Imperfect competition and congestion in the City

The probability that a worker chooses to commute to subcenter i is

P_i^w_|k =Prob U_i|_k ≥ Uj|_k ,j=1...n . Note that the choice probabilities are
independent of k and can be written as P_i^w , with

pw = pr ob {wi + μ^wεi ≥ wj + μ^wεj, j = 1...n} .

Using the fact that εi are double exponentially distributed:

^exp (wi)     .   ₁

------------γ---r-, i = 1...П.

P exp μj

-1 n         ×     '

Therefore, the choice probabilities for the labor market have a logit type. Note
that all the workers will select the job which offers the largest wage if the
heterogeneity parameter μ^w is zero. Otherwise, a worker may accept a reduced
wage in order to work for a firm which best fits his preferences. The average
expected number of workers in subcenter i is: NP_i^w .

3.2 Consumer choices

When a household is choosing in which subcenter k to shop, all the terms
(w_i + μ^wε_i) connected with the choice of employment are identical and therefore
do not affect their choice. In this case, we can rewrite the conditional utility of
shopping in k given the choice of workplace i as (see 8) :

U_k∣_i = ^{ω- p}k⁺ μd^εk^,

where Ωi = Ω + Wi + μ^wε⅛. The probability that a household located in the
center patronizes subcenter k is P_k^d_|i =Prob U_k|_i ≥ U_l|_i ,l=1...n . As before,
the choice probability P_k^d_|i is independent of the choice i and denoted by P_k^d.
We have Fjd = Pr ob {— p_k + μ^dε_k ≥ — pl + μ^dε_l, l = 1...n}. With the double
exponential distribution, we get:

3.3 Market clearing conditions

Recall that every household consumes one unit of the differentiated good and
that the production of every unit of the differentiated good requires one unit of
labour (provided by one household). Assuming that the labour market clears
(wages are flexible), the fraction of workers which decides to work at subcenter i
must be equal to the fraction of shoppers which patronize subcenter i, whatever

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