We can express the last functions simply as:
C(Dt)=uDt C(Lt)=zLt,
(4)
where u and z are positive real numbers.
A stochastic default cost must be added to the industrial costs:
where
and with
D ( Lt ) = 1 vL2,
∂D ( Lt ) ∂2 D ( Lt )
∂Lt > 0 ∂Lt2 < 0,
v = Vd + ed with E [ ed ]=0 E [ e 2d ]= σ2.
(5)
(6)
(7)
The cost functions we have assumed are constant over time, because with time-varying coefficients
no closed-form solution can be obtained. Anyway the model could be extended to study this
more general case. For the purposes of this study this further complication was unnecessary. The
only important problem that could be obscured by our assumption regards the effect of expected
inflation on the problem of the bank. But if prices are not sticky and expectations are rational,
expected inflation is fully incorporated in all interest rates. And under these assumptions the price
of factors of production adjusts instantaneously to the change of other prices. This implies that
cost functions are homogeneous of degree one with respect to inflation. With our assumptions,
marginal costs coefficients and interest rates are proportionally shifted by variations of the price
level.
2.0.3 The demand for deposits and the dynamic constraint
Deposits are demanded not just as a financial asset for portfolio allocation, but mainly because
banks provide depositors with transaction services. The market for payment services has always
been highly competitive, since commercial banks where competing with issuing banks (only later
state-owned central banks) that provide those services by means of bank-notes. In order to get
remunerated for the payment services that they provide by means of checks, bookkeeping entries
and credit cards, banks charge fees on the transactions undertaken. On the contrary, transactions
by means of banknotes, whose technology is much simpler and cheaper, do not need the payment
of fees. As a consequence, commercial banks have to attract depositors offering an interest rate
that banknotes do not pay. The technological developments of the 20th century have reduced the
competitive pressure from banknotes, whose role has become smaller. But new competitors have
come out. At the beginning of the twentieth century savings institutions, which were developed
initially exclusively to provide financial intermediation services, have been allowed to provide pay-
ment services by means of the gyro. Only later they have been allowed to issue loans, becoming in
all respect analogous to commercial banks. More recent technological developments have allowed
money market mutual funds and other financial intermediaries to provide many of the payment
services that banks provide at a low cost. As a consequence the need to pay interest rates has
increased.
On the other hand the relevance of transaction costs, (search costs in particular) in the market
for deposits has been shown by Flannery [11] and Hess [15], and is quite uncontroversial. Deposits
are increasingly described as quasi-fixed inputs. Since search costs allow the firm to charge non-
competitive prices,6 in presence of search costs monopolistic competition becomes the normal
market structure. This suggests that each bank does not suffer a strong competitive pressure from
other banks and can price deposits monopolistically.
We can conclude that the need to pay an interest rate on deposits comes from the competition
of intermediaries different from banks. These new competitors can in fact offer interest rates no
too far from the rates that bonds pay. As a consequence we will assume that each bank can set
6See Salop [21] and Salop and Stiglitz [22] and [23].