Income Distribution and Inequality Measurement in Economics

Pen's Parade

y axis: income; x axis: cumulative population share, with people ordered poorest → richest.

Quantile Function

Income corresponding to a given population percentile.

CDF (Cumulative Distribution Function)

x axis: income; y axis: cumulative population.

PDF (Probability Density Function)

y-axis represents the probability density; x axis: income.

Histogram

Shows frequency of observations across income ranges.

Power of Pictures

Graphical comparisons can reveal distributional differences that summary statistics may hide.

Robust Ranking

A ranking that holds across an entire family of reasonable evaluation functions.

Curve Crossing Problem

When distributional curves cross, dominance analysis cannot give a unanimous ranking.

Lorenz Curve (LC)

Cumulative income share against cumulative population share, poorest → richest.

45° Equality Line

Represents perfect equality.

Generalised Lorenz Curve (GLC)

Lorenz Curve × mean income; incorporates both mean income and inequality.

First-Order Welfare Dominance

One Pen's Parade lies everywhere above another → higher welfare for all relevant increasing welfare functions.

Poverty Dominance

One distribution has lower poverty for all poverty lines within a specified range.

TIP Poverty Dominance

One TIP curve lies nowhere below another → lower poverty for the relevant class of poverty measures.

Atkinson Theorem

With equal means, if one Lorenz Curve lies everywhere above another, all appropriate concave welfare functions prefer it.

Lorenz Consistency

An inequality index respects Lorenz dominance / Principle of Transfers.

Variance

Average squared deviation of incomes from the mean.

Coefficient of Variation (CV)

Standard deviation divided by mean.

Quantile Ratio

Ratio of incomes at two percentiles, e.g. (p90/p10).

Income Share Ratio

Ratio of income shares of two population groups, e.g. (S80/S20).

Palma Ratio

Income share of richest 10% ÷ poorest 40%.

Gini Index / Coefficient

Inequality measure based on the area between the Lorenz Curve and equality line; ranges from 0 (equality) to 1 (maximum inequality).

Atkinson Index

Inequality measure based on an explicit Social Welfare Function and inequality-aversion parameter (ε).

Inequality Aversion

Determines how much greater weight is placed on lower incomes in the Atkinson Index.

EDE (Equally Distributed Equivalent)

Equal income producing the same social welfare as the actual unequal distribution.

Generalised Entropy (GE)

Family of inequality measures satisfying major axioms and allowing additive decomposition.

GE Parameter

Determines which part of the income distribution the GE index is most sensitive to.

MLD / GE(0)

Mean Log Deviation; particularly sensitive to the bottom of the distribution.

Theil / GE(1)

Generalised Entropy measure relatively sensitive to the middle/top.

Bottom-Sensitive

Gives greater weight to differences among lower incomes.

Middle-Sensitive

Most responsive to differences around the middle.

Top-Sensitive

Gives greater weight to differences among high incomes.

Distributional Sensitivity

How strongly an index responds to changes in different parts of the distribution.

Anonymity / Symmetry

Inequality depends on income values, not who receives them.

Scale Invariance

Multiplying every income by the same proportion does not change inequality.

Replication Invariance

Replicating the entire population does not change inequality.

Principle of Transfers (PoT)

A transfer from a richer person to a poorer person should reduce inequality, provided their ranking does not reverse.

Transfer Sensitivity

Inequality measures may give greater importance to transfers involving poorer individuals.

Additive Decomposability

Total inequality can be divided into subgroup contributions, e.g. within-group + between-group inequality.

Mean-Preserving Transfer

Redistribution that changes the distribution while leaving mean income unchanged.

Mean-Controlled Inequality

Comparing inequality while holding mean income constant.

Poverty Line ((z))

Income/resource threshold below which someone is classified as poor.

Headcount Ratio (H / FGT0)

Proportion of the population below the poverty line.

Poverty Gap

Amount by which a poor person's income falls below the poverty line: gi=max(z−xi,0).

Normalised Poverty Gap

Poverty gap as a proportion of the poverty line: gi=max(z−xi / z,0).

Incidence

How many people are poor.

Intensity

How large the poverty gaps are.

Inequality Among the Poor

How unequally poverty gaps are distributed among poor people.

TIP Curve

Graph of cumulative poverty gaps per capita against cumulative population share.

SST Poverty Index

Poverty measure related to the area under the TIP curve.

Focus Axiom

Poverty should depend only on incomes of the poor; changes to incomes above the poverty line should not affect the measure.

Monotonicity

Poverty should increase when a poor person's income falls.

Equality Preference in Poverty

Greater equality among poor people is preferred, especially when poverty gaps are concentrated among the poorest.

FGT Family

Foster-Greer-Thorbecke poverty indices controlled by sensitivity parameter (β).

FGT0

Headcount Ratio; measures incidence.

FGT1

Poverty Gap Index; measures incidence + intensity.

FGT2

Poverty Severity Index; captures incidence + intensity + inequality among the poor.

FGT Parameter ((β))

Controls sensitivity to the poorest; higher (β) → greater weight on the poorest.

Poverty Severity

Gives greater weight to the poorest people.

Watts Index

Log-based poverty measure capturing the Three I's and allowing additive decomposition.

Sen Index

Rank-based poverty measure capturing the Three I's.

Equality Preference

Preference for greater equality, usually represented by concave utility.

Social Welfare Function (SWF)

Converts the distribution of income into an overall welfare assessment.

Sen Social Welfare Function

SWF=μ(1−G).

Atkinson Social Welfare

EDE=μ(1−Aε).

Welfare Foundation

Explicit connection between an inequality measure and a Social Welfare Function.

Value Judgment

Normative choice about whose income/welfare receives more importance.

Statistical Index

Inequality measure based mainly on statistical properties rather than an explicit welfare foundation.

Portfolio of Indices

Using multiple reasonable inequality measures to test whether conclusions are robust.

Income

A flow of resources received over a period.

Consumption

A flow measuring goods/services actually used over a period.

Wealth / Net Worth

A stock of accumulated resources at a point in time.

Stock

Measured at a point in time.

Flow

Measured over a period.

Budget Identity

I=C+S

Saving (S)

Income not consumed.

Change in Wealth

Saving + capital gains/losses + net capital transfers + other asset/liability changes.

Capital Gains/Losses

Changes in asset values caused by price changes.

Realised Capital Gains

Capital gain occurring when an asset is sold.

Accrued Capital Gains

Increase in asset value before the asset is sold.

Net Capital Transfers

Wealth transfers such as inheritances.

Income vs Consumption

Income measures potential command over resources; consumption measures actual use/enjoyment.

Income vs Wealth

Income measures resources received over time; wealth measures accumulated resources at a point in time.

ICW Divergence

The same person/household can have very different positions in income, consumption and wealth distributions.

Wealth Distribution

Generally substantially more unequal than income or consumption.

Unit of Analysis

Individual, household or family unit used for measurement.

Household

One or more people sharing living/cooking facilities.

Family / Benefit Unit

Single adult/couple plus dependent children.

Equal Sharing Assumption

Assumes household resources are equally shared after accounting for needs.

Person-Weighting

Gives each individual equal weight, effectively replicating household resources according to household size.

Life-Cycle Effects

Differences in income, consumption and wealth caused by stage of life.

Consumption Smoothing

Using saving, borrowing or transfers to maintain stable consumption over time.

Potential Command

Well-being depends on resources and opportunities available.

Actual Enjoyment

Well-being depends on what is actually consumed/used.

Original / Market Income

Employee, self-employment, rental and property income.

Gross Income

Original income + current transfers received.

Disposable Income

Gross income − current transfers paid, such as taxes/social contributions.

Adjusted Disposable Income

Disposable income + Social Transfers in Kind.

Post-Tax Income

Disposable income − indirect taxes.

Final Income

Post-tax income + Social Transfers in Kind.