There are two types of consumers: one half of consumers are type 1 (low type) and the other half are type 2 (high type). Type 1's demand curve is q1 = 4 P, while type 2's demand is given by q2 = 6 P. Consider a monopolist selling its product to these consumers. Assume that the marginal cost is equal to zero.

However, the firm does not know an individual consumer's type. It only knows that there are two types of consumers with demand curves given as above. Suppose that the firm can offer only a single two-part tariff, (T,P), where T is the lump-sum fee and P is the unit price.

(1) (10 point) If the firm serves only high type consumers, what should be the optimal two-part tariff, T and P ? Also, compute the resulting profit.

(2) (5 point) If the firm serves both types of consumers, what is the firm's profit when it offers (T,P)? Find the expression in terms of P only.

(3) (10 point) Compute P that maximizes the firm's profit in (2). Also, compute the optimal T, as well as the resulting profit.

(4) (5 point) Compare the profit from (1) and the profit from (3), and determine which two-part tariff will be chosen by the firm. Also, compute the social surplus from the two-part tariff chosen by the firm.