Problem $3(20$ pts$)$

The leading brewery on the West Coast $($labeled A$)$ has hired an OR analyst to analyze its market position.

It is particularly concerned about its major competitor $($labeled B$).$ The analyst believes that brand switching

can be modeled as a Markov chain using three states, with states A and B representing customers drinking

beer produced from the aforementioned breweries and State C representing all other brands. Data are taken

monthly, and the analyst has constructed the following $(1-$step$)$ transition matrix from past data.

$$

$$

A B C

A $0\mathrm{.}80\mathrm{.}150\mathrm{.}05$

B $0\mathrm{.}250\mathrm{.}70\mathrm{.}05$

C $0\mathrm{.}150\mathrm{.}050\mathrm{.}8$

$$

$$

Answer the following questions.

$(1).$ What are the steady$-$state market shares for the two major breweries? Show your steps.

$(2).$ Company A has received information from some sources that the new CEO of Company B is planning

to implement a new marketing strategy. According to the evaluation from a consultant company, with

sufficient time, this strategy will increase pAB to $0\mathrm{.}2$ but will not affect pAC $,$ and it will increase pCB to $0\mathrm{.}15$

and reduce pCC to $0\mathrm{.}7.$ Find out how much market share would lose by Company A if they do not react.