Chapter 1: Introduction
the Sen mean. The expected (absolute) difference between two incomes can be written as (a′ − a), where a′ is the expectation of their maximum and a is the expectation of their minimum. Because the mean b can be written as (a′ + a)/2, the difference (b − a) is half of the expected absolute difference between incomes, which confirms that (b − a)/b is an equivalent definition of the Gini coefficient. In other words, the Gini coefficient is the extent to which the Sen mean falls below the mean as a percentage of the mean. Atkinson’s class of inequality measures also takes the form I = (b − a)/b, where the upper-income standard b is also the mean, but now the lowerincome standard a is a general mean with parameter a < 1. This income standard focuses on lower incomes by raising each income to the a power, averaging across all the transformed incomes, then converting back to income space by raising the result to the power 1/a. A lower value of the parameter a yields an income standard that is more sensitive to lower incomes and is lower in value. This will be reflected in a higher value for (b − a)/b, so the percentage loss from the mean is seen to be higher. The final example is the family of generalized entropy measures, whose definition and properties vary with a parameter a. There are three distinct ranges for the parameter: a lower range where a < 1, an upper range where a > 1, and a limiting case where a = 1. When a < 1, the generalized entropy measures evaluate inequality in the same way as the Atkinson class of inequality measures (and, in fact, are monotonic transformations). For example, when a = 0, the measure is the mean log deviation or Theil’s second measure given by ln(b/a), where b is the arithmetic mean and a is the geometric mean. Atkinson’s version is (b − a)/b. Over the second range where a > 1, the general mean places greater weight on higher incomes and yields a representative income that is typically higher than the mean. If all incomes were equal, the general mean and the mean would be equal. However, when incomes are unequal, the general mean will rise above the mean. The extent to which this occurs is used by the measure to evaluate inequality. For example, the inequality measure obtained when a = 2 is (half) the squared coefficient of variation, that is, one-half of the variance over the squared mean. The general mean in this case is the Euclidean mean, which first squares all incomes, then averages the transformed incomes, and finally returns to income space by taking the square root. The Euclidean mean and the mean of the two-income distribution (4, 4) are both 4. Altering the distribution to (1, 7) raises the Euclidean mean to 5 but leaves its mean at 4.
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