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2018 | OriginalPaper | Buchkapitel

Aggregate Productivity and Productivity of the Aggregate: Connecting the Bottom-Up and Top-Down Approaches

verfasst von : Bert M. Balk

Erschienen in: Productivity and Inequality

Verlag: Springer International Publishing

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Abstract

Productivity analysis is carried out at various levels of aggregation. In microdata studies the emphasis is on individual firms (or plants), whereas in sectoral studies it is on (groupings of) industries. An industry is an ensemble of individual firms (decision making units) that may or may not interact with each other. In National Accounts terms this is symbolized by the fact that industry (aggregate) nominal value added is the simple sum of firm-specific nominal value added. From this viewpoint it is natural to expect there to be a relation between industry productivity and the firm-specific productivities. Yet, microdata researchers do not appear to pay much attention to the interpretation of the weighted means of firm-specific productivities they employ in their analyses. In this paper the consequences of this are explored, based on a review of the literature.

However, a structurally similar phenomenon happens in sectoral studies, where the productivity change of industries is compared to each other and to the productivity change of some next-higher aggregate, which is usually the (measurable part of the) economy. Though there must be a relation between sectoral and economy-level measures, in most publications by statistical agencies and academic researchers this aspect is more or less neglected.

The point of departure of this paper is that aggregate productivity should be interpreted as productivity of the aggregate. It is shown that this implies restrictive relations between the productivity measure, the set of weights, and the type of mean employed. For instance, value-added based total factor productivities and output based weights require a harmonic mean, if additivity is assumed.

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Vorheriges Kapitel Frontier Estimation of a Cost Function System Model with Local Least Squares: An Application to Dutch Secondary Education

Nächstes Kapitel Confidence Sets for Inequality Measures: Fieller-Type Methods

This section has been copied from Balk (2016). Though the time dimension does not play an explicit role in the present paper, the notation is retained for consistency.

“Consolidated” means that intra-unit 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 $C_{EMS}^{kkt} = R^{kkt} = 0$.

See Balk (2015, footnote 2) for the treatment of net taxes on intermediates.

On the relation between levels and indices, see Balk (2016, 21–28).

de Loecker and Konings (2006) noted that there is no clear consensus on the appropriate weights (shares) that should be used. In their own work they used employment based shares L ^kt/∑_k L ^kt to weigh value-added based total factor productivity indices $Q_{\mathit {VA}}^k(t,b)/Q_{KL}^k(t,b)$. We will return to this example.

PROD ^t can be considered as a 2-stage aggregation procedure: first PROD ^kt aggregates over basic inputs and outputs per production unit k, and then PROD ^t aggregates over all the units $k \in \mathcal {K}^t$. $\mathit {PROD}^{\mathcal {K}^{t}t}$ can be considered as a 1-stage aggregate of the same basic inputs and outputs. See Diewert (1980, 495–498) for a similar discussion in terms of variable profit (or, value added) functions and technological change (assuming continuous time and differentiability), and the PPI Manual (2004, Chapter 18) for the cases of revenue, intermediate-input-cost, and value-added based price indices. Notice the double role of the variable t in $\mathit {PROD}^{\mathcal {K}^{t}t}$.

Expression (18) is the model underlying GEAD-TFP as implemented by Calver and Murray (2016). Stated in our notation, instead of the right-hand side of expression (18) Basu and Fernald (2002) consider

$$\displaystyle \begin{aligned} \sum_{k \in \mathcal{K}^t}\frac{\mathit{VA}^{kt}}{\mathit{VA}^{\mathcal{K}^{t}t}}\mathit{TFPROD}_{VA}^{k}(t,b); \end{aligned}$$

that is, mean value-added based total factor productivity where the weights are nominal value-added shares. This, then, cannot be interpreted as value-added based total factor productivity of the ensemble, unless special conditions apply.

This paragraph has been inserted at the suggestion of a referee.

Notice that we are considering here additivity of production units, which is different from additivity of commodities as considered in Balk (2016, Section 4.2).

This is the model underlying the CSLS decomposition as implemented by Calver and Murray (2016).

This measure was also considered by Foster et al. (2001). Actually, two variants were considered, one where the labour unit is an hour worked and one where it is a worker. The geometric alternative was employed by Hyytinen and Maliranta (2013) for plants; labour quantity was thereby measured in full time equivalents.

Actually, their multi-factor productivity index, discussed in the extended version of this paper, can be seen as a special case of $\mathit {TFPROD}_{Y}^{k}(t,b)$.

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Titel: Aggregate Productivity and Productivity of the Aggregate: Connecting the Bottom-Up and Top-Down Approaches
verfasst von: Bert M. Balk
Verlag: Springer International Publishing
Buch: Productivity and Inequality
Print ISBN: 978-3-319-68677-6

Electronic ISBN: 978-3-319-68678-3

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-68678-3_5