Imperfect competition and congestion in the City



Hence, using the expression above, we obtain:


00 _ μd μ Λd Λw (1 - 2P) P (1 - P)
g = -(μw2 '' - μwJ h1 + λwP(1 - P)i3.


The sign of g00 is ambiguous (and note that without congestion g00 =0). We are


now ready to sign the expression (59).

We first compute the expression -2 (P0)2 + PP00 . We have


-2 (P0)2 + PP"´ =--P2(1P)-----3

(      (μw)2 [1 + ΛWwP (1 - P)]


Λw

1 + 2--P (1 - P)2

μw


< 0.


Furthermore, replacing the expression for g0 and after simplifications, we get:


(1-g0)=


1 + £)        P (1 - P )

1 + Λw P (1 - P )


> 0.


A combination of the last two expressions leads to:


-2(P0)2 + PP00 (1-g0)


P2(1 - P)


г                           ι 4

w)2 [1 + ΛWWP (1 - P)]


Λw

1 + 2—P (1 - P)2
μw             .


(1+£) + (ΛW+Λd) p (1 - P)
V   
μw      μw

We are ready to compute the second term of (59). After substitution, we
obtain:

P0g00P =


μd
w )3


μ Λd Λw
Ud
- μw)


(1 - 2P) P3 (1 - P)2
h1 + Λw p (1 - p )]4


Note that Ω = Ω1 - h (P0)3 < Ω1 (since P0 > 0), with:

Ω1 = (-2 (P0)2 + PP00) (1 - g0) + P0g00P.

We wish to show that Ω2 < O with:

Ω1


P2 (1 - P )
w)2 [1 + ΛWWP (1 - P)]4

Using the two expressions derived above, we get:

34



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