Imperfect competition and congestion in the City

μw´² P(1 - P)(1 - 2P)
h¹+μw P (1—P )]³

Second, we wish to compute g⁰ and g⁰⁰. Recall that the solution of the equation
P_i^w = P_i^d is unique and denoted by: p_i = g_i (w_i). We have:

so that, using the same reasoning as above:

dpi _    -Td P^d (1 - Pd)

----- —--:—;---------------—

dpi 1 + ΛdPi^d (1 - P

Differentiation of the expression P_i^w - P_i^d —0, as a function of wi leads to

dPw dPi^d dpi _₀
dwi     dpi ^. dwi     ^,

thus (using again the condition P_i^w — P_i^d):

Therefore:

Γ                                                  "I 2 ^,

[1 + Λ^w Piw (1 - Piw)]

with

'" (1 - 2Pi^ dPd g⁰ ×
μ^{d κ}         d dpi

Λ^w

1 + PW P^w (1 - P^w )
^μ

Λ^w        dP^w

- — (1 - 2P^w ) dP- ×
μ^w           dwi

After simplification, we get:

1 ^μ Λ^d Λ^w \
μ^w ∖μ^d μ^w )

(1-2P)P(1-P)
[1 + Λ^w P (1 - P )]

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