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I/O Equations

(http://faculty.washington.edu/krumme/systems/ioquations.html)

Accounting Equations (underlying the "transaction table")

X_i - m
S
j=1 x_ij = Y_i
x_ij = a_ij X_j
X_i - m
S
j=1 a_ij X_j = Y_i

X_i = Total output of industry "i"

x_ij = ($) amount of i's production absorbed by industry "j" as inputs

j=1 to m: given number of receiving industries (j) 1,2,3....m

Y = Final demand of industry "i".

a_ij = production coefficient indicating the amount of i's products
needed to produce one unit of j's product

Equations underlying (Interregional) "Direct Coefficients Table"

_rX_i - m
S
s=1 m
S
j=1 _rsx_ij = _rY_i
_rsa_ij = _rsx_ij
-------
_sX_j
_rX_i - m
S
s=1 m
S
j=1 _rsa_ij _sX_j = _rY_i

_rX_i = Total output of product "i" in region "r"

_rY_i = Final demand of product "i" in region "r"

r,s = 1, 2, 3, ... n regions

_rsa_ij = (inter)regional production coefficient:

The inputs from industry i in region r
to industry j in region s are some proportion
of the total production of product j in region s.

Equations underlying "Leontief" Tables of Direct, Indirect (+ Induced) Coefficients

X - AX = Y
(I - A)X = Y
X = (I - A)^-1 Y
X = BY
X_i = b_i1Y₁ + b_i2Y₂ + b_i3Y₃ + .... + b_imY_m

X = column vector of industrial gross output

Y = column vector of final demand

A = m x m matrix of direct coefficients a_ij

I = Identity matrix (matrix with "1" in the diagonal,
"0" in all other fields)

(I-A)^-1 = "Leontief Inverse (Matrix)" = B
(or B' if induced effects are included)

B (or B') = Table of direct and indirect
(and induced) coefficients b_ij (or b'_ij)

b_ij (or b'_ij) = "Leontief Coefficient"

(representing the direct and indirect [and "induced"]
requirements per unit of final demand for the output of sector j)

We can now multiply the inverse matrix B (or B') by a new future, independently derived or forecasted final demand vector to obtain the level of gross ouput for each industry (in each region) necessary to satisfy this final demand.

Use of Location Quotients (L.Q.s) and national coefficients for estimating regional coefficients (simple method)

For (supplying) industries (i) with L.Q. > 1 ----> r_ij = a_ij^nat

For (supplying) industries (i) with L.Q. < 1 ----> r_ij = a_ij^nat x LQ_i

Assumptions: a_ij^nat = a_ij^reg [=technical coefficients at the regional and national levels are identical]