-Diferensial Parsial- Diferensial Total- Chain rule- dll

dss

pdr

r

pdq

q

pdp

s

psrqfc

r

psrqfb

q

psrqfa

p

srqfp

dzz

ydx

x

ydy

z

yzxfb

x

yzxfa

y

zxfy

s

r

q

x

x

),,()

),,()

),,()

'

),,(.2

),()

),()'

),(.1

Diferensial Total

Fungsi dengan lebih dari satu variabel bebas juga dapat diturunkan lebih dari satu kali

Turunan parsial z = f (x,y) kalau kontinyu dapat mempunyai turunannya sendiri. empat turunan parsial :

2

222

2

2

,,,y

fdan

xy

f

yx

f

x

f

• Dapat dilambangkan fxx, fxy, fyx, dan fyy

• fxy = fyx

Cobb-Douglas production function (+=1)

Q = 96K0.3 L0.7

7.7.07.7.0 8.28963.0 KLKLK

QMPPK

3.03.03.03.0 2.67967.0 LKLKL

QMPPL

Market model

dPcQ

bPaQ

s

d 0,

0,

dc

ba

db

ca

bd

ac

d

b

c

a

A

AP

c

a

P

Q

d

b

CdPQ

abPQ

p

s

d

1

1

1

1

1

1

db

caP

db

bcadQ

• Market model

0,

0,

dcdPcQ

babPaQ

s

d

db

caP

0

0

01

01

2

2

db

ca

d

P

db

ca

b

P

dbc

P

dba

P

• Market model

0,

0,

dcdPcQ

babPaQ

s

d

0

0

0

0

2

2

db

cab

d

Q

db

cad

b

Q

db

b

c

Q

db

d

a

Q

db

bcadQ

0

db

b

c

Q0

1

dba

P

Q S1

DP

S0

Q S

DP

D1

0,

0,

dcdPcQ

babPaQ

s

d

0,

0,

dcdPcQ

babPaQ

s

d

02

db

cad

b

Q

Q

S0

D1

D0

P

Q0

Q1

02

db

ca

d

P

Q S0

DP

S1

Y = C + I0 + G0

C = a + b(Y-T); b = MPC (a > 0; 0 < b < 1)

T=d+tY; t = MPT (d > 0; 0 < t < 1)

Y=( a-bd+I+G)/(1-b+tb)C=(b(1-t)(I+G)+a-bd)/ (1-b+tb)T=(t(I+G)+ta+d(1-b))/ (1-b+tb)

btbG

Y

o

1

1

btb

tb

G

C

o

1

)1(

btb

t

G

T

o

1

∂x1/∂d1 = b11

2

1

2221

1211

2

1

d

d

b

bb

x

x

222221

112111

2

1

dbdb

dbdb

x

x

2

1

2212

2111

2

1

d

d

dxdx

dxdx

x

x

Use Jacobian determinants to test the existence of functional dependence between the functions /J/

Not limited to linear functions as /A/ (special case of /J/

If /J/ = 0 then the non-linear or linear functions are dependent and a solution does not exist.

2212

2111

xyxy

xyxyJ

22

11

dxx

ydx

x

ydy

2211 dxfdxfdy

dzz

ydx

x

ydydz

z

fdx

x

f

dzdz

dfdx

dx

df

yy

zxfzzxf

xx

zzxfzzxxfdy

zzxz

zxfzzxxfdy

f(x,z)ydy

y

x

y

x

atau

),(),(

),(),(

0),()z(x, fn ditambahka

),(),(lim

: sebagaikan didefinisi dari Total lDiferensia

lim

lim

0

0

0

0

Let Utility function U = U (x1, x2, …, xn)

Differentiation of U wrt x1..n

U/ xi is the marginal utility of the good xi

dxi is the change in consumption of good xi

nn

dxx

Udx

x

Udx

x

UdU

22

11

• Given a function y = f (x1, x2, …, xn)

• Total differential dy is:

• Total derivative of y with respect to x2 found by dividing both sides by dx2 (partial total derivative)

nn

nn

dxfdxfdxfdy

dxx

ydx

x

ydx

x

ydy

...2211

22

11

22

2

11

2 dx

dxff

dx

dxf

dx

dy nn

This is a case of two or more differentiable functions, in which each has a distinct independent variable.where z = f(g(x)), i.e.,z = f(y), i.e., z is a function of variable y and

y = g(x), i.e., y is a function of variable x

xgyfdx

dy

dy

dz

dx

dz

• If R = f(Q) and if Q = g(L)

LL MRPMPPMRLgQfdL

dQ

dQ

dR

dL

dR

dt

dy

y

z

dt

dx

x

z

dt

dz

sehingga

dyy

zdx

x

zdz

y(t)ytxxyxzz

dxdyxfuufydx

du

du

dy

dx

dy

dan )(dan ),(Kalau

hasilnya. mengalikan serta x keu dan u key sialkan mendiferen

dengan diperoleh /),(dan )(

z

t

x y

Pohon rantai

Kalau w = w(x,y,z) dan x = x(u,v), y = y(u,v), dan z = z(u,v), maka pohon rantai :

vw

u

z

z

w

u

y

y

w

u

x

x

w

u

w

sehingga

/untuk serupa rumusdengan

:

w

y

v

z

u

x

Kalau z = z(x,y), dan x = x(s), y = y(s), dan s = s(u,v), maka pohon rantai menjadi :

vz

u

s

ds

dy

y

z

ds

dx

x

z

u

z

/untuk serupa rumusdengan

z

x

u

s

y

v