We consider the symmetric case: βi = β, i = 1..n (the centers are on
average equally attractive from the worker perspective), hk = h, k = 1..n (all
differentiated goods have the same gross benefit), and ti = t, i = 1...n (all
subcenter are equally far away). In this case, the conditional indirect utility (5)
reduces to:
Uik = ω + wi - pk + μwεi + μdεk , (8)
where:
Ω = - (αwt + β) + θ (1 - β) + h - αdt + (1/N ) X ∏l - T. (9)
i=1...n
Note that this model requires information on the distribution of the match
values (εi and εk). The precise value of the match value of a given household
is unknown. In other words, the individuals are statistically independent and
nothing changes in the model at the aggregate level if the match values were
to change over time. As a consequence, the households are allowed to modify
their employment choice and the shopping choices over time provided that this
will not change the expected demand addressed to each firm and the expected
number of workers hired by each firm.
2.4 Profits of firms
Recall that Di denotes the demand addressed to Firm i (with in=1 Di = N), wi
the wage offered by Firm i and pi the price charged by Firm i for one unit of the
differentiated good. In the symmetric case, the marginal cost of intermediate
inputs is c = c1 + αht, i = 1...n (in the non-symmetric case, it is ci = c1 + αhti,
i = 1...n), and the marginal production cost is c + wi. The profit of Firm i is:
πi (w,p)=(pi - wi - c) Di - (F + S) , (10)
where w = (w1, ..., wn) and p =(p1 , ...,pn) are the wage and the price vectors.
3Householdchoices
3.1 The labour market choices
When a household is choosing in which subcenter i to work, all the terms
(-Pk + μdεk) are identical and therefore do not affect their choice of employ-
ment. Given the choice of location k for shopping, the utility of working in i
becomes (see 8):
Ui|k = ¾ + wi + μw εi,
where Ωk = Ω - pk + μdεk.