1 Introduction

Measuring the efficiency or productivity of entities, e.g., companies, hospitals, banks, departments, is an important task for academics, practitioners, and policy-makers. Indeed, a researcher’s interest often lies in the performance of a group, e.g., industry or a country, which in turn raises the question of a coherent aggregation of individual performance measures (scores) into a group performance measure (score). While it may appear simple, resolving such an aggregation question can be challenging because the individual units can be quite heterogeneous in many respects: in terms of the volumes or sizes of their outputs or inputs, or the mixes of outputs or inputs, or possibly in terms of the technologies deployed to produce those outputs from various inputs, etc. As a result, different proposals on how the aggregation can be implemented have been proposed in the literature.Footnote 1

One of the first proposals goes back to Farrell (1957), whose approach has gained a wide-spread popularity, both in theory and in the empirical work.Footnote 2 In particular, in his seminal work, Farrell (1957) introduced several concepts of productive efficiency. He showed how the cost efficiency can be decomposed into technical efficiency (input-oriented) and allocative (or price) efficiency theoretically and how to estimate them in practice via linear programs, the approach that later was coined as Data Envelopment Analysis (Charnes et al. (1978)). While focusing primarily on the individual efficiency, Farrell (1957) also proposed an aggregate efficiency measure—a weighted arithmetic average, where his (input-oriented) technical efficiency scores were weighted with output shares (in a single-output framework). This simple idea sparked a stream of research on aggregation in efficiency analysis, e.g., including Førsund & Hjalmarsson (1979), Li & Ng (1995),Blackorby and Russell (1999), Briec et al. (2003), Färe & Zelenyuk (2003), Färe et al. (2004), Färe & Zelenyuk (2005), Bogetoft and Wang (2005), Zelenyuk (2006), Mussard & Peypoch (2006), Cooper et al. (2007), Färe & Zelenyuk (2007), Li & Cheng (2007), Nesterenko & Zelenyuk (2007), Simar & Zelenyuk (2007), Färe et al. (2008), Pachkova (2009), Kuosmanen et al. (2010), Raa (2011), Mayer & Zelenyuk (2014, 2019), Karagiannis (2015), Karagiannis & Lovell (2015), Cherchye et al. (2016) Walheer (2018, 2019), Briec et al. (2021), to mention just a few.

In a nutshell, the goal of this paper is to further refine, develop and generalize this aggregation literature. Specifically, we develop new results that help to unify the two sub-streams—the aggregation of Debreu-Farrell-type efficiencies and the aggregation of directional distance functions—that so far seems to be more different than they actually are.

To be more specific, recall that the aggregation theorem from Koopmans (1957) can be used to obtain a closed-form solution for aggregation of the Debreu-Farrell-type efficiency measures, with derived weights for aggregating the individual measures from the economic micro-foundations (Färe & Zelenyuk, (2003)). The derived weights are based on the revenue shares for the output-oriented measure and on the cost shares for the input-oriented measure, which are generalizations of the ideas from Farrell (1957). Most of the works on aggregation since then focused on the Debreu-Farrell-type efficiency measures, which consider either input or output orientation.

On the other hand, the Debreu-Farrell-type measures were also shown to be special cases of the directional distance functions, introduced by Chambers et al. (1996, 1998), who elaborated on ideas from Luenberger (1992). Being more general, the directional distance function is more flexible, as it may simultaneously handle input and output directions and embeds Debreu-Farrell-type measures as special cases. Chambers et al. (1998) also introduced a new measure of profit efficiency and named it after Mark Nerlove. The resulting measure, Nerlovian profit indicator (NPI), compares the maximum attainable profit given the technology and input/output prices and the observed actual profit of a unit, and the direction of measurement chosen by the researcher.Footnote 3 Also utilizing Koopmans aggregation theorem, NPIs and directional distance functions can be coherently aggregated under certain conditions, most important of which is the requirement of the same direction of the aggregate measure and all the individual measures (Briec et al. (2003), Färe et al. (2008)). Furthermore, their aggregation does not involve aggregation weights, which is in contrast with the aggregation of the Debreu-Farrell-type measures.

Hence, the aggregation structure for the directional distance functions looks very different from the aggregation structure for the Debreu-Farrell-type measures, although the latter are special cases of the former and both approaches use some versions of the Koopmans (1957) aggregation theorem. This was quite puzzling and in this work we try to resolve this puzzle.

More specifically, the contribution of this paper is to develop new results that generalize the aggregation theory for efficiency measures based on directional distance functions, where we allow for choosing different directions for any individual as well as aggregate measures. Moreover, the new aggregation of the individual NPIs and directional distance functions now can involve different (and derived) weights, bringing this aggregation in line with the Debreu-Farrell-type measure aggregation. At the same time, it also embraces the previous results from Färe et al. (2008) as a special case in this generalized theory.

The rest of the paper is organized as follows. In the next section, we introduce notation and discuss existing results. In Sect. 3 we present the new theory. Section 4 discusses various special cases of selected directions in the NPIs. Section 5 discusses practical aspects of estimation via Data Envelopment Analysis. Section 6 concludes.

2 Preliminaries

Let \(x\in \mathbb {R}_{+}^{N}\) be a column vector of N inputs used by an agent or the so-called decision making unit, hereafter DMU (a person, a firm, a department, etc.) to produce \(y\in \mathbb {R}_{+}^{M}\), a column vector of M outputs. Let the actual realizations of the generic input–output allocation (x, y) for firm k be denoted by \((x^{k},y^{k})\). In the sub-sections below we outline the necessary definitions and existing results that we will need as building blocks for our further developments.

2.1 Individual characterizations

Suppose that all technologically feasible input–output combinations that can be achieved by a DMU k can be represented by a technology set, defined in generic terms asFootnote 4

$$\begin{aligned} \Psi ^{k}=\{(x,y):y\text { can be produced from }x\text { with technology available to DMU }k\}. \end{aligned}$$
(1)

We assume this set obeys the standard regularity conditions of production theory in economics and operations research.Footnote 5

Various functions can be utilized to characterize \(\Psi ^{k}\), one of the most general of which is the so-called directional distance function (DDF) \(D^{k}:\mathbb {R}_{+}^{N}\times \mathbb {R}_{+}^{M}\rightarrow \mathbb {R}\cup \{\infty \}\), defined conditionally on the directional vector \((-d_{x}^{k},d_{y}^{k})\in \) \(\mathbb {R}_{-}^{N}\times \mathbb {R}_{+}^{M}\backslash \{0\}\) and technology \(\Psi ^{k}\) defined asFootnote 6

$$\begin{aligned} D^{k}(x,y|d_{x}^{k},d_{y}^{k},\Psi ^{k})=\sup _{\beta }\{\beta \in \mathbb {R}:\,\,(x-\beta d_{x}^{k},y+\beta d_{y}^{k})\in \Psi ^{k}\}. \end{aligned}$$
(2)

Under the standard regularity conditions of production theory (see footnote 5), this function provides a complete functional characterization of a technology set \(\Psi ^{k}\) in the sense that \(\forall (x,y)\in \mathbb {R}_{+}^{N}\times \mathbb {R}_{+}^{M}\), we have

$$\begin{aligned} D^{k}(x,y|d_{x}^{k},d_{y}^{k},\Psi ^{k})\ge 0\iff (x,y)\in \Psi ^{k}. \end{aligned}$$
(3)

A myriad of specific choices about the directional vector can be made, resulting in different efficiency measures. In particular, note that the direction may be a function of (x, y). We also have a superscript k in the directional vector to emphasize that, at least in principle, it can be different for different DMUs, which is important for the generalization of the existing aggregation theory for DDFs. In special cases, a researcher may choose directions that do not depend on k or on x or on y or on any of them and we will dedicate some attention to such special cases separately (see Sect. 4).

Also note that \(D^{k}(x,y|d_{x},d_{y},\Psi ^{k})=0\) whenever \((x,y)\in \partial _{d}\Psi ^{k}\), where \(\partial _{d}\Psi ^{k}\) is the frontier of \(\Psi ^{k}\) with respect to the direction d (which may also depend on k), defined as

$$\begin{aligned} \partial _{d}\Psi ^{k}&=\{(x,y)\ :\ (x,y)\in \Psi ^{k},\ \nonumber \\&\ \ \ \ \ \ (x',y')\equiv (x,y)+\beta (-d_{x},d_{y})\notin \Psi ^{k},\forall \beta >0\}. \end{aligned}$$
(4)

The DDF is also known to be dual to the neoclassical profit function, defined as

$$\begin{aligned} \Pi ^{k}(p,w|\Psi ^{k})=\max _{x,y}\{py-wx:\ (x,y)\in \Psi ^{k}\}, \end{aligned}$$
(5)

where \(p\in \mathbb {R}_{+}^{M}\) is a row vector of output prices corresponding to y and \(w\in \mathbb {R}_{+}^{N}\) is a row vector of input prices corresponding to x.

Different profit efficiency measures can be defined based on (5).Footnote 7 A special case of interest here is the so-called Nerlovian profit indicator (NPI), given by

$$\begin{aligned} E^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k}):=\frac{\Pi ^{k}(p,w| \Psi ^{k})-(py-wx)}{pd_{y}^{k}+wd_{x}^{k}} \end{aligned}$$
(6)

assuming \(pd_{y}^{k}+wd_{x}^{k}\ne 0\). In the next section, we will often use the fact from duality theory stating that we have

$$\begin{aligned} E^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\ge D^{k}(x,y|d_{x}^{k},d_{y}^{k},\Psi ^{k}) \end{aligned}$$
(7)

for all \((-d_{x}^{k},d_{y}^{k})\in \mathbb {R}_{-}^{N} \times \mathbb {R}_{+}^{M}\backslash \{\text {0}_{N+M}\}\), all \((x,y)\in \mathbb {R}_{+}^{N+M}\) and all \((w,p)\in \mathbb {R}_{++}^{N+M}\).

Intuitively, this measure compares the maximal profit a DMU can get to the profit it actually obtained, and normalizes it by the factor (\(pd_{y}^{k}+wd_{x}^{k}\)) that accounts for the direction of measurement. For example, if \(d_{y}^{k}=y^{k}\) and \(d_{x}^{k}=0\), then the normalization factor is total revenue of DMU k and so the measure gives the percentage of potential (or unrealized) profit relative to total revenue. On the other hand, if \(d_{y}^{k}=0\) and \(d_{x}^{k}=x^{k}\), then the normalization factor is total cost of DMU k, hence the measure gives the percentage of potential (or unrealized) profit relative to this total cost.Footnote 8

The right-hand side of (7) represents the technical inefficiency, while the left-hand side represents the profit inefficiency, and so the gap between these two types of inefficiency represents an allocative inefficiency. This gap is usually closed by introducing a (directional) allocative efficiency measure, which we denote here as \(A^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\), defined as the difference between (6) and (2), thus leading to the following additive decomposition

$$\begin{aligned} E^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})=D^{k}(x,y|d_{x}^{k}, d_{y}^{k},\Psi ^{k})+A^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k}) \end{aligned}$$
(8)

for all \((-d_{x}^{k},d_{y}^{k})\in \mathbb {R}_{-}^{N}\times \mathbb {R}_{+}^{M} \backslash \{\text {0}_{N+M}\}\), all \((x,y)\in \mathbb {R}_{+}^{N+M}\) and all \((w,p)\in \mathbb {R}_{++}^{N+M}\). Intuitively, this allocative efficiency measure quantifies the gap between an allocation on the technology frontier and the best possible allocation in the sense of profit maximization (i.e., optimizing all inputs and all outputs given the input and output prices), and also accounting for the direction of measurement.

Once obtained (e.g., via Data Envelopment Analysis or other methods), a question arises on how to appropriately aggregate the individual efficiency estimates so that the resulting aggregate score is representing efficiency of a group well. We focus on this important question next.

2.2 Aggregate characterizations

2.2.1 Key definitions

Let us consider a group of n DMUs, indexed by \(k=1,...,n\). This can be an industry consisting of n firms, or a group of n employees, or a union of n countries or states. To make the notation more concise, let \(\tilde{x}=\sum _{k=1}^{n}x^{k}\) denote the vector of aggregate inputs of this group and \(\tilde{y}=\sum _{k=1}^{n}y^{k}\) denote the vector of aggregate outputs of this group. Also, denote \(\overline{x}=\tilde{x}/n\) and \(\overline{y}=\tilde{y}/n\). Following Koopmans (1957), we assume that the aggregate technology of a group of n DMUs is represented by the Minkowski summation of the individual technology sets, i.e.,

$$\begin{aligned} \tilde{\Psi }_{n}=\sum _{k=1}^{n}\Psi ^{k}. \end{aligned}$$
(9)

The properties of such technology set depend on the properties possessed by (e.g., due to the regularity conditions imposed on) each of the n individual technology sets, hence we keep the subscript n to indicate this dependency.Footnote 9

A complete functional characterization of the aggregate technology set \(\tilde{\Psi }_{n}\) can be given via the aggregate analogue of (2). Specifically, the aggregate directional distance function, \(D:\mathbb {R}_{+}^{N}\times \mathbb {R}_{+}^{M}\rightarrow \mathbb {R}\cup \{\infty \}\), is defined conditionally on the directional vector \((-g_{x},g_{y})\in \) \(\mathbb {R}_{-}^{N}\times \mathbb {R}_{+}^{M}\backslash \{0\}\) and the aggregate technology \(\tilde{\Psi }_{n}\) as

$$\begin{aligned} D(x,y|g_{x},g_{y},\tilde{\Psi }_{n})=\sup _{\beta }\{\beta \in \mathbb {R}:\ (x-\beta g_{x},y+\beta g_{y})\in \tilde{\Psi }_{n}\}. \end{aligned}$$
(10)

Note that here we denoted the directional vector with \((-g_{x},g_{y})\) deliberately, to emphasize that, at least in principle, the direction for the aggregate DDF may be different than the directions for the individual DDFs (which, recall, we denoted with \((-d_{x}^{k},d_{y}^{k})\), potentially specific for each individual k). This distinction is indeed important for generalizing the existing aggregation theory, which so far (to the best of our knowledge) has been developed only for a fixed direction common to all DMUs.

Given the aggregate DDF, the aggregate profit function for the group is then defined analogously to (6), i.e.,

$$\begin{aligned} \Pi (p,w|\tilde{\Psi }_{n})=\max _{x,y}\{py-wx:(x,y)\in \tilde{\Psi }_{n}\}, \end{aligned}$$
(11)

and the aggregate analogue of the NPI for the group is then defined as

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|g_{x},g_{y},\tilde{\Psi }_{n})=\frac{\Pi (p,w| \tilde{\Psi }_{n})-(p\tilde{y}-w\tilde{x})}{pg_{y}+wg_{x}}, \end{aligned}$$
(12)

assuming \(pg_{y}+wg_{x}\ne 0\).

In the next section we will relate these aggregate efficiency measures to the individual analogues.

2.2.2 Existing results

From the Koopmans (1957) theorem, we have

$$\begin{aligned} \Pi (p,w|\tilde{\Psi }_{n})=\sum _{k=1}^{n}\Pi ^{k}(p,w|\Psi ^{k}) \end{aligned}$$
(13)

for all \((w,p)\in \mathbb {R}_{++}^{N+M}\) and therefore we also have the following aggregation result for the NPIs (Färe et al. (2008)):

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|g_{x},g_{y},\tilde{\Psi }_{n})=\sum _{k=1}^{n} E^{k}(x^{k},y^{k},p,w|g_{x},g_{y},\Psi ^{k}). \end{aligned}$$
(14)

Importantly, note that this result requires that all firms face the same prices, which are the same as the prices used in the aggregate profit function. Furthermore, it also requires the directional vector for all the individual profit efficiency measures to be the same as for the aggregate profit efficiency measure, denoted here with (\(g_{x},g_{y})\). This direction may depend on x and y (as the notation emphasizes), yet in special cases it may also be chosen as fixed.

Moreover, due to the duality theory, now with respect to the aggregate technology (9), for all \((-g_{x},g_{y})\in \mathbb {R}_{-}^{N}\times \mathbb {R}_{+}^{M} \backslash \{\text {0}_{N+M}\}\), all \((x,y)\in \mathbb {R}_{+}^{N+M}\) and all \((w,p)\in \mathbb {R}_{++}^{N+M}\), we have

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|g_{x},g_{y},\tilde{\Psi }_{n})\ge D(\tilde{x},\tilde{y}|g_{x},g_{y},\tilde{\Psi }_{n}). \end{aligned}$$
(15)

Similarly as in the disaggregate case, one can close the inequality gap (15) by introducing a residual representing aggregate allocative efficiency, leading to the following decomposition:

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|g_{x},g_{y},\tilde{\Psi }_{n})=D(\tilde{x}, \tilde{y}|g_{x},g_{y},\tilde{\Psi }_{n})+A(\tilde{x}, \tilde{y},p,w|g_{x},g_{y},\tilde{\Psi }_{n}). \end{aligned}$$
(16)

Now, combining (16) with (14), we obtain

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|g_{x},g_{y},\tilde{\Psi }_{n})&= \sum _{k=1}^{n}D^{k}(x^{k},y^{k}|g_{x},g_{y},\Psi ^{k})\nonumber \\&+\sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|g_{x},g_{y},\Psi ^{k}), \end{aligned}$$
(17)

and therefore concluding (due to Briec et al. (2003), Färe et al. (2008) and with credits to insights from Jesus Pastor) that

$$\begin{aligned} D(\tilde{x},\tilde{y}|g_{x},g_{y},\tilde{\Psi }_{n})=\sum _{k=1}^{n} D^{k}(x^{k},y^{k}|g_{x},g_{y},\Psi ^{k}) \end{aligned}$$
(18)

if and only if

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|g_{x},g_{y},\tilde{\Psi }_{n})=\sum _{k=1}^{n} A^{k}(x^{k},y^{k},p,w|g_{x},g_{y},\Psi ^{k}). \end{aligned}$$
(19)

A few remarks are in order here before we proceed with the new developments. First, and foremost, note that this aggregation result requires the same direction for the aggregate DDF and for all the individual DDFs. This excludes such popular directions as \((-x^{k},y^{k})\) and infinitely many other possibilities.Footnote 10

Second, note that there are no aggregation weights in this aggregation system (16)–(19). This is different from earlier aggregation results for Debreu-Farrell-type efficiencies.Footnote 11 In the next sections we show that this result is a special case of a more general theory that we develop in this paper.

Third, equality (18) implies that the DDF must have an affine functional form. This latter point was clarified by Färe et al. (2008) via the Pexider functional equations argument and, in some sense, should be interpreted as that very ‘grain of salt’ that researchers must take when adopting the aggregation system (16)–(19). Due to this reasoning, (18) can also be understood as the first order approximation to the aggregation result for the DDFs, which holds exactly (due to Koopmans theorem (13)) for the NPIs (14) and also for the DDF in the special case when allocative inefficiency is absent.

In general, Färe et al. (2008) also showed that

$$\begin{aligned} D(\tilde{x},\tilde{y}|g_{x},g_{y},\tilde{\Psi }_{n})\ge \sum _{k=1}^{n} D^{k}(x^{k},y^{k}|g_{x},g_{y},\Psi ^{k}) \end{aligned}$$
(20)

and

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|g_{x},g_{y},\tilde{\Psi }_{n})\le \sum _{k=1}^{n}A^{k} (x^{k},y^{k},p,w|g_{x},g_{y},\Psi ^{k}), \end{aligned}$$
(21)

i.e., the sum of individual DDFs over a group of DMUs gives a lower bound for the aggregate DDF for this group, while the sum of individual directional allocative efficiency scores over a group of DMUs gives an upper bound for the directional aggregate allocative efficiency score, when all are conditioned by the same directional vector.

An intuition behind this result can be summarized as follows: The aggregate technology presumes possibility of reallocation of output plans and input endowments across the DMUs in the group and, therefore, the simple sum of the individual technical inefficiencies of a group must be no larger than the technical inefficiency defined on the aggregate technology. Indeed, all DMUs can be efficient individually, i.e., when considered separately, yet when some reallocation among firms is allowed then there might be potential for producing greater total output from the same total inputs, with the same individual technologies being aggregated. In turn, this result (20), together with (14), (15) and (16), implies the second part, i.e., about the allocative inefficiency, stated in (21). Importantly, note that this aggregation does not involve weights. This will change when we allow for a general direction, embracing the current aggregation result a special case.

3 New developments

Utilizing the concepts, definitions and notations outlined above, we are now ready to present the new results. In this section, for the sake of generality of the developed theory, we first present the results for the general aggregate direction and then (in Sect. 4) we consider a few popular special cases.Footnote 12 Intuitively, the aggregate direction improvement focuses on the efficiency improvement potential of a group of DMUs as a whole, rather than examining individual DMUs in isolation. It considers the overall aggregate direction in which the group of DMUs can collectively move to achieve higher efficiency levels.

To make notation more concise, let \(\tilde{d}_{x}=\sum _{k=1}^{n}d_{x}^{k}\) and \(\tilde{d}_{y}=\sum _{k=1}^{n}d_{y}^{k}\) and substitute these for \(g_{x}\) and \(g_{y}\), respectively, in (12) and (14), then we get

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n})&=\frac{\Pi (p,w|\tilde{\Psi }_{n})-(p\tilde{y}-w\tilde{x})}{p\tilde{d}_{y}+w\tilde{d}_{x}}\nonumber \\&=\sum _{k=1}^{n}\frac{\Pi ^{k}(p,w|\Psi ^{k})-(py^{k}-wx^{k})}{pd_{y}^{k}+wd_{x}^{k}}\times S_{d}^{k}\nonumber \\&=\sum _{k=1}^{n}E^{k}(x^{k},y^{k},p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}, \end{aligned}$$
(22)

where

$$\begin{aligned} S_{d}^{k}=\frac{pd_{y}^{k}+wd_{x}^{k}}{p\tilde{d}_{y}+w\tilde{d}_{x}}, \end{aligned}$$
(23)

is the share-weight for DMU k. Intuitively, this share weight can be understood as a formal reflection of the importance of the DMU k relative to other DMUs. These weights take into account the direction of efficiency measurement, where the relative importance of each input and output is set through the elements of the directional vector, which is then evaluated at the prices of the corresponding inputs and outputs.Footnote 13 As will be seen more clearly in the next section, the direction is playing a critical role in such weighting, attaching different economic meanings to the weights for different directions.

Moreover, after the substitution of (8), we get

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n})&=\sum _{k=1}^{n}(D^{k}(x,y|d_{x}^{k},d_{y}^{k},\Psi ^{k})\nonumber \\&\ \ \ \ \ \ +A^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k}))\times S_{d}^{k}\nonumber \\&=\sum _{k=1}^{n}D^{k}(x,y|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}\nonumber \\&\ \ \ \ \ \ +\sum _{k=1}^{n}A^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}. \end{aligned}$$
(24)

Moreover, recall again that due to duality theory, for all \((-\tilde{d}_{x},\tilde{d}_{y})\in \mathbb {R}_{-}^{N}\times \mathbb {R}_{+}^{M}\backslash \{\text {0}_{N+M}\}\), all \((x,y)\in \mathbb {R}_{+}^{N+M}\) and all \((w,p)\in \mathbb {R}_{++}^{N+M}\), we have

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n})\ge D(\tilde{x},\tilde{y}|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n}) \end{aligned}$$
(25)

and, similarly as in the disaggregate case, one can close the gap by introducing a residual representing an aggregate allocative efficiency measure, leading to the following decomposition:

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n})&=D(\tilde{x},\tilde{y}|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n})\nonumber \\&\ \ \ +A(\tilde{x},\tilde{y},p,w|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n}). \end{aligned}$$
(26)

Therefore, using similar logic as for the additive case, we have

$$\begin{aligned} D(\tilde{x},\tilde{y}|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n})=\sum _{k=1}^{n} D^{k}(x^{k},y^{k}|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k} \end{aligned}$$

if and only if

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|\tilde{d}_{x},\tilde{d}_{y}, \tilde{\Psi }_{n})=\sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}. \end{aligned}$$

A few remarks are in order here as well. First, and foremost, note that the aggregation of individual NPIs now involves weights. This is different from previous aggregation results in the literature about the NPIs and is somewhat similar to the literature on the aggregation of Debreu-Farrell-type efficiencies, Malmquist, and Hicks-Moorsteen productivity indexes (e.g., Färe & Zelenyuk, (2003), Mayer & Zelenyuk (2014, 2019), etc.).

Second, the weights here depend on the direction, and this appears to be completely novel to the literature, yet quite coherent with how one may expect given that the aggregation is for the objects that may depend (both quantitatively and qualitatively) on the direction. Recall that, intuitively, the directional vector plays the ‘normative’ role, in a sense being the ‘normalizer’ that reflects (and accounts for) the subjective choice of the researcher regarding what is more/less important in terms of improving, with certain relative proportions postulated through the elements of this vector. Hence, it is quite natural that such a ‘norming’ vector also feeds into the aggregation weights, as is seen in the formulas above, as also in formulas that follow.

Third, and another very important difference with previous literature, is that the individual NPEs are allowed to have their own directions, \((-d_{x}^{k},d_{y}^{k})\), which may vary widely across \(k=1,...,n\), yet then altogether they define the aggregate direction \((-g_{x},g_{y})=(-\sum _{k=1}^{n}d_{x}^{k},\sum _{k=1}^{n} d_{y}^{k})=(-\tilde{d}_{x},\tilde{d}_{y})\). Furthermore, the directions can also be fixed and common to all DMUs, as a special case, which we will consider in a subsequent section.

These three features make this new aggregation scheme quite different and much more general relative to the aggregation scheme for DDFs derived earlier in the literature. Indeed, here we encompass the latter as a special case of the new more general aggregation scheme.

It is also worth noting that, in contrast to the aggregation from Färe et al. (2008), it is possible to have either

$$\begin{aligned} D(\tilde{x},\tilde{y}|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n}) \ge \sum _{k=1}^{n}D^{k}(x^{k},y^{k}|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k} \end{aligned}$$

or

$$\begin{aligned} D(\tilde{x},\tilde{y}|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n}) \le \sum _{k=1}^{n}D^{k}(x^{k},y^{k}|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k} \end{aligned}$$

and therefore also have, respectively, either

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n})\le \sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k} \end{aligned}$$

or

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|\tilde{d}_{x},\tilde{d}_{y}, \tilde{\Psi }_{n})\ge \sum _{k=1}^{n}A^{k}(x^{k},y^{k}, p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}. \end{aligned}$$

That is, the weighted sum of individual DDFs over a group of DMUs (given by \(\sum _{k=1}^{n}D^{k}(x^{k},y^{k}|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}\)) may be above or below the aggregate DDF for this group (given by \(D(\tilde{x},\tilde{y}|\tilde{d}_{x},\tilde{d}_{y},\tilde{\Psi }_{n})\)). In turn, this will be balanced (due to (22)–(23)) by the opposite happening to the allocative efficiency: the weighted sum of individual directional allocative efficiency scores over the same group being, respectively, below or above the aggregate directional allocative efficiency score for this group. This is different from what we know from previous results (described in Sect. 2), yet similar to the one observed in the aggregation theory for Debreu-Farrell-type efficiency scores. Clearly, the reason for the difference is the weights, which may value some DDFs higher or lower than others, depending on the prices and the direction that influence the weights.

In the next section, we show how this new aggregation result relates and generalizes other aggregation results in the literature, including the one we briefed on in Sect. 2.

4 A few cases of special interest

4.1 Average direction

A useful property of the directional distance function that we will involve is the homogeneity of the direction, namely, for all \(\delta >0,(x,y)\in \mathbb {R}_{+}^{N+M}\) and all relevant directions, we haveFootnote 14

$$\begin{aligned} D^{k}(x,y|d_{x}^{k}/\delta ,d_{y}^{k}/\delta ,\Psi ^{k})/\delta =D^{k} (x,y|d_{x}^{k},d_{y}^{k},\Psi ^{k}) \end{aligned}$$
(27)

as well as

$$\begin{aligned} E^{k}(x,y,p,w|d_{x}^{k}/\delta ,d_{y}^{k}/\delta ,\Psi ^{k})/\delta =E^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k}) \end{aligned}$$
(28)

and

$$\begin{aligned} A^{k}(x,y,p,w|d_{x}^{k}/\delta ,d_{y}^{k}/\delta ,\Psi ^{k})/ \delta =A^{k}(x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k}). \end{aligned}$$
(29)

These properties are useful for deriving the results for the average direction, as we will consider next.

4.1.1 General returns to scale

Consider the average direction \((-\bar{d}_{x},\bar{d}_{y})\), where \(\bar{d}_{x}:=n^{-1}\sum _{k=1}^{n}d_{x}^{k}\) and \(\bar{d}_{y}:=n^{-1}\sum _{k=1}^{n}d_{y}^{k}\), and let us use the homogeneity properties (27), (28) and (29), letting \(\delta =n\), we can get from (16)

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})= D(\tilde{x},\tilde{y}|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})+A(\tilde{x}, \tilde{y},p,w|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n}), \end{aligned}$$
(30)

as well as from (22)

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})=n \sum _{k=1}^{n}E^{k}(x^{k},y^{k},p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k}) \times S_{d}^{k}, \end{aligned}$$
(31)

and so

$$\begin{aligned} D(\tilde{x},\tilde{y}|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})=n \sum _{k=1}^{n}D^{k}(x^{k},y^{k}|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k} \end{aligned}$$
(32)

if and only if

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})=n \sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}. \end{aligned}$$
(33)

4.1.2 Constant returns to scale

In the case of constant returns to scale (CRS) technology, formally defined as

$$\begin{aligned} \delta \Psi ^{k}= & {} \Psi ^{k},\ \forall \delta >0, \end{aligned}$$
(34)

we have another useful homogeneity property, namely

$$\begin{aligned} D^{k}(\delta x,\delta y|d_{x}^{k},d_{y}^{k},\Psi ^{k})/\delta =D^{k} (x,y|d_{x}^{k},d_{y}^{k},\Psi ^{k}),\ \forall \delta >0. \end{aligned}$$
(35)

Similarly, we have

$$\begin{aligned} E^{k}(\delta x,\delta y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})/\delta =E^{k} (x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k}),\ \forall \delta >0, \end{aligned}$$
(36)

and therefore

$$\begin{aligned} A^{k}(\delta x,\delta y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})/\delta =A^{k} (x,y,p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k}),\ \forall \delta >0. \end{aligned}$$
(37)

In its turn, this means that under CRS, we also have

$$\begin{aligned} E(\bar{x},\bar{y},p,w|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})=D(\bar{x}, \bar{y}|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})+A(\bar{x},\bar{y},p,w| \bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n}). \end{aligned}$$
(38)

Moreover, from (31), (32) and (33) we have

$$\begin{aligned} E(\bar{x},\bar{y},p,w|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})= \sum _{k=1}^{n}E^{k}(x^{k},y^{k},p,w|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}. \end{aligned}$$
(39)

And so, finally, we have a neat aggregation result for the directional distance functions under the CRS:

$$\begin{aligned} D(\bar{x},\bar{y}|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})= \sum _{k=1}^{n}D^{k}(x^{k},y^{k}|d_{x}^{k},d_{y}^{k},\Psi ^{k})\times S_{d}^{k}, \end{aligned}$$
(40)

which holds if and only if

$$\begin{aligned} A(\bar{x},\bar{y},p,w|\bar{d}_{x},\bar{d}_{y},\tilde{\Psi }_{n})= \sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|d_{x}^{k},d_{y}^{k}, \Psi ^{k})\times S_{d}^{k}. \end{aligned}$$
(41)

Note that on the right hand side we have individual allocations \((x^{k},y^{k})\) and individual directions \((-d_{x}^{k},d_{y}^{k})\), as well as individual technologies \(\Psi ^{k}\), for all k going from 1 to n, while on the left hand side, the aggregate functions have the average of those individual input–output allocations and the average of those individual directional vectors, as well as the aggregate technology \(\tilde{\Psi }_{n}\).

Furthermore, we can use the CRS definition (34), to replace \(\tilde{\Psi }_{n}\) in the Eqs. (38)–(41), with its equivalent under CRS, \(\bar{\Psi }_{n}\), defined as

$$\begin{aligned} \bar{\Psi }_{n}=\frac{1}{n}\sum _{k=1}^{n}\Psi ^{k}. \end{aligned}$$
(42)

Moreover, if all DMUs also face the same technology (as is often assumed in practice) that is CRS, i.e., if \(\Psi ^{k}=\Psi ,\ \forall k\), then we can replace \(\tilde{\Psi }_{n}\) or \(\bar{\Psi }_{n}\) in the Eqs. (38)–(41), with \(\Psi \) because we have

$$\begin{aligned} \bar{\Psi }_{n}=\frac{1}{n}n\Psi =\Psi . \end{aligned}$$
(43)

4.2 The special case of direction \((-x,y)\)

Here we will focus on another interesting case, when \((-d_{x},d_{y})=(-x,y)\), which is a popular direction in practice. In this case, we have

$$\begin{aligned} D^{k}(x,y|x,y,\Psi ^{k})=\sup _{\beta }\{\beta \in \mathbb {R}:\,\, \left( x(1-\beta ),y(1+\beta )\right) \in \Psi ^{k}\}. \end{aligned}$$
(44)

Thus, for this directional vector, \(D^{k}(x,y|x,y,\Psi ^{k})\) can be interpreted as the maximal percentage of feasible contraction of inputs and expansion of outputs. For this reason, this particular choice of directional vector also makes (44) a more convenient measure of (in)efficiency, in the sense that \(\forall (x,y)\in \mathbb {R}_{+}^{N}\times \mathbb {R}_{+}^{M}\), we have

$$\begin{aligned} 0\le D^{k}(x,y|x,y,\Psi ^{k})\le 1\iff (x,y)\in \Psi ^{k}, \end{aligned}$$
(45)

and \(D^{k}(x,y|x,y,\Psi ^{k})=0\) when \((x,y)\in \partial _{d}\Psi ^{k}\) as defined in (4) for \(d=(-x,y)\).

The NPI for an individual k, in this case, is given by

$$\begin{aligned} E^{k}(x,y,p,w|x,y,\Psi ^{k})=\frac{\Pi ^{k}(p,w|\Psi ^{k}) -(py-wx)}{py+wx} \end{aligned}$$
(46)

and, again due to duality, such that \(\forall (x,y)\in \mathbb {R}_{+}^{N}\times \mathbb {R}_{+}^{M}\) we have

$$\begin{aligned} E^{k}(x,y,p,w|x,y,\Psi ^{k})\ge D^{k}(x,y|x,y,\Psi ^{k}). \end{aligned}$$
(47)

It is worth noting that in this interesting case, when the directional vector is \((-x,y)\), the difference between the maximal feasible profit and the observed profit is normalized by DMUs ‘volume of activity’, measured by \((py+wx)\). This allows comparing the NPI and the technical efficiency (based on the directional distance function) for DMUs with different volumes of activity.

Moreover, in the special case when \(d_{y}^{k}=y^{k}\) and \(d_{x}^{k}=x^{k}\) we get a result that is easier to interpret in intuitive terms. Specifically, we have

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\tilde{x},\tilde{y},\tilde{\Psi }_{n}) =\sum _{k=1}^{n}E^{k}(x^{k},y^{k},p,w|x^{k},y^{k},\Psi ^{k})\times S_{xy}^{k} \end{aligned}$$
(48)

where

$$\begin{aligned} S_{xy}^{k}=\frac{py^{k}+wx^{k}}{\sum _{k=1}^{n}(py^{k}+wx^{k})}. \end{aligned}$$
(49)

Intuitively, \(S_{xy}^{k}\) can be interpreted as the volume shares of k’s DMU in the total volumes of n DMUs.

In a sense, the aggregation of Färe & Zelenyuk, (2003) can now be considered a special case of this more general result. Indeed, recall that in Färe & Zelenyuk, (2003) the weights were only involving the outputs or inputs (valued at their prices) for output orientations or input orientations, respectively. If the researcher wishes to consider both input and output directions, then it is natural to consider both of them also in the weights and valuing them at their prices seems natural intuitively, as well as supported by the duality theory which was used to derive this result.

Furthermore, note that we also have

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\tilde{x},\tilde{y},\tilde{\Psi }_{n})&= \sum _{k=1}^{n}(D^{k}(x^{k},y^{k}|x^{k},y^{k},\Psi ^{k})\nonumber \\&\quad +A^{k}(x^{k},y^{k},p,w|x^{k},y^{k},\Psi ^{k}))\times S_{xy}^{k}\nonumber \\&=\sum _{k=1}^{n}D^{k}(x^{k},y^{k}|x^{k},y^{k},\Psi ^{k})\times S_{xy}^{k}\nonumber \\&\quad +\sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|x^{k},y^{k},\Psi ^{k})\times S_{xy}^{k}. \end{aligned}$$
(50)

Moreover, recall again that due to the duality theory, for all \((-\tilde{x},\tilde{y})\in \mathbb {R}_{-}^{N}\times \mathbb {R}_{+}^{M}\backslash \{\text {0}_{N+M}\}\) and all \((w,p)\in \mathbb {R}_{++}^{N+M}\), we have

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\tilde{x},\tilde{y},\tilde{\Psi }_{n})\ge D(\tilde{x},\tilde{y}|\tilde{x},\tilde{y},\tilde{\Psi }_{n}) \end{aligned}$$
(51)

and, similarly as in the disaggregate case, one can close the gap in (51) by introducing a residual representing the aggregate allocative efficiency, thus leading to the following decomposition:

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\tilde{x},\tilde{y},\tilde{\Psi }_{n})= D(\tilde{x},\tilde{y}|\tilde{x},\tilde{y},\tilde{\Psi }_{n})+ A(\tilde{x},\tilde{y},p,w|\tilde{x},\tilde{y},\tilde{\Psi }_{n}). \end{aligned}$$
(52)

Therefore, using similar logic as we used above, we have

$$\begin{aligned} D(\tilde{x},\tilde{y}|\tilde{x},\tilde{y},\tilde{\Psi }_{n})= \sum _{k=1}^{n}D^{k}(x^{k},y^{k}|x^{k},y^{k},\Psi ^{k})\times S_{xy}^{k} \end{aligned}$$
(53)

if and only if

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|\tilde{x},\tilde{y},\tilde{\Psi }_{n})= \sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|x^{k},y^{k},\Psi ^{k})\times S_{xy}^{k}. \end{aligned}$$
(54)

4.3 The special case of direction \((-\bar{x},\bar{y})\)

Combining the results from the previous sub-sections, we can now see more clearly what happens when the direction is \((-g_{x},g_{y})=(-\sum _{k=1}^{n}d_{x}^{k}/n, \sum _{k=1}^{n}d_{y}^{k}/n)=(-\bar{x},\bar{y})\). Specifically, we have:

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\bar{x},\bar{y},\tilde{\Psi }_{n})=D(\tilde{x},\tilde{y}| \bar{x},\bar{y},\tilde{\Psi }_{n})+A(\tilde{x},\tilde{y},p,w|\bar{x}, \bar{y},\tilde{\Psi }_{n}), \end{aligned}$$
(55)

as well as

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\bar{x},\bar{y},\tilde{\Psi }_{n})=n \sum _{k=1}^{n}E^{k}(x^{k},y^{k},p,w|x^{k},y^{k},\Psi ^{k})\times S_{xy}^{k}, \end{aligned}$$
(56)

and so

$$\begin{aligned} D(\tilde{x},\tilde{y}|\bar{x},\bar{y},\tilde{\Psi }_{n})=n \sum _{k=1}^{n}D^{k}(x^{k},y^{k}|x^{k},y^{k},\Psi ^{k})\times S_{xy}^{k} \end{aligned}$$

if and only if

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|\bar{x},\bar{y},\tilde{\Psi }_{n})=n \sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|x^{k},y^{k},\Psi ^{k})\times S_{xy}^{k}. \end{aligned}$$

In words, on the right-hand side, we have efficiency measures with individual allocations \((x^{k},y^{k})\) benchmarked relative to individual technologies \(\Psi ^{k}\) along the directions \((x^{k},y^{k})\) for all DMUs, aggregated over all DMUs \(k\in \{1,...,n\}\) with the help of weights \(S_{xy}^{k}\) that represent the revenue share of k’s DMU in the total revenue of the group of n DMUs, all scaled by n. On the left-hand side, we have the aggregate functions representing the aggregate efficiency measures for the sums of those individual input–output allocations measured relative to the aggregate technology \(\tilde{\Psi }_{n}\) along the direction defined by the average of those individual directional vectors.

Furthermore, in the case of constant returns to scale technology, we have

$$\begin{aligned} E(\bar{x},\bar{y},p,w|\bar{x},\bar{y},\bar{\Psi }_{n})=D(\bar{x},\bar{y}| \bar{x},\bar{y},\bar{\Psi }_{n})+A(\bar{x},\bar{y},p,w|\bar{x},\bar{y},\bar{\Psi }_{n}). \end{aligned}$$
(57)

And hence, using (31), (32) and (33), we obtain

$$\begin{aligned} E(\bar{x},\bar{y},p,w|\bar{x},\bar{y},\tilde{\Psi }_{n})=\sum _{k=1}^{n} E^{k}(x^{k},y^{k},p,w|\bar{x},\bar{y},\Psi ^{k})\times S_{xy}^{k}. \end{aligned}$$
(58)

And, therefore we also have

$$\begin{aligned} D(\bar{x},\bar{y}|\bar{x},\bar{y},\bar{\Psi }_{n})=\sum _{k=1}^{n}D^{k} (x^{k},y^{k}|\bar{x},\bar{y},\Psi ^{k})\times S_{xy}^{k} \end{aligned}$$

if and only if

$$\begin{aligned} A(\bar{x},\bar{y},p,w|\bar{x},\bar{y},\bar{\Psi }_{n})=\sum _{k=1}^{n} A^{k}(x^{k},y^{k},p,w|\bar{x},\bar{y},\Psi ^{k})\times S_{xy}^{k}. \end{aligned}$$

That is, on the right-hand side we have measures for individual allocations \((x^{k},y^{k})\) benchmarked relative to individual technologies \(\Psi ^{k}\), for all k going from 1 to n, along the common directions defined by the average of these individual allocations, over all n DMUs. On the left-side, we have the analogous aggregate measures of efficiency based on the average of those individual input–output allocations and the average of those individual directional vectors, as well as the aggregate technology \(\tilde{\Psi }_{n}\), which here (under CRS) is the same as \(\bar{\Psi }_{n}\).

4.4 The special case of direction (0, y)

Another case of particular interest is when \((-d_{x},d_{y})=(0,y)\) which gives a one-to-one closed form relationship of the DDF with the so-called Debreu-Farrell measure of output oriented technical efficiency,Footnote 15 defined as

$$\begin{aligned} F^{k}(x,y|\Psi ^{k})=\sup _{\theta }\{\theta >0:\,\,(x,y\theta )\in \Psi ^{k}\}. \end{aligned}$$
(59)

This measure also possesses the property of complete characterization of technology, in the sense that \(\forall (x,y)\in \mathbb {R}_{+}^{N}\times \mathbb {R}_{+}^{M}\), we have

$$\begin{aligned} F^{k}(x,y|\Psi ^{k})\ge 1\iff (x,y)\in \Psi ^{k}. \end{aligned}$$
(60)

Meanwhile, \(F^{k}(x,y|\Psi ^{k})=1\) whenever \((x,y)\in \partial _{d}\Psi ^{k}\) as defined in (4) for \(d=(0,y)\). Moreover, in this case when \(d=(0,y)\), we have

$$\begin{aligned} D^{k}(x,y|0,y,\Psi ^{k})=F^{k}(x,y|\Psi ^{k})-1 \end{aligned}$$
(61)

i.e., for this specific directional vector, \(D^{k}(x,y|0,y,\Psi ^{k})\) can be interpreted as the maximal percentage of a feasible expansion of outputs, while keeping the inputs fixed.

In turn, the NPI for an individual k in this case (and assuming \(py\ne 0\)) would be

$$\begin{aligned} E^{k}(x,y,p,w|0,y,\Psi ^{k})=\frac{\Pi ^{k}(p,w|\Psi ^{k})+wx}{py}-1, \end{aligned}$$
(62)

which is one of the profit efficiency measures discussed by Färe et al. (2019), who called it Farrell-type profit efficiency measure. As before, due to duality theory, \(\forall (x,y)\in \mathbb {R}_{+}^{N}\times \mathbb {R}_{+}^{M}\) and \(\forall (w,p)\in \mathbb {R}_{++}^{N}\times \mathbb {R}_{++}^{M}\) we have

$$\begin{aligned} E^{k}(x,y,p,w|0,y,\Psi ^{k})\ge D^{k}(x,y|0,y,\Psi ^{k}), \end{aligned}$$
(63)

and, again, we can have a decomposition into technical and allocative efficiencies, now appearing with the Debreu-Farrell efficiency measure

$$\begin{aligned} E^{k}(x,y,p,w|0,y,\Psi ^{k})=F^{k}(x,y|\Psi ^{k})+A^{k}(x,y,p,w|0,y,\Psi ^{k})-1. \end{aligned}$$
(64)

It is worth noting here that, usually, the Debreu-Farrell efficiency measure appears decomposed in a multiplicative form, while in (64) it is in an additive form, where “−1” at the end serves as the “converter” of the multiplicative Debreu-Farrell measure into what we call the additively-multiplicative scale.

Furthermore, the aggregate result, therefore, reduces to

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|0,\tilde{y},\tilde{\Psi }_{n})=\sum _{k=1}^{n} E^{k}(x^{k},y^{k},p,w|0,y^{k},\Psi ^{k})\times S_{0y}^{k} \end{aligned}$$
(65)

or

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|0,\tilde{y},\tilde{\Psi }_{n})=\sum _{k=1}^{n} \frac{\Pi ^{k}(p,w|\Psi ^{k})+wx^{k}}{py^{k}}\times S_{0y}^{k}-1 \end{aligned}$$
(66)

where \(S_{0y}^{k}\) represents the revenue share of k’s DMU in the total revenue of the group of n DMUs and is the same weight as in the aggregation result of Färe & Zelenyuk, (2003), namely

$$\begin{aligned} S_{0y}^{k}=\frac{py^{k}}{\sum _{k=1}^{n}py^{k}}. \end{aligned}$$
(67)

Moreover, we also have the following relationship between the aggregate and the disaggregate efficiency measures

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|0,\tilde{y},\tilde{\Psi }_{n})&= \sum _{k=1}^{n}(D^{k}(x^{k},y^{k}|0,y^{k},\Psi ^{k})\nonumber \\&+A^{k}(x^{k},y^{k},p,w|0,y^{k},\Psi ^{k}))\times S_{0y}^{k}\nonumber \\&=\sum _{k=1}^{n}F^{k}(x^{k},y^{k}|\Psi ^{k})\times S_{0y}^{k}\nonumber \\&+\sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|0,y^{k},\Psi ^{k}) \times S_{0y}^{k}-1. \end{aligned}$$
(68)

since \(\sum _{k=1}^{n}S_{0y}^{k}=1\). And so, due to duality theory, we have

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|0,\tilde{y},\tilde{\Psi }_{n})\ge F(\tilde{x},\tilde{y}|\tilde{\Psi }_{n})-1 \end{aligned}$$
(69)

for all \((0,\tilde{y})\in \mathbb {R}_{-}^{N}\times \mathbb {R}_{+}^{M}\backslash \{\text {0}_{N+M}\}\), all \((\tilde{x},\tilde{y})\in \mathbb {R}_{+}^{N+M}\) and all \((w,p)\in \mathbb {R}_{++}^{N+M}\), where \(F(\tilde{x},\tilde{y}|\tilde{\Psi }_{n})\) is the aggregate analogue of (59), i.e.,

$$\begin{aligned} F(\tilde{x},\tilde{y}|\tilde{\Psi }_{n})=\sup _{\theta }\{\theta >0:\,\, (\tilde{x},\tilde{y}\theta )\in \tilde{\Psi }_{n}\}. \end{aligned}$$
(70)

This leads us to the following decomposition on the aggregate level involving the aggregate Debreu-Farrell efficiency measure

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|0,\tilde{y},\tilde{\Psi }_{n})=F(\tilde{x}, \tilde{y}|\tilde{\Psi }_{n})+A(\tilde{x},\tilde{y},p,w|0,\tilde{y},\tilde{\Psi }_{n})-1, \end{aligned}$$
(71)

which, finally, helps us arrive at the following aggregation result:

$$\begin{aligned} F(\tilde{x},\tilde{y}|\tilde{\Psi }_{n})=\sum _{k=1}^{n}F^{k}(x^{k}, y^{k}|\Psi ^{k})\times S_{0y}^{k} \end{aligned}$$
(72)

if and only if

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|0,\tilde{y},\tilde{\Psi }_{n})=\sum _{k=1}^{n} A^{k}(x^{k},y^{k},p,w|0,y^{k},\Psi ^{k})\times S_{0y}^{k}. \end{aligned}$$
(73)

That is, on the left-hand side, we have the aggregate functions based on the sum of those individual input–output allocations and the sum of those individual directional vectors, as well as the aggregate technology \(\tilde{\Psi }_{n}\). Meanwhile, on the right-hand side we have efficiency measures with individual allocations \((x^{k},y^{k})\) benchmarked relative to individual technologies \(\Psi ^{k}\) with the directions that are \((0,y^{k})\), aggregated over all DMUs \(k\in \{1,...,n\}\). Importantly, note that unlike before, while the aggregation of the Debreu-Farrell efficiency scores is done in the same way as in Färe & Zelenyuk, (2003), the aggregation of allocative efficiencies is now different. Specifically, the latter also involves the same weights \(S_{0y}^{k}\) that represent the revenue share of k’s DMU in the total revenue of the group of n DMUs. Moreover, the allocative efficiencies here are different—they are closing the gap (additively) between technical efficiency and profit efficiency (measures via NPIs) rather than between technical efficiency and revenue efficiency (and do so multiplicatively) as was in Färe & Zelenyuk, (2003). It is interesting, however, that the aggregation of the Debreu-Farrell efficiency scores is the same whether in earlier works or in this more general framework.

Further simplifications can be obtained if CRS is assumed, by replacing \(\tilde{\Psi }_{n}\) with \(\bar{\Psi }_{n}\) and the sums with averages.

4.5 The special case of direction \((0,\bar{y})\)

Combining the results from the previous sub-sections, we can now see what happens when the direction is \((-g_{x},g_{y})=(0,\sum _{k=1}^{n}d_{y}^{k}/n)=(0,\bar{y})\). Specifically, we get:

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|0,\bar{y},\tilde{\Psi }_{n})=F(\tilde{x}, \tilde{y}|\tilde{\Psi }_{n})+A(\tilde{x},\tilde{y},p,w|0,\bar{y},\tilde{\Psi }_{n})-1, \end{aligned}$$
(74)

as well as

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|0,\bar{y},\tilde{\Psi }_{n})=n\sum _{k=1}^{n} E^{k}(x^{k},y^{k},p,w|0,y^{k},\Psi ^{k})\times S_{0y}^{k}, \end{aligned}$$
(75)

and so

$$\begin{aligned} F(\tilde{x},\tilde{y}|\tilde{\Psi }_{n})=n\sum _{k=1}^{n}F^{k} (x^{k},y^{k}|\Psi ^{k})\times S_{0y}^{k} \end{aligned}$$
(76)

if and only if

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|0,\bar{y},\tilde{\Psi }_{n})=n\sum _{k=1}^{n} A^{k}(x^{k},y^{k},p,w|0,y^{k},\Psi ^{k})\times S_{0y}^{k}. \end{aligned}$$
(77)

In words, on the right-hand side we have efficiency measures with individual allocations \((x^{k},y^{k})\) benchmarked relative to the individual technologies \(\Psi ^{k}\) along the directions \((0,y^{k})\) for all DMUs, aggregated over all DMUs \(k\in \{1,...,n\}\) with a help of weights \(S_{0y}^{k}\) that represent the revenue share of k’s DMU in the total revenue of the group of n DMUs, all scaled by n. Meanwhile, on the left-hand side, we have the aggregate functions based on the sums of those individual input–output allocations measured relative to the aggregate technology \(\tilde{\Psi }_{n}\) along the direction defined by the average of those individual directional vectors, i.e., \((0,\bar{y})\).

Furthermore, in the case of constant returns to scale technology, we have

$$\begin{aligned} E(\bar{x},\bar{y},p,w|0,\bar{y},\bar{\Psi }_{n})=F(\bar{x},\bar{y}|\bar{\Psi }_{n})+A(\bar{x},\bar{y},p,w|0,\bar{y},\bar{\Psi }_{n})-1. \end{aligned}$$
(78)

and therefore, using (39), (40) and (41), we obtain

$$\begin{aligned} E(\bar{x},\bar{y},p,w|0,\bar{y},\bar{\Psi }_{n})=\sum _{k=1}^{n}E^{k}(x^{k}, y^{k},p,w|0,\bar{y},\Psi ^{k})\times S_{0y}^{k}. \end{aligned}$$
(79)

Hence, we can conclude that under constant returns to scale technology, we have

$$\begin{aligned} F(\bar{x},\bar{y}|\bar{\Psi }_{n})=\sum _{k=1}^{n}F^{k}(x^{k},y^{k} |\Psi ^{k})\times S_{0y}^{k}, \end{aligned}$$
(80)

if and only if

$$\begin{aligned} A(\bar{x},\bar{y},p,w|0,\bar{y},\bar{\Psi }_{n})=\sum _{k=1}^{n}A^{k}(x^{k}, y^{k},p,w|0,\bar{y},\Psi ^{k})\times S_{0y}^{k}. \end{aligned}$$
(81)

In words, on the right-hand side we again have measures with individual allocations \((x^{k},y^{k})\) benchmarked relative to individual technologies \(\Psi ^{k}\) with the directions \((0,y^{k})\) for all DMUs, aggregated over all DMUs \(k\in \{1,...,n\}\) with a help of weights \(S_{0y}^{k}\) that represent the revenue share of k’s DMU in the total revenue of the group of n DMUs. And, on the left-hand side of the equations, we have the aggregate functions based on the averages of those individual input–output allocations benchmarked relative to the aggregate technology \(\bar{\Psi }_{n}\) along the direction defined by the average of those individual directional vectors, i.e., \((0,\bar{y})\).

4.6 The special cases of directions \((-x,0)\) or \((-\bar{x},0)\)

The cases for the input-oriented directions, i.e., \((-x,0)\), \((-\tilde{x},0)\) and \((-\bar{x},0)\), are also interesting as they relate to the input oriented Debreu-Farrell efficiency measure. The developments are analogous to those in the previous sub-sections and so, for the sake of brevity, we leave them to the readers.

4.7 The special case of a fixed direction

Finally, here we will focus on the case when the direction is fixed and the same for all k, i.e., \((-d_{x}^{k},d_{y}^{k})=(-a,b)\ne 0\) for some constant vectors \(a\in \mathbb {R}_{+}^{N}\) and \(b\in \mathbb {R}_{+}^{M}\). This is the case considered in some detail in Briec et al. (2003) and Färe et al. (2008) among others. In principle, this can be any choice of fixed vectors in the sense that they do not vary with k or with (x, y). For example, this direction could be set to \((-1,1)\) or (0, 1) or \((-1,0)\), which appear to be very popular in practice.

In this special case we get results that are also easy to interpret in intuitive terms. Specifically, we get

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|\sum _{k=1}^{n}a,\sum _{k=1}^{n}b,\tilde{\Psi }_{n})= \sum _{k=1}^{n}E^{k}(x^{k},y^{k},p,w|a,b,\Psi ^{k})\times S_{a,b}^{k}, \end{aligned}$$
(82)

where note that due to the fixed nature of the direction, the individual weight, here denoted as \(S_{a,b}^{k}\), simplifies to

$$\begin{aligned} S_{a,b}^{k}=\frac{pb+wa}{n(pb+wa)}=\frac{1}{n},\quad \forall k=1,...,n. \end{aligned}$$
(83)

On the other hand, because \((\sum _{k=1}^{n}a,\sum _{k=1}^{n}b)=(na,nb)\) and so engaging (28), we can also say

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|na,nb,\tilde{\Psi }_{n})=\frac{1}{n} E(\tilde{x},\tilde{y},p,w|a,b,\tilde{\Psi }_{n}) \end{aligned}$$
(84)

and therefore, (82) becomes

$$\begin{aligned} \frac{1}{n}E(\tilde{x},\tilde{y},p,w|a,b,\tilde{\Psi }_{n})= \frac{1}{n}\sum _{k=1}^{n}E^{k}(x^{k},y^{k},p,w|a,b,\Psi ^{k}), \end{aligned}$$
(85)

which is equivalent to (14), hence establishing the connection to the previous results discussed in Section 2.

Furthermore, we also have

$$\begin{aligned} \frac{1}{n}&E(\tilde{x},\tilde{y},p,w|a,b,\tilde{\Psi }_{n})\nonumber \\&\ \ =\sum _{k=1}^{n}(D^{k}(x^{k},y^{k}|a,b,\Psi ^{k})+A^{k}(x^{k},y^{k}, p,w|a,b,\Psi ^{k}))\times S_{a,b}^{k}\nonumber \\&\ \ =\frac{1}{n}\sum _{k=1}^{n}D^{k}(x^{k},y^{k}|a,b,\Psi ^{k})+\, \frac{1}{n}\sum _{k=1}^{n}A^{k}(x^{k},y^{k},p,w|a,b,\Psi ^{k}). \end{aligned}$$
(86)

Moreover, recall again that due to duality theory, for all \((-a,b)\in \mathbb {R}_{-}^{N}\times \mathbb {R}_{+}^{M}\backslash \{\text {0}_{N+M}\}\), all \((\tilde{x},\tilde{y})\in \mathbb {R}_{+}^{N+M}\) and all \((w,p)\in \mathbb {R}_{++}^{N+M}\), we have

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|a,b,\tilde{\Psi }_{n})\ge D(\tilde{x},\tilde{y}|a,b,\tilde{\Psi }_{n}) \end{aligned}$$
(87)

and, similarly as in the disaggregate case, one can close the gap by introducing a residual representing aggregate directional allocative efficiency, leading to the following decomposition:

$$\begin{aligned} E(\tilde{x},\tilde{y},p,w|a,b,\tilde{\Psi }_{n})=D(\tilde{x},\tilde{y} |a,b,\tilde{\Psi }_{n})+A(\tilde{x},\tilde{y},p,w|a,b,\tilde{\Psi }_{n}) \end{aligned}$$
(88)

Therefore, using similar logic as we used above, we have

$$\begin{aligned} D(\tilde{x},\tilde{y}|a,b,\tilde{\Psi }_{n})=\sum _{k=1}^{n} D^{k}(x^{k},y^{k}|a,b,\Psi ^{k}) \end{aligned}$$
(89)

if and only if

$$\begin{aligned} A(\tilde{x},\tilde{y},p,w|a,b,\tilde{\Psi }_{n})=\sum _{k=1}^{n} A^{k}(x^{k},y^{k},p,w|a,b,\Psi ^{k}). \end{aligned}$$
(90)

This encompasses the earlier results as special cases of the more general aggregation theory we have developed here.

5 Practical matters

The theoretical efficiency measures discussed above can then be estimated using various methods, most popular of which appears to be Data Envelopment Analysis (DEA). Specifically, using input–output data \(\mathcal {S}_{n}=\{(x^{k},y^{k}):k=1,...,n\}\), for an allocation (x, y) and a direction \((-d_{x},d_{y})\), the DEA formulation for the directional distance function is

$$\begin{aligned} \hat{D}(x,y|d_{x},d_{y},\mathcal {S}_{n})\equiv & {} \underset{\beta ,\,z^{1},...,z^{n}}{\mathop {\max }}\,\beta ,\\ \text {s.t.}\\{} & {} \sum \limits _{k=1}^{n}z^{k}y_{m}^{k}\ge y_{m}+\beta d_{y_{m}},\ m=1,...,M,\\{} & {} \sum \limits _{k=1}^{n}z^{k}x_{l}^{k}\le x_{l}-\beta d_{x_{l}},\ l=1,...,N,\\{} & {} (z^{1},...,z^{n})\in \mathcal {Z},\\{} & {} \beta \ \text {free} \end{aligned}$$

where \(\mathcal {Z}\) is the set of permitted values of the intensity variables \((z^{1},...,z^{n})\) that defines the properties (and hence the shape) of the estimated technology set (e.g., see Sickles & Zelenyuk (2019)). In particular, if \(\mathcal {Z}=\{(z^{1},...,z^{n}):z^{k}\ge 0,\forall k\}\) then the constant returns to scale (CRS) is imposed. When \(\mathcal {Z}= \{(z^{1},...,z^{n}):z^{k}\ge 0,\forall k,\sum \nolimits _{k=1}^{n}z^{k}\le 1\}\) then the non-increasing returns (NIRS) to scale is imposed, and when \(\mathcal {Z}=\{(z^{1},...,z^{n}):z^{k}\ge 0,\forall k,\sum \nolimits _{k=1}^{n}z^{k}=1\}\) then variable returns to scale (VRS) is imposed. These three variants of DEA assume convexity, which sometimes can be too restrictive and can be relaxed by instead letting \(\mathcal {Z}=\{(z^{1},...,z^{n}):z^{k}\in \{0,1\},\forall k,\sum \limits _{k=1}^{n}z^{k}=1\}\), resulting in the so-called free disposal hull (FDH) estimator.Footnote 16

Similarly, the DEA formulation for the profit function, with some prices \((w,p)\), is given by

$$\begin{aligned} \hat{\Pi }(p,w|\mathcal {S}_{n})\equiv & {} \underset{{\begin{matrix}(x,y),\\ z^{1},...,z^{n} \end{matrix}}}{\mathop {\max }}\,\sum \limits _{m=1}^{M} p_{m}y_{m}-\sum \limits _{l=1}^{N}w_{l}x_{l},\\ \text {s.t.}\\{} & {} \sum \limits _{k=1}^{n}z^{k}y_{m}^{k}\ge y_{m},\ m=1,...,M,\\{} & {} \sum \limits _{k=1}^{n}z^{k}x_{l}^{k}\le x_{l},\ l=1,...,N,\\{} & {} (z^{1},...,z^{n})\in \mathcal {Z},\\{} & {} y_{m}\ge 0,\ m=1,...,M, \end{aligned}$$

where, again, \(\mathcal {Z}\) can be chosen to satisfy some desired properties on technology.Footnote 17

It is also worth noting that while these DEA formulations require all observations in \(\mathcal {S}_{n}\) be associated with the same technology, the aggregation theory we discussed and further developed above is more general and is allowing different firms to have different technologies,

Finally, note that the Koopmans (1957) theorem requires that all firms face the same prices, \((w,p)\in \mathbb {R}_{++}^{N+M}\) (which can be understood as the equilibrium prices). Under such a requirement, the linear programming computations of the individual profit functions for DEA and FDH can be avoided, because the optimization problem simplifies then to \(\hat{\Pi }^{j}(p,w)=\max _{k}\{\pi ^{k}:k=1,...,n\}\), for any firm \(j=1,...,n\), where \(\pi ^{k}\) is the actual profit for the observation k, i.e., \(\pi ^{k}=py^{k}-wx^{k}\) (Färe & Zelenyuk (2020)). In turn, this implies that for the DEA and FDH formulations (with the same prices and the same technology), we have the following solution for the aggregate profit function

$$\begin{aligned} \hat{\Pi }(p,w|\mathcal {S}_{n})=n\hat{\Pi }^{j}(p,w| \mathcal {S}_{n})=n\max _{k}\{\pi ^{k}:k=1,...,n\}. \end{aligned}$$

6 Conclusions

The paper presents a cohesive generalized framework for aggregation of the Nerlovian profit indicators and directional distance functions. The proposed aggregation framework allows for a more flexible choice of the direction in the Nerlovian profit indicators. We consider the general case and a few popular special cases of specific directions and discuss their practical implications (see Table 1). Furthermore, the proposed aggregation scheme uses aggregation weights, which brings it closer to the aggregation theory for the Debreu-Farrell-type efficiency measures. In line with the expectations, the weights depend on the selected directions. We expect that the results presented in this paper will further popularize the use of the Nerlovian profit indicators and directional distance measures, in general, and especially among empirical researchers and practitioners.

Table 1 Summary of key aggregation results considered in this study
Full size table

Among the natural avenues for further developments of this work are the extensions to productivity indexes,Footnote 18 the developments of the bootstrap approaches and the central limit theorems for such aggregate measures,Footnote 19 the context of networks in production analysis, and game theory context of efficiency analysis (e.g., see Briec et al. (2021)) to mention a few.