no(s) <nf (s) <no(s)+1; nf <nf (s) (41)
Therefore, the equilibrium number and the optimum number of subcenters
increases with congestion.
7 Optimum with congestion
7.1 Optimum number of subcenters with fixed road ca-
pacity and no congestion charging
The policy maker has three types of instruments available to improve the Nash
equilibrium. He can influence the number of subcenters, he can optimise the road
capacity and he can implement congestion charging. The three instruments are
to some extent substitutes and it may be necessary to use them all to achieve
the first best. We first analyse the gains we can achieve by regulating only the
number of subcenters. Next we add to this first instrument the optimisation of
road capacity. In a final section we discuss the use of congestion charging.
We have the same expression for the welfare function as in the case without
congestion (see equation (21). However, now the transportation costs are no
longer constant since they depend on the number of cars (and trucks) on the
roads and on the road capacities. Using the definition of transport costs, we
rewrite equation (21) in a way that separates the variable transport costs from
the other costs:
W (n,s) = Ψ - NN (F + K)+ (μd + μw¢ log (n) - (b)2 n N, (42)
where Ψ is given by(20)). We first consider the long-run equilibrium assuming
that the size of each road is determined administratively and not optimised (for
example, each road has two lanes inbound and two lanes outbound).
For the first-best optimal number of subcenters, we consider the first-order
condition dW (n, s) /dn =0. It leads to a unique maximum no (s) which solves:
(μd + μw¢ ɪ + δ μN} 2 - (F + K) = 0.
no(s) s no(s)
This is the same equation as (39). As a consequence, the optimal number of
subcenters corresponds to the lower bound proposed for the equilibrium number
of subcenters, so that there is excess entry: no (s) <nf (s). The optimal number
of subcenters increases with the level of congestion and no <no (s) where nois
the optimal number of subcenters in the absence of congestion. Note also that
f) < 0.
ds
Proposition 5 shows that, at the long run free entry equilibrium and at the
optimum, congestion induces more (and smaller) subcenters. Excessive entry
remains the norm but overentry still occurs and there is at most one subcenter
too many.
19