Imperfect competition and congestion in the City

	Capacity given LR equilibrium	Capacity given LR optimum	Optimum capacity LR optimum
No cong. no toll	n^f =1 +A	n^o = A	n^o = A
Cong. no toll	n^f <n^f (s) n(s) ≤ n^f (s) n^f (s) ≤ n^o(s) + 1	n^o < n(s)	no = A s(n) = ^αN √δ nξ
Cong. fine toll	n_c^f_p (s) <n^f (s)	n_c^o_p <ncp (s) ncp(s) < n(s)	n = A⁰ s(n) = ^αN √√^δ nξ 2

with

A = (μd+μw ) FNK
.A = (μ^d + μ^w ) F

The symmetric equilibria can be illustrated numerically with a simple example.
Consider a city center with 1 million inhabitants who all work 8h a day, consume
one unit of the differentiated good per day and work one hour per day for the
production of the differentiated good.

The price of non-differentiated labour is set to one and taken as numéraire.
The individuals make one shopping and one commuting trip every 10 days to
one of the subcenters that produces and sell the differentiated good. We assume
also that one standard truck delivers enough intermediate goods to produce 50
units of the differentiated good and that one truck has the same congestion effect
as 2 cars. By assumption, the fixed set-up cost per firm equals 75 000 units and
a subcentre also requires a fixed public input of 75 000 units. One unit of the
differentiated good requires an intermediate input that can be produced using
0.1 units of homogenous labour.¹⁰

All individuals spend half an hour in transportation per day if there is no
congestion (note that this includes the cost of renting the car). When there is
congestion we assume that, without tolls residents’ transport costs increase by
50% in the free entry equilibrium (without tolls, taxes and capacity expansion).
For δ ,the time cost parameter in the bottleneck model, we use the estimate of
[4] and choose a value of 0.2425 - this is to be understood as a travel time cost
and means that queueing and schedule delay costs are 24% of the wage. We
assume that the traffic is evenly distributed over three identical time periods
(morning peak, lunch peak and evening peak). Finally, we assume in the base
case that μ^d and μ^w both equal 0.2. Table 2 illustrates for these base case
values the number of subcenters (1) when there is no congestion, (2) when
there is congestion but no tolls, and (3) when there is congestion but first-best

¹⁰This means that each inhabitant works about half an hour per day in order to produce
and maintain the roads, the infrastructure in the subcenters and the production equipment.

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