Chapter 1: Introduction
curve for the headcount ratio traces the cumulative distribution function associated with the distribution (except that it takes its limits from the left rather than the right when it has jumps), so the poverty ordering is firstorder stochastic dominance. If one recalls the above discussion of stochastic dominance, this poverty ordering is equivalent to having a higher quantile function and also to having greater welfare according to every utilitarian welfare function with identical, increasing utility functions. The poverty curve associated with the headcount ratio is often called the poverty incidence curve. The poverty curve for the poverty gap measure is closely linked to the area beneath (or the integral of) the poverty incidence curve (or the cdf), which is another way of representing second-order stochastic dominance. Hence, the poverty ordering for the poverty gap measure is simply secondorder stochastic dominance. By the previous discussion, this means that the poverty ordering can also be represented by the generalized Lorenz curve, with a higher generalized Lorenz curve indicating unambiguously lower (or no higher) poverty according to the poverty gap measure. In addition, there is a useful welfare interpretation of this poverty ordering: it indicates higher welfare according to every utilitarian welfare function with identical and increasing utility function exhibiting diminishing marginal utility (Atkinson’s general class of welfare functions). The curve found by plotting the area beneath the poverty incidence curve for each income level z is often called the poverty deficit curve. The FGT index has a poverty curve that is closely linked with the area beneath the poverty deficit curve (or the double integral of the cdf), and hence its poverty ordering is linked to a refinement of second-order stochastic dominance called third-order stochastic dominance. This poverty ordering also has a welfare interpretation: higher welfare according to every utilitarian welfare function with identical and increasing utility function exhibiting diminishing and convex marginal utility. The final condition on the convexity of marginal utility ensures that the welfare function is more sensitive to transfers at the lower end of the distribution—a welfare version of the transfer sensitivity axiom. The curve found by plotting the area beneath the poverty deficit curve for each income level z is often called the poverty severity curve. Notice that the poverty orderings for the three FGT measures are nested in that if the headcount ratio’s ordering ranks two distributions, then the poverty gap’s ordering also ranks the distributions in the same way (but not
39