1. shito@seinan-gu.ac.jp
2020 6 25
•
•
•
•
y = f(x0, x1, x2, . . . , xn) ∂y/∂x0 x1, xn, . . . , xn
x1, xn, . . . , xn x0 x1, xn, . . . , xn
∆x0 → 0
Y = C + I0 + G0
C = C(Y, T0)
Y = C(Y, T0) + I0 + G0
Y ∗
= Y (I0, T0, G0) ∂Y ∗
/∂T0
2 Y ∗
= C(Y ∗
::
, T0) + I0 + G0
↑
Y ∗
T0, I0, G0
reduced form Y ∗
= f( )
⇓
C Y
1
2. www.seinan-gu.ac.jp/˜shito 2020 6 25 10:41
1 (differential)
(1)
y = f(x) dy/dx 2 dy dx
: ∆y ≡
: dy ≡ ≡ x
dy: y (the differential of y)
dx: x (the differential of x)
dy dx (differentiation)
(2) dy/dx
differentiation x y (differential) differentiation
dy/dx x (differentiation respect to x)
y x
(3)
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(4)
2 dy dx
y = f(x) : x 1% y %
x −→ x + dx
y −→ y + dy
x 1% y % ε
ε ≡
x → x + dx 1
+1
−→ 2 %
100
+1
−→ 101 %
(5)
: Q = Q(P)
P Q
=⇒ dQ/dP
dQ/dP
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(1) pp.220–227
(2) 8.1 1–5
• 1 (c) x (b)
• 1 (d)
2
y = f(x) (differential) dy = dy
dx dx
2
• 1 x1 1 y
• 2 x1 dx1 y
• 3 y x1 x2 2
=⇒ dy =
=
dy (total differential) dy (total differentiation)
x2 dx2 = 0
dy = f1dx1
↔
dy
dx1 x2
=
∂y
∂x1
y = f(x1, x2, , . . . , xn)
dy =
∂y
∂x1
dx1 +
∂y
∂x2
dx2 + · · · +
∂y
∂xn
dxn
= f1dx1 + f2dx2 + · · · + fndxn
=
n
i=1
fidxi
(1) pp.228–231
(2) 8.2 1–4
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3
1 d(cun
) = cnun−1
du
2 d(u ± v) = du ± dv
3 d(uv) = vdu + udv
4 d
u
v
=
vdu − udv
v2
(1) pp.231–233
(2) 8.3 1–3
4
y = f(x, w) x = g(w)
• 1
dy =
• 2 dw
dy
dw
=
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u y
y = f(x1, x2, u, v)
x1 = g(u, v)
x2 = h(u, v)
• ∂y/∂u
=⇒ x1 x2 v x1 x2 u y
y = f(g(u, v), h(u, v), u, v)
v u y u x1
x2
=⇒ dy dv = 0
• 1:
dy =
• 2: du dv = 0
dy
du dv=0
=
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(1) 45
Y = C + I0 + G0
C = C(Y, T0)
(a) G Y ∗
= Y (I0, T0, G0)
(b) T G
dG
(c) C = c(Y − T) + A c A ∂C/∂Y
∂C/∂T
(2) pp.234–239
(3) 8.4 1–5
5
45 2 (reduced
form) Y = · · · IS I(r) + G = S(Y ) + T
Y = · · ·
dY
dT dG=0
dY
dG dT =0
=⇒
(1) (Implicit function)
y = f(x)
y = 3x4
y = . . . (explicit)
⇓
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8. www.seinan-gu.ac.jp/˜shito 2020 6 25 10:41
y − 3x4
= 0
F(y, x) = 0
y = . . . y x (implicit)
y = · · ·
⇓
• y − 3x4
= 0 x y y = f(x)
• y = . . . F(y, x) = 0 y = . . .
x y y = f(x)
↑
IS I(r) = S(Y ) Y = . . .
r Y Y = f(r)
•
F(y, x) = x2
+ y2
− 9 = 0
↔ x2
+ y2
= 9
↔ y = ± 9 − x2
x y ±2
y = . . . -4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
上半分
下半分
y−
= −
√
9 − x2
y+
= +
√
9 − x2
y
F(y, x) = x2
+ y2
− 9 = 0
y > 0 y = f(x) = +
√
9 − x2
y < 0 y = g(x) = −
√
9 − x2
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•
F(y, x) = 0 dF = 0 =
F(y, x) = 0
Fy = 0 x
y
⇓
Fy = 0 x F(y, x) = 0 y
y = f(x)
F(y, x1, · · · , xm) = 0 (a) F
Fy F1 · · · Fm (b)
F(y, x1, · · · , xm) = 0 (y0, x10, · · · , xm0) Fy
(x10, · · · , xm0) m- N
N y y = f(x1, · · · , xm) x1, · · · , xm
y0 = f(x10, · · · , xm0)
N m (x1, · · · , xm)
F(y, x1, · · · , xm) = 0 F(y, x1, · · · , xm) = 0
f
f1, · · · , fm
x2
+ y2
− 9 = 0
(a) Fy = Fx =
(b) Fy Fy = 0
2 y = f(x)
f fx
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(2)
dy = −
Fx
Fy
dx ↔
dy
dx
= −
Fx
Fy
Fy = 0 dy
dx Fy = 0
(3)
(a) Fy = 0
(b) Fy = 0 f f
=⇒ y−
y
(c) y = f(x1, x2, · · · , xm)
y
f1, f2, · · · , fm
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(4)
• F(y, x1, · · · , xm) = 0 y = f(x1, · · · , xm) y
=⇒ ∂y/∂x1
• F(y, x1, · · · , xm) = 0 y
Fy = 0
∂y
∂x1
= fi, (i = 1, · · · , m)
Fy = 0 ∂y/∂xi
Fy = 0 ∂y/∂xi
= ⇐⇒ d = d
dF(y, x1, · · · , xm) = d0 = 0
←→ Fydy + F1dx1 + · · · + Fmdxm = 0
←→ Fy
dy
dx1
+ F1
dx1
dx1
+ F2
dx2
dx1
+ · · · + Fm
dxm
dx1
= 0
←→ Fy
dy
dx1
+ F1 + F2
dx2
dx1
+ · · · + Fm
dxm
dx1
= 0
dx2 = dx3 = · · · = dxm = 0 y x1
Fy
dy
dx1
+ F1 = 0
←→
dy
dx1
≡
∂y
∂x1
=
∂y
∂xi
= (i = 1, 2, · · · , m)
2 F(y, x) = 0
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1 F(y, x) = y − 3x4
= 0 dy/dx
2 F(Q, K, L) = 0 Q =
f(K, L) K MPK ≡
∂Q/∂K L MPL ≡ ∂Q/∂L F
F(Q, K, L) = 0 ∂K/∂L = −FL/FK
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15. www.seinan-gu.ac.jp/˜shito 2020 6 25 10:41
: Y, C, T
: I, G, A, T0, c, t
Y ∗
= f1
(I, G, A, T0, c, t)
C∗
= f2
(I, G, A, T0, c, t)
T∗
= f3
(I, G, A, T0, c, t)
(1) ∂Y ∗
/∂I
1 f1
(·)
∂Y ∗
∂I
=
∂f1
∂I
f1
(·) (1)
2 Y ∗
= · · · ∂Y ∗
/∂I
(1) F(·) = 0
Y − C − I − G = 0
C − c(Y − T) − A = 0
T − tY − T0 = 0
(2)
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F1
(Y, C, T, I, G, A, T0, c, t) = 0
F2
(Y, C, T, I, G, A, T0, c, t) = 0
F3
(Y, C, T, I, G, A, T0, c, t) = 0
∂Y ∗
/∂I I I
Y ∗
, C∗
, T∗
Y ∗
I
∂Y
∂I
=
II 16