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

lβ (1 — a )

⁽¹ + ^β⁾⁽¹^— Qt⁾

(38)

ltt+1

⁽¹^— Qt+1) Ak^α+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

lβ (1 — a )

^{(1 +}^β⁾⁽¹^— Qt )

(40)

qt+1 qt¹

-σ

)++1 g-^-^σ l[(1+ β)(1 — t,)]^α

¹^— Qt+1 A^[^lβ⁽¹^—^a^)]^a

(41)

((1 — tt ) a⁺^aka —l (¹^-^q^t Α¹^-^σ+a )

t^∣√1 l ∖ < l l /QM —— l lβ (1 -a ) A^a¹

^a¹^-^-^{[(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 q_t₊₁ with respect to k_t and q_t , we use the implicit function theorem.

Given the function F (q_t₊₁, k_t, q_t) = 0, with

F⁽ tt+1 ,kt,Qt ) = tt+ι tt¹^-

^qt +1 ^qt ^- l^[(1+^β⁾⁽¹^—^qt^)]^α

1 — qt+1 A [ lβ (1 — a )]^α

(1 — qt ) ^σ⁺^αkα-

I (1 -qt )¹^-^σ⁺^αA

1 — 1

a¹^-^- [(1+ρ)(1+ β)]¹^-^-

lβ (1 — a )
1+β

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