Bill Barriers, the president of MightySoft software company is about to introduce a new computer operating system called DoorKnobs. Because it is easier to swap files with people who have the same operating system, the amount people are willing to pay to have DoorKnobs on their computers is greater the larger they believe DoorKnobs’s market share to be.
The perceived market share for DoorKnobs is the fraction of all computers that the public believes is using DoorKnobs. When the price of DoorKnobs is p, then its actual market share is the fraction of all computer owners that would be willing to pay at least $p to have DoorKnobs installed on their computers. Market researchers have discovered that if DoorKnobs’s perceived market share is s and the price of DoorKnobs is $p, then its actual market share will be x, where x is related to the price p and perceived market share s by the formula
p = 256s(1 − x)
In the short run, MightySoft can influence the perceived market share of DoorKnobs by publicity, advertising, giving liquor and gifts to friendly journalists, and giving away copies in conspicuous ways. In the long run, the truth will emerge, and DoorKnobs’s perceived market share s must equal its actual market share x.
(a) If the perceived market share is s, then the demand curve for Door-Knobs is given by Equation 1. On the graph below, draw the demand curve relating price to actual market share in the case in which Door-Knobs’s perceived market share is _______. Label this curve _______.
(b) On the demand curve that you just drew with s = 1/2, mark a red dot on the point at which the actual market share of DoorKnobs is 1/2. (This is the point on the demand curve directly above x = 1/2.) What is the price at which half of the computer owners actually want to buy DoorKnobs, given that everybody believes that half of all computer owners want to buy DoorKnobs? _________.
(c) On the same graph, draw and label a separate demand curve for the case where DoorKnobs’s perceived market share s takes on each of the following values: s =1/8, 1/4, 3/4, 7/8, 1.
(d) On the demand curve for a perceived market share of s = 1/4, put a red dot on the point at which the actual market share of DoorKnobs is 1/4. (This is the point on this demand curve directly above x = 1/4.) If the perceived market share of DoorKnobs is 1/4, at what price is the actual market share of DoorKnobs also 1/4? ________.
(e) Just as you did for s = 1/2 and s = 1/4, make red marks on the demand curves corresponding to s = 1/8, 3/4, 7/8, and 1, showing the price at which the actual market share is s, given that the perceived market share is s.
(f) Let us now draw the long-run demand curve for DoorKnobs, where we assume that computer owners’ perceived market shares s are the same as the actual market shares x. If this is the case, it must be that s = x, so the demand curve is given by p = 256x(1 − x). On the graph above, plot a few points on this curve and sketch in an approximation of the curve.
(g) Suppose that MightySoft sets a price of $48 for DoorKnobs and sticks with that price. There are three different perceived market shares such that the fraction of consumers who would actually want to buy Door- Knobs for $48 is equal to the perceived market share. One such perceived market share is 0. What are the other two possibilities? _________.
(h) Suppose that by using its advertising and media influence, MightySoft can temporarily set its perceived market share at any number between 0 and 1. If DoorKnobs’s perceived market share is x and if MightySoft charges a price p = 256x(1−x), the actual market fraction will also be x and the earlier perceptions will be reinforced and maintained. Assuming that MightySoft chooses a perceived market share x and a price that makes the actual market share equal to the perceived market share, what market share x should MightySoft choose in order to maximize its revenue and what price should it charge in order to maintain this market share? (Revenue is px = 256x2(1 − x).) Use calculus and show your work. __________.