Question 1

What is a Markov chain, in the context of income mobility?

Accepted Answer

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.

Question 2

How do I use this simulation?

Accepted Answer

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 to the mathematically computed stationary distribution.

Question 3

What does the stationary distribution mean?

Accepted Answer

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 and shown as white tick marks over the live histogram bars for direct visual comparison.

Question 4

Why does the population always converge to the same stationary distribution?

Accepted Answer

As long as every state can eventually reach every other state and there are no strict cycles preventing return, 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 every quintile has a positive probability of transitioning to every other quintile, so the population histogram always settles to the same equilibrium shape.

Question 5

How is expected generations to escape the bottom quintile calculated?

Accepted Answer

For a household in quintile 1, the number of generations until it first transitions elsewhere is a geometric random variable with success probability equal to 1 minus the diagonal entry P[1][1]. The expected value is 1 divided by that success probability, so escape time is 1/(1-P[1][1]). This rises sharply as stickiness approaches 1, showing how small increases in persistence can dramatically extend expected time in poverty.

Question 6

How does this 2D version differ from the 3D version?

Accepted Answer

Both drive the exact same Markov-chain model — the same transition-matrix construction, the same power-iteration stationary distribution, the same geometric escape-time formula. The 2D version renders the quintile bands and histogram on a plain Canvas2D surface instead of a Three.js/WebGL scene, so it loads faster and runs on hardware without a GPU, while showing identical numbers.

📈 Social Mobility — Markov Chain Income Quintiles (2D)

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