Plant capacity notions: review, new definitions, and existence results at firm and industry levels

Abstract

This study investigates the existence of solutions for the key plant capacity utilisation (PCU) concepts using general nonparametric technologies. This is done via a theoretical review of existing and some new PCU concepts. Focusing on short-run and long-run output-oriented, attainable output-oriented, and input-oriented PCU notions, we first investigate the existence of solutions at the firm level. Under mild axioms, this question regarding the existence of solutions for these PCU concepts at the firm level is affirmatively answered under variable and constant returns to scale as well as under convex and nonconvex assumptions. However, short-run and long-run output-oriented and attainable output-oriented PCU concepts may not be implementable depending on certain conditions. There are no such reservations for the input-oriented PCU. Then, for this same range of PCU concepts, we explore the more difficult question as to the existence of solutions at the industry level. The output-oriented and attainable output-oriented PCU exist at the industry level under strict conditions: existence and attainability are interwoven at this level. The industry input-oriented PCU is always feasible at the industry model. This theoretical review is supplemented by a semi-systematic empirical review, and an empirical application. We conclude that input-oriented PCU is clearly the best concept.

Keywords:

• Data envelopment analysis
• nonparametric technology
• capacity utilisation

JEL codes:

• D24

View correction statement:

Correction

Acknowledgments

We thank Alecos Papadopoulos for a constructive remark that led to a new definition. We thank four most constructive referees for their helpful comments. The usual disclaimer applies.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data Availability Statement

The data of Fan, Li, and Weersink (1996) have been downloaded at: DOI: 10.1080/07350015.1996.10524675. This data set is made available as supplementarymaterial.

Correction Statement

This article was originally published with errors, which have now been corrected in the online version. Please see Correction (http://doi.org/10.1080/00207543.2023.2250204)

Notes

1 Throughout this contribution, ${\mathbb{R}}^d$ denotes the d-dimensional Euclidean space, and ${\mathbb{R}}_{+}^d$ denotes its non-negative orthant; lowercase boldface letters are used to denote vectors; all vectors are considered to be column vectors and vectors $\mathbf{0}$ denotes vector of zeroes; and for vectors $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^d$, the inequality $\mathbf{a}\ge \mathbf{b}$ ($\mathbf{a}>\mathbf{b}$) means that $a_i\ge b_i$ ($a_i>b_i$), for all $i=1,\dots ,d$.

2 Note that the convex VRS technology does not satisfy inaction.

3 The industry O-oriented PCU $IPC{U}_o^{SR}({\mathbf{x}}_p,{\mathbf{x}}_p^f,{\mathbf{y}}_p\mid \mathrm{\Lambda },\mathrm{\Gamma })$ is not formally defined because from a mathematical viewpoint it is not always well-defined. Therefore, we refrain from providing a formal definition. We only use the concept industry O-oriented PCU in an informal way, denoted by the symbol $IPC{U}_o^{SR}({\mathbf{x}}_p,{\mathbf{x}}_p^f,{\mathbf{y}}_p\mid \mathrm{\Lambda },\mathrm{\Gamma })$.

4 Scarf (1994, 114–115) mocks the possibility of a CRS technology: ‘Both linear programming and the Walrasian model of equilibrium make the fundamental assumption that the production possibility set displays constant or decreasing returns to scale; that there are no economies associated with production at a high scale. I find this an absurd assumption, contradicted by the most casual of observations. Taken literally, the assumption of constant returns to scale in production implies that if technical knowledge were universally available we could all trade only in factors of production, and assemble in our own backyards all of the manufactured goods whose services we would like to consume.’

5 The following empirical comparative studies of our own are not included in Table 3 because these are methodological in nature. CKVDW17 offer a numerical example of SR VRS I-oriented PCU notion and discuss an empirical illustration of SR VRS O- and I-oriented PCU concepts under C and NC. KSVDW19b report an empirical analysis of the SR VRS O- and attainable O-oriented PCU concepts under C and NC. CKVDW19 provide a detailed numerical example as well as an empirical illustration of SR and LR VRS O- and I-oriented PCU notions under C.

6 This boils down to a semi-systematic approach to a literature review (see Snyder 2019), which is sufficient for our purpose.

Additional information

Notes on contributors

Kristiaan Kerstens

Kristiaan Kerstens holds a Ph.D. from KUBrussel (nowadays KULeuven) and is a research professor with CNRS-LEM and a professor of economics with IÉSEG School of Management in Lille, France. His main research focus is on developing nonparametric methodology to analyse microeconomic production and portfolio behaviour. His work covers productivity indices and indicators, efficiency measurement (with a particular interest in nonconvexities), capacity utilisation, and multimomentportfolio optimisation.

Jafar Sadeghi

Jafar Sadeghi holds a PhD in applied mathematics with a specialisation in operations research from Kharazmi University in Tehran, Iran. Currently, he serves as a postdoctoral associate at the Ivey Business School, Western University, Canada. His research focuses on the fields of optimisation, efficiency, and productivity measurement, along with capacity utilisation.