Chapter 2: Income Standards, Inequality, and Poverty
change. This property ensures that the measure focuses on the poor incomes in evaluating poverty. In fact, focus ensures that the income distribution is censored at the poverty line before evaluating a society’s poverty. For example, suppose the initial income vector is x = ($1k, $2k, $50k, $70k) and the poverty line income is $6k. Thus, the third person and the fourth person are nonpoor. If the income of either the third or the fourth person increases, but the poverty line remains unaltered at $6k, then the society’s poverty level does not change.
Focus: If distribution x' is obtained from distribution x by increasing the income of a nonpoor person while the poverty line remains the same at z, then P(x'; z) = P(x; z). The next group of properties are dominance properties. The first of these properties requires that if the income of a poor person in a society increases, then the poverty level should register a fall, or at least it should not increase. There are two versions of this property. One is weak monotonicity, which requires that poverty should not increase because of an increase in a poor person’s income. The other is monotonicity, the stronger version, which requires that poverty should fall if a poor person’s income in the society increases. These two properties are the same as the two corresponding properties of income standards, except the ones introduced here are solely concerned with incomes of the poor. For example, suppose the initial income vector is x = ($1k, $2k, $50k, $70k) and the poverty line income is $6k so that the first two people are identified as poor. If a new vector x' is obtained by increasing the income of either the first or the second person, while the poverty line remains unchanged, then according to the weak monotonicity property, poverty should not be higher in x', and, according to the monotonicity property, poverty should be lower in x'. Weak Monotonicity: If distribution x' is obtained from distribution x by increasing the income of a poor person while keeping the poverty line unchanged at z, then P(x'; z) ≤ P(x; z). Monotonicity: If distribution x' is obtained from distribution x by increasing the income of a poor person while keeping the poverty line unchanged at z, then P(x'; z) < (x; z).
109