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



To better understand the effects of bank consolidation, we develop a model that allows
us to investigate the joint impact of mergers on credit market competition, banks’ demand
for reserves and the functioning of the interbank market. Banks raise deposits at date
0, and invest in long-term loans to entrepreneurs and liquid short-term reserves. On the
loan market banks compete in prices and retain some market power through differentiation.
Reserves are needed to cope with the uncertainty about depositors’ time of withdrawals. As
in Klein (1971) and Diamond and Dybvig (1983), deposits are stochastic as a fraction of
them is withdrawn prematurely at date 1. If deposit withdrawals (also, demand for liquidity)
exceed a bank’s reserve holdings, the bank incurs a cost to obtain from the interbank market
the liquidity needed to satisfy depositors. Thus, a bank’s demand for reserves depends on
its uncertainty about deposit withdrawals and the relative cost of refinancing, i.e., the ratio
of the cost of borrowing on the interbank market in case of liquidity shortage to the cost of
raising more deposits and keeping more reserves initially. The interbank market redistributes
reserves from banks with excess reserves to banks with shortages. However, when there
is aggregate excess demand on the interbank market, the central bank must intervene to
provide the missing liquidity and smooth out fluctuations in the banking system. Aggregate
liquidity supply and central bank intervention can be thought of in terms of private versus
public liquidity, in the spirit of Holmstrom and Tirole (1988). In this sense, the risk of
aggregate illiquidity and the expected liquidity needs represent the intensity with which
central banks monitor and intervene in the interbank market.

A merger affects banks’ behavior with respect to both reserve management and loan
market competition. As regards the former, the merger modifies the uncertainty about
deposit withdrawals, and creates an internal money market where the merged banks can
reshuffle reserves. Thus, besides the typical diversification effect related to the pooling of
idiosyncratic liquidity shocks, the merger induces an internalization effect, which increases
ceteris paribus the marginal value of each unit of reserves that can now be used to meet
withdrawals at any of the two banks. The demand for reserves of the merged banks balances
these two effects. We find that the internalization effect is stronger when the relative cost
of refinancing is low, while the diversification effect dominates and banks reduce reserve
holdings when the relative cost of refinancing is high. The intuition behind this result hinges
on the relationship between the marginal value of one unit of reserves, the initial value of



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