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

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Abstract

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.

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Notes

  1. The same reasoning applies to the revenue function and all variations of the profit function, except of course the long run profit function where the effect of convexity or not cannot logically be distinguished.

  2. Budish et al. (2014) claim that the current defects in market microstructure are linked to a continuous-time market design based on a continuous limit order book. Instead, they propose a discrete-time frequent batch auction.

  3. Bogetoft et al. (2000) is the first article maintaining convexity on input and output space, but relaxing convexity in the graph of technology without restrictions on the dimensionality of input and output spaces. Li (2019) adds variable returns to scale to this quasiconcavity of production, but only manages to treat the single output case. This framework is justified by appeal to, for instance, the law of diminishing marginal rates of substitution in inputs space, or to the idea of diminishing marginal rates of transformation in outputs space. But, one may question whether there really is a law of diminishing marginal rates of substitution in inputs space. For example, Brokken (1977) summarises three studies revealing that there are increasing marginal rates of substitution of grain for roughage in beef production: thus, establishing nonconvexities in input space.

  4. Just as a nonconvex or a convex technology yields the same profit function. This implies that convexity cannot logically be tested using the profit function (even though any variation on the long-run profit function would allow testing for convexity).

  5. If one is willing to accept CRS, then the plausibility of additivity and divisibility separately is at stake. First, perfect divisibility of inputs and/or outputs is probably the most debatable assumption. Many if not most operations management problems in industry and distribution involve some form of indivisibilities and input fixities resulting in complex integer and possibly nonlinear optimization problems. In general, all production processes seem to have some lower limit below which a process cannot possibly be scaled down realistically. Therefore, divisibility is highly questionable (see Scarf (1994) or Winter (2008) for more detailed criticisms). Second, while additivity is essential to define free entry and may seem plausible at first, it is not beyond criticism since it presupposes spatial separation and noninteraction (see Winter (2008)). Since additivity relates to the aggregation of results of activities occurring in geographically distinct places, transportation costs must be small in order to be safely ignored. When activities are close for transportation costs to be negligible, then the risk of production externalities looms when activities get “too close” to create interactions. Furthermore, location is important in the definition of quite some outputs (e.g., Italian and Californian lemons are considered different).

  6. The Hasenkamp (1976) results are at least partially attributable to the strong assumptions on input/output separability, among others.

  7. While some studies offer explicit comparisons, some results in Table 1 are based on own computations. In particular, some studies report cost efficiency ratios based on cost frontier estimates under convexity and nonconvexity. Taking a ratio of these efficiency ratios nets out the observed cost and reveals the difference in cost estimates under convexity and nonconvexity.

  8. These authors also report results for six size classes each time for solvent and insolvent firms separately: the difference varies between a minimum of 9.38% and a maximum of 29.63%.

  9. Nonconvex results are presented graphically. Our numbers are based on information provided by the author.

  10. For the dynamic profit frontiers we report the difference between convex and nonconvex results relative to nonconvex results. We thank F. Ang for providing these results in personal communication.

  11. Nonparametric deterministic frontier estimators that minimally extrapolate the data subject to some maintained axioms are labeled Data Envelopment Analysis (DEA) models in contrast to different (e.g., parametric) methodologies. The same moniker is also used to distinguish these convex models from the nonconvex ones in (5) that are sometimes labeled Free Disposal Hull (FDH) technologies. This use of monikers to denote both general methodologies and specific models is confusing.

  12. Substituting \(t_k = \delta z_k\) in (5), one rewrites the sum constraint on the activity vector. Realizing that the constraints on the scaling factor are integrated into the latter sum constraint, the LP appears.

  13. Badunenko et al. (2012) show that the reliability of efficiency scores of nonparametric, deterministic convex frontier estimation remains excellent when the ratio of the variation in efficiency to the variation in noise is low. Assuming this result would also be confirmed for nonconvex similar estimators and would also hold for the unexplored case of the cost function, this reinforces our argument to opt for a simple deterministic nonparametric frontier framework to test for convexity.

  14. To enhance comparability we use a common Sheather and Jones plug-in bandwidth for the convex and nonconvex data series to be compared.

  15. Indeed, in the single output case it may be intuitively clear that convex and nonconvex cost functions trace the same function at the extremes of the empirical range, but that in between the approximation depends on the precise structure of the sample. This intuition for the single output case is developed in the figures in Appendix C in Electronic Supplementary material by computing the models in Sect. 3.1 on a numerical example with two inputs and a single output. However, the situation is less clear in the multi-output case: convex and nonconvex cost functions need not have very much in common except at some of the extremes of the empirical range (see Appendix D in Electronic Supplementary material for figures computed on a numerical example with one input and two outputs).

  16. As pointed out by a referee, there is a multitude of approaches to compute the difference between two functions and thus to assess global fit.

  17. As pointed out by the same referee, to the best of our knowledge it remains an open question to which extent this better fit of the nonconvex cost function may lead to overfitting when predicting.

  18. Results in, e.g., Ray and Kim (1995) learn that accounting for nonregressive technical change increases technical efficiency, thereby improving the empirical fit. However, their panel data set covers a long time period, while ours have a short time dimension.

  19. This proposal related to the engineering economics literature is not without criticism (see Maddala and Roberts (1981)). More recently, Preckel and Hertel (1988) propose a linear program summary functions method (LPSF) that is equivalent to the pseudodata approach with an infinite sample.

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Authors and Affiliations

  1. IESEG School of Management, CNRS, Univ. Lille, UMR 9221-LEM, 3 rue de la Digue, F-59000, Lille, France

    Kristiaan Kerstens

  2. Research Unit MEES, KU Leuven, Warmoesberg 26, B-1000, Brussels, Belgium

    Ignace Van de Woestyne

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  1. Kristiaan Kerstens
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Correspondence to Kristiaan Kerstens.

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We acknowledge the most helpful comments of G. Cesaroni, P. Kerstens, F. Maier-Rigaud and M. Vardanyan on earlier versions. We thank conference and seminar participants in Auckland, Berlin, Bordeaux, Edinburgh, Harbin, Helsinki, Houston, Karaj, Palmerston North, Rennes, Shanghai, Taipei and Tianjin for useful comments on earlier drafts. We are most grateful to two referees of this journal. The usual disclaimer applies. Note that this contribution has circulated under different titles.

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Kerstens, K., Van de Woestyne, I. Cost functions are nonconvex in the outputs when the technology is nonconvex: convexification is not harmless. Ann Oper Res 305, 81–106 (2021). https://doi.org/10.1007/s10479-021-04069-1

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