Monopolistic Pricing in the Banking Industry: a Dynamic Model

β^t {Ptq +(1 - q)(ω_t - z) - u - α [(1 - q)²Dt - (1 - q)Ft + (1 - q)NW] +

¹  ^ɪ  ^q) { - ωt + α [(1 - q)Dt - Ft + NW]

+ ^γYK~Et^t'{ - ^ωt+¹ + α [(1 - q)Dt+1 - Ft+ι + NW] + υt+` + z}} = 0.

Rearranging and dividing everything by β^t we obtain a difference equation for F_t and D_t. Since
X_t = E_t [X_t] we can include everything under the expectation, from which we will from now on
omit the time index, since all expectations are at time t.

E∣^γβαFt +ι = αFt + ^γβ(1 - q)αD_t+` - α(1 - q)Dt + [β^γα - α] NW +

+κ [ρtq + (1 - q)(ωt - z) - u] + [1 - к(1 - q)][ωt - υ - z] - γγβ[ωt+` - Ut+` - z] [.    (64)

The next step is to simplify the resulting expression and to substitute for the value of Dt in
Equation (64), the expression of the dynamic constraint that is obtained from Equation (58).

E∣yγβαFt+ι = αFt + γγβ(1 - q)α [ɪ - (^^γ- q)^ Dt - ɪ - ʃ- q)^ Ft + ` +

⁺ 1 - (Г- q)кNW ⁺ gTX(1 g-"q+¹ ] - ^α(1 - q)D^{t +} (^βγγα - ^α)NW ⁺

+κ [ρtq + (1 - q)(ωt - z) - u] + [1 - κ(1 - q)][ωt - Ut - z] - γγβ[ωt+` - Ut+` - z] ∣,    (65)

and

^E{^^γβ I - (^α- ,)κF^t + ' = ^αF ⁺ ^^{γβ: 1} - ^q)^α{ I - (^β q)κ D ⁺

⁺1 - β- q)к^{NW +} g^β-^-1 g-"O' } - ^α( 1 - ^q)^{Dt +} <^β^^γα - ^α>^{NW +}

+κ [Ptq + (1 - q)Ut - u] + (^γβ - L)[Ut + ` - ωt+`] + (β^^γ - 1)z∖.          (66)

Dividing both sides of the former equation for α, and introducing the lag operator L, we obtain:

We can simplify the value of the second intercept term as:

Zt+1 = к [p_tq + (1 - q)Ut - u] + (^γβ - L) ∖υ_t + ` - ωt +`] + (β^γ - 1)z =

⁽^γ^{β - l}) [^rB+' ^{- r}D+' ^- a ^- b^rB+' +  . ^- cf+ι] + ^{к [(r}R ^{- r}D)q + ^{(1 -} q)^(rB ^{- r}D) ^- u^] +

^{+ ( β}^γ ^{- 1})^z = { ^γ^β (^{1 -} b ) ⁺ [^{(1 - q} )^{к -} (^{1 -} b )] ^L} ^rB+' ⁺

^-(γγ^{β - l})^et+' ^{+ к} [qρt ^{- r}D ^{- u}] ^{+ (β}^γ ^{- 1}) (^{z -} b) ⁽⁶9)

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