Table 14-2
The following data consists of a matrix of transition probabilities (P)of three competing retailers, the initial market share π(0).Assume that each state represents a retailer (Retailer 1, Retailer 2, Retailer 3, respectively)and the transition probabilities represent changes from one month to the next.
P = π(0)= (0.3, 0.6, 0.1)
-Using the data given in Table 14-2, find the market shares for the three retailers in month 2.
A)π(2)= (0.30, 0.60, 0.10)
B)π(2)= (0.55, 0.33, 0.12)
C)π(2)= (0.44, 0.43, 0.12)
D)π(2)= (0.55, 0.12, 0.33)

Question

Table 14-2
The following data consists of a matrix of transition probabilities (P)of three competing retailers, the initial market share π(0).Assume that each state represents a retailer (Retailer 1, Retailer 2, Retailer 3, respectively)and the transition probabilities represent changes from one month to the next.
P =  π(0)= (0.3, 0.6, 0.1)
-Using the data given in Table 14-2, find the market shares for the three retailers in month 2.
A)π(2)= (0.30, 0.60, 0.10) 
B)π(2)= (0.55, 0.33, 0.12) 
C)π(2)= (0.44, 0.43, 0.12) 
D)π(2)= (0.55, 0.12, 0.33)

Quizplus · Accepted Answer

To find the market shares in month 2, we need to multiply the initial market shares (π(0)) by the transition probability matrix (P) twice. 
π(1) = π(0) * P = (0.3, 0.6, 0.1) * 
| 0.7  0.2  0.1 |
| 0.3  0.5  0.2 |
| 0.1  0.3  0.6 |
= (0.28, 0.46, 0.26)
π(2) = π(1) * P = (0.28, 0.46, 0.26) * 
| 0.7  0.2  0.1 |
| 0.3  0.5  0.2 |
| 0.1  0.3  0.6 |
= (0.44, 0.43, 0.12)
Therefore, the market shares for the three retailers in month 2 are π(2) = (0.44, 0.43, 0.12), which is option C.

Table 14-2 The Following Data Consists of a Matrix of Transition Probabilities