International Series in Operations Research & Management Science

Transcription

1 International Series in Operations Research & Management Science Volume 157 Series Editor: Frederick S. Hillier Stanford University, CA, USA Special Editorial Consultant: Camille C. Price Stephen F. Austin, State University, TX, USA For further volumes:

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3 Peter Bogetoft Lars Otto Benchmarking with DEA, SFA, and R

4 Peter Bogetoft Department of Economics Copenhagen Business School CBS Porcelaenshaven 16 A 2000 Frederiksberg Denmark pb.eco@cbs.dk Lars Otto Institute of Food and Resource Economics University of Copenhagen Rolighedsvej Frederiksberg C Denmark lo@foi.dk ISSN ISBN e-isbn DOI / Springer New York Dordrecht Heidelberg London Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To Victoria, Rasmus, Stina, and Nete Mathias and Gerda

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7 Preface Subject This book covers recent advances in efficiency evaluation, most notably the Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) methods. It introduces the underlying theories, shows how to make the relevant calculations and discusses some applications. The aim is to make the reader aware of the pros and cons of the different methods and to train him or her on the proper usage of these methods in both standard and non-standard cases. Several software packages have been developed that can be used to solve some of the most common DEA and SFA models. In this book, however, we rely on R, a free software environment that can be used for optimization, statistical computing, and graphics. This program enables the reader to solve not only standard problems but also many other problem variants. Using R, one can focus on understanding the business case and developing a good model. One is not restricted to predefined models or to the use of a one-size-fits-all approach. There are several R routines that support the use of DEA and SFA models. While writing this book, we have also developed an R-package named Benchmarking that makes applications easy without limiting the variations in the models and calculations that innovative researchers and practitioners seek to use. Audience and style The intended audience includes graduate students, advanced consultants and practitioners with an interest in quantitative performance evaluation. This book uses mathematical formulations of models, assumptions, etc. Unlike original contributions on this subject, however, this book de-emphasizes formal proofs, partially by placing them in appendices or by referring to the original vii

8 viii Preface sources. Moreover, this book emphasizes the use of theories and interpretations of the mathematical formulations. A series of small examples and graphical illustrations will be presented. This text also combines formal models with less formal economic and organizational thinking. Moreover, it discusses numerous applications based on projects on which we have worked. This includes some large projects with significant practical effects: e.g., the design of benchmarking-based regulations for energy companies in different European countries or the development of merger decision support systems for competition authorities. Acknowledgements The book covers material that we have used in several courses. We gratefully acknowledge comments by students at Copenhagen University and Copenhagen Business School, Denmark; Universitat Autònoma de Barcelona, Spain; the University of New England, UNE, Australia; the Helsinki School of Economics and Business Administration, Finland; and the Yale School of Management, USA. In addition, we have benefitted from questions raised by participants in a series of industry courses that we presented in Denmark, Sweden, Germany, the Netherlands and Canada during the last five years. We would also like to acknowledge the influence of a large number of friends, coauthors and colleagues who have helped us to understand the basics of efficiency evaluation. At the risk of leaving out important individuals, we would like to acknowledge the contributions of Professors Per Agrell, Mette Asmild, Rajiv Banker, Robert Chambers, Tim Coelli, William W. Cooper, Philippe Van den Eeckaut, Rolf Färe, Finn Førsund, Shawna Grosskopf, Harold Fried, Emili Griffle-Tatje, Arne Henningsen, Pekka Korhonen, Jens Leth Hougaard, Knox Lovell, Ole Olsen, Niels Chr. Petersen, Leopold Simar, Jørgen Tind, Henry Tulkens, and Paul Wilson. Copenhagen, September 2010 Peter Bogetoft Lars Otto

9 Contents 1 Introduction to Benchmarking Whybenchmark Learning Coordination Motivation Idealevaluations KeyPerformanceIndicatorsandRatios Technology and efficiency Many inputs and outputs From effectiveness to efficiency Frontiermodels A simple taxonomy Prosandcons Software Summary Bibliographicnotes Efficiency Measures Introduction Setting Efficient production Farrell efficiency Non-discretionary inputs and outputs Using Farrell to rank firms FarrellandShepharddistancefunctions Directional efficiencymeasures Efficiencymeasureswithprices Cost and input allocative efficiency Revenue and output allocative efficiency Profitefficiency Dynamic efficiency ix

10 x Contents 2.8 Structural and network efficiency Choice between efficiencymeasures Summary Bibliographicnotes Appendix: More advanced material on efficiencymeasures The rationale of efficiency Axiomatic characterization of efficiencymeasures Production Models and Technology Introduction Setting The technology set Free disposability of input and output Convexity Freedisposalandconvex Scalingandadditivity Alternative descriptions of the technology Summary Bibliographicnotes Appendix: Distance functions and duality Data Envelopment Analysis DEA Introduction Setting Minimalextrapolation DEA technologies DEAprograms Peerunits NumericalexampleinR DEAasactivityanalysis Scale and allocative efficiency Scale efficiencyindea Allocative efficiencyindea Summary Bibliographicnotes Appendix:MoretechnicalmaterialonDEAmodels Why the T./ setswork Linearprogramming DEA cost and production functions Additional Topics in DEA Introduction Super-efficiency Non-discretionaryvariables Directional efficiencymeasures...121

11 Contents xi 5.5 Improving both inputs and outputs Slackconsiderations Measurementunits,valuesandmissingprices Dualprograms Maximinformulations Partialvalueinformation Establishingrelevantvaluerestrictions Applicationsofvaluerestrictions Summary Bibliographicnotes Appendix: Outliers Types of outliers Identifying outliers Datacloudmethod Finding outliers in R Statistical Analysis in DEA Introduction Asymptotictests Testforgroupdifferences Testofmodelassumptions The bootstrap method Confidenceinterval BootstrappinginDEA Naive bootstrap Smoothing Biasandbiascorrection Algorithm to bootstrap DEA Confidenceintervals NumericalexampleinR Interpretation of the bootstrap results One input, one output Two inputs Statistical tests using bootstrapping Summary Bibliographicnotes Appendix:Secondstageanalysis OrdinarylinearregressionsOLS Tobitregression NumericalexampleinR Problemswiththetwo-stepmethod...196

12 xii Contents 7 Stochastic Frontier Analysis SFA Introduction Parametricapproaches Ordinaryregressionmodels Deterministicfrontiermodels Stochasticfrontiermodels Normal and half normal distributions Maximum likelihood estimation Justificationforthemethod Numerical methods The likelihood function Actualestimation Efficiencyvariance ComparingOLSandSFA Firm-specificefficiency Firm-specificefficiencyintheadditivemodel Comparing DEA, SFA, and COLS efficiencies Summary Bibliographicnotes Appendix: Derivation of the log likelihood function Additional Topics in SFA Introduction Stochasticdistancefunctionmodels Estimatinganoutputdistancefunction Functionalforms Approximationoffunctions Homogeneousfunctions Thetranslogdistancefunction Stochasticcostfunction Statisticalinference Varianceofparameters Hypothesis testing using the t-test General likelihood ratio tests Is the variation in efficiency significant? Testforconstantreturnstoscale Rewrite the model: t-test Linear hypothesis Likelihood ratio test Other distributions of technical efficiency Biasedestimates Summary Bibliographicnotes...262

13 Contents xiii 9 MergerAnalysis Introduction Horizontalmergers Integrationgains Disintegrationgains Learning,harmonyandsizeeffects Organizationalrestructuring Rationaleoftheharmonymeasure Decomposition with a cost function ImplementationsinDEAandSFA NumericalexampleinR Mergersinaparametricmodel Technicalcomplication Methodological complication Practical application: Merger control in Dutch hospital industry Practicalapplication:MergersofNorwegianDSOs Controllability, transferability, and ex post efficiency Summary Bibliographicnotes Regulation and Contracting Introduction Classicalregulatorypackages Cost-recoveryregimes Fixedpriceregimes(price-cap,revenuecap,CPI-X) Yardstickregimes Franchiseauctions Applications Practicalapplication:DSOregulationinGermany Towardsamodernbenchmarkbasedregulation Revenuecapformula Benchmarkingrequirements Modeldevelopmentprocess Modelchoice Finalmodel DEAbasedincentiveschemes Interestsanddecisions Super-efficiencyinincentiveschemes Incentiveswithindividualnoise Incentiveswithadverseselection Dynamicincentives Biddingincentives Practicalapplication:DSOregulationinNorway Summary Bibliographicnotes...323

14 xiv Contents A Getting Started with R: A Quick Introduction A.1 Introduction A.2 Getting and installing R A.3 An introductory R session A.3.1 Packages A.3.2 Scripts A.3.3 FilesinR A.4 Changingtheappearanceofgraphs A.5 ReadingdataintoR A.5.1 ReadingdatafromExcel A.6 Benchmarking methods A.7 A firstrscriptforbenchmarking A.8 OtherpackagesforbenchmarkinginR A.9 Bibliographicnotes References Index...347

15 Acronyms and Symbols List of main abbreviations used in the text DEA Data Envelopment Analysis SFA Stochastic Frontier Analysis KPI Key Performance Indicators ADD Additive technology CRS Constant Returns to Scale DRS Decreasing Returns to Scale, formally non-increasing returns to scale (NIRS) FDH Free Disposability Hull FRH Free Replicability Hull IRS Increasing Returns to Scale, formally non-decreasing returns to scale (NDRS) VRS Variable Returns to Scale RTS Returns To Scale Lists of standard notation and symbols used in the mathematical formulations T Technology set K Number of firms, decision making units k Counter for firms x Input, often a vector y Output, often a vector m Number of inputs n Number of outputs i Counter for inputs j Counter for outputs w Input prices, often a vector p Output prices, often a vector E Input efficiency F Output efficiency G Graph efficiency xv

16 xvi Acronyms and Symbols R R C R CC EV exp Set of real numbers Set of non-negative real numbers Set of positive real numbers Expected value Exponential function