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



players, thus increasing the asymmetry among banks. Differently, in countries like Australia
and Germany, the weight of the largest banks hardly changed. So it may be possible that
the reserve channel may have dominated in those countries, thus leading to an improvement
in aggregate liquidity. A similar result may have occurred in Japan and the UK, where
-according to Table 1- consolidation has even led to a more symmetric banking system
and thus to a reversed functioning of the asymmetry channel.

Note that we primarily address the structural effects of bank mergers on loan compe-
tition, reserve holdings and aggregate liquidity, but in practice business cycles may affect
some of our variables. In particular, the relative cost of refinancing is affected by trading
conditions in the interbank market and the level of interest rates, and it may behave pro-
cyclically. This implies that bank mergers involving large banks may affect reserve holdings
and private liquidity more negatively in upturns than in downturns. Our results above have
therefore to be interpreted as the “average” (or structural) effects of mergers over time. It
would be interesting to extend our model in future research to explicitly cover macroeco-
nomic features and analyze in depth how reserve holdings and aggregate liquidity change
over the business cycle.

6 The Relationship between Competition and Aggregate Liq-
uidity

We now discuss more in detail how loan market competition and reserve choices interact
in determining loan rates and aggregate liquidity (for simplicity, here interpreted only as
expected aggregate liquidity needs), and how liquidity effects relate to competition effects.

At the individual bank level, the loan market equilibrium affects banks’ reserve holdings
(in absolute terms) by determining the amount of deposits required to finance loans, and
hence the size of liquidity demands at any given level of reserves. Equilibrium reserve
holdings determine banks’ financing costs —the sum of the expected cost of refinancing and
of the expected repayment to depositors—, and thereby influence the loan market equilibrium.
At the aggregate level, loan market competition affects the degree of asymmetry in banks’
balance sheets through the distribution of equilibrium loan market shares.

Table 2 summarizes the possible effects of mergers on both loan rates rL and expected
aggregate liquidity needs Ω, as described in Propositions 3, 5 and 6. The rows of the table

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