Diversity in the Workplace

Diversity in the Workplace

John Morgan and Felix Várdy

© 2006 International Monetary Fund

WP/06/237

IMF Working Paper

IMF Institute Diversity in the Workplace

Prepared by John Morgan and Felix Várdy 1

Authorized for distribution by Ling Hui Tan

October 2006

Abstract

This Working Paper should not be reported as representing the views of the IMF.

The views expressed in this Working Paper are those of the author(s) and do not necessarily represent those of the IMF or IMF policy. Working Papers describe research in progress by the author(s) and are published to elicit comments and to further debate.

We study a model where an employer, trying to fill a vacancy, engages in optimal sequential search by drawing from two subpopulations of candidates who differ in their \"discourse systems\": during an interview, a minority candidate with a discourse system not shared with the employer conveys a noisier unbiased signal of ability than does a majority candidate. We show that, when the employer is \"selective,\" minority candidates are underrepresented in the permanent workforce, fired at greater rates, and underrepresented among initial hires, even though the employer has no taste for discrimination and the populations are alike in their average ability. Furthermore, workplace diversity is increased if: (1) the cost of firing is reduced, (2) the cost of interviewing is increased, (3) the opportunity cost of leaving the

position unfilled is increased, or (4)the prior probability that a candidate can perform the job is increased. Indeed, if the prior probability is sufficiently high, or the cost of firing sufficiently low, then minority candidates may be overrepresented in the permanent workforce.

JEL Classification Numbers: D21, D63, D83, J71, J78.

Keywords: Diversity, sequential search, statistical discrimination, reverse discrimination,

discourse systems.

Authors’ E-Mail Addresses: morgan@haas.berkeley.edu and fvardy@imf.org John Morgan is at U.C. Berkeley. The authors would like to thank Mary Amity, Burkhard Drees, Bob Feldman, Andrew Feltenstein, Harold Houba, Keith Takeda and, especially, Johan Walden for extremely useful comments. Morgan gratefully acknowledges the financial support of the National Science Foundation.

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Contents

I.Introduction.................................II.RelatedLiterature..............................III.Model.....................................IV.OptimalSearchandHiring.........................V.PerformanceMetrics.............................VI.

PolicyImplications..............................

VII.Conclusions..................................Appendix:ProofsofLemmas,PropositionsandImplications..........References......................................

Figures1.HiringB:1󰀂GProbabilityRatiosofCompetentCandidatesofKindAversus

A1(q)

................2.

Over-1󰀂GandB1(q)..................UnderrepresentationofMinorities................

368111219222444

1518

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I.Introduction

Acentralsocial,political,andeconomicchallengeconfrontingtheEuropeanUniontodayarisesfromthetensionscreatedbythegrowthofimmigrantpopulations,particularlythosefrompredominantlyMuslimcountries.Thesetensionshavemanifestedthemselvesinsometimesdramaticfashion—themurderofTheovanGoghanditsaftermathintheNetherlands,thewidespreadunrestoverDanishanti-Muslimcartoons,andtheweeks-longviolenceandriotingintheoutskirtsofParisintheFallof2005.Manyhavearguedthattheseeventsaremeresymptomsofabroadunderlyingdiscontentcaused,inlargepart,byalackofeconomic

opportunities.Indeed,generallyhighunemploymentintheEU,oftenattributedtolabormarketrigidities,a¤ectsimmigrantpopulationsparticularlyseverely:

unemploymentratesforminoritiesremainstubbornlyhigherthanforthemajority,andgrowespeciallysevereduringeconomicdownturns.

WhataccountsforthedisparityintheemploymentexperiencesofEurope’s

majoritypopulationsversusitsminoritypopulations?Skillandagedi¤erencesaresurelypartoftheexplanation.Minoritypopulationsare,onaverage,lesseducatedandyoungerthanthemajority,andunemploymentratestendtobehigheramongthelow-skilledandtheyoung.Nevertheless,whiletheemploymentdisadvantageofminoritiesisreducedoncedi¤erencesineducationalattainmentandagearetakenintoaccount,itdoesnotdisappear.(See,e.g.,Tesser,Merens,andVanPraag,1999;andDagevos,2006).Moreover,thedisadvantagedoesnotdisappearovertimeeither:intheNetherlands,evensecond-generationMuslimsdisplayconsiderablyhigherunemploymentratesthantheirmajoritycounterparts.Infact,controllingforeducationandage,theemploymentdisadvantageofsecond-generationMuslimsisevengreaterthanthatofthe…rstgeneration(Dagevos,2006).

Ofcourse,itmaybethatemployerssimplyhaveatastefordiscriminationandthattheunderrepresentationofMuslimsintheEUworkforcere‡ectsthestrengthofthesetastes.Whileitishardtoruleoutthisexplanation,onewouldexpectthatthecostofindulginginatastefordiscriminationhasbeenraisedwiththeincreasedglobalizationoftheEUeconomy.Thus,onewouldexpecttoseetheunemploymentgapbetweenMuslimsandtherestofthepopulationshrinkthroughcompetitivepressures,when,infact,theoppositehasoccurredintheNetherlandsoverthepastcoupleofyears(Dagevos,2006).

Analternativeexplanationforhigherminorityunemploymentmaybegleanedfromtheinterculturalcommunicationandsociolinguisticsliteratures.(See,forinstance,ScollonandScollon,2001.)Accordingtothishypothesis,minorityjobcandidatesstruggletomakethemselvesunderstoodduetodi¤erencesin“discoursesystems.”Forexample,acandidate’sbehaviorduringajobinterviewmaybequiterevealingtoanemployeriftheysharethesamesocialorculturalbackground.Butiftheydonot,itcanbemuchharderfortheemployertoformanaccurateopinionabouttheapplicant.Inotherwords,thesignalsconveyedbyminoritiesduringinterviewsmay

-4-besogarbledthattheyfailtoconvince(majority)employersoftheirqualities,evenwhentheyareperfectlycompetentandemployershavenotastefordiscrimination.Incontrast,byvirtueofsharingthesamediscoursesystemasemployers,majorityjobcandidatesdonotfacethisproblem.Thus,forthemittendstobeeasiertoconveyanaccurateimpressionoftheirquality.Asaconsequence,minoritypopulations…ndgreaterdi¢cultyinsecuringemploymentthanmajoritypopulations.2

Thishypothesisraisesseveralquestions.Candi¤erencesindiscoursesystemsaloneexplaindi¤erencesinunemploymentratesbetweenmajorityandminority

populations,absentanydi¤erencesinunderlyingabilityofthetwopopulations?Ifso,whatpolicyprescriptionscouldremedythis?Shouldemploymentprotectionbeincreasedordecreased?Whataboutotherrigidities—arethesehelpfulorharmfultoworkplacediversity?Whataboutmacroimplications—cantheEUsimplygrowitselfoutoftheproblem?

Toexaminethesequestions,westudyamodelinwhichanemployertriesto…llavacancybysequentiallyinterviewingjobcandidatesfromapoolofpotentialemployees.Thepoolconsistsoftwosubpopulations.Onesubpopulationmaybethoughtofasthemajoritypopulation,theotherastheminoritypopulation.Theemployerhasnoinherenttastefordiscrimination,andtheonlythinghecaresaboutiswhetheracandidatecandothejob.Onaverage,candidatesfromboth

subpopulationsareequallylikelytobeabletodothejob.Thismeansthatthereisnoroleforthestandardtypeofstatisticaldiscriminationinourmodel.Candidatesdo,however,di¤erintheirdiscoursesystems.Themajoritypopulationhasthesamediscoursesystemastheemployer,whiletheminorityhasadi¤erentdiscoursesystem.Tocapturethisdi¤erence,wesupposethatwhentheemployerinterviewsaminoritycandidatehereceivesanoisiersignalofthatcandidate’strueabilitythanwhenheinterviewsamajoritycandidate.

Ourmainresultshowsthat,whenanemployeris“selective,”equilibriumalwaysentailsunderrepresentationoftheminoritypopulationinthepermanentworkforce.Here,“selective”meansthatcandidatesarehiredonlywhenthepost-interviewprobabilitythattheycandothejobexceedsthepriorprobability.More

surprisingly,whenanemployerissu¢ciently“unselective,”equilibriumentails

overrepresentationoftheminoritypopulation.Su¢ciently“unselective”meansthatacandidateishiredprovidedhedoesnotdisappointtoomuchduringtheinterview.Finally,regardlessoftheselectivityoftheemployer,the…ringrateofminoritycandidatesalwaysexceedsthatofmajoritycandidates.

Ofcourse,matchingthebackgroundoftheinterviewerwiththebackgroundofthecandidatewouldsolvethisproblem.However,moreoftenthannot,thismaybequitedi¢culttoimplement.First,inorganizationslackingdiversity,minoritiesarescarcetobeginwith.Second,itshouldnotbeforgottenthatthevariousminoritiesareculturallyhighlydiverse,thusrequiringaverycarefulmatchingbetweentheevaluatorandtheevaluee.Forinstance,whileaFrench-speakingWestAfricanandanAfrican-Americanarebothpeopleofcolor,itseemsquiteclearthattheydonotsharethesamediscoursesystem.

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-5-Theintuitionforthemainresultmaybeseeninthefollowingexample.Supposethatthepriorprobabilitythatarandomcandidatecandothejobis50%andassumethattheemployerisveryselective,suchthatonlycandidatesaboutwhomtheemployerisatleast95%certainaftertheinterviewthattheycandothejobarehired.Suchahighthresholdisoptimalwhen…ringcostsareveryhigh.Inthatcase,therelativeuninformativenessofaminoritycandidate’ssignalabouthisquali…cationsmakesitextremelyhardtochangetheemployer’s50%priorbeliefof“success”toaposteriorbeliefofatleast95%.Therefore,itisveryunlikelythataminoritycandidateisgoingto…lltheposition.Asaresult,selectivehiringpracticesleadtosevereunderrepresentationofminorities,eventhoughminoritiesareascompetentasthemajorityandemployersarenotprejudicedagainstthem.Ontheotherhand,iftheemployerisnotselectiveatall,suchthatanycandidateishiredprovidedthattheposteriorprobabilitythathecandothejobisnolessthan5%,thentherelativeuninformativenessofaminoritycandidate’ssignalabouthisquali…cationsisanadvantage.Itmakesitvirtuallyimpossiblefortheemployer’s50%priorbeliefofsuccesstobedowngradedtolessthan5%.Underthese

circumstances,virtuallyallminoritycandidatesaregivenachanceandremaininthejobiftheyturnouttobegood.Atthesametime,inrelativeterms,manymajoritycandidatesareturnedawayatthegate,becausetheinformativenessoftheirsignalsdoesmakesigni…cantbeliefrevisionspossible.Asweshow,thisleadsto“reversediscrimination:”minoritieswillbeoverrepresentedintheworkforcesofunselectiveemployers.Forsimilarreasons,themodelalsopredictsthatthedegreeofunderrepresentationofminoritiesdependsonthepriorprobabilitythatrandomcandidatescandothejob.Speci…cally,minoritieswillbemostseverely

underrepresentedinpositionsthatdemandrareskills,suchthattheemployer’spriorsareverypessimistic.Incontrast,minoritieswillbeoverrepresentedinpositionsthatnearlyanyonecando.

Next,themodelpredictsthattherelativerepresentationofminoritiesintheworkplacevariesoverthebusinesscycle.Speci…cally,ifemployersareatall

selective,diversityispredictedtobeprocyclical,increasingduringeconomicupturnsanddecreasingduringdownturns.Intuitively,whentheeconomyisbooming,

recruitingjobcandidatesismorecostly.Atthesametime,theopportunitycostofleavingthepositionun…lledishigher.Bothe¤ectsmaketheemployerlesspicky,encouragingemployersto“takeachance”onjobcandidateswhosequalityisuncertain.Thisreducestheunderrepresentationofminorities.

ThispredictionisroughlyconsistentwiththeDutchexperienceoverthepast

decade.Duringthesecondhalfofthe1990s,aperiodofrapideconomicexpansion,unemploymentamongMuslimminoritiesintheNetherlandsfellquitespectacularly,fromover30%in1995toaround9%in2001.Duringthesameperiod,the

unemploymentrateamongthenonimmigrantDutchfellfromaround6.5%to3%.Sincethen,thetrendhaslargelyreversed.By2005,unemploymentamongMuslimswasagainashighas24%,whileunemploymentamongthenonimmigrantDutchhadonlyrisento5%.(Dagevos,2006.)

-6-Finally,weturntopolicysolutionstothe“diversityproblem.”Ourmain…ndinginthisregardisthathigh…ringcostsharmdiversity.Intuitively,protectionsthatraisethecostof…ringleadtheemployertoguardmorevigilantlyagainstTypeIIerrors(hiringofincompetentcandidates).Theemployerachievesthisbybecomingmoreselective,whichexacerbatestheunderrepresentationofminorities.ThissuggeststhatEUlabormarketrigiditiessuchashighcostsof…ringcontributetotheeconomicandsocialexclusionofMuslimminoritiesinEurope.

Toconclude,themodelimpliesthatdi¤erencesindiscoursesystemscanindeedgeneratedi¤erencesinunemploymentacrossotherwisehomogeneouspopulations.Goingbeyondthemodel,itsuggestsafeedbacksystembetweenculturaland

economicbarrierstointegration:thelackofashareddiscoursesystemleadstofewopportunitiesforminoritiestolanddemandingjobswithselectiveemployers.Instead,minoritiesaremorelikelytobeunemployed,orstuckatthelowerendofthelabormarket.This,inturn,impliesthattheyarelesslikelytobeinclose

contactwiththedominantdiscoursesystemand,therefore,theculturalsegregationacrosspopulationsisself-reinforcingandmay,infact,hardenovertime,perhapsexplainingtheexperienceofsecond-generationMuslimsintheNetherlands.WhilethemodelpresentedinthispaperismotivatedbytheplightofimmigrantpopulationsintheEU,itdoesseemtohavewiderapplicability.Forinstance,inaU.S.context,themodelmaycastsomelightontheheateddiscussionaboutthelackofdiversityamongSupremeCourtlawclerks.WhenmembersofCongressaskedwhythejusticesdidnotcasttheirnetsmorewidelyto…ndmoreminority

candidates,theyrespondedthattheycouldilla¤ordtotakeachancethatevenoneoftheirclerksmightnotbeatopperformer.(Peppers,2006).Thislineofreasoning…tswellwithourmodel,inwhichhighcostsofmakingamistakeleadtosevereunderrepresentationofminorities.

Fromhere,thepaperproceedsasfollows.WebeginwithanoverviewoftherelatedliteratureinSectionII.InSectionIIIwedevelopthemodel.InSectionIVtheemployer’spayo¤-maximizinghiringstrategyisderived.InSectionVwestudytheconsequencesofoptimalhiringforminorityrepresentation.SectionVIdiscussespotentialpolicyresponsesandSectionVIIconcludes.WehaverelegatedmostproofstotheAppendix.

II.RelatedLiterature

ThenearestantecedenttothecurrentpaperisCornellandWelch(1996).CornellandWelchlookattheprobabilitythataminoritycandidateishiredwhenanemployerchoosesthebestprospectfroma…xednumberofcandidates.Asinourpaper,CornellandWelchassumethattheminoritypopulationisequallyskilledasthemajoritypopulationbutthat(majority)employersarebetteratassessingthequalityofmajoritycandidatesthanthatofminoritycandidates.Whenthenumber

-7-ofcandidatesislarge,theyshowthattheemployerisoverwhelminglymorelikelytohireamajoritycandidatethanaminoritycandidate.Theintuitionreliesonanorder-statisticargument:thehigheraccuracyofmajoritycandidateevaluations

makesthevarianceoftheirinferredqualitieshigherthanthevarianceoftheinferredqualityofminoritycandidates.Thismakesitmuchmorelikelythatoutliers—inparticular,the…rst-orderstatistic—comefromamajoritycandidatethanfromaminoritycandidate.Anditistheoutlierwhogetshired.Incontrast,ourresultsdonotrelyonorder-statistice¤ects.

Ourpaperdi¤ersfromCornellandWelchinanumberofways.Mostimportant,weemployasequentialsearchapproachinthespiritofMcCall(1970),ascomparedwithCornellandWelch’s…xed-sample-size-approachalongthelinesofStigler

(1961).Indeed,thedi¤erenceinthetwomodelsisanalogousto…xedsampleversussequentialsearchinthepricingliterature.Forasummaryofthedi¤erencesinthepredictionsandoptimalitybetweenthetwosee,e.g.,Baye,Morgan,andScholten(forthcoming).

Ouroptimalsequentialsearchapproachallowsustoexplicitlymodelandanalyzethee¤ectsofwhatCornellandWelchcall“exantescreening”versus“on-the-jobperformancemeasurement.”Also,wedi¤erentiatebetweenskilllevelsandshowthatthisdistinctionmattersinanimportantway:whilediscriminationofminoritiestendstobestrongforjobsthatrequirerareskills,itismuchlesssoforcommonskilljobs.Infact,if…ringcostsarelow,correspondingtocheap“on-the-jobperformancemeasurement,”minoritieswillbeoverrepresentedincommonskillpositions.Inabroadercontext,the…rsttoanalyzediscriminationfromaneconomic

perspectivewasBecker(1957).Hestudiedtheeconomicconsequencesofpeople’sintrinsicdislikeof(interactingwith)otherraces.SeeArrow(1998)forasurvey.Intheliterature,thiskindofdiscriminationisknownastaste-orpreference-baseddiscrimination.SomewhatrelatedtoourworkareBlack(1995),whoexaminesthismotiveinasearch-theoreticsetting,andRosen(1997),whocombinessearchwithamatch-speci…cpayo¤.

ClosertoourworkisthestatisticaldiscriminationliteraturebeginningwiththeseminalpaperofPhelps(1972).Inthisliterature,discriminationisinformationbased.Majorityandminoritypopulationsareassumedtodi¤erstatisticallywithrespecttosomerelevantcharacteristic,suchasaveragelaborproductivity.Becauseinterviewsandtestscanonlyimperfectlypredictthelaborproductivityofa

particularjobcandidate,belongingtoonesubpopulationoranotherisstatisticallysigni…cantandtakenintoaccountbyapotentialemployer.Itisusedasanimperfectproxyin‡uencingtheemployer’sbeliefaboutacandidate’sexpectedability,inadditiontotheinformationgatheredthroughinterviewsandtestscores.Morerecently,byendogenizinghumancapitalacquisition,CoateandLoury(1993)aswellasLundbergandStartz(1998)haveshownhowstatisticaldiscriminationcanariseevenwithexantehomogeneouspopulations.

-8-AignerandCain(1977)extendedPhelp’sanalysisbyshowingthatlowerwagescanresultnotonlyfromlowerexpectedproductivity,butalsofromhighervarianceininferredproductivity.Lessaccuratetesting,orhigherintrinsicqualityvariation,depresseswagesforhigh-scoringminoritiesandboostswagesforlow-scoring

minorities.Theintuitionbehindtheirresultissimilartotheintuitionunderlyingours:beliefsaboutthequalityofminoritycandidatesarelesssensitivethanthoseaboutmajoritycandidates.

Finally,ourworkisalsosomewhatrelatedtoother“language”theoriesof

discrimination,suchasLang(1986)orAthey,Avery,andZemsky(2000).Tensioninthesemodelsstemsfrominteractionsbetweenworkers,whereworkerswhospeakthesamelanguagearemoreproductive.Inourmodel,workerinteractionsdonotplayarole.Rather,wefocusonproblemsofcommunicationbetweenanemployerandpotentialjobcandidates.

III.Model

Westudyalabormarketsearchprobleminwhichtheemployerdoesthesearching.Inorderto…llavacancy,anemployertakesrandomdrawsatacostk>0perdrawfromapopulationofjobcandidateswiththepowerofthecontinuum.Eachdrawcanbethoughtofastheemployerconductingajobinterviewwithacandidate.Eachcandidatehastwocharacteristics:whatsubpopulationhebelongsto,whichisobservabletotheemployeratthetimeoftheinterview;andwhetherhecandothejob,whichonlybecomesobservableifthecandidateisactuallyhired.Weshallrefertotheformercharacteristicasacandidate’skindandtothelatterasacandidate’stype.

Acandidate’skindisdenotedby󰀆2fA;Bg.AfractionmAofthecandidatesisfromsubpopulationA;whichconsistsofmembersofthe“dominant”culture—i.e.,candidateswiththesamediscoursesystemastheemployer/evaluator.The

remainingfractionmB=1󰀂mAofthecandidatesisfromsubpopulationB;whichconsistsofmembersnotbelongingtothedominantculture.Asshorthandfordi¤erencesbetweenthedominantandnondominantcultures,weshallsometimesrefertocandidatesofkindAas“majority”candidatesandcandidatesofkindBas“minority”candidates—although,asthedescriptionabovemakesclear,majoritycandidatesdonotnecessarilyhavetobemorenumerousthanminoritycandidates.Acandidate’stype,denotedby󰀉;equals1ifhecandothejobandequalszeroifhecannot.Letp󰀆denotetheprobabilitythatarandomlydrawncandidateofkind󰀆candothejob;thatis,p󰀆󰀆Pr(󰀃=1j󰀆):Weassumethatthetwosubpopulationsareequallyquali…edtodothejob;thatis,

pA=pB=p:

-9-Hence,noneoftheresultsinthepaperaredrivenbydi¤erencesbetweenthetypedistributionsinthesubpopulations.

Inadvanceoftheinterview,theemployerdoesnotknow,ordoesnotactupon,acandidate’sminoritystatus.3However,attheinterviewstage,acandidate’skind—AorB—isperfectlyrevealedtotheemployerthroughsomeeasilyobservablecharacteristicsuchasdialectorskincolor.Inaddition,theinterviewalsorevealstotheemployerasignalS󰀆astothecompetenceofthecandidate,where

S󰀆=󰀉+\"󰀆:

Thatis,thesignalisequaltothecandidate’stype󰀉plusanerrorterm\"󰀆,whichisassumedtobeNormallydistributedwithzeromeanandvariance󰀊2󰀆:Thekeydi¤erencebetweencandidatesofdi¤erentkindsisthattheemployer…ndsiteasiertoassessthecompetenceofcandidatesfromthesameculturecomparedwiththosefromadi¤erentculture.Tomodelthisdi¤erence,weassumethat󰀊B>󰀊A.Thatis,fromtheperspectiveoftheemployer,thereismorenoiseinthesignalofaminoritycandidatethaninthesignalofamajoritycandidate.

Thetimingoftheemployer’sdecisionproblemisasfollows.Inperiod1,the

employerdrawsarandomcandidateandconductsaninterviewatatotalcostk.Onthebasisofthecandidate’sinterviewsignals,andtakingintoaccounthiskind󰀆,theemployercalculatesthecandidate’s“successprobability”q.Thatis,qistheemployer’sposteriorbeliefabouttheprobabilitythatthecandidatecandothejob.Givenq;theemployerthendecideswhethertohirethecandidate,andperiod1ends.Inperiod2andallsubsequentperiods,iftheemployerdidnothireinthepreviousperiod,heinterviewsanewcandidateandthegameproceedsasbefore.If,however,theemployerdidhireinthepreviousperiod,theemployee’stype󰀉isperfectly

revealedtotheemployer.Iftheemployeecandothejob—i.e.,󰀉=1—heisretainedforever,andallsearchceases.Inthatcase,theemployerenjoysapayo¤withanetpresentvalueofv>0:If,however,theemployeecannotdothejob—i.e.,

󰀉=0—thenbyretainingtheemployeetheemployerearnsapayo¤withanet

presentvalueof󰀂w<0:Alternatively,theemployercan…retheemployeeinperiod2andincuracostofc>0:Throughout,weassumethatcInreality,anemployermaybeabletoguessapotentialcandidate’sminoritystatusfromhisnameoraddress.Onthebasisofthatinformation,theemployermightdecidenottoinvitehimforaninterview.Eventhoughinmostcountriesthisisclearlyagainstthelaw,thereisevidencethatitdoeshappen.See,forexample,BertrandandMullainathan(2004).Theassumptioninourmodelisthatemployersdoabidebythelawand,therefore,donotdiscriminateinthisway.Technicallyspeaking,ourmodelisoneofundirectedsearch.

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-10-PosteriorBeliefs

Asweshallsee,theemployer’soptimalstrategyistoimposeasuccessprobabilitythreshold,q󰀅,whendecidingwhethertohireacandidate.Thatis,acandidateishiredifandonlyiftheprobabilitythathecandothejobisatleastq󰀅.Theoptimalthresholdturnsouttobethesameforbothkindsofcandidates.Itdependsontheposteriordistributionoftheemployer’sbeliefsastothecompetenceofacandidate.Thus,itisusefultosummarizekeyfeaturesofthisposteriordistribution.De…neq󰀆(s)tobetheemployer’sposteriorbeliefthatacandidateofkind󰀆withsignalscandothejob.Formally,

q󰀆(s)󰀆P(󰀃=1jS󰀆=s):

ByBayes’rule,wecanrewritethisexpressionas

󰀊󰀋󰀂1

p󰀋s󰀊󰀄

󰀊󰀋;q󰀆(s)=󰀊󰀋ss󰀂1

󰀋󰀊󰀄p+󰀋󰀊󰀄(1󰀂p)

where󰀋(󰀃)denotesthedensityofastandardNormalrandomvariable.

Itwillsometimesbeusefultodeterminethesignalrealizationscorrespondingtoagivensuccessprobabilityq;whichweshalldenotebys󰀆(q):Sinceq󰀆(s)isamonotonefunction,itisinvertibleintheextendedrealsands󰀆(q)iswell-de…ned.󰀄󰀅Usingthat󰀋(t)󰀆p1exp󰀂1t2;itmaybereadilyshownthat22󰀉󰀌󰀍1󰀂q1p

s󰀆(q)=󰀂󰀊2:󰀆ln2q1󰀂pPriortotherealizationofthesignalbutafterhavingobservedacandidate’skind,

thesuccessprobabilityQ󰀆=q󰀆(S󰀆)isarandomvariable.Now,letG󰀆(󰀃)denotethecumulativedistributionfunction(cdf)ofQ󰀆:Formally,

G󰀆(q)=Pr(Q󰀆󰀇q)

󰀍󰀌󰀍󰀌

s󰀆(q)󰀂1s󰀆(q)

+(1󰀂p)󰀄;=p󰀄

󰀊󰀆󰀊󰀆

where󰀄(󰀃)denotesthecdfofastandardNormaldistribution.Theassociated

densityofG󰀆(q)is

󰀌󰀌󰀍󰀌󰀍󰀍

s󰀆(q)󰀂1s󰀆(q)󰀊󰀆

g󰀆(q)=p󰀋+(1󰀂p)󰀋:

󰀊󰀆󰀊󰀆q(1󰀂q)Similarly,letG(󰀃)denotethecdfofsuccessprobabilityQpriortoobservingthe

candidate’skindorsignal,andg(󰀃)denotetheassociateddensity.Formally,

G(q)=(1󰀂mB)GA(q)+mBGB(q):

-11-Finally,itisusefultoestablishthefollowingstochasticdominancerelationsforG(󰀃)andG󰀆(󰀃):

Lemma1Forallp>p0,G(󰀃;p)…rst-orderstochasticallydominatesG(󰀃;p0).Thatis,

d

G(q)<0;forallq2(0;1):dpLemma2GA(󰀃)isamean-preservingspreadofGB(󰀃).And,forallmBIV.OptimalSearchandHiring

Inthissection,weshowthatthereexistsauniquesolutiontotheemployer’soptimizationproblem.Theoptimalhiringstrategyistosetanidenticalsuccessprobabilitythreshold,q󰀅,forallcandidates,irrespectiveoftheirkind.Thatis,afterobservingasignalsfromacandidateofkind󰀆;thecandidateishiredifandonlyiftheposteriorprobabilitythathecandothejob,q󰀆(s),isatleastq󰀅.

Toseethis,letV󰀅denotetheemployer’sexpectedpayo¤ifhefollowsanoptimalsearchandhiringstrategy.Inanyoptimalstrategy,theemployerhiresacandidateifandonlyifhisbeliefqthatthecandidatecandothejobissuchthatthepayo¤fromhiring,whichwedenotebyH(q;V󰀅),exceedsthepayo¤fromnothiringandmovingtothenextperiod.

Hence,wemaywritethevaluefunctionas

Z1

V󰀅=󰀄max[H(q;V󰀅);V󰀅]dG(q)󰀂k;

0

(1)

where

H(q;V󰀅)=qv+(1󰀂q)(󰀂c+V󰀅):

Notethat,accordingtoourtimingconvention,costkisincurredimmediately,while

thepayo¤fromhiring,H(q;V󰀅),isreceivedinthenextperiod.Thefollowingobservationiscrucial.

Lemma3UnderthestandardEuclideanmetric,equation(1)speci…esacontractionmappingT:R!RinV󰀅.

Lemma3,togetherwiththewell-knownContractionMappingTheorem(see,forexample,Stokey,Lucas,andPrescott,1989),impliesthatthereexistsaunique,optimalvalueV󰀅fortheemployer’sproblem.

-12-Sincetheemployer’sproblemisstationary,anystrategyattainingV󰀅mustbeathresholdstrategy(see,forexample,McCall,1970).Moreover,thethresholdmustbethesameforbothkindsofcandidates.Thereasonisthat,conditionalonq,acandidate’skind󰀆iscompletelyirrelevant:theonlythingthatmattersistheprobabilityofsuccessitself,andnotwhetherthecandidateismainstreamorminority.

Finally,itremainstoshowthatthethresholdstrategyattainingV󰀅isunique.

Underagenericthresholdstrategy,whichwedenotebyq,thevaluefunctiongiveninequation(1)reducesto

\"#Z1

󰀂󰀃󰀂󰀂󰀃󰀃󰀂󰀃󰀂󰀃Vq=󰀄GqVq+Hq;VqdG(q)󰀂k:

q

󰀂󰀃SubstitutingforHandsolvingforVq,weobtain

󰀂󰀃󰀄Vq=

R1

q

󰀂󰀃

Thus,theemployer’sproblemreducestochoosingqtomaximizeVq:Proposition1characterizestheuniqueoptimum.

Proposition1Theoptimalthresholdstrategy,q󰀅,istheuniqueinteriorsolutionto

󰀋󰀋1󰀂󰀄1󰀂q󰀄qdG(q)c

󰀅

󰀂󰀃󰀃:q=󰀂1󰀂󰀄Gq󰀅c+(1󰀂󰀄)v+k

R1

󰀊󰀊(qv+(1󰀂q)(󰀂c))dG(q)󰀂k

󰀊󰀋:R1

1󰀂󰀄1󰀂qqdG(q)

(2)

Thenextpropositionshowsthateverypossiblethresholdsuccessprobabilitycanbeanoptimum.

Proposition2Forallq2[0;1);thereexistparametervaluessuchthatq󰀅=q:

V.PerformanceMetrics

RecallthattheoptimalhiringstrategyestablishedinProposition1is“color-blind”inthesensethattheemployersetsthesamethresholdsuccessprobabilityforbothkindsofcandidates,andthattheoptimalhiringthresholdcanbeatanylevel:Inthissection,westudytheimplicationsofauniformhiringthresholdforobservableperformancemetricsofdiversity.

-13-PermanentWorkforceComposition

Perhapsthemostimportantperformancemetricofdiversityisthefractionof

minoritiesinthepermanentworkforceofanorganization,relativetotheirshareintheunderlyingpopulation.Intermsofourmodel,thiscorrespondstotheprobabilitythatapermanentlyhiredcandidateisaminority.

Formally,letr󰀆denotetheprobabilitythatthevacancyispermanently…lledbyacandidateofkind󰀆,whentheemployerusesthe,notnecessarilyoptimal,thresholdstrategyq.Then,r󰀆canbeexpressedrecursivelyasfollows.

r󰀆

󰀂󰀂󰀂󰀂󰀃󰀃󰀂󰀃󰀃󰀃

=m󰀆p1󰀂G󰀆qj󰀃=1+G󰀆qj󰀃=1r󰀆+(1󰀂p)r󰀆

󰀂󰀂󰀂󰀂󰀃󰀃󰀃󰀃

+(1󰀂m󰀆)1󰀂p1󰀂G󰀂󰀆qj󰀃󰀂󰀆=1r󰀆:

Wecanwritethisexpressionmuchmorecompactlyifwede…neG󰀆󰀉tobethe

probabilitythatacandidateofkind󰀆andtype󰀉inducesaposteriorsuccessprobabilitylessthanorequaltoq:Formally,

G󰀆󰀉󰀆G󰀆qj󰀃=󰀉:

Solvingforr󰀆,weobtain,inourmoreeconomicalnotation,

r󰀆=

m󰀆(1󰀂G󰀆1)

:

1󰀂m󰀆G󰀆1󰀂(1󰀂m󰀆)G󰀂󰀆1

󰀂

󰀃

Wewanttocompareminorityrepresentationintheworkplace,rB,withtheminorityshareoftheunderlyingpopulation,mB:Minoritiesareproportionally

r󰀄representedintheworkplacewhenm=1.Itiseasilyveri…edthatthisis󰀄

equivalenttotheconditionthatGA1=GB1.Inotherwords,minoritiesare

proportionallyrepresentedifandonlyiftheprobabilityofTypeIerror(rejectionofcompetentcandidates)isthesameforbothkindsofcandidates.WhendoesequalityofTypeIerrorhold?Lemma4Thereexistsauniquethreshold,q1󰀆

1

probabilityofTypeIerrorsisthesameforbothkindsofcandidates.

Unsurprisingly,theoptimalthresholdq󰀅giveninProposition1isgenericallynotequaltoq1:Thenextpropositionshowsthat,dependingontherelationshipbetweenq󰀅andq1,minoritiesmaybeunderoroverrepresentedintheworkplace.

1p2󰀅A󰀅B

1+1󰀂ep-14-Proposition3

1.Minoritiesareoverrepresentedintheworkplace(i.e.,02.Minoritiesareunderrepresented(i.e.,

rBmB

rBmB

>1)ifandonlyif

<1)ifandonlyifq1rBmB

3.Minoritiesareproportionatelyrepresented(i.e.,

󰀇1󰀈󰀅

q20;q:

=1)ifandonlyif

Ifq1minoritiesintheworkplacerelativetotheirshareintheunderlyingpopulation.Ontheotherhand,ifq1>q󰀅,thenitisthemajoritycandidateswhoaremoresubjecttoTypeIerror.Thisresultsinminoritycandidatesbeingoverrepresentedintheworkplace.Hence,theoutcomedependsonhow“choosy”theemployeris.

Thefollowing…gureillustrateshowthedi¤erenceinTypeIerrorsformajorityandminoritycandidatesvarieswiththethresholdstrategyoftheemployer.Itdisplays

1󰀂GA1ofhiringprobabilitiesforcompetentmajorityversuscompetenttheratio1󰀂GB1

minoritycandidatesasafunctionoftheemployer’sthresholdstrategyq.The

p

parametervaluesusedtodrawthe…gureare:p=:3;󰀊A=1;󰀊B=2.Noticethat󰀂󰀃

atlowthresholdsqworkforce,andthisdisparitygrowsasthethresholdincreasesfromq=0:Sincetheworkforceproportionsexactlyre‡ectthoseofthecandidatepopulationatq=q1;minorityoverrepresentationmustreverseitselfforasu¢cientlychoosyemployer.Inthe…gure,thedegreeofminorityoverrepresentationisgreatestatq=0:18anddeclinesthereafter.Forthresholdsq>q1,thee¤ectofthedi¤erenceinTypeIerrorscanbequitesevereforcompetentminoritycandidates.Bythetimethethresholdreaches0.7,acompetentmajoritycandidatestandsanalmost140timesbetterchanceofbeinghiredthanacompetentminoritycandidate.Indeed,asthe…gureshows,theratioofhiringprobabilitiesincreaseswithoutboundasthethresholdapproaches1.

-15-

Figure1.HiringProbabilityRatiosofCompetentCandidatesofKindAversus

1󰀂GA1(q)B:1.󰀂GB1(q)1.51401.41201001.3801.2601.1401200.900.10.20.300.40.50.60.7The…gureillustratesthatitbecomesexceedinglyunlikelythataminoritycandidatewill…llthepositionasthethresholdincreases.Putdi¤erently,theworkplacecompositionbecomesincreasinglyhomogeneous.Asweshowinthenext

proposition,thepositiverelationshipbetweenthechoosinessofanemployerandthehomogeneityoftheworkplaceisageneralpropertyofthemodel.

Proposition4Supposethattheemployeris“selective”initshiringpolicy,i.e.,q>p;then:

1.Astheemployerbecomesmoreselective,minorityrepresentationintheworkplacedecreases.Formally,rBisdecreasinginq.

2.Astheemployerbecomesarbitrarilyselective,minoritiesvanishfromtheworkplace.Formally,limq!1rB=0:

OnemaywonderwhatconditionsonprimitivesguaranteethatanemployerwillindeedbeselectiveinthesensedescribedinProposition4.Ausefullowerboundontheoptimalthresholdmaybederivedfromthecaseofa“myopic”employerwhoonlyderivesbene…toneperiodintothefuture.Suchanemployerwouldchoosea

c

“break-even”thresholdwherevq󰀂(1󰀂c)q=0or,equivalently,q=c+:Employersvwhovaluepayo¤sinperiodsbeyondthenextwilloptimallyraisethethreshold

-16-abovethebreak-evenleveltocapturesomeoftheoptionvalueofwaiting.Hence,

c

q󰀅>c+:Asaresult,asu¢cientconditionforanemployertobeselectiveisthatv

c:pWehaveshownthatdi¤erencesinTypeIerrorscanleadtogrossdi¤erencesbetweentheshareofminoritiesinthepermanentworkforcecomparedwiththeirshareofthecandidatepopulation.Giventhe“colorblind”thresholdstrategyoftheemployer,onemightspeculatethatthefractionofminoritiesamonginitialhireswouldre‡ecttheunderlyingpopulation.Asweshallsee,thisisnottypicallythecase.De…nethefractionofinitialhireswhoareofkind󰀆as

h󰀆=

m󰀆(1󰀂G󰀆)

:

m󰀆(1󰀂G󰀆)+m󰀂󰀆(1󰀂G󰀂󰀆)

Noticethattheprobabilitythatacandidateofkind󰀆willbehired,1󰀂G󰀆;consistsoftheprobabilityoftwoseparateevents:(i)thejointeventthatthecandidateiscompetentandpassestheinterview;and(ii)thejointeventthatthecandidateisincompetentandpassestheinterview.EventiiisequivalenttotheprobabilityofTypeIIerror.

Havingpreviouslyestablishedathreshold,q1,whereTypeIerrorisequalizedacrossthetwokindsofcandidates,itisusefultodeterminetheanalogousthresholdwhereTypeIIerrorisequalized.Thatis,de…neq0tobethethresholdsuchthat

GA0=GB0;

whichhasasitssolution

q0=

1+

1

1

1󰀂p󰀂2󰀅A󰀅B

ep>p:

Whenqbeinghiredthanincompetentmajoritycandidates,whileforq>q0theoppositeholds.Furthermore,noticethatthethresholdatwhichTypeIIerrorisequalizedalwaysliesabovethatwhereTypeIerrorisequalized.Thatis,q1Finally,weturnourattentiontothethreshold,q;,wheretheinitialhiring

proportionsareequaltotheunderlyingpopulationproportions.Thatis,q;solves

GA=GB:

UnlikeforthethresholdsforequalTypeIandTypeIIerrors,thereexistsnoclosed-formsolutionforq;.However,fromthefactthatGAisamean-preservingspreadofGB(Lemma2),itfollowsthatq;existsandisunique.Moreover,sinceq;representsatrade-o¤betweenTypeIandTypeIIerrors,q1-17-Aswasthecaseforthecompositionofthepermanentworkforce,dependingontheoptimalthresholdq󰀅,minoritiesmaybeunder-oroverrepresentedamonginitialhires.UsingargumentsidenticaltothoseinProposition3,itmaybereadilyshownthat

Proposition5

1.Minoritiesareoverrepresentedamonginitialhires(i.e.,02.Minoritiesareunderrepresented(i.e.,

hBmB

hBmB

>1)ifandonlyif

<1)ifandonlyifq;hBmB

3.Minoritiesareproportionatelyrepresented(i.e.,

󰀇;󰀈󰀅

q20;q:

=1)ifandonlyif

Itisinterestingtonotethat,sinceq1greater…ringratesleadtounderrepresentationinthepermanentworkforce.Wenowturntoformallyanalyzing…ringrates.FiringRates

Wesawthatminorityover-orunderrepresentationamonginitialhiresandinthepermanentworkforcedependsonthethresholdstrategyoftheemployer.Inthecaseof…ringrates,bycontrast,themodeldeliversunambiguouspredictions.Themainresultofthissectionisthatminorityhiresare…redathigherratesthanmajorityhiresforall(interior)thresholdstrategiesq2(0;1).

The…ringrateforhiresofkind󰀆isequaltotheprobabilitythatacandidateofkind󰀆isincompetentconditionalonhisbeinghiredinthe…rstplace.Formally,de…nethe…ringrateas

󰀂󰀃f󰀆=Pr󰀃=0jQ󰀆󰀈q(1󰀂G󰀆0)(1󰀂p)=:

1󰀂G󰀆Toseehow…ringratesre‡ectthetrade-o¤betweenTypeIandTypeIIerrors,itishelpfultowritef󰀆asfollows

f󰀆=

(1󰀂p)Pr(TypeII)

:

(1󰀂p)Pr(TypeII)+p(1󰀂Pr(TypeI))

Whenq1󰀇q󰀇q0;minoritiessu¤ergreaterTypeIandTypeIIerrorsthandomajorities.Asaconsequence,the…ringrateofminoritiesishigherthanformajorities.

-18-WhenqTypeIerrorisnowhigherformajoritiesthanforminorities.Asaconsequence,theorderingofmajorityandminority…ringratesbecomesambiguousanddependsontherelativemagnitudeofthetwotypesoferrors.Similarly,whenq>q0;TypeIIerrorissmallerforminoritiesthanformajoritiesbutTypeIerrorisgreater.Hence,alsointhiscase,theorderingcouldgoeitherway.Asthenextpropositionshows,however,thetrade-o¤betweenTypeIandTypeIIerrorsisalwaysresolvedinthedirectionofhigher…ringratesforminorities.4

Proposition6Forallq2(0;1),minorityhiresare…redatahigherratethanmajorityhires.Summary

Thefollowing…guresummarizesthevariousperformancemetricsofdiversityasafunctionofthesuccessprobabilitythresholdq.

Figure2.Over-andUnderrepresentationofMinorities.

Proposition6ignoresthecaseswhereq2f0;1gsince,forthesedegeneratecases,eithereveryoneishiredornooneishired,andthe…ringrateproblemistrivial.

4

-19-

VI.PolicyImplications

Inthissection,weexaminehowtheoptimalthreshold—and,byimplication,thediversitymetricsdescribedabove—varieswithchangesintheparametersofthemodel.Someoftheseparametersarelikelytobeunderpolicycontrol;hence,thereisthepossibilityofin‡uencingworkplacediversity.Throughoutthissection,weshallusetheterm“workplacediversity”asbeingsynonymouswiththeminority

rB

.Thecloserthisratioistounity,themorediverseistherepresentationratiomB

workplace.

DiversityandWorkerProtections

Therehasbeenconsiderablydebate,especiallyinEurope,overtheappropriatelevelofworkerprotectionsagainstsummarydismissal.ThemassstreetprotestsinFranceduringtheSpringof2006againstthecontratpremièreembaucheareasalientexample.Thisnewlawwouldhaveallowedforsummarydismissalofemployees

belowtheageof26duringthe…rsttwoyearsoftheircontract.Byreducingtheriskofhiring,itwashopedthatthecontratpremièreembauchewouldleadtoa

reductionintheveryhighyouthunemployment.Whetheritwouldhaveachieveditsgoalshallremainunknown,sincethelawwasretractedinresponsetotheprotests.InmanyU.S.organizations,therearevariousrestrictionsininterviewingpracticestoensurea“levelplaying…eld”betweenmajorityandminoritycandidates.Forexample,forNFLheadcoachingvacancies,theleagueruleisthataminimum

numberofminoritycandidatesmustbeinterviewedbefore…llingtheposition(NFL,2003).Similarly,theUniversityofCaliforniahasmanyrulesandrestrictions

governinginterviewingpracticestoensurefairness.Forexample,in…llingapositionattheUniversityofCalifornia,theinterviewerisobligatedto…lloutformsforeachoftheinterviewedindividualsstatingtheprecisereasonsthattheywerenotselectedfortheposition.(See,forexample,SearchActivityStatementUCI-AP-80,availableathttp://www.ap.uci.edu/Forms/APforms/UCI-AP-80.pdf.)

Intermsofourmodel,EUworkerprotectionpoliciesmaybethoughtofasincreasingthecostof…ring,c;whiletheUniversityofCaliforniainterviewingpracticesmaybethoughtofasincreasingthecostperinterview,k:Obviously,increasesinbothcandkraisethe“frictions”associatedwiththehiringprocess,yet,asweshallsee,theyhaveoppositeimplicationsfordiversity.

Implication1Supposethattheemployeris“selective”initshiringpolicy—i.e.,q󰀅>p—then:

1.Anincreaseinthecostof…ring,c,reducesworkplacediversity.2.Anincreaseinthecostofinterviewing,k,increasesworkplacediversity.

-20-Intuitively,raisingthecostof…ringincreasesthecostofTypeIIerrorsfortheemployer.Asaresult,hebecomesmorereluctanttotakeachanceonwhetheracandidatecandothejoband,consequently,raisesthethresholdforhiring.Aswehaveshownintheprevioussection,whentheemployerisatallselective,increasedhiringthresholdshavethee¤ectofdi¤erentiallyraisingTypeIerrorstothedisadvantageofminorities.Asaresult,workplacediversitydecreases.

Incontrast,raisingthecostofinterviewingmakesitmoreexpensiveforthe

employertobechoosy.Asaresult,theemployerlowershisthresholdforhiringandthis,inturn,reducesthedi¤erenceinTypeIerrorsbetweenminoritiesandmajorities.Asaresult,workplacediversityincreases.DiversityovertheBusinessCycle

Next,weconsiderhowtheemployer’soptimalthresholdvarieswiththebusinesscycle.Atapeakinthebusinesscycle,jobcandidatesbecomemorescarceand,hence,thecostofrecruitingincreases.Aswehaveshownabove,thishasthee¤ectofraisingworkplacediversity.Inaddition,thevalue-addedofacompetentemployeeisalsolikelytobehigheratthepeakofthebusinesscyclethanduringarecession.Intermsofourmodel,thiscorrespondstoanincreaseinv:

Implication2Supposethattheemployeris“selective”initshiringpolicy—i.e.,q󰀅>p—thendiversityisprocyclical.Formally,q󰀅isdecreasinginv(andk):Intuitively,asacompetentemployee’svalue-addedincreases,itbecomesmorecostlytoleavethepositionun…lled.Asaconsequence,theemployerismorewillingtotakeachancebyhiringpossiblyincompetentemployeesand,hence,theoptimal

thresholdfalls.Alowerthresholdreducesthedi¤erenceinTypeIerrorsbetweenminoritiesandmajorities.Consequently,workplacediversityincreases.As

mentionedintheintroduction,theprocyclicalityofdiversityisindeedconsistentwiththeDutchexperienceoverthelastdecade.DiversityandtheCostofCapital

Anothertestableimplicationofthemodelisthatvariationintheriskinessof…rmsleadstodi¤erencesinworkplacediversity.Ifweinterpretthediscountparameter󰀄asrepresentinganemployer’scostofcapital,whichpresumablyvarieswiththeriskinessofhisbusiness,thenwehavethefollowingimplication:

Implication3Supposethattheemployeris“selective”initshiringpolicy,i.e.,q󰀅>p;thenriskier…rmsaremorediverse.Formally,q󰀅isincreasingin󰀄:Intuitively,theoptionvalueofwaitingisworthlessforrisky…rmsthanforsafe…rms.Sincethedegreetowhichtheoptimalthresholdliesabovethebreak-even

-21-thresholdpositivelydependsonthisoptionvalue,theoptimalthresholdforariskier…rmislowerthanthatforalessrisky…rm.Inturn,thislowerthresholdreducesthedi¤erenceinTypeIerrorsbetweenminoritiesandmajorities,and,consequently,workplacediversityincreases.AsmentionedintheIntroduction,theprocyclicalityofdiversityisindeedconsistentwiththeDutchexperienceoverthepastdecade.DiversityandtheScarcityofCompetence

Aswehighlightedabove,thekeydeterminantofminorityover-or

underrepresentationistherelationshipbetweentheoptimalthresholdq󰀅andthethresholdsforequatingTypeIandTypeIIerrorsacrossthetwopopulations—q1andq0;respectively.Thesetwothresholdsbracketthepriorprobabilitythata

candidateiscompetent;thatis,q1Whenfewcandidatescandothejob—i.e.,whenpislow—theresultsofthe

interviewmustbesu¢cientlyconvincingtoinducetheemployertotakeachanceonthecandidategiventhecostsof…ring.Acandidatewithaverynoisysignalisgoingtohaveadi¢culttimeinmakingthiscase.Inthelimit,imagineasituationwhereBcandidateshavearbitrarilynoisysignalsandwheretheemployerisselective.Clearly,thereisvirtuallynopossibilityofovercomingtheemployer’spriorbeliefaboutthelowlikelihoodthatthecandidateisquali…ed.Incontrast,acandidatewithaveryprecisesignalfacesnosuchhandicap.Inthisextremecase,onewouldexpect(andthemodelpredicts)severeunderrepresentationofminoritycandidatesbothatthehiringstageandinthepermanentworkforce.

Bycontrast,whenmostcandidatescandothejob,i.e.,whenpishigh,animprecisesignalintheinterviewstagecanbeanadvantageforacandidate.Supposethatpissu¢cientlyhighsuchthattheemployerispredisposedtogivemostcandidatesachancetoprovethemselvesonthejob.Inthatcase,havinganarbitrarilynoisysignalvirtuallyguaranteesthatthecandidatewillnotgreatlydisappointintheinterviewand,hence,willbeo¤eredtheposition.Incontrast,amoreprecisesignalexposesthecandidatetoagreaterpossibilityofmakingabadimpressionintheinterviewandhencebeingrejectedforthejob—eveninthecasewherethe

candidateisinfactcompetent.Inthissituation,overrepresentationofminoritycandidates,bothinhiringandinthepermanentworkforce,isthemorelikelyoutcome.Thenextimplicationformalizesthisintuition.

Implication4Injobsthataresu¢cientlyselective,minoritieswillbe

underrepresented.Injobsthataresu¢cientlynonselective,minoritieswillbe

overrepresented.Formally,thereexists0rBrB

<1whileforallp2(p;1);>1:1mBmB

-22-

VII.Conclusions

Inthispaperwehaveinvestigatedtheimplicationsofassumingthatemployers…ndsiteasiertoevaluatemajorityjobcandidates,withwhomtheytendtosharea

culturalandsocialbackground,thanminorityjobcandidates,whosebackgroundisquitedi¤erentfromtheemployers’.Intermsofsociolinguistics,employersandminorityjobcandidatesfailtoshareadiscoursesystemenablingclearcommunication.

Wehaveshownthatthisbasicpremiseimpliesthatthereexistsatensionbetweenjobsecurity,scarcityofskills,andworkplacediversity.Whenjobsecurityishigh—i.e.,…ringnonperformingsta¤isexpensive—minoritiesarelikelytobe

severelyunderrepresentedinselectivepositions.Attheotherextremetheconverseholds.Whenjobsecurityislow,minoritiesareoverrepresentedinnonselectivepositions.Thesedistortionsoccureventhoughmajorityandminoritypopulationshaveidenticalskilllevels.

Onafundamentallevel,ourresultsaredrivenbyBayes’law,whichimpliesthatemployers’posteriorbeliefsaboutmajoritycandidatesrespondmorestronglytonewinformationthantheirbeliefsaboutminoritycandidates.Whentheinformationreceivedisbetterthanexpected,thishighbelief-sensitivityworkstotheadvantageofmajoritycandidates.Ontheotherhand,whentheinformationisworsethanexpected,highbelief-sensitivityworkstothedisadvantageofmajoritycandidates.Whiletheoccurrenceof“reversediscrimination”maybeinterestingfromatheoreticalperspective,fromapolicyperspective,theunderrepresentationofminoritiesinselectivepositionsseemsthemorerelevantmodelprediction.Giventhatminoritiesareindeedgrosslyunderrepresentedinmanyorganizations,whatcanbedoneaboutit?

Inourmodel,thelackofworkplacediversityarisesbecauseofapostulated

informationorcommunicationmismatchbetweenthemajorityemployer/interviewerandminorityjobcandidates.Obviously,matchingthebackgroundoftheinterviewerwiththebackgroundofthecandidatewouldsolvethisproblem.However,moreoftenthannot,thismaybequitedi¢culttoimplement.First,inorganizationslackingdiversity,minoritiesarescarcetobeginwith.Second,itshouldnotbeforgottenthatthevariousminoritiesareculturallyhighlydiverse,thusrequiringaverycarefulmatchingbetweentheevaluatorandtheevaluee.Forinstance,whileaFrench-speakingWestAfricanandanAfrican-Americanarebothpeopleofcolor,itseemsquiteclearthattheydonotsharethesamediscoursesystem.

Asecond,andprobablymorerealistic,optiontoincreaseworkplacediversityistolower…ringcosts.Wehaveshownthathighcostsof…ringinduceemployerstoimposeextremethresholdsuccessprobabilities.Therelativelylowinformativenessofminoritycandidates’signalsmakesitvirtuallyimpossibletopasssuchhigh

thresholds,irrespectiveoftheirskills.Thisisespeciallytrueforveryselectivejobs,

-23-whereemployers’priorbeliefsthatarandomcandidatecandothejobareverylow.When…ringcostsarereduced,thresholdsuccessprobabilitiescomedowntomorerealisticlevels.Thislessensthedisadvantageofminoritycandidates,levelstheplaying…eld,andleadstoamorediverseworkplace.Finally,policiesthatincreasethecostofinterviewing(andhenceoptimallyreduceselectivitybyemployers)arealsodiversityenhancing.5

Thisdiscussionwouldnotbecompletewithoutpointingoutthelimitationsofthemodel.Fromatechnicalstandpoint,onelimitationistheone-sidedsearch,orpartialequilibriumnatureoftheanalysis.Itmightbeworthwhileextendingthemodeltoageneralequilibriumframeworkinwhichcandidateschoosewhatkindofpositionstheyapplyto.Also,thebinarynatureofcompetenceinour

model—candidateseithercandothejobortheycannot—isclearlyrestrictive.Otherlimitationsareofalesstechnicalnature,suchastheassumptionsofequalaverageskilllevels,identical…ringcostsacrosssubpopulations,andno“naked”racismandno“directedsearch”onthepartoftheemployers.Also,wehave

assumedthatemployersonlycareabouttechnicalcompetence,andnotabouthowacandidate“…ts”intothedominantcultureoftheorganization.Someorevenalloftheseassumptionsdonotholdinpractice;however,mostrealisticdeviationspointinthesamedirection:towardsmoreratherthanlessdiscriminationthanpredictedbythemodel.Assuch,themodelputsalowerboundontheproblemandshowsthat,evenunderthebestofcircumstances,competentminoritycandidatesarelikelytohaveamuchhardertimesecuringacovetedjobthanequallycompetentmajoritycandidates,inparticularwhenjobsecurityishigh.

Outsideofpoliciesa¤ectingtheecomomicincentivesoftheemployer,policiesthatreduceofeliminatethedi¤erenceinsignalprecisionbetweentheminorityandnon-minoritycandidatesarediversityenhancingaswell.

5

-24-APPENDIX

Appendix:ProofsofLemmas,Propositionsand

Implications

ProofsofLemmas

Lemma1Forallp>p0,G(󰀃;p)…rst-orderstochasticallydominatesG(󰀃;p0).Thatis,

d

G(q)<0;forallq2(0;1):dpProof.Recallthat

G(q)=(1󰀂mB)GA(q)+mBGB(q);

where

G󰀆(q)=p󰀄

󰀆=A;B.Now,

dGdp󰀆

󰀌

s󰀆(q)󰀂1

󰀊󰀆

󰀍

+(1󰀂p)󰀄

󰀌

s󰀆(q)󰀊󰀆

󰀍

;

(q)=

󰀍

󰀌

󰀍󰀌󰀌󰀍󰀌󰀍󰀍

s󰀆(q)󰀂1s󰀆(q)s󰀆(q)󰀂1s󰀆(q)@s󰀆(q)=󰀄󰀂󰀄+p󰀋+(1󰀂p)󰀋<0;

󰀊󰀆󰀊󰀆󰀊󰀆󰀊󰀆@p

󰀊󰀋󰀊󰀋

@s󰀄(q󰀄)󰀊2s(q)󰀂1s(q)󰀄󰀄󰀄because@p=󰀂p(1󰀂<0and󰀄<󰀄󰀊󰀄.p)󰀊󰀄SinceG(q)isaconvexcombinationofGA(q)andGB(q),itfollowsthatforallq2(0;1):Thisprovesthelemma.

d

G(q)dp󰀌

<0

Lemma2GA(󰀃)isamean-preservingspreadofGB(󰀃).And,forallmBG(󰀃;mB)isamean-preservingspreadofG(󰀃;m0B).Proof.First,weverifythatEGA[QA]=EGB[QB]=p.Byde…nition,

EG󰀄[Q󰀆]=

Z

1

qg󰀆(q)dq;

0

where󰀆2fA;Bg.Changingtheintegrationvariablefromprobabilityqtosignals,wegetZ1

dq󰀆(s)

EG󰀄[Q󰀆]=q󰀆(s)g󰀆(s)ds;

ds󰀂1

-25-󰀂1

p󰀋(s

󰀅)

󰀄

󰀂1

󰀋(󰀅s)󰀋(sp(1󰀂p)󰀅󰀄)󰀄

󰀊󰀄(p󰀋(s󰀂1)+(1󰀂p)󰀋(s

󰀅󰀅APPENDIX

whereq󰀆(s)=p󰀋s󰀂1+(1󰀂p)󰀋s,=

(󰀅󰀄)(󰀅󰀄)󰀄󰀊󰀊󰀋󰀊󰀋󰀋s󰀂1s󰀊󰀄

g󰀆(s)=p󰀋󰀊󰀄+(1󰀂p)󰀋󰀊󰀄.Hence,q󰀄(s)(1󰀂q󰀄(s))Z1

dq󰀆(s)

q󰀆(s)g󰀆(s)EG󰀄[Q󰀆]=ds

ds󰀂1

󰀍Z1󰀌

s󰀂1󰀋=pds

󰀊󰀆󰀂1

=p:

@q󰀄(s)@s))󰀄

2and

ThisprovesthatEGA[QA]=EGB[QB]=p.Forlateruse,notethat

EG(󰀃;mB)[Q]=EG(󰀃;m0)[Q]=p.

B

ToprovethatGA(󰀃)isamean-preservingspreadofGB(󰀃)itnowsu¢cestoshowthat,ontheinterval(0;1),GB(󰀃)crossesGA(󰀃)onlyonceandfrombelow.Wedothisbyestablishingthatthedi¤erenceD(q)󰀆GA(q)󰀂GB(q)hastwoextrema:startingfromzeroatq=0,D(q)…rstreachingamaximum—atwhichD(q)isstrictlypositive—andthenaminimum—atwhichD(q)isstrictlynegative.Let

󰀅=ln

suchthat

󰀌

1󰀂qpq1󰀂p

󰀍

D=GA(q)󰀂GB(q)

󰀌1󰀍󰀌12󰀍󰀂2󰀂󰀊2󰀅󰀂󰀊AA󰀅=p󰀄+(1󰀂p)󰀄2󰀊A󰀊A󰀌1󰀍󰀌12󰀍󰀂2󰀂󰀊2󰀅󰀂󰀊BB󰀅󰀂p󰀄󰀂(1󰀂p)󰀄2:

󰀊B󰀊B

Relyingonthefactthat󰀅isamonotonefunctionofq,wenowaskwhen󰀌1󰀍󰀌12󰀍󰀂2󰀂󰀊2󰀅󰀂󰀊dDAA󰀅=󰀂󰀊Ap󰀋󰀂󰀊A(1󰀂p)󰀋2d󰀅󰀊A󰀊A

󰀌1󰀍󰀌12󰀍󰀂2󰀂󰀊2󰀂󰀊󰀅BB󰀅+󰀊Bp󰀋+󰀊B(1󰀂p)󰀋2󰀊B󰀊B

=0()

󰀊󰀂󰀊2󰀂1B󰀅2󰀊B

dDd󰀅=0:

󰀋+󰀊A

󰀊12󰀋=󰀂2󰀂󰀊A󰀅󰀊B

󰀋+󰀊A

=

e

󰀂1

2󰀅

2󰀂12󰀂󰀅B󰀃󰀅B

󰀋1󰀂p

󰀋p1󰀂p󰀋p󰀆2

++

e

󰀂1

2󰀅2󰀂12󰀂󰀅A󰀃󰀅A

󰀆2

󰀂󰀊2A󰀅2󰀊A

󰀅12󰀆212󰀂󰀅B󰀃

󰀅B1󰀂p󰀂2ep󰀅12󰀆2:12󰀂󰀅A󰀃󰀂2󰀅1󰀂pp

󰀊1

2󰀊1

󰀂󰀊2B󰀅󰀊B

󰀋󰀋e

A

-26-APPENDIX

Nowconsidertheright-handside,whichwedenoteby󰀅,asafunctionof󰀅.

e

1󰀂2󰀅󰀆

󰀅󰀅2󰀂12󰀂󰀅B󰀃󰀅B1󰀂󰀅2󰀃󰀂2A󰀅A

󰀆2󰀆2

++

1󰀂p

ep1󰀂pep󰀅

󰀂1

2󰀂12󰀅󰀅21

2󰀂󰀅B󰀃󰀅B122󰀂󰀅A󰀃󰀅A

󰀆2󰀆2

e

󰀂12=e

=e󰀅

󰀂1218󰀅

󰀅

14󰀅2B

2

+󰀊2B󰀅

1

󰀂1󰀅2󰀅2AB

󰀆

󰀆

+1

212

+󰀊2A󰀅4󰀅2A

+12(

2󰀊2A󰀂󰀊B

)󰀅

2

󰀆

󰀆

Thus,Dtakesonextremaatvaluesof󰀅thatsolve

󰀊A

=e󰀊B

Takinglogs,

󰀅󰀅

181

󰀂1󰀅2󰀅2AB

󰀆

+1

22

(󰀊2A󰀂󰀊B)󰀅

2

󰀆

:

󰀌󰀍

󰀃21󰀊A111󰀂22

󰀂2+ln=󰀊A󰀂󰀊B󰀅:

󰀊B8󰀊2󰀊2AB

Therefore,thesolutionsto󰀅arerootsofthefunction

󰀌󰀍

󰀃211󰀂2󰀊A112

󰀂+󰀊󰀂󰀊󰀅󰀂ln:AB28󰀊2󰀊2󰀊BABTheserootsare

󰀅=

󰀂11

;󰀅=󰀊B:B2󰀊A󰀊Bln󰀊2󰀊󰀊lnAB󰀊A󰀊A

Theexistenceofexactlytwodistinctrootsfor󰀅(andhenceforq)impliesthatGA

andGBcrosseachotherexactlyonce.ItremainstoverifythatGBcrossesGAfrombelowandnotfromabove.Now,

D=GA(q)󰀂GB(q)

=p(GA1󰀂GB1)+(1󰀂p)(GA0󰀂GB0):

Atq=q=Atq=q=

01

1

p2󰀅A󰀅B

1+1󰀂ep1;GA1󰀂GB1=0whileGA0󰀂GB0

1

p1+1󰀂ep󰀂2󰀅1󰀅AB

;GA0󰀂GB0=0whileGA1󰀂GB1

󰀂1󰀃

>0.Hence,Dq>0.

Now,becauseq1󰀂0󰀃

<0.Hence,Dq<0.

ThiscompletestheproofthatGA(󰀃)isamean-preservingspreadofGB(󰀃).Finally,toprovethatG(󰀃;mB)isamean-preservingspreadofG(󰀃;m0B)forallmBZqZq^^

G(q;mB)dq󰀂G(q;m0B)dq󰀇0

0

0

-27-forallq^2(0;1),withstrictinequalityforsomeq^.Now,

G(q;m0B)dqG(q;mB)dq󰀂

00

Zq^

0

(GA(q)󰀂GB(q))dq󰀇0;=(mB󰀂mB)

0

APPENDIX

Z

q^

Z

q^

wheretheweakinequalityforallq^,andthestrictinequalityforsomeq^,followfrom

thefactthatGB(󰀃)second-orderstochasticallydominatesGA(󰀃).Thiscompletestheproof.

Lemma3UnderthestandardEuclideanmetric,equation(1)speci…esacontractionmappingT:R!RinV󰀅.

Proof.LetT:R!Rbegivenby

Z1

T(v)=󰀄max[H(q;v);v]dG(q)󰀂k:

0

Then,wehavetoshowthatforallv;w2Randforsome0󰀇󰀃<1,

kT(v)󰀂T(w)k󰀇󰀃kv󰀂wk:

Now,kT(v)󰀂T(w)k=

󰀉󰀉Z1Z1

󰀉󰀉

󰀉max[H(q;w);w]dG(q)+k󰀉max[H(q;v);v]dG(q)󰀂k󰀂󰀄=󰀉󰀄󰀉

00

󰀉Z1󰀉󰀉󰀉󰀉=󰀄󰀉(max[H(q;v);v]󰀂max[H(q;w);w])dG(q)󰀉󰀉0

󰀉󰀉Z1

󰀉󰀉

󰀉(max[qv+(1󰀂q)(󰀂c+v);v]󰀂max[qv+(1󰀂q)(󰀂c+w);w])dG(q)󰀉=󰀄󰀉󰀉0󰀉Z1󰀉󰀉󰀉

󰀉=󰀄kv󰀂wk:󰀇󰀄󰀉(v󰀂w)dG(q)󰀉󰀉

0

Toseethatthelastinequalityholds,assume,withoutlossofgenerality,thatv>w.

Now,ifvqv>qw.Hence,

wmax[H(q;v);v]󰀂max[H(q;w);w]=H(q;v)󰀂H(q;w)

=(1󰀂q)(v󰀂w)-28-Ifv>H(q;v),thenv>H(q;w).Therefore,

max[H(q;v);v]󰀂max[H(q;w);w]=v󰀂max[H(q;w);w]Thiscompletestheproof.

Lemma4Thereexistsauniquethreshold,q1󰀆

1p2󰀅A󰀅B

1+1󰀂epAPPENDIX

1

probabilityoftypeIerrorsisthesameforbothkindsofcandidates.Proof.,

󰀄

,,,

󰀂󰀃󰀂󰀃

GA1q=GB1q

12󰀂󰀃󰀂󰀃sAq󰀂1sBq󰀂1=

󰀊A󰀊B󰀋󰀋󰀊󰀊󰀋󰀊󰀊󰀋pp11122

󰀂󰀊Alnq󰀂11󰀂p󰀂1󰀂󰀊Blnq󰀂11󰀂p󰀂12=

󰀊A󰀊B

q=

11+

1󰀂p2󰀅A󰀅B

ep1

! 󰀂󰀃!󰀂󰀃sAq󰀂1sBq󰀂1=󰀄

󰀊A󰀊B

:

Lemma5Supposeq>p:Then:

1.ThedistributionGA1dominatesGB1intermsofthelikelihoodratio.2.ThedistributionGA0dominatesGB0intermsofthelikelihoodratio.

lngA1

Proof.Toestablishthis,itissu¢cienttoshowthat@@󰀊@q>0.

󰀋󰀊s(q)󰀂1󰀊2

@ln󰀋󰀊q(1󰀂q)@2lngA1

=

@󰀊@q@󰀊@q

󰀍󰀌1s(q)󰀂12󰀂2(󰀅)󰀊e@2lnp1

q(1󰀂q)2󰀉=

@󰀊@q󰀊󰀋1󰀂qp

2lnq1󰀂p=󰀊>0;

q1󰀂q

2

-29-APPENDIX

wheretheinequalityholdssinceq>p:Theproofofpart2oftheLemmaisvirtuallyidentical.

Lemma6Supposeq>p:Then:

1.ThedistributionGA1dominatesGB1intermsofthehazardrate.2.ThedistributionGA0dominatesGB0intermsofthehazardrate.Proof.Lemma5impliesthat

gB1(q0)gA1(q0)

<

gB1(q)gA1(q)

forallpZ

Z1gA1(t)gB1(t)

dt>

qgB1(q)qgA1(q)

1󰀂GA1(q)1󰀂GB1(q)

>;

gA1(q)gB1(q)

1

or,equivalently,

gA1(q)gB1(q)

<:

1󰀂GA1(q)1󰀂GB1(q)

Theproofforpart2ofthelemmaisvirtuallyidentical.ProofsofPropositions

Proposition1Theoptimalthreshold,q󰀅,istheuniqueinteriorsolutionto

󰀋󰀋1󰀂󰀄1󰀂q󰀄qdG(q)c

󰀅

󰀂󰀃󰀃q=󰀂:

1󰀂󰀄Gq󰀅c+(1󰀂󰀄)v+k

R1

R1

q

󰀊󰀊Proof.Recallthat

󰀄󰀂󰀃Vq=

(qv+(1󰀂q)(󰀂c))dG(q)󰀂k

󰀊󰀋R1

1󰀂󰀄1󰀂qqdG(q)

󰀂󰀂󰀃󰀃R1R1

󰀄vqqdG(q)󰀂󰀄c1󰀂Gq+󰀄cqqdG(q)󰀂k

󰀊󰀋=:R1

1󰀂󰀄1󰀂qqdG(q)

-30-APPENDIX

Itisusefultorepresentthisasnumeratoranddenominatorcomponentsforpurposesofdi¤erentiation.Hence,de…ne

Z1

(qv+(1󰀂q)(󰀂c))dG(q)󰀂k;N󰀆󰀄

q

and

D󰀆1󰀂󰀄1󰀂

Z

1

qdG(q):

q

!

@V(q)

Thus,the…rst-ordernecessaryconditionforoptimality,@q=0;maybeexpressedas

DN0󰀂ND0

=0:

D2

Therefore,

󰀃󰀃󰀂󰀂󰀃󰀃󰀂󰀃󰀂󰀂󰀃󰀂@VqD󰀂󰀄gq(v+c)q󰀂c󰀂N󰀂󰀄qgq

=@qD2

󰀂󰀃󰀂D(v+c)q+Dc+Nq=󰀄gq:

D2

Hence,

andthisimpliesthat

󰀂D(v+c)q+Dc+Nq=0;

q󰀅=

Dc

:

D(v+c)󰀂N

SubstitutingforDandN,andsimplifying,wegetthefollowingimplicitcharacterizationofq󰀅:

󰀊󰀊󰀋󰀋R1

1󰀂󰀄1󰀂q󰀄qdG(q)c

󰀅

󰀋󰀋󰀊q=󰀊R1R1

1󰀂󰀄1󰀂q󰀄qdG(q)(v+c)󰀂󰀄q󰀄(qv+(1󰀂q)(󰀂c))dG(q)+k󰀊󰀊󰀋󰀋R1

1󰀂󰀄1󰀂q󰀄qdG(q)c

󰀂󰀃󰀃=󰀂;

1󰀂󰀄Gq󰀅c+(1󰀂󰀄)v+kandthisyieldstheexpressioninLemma1.

Havingderivedthenecessary…rst-orderconditionforaninteriorsolutionq󰀅2(0;1),wenowproveitsactualexistence.

Atq󰀅=0,LHSRHS.Hence,bycontinuityandtheintermediatevaluetheorem,theremustbeaq󰀅2(0;1)suchthatLHS=RHS.Next,weproveuniquenessbyshowingthatthereisatmostoneq󰀅2(0;1)thatsatis…esthenecessary…rst-ordercondition.

-31-APPENDIX

Toseethis,…rstnoticethatwemayrewritethe…rst-orderconditionasfollows:

!Z1

󰀂󰀃

q󰀅(c+(1󰀂󰀄)v+k)=c󰀂c󰀄1󰀂qdG(q)+󰀄Gq󰀅cq󰀅:

q󰀄

Integratingbyparts,weobtain

󰀅

q(c+(1󰀂󰀄)v+k)=c󰀂c󰀄

Addingandsubtractingc󰀄

Rq󰀄

0

Z

1

G(q)dq:

q󰀄G(q)dqtotheright-handsideyields

Z1Zq󰀄

q󰀅(c+(1󰀂󰀄)v+k)=c󰀂c󰀄G(q)dq+c󰀄G(q)dq:G(q)dq=1󰀂pandsubstituting,weobtain

󰀌󰀍Zq󰀄

G(q)dq:q󰀅(c+(1󰀂󰀄)v+k)=c(1󰀂󰀄)+c󰀄p+

0

0

Finally,notingthat

R1

00

Hence,

Zq󰀄

c󰀄(1󰀂󰀄)c+c󰀄p

+q󰀅=G(q)dq:

(c+(1󰀂󰀄)v+k)(c+(1󰀂󰀄)v+k)0

Notethattheright-handsideismonotonicallyincreasinginq󰀅ataspeed<1;forallq󰀅2(0;1).Thisimplies,however,thattheright-handsidecancrossthe

45-degreeline,whichcorrespondstotheleft-handside,atmostonce.Hence,thereisatmostoneq󰀅2(0;1)thatsatis…esthenecessary…rst-ordercondition.Finally,weshowthatattheuniqueinteriorq󰀅,thevaluefunctionreachesaglobal

󰀂󰀃

maximum.Thisfollowsfromtheobservationthatlimq!1Vq!󰀂1,andthat

@V(q)thereexistsan\">0suchthatforall00.Toseethatthelatterassertionisindeedtrue,recallthat

R1

󰀂󰀃󰀄q(qv+(1󰀂q)(󰀂c))dG(q)󰀂k

󰀊󰀋Vq=R1

1󰀂󰀄1󰀂qqdG(q)andthat

󰀂󰀃

whereNandDdenotethenumeratorandthedenominatorofVq,respectively.Nowwerewrite

@V(q)@q󰀂󰀃󰀂󰀃󰀂D(v+c)q+Dc+Nq@Vq

=󰀄gq;2@qDtoget

!󰀂󰀃󰀂󰀃󰀂󰀃c@VqVq󰀂(v+c)=󰀄gq+q:@qDD

-32-APPENDIX

Writteninthisform,itisobviousthat,forsu¢cientlysmallq>0,bothfactorsinthelastexpressionarestrictlypositive.Thisprovestheproposition.Proposition2Forallq2[0;1);thereexistparametervaluessuchthatq󰀅=q:Proof.Fixk=0:Inthatcase,theemployerwillalwayswishtoparticipatebyinterviewingcandidatesratherthaneschewingtheemploymentmarket.When

c=0;theright-handsideofequation(2)equalszero;hence,q󰀅=0:Whenc!1;theright-handsideofequation(2)goesto1asthefollowingargumentshows:

󰀋󰀋󰀊󰀊R1

1󰀂󰀄1󰀂q󰀄qdG(q)c

󰀂󰀃󰀃lim󰀂c!11󰀂󰀄Gq󰀅c+(1󰀂󰀄)v

󰀊󰀊󰀋󰀋R1

1󰀂󰀄1󰀂q󰀄dG(q)c

󰀂󰀃󰀃󰀈lim󰀂c!11󰀂󰀄Gq󰀅c+(1󰀂󰀄)v

󰀂󰀂󰀅󰀃󰀃1󰀂󰀄Gqc󰀂󰀃󰀃=lim󰀂=1:

c!11󰀂󰀄Gq󰀅c+(1󰀂󰀄)v

Hence,limc!1q󰀅=1:Finally,sincetheright-handsideofequation(2)iscontinuous

inc;itfollowsthatthereexistparametervaluessuchthatq󰀅=qforallq2[0;1):Proposition3

1.Minoritiesareoverrepresentedintheworkplace(i.e.,02.Minoritiesareunderrepresented(i.e.,

rBmB

rBmB

>1)ifandonlyif

<1)ifandonlyifq1rBmB

3.Minoritiesareproportionatelyrepresented(i.e.,

󰀈󰀇

q󰀅20;q1:

=1)ifandonlyif

rBProof.Underauniformthresholdsuccessprobabilityq,m=1ifandonlyifB󰀂󰀃󰀂󰀃GA1q=GB1q.AswesawinLemma4,thiscorrespondsto

q=q1=1󰀂p11.Toprovetheproposition,weshowthatatthecriticalpoint

1+

q1,raisingqleadstostrictunderrepresentationofminorities.Thatis,wecalculatethederivativeof

󰀌󰀍󰀌󰀍sA(q)󰀂1sB(q)󰀂1

GA1(q)󰀂GB1(q)=󰀄󰀂󰀄

󰀊A󰀊Bwithrespecttoq,evaluateitatq1=negative.

1

p2󰀅A󰀅B

1+1󰀂ep1pe2󰀅A󰀅B

andshowthatitisstrictly

-33-Thederivativeisequalto

gA1(q)󰀂gB1(q)=󰀋

APPENDIX

Multiplyingbyq(1󰀂q)andevaluatingatq1,weget

! 1󰀊󰀂󰀊! 1󰀊󰀂󰀊BABA󰀂2󰀊A󰀂1󰀂12󰀊B

󰀊A󰀂󰀋󰀊B=󰀋

󰀊A󰀊B

󰀍󰀌󰀍󰀌

1󰀊B+󰀊A1󰀊B+󰀊A

󰀊A󰀂󰀋󰀂󰀊B=󰀋󰀂

2󰀊A󰀊B2󰀊A󰀊B

󰀌󰀍1󰀊B+󰀊A

=(󰀊A󰀂󰀊B)󰀋<0:

2󰀊A󰀊BThisprovestheproposition.

󰀌

sA(q)󰀂1󰀊A

󰀍

󰀊A

󰀂󰀋

q(1󰀂q)

󰀌

sB(q)󰀂1󰀊B

󰀍

󰀊B

:

q(1󰀂q)

Proposition4Supposethattheemployeris“selective”initshiringpolicy,i.e.,q>p;then:

1.Astheemployerbecomesmoreselective,minorityrepresentationintheworkplacedecreases.Formally,rBisdecreasinginq.

2.Astheemployerbecomesarbitrarilyselective,minoritiesvanishfromtheworkplace.Formally,limq!1rB=0:

Proof.Toprovepart1,di¤erentiaterBwithrespecttoq:@rB@q

=

󰀂mBgB1(1󰀂mBGB1󰀂mAGA1)󰀂(󰀂mBgB1󰀂mAgA1)mB(1󰀂GB1)

(1󰀂mBGB1󰀂mAGA1)2mBmA(gA1(1󰀂GB1)󰀂gB1(1󰀂GA1))

:=2(1󰀂mBGB1󰀂mAGA1)

@rB@qNoticethatthesignofdependsonlyonthehazardratesofGA1andGB1.And

@rB@qbyLemma6itthenfollowsthat<0:

Toprovepart2oftheproposition,noticethat(viaL’Hôpital’srule)

mB

limrB=lim;gq!1q!1mB+mAA1

gB1andthislimitdependssolelyonthelimitofthelikelihoodratio,bereadilyshownthat:

󰀊󰀋sA(q)󰀂1󰀋󰀊A

󰀊AgA1

󰀋lim=lim󰀊sB(q)󰀂1q!1gB1q!1󰀋󰀊

󰀊B

B

gA1

.gB1

Finally,itmay

=lime

q!1

1

8󰀅2󰀅2AB

q2222

(4󰀊2A󰀊Bln(1󰀂q)󰀂1)(󰀊B󰀂󰀊A)󰀊A

!1:󰀊B

-34-Hence,

q!1

APPENDIX

limrB=0:

Proposition5Forallq2(0;1),minorityhiresare…redatahigherratethanmajorityhires.

Proof.Becausehiresare…redifandonlyiftheyturnouttobeincompetent,wehavetoprovethat

󰀃(1󰀂GA0)(1󰀂p)󰀂󰀃󰀂(1󰀂GB0)(1󰀂p)

<=Pr󰀉B=0jqB󰀈qPr󰀉A=0jqA󰀈q=

1󰀂GA1󰀂GB

forallq2(0;1).

Thisisequivalenttoshowingthat

1󰀂GA01󰀂GB0

<;

1󰀂GA1󰀂GB

orNow,

1󰀂GB1󰀂GA

<()

1󰀂GB01󰀂GA0

1󰀂pGB1󰀂(1󰀂p)GB0(1󰀂pGA1󰀂(1󰀂p)GA0)

<()

1󰀂GB01󰀂GA0

1󰀂GA11󰀂GB1

<:

1󰀂GB01󰀂GA0

󰀂󰀃󰀂󰀃

Hence,showingthatPr󰀉A=0jqA󰀈q󰀂G󰀄1showingthattheratioofgoodhiringdecisionsoverbadhiringdecisions,1,is1󰀂G󰀄0greaterforkindAhiresthanforkindBhires.Toprovethelatter,weshowthat

\"󰀂󰀃#d1󰀂G󰀆1q

󰀂󰀃<0:

d󰀊󰀆1󰀂G󰀆0q

󰀎󰀏1󰀂Gq󰀄1()d

Now,d󰀊

󰀄1󰀂G󰀄0(q)2R13d4qg󰀆1(q)dq5=R1d󰀊󰀆g(q)dqq󰀆0

2R1󰀊s(q)󰀂1󰀋3

󰀊󰀄󰀄

dq󰀊󰀄q(1󰀂q)d4q󰀋5:󰀊󰀋=R1󰀊󰀄d󰀊󰀆

󰀋s󰀄(q)dq

q

󰀊󰀄

q(1󰀂q)1󰀂GA1󰀂GB

<:

1󰀂GB01󰀂GA0

-35-ds󰀄(q)d󰀊󰀄󰀎1

2(s󰀄(q)󰀂2)=,󰀊󰀏󰀄1󰀂G󰀄1(q)APPENDIX

UsingthatthesignofZ

1

straightforwardalgebraleadstotheconclusionthat

dd󰀊󰀄

1󰀂G󰀄0(q)isequaltothesignof

1

g󰀆1(q)dq

q

Z

1

q

Z

s󰀆(q)(s󰀆(q)󰀂1)g󰀆0(q)dq󰀂

g󰀆0(q)dq

q

Z

1

s󰀆(q)(s󰀆(q)󰀂1)g󰀆1(q)dq:

q

Changingvariablesofintegrationfromqtos,weget

Z1Z1

@q󰀆(s)@q󰀆(s)g󰀆1(s)s(s󰀂1)g󰀆0(s)dsds

@s@ss󰀄(q)s󰀄(q)

Z1Z1

@q󰀆(s)@q󰀆(s)

󰀂g󰀆0(s)s(s󰀂1)g󰀆1(s)dsds:

@s@ss󰀄(q)s󰀄(q)

󰀄(s),Substitutingforg󰀆0,g󰀆1,and@q@s󰀌󰀌󰀍󰀌󰀍Z1󰀌󰀍Z1󰀍Z1Z1s󰀂1sss󰀂1󰀋s(s󰀂1)󰀋󰀋s(s󰀂1)󰀋dsds󰀂dsds:

󰀊󰀊󰀊󰀊󰀆󰀆󰀆󰀆s󰀄(q)s󰀄(q)s󰀄(q)s󰀄(q)

Expandings(s󰀂1),

Z

s󰀂1ss

dss2󰀋ds󰀂s󰀋ds󰀊󰀊󰀊󰀆󰀆󰀆s󰀄(q)s󰀄(q)s󰀄(q)

󰀌󰀍 Z1󰀌󰀍󰀌󰀍!Z1Z1

ss󰀂1s󰀂1

󰀂󰀋dss2󰀋ds󰀂s󰀋ds:

󰀊󰀊󰀊󰀆󰀆󰀆s󰀄(q)s󰀄(q)s󰀄(q)

󰀋

1

󰀌󰀍

Z

1

󰀌󰀍Z

1

󰀌󰀍

!

Writingintermsofconditionalexpectations,

󰀂󰀃!! 󰀂󰀃!!

󰀂󰀄2󰀂󰀃󰀅󰀄󰀂󰀃󰀅󰀃s󰀆q󰀂1s󰀆q

1󰀂󰀄1󰀂󰀄ES󰀆0jS󰀆0󰀈s󰀆q󰀂ES󰀆0jS󰀆0󰀈s󰀆q

󰀊󰀆󰀊󰀆

󰀂󰀃!! 󰀂󰀃!!

󰀂󰀄2󰀂󰀃󰀅󰀄󰀂󰀃󰀅󰀃s󰀆q󰀂1s󰀆q

1󰀂󰀄ES󰀆1jS󰀆1󰀈s󰀆q󰀂ES󰀆1jS󰀆1󰀈s󰀆q:󰀂1󰀂󰀄

󰀊󰀆󰀊󰀆Dividingbythecommonpositivefactor

󰀌1󰀂󰀄

󰀌

s󰀄(q)󰀂1󰀊󰀄

Now,themomentgeneratingfunction,mgf,ofaleft-truncatedstandardnormalrandomvariableUwithtruncationpointdis(see,forexample,HeckmanandHonoré,1990):󰀂12󰀃R11

pexp󰀂2udu12d󰀂󰀃2󰀉󰀃2󰀂1󰀃R11mgf(󰀃)=e:2dupexp󰀂u2d2󰀉󰀄2󰀂󰀃󰀅󰀄󰀂󰀃󰀅󰀄2󰀂󰀃󰀅󰀄󰀂󰀃󰀅

ES󰀆0jS󰀆0󰀈s󰀆q󰀂ES󰀆0jS󰀆0󰀈s󰀆q󰀂ES󰀆1jS󰀆1󰀈s󰀆q󰀂ES󰀆1jS󰀆1󰀈s󰀆q:

󰀍󰀍󰀌

1󰀂󰀄

󰀌

s󰀄(q)󰀊󰀄

󰀍󰀍

:

-36-Hence,

E[UjU󰀈d]=

@mgf

j󰀃=0@󰀃󰀋(d)=;

1󰀂󰀄(d)

APPENDIX

while

󰀄2󰀅@2mgfEUjU󰀈d=2j󰀃=0

@󰀃

@mgf

=1+dj󰀃=0

@󰀃d󰀋(d)

=1+:

1󰀂󰀄(d)

ForX󰀉N(󰀈;󰀊2),thisimplies

E[XjX󰀈d]=󰀈+

0

󰀊󰀋

1󰀂󰀄

󰀊d0󰀂󰀈󰀊Now,recallthatS󰀆0󰀉N(0;󰀊󰀆)andS󰀆1󰀉N(1;󰀊󰀆).Hence,

󰀄󰀄2󰀄󰀄2󰀂󰀃󰀅󰀂󰀃󰀅󰀂󰀃󰀅󰀂󰀃󰀅ES󰀆0jS󰀆0󰀈s󰀆q󰀂ES󰀆0jS󰀆0󰀈s󰀆q󰀂ES󰀆1jS󰀆1󰀈s󰀆q󰀂ES󰀆1jS󰀆1󰀈s󰀆q

󰀂󰀃

=󰀊2+s󰀆q󰀆

0

󰀊󰀆󰀋

󰀌󰀌s󰀄(q)󰀊󰀄

󰀊󰀋󰀄2󰀅020

󰀂d0󰀂󰀈󰀃+󰀈2:EXjX󰀈d=󰀊+(󰀈+d)

1󰀂󰀄󰀊󰀂d0󰀂󰀈󰀃󰀊󰀋󰀊d0󰀂󰀈󰀊󰀋1󰀂󰀄

s󰀄(q)󰀊󰀄

󰀍󰀊󰀆󰀋

󰀍󰀂

󰀌1󰀂󰀄

Dividingby󰀊󰀆andcollectingterms,weget

󰀌󰀍󰀌󰀍s󰀄(q)s󰀄(q)󰀂1󰀋󰀋󰀊󰀄󰀊󰀄󰀃󰀂󰀃󰀂󰀂󰀃

󰀌󰀍󰀂s󰀆q󰀌󰀍:s󰀆q󰀂1

s󰀄(q)s󰀄(q)󰀂11󰀂󰀄1󰀂󰀄󰀊󰀄󰀊󰀄

B2󰀂󰀂󰀃󰀃

B󰀂@󰀊󰀆+1+s󰀆q

󰀊󰀆󰀋

󰀌1󰀂󰀄

󰀌s󰀄(q)󰀂1󰀊󰀄

󰀌s󰀄(q)󰀊󰀄

s󰀄(q)󰀂1󰀊󰀄

󰀍s󰀄(q)󰀊󰀄

󰀍󰀍󰀊󰀆󰀋

󰀍+1󰀂1󰀂

󰀌1󰀂󰀄

󰀌s󰀄(q)󰀂1󰀊󰀄

s󰀄(q)󰀂1󰀊󰀄

󰀍1

C󰀍CA:

-37-Hence,thequestioniswhether

󰀂s󰀃󰀂1󰀃󰀋󰀊󰀋s󰀂

󰀊󰀂s󰀃󰀂s󰀂󰀃<0;(s󰀂1)s󰀂1

1󰀂󰀄󰀊1󰀂󰀄󰀊󰀂s󰀃󰀂s󰀂1󰀃s󰀂1󰀋󰀊s󰀋󰀊󰀂s󰀃󰀂󰀂1󰀃<0;

󰀊1󰀂󰀄󰀊󰀊1󰀂󰀄s󰀂󰀊by󰀇

󰀂s󰀃

󰀊APPENDIX

or

foralls2Rand󰀊>0.Denotehazardrate

s1󰀂󰀃(󰀅)s󰀋(󰀅)

.Theexpressionthenbecomes󰀊s󰀋󰀊

󰀂s󰀇

󰀌s󰀂1

󰀊

󰀍:

(s󰀂1)󰀇

Graphically,whens󰀂1<0;

Hence,foralls󰀂1<0,itisobviousthat

(s󰀂1)󰀇

󰀊s󰀋󰀊

󰀂s󰀇

󰀌s󰀂1󰀊

󰀍<0:

-38-Whens󰀂1>0;graphically,

APPENDIX

Here,inprinciple,itcouldgoeitherway.Now,fors󰀂1>0,

󰀍s󰀂1

(s󰀂1)󰀇󰀂s󰀇

󰀊󰀊󰀌󰀊󰀋󰀌󰀍󰀍󰀌󰀍

s󰀂1s󰀂1s

󰀂󰀇󰀂(s󰀂(s󰀂1))󰀇=(s󰀂1)󰀇󰀊󰀊󰀊

Z󰀇(s)Zs󰀊󰀋

󰀅x

󰀇󰀇󰀇󰀂1(l)dl󰀂dx;

s󰀂1󰀊󰀇(󰀅)s󰀂1

󰀊s󰀋

wheretheinequalityfollowsfromtheconvexityof󰀇

Changingthevariableofintegrationinthe…rsttermfromhazardrateltosignalx,

thelastexpressionbecomes

ZsZs󰀊󰀋

x@l

󰀇dx=xdx󰀂

@x󰀊s󰀂11

Zs󰀂Zs󰀊󰀋󰀊󰀋s

x0xx

=󰀇dx󰀂󰀇dx

󰀊󰀊󰀊1s󰀂1

Zs󰀂󰀋󰀊󰀊s󰀊x󰀋󰀋x0x=󰀇󰀂󰀇dx:

󰀊󰀊󰀊s󰀂1Finally,weshowthattheintegrand,whichwewriteas

z󰀇0(z)󰀂󰀇(z);

󰀂s󰀃

󰀊󰀌

.

-39-isnegativeforallz󰀈0.First,notethat

󰀇0󰀊s󰀋󰀊

=====

󰀂s󰀃#󰀋󰀊dd

󰀂s󰀃=ss󰀇d󰀊󰀊d󰀊1󰀂󰀄󰀊󰀂s󰀃󰀂󰀂s󰀃󰀃󰀂󰀃2ss

󰀂󰀊󰀋󰀊1󰀂󰀄󰀊+󰀋󰀊󰀂󰀂s󰀃󰀃21󰀂󰀄󰀊󰀂s󰀃󰀃!󰀂s󰀃 󰀂s󰀃s󰀂󰀋󰀊󰀂󰀊1󰀂󰀄󰀊󰀋󰀊

󰀂s󰀃󰀂󰀂s󰀃󰀃1󰀂󰀄󰀊1󰀂󰀄󰀊!󰀂s󰀃 󰀂s󰀃󰀋󰀊󰀋󰀊s

󰀂s󰀃󰀂s󰀃󰀂

󰀊1󰀂󰀄󰀊1󰀂󰀄󰀊󰀊s󰀋󰀊󰀊s󰀋s󰀋󰀇󰀇󰀂:󰀊󰀊󰀊

󰀊s󰀋

\"

APPENDIX

Hence,theintegrandcanbewrittenas

z󰀇0(z)󰀂󰀇(z)

=z󰀇(z)(󰀇(z)󰀂z)󰀂󰀇(z)=󰀇(z)(z(󰀇(z)󰀂z)󰀂1):

Dividingby󰀇(z),Thequestionbecomeswhether

z(󰀇(z)󰀂z)<1

forz󰀈0.

Now,notethat󰀇0(z)<1forallz,asthederivativeofthehazardrateofthestandardNormaldistributionconvergesto1frombelowwhenz!1.Hence,itsu¢cestoshowthat

z(󰀇(z)󰀂z)󰀇󰀇(z)(󰀇(z)󰀂z)=󰀇0(z):

Now,

z(󰀇(z)󰀂z)󰀇󰀇(z)(󰀇(z)󰀂z)

isequivalentto

0󰀇(󰀇(x)󰀂x)2;

wherethelastinequalityisobviouslytrue.

-40-ProofsofImplications

APPENDIX

Implication1Supposethattheemployeris“selective”initshiringpolicy,i.e.,q>p;then:

1.Anincreaseinthecostof…ring,c,reducesworkplacediversity.2.Anincreaseinthecostofinterviewing,k,increasesworkplacediversity.

Proof.Toestablishpart1oftheimplication,weshowthatq󰀅isincreasinginc:Recallthatoptimalityofthethresholdstrategyimpliesthat

󰀂󰀂󰀅󰀃󰀃󰀅󰀂󰀃󰀅

Vq󰀂vq+1󰀂qc=0:(3)Implicitlydi¤erentiatingwithrespecttocwhilenotingthat=0gives

󰀂󰀅󰀃󰀅󰀅

󰀂󰀂󰀅󰀃󰀃dq󰀃@Vq󰀅󰀂dqVq󰀂v+q+1󰀂q󰀅󰀂c=0:

dc@cdc

dq󰀄

;dc

@V(q󰀄)

@q󰀄Solvingfor

dq󰀅

=dc

Itiseasilycheckedthat

󰀌@V(q󰀄)@cToestablishthattheright-handsideofthisexpressionispositiverequiresthatweshowthat !!Z1

󰀂󰀂󰀅󰀃󰀃󰀅1󰀂󰀄Gqq󰀂1󰀂󰀄1󰀂qdG(q)<0:

q󰀄

Substitutingintotheexpressionforandsimplifying,oneobtains

01󰀂󰀃dq󰀅󰀄Gq󰀅󰀂1󰀊󰀋Aq󰀅+1:=@R1dc1󰀂󰀄1󰀂qdG(q)

q󰀄

R1󰀂󰀅󰀃󰀂󰀄q󰀄(1󰀂q)dG(q)@Vq

󰀊󰀋:=R1@c1󰀂󰀄1󰀂q󰀄qdG(q)

dq󰀄

dc󰀍󰀂1q󰀅+1

󰀂󰀃:

v+c󰀂Vq󰀅

Toseethis,noticethat

󰀂

󰀂󰀅󰀃󰀃󰀅󰀂󰀂󰀂󰀂󰀅󰀃󰀃󰀃󰀃󰀅

<1󰀂󰀄Gqq󰀂1󰀂󰀄1󰀂q1󰀂Gq

󰀂󰀃󰀅

=󰀂(1󰀂󰀄)1󰀂q<0:

󰀂

󰀂󰀅󰀃󰀃󰀅

1󰀂󰀄Gqq󰀂

1󰀂󰀄1󰀂

Z

1

qdG(q)

q󰀄!!

-41-APPENDIX

Toestablishpart2oftheimplication,weshowthatq󰀅isdecreasingink:Implicitly

@V(q󰀄)

di¤erentiatingequation(3)withrespecttokwhilenotingthat@q󰀄=0;weobtain

󰀂󰀃󰀃dq󰀅@Vq󰀅󰀅󰀂󰀂󰀅󰀃dq󰀅

+q󰀂c=0:Vq󰀂vdk@kdk

dq

󰀂󰀃:=

dkv+c󰀂Vq󰀅

󰀅

@V(q󰀄)󰀅

q@kSolvingfor

dq󰀄

;dkHence,

dq󰀄

dkand

@V(q󰀄)@khavethesamesign,whileitiseasilycheckedthat

@V(q󰀄)@k<0.

Implication2Diversityisprocyclical.Formally,q󰀅isdecreasinginv(andk):Proof.FromImplication1,wealreadyknowthatq󰀅isincreasingink.Implicitlydi¤erentiatingequation(3)withrespecttovwhilenotingthat@V(q󰀄)

=0;weobtain@q󰀄󰀂󰀂󰀅󰀃󰀃dq+Vq󰀂vdv:

󰀅

!󰀂󰀅󰀃@Vqdq󰀅

󰀅

󰀂1q󰀂c=0:@vdv

Solvingfor

dq󰀄

dv

Itiseasilycheckedthat

󰀍@V()

󰀂1q󰀅󰀅@vdq

󰀂󰀃:=

dvv+c󰀂Vq󰀅

q󰀄

󰀌Substitutingthisbackinto

R1󰀂󰀅󰀃󰀄q󰀄(q)dG(q)dVq

󰀊󰀋:=R1

dv1󰀂󰀄1󰀂q󰀄qdG(q)

dq󰀄dv

andsimplifying;oneobtains

󰀂󰀄󰀃1󰀄q󰀅R1

1󰀂󰀄1󰀂q󰀄qdG(q)

dqdv

󰀅

=󰀂<0:

󰀂󰀃v+c󰀂Vq󰀅

-42-APPENDIX

Implication3Supposethattheemployeris“selective”initshiringpolicy,i.e.,q󰀅>p;thenriskier…rmsaremorediverse.Formally,q󰀅isincreasingin󰀄:Proof.Implicitlydi¤erentiatingequation(3)withrespectto󰀄whilenotingthat@V(q󰀄)

=0;weobtain@q󰀄󰀃dq󰀂󰀂󰀅󰀃

+Vq󰀂vd󰀄

󰀅

!󰀂󰀅󰀃dq󰀅dVq󰀅

󰀂1q󰀂c=0:d󰀄d󰀄

q󰀄

Solvingfor

dq󰀄

;d󰀄

Itiseasilycheckedthat:

󰀂󰀃dVq󰀅Z(1󰀂󰀄X)+X(󰀄Z󰀂k)

;=

d󰀄(1󰀂󰀄X)2where

Z󰀆X󰀆

Toshowthat

dq󰀄

d󰀄

󰀍dV()

󰀂1q󰀅󰀅d󰀄dq

󰀂󰀃:=

d󰀄v+c󰀂Vq󰀅

󰀌Z

1

(qv+(1󰀂q)(󰀂c))dG(q)1󰀂

Z

1

q

qdG(q):

dV(q󰀄)

d󰀄

q

!

>0;itissu¢cienttoshowthat

󰀂1>0,or,equivalently,

Z(1󰀂󰀄X)+X(󰀄Z󰀂k)󰀂(1󰀂󰀄X)2>0:

Toseethis,simplifytheleft-handsideoftheaboveexpressionandrecallthat,sincetheemployer…ndsitoptimaltosearchinthe…rstplace,󰀄Z󰀂k󰀈0:Thisyields

Z󰀂Xk+(1󰀂X󰀄)2󰀈Z󰀂X󰀄Z+(1󰀂X󰀄)2=(1󰀂X󰀄)(Z+1󰀂X󰀄)>0;

wherethelastinequalityfollowsfromthefactthatZ>0andX,󰀄2(0;1):Implication4Injobsthataresu¢cientlyselective,minoritieswillbe

underrepresented.Injobsthataresu¢cientlynon-selective,minoritieswillbe

overrepresented.Formally,thereexists01:mBB

-43-APPENDIX

Proof.First,weestablishthatlimp\"1q󰀅<1andlimp#0q󰀅>0:Toseethis,notethatq󰀅ismonotoneinpsince,byimplicitlydi¤erentiatingequation(3);

dq

=dpv

󰀅

@V(q󰀄)󰀅

q@p

󰀂󰀃󰀂Vq󰀅+

c

>0;

wheretheinequalityfollowsfromthefactthatv>V(q󰀅)and,byLemma1,@V(q󰀄)

>0.@pSinceq󰀅isboundedandmonotonefunctionofpweknowthatbothlimitsmustexist.

Toestablishthatlimp\"1q󰀅<1;suppose,tothecontrary,thatlimp\"1q󰀅=1.Thentheright-handsideofequation(2)becomes

󰀊󰀊󰀋󰀋R1

1󰀂󰀄1󰀂1qdG(q)clim

p\"1(1󰀂󰀄G(1))c+(1󰀂󰀄)v+k

(1󰀂󰀄)c

=1;=

(1󰀂󰀄)c+(1󰀂󰀄)v+kwhichisacontradiction.

Toestablishthatlimp#0q󰀅>0;recallthatq󰀅isimplicitlyde…nedbyequation(2):Takinglimits,

󰀊󰀊󰀋󰀋R1

1󰀂󰀄1󰀂q󰀄qdG(q)c

󰀅

󰀂󰀃󰀃limq=lim󰀂p#01󰀂󰀄Gq󰀅p#0c+(1󰀂󰀄)v+k

(1󰀂󰀄)c

>0:>lim

p#0c+(1󰀂󰀄)v+kTocompletetheproof,itremainstoshowthatq0andq1aremonotoneinpwithlimitslimp#0q0=0andlimp#1q1=1:Monotonicitymaybereadilyveri…edby

di¤erentiatingtheexpressionsforq0andq1.Likewise,thelimitresultsaretrivialtoobtain.

-44-

References

Aigner,Dennis,andGlenCain,1977,“StatisticalTheoriesofDiscriminationin

LaborMarkets,”IndustrialandLaborRelationsReview,Vol.30:pp.175-87.Arrow,KennethJ.,1998,“WhatHasEconomicstoSayaboutRacial

Discrimination,”JournalofEconomicPerspectives,Vol.12,pp.91-100.Athey,Susan,ChristopherAvery,andPeterZemsky,2000,“Mentoringand

Diversity,”AmericanEconomicReview,vol.90(4),pp.765-786.Baye,M.R.,J.Morgan,andP.Scholten,“Information,Search,andPrice

Dispersion,”inHandbookofEconomicsandInformationSystems(T.Hendershott,ed.),Elsevier:Amsterdam,forthcoming.Becker,GaryS.,1957,“TheEconomicsofDiscrimination,”Chicago:Chicago

UniversityPress.Bertrand,Marianne,andSendhilMullainathan,2004.\"AreEmilyandGregMore

EmployablethanLakishaandJamal?AFieldExperimentonLaborMarketDiscrimination,\"AmericanEconomicReview,vol.94(4),pp.991-1013.Black,DanA.,1995,“DiscriminationinanEquilibriumSearchModel,”Journalof

LaborEconomics,Vol.13,pp.309-33.Coate,Stephen,andGlennLoury,1993,“WillA¢rmativeActionEliminate

NegativeStereotypes?”AmericanEconomicReview,Vol.83.Cornell,Bradford,andIvoWelch,1996,“Culture,InformationandScreeningDiscrimination,”JournalofPoliticalEconomy,Vol.104,pp.542-571.Dagevos,J.2006,“Hoge(jeugd)werkloosheidonderetnischeminderheden.Nieuwe

bevindingenuithetLAS-onderzoek”SociaalenCultureelPlanbureau:DenHaag.Heckman,James,andBoHonoré,1990,\"TheEmpiricalContentoftheRoy

Model,\"Econometrica58,pp.1121-1149.Lang,Kevin,1986,“ALanguageTheoryofDiscrimination,”QuarterlyJournalof

Economics,Vol.101,pp.363-382.LegalTimes,May25,2006,“HighCourtClerks:StillWhite,StillMale,”available

athttp://www.law.com/jsp/newswire_article.jsp?id=1148461530991.McCall,JohnJ.,1970,“EconomicsofInformationandJobSearch,\"Quarterly

JournalofEconomics,Vol.84,pp.113-26.Lundberg,ShellyJ.,andRichardStartz,1998,“OnthePersistenceofRacial

Inequality,”JournalofLaborEconomics,Vol.16,pp.292-323.

-45-NationalFootballLeague,December9,2003,“NFLissuesinterviewingguidelines,”

availableathttp://www.n‡.com/news/story/6908387.Pepper,ToddC.,2006,\"CourtiersoftheMarblePalace:TheRiseandIn‡uenceoftheSupremeCourtLawClerk,\"StanfordUniversityPress:PaloAlto,CA.Phelps,EdmundS.,1972,“TheStatisticalTheoryofRacismandSexism,”

AmericanEconomicReview,Vol.62:pp.659-61.Rosen,Asa,1997,“AnEquilibriumSearch-MatchingModelofDiscrimination,”

EuropeanEconomicReview,Vol.41,pp.1589-1613.Scollon,Ronald,andSuzanneScollon,2001,“InterculturalCommunication:A

DiscourseApproach,”BlackwellPublishers.Stigler,George,1961,“TheEconomicsofInformation,”JournalofPolitical

Economy,Vol.69,pp.213-25.Tesser,P.,J.Merens,andC.vanPraag,1999,“Rapportageminderheden1999:

positieinhetonderwijsenopdearbeidsmarkt,”SociaalenCultureelPlanbureau:DenHaag.