Question $3$:

The personal computer industry is very fast moving and technology provides motivation for customers to upgrade with new computers every few years. Brand loyalty is very important and companies try to do things to keep their customers happy. However, some current customers will switch to a different company. Three particular brands, Doorway, Bell, and Kumpaq, hold the major shares of the market. People who own Doorway computers will buy another Doorway in their next purchase $80\%$ of the time, while the rest will switch to the other companies in equal proportions. Owners of Bell computers will buy Bell again $90\%$ of the time, while $5\%$ will buy Doorway and $5\%$ will buy Kumpaq. About $70\%$ of the Kumpaq owners will make Kumpaq their next purchase while $20\%$ will buy Doorway and the rest will buy Bell.

At the present time, tachi of the thiee companies has an equal share of the market.

A$)$ What is the market share for the next period given the initial market shares?

B$)$ What long$-$run market shares for the three companies?