New York University - ECON 227        Urban Economics Spring 2013   Problem Set #3 Due Monday, Mar 11, 4pm Slip the Problem set under my office door, Econ Dept, 19 W 4th St, Rm 704 Do not put it into any boxes or mailboxes!   Assume the utility functions for two cities are identical and are given by  U=N - 0

Urban Economics Spring 2013

Problem Set #3

Due Monday, Mar 11, 4pm

Slip the Problem set under my office door, Econ Dept, 19 W 4^th St, Rm 704 Do not put it into any boxes or mailboxes!

1. Assume the utility functions for two cities are identical and are given by  U=N - 0.1*N², where N denotes the city’s population in million. 
   1. What is each city’s utility maximizing population?
   2. If each city had a population of 6.5 million people, how would these cities change their size? Assuming that the total population of 13 million cannot be changed, would there be a smaller and a larger city? Would there be three or more cities? Or would there be no change at all. Explain.  

1. 1. to find the utility-maximizing population calculate the derivative of U and set it equal to zero.

dU/dN=1-0.2N è N=5

1. 2. A population of 6.5m people is more the U-max population (downward sloping branch of U curve). Any point on the downward sloping part of the Ucurve is a stable equilibrium, i.e., leaving this point means losing. There would be no change.

2. Correct or false? Do not explain
   1. The K=6 principle of the Central Place Theory is reflected in the hexagonal market form (“hexa” in Greek means 6). FALSE
   2. In the K=3 principle of the Central Place Theory suggests that each central place serves exactly its own market plus 3 markets of adjacent cities.  FALSE (c) In the K=4 principle of the Central Place Theory suggests that each central place serves exactly its own market plus 3 markets of adjacent cities. CORRECT

3. Innovation and Growth:

Suppose a region’s workforce of 14 million is initially split equally between two cities, X and Y. The urban utility curve peaks at 4 million workers, and beyond that point the slope is constantly -$3 per million workers. The initial equilibrium utility is $60.  Suppose city X experiences technological innovation that shifts its utility curve upward by $12.

1. 1. Draw a pair of utility curves, one for X and one for Y, and label the positions immediately after the innovation (before any migration) as x for city X and y for city Y. Use arrows along the curves to indicate that migration that follows. Show the long-run equilibrium using x’ and y’ respectively.
   2. For the new equilibrium (after migration) calculate the utility and the population in each city.