6.4 Long run equilibria
For a fixed level of road capacity s, the free entry equilibrium with congestion
denoted by nf (s) solves πe =0. In order to study the free entry equilibrium, we
need to specify the fixed levy per firm S. As the default value, we use S = K .
This leads to a cubic equation, and its solution is not too illuminating. The
profit πe (s) is a decreasing function of the number of subcenters n. Given
that the equilibrium profit with congestion is larger than without, the free en-
try equilibrium with congestion involves more firms than without congestion:
nf (s) >nf.
We can find a lower bound (no (s) <nf (s)) and an upper bound for the
solution of (38). As lower bound, we use no (s) that is the solution of the
following equation:
(μd + μw ) N + δ μɪX — (F + K) = 0. (39)
no (s) s no (s)
We will show in the next section that no (s) is the optimal number of firms for
given road capacity and in the absence of congestion charging. Observe that
πe (no) > 0, no (s) <nf (s). So that equation 39 has a unique positive root:
no
n (s) = n + —

4δ
s (F + S)
+1
(40)
where no = (μd + μw)N/ (F + S), represents the optimum number of subcenters
without congestion (see equation (22)) provided that the firm pays the road
infrastructure cost (S = K).
As upper bound for nf (s) , we use no (s)+1.Wehave:
e o d w N δ αbN 2
π (n (s) + 1)= (μ + μ ) noη-)+ s (∖no (s) + J - (F + K ) .
Subtracting (39) from this equation, we get:
(no + 1) = δ (αN)2 I ----1---- -
< 0.
s (no +1)2 (no)2
As a consequence, nf (s) <no (s)+1.
Summarizing, for given road capacity, in the absence of road pricing and for
an infrastructure charge on firms S = K, we have an upper and lower bound for
the equilibrium number of subcenters where no (s) denotes the optimal number
of subcenters.
18
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