Duopoly and warranties

Imagine a model of a market in which there are two firms, 1 and 2, producing TV sets, and one consumer. The consumer has a unit demand for a TV set, which both firms can produce at a constant marginal cost c = 100. He knows that there is one high-quality firm and one low-quality firm, but he does not know which firm is which. He believes that firm 1 (resp. firm 2) is the high-quality firm with probability 1/2. A TV set can be either functional, in which case the consumer derives gross utility V = 500 from consuming it, or broken, in which case the consumer gets 0 utility. The consumer does not know whether the TV set is functional prior to purchasing it. We assume that a high-quality (resp. low-quality) firm produces a functional TV set with probability _H = 3/4 (resp. _L = 1/4.)

In the following questions, we assume that firms 1 and 2 simultaneously compete in prices, and denote these prices by p_i, i = 1, 2 .

(a)In this model, the consumer does not know the quality of the product before purchasing it. How do we call such a good?

(b)Assume that the utility of the consumer is the expected value of the product minus the product price and equal to zero if she does not buy the product. Write down the expected utility of purchasing a product of value V at price p with expected probability of purchasing a functional TV set equal to .

(c)Explain, in words, why the consumer has no way to infer a firm's quality just by observing its price. Calculate the expected net utility that the consumer derives from consuming firm i's product at price p_i, i = 1, 2.

(d)Using your answer to question (c), show that both firms set prices equal to marginal cost in equilibrium (to simplify the analysis, you may assume that firms are not allowed to price below cost: p_i c, i = 1,2)