Chapter 7.2, Problem 28E

Consider the isolated Swiss town of Andelfingen, in habited by 1,200 families. Each family takes a weekly shopping trip to the only grocery store in town, run by Mr. and Mrs. Wipf, until the day when a new, fancier (and cheaper) chain store, Migros, opens its doors. It is not expected that everybody will immediately run to the new store, but we do anticipate that 20% of those shopping at Wipf’s each week switch to Migros the following week. Some people who do switch miss the personal service (and the gossip) and switch back: We expect that 10% of those shopping at Migros each week go to Wipf’s the following week. The state of this town (as far as grocery shopping is concerned) can be represented by the vector $\stackrel{\to }{x}\left(t\right)=\left[\begin{array}{c}\omega \left(t\right)\\ m\left(t\right)\end{array}\right]$ ,where $\omega \left(t\right)$ and m(t) are the numbers of families shopping at Wipf’s and at Migros, respectively, t weeks after Migros opens. Suppose $\omega \left(0\right)=1,200$ and $m\left(0\right)=0$ .
a. Find a $2\times 2$ matrix A such that $\stackrel{\to }{x}\left(t+1\right)=A\stackrel{\to }{x}\left(t\right)$ . Verify that A is a positive transition matrix. See Exercise 25.
b. How many families will shop at each store after t weeks? Give closed formulas.
c. The Wipfs expect that they must close down when they have less than 250 customers a week. When does that happen?