Chapter 2: Income Standards, Inequality, and Poverty
24. Suppose that the per capita poverty gap measure is used with a relative poverty line that sets z = αμ for some α > 0. When does one distribution have a lower level of relative poverty for all α > 0? (Hint: Think Lorenz.) 25. We have already shown that the poverty measures are different from each other and differ in their sensitivity to a distribution. Please provide certain examples with illustrative distributions and poverty lines such that a. The SST index rises, but the three FGT indices fall. b. The headcount ratio rises, but the SST index, poverty gap measure, and squared gap measure fall. c. The poverty gap measure rises, but the headcount ratio, SST, and squared gap measures fall. d. The squared gap measure rises, but the headcount ratio, poverty gap measure, and SST measure fall.
Notes 1. For further discussion on the use of consumption expenditure data versus income data, see Atkinson and Micklewright (1983) and Grosh and Glewwe (2000). 2. For a more detailed discussion of some of these issues, see Deaton (1997). 3. For the concept and a more detailed discussion about the principle, see Pigou (1912, 24–25); Dalton (1920); Atkinson (1970); Dasgupta, Sen, and Starrett (1973); and Rothschild and Stiglitz (1973). 4. For further discussion of the concept, see Foster and Shorrocks (1991). – 5. Going forward in this book, we will use the notation WA(x) and x interchangeably. They both denote the mean of distribution (x). 6. The measure was originally proposed by Sen (1976b) and thus we named the income standard after him. See also Foster and Sen (1997). 7. A related property has been developed by Zheng (2007a). Called unit consistency, it has a weaker requirement than the scale invariance property. The unit consistency property requires that if one distribution is more unequal than another distribution, then just changing the unit of measurement keeps the former distribution more unequal than the latter. The property can be formally stated as follows: for any two distributions x and x', if I(x) < I(x'), then I(cx) < I(cx') for any c > 0.
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