Chapter 9.2, Problem 64E

Problems 63 and 64 require the use of a graphing calculator

Market share. Refer to Problem 63. The chemists at Acme Soap Company have developed a second new soap, called brand $Y$. Test-marketing this soap against the three established brands produces the following transition matrix:

$\begin{array}{l}SBDBSAX\\ P=\begin{array}{c}SB\\ DB\\ SA\\ X\end{array}\left[\begin{array}{cccc}.3& .2& .2& .3\\ .2& .2& .2& .4\\ .2& .2& .4& .2\\ .1& .2& .3& .4\end{array}\right]\end{array}$

Proceed as in Problem 63 to approximate the elements in the stationary matrix correct to two decimal places. If Acme decides to market brand $Y$, what is the long-run expected total market share for Standard Acme and brand $Y$ ? Should Acme market brand $X$ or brand $Y$ ?