Imperfect competition and congestion in the City

Proposition 5 Assume fixed road capacity and no congestion charging. In
the long run, congestion increases the equilibrium and the optimal number of
subcenters. If the fixed levy on firms exactly covers the infrastructure cost per
subcenter, the equilibrium number of firms is never smaller than the optimal
number of firms and there is at most one subcenter too many.⁸

7.2 Optimum number of subcenters and optimum road
capacities without road pricing

Assume now that the government can also determine the size of the roads. We
assume a linear construction technology. In this case, we can solve for the
optimum capacity, for a fixed number of subcenters.

W (n,s) = Ψ - n (F + ξ²s) + (μ^d + μ^w) log (n) - (b)^{2 δ} N, (43)
N                    ns

where the infrastructure cost is K = ξ²s,whereξ² represents the unit infrastruc-
ture cost.⁹ Higher transport capacity has two impacts in the economy. First, it
decreases the transportation cost (since there is congestion) and second it takes
away resources.

For n fixed, the optimal capacity solves: ∂W (n, s) /∂s =0, and this leads
to:

In the optimum, the total construction cost per individual is independent of the
number of subcenters and is equal to:

CC = N ξ²s^o(n) = bξ√δ.

The congestion cost for optimised capacity is aξy∕δ; and therefore equal to the
construction cost per individual.

Proposition 6 When road capacities are optimally chosen, the construction
technology is linear and when there is no congestion charging, the average con-
gestion cost per individual is equal to the construction cost per individual ξs∕δ.The
total construction cost is independent of the number of subcenters.

We then have for the welfare level:

c (n) = W [n, s^o(n)] = Ψ - nF + (μ^d + μ^w) log (n) - 2αξ√δ.    (45)

⁸ Of course, as in the non-congested case, there exist an optimal level of tax S which
decentralizes the social optimum.

⁹ Basically this amount to assume that the cost to construct two lanes is twice the cost of
constructing one lane. In a more refined version of the model the cost would be concave.

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