What Are the Axioms of Inequality Measures?

The axioms of inequality measures are fundamental properties that an ideal measure of income or wealth inequality should satisfy to be considered robust and meaningful. These principles ensure that an inequality measure behaves consistently and logically when changes occur in income distribution or population characteristics.

According to the provided reference, the five key axioms for inequality measures are:

1. Principle of Transfer (Pigou-Dalton Principle)
2. Scale Invariance
3. Translation Invariance
4. The Principle of Population
5. Decomposability

These axioms, established over time in economic theory, serve as benchmarks for evaluating the suitability and accuracy of various inequality indices like the Gini coefficient, Theil index, or Atkinson index.

Understanding the Core Axioms

Each axiom represents a desirable characteristic for an inequality measure, reflecting different aspects of how inequality should be conceptualized and measured.

1. Principle of Transfer (Pigou-Dalton Principle)

The Principle of Transfer, also known as the Pigou-Dalton Principle, is perhaps the most fundamental axiom. It states that an inequality measure should decrease if a transfer of income occurs from a richer individual to a poorer individual, provided that this transfer does not reverse their relative ranks (i.e., the richer person doesn't become poorer than the person they transferred money to).

• Practical Insight: This axiom captures the intuitive idea that moving resources from the "haves" to the "have-nots" generally reduces inequality.
• Example: If person A earns \$100 and person B earns \$20, and \$10 is transferred from A to B (so A has \$90, B has \$30), the inequality measure should register a decrease in inequality.

2. Scale Invariance

Scale Invariance (also known as relative income principle or homogeneity of degree zero) dictates that an inequality measure should remain unchanged if all incomes in a population are scaled by the same positive factor. This means if everyone's income doubles or halves, the relative inequality should not change.

• Practical Insight: This axiom is crucial for comparing inequality across different economies or over time, especially when national income levels change due to inflation or economic growth. It focuses on the distribution shape, not the absolute income level.
• Example: If a country's average income doubles across the board due to economic prosperity (everyone earns twice as much), the distribution of wealth relative to each other remains the same, and thus the inequality measure should yield the same result.

3. Translation Invariance

Translation Invariance (also known as absolute income principle) states that an inequality measure should remain unchanged if an equal amount is added to (or subtracted from) everyone's income. This implies that if all individuals receive an identical lump-sum bonus or face an identical lump-sum tax, the inequality measure should not change.

• Practical Insight: This axiom focuses on absolute differences. If inequality is measured by absolute gaps, then adding a constant to everyone's income doesn't change those gaps. However, it's important to note that many common measures like the Gini coefficient are not translation invariant, as adding a constant amount reduces relative inequality. Its inclusion as an axiom highlights different theoretical perspectives on what constitutes "invariance."
• Example: If every person in a society receives an extra \$500 through a universal basic income program, an inequality measure satisfying this axiom would report no change in inequality.

4. The Principle of Population

The Principle of Population asserts that the size of the population should not affect the measure of inequality. If an identical population group is replicated (e.g., two identical countries merge), the overall inequality measure should remain the same.

• Practical Insight: This ensures that the measure can be used to compare inequality across populations of different sizes without being biased by population counts. It focuses purely on the distribution within the population.
• Example: If a town with a certain income distribution is perfectly duplicated, creating a new, larger town with the exact same internal income structure, the inequality measure for the combined town should be identical to that of the original town.

5. Decomposability

Decomposability refers to the property where the total inequality of a population can be broken down into contributions from different subgroups within that population, plus the inequality between those subgroups.

• Practical Insight: This axiom is highly valuable for policy analysis as it allows economists and policymakers to identify which specific demographic, regional, or occupational groups contribute most to overall inequality. This helps in targeting interventions more effectively.
• Example: The total income inequality in a country could be decomposed into:
  • Inequality within urban areas.
  • Inequality within rural areas.
  • Inequality between urban and rural areas.

Summary of Axioms

The table below provides a quick overview of the key axioms discussed:

These axioms guide the construction and selection of appropriate inequality measures, ensuring they possess desirable properties for analyzing and comparing income and wealth distributions.