經濟數學方法
壹、 矩陣與行列式
∥定義: 階矩陣為一包括 列和 行的數字的方形排列，若以 A 代表
此矩陣，則
例:
分別為 4 和 2 矩陣
∥定義: 若 則
=C
例:
則
mn n m
mna
aaa
aaa
aaa
A ij
nmnn
m
m
)(
21
22221
11211
1113
3111
,
531
321
213
102
BA
3 4
mnijmnij bBaA )(,)(
mnijmnijij CbaBA )()(
mnijaA )(
31
52
12
,
11
23
12
BA
22
75
20
3111
5223
1122
BA
84
513
412
31
52
12
55
1015
510
)1(55 BABA
∥ 定義:若 A=( 為 矩陣，B=( 為 矩陣，則 A 和 B 的
乘積 AB 為 矩陣 C
例: 求 AB 及 BA
=
=
BA 無法計算
∥ 行列式: Cramer's Rule
已知
例:解下列聯立方程式:
AAA 2
11
23
12
2
22
46
24
11
23
12
11
23
12
)ija mn )ijb km
kn
130
112
001
,
102
210
BA
130
112
001
012
120
AB
)1(
12)1(
132
172
33 32
1212111 bXaXa
2222121 bXaXa
21122211
122221
2221
1211
222
121
*
1 aaaa
abab
aa
aa
ab
ab
X
21122211
121211
2221
1211
221
111
*
2 aaaa
baba
aa
aa
ba
ba
X
0
2
5
312
121
111
3
2
1
X
X
X
貳、微分
∥ 微分公式:
∥ 若
∥ 設 與 皆存在:
∥
∥
∥
∥ 鏈鎖律(chain rule):
設函數 f 與 g 皆可微分
∥ 反函數 (inverse function):
設函數 f 與 g 滿足 f(g(Y))=Y 函數 g 為 f 之反函數
g(f(X)=X 且 g=f
032
22
5
321
321
321
XXX
XXX
XXX
9
43
9
312
121
111
310
122
115
*
1
X
9
23
9
302
121
151
*
2
X
1
*
3 9
21
9
012
221
511
X
)(XfY
dX
dY
X
xfXXf
Xf
X
Y
X
)()(
lim)(
0
)(2
2
2
2
Xf
dX
Yd
X
Y
RXnXXfRXXXf nn ,)(,)( 1
)(Xf )(Xg
dX
Xdg
dX
Xdf
XgXf
dX
d )()(
)()(
dX
Xdf
Xg
dX
Xdg
XfXgXf
dX
d )(
)(
)(
)()()( 乘法公式
0)(,
)(
)()()()(
)(
)(
2
Xg
Xg
XgXfXgXf
Xg
Xf
dX
d 除法公式
)())(())(( XgXgfXgf
dX
d
1
XXff
YYff
))((
))((
1
1
∥ 偏微分:
例:
∥ 全微分:
例: TE=P Q
∥ 自然對數(e)與自然指數(ln):
性質: (1) 、
、
(2)
(3)設 存在
(4)
(5)
(6)
),(),( 211
1
21 XXfX
y
XXfy
),(),( 212
2
21 XXfX
y
XXfy
XYX
dX
d
623 2
),( 21 XXfy
2
21
1
dX
X
y
dX
X
y
dy
PdQdPdTE 2
0lim1)0()(
X
X
x efeXf
X
X
elim
XfXXf
X
lnlim0)1(ln)(
X
X
lnlim
XX ee
dX
d
f )()( )()( Xfee
dX
d XfXf
RYXeee YXYX ,,
X
X
e
e
1
0,
1
ln X
X
X
dX
d
x
y ex
lnx
1
1
(7)
(8)
(9)
(10)
(11)
(12) 且
(13)
∥ 切線與射線:
∥給定切線上任一點(X, Y)
∥ 射線角度值
∥函數的高階導數:
、
∥函數的臨界點及反曲點:
(一) 若
則 為函數 f 之臨界點
(二)
函數 f 在 為嚴格遞增
0,
1
X
X
Xn
dx
d
)(
(
)(
Xf
Xf
Xfn
dX
d
YXYX lnlnln
YX
Y
X
lnlnln
XYX Y lnln
Xe X ln Xe X ln
YXX eY ln
)(
)(
0
0 Xf
XX
Xfy
0
0tan
X
y
dX
dY
dX
d
dX
Yd
2
2
2
2
3
3
dX
Yd
dX
d
dX
yd
X
XfXXf
XX
XfXf
Xf
XXX
)()(
lim
)()(
lim)( 00
0
0
0
0
00
，不有在Xf或Xf，函數定義域DfX ))((0)()( 000
0XX
ba,
)()( 2121 XfX則fXX
(x0,y0)
y=f(x)
α
y
x
f/(x)>0
X1 X2
f(x2)
f(x1)
X
Y
a b
函數 f 在 為嚴格遞減
(三)
ba,
)()( 2121 XfX則fXX
0)( Xf 為上凹ba函數f在baX ,,
f/(x)<0
X1 X2
f(x2)
f(x1)
X
Y
a b
concave upward
f/(x)>0
f//(x)<0
X
Y
上凹 f/(x)<0
f//(x)>0
X
Y
上凹
concave downward
f/(x)<0
f//(x)<0
X
Y
下凹 f/(x)>0
f//(x)<0
X
Y
上凹反曲點
(inflection point)
上凹
下凹
x
y
C0
故 函數遞增遞減性， 函數凹性
(四)第一導數檢驗定理: 或
X<C X>C 切記
- + f(C)為局部極小值
+ - f(C)為局部極大值
- -
+ + f(C)為非局部極值
第二導數檢驗定理:
∥
∥
∥ 本定理失敗
參、積分
(一) 不定積分(Indefinite integral)
而 F 之導函數、F 為 f 之
反導數故 F 為 f 之反導數
∥ 性質: ∥
0)( Xf 為下凹ba函數f在baX ,,
f f
0)( Cf 不存在Cf )(
f
f
f
f
0)( Cf
為局部極小 值CfCf )(0)(
為局部極大 值CfCf )(0)(
0)( Cf
積分 值積分函數、dXX積分符號、f ::)(:
dXXf )(: )()( XfXF f為
)()()( 常數KXFdXXf
dXXgdXXfdXXgXf )()()()(
x
y
f(C1)
f(C2)
C2
局部
最小值
C1
局部
最大值
∥
∥
∥
∥
(二) 定積分 (definite integral)
∥ 性質: ∥
∥
∥
∥
∥
∥ X=a 被定義
∥
∥
C dXXfCdXXf )()(
)()( XfdXXf
dX
d
CXfdXXf
dX
d
)()(
CXndX
X
1
Cba )( abCdX
Cba dXXfCdXXf ba )()(
dXXgdXXfdXXgXf bababa )()()()(
)()()()()( 錯dXXgdXXfdXXgXf bababa
baCdXXfdXXfdXXf bccaba ,,)()()(
f在 0)( dXXfaa
dXXfdXXf ab
b
a )()(
0)(0)( dXXfX設f ba
x
y
f(x)
a b
dxxfba )(
肆、齊次函數與尤拉定理
(一) 階齊次函數 (homogeneous function of degree n)
∥ 定義:
若
則稱 階齊次函數
(二) 尤拉定理 (Euler Theorem)
∥定義:若
則
∥ 証明:
對入微分:
令
(三) 齊序函數 (同位函數) (homothetic function)
∥ 定義: (一階齊次函數的正單調上升轉換稱之)
若 為 1 且
稱之。
例: 若有齊次偏好，所得 1000 元，買 40 本書，60 張 CD,
當所得為 1500 時，而書，CD 價格不變，會買 60 本書，
90 張 CD
伍、古典規劃分析:最適化(Optimization)
(一) 未受限制下的極大與極小
∥ 單變數函數(X)
1. 極大:
n
),( 21 XXfy
0),,(),( 2121 XXfXXf
n
為nXXfy ),( 21
nDO為HXXfy ...),( 21
21
1 X
f
X
X
f
ny
2X
),(),( 2121 XXfXXf
n
),( 21
12
2
1
1
XXfn
X
X
fX
X
f n
:1 ),(: 21
2
21
1
XXfnX
X
f
X
X
f
),( 21 XXg 0 dg
df
f
),()),(( 2121 XXhxxgf則y
Max )(Xfy
.
I=1500I=1000
CD
book40 60
60
90
2. 極小:
(二) 多變數函數(
1.
∥
∥
∥ 有限制條件下之極值分析:
(
...0)( COFdXXfdy
0)( Xf
dX
dY
判斷選一個CO由SX
個解求得CO由FX
...
2...
*
2
*
1
...02 COSYd
MaxYXXf
dX
Yd
dXXfYd *112
2
22 0)(0)(
Min )(Xfy
0)(...
0)(...
XfCOS
XfCOF
), 21 XX
),( 21
)(
XXfYMax
Min
... COF 0dY
0),(),( 2211211 dXXXfdXXXf
*
2212
2
*
1211
1
0),(0
0),(0
XXXf
X
Y
XXXf
X
Y
... COS
正定Min全為正MatrixHessionYd
負定Max負正相間MatrixHessianYd
)(0
)(0
2
2
0,0
2221
1211
11
2221
1211 ff
ff
f
ff
ff
H
Max methodLagrangeXXfy ),( 21
)Min
..tS CXXg ),( 21
MaxStep :1 CXXgXXfXXL ),(),(),,( 212121
...:2 COFStep
0
1
X
L
0),(),( 211211 XXgXXf
*
1X
0
2
X
L
0),(),( 212212 XXgXXf
*
2X
正負相間(Max)
全為正 (Min)
陸、古典規劃分析應用:
Optimization
(1) max
Q
(2) min C=W
3 個主要
問題類型
(3) max f(x)
max U(x, y)
x or
∥ The Structure of an Optimization Problem
Max f(x) f(X)=objective function
X: choice variables
S: feasible set
solutions:
Important general problems about the solutions to any optimization
problem:
(1) Existence of Solutions
Propositions: An optimization problem always has a solution if
0
L 0),( 21 CXXg *
...:3 COSStep Ld 2 0
Boarder Hessian Matrix
021
222221221
112121111
gg
ggfgf
ggfgf
F
)()( QCPQQ
krL
KL,
),(.. LKFQts
yx,
0,0)( xxg Iypxp yx
sx
*X
Sxxfxf )()( *
(1) the objective function is “ continuous”
(2) the feasible set is “nonempty, close and bounded”
(2) Local and Global Optima
Prepositions: A local maximum is always a global maximum if
(1) the objective function is quasiconcave.
(2) the feasible set is convex.
(3) Uniqueness of Solution
Propositions: Given an optimization problems in which the
feasible set is convex and the objective function is nonconstant
and quasiconcave, a solution is unique if:
(1) the feasible set is strictly convex, or
(2) the objective function is strictly
quasiconcave, or
(3) both
(4) Interior and Boundary Optima
(5) Location of the Optimum
min
max f(x)
X R
(多變數)
∥ Multivarial Case
Gradient vector of f
Hessian of f
)(),()(:
),()(:
****
*
xBexxfxfSolutionLocal
SxxfxfSolutionGlobal
0
)(
dx
xdf
(max)0
(min)0)(
2
2
dx
xfd
21xx
)( 21xxfY
:
/
/
2
1
2
1
f
f
xf
xf
f
nn
n
ffn
ff
H
........1
......... 111
j
i
ij x
f
f
now, max f(
(負定)
(
∥ Quadratic Forms and their Signs
symmetric:
X A X=(
=
(1) Negative Semidefinite
(2) Negative definite
(3) Positive Semidefinite
(4) Positive definite
ex n=2
), 21 xx COF .. 01
x
f
21, xx 0
2
x
f
02
1
2
x
f 011 f
02
2
2
x
f
0)()()
2221
1211
21
2
2
2
2
2
1
2
ff
ff
即
xx
f
x
f
x
f
2
12121121122211 )(0 fffffff
2
12 )( f
nnn
n
aa
aa
A
1
11 1
jiij aa
mnnn
n
m X
X
aa
aa
XX 1
1
111
1 ..............
............
)........
n
i
n
tj
jiij xxa
1
RXAXX ,0
0,0 XAXX
RXAXX ,0
0,0 XAXX
2
1
2221
1211
21 )( x
x
aa
aa
XXAXX
=
=
=
-Negative definite
and
- Positive definite:
and
續 Hessian;
H is negative definite if
H is positive definite if
General Case
A =(
Negative definite:
MAX …….
Positive definite:
2
2222112
2
111 2 xaxxaxa
)()2( 2222
2
22
11
2
122
22
11
12
12
21
11
122
111 XaXa
a
X
a
a
xx
a
a
xa
2
2
11
2221
1211
2
2
11
12
111 )( Xa
aa
aa
x
a
a
xa
011 a 0
2221
1211
aa
aa
011 a 0
2221
1211
aa
aa
2
12121111 ,0 ffff
022 f
2
12121111 ,0 ffff
022 f
X X
nnnn
n
n
x
x
aa
aa
XX
i
....
....
............
....
)........
1111
1
011 a 0
2221
1211
aa
aa
nnni
n
n
aa
aa
...
.........
...
)1(
111
011 a 0
2221
1211
aa
aa
MIN……
∥ Optimizations: The unconstrained case
I. may f(
Min
Gradient Veotor
Hessian Matrix
Necessary conditions
Sufficient conditions Df=0
H is definite
f is concave (dx) H(dx)<0
convex >0
ex 1.
2.
0
...
.........
....
1
111
nnn
n
aa
aa
)x
............
)( 11
n
n
f
f
X
f
X
f
x
xf
Df
....
..
......
...
2221
11211
2
nnnt
n
ff
ff
fff
fDH
isHCOS
DfCOF
..
0..
tesemidefininegative
positive
nositive
negative
1
)()(max qcqpq
q
MCP
dp
d
0
00 2
2
2
2
dq
Cd
dq
d
xwxpfx
q
*)()(max
00 *
Wi
Xi
f
p
xi
WiVMPi
H is negative definite f is concave.
II. The Constrained Case
g( =b
Lagrangian Function:
L (
constraint gualification:
D
. Dg( 全微分
∥ Bordered Hessian
. for
x
max )( xf
)x 在有限制下，求最大點dxi
Xi
g
.....0
max ))(()(), bxgxfx
x, U
xi
xg
)(
0
Xi
L
0
)(
,,,2,1
Xi
g
Xi
xf
nI
xj
g
xi
g
xjf
xif
)(
)(
0)(
xgb
L
2
2
2 )()(
xd
xLd
xL
0))(()( 2 dxxLDxd T
xd 0) xdx
)(
...
............
......
......
),( 2
2
222
1
2
2
1
2
1
1
2
2
1
2
2
2
2 xLD
x
L
xx
L
x
L
xx
L
x
L
g
x
L
x
L
x
LL
xLDH
ninn
n
n
max
(min)
The naturally ordered principled mincrs of the bordered
(all be negative) guaslconcave
Hessianmatrix alternate in sign, the sign of the first being positive
ex min
Lagrangian funotion:
.
∥ Nonlinear Programming
Max f( inequality constraint
0
0
0
0
2
3
2
23
2
13
2
3
32
2
2
2
2
12
2
2
31
2
21
2
2
1
2
1
321
2
2
2
12
2
2
21
2
2
2
1
21
x
L
xx
L
xx
Lg
xx
L
x
L
xx
Lg
xx
L
xx
L
x
Lg
ggg
x
L
xx
L
g
xx
L
x
L
g
gg
w x
x
ts .. gxf )(
))((),( * gxfxwxL
x
max
MRTS
xj
f
xi
f
wj
wi
nI
xi
xf
wix
xi
L
,...2,10
)(
),(
*
**
0)(),( **
xfgx
L
)x
x
fs .. )(xg b 0)( xg
x 0 0 x
0
)(
*
*
*
xi
xf
X i
0* iX
Max f(x)
Langrangian Function:
Max
Ex
=
因有 ineguediy,
…. 所以要多考慮這些可能
ex
“∥” min
0
)( 2
xi
xf
x
ls.
)()(),(
1
n
i
xigixfxL
),,,( 1 nXXf
nxx
ts
...1
.. bxg )(
0,.......0,0 21 uxxx
),,,,,,,,,,( 2121 nn uuuxxxL
)...))()( 2211 nn xuxuxubxgxf
01
111
u
x
g
x
f
x
L
02
u
x
g
x
f
x
L
nnn
0,0**
L
0,0,0 11
1
11
xu
u
L
xu
0,0,0
nn
n
nn xuu
L
xu
2211 xwxw
yxx 21
01 x 02 x
22112122112121 )(),,,,( XMXMyxxxwxwMMXXL
四
種
可
能
情
況
檢查這些條件是否都符合
∥ ∥
限制式中 共有四種組合
Case 1 ( 代入 (2) 式)
Case 2 step2
(
step2
用第 2 種生產要素
Case 3 用第 1 種生產要素
Case 4
Ex
Kuhn-Tucker Formulation
Kuhn-Tucker Conditions
02211
21
2
2
11
1
,0
)3.....(0)(
)2.....(02
)1.(..........0
XMXM
yxx
L
Mw
X
L
Mw
X
L
0
1
1
U
L
0
2
2
U
L
0,0 21 XX
00,0 21 yXX 0, 22 x
,0,0 21 XX 222 0,0 WW
))1(,0, 11 式代入x
0
11112 ,
Wwyx
.....21 WW
....,0,0 1221 WWXX
212121 0,0,0 WWXX
yWyWyWWC 2121 ,min),,(
2121
2112
2121
WWyWyWC
WWyWyWC
WWifyWyWC
0)(..0)(.
)(min)(max
XgtsXgts
XfXf
))(()(),( bXgXfXL
0,0,0
0,0(max),0
1
(min)
1
LL
X
L
XX
X
L
i
i
i
i
i
2211min XWXW
.
(K-T conditions):
Utility Maximization Problem
max u(x, y)
x, y
Comparative Statics
Implicit Functions
Implicit Functions Theorem
If D= =
-totally differentiating the system
21, xx
yXX 21
0,0 21 XX
)( 212211 yXXXWXWL
0,0,0
1
111
1
X
L
XXW
X
L
0,0,0
2
2222
X
L
XXW
X
L
021
YXX
L
0 0
L
IyPxP yX
0,00,0 yxyx
).....(
.....,
,
0)...................(
0)....,........(
21
**
2
*
1
2121
2121
1
m
i
j
n
mn
n
mn
aaa且x
XXX
可求得equationsn
xxxxxxF
xxxf
0
1
11
1
n
n
n
n
ff
ff *
jX )......( m
ih
之影響為正或負a受即X
aj
xi
mi .....0* 1
*
*
0..........2 111
111
21
1
1 mmnnnn adfdfdxfdxfdxf
1
1
112211 .... dffdxfdxfdxf n
n
nn
n
n
nn
0
...
amdmn
f n
∥ ∥ ∥ ∥
D
<Cramer's Rule>
(無限制式) ex max
Totally differentiate with respect to
By Cramer's Rule
m
n
mn
n
n
mnn
m
n
mn
n
n
mmnn
n
n
n
n
n
d
d
ff
ff
dfdaf
dfdf
dx
dx
ff
ff
1
1
11
1
11
1
1
1
11
1
11
1
....
...
).....(
)......(
........
...........
idx ijD jd
22
2
/
/
xf
xf
D
Dx
jd
D
D
dx ij
j
iij
i
221121 ),(. XWXWXXfP
0
0
11
1
1
11
1
1
wpf
x
wpf
x
:1W
01
01
1
*
2
2
2
1
*
1
1
2
1
*
2
2
1
1
*
1
1
w
x
x
f
P
w
x
x
f
P
w
x
x
f
P
w
x
x
f
P
0
1
1
*
2
22
1
*
1
21
1
*
2
12
1
*
1
11
w
x
pf
w
x
pf
w
x
pf
w
x
pf
2221
1211
PfPf
PfPf
0
1
1
*
2
1
*
1
w
x
w
x
0
)(
0
1
2
122211
22
2221
1211
22
12
1
*
1
fffp
f
pfpf
pfpf
pf
pf
w
x
0
)(
0
1
2
122211
21
2221
1211
12
11
1
*
2
fffp
f
pfpf
pfpf
pf
pf
w
x
同理
(有限制式)
max
.
=
= (p
>0 <0
From
0
)( 2122211
11
2
*
2
fffp
f
w
x
0)12(
)( 2122211
12
2
*
1
fsign
fffp
f
w
x
1121 )( XWXXpf
21xx
22
XX
)()( 21121 XXXWXXPL
)(
),(
0
0
0
211
*
22
0
211
*
11
2
2
11
1
XPWXX
XPWXX
pf
X
L
wpf
X
L
022
0
XX
L
),( 0211
* XPW
1X
L
2X
L
H
2
2
2
21
2
2
21
2
1
2
1
21
0
xxx
xg
XXX
L
xx
g
x
g
x
g
2221
1211
0
01
10010
pfpf
pfpf
011 pf )011 p
0
0..
011max
H
C又又OF
fTL
totally differentiating with respect to w1:
By the Cramer's Rule:
把 算出，代入利潤函數中，即可得:…*profit Function
But 此題中 可直接代入 為 one decision 的問題，不需如此
麻煩。
0
0)(
0)(
2
0
2
**
2
*
12
1
*
2
*
11
2
1
XX
WXXPf
WXXPf
L
X
L
X
L
1
..
對W
COF
0
01
1
*
1
*
2
22
1
*
1
21
1
*
2
12
1
*
1
11
ww
x
pf
w
x
pf
w
x
pf
w
x
Pf
0
1
0
10
10
10100
1
*
1
1
*
1
1
*
2221
1211
w
x
w
x
w
pfpf
pfpf
0/
0/0
0/1
1121
1
*
11
1
*
11
1
*
pffw
x
pfw
x
pfw
x
1121 )( xwxxpf
),,()),,(),,,(( 021
*
11
0
21
0
21
*
1
* xpwxwxpwxpwxpf
和x*1
*
2x
0
22 XX
