1 Modèle de Solow avec croissance démographique

n
δ
F(Kt, Lt) = Kα
t(Lt)1−α

F(Kt, Lt) = (Kt)α(Lt)1−α
F(λKt, λLt) = (λKt)α(λLt)1−α
F(λKt, λLt) = λ(Kt)α(Lt)1−α
F(λKt, λLt) = λF (Kt, Lt)
F(Kt, Lt) = Kα(L)1−α
gk= 0 gk
gk
∼
=ln kt+1 −ln kt
∼
=ln Kt+1
Lt+1 −ln Kt Lt
gk= [ln Kt+1 −ln Kt]−[ln Lt+1 −ln Lt]
g˜
k= [ln Kt+1 −ln Lt+1]−[ln Kt−ln Lt]
gk=gK−gL=∆k
k=∆K K
−∆L
L
∆K =
I−δK
∆k
k=I−δK K
−n
∆k=I
L
−δK
L
−nK
L
∆k=i−(δ+n)k
Y=C+I
Y=S+I
I=S
s Y s
S=sY
∆k = sf(k) −(δ+ n)k
δk = 0
sf(k∗) = (δ+n)k∗

k∗=s
δ+n1
1−α
˜y∗=k∗α
y∗=s
δ+n
α
1−α
P mL =
∂F (K,L)
∂L = (1 −α)K Lα
Lt
Lt
n
k∗=s
δ+n1
1−α
gk=gy= 0
n
∆Y
Y=Yt+1−Yt Yt=gY
Yt+1
Yt= 1 + gY
gY
∼
=ln (Yt+1)−ln (Yt)y=Y
L
gy
∼
=ln (Yt+1)−ln (Yt)−[ln (Lt+1)−ln (Lt)] ∼ =gY−gL
gy= 0
gY=gy+gL= 0 + n=n n
gKgY=gK=n gy=gk= 0
Y=C+I
c=f(k)−i
c∗=f(k)∗ −i∗
c∗=f(k)∗ −(δ+n)k∗
∂c∗
∂k∗= 0 f0(k)∗= (δ+n)f0(k)∗=
αk∗α−1αk∗α−1= (δ+n)k∗=(δ+n)
α
1
α−1