Imperfect competition and congestion in the City

As in the no-congestion case, we need to compute the derivative dg(w_i) /dw_i ,
where pi = gi (wi) (see (14) in the non-congestion case). With congestion, the
probability that a consumer purchases good i is:

This equation reduces to (12), when the variable travel time is zero or in
the symmetric case (P_k^d =1/n). This is an implicit equation since the travel
time on route k depends on the total traffic on route k, which is an increasing
function of P_k^d (see equation (27)).

Since the travel costs depend on congestion, they cannot be assumed to be
symmetric. Indeed, when a firm deviates from a symmetric candidate equi-
librium, it will affect road use and travel costs. For example, a price cut in
subcenter i will increase the level of demand, labour supply and intermediate
inputs and therefore the level of congestion and the travel cost ti .

Using the implicit function theorem, we get:

dPd = - μd Pd (1 - Pd

^dPk     1 + Λd Pd (1 - Pd ,

Therefore, in the symmetric case P_i^d =1/n :

Note that the price sensitivity in the symmetric case decreases as the impact of
congestion measured by Λ^d (that contains α^d and α) gets larger. Congestion
decreases the incentive to cut prices, since a lower price implies more customers,
more workers and more intermediate deliveries and therefore more congestion,
which both reduce the benefit of the initial price cut. In fact the initial price
cut is compensated partially by congestion so that the firm is exchanging a
lower profit margin for more time losses rather than for more customers. With
an extremely high level of congestion (Λ^d →∞) the demand for one specific
variety is inelastic.

Similarly, for the labour market we have:

This expression reduces to (11) when the variable travel time is zero: when
there is no congestion. At a symmetric situation:

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