Consider two sellers 1 and 2 and a continuum of buyers. Seller i offers product i at price p_i and incurs zero marginal cost of production. Buyers are identical and derive utility u_i from one unit of each product. Thus their utility is u_i if they buy one unit of product i and zero units of product j, j ≠ i; it is u₁ + u₂ if they buy one unit of each product. Additional units do not give any extra utility. Each buyer has a budget y, which she cannot exceed. Each seller is assumed to prefer not to sell a unit rather than setting a zero price. Suppose that u₁ > u₂.

1. Derive the demand function of each buyer.

2. Consider the game in which sellers simultaneously set price. Characterize the Nash equilibrium of the game.

3. Determine consumer surplus and total surplus that realize in equilibrium. Is the equilibrium necessarily efficient or are there inefficiencies Explain.