Alternative output, input and income concepts for the production

7 De ﬁ nitions of output and input are key to studies of productivity analysis, as they are to the national accounts of countries. 8 This paper systematically reviews alternative de ﬁ nitions at production unit and aggregate levels, illustrating the different 9 perspectives that they provide on production and income, and making the case for their use in understanding different 10 aspects of ﬁ rm and country economic performance.


Introduction
This paper takes a new look at the production accounts in the international System of National Accounts with emphasis on alternative measures of output and primary input, with implications for the resulting alternative measures of productivity.Perhaps more importantly, the paper considers alternative measures of income generated by the production sector of an economy.
As well as their central use in informing macroeconomic policy, national accounts data on inputs and outputs for countries are used extensively in the academic literature on productivity; see for example Solow (1957), Jorgenson and Griliches (1967), Diewert and Fox (1999) and Fernald and Inklaar (2020).They are also used in the literature on efficiency analysis; see for example Färe et al. (1994) and Kumar and Russell (2002).Given their extensive use and broad acceptance as the authoritative source of information on economic performance, it is tempting to believe that all matters relating to national accounts have been settled by the international community.Yet the United Nations growing literature on "Beyond GDP" concepts nor on "GDP The rest of the paper is organised as follows.The next section introduces production accounting using a simplified context of a single production unit.Section 3 considers alternative net output, input and income concepts for a production unit, and Section 4 provides additional discussion about our accounting framework.Section 5 considers corresponding economy wide measures with multiple types of capital and Section 6 concludes.
2 Production unit accounting: the hicks and edwards and bell framework In order to simplify the notation, we consider a very simple model of production in this section for a single production unit that produces or uses only six types of goods and services during an accounting period t.A production unit could be a firm, a division of a firm or what national income accountants call an establishment.The establishment must be able to provide period by period accounting information about periodic revenues and costs as well as balance sheet information on the status of its asset holdings at the end of each accounting period.
Equation (1) below defines the production unit's pure profits in period t, ∏ t , using the Hicks, Edwards and Bell approach to production theory: The price and quantity variables appearing on the right hand side of (1) are defined as follows: K quantity of the capital stock held by the unit at the beginning of period t; r t period t cost of capital for the production unit.
Units of the total output Q t Y could be sold to domestic customers or could be exported.Later in the paper, this distinction will become important when we aggregate over producers but at present, we do not have to distinguish domestic sales from foreign sales.Similarly, units of the intermediate input and units of the investment good could be purchased from domestic suppliers or could be imported. 5    We note that prices and quantities of output, intermediate input, purchased investment goods and labour can in principle be observed by the accountant.However, the quantity and price of the production unit's beginning and end of period capita stocks, Q tÀ1 K , Q t K , P tÀ1 K and P t K , typically cannot be observed but must be estimated by the accountant.We will indicate how this can be done shortly.The production unit's period t cost of capital is denoted by r t on the right hand side of (1).If the production unit purchased its beginning of period t capital stock and financed this purchase by issuing a one period bond at the interest rate r t* in the amount equal to 203 204 205 Our next assumption relates period t total investment to 206 the beginning and end of period t capital stocks held by the 207 unit; i.e., we assume that the following equation holds:  Diewert (2014) for a more complete accounting model that deals with the financing of the initial capital stock and other financial transactions using the Hicksian accounting framework. 7This accounting convention dates back to Garske and Fells (1893).For a discussion of this convention, see Anthony (1973).Diewert and Fox (1999) attributed some of the fall in the worldwide fall in Total Factor Productivity during the 1970s to the problems associated with measuring income using historical cost accounting when inflation is high. 8"This [convention] accords with the assumption conventional in discrete compounding that flows occur at the end of each period."K.V. Peasnell (1981;56). 9If Q t II ¼ 0; there is no need to impute P t II .If Q t II > 0, then define P t II as the average cost of producing the internally manufactured investment goods.Typically, Q t II will be a small amount of total investment.If firms make very large infrastructure investments such as building pipelines or new natural gas liquefaction plants, then internally produced investments become important. 10The geometric model of depreciation was used by Jorgenson and Griliches (1967) in their classic study of the Total Factor Productivity of the U.S. economy.For additional materials on the geometric model of depreciation, see Jorgenson (1989Jorgenson ( ) (1996aJorgenson ( ) (1996b) ) and Schreyer (2001Schreyer ( ) (2009)).Schreyer (2009) and Balk (2011) both introduce a modification of the classical geometric depreciation model by assuming that this period's investment adds to the productive capital stock at the midpoint of the present period instead of at the end of the current period.This is a reasonable assumption but implementing it leads to extra complications in that we need to construct separate user costs for new investments and the depreciated capital stocks at the end of the accounting period.Also deferring depreciation of newly purchased capital stocks until the period after their purchase is consistent with accounting conventions; see Peasnell (1981).In any case, we assume that the national income accountant has estimates available for the beginning and end of period t prices of a new unit of the capital stock.

Journal of Productivity Analysis
These prices can be used to define the period t asset inflation rate i t using the following equation: and use (3) to eliminate Q t K from definition (1).This allows us to express period t pure profits ∏ t for the production unit as follows: 256 The period t user cost of capital U t which makes its 257 appearance in the second line of ( 5) is defined as follows: 12 investments and define conventional period t pure profits of 279 the production unit, Π tÃ , as follows:  Ijiri (1979) for a defence of historical cost accounting.
12 Babbage (1835; 287) described the user cost idea in words and Walras (1954;268-269) developed an explicit user cost formula (in 1874) as did the industrial engineer Church (1901;907-908).Alternative derivations of a user cost formula may be found in Jorgenson (1963Jorgenson ( ) (1989Jorgenson ( ) (1996b)), Griliches (1963;120), Christensen and Jorgenson (1969;302), Diewert (1974;504) and Diewert and Lawrence (2000;276). 13If the asset is a land or structure asset, then the use of this input may also be subject to a property tax.If the period t property tax rate τ t is a percentage of the beginning of the period value of the asset, then the user cost becomes the value of pure profits ∏ t : 296 297 298 To get the measure of production unit output that cor-299 responds to the income measure defined by ( 8), replace ∏ t in 300 (8) by the right hand side of (5).Period t Gross Domestic 301 Output, GDO t , is then defined as follows: 309 310 311 Suppose the following conditions hold: 312 313 314 Then it can be seen that our measure of gross output, 315 GDO t , is equal to Regular Value added, VA t .

316
The problem with the gross income measure, GDI t 317 defined by ( 8) is that it includes the value of depreciation as Subtract this term from 328 period t Gross Domestic Income to define the period t Net 329 Domestic Income, NDI t , generated by the production unit: In order to obtain the output measure NDO t that matches up with the net income measure NDI t defined by ( 12), substitute the right hand side of (5) to eliminate ∏ t from the second line in (12).We obtain the following expression for the Net Domestic Output NDO t produced by the production unit during period t: The second line of (13) tells us that period t Net Domestic Output is equal to the production unit's Comprehensive Value Added, CVA t , plus the production unit's period t gross investment, Q t I , less period t depreciation of the starting capital stock, δ t Q tÀ1 K , valued at the end of period capital stock price, The measure of net output defined by ( 13) looks reasonable enough.It adds the value of net investment (valued at the end of period price for units of the capital stock) to a comprehensive measure of value added produced by the production unit during period t.Thus this net output measure is consistent with Pigou's (1941;273−274) preference for an output measure that is consistent with maintaining the physical capital stock.However, the problem with the net output measures of output and income, NDO t and NDI t , is the fact that the income measure does not accurately measure the nominal income generated by the production unit over the period; NDI t omits the capital gains (or losses) that accrue to the initial capital stock held by the production unit.Adding these capital gains to NDI t leads to period t Comprehensive Net Domestic Income generated by the producer over period t, CNDI t , defined as follows: The first line in ( 14) tells us comprehensive net income is equal to payments to labour P t L Q t L plus interest and dividend payments to the owners of the production unit for tying up their capital for the period, r t P tÀ1 K Q tÀ1 K , plus any pure profits ∏ t that might have occurred. 16The second line in (14) tells 14 The production unit could be producing units of the capital stock and this production would be included in the definition of a firm's regular value added.However, purchases of units of the capital stock are not included in regular value added because the cost of purchased investment goods is capitalised and depreciated over time using normal accounting procedures.Comprehensive Value Added allows revenues from sales of the investment good and costs from purchases of the investment good to enter the net output aggregate. 15See Hicks (1946;174) (1973;155), Samuelson (1961) and Balk (2010) (2011) on alternative definitions of income and on the treatment of depreciation.See also Schreyer (2009;43-51) and Stiglitz et al. (2009) on net income measures in the System of National Accounts. 16Rymes (1968) (1983) defined r t P tÀ1 K Q tÀ1 K as waiting services and advocated replacing the user cost of capital by waiting services.The term "waiting" can be traced back to Marshall (1920;232): "And human nature being what it is, we are justified in speaking of the interest on capital as the reward of the sacrifice involved in the waiting for the enjoyment of material resources, because few people would save much without reward; just as we speak of wages as the reward of labour, because few people would work hard without reward".
us that CNDI t is equal to NDI t plus capital gains on the production unit's initial capital stock.
In order to determine the net output measure that matches up with the comprehensive measure of income defined by the first line in ( 14), we use the right hand side of (5) to eliminate ∏ t from the right hand side of ( 14).We obtain the following expression for period t Comprehensive Net Domestic Output, CNDO t for the production unit: The second last line in (15) tells us that our comprehensive measure of net domestic product for the production unit CNDO t is equal to comprehensive value added, CVA t , plus the value of the end of period capital stock, P t K Q t K , less the value of the beginning of the period capital stock, . This is a very straightforward definition of net (nominal) output.On the other hand, the net domestic measure of output, NDO t , is equal to CVA t plus the net change in the capital stock evaluated at end of period prices, Value Added from (10), and is the user cost of capital from (6): 18 17 This is the case where i t is negative. 18Balk (2010; S239-S247) introduced many more rows to Table 1 by decomposing user cost into four separate components and then shifting these components from the input column to the output column.We note that his decomposition of user cost into separate components is slightly different from our decomposition.Balk correctly includes property taxes in user cost so this adds the term τP tÀ1 K Q tÀ1 K to the income column in Table 1.Thus we regard property taxes paid by the Production Unit as a contribution to all of the income concepts defined in Table 1.Our r t is a gross rate of return that includes income taxes paid by the Production Unit so income taxes also contribute to all forms of income defined in Table 1.
Following Balk (2010), one can define (one plus) Productivity Growth (or Total Factor Productivity Growth) of the Production Unit in time period t relative to a base period 0, Prod t , as the Fisher (1922) quantity index of (net) outputs relating period t to period 0 divided by the corresponding Fisher quantity index of inputs. 19For each row in  Fisher output and input price and quantity indexes for period t relative to period 0 by P F p 0 ; p t ; y 0 ; y 1 ð Þ¼p t Á ½ y 0 p t Á y t =p 0 Á y 0 p 0 y t 1=2 and Q F p 0 ; p t ; y 0 ; y t ð Þ¼p 0 Á y t p t Á y t =p 0 ½ Áy 0 p t Á y 0 1=2 (for outputs) and P F w 0 ; w t ; x 0 ; x t ð Þ¼w t Á ½ x 0 w t Á x t =w 0 Á x 0 w 0 x t 1=2 and Q F w 0 ; w t ; x 0 ; x t ð Þ¼w 0 Á x t w t ½ Áx t =w 0 Á x 0 w t Á x 0 1=2 (for inputs).Thus Prod t ¼ Q F p 0 ; p t ; ð y 0 ; y t Þ=Q F w 0 ; w 1 ; x 0 ; x 1 ð Þ and Balk's (2010: S233) growth accounting decomposition into explanatory factors for the output/income concept defined by p t •y t is the following identity: The idea of defining TFP growth as an output index divided by an input index goes back to Jorgenson and Griliches (1967).Balk probably chose the Fisher index as his functional form for price and quantity indexes because of its superior axiomatic properties; see Diewert (1992).Balk's accounting approach to productivity measurement draws on Diewert (1990) and Diewert and Nakamura (2003) but is more general since Balk allows profits to be nonzero. 20Using the Törnqvist index number formula in place of the Fisher formula, Kohli (1990) was able to obtain a growth accounting decomposition that was more detailed, i.e., the output price index and the input quantity index were decomposed into individual price and quantity explanatory factors; see also Diewert and Morrison (1986).These authors assumed that profits were equal to zero. 21This section was added in response to the comments of the referees on an earlier draught. 22As was indicated in Section 2, the quantities are total amounts purchased or sold during period t and the corresponding prices are unit value prices.The use of unit value prices to aggregate over transactions made during the accounting period was recommended by early index number theorists; see Walsh (1901;96) and Fisher (1922;318).If units of the capital stock are sold during the accounting period, then Q t IP becomes net asset purchases and could become negative if asset sales are bigger than asset purchases. 23Accounting theorists have stressed the importance of using transactions data which are objective, reliable and reproducible; see Daines (1929;99-101) and Ijiri (1979) on objectivity, Canning (1929; 321) on reliability and Davidson et al. (1976;225) 4.2 Can neo-austrian profit be written as a flow?
A referee pointed out that our definition (1) of pure profit involved a mixture of stock and flow variables and one can ask whether pure profits can be rewritten purely in terms of flow variables.Using the geometric model of depreciation, we showed that pure profits Π t defined by ( 1) are equal to the expression on the right hand side of (5).Using definition (17) of period t cash flow CF t , (5) can be rewritten as follows: The user cost price of the beginning of the period capital stock, U t , was defined by ( 6).U t Q tÀ1 K is conceptually equal to the cost of renting the initial capital stock and hence is a flow variable.If we use ( 6) to decompose the user cost into its components, then we have the following decomposition: Thus the various components of the cost of using the initial capital stock can also be decomposed into flows.We also need to rewrite the final two terms on the right hand side of ( 19) in terms of flows that make sense.Replace total period t investment Q t I by the sum of own account investment Q t II and purchased investment Q t IP .This leads to the following equations: The term P t K Q t II is the imputed value of own account investment valued at the end of period price for a unit of the capital stock which is P t K .This term is a flow.The term IP is a revaluation term for purchased investment and hence is also a flow variable.This term will contribute to period t profits if the end of period price of an investment good, P t K , is greater than the within the period purchase price for the investment good, P t IP .The terms on the right hand side of ( 21) are flows so it is possible to interpret our measure of pure profits in terms of period t flows.
There is no explicit revaluation term for own account investment because there is no explicit purchase price for this type of investment.The cost of own account investment is included in intermediate input, labour and capital services that were used to produce Q t II .If these costs could be separated from the overall costs then these separated costs could be cumulated and divided by Q t II to give us an estimated (or imputed) price P t II .One 24 See Daines (1929;98) and Ijiri (1979;66). 25"The main problem is that when a reproducible capital input is purchased for use by a production unit at the beginning of an accounting period, we cannot simply charge the entire purchase cost to the period of purchase.Since the benefits of using the capital asset extend over more than one period, the initial purchase cost must be distributed somehow over the useful life of the asset.This is the fundamental problem of accounting."W. Erwin Diewert (2005a;480). 26In the accounting literature, our Cash Flow is roughly equivalent to Cash Flow from Operations.Our measure of Comprehensive Cash Flow includes (net) purchases of the investment good.Our comprehensive measure is not a truly comprehensive measure because it excludes transactions in financial markets that determine the production unit's cost of capital, r t .For models that integrate financial transactions into the Neo-Austrian model, see Diewert (2014) and Diewert et al. (2016). 27If the production unit sells part of its beginning of the period capital stock during period t, then Q IP t is interpreted as net (market) purchases of the investment good and if period t asset sales are bigger than asset purchases, then Q IP t becomes negative.
Note that that the flow decomposition defined by ( 21) 594 can be applied to our definition of Gross Domestic Output, 595 GDO t , defined by ( 9).Using ( 9), ( 17) and ( 21), we have:  gains on the initial capital stock (the term i t P tÀ1 K Q tÀ1 K to the value of net output: "The present Manual uses a notion of depreciation that does not encompass the changes in relative prices of assets.
There are several reasons for this.

•
The first reason is that it keeps the supply side and production perspective of the economy separate from the demand and consumer side.A measure of depreciation that captures the discounted value of capital used up in production and the investment needed to keep the productive capacity of the economy intact fits into a supply-side perspective.A consumer or demand side perspective can easily be added by considering wealth effects arising with the ownership of productive assets but it seems better to keep these effects separate rather than lumping them together in the first place. • The second reason is that present practice in OECD countries' national accounts corresponds to a notion of depreciation that excludes wealth effects.Also, if one wanted to bring real wealth effects into measures of depreciation, there is a question whether such effects should be integrated asymmetrically (capturing only expected real holding losses) or symmetrically (allowing also for real holding gains).However, we reiterate that different analytical questions may give rise to different treatment of relative price changes for capital goods.In particular, for the analysis of wealth effects and associated welfare considerations, it is meaningful to account for real price changes.Net income would then decline in the presence of expected holding losses and rise in the presence of expected holding gains."Schreyer (2009;51).
There is a third reason to exclude holding gains from a measure of net output: asset price inflation, i t , can be very large and positive (and negative) and thus the addition of the term i t P tÀ1 K Q tÀ1 K to the measure of net output can lead to an income measure that is extremely volatile.Our suggested solution to this volatility problem is to replace actual ex post asset price inflation rates by smoothed asset inflation rates. 30    Thus computing a nonvolatile measure of comprehensive net output requires two major imputation models: a model of depreciation and a model for smoothing asset prices.
It is unfortunate that a useful measure of comprehensive net income generated by a production unit requires so many imputations, but we believe it is important for statistical offices to provide a measure of comprehensive net income due to the increasing importance of land as a factor of 28 If the investment good is being produced by the production unit, then sales of the good would appear as a revenue item.Thus own account production is interpreted as production of the investment good for use by the production unit for its own use in the following period. 29For example, see Samuelson (1961), Schreyer (2009;43) and Balk (2010; S244) for discussions of this issue.Before the use of memory chips became widespread, measures of gross and net output tended to move in a proportional manner, so growth rates of gross and net domestic product were similar.However, Spant (2003) showed empirically that this similarity in growth rates no longer holds. 30This volatility problem shows up in the user cost of land which can easily become negative if ex post asset inflation rates are used as the i t .The use of smoothed asset price inflation rates in the user cost formula will tend to eliminate negative user costs; see Diewert and Fox (2018).
Equation ( 23) is consistent with the geometric model of depreciation if we set the period t depreciation rate δ t equal to zero.If Q t I > 0, then P t I is the purchase price for newly acquired land; if Q t I < 0, then P t I is the observed selling price for sold land.With these assumptions, pure profits for the PU are defined as follows: where the user cost of capital is defined as when the depreciation rate δ t = 0. Thus pure profits are equal to cash flow less the user cost of land plus the term which is equal to the end of period capital gains or losses on the (net) purchases of land made during the accounting period.
Typically, this capital gains term will be small.
Since the depreciation rate for land is zero, our measures of gross and net domestic output, GDO t and NDO t , will be equal.Thus for land investments, Table 1 in Section 3 becomes Table 2.
A number of points of interest emerge from a study of Table 2: • The asset inflation rate for land, i t , can exceed the reference cost of capital, r t , and so the user cost of capital in this case, r t À i t ½ P tÀ1 K , becomes a user benefit.

•
Our GDO concept differs from national accounts GDP by adding the asset revaluation term P t K À P t I À Á Q t I .As was mentioned above, for an individual production unit, this revaluation term will usually be small for an individual firm or sector.However, when we aggregate over production units in the national economy, the P t I Q t I terms will sum to zero, so effectively, we are adding the term P t K Q t I term to value added to obtain our Neo-Austrian measure of gross output.In many economies, agricultural land (which has a low price) is converted into commercial, industrial and residential land (which tends to have a much higher price).
Thus in aggregate, adding the terms P t K Q t I for the different types of land to value added will tend to give a significant boost to our measure of gross output.

U N
As in the previous section, we assume that geometric depreciation applies to each capital stock.Thus we assume that the following relationships between beginning and end of period capital stocks and total investment hold: Note that the period t geometric depreciation rate for the nth type of capital, δ tt n , depends on t and n but not on f.
Using these assumptions, it can be shown that we can obtain the following expression for the pure profits of Production where the user cost of capital stock component n for unit f is defined as Kfn and the capital stock asset inflation rates i t fn are defined by In what follows, we make the simplifying assumption that for each asset n, the inflation rate for each production unit is constant, i.e., we assume that for f = 1, …, F and n = 1, …, N.
For each class of the six quantity variables on the right hand side of (29), define the corresponding national aggregate by summing over production units.Thus Q t Yj Define the corresponding national unit value prices as follows: P t Yj In for n = 1, …, N; 35 and P t IPn The interpretation of the various macroeconomic concepts follows along the same lines as our discussions of the microeconomic concepts.However, there is a reduc-  37 Then it can be seen that 952 the following equality holds: 953 954 955 Now replace VA t in (32), which defined Gross Domestic 956 Output GDO t , and we obtain the following expression: 36 Jorgenson and Griliches (1972) noted the importance of using prices that producers face in productivity studies.If an output of a domestic producer is taxed, then the producer only gets the before tax price to add to revenue; if an imported good or service is taxed, then the producer faces the after tax price and the after tax value of the input should be added to producer cost. 37However, there is a problem with taxed intermediate inputs that are produced domestically and purchased by a domestic final demander.The tax revenue raised by this internal commodity tax does not cancel out as we aggregate over units.For more on the treatment of taxes in the production accounts, see Diewert (2006).
income.But depreciation is not a compo-319 nent of income that can be spent on the purchase of con-320 sumer goods and services.Thus the depreciation component 321 of user cost should be removed as a source of income and 322 transferred to the net output accounts; i.e., depreciation 323 should be treated as deduction from production unit rev-324 enues and be treated as a type of intertemporal intermediate 325 input. 15The period t value of depreciation (valued at end been recognized that measures of Gross 619 Domestic Output overstate the value to society of produc-620 tion because depreciation of the beginning of the period 621 capital stock is not deducted from measures of gross out-622 put. 29Thus from a theoretical point of view, deducting 623 depreciation from the measure of gross output has not been 624 controversial.However, adding capital gains (or losses) to a 625 measure of net output has been resisted by national income 626 accountants.Schreyer explained why the current System of 627 National Accounts does not add the value of (net) capital GDO t is essentially equal to standard 960 expenditure side GDP at producer prices except that gross 961 investment is valued at end of period prices instead of at the 962 average prices of investment transactions during period t. 963 Thus our economy wide various output and input measures 964 defined above can be computed using standard macroto the literature by making clear 975 the definitions and their relationships, highlighting how 976 each provides a different perspective.For example, each 977 definition of output (at both individual production unit and 978 aggregate levels) provides a different perspective of pro-979 duction.Use of price deflated versions of these output 980 concepts in productivity studies will typically lead to dif- Please ensure you fill out your response to the queries raised below and return this form along with your corrections Dear Author During the process of typesetting your article, the following queries have arisen.Please check your typeset proof carefully against the queries listed below and mark the necessary changes either directly on the proof/online grid or in the 'Author's response' area provided below Queries Details Required Author's Response AQ1 Reference Garske and Fells (1893) not listed into the reference list.Please add into the list else delete text citations.
The price of a new unit of the capital stock at the beginning of period t, P tÀ1 K , should be equal to the price of a new investment good at the beginning of period t.Note that this beginning of the period price is not necessarily equal to the period t market price of the investment good, P t IP , since CVA t , is defined as Regular Value Added, 14Thus period t CVA t is defined as:

Table 1
on reproducibility.
IP , from VA t and CF t , respectively.Using the 537 above definition of Value Added, definition (1) for period t 538 pure profits Π t of the production unit can be written as 539 follows: