Consider a monopolist who has one unit of a product. The outside option of not selling the product is c. This product has high quality s_H with probability λ and low quality s_L with probability 1 - λ. There are two consumers, each of whom has valuation v_H for high quality and v_L for low quality. We assume that v_H > c > v_L. Furthermore, we assume that λ_vH + (1 - λ) v_L > c. Consumers bid for the object using a second-price auction. Suppose that the firm learns its quality, but that consumers are initially uncertain about product quality.

1. Determine the equilibrium price of the game in which the firm first decided whether to post the item and consumers then make their bids.

2. Consider a two-period extension in which quality is constant over time. Suppose that consumers learn the quality after period 1-i.e., quality becomes public information-and that they bid for a second unit of the product (both consumers are also identical in period 2) in period 2. Characterize  the equilibrium of the two-period game in which in period 1a Nature draws quality, in 1b the firm becomes privately informed about quality and decides whether to offer its product, and in 1c consumers bid for the period-1 product and in period 2a consumers learn the true quality, in 2b the firm decides whether to offer the product, and in 2c consumers bid for the period-2 product. Compare your findings to (1).

3. Consider a modified model in which consumers are ex ante uninformed also about the existence of the product. Suppose that the firm can make the advertising expenditure A(∅) at the beginning of period 1, where ∅ is the probability that both consumers are informed about the existence of the product. A(∅) is assumed to be continuously differentiable and strictly convex with the appropriate limit properties such that you can restrict attention to interior solutions. Note that either both or none of the consumers become informed. Consider the game from (2) with the addition that when the firm decides whether to offer the product it also chooses ∅ and consumers observe A(∅). A consumer can only buy in period 2 if she has become informed about the existence of the product in period 1. Characterize the separating equilibrium that gives the highest profit to the high-quality firm.

4. Compare your result in (3) to a situation in which consumers observe the quality ex ante (but which is otherwise the same as in part 3).