B.3 Long-run equilibrium
Note that if n is log-concave, then n (n - 1) Γd is increasing is n, as well as
n (n - 1) Γw and therefore the markup pe - c - we is decreasing in n (see [2]).
As a consequence, πe = N(p nw—c) — (F + S) is decreasing at least in an
hyperbolic manner and there exists a unique free entry equilibrium given by:
1 [ μd + μw ʌ = (F + S)
n2 (n — 1) I R∞∞ h2 (x) Hn-2 (x) dx R∞∞ g2 (x) Gn-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 — ΛhPw)] NPw — (F + S).
The first-order condition is:
de<<w-w-<-p> = μ'dg/w ) — : Pw + r9,fWi) — Wi — c — 2Λhpiw] P = 0.
∂wi dwi i i dwi
Moreover, we have:
∂ 2ei(wi,w-i,p)
∂wi
μd2gi(wi)∖ Pw + 2 μdgi(wi)
∖ dw2 J i y dwi
—л dPw—2Λh μ dρw y
dwi dwi
+ rgi (wi) — wi — c — 2ΛhPiw ]
d2Piw
dwi2
We wish to show that any turning point is a maximum:
∂2πei (wi, wi, p)
Bw2
< 0.
FOC
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:
gi (wi) — wi — c — 2ΛhPiw
d dgi(wi)
y dwi
— 1 Piw
dP w
i
dwi
Using this expression, we obtain after simplifications:
31