Abstract
Using a standard definition of productivity growth, it is shown that a country may have higher productivity growth than another country in each sector, but may have a lower productivity growth rate overall. Also, it is shown that popular methods for aggregating firm/industry estimates of productivity growth have a serious problem in that productivity of all firms/industries can go up, but aggregate productivity can fall. This is not necessarily due to changes in the reallocation of resources across firms/industries. Hence, there are problems for the interpretation of previously published articles which use these methods. There can be inappropriate assessments of the cyclical properties of productivity, and the productivity impact of industry dynamics, micro-economic reforms and regulatory change. Index-number methods that avoid these aggregation problems are introduced.
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Notes
This result can be interpreted as a failure of “monotonicity in quantity relatives”; see Hill (2004).
In this case we start with the definition of productivity level in each sector as output divided by input. Some simple algebra leads to the conclusion that the natural way to aggregate then is to sum each of the productivity level using real input shares, then to take the ratio of the different period productivity aggregates to get productivity growth. However, the approach of Färe and Zelenyuk (2003) and Zelenyuk (2003) naturally leads to the use of output shares in aggregation.
It is worth laboring this point a little, as there seems to be a somewhat popular view that the use by the U.S. of a chained Fisher index to construct real GDP has solved all manner of aggregation and international comparison issues. Different countries will have different shares, and hence paradoxical results such as those illustrated above are possible. Chaining cannot help. Equation 11 has only two periods, whereas chaining can only be done when there are more than two periods, and always involves the direct comparison of two periods as in Eq. 11. Kohli (2003) highlights some other problems with the measurement of GDP, and international comparisons, that cannot be solved by the use of chaining and the use of the Fisher index formula.
In assessing a country’s growth performance, it is entirely legitimate to use that country’s weights in aggregation. These weights are clearly the most relevant for assessing their performance across time. However, as noted, problems subsequently arise when using the resulting indexes to make comparisons across countries, as countries typically do not have the same shares of their economies in every sector.
Also, it can be shown that invariance to units of measurement of this aggregation method only holds if shares in each sector are constant over time, or if weights sum to one. That is, the use of Domar weights (Domar 1961), which do not sum to one, would make this method sensitive to the units of measurement of inputs and outputs, unless these units are always the same for all goods.
Baily, Bartelsman and Haltiwanger (2001) is somewhat of a rare exception. They use labour productivity as their productivity concept and aggregate using an arithmetic approach with labour shares as weights. It is easy to show that this approach will not lead to monotonicity problems using an arithmetic approach, it is only when the weights are inconsistent with the input aggregate used in calculating productivity, which is the usual case in the literature.
Foster et al. (1998) use total factor productivity as their productivity concept, and aggregate using an arithmetic approach but with employment and gross output weights. Both are inconsistent with the input aggregate used in calculating productivity, and thus can result in monotonicity problems. Disney, Haskel and Heden (2003) also use total factor productivity, but aggregate using a geometric approach with employment and gross output weights. Again, monotonicity problems can arise in this case.
The aggregator function in Eq. 20 is sometimes referred to as a “Divisia” index. It is actually a discrete approximation to the continuous-time Divisia index.
It should be noted that it is possible for the Törnqvist index to fail the monotonicity property (Diewert 1987), but only under extreme conditions compared with the ease with which the indexes considered previously fail this condition. In addition, the Törnqvist index can provide a close (second-order) approximation to the Fisher Ideal index, which does satisfy monotonicity.
This is not a 50% increase (with aggregate output going from 20 to 30), because this assumes a Laspeyres quantity index, with prices normalised to one, as noted in Sect. 2. That is, Eq. 4 is implicitly being used to get the result that there was a 50% increase. The Törnqvist index is much preferable to a Laspeyres index. See e.g., Diewert (1987) and Reinsdorf and Dorfman (1999).
However, it can be shown that the same kind of productivity paradox as observed in Sect. 2 can occur using this method. That is, paradoxical results are possible when using Eq. 20 to perform cross-country (cross-regional) comparisons. This is due to each country using it’s own share weights in calculating it’s productivity growth. As noted in Sect. 2, multilateral index number techniques can be used to avoid such paradoxical results.
This is also true for \(\triangle TFP_G^{0,1}\) in Eq. 12.
This is analogous to reconciling nominal GDP growth with real GDP growth by multiplying the latter by the corresponding price index.
Kohli (2002) has shown how a Fisher Ideal index can be decomposed in a similar fashion to the Törnqvist index. However, as Kohli’s expression for the Fisher Ideal index differs from the Törnqvist index only in the weights that are used in aggregation, the entry and exit of firms causes the same problems.
The terminology “indicator” is used to distinguish this kind of index in differences from the usual ratio type of index number.
In this sense, it is very much like an additive version of the Fisher Ideal index, which is the geometric mean of Laspeyres and Paasche indexes. Alternatively, it can be thought of as an additive version of the Törnqvist index seen in the previous section.
However, it can be shown that the same kind of paradoxical result as in Sect. 2 can occur with this indicator used to aggregate productivity changes.
This is analogous to the relationship between \(\triangle TFP_G^{0,1}\) and \(\triangle TFP_T^{0,1}\) in Eq. 21.
This decomposition can be changed in various minor ways in order to get different interpretations. See, e.g. Balk (2003).
The real output measure of Bartelsman and Gray (1996, p. 13) is an implicit Laspeyres quantity index for most of the sample (and a hybrid quantity index for the rest of the sample), which they get by dividing the value of output by a price index. The description for the price index is given as follows: “For example, for the 1982 through 1987 period, the deflator is the geometric mean of the 4-digit deflator based on the 1982 make table weights and the 4-digit deflator based on the 1987 make table weights. For the years 1972 through 1982, the deflator is a geometric mean based on the 1972 and 1982 weights. Prior to 1972, the deflators are based on 1972 make table weights.” Baily, Bartelsman and Haltiwanger (2001) use this series as their measure of real output.
The use of Fisher index to construct the output aggregate does not solve this problem, as it is the geometric mean of two quantity indexes, as can be seen from Eq. 11. Chaining is not relevant here, because we are only considering two periods.
“TFP1” from the database is used in the following calculations. This is a four-factor definition of TFP (with factors energy+materials, capital, production workers and non-production workers), which is normalized to one in 1987 in the database.
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Acknowledgments
The author gratefully acknowledges the hospitality of the Universitat Autònoma de Barcelona, helpful comments from an anonymous referee, Bert M. Balk, Erwin Diewert, Ulrich Kohli, Nathan McLellan, Valentyn Zelenyuk, and seminar participants at: the Swiss National Bank; The Australian National University; LaTrobe University; Australasian Econometric Society Meeting, Brisbane; Centre for Efficiency and Productivity Analysis, University of New England; the Australasian Macroeconomics Workshop, Wellington; the Econometric Study Group Winter Meeting, Dunedin; and the University of Canterbury; and financial support from the Australian Research Council (DP0559033, LP0884095).
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Fox, K.J. Problems with (dis)aggregating productivity, and another productivity paradox. J Prod Anal 37, 249–259 (2012). https://doi.org/10.1007/s11123-011-0250-2
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DOI: https://doi.org/10.1007/s11123-011-0250-2