Imperfect competition and congestion in the City



dPiw 2i(wi,w-i,p)
dwi      ∂wi2


1-


FOC


( 2 (⅛)'


+ Piw


d2Piw
dw2


dgi(wi)A + dPjw d d2gi(wi)Pw
dwi J    dwi I dw2 i


- h


We wish to show that this expression Ω is negative given that dPW / dwi > 0.
To to that, when there is no ambiguity, in order to simplify expressions, we use
the following notations:


PPiw = Pid


P о W
dwi

P00 d2PW
dw2
i
0 _ dgi(wi)
g      dwi

00 _ d2gi(wi)
g ≡  dw2


Using these notations, we have equivalently:


Ω = (-2 (P0)2 + PP"ɔ (1 - g0) + P0 (g00P - h (P0)2} .       (59)

We now need to compute P0, P00, g0 and g00 at any point (i.e. not only at the
symmetric candidate equilibrium).

First let compute P0 and P00. Recall that


P = Piw


exp


∕wi -Λw pw

  μw


/Wj∙ -ΛwPjwλ
expp( —μwj)
1...n


We have, using again the implicit function theorem


μW p (1 p )

1 + λw P (1 - P ).

Note that:


∂P

Pо      ∂wi

P = i _ ∂P

1   ∂Pw


[P (1 - P)]0 = (1 - 2P) P0


Therefore, after simplifications, we get:


32




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