On value efficiency

Journal of Productivity Analysis, 24, 197–201, 2005 © 2005 Springer Science+Business Media, Inc. Manufactured in The Netherlands.

On the Interpretation of Value Efficiency PEKKA J. KORHONEN∗ Pekka.Korhonen@hkkk.fi Department of Business Technology, Helsinki School of Economics, P.O. Box 1210, 00101 Helsinki, Finland ¨ MIKKO J. SYRJANEN Gaia Group Oy, Bulevardi 6 A, 00120 Helsinki, Finland

Mikko.Syrjanen@iki.fi

Abstract Halme et al., Management Science, 45, 103–115 (1999) have proposed Value Efficiency Analysis as an approach to incorporate preference information in Data Envelopment Analysis (DEA). Generally, a value function includes only ordinal information and thus a value efficiency score does not provide information on the value difference. The score only describes the improvements in the input/output values that are needed in order to make the Decision-Making Unit (DMU) as preferred as the Most Preferred Solution (MPS). This Paper discusses two sets of additional assumptions that enable us to give the efficiency score a value difference interpretation. Keywords: data envelopment analysis, directional distance function, value function

1.

Introduction

Halme et al. (1999) proposed the concept of Value Efficiency as a way to incorporate preference information in Data Envelopment Analysis. The basic idea is to measure efficiency as a distance to an approximated indifference contour of a Decision-Maker’s (DM) value function. The distance is measured to the contour that passes through the Most Preferred Solution (MPS) in the Production Possibility Set (PPS). Although the value function is not explicitly specified, the DM is assumed to be able to identify the MPS, which maximizes the unknown value function over the PPS. Generally, a value function includes only ordinal information and thus a value efficiency score does not provide information on the value of a Decision-Making Unit (DMU) to the DM (except that the maximal value is achieved at the MPS). Hence, the score only describes the improvements in the input/output values that are needed in order to reach the contour at which the DMU is as preferred as the ∗

Corresponding author.

¨ KORHONEN AND SYRJANEN

198

MPS. However, by making more restrictive assumptions about the value function, value efficiency scores can be interpreted to provide value information as well. Our aim is to discuss those additional assumptions. Let us assume that we have n DMUs each consuming m inputs and producing p×n p outputs. X ∈ m×n and Y ∈ + are matrices representing the observed input + p and output measures for the DMUs. Vector y∈ + refers to outputs and x∈ m + to inputs. In the Data Envelopment Analysis (DEA) context, the PPS is defined as a set T = {(y, x) | y can be produced from x} = {(y, x) |x ≥ Xλ, y ≤ Yλ, λ ∈ }. In the case of the CCR model (Charnes et al., 1978)
= n+ and in the case of the BCC-model (Banker et al., 1984)
= {λ | 1T λ = 1, λ ∈ n+ }. Because, the value function and thus also the indifference contour are generally unknown, Halme et al. (1999) developed the requisite theory and the procedure to approximate the indifference curve of the unknown value function by the tangent cone at the MPS. The value function v(y, x) is assumed to be strictly increasing in y, strictly decreasing in x, and function u(y, −x) (= v(y, x)) is assumed to be pseudo-concave. In this case, there is a unique MPS (y∗ , x∗ )∈ T at which the value function v(y, x) of the DM has a maximum value v(y∗ , x∗ ). Moreover to simplify considerations in this paper, but without loss of generality, we assume that v is continuous. The value efficiency analysis provides the scores, which are lower bounds for true value inefficiency scores. For more theoretical considerations (see Halme et al., 1999). The use of value efficiency in practice is discussed in Korhonen et al. (2002). In this paper, we consider the interpretation of the value efficiency score at a conceptual level without discussing the approximation of the value function. Because v is continuous and strictly monotonic, for each existing unit (y0 , x0 ) ∈ T and for any (wy , wx ) ≥ 0, (wy , wx ) = 0, there exists such a parameter γ ≥ 0 that v(y 0 + γ w y , x 0 − γ wx ) = v(y∗ , x∗ ).

(1)

Parameter γ is called the weighted true value inefficiency score γ for point (y0 , x0 ). Vector (wy , wx ) determines the preassigned direction to the (weakly) efficient boundary of T. When the weighting vector or direction (wy , wx ) for the unit under consideration is (y0 , 0), the model is output oriented. Correspondingly, when the vector is (0, x0 ), the model is input oriented. These projections are called radial in the DEA literature. We may use so-called combined orientation by letting the weighting vector be (y0 , x0 ). For general discussion on the use of directional distance functions, see Chambers et al. (1998). The true value inefficiency score tells how much a DMU has to improve its current output and input values (y0 , x0 ) in the direction (wy , wx ) in order to be as preferred as the MPS. It is evident that γ > 0 iff the point (y0 , x0 ) is value inefficient. Without additional assumptions, the value inefficiency score represents just the (minimum) change in inputs and/or outputs (in the direction of (wy , wx )) required for achieving the value equal to the value at the MPS. Continuity, monotonicity, and pseudoconcavity assumptions alone do not provide enough information for

INTERPRETATION OF VALUE EFFICIENCY

199

evaluating the change in value. The assumptions are not even sufficient for ranking the units on the basis of their value inefficiency scores. Even if we assume that the value function has a cardinal interpretation, we have to make more specific assumptions concerning the functional form of the value function. However, some quite general assumptions enable us to have information about value changes as well. Next we study assumptions concerning the functional form of the value function, and demonstrate that in those cases value (in)efficiency score provides a possibility to evaluate value changes. Homogeneity assumptions that are widely used in economic analysis play an important role in the following considerations. An additional reason to use a homogeneity assumption is that the traditional Farrell efficiency measure is homogenous of degree 1 (F¨are and Lovell, 1978). Assumption 1. Assume that the value function v can be separated so that v(y,x) = f(y)/g(x) for all (y, x) ∈ T . Function f is assumed to be continuous, strictly increasing and homogenous of degree r > 0, and g is assumed to be continuous, strictly increasing and homogenous of degree s > 0. Note that the assumption implies that v is homogenous of degree r – s. Let us first consider a combined-oriented model, i.e. (wy , wx ) = (y0 , x0 ). Equation (1) can now be written as v(y0 + γ y0 , x0 − γ x0 ) = f (y0 + γ y0 )/g(x0 − γ x0 ) = f((1 + γ )y0 )/g((1 − γ )x0 ) = (1 + γ )r f(y0 )/[(1 − γ )s g(x0 )] = v(y∗ , x∗ ). Thus, we may calculate the ratio of the value at the current point to the value at the MPS: v(y0 , x0 )/v(y∗ , x∗ ) = f(y0 )/g(x0 )/[(1 + γ )r f(y0 )/[(1 − γ )s g(x0 )]] = (1 − γ )s /(1 + γ )r . Consider next an output-oriented model, i.e. (wy , wx ) =(y0 , 0). Equation (1) can now be written as v(y0 + γ y0 , x0 ) = f(y 0 + γ y 0 )/g(x 0 ) = f((1 + γ )y 0 )/g(x 0 ) = (1 + γ )r f(y 0 )/g(x 0 ) = v(y ∗ , x ∗ ). In this case, the ratio can be written as follows: v(y0 , x0 )/v(y ∗ , x ∗ ) = f(y 0 )/g(x 0 )/[(1 + γ )r f(y 0 )/g(x 0 )] = 1/(1 + γ )r . Note that in the output oriented case, we do not need to make assumptions about the homogeneity of g. Correspondingly, in the case of an input-oriented model: v(y 0 , x 0 )/v(y ∗ , x ∗ ) = f(y 0 )/g(x 0 )/[f(y 0 )/[(1 − γ )s g(x 0 )]] = (1 − γ )s . (Note that γ 0, αi > 0 and βj < 0 are constants. Clearly, the function v can be separated: v(y, x) =  
p
p
m −βj −1 αi βj αi
m y x = y x = f (y)/g(x), where f(y) is a strictly j =1 j i=1 j =1 i j i=1 i p increasing homogenous function of degree i=1 αi and g(x) is strictly increasing homogenous function of degree m j =1 −βj . Function v thus fulfills Assumption 1. If the inefficiency score for theoutput-oriented model is γ , the proportional value p the proportional value will be 1/(1 + γ )ω , where ω = i=1 αi > 0. Correspondingly,  for the input-oriented model is (1 − γ )δ , where δ = m −β j > 0. j =1 For instance, if we may assume that the value function is a Cobb–Douglas function of itsfunctional form, we do not need to specify it explicitly. If we assume than δ = m j =1 −βj = 1, then in the input-oriented case we may say that (in)efficiency score also stands for value change. Suppose next that the value function is of the form: v(y, x) = a  y − b x, where m p+ and b∈ a ∈ + . Assumption 2 is fulfilled, and the inefficiency score γ describes the value change relative to the change direction vector used in the projection. If (wy , wx ) ≥ 0, (wy , wx ) = 0, the proportional value change is γ , when the current point is projeced onto the efficient frontier in the direction (wy , wx ). 3.

Final Remarks

Value (in)efficiency score does not generally provide information about the value change, i.e. how much the value of the DM will change, if the current output and/or

INTERPRETATION OF VALUE EFFICIENCY

201

input values of a DMU change so that the unit becomes equally preferred to the most preferred unit. However, by making additional assumptions about the functional form of the value function, we may have useful information about the value change in projecting the output- and/or input-values onto the contour of the value function at the most preferred solution. From a practical point of view, it is very useful that we do not need to explicitly estimate a value function. To assume the functional form is enough.

Acknowledgment The authors would like to thank two referees and Prof. Rolf F¨are for their useful comments which helped us to improve the paper. In addition, we would like to thank late Prof. Seppo Salo for his input to the preliminary discussion on this topic. The research was supported by the Academy of Finland.

References Banker, R. D., A. Charnes and W. W. Cooper. (1984). “Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis.” Management Science 30, 1078–1092. Chambers, R. G., Y. Chung and R. F¨are. (1998). “Profit, Directional Distance Functions, and Nerlovian Efficiency.” Journal of Optimization Theory and Applications 98, 351–364. Charnes, A., W. W. Cooper and E. Rhodes. (1978). “Measuring Efficiency of Decision Making Units.” European Journal of Operational Research 2, 429–444. F¨are, R. and C. A. K. Lovell. (1978). “Measuring the Technical Efficiency of Production.” Journal of Economic Theory 19, 150–162. Halme, M., T. Joro, P. Korhonen, S. Salo and J. Wallenius. (1999). “A Value Efficiency Approach to Incorporating Preference Information in Data Envelopment Analysis.” Management Science 45, 103– 115. Korhonen, P., A. Siljam¨aki and M. Soismaa. (2002). “On the Use of Value Efficiency Analysis and Further Developments.” Journal of Productivity Analysis 17, 49–64.

View publication stats