Monopolistic Pricing in the Banking Industry: a Dynamic Model

Dt =---—----Dt- 1--^K----Ft +--^K----NW +--Xt----.

1 — κ (1 — q )         1 — κ (1 — q )      1 — κ (1 — q )        1 — κ (1 — q )

The second equation can be exposed as:

Ft = — Dt-₁ — ^{1 — K (1 —} q ) Dt + NW + Xt.
κκ   κ

YY- Dt — ^{1 — κ (1 — q} ) E[Dt +1] + NW + E^[X^t+1] = ^{1 — (1} - ^q)^κ
κ         κ                     κ        γY β

∖ YY D₁-1 — ^{1 — κ}<^{1 — q}) Dt + NW + Xt 1 + W   ∙, ) I 2 ^β I . I ^κ I Dt +

L κ t ¹ κ t ^{+     +} κ J ⁺             YY β              t ⁺

+ ^βγ_γ ^{+ (1} q)^{κ 1}NW + (1 — q) Xt + ^{1  (1 q})^κ E ∖¹ Zt +11.

YY β                      YY β     α

Putting all the terms in D on the left-hand side, we obtain the following expression:

₁₇ ^{(1 —} q) [yY^{β — [1 — (1 —} q)^κ]] _n 1 — (1 — q) κγγ

^—E----------■----------D^t--γγγβ~-D ¹ -

= ^{1 — (1 —} q)^κ NW + ^{1 — (1 —} q)^κ Xt + ^βγγ + ^{(1 —} q)^{κ — 1}NW +

Yγβ            Yγβ    κ         Yγβ

—NW + (1 — q) Xt — E^[Xt^+1] + ^{1 — (1} ~ q)^κ E ∖¹ Zt +1].
κ        Yγ β     α

It can be simplified as:

Yy    _{d —} 1 — (1 — q)κ D + ^{(1 —} q) ^γY^{β — [1 — (1 —} q)^κ]]      κ D +

1 — κ (1 — q ) t       γγ β      t               γγ β             1 — κ (1 — q ) t

--K----7 (1 — q)Xt +-----1----?E[Xt ₊₁] — ——E ∖¹ Zt₊₁]. (81)

1 — κ(1 — q)^{(    q})  t ⁺1 — κ(1 — q)  [ t^+1]   y_yβ  La  t⁺1J  ⁽)

The left-hand side of the equation has exactly the same structure as the one in Ft and clearly the
simplification is identical. We will now study the different parts of the right-end side, starting with
net worth.

—K_nw _ ^βι^γ - [(¹ - ^q)^κ - ^1]---^κ---NW +---^κ---NW.

Yy β               Yy β        1 — κ (1 — q )       1 — κ (1 — q )

Under a common denominator we obtain the following result:

^{—K [1 — κ (1 —} q )^{] — β}Yγ^κ + ^{[(1 —} q )^{κ — 1] κ} + Yy ^βκ     ₌ 0

YY β [1 — K (1 — q )]                           .

As a consequence the impact of net worth on deposits is null, as we would intuitively expect. The
direct effect of interest rates is due to:

1K                             1

^— YYβXt ^— 1 — K(1 — q) ^{(1 —} q)Xt ⁺ 1 — K(1 — q) E[Xt^+1].

Or:

YY^{βκ(1 —} q) ⁺ [^{1 — κ(1 —} q)^] X +      ¹ _x

Yy ^{β [1 — κ (1 —} q )^]        1   ^{1 — κ (1 —} q ) *⁺¹

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