# Consider a small geographic region with a total population of 1 million people. There are two towns,

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?