On Dictatorship, Economic Development and Stability

saddle.

11 + DetJ|. If ∖DetJ| > 1, the steady state is unstable. If ∖DetJ| < 1, the steady

state is a sink.

When σ < 1, there are two cases:

• If σ < 2 ,it can be shown that 0 < ^-^ [1 — q^r(2 — ^a + α )] < 1 and, therefore,
[trJ| < 11 + DetJ|.If ]DetJ| > 1, the steady state is unstable. If ]DetJ| < 1, the
steady state is a sink.

• If 1 > σ > 2, there are two subcases depending on ρ:

, „             2-¹         ,          _ _r .                            ,       .        , ,       „        ,     ,,      , .      _ ,,

a/ If q < -—, ^σ ₁., then _τσ~ [1 — q(2 — α + ^α)] > 1 and, therefore, [trJ| > 11 + DetJ|.

'             2—α (1 — σ )           ^1σl- ^v^            ^σ'^{,j                                 1      11               1}

The steady state is a saddle.

2_ ¹

b/ If q > -—, ^σ ₁., then _τσ~ [1 — q(2 — ^a + ^α)] < 1 and, therefore, [trJ| < 11 + DetJ|.

'               2α (1 σ )             ^1σl-                         ^σ'^,j

If ]DetJ| > 1, the steady state is unstable. If ]DetJ| < 1, the steady state is a sink.

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