Nonparametric analysis of educational costs

Theory and Methodology

Nonparametric analysis of educational costs

John Ruggiero *

Department of Economics and Finance, University of Dayton, 300 College Park, Dayton, OH 45469-2240, USA

Received 17 March 1997; accepted 3 June 1998

Abstract

Traditional economic analyses of the public sector that assume cost minimization are not consistent with political

models of bureaucracy. If costs are not minimized then estimated cost functions will be biased. The purpose of this

paper is to provide a ¯exible nonparametric technique based on Farrell-type e�ciency measures to estimate public

sector costs. Standard indices need to be modi®ed to ®t the special nature of public sector service provision which is

characterized by an in¯uence of exogenous variables on cost. A useful by-product is an index of environmental

harshness faced by local governments. For illustrative purposes, this technique is applied to a sample of New York

State school districts. It is found that nearly 64% of districts are cost ine�cient, spending on average $1200 per pupil

above the cost minimizing level. In addition, it is estimated that the average school district faces environmental cost of

over $1700 per pupil. Ó 1999 Elsevier Science B.V. All rights reserved.

Keywords: Data envelopment analysis; Educational costs

1. Introduction

During the last decade local governments havebeen under increasing ®scal stress. In this tightenvironment an understanding of the underlyingcost of public sector service provision has becomeimportant. Most cost analyses begin with the as-sumption that public sector decision makers min-imize cost subject to the production technology(see Darrough and Heineke [1] and Duncombeand Yinger [2].) While the assumption of cost

minimization is appropriate for long-run analysesof private sector production, it may not be forpublic sector applications where o�cials may havealternative goals (e.g., obtaining a higher salary,an excess sta�, and/or larger budgets).

Niskanen [3] postulated that public sector de-cision-makers have incentives to maximize theirbudgets in order to maximize utility. In the Nis-kanen model, o�cials receive higher utility by in-creasing their salary, perquisites of o�ce, power,patronage, and the services provided, all of whichlead to higher budgets. Thus, bureaucrats have anincentive to provide the level of outcomes thatyields the highest budget rather than the level thatminimizes cost. Niskanen [4] extended his original

European Journal of Operational Research 119 (1999) 605±612www.elsevier.com/locate/orms

* Tel.: +1 937 299 2550; fax: +1 937 229 2477; e-mail:

[email protected]

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 3 5 1 - 8

model to allow maximization of budgetary slack,i.e., the portion of the budget in excess of the costminimizing level (see also Migu�e and B�elanger [5]and Wycko� [6]). If bureaucratic supply modelsare correct then estimated public sector costequations that assume cost minimization confoundestimated minimum cost with budgetary slack (i.e.,ine�ciency), leading to potential biases. The pur-pose of this paper is to provide a ¯exible meth-odology to analyze public sector costs withoutimposing the cost minimizing assumption.

The technique employed in this paper wasconceptualized by Farrell [7] and developed byCharnes et al. [8] to analyze multiple output pro-duction frontiers. These mathematical program-ming models were extended by Grosskopf andYaisawarng [9] to analyze cost and economies ofscope in the provision of public sector services.The programs are solved to compare the expen-diture of a given local government with the ex-penditure of other local governments producingthe same level of services. If the local governmentis producing at the cost minimizing level then noother local government (or combination of localgovernments) can produce the same level of ser-vices with lower expenditures, ceteris paribus. Ifexpenditure is higher than the cost minimizinglevel, however, the Farrell-type e�ciency measurewill indicate cost ine�ciency.

A problem with previous applications is theassumption that the production and/or costs canbe represented by only one frontier (see Bessentand Bessent [10], Bessent et al. [11] and F�are et al.[12].) This suggests that all deviations from the``best practice'' frontier can be attributed to inef-®ciency. This is problematic because no allowancesare made for di�erences in resource prices andother exogenous environmental factors. Measuresof minimum cost are therefore contaminated bythe e�ect that the environment has on cost. Inorder to properly measure ine�ciency, it is neces-sary to control for environmental factors. Bankerand Morey [13] extended Data EnvelopmentAnalysis to allow exogenous factors by includingconvexity constraints on these ®xed inputs. Rug-giero [14] showed that the Banker and Moreymodel is inappropriate for public sector applica-tions because it assumes convexity with respect to

the non-discretionary inputs and developed analternative model.

This paper extends the public sector productionmodel of Ruggiero [14,15] to allow comparisons ofexpenditure levels of local governments whilecontrolling for the level of services, input pricesand environmental variables. One useful by-product of the methodology is an index whichshows the e�ect that the environment has on cost.This index can be converted to estimate the ex-penditure need (i.e., the amount of spending nec-essary to supply a given level of services) of localdecision-making units. For illustrative purposes,this modi®ed technique will be applied to NewYork State school districts in 1990±91 to deter-mine the potential dollar savings from eliminatingthe relative budgetary slack of cost ine�cientschool districts.

The rest of the paper is organized as follows.The next section presents the standard model ofpublic sector service provision that will serve as theconceptual basis for the frontier analysis. In thethird section, Farrell-type cost e�ciency measuresare developed for the public sector. This sectionhighlights the problems that may arise from failureto control environmental conditions and alsoprovides an index of relative expenditure need. Inthe fourth section, the methodology developed inthis paper is applied to New York State schooldistricts in school year 1990±91. The results indi-cate that the average school district spends $1.4million in excess of minimum cost, i.e., 9.5% ofapproved expenditure. In addition, the averagedistrict faces relative environmental cost of $1700per pupil, implying that the average district needsto spend $1700 per pupil more than districts withthe most favorable cost environment to providethe same level of services. The last section con-cludes with implications for future analyses andfor policy.

2. A model of public sector production and cost

A useful framework for estimating productionand cost of public sector service provision is pro-vided by the simple but powerful model of Brad-ford et al. [16]. Local government authorities are

606 J. Ruggiero / European Journal of Operational Research 119 (1999) 605±612

assumed to employ capital (K) and labor (L) in-puts to provide services (S) that are of interest tovoters. For example, school districts use teachersand other labor inputs, computers, books, etc., toprovide lessons in mathematics, reading, and othercontent areas. These instructional lessons aretransformed into standardized test scores that areof interest to the voters according to the followingfunction:

S � f �K; L; Z�; �1�where Z is a vector of environmental factors whichin¯uences the conversion of inputs into the servicesthat are provided. For example, for a given level ofinputs, students with limited English pro®ciencywill not do as well on a reading test. Note that thisrepresentations is consistent with the educationalproduction literature. For a further discussion, seeHanushek [17,18].

As shown by Bradford et al. [16], the followingpublic sector cost function results:

TC � C�SjW ; Z�: �2�This function relates the minimum level of ex-penditure necessary to provide a given level ofservices holding exogenous resource prices andenvironmental factors constant. Further, thisfunction provides the basis to analyze di�erencesin cost that arise when services are e�cientlyprovided. This equation reveals that the minimumcost of providing a given level of services will varywith factor prices and/or environmental factors.Eq. (2) can therefore be used to index the e�ectthat the environment has on cost for each decision-making unit. This index is de®ned as

qi �C�SijWi ; Zi�C�SijWj; Zj� ; �3�

where i references the decision-making unit underanalysis and j references a district with the mostfavorable environment. The numerator indicatesthe minimum cost necessary to provide servicesgiven resource prices and environmental factors.The denominator indicates the level of cost nec-essary to provide the same level of services giventhe most favorable cost environment. Conse-quently, qi P 1 measures the relative impact that

exogenous factors have on the cost of serviceprovision.

The cost function (2) was derived under thebehavioral assumption that the decision-makingunit minimizes cost. Most cost analyses proceed toestimate costs by specifying a function a priori,controlling for outcomes, and proxying cost withexpenditure. However, this formulation is notconsistent with Niskanen's bureaucracy model. Ifbureaucracy leads to budgetary slack then costminimization is an inappropriate assumption. Thebureaucratic model postulates that decision mak-ers have an incentive to increase expenditure be-yond the minimum cost level in order to obtainpecuniary bene®ts, leading to budgetary slack. Toaccommodate this distinction, the following de®-nition is introduced:

De®nition. A local government is said to be coste�cient if the observed level of expenditure isequal to the total minimum cost of providing theobserved level of services, given resource pricesand environmental conditions.

Cost e�ciency is then measured as

c�W ; Z� � TCE; �4�

where c indexes the level of cost e�ciency and E isthe observed level of expenditure. If local govern-ments are cost e�cient, then expenditure re¯ectsthe minimum cost of providing services and c� 1.Since minimum cost is a function of exogenousresource prices W and environment Z, it is clearthat cost e�ciency is also a function of these ex-ogenous variables.

Rearranging Eq. (4) yields the insight that ob-served expenditure is composed of the minimumcost of service provision and a cost e�ciency ad-justment factor:

E � TCc� 1

cC�SjW ; Z�: �5�

We would expect ceteris paribus a positive rela-tionship between the level of services provided andspending. In addition, holding service levels con-stant, harsher cost environments re¯ected byhigher resource prices and/or harsher environ-

J. Ruggiero / European Journal of Operational Research 119 (1999) 605±612 607

mental factors would lead to higher spending. Thisframework relaxes the assumption of cost mini-mization and allows expenditure to vary not onlywith cost factors but also with the level of e�-ciency.

3. Nonparametric estimation of public sector costs

Assume that each decision-making unit (DMU)is observed spending $E. Using the expenditurefunction (5), the empirical expenditure set can bede®ned as

E�W ; Z� � f�S;E�jE P C�SjW ; Z�g: �6�Eq. (6) provides a formal relationship between thelevel of services provided and the money spent toprovide those services. Piecewise-linear minimumcost curves are shown in Fig. 1 where it is assumedthat six municipalities H through M provide oneservice facing one of two cost environments ±municipalities L, M, and I face a more favorablecost environment (due to di�erential resourceprices and/or better environmental conditions)than municipalities H, J, and K.

The development of each cost frontier parallelsthe methodology presented in Grosskopf andYaisawarng [9] where e�ciency is analyzed in cost/

output space. In this case, however, multiplefrontiers arise out of di�erential resource pricesand/or environmental conditions faced by munic-ipalities. The distinction between the frontiers canbe extended to handle multiple resource prices andvariations in environmental conditions. In e�ect,the description of the cost correspondence devel-oped above allows for the existence of multiplecost frontiers arising from di�erences in exogenousfactors.

The cost e�ciency ci of each local government ican be measured with the following linear pro-gram:

ci � min ki s:t: �kiEi; Si� 2 E�Wi ; Zi�

� C�SijWi ; Zi�Ei

: �7�

The cost e�ciency of a local government is mea-sured by the ratio of the cost minimizing levelnecessary to provide a given level of services to theobserved expenditures used in providing theseservices. Fig. 1 highlights that the minimum levelof expenditure necessary to provide a given level ofservices may di�er for di�erent local governments.This is illustrated with local government H, whichspends EH to provide SH . Failure to control forexogenous factors leads to distorted e�ciencymeasurement: the cost e�ciency of H is CH /EH andnot CI /EH . Note that SH could not have been ob-tained by government H at cost CI due to therelatively harsh cost environment that it faces.Based on Eq. (3) an index for local government Hthat captures the e�ect that the exogenous envi-ronment has on the frontier would be

qH �CH

CI: �8�

In order to estimate qi for a given local gov-ernment i, it is necessary to arbitrarily specify anenvironment to facilitate comparison. A usefulcomparison would be the most favorable envi-ronment, determined from the outer envelope ofall expenditure service pairs. This entails an addi-tional linear program for each district; unlikeprogram (7), however, the exogenous variable re-strictions are removed. Thus, the linear program-ming models are solved with and without theFig. 1. Cost frontiers and cost e�ciency in the public sector.


environmental variables. Removing the exogenousvariable constraint leads to the existence of anunrestricted ``best-practice'' frontier and the fol-lowing index of best practice:

cUi � min kU

i s:t: �kUi Ei; Si� 2 EU : �9�

Note that Eq. (9) is a linear program similar to theoriginal DEA model. This index captures not onlyine�ciency but also exogenous cost e�ects. Forlocal government H shown in Fig. 1, cU

H � CI=EH .Clearly, cU

i 6 1 with cUi � 1 only if i can provide Si

at the minimum possible cost given the most fa-vorable cost environment. If local government i iscost ine�cient and/or faces a relatively harsh en-vironment, this index will be less than one. Toindex the e�ect the environment has on costs it isnecessary to control for cost ine�ciency. FromEqs. (3), (7) and (9), a measure qi of the e�ect thatthe environment has on the costs of providing agiven level of services is given by

qi �ci

cUi: �10�

For illustrative purposes, the measures introducedin this paper are estimated for the provision ofeducational services.

4. Empirical analysis of school districts

Indices for cost e�ciency and the e�ect that thecost environment has on costs were estimated for584 school districts for the school year 1990±91using the methodology developed above. Thissection describes the data and results. Table 1provides descriptive statistics of the variables usedin the analysis. District level approved operatingexpenses per pupil are used as the observed ex-penditure in this study. This measure includesteacher salaries, other instructional expenditure,and all other expenditure related to operation andmaintenance and excludes transportation expens-es. The cost of transportation will vary by type ofregion and was excluded to facilitate comparisonof districts among regions. In addition, most debtservice is excluded from approved operating ex-penses. The average level of expenditure per pupilfor the sample included was about $5836, which

translates into an average district budget of ap-proximately $14.6 million.

The measure of expenditure used above doesnot control for resource price di�erentials thatmay exist among school districts. To control forthe major resource price di�erential, a teachersalary index was estimated. This index was esti-mated adjusting for di�erences in teacher experi-ence, education, and certi®cation using multipleregression analysis. For a further discussion, seeDuncombe et al. [19].

While this index controls for di�erences thatmay exist in the price of labor, it is important alsoto control for di�erences in the price of capitalequipment and facilities that are used in educa-tional production. For this reason, a measure ofthe annual cost of capital was constructed frominformation on bond ratings. For a further dis-cussion see Duncombe et al. [19]. In addition, tocontrol for environmental e�ects, two socio-eco-nomic measures were used: the percentage of stu-dents with limited English pro®ciency and thepercentage of students that are minority. Theseexogenous variables re¯ect di�erences in cost thatmay result from the transformation of output intoservices.

A more fundamental data issue involves themeasurement of outcomes. Following most stud-ies, this paper uses standardized test scores asmeasures of the ®nal outcomes that are of interestto voters. In particular, service outcomes aremeasured by the school district's average testscores in two content areas, reading and mathe-matics. These tests are administered for the PupilEvaluation Program (PEP) and all sixth gradestudents are required to take them. Average scoresare scaled between 0 and 100. As Table 1 indicates,average scores for mathematics are lower than forreading, a common ®nding in educational pro-duction analyses.

The mathematical programming models (7) and(9) were run for each of the 584 school districts inthe sample to estimate cost ine�ciency and envi-ronmental e�ects on the costs of providing a givenlevel of services. As discussed above, the primaryadvantage of analyzing e�ciency from the costside rather than from the production side is theinterpretation of results. In addition, the method-


ology developed in this paper allows a dollar es-timate of ine�ciency and environmental e�ects.Results from the nonparametric analysis are re-ported in Tables 1 and 2.

As shown in Table 1, school districts are onaverage 86% e�cient (i.e., 14% ine�cient). Fur-ther, the average district needs to spend 52% morethan districts facing the most favorable environ-ment. This is suggestive of the potential distortionsthat are introduced when exogenous factors (in-cluding resource prices and environmental vari-ables) are not properly controlled. Table 2 showsthat cost e�cient districts (36% of the sample)spend $6197 per pupil. By de®nition, the averageminimal cost for these e�cient districts is $6197,i.e., there exists no relative budgetary slack. Theresults also indicate that 374 of the school districtsare relatively cost ine�cient. Half of the districts

spend on average approximately $1800 per pupil inexcess of the cost minimizing level, which amountsto nearly one-third of total spending. This be-comes alarming because highly ine�cient districtsrepresent approximately one third of the sample.

In addition, Table 2 presents results by envi-ronmental harshness. In the analysis, the exoge-nous cost factors were weighted nonparametricallyand combined into an overall index of environ-mental harshness. Districts were ordered accord-ing to this measure and averages were computedby quartile. The results suggest that the quartilecontaining districts with the most favorable envi-ronment are the most ine�cient, on averagespending $1458 per pupil in excess of cost mini-mization. The results further indicate that mini-mum cost is a function of the size of the schooldistrict. The quartile of schools facing the harshestenvironment have more than twice as many stu-dents on average as districts in the ®rst and secondquartiles. Districts facing the harshest environ-ment spend $8235 per pupil (approximately 65%more than all other districts); however, as shown,this results primarily from exogenous cost factorsand not cost ine�ciency. An interesting patternemerges with respect to budgetary slack per pupil.Districts in the quartile with the most favorableenvironment have the highest budgetary slack perpupil ($1458). As the environment becomesharsher, the amount of excessive spending de-creases. This implies that the amount of budgetaryslack that exists is a function of external ®scalpressure: districts facing relatively favorable envi-ronments may be less concerned with excessivespending. Somewhat surprisingly, however, theaverage budgetary slack per pupil increases fromthe third quartile to the harshest quartile.

5. Conclusions and policy implications

The objective of this paper has been to develop¯exible nonparametric methodology for estimatingcost and cost e�ciency in the provision of publicsector services. In order to do so, a distinction wasmade between the cost needed and the expenditureused to provide a given level of services. This papersuggests that the distinction between cost and ex-

Table 1

Descriptive statistics/e�ciency results (N� 584)

Variable Mean Standard

deviation

Expenditure ($/pupil) 5837 2037

Outcomes

Math (percent) 66.2 6.0

Reading (percent) 82.1 3.4

Resource prices

Labor 100.0 13.2

Capital 100.0 9.8

Environmental variables

Minority students (percent) 9.8 15.0

Limited english students (percent) 1.0 1.97

E�ciency results

Cost e�ciency 0.86 0.15

Combined e�ciency/exogenous e�ect 0.61 0.15

Exogenous cost index 1.52 0.55

There were 691 districts in New York State in 1990±91. Due to

missing observations, the sample was limited to 584 observa-

tions for which all data were available. The remaining sample

appears representative of the major regions in New York State.

The data were collected by the New York State Department of

Education. The data source for the outcome measures is the

``Comprehensive Assessment Report'' (CAR), available on

magnetic tape. The environmental factors used in this paper are

available from either the CAR or the ``Basic Education Data

System'' (BEDS).


penditure results from ine�ciency (resulting fromboth technical and allocative ine�ciency) in publicservice provision. The structure placed on the costcorrespondence is minimal and allows measure-ment of not only cost e�ciency, but also the im-pact of environmental factors on costs.Application of the methodology to school districtsin New York State reveals potentially large savingsfrom reducing ine�cient service provision. Evenwith a large amount of cost e�ciency, it is esti-mated that the environment has a signi®cant im-pact on the costs of service provision. Withexisting budget crises at federal and state levels,these results take on renewed importance at boththe state and local level. The indices created can beused to modify grant formulas to re¯ect di�eren-tial costs and e�ciency.

While this paper extends Data EnvelopmentAnalysis to measure cost e�ciency in the publicsector, more work needs to be done. This tech-nique o�ers a way to test bureaucratic modelsinvolving ine�ciency, and may provide a meansto relate bureaucratic theory to empirical ana-lyses of cost ine�ciency. Future analyses canalso use the method developed in this paper toidentify the extent of bias introduced into esti-mated cost equations by the assumption of cost

minimization. Future studies should seek ways ofadjusting grants-in-aid to discourage ine�cientbehavior.

References

[1] M. Darrough, J. Heineke, Law enforcement agencies as

multi-product ®rms: An econometric investigation of

production costs, Public Finance 34 (1979) 176±195.

[2] W. Duncombe, J. Yinger, An analysis of returns to scale in

public production, with an application to ®re protection,

Journal of Public Economics 52 (1993) 49±72.

[3] W.A. Niskanen, Bureaucracy and Representative Govern-

ment, Aldine-Atherton, Chicago, 1971.

[4] W.A. Niskanen, Bureaucrats and politicians, The Journal

of Law and Economics 18 (1975) 617±643.

[5] J.L. Migu�e, G. B�elanger, Toward a general theory of

managerial discretion, Public Choice 17 (1974) 27±43.

[6] G.P. Wycko�, Bureaucracy and the publicness of local

public goods, Public Choice 56 (1988) 271±284.

[7] M.J. Farrell, The measurement of productive e�ciency,

Journal of the Royal Statistical Society, Series A, General,

120 (1957) 253±281.

[8] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the

e�ciency of decision making units, European Journal of

Operational Research 2 (1978) 429±444.

[9] S. Grosskopf, S. Yaisawarng, Economies of scope in the

provision of local public services, National Tax Journal

XLIII (March) (1990) 61±74.

Table 2

Estimates of average 1990±91 school district costs and expenditures (N� 584) a

Classi®cation Number of

districts

Average

enrollment

Expenditure Cost Exogenous

cost c

Budgetary

slack

Cost e�ciency class

E�cient districts 210 3193 6197 6197 2638 0

Ine�cient districts

Low ine�ciency 187 2410 5822 5168 1909 654

High ine�ciency 187 1654 5440 3630 503 1810

Environmental class b

1 Most favorable 145 1457 4937 3479 72 1458

2 147 1677 4726 3920 640 806

3 147 3336 5462 5151 1885 311

4 Harshest 145 3326 8235 7648 4300 587

All districts 584 2450 5835 5046 1721 789

a Expenditure, cost, environmental cost, and budgetary slack are measured in dollars per pupil.b The environmental classes are de®ned into quartiles based on the ranking determined from environmental index (where quartile 1

consists of those districts with the most favorable environment).c Exogenous cost is measured for each district relative to the most favorable environment as discussed above.

Source: Computed by author.


[10] A. Bessent, E. Bessent, Determining the comparative

e�ciency of schools through data envelopment analysis,

Educational Administration Quarterly (1980) 57±75.

[11] A. Bessent, W. Bessent, J. Kennington, B. Reagan, An

application of mathematical programming to assess pro-

ductivity in the Houston independent school district,

Management Science 28 (1982) 1335±1366.

[12] R. F�are, S. Grosskopf, W. Weber, Measuring school

district performance, Public Finance Quarterly 17 (1989)

409±428.

[13] R.D. Banker, R. Morey, E�ciency analysis for exoge-

nously ®xed inputs and outputs, Operations Research 34

(1986) 513±521.

[14] J. Ruggiero, On the measurement of technical e�ciency in

the public sector, European Journal of Operational

Research 90 (1996) 553±565.

[15] J. Ruggiero, Measuring technical ine�ciency in the public

sector: An analysis of educational production, Review of

Economics and Statistics 78 (1996) 499±509.

[16] D. Bradford, R. Malt, W. Oates, The rising cost of local

public services, some evidences and re¯ections, National

Tax Journal 22 (1969) 185±202.

[17] E. Hanushek, Conceptual and empirical issues in the

estimation of educational production functions, The

Journal of Human Resources 14 (1979) 351±388.

[18] E. Hanushek, The economics of schooling: Production and

e�ciency in public schools, Journal of Economic Litera-

ture 24 (1986) 1141±1177.

[19] W. Duncombe, J. Miner, J. Ruggiero, Scale economies

and technical e�ciency in New York public schools,

Metropolitan Studies Program Series Paper No. 163, The

Maxwell School, Syracuse University, Syracuse, NY, 1993.


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Nonparametric analysis of educational costs