The Employment Impact of Differences in Dmand and Production

Appendix 4:   The Decomposition of the Employment

Matrix

Input-output tables and the results from them change over time. In the same spirit as growth
accounting, decomposing these variations allows us to identify and quantify the sources of
change. The problem with decomposing within the input-output approach is that the
decomposition is typically not unique. In our particular case, we are looking for a suitable
decomposition of changes in the employment matrix.

The employment matrix can be written as:

Nk^t=nk^tBk^tFk^t

where the subscript k denotes the country, the superscript t denotes the time period, N is the
employment matrix, B is the Leontief inverse, n is a diagonal matrix, whose main diagonal
contains sectoral productivity (ratio of employment to gross output), and F is a diagonal matrix
whose main diagonal contains final demand (an entry for each industry). To lighten the notation
we will drop the country subscript k, and index the time-periods as (19)70 and (19)90.

In the spirit of shift and share analysis, the difference in employment ∆N=N⁹⁰ -N⁷⁰can be
decomposed into the change in its three components: productivity, technical coefficients, and
final demand. There are many possible decompositions; in general if there are z components
there are z! possible (and equivalent) decompositions (in our case we have three components
and thus 3!=6 possible decompositions).

One of the possible decompositions could be as shown in the following equation:

∆N=∆nB⁷⁰F⁹⁰ +N⁷⁰∆BF⁹⁰ +N⁹⁰B⁷⁰∆F

but an alternative and equivalent decomposition could read as:

∆n = ∆nB⁹⁰F⁷⁰ + N⁹⁰∆BF⁹⁰ + N⁷⁰B⁹⁰∆F

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