Atkinson index¶

An inequality measure equal to the proportional mean income society could forgo while retaining the same equally distributed equivalent welfare at a chosen inequality-aversion parameter.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3355
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Welfare Economics → Economics & Finance
Aliases: Atkinson measure, Atkinson inequality measure

Core Idea¶

The value depends on the normative aversion parameter and income-domain conventions, especially zeros and negatives; comparisons across different parameters do not share one welfare judgment. Individual incomes enter an isoelastic social-welfare function, whose equally distributed equivalent is compared with arithmetic mean income to express the welfare loss attributable to inequality. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

Wasted-Cookie Counter

Imagine a bag of cookies shared very unevenly, so some kids get lots and some get almost none. Ask: how many cookies, shared perfectly evenly, would make the group just as well off? If far fewer even cookies would do just as well, the uneven sharing was wasting a lot, and the Atkinson index tells how much. How much you care about fairness changes the answer.

Cost-of-Unfairness Number

The Atkinson index is a number that measures how unequal incomes are in a group of people. It asks a clever question: what smaller amount of money, if everyone got exactly the same share, would make the group just as well off as the real, uneven incomes do? The gap between that equal amount and the real average is the part lost to inequality. There is also a setting that says how much you care about unfairness; if you care more, the same incomes count as more unequal. So the answer depends on that setting, and results using different settings can't be treated as the same judgment.

Welfare Lost to Inequality

The Atkinson index measures income inequality as a welfare loss. It starts from a social welfare function where each person's income adds to total welfare, but with diminishing returns controlled by an inequality-aversion parameter, epsilon. From that it computes the "equally distributed equivalent" income: the single income that, if everyone had it, would give the same total welfare as the actual distribution. The index is one minus the ratio of this equivalent income to the mean income, so it runs from 0 (perfect equality) upward toward 1. A larger epsilon means you care more about the worst-off, which makes the same distribution look more unequal. The value depends on epsilon and on how zero or negative incomes are handled, so comparing indices computed with different epsilons is not comparing one welfare judgment.

The Atkinson index is a normative inequality measure derived from an isoelastic (constant-elasticity) social-welfare function of individual incomes. From that function one computes the equally distributed equivalent income: the single income which, received by everyone, would yield the same social welfare as the observed distribution. The index is one minus the ratio of this equivalent to the arithmetic mean income, interpreted as the proportional welfare loss attributable to inequality. An inequality-aversion parameter epsilon governs how the welfare function weights low incomes; epsilon equal to one uses a geometric-mean limiting form, and higher epsilon places more emphasis on the bottom of the distribution. The value depends on this normative parameter and on income-domain conventions, since zero or negative incomes can make the formula undefined for some epsilons. Comparisons across different epsilon values do not share a single welfare judgment. A full specification states the population and income concept, weights, the mean, epsilon, the admissible income domain, the formula including limiting cases, normalization and decomposability properties.

Scope of Application¶

Atkinson index belongs to welfare economics and is useful where the analyst can specify the typed welfare economics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the population and income concept, weights and mean, inequality-aversion parameter epsilon, admissible income domain, equally distributed equivalent formula including epsilon one and limiting cases, normalization, decomposability and interpretation as welfare-equivalent loss are explicit. The scope is broad within that domain but bounded by the need for the population and income concept, weights and mean, inequality-aversion parameter epsilon, admissible income domain, equally distributed equivalent formula including epsilon one and limiting cases, normalization, decomposability and interpretation as welfare-equivalent loss are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the population and income concept, weights and mean, inequality-aversion parameter epsilon, admissible income domain, equally distributed equivalent formula including epsilon one and limiting cases, normalization, decomposability and interpretation as welfare-equivalent loss are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Atkinson index. Atkinson index compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed welfare economics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the population and income concept, weights and mean, inequality-aversion parameter epsilon, admissible income domain, equally distributed equivalent formula including epsilon one and limiting cases, normalization, decomposability and interpretation as welfare-equivalent loss are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of welfare economics because they reuse the typed welfare economics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Individual incomes enter an isoelastic social-welfare function, whose equally distributed equivalent is compared with arithmetic mean income to express the welfare loss attributable to inequality., and type the carrier, state every parameter and convention in the definition, test that the population and income concept, weights and mean, inequality-aversion parameter epsilon, admissible income domain, equally distributed equivalent formula including epsilon one and limiting cases, normalization, decomposability and interpretation as welfare-equivalent loss are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Atkinson index Domain-specific

Parents (1) — outgoing typed relations

Atkinson index is a kind of Measurement Prime

It remains its own entry because its identity is fixed by the population and income concept, weights and mean, inequality-aversion parameter epsilon, admissible income domain, equally distributed equivalent formula including epsilon one and limiting cases, normalization, decomposability and interpretation as welfare-equivalent loss.

Hierarchy path (1) — routes to 1 parentless root

Atkinson index → is a kind of Measurement

Neighborhood in Abstraction Space¶

Atkinson index sits in a crowded region of the domain-specific corpus (26^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Welfare Economics & Utility Theory (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25