Gini coefficient

Overview

The Gini coefficient is a summary measure of inequality for any nonnegative resource—most commonly income or wealth. It ranges from 0 (perfect equality: everyone has the same amount) to 1 (maximum inequality: one person has everything, all others have zero). Intuitively, it captures how unevenly a pie is shared. In microeconomics and applied settings, the Gini is used to compare inequality across regions, over time, or between groups; to evaluate how taxes, transfers, or pricing changes affect dispersion; and, in business, to assess concentration in pay, sales, or customer revenue.

Origins and Credit

Italian statistician and economist Corrado Gini introduced the coefficient in 1912 (“Variabilità e mutabilità”). He related it to the Lorenz curve—a graph that plots the cumulative share of income held by the bottom x% of the population—providing a geometric interpretation of inequality that remains standard.

Core Idea and Mechanics

The Gini is defined by the area between the Lorenz curve and the 45-degree line of perfect equality. Formally, if L(p) is the Lorenz curve (the share of total income held by the bottom p share of the population), then:

G = 1 − 2 ∫₀¹ L(p) dp.

In a discrete sample with individual incomes y_i (all nonnegative), mean income μ, and population size n, a widely used equivalent formula is:

G = (1 / (2 n² μ)) · Σ_i Σ_j | y_i − y_j |.

Interpretation and properties:

• Bounds and scale invariance: G ∈ [0,1]. Multiplying all incomes by the same factor does not change G.
• Anonymity and population invariance: G does not depend on individuals’ identities or the population size, only on the distribution.
• Pigou–Dalton principle: A mean-preserving transfer from a richer to a poorer person reduces G.
• Sensitivity profile: The Gini is most sensitive to changes in the middle of the distribution; measures like the Atkinson index (with high inequality aversion) place more weight on the bottom tail.

Beyond income, the same logic applies to concentration in other contexts, such as the distribution of sales across customers or defects across product lines (sometimes called a “Gini concentration” index).

Key Assumptions and Conditions

• Resources are measured consistently, nonnegatively, and over a comparable unit (individuals vs. households; pre- vs. post-tax; time period).
• For household-based comparisons, an equivalence scale (adjusting for household size) should be defined if the goal is individual welfare.
• Samples should represent the target population; top-coding (censoring very high values) and undercoverage can bias G downward.
• When resources can be zero or negative (e.g., net income after losses), the standard formulas require care; some variants or transformations are used to handle negatives.

Implications

• Policy evaluation: Changes in G help assess whether taxes, transfers, or wage policies compress or stretch the distribution. Combined with growth metrics, they inform inclusive growth analyses.
• Targeting and program design: Segment-specific Ginis (within and between groups) highlight where dispersion is most pronounced, guiding targeted interventions.
• Corporate applications: Firms can track a Gini of base pay or bonuses to monitor internal dispersion; in sales, a customer-revenue Gini quantifies concentration risk and informs account management and diversification strategies.
• Benchmarking and communication: Because G is unit-free and widely recognized, it aids clear cross-country or cross-division comparisons, provided the resource concept and unit of analysis are aligned.

Example in Practice

Suppose a company examines the distribution of annual pay across employees using quintiles (20% groups). The cumulative shares of total payroll for the bottom 20%, 40%, 60%, and 80% are 6%, 18%, 36%, and 62%, respectively; by definition, the bottom 100% have 100%. These points define an approximate Lorenz curve: (0,0), (0.2,0.06), (0.4,0.18), (0.6,0.36), (0.8,0.62), (1,1).

Approximating the area under the curve via trapezoids yields about 0.344. The Gini is then G ≈ 1 − 2 × 0.344 = 0.312. Interpretation: pay is moderately unequal (a Gini near 0.31), with the middle of the distribution contributing notably to dispersion. HR can track this over time as pay bands are adjusted, and can complement it with bottom-sensitive measures (e.g., Atkinson with a higher inequality aversion) to assess the impact of minimum pay policies.

Limitations and Common Misunderstandings

• Same Gini, different shapes: Different distributions can share the same G but differ in where inequality sits (bottom, middle, or top). Inspect Lorenz curves or quantiles alongside G.
• Not a poverty measure: G captures relative dispersion, not absolute living standards. A low-income society can have a low Gini and high poverty, and vice versa.
• Tail sensitivity and data issues: G is less sensitive to top-end changes than measures designed for tails; underreporting or top-coding high incomes can materially bias it.
• Decomposition limits: While G can be decomposed into within- and between-group components, it is not additively decomposable as cleanly as the Generalized Entropy class (e.g., Theil index). Interpret group decompositions with care.
• Negative values and zeros: The standard Gini is defined for nonnegative resources. With negatives, specialized adjustments are needed; naive application can yield misleading results.
• Unit-of-analysis choices matter: Switching between individuals and households, or pre- and post-tax concepts, can change G substantially; comparisons should keep definitions consistent.

Related Concepts

• Lorenz curve
• Atkinson index
• Theil index (Generalized Entropy)
• Pareto and heavy-tailed distributions
• Pigou–Dalton transfer principle
• Palma ratio