Amonopolist produces output with constant marginal cost equal to 1. there are two of consumers that are potentially in the market for the good. consumer a has inverse demand function pa(x) = 7 − x, and consumer b has inverse demand function pb(x) = 5 − x.

(i) what are the pareto efficient levels of consumption of the good by consumers a and b? call these x∗a and x∗b respectively.
(ii) if the monopolist can identify which consumer is which, and can charge them different prices, what price will each consumer be charged? what profit does the monopolist make?
(iii) if the monopolist can identify which consumer is which, and can charge a two-part-tariff to each, what price schedule will each consumer face? that is, what will be the fixed "entry fee" charged to each person, and what will be the marginal price per unit paid? what is the monopolist’s profit now?
(iv) suppose the monopolist cannot identify which consumer is which. what is the best pair of purchase options (xa, ra) and (xb, rb) the monopolist should offer? (hint: you’ll need to show that xb = 2.) what is the firm’s profit?