Income Inequality Simulator: Random Trades, the Gini Coefficient and the Pareto Tail

Trading wealth like colliding gas molecules

The Boltzmann-Pareto (or kinetic) wealth-exchange model treats an economy as a gas of agents who trade money the way molecules exchange energy in collisions: pick two agents at random, pick a random fraction of the combined pot, and reassign it between them.

for a random pair (i, j) with wealth (w_i, w_j) at a step:
  pool = w_i + w_j
  r    = random fraction, e.g. uniform(0,1)
  w_i' = r * pool
  w_j' = (1 - r) * pool          # total wealth conserved: w_i' + w_j' = w_i + w_j

Why perfectly fair trades still produce extreme inequality

Nothing in that exchange rule favours the richer or poorer agent — the split fraction r is symmetric and unbiased. Yet simulate thousands of these "fair coin-flip" trades among many agents and the wealth distribution reliably collapses toward extreme inequality, with most of the population converging toward near-zero wealth and almost all the wealth concentrated in a small remainder. This is a genuinely counter-intuitive result: unbiased, symmetric randomness alone, with only one conserved quantity (total wealth), is enough to generate stark inequality — no exploitation, luck-favouring bias or unequal starting conditions are required. The mechanism is that an agent who happens to lose several trades in a row has proportionally less to lose on the next unfair split of their own shrinking pool, while an agent on a winning streak has proportionally more to gain — a rich-get-richer effect that falls directly out of multiplicative, proportional randomness rather than being programmed in.

The Gini coefficient

Inequality in the resulting distribution is usually summarised with the Gini coefficient, derived from the Lorenz curve — a plot of the cumulative share of total wealth held by the poorest X% of the population. A perfectly equal society traces a straight diagonal line; the Gini coefficient is the area between the actual Lorenz curve and that diagonal, normalised to run from 0 (perfect equality) to 1 (one agent holds everything). The base model without any redistribution reliably drives the Gini coefficient up toward values around 0.5, higher than most real national economies, illustrating just how much of real-world inequality could arise from pure chance rather than differences in skill or effort.

The Pareto tail

The upper end of the resulting wealth distribution follows a power law — the Pareto distribution, first identified empirically by Vilfredo Pareto in 1896 when he noticed that roughly 80% of Italian land was owned by roughly 20% of the population, the origin of the popular "80/20 rule." A power-law tail means the probability of an agent holding wealth above some large value W falls off as a slow polynomial (P(w > W) ~ W^-alpha) rather than a fast exponential, which is why extreme wealth concentrations — far beyond what a bell-curve intuition would predict — are common in both this simple model and in real economic data on income and wealth.

Adding a saving propensity and taxation

Two extensions bring the toy model closer to real economies. Giving agents a saving propensity λ — a fraction of wealth each agent always keeps out of every trade — dramatically changes the equilibrium: with a uniform λ across agents the distribution develops a more realistic middle bulge (close to a Gamma distribution) before its Pareto tail, and agents with a higher λ (more conservative traders) systematically end up wealthier on average, since they simply expose less of their wealth to each round of random redistribution. Adding a flat redistributive tax — skimming a fraction of every transaction and redistributing it equally to all agents — directly counteracts the concentration mechanism and measurably lowers the equilibrium Gini coefficient, giving the simulator a simple, tunable lever for exploring how much of real-world redistribution policy would need to do to offset the model's spontaneous drift toward inequality.

Frequently asked questions

Try it live

Everything above runs in your browser — open Income Inequality Simulator and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.