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3 Short-Answer Questions (40 points)
1. In the light of the cu
ent pandemic and based on what you have learned this term, discuss
your own 2020 Canadian growth forecasts for: 1) the output per capita; 2) the physical
capital per capita; 3) the human capital per capital; 4) technology; 5) efficiency. Among
physical capital per capita, human capital per capita, technology and efficiency, which
factor(s) in 2020 will likely contribute the most and which one(s) will likely contribute the
least to output per capita growth in Canada? (20pts)
2. In your homework assignment 3, the average annual growth rate of productivity in Canada
from 2010 to 2017 was estimated at XXXXXXXXXX%). Is this result surprizing? What can
you infer about the growth rates of technology and efficiency in Canada from 2010 to 2017?
(10pts)
3. Are the economic effects of a pandemic different today than during Malthus time? Explain.
(10pts)
2 Quantitative Questions (60 points)
1. A Solow Growth Model with Productivity Growth (30 points)
1. Let us consider a Solow growth model endowed with a manufacturing sector (Y) and a R&D
sector (A). Let ?& denote the level of productivity defined at every period as the product of
technology ?& and efficiency ?&:
?& = ?&?&
In the R&D sector, the rate of technological progress from period t to t+1 is:
?&+, − ?&
?&
=
(?/)0
?
where ?/ stands for the fixed aggregate population of workers engaged in the R&D sector, ? > 0
epresents the cost of raising technology and ? ∈ (0,1) captures the diminishing marginal returns
to the labour input. Efficiency is assumed to grow at a constant rate:
?&+, = (1 + θ)?&
where θÎ(−1,+∞). The aggregate output of the manufacturing sector in period t is given by:
?& = ?&?&,=>??
where ? ∈ (0,1) is a production parameter, ?& stands for the physical capital in period t and ?&?
stands for the fixed aggregate population of workers engaged in the manufacturing sector. The
law of motion of the aggregate physical capital from period t to period t+1 can be written as:
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?&+, = ?& + (1 − δ)?&
where ?& denotes the aggregate investment level in physical capital, and ? ∈ (0,1) represents the
depreciation rate. In equili
ium: ?& = s?& where ? ∈ (0,1) denotes the saving or the investment
ate and the aggregate population of workers is allocated across sectors:
? = ?? + ?/
where ? stands for the fixed aggregate population of workers. Let ?ÃŽ(0,1) stands for the fraction
of workers engaged in the R&D sector:
? =
?
?
Let ?& stand for the income per worker at time t: ?&/? and let ?& represent the physical capital
per worker at time t: ?&/?.
a. Show that the growth rate of productivity can be decomposed as a simple weighted sum of the
growth rate of technology and the growth rate of efficiency. (2pts)
. Write down the production function of the manufacturing sector in per worker units and show
that the growth rate of the output per worker can be decomposed as a simple weighted sum of
the productivity growth rate and the physical capital per worker growth rate. (6pts)
c. Write-down an expression for the equili
ium growth rate of the physical capital per worker.
(2pts)
d. Show that the equili
ium growth rate of the output per workers from time t to t+1 is a function
of the output-physical capital ratio: ?&/?& given ?, b, ?, θ, ?,a, ?, δ. (4pts)
e. Use a two-dimensional diagram with ?&/?& on the horizontal axis to display the growth
elationships found in c and d. Show that the economy converges to a steady-state solution.
(6pts)
f. Derive the steady-state output-physical capital ratio: JK
LK
= JKMN
LKMN
= OJ
L
P
QQ
and the output per worker
growth rate at the steady-state solution? (6pts)
g. Using the two-dimensional of question e, show how a reduction in the efficiency growth rate
affects the steady-state solution? Explain. (4pts)
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2. A simple Two-Sector Model (20 points)
Let us consider a simple two-sector model with an agrarian sector (A) and a manufacturing
sector (Y). The aggregate output of the agrarian sector in period t is given by:
?&/ = ?&/?>(?&/),=>
where ? ∈ (0,1) is a production parameter, ?&/ is the productivity of the agrarian sector, ? stands
for a fixed stock of land and ?&/ stands for the aggregate population of workers engaged in the
agrarian sector. The aggregate output of the manufacturing sector in period t is given by:
?&? = ?&??&?
where ?&? is the productivity of the manufacturing sector, and ?&? stands for the aggregate
population of workers engaged in the manufacturing sector. In equili
ium at time t, the real
wage ?& is equal to the marginal product of labour and the aggregate population of workers is
allocated across sectors:
?& = ?&? + ?&/
where ?& denotes the aggregate population of workers.
a. At time t, derive the marginal product of labour schedule in the agrarian sector: ???&/, and the
marginal product of labour schedule in the manufacturing sector: ???&?. (2 pts)
. If workers in both sectors are paid their marginal products and are free to move across sectors,
then calculate how many workers will work in each sector at time t. Use a graph to illustrate your
esults. (6 pts)
c. What would happen over time to the allocation of workers across sectors if the aggregate
population of workers ?& increases? (4 pts)
d. What would happen if ?&? < (1 − ?)?&/?>?&=>? (4 pts)
e. According to this model, what are the two factors that could trigger industrialization? What are
the two factors that could delay industrialization? (4 pts)