The Employment Impact of Differences in Dmand and Production

1   The Input-Output Approach to Employment: the

Vertically Integrated Sector

This section lays out the framework for the analysis of employment within an input-output
system, using the concept of the vertically integrated sector (VIS). This will show how the
allocation of economy-wide employment across individual sectors as conventionally measured
relates to the VIS measure.

In the input-output framework total employment can be expressed as

N= n^' X

[1]

=n^' (I-A)^-1 F

where N is the (scalar) level of total employment, X and F are column vectors of gross output
and final demand for domestic output by sector,¹ (I - A)^-1 is the Leontief inverse matrix and n′
the row vector of labour requirements per unit of sectoral gross output². The first line of
equation [1] expresses total employment in terms of sectoral gross outputs and the associated
labour requirements within the sector. The second line of equation [1] uses the input-output
relationship of the Leontief multiplier, X = (I - A)^-1F to express total employment as a function
of final demands for domestically produced goods and services as transmitted through the inter-
industry (input-output) structure. Final demand for the output of sector i, Fi, gives rise to gross
output in (all) other sectors through the chain of intermediate purchases encapsulated in the
Leontief inverse, and therefore to employment in (all) other sectors.

The different basis for the allocation of economy-wide employment across sectors on the VIS
approach can be shown by the expansion of [1]:

¹ We use the terms ‘sector’ and ‘industry’ interchangeably throughout. The input-output tables available to us are on
an industry basis. Since it is demand for domestically produced goods that generates employment in the economy
concerned, final demand is defined throughout as final demand for the outputs of domestic industries, with the
input-output tables used also those for domestic outputs only.

² This is the reciprocal of labour productivity when the productivity measure is gross output per worker, rather
than value added; see ten Raa and Schettkat (2001).

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