Racial bias in police stops and searches: an economic analysis

European Journal of Political Economy Vol. 17 Ž2001. 17–37

Racial bias in police stops and searches: an economic analysis Vani K. Borooah ) School of Public Policy, Economics and Law, UniÕersity of Ulster, Newtownabbey BT37 0QB, Northern Ireland, UK Received 1 May 1999; received in revised form 1 March 2000; accepted 1 May 2000

Abstract The purpose of this paper is to provide an economic analysis of racial bias in police stops and searches. It develops a model of policing behaviour, which is used to define discrimination, clarify its nature and identify its sources. This paper identifies two sources of discrimination—bigotry and business necessity—and suggests how they might be identified in terms of the available data. Bigotry is always inefficient but discrimination based on business necessity makes for efficient policing. However, discrimination based on business necessity may be unacceptable on equity grounds and the paper explores the tension between efficient and equitable policing. q 2001 Elsevier Science B.V. All rights reserved. JEL classification: D6; H3; K4 Keywords: Racial bias; Police stop; Police search

1. Introduction AZero-toleranceB policing—under which no offence, however trivial, is allowed to go unpunished—is increasingly viewed as the most effective method of reducing crime rates ŽFarrington et al., 1986; Goldstein, 1990; Wilson and Petersilia, 1995; Kelling and Colis, 1996; Bratton, 1998.. This policing model, which has won admirers all over the world,1 has dramatically altered the way that the police go about their business. A major )

Tel.: q44-28-9036-6346; fax: q44-28-9036-6847. E-mail address: [email protected] ŽV.K. Borooah.. 1 Including the Prme Ministers of Britain and Australia, as reported in The Economist, 3 April 1999, p.13.

0176-2680r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 7 6 - 2 6 8 0 Ž 0 0 . 0 0 0 2 6 - 4

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V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

casualty has been the concept of Acommunity policingB: under zero-tolerance, as Massing Ž1998. notes, the New York Police Department, which pioneered the use of this concept of policing, was resolved, in direct contrast to the restraint usually counseled by community policing, to adopt a more aggressive stance in the community. A major instrument of aggression was the large scale stopping and searching of suspected offenders, with young black men being particular targets. In consequence, an unfortunate, but perhaps inevitable, consequence of zero-tolerance policing has been a rupture in relations between the police and the black community in New York. In England and Wales ŽE & W. too, the use of stop and search methods by the police ruffles racial sensitivities. Police officers in E & W, using their powers under the Police and Evidence Act of 1984 to stop and search suspected offenders Žhereafter, abbreviated to AstopsB . carried out over 1 million stops in 1998. Judging by Home Office data,2 the likelihood of being stopped was much greater for black and Asian persons than for persons who were white. In 1998, for example, 145 blacks and 45 Asians, but only 19 whites, in E & W were stopped per 1000 of their respective population ŽHome Office, 1998.. Therefore, there can be little doubt that there was a racial bias to these stops with the police in E & W discriminating against blacks and Asians in favour of whites. The fact of discrimination, nevertheless, leaves open the question of why such discrimination should arise. Without knowing its sources, one cannot address the problem of eradicating discrimination. The first purpose of this paper, which derives from this observation, is to develop, in Section 2, a model of policing behaviour which can be used to define discrimination, clarify its nature and identify its sources. This model is an adaptation of Longhofer and Peters’ Ž1998. model of discriminatory behaviour by mortgage lenders. Implicit in this is a parallel between the behaviour of lenders and that of the police. Both lenders and the police have to decide on whether to detain a AclientB or to allow himrher to proceed. For lenders, clients are loan applicants and detaining means refusing a loan; for the police, clients are persons who are out on the street and detaining means stopping and searching. In both cases, the decision to detain a person is based on the fear that if the person was allowed to proceed, an adverse outcome would follow: a loan would not be repaid or a crime would be committed. The decision on whether or not to detain a client is based on inferring, from certain observed characteristics of the client, the likelihood of that client AoffendingB. Lenders study the financial history and circumstances of their applicants while police officers observe a person’s age, sex, demeanour, behaviour and circumstances. In both cases, action is triggered if the likelihood of offending exceeds some threshold value: the lender rejects a loan application while the police stop a person. Discrimination arises if different groups are assigned different threshold values for triggering action. The sources of discrimination are essentially two: a lower Aaction-triggeringB threshold may be set for, say, black persons because the responsible authority—the police or

2 The UK Home Office has recently begun to publish data on the ethnicity of persons stopped by the police ŽHome Office, 1998.. The classification of ethnicity—into white, black, asian and other—depends upon the police officer’s judgement about the ethnicity of the person who was stopped.

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19

lenders—dislikes blacks. In this case, discrimination is based on bigotry. Alternatively, discrimination may be based upon the belief, whether justified or not, that black persons are, on average, more likely to offend, whether by defaulting on loans or by committing crimes, than persons from other groups. In this case, discrimination against blacks is AstatisticalB or, equivalently, based on Abusiness interestB. Needless to say, any given act of discrimination may contain elements of both bigotry and business interest, and indeed, the perception that there is a business interest to discrimination may itself stem from bigotry. Putting this last point to one side, Section 2 shows that statistical discrimination, untainted by bigotry, is optimal from a policing perspective because it maximises the number of arrests consequent upon a given number of persons stopped. However, statistical discrimination—carrying as it does the implication that Arace mattersB in determining the likelihood of a person being stopped—may be reprehensible to society. Instead, society may prefer its police to implement a Acolour-blindB policy. This then leads to the second purpose of the paper ŽSection 3. which is to define a socially optimum distribution of stops between different groups and then to derive from this a measure of the degree to which the existing distribution is Žsocially. sub-optimal. This analysis draws upon the method of Atkinson Ž1970. to develop an inequality measure for the distribution of stops which perfectly reflects the level of social welfare associated with that distribution. Section 4 explores, using Home Office Ž1998. data, the extent to which the police, in carrying out stops, discriminate against blacks and Asians and attempts to allocate the totality of discrimination into a AbigotryB and a Abusiness interestB component. Section 5 concludes the paper.

2. A model of police stops and searches It is assumed that persons by being on the street are potential candidates for being stopped by the police. Let V be the set of such persons. Associated with every person is an Aarrest-likelihoodB, u g w0,1x: u is the likelihood that a person if stopped would be arrested. This likelihood is assumed to be distributed across the persons in V according to the density function f Ž u ., with the cumulative density function F Ž z . s PrŽ u F z . s H0z f Ž u .d u . If u ) is the AthresholdB arrest-likelihood set by the police, then a person is stopped if and only if u ) u ) . The value of u ) will inter alia depend upon the resources available to the police for carrying out stops: the greater these resources are, the lower will be the value of u ) be. Unfortunately, a person’s u value is unobservable. What is observable to the police is an Aarrest-signalB whose strength is given by the value of r . It is assumed that the signal encapsulates all relevant and observable information about a person’s arrest-likelihood, u . Let g Ž r < u . be the conditional density of r given a value of u . Then if w Ž r , u . is the joint density, the unconditional density of r —termed as the Asignal-densityB function—observed by the police is hŽ r . s

HVw Ž r ,u . d u sHV g Ž r N u . f Ž u . d u .

Ž 1.

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V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

Let Õ Ž u < r . be the conditional density of u , where: Õ Ž u < r . s w Ž r , u .rhŽ r . s g Ž r < u . f Ž u .rhŽ r .. Then Õ Ž u < r . represents the posterior density of u and this posterior density is arrived at through a Bayesian updating of the prior density, f Ž u .. Then the expected value of u of a person is obtained by forming expectations using this posterior density E w u N r x s uˆ Ž r . s

HVu Õ Ž u N r . d u sHVu

g Ž r N u . f Ž u . rh Ž r . d u .

Ž 2.

The police then stop a person if and only if uˆ Ž r . ) u ) . It is important to stress that the police are not concerned with the value of r per se but rather with what this value implies for a person’s uˆŽ r . value. Two assumptions may be made: Ži. EuˆrE r ) 0; Žii. uˆŽ r . is 1–1 so that the inverse function r s r Ž uˆ . exists. The relation between uˆ and r is illustrated in Fig. 1 below. Corresponding to a threshold arrest-likelihood, u ) , there is a threshold signal-strength r ) such that a person is stopped if and only if hisrher r ) r ).

Fig. 1. The relationship between arrest-likelihood Ž u . and arrest-signal Ž r ..

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

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2.1. Defining and detecting discrimination Suppose that the set V can be subdivided on the basis of some observable characteristic—which in this paper is taken to be a person’s colour—into two mutually exclusive subsets, V B and V W where the subscripts B and W hereafter refer, respectively, to black and white. Then the police Žin implementing stops. are said to ) discriminate against black persons if r B) - r W that is, if the threshold signal for being stopped is weaker for blacks than for whites. There are two sources of discrimination, so defined. Ž1. Bigotry ŽBecker, 1971, 1993.. The police express their dislike of black persons by ) setting r B) - r W . The underlying assumption of the bigotry model is that the density function is the same for blacks as for whites and so that assigning a lower threshold r ) to blacks results in a larger proportion of blacks being stopped ŽFig. 1.. Ž2. Business Necessity ŽArrow, 1972; Phelps, 1972.. The police are neutral in their preferences between blacks and whites but they believe that the density function f Ž u . is different for blacks and whites. In particular, they believe that, relative to the white density function f W Ž u ., the black density function, f B Ž u ., is more concentrated in the upper tail so that the average arrest-likelihood Ždefined as: HV k u f k Ž u .d u , k s B,W. is greater for blacks than for whites ŽFig. 2.. Then the basis of discrimination is business necessity Žor statistical. since by discriminating against blacks, as Fig. 3 below shows, the police equalise the Žexpected. arrest-likelihood of the two groups. Hence, on this

Fig. 2. The density functions of arrest-likelihood for Blacks and Whites.

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V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

Fig. 3. Inter-group differences in the relation between arrest-likelihood Ž u . and arrest-signal Ž r ..

view, the equal treatment of blacks and whites demands that blacks be discriminated against. Under both bigotry and business necessity, the average likelihood of being stopped Žhereafter referred to as the Astop-rateB . is higher for blacks than for whites. These rates —n B and n W for blacks and whites, respectively—are defined as `

n k Ž r k) . s

Hr

) k

hk Ž r . d r ,

k s B,W.

Ž 3.

) ) Under bigotry h B Ž r . s h W Ž r ., but r B) - r W . Under business necessity, r B) - r W and h B Ž r . is more concentrated in the upper tail than h W Ž r .. Hence, regardless of the source of discrimination, n B ) n W . However, the fact that n B ) n W does not necessarily ) imply the existence of discrimination. It may be that r B) s r W , so that there is no Ž . discrimination, but that h B r is more concentrated in the upper tail than h W Ž r . with the consequence that n B ) n W . Denote by m B and m W the average likelihood of arrest following a stop—hereafter referred to as the Aarrest-rateB. The arrest-rate for blacks Žor whites. is simply the weighted average of the arrest-likelihoods of the black Žor white. persons who were

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

23

stopped, the weight being the density value as a ratio of the black Žor white. stop-rate. More formally m k Ž u k) .

Hu)u

u ) k

fk Ž u .

Hu)u

) k

du ,

k s B,W

Ž 4.

fk Ž u . du

where Em krEu k) ) 0. The difference between bigotry and business necessity is that under bigotry, a higher stop rate for blacks should result in a lower black arrest-rate ). ). Ž m B Ž u B) . - m W Ž u W , since u B) - u W while under business interest it should result in ). ). Žapproximately. equal arrest-rates Ž m B Ž u B) . s m W Ž u W , since u B) s u W . However, the fact that m B - m W does not necessarily imply the existence of discrimination: it may be ) that f Ž u . is different for blacks than for whites so that setting r B) s r W results in ) ) Ž . u B ) u W see Fig. 3 , and therefore, in a lower arrest-rate for blacks than for whites. Conversely, the fact m B s m W also does not necessarily imply the existence of discrimination: it may be that f Ž u . is the same for blacks and for whites so that setting ) ) Ž r B) s r W results in u B) s u W see Fig. 1., and therefore, in the same arrest-rate for blacks as for whites. The essence of the matter is that without a judgement as to whether the average likelihood of arrest, following a stop, is the same or is different for blacks and for whites, it is not possible to deduce from the configuration of black–white stop rates or arrest-rates as to whether discrimination is present or is absent. Section 2.2 focuses on the evidence that might help form such a judgement. 2.2. Differences in aÕerage arrest-likelihood between blacks and whites Table 1 below shows stop-rates Ž n k . and arrest-rates Ž m k . for three ethnic groups— whites ŽW., blacks ŽB. and Asians ŽA. —for 10 police areas in England.3 As the table shows, in every area the stop rate for blacks was considerably higher than that for whites. The pattern in respect of arrest-rates was however, more uneven. While in some Areas blacks were more likely than whites to be arrested, after a stop, there were other Areas where the black arrest-rate was the same, or less than, the white rate. Generally speaking, Asians were more likely to be stopped than whites—though not as likely to be stopped as blacks—but less likely to be arrested after a stop. Table 2 shows the total number of arrests and of custodial sentences handed out per 1000 of ethnic population4 in each of 10 police areas in England. Both the arrest and the sentencing figures show clear differences between blacks and whites: for example in 1998 in the London Metropolitan Area—in which nearly 60% of Britain’s black population lives—nearly 17% of the black population, in contrast to less than 5% of the white population, was arrested and nearly 1% of the black population, but less than 0.2% of the white population received custodial sentences. While there could well be a racial bias to police arrests and to judicial sentencing, taken at face value these figures do suggest that the 3

These were areas in which there was the largest concentration of blacks and Asians. In contrast to Table 1 which shows the number of ethnic arrests following from stops as a proportion of the total number of stops of that ethnicity. 4

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V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

Table 1 Stop and arrest rates by police area and ethnicity, 1997r98 Žper thousand of ethnic population, aged 10q. Area

Ethnicity Stops

Bed Hert Lan Lec Man Met Nott Tvl Wmd Wyk

Arrests

W

B

A

W

B

A

10 9 15 17 20 38 8 7 15 11

30 64 55 123 116 181 36 48 77 56

27 40 28 17 22 66 17 37 36 25

13.8 10.5 14.3 10.2 8.7 11.7 10.5 14.0 6.7 13.2

18.1 10.8 14.2 13.2 12.5 11.7 20.6 13.2 10.9 14.7

14.3 7.2 13.2 12.5 12.3 8.1 15.7 11.6 8.1 12.9

Notes: Ži. Stopss total number of persons stopped by ethnicity, per 1000 of ethnic population; Arrestss percentage of stopped persons who were arrested. Žii. Ws white; Bs black; A s Asian. Žiii. Beds Bedfordshire; Hert s Hertfordshire; Lan s Lancashire; Lec s Leceistershire; Man s Manchester; Met s Metropolitan London; Notts Nottinghamshire; Tvls Thames Valley; WmdsWest Midlands; Wyk sWest Yorkshire. Source: Statistics on Race and the Criminal Justice System, 1998, Home Office.

average likelihood of being arrested was higher, and the gravity of the offence for which conviction was secured was greater, for blacks than for whites.

Table 2 Arrests and custodial sentences by police area and ethnicity, 1997r98 Žper thousand of ethnic population, aged 10q. Area

Ethnicity Arrests

Bed Hert Lan Lec Man Met Nott Tvl Wmd Wyk

Sentences

W

B

A

W

B

A

46 24 53 34 43 44 42 34 49 50

138 171 196 231 182 162 185 217 211 206

76 58 73 43 69 50 66 108 82 79

1.9 1.0 1.6 1.7 2.3 1.5 2.2 0.7 1.7 2.2

9.4 7.3 11.8 17.9 13.4 7.9 13.6 5.1 9.7 9.8

4.3 2.3 1.0 2.2 1.5 1.1 2.6 1.7 1.6 3.1

Notes: Ži. Arrestss number of persons arrested by ethnicity, per 1000 of ethnic population; Sentencess number of persons given custodial sentences by ethnicity, per 1000 of ethnic population. Žii. Ws white; Bs black; A s Asian. Žiii. Beds Bedfordshire; Herts Hertfordshire; Lans Lancashire; Lec s Leceistershire; Mans Manchester; Mets Metropolitan London; Notts Nottinghamshire; Tvls Thames Valley; WmdsWest Midlands; Wyk sWest Yorkshire. Source: Statistics on Race and the Criminal Justice System, 1998, Home Office.

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

25

Under these circumstances, there might be potential for the police to discriminate, on ) then, as Fig. 3 grounds of Abusiness necessityB against blacks. If the police set r B) - r W ) ) ) shows, the effect on the relationship between u B and u W is indeterminate: given a r W ) ) ) ) ) ) ) value, r B s r , r B ) r and r B - r will respectively imply u B s u W , u B ) u W and ) Ž u B) - u )W . There would be Atoo muchB discrimination if u B) - u W since then the police supplement business necessity discrimination with bigotry. and there would be Atoo ) Ž littleB discrimination if u B) ) u W since then the police offset business necessity discrimination with a pro-black bias.. Pure business necessity Žor statistical. discrimina) : police discrimination against blacks is just enough to tion would result when u B) s u W equalise threshold arrest-likelihoods.5 2.3. Efficient policing Section 2.2 discussed inter alia the potential for statistical discrimination against blacks, arising from the fact that the expected arrest-likelihood of blacks was greater than that of whites: EB Ž u . s HV B u f B Ž u .d u ) E W Ž u . s HV W u f W Ž u .d u . This section discusses how such discrimination might make for AefficientB policing, and thereby, offer the police incentives to practice such discrimination. It is assumed that the overriding objective of the police is to Aclear upB crime and that their success in this regard is measured by the arrests that they make.6 In order to make these arrests, the police employ a number of instruments of which stops are but one. Suppose there are two arrest-generating activities—stops and AotherB —and that of the total police budget of $ R, $ R S and $ R O , respectively, are allotted to stops and to other activities. The problem before the manager is to set the threshold signals for blacks ) and whites— r B) and r W —at levels that will maximise the number of arrests that result from stopping and searching people. As Eq. Ž3. shows, this amounts to deciding on n B and n W , the likelihoods of black and white persons of being stopped. Suppose that r B) ) ) and r W are two arbitrary levels such that r B) - r W . Associated with these levels are )Ž ). ) ˆ two threshold arrest-likelihoods u k s u k r k , k s B,W Žsee Fig. 3.. From Eq. Ž4., this amounts to determining m B and m W , the likelihood of black and white persons of being arrested, if stopped. In effect, therefore, for k s B,W, the arrest rate m k is a decreasing function of the stop rate, n k . This follows because a lower r k) results in a higher n k ; however, a lower r k) is associated with a lower u k) , and in turn, a lower u k) is associated with a lower m k ; hence, m k and n k are inversely related. Consequently, one may write mk s mk Ž nk . 5

Ž 5.

) ) However, u B) s u W would not imply m B s m W . As Eq. Ž4. makes clear, even when u B) s u W , mB ) mW and the magnitude of the excess would depend on how much more f B Ž u ., compared to f W Ž u ., was concentrated in the upper tail. 6 Others would contest this assumption. For example, McConville et al. Ž1991, 1997. argue that the Aaims of stops and searches are often not to enforce law per se but to secure broader objectives: the imposition of order, the assertion of authority, the acquisition of informationB ŽMcConville et al., 1991, p.16..

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

26

where mXk s Em krEn k - 0. If the budget constraint allows a total of N persons to be stopped, then the problem is to choose n B and n W so as to maximise m s MrN. Now, the overall stop-rate, n, and the overall arrest rate, m, are given by n s NrP s n B Ž PB rP . q n W Ž P W rP . m s MrN s m B Ž NB rN . q m W Ž NW rN . W

s

Ý mk ksB

Nk

Pk

P

Pk

P

N

ž /ž /ž /

Ž 6a .

W

s

Ý m k n k pk n

Ž 6b .

ksB

where P B and P W are, respectively, the total number of blacks and whites in the population of P persons; NB and NW are, respectively, the total number of blacks and whites stopped by the police; and p B s PB rP and p W s P W rP are the proportions of the population that is, respectively, black and white. Given N and, therefore, n, the first-order conditions for maximising m are from Eq. Ž6b. EmrEn k s Ž mXk n k q m k . p k n s 0, ´ Ž 1 q n k . m k s 0,

k s B,W

´ mXk n k q m k s 0

Ž 7.

hk s mXk Ž n krm k . - 0

where is the elasticity of the arrest rate for group k with respect to its stop rate. In equilibrium, the stop rates n B and n W should be such that mB Ž nB . mW Ž nW .

s

1 q hW 1 q hB

Ž 8.

and if it is assumed that h B s h W , the number of arrests, resulting from a given number of stops, is maximised when the arrest rate is the same for blacks and for whites that is, ) when m B s m W . Now m B s m W implies u B) s u W , and if E B Ž u . s E W Ž u ., this implies ) ) ) r B s r W and therefore, n B s n W , and if EB Ž u . ) E W Ž u ., this implies r B) - r W , and therefore, n B ) n W .

3. Socially optimal policing The spectrum of choices that society faces, in terms of deciding on how its police should conduct stop and search operations, has two poles. At one extreme, society may require that, regardless of whether the person is black or white, the likelihood of being stopped should be the same Žthat is, n B s n W .. At the other extreme, society may require that, regardless of whether the person is black or white, the likelihood of being arrested, as a consequence of being stopped, should be the same Žthat is, m B s m W .. When the expected arrest-likelihood of blacks is the same as that for whites Žthat is, E B Ž u . s HV B u f B Ž u .d u s E W Ž u . s HV W u f W Ž u .d u . then, as Eqs. Ž3. and Ž4. show, equality of stop-rates implies equality of arrest-rates and vice-versa. However, when the expected arrest-likelihoods are different Žsay, E B Ž u . ) E W Ž u . as the previous section suggested. then there is a conflict between the two types of equality: stop rate equality, arising from setting blacks and whites the same threshold signal ŽAno discriminationB .

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

27

) would result in m B ) m W since u B) ) u W ; on the other hand, arrest rate equality, arising from setting blacks and whites the same threshold arrest-likelihood, would result Žsince ). r B) - r W in n B ) n W . The conflict between the two types of equality arises because they represent different perspectives to the welfare aspects of police stops. In carrying out stops, the police both confer a gain to, and impose a loss upon, society. Social gain arises from the fact that Žpotential or actual. offenders are apprehended. Social loss arises from the fact that innocent persons are harassed. High arrest rates are positively correlated with social gain Ždenoted G . while high stop rates are positively associated with social loss Ždenoted L.. The overall level of welfare, denoted W, is a function of the gain and the loss associated with police stops

W s W Ž G, L .

Ž 9.

where: EWrEG ) 0, EWrEL - 0 and L, the Social Loss Function, and G, the Social Gain Function, can be represented, in additively separable form as L s L Ž n B ,n W . s PB U Ž n B . q P W U Ž n W .

Ž 10a.

G s G Ž m B ,m W . s NB V Ž m B . q NW V Ž m W . .

Ž 10b.

The function UŽ.. G 0 in Eq. Ž10a. represents society’s valuation of the loss arising from a person in group k s B,W having a likelihood n k of being stopped and the function V Ž.. G 0 in Eq. Ž10b. represents society’s valuation of the gain arising from a person in group k having a likelihood m k of being arrested, if stopped. The functions UŽ.. and V Ž.. are invariant across the groups—society is colour-blind—with higher values of the functions representing, respectively, higher levels of loss and gain. Both functions are increasing in their arguments so that U X Ž n k ., V X Ž m k . ) 0, k s B,W. Furthermore, it is assumed that UŽ.. is strictly convex and that V Ž.. is strictly concave, so that marginal social loss increases with an increase in stop rates while marginal social gain decreases with an increase in arrest rates. The sum of the group specific losses is the social loss associated with a particular configuration of stop rates, n B , n W , while the sum of the group specific gains is the social gain associated with a particular configuration of arrest rates, m B , m W . Because of the strict convexity and concavity assumptions, social loss will be minimised when n B s n W and social gain will be maximised when m B s m W . There is thus a clear link —developed further in the subsequent discussion—between the degree of inequality in stop rates and arrest rates and levels of social loss and gain. As noted earlier, except when EB Ž u . s E B Ž u . it is impossible to simultaneously minimise loss and maximise gain. Let n ) G n and m ) F m represent, respectively, the stop and arrest rate which, if assigned equally to blacks and whites, would yield the same level of social loss and of social gain as the existing distribution of stop rates and of arrest rates.7 Then n ) may be termed the Aequally distributed loss equivalentB stop rate and m ) may be termed the 7

That is LŽ n ) ,n ) . s LŽ n B ,n W . and GŽ m ) ,m ) . sGŽ m B ,m W ..

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V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

Aequally distributed gain equivalentB arrest rate. Following Atkinson Ž1970., the inequality indices associated with the distribution of stop rates Ž n B ,n W . and of arrest rates Ž m B ,m W . are, respectively I s Ž n )rn . y 1 G 0 and

J s 1 y Ž m )rm . G 0.

Ž 11 .

In order to relate the values of the inequality indices, I and J, to, respectively, levels of social loss and gain, L and G, the reverse of the Atkinson transformation may be used ŽSen, 1998. to obtain the loss and the gain functions as L s nŽ 1 q I .

and

G s mŽ 1 y J .

Ž 12 .

The loss and gain functions in Eq. Ž12. have a natural interpretation: the loss from an overall stop rate Ž n. is increased, and the gain from an overall arrest rate Ž m. is reduced by the degree of inequality in their distribution between blacks and whites. Given an overall stop rate n and an overall arrest rate m, the inequality index I is a measure of the social loss, and the inequality index J is a measure of the social gain, associated with police stops. These ideas are illustrated below in Figs. 4 and 5. In Fig. 4, each point on QQ represents a Ž n B ,n W . combination that yields the same Žgiven. value of n and in Fig. 5 each point on RR represents a Ž m B ,m W . combination that yields the same Žgiven. value of m. The QQ and RR are, therefore, referred to as the Astop-possibilityB and Aarrest-possibilityB lines. From Eqs. Ž6a. and Ž6b., the slope of QQ is P W rPB and that of RR is NW rNB s Ž n W p W .rŽ n B p B .. Superimposed upon QQ, in Fig. 4, are the indifference curves associated with the loss function ŽEq. Ž10a.. and superimposed upon RR, in Fig. 5, are the indifference curves associated with the gain function ŽEq. Ž10b... Curves further away from the origin represent, in Fig. 4, higher levels of social loss and, in Fig. 5, higher levels of social gain. By assumption, the curves in Fig. 4 are concave, and those in Fig. 5 are convex, to the origin.

Fig. 4. Stop rates and social loss.

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

29

Fig. 5. Arrest rates and social gain.

From Eqs. Ž10a. and Ž10b., the slopes of the loss and the gain indifference curves are, respectively ELrEn W

s

ELrEn B

PB

aB

PW

aW

ž /ž /

and

EGrEm W EGrEm B

s

NB

bB

NW

bW

ž /ž /

Ž 13 .

where a k s EUŽ n k .rEn k and b k s EV Ž m k .rEm k , k s B,W. Social loss is minimised in Fig. 4, and social gain is maximised in Fig. 5, when the slopes of the respective indifference curves are equal to that of the relevant Apossibility linesB that is when PB

aB

PW

aW

NB

bB

NW

bW

PB

ž /ž / ž / s

PW

NB

ž /ž / ž / s

NW

´ aB s a W

Ž 14a.

´ bB s bW .

Ž 14b.

Since by convexity, the marginal losses a B and a W increase in n B and n W , and since by concavity, the marginal gains b B and b W increase in m B and m W , aB s a W ´ n B s n W and b B s b W ´ m B s m W . Therefore, in Figs. 4 and 5, the tangency between the indifference curve and the possibility line occurs at a Ž X . point on the 458 line. If, in Figs. 4 and 5, the outcomes with regard to black–white stop rates ŽFig. 4. and arrest rates ŽFig. 5. occurred at A, then this is Awelfare-equivalentB to CD. In Fig. 4, society is indifferent between the unequal distribution at A of a lower stop rate and the equal distribution at C, of a higher stop rate. The degree of inequality in the distribution of stop rates is, from Eq. Ž11., Ž CDrXY . y 1 and this is also the percentage amount by which social loss is greater than its minimum value at X. In Fig. 5, society is indifferent between the unequal distribution at A of a higher arrest rate and the equal distribution at C, of a lower stop rate. The degree of inequality in the distribution of arrest rates is,

30

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

from Eq. Ž11., 1 y Ž CDrXY . and this is also the percentage amount by which social gain is lower than its maximum value at X. To determine the values of I and J that yield the highest level of social welfare, W, suppose that the overall stop rate n is given and that the question is of deciding on values n B and n W consistent with this overall rate, n B G n W . In terms of Fig. 4, this involves a choice of point on QQ at, or below, X. Associated with any such point Žsay, A A. is a distribution n BA and n W and a level of inequality, I A. Associated with the A A A distribution n B and n W is, by Eq. Ž5., a distribution of arrest rates, m BA and m W and A A A also, by Eq. Ž6b., an overall arrest rate, m . Since associated with m B and m W is a level of inequality J A , it follows that JsJŽ I . ,

m s mŽ I .

Ž 15 .

where J X - 0 and mX ) 0. Using Eq. Ž12., Eq. Ž9. can be written in additive separable form as W s W Ž L Ž I . , G Ž m, J . . s G Ž I . y L Ž I .

Ž 16 .

where d Lrd I ) 0 and dGrd I ) 0 and Žit is assumed. d 2 Lrd I 2 ) 0 and d 2 Grd I 2 - 0. Social welfare is then maximised for the level of inequality, I ) , for which the marginal social loss and marginal social gain, from a small increment to inequality, are equal ŽFig. 6, lower panel.. The optimal level of inequality in the distribution of stop rates, I ) then implies an optimal level of inequality J ) s J Ž I ) . in the distribution of arrest rates ŽFig. 6, upper panel.. In order to give empirical life to the inequality indices I and J of Eq. Ž11., the functions UŽ.. and V Ž.. may be written in constant elasticity form as UŽ nk . s

´ n1q k

1q´

and V Ž m k . s

d m1y k

Ž 17 .

1yd

where a k s EUŽ n k .rEn k s n k ´ ŽEa krEn k .Ž n kra k . s ´ and b k s EV Ž m k .rEm k s m dk ´ ŽEb krEm k .Ž m krbk . s d . Consequently, the percentage change in the marginal losses and gains, following a percentage in the likelihood, is constant. Under these specific functional forms for UŽ.. and V Ž.. the inequality indices I and J become W

I s Ž n ) rn . y 1 s

Ý

pk

ksB

nk

ž /

J s 1 y Ž m rm . s 1 y

y1

n

W )

1q ´ 1r Ž1q ´ .

Ý sk ksB

mk

ž / m

1y d

Ž 18a.

1r Ž1y d .

Ž 18b.

where p k s PkrP and sk s NkrN are the shares of group k in, respectively, the total population and the total number of persons stopped. When ´ s 0, n ) s n and society is indifferent as to how a given number of stops is distributed, in terms of the likelihood of persons from the different groups being stopped. When d s 0, m ) s m and society is

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

31

Fig. 6. The optimal level of inequality in the inter-group distribution of stop rates.

indifferent as to how a given number of arrests is distributed, in terms of the likelihood of persons from the different groups being arrested, if stopped. When ´ ) 0, n ) ) n and I ) 0: the greater the value of the parameter, the greater the value of I and, therefore, the higher the level of social loss associated with the distribution of stops. When d ) 0, m ) ) m and J ) 0: the greater the value of the parameter d , the greater the value of J, and therefore, the lower the level of social gain associated with the distribution of arrests. In that sense, the parameters and d represent the degree to which society is averse to inequality in, respectively, the distribution of stops and of arrests.

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

32

4. Empirical results Tables 3 and 4 below show, respectively, values of the index I ŽEq. Ž18a.. and J of ŽEq. Ž18b.. for various values of the inequality-aversion parameters, and d , ranging from d s 0 to ´ , d s 2. The values of I and J, as discussed earlier, show, respectively, the percentage amounts by which the level of social loss, associated with a given distribution of stops, exceeded its minimum value and by which the level of social gain, associated with a given distribution of arrests, fell short of its maximum value. The amount of excess, or shortfall, depended upon society’s aversion to inequality. So, for example, when ´ s 2—so that society, in exchange for a 1 percentage point Žpp. reduction in the black stop rate, was prepared to increase the white stop rate by 0.3 pp, provided the new rates were equally distributed—the percentage excess over the minimum level of social loss varied from 21% in Bedfordshire to 69% in Thames Valley and in four of the remaining nine areas it exceeded 50%. However, when d s 2, the percentage shortfall from the maximum level of social gain was not more than 3%. This suggests that the police discriminated against blacks in stopping potential offenders but, as a consequence of such discrimination, the percentage of blacks and whites arrested was roughly equal. In other words, police discrimination against blacks was largely statistical Žthat is, motivated by Abusiness necessityB . and there was little evidence that such discrimination was supplemented by bigotry. Had the police, in addition to practicing statistical discrimination, also discriminated against blacks for reasons of bigotry then, as discussed in Section 2, the arrest rate of blacks would have been significantly lower and this would have resulted in a high J value. In operational terms, the preceding model suggests that there are four instruments whose levels the police need to establish: the overall stop rate or likelihood of being stopped Ž n. and the translation of this overall stop rate into group-specific likelihood of being stopped Ž n B , nA and n W . where the overall rate and the group-specific rates are connected through Eq. Ž6a., suitably extended to include Asians. The linch-pin of the model is a negative relation, as set out in Eq. Ž5., between a group’s arrest rate Ž m k . and

Table 3 Values of the inequality index, I for different values of the inequality aversion parameter

Bed Hert Lan Lec Man Met Nott Tvl Wmd Wyk

´ s0

´ s 0.4

´ s 0.8

´ s1

´ s1.5

´ s2

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.03 0.05 0.01 0.04 0.03 0.09 0.02 0.08 0.07 0.03

0.06 0.13 0.01 0.10 0.08 0.19 0.05 0.18 0.15 0.07

0.08 0.18 0.02 0.14 0.11 0.25 0.07 0.25 0.20 0.09

0.14 0.35 0.04 0.32 0.23 0.42 0.14 0.46 0.35 0.17

0.21 0.57 0.06 0.56 0.40 0.59 0.24 0.69 0.51 0.27

Note: I =100 is the percentage increase in social loss resulting from the actual distribution of stop rates between blacks and whites being different from a situation where black–white stop rates were the same.

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33

Table 4 Values of the inequality index, J for different values of the inequality aversion parameter

Bed Hert Lan Lec Man Met Nott Tvl Wmd Wyk

d s0

d s 0.4

d s 0.8

d s1

d s1.5

d s2

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.01 0.0 0.01 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.01 0.0 0.01 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.02 0.0 0.02 0.0

0.0 0.0 0.0 0.01 0.01 0.0 0.03 0.0 0.03 0.0

0.01 0.0 0.0 0.01 0.01 0.01 0.03 0.0 0.03 0.0

Note: J =100 is the percentage reduction in social gain resulting from the actual distribution of arrest rates between blacks and whites being different from a situation where black–white arrest rates were the same.

its stop rate Ž n k .. Intuitively, the likelihood of a person from group k being stopped is raised because the threshold arrest-signal Ž r k) . for persons from the group is lowered; lowering the threshold signal implies lowering the threshold arrest-likelihood Ž u k) .; consequently, the likelihood of persons from group k being arrested, after they have been stopped, falls or, in other words, the arrest rate for the group is reduced. This model is given econometric representation in terms of the following four equations and identity n i s a 0 q a 1 pi B q a 2 pi A q e i

Ž 19 .

n i B s b 0 q b1 n i q b2 m i B q u i

Ž 20 .

n i A s g 0 q g 1 n i q g 2 m i A q Õi

Ž 21 .

n i W s d 0 q d 1 n i q d 2 m i W q wi

Ž 22 .

n i ' n i B pi B q n i A pi A q n i W p W

Ž 23 .

where the subscript i extends over the 10 police areas in England Žsee Table 3. whose data was used to estimate the model and e i , u i , Õi and wi are error terms. Eq. Ž19. determines the overall stop rate, n i . It was specified on the hypothesis that the likelihood of a person being stopped would be higher Žthat is, the police would put more resources into stops and searches. in areas with high concentrations of ethnic persons:8 the expectation is that a 1 , a 2 ) 0. Eqs. Ž20., Ž21. and Ž22. represent AsharingB equations: they examine how the overall stop rate is allocated between the groups as AethnicspecificB likelihood of being stopped—n i B , n i A and n i W . In addition to depending on the overall stop, the ethnic stop rates also depend the group arrest rates, m i B , m i A and m i W —the expectation is that b 3 , g 3 , d 3 - 0 so that high arrest rates for each group, for reasons discussed above, would be associated with low stop rates for that group. 8

Note that since pi B q pi A q pi W s1, only two of the three variables can be included in this equation.

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V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

Eq. Ž19. was estimated using OLS and the estimates are shown in Table 5 below. The overall likelihood of being stopped Ž n. was higher in areas with greater concentrations of black persons Ž p B .; there was, however, no significant association Ž t s 0.3. between the presence of Asians in the population the overall stop rate. Accordingly, the estimates shown in Table 5 relate to Eq. Ž19. with a 2 s 0. These estimates show that inter-area variations in the proportion of the population that was black accounted for 75% of inter-area variations in the overall stop rate. The figure in parentheses is the t-value.The x 2 Ž1. value is the Cook–Weisberg test of the null hypothesis that the variance of e i was constant across the observations: this hypothesis could not be rejected. The F Ž3,5. value relates to the Ramsey RESET test of the null hypothesis that the equation had no omitted variables: this hypothesis too could not be rejected. The percentage of the population that was black and the overall likelihood of being stopped were—taking the average across the areas—respectively 2% and 1.7%. The estimates suggest that if the proportion of the population that was black increased by 1 pp then the overall likelihood of a person being stopped would increase by 0.5 pp. ŽEqs. Ž20., Ž21. and Ž22. were estimated—with the identity Eq. Ž23. imposed—using the method of Seemingly Unrelated Regressions ŽSURE. since this allowed the error terms Ž u i , Õi and wi . of the sharing equations to be correlated. The estimation results from the above model suggested that the following zero and equality restrictions might be imposed: Ži. d 2 s 0; Žii. g 1 s d 1. Restriction Ži. suggests that the relation for between m W and n W Žsee Eq. Ž5.. was very AflatB, while restriction Žii. suggests that a change in the overall stop rate has the same effect on the white and Asian stop rates. The joint hypothesis that these restrictions were validly imposed was not rejected with a F Ž2,27. s 0.64. Table 6 below shows the results from estimating Eqs. Ž20. – Ž22. with restrictions Ži. and Žii. imposed. A Breusch-Pagan test, with x 2 Ž6. s 15.4, suggested that the null hypothesis that the error terms were independently distributed could not be accepted: in other words, SURE was the appropriate method of estimation. The R 2 values show that the percentage of variation in the stop rates explained by the predictors ranged from 75% Ž nA . to 98% Ž n W . and the F values decisively rejected, for all the equations, the null hypothesis that the slope variables did not add to explanatory power. Both the black and Asian stop rate Ž n B and nA . were—as predicted by the model— negatively correlated with their respective arrest rates Ž m B and mA .. All three stop rates were positively related to the overall stop rate: a rise in the overall stop rate led to an increase in all the group stop rates, but the greatest increase was reserved for blacks. A 1 pp increase in the overall likelihood of being stopped would increase the likelihood of black persons being stopped by 2.7 pp and increase the likelihood of Asian and white persons being stopped by Žthe same. 0.7 pp. Under the Aefficient policingB scenario, discussed in the earlier section, group-specific stop rates would be set so as to equalise group-specific arrest rates. In order to compute

Table 5 n i s 0.735Ž2.7.q0.497Ž5.3. pi B R 2 s 0.75 x 2 Ž1. s 0.03 F Ž3,5. s 3.5 ns1.7% p B s 2%

V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

35

Table 6 n i B s8.068Ž2.5.q2.738Ž8.2.= n i y0.356Ž2.4.= m i B R 2 s 0.845 F Ž2,10. s 50.7 n B s 7.9% m B s14.0% n i A s 4.437Ž6.9.q0.679Ž34.1.= n i y0.210Ž4.1.= m i A R 2 s 0.744 F Ž2,10. s698.0 nA s 3.1% mA s11.8% n i W s 0.308Ž5.6.q0.679Ž34.1.= n i R 2 s 0.744 F Ž1,10. s1161.1 n W s1.5% m W s11.0%

these efficient stop rates, the estimates shown in Table 6 were used to compute the predicted values of n i B , n i A and n i W when, in each area, the arrest rate for each group was set equal to the overall arrest rate in that area: m i B s m i A s m i W s m i , where m i was the average arrest rate in that area. The results showed that there was little difference between the efficient stop rates and the actual stop rates. Aggregating across all the 10 police areas, black persons had a 7.9% likelihood of being stopped by the police; under efficient policing, this likelihood would rise to 8.7%. The corresponding rates for Asians were 3.1% Žactual. and 3.2% Žefficient. while for whites both the rates were 1.5%. This reinforces the conclusion arrived at earlier in the section, namely that the racial bias that the police in England displayed, in deciding on persons to stop, represented discrimination on grounds of Abusiness necessityB rather than discrimination because of bigotry.

5. Conclusions The issue of racial discrimination in police stops poses a puzzle not unlike that observed in the area of mortgage lending. Longhofer and Peters Ž1998. remarked that the puzzle about mortgage lending is why high rates of rejection of minority loan applicants co-existed with high rates of default by minority borrowers. In the area of police stops the puzzle, at least in England, is why the high likelihood of blacks, relative to whites, being stopped by the police co-exists with black arrest rates, which are no lower than that of whites. The answer in both cases is the same: discrimination on grounds of business necessity. If the likelihood of being stopped was the same for blacks and whites, then the likelihood of being arrested after a stop would be substantially higher for blacks. That is because the evidence suggests that—with all its flaws—the average arrest-likelihood of blacks is greater than that of whites. The police in England exploit this fact to stop a greater proportion of the black, than of the white, population. A corollary of such practice is that inequality in black–white stop rates co-exists with relative equality in black–white arrest rates. It is possible, none the less, that discrimination on grounds of business necessity could be supplemented with bigotry. The relevant question is how much of the observed discrimination against blacks by police stops is based on business necessity and how much is the result of bigotry? The conclusion of this paper was that the implementation of police stops in England, while undoubtedly discriminating against blacks, was largely free of bigotry. This is not to deny that racism exists among the police forces in England ŽMcPherson, 1999.; but it would be wishful

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V.K. Borooahr European Journal of Political Economy 17 (2001) 17–37

thinking to explain away the large imbalance in black–white stop rates, which exists in every police area in England, in terms of police racism. Rather, it is more accurate to view the high stop rate for blacks as the consequence of the police in England targeting their resources to achieve the maximum effect in terms of arrests. In turn, this observation raises the further question as to whether such targeting can be justified? If, adopting a civil libertarian perspective, one treats the disparate treatment of individuals, on the basis of their belonging to particular groups, as wrong then clearly statistical discrimination cannot be justified. Under such discrimination, the treatment of an individual is conditional upon the average behaviour of the group to which hershe belongs. The fact that the sins of the group visit themselves upon every member of the group may, to many persons, be distasteful. On the other hand, if one is prepared to tolerate disparate treatment on grounds of business necessity, then such discrimination may be defended on the grounds that it contributes to police efficiency. As Wilson Ž1989. observed, the essence of the stopping Žand then discharging or arresting. process is exercising judgement about who is likely to have committed a crime. In this process, the guiding principle is the unequal treatment of persons. As this paper has argued, society might not object to persons being treated unequally if some broader principle of fairness was adhered to. This broader principle might be, for example, that unequal treatment was untainted by racism and that it contributed positively to the efficiency of the business. The paper’s central contribution has been to show that, with respect to the treatment of blacks vis-a-vis whites, such an assurance can be plausibly given for police stops and searches in England.

Acknowledgements An earlier version of this paper was written while I was a Fellow at the International Centre for Economic Research at Torino and I am grateful to the Centre for supporting this work. This work was also presented at seminars at the Universities of Catania, Torino and Queensland; at the Annual Conference of the International Association for Research in Applied Psychology ŽBelgirate, 1999.; and at the Annual Conference of the European Public Choice Society ŽSiena, 2000.: I am grateful to participants at all these venues for their views and comments. Special thanks are due to Shanti Chakravarty for several discussions on this subject and to three anonymous referees of this journal for their most valuable comments.

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