A novel decomposition of aggregate total factor productivity change
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Abstract
An industry is an ensemble of individual firms (decision making units) which may or may not interact with each other. Similarly, an economy is an ensemble of industries. In National Accounts terms this is symbolized by the fact that the nominal value added produced by an industry or an economy is the simple sum of firm, or industryspecific nominal value added. From this viewpoint it is natural to expect that there is a relation between (aggregate) industry or economy productivity and the (disaggregate) firm or industryspecific productivities. In an earlier paper (Statistica Neerlandica 2015) three timesymmetric decompositions of aggregate valueaddedbased total factor productivity change were developed. In the present paper a fourth decomposition will be developed. A notable difference with the earlier paper is that the development is cast in terms of levels rather than indices. Various aspects of this new decomposition will be discussed and links with decompositions found in the literature unveiled. It turns out that one can dispense with the usual neoclassical assumptions.
Keywords
Productivity Aggregation Decomposition Domar weight Index number theoryJEL codes
C43 D24 O471 Introduction
This introduction^{1} sketches the context. The first article of this series, Balk (2010), considered productivity measurement for a single, consolidated production unit. In terms of levels, productivity is defined as real output divided by real input. Real output or input means nominal output or input deflated by some output or inputspecific price index, respectively. For the production unit considered, productivity change (through time) can then be measured as a difference or a ratio of productivities. In the latter case it appears that productivity change can also be defined directly as output quantity index divided by input quantity index.
The choice of the output and input concepts appears to be critical. Three main models can be distinguished: KLEMSY, KLVA, and KCF. Taking the composition of capital input cost into account, as set out in the companion paper Balk (2011), two more models can be added, namely KLNVA and KNCF. Assuming profit (defined as revenue minus total cost) to be equal to zero, or, what amounts to the same, replacing an exogenous interest rate by an endogenous rate, multiplies the number of models by two. And the introduction of a capital utilization rate further complicates the picture. Thus, there is a lot of choice here, with not unimportant empirical consequences, as illustrated by Vancauteren et al. (2012).
Production units exist at various levels of aggregation. We see plants, enterprises, industries, countries, to name just some types of production units materializing in analyses of productivity change. Usually such units appear, more or less naturally, arranged into higher level aggregates. For instance, a number of plants belonging to the same enterprise; a certain type of enterprises defining an industry; a number of industries defining the ‘measurable’ part of a national economy; national economies making up the world economy. It is not difficult to perceive several sorts of hierarchy here.
As in any of these situations the structure is the same—there is an ensemble of production units, and the ensemble itself may or may not be considered as a higher level production unit –, it is interesting to study the relation between aggregate productivity (change) and productivity (change) of the aggregate.
There are basically two approaches here. Balk (2016) reviews and discusses the socalled bottomup approach, the approach that takes an ensemble of individual production units as the fundamental frame of reference. The topdown approach is the subject of three other papers, namely Balk (2014) plus Dumagan and Balk (2016) on labour productivity, and Balk (2015) on total factor productivity. The connection between the two approaches is considered in Balk (2018a).
The present paper basically continues Balk (2015). In the 2015 paper three (time) symmetric decompositions of aggregate valueadded based total factor productivity change were developed. In the present paper a fourth decomposition will be developed. A notable difference with the earlier paper is that the development is cast in terms of levels rather than indices.
This paper unfolds as follows. Section 2 refreshes the accounting framework; nothing new there. Valueadded based total factor productivity is defined as real value added divided by real primary input; hence, Section 3 defines these two concepts. Section 4 shows that aggregate valueadded based total factor productivity change essentially consists of three components: a weighted mean of individual valueadded based total factor productivity changes, a factor reflecting reallocation between the production units, and a factor reflecting relative price changes at the input and output sides. Section 5 shows how the reallocation factor can be decomposed further into the contributions of the separate primary inputs. Section 6 shows how the decomposition derived in Section 4 changes if valueadded based productivity change is replaced by grossoutput based productivity change. Section 7 contains a key result: under mild restrictions on the relation between aggregate and individual deflators, if profit equals 0 then the reallocation factor vanishes, and aggregate valueadded based total factor productivity change equals the product of Domarweighted individual grossoutput based total factor productivity changes. In Section 8 we take a further step by assuming that the production units share the same timeinvariant production function. We then obtain a decomposition in terms of technical efficiency change, scale and mix effects.
2 Accounting framework
The commodities in the capital class K concern owned tangible and intangible assets, organized according to industry, type, and age class. Each production unit uses certain quantities of those assets, and the configuration of assets used is in general unique for the unit. Thus, again, for any production unit most of the asset cells are empty. Prices are defined as unit user costs and, hence, capital input cost \(C_L^{kt}\) is a sum of prices times quantities.
Finally, the commodities in the labour class L concern detailed types of labour. Though any production unit employs specific persons with certain capabilities, it is usually their hours of work that count. Corresponding prices are hourly wages. Like the capital assets, the persons employed by a certain production unit are unique for that unit. It is presupposed that, wherever necessary, imputations have been made for selfemployed workers. Henceforth, labour input cost \(C_L^{kt}\) is a sum of prices times quantities.
Total primary input cost is the sum of capital and labour input cost, \(C_{KL}^{kt} \equiv C_K^{kt} + C_L^{kt}\). Profit Π^{kt} is the balancing item and thus may be positive, negative, or zero. We are operating here outside the neoclassical framework where profit always equals zero due to the structural and behavioural assumptions involved.
3 Prerequisites
4 Decomposing valueadded based total factor productivity change
Going from (an earlier) period t′ to (a later) period t, individual TFP change is measured by the ratios \(TFPROD_{VA}^k(t,b)/TFPROD_{VA}^k(t{\prime},b)\)\((k \in {\cal{K}})\), and aggregate TFP change by \(TFPROD_{VA}^{\cal{K}}(t,b)/TFPROD_{VA}^{\cal{K}}(t{\prime},b)\). Can the last ratio be written as a function of all the productionunitspecific ratios?^{7} Balk (2015, expressions (20), (28), and (34)) developed three (timeperiod) symmetric decompositions of the aggregate TFP index. We will now show that there is a fourth decomposition.
5 Decomposing the reallocation factor into contributions of separate primary inputs
The reallocation factor ln RAL_{KL}(t, t′), as defined in the previous section, reads in terms of joint primary inputs capital (K) and labour (L). To see the contributions of these two input classes separately one needs some additional prerequisites.
Notice that expression (36) represents a productionunitspecific CobbDouglas aggregator function. This choice is not completely arbitrary, but its defense would require a separate paper. In conventional empirical work the α^{k}’s are estimated and not productionunitspecific.
6 Introducing grossoutput based total factor productivity change
At the righthand side of expressions (28), (29) and (30) we see weighted means of productionunitspecific valueadded based TFP change. As grossoutput (or revenue) stays closer to the actual operations of a production unit, we want to replace valueadded by grossoutput based TFP change.
An alternative decomposition of valueadded based TFP change in terms of grossoutput based TFP change plus some additional factors was obtained by Basu and Fernald (2002). It is possible to mimick their derivation in our setup; however, their avoidance of the Domar factor leads to a final expression which, though containing the same factors as our expression (55) – real primary input change and real intermediate input change—exhibits more complicated weights.

For each production unit, the revenuebased output quantity index is an MV index of the valueadded based output quantity index and the primary input quantity index.

For each production unit, the total input quantity index is an MV index of the primary input quantity index and the intermediate inputs quantity index.
The functional forms of the quantity indices for value added, primary input, and intermediate inputs are left unspecified. However, if these indices were themselves MV indices of the underlying price and quantity data then, due to the consistencyinaggregation of MV indices, both the revenuebased output quantity index and the total input quantity index would be MV indices of the underlying data.
Further, as Diewert (1978) has shown, at any given data point an MV index differentially approximates to the second order any other timesymmetric index, such as Fisher or Törnqvist. Thus, if for revenuebased output quantity and total input quantity instead of MV indices other timesymmetric indices were used, then the equality sign in expression (55) must be replaced by an approximation sign. In the limit, that is, if period t′ approaches period t, then appproximation tends to equality.^{12}
7 The zero profit case
It is important to consider what happens if for all the production units at any time period profit equals zero; that is, Π^{kt} = 0 \((k \in {\cal{K}})\). Such a situation materializes if the unit user cost of all the capital assets is based on endogenous interest rates (which, then, are productionunitspecific), or if actual profit is considered as cost of an additional input called enterpreneurial activity (the price of which, then, is productionunitspecific). Zero profit is easily seen to be equivalent to R^{kt} = C^{kt} or \(VA^{kt} = C_{KL}^{kt}\)\((k \in {\cal{K}})\).
It is useful to summarize our findings in the form of a theorem.
Theorem 1
In official statistical practice the assumptions concerning the use of MV and SV indices are not fulfilled because simpler indices such as Laspeyres or Fisher are used as deflators. Then expression (59) holds only approximately. The better the indices actually used approximate MV and SV indices the better the final approximation will be. As the accuracy of any approximation hinges on the variance, over time and over production units, of the underlying price and quantity data, closeness of the time periods compared and similarity of the production units involved are crucial for obtaining a good approximation.
8 Going beyond total factor productivity change
In the absence of such assumptions, the Solow residual is what it is. In order to make progress we need to decompose the residual into economically meaningful components representing technical efficiency change, technological change, scale effects, and input and output mix effects. For this we need to assume the existence of a timeperiodspecific technology to which the production units belonging to the ensemble \({\cal{K}}\) have access, with features so regular that analytical techniques can be used, and which can be estimated from available data. It is beyond the scope of this article to explore this topic further; the reader is referred to Balk and Zofío (2018).
9 Conclusion
A key element in any system of productivity statistics comprising various levels of aggregation (economy, industry, firm) is a relation connecting a productivity index at a certain level to those at lower levels. In this article such a relation was derived, without invoking any of the usual neoclassical assumptions (a technology exhibiting constant returns to scale, competitive input and output markets, optimizing behaviour of the agents, and perfect foresight), just by mathematically manipulating the various accounting relations. In the process also the famous Domar factor could be demystified to being nothing but a mathematical artefact.
Our key relation links higher level valueadded based productivity growth to a weighted sum of lower level productivity growth, a reallocation factor (reflecting the aggregate effect of lower level dynamics), and a relative price change factor. If zero profit is imposed, then the reallocation factor vanishes, and lower level valueadded based productivity growth can be replaced by Domar weighted grossoutput based productivity growth. Moreover, if the ‘correct’ deflators are used, then the relative price change factor also vanishes.
All this underscores the fact that by and large in empirical work, at various levels of aggregation, reallocation and relative price change tend to play a minor role visavis lower level productivity growth as such.
Footnotes
 1.
Adapted from the corresponding section of Balk (2018a).
 2.
 3.
“Consolidated” means that intraunit deliveries are netted out. At the industry level, in some parts of the literature this is called “sectoral”. At the economy level, “sectoral” output reduces to GDP plus imports, and “sectoral” intermediate input to imports. In terms of variables to be defined below, consolidation means that \(C_{EMS}^{kkt} = R^{kkt} = 0\).
 4.
This is a necessary but innocuous assumption. Only in exceptional cases value added is nonpositive, for instance when the accounting period is so short that revenue and intermediate inputs cost are booked in different periods. Value added is an accounting concept, without normative connotations. After all, value added must be used to pay for capital and labour expenses.
 5.
See Balk (2015, footnote 2) for the treatment of net taxes on intermediates.
 6.
If \({\cal{K}}\) is an economy and \({\Pi}^{{\cal{K}}t} = 0\) then this expression reduces to the familiar identity of gross domestic income and gross domestic product.
 7.
Recall that the logarithm of any such ratio, if in the neighbourhood of 1, can be interpreted as a growth rate.
 8.
The logarithmic mean is, for any two strictly positive real numbers a and b, defined by LM(a, b) ≡ (a − b)/ln(a/b) if a ≠ b and LM(a, a) ≡ a. It has the following properties: (1) min(a, b) ≤ LM(a, b) ≤ max(a, b); (2) LM(a, b) is continuous; (3) LM(λa, λb) = λLM(a, b) (λ > 0); (4) LM(a, b) = LM(b, a); (5) (ab)^{1/2} ≤ LM(a, b) ≤ (a + b)/2; (6) LM(a, 1) is concave. See Balk (2008) for details.
 9.
There is a large literature on the topic of reallocation, but no universal definition of the concept. Though the word ‘reallocation’ seems to have a normative undertone, in the present context it can best be read as ‘dynamics’: the process of (relative) growth and decline of production units.
 10.
The occurrence of such a factor in a decomposition of aggregate productivity change was discussed in Balk (2015, Section 7). The central argument is that “… even if at the level of individual commodities the price is the same for every buyer/seller then the ‘price’ of the composite input and output commodity will vary over the production units.”
 11.
An alternative interpretation in terms of primary inputs moving to production units whose output per unit of primary inputs, \(VA^{kt}/X_{KL}^k(t,b)\), is higher than average, \(VA^{{\cal{K}}t}/X_{KL}^{\cal{K}}(t,b)\), as suggested by Bollard et al. (2013), holds only if \(P_{KL}^k(t,b) = P_{KL}^{\cal{K}}(t,b)\)\((k \in {\cal{K}})\).
 12.
Diewert (2015) replaced the MV indices in the two expressions (47) and (51) by Laspeyres and Paasche indices, which are only firstorder differential approximations, and found that, under the zeroprofit condition discussed below, the ratio of valueadded based and grossoutput based TFP growth rates approximates the asymmetric Domar factors, R^{kt′}/VA^{kt′} and R^{kt}/VA^{kt}, respectively. Two further assumptions, namely that geometric means can be approximated by arithmetic means and that Laspeyres and Paasche revenuebased output quantity indices are equal, made it possible to obtain a similar result in the case of Fisher indices. It is left to the reader to judge whether Diewert’s derivation method is “much simpler” than mine. Using Australian data, Calver (2015) presents evidence on the variability of the Domar factors over industries and through time and on the accuracy of the approximations.
 13.
A consequence is that the covariance of valueadded based TFP growth and some other variable equals the covariance of grossoutput based TFP growth and this variable times the Domar factor. It is good to keep this in mind when meeting such covariances in the literature on firm dynamics.
 14.
An important part of the Petrin and Levinsohn (2012) article was devoted to an empirical comparison of the decomposition in expression (63), minus the relative price change factor, with a concept called ‘BHC productivity change’. However, the two concepts appear to measure different things, which makse a comparison rather meaningless.
Notes
Acknowledgements
The author thanks two referees whose comments, questions, and suggestions have led to several improvements.
Compliance with ethical standards
Conflict of interest
The author declares that he has no conflict of interest.
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