Chapter 2: Income Standards, Inequality, and Poverty
Example 2.9: Consider the six-person income vector x = ($80, $100, $800, $1,000, $50,000, $70,000), which is divided into two subgroups x' = ($80, $100, $50,000) and x" = ($800, $1,000, $70,000). Suppose the poverty line is z = $1,100, which is the same across both subgroups. Note that N' = 3, N" = 3, and N = 6, and, thus, N'/N = N"/N = 3/6 = 0.5. Then any additively decomposable poverty index can be expressed as P(x;$1,100) = 0.5P(x';$1,100) + 0.5P(x";$1,100).
Poverty and Income Standards The second way of understanding poverty measures is through the income standards discussed earlier. Like inequality measures, most poverty measures are based on a comparison between two income standards: a higher income standard b and a lower income standard a. However, there is a crucial difference between inequality measures and poverty measures. In inequality measures, the higher and lower income standards are two different income standards applied to the same income vector. In poverty measures, the higher and lower income standards are the same income standards applied to two different income vectors: one is the censored distribution and the other is the nonpoverty censored distribution. Recall that a censored distribution is obtained from an original income distribution by replacing the income of the nonpoor by the poverty line. The nonpoverty censored distribution is that income distribution where all incomes are equal to the poverty line income. It turns out that the higher income standard for poverty measures is the poverty line itself. Why is that so? This can be understood by the normalization property of income standards, which requires that if all incomes are equal in an income distribution, then an income standard of the distribution should be equal to that commonly held income. Because in a nonpoverty censored income distribution all incomes are equal to the poverty line, any income standard of the nonpoverty censored distribution should be equal to the poverty line itself, that is, b = z. Many well-known poverty measures take the form P = (z − a)/z or the form P = a/z or a monotonic transformation of either form. Commonly Used Poverty Measures In this section, we introduce various poverty measures that are in common use. We classify them into two categories. The first category lists basic
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