Directory UMM :Data Elmu:jurnal:S:Socio-Economic Planning Sciences:Vol34.Issue1.Mar2000:

Slacks and congestion: a comment

Rolf FaÈre

a,b,

*, Shawna Grosskopf

b a

Department of Agricultural and Resource Economics, Oregon State University, Corvallis, OR 97331-3612, USA b

Department of Economics, Oregon State University, Corvallis, OR 973331-3612, USA

Abstract

In this paper, we provide some comments on Cooper, Seiford and Zhu (this issue). We focus on clarifying the distinction between our approach to modeling technology and measuring congestion, di€erentiating between weak disposability and the law of variable proportions. We also show that whereas our congestion measure identi®es congestion only if associated strongly disposable technology solution has nonzero slacks, the converse is not true.#2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

In their paper ``A Uni®ed Additive Model Approach for Evaluating Ineciency and Congestion with Associated Measures in DEA,'' [1], Cooper, Seiford and Zhu (hereafter CSZ) study the relationship between what they call the FaÈre, Grosskopf and Lovell (hereafter FGL) [2,3] and the Brockett, Cooper, Shin and Wang (hereafter BCSW) [4] DEA approaches to measuring congestion. The two approaches di€er most fundamentally in the sense that FGL use a radial Farrell measure, while BCSW apply a slack-based approach to detect and measure congestion and to identify the input(s) responsible for congestion. However, both approaches detect congestion when it exists, (although the de®nitions of congestion are not identical), provide a measure of the magnitude of the e€ect of congestion, and identify the congesting factors.

In this comment, we address several points in the CSZ paper, including (i) their graphical examples and the role of weak disposability in our approach, (ii) the distinction between weak disposability and the law of variable proportions, and (iii) the relationship between slacks and our measure of congestion.

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2. Examples and weak disposability of inputs

To analyze the two examples described in Figs. 1 and 3 in CSZ, we ®rst introduce the two DEA models we use to identify congestion. They are distinguished by their treatment of disposability of inputs: one imposing weak disposability and the other strong disposability. To focus on input

disposability and congestion, we employ the input setsL(y),y,2RM_‡, where

L…y† ˆ fx: x can produceyg, …1†

x2RN_‡ denoting inputs and y2RM_‡ denoting outputs. Suppose there are k= 1,. . ._{, K}

observations or DMU:s, each characterized by their input±output vector (xk,yk)=

(xk1,. . .,xkN,yk1,. . .,ykN). From these DMU:s, we may construct the input requirement set for

observationk'_{which satis®es strong disposability of inputs (and variable returns to scale) as}

Lÿyk0jSˆ…x1, . . ., xN†: S_kK_ˆ1lkykmeyk0_m, mˆ1, . . ., M, SK

kˆ1lkxknExn,

nˆ1, . . ._{, N, S}K

kˆ1lk ˆ1,lke0,kˆ1, . . ., K :

…2†

Strong disposability of inputs as de®ned as

xex0, x02L…y†ˆ)x2L…y†, …3†

which states that if inputs are increased or stay the same (xex0), then outputs will not

decrease, i.e., x$L(y), meaning that x can produce the original output level y. The

inequalities on the inputs xn, n= 1,. . ., N in expression (2) show that the DEA model (2) has

strongly disposable inputs. The constraint SK_{k=1lk=1 on the intensity variables indicates that}

the technology exhibits variable returns to scale.

The de®nition of weak disposability of inputs requires

x2L…y†, ye1ˆ)yx2L…y†: …4†

This de®nition states that if x can produce y and if inputs are increased proportionally by

ye1, then the scaled input vectoryxcan also produce y. It follows that strong disposability of

inputs implies weak disposability, but weak disposability does not imply strong. The DEA model with weak disposability of inputs is

Lÿyk0jWˆ…x1, . . ., xN†: SKkˆ1lkykmeyk0_m, mˆ1, . . ., M, SK_kˆ1l_kx_kn=tˆx_n,

nˆ1, . . ._{, N, 0<t}E1SK_kˆ1lk ˆ1,lke0,kˆ1, . . ., K :

…5†

Here, the factor t allows for proportional scaling (increases) of observed input vectors or

their convex combinations as feasible elements ofL(yvW). Both weak and strong disposability

imply that isoquants will generally not be bounded in contrast to the isoquant depicted in Fig. 1 in CSZ. We have reproduced that ®gure here as Fig. 1a; in Fig. 1b we show that the unit isoquant for these data points would look like under weak and strong disposability of inputs. For this particular example, our approach would take the radial contraction of the input

bundle at G, for example, and project it to G' _{on the isoquant for both L(y}v_{S) and L(y}v_W).

There would be technical ineciency, but no congestion, since the Farrell measure is the same for both technologies. In our approach, congestion is identi®ed when the Farrell measures di€er for the two technologies.

We provide an example in which our approach would signal (and measure the magnitude) of our version of congestion in Fig. 1c. In this ®gure, the weakly disposable technology's isoquant goes through points A, B and C with the rays extending from B and C. The strongly disposable isoquant includes point A and the horizontal and vertical extensions from A. In this example, points B and C would be identi®ed with congestion, whereas D and A would not. D, however, would be associated with technical ineciency of 0A/0D.

on our two technologies. Alternatively, one could increase one input, holding the other constant (which is the way in which the law of variable proportions is usually stated). Then, the two technologies would again give di€erent scores (the strongly disposable technology would yield no solution) for points B, C and D. Although we are unclear about the meaning of Theorem 2 in CSZ, we would conjecture that our ®gure provides a counterexample.

Returning to our assumption of weak disposability, we would argue that this assumption is fundamental to an axiomatic approach to production theory. This axiomatic approach Ð which we have tried to integrate into the eciency measurement literature Ð follows the tradition of Shephard [6], and allows us to exploit duality theory. For example, this assumption is a necessary and sucient condition for the input distance function (the reciprocal of the Farrell measures we use to measure congestion) to completely characterize the technology, i.e.,

L…y† ˆx: Di…y, x†e1 , …6†

where

Di…y, x† ˆ sup

l: x=l2L…y† , …7†

is the input distance function. See FaÈre and Primont [3, p. 22] for details. The ability of the distance function to completely represent technology is of course a fundamental property that one would like to have satis®ed by the associated eciency measures. Among other things, it is required for the Mahler inequality [7] to hold, which is the basis of the Farrell type decompositions of cost (revenue) eciency into technical and allocative components. This provides an all important link to traditional economic concepts of cost, pro®t, etc.

We note that the isoquant depicted in Fig. 1a would also violate the monotonicity condition generally assumed in DEA models.

2.1. Slacks and congestion

In this section, we investigate the relationship between slacks and congestion. We will show that the existence of slacks does not imply the existence of congestion (in contrast to claim in CSZ). However, congestion does imply the existence of slacks in the solution to the Farrell measure computed relative to the strongly disposable technology.

Consider the following example of a technology consisting of 2 DMU:s (A and B) which

produce a single output y with two inputs x 1 and x2. We may compute the input-oriented

measure of congestion for DMU B by solving the following two optimization problems.

y_Bˆ miny

s:t: lA1‡lB2Ey2

lA1‡lB1Ey1

lA1‡lB1e1

b_Bˆ minb

s:t: lA1‡lB2ˆtb2

lA1‡lB1ˆtb1

lA1‡lB1e1

lA‡lBˆ1,lA, lBe0

0EtE1: …9†

The solution to Eq. (8) is yB=1 and to Eq. (9) is b

B=1(t=1). Thus, DMU B does not

exhibit congestion in our framework. It is clear from the data, however, that DMU B has one unit of slack (under strong disposability) in the ®rst input. Thus, the existence of slack does not imply congestion in our framework.

As a side issue, the example in Table 1 may also be used to provide a counterexample to Corollary 1 in CSZ. In particular,

b_Bˆ maxb

s:t: lA1‡lB2ˆt2

lA1‡lB1ˆt1

lA1‡lB1eb^ 1

lA‡lBˆ1,lA, lBe0

0EtE1 …10†

yields bB=1 and t=1 Ð no congestion Ð but there is of course still one unit of slack in

input x1. As this example indicates, the solution to (3') in CSZ with t=1 still allows for

positive slack to form part of the optimal solution to (1'_{) in CSZ.}

To show that if there is congestion in the sense in FGL [2,8] i.e., if

Table 1

Example of a technology

x1 x2 y

A 1 1 1

y=b<1, …11†

then the scaled input vector for DMU0, i.e., yx0 in problem (1) in CSZ, has some non-zero

slack. This result follows from FaÈre [5, p. 16], who showed that the input sets L(yvS) and

L(yvW) have the same ecient subset, i.e.,

In their paper ``A Uni®ed Additive Model Approach for Evaluating Ineciency and Congestion with Associated Measures in DEA'', [1], Cooper, Seiford and Zhu study the relationship between what they call the FaÈre, Grosskopf and Lovell [2,8] and the Brockett, Cooper, Shin and Wang [4] DEA approaches to measuring congestion. The two approaches di€er most fundamentally in the sense that FGL use a radial Farrell measure, while BCSW apply a slack-based approach to detect and measure congestion and to identify the input(s) responsible for congestion. However, both approaches detect congestion when it exists, (although the de®nitions of congestion are not identical), provide a measure of the magnitude of the e€ect of congestion, and identify the congestion factors.

Since we measure congestion as di€erences in technology under weak and strong disposability inputs, we spend some time de®ning and illustrating these production axioms. We point out that weak disposability implies ``non-satiation'', i.e., isoquants may be ``backward-bending'' Ð which is the essence of congestion Ð but may not become ``circles''. We have always taken an axiomatic approach to specifying the underlying production models we use to measure performance (including congestion). Weak disposability is the weakest restriction we can impose in order for the radial eciency measures we use to completely represent technology. Weak disposability is also essential for decomposing economic measures of eciency such as cost eciency into meaningful technical (including congestion) and allocative components.

We also note that weak disposability and the law of variable proportions are not synonomous, since the de®nition of the law of variable proportions requires that one input be ®xed. Weak disposability (and our radial measures of eciency) rely on scaling all inputs.

may well argue that the slacks are a part of what is generally meant by congestion, we take a di€erent view. We think of input congestion as a piece of a bigger decomposition of cost ineciency into technical (non-price related) ineciencies (including what we refer to as pure technical, scale and congestion ineciency) and allocative ineciency, which is price-related. Given strictly positive input prices, the loss in eciency attributed to slacks in a CSZ world would be attributed to allocative ineciency in an economic world.

Finally, we note that, although not discussed in these two papers, we have used our general approach to measuring congestion as the divergence between weakly and strongly disposable technologies in terms of output increasing eciency and output sets as well. This allows us to model joint production of good and bad outputs and measure losses due to restrictions on disposability such as environmental regulations.

References

[1] Cooper WW, Seiford LM, Zhu J. A uni®ed additive model approach for evaluating ineciency and congestion with associated measures in DEA, Socio-Econ. Plann Sci 1999; (This issue).

[2] FaÈre R, Grosskopf S, Lovell CAK. Production frontiers. Cambridge: Cambridge University Press, 1994. [3] FaÈre R, Primont D. Multi-Output production and duality: theory and applications. Boston: Kluwer Academic,

1995.

[4] Brockett PL, Cooper WW, Shin HC, Wang Y. Ineciency and congestion in Chinese production before and after the 1978 economic reforms. SocioEcon Plann Sci 1988;32:1±20.

[5] FaÈre R. Fundamentals of production theory. Berlin: Springer-Verlag, 1988.

[6] Shephard RW. Theory of cost and production functions. Princeton: Princeton University Press, 1970.

[7] Mahler K. Ein UÈbertragungsprinsip fuÈr Konvexe KoÈrper. Casopis pro PeÏstovaÏni Matematiky a Fysiky 1939;64:93±102.