Consider a Hotelling model in which two firms are located at the opposite ends of the unit interval and serve a unit mass of consumers, who are uniformly distributed on this interval. Each consumer has unit demand and her utility if she buys from …rm 1, located at 0, is r –τx - p₁, where x is the consumer’s location on the line, her utility if she buys from firm 2, located at 1, is r -τ (1- x) -p₂. Her utility if she does not buy at all is 0. For simplicity, both firms are assumed to have zero costs. The two firms compete by simultaneously setting their prices. Consumers fall into two categories: at every point on the line, a fraction λ with λ ≥ λ > 0 of consumers observe p₁ and p₂ and then decide whether to buy from firm 1, firm 2, or not to buy at all (these consumers behave as in a standard Hotelling model). A fraction 1- λ of consumers at every point x, do not observe p₁ and p₂ (i.e., they are “uninformed” about prices). Instead, each uninformed consumer forms an expectation about p₁ and p₂, and uses these expectations to choose whether to visit firm 1, firm 2, or none of the firms. Visiting one firm is possible at zero costs, visiting both firms is infeasible or prohibitively costy. If an uniformed consumer chooses to visit one of the two firms, she learns its actual price, and then either buys from that firm or does not buy at all. In equilibrium, the beliefs of uninformed consumers are correct. For simplicity, assume that r is sufficiently high to ensure that the market is fully covered for all values of λ.

1. Solve for the equilibrium when firms 1 and 2 choose p₁ and p₂, respectively.

2. Let us interpret λ as “market transparency”: An increase in λ makes the market “more transparent”. What happens to prices and what happens to consumer surplus when the market becomes more transparent? What is the intuition for your answer?

3. Suppose that a policy maker maximizes total surplus as the sum of consumer surplus and profits. Should the policy maker enforce high transparency or not? Explain the intuition for your answer.

4. Now suppose that consumers always observe firm 2’s price, p₂, but, as before, only a fraction λ of consumers observe p₁ while the others are uniformed and base their decision on their expectations regarding p₁, which are correct in equilibrium. Solve again for the Nash equilibrium. How does λ affect the profit of firm? Does it pay firm 1 to have non-transparent prices? Provide an intuition for your result.