Question

C 295-4. The leading brewery on the West Coast(labeled A) has hired an OR analyst to analyze its market position. It is particularly concerned about its major competitor (labeled B). The analyst believes that brand switching can be modeled as a Markov chain using three states, with states A and B representing customers drinking beer pro- duced from the aforementioned breweries and state C representing all other brands. Data are taken monthly, and the analyst has con- structed the following (one-step) transition matrix from past data. 0.8 0.25 0.15 0.15 0.7 0.05 0.05 0.05 0.8 What are the steady-state market shares for the two major breweries?

Answer 1

Given that AB C A B C 0.8 0.25 0.15 0.15 0.7 0.05 0.05 0.05 0.8 p = [0.8 0.25 0.15 0.15 0.7 0.05 0.05 0.05 0.8] q^p = q q^I or q(p - I) = 0 q([0.8 0.25 0.15 0.15 0.7 0.05 0.05 0.05 0.8] = [1 0 0 0 1 0 0 0 1) = 0 q([- 0 . 2 0 . 15 0.05 0.25 0.7 0.05 0.15 0.05 0.15 0.05 0.20) = 0 [q_1 q)2 q_3] = [- 0.2 0.15 0.05 0.025 - 0.3 0.05 0.15 0.05 0.30] = [0 0 0] = 0 .2 q-1 + 0.25 q_2 + 0.15 q_3 = 0 0.25 q_1 - 0.3 q-2 + 0.05 q_3 = 0 q_1 + q_2 + 1-3 - 1 [Because its a probability vector] solving then we get q-`1 = 0.5 u q_2 = 0.289 q-3 = 0.2 steady state market share