On Dictatorship, Economic Development and Stability



dictator’s power. Second, the dictator’s behavior has a significant impact on the long-term
level of aggregate output.

3.3 Phase diagram: a few graphical examples

To describe the dynamics graphically, we build a phase diagram using the two equations
of the dynamical system (20)-(22). Following the method shown by de la Croix and
Michel (2002), we characterize the set of points (
kt, τt) for which there is no change in kt
in equation (20). Solving the equation leads to:

1 - τt =


(1 + β )
β (1 - α ) A


kt1


(24)


This equation maps the points of the function kt+1 = kt . Its derivative with respect to k

is:

d(1 - τt)


(1 + β)



dkt


βA


>0


Then we characterize the set of points (kt, τt) for which there is no change in τt in equation

(22). Solving for this equation leads to:

τt =


α 1 + α ( σ-1)


α(σ


1)(1 -α )


1

(1 + ρ )1+α (σ-1)


l+ββ (1 - α ) A


α(σ


1)


1)


k 1 + α ( σ- 1)


(25)


This equation maps the points of the function τt+1 = τt . Its derivative with respect to k
is:

d(1 - τt)


α(σ


1)(1 - α)


dkt


1 + α(σ


1)


α1+α (σ


1)


(1 + ρ)


1)


ι+β (1 - α ) A


α(σ


l∙ α ( σ


1)


1)


kt


1)


23




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