Chapter 9.2, Problem 63E

Problems 63 and 64 require the use of a graphing calculator

Market share. Acme Soap Company markets one brand of soap, called Standard Acme $\left(SA\right)$, and Best Soap Company markets two brands. Standard Best $\left(SB\right)$ and Deluxe Best $\left(DB\right)$. Currently, Acme has $40\%$ of the market, and the remainder is divided equally between the two Best brands. Acme is considering the introduction of a second brand to get a larger share of the market. A proposed new brand, called brand $X$, was test-marketed in several large cities, producing the following transition matrix for the consumers' weekly buying habits:

$\begin{array}{l}SBDBSAX\\ P=\begin{array}{c}SB\\ DB\\ SA\\ X\end{array}\left[\begin{array}{cccc}.4& .1& .3& .2\\ .3& .2& .2& .3\\ .1& .2& .2& .5\\ .3& .3& .1& .3\end{array}\right]\end{array}$

Assuming that $P$ represents the consumers' buying habits over a long period of time, use this transition matrix and the initial-state matrix $S_0=\left[\begin{array}{cccc}.3& .3& .4& 0\end{array}\right]$ to compute successive state matrices in order to approximate the elements in the stationary matrix correct to two decimal places. If Acme decides to market this new soap, what is the long-run expected total market share for their two soaps?