19.23    It is useful to distinguish between measured productivity and technical progress in productivity analysis. Productivity statistics aim to measure technical progress or the efficiency of production. In practice, productivity changes are measured by the difference between the growth in the volume of output and the growth in the volume of inputs, reflecting more than just technical progress. Year-to-year changes also contain 'noise' that is distinct from the notion of technical progress; it is therefore advisable to examine productivity changes over an extended period to look through some of the short-term volatility.

19.24    Although, from a conceptual standpoint, MFP can be interpreted in various ways, a key interpretation of MFP is as disembodied technological change attributable to improved use of factor inputs. Embodied technological change represents advances in the design and quality of new capital and intermediate inputs. Disembodied technological change is generally interpreted as representing costless improvements or knowledge, for example, network effects or spillovers from diffusion of publicly available R&D, and benefits to factor inputs from organisational change or better management. These spillovers and other benefits to factor inputs are generally not quantifiable within the KLEMS growth accounting framework.

19.25    At the industry level, ABS publishes both gross output-based MFP and value added-based MFP – they are complementary. One advantage of the gross output-based MFP approach is that it is a natural output concept⁷⁵ and consistent with the traditional production theory linking output to primary as well as intermediate inputs. By comparison, the value added-based MFP approach assumes that the components of value added are separable from that of intermediate inputs.⁷⁶ ⁷⁷

19.26    For a given industry, the relationship between changes in gross output-based MFP and value added-based MFP can be approximated by:

    \(\large \Delta \ln GO\;MFP_i^{} \approx \frac{{V{A_i}}}{{G{O_i}}}\Delta \ln VA\;MFP_i^{}\)                                                                                                         

where \(\small{\frac{{V{A_i}}}{{G{O_i}}} , \Delta \ln GO\;MFP , \Delta \ln VA\;MFP}\) are the two period average of the ratio of nominal industry value added to nominal industry gross output, rate of change in gross output-based MFP, and rate of change in value added-based MFP, respectively.⁷⁸ Since this ratio is always less than unity, gross output-based MFP will always have less amplitude than value added-based MFP, i.e. rise less and fall less. However, the degree to which they differ varies from industry to industry, due to both the variation in each industry’s relative value-added proportion, as well as the degree to which the ratio changes over time.  At an aggregate level, the value-added concept is more appropriate as it removes inter–industry transfers.

19.27    In interpreting MFP, it should be noted that measured productivity growth could include factors other than technological change, for example adjustment costs, cyclical effects and measurement errors.⁷⁹ A limitation of MFP theory is that the assumptions of the neoclassical models do not necessarily hold in practice, which can affect the interpretation of the resulting estimates. For example, imperfect competition can result in gains from increasing market dominance being reflected as productivity gains. Additionally, in static models of production, such as the one used in estimating KLEMS MFP, capital is an exogenous input, which ignores dynamic feedback between MFP and capital. For example, if technological change increases output per person, the additional output per person may lead to further savings and investment and thus a rise in the capital–labour ratio. While traditional growth accounting identifies this induced effect as the contribution of capital growth, the effect can be attributed to an initial shift in technology. Therefore, MFP measures may understate the importance of productivity growth in contributing to output growth.

19.28    The methodology used in compiling the estimates implicitly assumes that the proportion of capital stock used in production (capital utilisation) does not change; therefore any real world change in the extent to which capital is utilised in production will be recorded as a change in productivity. Another assumption of the methodology is each hour of labour input is fully utilised in production. Further, improvements in output due to a firm’s ability to produce more output because of their size, that is, economies of scale, will also appear as a measured productivity improvement.

19.73    The quantity of capital services is estimated by assuming that capital services produced by an asset are proportional to the value of productive capital stock of the asset; that is:

    \(\large K_t = u_t PKS_t\)

where \(K_t \)is the quantity of capital services, and \(PKS_t \)is the productive capital stock and \(u_t\) is the capacity utilization rate.

19.74    The capacity utilization rate is assumed to be constant over time. This assumption has two implications. First, as \(u_t\) is constant and invariant to time, a change in the quantity of capital services delivered from any given capital asset is tantamount to a change in its productive capital stock. Second, variations in the utilization of the capital stock are not accounted for in the estimation of its capital services, and as a consequence changes in the capital services over time may reflect the impact of short-term business cycles, other than movements of capital input.

19.75    The productive capital stock estimates are derived from data on gross fixed capital formation (except inventories and land), using the PIM. The essence of this method is to transform all capital assets of different vintages into equivalent efficiency units and then add them up into an estimate of the productive capital stock. Chapter 14 provides a full description of the procedures used to derive the productive capital stock. Chapter 10 provides a full description of the data sources and procedures used to compile estimates of gross fixed capital formation.

19.76    In estimating the value of labour services, statisticians can directly observe labour rental prices as wage rates paid to workers. In the case of capital however, the rental prices for capital have to be imputed. The rental price reflects the price at which an investor is indifferent between two alternatives:

1. earning a nominal rate of return on a different investment; and
2. buying a capital asset, renting it out, collecting rent and selling it in the next period. 

19.77    A standard specification for the capital rental price in the absence of taxes is the arbitrage equation⁹⁰:  

     \(\large P_j,_{t-1}(1+i_t) = r_{j,t}+(1-δ_j)P_{j,t}\)         - - - - - - - (19.1)

where \(i_t\) is the nominal interest, \(P_{j,t-1}\) is the acquisition price of capital asset \(j\) at the beginning of the period, \(r_{j,t}\) is the rental price, \(P_{j,t}\) is the price of capital asset j at the end of the period and \(δ_{j,t}\) is the rate of economic depreciation.

19.78    This can be rearranged into the expression:

    \(\large r_{j,t}=i_tPj_{,t-1}+δ_{j,t}P_{j,t}- \pi _{j,t}\)         - - - - - - - (19.2)

where \(\pi_{j,t} = P_{j,t} - P_{j,t-1}\) is the asset-specific capital gains term.

19.79    Equation (19.2) shows that the capital rental price consists of three components: the rate of return to capital, the depreciation rate and the capital gain or loss due to revaluation. The industry dimension is supressed here.

19.80    When tax considerations are given to the measurement of capital rental prices (both capital income taxes and indirect business taxes), the tax-adjusted rental price equation becomes:

    \(\large r_{i,j,t}=T_{i,j,t}(i_{i,t}P_{i,j,t-1}+δ_{j,t}P_{i,j,t}-π_{i,j,t})+x_{i,t}P_{i,j,t-1}\)         - - - - - - - (19.3)

where \(i\) indexes industries, \(T_{i,j,t}\) is the income tax parameter and \(x_{i,t}\) is the effective net indirect tax rate on production. The description of data sources for constructing the tax parameter is provided in Annex C.

19.81    The rate of return to capital \(i_t\) can be estimated by either endogenously or exogenously. Under the endogenous approach, the total value of capital services in each industry is assumed to be equal to the compensation for all assets in that industry. The resulting internal rate of return exhausts capital income and is consistent with constant returns to scale. The nominal rate of return is the same for all assets in an industry but may vary across industries.

19.82    In the case of the exogenous approach, the nominal rate may equal the Treasury bond rate, or the dividend yield on a stock index. This method allows the value of capital income to deviate from property compensation, assuming imperfect competition and non-constant returns to scale. For a detailed discussion of these two alternative methods and associated sensitivity analysis, see Appendix 2 Sensitivity Analysis of Capital Inputs, in the Information paper, Experimental Estimates of Industry Multifactor Productivity.

19.83    The ABS follows the endogenous method in producing its official productivity estimates. For the corporate sector, iit, is solved for all assets in each industry by assuming that gross operating surplus, \(GOS_{it}\) equals the rental price multiplied by the real productive capital stock in each industry:

    \(\large GO{S_{i,t}} = \sum\limits_j {{r_{i,j,t}}{K_{i,j,t}}}\)         - - - - - - - (19.4)

and substituting for the rental price in equation (19.4) giving:

    \(\large GO{S_{i,t}} = \sum\limits_j {{K_{i,j,t}}\left( {{T_{i,j,t}}\left( {{i_{i,t}}{P_{i,j,t - 1}} + {\delta _j}{P_{i,j,t}} - {\pi _{i,j,t}}} \right) + {x_{i,t}}{P_{i,j,t - 1}}} \right)}\)         - - - - - - - (19.5)

so

    \(\large {i_{i,t}} = \frac{{GO{S_{i,t}} - \sum\nolimits_j {{K_{i,j,t}}\left( {{T_{i,j,t}}\left( {{\delta _j}{P_{i,j,t}} - {\pi _{i,j,t}}} \right) + {x_{i,t}}{P_{i,j,t - 1}}} \right)} }}{{\sum\nolimits_j {{K_{i,j,t}}{T_{i,j,t}}{P_{i,j,t - 1}}} }}\)         - - - - - - - (19.6)

19.84    To prevent negative rental prices, the ABS imposes a floor limit on the internal rate of return of CPI plus 4 per cent; otherwise, the endogenous rate is used.

19.85    The depreciation of a capital asset measures the change in its real economic value during the accounting period. The depreciation rates are derived using asset age-price profiles. The age-price profiles are constructed by using corresponding age-efficiency profiles, multiplied by a suitable discount rate (the ABS chooses a real discount rate at 4 per cent). See Chapter 14 for the detailed description of age-efficiency and age-price profiles and their roles in constructing various capital components.  

19.86    The capital gain or loss due to revaluation can be calculated as an asset-specific deflator or a general deflator. As defined in equation (19.3), the asset-specific capital term is used and calculated as the percentage change in the value of the asset in time t-1 relative to its value in time t. Alternatively, \(π_t\) can be replaced by a general price deflator such as the consumer price index. The former is preferred because it is able to account for the large changes in relative prices across heterogeneous asset classes and therefore reduces measurement errors. However, the disadvantage of using asset-specific deflators is that it often introduces volatility into the rental price equation. 

19.87    The elemental capital inputs are compiled at a detailed level. There are capital input measures for up to 16 asset types for the corporate and unincorporated entities for each of the 16 ANZSIC industry divisions that comprise the market sector. For each capital input there is a volume indicator of the flow of capital services, and a rental price that is used to weight the service flow with the service flows of other capital inputs. 

19.109    The capital and labour income shares, \( S_K\) and \(S_L\) are derived from the current price factor income accounts. For a given industry or aggregate, total income can be decomposed into:

• gross operating surplus (GOS) of corporations and general government;
• gross mixed income (GMI) of unincorporated firms;
• compensation of employees (COE); and
• taxes less subsidies on production and imports (IBT).

19.110    Note that total income includes the GOS of general government but not the GOS of dwellings owned by persons, as ownership of dwellings is excluded from the market sector.

19.111    Both GMI and IBT include capital and labour components. They can be further decomposed into income attributable to labour and capital, as described in the next two sections. Total income can be written as:

    \(Total\:Income = GOS + GMI\left( K \right) + GMI\left( L \right) + COE + IBT\left( K \right) + IBT\left( L \right)\)

where K and L are income attributable to capital and labour, respectively.

19.112    The income share of capital is thus:

    \(\large {S_K} = \frac{{GOS + GMI\left( K \right) + IBT\left( K \right)}}{{Total\:Income}}\)

and the income share of labour is:

    \(\large {S_L} = \frac{{COE + GMI\left( L \right) + IBT\left( L \right)}}{{Total\:Income}}\)

19.113    The labour and capital shares of income earned by unincorporated enterprises are subsumed into one national accounts aggregate: gross mixed income. The following procedure is used to impute labour and capital shares of this aggregate for each industry in the market sector.

19.114    An estimate of labour income is imputed by assuming that proprietors and unpaid helpers receive the same average compensation per hour as wage and salary earners. Similarly, an estimate of proprietors' capital income is derived by multiplying the unincorporated productive capital stock of each asset type by the corporate rental prices. These estimates are then scaled so they sum to the observed GMI. The capital and labour shares of GMI are the corresponding scaled estimates.

19.115    That is, the capital share of GMI is:

    \(\large {s_i}\sum\limits_j {{r_{c,i,j}}{K_{u,i,j}}}\)

where \(s_i\) is the scaling factor for industry \(i\);

 \(r_{c,i,j}\) is the corporate rental price of asset \(j\) in industry \(i\); and

\(K_{u,i,j}\) is the productive capital stock of asset \(j\) in industry \(i\) for unincorporated enterprises.

The labour share of GMI is:

    \(\large {s_i}{w_i}{H_{u,i}}\)

where \(s_i\) is (again) the scaling factor for industry \(i\)

\(w_i\) is the average hourly income for wage and salary earners in industry \(i\); and

\(H_{u,i} \)is the hours worked by proprietors and unpaid helpers in industry \(i\).

19.116    The scaling factor \(s_i\) for industry \(i\) is given by:

    \(\large {s_i} = \frac{{GMI}}{{{{\widehat {GMI}}_{u,i}}}}\)

and \({\widehat {GMI}_{u,i}}\) for each industry is imputed, based on the labour and capital cost as:

    \(\large {\widehat {GMI}_{u,i}} = {w_i}{H_{u,i}} + \sum\limits_j {{r_{c,i,j}}{K_{u,i,j}}}\)

19.117    Some taxes and subsidies on production and imports can be attributed solely to either capital or labour (for example, land tax and payroll tax). Such taxes and subsidies, however, make up only a small proportion of total net taxes. The capital and labour shares of net taxes are thus allocated proportionally, using the other income components attributable to labour and capital

19.118    The gross output income shares are derived similarly except that intermediate inputs need to be included:

    \(\begin{aligned} Total\:Income &= GOS + GMI\left( K \right) + GMI\left( L \right) + COE + IBT\left( K \right) + IBT\left( L \right)\\&\hspace{0.6cm}+ Intermediate\:inputs \end{aligned}\)

19.119    Thus, the income share of capital is:

    \(\large {Z_K} = \frac{{GOS + GMI\left( K \right) + IBT\left( K \right)}}{{Total\:Income}}\)

and the income share of labour is:

    \(\large {Z_L} = \frac{{COE + GMI\left( L \right) + IBT\left( L \right)}}{{Total\:Income}}\)

and the income share of intermediate inputs is:

    \(\large {Z_M} = \frac{{Intermediate\:inputs}}{{Total\:income}}\)

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

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 \(t\).

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.

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.

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:

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

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

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.

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.

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.

19.134    Stiroh⁹⁷ developed a decomposition framework to identify the industry sources of aggregate labour productivity growth, which becomes a standard method for analysing industry contributions to aggregate labour productivity growth⁹⁸ ⁹⁹. An ABS productivity research paper¹⁰⁰ has applied this method to link the market sector labour productivity growth to the industry sources. The decomposition formula is given as:

    \(\small {\begin{aligned}\ln \left( {\frac{{AL{P_t}}}{{AL{P_{t - 1}}}}} \right) &= \sum\limits_i {{w_{i,t}}\ln \left( {\frac{{LP_{i,t}^y}}{{LP_{i,t - 1}^y}}} \right)} + \left[ {\sum\limits_i {{w_{i,t}}\ln \left( {\frac{{{H_{i,t}}}}{{{H_{i,t - 1}}}}} \right) - \sum\limits_i {{h_{i,t}}\ln \left( {\frac{{{H_{i,t}}}}{{{H_{i,t - 1}}}}} \right)} } } \right] \\&\hspace{0.6cm}- \left[ {\sum\limits_i {{m_{i,t}}\left( {\ln \left( {\frac{{{X_{i,t}}}}{{{X_{i,t - 1}}}}} \right) - \ln \left( {\frac{{{Y_{i,t}}}}{{{Y_{i,t - 1}}}}} \right)} \right)} } \right] \end{aligned}}\)         - - - - - - - (19.17)

19.135    Using the value added concept of labour productivity, equation (19.17) can be simplified to

    \(\ln \left( {\frac{{AL{P_t}}}{{AL{P_{t - 1}}}}} \right) = \sum\limits_i {{w_{i,t}}\ln \left( {\frac{{LP_{i,t}^V}}{{LP_{i,t - 1}^V}}} \right)} + \left[ {\sum\limits_i {{w_{i,t}}\ln \left( {\frac{{{H_{i,t}}}}{{{H_{i,t - 1}}}}} \right) - \sum\limits_i {{h_{i,t}}\ln \left( {\frac{{{H_{i,t}}}}{{{H_{i,t - 1}}}}} \right)} } } \right]\)         - - - - - - - (19.17a)

where

    \(ALPt\) is the aggregate labour productivity (aggregate value added per hour);

    \(LP_{i,t}^{y}\) is the gross output labour productivity for industry \(i\);

    \(LP_{i,t}^v\) is the value added labour productivity for industry \(i\);

    \(w_{i,t}\) is the two-period average of industry \(i\)'s share in aggregate value added;

    \(m_{i,t}\) is the two-period average of the ratio of industry \(i\)’s intermediate input in aggregate value added;

    \(h_{i,t}\) is the industry \(i\)’s share in aggregate hours in period \(t-1\); and

    \(M\), \(Y\) and \(H\) stand for intermediate input, gross output and hours worked respectively. 

19.136    The first term in equation 19.17 is a 'direct productivity effect', which is equal to the weighted sum of industry gross output productivity growth rates, with the industry shares in total value added as weights. This term captures the impact of productivity growth in each industry. As industry labour productivity rises, the aggregate labour productivity also improves in proportion to industries’ shares in aggregate output.

19.137    The second term in equation 19.17 is a 'labour reallocation effect' that captures the impact on aggregate output of the shift of labour between low-productivity-level industries and high-productivity-level industries. Aggregate productivity growth depends not only on the rates of productivity within industries but also on changes in the composition of industries. Faster employment growth in high-productivity-level industries contributes to improvements in the aggregate labour productivity growth by increasing the size of aggregate output given the same quantity of hours worked.

19.138    The third term in equation 19.17 is the intermediate input intensity factor. As value added is defined as gross output minus intermediate input, the relative growth of intermediate inputs over gross output must be accounted for in aggregating industry gross output to reach aggregate output, which is a value added concept. For example, if growth of intermediate input usage is faster than that of gross output, the growth of value added is reduced and hence the growth rate of aggregate labour productivity declines. If less intermediate inputs are used for a given level of gross output, then more value added is created and hence aggregate labour productivity improves.

19.139    The 'direct productivity effect' in equation 19.17 can also be expressed in terms of capital services per hour (capital deepening), labour composition, and MFP. Moreover, to facilitate analysis of the productive contribution of information technologies, capital services per hour can be partitioned into IT and non-IT capital services per hour. That is

    \(\begin{aligned}\sum\limits_i {{w_{i,t}}} \ln \left( {\frac{{LP_{i,t}^V}}{{LP_{i,t - 1}^V}}} \right) &= \sum\limits_i {{w_{i,t}}} \left[ {\widetilde s_{i,t}^K\widetilde \alpha _{i,t}^{IT}\Delta \ln \left( {\frac{{K_{i,t}^{IT}}}{{{H_{i,t}}}}} \right) + \widetilde s_{i,t}^K\widetilde \alpha _{i,t}^N\Delta \ln \left( {\frac{{K_{i,t}^N}}{{{H_{i,t}}}}} \right) \\+ \widetilde s_{i,t}^L\Delta \ln {Q_{i,t}} + \Delta \ln A_{i,t}^v} \right]\end{aligned}\)         - - - - - - - (19.18)

Where \(\widetilde s_{i,t}^K\) is industry’s two period capital income share, \(\widetilde \alpha _{i,t}^{IT}\) is the IT share of industry capital (computers and software), \(\widetilde \alpha _{i,t}^{N}\) is the non-IT share of industry capital \(\widetilde s_{i,t}^L\) is industry’s two period labour income share, and \({Q_{i,t}}\) is labour composition. On the RHS of equation (19.18), capital services per hour are partitioned into the change in IT capital services per hour (that is, IT capital deepening from computers and software, \(\frac{{K_t^{IT}}}{{{H_t}}}\), change in non-IT capital deepening \(\frac{{K_t^N}}{{{H_t}}}\), the change in labour composition \(Q_t\), and the change in GVA based MFP, \(A_t^v\).

19A.2    When output is measured as value added and the inputs considered are labour and capital, output is modelled:

    \(\large {V_t} = {A_t}F\left( {{K_t},{L_t}} \right)\)         - - - - - - - (19A.1)

where

    \(V_t\) is real value added at time \(t\):

    \(A_t\) is multifactor productivity at time \(t\);

    \(F\) is the production function at time \(0\);

    \(K_t\) is the real capital input at time \(t\);

    \(L_t\) is the real labour input at time \(t\); and

    \(t\) is a continuous measure of time.

19A.3    Note that the production function \(F\) s not observable for the actual economy. Thus to measure productivity an expression for \(A_t\) not involving \(F\) must be derived. To do so, two assumptions are made about the production function \(F\). First, that it exhibits constant returns to scale. That is, for any positive \(λ∂\) the production function satisfies

    \(\large F\left( {\lambda K,\lambda L} \right) = \lambda F\left( {K,L} \right)\)

19A.4    In words, this means that (say) doubling both inputs will double the output. Second, we assume that the marginal returns to capital and labour equal their respective real market prices. That is, we assume that

    \(\large \frac{{\partial V}}{{\partial K}}\left( {{K_t},{L_t}} \right) = {r_t}\)

and

    \(\large \frac{{\partial V}}{{\partial L}}\left( {{K_t},{L_t}} \right) = {w_t}\)

where \(r_t\) is the real rental price of a unit of capital (at time \(t\)); and

\(w_t\) is the real wage rate for a unit of labour (at time \(t\))

19A.5     Now, differentiating \(V_t\) with respect to time gives:

    \(\large \begin{aligned} {\mathop V\limits^.} _t &= {{\mathop A\limits^. }_t}F\left( {{K_t},{L_t}} \right) + {A_t}\left( {\frac{{\partial F}}{{\partial K}}{{\mathop K\limits^. }_t} + \frac{{\partial F}}{{\partial L}}{{\mathop L\limits^. }_t}} \right) \\ &= \mathop {{{\mathop A\limits^. }_t}F\left( {{K_t},{L_t}} \right) + }\limits^. \frac{{\partial V_t}}{{\partial K}}{\mathop K\limits^.} _t + \frac{{\partial {V_t}}}{{\partial L}}{\mathop L\limits^.} _t \end{aligned}\)         - - - - - - - (19A.2)

where for any variable \(X\), \(\mathop{X}\limits^.\) denotes the derivative of a function \(X\) with respect to time.

19A.6    Now dividing equation (19A.2) by \(V_t\) gives:

    \(\large \begin{aligned} \frac{{\mathop{V}\limits^.}_t}{V_t}&=\frac{{\mathop{A}\limits^.}_t}{A_t}+\frac{\partial V_t}{\partial K}\frac{{\mathop{K}\limits^.}_t}{V_t}+\frac{\partial V_t}{\partial L}\frac{{\mathop{L}\limits^.}_t}{V_t} \\&=\frac{{\mathop{A}\limits^.}_t}{A_t}+\frac{\partial V_t}{\partial K}\frac{{\mathop{K}}_t}{V_t}\frac{{\mathop{K}\limits^.}_t}{K_t}+\frac{\partial V_t}{\partial L}\frac{L_t}{V_t}\frac{{\mathop{L}\limits^.}_t}{L_t} \end{aligned}\)         - - - - - - - (19A.3)

19A.7    Since we have assumed that the marginal products of capital and labour are equal to their respective real market prices, equation (19A.3) becomes 

    \(\large \frac{{{{\mathop V\limits^. }_t}}}{{{V_t}}} = \frac{{{{\mathop A\limits^. }_t}}}{{{A_t}}} + {S_{K,t}}\frac{{{{\mathop K\limits^. }_t}}}{{{K_t}}} + {S_{L,t}}\frac{{{{\mathop L\limits^. }_t}}}{{{L_t}}}\)         - - - - - - - (19A.4)

where

    \({S_{K,t}} = {r_t}\frac{{{K_t}}}{{{V_t}}}\)

    \({S_{L,t}} = {w_t}\frac{{{L_t}}}{{{V_t}}}\)

19A.8    Note that \(S_{K,t}\) and \(S_{L,t}\) are the (value added) income shares of capital and labour, respectively. As the production function exhibits constant returns to scale, income can be attributed to either capital or labour; that is:

    \({S_{K,t}} + {S_{L,t}} = 1\)

19A.9    To translate equation (19.4) into a discrete time equivalent, a Törnqvist index formula is chosen. Using a Törnqvist index follows international best practice. It is preferred to other index formulas due having desirable properties (from a microeconomic perspective) as shown by Diewert.¹⁰¹ In particular, an index of multifactor productivity is calculated using the equation:

    \(\large \ln \left( {\frac{{{A_t}}}{{{A_{t - 1}}}}} \right) = \ln \left( {\frac{{{V_t}}}{{{V_{t - 1}}}}} \right) - {\overline S _{K,t}}\ln \left( {\frac{{{K_t}}}{{{K_{t - 1}}}}} \right) - {\overline S _{L,t}}\ln \left( {\frac{{{L_t}}}{{{L_{t - 1}}}}} \right)\)         - - - - - - - (19A.5)

where

    \({\overline S _{K,t}} = \frac{1}{2}\left( {{S_{K,t}} + {S_{K,t - 1}}} \right)\)

and

    \({\overline S _{L,t}} = \frac{1}{2}\left( {{S_{L,t}} + {S_{L,t - 1}}} \right)\)

19A.10    Note that \(\ln(\frac{X_t}{X_{t-1}})\) is an approximation to the growth of \(X_t\) when this growth is small; that is:

    \(\large \ln \left( {\frac{{{X_t}}}{{{X_{ - 1}}}}} \right) \approx \frac{{{X_t} - {X_{t - 1}}}}{{{X_{t - 1}}}}\)

19A.11    Equation (19A.5) provides the standard growth accounting framework for growth in real value added. From this equation the contributions of MFP, capital, and labour to growth in value added are quantified:

• the contribution of capital is defined to be the growth rate of capital input times the capital share of value added \({\overline S _{K,t}}\Delta \ln {K_t}\),
• the contribution of labour is defined to be the growth rate of labour input times the labour share of the value added \({\overline S _{L,t}}\Delta \ln {L_t}\), and

• the contribution of multifactor productivity is defined as the residual

 \(\hspace{1.6cm} \large \ln \left( {\frac{{{V_t}}}{{{V_{t - 1}}}}} \right) - {\overline S _{K,t}}\ln \left( {\frac{{{K_t}}}{{{K_{t - 1}}}}} \right) - {\overline S _{L,t}}\ln \left( {\frac{{{L_t}}}{{{L_{t - 1}}}}} \right)\)

              that is, as the growth of value added not attributed to capital or labour.

19A.12    Note that when labour is measured as quality adjusted hours worked the contribution of labour can be further decomposed into the contributions of labour quality and hours worked.

19A.13    The gross output-based measure of MFP is an approach that includes the use of intermediate inputs as a source of output growth. For each industry, a production function postulated is as follows:

    \(\large {G_t} = {A^G}H\left( {{K_t}{L_t}{X_t}} \right)\)         - - - - - - - (19A.6)

where

    \(G_t\)= real output;

    \(K_t\) = real capital input;

    \(L_t\) = real labour input;

    \(X_t\) = real intermediate input;

    \(A_t^G\) = the gross output index of MFP, reflecting technological change, etc.;

    \(H(K_tL_tX_t)\) = a function of factor inputs [\(K_t\), \(L_t\) and \(X_t\)] defining the expected level of output at time \(t\), given the conditions of technology in the base period; and

    \(t\) = a continuous measure of time.

19A.14    For equation (19A.6), we make the assumptions of constant returns to scale and competitive equilibrium. Then differentiating with respect to time and dividing both sides by \(G_t\) it can be shown that

    \(\large \frac{{{{\mathop G\limits^. }_t}}}{{{G_t}}} = \frac{{\mathop {{A^G}}\limits^. }}{{{A^G}}} + {S_K}\frac{{{{\mathop K\limits^. }_t}}}{{{K_t}}} + {S_L}\frac{{{{\mathop L\limits^. }_t}}}{{{L_t}}} + {S_X}\frac{{{{\mathop X\limits^. }_t}}}{{{X_t}}}\)         - - - - - - - (19A.7)

where \(\mathop{G}\limits^.\), \(\mathop{K}\limits^.\), \(\mathop{L}\limits^.\) and \(\mathop{X}\limits^.\) are derivatives with respect to time:

    \(\large \mathop G\limits^. = \frac{{\partial G}}{{\partial t}},etc.\)

the weights \(S_K\), \(S_L\) and \(S_X\) are the output elasticities of the three inputs:

    \(\large {Z_K} = \frac{{\partial G}}{{\partial K}}\frac{K}{G},\)

    \(\large {Z_L} = \frac{{\partial G}}{{\partial L}}\frac{L}{G},and\)

    \(\large {Z_X} = \frac{{\partial G}}{{\partial X}}\frac{X}{G}\)

and weights \(Z_K\), \(Z_L\) and \(Z_X\) are the relative cost shares of capital, labour and intermediate inputs in the total cost:

    \(\large {Z_K} = \frac{{K{r_K}}}{{G{p_G}}},\)

    \(\large {Z_L} = \frac{{L{w_L}}}{{G{p_G}}},and\)

    \(\large {Z_X} = \frac{{X{p_X}}}{{G{p_G}}}\)

where

    \(r_K\) = the rental price of capital services;

    \(w_L\) = the price of labour;

    \(p_X\) = the price of intermediate inputs; and

    \(p_G\) = the price of gross output

19A.15    Equation (19A.7) can be rearranged to show that the growth rate of multifactor productivity is equal to the growth rate of the ratio of output to inputs as follows:

    \(\large \frac{{\mathop {{A^G}}\limits^. }}{{{A^G}}} = \frac{{\left( {\frac{{\mathop G\limits^. }}{I}} \right)}}{{\frac{G}{I}}}\)

where

    \(\large \frac{{\mathop I\limits^. }}{I} = {Z_K}\frac{{\mathop K\limits^. }}{K} + {Z_L}\frac{{\mathop L\limits^. }}{L} + {Z_X}\frac{{\mathop X\limits^. }}{X}\)

19A.16    This implies that productivity can be expressed as the ratio of output to a composite index of inputs:

    \(\large A_t^G = \frac{{{G_t}}}{{{I_t}}}\)         - - - - - - - (19A.8)

where the index \(I_t\) is computed as a Törnqvist index as follows:

    \(\large \frac{{{I_t}}}{{{I_{t - 1}}}} = {\left( {\frac{{{K_t}}}{{{K_{t - 1}}}}} \right)^{\left( {{Z_{Kt}} + {Z_{K\left( {t - 1} \right)}}} \right)/2}}{\left( {\frac{{{L_t}}}{{{L_{t - 1}}}}} \right)^{\left( {{Z_{Lt}} + {Z_{L\left( {t - 1} \right)}}} \right)/2}}{\left( {\frac{{{X_t}}}{{{X_{t - 1}}}}} \right)^{\left( {{Z_{Xt}} + {Z_{X\left( {t - 1} \right)}}} \right)/2}}\)

and \(Z_{Kt}\), \(Z_{Lt}\) and \(Z_{Xt}\) are the respective relative cost shares of capital, labour and intermediate inputs respectively. In the KLEMS growth accounting framework, the growth in intermediate inputs \((X_t/X_{t-1})\) is further partitioned into energy, materials and services. For a more detailed description of the KLEMS growth accounting framework, see Information Paper: Experimental Estimates of Industry Level KLEMS Multifactor Productivity, 2015.

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.

19C.1    The income tax parameter,\(T_{ijt}\) allows for the variation of income tax allowances according to different industries, asset types, and variations in allowances over time. Changes in corporate profit taxes over time are also allowed for. Corporate taxes aside, these provisions increase the after-tax returns on investment and lower the rental price of capital. For each industry \(i\), and asset type \(j\), \(T_{ijt}\) is expressed as:

    \(\large {T_{ijt}} = \frac{{1 - {u_t}{z_{ijt}} - {u_t}{a_{ijt}}}}{{1 - {u_t}}}\)

where

    \(u_t\) = the corporate profit tax rate;

    \(Z_{ijt}\) = the present discounted value of one dollar of depreciation allowances; and

    \(a_{ijt}\) = the additional allowance rate.

19C.2    The tax parameter reflects the differing tax circumstances that owners of capital face. The method adopted by ABS follows Jorgenson\(¹⁰²\) and Hall and Jorgenson¹⁰³ ¹⁰⁴ and reflects changes to:

• tax concessions;
• write off periods (i.e. tax lives);
• deductions allowable;
• allowable capital expenditure;
• special allowances; and
• amortisation of capital.

19C.3    For example, allowance is made for the differing depreciation and additional allowances available to specific industries and asset types over time.  These allowances tended to be more generous in the Agriculture, forestry and fishing, Mining, and Manufacturing industries, especially for certain types of equipment.  In addition, the Australian Taxation Office (ATO) allowed for faster depreciation rates over time through shorter effective tax lives. Since 1985, various research and development (R&D) tax concessions have been introduced to encourage increased investment in R&D by Australian companies. These concessions have had the effect of reducing rental prices on R&D considerably.

19C.4    The Corporate Profit Tax Rates (\(u\)) are obtained from the ATO website.

19C.5    The depreciation allowance (\(z\)) is the present discounted value (PDV) of the stream of deductions multiplied by the marginal tax rate applicable in that year. Asset lives and a nominal discount rate are used to determine the present discounted value of depreciation allowances. Prior to 1980, the average asset lives used to calculate capital stock for each asset type are used. After 1980, the asset life consistent with the shortest life within broad asset life bands specified by the ATO is used. Broad banding reduces the effective life of the asset. The nominal discount rate is based on the business overdraft rate published in the Reserve Bank Bulletin.  It assumes that the business overdraft rate applies to all borrowers for investment in equipment or structures and contains a risk premium (over and above government bonds). 

19C.6    Specific rulings on eligible depreciation allowances are obtained from the Master Tax Guides (MTG), ATO rulings published online, and Commonwealth Budget Papers. Of the two depreciation schedules permitted, the diminishing value method has been chosen. Prior to 10 May 2006, it allowed software and machinery and equipment assets to be geometrically depreciated at 150 per cent of the straight-line rate (the other schedule permitted). From 10 May 2006, the government introduced a 200 per cent diminishing balance rate for eligible new plant and equipment assets.

19C.7    From 1980, broad banded depreciation rates were introduced, allowing assets with effective lives over a particular band of years to depreciate at a certain rate. In 1996, for example, assets with a life of 0-3 years could be depreciated immediately, and assets with a life of 3 to 5 years could be depreciated at a prime cost rate of 40 per cent of its purchase price.

19C.8    In addition to broad banding, the Commonwealth Government allowed a loading factor of between 18 per cent and 20 per cent from 1990, depreciating some assets more quickly. Most equipment except motor vehicles was permitted to use loading factors.

19C.9    Double depreciation allowances were permitted for most assets for the period in 1974-76. Between 1 July 1974 and 30 June 1976, companies were allowed to depreciate new investment excluding motor vehicles at twice the stated rates. Once purchased, the asset continued to be depreciated at these accelerated rates until completely depreciated.  We treat this by doubling the loading factor which has the effect of doubling the depreciation rate.

19C.10    In 1980, the Commonwealth Government permitted a separate allowance for buildings. Depending on the year, a straight-line allowance of 2.5 per cent or 4 per cent was permitted. This allowance is treated in the same way as depreciation allowance in the tax parameter.

19C.11    On 1 July 2001, the government introduced the 'uniform capital allowances regime'. This regime replaced the special capital allowance provisions for the Mining industry. The regime applied to all depreciable assets except where specific provisions apply to R&D activities, investments in Australian films, or cars.

19C.12    In 2002, statutory effective life caps were introduced, allowing an accelerated depreciation for certain types of equipment. Specifically, statutory life caps halved the effective tax lives of aircraft (to 10 years) and buses and trucks (larger than 3.5 tonnes) to 7.5 years.

19C.16    The additional allowance rate (\(a\)) is an immediate write-off which results in tax savings (i.e. discounting is not required). The value of an allowance is the tax savings which is the product of the tax rate and the rate of the allowance. For example, if the allowance rate is 50 per cent and the profit tax rate is 30 per cent, then the company effectively saves 15 per cent of the purchase price of the asset in tax savings (\(30\%×50\% =u×a\)). Most equipment types have attracted an allowance of some kind.

19C.17    There are general allowances across all industries and special allowances. Special allowances vary widely according to asset type and time period. In 1996, for example, purchasers of machinery and equipment (other than motor vehicles) were permitted to deduct an additional ten per cent in the purchase year.

19C.18    Pro rata adjustments are made to align the dates of the tax law with the financial year, assuming that investment occurred evenly over the tax year. This leads to determining pro rata depreciation rates based on the portion of the year covered. 

19C.19    Some allowances may have not been taken into consideration because of the assets eligible may be at a finer detail than assets classes to which tax parameters can be assigned (i.e. the asset classification in the Perpetual Inventory Model), or because further research was needed. The ABS welcomes comments which may assist in improving the accuracy and fitness-for-purpose of tax parameters.