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Nonconvexity in Production and Cost Functions: An Exploratory and Selective Review*

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Handbook of Production Economics

Abstract

The purpose of this contribution is to provide an overview of developments in nonconvex production technologies and economic value functions, with special attention to the cost function. Apart from a somewhat selective review of theoretical issues, the emphasis is on whether the assumption of convexity makes a difference in practice. Anticipating our conclusion, we argue that traditional convex empirical results differ on average rather markedly from alternative nonconvex ones. This should make the discipline reconsider its traditional relationship with convexity in both theoretical and applied production analysis.

We acknowledge the most helpful comments of R. Chambers and G. Cesaroni on an earlier version. The usual disclaimer applies.

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Notes

  1. 1.

    Note that the convex VRS and NDRS technologies do not satisfy inaction.

  2. 2.

    Note that the use of enumeration for the basic nonconvex FDH production model (9) has been around in the literature for quite a while: examples include [49, 63, 122], among others.

  3. 3.

    This poor performance is related to the huge size of the LP formulation in Leleu [85].

  4. 4.

    This point was suggested to the authors by R. Chambers.

  5. 5.

    This decomposition ignores structural efficiency or congestion. Recently, an attempt was made to develop new methods to measure strong forms of hypercongestion for convex and nonconvex technologies alike in Briec et al. [31]. This new methodology is empirically illustrated in Briec et al. [32]. Abad and Briec [1] transpose this methodology toward the modeling of bad outputs using a by-production framework.

  6. 6.

    In case the study does not report cost estimates but rather overall efficiency ratios, one can obtain CC Γ(y, w) = C C,Γ(y, w)∕C NC,Γ(y, w) by taking the ratio of the corresponding overall efficiency ratios OE C(x, y, w)∕OE NC(x, y, w). The observed cost in each of the denominators of OE Λ(x, y, w) cancels out.

  7. 7.

    The Painlevé-Kuratowski lower [upper] limit (sometimes also called Peano limit) of the sequence of sets \(\{E_n\}_{n\in \mathbb {N}}\) is denoted Li n→∞ E n [Ls n→∞ E n]. For a set of points p for which there exists a sequence {p n} of points such that p n ∈ E n for all n and p =limn→∞ p n, a sequence \(\{E_n\}_{n\in \mathbb {N}}\) of subsets of \( {\mathbb {R}}^m\) is said to converge, in the Painlevé-Kuratowski sense, to a set E if Ls n→∞ E n = E = Li n→∞ E n, in which case we write E = Lim n→∞ E n.

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Authors and Affiliations

  1. University of Perpignan, LAMPS, Perpignan, France

    Walter Briec

  2. IESEG School of Management, CNRS, Université de Lille, UMR 9221-LEM, Lille, France

    Kristiaan Kerstens

  3. Research Unit MEES, KU Leuven, Brussel, Belgium

    Ignace Van de Woestyne

Authors
  1. Walter Briec
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  2. Kristiaan Kerstens
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  3. Ignace Van de Woestyne
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Correspondence to Kristiaan Kerstens .

Editor information

Editors and Affiliations

  1. Department of Economics, University of Connecticut, Storrs, CT, USA

    Subhash C. Ray

  2. Department of Agricultural and Resource Economics, University of Maryland, College Park, MD, USA

    Robert G. Chambers

  3. Department of Economics, Binghamton University, Binghamton, NY, USA

    Subal C. Kumbhakar

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Briec, W., Kerstens, K., Van de Woestyne, I. (2022). Nonconvexity in Production and Cost Functions: An Exploratory and Selective Review*. In: Ray, S.C., Chambers, R.G., Kumbhakar, S.C. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3455-8_15

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