Imperfect competition and congestion in the City

Ω2

Λ^w

1 + 2--P (1 - P)²

μ^w             .

(1+Л + ^λ ■ ^λ p (1 - P)

V   μ^wJ      μ^w

μ^d Λd Λ^w∖ ,
-—   _i--(1 — 2P )

μ^w ∖μ^d  μ^w )

P(1-P).

We can expand and regroup the terms to get:

Ω2 = -2 — P (1 - P)² - ^ʌwP (1 - P) - Λ + ^μdʌ

μ^w             (μw )²            к μ^wJ

- — P (1 - P)[1 + 2(1 - P)] - 2^{Л U}  + ^Л )p² (1 - P)³ .

μ^w            ’ ^{1 V} (           (μw)2          ^V ’

This shows, as required, that Ω² < 0 and therefore Ω¹ and Ω < 0. As a
consequence:

∂²πei (wi, w-i ,p)

∂wi

< 0.

FOC

Q.E.D.

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