Decomposable Poverty Measures

A POVERTY MEASURE is said to be decomposable if the poverty measure of a group is a weighted average of the poverty measures of the individuals in the group. An important property of decomposable poverty measures is that a ceteris paribus reduction in the poverty measure of a subgroup always decreases poverty of the population as a whole.

Decomposable poverty measures are particularly useful in poverty studies where a population is broken down into subgroups defined along ethnic, geographical, or other lines. We can use these measures to obtain the contribution of each subgroup to total poverty, and to estimate the effect of a change in subgroup poverty on total poverty.

Standard poverty indicators like the headcount ratio and the average poverty gap ratio are decomposable. However, these poverty measures violate some basic properties or axioms proposed by A.K. Sen, like the monotonicity axiom and the transfer axiom. On the other hand, the Sen poverty index and all its variants that rely on rank-order weighting are not decomposable. In particular, the indicators in that class fail to satisfy the property that an increase in subgroup poverty must increase total poverty.

In this context, an important contribution by J. Foster, Greer, and Thorbecke was to define a class of decomposable [Page 239]poverty measures that satisfy the basic axioms proposed by Sen. We describe here this class of decomposable indicators.

The Foster-Greer-Thorbecke index is a weighted sum of the poverty gap ratios of the poor. In contrast to the Sen index, the weights do not depend on the “ordering rank” of the poor, but on the poverty gap ratios themselves. In other words, the contribution of an individual to the poverty measure depends only on the distance between his income and the poverty line and not on the number of individuals who lie between him and the poverty line.

The Foster-Greer-Thorbecke index satisfies the monotonicity axiom (that is, a reduction in a poor person's income, holding other incomes constant, increases the poverty index) and the transfer axiom (that is, the index increases whenever a pure transfer is made from a poor person to someone with more income). Its decomposition shows that, in contrast to Sen-type indexes, this index satisfies the subgroup monotonicity axiom: if we change the incomes in a subgroup such that we reduce poverty in this subgroup, leaving the other subgroups the same, then total poverty in the population should decrease. Foster, Greer, and Thorbecke propose a generalization of this index.