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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
a_{ij} = production coefficient indicating the amount of i's
products 

_{r}X_{i} = Total output of product "i" in region "r"
_{r}Y_{i} = Final demand of product "i" in region "r" r,s = 1, 2, 3, ... n regions _{rs}a_{ij} = (inter)regional production coefficient: to industry j in region s are some proportion of the total production of product j in region s. 

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,
(IA)^{1} = "Leontief Inverse (Matrix)" = B
B (or B') = Table of direct and indirect b_{ij} (or b'_{ij}) = "Leontief Coefficient" 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.
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}
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Geog 350 
Econ & Bus Geog
2000 [econgeog@u.washington.edu]