Question

Two products A and B currently share the market with shares 50% and 50% each respectively. Each week some brand switching takes place. Of those who bought A the previous week, 60% buy it again whereas 40% switch over to B. Of those who bought B the previous week, 80% buy it again where as 20% switch over to A. Find their shares after one week and after two weeks. If the price war continues, when is the equilibrium reached?

Answer 1

Given that two products A and B have shared 50% and 50% respectively.

(A A) ⇒ those who bought A the previous week will buy it again = 60% = 0.6

(A B) ⇒ those who bought A the previous week will buy B now = 40% = 0.4

(B A) ⇒ those who bought B the previous week will switch to A = 20% = 0.2

(B B) ⇒ those who bought B will again buy B = 80% = 0.8

The transition probability matrix is given by

\[{\begin{matrix}\phantom{......}\begin{matrix}\text{A}&&\text{B}\end{matrix} \\ \text{T}=\begin{matrix}\text{A}\\\text{B}\end{matrix} \begin{pmatrix}0.6&0.4\\0.2&0.8\end{pmatrix}\\ \end{matrix}}\]

The current position of A and B in the market is

`(("A", "B"),(0.5, 0.5))`

After one week

The shares of A and B are given by

\[\begin{matrix} & \begin{matrix}\text{A}&&\text{B}\end{matrix} \\  & \begin{pmatrix}0.5&0.5\end{pmatrix}\\&&& \end{matrix} \begin{matrix}\phantom{..} \begin{matrix}\text{A}&&\text{B}\end{matrix} \\ \begin{matrix}\text{A}\\\text{B}\end{matrix}  \begin{pmatrix}0.6&0.4\\0.2&0.8\end{pmatrix}\\ \end{matrix}\]

= `(("A", "B"),(0.3 + 0.1, 0.2 + 0.4))`

= `(("A", "B"),(0.4, 0.6))`

So after one week the market share of A is `0.4/100 xx 100` = 40% and that of B is `0.6/100 xx 100` = 60%

After two weeks

The shares of A and B are given by

\[\begin{matrix} & \begin{matrix}\text{A}&&\text{B}\end{matrix} \\  & \begin{pmatrix}0.4&0.6\end{pmatrix}\\&&& \end{matrix} \begin{matrix}\phantom{..} \begin{matrix}\text{A}&&\text{B}\end{matrix} \\ \begin{matrix}\text{A}\\\text{B}\end{matrix}  \begin{pmatrix}0.6&0.4\\0.2&0.8\end{pmatrix}\\ \end{matrix}\]

= `(("A", "B"),(0.24 + 0.12, 0.16 + 0.48))`

= `(("A", "B"),(0.36, 0.64))`

Thus after two weeks, A will have 36% of shares and B will have 64% of shares.

As time goes, equilibrium will be reached in the long run.

At this point A + B = 1

We have

`(("A", "B")) ((0.6, 0.4),(0.2, 0.8)) = (("A", "B"))`

By matrix multiplication,

(0.6A + 0.2B 0.4A + 0.8B) = (A B)

Equating the corresponding elements,

0.6A + 0.2B = A

0.6A + 0.2(1 – A) = A ......(Using A + B = 1)

0.6A + 0.2 – 0.2A = A

0.2 = A – 0.4A

A = `0.2/0.6` = 0.33 = 33%

B = 1 – 0.33 = 0.67 = 67%

Thus the equilibrium is reached when the share of A is 33% and share of B is 67%.