Suppose that living in the countryside gives you a payoff of 1, while living in a city gives you a payoff that first increases with the number of people living in the city (agglomeration), and then declines after the number of people goes above a certain threshold. Let us write this payoff as

(5) Suppose that living in the countryside gives you a payoff of 1, while living in a city gives you a payoff that first increases with the number of people living in the city (agglomeration) and then declines after the number of people goes above a certain threshold. Let us write this payoff as r = An - n'/2, where n is the number of city dwellers. (a) If only one city can exist, trace out the set of equilibria as population size varies from ( to infinity. (b) What if one can indefinitely replicate cities, each with the same payoff function to living there as the one described above