## Abstract

In the empirical literature, the analysis of aggregate productivity dynamics using firm-level productivity has mostly been based on changes in the mean of log-productivity. This paper shows that there can be substantial quantitative and qualitative differences in the results relative to when the analysis is based on changes in the mean of productivity, and discusses the circumstances under which such differences are likely to happen. We use firm-level data for Portugal for the period 2006–2015 to illustrate the point. When the mean of productivity is used, we estimate that TFP and labor productivity for the whole economy increased by 17.7 percent and 5.2 percent, respectively, over this period. But, when the mean of log-productivity is used, these two productivity measures decline by 4.3 percent and 1.8 percent, respectively. Similarly disparate results are obtained for productivity decompositions regarding the contributions for productivity growth of surviving, entering and exiting firms.

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## Notes

- 1.
To the best of our knowledge, the single exception is the contribution by Melitz and Polanec (2015) who, besides the usual geometric mean, also conduct the analysis based on the arithmetic mean of productivity - this latter analysis is shown in the appendix and is not the main focus of their paper. In their case, despite some sizable quantitative differences, the results based on the two alternative means deliver qualitatively similar results for aggregate productivity dynamics.

- 2.
In the more general case, Jensen’s inequality states that

*g*(*E*(*X*)) ≧*E*[*g*(*X*)] for any concave function g(.). Here we stick to the special case of g(.)=ln(.), because we are concerned with the relationship between the arithmetic and geometric mean of X. - 3.
Remember that the coefficient of skewness of a random variable X is the third standardized moment, defined as \(SK=\frac{E[{(X-E(X))}^{3}]}{{(E[{(X-E(X))}^{2}])}^{3/2}}\).

- 4.
It is useful to note that the differences in average productivity growth implied by the differences in average productivity based on the arithmetic and the geometric means can be seen as a special case of the problems of using log-linearized models in econometrics. See the Appendix for the econometric details.

- 5.
- 6.
In each year, the firms operating in the economy may be classified into three types: firms that began the activity in that year (entrants or entering firms), firms that ceased activity in that year (exiters or exiting firms) and firms, which are active and survive to the next year (incumbents, survivors or surviving firms).

- 7.
Note that for surviving firms we have:

\({p}_{s,t}-{p}_{s,t-1}=\sum\limits _{i\in S}{\mu }_{i,t}^{s}{p}_{i,t}-\sum\limits_{i\in S}{\mu }_{i,t-1}^{s}{p}_{i,t-1}=\sum\limits_{i\in S}{\mu }_{i,t-1}^{s}{{\Delta }}{p}_{i,t}+\sum\limits_{i\in S}{p}_{i,t-1}{{\Delta }}{\mu }_{i,t}^{s}+\sum\limits_{i\in S}{{\Delta }}{\mu }_{i,t}^{s}{{\Delta }}{p}_{i,t}\)

- 8.
The only contribution addressing the decomposition of an arithmetic mean, that we are aware of, is the one by Melitz and Polanec (2015), but the authors look at \(({P}_{t}-{P}_{t-1})/{\overline{P}}_{t}\) where \({\overline{P}}_{t}=({P}_{t}+{P}_{t-1})/2\). The use of \({\overline{P}}_{t}\) in the denominator, instead of

*P*_{t−1}, changes the interpretation of the different components, as part of the “within”, “between”, “cross” and “entry” effects is introduced in the denominator through*P*_{t}. Thus, we stick to our definition (9) as a way of making the results as much comparable as possible to the ones of Eq. (8). - 9.
The derivation of Eq. (11) uses the fact that \({\sum }_{i\in S}ln({P}_{t-1}){{\Delta }}{\mu }_{i,t}^{s}=0\) and \({\sum }_{i\in S}{\mu }_{i,t}^{s}ln({P}_{t-1})={\sum }_{i\in E}{\mu }_{i,t}^{e}ln({P}_{t-1})={\sum }_{i\in X}{\mu }_{i,t}^{x}ln({P}_{t-1})=ln({P}_{t-1})\).

- 10.
This positive contribution will tend to be larger in the presence of a positive correlation between the weights,

*μ*_{i,t−1}, and the gaps,*α*_{i,t}− Δ*p*_{i,t}, (larger firms displaying larger productivity changes, on average). This component also increases with firm-level productivity changes: the difference,*α*_{i,t}− Δ*p*_{i,t}, which is always positive, increases with*α*_{i,t}. Suppose, for instance, that*α*_{i,t}= 10%. Then*α*_{i,t}− Δ*p*_{i,t}= 0.47 p.p. - 11.
Note also that

*λ*_{i,t−1}can be written as \({\lambda }_{i,t-1}=1+(\frac{{P}_{i,t-1}-{P}_{t-1}}{{P}_{t-1}}-ln(\frac{{P}_{i,t-1}}{{P}_{t-1}}))\)=1+ the difference between the arithmetic and the geometric measures of proportional deviations. - 12.
Because

*λ*_{i,t}is a positive U-shaped convex function, a large positive covariance between size,*μ*_{i,t}, and*λ*_{i,t}will be obtained if size and productivity are either positively or negatively correlated. In turn, a large \({\overline{\lambda }}_{e,t}\) will be obtained if productivity of entering firms is largely above or largely below*P*_{t−1}, the initial average aggregate productivity. - 13.
The production functions are estimated at the industry level using the Levinsohn-Petrin estimator (see Levinsohn and Petrin 2003), to account for the endogeneity of the regressors. Besides the Levinsohn-Petrin estimator, we tried other methods that also account for endogeneity of the regressors as the ones suggested in Wooldridge (2009) and Gandhi et al. (2016). However, both methods turned out to exhibit strong estimation convergence issues.

- 14.
- 15.
The composite input is defined as a geometric mean of inputs using estimated factor elasticities.

- 16.
The log transformation has been suggested in the literature as an alternative to trimming or winsorizing to deal with outliers. By permitting extreme values to be kept in the data, it avoids the uncertainty associated with the choice of the trimming or winsorizing thresholds. The use of the log transformation has also been suggested as way to correct for the skewness of positively skewed distributions (see Osborne 2002; Osborne and Overbay 2004). The log transformation compresses the distribution of the weights around the “average” firm, reducing the importance of the largest firms and increasing the importance of the smallest ones. In an economy characterized by the presence of many small firms and a few very large firms (right skewed distribution), this transformation may prevent aggregate productivity measures from being fully dominated by productivity developments of a small number of big firms. In summary, by using the shares of log composite input as weights, we generate aggregate productivity measures that are robust to measurement errors affecting inputs (employment, capital stock or intermediate inputs) and may be interpreted as yielding the productivity developments of a “typical” firm.

- 17.
According to information from the National Accounts, in 2010, agriculture, manufacturing, construction, utilities and services contribute 2.3, 13.8, 6.2, 3.9 and 73.8 percent for aggregate GDP, respectively. Thus, if anything, our dataset appears to be slightly skewed towards manufacturing and against the service sector. We note, however, that in contrast to the National Accounts, services in our dataset do not include information of the government sector, the financial sector and self-employment.

- 18.
Some examples: for industry “agriculture” we used the CPI item “bread and cereals”; for industry “animal production” we used the CPI item “animal production”; for industry “real estate renting” we used the CPI item “effective rents paid by tenants” and for industry “motor vehicle trade” we used the CPI item “motor vehicles”. In general, using proxies for industry-level price deflators can lead to the overestimation or the underestimation of productivity growth in some industries, and thus to a less precise analysis. However, it is hard to envision a situation in which this type of measurement error could be systematically related with the questions we are tackling in this paper. In particular, one should not expect significant consequences at the industry-level because the use of proxies does not change relative prices of the firms in the industry.

- 19.
The estimates of our productivity measures for the utilities sector are very erratic, displaying huge annual variations that can be as big as +30 percent (for TFP in 2007) or -30 percent (for value-added labour productivity in 2015). Cumulatively, between 2006 and 2015, the numbers also vary across our productivity measures beyond any sensible thresholds (+15 percent for TFP and −82 percent for value-added labour productivity) making it impossible to draw any interesting conclusions. For this reason, in what follows, figures for the “Total economy” include agriculture, manufacturing, services and construction, but exclude utilities (electricity, gas, and water services).

- 20.
Below, we will show that this exception for agriculture stems mainly from a very different contribution of the “within” effect. In contrast to the other sectors, the contribution of the “within” effect in agriculture is larger for the geometric mean, which results from a strong negative correlation between (size-adjusted) productivity levels and productivity changes, i.e., (size-adjusted) low productivity firms display, on average, higher productivity changes. We believe that the larger increase in productivity, recorded by low productivity firms (usually also smaller firms) might be the result of special incentive programs directed to surviving micro-, small- and medium-size firms, implemented during this period.

- 21.
This offsetting effect is consistent with the view that idiosyncratic productivity shocks induce changes in size and that changes in size, in turn, induce productivity changes, given decreasing within-firm returns. The negative cross term is also consistent with the idea that downsizing may be productivity enhancing.

- 22.
Note that the arithmetic and the geometric means deliver similar aggregate dynamics for labor productivity in agriculture, but the contributions of some of the components, like the “within” effect or the “entry” effect are very different. These results show that even similar aggregate productivity growth rates of the two means may hide very different productivity decompositions.

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The analyses, opinions and findings of this paper represent the views of the authors, which are not necessarily those of Banco de Portugal, the Eurosystem, the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

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## Appendix: The econometrics of the arithmetic and the geometric means

### Appendix: The econometrics of the arithmetic and the geometric means

In this Appendix we show that the differences in average productivity growth implied by the differences in average productivity based on the arithmetic and the geometric means can be seen as a special case of the problems of using log-linearized models in econometrics.

Let us assume that we have information on firm-level productivities for T+1 time periods (t = 0, 1, 2, ... T). As shown by Santos Silva and Tenreyro (2006), the arithmetic averages for each time period can be calculated by estimating by pseudo-Poisson maximum likelihood (PPML) the equation:

and the geometric averages can be calculated by estimating by OLS the following equation

where *I*_{j} is an indicator variable that takes the value 1 if *t* = *j* and 0 otherwise (j = 1, 2, .. T).

From these equations, the cumulative change in average aggregate productivity between periods 0 and t based on the arithmetic mean is given by \(exp({\hat{{\gamma }_{t}}}^{PPML})\), while the cumulative change in average aggregate productivity between periods 0 and t based on the geometric mean is given by \(exp({\hat{{\gamma }_{t}}}^{OLS})\).

Importantly, as shown in Santos Silva and Tenreyro (2006), if the error term *ϵ*_{it} is not i.i.d., the estimates of *α* and *γ*_{j} based on OLS and PPML will be different, and those based on OLS will be biased while those based on PPML will not. Because the assumption of i.i.d. error terms will be violated if 2^{nd} – or higher-order moments – of the distribution of log-productivity are not constant over time, it is very likely that the analysis of productivity dynamics based on the arithmetic and the geometric means of productivity will lead to different conclusions.

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Dias, D.A., Marques, C.R. From micro to macro: a note on the analysis of aggregate productivity dynamics using firm-level data.
*J Prod Anal* **56, **1–14 (2021). https://doi.org/10.1007/s11123-021-00601-4

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### JEL

- D24
- E32
- L25
- O47

### Keywords

- Jensen’s inequality
- productivity decomposition
- geometric mean