SITE MAP SEARCH! LIBRARY E & B GEOG RESOURCES A-Z INDEX GLOSSARY PASSWORD 350 PAGE CALENDAR ECON NEWS EXERCISES

## I/O Equations

### Accounting Equations (underlying the "transaction table")

1.  Xi - m S j=1 xij = Yi

2. xij = aij Xj

3.  Xi - m S j=1 aij Xj = Yi
Xi = Total output of industry "i"

xij = (\$) 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".

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

### Equations underlying (Interregional) "Direct Coefficients Table"

1.  rXi - m S s=1 m S j=1 rsxij = rYi

2.  rsaij = rsxij------- sXj

3.  rXi - m S s=1 m S j=1 rsaij sXj = rYi
rXi = Total output of product "i" in region "r"

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

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

rsaij = (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 Xi = bi1Y1 + bi2Y2 + bi3Y3 + .... + bimYm X = column vector of industrial gross output Y = column vector of final demand A = m x m matrix of direct coefficients aij 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 bij (or b'ij) bij (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 ----> rij = aijnat

For (supplying) industries (i) with L.Q. < 1 ----> rij = aijnat x LQi

Assumptions: aijnat = aijreg [=technical coefficients at the regional and national levels are identical]