Chapter 2: Income Standards, Inequality, and Poverty
Finally, a third curve depicts the general mean levels as parameter a varies. We call this curve a general mean curve. This curve has already been outlined in figure 2.6, where it is increasing in Îą; tends to the minimum income for very low a ; rises through the harmonic, geometric, arithmetic, and Euclidean means; and tends toward the maximum income as Îą becomes very large. Why is this curve useful? At the beginning of this subsection, an example showed that different generalized means may rank an income distribution differently. So the general mean curve is useful for determining (a) whether a given comparison of general means is robust to the choice of any income standard from the entire class of general means, and, if not, (b) which of the income standards is higher or lower. General mean curves are also related to the quantile function and the generalized Lorenz curve. A higher quantile function will always yield a higher general mean curve, and a higher generalized Lorenz curve will raise the general mean curve for a < 1, or the general means that favor the low incomes. The general mean curve concept will be particularly relevant to our later discussions of Atkinsonâ&#x20AC;&#x2122;s inequality measure. Growth Curves Some analyses go beyond the ordinal question (Which distribution is larger?) to consider the cardinal question: How much larger in percentage terms is one distribution than another? This question is especially salient when the two distributions are associated with the same population at two points in time. Thus, the second question follows: At what percentage rate did the income standard grow? The most common and well-known way of understanding growth is by the growth of per capita income or mean income. The arithmetic mean is the income standard involved in this case. However, the defining properties of an income standard ensure that its rate of growth is a meaningful number that can be compared with the growth rates of other income standards, either for robustness purposes or for an understanding of the quality of growth. As in our use of various curves in dominance analysis, we may also use different growth curves to understand how robust the growth of an income standard is and to understand whether the growth is of meaningful quality. A growth curve depicts the rates of growth across an entire class of income standards, in which the standards are ordered from lowest to highest.
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