Annex B Compiling quality-adjusted labour input indexes

19B.1    This annex provides a detailed description of how quality adjusted labour inputs (QALI) indexes are compiled for each market sector industry, and the market sector and twelve selected industries aggregates. Recall that QALI indexes can be written as a combination of labour composition and unadjusted hours worked. Census data are used to estimate labour composition for each industry. Then these estimates are combined with hours worked data for each industry, the market sector and twelve selected industries aggregates to obtain the corresponding QALI indexes.

19B.2    The general formula for calculating QALI indices is as follows. The workforce is partitioned into groups \(g_1,....., g_K\) for each year \(t\). This assumes that for each group \(g\) we have an hours worked index \(H_{g,t}\). Note that the sum of hours worked over each group:

    \(\large {H_t} = \sum\limits_{g = 1}^K {{H_{{g_j},t}}}\)

is the unadjusted hours worked index. It is further assumed that for each group \(g\) we have the average hourly income \(w_{g,t}\) . Then the QALI index \(\frac{L_t}{L_{t-1}}\) is given by:

    \(\large \frac{{{L_t}}}{{{L_{t - 1}}}} = \prod\limits_{g = 1}^K {{{\left( {\frac{{{H_{g,t}}}}{{{H_{g,t - 1}}}}} \right)}^{\left( {{S_{g,t}} + {S_{g,t - 1}}} \right)/2}}}\)         - - - - - - - (19B.1)

where

    \(\large {S_{g,t}} = \frac{{{w_{g,t}}{H_{g,t}}}}{{\sum\nolimits_{g = 1}^K {{w_{g,t}}{H_{g,t}}} }}\)

is income share of group \(g\) in year \(t\).

19B.3    Now labour composition is defined to be the ratio \(\frac{L_t}{H_t}\). For the equation above, we see that an index for labour composition is given by

    \(\large Q_t=\frac{L_t/H_t}{L_{t-1}/H_{t-1}}=\frac{L_t/L_{t-1}}{H_t/H_{t-1}}\)         - - - - - - - (19B.2)

The term \(Q_t\) is the compositional change (in year \(t\)).

19B.4    Let \({p_{g,t}} = \frac{{{H_{g,t}}}}{{{H_t}}}\) be the proportion of hours worked by group \(g\) in year \(t\). Then we can write \(Q_t \)as

    \(\large {Q_t} = \prod\limits_{g = 1}^K {{{\left( {\frac{{{P_{g,t}}}}{{{P_{g,t - 1}}}}} \right)}^{\left( {{{\widehat S}_{g,t}} + {{\widehat S}_{g,t - 1}}} \right)/2}}}\)         - - - - - - - (19B.3)

where

    \(\large {\widehat S_{g,t}} = \frac{{{w_{g,t}}{P_{g,t}}}}{{\sum\nolimits_{g - 1}^K {{w_{g,t}}{P_{g,t}}} }}\)

19B.5     To calculate the compositional changes from the census data the workforce is grouped by education, age, and sex (see Table 19.1). For education, there are four categories: Unqualified, Skilled Labour, Bachelor Degree, and Higher Degree; for age there are five categories: 15 to 24 years, 25 to 34 years, 35 to 44 years, 45 to 54 years, and 55 to 64 years; for sex there are two categories: Male and Female. Definitions of the education categories are given in Table 19.2. From the census data, we derive the proportion of hours worked and average hourly wage of workers with a given education level, age group, and sex (for all choices of education level, age group, and sex). Note that to take into account time spent in education, we restrict the age range of workers considered depending on the education category (see Table 19.2).

19B.6    Compositional changes for the whole economy are calculated from 1981 until the current year using census data. Census data are only available every five years (in 1981, 1986, etc.), so much data has to be interpolated. The years falling between census years are linearly interpolated for both \(p_{g,t}\) and \(w_{g,t}\). For example, the years 1982 to 1985 are defined as:

    \(\large {P_{g,1981 + t}} = {P_{g,1981}} + \frac{t}{5}\left( {{P_{g,1986}} - {P_{g,1981}}} \right)\)

and

    \(\large {w_{g,1981 + t}} = {w_{g,1981}} + \frac{t}{5}\left( {{w_{g,1986}} - {w_{g,1981}}} \right)\)

For \(t=1,2,3,4,...,Q_t\) is then calculated or years 1981 to the latest census year for which data is available. Finally, the compositional changes for years past the last census year are extrapolated using the following formula:

    \(\large {Q_{2006 + t}} = {Q_{2006}}{\left( {\frac{{{Q_{2006}}}}{{{Q_{2001}}}}} \right)^{t/5}}\)

19B.7    The extrapolation assumes that the yearly changes in compositional change past the last census year are equal to the (average) yearly change during the latest inter-census period.