Problem Set 3
FE411 Spring 2009
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.
B,D
Birth Ratenew
Birth Rate
Death Rate
yss,2
yss,1
y
L1
L2
L
The sudden shift (shown by an upward shift in the birth rate) implies that
population growth will suddenly be positive. As the population size grows, the
income per capita falls and slows down the growth rate of population. The final
point will be at a lower income per person level with no more population growth.
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Problem Set 3
FE411 Spring 2009
Rahman
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?
^
y 0 .
The steady-state level of income per worker is characterized by
Hence,
we must 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:
 AX 
ln( y )  ln( Ax)  ln 
  ln A  ln X  ln L,
 L 
^
^
^
^
d
ln( y )  y  A X  L .
dt
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, and productivity, A, is constant, it must be that the growth of income per
person equals the growth of population. Now, we plug the growth rate of
population into the equation relating population growth to income per capita to
arrive at the solution:
^
L0
y  100
 y ss  100.
100
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Problem Set 3
FE411 Spring 2009
Rahman
^
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.
Again, in steady-state growth in income must be zero. Thus we have that growth
in population must equal 0.1. Using this value to solve for steady-state income,
we have:
^
L  0.1 
y  100
 y ss  110.
100
The steady-state value is higher in this scenario. Due to consistent productivity
growth of 10%, the population can grow as well, leading to a higher level of
income per capita.
2) Solow when Population > Workers
Consider how the Solow growth model is affected when the number of workers differs
from the total population. Suppose that total output is produced according to the
production function:
Y  K  [(1  x) L]1
where K is capital, L is the population, and x is the fraction of the population that does
not work. The national savings rate is γ, the labor force grows at rate n, and capital
depreciates at rate δ.
a. Express output per capita (y = Y/L) as a function of capital per capita (k = K/L)
and x.
y  k  (1  x)1
b. Describe the steady-state of this economy by solving for yss and describing (in a
couple of sentences) how this case differs from our standard one.

  1
y ss  (1  x)

  n 
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Problem Set 3
FE411 Spring 2009
Rahman
The fact that there are fewer workers lowers the marginal product of
capital and , hence, acts like a negative technological shock that reduces
the amount of capital the economy reproduce in steady. This lowers the
steady-state level of output per person.
c. Suppose that some change in government policy reduces x (for example, the
retirement age is retroactively raised from 65 to 70 years of age). Describe (both
graphically and with words) how this change affects output both immediately and
over time. Is the steady-state effect on output larger or smaller than the
immediate effect? Explain.
As soon as x falls, output jumps up from its initial steady-state value. The
economy has the same amount of capital (since it takes time to adjust the
capital stock), but this capital is combined with more workers. At that
moment the economy is out of steady state: it has less capital than it
wants to match the increased number of workers in the economy. More
capital is accumulated, raising output even further than the original jump.
3) 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?
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Problem Set 3
FE411 Spring 2009
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 3
FE411 Spring 2009
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
4) 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 return to education for an individual with 9 years of education is:
(1.134)4 · (1.101)4 · (1.068)1 = 2.6.
In other words, if the wage of an individual with no education is W, the wage paid
to this individual is 2.6·W. 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:
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Problem Set 3
FE411 Spring 2009
Rahman
Payment to Human Capital / Total Payment =
2 .6  1
 0.615.
2 .6
The fraction of wages paid to human capital is 61.5%.
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 2
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:
y2000,ss
y1900,ss

h2000 3.16

 2.45.
h1900 1.29
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:
g  (2.45)
1
100
 1  0.009 .
We conclude that the annual average growth rate of output per worker is 0.9%.
5) 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.
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Problem Set 3
FE411 Spring 2009
Rahman
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.
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.
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Problem Set 3
FE411 Spring 2009
Rahman
6) 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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