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.
1. 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.
2. From the constraints above, show that q2 ≥ q1. (c) 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.
3. 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.