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

where the third term, Rrⁱ rI(xi — Ri)f (xi)dxi = RRi rI(δiDi — Ri)f (δi)dδi, indicates that if
Di

a bank’s demand for liquidity, xi = δiDi, exceeds its reserves Ri , the bank incurs the cost
r^I on each unit of liquidity needed to satisfy depositors from the interbank market. Thus,
as in Klein (1971), reserves are kept for precautionary reasons. In choosing the amount of
reserves Ri at date 0, a bank trades off the cost of satisfying the expected liquidity needs
at date 1, ^R_R^D_iⁱ r^I (xi — Ri)f(xi)dxi, with the cost r^D of raising more deposits and keeping
more reserves at date 0. As a consequence, a bank’s demand for reserves depends on the
uncertainty about deposit withdrawals and on the costs incurred to borrow liquidity at date
1 and of keeping reserves initially. The more uncertain the date 1 demand for liquidity xi
and the more costly raising liquidity at date 1 (i.e., the higher r^I), the higher is the demand
for reserves at date 0.

The following proposition characterizes the symmetric equilibrium in the status quo.

Proposition 1 The symmetric status quo equilibrium is characterized as follows:

1. Each bank sets a loan rate rL_q =   Nl_-1 + csq, where csq = c + л/rIrD;

^γ( ~N^ )

2. It has a loan market share L_sq = l;

3. If rI > rD, it keeps reserves Rsq = ^/TD — 1^ Lsq, and raises deposits D_sq

The differentiation on the loan market implies that the loan rate r_s^L_q exceeds the total
marginal cost csq via the mark up  Nl_-1 . This decreases with both the number of banks N

^γ(N^ )

and the loan substitutability parameter γ , while it increases with the level of loan demand
l. The total marginal cost includes the loan lending cost c and the marginal financing
cost л/rIrD, i.e., the sum of the expected cost of refinancing and of raising deposits.

Equilibrium reserve holdings R_sq balance the marginal benefit of reducing the expected
cost of refinancing with the marginal cost of increasing deposits, as explained above, and
they are positive as long as r^I >r^D . We restrict our attention to this plausible case.
Reserves increase with the demand for loans L_sq and with the interbank refinancing cost r^I ,
while they decrease with the deposit rate r^D . The intuition is simple. When the demand for
loans L_sq is high, banks face ceteris paribus higher deposit withdrawals at date 1 and wish
to hold more reserves to satisfy them. Similarly, when the ratio rrD is high, banks prefer to

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