On Dictatorship, Economic Development and Stability

3.3.2 Dynamics when σ > 1

If σ > 1, the functions (24) and (25) are both increasing in k and intersect twice if the
slope of (25) is steeper than the one of (24). We can check it by dividing the slope of (24)
by the slope of (25) and take the limits when k → 0 and k → +∞:

since ₁+α(_σ0-₁) > 0 when σ > 1, and

Therefore, the function (25) representing τ_t+1 = τ_t has a steeper slope than the function
(24) representing kt+1 = kt for small value of k . This is the inverse for large values of
k. This means that both functions start from 0, cross at the interior steady state and
diverge afterwards. We can see from the phase diagram that the interior steady state is
a saddle point.

3.3.3 Dynamics when σ = 1

If σ = 1 (logarithmic utility function), then the function (24) is increasing and the function
(25) is linear. They intersect once at the interior steady state. The steady state is a saddle
point.

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