Problem 2 (Third-degree price discrimination).

Suppose a monopolist sells its product to two groups of buyers: teenagers (T ) and adults (A). The inverse demand functions of these two groups are given by P_T (Q_T) = 3 Q_T and P_A(Q_A) = 5 Q_A, respectively. Note that in this question, the monopolist is able to distinguish between two groups, and can possibly charge different prices.

(a) Let MC = 1 be the marginal cost of production. If this monopolist charges a single price, what is the optimal price P?

(b) Still assume MC = 1. Now the monopolist charges two different prices to the groups. Find the optimal prices P_T^* and P_A^*.

(c) Instead of constant marginal cost, now suppose the cost function is C(Q) =1/2(Q_T + Q_A)². Is this problem still separable into two independent optimizations? What prices should the monopolist charge to these two groups?