Dynamic Competition in Parental Investment and Child’s Efforts
Implications for Intergenerational Mobility and Population Decline *Hyunjae Kang
Kyoto University, JapanFirst Version: November 2022
Current Version: April 2024

Abstract
Competition for a limited number of seats in prestigious college generates a“rat-race” equilibrium effects, leading to increased household investment. I construct and estimate a dynamic tournament model where each household chooses
the quality and hours of private tutoring as well as hours of student self-study.
The model rationalizes the high amount of parental investments in secondary
school despite their lower effects on academic achievement. The estimated model
is first used to evaluate the impact of household investments on intergenerational
persistence of earnings. I find that heterogeneity in parental investment significantly contributes to intergenerational earnings persistence. Removing childefforts amplifies this persistence by 30%, emphasizing the role of self-effort inmoderating the intergenerational link. Secondly, by leveraging the number ofcompetitors in the framework, I assess the heterogeneous impact of the rapidly
shrinking cohort size on the demand for parental investment.
JEL Classification Codes: D15, D64, I21,I22, I24, I26, J62
Keywords: Parental Investment, Child’s Efforts, Intergenerational Mobility, Student Competition* Institute of Economic Research, Kyoto University, Japan. I am immensely grateful to Steven Stern

and Juan Pantano for invaluable guidance and support. I would also like to thank Meta Brown, David
Wiczer, Gabriel Mihalache, Peter Arcidiacono, Michael Keane, Lance Lochner, John Rust, Todd Stinebrickner, Michèle Tertilt, Alexis Anagnostopoulos, Arda Aktas, Hugo Benitez-Silva, Juan Carlos Conesa,
Samuele Centorrino, Miguel Poblete-Cazenave, Sunghun Cho, Jihwan Do, Marcos Fernandes, Jeremy
Fox, Mateo Velásquez-Giraldo, Yujung Hwang, Fedor Iskhakov, Chinhui Juhn, Laura Karpuska, Jiyeon
Kim, Sunham Kim, Alfonso Flores Lagunes, Marc Claveria Mayol, Bob Millard, Mark Montgomery, José
Alfonso Muñoz-Alvarado, Victor Ronda, Warren Sanderson, Eran Shmaya, Takashi Unayama, Greg
Veramendi, Jiayi Wen, Sangha Yoon, Basit Zafar, Weilong Zhang, and Yiyi Zhou for helpful comments.
All errors are my own.

1

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

1.

1

Introduction

Competition motive is a pivotal driver of time investments by parents and the child.
Graduating from an elite university has a sizeable impact on labor market outcomes(Hoekstra 2009; MacLeod et al. 2017; Zimmerman 2019; Anelli 2020; Sekhri 2020; Guoand Leung 2021; Jia and Li 2021; Lee and Koh 2023), but seats for such prestigious
colleges are limited, and evidence suggests that colleges tend to not adjust the seats toaccommodate for increasing cohort size (Bound and Turner 2007). The scarcity of seats
in prestigious colleges leads to competition, potentially creating a “rat-race” scenariothat drives an increasing trend in parental investment (Ramey and Ramey 2010). In
the United States, parents frequently invest considerable time in supporting theirchildren’s extracurricular activities. In East-Asian countries, parents often allocate asignificant portion of their income to private tutoring, primarily aimed at securingadmission to prestigious universities, (Bray 1999, 2022), despite evidence suggesting
minimal impact on test score improvement (Ryu and Kang 2013; Kang and Park 2021).
Most previous work on parental investment does not include this competition aspect
into their framework.1,2
On the other hand, parental investment significantly contributes to the intergenerational transmission of earnings (Caucutt and Lochner 2020; Bolt et al. 2021a,b; Gayle,Golan and Soytas 2022; Yum 2022). It is natural to conjecture that children who havereceived more investment from parents are likely to achieve better future outcomessuch as better performance in the labor market (Becker and Tomes 1979; Guryan,
Hurst and Kearney 2008). Thus, parental investment potentially has important consequences on social mobility. Meanwhile, previous studies report the significant
impact of children’s own efforts on their educational outcomes (Stinebrickner andStinebrickner 2004; Del Boca, Monfardini and Nicoletti 2017). The self-effort of thechild is not responsive to parental background as much as parental investment isaffected by parental background.3 Despite the potential relevance, few studies havemodeled the interdependence of parental investment and children’s self-effort inshaping intergenerational mobility.
1

On the other hand, the competition between private and public schools is extensively studied in
the literature. See Epple and Romano (1998); Epple, Figlio and Romano (2004); Epple and Romano(2008); Epple, Romano and Urquiola (2017, 2021).
2
In Section 3.3, I state the reasons why I do not model competition between secondary schools.
3
I show related empirical evidence in Section 4.

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This paper investigates these two interrelated aspects of parental investment. First,
it seeks to shed light on the role of parental investment on intergenerational persistence or earnings. The inclusion of self-effort of the child, which is often ignored in
the literature, might amplify or offset the link between the two generations. Second,
the intensity of household competition is affected by the size of the cohort for thelimited number of college seats with different level of quality. If there are significant
changes in the number of competitors, the decision of parental investment and childeffort is likely to be affected. In fact, many developed countries face a drastic shift indemographic structure caused by a declining fertility rate, as shown in Figure 1. Little
is known about the consequence of the shift in the demographic structure on parental
investment.
To answer these questions, this paper builds and estimates a dynamic tournamentmodel using a unique longitudinal dataset that contains information on parental
investment, the child’s time allocation, and administrative test scores. I first document
the descriptive evidence regarding parental investments and the self-efforts of thechild. Second, motivated by empirical evidence, I build a dynamic model of a tournament which approximates the college admission competition among households.
I estimate the tournament model using Maximum Simulated Likelihood. In a seriesof graphs, I show that the model fits reasonably well with the data. Finally, I perform
counterfactual exercises using the estimated structural model. I quantify the impactof parental investment and the child’s self-efforts on intergenerational mobility. Then
I simulate the model to measure the effects of the shrinking cohort size on parental
investment.
This study uses Korean datasets and is based on institutional features of the country.4 Students are assigned to the middle schools within the residential education
district by lottery. As the distribution of school quality of secondary school is relatively
homogeneous, the private tutoring expenditure of parents stands out as a primarycontribution to the child’s future outcomes. The importance of the final test scorein college admissions helps to link the test score measure to the child’s labor market
outcomes. Such institutional characteristics offer a transparent environment in whichhousehold income is translated into the educational outcome of the child.
4

A number of countries share the institutional features, which I explain in Section 3.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

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Figure 1: Shrinking Cohort

Source: World Bank for data for China, Hongkong, Japan, Singapore and South Korea. Data for Taiwan
is drawn from United Nations World Population Prospects.

I start by documenting the descriptive evidence that provides the empirical basisof the dynamic tournament model. Two empirical facts show that competition withrespect to getting into a more prestigious college is the primary motivation for parental
investment. First, college ranking positively affects the growth of alumni’s income.
Using the Korean Labor Income and Panel Study, I estimate the effects of college-tier,a categorization of colleges in Korea based on their quality measured by alumni’s
income growth. Pooled OLS results suggest that there is a significant variation inlifetime income based on tier of the college from which workers graduate. The effects
are economically and statistically significant, controlling for CSAT score. This evidence
is consistent with the empirical studies on the effects of elite colleges on labor market
outcomes (Zimmerman 2019; Sekhri 2020; Jia and Li 2021).5 Second, the amountof parental investment drops substantially as students finish the college admission
process. This suggests that the purpose of parental investment is for their child to do
5
Using data on students majoring in science at University of California campuses, Arcidiacono,
Aucejo and Hotz (2016) show the mismatch between minority students’ preparedness and higher
ranked campuses decreases the likelihood of graduation.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

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well in the college admission competition rather than enhancing the child’s humancapital.
Also, another empirical fact suggests that parental investment and the child’s
self-efforts potentially have different implications for intergenerational mobility. As
expected, data show that parental background, especially household income, generates a significant variation in parental investment. On the other hand, self-efforts ofthe child, measured by hours of self-study, do not vary as much as parental investment
with different levels of parental income. At the same time, both parental investment
and the child’s self-efforts are expected to affect the child’s outcome. If parental investment and self-efforts are technological substitutes, an income-constrained household
can compensate for the lack of parental investment by increasing hours of self-study.
Thus, omitting self-efforts of the child might result in an exaggeration of intergenerational persistence of earnings. This suggests the importance of modeling bothparental investment and the child’s self-efforts in understanding the contribution of
educational investments to the intergenerational persistence of earnings.
Finally, the data show that households select the different levels of parental investment and child efforts over time based on their preconditions. Students’ time
allocations change considerably as they proceed to the later periods in secondary
school. Also, the exogenous characteristics of the household persistently affect theparental investment decisions throughout the secondary school periods.
Motivated by the empirical evidence, I develop and estimate an equilibrium dynamic tournament model of college admission competition. The model builds upon
the rank-order tournament model introduced by Lazear and Rosen (1981). The tournament structure is embedded into the model of altruistic households. The householdcares about the future outcome of the child, which is the result of the college admission
tournament. In every period, each household makes decisions of parental investment
and the level of the child’s self-efforts, and these two are inputs of the test score. To
capture the student’s persistence in test-taking skills, I allow the previous test score tohave its own direct effect in the production of test score (Cunha and Heckman 2007).
The model structure repeats until the final test score is produced, and students are
assigned to the college tiers based on their final test score. The college tier is the soledeterminant of the child’s lifetime income. This is an arguably reasonable assumption.
Using confidential job offers data provided by a conglomerate in South Korea, I show

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

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that the effects of college-tier on earnings is economically and statistically significant
controlling for effort during the college period.
The dynamic tournament model offers several features that help answer the research question of this paper. First, the rich heterogeneity of state variables and the
choice set and the specification of the test score production function help disentangle
the source of intergenerational persistence of earnings. Each household can simultaneously choose the quality of tutoring, hours of tutoring, and hours of self-study in themodel based on its state variables. Second, the rank-order feature of the tournamentmodel enables me to study the effects of the changes in cohort size and the role ofdisparity in college quality. Third, as I allow for the time-varying effects of the choices,
I can compare the effects of hours of self-study and hours of parental investment.
I estimate the model using Maximum Simulated Likelihood. The estimation results suggest that hours of self-study have stronger average marginal effects on thesubsequent test score than hours of tutoring. Both the marginal effects of parental
investments and self-study of the child decline over time. Also, the estimate of thesubstitution parameter of the production function suggests that parental investments
and hours of self-study are close to perfect substitutes. Compared to hours of tutoringand hours of self-study, there exists sizeable unobserved heterogeneity in the qualityof tutoring. Using local linear regression, I show that the estimated model fits the
sample reasonably well.
Using the estimated structural model, I first quantify the role of heterogeneity inhousehold income. I use the rank-rank slope, the slope between income percentiles of
two generations, as the measure of intergenerational persistence of earnings (Chetty
et al. 2014). Removing heterogeneity in the parental income during the adolescentperiod decreases the rank-rank slope by 47.2%. Next, I quantify the role of parental investments and the self-efforts of the child on intergenerational persistence of earnings.
I simulate the model by shutting down one of the choices. In particular, I compare
the changes in the rank-rank slope by shutting down the choice of (i) self-study of thechild or (ii) the parental investments for the child. Relative to the estimated model, therank-rank slope increases by 30.2% when the channel of self-study is shut down. Also,
the rank-rank slope decreases by 79.5% when the channel of parental investments
is shut down. The result of the quantification suggests that parental investment reinforces the intergenerational persistence of earnings and the self-study of the childmitigates it.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

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Next, to understand the effects of the shrinking cohort size on the choices ofhouseholds, I simulate the structural model using the projected number of high school
graduates and the assumptions on changes in the number of seats in colleges andchanges in the distribution of college quality. College admission competition is about
winning a limited seat within the cohort. The tournament model enables studying theeffects of changes in cohort size and the distribution of college quality. Based on themodel projection, as the size of the cohort shrinks by 52.5%, low-income households
spend more on private tutoring expenditure as cohort size decreases, while there isvirtually no change in the private tutoring expenditure of high-income households. As
the intensity of competition becomes weaker due to the decrease in the cohort-to-seatratio, low-income households have a higher probability of going to a better collegetier. With a higher probability of going to a better college tier, low-income households
have incentive to spend more on private tutoring.
The rest of the paper is organized as follows. I discuss the related literature andcontributions of this paper in Section 2. I describe the institutional features in Section
3. In Section 4, I document empirical facts that motivate the dynamic tournamentmodel. Section 5 introduces the tournament model. Section 6 explains the estimation
procedure, source of identification, and results. I present the counterfactual exercises
in Section 7 and conclude in Section 8.

2.

Related Literature and Contributions

This paper contributes to the burgeoning empirical literature seeking to understandthe source of intergenerational mobility. The literature of intergenerational mobility
has been focusing on reporting estimates of intergenerational persistence in earnings(Solon 1999; Mazumder 2005; Chetty, Hendren, Kline and Saez 2014; Adermon, Lindahl and Palme 2021). Only recently, there have appeared a few papers empiricallyinvestigating the mechanism that generates the intergenerational correlations in earnings (Lee and Seshadri 2019; Caucutt and Lochner 2020; Bolt, French, Maccuish andO’Dea 2021b; Daruich 2022; Gayle, Golan and Soytas 2022; Yum 2022).6 I contributeto this literature by building and estimating a dynamic model of a tournament thatincorporates parental investment and the self-efforts of the child. Previous studies
6

Relatedly, Björklund, Lindahl and Plug (2006) estimate the contribution of pre-natal factors on
intergenerational transmission of earnings utilizing data on adoptees in Sweden.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

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quantifying intergenerational mobility do not consider the self-efforts of the child in
their framework. Also, the novel feature of my paper is to incorporate the competition
among households to the dynamic structural model.
This paper relates to the large body of literature modeling post-birth parental
choice.7 Since Becker and Tomes (1979), economists have sought to understandhow parents allocate resources to their children and how such decisions affect thechild’s outcomes such as cognitive development (Doepke, Sorrenti and Zilibotti 2019).
Del Boca, Flinn and Wiswall (2014) build and estimate a dynamic model in which
parents jointly choose the amount of time investment, the amount of monetary investment, and the decision of labor supply participation. Doepke and Zilibotti (2017)
formulate a model of parenting styles. Agostinelli, Doepke, Sorrenti and Zilibotti (2020)
extend their work by combining the choice of parenting with the child’s peer formation.
Papers in this literature have recently started incorporating externalities into parental
choices.8 In particular, papers also model externalities by incorporating competition
among students. Ramey and Ramey (2010) are the first paper that rationalizes theincrease of parental time investment in the United States using a theoretical model ofcompetition for elite colleges. The two closest papers modeling student competition
are Bodoh-Creed and Hickman (2019) and Grau (2018). Bodoh-Creed and Hickman(2019) build a static structural model of an admission contest to study returns to precollege human capital investment in the United States and estimate their model. Also,
Grau (2018) builds a static tournament model, estimates it’s parameters and applies
the estimated model to the college competition in Chile.
The theoretical model in this paper is different from theirs in several ways. First, I
model the dynamic tournament allowing for uncertainties in the test score generationand the household choices.9 This helps estimate the effects of household investments
and capture how households self-select into the high and low level of investments.
Second, I model the channel of monetary and time investments, which has direct
implications for intergenerational transmission of earnings. Previous static models
abstract away from the measure of the resources and the efforts used for the college
admission competition. I propose a dynamic tournament model and suggest plausi7

See Chiappori, Salanié and Weiss (2017) for a model of joint decision of marriage and parental
investment.
8
a group of papers associates parental choices with social interactions (Agostinelli 2018; Agostinelli
et al. 2020; Boucher et al. 2022)
9
Outside the broad literature of economics of education, a handful of papers build and estimate
structural tournament models (Vukina and Zheng 2007; Chen and Shum 2010; Vukina and Zheng 2011).

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

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ble measures for the resources (household income) and the efforts (private tutoring
expenditure and hours of self-study) of the competition.
Another closely related paper is by Kim, Tertilt and Yum (2022), which studies thecause of the low fertility problem of South Korea. They propose a heterogeneousagents model of “status externality” based on the assumption that parents care about
the relative position of their children’s human capital compared to that of other children. The tournament model of this article complements their study by formallymodeling the dynamic competition with respect to getting into prestigious colleges.
The tournament structure can rationalize the underlying source of the status externalities in their paper.
Finally, this paper contributes to the literature on childhood investments and skill
development by estimating the effects of parental investment and the self-efforts of thechild in the adolescent period. Most previous work focuses on estimating the effectsof parental investment on child outcomes alone (for example, Cunha and Heckman
2007; Cunha, Heckman and Schennach 2010; Del Boca, Flinn and Wiswall 2014).10
These studies find declining effects of parental time investment over age. Several
studies estimate the effects of hours of self-study on academic achievements (e.g.,
Cooper, Robinson and Patall 2006; Stinebrickner and Stinebrickner 2008), but they do
not jointly estimate the effects of parental investments. Only recently, a few papers
have estimated models incorporating both parental investment and self-efforts of thechild. Del Boca, Monfardini and Nicoletti (2017) find that the effect of self-effort of thechild is stronger than the effect of the mother’s time investment during adolescence,and the effect of self-effort of the child increases over time. Del Boca, Flinn, Verriest
and Wiswall (2019) build a Stackelberg model of parent-child interaction and study
the effects of conditional-cash-transfers on child outcomes. Such a line of researchsuggests the importance of modeling both parental investment and self-efforts of thechild in studying the source of intergenerational mobility, which is the focus of thispaper.11,12
10

As this paper employs private tutoring expenditure as a measure of parental investment, it also
complements the literature of studies on private tutoring (Stevenson and Baker 1992; Cheo and Quah
2005; Tansel and Bircan Bodur 2005; Dang 2007; Ono 2007; Ryu and Kang 2013; Hof 2014; Kang andPark 2021).
11
Agostinelli and Sorrenti (2021) find that the trade-off between more household income from labor
supply and parental time investment is significant for disadvantaged families (mothers).
12
As college competition in reality uses actual test scores rather than unobserved skills of the student,
I do not apply the factor model techniques developed in the literature (see Cunha, Heckman and
Schennach (2010); Agostinelli and Wiswall (2016)).

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3.

9Key Institutional Features

As this paper utilizes Korean datasets, the theoretical framework and the identification
strategy are based on the country’s institutional features. In this section, I explain the
key institutional features of the country: the high-stakes college entrance exam, hierarchical college structure, homogeneous secondary schools, and an established private
tutoring market. While these institutional characteristics offer several advantages
in studying the research questions, a number of countries share these features. As I
describe the characteristics of the system, I explain the possibility of generalizationfor other countries.

3.1

High-Stakes College Entrance ExamIn Korea, the College Scholastic Ability Test (CSAT), the college entrance exam taking
place at the end of 12th grade, is the single most important factor for college admission.13 Students take Korean, Mathematics, English, and elective subjects. The exam
starts at 8:40 am and finishes at 5:45 pm. For this exam, take-offs and landings ofairplanes are suspended for 35 minutes during the English listening test. Firms andgovernment offices are encouraged to delay their workday by an hour to help students
avoid heavy traffic. All these suggest that the taking of the CSAT is a huge national
event. After the exam, students receive a scoresheet that contains a standardized scoreand a stanine score for each subject.14,15 Many educational consulting firms publish
the “cutoff sheet” that contains the firm’s prediction for the cutoffs for all colleges. The
predictions are largely consistent across the firms and are close to the actual cutoffs.
Based on the CSAT score and the predicted cutoffs, each student chooses up to threecolleges in which to apply. Based on the CSAT score and the quota, colleges determine
admission results for students. Several countries have their own high-stakes college
entrance exam. Gaokao of China is a representative example in that the ranking in
the exam is the most crucial factor in college admission. Other examples include
Yükseköğretim Kurumları Sınavı of Turkey, Exame Nacional do Ensino Médio of Brazil,
13

In South Korea, there has been a recent increase in the quota for the holistic review process, inwhich test score is not the only determinant for college admission. In 2019, 24.9% of total students wereadmitted through the holistic admission route (Bastedo 2021).
14
There was one exception in 2007 in which only stanine scores were available for the college admission process. The original standardized score system was restored in 2008. Han, Kang and Lee (2016)
estimate the changes in aggregate effort level of the students due to the grade scheme shift.
15
A stanine score is nine discrete scales ranging from a low of 1 to a high of 9.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

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Sijil Pelajaran Malaysia of Malayasia, and Ulttyq Biryńǵaı Testileý of Kazakhstan arehighly similar in terms of their importance in the college admission process. Baccalauréat of France is highly important for getting into grandes écoles, the group ofelite colleges of the country. The Scholastic Aptitude Test (SAT) of the United States isalso utilized as an important factor in college admission, but other components suchas high school grade-point-average and extra-curricular activities also matter.

3.2

Hierarchical College Structure and College-Tier

The institutional feature also prevalent in other countries is a hierarchical college
structure. In many countries including Korea, college quality is unequal in termsof alumni outcomes. Empirical studies report that graduating from an elite college
significantly affects a student’s future labor market outcomes.16 In South Korea, thecollege hierarchy has changed little (Kim and Lee 2006; Kim 2014). Starting from the
top institution, Seoul National University, the applicants’ preferences have been stablefor decades, and “SKY” is a well-known acronym that refers to the top three universities
in the country. In the 1980s, as the demand for elite colleges have increased, the SKY
universities have become too far of a reach for many people.17 Kim and Lee (2006)
study this hierarchical market structure of universities in Korea and show that a strong
university hierarchy is present in the country. They report that universities in thefirst three deciles strictly dominate the rest in terms of their measure of labor market
outcomes, private donations, quality of faculties, and physical facilities.
Motivated by the college hierarchy, I categorize colleges in Korea into four orderedtiers based on the “cutoff sheet” published by Jinhak (2022), one of the major education
consulting firms. Tier 1 includes the most prestigious universities. The cutoff of Tier
1 is around the top 1% of CSAT scores. Successively, the cutoffs of Tier 2 and 3 areapproximately the top 5%, and top 15% of the CSAT score distribution, respectively.18
Tier 4 is composed of graduates from 2 year colleges. Tier 5 is the residual tier thatabsorbs the rest of the students in the cohort. The member universities of each tier are
specifically reported in Appendix B. I use this categorization of college tiers throughout
16See, for example, Hoekstra (2009) for the United States, MacLeod et al. (2017) for Colombia, Zimmerman (2019) for Chile, Anelli (2020) for Italy, Sekhri (2020) for India, and Jia and Li (2021) for China.
17
In the late 1990s, the term “In Seoul” has appeared, which refers to a group of all universities in
Seoul. Anecdotally, Korean parents often say that they hope their children go to one of these “In Seoul”
universities.
18
The top 15% score is the cutoff for the “In Seoul” universities previously mentioned.

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this paper. In Section 4, I present empirical evidence suggesting the significant effectsof the college tier on post-graduation labor market outcomes.

3.3

Homogeneous Secondary School and Private Tutoring Market

Secondary schools in South Korea are homogeneous, which provides a transparent environment where private expenditure translates into students’ academic performance.
First, the curriculum of secondary school is uniform and under the strict control of
the Korean government. In addition to public schools, even private schools do nothave autonomy in terms of the curriculum and tuition.19 Second, as a result of theconsecutive school-equalization policies, the quality of education provided by schools
is similar.20 No schools are allowed to select students independently.21 In fact, school
assignments for middle school and high school are random within the residential
district for most regions. After graduating from primary school, students are assigned
to the middle schools within the residential education district by lottery.22
At the same time, 2.8% of GDP is spent on private tutoring activities for students byhouseholds in South Korea (Nam 2007). Parents spend 9% of their income on private
tutoring activities for their children, which is a significant amount of expenditure.23
The form of private tutoring varies. The most common form of private tutoring is
hagwon (or cram school), the private academic institutions students go to after regularschool hours. There are also one-on-one tutoring, group tutoring, and online classes.
The country has an established private tutoring market. With the centralized school
curriculum, private tutoring institutes are an effective substitute for parental timein teaching their kids. I use private tutoring as a measure of parental investment
throughout the paper.
19

One of the few decisions of private secondary schools in Korea is that they can independently hire
teachers. Park, Behrman and Choi (2013) provide evidence that the difference in the quality of teachers
is not significant between private and public secondary schools in Korea.
20
See Section II of Kim and Lee (2010) for a description of the history of school equalization policy. Asof 2010, the high school equalization policy has been adopted for all major cities in South Korea.
21
One exception is specialized high schools, which are not subject to the equalization policy. However,not like private schools in the United States, admission to specialized schools is mostly merit-based.
The enrollment for the specialized schools accounts for only 3% of total enrollment. I expect the
disparities due to the specialized high schools are captured by the household characteristics of thedataset.
22
Papers in the literature exploit this random assignment feature to estimate the effects of various
independent variables of interest on educational outcomes. See, for example, Kang (2007), Park,
Behrman and Choi (2013), and Park, Behrman and Choi (2018). Park, Behrman and Choi (2013) showthat the issue of non-compliers to the lottery policy is a minor concern.
23
See Bray (1999, 2021) for a comprehensive cross-country comparison of private tutoring.

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Two main features highlighting the education system of Korea are the homogeneous secondary schools and the fact that college admission relies heavily on the
final exam. This feature provides a transparent environment in which the householdincome is translated into the educational outcome of the child.

4.

Empirical Evidence

4.1

Data

I use the Korean Educational Longitudinal Study 2005 (KELS) for the main estimation procedure. To supplement the income information of KELS, I use the Korean
Labor Income and Panel Study (KLIPS) to supplement the college tier-specific lifetime
income.
4.1.1

Korean Educational Longitudinal Study

The choice of data is motivated by the main goals of the paper: (i) to quantify the roleof accumulated parental investment and student efforts on intergenerational mobility,and (ii) to account for the dynamic selection of the effort choices of the household.
Estimating the marginal effects of parental investment and hours of self-study in eachperiod is necessary to achieve the goals. The Korean Educational Longitudinal Study
2005 provides a rare combination of relevant data. The dataset includes information
on private tutoring expenditure, hours spent for private tutoring, hours spent for
self-study, income of the household, standardized test scores, and parental education.
Household income and private tutoring expenditure are collected each year. The
hours spent in tutoring activities and the hours spent for self-study are collected as aweekly average. There are five different measures of academic performance available
in the dataset. Academic performance in primary school is measured as an ordereddiscrete measure answered by the household. For 7th to 9th grades, the administrative
test scores are of achievement tests standardized at the national level. For 12th grade,the administrative College Scholastic Ability Test (CSAT) score is available. The actual
scores are available for the three achievement tests and the CSAT, which I treat ascontinuous variables.

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Table 1: Data Selection
Original Sample Size

6,908

Cause of Exclusion
Missing CSAT
3,310
Missing at least one period of Income
1,576
Zero Income
16
Missing Initial Test Score
40
Missing one of the parental education
59
Tutoring Expenditure greater than income
6
All choice variables missing
62
Implausible unit price of tutoring
47
Remaining Sample Size

1,792

Table 2: Sample Moments(a) Sample Moments: Other characteristics

Parental Education
6th grade Academic Performance

Mean
13.27
6.52

N

1792

Stdev
2.01
1.70

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.

The nationally representative dataset tracks 6,908 students (1st -year middle-school
students) sampled from the country’s 703,914 7th grade students. The students aretracked starting from 2005 when they are 7th graders. In the first stage of the survey,
the cohort is surveyed yearly up to 2012. In the second stage of the survey, namely
the college and the labor market period, the cohort is surveyed semi-annually up to2020, which is ten years after the cohort graduates from high school. The rules of
selection and their effects are reported in Table 1. The proportion of observations lostto missing the final test score is 0.48. Meanwhile, 99.9% of the students in the dataset
report that they applied for the final exam, which suggests that the missing final exam
score is not caused by the selection to take the final exam. In Appendix 9, I show thatthe effects of the selection do not result in severe differences in the sample moments.
The observations lost to missing income selection tend to have missing CSAT scores aswell. Importantly, I include households missing one of the choice variables: tutoring

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

14

expenditure, hours of tutoring, and hours of self-study. In the estimation section, I
explain the rules to simulate the missing choice variables.
Table 3: Sample Moments (Continued)(a) Sample Moments: 7th - 9th grades
School grade

7th

Tutoring Expenditure
Hours of Self-Study
Hours of Tutoring
IncomeTest Scores

Mean
25.8
5.48
11.37
370.4
323.03

N

1792

8th
Stdev
20.0
5.04
8.50
161.7
45.63

Mean
25.1
5.97
9.69
369.2
321.50

9th
Stdev
19.6
5.13
7.22
151.3
48.72

Mean
36.1
6.45
11.29
400.4
322.65

Stdev
31.0
5.27
9.90
169.9
48.45

(b) Sample Moments: 10th - 12th grades
School grade

10th

Tutoring Expenditure
Hours of Self-Study
Hours of Tutoring
IncomeTest Scores

Mean
38.3
7.65
7.40
406.9
-

N

1792

11th
Stdev
36.5
5.68
6.74
177.0
-

Mean
47.9
8.45
9.16
394.4
-

12th
Stdev
48.6
6.00
9.45
191.1
-

Mean
29.5
14.42
5.69
381.4
415.39

Stdev
41.7
9.14
7.89
171.4
62.46

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.

Table 2 and 3 present sample moments of KELS. While the average hours of selfstudy increase over time, the average hours of tutoring overall show a decreasing
trend. I revisit the implications of such changes in hours allocation in Section 4.5. The
moments of household income are stable over time. I use parental education datacollected in the first year of the survey, and I assume that parental education does notchange within the model period. This is a reasonable assumption given the relatively
short period of time in the data. In fact, information on parental education is collected
only in the first two years of the survey.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

4.1.2

15

Korean Labor Income and Panel Study

The college tier-specific lifetime income is inferred from the Korean Labor Income
and Panel Study (KLIPS). KLIPS is a panel dataset of representative Korean households
from 1998 to 2021. The dataset provides information on which college each worker
graduated from, her major, income history, and other demographic characteristics.
Using KLIPS, I generate the average lifetime income of the alumni for each college tierand complement the labor market information of KELS. In fact, KELS also provides
individual information on the early labor market outcomes of the sample. Still, boththe income data and the participation data have a substantial proportion of missing
data compared to KLIPS. Employing KLIPS is more useful in predicting alumni’s
lifetime income as it contains data on workers of age between 20 and 65.24

4.2

The Lifetime Income Differential

College ranking has a strong effect on the growth of alumni’s income.25 The effect issignificant controlling for CSAT score. Columns (1), (2), and (3) in Table 4 provide the
OLS estimates for the regression equations,

J
X
T ier
ln yit =(βj + δj · ageit )Di,j+ Zit γ + εyit

(1)

j=1
T ier
where Di,jis a dummy variable indicating that person i graduated from a tier jcollege, and Zit is the set of explanatory variables including age, squared age, birthyear, and gender of person i.26
24

The Lifelong Career Survey (LCS) by the Korea Research Institute for Vocational Education &Training (KRIVET) is an alternative dataset that could be used to generate the proxy of the prize of thetournament (Han, Kang and Lee 2016). For the purpose of this paper, KLIPS is preferred because it canrecover the age-specific income profile.
25
Lee and Koh (2023) reports that the alumni of Tier 1 colleges in Korea earn 50.5% more comparedto those from the bottom Tier group, based on their preferred specification and the tier definitions. The
lifetime income empirical exercise in this section is consistent with their findings, but the specificationdiffers to consistent with the structural model.
26
The purpose of the birth year dummy variable is to capture the cohort difference in workers’ income.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

16

Table 4: Log Income Regression(1)
Pooled OLS(2)Pooled OLS(3)Pooled OLS

(4)
RE

(5)
RE

(6)
RE

Top tier-1.931***(0.347)-1.684***(0.325)-2.958***(0.494)-1.671***(0.279)-1.491***(0.288)-2.194***(0.723)

Second Tier

-1.332***(0.350)-1.195***(0.296)-2.460(1.683)-1.409***(0.321)-1.364***(0.323)-1.908**(0.820)Third Tier

-0.864**(0.269)-0.817***(0.196)-1.549**(0.507)-0.958***(0.363)-1.075***(0.364)0.190(1.249)Fourth Tier

-0.895***(0.232)-0.618**(0.186)-1.954***(0.361)-0.727***(0.079)-0.524***(0.123)-0.425(0.436)age

0.092***(0.001)

0.092***(0.001)0.167***(0.018)0.095***(0.002)

0.095***(0.002)0.168***(0.040)Top tier × age

0.067***(0.009)0.065***(0.009)0.111***(0.014)0.058***(0.009)0.058***(0.009)0.105***(0.024)

Second Tier × age

0.050***(0.011)

0.052***(0.010)0.098(0.054)0.051***(0.010)0.055***(0.011)

0.095***(0.031)

Third Tier × age

0.032***(0.008)0.037***(0.007)0.058**(0.019)0.034***(0.012)0.044***(0.013)0.011(0.050)Fourth Tier × age

0.029***(0.007)0.027***(0.007)0.073***(0.015)0.022***(0.002)0.021***(0.002)0.038***(0.013)

N
Major
RE
CSAT

29599
No
No
No

29599
Yes
No
No

685
No
No
Yes

29599
No
Yes
No

29599
Yes
Yes
No

752
Yes
Yes
No

College Tier

Interactions

Source: Korea Labor Income and Panel Study 1998-2012, Korea Labor Institute.
Note: RE refers to “Random Effects.” Explanatory variables used in the regressions such as squared age,birth year, and gender are excluded from the table for brevity. The sample includes workers between 25
and 65 years old who work for wages or salary. I exclude workers who are born after 1992.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

17

Figure 2: Income Dynamics by College Tiers

Source: Korea Labor Income and Panel Study 1998-2012, Korea Labor Institute.
Note: The sample includes workers between 25 and 65 years old who work for wages or salary. I exclude
workers who are born after 1992. The figure has units of 1,000 KRW, which is about 0.85 USD. Annual
income is predicted using the Pooled-OLS estimates in column (1) of Table 4.

Note that the regression equation captures both the effect of graduating from a tierj college on the level and the growth of an alumnus’s income, respectively by βj and δj .
Columns (3) and (4) provide the estimates of the random effects model,
J
X
T ier
ln yit =(βj + δj · ageit )Di,j+ Zit γ + λyi + ηity
j=1

where λyi and ηity are the individual-specific and the idiosyncratic errors respectively.27
Columns (2) and (5) include the dummy variables of college-major, showing that theinclusion of major does not critically affect the main results of Columns (1) and (4),respectively. The Tier 1 dummy has the smallest estimate of intercept but the largest
estimate of age differential. Figure 2 presents the predicted annual income of alumni
using the estimates in Column (1) of Table 4. Before age 30, there is no economicallysignificant difference in terms of annual income. On the other hand, the gap becomessignificantly larger as people age. The effects are significant controlling for CSAT
27

Since the focus of the regression is the college tier, which is time-invariant, I do not consider the
fixed effects model.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

18

score, as can be seen in Columns (3) and (5) of Table 4.28 The estimation results are
consistent with the studies stressing the importance of using lifetime income in the
returns to schooling literature (Haider 2001; Tamborini et al. 2015; Nybom 2017). The
effects of parental investment on labor market outcomes through college reputation
would be underestimated if researchers narrow their focus to the early labor market
outcomes. The Pooled-OLS estimates in Column (1) of Table 1 are used in computing
college-specific lifetime income, which is a component of the dynamic tournamentmodel.

4.3

Competition Motives of Parental Investment

Competition with respect to getting into a more prestigious college is the primary
motivation of parental investment. First, data suggest that the demand for private
tutoring expenditure significantly drops as students finish the college admission
process. Figure 3 presents the change of tutoring expenditure and participation rateover time for the sample cohort of KELS. Both expenditure and participation of private
tutoring rapidly drop as soon as students graduate from high school, which suggeststhat the primary purpose of tutoring expenditure is associated with college admission.
If the purpose of tutoring expenditure was for enhancing the student’s human capital,
it is unlikely that most students would completely stop private tutoring activities upon
graduating from high school. Second, the number of seats at prestigious colleges islimited. Even with a very high final test score, students might not be able to go to a
top-tier college if the seats are filled with students with higher test scores. The scarcity
of seats at prestigious colleges and the fact that tutoring participation drops after
the college entrance exam show that competition is the key feature determining theparental investment decision of the household.
28

As CSAT performance is collected as a discrete variable in KLIPS, the estimation is different with
Regression Discontinuity Design.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

19

Figure 3: Private Tutoring Expenditure and Participation in Tutoring

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.
Note: I only include households who do not have missing information on the following variables:
tutoring expenditure, CSAT scores, and household income.

4.4

Parental background and child’s hours allocation

Compared to hours of tutoring, hours of self-study are less affected by parental income,which potentially has implications for intergenerational mobility. On the one hand,
the income elasticity of hours of tutoring is higher than the income elasticity of hours
of self-study. Figure 4 presents how hours of tutoring and hours of self-study varywith parental income when students are 7th, 8th, and 9th graders, using local linear
regression. The slope of hours of tutoring is much steeper than the slope of hours
of self-study, which shows that tutoring is an effort choice that is more responsive
to parents’ income. On the other hand, the covariation between hours of self-study
and parental education is higher than the covariation between hours of tutoring andparental education, conditional on other household characteristics. Figure 5 presents
how hours of tutoring and hours of self-study vary with parental education when
students are 7th, 8th, and 9th graders. Unlike household income, the effect of parental
education is higher on hours of self-study than the effect of parental education on
hours of tutoring.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

20

Figure 4: Income Gradient in Effort Decision

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.
Note: The gray regions are confidence bands with a significance level of 0.05. I only include households
who do not have missing information on the following variables: tutoring expenditure, CSAT scores,
and household income.

Figure 5: Parental Education and Efforts Allocation

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.
Note: In this graph, parental education is a categorical variable and based on the average years ofparents educationmi , which is defined as follows: Below High School if mi < 12, High School if mi = 12,
Some College if 12 < mi < 16, College Degree if mi = 16, and Graduate if mi > 16. I only include
households who do not have missing information on the following variables: tutoring expenditure,
CSAT scores, household income, and parental education.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

21

Parental education soaks up significant variation in hours of self-study, whichleaves a relatively small variation with parental income. Tables 5 and 6 presents thepooled OLS estimates of the regression equation,ln(1 + yit ) =β0 + β1 log(hhincit ) + β2 mi + ϵit(2)where hhincit is the income and mi is parental education of household i. Columns (1)through (3) present the results where yit is hours of self-study, and columns (4) through(6) present the results where yit is hours of tutoring. Columns (1) and (4) provide theestimates without including the average years of parents’ education, and Columns (2)and (5) provide the estimates with including the average years of parents’ education
to equation (2). Overall, hours of tutoring are explained more by parents’ income thanhours of self-study. Moreover, much of the covariation between hours of self-study
and income is absorbed after controlling for the average years of parents’ education.
Such empirical relationships suggest that different household backgrounds canlead to different allocations of effort choice. Thus, omitting one of the effort choices(parental investment or child effort) might result in biased estimates of intergenerational mobility, which calls for including both effort choices in the theoreticalframework.
Table 5: The Effects of Parental Background on the Hours Allocation

log(Income)(1)log(1+Study)(2)
log(1+Study)(3)
log(1+Study)

0.238***(0.022)0.152***(0.025)0.036(0.027)

Parental Edu
N
Year
FE

0.055***(0.007)
10454
Yes
No

10454
Yes
No

10454
Yes
Yes

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.
Note: log(1+Study) and log(1+Tutoring) refer to log of hours of self-study plus one and hours of tutoringplus one, respectively. I only include households who do not have missing information on the following
variables: tutoring expenditure, CSAT scores, and household income. Parental Educ indicates average
years of parents’ education.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

22

Table 6: The Effects of Parental Background on the Hours Allocation

log(Income)(4)
log(1+Tutoring)(5)
log(1+Tutoring)(6)
log(1+Tutoring)

0.677***(0.027)0.616***(0.030)0.269***(0.037)

Parental Edu
N
Year
FE

0.038***(0.008)
9431
Yes
No

9431
Yes
No

9423
Yes
Yes

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.
Note: log(1+Study) refer to log of hours of self-study plus one and hours of tutoring plus one, respectively.
I only include households who do not have missing information on the following variables: tutoring
expenditure, CSAT scores, and household income. Parental Educ indicates average years of parents’
education.

4.5

Dynamic effort allocation of households

Students’ time allocation of effort choices considerably changes as students proceed
to the later educational stages. Figure 6 presents how the average hours of self-study
and the average hours of tutoring change with students’ grade level. While the average
hours of tutoring shows a decreasing trend, the average hours of self-study shows anincreasing trend. In 12th grade, the average hours of self-study is almost three times
the average hours spent for tutoring. Such changes in time allocation suggest thatthe marginal effects of hours of self-study and tutoring expenditures on academic
outcomes might change over time.29
29

Several studies in the literature report that the effects of parental investment decrease with children’sage (Cunha et al. 2010; Del Boca et al. 2017). To the best of my knowledge, there is no study reportingthe changing effects of self-study over time.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

23

Figure 6: Dynamic Allocation of Efforts

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.
Note: I only include households who do not have missing information on the following variables:
tutoring expenditure, CSAT scores, and household income.

The initial conditions of the household persistently affect the parental investment
decisions throughout the secondary school periods. Figure 7 presents changes in the
average hours of tutoring expenditure over time differentiated by two of households’
pre-conditions: the initial academic performance and the initial parents’ income.
To see how these initial conditions affect the investment decision of households, I
present the changes in average tutoring expenditure of two sub-groups: the top 20%
and the bottom 20% of the ordered initial conditions. In particular, the solid lines of
Figure 7 connect the average tutoring expenditure of the highest 20% of households
classified by the two initial conditions. In the same manner, the dotted lines connect
the average tutoring expenditure of the bottom 20% of households. Figure 7 (a) shows
the increasing gap in tutoring expenditure between those who were in the top 20%of the test score in 6th grade and who were in the bottom 20% of the test score in
6th grade over time. In 7th grade, there is no significant difference between the twogroups in terms of tutoring expenditure. From 8th grade on, there is an evident gap
in tutoring expenditure between these two groups. Based on the average tutoring
expenditure in 12th grade, students who were in the top 20% of the test score in
7th grade increased their tutoring expenditure compared to when they were in 7th
grade. In comparison, the students who were in the lowest 20% of the test score in 7th
grade decreased their tutoring expenditure compared to when they were in 7th grade.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

24

Figure 7 (b) presents the average tutoring expenditure of high-income and low-income
groups. The gap is significant in 7th grade and becomes greater over time. On average,
high-income households’ tutoring expenditure increases in 12th grade compared towhen the students were in 7th grade. On the other hand, low-income households’
tutoring expenditure decreases on average compared to when the students were in
7th grade.
Figure 7: Dynamic of Parental Investment by Initial Conditions

Source: Korea Educational Longitudinal Study 2005, Korean Educational Development Institute.
Note: In this figure, academic performance is measured in 6th grade and used for subsequent years.
I only include households who do not have missing information on the following variables: tutoring
expenditure, CSAT scores, and household income.

The evidence suggests that households self-select into the different effort levels
based on their preconditions, and the allocation of the two efforts changes over time.
As suggested earlier, different effort choices might have different implications forintergenerational mobility. Capturing the changing behavior of the households is
crucial to get the correct quantification of the statistics of interest.

5.

A Dynamic Model of College Admission Tournament

Motivated by the empirical evidence, I build and estimate a dynamic model of competition where each household chooses the amount of parental investment and the levelof child’s efforts. The dynamic model is built upon the rank-order tournament first described by Lazear and Rosen (1981) and related to its applications in college admission
competition (Han, Kang and Lee 2016; Grau 2018; Tincani, Kosse and Miglino 2021).

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

5.1

25

Timeline

There exist N households in the dynamic tournament. Each household is composedof one student and the parents. I assume the household makes a unitary decision.
I abstract away from the intra-household decision-making process. The students
compete for the final prize against other students in the same cohort.
Figure 8 illustrates the timeline of the model. The model begins as the student of
the household enters into 7th grade, which is the first year of secondary school. Eachhousehold is born with the complete income stream {wit }Tt=1 , parental education mi ,and initial test score qi0 . Also, each household has a specific type k. Different types ofhouseholds have different person-specific characteristics that are unobserved by the
econometrician. I define them as λck , λxk , λsk and λqk , which affect marginal utility fromconsumption, disutility from hours of tutoring, disutility from hours of self-study, andlog of test score, respectively. Some households value non-academic goods such as
travel more than other households conditional on the observed characteristics (Lazear
1977). Such unobserved taste for consumption is captured by λck . Some households
prefer to encourage their child to study independently rather than send her to tutors,
which is captured by the relative size of λsk to λxk . Some students might be particularly
good or bad in taking exams, which would be captured by λqk .30

c , ηx ,Shocks ηitit

Born withs realizedηit

Test score(qi,t+1 )realizedwith εqitc , ηx ,Shocks ηiTiT
s realized
ηiT

Final Test
score (qi,T +1 )realizedqwith ηiT

income stream{wit }T
t=1 , anintiial test score
qi0 and a typet

t+1

Decision of tutoring(pit , xit ), and hours
of self-study (sit )T
Decision of tutoring(pit , xit ), and hours
of self-study (sit )T +1

Collegeadmission
resultdetermined

Figure 8: Model Timeline

At each time t, as the household enters into the period, the shock to the marginalutility of the consumption ηitc , the shock to the marginal disutility from the tutoring activities ηitx , and the shock to the marginal disutility from self-study ηits are realized. These shocks capture the unobserved time-varying components that are notqcxs
I introduce the joint distribution of the time-specific shocks (ηit
, ηit
, ηit
, and ηit) and the speciq
c x s
fication of type-specific unobserved heterogeneity (λk ,λk ,λk and λk ) when I explain the flow utility
component of the model.
30

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

26

accounted for by the deterministic components of the model. Based on those realized
shocks and the observed state variables, each household chooses the quality of tutoring pit , the hours spent on tutoring xit , and the hours of self-study sit to maximize its
value function. The choices are subject to budget and time constraints. Subsequently,
the test score qi,t+1 is produced with the realization of the test score shock. This processrepeats until the final test score qi,T +1 is generated.
Each student is assigned to a college tier based on the ranking of the final test score
and the fixed number of college seats in each tier. I denote nj as the fixed number ofseats for the j th college-tier. In particular, denoting n1 as the fixed number of seats forthe first college tier, the first n1 students are assigned to the top college tier, and thenext n2 students are assigned to the second tier. The process repeats until the (J − 1)thcollege tier is filled up with nJ−1 students so that all seats for the college tiers bind. The
bottom tier is a residual tier which is composed of students whose score is below thecutoff for the (J − 1)th college tier and the students who do not go to college.31 The
assigned college tier is the sole determinant of ex-post lifetime income.

5.2

The Preliminaries of the Tournament

Prize: Lifetime Income. The prize for going to a more prestigious college tier is ahigher expected lifetime income awarded to the student, which motivates the household to exert effort. There exist J college tiers that are characterized by expected
lifetime income vj . The tier-specific lifetime income vj is the discounted sum of thepredicted income of the graduates. In particular,
∗

vj =

T
X

β t−T ŷjt

t=T +1

where ŷjt is the estimated income of the alumni of college tier j in year t, T is the age
when the student graduates from college, T ∗ is the retirement age, and β is the discountfactor fixed to 0.95.32 I define ŷjt as the estimated tier-specific annual income at time t,
31

The implicit assumption regarding the bottom tier is that everyone graduates high school. The highschool drop-out rate in South Korea is less than 2%.
32
The average interest rate is around 5% for South Korea in 2010.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

27

which is predicted using Pooled-OLS estimates of Column (1) in Table 4.33 As tier 1 isdefined to be the top college tier, v decreases in j (i.e., v1 > v2 > ... > vJ−1 > vJ ).34
For the student of household i to obtain prize vj , her final test score qi,T +1 must
be above the cutoff for tier j and below the cutoff for the tier j − 1. In other words,
student i is placed in college tier j iff

Q̃j−1 > qi,T +1 ≥ Q̃jwhere Q̃j is the cutoff between college tier j and tier j + 1. The cutoff Q̃j is the test
Pscore of the Njth highest student in the sample, where Nj = jl=1 nl . Thus, {Q̃j }Jj=1
is where the competition enters the model. In order for a student to be in tier j or
better, she has to be above enough competitors by at least scoring the Njth highest
final test score. As qi,T +1 is a function of the effort choice of each household, {Q̃j }Jj=1 isendogenously determined by the competition across households. I assume that eachhousehold can correctly predict the final test score cutoffs.35
Assumption 1. Each household correctly guesses the set of final test score cutoffs {Q̃j }Jj=1 .
The facts that (i) college-tier is assigned solely using the final test score qi,T +1 and(ii) heterogeneity in college quality is the only variation of the lifetime income inthis framework imply that the final test score of a student essentially determines the
lifetime income of the student. That is, under the model environment, I assume
that there is no extra opportunity to improve one’s lifetime income once the college
entrance exam is over.
Assumption 2. The quality of the college one graduates from is the sole determinant of
one’s lifetime income.
This is an arguably reasonable assumption under the institutional setting of theinterest. I borrow the results of Kang, Kang, and Kim (2022) as supporting evidence for33

I assume no earnings in the college periods.
I confine the prize to pecuniary rewards and rule out other benefits from the model. One mightargue that the non-pecuniary value of attending an elite college should be considered part of the reward.
However, it is difficult to separately measure the non-pecuniary value of attending better colleges due
to data limitations. See Gong et al. (2019) for an empirical quantification of the consumption value ofcollege.
35
I assume away the inconsistency between the guessed cutoffs and the resulting cutoffs because theworking sample did not go through significant policy shock that might cause the difference between
the guessed and the resulting cutoffs. See Tincani, Kosse and Miglino (2021) for the case that resulting
cutoffs significantly deviate from the guessed cutoffs.
34

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

28

Assumption 2. Using a dataset of one of the big 5 companies of Korea, they find thatthe effect of college reputation dominates the effect of college GPA on receiving anoffer from the firm. Table 7 presents their estimates. Based on their probit estimates,
increases in college GPA by 10 leads to a 6% increase in getting an offer from one of
the subsidiary firms of the conglomerate. Meanwhile, graduating from one of the tier
1 colleges increases probability of getting an offer by 23% relative to graduating from acollege below tier 3.
Table 7: Job Offer Regression(1)
probitTier=1
Tier=2
Tier=3
ColGPAN

0.221***(0.039)0.056***(0.005)0.002(0.028)0.006***(0.002)
9132

Source: Confidential data of the conglomerate in late 2010s.
Note: The data are on the applicants to the subsidiary firms of the conglomerate for the latest three years.
Other explanatory variables include the subsidiary firm’s information and the applicants’ informationsuch as college major, age, and gender. The college GPA measured is scaled 0 to 100. ColGPA refers to
the average of standardized college GPA.

Parental Investment: One of the two modes of household effort is parental investment, which is embodied in private tutoring expenditure. Each household chooses
the unit price (quality) of tutoring pit and hours (quantity) of tutoring xit to increase
the child’s test score.36 The total amount of tutoring expenditure eit iseit = pit xit .
36

To the best of my knowledge, this is the first model to consider the quality and quantity of parental
monetary investment simultaneously.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

29

The tutoring expenditure is constrained under two dimensions. A household cannot
spend more tutoring expenditure than its income (i.e., eit ≤ wit ).37 Also, hours oftutoring are bounded by the child’s maximum available time, namely h. While theincome constraint is unequal among households, available hours for the child are
constant across all households.
Note that the time choice is solely about the time use of the child, which means I
do not model the time allocation of parents. The data suggest that, in the secondary
school periods, which the model concerns, the majority of parents do not teach theirchildren themselves in middle school periods, and very few parents use their timeto teach their child in the high school periods. A few potential explanations can begiven for this empirical fact. As students grow, the test materials become more andmore difficult to be taught by parents. Also, if there exists an established tutoringmarket, it would be a safer option for parents in terms of increasing student’s testscore. Note that the model concerns a regime with a high-stakes standardized test.
Full-time tutors would have a comparative advantage in preparing students for exams
over parents.
Child’s hours of self-study: Hours of self-study is the other household’s mode ofeffort in the tournament. Each household chooses how much time to allocate for hours
of self-study sit which is constrained by h. Unlike parental investment, the resource ofself-study does not vary over households as time is equally granted to everyone. The
taste for self-study, however, can be considerably heterogeneous across students. For
example, some students might prefer studying independently rather than re-learning
the same materials from the tutors. Others may prefer reviewing materials with tutors
rather than studying alone. I allow the taste for hours of self-study to vary by parental
education and the associated shock.
Test Score Production Function: The final test score is the result of accumulated
dynamic choices of the household along with its given initial conditions. The initialacademic performance qi1 is exogenously given and proxied by academic performancein primary school.38 The three choices affecting test scores are quality of tutoring
pit , hours of tutoring xit , and hours of self-study sit . I allow that the quantity (hours)
and quality (unit price) of the tutoring activity have different intensities in contributing to the test score production. Denoting κ as intensity of quality of tutoring, the37

I assume no borrowing.
Although an earlier measure of the initial child’s ability would be more desirable, this is the earliest
time period that the academic performance data are available.
38

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

30transformed tutoring input is specified asẽit =pκit xit1−κ

(3)where κ < 0.5 and follows decreasing returns to scale (DRS). The DRS restriction
is necessary to prevent the household from choosing an infinitesimal quantity oftutoring hours. If κ ≥ 0.5, the household always has an incentive to make pit greaterand xit smaller. The opposite case of a household choosing extremely large hours oftutoring does not occur as available time is restricted by h.
For each time t = 1, 2, ..., T , the test score qi,t+1 is produced following

qi,t+1 = g(θtq , qit , pit , xit , sit , ηitq , λqk )where ηitq is the test score shock, λqk is the type-specific error, and θq is the set of relevant
parameters for the test score production. The inclusion of the test score producedin the previous period, qit , allows that the previous test score has its own effects ingenerating subsequent test score (Cunha and Heckman 2007). Furthermore, I allow thesubset of production parameters to change across periods. The effect of the combined
efforts of the household is likely to change over time. As students grow older, thematerials taught become more advanced, which makes it harder for students withinsufficient background to catch up. Thus, private tutoring expenditure and hours
of self-study can be less effective in the later stages of education. In addition, therelative importance of each investment might change over time. For example, themarginal effects of parental investment might increase (decrease) while the effects
of self-study decrease (increase) over time. To reflect such changing effects, I let themarginal effects parameters νt , δpt , and δst be different for each period t = 1, 2, ...T .
For estimation, the production function g is a Constant Elasticity of Substitution(CES) production function and is specified asδqi,t+1 = At qitqtδet (1 + ẽit )ϕ + δst (1 + sit )ϕ

 νϕt

εqit

(4)

where At is total factor productivity, νt is the parameter of marginal effect of the
combined effort choices, and ϕ is the parameter governing substitution between
tutoring and self-study. The marginal effect of the total effort decision is captured by
νt , while the relative importance of the tutoring expenditure and hours of self-study

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

31

are captured by δet and δst , respectively. I define εqit as a combined shock of λqk and ηitq ,which is specified as ln εqit = λqk + ηitq .

5.3

Household

Flow Utility: The utility function of the unitary household is comprised of three parts:(i) the marginal utility from the household consumption cit , (ii) the marginal disutility
from hours spent on tutoring xit , and (iii) the marginal disutility from hours of selfstudy sit . I denote αc , αx , and αs as taste parameters for household consumption,
hours of tutoring, and hours of self-study, respectively. The taste parameters may
depend on the fixed characteristics of the household. I assume additive and separable
log utility, which is specified asu(cit , xit , sit , εit ) =αc εcit log(cit ) + αx εxit log(1 + xit ) + αs εsit log(1 + sit )(5)where εcit is the shock to the marginal utility from consumption, εxit is the shock to thedisutility from hours of tutoring, εsit is the shock to the disutility from hours of selfstudy, and εit = {εcit , εxit , εsit }. The shocks are distributed joint normal and separated
into the type-specific and the time-varying components. In particular, I denote λzk andηitz as type-specific and time-varying components of εzit (z = c, x, s), respectively. The
shocks are decomposed as     ln εcit
ηitc
λck
ηitc
 x   x   x x
ln εit  ηit  λk   =   +   , and ηit  ∼ N (0, Ωη )
ln εs  η s  λs η s 
 it   it   k 
 it 
qqq
ln εitηitλk
ηitq
where Ωη is the covariance matrix for the time-varying shocks.39 I assume that thecorrelations between the time-varying shocks ηitz (z = c, x, s, q) are 0.
Note that I do not specify the utility flow from the current test score. Each household is concerned solely about the final outcome, and the role of the current testscore is limited to the stepping stone for the final test score. That is, the current testscore affects the decision of the household only through the value of the future. The
39

In modeling the self-study shock, an alternative specification involves assuming that there exists
unobserved heterogeneity in terms of the productivity of hours of self-study. Such an assumption,
however, is computationally burdensome if the test score production function is CES.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

32

specification of future value is introduced with the recursive formulation at the end of
the subsection.
Terminal Value: Expected lifetime income is the terminal value of the model,
which drives the dynamic choices of the tournament model. With the tier-specificlifetime income vj , the expected lifetime income is a weighted sum,

J 
X


ln(vj ) ∗ P rob(ln Q̃j−1 ≥ ln qi,T +1 ≥ ln Q̃j ΓiT )

(6)

j=1

where P rob(ln Q̃j−1 ≥ ln qi,T +1 ≥ ln Q̃j ΓiT ) is the probability of getting into college tierj. The randomness of the admission probability comes from the test score shock ηitq .
Each student would have a different probability of going to a college tier j as they havedifferent characteristics affecting the evolution of the test scores. The disparity among
students in terms of going to each college tier leads to the discrepancies in expected
lifetime income, which generates the heterogenous incentives among households. The
higher expected lifetime income leads to bigger the terminal value of the household,which makes it more appealing for the parents to invest in the child.
The functional form of the expected lifetime income is determined by the test scoreshock εqit . With the log-transformation, the terminal value is specified asJ Xln(vj ) ∗ P rob(ln Q̃j−1 ≥ ln qi,T +1 ≥ ln Q̃j ΓiT )

j=1

=

J 
X
j=1



ln g̃ij−1
ln g̃i,j−1
ΓiT ) − Fq (ΓiT )ln(vj ) ∗ Fq (σq
σq

where ln g̃ij is the distance between the deterministic components of log final test scoreqof student i and the log cutoff of the college tier j (i.e. ln ḡij = ln Q̃j−1 − ln q[
i,T +1 − λk ),and F is the distribution of ηitq . I assume F follows normal distribution in the spirit ofrank-order tournament model (Lazear and Rosen 1981; Han et al. 2016; Grau 2018;
Tincani et al. 2021).40
Budget and Time Constraints: The choices of the household are restricted by thebudget and the time constraints. The budget constraint is given byq
One can also adopt a functional form that ηitfollows Generalized Extreme Value distribution which
results in a Tullock (2001) contest.
40

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

cit + pit xit ≤ wit

33

(7)

where wit is household income, and the time constraint is

xit + sit ≤ h(8)where h is student’s disposable time. I define h as the maximum time each student
can use every week, which is assumed to be 63.41
State Variables: There are observed and unobserved state variables in the dynamic
model. The set of observed state variables Zit includes the previous test score qit ,parental education mi , and the complete income stream {wit }Tt=1 . The set of unobserved state variables Ψit includes the set of unobserved shocks and the type specificheterogeneity. Based on the timeline, the time-varying shock regarding test score is
not an unobserved state variables. (i.e., Ψit = {ηitc , ηitx , ηits , λck , λxk , λsk , λqk }).
Information and Uncertainty: I assume a continuum of households. The continuum assumption is useful in that the information of other households can be summed
up as a distribution of households.
Assumption 3. The distribution of household is common knowledge.
As stated in Assumption 1, each household correctly anticipates the set of collegetier cutoffs {Q̃j }Jj=1 .42 They know the distribution of the final test scores in advanceand make dynamic choices based upon the perfect guess.
Assumption 4. Each household knows its complete wage stream.
There is no uncertainty in the income process. In fact, each household is assumed
to know its complete wage stream as the model begins. As this is a markov model, past
wages are irrelevant after conditioning on the remaining state variables. As depicted
in Figure 8, each household learns about the realization of the consumption shock ηitc ,the disutility shock to hours of tutoring ηitx , and disutility shock to hours of self-study
41

I assume each student can use 9 hours everyday for non-leisure activities other than hours spent inregular school42
In the static model of Grau (2018), Assumption 3 implies that the tournament participants cancorrectly guess the cutoffs. In my dynamic model, however, Assumption 3 does not guarantee theperfect foresight due to the presence of future shocks that each individual cannot predict.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

34

ηits at the beginning of each period. However, it does not know about the test scoreshock ηitq before it makes a decision. Therefore, it makes a set of choices based on thecxs
expectation over ηitq , ηi,t+1, ηi,t+1, and ηi,t+1, conditional on observed state variables andtype-specific unobserved heterogeneity.
Household Value Function: Building upon the model components, I describe thevalue function of the household. As stated earlier, each household chooses the unit
price (quality) of tutoring pit , hours of tutoring xit , and hours of self-study sit based on
the anticipation of future values. In particular, at each time t, the household i solvesVit (Zit , Ψit ) = max

pit ,xit ,sit

u(cit , xit , sit , εit ) + β qE

ηit, ηit
Vi,t+1 (Z i,t+1 , Ψi,t+1 Γit ) ,(9)subject to equation (4) and constraints (7) and (8), where Γit = {Zit , Ψit , {Q̄j }Jj=1 } isthe set of information before making the decision and ηit = {ηitc , ηitx , ηits } is the set ofunobserved time-varying shocks. Each household faces a tradeoff between current
flow utility and future payoffs. Each choice variable incurs costs associated with the
choice. In particular, investing more in parental investment (i.e., increasing pit orxit ) requires suffering more from the disutility from hours of tutoring and sacrificing
current consumption. Spending more time on hours of self-study leads to an increasein the disutility from hours of self-study. This dynamic incentive structure governs thedecision of the household.
At the final test stage (t = T ), where the tournament of the final score occurs, thevalue function is

ViT (ZiT , ΨiT ) =maxpiT ,xiT ,siT

+ αv

J
X


u(ciT , xiT , siT , εiT )ln(vj ) × P rob(ln Q̃j−1 ≥ ln qi,T +1 ≥ ln Q̃j ΓiT )

(10)

j=1

where αv is an altruism parameter. The altruism parameter measures the “exchange
rate” between the current household utility and the child’s future lifetime income.
All-in-all, each household makes a choice between the child’s lifetime income and its
flow utility. If the marginal value to the household is greater than the marginal loss of

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

35

flow utility of the household, it exerts more efforts using either parental investment,
the child’s self-efforts, or both.

5.4

Equilibrium of the Tournament

In this section, I define the dynamic equilibrium of the tournament model. Then
I prove the existence of the equilibrium using the Schauder Fixed-Point Theorem(Amir 1996; Fey 2008; Mertens and Judd 2018; Engers, Hartmann and Stern 2022). I
define a set k = {{Vt (pt , xt , st ; Zt , Ψt )}Tt=1 , {Q̃j }Jj=1 }, where {Vt (pt , xt , st ; Zt , Ψt )}Tt=1 is aset of value functions that are specified in equations (9) and (10), and {Q̃j }Jj=1 is theset of college-tier cutoffs. I define K as a set of all possible k.
Definition 1. Given the set of initial conditions
 and Assumptions 1, 2, and 3, aMarko-

vian equilibrium of the model is a vector k ∗ = {Vt∗ (pt , xt , st ; Zt , Ψt )}Tt=1 , {Q̃∗j }Jj=1 , whichis generated by the following process:1. I define Ka as a set of all possible combinations of choice variables {pt , xt , st }Tt=1 .
Given the set of initial conditions {qi1 , {wit }Tt=1 , mi }, a mapping ℵa maps K into
Ka (ℵa : K → Ka ), based on the value functions specified in equations (9) and(10). 43
2. I define Kb as possible distributions of the final test score qT +1 . A mapping ℵbmaps Ka into Kb (ℵb : Ka → Kb ), based on the test score production function
specified in equation (4) .
3. I define Kc as possible sets of resulting cutoffs {Q̌j }Jj=1 . Given the number ofseats for each college tier {nj }Jj=1 , a mapping ℵc maps the distribution of the final
test score qT +1 and {nj }Jj=1 into the set of cutoffs, {Q̌j }Jj=1 (ℵc : Kb → Kc ). The
mapping ℵc is based on the rules of college admission.
4. A mapping ℵd maps {Q̌j }Jj=1 into K (ℵd : Kc → K).
5. In equilibrium, the set of guessed cutoffs {Q̃j }Jj=1 match the set of realized cutoffs{Q̌j }Jj=1 .
43

The mapping ℵa involves backward recursion.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

36

Finally, I define a mapping ℵ : K → K. The mapping ℵ is the composition of submappings. In particular,ℵ =ℵa ◦ ℵb ◦ ℵc ◦ ℵd=ℵa (ℵb (ℵc (ℵd (k)))).
Lemma 2. The mapping ℵ is compact
Proof. [In Appendix A.1]
Lemma 3. The mapping ℵ is continuous
Proof. [In Appendix A.2]
Theorem 4. A Markovian equilibrium exists.
Proof. Previous results establish that K is a nonempty, compact, and closed subset ofa locally convex Hausdorff space. The map ℵ is continuous. Therefore, the set of fixed
points of ℵ is nonempty and compact. The mapping satisfies all the requirements of
Schauder Fixed-Point Theorem. Hence a fixed point exists.

5.5

Features of the Model

The dynamic tournament model offers several features that help answer the researchquestion of this paper. First, the rich heterogeneity of state variables and the choice set
and the specification of the test score production function help disentangle the sourceof intergenerational persistence of earnings. Each household can simultaneously
choose the quality of tutoring, hours of tutoring, and hours of self-study in the model
based on its state variables. The test score production function allows for a variety of
inputs: previous test score qit , quality of tutoring pit , hours of tutoring xit , and hours
of self-study sit . This specification enables me to separately quantify the impact ofparental investment, self-effort of the child, and other household characteristics onintergenerational persistence of earnings. In the model, each household can choose
the other mode of investment even if they are not allowed to use one of the options.
Thus, the model provides an opportunity to simulate the reaction of the household
when one of the choices is restricted. In Section 8, I quantify the role of each choice by

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

37

simulating the model with shutting down that particular choice from the model. Then
I compare the simulation results with when all choices are allowed.
Second, the rank-order feature of the tournament model enables me to study
the effects of the changes in cohort size and the role of disparity in college quality.
Since the tournament model is about obtaining a limited number of seats in the
better colleges, the model can be used to evaluate the exogenous changes in thenumber of competitors. As described in Section 2, countries with high private tutoring
expenditure face a sharp decrease in total fertility rate. The reduction of cohort sizedue to declining fertility means that there is a less number of competitors for higher
college tiers.44 The model provides an opportunity to simulate parental investment ofhouseholds if the cohort size is reduced. Additionally, the tournament model also canbe used in assessing the role of disparity in college quality on parental investment. The
well-known feature of the tournament model is that the size of the prize differential
affects the effort choice of the agents (Lazear and Rosen 1981). The prize differential
in my model is the income differential of the higher college-tier. Such a feature of
the tournament model captures the role of the distribution of college qualities on theinvestment decision of the household.
Third, as I allow for the time-varying effects of the choices, I can compare theeffects of hours of self-study and hours of parental investment.

6.

Estimation Strategy

The touranment term at the final period T generates heterogenous incentives forhouseholds with different state variabels. These features make the policy function
highly non-linear. The choice variables of the structural model involves the pairs ofinterior-interior, interior-corner, and corner-corner. In addition, the structural model
involves endogenous regressors. To this end, I estimate the parameters of the modelusing Maximum Simulated Likelihood. I describe the likelihood function and discuss
the sources of identification underlying the estimation procedure.
44

In Section 8 I show changes in the number of seats in colleges and the number of high school
graduates using administrative data of South Korea.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

6.1

38

The likelihood function

I denote θ as the set of parameters, Zit as the set of observed state variables, andλk as the set of unobserved type-specific characteristics. The individual likelihood
contribution of household i is

Li (θ|qi0 , {wit }Tt=1 , mi ) =

K 
X



ΠTt=1 Lit (θ|Zit , λk )Pr(type = k)

(11)

k=1

which is conditional on the initial test score qi0 , the income stream {wit }Tt=1 , andparental education mi . The time-specific likelihood contribution Lit (θ|Zit , λk ) can becharacterized in four different ways depending on the combination of the tutoringparticipation dummy variable dxit and self-study participation dummy variable dsit . In
particular,dxit dsit
Lit (θ|Zit , λk ) = f (pit , xit , sit , qit )dxit (1−dsit )× Pr(pit , xit , sit = 0) · fqit (qit |xit , sit = 0)(1−dxit )dsit
× Pr(xit = 0, sit ) · fqit (qit |xit = 0, sit )(1−dxit )(1−dsit )× Pr(xit = 0, sit = 0) · fqit (qit |xit = 0, sit = 0)where dxit = 1 means that household participates in tutoring at time t, and dsit = 1
means that the student of the household i has non-zero hours of self-study at time t.
The final form of the likelihood function is a sum of the log likelihood contributions,log L(θ) =N
X

log Li (θ|qi0 , {wit }Tt=1 , mi ).

i=1

The likelihood contributions of the choice variables are computed by transforming
the characterized expression of the shocks, using the Jacobian-transformation. In
particular, the time-specific likelihood contribution can be expressed as

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

39



q dx
c
x
s
) it
∂(e
ηit
, ηeit, ηeit, ηeitqcxsc (η̃ ) · fη x (η̃ ) · fη s (η̃ ) · f q (η̃ )|det
Lit (θ|Sit , λk ) = fηit|
ηit it
it
it
it
it
it
∂(pit , xit, sit , qit )Z (1−dsit )
q dx
c
x
) it
∂(η̃it, η̃it, η̃itqssxcc (η̃ ) · fη x (η̃ ) · fη s (η ) · f q (η̃ ) dη | det
fηit×|ηit it
it
it
it
it
it
it
∂(pit, xit , qit )
s
η̃it

 Z∞

Z



×
x (η c )−∞ η̃ititsq (1−dxsit )dit∂(η̃it, η̃it
)
q
s
x
c
x c
c
c (η ) · f l (η |η ) · fη s (η̃ ) · f q (η̃ ) dη dη | det
fηit|ηit itηit it it
it
it
it
it
it
∂(sit , qit ) Z∞ Z∞ Z∞ 
×

c
x
s
s
c
x
s
c
x
c
x
s
Pr{V00 (ηit
, ηit
, ηit
) > Vx0 (ηit), V00 (ηit
, ηit
, ηit
) > V0s (ηit, ηit), V00 (ηit
, ηit
, ηit
) > Vxs )}−∞ −∞ −∞(1−dxit )(1−dsit )q∂ η̃itqcxs|
× dηitdηitdηitf (η̃it)| det∂qi,t+1

where V00 is the value when xit = sit = 0, Vx0 is the value when xit > 0 and sit = 0, andV0s is the value when xit = 0 and sit > 0.45
To evaluate the integrals in the likelihood function, I use the Montecarlo simulation. Borsch-Supan, Hajivassiliou and Kotlikoff (1992) show that the MSL estimates
perform well under a moderate number of draws, such as 20, with an adoption of agood simulation method. To reduce the variance of simulation error, I use antithetic
acceleration (Geweke 1988; Stern 1997).
About 8.3% of the household-year observations are missing, creating “holes” in thehousehold data. I simulate the unobserved choice variables using the value functionof the model (Lavy, Palumbo and Stern 1998; Stinebrickner 1999; Sullivan 2009). In
particular, for each draw of the set of errors, I replace the unobserved choice variables
with the optimized choices that maximize the value function of the model. Also, forperiods 4 and 5, the test score data are unobserved. I simulate the unobserved testscores for each draw of test score error ηitq using equation (4). In Appendix D, I show thederivation of the density and probability I use for computing the likelihood function,and I explain the simulation of unobserved variables.

6.2

Identification

Parameters of the model can be classified into the productivity parameters associatedwith the test score function and the taste parameters that directly affect value function.
The productivity parameters in the test score production function are identified by45

For the case dxit = dsit = 0, I am working on a G.H.K type of simulation to reduce the variance of
simulation error.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

40

the covariation between the subsequent test score qi,t+1 and the inputs (qit , pit , xit , andsit ). As data of the inputs are available for each period, I can separately identify the
productivity parameters for each time t.
The taste parameters αc , αx , αs , and the altruism parameter αv affect the valuefunction, and do not directly affect the test score function. These parameters are theconstants for the likelihood contribution of the corresponding choice variables. I do
not differentiate the taste parameters for each period. The element of the covariancematrix of the shocks are identified in maximizing the log-likelihood contribution of
the associated shocks.
The exogenous variables in the model are the academic performance in primaryschool qi1 , the parental education mi , and the complete income stream of parents{wit }Tt=1 . The identifying assumption is the time varying shocks are orthogonal to theinitial conditions. In particular,{ηitc , ηitx , ηits , ηitq }Tt=1 ⊥T
qi1 , mi , {wit }t=1 .

7.

Estimation Results

7.1

Test score function parameters

Table 8 presents the estimates of test score production function specified in equation(4). The interpretation of the previous test score parameter is the same as the log-log
case of a linear regression equation. For example, for t = 2, a 1% increase in theprevious test score leads to a 0.98% increase in the subsequent test score controllingfor other inputs. The marginal effects of the effort parameters νt largely decline overtime. Especially in the final period, the effect plummets to 0.02. This estimate suggests
that, in the final period, the marginal effects of both the parental investment and hours
of self-study are significantly lower and it is more difficult to increase a test score withthe same amount of monetary or time investments.
Hours of self-study have stronger average marginal effects on the subsequent
test score than hours of tutoring. The marginal effects are computed by the partial
derivative of test score qi,t+1 with respect to either hours of tutoring (xit ) or hours ofself-study (sit ). I present the average marginal effects, which could vary by households
because the calculation of marginal effects involves the data of pit , xit , and sit . Figure

41

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

9 presents the comparison of the average marginal effects of hours of self-study andhours of tutoring. In almost all periods, hours of self-study have greater marginal
effects than hours of tutoring. Only in the final period, the average marginal effectof hours of tutoring is slightly larger than the average marginal effects of hours ofself-study. However, as seen in Figure 9, the difference of the marginal effects in the
final period is neither statistically nor economically significant.
Table 8: Parameter Estimates: Test score production function
Time-varying Parameters

t=1

t=2

t=3

t=4

t=5

t=6

Previous
Test Score (δqt )0.163(0.001)0.980(0.002)0.754(0.001)

0.731(0.001)0.474(0.009)0.426(0.001)

Effort Parameters (νt )0.556(0.001)0.659(0.014)0.420(0.009)0.360(0.001)0.150(0.002)0.020(0.001)

Share of tutoring
Expenditure (δet )0.472(0.007)0.485(0.005)0.480(0.011)

0.526(0.011)

0.592(0.012)0.737(0.015)Constants (δ0t )4.030(0.006)-1.124(0.001)0.627(0.001)1.205(0.001)

1.022(0.001)

4.077(0.004)

Time Invariant Parameters
Substitution Parameter (ϕ)

0.880(0.001)

Intensity of Private tutoring
Quality (κ)

0.145(0.002)

Note: Standard errors are in parentheses below estimates. Based on the CES test score function, shareof hours of self-study is implied by share of tutoring expenditure. (i.e., δst = 1 − δet ).

The early period investments have delayed effects through evolving test scores.
As seen in Figure 9, the marginal effect of hours of self-study are already stronger inthe earlier periods. Considering the delayed effects of hours of self-study through the
evolving test scores, hours of self-study have a strong effect on the final test score.
The estimate of the substitution parameter shows that the impact of parental
investments could be exaggerated in simulating the model if the child’s self-study is
not incorporated into the mechanism. The hours of self-study and hours of tutoring
are nearly perfect substitutes for each other based on the structural estimate. Table 8
includes the estimate of the substitution parameter ϕ, which is approximately 0.88.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

42

Suppose a researcher wants to conduct a counterfactual experiment of restricting
parental investment using a structural model without other options of investing in
the child. Each household must accept the restriction as given. Such an omission of
mechanism might result in an exaggeration of the effects of parental investment on
outcomes such as intergenerational persistence of earnings. This substitutability plays
an important role because the channel of hours of self-study provides a householdwith a restrictive income constraint an opportunity to exert efforts in the tournament.
Figure 9: Average Marginal Effects of Hours Allocation

Note: This figure presents the average of marginal effects of hours of self-study and hours of tutoringover time. Due to the functional form of the test score function, the marginal effects differ by eachindividual. The marginal effects are computed using the first order derivative with respect to hours ofself-study (sit ) or hours of tutoring (xit ) and the estimated parameters. The vertical interval at eachpoint indicates the standard deviation of the marginal effects.

7.2

Preference and shock parameters

Table 9 presents the estimates of the preference parameters and the shock parameters.
The preference parameters are components of equations (5), (9), and (10). For the
preference parameters, the estimates are relative estimates of the other preference
parameters. The altruism parameter is estimated as 1.018. To capture the observed
heterogeneity of the household, I allow the preference parameters to vary by parental
education. In particular, exp(τx Dipedu ) is multiplied to the disutility from hours of

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

43

tutoring αx and exp(τs Dipedu ) is multiplied to the disutility from hours of self-study,
where Dipedu is 1 for household whose average years of parental education is strictly
greater than 12. Table 9 (a) includes the estimates of the effects of parental educationon the preference parameters. Based on the estimates, parental education alleviates
the disutility to hours of self-study. Specifically, a child of a household whose average
education of parents is greater than 12 years feels more disutility of study by 0.005.
In contrast, the effect of parental education on mitigating disutility from hours oftutoring is not statisticially different from 0.
Table 9: Parameter Estimates: Preference and Shock Parameters
Estimate

Standard error

αc
αν
αx
αs

0.028
1.018
-0.006-0.005(0.000)(0.001)(0.000)(0.006)τxτs-0.001-0.005(0.002)(0.002)Preference Parameters
Taste for consumption
Altruism for the child’s future
Disutility from hours of tutoring
Disutility from hours of self-study

Parental education parameters
disutility from hours of tutoring
disutility from hours of self-study(a) Preference parameters
Standard Deviation of

Estimate

Standard Error

Test score shock

σηq

0.230

(0.000)Consumption shock

σηc

0.742

(0.014)

Study shock

σηs

0.549

(0.001)

Tutoring shock

σηl

0.472

(0.005)(b) Shock parametersP
1
Note: Standard errors are in parentheses. N ×T
logLi − Jacob) = −0.848.
×6 (The estimated standard deviations of unobserved shocks are overall modest, whichsuggests that the observed characteristics and the structural model capture a considerable proportion of heterogeneity in the data. The unobserved heterogeneity from

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

44

consumption is considerably different among the different types of households. This
could be due to the fact that I do not model the supply side of private tutoring, andthe price differences across regions are not captured by the deterministic parts of themodel.

7.3

Model Fit

To examine the goodness-of-fit of the structural model, I use a local linear regression
estimator to see how well the model prediction yb fits the actual data value y, fordependent variables y = p, x, s, q. Specifically, the expected value of data y conditionalon the model predicted value yb is E(y|by ) = κb0 (by ), whereκb0 (y)κb1 (y)

!
=

X
i

1
yi − ybi)K(byi − ybi!

!  −1  y − yb
1
i
i
yi ,
· K(
)
1 yi − ybibyi − ybi

and1
K(x) = √ exp(−0.5x2 )2π
is the kernel function with bandwidth b. The farther the kernel curve deviates fromthe 45 degree line, the less the model is successful in fitting the data.
Figures E.1 to E.5 present the sample fit of quality of tutoring, hours of tutoring,
hours of self-study, and test scores, respectively. Overall, the fits are very good. Whilethe quality of tutoring and the hours of self-study show excellent fits, hours of tutoring
are somewhat a little bit overpredicted. The level of the final test score, as depicted
in Figure E.5b, is somewhat overpredicted as well. This is due to the fact that theconstants of the test scores are part of the tournament model as specified in the tournament component of equation (10). Thus, the constant of the test scores cannot
be separated from the constants of all the other dependent variables’ likelihood contributions. Nevertheless, the model fits the distribution of the test scores very well,
as shown in Figure E.5a. As the tournament model is about the ranking of the final
test score, the distributions of the test scores are the major concern in simulating themodel, which is captured by the tournament model.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

45

8.

Counterfactual Analyses

8.1

Quantification and Decomposition of Intergenerational
Persistence of Earnings

The purpose of the quantification is to decompose the role of channels affecting
intergenerational persistence of earnings. Using the structural estimates, I simulate
the model under the counterfactual environments that help quantify the role of therelevant channels. Each simulation produces a different distribution of the final testscores, which leads to a different distribution of the predicted income of the child. I
define the predicted income of the child of household i as childinci . Two measuresof the intergenerational persistence of earnings are intergenerational elasticity ofearnings (IGE) and the rank-rank slope. In particular, the measure of IGE is thecoefficient of the regression equation,ln childinci =δ00 + δIGE ln hhinci + εi(12)where hhinci is the average of household i’s income over the periods. The estimateof IGE is δbIGE , and (1-δbIGE ) is the measure of intergenerational mobility (Black andDevereux 2010). On the other hand, the rank-rank slope (Chetty et al. 2014) is theestimate of the regression equation,

Ri =δ01 + δRR P i + υi

(13)where Ri is the percentile rank of the child income within the generation, and Pi is
the rank of the parental income within the generation. Although I present both IGE
and the rank-rank slope for each counterfactual simulation, the preferred estimate ofintergenerational persistence of earnings is the rank-rank slope. The IGE is sensitive
to the ratio of the income inequalities of the two generations.46 To minimize this issue,
the discussion is based on the results of estimates of the rank-rank slope, which ismore robust to the difference in the income variance across the generations.
Table 10 provides definitions of the counterfactual simulations. In particular,
In particular, δIGE = ρch σσlnlnchildinc, where σx is a standard deviation of data x and ρch is a correlation
hhinc
between ln childinci and ln hhinci .
46

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

46• BCF is the baseline of the counterfactuals where the model is simulated withouta counterfactual modification• OPI is the counterfactual where only parental investment is the means of thetournament model, and hours of self-study are excluded from the choice of thehousehold and fixed to 0.
• OSS is the counterfactual where only child’s self-study is the means of the tournament, and parental investment is excluded from the choice and fixed to 0.
• NIN is the counterfactual where all monthly net household income is fixed to
4,000,000 KRW (approximately 2800 USD).
Table 10: Definitions of Counterfactual Simulations

Benchmark Counterfactual (BCF)
Parental Investments (OPI)
Self-Study (OSS)
Without 6th grade Test (NST)
Without Household Inc (NIN)

Household Inc

6th grade Score

Parental Educ

Parental Investments

Child’s Self-Study

O
O
O
O
X

O
O
O
X
O

O
O
O
O
O

O
O
X
O
O

O
X
O
O
O

Note: For each simulation definition in the first column, the channel is either allowed (O) or shut down
(X). For example, in OPI, all channels of intergenerational transmission are allowed except for hours ofself-study.

Table 11 presents the estimates of the rank-rank slope and the intergenerationalelasticity of earnings under five different simulations. The estimated rank-rank slope
for the benchmark counterfactual (BCF) is 0.635.47 The OSS simulation is interestingon its own because it provides implications for the tutoring ban policy of China.
47

The estimated IGE with the benchmark model (BCF) is 0.269.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

47

Table 11: Intergenerational Persistence of Earnings under the Counterfactual Simulations(a) Rank-rank Slope Estimates

(1)
BCF

(2)
OPI

(3)
OSS

(4)
NIN

pincprctile

0.635***(0.018)0.827***(0.013)0.130***(0.023)0.341***(0.022)R-squared

0.403

0.684

0.017

0.116

(b) Intergenerational Elasticity of Earnings Estimates

(1)
BCF

(2)
OPI

(3)
OSS

(4)
NIN

log(hhinc)

0.269***(0.008)0.313***(0.005)0.068***(0.009)0.173***(0.011)R-squared

0.398

0.708

0.031

0.119

Note: Table (a) and (b) provide the estimates of equation (13) and (12), respectively. BCF is a benchmark
counterfactual. NIN is a simulation where household income is fixed to the mean. OPI is a simulation
where household can only use parental investment. OSS is a simulation where household can only use
hours of self-study.

The quantification exercise highlights several findings. First, removing heterogeneity in parental income decreases the rank-rank slope by 46.2%, which can be foundin the result of the NIN simulation in Column (5) in Table 11a. The result suggeststhat heterogeneity in parental income is responsible for a substantial part of the intergenerational persistence of earnings. Second, omitting hours of self-study leads tosignificant increases in the intergenerational persistence of earnings. The estimatedrank-rank slope under the OPI simulation is 0.827, which is greater than the benchmark simulation by 30.2%, as shown in Column (2) in Table 11a. At the same time,
when the channel of parental investment is removed, the rank-rank slope decreases by
79.5%, as shown in Column (3) in Table 11a. Such results suggest that while parental
investment reinforces the intergeneration persistence of earnings, the self-study of thechild mitigates it. Third, the effect of heterogeneity in the academic performance inprimary school on the intergenerational persistence of earnings is modest. To control
for the difference among students before 7th grade, I run the counterfactual simulations with fixing the academic performance in primary school and parental education,

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

48

which can be found in Table 12. For example, BCF’ is the same simulation as BCF
except that 6th-grade academic performance and parental education are fixed acrosshouseholds. The results are consistent with the original counterfactual simulations
that are conducted without fixing the household characteristics.
Table 12: Intergenerational Persistence of Earnings Fixing Initial Conditions(a) Rank-rank Slope Estimates

(1)
BCF’

(2)
OPI

(3)
OSS’

pincprctile

0.641***(0.018)0.834***(0.013)0.323***(0.022)R-squared

0.411

0.695

0.104

(b) Intergenerational Elasticity of Earnings Estimates

(1)
BCF’

(2)
OPI’

(3)
OSS’

logpinc

0.393***(0.011)0.312***(0.005)0.100***(0.007)R-squared

0.401

0.719

0.097

Note: Table (a) and (b) provide the estimates of equation (13) and (12), respectively. In order to assess
the importance of initial conditions, all simulations are conducted fixing the initial test score andparental education. Parental income is not fixed except for the NIN’ simualtion. BCF’ is the benchmark
counterfactual; NIN’ is a simulation where each household income is fixed to the mean; OPI’ is asimulation where each household can use only parental investment; and OSS’ is a simulation where
each household can use only hours of self-study. The linearity of the lines follows from the linearity of
the rank-rank equation.

Based on the simulation results of Section 8.1, which can be found in Table 11a, theprivate tutoring ban policy would decrease intergenerational persistence of earnings.
In other words, such a policy would increase intergenerational mobility. However,
the increase in mobility would come with the expense of an increase in consumption
inequality.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

49

Figure 10: Intergenerational Persistence of Earnings by Scenarios

Note: This graph is a visual representation of the rank-rank slope estimates described in equation (13).
Parent Income Rank is the average income over six years. Child Income Rank is computed based on thesimulated results of each scenario. BCF is the benchmark counterfactual; NIN is a simulation where
each household income is fixed to the mean; OPI is a simulation where each household can use only
parental investment; and OSS is a simulation where each household can use only hours of self-study.
The linearity of the lines follows from the linearity of the rank-rank equation.

8.2

Parental Investment Demand in Shrinking Cohorts

Exploiting the number of competitors in the tournament model, I simulate the demand of parental investment amidst a sharp cohort size reduction. Countries with ahigh amount of average parental monetary investments tend to experience extremely
low fertility rates. South Korea, China, Turkey, Singapore, and Taiwan are countries
where the demand for private tutoring is high (Bray 1999, 2021), and they are experiencing a drastic reduction in the size of the cohort as can be seen in Figure 1. The
reduction in the cohort size is equivalent to the reduction in the number of competitors in the college admission tournament. If there is a radical reduction in the numberof competitors, the degree of competition would be less fiercer given that the numberof seats for colleges do not change.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

50

Figure 11: Number of seats in colleges

Source: (1) Number of students by year, (2) number of graduates by year, and (3) number of entancequota in college all by Korean Educational Development InstituteNote: The number of high school graduates after 2022 is projected by the number of lower graders
assuming that drop out rates do not change.

Colleges have not decreased the number of seats amid the demographic shift.
Figure 11 presents the number of high school graduates up to 2033, the number ofseats of colleges in Korea, and colleges in Seoul up to 2022. The number of highschool graduates after 2022 is projected using the average dropout rates and thenumber of graduates from younger cohorts in 2022. The number of projected highschool graduates in 2033 is 289,216, which is only 44.5% of the number of high school
graduates in 2011. Meanwhile, colleges do not adjust the number of seats. Cohort sizehas been decreasing since around 2000, but colleges have not changed the number ofseats, as can be seen in Figure 11. Colleges in Seoul, which is the equivalent of Tier 1 toTier 3 colleges of the earlier classification, do not decrease the number of seats either.
The simulation is conducted under various scenarios. The three key exogenous
factors that characterize each simulation are: (i) the number of cohort sizes, (ii) theexit of existing colleges, and (iii) changes in the premium of upper college tiers. The
first factor can be plausibly projected using fertility rates and dropout rates. As it isempirically challenging to pin down the second and third factors, I evaluate the impactof each factor under the most plausible scenarios.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

51

Simulation I: Cohort Size Impact on Parental Investment Motivated by the cohort
reduction in South Korea, I simulate changes in the amount of parental investment
when the size of the cohort decreases by 50%. The effects of the cohort changes
are reflected through the increased seats of college tiers. As there are half of thecompetitors relative to the unchanged number of college seats, it is equivalent to
the number of seats for each tier doubling. For this simulation, I assume that theinequality of the college qualities remain same.
According to the simulation results, the amount of private tutoring expenditureslightly increases assuming the disparities in college quality do not change. Figures 12
presents the density of private tutoring expenditures of low income households (lowest
5%) and high income households (highest 5%) respectively. High income households
increase their total spending by 0.05% while low income households increase theirspending by 18.04%. The increase of parental investment might be driven by theassumption that college inequality does not change over time. As a result of cohort
reduction, students have a higher probability of going to a better college tier with lessfiercer competition. In turn, people who previously had no chance of getting into
an upper college tiers would have better chance. For low-income households, fewer
competitor leads to an increase in the probability of going to higher tier college. Low
income households have a higher marginal values from investing in the child, whichmakes them spend more.
Simulation II: Low-Tier Colleges Exit (TBU)
Simulation III: Reduced College Inequality To evaluate the effects of college inequality on parental investment, I change the values of {vj }Jj=1 , the earning prospectsof alumni for each tier. The relative size of v1 ,v2 ,v3 and v4 relative to v5 are 2.22, 1.95,
1.45, 1.25 respectively. In this simulation, I change them to 1.5, 1.4, 1.3, and 1.2 respectively. I set the percentiles that students need to make for each tier as the samewith the benchmark simulation. Figure 13 presents the resulting changes on parental
investment for the households. The decrease in college inequality leads to a substantial decrease in private tutoring expenditure of the entire households. Table 13
presents the quantification of the changes in private tutoring expenditure. I report
the changes amount of total private tutoring expenditure compared to the benchmark
simulation where I normalize private tutoring expenditure as 100. It can be seenthat the changes in college hierarchy results in a significant decrease in total private
tutoring expenditure.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

52

Figure 12: Cohort Simulation(a) Low Income Households(b) High Income Households

Note: Figure 13 displays simulated private tutoring expenditure for low income households (lowest 5%)
for each counterfactual scenario. Cohort Reduction 1 simulation is based on the scenario where thecohort size decrease by 50%.

Table 13: Changes in Simulated Private Tutoring Expenditure

BenchmarkEntire HH
100
Low Income HH 100
High Income HH 100

Cohort Reduction 1

Cohort Reduction 2

College Hierarchy

101.57114.38101.07114.06137.44102.77

76.13
68.63
83.98

Note: Table 12 presents the average simulated private tutoring expenditure for each counterfactual
scenario. To emphasize the change, expenditure of the benchmark case is standardized to 100. Cohort
Reduction 1 simulation is based on the scenario that the cohort size decreases by 50%. Cohort Reduction
2 simulation is based on the scenario that the cohort size decrease by 50% and the number of seats for
elite college increases. College Hierarchy simulation is based on the scenario that the relative size ofv1 ,v2 ,v3 and v4 relative to v5 are decreased to 1.5, 1.4, 1.3, and 1.2 respectively. The number of seats are
the same as the benchmark counterfactual in the College Hierarchy simulation.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

53

Figure 13: College Inequality Simulation

Note: Figure presents the average simulated private tutoring expenditure for each counterfactual
scenario. To emphasize the change, expenditure of the benchmark case is standardized to 100. Cohort
Reduction 1 simulation is based on the scenario that the cohort size decreases by 50%. Cohort Reduction
2 simulation is based on the scenario that the cohort size decrease by 50% and the number of seats for
elite college increases. College Hierarchy simulation is based on the scenario that the relative size ofv1 ,v2 ,v3 and v4 relative to v5 are decreased to 1.5, 1.4, 1.3, and 1.2 respectively. The number of seats are
the same as the benchmark counterfactual in the College Hierarchy simulation.

9.

Conclusion

College admission is a pivotal event in terms of lifetime income. A college one graduates from significantly affects lifetime income controlling for the results of the college
entrance exam. Each household competes for the limited seats in prestigious colleges
with excellent lifetime income prospects. Such competition for prestigious colleges isa channel where the income of parents translates into the future disparity in a child’s
income. Parental investment and a child’s self-efforts are two important means of thehousehold in this college admission competition.
The first goal of this paper is to quantify the role of parental investment in intergenerational persistence of earnings. While parental investment is highly responsive
to parental income, self-efforts of the child is not as responsive to parental income as

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

54

parental investment. Such an empirical relationship suggests the potential importanceof incorporating self-efforts of the child into the mechanism. I develop a dynamic
tournament model of college admission in which each household uses both private
tutoring expenditure and hours of self-study by the child. I estimate the structuralmodel using Maximum Simulated Likelihood. Using the estimated model, I quantify
the role of private tutoring expenditure, hours of self-study, and other household
characteristics. I find that heterogeneity in parental income in adolescent periods
accounts for 46% of intergenerational persistence of earnings. Parental investment
is responsible for a substantial portion of intergenerational persistence of earnings
controlling for the child’s efforts. Ignoring child’s efforts from the mechanism leads
to a significant increase in intergenerational persistence of earnings by 30%, whichsuggests the role of the self-effort as a mitigator of intergenerational persistence of
earnings.
Finally, in light of the recent fertility declines in developed countries, I assess theimpact of the rapidly shrinking cohort size on parental investment. Based on themodel projection, low income households spend money on private tutoring expenditure as cohort size decreases, while there is virtually no change in the parental
investment spending of high income households. Additionally, I evaluate the impact
of college inequality on parental investment. A counterfactual simulation suggeststhat a decrease in inequality in college income prospects leads to a significant decreasein parental investment.
The findings of this paper suggest two avenues for future research. First, this paper
does not allow the possibility of wealth transmission within the household. As Becker
and Tomes (1979) suggest, the transmission of capital can be an alternative way ofinheriting the income of the parents, especially when the child does not perform
well academically. Incorporating the channel of capital transmission within a family
requires at least decent data on wealth for more than one generation, which is not aneasy data requirement. Second, this paper does not allow for unobserved heterogeneity of labor income conditional on college quality, mainly due to the data limitations.
An efficiency analysis on the rat-race nature of the college admission competition
would be feasible with the addition of the channel. I leave this for future research.

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55

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63

Appendix A
Appendix A.1
Proof of Lemma 2: Compactness
Proof. A value function is the sum of flow utility and the discounted future value. The
flow utility term u(cit , xit , sit , εit ) is monotone in its arguments. Also, u is defined at thelower and upper bounds of cit , xit , sit . Thus, u(cit , xit , sit , εit ) is closed and bounded.
The expected future value EVt+1 is closed and bounded. For the final period, thetournament term described in equation (10) is closed and bounded because (i) the vj
term is finite and greater than 0, and (ii) P rob(ln Q̃j−1 ≥ ln qi,T +1 ≥ ln Q̃j ΓiT ) ∈ [0, 1].
Therefore, the choice-specific value function of the final period, Vit (Zit , Ψit ) is closed
and bounded for t = T . Following the backward recursion, Vit (Zit , Ψit ) is closed andbounded.

Appendix A.2
Proof of Lemma 3: Continuity
Proof. I start by showing that the value function Vit is continuous. To show Vit isR
continuous, It suffices to show that both u(cit , xit , sit , εit ) and η Vt+1 (Zt+1 , Ψt+1 )f (η)dηare continuous.
• Start from the final period and show that the final term is continous: Boundedright hand side. Left hand side is continous in its arguemnt. Use DominatedConvergence Theorem.
• Previous period same
• Then move on to the continuity of the mapping
Proof. One way to show the continuity of the expected value function is show that it issequentially continuous. For any sequence of the arguments of the value function,{Ztn , Ψtn } → {Zt0 , Ψt0 },we haveDYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

Znn
Vt+1 (Zt+1, Ψt+1
)dΨ →

Ψ

Z

64

0
0
Vt+1 (Zt+1, Ψt+1
)dΨ.

Ψ

c
x
s
Recall that Ψt+1 = {ηi,t+1, ηi,t+1, ηi,t+1, ηitq }. As the expectation of the unobserved shocks
R00, Ψt+1)dΨ
has finite expectation, the expected value term has finite expectation as well. Ψ Vt+1 (Zt+1

is continuous by the Dominated Convergence. Each supmapping is continuous asits elements are continuous. As each submapping is continuous. By induction, thecomposition of mapping is continuous. Therefore, ℵ is continuous.

Appendix B
List of the member collegesFirst Tier

Seoul National, Yonsei, Korea , Sogang, SKKU, Hanyang, KAIST, Pusan, Ewha, Postech

Second Tier

Choongang, Kyunghee, HUFS, University of Seoul, KU, Dongguk,
Kyongpook, Sookmyung, Ajou, Honggik, Inha, Hangkong, Kookmin,
Soongsil, Sejong, Dankook, Kwangwoon, Cheonnam, Seoul Industrial University

Third Tier

Myongji, Sangmyeong, Catholic, Choongam, Choongbook, Seongshin, Kyeongki
Kyongwon, Deoksong women, Dongdeok women, Dong-A, Bookyeong

Fourth Tier

The rest of the 2 year colleges

Fifth TierHigh school graduates

Appendix C
Define the first order conditions as∂∂ ln qi,t+1
EVi,t+1 (·, ·, ·)∂ ln qi,t+1
∂pit


∂
∂ ln qi,t+1
Vx =αc εcit ucx (cit ) + αx εxit uxx (xit ) + β
EVi,t+1 (·, ·, ·)∂ ln qi,t+1
∂xit


∂
∂
ln
q
i,t+1
Vs =αs εsit uss (sit ) + β
EVi,t+1 (·, ·, ·)∂ ln qi,t+1∂sit
Vp = αc εcit ucp (cit ) + β

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

65

∂V

p
∂p
∂w
= − ∂V
p
∂w

∂p

∂Vp
xit
=αc εcit∂w(wit − pit xit )2




∂ c
−xit
∂
∂
δ2t (1 + pit κ xit 1−κ )ϕ−1
∂Vp
κ−1
1−κ
= εit
+β
EVi,t+1 (·, ·, ·) νt
(κpit
xit
)
∂p
∂p (wit − pit xit )∂p ∂ ln qi,t+1[δ2t (1 + pit κ xit 1−κ )ϕ + δ3t (1 + sit )ϕ ]


2
∂2
δ2t (1 + pit κ xit 1−κ )ϕ−1
x3it
κ−1
1−κ
+
β
EV
(·,
·,
·)
ν
(κp
x
)
= − 2αc εcit
i,t+1
t
it
it
(wit − pit xit )3∂ 2 ln qi,t+1[δ2t (1 + pit κ xit 1−κ )ϕ + δ3t (1 + sit )ϕ ](2κ−2)(2−2κ)∂
δ 2 κ2 pit
ϕxit(1 + pit κ xit 1−κ )2ϕ−2
EVi,t+1 (·, ·, ·) νt − 2t(κpit κ−1 xit 1−κ )+β∂ ln qi,t+1[δ2t (1 + pit κ xit 1−κ )ϕ + δ3t (1 + sit )ϕ ]2(2κ−2)+

δ2t κ2 pit(ϕ − 1)x(2−2κ) (1 + pit κ xit 1−κ )ϕ−2
[δ2t (1 + pit κ xit 1−κ )ϕ + δ3t (1 + sit )ϕ ]

δ2t (1 + pit κ xit 1−κ )ϕ−1
κ−2 1−κ+((κ − 1)κpit
xit )[δ2t (1 + pit κ xit 1−κ )ϕ + δ3t (1 + sit )ϕ ]
2

∂V

!

∂V

∂p
> 0.
As ϕ < 1, κ < 0.5, and ∂ 2 ln∂qi,t+1 EVi,t+1 (·, ·, ·) < 0, ∂pp < 0 and ∂wp > 0, ∂w

Appendix DFirst-order conditions used for likelihood contribution
The goal of this section is to get a closed form expression of the shocks, which are thebuilding blocks of the likelihood function. I denote ucp (cit ) and ucx (cit ) as the first orderderivatives of uc (cit ) with respect to xit and pit respectively, and ulx (lit ) and uls (lit ) as thefirst order derivatives with respect to xit and sit respectively. The first order conditionsof the value function in equation (9) are∂ ln qi,t+1∂
1∂c: αc exp(ηit+ λck ) + β c
EVi,t+1 (Zi,t+1 (ln qi,t+1 (pit , xit , sit ), Ψi,t+1 )
= 0;
∂pit
up (cit ) ∂ ln qi,t+1∂pit

∂ ln qi,t+1∂∂kc:αc exp(ηit+ λck )ucx (cit ) + αx exp(ηit+ λck )ux
EVi,t+1 (Zi,t+1 (ln qi,t+1 (pit , xit , sit ), Ψi,t+1 )= 0;
x (xit ) + β∂xit∂ ln qi,t+1∂xit

∂ ln qi,t+1∂∂s: exp(ηit+ λsk )uss (sit ) + β
EVi,t+1 (Zi,t+1 (ln qi,t+1 (pit , xit , sit ), Ψi,t+1 )
= 0.
∂sit
∂ ln qi,t+1∂sitWith the functional form assumptions of log utility,

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

66

pit
;
wit − pit xit
xit
ucp (cit ) = −;wit − pit xit
1;
uss (sit ) =1 + sit1uxx (xit ) =1 + xitucx (cit ) = −And with the functional form of the test score function,

qi,t+1 = Ait qitδ1t δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕln qi,t+1 = ln Ait + δ1t ln qit + νϕtexp(λqk + ηitq )νln[δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕ ] + λqk + ηitqϕ∂ ln qi,t+1
δet (1 + pit κ xit 1−κ )ϕ−1
(κpit κ−1 xit 1−κ );=νt∂pit[δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕ ]∂ ln qi,t+1
δet (1 + pit κ xit 1−κ )ϕ−1
=νt((1 − κ)pit κ xit −κ );∂xit[δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕ ]∂ ln qi,t+1δst (1 + sit )ϕ−1
.
=νt
∂sit
[δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕ ]The first order conditions with respect to pit is characterized as

wit − pit xit∂cαc exp(ηit+ λck ) − β
EVi,t+1 (Zi,t+1 (ln qi,t+1 (pit , xit , sit )), Ψi,t+1 )xit∂ ln qi,t+1
δet (1 + pit κ xit 1−κ )ϕ−1× (κpit κ−1 xit 1−κ ) = 0.
× νt[δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕ ](14)

The first order conditions with respect to xit is characterized asc− αc exp(ηit+ λck )

× νt



pit
1
∂
x
x
+ αx exp(ηit + λk )+βEVi,t+1 (ln qi,t+1 (pit , xit , sit ), Ψi,t+1 )wit − pit xit
1 + xit∂ ln qi,t+1

δet (1 + pit κ xit 1−κ )ϕ−1× (1 − κ)pit κ xit −κ = 0.
[δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕ ]The first order conditions with respect to sit is characterized as(15)

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTSs
αs exp(ηit+ λsk )

× νt

67



∂
1
+β
EVi,t+1 (Zi,t+1 (ln qi,t+1 (pit , xit , sit )), Ψi,t+1 )1 + sit∂ ln qi,t+1δst (1 + sit )ϕ−1
= 0.
[δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕ ](16)

This difference between the previous period and the final period can be confusing.
For the final period,J Xln q̄j − ln qiT +1 − λqi
EVi,T +1 =v1 −ln(vj ) − ln(vj+1 ) Φ(
);
σq
j=1

J 
X
∂
1 ln q̄j − ln qiT +1 − λqi
EVi,T +1 (·, ·, ·) =ϕ(),
ln(vj ) − ln(vj+1 )∂ ln qi,T +1
σ
σ
q
q
j=1
while for t < T , EVit is an interpolated value function.

Computation of Likelihood Contribution(Case 1) (xit > 0 and sit > 0)
I define η̃itz for z = c, x, s as the particular realization of ηitz that satisfies the first orderconditions. The likelihood contribution for all-positive case is∂(eηitc , ηeitx , ηeits , ηeitq )|∂(pit , xit, sit , qit )∂(eηitc , ηeitx , ηeits , ηeitq )qcxs=ϕ(η̃it , η̃it , η̃it , η̃it ) · |det|∂(pit , xit, sit , qit )
 ′
 
ηeitc
ηeitc
 x
 



ηeit  −1 ηeitx ∂(eηitc , ηeitx , ηeits , ηeitq )
−4/2
−1/2




=(2π)
|det(Ω)|
exp − 0.5 s  Ω  s ] det∂(pit , xit, sit , qit )ηeit  4×4 ηeit 
ηeitq
ηeitq


f (pit , xit , sit , qit ) =f (η̃itc , η̃itx , η̃its , η̃itq ) · |det

1×4

4×1

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

68(Case 2) (xit > 0 and sit = 0)This is the case where household participate in tutoring, but have zero hours of selfstudy. First, I define the joint probability of such case, and separate the density of ηitcand ηitx out using Bayes’ theorem. I denote Axit ,sit =0 as the corresponding region thatthe joint integration of ηitc , ηitx , and ηits needs to be made.

Pr(pit , xit , sit = 0)= Pr(sit = 0|pit , xit )f (pit , xit )= Pr(ηits > η sit |η̃itc , η̃itx )fη (η̃itx , η̃itc )| det∂(η̃itc , η̃itx )|,∂(pit, xit )where η sit is the minimum value of ηits that leads to zero hours of self-study. I use thefirst order condition with respect to sit , equation (16), in computing the critical value.
With the zero correlation assumption between eta,∂(η̃itc , η̃itx )|∂(pit, xit )∂(η̃itc , η̃itx )= Pr(ηits > η sit )f (η̃itx )f (η̃itc )| det|∂(pit, xit )1 η̃itx 1 η̃itc∂(η̃itc , η̃itx )s= 1 − Φ(η it )ϕ( ) ϕ( )| det|,
σx σx σc σx∂(pit, xit )Pr(ηits > η sit |η̃itc , η̃itx )fη (η̃itx , η̃itc )| det

which is what I use for computing the likelihood contribution for (Case 2).

(Case 3) (xit = 0 and sit > 0)This is the case where household do not participate in tutoring, but do positive hours
of self-study. Since pit > 0 for all households, pit xit = 0 is equivalent to xit = 0. For
people who have xit = 0, I let them consider minimum quality of tutoring, p̄, which isequivalent the minimum market price.
Denote Axit =0 as the corresponding region that the joint integration of ηitc and ηitx
needs to be made. First, I separate out the marginal density of ηits using Bayes’ theorem,
which gives me

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

69

Pr(xit = 0, sit ) = Pr(xit = 0|sit )f (sit )= Pr(ηitc , ηitx ∈ Axit =0,sit |η̃its ) ·1 η̃its ∂ η̃its
ϕ( )||.
σs σs ∂(sit )As I assume there is no correlation between ηit ,1 η̃its ∂ η̃its
ϕ( )||σs σs ∂(sit )1 η̃ s ∂ η̃its= Pr(ηitc , ηitx ∈ Axit =0,sit ) · ϕ( it )||.
σs σs ∂(sit )Pr(ηitc , ηitx ∈ Axit =0,sit |η̃its ) ·Here, I use the first order condition, equation (16), in characterizing the unique valuesof η̃its . Define η xit as a minimum amount of shock that makes individual start doing zero
hours of tutoring. Again, with the assumption of no correlation between ηit ,1 η̃its ∂ η̃its|
ϕ( )|σs σs ∂(sit )1 η̃ s ∂ η̃its= Pr(ηitc , ηitx > η sit |η̃its )dηitx dηitc ϕ( it )||σs σs ∂(sit ) Z∞  Z∞
1 ηitx1 ηitc1 η̃its ∂ η̃itsxc=
ϕ( )dηitϕ( )dηit
ϕ( )|
|
σx σx
σc σc
σs σs ∂(sit )Pr(ηitc , ηitx ∈ Axit =0,sit ) ·−∞(η c )
ηx
it it Z∞ 
η xit (ηitc )1 ηitc1 η̃its ∂ η̃itsc|=1 − Φ()
ϕ( )dηit
ϕ( )|
σx
σc σc
σs σs ∂(sit )−∞(Case 4) (xit = 0 and sit = 0)This is the case where xit = 0 and sit = 0. To make the notation concise, I denote V00 as
the value when xit = sit = 0. Vx0 denotes the case x > 0 and s = 0. V0s denotes the casex = 0 and s > 0.

Pr(xit = 0, sit = 0) = Pr(ηitc , ηitx , ηits ∈ Axit =0,sit =0 )= Pr(V00 (ηitc , ηitx , ηits ) > Vx0 (ηits ), V00 (ηitc , ηitx , ηits ) > V0s (ηitc , ηitx ), V00 (ηitc , ηitx , ηits ) > Vxs ).

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

70

c
x
s
Pr(xit = 0, sit = 0) = Pr(ηit
, ηit
, ηit
∈ Axit =0,sit =0 )
 Z∞ Z∞ Z∞
c
x
s
s
c
x
s
c
x
c
x
s
1{V00 (ηit
, ηit
, ηit
) > Vx0 (ηit), V00 (ηit
, ηit
, ηit
) > V0s (ηit, ηit), V00 (ηit
, ηit
, ηit
) > Vxs )}=−∞ −∞ −∞
s
x
c
s
x
c
=f (ηit)f (ηit)f (ηit
)dηit
dηit
dηit



The integral does not have an analytical solution and needs to be simulated.
Simulation algorithm is(1) I draw an unconditional set of ηitr = {ηitcr , ηitxr , ηitsr }(2) Let household optimize their choices.
(3) Count the proportion of cases that household chooses xit = 0 and sit = 0
In particular define xr and sr such that

(xr , sr ) = arg max Vit (ηitc , ηitx , ηits )xit ,sit

ComputeR

1X
1(x∗, s∗ = 0).
R r=1
So

R

1X
Pr(xit = 0, sit = 0) ≈1(x∗, s∗ = 0).
R r=1

Simulation of unobserved variables
For each missing choice variables, I draw a set of corresponding error. For example, if
xit is missing for person i, the simulation algorithm is(1) I draw a simulation for the corresponding error. In this example, it is ηitxr(2) Let household optimize their choice

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

R

1X
x =
R r=1
r



71


.

arg max Vit (ηitc , ηitxr , ηits )xit ,sit

The optimized choice is used for computing likelihood function.
For missing test score, I draw a set of errors for ηitq . Then the unobserved test scoreis simulated using equation (4).

Appendix E
Figure E.1: Sample Fit: Quality of Tutoring

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

Figure E.2: Hours of Tutoring

Figure E.3: Sample Fit: Private Tutoring Expenditure72

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

Figure E.4: Sample Fit: Hours of self-study

73

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

Figure E.5: Sample Fit: Log Test Scores(a) Fit by distribution(b) Fit by level74