Consider the matrix of transition probabilities of a product available in the market in two brands A and B. | | A | B | | --- | --- | --- | | A | $0.9$ | $0.1$ | | B | $0.3$ | $0.7$ | Determine the market share of each brand in equilibrium position.

Consider the matrix of transition probabilities of a product available in the market in two brands A and B. |

Question 20 of 44

Consider the matrix of transition probabilities of a product available in the market in two brands A and B.

	A	B
A	$0.9$	$0.1$
B	$0.3$	$0.7$

Determine the market share of each brand in equilibrium position.

Puducherry TnboardTamil Nadu HSC (DGE) Commerce Board 2022Subjective· 3mImportance★★★★★

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Set the equilibrium state vector equal to itself after transition; solving $(a\;b)T=(a\;b)$ with $a+b=1$ gives $A=0.75,\ B=0.25$ .

Let the equilibrium shares be $a$ (for $A$ ) and $b$ (for $B$ ) with $a+b=1$ . The transition matrix is

$T=\begin{pmatrix}0.9 & 0.1\\ 0.3 & 0.7\end{pmatrix}.$

Equilibrium requires $(a\;\;b)T=(a\;\;b)$ :

$0.9a+0.3b=a,\qquad 0.1a+0.7b=b.$ …