Дыхта В.А. Динамические системы в экономике. Введение в анализ одномерных моделей
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f
0
(1) = f
0
(−1) = 2 > 1 ⇒ ¯x =
1 x
e
= −1 2
f(x)
x
t+1
− x
∗
= f(x
t
) − f(x
∗
) = f
0
(c
t
)(x
t
− x
∗
), (2.48)
c
t
x
t
x
∗
x
∗
|x
0
−x
∗
|
f(x) |x
t
−x
∗
|
f
0
(c
t
) f
0
(x
∗
)
y
t
=
x
t
−
x
∗
, a
=
f
0
(
c
t
)
.
(2
.
49)
y
t
y
t+1
= ay
t
. (2.50)
x
∗
x
∗
y
∗
= 0
|a| < 1 |a| > 1
x
∗
x
t+1
= f(x
t
) f(x) y
∗
= 0
y
∗
x
∗
y
∗
x
∗
•
•
•
•
d(x, y) x, y ∈
• x
t+1
= f(t, x
t
)
f(t, x) t ∈ Z
+
= {0, 1, . . .}, x ∈
⊆ R
Z
+
× f : Z
+
× →
• x
t+1
= f(t, x
t
)
x
t
t = 0, 1, . . .
• x
0
∈ S
x
0
x|
t=0
= x
0
• x
t+1
= f(x
t
)
f(x) f : S →
• x
t+1
= f(x
t
)
f : →
x
t
≡ x
∗
∈ f(x) = x
f(x
∗
) = x
∗
• x
∗
(x
∗
)
x
0
∈
x
∗
t → +∞
• x
∗
(x
∗
) = x
∗
• x
∗
(x
∗
)
x
∗
x
∗
x
∗
• x
∗
x
∗
x
∗
• x
∗
x
∗
x
∗
• x
∗
•
• f
|f(y) − f(x)| < |y − x| ∀x, y ∈ , x 6= y.
• f(x)
|f
0
(x)| < 1 f
• f
x
∗
∈
x
t+1
= f(x
t
)
• f
k ∈ (0, 1)
|f(y) − f(x)| ≤ k|y − x| ∀x, y ∈ .
•
• f x
∗
|f
0
(x
∗
)| < 1 x
∗
|f
0
(x
∗
)| > 1 x
∗
|f
0
(x
∗
)| = 1 x
∗
x
t+1
= 1/x
t
|a| < 1 f(x) = a ln(1+x
2
)
R x
n+1
=
f(x
n
) n → ∞
x
t+1
= ax
α
t
+ bx
t
S = R
+
a > 0 b > 0 0 < α < 1
x
n+1
= x
n
/[α + (1 − α)x
n
]
R
+
α ∈ (0, 1)
y
n
= 1/x
n
x
t+1
= ax
t
+ta
t
t → +∞
x
t+1
= ax
t
+ cy
t
+ d {y
t
}
y
t
= ay
t
+ b a 6= 0 a 6= 1
f : [a, b] → [a, b]
[a, b]
g(x) = x − f(x)
f(x) = 0
x
0
a
x
n+1
= x
n
−
f(x
n
)
f
0
(x
n
)
, n = 0, 1, . . . .
f(x)
f
0
(a) 6= 0 I a
x
0
∈ I
a
g(x) = x−f(x)/f
0
(x)
(b)
f(x) S x
∗
f
0
(x
∗
) = 1
f
00
(x
∗
) 6= 0 x
∗
f
00
(x
∗
) = 0 f
000
(x
∗
) > 0 x
∗
f
00
(x
∗
) = 0 f
000
(x
∗
) < 0 x
∗
f
00
(x
∗
) = 0 f(x)
x
∗
S f
00
(x
∗
) > 0 f
00
(x
∗
) <
0 f
00
(x
∗
) > 0 f
0
(x) x
∗
f
0
(x) > 1 S
I = (x
∗
, x
∗
+ δ) q 1 < q < f
0
(x)
•
Y
t
=
F (K
t
, L
t
)
•
(1 + n) L
t+1
= (1 + n)L
t
L
t
= (1 + n)
t
L
0
•
K
t+1
− K
t
= sY
t
− bK
t
,
s ∈ (0, 1)
b
F (K, L)
F (λK, λL) = λF (K, L) ∀λ ≥ 0
k
t
= K
t
/L
t
k
t+1
=
K
t+1
L
t+1
=
sF (K
t
, L
t
) + (1 − b)K
t
(1 + n)L
t
=
s
1 + n
f(k
t
) +
1 − b
1 + n
k
t
,
f(k) = F (k , 1),
Y
t
L
t
=
F (K
t
, L
t
)
L
t
= F
µ
K
t
L
t
, 1
¶
= F (k, 1).
k
t+1
= g(k
t
), g(k) =
s
1 + n
f(k) +
1 − b
1 + n
k. (2.51)
F (K, L) = AK
α
L
1−α
, A > 0, α ∈ (0, 1). (2.52)
f(k) = Ak
α
g(k) z = k k = 0
k
∗
> 0
k
0
> 0
k
z
k
∗
z = y(k)
z = k
g
R
±
R
+
R
+
R
+
g(k) = k
k(Ask
α−1
− b − n) = 0.
k
∗
=
µ
As
b + n
¶
1/(1−α)
. (2.53)
g
0
(k
∗
) = (1 − α)
b + n
1 + n
∈ (0, 1),
α ∈ (0, 1)
b ∈ (0, 1)
k
∗
g
0
(0+) = +∞
k ≥ ε, ε > 0
k
∗
k
0
> 0
0 < k
0
< k
∗
k
t+1
= g(k
t
) > k
t
g(k) > k k ∈ (0, k
∗
)
k
t+1
= g(k
t
) < g(k
∗
) = k
∗
g(k) k
∗
{k
t
} k
∗
k
∗
k
∗
k
0
> k
∗
K, Y, I = sY, C =
(1 −s)Y K
0
> 0 k
0
> 0
t
K
t
= k
t
L
t
≈ k
∗
(1 + n)
t
L
0
,
Y
t
= y
t
L
t
= Ak
α
t
L
t
≈ A(k
∗
)
α
(1 + n)
t
L
0
,
I
t
= i
t
L
t
= sy
t
L
t
≈ sA(k
∗
)
α
(1 + n)
t
L
0
,
C
t
= (1 − s)y
t
L
t
≈ (1 − s)A(k
∗
)
α
(1 + n)
t
L
0
,
K
0
= k
∗
L
0
k
t
≡ k
∗
s ∈ (0, 1)
k
∗
s k
∗
= k
∗
(s)
c
c = (1 − s)f(k)
k
∗
= k
∗
(s)
ϕ(s) = (1 − s)f(k
∗
(s)) s ∈ (0, 1), (2.54)
k
∗
(s) s
ϕ(s) = (1 − s)s
1/(1−α)
→ max
s∈(0,1)
.
s
∗
= α
1 − s
∗
=
1−α
k
∗∗
= k
∗
(s
∗
) =
µ
Aα
b + n
¶
1/(1−α)
c
∗
= (1 − s
∗
)f(k
∗∗
).
α
f(k)
f(k) k ≥
0
k
∗
g(k) > k k > 0
k
lim
k→0+
g(k)
k
> 1,
lim
k→0+
f(k)
k
>
b + n
s
. (2.55)
k > 0 f(k)
f
0
(0+) >
b + n
s
. (2.55
0
)
f
z = f(k)
k → +∞
lim
k→0+
g(k)
k
= 0. (2.56)
lim
k→0+
g(k)
k
=
1 − b
1 + n
> 1,
k
g(k) < k z = f(k) z = k
k
∗
> 0 f
lim
k→0+
f
0
(k) = 0. (2.56
0
)
k
∗
k
0
> 0
f
0
(0+) = +∞
k
∗
> 0
CES
Y = F (K, L) = Y
"
α
µ
K
K
¶
−γ
+ (1 − α)
µ
L
L
¶
−γ
#
−1/γ
, (2.57)