The Employment Impact of Differences in Dmand and Production

Appendix 2:   The Input - Output System

Consider the standard input-output system consisting of n industries, which can be partitioned
into two subsystems 1 and 2, which contain m and s industries respectively¹⁶. The economy can
then be represented as follows:

GO is a (m+s x 1) vector that denotes the gross output produced by each industry, A denotes
the matrix of input-output coefficients, whose generic element is defined as: aij = Xij/Xj. The
direct coefficient matrix A is partitioned in four sub-systems each identified by a superscript.
The subsystem identified by the superscript 11 (22) summarises the interaction within the
subsystem itself, and the one identified by the superscript 21 (12) summarises the interactions
between the sub-systems. F represents the (m+s x 1) vector of final demands (assumed to be
positive) also partitioned in the two sub-systems 1 and 2.

This system can be solved to yield the gross outputs needed to sustain a given level of final

demand:

where B denotes the Leontief inverse [B=(I-A)^- ].

Sub-systems 1 and 2 can be thought of as manufacturing and services.

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