Chapter 2: Income Standards, Inequality, and Poverty
What properties does a partial mean ratio satisfy? A partial mean ratio, as defined in equation (2.17), satisfies symmetry, normalization, population invariance, and scale invariance. Thus, a partial mean ratio satisfies all four invariance properties. What about the dominance properties? A quantile ratio satisfies the weak transfer principle but does not satisfy the transfer principle, transfer sensitivity, and subgroup consistency. It does not satisfy the transfer principle because some regressive and progressive transfers may leave the inequality measure unchanged, since a partial mean ratio does not consider the entire income distribution. Atkinson’s Class of Inequality Measures Atkinson’s class of inequality measures, developed by Sir Anthony Atkinson, is based on general means (see Atkinson 1970). All inequality measures in this family are constructed by comparing the arithmetic mean and another income standard from the family of general means. Recall that each measure’s formulation in the general means family depends on a parameter denoted by a, which can take any value between − ∞ and + ∞. In the Atkinson family of inequality measures, a is called the inequality aversion parameter. The lower the value of a, the higher a society’s aversion toward inequality. In other words, the more averse a society is toward inequality across the population, the more emphasis it gives to lower incomes in the distribution by choosing a lower value of a. The Atkinson class of inequality measures for a < 1 may be expressed as IA (x; a) =
WA (x) − WGM (x; a) W (x; a) = 1 − GM . WA (x) WA (x)
(2.18)
The Atkinson index of order a is the difference between the arithmetic mean and the general mean of order a divided by the arithmetic mean. Any Atkinson index lies between zero and one, and inequality increases as the index moves from zero to one. The minimum level of inequality, zero, is obtained when the total income is equally distributed across everyone in the society. Unlike the quantile ratios and the partial mean ratios, if IA(x; a) = 0 for any a < 1, then, by implication, the total income in the society is equally distributed. This is because any inequality measure in this family is constructed by considering the entire distribution.
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