Table 15-2
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 15-2, 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)
E) π(2) = (0.47, 0.40, 0.13)

Question

Table 15-2
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 15-2, 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) 
E) π(2) = (0.47, 0.40, 0.13)

Quizplus · Accepted Answer

Using the transition probabilities given, we can find the market shares for the three retailers in month 1 by multiplying the initial market shares by the transition matrix:
π(1) = (0.3, 0.6, 0.1) x 
       | 0.7  0.2  0.1 |
       | 0.3  0.6  0.1 |
       | 0.2  0.2  0.6 |
     = (0.32, 0.48, 0.20)
 
We can then find the market shares for the three retailers in month 2 by multiplying the market shares in month 1 by the transition matrix again:
π(2) = (0.32, 0.48, 0.20) x 
        | 0.7  0.2  0.1 |
        | 0.3  0.6  0.1 |
        | 0.2  0.2  0.6 |
      = (0.44, 0.43, 0.12)
      
Therefore, the best choice is C, as it matches the calculated market shares for the three retailers in month 2.

Table 15-2 The Following Data Consists of a Matrix of Transition Probabilities