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

Figure 2: Timing of the model

3 The Status Quo

In this section we characterize the equilibrium when all banks are identical. We start with
noting two features of the model. First, bank runs never occur in this model. The illiquidity
of loans together with r^D > 0 guarantees that depositors withdraw prematurely only if
hit by liquidity shocks. Second, we assume that the loan market is sufficiently profitable
(differentiated) for banks to borrow in the deposit and interbank markets. So, we can
directly focus on the date 0 maximization problem.

With these considerations in mind, at date 0 each bank i chooses the loan rate r_i^L and
the reserves Ri so as to maximize the following expected profit, where for simplicity the
intertemporal discount factor is normalized to one:

Ri                              Di

Πi = (r_i^L - c)Li +     r^IL (Ri - xi)f (xi)dxi -     r^IB (xi - Ri)f (xi)dxi - r^DDi(1 - E(δi)).

0                                  Ri

(8)

The first term in (8) represents the profit from the loan market, the second term is the
expected revenue from interbank lending at date 1 when the bank is in excess of reserves,
the third term is the expected cost of refinancing at date 1 when the bank faces a shortage
of reserves, and the fourth term is the expected repayment to depositors leaving their funds
until date 2. Taken together, the last two terms represent bank i’s financing costs.

For expositional convenience and without loss of generality, we set r^IL = 0 and denote
r^IB simply as r^I . (No qualitative result depends on this simplification, which also captures
the stylized fact that the interbank market is relatively ‘passive’ in that banks do not keep
reserves to make profits, but only to protect themselves against liquidity shocks.) This
simplifies (8) as follows:

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