Measurement of Poverty
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 T ⊆ S. 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:
The total poverty gap of the poor is denoted g, and is given by:
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*:
The income-gap ratio can also be expressed as:
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.