Skip to main content
SpringerLink
Log in

Nonparametric measures of returns to scale: an application to German water supply

  • Published:
Empirical Economics Aims and scope Submit manuscript

Abstract

The evaluation of market structures and the quantification of returns to scale in network industries usually are of high interest for researchers and policy makers. Regarding the debate on optimal market structures in German potable water supply, we use a cross-sectional sample of 364 German water utilities observed in 2006 to derive a nonparametric measure of scale elasticity for the water industry. The data sample is validated by applying a super-efficiency approach and a statistical testing procedure for outlier detection. Besides using a standard data envelopment analysis approach, a conditional efficiency approach is applied to account for the water utilities’ operating environments. The results indicate non-decreasing returns to scale for the majority of water utilities and constant or non-increasing returns for larger utilities. Optimal firm size is found to be generally larger than the current sample median firm size. Efficiency improvements could be realized by increases in firm sizes and through a consolidation of the industry.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Sauer (2006) also outlines the potential problems arising from the use of SFA for efficiency analysis and the necessity for checking the theoretical foundation of the estimation results.

  2. Simar and Wilson (2000) provide an overview of some approaches to statistical inference in DEA based on bootstrap methods.

  3. To our knowledge, only the studies by Førsund and Hjalmarsson (2004) on Swedish dairy farms and Erbetta and Rappuoli (2008) on Italian gas distribution companies also focus on the empirical quantification of scale elasticity in DEA.

  4. See De Witte and Marques (2011) for an example.

  5. Since water supply in Germany currently does not face an economic regulation of water tariffs, the Monopolies Commission recommended the introduction of an incentive regulation regime under the control of the Federal Network Agency. They recommended a regulation regime similar to the regulation of electricity and natural gas distribution companies (Monopolkommission 2010). See Hirschhausen et al. (2009) or Zschille and Walter (2012) for more information on German water supply and, e.g., Cullmann (2012) for more information about the incentive regulation regime in German electricity distribution.

  6. See Zschille and Walter (2012) for a more detailed overview over the potable water supply industry in Germany.

  7. In contrast to an output orientation, this is a valid assumption for water supply since the service areas of the water utilities and hence the water output levels can be considered as being fixed.

  8. The convexity assumption can be relaxed by employing the Free Disposal Hull (FDH) estimator that constructs a step-wise boundary for the empirical production set.

  9. Alternative partial frontier approaches like order-m (Cazals et al. 2002) are more robust against outlying or atypical observations as compared to full frontier approaches like DEA. However, for the nonparametric quantification of scale elasticity, it is necessary to rely on the convex DEA frontier. We thus prefer using the DEA approach. An alternative approach to the estimation of returns to scale in non-convex technologies has been proposed by Kerstens and Vanden Eeckaut (1999).

  10. Coelli et al. (2005), Thanassoulis et al. (2008), and De Witte and Marques (2010b) provide overviews of different approaches to outlier detection.

  11. We estimate an Epanechnikov Kernel for this purpose. For bandwidth selection, we follow the suggestion of Bădin et al. (2010) and estimate optimal conditional bandwidths using a least squares cross-validation approach.

  12. See Krivonozhko et al. (2004) or Førsund et al. (2007) for details on the direct approach to the calculation of scale elasticity in DEA.

  13. Following Banker et al. (1984), it is possible to use the peer units as a representation of inefficient DMUs on the frontier. However, as pointed out by Førsund et al. (2007), these peer units are usually corner points of the technology set where shadow prices as well as scale elasticities are not uniquely defined. Radial projections of inefficient units will usually be interior points of the technology set where scale elasticity is uniquely defined.

  14. There have been no major structural or regulatory changes in German water supply since 2006.

  15. On the average of the considered input and output variables, the excluded observations are smaller than the observations contained in the final sample. While noting the potential impact on final results, we have to exclude these observations due to missing information or erroneous data.

  16. We only focus on the potable water services provided by a company. We are, however, aware of potential scope effects in such multi-output companies, e.g., for the labor input due to a shared management overhead. We, however, assume this effect to be small since the variables in the sample explicitly represent the water activities of the companies.

  17. The available data do not allow for the specification of a cost function model or the consideration of further variables characterizing the water utilities’ production processes.

  18. Similarly, Thanassoulis (2000) recommends using a model focusing on the water utilities’ main activities: water quantities, connections, and network length. Since we assume network lengths to be a proxy variable for capital input in our model, the variable is not included as an output measure.

  19. The share of part-time employment in Germany usually is low. We thus assume the impact of part-time employment in our analysis to be low. In the case of multi-product companies providing other services like electricity or natural gas in addition to water supply, we only focus on the number of employees related to the water activities of a company.

  20. The amount of own water production is not included as an output variable since the share of purchased water is restricted to 20 %. With a Pearson correlation coefficient of 0.9984, the variable is furthermore highly correlated with the water deliveries to final customers and would thus not have any additional explanatory power.

  21. The low number of observations of small companies can be explained by the usually weak data availability for such companies since they are often part of a local administration.

  22. We only consider environmental variables that are assumed to be exogenous to the production technology and that can thus not be influenced by the firm.

  23. A similar assumption on the exogeneity of water losses is made by De Witte and Marques (2010a) and Zschille and Walter (2012).

  24. The minimum value of the share of water losses shown in Table 1 indicates that water utilities with very low shares of water losses of below 1 % are included in the sample. While this appears unrealistic from an engineering perspective, we observe a continuum of such water utilities and thus decide not to remove such water utilities from the sample.

  25. The consideration of further variables characterizing, e.g., water quality or topographical and climatic conditions is not possible since no such data are available.

  26. All calculations are conducted using the statistical software R with the additional packages “Benchmarking” version 0.18 by Bogetoft and Otto (2011), “FEAR” version 1.13 by Wilson (2008), and “np” version 0.40-3 by Hayfield and Racine (2008).

  27. Three identical observations are excluded under both outlier detection procedures.

  28. We are aware that the excluded observations might represent extreme best practices relevant for the final results. However, to insure the validity of the results, we decide to exclude these observations. On the average of the input and output variables, the excluded observations are smaller than the sample average.

  29. Corner points are fully radially efficient and slacks are zero.

  30. A differentiation of scale elasticity depending on the assumed orientation thus is not necessary. The upper bounds of the scale elasticity estimates are represented by black triangles, lower bounds by gray triangles. On the y-axis, the plots are truncated at a scale elasticity value of 3.0 for illustrating purposes.

  31. For interior points, input and output projections will usually lie at different parts of the DEA frontier and thus scale elasticity estimates differ, see Podinovski et al. (2009). The scatterplots are truncated at a scale elasticity value of 2.0 for illustratory purposes.

References

  • Abbott M, Cohen B (2009) Productivity and efficiency in the water industry. Util Policy 17(3–4):233–244

    Article  Google Scholar 

  • Andersen P, Petersen NC (1993) A prodcedure for ranking efficient units in data envelopment analysis. Manag Sci 39(10):1261–1264

    Article  Google Scholar 

  • Antonioli B, Filippini M (2001) The use of a variable cost function in the regulation of the Italian water industry. Util Policy 10(3–4):181–187

    Article  Google Scholar 

  • Ashton JK (2000) Cost efficiency in the UK water and sewerage industry. Appl Econ Lett 7(7):455–458

    Article  Google Scholar 

  • Ashton JK (2003) Capital utilisation and scale in the English and Welsh water industry. Serv Ind J 23(5):137–149

    Article  Google Scholar 

  • Ballance T, Saal DS, Reid S (2004) Investigation into evidence for economies of scale in the water and sewerage industry in England and Wales. Stone and Webster Consultants, London

    Google Scholar 

  • Banker RD (1984) Estimating most productive scale size using data envelopment analysis. Eur J Oper Res 17(1):35–44

    Article  Google Scholar 

  • Banker RD, Chang H (2006) The super-efficiency procedure for outlier detection, not for ranking efficient units. Eur J Oper Res 175(2):1311–1320

    Article  Google Scholar 

  • Banker RD, Gifford J (1988) A relative efficiency model for the evaluation of public nurse productivity. Carnegie Mellon University, Mimeo

    Google Scholar 

  • Banker RD, Thrall RM (1992) Estimation of returns to scale using data envelopment analysis. Eur J Oper Res 62(1):74–84

    Article  Google Scholar 

  • Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci 30(9):1078–1092

    Article  Google Scholar 

  • Baranzini A, Faust AK (2010) The cost structure of water utilities in Switzerland. Haute école de gestion de Genève, Centre de Recherche Appliquée en Gestion Cahier HES-SO/HEG-GE/C-10/5/1-CH

  • Bogetoft P, Otto L (2011) Benchmarking with DEA and SFA. R package version 0.18

  • Bottasso A, Conti M (2009) Scale economies, technology and technical change in the water industry: evidence from the English water only sector. Reg Sci Urban Econ 39(2):138–147

    Article  Google Scholar 

  • Bădin L, Daraio C, Simar L (2010) Optimal bandwidth selection for conditional efficiency measures: a data-driven approach. Eur J Oper Res 201(2):633–640

    Article  Google Scholar 

  • Bundesministerium für Wirtschaft und Arbeit (2005) Wasserleitfaden: Leitfaden zur Herausbildung leistungsstarker kommunaler und gemischtwirtschaftlicher Unternehmen der Wasserver- und Abwasserentsorgung. Bundesministerium für Wirtschaft und Arbeit Berlin Dokumentation 547

  • Bundesregierung (2010) Stellungnahme der Bundesregierung zum XVIII. Hauptgutachten der Monopolkommission 2008/2009. Drucksache 17/2600

  • Bundesverband der Energie- und Wasserwirtschaft (2008) 118. Wasserstatistik der Bundesrepublik Deutschland. wvgw Wirtschafts- und Verlagsgesellschaft Gas und Wasser mbH, Bonn

  • Cazals C, Florens JP, Simar L (2002) Nonparametric frontier estimation: a robust approach. J Econom 106(1):1–25

    Article  Google Scholar 

  • Coelli TJ, Walding S (2006) Performance measurement in the Australian water supply industry: a preliminary analysis. In: Coelli TJ, Lawrence D (eds) Performance measurement and regulation of network utilities, 1st edn. Edward Elgar, Cheltenham, pp 29–61

    Google Scholar 

  • Coelli TJ, Rao D, O’Donnell CJ, Battese GE (2005) An introduction to efficiency and productivity analysis, 2nd edn. Springer, New York

    Google Scholar 

  • Cullmann A (2012) Benchmarking and firm heterogeneity: a latent class analysis for German electricity distribution companies. Empir Econ 42(1):147–169

    Article  Google Scholar 

  • Daraio C, Simar L (2005) Introducing environmental variables in nonparametric frontier models: a probabilistic approach. J Prod Anal 24(1):93–121

    Article  Google Scholar 

  • Daraio C, Simar L (2007) Conditional nonparametric frontier models for convex and nonconvex technologies: a unifying approach. J Prod Anal 28(1–2):13–32

    Article  Google Scholar 

  • De Witte K, Dijkgraaf E (2010) Mean and bold? On separating merger economies from structural efficiency gains in the drinking water sector. J Oper Res Soc 61(2):222–234

    Article  Google Scholar 

  • De Witte K, Kortelainen M (2009) Blaming the exogenous environment? Conditional efficiency estimation with continuous and discrete exogenous variables. MPRA Paper 14034, University Library of Munich, Germany, http://ideas.repec.org/p/pra/mprapa/14034.html

  • De Witte K, Marques RC (2010a) Designing performance incentives, an international benchmark study in the water sector. Central Eur J Oper Res 18(2):189–220

    Article  Google Scholar 

  • De Witte K, Marques RC (2010b) Influential observations in frontier models, a robust non-oriented approach to the water sector. Ann Oper Res 181(1):377–392

    Article  Google Scholar 

  • De Witte K, Marques RC (2011) Big and beautiful? On non-parametrically measuring scale economies in non-convex technologies. J Prod Anal 35(3):213–226

    Article  Google Scholar 

  • Erbetta F, Rappuoli L (2008) Optimal scale in the Italian gas distribution industry using data envelopment analysis. Omega 36(2):325–336

    Article  Google Scholar 

  • Fabbri P, Fraquelli G (2000) Costs and structure of technology in the Italian water industry. Empirica 27(1):65–82

    Article  Google Scholar 

  • Färe R, Grosskopf S (1985) A nonparametric cost approach to scale efficiency. Scand J Econ 87(4):594–604

    Article  Google Scholar 

  • Färe R, Grosskopf S, Lovell CAK (1983) The structure of technical efficiency. Scand J Econ 85(2):181–190

    Article  Google Scholar 

  • Farrell M (1957) The measurement of productive efficiency. J R Stat Soc 120(3):253–281

    Google Scholar 

  • Farsi M, Fetz A, Filippini M (2008) Economies of scale and scope in multi-utilities. Energy J 29(4):123–143

    Article  Google Scholar 

  • Filippini M, Hrovatin N, Zorić J (2008) Cost efficiency of Slovenian water distribution utilities: an application of stochastic frontier models. J Prod Anal 29(2):169–182

    Article  Google Scholar 

  • Førsund FR, Hjalmarsson L (1979) Generalised Farrell measures of efficiency: an application to milk processing in Swedish dairy plants. Econ J 89(354):294–315

    Article  Google Scholar 

  • Førsund FR, Hjalmarsson L (2004) Calculating scale elasticity in DEA models. J Oper Res Soc 55(10):1023–1038

    Article  Google Scholar 

  • Førsund FR, Hjalmarsson L, Krivonozhko VE, Utkin OB (2007) Calculation of scale elasticites in DEA models: direct and indicrect approaches. J Prod Anal 28(1–2):45–56

    Article  Google Scholar 

  • Fraquelli G, Moiso V (2005) The management of cost efficiency in the Italian water industry. HERMES research center working paper 8

  • Fraquelli G, Piacenza M, Vannoni D (2004) Scope and scale economies in multi-utilities: evidence from gas, water and electricity combinations. Appl Econ 36(18):2045–2057

    Article  Google Scholar 

  • Frisch R (1965) Theory of production. D. Reidel Publishing, Dordrecht

    Book  Google Scholar 

  • Garcia S, Thomas A (2001) The structure of municipal water supply costs: application to a panel of French local communities. J Prod Anal 16(1):5–29

    Article  Google Scholar 

  • García-Sánchez IM (2006) Efficiency measurement in Spanish local government: the case of municipal water services. Rev Policy Res 23(2):355–372

    Article  Google Scholar 

  • Grosskopf S (1996) Statistical inference and nonparametric efficiency: a selective survey. J Prod Anal 7(2–3):161–176

    Article  Google Scholar 

  • Hanoch G (1970) Homotheticity in joint production. J Econ Theory 2(4):423–426

    Article  Google Scholar 

  • Hayfield T, Racine JS (2008) Nonparametric econometrics: the np package. J Stat Softw 27(5):1–32

    Google Scholar 

  • Hirschhausen Cv, Cullmann A, Walter M, Zschille M (2009) Fallende Preise in der Wasserwirtschaft: Hessen auf dem Vormarsch. DIW Berlin Wochenbericht, Berlin

    Google Scholar 

  • Jensen U (2000) Is it efficient to analyse efficiency rankings? Empir Econ 25(2):189–208

    Article  Google Scholar 

  • Kerstens K, Vanden Eeckaut P (1999) Estimating returns to scale using non-parametric deterministic technologies: a new method based on goodness-of-fit. Eur J Oper Res 113(1):206–214

    Article  Google Scholar 

  • Krivonozhko VE, Utkin OB, Volodin AV, Sablin IA, Patrin M (2004) Constructions of economic functions and calculations of marginal rates in DEA using parametric optimization methods. J Oper Res Soc 55(10):1049–1058

    Article  Google Scholar 

  • Kumbhakar SC, Tsionas EG (2008) Scale and efficiency measurement using a semiparametric stochastic frontier model: evidence from the US commercial banks. Empir Econ 34(3):585–602

    Article  Google Scholar 

  • Marques RC, De Witte K (2011) Is big better? On scale and scope economies in the Portuguese water sector. Econ Model 28(3):1009–1016

    Article  Google Scholar 

  • Martins R, Fortunato A, Coelho F (2006) Cost structure of the Portuguese water industry: a cubic cost function approach. Faculdade de Economia da Universidada de Coimbra, Grupo de Estudos Monetários e Financeiros (GEMF) working paper no. 9

  • Martins R, Coelho F, Fortunato A (2012) Water losses and hydrographical regions influence on the cost structure of the Portuguese water industry. J Prod Anal 38(1):81–94

    Article  Google Scholar 

  • Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, New York

    Google Scholar 

  • Monopolkommission (2010) Achtzehntes Hauptgutachten der Monopolkommission 2008/2009. Nomos Verlagsgesellschaft, Baden-Baden

    Google Scholar 

  • Panzar JC, Willig RD (1977) Economies of scale in multi-output production. Q J Econ 91(3):481–493

    Article  Google Scholar 

  • Pastor JT, Ruiz JL, Sirvent I (1999) A statistical test for detecting influential observations in DEA. Eur J Oper Res 115(3):542–554

    Article  Google Scholar 

  • Piacenza M, Vannoni D (2004) Choosing among alternative cost function specifications: an application to Italian multi-utilities. Econ Lett 82(3):415–422

    Article  Google Scholar 

  • Picazo-Tadeo A, Sáez-Fernández FJ, González-Gómez F (2009) The role of environmental factors in water utilities’ technical efficiency. Empirical evidence from Spanish companies. Appl Econ 41(5):615–628

    Article  Google Scholar 

  • Podinovski VV, Førsund FR, Krivonozhko VE (2009) A simple derivation of scale elasticity in data envelopment analysis. Eur J Oper Res 197(1):149–153

    Article  Google Scholar 

  • Racine JS (1997) Consistent significance testing for nonparametric regression. J Bus Econ Stat 15(3):369–379

    Google Scholar 

  • Saal DS, Parker D (2000) The impact of privatization and regulation on the water and sewerage industry in England and Wales: a translog cost function model. Manag Decis Econ 21(6):253–268

    Article  Google Scholar 

  • Saal DS, Parker D (2004) The comparative impact of privatization and regulation on productivity growth in the English and Welsh water and sewerage industry, 1985–1999. Int J Regul Gov 4(2):139–170

    Google Scholar 

  • Saal DS, Parker D (2006) Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier. In: Coelli TJ, Lawrence D (eds) Performance measurement and regulation of network utilities. Edward Elgar, Cheltenham

    Google Scholar 

  • Saal DS, Parker D, Weyman-Jones T (2007) Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985–2000. J Prod Anal 28(1):127–139

    Article  Google Scholar 

  • Saal DS, Arocena P, Maziotis A (2011) Economies of integration in the English and Welsh water only companies and the assessment of alternative unbundling policies, draft paper. Aston University Birmingham.

  • Saal DS, Arocena P, Maziotis A, Triebs T (2013) Scale and scope economies and the efficient vertical and horizontal configuration of the water industry: a survey of the literature. Rev Netw Econ 2013:1–37

    Google Scholar 

  • Sauer JF (2005a) The economics and efficiency of water supply infrastructure. Logos, Berlin

    Google Scholar 

  • Sauer JF (2005b) Economies of scale and firm size optimum in rural water supply. Water Resour Res 41(W11418):1–13

    Google Scholar 

  • Sauer JF (2006) Economic theory and econometric practice: parametric efficiency analysis. Empir Econ 31(4):1061–1087

    Article  Google Scholar 

  • Simar L, Wilson PW (2000) Statistical inference in nonparametric frontier models: the state of the art. J Prod Anal 13(1):49–78

    Article  Google Scholar 

  • Simar L, Wilson PW (2007) Estimation and inference in two-stage, semi-parametric models of production processes. J Econom 136(1):31–64

    Article  Google Scholar 

  • Statistisches Amt der DDR (1990) Statistisches Jahrbuch der Deutschen Demokratischen Republik, 1st edn. Rudolf Haufe, Berlin

    Google Scholar 

  • Statistisches Bundesamt (2009) Fachserie 19, Reihe 2.1: Öffentliche Wasserversorgung und Abwasserbeseitigung. Statistisches Bundesamt, Wiesbaden

    Google Scholar 

  • Thanassoulis E (2000) The use of data envelopment analysis in the regulation of UK water utilities: water distribution. Eur J Oper Res 126(2):436–453

    Article  Google Scholar 

  • Thanassoulis E, Portela MCS, Despić O (2008) Data enevelopment analysis: the mathematical programming approach to efficiency analysis. In: Fried HO, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency and productivity growth. Oxford University Press, New York

    Google Scholar 

  • Tupper HC, Resende M (2004) Efficiency and regulatory issues in the Brazilian water and sewage sector: an empirical study. Util Policy 12(1):29–40

    Article  Google Scholar 

  • Walter M, Cullmann A, Wand R, Zschille M (2009) Quo vadis efficiency analysis of water distribution? A comparative literature review. Util Policy 17(3–4):225–232

    Article  Google Scholar 

  • Wilson PW (2008) Fear 1.0: A software package for frontier efficiency analysis with R. Socio-Econ Plan Sci 42(4):247–254

    Article  Google Scholar 

  • Zschille M (2012) Consolidating the water industry: an analysis of the potential gains from horizontal integration in a conditional efficiency framework. Centre for Economic Policy Research London Discussion Paper Series 8737.

  • Zschille M, Walter M (2012) The performance of German water utilities: a (semi)-parametric analysis. Appl Econ 44(29):3749–3764

    Article  Google Scholar 

Download references

Acknowledgments

We thank the participants of the conference “4. Hallesches Kolloqium zur Kommunalen Wirtschaft” in November 2011 in Halle (Saale), Germany, and the participants of the conference “Contracts, Procurement, and Public–Private Arrangements” in May 2012 in Paris, France. In particular, we thank David Saal, Christian von Hirschhausen, Astrid Cullmann, and Maria Nieswand for discussions and suggestions. The usual disclaimer applies. This paper is produced as part of the project Growth and Sustainability Policies for Europe (GRASP), a Collaborative Project funded by the European Commission’s Seventh Research Framework Programme, Contract number 244725.

Author information

Authors and Affiliations

  1. DIW Berlin - German Institute for Economic Research, Mohrenstrasse 58, 10117 , Berlin, Germany

    Michael Zschille

Authors
  1. Michael Zschille
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Zschille.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zschille, M. Nonparametric measures of returns to scale: an application to German water supply. Empir Econ 47, 1029–1053 (2014). https://doi.org/10.1007/s00181-013-0775-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00181-013-0775-5

Keywords

JEL Classification

Search

Navigation