lOMoARcPSD|6803794
Intermediate macro
Assignment #2 solution
Assignment #2
Solution
Question 1
Convergence in two sets of countries. Go to the website containing the Penn World Table
(https://www.rug.nl/ggdc/productivity/pwt/) and collect data on real GDP (expenditure-side real
GDP at chained PPPs (in mil. 2011 USD)) and population from 1950 to 2017 for the United States,
France, Belgium, Italy, Ethiopia, Kenya, Nigeria, and Uganda. Then you can construct real GDP
per person for these countries. Define for each country for each year the ratio of its real GDP per
person to that of the United States for that year (so that this ratio will be equal to 1 for the United
States for all years.)
(1) Plot these ratios for France, Belgium, and Italy over the period for which you have data. Does
your data support the notion of convergence among France, Belgium, and Italy with the United
States? Give a brief explanation to rationalize your finding. (2 points)
Solution to Question 1 (1)
Q1 (1)
Convergence among Italy, France, Belgium and US
0.900
Real GDP per person ratio
0.800
0.700
0.600
0.500
0.400
0.300
0.200
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
2013
2016
year
Italy France Belgium
Figure 1: Convergence among Belgium, France, Italy and US
Q1 (2)
The results are shown in Figure 1. We can see that France, Belgium and Italy show convergence
with the UnitedConvergence amonghypothesis
States. The convergence Ethiopia, Kenya,
proposes thatNigeria,
real GDP per person among
Uganda and US
0.300
GDP per person raio
0.250
Downloaded by Felix Mungai ()
0.200
0.150
0.100
0.050
, lOMoARcPSD|6803794
Intermediate macro: assignment #2 solution 2
Q1 (1)
countries can reach the same balanced growth path if these countries share the same set of
parameters including the saving
Convergence rate, productivity
among growth
Italy, France, and etc. Among
Belgium the sample countries,
and US
it is reasonable to assume that they share these parameter values and therefore convergence
0.900
hypothesis holds within this group of countries.
Real GDP per person ratio
0.800
0.700 ratios for Ethiopia, Kenya, Nigeria, and Uganda. Does this data support the notion
(2) Plot these
of convergence
0.600 among Ethiopia, Kenya, Nigeria, and Uganda with the United States? Give a
brief explanation
0.500 to rationalize your finding. (2 points)
0.400 to Question 1 (2) The results are shown in Figure 2. Ethiopia, Kenya, Nigeria
Solution
0.300
and Uganda have not shown evidence of convergence with the United States from our results.
One explanation
0.200 could be that this group of countries does not share all parameter values with
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
2013
2016
the United States. Therefore, they may converge to a different balanced growth path compared
with the United States. This is consistent with yearthe conditional convergence hypothesis. An
alternative explanation to rationalize the economic performance in African countries is that
these countries save too little Italy
due to theFrance Belgium
need for consumption. With little saving, capital
accumulation is also slow. In this sense, this group of countries is in the “poverty trap” as a
result of the low saving rate. This can help explain the lack of convergence between this group
of countries
Q1 (2) and the United States.
Convergence among Ethiopia, Kenya, Nigeria,
Uganda and US
0.300
Real GDP per person raio
0.250
0.200
0.150
0.100
0.050
0.000
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
2013
2016
year
Uganda Nigeria Kenya Ethiopia
Figure 2: Convergence among Ethiopia, Kenya, Nigeria, Uganda and US
Question 2
Solow growth model. In this question, you will explore how changes in the saving rate and the rate
of technological progress affect an economy’s growth. In addition, you will examine how the golden
rule saving rate depends on the production function. Consider the Solow (neoclassical) growth model
with aggregate production function Y = K α (AN )1−α . Each period lasts a year.
Downloaded by Felix Mungai ()
, lOMoARcPSD|6803794
Intermediate macro: assignment #2 solution 3
(1) Using the parameter values in Table 1, calculate the steady-state values of capital per effective
worker k ≡ K/AN , output per effective worker y ≡ Y /AN and consumption per effective worker
c ≡ C/AN . Calculate the golden rule saving rate. Also calculate the growth rates of output
per worker and consumption per worker along the balanced growth path. (2 points)
α 1/3 gA 3.0%
δ 4% gN 1.0%
s 16%
Table 1: Benchmark Parameter Values – Solow model
Solution to Question 2 (1)
Let k ≡ K/AN and y ≡ Y /AN denote capital and output per effective worker. Then from the
aggregate production function
α
K α (AN )1−α
Y K
y= = = = kα
AN AN AN
and steady-state capital per effective worker k ∗ satisfies
s (k ∗ )α = (δ + gA + gN )k ∗
and 1
1−α
∗ s
k =
δ + gA + gN
Plugging in the numbers given in Table 1 we have
1
1−(1/3)
∗ 0.16
k = = 23/2 = 2.828
0.04 + 0.03 + 0.01
and
y ∗ = (k ∗ )α = 1.414
Consumption per effective worker is then
c∗ = (1 − s)y ∗ = (1 − 0.16)1.414 = 1.188
The golden rule saving rate sG is the saving rate that maximizes the steady-state consumption
per effective worker. Formally, sG maximizes
α
1−α
∗ ∗ s
c = (1 − s)y = (1 − s)
δ + gA + gN
The first-order condition with respect to s yields
1 −1/2
s (1 − s) − s1/2 = 0
2
The solution is sG = 1/3. The golden rule saving rate is 1/3.
Along the balanced growth path, output per worker Y /N and consumption per worker C/N
grow at the rate of productivity growth, in this case at 3%.
Downloaded by Felix Mungai ()
Intermediate macro
Assignment #2 solution
Assignment #2
Solution
Question 1
Convergence in two sets of countries. Go to the website containing the Penn World Table
(https://www.rug.nl/ggdc/productivity/pwt/) and collect data on real GDP (expenditure-side real
GDP at chained PPPs (in mil. 2011 USD)) and population from 1950 to 2017 for the United States,
France, Belgium, Italy, Ethiopia, Kenya, Nigeria, and Uganda. Then you can construct real GDP
per person for these countries. Define for each country for each year the ratio of its real GDP per
person to that of the United States for that year (so that this ratio will be equal to 1 for the United
States for all years.)
(1) Plot these ratios for France, Belgium, and Italy over the period for which you have data. Does
your data support the notion of convergence among France, Belgium, and Italy with the United
States? Give a brief explanation to rationalize your finding. (2 points)
Solution to Question 1 (1)
Q1 (1)
Convergence among Italy, France, Belgium and US
0.900
Real GDP per person ratio
0.800
0.700
0.600
0.500
0.400
0.300
0.200
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
2013
2016
year
Italy France Belgium
Figure 1: Convergence among Belgium, France, Italy and US
Q1 (2)
The results are shown in Figure 1. We can see that France, Belgium and Italy show convergence
with the UnitedConvergence amonghypothesis
States. The convergence Ethiopia, Kenya,
proposes thatNigeria,
real GDP per person among
Uganda and US
0.300
GDP per person raio
0.250
Downloaded by Felix Mungai ()
0.200
0.150
0.100
0.050
, lOMoARcPSD|6803794
Intermediate macro: assignment #2 solution 2
Q1 (1)
countries can reach the same balanced growth path if these countries share the same set of
parameters including the saving
Convergence rate, productivity
among growth
Italy, France, and etc. Among
Belgium the sample countries,
and US
it is reasonable to assume that they share these parameter values and therefore convergence
0.900
hypothesis holds within this group of countries.
Real GDP per person ratio
0.800
0.700 ratios for Ethiopia, Kenya, Nigeria, and Uganda. Does this data support the notion
(2) Plot these
of convergence
0.600 among Ethiopia, Kenya, Nigeria, and Uganda with the United States? Give a
brief explanation
0.500 to rationalize your finding. (2 points)
0.400 to Question 1 (2) The results are shown in Figure 2. Ethiopia, Kenya, Nigeria
Solution
0.300
and Uganda have not shown evidence of convergence with the United States from our results.
One explanation
0.200 could be that this group of countries does not share all parameter values with
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
2013
2016
the United States. Therefore, they may converge to a different balanced growth path compared
with the United States. This is consistent with yearthe conditional convergence hypothesis. An
alternative explanation to rationalize the economic performance in African countries is that
these countries save too little Italy
due to theFrance Belgium
need for consumption. With little saving, capital
accumulation is also slow. In this sense, this group of countries is in the “poverty trap” as a
result of the low saving rate. This can help explain the lack of convergence between this group
of countries
Q1 (2) and the United States.
Convergence among Ethiopia, Kenya, Nigeria,
Uganda and US
0.300
Real GDP per person raio
0.250
0.200
0.150
0.100
0.050
0.000
1950
1953
1956
1959
1962
1965
1968
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
2013
2016
year
Uganda Nigeria Kenya Ethiopia
Figure 2: Convergence among Ethiopia, Kenya, Nigeria, Uganda and US
Question 2
Solow growth model. In this question, you will explore how changes in the saving rate and the rate
of technological progress affect an economy’s growth. In addition, you will examine how the golden
rule saving rate depends on the production function. Consider the Solow (neoclassical) growth model
with aggregate production function Y = K α (AN )1−α . Each period lasts a year.
Downloaded by Felix Mungai ()
, lOMoARcPSD|6803794
Intermediate macro: assignment #2 solution 3
(1) Using the parameter values in Table 1, calculate the steady-state values of capital per effective
worker k ≡ K/AN , output per effective worker y ≡ Y /AN and consumption per effective worker
c ≡ C/AN . Calculate the golden rule saving rate. Also calculate the growth rates of output
per worker and consumption per worker along the balanced growth path. (2 points)
α 1/3 gA 3.0%
δ 4% gN 1.0%
s 16%
Table 1: Benchmark Parameter Values – Solow model
Solution to Question 2 (1)
Let k ≡ K/AN and y ≡ Y /AN denote capital and output per effective worker. Then from the
aggregate production function
α
K α (AN )1−α
Y K
y= = = = kα
AN AN AN
and steady-state capital per effective worker k ∗ satisfies
s (k ∗ )α = (δ + gA + gN )k ∗
and 1
1−α
∗ s
k =
δ + gA + gN
Plugging in the numbers given in Table 1 we have
1
1−(1/3)
∗ 0.16
k = = 23/2 = 2.828
0.04 + 0.03 + 0.01
and
y ∗ = (k ∗ )α = 1.414
Consumption per effective worker is then
c∗ = (1 − s)y ∗ = (1 − 0.16)1.414 = 1.188
The golden rule saving rate sG is the saving rate that maximizes the steady-state consumption
per effective worker. Formally, sG maximizes
α
1−α
∗ ∗ s
c = (1 − s)y = (1 − s)
δ + gA + gN
The first-order condition with respect to s yields
1 −1/2
s (1 − s) − s1/2 = 0
2
The solution is sG = 1/3. The golden rule saving rate is 1/3.
Along the balanced growth path, output per worker Y /N and consumption per worker C/N
grow at the rate of productivity growth, in this case at 3%.
Downloaded by Felix Mungai ()
