2 Construction of comparable capital stocks

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Both Canada and the United States use the perpetual inventory method (PIM) to estimate the capital stock of each asset type by assuming a geometric declining pattern. The capital accumulation equation using PIM is

where, Iⁱ_j,t is investment in asset type j in country i in period t. δ in Equation (1) is the geometric rate at which the efficiency of an asset declines over time.

The implementation of PIM requires estimates of the size and time profile of depreciation rates, gross investment time series and an initial level of capital stock, all of which will be discussed in turn below.

2.1 Estimates of depreciation

For the purposes of this paper, we make use of a geometric form of the depreciation rate—that is, we assume that the value of an asset depreciates at a constant rate per year. The productivity program in Canada derives estimates of the depreciation rate from data on the prices of used assets that are sold during their useful lives and data on the time when each asset is discarded (see Statistics Canada 2007). The United States also makes use of prices on used assets to estimate depreciation rates.

Nevertheless, the sources of the data differ. Statistics Canada receives its data directly from its investment survey that includes both positive prices and zero prices when assets are discarded at the end of their useful life. The United States has developed its data from a number of different sources—mainly trade data. These generally only contain prices of assets that are sold at a positive price. The U.S. method also has to make assumptions about the pattern and intensity of discards of assets at zero values because there is a selection bias problem if depreciation rates are estimated from only positive prices.

Both countries also derive constant geometric depreciation rates using data on the length of life of the asset—where data on used asset prices are not available but an estimate of the length of life is available.

This section discusses how depreciation estimates are commonly derived when estimates are available of the length of life (T). We focus on two specific forms of depreciation: straight-line and geometric. While our analytical interest rests with the latter, straight-line depreciation is a useful starting point, and it is applied extensively in a national accounting framework. In this section, the length of life of an asset is treated as non-stochastic—as known with certainty.

Straight-line patterns assume equal dollar value depreciation at all stages of an asset's lifecycle. Per-period depreciation for a dollar of investment takes the form

where T is service life. Although the dollar loss for straight-line depreciation is equal from period-to-period, the rate of depreciation—that is, the percent change in asset value from periodto- period—increases progressively over the course of an asset's service life. For a marginal dollar of investment, this rate is

Geometric depreciation represents a conceptual counterpoint to the straight-line case. Geometric profiles hold the rate of depreciation, not the period-to-period dollar amount, fixed over the course of an asset's service life.³ Geometric profiles are accelerated—with higher dollar depreciation in early periods—giving rise to the convex age-price profile.

Per-period depreciation is defined as

where δ is the constant (age invariant) rate of depreciation.

While the straight-line method can often be found in the accounting literature, the majority of empirical research in the North American productivity literature on asset depreciation has concentrated on the geometric form. In early studies, geometric patterns were often assumed. Evidence that geometric rates are generally appropriate for a wide range of asset types is found in Hulten and Wykoff (1981) and Koumanakos and Hwang (1988).⁴

In practice, geometric rates are analytically expedient for two reasons: (1) they can be estimated indirectly via accounting methods; and, (2) their constant-rate property allows them to be used as a proxy for the replacement rate in standard PIMs of capital stock. We address the first of these points below.

Direct estimates of δ can be derived from information on resale prices or on the length of life of the asset (T) . For many years, the latter method was the most common and T was determined from accounting information—often associated with tax laws. In the absence of sufficient price information, geometric rates can be calculated indirectly from estimates of the length of life (T) of an asset derived from the tax code as

where T is taken from the service life of the asset (the length of time over which it provides useful value) and DBR (referred to as the declining-balance rate) is chosen in a way that satisfies a certain concordance with the straight-line method (or any other method). The value of the DBR determines, other things equal, the extent to which asset values erode more rapidly early in the lifecycle (Fraumeni, 1997). Higher values of DBR bring about higher reductions in asset value earlier in service life, giving rise to more convex (i.e., accelerated) depreciation profiles.

Double-declining-balance rates (DDBRs)—which set the value of the DBR to 2—have been extensively used in practice. In their estimates of capital stock, Christensen and Jorgenson (1969) employ DDBRs to estimate rates of economic depreciation. Statistics Canada's productivity program at one time based its estimates of geometric depreciation on a double-declining rate. One advantage of the DDBR is that it provides a 'conceptual bridge' back to the straight-line case, anchoring the midpoints of the depreciation schedules at an equivalent age point. Indeed, the average depreciation rate in the straight-line case will match the constant rate derived from a DBR of 2.

To see this, we can examine a simple measure of central tendency. Defining μ as the midpoint of the geometric curve (the expected life of a dollar invested in an asset), then

from Equation (5), when δ is chosen as DBR/T

Description

Now also represents the midpoint of the linear depreciation schedule (the point at which a dollar is half-way depreciated) of an asset whose length of life is T. Thus, if the DBR in the geometric formula is set equal to 2, the linear depreciation world, often used by accountants, can be brought into congruency with a geometric world—so that an average dollar in the geometric world lasts the same amount of time as it takes a dollar to lose half its length of life, which is just the expected life of a dollar invested in an asset in the straight-line world.

When the rate of depreciation is calculated indirectly by Equation (5), an estimate of T is also required. When the estimate of T is based on ex ante expectations of service life, the depreciation rate can be described as ex ante. In Canada, service life estimates can be derived from the expectations of businesses regarding an asset's useful life. The Investment and Capital Stock Division captures in its annual investment survey the expected length of life on all new investments that are reported to Statistics Canada.

There has been considerable debate over whether the assumptions embodied in the calculation of geometric rates are empirically appropriate. Some researchers have questioned whether the high losses in asset value that are often observed early in asset life are consistent with constant, geometric rates. It should be stressed that constant rates do not, in and of themselves, preclude highly accelerated depreciation profiles; rather, the issue is simply whether these rates are, on net, sensible representations of the change in asset value in every period. A key aspect of this debate centres on choosing, by estimation or otherwise, an appropriate value for the DBR. Even if constant-rate, geometric age-price profiles are empirically justified, the choice of particular values for DBR and T is still at issue. If T is chosen from the tax code, the estimate thereof may differ from actual lives if the tax code does not use accurate length of lives—as it may deliberately do if it is trying to stimulate investment. For this reason, Statistics Canada derives a value of T from its investment survey. Admittedly, firms are required, in advance, to predict how long an asset may last—and may err in a systematic way. But the estimates of T derived in this matter can and have been checked against the evidence on discards and found to accord closely with the latter (Statistics Canada 2007).

Recent estimates of geometric depreciation used by the Bureau of Economic Analysis make use of a lower value for the declining-balance rate for many individual assets (DBR=1.65 for machinery and equipment [M&E] and 0.91 for structures). Based on the empirical research of Hulten and Wykoff (1981), these values will, other things being equal, produce lower rates of geometric depreciation than the double-declining case for the same value of T. But they need not do so if they are chosen along with T so as to produce correct values of the depreciation rate that are derived from used asset prices.

The basis for the Hulten-Wykoff estimates of the DBR warrant some discussion here. In a study for the Office of Tax Analysis of the Department of the Treasury, the authors generate direct estimates of geometric depreciation for those assets for which they had used asset price data, and then base estimates of δ for other assets (for assets for which no price information was available) on the geometric accounting method described by Equation (5), using arbitrary estimates of T that are essentially developed from the tax code. That is, they estimate DBR in the first stage for all assets where they have a direct estimate of δ and T and then apply this DBR in the second stage to those assets where they only have an estimate of T. This two-stage procedure enabled the authors to produce a set of depreciation estimates for asset classes used by the U.S. National Income and Product Accounts.

The first stage of this process yielded average DBR values of 1.65 for M&E and 0.91 for structures—average DBR values based on asset categories for which price information was directly available.⁵ In cases where no price information on other assets was available, the authors then combined these estimates of the DBR with asset-specific information on tax-code service life T to produce indirect estimates of δ .⁶ The estimates of DBR so produced were only meant to be useful for filling in their data set, not to be used for alternate estimates of T, such as those which Statistics Canada's survey produces from direct questions to firms on the expected length of life of assets.

The productivity program at Statistics Canada also follows a variant of this technique.⁷ It estimates δ directly from used asset and discard data for those assets where there are an adequate number of observations and a value of the DBR from these assets that produces the estimated δ from the ex ante length of life T that is yielded by its survey. The resulting estimates of DBR (2.1 for M&E and 2.3 for structures) are then applied to other asset classes where only information on asset life is available (mainly structures). While the DBR that is produced by these two methods differs, it should be noted that the differences are the result primarily of differences in the length of life that is used in the second stage and are not necessarily meaningful. Indeed, Statistics Canada (2007) shows that the depreciation rates for M&E in the two countries are essentially the same, despite the differences in the DBRs that are derived in this way.

For a comparison of Canadian and U.S. capital stocks, we can use several different methods. These are outlined in Table 1 as Methods 1, 2 and 3.

Method 1 takes the U.S. DBRs and applies them to the Canadian length-of-life data that are derived from the Canadian investment survey. As argued above, this is inappropriate for our purposes. These values were chosen in the U.S. studies so as to equate the estimated depreciation rates to the T's that are estimated from a wide variety of ad hoc sources in the United States. (Fraumeni 1997, Bureau of Economic Analysis 2003). Since Canadian ex ante T's are derived from the Annual Investment Survey from the Investment and Capital Stock Division, the U.S. DBRs do not generate the correct depreciation rates that are estimated from those assets that have an adequate quantity of data to directly estimate depreciation rates.

Method 2 makes use of the technique that essentially produces the depreciation rates that are derived directly and indirectly in each country. It employs used asset price data to derive depreciation rates directly for those assets for which used asset price data exist, then derives the DBRs for these assets using estimates of T and then applies these estimated DBRs to other assets where only estimates of T are available.

Method 3 is directly equivalent to Method 2. It uses the average DBRs that are obtained in Method 2.

When we use either Method 2 or Method 3, we are employing the respective depreciation rates that are imbedded in the two official statistical systems. There are reasons to be cautious about adopting this approach, which allows the rates to differ across the two countries. While the estimates of the depreciation rates in Canada and the United States are about the same for M&E, the rates of depreciation on structures are higher in Canada than in the United States (see Table 2).⁸

These cross-country differences in depreciation rates probably derive from differences in the quality of the data. The Canadian data have been collected in a systematic way from a survey that provides data on both used asset prices and discard patterns since 1987. The U.S. data are collected from a variety of sources, some of which are dated, and direct data on discards are rarely available.

For purposes of comparisons of growth rates across countries, slight differences in depreciation rates are not very important—at least not for the technique that is used in both countries that employs the internal rate of return. Statistics Canada (2007) reports that the differences in the depreciation rates that are produced by slightly different econometric techniques has only a minor effect on estimated rates of multifactor productivity (MFP) growth.

But for cross-country estimates of MFP levels, differences may be more important. We therefore chose to construct capital stocks for both Canada and the United States using one set of depreciation rates for purposes of comparability. For the majority of the results reported herein, we use the rates that have been derived from the Canadian data for both Canadian and U.S. capital stock. But we also make use of the U.S. rates to ask whether there is much of a difference between the two. We find these differences are minor.

Table 1
Depreciation patterns and methods used in Canada and the United States

Table 2
Bureau of Economic Analysis and Statistics Canada (productivity accounts) depreciation rates by asset type

2.2 Investment data

The underlying data used in generating the estimates of capital stocks in Canada are derived from investment data based on the North American Industry Classification System (NAICS) that is used in the MFP program of Statistics Canada (Baldwin, Gu and Yan 2007). These data contain investment in current dollars and chain Fisher volume indices over the 1926-to-2003 period for the 28 assets listed in Table 2.⁹

The main data source for estimating capital stock in the United States is investment data by industry based on NAICS over the 1901-to-2005 period.¹⁰ These data are obtained from the Bureau of Economic Analysis (BEA) and contain investment for 47 assets. The investment data for the U.S. government sector can only be divided into three assets—M&E, building structures and engineering structures.¹¹

For this paper, we divide our assets into four groups—engineering assets, buildings, noninformation and communications technology (ICT) M&E, and ICTM&E. Engineering assets provide the foundation capital for railways, utilities, oil and gas, and pipelines. Buildings house manufacturing plants, commercial offices, hotels, and retail and wholesale facilities. ICTM&E is defined here as including computers, telecommunications equipment and software. Non-ICTM&E is the remainder—some of which is also highly sophisticated and embodies computer automation.

2.3 Initial capital stocks

The level of capital stock is sensitive to the depreciation profiles and depreciation rates that are used to estimate it.¹² To develop comparable measures of capital stock, we have used geometric depreciation profiles and depreciation rates for Canada and the United States as explained in Table 1. The depreciation rates in the table are derived from Statistics Canada research that estimates depreciation profiles for a diverse set of assets that employs used asset prices derived from a Statistics Canada survey (Statistics Canada 2007). The resulting depreciation rates are on average the same as those used by the BEA for M&E. They are slightly higher for buildings and engineering construction.¹³

To estimate capital stock, we also need to choose an initial value of capital stock. For Canada, capital stock is estimated using the historical investment from 1926 to 2003. For the United States, capital stock is estimated using historical investment from 1901 to 2005. An initial capital stock in 1926 is chosen for Canada and in 1901 for the United States. That actual value chosen has little effect on capital stock estimates for the 1987-to-2003 period that is used in this paper.

2.4 Other data issues

2.4.1 Coverage of the finance, insurance, real estate and renting and leasing sector

While both Canada and the United States adhere to international standards (the 1993 System of National Accounts) in estimating gross domestic product (GDP), some differences in industrial coverage remain. One major difference is in the finance, insurance, real estate and renting and leasing (FIRE) sector. In the United States this sector includes the rent from rental and owneroccupied residential building. The FIRE sector in the Canadian Productivity Accounts (CPA) includes only the rent from rental residential buildings, but not the imputed rent from owneroccupied dwellings. In order to increase comparability, we have moved the imputed rent of owner-occupied dwellings from the U.S. FIRE sector to the U.S. non-business sector. This means that we have included investment in rental housing in the FIRE sector and included investment in owner-occupied dwellings in the non-business sector.

2.4.2 Definition of the business sector

In this paper, we will examine Canadian-U.S. differences in investment and capital intensities in the total economy and the business sector. The business sector covers the total economy less the non-business sector. The non-business sector in this paper includes the government sector and the health and education sectors. The data for Canada and the United States are both based on NAICS. As such, the government sector (NAICS 91) includes public schools and public hospitals. The private and non-profit schools and hospitals are included in the education and health services sectors of the business sector (NAICS 61 and 62).

It is sometimes argued that a comparison of the business sectors of Canada and the United States is problematic because most health and education activity in Canada is classified as part of the non-business sector (the government sector) since schools and hospitals are generally public in Canada. In contrast, a much smaller portion of health and education in the United States is classified as part of the non-business sector since the United States has a larger share of private and non-profit schools and hospitals. Therefore, for the purpose of comparability between Canada and the United States, we have included all health and education activity in the nonbusiness sector.¹⁴

2.4.3 Valuation of output

The next issue that needs to be addressed is the valuation of output for a Canadian-U.S. comparison of capital intensities. At the total economy level, both Canada and the United States produce estimates of total output (GDP) using similar price concepts—GDP measured at market prices. But this paper will move to the industry level, and here Canadian and U.S. practices differ. Statistics Canada values industry output at basic prices, while the U.S. BEA values industry output at market prices. The difference between the two measures of outputs is taxes and subsidies on products. The difference can be quite large for some industries. Baldwin et al. (2005) report that the value of output at market prices was about 7% higher than the value of output at basic prices in the Canadian business sector in 1999.

To derive comparable measures of output at the industry level for our comparisons of Canada and the United States, we have estimated output at basic prices for the U.S. industries that can be used to compare with the Canadian industry measures that are calculated in basic prices. To derive such a measure, we have subtracted net product taxes from industry output measures that are calculated at market prices. The BEA does not publish net product taxes at the industry level. Instead, it publishes the taxes on production and imports at the industry level, which include both product taxes and property taxes. In this paper, we have estimated taxes on product at the industry level as the difference between the taxes on production and imports and a measure of property taxes at the industry level, which is estimated using the industry distribution of net capital stock and total property taxes published by the BEA. A similar approach is used by the BEA to allocate the total property taxes among industries (Moyer et al. 2004).¹⁵

Alternatively, we can construct estimates of output at factor cost for the two countries as in Rao, Tang and Wang (2004) or estimates of output at market prices. Such measures are less comparable than the measures adopted here for the purpose of international comparison (Lal 2003). First, GDP at factor costs in the petroleum industries includes gasoline taxes in the United States while it does not in Canada. The use of GDP at factor costs will underestimate the output level in the Canadian petroleum industries relative to that of the United States. Second, GDP at market prices for the U.S. wholesale and retail trade sectors include taxes on all imports, and about 20% of GDP in those industries are product and import taxes in the United States. The use of GDP at market prices will show much lower levels of capital intensity in the Canadian wholesale and retail trade sectors relative to those in the United States.

2.4.4 Investment data

In this paper, we have taken the investment data in the two countries without making major modifications therein. The readers however should be aware that various issues with regards to comparability still remain.

First, it has been pointed out that concepts regarding the measurement of software investment in different countries influence some inter-country comparisons. Basically, the United States makes use of data on wages of software engineers to infer the investment that is being made in ownaccount software. Canada does the same and follows basically the same methodology.

Second, hedonic price indices are used by the United States for a number of high-tech products, while fewer European countries do the same. But Canada also uses hedonic price indices in most of the same industries and, while these indices differ from U.S. indices, the differences may be generally explained by movements in the Canadian-U.S. exchange rate.

Third, there may be differences in the classification of investments between buildings and engineering construction. Engineering construction is in effect those expenditures that are not easily reported either as M&E or as buildings. A number of industries—especially large projects like electrical generating facilities, dams, railway lines and pipelines—involve the type of civil engineering projects that fall within the ambit referred to as engineering construction. However, at the margin, there is always the possibility that it might be difficult to separate out buildings in some of these projects from the rest of the expenditure and that the practice of respondents to do so might vary across the two countries.

Fourth, it must be recognized that investment data for Canada and the United States are not perfectly comparable at the industry level. Both Canada and the United States collect detailed data on commodities for the economy as a whole. Some of these are classified as investment goods. Canada also has an investment survey that is used to spread the economy-wide totals derived from the commodity totals by industry of ownership of the capital. The BEA has, in the past, spread the total investment of M&E mainly by using occupational data (Haltiwanger 2006). Construction expenditure is obtained directly by industry from a special survey. All of this means that researchers need to be careful when comparing capital stock using fine levels of industry detail.

Fifth, when making a cross-country comparison of investment, one must always keep in mind that the division of expenditures on assets between new investment and repair expenditures may not be the same across countries. Firms buy new assets and repair old ones. While a statistical agency may define investments as those expenditures that extend the life of an asset by more than one year, in practice it may be difficult for a firm to make that distinction. Moreover, for tax reasons, a firm may have the incentive to expense expenditures that could be classified one way or the other. The route that is followed will depend on the vigilance of the tax authorities, and this may differ across countries.

 

^3. For an overview of the geometric distribution, see Hastings and Peacock (1975).

^4. For a survey of the empirical literature, see Fraumeni (1997); for a discussion of empirical methods, see Jorgenson (1994).

^5. As Hulten and Wykoff (1981: 94) note, the asset categories for which they were able to calculate depreciation rates directly from price information represent a substantial share of total National Income and Product Accounts investment expenditures—42% of investment in non-residential structures and 55% of investment in producers' durable equipment.

^6. For a useful discussion of the Hulten-Wykoff methodology, see Fraumeni (1997).

^7. We refer here to the capital stock estimates that are derived specifically for the productivity program at Statistics Canada. Other capital stock estimates are produced that serve other purposes in Statistics Canada. The Investment and Capital Stock Division produces capital stock estimates using a straight line method, a hyperbolic and a geometric method because various users have requested these alternatives. Prior to 2000, the geometric truncated model was produced using a double-declining balance rate. More recently, it adopted the U.S. declining-balance rates for its geometric series at the request of some users. The productivity group derives its own estimates of capital stock because of different requirements.

^8. The Bureau of Economic Analysis (BEA) depreciation rates in the table are implicit geometric depreciation rates. For those assets (such as computers and software) whose implicit depreciation rates changed over time, we take the average over the 1987-to-2002 period as the approximation to the BEA depreciation rates.

^9. The data for 1926 to 1960 contain investment for three assets (machinery and equipment, building structures and engineering structures). To obtain investment for the 28 assets over the 1926-to-1960 period, we assume that the share of investment by asset type over that period is the same as the one averaged over 1961, 1962 and 1963.

^10. See Lally (2004) for a discussion of the data.

^11. We make use of current dollar and constant dollar estimates from Statistics Canada and from the Bureau of Economic Analysis and do not impose a similar deflator for computers as is done in Schreyer (2005) for his comparison of North America and Europe since the latter is less likely to use hedonics for computers than is North America. However, both Canada and the United States use hedonics to estimate the price deflator for computers, and any differences between the two countries are therefore felt to reflect legitimate differences in market pricing behaviour.

^12. Growth rates may also be sensitive to these differences. For the effect on growth rates of capital in Canada using alternative assumptions on depreciation, see Gellatly, Tanguay and Yan (2002).

^13. Choosing Bureau of Economic Analysis rates across all categories does not change the results reported here in a material way. What is important is that both Canadian and U.S. capital stock be derived from a similar set of depreciation rates.

^14. The coverage of business sector industries is not entirely comparable in the two countries. Bureau of Economic Analysis output at the industry level includes the output of non-profit institutions, while the output in Canadian industries excludes the output of non-profit institutions. Both these components are fairly small.

^15. Total property taxes are published in the National Income and Product Accounts, Table 3.5.