StudentPortfoliosandthe
CollegeAdmissionsProblem*
†HectorChad
GregoryLewis
§LonesSmit
e
‡
h
Firstversion: July2003
*
Thisversion:
March9,2013
Abstract We develop a decentralized Bayesian
model of college admissions with two
rankedcolleges,heterogeneousstudentsandtworealisticmatchfrictions: students
find it costly to apply to college, and college evaluations of their applications is
uncertain. Students thus face a portfolio choice problem in their application
decision,whilecollegeschooseadmissionsstandardsthatactlikemarket-clearing
prices. Enrollmentateachcollegeisaffectedbythestandardsattheothercollege
throughstudentportfolioreallocation. Inequilibrium,student-collegesortingmay fail:
weaker students sometimes apply more aggressively, and the weaker college
mightimposehigherstandards. Applyingourframework,weanalyzeaffirmative
action,showinghowitinducesminorityapplicantstoconstructtheirapplication
portfoliosasiftheyweremajoritystudentsofhighercaliber.
Earlierversionswere called “The College Admissions Problemwith Uncertainty”and “A Supply
andDemandModeloftheCollegeAdmissionsProblem”.
WewouldliketothankPhilippKircher(CoEditor)andthreeanonymousrefereesfortheirhelpfulcommentsands
uggestions. GregLewisandLones Smith are grateful for the financial support of the National Science
Foundation. We have benefited from seminars at BU, UCLA, Georgetown, HBS, the 2006 Two-Sided
Matching Conference (Bonn),
2006SED(Vancouver),2006LatinAmericanEconometricSocietyMeetings(Mexico City),and2007
AmericanEconometricSociety Meetings (New Orleans),IowaState, Harvard/MIT,the 2009Atlanta NBER
Conference, and Concordia. ParagPathak and Philipp Kircher provideduseful discussions of our paper.
We are also grateful to John Bound and Brad Hershbein for providing us with student
collegeapplicationsdata.
†ArizonaStateUniversity,DepartmentofEconomics,Tempe,AZ85287.
‡HarvardUniversity,DepartmentofEconomics,Cambridge,MA02138.
§UniversityofWisconsin,DepartmentofEconomics,Madison,WI53706.
1 Introduction
Thecollegeadmissions processhaslatelybeentheobjectofmuchscrutiny, bothfrom
academicsandinthepopularpress. Thisinterestowesinparttothecompetitivenature
ofcollegeadmissions. Schoolssetadmissionsstandardstoattractthebeststudents,and
students in turn respond most judiciously in making their application decisions. This
paperexaminesthejointbehaviorofstudentsandcollegesinequilibrium.
1We
introduce anequilibrium model ofcollege admissions thatanalyzes the impact
oftwopreviouslyunexploredfrictionsintheapplicationprocess: studentsfinditcostly to apply
to college, and college evaluations of their applications is uncertain. As evidence
ofthenoiseintheprocess, observe thatadmissions ratesarewellbelow50%at
themostselective colleges, andbelow 75%atthemedian 4year college.This uncertainty
prevails conditional on the SAT. Even applicants with perfect SAT scores have nobetter
than a50%chance ofgetting into schools like Harvard, MIT and Princeton
(Avery,Glickman,Hoxby,andMetrick, 2004). So itis notsurprising thatmost
applicantsconstructthoughtfulportfoliosthatincludeboth“safety”and“stretch”schools.
2Despite
this uncertainty, the costly nature of applications substantially limits the
numberofapplicationssent—threeforthemedianstudent.3Afastgrowingempirical literature
confirms the impact of this friction: for instance, Pallais (2009) finds that when the ACT
allowed students to send an extra free application, 20% of ACT
testtakerstookadvantageofthisoption.-4Smallcostscanhavelargeeffectsonapplication
behavior because the marginal benefit of applying falls geometrically in the number of
applications. A student applying to identical four-year colleges with a typical 75%
acceptancerateseesthemarginalbenefittoher5thapplicationscaledby4=1/256:
Evenifattendingcollegethisyearisworth$20000,themarginalbenefitisonly$59.
4Figure
1 illustrates some motivating patterns in the application data. The right panel
demonstrates the importance of the applications frictions — for the chance of
matriculating at a private elite school is significantly higher for students who apply
widely.TheleftpanelplotsthenumberofapplicationsasafunctionoftheSATscore,
1Source:
Table329,DigestofEducationStatistics,NationalCenterforEducationStatistics.
HigherEducationResearchInstitute(HERI),usingalargenationallyrepresentativesurvey
ofcollegefreshmansince1966(dataalsousedtoconstructFigure1;seeonlineappendixfordetails).
3Steinberg(2010)reportedthatcollegeswhowaivedapplicationfeessawapplicationsskyrocket.
4AveryandKane(2004)studyaBostonprogramgivinglow-incomestudentsadviceonhow/where to apply; these
students matriculated at a higher rate than comparable students elsewhere. Similar results have later
been found in field experiments where students received application help: see
Bettinger,Long,Oreopoulos,andSanbonmatsu(2009)andCarrellandSacerdote(2012).
2Source:
1
Prob. Enrolls at Top Private University
0 .05 .1 .15
Number of Applications
23456
600 800 1000 1200 1400 1600 SAT Score
Applied in ’85-’89 Applied in ’90-’94 Applied in ’95-’98
1200 1300 1400 1500 1600 SAT Score
< Median Income, Apps <= 3 < Median Income, Apps > 3 > Median Income, Apps <= 3
> Median Income, Apps > 3
Fig ure1: ApplicationsandMatriculationbySAT.Th eleftpanelshowstheestimated
relationshipbetweenSATscoreandnumberofapplications. Therightpanelshowsthechance
ofmatriculatingatatopprivateuniversity,bynumberofapplicationsandhouseholdincome.
Estimationisbylocallinearregression;95%confidenceintervalsareshownasdottedlines.
smallbutlargerthanoneconsistentwithourtwofrictions.5
Anymodelthatfocusesonthesetwofrictions—costlyportfoliochoiceswithincompleteinf
ormation—mustdivergefromtheapproachofthecentralizedcollegematching literature
(GaleandShapley, 1962), for it expressly sidesteps such matching frictions.
Rather,weanalyzeanentirelydecentralizedmodelthatparallelstheactualprocess. It
affords sharp conclusions about two key decision margins: how colleges set
admission standardsandhowstudentsformulatetheirapplicationportfolios.
6Weassumethataheterogeneouspopulationofstudents,andtworankedcolleges—
one
better and one worse, respectively, called 1 and 2. Like the decentralized model
ofAveryandLevin(2010),thereisacontinuumofstudents;thisavoidscollegesfacing
aggregateuncertainty —otherwise, wait-listing isneeded, forinstance.Colleges seek
tofilltheircapacitywiththebeststudentspossible,butstudentcalibersareonlyimperfectlyobs
erved. Thetandemofcostlyapplicationsandyetnoisyevaluationsfeedsthe
intriguingconflictattheheartofthestudentchoiceproblem: shootfortheIvyLeague,
settleforthelocalstateschool,orapplytoboth. Asweshallsee,ourpaperformalizes the
critical roles played by stretch and safety schools. Meanwhile, college enrollments
areinterdependent,sincethestudents’portfoliosdependonthejointcollegeadmissions
5Data
fromUS News and World Reportshows that admissions ratesfall as college rank rises. So
largerportfoliosareparticularlyvaluableforhighSATstudentsapplyingtotopcolleges.
6Weomitmanyreal-worldelementslikefinancialaidandpeereffects,butthesearetypicallyruledout
incentralizedmatchingmodelstoo. Inanewpaper,AzevedoandLeshno(2012)assumeacontinuum of
students in a centralizedpaper in the spirit of GaleandShapley (1962), and find that it affordsa
characterizationofequilibriumintermsofsupplyanddemand—oneofourobservationstoo.
2
standards, and students accepted at the better college will not attend the lesser one.
Thisasymmetricinterdependenceleadstomanysurprisingresults.
Central toourpaper isaBayesian theorem characterizing how student acceptance
chancesatthecollegescovarywithstudentcaliber.
Wededucethatasastudent’scaliberrises,theratioofhisadmissionchancesatcollege1tocolle
ge2risesmonotonically.
Wearethusabletosolvetheequilibriuminthreestages,firstdeducinghowacceptance
chancestranslateintoapplicationportfolios,andthenseeinghowportfoliochoicesacross
studentcalibersrelate,foranypairofcollegeadmissionstandards;finally,wecompute
thederiveddemandforcollegeslots.
Weanalyzetheequilibriumoftheinducedadmissionsstandardsgameamongcollegesthrough
thelensofsupplyanddemand: Whena
collegeraisesitsstandards,itsenrollmentfallsbothbecausefewerstudentsmakethecut
—thestandards effect—andfewerwillapplyexante —theportfolioeffect. Treating
admissions standards as prices, these effects reinforce each other. In equilibrium, we
uncover a “law ofdemand”, in which a college’s enrollment falls in its standard. The
portfolioeffectincreasestheelasticityofthisdemandcurve.
AnalogoustoBertrandcompetitionwithdifferentiatedproducts,collegeswillchoose
admissions standards to fill their desired enrollment, taking rival standards and the
student portfolios as given. An equilibrium occurs when both markets clear and
studentsbehaveoptimally. Themodelfrictionsyieldsomenovelcomparativestatics. For
instance, the admissions standards at both colleges fall if college 2 raises its capacity,
whilelowerapplicationcostsateither schoolincreasetheadmissions standardsatthe
bettercollege. Wewillarguethatourequilibriumframeworkrationalizesthepatternof
changingcollegestandardsandadmissionratesrecentlydocumentedbyHoxby(2009).
Inamajorthrustofthepaper,weaskwhethersortingoccursinequilibrium: First,
dothebetterstudentsapplymore“aggressively”? Precisely,thebeststudentsapplyjust
tocollege1;weakerstudentsinsurebyapplyingtobothcolleges;evenweakeronesapply just
tocollege 2; and finally, theweakest apply nowhere. Such anapplication pattern
rationalizesthegeneralriseandfallthatweobserveintheleftpanelofFigure1. Second, does
the better college impose higher admissions standards? The answer to this latter
questionisnowhenthelessercollegeissufficientlysmall,forbyour“lawofdemand”,
college2continuestoraiseitsstandardsasitscapacityfalls. Failuresofstudentsorting
aremoresubtle: Thewillingnessofstudentseitherto(i)gambleonastretchschoolor(ii)
insurethemselveswithasafetyschoolmaynotbemonotoneintheirtypes. Conversely,
allequilibriaentailsortingwhenthecollegesdiffersufficientlyinqualityandthelower
3
rankedschoolisnottoosmall. Alltold,sortingproveselusivewithfrictions.7Thecollege
sortingfailuresthatweidentifyhaveproblematicimplicationsforrankingsbasedonthe
characteristics of matriculants, such as their SAT scores: Colleges that substantially
increasetheircapacityarepenalized,sincetheymustlowertheiradmissionstandards.
This paper takes very seriously the uncertainty that clouds the student admission
process. Students apply tocolleges, perhaps knowing their types, orperhaps ignorant
ofthem. Equallywell, colleges evaluatestudents tryingtogaugethefuturestars, and often
do not succeed. The best framework for analyzing this world therefore involves
two-sided incomplete information. In fact, we later formulate such a richer Bayesian
model, and argue that its predictions are well-approximated by ours where students
know their types, andcolleges observe noisy signals. The sortingfailures we claim, as
wellasthepositivetheoryofhowstudentsandcollegesreact,areinfactrobustfindings.
We conclude the paper with a topical foray into “affirmative action” for in-state
applicants,orotherpreferredapplicantgroups. Weshowthatcollegesimposedifferent
admissions standards so astoequate the “shadow values” ofapplicants fromdifferent
groups—aformofthirddegreepricediscrimination. This,inturn,affectshowstudents behave:
inasimple case, lowercaliberapplicants ofafavoredgroupbehave asifthey were higher
caliber applicants from a non-favored group. This is consistent with the
reductioninless-qualifiedminorityapplicationstoselectivepublicschoolsaftertheend
ofaffirmativeactioninCaliforniaandTexas,documentedinCardandKrueger(2005).
2 TheModel
A.AnOverview. Thepaperintroducesthreekeyfeatures—heterogeneousstudents, portfolio
choices with unit application costs, and noisy evaluations by colleges. We impose little
additional structure. Forinstance, we ignore the important and realistic
considerationofheterogeneousstudentpreferencesovercolleges,aswellaspeereffects.
8Acentralfeatureofouranalysisismodelingcollegeportfolioapplications.
Student
choiceistrivialifitiscostless,andinpractice,suchcostscanbequitehigh. Indeed,the
solepurposeoftheCommonApplicationistolowerthecostofmultipleapplications.
7Thisaddstotheliteratureondecentralizedfrictionalmatching—e.g.,ShimerandSmith(2000),
Smith (2006),
Chade (2006), and AndersonandSmith (2010). The student portfolio problem in the model is a special
case of ChadeandSmith (2006). In this sense, our paper also contributes to the directed search literature.
See BurdettandJudd (1983), Burdett,Shi,andWright (2001),
Albrecht,Gautier,andVroman(2003)andKircherandGalenianos(2006)).
8Thisgeneralapplicationformisusedbyalmost400colleges,andsimplifiescollegeapplications. It
4
Next, we assume noisy signals of student calibers. This informational friction
createsuncertaintyforstudents,andaBayesianfilteringproblemforcolleges. Itcaptures
thedifficultyfacedbymarketparticipants,withstudentschoosing“safetyschools”and
“stretchschools”,andcollegestryingtoinferthebeststudentsfromnoisysignals. Without
noise, sorting would be trivial: Better students would apply and be admitted to
bettercolleges,fortheircaliberwouldbecorrectlyinferredandtheywouldbeaccepted.
9Wealsomaketwootherkeymodelingchoices.
First,weassumejusttwocolleges,for
thesakeoftractability. Butasweargueintheconclusion,thisisthemostparsimonious
framework that captures all of our key findings. We also fix the capacity of the two
colleges. Thisisdefensibleintheshortrun,andsoitisbesttointerpretourmodelas
focusingonthe“shortrun”analysisofcollegeadmissions. Weexplorethesimultaneous
gameinwhichstudentsapplytocollege,andcollegesdecidewhomtoadmit.
In the interest of tractability, our analysis assumes that the colleges’ evaluations of
students are conditionally independent. This captures the case where students are
apprisedofallvariables(suchastheACT/SATortheirGPA)commontotheirapplicationsbefo
re applyingtocollege. Studentsareuncertainastohowtheseidiosyncratic elements such
as college-specific essays and interviews will be evaluated, but believe
thattheresultingsignalsareconditionallyindependentacrosscolleges. Werevisitthis
restrictioninSection6,andarguethatourmainresultsonsortingarerobust,andthat
wehaveanalyzedarepresentativecase.
and κ2
B. The Model. There are two colleges 1 and 2 with capacities κ1
1+κ2
9
5
, and a
unitmassofstudentswithcalibersxwhosedistributionhasapositivedensityf(x)over [0,8).
Non-trivialitydemands thatcollegecapacitybeinsufficient forallstudents, as κ<1.
Toavoidmanysubscripts,weshallalmostalwaysassumethatstudentspay
aseparatebutcommonapplicationcostc>0forthetwocolleges. Resultsaresimilar
withdifferentcosts. Allstudents prefercollege1. Everyone receives payoffv>0for attending
college 1, u∈ (0,v) for college 2, and zero payoff for not attending college.
Studentsmaximizeexpectedcollegepayofflessapplicationcosts. Collegesmaximizethe
totalcaliberoftheirstudentbodiessubjecttocapacityconstraints.
eliminatesidiosyncraticcollegerequirements,butretainsseparatecollegeapplicationfees.
Epple,Romano,andSieg (2006) analyze an equilibrium model that includes tuition as a choice
variable,peereffects,andstudentsthatdifferinabilityandincome. Undersinglecrossingconditions, they
obtainpositive sorting onability for eachincome level. Their model, however,does not include costly
applications or noise, thus precluding the portfolio effects we focus on and their implications forsorting.
While wedonotallowcollegestochoosetheirtuitionlevels,wedonotignorethe roleof
tuition,foronecansimplyinterpretthebenefitsofattendingcollegeasnetbenefits.
Students know their caliber, and colleges do not. Colleges 1 and 2 each just
observesanoisyconditionallyindependentsignalofeachapplicant’scaliber. Inparticular,
theydonotknowwhereelsestudentshaveapplied.
Signalssaredrawnfromaconditionaldensityfunctiong(s|x)onasubintervalofR,withcdfG(s|x).
Weassumethat g(s|x)iscontinuousandobeysthestrict
monotonelikelihoodratioproperty(MLRP).
Sog(t|x)/g(s|x)isincreasinginthestudent’stypexforallsignalst>s.
Students applysimultaneously toeither, both,orneithercollege,choosingforeach
caliberx,acollegeapplicationmenuS(x)in{Ø,{1},{2},{1,2}}. Collegeschoosethe set of
acceptable student signals. They intuitively should use admission standards to
maximizetheirobjectivefunctions,sothatcollegeiadmitsstudentsaboveathreshold signals i.
AppendixA.1provesthisgiventheMLRPproperty—despiteanacceptance curse
thatcollege2faces(asitmayacceptarejectofcollege1).
-s/xForafixed
admission standard, we want toensure thatvery high quality students
arealmostneverrejected,andverypoorstudentsarealmostalwaysrejected. Forthis, we
assume thatforafixed signals, we have G(s|x)→0asx→8 andG(s|x)→1 asx→0.
Forinstance, exponentially distributed signalshavethispropertyG(s|x)= 1-e'.
Moregenerally,thisobtainsforsignalsdrawnfromany“locationfamily”,in
whichtheconditionalcdfofsignalssisgivenbyG((s-x)/µ),foranysmoothcdfG
andµ>0—e.g.normal,logistic,Cauchy,oruniformlydistributedsignals. Thestrict
MLRPthenholdsifthedensityisstrictlylog-concave,i.e.,logGisstrictlyconcave.
C.Equilibrium. Weconsiderasimultaneousmovegamebycollegesandstudents.
10This yields the same equilibrium prediction as when students move first, as they are
atomless.Anequilibriumisatriple(S*(·),s* 1,s* 2)suchthat:
(a) Given(s* 1,s* 2),S*(x)is anoptimalcollegeapplicationportfolioforeachx,
(b) Given(S*(·),s* j),collegei’spayoffismaximizedbyadmissionsstandards* i.
Wea lsowishtoprecludetrivialequilibriainourmodelinwhichoneorbothcolleges
rejecteverybodywithaveryhighadmissionsthresholdandstudentsdonotapplythere.
Arobustequilibrium alsorequiresthatanycollegethatexpectstohaveexcesscapacity
setthelowestadmissionsthreshold. Sinceκ1+κ2<1,bothcollegeswillhaveapplicants.
10SeeAppendixA.2.
Alternatively,collegescouldmovefirst,committingtoanadmissionstandard.
Thisisarguablynotthecase,butregardless,ittooyieldsthesameequilibriauntilwestudyaffirmative
action(proofomitted). Intheinterestsofaunifiedtreatmentthroughoutthepaper,weproceedinthe
simultaneousmoveworld.
6
Inarobustsortingequilibrium,colleges’andstudents’strategiesaremonotone. This
means thatthe better college is more selective (s* 1>s* 2) and higher caliber students
areincreasinglyaggressiveintheirportfoliochoice: Theweakestapplynowhere;better
studentsapplytothe“easier”college2;evenbetterones“gamble”byapplyingalsoto
college1;thenexttierupshootan“insurance”applicationtocollege2;finally,thetop students
confidently just apply to college 1. Monotone strategies ensure the intuitive result that
the distribution of student calibers at college 1 first-order stochastically dominates that of
college 2 (see Claim 4 in Appendix A.7), so that all top student
quantilesarelargeratcollege1. Thisisthemostcompellingnotionofstudentsorting
inourenvironmentwithnoise(Chade,2006).
Ourconcernwitharobustsortingequilibriummaybemotivatedonefficiencygrounds.
Iftherearecomplementaritiesbetweenstudentcaliberandcollegequality,sothatwelfareismaxi
mizedbyassigningthebeststudentstothebestcolleges,anydecentralized matching system
must necessarily satisfy sorting to be (constrained) efficient. Since
formalizingthisideawouldaddnotationandofferlittleadditionalinsight,wehaveabstractedfrom
thesenormativeissuesandfocusedonthepositiveanalysisofthemodel.
D.CommonversusPrivateValues. Noticethatinourmodelcollegescareabout
thetruecaliberofastudentandnotaboutthesignalperse. Inotherwords,themodel
exhibitscommonvaluesonthesideofthecolleges. AppendixA.1showscollegesbehave
inexactlythesamewayiftheydonotcareaboutcaliberbutcareonlyaboutthesignal i.e.
theprivatevaluescase. Inthisinterpretation,studentsdifferintheirobservablesx
(knowntobothstudentsandcolleges),andalsointheir“fit”foreachcollegeo(known only to the
college). College payoffs depend on both observables and fit through the
signalsi=si(x,oii). UntiltheaffirmativeactionapplicationinSection7,all theresults
applytotheprivatevaluescaseaswell,albeitwithadifferentinterpretation.
3 EquilibriumAnalysisforStudents
3.1 TheStudentOptimization Problem
Webeginbysolvingfortheoptimalcollegeapplicationsetforagivenpairofadmission chances
at the two colleges. Consider the portfolio choice problem for a student with admission
chances 0 = a1,a= 1. The expected payoff of applying to both colleges is a1v+(1-a1)a22u.
The marginal benefit MBijofadding college itoaportfolioof
7
1.0
MB12 c Α1v Α2u MB21
0.8
c
1.0
0.8
SafetyStretch
0.6
C
B
2
0.6
0.4
0.4
0.2
0
.
2
C1
0.0 0.2 0.4 0.6 0.8 1.0 0.0
Α1 x
0.0 0.2 0.4 0.6 0.8 1.0 0.0
Α1 x
Figure 2: Optimal Decision Regions. The left
panel depicts (i) a dashed box, inside which
applying anywhere is dominated; (ii) the
indifference line for solo applications to colleges 1
and 2; and (iii) the marginal benefit curves MB12=
cand MB21= cfor adding colleges1or2.
Therightpanelshowstheoptimalapplicationregions.
Astudentintheblank regionFdoesnotapplytocollege.
Heappliestocollege2onlyintheverticalshadedregion
C2;
tobothcollegesinthehashedregionB,andtocollege1o
nlyinthehorizontalshadedregionC1.
= [a1v+(1-a1)a2u]-a1v=(1-a1)a2
= [a1v+(1-a1)a2u]-a2u=a1(v-a2u) (2)
1
2
collegejisthen:
MB21
u (1) MB
v= a2
u= a1
21
1v=c),andtobothcolleges(MB21
2u=c),andtobothcolleges(MB12
12
1
v>MB12
21=canda2u>MB21
8
=c,a1v=a2
12
Α2x
Α 2x
Theoptimalapplicationstrategyisthengivenbythefollowingrule:
(a) Applynowhereifcostsareprohibitive: c>au. (b) Apply just to
college 1, if it beats applying just to college 2 (a
TheleftpanelofFigure2depictsthreecriticalcurves: MB
2 =MB12,whena1v=a2u.
1
u. Allthreecurvesshareacrossingpoint,sinceMB
=c,MB
u), and nowhere(a<c,i.e.addingco
it beats applying just to college 1
nowhere(a<c,i.e.addingcollege1is
beats applying just to college 1 (M
justtocollege2(MB=c),forthen,thes
beatapplyingtonowhere,asa=cby(
illuminating and rigorous graphica
1.0
C2
0.8
0.6
0.4
B
C
1
0.2
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
0
.
0
Α
1
x
Figure3:
TheAcceptanceFunctionwithE
xponentialSignals.
Thefiguredepicts theacceptance
functionψ(a1)for thecase of
exponential signals. As their
caliber increases,
studentsapplytonowhere(F),colle
ge2only(C),bothcolleges(B)—spe
cifically,firstusing college 1 as a
stretch school, and later college 2
as a safety school — and finally
college 1 only(C12).
Studentbehavioristhereforemonot
onefortheacceptancefunctiondepi
cted.
Cases (a)–(d) partition the unit
square into (a1,a) regions
corresponding to the portfolio
choices (a)–(d), denoted
F,C2,B,C12, shaded in the right
panel of Figure 2.
Thissummarizestheportfolioch
oiceofastudentwithanyadmissi
onschances(a1,a2).
Inthemarginalimprovement
algorithmofChadeandSmith(20
06),astudentfirst
decides whether she should
apply anywhere. If so, she
asks which college is the best
singleton. In Figure 2 at the
left, college 1 is best right of
the line a1v = au, and college
2 is best left of it. Next, she
asks whether she should
apply anywhere else.
Intuitively,therearetwodistinctr
easonsforapplyingtobothcolleg
esthatwecannow parse: Either
college1 is a stretch school —
i.e., a gamble, added as a
lower-chance
higherpayoffoption—orcollege
2isasafetyschool,addedforinsu
rance. InFigure2,
thesearethepartsofregionBabo
veandbelowthelinea1v=a22u,re
spectively. The choice regions
obey some natural
comparative statics. The
application region
Cito either college increases in
Α2x
its payoff, in light of
expressions (1) and (2), and
the region B shifts right in the
college 2 payoff u, and left in
the college 1 payoff v. In
particular,ifastudentenjoysfixe
dacceptanceratesatthetwocoll
eges,acollegegrows less
attractive as the payoff of its
rival rises. Also, as the
application cost crises, the
jointapplicationregionBshrinks
andtheemptysetFgrows.
Althoughoutsideourmodel,l
etusbrieflyconsidernon-linearc
osts—forinstance, the second
application costs may be less
than c, possibly due to some
duplication of
9
forms, essays, etc. Weanalyze thisintheonline appendix, andshow thatregionBis
biggerandtheremainingregionssmallerthanwithconstantcosts. Interestingly,some
typeswhowouldsendnosingletonapplicationswouldnonethelessapplytobothcolleges.
3.2 Admission Chances and StudentCalibers
Wehavesolvedtheoptimizationforknownacceptancechances. Butwewishtopredict
theportfoliodecisionsofthestudents, despitetheendogenousacceptancechances. To
thisend,wenowderiveamappingfromstudenttypestostudentapplicationportfolios. Fix the
thresholds s1and s2set by college 1 and college 2. Student x’s acceptance chance at
college iis given by ai(x) = 1-G(s|x). Since a higher caliber student generates
stochastically higher signals, aii(x) increases in x. In fact, it is a smoothly monotoneonto
function—namely,itisstrictlyincreasinganddifferentiable,with0<
a1(x)<1,andthelimitbehaviorlimx→0a1(x)=0andlimx→8a1(x)=1.
Takingtheacceptancechancesasgiven,eachstudentofcaliberxfacestheportfolio
optimizationproblemof§3.1. She mustchoose asetS(x)ofcollegestoapply to,and
accepttheofferofthebestschoolthatadmitsher. WenowtranslatethesetsF,C2,B,C1
ofacceptancechancevectorsintocorrespondingsetsofcalibers,namely,F,C 2,B,C1.
Keytoourgraphicalanalysisisaquantile-quantilefunctionrelatingstudentadmissionchancesatthecolleges: Sincea(x)strictlyrisesinthestudent’stypex,astudent’s
admission chance a2ito college 2 is strictly increasing in his admission chance a1to
college 1. Inverting the admission chance in the type x, the inverse function ξ(a,s) is
the student type accepted with chance agiven the admission standard s, namely
a=1-G(s|ξ(a,s)). Thisyieldsanimplieddifferentiableacceptancefunction
a2 =ψ(a1,s 1,s 2)=1-G(s
2|ξ(a1,s 1)) (3)
1
Weproveintheappendixthattheacceptancefunctionrisesincollege1’sstandards
2
10
ndfallsincollege2’sstandards
a ,andtendsto0and1asthresholdsnearextremes.
ByFigure3,secantlinesdrawnfromtheoriginor(1,1)tosuccessivepoints
acceptancefunctiondecreaseinslope. Tothisend,saythatafunctionh:[0,1
the double secant property if h(a) is weakly increasing on [0,1] with h(0
h(1)=1,andthetwosecantslopesh(a)/aand(1-h(a))/(1-a)aremonotoneina
Thisdescriptionfullycaptureshowouracceptancechancesrelatetooneano
Theorem 1(The AcceptanceFunction) Theacceptancefunctiona2
=ψ(a1
1
2
1,s
1(x)andh(a1(x)).
2/s
1
1
1
-s/x
1)
= as1
13
14
=c
an
dM
B12
12
21
=1,thenMB21
=candMB12
2
1=crespectivelyforcea1
=1-(c/u)anda1
11
1
12
13Forifa2
14ForMB
)has
thedoublesecantproperty. Conversely,foranysmoothmonotoneontofunctiona(x),
andanyhwiththedoublesecantproperty,thereexistsacontinuoussignaldensityg(s|x)
withthestrictMLRP,andthresholdss,forwhichadmissionchancesofstudentx
tocolleges1and2area
Thisresultgivesacompletecharacterizationofhowstudent admissionchances attwo
ranked universities should compare. It says that if a student is so good that he is
guaranteed to get intocollege 1, then he is alsoa sure bet at college 2; likewise, ifhe
issobadthatcollege2surelyrejectshim,thencollege1followssuit. Moresubtly,we
arriveatthefollowingtestableimplicationaboutcollegeacceptancechances:
Corollary 1 Asastudent’scaliberrises,theratioofhisacceptancechancesatcollege1
tocollege2rises,asdoestheratioofhisrejectionchancesatcollege2tocollege1.
Foranexample,supposethatcalibersignalshavetheexponentialdensityg(s|x)=
(1/x)e. The acceptance function is then given by the increasing and concave
geometric function ψ(a, as seen in Figure 3 (as long as college 2 has a lower
admission standard). The acceptance function is closer to the diagonal when signals
are noisier, and farther from it with more accurate signals.For an extreme case, as
we approach the noiseless case, a student is either acceptable to neither college,
both colleges, or just college 2 (assuming that it has a lower admission standard).
The ψ functiontendstoafunctionpassingthrough(0,0),(0,1),and(1,1).
Sinceastudent’sdecisionproblemisunchangedbyaffinetransformationsofcostsand
benefits,wehenceforthassumeapayoffv=1ofcollege1; so,college2paysu∈(0,1).
Throughoutthepaper,wealsorealisticallyassumethatapplicationcostsarenottoohigh
relativetothecollegepayoffs—specifically,c<u(v-u)/v=u(1-u)andc<u/4. The
firstinequalityguaranteesthatthecurvesMB=cdonotcrosstwice
insidetheunitsquare.ThesecondinequalityensuresthattheMB21=ccurvecrosses below
the diagonal.If either inequality fails, the analysis trivializes since multiple
collegeapplicationsareimpossibleforsomeacceptancefunctions,astheyaretoocostly.
Specifically,fortheearlierlocation-scalefamily,ψ(a)risesinthesignalaccuracy1/µ(seePersico (2000)).
Easily, the acceptance function tends to the top of the box in Figure 3 as µ → 0, since
ψ(a)=1-G(-8)=1,andtothediagonalψ(a11-1)=1-G(0+G(1-a1))=a1asµ→8.
Thelimitfunctionisnotwell-defined: Ifastudent’stypeisknown,justthesethreepointsremain.
=c/(v-u). Now,1-(c/u)>c/(v-u)exactlywhenc<u(1-u)/v.
21 =chasnorootsonthediagonala
=a ifc>u/4.
11
4 EquilibriumAnalysisforColleges
4.1 A SupplyandDemand Approach
Each college imust choose an admission standard sas a best response to its rival’s
threshold sjiand the student portfolios. With a continuum of students, the resulting
enrollment Eiatcollegesi=1,2isanon-stochasticnumber:
E2( 1,s
2(x)f(x)dx+ B
a2(x)(1-a1(x))f(x)dx, (5)
s
E1( 1,s 22) = B∪C) a1(x)f(x)dx (4)
s
= C21a
onthestudentapplicationstrategy.
s
Tounderstand(4)and(5),observethatcaliberxstudentisadmitte
uppressingthedependenceofthesetsB,C1 chancea1(x),tocollege2withchancea(x),andfinallytocollege2b
a2(x)(1-a12(x)). Also, anyone thatcollege 1admits will enrolla
whilecollege2onlyenrollsthosewhoeitherdidnotapplyorgotreje
15If
we substitute optimal student portfolios into the enrollme
thentheybehavelikedemandcurveswheretheadmissionsstand
framework affordsanalogues tothe substitution and income e
Theadmissionrateofanycollegeobviouslyfallsinitsanticipateda
thestandardseffect. Butthereisacompoundingportfolioeffect —
fallsduetoanapplicationportfolioshift. Eachcollege’sapplicant
admissionsthreshold. Wethendeduceintheappendixthe“lawo
raisesitsadmissionstandard,thenitsenrollmentfalls. Becauseo
acollegefacesamoreelasticdemandforslotsthanpredicted pur
Aloweradmissionbarwillinviteapplicationsfromnewstudents.
Thelawofdemandappliesoutsidethetwocollegesetting. Fo
theadmissionsstandardatacollegerises. Absentanystudentpo
studentsmeetitstougheradmissionthreshold(thestandardseff
Theportfolioadjustmentreinforcesthiseffect. Thosewhohadm
toaddthiscollegetotheirportfoliosnowexciseit(theportfolioeffec
In consumer demand theory, the “price” of one good affe
15Theportfolioeffectmayactwithalag—forinstance,acollegemayunexpecte
standardsoneyear,andseetheirapplicantpoolexpandthenextyearwhenthis
12
16other,
and in the two goodworld, they are substitutes. Analogously, we prove in the
appendix,thatacollege’senrollmentrisesinitsrival’sadmissionstandard. Thisowes to a
portfolio spillover effect. If it grows tougher to gain admission to college i, then
thosewhoonlyappliedtoitsrivalcontinuetodoso,somewhowereapplyingtoboth
nowapplyjusttoj(which helps collegejwhenitisthelesser school),andalsosome
atthemarginwhoappliedjusttoinowalsoaddcollegejtotheirportfolios.
Since capacities imply vertical supply curves, we have justified a supply and
demandanalysis,inwhichthecollegesaresellingdifferentiatedproducts. Ignoringfornow
thepossibility thatsomecollegemightnotfillitscapacity, equilibrium withoutexcess
capacityrequiresthatbothmarketsclear:
κ1 =E1(s 1,s 2) and κ2
=E2(s 1,s 2) (6)
Sinceeachenrollment(demand)functionisfallinginitsownthreshold,wemayinvert
theseequations. Thisyieldsforeachschoolithethresholdthat“bestresponds”toits
rival’sadmissionsthresholdsjsoastofilltheircapacityκi:
2 =S2(s 1,κ2) (7)
s 1 =S1(s 2,κ1) and s
Given the discussion of the enrollment functions, we can treat Sias a “best response
function”ofcollegei. Itrises initsrival’sadmission standardandfallsinitsowncapacity.
Thatis,theadmissionsstandardsatthetwocollegesarestrategiccomplements.
Figure4depictsarobustequilibriumasacrossingoftheseincreasingfunctions.
By way of contrast, observe that without noise or without application costs, the
bettercollegeiscompletelyinsulatedfromtheactionsofitslesserrival—S1isvertical. The
equilibrium analysis is straightforward, and there is a unique robust equilibrium.
Ineithercase,theapplicantpoolofcollege1isindependentofwhatcollege2does. For when
the applicationsignal isnoiseless, just thetopstudents apply tocollege 1. And
whenapplicationsarefree,allstudentsapplytocollege1,andwillenrollifaccepted.
Withapplicationcostsandnoise,S1isupward-sloping,asapplicationpoolsdepend
onbothcollegethresholds. Whencollege2adjustsitsadmissionstandard,thestudent
incentivestogambleoncollege1areaffected. Thisfeedbackiscriticalinourpaper. It
16As
in consumer theory, complementarity may emerge with three or more goods available. With
rankedcolleges1,2,and3,college3maybeharmedbytougheradmissionsatcollege1,ifthisencourages
enoughapplicationsatcollege2.
13
S2
2
S2
E1
1
E0
2
E2
1
S1
E0
S
1
Figure4: College Responses and Equilibria. Inbothpanels,thefunctionsS1(solid)
andS2(dashed)givepairsofthresholdssothatcolleges1and2filltheircapacitiesinequilibrium.
Theleftpaneldepictsauniquerobuststableequilibrium,whiletherightpanelshowsa
casewithmultiplerobustequilibria. E0andE2arestable,whileE1isunstable.
leadstoaricherinteractionamongthecolleges,andperhapstomultiplerobustequilibria.
InFigure4,thefunctionS1issteeperthanS2atthecrossingpoint. Letuscallany
suchrobustequilibriumstable. Itisstableinthefollowingsense: Supposethatwhenever
enrollmentfallsbelowcapacity,thecollegeeasesitsadmissionstandards,andviceversa.
Then thisdynamic adjustment process pushes us back towardsthe equilibrium. Then
atthistheoreticallevel,admissionthresholdsactaspricesinaWalrasiantatonnement.
Unstable robust equilibria should be rare: They require that a college’s enrollment
respondsmoretotheotherschool’sadmissionstandardthanitsown.
17We
show in the appendix that a robust stable equilibrium exists, and in any such
equilibrium,college1fillsitscapacity,whilecollege2hasexcesscapacityifcollege1is
bigenough. Surprisingly,collegespacescangounfilleddespiteinsufficientcapacityfor
theapplicantpool. Ifcollege1is“toobig”relativetocollege2,thencollege2isleftwith
excesscapacity. Thereisexcessdemandforcollegeslots,andyetduetotheinformational
frictions,thereisalsoexcesssupplyofslotsatcollege2,evenat“zeroprice”.
Whencollege2hasexcesscapacity,itoptimallyacceptsallapplicants. Sincecollege1
maintainsanadmissionsstandard,collegebehaviorismonotone. Butthisforcesa 2=1
forallstudents,andsotheacceptancefunctiontraversesthetopsideoftheunitsquarein
Figure3. Inotherwords,asstudentcaliberrises,theloweststudentsapplytocollege{2},
higherstudentstobothcolleges,andthebeststudentsjustapplytocollege{1}. Letus
observeinpassingthatthisisarobustsortingequilibrium.
17College1cannothaveexcesscapacityinarobustequilibrium.
Forthenitmustsetthelowestpossiblestandards,whereuponallstudentswouldapplyandbeaccepted,violatingit
scapacityconstraint.
14
S2
S2
1
2
1' 1
2'
S1
E0
2
E0
E1
S1
E
1
Figure 5: Equilibrium Comparative Statics. The figure shows how the robust equilibriumis
affected by changing capacities κ1,κ2. ThebestresponsefunctionsS(solid) and
S2(dashed)aredrawn. Theleftpanelconsidersariseinκ1,shiftingS11left,therebylowering
bothcollegethresholds. Therightpaneldepictstheanalogousriseinκ2andshiftinS2.
18Sinceadmissionsstandardsarestrategiccomplements,multiplerobustequilibriaare
possible(rightpanelofFigure4).Insuchascenario,bothcollegesraisetheirstandards,
andyetstudentssendevenmoreapplications,andanotherrobustequilibriumarises.
4.2 Comparative Statics
We now continue toexplore the supply anddemand metaphor, andderive some basic
comparativestatics. Thepotentialmultiplicityofrobustequilibriamakesacomparative
staticsexercisedifficult. Butfortunately,ouranalysisappliestoall robuststableequilibria.
Indeed,greatercapacityateithercollegelowersbothcollegeadmissionsthresholds.
Thisresultspeakstotheequilibriumeffectsatplay. Greatercapacityatoneschool,or
anexogenousdownwardshiftinthe“demand”forslots,reducesthe“price”(admission
standard)atbothschools. ThegraphinFigure5provesthisassertion.
Forintuition,considerarobustsortingequilibrium,wherestudentsapplyasinFigure3.
Letcollege2raiseitscapacityκ(asintherightpanelofFigure5). Fixingthe admission standard
s12, this depresses s. The marginal student that was indifferent between
applyingtocollege2only(C22)andbothcolleges(B)nowpreferstoapplyto college2only.
Sofewerapplytocollege1. Giventhisportfolioshift,college1dropsits admissionstandards.
Theleftpaneldepictsthesamecomparativestaticforcollege1.
Unlikecollegecapacity,collegepayoffsorapplicationcostsaffectbothbestresponse
functions S1and S2. As a result, the comparative statics are not unambiguous, and
18Theonlineappendixcontainsasolvedexamplewithmultiplicity.
15
counter-intuitive resultsmayemerge. Supposethatthepayoffuofcollege2rises, and
assume fixed admission standards. Then as we have argued, region Bshifts right in
Figure3,therebyincreasingthedemandforcollege2. Atthesametime,thedemandfor
college1decreases,assomestudentsceasetosendstretchapplicationstothatcollege. These
forces lead to new best reply admission standards, namely an upward shift in the
S2locus and a leftward shift in S1. Depending on the strength of these shifts, the new
equilibrium admissions thresholds are either both higher, or both lower, or
loweratcollege1andhigheratcollege2higherthanbeforetheuincrease; wecannot
unambiguouslyascertainwhichcaseistrue.
Next,assumethattheapplicationcostcrises, perhapsduetoariseintheSATor
ACTcost,orthecommonapplicationfee. ThenS1shiftsleft—forcollege1mustdrop its
standards to counter the loss of stretch applications, since we have seen that the
regionBshrinks inFigure3. Ontheotherhand, S2shiftsambiguously inthecostc.
Forwithfixedstandards,wehaveseenthatcollege2losessafetyandsoloapplications,
butnolongerlosesasmanystudentswhosendstretchapplicationstocollege1.
Considerinsteadariseinjustonecollege’sapplicationcost,suchasacollegerequiring
alongeressayorimposingagreaterfee. Wearguethatinarobustsortingequilibrium,if
theapplicationcostateithercollegeslightlyfalls,thentheadmissionstandardatcollege1 rises
and its student caliber distribution stochastically worsens. Ifthe applicationcost at
college 2 falls, then more students apply, and it is forced to raise its standards. The
marginal benefit of a stretch application to college 1 thus rises. To counter this,
college1respondswithahigherstandard,butitstillgainsmoreapplicantsatitslower
end,anditscaliberdistributionstochasticallyworsens. Bycontrast,college2losesnot
onlyitsworststudents,butalsotoponesforwhomitwasinsurance;itscaliberchange
isambiguous. Graphically,afallintheapplicationcostatcollege2shiftsSupwards
withoutaffectingS12—astheapplicantpoolatcollege1dependsonlyindirectlyonthe
college2applicationcostviaportfolioeffectsasthecollege2standardchanges.
Theargumentismorecomplexwhentheapplicationcostatcollege1falls,sincethe
applicant pools at both colleges depend on it. The direct impact is an expansion in the
applicant poolatcollege 1—consisting ofthose who send stretch applications to
it—andacontractionintheapplicantpoolofcollege2,triggeringanincreaseinthe
admissionstandardatcollege1andadecreaseinthatofcollege2. Thechangesinthe
admissions thresholds lead to further portfoliostudent reallocation, and the net effect
isaprioriambiguous. Graphically,alowerapplicationcostatcollege1simultaneously
16
s
h
i
f
t
s
S
(rightwards)andS2
1
1(1-a2
17
(downwards).
Weshowintheappendixthattheportfolio effects at both colleges cancel out
for those calibers who send stretch applications to
college1;onbalance,itsadmissionstandardrises,andyetitsapplicantpoolwors
ens.
Thelogicunderlyingthissectiondoesnotessentiallydependontheassumpti
onthat therearetwocolleges.
Forinstance,whenevercollegeshaveoverlappingapplicantpools,
ariseincapacityateitherdepressestheadmissionstandardsatboth.
ConsiderthepositivetheoryofthissectioninlightofHoxby(2009).
Sheshowsthat during 1962-2007, the median college has become
significantly less selective, while at the same time, admissions have
become more competitive at the top 10% of colleges. Her explanation for
the fall in standards hinges on capacity: the number of freshman
placesperhigh-schoolgraduatehasbeenrisingsteadily.
ButasweillustrateinFigure5, highercapacity atallschools shoulddepress
standards atallschools, viaourspillover effect.
Asacountervailingforce,shearguesthatstudentshavesimultaneously
become more willing to enroll far from home, raising the relative payoff of
selective colleges. Proposition 7 of our online appendix speaks to this
insight: starting at a robust and stable sorting equilibrium, a perceived
increase in the value of an education at a top
schoolleadsittoraiseitsstandardsinsomenewrobustandstableequilibrium.
5 DoCollegesandStudentsSortinEquilibrium?
Casualempiricismsuggeststhatthebeststudentsapplytothebestcolleges,and
those collegesareinturnthemostselective.
Thislogicjustifiesrankingcollegesbasedontheir admissions standards.
Curiously, these claims arefalse without stronger assumptions.
Weidentifyandexplorethetwopossibletypesofsortingviolationsbystudents.
The first violation is seen in the left panel of Figure 6. By Corollary 1,
along the
acceptancefunction,highertypesenjoyahigherratioofadmissionschancesatc
ollege1 tocollege2.
Butthisdoesnotimplyahighermarginalbenefitau)ofapplying tocollege1.
Sotheacceptancefunctionmaymultiplyslicethroughthecurve(2),where
students are indifferent between a stretch application to college 2. This
yields a
nonmonotonesequenceofapplicationsetsF,{2},{1,2},{2},{1,2},{1}asthecalibe
rrises.
Thesecondviolationoccurswhencollegeadmissionsstandardsaresufficient
lyclose.
Forintheextremecase,whenstudentsentertainthesameadmissionchances
aatthe twocolleges,
themarginalbenefitofaddingasafetyapplication,namely(1-a)au,is
notmonotoneina(andthusnotincaliber,either).
TherightpanelofFigure6depicts
1.0
1.0
C2
0.8
0.6
B
0.8
B
0.4
0.2
C2
0.6
0.4
P
Q
C1
Α1 x
C1
Α1 x
0.0 0.2 0.4 0.6 0.8 1.0 0.0
0.2
0.0 0.2 0.4 0.6 0.8 1.0 0.0
Figure 6: Non-Monotone Behavior. In the left
panel, the signal structure induces a
piecewiselinearacceptancefunction.
Studentbehaviorisnon-monotone,sincethereareboth
lowandhighcaliberstudentswhoapplytocollege2only(C
),whileintermediateonesinsure byapplyingtoboth.
Intherightpanel,equalthresholdsatbothcollegesinduce
anacceptance function along the diagonal, a1= a22.
Student behavior is non-monotone, as both low and
high caliber students apply to college 1 only (C1),
while middling caliber students apply to both.
Suchanacceptancefunctionalsoariseswhencalibersig
nalsareverynoisy.
onesuchcase—wheretheapplicationsetsareF,{1},
{1,2},{1}ascaliberrises. This
violationcanevenoccurwhencollege2setsahigher
admissionsstandardthancollege1.
To rule out the first sorting violations, it
suffices that college 2 offer a low payoff (u<0.5).
We show in the Appendix that this ensures that
the marginal benefit of additionally applying to
college is increasing in caliber. The second sort
of violation
cannotoccurwhencollege1setsasufficientlyhigher
admissionstandardthancollege2. This in
particular happens when college 1 is sufficiently
smaller than college 2. The
thresholdcapacitywillingeneraldependonallthepri
mitivesoftheproblem: therival
capacity,applicationscost,payoffdifferentialandsi
gnalstructure.
Α2x
Α2x
Theorem 2(Non-Sorting and Sorting
inEquilibrium)
(a)Ifcollege2is“toogood”(i.e.,u>0.5),thenthereexis
tsacontinuousMLRPdensity
g(s|x)thatyieldsarobuststableequilibriumwithnonmonotonestudentbehavior.
(b)Ifcollege2issmallenoughrelativetocollege1,the
ncollege2setsahigheradmissions
standardthancollege1insomerobustequilibrium.
(c)Ifcollege1issmallenoughrelativetocollege2,and
college2isnottoogood(namely,
u=0.5),thenthereareonlyrobustsortingequilibriaan
dnocollegehasexcesscapacity.
Thechallengeinprovingthistheoremistoshowth
atallofnon-monotonebehavior
18
outlined above can happen in equilibrium. For part (a), we construct a robust
nonmonotoneequilibriumbystartingwiththeacceptancefunctiondepictedintheleftpanel
ofFigure6,whichconstrainstherelationshipsbetweenadmissionschancesacrosscolleges to
be some mapping a2(x) = h(a(x)). We then construct a particular acceptance
chancea11(x)sothattheinducedstudentbehaviorandacceptanceratesgiven(a 1,h(a)) equate
college capacities and enrollments. Finally, we show that these two mappings
satisfytherequirementsofTheorem1andthereforecanbegeneratedbyMLRPsignals and
monotone standards. For part (b), we show that that by perturbing a robust
equilibriumwithequaladmissionschancesbymakingcollege2smaller,wegetarobust
equilibriumwithnon-monotonestandards. Finally,part(c)turnsonshowingthatwhen
κ11isrelativelysmall,thecrossingofthebest-responsefunctionsmustoccuratapoint
wherecollege1setshighenoughstandardsthatlowcaliberstudentsdon’tapplythere.
19,20All
told, parts (a) and (b) show that sorting fails in some robust equilibria, which is
surprising given how well behaved the signal structure is. The take-away is that in
simultaneoussearchproblems,themarginalbenefitsbothofgamblingupandofinsuring
neednotbemonotoneintype,eveninequilibrium. Sortingcanfailinamorestriking sense that
undercuts how colleges are ranked: While colleges are often ranked based
onobservablecharacteristicsofmatriculatingstudents,suchastheaverageSATscore,
studentsattheworsecollegecaninfacthaveahighermeancaliber.
The sorting failures in this section exploit some but not allofthe structure ofour simple
two college world. In particular, assume two tiers of colleges, a top tier 1 and alesser tier
2. We prove in the online appendix thatstudent non-monotonicity result still persists in
the tiered college world. But college standards must be monotone, as essentially
arguedin ChadeandSmith (2006). Forifnot, then no student would ever
applytotier2,sinceitwouldofferalowerexpectedpayoff.
21For
an insightful counterpoint, consider what happens when students are limited to just
a single application, as it is sometimes the case.Recalling the left panel in
Figure2,weseethatthediagonallinea1u=a2,andtheindividualrationalequalities
19ThewidelyreadUSNewsandWorldReportcollegerankingsusessuchstatistics.
20ThiscanbeillustratedusingtherightpanelofFigure6.
Considerarobustequilibriumwithequal
admissionsstandardsatthetwocolleges. Iff(x)concentratesmostofits massontheintervaloflow calibers who
apply just to college1, then the averagecaliber ofstudents enrolledatcollege1 will be
strictlysmallerthanthatatcollege2. Thisexamplecanbeadjustedslightlysothatcollege2ismore
selectivethancollege1. (Seeouronlineappendixforafleshedoutexampleofthisphenomenon.)
21InBritain,allcollegeapplicationsgothroughUCAS,acentralizedclearinghouse. Amaximumof
fiveapplicationsisallowed.
19
a2u= a1= c, jointly partition the application space into three relevant parts. But
withanyacceptancefunctionwiththefallingsecantproperty,lowtypesapplynowhere, middle
types apply tocollege 2, and high types apply tocollege 1. It should come as nosurprise
thatthere isaunique robustequilibrium. Soitturns outthatthesorting
failuresinTheorem2(a)and(b)requireportfolioapplications.
6 GeneralIncompleteInformationAboutCalibers
A.WhenStudentsDoNotKnowtheirCalibers. Wehaveassumedthatcolleges observe
conditionally independent evaluations of the students’ true calibers. At the
oppositeextreme,onemightenvisionahypotheticalworldwherecollegesknowstudent
calibers,andstudentsseenoisyconditionallyindependentsignalsofthem. Yetobserve
thatthisisinformationallyequivalenttoaworldinwhichstudentsknowtheircalibers,
andcollegesobserveperfectlycorrelatedsignals. Foranystudentseesasignalequalto
t+“noise”,whilebothcollegesseethestudentcalibert.
This embedding suggests how we might capture the world in which students and
collegesalikeonlyseenoisyconditionallyiidsignalsofcalibers. Weargueintheappendix
thatthisisaspecialcaseofthefollowingworld:
(⋆)Studentsknowtheircalibersandcollegesobserveaffiliatednoisysignalsofthem.
Thus,wecanwithoutlossofgeneralityassumethatstudentsknowtheircalibersand
collegesvarybytheirsignalaffiliation. Observethatinthisworld,Theorem1remainsa
validdescriptionofhowtheunconditionalacceptancechancesatthetwocollegesrelate.
B.PerfectlyCorrelatedSignals. Thisbenchmarkishighlyinstructive. Suppose first that the
two colleges observe perfectly correlated signals of student calibers. As we mentioned
above, this is akin to observing the caliber of each applicant. The key
(counterfactual)featurehereisthatifastudentisacceptedbythemoreselectivecollege, then
so is he at the less selective one. This immediately implies that s1> s2in equilibrium,
forotherwisenoonewouldapplytocollege2. SocontrarytoTheorem2,
collegebehaviormustbemonotoneinanyequilibrium.
The analysis of this case differs in a few dimensions from §3.1. Since s 1 > s 2
1 +(a2 -a1 22)u-2c. Unlikebefore,
MB21 =(a2 -a1)u=c and MB12
=a1(1-u)=c (8)
, applyingtobothcollegesnowyieldspayoffa
Wes uppressthecaliberxargumentoftheunconditionalacceptancechanceai(x)atcollegei=1,2.
20
22
becauseadmissiontocollege1guaranteesadmissiontocollege2. Inthisinformational
world,bothoptimalityequationsarelinear,andthelatterisvertical(seeFigure7).
Assume thatcollege1issufficiently moreselective thancollege2. Thenthelowest
caliberapplicantsapplytocollege2—namely, thosewhoseadmissionchanceexceeds c/u.
Studentssogoodthattheiradmissionchanceatcollege1isatleastc/(1-u)add
astretchapplication,providedcollege2admitsthemwithchancec/u(1-u)ormore. In Figure 7,
this occurs when the acceptance function crosses above the intersection point of the
curves MB21= cand MB12= c.23Since the marginal benefit MBis
independentoftheadmissionchanceatcollege2,MB1212>cforallhighercalibers. But
monotone behavior for stronger caliber students requires another assumption.
Considerthemarginbetweenapplyingjusttocollege1,oraddingasafetyapplication.
Thetopcaliberstudentswillapplytocollege1only,sincetheiradmissionchanceisso high.
Butthe behavior ofslightly lesser student calibers is trickier, asthe acceptance
functioncanmultiplycrossthelineMB21=c.2425Underslightlystrongerassumptions,
theacceptancefunctionisconcave;thisprecludessuchperversemultiple-crossings,and
impliesmonotonestudentbehavior.
Consider now thepossibility ofnon-monotonestudent behavior. Absent aconcave
acceptancefunction,theprevioussortingfailureowingtomultiple-crossingsarises. But
evenwithaconcaveacceptancefunction,asortingfailurearisesifcollege1isnotsufficientlycho
osierthancollege2. Forthenasuitablydrawnconcaveacceptancefunction
couldconsecutivelyhitregionsC1,B,thenC1,andasortingfailureensues. Having explored the
impact of correlation on college-student sorting, we now flesh
outitseffectoncollegefeedbacks. SinceMB12isindependentoftheadmissionthreshold at
college 2 in (8), the pool of applicants to college 1 is unaffected by changes in s.
Hence,theS126locusisverticalovermostofitsdomain.2Thebettercollegeisinsulated from the
decisions of its weaker rival, and the setting is not as rich as our baseline
conditionallyiidcase. ItisobviousfromFigure7thattherobustequilibriumisunique.
C.AffiliatedCollegeEvaluations. Wenowturntothegeneralcaseofassumption
23Namely,
atthe mutualintersectionofregionsC 1,C2,andBinFigure7. Byinequality(13), this
holdsunderourhypothesisthatcollege1issufficientlychoosierthancollege2.
24ForasseeninFigure7,thatlinealsohasastrictlyfallingsecant.
25Concavityholdswhenever-Gx(s|x)islog-supermodular. Thisistruewhenwefurtherrestrictto
locationfamilies(liketheNormal)orscalefamilies(suchasexponential).
26ThelocusS12verticalifcollege1issufficientlymoreselectivethancollege2. Forthentheacceptance
functionishighenoughthatittraversesregionB,andsomestudentssendmultipleapplications. Ifnot,
thentheacceptancefunctioncouldhitCandthenC,bipassingregionB. Inthatcase,themarginal
applicanttocollege1dependsons2,an dhenceS11isnotvertical.
21
1.0
C2
0.8
0.6
S2
1
2
E0
B
C1
0.4
0.2
0.0 0.2 0.4 0.6 0.8 1.0 0.0
Α1 x
Figure7:
StudentBe
haviorwith
PerfectlyC
orrelatedSi
gnals.
Theshadedr
egions
depict the
optimal
portfolio
choices for
students
when
colleges
observe
perfectly
correlated
signals.
UnlikeFigur
e2,theMB12
=ccurvesep
aratingregio
nsBandC2is
vertical.
TheS1
curveattheri
ghtisvertical
(uptoapoint)
,sincecolleg
e
2nolongerim
posesanext
ernality
oncollege1
whentheset
S
1
ofcalibersse
ndingmultipl
eapplication
sisnonempt
y.
(⋆).
Eachstud
entknows
hiscaliber
x,andcoll
egessee
signalss1
,softhem,
withan
affiliatedj
ointdensi
tyg(s1,s2|
x).272Sinc
eaccepta
nceandre
jectionby
college1i
sgood
andbadn
ews,resp
ectively,it
intuitively
followsth
at
aA 2=a2
and aR 2
A =aR 2
2
(9)
H
e
r
e
,
a
A
2
A
2
>aare
the
respectiv
e
acceptan
M
B
2
1
=aR 2
2
2
A2
ce
chances
at
college 2
given
acceptan
ce
andreject
ionatcolle
ge1.
Forinstan
ce,
1=a2>aR
2withperf
ectlycorr
elation.
Butinthec
onditiona
llyiidcase
,college2i
sunaffect
edbythed
ecisionof
college1,
and
soa=a.
Since
these
areintuiti
vely
opposite
ends
ofanaffili
ationspe
ctrum,
wecallev
aluations
moreaffili
ated
ifthecond
itionalacc
eptancec
hanceaat
college2
ishigherf
oranygiv
enuncon
Α2x
ditionalac
ceptance
chancea.
Let’sfirsts
eehowa
ffiliationa
ffectsstu
dentappli
cations.
Inthismor
egeneral
setting,
=(1-a1)aR 2u and MB12
This
subsumes
the
marginal
analysis
forour
conditionall
y iid and
perfectly
correlated
cases:
(1)–(2) and
(8).
Relative to
these
benchmark
s, the
acceptanc
e curse (or
the
27Asisstan
dard,thism
eansthatg
obeysthe
monotonel
ikelihoodr
atioproper
tyforevery
fixedx.
22
=a1(1-aA 2u). (10)
“acceptanceblessing”)conferredbycollege1’stwopossibledecisionslessensthemarginal
gainofanextraapplicationtoeithercollege—duetoinequality(9). Moreintuitively, double
admission is more likely when signals are more affiliated. In our graph, this is
reflectedbyarightshiftofthecurveMB12=c,andaleftshiftofMB21=c. Inother words,
foranygiven collegeadmission standards, students send both fewer stretch and
safetyapplicationswhencollegeevaluationsaremoreaffiliated.
Wenextexplorehowaffiliatedevaluationsaffectscollegebehavior. Considerthebest
replylocusS1ofcollege1. Itisupward-slopingwithconditionallyiidcollegeevaluations, and
vertical with perfectly correlated evaluations. We argue that it is upward-sloping with
affiliated evaluations, and grows steeper as evaluations grow more affiliated. In
otherwords,ourbenchmarkconditionallyiidcasedeliversrobustresultsabouttwo-way
collegefeedbacks. Perfectlycorrelatedevaluationsthereforeignorestheeffectofthelesser
onthebettercollege,andsoislessreflectiveoftheaffiliatedcase.
We first show that the best response curve Sslopes upward with imperfect affiliation.
Forletcollege1’sadmissionstandards11rise. Thenitsunconditionalacceptance chance
a1falls for every student. The marginal student pondering a stretch applicationmust
thenfallinorderforcollege1tofillitscapacity(6). Optimality MB=c
in(10)nextrequiresthatthisstudent’sconditionalacceptancechanceaA 212fall. Thisonly
happensifhisunconditionalchancea2fallstoo—i.e.thestandardsrises. Next, college 1’s
best response curve S12slopes up more steeply when college evaluations are more
strongly affiliated. For as affiliation rises, the marginal student sees a greater fall in his
admission chance aA 2. So his unconditional chance afalls more too, and college 2’s
admission standard s22drops more than before (see Figure 7), as claimed.
Asanaside,sincerobustequilibriumisuniquewithperfectlycorrelatedcollege
evaluations,uniquenessintuitivelyholdsmoreoftenwhenweareclosertothisextreme.
Finally,weconsiderhowtheequilibriumsortingresultTheorem2changeswithaffiliation.
Byexamining(10),weseethataswetransitionfromconditionallyiidtoperfectly correlated
signals with increasing affiliation, the region of multiple applicationsshrinks
monotonically. Thissimpleinsighthasimportantimplicationsforsortingbehavior. By
standard continuity logic, forvery low orhigh affiliation, sorting obtains and fails exactly
as in the respective conditional independent or perfectly correlated cases. More
strongly, thenegativeresultinTheorem2(b)failsformoderatelyhighaffiliation: For then
nonmonotone college behavior is impossible since the locus MB21=cin (8) lies
strictlyabovethediagonal,andthusthesameholdsfor(10)withsufficientlyaffiliated
23
signals. Sotheacceptancefunctionwouldliebelowthediagonalifadmissionstandards were
inverted, and no student would ever apply to college 2. Lastly, the logic for the positive
sorting result of Theorem 2(c)is still valid: bothstudent and college
behavioraremonotoneifcollege2isnottoosmall andnottoogood,andiftheacceptance
functionisconcave—appealingtothelogicforperfectlycorrelatedsignals.
7 TheSpilloverEffectsof Affirmative Action
We now explore the effects of an affirmative action policy.28Slightly enriching our
model,wefirstassumethatafractionfoftheapplicantpoolbelongstoatargetgroup.
Thismaybeanunder-represented minority, butitmayalsobeamajoritygroup. For instance,
many states favor their own students at state colleges — Wisconsin public
collegescanhaveatmost25%out-of-statestudents. Justaswell,somecollegesstrongly
valueathletesorstudentsfromlow-incomebackgrounds. Weassumeacommoncaliber
distribution,sothatthereisnootherreasonfordifferentialtreatmentoftheapplicants.
Assume thatstudents honestly reporttheir“targetgroup”statusontheirapplications.
Reflectingthecolleges’desireforamorediversestudentbody,letcollegeiearna
bonuspi=0foreachenrolledtargetstudent. Collegesmaysetdifferentthresholdsfor
thetwogroups. Ifcollegeioffera“discount”∆itotargetapplicants,thentherespective
standardsfornon-targetandtargetgroupsare(s1,s2)and(s1-∆1,s2-∆2). Akinto
thirddegreepricediscrimination,nowcollegesequatetheshadowcostofcapacityacross
groupsforthemarginallyadmittedstudent. Soateachcollege,theexpectedpayoffof
themarginaladmitsfromthetwogroupsshouldcoincide—exceptatacornersolution,
whenacollegeadmitsallstudentsfromagroup.
Thisyieldstwonewequilibriumconditionsthataccountforthefactthatexpost,collegesbehaver
ationally,andequatetheir expectedvaluesoftargetandnon-targetapplicants.
Alongwithcollegemarketclearing
(6),equilibriumentailssolvingfourequationsinfourunknowns.
Theanalysisissimplerifweassumeprivatevalues,andwethusfocusonthiscase,
andlaterrevisitcommonvalues. Sincecollegescaredirectlyaboutthesignalobserved
withprivatevalues,equalizationofthemarginaladmitsofthetwogroupsi=1,2reduces
tosi=si-∆i+pi,andso∆i=pi. Thatis,the‘discount’affordedtoastudentfrom
thetargetgroupequalstheadditionalpayoffacollegeenjoysfromadmittingastudent
28For
recent treatment of complementary affirmative action issues, see Epple,Romano,andSieg
(2008),Hickman(2010),GroenandWhite(2004),andCursandSingell(2002).
24
fromthetargetversusthenon-targetgroup. Thus,collegepreferencesfortargetgroup
studentstranslatedirectlyintoadmissionstandarddiscountsforthatgroup.
Theorem 3(Affirmative Action) Assume private values, a robust stable equilibrium, and
p1= p2= 0. As the preference for a target group at one college rises, that college favors
those students and penalizes non-target students. As the preference
fortargetstudentsatbothcollegesrise,bothfavorthemandpenalizenon-targetstudents.
Observetheindirecteffectofstudentpreferences:
Nontargetstudentsfacestifferadmissionstandardssincetheshadowvalueofcapacityisnowhi
gher. Theassumptionthat p1= p2= 0 is important, as it precludes some complex feedback
effects that emerge whenthereisalreadyapreferencefortargetstudentsattheoutset.
Forasharperinsight,specializethesignaldistributiontothelocationfamilyG(s-x). Suppose
that both colleges exhibit identical preference pfor the target group. Since
∆=pforbothcolleges,theacceptancerelationisidentical forbothgroupsandgiven by
ψ(a1)=1-G(s2-s1+G-1(1-a1))(thediscounts cancel outin theargument of Gfor the target
group). This implies that caliber xfrom the target group applies
exactlyasiftheywereatypex+pfromthenon-targetgroup: atestableclaim.
If instead only one college, say 1, has a preference pforthe target group, then it
setsadiscount∆1=p11. Theacceptancerelationforthetargetgroupisthenψ(a)=
1-G(s2-s1+∆1+G-1(1-a12)),whichiseverywherebelowthatofthenon-target
groupψ(a2)=1-G(s2-s1+G-1(1-a1)). Asaresult,inarobustsortingequilibrium,
target-groupstudentswillgambleupandapplytocollege1moreoften,andinsurewith
anapplicationtocollege2lessoften,thannon-targetgroupstudents.
We have focused in this section on private values as it leads to a simpler analysis.
Consider the common values benchmark. Equalization of the expected payoff of the
marginaladmitsfromthetwogroupsateachcollegethenyieldsthericherconditions:
1 -∆1,target] = E[X|s
1
E[X+p1|s=s
2 -∆2,target,accepts] = E[X|s
i 2,non-target,accepts]
2|s=s
i
i
i
25
,non-target] E[X+p
=p
Here,Xisthestudentcaliber. Noticethatnolongerdowehave∆
nowdependsonp
;rather,thediscount
∆inanonlinearfashionviatheconditionalexpectations. As
before,alongwith(6),equilibriumamountstosolvingfourequationsinfourunknowns.
Let us explore how the first result in Theorem 3 changes for the common values
settingofthepaper. Assumearobustsortingequilibrium,aswellasanaturalstability
requirementonshadowvaluesofthegroupsofstudents(seeonlineappendix). Thenas
thepreferenceforatargetgroupatcollege1risesfromthenopreferencecasep 1=p2=
0,itfavorsthosetargetstudents, butnowcollege2insteadpenalizesthem. Butwhen
thepreferencefortargetstudentsattheworsecollege2rises,bothcollegesfavorthem.
29Inotherwords,anasymmetry
emergesundercommonvalues. Whenonlycollege1
favorsatargetgroupofstudents, college2mustcounterthiswithapenalty, owingto two
effects. First, the best favored students that previously applied to college 2 now
justapplytocollege1,andthusthepooloftargetapplicantsatcollege2worsens. This
portfolioeffectwaspresentwithprivatevalues. Second,college2confrontsanacceptance
curse. Astudentwhoenrollsthereeitheronlyappliedtoit,oralsoappliedtocollege1
—andwasrejected. Sotheeventthatastudentenrollsatcollege2isaworsesignalof
hiscaliberifcollege1hasfavoredthem.
8 ConcludingRemarks
Wehave formulated thecollege admissions problemfortworanked colleges with fixed
capacities in order to study the effects of two frictions in equilibrium. Student types
areheterogeneous andcolleges only partiallyobserve their types. College applications
arecostly,andstudentsthereforefaceanontrivialportfoliochoice. Thismodeladmits
atractableseparablesolutioninstages—studentportfoliosreflecttheadmissionstandards,an
dcollegesthencompeteasiftheywereBertrandduopolists. Thisframework is the only
equilibrium model that speaks to the recent empirical explorations of
studentapplicationbehavior(AveryandKane(2004);Pallais(2009);CarrellandSacerdote
(2012);HoxbyandAvery(2012)).
Wehavecharacterizedinatestablefashionhowstudentadmissionchancesco-moveas
theircalibersimprove,showinghowtheiroptimalportfoliochoicesoverstretchandsafety
schools differ. We have have discovered that even in this highly monotone matching
world, sorting of students and colleges fails absent stronger assumptions. For better
studentsneednotalwaysapplymoreaggressively: Iftheworsecollegeiseithertoogood
ortoosmall,ortheapplicationprocessisnoisyenough,onestudentmaygambleonthe
bettercollegewhileamoretalentedonedoesnot. Likewise,collegeadmissionsstandards
neednotreflecttheirquality—theworsecollegemayoptimallyimposehigherstandards
29Thisasymmetricresultshouldspeaktostudies,likeKane(1998),thatfoundthataffirmativeaction
fordisadvantagedminoritiesisgenerallyconfinedtoselectiveschools—aswethinkofcollege1.
26
30if
it is small enough. Large public schools might well be punished in college rankings
publicationsthatuseSATscoresofenrolledstudentsinrankingschools.
Turning to affirmative action, we predict that favored minority applicants apply
asambitiouslyasif theyweremajorityapplicantsofhighercaliber. CardandKrueger
(2005)investigatewhathappenedwhenaffirmativeactionwaseliminatedatstateschools in
California and Texas. They find a small but statistically significant drop in the probability
that minority applicants send their SAT scores to elite state schools, but find no such
effect for highly qualified minority applicants (those with high SATs or GPAs).
Thisisconsistentwithourfindingthatlowercaliberminorityapplicantssend
stretchapplicationsunderaffirmativeaction.
While the two college world is restrictive, it is the most parsimonious model with
portfolioeffects,stretchandsafetyschools,andadmissionstandardssetbycompeting
schools. Theportfolioeffectsinduceimportantandrealisticinterdependencies missing
fromallfrictionlessmodelsofstudent-collegematching. Sinceassortativematchingcan fail in
this setting, it can fail more generally. Our baseline model has assumed that
studentsperfectlyknowtheircalibersandcollegesonlyobservethemwithnoise,butwe
havearguedthattheportfolioeffectsandsortingfailuresmonotonicallydiminishaswe shift
toward the opposite extreme when colleges have superior information. And even
inthislimit,studentssendstretchandsafetyapplications,andsortingcanfail.
A Appendix: OmittedProofs
A.1 Colleges Optimally Employ Admissions Thresholds
Letχ(s)betheexpectedvalueofthestudent’scalibergiventhatheappliestocollegei,
hissignaliss,andheaccepts. Collegeioptimallyemploysathresholdruleif,andonly
if,χii(s)increasesins. Forcollege1thisisimmediate,sinceg(s|x)enjoystheMLRP property.
College2facesanacceptancecurse,andsoχ2(s)is:
xg(s|x)f(x)dx+ Bg( xG(s 1|x)g(s|x)f(x)dx (11)
s|x)f(x)dx+ B
C
2
G(s 1
7
χ2(s)= C2 2|x)g(s|x)f(x)dx Avery,Glickman,Hoxby,andMetrick (2004) develop an innovative revealed
30
preference college ranking based on enrollment decisions by students admitted at multiple sc
and public schools areindeed rankedhigherunder their approach. One difficulty for themis tha
initial application portfolios are endogenous. Explicitly modeling the application decision using
model of portfolio
choice—asinFu(2010)—maybehelpfulinresolvingtheeconometricinferenceproblemsthatresult.
B∪C22hwhere
we denoteby C312theset ofcalibersapplying to2only, andB thoseapplyingto
both.Write(11)asχ2(s)= B∪C(x|s)=
(I2Cxh22(x|s)dxusingindicatorfunctionnotation:(x)+I(IC2BG(s(t)+IB1G(s1|t))g(s|t)f(t)dt , (12)
|x))g(s|x)f(x) (x|s)hasth 2
2
eMLRP.Therefore,χ2
i
2
1(S),s
1(S)ands
1
henthe‘density’h2
(s)increasesins.
T
Noticethatthesameresultsobtainforanyincreasingfunctionχ(s)—inparticular,
ifitistheidentityfunction,asintheprivatevaluescase.
A.2 Simultaneous versus Sequential Timing
We claim that the subgame perfect equilibrium (SPE) outcomes of the two-stage
game withstudentsmovingfirstcoincidewiththeNashequilibriaoftheone-shotgame.
First,considertheoutcomeofaSPEofthetwo-stagegame,wherestudentschoose
applicationS=S(·)andthencollegeschoosestandardss(S). Thencolleges
mustbestrespondtoeachotherandtoS(whichisrealizedwhentheychoose). Also,
sincestudentscanforecasthowcollegeswouldrespondtoSinanSPEofthetwo-stage
game,theirapplicationsmustbestrespondtothestandardss(S)ands(S). Soitis
anequilibriumoftheone
hasmeasure zero, hecannot affectthe college
-shotgame.
standardsbyadjustinghisapplicationstrategy. Hence,anyequilibrium(S,s(S))of
Conversely, sincetheone-shotgameisalsotheoutcomeanSPEofthetwo-stagegame.
each student
A.3 AcceptanceFunctionShape: Proof of Theorem 1
1
>s
2
2
owetothecontinuityofourfunctionsai(x)in§3.2.
sinceG(s
31
oavoidduplication,weassumes
T
throughouttheproof.
(⇒)TheAcceptanceFunctionhastheDoubleSec
First,
1|x)iscontinuouslydifferentiableinx,theacceptancefunctioniscontinuous
on (0,1]. Given a= 1-G(s|ξ(a,s)), partial derivatives have positive
We assume that students employ pure strategies,which followsfromour analys
optimizationin§3.1. MeasurabilityofsetsBandC
28
slopesξa,ξs
= -Gx(s ∂ψ ∂s2 = -g(s
= -Gx(s
∂ψ ∂a1
>0.
∂ψ
∂s1
Differentiating(3),
2|ξ(a1,s 1 ))<0.
1,s
PropertiesofthecdfGimplyψ(0,s
1))and1-G(s|ξ(a1,s
1,sinces
1,s
(1 3)
1,s 2)=0andψ(1,s
1)>0
1))ξa(a1,s 2|ξ(a1,s
1)>0
1,s 2
1
1))ξs(a1,s 2|ξ(a1,s
1
>s 1))arethenstrictlylog-supermodularin(s,a1
2:
)=1. Thelimitsofψas
thresholdsapproachthesupremumandinfimumowetolimitpropertiesofG.
Now, G(s|x) and 1-G(s|x) are strictly log-supermodular in (s,x) since the density g(s|x) obeys the
strict MLRP. Since x = ξ(a1) is strictly increasing in a, G(s|ξ(a). So
thesecantslopesbelowstrictlyfallina
1-ψ(a1
= G(s 11|ξ(a2|ξ
1
(a1)) .
11|ξ(a2|ξ(a1)) an
d
ψ(a1) = 1-G(s )) 1-G(s
) 1-a
)) G(s
a1
(⇐) D eriving a Signal Distribution. Conversely, fix a function hwith the
doublesecantpropertyandasmoothlymonotoneontofunctiona1(x). Also,puta(x)=
h(a1(x)),sothata2(x)>a12(x). Wemustfindacontinuoussignaldensityg(s|x)with
thestrictMLRPandthresholdss1>s2thatrationalizethehastheacceptancefunction
consistentwiththesethresholdsandsignaldistribution.
Step1: ADiscreteSignalDistribution. Consideradiscretedistributionwith realizationsin{-1,0,1}:
g1(x)=a1(x),g0(x)=a2(x)-a1(x)andg-1(x)=1-a(x). Indeed,foreachcaliberx,gi2=0andsumto1.
ThisobeysthestrictMLRPbecause
g0(x)
a2(x)-a(x) a11(x) h(a1(x)) a1(x)
g1(x) = =
-1,
isstrictlydecreasingbythefirstsecantpropertyofh,and
g0(x)
a2(x)-a(x) 1-a21(x)
1-a1(x)
g-1(x) = =-1+
1-h(a1(x))
isstrictlyincreasinginxbythesecondsecantpropertyofh. Let the
1|x) = g-1
college thresholds be (s1,s2) = (0.5,-0.5). Then G(s
0(x) = 1-a1(x) and G(s
2|x) = g-1(x) = 1-a2(x). Rearranging yields a1
2|ξ(s 2|x). Invertinga1(x)andrecallingthata2
=h(a1
21|x)anda2
,a1
(x)+ g(x) =
1-G(s(x)=1-G(s), weobtaina=h(a1)=1-G(s1)) ,therebyshowingthathistheacceptance
29
functionconsistentwiththissignaldistributionandthresholds. Step 2: A Continuous Signal
Density. To create an atomless signal distrii={-1,0,1}gibution,wesmooththeatomsusingacarefullychosenkernelfunction.
Defineg(s|x)= (x)k(x,s-i)andletk(x,s)=1(|s|<0.5)(1+2sw(x)),wheretheweightingfunction
w(x)hasrange[0,1]. Thetransformationismass-preservingsinceasw(x)
transfersmasstoapoints>0itremovesthecorrespondingmassfrom-s.
Theweightingfunctiondeterminestheshapeofthesmoothing,sowenowfindconditionsonw(
x) suchthatthestrictMLRPholds. Considers0<s∈(-1.5,1.5),andsupposefirstthat
theyarebothclosetothesameatomiinthat|sj1-i|<0.5forj=0,1. Then
g(s1|x)
gi(x)(1+2(s1-i)w(x))
1+2(s1-i)w(x)
g(s0|x) = gi(x)(1+2(s0-i)w(x)) =
1+2(s0-i)w(x)
whichwillbestrictlyincreasinginxifw(x)isastrictlyincreasingfunction. Imposing
thisrestrictiononw(x),iftheyinheritmassfrompointsi<j,wehave
g(s1|x)
gk(x)(1+2(s-k)w(x))
g(s0|x) = gi1(x)(1+2(s0-i)w(x))
Asufficientconditionforthistobestrictlyincreasinginxalmosteverywhereisthatit isweakly
increasingwhens1-k=-0.5ands0-i=0.5(i.e.gki(x) g(x) (1-w(x)) (1+w(x))gkincreasing ∀k>i∀x). Sincei(x)
g(x)is strictly increasing, choosing a strictly increasing w(x) with
appropriatelyboundedderivativesachievesthis.
A.4 MonotoneStudentStrategies
Claim 1 Student behavior is monotone in caliber if (a) college 2 has payoff u=0.5,
and(b)college2imposesalowenoughadmissionsstandardrelativetocollege1.
By part (a), if a student applies to college 1, then any better student does too. By
part(b),ifastudentappliestocollege2,thenanyworsestudentappliesthereornowhere.
Theproofproceedsasfollows. Wefirstshowthat(i)u=0.5impliesthatifacaliber
appliestocollege1,thenanyhighercaliberappliesaswell. Second,we(ii)producea
sufficientconditionthatensuresthattheadmissionsthresholdatcollege2issufficiently lower
than that of college 1, so that if a caliber applies to college 2, then any lower
caliberwhoappliestocollegesendsanapplicationtocollege2,andcalibersatthelower
tailapplynowhere. Fromthesetworesults,monotonestudentbehaviorensues.
30
ProofofPart(i),Step1. Wefirstshowthattheacceptancefunctiona2
1=(1/u)(1-c/a1)(i.e., MB12
1=a1(1-a2
=ψ(a1
=1,(ii) MB12
2
[(1-ψ(a1
1
)u)=c.
=0andends ata1
1
) crosses au)=c)onlyoncewhenu=0.5. Since (i)theacceptance functionstartsata=c
starts at a=cand ends at a=c/(1-u), and (iii) both functions are continuous,
thereexistsacrossingpoint. Andtheyintersectwhena1(1-ψ(a1
)u)a1]'=1-uψ(a1)-a1uψ'(a1)>1-uψ(a1)-uψ(a1)=1-2uψ(a
)=1-2u=0,
falling in a1 (Theorem 1), i.e. ψ'(a1
where the first inequality exploits ψ(a1)/a1
12
1)/a1;thenexttwoinequalitiesuseψ(a1
whentheacceptancerelationhitsa2
1
1(x)
<a1(y) and a2(x) <a2
2(x)/a1
2
=(1-c/a1
2(y)/a1(y),andthusa2(y)u=a1
(x)u>a1
2
12
1,¯a2)=
=c/
u(1a1),i
.e.
MB
u(1
1-4c/u)/2
—pointPinthe
1,¯a2)if
21
)<
ψ(a)=1andu=0.5. SinceMBisrisingin a)/u,theintersectionisunique. Proof of Part (i), Step
2. We now show that Step 1 implies the following
singlecrossingpropertyintermsofx: ifcaliberxappliestocollege1(i.e.,if1∈S(x), then any
caliber y >xalso applies to college 1 (i.e., 1 ∈ S(y)). Suppose not; i.e., assume that
either S(y) = Ø or S(y) = {2}. If S(y) = Ø, then S(x) = Ø as well, as a(y), contradicting the
hypothesis that 1 ∈ S(x). If S(y)={2}, then there are two cases: S(x)={1}or S(x)={1,2}.
The first cannot occur, forby Theorem1a(x)>a(y)implies a(x),contradictingS(x)={1}.
Inturn,thesecondcaseisruledoutbythe
monotonicityofMBderivedabove,ascaliberyhasgreaterincentivesthanxtoadd
college1toitsportfolio,andthusS(y)={2}cannotbeoptimal.
ProofofPart(ii),Step1. Wefirstshowthatiftheacceptancefunctionpasses
abovethepoint(¯a1-4c/u)/2,(1 rightpanelofFigure6—thenthereisauniquecrossingofthea
cceptancefunctionand a=c. Now,theacceptancefunctionpassesabove(¯a
ψ(¯a1, 1,s 2)= ¯a2. (14)
s
1 ands 2. Rewrite(14)usingTheorem1ass
2 =η(s 1)<s
1
2
Thisconditionrelatess
, requiringalargeenough“wedge”betweenthestandardsofthetwocolleges.
Toshowthat(14)impliesauniquecrossing,considerthesecantofa
=c/u(1-a1
2 =c). Ithasanincreasingsecantifandonlyifa1
1
2
1
l
= c/ua1(1-a1) in a1. Notice also that MB
1l 2)=(1/2 1-4c/u/2,1/2
2
h 1,ah
2
) (thecurveMB/a=1/2. Toseethis, differentiate a= cintersects the
diagonala=a1atthepoints(a,a211-4c/u/2)and (a)=(1/2+ 1-4c/u/2,1/2+ 1-4c/u/2)>(
1/2,1/2).31
Condition(14)givesψ(al 1,s 1,s 2)>al 2. S inces 2 <s 1,wehaveψ(a1,s 1,s 2)=a2
=catorabove(ah 1
h=
1 ¯a1. Thus,theacceptancefunctioncrossesMB21
foralla1
1(x)<a2(y)/a1
21
,ah
2
2
).
Andsincea>1/2,thesecantofMB=cmustbeincreasingatanyintersectionwith
theacceptancefunction. Hence,theremustbeasinglecrossingpoint.
ProofofPart(ii),Step2. Wenowshowthatthissinglecrossingpropertyina
impliesanotherinx: Ifcaliberxappliestocollege2(i.e.,if2∈S(x)),thenanycaliber y <xthat
applies somewhere also applies to college 2 (i.e., 2 ∈ S(y) if S(y) =Ø). Suppose not; i.e.,
assume that S(y)={1}. Then there are two cases: S(x) = {2}or S(x)={1,2}.
Thefirstcannotoccur,forbyTheorem1a2(x)/a(y),and
thusa(x)u=a1(x)impliesa2(y)u>a1(y),contradictingS(x)={2}. Thesecondcase
isruledoutbythemonotonicityofMBgivencondition(14),ascaliberyhasgreater
incentivesthanxtoapplytocollege2,andthusS(y)={1}cannotbeoptimal. Finally,
S(y)=Øifa221(y)u<cby(14),whichhappensforlowcalibersbelowathreshold.
A.5 TheLaw of Demand
Claim 2 (TheFalling Demand Curve) Ifeithercollegeraisesits
admissionstandard,thenitsenrollmentfalls,andthusitsrival’senrollmentrises.
1 rises, since the argument for s 2
1
1
32
2
2,thena2(x)u-c=0anda
(x)u=a1
1
2
1
1,sincea1
We only consider the case when is similar. Also, we focus onthe nontrivial portfolioeffect in each case
s
standard effect ofan increaseinsisimmediate:
itlowersenrollmentincollege1,andraisesitincollege2. Proof Step1: The
at college 1 Shrinks. Whensrises, the acceptance relation shifts up b
thus the above type sets change as well. Fixacaliberx∈Corx∈F,sotha
/∈S(x).Wewillshowthatxcontinues
toapplyeithertocollege2onlyornowhere,andthusthepoolofapplicantsat
shrinksbecausea(x)declines. Ifx∈C2(x)fallswhilea(x),and
thiscontinuestoholdaftertheincreaseins(x)isconstant.
Andifx∈F,thenclearlycaliberxwillcontinuetoapplynowherewhensincrea
Step 2: The applicant pool at college 2 expands. It suffices to
showthatifanycaliberxthatappliestocollege2ats
2
1
1alsoappliesthereatahighers
2,
1
then a2
1
2(x)u=a1
32
32
. Fix
a caliber x∈Cor x∈B, so that2 ∈S(x). If x∈C(x)u-c=0 and
a(x);theseinequalitiescontinuetoholdaftersrises,sincea(x)fallswhile
Withaslightabuseofnotation,weletFdenotethesetofcalibersthatapplynowhere. Thesame
symbolwaspreviouslyusedtodenotetheanalogoussetina-space.
S2
S
E
1 1
L
2
S
1
Κ
1
S2
S1
S2 u Κ2
S1 L Κ2 S1 Κ1
1
2
E
Κ
2
S1
S1 u Κ1
>¯κ1(κ2 2(x)urises in s
1
2
1
1
a2(x) remains constant. And if x∈ B, then MB21
satisfyinglimc→0 ¯κ1(κ2,c)=1-κ2
1suchthatifκ
>¯κ1(κ2
(κ2,c)<1-κ2
1
1
1(κ2
2
∈(0,1),andletsL 1(κ2)be theuniquesolutiontoκ2
Fixanyκ2
2
1
(
κ
2(8,-8)=1,andE2(s
1(κ2)=E1
1(κ2
)
c
o
n
=E2(s
L 1(κ2
1
v o1-κ2
e
r 33
g
e
s
t
2
1
1,-8)increasingandcontinuousi
2
ns
(
sL
1
(
κ2
=E2(sL 1(κ2),-8),¯κ1(κ2
Figure8: Equilibrium Existence. Intheleftpanel,sinceκ1
= (1-a
),thebestresponse functionsSandS2donotinterse
Therightpanel depictstheproofofClaim3fortheca
(x))a
rises.
1
1
1
,c). We will choose the capacity ¯κ
given κ2
1
(
κ
2
,
encouragingcaliberxtoapplytocollege2. Sox/∈C∪Fevenafters
A.6 Existence of aRobustStable Equilibrium
Claim 3 (Existence) A robust stable equilibrium exists. College 1 fills its capacity.
Also,thereexists¯κ= ¯κ,c),thencollege2alsofillsitscapacityinanyrobustequilibrium.
Ifκ,c), thencollege2hasexcesscapacityinsomerobustequilibrium.
Proof: Fordefiniteness,wenowdenotetheinfimum(supremum)signalby-8(8). Definition
of ¯κso that when college 2 has no standards, both colleges exactly fill their
capacity. This borderline capacityislessthan1-κsinceapositivemass ofstudents
—perversely, thosewith
thehighestcalibers—appliesjusttocollege1,andsomearerejected.
,-8),i.e.,
when college 2 accepts everybody. (Existence and uniqueness of s) follows from
E(-8,-8)=0,E.) Definethethresholdcapacity ¯κ),-8). Li mitingBehaviorof ¯κ,c).
Sinceκ)equals1-κ
minus themassofstudents whoonlyappliedto,andwererejectedby, college1. This
massvanishesascvanishes,fortheneverybodyappliestobothcolleges. Therefore,the
threshold¯κ2ascgoestozero.
Robust Stable Equilibrium with Excess Capacity. Let κ1
l 1(κ1),
l 1(κ1)
theunique solution to κ1
=
¯κ1(κ2
=E1(s
l 1(κ1
1
= sL 1(κ2
L 1(κ2)an
1,s
dE2(s
(κ2
(-8,κ2
)<8,theuniquesolutiontoκ2
andS2
(-8,κ2
),w
hile
the
rec
anb
eex
ces
sca
pac
itya
tcol
leg
e2if
κ1
).
We
claimthatthereexistsarobuststableequilibriuminwhichcollege2acceptseverybody,
andcollege 1sets athreshold s,-8), which satisfies s). For since college 2 rejects
no one, s) fills college 1’s capacityexactly.
Theenrollmentatcollege2isthenE),-8)=κ)=
s2)isincreasingins1),sobyacceptingeverybodycollege2fillsas
muchcapacityasitcan. Thisrobustequilibriumistriviallystable,asSis‘flat’atthe
crossing point (see Figure 8, left panel). Moreover, if κ1>¯κ12), then college 2
has excesscapacityinthisequilibrium.
Robust Stable Equilibrium without Excess Capacity. Assume nowκ< ¯κ1(κ2).
WewillshowthatthecontinuousfunctionsS1andS2mustcrossatleastonce (i.e., a
robust equilibrium exists), and that the slope condition is met (i.e., it is stable).
First,inthiscasesL 1(κ2)<sl 1(κ1)or,equivalently, S-1 2)<S). Second,
asthestandardofcollege2goestoinfinity, college 1’sthresholdconverges to su
1(κ1)<8,theuniquesolutiontoκ1=E1(s1,8). Thisisthelargestthresholdthat college
1can set given κ1. Similarly, as the standard ofcollege 1goes to infinity,
college2’sthresholdconvergestosu 2(κ2),i.e.
thelargestthresholdthatcollege2cansetgivenκ2. Third,S1=Earecontinuous.
BytheIntermediateValueTheorem,theymustcrossatleastoncewiththeslopeconditio
nbeingsatisfied(seeFigure8,rightpanel). Thus,arobuststableequilibriumexists
when κ1<¯κ1(κ2). Moreover, inany robust equilibrium there isnoexcess capacity
at eithercollege,sincethebestresponsefunctionssatisfyS-1 2)<S1(-8,κ). Hence, a
robust stable equilibrium exists for any κ21∈ (0,1). Capacities are filled
whenκ1=¯κ1(κ2>¯κ1(κ2).
A.7 Stochastic Dominance in RobustSorting Equilibria
Claim 4 (Sortingand the Caliber Distribution) Inanyrobustsortingequilibrium,
thecaliberdistributionatcollege1first-orderstochasticallydominatesthatatcollege2.
Proof Step 1: Monotone Student Strategy Representation.
Amonotonestudentstrategyisrepresentedbythepartitionofthesetoftypes:
=[ξ2,ξB),B=[ξB,ξ1),C1
=[ξ1,8) (15)
34
F=[0,ξ2),C2
ξB
ξ
<ξB
2
2
<ξ1
2
1=c/[u(1-a1)](i.e.,MB21
=c),anda2
1
2.
Letf1
ands
)/u(i.e.,MB12
f2
B8
B
ξ[ξ1aB](x)f(x)
2
ξ
1
whereξ2 aredefinedbytheintersectionoftheacceptancefunctionwithc/u,
a=(1-c/a=c),respectively. Proof Step 2: Enrolled Caliber Densities. Fixs(x)and
f(x)bethedensitiesofcalibersenrolledatcolleges1and2,respectively. Formally,
2
1 (t)f(t)dt I[ξ
(x) = a
,8)(x) (1 6)
f1
2,ξ (x)a2(x)f(x)+(1-I[ξ2,ξB](x))a2(x)(1-aa(x))f(x) (s)f(s)ds+ ξ1
,ξ ](x),(17)
(x) = I
Ba2(s)(1-a11(s))f(s)ds I[ξ
H
H
i
,thenf1(xH)f2(xL)=f2(xH)f1(xL);i.e.,fi
(x )f (x )isequivalentto
whereIA
>xL istheindicatorfunctionofthesetA. ProofStep3: (x). Weshallshowthat,ifx
Log-Supermodularityoff
L
∈ [0,8),withx(x)islog-supermodular in (-i,x), oritsatisfies MLRP. The result follows
asMLRP implies thatthe cdfs are orderedbyfirst-orderstochasticdominance.
2 L
2 H
1 L
Using(16)and(17),f
(x
2L
a1HI[ξB,8)(xH ) I[ξ2,ξB](x L)a +(1-I[ξ2,ξB](xL))a2L(1-a1L) I[ξ2,ξ1](x)= a1LI[ξB,8)(xLI2BH2BH21LH), (18)
1
H )f
(x )=f
whereaij
H =ai(x
)
j
,ξ
[ξ ,ξ ](x
B
1
1
)a2H +(1-I[ξ
,ξ ](x
1H
))a2H (1-a1H
1L
a2L
)
1L
I[ξ,ξ ](x
a (1-a1H
,x
),or
H
),i=1,2,j=L,H. Itiseasytoshowthattheonlynontrivialcaseis
whenxL,x∈[ξB](inalltheothercases,eitherbothsidesarezero,oronlytheright sideis).
IfxL,x∈[ξ,ξ],then(18)becomesa(1-a)=a2H
2 |xL))G(s 1 |xL ). (19)
1 |xH ))(1-G(s
2
1
1
|xH
|xL
|xH))G(s
|xSince g(s| x) satisfies MLRP, it follows that G(s| x) is decreasing in x, and hence 1
|xH 1
G(sL)=G(s
1 |xL))(1-G(s
2 |xH )),
(1-G(s
|xH 1
))(1-G(s 2 |xL))=(1-G(s
2
1
ass >s
inarobustsortingequilibrium. Thus,(19)holds,therebyprovingthatf
first-orderstochasticallydominatesF2.
1
(
-G(s
1
). Next,
)
)
(1-G(s)= (1-G(s
i
islog-supermodularin(-i,x),andsoF
35
(x)
A.8 Comparative Statics: Changing Application Costs
Claim 5 Inarobuststablesortingequilibrium,iftheapplicationcostateithercollege
rises,thenatcollege1theadmissionstandardfallsandthecaliberdistributionstochasticallyimp
roves.
Proof: Assumearobuststablesortingequilibrium. Wemodify(15)fordifferentcosts:
ξ1isdefinedbyMB21=c2,ξBbyMB12=c1,andξ2bya2u=c. Step 1: An Increase in College 2’s
Cost. Ifc22rises,thenξ1drops,ξrises, and ξB2is unchanged; thus, the applicant
poolatcollege 2shrinks, andat college 1is unchanged. Sothe S2curve shifts down, while
S1remains unchanged. The functions
nowcrossatalowerthresholdpair,andsobothstandardss1,s2bothfall. Step 2: An Increase
in College 1’s Cost. Next consider an increase in c.
ThisraisesξB1,whichshrinkstheapplicantpoolatcollege1,andincreasestheenrollment at
college 2, at a fixed admission standard. This shifts S1left and Sup. While the
effectonthestandards2isambiguous,wenowdeducethats12falls. Differentiating(4)
and(5)withrespecttoc1,andusingCramer’sRule:
2
2)-(∂E1/∂c1)(∂E2/∂s
1 2)-(dE2/ds
1/dc1
1/
1)(dE1/ds 2
d
i=2,B,1
2
s
2(ξB|c
1
2
2
B
∂s 11 = (∂E2/∂c1) (∂E1/∂s ) (dE1)(dE2/ds) (20)
∂c
Sincetherobustequilibriumisstable,theslopeofS
issteeperthatofS
2 2 P2(ξi|s 2)-S2(C2)-S
i
=S
=P1(ξB|c1)<0,
dE2/ds
B ) >0, and dE1/ds
/
d
c
B)a1a2
1
'
2=
P1(ξB|s
slightly rises, then ξ
2)equals
B B|c1)P2(ξB|s
2)
>0. If c1
B|c1)-P1
,andthusthe denominatorispositive. LetP(ξ|y)betheportfoliodensityshifttocollegeiattypeξ
givenanincrementtostandardorcosty,andletS(A)betheown-standardseffectat
college2insetA. Thenparsetheenrollmentderivativesintotheportfolioandstandards
effects: dE(B)<0, dE= P
2
risesbysomed>0. Thus,college1losesmassf(ξ
)a1
B)a1a
2d'.
falls by some d
Thus,P1(ξB|s
2)P2(ξ
B)a1d'
1 2d]-[f(ξB)a1d][f(ξB)a1a2d']=0
[f(ξB)a1d'][f(ξB
dofstudents,andcollege2gains massf(ξdofstudentswhowouldhavegonetocollege1.
Likewise,ifsslightly rises, then ξ, and college 1 gains mass f(ξand college 2 loses
massf(ξ(ξ
)a a
Hence,thenumeratorin(20)reducesto
2)+P(ξ1|s
2)-S2(C2)-S2(B)]<0
36
-P1(ξB|c1)[P2(ξ2|s
St
e
p
3:
I
m
pr
o
v
e
m
e
nt
in
C
ol
le
g
e
1’
s
C
al
ib
er
Di
st
ri
b
ut
io
n.
In
a
ro
b
u
st
s
or
ti
n
g
e
q
ui
li
br
iu
m
,t
h
e
a
p
pl
ic
a
nt
p
o
ol
at
c
ol
le
g
e
1
c
o
n
si
st
s
of
c
al
ib
er
sx
∈[
ξ,
8)
.
Fr
o
m
th
el
a
st
p
ar
t,
a
n
yc
o
sti
n
cr
e
a
s
e
d
e
pr
e
ss
e
s
s1
Bi
n
e
q
ui
li
br
iu
m
.
Itf
ol
lo
w
st
h
at
ξr
is
e
s
in
e
q
ui
li
br
iu
m
—
si
n
c
e
c
ol
le
g
e
1
h
a
s
th
e
s
a
m
e
c
a
p
a
cit
y
a
s
b
ef
or
e,
if
it
is
to
h
a
v
e
lo
w
er
st
a
n
d
ar
d
s,i
t
m
u
st
al
s
o
h
a
v
ef
e
w
er
a
p
pl
ic
a
nt
s.
L
et
(ξ
0
B,
s0
1B
)b
et
h
e
ol
d
e
q
ui
li
br
iu
m
p
ai
r
a
n
d(
ξ1
B,
s1
1)t
h
e
n
e
w
o
n
e,
wi
th
ξ0
B
<
ξ1
B
a
n
d
s0
1>
s1
1.
T
h
e
nt
h
e
di
st
ri
b
ut
io
nf
u
n
cti
o
n
of
e
nr
ol
le
d
st
u
d
e
nt
s
at
c
ol
le
g
e
1
u
n
d
er
e
q
ui
li
br
iu
m
i=
0,
1i
s:
We must show F1
1
(x) = F0 1
8 ξi B(1-G(s|t))f(t)dt (x) for all x ∈ [ξi 11 B|t))f(t)dt,8). For any x, the denominators on both
sides equal k1, so cancel them. Now notice that 0 = F1 1(ξ1 B) < F0 1(ξ1 B) and limx→8F1
1(x) = limx→8F0 1(x) = 1. Since both functions are continuous in x, if ∂F1 1/∂x > ∂F0 1/∂x
for all x ∈ [ξ1 B,8), then F1 1(x) < F0 1(x). But this requires (1-G(s1 1|x))f(x)>(1-G(s0
1|x))f(x),whichfollowsfroms1 1<s0 1.
A.9 Sorting and Non-Sorting Equilibria: Proof of Theorem 2
Part(a): College2isTooGood. Weconstructacceptancechances(a1(x),h(a(x))) such that
student behavior is non-monotone, college enrollment equals capacity, and a 1(x) and
h((a11(x)) obey the requirements of Theorem 1. Then that theorem yields
existenceofasignaldistributionwithnon-monotoneequilibriumstudentbehavior.
Step1: TheAcceptanceFunctionandStudentBehavior. Whenu>0.5,
thesecantfromtheorigintoMB12=cfallsasatendstoc/(1-u)—asintheleft panel of Figure 6.
So for some z <c/(1-u), a line from the origin to (z,1) slices the MB121curve twice. Let
h:[0,1]→[0,1] be defined by h(a1)=a/zand on [0,z), andh(a1)=1fora11=z. Observe
thath(0)=0andh(1)=1,andthathisweakly
increasing,withbothh(a1)/a1and[1-h(a1)]/[1-a]weaklydecreasing,sohobeysthe
doublesecant property. Nowconsider student behavior. Movingalongtheacceptance
relation h, students with acceptance chance a11<(c¯z)/uwill apply nowhere; those in
the interval [(c¯z)/u,a ) will apply to college 2 only; those between [a,a] will apply to
both;thosein(a,c/(1-u))willapplytocollege2only,thosein[c/(1-u),1-(c/u)]
willapplytoboth,andthoseabove1-(c/u)willapplytocollege1;wherea,aarethe
pairofintersectionswiththeMB12curve. Hence,studentbehaviorisnon-monotone.
37
(1
S2
S2
4
1
1
1'
Η S1
25'
°
E1
E0
E0
2
2
E1
S1
S1
Figure 9: Existence of Sorting and Non-Sorting Robust Equilibria. We depict
theproofofTheorem2. Intheleftpanel,whenκ2falls,S2shiftsuptoS' 2,inducingarobust
non-sorting equilibrium at E1. In the right panel, when κ1falls, S1shifts right to S' 1. The
standardsatthenewrobustequilibriumE1obeys2<η(s1),i.e.,theequilibriumissorting.
Step2: Transformingthetypespace. Noticethatifonechoosesamonotone
functiona1(x),thenweinduceadistributionG(a)=P(a(X)<a)(i.e.,theprobability
thatarandomstudentXhasacceptance chanceatcollege1less thana). Conversely,
we can obtain any distribution G(a) by choosing a11(x) = G-1(F(x)). The resulting
functiona1(x)thusobtainedwillbesmooth,monotoneandontoprovidedthatGhasa
continuousandstrictlypositivedensityover[0,1].
Therefore,wecanrestateourproblem
aschoosingaGwiththesepropertiesandsuchthattheenrollmentequationshold.
aaadG(a)+ 1c 1-uadG(a)=κ1Step 3: Enrollment.
Theenrollmentequationsaregivenby:
a a ¯zdG(a)+ aa(1-a) a¯z
a¯z a
dG(a)+ c 1- c u
c¯z u
(1-a)dG(a)=κ2.
dG(a)+
¯z¯zdG(a)+ c1-u
1
u
WenowdecoupletheequationsbylettingG( a)-G(a)=G(1-(c/u))-G(c/(1-u))= ofor
some small o(e.g. by using a uniform density). It is clear that the resulting
individual integral equations have an infinite number of solutions for G(a) on
those regions. Choosingoneofthemcompletestheproof.
ForthenTheorem1yieldsasignal
densityg(s|x)andthresholdss1>s2suchthath(a1(x))istheacceptancefunction. Part
(b): College 2 is Too Small. The proof is constructive. Consider
=a1/u. Theacceptancefunctionevaluatedata1
ψ(c,s 1,s 2)<c/u. (21)
(a1,a2)=(c,c/u)onthelinea2
=c liesbelowc/uwhen
38
Wewillrestrictattentiontopairs(s
1,s 2
1,s 2)=
1). Thenenrollmentatcollege1isgivenby
8
1
E1(s
)suchthat(21)holds.
Inthiscase,anystudent who applies to
college starts by adding college 1 to his
portfolio, and this happens as
soonasa(x)=c,orwhenx=ξ(c,s
(1-G(s 1|x))f(x)dx,
1)
ξ(c,s
. Thus, for any capacity κ∈ (0,1), a unique threshold s1(κ1)solves κ1=E1(s1,s
which is independent of s 11functionis“vertical” when (21)holds, since
theapplicantpoolatcollege1doesnotdependoncollege2’sadmissionsthreshold
Theanalysisaboveallowsustorestrictattentiontofindingrobustequilibriawithin
thesetof(s1,s2)suchthats1=s1(κ1)ands2satisfiesψ(c,s1(κ1),s2)<c/u.
Enrollmentatcollege2isgivenby
E2( 1(κ1),s 2)= B G(s 1(κ1)|x)(1-G(s 2|x))f(x)dx,
s
1 (seeClaim2). Thus, κ2
2,andincreasingins
2(s
2
),κ2
1strictly falls in κ2, for any κ2
2 = s <¯κ2(κ1
1, let ¯κ2(κ1) = E2(s 33
),s 1(κ1
1(κ1
). Then (i) for any κ1
2
2
2
1(κ1
1(κ1)
2(κ1
Fix κ2
and s
1
1
2
1
,s
33
2)thatsatisfiesκ1
=E1(s
1,s 2),κ2
1
2
2/∂s
=E2(s
1,s 2),ands
2
<η(s
1
1
whichiscontinuous, decreasingins = E1(κ1),s2)yieldss2=S2(s1(κ1),whichisstrictlydecreasinginκ. Given κ1
equilibrium ensues if both colleges set the same threshold.Since S>
equilibrium exists with s∈ (0,1) and having(ii)non-monotonecollegeandstudentbehavior(Figure9,left). Pa
κ∈ (0,¯κ)], there is a unique robust Conditions for Equilibrium Sorting. We prove that there exists
equilibrium with s= s234),
=
(κ2
),smallerthanthebound¯κ1(κ2
1 ) >0 such that if κ1
κ
(κ2
1(κ
1(κ2
∂S
) and u= 0.5, then there are only
κ robust sorting
equilibriaandneithercollegehasexcesscapacity.
∈ (0,1). We first show that the robust stable equilibrium with no excess
capacityderivedinClaim3isalsosortingwhenthecapacityofcollege1issmallenough.
Moreprecisely,thereisathresholdκ)definedinthe
proofofClaim3,suchthatforallκ∈(0,κ)),thereisapairofadmissionsthresholds
(s/∂s)(i.e.,arobust sortingequilibrium),and∂S<1(i.e.,therobustequilibriumisstable).
The proof uses three easily-verified properties of the ηfunction implicitly defined
2 =s 1(κ1).
1,s 2)<c/uissatisfiedifs
Itisnotdifficulttoshowthatψ(c,s
34 Wearenotrulingouttheexistenceofanotherrobustequilibriumthatdoesnotsatisfy(21).
39
2
=η(s
1)→
8 as s
1
2
1,s
1
1
1
=E1(s 1,s 2)andκ2 =E2(s 1,s 2),with(∂S1/∂s
2)(∂S2/∂s 1)<1.
by inequality (14): (a) ηis strictly increasing; (b) s → 8; (c) s=η-1(s2)→-8ass→-8.
Foranyκ∈(0,¯κ1(κ22)),weknowfromClaim3thattheree
Claim 6 Thepair(s
1,s 2)isarobustsortingequilibriumwhenκ1
issufficientlysmall.
21
1
)={(s
1,s 2)|κ2
2(s =E2(s 1,s
2(s 2) and s 2 (s =η(s =η(s 1
Proof: LetM(κ
2
2
=E1
-1
1)foralls
2
2
1
2
2
1
1,κ
1(
s 1(κ ),ss 2
(s
,κ2
1
1
1
u2
(κ2)=¯κ1(κ
2
2
)}. Graphically, thisis
thesetofallpairsatwhichs=S1,κ)crossess2). IfM(κ2)=Øwe
aredone,forthens=S1,κ1)<η(s,including those at which κ(s,s) and κ2=E(s1,s2). To
see this, note that (i) s-1= η(s2) → -8 as s2→ -8, while we proved in Claim 3 that s=
S) convergestosl 1(κ2)>-8. Also,(ii)s2=η(s1)→8ass1→8,whileweprovedin
Claim3thats2=S2(s2)converges tosu 2(κ)<8. Pr operties(i)and(ii)reveal
thatifS2andηdonotintersect,thenS22iseverywherebelowη. IfM(κ2)=Ø,
let(s2(κ2))=supM(κ), which isfinite by property (b)of η(s1)and since
s2=S21)converges tosu 22goestoinfinity (see theproofofClaim3).
Now,asκgoestozero,s(κ12)<8 ass=S2,κ1)goestoinfinityforany
valueofs,forcollege1becomesincreasinglymoreselectivetofillitsdwindlingcapacity.
Sinces22isboundedabovebys(κ2),thereexistsathresholdκ1)suchthat,
forallκ1∈(0,κ1(κ2)),theaforementionedpair(s1,s)thatsatisfiesκ2=E)
andκ2=E2(s1,s2)isstrictlybiggerthan(ss 1(κ2),ss 2(κ)),therebyshowingthatitalso
satisfies s2<η(s1). Hence, arobust sorting stable equilibrium exists forany κand
κ1∈(0,κ1(κ22)),withbothcollegesfillingtheircapacities(seeFigure9,rightpanel).
Tofinishtheproof,noticethat,iftherearemultiplerobustequilibria,bothcolleges
filltheircapacityinallofthem(graphically,theconditionsoncapacitiesensurethatS 2
1
andS2 intersectareallbelowη).
startsaboveS1 forlowvaluesofs
1(κ 2
1
1
40
andeventuallyendsbelowit). Moreover,adjusting
theboundκ)downwardifneeded,allrobustequilibriaaresorting(graphically,for
κsufficientlysmall,thesetofpairsatwhichS
1
A.10 Affiliated Evaluations
Wenowshowthatifstudentsandcollegesseenoisyconditionallyiidsignalsofcalibers,
thenthisisformallyaspecialcaseof(⋆).
Lettbeastudent’s truecaliber, unknown tohimandcolleges. Ithasdensity p(t)
on[0,1]. AfterseeingthesignalrealizationX=x,drawnwithtype-dependentdensity
f(x|t),thestudentupdateshisbeliefstop(t|x)=f(x|t)p(t)/f(x). Ifastudentofcaliber
tappliestoacollege,thecollegeobservesasignalsdrawnwithdensityγ(s|t)andcdf
G(s|t)on[0,1]. Ifastudentappliestobothcolleges,thentheyobserveconditionallyiid signals.
Weassumethatf(x|t)andγ(s|t)obeythestrictMLRP.
1 0|x)=
γ(s1|t)γ(s2
Definetheconditionaljointdensityofsignalsg(s1,s2
10
(1-G(si|t))p(t|x)dt=
ai(x)=
0
1 s1i
γ(si|t)p(t|x)dsidt=
s1i
0 1
g(si,sj
|x)dsjdsi
01
|t)p(t|x)dt.
Noticethatgintegratesto1,andsoisavaliddensity. Also,asanintegralofproducts
oflog-supermodularfunctions,itinheritsthisproperty,byKarlinandRinott(1980). In
otherwords, thesignalsareaffiliated. Next, definethedensity f(x)=f(x|t)p(t)dt.
Wenowreinterpretthesignalxintheconditionaliidcaseasthestudenttruecaliber. To show
that this model is a special case of (⋆), we prove that student and college
optimizationshavethesamesolutionsasintheconditionaliidcase.
StudentBehavior. Itsufficestoexpressthechancesoftwoacceptanceeventsfor the
conditionaliidmodel without reference tothe type t, andthus as intheaffiliated model(⋆).
First,theunconditionalacceptancechanceatcollegei=1,2is
= 001G(s1|t)(1-G(s2Next,theprobabilityofbeingrejectedat1andacceptedat2is |t))p(t|x
)dt = 01s 10s s112 s12g(sγ(s1,s22|t)γ(s|x)ds11|t)p(t|x)sds2.1ds2dt
CollegeBehavior. Itlikewisesufficestoexpresstheenrollmentfunctionswithout
referencetothestudenttypet. Forinstance,forcollege1,
E1( 1,s 2) = 01
C1∪B f(x|t)dx
s11 γ(s1|t)ds1 p(t)dt
s
1 0 1
∪B
f(x)dx.
g(s1,s2|x)ds2ds1
=
C1
s1
Theanalysisofcollege2isanalogousandthusomitted.
41
= C1 ∪B s11 0 1
|t)f(x|t)p(t)dt
ds1dx
γ(s1
A
.1
1
A
ffi
r
m
at
iv
e
A
ct
io
n:
P
r
o
of
of
T
h
e
o
r
e
m
3
St
e
p
1:
E
q
ui
li
br
iu
m
C
o
n
di
ti
o
n
s.
H
er
e,
w
e
a
ss
u
m
e
a
c
o
nt
in
u
o
u
ss
ig
n
al
c
df
d
er
iv
at
iv
e
G
x.
Gi
v
e
n
a
n
y
di
sc
o
u
nt
p
ai
r(
∆1
,∆
2),
th
e
c
a
p
a
cit
y
e
q
u
at
io
n
s
wi
th
tw
o
gr
o
u
p
s
ar
e:
= fEt 1 (s 1 -∆1,s
2
= fEt 2 (s
2
1
-∆1,s
2
1:
-∆2)+(1-f)EN 1
(s 1,s
-∆2)+(1-f)EN 2
(s 1,s
2
2),
(23)
x
s
J ∂∆
1
f
κ1
) (22) κ
w
her
aretherespectivefractionsoftargetedandnon-targetedgroupsenrolledat
eEt
collegei,definedjustasin(4)and(5),forthesetsofsignals(15). Sincethesignaldensity g =
i,EN
i Gand its derivative Gare both continuous, all derivatives of the enrollment
function(usingLeibnitzrule)arecontinuoustoo.
Step 2: Single College Preference Case. Differentiating equations (22)
and(23)withrespectto∆
= i=1,2(-1)i+1 1∂
∂( 1∂Et i ) f ∂E∂(s2t 3-i-∆2) +(1-f)
∂s ∂EN 3-i
s
s
-∆1
2
2 =
∂ ∂ J
∂ ) f ∂E∂(s1t 3-i-∆1) +(1-f) -∆1
wherethedenominator,fromCramer’sRule,equals
∆
s
E
1
t
f
(
i
1
)i
=
1
,
2
i
1∂(s
∂
s
2
f
∂(
s
1∂Et 2
) +(1-f)
1
∂E
N1
-∆1
J = f ∂E∂(s 1t 1-∆f ∂E∂(s2t 11)
+(1-f)-∆2∂EN 1∂s) +(1-f)1
∂(s
is
p
o
sit
iv
ei
n
a
n
yr
o
b
u
st
st
a
bl
e
e
q
ui
li
br
iu
m
—
i.
e.
th
et
w
o
gr
o
u
p
v
er
si
o
n
of
th
e
2
-∆
∂s ∂EN 2
∂s
2
∂Et 2
2
f
) +(1-f)
∂EN
2
c
o
n
di
ti
o
n
th
at
th
e
sl
o
p
e
of
S
1e
xc
e
e
d
th
e
sl
o
p
e
of
S
2i
n
§
4
a
n
d
§
A.
8.
N
o
w,
∂s
1/
∂
∆
=
f>
0
a
n
d
∂s
2/
∂
∆1
=
0
w
h
e
n
∆1
=
∆2
1=
0,
b
e
c
a
u
s
et
h
e
d
er
iv
at
iv
e
s
of
th
ef
u
n
cti
o
n
Et
i,
E
N
iat
c
ol
le
g
e
si
=
1,
2
c
oi
n
ci
d
e.
T
h
u
s,
th
ef
e
e
d
b
a
ck
e
ff
e
ct
sv
a
ni
s
h
w
h
e
n
∆1
=
∆2
=
0,
a
n
d
ar
e
n
e
gl
ig
ib
le
in
a
n
ei
g
h
b
or
h
o
o
d
of
it,
b
yc
o
nt
in
ui
ty
of
th
e
e
nr
ol
l
m
e
nt
d
er
iv
at
iv
e
s.
Since ∆1 =p1, it follows that s
1)/∂p1|p1=0
1
g
o
e
s
u
1
increases while s1 -∆1
2
1
=
p
2
i
42
decreases when p1
=p=0
p,
a
s∂
(s
-∆
=f
-1
<
0.
T
h
e
a
n
al
ys
is
of
th
e
d
er
iv
at
iv
e
s
of
s,i
=
1,
2,
wi
th
re
s
p
e
ct
to
∆i
s
a
n
al
o
g
o
u
s.
St
e
p
3:
B
ot
h
C
ol
le
g
e
Pr
ef
er
e
n
c
e
C
a
s
e.
S
u
p
p
o
s
e
n
o
wt
h
at
p
= ∆2
i
i
-∆goesdown,therebyprovingtheresult.
43
i
nd thus ∆1
= ∆ = 0.aNow let pincrease. Replacing in the analysis above
∆,i=1,2,by∆anddifferentiating,yields,afterevaluatingtheexpressionat∆=0,
∂s/∂∆=f>0,i=1,2. Asbefores
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