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# Estimation formulae

Australian System of National Accounts: Concepts, Sources and Methods
Reference period
2020-21 financial year

## Capital services index

19.120    Capital services index for each industry is estimated (for both corporate and unincorporated entities) by weighting together the real growth in the productive capital stock of different assets, weighted together using the two-period average value share of each type of capital services. The indexes are compiled in the form of a Törnqvist index (i.e. the weighted geometric mean of the component growth rates). The quantity index of capital services in industry i is calculated as:

$$\large \ln \left( {\frac{{{K_{i,t}}}}{{{K_{i,t - 1}}}}} \right) = \sum\limits_j {{s_{K,i,j,t}}\ln \left( {\frac{{{K_{i,j,t}}}}{{{K_{i,j,t - 1}}}}} \right)}$$         - - - - - - - (19.7)

and the two-period average value share of each type of capital services is given by:

$$\large {s_{K,i,j,t}} = \frac{{\left( {\frac{{{r_{i,j,t}}{K_{i,j,t}}}}{{\sum\nolimits_j {{r_{i,j,t}}{K_{i,j,t}}} }} + \frac{{{r_{i,j,t - 1}}{K_{i,j,t - 1}}}}{{\sum\nolimits_j {{r_{i,j,t - 1}}{K_{i,j,t - 1}}} }}} \right)}}{2}$$

where

$$\ln \left( {\frac{{{K_{i,t}}}}{{{K_{i,t - 1}}}}} \right)$$ is the capital input growth rate for industry $$i$$ from period $$t-1$$ to period $$t$$, and

$$\ln \left( {\frac{{{K_{i,j,t}}}}{{{K_{i,j,t - 1}}}}} \right)$$ is the productive capital stock growth rate for capital asset $$j$$ in industry $$i$$, from period $$t-1$$ to period $$t$$.

19.121    The capital input growth rate for the market sector $$\ln \left( {\frac{{{K_{m,t}}}}{{{K_{m,t - 1}}}}} \right)$$, is given by:

$$\large \ln \left( {\frac{{{K_{m,t}}}}{{{K_{m,t - 1}}}}} \right) = \sum\limits_i {{s_{K,i,t}}\ln \left( {\frac{{{K_{i,t}}}}{{{K_{i,t - 1}}}}} \right)}$$         - - - - - - - (19.8)

and the two-period average value share of each type of capital services is given by:

$$\large {s_{K,mjt}} = \frac{{\left( {\frac{{GO{S_{i,t}} + GMI{{\left( K \right)}_{i,t}} + IBT{{\left( K \right)}_{i,t}}}}{{\sum\nolimits_i {GO{S_{i,t}} + GMI{{\left( K \right)}_{i,t}} + IBT{{\left( K \right)}_{i,t}}} }} + \frac{{GO{S_{i,t - 1}} + GMI{{\left( K \right)}_{i,t - 1}} + IBT{{\left( K \right)}_{i,t - 1}}}}{{\sum\nolimits_i {GO{S_{i,t - 1}} + GMI{{\left( K \right)}_{i,t - 1}} + IBT{{\left( K \right)}_{i,t - 1}}} }}} \right)}}{2}$$

19.122    That is, the industry capital income shares are derived as the proportion of $$GOS + GMI\left( K \right) + IBT\left( K \right)$$ in each industry to $$GOS + GMI\left( K \right) + IBT\left( K \right)$$for the market sector.⁹⁶

## Labour input index

19.123    Labour input index for each industry is calculated as a Törnqvist volume index of hours worked of different types of workers:

$$\large \ln \left( {\frac{{{L_{i,t}}}}{{{L_{i,t - 1}}}}} \right) = \sum\limits_j {{s_{L,ijt}}\ln \left( {\frac{{{H_{i,j,t}}}}{{{H_{i,j,t - 1}}}}} \right)}$$         - - - - - - - (19.9)

and the two-period average value share of each type of workers in the industry labour compensation is given by:

$$\large {s_{L,ijt}} = \frac{{\left( {\frac{{{W_{i,j,t}}{H_{i,j,t}}}}{{\sum\nolimits_j {{W_{i,j,t}}{H_{i,j,t}}} }} + \frac{{{W_{i,j,t - 1}}{H_{i,j,t - 1}}}}{{\sum\nolimits_j {{W_{i,j,t - 1}}{H_{i,j,t - 1}}} }}} \right)}}{2}$$

where

$$\ln \left( {\frac{{{L_{i,t}}}}{{{L_{i,t - 1}}}}} \right)$$ is the labour input growth rate for industry $$i$$, from period $$t-1$$ to period $$t$$;

$$\ln \left( {\frac{{{H_{i,j,t}}}}{{{H_{i,j,t - 1}}}}} \right)$$ is the growth rate of hours worked for $$j$$ the type of workers in industry $$i$$, from period $$t-1$$ to period $$t$$; and

$${W_{i,j,t}}$$ is the wage rate for $$j$$ the type of workers in industry $$i$$ at time $$t$$.

19.124    Labour input index for the market sector is calculated as a Törnqvist volume index of hours worked of different types of workers:

$$\large \ln \left( {\frac{{{L_{m,t}}}}{{{L_{m,t - 1}}}}} \right) = \sum\limits_j {{s_{L,mjt}}\ln \left( {\frac{{{H_{m,j,t}}}}{{{H_{m,j,t - 1}}}}} \right)}$$         - - - - - - - (19.10)

and the two-period average value share of each type of workers in the industry labour compensation is given by:

$$\large {s_{L,mjt}} = \frac{{\left( {\frac{{{W_{m,j,t}}{H_{m,j,t}}}}{{\sum\nolimits_j {{W_{m,j,t}}{H_{m,j,t}}} }} + \frac{{{W_{m,j,t - 1}}{H_{m,j,t - 1}}}}{{\sum\nolimits_j {{W_{m,j,t - 1}}{H_{m,j,t - 1}}} }}} \right)}}{2}$$

where

$$\ln \left( {\frac{{{L_{m,t}}}}{{{L_{m,t - 1}}}}} \right)$$ is the labour input growth rate for the market sector, from period $$t-1$$ to period $$t$$;

$$\ln \left( {\frac{{{H_{m,j,t}}}}{{{H_{m,j,t - 1}}}}} \right)$$ is the growth rate of hours worked for $$j$$ the type of workers in the market sector, from period $$t-1$$ to period $$t$$; and

$${W_{m,j,t}}$$ is the wage rate for $$j$$ the type of workers in the market sector at time $$t$$.

## Industry combined primary inputs index

19.125    Industry combined primary inputs index is calculated as a Törnqvist index of primary inputs - capital and labour:

$$\ln \left( {\frac{{I_{i,t}^{\left( V \right)}}}{{I_{i,t - 1}^{\left( V \right)}}}} \right) = \frac{1}{2}\left( {{v_{K,i,t}} + {v_{K,i,t - 1}}} \right)\ln \left( {\frac{{{K_{i,t}}}}{{{K_{i,t - 1}}}}} \right) + \frac{1}{2}\left( {{v_{L,i,t}} + {v_{L,i,t - 1}}} \right)\ln \left( {\frac{{{L_{i,t}}}}{{{L_{i,t - 1}}}}} \right)$$         - - - - - - - (19.11)

where

$$\ln \left( {\frac{{I_{i,t}^{\left( V \right)}}}{{I_{i,t - 1}^{\left( V \right)}}}} \right)$$ is primary input growth rate for industry $$i$$ from period $$t-1$$ to period $$t$$;

$$\ln \left( {\frac{{{K_{i,t}}}}{{{K_{i,t - 1}}}}} \right)$$ is the capital input growth rate for industry $$i$$;

$$\ln \left( {\frac{{{L_{i,t}}}}{{{L_{i,t - 1}}}}} \right)$$ is the labour input growth rate for industry $$i$$;

$$v_{K,i}$$ is the capital share in value-added based total industry income; and

$${v_{L,i}}$$ is the labour share in value-added based total industry income.

## Gross output based combined input index

19.126    The gross output based combined input index is constructed as:

\begin{aligned}\ln \left( {\frac{{I_{i,t}^{\left( G \right)}}}{{I_{i,t - 1}^{\left( G \right)}}}} \right) &= \frac{1}{2}\left( {{z_{K,i,t}} + {z_{K,i,t - 1}}} \right)\ln \left( {\frac{{{K_{i,t}}}}{{{K_{i,t - 1}}}}} \right) + \frac{1}{2}\left( {{z_{L,i,t}} + {z_{L,i,t - 1}}} \right)\ln \left( {\frac{{{L_{i,t}}}}{{{L_{i,t - 1}}}}} \right) \\&\hspace{0.6cm}+ \frac{1}{2}\left( {{z_{X,i,t}} + {z_{X,i,t - 1}}} \right)\ln \left( {\frac{{{X_{i,t}}}}{{{X_{i,t - 1}}}}} \right)\end{aligned}         - - - - - - - (19.12)

where

$$\ln \left( {\frac{{I_{i,t}^{\left( G \right)}}}{{I_{i,t - 1}^{\left( G \right)}}}} \right)$$ is the gross output based input growth rate for industry $$i$$ from period $$t-1$$ to period $$t$$;

$$\ln \left( {\frac{{{X_{i,t}}}}{{{X_{i,t - 1}}}}} \right)$$ is the intermediate input growth rate for industry $$i$$, and $$z_{K,i,t}$$$$z_{L,i,t}$$; and

$$z_{X,i,t}$$ are the capital, labour and intermediate input share of total industry income respectively.

## Market sector primary input index

19.127    The combined primary input index for the market sector is calculated as:

$$\ln \left( {\frac{{I_{M,t}^{\left( V \right)}}}{{I_{M,t - 1}^{\left( V \right)}}}} \right) = \frac{1}{2}\left( {{v_{K,t}} + {v_{K,t - 1}}} \right)\ln \left( {\frac{{{K_{M,t}}}}{{{K_{M,t - 1}}}}} \right) + \frac{1}{2}\left( {{v_{L,t}} + {v_{L,t - 1}}} \right)\ln \left( {\frac{{{L_{M,t}}}}{{{L_{M,t - 1}}}}} \right)$$         - - - - - - - (19.13)

where

$$\ln \left( {\frac{{{K_{M,t}}}}{{{K_{M,t - 1}}}}} \right)$$ is the capital input growth rate for the market sector;

$$\ln \left( {\frac{{{L_{M,t}}}}{{{L_{M,t - 1}}}}} \right)$$ is the labour input growth rate for the market sector, and

$${v_{K,t}}$$ and $${v_{L,t}}$$ is the capital and labour share respectively in total income in the market sector

19.128    The capital and labour income shares, $$v_K$$ and $$v_L$$ respectively, are defined below:

$$\large {v_{K,t}} = \frac{{\sum\nolimits_{i \in M} {\left( {GO{S_{i,t}} + GMI{{\left( K \right)}_{i,t}} + IBT{{\left( K \right)}_{i,t}}} \right)} }}{{\sum\nolimits_{i \in M} {\left( {CO{E_{i,t}} + GMI{{\left( L \right)}_{i,t}} + IBT{{\left( L \right)}_{i,t}}} \right)} + \sum\nolimits_{i \in M} {\left( {GO{S_{i,t}} + GMI{{\left( K \right)}_{i,t}} + IBT{{\left( K \right)}_{i,t}}} \right)} }}$$

$$\large v_{L,t} = \frac{{\sum\nolimits_{i \in M} {\left( {CO{E_{i,t}} + GMI{{\left( L \right)}_{i,t}} + IBT{{\left( L \right)}_{i,t}}} \right)} }}{{\sum\nolimits_{i \in M} {\left( {CO{E_{i,t}} + GMI{{\left( L \right)}_{i,t}} + IBT{{\left( L \right)}_{i,t}}} \right)} + \sum\nolimits_{i \in M} {\left( {GO{S_{i,t}} + GMI{{\left( K \right)}_{i,t}} + IBT{{\left( K \right)}_{i,t}}} \right)} }}$$

## Industry value added based MFP calculations

19.129    The industry value-added based MFP growth is calculated as the industry value added growth rate minus the industry combined primary input growth rate:

$$\large \ln \left( {\frac{{A_{i,t}^{\left( V \right)}}}{{A_{i,t - 1}^{\left( V \right)}}}} \right) = \ln \left( {\frac{{{V_{i,t}}}}{{{V_{i,t - 1}}}}} \right) - \ln \left( {\frac{{I_{i,t}^{\left( V \right)}}}{{I_{i,t - 1}^{\left( V \right)}}}} \right)$$         - - - - - - - (19.14)

where

$$\ln \left( {\frac{{A_{i,t}^{\left( V \right)}}}{{A_{i,t - 1}^{\left( V \right)}}}} \right)$$ is the industry value added MFP growth rate; and

$$\ln\left( {\frac{{{V_{i,t}}}}{{{V_{i,t - 1}}}}} \right)$$ is the industry value-added growth rate.

## Industry gross output based MFP calculations

19.130    Gross output-based MFP index is calculated as the industry gross output growth rate minus the industry gross output based combined input growth rate:

$$\large \ln \left( {\frac{{A_{i,t}^{\left( G \right)}}}{{A_{i,t - 1}^{\left( G \right)}}}} \right) = \ln \left( {\frac{{{G_{i,t}}}}{{{G_{i,t - 1}}}}} \right) - \ln \left( {\frac{{I_{i,t}^{\left( G \right)}}}{{I_{i,t - 1}^{\left( G \right)}}}} \right)$$         - - - - - - - (19.15)

where

$$\ln \left( {\frac{{A_{i,t}^{\left( G \right)}}}{{A_{i,t - 1}^{\left( G \right)}}}} \right)$$ is the industry gross output-based MFP growth rate; and

$$\ln \left( {\frac{{{G_{i,t}}}}{{{G_{i,t - 1}}}}} \right)$$ is the industry gross output growth rate.

## Multifactor productivity for the market sector

19.131    The market sector MFP is calculated as the market sector output growth rate minus the market sector combined input growth rate:

$$\large \ln \left( {\frac{{A_{M,t}^{\left( V \right)}}}{{A_{M,t - 1}^{\left( V \right)}}}} \right) = \ln \left( {\frac{{{V_{M,t}}}}{{{V_{M,t - 1}}}}} \right) - \ln \left( {\frac{{I_{M,t}^{\left( V \right)}}}{{I_{M,t - 1}^{\left( V \right)}}}} \right)$$         - - - - - - - (19.16)

where

$$\ln \left( {\frac{{A_{M,t}^{\left( V \right)}}}{{A_{M,t - 1}^{\left( V \right)}}}} \right)$$ is the market sector MFP growth rate; and

$$\ln \left( {\frac{{{V_{M,t}}}}{{{V_{M,t - 1}}}}} \right)$$ is the market sector output growth rate.

### Endnotes

1. A special case is that industry aggregation shares to the market sector are identical using either industry distribution of income or capital when the internal rate of return (IRR) is solved endogenously. In practice, a lower bound exogenous IRR of CPI+4% is frequently imposed and thus market sector capital services may differ using a pure capital aggregator.