because km >kc = ksq. Consider now the aggregate liquidity risk. When Dm = 2Dc, this
is given by
N
Φsq = prob
i=1
N
δiDsq > Rsq = prob(X0 < ksq)
i=1
in the status quo, and by
N
Φm = prob
i=1
N
δiDc >Rm + Rc = prob(X0
i=3
after the merger, where X0 = PN= NN. Since Km > ksq, it follows Φm < Φsq.
We can then express the expected aggregate liquidity needs in the status quo as
Ωsq = ∕,NDsq (Xsq — ksqNDsq)f (Xsq)d(Xsq) = NDsq (X0 - ksq)f (X0)d(X0).
ksq NDsq ksq
Applying the same logic, the post-merger expected aggregate liquidity needs are
Ωm = NDc [1 (X0 — Km)f(X0)d(X0)
Km
= NDsq (1 + (Km — ksq)) (X0 — Km)f(X0)d(X0),
Km
where we have used Dm =2Dc and Dm +(N — 2)Dc = NDc = NDsq +(Km — ksq)NDsq.
Given Km >ksq , we can write the expected aggregate liquidity needs as
Ωsq = NDsq
NDsq
1 (X0 — ksq)f(X0)d(X0)+ Km (X0 — ksq)f(X0)d(X0)
Km ksq
RK1m(X0 — Km)f(X0)d(X0)+(Km — ksq) RK1m f(X0)d(X0)+
RkKsqm (X0 — Km)f(X0)d(X0),
and, after rearranging and simplifying, we have
because (X — Km — 1) < 0. Analogous steps can be followed for the case rrD > ρ. Q.E.D.
Ωm ^-sq — NDsq
(Km
ksq) RK1m (X
Km 0
— ksq (Xsq —
— Km — 1)f(X0)d(X0)
Km)f(X0)d(X0)
<0
Proof of Proposition 5
Proposition 4 implies that if km = ksq, then Φm > Φsq and Ωm > Ωsq for any rrD > ρ. A
fortiori this must be true in equilibrium where km < ksq (Φm and Ωm are decreasing in Km,
which falls with km). Q.E.D.
37
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