W. Leontief put forward the linear function’s system
for relationship between supply and demand of economy by sectors, solved at
below:
?njXij + Yi = Xi
(1)
And ?niXij
+ Vj = Xj
(2)
Where: Xij present sector j used product i
as input; i,j = 1….n with n is number of sectors in input-output model; Yi is
final product of product i; Xi is gross output of product i (total demand of
product i) and Vj is value added of sector j.
Equation (3) shows: Total output = Intermediate demand
(for production) + Final demand (for consumption)
Equation (4) shows: Total input = Intermediate input
(for production) + Value added
Total output always equals to total input.
Put aij = Xij/Xjand
equation (1) we have:
?njaijXj +
Yi = Xi (3)
Rewrite the equation (3) to matrix form:
A.X + Y = X (4)
With: A = (aij)(nxn); Y = (Yi)(nx1);
X = (Xi)(nx1). The equation (4) is Leontief’s standard,
this equation can rewrite as follow:
X = (I – A)-1.Y
In this research the matrix A is divided by
sub-matrixes including ARR, ARS, ASRva ASS
Where: R, S are industries; R is the industry is
affected by increasing indirect tax; ARR is the matrix of
intermediate coefficients of r industry using its own product as input; ARS
is a matrix of intermediary coefficients for s industry using r product as
input; ASR is a matrix of intermediary coefficients for r industry
using s product as input; ASS is a matrix of intermediary
coefficients for s industry using its own product as input
We can rewrite
Leontief’s relation:
(5)
Or:
ARR.XR
+ ARS.XS + YR = XR (6)
ASS.XS
+ ASR.XR + YS = XS (7)
From
(6) and (7) we have:
XS
= (I – ASS)-1.(ASR.XR + YR) (8)
XR
= (I – ARR)-1.(ARS.XS + YS) (9)
Equation
(8) and (9) shows that output of industry is not only based on the final demand
but also depend on other sector’s productions. For example, ouput of R depend
on S’s production by ARS.XS, or output of S (XS)
depend on R’s production by ASR.XR.
Relationship
between S and R can be shown:
XS
= (I – ASS)-1.ASR.XR (10)
XR
= (I – ARR)-1.ARS.XS (11)
Or
?XS
= (I – ASS)-1.ASR.?XR (12)
?XR
= (I – ARR)-1.ARS.?XS (13)
Equation
(12), (13) show that the change in each industry can be led to the change in
other industries. Matrix (I – ASS)-1.ASR and
(I – ARR)-1.ARS show this relationship. This
equation is applied to quantify the output of industries that are not directly
affected by indirect tax increase are also reduced in the next production
cycle.
In
order to consider the effect of final demand of each industry to value added,
we put:
B
=
=
(14)
(15)
And
Or:
(16)
Equation
(16) indicates the spillover effect of final demand of R and S on value added.