On Dictatorship, Economic Development and Stability

A Proof of proposition 2

The dynamical system of the dictatorship without insurrection threat is characterized by
these two equations:

^kt +ι = 1 + β(1 - ^τt)(1 - ^α)A^k'_l'

(33)

(34)

_ α σ τtkf⁽¹^-^σ⁾⁽¹^-^α )

^T^t⁺¹ a ( σ-1)

/ ∖ 1 , ι 1 + a ( σ— 1) /ɔ . . . σ

^{(1 +}^ρ )^σ⁽¹^-^τt ) ^σ 1+β ⁽¹^-^α ) A

The linearized system is:

kt+1 ^- k

τt+1 - T

kt
τt

with

α 1++β(1 - τ)(1 - ^a)Ak^α¹

^- 1+β⁽¹^-^α ) A^{k a}

α¹⁺ - (1 -α)(1 -σ)τк^a⁽¹^—¹⁾⁽¹^—a)^—¹

1 z _β . ^a⁽^σ-¹) 1 + a(σ-1)

⁽¹⁺ρ)^σ ( ɪ⁽¹^-^α)^α) ^σ⁽¹^-^τ) ^σ

α 1 ka(¹ — 1)(¹ — a)(1+ ^{1 +}^a _σ ¹) τ)

1/ Я ^a^a⁽°~¹) 1 + a(σ—1),₁

(1+P)^σ ( 1⅛ (1 -α)A) ^σ (1 -τ) ^σ⁺¹

Replacing k and τ by their expressions, we can make some simplifications:

^- i⅛⁽¹^-^α ) A

∖ ^aβα(1 -α)A ∖ ¹^—a⁽¹⁺^β⁾⁽¹⁺^ρ ) J

^α⁽¹^-^α⁾⁽^σ-Γ )

1 - OL-

¹¹+ρ_____

β βa (1 — a) A ʌ 1 — a
⁽ (1+ β)(1+ρ) )

1+ P
α

⁽¹^-a⁾⁽σ^- 1) Z 1+ ρ _- β
σ ∖ α J