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Capital stocks, capital services, and depreciation: an integrated framework
Nicholas Oulton* and
Sylaja Srinivasan**
Working Paper no. 192
* Structural Economic Analysis Division, Monetary Analysis, Bank of England, Threadneedle Street, London, EC2R 8AH. E-mail: [email protected] ** Structural Economic Analysis Division, Monetary Analysis, Bank of England. E-mail: [email protected]
The views expressed in this paper are those of the authors, and not necessarily those of the Bank of England. We would like to thank Ian Bond, Hasan Bakhshi, Charles Bean, Simon Price and two anonymous referees for helpful comments and suggestions.
Copies of working papers may be obtained from Publications Group, Bank of England, Threadneedle Street, London, EC2R 8AH; telephone 020 7601 4030, fax 020 7601 3298, e-mail [email protected]
Working papers are also available at www.bankofengland.co.uk/wp/index.html The Bank of England’s working paper series is externally refereed.
© Bank of England 2003 ISSN 1368-5562

Contents

Abstract

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Summary

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1 Introduction

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Capital wealth and capital services

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Previous studies

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Plan of the paper

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2 Theory of capital measurement

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Asset prices and rental prices

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Aggregating over vintages

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Depreciation and decay

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Aggregating over asset types

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From theory to measurement

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Wealth measures of capital versus the VICS

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3 Depreciation and replacement

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The aggregate depreciation rate

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Straight-line as an alternative to geometric depreciation

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Obsolescence and the interpretation of depreciation

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Estimating depreciation in practice

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4 Capital stocks, VICS and depreciation: sources, methods and results

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Sources and methods for quarterly and annual estimates of the wealth stock and VICS 44

Estimates of capital stocks and VICS

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Estimates of aggregate depreciation

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5 Conclusions

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References

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Appendix A: Proofs of propositions in the text

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A.1. Proof that geometric depreciation implies geometric decay and of the converse 72

A.2. Proof that assets with proportionally high rental prices receive more weight

in a VICS than in a wealth measure

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A.3 Proof of proposition about real depreciation rate, dtR

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Appendix B: Data appendix

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Investment

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Real asset stocks

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Asset prices

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Tax/subsidy factor

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Rental prices

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Appendix C: A software investment series for the United Kingdom

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C.1 Revising the existing current-price series for software investment

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C.2 Updating the current-price series for software investment to 2001

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C.3 Constant-price series for software investment and the associated investment

price deflator

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Appendix D: Backing out non-computer investment from total investment

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D.1 Introduction

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D.2 The chain-linked solution

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D.3 Non-additivity

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Appendix E: Shares in wealth and profits and average growth rates of stocks,

1995 Q1-1999 Q4

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Abstract Neo-classical theory provides an integrated framework by means of which we can measure capital stocks, capital services and depreciation. In this paper the theory is set out and reviewed. The paper finds that the theory is quite robust and can deal with assets like computers that are subject to rapid obsolescence. Using the framework, estimates are presented of aggregate wealth, aggregate capital services and aggregate depreciation for the United Kingdom between 1979 Q1 and 2002 Q2, and the results are tested for sensitivity to the assumptions. We find that the principal source of uncertainty in estimating capital stocks and capital services relates to the treatment and measurement of investment in computers and software. Applying US methods for these assets to UK data has a substantial effect on the growth rate of capital services and on the ratio of depreciation to GDP. Key words: capital stocks, capital services, depreciation JEL classification: E22, O47
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Summary
This paper presents an integrated framework to measure capital stocks, capital services, and depreciation. The framework is integrated in two senses: first, our approach to measuring each of these variables is intellectually consistent; second, we use a common set of data for all three variables. Much of the difficulty of deriving good measures of aggregate capital, whether stocks or services, derives from two basic empirical facts. First, the relative prices of different types of asset are changing. Second, the pattern of investment is shifting towards assets with shorter economic lives. So we cannot treat capital as if it were composed of a single homogeneous good. To some extent, these two facts are aspects of the same important economic change: the shift in the pattern of investment towards information, communications and technology (ICT) assets. The relative prices of these assets are falling rapidly and their economic lives are much shorter than those of most other types of plant and machinery.
Theory
The wealth concept of capital, while appropriate for some purposes, is not the right one for a production function or for a measure of capacity utilisation. For the latter purposes, we need a measure of aggregate capital services. A second concept of aggregate capital, which will be called here the volume index of capital services (VICS), answers this need.
In principle, the VICS measures the flow of capital services derived from all the capital assets, of all types and all ages, that exist in a sector or in the whole economy. Methodologically, the main difference between the VICS and wealth-type measures of capital is the way in which different types and ages of assets are aggregated together. In the VICS, each item of capital is (in principle) weighted by its rental price. The rental price is the (usually notional) price that the user would have to pay to hire the asset for a period. By contrast, in wealth measures of the capital stock each item is weighted by the asset price.
An important practical implication of using a VICS rather than a wealth measure is that the VICS will give more weight to assets like computers and software for which the rental price is high in relation to the asset price.
We review the theory of, and empirical evidence on, depreciation. The assumption that depreciation is geometric greatly simplifies the theory and seems consistent with the (limited) facts. We also consider whether the geometric assumption is appropriate for assets like computers. Computers do not suffer much from physical wear and tear, but nevertheless have very short lives due to what is usually called ‘obsolescence’. We find that, in principle, our framework encompasses obsolescence. Nevertheless, we show that in practice depreciation rates may be somewhat overstated owing to failure to control fully for quality change.
Empirical measures of wealth and VICS
We adopt the geometric assumption in our empirical work for the United Kingdom. Because of the uncertainty about asset lives and the pattern of depreciation in the United Kingdom, we calculate wealth and VICS measures under a range of assumptions. We test the sensitivity of our
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results in three main ways. First, we compare results using both US and UK assumptions about asset lives. Second, we compare results based on a comparatively coarse breakdown of assets into four types only, with results derived from a more detailed breakdown in which computers and software are distinguished separately. Third, we compare the effect of US versus UK price indices for computers and software. Our results are for the whole economy and all fixed assets excluding dwellings, for the period 1979 Q1-2002 Q2. Our main findings for wealth and VICS are as follows:
1. Using the conventional National Accounts breakdown of assets into buildings (excluding dwellings), plant and machinery, vehicles, and intangibles, we find that the growth rates of wealth and the VICS are insensitive to variations in depreciation rates (ie, asset lives). In these experiments the rates for each asset are assumed constant over time.
2. However, the level of wealth is quite sensitive to variations in depreciation rates. 3. Still sticking with the conventional asset breakdown, wealth and VICS grew at similar rates
over the period as a whole. In the 1990s, the gap between the two measures widened a bit, with the growth rate of the VICS higher by about 0.1 percentage points per quarter. 4. The effect on the estimates of separating out computers and software is quite complex. First, much larger differences appear between the growth rates of VICS and wealth, of the order of 0.2-0.4 percentage points per quarter. Second, the growth rate of wealth tends to be slower, though that of the VICS is not necessarily faster. But when we apply the set of assumptions closest to US methods, the growth rate of the VICS is raised by 0.2 percentage points per quarter, relative to the VICS with computers and software included with other asset classes.
These results suggest that the treatment and measurement of investment in computers and software is an empirically important issue. It is common ground that the relative price of these assets has been falling, so it is in principle correct to separate them out explicitly – and it matters in practice. The conclusions about the growth rates of both VICS and wealth turn out also to be sensitive to the price index used for computers and to the way in which the level of software investment is measured.
The wealth and VICS estimates under a variety of assumptions can be downloaded from the Bank of England’s website (www.bankofengland.co.uk/workingpapers/capdata.xls).
The aggregate depreciation rate and the ratio of aggregate depreciation to GDP
We also estimate aggregate depreciation (capital consumption) for the same range of assumptions. We study the sensitivity of the aggregate depreciation rate and of the ratio of depreciation to GDP to the assumptions, and compare our estimates with ones derived from official data. We find:
1. Using the conventional asset breakdown and our assumptions about depreciation rates at the asset level, there is no tendency for the aggregate depreciation rate to rise over the past two decades.
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2. Separating out computers and software has less effect than one might have expected: even the use of US methods raises the aggregate rate by only about 1 percentage point, to 7% in 2000, and again there is no sign of an upward trend. The reason is that, even by 2000, the share of computers and software in wealth was only about 4% in the United Kingdom. By contrast and on a comparable basis, the aggregate depreciation rate in the United States has trended smoothly upwards since 1980, to reach nearly 9% in 2000. This illustrates the much greater scale of ICT investment in the United States.
3. Assumptions about asset lives have a large impact on the estimated ratio of depreciation to GDP. The official UK National Accounts measure has been drifting down fairly steadily since 1979. In 2001 it stood at 8%. Using shorter US asset lives and the conventional asset breakdown, the ratio was over 10% in the same year. Separating out ICT assets and using US methods, the ratio rises to nearly 13%, similar to the ratio in the United States. Interestingly, in neither country was there any upward trend in the ratio, except perhaps in the past couple of years. The reason is that, although the quantity of high-depreciation assets has been growing faster than GDP, this has been offset by their falling relative price.
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1 Introduction
Capital is an important part of the economy. Together with labour, it is a key factor of production, contributing to the output the economy can produce; changes in it – investment – constitute an element of demand in the economy; and it constitutes wealth, from which its owners obtain income in the form of profit.
But capital can be defined in different ways: in the context of production theory, the correct concept is the flow of capital services, whereas in the context of wealth, the correct concept is the present value of the returns accruing from the capital over its remaining productive life. And capital is difficult to measure. An economy’s capital is composed of different asset types and different vintages, and both the value of, and the services provided by, those assets change over time – eventually, to the point at which the asset has no further value or productive use. It is impossible in practice to measure those characteristics directly for each asset. So empirical measurement typically relies on measuring the rate at which new assets are acquired (gross investment) and the price of those new assets, and making a range of assumptions about how the quantity and value of older assets changes over time (loosely, depreciation).
In this paper we present an integrated framework to measure capital stocks, capital services, and depreciation, and apply it to the United Kingdom, illustrating the empirical differences which flow from the alternative concepts and different assumptions which can be made. The framework is integrated in two senses: first, our approach to measuring each of these variables is intellectually consistent; second, we use a common set of data for all three variables. Our approach is broadly neo-classical, in the tradition of Hall, Jorgenson, Griliches and Hulten. In the theoretical parts of this paper, we show that this framework is more robust than it is sometimes given credit for. Much of the difficulty of deriving good measures of aggregate capital, whether stocks or services, derives from two basic empirical facts. First, the relative prices of different types of asset are changing. Second, the pattern of investment is shifting towards assets with shorter economic lives. Because of these two facts, we cannot treat capital as if it were composed of a single homogeneous good. To some extent, though not entirely, these two facts are really aspects of the same important economic change: the shift in the pattern of investment towards information and communications technology (ICT) assets. The relative prices of these assets (at least on some measures) are falling rapidly and their economic lives are much shorter than those of most other types of plant and machinery.
Capital wealth and capital services
In current prices, the wealth represented by capital is just the sum of the values of the various asset stocks. Each stock is the cumulated sum of past investment, less the cumulated sum of depreciation (inclusive of retirement and scrapping), all revalued to current prices. In constant prices, the growth of wealth is a weighted average of the growth rates of the asset stocks, where the weights are the base-period shares of each asset in the value of wealth. Since the value of each asset is its price times its quantity, we refer to these kinds of weights as asset price weights.
Theory suggests that the wealth concept of capital, which we call for short the wealth stock or just wealth, is not the right one for a production function or for a measure of capacity utilisation.
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For the latter purposes, we need a measure of aggregate capital services. A second concept of aggregate capital, which will be called here the volume index of capital services (VICS), answers this need.(1)
In principle, the VICS measures the flow of capital services derived from all capital assets, of all types and all ages, that exist in a sector or in the whole economy. Methodologically, the main difference between the VICS and wealth-type measures of capital is the way in which different types and ages of assets are aggregated together. In the VICS, each item of capital is (in principle) weighted by its rental price. The rental price is the (usually notional) price that the user would have to pay to hire the asset for a period. By contrast, in wealth measures of the capital stock each item is weighted by the asset price. The two types of price are of course related: the price of an asset should equal the discounted present value of its expected future rental prices.
An important practical implication of using a VICS rather than a wealth measure is that the VICS will give more weight to assets for which the rental price is high in relation to the asset price. The rental price to asset price ratio is high when depreciation is high, due to a short service life, or when the asset price is falling, so that holding the asset incurs a capital loss. If the stocks of such assets are growing more rapidly than those of other types, then the VICS will be growing more rapidly than the wealth stock. This is likely to be particularly the case at the moment, with the increasing importance of computers and similar high-tech assets that are characterised by rapid depreciation and falling prices.
Previous studies
The wealth measure of the capital stock is the more firmly established and is the standard measure produced by national statistical authorities, including the Office for National Statistics (ONS) in the United Kingdom. Statistical agencies commonly estimate two different measures of the aggregate capital stock, known generally as the gross stock and the net stock. Several different asset types may be distinguished, eg buildings, plant and machinery, vehicles, etc. Conceptually, the gross stock of any asset is simply the sum of the past history of gross investment in that asset in constant prices, less the sum of past retirements. The aggregate gross stock is just the sum of the gross stocks of the different assets. The net stock differs from the gross stock in that allowance is also made for depreciation, often at a straight-line rate over each asset’s known or assumed service life.
In estimating stocks, statistical agencies nearly always employ what is called the perpetual inventory method (PIM). This starts with estimates of investment by asset and by industry or by sector. Capital stocks are then calculated by cumulating the flows of investment and subtracting estimated depreciation and retirements. Depreciation is not generally known directly, but is calculated by applying estimates of depreciation rates to the stocks. Depreciation rates may be based on asset lives (the straight-line method) or they may be deduced from econometric studies of new and second-hand asset prices (of which the best known are Hulten and Wykoff (1981a)
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(1) The OECD capital stock manual (OECD 2001b) uses the term ‘volume index of capital services’, from which we have coined the acronym VICS. The VICS is often called the productive capital stock (by contrast with the wealth stock), but this term is highly misleading since it not a stock at all but a flow.
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and (1981b)). Retirements are also not observed directly but can be calculated from estimates of the service lives of assets. Asset lives are usually derived from tax records and from surveys.(2)
Although the wealth concept is better known, the VICS concept is not new: it came to prominence in the seminal growth accounting study of Jorgenson and Griliches (1967) and was employed in subsequent studies by Jorgenson and his various collaborators, eg Jorgenson et al (1987) and Jorgenson and Stiroh (2000). The theory was set out in Jorgenson (1989); a related paper is Hall and Jorgenson (1967) on the cost of capital. Recently, the OECD has published a manual on capital measurement which contains a full discussion of the various concepts including the VICS, together with advice on how to measure it in practice (OECD (2001b)).
Versions of the VICS are already produced officially for the United States by the Bureau of Labor Statistics and for Australia by the Australian Bureau of Statistics. As far as the United Kingdom is concerned, unofficial versions of the VICS have previously been estimated by Oulton and O’Mahony (1994) for 128 industries within manufacturing (for three asset types: plant & machinery, buildings and vehicles) and by O’Mahony (1999) for 25 sectors covering the whole economy (for two asset types: plant & machinery and buildings). Oulton (2001a) contains annual estimates of the aggregate VICS incorporating explicit allowance for ICT assets. Earlier work at the Bank on the VICS is summarised in Oulton (2001b). Work is also currently under way at the ONS to produce a VICS on an experimental basis.
Plan of the paper
Sections 2 and 3 constitute the theoretical part of the paper. In Section 2 we start by reviewing the relevant part of capital theory. We discuss the relationship between asset prices and rental prices and show how this can be used to illuminate the twin issues of aggregating over vintages and aggregating over asset types. We also discuss the relationship between depreciation (how asset prices change with asset age) and what we call decay, which describes how the services of an asset change with age. Next, the equations of the two models used for estimating the VICS on quarterly and annual data are set out. These models make use of an important simplifying assumption, namely that depreciation is geometric. We compare the index number of the wealth measure with that of the VICS.
Section 3 is devoted to the related concepts of depreciation and replacement. Replacement is what must be spent to maintain the volume of capital services at the existing level, while depreciation is what must be spent to maintain the value of the capital stock at the existing level. We discuss the relationship between these two concepts and show that replacement and depreciation are equal when depreciation is geometric. We start by considering alternative measures of the aggregate depreciation rate. There are two broad classes of measure: nominal
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(2) Three other methods of estimating capital stocks have been employed. First, it is possible to do a sample survey or even a census of capital stocks. Such a survey has recently been done for the United Kingdom but no results have as yet been published (West and Clifton-Fearnside (1999)). Second, fire insurance values have been employed (Smith (1986)). Third, stock market values have been used (Hall (2001)). None of these methods has gained general acceptance, so they will not be considered further here. Also, stock market values can only yield a wealth measure, not a VICS. In the academic literature depreciation rates have also been derived as a by product of estimating a production function (Prucha (1997)) and scrapping has been estimated from company accounts (Wadhwani and Wall (1986)).
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Capital stocks, capital services, and depreciation: an