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 ) dw2∣d∏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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