Imperfect competition and congestion in the City

Vik = gik + ehk - βei - βθ,                        (3)

where gik represents the consumption of the homogenous good (whose marginal
utility is one), h_k is the direct utility of the consumption of one unit of the
differentiated good k, β_i is the disutility of labour in sub-center i and β is the
disutility of labour in the center.

We assume that households have an equal share of the total profit, _l=1...n π_l
and that the profit share is small. As a consequence, consumers take the profits
as given and the owner of a differentiated firm does not take into account the
impact of his pricing policy on his utility as a consumer or as a worker.² The
household budget constraint is

(wi - ^αwti) + θ + N χ X ^πι = (pk + ^αdtk) + gik + T.          (4)

l=1...n

According to identity (4), the revenue from supplying labour to subcenter i
minus the commuting cost plus the revenue from supplying labour to the center,
plus the share in total profits is equal to the cost of consumption, including
shopping cost, plus the cost of the homogeneous good plus the head tax. By
substitution of the budget constraint in (3), we get the indirect utility function:

Uik = (^wi - ^αwti) - ei + ^θ (1 - ^β) + hk - (^pk + ^αdtk^ + N X^ ^πl - T. (5)
l=1...n

To recognize the fact that the jobs in the differentiated industry are hetero-
geneous, we model the disutility of labour, β_i as a random variable:

βi = βi - μ^wεi,                               (6)

where μ^w > 0 is a scale parameter that measures employment heterogeneity
and εi are i.i.d. double exponentially distributed.³ The idiosyncratic terms εi
express the match values between the employments and the workers.

Similarly, the goods produced in the subcenter are differentiated from the
shoppers perspectives. We assume that:

hk = hk + μd^εk ^,                             (7)

where μ^d > 0 is a scale parameter and εi are i.i.d. double exponentially dis-
tributed.⁴

² This way we avoid one of the ma jor problems in general equilibrium with imperfect
competition. For a survey see [3]

³ The c.d.f. of the double exponential is F (x)=exp[- exp (-x)].

⁴ For symmetric distributions (such as for normal), this formulation is the same as h_k =
hk-μdεk. Later on, we use double exponential distribution which lead to the Logit model with
the specification (7). The specification hk = hk + μ^dεk, with double exponential distribution
leads to the reverse Logit, which is subtantiallly less tractable (see, [9]), and therefore not
considered here.

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