Monopoly

Suppose the monopolist has the following demand function P (q) = max{40 3q, 0}. The cost function is C(q) = q2. Compute the monopoly price and quantity, the Lerner index and the inefficiency in this market.

Suppose that all consumers have unit demand. They buy either 0 or 1 unit of the goods produced by the monopolist. All the consumers are identical and have S valuation (i.e. maximum willingness to pay) for the good. Show that monopoly pricing would not create a welfare loss.