Principles of Fairness: The Axioms of Inequality Measures

Ever wondered how economists look at the stark differences in income or wealth around the world? We see massive fortunes alongside deep poverty, particularly evident in countries like India, where the top 10% capture a significant share of national income. But how do we move from just noticing this gap to scientifically measuring it? The answer lies in the concept of inequality measures and, crucially, the foundational principles that validate them-the axioms of inequality measures. These axioms are the non-negotiable rules that any self-respecting measure must follow to be considered a genuine indicator of fairness and distribution.

The ultimate test of fairness: The Pigou-Dalton principle of transfers 🔗

Imagine a transfer of wealth. If you take a rupee from a billionaire and give it to a person living below the poverty line, does that make the society more equal or less? Intuitively, it seems clear: more equal. This fundamental moral and economic intuition is formally captured by the Pigou-Dalton Principle of Transfers, often considered the most important axiom.

What makes a transfer progressive? 🔗

The principle states that an inequality measure must:

• Decrease following a progressive transfer: A transfer of income from a relatively richer individual to a relatively poorer individual, provided the transfer is not so large that it reverses their ranks (the poor person doesn’t suddenly become richer than the rich person).
• Increase following a regressive transfer: A transfer from a poorer individual to a richer individual.

Think of it like a seesaw (see image below). Perfect equality is a perfectly level seesaw. A progressive transfer – shifting weight (income) toward the lighter end (the poor) – moves the seesaw closer to balance, thus reducing measured inequality. A good measure, like the Atkinson Index, will reflect this movement by showing a decrease in its value. Without this axiom, any index would fail to capture the very essence of what we mean by “inequality.”

Do we measure relative or absolute inequality? Scale and translation invariance 🔗

When comparing inequality over time or across different countries, two crucial questions arise:

1. If everyone’s income is doubled, does inequality change?
2. If a fixed, equal amount of money is added to everyone’s income, does inequality change?

The answers depend on whether the inequality measure satisfies Scale Invariance or Translation Invariance, which define whether we are measuring relative or absolute inequality.

Scale invariance (Relative inequality) 🔗

Scale Invariance (also called homogeneity) demands that if every income in a distribution is scaled up or down by the same proportion (e.g., everyone’s income doubles or is cut in half), the measured inequality should remain unchanged. Many popular measures, including the Gini Coefficient and the Generalized Entropy family of indices, satisfy this. These are considered relative inequality measures because they focus on the proportional gaps. For example, if Person A earns ₹10,000 and Person B earns ₹1,00,000 (a 10x difference), and then both incomes are doubled to ₹20,000 and ₹2,00,000 respectively, the *relative* gap remains 10x, and the inequality measure stays the same.

This is useful for cross-country comparisons, where income levels vastly differ. When we say income inequality is rising in a rapidly developing economy like India, we are typically discussing a relative measure, where the proportional gap is widening, even as average incomes rise.

Translation invariance (Absolute inequality) 🔗

In contrast, Translation Invariance dictates that if a fixed, equal amount of income is added to or subtracted from every individual, the measure of inequality must remain unchanged. These are absolute inequality measures. Consider the previous example: if ₹10,000 is added to everyone, the incomes become ₹20,000 and ₹1,10,000. The absolute difference remains ₹90,000, and an absolute measure would stay the same. However, the *relative* gap shrinks from 10x to 5.5x, meaning a scale-invariant measure would show a decrease in inequality. This highlights a critical, philosophical debate in economics: are we more concerned with relative gaps or absolute differences? The choice of axiom is a choice of what “fairness” fundamentally means.

The analytical powerhouse: The decomposability axiom 🔗

Policy-makers often need to drill down into the data. When the overall Gini coefficient for a country like India is high, they need to know: Is this driven by the gap between rural and urban populations? Or is it primarily due to inequality *within* certain states or social groups? This is where the Decomposability Axiom comes in as a powerful tool.

Breaking down the inequality puzzle 🔗

An inequality measure is said to be additively decomposable if the overall inequality can be expressed as the weighted sum of two components:

1. Within-group inequality: The inequality that exists *inside* each subgroup (e.g., inequality among the rural population and inequality among the urban population).
2. Between-group inequality: The inequality that exists *between* the subgroups (e.g., the difference in the average income of the rural group and the urban group).

This axiom is invaluable for analysis because it allows us to attribute total inequality to specific sources. If a measure is decomposable, a researcher can conclude, for example, that “70% of the inequality in the nation is due to differences between the states, and 30% is due to differences within the states.” This distinction helps prioritize policy efforts. Measures belonging to the Generalized Entropy class, such as the Theil Index, satisfy this property, making them favorites for in-depth policy analysis.

A symphony of principles 🔗

The axioms – Pigou-Dalton, Scale and Translation Invariance, and Decomposability – are more than just mathematical rules; they are the ethical foundation upon which we build our understanding of economic inequality. They force economists to be explicit about what they value in a distribution-whether it’s the non-negotiable goal of transfers from rich to poor, a focus on relative or absolute gaps, or the need to break down inequality by population groups. The choice of an inequality measure is ultimately a choice of which principles of fairness a society prioritizes.

What do you think? Given that India’s economy is growing, is it more important to focus on an inequality measure that is scale-invariant (focusing on proportional gaps) or translation-invariant (focusing on absolute gaps)? How might a measure that satisfies the Decomposability Axiom help the Indian government formulate targeted policies to address the disparity between the states?

You can see a deeper discussion of these criteria in the video below.

Find out more about the different criteria for measuring inequality in this video: [The Four criterion of Inequality Measurement | Anonymity, Relative Income, Population, and Dalton |]

This video discusses the four key criteria, including the Dalton Principle (Pigou-Dalton), that define a good measure of economic inequality.

References

1. https://en.wikipedia.org/wiki/Pigou%E2%80%93Dalton_principle
2. https://jncollegeonline.co.in/attendence/classnotes/files/1643792461.pdf
3. https://www.un.org/en/development/desa/policy/wess/wess_dev_issues/dsp_policy_02.pdf
4. https://www.researchgate.net/publication/24056527_A_General_Class_of_Additively_Decomposable_Inequality_Measures
5. https://www.researchgate.net/publication/236208490_Scale_vs_Translation_Invariant_Measures_of_Inequality_of_Opportunity_when_the_Outcome_is_Binary
6. https://grins.it/sites/default/files/2025-07/SPOKE3-DP362025-ISSN-GRINS.docx-combinato.pdf
7. https://www.youtube.com/watch?v=BreIKDFndY8