A firm sells a product which may be of high or low quality, s_H or s_L, respectively.  High quality is to occur with probability λ and low quality with probability 1- λ. There is a unit mass of consumers with unit demand and the same willingness-to-pay for the product of a particular quality. Consumers like high quality more than low quality- i.e., consumer valuations (= willingness- to-pay) satisfy r_H > r_L. The consumer valuation for high quality is larger than costs c, which is independent of quality. The marginal cost is assumed to satisfy that λr_H + (1 – λ) r_L > c. There are two groups of consumers. A share α is informed about product quality and a share 1 – α is uninformed; there is no communication between informed and uninformed consumers. We consider the following situation: First, Nature determines product quality. The quality is observed by the firm and a share α of consumers. Second, the firm sets the price of the product. Third, uninformed consumers update beliefs and all consumers make their purchasing decision.

1. Suppose that α = 1. Characterize the equilibrium (price, allocation, profit). Distinguish between case c < r_L and c > r_L.

2. Suppose that α = 0. Characterize the equilibrium (price, allocation, profit). Distinguish between case c < r_L and c > r_L.

3. Suppose that  α < 1 and c < r_L. Do there exist parameter constellations under which the same allocation as under (1) can be supported? If your answer is negative give a proof that shows that the outcome in (1) cannot be replicated for  α < 1. Otherwise, give the exact parameter range for  for which the outcome in (1) can be replicated.

4. Suppose that α > 0 and c > r_L. Do there exist parameter constellations under which the same allocation as under (2) can be supported? If your answer is negative give a proof that shows that the outcome in (2) cannot be replicated for α > 0. Otherwise, give the exact parameter range for  for which the outcome in (2) can be replicated.

5. Comment on what is happening in this market.

6. What kind of government interventions would matter in such a maket with α < 1? Discuss the welfare consequences of such government interventions (try to refer to (2)-(4)).