CCcp = NN ξ2sθ°p(n} = bξ↑H.
Therefore the self-financing result of Proposition 6 holds also with optimal
road pricing. Road pricing reduces the total construction costs by a factor vz2
compared to the no toll equilibrium.
8 Summary and a numerical illustration
We start by summarising the results obtained so far. Because the total demand
for the differentiated good is fixed, only two parameters matter for the welfare
analysis: the number of firms and the total transport costs. The total number of
firms depends on the profit margin of the firms in the Nash equilibrium. When
there is no congestion, in the equilibrium there is always one subcenter too many
(see first line in Table 1). The equilibrium and optimum numbers of subcenters
are always (increasing) linear functions of the same parameters A and A : A =
(μd + μ^ N (F + K)/ and A = (μd + μw¢ N/ F). More heterogeneity (on the
product or labour market) leads to a higher optimal number of subcenters.
Higher fixed production costs, lead to a lower optimum number of subcenters.
When the road size cannot be optimized, the public infrastructure cost also
points to a lower optimal number of subcenters.
When capacity is not infinite and congestion may occur, we need to distin-
guish the case with or without road capacity optimisation and with or without
optimal road tolling. We discuss first the case with given road capacity (columns
1 and 2 in Table 1). Without tolling and given road capacity, the short-run profit
margin is always larger in the presence of congestion so that the free-entry equi-
librium always entails more subcenters than in the situation without congestion
(see second line in Table 1). The free entry equilibrium with congestion has at
most one subcenter too many. Optimum congestion pricing can reduce but not
eliminate the additional profit margins due to congestion. This explains that
in equilibrium and with road capacity given, the equilibrium number of firms is
highest if there is no congestion pricing (see first column in Table 1).
Any number of subcenters can be implemented by choosing the right fixed
levy per firm. For the free-entry equilibrium computed in Table 1, we have
assumed that the fixed levy equals the infrastructure costs per firm (firms are
then responsable for the construction of the infrastructure). As can be seen in
Table 1, we need a fixed levy per firm higher than the infrastructure costs to
obtain the optimum number of subcenters. When the planner can optimally
choose the road capacity, she compares the welfare cost of congestion with the
marginal cost of capacity expansion. Without congestion tolling, the benefit of
road expansion will be larger than with road pricing. Indeed, in the case of fine
tolls, the optimum road capacity will be smaller by a factor 1/ √2.
TABLE 1: Long-Run optimum and equilibrium number of firms in the sym-
metric case under different congestion and policy assumptions
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