Question

Suppose that in Example 9 there were three rival companies R, S, and T so that the pattern of customer retention and switching is given by the information in the matrix A where $$A = \begin{bmatrix} \frac{1}{3} & \frac{1}{2} & \frac{1}{4} \\ \frac{2}{3} & \frac{1}{4} & \frac{1}{2} \\ 0 & \frac{1}{4} & \frac{1}{4} \end{bmatrix} \begin{matrix} R \\ S \\ T \end{matrix}$$ (a) If the initial market distribution is given by $$\mathbf{x_0} = \begin{bmatrix} \frac{1}{3} \\ \frac{1}{3} \\ \frac{1}{3} \end{bmatrix}$$ then determine the market distribution after 1 year; after 2 years. (b) Show that the stable market distribution is given by $$\mathbf{x} = \begin{bmatrix} \frac{21}{53} \\ \frac{24}{53} \\ \frac{8}{53} \end{bmatrix}$$ (c) Which company R, S, or T will gain the most market share over a long period of time (assuming that the retention and switching patterns remain the same)? Approximately what percent of the market was gained by this company?

          Suppose that in Example 9 there were three rival companies R, S, and T so that the pattern of customer retention and switching is given by the information in the matrix A where
$$A = \begin{bmatrix} \frac{1}{3} & \frac{1}{2} & \frac{1}{4} \\ \frac{2}{3} & \frac{1}{4} & \frac{1}{2} \\ 0 & \frac{1}{4} & \frac{1}{4} \end{bmatrix} \begin{matrix} R \\ S \\ T \end{matrix}$$
(a) If the initial market distribution is given by
$$\mathbf{x_0} = \begin{bmatrix} \frac{1}{3} \\ \frac{1}{3} \\ \frac{1}{3} \end{bmatrix}$$
then determine the market distribution after 1 year; after 2 years.
(b) Show that the stable market distribution is given by
$$\mathbf{x} = \begin{bmatrix} \frac{21}{53} \\ \frac{24}{53} \\ \frac{8}{53} \end{bmatrix}$$
(c) Which company R, S, or T will gain the most market share over a long period of time (assuming that the retention and switching patterns remain the same)? Approximately what percent of the market was gained by this company?

suppose that in example 9 there were three rival companies r s and t so that the pattern of customer retention and switching is given by the information in the matrix a where a beginbmatrix 34352

Added by Silvia T.

Question

Please give Ace some feedback