2.1 The multiregional I-O model
A multiregional I-O system describes all economic transactions among productive sectors and
among the regions considered. Formally, it can be written as:
|
X1I |
∖ A. I |
A12 I |
• ∙ ∙ I |
A1(N-1) I |
' ■ |
X1 |
^ FD1 ^ | ||||
|
-- X2 |
■ ■ I A21 I I |
.. I A22 I I |
.. I • ∙ ∙ I _ _ I |
.. I A2(N-1) I I |
A2 N |
"■ X2 |
■■ FD2 | ||||
|
-- |
= |
.. I ; I |
.. I ; I |
.. I ; I |
.. I ; I |
.. |
× |
T |
+ |
■■ ; | |
|
-- |
, I .. I |
.. I |
.. I |
.. I |
.. |
' ■■ |
; ■■ |
(1) | |||
|
XN-1 |
A( n-1)1 I |
A(N-1)2 I |
∙∙ ∙ I |
A(N-1)(N-1) I |
a( n-1) n |
X N-1 |
FD n-1 | ||||
|
-- X n _ |
■ ■ i _ A n 1 I |
.. I AN2I |
.. I ∙∙ ∙ I |
.. I AN(N-1) I |
A nn _ |
■■ _ X n _ |
■■ _ FDn _ |
where N is the number of the regions, X is output vector, A is the coefficient matrix and FD is a
final demand vector. In particular, Ass ( S = 1,..., N ) refers to flows of goods and services (per unit of
output) exchanged among sectors within region S. Ats (T, S = 1,., N ∧ T ≠ S) relates to export
flows of sectors of region T to sectors of region S, which equal import flows of sectors of region S
from sectors of region T.
More compactly, system (1) can be rewritten as a system of block matrices; i.e.:
X=AX+FD
(2)
(3)
As usual, the solution of the system is:
X=(I-A)-1FD
System (3) represents the multiregional I-O model which allows determining output variation in
the regions under study induced by a change in final demand. Output change takes account of both
direct and indirect effects generated by sector linkages within regions and spill-over and feedback
effects produced by interrelationships between the regions (Miller and Blair, 1985). To analyse
employment and income dynamics, system (3) has to be modified transforming goods and services
flows into employment and labour income flows, respectively. System (3) becomes:
E = E × FD (4)
Y = Y × FD (5)