Professor Debraj Ray 19 West 4th Street, Room 608 email: , homepage:

Professor Debraj Ray
19 West 4th Street, Room 608
Office Hours: Mondays 2.30–5.00pm
email: debraj.ray@nyu.edu, homepage: http://www.econ.nyu.edu/user/debraj/
Webpage for course: click on “Teaching”, and then on the course name.
Econ-UA 323
Development Economics
Problem Set 7
(1) Review the concepts of birth rates, death rates, and age distributions, and the way in
which these notions interact with one another. Construct an example of countries A and B,
where A has higher death rates than B in every age category, yet has an overall lower death
rate.
(2) Here is a specific example shows how birth and death rates affect age distributions.
Suppose there are just three ages; Y (young), M (middle-aged) and O (old). Suppose that
death at each age is given by the fractions dy , dm , with do = 1.
Finally, suppose that only M ’s give birth: let b be the fraction of newborns relative to the
entire M -population. Let ny (t), nm (t) and no (t) be the populations of the three ages. Make
sure you understand the following relationships (you don’t need to write anything for this):
ny (t + 1) = bnm (t),
nm (t + 1) = (1 − dy )ny (t), and no (t + 1) = (1 − dm )nm (t).
(a) Now suppose that the population is growing steadily at the rate of g, with pop shares of
each age group given by σy , σm and σo , and suppose that these shares are unchanging over
time. Using the above relationships, show that
(i) σy (1 + g) = bσm ,
(ii) σm (1 + g) = (1 − dy )σy , and
(iii) σo (1 + g) = (1 − dm )σm .
(b) Use part (a) to show that
σy
1 − dy
1 − dm
1+
+
= 1.
1+g
b
p
(c) Use equations (i) and (ii) of part (a) to show that 1 + g = b(1 − dy ), and then combine
with part (b) to conclude that
"
#
p
1 − dy
1 − dm
√
σy 1 +
+
= 1.
b
b
Now show that as the birth rate b goes up, or as the death rates dy and dm go up, the share
of the young, σy , must go up as well.
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(3) [A “Prisoner’s Dilemma” for population: Explain why each country might want to take
a pro-natalist stand for military or political reasons, but the combination of all countries
taking the same pro-natalist stance may make all countries worse off relative to a neutral
stance on population.
To answer this question I want you to read the example of the “Prisoner’s Dilemma” in the
game theory appendix of the main text, and then try and create an application of that to
the question above.
(4) We studied a model where a family wants one surviving child to provide old-age security.
Let us say that the probability of any one child living to look after its parents in old age is
1/2 (i.e., 50–50). However, the family wants this security level to be higher, say a probability
of q > 1/2.
(a) Describe the family’s fertility choices for different values of q, and examine the results for
different values of q.
(b) Calculate the expected number of surviving children for this family, under various values
of q.
(5) In the land of Oz, there are three inputs to production: capital, physical labor, and
mental labor. Men in Oz have more physical labor power than women, but both men and
women have the same amount of mental labor power.
(a) Who earns more in Oz, men or women? What do these differences depend upon?
(b) Now imagine that the technology is such that more capital raises the marginal product
of mental labor faster than it raises physical labor. As the economy of Oz grows over time,
its stock of physical capital is steadily increasing. How would you expect the relative wage
of men to women to change over time? Explain.
(c) Women have one unit of labor time that they can allocate between raising children and
being part of the workforce. How would this allocation be affected by the changes over time
that you found in your answer to (b)? Discuss the implications for fertility levels in the
population.
(6) True or False?
(a) A developing country is likely to have an overall death rate that is lower than that of a
developed country.
(b) The populations of Europe and North America grew at a combined rate between 1750
and 1900 that significantly exceeded the population growth rates of developing countries at
that time.
(c) If country A has a population growth rate that is lower than country B, then the average
woman in country A has less children than her counterpart in country B.
(d) Birth rates may be high even when death rates may be falling.
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(e) If total mortality among children remained constant, but the incidence of that mortality
shifted from late childhood to early childhood, fertility rates should decline.
(7) Suppose that families have a gender bias; that is, they have children until a son is born.
Suppose that at each birth, the probability of a child being a boy is 50-50.
(a) Will the country as a whole have more girl births than boy births (or vice versa) under
this targeting rule?
(b) Will larger families have more daughters or sons?
(c) If you have information on the sex and birth order of each child born to each family in
the village. How would you use the data to test your hypothesis that there is gender bias?
(8) This is a question on joint families, externalities, and fertility choice. Suppose that Ram
and Rani are the heads of a nuclear family, making their fertility decisions. For simplicity,
assume away gender bias and issues of child survival. The following table details the costs
and benefits (in dollars, say) of different numbers of children.
(a) Based on the information in the table, how many children would Ram and Rani have in
order to maximize their net benefit?
Number of children Total benefit ($) Additional cost
One
500
100
Two
750
100
Three
840
100
Four
890
100
Five
930
100
Six
950
100
Seven
960
100
Eight
960
100
(b) Now consider two identical nuclear families: Ram and Rani (as above), and Mohan and
Mona. Ram and Mohan are brothers and the two couples form a joint family. Both couples
have exactly the same costs and benefits of having children as in the table. Now suppose
that 50% of the upbringing costs of each child (e.g., child care) can be passed on to the other
family. Each couple makes independent decisions, taking only its own welfare into account.
Now how many children will each couple have?
(c) Explain the reason for this seemingly paradoxical result, using the concept of externalities,
and try and understand why larger families (either integrated across generations or between
siblings in the same generation), will tend to have a larger number of children per couple.