CESifo Economic Studies Advance Access published online on July 3, 2008
CESifo Economic Studies, doi:10.1093/cesifo/ifn022
Rental Prices, Rates of Return, Capital Aggregation and Productivity: Evidence from EU and US
* Groningen Growth and Development Centre, Faculty of Economics, University of Groningen, PO Box 800, 9700 AV, Groningen, The Netherlands, e-mail: a.a.erumban{at}rug.nl;
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Abstract |
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 Top Abstract 1 Introduction2 Growth accounting and...3 Measurement of rental...References |
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With the increasing importance of investment in information and communication technology, methods for measuring the contribution of capital to growth have re-assumed centre-stage in recent growth accounting literature. The importance of using capital service growth rates rather than capital stock growth rates has long been advocated, and has become mainstream practice. However, the choice for a particular rate of return in the derivation of capital service prices is not straightforward and has barely been researched. Using four alternative rental price models —based on both external and internal rates of return models—this article quantifies the differences in total factor productivity growth rates (TFPG) under different model assumptions. The differences in TFPG are also examined in terms of the inclusion of taxes and subsidies in the calculation of rental prices. Empirical analysis carried out for four EU countries and the US in 26 industries during 1979–2003 shows that the use of capital stock overestimates TFPG in most industries. Incorporation of taxes seems to have only modest effect. The magnitude of divergence generated by alternative rental price models-particularly between internal models- is quite low. The difference is seen to be relatively high between external rate of return models and internal rate of return models. (JEL Codes: E01,O47)
Key Words: Capital services • capital stock • EU • rental prices • rate of return • total factor productivity • US
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1 Introduction |
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TopAbstract1 Introduction2 Growth accounting and...3 Measurement of rental...References |
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The measurement of capital and its contribution to economic growth has been an important issue of attention in economics for long time. With the increasing importance of investment in Information and Communication Technologies (ICT), methods for measuring the contribution of capital to aggregate growth have reassumed centre-stage in growth accounting1 literature (Jorgenson, 2001; van Ark, Inklaar and McGuckin 2003; Jorgenson and Vu 2005; Inklaar, Timmer and van Ark 2008). Dividing the total capital into ICT capital and non-ICT capital, studies have attempted to delineate the contribution of these two different components to total growth using the growth accounting framework.2 One major issue in the quantification of capital's contribution to economic growth using this framework is related with the measurement of capital input. Though many studies still use aggregate capital stock as a measure of capital input (e.g. Caselli 2005; Jones and Olken 2005), it has long been advocated that the flow of capital services, rather than the capital stock, is appropriate for production and productivity analysis (Jorgenson 1963; Hall and Jorgenson 1967; Jorgenson and Griliches 1967).
The challenge in measuring capital services for growth accounting is associated with the implicit nature of the capital services; the quantity of capital services is not usually directly observable (Harper, Berndt and Wood 1989). Therefore, the empirical researcher has to rely on theoretical yardsticks to approximate the capital services. Following the theoretical arguments of Jorgenson (1963), Hall and Jorgenson (1967) and Jorgenson and Griliches (1967) capital services are usually derived using the estimated capital stock for the individual assets and the relevant user cost or rental price of capital, assuming that flows are in proportion to the stocks at individual asset level. However, there are different ways of measuring rental price of capital and thereby of obtaining capital service growth rates. The selection of any particular measure may significantly influence the calculated growth rates of capital and its contributions to output growth. The study by Harper, Berndt and Wood (1989) utilizes five alternative rental price models to evaluate the capital aggregation for US manufacturing industries over the period 1948–81. Their results show significant differences between different models.
The aim of this article, in line with Harper, Berndt and Wood (1989), is to examine whether the use of different rental price models in capital service aggregation produce significantly different capital services growth rates. This is important in the context of the recent surge in the growth accounting literature that examines the contribution of the ICT capital. While Harper, Berndt and Wood (1989) limit their study to the US only, we compare our results across 26 industry groups (Table 3) in four EU countries along with the US, over a more recent time period, 1979–2003. Further, we specifically examine the effect of alternative capital aggregation procedures on the measured TFPG. An attempt is also made to empirically understand the impact of including tax in the cost of capital equation on the growth rates of capital and thereby TFPG. It has been argued that the tax plays a major role in altering investment behavior (Hall and Jorgenson 1967), and therefore, taxation has to be incorporated in the measurement of capital service prices. However, most studies in the context of the EU have considered pretax rental prices in order to arrive at capital service growth rates.
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The article is organized into seven sections. In Section 2 we present a brief discussion on the analytical literature concerning growth accounting and capital aggregation. In Section 3 we discuss rental price formulation and the measurement of its components. Section 4 discusses the data used in the empirical analysis. Empirical results are discussed in Section 5, and finally Section 6 concludes the article.
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2 Growth accounting and capital aggregation |
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TopAbstract1 Introduction2 Growth accounting and...3 Measurement of rental...References |
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Assuming a competitive market and constant returns to scale, the standard growth accounting equation with value-added as output (Y) and labour (L) and capital (K) as inputs can be written as3:
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| (1) |
t is the share of labour in the value-added averaged over the two time periods, t and t – 1 (i.e. t = (at + at – 1)/2) and 
denotes the change in the relevant variable over the previous year. Equation (1) clearly shows that output growth is measured as a weighted sum of growth of labour and capital flows, where the weights are the shares of each input in the value added. This implies that in order to implement Equation (1) it is essential to calculate the growth rate of aggregate inputs and output.5 Ideally, one should be able to derive the growth in aggregate inputs as the cost share weighted sum of growth of inputs (Jorgenson, Gollop and Fraumeni 1987). In the case of labour such weights may be derived from the wage rates. The absence of an observable service price, however, makes it difficult to directly measure aggregate capital service growth rates.
Capital stock consists of different types of heterogeneous assets with specific capital service levels. Therefore, this heterogeneity has to be taken into account while aggregating capital services across various asset types. Assuming a strict proportionality between capital services and capital stocks at the level of individual assets6, Jorgenson (1963) and Jorgenson and Griliches (1967) have developed aggregate capital service measures that take into account the heterogeneity of assets. In accordance with their aggregation procedure, the growth rate of aggregate capital services may be measured as
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| (2) |
and 
where 
is the aggregate capital services in year t, n is the total number of assets, 
is the capital stock of asset i asset in year t, cit is the rental price of asset i in year t and 
it is the share of ith asset in total capital compensation in year t. it effectively incorporates the qualitative differences in the contribution of various asset types, as the capital composition changes. It is evident from Equation (2) that the two important components of the capital service measure are capital stock and the service prices (rental price) of capital. Hence, though the relevant measure of capital input in the productivity analysis is the flow of capital services, it is essential to have consistent measures of capital stock in productivity analyses in order to practically estimate capital service flows. The usual practice of measuring capital stock is based on a perpetual inventory method, i.e.
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| (3) |
i is the rate of depreciation for the ith capital asset and Iit is the real investment in asset i installed in the beginning of period t. Using Equations (2) and (3) the change in aggregate capital service growth rates may be measured as a weighted sum of the growth rates of asset specific capital stocks, with the weights being the relative compensation share of each asset.7 The remaining task, therefore, is to construct appropriate measures of capital service prices (cit)—rental prices or user cost of capital— in order to derive the relative compensation shares. |
3 Measurement of rental prices (c) |
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TopAbstract1 Introduction2 Growth accounting and...3 Measurement of rental...References |
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As in the case of the wage rate for labour, the rental price for capital represents the unit cost of using a capital good for a specified period of time (Jorgenson and Yun 1991). Capital services are delivered by capital goods over the course of their lifetime to their owners, without any recorder market transaction. Therefore, it is impossible to gauge the appropriate service prices from market transactions, making it essential for the researcher to impute the implicit rental price that the owners of capital assets pay themselves; hence it is often called as the user cost of capital (Harper, Berndt and Wood 1989).
Hall and Jorgenson (1967) have developed a methodology within the boundaries of neoclassical theory where the rental price of capital can be imputed from the relationship between price of a new asset and the discounted value of all future services derived from that asset (Jorgenson 1967; Christensen and Jorgenson 1969). In the absence of corporate taxes, they derive this relationship as,
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| (4) |
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| (5) |
is the expected capital gain, i.e. the gain or loss from holding an asset over time. Assuming a perfectly anticipated capital gain term, Christensen and Jorgenson (1969) have provided a discrete time derivation of the rental price equation,8
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| (6) |
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3.1 The role of taxes
The above derivation of rental prices based on the assumed correspondence between asset prices and service prices abstracts from any type of taxes. It is important to note here that the asset price–service price correspondence depends on the tax structure for property income generated by the asset. This is because taxes are assumed to play a major role in altering investment behavior, the premise being that entrepreneurs in pursuit of gain will be more attracted to purchasing capital goods if prices are low (Hall and Jorgenson 1967). Therefore, the derivation of rental prices of capital, which assumes strong correspondence with asset prices, should account for the impact of taxes. In this regard, the effect of tax policy on cost of capital has been subjected to empirical scrutiny (Hall and Jorgenson 1967).9 In line with Hall and Jorgenson (1967) and Harper, Berndt and Wood (1989), incorporating tax factors, Equation (6) can be re-written for i-th asset as10
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| (7) |
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| (8) |
3.2 The components of rental prices
Equation (7) clearly shows that the general formulation of the rental price comprises the nominal rate of return, the nominal cost of depreciation, the corporate taxation and the nominal gain from holding the asset for each accounting period, i.e. the capital gain, 
. A positive capital gain reduces the user cost of holding the asset, and therefore is subtracted, while a negative gain (or a loss) increases the user cost and therefore must be added, hence it has a negative sign. The empirical part of this article is concerned with the measurement of Equation (7) and therefore the three components of rental prices—rate of return, depreciation and capital gain. The measurement of these three components may be accomplished in different ways.
The issues associated with the measurement of the depreciation are well-discussed in economics (see Hulten and Wykoff 1981 and other papers in Hulten 1981). Depreciation measures the loss of the market value of a capital asset over time. It may vary over time and across countries and industries. However, considering the computational simplicity and the empirical support provided by Hulten and Wykoff (1981), the general practice in empirical literature is to assume a geometric depreciation rate. We do not delve much into this issue; rather opt to follow the general practice. Therefore, we concentrate on the other two components—the rate of return and the capital gain.
The rates of return (r)
The rate of return represents the opportunity cost of holding an asset. It may be considered either as a nominal rate of interest payment, if a loan was taken to acquire the asset or as the opportunity cost of employing capital elsewhere than in production (OECD 2001a). It may therefore be measured either as an external rate of return (ER) or an internal rate of return (IR). The external uses the ex ante approach, where the rate of return is represented by some exogenous rate such as interest rates on government bonds. The internal is a residual, or ex-post approach, where the rate of return may be calculated as a residual, given the value of capital compensation, depreciation and capital gains (see Table 1 for a brief picture of these two alternative measures in a comparative perspective).12 Clearly, there is a problem of choice between external and an internal measure of rate of return in the productivity analysis, and consequently this issue has been widely discussed in the literature (Berndt and Fuss 1986; Harper, Berndt and Wood 1989; Hulten 1990; Diewert 2001). Though there is no consensus on the matter, Berndt (1990) have shown that ex post rates which reflect realized marginal products are more theoretically consistent for productivity analysis. This is under the assumption that the capacity of ex post return to approximate realized marginal product is higher than that of ex ante rate of return, which however, have been questioned (Schreyer, Bignon and Dupont 2003). We use both these measures in order to understand the sensitivity of final growth rates of capital services to these choices. In what follows, we explain the procedure adopted in this article to derive the rates of return under these two assumptions.
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The IR
The IR is defined as a residual obtained after adjusting capital compensation for depreciation and capital gain. This approach, advocated by Hall and Jorgenson (1967), estimates the IR with the help of an accounting identity that assumes that the sum of rental payments to all assets is equal to the total capital compensation, i.e.
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| (9) |
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| (10) |
is the value added price, w is the wage rate and L is the labour input. The total capital income or property compensation is thus calculated as the nominal value added minus labour compensation, assuming a competitive market and constant returns to scale.13 It consists of pretax profits, capital consumption allowances, net interest, transfer payments, business subsidies, indirect taxes and the portion of firms’ income attributable to capital. Therefore, one can calculate the rate of return from the total property compensation after taking out the components of corporate taxes, indirect taxes and depreciation from capital compensation. Following Christensen and Jorgenson (1969), we measure IR as the residual of property income after adjusting for these variables as:
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| (11) |
The ER
As noted in Table 1, the internal approach is theoretically consistent under the assumptions of constant returns to scale, competitive markets and the equality of expected rate of return and realized rate of return. However, the implied assumption of the IR that this ex post measure represents the "realized marginal products" is contested. For instance, Schreyer, Bignon and Dupont (2003) argues that the ex post measures need not be necessarily the preferred measures of realized marginal productivity if, for instance the capital is rented by a producer at a given preagreed rental price to be paid at the end of the period. In addition, a practical problem arises when capital income in national accounts (gross operating surplus measured by M in Equation 11) becomes negative.14 In such cases, the measured rental prices using IR may also become negative, which is theoretically inconsistent.15 One way of eliminating such negative rental prices is to employ an external rate of return. The main problem associated with the external rate of return is the choice of a particular rate. There is no rule on which rate should be used; Diewert (1980) has suggested a wide range of rates. Diewert (2001) has suggested a 4 percent rate of return for OECD countries, under the assumption that producers face a real interest rate of around 4 percent. He suggests that this rate is consistent with long-run economy wide observed real rate of return for OECD countries that ranges 3–5 percent.16 In the present study, we use this 4 percent constant external rate of return (hereafter CER).
The capital gain
The second component of rental prices is the capital gain component, which is also a major component of IR (Equation 11). It measures capital gain or losses, or revaluation of an asset—the change in value that corresponds to a rise or fall in the price of that asset, independent of the effects of ageing. It compares the price of new capital assets in two periods, hence is independent of ageing. Despite its crucial role in the rental price specification, there are more than one way by which the capital gain can be incorporated in the measurement of rental prices. We include this component in our rental price and rate of return calculations under three assumptions.
Perfectly anticipated capital gain
As Equation (6) clearly shows one way of incorporating capital gain into the measurement of rental prices is to assume a perfectly anticipated capital gain (Christensen and Jorgenson 1969; Harper, Berndt and Wood 1989; Jorgenson and Yun 1991). Both in Equation (6) and in Equation (11), the expected capital gain is represented by annual realized capital gain or perfectly anticipated capital gain (that is 
). Following Harper, Berndt and Wood (1989), we call this IR model which includes perfectly anticipated capital gain is as internal "nominal" rate of return (INR) model.
Zero capital gain
A second possibility is to drop out this term from the user cost and rate of return equations. Following Jorgenson and Siebert (1968) and Harper, Berndt and Wood (1989), we also examine rental prices and IR that exclude capital gain term. In this case, we exclude the capital gain term from the rate of return as well. This rate of return model, which does not include capital gain term is called as internal "own" rate of return (IOR).17
Smoothed capital gain
Apart from the above two assumptions regarding capital gain in the rate of return (INR and IOR) and the rental prices, another possibility is to use a smoothed capital gain series. Such an approach will help reduce the volatility in rental prices and is therefore useful if price changes are highly volatile. In accordance with Harper, Berndt and Wood (1989), we propose using a smoothed capital gain series by employing a 3-year moving average. These smoothed capital gain measures are then substituted for the perfectly anticipated capital gain term in Equation (11), thus providing a third IR which we call as INR with smoothed capital gain (INRS).
Note that in all the models we assume that the internal rate of return is the same for all assets in an industry, while the CER is the same for all assets and industries. Also, the 4 percent external rate of return is assumed to be a real rate of return (net of capital gains), and therefore will be used in the rental price model that does not incorporate capital gain. All the three internal rates of returns are estimated with and without tax, thus providing six measures of rate of IR—INR, INR with tax (INRTX), INRS, INRS with tax (INRSTX), IOR and IOR with tax (IORTX). Similarly, these six measures of internal rates and the CER without tax and with tax (CERTX) are used in the calculation of rental prices, again under three assumptions—with annual capital gain, with smoothed capital gain and without capital gain. These are also derived with and without corporate and indirect tax terms (T and b) in order to understand the sensitivity of incorporating tax component. Thus finally we have eight different rental price models and corresponding capital service and productivity growth rates. Table 2 lists these eight rental price models.
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4 Data and variables
The data used in this study are based primarily on Inklaar, O’Mahony and Timmer (2005), and its updates available through Groningen Growth and Development Centre (GGDC) website. The study is conducted for four EU countries—Netherlands, France, Germany and UK—and the US using annual data for the period 1979–2003 for 26 industry groups listed in Table 3.
The empirical implementation in this article consists of the measurement of the capital services and the TFPG. Hence the major variables that enter into our analysis are inputs (labour and capital) and output. For the output data, the current price gross value added obtained from national accounts statistics is deflated using value added deflators with base 1995. Labour input is measured as the product of total hours worked and labour quality. In order to calculate TFPG using Equation (1) we also required having the share of labour and capital in value added. The labour share is calculated as labour compensation divided by value added, both in current prices, adjusted for self employed. And the capital share is derived as the residual, labour share. All these data are available through GGDC industry growth accounting database.18
Capital stock is estimated using a perpetual inventory method for six types of assets using Equation (3). The assets considered for this purpose are non-IT machinery, nonresidential structures, transport equipment, IT equipment, communication equipment and software. The capital service growth rates are then derived using Equation (2) applying alternative rental prices calculated using different rates of return models listed in Table 2. For this we needed to have data on depreciation rates and asset prices. A geometric rate of depreciation is assumed. The rates are based on Fraumeni (1997) and Jorgenson and Stiroh (2000) and are industry-specific.19 Investment price deflators with base year 1995 are used to represent the asset prices.
Apart from the input, the output and the price variables, we also require tax variables in order to derive rates of return and rental prices. As noted by Jorgenson and Yun (1991), appropriate tax rate to analyze the impact of tax incentives for investment is corporate income tax. Following Harper, Berndt and Wood (1989) we use the effective marginal rate of corporate tax, since our interest is in calculating the rental price. The marginal rate of effective taxation mirrors the incentive to invest. We derived the marginal rate of taxation applying Equation (8),20 using the data provided by Devereux, Griffith and Klemm (2002).21 They provide statutory tax rates on corporate income and the present discounted value of depreciation allowances. Time series data is available for 16 countries of European Union and the G7 for 1979–2005 period, of which we use the rates for the countries Netherlands, France, UK and the Germany. The statutory tax rates in these countries have shown a declining tendency over years, particularly during the recent years (Sørensen 2007). The depreciation allowances are provided for two types of assets—plant and machinery and industrial buildings. We have calculated the effective tax rate (Tit) using these two rates, in such a way that the rate for industrial buildings has been applied for nonresidential structures in our investment data, and the depreciation allowances for plant and machinery are used in all other assets including software that comes under the category of machinery.
The value added in our data consists of operating surplus, compensation to employees and the taxes on production. Therefore, while deriving capital compensation as value added minus labour compensation, we also need to subtract these taxes. The calculation of indirect taxes 
is made directly from National Accounts of selected countries. The data on taxes on production22 was gathered from the Source OECD National Accounts Database under components of value added. These values are divided by calculated aggregate capital stock in order to arrive at the tax rate (bit). However, these rates are not available for individual assets. Therefore, a common rate is derived by dividing total tax values by total (across all assets) capital stock, and this rate is applied to all assets. Note that in our empirical calculations, we have also measured rates of return and rental prices without including this measure of indirect tax, as this data includes taxes not only on capital but also on employees’ compensation. However, since the results are found to remain the same, they are not reported.
5 Empirical results
We have measured the rates of return and the rental prices using the four model assumptions, with and without corporate taxes (Table 2). Thus we have eight different measures of rental prices, rates of returns and the corresponding capital service and TFP growth rates. As we have mentioned earlier, in situations where capital compensation becomes negative, or there are high capital gains, it is possible to have negative rate of returns23 and rental prices. Such negative rental prices are theoretically inconsistent, as the theoretical assumption underlying the derivation of growth accounting framework requires input service prices to be positive. Table 4 provides the percentage of negative values that appeared for each models of rental prices. The table shows that the largest numbers of negative values appeared in the models that incorporate capital gain and tax. In the presence of large capital gains, it is possible to obtain negative rental prices in these models. Surprisingly, in the case of constant rates of return models, Germany has shown a positive number of negative values in tax-incorporated model. A detailed look at the data reveals that this is in the industry mining and quarrying, during the years 1996–2003. This may be attributed to the high subsidies given to this industry during this period. If we look at the taxes on production in German mining industry during this period, it is as high as (–) 102 percent of real value added in 1996. These taxes have shown a declining trend in later years, though they trickled down only to (–) 68 percent in 2003. Most negative rental prices in this industry are found to be in assets nonresidential structures, transport equipment and communication. In general, Germany and UK have witnessed the largest number of negative rental prices while the lowest is found in the US. In almost all the countries the largest numbers of negative rental prices are observed in the mining industry. Among the three internal models, IOR has produced the least number of negative rental prices, suggesting that the inclusion of capital gain causes the negative rental prices.
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We have calculated the rental share weighted capital service growth rates using Equation (2) for all 26 industry groups in each country separately, over the period 1979–2003 using alternative rental prices. The resulting capital service growth rates along with capital stock growth rates for the aggregate economy are plotted in Figure 1.24 It is quite clear that the capital services grow faster than capital stock in most years, thus leaving a gap between the levels of these two. This is due to the shift in the composition of capital; increasing share of short living assets will result in accelerated capital service growth, hence, following Harper, Berndt and Wood (1989), we call it a "composition effect". It can be also seen from the figure that the capital service growth rates produced by different models tend to differ, though not very significantly. While the difference is more prominent between internal and external models, particularly with those which do not include capital gain, it is trivial between internal models, in general. In UK and France, the capital services derived using CER models tend to grow at a slower pace compared to the internal models. This is more visible in the UK, which is in conformity with Oulton (2007)25. More importantly the inclusion of tax in the calculation of rates of return and rental price makes the story not very different. It caused very little difference in the calculated growth rates.
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The average composition effects measured as the difference between the growth rates of capital services and capital stock over the entire 1979–2003 period and two sub-periods 1979–95 and 1995–2003 are provided in Table 5. This effect is seen to be more prominent in the US followed by France and Netherlands and least in Germany. In France and in the Netherlands the effect is positive in all models during the entire period, while the US shows negative composition effect in the model with exogenous rate of return and tax and internal models without capital gain in the second period. In the UK the difference between capital service growth rates and capital stock growth rate is positive only during 1979–95 period, while it is negative in the later period and in Germany it is negative throughout. It may be noted that the larger composition effect could occur due to the shift towards equipment investment, particularly that of short-living equipments. Similarly, the decline in the relative price of an asset (for instance the decline in the price of equipment such as IT equipment relative to structures) would make the relative magnitude of capital gain term that is subtracted from the rental price equation smaller attributing it a larger weight in capital services (Harper, Berndt and Wood 1989). Both these factors would cause a larger capital service growth rate. In both UK and Germany, the share of construction in the total capital stock either remained the same or increased in the recent period, causing a negative composition effect (Figure 2). In all the countries the composition effect seems to have declined in the second period in most models. For instance, on average the composition effect in the US is 0.6 for the entire period, with 0.7 during 1979–95 and 0.15 during 1995–2003.
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The capital service growth rates obtained using alternative rental price models are plugged into Equation (1) in order to calculate the TFP growth. A comparison of the TFPG obtained from different models will help us understand how sensitive the measured TFP growth is to these different capital aggregation procedures. The effect of alternate aggregation procedure on aggregate economy TFPG is depicted in Figure 3a–d. The figures depict the difference in average TFPG for the entire period, 1979–2003 along with two sub periods, 1979–95 and 1995–2003 under various assumptions. As seen before, the composition effect is visible in the TFPG as well, indicating the importance of incorporating the asset heterogeneity (Figure 3a). In most countries capital stock tends to overestimate the TFPG. Exceptions to this phenomenon are Germany and the UK, particularly during the second period, 1995–2003. During 1979–2003, the composition effect is larger in the US, while during the later period it is higher in the UK. In the case of UK, however, the effect is positive suggesting a lower capital service growth rate. In Germany the effect is relatively low. In most cases the effect is seen to be larger in the exogenous rate of return model.
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It is also clear from the Figure 3b that the tax variable has shown generally very meagre effect on TFPG. In general the TFPG is low in the tax-incorporated models. However, the differences are quite marginal say less than 0.05 percentages. US is the only country where there is some difference observed when tax is incorporated throughout the period, that too only in the CER model. This model has produced lower TFPG in France as well during the second period, but the magnitude of the difference is lower than that of the US.
To compare alternative models (Figure 3c and 3d), it can be seen that there are modest differences between external and internal models. The external models tend to produce lower TFPG compared to the internal models. Notably the difference is larger when capital gains are excluded from internal models (i.e. IOR models). This is true in almost all the countries except in the UK, where in a large number of cases internal models produced higher productivity growth. The effect is seen to be relatively larger in the US, particularly during 1995–2003, followed by the Netherlands and France.
Between the different internal models there seems to exist no significant difference in most countries. Nevertheless, the inclusion of capital gain term in the internal models does make some differences at least in some countries (Figure 3d). For instance, in the US and in the UK, the difference between models with capital gain and without capital gain amounts up to more than – (0.15) percentage. In both these countries, the difference between INR with annual capital gain and INRS is also relatively larger, though the magnitude is trivial. This may suggest that difference in INR and IOR in these countries may be due to the volatility of investment prices that would affect the capital gain term. IOR model has generally shown a tendency to produce larger TFPG compared to the INR models. Thus the external models tend to be closer to internal models when capital gains are incorporated in the latter.
As we mentioned before, Harper, Berndt and Wood (1989) compare alternative rental prices in capital aggregation using the US data for the period 1974–81. They, however, do not examine the effect of incorporating tax in the rental price equation and also do not examine specifically the effect on TFPG. In Table 6, we compare our results on percentage negative rental prices and the capital composition effect with Harper, Berndt and Wood (1989). Similar to what we observe, they also found largest negative rental prices in INR with annual capital gain, followed by smoothed capital gain and without capital gain models. This is due to the capital gain effect on rental prices. In the case of composition effect, they found the largest positive effect in INRS, which is the same in our case as well. The ranking of these different models in terms of composition effect are also comparable, except that while they find a negative effect in CER model, we find a positive and larger effect. In general it seems that the composition effect has increased in all the models, when we compare our results with theirs for an earlier period.
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This aggregate picture, however, need not necessarily reflect the cross industry story. The results may change significantly across industries. We have presented the difference between capital stock and capital service growth rates in Table 7 and the difference in TFPG produced by different models in Tables 8–10
for 26 industry groups.
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As observed in the aggregate analysis, the industry picture also portrays an underestimation of capital input growth rate, when capital stock is used as a measure of capital input. It is evident from Table 7 that the US had the largest number of industries showing a higher composition effect. While the composition effect is positive and quite high in almost all industries in the Netherlands, France and the US, it is negative in a number of industries in the UK and Germany. This indicates that the capital service growth rates are lower than capital stock growth rates in these industries during the period 1979–2003. The largest negative effect is observed in agriculture and fishing sector in Germany and in business service sector in the UK. Agriculture and fishing sector has shown a negative capital composition effect in the US and the Netherlands as well. It is important to note that the extent of the difference between capital stock and capital service growth rates (both in positive and negative cases) is quite high in all the countries, thus indicating the importance of taking composition effect into account, while measuring capital input growth rates for the growth accounting purposes.
Table 8 presents the tax effect on TFPG across industries. The tax variable seems to show no effect on measured productivity growth in the Netherlands. However, it makes notable differences in the TFPG at least in some industries—mostly in service sector industries and in external rate of return models—in other EU countries. For instance, the external models show a difference of 0.1 percent or more in industries’ financial intermediation and business services in France and Germany and in construction and financial intermediation in the UK. Similarly, the internal models show differences in electricity generation and construction industries in Germany, and in business services in the UK—in the latter case only in those models that incorporate capital gain. The picture is slightly different in the US, where there are more industries where this difference is prominent. Again it is visible only in the external rate of return models. This includes industries’ food products, paper products, petroleum, chemicals, machinery, electricity generation, electrical equipment, construction, wholesale trade, financial intermediation and business service. However, a similar difference is not observed in the internal models; only one industry, financial intermediation, has shown a notable tax effect in the internal models. The observed differences are negative in most cases, indicating that the incorporation of tax variable reduces the estimated contribution of capital to output growth and hence produce a higher productivity growth. Thus, to summarize, the impact of tax is negligible in most industries in the internal models both in the EU and the US, while it is important in some service industries when the external models are considered in the EU countries. In the US, it is found to be important across the board when the external rate of return model is used in the rental price calculation. Thus, if we read these observations along with the negligible differences observed at the aggregate level, we may conclude that despite its theoretical importance, from an empirical point of view, the incorporation of tax makes no significant impact on the final growth rates of productivity in most industries and at the aggregate level. Thus, it appears that the tax variable has only a marginal effect, though it varies across industries, on capital growth rates (aggregated using rental price shares) and the resulting TFP growth rates in all the models in the EU, and in the internal models in the US.
Now, if we look at the differences between TFPG produced by alternative rental price models, we see a difference of more than 0.1 percent in a few industries in some countries (Tables 9
and 10). Within the three internal models—internal models with capital gain, without capital gain and with smoothed capital gain—there is hardly any noticeable difference between the first two models, thus suggesting that the impact of price volatility on rental prices is trivial during the period under consideration (Table 9). However, there exists some difference between models with capital gain and without capital gain. Both in the US and in the UK, for instance almost half the industries have shown a TFPG which is lower by more than 0.1 percent when the capital gain term is included. The largest difference is observed in industries’ mining and quarrying, communications, electricity, gas and water and transport and storage in the US. In the UK, industries that have shown lower TFPG in IOR models include communications, business services, financial intermediation and retail trade. Similar trend is seen in French electricity, gas and water generation sector. In effect, there seems to be some difference between the models that incorporate capital gain and those which do not, at the industry level, mostly in service sector industries in the UK and the US.
In Table 10, we provide the difference between internal and external models. Given the fact that there is hardly any difference between the two internal models that incorporate capital gain (INR and INRS), we have averaged the effect of these two, while the effect of IOR is presented separately. Again in the US and in the UK a large number of industries have shown difference between internal and external models, but this number is quite low in other countries. In all the countries financial intermediation has shown larger difference, accompanied by business services in the Netherlands, France, the UK and the US. In most countries the TFPG produced by the internal models tend to be larger than that of the external models in a large number of industries, with the possible exceptions of Germany and the UK. As we observed before, the IOR models show larger difference from the external models compared to INR models.
The above discussion based on visual inspection helps us understand the observed differences in TFPG under various assumptions. However, it would be interesting to see whether these observed differences are statistically significant. In order to test the statistical significance of these observed differences, we have estimated a linear regression equation, which takes the following form:
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| (12) |
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The results are in conformity with our preceding observations. Since we exclude the CER model from right hand side of the above specification, the intercept may be defined as the mean of the composition effect in the external rate of return model (without tax) during the period 1979–95. For all the countries the intercept is negative, though it is not significant in Germany and UK. As we noted before, these are the two countries where we observed low composition effect during 1979–95, particularly in the CER model. The negative sign shows that the capital stock significantly overestimates TFPG. The period dummy is positive for all countries, but significant only in UK and US. This coefficient should be interpreted with caution. It does not mean that the composition effect is higher in absolute magnitude during 1995–2003 compared to the previous period. Rather it implies that the negative capital composition effect on TFPG has declined significantly in both these countries during this period. In France and in the US, the tax variable tends to play a significant role only in the constant rate of return models, while it is not significant in the other countries. There is no sign of difference in the internal models. Both in France and in the US, internal models that does not incorporate capital gain component (IOR and IORTX) significantly differ from CER model. In both these countries, however, the coefficients of the IOR dummy are significant with and without tax, hence suggesting that the difference may not be due to the incorporation of tax, rather the capital gain component. In order to verify this, we have estimated the model introducing a tax dummy that takes 1 for tax-incorporated models.26 The results did not show any difference and are hence not reported. Thus, this validates our observations that, while the tax variable is negligible, some differences are observed between TFPG produced by internal and external models. Importantly, there is obvious difference between capital stock and capital service growth rates and the resultant TFP growth rates.
Thus, while the difference in TFPG produced between pretax and posttax (rate of return and rental price) models are quite trivial,27 there exists more difference between the TFPG produced by four different rental price models. The divergence is more between constant rates of return models and internal rates of return models, while it is relatively less between different internal models. This pattern holds across countries, though the number of industries registering differences is found to be high in the case of US. The question of how important these observed marginal differences are left to the empirical researcher. If one is interested in attributing significant importance for very minor point differences in productivity growth, these differences are important.
6 Conclusion
In this article we attempted to understand the effect of alternative rental prices in capital aggregation on the measured growth rates of capital input and thereby TFP. In order to arrive at some generally accepted measure of capital service growth, one requires reliable estimates of capital service prices or rental prices. The theoretical literature on rental prices has defined the concept well, though its measurement is still an empirical issue, as the researcher is left with many formulae. This article has used four alternative rental price models, measured using IR with annual capital gain, IR with smoothed capital gain, IR with no capital gain and an external constant rate of return, in calculating capital service growth rates. Moreover, we attempted to understand the effect of incorporating taxes into all these four models. The calculated capital service growth rates are used to calculate the TFPG using the growth accounting framework, in order to understand how sensitive the estimated TFPGs are to the use of different rental price models in capital aggregation.
Our results strengthen the case for using capital service growth rates instead of capital stock growth rates in growth accounting analysis. The composition effect, i.e. the difference between capital service and capital stock growth rates, is seen to be quite high in most industries in almost all the countries. This suggests that when the share of short living asset increases in total capital stock, the use of capital stock in productivity analysis will over estimate actual TFPG. As a prelude to capital service aggregation, we have examined the percentage of negative rental prices produced by alternative IR models. We have observed that the lowest number of negative rental prices is produced by the IOR model in almost all countries considered. This suggests that the incorporation of capital gain leads to negative rental prices.28
The inclusion of corporate tax in the rental price calculation influences the measured growth rates of capital services and productivity only in a very few industries. The observed differences between TFPG produced by two types of models are quite negligible in most cases. The difference is found to be visible only in the CER model both in the EU and the US. More importantly the difference is quite negligible at the aggregate level. Therefore, from an empirical point of view, it may not be possible to make a strong argument that the potential peril in using pretax rental prices in aggregating capital services for growth accounting analysis is very high, given this dataset and time period.
Nevertheless, there are marginal differences in the TFPG produced by alternative rental price models in some industries, though the magnitude of divergence is small. The TFPG generated by IR models and external rate of return models are found to differ relatively more. However, between the three internal rates of return models also we observed similar differences in the resultant productivity growth rates, though they were marginal in terms of quantitative magnitudes. The observed differences between different measures, however, may not be ignored because the productivity differences across countries are, sometimes, evaluated on a marginal scale. Moreover, if one opts to use a CER such as 4 percent in a disaggregate analysis, it is appealing to include corporate tax in the rental price calculation, as the inclusion of tax makes a difference in some service sector industries.
It is, therefore, quite difficult to conclude that one model is better than the other, from an empirical viewpoint. The good news is that the visible differences between alternate models are quite negligible, given our data. Which way one chooses to aggregate capital does play a role in studies of economic growth, particularly while analyzing the contribution of capital to growth, provided the researcher is so concerned about the marginal differences in growth rates.
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Acknowledgments |
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The author is thankful to Bart van Ark, Marcel Timmer, Robert Inklaar and Gerard Kuper for their comments and suggestions. Earlier versions of the article were presented at the CESifo conference on Productivity and Growth in Munich, at the 22nd annual meeting of the European Economics Association in Budapest and at the IARIW conference in Cork. The comments and suggestions by the participants of these conferences, in particular those of Theo Eicher at the CESifo conference and Barbara Fraumeni at the IAIRIW conference are also acknowledged. The usual disclaimer applies.
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Footnotes |
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1 Growth accounting allows the decomposition of total growth into the contribution of the growth of inputs and to a residual factor. This residual is considered to be a depiction of TFP and is often called Solow residual, as the theoretical premise of this framework is largely derived from Solow (1957).
2 There are also studies that tried to examine the US–EU productivity gap in terms of differences in ICT use, by classifying industries as ICT intensive and not. See for instance van Ark, Inklaar and McGuckin (2003) and Daveri (2004).
3 For a detailed discussion on growth accounting, see, among others, Jorgenson and Griliches (1967), Barro (1999) and Hulten (2000).
4 Inklaar, Timmer and van Ark (2008) provide a useful summary of the concept of TFP (see Box 1, Inklaar, Timmer and van Ark 2008).
5 See OECD (2001a and b)—the so-called capital and productivity manuals—and Timmer, O'Mahony and van Ark (2007) for discussions on the measurement of output and input for productivity analysis.
6 Note that even though one would assume proportionality between capital stock and capital service at individual asset level, the factor of proportionality would vary across asset types and over time depending upon the marginal productivity of each asset type. At the aggregate level, the capital services should take account of the differences in the service delivered by different asset types.
7 The economic rationale of using the rental share weights to aggregate capital services is that the investors expect to get more services in short time from an asset whose price is relatively high (or service life is relatively small). As one of the referees have pointed out, one could also account for the asset heterogeneity while calculating the contribution of capital to output growth, by entering each type of capital as a separate argument in the production function. This, however, will not be an appropriate solution in growth accounting, as it requires imposing the factor shares. Also note that the growth rate of aggregate capital stock may be calculated as ln (
) where 
. Such growth rate may differ significantly from the calculated share weighted growth rate using Equation (2), as the latter accounts for the asset heterogeneity. The difference between these two is often considered as a measure of capital quality or composition effect.
8 We avoid the details of this derivation. Interested readers may refer to Christensen and Jorgenson (1969).
9 Hall and Jorgenson (1967) conclude that tax policy highly influences the levels and timing of investment spending. They also note the crucial role of tax policy in changing the composition of investment; liberalization of depreciation rules has caused a shift away from equipment in the US, while the investment tax credit and depreciation guidelines have caused a shift towards equipment.
10 See Jorgenson and Sulliven (1981) for a detailed discussion on the derivation of the tax incorporated rental price formula.
11 Investment tax credit is a credit against tax liabilities in proportion to investment expenditure (Jorgenson and Yun 1991) aimed to encourage investment. It is equivalent to subsidies or investment grants aimed to offset tax liability.
12 See OECD (2001a) for a discussion of these alternatives.
13 Note that the sum of labour and capital compensation is identically equal to gross value added at factor cost under all the typical neo classical assumptions concerning a production function (Berndt and Hesse 1986).
14 In cases where there are high amount of net subsidies, or losses, value added may become less than labour compensation resulting in negative property compensation.
15 Negative user costs of capital can also arise even if the rate of return is positive, when there are large capital gains in the user cost formula.
16 Also see Oulton (2007) who make similar assumption in the OECD context.
17 Harper, Berndt and Wood. (1989) have shown that instead of asset-specific capital gain if one uses an average capital gain across assets, IR without capital gain will produce the same rental prices as that of INR and therefore IOR accommodates certain amount of capital gain. See Haper, Berndt and Wood (1989, pp. 350) for the derivation of this relationship.
18 For a detailed discussion on the data see Inklaar, O’Mahony and Timmer (2005).
19 See Inklaar, O'Mahony and Timmer (2005) for more discussion.
20 We assume investment tax credit, k, to be zero.
21 This data is available from Institute of Fiscal Studies website http://www.ifs.org.uk/corptax/internationaltaxdata.zip. A detailed description of the sources, definition and construction of these data is provided in Devereux, Griffith and Klemm (2002) and Devereux and Griffith (2003).
22 These data consist of taxes payable on goods and services when they are produced, delivered, sold, transferred or otherwise disposed of by their producers plus taxes and duties on imports that become payable when goods enter economic territory by crossing frontiers or when services are delivered to resident units by non-resident units. They also include other taxes on production, consisting mainly of taxes on the ownership or use of land, building or other assets used in production, or on the labour employed or on compensation paid to employees.
23 Barbara Fraumeni commented that the problem of negative rate of return may be handled by aggregating industry groups of similar characteristics. However, while we looked at the pattern of negative appearances we found that they are in industries not of near characteristics.
24 The total economy capital and TFP growth rates are derived as simple average across industries. One could also derive the aggregate TFPG as a weighted average of individual industry growth rates. However, since our interest is to understand the sensitivity of these growth rates to alternate model specifications, this choice does not matter much.
25 See Oulton (2007, Figure 2, pp. 311).
26 This regression takes the form 
where TX is the dummy that takes 1 for tax incorporated models and zero otherwise. Also note that we have included only three models, INR, INRS and IOR, despite whether they have incorporated tax or not, as we have introduced a tax dummy.
27 This may, however, not be considered as an argument for the exclusion of corporate taxes in investment analysis. The effect of differential taxes on investment is not tested in this article, rather from a strictly empirical perspective whether the tax makes any substantial difference in the conclusions regarding TFPG is examined. The cross-sectional implications of different tax regimes on investment behavior are beyond the scope of the present article.
28 As one of the referees has pointed out, the incidents of negative rates of return need not be a reason for dismissing a particular method. However, as we have clearly pointed out earlier, a negative rental price is inconsistent. Incidentally, it may be noted that the number of cases that has produced negative rates of return are much larger than the number of cases that produced negative rental prices.
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