Inequality from Fairness
Imagine a room of 200 people, each starting with exactly $100. Every round, two random people are paired and one gives the other a fraction of their wealth — a perfectly fair coin flip determines who pays whom. After a thousand rounds, the initial equality has vanished: some people have $500 while others have $5. No one cheated. No one had an advantage. Inequality emerged purely from the mathematics of random exchange.
The Boltzmann-Gibbs Distribution
Statistical physicists recognized this pattern immediately — it is the same exponential distribution that describes molecular energies in a gas. When a conserved quantity (money or energy) is randomly exchanged between agents (people or molecules), the maximum entropy distribution is exponential: P(w) = (1/W) e^(-w/W), where W is average wealth. This is not a coincidence; it is a fundamental result of statistical mechanics applied to economics.
The Lorenz Curve and Gini Coefficient
The Lorenz curve plots cumulative wealth share against population share, sorted from poorest to richest. Perfect equality is a 45-degree line; real distributions bow below it. The Gini coefficient — twice the area between the Lorenz curve and the equality line — quantifies this bow. Watch the Lorenz curve evolve in the simulation as the initially flat wealth distribution becomes exponential. The Gini converges to approximately 0.5, matching many real-world economies.
Policy Implications
This model delivers a profound insight: some degree of inequality is a mathematical inevitability of exchange economies, not evidence of systemic unfairness. However, real economies feature mechanisms that amplify inequality beyond the random-exchange baseline — compound returns on capital (Piketty's r > g), inheritance, and network effects. Understanding the baseline helps distinguish inevitable statistical inequality from policy-addressable structural inequality. The simulation shows what 'natural' inequality looks like — deviations above it are the target for economic policy.