slide 0

Maksimisasi utilitas dengan pendapatan

terbatas

� Utility Function

� Budget Constraint

121 2xxxU +=

6024 =+ xx

slide 1

� Lagrangian

602421

=+ xx

[ ]21121 24602 xxxxxL −−++= λ

� Necessary Conditions

042

02460

2

21

=−+=∂

∂

=−−=∂

∂

λ

λ

xL

xxL

slide 2

� Tentukan nilai x1 dan x2

02

042

1

1

2

1

=−=∂

∂

=−+=∂

λ

λ

xx

L

xx

Bentuk Umum

11 1 12 2 1 1

21 1 22 2 2 2

...

...

............................................

n n

n n

a x a x a x d

a x a x a x d

+ + + =

+ + + =

slide 3

1 1 2 2

............................................

...m m mn n ma x a x a x d+ + + =

Ukuran matriks

� Matrix [A] akan disebut berukuran mxn jika

mempunyai m baris dan n kolom

� Lambangnya adalah [A]mxn

� Elemennya disimbolkan dengan a , dimana i

slide 4

� Elemennya disimbolkan dengan aij, dimana i

merupakan urutan baris dan j urutan kolom.

CHAPTER 10 Aggregate Demand I

Matriks

Ax d=

11 1na a

A

=

K

M O M

1

2

x

xx

=

1

2

d

dd

= M

slide 5

1m mna a L

n

x

x

=

M

m

d

d

=

M

mxn ijA a = 1, 2,...,

1,2,...,

i m

j n

=

=

Solusi

Ax d=

1x A d

−=

slide 6

x A d=

Perkalian Matriks

mxn nxo mxoA B C=

[m x n] dan [n x o]syarat

hasil [m x o]

slide 7

hasil [m x o]

Contoh

1 3

28

40

A

=

5

9b

=

slide 8

( ) ( )( ) ( )( ) ( )

1 5 3 9 32

2 5 8 9 82

4 5 0 9 20

C

+

= + = +

Contoh

A =

2 3

1 1

1 0

and B =

1 1 1

1 0 2

slide 9

1 0 [3 x 2] [2 x 3]

A and B can be multiplied

=

=+=+=+

=+=+=+

=+=+=+

=

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

[3 x 3]

A =

2 3

1 1

1 0

and B =

1 1 1

1 0 2

Contoh

slide 10

=

=+=+=+

=+=+=+

=+=+=+

=

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

1 0 [3 x 2] [2 x 3]

[3 x 3]

Result is 3 x 3

Operasi matriks

� Perkalian skalar

3 1 21 77

0 5 0 35

− − =

slide 11

0 5 0 35

11 12 11 12

21 22 21 22

a a ca cac

a a ca ca

=

Keunikan matriks

AB BA≠

2 4 2 4−

slide 12

2 4

1 2A

=

2 4

1 2B

− =

−

240

60024

21

21

−=−+

−=+−−

λ

λ

xx

xx

042

02460

2

21

=−+=∂

∂

=−−=∂

∂

λ

λ

xx

L

xxL

slide 13

020

240

21

21

=−+

−=−+

λ

λ

xx

xx

02

042

1

1

2

1

=−=∂

∂

=−+=∂

λ

λ

xx

L

xx

Matriks

Ax d=

−

−− 60024 1x

slide 14

−=

−

−

0

2

201

410 2

1

λ

x

Operasi matriks

mxn nxo mxoA B C=

33xA =

−

−

=

−

−−

2

60

410

024 1

x

x

slide 15

13

13

xC

xB

=

=

−=

−

−

0

2

201

410 2

λ

x

Solusi

1x A d

−=

−

−−

−60024

1

1x

slide 16

−

−

−=

0

2

201

4102

1

λ

x

Inverse

1 1.

det .A adj A

A

− =

slide 17

det .A

Determinan

�Hanya matriks yang bujursangkar

(baris = kolom) yang mempunyai

determinan

slide 18

Matriks 2x2

11 12

21 22

a aA

a a

=

slide 19

11 22 12 21det A a a a a= −

Langkah 1: Tambahkan 2 kolom

yang pertama

Matriks 3x3

024 −− 24024 −−−−

slide 20

201

410

024

−

−

−−

0

1

2

1

0

4

201

410

024 −−

−

−

−−

Langkah 2: Jumlahkan perkalian

diagonal

Matriks 3x3

24024 −−−−

slide 21

0

1

2

1

0

4

201

410

024 −−

−

−

−−

[(-4x1x-2)+(-2x-4x1)+(0x0x0)]-[(1x1x0)+(0x-4x-4)+(-2x0x-2)]

Minors of a Matrix Determinant

� A minor Mi,j is a reduced determinant found

by omitting the ith row and jth column of a

larger determinant. For example:

slide 22

The Cofactor of Determinants

� A cofactor Ci,j is a minor Mi,j augmented with a

sign rule for the particular purpose of solving

matrix determinants. Cofactors are defined as

follows:

slide 23

201

410

024

−

−

−−

=A

slide 24

201

410

024

)1(11

11

−

−

−−

−= +C

20

4111

−

−=C

Adjoint

= 322212

312111

.

CCC

CCC

CCC

Aadj

slide 25

332313 CCC

Matrix Operations in Excel

slide 26

Select the

cells in

which the

answer

will

appear

Matrix Multiplication in Excel

1) Enter

“=mmult(“

2) Select the

cells of the

slide 27

cells of the

first matrix

3) Enter comma

“,”

4) Select the

cells of the

second matrix

5) Enter “)”

Matrix Multiplication in Excel

Enter these

three

key

strokes

at the

slide 28

at the

same

time:

control

shift

enter

Matrix Inversion in Excel

� Follow the same procedure

� Select cells in which answer is to be displayed

� Enter the formula: =minverse(

slide 29

� Select the cells containing the matrix to be

inverted

� Close parenthesis – type “)”

� Press three keys: Control, shift, enter

� matrix.xlsx

slide 30