Imperfect competition and congestion in the City



P00


μw´2 P(1 - P)(1 - 2P)
h1+μw P (1P )]3

Second, we wish to compute g0 and g00. Recall that the solution of the equation
Piw = Pid is unique and denoted by: pi = gi (wi). We have:

Pkd


exp


-pk-ΛdPkd
μd


P exp          ' ,

1...n


so that, using the same reasoning as above:

dpi _    -Td Pd (1 - Pd)

----- —--:—;---------------—

dpi 1 + ΛdPid (1 - P

Differentiation of the expression Piw - Pid —0, as a function of wi leads to

dPw dPid dpi _0
dwi     dpi . dwi     ,

thus (using again the condition PiwPid):

g0


dPw
dwi
^dPd
dpi


.   1 + Λd Pid (1 - d'

μw 1 + — Piw (1 - Piw )


(60)


Therefore:

g00


μd
μw


Γ                                                  "I 2 ,

[1 + Λw Piw (1 - Piw)]

with

'" (1 - 2Pi^ dPd g0 ×
μd κ         d dpi

Λw

1 + PW Pw (1 - Pw )
μ

Λw        dPw

- — (1 - 2Pw ) dP- ×
μw           dwi

Λd

1 + τd Pd (1 - Pd)
μ


After simplification, we get:

1 μ Λd Λw \
μw μd μw )

(1-2P)P(1-P)
[1 + Λw P (1 - P )]

33




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