On the Measurement and Trend of Inequality: Reply - Formby, John P.; Seaks, Terry G.; Smith, W. (American Economic Review, 2017)
John Formby, Terry Seaks, and W. Smith (hereafter FSS) argue that the P-Gini coefficient is affected by the arbitrary choice of the age to a degree which brings the validity of [the] age-related measure into question (FSS, 1989, p. 2). More pointedly, a sufficiently narrow age partition, the P-curve can always be driven to the L-curve. Convergence [of the P-Gini] to a nonzero estimate does not occur... (p. 4). These conclusions I believe result from a misapplication of Gastwirth's theorem on disaggregation, and a failure to observe statistical rules relating to sample size and sampling error. I will show that when these rules are observed, the value of the P-Gini does not converge to zero but properly reflects the relative importance of the nonlife-cycle factors affecting the distribution. When calculating the traditional L-Gini, the more disaggregation the better; the number and accuracy of the sample points are the only consideration since all are thrown into one conceptual box and compared in terms of income size. But if we try to identify the factors which account for income inequality in terms of age versus nonage related factors, we are setting up two conceptual boxes (the age-Gini and the P-Gini) and we are no longer simply dealing with a Gastwirth-type problem. Statistical considerations come into play; for example, we must have a sufficient number of sample points in each conceptual box in order to give a reliable estimate of the importance of each factor. The key-allocating device which I employ is the age-Gini, derived from the average age-income profile. The age-Gini shows the amount of inequality that would exist if all nonage-related sources of inequality were eliminated. When calculating this coefficient, the means of the age-groups are used in order to wash out all random and nonagerelated influences, but this separating device works well only if the means are based on large samples. Otherwise, sampling errors create spurious variation and impart an upward bias to the value of the age-Gini. FSS (p. 4) drive the age-Gini value up to the L-Gini by increasing the number of agegroups until they equal the number in the sample. Since the means of the age-groups are now based on samples of one, they become as erratic as the individual incomes, and impart the maximum upward bias to the age-Gini. It is true that the age-income profile (and the age-Gini) are conceptually refined by using smaller age intervals, but unless sample size is large compared to the number of age intervals, the gains from conceptual purification will be more than offset by the greater sampling errors of the age means. This kind of limitation is shared by many other statistical measures which do not thereby lose their validity or usefulness. Under what conditions will the true or limiting value of the P-Gini emerge? FSS in their footnote 3 state that there is no limiting value other than zero. Let us test this claim. Assume we have a scatter diagram of income (Y) and age (X), and wish to show average income in relation to age. We start with a finite number of age-groups and plot their mean incomes on the diagram. By continuously reducing the age interval and increasing sample size, we end up with a curve passing through the true means of infinitely small age intervals: this defines the average age-income profile. Since for each person we have data on income and age, we can with this curve (or an approximation of it) calculate the age-Gini and L-Gini without grouping for age or income. The age-income curve allows us to determine the mean income (u) at any given age and for all persons. *Department of Economics, Portland State University, P.O. Box 751, Portland, OR 97207.