Suppose that demand for DoorKnobs is as given in the previous problem, and assume that the perceived market share in any period is equal to the actual market share in the previous period. Then where x_t is the actual market share in period t, the equation p = 256x_{t − 1} (1 − x_t) is satisfied. Rearranging this equation, we find that x_t = 1 − (p/256x_{t − 1})  whenever p/256x_{t − 1} ≤ 1. If p/256x_{t − 1} ≤ 0, then x_t = 0. With this formula, if we know actual market share for any time period, we can calculate market share for the next period. 

Let us assume that DoorKnobs sets the price at p = $32 and never changes this price. 

(a) If the actual market share in the first period was 1/2, find the actual market share in the second period _________, the third period ________. Write down the actual market shares for the next few periods __________. Do they seem to be approaching a limit? If so, what _________?

(b) Notice that when price is held constant at p, if DoorKnobs’s market share converges to a constant x̅, it must be that x̅ = 1 − (p/256x̅). Solve this equation for x̅ in the case where p = $32. What do you make of the fact that there are two solutions?