βt {Ptq +(1 - q)(ωt - z) - u - α [(1 - q)2Dt - (1 - q)Ft + (1 - q)NW] +
1 ^ɪ q) { - ωt + α [(1 - q)Dt - Ft + NW]
+ υt + z} +
+ γYK~Ett'{ - ωt+1 + α [(1 - q)Dt+1 - Ft+ι + NW] + υt+` + z}} = 0.
(63)
Rearranging and dividing everything by βt we obtain a difference equation for Ft and Dt. Since
Xt = Et [Xt] 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)αDt+` - 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)Dt + (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- ,)κFt + ' = αF + ^γβ: 1 - q)α{ I - (β q)κ D +
+1 - β- q)кNW + gβ--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:
+E { 1 - (1 - q )κ
I ^γ βα
As before we end up
constraint:
Dt =
^γ
E [ Ft+'] = 1 (1 q )K Ft +
^γ β
+ β^γ + (1 - q ) к
^γ β
{κ [ Ptq + (1 - q ) Ut
with a
1 - к (1 - q )
Dt-1
(1 - q) [^γβ - [1 - (1 - q)κ]]
^γ β
Dt +
1NW + (1 - q )[ g 3 rD - g 4 rB ] +
u] + (^γβ - L) [Ut+' - ωt+'] + (β^γ - 1)zj.
(67)
system of two equations, the other is obtained from the dynamic
--κ----Ft +--K----NW +
1 - к (1 - q ) 1 - к (1 - q )
g3 rD
g4rtB
1 - к (1 - q )
= 0.
(68)
We can simplify the value of the second intercept term as:
Zt+1 = к [ptq + (1 - q)Ut - u] + (^γβ - L) ∖υt + ` - ωt +`] + (β^γ - 1)z =
(^γβ - l) [rB+' - rD+' - a - brB+' + . - cf+ι] + к [(rR - rD)q + (1 - q)(rB - rD) - u] +
+ ( β^γ - 1)z = { ^γβ (1 - b ) + [(1 - q )к - (1 - b )] L} rB+' +
-(γγβ - l)et+' + к [qρt - rD - u] + (β^γ - 1) (z - b) (69)
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