Consider a small geographic region with a total population of 1 million people. There are two towns, Alphaville and Betaville, in which each person can choose to live. For each person, the benefit...

Consider a small geographic region with a total population of 1 million people. There are two towns, Alphaville and Betaville, in which each person can choose to live. For each person, the benefit from living in a town increases for a while with the size of the town (because larger towns have more amenities and so on), but after a point it decreases (because of congestion and so on). If x is the fraction of the population that lives in the same town as you do, your payoff is given by

(a) Draw a graph like Figure 11.11, showing the benefits of living in the two towns, as the fraction living in one versus the other varies continuously from 0 to 1.

(b) Equilibrium is reached either when both towns are populated and their residents have equal payoffs or when one town—say Betaville—is totally depopulated, and the residents of the other town (Alphaville) get a higher payoff than would the very first person who seeks to populate Betaville. Use your graph to find all such equilibria.

(c) Now consider a dynamic process of adjustment whereby people gradually move toward the town whose residents currently enjoy a larger payoff than do the residents of the other town. Which of the equilibria identified in part (b) will be stable with these dynamics? Which ones will be unstable?