Imperfect competition and congestion in the City

Hence, using the expression above, we obtain:

00 _ μ^{d μ} Λ^d Λ^w ∖ (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(¹—P)-----3

(      (μ^w)² [1 + ΛW^wP (1 - P)]

Λ^w

1 + 2--P (1 - P)²

μ^w

< 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

(μ^w)² [1 + ΛWWP (1 - P)]

Λ^w

1 + 2—P (1 - P)²
μ^w             .

P⁰g⁰⁰P =

μ^d
(μ^w )³

^μ Λ^d Λ^w ∖
Ud^- μw)

(1 - 2P) P³ (1 - P)²
h¹ + Λ^{w p} (1 - ^p )]⁴

Ω¹