Racial Profiling, Fairness, and Effectiveness of Policingecondse.org/wp-content/uploads/2013/04/discrimination...Racial Pro® ling, Fairness, and Effectiveness of Policing ByNICOLAPERSICO*

Racial Proreg ling Fairness and Effectiveness of Policing

By NICOLA PERSICO

Citizens of two groups may engage in crime depending on their legal earningopportunities and on the probability of being audited Police audit citizens Policebehavior is fair if both groups are policed with the same intensity We provide exactconditions under which forcing the police to behave more fairly reduces the totalamount of crime These conditions are expressed as constraints on the quantilendashquantile plot of the distributions of legal earning opportunities in the two groupsWe also investigate the de nition of fairness when the cost of being searched re ectsthe stigma of being singled out by police (JEL J71 J78)

Law enforcement practices often have dispar-ate impact on different ethnic and racial groupsTo take one well-publicized example the cur-rent debate on racial pro ling has shown thatmotorists on highways are much more likely tobe searched by police looking for illegal drugs ifthe motorists are African-American Similar al-legations are made in connection with customssearches at airports and in a number of othersituations involving policing1 Such racial dis-parities are an undeniable reality of policing andare widely perceived to be discriminatory Pub-lic scrutiny has resulted in policy changes aimedat correcting the disparity The proposed reme-dies would reduce the extent to which racial orethnic characteristics can be used in policing2

However forbidding the police from usingsome characteristics may reduce the effective-ness of policing3 Often those who engage incertain criminal activities tend to share certaincharacteristics relating to speci c socioeco-nomic and ethnic backgrounds Forcing the po-lice to disregard such characteristics may lead toless effective policing and to increased crime4

These con icting considerations re ect a fun-damental tension between the principle of equaltreatment under the law and the practical de-mands of law enforcement This tension colorsthe current judicial position which tends touphold the use of racial and other characteristics

Department of Economics University of Pennsylvania3718 Locust Walk Philadelphia PA 19104 (e-mailpersicosscupennedu) URL httpwwwsscupennedu

persico This work develops a line of inquiry that stemsfrom joint work with John Knowles and Petra Todd I amgrateful to both of them Thanks also to Aldo Colussi forresearch assistance The input of several anonymous refer-ees helped improve the paper The nancial support of theNational Science Foundation under Grants SBR-9905564and SES-0078870 is gratefully acknowledged

1 See John Knowles et al (2001) for data that are rep-resentative of the disparity in highway searches Concerningairport searches John Gibeaut (1999) reports that of allpassengers who were strip-searched by customs of cials in1997 60 percent were black or Hispanic For a descriptionof the debasement connected with being bodily searched byairport customs of cials see Adedeji v United States 782F Supp 688 (D Mass) See Elijah Anderson (1990) for amore general view of the relationship between the policeand African-Americans

2 In response to charges of racial pro ling against NewJersey police the former attorney general of New Jersey

released a report which calls for a ruling by the New JerseySupreme Court holding that ldquorace may play no part in anof cerrsquos determination of whether a particular person isreasonably likely to be engaged in criminal activityrdquo (seePeter Verniero and Paul H Zoubek 1999 pp 52ndash53) PartV of the report lists a number of remedial actions aimed atavoiding that race be a factor in the decision to search amotorist

3 See Phyllis W Beck and Patricia A Daly (1999) for adiscussion of the public policy argument in favor of discre-tional searches of motorists

4 There are two logically distinct scenarios under whichmembers of one racial group may be searched dispropor-tionally often by racially unbiased police One is that race iscorrelated with other characteristics which are unobservableto police and are correlated with criminal behavior in thiscase race constitutes a proxy for other characteristics Thesecond scenario is that citizens are searched based on other(nonrace) characteristics which are observable to police andare more frequently present in that race The results pre-sented in this paper are valid under both scenarios althoughthe model is formally stated to describe the proxy scenarioit can readily be reinterpreted to account for the otherscenario

1472

as a factor in determining the likelihood that aperson is engaging in a crime as long as thisuse is reasonably related to law enforcementand is not a pretext for racial harassment5

Mindful of the established judicial line and ofpublic policy concerns for effectiveness of in-terdiction the courts have generally been cau-tious in imposing remedies on the police In sodoing the courts implicitly have placed lessweight on the plight of innocent citizens whobear a disproportionate amount of the policingeffort This position is defensible only insofar asthe two considerations fairness and effective-ness of policing are in con ict It is thereforeextremely important to establish whether a con- ict exists in practice

This paper studies the trade-off between fair-ness and effectiveness of interdiction in thecontext of a simple model in which citizens oftwo groups choose whether to engage in illegalactivities and police audit (or search) citizensThe model is too stylized to have direct policyimplications Despite this fact the model offersa more sophisticated view than is generallyaired in the public debate The rst point of thispaper is that in principle there need not be atrade-off Constraining police to behave in a fairmannermdashin our context searching both groupswith the same intensitymdashneed not per se resultin more crime This is because under veryplausible assumptions about the incentives ofpolice the equilibrium allocation does not co-incide with the crime-minimizing allocation

and so placing constraints on the behavior of thepolice may actually increase the effectiveness ofpolicing6

To understand this statement it is necessary tounderstand why in equilibrium the effectivenessof policing is generally not maximal Our as-sumption is that when choosing whom tosearch the objective of a police of cer is tomaximize the probability of uncovering crimi-nal behavior Suppose then for the sake of theargument that police can distinguish only twogroups A and W which partition the popula-tion In equilibrium it cannot be that one grouphas a lower fraction of criminals for otherwiseno police of cer would search members of thegroup with the lower fraction of criminals Butthen members of that group would respond bystepping up their illegal activity and then policewould want to search the group This equilibrat-ing process results in the key equilibrium con-dition in equilibrium the two groups must havethe same fraction of criminals7 Crucially thisequilibrium condition is independent of theelasticity of crime to policing In contrast theelasticity to policing plays a major role in de-termining the conditions for effective policingTo see this take any allocation (including theequilibrium one) and suppose an increase of 1percent in the search probability deters mem-bers of group W more than members of groupA then maximal effectiveness requires that 1percent of resources be transferred from search-ing group A to searching group W to takeadvantage of the higher deterrence effect ofpolicing on group W This shows that there is awedge between the equilibrium condition andthe effective allocation of policing effort8

5 See Randall Kennedy (1997) for an exposition of thisview and more recently Anthony C Thompson (1999) fora critical analysis It may seem surprising that the SupremeCourt would sanction such a narrow interpretation of theequal protection clause However racial pro ling and otherpolicing cases are often tried as fourth amendment cases(protection from unreasonable search and seizure) and notas fourteenth amendment cases (protection from racial dis-crimination) The distinction is crucial because the standardof evidence is lower in fourth amendment cases police areonly required to prove ldquoreasonablenessrdquo of their behaviorinstead of being subject to the ldquostrict scrutinyrdquo standard thatapplies in racial discrimination cases The Supreme Court nds that the fourth amendment permits pretextual stopsbased on objective justi cation and does not admit second-guessing of the police of cerrsquos subjective motivation [seeUS v Whren 517 US 806 116 SCt 1769 (1996)] Theappropriateness of applying fourth amendment law to racialpro ling cases is now hotly debated and intense publicscrutiny is changing the legal landscape in this area

6 In this paper effectiveness of interdiction is synoni-mous with ldquocrime minimizationrdquo This point is discussedfurther in Section I subsection A in terms of the generalityof the assumptions under which effectiveness and fairnessdo not coincide

7 Equalization of fraction guilty is predicted only forthose crimes which are the object of interdiction Further-more equalization of crime rates does not imply that equi-librium incarceration rates are the same for the two groupsthe group which is searched more intensely will have ahigher fraction of its members being incarcerated Finallythe crime rate is equalized only among those groups that aresearched in equilibrium These points are discussed in sub-section A

8 How this conclusion depends on police incentives isdiscussed in Section VIII

1473VOL 92 NO 5 PERSICO RACIAL PROFILING

Now let us see how constraining police to bemore fair can increase effectiveness of interdic-tion Keep the assumption that group W is moreresponsive to policing Suppose in equilibriumgroup W is policed with less intensity thenforcing the police to behave more fairly entailsincreasing the intensity with which group W ispoliced Since group W is more responsive topolicing this increases the effectiveness of in-terdiction In general if the elasticity to policingis lowest for the group which is policed moreintensely moving the system towards a morefair allocation increases the effectiveness ofpolicing in this case there is no trade-off be-tween fairness and effectiveness

In our model then constraining police mayor may not increase the effectiveness of inter-diction depending on the relative elasticity topolicing in the two groups We thus seek theo-retical conditions to compare that elasticityacross the two groups Measuring the elasticitydirectly is a dif cult exercise which at a mini-mum requires some sort of exogenous varia-tion in policing (see Steven D Levitt 1997who also reviews the literature) We outline analternative approach based on measuring theprimitives of our simple model of policing spe-ci cally legal earning opportunities The rea-son why earning opportunities matter is thatgiven a certain intensity of policing citizenswhose legal earning opportunities are higher areless likely to engage in crime In our stylizedformalization citizens above a given legal earn-ing threshold will not engage in criminal activ-ities and those below the threshold will Thusthe amount of criminal activitymdashand hence alsothe elasticity of crime to policingmdashdepends onthe distribution of legal earning opportunitiesThe advantage of this approach is that it doesnot require observing any variation in policingInstead the answer to the main questionmdashwhether the group which in equilibrium is po-liced more intensely is also the group in whichthe elasticity of crime to policing is highermdashliesin the shape of the distributions of legal earningopportunities in the two groups Mathemati-cally the key condition is expressed as a con-straint on the QQ (quantilendashquantile) plot ofthese distributions The QQ plot h( x) is easilyconstructed given the cumulative distributionfunctions (cdfrsquos) FA and FW of earning op-portunities in the two groups it is h( x) 5

FA2 1(FW( x)) We show that roughly speaking

if the function h has slope smaller than 1 thenin equilibrium group A has a higher elasticityof crime to policing and is policed more in-tensely As suggested above in these circum-stances moving slightly towards fairnessreduces the effectiveness of policing A similarcondition allows us to evaluate the conse-quences of large changes in fairness ie ofimplementing the completely fair outcome inwhich the two groups are policed with the sameintensity

At this point it is important to emphasize thatour modeling strategy will lead us to abstractfrom many factors that can affect the decision toengage in crime and focus on onemdashthe legalalternative to crime By isolating this factor ourmodel enables us to make progress with thetheoretical conditions that link fairness and ef-fectiveness of policing However many otherfactors also play important roles in the questionwe analyze One such factor is the intensity ofpunishment that citizens of different groups in-cur if found guilty If for example citizens ofgroup A expect to receive longer prison sen-tences if convicted relative to citizens of groupW the deterrence effect of policing will behigher within group A even if that group had thesame legal earning opportunities as group WSimilarly our conclusions are modi ed if theobjective function of police is not simply tocatch criminals but re ects some concern forthe overall crime level These generalizationsare investigated in Section VIII Extending themodel in these directions changes the form ofsome results but the QQ plot remains the keystatistic in the analysis Therefore it is interest-ing to seek empirical equivalents for the QQplot of legal earning opportunities

Although an agentrsquos legal earning opportu-nity may be unobservable if heshe decides toengage in crime and forgo legal occupation andso FA and FW are in principle unobservable wedo have surveys of earnings reported by thosecitizens who are not incarcerated We give con-ditions under which the QQ plot based on sur-veys of reported earnings coincides in theequilibrium of our stylized model with the QQplot of potential legal earnings which is theobject of interest Purely as an illustration ofthis methodology and with no direct policyapplication in mind we construct the QQ plot

1474 THE AMERICAN ECONOMIC REVIEW DECEMBER 2002

based on empirical distributions of reportedearnings by African-American and white maleresidents of metropolitan statistical areas in theUnited States Intriguingly we nd that this QQplot indeed appears to have slope smaller than 1(although the slope appears to be quite close to1 for plausible values of the parameters) In thecontext of our model a slope smaller than 1would imply that the elasticity to policing atequilibrium is higher in the A group than in theW group suggesting that if one were to movetoward fairness by constraining police to shiftresources from the A group to the W group thenthe total amount of crime would not fall Thusour back-of-the-envelope calculation cannotrule out the presence of a trade-off betweenfairness and effectiveness of policing Of coursewe do not imply that our stylized model is aclose approximation of the real-world problemthe purpose of the calculation is to illustratehow the methodologycan be applied to the data

Finally we explore the foundations of ournotion of fairness through an inquiry on thenature of the cost of being searched The notionof fairness adopted in this paper focuses onequating across groups the expected costs ofbeing searched This notion of fairness speaksto the concerns of critics of racial pro ling andembodies the idea that society should guaranteeequal treatment to citizens of all racial groupsThis notion is redistributional in nature becauseit is based on the idea that society can (andshould be prepared to) redistribute these costsacross racial groups We inquire whether thestigmatization associated with being singled outfor searchmdashas distinct from physical costs ofbeing searchedmdashcan be understood as a com-ponent of the cost to be redistributed To thisend we present a formal de nition of stigmati-zation which we call disrepute We de ne dis-repute as the rational updating about anindividualrsquos characteristics due to the knowl-edge that the individual is singled out forsearch The reason that disrepute is particularlysalient among the various costs of beingsearched is that this component is not a primi-tive of the model Because of this fact policemay be particularly limited in its ability to affectthis component of the cost To the extent thatdisrepute is perceived as an important compo-nent of the cost of being searched a policymaker might conceivably attempt to in uence

its magnitude andor its distributional impactHowever we show that the total amount ofdisrepute within a group is independent of thepolicersquos search behavior and consequentlychanges in interdiction strategy have no effecton the amount of disrepute suffered by anygroup For the purpose of distribution acrossgroups therefore our analysis suggests that wecan ignore that part of the cost of being searchedwhich is due to disrepute Of course this ndingshould not be taken to imply that a redistribu-tional notion of fairness is unjusti ed Disreputeis just one determinant of the cost of beingsearched To the extent that the cost of beingsearched re ects ldquoexogenousrdquo costs such asloss of time or other inconvenience associatedwith being searched it is perfectly reasonable toattempt to equate those costs across races Weview our result as cause for optimism in thesense that it allows us to focus attention on thesenonreputationalcomponents of the cost of beingsearched Such costs are arguably within thecontrol of a stringent remedial policy

Related Literature

The formal economic literature on crime andpolicing is vast starting from Gary S Becker(1968) and Isaac Ehrlich (1973) we refer toEhrlich (1996) for a survey In this literaturethe issue of fairness is generally posed in termsof relationship between crime and sanction Forexample in A Mitchell Polinsky and StevenShavell (1998) a sanction is ldquofairrdquo when it isproportional to the gravity of the act committedIn contrast our notion of fairness applies to thestrategy of policing and is a statement about theintensity with which two different groups arepoliced In a recent paper (Knowles et al 2001)a model similar to the one used here was em-ployed in an empirical analysis of highwaysearches for drugs That paper asks whether thepolice force is racially prejudiced and it doesnot address issues of fairness

We study the effects of placing constraints onpolice behavior to achieve outcomes that aremore fair This is a comparative statics exerciseand in this sense is related to the literature onaf rmative action (Stephen Coate and Glenn CLoury 1993 George Mailath et al 2000 An-drea Moro and Peter Norman 2000 Norman2000 That literature looks at the effects of


constraining the hiring behavior of rms Whileour analysis is different in many respects onefundamental difference is worth highlightingThe af rmative action literature builds on Ken-neth J Arrowrsquos (1973) model of statistical dis-crimination which analyzes a game withmultiple equilibria While that literature hasbeen very successful in highlighting a realisticforce leading to discrimination the comparativestatics conclusions about the effect of givenpolicies are often quali ed by the fact that theyapply to a speci c equilibrium In contrastwhile we take as given the heterogeneity be-tween groups and therefore do not explain howidentical groups can be treated differently inequilibrium our model does have a unique equi-librium and so our comparative statics results arenot subject to the multiple-equilibria criticism

There is a literature on the effect of penaltieson the crime rate This literature generally relieson exogenous variation in the interdiction effortor in the penalty to estimate the elasticity ofcrime to interdiction (see Levitt [1997] andAvner Bar-Ilan and Bruce Sacerdote [2001]who also provide a good review of the literatureon this topic) To our knowledge this literaturedoes not address the different elasticity of dif-ferent racial groups to crime An exception isBar-Ilan and Sacerdote (2001) who nd that asthe ne is increased for running a red light inIsrael the total decrease in tickets is muchlarger for Jews than for non-Jews

I Model

The following model is a stylized represen-tation of police activity and its impact on po-tential criminals The model is designed to moreclosely portray that area of interdiction which ishighly discretionary in nature in that it is thepolice who choose whom to investigate as op-posed to low-discretion interdiction in whichpolice basically respond to requests for inter-vention (for example domestic violence orrape) Examples of high-discretion interdic-tion are highway searches of vehicles ldquostopand friskrdquo neighborhood policing and airportsearches Allegations of racial disparities are mademainly in reference to high-discretion interdiction

We emphasize that attention is restricted tothe type of crimes that are the object of inter-diction for example transporting drugs in the

case of vehicular searches Thus although inwhat follows we will sometimes for brevityrefer to ldquocrimerdquo without qualifying its type themodel does not claim to have implications foraggregates such as the overall crime rate Themodel does however have implications for thefraction of motorists found guilty of transport-ing drugs on highways (if the model is appliedto vehicular searches)

There is a continuum of police of cers withmeasure 1 Police of cers search citizens andeach police of cer makes exactly S searchesWhen a citizen is searched who is engaged incriminal activity he is uncovered with proba-bility 1 A police of cer maximizes the numberof successful searches ie the probability ofuncovering criminals

There is a continuum of citizens partitionedinto two groups A and W with measures NA

and NW respectively Throughout the index r [A W will denote the group If a citizenengages in crime he receives utility (in mone-tary terms) of H 0 if not searched and J 0if searched If a citizen does not engage incrime he is legally employed and receives earn-ings x 0 independent of being searched

Every citizen knows his own legal earningopportunity x which is a realization from ran-dom variable Xr Denote with Fr and fr thecdf and density of Xr We assume that XA andXW are nonnegative and that their supports arebounded intervals including zero Notice thatthe distribution of legal earning opportunities isallowed to differ across groups A citizenrsquos le-gal earning opportunity x is unobservable to thepolice when they choose whether to search himThe police can however observe the group towhich the citizen belongs

In the aggregate there is a measure S ofsearches that is allocated between the twogroups Denote with Sr the measure of searchesof citizens of group r Feasibility requires thatSA 1 SW 5 S Denote with sr 5 SrNr the searchintensity in group r (ie the probability that arandom individual of group r is searched) Thefeasibility constraint can be expressed as

(1) NAsA 1 NWsW 5 S

Given a search intensity sr a citizen ofgroup r with legal earning opportunity x com-mits a crime if and only if


x sr J 1 ~1 2 sr H

5 sr ~J 2 H 1 H q~sr

The function q(s) represents the expected re-ward of being a criminal when the search inten-sity equals s The fraction of citizens of groupr that engages in criminal activity is Fr(q(sr))

We now give our de nition of effectivenessConsistent with the objective of this paper weuse a de nition of effectiveness that captureseffectiveness of interdiction9 When de ning ef-fectiveness of interdiction we should accountfor the costs and bene ts of the interdictionactivity In our model the cost of interdiction isS and is taken as given Therefore it is appro-priate for the next de nition to focus only on thebene ts of interdiction (ie the reduction incrime)

De nition 1 The outcome (sA sW) is moreeffective than the outcome (s9A s9W) if the totalnumber of citizens who commit crimes is lowerat (sA sW)

We now give our de nition of fairness whichshould be thought of as fairness of policing Anoutcome is completely fair if the two groups arepoliced with the same intensity By extensionmaking an outcome more fair means reducingthe difference between the intensity with whichthe two groups are policed

De nition 2 The outcome (sA sW) is morefair than the outcome (s9A s9W) if zsA 2 sWz zs9A 2 s9Wz The outcome (sA sW) is com-pletely fair if sA 5 sW

This de nition is designed to capture the con-cerns of those who criticize existing policingpractices It is consistent with the idea that thereis some (unmodeled) cost imposed on innocentcitizens who are searched and that it is desirable

to equalize the expected cost across races10

This de nition is essentially comparative sinceit focuses on the predicament of two citizenswho belong to different groups as such thisnotion of fairness cannot be discussed in amodel with only one group of citizens such asPolinsky and Shavell (1998)

II Equilibrium and Effectiveness of Interdiction

A Equilibrium Conditions

We use a superscript asterisk () to denoteequilibrium quantities A police of cer whomaximizes the number of successful searcheswill search only members of the group with thehighest fraction of criminals If the equilibriumis interior that is sA sW 0 a police of cermust be willing to search either group and soboth groups must have the same fraction ofcriminals Thus at an interior equilibrium (sAsW) the following condition must be veri ed

(2) FA ~q~sA 5 FW ~q~sW

If the equilibrium is not interior then aninequality will generally hold Whether theequilibrium is interior or not it is easy to seethat the value of the left- and right-hand sides ofequation (2) are uniquely determined in equi-librium The condition that pins them down isthat if a group is searched by the police then itmust have a fraction of criminals at least aslarge as the other group Leaving aside uninter-esting constellations of parameters11 this con-dition uniquely pins down the equilibrium intems of srsquos For instance if the equilibrium isinterior then obviously there is a unique pair(sA sW) which solves the equilibrium condi-tion (2) subject to the feasibility constraint (1)In this paper we restrict attention to constella-tions of parameters for which the equilibriumis interior that is both groups are searchedwith positive probability From the empiricalviewpoint this is a reasonable restriction To

9 In particular our de nition ignores the utility of citi-zens who commit a crime and so is not meant to capturePareto ef ciency While it may be interesting to discuss thewelfare effects of criminal activity and the merits of pun-ishing citizens for committing crimes this is not the focus ofthe debate Rather the public policy concern is for ef -ciency of interdiction

10 More on this in Section VII11 If S is very small or very large then in equilibrium all

citizens (of both races) commit crimes or no citizen com-mits a crime In this case a trivial multiplicity of equilibriaarises in terms of the police search intensity


guarantee interiority of equilibrium in the forth-coming analysis we will henceforth implicitlymaintain the following assumption

ASSUMPTION 1

FAX S

NA~J 2 H 1 HD FW ~H

Assumption 1 is used in the proof of Lemma 1and is more likely to be veri ed when S is large

An equilibrium prediction of our model isthat interdiction will be more intense for thegroup with more modest legal earning opportu-nities Indeed suppose that the distribution oflegal earning opportunities of one group saygroup W rst-order stochastically dominatethat of group A Then FW( x) FA( x) for all xso equation (2) requires sA $ sW Anothertighter equilibrium prediction is given by equa-tion (2) itself if the equilibrium is interior thesuccess rates of police searches should be equal-ized across groups The equalization is a directconsequence of our assumption that police max-imize successful searches12

Both these equilibrium predictions are con-sistent with the evidence gathered in a numberof interdiction environments In the case ofhighway interdiction Knowles et al (2001)present evidence that on Interstate 95 African-American motorists are roughly six times morelikely to be searched than white motorists andthat the success rate of searches is not signi -cantly different between the two groups (34percent for African-Americans and 32 percentfor whites) In that paper the success rate ofsearches is shown to be very similar (not sig-ni cantly different in the statistical sense)across other subgroups such as men andwomen (despite the fact that women aresearched much less frequently) motoristssearched at day and at night drivers of old andnew cars and drivers of own vehicles versusthird-party vehicles

Similarly in the case of precinct policing inNew York City Eliot Spitzer (1999) reports

African-Americans to be over six times morelikely to be ldquostopped and friskedrdquo than whitesbut the success rate of searches is quite similarin the two groups (11 percent versus 13 per-cent) Finally in the case of airport searches areport by US Customs concerning searches in1997ndash1998 reveals that 13 percent of thosesearched were African-Americans and 32 per-cent were whites13 and ldquo67 percent of whites63 percent of Blacks had contrabandrdquo14 Theevidence in these disparate instances of policingreveals a common pattern that is consistent withthe predictions of our model In particular the nding that success rates are similar across ra-cial groups is consistent with the assumptionthat police are indeed maximizing the successrate of their searches We take this evidence assupportive of our modeling assumptions

The nding of equal success rates of searchesmight seem to contradict the disparity in incar-ceration rates However equalization of successrates does not translate into equalization of in-carceration rates or probability of going to jailacross groups To verify this observe that thosewho are incarcerated in our model are the crim-inals who are searched by police Thus theequilibrium incarceration rate in group r equalssrFr(q(sr)) Since the term Fr(q(sr)) is in-dependent of r [see equation (2)] the incarcer-ation rate is highest in the group for which sr ishighest (ie the group that is searched moreintensely) To the extent that the interdictioneffort is more focused on African-Americans(as is the case for highway drug interdiction)the modelrsquos implication for incarceration ratesis consistent with the observed greater incarcer-ation rates within the African-American popu-lation relative to the white population

Finally we point out that in the equilibriumof our model it is possible that some groupshave a lower crime rate than others The modelpredicts equal crime rates only among thegroups that are searched To see this consider aricher model with more than two groups (sayold grandmothers and young males of bothgroups) and in which old grandmothers are

12 At an interior equilibrium individual police of cersare indifferent about the fraction of citizens of group W thatthey search and so they do not have a strict incentive toengage in racial pro ling The incentive is strict only out ofequilibrium

13 See Report on Personal Searches by the United StatesCustoms Service (US Customs Service p 6)

14 Cited from Deborah Ramirez et al (2000 p 10)


unlikely to engage in crime even if they are notpoliced at all In such a model police will onlysearch young males of both groups and will notsearch grandmothers The crime rate amonggrandmothers will be lower than among youngmales and the success rates of searches will beequalized among all groups that are searched(young males of group A and W) In light of thisdiscussion it may be best to interpret equation(2) as equating the success rate of searchesinstead of the aggregate crime rates

B Maximal Effectiveness Conditions

We denote maximal effectiveness (minimumcrime) outcomes by the superscript ldquoEFFrdquo Asocial planner who minimizes the total amountof crime subject to the feasibility constraintsolves

minsA sW

NA FA ~sA ~J 2 H 1 H

1 NW FW ~sW ~J 2 H 1 H

subject to NAsA 1 NWsW 5 S or using theconstraint to eliminate sW

minsA


1 NW FWX S 2 NAsA

NW~J 2 H 1 HD

The derivative with respect to sA is

(3) NA ~J 2 H fA ~sA ~J 2 H 1 H

2 fW ~sW ~J 2 H 1 H]

and equating to zero yields the rst-order con-dition for an interior optimum

(4) fA ~q~sAEFF 5 fW ~q~sW

EFF

This condition is necessary for maximal ef-fectiveness The densities fA and fW measure thesize of the population in the two groups whoselegal income is exactly equal to the expectedreturn to crime If the returns to crime areslightly altered fA and fW represent the rates atwhich members of the two groups will switch

from crime to legal activity or vice versa Wecan think of fA and fW as elasticities of crimewith respect to a change in policing At aninterior allocation that maximizes effectivenessthe elasticities are equal Condition (4) is dif-ferent from condition (2) and thus will generallynot hold in equilibrium therefore the equilib-rium allocation of policing effort is generallynot the most effective

III Benchmark The Symmetric Case

In this section we discuss the benchmark casewhere FA 5 FW that is the distribution of legalearning opportunities is the same in the twogroups We call this the symmetric case sincehere sA 5 sW (ie the equilibrium is symmet-ric and completely fair) Even in the symmetriccase in some circumstances the most effectiveallocation is very asymmetric and the symmet-ric (and hence completely fair) allocation min-imizes effectiveness This shows that the mosteffective allocation can be highly unfair andthat this feature does not depend on the hetero-geneity of income distributions (ie on the dif-ference between groups)15 We interpret thisresult as an indication that there is little logicalrelationship between the concept of effective-ness and properties of fairness

PROPOSITION 1 Suppose FA 5 FW [ FThen the equilibrium allocation is completelyfair If in addition F is a concave function on itssupport then the equilibrium allocation mini-mizes effectiveness and at the most effectiveallocation one of the two groups is either notpoliced at all or is policed with probability 1

PROOFSince FA 5 FW the equilibrium condition

(2) implies sA 5 sW so the equilibrium issymmetric and completely fair

To verify the second statement observe that

15 A similar conclusion that the ef cient allocation canbe unfair in a symmetric environment was reached byNorman (2000) in the context of a model of statisticaldiscrimination in the workplace a la Arrow (1973) Thererace-dependent (asymmetric) investment helps alleviate theinef ciency caused by workers who test badly and end upbeing assigned to low-skilled jobs despite having investedin human capital


since fA and fW are decreasing the only pair(sA sW) that solves the necessary condition (4)for an interior most effective allocation togetherwith the feasibility condition (1) is sA 5 sWConsider the second derivative of the plannerrsquosobjective function of Section II subsection Bevaluated at the symmetric allocation

NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW

Recall that maximizing effectiveness requiresminimizing the objective function Since by as-sumption f9A and f9W are negative the second-order conditions for effectiveness maximizationare not met by (sA sW) Indeed the allocation(sA sW) meets the conditions for effective-ness minimization Finally to show that themost effective allocation is a corner solutionobserve that (sA sW) is the unique solution toconditions (4) and (1) Since (sA sW) achievesa minimum the allocation that maximizes ef-fectiveness must be a corner solution16

The reason why it may be effective to moveaway from a symmetric allocation is as followsSuppose for simplicity the two groups haveequal size At the symmetric allocation sA 5sW [ s the size of the population who isexactly indifferent between their legal incomeand the expected return to crime is f(q(s)) inboth groups Suppose that a tiny bit of interdic-tion is shifted from group W to group A Ingroup A a fraction of size approximately laquo 3f(q(s) 2 laquo) will switch from crime to legalactivity while in group W a fraction of sizeapproximately laquo 3 f(q(s) 1 laquo) will switch

from legal activity to crime If f is decreasing(ie F is concave) fewer people take up crimethan switch away from it and so total crime isreduced relative to the symmetric allocation

The fact that the most effective allocation canbe completely lopsided is a result of our assump-tions on the technology of search and policing Ina more realistic model it is unlikely that the mosteffective allocation would entail complete absten-tion from policing one group of citizens We dis-cuss this issue in Section VIII subsection F

Proposition 1 highlights the fact that fairnessand effectiveness may be antithetical and thatthis is true for reasons that have nothing to dowith differences across groups Our main focushowever is on the effectiveness consequence ofperturbing the equilibrium allocation in the di-rection of fairness This question cannot be askedin the symmetric case since as we have seen theequilibrium outcome is already completely fairThis is the subject of the next sections

IV Small Adjustments Toward Fairness andEffectiveness of Interdiction

In this section we present conditions underwhich a small change from the equilibrium to-ward fairness (ie toward equalizing the searchintensities across groups) decreases effectivenessrelative to the equilibrium level To that end we rst introduce the notion of quantilendashquantile(QQ) plot The QQ plot is a construction of sta-tistics that nds applications in a number of elds descriptive statistics stochastic majoriza-tion and the theory of income inequality statis-tical inference and hypothesis testing17

De nition 3 h( x) 5 FA21(FW( x)) is the QQ

(quantilendashquantile) plot of FA against FW

The QQ plot is an increasing functionGiven any income level x with a given per-centile in the income distribution of group Wthe number h( x) indicates the income levelwith the same percentile in group A If for

16 An alternative method of proof would be to employinequalities coming from the concavity of F This methodof proof would not require F to be differentiable I amgrateful to an anonymous referee for pointing this out

17 See respectively M B Wilk and R Gnanadesikan(1968) Barry C Arnold (1987 p 47 ff) and Eric LLehmann (1988) A related notion the Ratio-at-Quantilesplot was employed by Albert Wohlstetter and SinclairColeman (1970) to describe income disparities betweenwhites and nonwhites


example 10 percent of members of group Whave income smaller than x then 10 percentof members of group A have income less thanh( x) In other words the function h( x)matches quantile x in group W to the samequantile h( x) in group A

We are interested in quantiles of the incomedistribution because citizens are assumed to com-pare the expected reward of engaging in crime totheir legal income If the quantiles of the incomedistribution are closely bunched together alarge fraction of the citizens has access to legalincomes that are close to each other and inparticular close to the income of those citizenswho are exactly indifferent between committingcrimes or not Thus a large fraction of citizensare close to indifferent between engaging incriminal activity or not and a small decrease inthe expected reward to crime will cause a largefraction of citizens to switch from criminal ac-tivity to the honest wage If instead the quan-tiles of the legal income distribution are spacedfar apart only a small fraction of the citizens areclose to indifferent between engaging in crimi-nal activity or not and so slightly changing theexpected rewards to crime only changes thebehavior of a small fraction of the population

Since we consider the experiment of redirect-ing interdiction power away from one grouptoward the other group what is relevant for us isthe relative spread of the quantiles If for ex-ample the quantiles of the legal income distri-bution in group W are more spread apart than ingroup A shifting interdiction power away fromgroup A will cause a large fraction of honestgroup-A citizens to switch to illegal activitywhile only few group-W criminals will switchto legal activities The relative spread of quan-tiles is related to the slope of h Suppose forexample that x and y represent the 10 and 11percent quantiles in the W group If h9( x) issmaller than 1 then x and y are farther apartthan the 10- and 11-percent quantiles in the Agroup In this case we should expect group Wto be less responsive to policing than group AThis is the idea behind the next lemma

LEMMA 1 Suppose FW rst-order stochasti-cally dominates FA Then making the equilib-rium allocation marginally more fair increasesthe total amount of crime if and only ifh9(q(sW)) 1

PROOFFirst observe that the equilibrium is interior

Indeed it is not an equilibrium to have sA 5S NA and sW 5 0 because then we would have

FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0

where the inequality follows from Assumption1 This inequality cannot hold in equilibriumsince then a police of cerrsquos best response wouldbe to search only group W and this is incon-sistent with sW 5 0 A similar argument showsthat it is not an equilibrium to have sW 5 S NWand sA 5 0 since by stochastic dominance

FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD

The (interior) equilibrium must satisfy con-dition (2) Leaving aside trivial cases where inequilibrium every citizen or no citizen is crim-inal and so marginal changes in policing inten-sity do not affect the total amount of crimecondition (2) can be written as

(5) q~sA 5 h~q~sW

Now given any x by de nition of h we haveFW( x) 5 FA(h( x)) and differentiating withrespect to x yields

fW ~x 5 h9~x 3 fA ~h~x

Substituting q(sW) for x and using (5) yields

(6) fW ~q~sW 5 h9~q~sW 3 fA ~q~sA

This equality must hold in equilibrium Nowevaluate expression (3) at the equilibrium anduse (6) to rewrite it as

(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW

This formula represents the derivative at theequilibrium point of the total amount of crimewith respect to sA If h9(q(sW)) 1 thisexpression is negative (remember that J 2 H isnegative) and so a small decrease in sA from


sA increases the total amount of crime Toconclude the proof we need to show that mov-ing from the equilibrium toward fairness entailsdecreasing sA

To this end note that since FW rst-orderstochastically dominates FA then h( x) x forall x But then equation (5) implies q(sA) q(sW) and so sA $ sW [remember that q(z) isa decreasing function] Thus the A group issearched more intensely than the W group inequilibrium and so moving from the equilib-rium toward fairness requires decreasing sA

Expression (7) re ects the intuition given be-fore Lemma 1 the magnitude of h9 measures theinef ciency of marginally shifting the focus ofinterdiction toward group A When h9 is small itpays to increase sA because the elasticity to po-licing at equilibrium is higher in the A group

De nition 4 FW is a stretch of FA if h9( x) 1 for all x

The notion of ldquostretchrdquo is novel in this paperIf FW is a stretch of FA quantiles that are farapart in XW will be close to each other in XAIntuitively the random variable XA is a ldquocom-pressionrdquo of XW because it is the image of XWthrough a function that concentrates or shrinksintervals in the domain Conversely it is appro-priate to think of XW as a stretched version ofXA It is easy to give examples of stretches AUniform (or Normal) distribution XW is astretch of any other Uniform (or Normal) dis-tribution XA with smaller variance18

The QQ plot can be used to express relationsof stochastic dominance It is immediate tocheck that FW rst-order stochastically domi-nates FA if and only if h( x) x for all x Butthe property that for all x h( x) x is impliedby the property that for all x h9( x) 119 In

other words if FW is a stretch of FA then afortiori FW rst-order stochastically dominatesFA (the converse is not true) This observationallows us to translate Lemma 1 into the follow-ing proposition

PROPOSITION 2 Suppose that FW is astretch of FA Then a marginal change fromthe equilibrium toward fairness decreaseseffectiveness

V Implementing Complete Fairness

Under the conditions of Proposition 2 smallchanges toward fairness unambiguously de-crease effectiveness we now turn to largechanges We give suf cient conditions underwhich implementing the completely fair out-come decreases effectiveness relative to theequilibrium point

PROPOSITION 3 Assume that FW rst-orderstochastically dominates FA and suppose thatthere are two numbers

q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q

Assume further that the equilibrium outcome(sA sW) is such that

q q(sW) q(sA) q

Then the completely fair outcome is less effectivethan the equilibrium outcome

PROOFThe proof of this proposition is along the

lines of Lemma 1 but is more involved and notvery insightful and is therefore relegated to theAppendix

Since minz [ [h(x)x] fA( z) fA(h( x)) condi-tion (8) is more stringent than simply requiringthat h9( x) 1 It is sometimes easy to checkwhether condition (8) holds20 However condi-

18 Indeed when XW is Uniform (or Normal) the randomvariable a 1 bXW is distributed as Uniform (or Normal)This means that when XA and XW are both Uniformly (orNormally) distributed random variables the QQ plot has alinear form h( x) 5 a 1 bx In this case h9 1 if and onlyif b 1 if and only if Var(XA) 5 b2Var(XW) Var(XW)The notion of stretch is related to some conditions that arisein the study of income mobility (Roland Benabou and EfeA Ok 2000) and of income inequality and taxation (UlfJakobsson 1976)

19 The implication makes use of the assumption that thein ma of the supports of FA and FW coincide

20 For instance suppose FA is a uniform distributionbetween 0 and K In this case any function h with h(0) 50 and derivative less than 1 satis es the conditions inProposition 3 for all x in [0 K] Indeed in this case


tion (8) can only hold on a limited interval [q

q ] This interval cannot encompass the entiresupport of the two distributions and in generalthere will exist values of S such that eitherq(sA) or q(sW) lie outside the interval [

q q ]

More importantly there generally exist valuesof S for which the conclusion of Proposition 3 isoverturned and there is no trade-off betweencomplete fairness and effectiveness This isshown in the next proposition

PROPOSITION 4 Assume FA(H) 5 1 orFW(H) 5 1 or both Assume further that thesuprema of the supports of FA and FW do notcoincide Then there are values of S such thatthe completely fair outcome is more effectivethan the equilibrium outcome

PROOFDenote with q the supremum of the support

of FA We can restate the assumption on thesuprema of the supports as FW(q) 1 5FA(q) (this is without loss of generality as wecan switch the labels A and W to ensure that thisproperty holds)

Denote with s 5 S (NA 1 NW) the searchintensity at the completely fair outcome Thevariation in crime due to the shift to completefairness is

(9) TV 5 NA FA ~q~s 2 FA ~q~sA

1 NW FW ~q~s 2 FW ~q~sW

Case FW(H) 1 In this case our assumptionsguarantee that FA(H) 5 1 Thus for S suf -ciently small we get a corner equilibrium inwhich sW 5 0 and sA 5 S NA ThereforesW s sA Further when S is suf cientlysmall q(sA) 5 sA( J 2 H) 1 H approachesH In turn H is larger than q since FA(H) 5 1Therefore for small S we must have q(sA) q In sum for S suf ciently small we have

q q~sA q~s q~sW 5 H

These inequalities imply that FW(q(s )) FW(H) and also since FA(q) 5 1 that

FA(q(sA)) 5 FA(q(s )) 5 1 ConsequentlyTV 0

Case FW(H) 5 1 In this case pick the maxi-mum value of S such that in equilibrium allcitizens engage in crime Formally this is thevalue of S giving rise to (sW sA) with theproperty that FW(q(sW)) 5 FA(q(sA)) 5 1and Fr(q(sr 1 laquo)) 1 for all laquo 0 and r 5A W By de nition of q we are guaranteed thatq 5 q(sA) q(s ) q(sW) Expression (9)becomes

TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0

It is important to notice that Proposition 4holds for most pairs of cdfrsquos including thosefor which h9 1 Comparing with Proposition2 one could infer that it is easier to predict theimpact on effectiveness of small changes infairness rather than the effects of a shift tocomplete fairness21 By showing that the con-clusions of Proposition 3 can be overturned fora very large set of cdfrsquos (by choosing S ap-propriately) Proposition 4 highlights the impor-tance of knowing whether the equilibrium valuesq(sr) lie inside the interval [

q q ] this is

additional information beyond the knowledgeof FA and FW which was not necessary forProposition 2

VI Recovering the Distribution of LegalEarning Opportunities

The QQ plot is constructed from the distri-butions of potential legal earning opportunitiesPotential legal earnings however are some-times unobservable This is because accordingto our model citizens with potential legal in-come below a threshold choose to engage incrime These citizens do not take advantage oftheir legal earning opportunities Some of thesecitizens will be caught and put in jail and are

fA( y)fA(h( x)) 5 1 for all y in [h( x) x] so long as x [[0 K]

21 Although in the proof of Proposition 4 we have chosenthe value of S such that at the completely fair outcome allcitizens of one group engage in crime this feature is notnecessary for the conclusion In fact it is possible to con-struct examples in which even as h9 1 (a) the com-pletely fair allocation is more ef cient than the equilibriumone and (b) at the completely fair allocation both groupshave a fraction of criminals between 0 and 1


thus missing from our data which raises theissue of truncation The remaining fraction ofthose who choose to become criminals escapesdetection When polled these people are un-likely to report as their income the potentiallegal incomemdashthe income that they would haveearned had they not engaged in crime If weassume that these ldquoluckyrdquo criminals report zeroincome when polled then we have a problem ofcensoring in our data

In this section we present a simple extensionof the model that is more realistic and deals withthese selection problems Under the assump-tions of this augmented model the QQ plot ofreported earnings can easily be adjusted to co-incide with the QQ plot of potential legal earn-ings Thus measurement of the QQ plot ofpotential legal earnings is unaffected by theselection problem even though the cdfrsquos them-selves are affected

Consider a world in which a fraction a ofcitizens of both groups is decent (ie will neverengage in crime no matter how low their legalearning opportunity) The remaining fraction(1 2 a) of citizens are strategic (ie will en-gage in crime if the expected return from crimeexceeds their legal income opportunity this isthe type of agent we have discussed until now)The police cannot distinguish whether a citizenis decent or strategic The base model in theprevious sections corresponds to the case wherea 5 0

Given a search intensity sr for group r thefraction of citizens of group r who engage incrime is (1 2 a) Fr(q(sr)) The analysis ofSection II goes through almost unchanged in theextended model and it is easy to see that

(i) For any a 0 the equilibrium and effec-tiveness conditions coincide with condi-tions (2) and (4)

(ii) Consequently the equilibrium and mosteffective allocations are independent of a

(iii) The analysis of Sections IV and V goesthrough verbatim

This means that also in this augmentedmodel our predictions concerning the relation-ship between effectiveness and fairness dependon the shape of the QQ plot of (unobservable)legal earning opportunities exactly as describedin Propositions 2ndash4 Therefore in terms of

equilibrium analysis the augmented model be-haves exactly like the case a 5 0

Furthermore the augmented model is morerealistic since it allows for a category of peoplewho do not engage in crime even when theirlegal earning opportunities are low This ex-tended model allows realistically to pose thequestion of recovering the distribution of legalearning opportunities To understand the keyissue observe that in the extreme case of a 5 0which is discussed in the preceding sections allcitizens with legal earning opportunities belowa threshold become criminals and do not reporttheir potential legal income If we assume thatthey report zero the resulting cdf of reportedincome has a at spot starting at zero This starkfeature is unrealistic and is unlikely to be veri- ed in the data

When a 0 the natural way to proceedwould be to recover the cdfrsquos of legal earningopportunities from the cdfrsquos of reported in-come However this exercise may not be easy ifwe do not know a and the equilibrium valuesq(sr) The next proposition offers a procedureto bypass this obstacle by directly computingthe QQ plot based on reported incomes Theresulting plot is guaranteed to coincide underthe assumptions of our model with the onebased on earning opportunities

PROPOSITION 5 Assume all citizens who en-gage in crime report earnings of zero while allcitizens who do not engage in crime realizetheir potential legal earnings and report themtruthfullyThen given any fraction a [ (0 1) ofdecent citizens the QQ plot of the reportedearnings equals in equilibrium the QQ plot ofpotential legal earnings

PROOFLet us compute Fr( x) the distribution of

reported earnings All citizens of group r withpotential legal earnings of x q(sr) do notengage in crime regardless of whether they aredecent or strategic They realize their potentiallegal earnings and report them truthfully Thismeans that Fr( x) 5 Fr( x) when x q(sr)

In contrast citizens of group r with an x q(sr) engage in crime when they are strategicand report zero income Thus for x q(sr)the fraction of citizens of group r who reportincome less than x is


F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr

The rst term represents the decent citizens whoreport their legal income the second term thosestrategic citizens who engage in crime and byassumption report zero income The functionFr( x) is continuous Furthermore the quantity(1 2 a) Fr(q(sr)) is equal to some constant bwhich due to the equilibrium condition is in-dependent of r We can therefore write

F r ~x 5 5 aF r ~x 1 b for x q~sr F r ~x for x q~sr

The inverse of Fr reads

(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr

F r2 1~ p for p F r ~q~sr

Denote the QQ plot of the reported incomedistributions by

h~x FA2 1~FW ~x

When x q(sW) we have FW( x) 5 FW( x)and so

h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x

where the second equality follows from (10) be-cause here FW(x) FW(q(sW)) 5 FA(q(sA))When x q(sW) then FW( x) 5 aFW( x) 1 band so

h~x 5 FA2 1~aFW ~x 1 b

5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x

where the second equality follows from (10)because here aFW( x) 1 b aFW(q(sW)) 1

b 5 FW(q(sW)) 5 FA(q(sA)) (the rst equal-ity follows from the de nition of b and thesecond from the equilibrium conditions) Thisshows that h( x) 5 h( x) for all x

Proposition 5 suggests a method which un-der the assumptions of our model can be em-ployed to recover the QQ plot of potential legalearnings starting from the distributions of re-ported earnings It suf ces to add to the samplethe proportion of people who are not sampledbecause incarcerated and count them as report-ing zero income Since we already assume thatldquoluckyrdquo criminals (who are in our sample) re-port zero income this modi cation achieves asituation in which all citizenswho engage in crimereport earnings of zero The modi ed sample isthen consistent with the assumptions of Proposi-tion 5 and so yields the correct QQ plot eventhough the distributions of reported income sufferfrom selection Notice that to implement this pro-cedure it is not necessary to know a or sr

As an illustration of this methodology we cancompute the QQ plot based on data from theMarch 1999 CPS supplement We take thecdfrsquos of the yearly earning distributions ofmales of age 15ndash55 who reside in metropolitanstatistical areas of the United States22 Fig-ure 1 presents the cdfrsquos of reported earnings(earnings are on the horizontal axis)

22 We exclude the disabled and the military personnelEarnings are given by the variable PEARNVAL and includetotal wage and salary self-employment earnings and anyfarm self-employment earnings

FIGURE 1 CUMULATIVE DISTRIBUTION FUNCTIONS OF

EARNINGS OF AFRICAN-AMERICANS (AArsquoS)AND WHITES (WrsquoS)


It is clear that the earnings of whites stochas-tically dominates those of African-AmericansFor each race we then add a fraction of zerosequal to the percentage of males who are incar-cerated and then compute the QQ plot23 Fig-ure 2 presents the QQ plot between earninglevels of zero and $100000 earnings in thewhite population are on the horizontal axis24

This picture suggests that the stretch require-ment is likely to be satis ed except perhaps inan interval of earnings for whites between10000 and 15000 In order to get a numer-ical derivative of the QQ plot that is stable we rst smooth the probability density functions(pdfrsquos) and then use the smoothed pdfrsquos toconstruct a smoothed QQ plot25 Figure 3 pre-sents the numerical derivative of the smoothedQQ plot

The picture suggests that the derivative of theQQ plot tends to be smaller than 1 at earningssmaller than $100000 for whites The deriva-tive approaches 1 around earnings of $12000for whites In the context of our stylized modelthis nding implies that if one were to movetoward fairness by constraining police to shift

resources from the A group to the W group thenthe total amount of crime would not fall Weemphasize that we should not interpret this as apolicy statement since the model is too generalto be applied to any speci c environment Weview this back-of-the-envelope computation asan illustration of our results

It is interesting to check whether condition(8) in Proposition 3 is satis ed That conditionis satis ed whenever

r~x h9~xfA ~h~x minz [ h~xx

fA ~z 1

Figure 4 plots this ratio (vertical axis) as afunction of white earnings (horizontal axis)

The ratio does not appear to be clearly below1 except at very low values of white earningsand clearly exceeds 1 for earnings greater than$50000 Thus condition (8) does not seem to

23 Of 100000 African-American (white) citizens 6838(990) are incarcerated according to data from the Depart-ment of Justice

24 In the interest of pictorial clarity we do not report theQQ plot for those whites with legal earnings above$100000 These individuals are few and the forces capturedby our model are unlikely to accurately describe their in-centives to engage in crime

25 The smoothed pdfrsquos are computed using a quartickernel density estimator with a bandwidth of $9000

FIGURE 2 QQ PLOT BASED ON EARNINGS OF

AFRICAN-AMERICANS AND WHITES

FIGURE 3 NUMERICAL DERIVATIVE OF THE QQ PLOT

FIGURE 4 PLOT OF r


be useful in this instance to sign the effect of aswitch to complete fairness

VII Disrepute as a Cost of Being Searched

In this paper we posit that it is desirable toequalize search intensities across groups Thisprinciple rests on the idea that being searchedby police entails a cost of time or distress andthat citizens of all groups should bear this costequally (in expectation)26 It may seem harm-less to include in this computation those costswhich relate to the loss of reputation of theindividual being searched This is to capture thestigmatization shame or disrepute that is at-tached to being singled out for search by thepolice

Being publicly singled out for search entails acost because external observers (bystandersneighbors etc) use the search event to infersomething about the criminal propensity of theindividual who is subject to search For in-stance if police have knowledge of the criminalrecord of an individual at the time of the searchand an external observer does not the observershould use the search event to update his opin-ion about the criminal record of the individualwho is searched27 Christopher Slobogin (1991)calls this notion ldquofalse stigmatizationrdquo and saysthat ldquo the innocent individual can legitimatelyclaim an interest in avoiding the stigma embar-rassment and inconvenience of a mistaken in-vestigationrdquo This interest is listed as a keyinterest of citizens in avoiding search andseizure

An interesting feature of disrepute as we havede ned it is that it is endogenous in the sensethat it is the result of Bayesian updating and isnot a primitive of the model The amount ofdisrepute that is attached to being searched willgenerally depend on the search strategy that

police use The fact that the loss of reputation isendogenous opens up the possibility that byemploying different search strategies the policemight reduce the total amount of discredit in theeconomy and achieve a Pareto improvementAlternatively it seems possible that by alteringtheir search strategy the police might transfersome of the burden across groups

However we argue that neither possibility istrue To esh out the argument let us establisha minimal model in which it is meaningful totalk about disrepute Since we wish to capture anotion of updating on the part of bystandersin addition to the observable characteristicldquogrouprdquo there must be some characteristicwhich is unobservable to the bystanders butobservable to police (and this characteristicmust be correlated with criminal behavior) De-note this characteristic by c For concretenesswe refer to this characteristic as the criminalrecord and we denote by Pr(c zr) its distributionin each group Since the police condition theirsearch behavior on c upon seeing an individualbeing searched the bystanders would updatetheir prior about the c of the individual beingsearched and hence about his criminal propen-sity Denote with s (c r) the probability ofbeing searched for a random individual withcriminal record c and group r

De ne the disrepute d of a member of groupr as

d~r Pr~r is criminal

2 Pr~r is criminalzsearched

where the event ldquor is criminalzsearchedrdquo de-notes the event that a randomly chosen individ-ual of group r is criminal conditional on beingsearched28 Thus we de ne the cost of disre-pute as the amount of updating that an observerdoes about an individual of group r upon seeingthat he is being searched This de nition cap-tures the humiliation connected with being pub-licly searched

Consistent with the notion of updating wemust also take into account the fact that an

26 These costs encompass ldquoobjectiverdquo and ldquosubjectiverdquointrusions as de ned in Michigan Department of Police vSitz 496 US 444 (1990)

27 We refer to the criminal record for concreteness Inhighway interdiction for example police often have knowl-edge of the criminal record because they radio in the driv-errsquos license of the individual who is stopped before decidingwhether to initiate a search Of course our analysis isequally relevant if police are more informed than the exter-nal observer about any characteristic of the individual whomay be searched

28 There is nothing special about the event ldquor is crimi-nalrdquo The argument in this section applies equally to anyevent of the form ldquothe individual of group r has a value ofc [ Erdquo where E is any set


observer updates about a random individualalso when that individual is not searched Indi-viduals who are not searched experience a cor-responding increase in status in the eye of anexternal observer We term this effect negativedisrepute and we denote it by d Formally

d ~r Pr~r is criminal

2 Pr~r is criminalznot searched

The total disrepute effect (positive and nega-tive) in group r is given by

(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0

where s (r) 5 yenc s (r c)Pr(c zr) denotes theprobability that a randomly chosen individual ofgroup r is searched Equality to zero followsfrom the fact that conditional probabilities aremartingales Notice that equation (11) holdstrue for any value of s (r) regardless of howthe search intensity is distributed betweengroups the disrepute effect has constant sum(zero) within a group29 This means that interms of disrepute the search strategy is neutralwithin group r changing the search strategycannot change the average disrepute within agroup

This neutrality result says that redistributingsearch intensity across groups has no effect onthe disrepute portion of the average (within agroup) cost of being searched Thus when weevaluate the effects of a certain search strategywe can ignore the distributive effects of disre-pute across groups This nding is in contrastwith the conventional view that ldquoreputationalrdquocosts are an important element of disparity inpolicing This view relies on a logical failure totake into account the complete effects of Bayes-ian updating (ie the effect on those who arenot searched) After taking into account the fulleffects of Bayesian updating a redistributivenotion of fairness seems harder to justify purelyon the basis of reputational costs

As mentioned in the introduction this line ofinquiry does not lead to questioning the appro-

priateness of refocusing search intensity fromAfrican-Americans to whites Disrepute is butone of the components of the cost of beingsearched To the extent that being searched in-volves important nonreputational costs (such asloss of time risk of being abused etc) it isperfectly reasonable to wish to equate thesecosts across races The point of our line ofinquiry is to suggest that if nonreputationalcosts are what causes the unfairness there isprobably room for improvement through reme-dial policies such as sensitivity training for po-lice These policies can reduce the cost of beingsearched but only if the cost is nonreputational

VIII Discussion of Modeling Choices

A The Objective Function of Police

An important assumption in our model is thatpolice choose whom to search so as to maxi-mize successful searches In the equilibriumanalysis this assumption is re ected in theequalization across groups of the success ratesof searches This equalization is observed in anumber of instances of interdiction includingvehicular searches ldquostop and friskrdquo neighbor-hood patrolling and airport searches as dis-cussed in Section II subsection A We view thisevidence as supportive of our assumptionsabout the motivation of police since equality ofthe crime rates across different categories ofcitizens is not likely to arise under other as-sumptions about the motivation of police

Additional support for the assumption thatpolice maximize successful searches is pro-vided for the case of highway interdiction bythe explicit reference to this point in Vernieroand Zoubek (1999) who describe the trooperrsquosincentive system as follows

[E]vidence has surfaced that minoritytroopers may also have been caught up ina system that rewards of cers based onthe quantity of drugs that they have dis-covered during routine traf c stops The typical trooper is an intelligent ra-tional ambitious and career-oriented pro-fessional who responds to the prospectof rewards and promotions as much as tothe threat of discipline and punishmentThe system of organizational rewards byde nition and design exerts a powerful

29 This phenomenon is purely a consequence of Bayes-ian updating and does not hinge on any assumption aboutthe behavior of either police or citizens


in uence on of cer performance and en-forcement priorities The State Policetherefore need to carefully examine theirsystem for awarding promotions and fa-vored duty assignments and we expectthat this will be one of the signi cantissues to be addressed in detail in futurereports of the Review Team It is none-theless important for us to note in thisReport that the perception persistedthroughout the ranks of the State Policemembers assigned to the Turnpike thatone of the best ways to gain distinction isto be aggressive in interdicting drugsThis point is best illustrated by theTrooper of the Year Award It was widelybelieved that this singular honor was re-served for the trooper who made the mostdrug arrests and the largest drug seizuresThis award sent a clear and strong mes-sage to the rank and le reinforcing thenotion that more common rewards andpromotions would be provided to trooperswho proved to be particularly adept atferreting out illicit drugs30

B Alternative Objectives of Police

A system of rewards based on successfulsearches helps motivate police to exert effort ina world where an individual police of cerrsquoseffort is too small to measurably affect the ag-gregate crime level On this basis we shouldbelieve that police incentives should be based atleast partly on success maximization and thatthese incentives will partly affect the racial dis-tribution of searches At the same time otherfactors may also play a role in the allocation ofpolice manpower across racial groups in citypolicing for example crime in certain wealthy(and disproportionately white) neighborhoodsmay have great political salience so thoseneighborhoods will be allocated more interdic-tion power and will experience lower crimerates It is therefore interesting to consider anextension of the model which re ects theseconsiderations

Consider a situation of city policing in whichpolice incentives are shaped by political pres-sure to stamp out crime committed in nonmi-nority neighborhoods Given any allocation of

police manpower across neighborhoods thebase model of Section I would apply withineach neighborhood If however neighborhoodsare segregated making the allocation more fairmay require shifting manpower across neigh-borhoods This is a case that is not contemplatedin our base model The model can neverthelessbe adapted to approximate this situation if weimagine two completely segregated neighbor-hoods the A and the W Citizens (criminals ornot) cannot escape their neighborhood Assumethat police are rewarded more highly for catch-ing a criminal belonging to neighborhood WUnder this assumption the equilibrium crimerate (and the success rate of searches) will nolonger be constant across the two groups In-stead the equilibrium crime rate will be higherin neighborhood A At the same time as long asthe rewards for catching a neighborhood-Wcriminal are not too much larger than those forcatching a neighborhood-A criminal the Wneighborhood will be searched with less inten-sity than the A neighborhood There is somerealistic appeal in this combination of highersearch intensity and higher crime rate (and suc-cess rate of searches) in the A neighborhoodStarting from this premise we ask what hap-pens to the crime rate if police are required tobehave more fairly

To answer this question denote with sr theequilibrium search intensities in the augmentedmodel in which police receive a higher rewardfor busting a neighborhood-W criminal Since atan interior equilibrium police must be indiffer-ent between searching a member of the twoneighborhoods the expected probability of suc-cess must be lower in neighborhood W that isFA(q(sA)) FW(q(sW)) This inequalitycan be rewritten as

q~sA h~q~sW

In addition we have posited that the equilib-rium search intensity in neighborhood W doesnot exceed that in neighborhood A or equiva-lently that

q~sA q~sW

Equipped with these inequalities let us writedown the total crime rate Assume for simplicitythat the two neighborhoods are of the same size30 Cited from Verniero and Zoubek (1999 pp 42ndash43)


(all the steps go through almost unchanged andthe conclusion is the same if we remove thissimplifying assumption) The crime rate is then

FA ~q~sA 1 FW ~q~sW

Transfering one unit of interdiction from the Ato the W neighborhood increases total crime ifand only if

fW ~q~sW fA ~q~sA

Since h(q(sW)) q(sA) q(sW) thiscondition will hold if

fW ~q~sW minz [ h~q~sWq~sW

fA ~z

Rewrite this inequality substituting x for q(sW)and divide through by fA(h(x)) to obtain

h9~x

minz [ h~xx

fA ~z

fA ~h~x

This condition coincides with condition (8)from Proposition 3 We have therefore shownthat in the augmented model in which policerewards are greater for busting neighborhood-Wcriminals condition (8) is a suf cient conditionfor there to exist a trade-off between fairnessand effectiveness

C Nonracially Biased Police

In this paper we assume that police are notracially biased (ie that they do not derivegreater utility from searching members of groupA rather than members of group W) We do thisfor two reasons First we are interested in howthe system of police incentives (maximizingsuccessful searches) results in disparate treat-ment and the theoretically clean way to addressthis question is to abstract from any bigotrySecond at least in some practical instances ofalleged disparate impact the unfairness is as-cribed to the incentive system and not directlyto racial bias on the part of police this is theposition in Verniero and Zoubek (1999 seetheir part III-D) and is also consistent with the ndings in Knowles et al (2001) Therefore wethink it is interesting to study our problem ab-stracting from the issue of bigotry If police

were racially biased then that would be anadditional factor affecting the equilibrium butin terms of the focus of this paper the presenceof racial bias on the part of the police need notnecessarily make it more effective to implementfairnessmdashalthough it would probably makefairness more desirable on moral grounds

In this connection it is worth emphasizingthat while our model focuses on the economicincentives to commit crime it ignores the pos-sibility that unfairness of policing per se encour-ages crime among those who are unfairlytreated According to Robert Agnewrsquos (1992)general strain theory the anticipated presenta-tion of negative stimuli results in strain Strainin turn causes individuals to resort to illegiti-mate ways to achieve their goals If we take thisviewpoint the expectation of being unfairlytreated by police can produce strain There isevidence that some groups anticipate beingfrustrated in the interaction with police (seeAnderson 1990) According to this view inter-diction power can be not a deterrent but a causeof crime if it is unfairly applied

D Differential Deterrence

We have assumed that the deterrence effect isidentical for both groups This is a reasonableassumption as long as crimersquos gain and punish-ment are similar across groups It is possiblehowever that convicted criminals from a givengroup are likely to incur more severe punish-ment (longer prison sentences) leading to asmaller value of J for that group On the otherhand the social stigma from being convictedpresumably also a component of J could beweaker in a given group leading to larger val-ues of J Along the same lines it could beargued that H is larger for one group whenbeing a (successful) criminal does not carrymuch social stigma in that group To allow forthe possible difference in the deterrence effectof policing we now explore what happens tothe key result in this paper Lemma 1 if weassume that J and H are group-speci c31

31 To some extent the effects mentioned above maycountervail themselves Consider for example a thoughtexperiment in which the social stigma attached to being acriminal decreases In our formalization this decrease willresult in higher values of both J and H but the difference


Consider an augmented model in which Jr

and Hr denote the group-speci c cost and ben-e t of crime but which is otherwise the same asthe one of Section I Denote by sr the equi-librium search intensities in this augmentedmodel The police maximization problem willstill require that in equilibrium the fraction ofcriminals is the same in the two groups Thusthe equilibrium search intensities must solve

FA ~qA ~sA 5 FW ~qW ~sW

where qr(s) 5 s ( Jr 2 Hr) 1 Hr One canthen replicate the analysis of Lemma 1 and ndthat its conclusion holds if and only if

(12) h9~qW ~sW zJA 2 HAzzJW 2 HWz

In other words marginally shifting the focus ofinterdiction towards the W group increases thetotal amount of crime if and only if this inequalityholds Condition (12) differs from the condition inLemma 1 in the right-hand side of the inequalitywhich is no longer necessarily equal to 1 Thequantity zJr 2 Hrz represents the drop in utility thata criminal of group r experiences if caught itcaptures the deterrence effect of policing on groupr If zJA 2 HAz 5 zJW 2 HWz the deterrence effectof policing is the same in both groups and con-dition (12) reduces to the one in Lemma 1 Ifhowever zJA 2 HAz zJW 2 HWz the prospectof being caught deters citizens of group W morethan citizens of group A In this case condition(12) is stronger (more restrictive) than the con-dition in Lemma 1 re ecting the fact that shift-ing the focus of interdiction towards the Wgroup is less likely to result in increased crime

Condition (12) shows how our key result ischanged with differential deterrence The factthat there is a change highlights the importanceof the maintained assumption of equal deter-rence At the same time from the methodolog-ical viewpoint we nd that the basic analysisstraightforwardly generalizes to this richer en-vironment and equation (12) con rms the cen-tral role played by the QQ plot even in the caseof differential deterrence

E Changing Characteristics

We have assumed that the composition of thegroups is xed and cannot be changed In thissubsection we consider the case in which crim-inals can change their characteristics to reducethe risk of being caught In a model with morethan two groups we could think of many waysin which criminals of a highly targeted groupmight attempt to disguise some suspicious char-acteristic (by driving certain types of cars bydressing differently etc) to blend in with mem-bers of a different group Even in our simplemodel with just two groups we could get thesame effect if one racial group is searched morefrequently by highway patrols drug traf ckerswho belong to that group may decide to employcouriers or ldquomulesrdquo of the other racial group inorder to reduce the probability of their cargobeing discovered32 Knowles et al (2001) pointout that the key equilibrium prediction of thebase model namely equality of success rates ofsearch is preserved in the more general modelin which characteristics can be changed

To see how the argument works in our two-group model imagine that members of group Acan at cost c change their appearance to that ofa group-W member Assume further that FAstochastically dominates FW so that in themodel with xed characteristics we have sA sW Paying c allows a group-A member toenjoy the lower search intensity sW instead ofsA If c is not too high all those group-Acitizens who engage in crime nd it pro table topay c and change their appearance In this caseit is no longer an equilibrium for police tosearch those who look like they belong to groupA since there are no criminals among thosecitizens This means that the search intensity mustbe different in the equilibrium of the model withchangeable characteristics Yet if (sA sW)was an interior equilibrium when characteristicsare xed then a fortiori the equilibrium will beinterior if citizens can change their appearance33

That is ldquointeriorityrdquo of equilibrium is maintained

J 2 H may not be affected This difference is what mattersfor the analysis as shown in what follows

32 We assume that the drug traf ckers as well as themules will go to jail if their cargo is found by police

33 Equivalently and perhaps easier to see if an alloca-tion (sA sW) is a noninterior equilibrium in the model inwhich citizens can change their appearance then a fortiori itis an equilibrium if citizens cannot change their appearance


if we allow citizens to change characteristicsInteriority implies that police must obtain thesame success rate in searching either groupThus that key implication of our model equal-ity of success rates of searches among groups(now among those citizens who appear to be-long to the different groups) holds even if weallow citizens to choose their characteristics34

The elasticity of crime to policing will de-pend on the opportunity for groups to disguisethemselves Nevertheless some of our resultsstill apply concerning the trade-off between ef-fectiveness and fairness To see why let usevaluate the effect of a small shift in policingintensity starting from the equilibrium of themodel with changeable characteristics It is eas-iest to reason starting from an allocation that isslightly more fair than the equilibrium one wewill then use continuity of the crime rate in thesearch intensity to draw implications about theequilibrium allocation At this slightly fairerallocation no group-A criminal nds it expedi-ent to change appearance In this respect thesituation is similar to the base model in whichcharacteristics cannot be changed However theallocation is more fair than the equilibrium inthat base model group A must be policed lessintensely and group W more intensely in orderto make group-A citizens almost indifferent be-tween switching groups We are then in anenvironment that is identical to the one analyzedin subsection B The same condition condition(8) from Proposition 3 will be suf cient toidentify a trade-off between effectiveness andfairness Under this condition a small increasein fairness comes at the cost of increasedcrime Using the fact that the crime ratevaries continuously with intensity of interdic-tion we then extend this result to a nearbyallocation the equilibrium one We concludethat if condition (8) is met making the equilib-rium slightly more fair will increase the crime

rate even if citizens can choose to switch awayfrom group A

F The Technology of Search

An important simplifying assumption under-lying our analysis is that search intensity can beshifted effortlessly across groups Perfect sub-stitutability is a strong assumption In particu-lar achieving a very high search intensity ofone group may be very costly in terms of policemanpower Imagine for example that trooperswanted to stop and search almost all male driv-ers To achieve that high level of search inten-sity troopers would have to continually monitortraf c 24 hours a day and that may be verycostly The last unit of search intensity of malesis probably more costly than (cannot beachieved by forgoing) just one unit of searchintensity of females To the extent that the costof achieving a given search intensity differsacross groups the success rates of searches willdepart from equalization across groups Ulti-mately the question of scale ef ciencies in po-licing is an empirical one The result inKnowles et al (2001) suggest that in highwayinterdiction scale effects are not suf cientlyimportant to undermine equality of successrates In other situations however scale ef -ciencies may well play an important role

G Achieving the Most Effective Allocation

It is important to recognize that this papertakes a positive approachmdashin the sense that ittakes as given a realistic incentive structure andinvestigates the comparative statics propertiesof the model and whether there is a con ictbetween various desirable properties that maybe required of allocations Because police areassumed not to care about deterrence but onlyabout successful searches in the equilibrium ofthe model crime is not minimized

The approach in this paper is not normativein the sense that we do not pursue the questionof how to achieve particular allocations Forexample the fact that the most effective (crime-minimizing) outcome is generally not achievedin equilibrium does not mean that in this setupthe most effective outcome cannot be imple-mented In our simple model the most effectiveallocation could be implementedmdashat the cost of

34 One may be interested in the complete description ofequilibrium in this case Denote the equilibrium searchintensities by (sW sA) Group-A criminals are indifferentbetween switching group or not so we must have sA( J 2H) 5 sW( J 2 H) 2 c Group-A criminals will randomlyswitch group with a probability that is designed to equatethe fraction of criminals in group A to the fraction ofcriminals among those citizens who look like group W Thischoice of probability makes police willing to search the twogroups with search intensities (sA sW)


giving race-dependent incentives to police Bygiving police higher rewards for successfulbusts on citizens of a particular race it is pos-sible to induce any allocation of search intensityon the part of police In particular it will bepossible to implement the most effective allo-cation and the completely fair allocation35 Thisargument shows that our model is simpli ed insuch a way that asking the implementationquestion is not interestingmdashalthough the modelis well suited to asking other questions A modelthat were to attempt an interesting analysis ofpolicing from the viewpoint of implementationwould probably have to provide foundations forthe costs and bene ts of interdiction as well as oflegal and illegal activities Such an analysis is notthe goal of the present work

Nevertheless it is interesting to inquire aboutthe reason for why police might be given incen-tives that are out of line with crime reduction Inaddition to the fact that minimizing crime mightrequire incentive schemes that are highly unfairand therefore politically risky we can offeranother possible reason the crime rate is some-times not a direct concern of the police forcewho does the policing For example only arelatively small fraction of the illegal drugstransported on I-95 originates from or is di-rected to Maryland Most of it originates fromFlorida and is directed to the large metropolitanareas of the Northeast One might thereforeexpect that the Maryland police force policingI-95 would only have a partial stake in the crimerate generated by the I-95 drug traf c In suchcases it is not hard to believe that a policedepartment would maximize the number ofdrug nds and place little emphasis on crimeminimization

H Notions of Effectiveness of Interdiction

We have taken the position that effectivenessof interdiction is measured by the total numberof citizens who commit crimes According tothis measure given a total number of crimeseffectiveness of interdiction is the same regard-

less of whether most of the crimes are commit-ted by one racial group or equally by both racialgroups However one may argue that if most ofthe crimes are committed by one racial groupthen depending on the type of crime membersof that group may be targets of crime moreoften This is a valid argument which suggeststhat one should also try to reduce the inequalityin the distribution of crime across groups Thisraises some interesting questions such as howmuch inequality in crime rates should be toler-ated across groups Going to one extreme andkeeping the crime rate completely homoge-neous across groups necessitates policing dif-ferent groups with different intensity (this is theoutcome that obtains in the equilibrium of ourmodel when police are unconstrained) Thiscritics of racial pro ling nd inappropriateHow to strike an appropriate balance betweenconsiderations of fairness in policing and dis-parities in crime rates across groups is an inter-esting question which we leave for futureresearch

IX Conclusion

The controversy on racial pro ling focuseson the fact that some racial groups are dispro-portionately the target of interdiction This fea-ture is not peculiar to highway interdiction itarises in many other policing situations such asneighborhood policing and customs searchesThe resulting racial disparities are unfortunatebut as discussed in the introduction are not perse illegal if they are an unavoidable by-productof effective policing The fundamental questiontherefore is whether there is necessarily a trade-off between fairness and effectiveness in polic-ing This question is coming to the fore onceagain in regards to the pro ling of airline pas-sengers At the moment airport security mostlyrelies on screening all passengers with metaldetectors (which incidentally means that womenand men are treated equally despite the fact thatfemale terrorists seem to be rare) To achievemore effective interdiction the US govern-ment is planning to establish a centralized database of ticket reservations to be mined forunusual or suspect patterns36 At the moment

35 However it is doubtful that it would be politicallyfeasible or ethically desirable to set up such incentiveschemes In addition as shown in Section III the ef cientoutcome may be very unfair We implicitly subscribe tosome of these considerations by focusing on fairness 36 See Robert OrsquoHarrow Jr (2002)


public opinion seems willing to tolerate someamount of pro ling in exchange for greatersecurity37

In this paper the trade-off between fairnessand effectiveness in policing has been investi-gated from a theoretical viewpoint by employ-ing a rational choice model of policing andcrime to study the effects of implementing fair-ness Our notion of fairness is appealing onnormative grounds and speaks to the concernsof critics of racial pro ling We have shown thatthe goals of fairness and effectiveness are notnecessarily in contrast sometimes forcing thepolice to behave more fairly can increase theeffectiveness of interdiction We have givenexact conditions under which within our simplemodel the contrast is present

These conditions are based on the distribu-tions of legal earning opportunities of citizensand are expressed as constraints on the QQ plotof these distributions Thus our model relatesthe trade-off between fairness and effectivenessof policing to income disparities across races Inour model it is possible directly to recover theQQ plot of the distributions of legal earningopportunities by using reported earnings so ourapproach has the potential to deliver quantita-tive implications We view the close relation-ship between the model and data as a desirablefeature38 At the same time we are aware thatwe have ignored many factors that are importantin the decision to engage in crime an issue towhich we will return

Finally we have suggested that a redistribu-tive notion of racial fairness such as the one weadopt (the average citizen should bear the sameexpected cost of being searched regardless ofhisher race) may not be meaningful when thecost of being searched re ects the stigmatiza-tion connected with being singled out forsearch Therefore we conclude that there maybe theoretically valid reasons to ignore the cost

of stigmatization in a discussion of racial fair-ness This conclusion may be cause for opti-mism The nding that the main costs of beingsearched is due to ldquophysicalrdquo aspects of thesearch (such as loss of time improper behaviorof police etc) is encouraging since aggressivepolicy action can presumably ameliorate theseproblems This is in contrast to the costs due tostigmatization which are endogenous to themodel and therefore potentially beyond thereach of policy instruments To the extent thatthe physical costs of being searched can bereduced by remedies such as better training forpolice remedial action can focus on police be-havior during the search rather than on con-straining the search strategy of police

One contribution of this paper is to give arigorous foundation to the idea that fairnessand effectiveness of policing are not neces-sarily in contrast and to show that due to aldquosecond bestrdquo argument constraining policebehavior may well increase effectiveness ofinterdiction On the other hand we have notshown that this is the case in practice Al-though we do not claim conclusively to haveanswered the question we hope that if theframework proposed here proves to be con-vincing it can provide an analytical founda-tion for the public debate

This paper has dealt with a very simple modelwhere crime is endogenous and the decision toengage in crime re ects a lack of legal earningopportunities To highlight the basic forces inthe simplest way we chose to focus on fewmodeling elements In most of the paper forexample race is the only characteristic that isobservable to police Also we have assumedthat the rewards to crime are independent ofonersquos legal earning opportunities whereas inreality the two may be correlated Analogoussimplifying assumptions are common to muchof the theoretical literature on discriminationand we have exploited the simpli cations toobtain theoretically clean results At the sametime we have ignored many interesting andpossibly controversial questions that wouldarise in a more complex model for example thequestion of the right de nition of ldquofairer inter-dictionrdquo in a world in which there are more thantwo characteristics Due to the many simpli -cations and to the generality of the model nopart of the analysis should be understood to

37 See Henry Weinstein et al (2001)38 Some interesting recent papers in the discrimination

literature (see especially Hanming Fang [2000] and Moroand Norman [2000]) have focused on estimating models ofstatistical discrimination a la Arrow (1973) In statisticaldiscrimination models individuals differ in their cost ofundertaking a productive investment Since this character-istic is unobservable estimating its distribution in the pop-ulation is a dif cult endeavor


have direct policy implications Any realisticsituation will be richer in detail than this styl-ized model and a number of additional consid-erations will be relevant in each case In eachspeci c situation of policing the additionalmodeling structure will likely enrich the in-sights of the simple model presented here Inorder to obtain policy implications the modelneeds to be tailored to speci c situations ofinterdiction

The model developed here could be appliedto tax auditing In this application the twogroups of citizens would represent groups oftaxpayers that are observably different Thesedifferent groups of citizens will reasonably dif-fer in their elasticity to auditing small taxpay-ers for example more than large corporationswho employ tax lawyers may dread the prospectof being audited The role of police would nowbe played by the IRS who is assumed to max-imize expected recovery There is a large liter-ature on auditing models (see eg ParkashChander and Louis L Wilde 1998) Most of itassumes that there is only one observable groupof taxpayers with the exception of SusanScotchmer (1987) who does not focus on thedisparity in auditing rates We hope that thequestions asked in the present paper can stimu-late further research into the issue of auditingwith different groups but we leave this topic forfuture research

APPENDIX

PROOF OF PROPOSITION 3Since FW rst-order stochastically dominates

FA we have sW s sA To construct thetotal variation in crime from implementing thefair outcome write

FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s

sA shyFA ~s~J 2 H 1 H

shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds

The total variation in crime [expression (9)]reads

TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds

Now denoting m 5 mins [ [s sA] fA(q(s)) the rst integral in the above expression is greaterthan s

sA m ds and so

(A1)

TV $ zJ 2 Hz 3 NA ~sA 2 s m

2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm

2 NW h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 sW

s

NW m


where to get the last equality we used the identity


NW(s 2 sW) 5 2NA(s 2 sA) To verify thisidentity write

NA ~s 2 sA 5 NAX sA NA 1 sW NW

NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA

and thus symmetrically

(A2) NW ~s 2 sW 5NA NW

NA 1 NW~sA 2 sW

5 2NA ~s 2 sA

Now let us return to expression (A1) from thede nition of m we have

m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z

where to get from the rst to the second line weuse the fact that the equilibrium (sA sW) isinterior as shown in the proof of Lemma 1Now x any s [ [sW s ] Since s $ sW andq(z) is a decreasing function we have q(s) q(sW) and so h(q(s)) h(q(sW)) Alsosince s s we have q(s) $ q(s ) Thus forany s [ [sW s ] we have

m $ minz [ h~q~sq~s

fA ~z

Substituting for m into expression (A1) we get

TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z

2 h9~q~sfA ~h~q~s ds

which is positive under the assumptions of theproposition This shows that crime increases asa result of implementing the completely fairoutcome

REFERENCES

Agnew Robert ldquoFoundation for a GeneralStrain Theoryrdquo Criminology February 199230(1) pp 47ndash87

Anderson Elijah StreetWise Race class andchange in an urban community ChicagoUniversity of Chicago Press 1990

Arnold Barry C Majorization and the Lorenzorder A brief introduction HeidelbergSpringer-Verlag Berlin 1987

Arrow Kenneth J ldquoThe Theory of Discrimina-tionrdquo in O Ashenfelter and A Rees edsDiscrimination in labor markets PrincetonNJ Princeton University Press 1973 pp 3ndash33

Bar-Ilan Avner and Sacerdote Bruce ldquoThe Re-sponse to Fines and Probability of Detectionin a Series of Experimentsrdquo National Bureauof Economic Research (Cambridge MA)Working Paper No 8638 December 2001

Beck Phyllis W and Daly Patricia A ldquoStateConstitutional Analysis of Pretext Stops Ra-cial Pro ling and Public Policy ConcernsrdquoTemple Law Review Fall 1999 72 pp 597ndash618

Becker Gary S ldquoCrime and Punishment AnEconomic Approachrdquo Journal of PoliticalEconomy MarchndashApril 1968 76(2) pp169ndash217

Benabou Roland and Ok Efe A ldquoMobility asProgressivity Ranking Income ProcessesAccording to Equality of OpportunityrdquoMimeo Princeton University 2000

Chander Parkash and Wilde Louis L ldquoA Gen-eral Characterization of Optimal Income TaxEnforcementrdquo Review of Economic StudiesJanuary 1998 65(1) pp 165ndash83

Coate Stephen and Loury Glenn C ldquoWillAf rmative-Action Policies Eliminate Nega-tive Stereotypesrdquo American Economic Re-view December 1993 83(5) pp 1220ndash40

Ehrlich Isaac ldquoParticipation in Illegitimate Ac-tivities A Theoretical and Empirical Investi-gationrdquo Journal of Political Economy MayndashJune 1973 81(3) pp 521ndash65

ldquoCrime Punishment and the Marketfor Offensesrdquo Journal of Economic Perspec-tives Winter 1996 10(1) pp 43ndash67

Fang Hanming ldquoDisentangling the CollegeWage Premium Estimating a Model withEndogenous Education Choicesrdquo MimeoYale University April 2000


Gibeaut John ldquoPro ling Marked for Humilia-tionrdquo American Bar Association JournalFebruary 1999 pp 46ndash47

Jakobsson Ulf ldquoOn the Measurement of theDegree of Progressivityrdquo Journal of PublicEconomics JanuaryndashFebruary 1976 5(1ndash2)pp 161ndash68

Kennedy Randall Race crime and the lawNew York Pantheon Books 1977

Knowles John Persico Nicola and Todd PetraldquoRacial Bias in Motor-Vehicle SearchesTheory and Evidencerdquo Journal of PoliticalEconomy February 2001 109(1) pp 203ndash29

Lehmann Eric L ldquoComparing Location Exper-imentsrdquo Annals of Statistics 1988 16(2) pp521ndash33

Levitt Steven D ldquoUsing Electoral Cycles in Po-lice Hiring to Estimate the Effect of Police onCrimerdquo American Economic Review June1997 87(3) pp 270ndash90

Mailath George J Samuelson Larry andShaked Avner ldquoEndogenous Inequality inIntegrated Labor Markets with Two-SidedSearchrdquo American Economic Review March2000 90(1) pp 46ndash72

Moro Andrea and Norman Peter ldquoAf rmativeAction in a Competitive Economyrdquo MimeoUniversity of Minnesota 2000

Norman Peter ldquoStatistical Discrimination andEf ciencyrdquo Mimeo University of Wiscon-sin June 2000

OrsquoHarrow Robert Jr ldquoIntricate Screening ofFliers in Works Database Raises PrivacyConcernsrdquo Washington Post February 12002 p A1

Polinsky A Mitchell and Shavell Steven ldquoTheFairness of Sanctions Some Implications forOptimal Enforcement Policyrdquo Harvard OlinCenter Discussion Paper No 247 December1998

Ramirez Deborah McDevitt Jack and Farrell

Amy A resource guide on racial pro lingdata collection systems Promising practicesand lessons learned US Department of Jus-tice Washington DC November 2000[Available online at httpwwwncjrsorgpdf les1bja184768pdf ]

Scotchmer Susan ldquoAudit Classes and Tax En-forcement Policyrdquo American Economic Re-view May 1987 (Papers and Proceedings)77(2) pp 229ndash33

Slobogin Christopher ldquoThe World Without aFourth Amendmentrdquo UCLA Law ReviewOctober 1991 39(1) pp 1ndash107

Spitzer Eliot A report to the people of the stateof New York from the of ce of the attorneygeneral New York Civil Rights Bureau De-cember 1 1999

Thompson Anthony C ldquoStopping the Usual Sus-pects Race and the Fourth AmendmentrdquoNew York University Law Review October1999 74(4) pp 956ndash1013

US Customs Service Report on personalsearches by the United States Customs Ser-vice personal search review commissionPersonal Search Review Commission Wash-ington DC [Available online at httpwwwcustomsgovpersonal_searchpdfsfullpdf ]

Verniero Peter and Zoubek Paul H Interim re-port of the state police review team regardingallegations of racial pro ling NJ AttorneyGeneralrsquos Of ce April 20 1999

Weinstein Henry Finegan Michael and Watan-ber Teresa ldquoRacial Pro ling Gains Supportas Search Tacticrdquo Los Angeles Times Sep-tember 24 2001 p A1

Wilk M B and Gnanadesikan R ldquoProbabilityPlotting Methods for the Analysis of DatardquoBiometrika March 1968 55(1) pp 1ndash17

Wohlstetter Albert and Coleman Sinclair ldquoRaceDifferences in Incomerdquo RAND PublicationR-578-OEO October 1970


as a factor in determining the likelihood that aperson is engaging in a crime as long as thisuse is reasonably related to law enforcementand is not a pretext for racial harassment5

Mindful of the established judicial line and ofpublic policy concerns for effectiveness of in-terdiction the courts have generally been cau-tious in imposing remedies on the police In sodoing the courts implicitly have placed lessweight on the plight of innocent citizens whobear a disproportionate amount of the policingeffort This position is defensible only insofar asthe two considerations fairness and effective-ness of policing are in con ict It is thereforeextremely important to establish whether a con- ict exists in practice

This paper studies the trade-off between fair-ness and effectiveness of interdiction in thecontext of a simple model in which citizens oftwo groups choose whether to engage in illegalactivities and police audit (or search) citizensThe model is too stylized to have direct policyimplications Despite this fact the model offersa more sophisticated view than is generallyaired in the public debate The rst point of thispaper is that in principle there need not be atrade-off Constraining police to behave in a fairmannermdashin our context searching both groupswith the same intensitymdashneed not per se resultin more crime This is because under veryplausible assumptions about the incentives ofpolice the equilibrium allocation does not co-incide with the crime-minimizing allocation

and so placing constraints on the behavior of thepolice may actually increase the effectiveness ofpolicing6

To understand this statement it is necessary tounderstand why in equilibrium the effectivenessof policing is generally not maximal Our as-sumption is that when choosing whom tosearch the objective of a police of cer is tomaximize the probability of uncovering crimi-nal behavior Suppose then for the sake of theargument that police can distinguish only twogroups A and W which partition the popula-tion In equilibrium it cannot be that one grouphas a lower fraction of criminals for otherwiseno police of cer would search members of thegroup with the lower fraction of criminals Butthen members of that group would respond bystepping up their illegal activity and then policewould want to search the group This equilibrat-ing process results in the key equilibrium con-dition in equilibrium the two groups must havethe same fraction of criminals7 Crucially thisequilibrium condition is independent of theelasticity of crime to policing In contrast theelasticity to policing plays a major role in de-termining the conditions for effective policingTo see this take any allocation (including theequilibrium one) and suppose an increase of 1percent in the search probability deters mem-bers of group W more than members of groupA then maximal effectiveness requires that 1percent of resources be transferred from search-ing group A to searching group W to takeadvantage of the higher deterrence effect ofpolicing on group W This shows that there is awedge between the equilibrium condition andthe effective allocation of policing effort8

5 See Randall Kennedy (1997) for an exposition of thisview and more recently Anthony C Thompson (1999) fora critical analysis It may seem surprising that the SupremeCourt would sanction such a narrow interpretation of theequal protection clause However racial pro ling and otherpolicing cases are often tried as fourth amendment cases(protection from unreasonable search and seizure) and notas fourteenth amendment cases (protection from racial dis-crimination) The distinction is crucial because the standardof evidence is lower in fourth amendment cases police areonly required to prove ldquoreasonablenessrdquo of their behaviorinstead of being subject to the ldquostrict scrutinyrdquo standard thatapplies in racial discrimination cases The Supreme Court nds that the fourth amendment permits pretextual stopsbased on objective justi cation and does not admit second-guessing of the police of cerrsquos subjective motivation [seeUS v Whren 517 US 806 116 SCt 1769 (1996)] Theappropriateness of applying fourth amendment law to racialpro ling cases is now hotly debated and intense publicscrutiny is changing the legal landscape in this area

6 In this paper effectiveness of interdiction is synoni-mous with ldquocrime minimizationrdquo This point is discussedfurther in Section I subsection A in terms of the generalityof the assumptions under which effectiveness and fairnessdo not coincide

7 Equalization of fraction guilty is predicted only forthose crimes which are the object of interdiction Further-more equalization of crime rates does not imply that equi-librium incarceration rates are the same for the two groupsthe group which is searched more intensely will have ahigher fraction of its members being incarcerated Finallythe crime rate is equalized only among those groups that aresearched in equilibrium These points are discussed in sub-section A

8 How this conclusion depends on police incentives isdiscussed in Section VIII












Related Literature






I Model









(1) NAsA 1 NWsW 5 S



x sr J 1 ~1 2 sr H




















ASSUMPTION 1

FAX S

NA~J 2 H 1 HD FW ~H















minsA sW




minsA


1 NW FWX S 2 NAsA

NW~J 2 H 1 HD


(3) NA ~J 2 H fA ~sA ~J 2 H 1 H

2 fW ~sW ~J 2 H 1 H]



EFF












NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW




















FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES
















































Related Literature






I Model









(1) NAsA 1 NWsW 5 S



x sr J 1 ~1 2 sr H




















ASSUMPTION 1

FAX S

NA~J 2 H 1 HD FW ~H















minsA sW




minsA


1 NW FWX S 2 NAsA

NW~J 2 H 1 HD


(3) NA ~J 2 H fA ~sA ~J 2 H 1 H

2 fW ~sW ~J 2 H 1 H]



EFF












NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW




















FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES









































Related Literature






I Model









(1) NAsA 1 NWsW 5 S



x sr J 1 ~1 2 sr H




















ASSUMPTION 1

FAX S

NA~J 2 H 1 HD FW ~H















minsA sW




minsA


1 NW FWX S 2 NAsA

NW~J 2 H 1 HD


(3) NA ~J 2 H fA ~sA ~J 2 H 1 H

2 fW ~sW ~J 2 H 1 H]



EFF












NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW




















FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES








































I Model









(1) NAsA 1 NWsW 5 S



x sr J 1 ~1 2 sr H




















ASSUMPTION 1

FAX S

NA~J 2 H 1 HD FW ~H















minsA sW




minsA


1 NW FWX S 2 NAsA

NW~J 2 H 1 HD


(3) NA ~J 2 H fA ~sA ~J 2 H 1 H

2 fW ~sW ~J 2 H 1 H]



EFF












NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW




















FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES






































x sr J 1 ~1 2 sr H




















ASSUMPTION 1

FAX S

NA~J 2 H 1 HD FW ~H















minsA sW




minsA


1 NW FWX S 2 NAsA

NW~J 2 H 1 HD


(3) NA ~J 2 H fA ~sA ~J 2 H 1 H

2 fW ~sW ~J 2 H 1 H]



EFF












NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW




















FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES







































ASSUMPTION 1

FAX S

NA~J 2 H 1 HD FW ~H















minsA sW




minsA


1 NW FWX S 2 NAsA

NW~J 2 H 1 HD


(3) NA ~J 2 H fA ~sA ~J 2 H 1 H

2 fW ~sW ~J 2 H 1 H]



EFF












NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW




















FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES









































minsA sW




minsA


1 NW FWX S 2 NAsA

NW~J 2 H 1 HD


(3) NA ~J 2 H fA ~sA ~J 2 H 1 H

2 fW ~sW ~J 2 H 1 H]



EFF












NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW




















FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES







































NA ~J 2 H2 f9A ~sA ~J 2 H 1 H

1NA

NWf9W X S 2 NAsA

NW~J 2 H 1 HD

5 NA ~J 2 H21

NAf9A ~q~sA

11

NWf9W ~q~sW




















FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES












































FA ~q~sA 5 FAX S

NA~J 2 H 1 HD

FW ~H 5 FW ~q~0


FA ~H $ FW ~H FWX S

NW~J 2 H 1 HD


(5) q~sA 5 h~q~sW






(7) NA ~J 2 HfA ~q~sA 1 2 h9~q~sW














q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES

















































q and q such that

(8)

h9~x

minz [ h~xx

fA ~z

fA ~h~xfor all x in

q q


q q(sW) q(sA) q











q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES








































q q ]









q q~sA q~s q~sW 5 H




TV 5 NA 1 2 1 1 NW FW ~q~s 2 1 0


q q ] this is























F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES






















































F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES






































F r ~x 5 aF r ~x 1 ~1 2 aF r ~q~sr




(10) F r2 1~ p

5 5 F r2 1X p 2 b

a D for p F r ~q~sr



h~x FA2 1~FW ~x


h~x 5 FA2 1~FW ~x

5 FA2 1~FW ~x

5 FA2 1~FW ~x 5 h~x



5 FA2 1X ~aFW ~x 1 b 2 b

a D5 FA

2 1~FW ~x 5 h~x















fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES












































fA ~z 1









FIGURE 4 PLOT OF r






















(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES


























































(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES










































(11) d~r 3 s ~r 1 d ~r 3 1 2 s ~r 5 0


















q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES












































q~sA h~q~sW


q~sA q~sW




FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES







































FA ~q~sA 1 FW ~q~sW


fW ~q~sW fA ~q~sA



fA ~z


h9~x

minz [ h~xx

fA ~z

fA ~h~x








































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES




































































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES





















































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES











































IX Conclusion
















APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES


















































APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES








































APPENDIX



FA ~s ~J 2 H 1 H 2 FA ~sA ~J 2 H 1 H

5 2s


shysds

5 zJ 2 Hz s

sA

fA ~q~s ds

Similarly

FW ~sW ~J 2 H 1 H 2 FW ~s ~J 2 H 1 H

5 zJ 2 Hz sW

s

fW ~q~s ds

5 zJ 2 Hz sW

s

h9~q~sfA ~h~q~s ds


TV 5 zJ 2 Hz 3 NA s

sA

fA ~q~s ds

2 NW s

W

s

h9~q~sfA ~h~q~s ds


sA m ds and so

(A1)


2 NW s

W

s

h9~q~sfA ~h~q~s ds

5 zJ 2 Hz 3 s

W

s

NA

sA 2 s

s 2 sWm


5 zJ 2 Hz 3 sW

s

NW m






NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES








































NA 1 NW2 sAD

5NA NW

NA 1 NW~sW 2 sA



NA 1 NW~sA 2 sW

5 2NA ~s 2 sA


m 5 minz [ q~sA q~s

fA ~z

5 minz [ h~q~sW q~s

fA ~z


m $ minz [ h~q~sq~s

fA ~z


TV $ zJ 2 HzNW 3 sW

s

minz [ h~q~sq~s

fA ~z



REFERENCES





























































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