Assume two urban clusters, X and Y, exhibit the same utility function Ux = Uy = 2N - 0.08N^2 , where N denotes population in million.

a) If both clusters have a population of Nx=Ny=20, what is their utility?

b) Now assume, cluster X is able to shift its utility curve up and now enjoys Ux = 2N - 0.07N^2 while Y remains on its old utility curve. Assuming no population change, what is X's new utility?

c) As a response of the shift of Ux, people from Y will now move to X. If one million people move - what is the resulting Ux and Uy? Are X and Y in equilibrium after the one million population change ?

d) How many people must move from Y to X in order to reach the spatial equilibrium? Be precise - down to two decimals. You can solve this either mathematically by solving a quadratic equation or with Excel tables, trying step by step. What is the resulting Ux and Uy