In a given city, the market for homeowner’s insurance is dominated by two companies: National Property and United Family. Currently, National Property insures 50% of homes in the city, United Family insures 30%, and the remainder are insured by a collection of smaller companies. United Family decides to offer rebates to increase its market share. This has the following effects on insurance purchases for the next several years: each year 25% of National Property’s customers switch to United Family and 10% switch to other companies; 10% of United Family’s customers switch to National Property and 5% switch to other companies; 15% of the customers of other companies switch to National Property and 35% switch to United Family.

(A) Draw a transition diagram.

(B) Write the transition matrix.

(C) What percentage of homes will be insured by National Property next year? The year after next?

(D) What percentage of homes will be insured by United Family next year? The year after next?