Social Mobility as a Markov Chain: Where Families End Up

Five boxes, and where children land

Split a population into five income quintiles — bottom fifth to top fifth — and ask a simple question: given which quintile a parent belonged to, what is the probability distribution of which quintile their adult child ends up in? Answering that question for every starting quintile produces a transition matrix, and treating each generation as one step of a Markov chain — where the next state depends only on the current state, not on the family's earlier history — turns intergenerational mobility into a clean, well-studied piece of applied probability.

Reading the transition matrix

Each row of the matrix corresponds to a parent's quintile, and each entry in that row gives the probability a child raised there lands in each of the five quintiles as an adult. The diagonal entries — the probability of landing in the same quintile as your parents — are the single most informative numbers in the whole matrix. A perfectly mobile society would have every row identical, with a child's outcome completely independent of their parents' quintile; a perfectly rigid, caste-like society would have a matrix that is the identity — every child ends up exactly where their parents started, with probability 1 on every diagonal entry and 0 everywhere else. Real societies sit somewhere between these extremes, and the specific shape of the matrix — is stickiness concentrated at the very top and bottom, or spread evenly? — matters as much as any single summary number.

P_ij = P(child in quintile j | parent in quintile i)      5×5 matrix, rows sum to 1

stationary distribution π solves:   π = π P
  (the long-run share of the population in each quintile,
   independent of where any specific lineage started)

found by iterating:   π_(n+1) = π_n · P     until it stops changing
  — this convergence speed is the chain's "mixing time":
    how many generations before your starting quintile stops mattering

The stationary distribution and mixing time

Apply the same transition matrix generation after generation, and — provided the chain is well-behaved, meaning it is possible eventually to reach any quintile from any other — the distribution of the population across quintiles converges to a unique stationary distribution π, the left eigenvector of the matrix associated with eigenvalue 1. This is the long-run equilibrium the whole population settles into, no matter which quintile any particular family line began in. How quickly the chain gets there — its mixing time — is itself informative: a highly sticky matrix, with strong diagonal persistence, takes many generations to forget a family's starting position, which is a mathematically precise way of saying mobility is low.

The Great Gatsby Curve

Economist Alan Krueger popularised, and Miles Corak extensively documented, an empirical cross-country pattern now known as the Great Gatsby Curve: countries with higher income inequality within a generation also tend to show lower intergenerational mobility — a child's economic outcome is more tightly tied to their parents' outcome in societies that are more unequal to begin with. It is a correlation observed across many countries rather than proof of a single causal mechanism, and researchers debate exactly which channels — access to education, neighbourhood effects, credit constraints, inherited wealth — do the most work in linking the two, but the pattern itself has replicated consistently enough to become a standard reference point in the mobility literature, including the large-scale U.S. tax-record studies led by Raj Chetty and collaborators at Opportunity Insights.

Frequently asked questions

Try it live

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