Inner and outer approximations of technology: A shadow profit approach

Omega 41 (2013) 868–871

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Inner and outer approximations of technology: A shadow profit approach Herve´ Leleu n ´SEG School of Management, 3 rue de la Digue, 59000 Lille, France CNRS/LEM and IE

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 October 2011 Accepted 13 November 2012 Available online 29 November 2012

In this paper we extend the taxonomy on inner and outer approximations to a technology by assuming that price data are not available. Mimicking Varian [Varian H., Econometrica 1984;52(3):579], we introduce a Weak Axiom of Shadow Profit Maximization (WASPM) to test if observed production plans are compatible with technically efficient behavior. If the test fails for an observed sample, we then characterize the maximal subset of observed production plans that meets WASPM and we derive lower and upper bounds on technical efficiency for production plans that are observed but not in this subset. We also derive linear programs to implement these bounds. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Technical efficiency Shadow prices Inefficiency bounds

1. Introduction In this paper, we extend the theoretical literature on nonparametric production analysis initiated by Afriat [1] and Hanoch and Rothschild [12] and further developed by Diewert and Parkan [8], Varian [18] and Banker and Maindiratta [3]. In particular, we add an outer bound for technical inefficiency to the taxonomy ¨ introduced by Fare and Li [10]. Aside from Leleu [13] and Ray [15], we depart from all earlier works in one major respect. We assume that firms0 input and output prices are unknown. We therefore restrict our attention to the productive behavior but we use the duality theory to define the technical efficiency as a shadow profit maximization behavior. This allows us to define inner and outer approximations to technology and to derive lower and upper bounds for technical inefficiency. When prices are known, Varian [18] introduces the Weak Axiom of Profit Maximization (WAPM) which is the direct consequence of the behavior of profit maximization. Here, we extend this axiom by considering shadow prices instead of observed prices, hence introducing the Weak Axiom of Shadow Profit Maximization (WASPM). Our postulate on unknown prices is first motivated by the fact that in many empirical works, prices are either difficult to obtain, they are subject to measurement errors, or they are not true economic prices but derived from accounting data (as emphasized by Cherchye and Van Puyenbroeck [7]). From a theoretical point of view, shadow prices are the relevant economic information if prices are not observed as pointed out by Russell [16], p. 124: ‘‘Even if market prices do not exist, notions of economic efficiency are not irrelevant to the analysis of technical efficiency: shadow prices, implicit in all production technologies, are relevant’’.

n

Tel.: þ33 3 20 54 58 92. E-mail address: [email protected]

0305-0483/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.omega.2012.11.001

As in earlier works, two main concerns are discussed here. We first deal with the consistency of a set of observed data with the postulated behavior. A general test based on the WASPM is developed to test for technically efficient behavior based on shadow profit maximization. Our second concern is related to the recoverability of the technology from observed behavior. While many approximations of the underlying production set can be consistent with a finite number of observations, we naturally seek bounds on the underlying production set and the chosen technical efficiency measure. Such approximations can be ¨ and Li [10]. found in Fare When prices are unknown, Leleu [13] and Ray [15] proposed measures of technical inefficiency which can be interpreted as overall efficiency measures based on the shadow profit maximization. These measures are therefore based on an outer approximation of the technology. While Leleu0 s approach relied on directional distance function, Ray selected two specific normalizations namely the revenue and the cost of the evaluated production plan equal to one. As such, the shadow profit of the evaluated firm is zero and the optimal shadow profit can be interpreted as a measure of the technical inefficiency. The choice of this specific normalization allows us to define an outer approximation and is convenient when solving the problem by enumeration. Yet it remains one among many others. In this paper, we consider a general class of normalizations based on the directional distance function. Moreover, Ray [15] gave an outer measure of efficiency but without a connected inner measure. This work mainly differs from Ray0 s work by giving such a pair of technical inefficiency measures based on interconnected inner and outer approximations of the technology. We therefore obtain a relevant interval for technical inefficiency as opposed to the traditional DEA approach which constructs a ‘point estimate’ of the technology. Our approach is complementary to super-efficiency models which give an interval for efficient production plans [2,6,9].

H. Leleu / Omega 41 (2013) 868–871

The rest of the paper unfolds as follows. In Section 2, we introduce the notion of shadow profit rationalization of a sample of observed data by deriving a necessary condition of shadow profit maximization. We also characterize any subset of the observed data that meets the axiom of shadow profit maximization. Section 3 is devoted to the derivation of lower and upper bounds for the technical inefficiency of observed data that do not meet WASPM while Section 4 presents an example. Final remarks conclude the paper.

Here we consider the situations where we are only given a set   of observed production plans xk ,yk ,kA K, where K is an index set. Prices are assumed to be unknown. Nevertheless, we still keep a dual approach to model efficient producer behavior. Instead of using a set of observed prices, we use shadow prices to test the compatibility of the observed data with the assumption of shadow profit maximization. Following Varian [18], we introduce an axiom which is the direct consequence of productive efficiency. We call it the Weak Axiom of Shadow Profit Maximization (WASPM). 0

Definition 1. An observed production plan k A K satisfies the Weak Axiom of Shadow Profit Maximization (WASPM) if and only if the following condition holds: such that

0

0

pyk wxk Zpyk wxk 8k A K

ð1Þ

Obviously, if all observed production plans satisfy WASPM then we cannot reject the assumption of technical efficiency as the economic behavior of firms. On the contrary, two interpretations of any departures from WASPM can be considered. First, we can see this negative result as an argument to refute the assumption of technical efficiency as the economic behavior of firms since WASPM is a necessary and sufficient condition for efficiency in production. This binary interpretation was the approach adopted by Varian [18] in his work on the Weak Axiom of Profit Maximization (WAPM). In later works, Varian [19,20] also considered that deviations from WAPM can be attributed to measurement errors in the input/output quantities or prices. Hence, he proposed to maintain the assumption of profit maximization if these deviations can be considered as small. Second, according to the literature on production inefficiencies [11], we can consider any departure (small or large) from WASPM as the presence of technical inefficiencies in the operations of some firms [3]. Either way, our definition of WASPM is less stringent than those of Varian [18] or Banker and Maindiratta [3] since it is obvious that WAPM implies WASPM but the converse is false. To measure any departure from WASPM, we first need to identify the subset of observations that satisfy WASPM. We denote by the set EðKÞ, the subset of K that satisfies WASPM: n 0 EðKÞ ¼ k A K : (ðp,wÞ Z 0

such that

0

0

pyk wxk Z pyk wxk 8k A K

pg o þwg i ¼ 1 as the normalization and we consider variable returns to scale technologies.1 0

Proposition 1. plan k A K belongs to the set EðKÞ if  A0 production  0 and only if C xk ,yk ¼ 0 where  0 0 C xk ,yk ¼ maxp ð3Þ p,w, p

 0    0 s:t: pyk wxk  pyk wxk Z p

8k A K

pg o þwg i ¼ 1

2. Consistency of observed data with WASPM

(ðp,wÞ Z 0

869

o

ð2Þ The identification of any production plan that belongs to the set EðKÞ can be easily done by solving a linear program. However, for solving this problem, we require a normalization since the shadow profit is homogenous of degree þ1 then the objective function of model (3) will be unbounded. Due to this fact, it is necessary to use a normalization condition on the shadow prices. We opt for a directional distance function framework by defining    o i the direction vector g ¼ g o ,g i a0, g ,g A ROþ xRIþ in the output/input space. Following Chambers et al. [4,5] we use

p Z 0,

w Z0 0

0

Proof. Assume first that Cðxk ,yk Þ ¼ 0, then we necessarily have: 0 0 0 pyk wxk Zpyk wxk 8kA K and thus, k A EðKÞ. Now, assume that 0 0 k k Cðx ,y Þ a 0, we necessarily have from the constraints: 0 0 Cðxk ,yk Þ 4 0 because the evaluated observation k’ belongs to K 0 and thus p ¼ 0 if k ¼ k . Therefore, a production plan exists, k A K 0 0 such that pyk wxk 4 pyk wxk since by virtue of optimality at 0 least one constraint will be binding. Hence, k 2 = EðKÞ. Clearly, the set EðKÞ can be characterized by sets of shadow prices ðp,wÞ because by definition at least one such set exists for each k A EðKÞ. We call the operation of finding such sets of admissible shadow prices as the shadow profit rationalization of observed data. Note that the cardinality of each admissible set (i.e. the number of shadow prices that characterizes EðKÞ) can be smaller than, equal to or greater than the cardinality of EðKÞ but it is sufficient for characterizing EðKÞ to find admissible sets with cardinality just equal to the cardinality of EðKÞ. We formally define any set P of admissible shadow prices and the class A of all admissible sets P. Definition 2. A set of shadow prices P characterizes the set EðKÞ if and only if: n  pk ,wk Z0, 8k ¼ 1,::, CardðEðK ÞÞ : P¼ o 0 0 0 pk yk wk xk Z pk yk wk xk , pk g o þ wk g i ¼ 1 8k,k A EðKÞ, ð4Þ A denotes the classof all admissible sets P of shadow prices and is defined by: A ¼ P : P characterizes EðKÞ :

3. Lower and upper bounds on technical inefficiency The production plans in EðKÞ are technically efficient. More precisely, they are weakly efficient because, as zero shadow prices are allowed for some inputs and outputs but not all, production plans in EðKÞ are not necessarily strongly efficient in the sense of Koopmans. There might be slacks in outputs and/or inputs. Obviously, we can deal with strong efficiency by adding constraints on shadow prices (e.g.p Z e, w Z e, e 40 in (3)) in order to make them strictly positive. Still we are interested in the production plans that are in K but not in EðKÞ. In particular, we seek lower and upper bounds on the technical inefficiency. In order to define these bounds more formally, we start with a dual characterization of the directional distance function due to Chambers et al. [5]. 1 Other returns to scale assumptions are easily derived by adding one of the 0 0 0 following constraints to problem (3): pyk wxk þ p Z 0 under NIRS, pyk  k0 k0 k0 wx þ p r 0 under NDRS, or py wx þ p ¼ 0 under CRS.

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H. Leleu / Omega 41 (2013) 868–871

Definition 3. The directional distance function is defined by  0 9 8 0
pðp,wÞ ¼

n o sup pyk wxk ðxk ,yk Þ,k A EðKÞ

ð6Þ

Since we deal with compact sets, the sup and inf will be replaced by min and max in the rest of the paper. The technical inefficiency computed by the directional distance function is the smallest difference between the observed and the maximal shadow profit that a firm can obtain from a particular admissible set P of shadow prices. Now, we are interested in defining lower and upper bounds on the technical inefficiency over all the admissible sets of shadow prices, i.e. over the class A. Proposition 2. The lower and upper bounds for the technical     inefficiency in the direction of g ¼ g o ,g i a 0, g o ,g i A ROþ xRIþ of 0 a production plan k evaluated over all admissible sets P A A are given respectively by the following pair of optimization problems:  0 0  n  0 0 o ¼ D xk ,yk ; g i ,g o ¼ min D xk ,yk ; g i ,g o PAA

  0  n o 0 min min pðp,wÞ pyk wxk : pg o þ wg i ¼ 1 PAA

ðp,wÞ A P

PAA

ðp,wÞ A P

ð8Þ

Clearly, the upper (resp. lower) bound on technical inefficiency is given by maximizing (resp. minimizing) the directional distance function over all admissible sets P. Again, we can solve these two optimization problems with two linear programs. The 0 lower bound on the technical inefficiency of a production plan k evaluated over all admissible sets P A A can be computed via the following linear program:  0 0  D xk ,yk ; g i ,g o ¼ min l ð9Þ p,w, l

8k A EðK



pg o þwg i ¼ 1 p Z 0,

Lower bound

Upper bound

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.0153 0 0.0249 0.0386 0 0 0.0502 0.0366 0.0679 0.0645 0.0571 0 0 0 0 0.0654 0 0.0427 0.0101 0

0.0414 0 0.0943 0.0542 0 0 0.0993 0.0599 0.0940 0.0697 0.1281 0 0 0 0 0.1179 0 0.0622 0.0479 0

guarantying the shadow profit rationalization of observed data by the first set of constraints.

4. An example

  0  n o 0 max min pðp,wÞ pyk wxk : pg o þ wg i ¼ 1

   0  0 s:t: pyk wxk  pyk wxk r l

Observations

ð7Þ

 0 0  n  0 0 o ¼ D þ xk ,yk ; g i ,g o ¼ max D xk ,yk ; g i ,g o

P AA

Table 1 Lower and upper bounds on technical inefficiency.

w Z0

In this section we compute the two bounds on technical inefficiency using the data from Banker and Maindiratta [3]. We use a directional distance function with a radial direction in the 0 0 0 input/output space ððg i ,g o Þ ¼ ðxk ,yk Þ,8k A KÞ. The lower bound corresponds to the traditional DEA–VRS inefficiency measure. The upper bound gives the maximal technical inefficiency measure compatible with a set of shadow prices that rationalizes the data (Table 1). We first note that some variability arises in the difference between the lower and upper bounds. For example, by considering observations 10 and 11, we note that the difference between the bounds could be small or large. Even if observation 11 is rated as more efficient than observation 10 within a DEA framework, the computation of the upper bound put this result into perspective. By relaxing the inner approximation of the technology, this observation doubles its inefficiency measure.

5. Conclusion

LP (9) is equivalent to a traditional DEA model for the directional distance function but written in a slightly different form. The upper bound on the technical inefficiency can be computed via the following linear program:  0 0  D þ xk ,yk ; g i ,g o max l ð10Þ pk ,W k , l

    s:t: pk yk wk xk  pk yk wk xk Z0 pk g o þ wk g i ¼ 1 8k A EðKÞ     0 0 pk yk wk xk  pk yk wk xk Z l

00

8k,k A EðK

8kA EðK





pk Z0 8kA EðKÞ wk Z 0 8k A EðKÞ Compared to LP (9), LP (10) seeks the set of shadow prices that leads to the maximization of the technical inefficiency while

We have introduced inner and outer approximations to a technology by assuming that price data are not available. When prices are known, Varian [18] and Banker and Maindiratta [3] derived bounds based on an economic rationalization principle namely, the Weak Axiom of Profit Maximization (WAPM). Here we extend their work by considering unknown prices and a technical rationalization principle based on shadow prices. Our approach should be useful because in many empirical studies, prices are either difficult to obtain, subject to measurement errors, or reflect more cost data than value information [7]. We therefore concentrate on quantities and we create bounds within which one can reasonably expect the true technical inefficiency to lie. Even if the bounds are not a ‘‘confidence interval’’ in a statistical sense as developed for example by Tsionas and Papadakis [17], they give an interval for technical inefficiency as opposed to the traditional DEA approach which constructs a ‘‘point estimate’’ of the technology.

H. Leleu / Omega 41 (2013) 868–871

References [1] Afriat S. Efficiency estimates of production functions. International Economic Review 1972;13:568–598. [2] Andersen P, Petersen NC. A procedure for ranking efficient units in data envelopment analysis. Management Science 1993;39:1261–1264. [3] Banker R, Maindiratta A. Nonparametric analysis of technical and allocative efficiencies in production. Econometrica 1988;56(6):1333–1354. ¨ [4] Chambers R, Chung Y, Fare R. Benefit and distance functions. Journal of Economic Theory 1996;70(2):407–419. ¨ [5] Chambers R, Chung Y, Fare R. Profit, directional distance functions, and Nerlovian efficiency. Journal of Optimization Theory and Applications 1998;98(2):351–364. [6] Chen Y, Du J, Huo J. Super-efficiency based on a modified directional distance function. Omega 2013;41:621–625. [7] Cherchye L, Van Puyenbroeck T. Profit efficiency analysis under limited information with an application to German farm types. Omega 2007;35:335–349. [8] Diewert W, Parkan C. Linear programming tests of regularity conditions for production functions. In: Eichhorn W, Henn K, Neumann K, Shephard. R, ¨ editors. Quantitative Studies on Production and Prices. Wurzburg-Wien: Physica-Verlag; 1983. p. 131–158. [9] Fang HH, Lee HS, Hwang SN, Chung CC. A slacks-based measure of superefficiency in data envelopment analysis: an alternative approach. Omega 2013;41:731–734.

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¨ [10] Fare R, Li SK. Inner and outer approximations of technology: a data envelopment analysis approach. European Journal of Operational Research 1998;105:622–625. [11] Farrell MJ. The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A 1957;120:253–290. [12] Hanoch G, Rothschild M. Testing asumptions of production theory: a nonparametric approach. Journal of Political Economy 1972;79:256–275. [13] Leleu H. The weak axiom of shadow profit maximization, nonparametric tests of regularity and inner and outer approximations of technology. In: Proceedings of the communication at the North American productivity workshop 2004, University of Toronto, Toronto, Canada, June 2004, p. 22–5. [15] Ray SC. Shadow profit maximization and a measure of overall inefficiency. Journal of Productivity Analysis 2007;27:231–236. [16] Russell R. Measures of technical efficiency. Journal of Economic Theory 1985;35(1):109–126. [17] Tsionas EG, Papadakis EN. A Bayesian approach to statistical inference in stochastic DEA. Omega 2010;38:309–314. [18] Varian H. The nonparametric approach to production analysis. Econometrica 1984;52(3):579–597. [19] Varian H. Nonparametric analysis of optimizing behavior with measurement error. Journal of Econometrics 1985;30:445–458. [20] Varian H. Goodness-of-fit in optimizing models. Journal of Econometrics 1990;46:125–140.