Monopolistic Pricing in the Banking Industry: a Dynamic Model

Appendix

Appendix: solution of the model with contemporaneous feedback

The Lagrangian of the problem is the following:

t = ^ ^{β t} Il - - [(¹ - q ) Dt ^- Ft I NW ] I--ɪtɪ I -rB ^{- r}D I EL }[(^{1 -} q ) Dt ^{- F}t ^{+ nw} ] +

t=0

- 2^v [(1 - q)Dt - Ft + NW]² - uDt - z [(1 - q)Dt - Ft + NW] + VtFt + PtqDt +

-μt{Dt - γγDt-1 - к [(1 - q)Dt - Ft + NW] - g3rD + g4rB }}. (55)

The first order conditions are:

——— = ^βt+jIlV⁺~)∣^(1^-q)Dt+j^-Ft+j ^{+ nw} 1--( ^{t+j -}r-rj j ^{+ r}D₊ j⁺z^-eL₊ j ⁺ Vt+j^-κμt+j} = 0,

UFt + j           к ∖ O b L                              J O O                                          J

(56)

∂t

U

= ^βt+jEt+j [^{(1 - q}) [ ( O⁺' ⁺ b^rB+ j ^{- r}D+ j ⁺ eL+ j 1 ^- ∣^{v +} O)[^{(1 - q})²D_t+j ⁺

- (1 - q) Ft+j + (1 - q) NW] - u - (1 - q) z + ρ_t+j q +
^- [l ^- к^{(1 -} qH μt+j + ^βγγ^e[μt+j+1]j = ^0,

∂t

∂^μt+j

^-Dt+j ⁺ YY^Dt+j-1 ^{+ kE} [^{(1 - q})^Dt+j ^{- F}t+j ^{+ nw} 1 ^{+ g}3^rD+ j ^{- g}4^rB+ j = 0.

(57)

(58)

For j = T, Condition (57) implies:

^βt+τe|^{(1 - q}) [ ( ^t+ ^t') ⁺ b^rB+ _τ ^{- r}D+ _τ ⁺ eL+ T1 ^- ∣^{v +} O)[^{(1 - q)2D}t+T ⁺

-(1 - q)Ft+T + (1 - q)NW - u - (1 - q)z + ρ_t+τq - μt+τ f = 0.

(59)

The transversality condition of the problem is:

_τlim β^t+τEt ∣(1 - q)ωt+τ - a [(1 - q)²Dt+τ - (1 - q)Ft+τ + (1 - q)NW] - u - (1 - q)z +
+Pt+τq} - κ I - ^ωt+τ + v [(1 - q)Dt+τ - Ft+τ + NW1 + Vt+τ }} = 0.

(60)

To keep the notation simpler, we omit from now on the index j , to show it only when it will be

necessary, in the final solutions.

Besides we have used the definitions

b^β+b¹') = a and a b' +

drB - rD + EL = ω_t. From Equation (56) we can obtain an equation that can be solved in order

to get a solution for the multiplier μ_t.

(61)

+ ^γγβEt I a [(1 - q)Dt +1
κ I L

- Ft +1 + NW^ - ( ωt+1 - z ) + Vt+ι∣∣

= 0.

(62)

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