Monopolistic Pricing in the Banking Industry: a Dynamic Model

Since we know that X_{t +}ι = X_t + eD₊1 — eB₊1, we can write:

- ^γY^βκ(ɪ - ^q) ⁺ [ɪ - ^κ(ɪ - ^q)] χ + ^χt ⁺ ⅜1 - ⅜1

Yy^βI - κ⁽1 - q^{)]        t} ɪ - к⁽1 - q)

And, sinceE_t [e_t-₁] = 0:

^-Yy^βκ(1 - q) - I - ^κ(1 - q^)] + γγ^β _χ

Yyβ[ɪ - к(ɪ - q)]             ¹¹

It can be simplified as:

The other intercept term is - -γ~~₁E [ɪZ_t-ɪ]. We can finally write the value of D_t as:

E[Dt +ι] - [ɪ + γγ ] Dt + ɪDt-₁ =      -ɪX_t - ~^~E[ ɪ Z_t+11.

Lγγ β     J β          γγ β       γγ β La    J

Deposits

We can obtain the value of Dt from:

E[Dt +1] - [ɪ + γγ ] Dt + ɪDt-1 = ^γγβ-1 Xt - E [ ɪ Zt₊₁1.
Lγγ β     J β          γγ β       γγ β La    J

The solution is given by:

∞                        ∞ -Ixi

^Dt+j+1 = γγβ^{Dt+j +} βYY ∑ (YY) ^Et+i [   O¹ ] - βγγa(ɪ - Yy)^{C +}

- Y-¹ ∑ ( γY ) ^lE^t-+i [ Xt+j+i+1],

Under the assumption that interest rates follow a random walk process, we can for simplicity
assume that deterministic component of the rates remain constant and treat the values of the rates
at time t and t + ɪ that enter in the solution as constants. In this case the result is the following:

D^{t+j +1} = γγΓβ^Dt+j - βγ_γa(ɪ - γ_γ) [(ɪ - K^)(γγβ - ^H) ⁺ (ɪ - ^q)^κH][^rB+j+^{1 + COV} (^rB, ɪ)] ⁺

- βγγa(ɪ - γγ) ^K(^{rR+ j}q - ^{rD+ j}) ⁺ γ_γβ(1 - γγ) (g^3rD^+j - g^4rB^+j) - βγγa(ɪ - γγ) C ⁺

- j^γγβ - ɪ)K COV (L^d, ɪ ); (86)
βγY (ɪ - γY )           a

it can finally be simplified as:

ɪ         ^κ[(ɪ^- d)⁽γγ^{β- h) + (1 -} q)^kH]^{+ (}γγ^β- ɪ)^ag4^h _b

^{Dt +} ’ ⁺' = γγβ ^{Dt + j +}                        a T)                   ^r‘ ⁺ ’ ' ⁺

κ R     κ² + a(γγβ - ɪ)g3 d  _i4 ^(βγγ - ɪ) [^Z - ^a] - ^κ^}

~rrt q+ +

βγγa(γγ - ɪ)   ⁺ j      βγγa(γγ - ɪ)    ⁺ j         βγγa(γγ - ɪ)

+ d----K----U [(ɪ - ^k)(Yyβ - ɪ) + (ɪ - q)K] COV(r^B, ɪ) - ^(γγβ ɪ)K COV(L^d, ɪ ). (87)

βγγa(γγ - ɪ) L ∖ b/                     a β a/   βγγ(γγ - ɪ) ∖    a/

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