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 5
Note: A problem marked with an [E] is in the format of a possible exam question. Other
problems not so marked are just as useful. Some have open-ended answers, and some are
designed just to get you to re-read the chapters in DE or the handouts.
(1) [E] The economy of ShortLife has two kinds of jobs, which are the only sources of income
for the people. One kind of job pays $200, the other pays $100. Individuals in this economy
live for two years. In each year, only half the population can manage to get the high-paying
job. The other half has to be content with the low-paying one. At the end of each year,
everybody is fired from existing positions, and those people assigned to the high-paying job
next year are chosen randomly. This means that at any date, each person, irrespective of
past earnings, has probability 1/2 of being selected for the high-paying job.
(a) Calculate the Gini coefficient based on people’s incomes in any one particular period and
show that it suggests a good deal of inequality. Now calculate each person’s average per
period lifetime income and compute the Gini coefficient based on these incomes. Does the
latter measure suggest more or less inequality? Explain why.
(b) Now change the scenario somewhat. Suppose that a person holding a job of one type has
probability 3/4 of having the same kind of job next year. Calculate the expected lifetime
income (per year average) of a person who currently has a high-paying job, and do the same
for a person with a low-paying job. Compute the Gini coefficient based on these expected per
period incomes and compare it with the measure obtained in case (a). Explain the difference
you observe.
(c) Generalize this idea by assuming that with probability p you hold your current job, and
with probability 1 − p you change it. Find a formula for inequality as measured by the Gini
coefficient for each p, examine how it changes with p, and explain your answer intuitively.
(2) [E] Draw Lorenz curves and calculate the Gini coefficient and the coefficient of variation
for the income distributions (a)–(f). In each situation, the first set of numbers represents the
various incomes, whereas the second set of numbers represents the number of people earning
each of these incomes:
(a) (100, 200, 300, 400); (50, 25, 75, 25)
(b) (200, 400, 600, 800); (50, 25, 75, 25)
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(c) (200, 400, 600, 800); (100, 50, 150, 50)
(d) (200, 400, 600, 800); (125, 25, 125, 50)
(e) (100, 200, 300, 400); (50, 15, 95, 15)
(f) (100, 200, 300, 400), (50, 35, 55, 35).
[Try to understand the implicit transfers that move you from one income distribution to the
other (except for the first three, which should turn out to have the same inequality — why?).]
(3) [E] The economy of Nintendo has ten people. Three of them live in the modern sector
of Nintendo and earn $2000 per month. The rest live in the traditional sector and earn only
$1000 per month. One day, two new modern sector jobs open up and two people from the
traditional sector move to the modern sector.
(a) Show that the Lorenz curves of the income distributions before and after must cross. Do
this in two ways: (i) by graphing the Lorenz curves and (ii) by first expressing both income
distributions as divisions of a cake of size 1, and then showing that the two distributions are
linked by “opposing” Dalton transfers.
(b) Calculate the Gini coefficients and the coefficients of variation of the two distributions.
(4) A well-known measure of economic inequality is the log variance. It is measured by
m
1X
LV =
ni [ln yi − ln µ]2 .
n
i=1
Show that this measure satisfies anonymity, the population principle and the relative income
principle, but fails to be Lorenz-consistent.
(5) A beautiful property of the Gini coefficient is that it turns out to be proportional to the
area between the Lorenz curve of an income distribution and the 450 line. (It is actually
equal to the ratio of this area relative to the area of the triangle formed below the 450 line.
Try and prove this for the case in which there are just two income classes, with a proportion
n earning a fraction a of total income, and the remainder 1−n earning the remaining fraction
1 − a of income.
(6) Are the following statements true, false, or uncertain? In each case, back up your answer
with a brief, but precise explanation.
(a) The Kuznets ratios satisfy the Dalton transfer principle.
(b) If the Lorenz curve of two situations do not cross, the Gini coefficient and the coefficient
of variation cannot disagree.
(c) If a relatively poor person loses income to a relatively rich person, the mean absolute
deviation must rise.
(d) The Lorenz curve must necessarily lie in the lower triangle of the diagram, bounded by
the 45◦ line at the top and the axes at the bottom.
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(e) The ethical principles of inequality measurement — anonymity, population, relative income, and transfers — are enough to compare any two income distributions in terms of
relative inequality.
(f) If everybody’s income increases by a constant dollar amount, inequality must fall.
(7) Suppose that there are three individuals with three wealth levels in the economy. Denote
the wealth levels by A, B, and C, and suppose that A < B < C.
(a) Suppose that the person with wealth level A earns an annual income of wA , and saves a
fraction sA of it. If the rate of interest on asset holdings is r, write down a formula for this
person’s wealth next year.
(b) Show that if income earnings of each individual is proportional to wealth (that is,
wB /wA = B/A and wC /wB = C/B), and if the savings rate is the same across individuals, inequality of wealth next year must be the same as inequality this year, as measured
by the Lorenz curve.
(c) Retain the same assumptions on income and wealth as in part (b) but now suppose that
the savings rates satisfy sA < sB < sC . Now how does the Lorenz curve for wealth next year
compare with its counterpart for the current year?
(d) Carry out the same exercise as in part (c), but now assume that all wages are equal and
so are savings rates.
(8) This problem tests your understanding of the effect of the income distribution on savings
rates.
The economy of Sonrisa has people in three income categories: poor, middle class, and rich.
The poor earn $500 per year and have to spend it all to meet their consumption needs. The
middle class earn $2,000 per year, of which $1,500 is spent and the rest saved. The rich earn
$10,000 per year, and consume 80% of it, saving the rest.
(a) Calculate the overall savings rate in Sonrisa if 20% of the people are poor and 50% are
in the middle class.
(b) Suppose that all growth occurs by moving people from the poor category to the middleclass category. Will the savings rate rise over time or fall? Using the Harrod–Domar model
and assuming that population growth is zero and all other variables are exogenous, predict
whether the resulting growth rate will rise or fall over time.
(c) Outline another growth scenario where the rate of growth changes over time in a way
opposite to that of (b).
(d) Understand well that this question asks you about how growth rates are changing. In
the simple Harrod–Domar model, the growth rate is constant over time because the savings
rate is presumed to be unchanging with the level of income. Do you understand why matters
are different here?