I. Introduction, 38. — II. Review and development of Stigler's original model, 39. — III. The effect of dependence among the prices offered, 44. — IV. The sequential rule for Stigler's model, 45. — V. Implications and comments, 49.

THE MEASUREMENT of economic inequality is a timely and important topic. Often the Gini index or the entire Lorenz curve is used; however, the relative mean deviation (or Pietra ratio) has been used by Schutz [9] and Budd [1] to study United States data. Eltet6 and Frigyes [2] developed new measures to aid in their analysis of Hungary's income distribution, and Kondor [6] has shown that these new indices are related to the relative mean deviation. In order to draw valid conclusions from actual samples, one needs to know the sampling distribution of the statistic used to estimate the measure of inequality. The purpose of the present paper is to adapt methods used by the author [3 and 4] in another context to obtain the large sample theory of the mean deviation, Pietra ratio, and the measures of Eltet6 and Frigyes. Since several of these measures estimate some parameters of the underlying income distribution function, the asymptotic theory of the estimators is more complicated than might appear at first glance.

[This paper presents a method for obtaining the large sample distribution of the mean deviation and related measures of income inequality.]

M OST of the measures of income inequality are derived from the Lorenz curve; indeed Morgan (1962) states that the Gini index is the best single measure of inequality. The present article reviews some of the theoretical properties of the Lorenz curve, relates them to characteristics of the frequency function underlying the income distribution and develops methods for obtaining accurate bounds on the Gini index which do not depend on curve fitting. In the process we should also like to lay to rest some myths concerning the Gini index such as: (a) its relative insensitivity (Rltet6 and Frigyes, 1968), (b) difficulty in computation (1968), and (c) problems related to the inclusion of negative incomes (Budd, 1970). The basic idea of our approach is to obtain upper and lower bounds to the Gini index from data which are grouped in intervals and the mean income in each interval is known. The usual method (Morgan, 1962) of estimating the Gini index yields a lower bound by assuming that all incomes in any interval equal the average income. We derive an upper bound to the grouping correction (Goldsmith, et al., 1954, p. 10) and hence to the Gini index by distributing the income to maximize the spread within each group. On the 1967 Internal Revenue Service tax data, the difference between our bounds is less than 0.006. As most income distributions come from a frequency function (density) which decreases in the large income range, we develop improved bounds for the Gini index based on this assumption. Fortunately, this assumption can be checked from the data so that we can use the sharper bounds only for the appropriate intervals. Using this second method the difference between our bounds is ? .002. Because Soltow (1965) detects a change in the Gini index of 0.8 of one per cent or about 0.003 or 0.004, our bound seems quite adequate. In section VI we extend our method to obtain upper and lower curves for the Lorenz curve. After reviewing the basic properties of the Lorenz curve we proceed to derive bounds on the mean difference and Gini index. In section IV we analyze an actual sample and show that the method used by the Census Bureau (1967) often leads to estimates which are outside the mathematically possible bounds we derived. Finally, in an appendix we show that the Pareto law does not give a good fit to current United States tax data.

ECONOMISTS OFTEN SUMMARIZE the income distribution by the Lorenz curve and Gini index. A variety of parametric methods (e.g., [1 and 8]) have been developed to estimate these measures from the grouped income data governments provide (e.g., [3 and 12]). Previously, one of the authors developed a distribution-free approach [5] which yielded accurate bounds on the Gini index. While analogous bounds on the Lorenz curve can be obtained [5 and 10], the resulting curve is not smooth so a method of interpolation is needed. The purpose of this paper is to adapt an old technique of numerical analysis, Hermite interpolation [7 and 13], to our problem and to show that it usually works in theory and in practice. Our paper was motivated by the work of Brittain [2] who also used numerical methods. Unfortunately, his procedure often resulted in estimates of the Gini index which were inconsistent with the above-mentioned bounds. Although the piecewise Hermite interpolation yielded accurate estimates of the Gini index, it is not always convex as the Lorenz curve must be. Section 5 states conditions for the interpolated curve to be convex or at least increasing over an interval. While these conditions are usually satisfied by real data, a theoretical example illustrates how an error may arise.

[This paper considers the interpolation of percentiles when the value of the c.d.f. F(x) is specified at n points. Upper and lower bounds on the percentile are obtained assuming that data is generated by a unimodal density. Sharper bounds are derived when the average of the observations in each interval is given. In addition to improving the standard linear interpolation, our results indicate that much information can be gained by reporting group means as well as frequency counts.]