(Big profit) A consumer has utility function for the quantity of a commodity given by U(q) = q -½q² - pq, where p is the unit price.

(a) If a monopolist with zero marginal cost charges a fixed unit price, what is the maximum producer surplus?

(b) Suppose that the monopolist charges according to the price schedule p = K - Lq. Show that by proper choice of K and L, the producer can get arbitrarily close to double the surplus of part (a).

Hint: Solve this problem graphically, noting that p(q) = K - Lq, but the quantity q satisfies 1 - q = K - 2Lq. A further hint is that K should be nearly 1.