Watch a population of households move between five income quintiles generation after generation, governed by a transition matrix. Adjust how "sticky" income is across generations and see the population converge to a stationary distribution.
A Markov chain model of intergenerational income mobility. Each household's quintile next generation depends only on its current quintile, via a row-stochastic transition matrix. Regardless of starting distribution, repeated application of the matrix converges to a unique stationary distribution — the long-run equilibrium share of households in each quintile.
Drag the Stickiness slider toward 1 to make income highly persistent across generations (low mobility, dynastic wealth); drag it toward 0 for near-total reshuffling each generation (high mobility). Watch the white markers (stationary distribution) versus the live colored bars, and track the average number of generations needed to escape the bottom quintile.
Economists estimate real intergenerational income persistence using very similar transition matrices built from tax and survey panel data. Countries with markedly different sticky-matrix diagonals — such as Denmark versus the United States — form the basis of the "Great Gatsby Curve" linking inequality to low mobility.
This simulation models intergenerational income mobility as a discrete-time Markov chain over five income quintiles. Each household is characterised entirely by which quintile it occupies; every generation, it transitions to a new quintile according to fixed probabilities given by a transition matrix, with no memory of earlier generations. This simplification, first applied to income mobility by economists in the 1970s, still underlies much of the empirical mobility literature, including studies using tax-record panel data from Scandinavia, the UK and the United States to estimate real transition matrices and compare mobility across countries.
The simulation exposes two key theoretical results. First, any transition matrix with strictly positive entries has a unique stationary distribution that the population converges to regardless of its starting configuration — a consequence of the Perron-Frobenius theorem. Second, the expected number of generations needed to leave the bottom quintile can be computed directly from the matrix's persistence parameter, giving a concrete, interpretable measure of "stickiness" at the bottom of the income distribution, closely related to the real-world concept measured in the "Great Gatsby Curve" literature linking inequality and mobility.
A Markov chain is a mathematical model in which a system moves between a finite set of states, and the probability of moving to any given state next depends only on the current state, not on the history that preceded it. Here the states are the five income quintiles, and the transition matrix encodes the probability that a household in quintile i ends up in quintile j the following generation, allowing researchers to summarise complex mobility patterns in a single compact matrix.
Households start randomly distributed across the five quintile bands on the left. Each "generation," every dot resamples its quintile according to the transition matrix and animates to its new band. Raise the Stickiness slider to make income highly persistent across generations, or lower it toward zero for near-complete reshuffling. The right-hand panel compares the live population histogram (colored bars) to the mathematically computed stationary distribution (white markers).
The stationary distribution is the unique population split across quintiles that remains unchanged after one more application of the transition matrix — it is the long-run equilibrium the system converges to no matter how the population started out. It is computed here via power iteration, repeatedly multiplying an initial distribution vector by the matrix until it stops changing, and is shown as white tick marks over the live histogram bars for direct visual comparison.
As long as every state can eventually reach every other state (the chain is irreducible) and there are no strict cycles preventing return at arbitrary times (it is aperiodic), the Perron-Frobenius theorem guarantees a unique stationary distribution, and repeated application of the transition matrix from any starting point converges to it. In this simulation, because every quintile has a positive probability of transitioning to every other quintile, both conditions are automatically satisfied, so the population histogram always settles to the same equilibrium shape regardless of the initial random assignment.
For a household currently in quintile 1, the number of generations until it first transitions to a different quintile is a geometric random variable with success probability equal to 1 minus the diagonal entry P[1][1] (the probability of staying in quintile 1). The expected value of a geometric distribution is one divided by its success probability, so the expected escape time is 1 / (1 − P[1][1]). This value rises sharply as stickiness approaches 1, illustrating how small increases in persistence can dramatically extend the number of generations a family expects to remain in poverty.
Empirical mobility researchers construct transition matrices from longitudinal tax and survey data linking parents' and children's income ranks, most famously in the "Opportunity Insights" project led by Raj Chetty and collaborators, which tracks mobility by U.S. county using millions of anonymised tax records. International comparisons of these matrices underlie the "Great Gatsby Curve," which shows that countries with higher income inequality tend to also have stickier, less mobile transition matrices, meaning income persists more strongly across generations.
Absolute mobility asks whether children earn more in real terms than their parents did, driven largely by overall economic growth. Relative mobility, which this simulation focuses on, asks whether a child's rank within the income distribution differs from their parent's rank, independent of growth. A society can have high absolute mobility (everyone gets richer) while still having low relative mobility (rank order barely changes), which is why economists track both measures separately when assessing fairness of opportunity.
Not necessarily by itself, but high persistence is generally interpreted as evidence that factors outside a person's control — inherited wealth, neighbourhood effects, unequal access to education — play an outsized role in determining life outcomes. A perfectly mobile society (stickiness near zero) would imply outcomes are essentially random with respect to family background, which is also not obviously "fair." Most economists therefore treat mobility not as inherently good or bad, but as one input, alongside absolute living standards and inequality levels, into a broader assessment of economic opportunity.
Assuming transitions depend only on the current quintile ignores many real-world features: mobility rates differ by initial wealth level, geography, race and family structure, transition probabilities are unlikely to stay constant over long historical periods, and treating income as five discrete bins loses information relative to continuous income data. Real mobility research increasingly uses continuous income rank correlations and machine-learning models incorporating dozens of covariates, but the discrete Markov chain remains a valuable, easily interpretable first approximation.