Imperfect competition and congestion in the City

intermediate input from the center to the subcenters. The total usage on road
i, expressed in car equivalent, is:

ρi = N [(κα^h + α^d) Pd + α^wp^w] .                (25)

where κ represents the car equivalent of a truck carrying the homogenous good.
The α⁰ s can also to some extent take into account the difference in timing of
the three types of road use.

We assume that the relation between travel cost, t_i , and total activity on
the road i, ρ_i , is given by:

ti = t + δ^ρi
s

where s is the capacity of the road measured in car equivalent (for the time
period considered in the model), and δ is a coefficient that translates waiting
time and schedule delay costs in equivalent queueing time. The first term rep-
resents the transport time in the absence of any congestion. The second term
in (26) represents the variable travel cost. This expression is the reduced form
of the bottleneck model [4] where road users decide on their trip timing (with
no congestion pricing).

Recall the market clearing condition (13): P_i^d = P_i^w . Equation (26) reduces
to:

NN

t_i = t + δ—αP_i^d = t + δ—αPw,

where α ≡ κα^h + α^d + α^w . In the symmetric case, P_i^w = P_i^d =1/n and the
travel cost, denoted by t^e , is the same on all routes:

te = t + δ N α.

ns

6.2 Demand for goods and supply of labour

With congestion, the indirect utility of a consumer working at i and consuming
at k is Uik = Ωik + wi — pk + μ^wεi + μ^dεk, where:

⅛k = ^{( —α}w^tk ^{- β}) ^{+ θ (1 - β}) ^{+ h - α}d^ti ^{+ (1} /N ) X X ^πl ^- T.

i=1...n

Using the same notation as in the non-congested case, this expression can we
written as:

Uik = Ω — Λ^w P^w — Λ^dP^d + Wi — Pk + μ^w εi + μ^dε_fc,         (29)

where Ω is given by equation (9), Λ^w = α^wδNα and Λ^d = α^dδNα.

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