Estimating most productive scale size using data envelopment analysis

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1 Estimating most productive scale size using data envelopment analysis 35 Rajiv D. BANKER Carnegie. Mellon Untt,erslt.v. School o/urban and Public Affairs. Pittsburgh. PA 15213, U.S.A. Received February 1982 Revised February 1983 The relation between the most productive scale size (mpss) for particular input and output mixes and returns to scale for multiple-inputs multiple-outputs situations is explicitly developed. This relation is then employed to extend the applications of Data Envelopment Analysis (DEA) introduced by Charnes. Cooper and Rhodes (CCR) to the estimation of most productive scale sizes for convex production possibility sets. It is then shown that in addition to productive inefficiencies at the actual scale size. the CCR efficiency measure also reflects any inefficiencies due to divergence from the most productive scale size. Two illustrations of the practical applications of these results to the estimation of most productive scale sizes and returns to scale for hospitals and stem-electric generation plants are also provided to emphasize the advantage of this method in examining specific segments of the efficient production surface. 1. Introduction In the estimation of production correspondence, the question of determining the optimal scale for the production process is often of considerable interest. This is evidenced, for instance, by studies of production and cost functions for hospitals [20] and for electricity generating plants [16]. For a single-input single-output case, the most productive scale size is simply that scale for which the average productivity measured by the ratio of total output to total input is maximized. On the other hand. at the optimal scale size, the marginal productivity is equal to the ratio of the output price to the input price. The concept of average productivity is commonly extended to the case of multiple inputs by Helpful comments by Professors A. C'harne.~ and W.W. Cooper are gratefully acknowledged. North-Holland European Journal of Operational Research 17 ( the use of input prices to aggregate the multiple inputs, and by the estimation of the correspondence between the total input cost and the output. But input prices are affected by many factors other than the pure technological correspondence between the inputs and the outputs that characterize the production process. Prices are likely to be more volatile than the pure technological characteristics, and therefore, estimation of merely the cost function is likely to retain its relevance for managerial and policy decisions for a shorter period than the estimation of the purely technological relation between the physical quantities of inputs and outputs. It is useful, therefore, to distinguish between the problem of determining the minimum cost mix of inputs and outputs on the basis of their relative prices, and the problem of determining the most productive scale size ( = mpss) for particular input and output mixes. In other words, for each input and output mix there corresponds a mpss, while the overall optimal scale size depends on the prevailing prices. The former is related to the concept of returns to scale, while the latter is associated with economies of scale. The focus of this paper will be on the problem of estimation of mpss for different input and output mixes. We shall first develop a rigorous generalization of the concepts of returns to scale and most productive scale size to the case of multiple inputs and multiple outputs. We shall then extend the applications of Data Envelopment Analysis (= DEA) introduced by Charnes, Cooper and Rhodes (hereinafter referred to as CCR). In [12], CCR provide a new mathematical programming approach for measuring the relative efficiency of a Decision Making Unit (= DMU) that extends to situations involving multiple inputs and outputs. The CCR approach has been employed to estimate individual DMU managerial inefficiency, program inefficiency and for setting efficient targets for inefficient DMUs. In an important paper [5], Banker, Charnes and Cooper (hereafter referred to as BCC) formalize the links of DEA with the estimation of efficient /~4/$3.00 L 1984, Elsevier Science Puhlisher~ B.V. (North-Holland)

2 36 R.D. Banker / Estimating most productive scale size production frontiers from observed data on inputs and outputs levels for several DMU's representing a particular industry. They postulate certain basic properties, such as convexity, for the production possibility set, and derive a characterization for this set. They also provide linear programming formulations to estimate various characteristics of the efficient production function, such as rates of substitution, rates of transformation, marginal productivities and local returns to scale. Banker and Maindiratta [10] further develop the link between DEA and recently-developed nonparametric methods in econometrics for estimating production correspondences and for testing consistency of production and price data with a postulated profit maximization behavior. DEA techniques for estimating efficient production frontiers, therefore, provide a promising mathematical programming alternative to the usual parametric statistical methods used for this purpose. DEA enables us to estimate the efficient production correspondence without requiring restrictive assumptions about the underlying production technology. This is in contrast to the classical estimation methods that estimate production correspondence using a pre-specified parametric functional form which involves implicit assumptions about the nature of the underlying production technology. Banker, Charnes, Cooper and Maindiratta [7] describe a simulation study that demonstrates the superior results obtained when DEA methods are used for estimating production frontiers as compared to when parametric translog function estimation methods are employed. The DEA techniques are also more flexible because they enable us to examine differences in production characteristics in difficult segments of the production possibility set. This paper is structured as follows. In Section 2, we first provide a rigorous generalization of the concept of returns to scale for a multiple-outputs and multiple-inputs technology. We also develop the relation between the returns to scale for a given mix of inputs and outputs and the most productive scale size (mpss) for that mix. In Section 3, we illustrate how these conceptual constructs may be measured by examining a simple diagram representing a two-dimensional section of a production possibility set for a mix of inputs and a mix of outputs. In Section 4, we address the question of estimating returns to scale and mpss when the production possibility set is assumed to satisfy the convexity and other postulates of DEA. We provide explicit characterizations for the mpss and the nature of the returns to scale. We also prove that the CCR efficiency measure is equal to one only if the DMU represents a mpss. Finally, in Section 5, we briefly describe two studies where these concepts and methods for estimating returns to scale and mpss were employed. The estimation of returns to scale and mpss is not the sole objective of either of the two empirical studies, and therefore, they are presented here only to illustrate some practical applications of the concepts developed in this paper. 2. Returns to scale and most productive scale size The concept of returns to scale is directly related to the estimation of the most productive scale size (-- mpss). Menger [23] distinguishes between diminishing marginal productivity and diminishing returns to scale. The latter is expressed in terms of proportionate changes in output and inputs. A production correspondence is said to exhibit increasing returns to scale if an increase in all inputs (keeping mix constant) results in a greater than proportionate increase in the output; and decreasing returns to scale if the increase in the output is less than the proportionate increase in all the inputs (see [21, p. 131].) To formally develop and extend this concept to the case of multiple outputs, we first define production possibly set T as T = { ( X, Y) [the output vector Y >/0 can be produced from the input vector X >1 0}. (1) Returns to scale at a point (X, Y) on the efficient surface of the production possibility set can then be expressed in terms of a quantity p defined by p= lim a(fl)-1,,_, #-1 (2) where a(,a) = max{ al(bx, ar) ~ T},~>O. Thus, p > 1 would imply that the efficient production correspondence exhibits increasing returns to scale, since a change in the inputs (keeping input mix constant) causes a greater than proportionate

3 R.D. Banker / Estimating most productwe scale stze 37 change in the outputs (keeping output mix constant). Similarly, diminishing returns to scale and constant returns to scale would correspond to values of p less than one, and equal to one respectively. Note that the concept of returns to scale is defined in terms of the efficient production surface in a small neighborhood around the given point (X, Y). It can also be easily verified that the above definition is a direct generalization of the concept of returns to scale commonly defined for the case of only a single output. We next turn to the concept of mpss and its relation to returns to scale in the context of multiple-inputs multiple-outputs situations described above. As noted in the preceding section, for each input and output mix there corresponds mpss, and only by employing knowledge of input and output prices or a similar trade-off mechanism can we determine an optimal scale and mix for the technology. The mpss for a given input and output mix is the scale size at which the outputs produced 'per unit' of the inputs is maximized. Thus, a production possibility (X~, y,)~ T represents a mpss if and only if for all production possibilities (fix,, ay,)~ Twe have a/fl < The concept of mpss is thus based on the comparison of average productivities. In order.to maximize the average productivity, one would increase the scale size if increasing returns to scale were prevailing, and decrease the scale size if decreasing returns to scale were prevailing. This relation can be formalized in terms of Proposition 1. Proposition I. 1]" a production possibility ( X,, Y, ) T represents a mpss for the input and output mixes represented by the vectors X 5 and Y, respectively, and if ( X,, Y, ) is neither the smallest nor the largest production possibility for these input and output mixes, then the production correspondence exhibits non-decreasing returns to scale at production possibilities a little smaller than ( X,, Y,) and non-increasing returns to scale at production possibilities a little larger than (X~, Y~). Further, constant returns to scale prevail at ( X~, Y, ). Proof. Consider a neighboring point (fix,, a(fl)y,)~ 7", where a(fl)= max{al(/3x,, ay,)~ T}. Since (X,,Y,) represents mpss, and (fix, a(fl)y,) ~ T, a(o)/g <~ 1. Then for the case where fl < 1, we have Therefore, /3-I I> I, sincefl-i <0. lim a(/3)-i._,_, /3-1 :>l (3) where ( > 0 is a sufficiently small quantity. Thus, for a production possibility a little smaller than ( X,, Y~), the returns to scale must be non-decreasing. Similarly for the case where B > 1, we have o,(b)- I a(/3)~/3~ /~-I ~<I, sincefl-i>o, and therefore lim a( fl ) - 1 ~< 1 (4) ~-l, /3-1 for sufficiently small c > 0. Thus, for a production possibility a little larger than (X, Ys), the returns to scale must be non-increasing. Finally, if the returns to scale function is defined at (X,, Y,), it follows from (3) and (4) that lim a( /3 ) - 1 = Thus. constant returns to scale prevail at (X,, Y~). 3. Motivating illustration In this section we shall consider a simple example to illustrate the concept of mpss defined in the preceding section and to motivate the theoretical developments presented in the next section that provide a method for the empirical estimation of mpss. We shall consider, in particular, production possibilities having the same input and output mixes, given by the vectors X 0 and Y0 respectively. Figure 1 displays a two-dimensional section of a production possibility set T by the plane given by X=xX o. and Y=yY0, where x and y are scalar quantities. In this example, the point E represents the production possibility that maximizes the 'average productivity', measured by the slope of the line

4 38 R.D. Banker / Estimating most productive scale size YYo.! M / jr 1, 7- / Section of Production Possibility Set T / // 0 L V xx o Fig. 1. Two-dimensional section of the production possibility set T by the plane determined by X - xx o and Y =.v Y0. OE (or the ratioye : xe), for the mix of outputs Y0 and inputs X 0. Therefore, E represents mpss for the given mix. Also constant returns to scale prevail at E since (dy/dx).(x/y)= 1. At a point such as B on the efficient production surface, the average productivity measured by the slope of the line OB (or the ratio YB: xb) is less than the slope of the line OE. Also, increasing returns to scale prevail at B since (dy/dx). (x/.v)> 1. Similarly, at point C, the average productivity is less than at the point E, and decreasing returns to scale prevail since (d y/dx ).( x/y ) < 1. Since E represents the production possibility with the largest average productivity for the given mix of inputs and outputs, we can estimate the inefficiency of a point such as A by comparing it with E. Therefore, we measure the efficiency of A by the ratio t ya/xa _ OM/MA ha ye/xe OR/RE We let k = OM/OR =.YJYe represent the divergence of the scale size at A from the mpss at E.Since MN = k RE, we have MN k x e ha = M,4 = x A Therefore, we can now write the most productive I See [1] for a detailed analysis. j/ scale sizes as follows: (1) measured for inputs: x E = ( h A/k ). x A, (2) measured for outputs: YL = k- 1 "YA. The efficiency measure h A is equivalent to the CCR efficiency measure. In the next section we shall describe how the CCR efficiency analysis can be employed in the manner described above to estimate mpss. Note also that the CCR efficiency measure captures both scale inefficiency due to divergence from mpss, and productive inefficiency at the given scale of operations, this becomes apparent when we write ha= ya/xz = )'A/XA )'n/xb Ye/XE Ys/Xs )'E/XE" or equivalently, MN MB MN h "~ = -MA = M A M B " The ratio MB:MA measures the efficiency-" of the point A relative to the point B, which is the most productive production possibility with the same scale size. In [5], Banker, Charnes and Cooper describe how we can estimate this component of the overall CCR efficiency measure. On the other hand, the ratio MN: MB measures the efficiency due to the divergence of the actual scale size from the mpss. 4. Data envelopment analysis In this section we shall extend Data Envelopment Analysis (DEA) to the estimation of most productive scale size (= mpss). These developments will also enable us to identify the two kinds of inefficiencies of DMUs that are reflected in the CCR efficiency measure. In [13,15] CCR define this new efficiency measure for a specific DMU o in terms of the following fractional programming problem: maximize ~ Ur);O / ~ V,X,., (5) r-i I-1 2 The ratio MB: MA represents the 'input" efficiency measure for A, and it will not be the same as the "output" efficiency measure except for a constant returns to scale technology. See [17].

5 R.D. Banker / Estimating most productive scale size 39 subject to A Y'. u,y~/ ~ v,x,, r-- I t~ ] tlr, V I > O. ~<1, j=l... n, Here the x,j and )~ represent the observed values for the i m inputs and r= 1... s outputs for each of j = 1... n DMUs. The CCR measure evaluates the efficiency of a specific DMU o relative to the other DMUs. CCR [12] also show that the above fractional programming problem can be reduced to an equivalent linear programming problem. We shall prefer working with the dual of this linear programming problem, which we represent as minimize ho -, E s:0+ S,o +, subject to i J~l r=l t-i X~x,j+s,0--h0x,0, i= l... m, '~" )t,);j - s20 =Yr0, r = 1... S, /~1 X 1, Slo, ~ Co,ho>~O, (6) and where c > 0 is a small non-archimedean quantity. In the preceding description of the CCR efficiency measure, we have not made any assumptions about the underlying production technology. Following Banker, Charnes and Cooper [4], we shall now introduce three postulates to characterize the production possibility set T. Postulate 1 (Convexity). If (X r Y~)~ T, j = 1... n, and ~1 >~ 0 are non-negative scalars such that.,'-i then (~_~ ~ ~ T. Postulate 2 ( Inefficiency postulate ). (a) If(X, Y)~ T and X>~ X, then ( X, Y)~ T. (b) If(X, Y)~ T and Y <~ Y, then ( X, P)~ T. Postulate 3 (Minimum extrapolation). T is the intersection set of all 7" satisfying the earlier postulates, and subject to the condition that each of the observed vectors ( Xj, 1Ii ) ~ ~' J = 1... n. IfT satisfies the above three postulates, then T can be expressed as T={(X, Y)IX>~ ~#jxj, Y<~ ig,y,, J~l J'l For the purpose of the analysis that follows, we shall assume that T satisfies the three postulates. The CCR measure captures not only the productive inefficiency of a DMU at its actual scale size, but also any inefficiency due to its actual scale size being different from the mpss. This becomes evident from Proposition 2. Proposition 2. The Archimedean component of the CCR efficiency rating, i.e. hg in the optimal solution to (6), for a DMU 0 is equal to one if and only if it represents mpss. Proof. Suppose (X 0, Y0) is not mpss. Therefore, there exists some (X, Y)~ T such that X=[3X o, Y= ay o and a/fl> 1. Furthermore. since ( X, Y) T, we have X>~ E #,Xj and Y~< gjy~ (8) y-i /'1 J~l for some #j >/0 and F.~_ ~#j = 1. Let Xj = #j/a, then from (8) we have l- I and xjx,.< Ot Ilt xjr, >~- Y= Yo. Ot t- I Xo Putting h o = ~/a we see that all the constraints of (6) are satisfied. But since h o = fl/a < 1 represents

6 40 R.D. Banker / Estimating most productive scale size a feasible solution of the minimization problem in (6), we must have hg < 1. Thus, we have proved that h~ = 1 =, (X o, Y0) is mpss. Next we shall assume hg < 1. Let ~,~ be the corresponding values of hj in the optimal solution to the programming problem in (6). Then, J--I Define new variables J--1 to. <9) procedure. We formalize this in terms of the following two corollaries: Corollary 1. Local increasing returns to scale correspond to k~ > 1, and local decreasing returns to scale correspond to k ~ < 1. The proof follows directly from the construction of k~ in the proof of Proposition 2. Corollary 2. k~ "'o, k--~o Yo ~ T and is mpss. and J-I tt~=xjk~>10, j=l... n, (10) so that 1-1 g~--1. Further let a = 1/k~ and fl= ho/k o. From (9) and (10), we then have j-i J-I j-1 tq x, < Xo, gtyj >i ay o, (11) Therefore, using (7) we have (fix o, ayo)~ T and also a 1/k~ 1 --= =-->1. B ho/ko hg Therefore ()to, Y0) cannot be a mpss. Thus, we have also proved that (Xo, Yo) is mpss ~ h~ = 1. The transformation defining the new variable k~ in the latter half of the above proof suggests how we may estimate the local returns to scale and the mpss for the input mixes corresponding to DMUs for which hg < 1 in the CCR evaluation tt tt The proof follows directly from the fact that in the CCR efficiency evaluation of the production possibility ((hg/k~)x o, (1/k~)Yo), the Archimedean component of the efficiency measure is necessarily equal to one. By virtue of the first part of Proposition 2, it follows that this production possibility must also be mpss. This corollary thus enables us to estimate the mpss corresponding to any given input and output mixes represented by the vectors X 0 and Y0 respectively, and the value k~ = E~. 1X~ provides a measure of the divergence of the actual scale size from the mpss for the given input and output mixes. Even though the production possibility ((h~/k~)x o, (1/k~)Yo) represents a mpss, it may not represent a point on the efficient production surface. Analogous to the preceding corollary, we have Corollary 3. Corollary 3. The production possibility ( h~x -s~*k~ ' Y +S ')k~ represent a mpss and lies on the efficient production surface, where S~ and S o represent the vectors of input and output slacks, respectively. The above results have been derived from the representation of the CCR efficiency evaluation model as in formulation (6) to compute the input efficiency h~. An equivalent representation to compute the output inefficiency g~' (which is equal to the reciprocal of h~) is as follows: maximize go + ~ si; + s~- o, (12) i

7 R.D. Banker / Estimating most producm,e scale size 41 subject to bt~x,~+ S,o = X,o, i= l... m, I" 1 i #~))- sro - = goy, o, r= l... s, J--1 ~,, s,*o, s2o, go >1 0, and where c > 0 is a small non-archimedean quantity. Proceeding as before we have the following corollaries: Corollary. 4. (Ix g~y )~T q~ o, q~ tl ~t and represents a mpss. where q~ = ~,j. d~. Corollary 5. ( x -S 'q~ " g~y +S *)q~ represents a mpss and lies on the efficient production surface. Finally, if there exists a unique mpss for the input and output mixes represented by (X o. Yo), then the relation between the weights?~7 for the referent DMUs in the programming problem in (6), and the weights #7 for the referent DMUs in the programming, problem in (12) is given by Corollary 6. Corollary 6. J~l J=l This result follows directly from a comparison of the expressions for the mpss. corresponding to the input and output mixes given by (X o, Y0), in Corollary 1 and Corollary Some practical applications Data envelopment analysis has opened up several interesting possibilities for the estimation of production correspondences. This approach does not require any stringent assumptions about the underlying production correspondence, unlike the classical approaches for estimating production functions which necessarily assume several restrictive properties for the production correspondence that are implicit in the use of a prespecified parametric form for the estimation. Furthermore, as the developments in [3] and [4] and in this paper indicate, the assumption that the production possibility set satisfies convexity and other basic postulates, allows us to further extend the CCR efficiency analysis to estimate other characteristics of the production correspondence. The approach outlined here differs in a fundamental manner from the classical approaches. Instead of first specifying a parametric functional form and making further assumptions about the distribution of the disturbance term, we estimate the production correspondence based on minimum assumptions about the characteristics of the production possibility set itself. Having thus obtained the estimates of the production correspondence. we can then test for variations in these values over different segments of the production possibility set. Such an approach is illustrated by the analysis described in [8], where the results presented in this paper were employed to estimate the mpss for different output mixes using the available data for 117 North Carolina hospitals. The production m~xtel specified three outputs: patient-days for patients less than 14 years old, between 14 and 65 years in age, and more than 65 years old, respectively, and four inputs: (1) nursing service hours, (2) general service hours, (3) ancillary service hours and (4) number of beds. The production correspondence was first estimated by using the usual econometric methods to fit a translog parametric function, Statistical tests with this model indicated that a constant return to scale hypothesis could not be rejected. DEA was then employed to estimate mpss for the 117 observations. Welch's [27] mean test indicated that the mpss (measured in terms of number of beds) was significantly greater for the 29 hospitals with the highest proportion of patient-days for patients aged less than 14 years than the 29 hospitals with the lowest proportion of such patient-days. This

8 42 R.D. Banker / Estimating most productive scale size result was confirmed by the Mann-Whitney [22] test. Similar results were obtained when the mpss for the 29 hospitals with the highest proportion of patient-days for patients aged between 14 and 65 years were compared with the mpss for the 29 hospitals with the lowest proportions of such patient-days. The average mpss for the hospitals with a high proportion of patient-days for patients aged below 65 years was approximately 200 beds, indicating that increasing returns to scale could be exploited up to a capacity of about 200 beds, and that decreasing returns set in thereafter. On the other hand, for the 29 hospitals with the highest proportion of patient-days for patients aged above 65 years, the mpss was significantly less than the mpss for the 29 hospitals with the lowest proportion of such patient-days. See Table 1 for detailed results of the statistical tests. The mean mpss for the hospitals with the highest proportion of patient-days for patients aged above 65 years was 76 beds, indicating that decreasing returns to scale set in at much lower capacity levels for hospitals specializing in older patients. It was thus possible to identify different returns to scale in different segments of the production surface, which was not possible using an aggregate econometric estimation procedure with a pre-specified parametric functional form. The next illustration pertains to a part of the analysis described in [9] of production data for 591 steam-electric generation plants in the United States. The preduction model specified two outputs: net generation and peak demand, and three inputs: plant cost, labor and related cost, and fuel consumption. The mpss was estimated for the output-input mixes represented by the production data for each of the 591 electric plants. The observations were divided into three groups on the basis of the ratio of net generation to peak demand. The mean mpss for the 197 plants with the highest such ratios were KWH net generation and MW peak demand. These were significantly greater than the mean mpss of KWH net generation and MW peak demand for the 197 plants with the lowest net generation to peak demand ratios. Mann- Whitney tests indicated similar differences in the mpss for the two groups. See Table 2 for detailed results of these statistical tests. The above two empirical studies are not the principal subject matter of this paper, and the estimation of mpss is not the sole objective of either of the two studies. But these applications provide a good way to conclude this paper because they illustrate how DEA may be employed to examine specific characteristics of a production correspondence that may be different for its several segments. Table 1 Average mpss measured in terms of number of beds ~ Proportion of Proportion of Proportion of patient days below patient days between patient days above 14 years 14 and 65 years 65 years Highest Lowest 29 hosp. 29 hosp. Mean Highest Lowest Highest Lowest 29 hosp. 29 hosp. 29 hosp. 29 hosp Welch's mean test: t b d.f. " 42 p < Median Mann- Whimey test: p < , < < < < O.O00l mpss = (h~/f)~l~) x (number of beds). d.f. - [(s~/n,)+(s~/n 2)]2/((s~/n,)2/(n, - l)+(s~/n 2)2/1n 2 - l)).

9 R.D. Banker / Estimating most productive scale size 43 Table 2 Average mpss for steam generation electric plants mpss measured for net generation (106 KWH) mpss measured for peak demand (MW) 197 plants with 197 plants with 197 plants with 197 plants with highest ratio of lowest ratio of highest ratio of lowest ratio of net gen.: peak dd. net gen.: peak dd. net gen.: peak dd. net gen.: peak dd. Mean Welch's mean test t d.f p < < Median Mann-Whitney test p < < mpss -- (ho/y" _ * ~gllh ~ j, ).(net generation or peak demand). References [1] R.D. Banker, A game theoretic approach to measuring efficiency, European J. Operational Res. 5 (1980) [2] R.D. Banker, Studies in cost allocation and efficiency evaluation, Doctoral Thesis, Harvard University. Graduate School of Business Administration (1980). [3] R.D. Banker, Efficiency analysis for exogeneously fixed inputs, Mimeo (1984). [4] R.D. Banker. A. Charnes and W.W. Cooper, A production economics approach to efficiency measurement, Carnegie- Mellon University, School of Urban and Public Affairs, Working Paper 81-4 (1981). [5] R.D. Baker, A. Charnes and W.W. Cooper. Estimation of technical and scale inefficiencies in data envelopment analysis, Management Set., to appear. 16] R.D. Banker. A. Charnes, W.W. Cooper and A. Schinnar, A bi-extremal principle for frontier estimation and efficiency evaluation, Management Sol. ( 1981 ). [71 R.D. Banker, A. Charnes, W.W. Cooper and A. Maindiratta, A comparison of DEA and translog estimates of production frontiers using simulated observations from a known technology, in Current Issues in Producttl:ttv Analysis, to appear. [8] R.D. Banker. R.F. Conrad and R.P. Strauss, An application of data envelopment analysis to the empirical investigation of a hospital production function, in Current Issues in Productivity Analysts, to appear. [9] R.D. Banker and A. Maindiratta, A nonparametric estimation of production functions and efficiencies of steamelectric generation plants, Carnegie-mellon University. School of Urban and Public Affairs, Working Paper (1982). [101 R.D. Banker and A. Maindiratta, Nonparametric estimation of production frontiers, Mimeo (1984). I111 A. Charnes and W.W. Cooper, An efficiency opening for managerial accounting in not-for-profit entities, in: P. Holzer, Ed., Proceedings of a Conference on Managerial Accounting (Urbana, University of Illinois, Department of Accounting, 1980). [12] A. Charnes, W.W. Cooper and E. Rhodes, Measuring the efficiency of decision making umts, European J. Operational Res. 2 (1978) [13] A. Charnes, W.W. Cooper and E. Rhodes, Short communication: measuring the efficiency of decision making units. European J. Operational Res. 3 (1979) 339. [14] A. Charnes, W.W. Cooper and E. Rhodes, Evaluating program and managerial efficiency, Management Set. (1981). [15] A. Charnes, W.W. Cooper, A. Lewin. R. Morey and J. Rousseau, Efficiency analysis with non-discretionary resources, Center for Cybernetic Studies, the University of Texas at Austin, Research Report 379 ( [16] T.G. Cowing and V.K. Smith, The estimation of a production technology: a survey of econometric analyses of stem-electric generation, Land Economics { 1978) [171 R. Fare and C.A.K. Lovell. Measuring the technical efficiency of production, J. Econom. Theory (1978) [18] M.J. Farrell, The measurement of productive efficiency. J. Roy. Statist. Soc. Set. A (1957) [19] M.J. Farrell and M. Fieldhouse, Estimating efficient production frontiers under increasing returns to scale, J. Roy. Statist. Soc. Set. A (1962) [20] M.S. Feldstein, Econometric studies of health economics, in: M. lntriligator and D. Kendrick, Eds.. Frontiers of Quantitative Economics (North-Holland, Amsterdam 1974) K. Lancaster, Mathematical Economics (MacMillan, New York, 1968). [22] M.B. Mann and D.R. Whitney, On a test of whether one of two variables is stochastically larger than the other, Ann. Math. Statist. 18 (1947) [23] K. Menger. The laws of return: a study in recta-economics, in: O. Morgenstern, Ed., Economic Actioity Analysis Part 111 (Wiley, New York, 1954). [24] J.C. Panzar and R.D. Willig, Economies of scale in multioutput production. Quart. J. Econom. (1977)

10 44 R.D. Banker / Estimatin 8 most productive scale size [25] R.W. Shephard, Cost and Productwn Functions (Princeton University Press, Princeton, MJ, 1953). [26] R.W. Shephard, The Theory of Cost and Producnon Functions (Princeton University Press, Princeton. N J, 1970). [27] B.L. Welch, The significance of the difference between two means when the population variances are unequal, Biometrika 29 (1937) [28] F. Wilcoxon0 Individual comparisons by ranking methods, Biometrics Bull. (1947)