Cost functions are nonconvex in the outputs when the technology is nonconvex: convexification is not harmless

This contribution focuses on testing the empirical impact of the convexity assumption in estimating costs using nonparametric specifications of technology and cost functions. Apart from reviewing the scant available evidence, the empirical results based on two publicly available data sets reveal the effect of the convexity axiom on cost function estimates: cost estimates based on convex technologies turn out to be on average between 21% and 38% lower than those computed on nonconvex technologies. These differences are statistically significant when comparing kernel densities and can be illustrated using sections of the cost function estimates along some output dimension. Finally, also the characterization of returns to scale and economies of scale using production and cost functions for individual units yields conflicting results for between 19% and 31% of individual observations. The theoretical known potential impact as well as these empirical results should make us reconsider convexity in empirical production analysis: clearly, convexity is not harmless.

[1]  S. Wibe,et al.  Engineering production functions : a survey , 1984 .

[2]  Scott E. Atkinson,et al.  Feasible Estimation of Firm-Specific Allocative Inefficiency through Bayesian Numerical Methods , 2009 .

[3]  Jing Liu,et al.  A survey of scheduling problems with setup times or costs , 2008, Eur. J. Oper. Res..

[4]  Kristiaan Kerstens,et al.  Total factor productivity growth and convergence in the petroleum industry : empirical analysis testing for convexity , 2012 .

[5]  N. M. Hung,et al.  Non-convex aggregate technology and optimal economic growth , 2009 .

[6]  Herbert E. Scarf.  Social choice and public decision making: Testing for optimality in the absence of convexity , 1986 .

[7]  P. W. Wilson,et al.  Statistical Inference in Nonparametric Frontier Models: The State of the Art , 1999 .

[8]  Rahul Savani.  High-Frequency Trading: The Faster, the Better? , 2012, IEEE Intelligent Systems.

[9]  V. V. Podinovski.  Local and global returns to scale in performance measurement , 2004, J. Oper. Res. Soc..

[10]  A. V. Volodin,et al.  Constructions of economic functions and calculations of marginal rates in DEA using parametric optimization methods , 2004, J. Oper. Res. Soc..

[11]  Kristiaan Kerstens,et al.  Estimating returns to scale using non-parametric deterministic technologies: A new method based on goodness-of-fit , 1999, Eur. J. Oper. Res..

[12]  Benjamin E. Hermalin,et al.  A Comparison of Nonparametric Methods to Measure Efficiency in the Savings and Loan Industry , 1994 .

[13]  R. Tiffin,et al.  On the Opportunity Cost of Crop Diversification , 2018 .

[14]  W. Diewert,et al.  Linear Programming Tests of Regularity Conditions for Production Functions , 1983 .

[15]  A. Oude Lansink,et al.  The impact of inefficiency on diversification , 2015 .

[16]  Sidney G. Winter,et al.  Scaling Heuristics Shape Technology! Should Economic Theory Take Notice? , 2008 .

[17]  G. Hasenkamp,et al.  A study of multiple-output production functions: Klein's railroad study revisited , 1976 .

[18]  Approximating Linear Programs with Summary Functions: Pseudodata with an Infinite Sample , 1988 .

[19]  Kurt Jörnsten,et al.  Equilibrium prices supported by dual price functions in markets with non-convexities , 2008, Eur. J. Oper. Res..

[20]  Robert Crouchley,et al.  Stochastic frontier estimation of a CES cost function: the case of higher education in Britain , 2002 .

[21]  R. F. Brokken.  THE CASE OF A QUEER ISOQUANT: INCREASING MARGINAL RATES OF SUBSTITUTION OF GRAIN FOR ROUGHAGE IN CATTLE FINISHING , 1977 .

[22]  Kristiaan Kerstens,et al.  Non-convex Technologies and Cost Functions: Definitions, Duality and Nonparametric Tests of Convexity , 2004 .

[23]  Ury Passy,et al.  Explicit representation of the two-dimensional section of a production possibility set , 1994 .

[24]  Kristiaan Kerstens,et al.  Global and local scale characteristics in convex and nonconvex nonparametric technologies: A first empirical exploration , 2017, Eur. J. Oper. Res..

[25]  J. Cummins,et al.  Comparison of Frontier Efficiency Methods: An Application to the U.S. Life Insurance Industry , 1998 .

[26]  Robin C. Sickles,et al.  Time Series Analysis of Deregulatory Dynamics and Technical Efficiency: The Case of the U.S. Airline Industry , 2000 .

[27]  Subhash C. Ray,et al.  Cost efficiency in the US steel industry: A nonparametric analysis using data envelopment analysis , 1995 .

[28]  D. Prior,et al.  On the determinants of local government performance: A two-stage nonparametric approach , 2007 .

[29]  Andrea Resti.  Evaluating the cost-efficiency of the Italian Banking System: What can be learned from the joint application of parametric and non-parametric techniques , 1997 .

[30]  S. Afriat.  Efficiency Estimation of Production Function , 1972 .

[31]  R. Shepherd.  Theory of cost and production functions , 1970 .

[32]  Giovanni Cesaroni,et al.  Average-cost efficiency and optimal scale sizes in non-parametric analysis , 2015, Eur. J. Oper. Res..

[33]  A. Ichino,et al.  Time Allocation and Task Juggling , 2014 .

[34]  Ilya Segal,et al.  Solutions manual for Microeconomic theory : Mas-Colell, Whinston and Green , 1997 .

[35]  Rolf Färe,et al.  Fundamentals of production theory , 1988 .

[36]  Kristiaan Kerstens,et al.  Incentive Regulation and the Role of Convexity in Benchmarking Electricity Distribution: Economists Versus Engineers , 2008 .

[37]  P. Romer.  Are Nonconvexities Important for Understanding Growth? , 1990 .

[38]  Yanqin Fan,et al.  On goodness-of-fit tests for weakly dependent processes using kernel method , 1999 .

[39]  Emili Tortosa‐Ausina,et al.  Evaluating local government performance in times of crisis , 2018, Local Government Studies.

[40]  M. Ivaldi,et al.  Comparing Fourier and Translog Specifications of Multiproduct Technology: Evidence from an Incomplete Panel of French Farmers , 1996 .

[41]  Ole Bent Olesen,et al.  Stochastic Data Envelopment Analysis - A review , 2016, Eur. J. Oper. Res..

[42]  Subhash C. Ray.  Data Envelopment Analysis by Subhash C. Ray , 2004 .

[43]  Herbert E. Scarf,et al.  The Allocation of Resources in the Presence of Indivisibilities , 1994 .

[44]  Subhash C. Ray.  Weak Axiom of Cost Dominance: A Nonparametric Test of Cost Efficiency Without Input Quantity Data , 1997 .

[45]  Qi Li.  Nonparametric testing of closeness between two unknown distribution functions , 1996 .

[46]  Kristiaan Kerstens,et al.  Estimating scale economies in non-convex production models , 2017, J. Oper. Res. Soc..

[47]  Victor V. Podinovski,et al.  Efficiency and Global Scale Characteristics on the “No Free Lunch” Assumption Only , 2004 .

[48]  K. Kerstens,et al.  Cost efficiency of Belgian local governments: A comparative analysis of FDH, DEA, and econometric approaches , 1996 .

[49]  G. Hall,et al.  The Response of Prices, Sales, and Output to Temporary Changes in Demand , 2005 .

[50]  Jean-Paul Chavas,et al.  A Generalized Distance Function and the Analysis of Production Efficiency , 1999 .

[51]  J. Tind,et al.  Convex Input and Output Projections of Nonconvex Production Possibility Sets , 2000 .

[52]  Eric Budish,et al.  Implementation Details for Frequent Batch Auctions: Slowing Down Markets to the Blink of an Eye † , 2014 .

[53]  Sung-ko Li.  A nonparametric test of quasiconcave production function with variable returns to scale , 2019, Journal of Mathematical Economics.

[54]  Daniel McFadden,et al.  A Survey of Functional Forms in the Economic Analysis of Production , 1978 .

[55]  Emili Tortosa‐Ausina,et al.  Which estimator to measure local governments’ cost efficiency? The case of Spanish municipalities , 2019, SERIEs.

[56]  Subhash C. Ray,et al.  Data Envelopment Analysis: Theory and Techniques for Economics and Operations Research , 2004 .

[57]  Johannes Sauer.  Economic Theory and Econometric Practice: Parametric Efficiency Analysis , 2005 .

[58]  Christian A. Ruzzier,et al.  The Case for International Coordination of Electricity Regulation: Evidence from the Measurement of Efficiency in South America , 2002 .

[59]  William A. Barnett,et al.  Fellow’s Opinion Article: Tastes and Technology: Curvature is not Sufficient for Regularity , 2001 .

[60]  Kristiaan Kerstens,et al.  Convex and nonconvex input-oriented technical and economic capacity measures: An empirical comparison , 2019, Eur. J. Oper. Res..

[61]  Statistical Cost Analysis Re-Revisited , 1981 .

[62]  H. Varian.  The Nonparametric Approach to Production Analysis , 1984 .

[63]  Herbert E. Scarf.  Testing for optimality in the absence of convexity , 2008 .

[64]  Lawrence R. Klein,et al.  Some Theoretical Issues in the Measurement of Capacity , 1960 .

[65]  S. Kumbhakar,et al.  When, where and how to perform efficiency estimation , 2011, SSRN Electronic Journal.

[66]  M. J. Farrell,et al.  The Convexity Assumption in the Theory of Competitive Markets , 1959, Journal of Political Economy.

[67]  Steven T. Hackman,et al.  Production Economics: Integrating the Microeconomic and Engineering Perspectives , 2007 .