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 nex_t period. Let us assume that DoorKnobs sets the price at p = $32 and never changes this price. (To answer the following questions, you will find a calculator useful.)

(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/256 x̅). Solve this equation for x̅ in the case where p = $32. What do you make of the fact that there are two solutions?