Properties of Inefficiency Indexes on Input, Output Space

Transcription

1 Properties of Inefficiency Indexes on Input, Output Space by R. Robert Russell University of California, Riverside, and University of New South Wales and William Schworm University of New South Wales September 2010 We thank Erwin Diewert, Rolf Färe, Daniel Primont and the referees for suggestions that substantially improved the paper. The paper has also benefitted from discussions with conference participants at the 2007 European Workshop on Efficiency and Productivity Analysis in Lille, the 2009 Economic Measurement Group Workshop at the University of New South Wales, and the 2010 Efficiency and Productivity Workshop at the Auckland University of Technology.

2 Abstract We analyze efficiency measurement in the full input, output space. We posit four types of axioms: indication (of efficient production bundles), monotonicity, independence of units of measurement, and continuity (in technologies as well as input and output quantities). Impossibility results establish a tension between indication and continuity. We focus on seven well-known inefficiency indexes from the operations-research and economics literature, establishing the properties they satisfy and do not satisfy on a general class of technologies satisfying minimal regularity conditions and on the subset of these technologies satisfying convexity. We also discuss several other indexes that are dominated by or very similar to these seven indexes. The set of properties satisfied by these indexes elucidates the trade-offs faced in selecting among the indexes. JEL classification: C43; C61; D24. Keywords: Technical efficiency indexes; technical efficiency axioms.

3 1. Introduction. Axiomatic analysis of efficiency measurement began with Färe and Lovell [1978], who proposed three axioms that an input-based efficiency index should satisfy: indication (the index is equal to one if and only if the input vector is efficient in the sense of Koopmans [1951]), monotonicity (an increase in any input, holding other inputs as well as all outputs constant, reduces the value of the index), and homogeneity (e.g., doubling all inputs, holding outputs constant, cuts the value of the index in half). Subsequently, Russell [1985, 1988, 1990] clarified the initial Färe and Lovell axioms and introduced two additional axioms for input-based efficiency measurement: invariance with respect to units of measurement (also known as commensurability) and continuity (in technologies as well as input and output quantities). 1 In recent years, empirical research on efficiency (or inefficiency) measurement has focused much more on measurement in the full space of inputs and outputs, which we refer to as input, output space. Numerous papers have proposed specific indexes satisfying certain properties. To our knowledge, however, no systematic comparison of indexes defined on this space has been carried out. In this paper, we provide an axiomatic analysis of inefficiency measurement in the full input, output space, referred to as graph efficiency measurement by Färe, Grosskopf, and Lovell [1985]. We posit four types of axioms: indication, monotonicity, independence of units of measurement, and continuity. 2 We analyze seven well-known inefficiency indexes from the operations-research as well as the economics literature: the hyperbolic index (Färe, Grosskopf, and Lovell [1985]), the Färe-Grosskopf-Lovell (FGL) index (Färe, Grosskopf, and Lovell [1985]), the weighted additive (slacks-based) index (Charnes, Cooper, Golany, Seiford, and Stutz [1985]), the quantity-weighted additive (slacks-based) index (Charnes, Cooper, Golany, Seiford, and Stutz [1985]), the directional distance index (Luenberger [1992], Chung, Färe, and Grosskopf [1997], and Chambers, Chung, and Färe [1998]), the Briec index (Briec [1997]), and the quantity-weighted Holder distance function (Briec [1998]). Many elaborations of these indexes have appeared in the literature and we discuss a few of these by comparing them with those above. 1 Although this early literature focused on input-oriented efficiency indexes, each of these axioms has an obvious analog for output-oriented efficiency indexes (e.g., homogeneity implies that halving all output quantities, holding input quantities constant, cuts the value of the index in half). 2 Homogeneity is not considered, since it is not obvious how this property should be extended to input, output space. We have formulated some possible extensions, but none is satisfied by any of the indexes we consider. Another axiom that has received considerable attention in the operations research literature is translation invariance: invariance with respect to affine transformations of the variables (see, e.g., Lovell and Pastor [1995]). This property is important for computational reasons when the data contain zero or negative values. It is not, however, important for our analysis, which focuses on intrinsic appeal of the axioms rather than on computational facility. Moreover, we consider only positive values of inputs and outputs in this paper.

4 In order to encompass technologies estimated by a variety of techniques, we do not restrict technologies to those generated by popular empirical methods of measuring inefficiency. Rather, the class of permissible technologies is general and in particular is not restricted to those generated by Data Envelopment Analysis (DEA) or stochastic frontier procedures. Our analysis applies to technologies constructed by a variety of techniques including DEA and stochastic frontier methods as well as non-parametric procedures. Given the importance of the convexity restriction in both theoretical and empirical research, we also consider the properties of inefficiency indexes restricted to the particular class of convex technologies. Many inefficiency indexes were initially defined in the literature for restricted classes of technologies. Those we have selected for study may be extended to the general class of technologies. Three of the indexes the directional distance index, the quantity-weighted Holder index, and the weighted additive index are classes of indexes that depend on parameters. We restrict the parameters if this improves the performance of the index with respect to our axioms. This restriction is discussed in more detail when we define the indexes. Although our axioms seem to be natural requirements for an inefficiency index, we present an impossibility result demonstrating that no index can satisfy all of the axioms, even when we restrict the space of technologies to those that are convex. Specifically, no inefficiency index can satisfy both indication and continuity (in either quantities or technologies when the latter are not required to be convex and in technologies when they are convex). Thus, the set of all (undominated) indexes can be partitioned into those satisfying indication and those satisfying continuity. Our full axiomatic analysis shows that, among those indexes satisfying continuity (the hyperbolic index, the directional distance index, the Briec index, and the quantityweighted Holder index), no trade-offs remain; they can be treated as an equivalence class in terms of our axiom system. Among those indexes satisfying indication, the weighted additive index dominates the Färe-Grosskopf-Lovell and the quantity-weighted additive indexes, since the latter do not satisfy monotonicity whereas the former does. Restricting technologies to be convex, we show that all of the indexes satisfy continuity in quantities. However, there are still two groups of indexes: those that are continuous in technologies and those that satisfy indication. All of the indexes that are continuous in technologies share the same properties while, among those that satisfy indication, the weighted additive index is dominant. Section 2 introduces the notation and describes the general assumptions on technologies. Section 3 defines an inefficiency index and discusses the seven indexes that we feature, while Section 4 describes the axioms we use to evaluate the indexes. Section 5 presents our results: Theorem 1 establishes the impossibility of satisfying both indication and continuity, and Theorems 2 and 3 catalog the performance of each of the seven indexes in terms of our axiomatic structure. The proofs of Theorems 2 and 3 are presented

5 in the Appendix. Section 6 discusses the implications of these results for some additional inefficiency indexes. Section 7 concludes. 2. Preliminaries. A firm (or other production unit) uses n inputs to produce m outputs with production vectors, denoted x, y, contained in the input, output space R n+m +. Denote the origin of this space by 0 [n], 0 [m] and the unit vector by 1 [n], 1 [m]. The firm s technology set T R n+m + contains feasible production vectors. The boundary of T is denoted by 3 T = { x, y T x, y x, ȳ = x, ȳ / T }. (2.1) A production vector x, y T is technologically efficient (in the sense of Koopmans [1951]) if x, y > x, ȳ implies x, ȳ / T. The efficient subset of T, denoted Eff(T), is a subset of T. Denote the output possibility set for given x by P(x) = { y R m + x, y T } and the input requirement set for given y by L(y) = {x R n + x, y T }.4 Most of the theoretical literature on technical efficiency measurement has focused on a general class of technologies satisfying only very weak regularity conditions: namely, the collection of non-empty, closed technology sets that satisfy the following conditions: 5 (i) x, y T and x, ȳ > x, y implies x, ȳ T (free disposability of inputs and outputs), (ii) y > 0 [m] implies 0 [n], y / T, and (iii) the production possibility set, P(x), is non-empty and bounded for all x R n +. We denote by T the set of non-empty, closed technologies satisfying these conditions. Some theoretical, as well as empirical, studies consider the subset of T satisfying convexity. Denote this subset by T C. 3. Inefficiency Indexes. A technological inefficiency index measures the distance 6 from the production vector to the frontier of T or to its efficient subset Eff(T). Typically, the production point 3 Vector notation: x x if xi x i for all i; x > x if x i x i for all i and x x; and x x if x i > x i for all i. 4 That is, P : R n + 2 Rm + is the output correspondence and L : R m + 2 Rn + is the input correspondence. 5 All but free disposability of these conditions are necessary to guarantee that our efficiency indexes are well defined. Free disposability could be dispensed with (theoretically); the only change that would be needed in what follows would be to redefine the inefficiency indexes on the free-disposal hull of T rather than on T itself (as in Russell [1988] for input-based efficiency indexes). Note, finally, that the free disposability assumption implies connectedness of the technology, a property that we exploit. 6 We surround distance with quotation marks to underscore the informal nature of this notion, which is not consistent with the formal mathematical concept of distance. A quasi-exception is the quantityweighted Holder inefficiency index, which is based on a mathematical distance function.

6 is compared to a particular point the reference vector on the boundary or the efficient subset of T. The principal issues addressed in formulating a specific inefficiency index are (i) the selection of the reference vector corresponding to any production vector and (ii) the specification of the distance between the production vector and the reference vector. Many of the inefficiency indexes in the literature are not well defined or have unacceptable properties at the boundary of input, output space. To avoid a loss of focus on the basic structures of these indexes and the relationships among them, we restrict our attention here to strictly positive quantities. 7 The technology, however, does contain points on the boundary of R n+m +. Define an inefficiency index as a real-valued mapping I, with image I(x, y, T), that assigns an inefficiency value to a production vector and technology set. As an inefficiency index is defined only for feasible production vectors, the domain is defined by Ξ = { x, y, T x, y T T T }. (3.1) If the inefficiency index is restricted to convex technologies, the domain is Ξ C = { x, y, T x, y T T T C }. (3.2) The range of I is [0, + ) and the effective range, denoted R(I) [0, + ), varies across specific indexes. Many of the indexes we consider were originally defined on the particular subset of T generated by mathematical programming methods of constructing technology sets on a finite set of data points. This method, commonly referred to as Data Envelopment Analysis (DEA), generates convex polyhedral reference technologies (i.e., intersections of finite numbers of half-spaces). 8 Many of these indexes, however, can be applied to the more general class of technologies T. 9 To our knowledge, the first formulation of an inefficiency index in the full input, output space, attributable to Färe, Grosskopf, and Lovell [1985, pp ], 10 is the hyperbolic inefficiency index, defined by I H (x, y, T) = max {λ x/λ, λy T }. (3.3) 7 We do not mean to imply that these boundary issues are unimportant; on the contrary, since data sets may contain zero values of input or output quantities, these boundary issues need to be dealt with in some circumstances. See Russell and Schworm [2009a] for an analysis of boundary problems. 8 See Charnes, Cooper, Lewin, and Seiford [1994]. 9 The Range Adjusted Measure of Inefficiency (RAM), attributable to Cooper, Park, and Pastor [1999], depends critically on the DEA construction of the technology and hence cannot be extended to the general class of technologies. This index is discussed in Section They refer to this index as the Farrell Graph Measure of Technical Efficiency and give it additional attention in Färe, Grosskopf, and Lovell [1994, Ch. 8].

7 This index contracts inputs and expands outputs along a (particular) hyperbolic path to the frontier and maps into the [1,+ ) interval. The directional distance inefficiency index, I DD, adapted from the shortage function of Luenberger [1992] to the measurement of inefficiency by Chung, Färe, and Grosskopf [1997], 11 is defined by I DD (x, y, T, g) = max { λ x λg x, y + λg y T }, (3.4) where g = g x, g y R n+m ++ and the units of each element of g are the same as the corresponding element of x, y. This index measures the feasible contraction/expansion in the direction g and maps into R +. We have restricted the parameters of the directional distance index in two ways: we require g 0 [n], 0 [m] and we require the units of g to be the same as those of the production vector. Although Luenberger [1992] and others restrict the direction g only to the non-negative orthant, we choose to restrict g to the positive orthant because this enhances the number of axioms notably continuity axioms that the directional distance index satisfies. Also, by our choice of units for g, we ensure that the index is unit independent. The directional distance index is actually a class of indexes parameterized by the distance vector g, and we choose to examine only a subset of this class of indexes in order to improve its properties. Briec [1997] proposes an alternative to the directional distance function by using the definition of I DD with the direction g = x, y. The Briec inefficiency index, defined by { I B (x, y, T) = max λ (1 λ)x, (1 + λ)y } T, (3.5) maps into the [0, 1] interval. 12 Figure 1 illustrates the paths followed by I H, I B, and I DD indexes for a (non-convex) technology with one input and one output. The technology set is the shaded area, with the boundary given by the thick solid line. Three initial production vectors are displayed (x, y), (x, y ), and (x, y ) with the arrows indicating the reference point. Note that the paths for I H and I B are not parallel while the paths of I DD are parallel. 11 See also Chambers, Chung, and Färe [1996], Cherchye, Kuosmanen, and Post [2001], and Färe and Grosskopf [2000]. A precursor of this literature is Diewert [1983], where the directional distance function is introduced to measure waste in production. 12 Briec and others argue that this inefficiency index is a special case of the directional distance function, with g = x, y ; this is not strictly correct, since the formal definition of the directional distance function specifies a direction g that is independent of x, y. In fact, if one were to specify a generalized directional distance function as in (3.4) but with g being a function of x, y, then the standard directional distance inefficiency index (3.4) and the Briec inefficiency index (3.5) would each be special cases of the generalized directional inefficiency index.

8 Figure 1: The Hyperbolic, Directional Distance, and Briec Inefficiency Indexes. y I H : I B : I DD : T x, y x, y x, y x The three indexes described above depend on a specified path from an initial point to the boundary of the technology set. Alternatively, Briec [1998] proposes an index that is built on an explicit mathematical distance function. He restricts technologies to be convex, but the index is easily extended to T. While he introduces three different indexes, we focus on the quantity-weighted Holder index, which satisfies unit independence. The quantity-weighted Holder distance function, measuring the distance from a production vector to the boundary of the technology set, T, is given by I QWH (x, y, T) = min {h(x, y, x x, y ȳ) x, ȳ T } (3.6) x,ȳ where n m h(x, y, u, v) = u i /x i k + v j /y j k i=1 j=1 1/k (3.7) for k [ 1, ) and { h(x, y, u, v) = max max i } { u i /x i }, max { v j /y j } j (3.8)

9 Figure 2: The Quantity-Weighted Holder Inefficiency Index. y x, y x, y T x, y x for k =. The quantity-weighted Holder inefficiency index maps into the [0, + ) interval. Figure 2 illustrates the paths followed by I QWH starting at three different production vectors for the case with k = 2. Färe, Grosskopf, and Lovell [1985, pp ] have formulated an extension of the input-based Färe-Lovell [1978] efficiency index to the full input, output space: 13,14 { i E FGL (x, y, T) = min α i + j β } j α, β Ω(x, y, T), (3.9) α,β n + m where Ω(x, y, T) = { α, β α x, y β T 0 [n] α 1 [n] 0 [m] β 1 [m] }, (3.10) 13 For obscure historical reasons, they refer to this index as the Russell Graph Measure of Technical Efficiency. 14 Russell and Schworm [2009a] have recently modified this index to eliminate critical problems at the boundary of output space. Since we are considering production vectors that are strictly positive, these problems do not affect us here.

10 α x = α 1 x 1,...,α n x n, and y β = y 1 /β 1,..., y m /β m, for α = α 1,...,α n and β = β 1,...,β m. 15 To make the index comparable to the other indexes in terms of the axioms that follow, we define the FGL inefficiency index by I FGL (x, y, T) = 1 E FGL (x, y, T), (3.11) with range [0, 1]. 16 The FGL index is (the negative of) an average of the coordinate-wise maximal contractions of inputs and expansions of outputs. The weighted additive index (Cooper and Pastor [1995] 17 ), defined on DEA technologies, can be extended to the class of technologies T as follows: I WA (x, y, T) = max s,t u i s i + i j v j t j s, t Γ(x, y, T), (3.12) where u R m ++ and v Rm ++ are pre-specified weights with the units of u i and v j equal to, respectively, the inverse of the units of x i and y j and Γ(x, y, T) = { s, t x s, y + t T s 0 [n] t 0 [m] }. (3.13) This index minimizes the weighted sum of the input and output slacks and maps into the [0, + ) interval. We have restricted the weights in the definition of I WA to be strictly positive in order to improve its performance with respect to our axioms. Without this restriction, this index fails to satisfy both the indication property and the monotonicity property to be discussed below. In the original paper introducing the well-known additive index, 18 Charnes, Cooper, Golany, Seiford, and Stutz [1985] propose a quantity-weighted modification: I QWA (x, y, T) = max s,t i s i x i + j t j s, t Γ(x, y, T) y j, (3.14) where Γ is defined in (3.13). This index is similar to the weighted additive index with the weights replaced by the inverses of the corresponding input and output quantity variables. It also maps into the [0, + ) interval. 15 Although Ω(x, y, T) is not a closed set, the min in (3.9) exists, owing to our restriction that y 0 [m] and our assumption that the production set P(x) is bounded for all x R n An alternative that satisfies the same properties is IFGL (x, y, T) = [ E FGL (x, y, T)] 1, with range [1, + ). 17 See also Cooper, Park, and Pastor [1999]. 18 The additive index is the special case of the weighted additive index with all weights equal to 1. We do not consider this index in our basic analysis since it has been superseded (and is dominated) by the weighted versions.

11 Figure 3: The FGL, Weighted Additive, and Quantity-Weighted Additive Inefficiency Indexes. y I FGL : I WA : I QWA : T x, y x, y x, y x The I FGL, I WA, and I QWA indexes contract all inputs and expand all outputs until an efficient production vector is achieved. Therefore, as illustrated in Figure 3, all feasible input, output vectors are compared to efficient points. Many other (in)efficiency indexes have been formulated in the literature, but we have chosen not to include them in our basic analysis because (i) they are defined only for special classes of technologies such as the DEA class and cannot be straightforwardly extended to the general class T, (ii) they are dominated, in the context of our axiomatic structure, by indexes that we do consider, or (iii) they are similar to, or even equivalent to, an index that we do analyze and hence have identical axiomatic properties. Several of these indexes are discussed briefly in Section Axioms. We propose four types of axioms as suitable for inefficiency measures defined on the full space of inputs and outputs. Two of the axioms indication and monotonicity are obvious extensions of axioms proposed by Färe and Lovell [1978] for input-based measures

12 of efficiency. We also consider a weaker version of monotonicity. We posit an axiom of invariance with respect to changes of units of measurement and continuity axioms in production vectors and in technologies, as proposed by Russell [1990] for input-oriented efficiency indexes. The most basic axiom requires that an inefficiency index distinguish between inefficient and efficient production vectors: 19 Indication of Efficiency (I): There exists a θ 0 such that I(x, y, T) θ for all x, y, T Ξ and I(x, y, T) = θ if and only if x, y Eff(T). 20 The two monotonicity axioms are as follows: Monotonicity (M): For all pairs x, y, T Ξ and x, y, T Ξ satisfying x, y < x, y, I(x, y, T) < I(x, y, T). Weak Monotonicity (WM): For all pairs x, y, T Ξ and x, y, T Ξ satisfying x, y < x, y, I(x, y, T) I(x, y, T). Our unit independence axiom is an extension to input, output space of that proposed by Russell [1988] for input-based efficiency indexes: 21 Unit Independence (UI): For all x, y, T Ξ and for all ω x R n ++ and ω y R m ++, if ˆx = ω x x, ŷ = ω y y and T = {(ˆx, ŷ) (ˆx ω x, ŷ ω y ) T }, (4.1) then I(x, y, T) = I(ˆx, ŷ, T). (4.2) This axiom requires that unit changes have no effect on the inefficiency index. We consider three continuity axioms. Russell [1990] argued (page 256) that continuity is a compelling property, for it provides assurance that small errors of measurement (of, e.g., input or output quantities) result only in small errors of efficiency measurement. If the technology is constructed from data on production vectors, the argument for continuity in the technology is perhaps even more compelling. We therefore believe the strongest of these continuity axioms continuity in both input and output quantities and in technologies is a desirable property for an inefficiency index. Since some standard inefficiency indexes do not satisfy this strong continuity property, we consider weaker versions as well. 19 We introduce these axioms in the context of the general set of technologies T ; they are modified for the class of convex technologies T C by replacing Ξ with Ξ C. 20 In the operations research literature (e.g., Cooper, Park, and Pastor [1999]), an index satisfying (I) is said to be comprehensive. 21 Following the nomenclature of Eichhorn and Voeller [1976] in their axiomatic analysis of price and quantity indexes, Russell [1988] referred to this property as commensurability.

13 Continuity in production vectors (C x, y ): I is continuous in x, y. Continuity in technologies (C T): I is continuous in T. 22 Joint continuity (C x, y, T ): I is continuous in x, y, T. 5. Results. A fundamental incompatibility among our axioms is encapsulated in the following result. Theorem 1: (a) There does not exist an inefficiency index satisfying (I) and (C x, y ) on T. (b) There does not exist an inefficiency index satisfying (I) and (C T) on T C (nor, a fortiori, on T ). Proof: (a) In Figure 4, x ν, y ν x o, y o. As x ν, y ν is efficient for all ν and x o, y o is inefficient, (I) implies I(x ν, y ν, T) = θ for all ν and I(x o, y o ) > θ, violating (C x, y ). Figure 4: Incompatibility of (I) and (C x, y ). y x ν, y ν x 0, y 0 I(x ν, y ν, T) = θ, I(x 0, y 0, T) > θ x ν, y ν x 0, y 0 T x 22 As in Russell [1990], we adopt the topology of closed convergence on T.

14 (b) Consider the sequence of technologies given by T ν = { x 1, x 2, y R 3 + x1, x 2, y λ 1 1, 2, 1 + λ 2 α ν, 1, 1, λ 1, λ 2 R 2 +, λ 1 + λ 2 1 }, (5.1) where {α ν } is a sequence in (1,2) converging to 1. Thus, T ν converges to T o = { x 1, x 2, y R 3 + x 1, x 2, y λ 1, 1, 1, λ [0, 1] }. (5.2) Figure 5 displays the unit-output input requirement sets for T ν and T o. As the intersection of closed half-spaces, these technologies are closed and convex. Moreover, the inequalities in the constraints imply free disposability of inputs and outputs. 23 Figure 5: Incompatibility of (I) and (C T). x 2 x ν 1, 1, L ν (1) L o (1) I(1, 2, 1, T ν ) = θ I(1, 2, 1, T o ) > θ 2 x L ν (1) 1 x 0 L o (1) x ν 1 2 x 1 23 In fact, the technologies in the sequence are DEA technologies with non-increasing returns to scale constructed with two data points, 1, 2, 1 and α ν, 1, 1, and T o is a DEA technology with one data point, 1, 1, 1.

15 The production bundle x, y = 1, 2, 1 is an efficient point in the technology T ν for all ν but an inefficient point in the technology T o (since it is dominated by the bundle 1, 1, 1 ). If the index satisfies (I), it takes the value θ at ( 1, 2, 1, T ν ) for all ν but takes a value greater than θ at ( 1, 2, 1, T o ). Theorem 1 indicates that the set of all efficiency indexes defined on technologies T can be partitioned into three subsets: (1) those satisfying indication and hence violating continuity, (2) those satisfying continuity and hence violating indication, and (3) those that violate both indication and continuity. Since we exclude from consideration indexes that are dominated in terms of our axioms, we are left with two classes of indexes: those satisfying (I) and those satisfying (C x, y, T ). The properties satisfied by the seven inefficiency indexes on the set of all technologies are spelled out in the following theorem (proved in the Appendix): Theorem 2: On the set of technologies T, I H, I DD, I B, and I QWH satisfy (WM), (UI), and (C x, y, T ) but fail to satisfy (I) and (M). 24 I FGL and I QWA satisfy (I), (WM), and (UI) but fail to satisfy (M), (C x, y ), and (C T). I WA satisfies (I), (M), and (UI) but fails to satisfy (C x, y ), and (C T). Abstracting for the moment from the properties of the weighted additive index, Theorem 2 partitions the indexes into two classes: those satisfying indication but not continuity and those satisfying continuity but not indication. All indexes satisfying continuity satisfy identical properties and can be treated as equivalent in terms of our axioms. Among the indexes satisfying indication (and not continuity), the weighted additive index dominates the other indexes since it alone satisfies monotonicity. Restriction to the set of convex technologies T C yields additional possibilities only with respect to continuity in quantities, as shown in the following theorem (proved in the Appendix): Theorem 3: On the set of technologies T C, I H, I DD, I B, and I QWH satisfy (WM), (UI), and (C x, y, T ) but fail to satisfy (I) and (M). I FGL and I QWA satisfy (I), (WM), (UI), and (C x, y ) but fails to satisfy (M) and (C T). I WA satisfies (I), (M), (UI), and (C x, y ) but fails to satisfy (C T). 24 Luenberger [1992] did not address continuity issues in his paper introducing the shortage function. (In their papers on benefit functions, Luenberger [1992] and Briec and Garderes [2004] did prove some continuity results, but these results do not generalize to shortage functions.) Later, Chambers, Chung, and Färe [1998] and Bonnisseau and Crettez [2007] were able to establish only upper semicontinuity of the shortage function; this is because they restricted the direction g only to be non-negative,whereas we restrict it to be strictly positive.

16 For convex technologies, all indexes satisfy (C x, y ), but no other properties are altered when technologies are restricted to this subset. The indexes divide into two groups: those that satisfy (C T) and those that satisfy (I). All other aspects of the indexes are the same as in the general case. We emphasize that these results depend on the class of technologies and the particular restrictions on parameters that have been selected. For example, if we were to broaden the class of directions for the directional distance function to g 0 [n+m], then I DD would no longer satisfy (C x, y ), even for convex technologies. Similarly, if parameters were selected without the specific units we impose for I DD and I WA, these indexes would not satisfy unit independence. Our assumption about the units of the parameters for I DD and I WA requires that researchers adjust parameters appropriately in response to any unit changes. In an earlier version of this paper (Russell and Schworm [2009b]) we formalized this distinction by distinguishing between unit invariance automatic satisfaction of (UI) and unit scalability invariance with respect to unit changes only after adjusting coefficients like the weights in the weighted additive index. Rather than maintaining this distinction, we have decided to restrict parameters to provide the best performance for each index. 6. Discussion of Additional Indexes. As noted in the closing paragraph of Section 3, a number of additional inefficiency indexes formulated in the literature are not included in the statement of Theorems 2 and 3. This theorem does, however, have implications, discussed briefly in this section, for these alternative indexes. To facilitate comparison of these indexes, we express them in terms of the sets, Ω(x, y, T) and Γ(x, y, T), defined by (3.10) and (3.13). The set Γ(x, y, T) is the set of feasible additive slacks while Ω(x, y, T) is the set of feasible proportional slacks. These are alternative but equivalent representations of the difference between an initial production vector and its reference vector. Their relationships can be expressed as follows: s i = (1 α i )x i, i = 1,..., n, (6.1) and t i = (1 β j )y j /β j, j = 1,..., m. (6.2) Although the weighted additive and the quantity-weighted additive indexes were initially proposed for DEA technologies, they can be defined as we have done above for the general class of technologies T without modification. Some indexes, however, depend critically on the structure of the DEA framework. An example is the Range Adjusted Measure (RAM) of Inefficiency, a modification by Cooper, Park, and Pastor [1999] of the additive index to achieve unit independence. It depends explicitly on the quantity data

17 for K decision making units : { x ik } K k=1, i = 1,...,n, and { } K y jk, j = 1,...,m. k=1 The RAM index is defined by I RAM (x, y, T) = max s,t i s i x i + j t j y j s, t Γ(x, y, T), (6.3) where and x i = max { x ik } K k=1 min { x ik } K k=1, i = 1,...,n, (6.4) k k y j = max k { } K yjk k=1 min { } K yjk, j = 1,..., m. (6.5) k k=1 For this index to be well defined, the set of possible production vectors must be bounded so that x i for i = 1,..., n and y j for j = 1,...,m are defined. This cannot be achieved for the general class of technologies T. Some indexes have been excluded from Theorems 2 and 3 because they are dominated by other indexes in terms of our axiomatic structure. An interesting example is a generalized hyperbolic measure, introduced in Section 5.7 of Färe, Grosskopf, and Lovell [1985]: I GH (x, y, T) = max λ x,λ y {λ x + λ y x/λ x, λ y y T }. (6.6) This index is based on two parameters, one to contract x and one to expand y. It combines some of the properties of (a) the hyperbolic index and the Briec index, each based on a single parameter, and (b) the FGL, weighted additive, and quantity-weighted additive indexes, each based on n+ m parameters to achieve coordinate-wise contractions of x and coordinate-wise expansions of y. This index fails to satisfy either indication or continuity in either the production vector or the technology and fails to satisfy monotonicity (but is weakly monotonic and unit independent); it, therefore, is dominated by all of the indexes considered in Theorems 2 and Pastor, Ruiz, and Sirvent [1999] define an index for DEA technologies and treat it as a modification of the FGL index. Their formulation is 26 I PRS (x, y, T) = min α,β { 1m 1 n j 1 β j i α i } α, β Ω(x, y, T). (6.7) 25 Proofs of violation of identification and monotonicity are easily adapted from the corresponding proofs of these properties by the hyperbolic, Briec, and directional distance indexes in the appendix; proof of violation of continuity is easily adapted from the proof of the same for the FGL index in the Appendix. 26 Pastor, Ruiz, and Sirvent define an efficiency index, which we invert to obtain an inefficiency index. Also, we use a different but equivalent parameterization.

18 This index is also similar to the FGL index in that it begins with the vector of coordinatewise input contractions and output expansions, but the PRS index is a ratio of the average output expansions and the average input contractions. The axiomatic properties of this measure are identical to those of the FGL index. 27 The Measure of Inefficiency Proportions, as described in Cooper, Park, and Pastor [1999], is explicitly identical to I QWA and hence does not need separate treatment. The Measure of Efficiency Proportions, proposed by Banker and Cooper [1994] for DEA technologies, can be defined for the class of technologies T by I MEP (x, y, T) = max s,t 1 1 s i + t j s, t Γ(x, y, T) n + m x i i y j j + t j. (6.8) Substitution for s i, i = 1,..., n, from (6.1) and for t j, j = 1,...,m, from (6.2) and some algebraic manipulation shows that this representation is equivalent to the I FGL index. Tone [2001] introduced a slacks-based measure (SBM) of efficiency in an additive DEA model. This index can be extended to an efficiency index defined for all technologies as follows: E SBM (x, y, T) = min s,t 1 1 n 1 + m 1 i s i x i t j y j j s, t Γ(x, y, T). (6.9) As already shown by Pastor, Ruiz, and Sirvent [1999, p. 599], a simple rearrangement of terms shows that 1 E SBM (x, y, T) is equivalent to the I PRS index and hence does not need independent treatment. Fukuyama and Weber [2009] have recently introduced a directional slacks-based measure of technical efficiency in a DEA context. This index can be extended to the full space of technologies T as follows: E FW (x, y, T) = min s,t 1 n i s i + 1 g xi m j t j s, t Γ(x, y, T) g yj, (6.10) where, in an obvious notation, the divisors of the slacks are elements of the directional vector g R n+m ++. It is clear from inspection that this index is equivalent to the weighted additive index (although this specification provides a novel interpretation of the weights). 7. Conclusion. The results in Sections 5 6 are summarized in Figure 6. The branches indicate satisfaction of the appropriate axiom. The first node reflects the dilemma posed by Theorem 27 Again, this can be seen by following the same arguments as in the proofs of the properties of IFGL in the Appendix.

19 1. For the general class of technologies T, C indicates satisfaction of continuity in either quantities or technologies (indexes that satisfy either in fact satisfy both). For the convex subset of technologies T C, C indicates satisfaction of continuity in technologies (since all of the indexes satisfy continuity in quantities on this restricted class of technologies). (The RAM index and the generalized hyperbolic index are omitted from the diagram because the former cannot be adapted to the general set of technologies T and the latter satisfies neither indication nor continuity.) Figure 6: Choices Among Inefficiency Indexes. C I H, I B, I DD, I QWH Theorem 1 I Yes I WA = I FW Monotonicity No I FGL = I MEP, I QWA, I PRS = I SBM Our axiomatic structure leads to the following conclusions: The weighted additive index dominates the FGL, PRS, and quantity-weighted additive indexes (and of course their equivalents), since it satisfies monotonicity. Owing to the incompatibility of the indication and continuity axioms, trade-offs remain between the weighted additive index and the other four indexes: 28 (i) If identification is essential, choose the weighted additive, FGL, PRS, or quantityweighted additive inefficiency index. (ii) If identification and monotonicity are essential, choose the weighted additive index. (iii) If continuity is essential, choose the hyperbolic index, the directional distance index, the Briec index, or the quantity-weighted Holder index. 28 Only the unit independence axiom fails to discriminate among the indexes, but that is because we have explicitly omitted from consideration indexes that violate (UI) but have been superceded by indexes that explicitly correct for this deficiency (e.g., the additive index and Briec s [1998] basic Holder distance function).

20 Appendix While numerous results on the properties of various indexes have been reported in the literature, few have pertained to general technologies. Many of the proofs, moreover, exploit the special properties of restricted technologies, especially those in the DEA class. We therefore present proofs of most of the claims in Theorems 2 and 3, even in some cases where similar results have been reported in the literature. Our de novo approach has the added advantage of presenting proofs that apply to several indexes rather than needlessly exploiting the special properties of each index. Proof of Theorem 2: The proof is divided into four parts: (i) indication, (ii) monotonicity, (iii) unit independence, and (iv) continuity. Indication. It is clear that the only candidates for θ in the definition of (I) are the minimal values in the ranges of the inefficiency index mappings: 0 for I DD, I B, I FGL, I QWA, I WA, and I QWH and 1 for I H. That I H, I DD, I B and I QWH fail to satisfy (I) is obvious from Figures 1 and In each case, the production vector x, ȳ is inefficient but I( x, ȳ, T) = θ. To show that I FGL satisfies indication, first suppose that x, y T, with x, y 0 [n+m], is not efficient so that there exists a production vector x, y T satisfying x, y 0 [n+m] and x, y < x, y. Then their exists an α and a β satisfying 0 [n] < α 1 [n] and 0 [m] β 1 [m] and either α < 1 [n] or β < 1 [m] such that x = α x and y = y β. Since α, β Ω(x, y, T), we have I FGL (x, y, T) 1 i α i + j β j n + m > 0. (A.1) Next suppose that I FGL (x, y, T) > 0. Then either 0 [n] < α < 1 [n] or 0 [m] β < 1 [m], so that there exists a point x, y T, with x = α x and y = y β, satisfying x, y T, x, y 0 [n+m], and x, y < x, y. Therefore, (x, y) is inefficient, and I FGL satisfies indication. To establish satisfaction of (I) (and other properties) by I QWA, it is useful to rewrite it in a more convenient form. Substituting for s i, i = 1,..., n, from (6.1) and for 29 And follows from Theorem 1 and the proofs of continuity of these indexes below.

21 t j, j = 1,...,m, from (6.2),we arrive at I QWA (x, y, T) = max n α i + i j = n m min α i α,β i j 1 β j m α, β Ω(x, y, T) 1 α, β Ω(x, y, T) β j. (A.2) This reformulated quantity-weighted additive index, featuring a vector of coordinatewise input contractions and output expansions, is very similar to the FGL index, and an argument analogous to that used in the proof of (I) for I FGL goes through for I QWA. We next sketch the analogous proof for satisfaction of (I) by I WA. If x, y is inefficient, there exists an s, t Γ(x, y, T) such that either 0 [n] < s or 0 [n] < t, which implies that I WA (x, y, T) > 0. If I WA (x, y, T) > 0, there exists an s, t Γ(x, y, T) satisfying either s > 0 [n] or t > 0 [n], in which case x, y is inefficient. Monotonicity. Failure of I H, I DD, I B, and I QWH to satisfy (M) is obvious from Figures 1 and 2. In each figure, these indexes have their minimal value θ at all points in T. Since T in Figures 1 and 2 includes points x, y and x, y with x, y < x, y, this contradicts (M). Briec [1998] shows that I QWH satisfies (WM). To establish (WM) for I H, note that x/λ, λy T and x, y < x, y implies, using free disposability, that x /λ, λy T; hence I H (x, y, T) I H (x, y, T). Proofs of (WM) for I DD and I B are analogous. To prove that I QWA and I FGL are not monotonic, we develop the (single output, two input) technology defined by the following production function: f(x 1, x 2 ) = { x2 if x 2 γ γ + min { x 1, x 2 γ } if γ x 2 µ γ + min { x 1, µ γ } if x 2 µ (A.3) This is the DEA technology generated by the two production vectors 0, γ, γ and 0, µ, µ with the assumption of non-increasing returns. Representative isoquants and input requirement sets for three outputs satisfying y < γ < µ are displayed in Figure 7. Since T is a DEA technology with non-increasing returns, it is an intersection of a finite collection of closed half-spaces so that T is closed and convex. For points such as x and ˆx in Figure 7, a simple calculation shows that, while x < ˆx, I FGL ( x, y, T) = I FGL (ˆx, y, T) = 2/3 and I QWA ( x, y, T) = I QWA (ˆx, y, T) = 1. Thus, both indexes violate monotonicity.

22 Figure 7: The FGL and Quantity-Weighted Additive Inefficiency Indexes Are Not Monotonic. x 2 µ L(µ) γ L(γ) x ˆx L(y) x 1 To prove that I FGL satisfies weak monotonicity, let x, y < x, y and note that Ω(x, y, T) Ω(x, y, T). The definition of E FGL in (3.9) implies that E FGL (x, y, T) E FGL (x, y, T) so that I FGL (x, y, T) I FGL (x, y, T). An analogous argument using (A.2) shows that I QWA satisfies (WM). To prove that I WA satisfies monotonicity, let x, y < x, y. Let s, t be the solution to (3.12) for x, y so that I WA (x, y, T) = i u is i + j v jt j. Define x, ȳ = x, y + s, t so that x, ȳ T. Define s, t = x, y x, ȳ and note that s, t Γ(x, y, T) and s, t > s, t. Therefore, I WA (x, y, T) i u i s i + j v j t j > i u i s i + j v j t j = I WA (x, y, T), (A.4) since u 0 and v 0. This proves the result. Unit independence. Briec [1998] and Balk [1998] have shown that I QWH and I DD, respectively, satisfy (UI). Proofs that (UI) is satisfied are straightforward and nearly identical for I H, I B,

23 I FGL, and I QWA. We therefore write out a formal proof only for I H and I WA. Given a transformation of the units of x and y, we have I H (ˆx, ŷ, T) { = max λ ˆx/λ, λŷ T } { = max λ ω x x/λ, λω y y T } = max { λ (ω x x) ω x /λ, λ(ω y y) ω y T } = I H (x, y, T). (A.5) To show that I WA satisfies (UI), consider a transformation of the units, ˆx = ω x x, ŷ = ω y y, û = ω u u and ˆv = ω v v. Then ( I WA ˆx, ŷ, T, ) û, ˆv = max s,t = max s,t = max s,t { û s + ˆv t ˆx s, ŷ + t T } { u ωx s + v ω y t (ˆx s) ω x, (ŷ + t) ω y T } { u s ωx + v t ω y ˆx ω x s ω x, ŷ ω y + t ω y T } =: max s,y { u s + v t x s, y + t T } = I WA ( x, y, T, u, v ). (A.6) Continuity. The failure of I FGL, I WA, and I QWA to satisfy continuity in either quantities or technologies is demonstrated by the examples in Figures 4 and To prove joint continuity of I H, I DD, and I B, consider a sequence {x ν, y ν, T ν } converging to {x o, y o, T o }. Denote this set of indexes by I = {I H, I DD, I B } and an arbitrary element of I by I G. Let λ ν = I G (x ν, y ν, T ν ) and λ o = I G (x o, y o, T o ) for all I G I and define w ν = ω ν x, ων y by and w o = w o x, wo y by w ν = x ν /λ ν, λ ν y ν if I G = I H w ν = x ν λ ν g x, y ν + λ ν g y if I G = I DD (A.7) w ν = (1 λ ν )x ν, (1 + λ ν )y ν if I G = I B w o = x/λ o, λ o y o if I G = I H w o = x o λ o g x, y o + λ o g y if I G = I DD w o = (1 λ o )x o, (1 + λ o )y o if I G = I B. (A.8) 30 Failure of IA and I WA to satisfy continuity also follows from Theorem 1 and satisfaction of (I) by these indexes.

24 The effective domain of I G is bounded for I G = I B. Also, λ ν is bounded from below at 1 for I G = I H and at 0 for I G = I DD. Given x ν x o and T ν T o, it follows that P(x ν ) P(x o ). By assumption, P(x o ) is bounded; hence, for an arbitrary δ > 0, there exists a ν such that P(x ν ) is a subset of the cube {y R n + y k δ k} for all ν > ν. Moreover, for arbitrary ǫ, there exists a ν such that y ν N ǫ (y o ) for all ν > ν. Consequently, for each I G I, w ν is bounded for all ν > max{ν, ν }. It follows, using (A.7), that, for each I G I, λ ν is bounded for all ν > ν. The strategy is to show that, for arbitrary ǫ, there exists a ˆν such that λ o ǫ < λ ν < λ o + ǫ (A.9) ν > ˆν. To prove the second inequality, suppose the contrary so that λ ν λ o +ǫ for infinitely many ν and some ǫ > 0. As λ ν is bounded for sufficiently large ν, this sequence has a convergent subsequence: λ ν k λ > λ o. This in turn implies that w ν k w = ω x, ω y satisfying w = x o / λ, λy o if I G = I H w = x o λg x, y o + λg y if I G = I DD w = (1 λ)x o, (1 + λ)y o if I G = I B (A.10) Since w ν k T ν k for all ν k and T ν T o, we have w T o. Along with the definition of λ o, this implies that λ λ o, a contradiction. To prove the first inequality, suppose the contrary so that λ ν λ o ǫ for infinitely many ν and some ǫ > 0. As λ ν is bounded for sufficiently large ν, this sequence has a convergent subsequence: λ ν k λ < λ o. As T ν T o, this in turn implies that w ν k, a boundary point in T ν k, converges to a boundary point, w, defined as in (A.10). A comparison of (A.8) and (A.10), using λ < λ o implies that w x wx o and w y w y. As w and w o are boundary points of T o, this violates the free disposability assumption. (Note that this last step exploits, for I G = I DD, the restriction that g x, g y R n+m ++.) This completes the proof that the indexes I H, I DD, and I B are continuous. To prove continuity of I QWH, we exploit two salient properties of the Holder norm h defined (on vector differences) in (3.7) and (3.8) and used in the definition of the index (3.6). These properties are continuity and satisfaction of the triangle inequality. To simplify notation, set z = x, y. Define the sets W o and W ν by W o = { w o T o h(z o, z o w o ) = I QWH (z o, T o ) } (A.11) and W ν = { w ν T ν h(z ν, z ν w ν ) = I QWH (z ν, T ν ) } (A.12)

25 so that W ν (W o ) is the set of solutions for the minimization problem defining I QWH (z ν, T ν ) (I QWH (z o, T o )). Suppose the contrary that I QWH is not continuous in z, T. Then, there is a sequence {z ν, T ν } in Ξ converging to z o, T o Ξ and an ǫ > 0 such that either I QWH (z ν, T ν ) I QWH (z o, T o ) + ǫ (A.13) or I QWH (z ν, T ν ) I QWH (z o, T o ) ǫ (A.14) for infinitely many ν. We first establish a contradiction of (A.13). Since h is continuous and z ν z o as ν, h(z ν, z ν z o ) converges to h(z o, 0) = 0. As T ν T o and hence T ν T o, there exists a sequence {u ν } T ν converging to any w o W o. Consequently, h(z ν, u ν w o ) converges to h(z o, 0) = 0. By the triangle inequality h(z ν, z ν u ν ) h(z ν, z o w o ) + h(z ν, z ν z o ) + h(z ν, w o u ν ). (A.15) Since the last two terms converge to zero, for any ǫ 1 > 0, there exists a ν such that h(z ν, z ν u ν ) < h(z ν, z o w o ) + ǫ 1 (A.16) for all ν > ν. Since u ν is in the feasible set in (3.6), I QWH (z ν, T ν ) h(z ν, z ν u ν ) for all ν. Also, I QWH (z o, T o ) = lim ν h(z ν, z o w o ) so that for any ǫ 2 > 0, there exists an ν such that I QWH (z o, T o ) h(z ν, z o w o ) ǫ 2 for ν > ν. Therefore, (A.16) implies I QWH (z ν, T ν ) I QWH (z o, T o ) + ǫ 1 + ǫ 2 (A.17) for all ν > max {ν, ν }. Choose ǫ 1 and ǫ 2 so that ǫ 1 +ǫ 2 ǫ and this contradicts (A.13). Now, assume there is a sequence { z ν } converging to z o with an infinite number of terms satisfying (A.14). Let {z ν k } denote the subsequence with each element satisfying (A.14) so that h(z ν k, z ν k w ν k ) < h(z o, z o w o ) for all k. To show the sequence is bounded, use the triangle inequality (A.18) h(z ν k, w o w ν k ) h(z ν k, w o z o ) + h(z ν k, z o z ν k ) + h(z ν k, z ν k w ν k ). (A.19) and using (A.18) implies h(z ν k, w o w ν k ) < h(z ν k, w o z o ) + h(z ν k, z o z ν k ) + h(z o, z o w o ). (A.20)

26 Since z ν k z o, for any ǫ > 0 there is a ν such that h(z o, w o w ν k ) 2h(z o, w o z o ) + ǫ (A.21) for all ν > ν. Therefore, each term w ν k for ν > ν is contained in a lower level set of h(z o, w o w ν k) which is bounded. Since the sequence { w ν k } is bounded, there is a converging subsequence, say {w ν k t }, with limit denoted w. Since z ν k t z o, w ν k t w, and h is continuous, and (A.18) implies lim t h(zν k t, z ν k t w ν k t ) = h(z o, z o w), h(z o, z o w) < h(z o, z o w o ). (A.22) (A.23) Since T ν T o, w T o and w is feasible in (3.6), (A.23) contradicts the optimality of w o. Therefore, neither (A.13) nor (A.14) hold, so that I QWH (z ν, T ν ) I QWH (z o, T o ). Proof of Theorem 3: Indication and monotonicity. Proof that I H, I DD, I B, and I QWH fail to satisfy (I) and (M) when the domain is restricted to the set of convex technologies is essentially identical to the proof for the general case: the same counterexamples go through if we simply take the convex hull of the technologies in Figures 1 and 2. Similarly, the technology in the counterexample (equation (A.3) and Figure 7) showing that I FGL and I QWA fail to satisfy (M) is convex. Continuity Since, from Theorem 3, I FGL, I WA, and I QWA satisfy (I), part (b) of Theorem 1 implies that each must violate (C T). It remains to prove that I FGL, I WA, and I QWA satisfy (C x, y ) on convex technologies. First note that these three indexes can be re-written as follows: { i I FGL (x, y, T) = max 1 x i/x i + j y j/ȳ j x,ȳ n + m } x, ȳ Φ(x, y, T), (A.24) I WA (x, y, T) = max x,ȳ u i (x i x i ) + i j v j (ȳ j y j ) x, ȳ Φ(x, y, T), (A.25)