Mean Variance Optimization of Non-Linear Systems and Worst-case Analysis



{  ,  . rI(3-6km+4k3)+3rD f I D D

βc + -i6(i-kβm— for rI 3rD

(26)

βc + 4rI 6       3         for rI > 3rD,

6(1-km)

respectively. Using the expressions for km and kc in (20) and (21), those for cc and cm in
(25) and (26), D
m = Rm + L1 + L2 and Dc = Rc + Lc, we can write the expected profits
for the merged banks and competitors when reserves are chosen optimally as

Πm = r1LL1 + r2LL2 - cm(L1 + L2)

Πc = (r1L - cc)Lc,

where

Lm = L1 + L2 =


l-γ


r1L


1N

-1 X

N ʌ

j=1

+ L-γ


1N

-1 X rL
j
j=1


(27)


and Lc is given by (2). The first order conditions are then given by


■               ∂LL1 ɪ L L ^L2_

l L = Lh + (r1 - cm) ^TΓL + (r2 - cm) ^TΓL = 0 for h = 1, 2

∂rh                   ∂rh             ∂rh


(28)


—c = Lc + (riL cc)    = 0 for i = 3...N.

∂rL     c i c ∂rL                ... .


(29)


We look at the post-merger equilibrium where r1L = r2L = rmL and riL = rcL . Substituting


(27) in (28) and (2) in (29), we obtain the best response functions as


L l cm rL
rL =---.r „ + — + -c-
m   2γ( nN2 )    2    2


(30)


L l N1      2 L

rc = γ( N+1 )+( N + 1)cc + N + 1 rm.                    (31)


Solving (30) and (31) gives the post-merger equilibrium loan rates rmL and rcL . Substituting
r
mL and rcL respectively in (27) and in (2) gives the equilibrium Lm and Lc. Analogously, we
derive D
m and Dc.                                                          Q.E.D.


Proof of Corollary 3


33




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