Long-run share of active members

Question

A gym membership is either active or lapsed each month, and next month depends only on this month: - If active now, it lapses next month with probability 0.2 (and stays active with probability 0.8). - If lapsed now, it becomes active next month with probability 0.3 (and stays lapsed with probability 0.7). In the long run, what fraction of months is a membership active?

Accepted Answer

We want the steady shares (\piA, \piL) that stop changing month to month. Balance the flow out of "active" against the flow in. Flow leaving active: \piA 	imes 0.2. Flow entering active: \piL 	imes 0.3. Balance: \piA 	imes 0.2 = \piL 	imes 0.3 \implies \piA = 1.5\,\piL. Since the shares add to 1: 1.5\piL + \piL = 1 \implies 2.5\,\piL = 1 \implies \piL = 0.4, \quad \piA = 0.6. So a membership is active three fifths of the time, or 60\%. Two-state shortcut for the active state: it is entered with probability b = 0.3 and left with probability a = 0.2, giving \frac{b}{a+b} = \frac{0.3}{0.5} =…

Long-run share of active members

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