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Problem Set 2
MSU EC 410
Prof. Ahlin
due 2/25/16
1. Consider a country in which Y = 200 K2/5N3/5. Assume in this country they save 20% of
their income, population grows at 3% per year, and depreciation of capital occurs at 10% per
year. Use the Solow model.
a. Compare the effectiveness of i) a 50% increase in the savings rate (to 30%), ii) a 67% decline
in the population growth rate (to 1%), and iii) a 10% increase in productivity (to 220). That is,
for each, give the percent by which it increases long-run average income (y*) and long-run
average consumption (c*).
b. Give one policy each that could be undertaken to accomplish i)-iii). Which policy has the
greatest impact on long-run well-being (assuming each policy has zero costs)?
2. Imagine the following goal of Lenin/Stalin at the beginning of the Soviet regime in Russia: to
overtake (i.e. equal) and surpass the world’s industrialized economies in terms of GDP per
capita. To achieve this goal, the main instrument of control is the fraction of national production
that is devoted to building the nation’s productive capacity: new machines, factories,
transportation equipment, and roads (i.e. investment as a share of GDP). The rest of national
production is used for consumer items like clothing and food. The country begins with relatively
little capital, being mostly rural and non-industrialized. Assume each of the following:
GDP per capita starts in Russia at $300/year.
The world’s industrialized economies start with GDP per capita of $5000/year.
Population growth rates are 2% everywhere in the world.
All capital depreciates at 8% per year.
Assume the basic growth framework of Solow. Further assume y = 400k1/3 in Russia.
a. Solve for Soviet long-run GDP per capita (y*), as a function of its savings rate. (That is, your
expression for y* should include the variable s in it, and no other variables.)
For parts b.-d., assume long-run GDP per capita equals $10,000 in the industrialized countries,
and that they are saving 10% of income.
b. What fraction of national output should be devoted by the Soviet Union to building new
capital goods in order to overtake, i.e. equal, the industrialized nations’ GDP per capita in the
long-run? What fraction is left for consumer items?
c. What fraction of national output should be devoted by the Soviet Union to building new
capital goods in order to surpass, i.e. double, the industrialized nations’ GDP per capita in the
long-run? What fraction is left for consumer items?
d. In the long run, what is the ratio of Soviet GDP per capita to GDP per capita in the
industrialized countries, and what is the ratio of Soviet consumption per capita to consumption
per capita in industrialized countries if the Soviet Union achieves the goal of part b? of part c?
(That is, find four ratios.)
e. Comparing the outcomes of part d. of this question to the same question answered using the
Harrod-Domar model (5e. from problem set 1), which model produces a more optimistic outlook
for achieving, by saving and investing at high rates, the Soviet goals of overtaking and
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surpassing industrialized countries’ living standards? Remember that living standards are best
measured by consumption here.
3. a. Use the Solow model to determine the a) instantaneous impact on GDP per capita, b)
instantaneous impact on consumption per capita, c) impact on long-run GDP per capita, d)
impact on long-run consumption per capita, e) impact on long-run GDP per capita growth rate,
and f) impact on long-run GDP growth rate of a one-time and instantaneous increase in the
population Nt, through wartime refugee immigration, say. Assume the country begins at its
steady state value of k* before this event occurs. Justify your answer by use of graph and/or
equation. [Hint: this should not be considered a change in n, since the population continues to
grow at rate n after the one-time jump; it should be modeled as a one-time jump in Nt.]
b. Graph the path of yt and ct against time for the event analyzed in part a.
c. Use the Harrod-Domar model to determine the a) instantaneous impact on GDP per capita, b)
instantaneous impact on consumption per capita, c) impact on long-run GDP per capita growth
rate, and d) impact on long-run GDP growth rate of a one-time and instantaneous increase in the
population N, through wartime refugee immigration, say. Justify your answer by use of graph
and/or equation.
d. Graph the path of yt and ct (or ln yt and ln ct, which are typically linear) against time for the
event analyzed in part c.
4. Consider the Solow model with total factor productivity At constantly growing at rate g>0.
a. Determine the a) instantaneous impact on GDP per capita, b) instantaneous impact on
consumption per capita, c) long-run impact on GDP per capita (i.e. compare the level of GDP per
capita with and without the parameter change, in the long-run), d) long-run impact on
consumption per capita (i.e. compare the level of consumption per capita with and without the
parameter change, in the long-run), and e) impact on long-run GDP per capita growth rate of a
one-time and instantaneous increase (jump) in productivity At, through a significant and nonrepeatable invention. Assume the country begins at its “steady state value” of k* before this
event occurs. Justify your answer by use of graph and/or equation. [Hint: this should not be
considered a change in g, since productivity resumes growth at rate g after the one-time jump; it
should be modeled as a one-time jump in At.]
b. Graph the path of yt and ct against time (or better yet, ln(yt) and ln(ct), which will be linear)
for the event analyzed in part a.
c. Repeat parts a&b for a permanent, instantaneous increase in the growth rate of
productivity, g. Treat this simply as an increase in trend growth with no change in the “steady
state” value of k or y.
5a.
Use the H-augmented Solow model to determine the a) instantaneous impact on GDP per
capita, b) instantaneous impact on consumption per capita, c) long-run impact on GDP per
capita, d) long-run impact on consumption per capita, e) impact on long-run GDP per capita
growth rate, and f) impact on long-run GDP growth rate of a permanent and instantaneous
increase in the fraction of national resources devoted to investment in human capital, q. Assume
the country begins at its steady state values of k* and h* before this event occurs. Justify your
answer by use of graph and/or equation.
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5b.
How does each answer compare to the answer the original Solow model would give when
s increases, both qualitatively (whether the amount goes up or down) and quantitatively (the
amount by which it goes up or down)?
6. Growth Simulations. See GrowthSimulationQuestion.xlsx posted on D2L. Fill in 200 years
of data using the H-D model, the Solow model, and the H-Solow model using the functions and
parameters given in the “GrowthCalculations” worksheet. The savings rate in all cases increases
to 30% at year 25. The H-D model is columns A-K, Solow is columns M-U, and H-Solow is
columns W-AE. Specifically:
3.1. For the H-D model, A=0.25, n=0.01, d=0.04, and s=0.2. Capital per person starts at $4000.
Fill out k, y, ln(y), c, ln(c), actual investment, break-even investment, k, and gy for 200 years.
3.2. For the Solow model, f(k) = k1/3, A=50, n=0.005, d=0.02, and s=0.2. Capital per person
starts at $8000. Fill out k, y, c, actual investment, break-even investment, k, and gy for 200
years.
3.3. For the H-Solow model, f(k,h) = k1/3h1/3, A=5, n=0.01, d=0.04, q = 0.1, and s=0.2. Physical
capital per person starts at $4000, human capital per person starts at $2000. Fill out k, h, y, c, k,
h, and gy for 200 years.
Note that in all cases, the savings rate s switches to 0.3 at the 25th year. Make sure to
incorporate this in your answers. [Hint: it will only affect the consumption formula and the
actual investment column formula for the H-D and Solow models, and it will only affect the
consumption formula and the k column formula for the H-Solow model.]
[Hint: Of course, you need only specify each column’s formula once, then copy and paste down
the column for all the years. The formulas are pretty straightforward, and can be found by
looking back at the key equations for each model. Remember that next year’s k is determined
from this year’s k plus k. It is simplest for actual investment not to recalculate income, but
simply use the fact that actual investment equals a fixed fraction of income, sy in the case of
physical capital and qy in the case of human capital.]
a. Give the income and consumption levels in year 200 for each of the three models. In which
model is the increase in s most effective? In which model is it least effective? Justify your
answer.
b. Look at the graphs for the three models (which are in the other worksheets and should be
filled out automatically from the data you generate in the GrowthCalculations worksheet). Look
at both H-D graphs, but focus on the one using logs. Discuss one significant way in which all
three models’ graphs are similar. How do the Solow and H-Solow graphs differ?
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H-D model
f(k)=k
s=0.20
n=0.01
d=0.04
A=0.25
s'=0.3
year
k
0
1
2
3
4
5
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10
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25
26
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28
4000,00
y
ln(y)
c
ln(c)
actual inv.
break-even inv.
Dk
29
30
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32
33
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200
gy
NA
Solow model
1/3
f(k)=k
s=0.2
n=0.005
d=0.02
A=50
s'=0.3
year
k
0
1
2
3
4
5
6
7
8
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10
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12
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14
15
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25
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27
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8000,00
y
c
actual inv.
break-even inv.
29
30
31
32
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34
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200
Dk
gy
NA
H-Solow model
1/3 1/3
f(k,h)=k h
s=0.2
q=0.1
n=0.01
d=0.04
A=5
s'=0.3
year
k
h
0
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4000,00
2000,00
y
c
Dk
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200
Dh
gy
NA
Harrod-Domar growth
1,20
1,00
Income per capita
0,80
Income per
capita
Consumption
per capita
0,60
0,40
0,20
0,00
1
9
17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193 201
Years
Harrod-Domar growth
1,20
ln(Income per
capita)
1,00
ln(Consumption per
capita)
ln(Income per capita), ln( Consumption per capita)
0,80
0,60
0,40
0,20
0,00
1
9
17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193 201
Years
Solow growth
1,20
1,00
Income per capita
Consumption per
capita
Income per capita, Consumption per capita
0,80
0,60
0,40
0,20
0,00
1
9
17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193 201
Years
H-Solow growth
1,20
1,00
Income per capita
Income per capita, Consumption per capita
0,80
Consumption per
capita
0,60
0,40
0,20
0,00
1
9
17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193 201
Years