On Dictatorship, Economic Development and Stability



there exists an implicit function qt+1 = f (kt ,qt) if F has continuous partial derivatives
Fqt, 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

(1 )
(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 + α)k'   A(1 σ-qt)

ω 4                        -1----------------

α (1+ ρ )- Υ


(q,k,q)


where,

Υ = (αqt)1 [(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 )qt       qt    (1 )   1 -qt / (1 -qt+1)a[(1 )]α

l [(1+β )(1 -qt )]α

a [ (1 )] α (1 -qt +1)2

Therefore, the Jacobian matrix is:

42



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