The small and faraway island of Imaginola is divided into two countries, Freeland and Episcopia. Because of the local geography, each country has a single main road, with its population distributed evenly along it. The population of each country is sufficient to support, among other services, two shoe shops. All the shoe shops being excellent, the habit of the populace is to patronize the shop nearer to wear they live, except that if they find two shoe shops within sight of each other they will choose one in some idiosyncratic manner, which can be treated as random for the purposes of analysis. The topic of this post is where we expect the shoe shops to be situated.
In Freeland, the shopkeepers choose freely where to put their shops, and naturally seek to maximize their custom. The result is that the shops have converged to occupy nearby locations half way along the road. Each shop gets half the custom, and shoppers have to travel on average a quarter of the length of the road to buy shoes. Neither shop can move further away from the other without ceding some of its customers.
In Episcopia, government officials use fine control of the business rates system to encourage shopkeepers to position themselves where they can best serve the populace. The result is that the two shoe shops there are to be found a quarter of the way along the road from each end. Shoppers have to travel on average an eighth of the length of the road to buy shoes.
Whereas in Freeland the citizens congratulate themselves on the fine shopping experience afforded by its free markets, in Episcopia shopping is more convenient, the roads are less congested, and the citizens have more time for productive labour. The economy is booming and there may be a need for a third shoe shop. The interested reader can amuse herself working out where three shops will put themselves in Freeland (should it ever need them), and where the officials will encourage three shops to go in Episcopia.
I should mention that I have assumed that the start up costs (in shoe stocks and staffing) are sufficient that the shopkeepers in Freeland have no fear of a new store opening up, hoping to drive one of them out of business. If that were a factor, the positioning of the shops would be different.