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 - rD I EL }[(1 - q ) Dt - Ft + nw ] +

t=0

- 2v [(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 + rD+ 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+' + brB+ j - rD+ j + eL+ j 1 -v + O)[(1 - q)2Dt+j +


- (1 - q) Ft+j + (1 - q) NW] - u - (1 - q) z + ρt+j q +
- [l - к(1 - qH μt+j + βγγe[μt+j+1]j = 0,


∂t

μt+j


-Dt+j + YYDt+j-1 + kE [(1 - q)Dt+j - Ft+j + nw 1 + g3rD+ j - g4rB+ j = 0.


(57)

(58)


For j = T, Condition (57) implies:


βt+τe|(1 - q) [ ( t+ t') + brB+ τ - rD+ τ + eL+ T1 -v + O)[(1 - q)2Dt+T +

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


(59)


The transversality condition of the problem is:


τlim βt+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


bβ+b1') = a and a b' +


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


to get a solution for the multiplier μt.


μt = 11 a [(1 - q)Dt - Ft + NW1 - ωt + z + Vt}.

(61)


This value can be substituted in the other first order condition, shown in equation (57), that
provides the Euler equation of the system:

βt IPtq +(1 - q)(ωt - z) - u - a [(1 - q)2Dt - (1 - q)Ft + (1 - q)NW] +

----κ(K—q) I a [(1 - q ) Dt - Ft + NWt 1 - ( ωt - z ) + Vt} +

+ γγβEt I a [(1 - q)Dt +1
κ I L


- Ft +1 + NW^ - ( ωt+1 - z ) + Vt+ι∣∣


= 0.


(62)


22




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