Quantitative Problem using the Penn World Data Set (35 Points)
Let us consider that the aggregate output/income in country � ∈ {���, ���} at time � can be
written as:
�!
" = /�!
"
1

.%

/�"
�!
" 1

.&

where �!
" stands for the physical capital in country � at time �, 0.3 is the physical capital’s
share of output, �" represents the time allocated to work in country � and �!
" is the aggregate
number of workers in country � at time � which is a fraction �" ∈ (0,1) of the aggregate
population size �!
"
:
�!
" = �"
�!
"
The aggregate population grows at a constant rate �" ∈ (−1, +∞):
�!'( " − �!
" = �"
�!
"
The equilibrium law of motion for the physical capita per capita from time t to time t+1 in
country � can be written as:
�!'( " = �"
�!
" + (1 − �"
)�!
"
where �" ∈ (0,1) represents the investment rate in country � and �" ∈ (0,1) is the depreciation
rate in country �.
Let �!
" ≡ �!
"
/�!
" denote the output/income per capita and �!
" ≡ �!
"
/�!
" stands for the physical
capital per capita.
a. Write-down both the production function and the equilibrium law of motion for the
physical capital in per capita units: (10 points)
b. Derive the steady-state output/income per capita expression for country �. (5 points)
2
Using the Penn World table 10.0: https://www.rug.nl/ggdc/productivity/pwt/ get the
following data for both Canada and Mexico:
“Expenditure-side real GDP at current PPPs (in mil. 2017US$)” denoted by rgdpe and
representing variable �.
“Population (in millions)” denoted by pop and representing variable �.
“Number of persons engaged (in millions)” denoted by emp and representing variable �.
“Average depreciation rate of the capital stock” denoted by delta and representing parameter �.
“Average annual hours worked by persons engaged” denoted by avh and representing parameter
�.
“Share of gross capital formation at current PPPs” denoted by csh_i and representing parameter
�.
c. Calculate the real GDP per capita in both Canada and Mexico in 2019 and the real GDP
per capita gap in 2019 between these two countries. (5 points)
d. Calculate the average fraction of the total population employed: �, the average time
allocated to work: �, the average rate of depreciation: �, the average investment rate: �,
in both Canada and Mexico from 2010 to 2019. (8 points)
e. Calculate the average annual growth rate of the aggregate population: � in both Canada
and Mexico from 2010 to 2019. (2 points)
f. Using the average figures found in questions d. and e., derive using the steady state
formula found in b. the ratio of the output per capita in Canada to the output per capita
in Mexico predicted by the Solow growth model. (5 points)