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

The following proposition describes the post-merger equilibrium with symmetric behav-
ior within the ‘coalition’ (merger) and among competitors.

Proposition 3 The post-merger equilibrium with r₁^L = r₂^L = r_m^L and r_i^L = r_c^L for i =3,...,N
is characterized as follows:

Each merged bank sets a loan rate rm = (^2——1) 2γ + (NN-—cc + (^—+1)cm, and each

LonTtirNNf^-1' Lf^,∕s L~ⁱ — N —■—¹I — -I- (ʌ——с -I- --г* '

cumpeutui sets I c — I — _-2 I γ +  — cc + — cm ;

The merged banks have a total loan market share L_m — (2—- ¹ ¢ l+γ(^— 1—(2^{— 2}) (cc-c_m),
and each competitor has L_c — ———-)2) l — γ⁽——^—— (c_c — c_m);

3. The merged banks raise total deposits D_m — ₁ ¹, L_m, and each competitor raises
1—k_m

D = ¹ L ∙

D^c    1-kc L^c ;

where cm, cc are the total marginal costs of the merged banks and of the competitors,
and km and kc are their respective optimal reserve-deposit ratios.⁹

Since banks compete in strategic complements, the merged banks drive the loan rate
movements in the market and the competitors move in the same direction. The effect of a
merger on loan rates depends on the relative strength of a market power effect and a cost
efficiency effect.Thefirst refers to the higher market power banks enjoy after a merger
because of the lower number of active banks (which reduces from N to N — 1). The second
derives from the lower total marginal costs cm that the merged banks enjoy relative to
competitors. Post-merger equilibrium loan rates increase when the merger induces small
cost advantages relative to the increase in market power, whereas they decrease otherwise.

Loan market shares across banks change in line with loan rates. As the merged banks
change their loan rates by more than competitors, their total loan market share shrinks
when loan rates increase and it expands otherwise, i.e., Lm < 2Lsq < 2Lc when r_m^L > r_c^L,
and Lm > 2Lsq > 2Lc otherwise.

The modification of loan market shares together with the change in the optimal reserve-
deposit ratio described in Proposition 2 determines the effects on the size of banks’ balance
sheets (as measured by the amount of deposits). In the present set-up the merger breaks

⁹ The expressions for cm, cc are in the proof of this proposition; those for km and kc are, respectively, in
the proof of Proposition 2 and in equation (10).

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