Microsoft Word - ECON3810F20HW3.docx
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UNIVERSITY OF GUELPH
Gordon S. LANG School of Business and Economics
Department of Economics and Finance
ECON*3810: ADVANCED MACROECONOMICS
Fall Semester 2020
Homework Assignment 3
(Due: 11:59PM EST on November 26th 2020)
I. A Simple Two-Country Model (55 points)
Let us consider a two-country model with two production sectors in which ?"
# denotes the level of
productivity or technology at time t in country j for ??{1,2}. The country with the highest A is the
technology leader (L) and the country with the lowest A is the technology follower (F). If both countries
haves the same level of productivity, then they are both leaders.
The rate of technological progress from time t to t+1 in country j is:
?"+,
# − ?"
#
?"
# =
?"
#
?"
# ?
#
where ?"
# is the cost of raising technology at time t in country j, ?"
#? (0,1) stands for the fraction
(expressed in decimal form) of workers engaged in the R&D sector at time t in country j and ?# is the
fixed size of the total population of workers in country j.
Let us assume that the cost of inventing is fixed:
?"
# = 1
Copying is only available to the follower country. The cost of copying at time t depends on the cost of
inventing and the technology gap between the leader and the follower at time t:
?"
# =
1
6?"
7
?"8
9
:.<
Country 1 has a total population of workers of size 2 and Country 2 has a total population of workers of
size 1. At every period, let us assume that 2% of the population of workers in country 1 is engaged in the
R&D sector and 6% of the worker population in country 2 is engaged in the R&D sector. The aggregate
output in country j at time t is given by:
?"
# = ?"
#(1 − ?"
#)?#
At time 0, Let us assume that each county has the same initial level of technology: ?:, = ?:>.
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a. Write down the production function in per worker units for each country at time t. Which
country has the highest income per worker at time 0? (5 points)
. Write down the rate of technological progress from time 0 to time 1 for each country.
Which country is the technology leader at time 1? (5 points)
c. Write down the rate of technological progress from time 1 to time 2 for each country. (5
points)
d. Calculate the steady-state technology gap between the leader and the follower and the
growth rate of productivity in each country. (7.5 points)
e. Calculate the steady-state output per worker gap between country 1 and country 2.
Which country has the highest living standard? (7.5 points)
f. Let us assume that the steady-state solution computed in questions d. and e. prevails up
to time t-1. Starting at time t, the fraction of the labour force engaged in the R&D sector
in country 2 is permanently increased from 6% to 8% while the fraction of the labour
force engaged in the R&D sector in country 1 remains at 2%. Compute the productivity
growth rate and the output per worker growth rate from time t-1 to t, from time t to t+1
and from time t+1 to t+2 in country XXXXXXXXXXpoints)
g. How many periods will it take before the output per worker in country 2 coincides with
the level it would have reached if the fraction of the labour force engaged in the R&D
sector had remained constant at 6%? (5 points)
h. Derive the new steady-state productivity gap between the leader and the follower, the
growth rate of productivity in each country, and the new steady-state output per worker
gap between country 1 and country 2 if the leader has now 8% of its population
engaged in the R&D sector. (10 points)
II. Growth Accounting Exercise (45 points)
Let us consider that the aggregate output/income ?" is produced at every time t according to
the following Co
-Douglas production function:
?" = ?"?"
,/A(ℎ"?"?")
A
where ?">0 stands for the aggregate productivity level at time t, ?" stands for the
aggregate physical capital at time t, ℎ" stands for the human capital per worker at time t, ?"
stands for the hours worked at time t and ?" denotes the number of workers at time t.
a. Write-down the production function in per worker units. (5 points)
. Under the assumptions that the growth rate of the output per worker, the growth rate of
aggregate productivity, the growth rate of the physical capital per worker, the growth
ate of the human capital per worker and the growth rate of the hours worked are
elatively small, then show that the growth rate of the output per worker can be
decomposed as a simple weighted sum of the other four growth rates. (10 points)
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Using the Penn World table version 9.1: (https:
www.rug.nl/ggdc/productivity/pwt/),
download from 1990 to 2017 the following data for Canada:
“Output-side real GDP at chained PPPs (in mil. 2011US$)” denoted by rgdpo.
“Capital services at cu
ent PPPs (in mil. 2011US$)” denoted by ck.
“Number of persons engaged (in millions)” denoted by emp.
“Average annual hours worked by persons engaged” denoted by vh.
“Human capital index, based on years of schooling and returns to education” denoted by hc.
“Price level of the capital services, price level of USA=1” denoted by pl_k.
c. Using the above data, calculate for Canada from 1990 to 1999, from 2000 to 2009 and
from 2010 to 2017 the average annual growth rate of the real GDP per worker defined
as: rgdpo/emp, the average annual growth rate of the real physical capital per worker
defined as ck/(pl_k × emp), the average annual growth rate of the human capita index:
hc and the average annual growth rate for the average annual hours worked by
persons engaged: vh. (12 points)
d. From the growth decomposition equation found in b. and your answers to c., derive the
average annual growth rate of productivity from 1990 to 1999, from 2000 to 2009 and
from 2010 to 2017 in Canada. (6 points)
e. Among the following 4 factors: productivity, physical capital per worker, human capital,
and hours worked, which one contributed the most and which one contributed the least
to output per worker growth from 1990 to 1999, from 2000 to 2009 and from 2010 to
2014 in Canada? (12 points)