# The measurement of capital input

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

19.72    The measurement of capital input is concerned with estimating the contribution of capital to the production process; that is, the flow of capital services from the capital stock used in the production process. Capital services have both quantity and price dimensions. The quantity of capital services represents hours a machine is used or months a building is occupied. The price dimension, called the rental price, represents an hourly rate for using the machine or a monthly rate for occupying a building.

## Productive capital stock and quantity of capital services

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.

## Imputing rental prices

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.

## Capital service flows for fixed assets

19.88    The estimates of fixed assets from the PIM that are used to derive MFP are:

• machinery and equipment: computers and computer peripherals; electronic and electrical machinery and communications equipment; industrial machinery and equipment; road vehicles; other transport equipment; and other equipment;
• non-dwelling construction;
• ownership transfer costs of non-dwelling construction;
• intellectual property products: computer software; research and development; mineral and petroleum exploration; and artistic originals (Film and TV; music; and literary);
• orchards, plantations and vineyards; and
• livestock.

19.89    Ownership transfer costs relating to non-dwelling construction are allocated to industry using industry proportions of chain volume non-dwelling construction by industry. This approach assumes that the proportion of ownership transfer costs to non-dwelling construction at a point in time does not vary between industries.

## Inventories

19.90    Volume estimates for the stock of inventory items are obtained for Divisions A to I (see Chapter 10 for more details). They are non-capitalised assets that are used up in the productive process and collected according to three categories:

• inventories of raw materials, including materials and fuels, spare parts designated for use in fixed assets, containers and packaging materials. Inventories of fuels for sale are included in inventories of finished goods;
• inventories of work-in-progress, including partially processed or fabricated goods which will be further processed prior to sale, and general work-in-progress less payments billed. Prepayments are excluded;
• inventories of finished goods, including goods manufactured or processed which are ready for sale, goods purchased from other businesses which are ready for resale without further processing, and fuels for sale. Hired goods, inventories of land, and rented or leased buildings are excluded.

## Land

19.91    Land can be further classified as either agricultural (for ANZSIC Division A) or non-agricultural (for the other ANZSIC divisions). Volume estimates of agricultural and non-agricultural land and the corresponding rental prices are constructed separately.

19.92    The volume estimate for agricultural land is derived starting with a nominal estimate obtained from the National Balance Sheet in the ASNA. In particular, the value of 'Rural' land in the reference year is used. As there is no suitable price index for agricultural land, its volume is assumed to be constant over time. The estimates of land values are discussed further in Chapter 17.

19.93    Similarly, a volume estimate of non-agricultural land is derived starting with a nominal estimate of the market sector's non-agricultural land in the reference year. A benchmark estimate is obtained from the National Balance Sheet by multiplying total 'Commercial' and 'Other' land by the proportion of the stock of non-dwelling construction in the market sector. This estimate is then split by industry proportionally using the productive capital stock of non-dwelling construction in the reference year. Then for a given industry, the volume estimate is constructed by assuming that its growth rate is half the growth rate of the industry's real productive capital stock of non-dwelling construction.

19.94    To calculate the rental prices for land, proxy price indicators are used as suitable land price indexes are not available. For agricultural land, the total investment deflator for Agriculture, forestry and fishing is used for the years prior to 1995-96, and the All groups CPI thereafter. For non-agricultural land, the index is based on the weighted aggregation of commercial and industrial rent indexes for Australia's main capital cities, provided by a private sector contractor.

## Operating leases and finance leases

19.95    The ABS classifies the use of capital as an intermediate input of the lessee when the capital is rented under an operational lease arrangement from a firm primarily operating in another industry. For example, a construction company may lease a crane from the rental and hiring industry, which is recorded as a service component in the intermediate inputs of the lessee and as capital services held by the lessor. If the proportion of the capital that is leased is changing it can affect value added productivity growth estimates. A reduction in the percentage of capital held within an industry over time, such as when a firm leases rather than purchases capital, would understate growth in the capital service index, which would have the effect of overstating value added MFP growth. For capital held under a long-term finance lease, the capital is treated as capital owned by the lessee and included in the productive capital stock estimates of the lessee industry.

### Endnotes

1. Jorgenson, Dale W., Mun S. Ho, and Kevin J. Stiroh (2005) Information Technology and the American Growth Resurgence. Cambridge, MA: MIT Press.