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Clarifying the confusion in class
Juanpa Nicolini
August 14, 2017
The general solution for the neoclassical growth model is
Ú(ct)
βÚ(ct+1)
= Afk(kt+1) + 1− δ
ct = Af(kt) + (1− δ)kt − kt+1
Under the following assumptions
δ = 0, U(c) = ln c, and f(k) = kα,
they become
ct+1
βct
= Aαkα−1t+1 (1)
ct = Ak
α
t − kt+1 (2)
Proposition: The sequences
ct = (1− αβ)Ak
α
t (3)
kt+1 = αβAk
α
t (4)
are a solution to the system of equations in 1 and 2.
Pf: Using 3 in 1, we obtain
Akαt+1
βAkαt
= Aαkα−1t+1
or
Akt+1
βAkαt
= Aα
which holds, because of 4. Finally, using 3 in 2, we obtain
(1− αβ)Akαt = Ak
α
t − kt+1
which implies
αβAkαt = kt+1
which holds because of 4.
Regarding the confusion created in class.
• Using equation 3 for two consecutive periods, it can be show that the
solution satisfies
ct+1
ct
=
kαt+1
kαt
as claimed by some of you.
• Also using 1 and 4, we obtain
ct+1
ct
= βAα
kαt+1
kt+1
=
βαAkαt+1
kt+1
=
kt+2
kt+1
which is NOT what I claimed in class.
• In class, I claimed that
ct+1
ct
=
kt+1
kt