A petrol station owner is considering the effect on his business (Superpet) of a new petrol station (Global) which has opened just down the road. Currently (of the total market shared between Superpet and Global) Superpet has 80% of the market and Global has 20%. Analysis over the last week has indicated the following probabilities for customers switching the station they stop at each week: To Superpet Global From Superpet 0.75 0.25 Global 0.55 0.45 •What will be the expected market share for Superpet and Global after another two weeks have past? •What would be the long-run prediction for the expected market share for Superpet and Global?
The correct answer and explanation is :
The analysis of the petrol station market dynamics reveals the following results:
- Market Share After Two Weeks:
The initial market shares are 80% for Superpet and 20% for Global. By applying the transition matrix over two weeks, the expected market share for Superpet and Global after two weeks will be:
- Superpet: 76.75%
- Global: 23.25%
- Long-Run Market Share:
The long-run market share is determined by the stationary distribution of the transition matrix. This is found by solving for the eigenvector corresponding to the eigenvalue of 1. In this case, the long-run market shares are:
- Superpet: 72.73%
- Global: 27.27%
Explanation:
The transition matrix captures the probabilities of customers switching between the two stations each week:
- Superpet retains 75% of its customers, while 25% switch to Global.
- Global retains 45% of its customers, while 55% switch to Superpet.
By applying this matrix iteratively, the market shares evolve. After two weeks, Superpet’s market share slightly decreases, while Global’s share increases. The long-run market shares are stable and represent the equilibrium state, where the proportions of customers between the two stations no longer change.
The chart visualizes the progression of market shares from the initial state to after two weeks and the long-run equilibrium. Over time, Global gains more market share, but the system reaches a stable distribution due to the influence of the transition probabilities.