Solution of the system
Defining Xt = g3rD - g4rB, the first equation can be rewritten as:
Dt =
Yy β
(1 - q) [yYβ - [1 - (1 - q)«]]
βγγ + [(1 - q ) «
E [ Ft+ι]
- 1]
(1 - q ) [yy β - [1 - (1 - q ) «
________[1 - (1 - q ) «]________f +
(1 - q)[yYβ - [1 - (1 - q)«]] t
] NW -
Yy β
[yy β - [1 - (1 - q)«
Xt +
________1 - (1 - q ) «________E [ ɪ z ]
(1 - q)[yYβ - [1 - (1 - q)«]] ∣-α t+1-∣'
(70)
Substituting the former in the other equation, we obtain a second order difference equation in
E[Ft].
Yy β
(1 - q) [yY β - [1 - (1 - q) «
E [ Ft +1]
[1 - (1 - q ) «] cι ,
--------;:—z----------------------fΓ+ +
(1 - q)[yYβ - [1 - (1 - q)«]]
βγγ + [(1 - q)« -1]
(1 - q ) [yY β - [1 - (1 - q ) « «
NW -
Yy β
[yY β - [1 - (1 - q ) «]
Xt +
Yy
Yy β
1 - «(1 - q ) (1 - q ) [yY β - [1 - (1 - q ) «]
Ft
Yy
Yy β
1 - «(1 - q) [yYβ - [1 - (1 - q)«]]
Xt-1
________1 - (1 - q ) «________E [ ɪ z ] =
(1 - q)[yYβ - [1 - (1 - q)«]] ∣-α t+1-∣
Yy
[1 - (1 - q ) «]
1 - «(1 - q) (1 - q) [yY β - [1 - (1 - q) «
Ft-1 +
Yy
βγγ + [(1 - q)« -1]
1 - «(1 - q ) (1 - q ) [yY β - [1 - (1 - q ) « ]
NW +
Yy
1 - (1 - q ) «
1 - «(1 - q) (1 - q)α[γYβ - [1 - (1 - q)«]]
+
-i---«---FFt + i---«---^W + ---X----. (71)
1 - «(1 - q) 1 - «(1 - q) 1 - «(1 - q)
It can be expressed as:
E [ Ft +1 ]
[1 - (1 - q ) « ]r, ,
-------F----Ft+
Yy β
«
1 - « (1 - q )
(1 - q) [yYβ - [1 - (1 - q)«]]
1 - «YY - q) Ft + 1
Yy
Yy β
[1 - (1 - q)«]
- « (1 - q ) γγ β
Ft +
Ft-1 =
(1 - q) [yYβ - [1 - (1 - q)«]]
Yy β
Yy
C--γ
L 1 - « (1 - q )
Yy β
+1
βγγ + [(1 - q) « - 1]
(1 - q) [yYβ - [1 - (1 - q) «
NW + ɪ
Yy
«(1 - q) [yYβ - [1 - (1 - q)«]]
1 - (1 - q ) «
_ _ _
1 - «(1 - q) (1 - q) [yY β - [1 - (1 - q) «
Xt-1 +
Yy β
[yyβ - [1 - (1 - q)«]]
Xt + τ
«
« (1 - q )
1
NW +
- «(1 - q)
Xt +
E [1 ] Zt + ∩----ʧ 12 - (1 - q)«----∏TE [1 Zt +1] . (72)
LaJ (1 - q) yYβ - [1 - (1 - q)«]J La -l
We now study separately the left-hand side.
E [ Ft+ι]
[1 - (1 - q)«]
1 - «YY - q )Ft + 1
Yy β
Yy
Ft + I----«----1
1 - «(1 - q)
[1 - (1 - q)«]
- « (1 - q ) γγ β
(1 - q) [yYβ - [1 - (1 - q)«]]
Yy β
Ft +
Ft-1 = E[Ft +ι] - [—- + γγ] Ft + -ιl- 1. (73)
Lγγβ J β
24
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