Suppose there are two groups of consumers, group 1 of size and group 2 of size

Suppose there are two groups of consumers, group 1 of size α and group 2 of size 2 - α. Consumers have unit demand. Consumers in group 1 have willingness-to-pay for a high-quality good equal to 3 and for a low-quality good equal to 2. Consumers in group 2 have willingness-to-pay equal to 2 independent of the quality of the good. A monopolist sells the product to consumers. With probability 1/2 it produces  high quality and with the remaining probability 1 / 2 it produces low quality.  High quality is produced at unit cost c = 3 / 2 and low quality is produced at 0 unit cost.

1. Determine profit-maximizing prices for each type of the firm under the assumption that consumer observe the quality before purchase.

2. Suppose now that consumers only observe price but not quality prior to purchase. Consider the case α = 1. Does a separating equilibrium exist? If your answer is affirmative characterize the separating equilibrium which is best for the high-quality firm. Do not forget to specify consumer beliefs.

3. What happens if α = 3 / 2? Does a separating equilibrium exist? If your answer is affirmative characterize the equilibrium which is best for the high-quality firm. Again do not forget to specify consumer beliefs.