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

I/O Equations

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


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

  1. X - AX = Y


  2. (I - A)X = Y


  3. X = (I - A)-1 Y


  4. X = BY


  5. 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]


Return to Input-Output || Geog 350 || Econ & Bus Geog
2000 [econgeog@u.washington.edu]