Purchasing seats in school choice and inequality

Wang, Tong; Zhou, Congyi
A icle
Pu chasing sea s in school choice and inequali y
Quan i a i e Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Wang, Tong; Zhou, Congyi (2024) : Pu chasing sea s in school choice and
inequali y, Quan i a i e Economics, ISSN 1759-7331, The Econome ic Socie y, New Ha en, CT, Vol.
15, Iss. 4, pp. 1151-1195,
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Quan i a i e Economics 15 (2024), 1151–1195 1759-7331/20241151
Pu chasing sea s in school choice and inequali y
Tong Wang
College o In e na ional Rela ions, Ri sumeikan Uni e si y
Congyi Zhou
Wil Depa men o Poli ics, New Yo k Uni e si y
We s udy a mechanism ha gi es s uden s he op ion o paying highe ui ion o
a end hei p e e ed schools. This sea -pu chasing mechanism is nei he s a -
egyp oo no s able. Ou pape combines adminis a i e and su ey da a o es i-
ma e s uden s’ p e e ences and conduc s wel a e analysis. We ind ha changing
om a de e ed accep ance mechanism o he cade -op imal s able mechanism
educes s uden s’ wel a e bu ha adop ing he obse ed sea -pu chasing mech-
anism alle ia es his wel a e loss. Mo eo e , s uden s om a luen communi ies
p e e o pay highe ui ion o s ay a p e e ed schools, while hose om less a -
luen communi ies a e mo e likely be p iced ou o lowe -quali y schools.
Keywo ds. School choice, ma ke design, pu chasing sea , inequali y.
JEL classi ica ion. C78, D82, I21, I28.
1. In oduc ion
The analysis o cen alized school choice mechanisms has become a key ocus o e-
sea ch in ma ke design (Abdulkadi oglu and Sönmez (2003)). Kelso J . and C aw o d
(1982)andHa ield, William, and Milg om (2005) ha e buil he connec ion be ween
auc ion and ma ching by in oducing he ma ching wi h con ac s model. Since hen,
analyzing how indi iduals espond o a “p ice menu” o an indi idual good in ma ch-
ing ma ke s has a ac ed g owing in e es . Theo e ical analysis has been used o add ess
his ques ion in p ac ice (Sönmez and Swi ze (2013); Bi o, Hassidim, Romm, Sho e ,
and So ago (2022)). Howe e , no clea empi ical analysis has disen angled indi idual
beha io s unde he ma ching model wi h mone a y ans e .
In ex an li e a u e on he school choice p oblem, he e ec o mone a y ans e s
be ween s uden s and schools is seldom conside ed because public schools ei he o -
e ee educa ion o ha e a ixed (and usually low) ui ion ee. Ye unlike public school
Tong Wang: [email p o ec ed]
Congyi Zhou: [email p o ec ed]
We would like o hank A ila Abdulkadi oglu, Hideo Akabayashi, Daniel Ba on, Dan Black, E ic Budish,
Yinghua He, Kohei Kawamu a, Jacob Leshno, Ha uko Noguchi, Elena P age , Shigehi o Se izawa o help ul
commen s and wo kshop and con e ence pa icipan s a Keio Uni e si y, No hwes e n Kellogg S a egy
Lunch, Osaka Uni e si y, Uni e si y o Chicago Indus ial O ganiza ion Lunch, Uni e si y o Tokyo, Waseda
Uni e si y, AEA annual con e ence, EEA annual con e ence, IIOC con e ence, and APPAM annual con e -
ences.
©2024 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h p://qeconomics.o g.h ps://doi.o g/10.3982/QE2220
1152 Wang and Zhou Quan i a i e Economics 15 (2024)
choice sys ems in o he coun ies, many Chinese ci ies ha e—s a ing in he 1990s—
o e ed s uden s he op ion o paying highe ui ion and he eby gaining admission o
public schools.1This p ocedu e is e e ed o as he Ze Xiao (ZX) policy.2The ZX policy
is a p ac ical applica ion o he ma ching wi h con ac s model (Kelso J . and C aw o d
(1982); Ha ield, William, and Milg om (2008,2010); Ha ield, Komine s, and Wes kamp
(2017)). Howe e , his policy p o oked con o e sy because i was pe cei ed as an un-
ai policy o amilies ha canno a o d highe ui ion (Shen and Wu (2006)). The con-
o e sy las ed o mo e han a decade and was somewha de used in 2012, when he
Minis y o Educa ion announced he es ic ions on he ZX policy and eques ed ha
public high schools s op using i wi hin 3 yea s.3
We exploi a new da a se co e ing high school admissions o he pe iod 2012–
2014 in a la ge Chinese ci y.4By combining hese admission eco ds wi h da a om a
2014 su ey, we ill wo aspec s o he esea ch gap in educa ion policy and ma ke de-
sign. Fi s , we p o ide empi ical e idence o unde s and s uden s’ s a egic beha io s in
ma ching wi h con ac s heo y. Second, conside ing ha he ZX policy di ec ly in ol es
mone a y ans e s be ween s uden s and schools, e alua ing his policy helps us unde -
s and whe he o e ing a “p ice menu” in a cen alized school choice p ocedu e would
inc ease he inequali y in educa ion among s uden s om di e en backg ounds.
The high schools in ou ocal ci y adop ed he ypical ZX policy o hei admissions
p ocedu e un il ha policy was canceled in 2014 (see Supplemen al Appendix I (Wang
and Zhou (2024)) o de ails o he ZX policy in a ious Chinese ci ies). The ZX policy
speci ied he basic and highe ui ion le els (i.e., he “p ice menu”) and he numbe o
sea s o sale in each school (i.e., he ZX quo a), and was ully con olled by he ci y go -
e nmen . F om 2008 o 2013, Chinese pa allel pu chasing sea s (CPPS) mechanism was
used o assign s uden s o schools; i was an indi ec ex ension o he Chinese pa allel
(CP) mechanism (Chen and Kes en (2017)).
The CPPS mechanism is no a di ec mechanism. When anking a ious schools, a
po en ial s uden ’s ank-o de ed lis (ROL) needs o indica e whe he she is willing o
pay highe ui ion o each school ha would o he wise deny he admission. This mech-
anism has some undesi able ea u es. I is no s a egy-p oo , which allows s uden s
o “game” he sys em by mis epo ing hei ue p e e ences wi h espec o schools.5
Mo eo e , he equilib ium ou comes o his mechanism can be ine icien and uns a-
ble. One way o o e come hese impe ec ions—while e aining he op ion o pu chase
admission—is o adop he cade -op imal s able mechanism (COSM) and i s a ia ion,
he COSM-BRADSO mechanism p oposed by Sönmez and Swi ze (2013)andG een-
be g, Pa hak, and Sonmez (2021). These mechanisms a e he ex ension o he s uden -
1Zhu Kaixuan, hen chai man o he s a e educa ion commission, publicly add essed he sea -
pu chasing p oblem in public schools. In 1995 he a gued, in he People’s Daily, agains paying highe ui ion
o pu chase admission o compulso y educa ion.
2“Ze Xiao” is Chinese o “school selec ion.”.
3Many ci ies, including Shanghai (which ceased using he ZX policy in 2012), Beijing (2014), and Shen-
zhen and Tianjin (2015), ceased he policy o high school admissions.
4Con iden iali y es ic ions p e en his ci y om being iden i ied by name.
5The ue p e e ence wi h espec o schools is e e ed as s uden s’ p e e ences on schools wi hou con-
side ing he ui ion.
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1153
p oposing de e ed accep ance (DA) mechanism (Gale and Shapley (1962)), ensu ing s a-
bili y and s a egyp oo ness, whe ein submi ing ue p e e ences is a weakly dominan
s a egy. Since he COSM-BRADSO mechanism is be e ela ed o he Chinese mecha-
nism in p ac ice (i is o mally de ined in Sec ion 3), we ocus on his mechanism and
use he COSM o deno e COSM-BRADSO he ea e .
The heo e ical p ope ies o hese mechanisms mo i a e us o in es iga e eal-
wo ld s uden beha io and wel a e consequences. One di icul y wi h any empi ical
analysis o he school choice p oblem is es ima ing s uden s’ p e e ences when only
he submi ed applica ions can be obse ed and he adop ed mechanism is no s a -
egyp oo . Ou su ey, which co e ed nea ly hal o hose who g adua ed om middle
school in 2014, aimed o unco e s uden s’ ue p e e ences and he eby o some ex-
en , sol e he p oblems associa ed wi h assessing hose p e e ences in he p esence o
s a egic beha io .
The i s con ibu ion o ou analysis is ha s uden s ha e he e ogeneous p e e -
ences on he “p ice menu” o schools. We es ima e s uden s’ p e e ences in wo s eps. In
he i s s ep, we use su ey esul s o es ima e s uden s’ p e e ences o e schools wi h-
ou conside ing he s a egic beha io in ROLs. Gi en ha he ZX policy ceased and all
s uden s paid he same basic ui ion a e 2013, he su ey da a canno be used o iden-
i y any ZX– ela ed pa ame e s (e.g., ui ion). In he second s ep, we use he ROLs sub-
mi ed in 2012 and 2013 o es ima e o he pa ame e s. In his s ep, we assume ha s u-
den s ha e homogeneous belie s abou he likelihood o being admi ed o each school
and ha hey y o maximize hei expec ed u ili ies in a a ional manne . Ou es ima ed
esul s indica e ha a one-uni inc ease in a school’s posi i e epu a ion (see Sec ion 4.2
o he de ini ion) is associa ed wi h he willingness o high-sco ing s uden s om com-
muni ies wi h high housing p ices o pay an addi ional 296 yuan—o abou US$48.5— o
a end ha school. In con as , high-sco ing s uden s om communi ies wi h low hous-
ing p ices a e willing o pay only 184 yuan o he p i ilege. The compe i ion o college
admission in China is ie ce and in ense. Ou esul s indica e ha s uden s om ich
communi ies ha e highe desi es o “consume” he high-quali y schools ha migh help
hem a ain admissions o colleges compa ed wi h o he s.
Ou second con ibu ion is he e alua ion o he wel a e consequences o di e en
mechanisms. We use he simula ed ma ching ou comes unde he DA mechanism as a
benchma k, hen we measu e he wel a e change when he DA mechanism is eplaced
by a sea -pu chasing mechanism. This eplacemen could a ec a s uden ’s wel a e in
a ious ways. Fi s , a s uden may ake ad an age o he ZX policy o a end a p e e ed
school by paying highe ui ion, which may inc ease he wel a e. Second, i his s uden ’s
sco e is high enough, hen she may s ay a he same school and pay he no mal ui ion.
O he wise, she will su e a wel a e loss by ei he paying highe ui ion o sa e he sea
in his school o being p iced ou and a ending a less p e e ed school.
Ou coun e ac ual analysis indica es ha when he DA mechanism is eplaced wi h
he COSM, s uden wel a e is educed (on a e age) by 30 yuan when 10% o he sea s
a e ese ed o sale ( e e ed o as he ZX quo a). The wel a e loss inc eases o 56 yuan
when he ZX quo a is inc eased o 30%. These esul s e lec ha he di ec in luence
o he sea -pu chasing op ion dec eases s uden s’ o al wel a e gi en ha bo h o he
1154 Wang and Zhou Quan i a i e Economics 15 (2024)
wo mechanisms in ques ion a e s a egyp oo . I he DA mechanism is ins ead eplaced
by he CPPS mechanism, he a e age wel a e inc eases sligh ly by 6 yuan when he ZX
quo a is 10%, and he wel a e loss due o pu chasing sea s is 118 yuan when he ZX quo a
is 30%. The eason is ha mo e s uden s can a end hei p e e ed schools by gaming
he sys em. These esul s e eal an in e es ing phenomenon: I he ZX quo a is limi ed,
lea ing oom o s uden s o game he sys em may educe he a e age wel a e loss. How-
e e , when he ZX quo a is la ge , he wel a e loss is much la ge .
Meanwhile, s uden s om di e en communi ies eac di e en ly o sea -
pu chasing mechanisms. When su e ing a wel a e loss, s uden s om high-housing-
p ice communi ies p e e o pay highe ui ion o keep sea s a he same schools. How-
e e , hose om low-housing-p ice communi ies would a he be p iced ou o less p e-
e ed schools. Among he ew s uden s who a end mo e p e e ed schools by aking
ad an age o he ZX policy, mos a e om ich communi ies. In e es ingly, when he ZX
quo a is la ge, al hough mo e s uden s om poo communi ies a e p iced ou , high-
sco ing s uden s (app oxima ely 10%) om hese communi ies exhibi a s onge in-
cen i e o pay highe ui ion o s ay in be e schools han medium- and low-sco ing
s uden s (app oxima ely 1%). In summa y, ou esul s imply ha he ZX policy inc eases
inequali y among s uden s in e ms o hei u u e educa ional oppo uni ies, and is no
de e mined solely by hei wel a e measu e in mone a y e ms. Compe i i e s uden s
(high-sco ing s uden s), speci ically hose om poo communi ies, show a s ong in-
cen i e o a end be e schools compa ed o o he s uden g oups.
Finally, we in es iga e he impac o he ZX policy on schools. In China, an in ense
compe i ion exis s among high schools ega ding admissions. Schools, which su e a
wel a e loss unde e o ms, ha e a s ong incen i e o block such e o ms. The e o e, ou
analysis o he policy impac on schools may p o ide e e ences o policymake s abou
po en ial di icul ies om e o ms. We measu e he impac on high schools in e ms o :
(a) he quali y o admi ed s uden s and (b) he p o i de i ed om collec ing s uden
ui ion. The sea -pu chasing op ion helps uppe - ie schools collec signi ican ly mo e
ui ion wi h only a limi ed decline ( ela i e o he DA mechanism) in he quali y o hei
admi ed s uden s. Ye o o he schools, he sea -pu chasing op ion leads o mo e un-
ce ain y abou bo h collec ed ui ion and he esul ing quali y o hei admi ed s u-
den s.
This pape is closely ela ed o he heo e ical wo k o Sönmez (2013), Sönmez
and Swi ze (2013), and G eenbe g, Pa hak, and Sonmez (2021), who in es iga e cade -
b anch ma ching in he US mili a y. We ex end he heo e ical esul s and complemen
hese ou comes by o e ing an empi ical analysis. Ou wo k is also di ec ly ela ed o he
ex ensi e heo e ical li e a u e on he cen alized school choice p oblem.6Mo e speci -
ically, he e is a g owing li e a u e ha discusses he ole o mul i-le el inancial aid in
he school choice p oblem (Hassidim, Romm, and Sho e (2016)). Hassidim, Romm,
and Sho e (2017) disco e ha , in a ma ching p ocedu e o Is aeli Mas e ’s p og ams
in psychology, many applican s make he mis ake o highly anking p og ams ha o e
less inancial aid.
6See Pa hak (2011) o a su ey on he school choice p oblem om he pe spec i e o ma ke design.

Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1155
The esea ch unde aken he e con ibu es o a g owing body o empi ical wo k on
he school choice mechanism. Aga wal and Budish (2021) e iew he de elopmen o
s uc u al es ima es o ma ke design models. One s and o ha li e a u e uses he
p e e ences epo ed unde nons a egyp oo mechanisms o es ima e s uden s’ p e -
e ences (Hwang (2015); He (2016); Aga wal and Somaini (2018); Calsamiglia, Fu, and
Güell (2020)). O he pape s ocus on s a egyp oo mechanisms. Abdulkadi o˘
glu, Aga -
wal, and Pa hak (2017) ea p e e ences epo ed unde he DA mechanism as s uden s’
ue p e e ences and hen use hose p e e ences o analyze he demand o pa icula
schools in New Yo k Ci y. Fack, G ene , and He (2019) p opose an app oach o es ima -
ing p e e ences ha does no equi e u h- elling o be he unique equilib ium unde
he DA mechanism. Se e al empi ical pape s (e.g., Bu gess, G ea es, Vignoles, and Wil-
son (2014); Akyol and K ishna (2017); Wang and Zhou (2020); Ajayi (2022)) bea simila -
i ies o ou s ic p io i y se ing. O he s begin o in es iga e he e ec o ans e s in he
ma ke design (Aga wal (2015,2017); Bobba, Ede e , Leon-Cilio a, Neilson, and Nieddu
(2021)).
The e is an inc easing use o su ey da a in schola ly esea ch explo ing s a egic
beha io unde ma ching mechanisms. Budish and Can illon (2012)conduc asu ey
on s uden s’ p e e ences o o e ed cou ses o s udy he cou se alloca ion mechanism
a Ha a d Business School, and Rees-Jones (2018) p o ide su ey-based e idence o
p e e ence mis ep esen a ion. Bu gess e al. (2014) use su ey da a o di ec ly assess he
p e e ences o s uden s ega ding schools. Su eys a e also used by De Haan, Gau ie ,
Oos e beek, and Van de Klaauw (2023) o analyze he Bos on mechanism’s de iciencies
and by Kapo , Neilson, and Zimme man (2020) o s udy he e ogeneous belie s in he
school choice p oblem.
Ou es ima ion o s uden s’ p e e ences also unde sco es he impo ance o consid-
e ing he ca dinal p e e ence. Abdulkadi o˘
glu, Che, and Yasuda (2011b,2015) sugges
ha om an ex an e pe spec i e, when schools ha e coa se p e e ences o s uden s
coupled wi h a symme ic ie-b eaking ule, s uden s could a e be e unde he Bos on
mechanism han unde he DA mechanism, as assessed by hei ca dinal p e e ences.
Ou analysis inds ha om an ex pos pe spec i e, when schools ha e s ic p io i ies
o s uden s, a manipulable mechanism such as he CPPS can s ill yield highe a e age
s uden wel a e o some ypes o s uden s han he DA mechanism, especially when he
numbe o sea s o sale is limi ed.
The emainde o his pape p oceeds as ollows. Sec ion 2p o ides de ails on he
local ZX policy’s backg ound. In Sec ion 3, we p esen school choice mechanisms ha
inco po a e sea -pu chasing op ions. Sec ion 4desc ibes ou da a and he summa y
s a is ics. We p esen he empi ical model and ou es ima es o s uden s’ p e e ences in
Sec ion 5, and in Sec ion 6, we conduc coun e ac ual expe imen s ac oss mechanisms.
Sec ion 7concludes wi h a summa y o ou indings.
2. Backg ound on high school admissions
The high schools in ou ocal ci y can be ca ego ized in o a ious ypes based on hei
educa ional goals o s uden s a e comple ing middle school. These ypes include gen-
1156 Wang and Zhou Quan i a i e Economics 15 (2024)
e al high schools, which can be public o p i a e, aimed a p epa ing s uden s o col-
leges and uni e si ies wi hin China. Addi ionally, he e a e o eign language schools ha
ocus on p epa ing s uden s o s udies a o eign ins i u ions. Fine a s schools ca e o
s uden s aspi ing o a end ine a s colleges in China. Las ly, oca ional schools p epa e
s uden s wi h skills necessa y o he labo ma ke .
The Ci y Educa ion Bu eau (he ea e e e ed o as “ he Bu eau”) manda es ha all
schools, ega dless o ype o owne ship, pa icipa e in he cen alized admission sys em
o middle school g adua es. Mo eo e , each s uden going h ough his p ocess mus
egis e a he school assigned by he sys em. Thus, no o he op ions a e a ailable o
s uden s wishing o con inue hei educa ion in his ci y.
Annually, a he end o Ma ch, he Bu eau announces an admissions plan de ailing
he admission quo a o each school. The quo a o each public high school jconsis s
o h ee pa s: he quo a o ea ly admission (qe
j),7 he quo a o no mal admission (qa
j),
and he quo a o he ZX policy (qz
j). The Bu eau, no he schools, con ols hese quo-
as. S uden s admi ed unde any ca ego y ecei e iden ical educa ion wi hin each high
school. In mid-May, s uden s mus submi hei ank-o de ed lis s (ROLs) o p e e ed
schools. All s uden s hen ake he cen alized high school en ance exam in ea ly June.
F om 2012 o 2014, he maximum sco e o his exam was 665.8A e g ading, a cen al-
ized ma ching mechanism assigns s uden s o schools. No ably, all schools en o ce he
same s ic p io i y based on exam sco es du ing s uden admission.
Local public high schools a e pi o al in p epa ing s uden s o college in China. Fo
many, en e ing a public high school is hei sole oppo uni y o highe educa ion. How-
e e , high school educa ion in China ex ends beyond compulso y le els, and local pub-
lic high schools can accommoda e ewe han hal o all middle school g adua es. Be o e
he ma ching p ocedu e, he Bu eau es ablishes and announces a public high school ad-
mission h eshold (he ea e “ he h eshold”) based on sco e dis ibu ion and sea a ail-
abili y. Only s uden s sco ing abo e his h eshold a e eligible o admission o public
high schools. This h eshold ensu es he numbe o quali ied s uden s does no exceed
a ailable sea s in public high schools.
S uden s can lis up o h ee schools on hei ROL and indica e hei choice o he
ZX op ion o each. They also need o s a e i hey will accep a andom assignmen i
ejec ed by hei chosen schools. Since 2008, he CPPS mechanism—wi h pe manency-
execu ion pe iods (2, 1)—has been employed o assign s uden s (de ails on ma ching
algo i hms a e desc ibed in Sec ion 3). This mechanism concludes a e conside ing
each s uden ’s h ee choices.9S uden s admi ed wi h only basic ui ion ees a e e e ed
o as no mal s uden s, while hose admi ed wi h highe ui ion ees a e ZX s uden s.
Unma ched s uden s open o andom assignmen a e andomly placed in public high
schools wi h acancies. The es explo e al e na i es o con inue educa ion o join he
wo k o ce.
7S uden s eligible o ea ly admission a e de e mined h ough a sepa a e p ocedu e, no di ec ly impac -
ing he no mal admission p ocess, so hey a e excluded om his analysis.
8Be o e 2012, he highes sco e was 650, and i inc eased o 780 a e 2014.
9This mechanism is a cons ained mechanism as desc ibed by Hae inge and Klijn (2009).
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1157
The ZX policy was designed exclusi ely o public high schools and no o o he
school ypes.10 Tui ion o public high schools is based on he s uden ’s exam sco e.
Since sco es a e he only admission c i e ia, schools se a cu o o no mally admi ed
s uden s. No mal s uden s pay an annual ui ion o 1600 yuan ( oughly $260 in 2013),
ensu ing a o dabili y. ZX s uden s’ ui ion depends on hei sco e: sco es wi hin 10
poin s o he cu o esul in a 3333.3 yuan annual ee; sco es 11–20 poin s abo e he cu -
o pay 5000 yuan; and sco es 21–30 poin s abo e he cu o pay 6000 yuan.11 No school
can admi a ZX s uden wi h a sco e mo e han 30 poin s below i s cu o . On hei ROLs,
s uden s can only answe “yes” o “no” o he ZX op ion, meaning hey canno selec a
speci ic ZX ui ion bu mus accep he en i e package i admi ed as ZX s uden s. Fol-
lowing he Minis y o Educa ion’s di ec i e, he local educa ion bu eau ceased he ZX
op ion pos he 2013 admissions p ocess.
3. Chinese pa allel pu chasing sea s (CPPS) mechanism
In his sec ion, we p o ide a o mal de ini ion o a school choice p oblem ha inco -
po a es he op ion o pu chasing sea s. We conside a ini e se o s uden s, deno ed by
I, and a ini e se o schools, deno ed by J∪∅,whe e∅ ep esen s he si ua ion whe e
a s uden does no a end any school. Each school o e s wo ui ion le els: c0and c1,
whe e c0is he basic ui ion paid by no mal s uden s, and c1is he highe ui ion o
ZX s uden s.12 Each school jhas wo quo as, qa
jand qz
j, o no mal and ZX s uden s,
espec i ely. I holds ha j∈J(qa
j+qz
j)≥n,whe en ep esen s he o al numbe o s u-
den s. Each s uden ihas a s ic p e e ence, deno ed by πi, o e schools and ui ion.
The no a ion (j,c0)πi(j,c1)indica es ha s uden is ic ly p e e s o pay he basic u-
i ion o a sea in school j a he han paying he highe ui ion o he same school. All
schools employ a uni o m s ic p io i y anking, deno ed by , o o de s uden s based
on hei exam sco es. When s uden iis alloca ed a sea in school jby paying ui ion
c, hepai (j,c)is e med as s uden i’s assignmen . A ma ching Xis de ined as a col-
lec ion o s uden - o-assignmen pai ings ha sa is ies wo condi ions: (a) each s uden
has only one assignmen , and (b) no school admi s mo e s uden s han i s o al quo a.
We deno e he se o all ma ching ou comes as X.
Amechanism is de ined as a s a egy space i o each s uden i, accompanied by an
ou come unc ion ψ:(i1×i2×···×in)→X, which selec s a ma ching ou come o
each s a egy ec o a∈(i1×i2×···×in).Adi ec mechanism is a unc ion ψ ha
selec s a ma ching ou come o each p e e ence p o ile.
10The college admission a e o he bes p i a e high school is below 1%, making i lowe han e en low-
quali y public high schools. P i a e schools cha ge a egula ed la ui ion ee, wi h admission cu o s equal
o he public high school h eshold e e y yea . Essen ially, p i a e high schools mainly ca e o s uden s
sco ing below he public high school h eshold bu s ill wishing o con inue hei high school educa ion.
Ou analysis does no del e in o hese schools.
11ZX s uden s pay a lump-sum o all h ee high school yea s, unlike no mal s uden s who pay annually.
12The model can be eadily expanded o inco po a e mul iple le els o ui ion, as discussed in Sönmez
and Swi ze (2013). Howe e , in ou ocal ci y, ZX s uden s a e p esen ed wi h a singula ZX ui ion package
and can only decide whe he o accep o ejec i , a he han selec ing a speci ic ui ion le el. Fo he sake
o simplici y, we ocus on wo ui ion le els in ou model.
1158 Wang and Zhou Quan i a i e Economics 15 (2024)
The CPPS mechanism is an ex ension o he Chinese pa allel (CP) mechanism (Chen
and Kes en (2017)). Howe e , unlike he CP mechanism, he CPPS mechanism is no a
di ec mechanism. Speci ically, unde he CPPS mechanism, each s uden is equi ed o
p o ide (i) he anked p e e ences o schools and (ii) indica e, o each anked school,
whe he she would op o he ZX op ion (i.e., paying highe ui ion) o a end ha school
i she is no assigned a sea as a no mal s uden .
Unde he CPPS mechanism, schools alloca e he no mal sea s based on he no mal
p io i y (). Fo he alloca ion o ZX sea s, each school jemploys he ZX p io i y +,
which is cons uc ed as ollows: All applican s o ZX sea s a school ja e di ided in o
wo g oups: he ZX-quali ied g oup Aj, comp ising s uden s who op o he ZX op ion
o school jand mee he p ede e mined quali ica ion ule ( ela ed o he school’s no -
mal p io i y) and he emaining applican s in g oup Bj. When school jcompa es wo
s uden s iand i, he ollowing ules apply: I i∈Ajand i∈Bj, heni+i, indica ing
ha s uden iis gi en highe p io i y o e i.I i,i∈Ajo i,i∈Bj, heni+ii and
only i ii. In o he wo ds, school j’s p e e ence is solely de e mined by s uden s’ exam
sco es in his case.
The ZX p io i y indica es ha a quali ied ZX applican has a highe p io i y o e-
cei ing a ZX sea compa ed o an applican who ei he does no choose he ZX op ion
o selec s he ZX op ion bu does no mee he quali ica ion c i e ia. In all o he cases,
school ju ilizes he no mal p io i y o alloca e ZX sea s.13 As men ioned in Sec ion 2
conce ning ou ocal ci y, a ZX applican is conside ed quali ied o a ZX sea only i he
exam sco e alls wi hin a ange o 30 poin s below he school’s no mal admission cu -
o (G oup A). S uden s whose sco es a e mo e han 30 poin s below he cu o a e no
quali ied o admission as ZX s uden s. In essence, op ing o he ZX op ion can gi e a
s uden he p i ilege o ake a ZX sea unde he condi ion ha he sco e is no oo low.
The CPPS mechanism wi h a pe manency-execu ion pe iod ec o , e=(e1,e2,),
selec s he ma ching ou come as desc ibed below.
Round 1:
•E e y s uden applies o he i s choice. Each school japplies he no mal p io i y
o en a i ely ese e he op qa
japplican s in he no mal pool. Among he emaining
applican s, he school en a i ely ese es he op qz
japplican s in i s ZX pool, ollowing
he ZX p io i y. All o he applican s a e ejec ed.
In gene al:
•Any ejec ed s uden iwho has no ye applied o he (e1) h-choice school applies
o he nex -p e e ed school. A s uden who has been ejec ed by all he i s e1choices
does no apply o any o he school un il he nex ound. Each school je alua es he new
applican s, along wi h hose al eady held in he no mal pool, and en a i ely ese es he
op qa
japplican s in i s no mal pool based on he no mal p io i y. Subsequen ly, school
jconside s all emaining applican s, along wi h hose al eady held in i s ZX pool, and
13All o he cases encompass he ollowing scena ios: (1) when bo h applican s choose he ZX op ion
and a e quali ied, (2) when bo h applican s choose he ZX op ion bu nei he a e quali ied, (3) when one
applican chooses he ZX op ion bu is no quali ied, and he o he applican does no choose he ZX op ion,
and (4) when nei he o hem chooses he ZX op ion.
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1165
s uden s’ p e e ences among lowe - ie schools.27 When conduc ing he coun e ac ual
analysis o assess he wel a e o low-sco ing s uden s, unde s anding hei p e e ences
ega ding a ainable schools is mo e aluable han hei p e e ences o a o i e bu
una ainable schools. In ou su ey, all schools ecei ed subs an ial ep esen a ion in
s uden s’ esponses. I is no ewo hy ha each o he emaining h ee schools was cho-
sen by mo e han 100 low-sco ing s uden s. In con as , hese ins i u ions we e seldom
men ioned by hei high-sco ing coun e pa s (Table B.2 in Supplemen al Appendix B).
Figu e 1shows he a e age admission cu o s o schools chosen by s uden s bo h
in he su ey and in hei ROLs.28 S uden s a e ca ego ized in o ou g oups based on
hei sco e pe cen iles. In he su ey, high-sco ing s uden s (wi h exam sco es abo e he
90 h pe cen ile) ha e a e age school cu o s o 606.1 and 599.4 o hei i s and sec-
ond choices, espec i ely. The a e age cu o o hi d choices is 593.2, which is 6 poin s
lowe . Addi ionally, he su ey e eals ha he cu o gaps be ween he hi d and ou h
choices, and be ween he ou h and i h choices, a e 5 and 9 poin s, espec i ely. S u-
den s in he o he h ee sco e pe cen ile g oups exhibi simila pa e ns. Wi hin each
g oup, he gap in a e age cu o s be ween consecu i e choices is a ound 6 poin s and
ne e su passes 10 poin s. When compa ing be ween g oups, he a e age cu o o he
i s choices o s uden s in he 80 h o 90 h pe cen iles is 6 poin s lowe han ha o he
highes decile o s uden s. Fu he mo e, his a e age cu o dec eases by an addi ional
9 poin s ( o 591) o s uden s in he 70 h o 80 h pe cen ile ange. S uden s below he
70 h pe cen ile o exam sco es ha e an a e age i s -choice cu o o 585. Wi h each ad-
di ional choice, he a e age cu o s dec ease simila ly (a a a e o 4 o 10 poin s) based
on exam sco es.
The obse ed dec ease in he a e age cu o o s uden s’ i s choices as hei sco es
dec ease sugges s ha he su eyed s uden s p o ided u h ul esponses by lis ing and
anking schools o which hey had ealis ic admission chances. The gaps be ween con-
secu i e choices wi hin g oups in he su ey indica e ha s uden s’ p e e ences o
schools dec ease wi h he popula i y o hose schools. Fo example, in 2014, he gaps in
consecu i e cu o s o wo sough -a e schools anged om 3 o 9 poin s. Addi ionally,
he small cu o gaps (4 o 10 poin s) be ween consecu i e choices wi hin each g oup
implies ha he p e e ences epo ed in he su ey a e eliable enough o be iewed as
he s uden s’ ue p e e ences.
In he ank-o de ed lis s, he a e age cu o s o he i s choices o s uden s whose
exam sco es we e abo e he 70 h pe cen ile nea ly coincide wi h he co esponding
pa s in he su ey. Howe e , he a e age cu o s o he i s choices o low-sco ing s u-
den s (i.e., wi h exam sco es below he 70 h pe cen ile) a e 6 poin s lowe han ha in
he su ey. No ably, he gap be ween he i s and second choices inc eases signi ican ly
as exam sco es decline. While he gap in a e age cu o s be ween he i s and second
choices o he op 10% o s uden s emains almos he same as in he su ey, i in-
c eases o 19 poin s o s uden s in he 80 h o 90 h pe cen ile ange, and app oxima ely
27Fo example, suppose he e a e ou schools, namely A, B, C, and D. Mos s uden s’ mos a o i e school
is A, and B is he second a o i e, hen we can only in e ha s uden s p e e A o B o C and D, bu i is
di icul o ell s uden s’ p e e ence be ween C and D.
28The co esponding able can be ound in Supplemen al Appendix B.

1166 Wang and Zhou Quan i a i e Economics 15 (2024)
Figu e 1. A e age admission cu o s o schools: Su ey e sus ROLs. No es:They-axis ep e-
sen s absolu e sco es, and he x-axis ep esen s ou s uden g oups ca ego ized by exam sco es
in pe cen ile. The do ed cu es ep esen he a e age cu o s o he chosen schools in he su ey.
The solid cu es ep esen he a e age cu o s o schools in he ROLs. The h eshold o public
high school admission is 535 (60.95 pe cen ile) in 2014.
25 poin s o he wo g oups o low-sco ing s uden s. Fu he mo e, he a e age cu o s
o hi d choices emain consis en ly a ound he 535 h eshold ac oss all g oups in he
ROLs.
When compa ed o he su ey da a, he signi ican gaps be ween consecu i e
choices in he ROLs indica e s uden s’ s a egic beha io in hei submi ed p e e ences:
main aining a subs an ial gap be ween choices wi h he in en ion o inc easing hei
chances o admission o some school.29 The co espondence be ween he i s choices
in he su ey and he ROLs implies ha s uden s p e e o apply o hei a o i e a -
ainable schools. This coincidence, along wi h he small cu o gaps among choices e-
po ed in he su ey, p o ides u he e idence ha he su eyed s uden s accu a ely
epo ed hei i e a o i e a ainable schools. Howe e , i is e iden ha s uden s, espe-
cially hose ou side he op-sco ing g oup, s a egically manipula e hei epo ed p e -
e ences in he ROLs o inc ease hei o e all likelihood o admission, ha is, pa icula ly
when aced wi h ejec ion om hei op choices. Thus, he second choices in he ROLs
o s uden s in 80 h–90 h pe cen ile ( esp., 70 h–80 h pe cen ile) closely esemble hei
ou h ( esp., i h) choices in he su ey. Fu he mo e, a majo i y o s uden s (ac oss all
ou g oups) selec ed a le o e school as hei hi d choice because he ROL is es ic ed
o only h ee choices.
One d awback o a nons a egyp oo mechanism is ha s uden s who s a egically
modi y hei ROLs may exploi nai e s uden s who e eal hei ue p e e ences (Pa hak
29This inding is consis en wi h he li e a u e ha sugges s s uden s beha e s a egically unde non-
s a egyp oo mechanisms (see, e.g., Abdulkadi o˘
glu, Pa hak, and Ro h (2005); Chen and Sönmez (2006);
Abdulkadi oˇ
glu, Che, Pa hak, Ro h, and Te cieux (2020)).
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1167
and Sönmez (2008)).30 We di ec ly compa e he schools lis ed in he su ey and in he
ROLs. Only 20 s uden s (1.38% o all obse a ions) submi ed ROLs ha ma ched hei
su ey lis s. Howe e , i is possible ha he numbe o naï e s uden s is e en smalle ,
as epo ing ue p e e ences could be a weakly dominan s a egy o some s uden s,
especially hose in he op-sco ing g oups. Addi ionally, some s uden s may exhibi isk
a e sion by epo ing hei ue p e e ences. In bo h cases, hese s uden s a e no nai e
playe s in he game. The e o e, he 1.38% igu e can be conside ed an uppe bound o
he numbe o nai e s uden s. These indings a e consis en wi h p e ious esea ch, sug-
ges ing ha only a small numbe o s uden s submi a ank-o de ed lis wi hou s a egic
conside a ions, pa icula ly when a s ic c i e ion is used o s uden assignmen .31
4.4 Housing p ices and admission dis ibu ions
Ou da a se does no con ain any indi idual- o household- ela ed in o ma ion, such as
household income. Addi ionally, he ci y does no p o ide subci y le el agg ega e in o -
ma ion ega ding local esiden s’ income.32 To analyze whe he he ZX policy has he -
e ogeneous e ec s on households wi h a ying economic s a uses, we ma ch s uden s’
home add esses wi h he local housing ma ke in o ma ion as p oxy. U ban economis s
ha e p e iously examined he posi i e co ela ion be ween local housing p ices and
esiden s’ income (Goodman (1988); Hwang and Quigley (2006)). The assump ion ha
highe -income amilies eside in a eas wi h highe housing p ices has also been em-
ployed in economic analysis (Ab aham and Hende sho (1996); Capozza, Hende sho ,
Mack, and Maye (2002)). Simila ly, s udies in China ha e de ec ed a posi i e ela ion-
ship be ween housing p ice and esiden s’ income (Zhang, Jia, and Yang (2016)). In ou
s udy, he ocal ci y is di ided in o 85 communi ies, which a e nei he go e nmen ad-
minis a i e uni s no school dis ic s. This di ision is based on he local housing ma ke
and adi ional li ing a eas iden i ied by a eal es a e websi e (Wang and Zhou (2024)).33
Local public schools ecei e ull unding om he ci y-le el go e nmen , and s uden s
a e no es ic ed o speci ic zones when choosing high schools. The e o e, he e is no
di ec connec ion be ween housing p ices and school quali y.
The median housing p ice in hese communi ies is 13,636 yuan/m2,34 wi h he high-
es p ice being 30,405 yuan/m2, and he lowes p ice being 3968 yuan/m2(Figu e 2).35
To simpli y he analysis, we classi y communi ies wi h housing p ices abo e he hi d
30Calsamiglia, Fu, and Güell (2020) indica e ha , in Ba celona’s local school choice se ing, he p opo -
ion o such nai e s uden s is less han 4%.
31In ou con ex , unlike si ua ions whe e s uden s a e assigned based on coa se c i e ia (e.g., walking
zones o siblings), high school admission o e s no sa e choice o s uden s un il hei exam sco es a e
known. Consequen ly, es ima ing hei exam sco es becomes a s uden ’s ini ial s a egic mo e. The e o e,
one can expec an ex emely low pe cen age o nai e s uden s when subjec ed o admission p ocedu es like
hose desc ibed he e.
32The ci y le el annual pe capi a disposable income o an u ban household in 2013 was 35,227 yuan
(≈5775 US dolla ).
33The au ho s ha e collec ed and p ocessed he da a on local housing p ices om h ps://www.58.com/.
This da a se can be downloaded om he eplica ion package. h ps://doi.o g/10.5281/zenodo.12735964
341m2is equal o 10.76 sq. .
35The a e age housing p ice o his ci y was 12,187 yuan/m2in 2014.
1168 Wang and Zhou Quan i a i e Economics 15 (2024)
Figu e 2. Dis ibu ion o he housing p ice. No es: This is he his og am o communi ies’ hous-
ing p ices in 2014. The uni o he x-axis is 10,000 yuan/m2. The uni o he y-axis is he numbe
o communi ies. Each bin ep esen s 2500 yuan/m2excep ha he i s bin includes he hous-
ing p ice lowe han 5000 yuan/m2and he las one includes he housing p ice abo e 25,000
yuan/m2.
qua ile as high housing p ice (HHP) communi ies,36 communi ies wi h housing p ices
below he i s qua ile as low housing p ice (LHP) communi ies,37 and he emaining
communi ies as mode a e housing p ice (MHP) communi ies.
On a e age, each communi y has 54.48 s uden s, wi h a s anda d de ia ion o 26.
App oxima ely 35.6% o he s uden s li e in HHP communi ies, 49.8% li e in MHP com-
muni ies, and he emaining 14.7% come om LHP communi ies. A e he cancella ion
o he ZX policy, he pe cen age o admissions om HHP communi ies o uppe - ie
schools in 2014 was 38.2% (Table 3). This numbe was lowe han he pe cen ages in
2012 (41.9%) and in 2013 (45.8%). Ins ead, uppe - ie schools admi ed mo e s uden s
om MHP and LHP communi ies. These changes canno be simply a ibu ed o he
luc ua ion o exam pe o mance (Table 4), as 43.7% o high-sco ing s uden s (sco ing
abo e he 90 h pe cen ile) we e om HHP a eas in 2014, compa ed o 46.7% in 2013 and
42% in 2012.38
P io o 2014, uppe - ie schools admi ed a compa able pe cen age o s uden s om
HHP and MHP communi ies. Howe e , he HHP a ea accoun ed o a la ge p opo ion
o ZX s uden s compa ed o he combined con ibu ions om MHP and LHP commu-
ni ies. Con e sely, mo e han hal o he high-sco ing s uden s came om he MHP and
LHP communi ies.
These summa y s a is ics a he agg ega e le el indica e ha he cancella ion o he
ZX policy has p o ided s uden s om mode a e and low housing p ice a eas wi h in-
c eased oppo uni ies o en oll in well- ega ded schools. Howe e , he impac o his
policy change on middle- and lowe - ie schools is no as e iden as i is on uppe - ie
36Communi ies wi h housing p ice g ea e o equal o 16,161 yuan/m2.
37Communi ies wi h housing p ice less han 10,609 yuan/m2.
38The uppe - ie schools’ admission cu o was se a he 93 d pe cen ile.
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1169
Table 3. Communi y admission dis ibu ion (%).
Uppe -Tie Schools Middle-Tie Schools Lowe -Tie Schools
2012 2013 2014 2012 2013 2014 2012 2013 2014
HHP communi ies. 41.9 (13.3) 45.8 (14.5) 38.2 31.4 (7.8) 38.9 (10.7) 36.9 26.8 (0) 31.0 (0) 30.9
MHP communi ies 46.5 (9.3) 42.0 (8.3) 48.2 54.7 (11.3) 47.8 (10.9) 49.3 53.7 (0) 48.0 (0) 51.3
LHP communi ies 11.4 (1.0) 12.1 (1.1) 13.6 14.0 (1.7) 13.2 (2) 13.9 19.4 (0) 20.8 (0) 17.8
No e: This able indica es he dis ibu ion o admi ed s uden s om di e en communi ies. In each column, he i s num-
be ep esen s he pe cen age o s uden s who li e in high, mode a e, o low housing p ice communi ies; he numbe in he
pa en hesis ep esen s he pe cen age o s uden s who a e he ZX s uden s in he co esponding communi ies.
schools. To u he analyze all hese ques ions, we es ima e s uden s’ p e e ences in he
nex sec ion.
5. Empi ical model and p e e ence es ima e
To es ima e s uden s’ p e e ences, we adjus he ui ion ee s uc u e, based on he local
admission ule, as desc ibed in he school choice p oblem om Sec ion 3. Recall ha
he e is a se o ui ion ees C={c0,c1,c2,c3},whe ec0is he basic ui ion o no mal
s uden s while c1,c2and c3a e he highe ui ion amoun s paid by ZX s uden s; he e,
c0<c
1<c
2<c
3.
S uden i’s (indi ec ) u ili y om being assigned o public high school jwi h ui ion
cij ∈Cis
ui,j=
l
βlyl
j+
w
βwxw
iyw
j+βDdij +
k
αkxk
i(cij −c0)+εij (1)
and ha he u ili y om being assigned o nonpublic high school ois
ui,o=Fo+εio.(2)
He e, {yj} ep esen s a ec o o obse ed cha ac e is ics o school j;{xi}is a ec o o
s uden i’s obse ed cha ac e is ics; dij is he home-school dis ance;39 Fois he ixed
e ec o nonpublic high schools; and εij and εio a e i’s idiosync a ic as e o ( espec-
i ely) public high school jand nonpublic high schools. In he es ima e, we assume ha
Table 4. Communi y sco e dis ibu ion (%).
High-Sco ing S uden s Median-Sco ing S uden s Low-Sco ing S uden s
2012 2013 2014 2012 2013 2014 2012 2013 2014
HHP communi ies 42.1 46.7 43.7 29.8 37.2 34.1 29.2 33.2 31.7
MHP communi ies 48.9 44.5 46.4 55.0 47.6 50.5 52.2 46.6 49.9
LHP communi ies 9.0 8.8 9.9 15.2 15.2 15.4 18.7 20.2 18.5
No e: This able decomposes each sco ing g oup in o a di e en housing p ice.
39The oad dis ance dij is calcula ed ia Google Maps by inpu ing he ocal school’s add ess and he
s uden ’s home add ess.
1170 Wang and Zhou Quan i a i e Economics 15 (2024)
he home-school dis ance is addi i ely sepa able and independen o unobse ed s u-
den s’ p e e ences; in addi ion, we no malize he coe icien o dij o he home-school
dis ance o be −1.40
The u ili y unc ion o s uden s in Equa ion (2) is simila o ha in Abdulkadi o˘
glu,
Aga wal, and Pa hak (2017)andAga wal and Somaini (2018), wi h he excep ion ha
we do no p esen he andom coe icien model o es ima ing s uden s’ he e ogeneous
p e e ences o obse ed school cha ac e is ics due o limi ed a ia ion in ou da a. In
China, he p ima y goal o gene al high schools is o p epa e s uden s o he college
en ance exams. Apa om epu a ion, he obse ed cha ac e is ics o schools, such
as acili ies, a e ai ly homogeneous. The eaching p og ams a e ully con olled by he
local educa ion bu eau. Addi ionally, s uden s who a e quali ied o local public high
schools exhibi simila p e e ences o schools (see Supplemen al Appendix H o de ails
o s uden s’ su ey esponses).41
Consis en wi h Abdulkadi o˘
glu, Aga wal, and Pa hak (2017), we do no explici ly
model an ou side op ion. I is because no ou side op ion can be obse ed in he cu -
en admission eco d, as men ioned in Sec ion 2. In addi ion, we make he ollowing
assump ion.
Assump ion 1. The e ms εij and εio a e independen o he explana o y a iables,xi,
yj,dij ,C,and Fo.Bo h εij and εio a e independen and iden ically dis ibu ed (i.i.d.)
and exhibi a ype I ex eme alue dis ibu ion wi h cumula i e dis ibu ion unc ion
(CDF)F(ε).
Ou es ima ion o s uden s’ p e e ences employs bo h adminis a i e and su ey
da a. A key bene i o he su ey da a is i s abili y o yield es ima es wi hou ac o ing
in he s a egic beha io s s uden s migh exhibi when submi ing hei ROLs. Howe e ,
he su ey da a canno p o ide insigh s in o s uden s’ p e e ences ega ding ZX op ions
because he ZX policy was discon inued a e 2013. Consequen ly, in 2014, all s uden s
paid he same basic ui ion o all public high schools. As a esul , he su ey da a alone
canno es ima e he pa ame e s αkin equa ion (1). The e o e, we di ide ou es ima ion
p ocedu e in o wo s eps. Fi s , we use he su ey da a om 2014 o es ima e he ec o
o pa ame e s un ela ed o he ZX op ion, ha is, {β}. Second, we es ima e he ec o {α}
o pa ame e s ela ed o he ZX policy using he s uden ROLs submi ed p io o 2014.
5.1 S ep one: Es ima ing he non-ZX- ela ed pa ame e s β
In his s ep, we ocus on he su ey da a wi hou conside ing s uden s’ s a egic beha -
io when submi ing hei ROLs. Each su eyed s uden anked i e schools she belie ed
she had a ealis ic chance o a ending. This selec ion p ocess implies ha he s uden
40Unlike admission o elemen a y and middle schools, he high school admission p ocedu e does no
conside he loca ions o school dis ic s o homes. Hence, we assume ha , in his ci y, he school choice
mechanism does no di ec ly in luence esiden ial decisions o local housing p ices.
41To a oid choosing he w ong empi ical model, we conside an al e na i e andom coe icien model
and compa e he esul ing es ima es. Howe e , he andom coe icien model pe o ms wo se han he non-
andom coe icien model in bo h wi hin-sample and ou -o -sample es s.

Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1171
i s iden i ies he schools o which admission is easible and hen anks hem acco d-
ingly. This p ocedu e complica es he cons uc ion o he model ha explains how hese
middle school g adua es ini ially selec schools. Fo ins ance, i a school wi h a high ad-
mission cu o is no included in a su eyed s uden ’s lis , i becomes challenging o dis-
inguish be ween wo possibili ies: (a) he s uden p e e s he lis ed schools o e he
high-cu o school, o (b) he s uden belie es ha admission o he high-cu o school
is no possible. Based on he e idence p esen ed in Sec ion 4.3, we conclude ha he su -
ey esponses e lec s uden s’ ue p e e ences, condi ioned on hei belie s abou he
possibili y o admission. To simpli y he es ima ion p ocess, we ocus solely on he ank-
ings o he lis ed schools in he su ey and do no conside unlis ed schools. In o he
wo ds, we do no a emp o in e he ela i e ankings o lis ed and unlis ed schools.
While his app oach may esul in a less e icien es ima e, i emains consis en when
su eyed s uden s epo hei ue ankings. Fo example, conside a s uden iwi h a
sco e swho lis s school jbe o e school jin he su ey. The admission cu o s o hese
wo schools a e deno ed as ¯
Sjand ¯
Sj, espec i ely. Then he p obabili y ha ip e e s j
o jcondi ional on bo h schools being a ainable o he is P (ui,j>u
i,j|s> ¯
Sj∩s> ¯
Sj).
This condi ional p obabili y equals he uncondi ional p obabili y, ha is,
P (ui,j>u
i,j|s> ¯
Sj∩s> ¯
Sj)=P (ui,j>u
i,j∩s> ¯
Sj∩s>¯
Sj)
P (s>¯
Sj∩s>¯
Sj)
=P (ui,j>u
i,j)P (s>¯
Sj∩s> ¯
Sj)
P (s>¯
Sj∩s> ¯
Sj)
=P (ui,j>u
i,j)
The second equali y a ises om he ac ha s uden s’ belie s abou admission p ob-
abili ies only a ec whe he esea che s can obse e s uden s’ p e e ences in he su ey,
bu do no al e he ela i e posi ions o hese p e e ences. Fo ins ance, suppose s uden
ihas a ue p e e ence o all schools as j1πij2πij3πi···, i.e., jkπijkwhen k<k
.I he
selec ed and anked schools in he su ey a e j2πij7πij9, he ela i e ank o any wo o
hese schools s ill p ese es he ela ionship: jkπijkwi h k<k
, ega dless o how hese
schools a e selec ed in o he su ey. The ankings a e independen o he se o schools
selec ed when he op i e schools in a easible se a e chosen. We assume ha he p o-
cess o selec ing he easible se is independen o p e e ences. The equa ion indica es
ha one’s ela i e p e e ence o e any wo schools is independen o he se o schools
picked in he su ey. No e ha we do no assume he selec ion o easible schools in he
su ey is independen o s uden s’ sco es o ha s uden s wi h he same sco es ank he
same se o schools.
Gi en Assump ion 1, while e e ing o he su ey da a, we use he ank-o de ed logi
model (Beggs, Ca dell, and Hausman (1981)) o es ima e β.42 Gi en a su eyed s uden
i’s anked school lis (j1,,jl)o leng h l≤5, we conclude ha j1is he a o i e school
among all he lschools on he su ey lis , ha j2is he second- a o i e school, and so
42Because cij =c0in his s ep, αdoes no appea in he u ili y unc ion.
1172 Wang and Zhou Quan i a i e Economics 15 (2024)
on. The join p obabili y o hese choices is
P (ui,j1>u
i,j2>···>u
i,jl)=
l−1

k=1
eμi,jk
eμi,jk+eμi,jk+1+···+eμi,jl,(3)
whe e μi,jis he de e minis ic componen o ui,jo ui,o.43 Then he log-likelihood unc-
ion can be w i en as
logL1(β)=
n

i=1
li−1

k=1
μi,jk−
n

i=1
li−1

k=1
logli

s=k
eμi,js.(4)
Now we can es ima e βusing maximum likelihood es ima ion.44
5.2 S ep wo: Es ima ing he ZX- ela ed pa ame e s α
In he second s ep, we es ima e αwhile conside ing s uden s’ s a egic beha io in he
admission p ocedu e. A e plugging he es ima ed ˆ
βin o equa ions (1)and(2), we can
ew i e s uden i’s u ili y unc ion as
ui,j=ˆ
ui,j+
k
αkxk
i(cij −c0)+εij ,(5)
ui,o=ˆ
Fo+εio,(6)
whe e ˆ
ui,j=lˆ
βlyl
j+wˆ
βwxw
iyw
j+ˆ
βDdij .
Calsamiglia, Fu, and Güell (2020) ind ha 96% o s uden s in Ba celona a e s a egic
playe s, and Abdulkadi o˘
glu, Aga wal, and Pa hak (2017) use s uden s’ epo ed p e e -
ences as hei ue p e e ences unde he DA mechanism o es ima e he pa ame e s
based on he assump ion ha s uden s maximize hei expec ed u ili ies gi en hei
belie s abou admission p obabili y (see Sec ion VI.A in Abdulkadi o˘
glu, Aga wal, and
Pa hak (2017)). The e o e, in ligh o he e idence ( om Sec ion 4.3) ha ew s uden s
epo hei ue p e e ences when submi ing ROLs, we model hei s a egic beha io
by assuming ha hey submi ROLs ha a e op imal gi en hei belie s abou hei like-
lihood o admission. The assump ion o a ional expec a ions will be u he suppo ed
when we calcula e he likelihood unc ion. Ou analysis e eals ha only a small num-
be o s uden s (1–2%) adop weakly domina ed s a egies in hei ROLs. The li e a u e
sugges s ha s uden s may hold he e ogeneous belie s abou hese p obabili ies (Ka-
po , Neilson, and Zimme man (2020)) o make mis akes in hei ROLs (Hwang (2015);
A emo , Che, and He (2017)). Howe e , de ining “mis akes” in ou adminis a i e da a
is challenging since s uden s submi hei ROLs be o e aking he exams, and a s uden
may accu a ely es ima e admission cu o s bu ace unce ain y abou hei exam pe -
o mance. The e o e, we make he ollowing assump ion.
43Mo e p ecisely, μi,j=lβlyl
j+wβwxw
iyw
j+βDdij when jis a public high school and μi,j=Fowhen
jis no a public high school.
44We assume ha he u ili y unc ion has an addi i ely sepa able o m. I is hus easy o demons a e ha
logL1is globally conca e in β— om which i ollows ha he e exis s a unique maximum o he likelihood
unc ion.
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1173
Assump ion 2. S uden s a e ully in o med abou hei own p e e ences,and hey max-
imize hei expec ed u ili y in a a ional manne .
S uden s’ decision p oblem Ini ially, s uden isubmi s he ROL ai={(j1, 1),
(j2, 2),(j3, 3), };he e k∈{0, 1}indica es whe he iselec s he ZX op ion o he k h
choice jk,and ∈{0, 1}indica es whe he iaccep s a andom assignmen i she is e-
jec ed by all h ee o he chosen schools.45 Subsequen ly, each s uden akes he en-
ance exam and ecei es a sco e si. The decision p oblem o s uden iis o maximize
he expec ed payo by selec ing he bes choice ai om he se o all possible choices
Ai.
max
ai∈Ai
EU(ai,si)(7)
Since he choice se o each indi idual is la ge han 100,000, we ollow he back-
wa d induc ion me hod de eloped by Calsamiglia, Fu, and Güell (2020) o add ess he
cu se o dimensionali y o he empi ical school choice p oblem. This app oach ollows
he logic ha al hough he en i e ROL is submi ed all a once, he anked schools on he
lis a e conside ed sequen ially in he p ocedu e. The k h lis ed school and i s ZX op ion
a e ele an only i he s uden is ejec ed by all p e iously lis ed schools. The e o e, he
k h choice should be he s uden ’s bes choice condi ional on eaching his s age, and
consequen ly, he p oblem can be sol ed ia backwa d induc ion. The s uden ’s decision
p oblem can be ew i en as
Vk(si)=max
(jk, k)¯
Pk
i·Uk
i+1−¯
Pk
i·ˆ
I·Vk+1(si)(8)
wi h k∈{1, 2, 3},and
V4(si)=max
∈{0,1}I =1·EUl
i+(1−I =1)·
ui(9)
Equa ion (8) indica es ha s uden ineeds o choose school jkand i s ZX op ion
k o maximize he alue unc ion Vk.¯
Pk
iand Uk
ia e ec o s o p obabili ies and u ili-
ies, espec i ely, associa ed wi h ia ending school jka di e en ui ion le els.46 He e,
ˆ
I=(1, 1, 1, 1).Equa ion(9) indica es ha i s uden iis ejec ed by all h ee choices, she
needs o decide whe he o accep a andom assignmen o a le o e school. EUl
i ep e-
sen s he expec ed payo when iis andomly assigned o a le o e school, 
ui ep esen s
he payo when ihas los all chances o a end a school in he ma ching p ocedu e, and
I =1is an indica o unc ion o whe he s uden chooses o accep he andom assign-
men .
45We d op subsc ip i o schools and ZX op ions o simplici y.
46Mo e p ecisely, ¯
Pk
i=(Pk
i,c0,Ik·Pk
i,c1,Ik·Pk
i,c2,Ik·Pk
i,c3).Pk
i,c ep esen s he p obabili y o s uden ibe-
ing admi ed o he k h choice wi h ui ion c , gi en ha she has been ejec ed by he p e ious choices
and/o ui ion le els. Ikis an indica o unc ion ha de e mines whe he ichooses he ZX op ion o jk.
Uk
i=(ui,jk,c0,ui,jk,c1,ui,jk,c2,ui,jk,c3). I is wo h no ing ha we sligh ly abuse he no a ion he e: i s uden
ichooses a nonpublic school a any posi ion, we can eplace ui,j,c0in equa ion (8)wi hui,oas de ined in
equa ion (6) and se k=0.
1174 Wang and Zhou Quan i a i e Economics 15 (2024)
Admission p obabili ies and belie s Gi en s uden i’s ROL ai, he condi ional p obabil-
i y o he being admi ed o he k h choice as a no mal s uden is
Pk
i,c0=P ¯
Sjk≤si|iis ejec ed by jk−1. (10)
He e, ¯
Sjk ep esen s he cu o o jk.47 Equa ion (10) indica es ha ia ends he k h
choice school as a no mal s uden i and only i he sco e is no less han he school’s
cu o condi ional on being ejec ed by k−1 h choice, ha is, si<¯
Sjk−1i idoes no
choose he ZX op ion o jk−1o si<¯
Sjk−1−30 o he wise.48
The condi ional p obabili y o s uden ibeing admi ed o jkas a ZX s uden , wi h
ui ion c whe e ∈{1, 2, 3},is
Pk
i,c =P ¯
Sjk−10 ≤si|si<¯
Sjk−10( −1). (11)
F om he pe spec i e o s uden i, we assume ha she an icipa es he exam sco e o
be mi+ηi;he e,mi ep esen s ei he i’s mock exam sco e o he ue abili y (by which
she es ima es he exam sco e) and ηiis he unce ain y. We assume ha ηiis i.i.d. and
dis ibu ed no mally as N(0, δ).No e ha micanno be di ec ly obse ed om he da a.
Ins ead, we use he s uden ’s ac ual exam sco e sias he p oxy o mi. We simpli y ou es-
ima ion p ocess by se ing δ=20, which is 3% o he ull ma k.49 A e we eplace siwi h
si+ηiin equa ion (10) o(11), he admission p obabili ies can be exp essed as he CDF
o he s anda d no mal dis ibu ion (see Supplemen al Appendix D o he unc ional
o ms).
S uden s assess hei chances o being admi ed o each school be o e submi ing
hei ROLs. In line wi h much o he school choice li e a u e, we assume ha s uden s
conside admission p obabili ies as exogenous, meaning hey can p ecisely o ecas he
equilib ium admission cu o s.50 While his assump ion is no wi hou i s complexi ies
and limi a ions, i enables us o es ima e s uden s’ p e e ences wi hou he need o ex-
plici ly sol e o he equilib ium, which can be nonunique in many nons a egyp oo
ma ching mechanisms. Sol ing o he equilib ium and selec ing he app op ia e one
can be compu a ionally in ensi e. The p ice- aking assump ion o e s he ad an age o
simpli ying he analysis and is commonly employed in he li e a u e o his eason.
In ou s udy, he admission cu o s o schools a e made public a e he annual ad-
mission season. Analyzing he p e ious yea ’s da a, we obse ed ha he majo i y o
popula schools’ cu o sco es (wi h one excep ion) in he pe iod be ween 2011 and 2013
47No e ha because he CPPS mechanism is an ex ension o he CP mechanism wi h an execu i e pe iod
(2,1), he schools’ cu o s used o he i s wo choices a e gene a ed a e conside ing all s uden s’ second
choices. Fo he hi d choice, he schools’ cu o is gene a ed a e conside ing all s uden s’ hi d choices.
The simila calcula ion app oach o he gene al Chinese pa allel mechanism can be ound in Calsamiglia,
Fu, and Güell (2020) and hei Supplemen al Appendix.
48When k=1, equa ion (10) becomes he uncondi ional p obabili y: P (¯
Sj1≤si).
49The es ima ed esul s when δ=13.3 (2% o he ull ma k), when δ=26.6 (4% o he ull ma k), and
when δ=33.35 (5% o he ull ma k) a e epo ed in Sec ion 5.4.
50Abdulkadi o˘
glu, Ang is , Dyna ski, Kane, and Pa hak (2011a); Aze edo and Ha ield (2018); Kojima
(2017); Aga wal and Somaini (2018); and Calsamiglia, Fu, and Güell (2020).
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1181
Table 6. Admission cu o s.
Wi hin Sample Ou o Sample
2012 2013 2012 2014
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
School T ue P edic ed Di T ue P edic ed Di . T ue P edic ed Di . T ue P edic ed Di .
141 607.0 603.4 3.6 604.0 598.1 5.9 607.0 603.2 3.8 605.0 600.2 4.8
142 535.0 535.0 0.0 530.0 530.0 0.0 535.0 535.0 0.0 535.0 535.0 0.0
147 555.5 555.0 0.5 552.5 558.0 −5.5 555.5 551.6 3.9 558.0 555.9 2.1
167 592.5 592.3 0.2 590.0 587.2 2.8 592.5 591.8 0.7 593.5 589.6 3.9
173 535.0 535.0 0.0 530.0 530.0 0.0 535.0 535.0 0.0 552.0 547.3 4.7
177 597.0 591.7 5.3 590.5 585.8 4.7 597.0 591.2 5.8 600.5 597.3 3.2
179 571.5 570.7 0.8 565.0 567.0 −2.0 571.5 570.4 1.1 573.5 568.2 5.3
181 535.0 535.0 0.0 530.0 530.0 0.0 535.0 535.0 0.0 535.0 535.0 0.0
183 617.0 613.0 4.0 611.0 606.5 4.5 617.0 613.0 4.0 611.0 609.6 1.4
184 535.0 535.0 0.0 530.0 530.0 0.0 535.0 535.0 0.0 535.0 535.0 0.0
185 583.0 579.5 3.5 580.0 574.0 6.0 583.0 579.7 3.3 583.0 576.5 6.5
186 583.0 576.7 6.3 578.0 571.8 6.2 583.0 575.9 7.1 576.0 573.3 2.7
187 599.5 598.5 1.0 594.5 594.9 −0.4 599.5 598.3 1.2 596.5 593.3 3.2
188 571.5 583.7 −12.2 575.0 571.7 3.3 571.5 582.2 −10.7 580.0 577.2 2.8
28†594.5 587.0 7.5 589.0 579.3 9.7 594.5 586.4 8.1
165†608.5 613.5 −5.0 605.5 608.6 −3.1 608.5 613.5 −5 609.0 607.8 1.2
166†595.0 594.5 0.5
169†599.0 596.6 2.4 604.0 603.7 0.3
180†576.5 577.9 −1.4 584.5 588.6 −4.1
200†607.0 607.8 −0.8
No e: This able indica es he wi hin- and ou -o -sample es s o he schools’ cu o s, using he es ima ed coe icien s in
column 3 o Table 5. The ull ma k is 665. The h eshold is 535 in 2012 and 2014, and 530 in 2013. †indica es he special class.
The numbe o special classes a ies wi h yea s.
s ands o eason ha hei main in o ma ion sou ce leans on da a om p e ious yea s
(adap i e expec a ions). In he second panel o Table 7,incolumn1,wep esen hepa-
ame e s es ima ed in ela ion o he ZX op ion, assuming s uden s base hei es ima ed
chances o admission on he p io yea ’s cu o s.63 These esul s align closely wi h hose
s emming om a ional expec a ions. This alignmen indica es consis en s abili y in
admission cu o s o e he yea s and a i ms ha s uden s’ pe cep ions, o med using
he p eceding yea ’s cu o s, yield ai ly accu a e es ima es.
Unce ain y ied o hei exam sco es can shape he s a egic beha io o s uden s.
In he second panel o Table 7, columns 2–4 in he second panel o Table 7showcase
pa ame e s es ima ed in ela ion o he ZX op ion. He e, he s anda d de ia ion δo he
exam sco es is designa ed as 13.3 (2% o he ull ma k), 26.6 (4% o he ull ma k), and
33.35 (5% o he ull ma k), espec i ely. These esul s exhibi a simila pa e n o he
indings p esen ed in Table 5.
63No e ha ou es ima ion o he non-ZX- ela ed pa ame e s does no ely on any assump ions abou
s uden belie s.

1182 Wang and Zhou Quan i a i e Economics 15 (2024)
Table 7. Es ima ed Pa ame e s in he Robus ness Check
Repu a ion ×HS 0.225 Capaci y ×HS 0.365
(0.054) (0.545)
Repu a ion ×MS 0.094 Capaci y ×MS −0.808
(0.022) (0.248)
Repu a ion ×LS 0.052 Capaci y ×LS −0.569
(0.023) (0.176)
Special class ×HS −6.625 Dis ance −1
(2.312)
Special class ×MS 1.060 Dis ance ×Male 0.794
(2.072) (0.036)
Special class ×LS 6.377 Do m 4.893
(8.139) (1.209)
Sco e ange −0.043 Do m×Male 0.791
(0.445) (0.318)
Sco e ange ×Male 0.534
(0.554)
Same dis ic −1.888
(0.234)
Same dis ic ×Male 1.794
(0.298)
Non-public high school −9.113 School Fixed E ec Y
(1.192)
(1) (2) (3) (4)
Cos ×HS −1.242 −1.011 −1.162 −1.163
(0.008) (0.017) (0.009) (0.010)
Cos ×MS −1.430 −1.171 −1.391 −1.391
(0.006) (0.015) (0.013) (0.013)
Cos ×LS −1.580 −1.423 −1.674 −1.673
(0.007) (0.015) (0.012) (0.012)
Cos ×HHP −0.671 −0.635 −0.694 −0.694
(0.006) (0.012) (0.013) (0.013)
Cos ×MHP −1.051 −1.044 −1.121 −1.121
(0.007) (0.012) (0.010) (0.012)
Cos ×LHP −1.992 −1.801 −1.957 −1.955
(0.008) (0.018) (0.015) (0.013)
School Fixed E ec Y Y Y Y
No e: The i s panel is he es ima ed esul s based on he admi ed s uden quali ies om p e ious yea s as he school
epu a ion measu e. The second panel is he es ima ed esul s o di e en assump ions abou s uden s’ beha io s in ROLs.
Column 1 ep esen s he adap i e expec a ion assump ion. Column 2–4 ep esen s he s.d. o he unce ain y o exam sco e
a e 13.3, 26.6, and 33.35, espec i ely. S anda d e o s a e epo ed in pa en heses. Dis ance is measu ed by kilome e . The
coe icien o emale’s a i ude o home-school dis ance is no malized o -1. Cos ( ui ion) is measu ed by 1000 yuan. “HS,”
“MS,” and “LS” ep esen high-, medium-, and low-sco ing s uden s, espec i ely. “HHP,” “MHP,” and “LHP” ep esen s uden s
om high-, mode a e-, and low-housing p ice communi ies, espec i ely.
6. Coun e ac ual analysis
To assess he impac o he ZX policy on s uden wel a e, we conduc simula ions o
compa e di e en mechanisms. Speci ically, we use wo benchma k mechanisms o
compa ison: he DA mechanism and he cade -op imal s able mechanism (COSM).
Economis s widely ecommend he DA mechanism, known o i s s a egyp oo na u e.
Unde he DA mechanism, s uden s ha e a weakly dominan s a egy o u h ully e-
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1183
po ing hei p e e ences. Empi ical s udies sugges ha s uden s’ s a egies in p ac ice
align closely wi h heo e ical p edic ions (Abdulkadi o˘
glu, Pa hak, and Ro h (2009); Ab-
dulkadi o˘
glu, Aga wal, and Pa hak (2017); Pa hak and Shi (2021)). The e o e, he DA
mechanism is commonly used as a benchma k in coun e ac ual analyses o di e en
mechanisms (Aga wal and Somaini (2018)). Pa hak and Shi (2021) analyze he e ec i e-
ness o such coun e ac ual analyses o he school choice p oblem. Following a simila
app oach o Calsamiglia, Fu, and Güell (2020), we simula e he s uden s’ ue- elling
s a egy unde he DA mechanism o compa e he wel a e om o he mechanisms. The
COSM, ano he benchma k mechanism, is an ex ension o he DA mechanism. Unde
he COSM, playe s s ill ha e a weakly dominan s a egy o u h ully epo hei p e -
e ences. Sönmez and Swi ze (2013) p o ide e idence ha he s a egies o US mili a y
cade s la gely con o m o he heo e ical p edic ions. The e o e, we assume ha s u-
den s will u h ully epo hei p e e ences in hei ROLs unde he COSM mechanism.
Based on he es ima ed p e e ences, we simula e s uden s’ applica ion lis s. Fo he
simula ion, we u ilize he s uden and school p o iles om he 2014 adminis a i e da a.
Gi en he absence o ZX s uden s ha yea , we ea he no mal admission quo a as he
school’s o al capaci y. To analyze he wel a e impac o di e en ZX quo as, we con-
duc expe imen s unde wo se ups: one wi h he ZX quo a ep esen ing 10% o he o al
quo a and ano he wi h he ZX quo a ep esen ing 30% o he o al quo a when he ocal
mechanism allows he op ion o pu chase sea s.64
Fo bo h he DA and COSM mechanisms, we assume ha s uden s’ ROLs e lec hei
genuine p e e ences. Unde he CPPS mechanism, we c ea e ROLs ha e lec each s u-
den ’s bes esponse in equilib ium. Speci ically, we begin wi h he “ elling he u h”
s a egy as he ini ial poin and i e a i ely calcula e each s uden ’s bes esponse while
keeping all o he s uden s’ s a egies ixed. I a s uden has an incen i e o adop a new
s a egy, we eplace hei old s a egy wi h his new one and ecalcula e he ma ching
ou come, se ing i as he new s a ing poin . We epea his i e a ion un il no s uden
de ia es o a new s a egy (see Supplemen al Appendix E o de ails). We hen ca y
ou 5000 simula ions, wi h each s uden subjec ed o a unique ec o o andom u il-
i y shocks.
We conside wo compa isons in his sec ion. Fi s , we use he ma ching ou come
unde he DA mechanism as ou benchma k. Then we eplace he DA mechanism wi h
he COSM, which se es as an al e na i e s a egyp oo and s able mechanism. Wi hou
conside ing s uden s’ s a egies, his compa ison e alua es he sea -selling policy i sel ,
and using di e en ZX quo as u he enables he s udy o wel a e unde a ious policies.
To e alua e he mechanisms ac ually adop ed, we also analyze he wel a e changes when
he CPPS mechanism eplaces he DA mechanism. This eplacemen may e lec how
s uden s’ s a egic beha io s may change unde di e en mechanisms and he wel a e
consequences. These wo compa isons p o ide a comp ehensi e analysis o he en i e
ZX policy. The ansi ion om he DA mechanism o he COSM e alua es he e ec s
o implemen ing a p ice menu in he sea -pu chasing policy, while he ansi ion om
he DA mechanism o he CPPS mechanism assesses he in luence o s uden s’ s a egic
beha io s unde he CPPS.
64The local go e nmen equi ed ha no school could admi ZX s uden s o aling mo e han 20% o i s
capaci y.
1184 Wang and Zhou Quan i a i e Economics 15 (2024)
Figu e 3. Wel a e change. No es: These igu es ep esen he wel a e change when DA is e-
placed by ano he mechanism measu ed by he wel a e-equalizing ui ion adjus men . The
y-axis ep esen s he change o yuan, and he x-axis ep esen s he ZX quo a. To keep he wel a e
le el unde he DA mechanism, a posi i e posi ion indica es a s uden needs o pay addi ional
ui ion (loss), a nega i e posi ion indica es a s uden ecei es a ui ion deduc ion (gain). “HHP,”
“MHP,” and “LHP” ep esen s uden s om high-, mode a e-, and low-housing p ice communi-
ies.
S uden s’ wel a e
Fo each es ed mechanism, we employ he wel a e-equalizing ui ion adjus men
yuan, as p oposed by Calsamiglia, Fu, and Güell (2020), o quan i y he wel a e change.
This adjus men ep esen s he ui ion amoun ha a s uden would need o pay (o be
c edi ed) unde he DA mechanism o a ain he same u ili y le el as unde he eplace-
men mechanism being e alua ed.65
Fi s , we examine he compa ison be ween he DA mechanism and he COSM,
which di ec ly e alua es he ZX policy wi hou conside ing s uden s’ s a egies. As
showninFigu e3(a), when he DA mechanism is eplaced by he COSM, he a e age
wel a e o s uden s dec eases as he ZX quo a inc eases. On a e age, s uden s unde he
DA mechanism need o pay an addi ional 30 yuan o each he same u ili y le el as un-
de he COSM when he ZX quo a is 10% o he o al quo a. This loss inc eases o 57 yuan
when he ZX quo a ises o 30%. S uden s om di e en communi ies exhibi a simila
end o wel a e loss as he ZX quo a inc eases, wi h s uden s om he LHP communi ies
expe iencing ela i ely less loss compa ed o s uden s om o he communi ies.
To u he analyze he impac o he ZX policy on di e en s uden s, we examine he
school assignmen s when he DA mechanism is eplaced by he COSM. The i s panel
o Table 8shows ha wi h a 10% ZX quo a, app oxima ely 1.8% o 2.2% o s uden s om
he HHP and MHP communi ies choose o pay highe ui ion o secu e hei sea s in he
65All o he pa ame e s (excep o ui ion) emain ixed. Fo mally, le uij =U(cij )be i’s u ili y de i ed
om admi ance o school jwhen paying ui ion cij unde he DA mechanism. I ha mechanism is e-
placed by he ocal new mechanism—in which case s uden iis assigned o school jand achie es u il-
i y uij— hen he wel a e-equalizing ui ion adjus men (yuan) is he solu ion o U(cij +yuan)=uij .
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1185
same schools, while only 0.7% o s uden s om he LHP communi ies make he same
choice. Unde he same ZX quo a, a ound 4% o s uden s in each communi y a e unable
o a end hei mo e p e e ed schools due o he inc eased compe i ion o sea s. On
he o he hand, 4.8% o s uden s om he HHP communi ies a e able o secu e spo s
in hei mo e p e e ed schools, a highe pe cen age compa ed o s uden s om o he
communi ies (4% in he MHP communi ies and 3.5% in he LHP communi ies). When
he ZX quo a is inc eased o 30%, an addi ional 2% o 3% o s uden s om each com-
muni y choose o pay highe ui ion o s ay in he same schools, and a simila inc ease
is obse ed o s uden s who a e unable o a end hei mo e p e e ed schools and a e
ins ead assigned o less p e e ed ones. Howe e , 9% o s uden s om he HHP commu-
ni ies may ge in o hei mo e p e e ed schools by paying highe ui ion, and only 6%
o s uden s om he LHP and MHP communi ies can ake ad an age o he ZX policy in
hesameway.
In each ype o communi y, he impac o he ZX policy a ies o s uden s wi h di -
e en sco es. When he ZX quo a inc eases om 10% o 30%, high-sco ing s uden s om
he LHP a ea a e he mos a ec ed. A ound 14% o hese s uden s choose o pay highe
ui ion o secu e hei sea s a he same schools, while 21% o hem a e unable o a -
end hei mo e p e e ed schools and a e p iced ou . Ano he in e es ing inding is ha
he ZX policy has a signi ican in luence on medium-sco ing s uden s ac oss all com-
muni ies. Compa ed o high- and low-sco ing s uden s, a la ge p opo ion o medium-
sco ing s uden s choose o pay highe ui ion o a end hei p e e ed schools unde he
ZX policy. Howe e , a he same ime, many medium-sco ing s uden s a e also a ec ed
nega i ely and a e p iced ou , esul ing in hem being assigned o hei less p e e ed
schools.
When he DA is eplaced by he COSM, high-sco ing s uden s expe ience a signi -
ican wel a e loss unde he COSM, p ima ily because mos o hem we e al eady as-
signed o hei mos p e e ed schools unde he DA mechanism. Unde he COSM,
hese s uden s mus ei he pay highe ui ion o secu e hei sea s in he same schools
o ace being p iced ou and assigned o less p e e ed schools. In con as , medium-
sco ing s uden s a e in luenced in di e en ways. They ha e a highe p obabili y o a -
ending hei p e e ed schools by paying highe ui ion, bu hey a e also suscep ible o
being p iced ou and assigned o hei less p e e ed schools.
The es ima ed coe icien s sugges ha he willingness o pay highe ui ion in-
c eases wi h bo h s uden s’ sco es and housing p ices. As a esul , when he ZX quo a
is inc eased, he high-sco ing s uden s om he LHP ha e a s onge incen i e o secu e
hei sea s compa ed o s uden s om he same communi ies; hence a la ge p opo ion
o hem choose o pay highe ui ion. Howe e , hey a e also mo e likely o be p iced
ou since hei incen i e o pay highe ui ion is lowe han ha o high-sco ing s uden s
om a luen communi ies. Consequen ly, a la ge p opo ion o high-sco ing s uden s
om LHP communi ies a e p iced ou . The acan sea s le by hese high-sco ing s u-
den s a e mos ly occupied by medium-sco ing s uden s om a ious communi ies. A
he same ime, a compa able numbe o medium-sco ing s uden s a e also p iced ou
due o he inc eased ZX quo a. The e o e, he in luence o he ZX policy on medium-
sco ing s uden s is subs an ial in bo h di ec ions.
1186 Wang and Zhou Quan i a i e Economics 15 (2024)
Table 8. Changes o ma ching assignmen s unde he pu chasing sea s op ion(%).
ZX Quo a 10% ZX quo a 30%
Same Be e Wo se Same Be e Wo se
ZX No mal ZX No mal ZX ZX No mal ZX No mal ZX
DA-COSM
HHP 2.2 0.0 4.8 3.9 0.0 4.3 0.0 9.0 5.9 0.0
HS 2.2 0.0 2.6 3.2 0.0 6.6 0.0 7.6 7.2 0.0
MS 3.7 0.0 7.7 6.7 0.0 4.8 0.0 12.1 8.1 0.0
LS 0.4 0.0 4.2 0.9 0.0 0.3 0.1 7.0 0.8 0.0
MHP 1.8 0.0 4.0 4.1 0.0 3.3 0.0 6.6 7.1 0.0
HS 2.1 0.0 3.0 2.4 0.0 6.0 0.0 6.5 9.6 0.0
MS 3.4 0.0 6.2 8.9 0.0 4.6 0.0 10.0 11.6 0.0
LS 0.1 0.0 2.8 1.1 0.0 0.1 0.1 3.6 0.9 0.0
LHP 0.7 0.0 3.5 4.6 0.0 2.4 0.0 5.9 7.4 0.0
HS 2.0 0.0 0.1 6.7 0.0 14.3 0.0 0.4 21.6 0.0
MS 1.3 0.0 9.4 10.3 0.0 2.2 0.0 15.7 13.3 0.0
LS 0.0 0.0 0.4 0.4 0.0 0.0 0.0 0.6 0.4 0.0
DA-CPPS
HHP 0.6 3.4 5.7 10.5 0.0 6.2 2.0 10.3 10.8 0.0
HS 1.0 0.0 3.4 9.7 0.0 8.5 0.0 9.5 10.4 0.1
MS 0.5 6.5 9.7 16.8 0.0 8.0 3.6 14.1 18.6 0.0
LS 0.0 4.2 3.8 2.8 0.0 0.1 3.0 6.2 0.6 0.0
MHP 0.5 4.3 4.1 8.0 0.0 3.8 2.9 7.0 11.1 0.0
HS 1.2 0.0 3.4 7.7 0.0 7.4 0.0 7.4 13.7 0.1
MS 0.3 8.1 7.8 15.0 0.0 4.8 5.2 11.5 20.2 0.0
LS 0.0 4.4 1.4 1.9 0.0 0.0 2.9 2.6 0.6 0.0
LHP 0.0 4.0 3.0 7.0 0.0 2.2 2.8 4.6 10.3 0.0
HS 0.1 0.0 0.0 9.4 0.0 16.8 0.0 1.1 22.3 0.1
MS 0.0 8.5 7.9 15.7 0.0 1.0 5.9 11.7 21.5 0.0
LS 0.0 1.9 0.3 0.8 0.0 0.0 1.4 0.5 0.3 0.0
No e: This able indica es he pe cen age change in he numbe o s uden s whose assignmen s a e di e en unde he
pu chasing sea s op ion, when he DA mechanism is eplaced by he COSM and CPPS mechanisms. When DA is eplaced by
ano he mechanism, “Same” means he s uden is assigned o he same school, “Be e ” ep esen s he s uden is assigned o a
mo e p e e ed school, and “Wo se” ep esen s he s uden is assigned o a less p e e ed school. “ZX” and “No mal” ep esen s
he s uden pays he basic and highe ui ion, espec i ely. “HHP,” “MHP,” and “LHP” ep esen s uden s om high-, mode a e-,
and low-housing p ice communi ies, espec i ely. “HS,” “MS,” and “LS” ep esen high-, medium-, and low-sco ing s uden s,
espec i ely.
Table 9iden i ies he pe cen age o “winne s” (s uden s whose wel a e inc eases)
and “lose s” (s uden s whose wel a e dec eases) when he DA mechanism is eplaced.
Rega dless o he ZX quo a, he p opo ion o winne s unde he COSM ne e exceeds
8.7% o any s uden g oup. Howe e , he p opo ion o lose s exceeds ha o winne s in
all cases.
When he ZX quo a is inc eased om 10% o 30%, he p opo ions o lose s in he
HHP and MHP communi ies expe ience a subs an ial ise, while he change is ela i ely
small o he LHP communi ies. The analysis o he wel a e change in each ype o com-
muni y (Table F.2) u he con i ms he explana ion o he e ec o he ZX policy. Mo e
han 35% o high-sco ing s uden s om poo communi ies become lose s and expe i-
ence an a e age wel a e loss o 1587 yuan when he ZX quo a inc eases. This wel a e

Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1187
loss p ima ily a ises om a la ge p opo ion o s uden s who mus ei he pay highe u-
i ion o secu e hei sea s o be p iced ou . The a e age wel a e gain o medium-sco ing
s uden s is sligh ly highe han he losses, excep o one communi y.
In summa y, when he DA mechanism is eplaced by he COSM, he a e age impac
on s uden s om di e en esiden ial a eas is simila . The numbe o s uden s who ex-
pe ience a wel a e loss due o he ZX policy is highe han he numbe o s uden s who
po en ially bene i om i , and his loss is magni ied as he ZX quo a inc eases. How-
e e , s uden s’ eac ions o he policy a y. Medium-sco ing s uden s a e he mos a -
ec ed g oup by he ZX policy. They a e mo e likely han low-sco ing s uden s o a end
hei p e e ed schools by paying highe ui ion, bu hey a e also mo e suscep ible o
being p iced ou and assigned o less p e e ed schools. Top-pe o ming s uden s om
economically disad an aged communi ies bea he g ea es bu den unde he ZX pol-
icy. A signi ican p opo ion o hem ei he ha e o pay highe ui ion o secu e sea s in
hei p e e ed schools o a e p iced ou al oge he . Compa a i ely, s uden s om a lu-
en communi ies a e mo e likely o s ay a hei desi ed schools by paying highe ui ion,
while s uden s om o he communi ies a e mo e likely o be p iced ou and assigned o
less p e e ed schools. These indings indica e ha while he ZX policy in ensi ies edu-
ca ional inequali y among s uden s, i s o e all impac on hei wel a e, when assessed
in mone a y e ms, is no as p onounced.
Nex , we in es iga e how he p ac ical implemen a ion o he mechanism may im-
pac s uden s’ wel a e and hei s a egic beha io s. When he DA mechanism is e-
placed by he CPPS mechanism, he changes in s uden wel a e exhibi a simila pa -
e n as obse ed in he COSM case, bu wi h a no able di e ence (Figu e 3b). S uden s
om he MHP and LHP communi ies expe ience a wel a e gain o 22 yuan and 63 yuan,
espec i ely, when he ZX quo a is 10%.66
Howe e , as he ZX quo a inc eases o 30%, all s uden g oups ace wel a e losses,
pa icula ly s uden s om he HHP and MHP communi ies. S uden s om he HHP
communi ies endu e a wel a e loss equi alen o a 216 yuan inc ease in ui ion, while
s uden s om he MHP communi ies expe ience a loss o 128 yuan.
The second panel o Table 8explains he easons o he imp o ed s uden wel a e
unde he CPPS mechanism compa ed o he COSM a lowe ZX quo as. Unlike unde
he COSM, a signi ican ly lowe p opo ion o s uden s choose o sa e hei sea s in he
same schools by paying highe ui ion unde he CPPS mechanism, ega dless o s u-
den g oups. Meanwhile, mo e s uden s a e p iced ou o hei less p e e ed schools
unde he CPPS mechanism compa ed o he COSM. Addi ionally, a posi i e numbe o
s uden s om e e y ype o communi y a e able o secu e spo s in hei mo e p e e ed
schools wi hou paying ex a ui ion, which is no he case unde he COSM. Howe e ,
when he ZX quo a is inc eased o 30%, a la ge numbe o s uden s, pa icula ly hose
66Abdulkadi o˘
glu, Che, and Yasuda (2011b,2015) sugges ha om an ex an e pe spec i e, when schools
ha e coa se p e e ences o s uden s coupled wi h a symme ic ie-b eaking ule, s uden s could a e be e
unde he Bos on mechanism han unde he DA mechanism, as assessed by hei ca dinal p e e ences. In
con as , ou esul s show ha when schools ha e s ic p io i ies o s uden s, a manipulable mechanism
such as he CPPS can s ill yield highe a e age s uden wel a e o some ypes o s uden s han he DA
mechanism om an ex pos pe spec i e.
1188 Wang and Zhou Quan i a i e Economics 15 (2024)
om he HHP and MHP communi ies, choose o pay highe ui ion o s ay in he same
schools. Simul aneously, mo e s uden s ac oss all communi ies a e p iced ou compa ed
o he COSM. Consequen ly, all s uden s expe ience g ea e wel a e losses when he ZX
quo a is high.
Table 9 u he con i ms ha he numbe o winne s and lose s inc eases in all s u-
den g oups. These indings sugges ha when he CPPS mechanism is used o eplace
he DA mechanism, s uden s may ha e mo e oppo uni ies o s a egically manipula e
hei p e e ences, leading o g ea e a ia ions in s uden wel a e ac oss di e en com-
muni ies.
To u he in es iga e s uden s’ s a egic beha io s unde he CPPS mechanism, we
also simula e he s uden s’ s a egies unde he Chinese pa allel mechanism as an in e -
media e s ep. Fo high-sco ing s uden s om HHP communi ies, hei i s choice unde
he Chinese pa allel mechanism is, on a e age, hei 1.12 choice in hei ue p e e ence
(Table F.3). Howe e , unde he CPPS mechanism, hei i s choice mo es sligh ly close
o hei ue i s choice a 1.01. This indica es ha mo e high-sco ing s uden s a e in-
clined o choose hei ue i s choice unde he CPPS mechanism. Addi ionally, almos
50% o hese s uden s choose he ZX op ions o hei i s choice. The a e age second
choice o his g oup is simila o hei hi d choice in hei ue p e e ence unde bo h
he Chinese pa allel and CPPS mechanisms. Howe e , unde he CPPS mechanism, 80%
o s uden s op o he ZX op ions o hei second choices.
S uden s’ i s choice unde he CPPS mechanism shows a sligh upwa d shi o
high- and medium-sco ing s uden s in all communi ies, bu a sligh downwa d shi o
low-sco ing s uden s. High-sco ing s uden s exhibi a highe likelihood o choosing he
ZX op ions, pa icula ly o hei second choices, and his pa e n dec eases wi h hous-
ing p ice. Meanwhile, low-sco ing s uden s a e less inclined o choose he ZX op ions,
especially o hei second choices. Howe e , when he ZX quo a ises om 10% o 30%,
he change in s a egic beha io s is no subs an ial. This phenomenon indica es ha
s uden s’ s a egic beha io unde he CPPS mechanism imp o es hei wel a e unde a
low ZX quo a. Howe e , he same s a egies lead o a subs an ial wel a e loss when he
ZX quo a is inc eased.
6.1 Impac on schools
In his inal sec ion, we examine he impac o he ZX policy on schools, conside ing wo
ac o s: he quali y o admi ed s uden s and he ui ion collec ed by schools. Schools
ace a ade-o in implemen ing his policy. On one hand, allowing s uden s o buy sea s
may inc ease he schools’ income. On he o he hand, sea pu chasing can lead o he
dispe sion o high-quali y s uden s ac oss di e en schools. Unde he ZX policy, some
high-sco ing s uden s who migh ha e a ended uppe - ie schools unde he DA mech-
anism may be p iced ou and end up in middle- ie schools i hey choose no o pay he
highe ui ion. Con e sely, some low-sco ing s uden s who a e willing o pay mo e u-
i ion o hei p e e ed schools may displace high-sco ing s uden s and secu e sea s in
hose schools. As a esul , uppe - ie schools may collec mo e ui ion bu expe ience a
decline in he o e all quali y o admi ed s uden s. While middle- ie schools may admi
mo e high-quali y s uden s.
Quan i a i e Economics 15 (2024) Pu chasing sea s in school choice and inequali y 1189
Table 9. Winne s and lose s.
DA-COSM DA-CPPS
10% 30% 10% 30%
WLWLWLWL
HHP % 4.5 8.3 8.7 14.3 8.4 14.9 8.4 27.6
MHP % 4.2 7.2 7.3 12.9 8.7 10.7 8.3 20.8
LHP % 3.9 4.0 5.8 6.6 7.2 5.8 6.2 9.4
To al % 3.9 6.3 6.9 10.9 7.8 10 7.3 18.6
HHP 895 −1040 1043 −1147 1297 −1148 1153 −1138
MHP 1064 −1106 1170 −1203 1459 −980 1307 −1142
LHP 734 −943 766 −1182 1477 −754 1253 −1026
To al 924 −1052 1036 −1076 1399 −1021 1237 −1124
No e: The i s panel o his able indica es he pe cen age change in he numbe o s uden s whose u ili ies inc ease (win-
ne s) o dec ease (lose s) when he DA mechanism is eplaced by he COSM and CPPS mechanisms. The second panel indica es
he wel a e change measu ed by yuan. “W” ep esen s winne s, and “L” ep esen s lose s. Fo each mechanism change, u il-
i y changes a e measu ed in h ee scena ios in which he ZX quo as a e 10% and 30% o he o al quo as. “HHP” ep esen s
s uden s om high housing p ice communi ies, ‘MHP” ep esen s s uden s om mode a e housing p ice communi ies, and
“LHP” s uden s om low housing p ice communi ies.
Fo uppe - ie school #183 (F.4 in Supplemen al Appendix F), he collec ed ees in-
c ease p opo ionally wi h he ZX quo a when he DA mechanism is eplaced by he
COSM. When he CPPS is adop ed, his school may gain e en mo e in e ms o ui ion
collec ion, wi h he gain exceeding 40% when he ZX quo a is 30%. Impo an ly, when
he DA mechanism is eplaced by ei he he COSM o he CPPS, his school expe iences
only a negligible decline in s uden quali y. Conside ing he indings o o he uppe -
ie schools (see Table F.4 in Supplemen al Appendix F), i becomes e iden ha he e
is a signi ican demand o eli e schools. This allows hem o p o i subs an ially om
selling sea s wi hou comp omising he quali y o admi ed s uden s.
Fo middle- ie school #179, he sea -pu chasing op ion has he po en ial o gen-
e a e p o i s. The impac on s uden quali y can a y depending on he mechanism
adop ed. When he ZX quo a inc eases o 30% unde he COSM, s uden quali y sligh ly
dec eases compa ed o i s le el unde he DA mechanism. On he o he hand, i he DA
mechanism is eplaced by he CPPS mechanism, he e is a small inc ease in s uden
quali y. Consequen ly, hese schools may expe ience signi ican a ia ions in he quali y
o hei admi ed s uden s, wi h some expe iencing posi i e changes and o he s nega-
i e changes.
7. Conclusion
Ou pape examines a con en ious bu p e iously o e looked Chinese school choice pol-
icy, Ze Xiao. This policy allows s uden s o “pu chase” sea s a hei desi ed schools by
paying highe ui ion. We ind ha he co esponding ma ching mechanisms employed
in his policy a e no s a egyp oo and may lead o uns able ou comes. We combine da a
om high school admission eco ds wi h su ey esponses om s uden s in China o
es ima e hei p e e ences o schools and ui ion. Ou indings e eal ha high-sco ing
s uden s a e mo e willing han o he s uden s o incu addi ional cos s, such as highe
1190 Wang and Zhou Quan i a i e Economics 15 (2024)
ui ion ees, o a end hei p e e ed schools. Fu he mo e, s uden s om communi ies
wi h high housing p ices a e less mo i a ed o bea he inancial bu den o highe ui ion
compa ed o s uden s om communi ies wi h low housing p ices.
Using es ima ed p e e ences, we conduc coun e ac ual expe imen s o e alua e
he wel a e consequences o he ZX policy. We ind ha , when he s a egy-p oo COSM
eplaces he DA mechanism, s uden s’ wel a e dec eases ac oss all s uden g oups. How-
e e , when he DA mechanism is subs i u ed by a non-s a egyp oo mechanism like
CPPS, i may alle ia e he wel a e losses, pa icula ly when he ZX quo a is low. This is
because mo e s uden s can exploi he sys em o secu e admission o hei p e e ed
schools.
When expe iencing a wel a e loss, s uden s om high housing p ice communi ies
end o op o paying highe ui ion o e ain hei sea s a he same schools. S uden s
om communi ies wi h low housing p ices a e mo e inclined o be p iced ou and se le
o less p e e ed schools. As he ZX quo a inc eases, high-sco ing s uden s om eco-
nomically disad an aged communi ies demons a e a g ea e mo i a ion o pay highe
ui ion in o de o emain in highe - anked schools compa ed o medium- and low-
sco ing s uden s.
F om he school’s poin o iew, he sea -pu chasing op ion p o es bene icial o
uppe - ie schools as i enables hem o collec a subs an ial amoun o addi ional u-
i ion while expe iencing only a mino decline in he quali y o admi ed s uden s com-
pa ed o he DA mechanism. Howe e , o o he schools, he sea -pu chasing op ion
in oduces g ea e unce ain y ega ding bo h he amoun o collec ed ui ion and he
quali y o s uden s admi ed.
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