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)^κκ 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)^κ¹NW + (1 — q) Xt + ^{1 (1}^q)^κ E ∖¹ Zt +11.

YY β YY β α

(77)

(78)

(79)

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

— 1 ^κ⁽¹^q ) E [ Dt +1∣ + ^γ^γ Dt +
κκ

¹^—⁽¹^— q )^κ¹^—^κ⁽¹^— q ) D _l

-----D--Dt ⁺

YY β κ

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 + ^βγ^γ + ⁽¹^— q)^κ^—¹NW +

Yγβ Yγβ κ Yγβ

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

(80)

It can be simplified as:

E[D_t₊₁] —

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

⁺ βD^t¹

—NW

Yγβ

ɪ Xt

Yγβ

βγγ — [(1 — q ) κ — 1] κ

Yy β 1 — κ (1 — q )

— κ (1 — q )

NW +

--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.

(82)

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 )^]^—^βYγ^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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