Translation efficiency and directionally optimal scale

Abstract

The aim of this paper is to show that instead of choosing the benchmark technology solely by means of maximal average productivities, a new whole range of options is opened up by using for this purpose the direction vector. By choosing any particular direction vector to evaluate technical efficiency we in fact predetermine by means of the relevant reflection vector the ray of average productivities we want to achieve. Then we may measure the extent of producing an inefficiently small (large) output by means of what we refer to as translation efficiency by comparing the average productivities of the projection of the observed input–output combination onto the best practice frontier and that of the graph translation homothetic technology. We can also determine the optimal production scale, which we refer to as directionally optimal scale, and which is defined conditionally on this vector of average productivities.

1 Introduction

In performance evaluation, the benchmark technology has the role of a “diagnostic device” (Førsund 1996, p. 283) inspecting whether it would be advantageous for firms (or units of assessment in general) to change scale size so as to improve further their productive efficiency, after technical inefficiency, measured with respect to best practice frontier, has been taken into account. The benchmark technology in the radial (proportional) efficiency framework exhibits constant returns to scale (CRS) for the whole range of input quantities. Thus when a firm is not operating under CRS it can adjust accordingly its scale size to exploit returns to scale. To account for this aspect of inefficiency, Førsund and Hjalmarsson (1979) introduced the notion of scale efficiency to measure the extent of producing an inefficiently small (large) output in the region of increasing (decreasing) returns to scale.

Related to scale efficiency is the concept of technically optimal scale (TOS) when we refer to single-output technologies (Frisch 1965) or of the most productive scale size (MPSS) when we refer to multi-output technologies (Banker 1984), which for proportional variation of inputs and outputs (i.e., along a given input and output mix) is determined by the maximal ray average productivity attained at any point on the production frontier. This in turn is associated with the CRS benchmark technology and by definition implies scale efficiency.

The issue of optimal production scale is of pivotal importance for the restructuring of regulated or state-owned industries as well as many traditional public sector activities (Førsund and Hjalmarsson 2004), and policy recommendations based on the TOS or the MPSS may have serious implications for setting scale efficient targets. The only empirical study of scale efficiency targets that we are aware of concerns the case of container ports in the Asia–Pacific region (see Hung, Lu and Wang 2010). Its estimated scale efficiency targets, based on the TOS, may be appropriate within the context of country policies, such as those of China for example, for expanding-only-development-policy with government support, but are there of any relevance if international safety regulations regarding container berths and terminal length have to be considered? Probably not, as the provided guidelines for the policy-makers to leverage their resource optimal scale reflects maximization of average productivity rather than complying with safety regulations. In addition, the TOS or the MPSS were criticized as being unrealistic for firms facing increasing returns to scale (Appa and Yue 1999) and instead the minimum MPSS was proposed in terms of the best returns to scale (BRS) model; see also Zhu (2000). These suggest some doubts in the careless use of TOP or MPSS in analyzing policy or regulation questions.

On the other hand, in the directional efficiency framework, we may go beyond the CRS benchmark technology and use graph translation homotheticity (GTH) (see Chambers 2002) to define an alternative benchmark technology. Instead of choosing the benchmark technology solely my means of maximal average productivities, a new whole range of options is opened up by using for this purpose the direction vector, which is also essential for estimating technical efficiency.¹ Specifically, by choosing a particular direction vector to evaluate technical efficiency we in fact predetermine by means of the relevant reflection vector the ray of average productivities that we want to achieve.² Then we may measure the extent of producing an inefficiently small (large) output by means of what we refer to as translation efficiency, by comparing the average productivities of the observed input–output combination and that of the GTH technology. We can also determine the optimal production scale, which we refer to as directionally optimal scale (DOS), defined conditionally on this vector of average productivities. Thus DOS provide an answer on how much firm’s production scale may be changed in order to achieve the value of average productivities given by the slope of the reflection vector. If this happens to be the same as maximal average productivities, DOS coincides with TOS or MPSS.

We may think of several cases where the notion of DOS may be useful. For regulation or policy purposes, for example, it may sometimes be more appropriate to determine the optimal farm size conditionally on some acceptable social criteria in terms of a sustainable or more generally, an environmentally appropriate yield level instead of the maximal realized yields.³ Similarly, when there are safety regulations, container ports’ optimal scale size should be in accordance with these regulations. On the other hand, the process of restructuring its branches, managers may be interesting in determining their optimal scale size conditionally on given sales targets set by headquarters. In addition, in the case of regulations related to pollution generating industries, regarding emission standards per unit of output provide more accurate guidelines for analyzing optimal scale compared to technically permissible bad-to-good-output ratio. These alternative formulations are implemented by choosing different direction vectors along which performance is evaluated.

The rest of this paper unfolds as follows: in the next section we briefly present the properties of the directional technology distance function and in particular, those related to constant returns to scale and graph translation homotheticity. In the third section, we reconsider the concept of scale efficiency within the framework of the directional technology distance function and in the fourth section, we develop the concepts of translation efficiency and DOS and we explain their relation to scale efficiency and TOS or MPSS. In the fifth section, we relate translation efficiency with Balk, Färe and Karagiannis (2015) translation elasticity. Concluding remarks follow in the last section.

2 Preliminaries

Let $x \in R_{ + }^{n}$ denote a vector of inputs and $y \in R_{ + }^{m}$ a vector of outputs with $w \in R_{ + + }^{n}$ and $p \in R_{ + + }^{m}$ being their corresponding price vectors. Technology is defined in terms of the production possibility set $T = \{ (x,y):x\;can\;produce\;y\}$, which is closed, allows for no free lunch, namely $(0^{n} ,y) \in T \Leftrightarrow y = 0^{m}$, inaction, namely $(0^{n} ,0^{m} ) \in T$, and free disposability of inputs and outputs, and it is convex. The directional technology distance function, which is a variation of Luenberger’s (1992) shortage function, is defined as:

$$ \vec{D}^{T} (x,y;g_{x} ,g_{y} ) = \sup \left\{ {\beta :\left( {x - \beta g_{x} ,y + \beta g_{y} } \right) \in T} \right\} \ge 0 $$

where $(g_{x} ,g_{y} ) \in R_{ + }^{n} \times R_{ + }^{m}$ is the direction vector.⁴ The directional technology distance function has the following properties (Chambers, Chung and Färe 1998):

• 1. Representation property: $\vec{D}^{T} (x,y;g_{x} ,g_{y} ) \ge 0$ if and only if $(x,y) \in T$ assuming that $(x,y)$ are freely disposable;

• 2. Monotonicity property: (a) $\vec{D}^{T} (\tilde{x},y;g_{x} ,g_{y} ) \ge \vec{D}^{T} (x,y;g_{x} ,g_{y} )$ for $\tilde{x} \ge x$ if inputs are freely disposable and (b) $\vec{D}^{T} (x,\tilde{y};g_{x} ,g_{y} ) \le \vec{D}^{T} (x,y;g_{x} ,g_{y} )$ for $\tilde{y} \ge y$ if outputs are freely disposable;

• 3. Curvature Property: $\vec{D}^{T} (x,y;g_{x} ,g_{y} )$ is concave in x and y if T is convex;

• 4. Translation property: $\vec{D}^{T} \left( {x - \alpha g_{x} ,y + \alpha g_{y} ;g_{x} ,g_{y} } \right) = \vec{D}^{T} \left( {x,y;g_{x} .g_{y} } \right) - \alpha \quad \forall \alpha \in R;$

• 5. Homogeneity property: $\vec{D}^{T} (x,y;\lambda g_{x} ,\lambda g_{y} ) = \lambda^{ - 1} \vec{D}^{T} (x,y;g_{x} ,g_{y} )$ for $\lambda > 0$.

The following two properties are related to the structure of production: first, technology is characterized by constant returns to scale and $\vec{D}^{T} (x,y;g_{x} ,g_{y} )$ is homogenous of degree 1 in $(x,y)$ if $\vec{D}^{T} (\lambda x,\lambda y;g_{x} ,g_{y} ) = \lambda \vec{D}^{T} (x,y;g_{x} ,g_{y} )$ for $\lambda \ge 0$ (Briec, Dervaux and Leleu 2003). Second, technology exhibits graph translation homotheticity if $\vec{D}^{T} (x + \gamma g_{x} ,y + \gamma g_{y} ;g_{x} ,g_{y} ) = \vec{D}^{T} (x,y;g_{x} ,g_{y} )$ for $\gamma \ge 0$ (Chambers 2002).

3 Measuring scale efficiency

For any given output or input level, radial scale efficiency measures the gap between the benchmark and the best practice technology, defined respectively by means of a constant-returns-to-scale (CRS) and a variable-returns-to-scale (VRS) frontier; see Førsund and Hjalmarsson (1979), Färe, Grosskopf and Lovell (1983) and Banker, Charnes and Cooper (1984). In the radial efficiency framework, scale efficiency measures are distinguished into input- and output-oriented. The output-oriented scale efficiency index is given by the ratio of the output-oriented technical efficiency index under CRS and VRS. In terms of Fig. 1, the output-oriented scale efficiency index is given by the vertical distance between the CRS and the VRS frontiers $\hat{F}$ and F at the level of the observed input quantity. For a firm with an input–output combination as such depicted by point A, the output-oriented scale efficiency index is given as: $SE^{O} = TE_{C}^{O} /TE_{V}^{O} = x^{\prime}B/x^{\prime}C$ and thus there is a multiplicative decomposition of $TE_{C}^{O}$ into $TE_{V}^{O}$ and $SE^{O}$. Accordingly, the input-oriented scale efficiency index is given by the horizontal difference between the CRS and the VRS frontiers at the level of observed output.

Fig. 1

Measuring scale efficiency with directional technology distance function

In the directional efficiency framework, the scale efficiency indicator may be defined as the distance between the CRS and VRS frontiers along the direction vector $g = (g_{x} ,g_{y} )$.⁵ Based on the translation property we can formally proved what Epure, Kestens and Prior (2011) have mentioned in passing, namely that

$$ \mathop {SE}\limits^{ \to } = \mathop {TE_{C} }\limits^{ \to } - \mathop {TE_{V} }\limits^{ \to } $$

(1)

which provides an additive decomposition of $\mathop {TE_{C} }\limits^{ \to }$ into $\mathop {TE_{V} }\limits^{ \to }$ and $\mathop {SE}\limits^{ \to }$. This measure is obtained by relating the efficiency indicator of point A in Fig. 1 with respect to both the CRS and the VRS frontiers and that of point D relative to the VRS frontier. Point D gives the technically efficient input–output combination of point A along the direction vector $g = (g_{x} ,g_{y} )$ according to the best practice frontier.

Consider an input–output combination $(x^{\prime } ,y^{\prime } )$ depicted by point A in Fig. 1. Then the technical efficiency indicator defined with respect to the best practice (i.e., the VRS) technology is given as:

$$ \mathop {TE_{V} }\limits^{ \to } = \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) = \mathop {\max }\limits_{\beta } \left\{ {\beta :\left( {x^{\prime} - \beta g_{x} ,y^{\prime} + \beta g_{y} } \right) \in \overline{T}^{\prime}} \right\} $$

(2)

where $\overline{T}^{\prime} = \left\{ {(x,y):x \ge \lambda x^{\prime},0 \le y \le \lambda y^{\prime},\lambda = 1} \right\}$ (Briec, Kerstens and Vanden Eeckaut, 2004).⁶ Using the direction vector $g = (g_{x} ,g_{y} )$, $T\vec{E}_{V}$ translates the input–output vector $(x^{\prime } ,y^{\prime } )$ at point A onto the boundary of the best practice technology F⁷ and in particular, at point D with the technically efficient input–output vector being $(x^{*} ,y^{*} ) =$$(x - \vec{D}(x,y;g_{x} ,g_{y} )g_{x} ,y + \vec{D}(x,y;g_{x} ,g_{y} )g_{y} )$ and thus, $T\vec{E}_{V}$ is equal to ${\text{OD}}^{\prime } /{\text{OF}}^{\prime } = {\text{AD}}/{\text{AF}}$. On the other hand, the technical efficiency indicator defined with respect to the CRS benchmark frontier is given as:

$$ \mathop {TE_{C} }\limits^{ \to } = \vec{D}_{C} (x^{\prime},y^{\prime};g_{x} ,g_{y} ) = \mathop {\max }\limits_{\delta } \left\{ {\delta :\left( {x^{\prime} - \delta g_{x} ,y^{\prime} + \delta g_{y} } \right) \in \hat{T}^{\prime}} \right\} $$

(3)

where $\hat{T}^{\prime} = \left\{ {(x,y):x \ge \lambda x^{\prime},0 \le y \le \lambda y^{\prime},\lambda \ge 0} \right\}$ (Briec, Kerstens and Vanden Eeckaut, 2004).⁸ Using the direction vector $g = (g_{x} ,g_{y} )$, $T\vec{E}_{C}$ translates the input–output vector $(x^{\prime},y^{\prime})$ at point A onto the boundary of the CRS benchmark technology $\hat{F}$⁹ and in particular, at point E with the technically efficient input–output vector being $(x^{o} ,y^{o} ) = (x - \vec{D}_{C} (x,y;g_{x} ,g_{y} )g_{x} ,y + \vec{D}_{C} (x,y;g_{x} ,g_{y} )g_{y} )$ and thus, $T\vec{E}_{C}$ is equal to ${\text{OE}}^{\prime } /{\text{OF}}^{\prime } = {\text{AE}}/{\text{AF}}$.

Then the scale efficiency indicator is given by the difference between the technical efficiency indicator defined with respect to the CRS benchmark technology and that defined with respect to the best practice technology. This is equivalent to evaluating the efficiency of the translated input–output vector $(x^{*} ,y^{*} )$ with respect to the benchmark frontier. That is, from the observed input–output vector we remove any inefficiency that is due to purely technical matters by translating it onto the boundary of the best practice frontier and then we account for any remaining inefficiency due to non-optimal scale by gauging the efficiency of the translated input–output vector with respect to the CRS benchmark frontier.¹⁰ Then, it can be shown that the corresponding efficiency indicator, which is given as:

$$ \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) = \mathop {\max }\limits_{\gamma } \left\{ {\gamma :\left( {x^{*} - \gamma g_{x} ,y^{*} + \gamma g_{y} } \right) \in \hat{T}^{\prime}} \right\} $$

(4)

is equal to $\mathop {TE_{C} }\limits^{ \to } - \mathop {TE_{V} }\limits^{ \to }$ and thus, it coincides with $\mathop {SE}\limits^{ \to }$. In terms of Fig. 1, $\vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) = {\text{OE}}^{\prime \prime } /{\text{OF}}^{\prime } = {\text{DE}}/{\text{DF}}^{\prime \prime }$ = ${\text{E}}^{\prime } {\text{D}}^{\prime } /{\text{OF}}^{\prime } = {\text{ED}}/{\text{AF}}$ since ${\text{ED}} = {\text{E}}^{\prime } {\text{D}}^{\prime } = {\text{OE}}^{\prime }$ and ${\text{OF}}^{\prime } = {\text{AF}} = {\text{DF}}^{\prime \prime }$. Then, one can verify that $S\vec{E}$ is given as ED/AF = (AE-AD)/AF or ${\text{D}}^{\prime } {\text{E}}^{\prime } /{\text{OF}}^{\prime } = \left( {{\text{OE}}^{\prime } - {\text{OD}}^{\prime } } \right)/{\text{OF}}^{\prime }$. Now at point E the efficient input–output vector is

$$ \begin{gathered} (x^{o} ,y^{o} ) = (x^{*} - \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} )g_{x} ,y^{*} + \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} )g_{y} ) \hfill \\ \quad \quad \quad \; = (x^{\prime} - \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} )g_{x} - \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} )g_{x} ,y^{\prime} \hfill \\ \quad \quad \quad \quad + \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} )g_{y} + \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} )g_{y} ) \hfill \\ \quad \quad \quad \; = (x^{\prime} - \left\{ {\vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) + \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} )} \right\}g_{x} ,y^{\prime} \hfill \\ \quad \quad \quad \quad + \left\{ {\vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) + \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} )} \right\}g_{y} ) \hfill \\ \end{gathered} $$

which implies that $\vec{D}_{C} (x^{\prime},y^{\prime};g_{x} ,g_{y} ) = \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) + \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} )$.

A formal proof of this relationship follows immediately from the translation property of the directional technology distance function:

$$ \begin{gathered} \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) = \vec{D}_{C} (x^{\prime} - \beta g_{x} ,y^{\prime} + \beta g_{y} ;g_{x} ,g_{y} ) \hfill \\ \quad \quad \quad \quad \quad \quad \quad \;\; = \vec{D}_{C} (x^{\prime},y^{\prime};g_{x} ,g_{y} ) - \beta \hfill \\ \quad \quad \quad \quad \quad \quad \quad \;\; = \vec{D}_{C} (x^{\prime},y^{\prime};g_{x} ,g_{y} ) - \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) \hfill \\ \end{gathered} $$

(5)

where the second equality follows from the translation property and the third is based on (2). Thus, $S\vec{E} = \vec{D}_{C} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) \ge 0$, with the equality indicating scale efficiency. As with the radial efficiency measures, the MPSS is identified with scale efficiency, i.e., $S\vec{E} = 0$, and thus it requires $\mathop {TE_{C} }\limits^{ \to } = \mathop {TE_{V} }\limits^{ \to }$.

As with technical efficiency measures, for the directional distance functions there is no need to define input- or output-oriented scale efficiency. In addition, the proposed scale efficiency indicator is readily applicable to directional input and output distance functions as well as to translation technology, input and output functions. In addition, for particular direction vectors, i.e., $g = (x,0)$ and $g = (0,y)$, it becomes respectively the input-oriented and the output-orientd radial scale efficiecny measures.

4 Measuring translation efficiency

We define translation efficiency to be the difference between average productivity at the frontier point of an observed input–output combination and reference productivity at the frontier. The latter is related to the GTH benchmark technology, represented with a line parallel to the reflection vector $(h_{x} ,h_{y} )$, the slope of which corresponds to what may be called “reference average product”. If, as in Fig. 2, the direction vector is located in the fourth quadrant, indicating that outputs are expanding and inputs are contracting as we move towards the frontier, $h = (h_{x} ,h_{y} )$ is obtained by reflecting the direction vector across the vertical axis. Then, we may think of three alternatives regarding the position of GTH depending on the relationship between the maximum feasible and the reference average product. The one depicted in the upper panel of Fig. 2 reflects the case of a positive minimum input scale below which production is not possible. This means that some fixed or sunk (e.g. entry) costs should be accrued before production actually starts. In this case the reference average product is greater than the maximum feasible average product. On the other hand, when the reference average product is less than the maximum feasible average product, we have a GTH technology allowing for the possibility of “free lunch”; see the lower panel of Fig. 2. It is also possible that the choice of the direction vector is such that the GTH and the CRS technologies coincide, as in the middle panel of Fig. 2, and then the same input–output combination determines the point of local constant returns to scale and local constant returns to translation.

Fig. 2

Frontier, CRS and GTH technologies

Alternatively, the reference average product may be seen as the maximum feasible average product of a new Cartesian coordinate system, from the origin of which the GTH technology passes. This may be portrayed by means of a rightward parallel shift of the vertical axis in the upper panel of Fig. 2 and a leftward parallel shift of the vertical axis in the lower panel.¹¹ Then, the DOS may be viewed as the TOS that a production unit may attain under the new Cartesian coordinate system.

Translation efficiency is defined as the distance between the GTH and VRS frontiers along the direction vector $g = (g_{x} ,g_{y} )$.¹² Based on the translation property, we can formally proved that

$$ \mathop {GE}\limits^{ \to } = \mathop {TE_{G} }\limits^{ \to } - \mathop {TE_{V} }\limits^{ \to } $$

(6)

which provides an additive decomposition of $\mathop {TE_{G} }\limits^{ \to }$ into $\mathop {TE_{V} }\limits^{ \to }$ and $\mathop {GE}\limits^{ \to }$. For the graphical exposition of translation efficiency we consider without loss of generality the case of GTH technology with a positive minimum input scale. Then $\mathop {GE}\limits^{ \to }$ is obtained by relating the technical efficiency indicator of point A in Fig. 3 with respect to both the GTH and the VRS frontiers and that of point D relative to the VRS frontier. Point D gives the technically efficient input–output combination of point A along the direction vector $g = (g_{x} ,g_{y} )$ according to the best practice frontier.

Fig. 3

Measuring translation efficiency with directional technology distance function

Consider an input–output combination $(x^{\prime},y^{\prime})$ depicted by point A in Fig. 3. Then, the technical efficiency indicator defined with respect to the GTH benchmark technology is given as:

$$ \mathop {TE_{G} }\limits^{ \to } = \vec{D}_{G} (x^{\prime},y^{\prime};g_{x} ,g_{y} ) = \mathop {\max }\limits_{\theta } \left\{ {\theta :\left( {x^{\prime} - \theta g_{x} ,y^{\prime} + \theta g_{y} } \right) \in \tilde{T}^{\prime}} \right\} $$

(7)

with $\tilde{T}^{\prime} = \gamma \left\{ {\left( {x,y} \right):x \ge x^{\prime} - \gamma g_{x} ,0 \le y \le y^{\prime} + \gamma g_{y} ,\gamma \ge 0} \right\}$.¹³ Using the direction vector $g = (g_{x} ,g_{y} )$, $T\vec{E}_{G}$ translates $(x^{\prime},y^{\prime})$ at point A onto the boundary of the GTH benchmark technology and in particular, at point E with the technically efficient input–output vector being $(x^{a} ,y^{a} ) = (x^{\prime} - \vec{D}_{G} (x^{\prime},y^{\prime};g_{x} ,g_{y} )g_{x} ,y^{\prime} + \vec{D}_{G} (x^{\prime},y^{\prime};g_{x} ,g_{y} )g_{y} )$ and thus, $T\vec{E}_{G}$ is equal to ${\text{OE}}^{\prime } /{\text{OF}}^{\prime } = {\text{AE}}/{\text{AF}}$.

Then, translation efficiency is given by the difference between the technical efficiency indicator defined with respect to the GTH benchmark technology and that defined with respect to the best practice technology. This is equivalent to evaluating the efficiency of the translated input–output vector $(x^{*} ,y^{*} )$ with respect to the benchmark frontier. That is, from the observed input–output vector we remove any inefficiency that is due to purely technical matters by translating it onto the boundary of the best practice frontier and then we account for any remaining inefficiency due to non-optimal scale by gauging the efficiency of the translated input–output vector with respect to the GTH benchmark frontier. Then, it can be shown that the corresponding efficiency indicator, which is given as:

$$ \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) = \mathop {\max }\limits_{\varphi } \left\{ {\varphi :\left( {x^{*} - \varphi g_{x} ,y^{*} + \varphi g_{y} } \right) \in \tilde{T}^{\prime}} \right\} $$

(8)

is equal to $\mathop {TE_{G} }\limits^{ \to } - \mathop {TE_{V} }\limits^{ \to }$ and thus, it coincides with $\mathop {GE}\limits^{ \to }$. In terms of Fig. 3, $\vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) = {\text{OE}}^{\prime \prime } /{\text{OF}}^{\prime } = {\text{DE}}/{\text{DF}}^{\prime \prime } = {\text{E}}^{\prime } {\text{D}}^{\prime } /{\text{OF}}^{\prime } = {\text{ED}}/{\text{AF}}$ since ${\text{ED}} = {\text{E}}^{\prime } {\text{D}}^{\prime } = {\text{OE}}^{\prime }$ and ${\text{OF}}^{\prime } = {\text{AF}} = {\text{DF}}^{\prime \prime }$. Then, one can verify that $\mathop {GE}\limits^{ \to }$ is given as ${\text{ED}}/{\text{AF}}\, = \,\left( {{\text{AE}} - {\text{AD}}} \right)/{\text{AF}}$ or ${\text{D}}^{\prime } {\text{E}}^{\prime } /{\text{OF}}^{\prime } = \left( {{\text{OE}}^{\prime } - {\text{OD}}^{\prime } } \right)/{\text{OF}}^{\prime }$. Now at point E the efficient input–output vector is

$$ \begin{gathered} (x^{o} ,y^{o} ) = (x^{*} - \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} )g_{x} ,y^{*} + \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} )g_{y} ) \hfill \\ \quad \quad \quad \;\; = (x^{\prime} - \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} )g_{x} - \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} )g_{x} ,y^{\prime} \hfill \\ \quad \quad \quad \quad + \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} )g_{y} + \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} )g_{y} ) \hfill \\ \quad \quad \quad \;\; = (x^{\prime} - \left\{ {\vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) + \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} )} \right\}g_{x} ,y^{\prime} \hfill \\ \quad \quad \quad \quad + \left\{ {\vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) + \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} )} \right\}g_{y} ) \hfill \\ \end{gathered} $$

which implies that $\vec{D}_{G} (x^{\prime},y^{\prime};g_{x} ,g_{y} ) = \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) + \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} )$.

A formal proof of this relationship follows immediately from the translation property of the directional technology distance function:

$$ \begin{gathered} \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) = \vec{D}_{G} (x^{\prime} - \beta g_{x} ,y^{\prime} + \beta g_{y} ;g_{x} ,g_{y} ) \hfill \\ \quad \quad \quad \quad \quad \quad \quad \;\; = \vec{D}_{G} (x^{\prime},y^{\prime};g_{x} ,g_{y} ) - \beta \hfill \\ \quad \quad \quad \quad \quad \quad \quad \;\; = \vec{D}_{G} (x^{\prime},y^{\prime};g_{x} ,g_{y} ) - \vec{D}(x^{\prime},y^{\prime};g_{x} ,g_{y} ) \hfill \\ \end{gathered} $$

(9)

where the second equality follows from the translation property and the third is based on (2). Thus, $G\vec{E} = \vec{D}_{G} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) \ge 0$, with the equality indicating translation efficiency. In addition, the DOS is identified by translation efficiency, i.e., $G\vec{E} = 0$, which in turns requires $\mathop {TE_{G} }\limits^{ \to } = \mathop {TE_{V} }\limits^{ \to }$.

Notice that the proposed translation efficiency measure is readily applicable to directional input and output distance functions as well as to translation technology, input and output functions.

In the case that the choice of the direction vector is such that the GHT and the CRS technologies coincide (see Fig. 2, middle panel), we have $\mathop {TE_{G} }\limits^{ \to } = \mathop {TE_{C} }\limits^{ \to }$ and as a result, $G\vec{E} = S\vec{E}$. In other words, in a directional distance function framework, scale efficiency may be viewed a special case of translation efficiency resulting from a particular choice of the direction vector, namely that reflecting the maximum feasible average product. In fact, for any given input and output data set, there is a (infinite) number of translation efficiency indicators (depending upon the chosen direction vector) one of which coincides with scale efficiency. In that case, DOS coincides with MPSS. Thus, in a directional distance function framework, MPSS may be viewed as a special case of DOS resulting from a particular choice of the direction vector. The former is defined with respect to the maximal ray average productivity while the latter with respect to the reference average product, which depends upon the chosen direction vector.

In contrast to the radial framework, in the directional distance function setup there is a different DOS for each direction vector and the closer a firm is to the particular DOS the higher is its translation efficiency. Each firm may choose the direction vector that maximizes its translation efficiency. This would be the direction vector that results in the minimum difference between $\mathop {TE_{G} }\limits^{ \to }$ and $\mathop {TE_{V} }\limits^{ \to }$. Alternatively, a decision maker or a social planner may choose the direction vector that maximizes aggregate or average translation efficiency.¹⁴

5 Translation efficiency and translation elasticity

In this section we show that translation efficiency is related to translation elasticity in a manner analogous to that scale efficiency is related to scale elasticity (see Førsund and Hjalmarsson 1979; Banker 1984; Färe, Grosskopf and Lovell 1988) in radial efficiency models. Translation elasticity is one of the directional scale elasticities introduced by Balk, Färe and Karagiannis (2015), for which changes in input and output quantities are computed along the same direction as technical efficiency is measured. Suppose that by adding a certain number of units of the input reference vector ($\alpha g_{x}$) to the input vector it is possible to increase the output vector by a (probably) different number of units of the output reference vector ($\rho g_{y}$) in such a way that the resulting input–output combination remains feasible; that is¹⁵:

$$ \vec{D}^{T} (x + \alpha g_{x} ,y + \rho g_{y} ;g_{x} ,g_{y} ) = \vec{D}^{T} (x^{*} ,y^{*} ;g_{x} ,g_{y} ) = 0 $$

(10)

where $x^{*} = x + \alpha g_{x}$, $y^{*} = y + \rho g_{y}$, $\alpha$ is the input translation factor, $\rho$ is the output translation factor, and the last equality follows from the fact that translation elasticity is a frontier measure.¹⁶ By assuming that the directional technology distance function is continuous and differentiable, the directional derivative (see Rockafellar 1970, p. 241) of (10) evaluated at $\alpha = \rho = 0$ is¹⁷:

$$ \sum\limits_{i = 1}^{n} {\frac{{\partial \vec{D}^{T} }}{{\partial x_{i}^{{}} }}g_{x}^{i} } + \sum\limits_{j = 1}^{m} {\frac{{\partial \vec{D}^{T} }}{{\partial y_{j}^{{}} }}} g_{y}^{j} \frac{\partial \rho }{{\partial \alpha }} = 0 $$

(11)

Then, a measure of local returns to translation, given as the marginal change in the output translation factor due to a marginal change in the input translation factor, is obtained by means of the translation elasticity, i.e.:

$$ \theta = \frac{\partial \rho }{{\partial \alpha }} = - \frac{{\sum\limits_{i = 1}^{n} {\frac{{\partial \vec{D}^{T} }}{{\partial x_{i} }}g_{x}^{i} } }}{{\sum\limits_{j = 1}^{m} {\frac{{\partial \vec{D}^{T} }}{{\partial y_{j} }}g_{y}^{j} } }} = - \frac{{\left( {\nabla_{x} \vec{D}^{T} } \right)^{\prime } g_{x}^{{}} }}{{\left( {\nabla_{y} \vec{D}^{T} } \right)^{\prime } g_{y}^{{}} }} $$

(12)

The translation elasticity determines the maximal number of times the output direction vector is allowed by the technology to be added to output quantities when the input direction vector has been added a particular number of times to input quantities. If the former is greater (less) than the latter then technology exhibits increasing (decreasing) returns to translation.

From the duality between the profit and the directional technology distance functions we have (Chambers, Chung and Färe 1998):

$$ \pi (p,w) = \mathop {\max }\limits_{x,y} \left\{ {p^{\prime}y - w^{\prime}x + \vec{D}^{T} (x,y;g_{x} ,g_{y} )\left( {p^{\prime}g_{y} + w^{\prime}g_{x} } \right)} \right\} $$

The first-order conditions of this problem are: $- w + \nabla_{x} \vec{D}^{T} (p^{\prime}g_{y} + w^{\prime}g_{x} ) = 0$ and $p + \nabla_{y} \vec{D}^{T} (p^{\prime}g_{y} + w^{\prime}g_{x} ) = 0$. By substituting them into (12), one can verify that $\theta = w^{\prime}g_{x} /p^{\prime}g_{y}$, namely that the translation elasticity is equal to the relative value of the input and the output direction. Thus, a “return to the dollar” interpretation may be given to the translation elasticity analogous to the cost/revenue interpretation of the scale elasticity.

From (12), constant returns to translation in the direction of $(g_{x} ,g_{y} )$ means that $\theta = 1$. This in turn implies $(\nabla_{x} \vec{D}^{T} )^{\prime}g_{x} + (\nabla_{y} \vec{D}^{T} )^{\prime}g_{y} = 0$, which is the condition for graph translation homotheticity (see Chambers 2002). In this case, the maximal number of times the output direction vector is allowed by the technology to be added to output quantities is exactly equal to the number of times the input direction vector has been added to input quantities. Moreover, $w^{\prime}g_{x} = p^{\prime}g_{y}$ and thus the Lagrangian multiplier in the profit maximization problem is equal to $2w^{\prime}g_{x}$ or $2p^{\prime}g_{y}$. In addition, graph translation homotheticity and the translation property of the directional technology distance function imply that $(\nabla_{x} \vec{D}^{T} )^{\prime } g_{x} = 1/2$ and $(\nabla_{y} \vec{D}^{T} )^{\prime } g_{y} = - 1/2$.

By combining the above with the results of the previous section, we can infer that graph translation homotheticity is associated with both translation efficiency, i.e., $G\vec{E} = 0$, and constant returns to translation, i.e., $\theta = 1$.

6 Concluding remarks

In this paper we introduce two new concepts in efficiency analysis, namely that of the translation efficiency and of the directionally optimal scale. Translation efficiency is given by the difference between two technical efficiency indicators, one defined with respect to the graph translation homothetic technology and the other with respect to the best practice technology. Translation efficiency may be seen as a generalization of the concept of scale efficiency in the directional distance function framework. On the other hand, for additive variation in inputs and outputs along a particular direction vector, the directionally optimal scale is determined by means of the reference average product, i.e., the maximal average productivity for the particular direction vector. The directionally optimal scale may be seen as a generalization of the concept of MPSS in the directional distance function framework.