On Dictatorship, Economic Development and Stability

current income tax rate and the current capital stock. The system of these three equations
is computed by eliminating the Lagrange multipliers from the equations (16)-(18), and
by solving the new system to find the expressions of k_t+1 , τ_t+1 and q_t+1 .

4.1 Steady states

Proposition 3 Any politically stable dictatorship characterized by the dynamic system
(26)-(28) admits a unique steady state.

Proof:

By solving the system (26)-(28) at the steady state, we obtain the following unique solu-
tion:

(1 + ρ )[ β (1 — α ) l ]² + αβ (1 — α ) l
α (1 + β )

l (β(1 + ρ)(1 — α)l + α)^1-α

¹                       α         ∖ α

A (α)^{1 -α} (^β⅛+β))

1

1 ₊      ^α,----

β (1+ ρ )(1 -α ) /

Hence, the dictator’s steady state consumption is:

__     [β(1 — α)l]^2-α AA A ∙ αA ^j1 -Λ ^μβ(1 — α)l ∖ ^α l

g = [β(1 — α)l + Aα]^{2(1 -α}) ∖α ⁺ [β(1 — α)lJ ) \ 1 + β ^— ~ A2â

If the insurrection constraint is not binding, then the steady state solution is the same as
in proposition 1.¥

Proposition 3 gives the steady state of dictatorship that is politically stable. The insurrec-
tion constraint makes the rate of predation, q , one of the variables determining the steady

28

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