Question

1. Spatial Indifference There are two cities, San Francisco and Eugene, and 100 identical workers who can choose to live in either city. Suppose the indirect utility of living in either city is given by: V=W-R Where W, is the wage of living in either city j and R, is the rent in city j. Wages in Eugene are $200 and wages in San Francisco are $400. Rents in Eugene, Re are given by: R=30+LE, where Le is the number of workers in Eugene, and rents in San Francisco, Rs are given by: Rs=60+2Ls, where Ls is the number of workers in San Francisco. a. If half the workers live in San Francisco, and half the workers live in Eugene, what is the indirect utility of living in each city? b. Would this allocation, with half the workers in each city, be a spatial equilibrium? Why or why not?

Answer 1

Spatial indifference.

(a)

There are 100 identical workers. Half the workers live in San Francisco and Half the workers in Eugene.

So, 50 workers live in Eugene and 50 workers live in San Francisco.

City Eugene.

Wage rate in Eugene (We) = $200.

Rents in Eugene (Re) = 30 + Le
Rents in Eugene (Re) = 30 + 50.

Rents in Eugene (Re) = 80.

Indirect utility of living in Eugene (Ve) = We - Re

Ve = 200 - 80.

Ve = 120.

The indirect utility of living in Eugene is 120.

City San Francisco

Wage rate in San Francisco (Ws) = $400.

Rents in San Francisco (Rs) = 60 + 2Ls
Rents in San Francisco (Rs) = 60 + 2(50).

Rents in San Francisco (Rs) = 160.

Indirect utility of living in San Francisco (Vs) = Ws - Rs

Vs = 400 - 160

Vs = 240

The indirect utility of living in San Francisco is 240

(b)

The given allocation (i.e., 50 workers in Eugene and 50 workers in San Francisco) is not a spatial equilibrium allocation equilibrium because the indirect utility of living in both cities are not same.

The indirect utility of living in Eugene is 120.

The indirect utility of living in San Francisco is 240