Department of Economics Working Paper Series

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1 Department of Economics Working Paper Series Comparing Input- and Output-Oriented Measures of Technical Efficiency to Determine Local Returns to Scale in DEA Models Subhash C. Ray University of Connecticut Working Paper September Mansfield Road, Unit 063 Storrs, CT Phone: (860) Fa: (860) This working paper is indeed on RePEc,

2 Abstract This paper shows how one can infer the nature of local returns to scale at the input- or output-oriented efficient projection of a technically inefficient inputoutput bundle, when the input- and output-oriented measures of efficiency differ. Journal of Economic Literature Classification: D2, C6 Keywords: Most Productive Scale Size; Conve Technologies, Nonparametric Efficiency Analysis

3 COMPARING INPUT- AND OUTPUT-ORIENTED MEASURES OF TECHNICAL EFFICIENCY TO DETERMINE LOCAL RETURNS TO SCALE IN DEA MODELS It is well known in the nonparametric Data Envelopment Analysis (DEA) literature that when the technology ehibits 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). There are three alternative approaches to identifying the nature of local returns to scale at any given inputoutput 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 (985). The objective of this methodological note is to show how a comparison of the two different efficiency measures reveals the nature of (local) returns to scale at the input-oriented projection of an inefficient input-output bundle on to the frontier of the production possibility set. The paper unfolds as follows. We start with some basic concepts and definitions. Net we prove a lemma showing that if the production possibility set is conve, increasing returns to scale (IRS) cannot follow diminishing returns to scale. Net we prove the main result for both single-output single input and multioutput multi-input technologies. Basic Concepts and Definitions: The production technology faced by firms in an industry producing output vectors () can be described by the production possibility set ( from input vectors + + T = {(, : R ; y R ; y can be produced from }. () n m An input-output bundle (, is considered feasible if and only if (, T. possibility set (also known as the graph of the technolog is The frontier of the production G = {(, :(, T ; > (, T ; β < ( β, T}. α (2) The input-oriented technical efficiency of a feasible input-output bundle (, is τ = θ = minθ :( θ,. (2) Similarly, the output-oriented technical efficiency of the same bundle is See Ray (2004; chapter 3) for an eposition of the alternative approaches and their equivalence. 2

4 τ y =, where ϕ ϕ = maϕ:(, ϕ Obviously, θ and ϕ.. (3) Banker (984) generalized Frisch s (963) concept of the technically optimal production scale to define a most productive scale size (MPSS) in the contet of multiple-input multiple-output technology. An inputoutput bundle (,y ) is a most productive scale size if, for any non-negative scalars α and β such that (β, αy ) G, <. Thus, (,y ) is a point on the frontier of the production possibility set where average β α productivity attains a maimum in the single-output single-input case or ray average productivity reaches a maimum in the multiple-output multiple-input case. Locally constant returns to scale holds at an input-output bundle (, G, ε >0 (however small) such that ( β, and α = β, where β = ± ε. Locally increasing returns to scale holds if ( β, and α > β, where β = + ε. Similarly, locally diminishing returns to scale holds if ( β, and α < β, where β = + ε. if there is a real number Banker (984) has shown that locally constant returns to scale holds at an MPSS. It is possible, however, that the maimum (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 2. (Insert Figure here) In Figure, the two aes 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 aes show the input and output scale for some given input- and output-mi. 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 ehibit locally increasing returns to scale and 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 is its input-oriented technically efficient projection on to the graph. Similarly, the point R y is its output oriented projection. 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 hold s at very scale greater than the largest MPSS. 2 See also Banker and Thrall (992) and Banker et al. (2004) 3

5 Lemma: For any conve productivity possibility set T, if there eist non-negative scalars α and β such that α >β >, and both (, ( δ, γ, γ > δ. Proof: Because (, and and (,, ( β, λ ( 0,),(( λ + ( λ) β ), ( λ + ( λ) α) λ ( λ) β = δ. + Further, define µ λ + ( λ) α. β then for every γ and δ such that <δ<β and are both feasible, by conveity of T, for every is also feasible. Now select λ such that = Using these notations, ( δ, µ T. ( δ, γ, γ µ. However, because α > β, µ > δ. Hence, γ > δ. But, because An implication of this lemma is that, when t he production possibilit y set is conve, if the technology ehibits locally diminishing returns to scale at smaller input scale, it cannot ehibit increasing returns at a bigger input scale. This is easily understood in the single-input ingle-output case. When both and y are scalars, average productivity at (, is y y and at ( β, it is. α Thus, when, average β α > β productivity has increased. The above lemma implies that for every input level in between and β, average productivity is greater than y. Thus, average productivity could not first decline and then increase as the input level increased from to β. Two results follow immediately. Fi rst, locally increasing returns to scale holds at every input-output bundle (, G that is smaller than the smallest MPSS. Second, locally diminishing returns to scale holds at every input-output bundle (, G that is greater than the largest MPSS. To see this, let =b and y = ay, w here (, y ) is the smallest MPSS for the given input and output mi. Because (, is not an MPSS, a b <. Further, assume that b <. De fine ( > ) β and α =. Then (, y α ) = (β, and >. = b Because ray average productivity is higher at a larger input scale, by virtue of the lemma, locally increasing returns to scale holds at (,. Net assum e that b >. Again, because (, is not an MPSS, b a a < β. That is ray average productivity has fallen as the input scale is increased from to = b. Then, by virtue to of the lemma, ray average product could not be any higher than a b at a slightly greater input scale, = (+ε). But, because (, 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 i nput scale is slight increased from. Hence, locally diminishing returns to scale holds at every (,, when is larger than the largest MPSS. We may now present the main result of this note relating the input- and output-oriented technical efficiencies of an inefficient bundle ( 0, y 0 ). 4

6 Theorem : 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 projection of ( 0, y 0 ). Proof: The input-oriented projection of the bundle ( 0, y 0 ) on to G is (θ 0, y 0 ) where θ is the measured level of input-oriented technical efficiency (τ ). Similarly, the output-oriented projection is ( 0, φ y 0 ) and τ = the output-oriented technical efficiency is. Define the input bundle y ϕ θ 0 0 = and the output bundle y = ϕ y. Note that both the input-outpu t bundles (, y ) and ( 0, y 0 ) are in G. Further, ( 0, y ) can also be epressed as (, αy ) β where β = and α = ϕ.now, τ = θ > τ = implies α = ϕ > β =. Note that, by construction, θ < and y ϕ 0 us, ray ivity is r input scale β. Hence, by virtue of the β >. Th average product higher At the bigge lemma, locally increasing returns to scale holds at the input-oriented efficient projection ( 0, y 0 ). θ θ Theorem 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 projection of ( 0, y 0 ). Proof: Now suppose that τ y > τ. That is, > implying that α = ϕ < β =. In other words, ray θ ϕ θ average productivity is lower at the output-oriented projection ( 0, φ y 0 ) than at the input-oriented projection (θ 0, y 0 ). Hence, locally diminishing returns to scale holds at ( 0, φ y 0 ). Conclusion: An implication of the above is that when input-oriented technical efficiency is higher than the output-oriented, the firm would need to increase its output scale in order to attain the most productive scale size, once input-inefficiency is eliminated. Similarly, if output-efficiency is higher, the firm needs to scale down after eliminating output inefficiency 3. One limitation of the methodology proposed here, however, is that it can be applied only when τ τ y. References: Banker, R.D. (984), Estimating the Most Productive Scale Size Using Data Envelopment Analysis, European Journal of Operational Research 7: (Jul 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), Banker, R.D., W.W. Cooper, L. M. Seiford, R.M. Thrall, and J. Zhu (2004) Returns to Scale in different 3 This was pointed out to me by Subal Kumbhakar. 5

7 DEA Models ; European Journal of Operational Research 54: Banker, R.D. and R.M. Thrall (992) Estimating Most Productive Scale Size using Data Envelopment Analysis, European Journal of Operational Research,62, Fare, R., S. Grosskopf, and C.A.K. Lovell (985) The Measurement of Efficiency of Production Boston: Kluwer-Nijhoff. Frisch, R. (965), Theory of Production. Chicago: Rand McNally and Company. Ray, S.C. (2004) Data Envelopment Analysis: Theory and Techniques for Economics and Operations Research New York: Cambridge University Press 6

8 Output (level/scale) φy 0 Ry E F D C y 0 B R R( 0,y 0 ) O A θ 0 0 Input (level/scale) Figure. Graph of the Technology and MPSS 7