Imperfect competition and congestion in the City

_d PR-∞∞g(χ)g ⅛^j + χ) Q g (⅛W^k + x>

- μ-j=--------7------k^j-₇------<0- < 0. (54)

^μw PR-∞∞ h (x) h (p^j-pi + x) Q H (p^k-pi + x)dx
j≠i               ^{4 M}        k k≠i,j ^           ⁷

Note that, at the symmetric equilibrium:

dP^w        (n  1)  ^∞

----      =------- I g (x) G    (x) dx,i = 1...n.         (55)

dWi ISym      μ^w   ./ ∞                     ,

Therefore:

dr_i(w_i)           μ^d Γ^w

------ =---T
^dwi |Sym     μ^{w γ-}

where Γ^w = ^R_-^∞_∞ g² (x) G^n-2 (x) dx and Γ^- = ^R_-^∞_∞ h² (x) H^n-2 (x) dx.  The

profit function is now:

πei (wi ,w-i ,p)= [ri (wi) - wi - c] NP_i^w - (F + S) .          (57)

B.2 Short-run equilibrium

The best reply of subcenter i is (seeing w.l.o.g. N =1)

^dei(^wi,^w-i,P) _ d dri (wi) Λ P w _{( ( }}         ʌ dPw

dWi =V dWi  ^- 1) Pi ⁺ (^ri (^wi) ^{- w}i ^- c) dWi .

Thus, using (55) and (55) we get:

- μ~ ^rw + 1^ ¹ + (pe - We - c) (n ^- 1) Γ^w = 0         (58)

V^w Γ^-   J n   p           μ^{w                v 7}

Or:

i        i 1     μ^-d   μ^w∖

^pe = c^+we⁺ nτn-!) Vr-⁺ H.

Note that, for the double exponential distribution Γ^d = Γ^w =1 n² , then
we get the formula: (17), as expected. We have therefore proved the following
result:

Proposition 8 Consider a differentiated labour and product labour. with i.i.d.
preferences with density function h (.) for the product market and g (.) for the
labour market Then, there exists a unique symmetric Nash equilibrium in prices
and wages given by:

^μ________μ^-__.__μ^w________ʌ

\ R-∞∞ h2 (x) H n^-2 (x) dx   R∞∞ 92 (x) Gn^-2 (x) dx J

30

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