Overview
A Lorenz curve is a graphical way to show how a resource (most commonly income or wealth) is distributed across a population. It was introduced by Max Lorenz (1905). The horizontal axis shows cumulative population percentiles (from poorest to richest) and the vertical axis shows cumulative share of income (or wealth). A 45° diagonal (the line of equality) represents perfect equality; the Lorenz curve lies beneath it, and the farther it is from that line, the greater the inequality.
Key components
– Line of equality (45° diagonal): perfect equality — each percentile has the same share.
– Lorenz curve: cumulative share of income (vertical) vs cumulative share of population (horizontal).
– Gini coefficient: a single-number summary of inequality derived from the Lorenz curve (ranges from 0 = perfect equality to 1 = maximal inequality; can behave unusually with negative incomes or wealth).
Why the Lorenz curve matters
– Visualizes which parts of the population hold more or less of the total income/wealth.
– Provides richer information than a single index alone (e.g., two distributions can have the same Gini but different shapes).
– Useful for policy analysis (tax design, redistributive policy), comparative studies (countries, regions), and tracking change over time.
How the Lorenz curve measures inequality
– Plot cumulative population share (x-axis) against cumulative income share (y-axis).
– If the Lorenz curve is close to the diagonal — inequality is low. If it bows far below — inequality is higher.
– The Gini coefficient equals the area between the line of equality and the Lorenz curve divided by the total area under the line of equality (commonly computed via trapezoidal approximation).
Practical steps to create a Lorenz curve and compute the Gini coefficient
Step 1 — Collect and prepare data
– Obtain individual or grouped income/wealth data (tax records, household surveys, administrative data).
– Clean data (handle negatives, zero incomes, outliers as appropriate).
– Sort observations by income from lowest to highest.
Step 2 — Compute shares and cumulative shares
– For each observation or group i:
• population share p_i = (count in i) / (total population)
• income share y_i = (income in i) / (total income)
– Compute cumulative population shares P_k = Σ_{i=1..k} p_i and cumulative income shares Y_k = Σ_{i=1..k} y_i. Set P_0 = Y_0 = 0.
Step 3 — Plot the Lorenz curve
– Horizontal axis: cumulative population share P_k (0 → 1).
– Vertical axis: cumulative income share Y_k (0 → 1).
– Plot points (P_k, Y_k) and connect with lines (piecewise linear) to form the Lorenz curve. Add the diagonal line y = x as the line of equality.
Step 4 — Compute the Gini coefficient (trapezoid formula)
– Calculate the area under the Lorenz curve using trapezoids:
A = Σ_{k=1..n} (P_k − P_{k−1}) * (Y_k + Y_{k−1}) / 2
– The Gini coefficient G = 1 − 2 * A
– (Equivalent: G = (Area between diagonal and Lorenz curve) / 0.5)
– Note: special handling is needed if income/wealth contains negative values — Lorenz and Gini behave differently.
Simple numeric example
Five individuals with incomes: [10, 20, 30, 40, 100] (total = 200).
– Population share each p_i = 0.2.
– Income shares y_i = [0.05, 0.10, 0.15, 0.20, 0.50].
– Cumulative P_k: [0.2, 0.4, 0.6, 0.8, 1.0]. Cumulative Y_k: [0.05, 0.15, 0.30, 0.50, 1.0].
– Area under Lorenz curve (A) via trapezoids = 0.30.
– Gini = 1 − 2*A = 1 − 0.6 = 0.40.
Interpretation: a Gini of 0.40 indicates moderate inequality in this (toy) population.
Practical tools and code examples
– Excel/Sheets:
• Sort incomes, compute shares and cumulative shares, create an XY (scatter) plot with lines; add y = x for equality.
– Python (pseudocode):
• Sort incomes; compute p_i and y_i; compute cumulative sums; use matplotlib to plot; compute A via trapezoid rule (numpy.trapz) and G = 1 − 2*A.
– Statistical packages (R, Stata) have built-in functions to compute Lorenz curves and Gini coefficients.
Advantages of the Lorenz curve
– Intuitive visual representation of distribution across all percentiles.
– Shows where inequality is concentrated (bottom, middle, top).
– Enables decomposition and comparison by region, subgroup, and over time.
– Preserves anonymity of individuals (aggregate curve).
Disadvantages and limitations
– Requires good-quality, sufficiently granular data; sampling error and underreporting can bias results.
– Building a continuous curve from grouped data needs interpolation, which introduces assumptions.
– Lorenz/Gini alone may not capture all aspects of inequality (e.g., changes at different parts of distribution may cancel out in a single index).
– Special cases (negative income/wealth) complicate interpretation—Lorenz curve may dip below the horizontal axis and Gini may exceed 1 under certain definitions.
– Comparisons across populations with different means or demographic structures require care.
Tips for interpretation and use
– Overlay Lorenz curves from different years to see which groups gained or lost share.
– Combine Lorenz analysis with other measures (Atkinson index, Theil index) to capture sensitivity to different parts of distribution.
– When comparing countries, check data comparability (definitions, income concept, tax vs market income).
– Zoom in on tails (bottom/top percentiles) if policy interest focuses on poverty or top-end concentration.
– Use confidence intervals or bootstrap methods when using sample survey data to show uncertainty.
Who uses Lorenz curves
– Economists and researchers studying distributional outcomes.
– Governments and tax authorities designing progressive taxation or social programs.
– International organizations (World Bank, OECD, UN) for cross-country comparisons.
– NGOs and think tanks assessing inequality and advocating policy.
When to be careful
– With small sample sizes or grouped data, the fitted curve between points is an interpolation — don’t overinterpret fine-grained features.
– Negative or zero income/wealth require special handling; standard formulas assume non-negative values.
– Two distributions with identical Gini coefficients may have very different Lorenz curves — always look at the curve (not only the index).
The bottom line
The Lorenz curve is a fundamental, visual tool to describe how income or wealth is distributed across a population. Paired with the Gini coefficient, it facilitates comparisons and helps policymakers identify where inequality is concentrated. Its usefulness depends on data quality and careful interpretation; it is best used together with other inequality measures and decomposition analyses.
Reference
– Investopedia, “Lorenz Curve” —
Editor’s note: The following topics are reserved for upcoming updates and will be expanded with detailed examples and datasets.