different component of the portfolio is not uncontroversial. This is not surprising, though, because
complementarities and economies of scope do not arise between the provision of deposit services
and loans. They arise between the two separate economic functions that banks fulfil: the provision
of payment services and financial intermediation. The empirical analysis is complicated by the fact
that revenues of one service are often confused with revenues or costs of the other and vice-versa.
We choose to describe the cost of servicing deposits and loans as linear in the quantity. We
assume that the bookkeeping transaction function implies a deterministic industrial cost that the
bank has to incur in order to provide bookkeeping services to depositors and borrowers. The cost of
checks clearing and other desk operation is in fact linked to the number of transactions made by the
customers, and we can for simplicity assume that they are proportional to the amount of deposits
and loans.4 In general, on there are no obvious reasons for the industrial costs to be convex. On
the contrary, they might be concave, because of the presence of some fixed costs. But since we
choose an infinite horizon problem, we can disregard the eventual relevance of fixed costs, which
in the banking sector, anyway, are not overwhelming. The large empirical literature regarding
the existence of scale economies in the banking system (economies that besides could be linked
to portfolio management function) has not reached undisputed conclusions.5 The simultaneous
survival of banks of different size in almost every country shows that economies of scale certainly
are not overwhelming.
The assumptions regarding the structure of costs linked with the provision of financial interme-
diation services are crucial. Banks are normally assumed to face two kinds of costs, default costs
and liquidity costs. Both liquidity and default costs are to be assumed as stochastic. They are in
fact essentially due to the uncertainty regarding shocks that may hit borrowers or depositors. The
first might cause the impossibility of borrowers to refund loans, the second may cause a bank run.
We choose to disregard the importance of liquidity costs, focusing just on default costs, because
their introduction would not change in a relevant way the analysis we want to develop.
The fundamental assumption of the model is that the default cost function is quadratic. This
assumption implies that the returns on the investment in information are decreasing. Banks cannot
increase direct lending at will without reducing the efficiency of their monitoring and screening
processes. Increasing direct lending indefinitely sooner or later they would finance investment
projects of decreasing quality, taking a higher risk without a proportional increase in the return.
They would end up not pricing risk properly. We assume that the default cost affects loans only.
This seems counterfactual since banks can hold corporate bonds as well as gilts. This does not
imply that there are no defaults on bonds, but that the market is efficient and prices risk correctly.
Any agent can take as much market risk as he desires at the market price for risk, as assumed
by the CAPM model. So banks can buy bonds without incurring in non-linear default costs
because they just buy market risk, and we assume that no bank is large enough to affect returns.
The decreasing returns are just with the banks own activity, the pricing of uncertain investments,
whose information is not common knowledge and has not been disclosed to the market. Investments
whose risk is virtually unknown so that the market can not price it.
It might be reasonable to assume a non linear cost function only if assets holding are allowed
to be negative, so that the bank can borrow issuing bonds in order to lend more. In this case the
non-linearity would imply that the cost of borrowing has to be increasing with the quantity. This
case can be studied with the same framework, but the results would not be radically different. In
order to simplify the notation we do not specify neither a linear default cost nor the transaction
cost of bonds. With no loss of generality we can define the returns on bonds as net of default and
transaction costs.
Formally:
∂C(Dt) > 0 ∂2CDt) = 0; ∂C(Lt) > 0 ∂2CL) =0.
(3)
∂Dt ∂Dt2 ∂Lt ∂Lt2
4A detailed study of the industrial costs of deposit is provided by Osborne [19], and our assumptions are com-
patible with it.
5 The most recent empirical evidence regarding the return to scale of banks is in Weelock and Wilson [29]. They
showed that after 1985 there is evidence of increasing returns to scale for small and medium size banks, while the
restriction of constant returns to scale could not be rejected for large banks. The finding of relevant return to scale
is probably due to the progressive deregulation of the banking sector.