# Suppose Johnson Dry Cleaning (J) has a 40% market share initially, while NorthClean (N) has a 60% market share

215 points

Suppose Johnson Dry Cleaning (J) has a 40% market share initially, while NorthClean (N) has a 60% market share. Use the corresponding initial probability vector and the given weekly transition matrix to find the share of the market for each firm after each of the following time periods. Assume that customers bring in one batch of dry cleaning per week.

\begin{bmatrix}0.5 & 0.5\\0.65 & 0.35\end{bmatrix}

Row 1: J

Row 2: N

Col 1: J

Col 2: N

(a) 1 week

(b) 2 weeks

(c) 3 weeks

(d) 4 weeks

Suppose Johnson
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9.9k points

We are given initial market shares of 40% for J and 60% for N, use this information to write the initial probability vector $X_0$,

$$X_0 = \begin{bmatrix}0.4 & 0.6\end{bmatrix}$$

for transitions matrices $A$, $A^k$ gives the probabilities of a transition from one state to another in k repetitions of an experiment. So if the initial probability vector is $X_0$, then the probability vector after k repetitions will be $X_0 \cdot A^k$.

For part (a) then,

$$X_0 \cdot P = \begin{bmatrix}0.4 & 0.6\end{bmatrix} \begin{bmatrix}0.5 & 0.5\\0.65 & 0.35\end{bmatrix} = \begin{bmatrix}0.59 & 0.41\end{bmatrix}$$

where col 1 represents the probability (59%) of Johnson's share of the market after 1 week, and col 2 represents the probability (41%) of NorthClean's share of the market after 1 week.

for part (b) find $X_0 \cdot P^2$,

$$P^2 = \begin{bmatrix}0.5 & 0.5\\0.65 & 0.35\end{bmatrix} \begin{bmatrix}0.5 & 0.5\\0.65 & 0.35\end{bmatrix} = \begin{bmatrix}0.575 & 0.425\\0.5525 & 0.4475\end{bmatrix}$$

and,

$$X_0 \cdot P^2 = \begin{bmatrix}0.4 & 0.6\end{bmatrix} \begin{bmatrix}0.575 & 0.425\\0.5525 & 0.4475\end{bmatrix} = \begin{bmatrix}0.5615 & 0.4385\end{bmatrix}$$

where col 1 represents the probability (56.15%) of Johnson's share of the market after 2 weeks, and col 2 represents the probability (43.85%) of NorthClean's share of the market after 2 weeks.

for parts (c) and (d) compute $X_0 \cdot P^3$ and $X_0 \cdot P^4$ respectively and proceed in the same manner as parts (a) and (b)

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