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

Denote as f(k_m) the LHS of (19) and as f(k_sq) the LHS of (21). Plotting f(k_m)and
f (k_sq) for k_sq and k_m between 0 and 1, we get Figure 4.

[FIGURE 4 ABOUT HERE]

The curves f (km) and f (ksq) cross only once at ksq = km = 8. Substituting this value in
(19) or (21) gives ksq = km when r⅛ = 694 ≡ ρ. Thus, km > ksq if r⅛∙ < ρ, and km < ksq
otherwise.                                                                         Q.E.D.

Proof of Corollary 2

From the last two terms in (8), we can express the financing costs of competitors as

r^I    L²      r^D

T(RC+L) ⁺ TR ⁺ Lc>∙

Using DRcc = k_c and Dc = 1 — k_c in (22) and rearranging terms, we obtain

rI (1 — k_c)² + r^D
2(1 — kc)

Analogously, from the last two terms in (11), using DRm = km and Dm = 1 - km, we obtain
the financing costs of the merged banks as

It is easy to check that when the merged banks set km at the level which is optimal for
competitors, the financing costs of the merged banks are always lower than the ones of the
competitors. A fortiori this must be true when they set km to minimize their financial costs.
Q.E.D.

Proof of Proposition 3

The merged banks choose r₁^L and r₂^L to maximize (11) while competitors choose r_i^L to
maximize (8) where the subscript i is now c. Define from the financing costs in Corollary 2
((23) and (24)) the total marginal costs of the competitors and the merged banks as

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