Problem Set 4
FE411 Fall 2014
Rahman
Some Answers
1) The Malthusian Model
Consider the Malthusian model, as described by Weil in section 4.1.
a. Suppose that the economy is in steady state when suddenly there is a change in
cultural attitudes toward parenthood. For a given income, people now want to
have more children. Draw graphs showing the growth rates of population and
income per capita over time.
Graphs should essentially look like those in figure 4.3.
b. Let’s look at a Malthusian model with some actual functions. Let’s say that the
growth rate of population is given by:
^
L
y  100
100
Let X be the total quantity of land in the economy, which is fixed. Let x then be
the quantity of land per capita. The function that relates land per capita and
income per capita is simply
y  Ax
where A is just a measure of productivity.
Suppose that A is constant. What will the steady-state level of income per capita
be?
Page 1 of 8
Problem Set 4
FE411 Fall 2014
Rahman
The steady-state level of income per worker is characterized by yˆ  0. Hence, we must
first find the relationship among the growth rate of income per worker, productivity, and
population.
By taking natural logs of the production function and taking the derivative with respect
to time, we get:
X
A

ln
()
y
ln
(A
x
)
ln
ln
(A
)
ln
(X
)
ln
()
L
,
L
 
d
d
ln
()
y ln
(A
)
ln
(X
)
ln
()
L
,
d
t
d
t
ˆ
ˆ
ˆ
y
AXL
.
Therefore, the growth rate of income per worker must equal the growth rate of
productivity plus the growth rate of land minus the population growth rate. Since land X,
ˆ  A 0. These values tell us that yˆ  Lˆ , and because
and productivity A, are constant, X
we are interested in the steady-state value of income per capita, yˆ  0  Lˆ. Now, we plug
the growth rate of population into the equation that relates population growth to incomeper-capita to arrive at our solution.
ˆ 0y100,
L
100
yss 100.
^
Now suppose that A grows at a rate of 10% per year (that is, A = 0.1). What will
be the steady-state level of income per capita? Explain what is going on.
Referring back to our equation relating the growth rate of income per capita,
productivity, land, and population, we can set Xˆ  0 and Aˆ  0.1.
ˆ
ˆ 0
ˆ
ˆ
y
AXL
.1

L
.
In the steady-state level of income per worker, ŷ must equal zero. Thus, we have that L̂
must equal 0.1. Using this value to solve for yss in the equation relating population
growth and income per capita,
Page 2 of 8
Problem Set 4
FE411 Fall 2014
Rahman
ˆ0.1y100,
L
100
yss 110.
The steady-state value is higher in this scenario. Due to consistent productivity growth
of
10 percent, the population can grow as well, leading to a higher level of income per
capita.
2) Composition Effects of Population Growth
Suppose that the world has only two countries. The following table gives data on their
populations and GDP per capita. It also shows the growth rates of population and GDP.
The growth rates of population and GDP per captia in each country never change.
Country
Population in
2000
GDP per
Capita in 2000
Country A
Country B
1,000,000
1,000,000
1000
1000
Growth Rate
of Population
(% per year)
0
2
Growth Rate of
GDP per Capita
(% per year)
2
0
a. What will the growth rate of world population be in the year 2000? Following
2000, will the growth rate of world population rise or fall? Explain why. Draw a
graph showing the growth rate of world population starting in 2000 and
continuing in to the future. Toward what growth rate does the world population
move in the long run?
Page 3 of 8
Problem Set 4
FE411 Fall 2014
Rahman
Growth Rate of World Population
2%
1%
time
b. Draw a similar graph showing the growth rate of total world GDP.
Growth Rate of Total World GDP
2%
1%
time
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Problem Set 4
FE411 Fall 2014
Rahman
c. Draw a similar graph shoing the growth rate of average GDP per capita in the
world.
Growth Rate of Average GDP per Capita
1%
0%
time
Page 5 of 8
Problem Set 4
FE411 Fall 2014
Rahman
3) Education as a Form of Human Capital Accumulation
Read 6.2 and 6.3 of the Weil text, which describes how years of education translates in to
human capital, and how his human capital in turn translates into additional income. Use
the insights from these sections to answer the following:
a. What fraction of wages is due to human capital for a worker who has nine years
of education?
The wage for an individual with nine years of education, relative to one with no education, is
(1.134)4  (1.101)4  (1.068)1  2.60.
In other words, if the wage of an individual with no education is W, the wage paid to this
individual
is 2.6W. Therefore, we can conclude that the payment to raw labor is W and the payment to
human capital is the difference between the total wage and the wage for raw labor. To find
the fraction, we compute,
Payment to Human Capital/Total Payment 
2.6  1
 0.615.
2.6
The fraction of wages paid to human capital is 61.5 percent.
b. In a certain country, everyone in the labor force in the year 2000 had 12 years of
education. In 1900 everyone in the labor force had 2 years of education. What
was the annual growth in income per worker that was due to the increase in
education?
The relative return to 12 years of schooling is 3.16, and the relative return to two years of
schooling is (1.134)2 or 1.29. Writing the steady-state ratio for one country over time and
denoting h1900  1.29 and h2000  3.16, we get:
y
.
1
6
2
0
0
0
,s
s h
2
0
0
0 3

 
2
.
4
5
.
y
.
2
9
1
9
0
0
,s
s h
1
9
0
0 1
Thus, the ratio of steady-state output per worker for this country over time is 2.45. If over
100 years, the steady-state output has increased by a factor of 2.45, we can solve for the
growth rate, g, by the following calculation.
1
1
0
0
g

(
2
.4
5
)
10
.0
0
9
.
We conclude that the annual average growth rate of output per worker is 0.9 percent.
Page 6 of 8
Problem Set 4
FE411 Fall 2014
Rahman
4) An Alternative Production Function with Human Capital
Consider a Cobb-Douglas production function with three inputs. K is capital (the number
of machines), L is labor (the number of workers), and H is human capital (the number of
college degrees among the workers). The production function is:
Y  K 1/ 3 L1/ 3 H 1/ 3
a. Derive an expression for the marginal product of labor. How does an increase in
the amount of human capital affect the marginal product of labor?
MPL = 1/3*K1/3H1/3 L-2/3. Increase in human capital raises MPL.
b. Derive an expression for the marginal product of human capital. How does an
increase in the amount of human capital affect the marginal product of human
capital?
MPH = 1/3*K1/3H-2/3 L1/3. Increase in human capital lowers MPH.
c. What is the income share paid to labor? What is the income share paid to human
capital? In the national income accounts of this economy, what share of total
income do you think all workers would appear to receive?
1/3, 1/3, and 2/3, respectively.
d. Say an unskilled worker earns the marginal product of labor, whereas a skilled
worker earns the marginal product of labor plus the marginal product of human
capital. Using your answers to (a) and (b), find the ratio of the skilled wage to the
unskilled wage. How dies an increase in the amount of human capital affect this
ratio? Explain.
Wskilled/Wunskilled = (MPL + MPH) / MPL = 1 + (L/H). When H increases,
this ratio falls because the diminishing returns to human capital lower its
return, while at the same time increasing the marginal product of
unskilled workers.
Page 7 of 8
Problem Set 4
FE411 Fall 2014
Rahman
e. Some people advocate government funding of college scholarships as a way of
creating a more egalitarian society. Others argue that scholarships help only those
who are able to go to college. Do your answers to the preceding questions shed
light on this debate?
If more college scholarships increase H, then it does lead to a more
egalitarian society. The policy lowers the returns to education,
decreasing the gap between the wages of more and less educated workers.
More importantly, the policy even raises the absolute wage of unskilled
workers because their marginal product rises when the number of skilled
workers rises.
5) Physical and Human Capital as Perfect Complements
Consider two countries, Sri Lanka and Japan. Both countries have a production function
per person where physical capital per person (k) and human capital per person (h) are the
only two inputs in production. Specifically, the production function per person is:
y  min[ k , h]
Suppose Sri Lanka has a low rate of physical capital investment, a large h, and a very low
k. Japan on the other hand as a high rate of physical capital investment, a large h, and a
moderate amount of k. Both countries are the same in all other respects.
a. Which country do you predict will grow faster in the short term? Why?
Japan. Because both countries have large h’s, we would expect physical
capital per person to be the relatively scare factor, so only growth in k
will allow growth in y. Thus in the short run, Japan should grow faster
because its investment rate in physical capital is higher.
b. Which country do you predict will grow faster in the long term? Why?
Sri Lanka. Once Japan has large levels of both k and h, its growth stops once k = h, further increases in k does nothing to increase y. Since Sri
Lanka starts with such an imbalanced situation, where k <<< h, it can
grow for a much longer period of time by increasing k though physical
capital investment.
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