A monopolist faces a consumer who it believes to be of type H with probability 13 and of type L with probability 23: Type H has utility function UH = 2pq tand type L has utility function UL =pqt; where q is the quality of a particular good the monopolist sells and t is the monetary payment for the good.
The monopolist makes offer(s) to the consumer on a take-it-or-leave-it basis. It costs the monopolist $q to produce q units of quality. There is no Öxed cost.
The reservation utility of both consumer types is zero.The consumer maximizes expected utility and the monopolist maximizes its expected proÖt. The monopolist knows all of the above, but cannot identify the consumerís type directly.
(a) Suppose the monopolist o§ers a two part tariff to the consumer and produces afterthe order. What is the expected proÖt maximizing entry fee F and price p per unit?(At the optimum, the monopolist may Önd it proÖtable not to serve a certain type.)
(b) Suppose the monopolist has already produced a product with quality q . Supposeit o§ers the product to the consumer in exchange for a payment r: What is the expected proÖt maximizing r? Assume that the consumer purchases the object ifr is equal to the valuation of the consumer.1
(c) In the answer to b; what level of q should the monopolist choose if it wants tomaximize expected proÖt by offering the product for some r?
(d) From now on, suppose there are two consumers whose types are drawn independently from the distribution above: The consumers know their own types but not the other consumer ís type. Suppose by a capacity constraint, the producer can produce only one unit of output (of different quality). How do your answers to band c:change in this case?
(e) Suppose the monopolist produces an output with quality q and auctions it of to the two consumers in an ascending English auction. If the bids are equal, the good is randomly allocated to the consumers with equal probability. What is the expected revenue for the monopolist?
(f) Given the result in e, what is the expected proÖt maximizing q if the monopolist auctions o§ the unit of output in an ascending English auction?
(g) Compare and interpret the expected proÖts of the monopolist in a; c:d:f: How do the informational requirements for the monopolist vary?