A simple method for measuring inequality

The paper addresses a core measurement problem in inequality metrics: the widely used Gini index, derived from the Lorenz curve, summarizes the entire distribution in a single statistic but is less sensitive to changes at the tails, and countries with intersecting Lorenz curves can be ranked ambiguously by Gini. Alternative measures such as the Atkinson index can deliver a complete ranking but depend on a normative inequality aversion parameter and the choice of social welfare function, which may vary across contexts. Inter-decile ratios like the share of the top 10% to the bottom 10% (T10/B10) capture tail disparities but ignore differences in the middle of the distribution. The research question is whether a simple composite index, using only widely available aggregate statistics, can jointly capture tail and overall distributional inequality, distinguish countries with the same Gini but different tail gaps (and vice versa), and detect dynamics when Gini is stable but tail gaps widen. The authors propose and test such an index using cross-country income distribution data.

The Gini index (Gini, 1912; Lorenz, 1905; Gini, 2005) is prevalent across socioeconomics and numerous scientific fields (Eliazar and Sokolov, 2012; Eliazar, 2018) including astrophysics, ecology, finance, engineering, health, and transport. Despite its interpretability and data availability, Gini’s rankings can disagree when Lorenz curves intersect and it is less sensitive to tail changes. The Atkinson index (Atkinson, 1970) introduces an inequality aversion parameter ε and the concept of equally distributed equivalent income, yielding complete rankings but embedding normative choices (Atkinson and Bourguignon, 2015; Cowell, 2011; McGregor et al., 2019). Generalized entropy (GE) measures and Theil-type indices (Theil, 1967; Bellù and Liberati, 2006) offer decomposability and sensitivity tuning but likewise require parameter choices. Tail-focused metrics (e.g., P90/P10, T10/B10) highlight disparities between richest and poorest deciles (Palma, 2011; OECD datasets) but omit mid-distribution information. This study positions its contribution as a simple, parameter-standardized composite measure combining Gini with top/bottom income shares to mitigate these limitations without requiring microdata.

Overview: The authors construct a composite inequality index I_i for country i using two components: the Gini coefficient (Gini_i) and a transformed function of the income share ratio between the top 10% (T10) and bottom 10% (B10). The aim is to balance sensitivity to tails (via T10/B10) with whole-distribution information (via Gini), while keeping the index bounded in [0,1] and simple to compute from widely available data. Components and transformation: Let R_i = T10_i / B10_i denote the ratio of the income share of the top 10% to that of the bottom 10%. Define H_i = 1 − (B10_i / T10_i)^α = 1 − R_i^{−α}. The parameter α ∈ (0,1) scales H_i to be commensurate with Gini_i so that both contribute comparably to the composite index. Calibration of α: Using annual data from the World Bank and OECD IDD for 2005–2015, α is empirically calibrated from the relationship avg(Gini) ≈ 1 − avg(R)^{−α}. Year-by-year estimates yield α in the ranges 0.197–0.207 (World Bank) and 0.271–0.281 (OECD IDD). Averaging across samples gives ᾱ = 0.239. For simplicity and standardization across samples and years, the authors fix α = 0.25 (¼), which is close to ᾱ and easy to compute (equivalent to taking two successive square roots). Composite inequality index: With α fixed at 0.25, compute H_i = 1 − (B10_i / T10_i)^{1/4}. The composite inequality index is then defined to lie in [0,1] by combining Gini_i and H_i via a Euclidean-type aggregation normalized by the number of components. In the generalized form for K components, I_i = sqrt(Gini_i^2 + Σ_j H_{j,i}^2) / sqrt(K), ensuring 0 ≤ I_i ≤ 1. In the baseline case with two components (Gini and one H term), K = 2 so I_i = sqrt(Gini_i^2 + H_i^2) / sqrt(2). The paper also presents the index in equivalent notational forms and discusses an earlier expression; the final recommended implementation uses α = 0.25 and the normalized Euclidean aggregation. Data and implementation: Inputs are (i) Gini_i from the World Bank/OECD IDD; (ii) T10_i and B10_i shares from the same sources to compute R_i and H_i. The main demonstrations use cross-country data for 2015 (75 countries from the World Bank; 35 countries from the OECD IDD). Time-series (2005–2015) are used to estimate α and to test dynamic sensitivity (e.g., stable Gini with rising T10/B10). No microdata are required. Extensions: To incorporate mid-distribution sensitivity, the method generalizes to multiple inter-percentile ratios P(100−x)/Px (e.g., P90/P10, P80/P20, P70/P30, P60/P40, P50/P50), defining corresponding H_{jx} terms with their α values (or a standardized α) and aggregating: I_i = sqrt(Gini_i^2 + Σ_{j=1..N} H_{jx,i}^2) / sqrt(N+1). The number of H terms depends on data availability. Alternative (not recommended) index: For reference, the authors note an unbounded alternative: Alternative index_i = (Gini_i in percent)^2 + (T10_i/B10_i)^2 / 100, which ranges from 0.01 to ∞ and is harder to interpret and compare.

• Correlations: Across 2005–2015, the Gini, H_i, and T10/B10 are highly correlated in expected directions: Gini positively with H_i; T10/B10 negatively with Gini and H_i (absolute correlation coefficients > 0.900). • Calibration of α: Yearly α estimates are 0.197–0.207 (World Bank) and 0.271–0.281 (OECD IDD). The across-sample mean ᾱ = 0.239; the study standardizes α = 0.25 for practical use. • Discrimination when Gini is equal but tails differ: In 2015 (World Bank), Greece and Thailand have the same Gini (0.360) but different T10/B10 (13.79 vs 8.88). The proposed index ranks Greece as more unequal: I = 0.425 (Greece) vs 0.391 (Thailand). Sixty-two of 75 countries change rank when switching from Gini to I; 13 remain unchanged. • Discrimination when T10/B10 is equal but Gini differs: Malta and Slovak Republic share T10/B10 = 6.74 but have different Gini (0.294 vs 0.265). The index reflects higher inequality in Malta: I = 0.339 (Malta) vs 0.327 (Slovak Republic). • Dynamic sensitivity: Mexico (World Bank) 2008–2014 shows a roughly stable Gini (~0.453) while T10/B10 rises (17.6 to 18.6), with I capturing increasing inequality. Similarly, Italy (OECD IDD) shows stable Gini with an increasing T10/B10 trend; I increases accordingly. • Broad applicability: Demonstrations for 75 countries (World Bank) and 35 countries (OECD IDD) in 2015 show the index’s capacity to refine cross-country rankings and capture distributional dynamics using readily available statistics.

The composite index directly addresses two complementary limitations: Gini’s lower sensitivity at the distribution tails and inter-decile ratios’ disregard for the distribution’s middle. By combining Gini with a transformed T10/B10-derived term H_i and normalizing to [0,1], the index preserves interpretability while incorporating tail information. Empirical demonstrations show that the index reorders countries that appear identical under Gini when their top/bottom income shares differ, and it differentiates countries with identical T10/B10 when overall distributional shape (reflected in Gini) differs. Time-series results illustrate that the index captures rising tail disparities even when Gini is flat, providing a more nuanced monitoring tool for inequality trends. The approach is data-light (no microdata), scalable across settings, and generalizable via additional inter-percentile ratios to cover the full Lorenz curve if desired. This enhances relevance for policy analysis where both tail inequality and overall dispersion matter, and for broader scientific applications that require a bounded, simple heterogeneity metric for non-negative size distributions.

The study introduces a simple, standardized composite inequality index that integrates the Gini coefficient with top-versus-bottom income share information to overcome limitations of each component metric in isolation. Using cross-country data (World Bank and OECD IDD, 2005–2015), the index differentiates countries with equal Gini but different tail gaps and vice versa, and detects increasing inequality when Gini is stable but T10/B10 rises. The index requires only widely available aggregate indicators and is bounded in [0,1], facilitating interpretation and cross-country comparisons. The authors suggest extending the index by incorporating multiple inter-percentile ratios (e.g., P90/P10, P80/P20, etc.) to better reflect the entire distribution, and they highlight potential applications beyond socioeconomics to any domain involving size distributions of non-negative quantities. Future research could explore alternative weightings, sensitivity to different α choices, and validation against microdata-based measures.

• Different Lorenz curves can yield the same composite index value: distinct combinations of Gini and T10/B10 can produce identical I, implying residual differences in distributional shape not captured by the two-component index (examples: Belgium vs Serbia; Estonia vs Portugal). • Dependence on transformation parameter: Although standardized at α = 0.25 for practicality, results depend on the chosen α; while close to the empirically estimated ᾱ, alternative α values could alter sensitivity. • Coverage of the distribution: Using only T10/B10 emphasizes tails and may still miss mid-distribution differences; the authors propose adding more inter-percentile ratios to mitigate this. • Typological and data limitations: The approach relies on the accuracy and availability of aggregate indicators (Gini, T10, B10) from international databases; measurement error or definitional differences may affect comparability.