Abstract
Using the new Bureau of Economic Analysis (BEA) Integrated Macroeconomic Accounts as well as other BEA data, we construct productivity accounts for two key sectors of the US economy: the Corporate Nonfinancial Sector (Sector 1) and the Noncorporate Nonfinancial Sector (Sector 2). Calculating user costs of capital based on, alternatively, ex post and predicted asset price inflation rates, we provide alternative estimates for capital services and Total Factor Productivity growth for the two sectors. Rates of return on assets employed are also reported for both sectors. In addition, we compare rates of return on assets employed and TFP growth rates when the land and inventory components are withdrawn from the asset base. Finally, implications for labour and capital shares from using alternative income concepts are explored.
The first author, W. Erwin Diewert, gratefully acknowledges the financial support of the SSHRC of Canada, and both authors, W. Erwin Diewert and Kevin J. Fox, gratefully acknowledge the financial support of the Australian Research Council (LP0884095, DP150100830).
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- 1.
The Appendix in Diewert and Fox (2016) explains in detail how we used the Integrated Macroeconomic Accounts to construct our data set for the two sectors of the US economy.
- 2.
- 3.
- 4.
The assumption of Eq. (2) allows us to replace the initial production possibilities set St with a new set St* which is the feasible set of (qO t, qI,qL,qK).
- 5.
This simple discrete time derivation of a user cost (as the net cost of purchasing the durable good at the beginning of the period and selling the depreciated good at an interest rate discounted price at the end of the accounting period) was developed by Diewert (1974; 504, 1980; 472–473, 1992; 194). Simplified user cost formulae (the relationship between the rental price of a durable input to its stock price) date back to Babbage (1835; 287) and to Walras (1954; 268–269). The original version of Walras in French was published in 1874. The early industrial engineer, Church (1901; 907–909) also developed a simplified user cost formula.
- 6.
Assuming that all of the flow transactions within the accounting period are realized at the end of each period is consistent with traditional accounting treatments of assets at the beginning and end of the accounting period and the cash flows that occur during the period; see Peasnell (1981; 56). The idea of anti-discounting to the end of the period to form end of period user costs uK t (as opposed to the usual discounted to the beginning of period user costs fK t) was explicitly suggested by Diewert (2005a, b; 485). Anti-discounting is implicit in the derivation of the user cost of an asset using the geometric model of depreciation that was made by Christensen and Jorgenson (1969; 302).
- 7.
We have ignored tax complications in deriving (6). Any specific capital taxes (such as property taxes on real estate assets) should be added to the user cost formula for the relevant assets. In our empirical work, we were not able to obtain a breakdown of property taxes into land and structure components and so property tax rates are missing in our user costs that we construct in the following sections of this study. Business income taxes that fall on the gross return to the asset base can be absorbed into the cost of capital, rt, so that rt can be interpreted as the before income tax gross return to the asset base used by the production unit. For material on the construction of user costs for more complex systems of business income taxation, see Diewert (1992) and Jorgenson (1996).
- 8.
- 9.
Of course, the problem with using ex ante user costs is that there are many methods that could be used to predict asset inflation rates and these different methods could generate very different user costs. For empirical evidence on this point, see Harper, Berndt and Wood (1989), Diewert (2005a) and Schreyer (2012).
- 10.
The problem with the exogenous method is that it is difficult to determine exactly the appropriate external cost of financial capital. In particular, it is difficult to estimate the risk premium that is associated with investing in a production unit that generates variable ex post rates of return on its asset base over time. Nevertheless, the exogenous method is probably the preferred method from a theoretical point of view. These issues are discussed more fully in Schreyer, Diewert and Harrison (2005) and Schreyer (2009, 2012).
- 11.
The Bureau of Labor Statistics in the U.S. was the first to introduce an official program to measure Multifactor Productivity or Total Factor Productivity in 1983; see Dean and Harper (2001). Other countries with TFP programs now include Canada, Australia, the UK and New Zealand.
- 12.
- 13.
Period t predicted prices for output, intermediate input and labour, say \( {{\mathrm{P}}_{\mathrm{O}}}^{{\mathrm{t}}^{\ast }} \), \( {{\mathrm{P}}_{\mathrm{I}}}^{{\mathrm{t}}^{\ast }} \) and \( {{\mathrm{P}}_{\mathrm{L}}}^{{\mathrm{t}}^{\ast }} \), should be used in equation (11) in order to calculate the period t predicted rate of return, rt*, instead of the actual ex post prices for output, intermediate input and labour, PO t, PI t and PL t. However, it is the usual convention in production theory to assume that actual ex post unit value prices for variable outputs and inputs are equal to their predicted counterparts.
- 14.
There is a problem with interpreting these smoothed user costs as rental prices that might be anticipated at the beginning of the accounting period. When there is a severe recession in the economy in say period t, both rt defined by solving (9) and rt* defined by solving (11) will become unusually low (or even negative) and it is unlikely that the resulting low (or negative) user costs defined by (10) could be anticipated in practice. This limitation of our analysis should be kept in mind, particularly when looking at the user costs for 2008. This suggests that exogenous estimates for the cost of capital may be a more appropriate strategy for forming user costs that more closely approximate rental prices. If an exogenous rt* is used, then equation (11) will not hold in general and it will be necessary to include pure profits (or losses) as a balancing item in the SNA. However, we do not pursue this line of inquiry in the present study.
- 15.
However, to make the accounting precisely consistent with the Austrian model of production, we require that the price used to value gross investments in asset n during period, PGIn t, be equal to the end of period t imputed value for a unit of the nth capital stock. Setting PKn t + 1 = PGIn t will ensure consistency. In our empirical work, we used the BEA end of period price for reproducible units of the capital stock which may be slightly different from the corresponding investment price for the asset.
- 16.
The BEA in particular does include the value of inventory change as part of the gross output of an industry. However, they may not value the change in inventories at end of period prices of the inventory item and so again there may be a slight inconsistency in our empirical work due to this pricing difference. For a more complete treatment of the accounting problems associated with the treatment of inventories in the Austrian model of production, see Diewert (2005b).
- 17.
Suppose some land is purchased during period t at the price \( {{\mathrm{P}}_{\mathrm{Kn}}}^{{\mathrm{t}}^{\ast }} \) where this purchase price is not equal to the end of period price of land, PKn t + 1. The quantity of new land purchased will be equal to qKn t + 1 − qKn t. Then the term \( -{{\mathrm{P}}_{\mathrm{Kn}}}^{{\mathrm{t}}^{\ast }}\left({{\mathrm{q}}_{\mathrm{Kn}}}^{\mathrm{t}+1}-{{\mathrm{q}}_{\mathrm{Kn}}}^{\mathrm{t}}\right) \) should be added to the right hand side of (12) as a purchase of a primary input (a cost item) and at the same time, we should add the term PKn t + 1(qKn t + 1 − qKn t) to the right hand side of (12) to value this land purchase at the end of period t price of this type of land (a revenue item). Thus in principle, we should add the term \( \left({{\mathrm{P}}_{\mathrm{Kn}}}^{\mathrm{t}+1}-{{\mathrm{P}}_{\mathrm{Kn}}}^{{\mathrm{t}}^{\ast }}\right)\left({{\mathrm{q}}_{\mathrm{Kn}}}^{\mathrm{t}+1}-{{\mathrm{q}}_{\mathrm{Kn}}}^{\mathrm{t}}\right) \) to the right hand side of (12). If some land is sold during the period at the price \( {{\mathrm{P}}_{\mathrm{Kn}}}^{{\mathrm{t}}^{\ast }} \), then qKn t + 1 − qKn t is negative and is equal to minus the quantity sold. In this case, we should still add the term \( \left({{\mathrm{P}}_{\mathrm{Kn}}}^{\mathrm{t}+1}-{{\mathrm{P}}_{\mathrm{Kn}}}^{{\mathrm{t}}^{\ast }}\right)\left({{\mathrm{q}}_{\mathrm{Kn}}}^{\mathrm{t}+1}-{{\mathrm{q}}_{\mathrm{Kn}}}^{\mathrm{t}}\right) \) to the right hand side of (12) to make the accounting consistent with our Austrian model of production. In our empirical work, we did not make these adjustments to the accounting identity given by (12); we simply assumed that \( {{\mathrm{P}}_{\mathrm{Kn}}}^{{\mathrm{t}}^{\ast }} \) is equal to our end of period price for the asset, PKn t + 1.
- 18.
- 19.
See Harberger (1998) on the importance of the rate of return on assets.
- 20.
This data base is described in more detail in the Appendix of Diewert and Fox (2016).
- 21.
This series was normalized to equal 1 in 1960. Note that the Sector 1 wage rate series PL1 t is also normalized to equal 1 in 1960.
- 22.
We constructed chained Fisher land price and quantity indexes for Sector 1 and then compared the value of land to value added and the quantity of land to the quantity of output. The nominal land to output ratio went from 36.7% in 1960 to a peak of 51.2% in 2006, declined to 22.0% in 2012 and finished up in 2014 at 30.4%. The corresponding real land to output ratio declined steadily from 36.7% in 1960 to 9.8% in 2014. The inclusion or exclusion of land from the productive asset base does make a significant difference to capital output ratios.
- 23.
The BEA Fixed Asset Tables are organized somewhat differently for the Nonfinancial Noncorporate Sector as compared to Sector 1, with a decomposition of Sector 2 into subsectors. This led us to organize the capital stock data for Sector 2 into fourteen rather than nine components.
- 24.
The average share of residential land in Sector 2 value of the capital stock is 28.2%, farm land is 16.4% and commercial (nonresidential and nonfarm) land is 7.1%. Thus the overall average land share in the total value of Sector 2 assets is 51.6% and for reproducible assets is 48.4%. The average land share of asset value in Sector 1 is only 14.4% and the corresponding reproducible asset share is 85.6%.
- 25.
The nonzero depreciation rates for assets n = 1,2,3,4 used in Sector 1 are listed in Table A10 in the Appendix of Diewert and Fox (2016).
- 26.
It may be that the length of our moving average process is too long or that better methods for predicting asset prices one year hence could be devised. However, our goal is to obtain user costs that could approximate one year rental prices for assets used in production (when they exist). Since observed rental prices are relatively smooth, our suggested method for generating predicted asset prices does lead to relatively smooth user costs as will be seen later.
- 27.
Tabulated data for the series in this and following figures are available in Diewert and Fox (2016).
- 28.
Note that our expected real rate of return on Sector 1 assets has been fairly stable over the period 1982–2014. \( {{\mathrm{R}}_1}^{{\mathrm{t}}^{\ast }} \) ranged between 4.62% (1990) and 9.33% (1997) over this period.
- 29.
The average corporate income tax paid by the nonfinancial corporate sector on assets during our sample period as a percentage of the asset base is 1.98% per year; see the series VTI1 t in Appendix Table A3 of Diewert and Fox (2016).
- 30.
The nonzero depreciation rates for assets n = 1,...,9 used in Sector 2 are listed in Table A11 in the Appendix of Diewert and Fox (2016).
- 31.
The reason why nominal and real ex post rates of return on assets are much higher in Sector 2 compared to Sector 1 can be explained by the fact that production in Sector 2 is highly land intensive and land inflation rates are much higher than inflation rates for other assets.
- 32.
The average business income tax paid by the nonfinancial noncorporate sector on assets during our sample period as a percentage of the asset base is only 0.15% per year; see the series VT12 t in Appendix Table A3 of Diewert and Fox (2016). This income tax rate for Sector 2 seems to be too low to be true!
- 33.
Thus the use of Jorgensonian user costs is not recommended in econometric studies where cost functions are estimated or where production functions are estimated using inverse factor demand equations as additional estimating equations.
- 34.
This formula was attributed to Törnqvist (1936) by Jorgenson and Griliches (1972; 83) as a discrete time approximation to the continuous time Divisia indexes that Jorgenson and Griliches (1967, 1972) advocated for aggregating inputs and outputs in productivity studies. The formula does not explicitly appear in Törnqvist (1936) but it is explicit in a follow up paper co-authored by Törnqvist; see Törnqvist and Törnqvist (1937). The formula was derived in an instructive manner by Theil (1967; 136–137) and so it is also known as the Törnqvist-Theil formula. Jorgenson and Nishimizu (1982) called the index the translog index. Diewert (1976; 118–129), Diewert and Morrison (1986) and Kohli (1990) related Törnqvist price and quantity indexes to various translog functional forms for cost, revenue and production functions.
- 35.
See Diewert (2014b) for a detailed explanation of the methodology and an application to US data. The land data used in this earlier study was of lower quality than the land data used in the current study.
- 36.
These series are normalized to equal one in 1960.
- 37.
These series were also normalized to equal one in 1960. The price and value of labour input for Sector 1 in year t, PL1 t and VL1 t, are listed in Table 1. Define the quantity of labour used in Sector 1 in year t as QL1 t ≡ VL1 t/PL1 t. Thus we added PL1 t and QL1 t to our user costs and capital stock quantities to form the overall chained Törnqvist input quantity indexes.
- 38.
Using Jorgensonian user costs, we find that the sample average input cost shares of labour, land services and reproducible capital stock services in Sector 1 were 68.6%, 2.1% and 29.3%. The sample average cost shares of residential, farm and commercial land (assets 5, 6 and 7) were only 0.05%, 0.17% and 1.85%.
- 39.
Note that QXJ1 t and QXP1 t (and TFPJ1 t and TFPP1 t) cannot be distinguished in Figure 11.
- 40.
The last “decade” covers only the years 2010–2014.
- 41.
Using predicted user costs, the corresponding decade by decade geometric average rates of TFP growth in Sector 1 were as follows: 2.59%, 1.26%, 1.49%, 2.02%, 1.16% and 1.72% per year.
- 42.
See Diewert and Fox (2017) on potential sources of the productivity slowdown.
- 43.
These series are normalized to equal one in 1960 when they are listed in Table 13. The input price and quantity series used in the index number formula for QKJ2 t and QKP2 t are the u2, n t and \( {{\mathrm{u}}_{2,\mathrm{n}}}^{{\mathrm{t}}^{\ast }} \) listed in Tables 9 and 11 respectively and the corresponding quantity series QK2, n t are described in Table 5.
- 44.
These series were also normalized to equal one in 1960. The price and value of labour input for Sector 1 in year t, PL2 t and VL2 t, are listed in Table 2. Define the quantity of labour used in Sector 2 in year t as QL2 t ≡ VL2 t/PL2 t. Thus we added PL2 t and QL2 t to our user costs and capital stock quantities to form the overall chained Törnqvist input quantity indexes.
- 45.
However, the predicted asset price inflation rates are on average quite close to the average ex post asset price inflation rates. Thus on average, the two sets of user costs are similar, giving rise to similar trends in the two sets of capital service prices.
- 46.
Using Jorgensonian and predicted user costs, we find that the sample average input cost shares of labour and capital services were 56.7% and 43.3%. Using Jorgensonian user costs, the sample average cost shares of residential, farm and commercial land services (assets 10, 11 and 12) were 7.51%, 4.44% and 2.43%. Using predicted user costs, the sample average input cost shares for assets 10, 11 and 12 were 8.04%, 4.05% and 2.44%. These input cost shares for land are low compared to the share of land assets in total asset value: the average overall land share of total asset value was 51.6% while reproducible assets contributed 48.4% of total asset value. The average shares of the three types of land in total asset value were 28.2%, 16.4% and 7.1%. The user cost shares of capital services for land are lower than their corresponding asset value shares because the high land price inflation terms dramatically reduce land user costs relative to their asset prices.
- 47.
These trends in the prices and quantities of labour and capital input into Sector 2 indicate the presence of labour saving technical progress in this sector.
- 48.
Note that QXJ2 t and QXP2 t (and TFPJ2 t and TFPP2 t) can hardly be distinguished in Figure 12.
- 49.
Again, the last “decade” covers only the years 2010–2014.
- 50.
Using predicted user costs, the corresponding decade by decade geometric average rates of predicted TFP growth, TFPP2 t, were as follows, with the corresponding Jorgensonian rates of growth in brackets: 2.28% (2.32), 0.39% (0.41), 0.49% (0.64), 1.35% (1.29), 1.37% (1.22) and 1.80% (1.81) per year. Note that the difference is particularly large for the 2000s.
- 51.
See the EUKLEMS and World KLEMS data bases on line; Jorgenson and Timmer (2016).
- 52.
To recover the un-normalized QL1 t, multiply the listed QL1 t series by the value of labour input in Sector 1 for 1960, which is 180.4. To recover the four un-normalized capital services series, multiply QKJ1 t, QK1M t, QK1IM t and QK1LIM t by the Gross Operating Surplus for Sector 1 for 1960, which is 75.5.
- 53.
To recover the un-normalized QL2 t, multiply the listed QL2 t series by the value of labour input in Sector 2 for 1960, which is 76.6. To recover the four un-normalized capital services series, multiply QKJ2 t, QK2M t, QK2IM t and QK2LIM t by the Gross Operating Surplus for Sector 2 for 1960, which is 30.8.
- 54.
- 55.
- 56.
Jorgensonian ex post rates of return are appropriate in this context; see Sect. 5.
- 57.
See Appendix A8 of Diewert and Fox (2016).
- 58.
See tables A1 and A9 of Diewert and Fox (2016).
- 59.
The value added shares are the same as those in Table 1.
- 60.
Labour shares are of course a mirror image of these capital shares.
- 61.
The value added shares are the same as those in Table 2.
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Diewert, W.E., Fox, K.J. (2018). Alternative User Costs, Productivity and Inequality in US Business Sectors. In: Greene, W., Khalaf, L., Makdissi, P., Sickles, R., Veall, M., Voia, MC. (eds) Productivity and Inequality. NAPW 2016. Springer Proceedings in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-68678-3_2
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