there exists an implicit function qt+1 = f (kt ,qt) if F has continuous partial derivatives
Fqt+ι, Fkt, Fqt, and if ∂F∕∂qt+1 = 0. Both conditions are verified if qt+1 = 1 and qt = 0.
As only the values between 0 and 1 are of interest, we can define an interval ]0, 1[ for q
to ensure that F admits the implicit function qt +1 = f (kt,qt).
The linearized system of (40)-(41) is the following:
kt+1 - k
qt+1 - k
= J.
kt
qt
with
lβ (1 -α )
(1+ β )(1 -qt )2
A (1 -qt+1)2(1 -qt ) 1 - + αkcΓ1
-1
(1+ρ )- Υ
σ I ( , ., . l ( η-σ + α )
a(1 -qt+1)2(1 -qt)- (τ-σ + α)k' A(1 σ-qt)
ω 4 -1----------------
α (1+ ρ )- Υ
(q,k,q)
where,
Υ = (αqt)1-σ [lβ(1 - α)]- 1-σ i q ■ A. ' ' - l (1 - qt)α)
σqt+1 _ (1 -qt)α f σ___αqt ´ qt+1l(1+β)α
(1 -σ)qt qt ∖(1 -σ) 1 -qt / (1 -qt+1)a[lβ(1 -α)]α
l [(1+β )(1 -qt )]α
a [ lβ (1 -α )] α (1 -qt +1)2
Therefore, the Jacobian matrix is:
42