(Alternate product competition) Firm A produces product A. Firm B produces an alternative product B. Firm A sets the price of product A at pA = 3 and firm B sets the price of product B at pB = 6. The willingness-to-pay values for the two products are distributed evenly on the rectangle shown in figure 8.11, with a total number of people equal to 7 × 12 = 84. (The exact positions are not known. Each person's position is random, but uniform over the rectangle, and independent of everyone else's.) The WTP values for the two products correspond to a person's coordinates in the rectangle. (A person at the upper-right-hand corner has WTPA = 7 and WTPB = 12.) A person might buy A or B, but never both.
(a) Show the region in the square corresponding to people who purchase A, and another region for those who purchase B.
(b) Compute the areas of these two regions.
(c) Assuming zero marginal cost, find the expected producer surpluses πA and πB.