Problem
A seller of a good chooses its price (p ≥ 0) and quality (q ≥ 0). The cost of quality q for the seller is C(q) = q 2 .
A buyer can be either of type 1 or type 2. If a type 1 buyer purchases the good of quality q at price p, its net utility is 2q - p. If a type 2 buyer purchases the good of quality q at price p, its net utility is 3q - p. Any buyer who does not purchase the good gets zero net utility.
The seller knows the fraction 1/2 of buyers is type 1 while the remaining 1/2 is type 2.
i. Suppose the seller offers a menu of price-quality pairs ((p1, q1), (p2, q2)) where (pt, qt) is intended for type t for t = 1,2. For the two types, write down the individual rationality constraints IR1,IR2 and the incentive compatibility constraints IC1,IC2.
ii. From the constraints above, show that q2 ≥ q1.
iii. We know that at any menu that maximizes profit of the seller: IR1, IC2 hold with equality and IR2, IC1 can be ignored (you don't have to prove these results). Using these results, determine menu that maximizes profit of the seller.