If the demand function for the DoorKnobs operating system is related to perceived market share s and actual market share x by the equation p = 512s(1 - x), then in the long run, the highest price at which DoorKnobs could sustain a market share of 3/4 is
A)$256.
B)$128.
C)$113.78.
D)$96.
E)$81.92.

Question

Quizplus · Accepted Answer

In the long run, equilibrium is achieved where the actual market share x is equal to the perceived market share s (i.e., x = s). Thus, we have:

p = 512s(1 - s)

To find the highest price at which DoorKnobs could sustain a market share of 3/4 (i.e., s = 0.75), we need to find the maximum value of p that satisfies the above equation.

Taking the derivative of p with respect to s, we get:

dp/ds = 512(1 - 2s)

Setting dp/ds = 0 to find the maximum, we get:

1 - 2s = 0
s = 1/2

This means that the maximum value of p occurs when s = 1/2. Substituting s = 1/2 into the demand function, we get:

p = 512(1/2)(1 - 1/2) = 256

Therefore, the highest price at which DoorKnobs could sustain a market share of 3/4 is $256. The correct answer is (D).

If the Demand Function for the DoorKnobs Operating System Is