# Poverty and Famines: An Essay on Entitlement and Deprivation

By Amartya Sen | Go to book overview

Appendix C
Measurement of Poverty

C.1. POVERTY GAPS AND HEAD COUNTS

S is the set of people in a community of n people. Person i's income yi, and those whose incomes are no higher than π (the poverty line) are poor, making up the set TS. The poor, q in number, are ranked according to income, and person i in T has the rank r(i), being r(i)th richest among the poor. Equi-incomed persons are ranked in any arbitrary order, but once the ranking has been done, r(i) is, in fact, a strict ordering.

The poverty gap of person i in T is gi, given by:

 (C1) gi=π-yi.

The total poverty gap of the poor is denoted g, and is given by:

(C2)

The two standard measures of poverty are the head-count ratio H and the income-gap ratio I, given respectively by:

 (C3) H = q/n
 (C4) I = g/qπ.

Denote the mean income of the poor as y* and their mean poverty gap as g*:

(C5)

 (C6) g*=π-y*=g/q.

The income-gap ratio can also be expressed as:

 (C4*) I=g*/π.

Consider now the following axioms of legitimacy of poverty measures. Take x and y as two n-vectors of income with xi and yi the incomes of person i in the two cases, respectively, and let the poverty measures be such that x and y yield values P(x) and P(y) respectively (given π and S). In all the axioms proposed in this section the set S of people and π the poverty-line income are assumed to be given. T(x) and T(y) are the poor in S respectively for x and y.

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