Centre for Efficiency and Productivity Analysis

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1 Centre for Efficiency and Productivity Analysis Working Paper Series No. WP02/2013 Scale Efficiency and Homotheticity: Equivalence of Primal and Dual Measures Valentin Zelenyuk Date: February 2013 School of Economics University of Queensland St. Lucia, Qld Australia ISSN No

2 Scale Efficiency and Homotheticity: Equivalence of Primal and Dual Measures Valentin Zelenyuk 27 February 2012 Abstract We address the issue of equivalence of primal and dual measures of scale efficiency in general production theory framework. We find that particular types of homotheticity of technologies, which we refer to here as scale homotheticity, provide necessary and sufficient condition for such equivalence. We also identify the case when the scale homotheticity is equivalent to the homothetic structure from Shephard (1970). Keywords: Production theory, Scale Efficiency, Homotheticity, Duality theory. JEL Codes: D24. School of Economics and Centre for Efficiency and Productivity Analysis (CEPA) at The University of Queensland, Australia; address: 530, Colin Clark Building (39), St Lucia, Brisbane, Qld 4072, AUSTRALIA; tel: ; Acknowledgment: I would like to thank Erwin Diewert, Rolf Färe, John Farrell, Shawna Grosskopf, Knox Lovell, Antonio Peyrache and Prasada Rao for their valuable comments. I also thank the audience where this paper was presented and, in particular, participants of my seminar at the University of Auckland and at the the Productivity Workshop of CEPA. I especially thank anonymous referees and the editor for their valuable comments; they helped improving this paper substantially. I remain to be solely responsible for my views expressed in this article. Note: This paper is a revised version of WP09/2011 of CEPA Working Papers Series (University of Queensland, School of Economics). The main update is the new section 4, new proposition 3 and its proof in appendix. 1

3 1 Introduction Data limitation in empirical economics studies usually stands out as a rule rather than an exception. In applied production analysis, for example, data on all inputs used in a production are often hard or even impossible to obtain for an independent researcher. With a help of duality theory in economics (e.g., due to Shephard, 1953, 1970), vital economic information about production technology can be retrieved, for example, from data on input prices that firms faced and output levels these firms produced and the resulted total costs these firms incurred. Such information sometimes is more readily available to researchers than the primal (i.e., input-output) data, enabling researchers to estimate firms cost functions, obtaining an alternative (and complete) dual characterization of production technologies. In other contexts, a researcher might only have information on output prices that firms faced, the levels of inputs they used and the resulted total revenues these firms obtained, and then the duality theory can be used to estimate firms revenue functions that provide yet another complete characterization of production technologies. It is known that while giving complete characterizations of technology, the various primal and dual approaches to measuring the same issue in production (e.g., such as scale economies) do not always yield the same results, yet they may provide equivalent information, under some conditions. For example, in general, given some regularity conditions, the dual scale elasticity measures (based on cost, revenue or profit functions) provide the same information about scale economies of a technology as the primal scale elasticity measures, if the latter are evaluated at certain optimal allocations (w.r.t. cost, revenue or profit functions). 1 While such conditions of optimality might be natural in the context when firms are assumed to be fully efficient, such conditions might be incoherent with the contexts where one wants to allow for and to measure various types 1 E.g., see Färe et al. (1986), Färe and Primont (1995), Zelenyuk (2011a,b) for details. 2

4 of inefficiency of firms and its sources, as is the context of our study. In the theoretical production efficiency framework, a methodology to retrieve useful technology information about scale using the cost-output data instead of input-output data goes back to at least the work of Färe and Grosskopf (1985). Specifically, in their seminal paper, authors provided conditions for equivalence of the primal and dual scale efficiency measures in the context of the Data Envelopment Analysis (DEA) estimator (or activity analysis models). This condition requires that the input allocative efficiency estimated under the assumption of constant returns to scale (CRS) is equal to that estimated under the assumption of variable returns to scale (VRS). This same condition, informally, was also envisioned in Seitz (1970), and so we refer to it here as Färe-Grosskopf-Seitz condition. The main goal of this paper is to theoretically determine what type of technology, if any, is consistent with the Färe-Grosskopf-Seitz condition, or other conditions that ensure equivalence of the primal and dual measures of scale efficiency. In economic theory literature one may notice that various types of homotheticity conditions on technology help establishing useful relationships of different measures of efficiency and productivity (e.g., see Chambers and Färe (1994, 1998), Färe et al. (2001), Färe and Li (2001), etc.). What we find in our context is that, also, particular types of homotheticity, which we call here as scale homotheticity, provide necessary and sufficient condition for the equivalence relationships we are interested in this work. We also identify the case when the scale homotheticity is equivalent to technologies with homothetic structures from Shephard (1970). This work is theoretical yet it carries very important practical implications. Indeed, empirical researchers often choose to use the primal approach in some cases and the dual approach in others and rarely or never question the issue of equivalence of the results between these two approaches mainly because decisions on the approach are 3

5 often driven by data availability. As a result, until now it has been practically infeasible to check whether the conclusions about the scale efficiency would be the same or not depending on whether a researcher were to use the primal or the dual approach, when having data for only one of these approaches. In other words, to ensure the equivalence, one could assume that Färe-Grosskopf-Seitz condition holds, but testing for relevance of this assumption so far appeared to require data for both the dual and the primal approaches. This work expands the possibility for the empirical researchers by theoretically unveiling the type of technology that is equivalent to Färe-Grosskopf-Seitz condition and so is equivalent to the case when the primal and dual approaches provide equivalent information about scale efficiency. Importantly, this condition on technology can be equivalently defined or characterized (and thus tested) via only the dual or only the primal approach/data. As a result, this paper provides theoretical foundation for developing an empirical test for equivalence of the primal and dual approaches to scale efficiency as a test for a scale homotheticity property of technology that requires only the primal or only the dual approach/data. The paper is organized as follows. In section 2 we briefly outline theoretical framework of technology characterization and efficiency measurement. In section 3 we outline one of the main results of the paper, for the input oriented case, showing that scale homothetic technology provides necessary and sufficient condition for the equivalence of the primal and dual measures. In section 4 we relate the scale homotheticity to a more general notion of homotheticity. Section 5 provides some concluding remarks and, in particular, also mentions several possible extensions of this work. Finally, the proofs of the main results of the paper are laid out in Appendix. 4

6 2 Theoretical Framework: Input Oriented Case In order to obtain general results, potentially applicable to any industry, we look at the issue from a general theoretical perspective using the framework sparked by Shephard (1953, 1970) and further elaborated in Färe et al. (1994) and Färe and Primont (1995). In particular, we assume that a production technology of interest can be characterized via technology set T, defined in general terms as T = { (x, y) R N + R M + : y is producible from x }. (2.1) Equivalently, we can also characterize technology via the input correspondence L : R M + 2 RN + that assigns to each output vector y R M + the subset of all input vectors x R N + that can produce this particular y, i.e., L (y) = { x R N + : (x, y) T }, y R M +. (2.2) We assume that technology is regular, i.e., satisfies standard regularity conditions of production theory (e.g., see Färe and Primont, 1995), namely: - technology set T is closed; - for any finite x R N +, the output set P (x) = { y R M + : (x, y) T } is bounded; - there is no free lunch, i.e., (0 N, y) / T, y 0 M ; - producing nothing is possible, i.e., (x, 0 M ) T, x R N +; - free disposability of outputs and free disposability of inputs hold, i.e., (x o, y o ) T (x, y) T, y y o, x x o. To conveniently characterize technology set T and to measure technical efficiency, we use implicit function D i : R M + R N + R 1 + {+ }, defined as D i (y, x) = sup {λ R ++ : x/λ L(y)} (2.3) 5

7 which is known as Shephard s input oriented distance function. The reciprocal of (2.3) can be used to define the Farrell s input oriented measure of technical efficiency, which we denote with T E i (y, x) = 1/D i (y, x). Incidentally, note that for any x L(y) we have T E i (y, x) (0, 1], i.e., T E i (y, x) gives a score between 0 and 1, with 1 standing for full or 100% technical efficiency level (in the input oriented case). Given standard regularity conditions about T, both functions completely characterize T (and L), in the sense that, 2 Di(y, x) 1 x L(y), y R M +. (2.4) To facilitate the measurement of scale efficiency in a general theoretical context, let Ť be the CRS-hypothetical (or counterfactual) technology defined as Ť = {δ(x, y) : (x, y) T, δ > 0}. (2.5) That is, Ť is a set generated on T as the conical closure of T. Intuitively, Ť can be understood as the smallest CRS-hypothetical (or CRS-counterfactual) technology set that includes the actual technology set T. Thus, the upper boundary or technological frontier of Ť would be just tangent with that of T at least at one point and such tangent point(s) can be called as the best-possible-scale allocations of (x, y). Note that for some technologies, such best scale allocations may be not unique as well as there might be uncountably infinite number of them. Clearly, if the actual technology exhibits CRS, i.e., T = δt, δ > 0, then (and only then), by construction, we have T = Ť. For technical purposes, to make such measurement of scale efficiency possible, we assume that R N+M + \Ť. This type of measurement is coherent with the approach used in DEA, but also can be used with other approaches, such as stochastic frontier analysis (SFA), or a mixture of the two, such as stochastic-dea, etc. 2 For the properties of the Shephard s distance function see Shephard (1953, 1970). 6

8 Given the set Ť, we now can define its level sets in the input space at particular levels of outputs, which we denote with L(y crs), defined similarly to (2.2), as L(y crs) = { x R N + : (x, y) Ť }, y R M +. (2.6) So, by construction, L(y crs) is an equivalent characterization of the CRS-hypothetical technology set Ť, in the sense that we have x L(y crs), y R M + (x, y) Ť. (2.7) Furthermore, the input oriented Shephard-type distance function with respect to the CRS-hypothetical technology for an allocation (x, y) would be given by D i (y, x crs) = sup {λ R ++ : x/λ L(y crs)}, y R M +. (2.8) Properties of this function can be derived analogously to those of (2.3), with an exception that D i (y, x crs) will also be homogenous of degree -1 in y. The Farrell-type input oriented measure of technical efficiency w.r.t. CRS-hypothetical technology is defined as T E i (y, x crs) = 1/D i (y, x crs). Clearly, for any x L(y) we also have T E i (y, x crs) (0, 1], i.e., this measure of efficiency also gives a score between 0 and 1, with 1 standing for full or 100% technical efficiency level, but now w.r.t. the CRShypothetical technology rather than the original technology. The input oriented scale efficiency measure, can now be defined as 3 SE i (y, x) = D i(y, x) D i (y, x crs) = T E i(y, x crs) T E i (y, x), x L(y), y R M +. (2.9) 3 The origins of this measure go back to at least Førsund and Hjalmarsson (1979). For this and other ways of measuring scale issues see, for example, Banker et al. (1984), Färe and Grosskopf (1985), Färe et al. (1986), Førsund (1996), Zelenyuk (2011a,b) to mention just a few. 7

9 Because T Ť, then (x, y) T we also have T E i(y, x crs) T E i (y, x) and so SE i (y, x) (0, 1], i.e., this measure of efficiency also gives a score between 0 and 1, with 1 standing for full or 100% scale efficiency level (for input oriented case). The geometric intuition of this measure of scale efficiency is illustrated in Figure 1, where for a firm at point D (or point A) it is given by the relative distance between points A and B, measured as CB/CA. The approach to efficiency measurement presented just above is based on the inputoutput data, (x, y), and is referred to as the primal approach (for input orientation). To outline a dual, cost-oriented approach, let w be a row-vector of strictly positive input prices corresponding to each element of x, and define the (minimal) cost function as C(y, w) = min x {wx : x L(y)}, y R M + : L(y), w R N ++. (2.10) With extra assumption that L(y) is convex for all y R M +, a dual analogue of (2.4) is given by L(y) = { x R N + : } C(y, w) wx 1, w RN ++, y R M + (2.11) i.e., technology can be completely characterized by the cost function (2.10), provided that technology is regular and L(y) is convex (see Färe and Primont (1995) for details). The cost function in (2.10) can now be used to obtain the cost efficiency measure, a dual analogue of (2.3), via CE(x, y, w) = C(y, w) wx, x L(y), y RM +, w R N ++. (2.12) And therefore, for any allocation x L(y), y R M +, and prices w R N ++, the cost-scale 8

10 efficiency can be measured as SE c (y, w) = CE(x, y, w crs) CE(x, y, w) = C(y, w crs), (2.13) C(y, w) where C(y, w crs) is the cost function w.r.t. the CRS-hypothetical technology, i.e., C(y, w crs) = min x {wx : x L(y crs)}, y R M + : L(y), w R N ++, (2.14) while the associated cost efficiency w.r.t. the CRS-hypothetical technology is CE(x, y, w crs) = C(y, w crs), x L(y), y R M +, w R N wx ++. (2.15) As it is for the other efficiency measures defined above, for any x L(y) we have CE(x, y, w) (0, 1] and CE(x, y, w crs) (0, 1], and since T Ť, then (x, y) T we also have CE(x, y, w crs) CE(x, y, w) and, as a result, SE c (y, w) (0, 1]. That is, all these cost-based efficiency measures also give scores between 0 and 1, with 1 standing for full or 100% level of certain type of efficiency. The geometric intuition of this measure of scale efficiency is illustrated in Figure 2, where for a firm at point D (or point A) it is given by the relative distance between points A and B, measured as CB/CA. Given convexity of L(y), the relationship between the dual and primal approaches is obtained through the duality theory between D i (y, x) and C(y, w). In particular, the Mahler s inequality (see Färe and Grosskopf (2000)) tells us that C(y, w) wx 1 D i (y, x), x L(y), y RM +, w R N ++. (2.16) This inequality can be closed by defining the allocative efficiency measure as a multi- 9

11 plicative residual that turns (2.16) into equality, i.e., AE i (y, x, w) = CE(x, y, w) T E i (y, x), x L(y), y RM +, w R N ++. (2.17) Analogously, we can also write that C(y, w crs) wx 1 D i (y, x crs), x L(y), y RM +, w R N ++, (2.18) and so one can define the allocative efficiency measure w.r.t. CRS-hypothetical technology, as a multiplicative residual that turns (2.18) into equality, i.e., AE i (y, x, w crs) = CE(x, y, w crs) T E i (y, x crs), x L(y), y RM +, w R N ++. (2.19) Clearly, because the two measures of scale efficiency, SE i (y, x) and SE c (y, w), are functions of different variables, in general they may yield different efficiency scores. There is, however, a special case when they are equal. Specifically, one can observe that from (2.9), (2.13), (2.17) and (2.19) it follows that SE c (y, w) = SE i (y, x) ASE i (y, x, w), x L(y), y R M +, w R N ++, (2.20) where ASE i (y, x, w) is the allocative scale efficiency measure, defined as ASE i (y, x, w) = AE i(y, x, w crs), x L(y), y R M AE i (y, x, w) ++, w R N ++. (2.21) Therefore, for any allocation x L(y), y R M +, w R N ++, we can say SE i (y, x) = SE c (y, w) if and only if AE i (y, x, w crs) = AE i (y, x, w). (2.22) 10

12 The result in (2.22) was earlier reached by Färe and Grosskopf (1985) in the context of DEA (or activity analysis) modeling framework. An early reference for an intuitive explanation of this case is also found in Seitz (1970), so we refer to (2.22) as the Färe- Grosskopf-Seitz condition (for the input oriented context). Importantly, note that Färe- Grosskopf-Seitz condition is stated in terms of both primal and dual data, i.e., the same data as required to obtain SE i (y, x) and SE c (y, w) for comparing their equality. So, it is very desirable to find a deeper condition, e.g., a condition on the level of technology characterization that ensures Färe-Grosskopf-Seitz condition holds but requires only primal or only dual data to be verified. A formal answer to this and related questions are given in the next sections. 3 Scale Homotheticity Consider a technology that satisfies condition referred to here as the input scale homotheticity (ISH), formally defined as technology where L(y) = G(y)L(y crs), y R M + (3.1) where G(y) is a finite real-valued lower semi-continuous function G : R M + [1, ) coherent with regularity conditions on technology. Intuitively, the structure of technology of the type given by (3.1) postulates that the input requirement set can be decomposed (in the multiplicative way) into the CRS-hypothetical input requirement set, constructed from the original L(y), and an appropriate scaling factor G(y) that, in general, may depend only on the scale and the mix of outputs described by y. To see what implications ISH has towards the relationship between the primal and dual measures of scale efficiency, we first state the following two general results (proofs are outlined in the Appendix). 11

13 Proposition 1. A regular technology is input scale homothetic, as defined in (3.1), if and only if D i (y, x) = (G(y)) 1 D i (y, x crs), (x, y) R N+M +. (3.2) Intuitively, Proposition 1 states that if a regular technology is input scale homothetic then, and only then, the Farrell s input oriented measure of technical efficiency w.r.t. this technology can be multiplicatively decomposed into two parts: this same measure but w.r.t. the CRS-hypothetical technology and the scaling factor given by the same function (G(y)) used to characterize the ISH property of the technology. Figure 3 provides some geometric intuition, illustrating that ISH implies that the relative distances between L(y) and L(y crs), along a ray from the origin, are the same at any point x L(y) (although may vary with y). For example, ratios such as OA/OB, OE/OF are equal, as confirmed by Proposition 1. Naturally, a ISH property must have a similar implication onto the dual characterization of L(y), which is summarized in the next proposition. Proposition 2. A regular technology with convex L(y) is input scale homothetic, as defined in (3.1), if and only if C(y, w) = G(y)C(y, w crs), w R N ++, y R M + : L(y). (3.3) In words, Proposition 2 says that if a regular technology is input scale homothetic with the scaling factor given by G(y) then, and only then, the cost function w.r.t. 12

14 this technology can be multiplicatively decomposed into the cost function w.r.t. the CRS-hypothetical technology and this scaling factor G(y). In turn, combining (3.2) and (3.3), implies the following important result. Corollary. A regular technology with convex L(y) is input scale homothetic, as defined in (3.1), if and only if SE i (y, x) = SE c (y, w) = 1 G(y), w RN ++, y R M + : L(y), (3.4) which is equivalent to the Färe-Grosskopf-Seitz condition, i.e., when AE i (y, x, w crs) = AE i (y, x, w) for any x L(y), y R M +, w R N ++. In words, the corollary above says that in the input oriented case, the equivalence of the dual and primal scale efficiency measures for all technologically feasible allocations (i.e., the case when the Färe-Grosskopf-Seitz condition holds) is achieved if and only if technology is input scale homothetic. Moreover, notice that in such a case, both scale efficiency measures would be equal to the reciprocal of G(y), where the latter is defined in (3.1) the function reflecting the scale of the production activity of such technology. In terms of Figure 3, equivalence of ISH and the Färe-Grosskopf-Seitz condition implies equality of such ratios as OC/OE and OD/OF. Few more remarks about the results above might be useful here. Remark 1. It is worth noting that if L(y) is not convex, then ISH still implies equivalence of the dual and primal scale efficiency measures, i.e., (3.4) is still true. In other words, ISH is a sufficient condition for the equivalence of the dual and primal measures of scale efficiency for any regular technology but becomes necessary and sufficient condition when, in addition, L(y) is convex. (This is a reason Figure 3 gives example with non-convex L(y).) 13

15 Remark 2. The input scale homotheticity condition on technology is somewhat restrictive to assume, yet the results attained above suggest that it is the minimal condition that guarantees equality of the primal and dual measures of scale efficiency (for any feasible allocation) in the input oriented framework, for any regular technology with convex L(y). Remark 3. The conclusions reached above assume global ISH and therefore provide global results about the equivalence. However, if global ISH is viewed as too restrictive, a researcher may consider imposing ISH (and convexity of L(y), if needed) only locally, at some ranges of y that are of particular interest, or as an approximation of some degree. Remark 4. Although fairly restrictive, the ISH technology is still less restrictive than the CRS technology and, in fact, is a generalization of it, since any CRS technology is necessarily an ISH technology but the reverse is not true. This implies that many models where CRS was assumed for simplicity can potentially be extended to a more general context of scale homotheticity, which may still allow for a lot of simplifications of derivations, similar to those inherited by CRS context, yet preserving the scale component ignored or assumed away in the CRS case. Remark 5. Note that any regular technology with one input (and any finite number of outputs) is necessarily ISH, provided that R N+M + \Ť. Remark 6. For the case of output orientation, the output scale homotheticity (OSH), can be defined similarly to (3.1) but on output sets P (x), and it would be necessary and sufficient condition for the equivalence of the primal and dual measures of scale efficiency defined in terms of the output oriented Shephard s distance functions (or Farrell efficiency measures) and revenue functions, respectively (see Zelenyuk (2011c) for details). 14

16 4 Relationship to Homothetic Structures In this section we establish relationship of the scale homotheticity with a more general homotheticity found in Shephard (1970). Focusing on the input orientation, recall (e.g., from p. 255 of Shephard (1970)) that technology has homothetic input structure (HIS) if and only if L(y) can be characterized as L(y) = {x : F (f(x)) q(y), x R N +}, y R M + (4.1) where q : R M + R + is a finite, real-valued, lower-semi-continuous, non-decreasing function, and f : R N + R + is a finite real-valued, upper-semi-continuous, linearly homogeneous and non-decreasing function, such that both q and f satisfy the regularity conditions mentioned above, while F ( ) is a transform of f(x). 4 Incidentally, note that HIS is equivalent to saying that D i (y, x) = f(x)/ψ (q(y)), (4.2) where Ψ : R + R + is an inverse of the transform F. 5 Moreover, due to duality of w and x (when L(y) is convex), HIS is also equivalent to saying that C(y, w) = Ψ (q(y)) c(w), (4.3) where c(w) is a linearly homogenous function satisfying properties of a cost function. 6 4 To be more precise, q and f must also satisfy the following properties: q(0) = 0; q(y) > 0 if y 0; q(y n ) + for { y n } + ; f(0) = 0; and for any x > 0 or x 0 such that f(δx) > 0, for some scalar δ > 0, we have f(δx) + as δ +. To preserve convexity of L(y), we also need q(y) being quasi-convex onr M + and f(x) being quasi-concave on R N +. Also, the transform F : R + R + means it is a finite real-valued non-negative, upper semi-continuous and non-decreasing function such that F (0) = 0. See Shephard (1970) for details of these properties. 5 To be precise, Ψ(z) = min{v : F (v) z}, z 0, and so it is a finite real-valued non-negative, lower semi-continuous and non-decreasing function such that Ψ(0) = 0 and Ψ(z) > 0, for z > 0. 6 More precisely, note that c(w) = min x {wx : f(x) 1}. 15

17 To see that, in general, the two technologies, ISH and HIS, are not equivalent it suffices to provide an example where one does not imply the other. To do so, consider a technology given by (4.1) where f(x) = N and q(y) = M i=1 i=1 (xα 1 ( ) y β y β M M, β i (0, 1) : 1... x α N N ), α i (0, 1) : N i=1 α i = 1 M i=1 β i 1 and so the technology has homothetic input structure, yet the technology is not input scale homothetic. Indeed, ISH fails to satisfy here because for this type of technology we have R + \Ť =, implying non-existence of a real-valued function G : RM + [1, ) to ensure L(y) = G(y)L(y crs), y R M +. So, in general, HIS is not a sufficient condition for ISH. Although the arguments above conclude that HIS and ISH are not equivalent, the two classes of technologies are still closely related. More precisely, ISH is equivalent to HIS for the restricted class of technologies, as we state in the next proposition (see Appendix for a proof). 7 Proposition 3. In the class of regular technologies where R N+M + \Ť, technology is input scale homothetic if and only if this technology has homothetic input structure. In words, proposition above states that the well-known homothetic input structure can be equivalently represented in terms of the input scale homotheticity, as long as R N+M + \Ť, which gives a convenient decomposition of L(y), D i(y, x) and C(y, w) into their CRS counterparts and the scale function G(y). This is a useful interpretation of a class of homothetic structures. Moreover, combining this proposition with the results in the previous section, we conclude that for technologies satisfying R N+M + \Ť, the homothetic input structure is the necessary and sufficient condition for the equivalence of the input oriented primal and dual scale efficiency measures. 7 I thank anonymous referee for hinting me towards establishing this more general result than the one I had in the first version of this paper. 16

18 Furthermore, note that while the equivalence is established for a restricted class of regular technologies, those satisfying R N+M + \Ť, this restriction is merely a feasibility restriction, which makes the measures of scale efficiency being well-defined and feasible to estimate. Noteworthy, this restriction is satisfied automatically (by construction) when using standard DEA-type estimators, yet may need to be imposed when using other estimators, such as SFA. Finally, similar statements can be established for equivalence of the output scale homotheticity and the homothetic output structure from Shephard (1970). 5 Concluding Remarks In this paper we showed that the equivalence of primal and dual scale efficiency measures (or the Färe-Grosskopf-Seitz condition) holds if and only if technology is of a peculiar type scale homothetic. Specifically, under certain regularity conditions, the input scale homotheticity is necessary and sufficient condition for equivalence of primal and dual scale efficiency measures in the input oriented case, while the output scale homotheticity is necessary and sufficient condition for equivalence of primal and dual scale efficiency measures in the output oriented case. Our work also opens several directions for further research. One natural direction is to develop empirical tests for whether technology is scale homothetic, which in turn would help establishing equivalence of primal and dual scale efficiency measures. Till now, empirical researchers, while choosing the primal approach in some cases and the dual approach in others, hardly ever question the issue of equivalence of the results between these two approaches usually because decisions on the approach are based on data availability. To ensure the equivalence, one could assume that Färe-Grosskopf- Seitz condition holds but testing for relevance of this assumption was so far deemed to 17

19 require data for both the dual and the primal approaches. The present work expands the possibility for the empirical researchers by theoretically unveiling the type of technology that is equivalent to Färe-Grosskopf-Seitz condition and this condition can be equivalently defined or characterized and thus tested via only the dual or only the primal approach/data. As a result, this paper provides theoretical foundation, with a welldefined if and only if condition, for developing an empirical test for equivalence of the primal and the dual approaches to scale efficiency. Importantly, the developed theory implies that such a test would require only the primal or only the dual approach/data. Development of such a test could be approached using similar developments for other types of homotheticity (e.g., in Färe et al. (2001), Färe and Li (2001)) as well as using recent developments on applications of non-parametric statistical methods in efficiency analysis, or both. 8 This is a subject in itself and so is left for future research. Another direction of future research is about theoretical investigations of how restrictive are the assumptions of scale homotheticity, and what are their relationships to other types of separability (Hicks-neutrality, ray-homotheticity, etc.), e.g., as those investigated and cataloged by Chambers and Färe (1994) and Färe et al. (2001). It might be also worth looking at applications of scale homotheticity to other related areas. For example, a natural direction would be the area of productivity indexes, where the total productivity change is often decomposed into change in technology and change in efficiency and the latter component is often decomposed into change in pure-technical efficiency and change in scale efficiency. Thus, naturally, the (input or output) scale homotheticity condition may help relating the primal and dual measures of scale efficiency change. Moreover, scale homotheticity can be used to derive new relationships between various productivity indexes, or for simplifying. Overall, we hope that this article will stimulate work in these and other interesting directions of research. 8 E.g., see recent works of Simar and Wilson (2007, 2011) and Simar and Zelenyuk (2006, 2007, 2011) and references cited therein. 18

20 6 Appendix Proof of Proposition 1. Assume a regular technology given by L(y), y R M + is input scale homothetic, i.e., L(y) = G(y)L(y crs), y R M +, G(y) 1, then (x, y) R N+M + we have D i (y, x) = sup {λ R ++ : x/λ L(y)} λ = sup λ {λ R ++ : x/λ G(y)L(y crs)} { ( ) x λg(y) R ++ : λg(y) {ˆλ R++ : x/ˆλ L(y crs)} = (G(y)) 1 inf λ = (G(y)) 1 inf ˆλ = (G(y)) 1 D i (y, x crs). } L(y crs) (where ˆλ = λg(y), y R M + ) To prove the converse, assume that for a regular technology characterized by L(y) we have D i (y, x) = (G(y)) 1 D i (y, x crs), (x, y) R N+M +, for some finite real-valued, lower semi-continuous function G : R M + R + such that G(y) 1. Then it must be also true that for all y R M + we also have L(y) = { x R N + : D i (y, x) 1 } = { x R N + : (G(y)) 1 D i (y, x crs) 1 } = G(y) { x/g(y) R N + : D i (y, x/g(y) crs) 1 } = G(y) {ˆx R N + : D i (y, ˆx crs) 1 } = G(y)L(y crs), where we used the complete characterization property (2.4), the linear homogeneity property of D i (y, x), namely that D i (y, αx crs) = D i (y, x crs)α, α > 0 and made ˆx = x/g(y), y R M +. Q.E.D. 19

21 Proof of Proposition 2. Assume a regular technology characterized by L(y), y R M + is also ISH, i.e., L(y) = G(y)L(y crs), for some G(y) 1 and y R M +, then for all w R N ++and for all y R M + such that L(y) we must also have C(y, w) = min{wx : x L(y)} x = min x = G(y) min x {wx : x G(y)L(y crs)} { w x G(y) : x G(y) L(y crs) = G(y) min ˆx {wˆx : ˆx L(y crs)} (where ˆx = x/g(y), y R M + ) = G(y)C(y, w crs). } To prove the converse, assume that for a regular technology characterized by L(y), y R M + we also have C(y, w) = G(y)C(y, w crs) for some G(y) 1, (x, y) R N+M +, and assume that L(y) is convex, then from the duality theory result (2.11), it must be true that for all y R M + we also have L(y) = { } x R N + : wx C(y, w), w R N ++ = { } x R N + : wx G(y)C(y, w crs), w R N ++ { x = G(y) G(y) RN + : w x } G(y) C(y, w crs), w RN ++ = G(y) {ˆx R N + : wˆx C(y, w crs), w R N ++} (where ˆx = x/g(y), y R M + ) = G(y)L(y crs). Note that convexity of L(y) is only required to prove the converse part (i.e., the necessity of ISH), and C(y, w) = G(y)C(y, w crs) for some G(y) 1 and (x, y) R N+M + is ensured by ISH whether L(y) is convex or not, for any regular technology. Q.E.D. 20

22 Proof of Proposition 3. Necessity: Suppose technology satisfies input scale homotheticity, then (and only then), due to (3.4), we have S i (y, x) = S c (y, w), (x, y) R N+M +, which in turn implies that S i (y, x) must be independent of x, while S c (y, w) must be independent of w. This means that there exist some functions φ(x), ψ(x), h(y), and ȟ(y) such that D i (y, x) = φ(x)/h(y) and D i (y, x crs) = φ(x)/ȟ(y) as well as C(y, w) = ψ(w)h(y) and C(y, w crs) = ψ(w)ȟ(y) and so that SE i(y, x) = ȟ(y)/h(y) = SE c(y, w). What is left is to show that φ(x), h(y), ψ(w) and ȟ(y) satisfy properties required by the definition of homothetic input structure. Let us first show that φ(x) is a upper semi-continuous, linearly homogeneous and non-decreasing function such that φ(0) = 0, and for any x 0 such that φ(δx) > 0, for some scalar δ > 0, we have φ(δx) + as δ +. To do so, note that D i (y, x) is a finite real-valued continuous and linearly homogeneous in x on R N + and therefore because D i (y, x) = φ(x)/h(y) these same properties must also be shared by φ(x). 9 Similarly, because D i (y, 0) = 0, we also have φ(0) = 0. Moreover, for any x 0 such that φ(δx) > 0, for some scalar δ > 0, we will have φ(δx) + as δ +, because for any x 0 such that D i (y, xδ) > 0 we have D i (y, xδ) 0 when δ +. Furthermore, let us show that h(y) is a finite real-valued lower semi-continuous nondecreasing function such that h(0) = 0, and h(y) > 0 if y 0, while h(y n ) + for { y n } +. To do so, recall that C(y, w) is a finite, lower semi-continuous, and nondecreasing in y on R M +, and so, because C(y, w) = ψ(w)h(y), the same properties should hold for h(y). Also, because D i (y, x) > 0 when y 0, it must be true that h(y) > 0 if y 0, since we have D i (y, x) = φ(x)/h(y). Similarly, because D i (y, x) + for y 0, we have h(y) 0, when y 0. Analogously, h(y n ) + for { y n } +. Next, since we involve convexity of L(y), we need to show that h(y) is quasi-convex 9 See Shephard (1970) for the proofs of these and other properties of D i (y, x) and of C(y, w) that we mention here. 21

23 on R M + and φ(x) is quasi-concave on R N +. These are established by noting that D i (y, x) is concave and non-decreasing in x on R N + and is quasi-concave in y on R M +, which in turn implies that, to ensure that D i (y, x) = φ(x)/h(y) holds, φ(x) must be quasi-concave on R N +, while 1/h(y) is quasi-concave on R M + and so h(y) is quasi-convex on R M +. (One may also establish similar properties of ψ(x) and ȟ(y) by a similar argument). Combining all these conclusions, we conclude that (4.2) is satisfied, with all the required properties for the involved functions, and so technology (and its CRS-hypothetical counterpart) has homothetic input structure. Sufficiency: If technology is regular and satisfies R N+M + \Ť, then D i(y, x crs) < and C(y, w crs) < for any (x, y) T where D i (y, x) < and C(y, w) <. This ensures existence of SE i (y, x) and SE c (y, w). Now, suppose technology has input homothetic structure defined in (4.1). Then, (4.2) and (4.3) must hold and, moreover, we must also have D i (y, x crs) = f(x)/ˇq(y) (6.1) and C(y, w crs) = c(w)ˇq(y). (6.2) where ˇq(y) = inf δ {δψ (q(y/δ)), δ R ++ }, Ψ(z) = min{v : F (v) z}, z 0 and c(w) = min x {wx : f(x) 1}. Specifically, (6.1) is true because 22

24 D i (y, x crs) = { sup λ R++ : (x/λ, y) Ť } λ = sup {λ R ++ : (x/λ, y) δt, δ R ++ } λ = sup {λ R ++ : F (f (x/(λδ))) q(y/δ), δ R ++ } λ = sup {λ R ++ : f (x/(λδ)) Ψ (q(y/δ)), δ R ++ } λ = sup {λ R ++ : f (x) /(λδ) Ψ (q(y/δ)), δ R ++ } λ = sup λ = sup λ {λ R ++ : f(x)/ (δψ (q(y/δ))) λ, δ R ++ } { λ R ++ {δψ (q(y/δ)), δ R ++} λ : f(x)/ inf δ = f(x) ˇq(y), ( where ˇq(y) = inf δ {δψ (q(y/δ)), δ R ++} ). } and, in turn, (6.2) is true because C(y, w crs) = min x {wx : (x, y) Ť } = min x {wx : D i (y, x crs) 1} = min x {wx : f(x) ˇq(y) 1} = ˇq(y) min x {wx/ˇq(y) : f(x/ˇq(y)) 1} = ˇq(y) min {wˆx : f(ˆx) 1} (where ˆx = x/ˇq(y), y R M + ) ˆx ( ) = ˇq(y)c(w), where c(w) = min {wˆx : f(ˆx) 1}. ˆx Now, combining (6.1) with (4.2) and (6.2) with (4.3), we get SE i (y, x) = D i(y, x) D i (y, x crs) = ˇq(y) q(y) = C i(y, w crs) C i (y, w) = SE c (y, w) 23

25 which in turn implies technology is input scale homothetic, with G(y) = Ψ (q(y)) /ˇq(y) satisfying the required properties. Indeed, because ˇq(y) = inf δ {δψ (q(y/δ)), δ R ++ } we always have ˇq(y) Ψ (q(y)), y R M +, thus ensuring that G(y) 1. Finally, note that q(y) is a finite real-valued lower semi-continuous on R M +, while Ψ ( ) a finite realvalued lower semi-continuous on R + and so these same properties must be also shared by ˇq(y), which was defined as ˇq(y) = inf δ {δψ (q(y/δ)), δ R ++ }. All these, in turn, ensures that G(y) = Ψ (q(y)) /ˇq(y) is also a finite real-valued lower semi-continuous function on [1, + ). Q.E.D. 24

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29 C B A D Figure 1. Primal scale efficiency measure: e.g., ( )= CB/CA for a firm at point D or A.

30 A D B C Figure 2. Dual (cost) scale efficiency measure: e.g., for a firm at point D or A.

31 B F A D E C O Figure 3. Input Scale Homotheticity: ISH implies the relative distances between and, along a ray from the origin, are the same at any point, although may depend on.