Chapter 2: Income Standards, Inequality, and Poverty
• Invariance properties leave poverty measures invariant to certain changes in the dataset. Properties in the invariance category are symmetry, normalization, population invariance, scale invariance, and focus. • Dominance properties cause a poverty measure to change in a particular direction. Properties in the dominance category are monotonicity, transfer principle, transfer sensitivity, and subgroup consistency. Six of these properties—symmetry, population invariance, scale invariance, focus, monotonicity, and transfer principle—are called basic properties. Many of these properties are analogous to the corresponding properties of income standards and inequality measures.11 The first invariance property, symmetry, requires that switching the income levels of two people while the poverty line remains the same leaves poverty unchanged. In other words, a person should not be given priority on the basis of his or her identity when evaluating the level of poverty within a society. Formally, it requires that the poverty measure of distribution x be equal to the poverty measure of another distribution x', if x' is obtained from x by a permutation of incomes without changing the poverty line. For example, recall the four-person income vector ($1k, $2k, $50k, $70k). If the poverty line is z = $10k, then the first two people are poor and the last two people are nonpoor. Now, if the income of the first and the fourth individuals are switched, the new income vector becomes x' = ($70k, $2k, $50k, $1k). This new vector x' is said to be obtained from vector x by a permutation of incomes.
Symmetry: If distribution x' is obtained from distribution x by permutation of incomes and the poverty line z remains the same, then P(x'; z) = P(x; z). The second invariance property, normalization, requires that the poverty measure be zero if no one’s income in the society is less than the poverty line. This is a natural property. For example, if the income vector of the four-person society is ($1k, $2k, $50k, $70k), but the poverty line in this case is $1k, then any poverty measure should be 0, reflecting that there are no poor in the society.
Normalization: For any income distribution x and poverty line z, if min{x} ≥ z, then P(x; z) = 0.
107