A single monopolistic seller sells its products to buyers that can be of one of two types i{H,L} , with H>L . The probability that an agent is of type H is . A buyer of type i receives utility equal to iqq2/2t if he purchases q units of the good for a total payment of t. It costs the seller a constant cost $c per unit to produce the good. Moreover, c<L . Each buyer receives a reservation utility equal to zero if the buyer does not purchase anything from the seller.

(a) Assume the seller can observe each buyer's type and can force each type i to choose between a contract (qi,ti) or else buying nothing. What is the seller's optimal contract (qi,ti) for each type i ?

(b) Suppose buyers' types are not observable to the monopolist. The firm can offer a menu of two contracts {(qL,tL),(qH,tH)} to buyers. If a buyer selects contract (qi,ti) then s/he is entitled to receive qi units of the product by paying ti to the seller regardless of the buyer's true type. Find the optimal set of contracts that maximize the seller's profits.