APPENDIX 2 SENSITIVITY ANALYSIS OF CAPITAL INPUTS
SENSITIVITY ANALYSIS OF CAPITAL INPUTS
The measure of capital input required for the calculation of MFP creates particular difficulty for the statistician. In principle there is a requirement to measure the volume of capital utilised in delivering output. In much of the productivity literature these flows of capital are described as capital services. However, neither the value nor volume of these services are directly available from business accounts. Therefore it is necessary to estimate these flows indirectly using modelling approaches.
The method used by the ABS to estimate capital inputs involves the use of a perpetual inventory model. This model involves the accumulation, by industry and asset type, of the additions to the capital stock as recorded by the Gross Fixed Capital Formation (GFCF) estimates recorded in the Australian System of National Accounts (ASNA). These additions to the stock are adjusted to account for retirements and other removals from the capital stock. As there is little direct empirical information available in relation to these withdrawals from the capital stock it is necessary to make some assumptions about how long each asset will remain in production.
GFCF is recorded for aggregated classes of assets. Even if detailed GFCF were available, additional modelling would be required for each asset, such as retirement distributions, mean asset lives and efficiency profiles. By using asset classes the modelling process is representative of an average asset. However, this raises a number of issues because the asset class is not as homogenous as it is assumed. Hulten (1999) gives the example of different working lives for the many types of machine tools, some with short working lives and some with long working lives. As Hulten (1999) points out, by placing these heterogeneous assets into a single, broad class, the model effectively treats the different assets as though they were one. Changes in the mix of assets contained within the aggregate can affect the quality of the modelled results.
For the purpose of productivity analysis it is important to recognise that the capacity of an asset to deliver services (its efficiency) will generally decline over time. In order to account for declining efficiency the model applies assumptions regarding the age-efficiency profile for each asset. This means that an older asset will make less of a contribution to the capital stock than a newer asset of the same type. The estimate of capital stock derived in this way is known as the productive capital stock and is derived for each asset class. A flow measure of capital services is estimated for an industry that reflects the different mix of assets used, and is based on the productive capacity of capital.
Capital services for an individual industry are created by aggregating different vintages of the same type of asset, and then aggregating different assets using rental prices as weights to form an aggregate index. The capital services produced by an asset over its life are not usually observed, however, they may be approximated by assuming that capital services are directly proportional to the productive capital value of the asset. This relationship is fixed over the asset's life, but does vary between asset types and even between different vintages of the same asset, since it depends on the expected life of the asset and the rate of decline in the asset's efficiency, which may change over time.
Capital services (inputs) are assumed to change at the same rate as the movement in the volume of this productive capital stock. It is implicitly assumed that there is a constant utilisation of this stock, that is, no adjustment is made to the movements to account for any real world change in capacity utilisation of these assets due either to the business cycle or technological change.
Currently the ABS produces industry capital services indexes and these are used as the capital input for industry and market sector MFP estimates. The following section describes the way in which capital services indexes are compiled and the implications for industry MFP estimates.
AGGREGATE CAPITAL SERVICES INDEXES
In order to combine the individual indexes of capital services for each asset type into a single index for the industry or market sector it is necessary to aggregate the elemental indexes. This is achieved by weighting them together via the use of their respective price weights. As the prices of capital services are not normally observable on the market it is necessary to derive them using economic theory. These price weights are also known as the user cost of capital or rental prices. A key focus of the remainder of this appendix is on the user cost of capital equation and the sensitivity of the aggregate capital services indexes to changes in the rate of return component of the equation. The ABS currently uses a combination of endogenous and exogenous rates of return in the user cost of capital equation. The sensitivity of capital services indexes to changes in mean asset lives is also investigated.
The user cost of capital equation in its most basic form is comprised of three components: depreciation of the asset, a rate of return reflecting financing costs, and a capital gain/loss component. In practice the equation also includes a corporate income tax component, tax depreciation allowances, investment credits and indirect taxes. The user cost equation (footnote 1) used by the ABS is as follows:
Where:
i = industry
j = asset type
t = discrete time period
T = income tax parameter
r = rate of return
p = price for new capital goods
d = depreciation rate
x = effective average non-income tax rate on production.
RATES OF RETURN
Rates of return may be calculated in one of two ways. First, by using an endogenous rate of return which is represented by the internal rate of return for the industry. Using an endogenous rate of return to calculate user costs of capital imposes some implicit assumptions, namely that the underlying production function exhibits constant returns to scale, that markets are competitive, and that the expected return is the same as the realised return (OECD 2001b). Also, using an endogenous rate of return imposes the same rate of return for all asset types within an industry. This is done by equating non-labour income to capital rent, where capital rent is defined as the sum across all assets of their rental price multiplied by the real productive capital stock. The rate of return is then estimated by equating this to the non-labour income (BLS 1983, Hall and Jorgenson 1967).
There are a few issues surrounding the calculation of endogenous rates of return. The first issue is whether the user cost should be in real or nominal terms. OECD (2001b) indicates that the user cost should be in nominal terms, as the user cost of capital is analogous to compensation of employees. The second issue is whether non-labour income should be on a gross or net basis. The third issue is whether the productive capital stock or the net capital stock should be used to estimate the endogenous rate of return. Currently real productive capital stock is used in deriving ABS MFP estimates, rather than a nominal productive or net capital stock, and this is why the rates of return presented in tables A2.1 and A2.2 are relatively high. The ABS will consider changing its approach at a suitable time in the future.
The second approach to calculating rates of return is to use an exogenous rate of return such as the interest rate on government bonds. Using an exogenous rate may lead to a difference between the calculated capital rent and capital income. Capital income differs from capital rent as capital income is defined as the sum of gross operating surplus (GOS) and the proportion of gross mixed income that would represent a return on the owner's capital.
The endogenous approach is an ex post approach to calculating the rate of return as it calculates the rate of return after the results of the investment decision are known. A practical issue involved in using an endogenous rate of return is that when capital income is small, the associated internal rate of return will be small. The second approach or the exogenous approach could be considered an ex ante approach to calculating the rate of return because it is the expected return on an investment decision.
The rental price equation employed here can give rise to negative rental prices when there are large changes in capital gains and losses. This is a problem for estimating capital services indexes because rental prices are used to form weights, and a negative weight causes problems in forming an aggregate index. Currently the ABS uses a combination of endogenous and exogenous rates of return when estimating capital services estimates to overcome the problem of negative rental prices in some industries in some years. When rates of return are low the ABS applies a floor to the rate of return equal to 4 per cent plus the consumer price index (CPI), which could be considered an exogenous rate. If the endogenous rate is greater than or equal to this floor then the endogenous rate is used in the user cost equation. If the derived endogenous rate is less than the set exogenous rate of 4 per cent plus the CPI, then the exogenous rate is used. However, applying this floor does not prevent negative user costs in all cases.
Tables A2.1 and A2.2 show the average rates of return for the entire period by industry calculated using the exogenous approach, the endogenous approach and the rates of return actually used to produce the estimates in this paper (referred to as 'current' in the tables). The deviations from the average exogenous rate of return are also shown. The tables are split into two periods covering roughly the last 40 years, where table A2.1 is from 1964-65 to 1984-85 and table A2.2 is from 1984-85 to 2005-06.
The tables show that the average exogenous and endogenous rate of return can differ substantially for each industry. For the majority of industries the average endogenous rate of return is less than the average exogenous rate of return. By definition, the current approach used by the ABS produces average rates of return that are higher than the exogenous rate of return, however, for the majority of industries, the deviations from the exogenous rate of return are not significant.
There are some industries, such as Electricity, gas & water that on a year to year basis have an endogenous rate of return that is consistently below the minimum exogenous rate of return. In contrast, there are other industries, such as Agriculture, where there are some years that the endogenous rate of return is significantly higher than the exogenous rate of return.
A2.1 Average rates of return and derivations from the exogenous rate of return, Market sector industries - 1964-65 to 1984-85 |
| |
| Average rates of return | Deviations from exogenous rate of return | |
| Exogenous | Endogenous | Current | Endogenous | Current | |
| (1) | (2) | (3) | (4) | (5) | |
| |
Agriculture, forestry & fishing | 11.8 | 7.2 | 13.3 | -4.6 | 1.5 | |
Mining | 11.8 | 15.3 | 15.5 | 3.5 | 3.7 | |
Manufacturing | 11.8 | 14.4 | 15.4 | 2.6 | 3.6 | |
Electricity, gas & water | 11.8 | 7.2 | 11.8 | -4.6 | - | |
Construction | 11.8 | 14.9 | 15.2 | 3.1 | 3.4 | |
Wholesale trade | 11.8 | 10.9 | 12.4 | -0.8 | 0.6 | |
Retail trade | 11.8 | 9.7 | 11.9 | -2.1 | 0.2 | |
Accommodation, cafes & restaurants | 11.8 | 9.3 | 11.8 | -2.5 | - | |
Transport & storage | 11.8 | 5.9 | 11.8 | -5.9 | - | |
Communication services | 11.8 | 10.6 | 12.4 | -1.1 | 0.6 | |
Finance & insurance | 11.8 | 14.2 | 15.9 | 2.4 | 4.1 | |
Cultural & recreational services | 11.8 | 7.6 | 11.8 | -4.2 | - | |
| |
- nil or rounded to zero (including null cells) |
A2.2 Average rates of return and derivations from the exogenous rate of return, Market sector industries - 1984-85 to 2005-06 |
| |
| Average rates of return | Deviations from exogenous rate of return | |
| Exogenous | Endogenous | Current | Endogenous | Current | |
| (1) | (2) | (3) | (4) | (5) | |
| |
Agriculture, forestry & fishing | 8.0 | 1.6 | 8.3 | -6.3 | 0.4 | |
Mining | 8.0 | 7.1 | 8.9 | -0.8 | 0.9 | |
Manufacturing | 8.0 | 7.6 | 8.8 | -0.7 | 0.8 | |
Electricity, gas & water | 8.0 | 3.6 | 8.0 | -4.3 | - | |
Construction | 8.0 | 13.6 | 14.3 | 5.6 | 6.3 | |
Wholesale trade | 8.0 | 6.1 | 8.2 | -1.9 | 0.3 | |
Retail trade | 8.0 | 6.3 | 8.2 | -1.6 | 0.2 | |
Accommodation, cafes & restaurants | 8.0 | 4.4 | 8.0 | -3.6 | 0.1 | |
Transport & storage | 8.0 | 1.1 | 8.0 | -6.8 | - | |
Communication services | 8.0 | 7.8 | 8.7 | -0.2 | 0.8 | |
Finance & insurance | 8.0 | 8.0 | 10.5 | - | 2.5 | |
Cultural & recreational services | 8.0 | 1.6 | 8.0 | -6.4 | - | |
| |
- nil or rounded to zero (including null cells) |
ASSET PRICE DEFLATOR
Another key component of the user cost equation is the choice of price deflator. Currently, the ABS uses ex post asset specific price changes in estimating holding gains, but in some cases this has led to negative rental prices for particular assets, due large changes in the asset price. To avoid this issue the current practice is to set any negative rental price to a very small positive number (0.001). By adjusting the negative rental prices in this way the capital stock weights for that asset return to positive values and the weights of the remaining assets are also adjusted. As a consequence, the corresponding capital services index also returns to a 'reasonable' level. Further research will attempt to identify improved methods of dealing with negative rental prices.
One approach that might overcome the negative rental prices is to consider that the user cost equation is an expectations model. The ABS's current approach to user costs is that the variables are measured ex post rather than ex ante. By shifting to an ex ante approach and assuming that businesses generally base their expectations of holding gains on movements in the CPI, the CPI would then replace the price deflator (p_{ijt}) in the user cost of capital equation above. However, there would be some assets, such as computers, where expectations of price change would likely be different to the general level of inflation, and different price deflators may be required. This is an area for further investigation.
Given the possible combinations of rates of return and price deflators the following combinations were tested - the results of which are shown in figures A2.3 to A2.14:
- Current ABS approach (as described above)
- Exogenous rate of return with separate asset price deflators
- Exogenous rate of return with the CPI as the price deflator
- Endogenous rate of return with separate asset price deflators
- Endogenous rate of return with the CPI as the price deflator.
Capital services for the market sector industries, using different rates of return (2004-05 = 100)
A2.3 AGRICULTURE, FORESTRY & FISHING
A2.4 MINING
A2.5 MANUFACTURING
A2.6 ELECTRICITY, GAS & WATER
A2.7 CONSTRUCTION
A2.8 WHOLESALE TRADE
A2.9 RETAIL TRADE
A2.10 ACCOMMODATION, CAFES & RESTAURANTS
A2.11 COMMUNICATIONS SERVICES
A2.12 TRANSPORT & STORAGE
A2.13 FINANCE & INSURANCE
A2.14 CULTURAL & RECREATIONAL SERVICES
The data show that for Mining and to a lesser degree Manufacturing, the approach used has little effect on the capital services index. With the exception of Agriculture, forestry & fishing and Transport & storage, the rest of the industries also show minimal differences in the choice between an exogenous and endogenous rate of return, but there is more notable impact when the CPI price deflator is used. For Agriculture, forestry & fishing and Transport & storage the graphs highlight problems with negative rental prices.
Figure A2.3 shows that for the Agriculture, forestry & fishing industry the exogenous capital services curve does not exhibit the same pattern as the other series. The reason behind this is the volatility in land prices earlier in the series. This leads to a negative rental price for land when an exogenous rate of return equal to 4 per cent plus the CPI is used. In this instance the negative rental price was not set to 0.001. For the Agriculture, forestry & fishing industry, land contributes significantly to the overall capital stock and has fallen in price over the period. Consequently, the aggregation to total capital rent (rental price multiplied by productive capital stock) gives a negative value for land and for total assets. This means that the asset weights for all assets other than land become negative. The rental price weight is positive for land because its negative capital rent is divided by a negative capital rent for the industry. While the other asset's rental price weights are negative because their positive capital rent is divided by the negative total.
However, land does not contribute directly to the capital services index as the productive capital stock of land does not change over time. Therefore, it is not the large weights for land themselves that lead to the wayward capital services index such as the one shown in figure A2.3 but their distorting effect on the weights for other assets. While this effect is shown occurring for exogenous rates it also occurs for endogenous rates but a further adjustment has been made to the rental price in this case, that is setting the negative rental price to 0.001.
For the Transport & storage industry negative rental prices occur using an endogenous rate of return when the asset price deflator is the CPI. The negative rental price occurs across a number of assets over various years, with the most common being land and non-dwelling construction.
Tables A2.15 and A2.16 show the growth rates of capital services indexes for selected time periods based on exogenous, endogenous and current rates of return. For the majority of time periods and industries, the growth rates do not differ substantially with the choice of rate of return.
A2.15 Annual average growth in capital services index, Market sector industries |
| |
| 1985-86 to 1995-96 | 1995-96 to 2005-06 | 1985-86 to 2005-06 | |
| Exogenous | Endogenous | Exogenous | Endogenous | Exogenous | Endogenous | |
| |
Agriculture, forestry & fishing | -4.6 | 0.3 | 0.3 | 0.4 | -2.2 | 0.3 | |
Mining | 3.6 | 3.7 | 4.8 | 4.7 | 4.2 | 4.2 | |
Manufacturing | 2.7 | 2.8 | 3.0 | 3.1 | 2.9 | 2.9 | |
Electricity, gas & water | 0.8 | 0.6 | 3.0 | 4.0 | 1.9 | 2.3 | |
Construction | 4.9 | 4.9 | 3.6 | 3.1 | 4.2 | 4.0 | |
Wholesale trade | 3.5 | 3.9 | 5.8 | 6.0 | 4.6 | 5.0 | |
Retail trade | 3.2 | 3.8 | 5.4 | 5.6 | 4.3 | 4.7 | |
Accommodation, cafes & restaurants | 5.6 | 6.6 | 4.3 | 4.3 | 4.9 | 5.4 | |
Transport & storage | 2.3 | 3.9 | 3.8 | 5.7 | 3.0 | 4.8 | |
Communication services | 6.5 | 6.6 | 5.9 | 5.8 | 6.2 | 6.2 | |
Finance & insurance | 5.9 | 6.8 | 5.7 | 5.1 | 5.8 | 5.9 | |
Cultural & recreational services | 5.3 | 5.8 | 6.1 | 6.9 | 5.7 | 6.4 | |
| |
With the exception of Agriculture, forestry & fishing, and Transport & storage, the growth rates of the capital services indexes do not differ substantially with the choice of rate of return or the asset price deflator. The subsequent impact on growth rates of MFP estimates will also be relatively small, assuming every thing else remains constant.
A2.16 Annual average growth in current capital services measures, Market sector industries |
| |
| 1985-86 to 1995-96 | 1995-96 to 2005-06 | 1985-86 to 2005-06 | |
| % | % | % | |
| |
Agriculture, forestry & fishing | 0.2 | 0.5 | 0.3 | |
Mining | 3.6 | 4.7 | 4.2 | |
Manufacturing | 2.8 | 3.1 | 2.9 | |
Electricity, gas & water | 0.8 | 3.0 | 1.9 | |
Construction | 4.8 | 3.1 | 3.9 | |
Wholesale trade | 3.6 | 5.8 | 4.7 | |
Retail trade | 3.3 | 5.5 | 4.4 | |
Accommodation, cafes & restaurants | 6.3 | 4.1 | 5.2 | |
Transport & storage | 2.3 | 3.7 | 3.0 | |
Communication services | 6.4 | 5.7 | 6.1 | |
Finance & insurance | 6.1 | 5.0 | 5.5 | |
Cultural & recreational services | 5.3 | 6.1 | 5.7 | |
| |
As figures A2.3 to A2.14 show, the use of the CPI shows that for the majority of industries, growth in the capital services index was slower, which would lead to faster growth in MFP as opposed to using the existing asset price deflators. The reason for this slower growth is that assets with falling price deflators, e.g. software and computers, have a smaller rental price weight using the CPI. Since these two assets generally had the fastest growth in productive capital stock, the smaller weight means a flatter capital services index as the other slower growing assets receive a greater weight when using the CPI. As mentioned earlier, at least for these assets which show persistent price falls, the use of the CPI may not be appropriate.
MEAN ASSET LIVES
Mean asset lives are an important component in the measurement of capital stock. Asset lives are influenced by a number of variables including changes in technology, quality, and the rate of use. The choice of mean asset lives impacts on capital stock estimates through the Perpetual Inventory Method (PIM), and has flow through effects to capital services estimates and ultimately, MFP estimates.
The ABS has updated mean asset lives for some assets as more information has become available, although the most recent review was conducted in 1996-97. To test the sensitivity of any potential changes to the mean asset lives on capital services estimates and subsequent MFP estimates, changes were made to the mean asset lives for computers, computer software and non-dwelling construction (footnote 2). For computer hardware and software, recent tax lives, sourced from the Australian Master Tax Guide, 2006 (CCH 2006), were used.
In 1996-97, the mean asset life for computers, in-house software, and purchased software were 4.9 years, 6 years and 4 years, respectively. Figures A2.17 to A2.28 compare the currently published capital services estimates to a capital service estimate using an updated mean asset life of 4 years for computers, and a mean asset life of 4 years for all software. Another capital service index is also shown in the figures using the updated computer and software lives and updated mean asset lives for non-dwelling construction, where the mean asset lives have been decreased by 20 years, from on average 50 years to 30 years. The adjusted asset lives for computers and software have been applied from 1990 to 2006 to allow a long enough time series to capture any potential change in capital services, and the adjusted mean asset lives for non-dwelling construction have been applied for the whole time series.
Capital services for the market sector industries, using different mean asset lives (2004-05 = 100)
A2.17 AGRICULTURE, FORESTRY & FISHING
A2.18 MINING
A2.19 MANUFACTURING
A2.20 ELECTRICITY, GAS & WATER
A2.21 CONSTRUCTION
A2.22 WHOLESALE TRADE
A2.23 RETAIL TRADE
A2.24 ACCOMMODATION, CAFES & RESTAURANTS
A2.25 COMMUNICATIONS
A2.26 TRANSPORT & STORAGE
A2.27 FINANCE & INSURANCE
A2.28 CULTURAL & RECREATIONAL SERVICES
The results in figures A2.17 to A2.28 show that changing the mean asset lives for computers and software has very little impact on capital services estimates in any industry in the market sector. Even the more sizeable changes to the mean asset lives of non-dwelling construction had little effect on capital service estimates at the industry level for most industries, indicating that capital services, and hence MFP estimates, are not particularly sensitive to changes in mean asset lives.
CONCLUSIONS
Several aspects of the capital services model were tested for sensitivity of the assumptions. The results show that there were only minimal differences to growth in capital services in respect of the choice of rate of return. The other aspect of the rental price equation that was considered was the choice of price deflator used to estimate capital gains. The choice between an asset price deflator and the CPI showed some differences in growth rates for capital services, but did not change the underlying growth patterns. However, this did not completely resolve the issue of negative rental prices as they also occurred in the Transport & Storage industry when the CPI was used as the asset price deflator.
The other aspect that was sensitivity tested was mean asset lives. The results were that the capital services index is not particularly sensitive to the choice of mean asset lives.
1 For a complete derivation of this equation see Hall and Jorgenson (1967). (back)
2 Retirement distributions are the extent to which assets are retired before, on, or after the mean asset lives and the current approach used by the ABS assumes a Winfrey S3 probability distribution for most asset types. The choice of mean asset lives used in this sensitivity analysis is restricted to a set of mean asset lives for which a Winfrey S3 probability function has already been derived. (back)
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