Question

There are two cities, San F. 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_j =W_j -R_j

Where W_j is the wage of living in either city j and R_j is the rent in city j.

Wages in Eugene are $200 and wages in San F. are $400.

Rents in Eugene, R_e are given by: R_e = 30 + L_e, where Le is the number of workers in Eugene, and rents in San F., R_s are given by: R_s = 60 + 2L_s, where L_s is the number of workers in San F..

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?

c. In spatial equilibrium, how many workers will live in each city?

**Please answer I will rate!

Answer 1

(a) There are 100 identical workers who can choose to live in either city (i.e., San. F and Eugene)

Let us suppose half the workers live in San Francisco and Half the workers live in Eugene.

Wage in Eugene (We) = 200

Rent in Eugene (Re) = 30 + Le

Rent in Eugene (Re) = 30 + 50 (note: Le = 50, half the workers live in Eugene)

Rent in Eugene (Re) = 80.

Indirect Utility (Ve) = We - Re

Indirect Utility (Ve) = 200 - 80

Indirect Utility (Ve) = 120

Hence indirect utility in Eugene is 120.

Wage in San Francisco (Ws) = 400

Rent in San Francisco (Rs) = 60 + 2Ls

Rent in San Francisco (Rs) = 60 + (2) (50) (note: Ls = 50, half the workers live in San Francisco)

Rent in San Francisco (Rs) = 160.

Indirect Utility (Vs) = Ws - Rs

Indirect Utility (Vs) = 400 - 160

Indirect Utility (Vs) = 240

Hence indirect utility in San Francisco is 240

(b) The allocation of half half workers in each city (i.e., 50 workers in Eugene and 50 worksrs in San Francisco) is not a spatial equilibrium because the indirect utility for both the cities are not equal.

(c) For Spatial equilibrium, the indirect utility in both cities must be the same.

Indirect Utility in San Francisco (Vs) = Indirect utility in Eugene (Ve)

Vs = Ve

Ws - Rs = We - Re

400 - (60 + 2 Ls) = 200 - (30 + Le)

400 - 60 - 2Ls = 200 - 30 - Le

340 - 2Ls = 170 - Le

Note: Le + Ls = 100

Le = 100 - Ls

So, 340 - 2Ls = 170 - (100 - Ls)

340 - 2Ls = 170 - 100 + Ls

340 - 2Ls = 70 + Ls

340 - 70 = Ls + 2Ls

270 = 3Ls

Ls = 270 / 3

Ls = 90

and, Le = 100 - Ls

Le = 100 - 90

Le = 10

For spatial equilibrium, 90 workers should live in San Francisco and 10 workers should live in Eugene.