Imperfect competition and congestion in the City

B.3 Long-run equilibrium

Note that if n is log-concave, then n (n - 1) Γ^d is increasing is n, as well as
n (n - 1) Γ^w and therefore the markup p^e - c - w^e is decreasing in n (see [2]).
As a consequence, π^e = N(^p n^w—^c) — (F + S) is decreasing at least in an
hyperbolic manner and there exists a unique free entry equilibrium given by:

1    [         μ^d         ₊         μ^w         ʌ = (F + S)

n² (n — 1) I R∞∞ h² (x) H^n-2 (x) dx   R∞∞ g² (x) G^n-2 (x) dx j      N

C Proof of Proposition 4

The profit function is (where N is normalized to one, w.l.o.g.):
ei(wi,w-_i,p) = £gi(wi) — Wi — c — Λ^hPw)] NPw — (F + S).

The first-order condition is:

^de<<^w-^w-<-^p> = μ'd^g/^w ) — _: Pw + r₉,_fWi) — Wi — c — 2Λhp_iw^] P = 0.

∂wi             dwi          ^{i                              i}   dwi

Moreover, we have:

If this condition is satisfied everywhere, the profit function πei (wi, w-i, p) is
quasi-concave, and the candidate symmetric equilibrium is Nash.

Note that, the first-order condition equation can be rewritten as:

Using this expression, we obtain after simplifications:

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