Chapter 2: Income Standards, Inequality, and Poverty
Another interesting property of WGM is its monotonic relationship with parameter a, which requires that the value of WGM increase as a rises and decrease as a falls. A lower a gives more emphasis to lower values within a distribution and thus causes WGM to fall. Conversely, a higher a gives more emphasis to higher values within a distribution, causing the value of WGM to rise. Technically speaking, WGM(x; a) < WGM(x; a ') for any a < a '. We refer to this property of general means as increasingness to a. It follows from this property that WE(x) ≥ WA(x) ≥ WG(x) ≥ WH(x). There is an exception, however, when the values of general means do not change as a changes, and this happens when a distribution is degenerate. A society’s income distribution is degenerate if all people in that society have equal incomes. For a degenerate income distribution, all general means are equal; that is, WGM(x; a) = WGM(x; a ') for all a ≠ a '. Invariance of general means to degenerate distribution is another way of ensuring that they satisfy the normalization property. Given that a ranges from − ∞ to + ∞, what is the range of WGM? Unlike the value of a, however, WGM is not unbounded. Rather, it has a lower bound and an upper bound. When a decreases and approaches − ∞, WGM(x; a) converges to the minimum element in x. The society’s income standard in this case is nothing, but the poorest person’s income is x1. In contrast, when a increases and approaches + ∞, WGM(x; a) converges toward the maximum element in x, and the society’s income standard equals the income of the richest person, xN. Notice, however, that unlike the other general means, these two extreme income standards—WGM(x; − ∞) and WGM(x; + ∞) —are not sensitive to the entire distribution. That is, if any element in x other than x1 and xN changes, these two income standards do not reflect that change. Figure 2.6 describes the relationship between the family of generalized means and parameter a. As already discussed, the general mean is the arithmetic mean at a = 1, the geometric mean at a = 0, the harmonic mean at a = −1, and the Euclidean mean at a = 2. Values of general means increase with parameter a. They are bounded below by x1 = min{x} and are bounded above by xN = max{x}. One feature we should note carefully is that the general means are undefined for a < 0 when there is at least one nonpositive element in an income vector. For example, if an element of x is 0, then for a = −1, we have (0)−1 = 1/0. Therefore, one requirement for any measure in this family with a < 0 is that all elements in x be strictly positive.
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