2.1.4 The interest rate on loans
The interest rate on loans can easily be obtained substituting the solution (44) for the quantity of
loans in the demand condition. For simplicity we use the solution of (26), but the general result
could easily obtained following the same procedure:
rL+ j=bEt++j {drB+ j+
[(1 - d) (γYβ - L) + (1 - q) κL] rB+ j + 1
γY βα
+ G ∣,
(46)
where
G = a + ηt+j
YY βLt+j
ɪ NW +
γYβ
κ(rR+ j q
rD
rt+j
γY βα
) ( βYY
- +---
- 1)Z* - a] -
γY βα
κu
— +
(1 - τ) (yyβ - 1) + (1 - q)
γY βα
κ1
-ævp,-)
COV ( LD, 1 )};
βγY α
(47)
and L is the lag operator. When the rate on loans is set monopolistically, interest rates on loans
depend on contemporaneous and expected rates on bonds. Interest rates on bonds have a strong
impact on the rates on loans.
It is useful to show the result when the bank has no market power and the rate on loans is
taken as exogenous, in order to separate the effect of market power from the supply side effect due
to the presence of bonds in the portfolio. Considering the bank as representative of the sector and
aggregating we would obtain:
L
rt+j+1
1 - γYβv rL I (1 - q) κ rB + vG
γγβ rt+j + YYβ rt+j + vG ,
(48)
where
G' =--Ldjt+j
YYβ
ɪ NW +
γYβ
κ(rR+ j q
rD
rt+j
γYβv
∑u. + (^-sHκoov(rB,i) -
YY βv v
γγYβ-1 COV(LD.i)}. (49)
An interest rate shock would decrease the contemporaneous value of the rate on loans but increase
the value of next period. We can see that in general higher interest rates on bonds tend to increase
the rate on loans since the difference between the two periods rates on loans is positive. It is
important to observe that the impact is in this case quite small, indicating that the rate on loans is
much more sluggish than the rate on bonds. The default cost coefficient, v , reduces the dependence
on the lagged value of the interest rate, but has no influence on the coefficient of the rate on bonds.
The equilibrium level of the interest rate on deposits can be obtained in the same way, consid-
ering the equilibrium rate for which demand and supply are equated. It can easily be realised how
the rate on deposits is sub ject to contrasting demand and supply effects, whose net effect is not
obvious on a priori ground.
2.1.5 Cournot competition
Assuming the presence of different banks in the market and that each bank’s cost structure is
common knowledge, the model developed in the former sections can be structured as a Cournot
model. The problem of every individual bank would in this case include the market share as
an unknown of the problem, and it would take into account the result of the same optimisation
problem performed by the others banks. We would now have n firms facing their n maximization
problems, that include the problems of the competitors in the price setting equation. And each
individual firm’s problem would now include as an unknown the value of the market shares ψ = L-
Lj
and θ = D. The n equations would provide the optimal supply functions. The condition of
Dj
aggregation of the loan and deposits supply schedules provides the two extra equations that allow
closing the system:
where n is the number of firms.
L = Lj, D = Dj,
(50)
j=1
j=1
14