Consider the horizontal quality model on the unit interval from 0 to 1. There are N consumers located uniformly along the interval. There are two firms, with zero marginal costs, initially located at 0 to 1. Consumers will buy one unit of the good from the lowest cost retailer as long as the effective price is below V. They have transportation costs of t getting from their location to the store and back.
1. If the firms sell to the whole market, derive the function governing demand to each firm as a function of the two prices.
2. Solve for the equilibrium price if both firms sell to the whole market.
3. What condition do we have to check to ensure that the firms sell to the whole market?
4. Solve for the equilibrium price if both firms only sell to a part of the market.
5. Suppose that firm 1 is located at a and firm 2 is located a b (without loss of generality, let 0 ≤ a < b ≤ 1). Show the profit function for each firm as a function of their prices and the location of both firms.