Demography/Economics 175
Professor Ronald Lee
Spring 2012
Problem Set #4
Due: Tuesday, March 20th by 2:10 pm
** Late assignments will not be accepted *
Answer Key
Questions about this problem set?
If you encounter any problems, please feel free to post a question on the course’s
Piazza site (search for Demog 175 on www.piazza.com/berkeley), or send an email
with your question to 175gsi2012@gmail.com.
Population and Land Size
1. Model Setup
For this question, please make the following assumptions:
• There is one output: food
• There are two inputs: land and labor
• There is no change in technology
• There is no capital accumulation
• There are diminishing returns to labor on a fixed supply of land after some
threshold population size is reached, and until that point the returns to labor
are constant.
• There is no distinction between working age population and total
population
• Labor supply does not depend upon wage levels
In 1626, Peter Minuit, the Director-General of the Dutch colony of New Netherland
purchased Manhattan Island from the Native American Lenape people for a price of
60 guilders. In the analysis that follows, assume that, at the time of purchase,
Manhattan was uninhabited and that the island would be settled in the near future
by Dutch colonists.
a) Draw a graph that plots Manhattan’s population on the x-axis against total
output on the y-axis. Draw a second graph that depicts the marginal product of
labor and the average product of labor for Manhattan’s population. The population
should be plotted on the x-axis against output per worker on the y-axis. Clearly
label all axes, curves and the point, P*, at which diminishing marginal returns to
labor set in.
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Note: On the bottom graph, the line should be linear in the range, (0,P*) and
diminishing at the margin thereafter. The APL must be above the MPL.
b) Assume that Manhattan’s economy is competitive such that the wage is given by
the marginal product of labor. For the population size, P*, what part of total output
will accrue to landowners in the form of rents? Briefly explain why this is the case.
Zero. This is because, at this population size, land is relatively abundant and labor
is relatively scarce.
c) Can Manhattan’s average product of labor (APL) fall below its marginal product of
labor (MPL)? Briefly explain why or why not.
No. With diminishing marginal returns to labor, each immigrant to the island adds
to the island’s total output by a smaller amount than the immigrant who came
before him. While the APL is an average of the productivity of every worker, the
MPL only captures the productivity of the last worker.
d) Now assume that every Manhattan resident needs a given amount of food, s, to
survive and reproduce. Draw a graph that depicts the maximum population size
that can be sustained on Manhattan.
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e) Name a condition which gives the maximum sustainable population size under
egalitarian income distribution assuming every resident needs an amount of food, s,
to survive.
s = APL
f) In an economy in which there is no government redistribution of income from
landowners to workers, what will happen to the population when APL > s > MPL? If
the population changes, what will be the new equilibrium? Briefly explain your
answer.
If there is redistribution in the economy (from landowners to workers), the population size is sustainable as landowners can subsidize workers’ wages so that they are equal (or higher than) the subsistence level. Without redistribution, the population size is not sustainable and must, in the long run, shrink until the MPL=s. This is because workers cannot survive when the wage is below subsistence. 3
g) Using your graph, name the condition under which the average standard of living
in Manhattan is maximized. [2 points]
The average standard of living is conceptually equivalent to the GDP per capita of this economy. Thus, per capita Income is maximized in the region at which APL = MPL. In other words, this is the region between (0, P*). h) Using your graph, identify a region of population sizes within which the one that
maximizes Manhattan’s military power is located. [2 points]
This is the region between the point at which s=MPL and s=APL. Military power requires both people and equipment. While people are maximized at s=APL, equipment Is maximized where total surplus is highest at s=MPL. 2. The Malthusian Model
Two economists, Paula and Elise have been hired to discuss the economic
consequences of population growth for the hypothetical country of Florin. The
following are excerpts from their debate. Using your knowledge of the model of
population size and land use presented in lecture as well as Malthus’ theory of
population dynamics, you are asked to provide responses to several of their
statements listed below. In the questions that follow, assume that the economy of
Florin operates under competitive labor markets.
a) Elise states: “I am in favor of government policies that incentivize a lower level of
fertility. When population grows, wages fall. Therefore, population control is vital to
maintaining our current standard of living.” Assuming that the capital stock and
technology are not changing appreciably, name a condition that is required for
Elise’s statement to be true. [3 points]
Elise’s statement is only true if we assume that there are diminishing marginal returns to labor.
b) Paula states: “I believe it is important to distinguish between an increase in
population that is due to a one-time shock, such as the one-time arrival of a large
group of immigrants, and an increase in population that is due to a permanent
increase in fertility.” Using Malthus’s theory as a guide, is Paula’s statement correct
or has she made an error? Will population growth that is due to a temporary rise in
immigration really lead to a different wage level than population growth that arises
from a permanent change in the fertility level? Briefly explain. [4 points]
Under Malthus’ assumption, Paula is indeed correct. A one-­‐time shock to the population that does not change the fertility or mortality curves does not, in the long run, lead to a change in the wage. On the other hand, a permanent increase in the fertility schedule will lead to a larger population size and a lower wage.
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