Suppose that two firms with constant marginal costs compete in prices in a homogeneous product market. All consumers have unit demand and a willingness to-pay r. A share α of consumers is informed about the prices in the market. The share (1 – α) = 2 goes to firm i = 1, 2 and decides whether to buy (these consumers do not know that a product from firm j ≠ i exists). Firms set prices and then consumers make their consumption decisions.

1. Show that there does not exist a symmetric Nash equilibrium in pure strategies.

2. Characterize equilibrium prices in the unique mixed-strategy Nash equilibrium.

3. How do prices change if α is increased?

4. How do equilibrium profits change as (Calculate 2π^* / 2α)