Monopolistic Pricing in the Banking Industry: a Dynamic Model



Since we know that Xt +ι = Xt + eD+1 — eB+1, we can write:

- γYβκ- q) + - κ- q)] χ + χt + ⅜1 - ⅜1

YyβI - κ(1 - q)]        t ɪ - к(1 - q)

And, sinceEt [et-1] = 0:

-Yyβκ(1 - q) - I - κ(1 - q)] + γγβ χ

Yyβ- к- q)]             11

It can be simplified as:

Yy β - ɪ
Yy β


Xt.


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

E[Dt] - [ɪ + γγ ] Dt + ɪDt-1 =      Xt - ~^~E[ ɪ Zt+11.

(83)


(84)


(85)


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.
L
γγ β     J β          γγ β       γγ β La    J

The solution is given by:

∞                        ∞ -Ixi

Dt+j+1 = γγβDt+j + βYY ∑ (YY) Et+i [   O1 ] - βγγa- Yy)C +

- Y-1 ∑ ( γY ) lEt-+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:

Dt+j +1 = γγΓβDt+j - βγγa- γγ) [(ɪ - K)(γγβ - H) + - q)κH][rB+j+1 + COV (rB, ɪ)] +

- βγγa- γγ) K(rR+ jq - rD+ j) + γγβ(1 - γγ) (g3rD+j - g4rB+j) - βγγa- γγ) C +

- jγγβ - ɪ)K COV (Ld, ɪ ); (86)
βγY- γY )           a

it can finally be simplified as:

ɪ         κ[(ɪ- d)(γγβ- h) + (1 - q)kH]+ (γγβ- ɪ)ag4h 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(rB, ɪ) - (γγβ ɪ)K COV(Ld, ɪ ). (87)

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

27



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