Lorenz Curve

Macro EC417_Macroeconomics

The Lorenz Curve is a graphical representation of income or wealth inequality in a population. It’s a fundamental tool for visualizing how a resource (income, wealth, consumption) is distributed across a population.

How It Works

The Lorenz curve plots:

• X-axis: Cumulative share of the population (from poorest to richest), ranging from 0 to 1
• Y-axis: Cumulative share of total income/wealth held by that population share

Key Features

1. The 45-degree line (Line of Equality): This represents perfect equality—if the poorest 20% of people have 20% of income, the poorest 50% have 50% of income, etc.

2. The Lorenz curve: Always lies on or below the 45-degree line. The more it bows downward (away from the diagonal), the greater the inequality.

3. Perfect inequality: If one person has everything and everyone else has nothing, the Lorenz curve hugs the x-axis until it reaches 100% of the population, then jumps to 100% of income.

Mathematical Formulation

If we order individuals by income $y_i$ where $i=1,2,...,n$ (from poorest to richest), the Lorenz curve $L(p)$ at population share $p\in [0,1]$ is:

$L(p)=\frac{{\sum }_{i=1}^{np}{y}_i}{{\sum }_{i=1}^n{y}_i}$

This gives the share of total income held by the poorest $p$ fraction of the population.

Connection to Your Course

In the heterogeneous agent models you’re studying (like the income fluctuation problem in Lecture 7), the Lorenz curve helps visualize the cross-sectional wealth distribution that emerges from idiosyncratic income shocks. The Gini coefficient—a measure of inequality—is calculated as twice the area between the 45-degree line and the Lorenz curve.

Does this help clarify the concept? Would you like me to explain how it connects to specific models you’re studying?