# A General Input Distance Function Based on Opportunity Costs

Authors: Juan Aparicio; Jesus T. Pastor
Journal: Advances in Decision Sciences
 (2011)

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## Abstract

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.


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## Body

## 1. Introduction

The theory of duality has acquired great popularity in microeconomics [1–4]. Duality theory has allowed to state the most common alternative ways of representing preferences and technologies, such as indirect utility and expenditure functions, cost and distance functions, and so forth. Having different ways to describe a technology seems very suitable since some types of mathematical arguments are easier to demonstrate by using, for example, a cost function instead of a distance function, that is, a direct representation of the technology [1, page 81]. (A firm produces outputs from a set of inputs. In order to analyze firm choices, it is necessary a convenient way to summarize the production possibilities of the firm, that is, which combinations (vectors) of inputs and outputs are feasible. A technology is the set of all these feasible combinations [1].) Both the cost function and the distance function are, by definition, optimization problems. The theory of duality studies under which conditions these two optimization problems are related.Distance functions are natural representations of multiple-output and multiple-input technologies. Shephard [5] was the first to define a distance function in a production context. In particular, the Shephard input distance function measures the largest radial contraction of an input vector consistent with remaining technically feasible. Additionally, the Shephard input distance function is of common use in production theory and it also has a dual relation to the cost function. Nevertheless, the Shephard input distance function is not the unique known distance function in the literature.During the last two decades, Luenberger [6], Chambers et al. [7], Briec and Lesourd [8] and Briec and Gardères [9] have proposed some new achievements with respect to duality theory and distance functions. To be precise, Luenberger [6] introduced the concept of benefit function as a representation of the amount that a consumer is willing to trade, in terms of a specific reference commodity bundle g, for moving from a utility threshold to a consumption bundle. Later, Chambers et al. [7] redefined the benefit function as an inefficiency measure, introducing the notion of directional input distance function allowing the move from consumer theory to production theory. Chambers et al. showed how the directional input distance function encompasses, among others, the Shephard input distance function. They derived also a dual correspondence between the directional input distance function and the cost function that generalizes all previous dual relationships. Subsequently, Briec and Lesourd [8] introduced the notion of Hölder metric distance function, and additionally an input oriented version, relating the concept of inefficiency and the concept of distance in topology. Following the work by Chambers et al. [7], Briec and Lesourd [8, pages 30–32] derived a duality result based upon the cost function and the input Hölder metric distance function. Another interesting related recent paper is Briec and Gardères [9], who tried to generalize Luenberger’s benefit function in the context of consumer theory. Their generalized benefit function is intimately related to topological norms, which somehow represents a drawback. In fact, they cannot encompass the case of the benefit function when the reference vector g has some zero components. (Given two vectors z,g∈R+n, we define ∥z∥g:=zTg. It is straightforward to prove that if each component of g is strictly positive, then ∥z∥g is a norm. However, if vector g has some zero components, then it is easy to find a numerical example where “∥z∥g=0 if and only if z is the null vector”, a basic property of any norm (see [10]), does not hold.)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. [7].One precursor of this paper is the work by Debreu [11]. Debreu introduced a well-known radial efficiency measure termed “coefficient of resource utilization.” Nevertheless, he derived this coefficient from a much less well-known “dead loss function,” that characterizes the monetary value sacrificed due to inefficiency; that is, it measures an opportunity cost. (The opportunity cost of an action is what you give up to obtain as a consequence of this action. When making any decision, decision makers should be aware of the opportunity costs that accompany each possible action (see [12]).) The minimization problem originally proposed by Debreu to measure such opportunity cost was Minz,pz{pz(z0-z)}, where z0 is a vector representing the actual allocation of resources, z is a vector belonging to the set of optimal allocations (the isoquant), and pz is a vector of the corresponding set of shadow price vectors for z (see [11, page 284]). Debreu pointed out that “pz is affected by an arbitrary positive scalar.” The influence of this multiplicative scalar means that the minimum value can be driven to zero by appropriately scaling all components of pz. In order to eliminate this problem, Debreu proposed to divide the objective function by a price index, reformulating the original problem as Minz,pz{pz(z0-z)/pzz0} or, equivalently, as Minz,pz{pz(z0-z):pzz0=1}. Then, Debreu proved that z*=ρ·z0 is always an optimal solution to the above minimization problem, where the scalar ρ (0<ρ≤1) is known as Debreu’s coefficient of resource utilization.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 Section2, we list the usual requirements that the production set must satisfy and we define a general input distance function. This measure is formulated in a generic way since the normalization restrictions regarding the set of feasible shadow prices are not specifically formulated, allowing us to encompass a large class of normalization constraints. In Section 3, we study the basic properties of the general distance function. Section 4 is devoted to stating that the new distance function can be recovered from the cost function. Additionally, we will show that the general distance function encompasses the Shephard input distance function, the directional input distance function, and the input Hölder metric distance function, among others. In Section 5, we show that the sign of the new distance function is really significant to characterize the feasible allocations. Section 6 shows that Shephard’s duality theorem and other similar results are special cases of a more general duality result. Finally, Section 7 concludes.


## 2. The Input Correspondence and a General Input Distance Function

LetR+n and R++n denote the nonnegative orthant and the positive orthant in the n-dimensional space, respectively. Let 0n denote a vector of n zeros. A vector of m inputs is denoted as x=(x1,…,xm)∈R+m, and a vector of s outputs is denoted as y=(y1,…,ys)∈R+s. Furthermore, pTx denotes the inner product of the vectors p and x. ∥·∥ denotes an arbitrary norm in R+n. Let also 2R+m denote the class of all subsets of R+m.For each output vectory, we can define L(y)={x∈R+m:xyields at least y}, the set of feasible inputs. This characterizes the input correspondence L:R+s→2R+m, which maps each y in R+s to an input set, L(y)⊂R+m, known as the input requirement set.It is assumed here that the input correspondenceL satisfies the following subset of axioms as suggested by Färe et al. [13, pages 23–27]:(P1)
y∈R+s,y≠0s⇒0m∉L(y), and L(0s)=R+m;(P2)
Let{yn} be a sequence such that limn→∞∥yn∥=+∞. Then, ⋂nL(yn)=∅;(P3)
x∈L(y),u≥x⇒u∈L(y);(P4)
L(y) is closed for ally∈R+s;(P5)
L(y) is a convex set for ally∈R+s.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 ofL(y) as a subset of the input requirement set. Postulate P5 states that, given any two input vectors in L(y), the line segment joining them is also in L(y). Although these are the usual general postulates in order to define an input correspondence in the literature, we want to point out that we do not really need P2 to prove the results that appear in this paper.We assume the usual axioms because we want to generalize the most famous distance functions, which have been developed by assuming all these postulates.Giveny∈R+s and p∈R+m, a vector of input prices, in L(y) the input vector x is chosen so as to minimize costs. Therefore, we assume cost minimizing behaviour throughout the paper. In other words, the firm takes prices and outputs as fixed and selects a feasible input vector belonging to L(y) which minimizes their cost. The resulting (optimum) cost is a function of y and p which we denote here by C(y,p).Definition 2.1.
Giveny∈R+s and p∈R+m, the cost function is defined as
(2.1)C(y,p)=min⁡x{pTx:x∈L(y)}.
(We assume that (2.1) attains its minimum in the set L(y). There exist several sufficient conditions in the literature which assure such result. For example, Shephard [14, page 223] assumed that the subset of Pareto-efficient points of L(y) is bounded. A special case is when the technology is a polyhedral, as in a Data Envelopment Analysis (DEA) context. In this case, we do not need any assumption since (2.1) always attains its minimum in the set L(y) (see [15, page 130]).)
The price vectorp need not be strictly positive in the above definition. When pi=0 for some i=1,…,m, the corresponding factor of production is a free item of goods or service, situation of minor interest in economics but considered in the last definition. The same can be said regarding the case L(y)=∅. Indeed, we focus our attention on a specific set of output vectors: DomL={y∈R+s:L(y)≠∅}, that is, the set of producible outputs. On the other hand, if L(y)=∅, then we define C(y,p)=+∞, as usual. In addition, let us observe that C(y,p)≥0 since p∈R+m and L(y)⊂R+m.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 ofL(y), as defined in what follows.Definition 2.2.
The isoquant ofL(y) is defined by
(2.2)IsoqL(y)={x:x∈L(y),λ<1⟹λx∉L(y)},y∈R+s.
Using Definition 2.2, we say that (x,y) is input-isoquant efficient if x belongs to the isoquant of L(y). Due to the assumed postulates (specially P3), if y∈DomL and p∈R+m, then the optimal solution(s) of problem (2.1) are achieved necessarily at some input-isoquant efficient point of L(y).Now, we are ready to introduce the general input distance function. This new notion is defined with respect to a given normalization set denoted asNS. Nevertheless, before introducing the aforementioned notion, we need to introduce some notation.Definition 2.3.
Letx̅∈IsoqL(y). Then, p̅∈R+m∖{0m} is called a shadow price vector for x̅ if and only if  p̅Tx̅≤p̅Tu for all u∈L(y).By postulates P4 and P5 and applying the separating hyperplane theorem, we know that for eachx̅∈IsoqL(y) there exists at least a shadow price vector p̅∈R+m∖{0m}.Definition 2.4.
Let(x,y)∈R+m×R+s be an input-output vector. Let L(y) be an input requirement set satisfying P1–P5. And let also NS be a subset of R+m. Then, the function GI:R+m×R+s×2R+m→[-∞,+∞] defined as
(2.3)GI(x,y;NS)=inf⁡x̅,p̅{p̅T(x-x̅):x̅∈IsoqL(y),p̅∈Qx̅∩NS},
where Qx̅={p∈R+m:pis a shadow price vector ofx̅}, is called the general input distance function.
As a referee points out, the formulation ofGI(x,y;NS) could be easier if we remove the constraint x̅∈IsoqL(y) in (2.3), deriving an equivalent expression. It is due to the fact that it follows already by the supposition that p̅ is a shadow price of x̅. However, in order to follow a formulation similar to that used by Debreu [11] to define the dead loss function, we will use (2.3) hereafter.This general input distance function can be interpreted as a measure of the distance fromx∈R+m to the boundary of L(y). To get a distance with an economic meaning, we evaluate (x-x̅) times the shadow price vector p̅, associated with x̅∈IsoqL(y). In economics terms, the general input distance function represents the monetary value sacrificed due to technical inefficiency. In this sense, the general input distance function could be understood as an opportunity cost for firms.Obviously,GI will take different values depending on the structure of the normalization set NS and the vector (x,y). In particular, if (⋃x̅∈IsoqL(y)Qx̅)∩NS=∅, then we set LI(x,y;NS)=+∞, as usual. Nevertheless, from now on, we will assume in our results that (⋃x̅∈IsoqL(y)Qx̅)∩NS≠∅, which guarantees GI(x,y;NS)<+∞.The above general input distance function has the same arbitrary multiplicative scalar problem pointed out by Debreu [11]. Debreu showed that “pz is affected by an arbitrary positive scalar” in his formulation of the dead loss function. The influence of this multiplicative scalar in our formulation means that the minimum value can be driven to zero by appropriately scaling all components of p̅. Therefore, the set NS in the definition of GI should satisfy some property which avoids this problem in practice. To this end, the next condition must hold.(C1)
NS is a closed set and 0m∉NS; that is, NS is bounded away from zero.Obviously, any setNS⊂R+m that satisfies C1 avoids the arbitrary multiplicative scalar problem in (2.3). In fact, it is a sufficient, but not necessary, condition. Since the vector 0m does not belong to the closed set NS, the distance from 0m to NS is strictly positive and we cannot achieve 0m scaling any p∈NS.Another regularity condition on the setNS, which will be used below in some results, states that NS must contain at least a “representative” of each ray that belongs to the cone R+m.(C2)
for allp∈R+m∖{0m}there existsk>0 such that kp∈NS.


## 3. Properties of the General Input Distance Function

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.Proposition 3.1.
Let(x,y)∈R+m×R+s and u∈R+m. Let L(y) be an input requirement set that satisfies P1–P5. Let NS be a subset of R+m that satisfies C1. Then, the function GI satisfies the following properties:(a)
u≥x⇒GI(u,y;NS)≥GI(x,y;NS);(b)
GI(x,y;NS) is concave in x;(c)
GI(x,y;NS) is continuous with respect to x on each open convex subset of R+m in which the general input distance function is finite;(d)
letx∈R+m be a vector such that GI(x,y;NS) is finite. For each u∈R+m, there exists the one-sided directional derivative of the general input distance function at x with respect to the vector u.Proof.
(a)u≥x⇒u-x̅≥x-x̅, for allx̅∈R+m. Then, we have that p̅T(u-x̅)≥p̅T(x-x̅), for allx̅∈IsoqL(y), for allp̅∈Qx̅∩NS. Therefore, GI(u,y;NS)≥GI(x,y;NS).
(b) Letx1,x2∈R+m and λ∈[0,1]. It holds that
(3.1)p̅T(λx1+(1-λ)x2-x̅)=λ[p̅T(x1-x̅)]+(1-λ)[p̅T(x2-x̅)],∀x̅∈IsoqL(y),∀p̅∈Qx̅∩NS
since x̅=λx̅+(1-λ)x̅. Then, since p̅T(x1-x̅)≥GI(x1,y;NS) and p̅T(x2-x̅)≥GI(x2,y;NS), for allx̅∈IsoqL(y) and for allp̅∈Qx̅∩NS, we have that
(3.2)GI(λx1+(1-λ)x2,y;NS)=inf⁡x̅,p̅{p̅T((λx1+(1-λ)x2)-x̅):x̅∈IsoqL(y),p̅∈Qx̅∩NS}≥λGI(x1,y;NS)+(1-λ)GI(x2,y;NS).(c) and (d) The concavity of the general input distance function implies directly these properties (see [15, page 62 for (c)] and [16, page 214 for (d)]). This concludes the proof.


## 4. How to Recover the General Input Distance Function from the Cost Function

Shephard [5, 14] developed a great deal of duality theory. In particular, he established a duality relationship between the cost function and the Shephard input distance function. In this section, we prove that the general input distance function can be recovered from the cost function, establishing a new duality result. Later, we will also show that the general input distance function encompasses a wide family of well-known distance functions.Theorem 4.1.
Let(x,y)∈R+m×DomL. Let L(y) be an input requirement set that satisfies P1–P5. Let NS be a subset of R+m that satisfies C1. Then,
(4.1)GI(x,y;NS)=inf⁡p∈R+m{pTx-C(y,p):p∈NS}.Proof.
For anyx̅∈IsoqL(y) and p̅∈Qx̅∩NS, we have that p̅T(x-x̅)=p̅Tx-p̅Tx̅=p̅Tx-C(y,p̅), where the last equality holds because p̅ is a shadow price vector of x̅. In addition, (⋃x̅∈IsoqL(y)Qx̅)∩NS⊂NS. Therefore, it is apparent that infp∈R+m{pTx-C(y,p):p∈NS}≤GI(x,y;NS).
To prove the other inequality, sincep̅∈R+m, we have that there existsx̅∈IsoqL(y) such that C(y,p̅)=p̅Tx̅. In other words, p̅∈Qx̅∩NS. Additionally, we have that p̅Tx-C(y,p̅)=p̅T(x-x̅). Then, p̅T(x-x̅)≥GI(x,y;NS). Finally, by the definition of infimum, we have that infp{pTx-C(y,p):p∈NS}≥GI(x,y;NS).Theorem4.1 states a first duality result: the general input distance function can be recovered from the cost function. The main implication of this result is that several interesting specific normalization sets satisfy their hypothesis. Hence, it will allow us to derive important relations between the general input distance function and some well-known distance functions.Briec and Lesourd [8, page 31] defined the input Hölder metric distance function (it is denoted here as DTt,i(x,y), where ℓq is the dual space of ℓt with 1/t+1/q=1) and related it to the cost function. This distance function is a special case of the general input distance function, introduced in this paper, as we show next. If we consider as normalization set NS={p∈R+m:∥p∥q≥1}, then we get, by Theorem 4.1, GI(x,y;NS)=infp{pTx-C(y,p):∥p∥q≥1}=DTt,i(x,y), where the last equality holds thanks to Proposition  4.1 in Briec and Lesourd [8]. The conclusion is that GI encompasses the input Hölder metric distance function.Other special case of the general input distance function which is the interest is the directional input distance function [7]. In particular, when we consider NS={p∈R+m:pTg=1}, with g∈R+m a nonzero vector, we get GI(x,y;NS)=infp{pTx-C(y,p):pTg=1} thanks to Theorem 4.1. Chambers et al. [7, page 413] proved that the directional input distance function (it is denoted here as D⃗i(x,y;g)) can be recovered from the cost function by means of infp{pTx-C(y,p):pTg=1}. As a direct consequence, we get that GI(x,y;NS)=D⃗i(x,y;g).On the other hand, it is well known that the directional input distance function generalizes the Shephard input distance function (see [7, page 411]) when g=x. In this sense and thanks to the above paragraph, the general input distance function encompasses the Shephard input distance function as well as other families of distance functions.


## 5. Characterization of the Input Requirement Set

One of the main uses of the distance functions is to characterize whether an input vectorx∈R+m belongs or not to L(y), assuming that  y is producible. Next we characterize the input requirement set by means of the value of the general input distance function. The proof is based on the next two lemmas, which need different hypothesis.Lemma 5.1.
Let(x,y)∈R+m×DomL(y). Let L(y) be an input requirement set that satisfies P1–P5. Let NS be a subset of R+m. Then, if x∈L(y), then one has that GI(x,y;NS)≥0.Proof.
for allx̅∈IsoqL(y) and for allp̅∈Qx̅∩NS, we have, by definition of shadow price vector, p̅Tx̅=C(y,p̅). In addition, we know that p̅Tx≥C(y,p̅) since x∈L(y). Therefore, p̅T(x-x̅)=p̅Tx-p̅Tx̅≥0for allx̅∈IsoqL(y) and for allp̅∈Qx̅∩NS. Finally, by  (2.3), GI(x,y;NS)≥0.Lemma 5.2.
Let(x,y)∈R+m×DomL. Let L(y) be an input requirement set that satisfies P1–P5. Let NS be a subset of R+m that satisfies C1 and C2. Then, if x∉L(y), then one has that GI(x,y;NS)<0.Proof.
Applying the separation of a convex set and a point theorem, we get that there existsp∈R+m∖0m such that pTx<C(y,p). By C2 there existsk>0 such that p̅=kp∈NS. Moreover, there exists x̅∈L(y) (in fact, x̅∈IsoqL(y)) which satisfies pTx̅=C(y,p), that is, p̅∈Qx̅. Hence, p̅T(x-x̅)=k[pTx-pTx̅]<0. Finally, GI(x,y;NS)<0 since we are seeking an infimum in (2.3).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.Proposition 5.3.
Let(x,y)∈R+m×DomL. Let L(y) be an input correspondence that satisfies P1–P5. Let NS be a subset of R+m that satisfies C1 and C2. Then, x∈L(y) if and only if  GI(x,y;NS)≥0.Finally, the following proposition states the intuitive result thatGI(x,y;NS)=0 is a necessary condition for input-isoquant efficiency.Proposition 5.4.
Let(x,y)∈R+m×DomL. Under the same hypothesis of Proposition 5.3, if (x,y) is input-isoquant efficient, then GI(x,y;NS)=0.Proof.
Let(x,y)∈R+m×DomL be an input-isoquant efficient point of the input requirement set L(y). Then, it must exist p∈R+m∖{0m} such that p∈Qx. Then, by C2, there existsk>0 such that kp∈NS and, consequently, kp∈Qx∩NS. This implies that (kp)T(x-x)=0≥GI(x,y;NS), by the definition of the general input distance function. Finally, GI(x,y;NS)≥0 since Lemma 5.1 holds and, necessarily, GI(x,y;NS)=0.


## 6. How to Recover the Cost Function from the General Input Distance Function

The dual “general” relationship (4.1) says that the general input distance function can be recovered from the cost function. This section is devoted to study the converse result, that is, how to derive the cost function from the general input distance function. Additionally, we will show that this result generalizes all previous dual connections between distance functions and the cost function.Lemma 6.1.
Let(x,y)∈R+m×DomL. Let L(y) be an input requirement set that satisfies P1–P5. Let NS be a subset of R+m that satisfies C1. Let also p∈NS. Then,
(6.1)C(y,p)=inf⁡x∈R+m{pTx-GI(x,y;NS)}.Proof.
By Theorem4.1, we have that GI(x,y;NS)≤pTx-C(y,p), for allx∈R+m. It implies that C(y,p)≤pTx-GI(x,y;NS), for all x∈R+m. Hence,
(6.2)C(y,p)≤inf⁡x∈R+m{pTx-GI(x,y;NS)}.
To prove the converse inequality, given any ε>0there existsx∈L(y) such that pTx≤C(y,p)+ε since C(y,p)>-∞ (in fact, C(y,p)≥0). Since we know that GI(x,y;NS)≥0 by Lemma 5.1, pTx-GI(x,y;NS)≤C(y,p)+ε. Now, through the definition of ε, infx∈L(y){pTx-GI(x,y;NS)}≤C(y,p). Finally, since L(y)⊂R+m, we have that infx∈R+m{pTx-GI(x,y;NS)}≤C(y,p).When a price vectorp does not belong to the normalization set NS, the above result is true if there exists at least a representative vector in the ray associated with p which belongs to NS. This idea is formalized by the following theorem.Theorem 6.2.
Let(x,y)∈R+m×DomL. Let L(y) be an input requirement set that satisfies P1–P5. Let NS be a subset of R+m that satisfies C1. Let also p∈R+m be such that there existsk>0 with kp∈NS. Then,
(6.3)C(y,p)=inf⁡x∈R+m{pTx-k-1GI(x,y;NS)}.Proof.
Sincekp∈NS, by Lemma 6.1, C(y,kp)=infx∈R+m{(kp)Tx-GI(x,y;NS)}. By homogeneity of degree +1 of the cost function, C(y,kp)=kC(y,p). It implies that
(6.4)C(y,p)=k-1C(y,kp)=k-1inf⁡x∈R+m{(kp)Tx-GI(x,y;NS)}=inf⁡x∈R+m{pTx-k-1GI(x,y;NS)}.Next, we apply Theorem6.2 in order to get the well-known dual connections that allow to recover the cost function from a specific distance function, as stated by Shephard [5, 14], Chambers et al. [7], and Briec and Lesourd [8].First, we study the case of the input Hölder metric distance function of Briec and Lesourd [8]. Given any p∈R+m∖{0m}, k:=∥p∥q-1>0 yields that kp∈NS={p∈R+m:∥p∥q≥1}. As a consequence, we can apply Theorem 6.2 and that GI(x,y;NS)=DTt,i(x,y) (see Section 4) to get C(y,p)=infx∈R+m{pTx-∥p∥qDTt,i(x,y)}. This result is similar, but not identical, to Proposition 4.1 (a) due to Briec and Lesourd [8, page 31]. (C(y,p)=infx{pTx+DTt,i(x,y):x∈L(y)} in the paper of Briec and Lesourd [8]) Briec and Lesourd need to know DTt,i(x,y) and L(y) to derive the cost function of the firm. Unlike them, we only need to know DTt,i(x,y) to recover the same cost function. In this sense, our result seems closer to the spirit of the duality in microeconomics.In the case of the directional input distance function of Chambers et al. [7], taking into account that given any p∈R++m and g∈R+m∖{0m}, k:=(pTg)-1 yields that kp∈NS={p∈R+m:pTg=1}, we have that C(y,p)=infx∈R+m{pTx-pTgD⃗i(x,y;g)} by applying directly Theorem 6.2 and realizing that GI(x,y;NS)=D⃗i(x,y;g) (see Section 4). This derived result was first stated in Chambers et al. [7, page 413]. Hence, the dual correspondence developed by Chambers et al. between the cost function and the directional input distance function is a particular case of the general duality result (6.3).Finally, takingg=x, it is possible to deduce Shephard’s duality result (Di(x,y) denotes here the Shephard input distance function) C(y,p)=infx∈R+m{pTx/Di(x,y)} [14, Chapter 8] from C(y,p)=infx∈R+m{pTx-pTgD⃗i(x,y;g)}, as Chambers et al. [7] showed in their paper. Therefore, Theorem 6.2 encompasses Shephard’s dual correspondence as well.


## 7. Conclusions

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 [17, 18]).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 ofm inputs and s outputs, then it is a polyhedron.) Hence, the input-oriented BCC and the input-oriented CCR DEA models collapse in to the general input distance function (see for more details [19]).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 [6]. So, we believe that this line may be a good avenue for further follow-up research.



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