On Dictatorship, Economic Development and Stability

Using the expression of Υ, the determinant becomes:

DetJ

τ⅛⁽¹^— q)¹^σ⁺¹⁺^a^{[(1 +} ρ )^β⁽¹^—^α )^l^]¹^-^σ

( αq)¹^-^σ

(1 -q)2 A [ lβ (1 -a )]^α

(1+β )^α

— l (1 — [)^a)

_-_ (1 — [) AA⁺¹⁺^a ( ɪ ) -

_α-101⅛ _l₍₁_— [a (--q)²^A[lβ(¹ -a)]^α^{α q}¹⁽¹^q) l (1+ β) αl (1 -q)α

-1

Simplifying and replacing (1 — q)^a by its steady state expression yields

DetJ = -

αq

⁽¹^—^[)

(1 -q)² Aa^-^α [lβ(1 -a)]^α [β(1+ ρ)(1 -a)l+a]^α

l (1+β )^α

-1

Replacing (1 — q) by its steady state expression, the large term in the denominator can

be identified as (1 — q)/(1 — τ). Finally, after simplification, we obtain:

DetJ =

-αq[

(44)

Condition 1: if q ∈ ] ₁ +_a , ^-α[, then ∖DetJ| > 1;

Condition 2: if q ∈ ] ₁ ∖_ι, 1[, then ∖DetJ| < 1;

where q is a function of the parameters ρ and l ; τ is a function of the parameters l, ρ and

A (see equation (29)); and 0 < τ,q < 1.

As for the trace, rearranging (42) yields
where 1 — τ = _a(₁_-^l-)^_α (see equation (36)).

trJ = Ω + J

_ ^μ α ∖ ^a (¹ — q)^-⁺^a^[^a

σ y(1 + ρ ) β (1 — α ) l + α J ∣

1+a (1 - σ ) (1 - ∙

aσ

(1 + ρ ) ^-1σ Υ

τ)(^σ+a(1 -σ)) ´
aσ I

The second additive term of the trace can be simplified in the same way as the determinant.

After some computations, we obtain: