Inequality Measures: The Kolkata Index in Comparison With Other Measures

Abstract

We provide a survey of the Kolkata index of social inequality, focusing in particular on income inequality. Based on the observation that inequality functions (such as the Lorenz function), giving the measures of income or wealth against that of the population, to be generally nonlinear, we show that the fixed point (like Kolkata index k) of such a nonlinear function (or related, like the complementary Lorenz function) offer better measure of inequality than the average quantities (like Gini index). Indeed the Kolkata index can be viewed as a generalized Hirsch index for a normalized inequality function and gives the fraction k of the total wealth possessed by the rich 1−k fraction of the population. We analyze the structures of the inequality indices for both continuous and discrete income distributions. We also compare the Kolkata index to some other measures like the Gini coefficient and the Pietra index. Lastly, we provide some empirical studies which illustrate the differences between the Kolkata index and the Gini coefficient.

Evolutionary Dynamics of Social Inequality and Coincidence of Gini and Kolkata indices under Unrestricted Competition

“…The Gini coefficient is the ratio of the area that lies between the line of equality and Lorenz curve over the total area under the line of equality (Gini index g = 2S, S represents area of shaded region). The complementary Lorenz function L c (p) ≡ 1 − L(p) (see e.g.,[4]) is represented by the green line. The Kolkata index k is given the ordinate value of the intersecting point of the Lorenz curve and the diagonal perpendicular to the equality line (implying L c (k) = k, or k corresponds to the fixed point of the complementary Lorenz function).…”

Kinetic Exchange Income Distribution Models with Saving Propensities: Inequality Indices and Self-Organised Poverty Level

“…1) 1 − L(k) = k, that k fraction of wealth is being possessed by (1 − k) fraction of the richest population. As such, it gives a quantitative generalization of the approximately established (phenomenological) 80-20 law of Pareto (see e.g., [14]), indicating that typically about 80% wealth is possessed by only 20% of the richest population in any economy. Now defining the complementary Lorenz function L c (x) ≡ [1 − L(x)], one gets k as its (nontrivial) fixed point (while Lorenz function L(x) itself has trivial fixed points at x = 0 and 1).…”

“…The resulting inequalities can be measured here by determining the Most Probable Income (MPI), given by the location of maximum value of the distribution P (m), or by the location of the SOPL, below which P (m) = 0, together with the determination of the values of Gini (g) and Kolkata (k) indices (see e.g., [13,14]). Both the indices, Gini (oldest and most popular one) and Kolkata (introduced in [15], see [14] for a recent review), are based on the Lorenz curve or function (see [13,14]) L(x), giving the cumulative fraction (L = m 0 mP (m)dm/ [ ∞ 0 mP (m)dm]) of (total accumulated) income or wealth possessed by the fraction (x = m 0 P (m)dm/[ ∞ 0 P (m)dm]) of the population, when counted from the poorest to the richest (see Fig. 1).…”

Near universal values of social inequality indices in self-organized critical models

“…Another recently introduced inequality index, namely the Kolkata (k) index [3], can be defined as the nontrivial fixed point of the complementary Lorenz function L(p) ≡ 1 − L(p): L(k) = k. It says, (1 − k) fraction of people possess k fraction of wealth (k = 1/2 corresponds to perfect equality and k = 1 corresponds to extreme inequality. As such, k index quantifies and generalizes (see e.g., [4]) the (more than a) century old 80-20 law (k = 0.8) of Pareto [5]. Extensive analysis of social data (see e.g., [6,7]) indicated that in extremely competitive situations, the indices k and g equals each other in magnitude and becomes about 0.87.…”