Chapter 2: Income Standards, Inequality, and Poverty
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⎛ 1 N ⎛ z − x* ⎞ 2 ⎞ 2 n * PMG (x; z) = WE (g ) = P = ⎜ ∑ ⎜ ⎟⎠ ⎟ . N z ⎝ ⎝ n =1 ⎠ 1 2 SG
(2.50)
There is another interpretation of the mean gap measure: P'FGT(x;z,2). Because the mean poverty gap is a monotonic transformation of the squared gap measure, it satisfies all the properties that are satisfied by the squared gap measure except the additive decomposability. One advantage of the mean gap measure compared with the squared gap measure is that the values of the mean gap measure are commensurate with the values of the poverty gap measure as discussed using equation (2.49). Values of the squared gap measure tend to be much smaller than the poverty gap measure, and these numbers are not comparable to each other. Unlike the squared gap measure, values of the mean gap measure tend to be higher than those of the poverty gap measure, because it uses the Euclidean mean instead of the arithmetic mean. For example, for the fourperson income vector x = ($800, $1,000, $50,000, $70,000) and poverty line z = $1,100, the poverty gap measure is 0.09, whereas the mean gap measure is (0.02)1/2 = 0.14. However, had the income of the poor been equally distributed, the income vector would have been x' = ($800, $1,000, $50,000, $70,000), and the poverty gap measure would remain the same as that of x (that is, 0.09), but the mean gap measure would be 0.13. Like the squared gap measure, the mean gap measure also lies between zero and one. Moreover, this measure has an interesting relationship with the average normalized income shortfall. When everyone in a society is poor, but there is no inequality, then the squared gap measure is equal to the average normalized income shortfall among the poor because CV = 0 and PH = 1. Thus, PMG = PSG = PH ⎡⎣P12G + z (1 − P1G )2 IGE (xa; z)⎤⎦ = P12G = P1G . (2.51) Clark-Hemming-Ulph-Chakravarty (CHUC) Family of Indices The final measure in our discussion of poverty measures is the ClarkHemming-Ulph-Chakravarty (CHUC) family of indices (see Clark, Hemming, and Ulph 1981; Chakravarty 1983). This family is an extension of the Watts index. The CHUC index is based on the generalized mean
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