Income Inequality Analysis

1. Quantiles

Quantiles divide data into equal groups (percentiles).
- Example: Median (50th percentile), Quartiles (25th, 50th, 75th), Quintiles (20%, 40%...), Deciles (10%, 20%...).

2. Quantile Shares

Show share of total income by each quantile group.
Example: If top 10% earn 50% of total income → high concentration.

3. Cumulative Income Shares & Lorenz Curve

Cumulative Income Shares: Proportion of total income held up to each income level (sorted).
Lorenz Curve: Plots cumulative income shares vs cumulative population shares.
- Equality line = perfect equality.
- Further curve is from line → higher inequality.

Python Example (Lorenz Curve):

import numpy as np
import matplotlib.pyplot as plt

income = np.array([25000,30000,35000,40000,45000,
                   50000,60000,70000,80000,100000])
sorted_income = np.sort(income)

cum_income = np.cumsum(sorted_income) / np.sum(sorted_income)
cum_pop = np.linspace(0,1,len(sorted_income))
equality_line = np.linspace(0,1,len(sorted_income))

plt.plot(cum_pop, cum_income, label='Lorenz Curve')
plt.plot(cum_pop, equality_line, '--', color='red', label='Equality Line')
plt.fill_between(cum_pop, cum_income, equality_line, alpha=0.5)
plt.legend(); plt.show()

4. Gini Coefficient

Measure of inequality (0 = equality, 1 = max inequality).
Based on Lorenz curve:

Python Example (Gini):


Gini: 0.4666666666666667

5. Properties of Good Inequality Measures

Scale invariance: unaffected by unit changes.
Anonymity: independent of identity of individuals.
Population independence: unaffected by population size.
Transfer principle: inequality decreases if resources move from rich → poor.
Decomposability: allows separating within-group & between-group inequality.
Mathematical properties: continuity, boundedness, etc.
Interpretability: easy to understand.

Note: No single measure satisfies all; multiple measures should be used.

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