CATEGORICAL AND AMBIGUOUS CHARACTERIZATION OF RETUNS TO SCALE PROPERTIES OF OBSERVED INPUT-OUTPUT BUNDLES IN DATA ENVELOPMENT ANALYSIS Subhash C Ray Department of Economics University of Connecticut Storrs CT 06269-063 subhash.ray@uconn.edu (Abstract) This paper shows how one can determine the returns to scale properties of every feasible input-output bundle efficient or otherwise. Specifically, we identify different regions within the production possibility set where increasing, constant, or diminishing returns to scale hold with respect to both inputs and outputs. We also identify regions where the nature of returns to scale is ambiguous and depends on the direction of proection on to the frontier. It is also shown that when the production possibility set is convex, a comparison of the input- and output-oriented efficiency measures reveals the nature of (local) returns to scale when increasing returns prevails at the input-oriented proection or diminishing returns holds at the output-oriented proection of an inefficient input-output bundle. Keywords: Most Productive Scale Size; Convex Technologies, Nonparametric Efficiency Analysis JEL Classification Codes: D2; C6 Revised: April 2009 The paper has benefited from insightful comments from Subal Kumbhakar and from two referees on an earlier version of the manuscript. The usual disclaimer applies.
CATEGORICAL AND AMBIGUOUS CHARACTERIZATION OF RETUNS TO SCALE PROPERTIES OF INPUT-OUTPUT BUNDLES IN DATA ENVELOPMENT ANALYSIS In the nonparametric Data Envelopment Analysis (DEA) literature, there are three alternative (but equivalent) approaches to identifying the nature of local returns to scale at any given input-output bundle on the frontier of the production possibility set: (i) a primal approach due to Banker (984), (ii) a dual approach due to Banker, Charnes, and Cooper (984), and (iii) a nesting approach due to Färe, Grosskopf, and Lovell (FGL) (985). In a later paper, Kerstens and Vanden Eeckaut (KVE) (999) combined FGL s nesting approach with Varian s (990) concept of goodness-of-fit in nonparametric models to propose a slightly different method of determining the nature of local returns to scale. It is well known that when the technology exhibits constant returns to scale (CRS) globally, input- and output oriented radial measures of technical efficiency are identical. By implication, if this equality does not hold for every input-output bundle, the technology is characterized by variable returns to scale (VRS). In their paper on Generalized Measures of Farrell Efficiency, Førsund and Halmarsson (979) invoked Frisch s (965) second law of beam variation to note that whenever the input-oriented radial measure of technical efficiency is greater (less) than the corresponding output-oriented measure, the scale elasticity is greater (less) than unity. As noted by KVE (999), the Førsund and Halmarsson paper has received far less attention than it deserves in the subsequent returns to scale and scale efficiency literature. It is to be noted, however, that their scale elasticity is a weighted average of point elasticities along the segment of the production frontier that lies between the input-and the output-oriented proections of an inefficient inputoutput bundle. As such, this elasticity measure does not provide any unequivocal information about the local returns to scale at the input- or the output-oriented proection. When a firm is found to be operating in the region of increasing returns to scale, an implied udgment is that it is smaller than its optimal size. Similarly, a firm operating in the region of diminishing returns to scale is considered to be too large. However, the concept of local returns to scale is meaningful only for efficient input-output bundles. For an inefficient bundle, one must consider its efficient proection on to the frontier. It is often the case that the input- and the output-oriented proections lie in two different regions of the frontier one exhibiting increasing and the other exhibiting diminishing returns to scale. This can be a source of confusion about whether the firm is too small or too large. Such confusion arises mainly because returns to scale properties are evaluated at an efficient proection, but udgment about the size of the firm is usually made about its actual input-output bundle. The obectives of this paper are twofold. First, we show how one can determine the returns to scale properties of every feasible input-output bundle efficient or otherwise. Specifically, we identify different regions within the production possibility set where increasing, constant, or diminishing returns to scale hold See Ray (2004, chapter 3). 2
with respect to both inputs and outputs. We also identify regions where the nature of returns to scale is ambiguous and depends on the direction of proection on to the frontier. The second contribution of this paper is to show that, when the production possibility set is convex, a comparison of the input- and outputoriented efficiency measures reveals the nature of (local) returns to scale when increasing returns prevails at the input-oriented proection or diminishing returns holds at the output-oriented proection of an inefficient input-output bundle. The paper unfolds as follows. Section 2 provides some basic concepts and definitions and proves a lemma showing that if the production possibility set is convex, increasing returns to scale (IRS) cannot follow diminishing returns to scale. An implication of this lemma is that increasing returns to scale prevails at all points on the frontier until a most productive scale size (MPSS) is attained. Similarly, diminishing returns holds at all efficient bundles larger than the (largest) MPSS. Section 3 describes how a DEA model can be utilized to find the MPSS for any feasible input-output bundle and what we can conclude about the observed size of a firm. Section 4 shows that, in some cases, a comparison of input- and output-oriented measures of technical efficiency of an inefficient bundle can reveal the nature of returns to scale at the input- or the output-oriented proection. Section 5 summarizes the main conclusions. 2. The Methodological Background Basic Concepts and Definitions: The production technology faced by firms in an industry producing output vectors ( from input vectors (x) can be described by the production possibility set T = {( x, : xr ; yr ; y can be produced from x}. () n m An input-output bundle (x, is considered feasible if and only if (, T. x Following the convention in DEA, we assume that the production possibility set is convex and also that both inputs and outputs are freely disposable. The frontier of the production possibility set (also known as the graph of the technolog is G ( x, :( x, T; ( x, T; ( x, T. (2) The input-oriented technical efficiency of a feasible input-output bundle (x, is min :( x, G. (3) x Similarly, the output-oriented technical efficiency of the same bundle is, where y max:( x, G. (4) Obviously, and. 3
Banker (984) generalized Frisch s (963) concept of the technically optimal production scale to define a most productive scale size (MPSS) in the context of multiple-input multiple-output technology 2. An inputoutput bundle (x,y ) G is a most productive scale size if, for any non-negative scalars α and β such that (βx, αy ) G,. Thus, (x,y ) is a point on the frontier of the production possibility set where average productivity attains a maximum in the single-output single-input case or ray average productivity reaches a maximum in the multiple-output multiple-input case. Locally constant returns to scale holds at an input-output bundle ( x, G, if there is a real number 0 (however small) such that ( x, G and, for. Locally increasing returns to scale holds if ( x, G and, for. Similarly, locally diminishing returns to scale holds if ( x, G and, for. Banker (984) has shown that locally constant returns to scale holds at an MPSS. It is possible, however, that the maximum (ra average productivity is attained at multiple levels (scales) of input. For such technologies, locally constant returns to scale holds at every input (bundle) within this range. In such cases, of particular interest are the smallest and the largest MPSS bundles. We now prove the following lemma to show that locally increasing returns holds at every scale smaller than the smallest MPSS and locally diminishing returns holds at every scale greater than the largest MPSS. Lemma: For any convex productivity possibility set T, if there exist non-negative scalars α and β such that α >β >, and both ( x, and ( x, G, then for every γ and δ such that <δ<β and ( x, G. Proof: Because ( x, and ( x, are both feasible, by convexity of T, for every ( 0,),(( ( ) ) x, ( ( ) ) is also feasible. Now select such that ( ). Further, define ( ). Using these notations, ( x, T. But, because ( x, G,. However, because,. Hence,. An implication of this lemma is that, when the production possibility set is convex, if the technology exhibits locally diminishing returns to scale at smaller input scale, it cannot exhibit increasing returns at a bigger input scale. This is easily understood in the single-input single-output case. When both x and y are 2 See also Banker and Thrall (992) and Banker et al. (2004). 4
scalars, average productivity at ( x, is y x and at ( x, y it is. Thus, when, average x productivity has increased. The above lemma implies that for every input level x in between x and x, average productivity is greater than y x. Thus, average productivity could not first decline and then increase as the input level increased from x to x. Two results follow immediately. First, locally increasing returns to scale holds at every input-output bundle (x, G that is smaller than the smallest MPSS. Second, locally diminishing returns to scale holds at every input-output bundle (x, G that is greater than the largest MPSS. To see this, let x =bx and y = ay, where (x, y ) is the smallest MPSS for the given input and output mix. Because (x, is not an MPSS, a. Further, assume that b <. Define ( ) and. Then (x, y ) = (βx, α and. b b Because ray average productivity is higher at a larger input scale, by virtue of the lemma, locally increasing returns to scale holds at (x,. Next assume that b >. Again, because (x, is not an MPSS, b a <. That is ray average productivity has fallen as the input scale is increased from x to x = bx. Then, by virtue of the lemma, ray average product could not be any higher than b a at a slightly greater input scale, a x (+ε)x. But, because (x, is not an MPSS, ray average product cannot remain constant as the input scale is slightly increased. Hence, ray average product must fall as the input scale is slight increased from x. Thus, locally diminishing returns to scale holds at every (x, G, when x is larger than the largest MPSS. (Insert Figure here) In Figure, the two axes measure the levels of input and output in the one input-one output case and the graph is merely the production function. In the multiple-output multiple-input case, the two axes show the input and output scale for some given input- and output-mix. The broken line ABCDEF in Figure shows the graph of some hypothetical technology. All points between (and including C and D) represent an MPSS. In this figure, C represents the smallest and D the largest MPSS. All points to the left of C exhibit locally increasing returns to scale. Locally diminishing returns to scale holds at any point to the right of D. For the inefficient input-output bundle shown by the point R, R x is its input-oriented technically efficient proection on to the graph. Increasing returns to scale holds at R x. Similarly, the point R y is its output oriented proection where diminishing returns to scale holds. 3. Using the Most Productive Scale Size for a Benchmark In the DEA literature on scale efficiency, the main interest has been on testing whether a specific inputoutput bundle is an MPSS. Surprisingly, there has been little interest in finding out what is an MPSS for that input-output bundle when the answer is in the negative. In this section, it is shown how a DEA model 5
currently available in the literature 3 can be used not only to determine whether an input-output bundle (x 0, y 0 ) is an MPSS but also to identify the bundle (x, y ) which is an MPSS for (x 0, y 0 ). For this, we use the observed input-output data (x, y ) (=,2,,N) to solve the following DEA problem considered by Cooper, Thompson, and Thrall (996): Subect to Maximize 0 x x ; y y 0 ; (5) ;,, (,2,..., N) 0. Because (x 0, y 0 ) is assumed to be a feasible input-output bundle, ( ) is a feasible solution for this problem. Hence, the optimal value is always greater than or equal to.when exceeds unity, we know that (x 0, y 0 0 0 ) is not an MPSS. But we can also conclude that ( x, y ) is an MPSS. We first assume that the MPSS is unique and deal with the case of multiple MPSS presently. When the bundle (x 0, y 0 ) is not itself an MPSS, so that. possibilities: (i) ; (ii) ; and (iii). There are three distinct (Insert Figure 2 here) Figure 2 shows the three different cases graphically. It should be noted that in this diagram (as would be the case in Figure for a multi-output multi-input case) the broken line ABCDE is a fixed-proportions graph of the technology and is defined as: 0 0 0 0 G ( x, y ) (, ) :( x, y ) G;, 0. (6) The point C (α= α, β=β ) is the unique MPSS. The actual input-output bundle (x 0, y 0 ), can be represented by the point (α=, β= ) in this diagram. At every point in region () to the southwest of C, both the input and output scales are smaller than the MPSS (i.e., α and β ). Therefore, if the actual bundle (α=, β=) lies in this region, the firm is unambiguously exhibiting increasing returns to scale. If it is an efficient bundle lying on the frontier, increasing returns holds at that point. If it is an inefficient bundle that lies below the 3 See, for example, Cooper, Seiford, and Tone (2004). 6
frontier, increasing returns holds at both the input- and the output-oriented proections. In a perfectly analogous way, at every point in region (2) to the northeast of C, the input and output scales are both larger than the MPSS. Hence, if (α=, β=) falls in this area, the observed bundle exhibits categorically diminishing returns to scale. Points in region (3) to the southeast of C, however, are problematic. On the one hand, for any point located in this area, the input scale is larger than the MPSS (β ). This implies diminishing returns to the input scale as is evident from the fact that an output-oriented efficient proection would lie to the right of C on the frontier. At the same time, the output scale is smaller than the MPSS (α ) and the input-oriented oriented proection is to the left of C implying increasing returns to the output scale. This is exactly the situation shown by the point R in Figure. It is an inherent dilemma in technologies where variable returns to scale apply: not only the measured technical efficiency but also the returns to scale characterization depends on whether the analysis is output- or input-oriented. Before we move on to the case of multiple MPSS, it would be useful to take a closer look at the DEA problem in (5). As such, the obective function is nonlinear. However, it can be easily transformed into a linear programming problem. Define t and μ = tλ ( =, 2,,N). Note that non-negativity of β and λ s ensures that t and μ s are also non-negative. Problem (5) can, therefore, be reformulated as the following linear programming problem: Maximize Subect to 0 x x ; 0 y y ; (7) t; t, (,2,..., N) 0. From the optimal solution of this problem we can derive and. One can then infer the nature of returns to scale from these values of α and β. An interesting point to note here is that because t has no restriction other than non-negativity, (6) actually is an output oriented CCR problem. Hence, when at the optimal solution of this problem, we can conclude that β is less than unity and diminishing returns to scale holds at the output-oriented proection on to the VRS frontier. This is what can also be readily derived from Banker s (984) primal approach with an output-oriented DEA model. But without a corresponding value of α as well, we cannot conclude anything about returns to scale at the input-oriented proection. t t 7
Next we consider the possibility of multiple MPSS. This is depicted graphically in Figure 3. Here both C and C 2 are MPSS and so are their convex combinations lying on the line segment connecting them. At C, (, ) is the smallest MPSS. Similarly, ( 2, 2 ) at C 2 is the largest MPSS. It is obvious that when (6) has a unique optimal solution (in particular, t is unique), there cannot be multiple MPSS. For multiple optimal solutions, the largest across all optimal solutions of (6) corresponds to the smallest t MPSS,. Similarly, 2 corresponds to the smallest t at an optimal solution. Note that across all optimal solutions the value of the obective function is the same ( ). Hence,, t where s.t. t max 0 x x ; 0 y y ; (8) (,2,..., N) 0. Similarly,, where 2 2 t t min 2 s.t. 0 x x ; 0 y y ; (9) Once (,2,..., N) 0. and 2 have been determined from (8) and (9), the corresponding values of are readily obtained as and. 2 2 (Insert Figure 3 here) As shown in Figure 3, the set of output-input scales (α, β) for which the input-output bundles (βx 0,αy 0 ) are feasible can be partitioned into six different regions defined below: 8
(i) In region () towards the southwest of the smallest MPSS (C ), ( ; ). When (x 0, y 0 ) falls in this region,. Hence, increasing returns to scale holds unambiguously. (ii) In region (2) to the northeast of the largest MPSS (C 2 ), ( ; ). If (x 0, y 0 ) falls in 2 2 this region,. Diminishing returns to scale holds unambiguously in this region. (iii) In region (3), 2 while 2. Points in this region lie between the smallest and the largest MPSS. It is interesting to note, that even if the point (α=, β= ) is not technically efficient and lies below the C C 2 line, both the input- and the output-oriented proection of the inefficient bundle will fall in the region of constant returns to scale. Thus, there is no scale inefficiency in this region even though there may be technical inefficiency,. (iv) In region (4), 2 ;. When the actual input-output bundle lies here, 2. The input bundle x 0 is larger than the largest MPSS hence the output oriented proection falls in the area of diminishing returns. At the same time, the actual output bundle is smaller than the smallest MPSS. Hence, increasing returns to scale holds at the input oriented proection. Thus, returns to scale cannot be unambiguously defined at the actual input-output bundle. (v) In region (5a), but. When the actual input-output bundle lies here, y 0 2 is smaller than the smallest MPSS and the input oriented proection falls in the area of increasing returns. At the same time, the actual input bundle lies between the smallest and the largest MPSS. Hence, constant returns to scale holds at the output oriented proection. Here also the returns to scale characterization depends on the orientation. (vi) In region (5b), 2 while 2. When the actual input-output bundle lies here, x 0 is larger than the largest MPSS. Hence the output oriented proection falls in the area of diminishing returns. At the same time, the actual output bundle lies between the smallest and the largest MPSS. Hence, constant returns to scale holds at the input oriented proection. Here the input bundle is too large. But the actual output bundle, if produced from the technically efficient input bundle would correspond to an MPSS.. 4. Comparing Input- and Output-oriented Measures of Technical Efficiency We now show what one can learn about the returns to scale properties of the input-output bundle (x 0, y 0 ) of an inefficient firm by comparing its input- and output-oriented technical efficiencies.. Proposition : If the input-oriented technical efficiency is greater than the output-oriented technical efficiency, then locally increasing returns to scale holds at the efficient input-oriented proection of (x 0, y 0 ). 9
Proof: The input-oriented proection of the bundle (x 0, y 0 ) on to G is (θ x 0, y 0 ) where θ is the measured level of input-oriented technical efficiency (τ x ). Similarly, the output-oriented proection is (x 0, φ y 0 ) and 0 0 the output-oriented technical efficiency is. Define the input bundle and the output y x x 0 0 bundle y y. Note that both the input-output bundles ( x 0, y 0 ) and ( x 0, y 0 ) are in G. Further, ( 0, y 0 0 0 x ) can also be expressed as ( x, y ) where and. Now, x y implies. Note that, by construction, and. Thus, ray average productivity is higher at the bigger input scale 0 x. Hence, by virtue of the lemma, locally increasing returns to scale holds at the input-oriented efficient proection ( x 0, y 0 ). Proposition 2: If the output-oriented technical efficiency is greater than the input-oriented technical efficiency, then locally diminishing returns to scale holds at the efficient output-oriented proection of (x 0, y 0 ). Proof: Now suppose that τ y > τ x. That is, implying that. In other words, ray average productivity is lower at the output-oriented proection (x 0, φ y 0 ) than at the input-oriented proection (θ x 0, y 0 ). Hence, locally diminishing returns to scale holds at (x 0, φ y 0 ). It is obvious that compared to the procedure described in the previous section, this method of simply comparing the output- and input-oriented technical efficiencies to determine the nature of returns to scale has many limitations. First, it cannot be applied when the two efficiency measures are equal. Second, even though it can identify increasing returns at the input-oriented proection, one cannot detect increasing returns at the output-oriented proection. Similarly, it fails to determine if diminishing returns holds at the input-oriented proection. In spite of these limitations, this procedure has some advantages. First, this procedure is not affected by the presence of multiple optimal solutions of the relevant DEA models. Second, knowing that increasing returns holds at the input-oriented proection is by itself useful information because it implies that as an output target the bundle y 0 is too small. Similarly, when diminishing returns holds at the output-oriented proection, we can conclude that the input bundle is too large 4. 5. Summary 4 One may note in passing that the procedure proposed in Section 3 uses a CCR model. By contrast, in this section we use a BCC model. 0
When both the actual input and output bundles of a firm are smaller than their (smallest) MPSS, the firm is unambiguously too small and an increase in size will lead to higher productivity. Conversely, when both bundles are larger than the (largest) MPSS, the firm is too big. In some cases, the observed output scale is too small but the input scale is not. There are other cases, where the opposite is true. Also, comparison of input- and output-oriented technical efficiency measures can identify if the actual output scale is too small or the actual input scale is too large.
References: Banker, R.D. (984), Estimating the Most Productive Scale Size Using Data Envelopment Analysis, European Journal of Operational Research 7: (Jul 35-44. Banker, R.D., A. Charnes, and W.W. Cooper (984), Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis, Management Science, 30:9 (September), 078-92. Banker, R.D., W.W. Cooper, L. M. Seiford, R.M. Thrall, and J. Zhu (2004) Returns to Scale in different DEA Models ; European Journal of Operational Research 54: 345-362. Banker, R.D. and R.M. Thrall (992) Estimating Most Productive Scale Size using Data Envelopment Analysis, European Journal of Operational Research,62, 74-84. Cooper, W.W., R.G. Thompson, and R.M. Thrall (996) Introduction: Extensions and New Developments in DEA ; Annals of Operations Research, 66, pp 3-45. Cooper, W.W., L. Seiford, and K. Tone (2004) Data Envelopment Analysis: A Comprehensive Text with Uses, Example Applications, References, and DEA Solver Software Norwell, MA: Kluwer Academic Publishers. Fare, R., S. Grosskopf, and C.A.K. Lovell (985) The Measurement of Efficiency of Production Boston: Kluwer-Nihoff. Førsund, F. and L.Halmarsson (979) Generalized Farrell Measures of Efficiency: An Application to Milk Processing in Swedish Dairy Plants ; The Economic Journal;(89) 354, 294-35. Frisch, R. (965) Theory of Production. Chicago: Rand McNally and Company. Kerstens, K. and P. Vanden Eeckaut (999) Estimating Returns to Scale Using Nonparametric Deterministic Technologies: A New Method Based on Goodness-of-Fit ; European Journal of Operational Research, 3(), 206-24. Podinovski, V. (2004) Local and Global Returns to Scale in performance Measurement ; Journal of the Operations Research Society (2004) 55, 70-78. Ray, S.C. (2004) Data Envelopment Analysis: Theory and Techniques for Economics and Operations Research New York: Cambridge University Press Varian, H.R. (990), Goodness-of-Fit in Optimizing Models, Journal of Econometrics 46, 25-40. 2
Output (level/scale) φy 0 Ry E F D C Rx R(x 0,y 0 ) y 0 B O A θx 0 x 0 Input (level/scale) Figure. Graph of the Technology and MPSS 3
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Output Scale (α) D E α 2 C 2 2 α C 3 5b 5a 4 B 0 A β β 2 Input Scale (β) Figure 3: Multiple MPSS & Regions of Increasing, Constant, Decreasing, and Ambiguous Returns to Scale 5