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

reserves internally, which changes how they insure against liquidity risk. Second, this ‘in-
ternal money market’ gives them a financing cost advantage, whose size is endogenously
determined. Third, the merged banks may enjoy cost efficiencies that reduce their lending
costs to β, where β < 1. Fourth, they gain market power in setting loan rates. All these
factors affect banks’ equilibrium balance sheets and, in turn, the demand and supply of
liquidity. We begin with discussing how the merger modifies banks’ reserve holdings, and
then we turn to its effects on costs and loan market competition.

4.1 Internal Money Market and Choice of Reserves

We note first that the merger does not affect the optimal reserve-deposit ratio of the N - 2
competitors. As they have the same cost structure as in the status quo, they still choose
their reserve-deposit ratios according to (10), i.e., k_c = k_sq. This implies also that they have
the same per-unit financing costs r/r^lr^D as in the status quo (from Proposition 1).

By contrast, the merged banks, say bank 1 and bank 2, choose a different reserve-deposit
ratio, because the merger modifies the distribution of their liquidity shocks and also allows
them to pool their reserves to meet the total demand for liquidity. Thus, as long as the
two banks continue to raise deposits in two separate regions, the merger leaves room for
an internal money market in which they can reshuffle reserves according to their respective
needs. For simplicity, we assume a ‘perfect’ internal money market, so that exchanging
reserves internally involves no cost, but all qualitative results go through as long as the
internal money market is less costly than the interbank one. Proceeding in this way is
motivated by recent empirical research suggesting that internal capital markets function
relatively efficiently (see, e.g., Graham et al., 2002; Houston et al.,1997; and Campello,
2002).

Let x_m = δ1D1 + δ2D2 be the total demand for liquidity of the merged banks at date
1, R_m = R1 + R2 be their total reserves and D_m = D1 + D2 be their total deposits. The

combined profits of the merged banks are then given by

-r^D [D1(1 - E(δ1)) + D2(1 - E(δ2))] .

The first two terms in (11) represent the combined profits from the loan market, with β
reflecting potential efficiency gains in the form of reduced loan lending costs, the third term

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