Can a Poverty Index be Both Relative and Absolute?

Abstract

BEGINNING WITH THE WORK OF A. K. SEN (1976) poverty research has made extensive use of axioms to construct and evaluate distribution-sensitive poverty measures. An index is distribution-sensitive if it satisfies Sen's (1976) weak transfer axiom; poverty decreases when income is transferred from a high income poor person to a low income poor person. A distribution-sensitive measure is appealing because it reflects the extent of deprivation among the poor and incorporates aspects of poverty that are ignored by standard measures such as the headcount ratio and poverty gap ratio. In addition to the weak transfer axiom, a number of other axioms that have considerable intuitive appeal when considered in isolation have been identified. However, several researchers (e.g., Thon (1979), Kundu and Smith (1983), and Donaldson and Weymark (1986)) have shown that in some cases it is impossible to devise a poverty index that is consistent with the joint implications of otherwise reasonable axioms. Foster (1984, p. 233) interprets the difficulties caused by the combined effects of two or more axioms . . as a warning not to measure too many aspects of poverty at the same time. As a consequence, it is now generally recognized that poverty axioms can have joint effects and it is important to explore the implications of such influences. In a comprehensive investigation of subgroup consistent poverty measures, Foster and Shorrocks (1991) examine the interaction between the scale invariance and translation invariance axioms.2 The first axiom requires a poverty index to be unaffected by the doubling of all incomes and the poverty line. The second axiom requires a poverty index to be unaffected by equal increments to each income and to the poverty line.3 Poverty indices that satisfy the first axiom are considered to be poverty while those satisfying the second are poverty indices. If a distribution-sensitive index satisfying both axioms could be found it would provide a single measure of relative and absolute poverty and have considerable intuitive appeal in poverty measurement. But, one implication of Foster and Shorrocks' (1991) work (Proposition 6) is that for the class of poverty measures they consider, a distribution-sensitive poverty index that is both relative and absolute cannot exist. This note follows Foster and Shorrocks (1991) but concentrates on the implications of the joint effects of scale and translation invariance. We drop the assumption of subgroup consistency and derive a proposition that has several important implications. First, for all poverty indices, only those relying upon simple headcounts of the poor satisfy both the scale and translation invariance axioms. This result clearly illustrates the strong restrictions imposed by combining scale and translation invariance with other standard axioms; there can be no distribution-sensitive poverty index that is relative and absolute. Second, the result provides a simple characterization of the entire class of the headcount-related poverty indices (defined below). Third, the result leads to a direct proof of the result