Long-run share on the Pro tier

Question

Each month a customer sits on one of three subscription tiers, Basic, Plus, or Pro. Next month depends only on this month:

From Basic: to Basic $0.6$ , to Plus $0.4$ , to Pro $0$ .
From Plus: to Basic $0.2$ , to Plus $0.5$ , to Pro $0.3$ .
From Pro: to Basic $0.1$ , to Plus $0.3$ , to Pro $0.6$ .

In the long run, what fraction of customers sit on the Pro tier?

Accepted Answer

We want the steady long-run shares (\piB, \piP, \piR) for Basic, Plus, Pro that stop changing month to month. The rule is that each share equals everything flowing into it: for each state, add up (share of every origin) times (chance it sends a customer here). Pro: the only ways to be on Pro next month are to come from Plus (0.3) or stay on Pro (0.6): \piR = 0.3\,\piP + 0.6\,\piR \implies 0.4\,\piR = 0.3\,\piP \implies \piR = 0.75\,\piP. Basic: arrivals are stay-on-Basic (0.6), from Plus (0.2), from Pro (0.1): \piB = 0.6\,\piB + 0.2\,\piP + 0.1\,\piR \implies 0.4\,\piB = 0.2\,\piP + 0.1\,\piR…

Long-run share on the Pro tier

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