Monopolistic Pricing in the Banking Industry: a Dynamic Model

growth is negative, the bank does not tend to disappear only if the expected return on equities is
negative too and larger in absolute value than the rate of growth. The rate of growth of income
is the fundamental variable for the problem of the bank as long as the demand for deposits is a
function of nominal income. For simplicity we have assumed that deposits change as income, but
the conclusions would hold anyway.

If markets are efficient and shareholders are for simplicity considered to be risk-neutral, the
standard assumption is that the bank uses the expected return on equities as a discount rate. As
long as wages and rents do not adjust instantaneously this condition should normally be guaranteed.
In this case in fact profits are more volatile than income. And as a consequence the expected return
on equities should always be in absolute terms larger than the expected rate of growth of income.

2.1.1 General solution

The Rational Expectations Equilibrium of the system can be obtained substituting one of the
equations in the other. Substituting the first in the second we obtain an equation for the stock of
bonds:¹⁵

E[Ft+ι] - [ɪ + γγ] Ft + 1 Ft-1 = [^{1  YY}] ^βγγY^{f  1} NW +

γYβ              β                  γYβ

Following the same procedure, we can write the value of Dt as:

E[Dt +1] - [ɪ + Yy ] Dt + 1 Dt-1 =       Xt - -EE[ 1 Zt₊₁] .          (32)

γYγ          γ        γYγ      γYγ  α

2.1.2 The portfolio of bonds

Using the expectation lag operator H, such that H^-jEs-1xs = Es-1 xs+j, the left hand side of the
equation can be expressed as:

E[Ft+ι] - E[ɪ + γγ]Ft + 1 Ft-1 = (1 - λιH)(1 - λ2H)E[Ft+1].            (33)

YYγ         γ

Where λ1 and λ2 are the roots of the system. We already know their values, but they could have
easily been obtained realizing that the right hand side can be rewritten as

¹⁵ As shown in the appendix.

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