Дыхта В.А. Динамические системы в экономике. Введение в анализ одномерных моделей
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Y K L Y = F (K, L) α ∈
(0, 1) γ > 0
CES
1/(1 + γ) < 1
γ −→ 0 CES
γ −→ +∞
Y = F
∞
(K, L) = Y min
½
K
K
,
L
L
¾
CES
CES
f(k) = y
"
α
µ
k
k
¶
−γ
+ 1 − α
#
−1/γ
,
y = Y /L k = K/L
f
0
(0+) = α
−1/γ
y
k
s > α
1/γ
k
y
(b + n). (2.58)
CES
Y
t+1
− Y
t
Y
t
= a + b
µ
K
t+1
− K
t
K
t
¶
(2.59)
a > 0, 0 < b < 1.
I
t
= sY
t
s > 0
K
t+1
− K
t
= sY
t
(2.60)
Y
t+1
= (1 + a)Y
t
+ bs
Y
2
t
K
t
(2.61)
Y
0
> 0 K
0
> 0 Y
t
K
t
Y
t+1
> (1+a)Y
t
Y
t
> (1+a)
t
Y
a
Y
t
−→ +∞ t −→ +∞ K
t+1
> sY
t
K
t
−→ +∞
x
t
=
Y
t
/K
t
x
t+1
=
Y
t+1
K
t+1
=
1
K
t
+ sY
t
[(1 + a)Y
t
+ bsY
t
x
t
] ,
K
t
x
t+1
= f(x
t
), f(x) =
(1 + a)x + bsx
2
1 + sx
. (2.62)
f
+
x = 0 x
∗
= a/(1 − b)s > 0
f
f
0
(x) =
1 + a + 2sbx + s
2
bx
2
(1 + sx)
2
. (2.63)
f
0
(x) = f
0
(0) = 1 + a > 1
x = 0
f
0
(x) =
1 + a − 2b + b
2
+ a
2
b − ab
2
(1 + a − b)
2
=
1 − b + ab
1 − b + a
.
a > 0 b ∈ (0, 1)
x
∗
x
∗
x > 0
f
0
(x) > 0 x ≥ 0 f (x)
f(x) < f(x
∗
) = x
∗
x ∈ (0, x
∗
),
f(x) > f(x
∗
) = x
∗
x > x
∗
.
x
0
< x
∗
, x
t
< x
∗
∀ t > 0,
x
0
> x
∗
, x
t
> x
∗
∀ t > 0.
x
t+1
− x
t
=
1
1 + sx
t
[a − s(1 − b)x
t
] x
t
.
x
t
< x
∗
x
t
> x
∗
x
0
< x
∗
x
∗
x
0
> x
∗
x
∗
t −→ +∞
x
∗
S(x
∗
) = (0, +∞) x
∗
Y
0
> 0 K
0
> 0 Y
t
/K
t
x
∗
t → +∞
≈
Y
S
t+1
=
Y
D
t
Y
S
t+1
− Y
S
t
= α(Y
D
t
− Y
S
t
), (2.64)
α > 0
α = 1
C
t
= C
0
+cY
S
t
(I
0
) I
t
= I
0
+ ∆I
Y
D
t
= C
t
+ I
t
= C
0
+ cY
S
t
+ I
0
+ ∆I
Y
S
t+1
= Y
S
t
+ α[(c − 1)Y
S
t
+ C
0
+ ∆I]. (2.65)
Y
∗
=
∆I + I
0
+ C
0
1 − c
y
t
= Y
S
t
− Y
∗
y
t+1
= ay
t
, a = 1 − α(c − 1).
y
∗
= 0
|a| < 1
0 < α(1 − c) < 2, α < 2/(1 − ),
α > 0 α = 1
c ∈ (0, 1)
α
0 < α < 1/(1 − )
1/(1 − c) < α < 2/(1 − c)
α
c
Y
S
t+1
= Y
S
t
exp[α(Y
D
t
− Y
S
t
)] (2.66)
exp(z) = e
z
Y
D
t
= Y
∗
+ c[Y
S
t
− Y
∗
]
Y
S
t+1
= Y
S
t
exp[α(1 − c)(Y
∗
− Y
S
t
)].
a = α(1 − c), A = exp(aY
∗
),
Y
S
t+1
= Y
S
t
A exp(−aY
S
t
). (2.67)
x
t
= aY
S
t
,
x
t+1
= f(x
t
), f(x) = Ax exp(−x). (2.68)
Q
Dt
Q
St
P
t
P
t+1
= P
t
exp[α(Q
Dt
− Q
St
)], (2.69)
α > 0
Q
Dt
= a − bP
t
= Q
∗
− b(P
t
− P
∗
),
Q
St
= s + dP
t
= Q
∗
+ d(P
t
− P
∗
),
P
∗
=
a − c
b + d
, Q
∗
= a − bP
∗
= c + dP
∗
P
t+1
= P
t
exp[−α(b + d)(P
t
− P
∗
)]. (2.70)
A = exp[α(b + d)P
∗
]
x
t
= α(b + d)P
t
x P
x
t+1
= f(x
t
), f(x) = Axe
−x
, A > 0.
+
= [0, +∞) f
+
+
f(x) = x x = 0 x
∗
= ln A
+
A > 1
A > 0 A ∈ (0, 1)
f
0
(x) = Ae
−x
(1 − x), f
00
(x) = −Ae
−x
(2 − x).
f(x)
x = 1 f
0
(x)
+
x = 2 x = 0
f
0
min
= f
0
(2) = −Ae
−2
, f
0
max
= f
0
(0) = A (2.71)
lim
x→+∞
f(x) = 0, lim
x→+∞
f
0
(x) = 0.
f
f
0
1 < A < e
y
x
y = x
y = f (x)
x
∗
1 2
e < A < e
2
y
x
y = x
y = f (x)
x
∗
1 2
A > e
2
y
x
y = x
y = f (x)
x
∗
1 2
A
A ∈ (0, 1) |f(x)| < 1
+
f
x = 0
A = 1
x
t+1
= x
t
e
−x
t
< x
t
e
−x
∈ (0, 1) x > 0
x
0
> 0
x = 0
A > 1 x = 0
x
∗
= ln A f
0
(x
∗
) = 1 − x
∗
−1 < 1 − ln A < 1 A ∈ (1, e
2
)
x
∗
∈ (0, 2) A > e
2
x
0
> 0 x
∗
x
0
∈ (0, x
∗
) {x
t
}
f(x) > x x < x
∗
x
∗
= f(x
∗
) x
t
−→ x
∗
t −→ +∞
x
0
x
t+1
= λx
t
(1 − x
t
), (2.72)
λ > 0
f(x) = λx(1 − x),
[0, 1] λ ≤ 4
x
t+1
= f(x
t
), (2.73)
⊂ R
{x
t
}
p x
t+p
= x
t
t = 0, 1, 2, . . .
p p
p
p
p p
f(x)
t → +∞ f(x)
x
t+1
=
−x
t
f
p
p
f
f
f
0
(x) = f(x), f
1
(x) = f
¡
f(x)
¢
, . . . ,
f
n
(x) = f
¡
(f
n−1
(x)
¢
f n .
x
0
{x
0
, f
0
(x
0
), f
1
(x
0
), . . .}.