## Abstract

The study applies the probabilistic framework of nonparametric frontier estimation to model the effect of competitive conditions on sectors’ production efficiency levels. We utilize conditional order-m robust frontiers to model the dynamic effects of competition on a sample of U.S. manufacturing sectors over the period 1958–2009. Contrary to the existing studies, we apply for the first time in the Industrial Organization literature the latest advances of robust nonparametric frontier analysis to disentagle the dynamic effects alongside the effects of competition on sectors’ productive efficiency levels. The results derived from the time-dependent robust conditional estimators unveil a non-linear relationship between product market competition and productive efficiency. Our findings suggest that for higher competition levels the effect is positive up to a certain threshold after which the effect becomes negative.

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## Notes

- 1.
For interesting studies using the probabilistic framework of efficiency measurement see the studies by Roudaut and Vanhems (2012), Balaguer-Coll et al. (2013), De Witte and Kortelainen (2013), Tzeremes (2015), Zschille (2015), Cordero et al. (2017), Huiban et al. (2018) and Mastromarco et al. (2019).

- 2.
The European Commission is unlikely to identify horizontal competition concerns in a market with a post-merger HHI below 1 000. Moreover, the Commission is also unlikely to identify horizontal competition concerns in a merger with a post-merger HHI between 1000 and 2000 and a delta below 250, or a merger with a post-merger HHI above 2 000 and a delta below 150.

- 3.
Simar and Wilson (2011) and Daraio et al., (2018) have asserted that many DEA based studies have performed a second stage regression type analysis assuming in an ad hoc manner that the “

*separability*” condition holds. However, such a strong and restrictive assumption can lead to miss estimation of how truly the environmental factors might affect the production process. - 4.
Equations (1) to (7) referred to the probabilistic theoretical framework which is the basis both for the full and partial frontiers.

- 5.
- 6.
In our analysis we are using the mean values of order-m estimators in order to derive to the diachronic representation at the tree digit level. However, for the DEA estimators the relative literature (Färe and Zelenyuk, 2003; 2005; 2007; Zelenyuk, 2006) has developed several aggregated techniques that can be applied.

- 7.
The m value represents the number of sectors under which the evaluated sector is benchmarked. As a result the estimated robust efficiency measure the expected value of the maximum of the output of m sectors drawn randomly from the populations of sectors using less inputs compared to the evaluated sector. According to Mastromarco and Simar (2018) when we set

*m*= 1 then estimated order-m frontier represents sectors’ average production function. - 8.
We have noticed that for m values greater than 50 the order-m efficiency scores convergence to the full frontiers estimate. For a direct comparison of conditional efficiencies under the full and partial frontiers see Appendix.

- 9.
Highly competitive conditions are indicated by values less than 100. Competitive conditions are indicated by values greater than 1500; Moderate competitive conditions are indicated by values between 1500 and 2500 and low competition is indicated for values greater than 2500 (DOJ, 2010).

- 10.
This can be attributed to the fact that all capital-intensive sectors require high levels of starting cost. As a consequence, the number of new entrants and subsequently the level of potential competition is relatively less compared to any labor intensive sector.

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## Acknowledgements

We are grateful to Victor Podinovski (Editor) for giving us the opportunity to revise our work, the associate editor and the three anonymous reviewers of this journal for their fruitful comments and suggestions made that enhanced the merit of this study. All the remaining errors remain with the authors. The usual disclaimer applies.

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## Appendix

### Appendix

*A1: Computational aspects among full and partial frontiers*

Under the probabilistic framework (Daraio and Simar 2005, 2007a, 2007b), the effect of the competitive conditions *W* and time *T*, can be represented as:

Then we can estimate sectors’ time-dependent conditional efficiency level applying the following LP of the DEA estimator under the variable returns to scale (VRS) assumption as:

Where ℑ(*w*,*t*) = {*j* = (*i*, *v*)|*w* − *h*_{w} < *w*_{i,v} < *w* + *h*_{w}; *t* − *h*_{t} < *v* < *t**+**h*_{t}}. Based on Mastromarco and Simar (2015, p.829) we can estimate the conditional distribution \(\widehat \Phi _{\left. Y \right|X,W}^t\left( {\left. y \right|x,w} \right)\) as:

It must be noted that *h* represents the smoothing and K(·) is a kernel with compact support (for Matlab codes see Bădin et al. 2010, 2019). The DEA estimator presented in Eq. (17) has been applied by Polemis and Tzeremes (2019) assuming that the technology is convex (concave). When assuming non-convex (non-concave) technology the time-dependent conditional order-m estimator can be computed as:

Daraio and Simar (2007b) provide a Monte Carlo algorithm to solve the conditional order-m estimators, however, for large *m* values the estimator can be calculated via the numerical integration (for computational issues also see the Matlab codes provided by Daraio et al. 2020).

*A2: Comparison of the results among the full and partial frontiers*

Figure 7 presents the density plots of the conditional estimators under which imply convexity (concavity) assumption and those which imply the non-convexity (non-concavity) assumption. Specifically, as in the study by Polemis and Tzeremes (2019) the left panel represent the \(\widehat {\it{\varrho }}_t\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)\) estimator (red line) whereas, the black line represents the estimator used in this study \(\widehat \vartheta _{t,m}\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)\). Finally, the right panel represents the Q-ratios under the full frontiers \(Q_{vrs} = \frac{{\widehat {\it{\varrho }}_t\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)}}{{\widehat {\it{\varrho }}\left( {x,y} \right)}}\) and under the partial frontiers \(Q_m = \frac{{\widehat \vartheta _{t,m}\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)}}{{\widehat \vartheta \left( {{\mathrm{x}},{\mathrm{y}}} \right)}}\).

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Polemis, M.L., Stengos, T. & Tzeremes, N.G. Modeling the effect of competition on US manufacturing sectors’ efficiency: an order-m frontier analysis.
*J Prod Anal* (2020). https://doi.org/10.1007/s11123-020-00583-9

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### Keywords

- Conditional efficiency
- Probabilistic frontier analysis
- Order-m estimators
- Product Market Competition
- Manufacturing

### JEL codes

- L60
- C14
- O14