There are several distance function definitions in a general production framework, including Data Envelopment Analysis, which can be used to describe the production technology and to define corresponding measures of technical efficiency (notably the Shephard and the directional distance functions). This paper introduces a generalisation of the distance function concept based on the idea of minimizing firm's opportunity cost. We further state a general dual correspondence between the cost function and this new general distance function, which encompasses all previously published duality results. All our results also hold under the assumption that we work in a Data Envelopment Analysis context.

The theory of duality has acquired great popularity in microeconomics [

Distance functions are natural representations of multiple-output and multiple-input technologies. Shephard [

During the last two decades, Luenberger [

Since there are different families of distance functions in the literature, this paper is devoted to encompass all these alternatives in a unique general family. Additionally, we are interested in enlarging the measurement possibilities provided by the distance functions which currently exist. To achieve these aims, we introduce in this paper a general input distance function and study its mathematical properties. In this respect, we will focus the analysis on production theory and considering a model of cost-minimizing behaviour as in Chambers et al. [

One precursor of this paper is the work by Debreu [

Inspired by the Debreu’s work, in this paper, we minimize the dead loss function to evaluate the technical inefficiency of any producer, but considering a wide set of normalization conditions on the shadow prices instead of using a price index. Following this line, we will define the main concept of this work: the general input distance function. This new notion measures the opportunity cost associated to perform inefficiently, since it is based, by definition, on the Debreu’s dead loss function. Moreover, there are numerous distance functions in the literature and we will show that the general distance function encompasses all these alternatives. In this sense, we make some order in this complexity.

The paper unfolds as follows. In Section

Let

For each output vector

It is assumed here that the input correspondence

Let

Postulate P1 states that positive output vectors cannot be obtained from a null input vector and any nonnegative input vector yields at least the null output vector. P2 states that finite inputs cannot produce infinite outputs. P3 implies strong disposability of inputs. P4 is assumed in order to be able to define the boundary of

We assume the usual axioms because we want to generalize the most famous distance functions, which have been developed by assuming all these postulates.

Given

Given

The price vector

Obviously, not all input vectors belonging to an input requirement set are technologically efficient. Firms usually want to use the smallest levels of inputs to produce a given output vector. In fact, doing otherwise would be wasteful. In this respect, the measurement of inefficiency is necessary to compare the actual performance with respect to a certain reference set of the input requirement set. We are really referring to the boundary or isoquant of

The isoquant of

Now, we are ready to introduce the general input distance function. This new notion is defined with respect to a given normalization set denoted as

Let

By postulates P4 and P5 and applying the separating hyperplane theorem, we know that for each

Let

As a referee points out, the formulation of

This general input distance function can be interpreted as a measure of the distance from

Obviously,

The above general input distance function has the same arbitrary multiplicative scalar problem pointed out by Debreu [

Obviously, any set

Another regularity condition on the set

This section shows the main properties of the general input distance function. It satisfies the weak monotonicity condition with respect to the inputs and is a continuous concave function as well as one-sided directional differentiable.

Let

let

(a)

(b) Let

Shephard [

Let

For any

To prove the other inequality, since

Theorem

Briec and Lesourd [

Other special case of the general input distance function which is the interest is the directional input distance function [

On the other hand, it is well known that the directional input distance function generalizes the Shephard input distance function (see [

One of the main uses of the distance functions is to characterize whether an input vector

Let

Let

Applying the separation of a convex set and a point theorem, we get that there exists

The following result is derived from the last two lemmas. As can be seen, the sign of the general input distance function allows us to characterize the input requirement set.

Let

Finally, the following proposition states the intuitive result that

Let

Let

The dual “general” relationship (

Let

By Theorem

When a price vector

Let

Since

Next, we apply Theorem

First, we study the case of the input Hölder metric distance function of Briec and Lesourd [

In the case of the directional input distance function of Chambers et al. [

Finally, taking

This paper has introduced a general input distance function and has shown that it encompasses other existing distance functions: the Shephard input distance function, the directional input distance function, and the input Hölder metric distance function. After developing its properties, we have outlined a series of general duality results that represent a generalization of the well-known relations between the cost function and different distance functions.

Thanks to the general distance function and since all the revised input distance functions have the same structure, it would be possible to study globally the achievement of certain properties, for example, units invariance. Additionally, we could derive generic economic relationships from the general input distance function (e.g., a general Fenchel-Mahler inequality for measuring profit inefficiency; see for more details [

As a byproduct, all our results also hold in a Data Envelopment Analysis (DEA) context, since this type of polyhedral technologies satisfy the postulates that we assume. (A polyhedral technology is a technology such that if it is represented in the space of

Flexibility is one of the features of the proposed general approach. In this respect, simply by varying the normalization set in the general framework we are able to derive new input distance functions in terms of opportunity costs. Nevertheless, the selection of a specific normalization condition deserves further research.

An additional potential extension of this paper is to focus on consumer theory instead of production theory and try to generalize the Luenberger benefit function [

The authors thank an anonymous referee for providing constructive comments and help in improving the contents and the presentation of this paper. They are grateful to the Ministerio de Ciencia e Innovación, Spain, and to the Conselleria de Educación, Generalitat Valenciana, for supporting this research with Grants MTM2009-10479 and ACOMP/2011/115, respectively.