The rate on bonds is assumed to be set exogenously, and the bank is assumed to be price taker.
Reserves are assumed to provide (eventually) a return equal to rtR. Since banks are normally
compelled to hold reserves in form of cash, or non-interest bearing deposits at the central bank,
the net return ρt = rtR - rtD on reserves is usually negative, since the return is zero, but the bank
has to bear the costs of the proportional share of deposits.
2.0.6 Monopolistic pricing
The profit function that results from the previous assumptions is the following:
∞1
π = ∑j βt [(rL - rD)Lt + υtFt + PtRt- 2vL2 - utDt - zLt^. (17)
t=0 2
Its logical structure is very simple: revenues come from the interest rates spreads, the costs that
must be detracted are the cost functions previously defined. The importance of search costs in the
market for deposits and the relevance of relationship lending in the market for loans create the
need of a monopolistic model. As a consequence, the bank takes into account the demand schedule
for deposits and loans in its maximization problem. The demand for loans is introduced in the
model in a standard way, obtaining an inverse demand function and substituting its value for the
value of the interest rate on loans.
1 a(Yt ) d
rL = - τ Lt + / + B + ⅛ (18)
b bb
The peculiarity of the model lays in the way the deposit demand schedule is introduced in the
model. Because of the presence of the quantity of deposits in two different periods of time, the
demand schedule introduces another unknown in the problem, which becomes a dynamic problem.
The information provided by this equation cannot be used to eliminate the interest rate on deposits,
because it must be used to solve for the two quantities. The bank has to choose: it can either get
rid of the rate on deposits and solve for the quantity of deposits in just one period, or solve for
both quantities and treat the dynamic of the interest rate as exogenous. The correct solution is
the second. Solving for the quantity of loans, from:
Dt = γY Dt-1 + g3rtD - g4rtB + κLt, (19)
the following can be obtained:
Lt = κ {Dt - YYDt-1 - g3rD + g4rB }• (20)
Substituting this function for Lt in the profit function, we can observe that the quadratic cost on
loans works as a quadratic adjustment cost on deposits. Our model becomes formally identical
to a standard dynamic model, but its structure is much simper than the structure of any other
dynamic models of banking.
Rather than solving the model for Lt , it is convenient to use the budget constraint differently,
forming a Lagrangian. It becomes possible in this way to solve the model for the other two variables,
obtaining as a the solution, both the optimal size of the portfolio, and the optimal composition of
the portfolio of assets. Making a further assumption, it is possible to obtain the rate on deposits
from the solution on the quantities, as it happens in the perfectly competitive models.
2.1 Intertemporal maximization
The firm maximizes its expected profits over an infinite horizon period. The budget constraint
is implemented by substitution. The problem of the banking firm, for every pair of positive real
numbers (v, u), can be expressed as:
M-x
{Ft,Dt}0∞
∏ = ∑βt [(rL - rD)Lt + UtFt + PtRt - 1 vL2 - UtDt - ZLt],
t=0 2
(21)