{ "cells": [ { "cell_type": "raw", "id": "2e07f024", "metadata": {}, "source": [ "---\n", "title: \"What the Gini coefficient measures: a playground\"\n", "subtitle: \"Build a toy economy, stretch it, and watch inequality change.\"\n", "description: \"Drag a slider and watch inequality change.\"\n", "date: 2026-10-18\n", "type: tutorial\n", "nb: NB-006\n", "categories: [inequality, python, excel, how-to]\n", "featured: true\n", "interactive: true\n", "note: \"Simulated incomes, real method.\"\n", "key-points:\n", " - \"What a Lorenz curve shows, and how the Gini coefficient reads off it.\"\n", " - \"Why the Gini rises fast at first, then slows as inequality grows.\"\n", " - \"Why two economies with the same Gini can still split income differently.\"\n", "notebook: index.ipynb\n", "excel: replication.xlsx\n", "sources:\n", " - name: \"No outside data: incomes are simulated in the notebook (log-normal, seed 42).\"\n", "hero-data: hero.csv\n", "hero-title: \"Lorenz curves for three toy economies\"\n", "hero-x: \"people, poorest first\"\n", "hero-y: \"share of income\"\n", "hero-format: percent\n", "hero-note: \"Simulated incomes. More sag, more inequality.\"\n", "---" ] }, { "cell_type": "markdown", "id": "145ab8b8", "metadata": {}, "source": [ "## Stretch an economy, watch the curve sag\n", "\n", "I drew 10,000 incomes with one dial, the spread, that sets how unequal the economy is. Line everyone up from poorest to richest and add up their income as you go: that's the Lorenz curve. The shaded gap under the diagonal is what the Gini coefficient measures." ] }, { "cell_type": "code", "execution_count": 1, "id": "422969f0", "metadata": { "execution": { "iopub.execute_input": "2026-10-18T18:23:52.940480Z", "iopub.status.busy": "2026-10-18T18:23:52.940138Z", "iopub.status.idle": "2026-10-18T18:23:54.651998Z", "shell.execute_reply": "2026-10-18T18:23:54.650600Z" } }, "outputs": [], "source": [ "import numpy as np\n", "import pandas as pd\n", "import plotly.graph_objects as go\n", "import plotly.io as pio\n", "\n", "pio.renderers.default = \"plotly_mimetype+notebook_connected\"\n", "GOLD, SLATE, GREY, INK = \"#D99A1E\", \"#4F6D9A\", \"#8A8F98\", \"#4A4D55\"\n", "\n", "def toy_economy(spread, n=10_000, seed=42):\n", " \"\"\"Log-normal incomes. A bigger spread means a longer tail and more inequality.\"\"\"\n", " rng = np.random.default_rng(seed)\n", " return rng.lognormal(mean=10.8, sigma=spread, size=n)\n", "\n", "def lorenz(incomes):\n", " \"\"\"Cumulative share of people (poorest first) against their cumulative share of income.\"\"\"\n", " x = np.sort(incomes)\n", " people = np.arange(len(x) + 1) / len(x)\n", " income = np.concatenate([[0], np.cumsum(x) / x.sum()])\n", " return people, income\n", "\n", "def gini(incomes):\n", " \"\"\"One minus twice the area under the Lorenz curve.\"\"\"\n", " people, income = lorenz(incomes)\n", " return 1 - 2 * np.trapezoid(income, people)" ] }, { "cell_type": "markdown", "id": "f2fd34b2", "metadata": {}, "source": [ "Drag the slider to stretch the economy. Hover the curve to read any point." ] }, { "cell_type": "code", "execution_count": 2, "id": "20819f7d", "metadata": { "execution": { "iopub.execute_input": "2026-10-18T18:23:54.654792Z", "iopub.status.busy": "2026-10-18T18:23:54.654516Z", "iopub.status.idle": "2026-10-18T18:23:55.083024Z", "shell.execute_reply": "2026-10-18T18:23:55.081546Z" } }, "outputs": [ { "data": { "text/html": [ " \n", " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/vnd.plotly.v1+json": { "data": [ { "hoverinfo": "skip", "line": { "color": "#8A8F98", "dash": "dash", "width": 1.5 }, "mode": "lines", "name": "Perfect equality", "type": "scatter", "x": { "bdata": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "spreads = np.round(np.arange(0.2, 1.21, 0.1), 1)\n", "grid = np.linspace(0, 1, 101)\n", "\n", "fig = go.Figure()\n", "fig.add_trace(go.Scatter(x=grid, y=grid, mode=\"lines\", name=\"Perfect equality\",\n", " line=dict(color=GREY, dash=\"dash\", width=1.5), hoverinfo=\"skip\"))\n", "for i, s in enumerate(spreads):\n", " people, income = lorenz(toy_economy(s))\n", " curve = np.interp(grid, people, income)\n", " fig.add_trace(go.Scatter(\n", " x=grid, y=curve, mode=\"lines\", name=f\"spread {s}\", visible=(s == 0.6),\n", " line=dict(color=GOLD, width=3), fill=\"tonexty\", fillcolor=\"rgba(217,154,30,0.18)\",\n", " hovertemplate=\"The poorest %{x:.0%} earn %{y:.1%} of all income\"))\n", "\n", "steps = []\n", "for i, s in enumerate(spreads):\n", " g = gini(toy_economy(s))\n", " visible = [True] + [j == i for j in range(len(spreads))]\n", " steps.append(dict(method=\"update\", label=f\"{s}\",\n", " args=[{\"visible\": visible}, {\"title.text\": f\"Spread {s}: Gini = {g:.2f}\"}]))\n", "\n", "start = int(np.where(spreads == 0.6)[0][0])\n", "fig.update_layout(\n", " title=dict(text=steps[start][\"args\"][1][\"title.text\"], x=0, font=dict(size=16)),\n", " sliders=[dict(active=start, steps=steps, currentvalue=dict(prefix=\"Spread: \"), pad=dict(t=40))],\n", " xaxis=dict(title=\"Share of people, poorest first\", tickformat=\".0%\", range=[0, 1]),\n", " yaxis=dict(title=\"Share of income\", tickformat=\".0%\", range=[0, 1]),\n", " showlegend=False, template=\"simple_white\", height=520, margin=dict(l=60, r=20, t=60, b=40),\n", " font=dict(family=\"IBM Plex Sans, sans-serif\"),\n", ")\n", "fig.show()" ] }, { "cell_type": "markdown", "id": "0c6404c1", "metadata": {}, "source": [ "## One number per economy\n", "\n", "More spread means a higher Gini, but not in a straight line. The first bit of stretching moves it fast, then it slows. For reference, most rich countries report Gini coefficients for disposable income between about 0.25 and 0.40." ] }, { "cell_type": "code", "execution_count": 3, "id": "728be126", "metadata": { "execution": { "iopub.execute_input": "2026-10-18T18:23:55.085789Z", "iopub.status.busy": "2026-10-18T18:23:55.085529Z", "iopub.status.idle": "2026-10-18T18:23:55.227918Z", "shell.execute_reply": "2026-10-18T18:23:55.225875Z" } }, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "data": [ { "hovertemplate": "Spread %{x}
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\n", "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "table = pd.DataFrame({\"spread\": spreads, \"gini\": [gini(toy_economy(s)) for s in spreads]})\n", "\n", "fig2 = go.Figure(go.Scatter(\n", " x=table[\"spread\"], y=table[\"gini\"], mode=\"lines+markers\",\n", " line=dict(color=GOLD, width=3), marker=dict(size=8, color=GOLD),\n", " hovertemplate=\"Spread %{x}
Gini %{y:.2f}\"))\n", "fig2.update_layout(\n", " xaxis=dict(title=\"Spread of the toy economy\"), yaxis=dict(title=\"Gini coefficient\", range=[0, 0.65]),\n", " template=\"simple_white\", height=380, margin=dict(l=60, r=20, t=20, b=40),\n", " font=dict(family=\"IBM Plex Sans, sans-serif\"),\n", ")\n", "fig2.show()" ] }, { "cell_type": "markdown", "id": "be2c28d5", "metadata": {}, "source": [ "## Who gets what\n", "\n", "In the most stretched toy economy, the richest fifth takes over half of all income. Two economies can share a Gini and still split income differently, so it pays to look at the pieces too." ] }, { "cell_type": "code", "execution_count": 4, "id": "d7ec09f0", "metadata": { "execution": { "iopub.execute_input": "2026-10-18T18:23:55.231386Z", "iopub.status.busy": "2026-10-18T18:23:55.231087Z", "iopub.status.idle": "2026-10-18T18:23:55.274852Z", "shell.execute_reply": "2026-10-18T18:23:55.273457Z" } }, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "data": [ { "hovertemplate": "%{x}: %{y:.1%} of income", "marker": { "color": "#8A8F98" }, "name": "Spread 0.3", "type": "bar", "x": [ "Poorest 20%", "Second", "Middle", "Fourth", "Richest 20%" ], "y": [ 0.12624781925950268, 0.16358620445561137, 0.19122825945300184, 0.22409267156990262, 0.29484504526198135 ] }, { "hovertemplate": "%{x}: %{y:.1%} of income", "marker": { "color": "#4F6D9A" }, "name": "Spread 0.6", "type": "bar", "x": [ "Poorest 20%", "Second", "Middle", "Fourth", "Richest 20%" ], "y": [ 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\n", "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def quintile_shares(incomes):\n", " x = np.sort(incomes)\n", " return [chunk.sum() / x.sum() for chunk in np.array_split(x, 5)]\n", "\n", "labels = [\"Poorest 20%\", \"Second\", \"Middle\", \"Fourth\", \"Richest 20%\"]\n", "fig3 = go.Figure()\n", "for s, color in [(0.3, GREY), (0.6, SLATE), (1.0, GOLD)]:\n", " fig3.add_trace(go.Bar(x=labels, y=quintile_shares(toy_economy(s)), name=f\"Spread {s}\",\n", " marker_color=color, hovertemplate=\"%{x}: %{y:.1%} of income\"))\n", "fig3.update_layout(barmode=\"group\", yaxis=dict(title=\"Share of income\", tickformat=\".0%\"),\n", " template=\"simple_white\", height=400, margin=dict(l=60, r=20, t=20, b=40),\n", " legend=dict(orientation=\"h\", y=1.08), font=dict(family=\"IBM Plex Sans, sans-serif\"))\n", "fig3.show()" ] }, { "cell_type": "code", "execution_count": 5, "id": "3a06e604", "metadata": { "execution": { "iopub.execute_input": "2026-10-18T18:23:55.277804Z", "iopub.status.busy": "2026-10-18T18:23:55.277545Z", "iopub.status.idle": "2026-10-18T18:23:55.293513Z", "shell.execute_reply": "2026-10-18T18:23:55.292154Z" } }, "outputs": [], "source": [ "#| include: false\n", "# Writes the small file the home page uses to draw this piece's hero chart.\n", "rows = []\n", "for s in [1.0, 0.6, 0.3]:\n", " people, income = lorenz(toy_economy(s))\n", " pts = np.linspace(0, 1, 21)\n", " name = f\"Gini {gini(toy_economy(s)):.2f}\"\n", " rows += [{\"series\": name, \"x\": round(p, 3), \"y\": round(float(np.interp(p, people, income)), 4)} for p in pts]\n", "pd.DataFrame(rows).to_csv(\"hero.csv\", index=False)" ] }, { "cell_type": "markdown", "id": "8bb006a6", "metadata": {}, "source": [ "## What this doesn't answer\n", "\n", "- Real Gini numbers depend on choices this toy skips: income before or after taxes and transfers, household or person, and how you adjust for household size.\n", "- A log-normal may miss the very top of real income data, where the richest pull further away.\n", "- I still don't know how much country rankings move once everyone uses the same definition. If you know a good dataset, tell me." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.16" } }, "nbformat": 4, "nbformat_minor": 5 }