Imagine a duopoly in which firms produce an "identical good" for consumers who live along a 100-mile highway. We imagine (in the spirit of problem 10 in chapter 9) that consumers are distributed uniformly along the highway at a density of ten consumers per mile. Each consumer has reservation price $100 for one unit of the good, but each pays a cost getting to and. from the store that must be added to the price of the good when comparing against the reservation price.

The good costs each of the two firms $1 per unit to produce. Each firm is allowed to open one store somewhere along the highway. If the firm is located d miles from a consumer, the consumer pays $(d/50)2 for the round-trip from home to the store.and back. The two firms locate their stores (which serves to differentiate·their goods in the minds of consumers, if they don't locate one on top of the other) and then they compete with Bertrand conjectures.

What is the equilibrium in this case? (We have been a bit imprecise in specifying the model, so be very clear what assumptions you are making. If you like a challenge, consider what would happen if travel costs were linear instead of quadratic. If you do this, be prepared for enormous frustrations.)