Purchasing seats in school choice and inequality

Wang, Tong; Zhou, Congyi Article Purchasing seats in school choice and inequality Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Wang, Tong; Zhou, Congyi (2024) : Purchasing seats in school choice and inequality, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 15, Iss. 4, pp. 1151-1195, https://doi.org/10.3982/QE2220 This Version is available at: https://hdl.handle.net/10419/320324 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 15 (2024), 1151–1195 1759-7331/20241151 Purchasing seats in school choice and inequality Tong Wang College of International Relations, Ritsumeikan University Congyi Zhou Wilf Department of Politics, New York University We study a mechanism that gives students the option of paying higher tuition to attend their preferred schools. This seat-purchasing mechanism is neither strategyproof nor stable. Our paper combines administrative and survey data to estimate students’ preferences and conducts welfare analysis. We find that changing from a deferred acceptance mechanism to the cadet-optimal stable mechanism reduces students’ welfare but that adopting the observed seat-purchasing mechanism alleviates this welfare loss. Moreover, students from affluent communities prefer to pay higher tuition to stay at preferred schools, while those from less affluent communities are more likely be priced out to lower-quality schools. Keywords. School choice, market design, purchasing seat, inequality. JEL classification. C78, D82, I21, I28. 1. Introduction The analysis of centralized school choice mechanisms has become a key focus of research in market design (Abdulkadiroglu and Sönmez (2003)). Kelso Jr. and Crawford (1982)andHatfield, William, and Milgrom (2005) have built the connection between auction and matching by introducing the matching with contracts model. Since then, analyzing how individuals respond to a “price menu” for an individual good in matching markets has attracted growing interest. Theoretical analysis has been used to address this question in practice (Sönmez and Switzer (2013); Biro, Hassidim, Romm, Shorrer, and Sovago (2022)). However, no clear empirical analysis has disentangled individual behaviors under the matching model with monetary transfer. In extant literature on the school choice problem, the effect of monetary transfers between students and schools is seldom considered because public schools either offer free education or have a fixed (and usually low) tuition fee. Yet unlike public school Tong Wang: [email protected] Congyi Zhou: [email protected] We would like to thank Atila Abdulkadiroglu, Hideo Akabayashi, Daniel Barron, Dan Black, Eric Budish, Yinghua He, Kohei Kawamura, Jacob Leshno, Haruko Noguchi, Elena Prager, Shigehiro Serizawa for helpful comments and workshop and conference participants at Keio University, Northwestern Kellogg Strategy Lunch, Osaka University, University of Chicago Industrial Organization Lunch, University of Tokyo, Waseda University, AEA annual conference, EEA annual conference, IIOC conference, and APPAM annual conferences. ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE2220 1152 Wang and Zhou Quantitative Economics 15 (2024) choice systems in other countries, many Chinese cities have—starting in the 1990s— offered students the option of paying higher tuition and thereby gaining admission to public schools.1This procedure is referred to as the Ze Xiao (ZX) policy.2The ZX policy is a practical application of the matching with contracts model (Kelso Jr. and Crawford (1982); Hatfield, William, and Milgrom (2008,2010); Hatfield, Kominers, and Westkamp (2017)). However, this policy provoked controversy because it was perceived as an unfair policy to families that cannot afford higher tuition (Shen and Wu (2006)). The controversy lasted for more than a decade and was somewhat defused in 2012, when the Ministry of Education announced the restrictions on the ZX policy and requested that public high schools stop using it within 3 years.3 We exploit a new data set covering high school admissions for the period 2012– 2014 in a large Chinese city.4By combining these admission records with data from a 2014 survey, we fill two aspects of the research gap in education policy and market design. First, we provide empirical evidence to understand students’ strategic behaviors in matching with contracts theory. Second, considering that the ZX policy directly involves monetary transfers between students and schools, evaluating this policy helps us understand whether offering a “price menu” in a centralized school choice procedure would increase the inequality in education among students from different backgrounds. The high schools in our focal city adopted the typical ZX policy for their admissions procedure until that policy was canceled in 2014 (see Supplemental Appendix I (Wang and Zhou (2024)) for details of the ZX policy in various Chinese cities). The ZX policy specified the basic and higher tuition levels (i.e., the “price menu”) and the number of seats for sale in each school (i.e., the ZX quota), and was fully controlled by the city government. From 2008 to 2013, Chinese parallel purchasing seats (CPPS) mechanism was used to assign students to schools; it was an indirect extension of the Chinese parallel (CP) mechanism (Chen and Kesten (2017)). The CPPS mechanism is not a direct mechanism. When ranking various schools, a potential student’s rank-ordered list (ROL) needs to indicate whether she is willing to pay higher tuition to each school that would otherwise deny her admission. This mechanism has some undesirable features. It is not strategy-proof, which allows students to “game” the system by misreporting their true preferences with respect to schools.5 Moreover, the equilibrium outcomes of this mechanism can be inefficient and unstable. One way to overcome these imperfections—while retaining the option to purchase admission—is to adopt the cadet-optimal stable mechanism (COSM) and its variation, the COSM-BRADSO mechanism proposed by Sönmez and Switzer (2013)andGreenberg, Pathak, and Sonmez (2021). These mechanisms are the extension of the student1Zhu Kaixuan, then chairman of the state education commission, publicly addressed the seatpurchasing problem in public schools. In 1995 he argued, in the People’s Daily, against paying higher tuition to purchase admission to compulsory education. 2“Ze Xiao” is Chinese for “school selection.”. 3Many cities, including Shanghai (which ceased using the ZX policy in 2012), Beijing (2014), and Shenzhen and Tianjin (2015), ceased the policy for high school admissions. 4Confidentiality restrictions prevent this city from being identified by name. 5The true preference with respect to schools is referred as students’ preferences on schools without considering the tuition. Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1153 proposing deferred acceptance (DA) mechanism (Gale and Shapley (1962)), ensuring stability and strategyproofness, wherein submitting true preferences is a weakly dominant strategy. Since the COSM-BRADSO mechanism is better related to the Chinese mechanism in practice (it is formally defined in Section 3), we focus on this mechanism and use the COSM to denote COSM-BRADSO hereafter. The theoretical properties of these mechanisms motivate us to investigate realworld student behavior and welfare consequences. One difficulty with any empirical analysis of the school choice problem is estimating students’ preferences when only the submitted applications can be observed and the adopted mechanism is not strategyproof. Our survey, which covered nearly half of those who graduated from middle school in 2014, aimed to uncover students’ true preferences and thereby to some extent, solve the problems associated with assessing those preferences in the presence of strategic behavior. The first contribution of our analysis is that students have heterogeneous preferences on the “price menu” of schools. We estimate students’ preferences in two steps. In the first step, we use survey results to estimate students’ preferences over schools without considering the strategic behavior in ROLs. Given that the ZX policy ceased and all students paid the same basic tuition after 2013, the survey data cannot be used to identify any ZX–related parameters (e.g., tuition). In the second step, we use the ROLs submitted in 2012 and 2013 to estimate other parameters. In this step, we assume that students have homogeneous beliefs about the likelihood of being admitted to each school and that they try to maximize their expected utilities in a rational manner. Our estimated results indicate that a one-unit increase in a school’s positive reputation (see Section 4.2 for the definition) is associated with the willingness of high-scoring students from communities with high housing prices to pay an additional 296 yuan—or about US$48.5—to attend that school. In contrast, high-scoring students from communities with low housing prices are willing to pay only 184 yuan for the privilege. The competition for college admission in China is fierce and intense. Our results indicate that students from rich communities have higher desires to “consume” the high-quality schools that might help them attain admissions to colleges compared with others. Our second contribution is the evaluation of the welfare consequences of different mechanisms. We use the simulated matching outcomes under the DA mechanism as a benchmark, then we measure the welfare change when the DA mechanism is replaced by a seat-purchasing mechanism. This replacement could affect a student’s welfare in various ways. First, a student may take advantage of the ZX policy to attend a preferred school by paying higher tuition, which may increase her welfare. Second, if this student’s score is high enough, then she may stay at the same school and pay the normal tuition. Otherwise, she will suffer a welfare loss by either paying higher tuition to save her seat in this school or being priced out and attending a less preferred school. Our counterfactual analysis indicates that when the DA mechanism is replaced with the COSM, student welfare is reduced (on average) by 30 yuan when 10% of the seats are reserved for sale (referred to as the ZX quota). The welfare loss increases to 56 yuan when the ZX quota is increased to 30%. These results reflect that the direct influence of the seat-purchasing option decreases students’ total welfare given that both of the 1154 Wang and Zhou Quantitative Economics 15 (2024) two mechanisms in question are strategyproof. If the DA mechanism is instead replaced by the CPPS mechanism, the average welfare increases slightly by 6 yuan when the ZX quota is 10%, and the welfare loss due to purchasing seats is 118 yuan when the ZX quota is 30%. The reason is that more students can attend their preferred schools by gaming the system. These results reveal an interesting phenomenon: If the ZX quota is limited, leaving room for students to game the system may reduce the average welfare loss. However, when the ZX quota is larger, the welfare loss is much larger. Meanwhile, students from different communities react differently to seatpurchasing mechanisms. When suffering a welfare loss, students from high-housingprice communities prefer to pay higher tuition to keep seats at the same schools. However, those from low-housing-price communities would rather be priced out to less preferred schools. Among the few students who attend more preferred schools by taking advantage of the ZX policy, most are from rich communities. Interestingly, when the ZX quota is large, although more students from poor communities are priced out, highscoring students (approximately 10%) from these communities exhibit a stronger incentive to pay higher tuition to stay in better schools than mediumand low-scoring students (approximately 1%). In summary, our results imply that the ZX policy increases inequality among students in terms of their future educational opportunities, and is not determined solely by their welfare measure in monetary terms. Competitive students (high-scoring students), specifically those from poor communities, show a strong incentive to attend better schools compared to other student groups. Finally, we investigate the impact of the ZX policy on schools. In China, an intense competition exists among high schools regarding admissions. Schools, which suffer a welfare loss under reforms, have a strong incentive to block such reforms. Therefore, our analysis of the policy impact on schools may provide references for policymakers about potential difficulties from reforms. We measure the impact on high schools in terms of: (a) the quality of admitted students and (b) the profit derived from collecting student tuition. The seat-purchasing option helps upper-tier schools collect significantly more tuition with only a limited decline (relative to the DA mechanism) in the quality of their admitted students. Yet for other schools, the seat-purchasing option leads to more uncertainty about both collected tuition and the resulting quality of their admitted students. This paper is closely related to the theoretical work of Sönmez (2013), Sönmez and Switzer (2013), and Greenberg, Pathak, and Sonmez (2021), who investigate cadetbranch matching in the US military. We extend the theoretical results and complement these outcomes by offering an empirical analysis. Our work is also directly related to the extensive theoretical literature on the centralized school choice problem.6More specifically, there is a growing literature that discusses the role of multi-level financial aid in the school choice problem (Hassidim, Romm, and Shorrer (2016)). Hassidim, Romm, and Shorrer (2017) discover that, in a matching procedure for Israeli Master’s programs in psychology, many applicants make the mistake of highly ranking programs that offer less financial aid. 6See Pathak (2011) for a survey on the school choice problem from the perspective of market design. Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1155 The research undertaken here contributes to a growing body of empirical work on the school choice mechanism. Agarwal and Budish (2021) review the development of structural estimates of market design models. One strand of that literature uses the preferences reported under nonstrategyproof mechanisms to estimate students’ preferences (Hwang (2015); He (2016); Agarwal and Somaini (2018); Calsamiglia, Fu, and Güell (2020)). Other papers focus on strategyproof mechanisms. Abdulkadiro˘ glu, Agarwal, and Pathak (2017) treat preferences reported under the DA mechanism as students’ true preferences and then use those preferences to analyze the demand for particular schools in New York City. Fack, Grenet, and He (2019) propose an approach for estimating preferences that does not require truth-telling to be the unique equilibrium under the DA mechanism. Several empirical papers (e.g., Burgess, Greaves, Vignoles, and Wilson (2014); Akyol and Krishna (2017); Wang and Zhou (2020); Ajayi (2022)) bear similarities to our strict priority setting. Others begin to investigate the effect of transfers in the market design (Agarwal (2015,2017); Bobba, Ederer, Leon-Ciliotta, Neilson, and Nieddu (2021)). There is an increasing use of survey data in scholarly research exploring strategic behavior under matching mechanisms. Budish and Cantillon (2012)conductasurvey on students’ preferences for offered courses to study the course allocation mechanism at Harvard Business School, and Rees-Jones (2018) provide survey-based evidence of preference misrepresentation. Burgess et al. (2014) use survey data to directly assess the preferences of students regarding schools. Surveys are also used by De Haan, Gautier, Oosterbeek, and Van der Klaauw (2023) to analyze the Boston mechanism’s deficiencies and by Kapor, Neilson, and Zimmerman (2020) to study heterogeneous beliefs in the school choice problem. Our estimation of students’ preferences also underscores the importance of considering the cardinal preference. Abdulkadiro˘ glu, Che, and Yasuda (2011b,2015) suggest that from an ex ante perspective, when schools have coarse preferences for students coupled with a symmetric tie-breaking rule, students could fare better under the Boston mechanism than under the DA mechanism, as assessed by their cardinal preferences. Our analysis finds that from an ex post perspective, when schools have strict priorities for students, a manipulable mechanism such as the CPPS can still yield higher average student welfare for some types of students than the DA mechanism, especially when the number of seats for sale is limited. The remainder of this paper proceeds as follows. Section 2provides details on the local ZX policy’s background. In Section 3, we present school choice mechanisms that incorporate seat-purchasing options. Section 4describes our data and the summary statistics. We present the empirical model and our estimates of students’ preferences in Section 5, and in Section 6, we conduct counterfactual experiments across mechanisms. Section 7concludes with a summary of our findings. 2. Background on high school admissions The high schools in our focal city can be categorized into various types based on their educational goals for students after completing middle school. These types include gen- 1156 Wang and Zhou Quantitative Economics 15 (2024) eral high schools, which can be public or private, aimed at preparing students for colleges and universities within China. Additionally, there are foreign language schools that focus on preparing students for studies at foreign institutions. Fine arts schools cater to students aspiring to attend fine arts colleges in China. Lastly, vocational schools prepare students with skills necessary for the labor market. The City Education Bureau (hereafter referred to as “the Bureau”) mandates that all schools, regardless of type or ownership, participate in the centralized admission system for middle school graduates. Moreover, each student going through this process must register at the school assigned by the system. Thus, no other options are available for students wishing to continue their education in this city. Annually, at the end of March, the Bureau announces an admissions plan detailing the admission quota to each school. The quota for each public high school jconsists of three parts: the quota for early admission (qe j),7the quota for normal admission (qa j), and the quota for the ZX policy (qz j). The Bureau, not the schools, controls these quotas. Students admitted under any category receive identical education within each high school. In mid-May, students must submit their rank-ordered lists (ROLs) of preferred schools. All students then take the centralized high school entrance exam in early June. From 2012 to 2014, the maximum score for this exam was 665.8After grading, a centralized matching mechanism assigns students to schools. Notably, all schools enforce the same strict priority based on exam scores during student admission. Local public high schools are pivotal in preparing students for college in China. For many, entering a public high school is their sole opportunity for higher education. However, high school education in China extends beyond compulsory levels, and local public high schools can accommodate fewer than half of all middle school graduates. Before the matching procedure, the Bureau establishes and announces a public high school admission threshold (hereafter “the threshold”) based on score distribution and seat availability. Only students scoring above this threshold are eligible for admission to public high schools. This threshold ensures the number of qualified students does not exceed available seats in public high schools. Students can list up to three schools on their ROL and indicate their choice of the ZX option for each. They also need to state if they will accept a random assignment if rejected by their chosen schools. Since 2008, the CPPS mechanism—with permanencyexecution periods (2, 1)—has been employed to assign students (details on matching algorithms are described in Section 3). This mechanism concludes after considering each student’s three choices.9Students admitted with only basic tuition fees are referred to as normal students, while those admitted with higher tuition fees are ZX students. Unmatched students open to random assignment are randomly placed in public high schools with vacancies. The rest explore alternatives to continue education or join the workforce. 7Students eligible for early admission are determined through a separate procedure, not directly impacting the normal admission process, so they are excluded from this analysis. 8Before 2012, the highest score was 650, and it increased to 780 after 2014. 9This mechanism is a constrained mechanism as described by Haeringer and Klijn (2009). Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1157 The ZX policy was designed exclusively for public high schools and not for other school types.10 Tuition for public high schools is based on the student’s exam score. Since scores are the only admission criteria, schools set a cutoff for normally admitted students. Normal students pay an annual tuition of 1600 yuan (roughly $260 in 2013), ensuring affordability. ZX students’ tuition depends on their score: scores within 10 points of the cutoff result in a 3333.3 yuan annual fee; scores 11–20 points above the cutoff pay 5000 yuan; and scores 21–30 points above the cutoff pay 6000 yuan.11 No school can admit a ZX student with a score more than 30 points below its cutoff. On their ROLs, students can only answer “yes” or “no” to the ZX option, meaning they cannot select a specific ZX tuition but must accept the entire package if admitted as ZX students. Following the Ministry of Education’s directive, the local education bureau ceased the ZX option post the 2013 admissions process. 3. Chinese parallel purchasing seats (CPPS) mechanism In this section, we provide a formal definition of a school choice problem that incorporates the option of purchasing seats. We consider a finite set of students, denoted by I, and a finite set of schools, denoted by J∪∅,where∅represents the situation where a student does not attend any school. Each school offers two tuition levels: c0and c1, where c0is the basic tuition paid by normal students, and c1is the higher tuition for ZX students.12 Each school jhas two quotas, qa jand qz j, for normal and ZX students, respectively. It holds that j∈J(qa j+qz j)≥n,wherenrepresents the total number of students. Each student ihas a strict preference, denoted by πi, over schools and tuition. The notation (j,c0)πi(j,c1)indicates that student istrictly prefers to pay the basic tuition for a seat in school jrather than paying the higher tuition for the same school. All schools employ a uniform strict priority ranking, denoted by , to order students based on their exam scores. When student iis allocated a seat in school jby paying tuition c,thepair(j,c)is termed as student i’s assignment. A matching Xis defined as a collection of student-to-assignment pairings that satisfies two conditions: (a) each student has only one assignment, and (b) no school admits more students than its total quota. We denote the set of all matching outcomes as X. Amechanism is defined as a strategy space ifor each student i, accompanied by an outcome function ψ:(i1×i2×···×in)→X, which selects a matching outcome for each strategy vector a∈(i1×i2×···×in).Adirect mechanism is a function ψthat selects a matching outcome for each preference profile. 10The college admission rate for the best private high school is below 1%, making it lower than even lowquality public high schools. Private schools charge a regulated flat tuition fee, with admission cutoffs equal to the public high school threshold every year. Essentially, private high schools mainly cater to students scoring below the public high school threshold but still wishing to continue their high school education. Our analysis does not delve into these schools. 11ZX students pay a lump-sum for all three high school years, unlike normal students who pay annually. 12The model can be readily expanded to incorporate multiple levels of tuition, as discussed in Sönmez and Switzer (2013). However, in our focal city, ZX students are presented with a singular ZX tuition package and can only decide whether to accept or reject it, rather than selecting a specific tuition level. For the sake of simplicity, we focus on two tuition levels in our model. 1158 Wang and Zhou Quantitative Economics 15 (2024) The CPPS mechanism is an extension of the Chinese parallel (CP) mechanism (Chen and Kesten (2017)). However, unlike the CP mechanism, the CPPS mechanism is not a direct mechanism. Specifically, under the CPPS mechanism, each student is required to provide (i) her ranked preferences for schools and (ii) indicate, for each ranked school, whether she would opt for the ZX option (i.e., paying higher tuition) to attend that school if she is not assigned a seat as a normal student. Under the CPPS mechanism, schools allocate the normal seats based on the normal priority (). For the allocation of ZX seats, each school jemploys the ZX priority +, which is constructed as follows: All applicants for ZX seats at school jare divided into two groups: the ZX-qualified group Aj, comprising students who opt for the ZX option for school jand meet the predetermined qualification rule (related to the school’s normal priority) and the remaining applicants in group Bj. When school jcompares two students iand i, the following rules apply: If i∈Ajand i∈Bj,theni+i, indicating that student iis given higher priority over i.Ifi,i∈Ajor i,i∈Bj,theni+iif and only if ii. In other words, school j’s preference is solely determined by students’ exam scores in this case. The ZX priority indicates that a qualified ZX applicant has a higher priority for receiving a ZX seat compared to an applicant who either does not choose the ZX option or selects the ZX option but does not meet the qualification criteria. In all other cases, school jutilizes the normal priority to allocate ZX seats.13 As mentioned in Section 2 concerning our focal city, a ZX applicant is considered qualified for a ZX seat only if her exam score falls within a range of 30 points below the school’s normal admission cutoff (Group A). Students whose scores are more than 30 points below the cutoff are not qualified for admission as ZX students. In essence, opting for the ZX option can give a student the privilege to take a ZX seat under the condition that her score is not too low. The CPPS mechanism with a permanency-execution period vector, e=(e1,e2,), selects the matching outcome as described below. Round 1: •Every student applies to her first choice. Each school japplies the normal priority to tentatively reserve the top qa japplicants in the normal pool. Among the remaining applicants, the school tentatively reserves the top qz japplicants in its ZX pool, following the ZX priority. All other applicants are rejected. In general: •Any rejected student iwho has not yet applied to her (e1)th-choice school applies to her next-preferred school. A student who has been rejected by all her first e1choices does not apply to any other school until the next round. Each school jevaluates the new applicants, along with those already held in the normal pool, and tentatively reserves the top qa japplicants in its normal pool based on the normal priority. Subsequently, school jconsiders all remaining applicants, along with those already held in its ZX pool, and 13All other cases encompass the following scenarios: (1) when both applicants choose the ZX option and are qualified, (2) when both applicants choose the ZX option but neither are qualified, (3) when one applicant chooses the ZX option but is not qualified, and the other applicant does not choose the ZX option, and (4) when neither of them chooses the ZX option. Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1165 students’ preferences among lower-tier schools.27 When conducting the counterfactual analysis to assess the welfare of low-scoring students, understanding their preferences regarding attainable schools is more valuable than their preferences for favorite but unattainable schools. In our survey, all schools received substantial representation in students’ responses. It is noteworthy that each of the remaining three schools was chosen by more than 100 low-scoring students. In contrast, these institutions were seldom mentioned by their high-scoring counterparts (Table B.2 in Supplemental Appendix B). Figure 1shows the average admission cutoffs of schools chosen by students both in the survey and in their ROLs.28 Students are categorized into four groups based on their score percentiles. In the survey, high-scoring students (with exam scores above the 90th percentile) have average school cutoffs of 606.1 and 599.4 for their first and second choices, respectively. The average cutoff for third choices is 593.2, which is 6 points lower. Additionally, the survey reveals that the cutoff gaps between the third and fourth choices, and between the fourth and fifth choices, are 5 and 9 points, respectively. Students in the other three score percentile groups exhibit similar patterns. Within each group, the gap in average cutoffs between consecutive choices is around 6 points and never surpasses 10 points. When comparing between groups, the average cutoff for the first choices of students in the 80th to 90th percentiles is 6 points lower than that of the highest decile of students. Furthermore, this average cutoff decreases by an additional 9 points (to 591) for students in the 70th to 80th percentile range. Students below the 70th percentile of exam scores have an average first-choice cutoff of 585. With each additional choice, the average cutoffs decrease similarly (at a rate of 4 to 10 points) based on exam scores. The observed decrease in the average cutoff for students’ first choices as their scores decrease suggests that the surveyed students provided truthful responses by listing and ranking schools to which they had realistic admission chances. The gaps between consecutive choices within groups in the survey indicate that students’ preferences for schools decrease with the popularity of those schools. For example, in 2014, the gaps in consecutive cutoffs for two sought-after schools ranged from 3 to 9 points. Additionally, the small cutoff gaps (4 to 10 points) between consecutive choices within each group implies that the preferences reported in the survey are reliable enough to be viewed as the students’ true preferences. In the rank-ordered lists, the average cutoffs for the first choices of students whose exam scores were above the 70th percentile nearly coincide with the corresponding parts in the survey. However, the average cutoffs for the first choices of low-scoring students (i.e., with exam scores below the 70th percentile) are 6 points lower than that in the survey. Notably, the gap between the first and second choices increases significantly as exam scores decline. While the gap in average cutoffs between the first and second choices for the top 10% of students remains almost the same as in the survey, it increases to 19 points for students in the 80th to 90th percentile range, and approximately 27For example, suppose there are four schools, namely A, B, C, and D. Most students’ most favorite school is A, and B is the second favorite, then we can only infer that students prefer A to B to C and D, but it is difficult to tell students’ preference between C and D. 28The corresponding table can be found in Supplemental Appendix B. 1166 Wang and Zhou Quantitative Economics 15 (2024) Figure 1. Average admission cutoffs of schools: Survey versus ROLs. Notes:They-axisrepresents absolute scores, and the x-axis represents four student groups categorized by exam scores in percentile. The dotted curves represent the average cutoffs of the chosen schools in the survey. The solid curves represent the average cutoffs of schools in the ROLs. The threshold for public high school admission is 535 (60.95 percentile) in 2014. 25 points for the two groups of low-scoring students. Furthermore, the average cutoffs for third choices remain consistently around the 535 threshold across all groups in the ROLs. When compared to the survey data, the significant gaps between consecutive choices in the ROLs indicate students’ strategic behavior in their submitted preferences: maintaining a substantial gap between choices with the intention of increasing their chances of admission to some school.29 The correspondence between the first choices in the survey and the ROLs implies that students prefer to apply to their favorite attainable schools. This coincidence, along with the small cutoff gaps among choices reported in the survey, provides further evidence that the surveyed students accurately reported their five favorite attainable schools. However, it is evident that students, especially those outside the top-scoring group, strategically manipulate their reported preferences in the ROLs to increase their overall likelihood of admission, that is, particularly when faced with rejection from their top choices. Thus, the second choices in the ROLs for students in 80th–90th percentile (resp., 70th–80th percentile) closely resemble their fourth (resp., fifth) choices in the survey. Furthermore, a majority of students (across all four groups) selected a leftover school as their third choice because the ROL is restricted to only three choices. One drawback of a nonstrategyproof mechanism is that students who strategically modify their ROLs may exploit naive students who reveal their true preferences (Pathak 29This finding is consistent with the literature that suggests students behave strategically under nonstrategyproof mechanisms (see, e.g., Abdulkadiro˘ glu, Pathak, and Roth (2005); Chen and Sönmez (2006); Abdulkadiroˇ glu, Che, Pathak, Roth, and Tercieux (2020)). Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1167 and Sönmez (2008)).30 We directly compare the schools listed in the survey and in the ROLs. Only 20 students (1.38% of all observations) submitted ROLs that matched their survey lists. However, it is possible that the number of naïve students is even smaller, as reporting true preferences could be a weakly dominant strategy for some students, especially those in the top-scoring groups. Additionally, some students may exhibit risk aversion by reporting their true preferences. In both cases, these students are not naive players in the game. Therefore, the 1.38% figure can be considered an upper bound for the number of naive students. These findings are consistent with previous research, suggesting that only a small number of students submit a rank-ordered list without strategic considerations, particularly when a strict criterion is used for student assignment.31 4.4 Housing prices and admission distributions Our data set does not contain any individualor household-related information, such as household income. Additionally, the city does not provide subcity level aggregate information regarding local residents’ income.32 To analyze whether the ZX policy has heterogeneous effects on households with varying economic statuses, we match students’ home addresses with the local housing market information as proxy. Urban economists have previously examined the positive correlation between local housing prices and residents’ income (Goodman (1988); Hwang and Quigley (2006)). The assumption that higher-income families reside in areas with higher housing prices has also been employed in economic analysis (Abraham and Hendershott (1996); Capozza, Hendershott, Mack, and Mayer (2002)). Similarly, studies in China have detected a positive relationship between housing price and residents’ income (Zhang, Jia, and Yang (2016)). In our study, the focal city is divided into 85 communities, which are neither government administrative units nor school districts. This division is based on the local housing market and traditional living areas identified by a real estate website (Wang and Zhou (2024)).33 Local public schools receive full funding from the city-level government, and students are not restricted to specific zones when choosing high schools. Therefore, there is no direct connection between housing prices and school quality. The median housing price in these communities is 13,636 yuan/m2,34 with the highest price being 30,405 yuan/m2, and the lowest price being 3968 yuan/m2(Figure 2).35 To simplify the analysis, we classify communities with housing prices above the third 30Calsamiglia, Fu, and Güell (2020) indicate that, in Barcelona’s local school choice setting, the proportion of such naive students is less than 4%. 31In our context, unlike situations where students are assigned based on coarser criteria (e.g., walking zones or siblings), high school admission offers no safe choice for students until their exam scores are known. Consequently, estimating their exam scores becomes a student’s initial strategic move. Therefore, one can expect an extremely low percentage of naive students when subjected to admission procedures like those described here. 32The city level annual per capita disposable income of an urban household in 2013 was 35,227 yuan (≈5775 US dollar). 33The authors have collected and processed the data on local housing prices from https://www.58.com/. This data set can be downloaded from the replication package. https://doi.org/10.5281/zenodo.12735964 341m2is equal to 10.76 sq.ft. 35The average housing price of this city was 12,187 yuan/m2in 2014. 1168 Wang and Zhou Quantitative Economics 15 (2024) Figure 2. Distribution of the housing price. Notes: This is the histogram of communities’ housing prices in 2014. The unit of the x-axis is 10,000 yuan/m2. The unit of the y-axis is the number of communities. Each bin represents 2500 yuan/m2except that the first bin includes the housing price lower than 5000 yuan/m2and the last one includes the housing price above 25,000 yuan/m2. quartile as high housing price (HHP) communities,36 communities with housing prices below the first quartile as low housing price (LHP) communities,37 and the remaining communities as moderate housing price (MHP) communities. On average, each community has 54.48 students, with a standard deviation of 26. Approximately 35.6% of the students live in HHP communities, 49.8% live in MHP communities, and the remaining 14.7% come from LHP communities. After the cancellation of the ZX policy, the percentage of admissions from HHP communities to upper-tier schools in 2014 was 38.2% (Table 3). This number was lower than the percentages in 2012 (41.9%) and in 2013 (45.8%). Instead, upper-tier schools admitted more students from MHP and LHP communities. These changes cannot be simply attributed to the fluctuation of exam performance (Table 4), as 43.7% of high-scoring students (scoring above the 90th percentile) were from HHP areas in 2014, compared to 46.7% in 2013 and 42% in 2012.38 Prior to 2014, upper-tier schools admitted a comparable percentage of students from HHP and MHP communities. However, the HHP area accounted for a larger proportion of ZX students compared to the combined contributions from MHP and LHP communities. Conversely, more than half of the high-scoring students came from the MHP and LHP communities. These summary statistics at the aggregate level indicate that the cancellation of the ZX policy has provided students from moderate and low housing price areas with increased opportunities to enroll in well-regarded schools. However, the impact of this policy change on middleand lower-tier schools is not as evident as it is on upper-tier 36Communities with housing price greater or equal to 16,161 yuan/m2. 37Communities with housing price less than 10,609 yuan/m2. 38The upper-tier schools’ admission cutoff was set at the 93rd percentile. Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1169 Table 3. Community admission distribution (%). Upper-Tier Schools Middle-Tier Schools Lower-Tier Schools 2012 2013 2014 2012 2013 2014 2012 2013 2014 HHP communities. 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 communities 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 communities 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 Note: This table indicates the distribution of admitted students from different communities. In each column, the first number represents the percentage of students who live in high, moderate, or low housing price communities; the number in the parenthesis represents the percentage of students who are the ZX students in the corresponding communities. schools. To further analyze all these questions, we estimate students’ preferences in the next section. 5. Empirical model and preference estimate To estimate students’ preferences, we adjust the tuition fee structure, based on the local admission rule, as described in the school choice problem from Section 3. Recall that there is a set of tuition fees C={c0,c1,c2,c3},wherec0is the basic tuition for normal students while c1,c2and c3are the higher tuition amounts paid by ZX students; here, c0<c 1<c 2<c 3. Student i’s (indirect) utility from being assigned to public high school jwith tuition cij ∈Cis ui,j= l βlyl j+ w βwxw iyw j+βDdij + k αkxk i(cij −c0)+εij (1) and that the utility from being assigned to nonpublic high school ois ui,o=Fo+εio.(2) Here, {yj}represents a vector of observed characteristics for school j;{xi}is a vector of student i’s observed characteristics; dij is the home-school distance;39 Fois the fixed effect of nonpublic high schools; and εij and εio are i’s idiosyncratic taste for (respectively) public high school jand nonpublic high schools. In the estimate, we assume that Table 4. Community score distribution (%). High-Scoring Students Median-Scoring Students Low-Scoring Students 2012 2013 2014 2012 2013 2014 2012 2013 2014 HHP communities 42.1 46.7 43.7 29.8 37.2 34.1 29.2 33.2 31.7 MHP communities 48.9 44.5 46.4 55.0 47.6 50.5 52.2 46.6 49.9 LHP communities 9.0 8.8 9.9 15.2 15.2 15.4 18.7 20.2 18.5 Note: This table decomposes each scoring group into a different housing price. 39The road distance dij is calculated via Google Maps by inputting the focal school’s address and the student’s home address. 1170 Wang and Zhou Quantitative Economics 15 (2024) the home-school distance is additively separable and independent of unobserved students’ preferences; in addition, we normalize the coefficient of dij for the home-school distance to be −1.40 The utility function of students in Equation (2) is similar to that in Abdulkadiro˘ glu, Agarwal, and Pathak (2017)andAgarwal and Somaini (2018), with the exception that we do not present the random coefficient model for estimating students’ heterogeneous preferences for observed school characteristics due to limited variation in our data. In China, the primary goal of general high schools is to prepare students for the college entrance exams. Apart from reputation, the observed characteristics of schools, such as facilities, are fairly homogeneous. The teaching programs are fully controlled by the local education bureau. Additionally, students who are qualified for local public high schools exhibit similar preferences for schools (see Supplemental Appendix H for details of students’ survey responses).41 Consistent with Abdulkadiro˘ glu, Agarwal, and Pathak (2017), we do not explicitly model an outside option. It is because no outside option can be observed in the current admission record, as mentioned in Section 2. In addition, we make the following assumption. Assumption 1. The terms εij and εio are independent of the explanatory variables,xi, yj,dij ,C,and Fo.Both εij and εio are independent and identically distributed (i.i.d.) and exhibit a type I extreme value distribution with cumulative distribution function (CDF)F(ε). Our estimation of students’ preferences employs both administrative and survey data. A key benefit of the survey data is its ability to yield estimates without factoring in the strategic behaviors students might exhibit when submitting their ROLs. However, the survey data cannot provide insights into students’ preferences regarding ZX options because the ZX policy was discontinued after 2013. Consequently, in 2014, all students paid the same basic tuition for all public high schools. As a result, the survey data alone cannot estimate the parameters αkin equation (1). Therefore, we divide our estimation procedure into two steps. First, we use the survey data from 2014 to estimate the vector of parameters unrelated to the ZX option, that is, {β}. Second, we estimate the vector {α} of parameters related to the ZX policy using the student ROLs submitted prior to 2014. 5.1 Step one: Estimating the non-ZX-related parameters β In this step, we focus on the survey data without considering students’ strategic behavior when submitting their ROLs. Each surveyed student ranked five schools she believed she had a realistic chance of attending. This selection process implies that the student 40Unlike admission to elementary and middle schools, the high school admission procedure does not consider the locations of school districts or homes. Hence, we assume that, in this city, the school choice mechanism does not directly influence residential decisions or local housing prices. 41To avoid choosing the wrong empirical model, we consider an alternative random coefficient model and compare the resulting estimates. However, the random coefficient model performs worse than the nonrandom coefficient model in both within-sample and out-of-sample tests. Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1171 first identifies the schools for which admission is feasible and then ranks them accordingly. This procedure complicates the construction of the model that explains how these middle school graduates initially select schools. For instance, if a school with a high admission cutoff is not included in a surveyed student’s list, it becomes challenging to distinguish between two possibilities: (a) the student prefers the listed schools over the high-cutoff school, or (b) the student believes that admission to the high-cutoff school is not possible. Based on the evidence presented in Section 4.3, we conclude that the survey responses reflect students’ true preferences, conditioned on their beliefs about the possibility of admission. To simplify the estimation process, we focus solely on the rankings of the listed schools in the survey and do not consider unlisted schools. In other words, we do not attempt to infer the relative rankings of listed and unlisted schools. While this approach may result in a less efficient estimate, it remains consistent when surveyed students report their true rankings. For example, consider a student iwith a score swho lists school jbefore school jin her survey. The admission cutoffs for these two schools are denoted as ¯ Sjand ¯ Sj, respectively. Then the probability that iprefers j to jconditional on both schools being attainable for her is Pr(ui,j>u i,j|s> ¯ Sj∩s> ¯ Sj). This conditional probability equals the unconditional probability, that is, Pr(ui,j>u i,j|s> ¯ Sj∩s> ¯ Sj)=Pr(ui,j>u i,j∩s> ¯ Sj∩s>¯ Sj) Pr(s>¯ Sj∩s>¯ Sj) =Pr(ui,j>u i,j)Pr(s>¯ Sj∩s> ¯ Sj) Pr(s>¯ Sj∩s> ¯ Sj) =Pr(ui,j>u i,j) The second equality arises from the fact that students’ beliefs about admission probabilities only affect whether researchers can observe students’ preferences in the survey, but do not alter the relative positions of these preferences. For instance, suppose student ihas a true preference for all schools as j1πij2πij3πi···, i.e., jkπijkwhen k<k .Ifthe selected and ranked schools in the survey are j2πij7πij9, the relative rank of any two of these schools still preserves the relationship: jkπijkwith k<k , regardless of how these schools are selected into the survey. The rankings are independent of the set of schools selected when the top five schools in a feasible set are chosen. We assume that the process of selecting the feasible set is independent of preferences. The equation indicates that one’s relative preference over any two schools is independent of the set of schools picked in the survey. Note that we do not assume the selection of feasible schools in the survey is independent of students’ scores or that students with the same scores rank the same set of schools. Given Assumption 1, while referring to the survey data, we use the rank-ordered logit model (Beggs, Cardell, and Hausman (1981)) to estimate β.42 Given a surveyed student i’s ranked school list (j1,,jl)of length l≤5, we conclude that j1is her favorite school among all the lschools on her survey list, that j2is her second-favorite school, and so 42Because cij =c0in this step, αdoes not appear in the utility function. 1172 Wang and Zhou Quantitative Economics 15 (2024) on. The joint probability of these choices is Pr(ui,j1>u i,j2>···>u i,jl)= l−1  k=1 eμi,jk eμi,jk+eμi,jk+1+···+eμi,jl,(3) where μi,jis the deterministic component of ui,jor ui,o.43 Then the log-likelihood function can be written 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 estimate βusing maximum likelihood estimation.44 5.2 Step two: Estimating the ZX-related parameters α In the second step, we estimate αwhile considering students’ strategic behavior in the admission procedure. After plugging the estimated ˆ βinto equations (1)and(2), we can rewrite student i’s utility function as ui,j=ˆ ui,j+ k αkxk i(cij −c0)+εij ,(5) ui,o=ˆ Fo+εio,(6) where ˆ ui,j=lˆ βlyl j+wˆ βwxw iyw j+ˆ βDdij . Calsamiglia, Fu, and Güell (2020) find that 96% of students in Barcelona are strategic players, and Abdulkadiro˘ glu, Agarwal, and Pathak (2017) use students’ reported preferences as their true preferences under the DA mechanism to estimate the parameters based on the assumption that students maximize their expected utilities given their beliefs about admission probability (see Section VI.A in Abdulkadiro˘ glu, Agarwal, and Pathak (2017)). Therefore, in light of the evidence (from Section 4.3) that few students report their true preferences when submitting ROLs, we model their strategic behavior by assuming that they submit ROLs that are optimal given their beliefs about their likelihood of admission. The assumption of rational expectations will be further supported when we calculate the likelihood function. Our analysis reveals that only a small number of students (1–2%) adopt weakly dominated strategies in their ROLs. The literature suggests that students may hold heterogeneous beliefs about these probabilities (Kapor, Neilson, and Zimmerman (2020)) or make mistakes in their ROLs (Hwang (2015); Artemov, Che, and He (2017)). However, defining “mistakes” in our administrative data is challenging since students submit their ROLs before taking the exams, and a student may accurately estimate admission cutoffs but face uncertainty about their exam performance. Therefore, we make the following assumption. 43More precisely, μi,j=lβlyl j+wβwxw iyw j+βDdij when jis a public high school and μi,j=Fowhen jis not a public high school. 44We assume that the utility function has an additively separable form. It is thus easy to demonstrate that logL1is globally concave in β—from which it follows that there exists a unique maximum of the likelihood function. Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1173 Assumption 2. Students are fully informed about their own preferences,and they maximize their expected utility in a rational manner. Students’ decision problem Initially, student isubmits the ROL ai={(j1,v1), (j2,v2),(j3,v3),r};herevk∈{0, 1}indicates whether iselects the ZX option for her kth choice jk,andr∈{0, 1}indicates whether iaccepts a random assignment if she is rejected by all three of her chosen schools.45 Subsequently, each student takes the entrance exam and receives a score si. The decision problem for student iis to maximize her expected payoff by selecting the best choice aifrom the set of all possible choices Ai. max ai∈Ai EU(ai,si)(7) Since the choice set for each individual is larger than 100,000, we follow the backward induction method developed by Calsamiglia, Fu, and Güell (2020) to address the curse of dimensionality for the empirical school choice problem. This approach follows the logic that although the entire ROL is submitted all at once, the ranked schools on the list are considered sequentially in the procedure. The kth listed school and its ZX option are relevant only if the student is rejected by all previously listed schools. Therefore, the kth choice should be the student’s best choice conditional on reaching this stage, and consequently, the problem can be solved via backward induction. The student’s decision problem can be rewritten as Vk(si)=max (jk,vk)¯ Pk i·Uk i+1−¯ Pk i·ˆ I·Vk+1(si)(8) with k∈{1, 2, 3},and V4(si)=max t∈{0,1}Ir=1·EUl i+(1−Ir=1)· ui(9) Equation (8) indicates that student ineeds to choose school jkand its ZX option vkto maximize her value function Vk.¯ Pk iand Uk iare vectors of probabilities and utilities, respectively, associated with iattending school jkat different tuition levels.46 Here, ˆ I=(1, 1, 1, 1).Equation(9) indicates that if student iis rejected by all three choices, she needs to decide whether to accept a random assignment to a leftover school. EUl irepresents the expected payoff when iis randomly assigned to a leftover school,  uirepresents the payoff when ihas lost all chances to attend a school in the matching procedure, and Ir=1is an indicator function of whether student chooses to accept the random assignment. 45We drop subscript ifor schools and ZX options for simplicity. 46More precisely, ¯ Pk i=(Pk i,c0,Ik·Pk i,c1,Ik·Pk i,c2,Ik·Pk i,c3).Pk i,ctrepresents the probability of student ibeing admitted to her kth choice with tuition ct, given that she has been rejected by her previous choices and/or tuition levels. Ikis an indicator function that determines whether ichooses the ZX option for jk. Uk i=(ui,jk,c0,ui,jk,c1,ui,jk,c2,ui,jk,c3). It is worth noting that we slightly abuse the notation here: if student ichooses a nonpublic school at any position, we can replace ui,j,c0in equation (8)withui,oas defined in equation (6) and set vk=0. 1174 Wang and Zhou Quantitative Economics 15 (2024) Admission probabilities and beliefs Given student i’s ROL ai, the conditional probability of her being admitted to her kth choice as a normal student is Pk i,c0=Pr¯ Sjk≤si|iis rejected by jk−1. (10) Here, ¯ Sjkrepresents the cutoff of jk.47 Equation (10) indicates that iattends the kth choice school as a normal student if and only if her score is no less than the school’s cutoff conditional on being rejected by k−1th choice, that is, si<¯ Sjk−1if idoes not choose the ZX option of jk−1or si<¯ Sjk−1−30 otherwise.48 The conditional probability of student ibeing admitted to jkas a ZX student, with tuition ctwhere t∈{1, 2, 3},is Pk i,ct=Pr¯ Sjk−10t≤si|si<¯ Sjk−10(t−1). (11) From the perspective of student i, we assume that she anticipates her exam score to be mi+ηi;here,mirepresents either i’s mock exam score or her true ability (by which she estimates her exam score) and ηiis the uncertainty. We assume that ηiis i.i.d. and distributed normally as N(0, δ).Notethatmicannot be directly observed from the data. Instead, we use the student’s actual exam score sias the proxy of mi. We simplify our estimation process by setting δ=20, which is 3% of the full mark.49 After we replace siwith si+ηiin equation (10)to(11), the admission probabilities can be expressed as the CDF of the standard normal distribution (see Supplemental Appendix D for the functional forms). Students assess their chances of being admitted to each school before submitting their ROLs. In line with much of the school choice literature, we assume that students consider admission probabilities as exogenous, meaning they can precisely forecast the equilibrium admission cutoffs.50 While this assumption is not without its complexities and limitations, it enables us to estimate students’ preferences without the need to explicitly solve for the equilibrium, which can be nonunique in many nonstrategyproof matching mechanisms. Solving for the equilibrium and selecting the appropriate one can be computationally intensive. The price-taking assumption offers the advantage of simplifying the analysis and is commonly employed in the literature for this reason. In our study, the admission cutoffs of schools are made public after the annual admission season. Analyzing the previous year’s data, we observed that the majority of popular schools’ cutoff scores (with one exception) in the period between 2011 and 2013 47Note that because the CPPS mechanism is an extension of the CP mechanism with an executive period (2,1), the schools’ cutoffs used for the first two choices are generated after considering all students’ second choices. For the third choice, the schools’ cutoff is generated after considering all students’ third choices. The similar calculation approach for the general Chinese parallel mechanism can be found in Calsamiglia, Fu, and Güell (2020) and their Supplemental Appendix. 48When k=1, equation (10) becomes the unconditional probability: Pr(¯ Sj1≤si). 49The estimated results when δ=13.3 (2% of the full mark), when δ=26.6 (4% of the full mark), and when δ=33.35 (5% of the full mark) are reported in Section 5.4. 50Abdulkadiro˘ glu, Angrist, Dynarski, Kane, and Pathak (2011a); Azevedo and Hatfield (2018); Kojima (2017); Agarwal and Somaini (2018); and Calsamiglia, Fu, and Güell (2020). Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1181 Table 6. Admission cutoffs. Within Sample Out of Sample 2012 2013 2012 2014 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) School True Predicted Diff True Predicted Diff. True Predicted Diff. True Predicted Diff. 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 Note: This table indicates the withinand out-of-sample tests for the schools’ cutoffs, using the estimated coefficients in column 3 of Table 5. The full mark is 665. The threshold is 535 in 2012 and 2014, and 530 in 2013. †indicates the special class. The number of special classes varies with years. stands to reason that their main information source leans on data from previous years (adaptive expectations). In the second panel of Table 7,incolumn1,wepresenttheparameters estimated in relation to the ZX option, assuming students base their estimated chances of admission on the prior year’s cutoffs.63 These results align closely with those stemming from rational expectations. This alignment indicates consistent stability in admission cutoffs over the years and affirms that students’ perceptions, formed using the preceding year’s cutoffs, yield fairly accurate estimates. Uncertainty tied to their exam scores can shape the strategic behavior of students. In the second panel of Table 7, columns 2–4 in the second panel of Table 7showcase parameters estimated in relation to the ZX option. Here, the standard deviation δof the exam scores is designated as 13.3 (2% of the full mark), 26.6 (4% of the full mark), and 33.35 (5% of the full mark), respectively. These results exhibit a similar pattern to the findings presented in Table 5. 63Note that our estimation of the non-ZX-related parameters does not rely on any assumptions about student beliefs. 1182 Wang and Zhou Quantitative Economics 15 (2024) Table 7. Estimated Parameters in the Robustness Check Reputation ×HS 0.225 Capacity ×HS 0.365 (0.054) (0.545) Reputation ×MS 0.094 Capacity ×MS −0.808 (0.022) (0.248) Reputation ×LS 0.052 Capacity ×LS −0.569 (0.023) (0.176) Special class ×HS −6.625 Distance −1 (2.312) Special class ×MS 1.060 Distance ×Male 0.794 (2.072) (0.036) Special class ×LS 6.377 Dorm 4.893 (8.139) (1.209) Score range −0.043 Dorm×Male 0.791 (0.445) (0.318) Score range ×Male 0.534 (0.554) Same district −1.888 (0.234) Same district ×Male 1.794 (0.298) Non-public high school −9.113 School Fixed Effect Y (1.192) (1) (2) (3) (4) Cost ×HS −1.242 −1.011 −1.162 −1.163 (0.008) (0.017) (0.009) (0.010) Cost ×MS −1.430 −1.171 −1.391 −1.391 (0.006) (0.015) (0.013) (0.013) Cost ×LS −1.580 −1.423 −1.674 −1.673 (0.007) (0.015) (0.012) (0.012) Cost ×HHP −0.671 −0.635 −0.694 −0.694 (0.006) (0.012) (0.013) (0.013) Cost ×MHP −1.051 −1.044 −1.121 −1.121 (0.007) (0.012) (0.010) (0.012) Cost ×LHP −1.992 −1.801 −1.957 −1.955 (0.008) (0.018) (0.015) (0.013) School Fixed Effect Y Y Y Y Note: The first panel is the estimated results based on the admitted student qualities from previous years as the school reputation measure. The second panel is the estimated results for different assumptions about students’ behaviors in ROLs. Column 1 represents the adaptive expectation assumption. Column 2–4 represents the s.d. of the uncertainty of exam score are 13.3, 26.6, and 33.35, respectively. Standard errors are reported in parentheses. Distance is measured by kilometer. The coefficient of female’s attitude to home-school distance is normalized to -1. Cost(tuition) is measured by 1000 yuan. “HS,” “MS,” and “LS” represent high-, medium-, and low-scoring students, respectively. “HHP,” “MHP,” and “LHP” represent students from high-, moderate-, and low-housing price communities, respectively. 6. Counterfactual analysis To assess the impact of the ZX policy on student welfare, we conduct simulations to compare different mechanisms. Specifically, we use two benchmark mechanisms for comparison: the DA mechanism and the cadet-optimal stable mechanism (COSM). Economists widely recommend the DA mechanism, known for its strategyproof nature. Under the DA mechanism, students have a weakly dominant strategy of truthfully re- Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1183 porting their preferences. Empirical studies suggest that students’ strategies in practice align closely with theoretical predictions (Abdulkadiro˘ glu, Pathak, and Roth (2009); Abdulkadiro˘ glu, Agarwal, and Pathak (2017); Pathak and Shi (2021)). Therefore, the DA mechanism is commonly used as a benchmark in counterfactual analyses of different mechanisms (Agarwal and Somaini (2018)). Pathak and Shi (2021) analyze the effectiveness of such counterfactual analyses of the school choice problem. Following a similar approach to Calsamiglia, Fu, and Güell (2020), we simulate the students’ true-telling strategy under the DA mechanism to compare the welfare from other mechanisms. The COSM, another benchmark mechanism, is an extension of the DA mechanism. Under the COSM, players still have a weakly dominant strategy to truthfully report their preferences. Sönmez and Switzer (2013) provide evidence that the strategies of US military cadets largely conform to the theoretical predictions. Therefore, we assume that students will truthfully report their preferences in their ROLs under the COSM mechanism. Based on the estimated preferences, we simulate students’ application lists. For the simulation, we utilize the student and school profiles from the 2014 administrative data. Given the absence of ZX students that year, we treat the normal admission quota as the school’s total capacity. To analyze the welfare impact of different ZX quotas, we conduct experiments under two setups: one with the ZX quota representing 10% of the total quota and another with the ZX quota representing 30% of the total quota when the focal mechanism allows the option to purchase seats.64 For both the DA and COSM mechanisms, we assume that students’ ROLs reflect their genuine preferences. Under the CPPS mechanism, we create ROLs that reflect each student’s best response in equilibrium. Specifically, we begin with the “telling the truth” strategy as the initial point and iteratively calculate each student’s best response while keeping all other students’ strategies fixed. If a student has an incentive to adopt a new strategy, we replace their old strategy with this new one and recalculate the matching outcome, setting it as the new starting point. We repeat this iteration until no student deviates to a new strategy (see Supplemental Appendix E for details). We then carry out 5000 simulations, with each student subjected to a unique vector of random utility shocks. We consider two comparisons in this section. First, we use the matching outcome under the DA mechanism as our benchmark. Then we replace the DA mechanism with the COSM, which serves as an alternative strategyproof and stable mechanism. Without considering students’ strategies, this comparison evaluates the seat-selling policy itself, and using different ZX quotas further enables the study of welfare under various policies. To evaluate the mechanisms actually adopted, we also analyze the welfare changes when the CPPS mechanism replaces the DA mechanism. This replacement may reflect how students’ strategic behaviors may change under different mechanisms and the welfare consequences. These two comparisons provide a comprehensive analysis of the entire ZX policy. The transition from the DA mechanism to the COSM evaluates the effects of implementing a price menu in the seat-purchasing policy, while the transition from the DA mechanism to the CPPS mechanism assesses the influence of students’ strategic behaviors under the CPPS. 64The local government required that no school could admit ZX students totaling more than 20% of its capacity. 1184 Wang and Zhou Quantitative Economics 15 (2024) Figure 3. Welfare change. Notes: These figures represent the welfare change when DA is replaced by another mechanism measured by the welfare-equalizing tuition adjustment. The y-axis represents the change of yuan, and the x-axis represents the ZX quota. To keep the welfare level under the DA mechanism, a positive position indicates a student needs to pay additional tuition (loss), a negative position indicates a student receives a tuition deduction (gain). “HHP,” “MHP,” and “LHP” represent students from high-, moderate-, and low-housing price communities. Students’ welfare For each tested mechanism, we employ the welfare-equalizing tuition adjustment yuan, as proposed by Calsamiglia, Fu, and Güell (2020), to quantify the welfare change. This adjustment represents the tuition amount that a student would need to pay (or be credited) under the DA mechanism to attain the same utility level as under the replacement mechanism being evaluated.65 First, we examine the comparison between the DA mechanism and the COSM, which directly evaluates the ZX policy without considering students’ strategies. As showninFigure3(a), when the DA mechanism is replaced by the COSM, the average welfare of students decreases as the ZX quota increases. On average, students under the DA mechanism need to pay an additional 30 yuan to reach the same utility level as under the COSM when the ZX quota is 10% of the total quota. This loss increases to 57 yuan when the ZX quota rises to 30%. Students from different communities exhibit a similar trend of welfare loss as the ZX quota increases, with students from the LHP communities experiencing relatively less loss compared to students from other communities. To further analyze the impact of the ZX policy on different students, we examine the school assignments when the DA mechanism is replaced by the COSM. The first panel of Table 8shows that with a 10% ZX quota, approximately 1.8% to 2.2% of students from the HHP and MHP communities choose to pay higher tuition to secure their seats in the 65All other parameters (except for tuition) remain fixed. Formally, let uij =U(cij )be i’s utility derived from admittance to school jwhen paying tuition cij under the DA mechanism. If that mechanism is replaced by the focal new mechanism—in which case student iis assigned to school jand achieves utility uij—then the welfare-equalizing tuition adjustment (yuan) is the solution to U(cij +yuan)=uij . Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1185 same schools, while only 0.7% of students from the LHP communities make the same choice. Under the same ZX quota, around 4% of students in each community are unable to attend their more preferred schools due to the increased competition for seats. On the other hand, 4.8% of students from the HHP communities are able to secure spots in their more preferred schools, a higher percentage compared to students from other communities (4% in the MHP communities and 3.5% in the LHP communities). When the ZX quota is increased to 30%, an additional 2% to 3% of students from each community choose to pay higher tuition to stay in the same schools, and a similar increase is observed for students who are unable to attend their more preferred schools and are instead assigned to less preferred ones. However, 9% of students from the HHP communities may get into their more preferred schools by paying higher tuition, and only 6% of students from the LHP and MHP communities can take advantage of the ZX policy in thesameway. In each type of community, the impact of the ZX policy varies for students with different scores. When the ZX quota increases from 10% to 30%, high-scoring students from the LHP area are the most affected. Around 14% of these students choose to pay higher tuition to secure their seats at the same schools, while 21% of them are unable to attend their more preferred schools and are priced out. Another interesting finding is that the ZX policy has a significant influence on medium-scoring students across all communities. Compared to highand low-scoring students, a larger proportion of mediumscoring students choose to pay higher tuition to attend their preferred schools under the ZX policy. However, at the same time, many medium-scoring students are also affected negatively and are priced out, resulting in them being assigned to their less preferred schools. When the DA is replaced by the COSM, high-scoring students experience a significant welfare loss under the COSM, primarily because most of them were already assigned to their most preferred schools under the DA mechanism. Under the COSM, these students must either pay higher tuition to secure their seats in the same schools or face being priced out and assigned to less preferred schools. In contrast, mediumscoring students are influenced in different ways. They have a higher probability of attending their preferred schools by paying higher tuition, but they are also susceptible to being priced out and assigned to their less preferred schools. The estimated coefficients suggest that the willingness to pay higher tuition increases with both students’ scores and housing prices. As a result, when the ZX quota is increased, the high-scoring students from the LHP have a stronger incentive to secure their seats compared to students from the same communities; hence a large proportion of them choose to pay higher tuition. However, they are also more likely to be priced out since their incentive to pay higher tuition is lower than that of high-scoring students from affluent communities. Consequently, a larger proportion of high-scoring students from LHP communities are priced out. The vacant seats left by these high-scoring students are mostly occupied by medium-scoring students from various communities. At the same time, a comparable number of medium-scoring students are also priced out due to the increased ZX quota. Therefore, the influence of the ZX policy on mediumscoring students is substantial in both directions. 1186 Wang and Zhou Quantitative Economics 15 (2024) Table 8. Changes of matching assignments under the purchasing seats option(%). ZX Quota 10% ZX quota 30% Same Better Worse Same Better Worse ZX Normal ZX Normal ZX ZX Normal ZX Normal 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 Note: This table indicates the percentage change in the number of students whose assignments are different under the purchasing seats option, when the DA mechanism is replaced by the COSM and CPPS mechanisms. When DA is replaced by another mechanism, “Same” means the student is assigned to the same school, “Better” represents the student is assigned to a more preferred school, and “Worse” represents the student is assigned to a less preferred school. “ZX” and “Normal”represents the student pays the basic and higher tuition, respectively. “HHP,” “MHP,” and “LHP” represent students from high-, moderate-, and low-housing price communities, respectively. “HS,” “MS,” and “LS” represent high-, medium-, and low-scoring students, respectively. Table 9identifies the percentage of “winners” (students whose welfare increases) and “losers” (students whose welfare decreases) when the DA mechanism is replaced. Regardless of the ZX quota, the proportion of winners under the COSM never exceeds 8.7% for any student group. However, the proportion of losers exceeds that of winners in all cases. When the ZX quota is increased from 10% to 30%, the proportions of losers in the HHP and MHP communities experience a substantial rise, while the change is relatively small for the LHP communities. The analysis of the welfare change in each type of community (Table F.2) further confirms the explanation of the effect of the ZX policy. More than 35% of high-scoring students from poor communities become losers and experience an average welfare loss of 1587 yuan when the ZX quota increases. This welfare Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1187 loss primarily arises from a large proportion of students who must either pay higher tuition to secure their seats or be priced out. The average welfare gain for medium-scoring students is slightly higher than the losses, except for one community. In summary, when the DA mechanism is replaced by the COSM, the average impact on students from different residential areas is similar. The number of students who experience a welfare loss due to the ZX policy is higher than the number of students who potentially benefit from it, and this loss is magnified as the ZX quota increases. However, students’ reactions to the policy vary. Medium-scoring students are the most affected group by the ZX policy. They are more likely than low-scoring students to attend their preferred schools by paying higher tuition, but they are also more susceptible to being priced out and assigned to less preferred schools. Top-performing students from economically disadvantaged communities bear the greatest burden under the ZX policy. A significant proportion of them either have to pay higher tuition to secure seats in their preferred schools or are priced out altogether. Comparatively, students from affluent communities are more likely to stay at their desired schools by paying higher tuition, while students from other communities are more likely to be priced out and assigned to less preferred schools. These findings indicate that while the ZX policy intensifies educational inequality among students, its overall impact on their welfare, when assessed in monetary terms, is not as pronounced. Next, we investigate how the practical implementation of the mechanism may impact students’ welfare and their strategic behaviors. When the DA mechanism is replaced by the CPPS mechanism, the changes in student welfare exhibit a similar pattern as observed in the COSM case, but with a notable difference (Figure 3b). Students from the MHP and LHP communities experience a welfare gain of 22 yuan and 63 yuan, respectively, when the ZX quota is 10%.66 However, as the ZX quota increases to 30%, all student groups face welfare losses, particularly students from the HHP and MHP communities. Students from the HHP communities endure a welfare loss equivalent to a 216 yuan increase in tuition, while students from the MHP communities experience a loss of 128 yuan. The second panel of Table 8explains the reasons for the improved student welfare under the CPPS mechanism compared to the COSM at lower ZX quotas. Unlike under the COSM, a significantly lower proportion of students choose to save their seats in the same schools by paying higher tuition under the CPPS mechanism, regardless of student groups. Meanwhile, more students are priced out to their less preferred schools under the CPPS mechanism compared to the COSM. Additionally, a positive number of students from every type of community are able to secure spots in their more preferred schools without paying extra tuition, which is not the case under the COSM. However, when the ZX quota is increased to 30%, a larger number of students, particularly those 66Abdulkadiro˘ glu, Che, and Yasuda (2011b,2015) suggest that from an ex ante perspective, when schools have coarse preferences for students coupled with a symmetric tie-breaking rule, students could fare better under the Boston mechanism than under the DA mechanism, as assessed by their cardinal preferences. In contrast, our results show that when schools have strict priorities for students, a manipulable mechanism such as the CPPS can still yield higher average student welfare for some types of students than the DA mechanism from an ex post perspective. 1188 Wang and Zhou Quantitative Economics 15 (2024) from the HHP and MHP communities, choose to pay higher tuition to stay in the same schools. Simultaneously, more students across all communities are priced out compared to the COSM. Consequently, all students experience greater welfare losses when the ZX quota is high. Table 9further confirms that the number of winners and losers increases in all student groups. These findings suggest that when the CPPS mechanism is used to replace the DA mechanism, students may have more opportunities to strategically manipulate their preferences, leading to greater variations in student welfare across different communities. To further investigate students’ strategic behaviors under the CPPS mechanism, we also simulate the students’ strategies under the Chinese parallel mechanism as an intermediate step. For high-scoring students from HHP communities, their first choice under the Chinese parallel mechanism is, on average, their 1.12 choice in their true preference (Table F.3). However, under the CPPS mechanism, their first choice moves slightly closer to their true first choice at 1.01. This indicates that more high-scoring students are inclined to choose their true first choice under the CPPS mechanism. Additionally, almost 50% of these students choose the ZX options for their first choice. The average second choice for this group is similar to their third choice in their true preference under both the Chinese parallel and CPPS mechanisms. However, under the CPPS mechanism, 80% of students opt for the ZX options for their second choices. Students’ first choice under the CPPS mechanism shows a slight upward shift for highand medium-scoring students in all communities, but a slight downward shift for low-scoring students. High-scoring students exhibit a higher likelihood of choosing the ZX options, particularly for their second choices, and this pattern decreases with housing price. Meanwhile, low-scoring students are less inclined to choose the ZX options, especially for their second choices. However, when the ZX quota rises from 10% to 30%, the change in strategic behaviors is not substantial. This phenomenon indicates that students’ strategic behavior under the CPPS mechanism improves their welfare under a low ZX quota. However, the same strategies lead to a substantial welfare loss when the ZX quota is increased. 6.1 Impact on schools In this final section, we examine the impact of the ZX policy on schools, considering two factors: the quality of admitted students and the tuition collected by schools. Schools face a trade-off in implementing this policy. On one hand, allowing students to buy seats may increase the schools’ income. On the other hand, seat purchasing can lead to the dispersion of high-quality students across different schools. Under the ZX policy, some high-scoring students who might have attended upper-tier schools under the DA mechanism may be priced out and end up in middle-tier schools if they choose not to pay the higher tuition. Conversely, some low-scoring students who are willing to pay more tuition for their preferred schools may displace high-scoring students and secure seats in those schools. As a result, upper-tier schools may collect more tuition but experience a decline in the overall quality of admitted students. While middle-tier schools may admit more high-quality students. Quantitative Economics 15 (2024) Purchasing seats in school choice and inequality 1189 Table 9. Winners and losers. 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 Total % 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 Total 924 −1052 1036 −1076 1399 −1021 1237 −1124 Note: The first panel of this table indicates the percentage change in the number of students whose utilities increase (winners) or decrease (losers) when the DA mechanism is replaced by the COSM and CPPS mechanisms. The second panel indicates the welfare change measured by yuan. “W” represents winners, and “L” represents losers. For each mechanism change, utility changes are measured in three scenarios in which the ZX quotas are 10% and 30% of the total quotas. “HHP” represents students from high housing price communities, ‘MHP” represents students from moderate housing price communities, and “LHP” students from low housing price communities. For upper-tier school #183 (F.4 in Supplemental Appendix F), the collected fees increase proportionally with the ZX quota when the DA mechanism is replaced by the COSM. When the CPPS is adopted, this school may gain even more in terms of tuition collection, with the gain exceeding 40% when the ZX quota is 30%. Importantly, when the DA mechanism is replaced by either the COSM or the CPPS, this school experiences only a negligible decline in student quality. Considering the findings for other uppertier schools (see Table F.4 in Supplemental Appendix F), it becomes evident that there is a significant demand for elite schools. This allows them to profit substantially from selling seats without compromising the quality of admitted students. For middle-tier school #179, the seat-purchasing option has the potential to generate profits. The impact on student quality can vary depending on the mechanism adopted. When the ZX quota increases to 30% under the COSM, student quality slightly decreases compared to its level under the DA mechanism. On the other hand, if the DA mechanism is replaced by the CPPS mechanism, there is a small increase in student quality. Consequently, these schools may experience significant variations in the quality of their admitted students, with some experiencing positive changes and others negative changes. 7. Conclusion Our paper examines a contentious but previously overlooked Chinese school choice policy, Ze Xiao. This policy allows students to “purchase” seats at their desired schools by paying higher tuition. We find that the corresponding matching mechanisms employed in this policy are not strategyproof and may lead to unstable outcomes. We combine data from high school admission records with survey responses from students in China to estimate their preferences for schools and tuition. Our findings reveal that high-scoring students are more willing than other students to incur additional costs, such as higher 1190 Wang and Zhou Quantitative Economics 15 (2024) tuition fees, to attend their preferred schools. Furthermore, students from communities with high housing prices are less motivated to bear the financial burden of higher tuition compared to students from communities with low housing prices. Using estimated preferences, we conduct counterfactual experiments to evaluate the welfare consequences of the ZX policy. We find that, when the strategy-proof COSM replaces the DA mechanism, students’ welfare decreases across all student groups. However, when the DA mechanism is substituted by a non-strategyproof mechanism like CPPS, it may alleviate the welfare losses, particularly when the ZX quota is low. This is because more students can exploit the system to secure admission to their preferred schools. When experiencing a welfare loss, students from high housing price communities tend to opt for paying higher tuition to retain their seats at the same schools. Students from communities with low housing prices are more inclined to be priced out and settle for less preferred schools. As the ZX quota increases, high-scoring students from economically disadvantaged communities demonstrate a greater motivation to pay higher tuition in order to remain in higher-ranked schools compared to mediumand lowscoring students. From the school’s point of view, the seat-purchasing option proves beneficial for upper-tier schools as it enables them to collect a substantial amount of additional tuition while experiencing only a minor decline in the quality of admitted students compared to the DA mechanism. However, for other schools, the seat-purchasing option introduces greater uncertainty regarding both the amount of collected tuition and the quality of students admitted. References Abdulkadiro˘ glu, Atila, Nikhil Agarwal, and Parag A. Pathak (2017), “The welfare effects of coordinated assignment: Evidence from the New York city high school match.” American Economic Review, 107 (12), 3635–3689. [1155,1170,1172,1183] Abdulkadiro˘ glu, Atila, Joshua D. Angrist, Susan M. Dynarski, Thomas J. Kane, and Parag A. Pathak (2011a), “Accountability and flexibility in public schools: Evidence from Boston’leveling the playing charters and pilots.” The Quarterly Journal of Economics, 126 (2), 699–748. 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