝒌∗ = (
)𝟏−𝜶
𝒏+𝜹
[5 marks]
In steady state:
𝑦 = 𝑘𝛼
𝜶
𝒔
𝒚∗ = (
)𝟏−𝜶
𝒏+𝜹
[3 marks]
According to the Solow steady state, countries with higher levels of investment, low population
growth rates and low rates of capital depreciation have higher incomes per worker. As such,
increasing population growth reduces the capital per worker accumulation.
[2 marks]
The high population growth rate in China during the late 1970s may have contributed to the lower
incomes per worker. The government was struggling to sustain the population levels. Therefore,
the one-child policy was implemented for the purpose of reducing the rapid population growth
rate so that the capital accumulation can increase and the population levels will become sustainable
in the future.
[3 marks]
b) Nearly three decades later, China’s one-child policy was relaxed. Reconcile
this decision with the economic implication1of the steady state output per worker in
Romer’s endogenous growth model. [10 marks]
I expect the students to show and explain the steady state in Romer’s model with production
function stated, and link to relaxation of one-child policy.
In Romer’s model, the production function is given as:
which describes how capital stock (K) and labour (L) combine to produce output (Y) using the
stock of ideas (A).
[1 marks]
In the steady state, output per worker is given as:
[2 marks]
According to Romer, economies that have higher rates of capital accumulation, higher levels of
technology and low capital depreciation rates will be richer. However, Solow’s negative
population growth effect on output per worker is countered in Romer’s model with the notion that
technology is captured in ideas embedded in people. More populous economies have greater
potential innovators with ideas. Therefore the more researchers, the more ideas, which increases
the productivity of the economy.
[2 marks]
The one-child policy in China may have resulted in a slowdown of the productivity due to a
shrinking labour force and lack of innovation (new ideas to increase technology and productivity)
as there were fewer children being born. Spillover consequences included a growing proportion
of elderly people that needed support from a depleted younger generation, putting strain on
economy’s resources. (students can discuss other consequences, but need to link to them to less
potential for ideas in economy and low productivity)
[5 marks]
2a) Making use of the quantity-quality framework and diagram/s (provide
necessary equations), illustrate and explain why population growth rises with income
when income is low and falls at higher income levels.
[15 marks]
I expect students to explain the quantity-quality framework using necessary
equations, as well as provide one or both of the diagrams below to answer the
question.
The quantity-quality framework theory is an economic description of family decisions
regarding the number of children to have and the education they receive.
The budget constraint is as follows:
where the budget (y) depends on subsistence consumption (c), resources spent on
2 spent on educating children (E).
children such as food and clothing (M) and amount
The utility of the family is given as:
𝑀
where utility (V) depends on number of kids (𝑚 = 𝜂 𝑦 ) in relation to resources available,
and amount of education that each child has ( 𝑢 = 𝐸 + ̅̅̅
𝑢) in relation to resources and
initial education that child may acquire outside of formal schooling (i.e. basic skills).
Rewrite utility function as:
Substitute the M with budget constraint:
Solve utility function w.r.t education (E):
[8 marks]
Families will increase spending on education as incomes increase. This quantity-quality
framework may explain the population transition dynamics observed in countries that are
moving out of the Malthusian stagnation into sustained growth era. In the Malthusian
period, there is an income effect. Population growth rises with income per capita.
However with increasing technology (g), the incentive for education has risen such that
families’ fertility decisions have shifted from quantity to quality of children (i.e.
substitution effect between number of children and educating them). As such, population
growth declines with rising income per capita. This dynamic is explained in the following
diagram/s.
[5 marks]
[2 marks for any or both diagrams]
3 in terms of population growth and
b) In your opinion, where is South Africa
income transition dynamics? [10 marks]
I expect students to know basic statistics about South Africa. It was discussed in class.
South Africa exhibits a negative correlation between income per capita and population growth
implying that the economy is transitioning from Malthusian stagnation into a sustained growth
era. Population growth is decreasing as income per capita rises suggesting that a substitution effect
is taking place as families trade number of children in favour of educating them. South Africa is
currently among the countries with low fertility rates at 2-3 children.
[10 marks]
(students may also use a graph to illustrate correlation)
3) In the model of globalisation and trade, the output per worker along the
balanced growth path is given by :
a) Explain the steady state output in detail. [15 marks]
I expect the students to explain each component in detail given the production function.
The production function is given as:
[2 marks]
Where output is produced using capital stock(K), skilled labour (hL), as well as imported
intermediate goods.
[1 mark]
According to the steady state:
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Component 1 – Solow’s model (countries with higher levels of investment, technology and lower
population growth and lower rates of capital depreciation will be richer).
Component 2 – countries with skilled labour (human capital accumulation) to use the technology
in the economies will be richer. Countries that also spend more time acquiring skills are better
equipped to use the technology in the economy.
Component 3 – countries that are open are able to import technologies to improve their
productivity. However, it is a combination of having available technologies (domestic or
imported) and the skills to use the technologies that will contribute to productivity.
Component 4 – world technological frontier capturing technological change over time. As long as
technology keeps progressing, growth in output per worker continues to rise.
[3 marks each component]
b) Reconcile the results with the Financial Times’ article on the current problem
facing Germany’s small and medium sized manufacturing companies. [10
marks]
I expect the students to link the benefits of having a skilled labour and open country
to the problem facing Germany’s manufacturing sector as per the article.
The article states that Germany’s small and medium-sized manufacturing companies are
facing loss in production capacity due to skills shortages in the country. As a result of the
country’s low population growth rates, they have been unable to fill the job vacancies
available in the sector. This has resulted in a shrinking labour force and slowdown in
economic performance. One of the government’s solution to deal with the skills shortage
was to take advantage of the influx of refugees (e.g. Syrian refugees due to crisis) and
absorb the skilled ones into the manufacturing sector. And even if the migrants were
unskilled, Germany countered this by offering apprenticeship training. This solution is
possible because Germany is an open country. If it had been a closed country, they would
be struggling to address their skills shortage causing further delay in productivity to the
manufacturing sector. Another solution was to increase the retirement age so that the
manufacturing sector can retain the skilled workers to transfer knowledge to the young
apprentices being hired. This relates to the technology diffusion highlighted in the model.
[10 marks]
4a) Using the Solow model augmented with nonrenewable resources, compute
the growth rate of output per worker along the balanced growth path and interpret the
result. [15 marks]
I expect the students to compute the steady state and explain result.
The production function is given as:
Where output is produced using capital stock (K), labour (L), technology (B) and energy input
from nonrenewable resources.
The total stock of energy remaining in the economy
5 depletes over time at the rate at which
the nonrenewable resource is being used:
Therefore the amount of energy used in production each period is:
Dividing the production function by (𝑌 𝛼 ) and substituting the energy (E):
Taking the logs and derivatives of production function:
where 𝑔 = 𝑔𝐵 /(1 − 𝛼) 𝑎𝑛𝑑 𝛾̅ = 𝛾/(1 − 𝛼).
Along the balanced growth path, the growth rate of output per worker is:
[10 marks]
Faster population growth leads to increased pressure on finite resource stock, reducing per
capita growth. An increase in the depletion rate (𝑠𝐸 ) reduces the long-run growth rate of the
economy. Resources are used up more quickly, leaving a smaller resource stock and therefore
lower output each period.
[5 marks]
b) Based on the results obtained above in (4a), reconcile with the empirical
evidence on prices of finite resources, per capita energy use and air pollutants in
the U.S. [10 marks]
I expect students to explain that empirical evidence on prices of finite resources
and air pollutants in US is contrary to what the model states.
According to Solow’s model with nonrenewable resources, there is a delay in the
growth of output due to the drag by population growth putting strain on land and
finite resources such as coal and oil. The model therefore implies that we should
observe an increase in the energy use of these finite resources and an increase in their
prices (demand is higher than supply of finite resources). In addition, we should also
observe increase in air pollution due to increased usage of these resources.
However, the empirical evidence shows that while per capita energy use has
increased over time, the prices of the nonrenewable resources are declining, as well
as the air pollutants.
One of the reasons to explain this contradictory evidence is the increase in technology
which has assisted in producing cleaner and more efficient ways of using
nonrenewable resources, as well as substituting nonrenewable resources for
6 energy. In addition, as incomes per
renewable ones such as solar, wind and water
capita continue to rise, pollution begins to fall as the economies find advanced ways
to use resources. The Constant Elasticity of Substitution production function also
explains the contradictory evidence, highlighting the possibility of substituting
nonrenewable resources for renewable ones.
[10 marks]
7