EXERCISE GUIDE
SEMINAR 1
DARIO DEBOWICZ
Seminar Number 1 – Poverty & Inequality Indexes
This Seminar seeks to refresh the knowledge on poverty & inequality
indicators, specifically in relation to their calculation and interpretation. Use
of an Excel spreadsheet is suggested as a time-saving devise.
A– Poverty Indexes
Formula for FGT Poverty Indexes:
1 M  ( z  yi ) 
P   
N i 1  z 

P(0) = Poverty Headcount Ratio (HCR)
P(1) = Poverty Gap Ratio
P(2) = Squared Poverty Gap Ratio
where
z is the poverty line
yi is the income of person i
N is the number of people in the population,
M (k in the Lecture Note) is the number of poor people
We start with two economies similar to economies A and B. The
economy of INCRILAND has 4 people, and their incomes are: 1, 2, 3, 5
Incriland’s pesos (from now on, pesos). Besides, the economy of
EQPOORLAND has also 4 people (what a coincidence!) with incomes
2,2,2,5. How much is national income in each of these economies?
o Prepare a graph putting individuals in X axis and incomes in Y
axis for each economy. Advise: order individuals by increasing
income when going from origin to the right.
o Graph a poverty line of 4 pesos (as a horizontal segment) for
each economy
o Identify the number of poor people in each economy
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EXERCISE GUIDE
SEMINAR 1
DARIO DEBOWICZ
o Calculate the Poverty Headcount Ratio Index. How do we
interpret this number?
o For each economy, identify graphically the income that each
poor person needs to get to the poverty line (an “individual
poverty gap”), and write down how many pesos each poor
person needs to get to the line.
o Calculate the Poverty Gap for INCRILAND &
EQPOORLAND. How do we interpret these results?
o Calculate the square of the poverty gap for each individual.
o How are the squared poverty gaps of the poorest in comparison
to the ones of the other poor people? Are these differences
larger than the ones for (simple) poverty gaps?
o Calculate the squared poverty gap for the whole population of
INCRILAND & EQPOORLAND. How do we interpret these
figures?
A - ADDITIONAL EXERCISES (time allowing)
o Why is it said that P(2) is a measure of inequality among the
poor? Answer this considering formula from Ray page 291:
P(2) = HCR . [ IGR2 + (1-IGR)2Cp 2 ]
Where Cp is the coefficient of variation among the set of poor
people & IGR is the income gap ratio, where IGR increases with
PGR (they just differ in how the normalization is done, as you can
see in Ray’s page 255).
o If the Ministry of Economy tells you to reduce the Poverty
Headcount Index as much as possible, and gives you a budget
of $5 (stinky Ministry?)… how would you allocate it? Now,
how would you allocate this budget if the Ministry had just told
you to “reduce poverty”? What can you conclude from this?
o The Weak Transfers Principle states that if a poor person
transfers money to someone who is more poor, then the poverty
indicator should decrease. Do the FGT measures reflect this?
o Principle of Transfer Sensitivity: a given regressive transfer
between two poor people must matter more if both (starting)
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EXERCISE GUIDE
SEMINAR 1
DARIO DEBOWICZ
incomes of the persons involved are reduced equally. Can you
show that the P(2) indicators does not reflect this principle?
(Ray p. 291)
o Decomposability: can you show with an example that the FGT
indicators are decomposable? (Ray p. 292)
B– Inequality Indexes


COEFVAR = 


G
(y
i
1
_
2n y (n  1)
_

 y) 2 

/ y
n


i
n
n
 y
i 1 j 1
i
 yj
As in last part, we start with two economies similar to economies A and
B from the Ravallion’s example (see lecture note). The economy of
INCRILAND has 4 people, and their incomes are: 1, 2, 3, 5 Incriland’s
pesos (from now on, pesos). Besides, the economy of EQPOORLAND
has also 4 people (what a coincidence!) with incomes 2,2,2,5.
o Start from the graph where you put individuals in X axis and
incomes in Y axis for each economy, ordering individuals by
increasing income when going from origin to the right.
o Graph a Lorenz curve for each economy
o Calculate the Coefficient of Variation and the Gini coefficient.
Can you categorically say that there is more inequality in
INCRILAND than in EQPOORLAND?
o Can you show using the formula why the Gini can be
interpreted as the expected difference in the incomes of two
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EXERCISE GUIDE
SEMINAR 1
DARIO DEBOWICZ
persons randomly taken from the population (normalized by
average income)? Could you give a numerical example to
illustrate this?
o Lorenz consistency states that for two distributions, if the
Lorenz curve of the first one is always to the right of the second
one, the inequality indicator of the first one will be larger.
o The Dalton principle states that “if one income distribution can
be achieved from another by constructing a sequence of
regressive transfers, then the former distribution must be
deemed more unequal than the latter. (Regreessive transfer: two
individuals i and j, originally yiyj and individual i transfers to
j.) (Ray ch. 6). Does the Lorenz consistency principle reflects
the Dalton principle?
B - ADDITIONAL EXERCISES (time allowing)
Are the following statements true, false, or uncertain? In each
case, back up your answer with a brief, but precise explanation.
a) The Lorenz curve must necessarily lie in the lower triangle
of the diagram, bounded by the 45 degrees line at the top
and the axes at the bottom.
b) 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.
c) If everybody’s income increases by a constant dollar
amount, inequality must fall.
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