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Productivity estimation and the size-efficiency relationshipin English and Welsh police forces

An application of data envelopment analysis and multiplediscriminant analysis

Leigh Drake*, Richard Simper

Department of Economics, Loughborough University, Loughborough, LE11 3TU, England

Abstract

This article utilizes data envelopment analysis (DEA) to estimate the productivity of the Englishand Welsh police forces and to determine whether there are categorical scale effects in policingusing multiple discriminant analysis (MDA). The article demonstrates that by using DEAefficiency results it is possible to make inferences about the optimal size and structure of theEnglish and Welsh police forces. In terms of individual force efficiency, the DEA results suggestthat the Surrey police force appears to be 38% less efficient than its efficient reference set and thatonly three police forces (Cleveland, Dorset, and Leicestershire) are consistently efficient. © 2000Elsevier Science Inc. All rights reserved.

1. Introduction

During the Thatcher/Major Conservative governments, state services were restructured sothat they utilized business techniques in creating “value for money” (see Her Majesty’sInspector of Constabulary, 1995). The reform of the police service instigated by the Con-servative government was in response to the steady increase in crime since 1979, thedisproportionate increase in the fear of crime, and the increasing cost of the police servicein real terms, from over £1 billion in 1979–1980 to nearly £7 billion in 1996–1997 (source:Home Office). These factors led to an inspection and review of the police under the

* Corresponding author. Tel.:144-0-1509-222709; fax:144-0-1509-2233910.E-mail addresses:[email protected] (L. Drake), [email protected] (R. Simper)

International Review of Law and Economics 20 (2000) 53–73

0144-8188/00/$ – see front matter © 2000 Elsevier Science Inc. All rights reserved.PII: S0144-8188(00)00021-1

Conservative government, which included agencies such as Her Majesty’s Inspector ofConstabulary (HMIC) and the Audit Commission, and the introduction of various publiccharters, including the Citizen’s Charter and the Victim’s Charter (for a discussion, seeStephens, 1994; Sullivan, 1998).

This comprehensive review resulted in various publications concerned with efficiency inthe service and included Audit Commission (1990), Home Office (1993), Police ResearchGroup (1993), and the report by Sheehy (1993), which led to recommendations included inThe Police and Magistrates’ Courts’ Act of 1994. One of the main recommendations of theSheehy report was to change the nature of police management from a public to a business-oriented organization and to introduce efficiency targets that were coordinated with localpolice authorities. Sullivan (1998) argues that the police reform of the 1990s led to themanagerialism of the service. That is, “managerialism referred to the belief that all stateservices do better when reconceived and restructured in terms of the business community’svalues of efficiency and effectiveness” (p. 307). The government’s concept of “value formoney” in the police service has led us to posit a socioeconomic model of the modern policeforce. That is, we introduce a methodology that is based on the reorganization of the policeforce that was begun in the Thatcher/Major government’s reforms but that is set within theconcept of the economics of the firm.

The new Labour government has carried on this agenda of ensuring efficiency in thepolice force (Home Office Inspectorate of Constabulary, 1998). The report reiterated theprevious Conservative government’s efficiency drive in the police service with the HMICarguing that, “police managers need to work harder to ensure that VFM [value for money]is achieved, for competitive pressure has to be created internally. The costing of activity withsubsequent measurement and comparison of performance provide the means by which suchencouragement is given” (p. 8, paragraph 10).

This article utilizes data envelopment analysis (DEA) to estimate the relative efficiency ofthe English and Welsh police forces. To determine whether there are categorical size effectsin policing, we also utilize multiple discriminant analysis (MDA). To the authors’ knowl-edge, this is the first article to examine the relative efficiency of the English and Welsh policeforces. The article is organized as follows. In Section 2, we discuss the methodology utilizedin our DEA analysis of police forces and provide details on the variables and data sources.Section 3 presents the results of the DEA efficiency rankings together with a discussion ofhow certain forces have fared over the 1992–1993 to 1996–1997 sample period. In Section4, we undertake MDA tests to discover whether there are categorical size effects that candiscriminate between police forces in the sense that one size of force is likely to be moreefficient than another. We conclude the article with Section 5.

2. Methodology and data

The term DEA was coined by Charnes et al. (1978) and is a linear-programming techniquefor constructing extremal, piecewise frontiers that were originally developed by Farrell(1957). The constructed relative efficiency frontiers are nonstatistical or nonparametric in the

54 L. Drake, R. Simper / International Review of Law and Economics 20 (2000) 53–73

sense that they are constructed through the envelopment of the decision-making units(DMUs) with the “best practice” DMUs forming the nonparametric frontier.

DEA is a leading analytical technique for measuring relative efficiency and has beenwidely used by both academics and practitioners in evaluating the efficiency of DMUs withinan organization or industry in terms of converting resources/inputs into outputs. The tech-nique was originally developed to determine performance measures in non-profit-makingorganizations where the usual monetary criteria of return on assets/capital, for example, werenot appropriate. Hence, DEA has been widely used for relative performance measurement inpublic sector services such as education (Chalos & Cherian, 1995; Sarrico et al., 1997),health services (see SalinasJime´nez & Smith, 1995, for an example assessing primary-careperformance in the English Family Health Service Authorities, and Thanassoulis et al., 1996,for an example using data concerned with prenatal care in England), and criminal courts (seePedrajaChaparro & SalinasJime´nez, 1996, for an example of Spanish court efficiency).Although DEA is ideally suited to the examination of the relative efficiency of law enforce-ment units, to our knowledge, this is the first study to apply this technique to the analysis ofrelative police force efficiency.

A particular advantage of nonparametric techniques such as DEA, relative to statistical orparametric techniques such as stochastic frontier analysis (Drake & Weyman-Jones, 1996;Ferrier & Lovell, 1990), is that the latter must assume a particular functional form thatcharacterizes the relevant economic production function or cost function. Hence, any result-ant efficiency scores will be partially dependent on how accurately the chosen functionalform represents the true production relationship (i.e., the relationship between inputs/resources and outputs). As DEA is nonparametric and envelops the input/output data of theDMUs under consideration, the derived efficiency results do not suffer from this problem offunctional form dependency.

The use of DEA is not confined to public sector enterprises, however. DEA can be appliedto any organization/industry in which a reasonably homogenous set of DMUs use the sameset of resources, possibly in different combinations, to produce an identifiable range ofoutputs or “deliverables,” again possibly in different combinations. In this context, DEA hasbeen applied to the analysis of individual building societies and banks within the U.K.financial sector (Drake & Weyman-Jones, 1992, 1996; Drake, 1997), to the relative effi-ciency of hotels within a hotel chain (Johns et al., 1997), and to the analysis of the relativeefficiency of the individual bank branches of a U.K. clearing bank (Drake & Howcroft,1994).

2.1. Measuring relative efficiency using DEA

Within the methodological framework of DEA it is possible to decompose the relativeefficiency performance of DMUs into the categories initially suggested by Farrell (1957),and later elaborated on by Banker et al. (1984) and Fare et al. (1985). Farrell’s categories arebest illustrated, for the single-output/two-input case in the unit isoquant diagram (Fig. 1)where the unit isoquant (yy) shows the various combinations of the two inputs (x1, x2) thatcan be used to produce one unit of the single output (y). The firm at E is productively (oroverall) efficient in choosing the cost-minimizing production process given the relative input

55L. Drake, R. Simper / International Review of Law and Economics 20 (2000) 53–73

prices represented by the slope of WW’. A DMU at Q is allocatively inefficient in choosingan inappropriate input mix, while a DMU at R is both allocatively inefficient (in the ratioOP/OQ) and technically inefficient (in the ratio OQ/OR) because it requires an excessiveamount of both inputs, x, compared with a firm at Q producing the same level of output, y.

The use of the unit isoquant implies the assumption of constant returns to scale. Howevera firm using more of both inputs than the combination represented by Q may experienceeither increasing or decreasing returns to scale so that, in general, the technical efficiencyratio OQ/OR may be further decomposed into scale efficiency, OQ/OS, and pure technicalefficiency, OS/OR, with point Q in Fig. 1 representing the case of constant returns to scale.The former arises because the firm is at an input-output combination that differs from theequivalent constant returns-to-scale situation. Only the latter pure technical efficiency rep-resents the failure of the firm to extract the maximum output from its adopted input levelsand, hence, may be thought of as measuring the unproductive use of resources. In summary,

productive efficiency5 allocative efficiency3 scale efficiency3 pure technical efficiency

OP/OR5 @OP/OQ# 3 @OQ/OS# 3 @OS/OR#. (1)

Due to the difficulties in accurately measuring all input prices in public sector services suchas the police force, this article does not consider allocative efficiency. Hence, concentratingon overall technical efficiency, Farrell (1957) suggested constructing, for each observedDMU, a pessimistic, piecewise, linear approximation to the isoquant, using activity analysisapplied to the observed sample of DMUs in the organization/industry in question. Thisproduces a relative rather than an absolute measure of efficiency because the DMUs on the

Fig. 1. Farrell efficiency.


piecewise, linear isoquant constructed from the boundary of the set of observations aredefined to be the efficient DMUs.

Subsequent developments have extended this mathematical linear-programming approach.If there are n DMUs in the industry, all the observed inputs and outputs are represented bythe n-column matrices X and Y. The input requirement set, or reference technology, can thenbe represented by the free disposal convex hull of the observations, i.e., the smallest convexset containing the observations consistent with the assumption that having less of an inputcannot increase output. We do this by choosing weighting vectors,l (one for each firm), toapply to the columns of X and Y to show that firm’s efficiency performance in the best light.

For each DMU in turn, using x and y to represent its particular observed inputs andoutputs, pure technical efficiency is calculated by solving the problem of finding the lowestmultiplicative factor,u, which must be applied to the firm’s use of inputs, x, to ensure thatit is still a member of the input requirements set or reference technology. That is, choose

$u, l% to min u, such that ux $ l9X

li $ 0, (li 5 1,y # l9Yi 5 1, . . . , n. (2)

To determine scale efficiency, we solve the technical efficiency problem (2) without theconstraint that the input requirements set be convex; i.e., we drop the constraintSli 5 1.This permits scaled-up or down-input combinations to be part of the production possibilityset of the DMUs. Fig. 2 illustrates this for the case of a single input and a single output.

In Fig. 2, the production possibility set under constant returns to scale is the region to theright of the ray, OC, through the leftmost input-output observation. Any scaled-up or

Fig. 2. SEandPTE.


scaled-down versions of the observations are also in the production possibility set under thisassumption of constant returns to scale. Imposing the convexity constraintSli 5 1 ensuresthat the production possibility set is the area to the right of the piecewise linear frontier VV’,which does not assume constant returns to scale, but allows for the possibility of increasingreturns to scale at low output levels and decreasing returns at high output levels. Theresulting overall technical and pure technical efficiency ratios, AQ/AR and AS/AR, areillustrated for one of the observations. Scale efficiency is the ratio of the two results.

In the case of program (2), the efficiency ratios with and without the convexity constraintmay be labeledup anduo, and scale efficiencyus is thenuo/up. In the subsequent results, werefer to overall technical efficiency asOE, pure technical efficiency asPTE, and scaleefficiency asSE. As explained above, it follows that:

OE 5 PTE3 SE, andSE5 OE/PTE

Using DEA to derive a measure ofOE, but also to decompose the results into the componentsof PTE andSE, allows us to examine not only the effectiveness of the use of resources inpolicing (PTE) but also to gain an insight into the relationship between efficiency and the sizeof police forces (SE). All economic organizations that use resources to produce outputs areprone to output ranges that display, first, increasing, then constant, and finally decreasingreturns to scale. Obtaining this type of information about English and Welsh police forcesmay enable us to shed some light on the optimal size and structure of police forces from theperspective of economic efficiency, although it is recognized that there will be many otherfactors that inevitably impinge on the size and structure of forces.

2.2. The identification of inputs and outputs in policing

The measurement of the police force in its actions and activities is complex because itinvolves many accountable and nonaccountable services. For example, Redshaw et al. (1997)argue that policing consists of the “prevention and detection of crime and the maintenanceof public order, but it also embraces a social service role such as welfare and the preventionof crime” (p. 284). Byrne et al. (1996) differentiate between two main police functions:traditional law enforcement, which includes the prevention and repression of crime; andpublic service duties, including the regulation of noncriminal activities. However, thecomplications of measurement rest not with the inputs of the police but with their outputs.That is, the former can be grouped as if the service was a firm and, therefore, include laborand various capital costs.

In our model, we break down the inputs of each police force into four distinct categories,as outlined in the Chartered Institute of Public Finance and Accountancy Police ForceStatistics. The first input in our estimation methodology is employment costs. This is the totalcost of the employed staff of each police force, which includes all police officer ranks, trafficwardens, civilian staff, and other staff development expenses that occur on a daily basis. Wehave included civilian staff in the summation of police staff costs because the demarcationbetween the police function and the civilian involvement in policing has become ever moreblurred. In a recent report by Her Majesty’s Inspectorate for the Constabulary (1998), forexample, the employment of civilian staff was thought to lead to an enhancement of


“efficiency and effectiveness,” and the report revealed that civilian staff represented approx-imately 30% of total staff employed in the service in 1995–1996. Furthermore, the reportargues that “the classification of roles into police/civilian was in itself a redundant concept.Instead, it would be more appropriate to shift the focus to the actual cost of delivering aservice function. . .” (HMIC, 1998, p. 55, paragraph 2.48). We believe, therefore, that a totallabor cost variable should be utilized as an input, as many of the functions once whollyundertaken by the police are now beginning to be undertaken by civilian staff.

The second input is premises-related expenses, which is the sum of all premises expensesand covers the general daily running costs including repair and maintenance. The third inputis transport-related expenses, which includes the running costs and repairs of police vehicles.Finally, the fourth input is capital and other costs. This latter variable includes capital-financing costs and all those costs associated with equipment bought for internal use such asinformation technology, communications, and furniture, and also includes contracted-in andcontracted-out services. This variable has been noted as one that could lead to greaterpressure on future capital expenditure due to the need to update information technologyfacilities so that police forces have the latest equipment. In total, the average annualpercapita expenditure of all forces in England and Wales on capital equipment has increasedfrom £67 in 1987–1988 to £123 in 1995–1996 (HMIC, 1998).

A major problem inherent in measuring the efficiency of the police service is how toquantify the role of the police in society. There have been many different measurable outputsthat have been advanced as useful in compiling efficiency rankings. The first relate tosurveys, where some authors have argued that surveys on the evaluation of police perfor-mance “provide more easily quantified measures that dominate HMIC requirements andthat. . .can lead to improvements in policing” (Redshaw et al., 1997, p. 284).

It also has been argued, however, that it would be incorrect to survey the public aboutpolice service actions as this would introduce bias when using qualitative judgments on howwell the police service operates. For example, Shaw & Williamson (1972) argued that youngpeople and the working class rated the service lower than did older people and the middleclasses. Recently, Waddington & Braddock (1991) found that white and Asian youths hadmixed views of how the police operate, whether as “guardians” or “bullies,” but that blackyouths tended “to favour the ‘bullies’ perception” (p. 39). The authors concluded that “whatdistinguishes the races is not the absence of some whites and Asians who regard the policeas ‘bullies,’ but the virtual absence amongst their black counterparts of any conception ofpolice as ‘guardians’” (p. 39). These problems and socioeconomic stereotypes imply thatsurveys could lead to a misinterpretation by the public of police functions.

It is for the above reasons that in addressing the issue of the use of survey data as possibleperformance indicators of police functions, the HMIC (1998) report “What Price Policing?”concluded that “surveys are an imperfect measure and are affected by sample size, surveymethodology and the nature of the population targeted” (p. 85, paragraph 2.167). Skogan(1996) also argued that local surveys were fraught with difficulties because there areinter-area differences. In a Greater Manchester police survey across 13 districts, for example,Skogan found “that the percentage of residents rating ‘burglary and theft’ the ‘single mostserious problem’ in their area ranged from 2% to 22%. The range for ‘street crime’ was from


less than 1% to 22%, car crime 13% to 28%, and ‘young people hanging/drilling around’from 5% to 24%” (p. 427).

In addition, a study by Redshaw et al. (1997) surveyed police officers, neighborhoodwatch coordinators, and members of the public and found that when asked to rank 37 jobsthat the police are asked to perform, they responded by ranking the top three as 1) respondimmediately to emergencies, 2) detect and arrest offenders, and 3) investigate crime, all threeof which relate to the classification of a reactive police force variable. However, the authorsnote that “even where activities appear to have no immediate ‘crime control’ payoffs, thereis widespread acceptance that the British tradition of local, community-based, service-oriented, policing needs to be preserved” (p. 300). It is hoped that in future research we willbe able to include in our model a variable that can proxy this all-important function ofpolicing.1 For the above reasons, and because of the lack of quantifiable data on other policefunctions, we use traditional outputs associated with response/reactive policing.

The response/reactive methodology of measuring policing can be found in a number ofstudies, including Todd & Ramanathan (1994) and Byrne et al. (1996), who argue that eventhough half of the police’s community work cannot be modeled, a production function canstill be estimated. They break down police activities into crime prevention “where crime iscontemplated but not committed,” and crime repression, where the “crime has occurred,” andthey use an argument from Schmidt & Witte (1984) that any criminal is likely to assess theprobability of getting caught after committing a crime. Todd & Ramanathan (1994) also statethat outputs should be a measure of activity, such as the number of arrests made, and that“employee allocations are explained marginally better by background demand for services,. . .” (p. 131).2 Hence, the probability of arrest is linked to the number of arrests in a policeforce and, in particular, to the number of convictions. For this reason we feel that the clear-uprate used in this study is a good proxy for the preventive methods used by the police.

We also note the criticisms of Walker (1992) in using clear-up rates as an output variableand have, therefore, split our sample police forces into Metropolitan English, Welsh, and theMetropolitan and London police forces, and into four size groupings.3 This will allowcomparisons of forces that are closely linked by geographic circumstance and economic size,

1 Jackson (1992) found, using data from the United States, that a sizeable proportion of the cost of policingcould be attributed to other factors, such as the decline in the demographic and socioeconomic bases of manycities. Most importantly, these factors, even when held constant, still led to increases in fiscal expenditure as the“police are called upon to manage the social threats that rise from the ashes of social decay” (p. 202).

2 O’Brien (1996) has argued that there is some level of police discretion in reporting or recording criminalincidences. Hence, instead of using recorded crime statistics for a variety of crimes, it would be better to considerserious crimes such as murder, where there is evidence (a body) and with which, therefore, the assessment ofpolice productivity can be better assessed. This methodology is not used in our estimation because it wouldexclude a considerable number of other crimes, which constitute a higher proportion of crime in the UnitedKingdom than in the United States, as represented by the study of O’Brien.

3 Walker (1992) believes that clear-up rates can be misleading and forcibly argues that they “should not becommended as performance indicators by which to judge the policing service delivered to the public. Compar-isons between forces in these rates are invidious, and they may lead to inefficient and even possibly corruptpractices” (p. 305).


and will mitigate any possible bias in our analysis of police forces and the results presentedin Section 4.

The second output variable is the total number of traffic offenses that the police andcontracted civilian staff (such as traffic wardens) deal with in a year, which includesprosecutions, the number of written warnings, and fixed-penalty fines. This is an importantvariable as it measures the effect on policing of the 6% increase in registered vehicles (from21.6 million in 1988 to 22.9 million in 1995–1996) and the associated increased trafficproblems encountered by the police. Furthermore, in line with the response/reactive meth-odology it would be expected that increases in the number of recorded traffic offenses would,ceteris paribus, tend to reduce theper capitanumber of traffic offenses.

In recent years, the government has implemented a strict drunk-driving campaign, whichcan take up police time with respect to performing breathalyzer test on drivers. In fact, therehas been a 76% increase in breathalyzer tests since 1988, and the 781,100 tests carried outby police in 1996–1997 is the largest number of tests since breathalyzer tests were intro-duced in 1967 (source: Home Office). We would expect that, as more people have breatha-lyzer tests administered to them, serious road accidents would be likely to drop, therebyfreeing up more police time for other activities. As mentioned above, we would also expectthat increased administration of breathalyzer tests would act as a deterrent to drunk drivingand, hence, should,ceteris paribus, ultimately reduce the level ofper capitadrunk-drivingoffenses. Following the methodology of Byrne et al. (1996), this action can be classified asa reactive approach to reducing car accidents, and so, the total number of breathalyzer testsconstitutes our final output variable. The next section discusses the results from the DEA andMDA using the methodology outlined above.

3. DEA results

The DEA results forOE, PTE, andSEfor the English and Welsh police forces are detailedin Tables 1, 2, and 3. The corresponding results for the London and Metropolitan policeforces are given in Tables 4, 5, and 6. It is important to note, however, that all the efficiencyscores were derived by contrasting each police force with all its peers, although we electedto summarize the results separately for the English, Welsh, and Metropolitan police forces.As can be seen from the tables, we produce DEA relative efficiency scores for each year from1992–1993 to 1996–1997 and also provide details of the mean efficiency scores for eachpolice force over these years.

With respect to the DEA efficiency results, the ratios discussed previously typicallyproduce efficiency scores of unity for efficient DMUs and less than unity for inefficient units.We choose to use a score of 100 for efficient units, however, as this permits a readyinterpretation of the degree of inefficiency in percentage terms. In Table 1, for example,Surrey appears to have the least efficient force with an averageOE score of 62.39 over the5-year period and with individual year scores ranging from 43.12 in 1992–1993 to 81.43 in1993–1994. The interpretation of these results is that, on average, the Surrey force is around38% less efficient than its efficient reference set forces (those forces that form the relevantfrontier and have scores of 100) in terms of translating its available resources into the


specified outputs. Furthermore, in 1992–1993 this relative degree of inefficiency was as highas 57%. If we analyze thePTEandSEresults for Surrey, we can gain some insight into thesource of this level of inefficiency. The meanPTE score is 69.34, for example, while themeanSEscore is 89.27. This suggests that the bulk of the inefficiency is not caused by afailure to operate under constant returns to scale. In fact, theSEscore in most years is over90, suggesting that the Surrey force is not too far removed from the constant returns region

Table 1DEA of English and Welsh police forces OE resultsa

1992–1993 1993–1994 1994–1995 1995–1996 1996–1997 Mean

Non-metropolitan EnglandAvon andSomerset

93.46 95.31 77.45 81.95 83.83 86.40

Bedfordshire 100 80.58 88.12 82.04 100 90.15Cambridgeshire 100 100 100 90.26 96.07 97.27Cheshire 78.81 76.35 69.59 83.88 81.18 77.96Cleveland 100 100 100 100 100 100Cumbria 88.97 86.7 79.43 95.76 85.71 87.31Derbyshire 85.94 100 95.61 96.85 100 95.68Devon andCornwall

89.65 78.89 91.9 96.65 93.53 90.12

Dorset 100 100 100 100 100 100Durham 64.08 84.59 82.05 77.07 60.85 73.73Essex 87.21 88.02 N/Ab N/A 81.15 85.46Gloucestershire 100 91.41 91.06 98.42 100 96.18Hampshire 87.54 89.65 86.38 100 100 92.71Hertfordshire 98.54 88.37 80.40 82.68 88.55 87.71Humberside 60.39 64.56 72.02 90.82 78.14 73.19Kent 74.75 80.61 79.39 77.92 75.81 77.70Lancashire 66.91 82.77 73.89 87.12 100 82.14Leicestershire 100 100 100 100 100 100Lincolnshire 77.48 100 N/A 100 100 94.37Norfolk 93.34 77.47 78.09 74.47 87.65 82.0Northamptonshire 100 100 100 100 98.98 99.80North Yorkshire 71.15 78.93 70.82 69.79 75.37 73.21Nottinghamshire 68.79 100 100 100 100 93.76Staffordshire 87.33 92.83 83.33 93.62 100 91.42Suffolk 79.08 80.33 72.90 100 85.79 83.62Surrey 43.12 81.43 62.15 65.36 59.89 62.39Sussex 94.69 93.93 79.14 87.69 96.01 90.29Thames Valley 100 100 100 87.15 84.98 94.43Warwickshire 76.17 77.17 83.28 97.49 93.94 85.61West Mercia 83.13 86.72 81.93 83.31 78.66 82.75Wiltshire 78.85 87.59 75.25 98.41 79.12 83.84

WalesDyfed-Powys 75.99 82.45 76.10 83.66 87.81 81.20Gwent 72.11 100 100 100 100 94.42North Wales 86.80 97.14 100 81.69 83.41 89.81South Wales 92.70 100 100 100 100 98.54

a Data for the Essex (1994–1995 and 1995–1996), and Lincolnshire (1994–1995) police forces were unavail-able.

b N/A, not available.


of operation. The meanPTE score of 69.34, however, suggests that the main factor behindthe observed low overall efficiency levels is a failure to utilize resources effectively.Specifically, the mean figure of 69.34 suggests that the Surrey force should be able to reducetheir use of resources by around 31%, on average, across the range of inputs withoutadversely affecting the capacity of the force to deliver the observed outputs. It is possible tomake this type of assertion because the DEA results tell us that, in comparison with the

Table 2DEA of English and Welsh police forces PTE resultsa

1992–1993 1993–1994 1994–1995 1995–1996 1996–1997 Mean


100 100 82.73 83.03 85.00 90.15

Bedfordshire 100 95.19 100 93.56 100 97.75Cambridgeshire 100 100 100 96.56 100 99.31Cheshire 81.61 77.11 69.88 84.72 84.49 79.56Cleveland 100 100 100 100 100 100Cumbria 98.59 97.15 94.60 100 100 98.07Derbyshire 94.58 100 97.58 100 100 98.43Devon andCornwall

96.78 87.54 100 98.98 94.59 95.58

Dorset 100 100 100 100 100 100Durham 88.94 84.59 82.25 78.41 78.07 82.45Essex 89.42 90.68 N/Ab N/A 81.48 87.19Gloucestershire 100 100 100 100 100 100Hampshire 97.24 96.77 100 100 100 98.80Hertfordshire 99.97 88.81 81.96 83.57 88.87 88.64Humberside 61.92 69.02 72.19 92.61 78.60 74.87Kent 79.45 80.66 83.78 89.00 94.03 85.38Lancashire 75.00 82.79 76.69 91.72 100 85.24Leicestershire 100 100 100 100 100 100Lincolnshire 88.72 100 N/A 100 100 97.18Norfolk 96.34 82.61 80.58 77.52 88.81 85.17Northamptonshire 100 100 100 100 99.06 99.81North Yorkshire 80.21 83.48 80.15 77.90 83.65 81.08Nottinghamshire 69.23 100 100 100 100 93.85Staffordshire 95.90 92.89 83.34 94.15 100 93.26Suffolk 90.16 92.66 91.48 100 100 94.86Surrey 61.86 83.46 66.77 68.92 65.68 69.34Sussex 100 100 86.09 91.17 96.23 94.70Thames Valley 100 100 100 87.99 91.61 95.92Warwickshire 94.40 98.71 100 100 100 98.62West Mercia 84.00 88.15 81.99 83.40 80.18 83.54Wiltshire 90.55 97.71 91.19 100 100 95.89

WalesDyfed-Powys 100 100 100 100 100 100Gwent 100 100 100 100 100 100North Wales 90.27 98.04 100 89.34 88.57 93.24South Wales 100 100 100 100 100 100




Surrey force, other forces with similar input and output configurations are using, on average,31% fewer inputs to deliver similar output levels.

If we turn now to Tables 4, 5 and 6, we see that the observedOE levels of the Metropolitanpolice force require a different interpretation. Table 4 indicates that the Metropolitan forcehas a meanOE score of only 57.52, which is the lowest of any force in the sample. With theexception of 1993–1994, the figures range from 30.56 in 1992–1993 to 62.06 in 1994–1995.

Table 3DEA of English and Welsh police forces SE resultsa

1992–1993 1993–1994 1994–1995 1995–1996 1996–1997 Mean


93.46 95.31 93.62 98.69 98.63 95.94

Bedfordshire 100 84.65 88.12 87.69 100 92.09Cambridgeshire 100 100 100 93.47 96.07 97.91Cheshire 96.57 99.01 99.59 99.01 96.08 98.051Cleveland 100 100 100 100 100 100Cumbria 90.24 89.24 83.96 95.76 85.71 88.98Derbyshire 90.86 100 97.98 96.85 100 97.14Devon andCornwall

92.63 90.12 91.90 97.64 98.88 94.24

Dorset 100 100 100 100 100 100Durham 72.05 100 99.76 98.29 77.94 89.61Essex 97.53 97.07 N/Ab N/A 99.59 98.06Gloucestershire 100 91.41 91.06 98.42 100 96.18Hampshire 90.02 92.64 86.38 100 100 93.81Hertfordshire 98.57 99.50 98.10 98.94 99.64 98.95Humberside 97.53 93.54 99.76 98.07 99.41 97.66Kent 94.08 99.94 94.76 87.55 80.62 91.39Lancashire 89.21 99.98 96.35 94.98 100 96.10Leicestershire 100 100 100 100 100 100Lincolnshire 87.33 100 N/A 100 100 96.83Norfolk 96.89 93.78 96.91 96.07 98.69 96.47Northamptonshire 100 100 100 100 99.92 99.98North Yorkshire 88.70 94.55 88.36 89.58 90.10 90.26Nottinghamshire 99.36 100 100 100 100 99.87Staffordshire 91.06 99.94 99.99 99.44 100 98.08Suffolk 87.71 86.69 79.69 100 85.79 87.98Surrey 69.71 97.57 93.08 94.83 91.181 89.27Sussex 94.69 93.93 91.93 96.18 99.771 95.30Thames Valley 100 100 100 99.05 92.77 98.36Warwickshire 80.689 78.18 83.28 97.49 93.94 86.72West Mercia 98.96 98.38 99.93 99.89 98.10 99.05Wiltshire 87.079 89.64 82.52 98.41 79.12 87.35

WalesDyfed-Powys 75.99 82.45 76.10 83.66 87.81 81.20Gwent 72.11 100 100 100 100 94.42North Wales 96.156 99.08 100 91.41 94.17 96.17South Wales 92.70 100 100 100 100 98.54




It would be inappropriate to label the Metropolitan as a highly inefficient police force,however, as Table 5 indicates that the correspondingPTEscores are 100 in each year of thestudy. This suggests that, given the scale of the Metropolitan’s operations, it is a highlyefficient police force with no obvious inefficiencies in resource utilization. In contrast, Table6 reveals an averageSE score of only 57.52, confirming that all of the observedOE isassociated with scale effects. Given that the Metropolitan is the largest force in the country,this result strongly suggests that there are significant diseconomies of scale at work withrespect to large police force operations. As in other large organizations, this is probablyattributable to the extra bureaucracy and layers of management structure that tend toaccompany large scale operations.

Further statistical examination of the relative efficiency results is undertaken in Section 4,but it is interesting to note from Table 7 that the meanSElevels for the largest and smallestpolice forces (86.99 and 85.85, respectively) are considerably lower than those of theintermediate-size forces, staff group 2 (95.11) and staff group 3 (96.23). This is notsurprising because we would expect that a large proportion of staff group 1 forces wouldexhibit increasing returns to scale, while the majority of staff group 4 forces would exhibit

Table 4DEA of London and Metropolitan police force OE results

1992–1993 1993–1994 1994–1995 1995–1996 1996–1997 Mean

MetropolitanGreater

Manchester90.32 100 100 100 100 98.06

Merseyside 69.33 66.75 60.87 76.01 72.64 69.12South Yorkshire 74.61 76.68 69.66 72.99 78.70 74.53Northumbria 73.41 60.96 69.76 73.21 76.05 70.68West Midlands 18.23 74.54 68.68 71.58 74.28 61.46West Yorkshire 65.49 69.65 70.99 74.51 77.08 71.54

LondonCity 100 100 75.00 73.21 71.22 83.89Metropolitan 30.56 99.35 62.06 42.24 53.41 57.52

Table 5DEA of London and Metropolitan police force PTE results

1992–1993 1993–1994 1994–1995 1995–1996 1996–1997 Mean

MetropolitanGreater

Manchester100 100 100 100 100 100

Merseyside 81.64 71.55 87.74 92.00 74.50 81.49South Yorkshire 81.21 82.48 74.72 76.85 79.16 78.88Northumbria 80.65 61.07 80.79 82.74 78.09 76.67West Midlands 18.82 74.54 82.05 100 100 75.08West Yorkshire 78.73 69.73 77.40 100 100 85.17

LondonCity 100 100 100 100 100 100Metropolitan 100 100 100 100 100 100


diseconomies of scale. Hence, both sets of police forces would haveSEscores well below1. In contrast, staff groups 2 and 3 would be expected to be operating much closer to, if notat, the constant-returns region of the production relationship and would, therefore, haveSEscores at, or closer to, unity. The fact that staff group 3 exhibits the highest meanSEscore(96.23), with by far the lowest standard deviation (4.04), strongly suggests that police forcesin this size band and staff group are close to the optimum in terms of scale efficiency.Clearly, this type of information could prove highly informative in the context of anyproposed restructuring of police forces such as the merging of forces or the redrawing ofpolice force boundaries, etc.

Interestingly, Table 7 indicates that the smallest forces (those in staff group 1) appear tobe the most technically efficient, with a meanPTE score of 99.64 and a standard deviationof only 1.30. Furthermore, the meanPTEscores appear to decline with size as the scores for

Table 6DEA of London and Metropolitan police forces SE results

1992–1993 1993–1994 1994–1995 1995–1996 1996–1997 Mean

MetropolitanGreater

Manchester90.32 100 100 100 100 98.06

Merseyside 84.92 93.29 69.38 82.62 97.50 85.54South Yorkshire 91.87 92.97 93.23 94.98 99.42 94.49Northumbria 91.02 99.82 86.35 88.48 97.39 92.61West Midlands 96.86 100 83.71 71.58 74.28 85.29West Yorkshire 83.18 99.89 91.72 74.51 77.08 85.28

LondonCity 100 100 75 73.21 71.22 83.89Metropolitan 30.56 99.35 62.06 42.24 53.41 57.52

Table 7Group descriptive statistics and the test for equality of group means between different staff groups,1992–1993 to 1996–1997

Dependent variable Independent variables SamplesizeSE PTE OE

Group meansStaff group 1 85.85 99.64 85.56 19Staff group 2 95.11 91.73 87.43 101Staff group 3 96.23 90.57 87.17 53Staff group 4 86.99 88.36 76.13 39

Total 93.07 91.52 85.12 212Standard

deviationsStaff group 1 11.02 1.30 11.23 19Staff group 2 6.61 10.44 12.73 101Staff group 3 4.04 9.72 10.24 53Staff group 4 16.87 15.49 18.89 39

Total 10.12 11.24 14.01 212


staff groups 2, 3, and 4 are 91.73, 90.57, and 88.36, respectively. This suggests that, leavingaside the issue of the scale of operations, effective resource usage and cost control are easierto accomplish in smaller police forces than in larger ones.

Hence, we appear to have an interesting dichotomy in the sense that levels ofPTEappearto decline with police force size, but there is clear evidence of an inverted U-shapedrelationship with respect to scale efficiency. The latter is suggestive of the classic U-shapedaverage-cost curve that is typically attributed to increasing, and eventually decreasing,returns to scale. Indeed, it is interesting to note that the meanSEscores support the notionof a “saucer-shaped” average cost curve for policing in the sense that there appear to besubstantial increasing and decreasing returns to scale in evidence at the extreme ends of thesize spectrum, but a relatively large region of constant returns or modest economies/diseconomies of scale at intermediate size ranges. Although this is a very common findingin economic studies of industrial production, it is a particularly interesting result to find thatthe same economic production relationship appears to hold good in public sector servicessuch as policing.

Clearly, the apparent tradeoff betweenPTE and SE presents particular problems in thecontext of decisions over police force management and structure, and it warrants furtherresearch and investigation. A simple way of characterizing the problem is to think of theSEresults (and the corresponding notional average cost curve) as revealing the minimum levelof average costs that could be attained for any given scale of output, provided that allresources are used effectively. Furthermore, this information reveals the relationship betweensize and efficiency, or size and unit costs. Economists frequently refer to this as revealing theminimum efficient scale of operation, i.e., the minimum level of output that exhausts alleconomies of scale. Our results suggest that this would be at staff group 2 or 3.

As Liebestein (1966) pointed out, however, these notional minimum costs at each givenscale of operation are not always realized due to various factors such as managerialinefficiency, for example. He referred to this failure to realize the minimum possible unitcosts as “X-inefficiency” and the DEA analog to this is ourPTEresults, which show whetherresources are being used at their maximum efficiency for any given scale of output. Clearly,a failure to utilize resources at their maximum efficiency would result in unit costs exceedingtheir potential minimum. Our finding thatPTEdeclines with the scale of output has powerfulimplications because it suggests that X-inefficiency increases with size and, hence, that thewedge between minimum and actual unit costs will be increasing, while at the same timeminimum unit costs are actually declining with size up to a point.

Our results suggest, therefore, that to enhance the overall efficiency of the English andWelsh police forces two approaches are necessary. First, consideration must be given tosome structural reorganization such that individual forces are operating closer to the mini-mum efficient scale, which appears to be staff groups 2 to 3. Second, the apparent problemof worsening X-inefficiencies must be investigated and tackled in larger police forces. WhileDEA can provide valuable insights into the reductions in inputs necessary to achievePTEforgiven output levels (using comparisons with police forces that have efficient reference sets),it seems likely that a review of management and staffing structures in the larger police forcesmay be required. It is interesting to note in this respect that only three police forces,Cleveland, Dorset, and Leicestershire, are consistently efficient in terms of bothSEandPTE


(and, necessarily,OE). Hence, these forces will tend to form part of the efficient referenceset of police forces for a large number of inefficient forces, and a detailed comparison ofthese “best-practice” forces with the less efficient units could provide very useful informationin any reorganization/restructuring process.

So far we have not addressed the issue of the statistical significance of the differences inour efficiency scores across staff size groups. We rectify this in the next section usinganalysis of variance (ANOVA) and discriminant analysis techniques.

4. MDA results

To assess the DEA results further, we adopt a dual post-hypothesis testing strategy thatutilizes ANOVA and MDA. Both of these statistical techniques allow us to determinewhether there are any significant differences between grouped police forces (see Hair et al.,1995, for an introduction). In this analysis, the categorical variable partitions the policeforces into four groups that are determined by the number of total police and civilian staffoperating in each force. This allows us to determine, for example, if large police forces (bytotal staff employed) displaySE, PTE, or OE scores that are significantly better (or worse)than their smaller counterparts. If a police force has 0 to 1,500 total staff, it is a member ofstaff group 1; between 1,501 to 3,000, they are a member of staff group 2; between 3,001 to4,500, they are a member of staff group 3; and above 4,501, they are a member of staff group4. To ensure that we follow at least the minimum requirements necessary for MDA, we havestacked the 5 years before estimation. Table 7 gives the total 1992–1993 to 1996–1997stacked grouped summary statistics for the three independent variables,OE, PTE, andSE.

As outlined previously, in terms ofSEstaff group 3 has the highest mean value, and thelowest standard deviation, while forPTEstaff group 1 has the highest mean value with thesmaller standard deviation. The results forOE reveal that staff groups 1, 2, and 3 are veryclose in terms of the overall mean rankings and their deviation. However, staff group 4 hasthe lowest overall mean value with the largest standard deviations. This can be attributed tothe wide variations inOE for the West Midlands and the Metropolitan police force that areevident in Table 4.

The estimation analysis that is followed in this study involves further testing proceduresafter the DEA estimation. The first stage is the estimation of an ANOVA, a univariate test,where the dependent variables arePTE and SE, and the independent variable is the cate-gorical staff group.4 The null hypothesis under interest is that each mean associated with thestaff group is equal. As can be seen from Table 8, the F-statistic is greater than the criticalvalue, and we can conclude that there is a statistically significant size difference associatedwith our two measures of efficiency. However, we do not know whether the differences arebetween staff groups 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, or, finally, 3 and 4.

MDA is much like ANOVA, but in this case the dependent variable is the categorical staff

4 As OE is a product of the multiplication ofPTEandSE, it is excluded in the second-stage estimation due toproblems of multicollinearity.


group, and the independent variables arePTE andSE. The reason for estimating the MDAis that it offers an alternative insight into the ANOVA results found in Table 8. For example,if we are trying to predict to which staff group a police force should belong, given a valuefor PTEor SE, then MDA will derive the linear combination of the two independent variablesthat would discriminate best between the staff groups. MDA distinguishes between thegroups by multiplyingPTE and SE by their corresponding weights and then adds theseproducts together giving a single discriminant score for each police force. After averagingeach discriminant score in each staff group, we obtain the centroid, which we can use tocompare how “far apart” the staff groups are. In this case, our hypothesis of equal means forthe staff groups are based on comparing the distribution of the discriminant scores. The testanalysis is such that “if the overlap in the distribution is small, the discriminant functionseparates the groups well. If the overlap is large, the function is a poor discriminator betweenthe groups” (Hair et al., 1992).5

Before discussing the MDA results, we need to compare the hit ratio with the maximum-chance and proportional-chance criteria to assess the predictive accuracy of the function. Themaximum chance criterion is calculated as the probability of correctly classifying all scoresby placing them in the staff group with the greatest probability of occurrence, which in thismodel is 48%. However, with unequal groups, we can calculate a proportional-chancecriteria, which in this model equals 33.34%. Hence, as our hit ratio (49.1%) exceeds themaximum-chance and proportional-chance criteria, we can conclude that the MDA model isvalid based on these measures. We also checked our model using Press’s Q-statistic, whichtests whether the staff group classification by MDA would exceed those classifications ifcarried out by chance. Having a total of 104 predicted group memberships correctlyclassified, the estimate of Press’s Q value equals 62.89%, which is significant at the 5%critical level. Therefore, utilizing the results obtained from the maximum-chance criteria, theproportional-chance criteria, and the hit ratio, we can conclude that the MDA model is betterat predicting staff group membership than if the prediction is carried out by chance.

Table 9 provides the overall MDA results and indicates that the discriminant functions arehighly significant, as measured by Wilksl and thex2 statistics. Overall, the first function

5 MDA was estimated using SPSS version 8 (SPSS Inc; Chicago, IL), with the stepwise Mahalanobis distance,Fisher’s function coefficients method, a decision rule of F being between 0.05 and 0.15, prior probabilitiescomputed from group size, and the use of the within-groups covariance matrix.

Table 8Test for equality of staff group means

Independentvariables

Wilksla

UnivariateF ratiob

Significance

OE 0.905 7.246 0.000PTE 0.936 4.708 0.003SE 0.843 12.880 0.000

a Degrees of Freedom, 3.b Degrees of Freedom, 208.


accounts for 74.4% of the variance, and the second function accounts for 25.6%. However,the functions display a low canonical correlation of 0.39 and 0.25, respectively; that is,15.21% and 6.25% of the variance in the dependent variable can be explained by this model(in regression models this is the R2 statistic). In this context, although the latter figures arerelatively low, we must remember that MDA involves the process of determining which staffgroup a police force should be included in. However, it cannot take into account sociologicaland political factors that cannot be included in the calculation but that will have an effect onhow large a police force will be and, therefore, on its staff group classification.

We have found above that the discriminant function is able distinguish between thedependent variables and that there are unequal staff group mean values. A first check ofwhether there are indeed staff group mean differences is shown in Table 10, where we cansee that none of the group centroids are equal in value. That is, it appears that the first andsecond discriminant functions significantly discriminates between all groups. To determinewhich pairs of group means are significantly different, we estimate a second-stage ANOVAthat allows us to calculate post-hypothesis pairwise tests: TUKEY HSD, Scheffe, and leastsignificant difference (see Sharma, 1996, for an introduction to this stage of analysis).

Table 11 gives the ANOVA results for the discriminant scores and the post-hypothesispairwise tests. The initial analysis shows that the discriminant scores are significantlydifferent across the staff group means and that the Scheffe test (our preferred post-hypothesis

Table 9Multivariate results for four staff group discriminant analysis

Function Eigen-value

% of Variance Canonicalcorrelation

Wilksl

x2 df Signif-icanceFunction Cumulative

1 0.188a 74.4 74.4 0.397 0.791 48.792 6 0.0002 0.065a 25.6 100.0 0.246 0.939 13.038 2 0.001

Independent Variables

StandardisedDiscriminantCoefficientsFunction 1

CanonicalFunction

Function 2

Structure Matrixb

Function 1 Function 2

PTE 20.121 0.994 0.993c 0.121SE 0.988 0.162 20.162 0.987c

Classification function coefficients-

Fisher’s Linear Discriminant Functions

Independent Variables Police Staff Groups

1 2 3 4

PTE 0.867 0.805 0.795 0.773SE 1.022 1.125 1.137 1.030Constant 289.466 291.137 292.112 280.666

a First two canonical discriminant functions were used in the analysis.b Pooled within-groups correlations between discriminating variables and standard canonical discriminant

functions. Variables ordered by absolute size of correlation within function.c Largest absolute correlation between each variable and any discriminant function.


pairwise test) shows that function Z1 (which corresponds toPTE) is significantly differentacross staff groups, with the exception of 1 and 4 and 2 and 3, and that function Z2 (whichcorresponds withSE) shows that only staff groups 1 and 4 are significantly different. Thislatter result corresponds well with our earlier analysis in the sense that staff groups 2 and 3appear to be operating closer to the constant-returns region of production and, hence, wouldnot be expected to exhibit significant differences inSEscores. Staff groups 1 and 4, however,would be expected to exhibit increasing and decreasing returns to scale, respectively, andhence might be expected to produce significant differences inSEscores.

The results forPTE, however, suggest a more complicated story than is apparent in Table7 (which suggested thatPTE scores declined with police size). This further statisticalanalysis suggests that staff groups 1 and 4 tend to be more technically efficient, with staffgroups 2 and 3 being less efficient but not significantly different inPTE terms. Thisinterpretation is supported by the very high meanPTEscore for police staff group 1 (99.64)and by the very high meanPTEscores of some large police forces such as the Metropolitanand Greater Manchester forces. The relatively low meanPTEscore for staff group 4 (88.36)is probably explained by the presence of outliers. This is supported by the relatively highstandard deviation of 15.49 evident in Table 7.

Table 11ANOVA and post hocmultiple comparison tests

Dependent variable ANOVA

F-statistic Significance

Z1 13.004 0.000Z2 4.485 0.004

Test criteria

Tukey HSD Scheffe LSD

Z1 1–2, 1–3, 2–4, 3–4 1–2, 1–3, 2–4, 3–4 1–2, 1–3, 2–4, 3–4Z2 1–4 1–4 1–2, 1–3, 1–4, 2–4

Table 10Functions at group centroids

Police staff group Function

1 2

Staff group 1 20.851 0.611Staff group 2 0.214 0.005Staff group 3 0.344 20.003Staff group 4 20.607 20.393


5. Conclusions

This article is the first to use DEA to examine the relative efficiency of the English andWelsh police forces. Relatively few forces were identified as being consistently efficientthroughout the sample period. The “best-practice” police forces could be used as valuablecomparitors in any attempt to restructure police forces to improve productivity and effi-ciency. The study revealed important information concerning the size-efficiency relationshipin English and Welsh policing. Specifically, evidence of significant increasing and decreas-ing returns was found at the extremes of the size spectrum, and theSEresults were supportiveof a “saucer-shaped” average cost curve in policing. This was confirmed by subsequentANOVA and MDA analysis. Furthermore, theSE results suggest that the optimal size ofpolice forces in England and Wales is at staff group 2 or 3.

Interestingly, however, thePTE scores suggested a very different size-efficient relation-ship for X-efficiency. Specifically, the smallest and largest forces tend to produce relativelyhigher PTE scores than the intermediate size forces, although staff group 1 showed thegreatest consistency with respect to high levels of X-efficiency. The clear differencesbetween theSEandPTE results suggest that the process of enhancing overall police forceefficiencies in England and Wales will necessarily be difficult and complex. Nevertheless,this article demonstrates that DEA can produce highly informative results that could, inconjunction with other types of analysis, potentially influence the design of efficiency-enhancing reforms in English and Welsh policing.

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