Suppose that there are two stores, A and B, that sell homogenous good. Suppose that the two stores are located on the real line: store A is at 0, while store B is at 1. All the potential customers for the good are also located on the real line, all of them between 0 and 1, and each customer has the same reservation utility of 1.
Now, to purchase the good from either store, each customer has to walk to it, and the marginal cost of walking is c: if a customer is located at x (between 0 and 1), and he buys from the store A at a price PA, his total cost of buying the good is PA + xc. Likewise, if he buys from the store B at a price PB , his total cost is PB + (1-x) c. He will buy at the store for which the cost is lower, as long as the cost is below 1, i.e., his reservation utility.
a) Derive the demand functions for the two stores.
b) Suppose that the marginal cost of producing the good is zero, for each firm. Find the Bertrand equilibrium price of each firm.