It is not difficult to navigate in Lonely-Line City: a single street runs from kilometer 0 to kilometer 1 along which 100 inhabitants are equidistantly distributed.  [Approximate the consumer distribution by a continuum on [0, 1] with a mass of 100.] To keep the place residential the local government has decided that no shops are allowed within the city limits. As it happens, there exists one shop at each boundary of the city [one at point 0 and one at point 1]. Each morning each inhabitant drinks one liter of fresh milk. Assume that transporting one liter of milk costs t cents per kilometer (this is the dis-utility incurred by an inhabitant if he walks or the cost for the shop for delivery), each shop pays a wholesale price of c cents per liter.

1. Suppose that each shop i sells one liter at price pi at the shop and that all inhabitants get up each morning and walk to one of the two shops to get the milk. What is the price set by each of the two shops, what are the shops profits? [Characterize the Nash equilibrium of the corresponding game!]

2. Suppose that instead of consumers walking to one of the shops, both shops have a delivery service and that shops set a price that depends on the address of the inhabitant who buys. What are the prices charged by the shops, what are the shops’ profits? [Characterize the Nash equilibrium of the corresponding game in which shops simultaneously set prices p_i! Illustrate your analysis by a figure!] Compare your findings to those in (1.).

3. Return to the situation in (1.) but suppose that shops sometimes do not have fresh milk available and that inhabitants only make the walk if they know that they get the milk for sure. Therefore, each shop can buy the right to use the city’s public speakers to advertise the availability of the milk. There is time for two ads. The inhabitants of Lonely-Line City, however, do not always pay attention to the ads. Each inhabitant listens to ad 1 with a 50% chance and to ad 2 also with a 50 % chance. Assume furthermore that for each inhabitant the probability to listen to ad 2 is independent of whether he or she has listened to ad 1. Consequently, there is a 25% chance that an inhabitant listens to ad 1 only, a 25% chance that an inhabitant listens to ad 2 only, a 25% chance that an inhabitant listens to ads 1 and 2, and a 25% chance that an inhabitant listens to none of the ads.

4. Consider a day at which both shops have milk available. Shops have the following two options: (a) they jointly announce the availability of milk in each ad, i.e., both ads contain information on both shops, (b) ad 1 contains information on shop 1, ad 2 contains information on shop 2. Determine the equilibrium in each of the two cases (the advertising costs are assumed to be the same in each case). Which option do shops prefer? [Characterize the equilibrium for options 1 and 2. In each case, you can assume that parameter constellations are such that first-order conditions of profits maximization characterize the equilibrium.] Provide an intuition for your result.