Imperfect competition and congestion in the City



APPENDIXES

A Proof of Proposition 1

It suffices to show that the profit function is quasi-concave. Since there exists a
candidate equilibrium, quasi-concavity is sufficient to guarantee that this candi-
date equilibrium is Nash. We prove below that at any extremum, the function
is concave.

At any extremum, the first-order condition is satisfied:

1 d∏i

N dwi


- (μW + 1) + (fi(wi) - wi - c) (1 μ^ ) Pw.

The corresponding second-order condition is:

1 μw     d2πi


N Piw (1 - Piw)    '     =0

I dwi


-2 μ μw+1 )+(fi(wi) - wi - c) (1 - 2Pw) μw


But, using the fist order condition, we get:


1 μw     d2πi


NPiw (1 - Piw ) dw2d∏i =o
dwi


d dd   (μw + l´

-2 (μw>μ, (1 - 2Piw )


(μw+0 (-


2+


(1 - 2Pw ) ʌ

(1 - Piw ) Г


or


1 μw d2πi

NPwwdW2 I d∏i
I dwi


+0


<0


Therefore, any turning point, where dπei /dwi =0is such that it is a maximum.
As a consequence, the profit function is quasi-concave, and the symmetric can-
didate equilibrium is a Nash Equilibrium. Q.E.D.

B I.I.D. preferences

B.1 Computation of the candidate price equilibrium

Labour market choices

The choice probabilities of subcenter i is given as before by:

Piw = Pr ob ©wi + μwγi wj + μwYj, j = 1...n} .

28



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