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 1.
A) π(1) = (0.09, 0.42, 0.49)
B) π(1) = (0.55, 0.33, 0.12)
C) π(1) = (0.18, 0.12, 0.70)
D) π(1) = (0.55, 0.12, 0.33)
E) π(1) = (0.33, 0.33, 0.33)

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 1.
A) π(1) = (0.09, 0.42, 0.49) 
B) π(1) = (0.55, 0.33, 0.12) 
C) π(1) = (0.18, 0.12, 0.70) 
D) π(1) = (0.55, 0.12, 0.33) 
E) π(1) = (0.33, 0.33, 0.33)

Quizplus · Accepted Answer

To find the market shares for the three retailers in month 1, we need to multiply the transition probabilities by the initial market share vector.

π(1) = P * π(0)

= (0.7 * 0.3) + (0.2 * 0.6) + (0.1 * 0.1)   (For Retailer 1)
  (0.2 * 0.3) + (0.6 * 0.6) + (0.2 * 0.1)   (For Retailer 2)
  (0.1 * 0.3) + (0.2 * 0.6) + (0.7 * 0.1)   (For Retailer 3)

= (0.55, 0.33, 0.12)

Therefore, the correct choice is B.

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