Long-run share of poetry weeks

Question

Each week a reader picks a genre, fiction, nonfiction, or poetry, and the choice depends only on last week's: - From fiction: to fiction 0.6, to nonfiction 0.3, to poetry 0.1. - From nonfiction: to fiction 0.4, to nonfiction 0.4, to poetry 0.2. - From poetry: to fiction 0.2, to nonfiction 0.6, to poetry 0.2. In the long run, what fraction of weeks does the reader choose poetry?

Accepted Answer

We want the steady shares (\piF, \piN, \piP) that stop changing week to week. Each share equals its total inflow. Nonfiction: arrivals are from fiction (0.3), stay (0.4), from poetry (0.6): \piN = 0.3\,\piF + 0.4\,\piN + 0.6\,\piP \implies 0.6\,\piN = 0.3\,\piF + 0.6\,\piP \implies \piN = 0.5\,\piF + \piP. Fiction: arrivals are stay (0.6), from nonfiction (0.4), from poetry (0.2): \piF = 0.6\,\piF + 0.4\,\piN + 0.2\,\piP \implies 0.4\,\piF = 0.4\,\piN + 0.2\,\piP \implies \piF = \piN + 0.5\,\piP. Substitute \piN = 0.5\,\piF + \piP into the fiction line: \piF = (0.5\,\piF + \piP) + 0.5\,\piP =…

Long-run share of poetry weeks

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