Teknik Iterasi Penyelesaian Model Multisektor Dinamis

Ekonomi dan Keuangan Indones ia Vo lume X L I V N o m o r 4 , 1 9 9 6

Teknik Iterasi Penyelesaian Model Multisektor Dinamis

Edison Hulu

Abstract

All the multisector models start with the input-output model, which captures sectoral interdependence arising from the floiv of intermediate goods among sector. Even with its strong linearity assumptions, input-output models and their extensions represent powerful tools for applied eqidlibrium analysis. For industry studies, it is useful to distinguish its applications, such as the analysis of the economic structure, formulation of the program action, and prediction of future event.

In terms of the mathematical structure or the methodology of the model, the development of the input-output models are two ways, the static and dynamic formulation. In the static input-output model, all the component of final demand are treated as exogenous variables. The dynamic model treats investment demand as endogenous and incorporates two fundamental assumption about technology ajid capital. These assumptions are (i) fixed incremental capital-output ratios by sectors, and (ii) sectoral capital stocks have a fixed compositional structure by sector of origin.

In this paper the author deals with the dynamic input-output model, particularly in the solution procedure of the model. In finding the solution of the model, the inverse of the capital matrix is assumed to exist. In fact, because many sectors do not produce capital goods, it means that some of the diagonal elements of the capital matrix in the dynamic input model will be zero, then the solution of the nwdel does not exist. This problem is called the singularity problem associated with the difference equation of Leontief s multisector dynamic model.

Some of the existing studies such as Kenderick (1972), Kreijger and Neudecker (1976), Livesey (1976), Schinnar (1978) are concerned with solving the

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smgularity of the model, but the results are still debated. In this paper the author examine one approach of solving the dynamic input-output model by using the Seidel's iterative calculation. This approach will not produce unique solution that converge toward the stability condition.

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Tehnik Iterasi Penyelesaian Mcxiel Multisektor Dinamis

I . PENDAHULUAN

M o d e l m u l t i s e k t o r t e r a p a n , b a i k s t a t i s m a u p u n d i n a m i s d i m u l a i d e n g a n m o d e l i n p u t - o u t p u t y a n g u n t u k p e r t a m a k a l i d i a p l i k a s i k a n d i A m e r i k a S e r i k a t p a d a t a h u n 1 9 4 7 d e n g a n P r o f e s o r W a s s i l y L e o n t i e f s e b a g a i k o o r d i n a t o r p r o y e k d a n s e k a l i g u s s e b a g a i p e n c i p t a f o r m u l a s i m o d e l t e r s e b u t . S e j a k s a a t i t u s a m p a i k i n i , b a n y a k n e g a r a y a n g t e l a b m e i a k u k a n p e n e l i t i a n g t m a m e n g u m p u l k a n d a t a ( t a b e l input-output) p e n d u k u n g m o d e l t e r s e b u t . I n d o n e s i a t i d a k k e t i n g g a l a n d a l a m p e r k e m b a n g a n t e r s e b u t , d a l a m h a l i n i B i r o P u s a t S t a t i s t i k ( B P S ) t e l a b m e m p u b l i k a s i k a n t a b e l i n p u t - o u t p u t I n d o n e s i a s e k a l i d a l a m l i m a t a b u n d a n d i m u l a i s e j a k t a b u n 1 9 7 1 , a t a u t e p a t n y a a d a i a b t a b u n 1 9 7 1 , 1 9 7 5 , 1 9 8 0 , 1 9 8 5 , d a n 1 9 9 0 . D i s a m p i n g i t u , B P S t e l a b m e i a k u k a n p e n e l i t i a n t a b e l i n p u t - o u t p u t d a e r a h p a d a b e b e r a p a p r o p i n s i d i I n d o n e s i a . S i n g k a t n y a , p e n g e m b a n g a n p e n g a p l i k a s i a n m o d e l i n p u t - o u t p u t s a n g a t p e s a t , t i d a k h a n y a d i n e g a r a -n e g a r a m a j u t e t a p i j u g a d i n e g a r a - n e g a r a b e r k e m b a n g .

M e n g a p a p e r k e m b a n g a n p e n g a p l i k a s i a n m o d e l i n p u t - o u t p u t s a n g a t p e s a t ? A d a b e b e r a p a a l a s a n , a n t a r a l a i n , y a i t u ( a ) p a r a p r a k t i s i s e m a k i n m e n y a d a r i b a b w a p e r k e m b a n g a n k e g i a t a n e k o n o m i s a n g a t p e s a t , s e b i n g g a d a l a m m e n g a m b i l k e p u t u s a n - k e p u t u s a n p e n t i n g s e m a k i n d i p e r l u k a n k e h a t i - h a t i a n , k o n s e k u e n s i n y a i a l a h a l a t p e n g a n a l i s i s a n s e m a k i n d i b u t u h k a n , s e b a g a i s a l a b s a t u p i l i b a n a d a i a b m o d e l i n p u t -o u t p u t , ( b ) p a r a p e m b u a t m o d e l m e n y a d a r i b a h w a m o d e l i n p u t - o u t p u t s e c a r a i m p l i s i t m e n c a k u p i n d i k a t o r m a k r o e k o n o m i K e y n e s i a n d a n a n a l i s i s k e t e r k a i t a n a n t a r - s e k t o r i n d u s t r i , s e b i n g g a f u n g s i n y a l e b i h l u a s d i b a n d i n g k a n d e n g a n a n a l i s i s p e n g g a n d a m a k r o e k o n o m i a g r e g a t , ( c ) m o d e l i n p u t - o u t p u t d a p a t d i g i m a k a n u n t u k . k e p e r l u a n p e r e n c a n a a n d a e r a h , p e r e n c a n a a n k e p e r l u a n e n e r g i , a n a l i s i s d a m p a k l i n g k u n g a n , a n a l i s i s j a r i n g a n t r a n s p o r t a s i , a n a l i s i s p e m b a n g u n a n b e r k e l a n j u t a n , d a n s e b a g a i n y a , ( d ) m o d e l input-output a d a i a b a w a l p e r m u l a a n p e n g e m b a n g a n m o d e l m u l t i s e k t o r e k o n o m i , s e b i n g g a e k s i s t e n s i m o d e l i n i m e r u p a k a n f o n d a s i p e n g e m b a n g a n m o d e l - m o d e l m u l t i s e k t o r t e r a p a n y a n g I e b i b l u a s c a k u p a n n y a , s e p e r t i m o d e l S A M (Social Accounting Matrix), m o d e l C G E (Computable General Equilibrium).

D i i i b a t d a r i s i s i d i m e n s i w a k t u , p e n g e m b a n g a n m o d e l input-output d a p a t d i b e d a k a n a t a s d u a s t r u k t u r , y a i t u m o d e l s t a t i s d a n m o d e l d i n a m i s . D a l a m m o d e l s t a t i s , v a r i a b e l p e r m i n t a a n a k b i r , s e p e r t i k o n s u m s i

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r u m a h t a n g g a , k o n s u m s i p e m e r i n t a h , i n v e s t a s i , d a n e k s p o r d i p a n d a n g s e b a g a i v a r i a b e l e k s o g e n . F a e d a h a n a l i t i s m o d e l s t a t i s y a i t u m e l a l u i p e n g g a n d a input-output d a p a t d i h i t u n g b e r a p a s a t u a n o u t p u t d i t i n g k a t k a n i m t u k m e n d u k u n g p e r m i n t a a n a k h i r , d a n b e r a p a s a t u a n i n p u t p r i m e r ( t e n a g a k e r j a , m o d a l , t a n a h , d a n i n p u t p r i m e r l a i n n y a ) d i p e r l u k a n u n t u k m e n d u k u n g p e n i n g k a t a n o u t p u t p a d a s e t i a p s e k t o r i n d u s t r i . D a l a m m o d e l d i n a m i s , s a l a h s a t u v a r i a b e l p e r m i n t a a n a k h i r d i j a d i k a n s e b a g a i v a r i a b e l e n d o g e n , y a i t u i n v e s t a s i . F a e d a h a n a l i t i s d a r i p a d a m o d e l input-output d i n a m i s , y a i t u m e r e n c a n a k a n a l o k a s i i n v e s t a s i p a d a b e r b a g a i s e k t o r i n d u s t r i y a n g m e n g h a s i l k a n n i l a i t a m b a h n a s i o n a l y a n g p a l i n g o p t i m a l .

D a l a m m a k a l a h i n i , p e n u l i s m e m b e r i p e r h a t i k a n k h u s u s t e r h a d a p m o d e l i n p u t - o u t p u t d i n a m i s . M e m p e r b a t i k a n s t r u k t u r p e r s a m a a n m o d e l i n p u t - o u t p u t d i n a m i s f o r m u l a s i L e o n t i e f , d a l a m p r o s e s p e n y e l e s a i a n n y a m e n g b a d a p i b e b e r a p a m a s a l a h . A n t a r a l a i n , y a n g p a l i n g b a k i k i y a i t u , s a l a h s a t u m a t r i k s ( m a t r i k s m o d a l ) , d a l a m d u n i a n y a t a m e m i l i k i p e l u a n g u n t u k t i d a k m u n g k i n d i i n v e r s j i k a m e m a k a i p e n y e l e s a i a n p e r s a m a a n d i f f e r e n s i a l . K a r e n a t i d a k s e m u a s e k t o r m e n g h a s i l k a n b a r a n g m o d a l , s e m e n t a r a j u m l a h b a r i s d a n k o l o m m a t r i k s m o d a l s a m a d e n g a n b a n y a k n y a j u m l a h s e k t o r d a l a m p e r e k o n o m i a n n e g a r a , s e h i n g g a e l e m e n d i a g o n a l m a t r i k s t e r s e b u t a d a l a h n o l , k h u s u s n y a b a g i s e k t o r y a n g t i d a k m e n g h a s i l k a n b a r a n g m o d a l . O l e h k a r e n a i t u , m e n u r u t s t u d i y a n g a d a , s u l i t d i a p b k a s i k a n m o d e l i n p u t - o u t p u t d i n a m i s j i k a m e m a k a i p e n y e l e s a i a n t e r s e b u t d i a t a s . D a r i b e b e r a p a s t u d i y a n g a d a d i k e t a h u i b a b w a K e n d e r i c k ( 1 9 7 2 ) , K r e i j g e r a n d N e u d e c k e r ( 1 9 7 6 ) , L i v e s e y ( 1 9 7 6 ) , S c h i n n a r ( 1 9 7 8 ) t e l a b b e r u s a h a m e n c a r i s o l u s i p e n y e l e s a i a n m a s a l a h i n v e r s m a t r i k s d a l a m m o d e l d i n a m i s L e o n t i e f , n a m u n h a s i l n y a m a s i b b e l u m m e m u a s k a n .

M e n g a t a s i m a s a l a h t e r s e b u t d i a t a s , p e n y e l e s a i a n m o d e l i n p u t -o u t p u t d i n a m i s d a p a t d i t e m p u h d e n g a n m e n g g u n a k a n p e n d e k a t a n p e n y e l e s a i a n i t e r a s i S e i d e l . P e n d e k a t a n i t e r a s i t e r s e b u t t e l a b d i a p l i k a s i k a n p a d a b e b e r a p a k a s u s p e n g e s t i m a s i p e r u b a h a n t e k n o i o g i d a l a m m o d e l i n p u t - o u t p u t , s e p e r t i A l m o n , d a n Y u k i o K a n e k o . D a l a m m a k a l a h i n i a k a n d i j e l a s k a n m e t o d e i t e r a s i S e i d e l s e c a r a t e o r e t i k d a n a k a n d i t u n j u k k a n b e b e r a p a k e l e m a b a n n y a d a n s e k a l i g u s s e b a g a i s a l a h s a t u o b j e k s t u d i l a n j u t a n d a l a m p e n y e m p u m a a n p e n y e l e s a i a n m o d e l m u l t i s e k t o r d i n a m i s f o r m u l a s i L e o n t i e f .

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Tehnik Iterasi Penyelesaian Model Multisektor Dinamis

I I . PEMBENTUKAN M O D E L INPUT-OUTPUT DINAMIS

S t r u k t u r p e r m i n t a a n t e r h a d a p o u t p u t d a l a m m o d e l i n p u t - o u t p u t s t a t i s d a p a t d i t u n j u k k a n d a l a m s e b u a h p e r s a m a a n l i n i e r , y a i t u :

X, = A , X , + f , - m , (2.1)

d i m a n a :

X, = v e k t o r k o l o m b e r u k u r a n n x l y a n g m e m m j u k k a n o u t p u t s e k t o r -s e k t o r i n d u s t r i p a d a p e r i o d e t ,

A, = m a t r i k s b e r u k u r a n n x n y a n g m e n u n j u k k a n k o e f i s i e n i n p u t - o u t p u t p a d a p e r i o d e t ,

f , = v e k t o r k o l o m b e r u k u r a n n x l y a n g m e m m j u k k a n p e r m i n t a a n a k b i r t e r b a d a p o u t p u t s e k t o r - s e k t o r i n d u s t r i p a d a p e r i o d e t , j

m , = v e k t o r k o l o m b e r u k u r a n n x l y a n g m e n u n j u k k a n i m p o r k o m o d i t a s y a n g s e j e n i s d e n g a n o u t p u t s e k t o r - s e k t o r i n d u s t r i d a l a m n e g e r i p a d a p e r i o d e t .

E l e m e n m a t r i k s A , p a d a p e r s a m a a n ( 2 . 1 ) d i b e r i n o t a s i a i j ,, u n t u k i , j = 1 , 2 , . . . , n , d i m a n a n m e m m j u k k a n b a n y a k n y a s e k t o r i n d u s t r i , a d a i a b k o e f i s i e n i n p u t - o u t p u t y a n g d i p e r o l e h d e n g a n m e n g g i m a k a n r u m u s , y a i t u :

Xij.t Hij,, - — ; 1 , 2 , . . , n

Xj. . (2.2)

d i m a n a , Xjj , = o u t p u t s e k t o r i n d u s t r i k e - i y a n g d i m i n t a s e k t o r i n d u s t r i k e -j i m t u k d i p e r g u n a k a n s e b a g a i i n p u t p a d a p e r i o d e t , d a n Xj = o u t p u t s e k t o r k e - j p a d a p e r i o d e t .

V e k t o r f , p a d a p e r s a m a a n ( 2 . 1 ) d a p a t d i u r a i k a n m e n j a d i b e b e r a p a b a g i a n , y a i t u p e r m i n t a a n u n t u k i n v e s t a s i ( b , ) , d a n p e r m i n t a a n u n t u k k o n s u m s i r u m a h t a n g g a d a n p e m e r i n t a b ( d , ) , s e r t a e k s p o r ( e , ) , a t a u d a l a m s e b u a h p e r s a m a a n l i n i e r d a p a t d i t u n j u k k a n , y a i t u :

f, = h.+d. + e,. (2.3)

P e r s a m a a n ( 2 . 3 ) d i s u b s t i t u s i k a n k e d a l a m p e r s a m a a n ( 2 . 1 ) d a n d i p e r o l e h :

X, = A, X, + h, + d, + e, - m,. (2.4)

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V e k t o r m , p a d a p e r s a m a a n ( 2 . 4 ) d a p a t d i d e f i n i s i k a n s e b a g a i r a s i o t e r b a d a p o u t p u t p a d a p e r i o d e t , a t a u d a l a m s e b u a b s u s i m a n p e r s a m a a n d i t i m j u k k a n s e b a g a i :

m , = M,x, , (2.5)

d i m a n a .

M, =

mil 0

0 m22

0 0 ITln

(2.6)

d a n .

m J.t Xj..

; j = L 2 , . . , n (2.7)

d i m a n a , m j j = k o e f i s i e n i m p o r u n t u k s e k t o r k e - j , r r i j , = i m p o r b a r a n g y a n g s e j e n i s d e n g a n k o m o d i t a s p r o d u k s i s e k t o r k e - j , d a n X|, = o u t p u t s e k t o r k e -j p a d a p e r i o d e t .

K e o f i s i e n n i j j y a n g d i p e r o l e h p a d a p e r s a m a a n ( 2 . 7 ) m u n g k i n n i l a i n y a l e b i h b e s a r d a r i s a t u . S e a n d a i n y a n i l a i k o e f i s i e n i m p o r l e b i h b e s a r d a r i s a t u , m a k a f o r m u l a s i u n t u k m e n d a p a t k a n k o e f i s i e n t e r s e b u t a d a l a h :

mjj. -Xj.i - Cj,,

X j . , - C j , , + mj. , ; j = l , 2 , . . , n (2.8)

d i m a n a , e, , = e k s p o r s e k t o r k e - j k e l u a r n e g e r i . R u m u s p a d a p e r s a m a a n ( 2 . 8 ) a d a l a h r a s i o a n t a r a p e r m i n t a a n d a l a m n e g e r i d a n p e n a w a r a n b a r a n g d a l a m n e g e r i y a n g b e r s u m b e r d a r i p r o d u k s i d a l a m n e g e r i d a n i m p o r d a r i l u a r n e g e r i . D e n g a n m e m a k a i r u m u s p a d a p e r s a m a a n ( 2 . 8 ) m a k a k o e f i s i e n i m p o r y a n g d i h a s i l k a n a d a i a b b i l a n g a n p o s i t i f I e b i b k e c i l a t a u s a m a d e n g a n s a t u .

P e r s a m a a n ( 2 . 5 ) d i s u b s t i t u s i k a n k e d a l a m p e r s a m a a n ( 2 . 4 ) d a n d i p e r o l e b :

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X, = A , X, + M, X, + h , + d , + e,. (2.9)

V e k t o r h , p a d a p e r s a m a a n ( 2 . 9 ) d i d e f i n i s i k a n s e b a g a i b a s i l p e r k a l i a n a n t a r a k o e f i s i e n p e r t a m b a b a n m o d a l ( m a t r i k s G b e r u k u r a n n x n ) d a n p e r t a m b a b a n o u t p u t (Ax„ y a i t u s e b u a b v e k t o r k o l o m b e r u k u r a n n x l ) , a t a u d a l a m s e b u a b p e r s a m a a n d a p a t d i t i m j u k k a n , y a i t u :

h , = G . A x , , ( 2 . 1 0 )

d a n e l e m e n m a t r i k s G , y a i t u :

§11 § 1 2

G = g 2 . g ; 22

g . n

g 2 „

d a n

k e m u d i a n .

_ g n l g n 2 ••• Onn_

Zj., = SjgyAxj. , ; i , j = l ,2 , . . . ,n

Zj,t - K j . t - K j . M ; j = l , 2 , . . . ,n

( 2 . 1 1 )

(2.12)

(2.13)

d i m a n a , K j = p e r s e d i a a n m o d a l p a d a s e k t o r k e - j , d a n Zj = p e r t a m b a b a n p e r s e d i a a n m o d a l p a d a s e k t o r k e - j d a l a m p e r i o d e t - 1 d a n t a t a u i n v e s t a s i y a n g d i t a n a m k a n p a d a s e k t o r k e - j .

P e r s a m a a n ( 2 . 1 0 ) d i s u b s t i t u s i k a n k e d a l a m p e r s a m a a n ( 2 . 9 ) , d a n d i p e r o l e b :

X, = A , x , + M,x, + G(Ax,) + d , + e , . ( 2 . 1 4 )

N i l a i v e k t o r Ax, p a d a p e r s a m a a n ( 2 . 1 4 ) a d a i a b :

Ax, = x,-i-x, ( 2 . 1 5 )

k e m u d i a n , d i s u b s t i t u s i k a n k e d a l a m p e r s a m a a n ( 2 . 1 4 ) , d a n d i p e r o l e b :

X, = A, X, + M, X, + G(x,+i - X, ) + d, + e,. (2.16)

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P e r s a m a a n ( 2 . 1 6 ) d i s e l e s a i k a n t e r h a d a p x , , . , , d a n d i p e r o l e h :

x,A, = [I+G-'(I-A, + M,)]x,-G-'(d, + e , ) . ( 2 . 1 7 )

P e r s a m a a n ( 2 . 1 7 ) a d a l a h d a s a r f o r m u l a s i m o d e l i n p u t - o u t p u t d i n a m i s d e n g a n c i r i y a i t u p e r s a m a a n d i f f e r e n s i a l t i n g k a t d u a a t a u b e r p e r i o d e g a n d a .

P e r s a m a a n ( 2 . 1 7 ) , m e m a k a i m e t o d e p e n y e l e s a i a n d i f f e r e n s i a l , d a p a t d i s e l e s a i k a n m e l a l u i d u a t a b a p p e n y e l e s a i a n , y a i t u ( i ) p e n y e l e s a i a n s t a t i s (static solution), d a n ( i i ) p e n y e l e s a i a n d i n a m i s (dynamic solution). D a l a m p e n y e l e s a i a n s t a t i s d i g i m a k a n a n d a i a n s t a t i s , y a i t u n i l a i v e k t o r x p a d a s e t i a p p e r i o d e a d a i a b t e t a p , a t a u a n d a i a n t e r s e b u t d a p a t d i n y a t a k a n d a l a m s e b u a b p e r s a m a a n , y a i t u :

x,+i = X , = X (2.18)

d a n p e n y e l e s a i a n d i n a m i s , y a i t u :

x . A i = X - I - u,+i ( 2 . 1 9 )

d i m a n a , u , + , a d a i a b s e b u a b v e k t o r k o l o m , y a n g m e n u n j u k k a n v a r i a s i ( s i m p a n g a n ) a n t a r a n i l a i x , + i d a n n i l a i r a t a - r a t a v e k t o r x d a l a m j a n g k a w a k t u o b s e r v a s i , a t a u :

u,+i = x , A i - X (2.20)

J i k a m e t o d e p e n y e l e s a i a n s t a t i s y a n g d i t u n j u k k a n p a d a p e r s a m a a n ( 2 . 1 8 ) d i s u b s t i t u s i k a n k e d a l a m p e r s a m a a n ( 2 . 1 7 ) , d a n d i p e r o l e b b a s i l p e n y e l e s a i a n s t a t i s m o d e l d i n a m i s p a d a p e r s a m a a n ( 2 . 1 7 ) , y a i t u :

X = [I-A, + M , ] ' ' ( d , + e , ) . ( 2 . 2 1 )

K e m u d i a n , a n d a i a n p e n y e l e s a i a n d i n a m i s y a n g d i t i m j u k k a n p a d a p e r s a m a a n ( 2 . 1 9 ) d i s u b s t i t u s i k a n k e d a i a m p e r s a m a a n ( 2 . 1 7 ) d a n d i p e r o l e b :

u , A i - [I+G-'(I-A, + M , ) ] u , - 0 . ( 2 . 2 2 )

T a m p a k b a b w a p e r s a m a a n d a r i p e n y i m p a n g a n v e k t o r x , t e r b a d a p n i l a i r a t a - r a t a v e k t o r x b e r s a n g k u t a n a d a i a b t i d a k b e r b e d a d e n g a n b e n t u k m o d e l p e n y e l e s a i a n p e r s a m a a n d i f e r e n s i a l b o m o g e n t i n g k a t d u a . P a d a

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l a z i m n y a , p e n y e l e s a i a n p e r s a m a a n d i f f e r e n s i a l h o m o g e n , i m t u k k a s u s u„ y a i t u :

U , = b ' U o ( 2 . 2 3 )

d i m a n a , b a d a i a b s k a l a r y a n g m e n u n j u k k a n p e r t u m b u b a n p e n y i m p a n g a n v e k t o r x , t e r b a d a p n i l a i r a t a - r a t a v e k t o r x b e r s a n g k u t a n , d a n d i a s u m s i k a n t i d a k m e n g a l a m i p e r u b a h a n p a d a s e t i a p p e r i o d e w a k t u , d a n

Uo = b ' ( X o - x ) . ( 2 . 2 4 )

S e j a l a n d e n g a n f o n n u l a s i p a d a p e r s a m a a n ( 2 . 2 4 ) , m a k a n i l a i u , ^ , y a i t u :

u , H = b ' ^ ' ( X o - x ) . ( 2 . 2 5 )

J i k a d i k e t a h u i s k a l a r b p a d a p e r s a m a a n ( 2 . 2 5 ) , m a k a m o d e l d i n a m i s p a d a p e r s a m a a n ( 2 . 1 9 ) d a p a t d i p e r o l e b b a s i l p e n y e l e s a i a n . U n t u k m e n d a p a t k a n s k a l a r b d i a p l i k a s i k a n m e t o d e p e n y e l e s a i a n k h u s u s {particular solution), y a i t u : I

u , = m b ' ( 2 . 2 6 )

d i m a n a , m = s e b u a h v e k t o r k o l o m b e r u k u r a n n x l . K e m u d i a n , p e r s a m a a n ( 2 . 2 6 ) d i s u b s t i t u s i k a n k e d a l a m p e r s a m a a n ( 2 . 2 2 ) , d a n d i p e r o l e b :

I ( m ) b ' * ' - B(m )b ' = 0 (2.27)

d i m a n a , B = I + G ' ( I - A - i - M ) .

T e r b a d a p p e r s a m a a n ( 2 . 2 7 ) d i k a l i k a n d e n g a n b ', d a n s t r u k t u r p e r s a m a a n t e r s e b u t b e r u b a h m e n j a d i :

I ( m ) b - B ( m ) = 0

a t a u

[ b l - B ] m = 0 . ( 2 . 2 6 )

A n d a i k a n W = b l - B , d a n d e t e r m i n a n m a t r i k s W t i d a k s a m a d e n g a n n o l , d i m a n a W a d a l a h s e b u a h m a t r i k s b e r u k u r a n n x n , m a k a d a r i d e t e r m i n a n

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Hull!

m a t r i k s t e r s e b u t d a p a t d i s u s i m p e r s a m a a n c i r i {charateristic equation), a t a u s e c a r a u m u m d i t i m j u k k a n , y a i t u :

/ l ,b" + A2b"-'+...+ / l„ . , = 0. (2.29)

N i l a i X d i k e t a h u i , m a k a n i l a i b d a p a t d i h i t u n g , y a n g d i k e n a l d e n g a n i s t i l a b a k a r c i r i {charateristic roots).

U n t u k m e n g b i t u n g e l e m e n v e k t o r m p a d a p e r s a m a a n ( 2 . 2 6 ) , d a p a t d i l a k u k a n d e n g a n m e n g a n d a i k a n b a b w a d i k e t a h u i e l e m e n v e k t o r x p a d a p e r i o d e t = l d a n t = 0 , y a i t u :

X o = X + U o

d a n

U o = mb°,

m a k a :

X o = x + m (2.30)

d a n

X i = X + mb. ( 2 . 3 2 )

J i k a p e r s a m a a n ( 2 . 3 0 ) d a n ( 2 . 3 1 ) s a l i n g d i s u b s t i t u s i k a n , s e b i n g g a d i p e r o l e b :

X o - m = xi-mb

a t a u

x , - X o = m(Ib-I),

s e b i n g g a ,

• m = ( x , - X o ) ( I b - I ) ' ' , ( 2 . 3 2 )

K a r e n a v e k t o r m b i s a d i s e l e s a i k a n d e n g a n m e n g g u n a k a n m e t o d e p e n y e l e s a i a n y a n g d i t u n j u k k a n p a d a p e r s a m a a n ( 2 . 3 2 ) , m a k a h a s i l p e n y e l e s a i a n m o d e l d i n a m i s y a n g d i t u n j u k k a n p a d a p e r s a m a a n ( 2 . 2 9 ) d a p a t d i p e r o l e h , y a i t u :

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Tehnik llerasi Penyelesaian Model Multisektor Dinamis

x,., = [I-A, + M , r ' (d ,+e. ) + (x,-Xo)(Ib-I)-'b'"'. ( 2 . 3 3 )

M o d e l p a d a p e r s a m a a n ( 2 . 3 3 ) a d a l a h p e n y e l e s a i a n m o d e l i n p u t - o u t p u t d i n a m i s m e m a k a i p e n y e l e s a i a n p e r s a m a a n d i f e r e n s i a l h o m o g e n t i n g k a t d u a . J i k a d i k e t a h u i , n i l a i v e k t o r d,,^, d a n e, . , , , m a k a d a p a t d i h i t i m g n i l a i X,,., i m t u k m e n d u k u n g p e n i n g k a t a n p e r m i n t a a n a k h i r , s e p e r t i k o n s u m s i r u m a h t a n g g a , k o n s u m s i p e m e r i n t a h , d a n e k s p o r p a d a p e r i o d e t + 1 .

J i k a d i p e r h a t i k a n s e c a r a c e r m a t , p r o s e s p e n y e l e s a i a n m o d e l i n p u t -o u t p u t d i n a m i k p a d a p e r s a m a a n ( 2 . 3 3 ) , t e m y a t a k i m c i n y a t e r l e t a k p a d a p e r s a m a a n ( 2 . 2 7 ) y a i t u d i w a k i l i o l e h m a t r i k s B = I + G . j ( I - A , + M t ) . J i k a m a t r i k s G d a p a t d i i n v e r s , m a k a m o d e l p e r s a m a a n d i n a m i k y a n g d i t u n j u k k a n p a d a p e r s a m a a n ( 2 . 3 3 ) d a p a t t e r p a k a i , t e t a p i , t i d a k a d a j a m i n a n b a h w a m a t r i k s G s e l a l u d a p a t d i i n v e r s . S a n g a t m i m g k i n b a h w a m a t r i k s G d e t e r m i n a n n y a s a m a d e n g a n n o l , a t a u m a t r i k s t e r s e b u t t i d a k d a p a t d i i n v e r s . D a l a m p r a k t e k , m a t r i k s G a d a l a h m a t r i k s p e r t a m b a h a n m o d a l , d a n u k u r a n m a t r i k s G s a m a b e s a m y a d e n g a n j u m l a h s e k t o r e k o n o m i d a l a m p e r e k o n o m i a n n e g a r a . D a l a m k e n y a t a a n , t i d a k s e m u a s e k t o r e k o n o m i m e n g h a s i l k a n b a r a n g m o d a l , d a n s a n g a t l a b m u n g k i n d a l a m p e r i o d e t e r t e n t u t i d a k a d a p e r t a m b a b a n m o d a l p a d a s e k t o r t e r t e n t u . D e n g a n d e m i k i a n , s a l a b s a t u e l e m e n d i a g o n a l d a r i m a t r i k s G s a n g a t m u n g k i n n i h i l a t a u s a m a d e n g a n n o l . A r t i n y a , m e t o d e p e n y e l e s a i a n m o d e l i n p u t - o u t p u t d i n a m i s m a s i b k u r a n g m e m a d a i j i k a m e n g g u n a k a n m e t o d e p e n y e l e s a i a n p e r s a m a a n d i f e r e n s i a l b o m o g e n .

I I I . P E N Y E L E S A I A N M O D E L DINAMIS M E M A K A I M E T O D E I T E R A S I S E I D E L

M e t o d e p e n d e k a t a n i t e r a s i S e i d e l a d a i a b s a t u p e n d e k a t a n y a n g d a p a t d i p e r g u n a k a n u n t u k m e n y e l e s a i k a n m o d e l i n p u t - o u t p u t d i n a m i s . D e n g a n d i m u n g k i n k a n n y a d i s e l e s a i k a n m o d e l i n p u t - o u t p u t d i n a m i s , m a k a m o d e l t e r s e b u t d a p a t d i p e r g u n a k a n u n t u k k e p e r l u a n p e r e n c a n a a n e k o n o m i , k h u s u s n y a s e b a g a i a l a t p r e d i k s i .

M o d e l i n p u t - o u t p u t d i n a m i s d i s u s u n d e n g a n a n d a i a n b a h w a Ax,=x,,.]-x„ s e p e r t i y a n g d i t u n j u k k a n p a d a p e r s a m a a n ( 2 . 1 5 ) . M o d e l t e r s e b u t d i k e n a l d e n g a n i s t i l a h m o d e l d i n a m i s d e n g a n d i m e n s i w a k t u k e d e p a n {the forward dynamic input-output model). M o d e l i n p u t - o u t p u t y a n g d i m u n g k i n k a n d i t e r a p k a n m e t o d e i t e r a s i S e i d e l , i a l a b m o d e l i n p u t -o u t p u t d e n g a n d i m e n s i w a k t u k e b e l a k a n g {the backward dynamic input-

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output model). • •

M o d e l i n p u t - o u t p u t d e n g a n d i m e n s i w a k t u k e b e l a k a n g d i s u s u n d e n g a n a n d a i a n , y a i t u :

A x , = X , - x , . i (3.1)

P e r s a m a a n (3.1) d i s u b s t i t u s i k a n k e d a l a m p e r s a m a a n (2.14), d e n g a n k e t e n t u a n b a h w a Mp(,=m„ s e h i n g g a d i p e r o l e h :

X , = A , X , + m , X , + G ( x , - x , . i ) + d , + e,. (3.2)

V e k t o r m , p a d a p e r s a m a a n (3.2) d i d e f i n i s i k a n s e b a g a i :

m , = M , [ A , x , + G ( x , - x , . , ) + d , ] . (3.3)

M a t r i k s M , p a d a p e r s a m a a n (3.3) a d a i a b s e b u a b m a t r i k s d i a g o n a l y a n g b e r u k u r a n n x n d a n e l e m e n n y a y a i t u m j , . K o e f i s i e n i m p o r s e b a g a i e l e m e n p a d a d i a g o n a l m a t r i k s M , d i p e r o l e b d e n g a n m e n g g u n a k a n r u m u s , y a i t u :

m j m j , , = ri: ^ V » r ; T T ' I J = 1 , 2 , . . . , n . (3.3)

Z j a y , , X j . , + L , , ( X j . , - Xj . , . i ) + d ,

P e r s a m a a n (3.3) d i s u b s t i t u s i k a n k e d a l a m p e r s a m a a n (3.2), d a n d i p e r o l e h :

X , = A , x , - M , [ A , x , + G ( x , - x , . , ) + d , ] + G ( x , - x , . , ) + d ,+e , . (3 .5 )

S e l a n j u t n y a , p e r s a m a a n (3.5) d i s e l e s a i k a n t e r b a d a p x„ d a n d i p e r o l e b :

X, = [ D r ' [ M , - I ] G , x , . , - [ D r ' ( I - M , ) ( d , + e , ) ; ^̂ ^̂

D = I - [ A , - G , ] [ I - M , ] . ^

M o d e l p a d a p e r s a m a a n (3.6) a d a i a b m o d e l i n p u t - o u t p u t d i n a m i s y a n g d i s u s u n b e r d a s a r k a n m o d e l b e r d i m e n s i w a k t u k e b e l a k a n g . P a d a m o d e l t e r s e b u t , m a t r i k s D d a p a t d i i n v e r s . M o d e l t e r s e b u t t i d a k d i s e l e s a i k a n m e m a k a i p e n y e l e s a i a n d i f f e r e n s i a l , t e t a p i m e n g g i m a k a n p e n d e k a t a n p e n y e l e s a i a n i t e r a s i S e i d e l . A d a b e b e r a p a i n f o r m a s i y a n g d i k e t a h u i d a l a m m e n g a p l i k a s i k a n m e t o d e i t e r a s i S e i d e l , y a i t u e l e m e n v e k t o r x , d a n x , . , y a n g m e n u n j u k k a n o u t p u t d a r i n s e k t o r e k o n o m i p a d a p e r i o d e t d a n p e r i o d e t - 1 , v e k t o r d , d a n e , y a n g m e n u n j u k k a n p e r m i n t a a n a k b i r u n t u k k o n s u m s i r u m a h t a n g g a , k o n s u m s i p e m e r i n t a b , d a n e k s p o r .

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s e r t a e l e m e n m a t r i k s G , d a n e l e m e n m a t r i k s A , .

P r o s e s p e r h i t u n g a n i t e r a s i S e i d e l , s e c a r a r i n c i , a k a n d i p a p a r k a n p a d a u r a i a n b e r i k u t i n i . C a r a p e r h i t v m g a n y a i t u m e l a l u i s u a t u p r o s e s y a n g d i l a k u k a n s e c a r a b e r t a b a p , y a n g d i s e b u t d e n g a n t a b a p p e r t a m a , t a b a p k e d u a , d a n s e t e r u s n y a .

S e b e l u m d i j e l a s k a n p r o s e s p e r h i t u n g a n i t e r a s i S e i d e l , u n t u k m e m u d a b k a n p e n j e l a s a n , t e r l e b i b d a b u l u d i t u n j u k k a n i n f o r m a s i y a n g d i p e r l u k a n d a l a m m e n y e l e s a i k a n m o d e l i n p u t - o u t p u t d i n a m i s , y a i t u e l e m e n m a t r i k s A ( k o e f i s i e n i n p u t - o u t p u t ) , e l e m e n m a t r i k s M ( k o e f i s i e n i m p o r ) , e l e m e n m a t r i k s G ( k o e f i s i e n p e r t a m b a b a n m o d a l ) , v e k t o r x ( o u t p u t d a r i n s e k t o r e k o n o m i ) , v e k t o r d ( p e r m i n t a a n a k b i r d a l a m n e g e r i , d a n v e k t o r e ( e k s p o r ) . N o t a s i d a r i e l e m e n - e l e m e n m a t r i k s d a n v e k t o r t e r s e b u t d i t i m j u k k a n s e b a g a i b e r i k u t :

a n ai2 • • a i„ mi 0 . 0"

321 322 • • a2„ 0 m2 . 0 A = , M =

.a„ i an2 • a n n _ . 0 0 .

"g.. g l 2 • • g.„'

G = g2> g22 • • g 2 „

G =

g n 2 • •• g„„.

X =

X | d, e i

X 2 d 2

, d = , e =

X n d„ e„

M a t r i k s A , M , d a n G , s e r t a v e k t o r x , d , d a n e , t e r s e d i a p a l i n g t i d a k d a l a m d u a p e r i o d e , y a i t u p e r i o d e t d a n p e r i o d e t - 1 .

I t e r a s i t a b a p p e r t a m a u n t u k s e k t o r k e - 1 , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

3 0 9

Hulu

X ( ^ " " ^ i ) b i j X j . , - ( l - m , ) b n X i , , - (3.7) j=2

n

X ( 1 - m,) b l J Xj . i . i + ( 1 - mi) d,., + e,.,]. J=2

I t e r a s i t a h a p p e r t a m a i m t u k s e k t o r k e - 2 , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

= 1 n m l 4 - g J ( l -"^2)a2ixi" + i - ( l -m2)(a22 + g22)

n

X ( 1 " 1̂ 2) 32 j Xj , - ( 1 - m2) b2i x ' l " +

( 3 . 8 )

X • "̂ 2) b2 J X j , - 5 ] (1 - m2) b2 j X j . , -j=3 j=l n

X ( 1 - "̂ 2) b2j X j . , . i + ( 1 - m2)d2., + 62.,]. j=3

I t e r a s i t a h a p p e r t a m a u n t u k s e k t o r k e - i ( i < n ) , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

= l - ( l - m k ^ g „ ) | ^ ^ - ' > - - - " ^

X ( l - m , ) a , j X j , , - X ( l - m i ) b , j x y > + j - i (3.9)

n n-l

X ( 1 - m , ) b i j X j , , - X ( l - m i ) b i j X j , , -

X ( 1 - m i ) b i j X j , , . , + ( 1 - m i ) d i . , + e i , , ] .

I t e r a s i t a h a p p e r t a m a u n t u k s e k t o r k e - n , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

310


Xn 1

[ X ( l - m „ ) a „ j x ( " + l - ( l - m „ X a n n + g„„) H

X ( l - m n ) b „ j x ( " - (3.10) j = i

n

X ( 1 - ni„) bnj Xj., + ( 1 - m„) d„., + e„,,]. j = i

S e t e l a h d i l a k u k a n p e r h i t u n g a n i t e r a s i t a h a p p e r t a m a u n t u k s e k t o r l , 2 , i , d a n n , s e l a n j u t n y a a k a n d i l a k u k a n i t e r a s i t a h a p k e d u a . P a d a i t e r a s i t a h a p k e d u a , s e b u a h k o n s t a n t a ttj d i t a m b a h k a n p a d a d , . , , d i m a n a a^, y a i t u :

tzj = S j gjj (x'j'* - Xj . , ) ; i , j = 1 ,2, . . , n. (3.11)

I t e r a s i t a b a p k e d u a u n t u k s e k t o r k e - 1 i m t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , y a i t u :

T - T j 7 — 7 - 7 ( 1 - m , ) [ X a , jx(" l - ( l - m , ) ( a , i + g,,) p _

(3.12)

' +

Xb.jx ' j '>-b„x' ,"-j - 2

X b ' j X j . t + ai + di.,+ (- )ei.,]. j=2 1 - mi

I t e r a s i t a b a p k e d u a s e k t o r k e - 2 , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

1

l - ( l -m2)(a22 + g22)

i a 2 j x V > - b 2 , x P ' +

( l -m2)[a2 ,xr» +

j=3 ( 3 . 2 3 )

X b 2 , x ) " - X b 2 , x r -j=3 J-1

n J

X b : , X j . , + a2 + d2., + (-; )62,,]. I - m2 j-=3

I t e r a s i t a h a p k e d u a u n t u k s e k t o r k e - i ( i < n ) , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

3 1 1

Hulu

i-1 X a u x < ' > - X b i j x f + X b i j x f - X b , x V ' - (3.14) )=r\ j=l j=i+l j=l

X bij X j . , + a, + d i . , + ( - C i . , ] . 1 - n i i

I t e r a s i t a h a p k e d u a u n t u k s e k t o r k e - n , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

l-(l-m„)(a„„ + g j 7 1 '

(3.25) n - l

n - l 2 X b" j X u , ' + a„ + d„., + (- e„.,]. j = i 1 - m,,

S e b e l u m d i l a n j u t k a n k e p a d a i t e r a s i t a b a p k e t i g a , t e r l e b i b d a b u l u d i s e s u a i k a n d e f i n i s i k o n s t a n t a ttj y a n g d i t a m b a h k a n p a d a d , , m e n j a d i :

= Ejgy(xf-x<j'>); i , j = l ,2 , . . ,n . (3.26)

I t e r a s i t a b a p k e t i g a u n t u k s e k t o r k e - 1 u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , y a i t u :

n

X b . j x f - b „ x P - (3.27) i-2

X b , j x 5 ' > + al'> + d , , + ( - - ^ ) e , . J . i - j 1 - m. mi

I t e r a s i t a b a p k e t i g a s e k t o r k e - 2 , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

3 1 2

Tehnik llerasi Penyelesaian Model Multisektor Dinamis

- ( l - m 2 ) [ a 2 , x r V + l - ( l -m2)(a22 + g22)

I a 2 j x f - b 2 , x r ' + X b : j x 7 - X b 2 j x f - (3.18) r - 3 J=3 j = i

Xb2jXj ( l )+a '2 ' ' + d 2 , , + ( T )e2,,]. ;=j 1 - ni2 j=3

I t e r a s i t a h a p k e t i g a u n t u k s e k t o r k e - i ( i < n ) , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

1

l-(l-m,)(a„ + g j i )

X a , j x f - I b , x 7 + X b n x 7 - X b y x f - (3.19) j=i+l j=l j=i - t- l j=l

X b . i x 7 + a ! " + d,, + ( - - ! - e , , ] . j = i A i 1 - m,

I t e r a s i t a h a p k e t i g a u n t u k s e k t o r k e - n , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

xl,̂ > = — — T ( l - m „ ) [ i ; a „ j x f > + l-(l-m„)(a,„, + g j

( 3 . 2 0 )

7r l - m „

D a l a m u r a i a n d i a t a s t e l a h d i j e l a s k a n t i g a t a h a p a n p r o s e s p e r h i t u n g a n i t e r a s i . P e r t a n y a a n n y a i a l a h b e r a p a k a l i p r o s e s p e r h i t u n g a n i t e r a s i b a r u s d i l a k u k a n i m t u k m e n d a p a t k a n b a s i l p e r h i t u n g a n y a n g v a l i d d a n d i n y a t a k a n a k u r a t . |

U n t u k m e n j a w a b p e r t a n y a a n t e r s e b u t d i a t a s d i p e r l u k a n u k u r a n p e n g u k u r a n v a l i d i t a s h a s i l e s t i m a s i , y a i t u :

/ ? j % = . , , ( k ) . y ( k - n X j X j

vl ' ' - ' )

1 0 0 % ; j = l , 2 , . . . , n (3.2U

3 1 3

Hulu

d i m a n a , k = b a n y a k n y a i t e r a s i , d a n Pj = s e l i s i h n i l a i o u t p u t s e k t o r k e - j a n t a r a d u a p e r i o d e i t e r a s i . A r t i n y a i a l a h j i k a s e l i s i h a n t a r a e l e m e n v e k t o r o u t p u t p a d a t a h a p i t e r a s i t e r t e n t u d a n h a s i l i t e r a s i p a d a t a h a p i t e r a s i s e b e l u n m y a n i l a i n y a s a n g a t k e c i l , s e k i t a r 0 , 0 0 1 % , u n t u k s e m u a s e k t o r , m a k a h a s i l p e r h i t v m g a n s u d a h d i a n g g a p c u k u p v a l i d . I n t e r p r e t a s i n y a i a l a h n i l a i o u t p u t y a n g d i h a s i l k a n s u d a h m e n d e k a t i s t a b i l , a t a u s e k a l i p u n p r o s e s p e r h i t u n g a n i t e r a s i d i l a k u k a n , b a s i l e s t i m a s i y a n g d i h a s i l k a n d i p a n d a n g t i d a k b e m a s . B e r a p a k a l i i t e r a s i d i l a k u k a n u n t u k m e n d a p a t k a n P j = 0 , 0 0 1 % , s u l i t d i t e n t u k a n , k a r e n a t e r g a n t u n g k e p a d a s t r u k t u r a n g k a - a n g k a s e b a g a i d a t a p e n d u k v m g , d a n b a n y a k n y a s e k t o r e k o n o m i . D e n g a n p e r k e m b a n g a n t e k n o i o g i k o m p u t e r y a n g s a n g a t p e s a t d e w a s a i n i , p r o s e s p e r h i t u n g a n i t e r a s i tidak m e n g b a d a p i k e n d a l a s e r i u s i m t u k m e n d a p a t k a n h a s i l p e r h i t u n g a n y a n g p a l i n g a k u r a t .

K e m u d i a n , a p a f a e d a h p e m a k a i a n t e k n i k i t e r a s i S e i d e l ? M i s a l k a n s a j a k i t a s u d a b m e n c a p a i t a b a p i t e r a s i k e - k , d a n t e p a t m e m i l i k i a n g k a pj = 0 , 0 0 1 % , u n t u k j = l , 2 , . . . , n . S e b e l u m d i j e l a s k a n f a e d a h a n a l i t i s n y a , t e r l e b i b d a b u l u d i j e l a s k a n f o r m u l a s i u m u m u n t u k m e n d a p a t k a n o u t p u t d a r i s e t i a p s e k t o r e k o n o m i p a d a t a h a p i t e r a s i k e - k .

I t e r a s i t a h a p k e - k u n t u k s e k t o r k e - 1 u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , y a i t u :

l - ( l - m , ) ( a i , + g | , )

X b u x f - " > - b n x r > - ( 3 . 2 2 )

Xb.jx7-^ + aS''> + d,, + ( 7 ^ ) e , , , ] . j=; 1 - mi

I t e r a s i t a h a p k e - k s e k t o r k e - 2 , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

= 7 7 1 r ^ ( ' - m : ) [ a : , x r ' + 1 - ( 1 - m : ) ( a : : + g^i)

I a 2 j x f " - b : , x'r + I b 2 j X < > - " - I b : j x 7 ' > -j=3 j=3 j - l

X b : I X , ( k - 2)+ar + d : , + ( — ! — ) 6 2 , , ] .

j - 3 I - m .

( 3 . 2 3 )

3 1 4


I t e r a s i t a h a p k e t i g a u n t u k s e k t o r k e - i ( i < n ) , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

i-(i-mi)(aii + gii) j=i

Z a, x7'> - i b, xf' + Z h x f " - Z bu xf - (324) j-i+i j=i j=i-i r i

Z b i j x r + a!^' + d|, + ( - - ^ e , , , ] . j=iAi 1 - nii

I t e r a s i t a h a p k e t i g a i m t u k s e k t o r k e - n , u n t u k m e n g e s t i m a s i o u t p u t s e k t o r b e r s a n g k u t a n , d a n d a p a t d i n y a t a k a n d a l a m r u m u s , y a i t u :

1 "-' = 7 7 V xo , ( l - m „ ) [ Z a > o x f +

l-(l-m„)(a„„ + g j H (325)

Z b»i x f - Z b H i x 7 " + + d„., + ( - - ^ e,,..]. j=i r i (- m,,

P e r s a m a a n ( 3 . 2 2 ) s a m p a i d e n g a n p e r s a m a a n ( 3 . 2 5 ) a d a l a h f o r m u l a s i u m u m t e k n i k p e n y e l e s a i a n m o d e l i n p u t - o u t p u t d i n a m i s m e n g g u n a k a n p e n d e k a t a n i t e r a s i S e i d e l . K e g u n a a n p e r s a m a n ( 3 . 2 2 ) d a n ( 3 . 2 5 ) a d a l a h p a d a t i a p - t i a p p e r s a m a a n t e r s e b u t t e r d a p a t v a r i a b e l e k s o g e n p e r m i n t a a n a k h i r y a i t u d i w a k i l i o l e h v e k t o r d , . J i k a e l e m e n v e k t o r d , m e n g a l a m i p e r u b a h a n ( m u n g k i n d i t a k s i r m e m a k a i m e t o d e e k o n o m e t r i k a , d a n d i p e r o l e h d,+i, m i s a l n y a ) , m a k a d e n g a n m e n g g u n a k a n p e r s a m a a n ( 3 . 2 2 ) s a m p a i d e n g a n ( 3 . 2 5 ) d a p a t d i p e r o l e h n i l a i o u t p u t y a n g s e h a r u s n y a d i t i n g k a t k a n d a l a m m e n d u k u n g p e n i n g k a t a n p e r m i n t a a n a k h i r t e r s e b u t .

D a r i u r a i a n d i a t a s d e n g a n j e l a s d a p a t d i n y a t a k a n b a b w a d e n g a n m e n g g u n a k a n m e t o d e i t e r a s i S e i d e l , t i d a k m u n c u i m a s a l a b - m a s a l a b k e t i a d a a n b a s i l s o l u s i m o d e l i n p u t - o u t p u t d i n a m i s . D e n g a n d e m i k i a n , m o d e l i n p u t - o u t p u t d i n a m i s , t i d a k b a n y a d o k u m e n y a n g s u l i t d i a p l i k a s i k a n u n t u k k e p e r l u a n p r a k t i s , t e t a p i d a p a t d i a p l i k a s i k a n k a r e n a s o l u s i m o d e l d i j a m i n a d a d e n g a n m e n g g i m a k a n p e n d e k a t a n i t e r a s i S e i d e l .

315

Hulu

IV. CATATAN UNTUK STUDI LANJUTAN

B e r d a s a r k a n s t r u k t u r m o d e l i n p u t - o u t p u t d i n a m i s y a n g t e l a h d i p a p a r k a n p a d a b a b i n i , t e m y a t a m o d e l t e r s e b u t a d a l a h s e b u a h p e r s a m a a n d i f e r e n s i a l . J i k a d i g i m a k a n p e n d e k a t a n p e n y e l e s a i a n m a t r i k s , m a k a b e s a r k e m u n g k i n a n a k a n m u n c u i b e b e r a p a m a s a l a h d a l a m m e n d a p a t k a n h a s i l p e n y e l e s a i a n m o d e l , y a i t u m e m i l i k i p e l u a n g s e b u a h m a t r i k s t i d a k d a p a t d i i n v e r s . S a l a h s a t u c a r a u n t u k m e n g a t a s i m a s a l a h t e r s e b u t i a l a b m e n g g u n a k a n p e n d e k a t a n p e n y e l e s a i a n i t e r a s i S e i d e l . N a m u n h a s i l n y a m a s i h k u r a n g s e m p u r n a , k a r e n a h a s i l n y a t i d a k t i m g g a l (unique), d i s e b a b k a n k a r e n a p r o s e s i t e r a s i i t u s e n d i r i t i d a k d i t u j u k a n u n t u k m e n d a p a t k a n h a s i l y a n g t i m g g a l , m e l a i n k a n s a l a b s a t u t e k n i k u n t u k m e n d e k a t i p o p u l a s i y a n g d i o b s e r v a s i .

O l e h k a r e n a i t u , s t u d i d a l a m u p a y a m e n e m u k a n s e b u a h m e t o d e u n t u k m e n y e l e s a i k a n m o d e l - m o d e l e k o n o m i d i n a m i s m e r u p a k a n s a l a b s a t u t o p i k y a n g m e n a r i k u n t u k d i d i s k u s i k a n . D a y a t a r i k n y a t i d a k h a n y a t e r l e t a k p a d a p e n g e m b a n g a n t e o r e t i s s e m a t a t e t a p i j u g a m e n g i n g a t d e w a s a i n i , t i d a k b a n y a p a r a i l m u w a n n a m u n j u g a p a r a p r a k t i s i s e m a k i n s a d a r a k a n p e n t i n g n y a a l a t a n a l i t i s d a l a m m e n y e d i a k a n i n f o r m a s i p e n u n j a n g p e n g a m b i l a n k e p u t u s a n e k o n o m i . D a l a m d u n i a n y a t a , k e g i a t a n e k o n o m i t i d a k s t a t i s , m e l a i n k a n s e l a l u d i n a m i s . D e n g a n d e m i k i a n , m o d e l - m o d e l d i n a m i s s a n g a t d i p e r l u k a n b a i k d a l a m m e n j e l a s k a n f a k t a m a u p u n d a l a m m e i a k u k a n p r e d i k s i .

3 1 6

I


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Teknik Iterasi Penyelesaian Model Multisektor Dinamis

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