Suppose demographic studies show that each year about 5% of the city’s population moves to the suburbs (and 95% remains in the city), while 3% of the suburban population moves to the city (and 97% remains in the suburbs). It is known that the population in 2015 was 600,000 in the city and 400,000 in the suburbs. Compute the population in the city for the year 2017.

Question

BlackTom AI · Accepted Answer

To model the population shift, treat the city and suburbs as two compartments with transfers between them each year.

First, understand the yearly dynamics: 5% of the city moves to the suburbs, so the city retains 95% of its population each year. Simultaneously, 3% of the suburban population moves to the city. This gives a transition set:
City next year = 0.95*(City this year) + 0.03*(Suburbs this year).
Suburbs next year = 0.97*(Suburbs this year) + 0.05*(City this year).

Start with the 2015 data: City = 600,000; Suburbs = 400,000.

Year 2016 calculations:
City_2016 = 0.95*600,000 + 0.03*400,000 = 570,000 + 12,000 = 582,000.
Suburbs_2016 = 0.97*400,000 + 0.05*600,000 = 388,000 + 30,000 = 418,000.

Year 2017 calculations (apply the same transitions to the 2016 values):
City_2017 = 0.95*582,000 + 0.03*418,000 = 552,900 + 12,540 = 565,440.
Suburbs_2017 = 0.97*418,000 + 0.05*582,000 = 404,460 + 29,100 = 433,560.

Thus, the computed city population for 2017 is 565,440.

If you compare alternative thinking, you could set up the 2x2 transfer matrix and multiply it by the initial vector (City, Suburbs) twice; you should arrive at the same result, confirming consistency across methods.

As a quick sanity check, city population declines from 600k to about 565k over two years, which aligns with a net 2–3% annual outward flow given the rates, reinforcing the plausibility of the result.

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