Monopolistic Pricing in the Banking Industry: a Dynamic Model

s.t.

Lt + Ft + Rt = Dt + NW,                          (22)

Rt = qDt ,                                        (23)

L_t = a - br_t^L + dr_t^B + η_t ,                                (24)

and

Dt = γY Dt-1 + κLt + g3r_t^D - g4r_t^B .                         (25)

The discount factor β^t represents the temporal perspective of the bank: β^t = ɑ+^)_t, where we
have assumed that r^E, the return on equities, is the discount rate that the bank applies.¹³ Under
our assumption deposits are the state variable of the problem, while Ft is the control variable of
the bank.

From the Euler equations of the problem, defining ɑ^+²^ = α, the following difference equation
can be obtained:

1 - (1 - q)κ_c, _i ⁽¹ - q)^γYγ^{β -} [^{1 - (1 -} q)^κ]}_{rη l}

YY β   Ft ⁺ ---------~β---------Dt ⁺

+ ^βγγ    I -^q )^{κ 1} _ot, + (1 - q ) Xt + ¹ (1 - ^q )^κ E [¹ Zt ₊1

γ_Y β                                γ_Y β α

where:¹⁴

Zt+1 = [(^{1 -} b) ⁽Yy^{β -} L) ^{+ (1 -} q)^κL] ^rB+ι ^{- (}Yy^{β -} L)eL+₁

+^κ [rRq ^- rD ^{- u}] +(^βγγ ^{- 1})(z ^- b)^,               (²7)

Xt = g3r_t^D - g4r_t^B .                                        (28)

Equation (26) together with the original dynamic constraint, which we rewrite, form a system of
difference equations:

= ---^γγ----D_t-1--^K----F_t +--^K----NW +--¹----X_t

1 - κ (1 - q )         1 - κ (1 - q )      1 - κ (1 - q )        1 - κ (1 - q )

Stability conditions

It can be shown that the eigenvalues of the system are:

λ 1 = -ɪ- and λ 2 = γγ.                            (30)

βYγ

Necessary and sufficient condition for the two eigenvalues to be one smaller the other larger
than one are the following:

Y γγ > 1    and   βγγ < 1    or    1+⅛ < 1    or   gγ < rE

ɪ YY < 1   and   βγγ > 1   or   —Е > 1   or   rE < -gγ

This condition is necessary, but not sufficient, in order to assure the saddle path stability of the
system. Here gγ is the rate of growth of nominal GDP, and we have made the strong assumption
of risk neutrality for the investors.

The meaning of this condition is that the stability of the banking system depends on the
relationship between the rate of growth of the demand for deposits and the interest rate that the
bank uses as a discount factor. If the discount factor is smaller than the rate of growth, and the
second is positive, the dimensions of the bank tend to increase indefinitely. If the nominal rate of

¹³Assuming investors to be risk neutral.

¹⁴Defining eL = ∣η_t.

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