Transition Matrices & Long-Run Behavior

You’ll be able to

Build a transition matrix from stated probabilities of moving between states
Apply a transition matrix repeatedly to project a distribution forward
Identify a steady-state distribution and explain what it does and does not predict

Encoding movement between states

Suppose each year 85% of a city's residents stay and 15% move to the suburbs, while 8% of suburban residents move to the city and 92% stay. A transition matrix stores those four numbers, and multiplying it by the current distribution gives next year's. Repeating the multiplication projects as far forward as you like — the whole dynamic collapses into one matrix.

Transition matrix and projection

T = [[stay_city, from_suburb], [to_suburb, stay_suburb]] · next = T · current · after n steps: Tⁿ · initial

Each **column** must sum to 1 in this convention: everyone currently in a state has to go somewhere. Checking the column sums catches setup errors immediately.

Worked example

City population 600,000; suburbs 400,000. Each year 15% of city residents move out and 8% of suburbanites move in. Project one and two years ahead.

1.T = [[0.85, 0.08], [0.15, 0.92]]. Column sums: 0.85 + 0.15 = 1 ✓ and 0.08 + 0.92 = 1 ✓.
2.Year 1 city: 0.85(600,000) + 0.08(400,000) = 510,000 + 32,000 = 542,000.
3.Year 1 suburbs: 0.15(600,000) + 0.92(400,000) = 90,000 + 368,000 = 458,000.
4.Total check: 542,000 + 458,000 = 1,000,000 ✓ — nobody was created or lost.
5.Year 2 city: 0.85(542,000) + 0.08(458,000) = 460,700 + 36,640 = 497,340.
6.Year 2 suburbs: 0.15(542,000) + 0.92(458,000) = 81,300 + 421,360 = 502,660.

Answer: After one year: 542,000 city and 458,000 suburbs. After two: about 497,340 city and 502,660 suburbs. The total stays at one million every year, which is the invariant to check at each step.

The steady state

Iterate long enough and the distribution usually stops changing — it reaches a steady state where T·X = X. The flows have not stopped; they have balanced, with as many people moving each way. Finding it means solving T·X = X together with the requirement that the components sum to the total, which is what pins down the scale.

Worked example

Find the steady-state distribution for T = [[0.85, 0.08], [0.15, 0.92]] with a total population of 1,000,000.

1.Let c and s be the steady-state city and suburb populations. Steady state means the flows balance.
2.City equation: 0.85c + 0.08s = c, so 0.08s = 0.15c.
3.That gives s = 0.15c/0.08 = 1.875c.
4.Constraint: c + s = 1,000,000, so c + 1.875c = 2.875c = 1,000,000.
5.c ≈ 347,826 and s ≈ 652,174.
6.Verify: 0.85(347,826) + 0.08(652,174) = 295,652 + 52,174 = 347,826 ✓.

Answer: The steady state is about 347,826 in the city and 652,174 in the suburbs. Note that the balance condition 0.08s = 0.15c says exactly that the two flows are equal — about 52,174 people move each way every year, so the totals hold steady while individuals keep moving.

Tip

The steady state does not depend on the starting distribution — only on T and the total. Start with everyone in the city or everyone in the suburbs and you converge to the same split. That independence is the most useful property of these models.

Checkpoint

A transition matrix column reads 0.7 and 0.4. What is wrong?

What the model assumes, and where it fails

A transition matrix assumes the probabilities are constant over time and that where you go next depends only on where you are now, not on your history. Both assumptions are strong. Real migration rates respond to housing prices, employment and policy, so they drift. And a household that just moved may be far less likely to move again than the matrix allows. The steady state is a statement about the model's long-run behavior, not a forecast — and stating that limitation is part of using it honestly.

Watch out

Reaching a steady state does not mean movement has stopped. Individuals keep transitioning at the same rates; only the aggregate counts are stable. Confusing "the numbers stopped changing" with "the process stopped" misreads what the model says.

Checkpoint

A market-share model reaches a steady state where Brand A holds 40%. What does this mean?

Answer the 2 checkpoints as you read.