Chapter 2: Income Standards, Inequality, and Poverty
measure is transfer neutral. However, unlike the SST index, it satisfies subgroup consistency because it is additively decomposable. Foster-Greer-Thorbecke (FGT) Family of Indices This family of measures was proposed by Foster, Greer, and Thorbecke (1984). The FGT family of measures has the following formulation: a
PFGT
* N ⎛ z − xj ⎞ 1 = (x; z,a ) = ∑ n =1 ⎜ ⎟ , N ⎝ z ⎠
(2.47)
where a ≥ 0. The parameter a can be interpreted as the inequality aversion parameter among the poor, which is conceptually the same as that for Atkinson’s class of inequality measures. As a increases, a society’s aversion toward inequality among the poor increases. Notice that there is a minor difference between parameter a in this case and parameter a in Atkinson’s class of inequality measures, where a lower value of a leads to greater aversion toward inequality. This difference exists because inequality is measured in the income space and poverty is measured in the normalized gap space, where large gaps imply worse situations. Measures in the FGT family take the form of various well-known poverty measures introduced earlier for different values of a. For example, for a = 0, the formulation in equation (2.45) becomes the headcount ratio because (z − x*n/z)0 =1 when xn < z and because (z − x*n/z)0 = 0 when xn ≥ z. Thus, PFGT(x;z,0) = q/N = PH(x;z). For a = 1, the formula becomes the poverty gap measure, which is the average of all normalized income shortfalls. For a = 2, the formula is the squared gap measure, which is the average of the square of all normalized income shortfalls. As a increases and becomes very large, PFGT approaches a Rawlsian measure17 placing more emphasis on the largest normalized income gap of the poorest person. However, note that the value of PFGT for any distribution decreases as a increases, and, for a very large a, the overall value of PFGT may be infinitesimally small. This occurrence can be verified by expressing the FGT formulation in equation (2.47) in general mean form using equation (2.3) as follows: PFGT(x;z,a) = [WGM(g∗; a)]a for a > 0.
(2.48)
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