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-α
(24)
This equation maps the points of the function kt+1 = kt . Its derivative with respect to k
is:
d(1 - τt)
(1 + β)
-α
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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