Dynamic Competition in Parental Investment and Child’s Efforts *Hyunjae Kang
Kyoto University, JapanFirst Version: November 2022
Current Version: September 2024

Abstract
Competition for a limited number of seats in prestigious colleges generates
a “rat-race” equilibrium effect, leading to increased household investment. I
construct and estimate a dynamic tournament model in which each householdchooses the quality and hours of private tutoring as well as hours of student
self-study. The model rationalizes the high amount of parental investment insecondary school despite its low effects on academic achievement. I use theestimated model to quantify the importance of household choices on the intergenerational persistence of earnings. Removing child efforts amplifies this persistenceby 36%, emphasizing the role of self-effort in moderating the intergenerational
link. By leveraging the number of seats in the tournament, I find that increasing
the number of seats in elite colleges by 50% leads to a decrease in the average
private tutoring expenditure by 14%. Declining cohort size leads to a modest
decrease in private tutoring expenditure unless there is a significant decrease in
the elite college premium.
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 their invaluable guidance and support. I would also like to thank Meta Brown,
David Wiczer, Gabriel Mihalache, Peter Arcidiacono, Michael Keane, Lance Lochner, Olivier De Groote,Nirav Mehta, John Rust, Todd Stinebrickner, Michela Tincani, Michèle Tertilt, Minchul Yum, Sunham
Kim, Changhui Kang, Alexis Anagnostopoulos, Arda Aktas, Minji Bang, 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, Shinhyuk Kang,
Laura Karpuska, Jiyeon Kim, Alfonso Flores Lagunes, Marc Claveria, Bob Millard, Mark Montgomery,José Alfonso Muñoz, 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 parental investments. Graduating froman elite university has a sizeable impact on labor market outcomes (Hoekstra 2009;
MacLeod et al. 2017; Zimmerman 2019; Anelli 2020; Sekhri 2020; Guo and Leung 2021;
Jia and Li 2021; Lee and Koh 2023), but seats for such prestigious colleges are limited.
The scarcity of seats in prestigious colleges leads to competition, potentially creatinga “rat-race” scenario that drives an increasing trend in parental investment (Rameyand Ramey 2010). In the United States, parents frequently invest considerable time insupporting their children’s extracurricular activities. In East-Asian countries, parents
often allocate a significant portion of their income to private tutoring, primarily aimed
at securing admission to prestigious universities, (Bray 1999, 2022), despite evidence
suggesting minimal impact on test score improvement (Ryu and Kang 2013; Kangand Park 2021). Most previous work on parental investment does not include thiscompetition 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 have received more investment from parents are likely to achieve better future outcomes suchas better performance in the labor market (Becker and Tomes 1979; Guryan, Hurst and
Kearney 2008). Thus, parental investment potentially has important consequences onsocial mobility. Meanwhile, previous studies report the significant impact of children’s
own efforts on their educational outcomes (Stinebrickner and Stinebrickner 2004;
Del Boca, Monfardini and Nicoletti 2017; Fu and Mehta 2018; Todd and Wolpin 2018;
De Groote 2023; Del Boca, Flinn, Verriest and Wiswall 2023). 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.
This paper investigates these two interrelated aspects of parental investment. First,
the intensity of household competition is affected by the limited number of college
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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

2seats with different level of quality and the number of competitors. If there are significant changes in the number of competitors, the decision of parental investment
and child effort is likely to be affected. Relatedly, many developed countries face adrastic shift in demographic structure caused by a declining fertility rate, as shownin Figure 1. At the same time, empirical evidence suggests that colleges tend to not
adjust the seats to accommodate for increasing cohort size (Bound and Turner 2007).
Little is known about the impact of the number of seats in elite colleges and the shift indemographic structure on parental investment. Second, using the college admission
competition set-up, this paper seeks to shed light on the role of parental investment
in intergenerational persistence or earnings. The inclusion of self-effort of the child,
which is often overlooked in the literature, might amplify or offset the link between
the two generations.
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 investigate the impacts of relaxing elite college constraints onparental investments.
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

3

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; Lee and Koh 2023).5 Second,
the amount of parental investment drops substantially as students finish the college
admission process. This suggests that the purpose of parental investment is for theirchild to do well in the college admission competition rather than enhancing the child’s
human capital.
Also, another empirical fact suggests that parental investment and the child’s
self-efforts potentially have different implications for intergenerational mobility. As
5Using 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

4

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.
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, based on its state variable, each household makes
decisions regarding parental investment and the level of the child’s self-efforts, which
produces a subsequent test score. The model structure repeats until the final test scoreis produced, and students are assigned to the college tiers based on the ranking of the
final test score and the number of seats for each college tier. The college tier is the soledeterminant of the child’s lifetime income.6
Two main features of the dynamic tournament model help answer the researchquestion of this paper. First, the rich heterogeneity of state variables and the choiceset, along with the specification of the test score production function, help disentangle
the source of intergenerational persistence of earnings. Second, the rank-order featureof the tournament model enables me to study the effects of the changes in the numberof seats in colleges and the disparity in college quality on 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 the subsequent test score than hours of tutoring, while the marginal effects of both investments
decline over time. Additionally, the estimate of the substitution parameter of the pro6

This is an arguably reasonable assumption. In Section 5.2, using confidential job offers data
provided by a conglomerate in South Korea, I show that the effects of college-tier on earnings are
economically and statistically significant controlling for effort during the college period.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

5

duction function suggests that parental investments and hours of self-study are close
to perfect substitutes. Unlike disutility from hours of tutoring, parental education has
significant effects in reducing disutility from hours of self-study. Using local linear
regression, I show that the estimated model fits the sample reasonably well.
Using the estimated structural model, I quantify the impact of household heterogeneity and choices by alternatively shutting down each channel one at a time.
Removing heterogeneity in the parental income during the adolescent period decreases the rank-rank slope by 47.2%. When the channel of self-study is shut downThe
rank-rank slope increases by 30.2%. The result of the quantification suggests thatparental investment reinforces the intergenerational persistence of earnings and theself-study of the child mitigates it.
Next, I use the model to investigate the consequences of relaxing the seat constraint
in elite colleges on the choices of households. First, I simulate the model under thealternative scenario where the number of seats in elite colleges expands.7 A 50%
increase in the number of seats of elite colleges leads to a 14% decrease in the average
private tutoring expenditure. Second, I simulate the model under the scenario where
the size of the cohort is reduced by half, which is based on the projected number of
12th-grade students in Korea in 2033. Shrinking cohort size does not lead to a decreasein private tutoring expenditure unless there is a significant decrease in the elite college
premium. As the cohort-to-seat ratio decreases, students have an increased chance ofgoing to a better college tier, which increases their incentives for spending 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 relates to the large body of literature on post-birth parental choice (Becker
and Tomes 1979; Del Boca, Flinn and Wiswall 2014; Doepke and Zilibotti 2017; Bolt,
7

The elite colleges refer to a group of colleges which I define as Tier 1 in Section 3.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

6

French, Maccuish and O’Dea 2021b). Papers in this literature have recently started
incorporating competition into parental choices.8 Ramey and Ramey (2010) are thefirst paper that rationalizes the increase of parental time investment in the UnitedStates using a theoretical model of competition for elite colleges. Bodoh-Creed andHickman (2019) build a static structural model of an admission contest to study
returns to pre-college human capital investment in the United States and estimate
their model.9,10 A closely related paper on the same context is by Kim, Tertilt andYum (2024), which studies the cause of the low fertility problem of South Korea. They
propose a model of “status externality” based on the assumption that parents careabout the relative position of their children’s human capital compared to that of otherchildren. The tournament model of my paper complements their study by formallymodeling the dynamic competition with respect to getting into prestigious colleges,
which rationalizes underlying source of the status externalities in their paper.11 Also,
the number of seats and the equilibrium cutoffs of the tournament model enable me
to investigate the impact of the college seats’ constraint on the demand for parental
investments.
This paper naturally relates to the theoretical and empirical literature on parental
investment and its intergenerational implications (Lee and Seshadri 2019; Caucutt andLochner 2020; Bolt, French, Maccuish and O’Dea 2021b; Daruich 2022; Gayle, Golan
and Soytas 2022; Yum 2022). In particular, Del Boca, Flinn and Wiswall (2014) buildand estimate a dynamic model of parental investment and cognitive development,
which allows them to separately identify the different effects of parental time andmonetary investments. Subsequently, emphasizing the role of child’s self-investment,
Del Boca, Flinn, Verriest and Wiswall (2023) build a Stackelberg model of parentchild interaction and investigate the effects of conditional cash transfers on childoutcomes. De Groote (2023) quantifies the role of students’ efforts in the academic
tracking system using a dynamic model. My paper contributes to the literature by
jointly modeling the dynamic decisions of parental investment and the self-efforts
8

A group of papers associates parental choices with social interactions (Agostinelli 2018; Agostinelli,
Doepke, Sorrenti and Zilibotti 2023; Boucher, Bello, Panebianco, Verdier and Zenou 2022)
9
While abstracting from the notion of parental investment, Grau (2018) builds and estimates a static
tournament model to study the college competition in Chile.
10
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).
11
Gu and Zhang (2024) is another recent paper modeling the college admission competition using
heterogeneous agents framework. However, they do not model the thresholds for the college admission
as equilibrium objects of the students’ competition.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

7of the child. The estimated model quantifies the impact of parental investment andthe child’s self-efforts on the intergenerational persistence of earnings, providing
novel insights about the mechanism that generates intergenerational correlations inearnings.
Finally, this paper contributes to the literature on childhood investments and skill
development by estimating the age-specific effects of parental investment and the selfefforts of the child during adolescence.12 Most previous work focuses on estimating
the effects of parental investment on child outcomes alone (for example, Cunha andHeckman 2007; Cunha, Heckman and Schennach 2010; Del Boca, Flinn and Wiswall2014).13 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; Fu and
Mehta 2018; Todd and Wolpin 2018). The production function estimates in my paper
add to this literature by providing age-specific estimates of the effects of parental
investment and self-efforts (Del Boca, Monfardini and Nicoletti 2017; Del Boca, Flinn,
Verriest and Wiswall 2023), and offer novel evidence on their substitutability.

3.

Key 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.
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)).
13
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).

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

3.1

8High-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.14 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.15,16 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. Baccalauréat of France ishighly important for getting into grandes écoles, the group of elite colleges of thecountry.17 The Scholastic Aptitude Test (SAT) of the United States is also utilized as
an important factor in college admission, but other components such as 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
14

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).
15
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.
16
A stanine score is nine discrete scales ranging from a low of 1 to a high of 9.
17
Other examples include Yükseköğretim Kurumları Sınavı of Turkey, Exame Nacional do EnsinoMédio of Brazil, Sijil Pelajaran Malaysia of Malayasia, and Ulttyq Biryńǵaı Testileý of Kazakhstan arehighly similar in terms of their importance in the college admission process.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

9significantly affects a student’s future labor market outcomes.18 In South Korea, thecollege hierarchy has changed little (Kim and Lee 2006; Kim 2014). Starting fromthe top institution, Seoul National University, the applicants’ preferences have beenstable for decades, and “SKY” is a well-known acronym that refers to the top three
universities in the country.19,20
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.21
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
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.22 Second, as a result of theconsecutive school-equalization policies, the quality of education provided by schools
18

See, 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.
19
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.
20
Kim and Lee (2006) study this hierarchical market structure of universities in Korea and show thata strong university hierarchy is present in the country. They report that universities in the first threedeciles strictly dominate the rest in terms of their measure of labor market outcomes, private donations,
quality of faculties, and physical facilities.
21
The top 15% score is the cutoff for the “In Seoul” universities previously mentioned.
22
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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

10

is similar.23 No schools are allowed to select students independently.24 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.25
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.26
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.
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: Korean Educational Longitudinal Study 2005

I use the Korean Educational Longitudinal Study 2005 (KELS) for the main estimation
procedure. The choice of data is motivated by the main goals of the paper: (i) toquantify the role of accumulated parental investment and student efforts on intergen23

See Section II of Kim and Lee (2010) for a description of the history of school equalization policy.
As of 2010, the high school equalization policy has been adopted for all major cities in South Korea.
24
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 meritbased. 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.
25
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.
26
See Bray (1999, 2021) for a comprehensive cross-country comparison of private tutoring.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

11

erational mobility, and (ii) to account for the dynamic selection of the effort choicesof the household in the competition. Estimating the marginal effects of parental investment and hours of self-study in each period is necessary to achieve the goals. The
Korean Educational Longitudinal Study 2005 includes information on private tutoring expenditure, hours spent for private tutoring, hours spent for self-study, incomeof the household, standardized test scores, and parental education, which is a rarecombination for one dataset. Household income and private tutoring expenditureare collected each year. The hours spent in tutoring activities and the hours spentfor self-study are collected as a weekly average. There are five different measures ofacademic performance available in the dataset. Academic performance in primaryschool is measured as an ordered discrete 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 as continuous variables.
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 thecollege and the labor market period, the cohort is surveyed semi-annually up to 2020,
which is ten years after the cohort graduates from high school. The rules of selection
and their effects are reported in Table C.1.27 In addition, I include households missing
one of the choice variables: tutoring expenditure, hours of tutoring, and hours ofself-study. In the estimation section, I explain the rules to simulate the missing choicevariables.
27

The proportion of observations lost to missing the final test score is 0.48. Meanwhile, 99.9% of thestudents 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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

12

Table 1: Sample Moments(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

9th

Mean
25.1
5.97
9.69
369.2
321.50

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
-

12th

Mean
47.9
8.45
9.16
394.4
-

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.
Note: The unit for measuring private tutoring expenditure is 10,000 KRW, approximately 8 USD, based on the average exchange
rate during the data collection periods. Hours data are weekly measures.

Table 2: Sample Moments (Continued)(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.

Table 1 and 2 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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

13

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.28
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. The
description of KLIPS is in Appendix C.1.

4.2

The Lifetime Income Differential
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 areborn 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 3.

College ranking has a strong effect on the growth of alumni’s income.29 The effect issignificant even after controlling for CSAT score. Columns (1), (2), and (3) in Table 3
provide the OLS estimates for the regression equations,28

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.
29
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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

14

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.30
Table 3: Log Income Regression(1)
Pooled OLS(2)Pooled OLS(3)Pooled OLS

(4)
RE

(5)
RE

(6)
RE

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

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.

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
30

A full set of estimates are available in Appendix C. 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

15

where λyi and ηity are the individual-specific and the idiosyncratic errors respectively.31
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 3. 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
score, as can be seen in Columns (3) and (5) of Table 3.32 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
31

Since the focus of the regression is the college tier, which is time-invariant, I do not consider the
fixed effects model.
32
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

16

the college entrance exam show that competition is the key feature determining theparental investment decision of the household.
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

17

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.

Parental education soaks up significant variation in hours of self-study, whichleaves a relatively small variation with parental income. Tables 4 and 5 presents thepooled OLS estimates of the regression equation,

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS18ln(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 4: 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

19

Table 5: 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
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.

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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

20

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.33 In addition, I discuss how parental investments
allocation differs based on their initial conditions in Appendix C.3.
The evidence suggests that the allocation of the two efforts changes over time andthat households self-select into the different effort levels based on their preconditions.
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).

5.1

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 7 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 qi1 . Also, each household has a specific type k. Different types
of households have different type-specific characteristics that are unobserved by the
econometrician. I define them as λck , λxk , λsk and λqk , which affect marginal utility from33

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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

21

consumption, 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 .34

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
qi1 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 7: 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 notaccounted 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)thqcxs
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.
34

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

22

college 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.35 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 0.95.36 I define ŷjt as the estimated tier-specific annual income at time t,which is predicted using Pooled-OLS estimates of Column (1) in Table 3.37 As tier 1 isdefined to be the top college tier, v decreases in j (i.e., v1 > v2 > ... > vJ−1 > vJ ).38
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
35

The implicit assumption regarding the bottom tier is that everyone graduates high school. The
high school drop-out rate in South Korea is less than 2%.
36
The average interest rate is around 5% for South Korea in 2010.
37
I assume no earnings in the college periods.
38
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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

23

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.39
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 (2023) as supporting evidence forAssumption 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 an
final interview request from the company. Table 6 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, graduatingfrom one of the tier 1 colleges increases probability of getting an offer by 23% relative
to graduating from a college below tier 3.
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.40 The total amount of tutoring expenditure eit iseit = pit xit .
39

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.
40
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

24

Table 6: Final Interview Request Regression (Probit)(1)
Tier=1
Tier=2
Tier=3
ColGPAN
Major
Company
ClusterSE

0.327***(0.038)0.029(0.044)-0.123(0.096)0.006**(0.002)
9138
No
Yes
No

(2)

(3)

0.737*** 0.760***(0.103)(0.113)0.140*** 0.161***(0.030)(0.035)-0.018-0.036(0.127)(0.109)0.021*** 0.022***(0.007)(0.006)
9132
Yes
No
Yes

9132
Yes
Yes
Yes

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.

The tutoring expenditure is constrained under two dimensions. A household cannot
spend more tutoring expenditure than its income (i.e., eit ≤ wit ).41 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.
41

I assume no borrowing.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

25

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.42 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, thetransformed tutoring input is specified asẽit =pκit x1−κit(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 in
42

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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

26

generating 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



ϕ

ϕ

δet (1 + ẽit ) + δst (1 + sit )

 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
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 threeparts: (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 ofself-study 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)

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

27

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, q), respectively. The
shocks are decomposed as     ln εcit
ηitc
λck
ηitc
 x   x   x x
ln εit  ηit  λk ηit η     
ln εs  = η s  + λs  , and η s  ∼ N (0, Ω ) it   it   k 
 it 
qqq
ln εitηitλk
ηitq
where Ωη is the covariance matrix for the time-varying shocks.43 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
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
43

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

28

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 g̃i,j−1
ln g̃ij−1ln(vj ) ∗ Fq (ΓiT ) − Fq (
ΓiT )
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, Kang and Lee 2016; Grau
2018; Tincani, Kosse and Miglino 2021).44
Budget and Time Constraints: The choices of the household are restricted by thebudget and the time constraints. The budget constraint is given by

cit + pit xit ≤ wit(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 63.45
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 specificq
One can also adopt a functional form that ηitfollows Generalized Extreme Value distribution which
results in a Tullock (2001) contest.
45
I assume each student can use 9 hours everyday for non-leisure activities other than hours spent
in regular school
44

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

29

heterogeneity. 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 .46 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 7, 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-studyηits 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 the46

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

30

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 term appears, the valuefunction 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 offlow 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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

31

Ka (ℵa : K → Ka ), based on the value functions specified in equations (9) and(10). 47
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 .
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.
47

The mapping ℵa involves backward recursion.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

6.

32

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.

6.1

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|qi1 , {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 qi1 , 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)

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

33

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|qi1 , {wit }Tt=1 , mi ).

i=1

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 by
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 7 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

34

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

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.
Table 7: 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 )0.161(0.001)

0.972(0.002)0.741(0.001)0.701(0.001)

0.445(0.009)0.415(0.001)

Effort Parameters (νt )0.575(0.001)0.710(0.014)0.515(0.009)0.350(0.001)0.152(0.002)0.017(0.001)

Share of tutoring
Expenditure (δet )

0.481(0.007)0.485(0.005)0.477(0.011)

0.575(0.011)

0.555(0.012)0.716(0.015)Constants (δ0t )4.031(0.006)-1.125(0.001)0.540(0.001)

0.936(0.001)

0.863(0.001)

4.484(0.004)

Time Invariant Parameters
Substitution Parameter (ϕ)

0.869(0.001)

Intensity of Private tutoring
Quality (κ)

0.154(0.002)

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

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
8 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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

35

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 8, the difference of the marginal effects in the
final period is neither statistically nor economically significant.
The early period investments have delayed effects through evolving test scores.
As seen in Figure 8, 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.
Figure 8: 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.

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 7
includes the estimate of the substitution parameter ϕ, which is approximately 0.88.
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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

36

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.

7.2

Preference and shock parameters
Table 8: Parameter Estimates: Preference and Shock Parameters
Estimate

Standard error

αc
αν
αx
αs

0.028
1.040
-0.006-0.005(0.000)(0.001)(0.000)(0.006)τxτs-1e-05-0.001(2e-05)(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

0.248

(0.000)Consumption shock

σηc

0.745

(0.014)

Study disutility shock

σηs

0.534

(0.001)

Tutoring disutility shock

σηx

0.521

(0.005)(b) Shock parameters
Note: Standard
errors are computed using delta method and are in parentheses below estimates.
P
1
(
logL
−
Jacob) = −0.848.
i
N ×T ×6

Table 8 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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

37

education. In particular, exp(τx Dipedu ) is multiplied to the disutility from hours of 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 greaterthan 12. Table 8 (a) includes the estimates of the effects of parental education on
the preference parameters. Based on the estimates, parental education alleviates thedisutility 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.
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 fromconsumption 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 = e, p, x, s, q. 48 Figures F.1 to F.5 present the sample fit oftutoring expenditure, hours of tutoring, quality of tutoring, hours of self-study, andtest scores, respectively. Overall, the fits are very good. While the hours of tutoring
and the hours of self-study show excellent fits, the quality of tutoring is somewhat
overpredicted. Also, while the level of the final test score fits reasonably well, the levelof the earlier test scores are also somewhat underpredicted. This is due to the factthat the constants 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 scorescannot 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 F.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.
48

Appendix G includes the specification of local linear regression used to judge the fit.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

38

8.

Counterfactual Analyses

8.1

Decomposition of Intergenerational Persistence of Earnings
Figure 9: Intergenerational Persistence of Earnings by Scenarios

Note: This graph presents local linear curves that fit child income rank and parental income rank under
different counterfactual scenarios. Parent Income Rank is the average income over six years. Child
Income Rank is computed based on the simulated results of each scenario. 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 purpose of the quantification exercise 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
impact of the relevant channels. As the cutoffs are unknown in this counterfactual case,
I simulate the household behavior until their guess of the cutoffs becomes identicalwith the resulting cutoffs.49 Each simulation produces a different distribution of the49

I use this algorithm again in the college constraint counterfactual analyses described in thesubsequent subsection.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

39

final test scores, which leads to a different distribution of the predicted income of thechild. I define the predicted income of the child of household i as childinci .
The rank-rank slope (Chetty et al. 2014) is the estimate of the regression equation,
Ri =δ01 + δRR P i + υi(12)where Ri is the percentile rank of the child income within the generation, and Pi is therank of the parental income within the generation. 50
Each counterfactual scenario is defined in the following way.
• BCF is the status quo where the model is simulated without a 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 0.52
• NIN is the counterfactual where all monthly net household income is fixed tothe average household income, which is about 4,000,000 KRW (approximately
3000 USD).
Table 9 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 0.572.53
The quantification exercise highlights several findings. First, removing heterogeneity in parental income decreases the rank-rank slope by 43.5%, which can befound in the result of the NIN simulation in Column (5) in Table 9a. The result suggests that heterogeneity in parental income is responsible for a substantial part ofthe intergenerational persistence of earnings. Second, omitting hours of self-study
50

Although I present both IGE and the rank-rank slope for each counterfactual simulation, thepreferred estimate of intergenerational persistence of earnings is the rank-rank slope. The IGE is
sensitive to the ratio of the income inequalities of the two generations.51 To minimize this issue, thediscussion is based on the results of estimates of the rank-rank slope, which is more robust to thedifference in the income variance across the generations.
52
The OSS simulation is equivalent to China’s tutoring ban policy in that it completely prohibits
private tutoring activities. Gu and Zhang (2024) evaluate the tutoring ban policy using their macroeconomic model.
53
The estimated IGE with the benchmark model (BCF) is 0.231, which aligns with the intergenerational elasticity of earnings estimates in the literature (Ueda 2013).

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

40

leads to significant increases in the intergenerational persistence of earnings. The
estimated rank-rank slope under the OPI simulation is 0.782, which is greater than thebenchmark simulation by 36.7%, as shown in Column (2) in Table 9a. At the same time,
when the channel of parental investment is removed, the rank-rank slope decreases
by 66.4%, as shown in Column (3) in Table 9a. Such results suggest that while parental
investment reinforces the intergenerational persistence of earnings, the self-studyof the child mitigates it. This finding is consistent with the substitution parameter
estimate in Table 7, which suggests that self-study serves as a substitute for parental
investment.54
Table 9: Intergenerational Persistence of Earnings under the Counterfactual Simulations(a) Rank-Rank Slope Estimates

(1)
BCF

(2)
OPI

(3)
OSS

(4)
NST

(5)
NIN

(6)
NED

pincprctile

0.572***(0.019)0.782***(0.015)0.192***(0.023)0.572***(0.019)0.323***(0.022)0.573***(0.019)R-squared

0.327

0.611

0.037

0.327

0.105

0.328

(b) Intergenerational Elasticity of Earnings Estimates

(1)
BCF

(2)
OPI

(3)
OSS

(4)
NST

(5)
NIN

(6)
NED

log(hhinc)

0.231***(0.008)0.254***(0.005)0.074***(0.008)0.231***(0.008)0.166***(0.011)0.233***(0.008)R-squared

0.302

0.624

0.051

0.303

0.110

0.305

Note: Table (a) and (b) provide the estimates of rank-rank slope and intergenerational elasticty ofearnings, respectively. BCF is a benchmark counterfactual. NIN is a simulation where householdincome 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.

8.2

Relaxing College Constraints

Leveraging the number of seats and the tier-specific returns in the tournament model,
I simulate the demand for parental investment under two counterfactual scenarios: (i)54

Appendix G provides results fixing other initial conditions.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

41

a 50% increase in the number of seats in Tier 1 colleges, and (ii) a 50% reduction incohort size.
Table 10: Description of College Constraint Simulation# Seats
Status Quo
Simulation I
Simulation II

n1
1.5n1
2n1

Tier 1Prize# Seatsv1
v1 (1 − R)
v1 (1 − R)

n2
n2
2n2

Tier 2Prize

v2
O1
v2 − 2n2v2 (1 − R)

# Seatsn3
n3
2n3

Tier 3Prize# Seatsv3
O1
v3 − 4n3
v3 (1 − R)

n4
n4
2n4

Tier 4Prize

v4O1
v4 − 4n4v4 (1 − R)

Tier 5# Seats
Prize

n5
n5
n5

v5
v5
v5 − On52

Note: For each simulation, the overflow of lifetime income is defined as O1 = v1 (1 − r1 )(n1 + n′1 ) + v1 n1 and O2 = 2(1 −PPR) 4j=1 nj vj − 4j=1 nj vj

An increase in the number of seats across college tiers can either raise or lower theoverall demand for private tutoring. Expanding the number of seats makes private
tutoring more valuable for students who previously had a lower probability of enteringa higher tier, as their chances of admission increase. On the other hand, if the increase
in seats sufficiently boosts a student’s chances of entering into the upper tier, thevalue of private tutoring may decline. The average expenditure on private tutoring
will decrease if the second effect outweighs the first, and increase if the first effect isstronger.
To account for the zero-sum game nature of the college admission competition, it isnecessary to make assumptions about (i) the trade-offs between an additional numberof tier-specific seats and the marginal decrease in tier-specific returns, and (ii) how todeal with the increased total lifetime income relative to the status quo. I define Rj as
the loss of return for Tier j. Specifically, if there is an n′j increase for the seat numbersof Tier j, the sum of future lifetime income of Tier j alumni is (nj + n′j )(vj (1 − Rj )). The“overflow” of the aggregated lifetime income caused by changes in all tiers, is defined
as

O=

J
X

vj (1 − Rj )(nj + n′j ) −

j=1

J
X

vj nj .

j=1

Two counterfactual scenarios are characterized by different combinations of n′j , Rj ,and O, which are described in Table 10.
Simulation I: Relaxation of the Elite College Constraint In this simulation, I
increase the number of seats in Tier 1 by 50% fixing the number of seats of other tiers(n′1 = 1.5 and nj = 0 for all other js). Only the Tier 1 burdens the cost of increasing

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

42

seats, thus the tier-specific lifetime income is decreased to v1 (1 − R1 ). Then, theoverflow of Simulation I is defined as O1 = v1 (1 − r1 )(n1 + n′1 ) − v1 n1 . Half of theoverflow of the future lifetime income is subtracted from the future lifetime incomeof Tier 2, and the rest is equally burdened by Tier 3 and 4. That is, v2 is changed to
O1
O1
O1
v2 − 2n, and v3 and v4 are changed to v3 − 4nand v4 − 4n
, respectively. Tier 5 stays
2
3
4

the same.
Figure 10: Simulation I: Elite College Constraints(a) Simulation with R1 = 0.9(b) Simulation with changing R1

Note: Graph (a) depicts the distribution of the simulated monthly private tutoring expenditure when
the assumed return R1 is 0.9, compared to the status quo. Graph (b) shows the distribution of thesimulated monthly private tutoring expenditure for R1 = 1, 0.9 and the status quo.

Table 11 presents the results of the college constraint simulation. An increase in
the number of Tier 1 seats by 50% leads to a decrease in the average private tutoring
expenditure by 15.5% to 16.3% when Rj values are assumed to be 1 and 0.9, respectively.
Figure 10a shows the distribution of the average monthly private tutoring expenditure
for the expansion scenario and the status quo. Compared to the status quo, thedistribution shifts to the left. These results suggest that expanding the number ofseats in the elite college overall leads to a decrease in the average private tutoring
expenditure.
Simulation II: Cohort Size Reduction Motivated by the cohort reduction in South
Korea, I simulate changes in the amount of parental investment when the size of thecohort decreases by half. The effects of the cohort changes are reflected through theincreased seats of college tiers. As there are half of the competitors relative to the

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

43

unchanged number of college seats, it is equivalent to that the number of seats foreach tier doubles.
In this simulation, I increase the number of seats in Tier 1 to 4 by 100% (n′j = nj andn5 = 0). Tier 1 to 4 burdens the cost of increasing seats, thus the tier-specific lifetime
income is decreased to vj (1 − R) for j = 1, 2, 3, 4. Then, the overflow of Simulation IIP
P
is defined as O2 = 2(1 − R) 4j=1 nj vj − 4j=1 nj vj . The overflow of the future lifetime
income is subtracted from the future lifetime income of Tier 5. That is, v5 is changed
to v5 − On52 .
Figure 11: Simulation II: Shrinking Cohort(a) Simulation with R = 0.9(b) Simulation with changing R

Note: Graph (a) depicts the distribution of the simulated monthly private tutoring expenditure when
the assumed return R1 is 0.75, compared to the status quo. Graph (b) shows the distribution of thesimulated monthly private tutoring expenditure for R1 = 0.9, 0.8, 0.75 and the status quo.

A decrease in cohort size does not necessarily lead to a reduction in average private tutoring expenditure unless there is a significant decrease in the returns from
graduating from a better-tier college. Figure 11b presents the average private tutoring
expenditure when the seat-to-cohort ratio doubles for R = 0.90, 0.80, 0.75. When there
is only a 5% decrease in R, there is a substantial increase in average private tutoring
expenditure. Figure 11a presents the simulation results for R = 0.90. Compared to thestatus quo, total private tutoring decreases only by 4%. Private tutoring expendituresignificantly decreases only when there are significant decrease in alumni for all tiers.
When there are 25% decrease in all college tiers, private tutoring significantly drops.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

44

Table 11: Changes in choice variables under the college-constraint simulation

Status quo
Simulation I

Simulation II

Tutoring Expenditure

Hours of Self-study

100

100

Increasing Elite College Seats
R=1
R=0.9

84.5
83.7

85.1
99.5

Shrinking Cohort Size
R=0.9
R=0.8
R=0.75

96.0
86.4
39.1

141.1
90.3
61.6

Note: Both tutoring expenditure and hours of self-study are the aggregated values through all time
periods. I standardize the value by setting the status quo values as 100.

9.

Conclusion

I develop and estimate a dynamic tournament model of college admissions in which
each household utilizes both private tutoring expenditures and the child’s hours ofself-study. [State the key mechanism of the model]
Using the estimated model, I quantify the roles of private tutoring expenditures,
hours of self-study, and other household characteristics. I find that heterogeneity inparental income during adolescence accounts for 46
I also use the model to evaluate the impact of relaxing college admission constraints. Expanding the number of seats in elite colleges leads to a 15% decreasein average private tutoring expenditures. The estimated model shows that a sharp
decline in cohort size does not necessarily reduce the demand for private tutoring,unless accompanied by a significant decline in the returns to graduating from anupper-tier college.
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 notan easy data requirement. Second, this paper does not allow for unobserved hetero-

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

45

geneity of labor income conditional on college quality, mainly due to the limitationsof micro-data. 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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

46

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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

53

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

has finite expectation, the expected value term has finite expectation as well.
R00
V (Zt+1, Ψt+1)dΨ is continuous by the Dominated Convergence. Each supmappingΨ t+1
is continuous as its elements are continuous. As each submapping is continuous. By
induction, the composition of mapping is continuous. Therefore, ℵ is continuous.

Appendix B
Colleges for each college tier
List of the member collegesFirst Tier
Second Tier

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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

54

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 65.55
Selection Rules and Effects
Table C.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

Additional Descriptive Evidence: Dynamic Allocation of Effort choices
The initial conditions of the household persistently affect the parental investment
decisions throughout the secondary school periods. Figure A.1 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 A.1 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 A.1 (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 the55

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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

55

two groups 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. Figure
A.1 (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 A.1: 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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

56

Liftime Income Estimates
(1)

(2)

(3)

(4)

(5)

(6)

Pooled OLS

Pooled OLS

Pooled OLS

RE

RE

RE

-1.931***-1.684***-2.958***-1.671***-1.491***-2.194***(0.347)(0.325)(0.494)(0.279)(0.288)(0.723)-1.332***-1.195***-2.460-1.409***-1.364***-1.908**(0.350)(0.296)(1.683)(0.321)(0.323)(0.820)-0.864**-0.817***-1.549**-0.958***-1.075***0.190(0.269)(0.196)(0.507)(0.363)(0.364)(1.249)-0.895***-0.618**-1.954***-0.727***-0.524***-0.425(0.232)(0.186)(0.361)(0.079)(0.123)(0.436)

0.092***

0.092***

0.167***

0.095***

0.095***

0.168***(0.001)(0.001)(0.018)(0.002)(0.002)(0.040)0.067***

0.065***

0.111***

0.058***

0.058***

0.105***(0.009)(0.009)(0.014)(0.009)(0.009)(0.024)0.050***

0.052***

0.098

0.051***

0.055***

0.095***(0.011)(0.010)(0.054)(0.010)(0.011)(0.031)0.032***

0.037***

0.058**

0.034***

0.044***

0.011(0.008)(0.007)(0.019)(0.012)(0.013)(0.050)0.029***

0.027***

0.073***

0.022***

0.021***

0.038***(0.007)(0.007)(0.015)(0.002)(0.002)(0.013)

N

29599

29599

685

29599

29599

752

Major

No

Yes

No

No

Yes

Yes

RE

No

No

No

Yes

Yes

Yes

CSAT

No

No

Yes

No

No

No

College Tier
Top tier

Second Tier

Third Tier

Fourth Tier

age

Interactions
Top tier × age

Second Tier × age

Third Tier × age

Fourth Tier × age

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

57

Appendix D
Define the first order conditions as∂∂ ln qi,t+1
EVi,t+1 (·, ·, ·)∂ ln qi,t+1
∂pit


∂
∂ ln qi,t+1c cx x
Vx =αc εit ux (cit ) + αx εit ux (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



∂V

p
∂p
∂w
= − ∂V
p
∂w

∂p

∂Vp
xit
=αc εcit∂w(wit − pit xit )2




∂
∂Vp
∂
−xit
∂
δ2t (1 + pit κ xit 1−κ )ϕ−1
κ−11−κ= εcit+βEVi,t+1 (·, ·, ·) νt
(κp
x
)
it
it
∂p
∂p (wit − pit xit )∂p ∂ ln qi,t+1[δ2t (1 + pit κ xit 1−κ )ϕ + δ3t (1 + sit )ϕ ]

2

∂2
x3it
δ2t (1 + pit κ xit 1−κ )ϕ−1
c
κ−1
1−κ
= − 2αc εit
+β 2
(κpit
xit
)
EVi,t+1 (·, ·, ·) νt(wit − pit xit )3∂ 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+β(κpit κ−1 xit 1−κ )EVi,t+1 (·, ·, ·) νt − 2t
∂ 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
As ϕ < 1, κ < 0.5, and ∂ 2 ln∂qi,t+1 EVi,t+1 (·, ·, ·) < 0, ∂pp < 0 and ∂wp > 0, ∂w> 0.

Appendix E
Details of the Likelihood Function
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

58



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 )−∞ η̃itits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 > 0.56
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 the next subsection, I
show the derivation of the density and probability I use for computing the likelihood
function, and I explain the simulation of unobserved variables.

First-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 the56

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

59

first 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∂∂ck:αc exp(ηitEVi,t+1 (Zi,t+1 (ln qi,t+1 (pit , xit , sit ), Ψi,t+1 )= 0;+ λck )ucx (cit ) + αx exp(ηit+ λck )ux
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,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 + ηitqϕδet (1 + pit κ xit 1−κ )ϕ−1
∂ ln qi,t+1=νt(κpit κ−1 xit 1−κ );∂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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTSc
αc exp(ηit+ λck ) − β

× νt

60



∂
wit − pit xit
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.
[δet (1 + pit κ xit 1−κ )ϕ + δst (1 + sit )ϕ ](13)

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

× νt



∂
1
pit
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 ](14)

The first order conditions with respect to sit is characterized ass
αs exp(ηit+ λsk )

× νt



∂
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 ](15)

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 (·, ·, ·) =ln(vj ) − ln(vj+1 )ϕ(),∂ 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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

61


∂(e
ηitc , 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

(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 (15), in computing the critical value.
With the zero correlation assumption between eta,∂(η̃itc , η̃itx )
s
s
c
x
x
c
Pr(ηit > η it |η̃it , η̃it )fη (η̃it , η̃it )| det|∂(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 )which is what I use for computing the likelihood contribution for (Case 2).

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

62(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

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 (15), 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 )−∞DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

63(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 ).

cxsPr(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

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

64

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

R

1X
x =
R r=1
r




.

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 F
The expected value of data y conditional on the model predicted value yb is E(y|by) =κb0 (by ), whereκb0 (y)κb1 (y)

!
=

X
i

1
yi − ybiK()
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.

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

Figure F.1: Sample Fit: Private Tutoring Expenditure

Figure F.2: Hours of Tutoring

65

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

Figure F.3: Sample Fit: Quality of Tutoring

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

66

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

Figure F.5: Sample Fit: Log Test Scores(a) Fit by distribution(b) Fit by level

67

DYNAMIC COMPETITION IN PARENTAL INVESTMENT AND CHILD’S EFFORTS

68

Appendix GIntergenerational persistence of earnings
Table G.5: Intergenerational Persistence of Earnings Fixing Initial Conditions
The effect of heterogeneity in the academic performance in primary school on theintergenerational 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, which can befound in Table G.5. For example, BCF’ is the same simulation as BCF except that 6thgrade academic performance and parental education are fixed across households. The
results are consistent with the original counterfactual simulations that are conductedwithout fixing the household characteristics.
(a) Rank-rank Slope Estimates

(1)
BCF’

(2)
OPI

(3)
OSS’

pincprctile

0.573***(0.019)0.715***(0.017)0.380***(0.022)R-squared

0.329

0.511

0.144

(b) Intergenerational Elasticity of Earnings Estimates

(1)
BCF’

(2)
OPI’

(3)
OSS’

logpinc0.232***(0.008)0.230***(0.005)0.139***(0.008)R-squared

0.304

0.539

0.137

Note: Table (a) and (b) provide the estimates of rank-rank slope and intergenerational elasticty ofearnings, respectively. In order to assess the importance of initial conditions, all simulations are
conducted fixing the initial test score and parental education. Parental income is not fixed except
for the NIN’ simualtion. BCF’ is the benchmark counterfactual; NIN’ is a simulation where eachhousehold 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.