A Unified Approach to Measuring Poverty and Inequality
However, if generalized Lorenz curves cross, then the final judgment is contingent on which monotonic income standard satisfying the transfer principle is employed. Notice that when quantile functions can rank two distributions, generalized Lorenz curves must rank them in the same way, because a higher quantile function ensures that the area beneath it is also greater. However, even when quantile functions cross, generalized Lorenz curves may be able to rank the two distributions. We will use these two curves and their stochastic dominance rankings later in discussing inequality and poverty measurement. A final curve depicts the general mean levels as the parameter a varies. Given the properties of the general means, this curve is increasing in a and tends to the minimum income for very low a and rises through the harmonic, geometric, arithmetic, and Euclidean means, tending toward the maximum income as a becomes very large. A higher quantile function will raise the general mean curve. A higher generalized Lorenz curve will raise the general mean curve for a < 1 or the general means that favor the low incomes. The curve is useful for determining whether a given comparison of general means is robust, and if not, which of the income standards are higher or lower. It will also be particularly relevant to discussions of inequality in later sections. Growth Curves Some analyses go beyond the question of which distribution is larger to consider the question of 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. Then the next question becomes at what percentage rate did the income standard grow. Such growth is most often defined by income per capita, or the mean income. 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. A growth curve depicts the rate of growth across an entire class of income standards, where the standards are ordered from lowest to highest. Each of the dominance curves described above suggests an associated growth curve.
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