Unit 4: Poverty and Inequality - Study Resource

Poverty Axioms

Poverty axioms are fundamental mathematical and logical criteria that a reliable index of poverty must satisfy. Introduced prominently by Amartya Sen (1976), these axioms ensure that a poverty measure accurately reflects changes in the well-being of the poor.

Key Poverty Axioms

• Focus Axiom: The poverty index must depend solely on the incomes of individuals who are poor (those below the poverty line z). Changes in the income of non-poor individuals should not affect the poverty measure.
• Monotonicity Axiom: Ceteris paribus, a reduction in the income of any person below the poverty line must strictly increase the poverty index.
• Transfer Axiom: Ceteris paribus, a transfer of income from a poor person to any richer person (whether poor or non-poor) must strictly increase the poverty index.
• Symmetry (Anonymity) Axiom: The poverty index must be independent of individual identities. Swapping the incomes of any two individuals should leave the index unchanged.
• Population Invariance (Replication) Axiom: Merging two or more identical populations should leave the overall poverty index unchanged.
• Subgroup Decomposability: Overall poverty should be expressible as a weighted average of sub-group poverty levels, weighted by their respective population shares.

Measurement of Poverty and Poverty Gap

Poverty measurement requires defining a threshold called the poverty line (z), which represents the minimum income or consumption necessary to meet basic living requirements.

1. Headcount Ratio (H)

The Headcount Ratio measures the proportion of the population living below the poverty line.

H = q / N

Where:

• q = number of poor individuals (whose income y_i < z)
• N = total population

Limitations: Violates both the Monotonicity and Transfer axioms. It treats all poor individuals equally regardless of how far below the poverty line they fall.

2. Poverty Gap Index (PGI)

The Poverty Gap Index measures the depth of poverty by calculating the average normalized shortfall of the poor from the poverty line across the entire population.

PGI = (1 / N) × ∑_i=1^q [(z - y_i) / z]

Alternatively, PGI = H × I, where I is the Income Gap Ratio defined as:

I = (z - μ_p) / z

Where μ_p is the mean income of the poor population.

Properties: Satisfies the Monotonicity Axiom (an income loss by a poor person increases the gap), but fails the Transfer Axiom because it treats a transfer between poor individuals as neutral.

3. Squared Poverty Gap Index (FGT₂)

Proposed by Foster, Greer, and Thorbecke (FGT), the squared poverty gap measures the severity of poverty by weighting each poor individual's shortfall by itself.

FGT₂ = (1 / N) × ∑_i=1^q [(z - y_i) / z]²

Properties: Satisfies Focus, Monotonicity, and Transfer axioms, as well as Subgroup Decomposability.

Comparison of Poverty Measures

Sen's Measure of Poverty

Amartya Sen (1976) developed an axiomatic poverty measure that unifies three distinct dimensions of poverty: the proportion of poor people, the depth of their deprivation, and the inequality of income distribution among the poor.

Mathematical Formulation

S = H × [I + (1 - I) × G_p]

Where:

• H = Headcount Ratio (q / N)
• I = Income Gap Ratio [(z - μ_p) / z]
• G_p = Gini Coefficient of income distribution among the poor

Components and Interpretation

• If income among the poor is distributed completely equally (G_p = 0), Sen's measure reduces to:

  S = H × I

• If all poor individuals have zero income (I = 1), then:

  S = H

Sen's measure was revolutionary because it was the first composite index to satisfy both the Monotonicity Axiom and the Transfer Axiom simultaneously.

Axioms of Inequality Index

An inequality index is a mathematical function I(y) that assigns a real number to an income vector y = (y₁, y₂, ..., y_N). To be economically meaningful, an inequality measure must satisfy specific axioms.

Core Axioms of Inequality

• Anonymity (Symmetry) Axiom: The inequality value does not depend on who earns which income. Permuting the income values among individuals leaves the index unchanged: I(y) = I(P × y), where P is a permutation matrix.
• Scale Invariance (Mean Independence): If everyone's income is multiplied by a positive scalar c > 0, overall inequality remains unchanged: I(c × y) = I(y).
• Population Invariance (Replication Axiom): If an income distribution is replicated k times, inequality remains identical: I(y) = I(y, y, ..., y).
• Pigou-Dalton Transfer Principle: A progressive transfer (a transfer of income from a richer person to a poorer person that does not reverse their relative ranking) strictly reduces measured inequality.
• Decomposability Axiom: Total inequality in a population can be broken down into the sum of within-group inequality and between-group inequality.

Measures of Inequality

1. Range and Relative Range

The Range measures the difference between the maximum and minimum income in a population.

Relative Range = (y_max - y_min) / μ

Limitation: Ignores income distribution between the extremes and violates the Pigou-Dalton Transfer Principle.

2. Variance and Coefficient of Variation (CV)

Variance measures the dispersion around mean income μ. The Coefficient of Variation normalizes variance by the mean:

CV = σ / μ

Where σ is the standard deviation of income.

3. Lorenz Curve

The Lorenz Curve is a graphical representation of income distribution. The horizontal axis represents the cumulative percentage of population (ranked from poorest to richest), and the vertical axis represents the cumulative percentage of total national income.

• Line of Absolute Equality: A 45-degree diagonal line where x% of the population earns x% of national income.
• The further the Lorenz Curve bows away from the 45-degree line, the higher the inequality.

4. Gini Coefficient (G)

The Gini Coefficient is defined directly from the Lorenz Curve. It measures the ratio of the area between the 45-degree line and the Lorenz Curve (Area A) to the total area under the 45-degree line (Area A + B).

G = A / (A + B) = 2A

Gini formula for ordered individual incomes y₁ ≤ y₂ ≤ ... ≤ y_N:

G = [2 / (N² × μ)] × ∑_i=1^N [i × (y_i - μ)]

Range: 0 (perfect equality) to 1 (perfect inequality).

5. Atkinson Index

An explicit normative measure based on a social welfare function, incorporating an inequality aversion parameter ε ≥ 0:

A = 1 - (y_e / μ)

Where y_e is the equally distributed equivalent income.

Summary Table of Inequality Measures

Inequality and Economic Development

The Kuznets Inverted-U Hypothesis

Simon Kuznets (1955) hypothesized that as an economy develops, market forces first increase and then decrease economic inequality.

Inequality = f(GDP per capita)

When plotted, the relationship traces an inverted-U shape curve:

Mechanisms Behind the Curve

• Structural Transformation: Rural-to-urban migration (Lewis model dynamics).
• Human Capital Expansion: Access to education reduces the skill premium.
• Democratization & Welfare State: Political pressure leads to progressive taxation and safety nets.

Gender Gap in Poverty

The gender gap in poverty refers to the disproportionate representation of women among the world's poor, often described as the feminization of poverty.

Key Drivers

• Intra-Household Allocation Bias: Standard income and poverty metrics assume equal sharing within households. In reality, women and girls frequently receive fewer resources (food, health, education) within poor households.
• Labor Market Discrimination: Women face systemic wage gaps, vertical segregation, and overrepresentation in informal or precarious employment.
• Unpaid Care Work: Women carry a disproportionate burden of uncompensated household labor and caregiving, restricting their earnings capacity.
• Asset and Property Restrictions: Legal and social barriers limit women's land ownership, inheritance rights, and access to formal credit.

Human Poverty Index (HPI)

Introduced by the United Nations Development Programme (UNDP) in 1997 in its Human Development Report, the Human Poverty Index (HPI) measures deprivation in basic dimensions of human development rather than monetary income alone.

1. HPI-1 (For Developing Countries)

HPI-1 focuses on deprivation across three essential dimensions:

• Longevity (P₁): Vulnerability to death at an early age, measured by the percentage of people not expected to survive to age 40.
• Knowledge (P₂): Exclusion from the world of reading and communication, measured by the adult illiteracy rate.
• Decent Standard of Living (P₃): Composite variable comprising:
  • Percentage of population without uncontained access to safe water (P₃₁).
  • Percentage of children under five who are underweight for their age (P₃₂).

Formula for HPI-1:

HPI-1 = [ (1 / 3) × (P₁³ + P₂³ + P₃³) ]^1/3

2. HPI-2 (For Developed OECD Countries)

HPI-2 incorporates social exclusion as a critical fourth dimension:

• Longevity (P₁): Percentage of population not expected to survive to age 60.
• Knowledge (P₂): Percentage of adults lacking functional literacy skills.
• Decent Standard of Living (P₃): Percentage of people living below the relative poverty line (less than 50% of median adjusted household disposable income).
• Social Exclusion (P₄): Long-term unemployment rate (unemployed 12 months or more).

Formula for HPI-2:

HPI-2 = [ (1 / 4) × (P₁³ + P₂³ + P₃³ + P₄³) ]^1/3

Comparison Table: HPI-1 vs. HPI-2