Distinguishing Income and Wealth
Income refers to the flow of money received over a specific period (e.g., wages, dividends, interest). Wealth, conversely, is a stock variable-the net value of assets owned at a point in time (e.g., real estate, equities, minus debt). Wealth inequality is inherently more pronounced than income inequality because wealth accumulates over generations through compound returns and inheritance, a dynamic formalized in Thomas Piketty’s $r > g$ hypothesis.
The Lorenz Curve
The Lorenz Curve is a graphical representation of distribution. The x-axis plots the cumulative percentage of the population (ranked from poorest to richest), while the y-axis plots the cumulative percentage of income or wealth they hold.
A perfectly equal society follows the 45-degree line of perfect equality (where the bottom 50% owns exactly 50% of the wealth). The further the Lorenz curve sags below this line, the greater the inequality.
The Gini Coefficient Formula
The Gini coefficient quantifies this gap. It is the ratio of the area between the line of perfect equality and the Lorenz curve ($A$) to the total area under the line of perfect equality ($A + B$):
$$ G = \frac{A}{A + B} $$
Since $A + B = 0.5$ in a standardized unit square, $G = 2A$. The coefficient ranges from 0 (perfect equality) to 1 (perfect inequality, where one person holds everything).
Worked Numerical Example
Consider a simplified 3-person economy with incomes of $10k, $30k, and $60k. Total income = $100k.
- Bottom 33.3% earns 10% of total income.
- Bottom 66.7% earns 40% (10% + 30%) of total income.
- 100% earns 100%.
Using the trapezoidal rule to calculate area $B$ under the Lorenz curve:
$B = (0.333 \times 0.10 / 2) + (0.333 \times (0.10 + 0.40) / 2) + (0.334 \times (0.40 + 1.00) / 2) = 0.01665 + 0.08325 + 0.2338 = 0.3337$
$A = 0.5 – 0.3337 = 0.1663$
$G = 2 \times 0.1663 = 0.3326$.
FAQs
Q: Why is wealth inequality always higher than income inequality?
A: Wealth can be negative (due to debt) and compounds over time, whereas income cannot be negative and is bounded by human labor hours.