Microsoft Word - ECON3810MDW21.docx
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Advanced MACROECONOMICS
ECON*3810.01
Midterm Exam
(Due on March 11th at 8PM EST)
Special Instructions:
1. The exam questions have to remain confidential at all time.
2. You are allowed to use your class notes, the instructor’s note, your textbook and a calculator.
3. Plagiarism, communication between students o
and with others about the exam subject o
and
questions will not be tolerated and will be subject to severe disciplinary actions.
4. For the quantitative exercises, you will be judged on the quality of your explanations as much as
on your final answers.
5. For the short answer questions, you will be judged on the quality of your argumentations as
much as on your final answers.
6. Your answers must be submitted in a single PDF file with YOUR LAST NAME using Dropbox on
Courselink before 8PM on March 11th.
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2 Short-Answer Questions (40 points)
1. Relaxing the Malthusian Assumptions (24 points)
Let us consider a simple Malthusian model with no productivity growth in which Assumption 1 holds
ut Assumption 2 is reversed in the sense that the income per capita has a negative effect on
population growth. Under which condition a steady-state solution still exists? What would happen
over time to such an economy with an initial level of income per capita below its steady-state
value? What would happen over time to such an economy with an initial level of income per capita
above its steady-state value? What would happen over time to such an economy initially located
at its steady-state solution and experiencing a one-time increase in productivity? If Assumption 1
is also reversed in the sense that a larger population raises the income per capita, then what would
happen in the short-run and in the long-run to an economy (with Assumption 2 reversed) initially
located at its steady-state solution and experiencing a one-time increase in productivity? Explain
and use graphs. (24 points)
2. The Solow Model with Government (16 points)
Using a simple Solow growth model with population growth, describe two possible government
policies to raise the long-run living standard? What would be the most effective method to finance
such policies and its potential drawbacks. Explain and/or use graphs. (16 points)
2 Quantitative Questions (60 points)
1. A Solow Growth Model with Physical and Human Capital Accumulation (42 points)
Let us consider a Solow growth model augmented with human capital. The aggregate
output/income ?" at every time t is produced according to the following production function:
?" = ??"&'( )â„Ž"
+?"-
(
where ? > 0, ?, ? ∈ (0,1) are production parameters, ?" stands for the aggregate physical capital,
ℎ" represents the human capital per worker: ℎ" ≡ ?"/?", ?" denotes the aggregate level of
human capital and ?" is the aggregate number of workers. At every time t, the aggregate number
of workers is assumed to be a fraction ? ∈ (0,1) of the aggregate population size denoted by ?"
which grows at a constant rate ? > 0:
?">& = (1 + ?) ?"
The change in the aggregate physical capital from time t to time t+1 can be written as:
?">& − ?" = ?"B − δ?"
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where ? ∈ (0,1) represents the depreciation rate and ?"B denotes the aggregate investment in
physical capital which in equili
ium is assumed to be a constant fraction ε Î(0,1) of the aggregate
output/income:
?"B = ε?"
The change in the aggregate human capital from time t to time t+1 can be written as:
?">& − ?" = ?"F − δ?"
where ?"F denotes the aggregate investment human capital which in equili
ium is assumed to be
a constant fraction µ Î(0,1) of the aggregate output/income:
?"F = µ?"
At every time t, the goods’ market is in equili
ium if:
?" = ?" + ?"B + ?"F
where ?" denotes aggregate consumption at time t.
Let ?" ≡ ?"/?" denotes the income/output per capita at time t, ?" ≡ ?"/?" stands for the
physical capital per capita and ?" ≡ ?"/?" is for the physical capital per capita.
a. At what rate does the aggregate population of workers grow? (4 points)
. Show that the output per capita at time t can be written as a function of the physical
capital per capita at time t and the human capital per worker at time t. (4 points)
c. Show that in equili
ium, the physical capital per capita at time t+1: ?">& can be written
as a function of ?" and â„Ž": (4 points)
d. Show that in equili
ium, the human capital per worker at time t+1: â„Ž">& can be written
as a function of ?" and â„Ž": (4 points)
e. Does a steady-state physical capital per capita solution: ?">& = ?" = ?LL also require
the human capital per worker to be at the steady-state: â„Ž">& = â„Ž" = â„ŽLL? Explain (6
points)
f. Derive the steady-state formulas for the physical capital per capita, the human capital
per worker, the output/income per capita and the consumption per capita. (12 points)
g. How does ε and µ affet the steady-state income per capita? (4 points)
h. Under the assumption that ε=µ, derive the investment rate that maximizes the steady-
state consumption per capita. (4 points)
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2. Quantitative Effects of Population Aging (18 points)
a. Show that the GDP per capita at time t can be written as the product of the GDP per worker
at time t, the employment rate at time t and the working age fraction of the population at time
t. (4 points)
. Show that the GDP per capita growth rate from time t to t+1 can be approximated as the sum
of the GDP per worker growth rate from time t to t+1 and the growth rate of the working age
fraction of the population from time t to t+1 if the employment rate remains constant. (6
points)
c. In 2021, 65% of the population in Canada was of working age. In 2034, 60% of the Canadian
population is expected to be of working age. Calculate the average annual growth rate of the
working age fraction of the population from time 2021 to XXXXXXXXXXpoints)
d. Using you answers to b. and c. how much does the GDP per worker has to grow on average
per year until 2034 to keep the GDP per capita constant? (4 points)