On Dictatorship, Economic Development and Stability



Plugging in (35) the expression of τt given by (36), and plugging (36) in (37), we obtain
a system of two equations,

kt+1


(1 — a )

(1 + β )(1 Qt)


(38)


ltt+1

(1 Qt+1) Akα+1


[(1 + ρ )(1 + β )]1 - - Ttkq (1 t) )1 - -

1   -~- q a— 1^—

a1 -σ q- - kt+1 -


(39)


Using (36) and (38) to rearrange the expression (39), we obtain a system equivalent to


(35)-(37):


kt+1


(1 — a )

(1 + β )(1 Qt )


(40)


qt+1 qt1



)++1 g--σ l[(1+ β)(1 — t,)]α

1 Qt+1   A[ (1 a )] a

(41)

((1 — tt ) a+aka l (1 -qt Α1-σ+a )

t^∣√1 l   ∖ < l l /QM —— l lβ (1 -a ) Aa 1

a 1 -- [(1 + P)(1 + β)] 1 --    1. β )

Then we take a first-order Taylor expansion around the steady state. To compute the
partial derivatives of
qt+1 with respect to kt and qt , we use the implicit function theorem.

Given the function F (qt+1, kt, qt) = 0, with

F( tt+1 ,kt,Qt ) = tttt1 -


-

qt +1 qt - l[(1+ β)(1 qt)]α


1 — qt+1    A [ (1 — a )]α


(1 — qt )   σ + αkα-


I (1 -qt )1-σ + α
A


1                   — 1

a1 -- [(1+ρ)(1+ β)]1 --


(1 — a )
1+β


41




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