DC DC N
Var (Xm) = -mVar(δι) + 4'Var(δm) + ∑-c2Var(δi)
4 4 i=3
DC
Var(δi) -m + ∑Dc2
because Var(δ1) = Var(δ2) = Var(δi).
tains PiC=1
Dm + PN=3 -C] > P
i=3
Since Dm + PiN=3 Dc = PiN=1 Dsq, one ob-
iN=1 Ds2q by Lagrangian maximization. Hence, it is al-
ways V ar (Xm) >Var(Xsq). Since f(Xsq) and f(Xm) are well behaved (they approach
a normal distribution), they intersect only in two points.11 This, along with the sym-
metry of the two density functions around the same mean E[Xm] = E[Xsq] = Dtot and
Var(Xm ) >Var(Xsq ), implies
Φsq = Pr(Xsq > Rtot) > Φm = Pr(Xm > Rtot) for any Rtot < -tot,
and vice versa for Rtot > DCot. Using Proposition 1, Rtot = NRsq, and (1), we obtain that
Rtot < D2ot if rD < 4 ≡ σ. The first statement follows.
Using the definition in (7), we have
Ωm - --sq
Dtot
tot (Xm
Dtot
Rtot)fm (Xm)d(Xm) - (Xsq
Rtot
Rtot)fsq(Xsq)d(Xsq)
DtotXmfm(Xm)d(Xm) - DtotXsqfsq(Xsq)d(Xsq)
Rtot Rtot
-Rtot(1
Fm(Rtot)) +Rtot(1
Fsq(Rtot)).
Deriving it with respect to Rtot gives
d(Ωm
Ωsq )
dRtot
-Rtotfm (Rtot) + Rtotfsq(Rtot) - (1
Fm(Rtot))
+Rtotfm (Rtot) + (1 - Fsq (Rtot))
Fm(Rtot) - Fsq (Rtot).
Rtotfsq(Rtot)
As showed earlier, Fm(Rtot) - Fsq (Rtot ) > 0 for Rtot < Dot and Fm(Rtot)
for Rtot > DCot. Also, Fm(0) = Fsq(0) = 0 and Fm(Rtot) = Fsq(Rtot) = 0. This implies
Fsq(Rtot) < 0
Ωm — Ωsq > 0 for all Rtot ∈ [0, Dtot]. The second statement follows.
Q.E.D.
Proof of Lemma 3
Suppose first -r-D < ρ. In this range, the aggregate reserve/deposit ratio in the status quo
(which coincides with the individual banks’ deposit ratio) is smaller than the one after
merger; i.e.,
k = Rsq = P=ι Rsq < K_
ksq Dsq NDsq <Km
11A formal proof that this is the case is in Manzanares (2002).
36
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