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)Ak'l'
(33)
(34)
_ α σ τtkf(1 -σ )(1 -α )
Tt +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
J.
kt
τt
with
α 1++β(1 - τ)(1 - a)Akα 1
- 1+β(1 - α ) Ak a
α1+ - (1 -α)(1 -σ)τкa(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
∖ a
βα(1 -α)A ∖ 1—a
(1+ β )(1+ ρ ) J
J=
α (1 - α )( σ-Γ )
1 - OL-
1 1+ρ_____
1
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
39