180 160 LA SF 140 120 100 Utility 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -LA 32 44 56 68 80 92 104 116 128 140 128 116 104 92 80 68 56 44 32 20 8 SF 50 70 90 110 130 150 170 150 130 110 90 70 50 30 10 Population in millions a. CA has a total population of 18 million, and only has two cities SF and LA. For each of the three initial equilibrium population allocations listed below: (PSF = 5, PLA=13), (PSF = 9, PLA=9), (PSF = 12, PLA=6 ) (i) Determine if each allocation is an equilibrium or not? (ii) If an equilibrium, determine whether stable or unstable. (iii) If not an equilibrium (and everyone is free to move, i.e. no costs of moving), what would the population levels be in equilibrium, in each city? b. If you were a benevolent urban planner working for CA, and were asked to calculate the optimum highest utility) stable equilibrium distribution of population across SF and LA, how would you distribute the total population of 18M across the two cities? Eye-balling is acceptable if you explain very clearly. [Conjecture on stability (which side of the urban curves to look for) carefully before you start calculating, review CS5 appendix Slides] Additional questions [Independent from parts a and b above, i.e. no total population restrictions]: c. How many equilibria distribution of populations (across SF and LA) can you find where the utility levels are 100 in both in SF and LA? What are the population levels in each of your equilibria? d. How many of the equilibria you found are stable vs unstable? Explain