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An indirect approach to the evaluation of producer performance
Journal
of Public
Economics
37 (1988) 71-89.
North-Holland
AN INDIRECT APPROACH TO THE EVALUATION PRODUCER PERFORMANCE R. FARE
OF
and S. GROSSKOPF
Department of Economics, Southern Illinois University, Carbondale, IL 62901, USA
C.A.K. LOVELL* Department of Economics, University of North Carolina, Chapel Hill, NC 27514, USA Received
April 1986, revised version
received
December
1987
1. Introduction In this paper we suggest a somewhat new analytical approach to producer performance evaluation. The suggested approach is ‘somewhat’ new in that it synthesizes existing analytical techniques from three areas, and uses them to look at performance evaluation in a new way. The approach is particularly relevant to public sector or service sector contexts in which the performance of alternative providers of a public service is to be evaluated. In these contexts any of the following conditions may hold: (1) the standard profit maximizing, cost minimizing or revenue maximizing paradigms are inappropriate; (2) prices of some services being provided, or of some resources being consumed, are either nonexistent or unobserved; (3) observed prices are not exogenous, due to monopoly or monopsony power. Examples abound. It is frequently argued that the provision of medical, legal, educational, transportation, employment and many other services, by public or private agencies, is driven by goals other than the maximization of profit. Moreover these and many other services, such as police and fire protection, do not always have market prices with which to value them. In addition, the lack of competition in the provision of many such services, and thus in the employment of specialized resources used in their provision, makes it dangerous to use existing but distorted market prices to compare the performance of providers on the basis of revenue or cost, much less profit. *The authors
are grateful
004772727/88/$3.50
0
to A.B. Atkinson
and two referees for their helpful comments.
1988, Elsevier Science Publishers
B.V. (North-Holland)
72
R. F&e et al., Evaluation of producer performance
Yet many services are provided by a number of suppliers, public or private or both, and it is important to be able to evaluate the performance of each supplier, or of each type of supplier. The use of the profit maximization paradigm, with known, parametric, exogenous service and resource prices, is frequently inappropriate for such a purpose. So too, to a somewhat lesser degree, are the assumptions of cost minimization with known resource prices, or revenue maximization with known service prices.’ No more appealing, however, is the opposite extreme in which suppliers are evaluated solely on the basis of resource usage and service provision, with no reliance placed on either observable prices or economic motivation.2 What is desired is an intermediate approach to producer performance evaluation, one which avoids the agnosticism of the latter approach by using available prices, and which avoids the opposite extreme of imposing inappropriate behavioral goals. The development of such an approach is our objective. The basic analytical tool is the indirect production function developed by Shephard (1974). The cost indirect production correspondence shows the set of output vectors obtainable from any given budget. The return indirect production correspondence shows the set of input vectors capable of generating any given revenue. The two indirect correspondences thus provide the basis for an evaluation of producer performance on the basis of service benefits measured by quantities supplied, or on the basis of resource costs measured by quantities employed. For this reason Shephard (1974, preface) argued that the indirect production function ‘is the foundation for a costbenefit analysis’. Quite apart from its role in a benefitxost framework, however, the indirect production function is consistent with axiomatic production theory and has useful dual representations. In fact, the distance functions developed by Shephard (1953, 1970) as such a dual representation are closely related to the second analytical tool we use, the family of efficiency measures originally proposed by Debreu (1951) and Farrell (1957). The significance of the application of efficiency measures to indirect production functions is that they provide a method for calculating frontier or best practice benefits or costs relative to the technology. This analytical approach to benelitxost comparisons provides the foundation for the intermediate approach to producer performance evaluation we seek. ‘Nonetheless, many performance evaluations, particularly those of a ‘public vs. private’ nature, are based on such paradigms. A critical survey of these studies, from airlines to weather forecasting, is provided by Borcherding, Pommerehne and Schneider (1982). 2The data envelopment analysis (DEA) literature correctly points out that much public sector production is characterized by the absence of the profit motive and the absence of service prices. The DEA solution, to ignore all prices and all behavioral motivation, and to evaluate performance solely on the basis of the efficiency of the physical transformation process, seems a bit extreme, however. See Charnes, Cooper and Rhodes (1978) for an early statement of the DEA approach to performance evaluation.
R. Fiire et al., Evaluation of producer performance
13
The third analytical technique we use is the nonparametric approach to the construction of indirect production frontiers and the measurement of efficiency relative to them. The nonparametric approach is simple, yet very flexible. Indirect production frontiers are derived by enveloping the data while ensuring that very general regularity conditions implied by the axioms are satisfied. No parametric functional form unduly restricting scale or substitution properties is imposed. A family of benefit-cost efficiency measures is calculated for each observation as solutions to a sequence of linear programming problems. Our analytical approach is, as we have argued, perhaps most valuable as a means of evaluating the performance of providers of public services. It is, in the terminology of Diewert (1983, 1986), a partial equilibrium approach, since we do not incorporate consumers or other sectors of the economy. It is in spirit related to Diewert’s ‘productive efficiency project evaluation criterion’, although we allow for multiple outputs and we do not use a profit function (for reasons mentioned above) to model technology. Although we derive benefit effectiveness criteria and expense effectiveness criteria, we do not suggest our approach as a way of doing conventional benefit-cost analysis or project evaluations, which typically involve the assessment of a small number of large, heterogeneous, indivisible investments. For these tasks the reader is referred to Diewert (1983, 1986) and the references cited therein. The paper unfolds as follows. Section 2 considers the cost indirect production correspondence and the measurement of benefit effectiveness. This framework is particularly appropriate to producer performance evaluation when resource usage can be reliably compared on the basis of cost, but benefits (outputs produced or services provided) cannot be priced reliably enough to allow revenue comparisons. An example is provided by the attempt to evaluate the performance of school districts whose budgets can be measured, but whose educational outputs can be measured but not priced.3 Section 3 considers the return indirect production correspondence and the measurement of cost effectiveness. This framework is suited to producer performance evaluation when benefits can be priced reliably enough to make revenue comparisons, but resource usage is measured in physical terms. An example is provided by an attempt to evaluate the performance of multipleuse public lands for which revenues can be measured but not all resources are priced.4 In section 4 we provide a brief example illustrating how the cost indirect production correspondence works in practice. The empirical problem is to compare the performance of a group of public school districts 3Bessent, Bessent, Kennington and Reagan (1982) adopt a DEA approach to public school district performance evaluation, in which test scores are outputs, student characteristics and school resources are inputs, and budgets are ignored. 4Hyde (1980) evaluates multiple-use forest management on public and private land in the United States without resorting to an explicit optimization model of the sort proposed here.
14
R. F&e et al., Evaluation of producer performance
in the state of Missouri. The results of the experiment include a ranking of districts on the basis of their ability to convert budgets into educational achievement, and they also include additional information resulting from a linear programming formulation of the problem. Section 5 concludes.
2. Cost indirect effkiency The first objective is to derive a family of efficiency measures that can be used to gauge the performance of individual producers relative to other producers in the sample. Performance is gauged by output quantities produced, given a fixed budget, input prices, and production technology. For this purpose, technology is represented most usefully by the cost indirect production correspondence. The second objective is to provide a computational method for calculating the family of efficiency measures. The method uses linear programming techniques to construct a nonparametric cost indirect production frontier, and to calculate efftciency relative to that frontier. Suppose there are j = 1,. . ., J producers using input vector xj~ R: to produce output vector uj~ R”,. Each producer faces input price vector and budget level cj>O. Since input price vectors and budgets are #EC+ indexed by j, producers are allowed to face different input prices and to have different budgets. Denote the J x m matrix of observed outputs by M and the J x n matrix of observed inputs by N, in which case the direct output correspondence can be written J
u:u~zM,zN~x,z~R~,
c j=l
z,.=} 1 ,
(2.1)
where z is a vector of intensity variables familiar from activity analysis. P(x) is the set of all technologically feasible output vectors with input vector x. The production technology described in (2.1) satisfies strong disposability of both outputs and inputs, an exhibits variable returns to scale. In addition, if every column sum and every row sum of both M and N are strictly positive, then the direct output correspondence x+P(x) in (2.1) is sufficiently wellbehaved for a number of duality theorems to go through.5 Specifically, we can represent production technology with the cost indirect output correspondence:
SFor details see Karlin (1959, pp. 338-343) or Shephard (1970, pp. 283-292). The scale and disposability properties of the output correspondence (2.1), and (3.1) below, are discussed in Grosskopf (1986).
R. Fiire et al., Evaluationof producer performance
75
J
(p/c)-+P(p/c) =
u: u 5 -
zitf,zN~x,
px SC, z E R:, c
j=l
Zi=]1
(2.2) Thus P(p/c) is the set of all technologically feasible output vectors attainable at budget-deflated input prices (p/c). Whereas P(x) is a resource-constrained output set, P(p/c) is a budget-constrained output set. It can be shown that the cost indirect output correspondence (2.2) is closed, bounded, convex, and satisfies strong disposability of outputs.6 Given this representation of technology, we can now evaluate the performance of individual producers by comparing their outputs with outputs located on the boundary of the cost indirect output set P(p/c). The measures we use to gauge the performance of individual producers are modifications of the family of efficiency measures introduced by Debreu and Farrell. We begin with: Definition.
The function
T,($lcj, uj) = max {8: BujE P($/cj)}
(2.3)
is called the cost indirect technical eficiency
measure.
For observation j the cost indirect technical efficiency measure computes the ratio of the largest feasible radial expansion of uj to uj itself, where feasibility is defined by the cost indirect output set P(p’/cj). Thus T&+/c’, u’) = IIT,($/cj,u’) . u’pp( is a budget-constrained measure of technical efticiency.7 The properties of T’,(p/c,u) are summarized in Proposition.
Given P(p/c), then for each j=
1,. . . , J,
‘?See Shephard (1974) and Shephard and Fare (1980) for proofs of these and other properties of the cost indirect output correspondence. Note also that if all producers face the same input price vector, the cost indirect output correspondence (2.2) can be simplified further, by using observed costs in place of observed input prices and quantities to obtain: (p/c)-+P’(p/c)= u:u~zM,zC~c,z~R:, i
$,zj=l}*
(2.2’)
where C is the J x 1 vector of observed costs. 7The fact that T,($lc’,u’) is a budget-constrained efficiency measure is what distinguishes it from the Debreu-Farrell resource-constrained efficiency measure F,(xj, u’) discussed in Fare, Grosskopf and Love11 (1985, ch. 4). It is also what makes T,(p’/ti,u’) similar to an index of economic effectiveness E&c’, uj, t’), t measuring time required to provide the service, proposed by Eichhorn (1972, 1978).
R.
Fire
et al.,
Evaluation
ofproducer
performance
To. 1
T,($/c’, 6U’)= 6 - 1T&+/e’, d), s > 0,
To.2
l~T,($/c’,u’)<+co,
TO.3
T,( $,‘d, u’) = 1+-w’ E Isoq P( p+‘) = (u: u E
P(f+/c’),6u 4 P($/cl)
(2.4)
if 6 > 1).
Thus T,(p/c, u) is a function homogeneous of degree - 1 in the output vector that attains its minimum value of unity if and only if the output vector belongs to the cost indirect output frontier Isoq P(p/c).* Implementation of the cost indirect technical efficiency measure requires specification of the cost indirect output correspondence (2.2). A nonparametric specification of that correspondence, together with a calculation of the efficiency measure for each observation, is provided by the constraints of, and the solution to, the linear programming problem: T,($lcj, uj) = max 13
(z,e.X) s.t.
Bu’~zM,
j$lzjcl.
(2.5)
Solving this problem J times generates a measure of cost indirect technical efficiency for every producer in the sample. T,($lcj,uj) provides a means of comparing the performance of producers in terms of their budget-constrained ‘real’ benefits, namely their output quantities. Such a procedure is particularly attractive when not all outputs are priced, or when not all output prices are observed accurately enough to allow comparisons across producers, or when differential market power distorts some output prices. However if reliable output prices are available, *Proposition (2.4) has the same mathematical structure as Theorem (4.2.5) in Fiire, Grosskopf and Love11 (1983, even though the latter deals with direct (rather than cost indirect) output technical efficiency measurement. Consequently, a proof of Proposition (2.4) here can be deduced from the proof of Theorem (4.2.5) there.
17
R. F&e et al., Evaluation of producer performance
then benefits can be measured in monetary as well as in physical terms. In this event the foregoing analysis can be extended to evaluate performance on the basis of budget-constrained returns. Suppose now that for each producer j=l , . . . , J we observe input vector xi, output vector uj, input price vector p’ and budget ei as before, and, in addition, we observe output price vector riERm++. In this case we can define a maximum return function relative to the cost indirect output correspondence as: &p/c, r) = max (ru: u E Q/c)>. ”
(2.6)
R(p/c,r) shows the maximum return obtainable from budget c given prices I and p for outputs and inputs. It is a budget-constrained revenue function. For each producer maximum return can be calculated as the solution to: R(p’lcj, ri) = max r’u
(z,“9X1 s.t. u szikf,
i
zj=l.
(2.7)
j=l We are now in a position to evaluate the performance comparing their observed and maximum returns. Definition.
of producers
by
The function
O,($/cj, r’, uj) = R(p’/cj, ri)/rjuj is called the cost indirect revenue eficiency
V-8) measure.
For observation j the cost indirect revenue efficiency measure is the ratio of the maximum feasible return at output prices ri and cost deflated input prices $/cl to actual receipts, where feasibility is once again defined relative to the cost indirect output set P(p/c). Thus O,,(~/c’,r’,u’) is a budget-
R. F&e et al., Evaluationof producer performance
78
constrained measure of revenue effkiency.9 The properties indirect revenue efficiency measure are summarized in: Proposition.l” 0,.
Given P(p/c),
1 O,($/cj,
then for each j=
r’, 6u’) = 6 - ‘O,(p’/c’,
0,. 2
1 5 O,($lc’, r’, uj) < + cc
00.3
O&‘/C?, r’, u’) = 1 ++ uj
of the cost
1,. . . , J, r’, u’), 6 > 0,
(2.9) solves (2.7)
Note that O,(p/c,r, u) depends on both output quantities and output prices, whereas T’,(p/c,u) is independent of output prices. We can isolate the output price-dependent component of Oo(p/c, r, u) by introducing: Definition.
The function
A,($lcj,
6, u’) = O,($lcj,
r’, uj)/T,($/cj,
uj)
(2.10)
is called the cost indirect allocative efficiency measure. For observation j the cost indirect allocative effkiency measure gauges the ability of the producer to produce the optimal mix of outputs in light of prevailing output prices. The properties of the cost indirect allocative efficiency measure are summarized in: Proposition.11
Given P(p/c),
then for each j= 1,. . . , J,
A,. 1
A&+]&, rj, 6u’) = A,($lc’, r’, u’), 6 > 0,
A,.2
l~&($lcj,rj,uj)<
9The fact distinguishes
that O,(p’/c’,r’,u’) it from Farrell’s
+co,
(2.11)
is a budget-constrained revemx efficiency measure is what resource-constrained revenue efficiency measure 0,(x’, u’,Tj) =
R(x’, r’)/r’u’. loThe proof of Proposition (2.9) is similar to the proof of Theorem (4.4.7) in Fiire, Grosskopf and Lovell (1985), although the latter deals with direct revenue efficiency measurement. “The proof of parts A,. 1 and A,,.2 of Proposition (2.11) follow proofs of similar parts of Theorem (4.6.4) in Fire, Grosskopf and Love11 (1985), although the latter deals with direct output allocative efficiency measurement. To prove A,,. 3 assume 362 1 such that 6~’ solves (2.7). Then R(p’lc’, r’) = r’6u’. Thus, from Definitions (2.3) and (2.10), A,($/c’, r’, u’) = 1. Coversely, assume that A,@]&, rj, u’) = 1. Then R(p’/c-‘,r’) = IJ(T,($/c’, u’) d). Since T,(p’lc’, u’) 11, A,. 3 holds. Q.E.D.
R. Fire et al., Evaluation of producer performance
‘40.3
79
A,($/c’,r’,u’)=le,36~1 such that Su’ solves (2.7).
From Definitions (2.5) and (2.10) we obtain the following multiplicative decomposition of the cost indirect revenue efficiency measure defined in (2.8): 0,($/d,
rj, d) = A,($lcj,
rj, d) . T&+/c’, u’).
(2.12)
Thus the ability to generate maximum return in a budget-constrained environment is the product of the ability to maximize output and the ability to select the optimal output mix. Departures from cost indirect maximum return as defined by R(p/c,r) in (2.6) must be technical or allocative in nature, and the decomposition described in (2.12) can be calculated empirically, using linear programming techniques. If output prices are known, R($/cj,rj) can be calculated from (2.7). OO($lcj,ri,uj) is calculated from (2.8), T,($lcj, uj) is calculated from (2.5), and A,($/&, rj, u’) is obtained residually from (2.10). The cost indirect revenue efficiency measure O,($lcj,rj,d) evaluates producer performance by comparing budget-constrained maximum returns to observed receipts. In some circumstances it may be preferable to compare budget-constrained maximum returns to observed budgets, and to rank producers on the basis of their benefit effectiveness ratiosI R($lcj, &)/cj, j= 1,. . . ,J. In contrast to a conventional revenue-cost ratio riuj/cj, a benefieffectiveness ratio eliminates technical and allocative inefficiency from the numerator, and compares producers on the basis of their potential for converting budgets into revenues. If output prices are not known, only T,($/cj,uj) can be calculated, and producers can be ranked on the basis of their ability to convert budgets into real services. They can also be compared in at least one additional way. The set of product prices for which a producer has a benefit-effectiveness ratio at least as great as unity can be written: S’o=(rj:R(pilcj,rj)~cj),
j=l,...,
J.
(2.13)
Producers can be compared on the basis of their sets Sj,. If Sjo c Sk for any two producers indexed j and k, then producer k is ranked ahead of producer 12The benefit effectiveness ratio is particularly attractive in circumstances in which a public agency oversees the operation of a number of service providers. Examples are state departments of education overseeing many school districts, state departments of justice supervising local courts, and the like. If individual school districts or courts exhibit relatively low benefit effectiveness ratios, this provides the oversight agency with good reason to consider merger of inefficiently small districts or division of inefticiently large districts in order to more closely approximate most productive scale size in all districts.
R. Fiire et al., Evaluation of producer performance
80
j on the basis of this shadow benefit-effectiveness criterion. If S(, #St and Sk, Q Si,, then producer j is ranked ahead of producer k for some subset of output price vectors, and producer k is ranked ahead of producer j for some other subset of output price vectors, on the basis of this shadow benefieffectiveness criterion. 3. Return indirect efficiency
The analysis of this section is similar in structure to that of the previous section, the essential difference being in the information presumed available to the analyst. Here we assume that output or service prices are known, and that it is desired to gauge the performance of producers on the basis of resources consumed in order to generate certain revenues. For this purpose technology is represented by the return indirect production correspondence, and efficiency measurement is input-based. Notation remains unchanged. For producers j = 1,. . . , J, inputs xj~ R”, are used to produce outputs USER”, that are valued at prices rjE R”, +. Revenue constraints are Rj>O. The direct input correspondence can be written: J
x:u~zM,zN~x,z~R:,
c j=l
z,.=} 1 ,
(3.1)
where z, A4 and N are as defined previously. The input set L(u) shows the set of all input vectors capable of producing output vector u. Production technology can also be represented by the return indirect input correspondence:
={x:x~L(u),(r/R)uzl}.
(3.2)
Whereas L(u) is an output-constrained input set, L(r/R) is a revenueconstrained input set, showing the set of all input vectors capable of generating revenue R at output prices r. The return indirect input correspondence is closed, convex, and satisfies strong input disposability.r3 ‘3Proofs of these properties are analogous to the proofs of Shephard (1974) and Shephard and Fire (1980) for similar properties of the cost indirect output correspondence. If all producers face the same output price vector, the return indirect input correspondence can be simpliIied to: (r/R)-*L’(r/R)=
x:R~zR,zN
where R is the J x 1 vector of observed
revenues.
(3.2’)
R. Fiire et al., Evaluation of producer performance
Producer performance is evaluated by comparing with efficient input usage located on the boundary input set by means of: Definition.
81
observed input usage of the return indirect
The function
T(&/Rj, xj) =min
{A: AXLEL(ljlR’)}
(3.3)
is called the return indirect technical efficiency measure. For producer j the return indirect technical efficiency measure computes the ratio of the smallest feasible radial contraction of xi to xj itself, feasibility being defined by the return indirect input set L(&/Rj). Thus IT;(fi/Rj,xj) is a revenue-constrained measure of technical efficiency.i4 The return indirect technical efficiency measure has properties summarized in: Proposition.
Given L(r/R), then for each j= 1,. . . , J,
7;. 1
7;(r’/Rj, Ax’) = 1~ ’ T(rj/R’, x’), A> 0,
T. 2
0 < 7;(r’lR’, x’) 5 1,
T. 3
T(ljlRj, xj) = 1 c) xj~ Isoq L(rj/Rj) = {x: x E L(rj/Rj), Ax 4 L(r’/Rj)
(3.4)
if A< 1).
Thus T(rjlRj,xj) is a function homogeneous of degree - 1 in the input vector that attains its maximum value of unity if and only if the input vector belongs to the return indirect input frontier Isoq L(rj/Rj). Proofs follow those of (2.4). T(r’/R’,xj) can be computed for observation j as the solution to the linear programming problem: T(+jRj, xj) = min A (2.=,U) s.t. U~ZM, 14This is what distinguishes 7;:(r’/Rj,xj) from the Debreu-Farrell of technical efficiency Fi(uj, xj).
output-constrained
measure
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R. Fiire et al., Evaluation of producer performance
zNj
fi
ix’,
‘j’l.
(3.5)
7Jrj/Rj,xj) provides a means of comparing the performance of producers on the basis of their revenue-constrained ‘real’ resource consumption, and is appropriate when not all resources are reliably priced. However, if resources are reliably priced, the analysis can be extended to provide a comparison of producers on the basis of their return-constrained costs. Suppose for each producer we observe output vector uj, input vector xi, output price vector ri and revenue constraint Rj as before. If we also observe input price vector $ER”+ +, then we can define a minimum expense function relative to the revenue indirect input correspondence as: C(r/R,p) =min {px: x E L(r/R)}. X
(3.61
C(r/R,p) shows the minimum expense required to generate revenue R at output prices r and input prices p; it is a revenue-constrained cost function. For each producer C(r/R,p) can be calculated as the solution to: C(rj/Rj, $) = min $x (2.X,U)
zNj x,
ZER:,
j$lzj=l. Once C(r’/Rj,$) the basis of:
is calculated,
producer
(3.7)
performance
can be evaluated
on
83
R. Fiire et al., Evaluationof producer performance
Definition.
The function
Oi(r’lRj, $, x’) = C(ri/Rj, $j/$x’
(3.8)
is called the return indirect cost eficiency
measure.
Given (ti/Rj,p’), this measure compares producers on the basis of their ability to select expense-minimizing input vectors. Thus, O,(fl]Rj,$, xj) is a revenue-constrained measure of cost effkiency.ls Its properties are summarized in: Given L_(r/R), then for each j =
Proposition. Oi.
1,. . . , J,
1 Oi(rj/Rj, p’ 2~‘) = I - ’ OJr'JR', p’, x’),
Oi. 2
0 < Oi(r’lR’, $,
Oi. 3
O,(r’/Rj,#,
X’) ~
A>
0,
1,
xj) = 1 ++ xj
(3.9) solves
(3.7).
Proofs follow those of Proposition (2.9). O,(r/R,p, x) depends on input prices, and 7Jr/R, x) does not. The input price-dependent component of Oi(r/R, p, x) can be isolated by introducing: Definition.
The function
A,(r’/Rj, p’, xj) = Oi(r’/Rj, $, xj)‘z(rj/R’,
is called the return indirect allocative eficiency
x’)
(3.10)
measure.
For producer j the return indirect allocative effkiency measure judges the ability to use inputs in the correct proportion in light of observed input prices. The properties of this measure are given in: Given L(r/R), then for each j = 1,. . . , J,
Proposition. Ai.
1 A,(r’/R’,$,
2.x’) = A,(rj/Rj, $, xj), 2 >O,
Ai. 2
0 < Ai(~/R’, $,
Ai.
A,(r’/R’, $, x’) = 1~
3
X’) ~
ISThis is what distinguishes O,($/Rj,$,x’) O,(uj,xj, $) = C(u’,$1/$x’.
measure
(3.11)
1, 31 E (0, 11 from
Farrell’s
output-constrained
cost
efkiency
R. F&e et al., Evaluation of producer performance
such that ix’ solves (3.7).
Proofs follow those of Proposition (2.11). From Definitions (3.3) and (3.10) we obtain the following multiplicative decomposition of the return indirect cost efficiency measure defined in (3.8):
Oi(r'lR',$, X')'Ai(r'fR',$, X’)’ ~(r'/R', X’).
(3.12)
Thus the ability to produce at minimum cost in a revenue-constrained environment is the product of the ability to select the optimal input mix and the ability to minimize input usage. T(ljlRj,x') is calculated by means of is calculated from (3.8) and (3.7), and Ai(r'/R'p$y X’) is (3.5), Oi(G/R'yp',X') calculated from (3.10). As in the previous section we may wish to augment, or replace, O,(ljlRj,p', x')with the expense effectiveness ratio RJ/C(ljlRf,p$ which compares producers on the basis of their potential for generating revenue at minimum expense. Finally, if input prices are not observed, only T,(ti/Rj,X'j can be calculated, and producers can be evaluated on the basis of their revenue-constrained technical efficiency. They can also be evaluated on the basis of the set of input prices for which their expense effectiveness ratios are at least as great as unity by means of
$={p':R'2C(ti/R',$)}, j=l,...,J. If Si c $ for any two producers j and k, then producer k is evaluated higher than producer j on the basis of this shadow expense effectiveness criterion. Alternatively, there may exist a subset of input prices for which producer k is ranked ahead of producer j, and another subset of input prices for which the opposite is true.
4.An empirical example In order to make the notion of indirect efficiency measurement more concrete, we now provide a simple empirical illustration of our approach. As we mentioned in the introduction, the cost indirect production correspondence and related notions introduced in section 2 could provide a useful set of tools in evaluating the performance of individual school districts. School districts face budget constraints and have some freedom in allocating their budgets, and they produce outputs which have no obvious market prices. Yet there is keen interest in monitoring and improving school district performance, which we measure here as the conversion of budgets into educational achievement.
R. Fiire et al., Evaluation of producer performance
85
Our application, due to data limitations, is restricted largely to the calculation of cost indirect technical efficiency scores for a sample of 40 Missouri public school districts in the St. Louis area during the school year 1985-86. The state of Missouri recently instituted a statewide basic skills testing program administered to all eighth grade students, who are tested for basic skills in reading (READ), mathematics (MATH) and government and economics (GVEC). The number of students passing each test serves as the three educational achievement outputs. Two inputs are specified: the number of eighth grade teachers, and the number of eighth grade students. We treat teachers as a variable input and students as a fixed input which school districts cannot change during the school year. Cost indirect technical efficiency is calculated for each school district by solving the following modification to the linear programming problem (2.5):
T,($,/c{,x{,uj) = max 8 (8,I. X”) s.t. 8~’ 5 zM,
j~lzj=l~
(4.1)
where Nr and N, represent partitions of the input matrix N into fixed and variable components, respectively, with a corresponding partition of the input vector x into xr and x,. The budget constraint is expressed in terms of expenditure on variable inputs, c,. Problem (4.1) is solved J = 40 times, once for each school district in the sample. Optimal values of 0, measure cost indirect technical efficiency. Optimal values of Zj identify school districts in the basic solution, districts relative to which the performance of each district is compared. Results are summarized in table 1. Eleven school districts have T,($,/& xj, d) = 1, and they define the ‘best practice’ cost indirect production frontier. The remaining school districts have room for improvement relative to this frontier. Consider, for example, school district 1, with an efficiency
86
R. Fire et al., Evaluation Table Cost
School district 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37” 38 39 40
indirect
technical
efficiency
Efficiency score T,t$,/c’,, x::, u’,
-
of producer performance 1
Missouri 1985-86.
Outputs
school
with positive
READ
MATH
districts,
St.
Louis
area,
slack GVEC
Districts in basic solution
1.054 1.001 1.034 1.067 1.000 1.476 1.579 1.042 1.000 1.000 1.000 1.024 1.020 1.087 1.058 1.087 1.013 1.027 1.000 l.OfKl 1.025 1.000 l.OQO l.COO 1.059 1.025 1.051 1.000 1.161 1.037 1.065 1.133 1.118 1.020 1.109 1.067 _
9, 19, 20, 22, 23 2, 9, 10 10, 11, 19, 22 10,ll 5, 10, 11 10, 22 10, 11 5, 10 11,22 9 10 9, 10, 11 11, 19, 22, 23 19, 23, 40 10, 11 19 19,40 10, 11 19, 20, 24, 40 19, 24, 40 19 20 10, 19, 22 22 23 24 10, 11, 19, 22 9, 10, 11 10, 19 28 10, 11, 28 9, 11, 19, 22 19, 28 10, 11 19,40 19, 23, 40 11,19 11,19
1.073 1.050 l.ooO
11,19 10, 11, 19, 28 40
*This observation
caused
a cycling
problem.
score of 1.054. Given its budget and the size of its student body, this district should be able to proportionately increase the number of students passing
R. Fiire et al., Evaluation of producer performance
87
each test by 5.4 percent. In addition to this equiproportionate increase in all three outputs, two outputs (READ and GVEC) have positive slack and can be increased still further. The school districts on the segment of the cost indirect production frontier to which school district 1 is compared are districts 9, 19, 20, 22 and 23. All of this suggests not just that school district 1 is relatively inefficient in converting its budget into student achievement, but also that it might concentrate on improving reading and government economics skills, perhaps using school districts 9, 19, 20, 22 and 23 as models. We warn that these conclusions are drawn with caution, and are meant to serve as an illustration of the power of the method rather than as an indepth study of school district performance. Such a study would require a more detailed data base including, at the very least, other fixed and variable inputs. Nonetheless, we would argue that our approach offers an appropriate methodology for such an undertaking: school districts are naturally viewed as budget-constrained (rather than resource-constrained as in the conventional approach) multiple output maximizers whose performance is to be evaluated relative to that of their peers. The computational technique imposes no unwarranted structure on the transformation process, and generates subsidiary information on output slack and basic solution identity that may be of as much value to policy-makers as the cost indirect efficiency scores themselves.
5. Conclusions
The purpose of this paper has been to propose a new way of looking at productive efftciency, and of comparing producers, particularly those operating in the public service sector of the economy. The analysis is based on Shephard’s indirect production function, the cost indirect output correspondence in section 2 and the return indirect input correspondence in section 3. In contrast to traditional analysis using Farrell’s resource-constrained, output-based efficiency measures and output-constrained, input-based efficiency measures, our analysis based on Shephard’s indirect production function generates budget-constrained, output-based efficiency measures and revenue-constrained, input based efficiency measures. Efficiency analysis based on indirect production correspondences is particularly relevant in two circumstances. The cost indirect efficiency analysis of section 2 is appropriate when producers are compared on the basis of their ability to transform budgets into benefits, the more so when benefits must be measured in physical terms because they are not priced, or are priced unreliably. The return indirect efficiency analysis of section 3 is appropriate when producers are compared on the basis of their ability to generate returns
88
R. F&e et al., Evaluation of producer performance
at minimum resource cost, particularly when resources are measured in physical terms but cannot be priced reliably. It is of interest to know when, if ever, the cost indirect efficiency measures of section 2 coincide with the return indirect efficiency measures of section 3. They rarely do coincide, and there is rarely reason for them even to provide consistent rankings of producers, essentially because they use different criteria to evaluate producer performance. Even if we observe all input and output prices, so that it is possible to calculate all cost and return indirect efficiency measures, they can be expected to diverge for a number of reasons. First, nonparametric prices will cause cost indirect efficiency measures to behave differently than return indirect efficiency measures. Even in the presence of parametric prices, however, T,($lc’, uj) #(Y&($/P, xj))- ’ unless technology is homogeneous of degree + 1, a feature particularly unlikely to characterize public service provision. Moreover, since input allocative efticiency has nothing whatsoever to do with output allocative efficiency, in general A,(p’lc’, r’, u’) #(Ai(I’/R’,$, xl))- ‘. Consequently, O,($lcj, r’, uj) # (Oi(r’/R’,$, x3)-’ and rankings based on the two criteria can differ as well, since the two types of allocative efficiency refer to performance in different markets. Finally, benefit effectiveness ratios and expense effectiveness ratios need not coincide, or even provide consistent rankings of producer performance, since observed expenses $xj and receipts r’uj involve both types of inefficiency. We have shown how to calculate the three cost indirect and the three return indirect efficiency measures, as well as the associated benefit and expense effectiveness ratios. The linear programming techniques suggested have three virtues: they are easy to implement, they do not impose unwarranted structure on production technology, and they provide ancillary information that enhances the value of the indirect efficiency measures. We have not shown how to calculate the price sets Si, and Si required to establish the shadow benefit and expense effectiveness criteria. However it appears that these sets also can be calculated using linear programming techniques, extending a line of inquiry initiated by Fare and Grosskopf (1987).
References Bessent, A., W. Bessent, J. Kennington and B. Reagan, 1982, An application of mathematical programming to assess productivity in the Houston independent school district, Management Science 28, no. 12, Dec., 1355-1367. Borcherding, T.E., W.W. Pommerehne and F. Schneider, 1982, Comparing the efftciency of private and public production: The evidence from five countries, Zeitschrift fur Nationaliikonomie 42, suppl. 2, 127-156. Charnes, A., W.W. Cooper and E. Rhodes, 1978, Measuring the efficiency of decision making units, European Journal of Operational Research 2, no. 6, Nov., 429-444. Debreu, G., 1951, The coefficient of resource utilization, Econometrica 19, no. 3, July, 273-292.
R. F&e et al., Evaluation @producer
performance
89
Diewert, W.E., 1983, Cost-benefit analysis and project evaluation: A comparison of alternative approaches, Journal of Public Economics 22, no. 3, Dec., 265-302. Diewert, W.E., 1986, The measurement of the economic benefits of infrastructure services, Lecture notes in economics and mathematical systems, vol. 278 (Springer-Verlag, New York, Berlin, Tokyo). Eichhorn, W., 1972, Effektivitlt von produktionsverfahren, Methods of Operations Research 12, 98-l 15. Eichhorn, W., 1978, Functional equations in economics (Addison-Wesley, Reading, MA). Fare, R. and S. Grosskopf, 1987, Measuring shadow price efficiency, in: A. Dogramaci and R. Fare, eds., Application of modern production theory: Efficiency and productivity (KluwerNijhoff, Boston, MA). Fare, R., S. Grosskopf and C.A.K. Lovell, 1985, The measurement of efficiency of production (Kluwer-Nijhoff, Boston, MA). Farrell, M.J., 1957, The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A, General 120, part 3, 253-281. Grosskopf, S., 1986, The role of the reference technology in measuring productive efficiency, The Economic Journal 96, no. 382,499-513. Hyde, W.F., 1980, Timber supply, land allocation, and economic efftciency (The Johns Hopkins Press for Resources for the Future, Baltimore and London). Karlin, S., 1959, Mathematical methods and theory in games, programming and economics (Addison-Wesley, Reading, MA). Shephard, R.W., 1953, Cost and production functions (Princeton University Press, Princeton, NJ). Shephard, R.W., 1970, Theory of cost and production functions (Princeton University Press, Princeton, NJ). Shephard, R.W., 1974, Indirect production functions (Verlag Anton Hain, Meisenheim Am Glad). Shephard, R.W. and R. Fare, 1980, Dynamic theory of production correspondences (Verlag Anton Hain, Konigstein).
of Public
Economics
37 (1988) 71-89.
North-Holland
AN INDIRECT APPROACH TO THE EVALUATION PRODUCER PERFORMANCE R. FARE
OF
and S. GROSSKOPF
Department of Economics, Southern Illinois University, Carbondale, IL 62901, USA
C.A.K. LOVELL* Department of Economics, University of North Carolina, Chapel Hill, NC 27514, USA Received
April 1986, revised version
received
December
1987
1. Introduction In this paper we suggest a somewhat new analytical approach to producer performance evaluation. The suggested approach is ‘somewhat’ new in that it synthesizes existing analytical techniques from three areas, and uses them to look at performance evaluation in a new way. The approach is particularly relevant to public sector or service sector contexts in which the performance of alternative providers of a public service is to be evaluated. In these contexts any of the following conditions may hold: (1) the standard profit maximizing, cost minimizing or revenue maximizing paradigms are inappropriate; (2) prices of some services being provided, or of some resources being consumed, are either nonexistent or unobserved; (3) observed prices are not exogenous, due to monopoly or monopsony power. Examples abound. It is frequently argued that the provision of medical, legal, educational, transportation, employment and many other services, by public or private agencies, is driven by goals other than the maximization of profit. Moreover these and many other services, such as police and fire protection, do not always have market prices with which to value them. In addition, the lack of competition in the provision of many such services, and thus in the employment of specialized resources used in their provision, makes it dangerous to use existing but distorted market prices to compare the performance of providers on the basis of revenue or cost, much less profit. *The authors
are grateful
004772727/88/$3.50
0
to A.B. Atkinson
and two referees for their helpful comments.
1988, Elsevier Science Publishers
B.V. (North-Holland)
72
R. F&e et al., Evaluation of producer performance
Yet many services are provided by a number of suppliers, public or private or both, and it is important to be able to evaluate the performance of each supplier, or of each type of supplier. The use of the profit maximization paradigm, with known, parametric, exogenous service and resource prices, is frequently inappropriate for such a purpose. So too, to a somewhat lesser degree, are the assumptions of cost minimization with known resource prices, or revenue maximization with known service prices.’ No more appealing, however, is the opposite extreme in which suppliers are evaluated solely on the basis of resource usage and service provision, with no reliance placed on either observable prices or economic motivation.2 What is desired is an intermediate approach to producer performance evaluation, one which avoids the agnosticism of the latter approach by using available prices, and which avoids the opposite extreme of imposing inappropriate behavioral goals. The development of such an approach is our objective. The basic analytical tool is the indirect production function developed by Shephard (1974). The cost indirect production correspondence shows the set of output vectors obtainable from any given budget. The return indirect production correspondence shows the set of input vectors capable of generating any given revenue. The two indirect correspondences thus provide the basis for an evaluation of producer performance on the basis of service benefits measured by quantities supplied, or on the basis of resource costs measured by quantities employed. For this reason Shephard (1974, preface) argued that the indirect production function ‘is the foundation for a costbenefit analysis’. Quite apart from its role in a benefitxost framework, however, the indirect production function is consistent with axiomatic production theory and has useful dual representations. In fact, the distance functions developed by Shephard (1953, 1970) as such a dual representation are closely related to the second analytical tool we use, the family of efficiency measures originally proposed by Debreu (1951) and Farrell (1957). The significance of the application of efficiency measures to indirect production functions is that they provide a method for calculating frontier or best practice benefits or costs relative to the technology. This analytical approach to benelitxost comparisons provides the foundation for the intermediate approach to producer performance evaluation we seek. ‘Nonetheless, many performance evaluations, particularly those of a ‘public vs. private’ nature, are based on such paradigms. A critical survey of these studies, from airlines to weather forecasting, is provided by Borcherding, Pommerehne and Schneider (1982). 2The data envelopment analysis (DEA) literature correctly points out that much public sector production is characterized by the absence of the profit motive and the absence of service prices. The DEA solution, to ignore all prices and all behavioral motivation, and to evaluate performance solely on the basis of the efficiency of the physical transformation process, seems a bit extreme, however. See Charnes, Cooper and Rhodes (1978) for an early statement of the DEA approach to performance evaluation.
R. Fiire et al., Evaluation of producer performance
13
The third analytical technique we use is the nonparametric approach to the construction of indirect production frontiers and the measurement of efficiency relative to them. The nonparametric approach is simple, yet very flexible. Indirect production frontiers are derived by enveloping the data while ensuring that very general regularity conditions implied by the axioms are satisfied. No parametric functional form unduly restricting scale or substitution properties is imposed. A family of benefit-cost efficiency measures is calculated for each observation as solutions to a sequence of linear programming problems. Our analytical approach is, as we have argued, perhaps most valuable as a means of evaluating the performance of providers of public services. It is, in the terminology of Diewert (1983, 1986), a partial equilibrium approach, since we do not incorporate consumers or other sectors of the economy. It is in spirit related to Diewert’s ‘productive efficiency project evaluation criterion’, although we allow for multiple outputs and we do not use a profit function (for reasons mentioned above) to model technology. Although we derive benefit effectiveness criteria and expense effectiveness criteria, we do not suggest our approach as a way of doing conventional benefit-cost analysis or project evaluations, which typically involve the assessment of a small number of large, heterogeneous, indivisible investments. For these tasks the reader is referred to Diewert (1983, 1986) and the references cited therein. The paper unfolds as follows. Section 2 considers the cost indirect production correspondence and the measurement of benefit effectiveness. This framework is particularly appropriate to producer performance evaluation when resource usage can be reliably compared on the basis of cost, but benefits (outputs produced or services provided) cannot be priced reliably enough to allow revenue comparisons. An example is provided by the attempt to evaluate the performance of school districts whose budgets can be measured, but whose educational outputs can be measured but not priced.3 Section 3 considers the return indirect production correspondence and the measurement of cost effectiveness. This framework is suited to producer performance evaluation when benefits can be priced reliably enough to make revenue comparisons, but resource usage is measured in physical terms. An example is provided by an attempt to evaluate the performance of multipleuse public lands for which revenues can be measured but not all resources are priced.4 In section 4 we provide a brief example illustrating how the cost indirect production correspondence works in practice. The empirical problem is to compare the performance of a group of public school districts 3Bessent, Bessent, Kennington and Reagan (1982) adopt a DEA approach to public school district performance evaluation, in which test scores are outputs, student characteristics and school resources are inputs, and budgets are ignored. 4Hyde (1980) evaluates multiple-use forest management on public and private land in the United States without resorting to an explicit optimization model of the sort proposed here.
14
R. F&e et al., Evaluation of producer performance
in the state of Missouri. The results of the experiment include a ranking of districts on the basis of their ability to convert budgets into educational achievement, and they also include additional information resulting from a linear programming formulation of the problem. Section 5 concludes.
2. Cost indirect effkiency The first objective is to derive a family of efficiency measures that can be used to gauge the performance of individual producers relative to other producers in the sample. Performance is gauged by output quantities produced, given a fixed budget, input prices, and production technology. For this purpose, technology is represented most usefully by the cost indirect production correspondence. The second objective is to provide a computational method for calculating the family of efficiency measures. The method uses linear programming techniques to construct a nonparametric cost indirect production frontier, and to calculate efftciency relative to that frontier. Suppose there are j = 1,. . ., J producers using input vector xj~ R: to produce output vector uj~ R”,. Each producer faces input price vector and budget level cj>O. Since input price vectors and budgets are #EC+ indexed by j, producers are allowed to face different input prices and to have different budgets. Denote the J x m matrix of observed outputs by M and the J x n matrix of observed inputs by N, in which case the direct output correspondence can be written J
u:u~zM,zN~x,z~R~,
c j=l
z,.=} 1 ,
(2.1)
where z is a vector of intensity variables familiar from activity analysis. P(x) is the set of all technologically feasible output vectors with input vector x. The production technology described in (2.1) satisfies strong disposability of both outputs and inputs, an exhibits variable returns to scale. In addition, if every column sum and every row sum of both M and N are strictly positive, then the direct output correspondence x+P(x) in (2.1) is sufficiently wellbehaved for a number of duality theorems to go through.5 Specifically, we can represent production technology with the cost indirect output correspondence:
SFor details see Karlin (1959, pp. 338-343) or Shephard (1970, pp. 283-292). The scale and disposability properties of the output correspondence (2.1), and (3.1) below, are discussed in Grosskopf (1986).
R. Fiire et al., Evaluationof producer performance
75
J
(p/c)-+P(p/c) =
u: u 5 -
zitf,zN~x,
px SC, z E R:, c
j=l
Zi=]1
(2.2) Thus P(p/c) is the set of all technologically feasible output vectors attainable at budget-deflated input prices (p/c). Whereas P(x) is a resource-constrained output set, P(p/c) is a budget-constrained output set. It can be shown that the cost indirect output correspondence (2.2) is closed, bounded, convex, and satisfies strong disposability of outputs.6 Given this representation of technology, we can now evaluate the performance of individual producers by comparing their outputs with outputs located on the boundary of the cost indirect output set P(p/c). The measures we use to gauge the performance of individual producers are modifications of the family of efficiency measures introduced by Debreu and Farrell. We begin with: Definition.
The function
T,($lcj, uj) = max {8: BujE P($/cj)}
(2.3)
is called the cost indirect technical eficiency
measure.
For observation j the cost indirect technical efficiency measure computes the ratio of the largest feasible radial expansion of uj to uj itself, where feasibility is defined by the cost indirect output set P(p’/cj). Thus T&+/c’, u’) = IIT,($/cj,u’) . u’pp( is a budget-constrained measure of technical efticiency.7 The properties of T’,(p/c,u) are summarized in Proposition.
Given P(p/c), then for each j=
1,. . . , J,
‘?See Shephard (1974) and Shephard and Fare (1980) for proofs of these and other properties of the cost indirect output correspondence. Note also that if all producers face the same input price vector, the cost indirect output correspondence (2.2) can be simplified further, by using observed costs in place of observed input prices and quantities to obtain: (p/c)-+P’(p/c)= u:u~zM,zC~c,z~R:, i
$,zj=l}*
(2.2’)
where C is the J x 1 vector of observed costs. 7The fact that T,($lc’,u’) is a budget-constrained efficiency measure is what distinguishes it from the Debreu-Farrell resource-constrained efficiency measure F,(xj, u’) discussed in Fare, Grosskopf and Love11 (1985, ch. 4). It is also what makes T,(p’/ti,u’) similar to an index of economic effectiveness E&c’, uj, t’), t measuring time required to provide the service, proposed by Eichhorn (1972, 1978).
R.
Fire
et al.,
Evaluation
ofproducer
performance
To. 1
T,($/c’, 6U’)= 6 - 1T&+/e’, d), s > 0,
To.2
l~T,($/c’,u’)<+co,
TO.3
T,( $,‘d, u’) = 1+-w’ E Isoq P( p+‘) = (u: u E
P(f+/c’),6u 4 P($/cl)
(2.4)
if 6 > 1).
Thus T,(p/c, u) is a function homogeneous of degree - 1 in the output vector that attains its minimum value of unity if and only if the output vector belongs to the cost indirect output frontier Isoq P(p/c).* Implementation of the cost indirect technical efficiency measure requires specification of the cost indirect output correspondence (2.2). A nonparametric specification of that correspondence, together with a calculation of the efficiency measure for each observation, is provided by the constraints of, and the solution to, the linear programming problem: T,($lcj, uj) = max 13
(z,e.X) s.t.
Bu’~zM,
j$lzjcl.
(2.5)
Solving this problem J times generates a measure of cost indirect technical efficiency for every producer in the sample. T,($lcj,uj) provides a means of comparing the performance of producers in terms of their budget-constrained ‘real’ benefits, namely their output quantities. Such a procedure is particularly attractive when not all outputs are priced, or when not all output prices are observed accurately enough to allow comparisons across producers, or when differential market power distorts some output prices. However if reliable output prices are available, *Proposition (2.4) has the same mathematical structure as Theorem (4.2.5) in Fiire, Grosskopf and Love11 (1983, even though the latter deals with direct (rather than cost indirect) output technical efficiency measurement. Consequently, a proof of Proposition (2.4) here can be deduced from the proof of Theorem (4.2.5) there.
17
R. F&e et al., Evaluation of producer performance
then benefits can be measured in monetary as well as in physical terms. In this event the foregoing analysis can be extended to evaluate performance on the basis of budget-constrained returns. Suppose now that for each producer j=l , . . . , J we observe input vector xi, output vector uj, input price vector p’ and budget ei as before, and, in addition, we observe output price vector riERm++. In this case we can define a maximum return function relative to the cost indirect output correspondence as: &p/c, r) = max (ru: u E Q/c)>. ”
(2.6)
R(p/c,r) shows the maximum return obtainable from budget c given prices I and p for outputs and inputs. It is a budget-constrained revenue function. For each producer maximum return can be calculated as the solution to: R(p’lcj, ri) = max r’u
(z,“9X1 s.t. u szikf,
i
zj=l.
(2.7)
j=l We are now in a position to evaluate the performance comparing their observed and maximum returns. Definition.
of producers
by
The function
O,($/cj, r’, uj) = R(p’/cj, ri)/rjuj is called the cost indirect revenue eficiency
V-8) measure.
For observation j the cost indirect revenue efficiency measure is the ratio of the maximum feasible return at output prices ri and cost deflated input prices $/cl to actual receipts, where feasibility is once again defined relative to the cost indirect output set P(p/c). Thus O,,(~/c’,r’,u’) is a budget-
R. F&e et al., Evaluationof producer performance
78
constrained measure of revenue effkiency.9 The properties indirect revenue efficiency measure are summarized in: Proposition.l” 0,.
Given P(p/c),
1 O,($/cj,
then for each j=
r’, 6u’) = 6 - ‘O,(p’/c’,
0,. 2
1 5 O,($lc’, r’, uj) < + cc
00.3
O&‘/C?, r’, u’) = 1 ++ uj
of the cost
1,. . . , J, r’, u’), 6 > 0,
(2.9) solves (2.7)
Note that O,(p/c,r, u) depends on both output quantities and output prices, whereas T’,(p/c,u) is independent of output prices. We can isolate the output price-dependent component of Oo(p/c, r, u) by introducing: Definition.
The function
A,($lcj,
6, u’) = O,($lcj,
r’, uj)/T,($/cj,
uj)
(2.10)
is called the cost indirect allocative efficiency measure. For observation j the cost indirect allocative effkiency measure gauges the ability of the producer to produce the optimal mix of outputs in light of prevailing output prices. The properties of the cost indirect allocative efficiency measure are summarized in: Proposition.11
Given P(p/c),
then for each j= 1,. . . , J,
A,. 1
A&+]&, rj, 6u’) = A,($lc’, r’, u’), 6 > 0,
A,.2
l~&($lcj,rj,uj)<
9The fact distinguishes
that O,(p’/c’,r’,u’) it from Farrell’s
+co,
(2.11)
is a budget-constrained revemx efficiency measure is what resource-constrained revenue efficiency measure 0,(x’, u’,Tj) =
R(x’, r’)/r’u’. loThe proof of Proposition (2.9) is similar to the proof of Theorem (4.4.7) in Fiire, Grosskopf and Lovell (1985), although the latter deals with direct revenue efficiency measurement. “The proof of parts A,. 1 and A,,.2 of Proposition (2.11) follow proofs of similar parts of Theorem (4.6.4) in Fire, Grosskopf and Love11 (1985), although the latter deals with direct output allocative efficiency measurement. To prove A,,. 3 assume 362 1 such that 6~’ solves (2.7). Then R(p’lc’, r’) = r’6u’. Thus, from Definitions (2.3) and (2.10), A,($/c’, r’, u’) = 1. Coversely, assume that A,@]&, rj, u’) = 1. Then R(p’/c-‘,r’) = IJ(T,($/c’, u’) d). Since T,(p’lc’, u’) 11, A,. 3 holds. Q.E.D.
R. Fire et al., Evaluation of producer performance
‘40.3
79
A,($/c’,r’,u’)=le,36~1 such that Su’ solves (2.7).
From Definitions (2.5) and (2.10) we obtain the following multiplicative decomposition of the cost indirect revenue efficiency measure defined in (2.8): 0,($/d,
rj, d) = A,($lcj,
rj, d) . T&+/c’, u’).
(2.12)
Thus the ability to generate maximum return in a budget-constrained environment is the product of the ability to maximize output and the ability to select the optimal output mix. Departures from cost indirect maximum return as defined by R(p/c,r) in (2.6) must be technical or allocative in nature, and the decomposition described in (2.12) can be calculated empirically, using linear programming techniques. If output prices are known, R($/cj,rj) can be calculated from (2.7). OO($lcj,ri,uj) is calculated from (2.8), T,($lcj, uj) is calculated from (2.5), and A,($/&, rj, u’) is obtained residually from (2.10). The cost indirect revenue efficiency measure O,($lcj,rj,d) evaluates producer performance by comparing budget-constrained maximum returns to observed receipts. In some circumstances it may be preferable to compare budget-constrained maximum returns to observed budgets, and to rank producers on the basis of their benefit effectiveness ratiosI R($lcj, &)/cj, j= 1,. . . ,J. In contrast to a conventional revenue-cost ratio riuj/cj, a benefieffectiveness ratio eliminates technical and allocative inefficiency from the numerator, and compares producers on the basis of their potential for converting budgets into revenues. If output prices are not known, only T,($/cj,uj) can be calculated, and producers can be ranked on the basis of their ability to convert budgets into real services. They can also be compared in at least one additional way. The set of product prices for which a producer has a benefit-effectiveness ratio at least as great as unity can be written: S’o=(rj:R(pilcj,rj)~cj),
j=l,...,
J.
(2.13)
Producers can be compared on the basis of their sets Sj,. If Sjo c Sk for any two producers indexed j and k, then producer k is ranked ahead of producer 12The benefit effectiveness ratio is particularly attractive in circumstances in which a public agency oversees the operation of a number of service providers. Examples are state departments of education overseeing many school districts, state departments of justice supervising local courts, and the like. If individual school districts or courts exhibit relatively low benefit effectiveness ratios, this provides the oversight agency with good reason to consider merger of inefficiently small districts or division of inefticiently large districts in order to more closely approximate most productive scale size in all districts.
R. Fiire et al., Evaluation of producer performance
80
j on the basis of this shadow benefit-effectiveness criterion. If S(, #St and Sk, Q Si,, then producer j is ranked ahead of producer k for some subset of output price vectors, and producer k is ranked ahead of producer j for some other subset of output price vectors, on the basis of this shadow benefieffectiveness criterion. 3. Return indirect efficiency
The analysis of this section is similar in structure to that of the previous section, the essential difference being in the information presumed available to the analyst. Here we assume that output or service prices are known, and that it is desired to gauge the performance of producers on the basis of resources consumed in order to generate certain revenues. For this purpose technology is represented by the return indirect production correspondence, and efficiency measurement is input-based. Notation remains unchanged. For producers j = 1,. . . , J, inputs xj~ R”, are used to produce outputs USER”, that are valued at prices rjE R”, +. Revenue constraints are Rj>O. The direct input correspondence can be written: J
x:u~zM,zN~x,z~R:,
c j=l
z,.=} 1 ,
(3.1)
where z, A4 and N are as defined previously. The input set L(u) shows the set of all input vectors capable of producing output vector u. Production technology can also be represented by the return indirect input correspondence:
={x:x~L(u),(r/R)uzl}.
(3.2)
Whereas L(u) is an output-constrained input set, L(r/R) is a revenueconstrained input set, showing the set of all input vectors capable of generating revenue R at output prices r. The return indirect input correspondence is closed, convex, and satisfies strong input disposability.r3 ‘3Proofs of these properties are analogous to the proofs of Shephard (1974) and Shephard and Fire (1980) for similar properties of the cost indirect output correspondence. If all producers face the same output price vector, the return indirect input correspondence can be simpliIied to: (r/R)-*L’(r/R)=
x:R~zR,zN
where R is the J x 1 vector of observed
revenues.
(3.2’)
R. Fiire et al., Evaluation of producer performance
Producer performance is evaluated by comparing with efficient input usage located on the boundary input set by means of: Definition.
81
observed input usage of the return indirect
The function
T(&/Rj, xj) =min
{A: AXLEL(ljlR’)}
(3.3)
is called the return indirect technical efficiency measure. For producer j the return indirect technical efficiency measure computes the ratio of the smallest feasible radial contraction of xi to xj itself, feasibility being defined by the return indirect input set L(&/Rj). Thus IT;(fi/Rj,xj) is a revenue-constrained measure of technical efficiency.i4 The return indirect technical efficiency measure has properties summarized in: Proposition.
Given L(r/R), then for each j= 1,. . . , J,
7;. 1
7;(r’/Rj, Ax’) = 1~ ’ T(rj/R’, x’), A> 0,
T. 2
0 < 7;(r’lR’, x’) 5 1,
T. 3
T(ljlRj, xj) = 1 c) xj~ Isoq L(rj/Rj) = {x: x E L(rj/Rj), Ax 4 L(r’/Rj)
(3.4)
if A< 1).
Thus T(rjlRj,xj) is a function homogeneous of degree - 1 in the input vector that attains its maximum value of unity if and only if the input vector belongs to the return indirect input frontier Isoq L(rj/Rj). Proofs follow those of (2.4). T(r’/R’,xj) can be computed for observation j as the solution to the linear programming problem: T(+jRj, xj) = min A (2.=,U) s.t. U~ZM, 14This is what distinguishes 7;:(r’/Rj,xj) from the Debreu-Farrell of technical efficiency Fi(uj, xj).
output-constrained
measure
82
R. Fiire et al., Evaluation of producer performance
zNj
fi
ix’,
‘j’l.
(3.5)
7Jrj/Rj,xj) provides a means of comparing the performance of producers on the basis of their revenue-constrained ‘real’ resource consumption, and is appropriate when not all resources are reliably priced. However, if resources are reliably priced, the analysis can be extended to provide a comparison of producers on the basis of their return-constrained costs. Suppose for each producer we observe output vector uj, input vector xi, output price vector ri and revenue constraint Rj as before. If we also observe input price vector $ER”+ +, then we can define a minimum expense function relative to the revenue indirect input correspondence as: C(r/R,p) =min {px: x E L(r/R)}. X
(3.61
C(r/R,p) shows the minimum expense required to generate revenue R at output prices r and input prices p; it is a revenue-constrained cost function. For each producer C(r/R,p) can be calculated as the solution to: C(rj/Rj, $) = min $x (2.X,U)
zNj x,
ZER:,
j$lzj=l. Once C(r’/Rj,$) the basis of:
is calculated,
producer
(3.7)
performance
can be evaluated
on
83
R. Fiire et al., Evaluationof producer performance
Definition.
The function
Oi(r’lRj, $, x’) = C(ri/Rj, $j/$x’
(3.8)
is called the return indirect cost eficiency
measure.
Given (ti/Rj,p’), this measure compares producers on the basis of their ability to select expense-minimizing input vectors. Thus, O,(fl]Rj,$, xj) is a revenue-constrained measure of cost effkiency.ls Its properties are summarized in: Given L_(r/R), then for each j =
Proposition. Oi.
1,. . . , J,
1 Oi(rj/Rj, p’ 2~‘) = I - ’ OJr'JR', p’, x’),
Oi. 2
0 < Oi(r’lR’, $,
Oi. 3
O,(r’/Rj,#,
X’) ~
A>
0,
1,
xj) = 1 ++ xj
(3.9) solves
(3.7).
Proofs follow those of Proposition (2.9). O,(r/R,p, x) depends on input prices, and 7Jr/R, x) does not. The input price-dependent component of Oi(r/R, p, x) can be isolated by introducing: Definition.
The function
A,(r’/Rj, p’, xj) = Oi(r’/Rj, $, xj)‘z(rj/R’,
is called the return indirect allocative eficiency
x’)
(3.10)
measure.
For producer j the return indirect allocative effkiency measure judges the ability to use inputs in the correct proportion in light of observed input prices. The properties of this measure are given in: Given L(r/R), then for each j = 1,. . . , J,
Proposition. Ai.
1 A,(r’/R’,$,
2.x’) = A,(rj/Rj, $, xj), 2 >O,
Ai. 2
0 < Ai(~/R’, $,
Ai.
A,(r’/R’, $, x’) = 1~
3
X’) ~
ISThis is what distinguishes O,($/Rj,$,x’) O,(uj,xj, $) = C(u’,$1/$x’.
measure
(3.11)
1, 31 E (0, 11 from
Farrell’s
output-constrained
cost
efkiency
R. F&e et al., Evaluation of producer performance
such that ix’ solves (3.7).
Proofs follow those of Proposition (2.11). From Definitions (3.3) and (3.10) we obtain the following multiplicative decomposition of the return indirect cost efficiency measure defined in (3.8):
Oi(r'lR',$, X')'Ai(r'fR',$, X’)’ ~(r'/R', X’).
(3.12)
Thus the ability to produce at minimum cost in a revenue-constrained environment is the product of the ability to select the optimal input mix and the ability to minimize input usage. T(ljlRj,x') is calculated by means of is calculated from (3.8) and (3.7), and Ai(r'/R'p$y X’) is (3.5), Oi(G/R'yp',X') calculated from (3.10). As in the previous section we may wish to augment, or replace, O,(ljlRj,p', x')with the expense effectiveness ratio RJ/C(ljlRf,p$ which compares producers on the basis of their potential for generating revenue at minimum expense. Finally, if input prices are not observed, only T,(ti/Rj,X'j can be calculated, and producers can be evaluated on the basis of their revenue-constrained technical efficiency. They can also be evaluated on the basis of the set of input prices for which their expense effectiveness ratios are at least as great as unity by means of
$={p':R'2C(ti/R',$)}, j=l,...,J. If Si c $ for any two producers j and k, then producer k is evaluated higher than producer j on the basis of this shadow expense effectiveness criterion. Alternatively, there may exist a subset of input prices for which producer k is ranked ahead of producer j, and another subset of input prices for which the opposite is true.
4.An empirical example In order to make the notion of indirect efficiency measurement more concrete, we now provide a simple empirical illustration of our approach. As we mentioned in the introduction, the cost indirect production correspondence and related notions introduced in section 2 could provide a useful set of tools in evaluating the performance of individual school districts. School districts face budget constraints and have some freedom in allocating their budgets, and they produce outputs which have no obvious market prices. Yet there is keen interest in monitoring and improving school district performance, which we measure here as the conversion of budgets into educational achievement.
R. Fiire et al., Evaluation of producer performance
85
Our application, due to data limitations, is restricted largely to the calculation of cost indirect technical efficiency scores for a sample of 40 Missouri public school districts in the St. Louis area during the school year 1985-86. The state of Missouri recently instituted a statewide basic skills testing program administered to all eighth grade students, who are tested for basic skills in reading (READ), mathematics (MATH) and government and economics (GVEC). The number of students passing each test serves as the three educational achievement outputs. Two inputs are specified: the number of eighth grade teachers, and the number of eighth grade students. We treat teachers as a variable input and students as a fixed input which school districts cannot change during the school year. Cost indirect technical efficiency is calculated for each school district by solving the following modification to the linear programming problem (2.5):
T,($,/c{,x{,uj) = max 8 (8,I. X”) s.t. 8~’ 5 zM,
j~lzj=l~
(4.1)
where Nr and N, represent partitions of the input matrix N into fixed and variable components, respectively, with a corresponding partition of the input vector x into xr and x,. The budget constraint is expressed in terms of expenditure on variable inputs, c,. Problem (4.1) is solved J = 40 times, once for each school district in the sample. Optimal values of 0, measure cost indirect technical efficiency. Optimal values of Zj identify school districts in the basic solution, districts relative to which the performance of each district is compared. Results are summarized in table 1. Eleven school districts have T,($,/& xj, d) = 1, and they define the ‘best practice’ cost indirect production frontier. The remaining school districts have room for improvement relative to this frontier. Consider, for example, school district 1, with an efficiency
86
R. Fire et al., Evaluation Table Cost
School district 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37” 38 39 40
indirect
technical
efficiency
Efficiency score T,t$,/c’,, x::, u’,
-
of producer performance 1
Missouri 1985-86.
Outputs
school
with positive
READ
MATH
districts,
St.
Louis
area,
slack GVEC
Districts in basic solution
1.054 1.001 1.034 1.067 1.000 1.476 1.579 1.042 1.000 1.000 1.000 1.024 1.020 1.087 1.058 1.087 1.013 1.027 1.000 l.OfKl 1.025 1.000 l.OQO l.COO 1.059 1.025 1.051 1.000 1.161 1.037 1.065 1.133 1.118 1.020 1.109 1.067 _
9, 19, 20, 22, 23 2, 9, 10 10, 11, 19, 22 10,ll 5, 10, 11 10, 22 10, 11 5, 10 11,22 9 10 9, 10, 11 11, 19, 22, 23 19, 23, 40 10, 11 19 19,40 10, 11 19, 20, 24, 40 19, 24, 40 19 20 10, 19, 22 22 23 24 10, 11, 19, 22 9, 10, 11 10, 19 28 10, 11, 28 9, 11, 19, 22 19, 28 10, 11 19,40 19, 23, 40 11,19 11,19
1.073 1.050 l.ooO
11,19 10, 11, 19, 28 40
*This observation
caused
a cycling
problem.
score of 1.054. Given its budget and the size of its student body, this district should be able to proportionately increase the number of students passing
R. Fiire et al., Evaluation of producer performance
87
each test by 5.4 percent. In addition to this equiproportionate increase in all three outputs, two outputs (READ and GVEC) have positive slack and can be increased still further. The school districts on the segment of the cost indirect production frontier to which school district 1 is compared are districts 9, 19, 20, 22 and 23. All of this suggests not just that school district 1 is relatively inefficient in converting its budget into student achievement, but also that it might concentrate on improving reading and government economics skills, perhaps using school districts 9, 19, 20, 22 and 23 as models. We warn that these conclusions are drawn with caution, and are meant to serve as an illustration of the power of the method rather than as an indepth study of school district performance. Such a study would require a more detailed data base including, at the very least, other fixed and variable inputs. Nonetheless, we would argue that our approach offers an appropriate methodology for such an undertaking: school districts are naturally viewed as budget-constrained (rather than resource-constrained as in the conventional approach) multiple output maximizers whose performance is to be evaluated relative to that of their peers. The computational technique imposes no unwarranted structure on the transformation process, and generates subsidiary information on output slack and basic solution identity that may be of as much value to policy-makers as the cost indirect efficiency scores themselves.
5. Conclusions
The purpose of this paper has been to propose a new way of looking at productive efftciency, and of comparing producers, particularly those operating in the public service sector of the economy. The analysis is based on Shephard’s indirect production function, the cost indirect output correspondence in section 2 and the return indirect input correspondence in section 3. In contrast to traditional analysis using Farrell’s resource-constrained, output-based efficiency measures and output-constrained, input-based efficiency measures, our analysis based on Shephard’s indirect production function generates budget-constrained, output-based efficiency measures and revenue-constrained, input based efficiency measures. Efficiency analysis based on indirect production correspondences is particularly relevant in two circumstances. The cost indirect efficiency analysis of section 2 is appropriate when producers are compared on the basis of their ability to transform budgets into benefits, the more so when benefits must be measured in physical terms because they are not priced, or are priced unreliably. The return indirect efficiency analysis of section 3 is appropriate when producers are compared on the basis of their ability to generate returns
88
R. F&e et al., Evaluation of producer performance
at minimum resource cost, particularly when resources are measured in physical terms but cannot be priced reliably. It is of interest to know when, if ever, the cost indirect efficiency measures of section 2 coincide with the return indirect efficiency measures of section 3. They rarely do coincide, and there is rarely reason for them even to provide consistent rankings of producers, essentially because they use different criteria to evaluate producer performance. Even if we observe all input and output prices, so that it is possible to calculate all cost and return indirect efficiency measures, they can be expected to diverge for a number of reasons. First, nonparametric prices will cause cost indirect efficiency measures to behave differently than return indirect efficiency measures. Even in the presence of parametric prices, however, T,($lc’, uj) #(Y&($/P, xj))- ’ unless technology is homogeneous of degree + 1, a feature particularly unlikely to characterize public service provision. Moreover, since input allocative efticiency has nothing whatsoever to do with output allocative efficiency, in general A,(p’lc’, r’, u’) #(Ai(I’/R’,$, xl))- ‘. Consequently, O,($lcj, r’, uj) # (Oi(r’/R’,$, x3)-’ and rankings based on the two criteria can differ as well, since the two types of allocative efficiency refer to performance in different markets. Finally, benefit effectiveness ratios and expense effectiveness ratios need not coincide, or even provide consistent rankings of producer performance, since observed expenses $xj and receipts r’uj involve both types of inefficiency. We have shown how to calculate the three cost indirect and the three return indirect efficiency measures, as well as the associated benefit and expense effectiveness ratios. The linear programming techniques suggested have three virtues: they are easy to implement, they do not impose unwarranted structure on production technology, and they provide ancillary information that enhances the value of the indirect efficiency measures. We have not shown how to calculate the price sets Si, and Si required to establish the shadow benefit and expense effectiveness criteria. However it appears that these sets also can be calculated using linear programming techniques, extending a line of inquiry initiated by Fare and Grosskopf (1987).
References Bessent, A., W. Bessent, J. Kennington and B. Reagan, 1982, An application of mathematical programming to assess productivity in the Houston independent school district, Management Science 28, no. 12, Dec., 1355-1367. Borcherding, T.E., W.W. Pommerehne and F. Schneider, 1982, Comparing the efftciency of private and public production: The evidence from five countries, Zeitschrift fur Nationaliikonomie 42, suppl. 2, 127-156. Charnes, A., W.W. Cooper and E. Rhodes, 1978, Measuring the efficiency of decision making units, European Journal of Operational Research 2, no. 6, Nov., 429-444. Debreu, G., 1951, The coefficient of resource utilization, Econometrica 19, no. 3, July, 273-292.
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performance
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Diewert, W.E., 1983, Cost-benefit analysis and project evaluation: A comparison of alternative approaches, Journal of Public Economics 22, no. 3, Dec., 265-302. Diewert, W.E., 1986, The measurement of the economic benefits of infrastructure services, Lecture notes in economics and mathematical systems, vol. 278 (Springer-Verlag, New York, Berlin, Tokyo). Eichhorn, W., 1972, Effektivitlt von produktionsverfahren, Methods of Operations Research 12, 98-l 15. Eichhorn, W., 1978, Functional equations in economics (Addison-Wesley, Reading, MA). Fare, R. and S. Grosskopf, 1987, Measuring shadow price efficiency, in: A. Dogramaci and R. Fare, eds., Application of modern production theory: Efficiency and productivity (KluwerNijhoff, Boston, MA). Fare, R., S. Grosskopf and C.A.K. Lovell, 1985, The measurement of efficiency of production (Kluwer-Nijhoff, Boston, MA). Farrell, M.J., 1957, The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A, General 120, part 3, 253-281. Grosskopf, S., 1986, The role of the reference technology in measuring productive efficiency, The Economic Journal 96, no. 382,499-513. Hyde, W.F., 1980, Timber supply, land allocation, and economic efftciency (The Johns Hopkins Press for Resources for the Future, Baltimore and London). Karlin, S., 1959, Mathematical methods and theory in games, programming and economics (Addison-Wesley, Reading, MA). Shephard, R.W., 1953, Cost and production functions (Princeton University Press, Princeton, NJ). Shephard, R.W., 1970, Theory of cost and production functions (Princeton University Press, Princeton, NJ). Shephard, R.W., 1974, Indirect production functions (Verlag Anton Hain, Meisenheim Am Glad). Shephard, R.W. and R. Fare, 1980, Dynamic theory of production correspondences (Verlag Anton Hain, Konigstein).