Essays in Macroeconomics
A DISSERTATION
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Shijun Gu
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
Doctor of Philosophy
Ellen McGrattan, Advisor
July, 2020
© Shijun Gu 2020
ALL RIGHTS RESERVED
Acknowledgements
I am indebted to Anmol Bhandari, Ellen McGrattan, and Joseph Mullins for their
invaluable guidance and support throughout this thesis. Additionally, I am very
grateful to Hengjie Ai, Kyle Herkenhoff, and Loukas Karabarbounis for their insightful
comments. Finally, I would like to thank participants in the Macro Workshop at the
University of Minnesota and Lichen Zhang for many helpful discussions.
i
Dedication
To my parents, Hua Wu and Renrong Gu.
ii
Abstract
This dissertation consists of three chapters. In the first, I perform an empirical
analysis of China’s college expansion policy. First, I estimate the impact of a sub-
stantial increase in the college-educated labor supply on the college wage premium. I
find that the return to college education is stable within a decade after college expan-
sion. Second, I investigate how college expansion has affected pre-college education
expenditure on children and their educational outcomes. The main finding is that the
magnitude of the effect depends crucially on parents’ socioeconomic backgrounds.
In the second chapter, I quantitatively evaluate how China’s public college expan-
sion program impacts human capital investment in children and inequality in the long
run. To this end, I introduce a heterogeneous-agent overlapping-generations model
in which altruistic parents invest in their children’s pre-college education, which can
raise their children’s future working efficiency and their chance of passing the College
Entrance Examination. I find that the increases in college attainment, human capi-
tal, and ex ante welfare are substantial but unevenly distributed, with disadvantaged
children benefiting the least from the existing policy. The simulation also reveals that
the reason for the unequal outcomes is that college expansion primarily incentivizes
wealthy parents to spend more on their children’s education, which is consistent with
the empirical evidence.
Finally, the third chapter (joint with Lichen Zhang) studies the driving forces
behind the decline in the formation of new businesses in the U.S. since the 1980s and
investigates their macroeconomic implications. We devote our attention to two forces:
changes in entry costs and the persistence of shocks to productivity. We develop a
quantitative general equilibrium model of entrepreneurship to identify and quantify
their relative importance in explaining the observed declines in new business creation.
We find that the relative contribution of higher entry cost is 1.5 to 2 times larger than
that of the higher persistence of shocks. Moreover, the increases in entry cost have
compelled entrepreneurs to pay 15% more in terms of their first year’s profit to start
a business.
iii
Contents
Acknowledgements i
Dedication ii
Abstract iii
List of Tables vii
List of Figures viii
1 An Empirical Analysis of College Expansion Policies in China 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Institutional Background . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 College Enrollment and the Wage Premium . . . . . . . . . . . . . . 5
1.3.1 College Enrollment . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 The College Wage Premium . . . . . . . . . . . . . . . . . . . 9
1.4 Education Expenditure and Outcomes . . . . . . . . . . . . . . . . . 11
1.4.1 The Aggregate Impact of College Expansion . . . . . . . . . . 11
1.4.2 Education Expenditure and Parents’ Characteristics. . . . . . 13
1.4.3 Education Expenditure and Developmental Stages. . . . . . . 13
1.4.4 Education Outcomes and Parents’ Characteristics . . . . . . . 16
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 The Impact of College Expansion on Human Capital Investment and
Inequality 19
iv
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Demographics and Timing . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Aggregate Production Function . . . . . . . . . . . . . . . . . 26
2.2.3 Preference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.4 Human Capital Formation . . . . . . . . . . . . . . . . . . . 27
2.2.5 Labor Earnings . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.6 Government Policies . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.7 Recursive Problems . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.8 Definition of Equilibrium . . . . . . . . . . . . . . . . . . . . . 34
2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Model Parameterization . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 Performance of the Benchmark Model . . . . . . . . . . . . . . 46
2.4 Policy Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.1 Descriptions of the Policies . . . . . . . . . . . . . . . . . . . . 48
2.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Understanding the Secular Decline in New Business Creation 62
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 Stationary Equilibrium . . . . . . . . . . . . . . . . . . . . . 72
3.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.1 Externally Calibrated Parameters . . . . . . . . . . . . . . . 75
3.4.2 Internally Calibrated Parameters . . . . . . . . . . . . . . . . 76
3.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.1 Understanding the Factors in Isolation . . . . . . . . . . . . . 80
3.5.2 Identification and Decomposition . . . . . . . . . . . . . . . . 81
3.5.3 TFP and Welfare . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.4 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
v
3.6 Impact of Credit Shocks . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References 93
A Appendix to Chapter 1 100
A.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.1.1 Education Expenditure . . . . . . . . . . . . . . . . . . . . . . 100
A.1.2 College Wage Premium . . . . . . . . . . . . . . . . . . . . . . 101
A.2 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B Appendix to Chapter 2 103
B.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B.1.1 Test Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B.1.2 Admission Probability . . . . . . . . . . . . . . . . . . . . . . 105
B.2 Identifying Dynamic Complementarity Parameters . . . . . . . . . . . 105
B.2.1 Description of Environment . . . . . . . . . . . . . . . . . . . 105
B.2.2 Household Problem . . . . . . . . . . . . . . . . . . . . . . . . 107
B.2.3 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . 108
B.2.4 Link to the Full Model . . . . . . . . . . . . . . . . . . . . . . 109
B.3 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
C Appendix to Chapter 3 114
C.1 Computational Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 114
C.1.1 Household Problem . . . . . . . . . . . . . . . . . . . . . . . . 114
C.1.2 Transition Dynamics . . . . . . . . . . . . . . . . . . . . . . . 118
vi
List of Tables
1.1 Descriptive Statistics (UHS) . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Effect of Household Income on Education Expenditure . . . . . . . . 16
2.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Estimate Return to Skill . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Key Moments: Model vs. China Data . . . . . . . . . . . . . . . . . 45
2.4 Non-targeted Moments . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 Human Capital (Test Scores) Distribution . . . . . . . . . . . . . . . 48
2.6 Aggregate Effects of Education Policies . . . . . . . . . . . . . . . . 53
2.7 Results Decomposition (with Existing Policy) . . . . . . . . . . . . . 55
2.8 Distributional Implications of Policies . . . . . . . . . . . . . . . . . 60
3.1 Parameter values set externally . . . . . . . . . . . . . . . . . . . . . 76
3.2 Parameter values calibrated internally . . . . . . . . . . . . . . . . . 78
3.3 Qualitative experiment results . . . . . . . . . . . . . . . . . . . . . 80
3.4 Entry Cost and Persistence of Shocks . . . . . . . . . . . . . . . . . 81
3.5 Entry Cost in Real Terms . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6 Relative Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.7 TFP and Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.8 Robustness Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
B.1 Descriptive Statistics (College Entrance Exam) . . . . . . . . . . . . 105
B.2 Estimation of Exam Admission Probability . . . . . . . . . . . . . . 106
B.3 Robustness: Skill-Biased Technological Change . . . . . . . . . . . . 111
vii
List of Figures
1.1 Four-year College Admission and Exam Attendance . . . . . . . . . . 6
1.2 Four-year College Attainment Rates by Age (2005-2015) . . . . . . . 8
1.3 The College Wage Premium since College Expansion . . . . . . . . . 10
1.4 Education Expenditure by Parents’ Characteristics . . . . . . . . . . 14
1.5 Education Expenditure by Children’s Developmental Stage . . . . . 15
1.6 College Attainment of Children by Parents’ Characteristics . . . . . 17
2.1 Timeline in the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 College Entrance Examination Admission Policy Function . . . . . . 40
2.3 Edu. Investment and Outcomes in Relation to Human Capital of Par-
ents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1 Transition Dynamics: Entrepreneurs’ Dynamics . . . . . . . . . . . . 88
3.2 Transition Dynamics: Macro Variables . . . . . . . . . . . . . . . . . 90
A.1 College Attainment of Children by Parents’ Characteristics . . . . . 102
B.1 Calibration of Dynamic Complementarity Parameters . . . . . . . . 110
B.2 Robustness: Impact of Skill-Biased Technical Changes on Distribution 113
viii
Chapter 1
An Empirical Analysis of College
Expansion Policies in China
1.1 Introduction
Since 1999, China has implemented a large-scale public college expansion program.
This policy has led to an additional three million students passing the College En-
trance Examination and attending four-year colleges every year. From 1998 to 2015,
the enrollment of new undergraduate students grew by 11% annually, from 0.65 mil-
lion in 1998 to 3.89 million in 2015. Meanwhile, the college admission rate also rose
dramatically, which suggests that the exam is less selective than before. In 1998, only
about 20% of exam takers could enroll in a four-year college, and the rate has doubled
in recent years.
This chapter first demonstrates that college expansion has substantially increased
college enrollment, which due not only to a higher admission rate, but also to the
higher rate of taking the entrance exam. Specifically, 62% took the exam at age 18 in
2015, which is 45% higher than two decades earlier. Moreover, a direct consequence
of a large-scale increase in college capacity is that younger cohorts have a significantly
higher college attainment rate than older groups.
Next, I empirically analyze the labor market effect of college expansion. In partic-
ular, I focus on estimating changes in the college wage premium, because it is unclear
1
whether a sizable rise in the college-educated labor supply has caused a decline in
returns to education in China. I find that although millions of college graduates have
entered the labor force since college expansion, the college wage premium was stable
until 2009. This suggests that the technological progress that favors college graduates
on the demand side has mitigated the change on the supply side. However, survey
data collected in 2013 show that the college wage premium has moderately declined.
Therefore, newer versions of survey data are required to determine whether we have
witnessed a turning point.
Finally, using household-level survey data, I investigate how college expansion has
affected pre-college education expenditure on children and their educational outcomes.
In particular, I focus on understanding their relationship with parental characteris-
tics, including income and education background. The main findings are summarized
as follows. (1) Pre-college education expenditure increased by 38% during 2002-2009,
but the share of education expenditure in household income declined during this
period. (2) Education investment in children is associated with their parents’ socioe-
conomic background, and the gap in education expenditure between parent groups
(rich vs. poor and college-educated vs. non-college-educated) became larger during
2002-2009. (3) The share of education expenditure in household income increases in
children’s age, and household income has a stronger effect on education expenditure
on children during early childhood. (4) Children of high-income and college-educated
parents are more likely to attend four-year colleges. (5) After college expansion,
there is a substantial but unequal increase in education expenditure and in children’s
college attainment across parent socioeconomic groups, with disadvantaged children
benefiting least from the policy.
Related Literature. This chapter contributes to the empirical literature that ex-
amines the impact of college expansion and the college admission system in China.
Zhang, Zhao, Park, and Song (2005) document a dramatic increase in return to edu-
cation and college wage premium prior to the college expansion policy in China. Li,
Whalley, and Xing (2014) examine the short-run labor market outcomes of college ex-
pansion and find that it increased the unemployment rate for young college graduates.
Ge and Yang (2014) document the changes in China’s wage structure and estimate
2
the labor market effects of the upsurge in educational attainment. Meng (2012) re-
view the labor market outcomes of various reform implemented in China. Jia and Li
(2016) find that access to elite education through the College Entrance Examination
does not guarantee one’s entry into the elite class, or alter the intergenerational link
between parents’ status and children’s status. Lee and Malin (2013) quantify a novel
channel through which expanded access to higher education can reduce labor market
friction in China.
The novel aspect of my empirical analysis is that I demonstrate a connection be-
tween the college expansion policy and household education expenditure on children,
which is absent in the related literature. This finding is crucial, because both the ac-
cessibility of higher education and the college return play essential roles in pre-college
educational investment. Furthermore, early childhood human capital investment,
which shapes children’s ability and influences their educational attainment, has an
endogenous impact on inequality, intergenerational mobility, and social welfare.
The rest of the chapter is organized as follows. Section 1.2 introduces the in-
stitutional background of China’s higher education system and its college expansion
program. Section 1.3 presents the empirical evidence on China’s college-educated
labor supply and college wage premium. Section 1.4 documents the effect of col-
lege expansion on education expenditure on children and their educational outcomes
and examines the connection with parental socioeconomic background. Section 1.5
concludes.
1.2 Institutional Background
This section briefly introduces the institutional background of China’s college sys-
tem, selection mechanism, and education reform. The main purpose is to motivate
the empirical investigation on the effect of college expansion on labor market and
educational investment on children.
Higher education system and types of college. In 2015, the total number of
Chinese national higher institutions was 2,845, most of which were funded by the
government. Private universities in China are a complement to public universities to
3
meet the needs of those students who failed in their College Entrance Examination
but could not afford the tuition fees to study abroad.
Students are sorted into different public universities through the college entrance
exam. The public university has two tiers: regular university (four-year college), and
vocational college (three-year college). Test takers are required to receive much higher
scores on the exam to attend regular universities. This feature suggests that, other
things held constant, parents have to spend more on their children’s education to
make children more likely to enter a four-year college. As a result, skill accumulation
plays a more crucial role in determining the four-year college admission probability.1
Zhang, Zhao, Park, and Song (2005) estimate that the return to completing vocational
college relative to high-school graduates was 17.8% in 2001, whereas regular university
graduates earned 37.3% more than high-school graduates. This evidence suggests that
an individual’s labor market result can be very different if they attend a four-year,
rather than a three-year college.
College Entrance Examination. The National College Entrance Examination is
a standardized test taken by high-school graduates who want to pursue their un-
dergraduate studies at college. This exam is uniformly designed by the Ministry of
Education of China based on the curriculum of (senior) high school and is held an-
nually in summer. Since the test is knowledge-based and highly selective, students
normally spend their entire high-school years to prepare for the exam.2 All of the stu-
dents take tests on Chinese, English, and mathematics. In addition, students choose
between two concentrations, either the social science-oriented area or the natural
science-oriented area.
College admission entirely depends on students’ performance on the College En-
trance Examination. Students are required to apply for their intended university
prior to the exam. The authority will announce official cutoff scores for universities
1In recent years, the total college admission rate (including both three- and four-year colleges)
in China has been as high as 80%, whereas the admission rate of four-year colleges has been only
40%. Therefore, whether or not one can attend college indicates more about her education choice,
but whether or not one can attend the four-year college indicates more about her skill level.
2Since high-school education mainly serves as a mechanism to select college students, it does
not teach students much practical knowledge for working. That is why Li, Liu, and Zhang (2012)
estimate that the return to high-school education is low in China.
4
in different tiers after all the exams have been graded. Each university also announces
a cutoff score (must be higher than the official cutoff) based on the number of appli-
cants and vacancies. If a student’s test score can make the official cutoff, as well as
the cutoff of the university to which they applied, they will be enrolled in that insti-
tution. Although students can apply for several universities by filling a list of ordered
preferences, many good institutions only admit students who apply to it as their first
choice, which means a student with a test score higher than the official cutoff may
end up failing in the test3 if they cannot make their first-choice university’s cutoff.
College expansion. The four-year college admission rate in 1998 was 20.4%. In
1999, the Chinese government suddenly announced its first-wave college expansion
plan, and the admission rate dramatically increased to 32.5%.4 The second wave
college expansion is conducted in the late 2000s, which ultimately raises the admission
rate of four-year college to around 40%.
Throughout the last two decades, college capacity has been continuously expand-
ing in China. From 1998 to 2015, the enrollment of new undergraduate students
grew by 11.1% annually, increasing from 0.65 million in 1998 to 3.89 million in 2015.
In recent years, the government has begun to control the rapid expansion of college
education.
1.3 College Enrollment and the Wage Premium
This section presents the main results of the paper. First, I empirically illustrate
how college expansion has affected China’s college enrollment and college-educated
labor supply. Second, I estimate changes in the college wage premium since college
expansion.
3Because this paper defines college as a four-year institution (regular university), attending a
three-year college (vocational college) is also considered failing the test.
4The policy was announced just four months before the exam, but the exam preparation takes
years. Hence, the policy did not boost the number of exam takers.
5
1.3.1 College Enrollment
In this subsection, I provide two sets of empirical evidence to demonstrate how college
expansion has affected college enrollment and the college-educated labor supply. First,
I show that both the number of entrants to four-year colleges and the number of
College Entrance Examination takers have substantially increased since 1999. Second,
I show that younger cohorts have a significantly higher college attainment rate than
older cohorts, due to the expansion in college capacity over the last two decades.
College admission and exam attendance. To identify the effect of the college
expansion policy on education choice and college capacity, I examine aggregate ed-
ucational statistics that report exam attendance and college admission over three
decades.
Figure 1.1: Four-year College Admission and Exam Attendance
Note: Data source: various issues of China Statistical Yearbook and Educational Statistical
Yearbook of China.
6
As shown in Figure 1.1 (A), the college expansion policy has led to an additional
3 million students passing the College Entrance Examination and attending four-year
colleges every year, and the annual number of college entrants rose from 30% to 300%
of its U.S. counterpart. Figure 1.1 (B) shows that the number of exam takers has
dramatically increased since the early 2000s, and Figure 1.1 (D) shows that the exam
attendance rate rises from 17% to 62%. This demonstrates that an increasing fraction
of high school graduates prefers going to college over immediately entering the labor
force.5
Figure 1.1 (C) shows that immediately following the announcement of college
expansion in 1999, the four-year college admission rate increased by about twelve
percentage points. However, due to the rapid increase of exam takers, the admission
rate fell gradually to 27% in 2006. As a result of the second wave of the college
expansion policy, in recent years, the admission rates are stable around 40%.
College attainment by age. An immediate consequence of college expansion is
that younger cohorts have a substantially higher four-year college attainment rate
than older groups. Figure 1.2 plots college attainment rates against age cohort. The
data come from two mini censuses (the 1% Population Survey) conducted by the
National Bureau of Statistics of China in 2005 and 2015.
As shown in Figure 1.3 (A), at the national level, the four-year college attainment
rate of individuals at age 20 increased from 5% to 30% within a decade. This suggests
that over half of the college attainment rate’s rise is the result of the second wave of
college expansion since the late 2000s.
It is also notable that the college attainment rate varies significantly across urban
and rural areas, as shown in Figures 1.2 (B), (C), and (D). Here, I provide three
potential explanations for the phenomenon, although which plays the most critical
role is an open question. First, students from urban areas have greater access to
high-quality education resources, which helps them perform a better outcome on the
College Entrance Examination. Second, college education is more valuable and afford-
able for students (also for their parents) from urban areas, which can influence their
5As indicated in Section 1.2, since the college entrance examination is a knowledge-based test,
attending the exam suggests that the college is both optimal and affordable for an individual.
7
Figure 1.2: Four-year College Attainment Rates by Age (2005-2015)
Census 2005
Census 2015
Note: Data source: 1% Population Survey (2005, 2015).
8
college choices. Third, a large number of rural students, after completing their college
education, move to urban areas, where more jobs and entrepreneurial opportunities
are available.
1.3.2 The College Wage Premium
I use three datasets to estimate China’s college wage premium. The Urban House-
hold Survey (UHS) collects household information for eight consecutive years from
2002 to 2009.6 However, it may not fully reveal the labor market effect of college
expansion, because only a few cohorts of college graduates (who took the exam after
1999) had entered the labor force in 2009. The Chinese Household Income Project
(CHIP) collects individuals’ information from 1999 to 2013. However, the survey is
only conducted every 3 or 5 years. Additionally, the UHS and CHIP only include
households in several representative provinces or cities in China. The 1% Population
Survey (2005) contains information about demographic characteristics and income
for over ten million individuals living in all provinces. However, the information on
income is omitted in a similar survey conducted in 2015, which makes it impossible
to track changes in the college wage premium.
Given each dataset’s strengths and weaknesses, I validate my estimation of the
college wage premium using the following strategy. The estimation obtained from the
UHS will be considered to be the baseline. Then, I check whether the estimates ob-
tained from the 2002, 2007, and 2008 waves of the CHIP are consistent with those from
the UHS. If so, the estimate produced by CHIP2013 will be informative. Moreover,
results from the two small-sample surveys (the UHS and the CHIP) can be validated
if the large-sample survey (the 1% Population Survey) yields a similar result.
Let we,j be the annual wage of individual with education level e and age j . For
each observed year, I run an ordinary least-squares regression of wages on a college
education dummy (e), and a cubic polynomial in potential experience (age minus
6I do not have access to earlier waves since UHS is a restricted dataset. The rotating structure
of the UHS enables the construction of a two or three-year panel. One can also treat the UHS as a
repeated cross-section survey. Ding and He (2018) discuss more details about UHS’s design.
9
years of education minus 6) L(j) :
log(we,j) = β0 + β1e+ L(j) + ,
where  is an error term.7 In this specification, the regression coefficient β1 captures
the college wage premium. Notice that I separately check the case in which college is
defined as: (1) a four-year institution or (2) both three-year and four-year institutions.
Figure 1.3: The College Wage Premium since College Expansion
CHIP
UHS
Census
Note: Data source: UHS (2002-2009), CHIP (1999, 2002, 2007, 2008, 2013), and 1%
Population Survey (2005).
The main findings are displayed in Figure 1.3. Panel (A) shows that the four-
year college wage premium calculated from the UHS over 2002-2009 is stable around
0.55, and Panel (B) shows that the (three-year and four-year) college wage premium
over the same period moderately increases from 0.48 to 0.53. The four-year college
wage premium produced by the CHIP is slightly lower than that of the UHS, but the
three-year college wage premium is very close. In addition, the 1% Population Survey
yields a higher college wage premium in 2005 than the UHS, but the magnitude of
the difference is small.
7I also add a province dummy and a gender dummy to control individual’s idiosyncratic charac-
teristics.
10
Next, I summarize the main implications of the empirical estimation. First, even
though millions of college graduates have entered the labor force since college expan-
sion, the college wage premium does not move much until 2009. This suggests that
there is a technological change on the demand side that accompanies an increase in
the supply of college-educated labor, which substantially increases the job vacancies
for college graduates. Second, the estimate from CHIP2013 shows that the college
wage premium moderately declines. However, in order to check whether the second
wave of college expansion has led to a significant fall in the college wage premium,
I would need access to newer versions of the UHS or CHIP, which are temporarily
unavailable.
1.4 Education Expenditure and Outcomes
From 2002 to 2009, the UHS collected detailed information about individual and
household income, consumption expenditure (including education expenditure), and
demographic characteristics. I have access to the portion that covers nine provinces,
representing over 14,000 households and 60,000 individuals per year.
With the UHS, I can examine how pre-college household education expenditure
on children depends on the parents’ characteristics (e.g., income and education level)
and children’s developmental stages. These sets of evidence are helpful in identifying
the patterns of Chinese households’ early-childhood educational investment, as well as
constructing moments for estimating the skill formation technology. Unfortunately,
the UHS’s rotating structure does not allow me to track human capital investment
on children and their education outcomes over a long period. However, it does allow
me to observe children’s terminal education attainments and their respective parents’
income and education levels, which can be informative for empirical analysis.
1.4.1 The Aggregate Impact of College Expansion
Table 1.1 summarizes statistics related to changes in household income, education
expenditure, and education attainment from 2002 to 2009. I restrict the sample to
households with one child, since this is the most common family structure for urban
11
Chinese households. All of the following results are robust if I relax this restriction.
For details on sample selection, see Appendix A.1.
Table 1.1: Descriptive Statistics (UHS)
Variable 2002 2009
Education expenditure (RMB)
All households 1,345 1,575
All with positive expenditure 1,996 2,747
Disposable income (RMB)
Households, all 22,334 42,038
Households, tercile 1 10,408 18,619
Households, tercile 2 19,078 35,244
Households, tercile 3 37,518 72,257
Variable 1993-1998 2001-2005
Percentage of college graduates
Child, four-year institution 18.04 38.21
Child, three- and four-year institution 51.63 71.50
Parent, four-year institution 6.57 8.77
Parent, three- and four-year institution 30.24 25.95
Note: Data source: UHS (2002,2009). This table shows unweighted averages of selected
characteristics. All RMB amounts are deflated to 2002 values. Sample restricted to urban
households with an only child.
Household pre-college education expenditure and disposable income both increased
rapidly over this period. In particular, average real disposable income increased by
188%, while real college tuition only increased by 51%. College, therefore, became
substantially more affordable for Chinese families. Moreover, average education ex-
penditure increased by 38% for households that reported positive spending on chil-
dren’s education. This pattern indicates that the share of education expenditure in
household income declined during this period.
Next, I show how to derive the impact of college expansion on education attain-
ment of children from the UHS.8 As I stressed earlier, UHS2002 is the earliest dataset
available to me, but the college expansion policy was implemented in 1999. Therefore,
I can only infer the fraction of college-educated children who took the exam before
8In UHS, students who are currently in universities are classified as college-educated individuals.
12
the policy change. Specifically, when I calculate the percentage of college-educated
children before college expansion, I only include households whose children were above
age 22 (and below age 26) in 2002, since they took the College Entrance Examination
between 1993 and 1998 and hence are unaffected by the college expansion program.
The last four rows of Table 1.1 show that as a result of the college expansion
policy, the fraction of children who entered college between 2001 and 2005 increased
by approximately 20 percentage points, relative to those who entered college between
1993 and 1998.9 Furthermore, immediately after college expansion, over 70% of chil-
dren went to three-year or four-year institutions. This demonstrates that college was
no longer very selective in urban areas.
1.4.2 Education Expenditure and Parents’ Characteristics.
The evidence presented in this subsection highlights two important features of pre-
college educational expenditure on children. First, education investment is associated
with the parents’ socioeconomic background. For example, the light bars in Fig-
ure 1.4 (left panel) show that high-income parents (top income tercile) spent about
RMB1,500 (144%) more on their children than low-income parents (bottom income
tercile) in 2002. The light bars in the right panel show that college-educated par-
ents spent approximately RMB860 (54%) more than non-college-educated parents in
2002. Second, the gap in education expenditure became larger between 2002 and
2009. High-income parents increased their education expenditure on children by 56%
(vs. a 33% rise for low-income parents), and college-educated parents increased their
education expenditure by 48% (vs. a 37% rise for non-college-educated parents).
1.4.3 Education Expenditure and Developmental Stages.
Figure 1.5 displays pre-college education expenditure (in % of household income) in
relation to the age of the child.10 It increases from 6% (preschool stage) to 13%
9Since the dataset does not cover rural households, the numbers displayed in the table are not
nationally representative.
10Since this chapter focuses on the pre-college educational investment decisions, I do not report
the statistics for children above age 17. Choukhmane, Coeurdacier, and Jin (2017) use the data from
CHIP2002 and plot a similar graph which contains the information about educational spending on
13
Figure 1.4: Education Expenditure by Parents’ Characteristics
Note: Data source: UHS (2002,2009). Sample restricted to urban households with an only
child.
(high school stage). Spending on tuition and extracurricular training increase the
most rapidly as children age, especially for children above age 14. This is mainly
because Chinese children receive a 9-year compulsory (almost free) education. Ad-
ditionally, extracurricular training to prepare for the College Entrance Examination
are incrementally important as children reach the high school stage.
Next, I estimate the stage-dependent effects of household income on pre-college
education expenditure. Using the subscripts i and t to index the individual household
and year, I run the following regression
log(EXPi,t) = α + βlog(INCi,t) + Y eart +HHi + i,t
respectively for each developmental stage j,11 where INC is household disposable
income, EXP is household education expenditure, fixed-effect Y ear controls for time
effects, HH controls for time-invariant household characteristics, and  is the error
post-college children.
11Here, j = 1 corresponds to preschool stage (age 2-5), j = 2 corresponds to early elementary
school stage (age 6-9), j = 3 corresponds to late elementary school stage (age 10-13), and j = 4
corresponds to high-school stage (age 14-17).
14
Figure 1.5: Education Expenditure by Children’s Developmental Stage
Note: Data source: UHS (2002). Sample restricted to urban households with an only child.
term.12 The regression coefficient β captures the impact of within-household devia-
tions in disposable income.
Table 1.2 summarizes the estimation of β at stage j using various regression spec-
ifications. In the first two specifications, I construct a 2- and 3-year short-panel,
respectively, and include year and household dummies. In the last specification, I run
a cross-sectional regression using 2002 UHS data.
The coefficient of interest is β, which reports how household income impacts edu-
cation spendings on children. The results presented in Table 1.2 show that education
expenditure is more elastic to changes in household disposable income at early child-
hood developmental stages. For example, column (1) shows that from 2002 to 2004,
a 1% increase in household income increases predicted education expenditure by ap-
proximately 0.75% at the first and second developmental stages, and by only 0.55%
at the last stage. The results are robust in the 2-year panel and cross-sectional re-
gressions, as shown in columns (2) and (3).
12For each household, I can observe its information for three consecutive years. If the child age
jumps across two developmental stages during this time span, I drop them from the fixed-effect
estimation.
15
Table 1.2: Effect of Household Income on Education Expenditure
Dep. variable: log-education expenditure
(1) (2) (3)
Indep. variable 2002-2004 2002-2003 2002
log-income (stage 1)
0.747∗∗∗
(0.282)
0.953∗∗∗
(0.214)
0.858∗∗∗
(0.073)
log-income (stage 2)
0.756∗∗∗
(0.140)
0.557∗∗∗
(0.130)
0.649∗∗∗
(0.047)
log-income (stage 3)
0.689∗∗∗
(0.118)
0.645∗∗∗
(0.117)
0.617∗∗∗
(0.040)
log-income (stage 4)
0.551∗∗∗
(0.128)
0.457∗∗∗
(0.107)
0.563∗∗∗
(0.047)
FE Y Y N
Note: Data source: UHS (2002-2004). Sample restricted to urban households with an only
child. Standard errors are reported in parentheses. *,**,*** denotes statistical significance
at 10, 5, and 1 percent, respectively.
1.4.4 Education Outcomes and Parents’ Characteristics
Figure 1.6 shows the education outcomes of children (proxied by the probability of
earning a terminal degree from a four-year college) in relation to the parents’ income
and education level prior to the college expansion policy.13 The light bars in the
left graph show that the children of the high-income (top tercile) parents are about
10 percentage points more likely to earn a college degree than those of middle- and
low-income (middle and bottom terciles) parents. The light bars in the right graph
show that the children of college-educated parents are about 16 percentage points
more likely to earn a college degree than those of non-college-educated parents.
After the college expansion program, it is notable that for middle- and high-
income parents, their children’s probability of attending college increases by over 23
to 26 percentage points (vs. an 8 percentage point rise for low-income parents).
For college-educated parents, their children are 40 percentage points more likely to
earn a college degree (vs. a 16 percentage point rise for non-college-educated parents).
13Here I am referring to individuals who took the College Entrance Examination between 1993
and 1998 (derived from UHS2002). For comparison, I also display the statistics for those who took
the exam between 2001 and 2005 (derived from UHS2009).
16
Figure 1.6: College Attainment of Children by Parents’ Characteristics
Note: Data source: UHS (2002, 2009). College is defined as four-year institutions. Sample
restricted to urban households with an only child.
This empirical evidence suggests that children with high-income and college-educated
parents benefit the most from the policy intervention, since their admission probability
is substantially higher than before. For a robustness test in which I define college as
three- and four-year institutions, see Appendix A.2.
The data demonstrate that changes in children’s college attendance rate vary sig-
nificantly across their parents’ income and education groups. However, for exam
takers who took the exam between 2001 and 2005, their parents had a short window
during which they could change their parental investments in response to the policy
implemented in 1999. Therefore, the empirical results may underestimate the aggre-
gate and distributional impact of college expansion. Model-based policy analysis has
a better predictive power for the long-run effect, since in that case, parents can take
the new policy environment into account as soon as the child is born.
17
1.5 Conclusion
In this chapter, I perform an empirical analysis of China’s college expansion policy.
First, I estimate the impact of a substantial increase in the college-educated labor sup-
ply on the college wage premium. I find that the return to college education is stable
within a decade after college expansion. Second, I investigate how college expansion
has affected pre-college education expenditure on children and their educational out-
comes. The main finding is that the magnitude of the effect depends crucially on
parents’ socioeconomic backgrounds. In particular, disadvantaged children, whose
parents are at the lower end of the income and education groups, benefits the least
from the college expansion policy. It is notable that the available data only cover
household information between 2002 and 2009. As a result, how college expansion
affects the college wage premium and the patterns of household education expenses
in recent years are open questions for future research.
18
Chapter 2
The Impact of College Expansion
on Human Capital Investment and
Inequality
2.1 Introduction
Individuals living in developing countries face difficulties in gaining access to higher
education due to the limited capacity of public college.1 Since education attainment
and skill accumulation are complementary, this situation can substantially distort
intergenerational investment in human capital and lead to slow growth in labor pro-
ductivity. Therefore, government intervention in the market for higher education is
crucial to promote economic development.
Since 1999, China has implemented a large-scale public college expansion pro-
gram. This policy has led to an additional three million students passing the College
Entrance Examination and attending four-year colleges every year. Meanwhile, col-
lege becomes more affordable for households due to the rapid growth in disposable
income. The number of exam takers has dramatically increased since the early 2000s,
which reflects that an increasing fraction of population prefer going to college over
1Private institutions in developing countries are characterized by their poor quality of teaching
and expensive tuition. As a result, public college is the only affordable channel through which people
can acquire a high-quality college education.
19
immediately entering the labor force.
As the aggregate capacity of public college constrains college attendance and the
exam-based selection scheme favors the test takers with high scores (human capital),
pre-college parental investments2 can relax the constraint at the individual level by
raising children’s probability of passing the test. If the college expansion policy can
further reward children with high skills by increasing their admission chances, parents
will have more incentives to make education expenditure on their children. The goal
of this paper is to quantify how much college expansion affects long-run intergener-
ational human capital investment and to examine the impact of the policy change
on the educational outcomes of children with different socioeconomic backgrounds.
Moreover, this investigation also provides a framework to analyze the effects of alter-
native education policies, including government subsidies on childhood development.
To these ends, I build an overlapping-generations framework where altruistic par-
ents invest in their children’s skills before they go to college and make the decision on
whether or not their children should take the College Entrance Examination. Specif-
ically, I incorporate childhood development into an otherwise standard incomplete
market model. Parents augment their children’s human capital children through
multiple-period skill investments according to a technology that features dynamic
complementarity and self-productivity. A critical element of this model is that inter-
generational skill investment can increase not only children’s future labor efficiency
but also their probability of passing the College Entrance Examination. To specify
how each level of human capital (proxied by test score) is associated with the proba-
bility of college admission, I estimate an admission policy function that resembles the
college selection scheme of China. I find that with a score below the average, a test
taker has a very low probability of attending college. However, if the score is above
the average, the probability of passing increases significantly with higher test scores.
The nonlinear correlation between the admission probability and human capital plays
an essential role in the analysis of the distributional effects of the college expansion
program.
2Throughout this paper, pre-college parental investments can accumulate children’s human cap-
ital according to a skill formation technology. Although parents also cover college tuition for their
children, investing in college only enables children to earn the college wage rate (after they enter the
labor force) rather than raise their human capital.
20
I estimate the model to match both macro and micro moments constructed using
Chinese data. The crucial part is the estimation of skill formation technology, which
combines the current-period child’s skill, parent’s skill, as well as monetary invest-
ment to produce the next-period child’s human capital. The model requires me to
specify how current-stage investment in children’s education is complementary to that
in the previous stage. I recover the parameters controlling the degrees of dynamic
complementarity at different childhood developmental stages via indirect inference.
Specifically, for each developmental stage, I first pin down the share of the child’s
skill in the production function by matching the corresponding average household
education expenditure to income ratio. Then I design an auxiliary model to highlight
data patterns on the stage-dependent effects of household income on education expen-
diture on children. The model parameters are chosen by matching model-predicted
auxiliary coefficients to their empirical counterparts.
To quantify the long-run macroeconomic and distributional consequences of col-
lege expansion programs, I lower the tuition-to-income ratio by 22 percentage points,
which reflects the college costs in 2015, and re-estimate an admission policy function
using the post-reform data. Additionally, I feed in a skill-biased technological change
that perfectly counterbalances the decline in college wage premium resulting from the
general equilibrium effect. I find that the existing policy yields an education expen-
diture increase of 16.5% and ex ante welfare gains of 17.2%. Meanwhile, inequality
in human capital and intergenerational persistence in education attainments rise sig-
nificantly. To gain some insights on these unequal outcomes, I explore how changes
in parental investments are correlated with the human capital and education groups
of parents. I find that consistent with the short-run evidence displayed in the data,
high-skill and college-educated parents’ education expenditure are more elastic to the
policy change following the college expansion in the long run. In turn, their children’s
human capital, as well as the college admission probability, increases more than in the
case of disadvantaged children. As a result, parents with high human capital and a
college degree are more likely to transmit their socioeconomic status to their children
than before, which leads to a widening income gap between rich and poor and more
persistent skill and schooling across generations.
To explore a more efficient way for the government to implement education poli-
21
cies, I use the calibrated model to conduct a counterfactual policy exercise. I analyze
an economy in which the government implements a targeted early childhood inter-
vention. This program subsidizes 60% of the education expenditure made by the
parents of disadvantaged children whose ages are between 2 to 9. The expenditure
is financed by raising college tuition. I find that this policy yields an education ex-
penditure increase of 29.4% and ex ante welfare gains of 26.1%, both of which are
significantly higher than the numbers under the existing policy. Furthermore, since
the program exclusively targets disadvantaged children, it redistributes human capital
and welfare gains to children of parents with low human capital. Consequently, the
remediation policy can generate a substantial decline in inequality and intergenera-
tional persistence, which suggests that policy makers should consider diverting a part
of the government subsidy on college tuition to support early childhood development
for disadvantaged children.
Related Literature. There is extensive literature that quantitatively evaluates the
effects of higher education policies. Garriga and Keightley (2007), Meghir, Abbott,
Gallipoli, and Violante (forthcoming) study the impact of various financial aid pro-
grams in the US under a general equilibrium context. Krueger and Ludwig (2016)
focus on characterizing the optimal fiscal and education policy mix. Boha´cek and
Kapicka (2016) explore whether the variation in education and tax policies can ex-
plain differences in educational outcomes in the US and Europe. Kindermann (2012)
proposes a reform that would transform the publicly funded college education system
to a privately funded system. Among the papers that study higher education policies
in developed economies, the public education sector can accommodate everyone who
chooses to study in college. This paper complements this literature by examining the
impact of education policies in China, where the capacity of public college constrains
college attendance. The main issue of education reform is hence expanding the size
of the college. Additionally, all of the literature mentioned above ignores the effect of
education policies on parental investments. This paper endogenizes the skill forma-
tion process of children and studies the distributional impact of college expansion on
children’s development.
My work contributes to the literature which examines the impact of college ex-
22
pansion in China on skill selection. Ma (2014) quantitatively studies the impact of
college expansion on college wage premiums for different age groups. In her model,
whether or not an individual can go to college depends on the exogenous ability draw.
In my model, a child’s performance in the College Entrance Examination depends on
the level of human capital accumulated by their parents. Through explicitly model-
ing the intergenerational transmission of skill, the richer model allows me to focus on
skill formation and study how college expansion affects the educational outcomes of
children whose parents differ in their socioeconomic background. Additionally, Feng
(2019) develops a discrete college choice model, estimates a college admission proba-
bility function, and studies heterogeneous effects of college expansion on individuals
in different cohorts.
There are two building blocks of my model. The first one is a heterogeneous-
agent framework where individuals are facing uninsurable idiosyncratic earnings risk
introduced by Bewley (1986), Huggett (1993), and Aiyagari (1994). The second is
the framework of intergenerational linkage, beginning with the work of Becker and
Tomes (1979). The model of this paper is most closely related to Daruich (2018),
which incorporates intergenerational human capital investment and education choice
into individuals’ life-cycle. Relative to the prior work, this paper highlights a new
channel that incentivizes parents to invest in children’s education. That is, since the
capacity of college constrains college attendance, skill accumulation can make college
easier to get into by increasing children’s college admission probability. With this
mechanism in place, college expansion will significantly affect parental investments
because it impacts the aggregate tightness of the capacity constraint.
The estimation of the skill formation function is essential in previous literature
(e.g., Agostinelli and Wiswall, 2016; Cunha, Heckman, and Schennach, 2010; Cunha,
2013; Lee and Seshadri, 2019; Moschini, 2019; Daruich, 2018; Mullins, 2019; Caucutt
and Lochner, forthcoming) since the technology specifies how parental investments
map onto children’s outcomes.3 In this paper, I present a human capital produc-
tion function consistent with two properties regarding childhood development: dy-
3As highlighted by several papers mentioned above, parental time is also an important input for
developing skills of children. However, monetary investment is the only endogenous input in the
skill formation technology.
23
namic complementarity and self-productivity. I estimate skill formation technology
using Chinese household-level education expenditure data. The dataset highlights
the empirical patterns on the effects of household income on education expenditure
on children. An indirect inference approach is then put into use to pin down the
stage-dependent dynamic complementarity parameters.
This paper also contributes to the recent literature which studies the Chinese
economy by endogenizing human capital investment in a quantitative framework.
Choukhmane, Coeurdacier, and Jin (2017) study the contribution of the ’one-child
policy’ to the rise in China’s household saving rate and education investment in recent
decades. Doesey, Li, and Yang (2019) use a model featuring endogenous human
capital investment to examine the role of demographics and industrial policies in
accounting for the rise of the savings rate and economic growth in China. Both papers
focus on quantifying the linkage between the changing demographic structure and skill
investment. Relative to their work, college expansion in my paper is the key driving
force that leads to an increase in educational expenditure on children. Additionally,
different from prior work, this paper focuses on the distributional consequences of the
policy on human capital accumulation and welfare.
The rest of the paper is organized as follows. Section 2.2 lays out the life-cycle
model. Section 2.3 describes the parameterization and model fit. Sections 2.4 present
the quantitative results from the existing college expansion policy and counterfactual
policies, respectively. Finally, Section 2.5 concludes.
2.2 Model
I study an overlapping-generations model in which parents, for altruistic purposes,
choose intergenerational human capital investment and choose whether or not their
children will take the College Entrance Examination. A skill formation technology
maps the monetary investment to child development outcomes. Accumulating human
capital can raise a child’s future labor productivity as well as her probability of pass-
ing the test. Whether the child can earn a college degree depends not only on the
endogenous education choice but also on the government admission policy because
the public college has a limited capacity. Once the education outcome (whether to
24
attend college) is realized, the individual’s human capital becomes fixed for the rest of
her life. Except for the intergenerational linkages through human capital investment,
households solve a standard consumption-savings problem for maximizing lifetime
utilities. I abstract from modeling the retirement stage. As a result, individuals exit
the economy immediately after the end of the working stage. A representative firm
combines capital, college labor, and non-college labor to produce the final consump-
tion good.
2.2.1 Demographics and Timing
Figure 2.1: Timeline in the Model
Work
s = nc
18 22
Have
Child
30
Exam
choice
46 58
Exit
Investment in child
Born
2 6 10 14 18
Work
s = c
22
College
Time is discrete, and one model period is four years. There is no mortality risk.
There are J = 11 overlapping generations, each with a unit mass of households. The
demographic structure is assumed to be uniform. Let j denote the model period4
of parents (adults). Before entering the labor force, children live with their parents
and do not make any economic decisions. Individuals become economically active
at different stages, depending on their education level. College-educated individuals
spend the first period (age 18-21) in college and start working as college labor in period
2. Non-college individuals enter the labor market in period 1. At the beginning of
period 4, all of the individuals give birth to a child (age 2). From period 4 to 7,
parents make monetary investments in their children’s human capital. At period 8,
they decide whether or not to support their children taking the College Entrance
4Model period j represent the stage of the life-cycle between ages [14 + 4j, 17 + 4j].
25
Examination. Whether a child can attend college also depends on the result of the
test. Parents pay tuition and living expenses for their children during their college
stage. After their children become independent, parents continue to work until the
end of period 11 (age 61).
2.2.2 Aggregate Production Function
An individual’s education outcome is endogenously determined before she becomes
independent. The education level s falls into the set s ∈ {nc, c}, where nc denotes
non-college-educated individuals, and c denotes college-educated individuals. I as-
sume that college and non-college-educated labor are imperfectly substitutable in
production. Let Hs denote the aggregate effective labor of education level s, mea-
sured in efficiency units. The total labor efficiency units aggregate across education
groups, which is given by
H = (Φ(AcHc)
Ω + (1− Φ)HΩnc)
1
Ω ,
where Ac captures a skill-biased change in productivity that directly affects the rela-
tive contribution of college-educated workers to output, 1
1−Ω is the elasticity of sub-
stitution between non-college and college-educated labor, and Φ denotes the share of
college educated labor in production. The college wage premium will be endogenously
determined by the supply of college workers, conditional on Ω < 1.
A representative firm combines the aggregate labor and physical capital to produce
final output according to a Cobb-Douglas production technology
Y = F (K,H) = KΛH1−Λ = KΛt (Φ(AcHc)
Ω + (1− Φ)HΩnc)
1−Λ
Ω , (2.1)
where Λ measures the capital share. The firm chooses two types of labor and capital
to solve
max
Hc,Hnc,K
Y − wcHc − wncHnc − (r + ξ + δ)K, (2.2)
where ws is the wage per efficiency unit of labor of skill s, r is the capital rental rate, ξ
is the intermediation cost in capital market,5 and δ is the depreciation rate of capital.
5The intermediation cost captures the fact that the return to capital in China is puzzlingly
26
2.2.3 Preference
Individuals derive utility from consumption of all household members that are repre-
sentable by a expected lifetime utility function
E1
J∑
j=1
βj−1u(
cj
1 + 1cζ
), (2.3)
where cj is the total consumption in period j, 1c is an indicator function taking the
value one during the period when the child is living with their parents,6 and ζ is
a parents equivalence parameter. Labor hours are assumed to be supplied inelasti-
cally. Expectations are taken with respect to the stochastic processes governing labor
productivity risk.
Additionally, at the period j = 8, the child’s expected lifetime utility enters the
parental lifetime utility function with a weight ν, which measures the strength of
parental altruism.
2.2.4 Human Capital Formation
Parents accumulate their children’s human capital through multiple-period skill in-
vestments according to a technology that features dynamic complementarity (i.e.,
human capital produced at one stage can raise the investment at subsequent stages)
and self-productivity (i.e., human capital produced at one stage is input in the next
stage).
Let hp denote the human capital of parents
7 and let hj,c denote the human cap-
ital stock of their child in period j, which is endogenously affected by parental skill
investment for four model periods (j ∈ [4, 7]). Once an individual enters the labor
force and becomes economically active, her human capital becomes fixed during the
high, whereas the deposit rate at State-owned banks are set to be low by the government. Song,
Storesletten, and Zilibotti (2011) interprets this cost as the operational costs, red tape, etc.
6Note that college-educated children leave their families a period after the non-college-educated
children in the same cohort, which implies that parents have to raise the children in college in period
8, which in turn increases the opportunity cost for supporting their children’s college education.
7At the first three model periods, individuals do not have their children yet, but they are still
regarded as parents in this model.
27
entire working stage.
I assume that every child is born with the same level of human capital endowment
h. The evolution of each child’s skills hj,c over time is determined by a human capital
production function. The next-period child’s human capital hj+1,c depends on her
parent’s human capital hp, her current stock of human capital hj,c, and the parental
investment m. I specify the technology of human capital formation as:
hj+1,c = ψh
w
p
[
αjh
ρj
j,c + (1− αj)mρjj
] 1−w
ρj
, (2.4)
where ψ is an anchor that transforms the child’s human capital into adult outcomes,
w captures the contribution of parental human capital in the child’s skill formation,
αj represents the share of current stock of the child’s human capital, and ρj measures
the complementarity between the investment in period j − 1 and j. The technology
parameters ρj and αj can vary across investment stages, which is well appreciated
in recent literature. Skills are persistent over generations because high-skill parents
have more resources to invest in their children’s human capital and they are more
productive in educating their children.
2.2.5 Labor Earnings
An individual with education attainment s, human capital stock hp , and idiosyncratic
labor productivity shock  earns a labor income
wsη(hp, ), (2.5)
where ws is the wage rate of type-s human capital, and η(hp, ) is an efficiency units
function depending on individual’s human capital and labor productivity shock. The
wage rate is endogenously determined in a competitive labor market. There is no
on-the-job human capital accumulation.
The idiosyncratic labor efficiency process η(hp, ) is specified as
log(η(hp, )) = λlog(hp) + ¯ (2.6)
28
¯′ = ρ¯+ z, σz
iid∼ N(0, σz)
where ¯ = log(),8 and λ is a parameter controlling the wage-skill gradient.
2.2.6 Government Policies
Education policies. The government has two education policy tools. First, be-
cause the demand for public college is greater than its capacity, the government holds
a College Entrance Examination to decide who is eligible for attending four-year col-
leges. As I discussed in Section 1.2 in Chapter 1, the probability of passing the test
depends only on test scores. In this paper, I assume that test scores are perfectly
correlated with test takers’ stock of human capital. I specify that for a test taker
with human capital hc, χ(hc) is the probability of admission, where χ(·) is a strictly
increasing admission function that will be estimated using micro-level admission data.
Second, I assume that a fraction θ of the college tuition is borne by the government.
Suppose the per-student cost of college is κ. Then the tuition net of subsidy is
(1− θ)κ. The government can commit to subsidizing tuition for all the students who
have passed the College Entrance Examination during their entire college stage.
Labor income tax. Labor income taxes are progressive. The total amount of labor
income taxes paid takes the following form
T (y) = τ¯ymax(y − d, 0), (2.7)
where y is labor income, d is the amount of income tax deduction, and τ¯y is the income
tax rate applied to the taxable labor income.
2.2.7 Recursive Problems
Next, I lay out the dynamic individual problems at the different stages in the life
cycle recursively.
8The productivity shock  is mean-reverting and follows a Markov chain with states ε =
{1, 2, ..., M} and transitions pi(′|) > 0.
29
Problem at j = 1, 2. Right after their independence, before having children, indi-
viduals choose how much to consume c and how much to save a. Households are not
allowed to borrow, so the next period asset (a′) is non-negative. The value function of
individuals of age j, with education level s, human capital stock hp, labor productivity
shock , asset a reads as
V sj (hp, , a) = max
c,a′
{
u(c) + β
∑
′
pi(′|)V sj+1(hp, ′, a′)
}
(2.8)
subject to the budget constraint
c+ a′ = (1 + r)a+ y − T (y)
y = wsη(hp, ), c ≥ 0, a′ ≥ 0.
I assume that all of the individuals start their life with a = 0, which means
there is no inter vivos financial wealth transfer from parent to child. The initial labor
productivity is drawn from an invariant distribution pi(). Note that college graduates
start solving this dynamic problem in period 2 because they spend period 1 in college.
Problem at j = 3. In period 3, individuals solve the same problem as before. I
lay out the problem separately because the parent will have a child in period 4, so
the child’s initial endowment of human capital h′c = h enters the continuation value
of the Bellman equation in period 3. The value function of individuals is given by
V sj (hp, , a) = max
c,a′
{
u(c) + β
∑
′
pi(′|)V sj+1(hp, ′, a′, h′c)
}
(2.9)
subject to the budget constraint
c+ a′ = (1 + r)a+ y − T (y)
y = wsη(hp, ), c ≥ 0, a′ ≥ 0.
Importantly, under this specification, children do not differ in their innate ability
or initial endowment of human capital. As a result, all of the post-independence
30
variation in human capital is due to the differences in the intensity of monetary
investment and the parents’ human capital.
Problem at j = 4, 5, 6, 7. At the beginning of period 4, the child is born and the
state space of the parent is expanded to include the child’s human capital hc. During
these four periods, in addition to the standard choices of consumption, savings, the
altruistic parent also decides how much money m to invest in child’s development
of skill. Then using the technology specified in Subsection 2.2.4, the next-period
human capital of child h′c is produced by combining the current-period child’s skill
hc, parent’s human capital hp, and education expenditure m. The dynamic problem
becomes
V sj (hp, , a, hc) = max
c,a′,m
{
u(
c
1 + ζ
) + β
∑
′
pi(′|)V sj+1(hp, ′, a′, h′c)
}
(2.10)
subject to the budget constraint
c+ a′ +m = (1 + r)a+ y − T (y)
y = wsη(hp, ), c ≥ 0, a′ ≥ 0, m ≥ 0
h′c = ψh
w
p
[
αjh
ρj
c + (1− αj)mρjj
] 1−w
ρj
.
Notice that since children live with their parents during these periods, I use a
consumption equivalence parameter ζ to capture the fact that parents should transfer
some additional resources to child. Also, it borrowing against the current stock of
human capital is not allowed, which implies that money investment must be non-
negative.
Problem at j = 8. At period 8, the final education outcome of children is realized.
A critical feature of the model is that parents will decide whether their children should
take or skip the College Entrance Examination. If they choose to have their children
take the exam, then their children can attend college as long as they pass the test.
Additionally, I assume that taking the test requires an upfront cost κe paid by parents.
31
Her parents will also cover tuitions (1 − θ)κ as well as living expenses (captured by
ζ) during their children’s college stage. If parents choose to have their children skip
the exam, or their children fail the test, then the children will immediately leave their
family and enter the labor market as non-college worker. I assume that the exam
choice is irreversible, and that the College Entrance Examination can be taken only
once.
I first lay out the dynamic problems solved by parents conditional on the exam
choice being and the exam result being. Let V sj (·|c) denote the value of an indi-
vidual with education level s if her child passes the exam and attends college. The
maximization problem is given by
V sj (hp, , a, hc|c) = max
c,a′
{
u(
c
1 + ζ
) + β
∑
′
pi(′|)V sj+1(hp, ′, a′)︸ ︷︷ ︸
parent′s value
+ νE′ [V c2 (hc, ′, 0)]︸ ︷︷ ︸
child′s value
}
(2.11)
subject to the budget constraint
c+ a′ + (1− θ)κ+ κe = (1 + r)a+ y − T (y)
y = wsη(hp, ), c ≥ 0, a′ ≥ 0
where ζ captures the fact that college students still live with their parents during the
college stage. The exam cost κe is in the parent’s budget constraint because children
must have passed the test before going to college. In additional, I assume that the
exam cost only gives children the knowledge that is specific to the test. Consequently,
children’s human capital is not affected by whether or not their parents pay the
upfront cost.
Notice that in the next period, the child will leave the family. As a result, her
human capital hc no longer appears in her parent’s state space in period 9. In addition,
the child’s initial value function V c2 enters the parent’s value function at period 8. The
superscript c indicates that the child is college labor, and the subscript j = 2 implies
that the child will not become independent until the next period since she will spend
one more period in college. In turn, the labor productivity shock ′, which is drawn
from an invariant distribution, will be realized in next period. The parameter ν
32
controls the strength of altruism.
Let V sj (·|nc) denotes the value of an individual with education level s if her child
does not attend college. The maximization problem is given by
V sj (hp, , a, hc|nc,1e) = max
c,a′
{
u(c) + β
∑
′
pi(′|)V sj+1(hp, ′, a′)︸ ︷︷ ︸
parent′s value
+ νE[V nc1 (hc, , 0)]︸ ︷︷ ︸
child′s value
}
(2.12)
subject to the budget constraint
c+ a′ + 1eκe = (1 + r)a+ y − T (y)
y = wsη(hp, ), c ≥ 0, a′ ≥ 0
where 1e denotes an indicator function taking the value one if the child takes the
College Entrance Examination.
The child’s initial value function V nc1 still enters the parent’s value function at
j = 8. The superscript nc indicates that the child is a non-college labor, and the
subscript j = 1 implies that the child will become independent and enter the labor
force immediately. As a result, the labor productivity shock  is realized in the same
period.
Next, I present a discrete choice problem in which parents decide whether to sup-
port their children to take the College Entrance Examination. A fraction of children
will take the test following their parents’ decision. Since public college has a limited
capacity, the government has to use a rationing scheme to select college students from
an admission pool. As discussed in Section 1.2 in Chapter 1, the admission system
does not perfectly sort students into college according to their test scores. This fea-
ture will be captured by a reduced-form admission policy χ(hc), which is a function
of children’s human capital hc.
For a parent with education level s, before the exam outcome is realized, the
ex ante value of supporting her child taking the exam V̂ sj is obtained by taking the
expectation over the value of passing the exam V sj (·|c) and the value of failing in the
33
exam V sj (·|nc), which is given by
V̂ sj (hp, , a, hc)︸ ︷︷ ︸
take exam
= χ(hc)V
s
j (hp, , a, hc|c)︸ ︷︷ ︸
success
+ (1− χ(hc))V sj (hp, , a, hc|nc, 1)︸ ︷︷ ︸
failure
(2.13)
where 1e = 1 indicates that her child took the College Entrance Examination but
ended up with a failure result.
Let V sj denote the value of an individual who can choose between have their
children skip the exam (enter the labor force), and have their children take the exam.
The discrete choice problem is given by
V sj (hp, , a, hc) = max
{
V sj (hp, , a, hc|nc, 0)︸ ︷︷ ︸
skip exam
, V̂ sj (hp, , a, hc)︸ ︷︷ ︸
take exam
}
(2.14)
where 1e = 0 indicates that the child skips the college entrance exam.
Note that by choosing to have their children skip the exam, parents can avoid
paying the upfront cost of the test. As a result, the value of skipping the exam is
strictly greater than the value of taking but failing the exam. However, every parent
who chooses to have her child take the exam (strictly) prefers to support her child
attending college than skipping the exam. Therefore, failing the exam will lead to a
deviation from the optimal decision path. To increase the probability of passing the
test, parents have incentives to invest in their children’s human capital. This channel
only exists in the environment where the college attainment is constrained by the
capacity of the college.
Problem at j = 9, 10, 11. Starting from period 9 all of the children leave their fam-
ily. Thus, before exiting the economy at period 11, parents only choose consumption
and savings. The dynamic problem will be the same as the that in periods 1 and 2.
2.2.8 Definition of Equilibrium
Let xj ∈ Xsj denote the vector of state variables of an individual in period j and com-
pleted education s. Let µsj be the corresponding measures over Borel sigma-algebras
defined using those state spaces. A stationary recursive competitive equilibrium for
34
this economy is a collection of (i) value functions {Vj(xj)} and V̂ s8 (x8); (ii) policy func-
tions for consumption and savings {cj(xj), a′j(xj)}, education expenditure {mj(xj)}
and exam choice {1e(x8)}; (iii) aggregate capital and labor inputs {K,Hc, Hnc}; (iv)
tax policy {τ, d} and admission policy χ; (v) measures for parents µ = {µsj}, such
that:
1. Given prices, the policy functions solve the dynamic programming problems
described in Subsection 2.2.7 and {Vj(xj)} and V̂ s8 (x8) are the associated value
functions;
2. Given prices, aggregate capital and labor inputs solve the representative firm’s
problem;
3. Labor market for each education group clears:
Hc =
11∑
j=2
∫
Xcj
jhc,jdµ
c
j
Hnc =
11∑
j=1
∫
Xncj
jhnc,jdµ
nc
j ,
where Hs denotes the aggregate effective labor supply of education level s.
4. Capital market clears:
K =
11∑
j=2
∫
Xcj
ja
′
j(xj)dµ
c
j +
11∑
j=1
∫
Xncj
ja
′
j(xj)dµ
nc
j ,
where the first term is the aggregate savings of college-educated parents, and
the second term is the aggregate savings of non-college-educated parents.
5. Good market clears;
6. The distribution of µ is stationary:
µ(x) =
∫
Γ(x)dµ(x),
35
where Γ(·) denote the aggregate law of motion of x = {xj}, which is derived
from the individuals’ policy functions.
2.3 Calibration
In this section I describe how I parametrize the model. Some parameters are cal-
ibrated externally or taken from other literature, others are jointly estimated from
the simulation of the model. To do so, I numerically solve the decision rules of in-
dividuals, approximate the distribution of this economy, and then adjust parameters
to minimize the distance between model moments and the data counterparts. Af-
ter calibrating the model, I check the performance of the model by comparing the
non-targeted model moments with the data.
2.3.1 Model Parameterization
The benchmark model is calibrated to the China data. The model period is four
years. Individuals enter the economy at period 1 [age 18], give birth to child at
period 4 [age 30], make college choice for their children at period 8 [age 46] and exit
from the economy at the end of period 11 [age 61].
Preference. I specify the period utility over parent and child consumption as
u(c) =
( c
1+1cζ
)1−σ
1− σ .
I choose a coefficient of relative risk aversion of σ = 1.50, and an adult consumption
equivalence scale ζ = 0.33 following the standard literature.9 Then the discount rate
β is chosen to generate an annual real interest rate of 4%, and a capital-output ratio
(in 1998) of 1.57, which is calculate by Bai, Hsieh, and Qian (2006). This requires an
annual discount rate β = β1/4 = 0.97. In addition, I choose ν to match that 29.6% of
9This paper implicitly assumes a two-parent and one-child family structure. The scales from
Organization for Economic Cooperation and Development (OECD) assign a value of 1 for the first
adult, and 0.5 for the subsequent adults. By setting ζ = 0.33, I assume a child is equivalent to 0.5
of the first adult in terms of consumption expenditure following Doesey, Li, and Yang (2019).
36
Table 2.1: Model Parameters
Parameters Calibrated outside the Model
Parameter Value
Risk aversion σ 1.50
Capital share in production Λ 0.41
Labor elasticity of substitution Ω 0.74
Skill-biased technology Ac 1.00
Depreciation rate of physical capital δ 0.11
Intermediation cost in financial sector ξ 0.12
Government subsidy on college tuition θ 0.78
Persistence of labor productivity shocks ρz 0.86
Variance of labor productivity shocks σ2z 0.06
Return to human capital λ 0.51
College admission policy χ(·) see text
Parameters Calibrated (Jointly) Inside the Model
Preference
Annual discount rate β 0.97
Altruism parameter ν 0.26
Aggregate production function
Share of skill labor Φ 0.41
Tuition and fees
Tuitions paid by parents κ 0.62
Upfront cost of test κe 0.04
Tax system
Labor income tax deductibles d 2.60
Labor tax rate τ¯y 0.22
Human capital formation technology
Total factor productivity of skill formation ψ 1.28
Self-productivity of child’s human capital α4, α5, α6, α7 see text
Complementarity parameter ρ4, ρ5, ρ6, ρ7 see text
Parent’s human capital share ω 0.18
37
children whose parents are college-educated attend four-year colleges in 1998, which
requires ν = 0.26. In addition, 1c takes the value of one in period 4 to 7 for all
parents, and in period 8 for parents whose children attend college.
Aggregate production function. I choose a capital share to be Λ = 0.41, fol-
lowing Bai and Qian (2010), which combines the China’s labor share computed with
GDP by income approach at provincial level into one series. The elasticity of substi-
tution between college and non-college labor efficiency units is set to be Ω = 0.74.10
At the initial steady state equilibrium, I normalize the skill-biased technology pa-
rameter Ac = 1. The depreciation of physical capital is set to be δ = 0.11, and the
intermediation cost is set to be ξ = 0.12 following Bai, Hsieh, and Qian (2006).11
The share parameter Φ is chosen to target the college wage premium of 0.50, which
requires Φ = 0.41.
Admission policy. My estimation strategy for college admission probability as-
sumes that test scores can precisely reflect the test takers’ human capital. Further-
more, it is the distance between a test taker’s human capital and the average human
capital of test takers that pins down her probability of admission.12 When I solve the
model and compute the stationary distribution, I use the method of linear interpola-
tion to approximate the admission probability off the known data points.
In order to approximate how exam passing probability depends on the human
capital of children (χ(hc)), I first normalize the raw scores by calculating their log
difference from the mean score of the exam-taking year. This exercise is motivated by
the consideration that the level of difficulty of the exam may vary across test-taking
years. With the normalization, the same adjusted score will correspond to the equal
probability of college admission in different years.
10The estimation strategy is following Katz and Murphy (1992). Related literature has estimated
the elasticity of substitution between college and non-college labor in China (e.g., Ge and Yang,
2014; Feng, 2019).
11The cost is chosen so that the net rate of return to capital is 16% in China since the long-run
real interest rate is 4%.
12As a result, before parents decide whether or not their children should take the test, they have
internalized the distribution of human capital since the distribution matters for the probability of
passing the test.
38
Next, I divide the scoring support into ten equal-length intervals and calculate the
mean score for each interval. For the samples in each range, I compute the fraction of
test takers who have a four-year college degree. I repeat this exercise for the test takers
who took the College Entrance Examination between 1989 and 1998, and between
2008 and 2012, respectively. I separately estimate two functions because the college
expansion policy was implemented in 1999. Therefore, I assume that the government
has adopted a new admission policy after college expansion.13 I provide more details
on sample selection and score adjustment in Appendix B.1.1.
Figure 2.2 plots the admission probability of passing the College Entrance Exam-
ination against the normalized test score in the college entrance exam.14 The figure
displays that with a test score below the mean score, one has very little chance to
pass the exam. Once the test score reaches the average, the probability of admission
sharply increases.15 This empirical observation suggests that the return to human
capital investment is increasing at a faster rate within the sensitive region, where the
admission chance grows more rapidly. Additionally, the college expansion policy gives
test takers whose score is 0.1 to 0.2 log point above the average a significantly higher
probability of attending college. However, the policy is unhelpful for the test takers
whose score is below the average score.
College tuition. The tuition paid by parents for their children’s college education
κ0 = (1 − θ)κ and the share of expenditure borne by the household 1 − θ in the
benchmark model is chosen to match the tuition to average household income ratio,
κ0
y
, and as a fraction of average per student costs of college education, κ
κ
.
To calculate the corresponding numbers from the data, I turn to the China Sta-
tistical Yearbook (CSY) and the China Statistical Yearbook of Education Fundings
(CSYEF). CSYEF reports that the average yearly tuition for a four-year college in
13I drop the observations on test takers who took the test between 1999 and 2007 because college
in China has been gradually expanding. Since I am interested in long-run outcomes of college
expansion, I need to ensure that the post-reform admission policy is estimated using the information
on recent test takers.
14Since the admission policy function displays a strong non-linearity, I cannot adopt a linear
probability model (Donovan and Herrington, 2019) to estimate the probability of passing the college
entrance exam.
15Column (1) of Table B.2 in Appendix B.2 shows the normalized human capital (test scores) and
the corresponding admission probability in the baseline economy.
39
Figure 2.2: College Entrance Examination Admission Policy Function
Note: Data source: CHIP2013. This figure plots the probability of passing the College
Entrance Examination against the normalized test score. A circle or diamond marker rep-
resents the admission probability among individuals in a scoring bin. Blue dashed line
and green dash-dot line reflects the admission policy function before (1989-1998) and after
(2008-2012) the implementation of the college expansion policy, respectively.
40
1998-2001 is RMB3,114. I assume that miscellaneous fees (textbook, supplies, room,
and transportation) tied to college is 30% of out-of-pocket tuition. According to CSY,
the average household income during this time span is RMB10,719. Thus, I have
κ0 × 1.30
y
=
3, 114× 1.30
10, 719
= 0.38
and the cost parameter κ = 0.62 is calibrated at the equilibrium of the benchmark
model.
Furthermore, CSYEF reports that the average yearly per student costs of college
education in 1998-2001 is RMB13,963. Therefore,
(1− θ) = κ0
κ
=
3, 114
13, 963
= 0.22,
which requires θ = 0.78.
Tax system. Labor income tax is the only tax in the model since it can distort
college attendance. The tax function is assumed to be
T (y) = τ¯ymax(y − d, 0),
which means there are two parameters to be calibrated. In practice, I choose the
income tax deductible d = 2.58 to capture the fact that the labor income tax only
applies on 15% of workers. Then I choose the income tax rate τ¯y = 22.4% to match
the total labor income tax revenue to GDP ratio (1.5%).
Labor earnings. I run a fixed effect regression to control the cohort-invariant test
takers’ characteristics to estimate the impact of human capital on labor income, which
is controlled by the parameter λ. I first filter out experience effects from the log wage
observations in the CHIP2013. Here I assume that any residual unobserved error
term is uncorrelated with human capital. Next following Hendricks and Schoellman
(2014), I estimate an OLS regression of log individual wages on log test scores, a
41
school dummy, and a cohort dummy. The estimation equation is given by
log(Income) = λlog(Score) + School + Cohort+ 
where Income is individual’s labor income, Score is individual’s adjusted test score,
School is an indicator variable that takes a value of one if the individual is college-
educated, Cohort is the cohort dummy indicating the year of test, and λ captures the
return to human capital (skill gradient).16
Table 2.2: Estimate Return to Skill
Dep. variable: log-wages
Indep. variable
Test score
0.512∗∗∗
(0.083)
0.525∗∗∗
(0.083)
College education
0.228∗∗∗
(0.034)
0.225∗∗∗
(0.033)
Cohort effect Y N
Observations 2,089 2,089
R-squared 0.111 0.06
Note: Data source: CHIP2013. The omitted school category is non-college graduate. Stan-
dard errors are reported in parentheses. *,**,*** denotes statistical significance at 10, 5,
and 1 percent, respectively.
The estimating results are reported in Table 2.2. A one standard deviation rise in
human capital (i.e., test score) raises log-wages by 51%. As a result, I set λ = 0.51.
The AR(1) process has a yearly persistence of 0.86 and an yearly innovation
variance of 0.06, which is consistent with estimation of Yu and Zhu (2013), and
16Recent literature, such as Meghir, Abbott, Gallipoli, and Violante (forthcoming) and Daruich
(2018), has exploited a similar method by using the US Armed Forces Qualification Test (AFQT)
score to proxy individual’s skills. Different from this paper, they estimate the skill gradient separately
for individuals with different educational backgrounds. This strategy is motivated by the fact that
the return to skill is higher among college-educated individuals than among other groups. In other
words, the education level and human capital investment are complementary. I do not adopt their
approach because there is an apparent selection bias associated with the observations of test scores
in CHIP2013. Since the College Entrance Examination is a knowledge-based test requiring a lengthy
learning period, a large fraction of individuals with low human capital never takes the exam because
the passing probability for them is extremely low. As a result, the estimation of skill gradient for
non-college graduates will be biased since the sample does not include those who skip the exam.
42
I˙mrohorog˘lu and Zhao (2018). Since one period is four years in my model, I can
recover ρz = 0.55, and σz = 0.42. I follow Kopecky and Suen (2010) and use the
method of Rouwenhorst (1995a) to approximate this process with a discrete Markov
transition matrix.
Human capital formation. Following Cunha, Heckman, and Schennach (2010), I
assume that child’s next-period human capital depends on her current stock of human
capital, parent’s human capital, and parental investment. Also I assume that there
is a CES aggregator that combines current human capital of child and investment
made by her parent
X =
[
αjh
ρj
c + (1− αj)mρj
] 1
ρj
where αj indicates the share of self-productivity of child’s human capital, and ρj
represents elasticity of substitution at the developmental stage j.
Next, I assume X is combined with parent’s human capital to produce next period
child’s human capital according to a Cobb-Douglas technology
h′c = ψh
ω
pX
1−ω
where the anchor parameter ψ and share parameter ω are independent from child’s
age.
My estimation strategy can be explained as follows. The household-level data
available to me only provide detailed information about household education expen-
diture on their children. However, it does not allow me to track how parental invest-
ments are linked with children’s outcomes over a long period. Therefore, I discipline
the skill formation technology by connecting the parental characteristics, including
their income and education levels, to household education expenditure. Furthermore,
I choose the altruism parameter by connection children’s terminal education attain-
ments to their associated parents’ education levels.
I choose a set of αj to target the average education expenditure to household
income ratio in period j, which requires α4 = 0.60, α5 = 0.81, α6 = 0.91, and
α7 = 0.95. This estimation result reflects children’s current-period human capital is
43
more important as they age. In other words, it is more difficult for parents to raise
their children’s human capital using monetary investment at the later developmental
stages.
Since the UHS dataset does not record any information about children’s educa-
tion outcomes during their early childhood, it is impossible to adopt the approach
employed by Lee and Seshadri (2019), which recovers the dynamic complementarity
parameters by directly estimating the skill formation technology. Instead, I estimate
these parameters via indirect inference, where I design an auxiliary model to highlight
data patterns that are key for identification.
The auxiliary model is displayed in Subsection 1.4.3 in Chapter 1, where I estimate
the effects of household income on pre-college education expenditure for each devel-
opmental stage by running a set of fixed-effect regressions. To estimate the dynamic
complementarity parameters, I repeatedly simulate the life-cycle model and search
for ρj for four different stages respectively to ensure that the model-predicted effects
of household income on education expenditure can match their data counterparts.
Next, I explain the identification strategy in an intuitive way. The crucial argu-
ment is that the stage-dependent effects of household income on pre-college education
expenditure on children can inform me of the degrees of the complementarity between
investments in period j − 1 and j. Specifically, a more complementary relationship
between current-period children’s skills and education expenditure is associated with
a stronger effect of household income on monetary investments. In Appendix B.2,
I show precisely how this logic works in a two-period model before quantitatively
characterizing how model-predicted regression coefficients are sensitive to the choice
of ρj. Here I summarize the key reasoning steps.
First, with a more substitutable (complementary) relationship between the current
stock of children’s human capital and parental investment, the optimal investment
in human capital is lower (higher) when the household income is low. However,
as household income increases, the optimal human capital investment grows more
rapidly (slowly) in the substitutable (complementary) case. Therefore, if I find in the
developmental stage j that household education expenditure grows more significantly
with the household income, it implies that the skill formation technology at this stage
features a more substitutable (complementary) relationship between current-period
44
skill of children and monetary investment. Putting this intuition into practice, the
auxiliary regression will generate a larger (smaller) regression coefficient in the stage
where the current human capital of children is more substitutable (complementary)
to the parental investments.
Table 2.3: Key Moments: Model vs. China Data
Description China Data Model
Aggregates
Capital-output ratio 1.57 1.57
Share of exam taker 0.17 0.17
College wage premium 0.50 0.50
Tuitions to income ratio 0.38 0.38
Taxation
Frac. of labor income tax payer 0.15 0.12
Labor income tax revenue to GDP 0.02 0.02
Skill formation
Avg. human capital (exam taker) 1.00 1.00
Interg. corr. of college education 0.30 0.29
Edu. exp. to income ratio at j 0.06, 0.07, 0.09, 0.13 0.05, 0.07, 0.09, 0.13
Reg. coeff. log(mj) on log(yj) 0.74, 0.75, 0.68, 0.56 0.74, 075, 0.68, 0.57
Corr. coeff. between m and hp 0.17 0.17
I estimate that ρ4 = 0.17, ρ5 = −0.16, ρ6 = −0.66, and ρ7 = −1.19. The main
finding is that it is easier to shape children’s skill through money investment at the
early stage of the human capital development, because the elasticity of substitution
determined by ρj is larger the younger the children. Meanwhile, high early childhood
investment in human capital will induce a larger amount of investment at the late
developmental stages due to the complementary effect. This finding is consistent with
the estimation implemented by Cunha, Heckman, and Schennach (2010).
Parent’s human capital share. I choose ω = 0.18 to target the correlation coef-
ficient between the parent’s human capital17 and education investment corr(hp,m) =
0.17. With a high ω, high-skill parents can augment their children’s human capital
17Here, I use parent’s years of schooling to proxy the parent’s skill.
45
with low monetary investments, which makes the two inputs (hp and m) less cor-
related. Lastly, the anchor parameter is calibrated jointly with other parameters to
ensure that the average human capital (exam taker) is normalized to 1, which requires
ψ = 1.28.
Table 2.1 summarizes the parameters that I calibrate independently (top panel)
and those that are calibrated jointly in equilibrium (bottom panel) to match the
moments shown in Table 2.3.
2.3.2 Performance of the Benchmark Model
Before moving on to the education policy analysis, I discuss whether the parame-
terized model provides an acceptable description of the China economy along the
dimensions relevant for the current analysis.
Table 2.4: Non-targeted Moments
Data Model Source
Standard Deviation
Log-income 0.60 0.53 UHS
Log-consumption 0.61 0.42 UHS
Log-skill of exam takers 0.20 0.17 CHIP
Avg. Education Expenditure
Income tercile Q1 -0.47 -0.49 UHS
Income tercile Q2 -0.11 -0.05 UHS
Income tercile Q3 0.39 0.37 UHS
College-educated parents 0.33 0.40 UHS
Non-college-educated parents -0.07 -0.02 UHS
Note: All of the model moments are computed at the steady state. All the data counterparts
are computed using the moments obtained from UHS2002, except for the standard deviation
of skills of test taker, which is computed using test takers information from CHIP2013.
Inequality. Since one of the main focuses of this paper is inequality, I first check
if the estimated model can reproduce the qualitative features in those dimensions as
shown in data. Table 2.4 reports various non-targeted moments generated from the
baseline model.
46
First, the upper panel of Table 2.4 reports that the household income and con-
sumption inequality measured in the standard deviation of log earnings is 0.60 and
0.61 in the data, respectively. The model produces a slightly lower income inequal-
ity, but a significantly lower consumption inequality. As pointed out in Ding and
He (2018), one of the puzzling observations in Chinese micro-level data is that the
consumption inequality closely tracks with income inequality over time.
Second, Table 2.4 displays that the standard deviation of log human capital of test
takers (proxied by test scores) in China is 0.20. The model counterparts are slightly
lower but still reasonably close to the empirical values. The model generates a lower
dispersion in test scores because it does not capture that in the real world, a small
number of children with zero probability of admission still take the College Entrance
Examination by paying an upfront cost.18
Education expenditure. The next set of moments I need to check is how educa-
tion expenditure vary across parents in different income and education groups. The
lower panel Table 2.4 reports that households in the bottom income tercile spend 49%
log points less than the average on education expenditure, and the model counterpart
is 47%. It also reports that college-educated households spend 33% more than the
average, and the model counterpart is 40%. Therefore, the simulated model does
a good job of matching the variation of education expenditure across income and
education groups.
Human capital distribution. To see whether the simulated model can reproduce
the patterns of human capital distribution, I compare the model-generated distribu-
tion with the Chinese data. Table 2.5 reports the statistics reflecting the distribution
of log test scores, which is used as a proxy for human capital.
In the data, a test taker within the top 5% of the human capital (hc) distribution
(among test takers) should at least have a test score of 27% better than the average.
The model counterpart is 29%, which means model-generated human capital distri-
bution of the top percentiles is quite close to the data. In the data, test takers with a
18This puzzle can be reconciled if I incorporate some uncertainties to the test performance, which
means that the test score may not be perfectly correlated with children’s human capital. In that
case, some children with low human capital may set to gamble on the test.
47
Table 2.5: Human Capital (Test Scores) Distribution
Test Scores of Top X%
X: 50 20 10 5 1
China Data 0.04 0.17 0.23 0.27 0.38
Model -0.08 0.09 0.20 0.29 0.41
Note: Data source: CHIP2013. The test scores are normalized by taking their log differences
from the average score.
median score should have a human capital level of 4% better than the average. The
model counterpart is -8%. This implies that the required human capital for being a
top 50% test taker is below the data.
2.4 Policy Evaluations
With estimates of skill formation technology, the wage function, and the admission
policy, I turn now to the impact of various policy environments. By feeding the
existing college expansion policy into the calibrated model, I can compare the model-
generated long-run household behaviors with the short-run evidence shown in the
data. Furthermore, to better understand the mechanisms that contribute to the
effects of college expansion on macro variables, I quantify the impact of each channel
through a sequential decomposition. Finally, I propose a counterfactual policy and ask
the following question: Can government expenditure on education be implemented
in a more efficient way to improve social welfare and reduce inequality? To address
this question, I divert a part of the government subsidy for college tuition to support
early childhood development for disadvantaged children and compare the aggregate
and distributional implications with those of the existing policy.
2.4.1 Descriptions of the Policies
This subsection describes how I incorporate the existing policy and the counterfactual
policy in the calibrated model framework. Let P = {0, 1, 2} denote the policy intro-
duced, with P = 0 being the baseline economy, P = 1 the economy with the current
48
college expansion policy, and P = 2 the counterfactual economy with the childhood
education subsidy program.
The current college expansion policy. To evaluate the impacts of the current
reform and provide a benchmark for the counterfactual experiment, I analyze the
aggregate and distributional effects of the existing college expansion program (P = 1).
There are three main differences between the benchmark Chinese economy and the
current reform economy: (i) due to rapid income growth, the tuition-to-income ratio
declines dramatically; (ii) the government adopts a new admission policy function;
and (iii) a skill-biased technological change increases the demand for college-educated
workers.
To capture the decline in tuition-to-household-income ratio, I enter a new tuition
parameter, κ1 = 0.26. As the average household income goes up due to college
expansion, κ1 is endogenously pinned down to match the tuition-to-income ratio (0.14)
in 2015. I assume that the share of college costs borne by the government is fixed at
θ1 = θ = 0.78. The per student costs of college in the model therefore is
κ¯1
1− θ1 =
0.26
1− 0.78 = 1.19,
and the amount of government spendings on college education under the current
policy is
EX1 = 1.19θ1
∫
1e=1
χ1(hc)dµ8,
where 1e denotes an indicator function taking the value one if the child takes the
college entrance exam.
Furthermore, as shown in column (2) of Table B.2 in Appendix B.2, the gov-
ernment adopts a new admission policy function χ1(hc). Relative to the pre-reform
economy, the new policy primarily increases the admission opportunity for test takers
whose scores are above the average.
Finally, there is a skill-biased technological change that affects the demand side
of labor. Of course, it is difficult to predict the magnitude of technical change in
the long run. In this exercise, I choose Ac = 1.54 to ensure that the existing college
expansion policy generates the same college wage premium (0.50) in the final steady
49
state as in the baseline economy. Appendix B.3 reports the results from two robustness
experiments, in which the long-run outcomes are affected by two levels of skill-biased
technological changes.
The early childhood subsidy program. In this case, I analyze an economy in
which the government implements a targeted early childhood intervention (P = 2).
The aim of this experiment is to explore a more efficient way for the government to
implement education policies.
The subsidy targets children whose parents are in the bottom quartile of the dis-
tribution of income. I do not use the parent or child’s human capital as an eligibility
criterion, because in practice it could be the private information of individuals. How-
ever, since income level largely depends on parent human capital, I can still use this
rule to identify disadvantaged families. In addition, the program only targets chil-
dren at the first and second developmental stages (j = 4, 5). This is motivated by
the fact that educational intervention at early developmental stages is more effective,
since human capital investment is more complementary to children’s human capital
at the later stages. I assume that the government subsidizes 60% of the education ex-
penditure made by the parents of disadvantaged children. The subsidizing program
is financed by increasing college tuition. I also assume that the admission policy
function and technological change are identical to those in P = 1.
Let 1g denote an indicator function taking the value one if the parent is eligible
for the child education subsidy. The budget constraint of parents becomes
c+ a′ + 1g0.4m+ (1− 1g)m = (1 + r)a+ y − T (y).
I assume the government expenditure on college subsidy and early childhood sub-
sidy equals the total amount of college subsidy under the existing education policy.
To do so, I search for the out-of-pocket tuition κ¯2 that can balance the government
budget in this economy. Since the tuition is adjusted, the new share of college costs
borne by the government is
θ2 = 1− κ¯2
1.19
where 1.19 is the per student cost of college.
50
The government expenditure are given by EX2, where:
EX2 = 0.6
5∑
j=4
∫
1g=1
mj(xj)dµj + 1.19θ2
∫
1e=1
χ1(hc)dµ8,
where the first term aggregates the government spending on the early childhood de-
velopment subsidy, and the second term aggregate the subsidy for college education
with a new level of tuition in place.
In the steady state equilibrium, the new tuition is such that the government budget
is balanced:
EX2 = EX1,
which requires κ¯2 = 0.50.
2.4.2 Results
When evaluating the aggregate impact of policies, I am interested in the following
three outcomes in particular. First, how does human capital investment respond to
the policy change? Second, how do education outcomes, including the human capital
of children and the education level of children, change as the result of the policy
change? Third, to what extent can the welfare of households improve due to the
policy change? When analyzing the distributional effects of policies, my attention will
be devoted to the distribution of the increase in human capital, college attainment,
and welfare gains across children from different socioeconomic backgrounds.
The welfare changes at the individual level are expressed in terms of consumption-
equivalent variation, which allows me to quantify the gains and losses experienced by
different groups in the population. Specifically, for a parent whose human capital is
hp and education level is s, before any realization of income shocks, I compute the
expected lifetime welfare for her child, which is denoted by V s˜P (h˜p).
19 The conditional
welfare change of children in the economy P relative to the baseline economy can be
19Although V s˜P (h˜p) measures the ex-ante welfare of a child, the notation ∼ indicates that the state
variable hp and s reflect the human capital and education level of her parent.
51
expressed as:
∆W s˜P(h˜p) =
[
V s˜P (h˜p)
V s˜0 (h˜p)
] 1
1−σ
− 1. (2.15)
In this way, I can compute the heterogenous impacts of policy on the welfare of
children whose parents are different in their human capital hp and education level s.
Next, I also calculate the aggregate-level welfare changes under the veil of igno-
rance by assuming that the planner weights every agent in the stationary distribution
equally. The ex ante welfare changes in consumption-equivalent units are computed
as follows:
∆WP =
[∫
V sP (hp)dµ
s
P∫
V s0 (hp)dµ
s
0
] 1
1−σ
− 1. (2.16)
The social welfare in the economy P is obtained by integrating over the stationary
distribution of human capital and education levels of parents. The total social welfare
changes come from two sources: (i) changes in the expected welfare at each state and
(ii) changes in the distribution over different states due to policy changes.
Macro outcomes. In this subsection, I analyze the effects of the existing college
expansion policy and the counterfactual childhood development subsidy program. I
compare the aggregate outcomes in the stationary equilibria of these two economies
with those in the benchmark, and report the results in Table 2.6.
With respect to the effects of the current policy, column (2) of Table 2.6 shows
that aggregate quantities increase across the board. Starting with the most direct
effect of the reform, due to the expansion in college capacity and decline in the
tuition-to-income ratio, the fraction of test takers and college graduates increases
by 17.1 and 10.1 percentage points,20 respectively. Furthermore, the existing policy
motivates parents to spend 16.5% more on their children’s education. The 16.1% rise
in aggregate human capital is due partly to the rise in pre-college parent investments
20In the data, the fraction of test takers and college graduates increased by 45% and 21%, respec-
tively, in 2015, which implies that the existing college expansion policy and explain about half of the
rise in the exam-taking rate and college attendance rate from 1998 to 2015. Other factors, including
the changes in family structure, job-specific skill requirement, and college wage premium, can also
affect exam choices and college attainments. However, this model does not take those driving forces
into account.
52
Table 2.6: Aggregate Effects of Education Policies
(1) (2) (3)
Baseline
Existing Childhood
Policy Subsidy
Level Change Change
(a). Aggregate
Test taker share 17.19% 17.1pp 20.0pp
College share 4.73% 10.1pp 11.1pp
College wage premium 0.50 0.0% 5.5%
Edu. expenditure 0.14 16.5% 29.4%
Human capital, all 0.72 16.1% 30.0%
Human capital, test takers 1.00 13.0% 14.0%
Labor income 1.62 17.9% 25.3%
Output 0.60 29.0% 44.1%
Welfare -13.51 17.2% 26.1%
(b). Std. Deviation
Edu. expenditure 0.63 3.6p -7.4p
Human capital 0.28 5.2p -2.6p
Labor income 0.53 1.2p -0.4p
(c). Persistence
Human capital 0.80 5.3p -0.4p
College education 28.91% 25.0pp 18.5pp
Note: The table shows the baseline and simulated results related to the macroeconomic
variables. Column (1) corresponds to the level in the baseline economy. Column (2) and
(3) correspond to the (percentage point, percentage, or point) changes after implementing the
existing and counterfactual policy relative to the baseline economy. The welfare in entry (a)
is measured by consumption-equivalent units. All the variables in entry (b) are in log scale.
The persistence in human capital in entry (c) is measured by computing the correlation
coefficient between parents’ and children’s human capital (in log).
53
and partly to the parents’ increased human capital, which causes them to develop
their children’s skills more efficiently. Finally, the reform leads to a 29.0% increase in
output and a 17.2% improvement in ex ante welfare, which are primarily driven by
the higher aggregate human capital and labor income, respectively.
Meanwhile, because the education expenditure becomes more dispersed (3.6 log
points) than the baseline economy, the standard deviation in human capital and in-
come increases by 5.2 and 1.2 log points, respectively. In addition, the correlation
between parents’ and children’s human capital increases by 5.3 points. These results
suggest that although the existing college expansion program generates substantial
human capital and welfare gains, it also increases inequality and reduces intergener-
ational mobility across generations.
With respect to the effects induced by the childhood development subsidy pro-
gram, column (3) of Table 2.6 shows that the counterfactual reform leads to additional
gains in all of the aggregate quantities relative to the existing policy. In particular,
the rise in the share of test takers increases from 17.1 to 20.0 percentage points. This
change is the result of two opposing forces. First, college tuition increases by 92% to
finance the childhood subsidy program, which renders college less appealing for poor
parents. However, low-income parents, incentivized by the subsidy policy, increase
their education expenditure on children. For disadvantaged children, since both their
human capital and their exam-passing probability go up, their parents are more likely
to choose to have their children take the exam.
Moreover, compared with the existing college expansion policy, the childhood
subsidy program I proposed can also promote growth in aggregate human capital
(from 16.1% to 30.0%); output (from 29.0% to 44.1%); and ex ante welfare (from
17.2% to 26.1%). More importantly, as the subsidy policy exclusively targets human
capital investment in disadvantaged children, the standard deviation of education
expenditure drops by 7.4 log points. Since parents with low human capital and a
low education level are induced to spend more on their children’s education, the
dispersion in human capital and intergenerational correlation in human capital drops
by 2.5 and 0.4 log points, respectively. These results suggest that supporting the
childhood development of disadvantaged children can not only raise social welfare
but also mitigate inequality and improve social mobility in the long run.
54
The existing college expansion policy affects aggregate variables through four main
channels: (i) college becomes more affordable for parents, (ii) admission probability,
given a human capital level, is higher, (iii) a general equilibrium effect moves prices,
and (iv) skill-biased technological change affects the demand side. Although these four
channels interact with each other and cannot be perfectly disentangled, this exercise
aims to gain insight into their relative importance through a sequential decomposition.
Table 2.7 reports a decomposition of the four effects on the change in macro variables
with the current college expansion policy.
Table 2.7: Results Decomposition (with Existing Policy)
(1) (2) (3) (4)
Total Due Decline Rise in G.E. Skill-biased
chg. to: in tuition capacity effect tech. chg.
(a). Aggregate
Test takers share 17.1pp -4.2pp 22.2pp -17.9pp 17.0pp
College share 10.1pp 0.0pp 10.4pp -7.9pp 7.5pp
Wage premium 0.00% 18.5% -10.3% -31.3% 23.1%
Edu. expenditure 16.5% 3.3% 9.5% -13.1% 16.9%
Human capital 16.1% 3.1% 9.2% -12.6% 16.5%
Labor income 17.9% 1.2% 7.4% -7.7% 17.0%
Output 29.0% 3.4% 9.1% -12.2% 28.8%
Welfare 17.2% 0.8% 6.2% -5.6% 15.8%
(b). St. Dev.
Edu. expenditure 3.6p 1.5p 2.1p -4.0p 4.0p
Human capital 5.2p 2.8p 3.5p -7.5p 6.4p
Labor income 1.2p 0.5p 1.7p -2.6p 1.6p
(c). Persistence
Human capital 5.3p 3.1p 3.1p -7.9p 7.1p
College education 25.0pp 16.9pp 6.5pp -26.1pp 24.5pp
Note: This table presents the results from the decomposition exercise. Column (1) reports the
effects of a 22 percentage points drop in tuition-to-income ratio, fixing the college capacity,
equilibrium prices, and skill-biased technology. Column (2) reports the incremental effects
from column (1) when the college capacity constraint is relaxed, while still fixing the prices
and technology. Column (3) reports the incremental effects from column (2) when prices
are adjusted, while the technology remains unchanged. Column (4) reports the incremental
effects from column (3) when a calibrated skill-biased technological change is fed in to the
model.
55
Column (1) shows that a decline in the tuition-to-income ratio on its own (fixing
college capacity21, prices, and technology) would have little impact on most aggre-
gate variables. In particular, the share of test takers goes down by 4.2 percentage
points, even though college is significantly cheaper. The reason for this decrease can
be explained as follows: If the college education supplies elastically, more affordable
college can induce more parents to support their children in attending college. How-
ever, due to the college capacity constraint, an increase in demand for college only
makes college admission more competitive. A lower probability of admission, in turn,
leads to a lower return from exam-taking. As a result, more parents avoid paying the
upfront cost (κe) by choosing to have their children skip the test.
Furthermore, the modest rise in aggregate human capital is mainly driven by a
behavioral education expenditure response by wealthy parents. As the test becomes
more selective than the baseline economy, rich parents must increase their parental
investment to secure their children’s college attendance. Finally, the college wage
premium rises substantially by 18.5%. Under the partial equilibrium environment,
the relative skill price remains unchanged. Therefore, the increase in the college wage
premium is only accounted for by the increased in human capital sorting between
college- and non-college individuals.
Column (2), together with column (1), presents the aggregate effects of college
expansion with fixed prices and technology. The dramatic rise in the share of test
takers (22 percentage points) is not surprising, given the substantial relaxation of the
college capacity constraint. Moreover, a comparison of columns (1) and (2) clearly
shows that the rise in aggregate quantities—of human capital, labor income, and
welfare—is mainly due to the capacity expansion and not the lower tuition-to-income
ratio. In addition, these two shocks are equally important in explaining the rise in
inequality and intergenerational persistence in human capital.
Column (3) shows that if equilibrium prices are adjusted, most of the gains in
aggregate variables will be offset by the general equilibrium effect. As the college
wage premium falls by 31.3%, parents are disincentivized to invest in their children’s
21Here, I adopt a counterfactual admission policy function max(χ1(hc)−Ξ, 0), where Ξ = 0.066 is
endogenously determined to ensure that the share of college individuals is fixed. This is a reduced-
form way to mimic a more selective admission process when college is more affordable.
56
skill development and college education. Column (4) shows that with a skill-biased
technological change that perfectly counterbalances the three effects that affects the
college wage premium, the exam-taking rate and college attendance rate rise back
to the levels with prices fixed. Furthermore, aggregate education expenditure and
human capital are slightly higher than those under the partial equilibrium exercise.
Lastly, due to the rising demand for labor inputs, the increase in skill prices leads to
substantial gains in labor income and welfare.
Distributional effects. The aggregate statistics reveal that the current college
expansion policy can contribute to rising inequality, while the childhood subsidy pro-
gram can mitigate this effect. However, the channel through which the education
policies affect inequality and intergenerational mobility is not clear. To illustrate the
mechanism, in this subsection I show how changes in education investments and out-
comes vary across parents who are heterogeneous in their human capital and education
levels.
Figure 2.3 plots the simulated results related to parental investments and educa-
tion outcomes against percentiles of parents’ human capital. Specifically, Panels (A)
and (B) of Figure 2.3 plot the average lifetime pre-college education expenditure and
the average human capital of children, respectively, against percentiles of the distri-
bution of parent human capital. Panels (C) and (D) of Figure 2.3 plot the fraction of
the associated children who take the college entrance exam, and the fraction of associ-
ated children who pass the test, respectively, against percentiles of the distribution of
parent human capital. The solid blue line displays the results in the baseline economy,
the dashed green line displays the results under the current policy, and the dash-dot
black line displays the results under the childhood development subsidy program.
Panel (A) of Figure 2.3 reveals that the existing college expansion policy (dashed
green line) significantly raises the educational investment of parents whose human
capital is above the 50th percentile. In contrast, parents whose human capital is
below the median level only slightly increase their human capital investment on chil-
dren. Consequently, as shown in Panel (B) of Figure 2.3, the increments in children’s
human capital that result from the current policy increase with the human capital of
parents. Panel (C) of Figure 2.3 shows that since college is more affordable than in
57
Figure 2.3: Edu. Investment and Outcomes in Relation to Human Capital of Parents
Note: This figure presents how life-time education expenditure (Panel (A)), average hu-
man capital of children (Panel (B)), fraction of test takers (Panel (C), and fraction of
college-educated children (Panel (D)) depend on parents’ human capital. The horizontal
axis corresponds to the human capital percentiles of parents.
58
the baseline economy, most parents (except for the bottom 20%) are more likely to
choose to have their children take the exam. However, Panel (D) of Figure 2.3 reveals
that the education outcome measured by the probability of college admission varies
substantially across the human capital groups, with the children of low-skill parents
gaining the least. The uneven outcomes can be explained as follows: The children
of low-skill parents only have a slight gain in their human capital compared with the
baseline economy. As a result, their admission possibility is still slim. The children
of parents with high human capital, in contrast, take full advantage of the college
expansion, since both their human capital and admission probability dramatically
increases.
It is worth noting that the implications from the simulated model are consistent
with the empirical evidence documented in Subsection 1.4.4 in Chapter 1. The quan-
titative exercise, as stated above, illustrates the channel through which the college
expansion policy leads to heterogeneous impacts on children from different socioe-
conomic backgrounds. For disadvantaged children whose human capital (test score)
is distant from the average of test takers, as shown in Figure 2.2, additional invest-
ments in human capital do not spur an immediate rise in the probability of admission.
Furthermore, college expansion also does not change the situation.22 Consequently,
low-skill parents, facing the new admission policy function, are disincentivized to
spend more on their children’s education. In contrast, for high-skill parents, their
children’s human capital can reach the average level before the reform. College ex-
pansion rewards additional skill accumulation for children with above-average human
capital by increasing their admission probability.
In the economy with a childhood subsidy (dash-dot black line), as shown in Panel
(A) of Figure 2.3 , the key difference is at the bottom: Low-skill parents significantly
increase their education expenditure on children. As a result, as shown in Panel (B)
of Figure 2.3 , the increments in children’s human capital are more evenly distributed
across parents with different human capital. The college attendance rate of disad-
vantaged children, in turn, goes up relative to that in the economy with the current
22The key observation from the estimation of college admission policies in Subsection ?? is that
only test takers whose scores are above the average have a significantly higher probability of passing
the exam following college expansion.
59
policy in place. Meanwhile, Panel (C) of Figure 2.3 also shows that fewer children of
high-skill parents take the exam, which is a result of the higher college tuition.
Table 2.8 show how education outcomes and welfare changes are distributed across
children whose parents differ in their education levels. Column (1) corresponds to
the baseline implications of the model. Columns (2) and (3) present the change in
education outcomes and welfare due to introduction of the current college expansion
policy and the counterfactual childhood subsidy program.
Table 2.8: Distributional Implications of Policies
(1) (2) (3)
Baseline Existing Policy Child Subsidy
Parent with a
No Yes No Yes No Yes
college degree
Level Change Change
Child
Test taker share 15.16% 58.01% 11.9pp 18.4pp 16.2pp 10.3pp
College share 3.53% 28.91% 4.5pp 25.0pp 6.3pp 18.5pp
Human capital 0.70 1.10 8.5% 15.2% 25.0% 13.9%
Welfare -12.35 -11.76 2.8% 21.7% 13.4% 19.4%
Note: This table displays how children’s education outcomes and welfare depend on the
associated parents with different education levels. Column (1) corresponds to the initial
steady state. Column (2) and (3) corresponds to the percentage points or percentage changes
after implementing the existing and counterfactual policies relative to the baseline economy.
Column (2) of Table 2.8 shows that in the economy with the existing policy in
place, the children of college-educated parents increase their college attendance rates,
human capital, and welfare more than the children of non-college-educated parents.
Therefore, the policy gives rise to more persistent schooling across generations. Col-
umn (3) of Table 2.8 shows that the childhood subsidy program redistributes the
human capital and welfare gains away from children of college-educated parents to
children of non-college-educated parents. The reason for this is in line with the pre-
vious explanations, since college-educated parents are associated with high human
capital.
60
2.5 Conclusion
In this paper, I quantitatively investigate how China’s college expansion program im-
pacts parental investment and inequality in the long run. To this end, I present a
heterogeneous-agent overlapping-generations model in which altruistic parents invest
in their children’s education, and estimate the model using Chinese household-level
data. I use the model to examine the aggregate and distributional effects of the higher
education reform. The main finding is that the increase in college attainment, human
capital, and ex ante welfare are substantial but unevenly distributed, with disadvan-
taged children benefiting the least from the existing policy. The numerical exercise
reveals that high-skill parents’ education expenditure are more elastic to the policy
change following the college expansion. As a result, their children can accumulate
higher human capital, and hence are more likely to gain access to college, which leads
to a widening income gap and more persistent schooling across generations. Next, I
use the model to study the effects of an early childhood development program, which
subsidizes 60% of the education expenditure by the eligible parents. The additional
spending is financed by raising college tuition. I show that the remediation policy can
generate substantial welfare gains and a decline in inequality relative to the existing
policy, which suggests that government expenditure on education subsidies can be
implemented more efficiently.
61
Chapter 3
Understanding the Secular Decline
in New Business Creation
3.1 Introduction
A growing body of research documents a significant drop in the formation of new
businesses in the United States since the early 1980s. This decline in the creation of
new businesses is at the center of the decline in the overall dynamism experienced
by the U.S. economy in the last last three decades, because startups and young firms
contribute to job creation, productivity, and economic growth (Decker et al. (2014)).
Recent papers have sparked heated debate regarding reasons for the secular decline
in new business creation. A possible explanation is that the cost of starting a firm is
increasing as a result, for example, of the growth of occupational licensing (Kleiner and
Krueger (2013)) or weaker anti-trust enforcement, which would otherwise erect entry
barriers (Gutierrez and Philippon (2018)). Another possible reason is that shocks to
firms’ productivity have become more serially correlated over time than before, in
the sense that it is less likely that an unproductive firm will become a productive
one. This stems, for example, from better intellectual property protection (Akcigit
and Ates (2019b)), which makes it harder for less productive firms to use ideas from
highly productive, successful firms. Consequently, potential startups may not choose
to enter the market, since they believe it is less likely that they will succeed. These
62
two forces cannot be observed directly from the data; this requires for a framework,
assisted by a set of relevant moments, in order to infer them indirectly. Moreover,
the implications of these two forces can be vastly different. If the decline in new
business creation is mainly due to higher entry cost, it essentially reduces overall
welfare and total factor productivity (TFP). If, on the other hand, it is the fact that
shocks to firms’ productivity become more persistent over time accounts for the largest
part of the decline, overall welfare and TFP can be even higher than before. The
intuition is that more persistent shocks to productivity help undo the misallocation of
production factors in the presence of financial friction through entrepreneurs’ stronger
self-financing motives, as emphasized by Buera and Shin (2011) and Moll (2014).
The goal of this paper is to identify and quantify the relative contributions of
higher entry cost and higher persistence of shocks to the observed declines in new
business creation, relying on a set of moments regarding business dynamism. More
specifically, the two key moments we use are the dispersion of employment growth
rate and the relative size of entrants. Our intuition is that if the decline in firm entry
is due to a higher entry cost, entrants should become larger relative to incumbents.
Likewise, if the decline in firm entry is due to higher persistence of productivity
shocks, we should observe a lower dispersion of the employment growth rate; this
is because higher persistence means that large firms continue to be large and small
firms continue to be small. In the data, dispersion of the employment growth rate at
both firm level and establishment level has dropped dramatically over the last three
decades (Decker et al. (2016)), while the relative size of entrants remains fairly stable
and has even increased slightly since the early 2000s (Hopenhayn et al. (2019)). This
suggests that the decline in firm entry is likely to be driven by the two forces jointly.
To further test and examine these insights, we first develop a quantitative gen-
eral equilibrium model of entrepreneurship that features occupational choices. In
the model, an individual must decide in each period whether to be a worker or an
entrepreneur conditional on their assets, productivity in running a business (i.e., en-
trepreneurial productivity), and productivity in working for someone else. A new
business is created if an individual who was previously a worker chooses to become
an entrepreneur. If a worker becomes an entrepreneur, she must pay a fixed entry
cost. Each agent’s entrepreneurial productivity and working productivity are two
63
independent Markov processes disciplined by the data.
The two forces that contribute to the decline in new business creation we focus
on in this paper manifest as an increase in the fixed entry cost and an increase in the
persistence of entrepreneurial productivity shocks respectively. While it is straight-
forward that a higher fixed entry cost causes fewer workers to choose to become
entrepreneurs, it is not self-evident that the higher persistence of entrepreneurial pro-
ductivity shocks also leads to less entry. The reason is as follows. Given the level of
assets and working productivity, only agents with relatively high entrepreneurial pro-
ductivity choose to become an entrepreneur, so potential entrepreneurs, i.e., workers,
are those with low entrepreneurial productivity. With more persistent entrepreneurial
productivity shocks, potential entrants know that once they become entrepreneurs,
they are more likely to remain in a state of low productivity; therefore, they are
reluctant to enter the market.
We calibrate the model to the U.S. business sector under the assumption that it
was at the steady state in the early 1980s to match several key moments, especially the
entry rate, the employment share of entrants, dispersion of the employment growth
rate, and the relative size of entrants. Based on our baseline calibration, entrants
must pay an entry cost equal to their first year’s average profit. The entry cost is also
equal to around 79% of the average entrepreneurial income and six times that of the
average labor income.
We then use this calibrated model to quantitatively infer which force—higher
persistence of productivity shocks or higher entry cost—plays a more important role
in driving the observed declines in new business creation. To achieve this goal, we
perform our quantitative analysis in two steps. First, we increase the persistence of
entrepreneurial productivity shocks and the entry cost in the model to match the
decline in the employment share of entrants, 1 then check the movement of the other
key moments (e.g., the dispersion of employment growth rate) we use to discipline
model parameters. If one or more movements of the moments are not consistent
with their empirical counterpart, we can at least say that the factor that drives the
1Alternatively, we can increase the persistence of entrepreneurial productivity shocks and the
entry cost in the model to match the decline in entry rate of entrepreneurs, which makes no difference.
We choose to match the decline in the employment share of entrants due to technical reasons, which
will be explained in Section 3.5.
64
movement may not solely account for the decline in new business creation. Our finding
is that after increasing the persistence of entrepreneurial productivity shocks while
holding all other parameters fixed, dispersion of the employment growth rate declines
(consistent with the data) and the relative size of entrants also declines (inconsistent
with the data). If we increase the entry cost to match decline in the employment share
of entrants while holding all other parameters fixed, dispersion of the employment
growth rate increases (inconsistent with the data) and the relative size of entrants
also increases (inconsistent with the data). This implies that the rising persistence
of shocks or rising entry cost cannot account for all of the entire decline in the new
business creation.
Second, we recalibrate the persistence of the entrepreneurial productivity process
and the entry cost to match five moments during the 2010s simultaneously: the annual
entry rate of entrepreneurs, the employment share of entrants, the entrepreneurial
employment share, dispersion of the employment growth rate, and the relative size
of entrants. While increasing the persistence of entrepreneurial productivity shocks
and increasing the entry cost both contribute to the decline in entry rate as well
as the employment share of startups, they yield opposite predictions regarding the
other three moments. For example, higher persistence leads to a lower dispersion
of the employment growth rate, but higher entry cost generates higher dispersion.
Therefore, these five moments are balanced with each other, providing us with a new
estimation on the persistence of shocks and entry cost in the 2010s.
We find that the persistence of productivity shocks and the entry cost both increase
substantially. After decomposing the changes, we find that the relative contribution of
higher entry cost is more than 1.5 times that of the higher persistence of idiosyncratic
entrepreneurial productivity shocks to the decline in the entry rate of entrepreneurs,
and around twice the decline in the employment share of startups. Moreover, the
entry cost paid by start-ups in the 2010s becomes 1.15 times the average profit of
entrants, 96.3% of the average entrepreneurial income, and 8.8 times the average
labor income. This means that increases in the entry cost cause entrepreneurs to
pay 15% more in terms of their first year’s profit, 22% more in terms of the average
entrepreneurial income, and 33% more in terms of the average labor income to start
a business.
65
In terms of welfare and TFP, our quantitative results show that the increased entry
cost and persistence of shocks calibrated to the 2010s jointly lead to a 2.8% decline
in entrepreneurial TFP and a 2.0% decline in consumption-equivalent welfare. More-
over, higher entry cost alone reduces both entrepreneurial TFP and consumption-
equivalent welfare, while higher persistence of productivity shocks alone generates
a higher level of entrepreneurial TFP and consumption-equivalent welfare. These
results are consistent with those of Buera and Shin (2011) and Moll (2014), who con-
sider a model similar to ours (their model also features heterogeneous entrepreneurs
and an imperfect capital rental market). That is, sufficiently persistent shocks imply
that steady-state productivity losses and the welfare cost of market incompleteness
are relatively small.
Our results do not mean—and are far from implying—that the increased entry
barriers and higher persistence of idiosyncratic productivity shocks are the only two
possible drivers of the observed decline in the creation of new businesses. However,
our analysis, however, sheds light on the relative importance of the two factors in
contributing to the declining firm entry and employment share of entrants. Identifying
their relative importance is important, since although both higher entry cost and
higher persistence of shocks lead to observed declines in new business creation, they
have divergent impacts on TFP and welfare. Because our work suggests a more
important role played by higher entry cost—which reduces TFP and welfare—this
should be a concern for policy makers.
Finally, we study the implications of higher entry cost and higher persistence
of shocks on how firms respond to a negative credit shock that mimics a financial
crisis in transitional dynamics, based on our calibration results. Suppose there is a
credit shock that suddenly tightens the collateral constraints of entrepreneurs and
then recovers to the pre-crisis state. Our key finding is that given the path of credit
shocks, the set of parameters with higher entry cost and higher persistence of shocks
that are calibrated to the data from the 2010s generates a slower transition of the
stock of entrepreneurs to the pre-crisis level. This result provides insights on the slow
recovery from the Great Recession compared with the previous recessions. It is well-
known that both the Great Recession and the 1980-1982 recession (or “Double Dip
Recession”) were accompanied by large drops in the number of firms, but the recovery
66
in the number of firms has been much more sluggish since the Great Recession. The
insights based on our results are that the entry barrier in the late 2000s is much
larger than that in the early 1980s, which makes it harder for an entrepreneur who
previously exited the market due to a credit crunch to re-enter the market.
Related Literature. This paper is related to a growing literature on the causes and
consequences of the secular decline in firm creation and entrepreneurship observed in
the United States since the early 1980s. Several papers have documented a decrease
in the share of economic activity accounted for by small and new businesses in the
United States (see, for example, Haltiwanger et al. (2012); Decker et al. (2014); and
Pugsley and Sahin (2019), among others).
There is not a definitive explanation for the decline in new business creation, but
the literature has proposed several potential candidates. For example, Hopenhayn
et al. (2019) argue that the decline in the growth rate of the labor force participation
observed in the data is at the heart of the decline in the formation of new businesses.
Engbom (2019) examines how the aging of the U.S. population reduces individuals’
incentives to start new firms. Salgado (2018) shows that skill-biased technical changes
and the decrease in the cost of capital goods can account for a significant fraction of the
decline in entrepreneurship. Akcigit and Ates (2019a) quantitatively investigate which
force plays a dominant role in a group of candidates (i.e., lower effective corporate tax,
higher R&D subsidy, higher entry cost, and lower knowledge diffusion from frontier
firms and lagged ones) in explaining the decline in firm entry and the slowdown in the
overall dynamism (e.g. increased concentration) of the U.S. economy. They conclude
that the decline in knowledge diffusion is the most important driver, which may
explain the higher persistence of productivity shocks we emphasized in our paper.
Our work is complementary to two papers, by Buera and Shin (2011) and Moll
(2014), that feature models (i.e., a Bewley-Aiyagari-styled heterogeneous agent model
with production and an imperfect capital rental market) quite similar to the one we
study; they also focus on the persistence of idiosyncratic productivity shocks. Buera
and Shin (2011) show that the overall welfare cost of market incompleteness can be
increasing, decreasing, or even non-monotone in shock persistence, depending on the
relative strengths of its two components: the cost of a lack of insurance and the cost
67
of imperfect capital markets. The reason is that more persistent shocks are harder to
self-insure but simultaneously lead to better allocation of production factors through
entrepreneurs’ self-financing. Based on similar logic, Moll (2014) argues that the
persistence of idiosyncratic productivity shocks determines the size of steady-state
productivity losses and shows that more persistent shocks lead to smaller steady-
state losses but slower transitions. In our paper, we show that higher persistence
of shocks hinders new business creation in the steady state, and leads to a quicker
transition of firm entry and the employment share of entrants.
Finally, we join the literature that examines the reasons for lack of firm entry
and the slow recovery of the number of firms after the Great Recession. Clementi
et al. (2014); Khan and Thomas (2013); and Siemer (2014) develop a quantitative
heterogeneous firm model that features borrowing constraints and showing that lack
of entry is a consequence of credit crunches, since credit tightening directly affects
small and young firms the most. The results of our work indicate that the slow
recovery of the number of firms after the Great Recession may be due to a higher
entry cost compared with the level in the early 1980s.
The paper is organized as follows. Section 3.2 briefly discusses the data we use in
this paper. Section 3.3 describes the model set-up and defines a recursive competitive
equilibrium. Section 3.4 discusses the calibration. In Section 3.5, we report the
main results of this paper. Section 3.6 discusses the implications of the main results
reported in Section 3.5 on firms’ response to a credit shock that mimics a financial
crisis. Section 3.7 concludes the paper.
3.2 Data
As emphasized in Decker et al. (2014), measuring entrepreneurship and its economic
effects is difficult. There are two major strands of literature to follow. The first strand
of literature (e.g. Decker et al. (2014), Haltiwanger et al. (2012)) uses firm-level data
or establishment data to measure it and define entrepreneurs as a particular type of
firms based on their age and size (in terms of number of employees). Since available
government datasets on the U.S. firms do not have a specific entry for“entrepreneurs.”
but have traditionally contained information about the size and age of firms, some
68
observers have written or spoken as if small and young businesses are synonymous
with entrepreneurs. We also notice that there are several recent papers defining
entrepreneurship based on the legal form of the business organizations (e.g. Bhandari
and McGrattan (2019), Dyrda and Pugsley (2019)). For example, Dyrda and Pugsley
(2019) defines entrepreneurial income as the income from pass-through entities (i.e.
sole proprietorships, partnerships, and S corporate firms). The second strand of
literature (e.g. Quadrini (2000), Cagetti and DeNardi (2006)) uses household-level
data such as PSID and SCF and define entrepreneurs as a type of households based
on whether they own a business or are self-employed. 2
In this paper, we follow the first strand of literature on measuring entrepreneur-
ship. We define entrepreneurs as establishments with less than 20 employees using
data from the Business Dynamic Statistics (BDS). Our reasoning is as follows. Es-
tablishments with less than 20 employees are relatively small businesses. The reason
for choosing 20 employees as a cutoff is that based on Dyrda and Pugsley (2019),
the average size of establishment with legal form of sole proprietorship and partner-
ship is around 6 employees, and by choosing 20 employees as a cutoff for defining
entrepreneurship, we get the average size of entrepreneurs closer to 6.
The following are the definitions of the moments that we are going to use in
this paper. We compute the annual entry rate of entrepreneurs as the number of
establishment with age 0 and size smaller than 20 employees divided by the number
of establishments with size smaller than 20 employees. We compute the employment
share of startups as the total employment of establishments with age 0 and size smaller
than 20 employees divided by the total employment of all the employer establishments
in BDS for a specific year. We compute the entrepreneurial employment share as the
total employment of establishments with size smaller than 20 employees divided by
2Even among papers which use household-level data to define entrepreneurs, there is little con-
sensus about which households or individuals should be classified as such. For example, Evans and
Leighton (1989) considers as entrepreneurs those that are self-employed, Hurst and Lusardi (2004) all
those households that own a business, whereas Gentry and Hubbard (2004) defines as entrepreneurs
all those business owners with businesses with a total market value of $5,000 or more. Quadrini
(2000) considers both, business owners and self-employed as entrepreneurs. Cagetti and DeNardi
(2006) define entrepreneurs as those self-employed business owners that have an active management
in the firm. Salgado (2018) thus refer to four classifications of entrepreneurs that encompass the
different alternatives considered in the literature.
69
the total employment of all the employer establishments in BDS for a specific year.
The average size of entrant entrepreneurs relative to the incumbent entrepreneurs
is computed as the average size of establishments with age 0 and size smaller than
20 employees divided by the average size of establishments with size smaller than
20 employees for a specific year. Finally, the dispersion of employment growth rate
is computed as the standard deviation of the employment growth rate for all the
continuing firms (i.e. (li,t−li,t−1)/[0.5×(li,t+li,t−1)] where lit denotes the employment
of firm i in year t). Since we do not have firm-level or establishment-level census
data, we draw this moment from Decker et al. (2016) that uses Longitudinal Business
Dynamics (LBD) data to compute the standard deviation of employment growth rate
for all the employer establishments for the years 1980-2014. 3
3.3 Model
We consider a model of entrepreneurship based on Buera et al. (2011) and Buera and
Shin (2013) but augmented with an entry cost and a corporate production sector as
in Quadrini (2000).
We model an economy populated by a continuum of individuals, who are het-
erogenous with respect to their wealth (or assets), previous occupational status (so
that new entrepreneurs are different from incumbents), entrepreneurial productivity,
and working productivity. In each period, an individual chooses whether to work
for a wage or to operate an individual-specific technology (entrepreneurship). One
entrepreneur can operate only one production unit (establishment) in a given pe-
riod. Entrepreneurial ideas are inalienable, and there is no market for managers or
entrepreneurial talent to be traded. If a previously worker becomes an entrepreneur,
she needs to pay a fixed entry cost.
Output is produced by both entrepreneurs and a representative corporate firm. A
zero-profit financial intermediary borrows from households with positive savings to
3We admit that there is discrepancy between this moment and the other moments used in this
paper. The ideal moment should be the standard deviation of employment growth rate for all the
continuing employer establishments with size smaller than 20 employees. However, since we do not
have LBD data at hand, this is the best moment on dispersion of employment growth rate that we
can use for now.
70
supply productive capital for entrepreneurs and the corporate sector.
3.3.1 Environment
Heterogeneity and Demographics. Consider an economy with a continuum of
individuals of measure one. Individuals live infinitely and are heterogenous in their
asset at, previous occupational status dt−1, worker ability t, and entrepreneurial
productivity zt. Both the worker ability and the entrepreneurial productivity follows
a Markov process, and the two processes are independent to each other. There is no
population growth and no aggregate uncertainty either.
Preferences. Individual preferences are described by the following expected utility
function over sequences of consumption ct:
U(C) = E0
[ ∞∑
t=0
βtu(ct)
]
(3.1)
where β is the discount factor, which is smaller than one. The expectation is taken
over the realizations of the worker ability t and the entrepreneurial productivity zt.
We choose a period utility function that has a constant relative risk aversion. That
is, u(ct) =
c1−σt −1
1−σ .
Occupational Choice and Production Technology. At the beginning of each
period, after the realization of shocks, an individual chooses whether to operate his
own business or work for a wage (labor is indivisible). If the individual decides to
be a worker, she receives an income of wtt where t is an idiosyncratic, positively
autocorrelated shock, and wt is the market wage rate in period t. A worker cannot
borrow but can save in a risk-free asset, at, with return rt. If the individual chooses
to be an entrepreneur, she gains access to a productive technology that uses her own
entrepreneurial ability zt, capital kt, and labor lt, based on a decreasing-return-to-
scale technology:
ztf(kt, lt) = zt(k
α
t l
1−α
t )
γ (3.2)
71
where γ < 1 is the span-of-control parameter. A share γ of output goes to factor of
inputs. Out of this, a fraction of α is going to capital and 1− α going to labor.
In reality, a large fraction of firms are not managed by households weighing the
cost and benefit of running their own business or working in someone else’s company.
Therefore, as in Quadrini (2000) and Cagetti and DeNardi (2006), we model a second
sector of production populated by a large number of homogeneous firms which we refer
to as the non-entrepreneurial, or corporate sector. Firms in this sector are operating
a constant returns to scale production technology given by
AF (KC,t, LC,t) = AK
θ
C,tL
1−θ
C,t (3.3)
where A is the time-invariant corporate productivity, which will be normalized to
1, while KC,t, LC,t are corporate capital and labor demand respectively. Corporate
production does not involve fixed costs. Both sectors produce the same good, and in
both sectors capital depreciates at the same rate.
Financial Market. Productive capital is the only asset in the economy. There is
a perfectly competitive financial intermediary that receives deposits and rents out
capital to entrepreneurs. The return on deposited assets, i.e. the interest rate in the
economy, is rt. The zero-profit condition of the intermediary implies that the rental
price of capital is rt + δ, where δ is the depreciation rate.
3.3.2 Stationary Equilibrium
Recursive Problem of Individuals. At the beginning of the period, each in-
dividual is characterized by her asset a, working productivity , and entrepreneurial
productivity zt, previous occupational status d−, where d− = 0 identifies a worker and
d− = 1 an entrepreneur. Then, an individual solves the occupation choice problem
given by
v(a, , z, d−) = max
d∈{0,1}
{
(1− d)vW (a, , z, d−) + dvE(a, , z, d−)
}
, (3.4)
72
where vW is the value of being a worker, vE is the value of being an entrepreneur,
and d denotes an individual’s current-period occupation.
The problem solved by current-period workers is given by
vW (a, , z, d−) = max
c,a′≥0
{
u(c) + β
∑
′,z′
P (′, z′|, z)v(a′, ′, z′, d = 0)
}
, (3.5)
subject to
c+ a′ = (1 + r)a+ w,
and also to the laws of motion of t and zt, and the law of motion of the distribution
of individuals over idiosyncratic states. Additionally, P (·|·) is transition probabilities,
v is continuation value, and d = 0 indicates that the individual will enter next period
as a worker before choosing a new occupation.
The problem solved by current-period entrepreneurs is given by
vE(a, , z, d−) = max
c,a′≥0
{
u(c) + β
∑
′,z′
P (′, z′|, z)v(a′, ′, z′, d = 1)
}
, (3.6)
subject to
c+ a′ + I(d− = 0)κ = (1 + r)a+ pi(a, z),
and also to the laws of motion of t and zt, and the law of motion of the distribution
of individuals over idiosyncratic states. Additionally, P (·|·) is transition probabilities,
and v is continuation value, and d = 1 indicates that the individual will enter next
period as an entrepreneur before choosing a new occupation. The profit function is
given by
pi(a, z) = max
0≤k≤λa,l≥0
{zf(k, l)− wl − (r + δ)k} .
Here, I(d− = 0) is an indicator function which is equal to 1 if the individual was
a worker in the previous period, i.e. d− = 0, and is equal to zero otherwise. This
function captures the assumption that the fixed cost of creating a firm is paid only
by those individuals transitioning from a worker to an entrepreneur.
Note that we focus on within-period borrowing, or capital rental for production
purposes. We do not allow borrowing for inter-temporal consumption smoothing,
73
which translates into a′ ≥ 0. Several papers have documented the importance of
borrowing constraints to the decision to become an entrepreneur.4. Here, we assume
that entrepreneurs’ capital rental k is limited by a multiple of the collateral, i.e.
k ≤ λa.5
Problem of Corporate Sector. The problem of the corporate (non-entrepreneurial)
sector is simple and is given by
piC = max
KC ,LC≥0
{AF (KC , LC)− wLC − (r + δ)KC} . (3.7)
Definition of Equilibrium. A stationary recursive competitive equilibrium is value
functions v, vE , and vW ; individual’s policy functions c, a′, and d; entrepreneur’s
factor demand k and l; corporate sector’s factor demand KC and LC ; prices r and w;
and a distribution (µ) over individual wealth (a), working ability (), entrepreneurial
ability (z), previous occupation status (d−) such that
1. given prices, the policy functions—namely, c, a′, d, k, l—solve dynamic pro-
gramming problems associated with value functions v, vE, vW ;
2. given prices, corporate sector’s factor demand—namely, Kc and Lc—solve cor-
porate firm’s optimization problem;
3. the asset market clears∫
a′(a, , z, d−)dµ = KC +
∫
d(a,,z,d−)=1
k(a, , z, d−)dµ; (3.8)
4See, for instance, Evans and Jovanovic (1989), Quadrini (2000), Hurst and Lusardi (2004), or
Cagetti and DeNardi (2006).
5Alternatively, we can have entrepreneurs own capital k and face a constraint on borrowing
leverage, i.e. b′ ≤ λ−1λ k′ . Defining b = k − a, with the understanding that b < 0 denotes savings,
and assuming that λ capital and debt (k, b) are chosen after the realizations of idiosyncratic shocks
(, z) give us an equivalent problem to the one specified in our paper. This assumption that producer
profits are a function solely of its net worth or wealth at, not of capital k and debt b in isolation,
helps to reduce the dimension of the problem and simplifies the computation, which is widely used
in the literature (e.g. Gavazza et al. (2018); Midrigan and Xu (2014)).
74
4. the labor market clears∫
d(a,,z,d−)=0
dµ = LC +
∫
d(a,,z,d−)=1
l(a, , z, d−)dµ; (3.9)
5. the distribution of individuals over states (a, , z, d−) are invariant, i.e.,
µ(a, , z, d−) = Ψ(µ(a, , z, d−)), (3.10)
where Ψ depends on optimal policy of a′ and d as well as the law of motion of
 and z.
3.4 Calibration
We numerically solve the model by using non-linear methods, and find a stationary
equilibrium where individual decisions are consistent with market clearing prices.6
We begin with the subset of parameters calibrated externally, and then consider those
estimated within the model. Calibrated parameters are chosen such that the model
resembles the U.S. economy around early 1980s when the U.S. business dynamism
starts to decline. Thus, data moments are averages over 1980-1985 unless otherwise
specified.
3.4.1 Externally Calibrated Parameters
To maintain the tractability of the calibration, we take some parameters directly from
the literature. The time period in this model is equal to a year. We take a standard
value of 1.5 for the coefficient of risk aversion of the households’ utility function and
the span of control parameter of 0.79 following Buera et al. (2011). The capital share
parameter of corporate firms’ production function is set to be 0.35 matching the labor
income share of corporate sector in early 1980s. For simplicity, we make the value
of the capital share parameter of entrepreneurs equal to that of the corporate sector.
Taking the scale of production γ into consideration leads to a capital share αγ = 0.28,
6See Appendix for computation details.
75
which is close to the value used in the literature (e.g. Buera et al. (2011), Cagetti and
DeNardi (2006)). The capital depreciation rate is set to be 6% based on the BEA
fixed asset tables taking both physical capital and BEA-measured intangible capital
(or IPP capital) into consideration.
The working productivity  is assumed to follow an autoregressive process with
normal innovations, log′ = ρlog + e with e ∼ N(0, σ). Since the main focus of
this paper is on the producers’ side, we take the parameters that govern the process of
working productivity, i.e. (ρ, σ) from Bhandari and McGrattan (2019), which is also
consistent with the estimated wage processes of Low et al. (2010) for U.S. households
in U.S. Census and the Survey of Income and Program Participation (SIPP). Table
3.1 summarizes these parameter values.
Table 3.1: Parameter values set externally
Parameter Value Source/Target
Curvature of utility function σ 1.50 BKS 2011
Entrepreneur capital share α 0.35 -
Entrepreneur scale of production γ 0.79 BKS 2011
Corporate capital share θ 0.35 Corp. labor income share
Persistence of  shocks ρ 0.70 Bhandari and McGrattan 2019
SD of  shocks σ 0.16 Bhandari and McGrattan 2019
Capital depreciation rate δ 0.06 BEA fixed asset tables
3.4.2 Internally Calibrated Parameters
The entrepreneurial productivity z is assumed to follow a discretized version of an au-
toregressive process with normal innovations, logz′ = ρzlogz + ez with ez ∼ N(0, σz).
In particular, we approximate this autoregressive process with a 9-state Markov
chain following the procedure of Rouwenhorst (1995b).7 We have six parameters
(β,A, ρz, σz, κ, λ) left to be calibrated within the model. Table 3.2 lists the results.
Even though every targeted moment is determined simultaneously by all parame-
7We choose the Rouwenhorst’s method over the Tauchen’s method (see Tauchen (1986)) since
Rouwenhorst method has better performance in the case of highly persistent shocks.
76
ters, in what follows we discuss each of them in relation to the parameter for which,
intuitively, that moment yields the most identification power.
Discount factor β is chosen such that the annual real interest rate is 4%. The pro-
ductivity of the representative corporate firm A is calibrated to match the aggregate
employment of all the entrepreneurs as a share of total employment (i.e. the sum
of employment of both corporate sector and entrepreneurial sector).8 We calibrate
the collateral parameter λ to match the aggregate debt-to-value-added ratio for the
non-corporate sector in the data.9
The rest three parameters (ρz, σz, κ) are calibrated jointly to match three impor-
tant moments that regards the dynamism of the U.S. economy in the early 1980s. The
first one is the annual entry rate of entrepreneurs. In the model, the moment in period
t is calculated as the number of entrants (i.e. entrepreneurs in period t who are work-
ers in period t−1) as a share of total number of entrepreneurs in period t. Increasing
the persistence of entrepreneurial productivity process ρz and increasing the fixed
entry cost κ respectively both leads to a reduction in the entry rate of entrepreneurs.
While it is straightforward that a higher κ renders less workers choose to become en-
trepreneurs, it is not self-evident that a higher ρz also leads to less entry. The reason
is as follows. Given the level of assets and working productivity, only agents with
relatively high entrepreneurial productivity choose to become an entrepreneur, so po-
tential entrepreneurs, i.e. workers, are those with low entrepreneurial productivity.
With more persistent entrepreneurial productivity shocks, potential entrants know
that once they become entrepreneurs they are more likely to remain low productivity,
so they choose not to enter the market. 10
8We calibrate A to match the entrepreneurial employment share since we assume the mean of
log of entrepreneurial productivity shocks z is zero. Alternatively, we can normalize A to one
and calibrate the mean of log z to target the entrepreneurial employment share, which makes no
difference.
9The data on debt is liability level of loans for non-financial non-corporate business
sector, obtained from Z.1 Financial Accounts of the U.S. of the Board of Governors of
the Federal Reserve System, retrieved from FRED, Federal Reserve Bank of St. Louis;
https://fred.stlouisfed.org/series/NNBLL.
10Due the general equilibrium (GE) effect, an increase in ρz leads to a higher equilibrium wage
(since with more persistent shocks, entrepreneurs have stronger motivation to do self-financing to get
rid of the collateral constraint so that they can choose capital and labor optimally, which increases
labor demand), and a higher wage means being a worker becomes more attractive, so one may wonder
whether it is mainly due to the GE effect or the reason specified above. Based on our computation,
77
The second moment is the standard deviation of employment growth rate for con-
tinuing establishments (excluding entrants and exiters). We draw this moment from
Decker et al. (2016). Decker et al. (2016) uses LBD data and calculates an establish-
ment i’s employment growth rate in year t as (li,t − li,t−1) / [0.5× (li,t + li,t−1)]. In the
model, we are consistent with this definition to compute the moment of the standard
deviation of employment growth rate for all the entrepreneurs. The third moment
is the average size of entrants relative to the average size of all the entrepreneurs in
the economy. Note that although we are not directly targeting the employment share
of entrants, which is another important moment regarding business dynamism and
features a secular decline in recent decades, since we are matching the entry rate, the
employment share of entrepreneurs, and the relative size of entrants, then the em-
ployment share of entrants are automatically matched.11 The employment share of
entrants in the model is equal to 0.021 which is very closer to its empirical counterpart
0.022.
Table 3.2: Parameter values calibrated internally
Parameter Value Moment Data Model
Discount factor β 0.943 Annual risk-free rate 0.040 0.040
Corporate productivity A 1.326 Entrep. empl. share 0.272 0.272
Persistence of z shocks ρz 0.915 Annual entry rate 0.125 0.125
SD of z shocks σz 0.236 SD of empl. growth 0.634 0.633
Entry cost κ 12.17 Rel. size of entrants 0.630 0.630
Collateral parameter λ 5.431 Debt to GDP (entrep.) 1.355 1.357
we find that the reason specified above is the dominant one. The GE effect only slightly strengthen
the results on the decline. Suppose we increase ρz from 0.915 to 0.940. With GE, the entry rate
declines from 12.5% to 10.76%, while without GE, i.e. fixing the wage, the entry rate declines from
12.5% to 10.89%.
11More specifically, the equation {entry rate} × {startups relative size} = {employment share of
startups} × {employment share of entrepreneurs} always holds. The reasons are as follows. The
LHS ≡ NsNe × lsle = LsLe = LsL /LeL ≡ RHS where Ns denotes the number of startups, Ne denotes the
number of entrepreneurs (so NsNe means the entry rate of entrepreneurs), ls denotes the average size
of startups, le denotes the average size of entrepreneurs (so
ls
le
means the relative size of entrants), Ls
denotes the employment of startups, and Ls denotes the employment of entrepreneurs (so
Ls
L and
Le
L
mean the employment share of startups and the employment share of entrepreneurs respectively).
This equation thus implies that if any three of the four moments are matched to the data perfectly,
the rest one will be automatically matched.
78
Increasing ρz while fixing the unconditional distribution of the idiosyncratic pro-
ductivity shocks reduces the standard deviation of employment growth rate as well as
the relative size of entrants. Increasing κ raises both the relative size of entrants and
also the standard deviation of employment growth rate. Thus, although an increase
in the persistence of entrepreneurial productivity process ρz and the entry cost κ both
contribute to a decline in entry rate, the other two moments, i.e. the dispersion of
employment growth rate and the relative size of startups, can help us identify the
parameters governing the productivity process and the entry cost parameter sepa-
rately. Based on our baseline calibration, we obtain a ρz of 0.915.
12 Moreover, we
find that the entry cost in early 1980s faced by firms equals the average income of
entrants, 78.9% of the average entrepreneurial income, and six times of the average
labor income.
3.5 Main Results
This section consists of three parts. In Subsection 3.5.1, we show that higher entry
cost and more persistent productivity shocks cannot solely account for the decline
in new business creation. In Subsection 3.5.2, we use the calibrated model from
Section 3.4 to measure changes in entry cost and the persistence of shocks from the
1980s to 2010s. We then identify and quantify how changes in entry cost and the
persistence of idiosyncratic productivity shocks contribute to the observed declines.
Moreover, we check the implications of changes in these two factors on the aggregate
productivity and welfare in Subsection 3.5.3 in terms of the creation of new businesses.
In Subsection 3.5.4, we analyze the robustness of our main results.
12Although the persistence of idiosyncratic productivity shocks is considered an important param-
eter, there is no consensus on the estimated value of it. The calibrated value obtained in this paper is
not far away from the existing literature that uses establishment-level or firm-level data to estimate
ρz. For example, Cooper and Haltiwanger (2006) estimate it to be 0.981. Lee and Mukoyama (2015)
estimate it be 0.843 or 0.956 depending on the specification used for estimation.
79
3.5.1 Understanding the Factors in Isolation
We first change the entry cost κ and the persistence of productivity shocks ρz (while
fixing the unconditional distribution of shocks—i.e., fixing σz/
√
1− ρ2z by adjusting
the standard deviation σz, as in Buera and Buera and Shin (2011) and Moll (2014))
to match the decreased employment share of startups in the data for the 2010s re-
spectively.13 We increase ρz from the baseline value 0.915 to 0.976 and κ from 12.17
to 26.10 in order to match the overall decline in the employment share of startups.
Table 3.3 summarizes the key results. It shows the observed change in each variable
and compares them with their model counterparts in each experiment.
Table 3.3: Qualitative experiment results
Data ρz = 0.976 κ = 26.1
(1) (2) (3)
Start End Start End Start End
Entry rate 0.125 0.077 0.125 0.063 0.125 0.091
Empl. share of startups 0.022 0.013 0.022 0.013 0.022 0.013
Entrep. empl. share 0.272 0.250 0.272 0.532 0.272 0.159
SD of empl. growth rate 0.634 0.535 0.634 0.330 0.634 0.666
Relative size of entrants 0.630 0.635 0.630 0.386 0.630 0.905
A few observations stand out. First, despite the fact that an increase in the
persistence of shocks leads to a lower entry rate of entrepreneurs and a lower standard
deviation of the employment growth rate, both of which are consistent with the data
qualitatively, changes in the entrepreneurial employment share and the relative size
of entrants are not consistent with the data. Second, the directions of change in the
entry rate and entrepreneurial employment share resulting from higher entry cost κ
are consistent with the data. However, the higher entry cost generates an increase in
13Alternatively, we can increase κ and ρz to match the decreased entry rate of entrepreneurs
respectively, but in that case, we find that we are not able to the increase in entry cost (column
3) to match the entire decline in firm entry. Even though we increase κ to a very large number,
e.g. ten times of the value of κ in the initial steady state, the entry rate is stable around 0.082. In
that case, only financially unconstrained agents with highest entrepreneurial productivity choose to
become or keep being an entrepreneur. However, if we keep increasing κ, the entry rate will jump
to zero from 0.082. We thus increase κ to match the decline in the employment share of startups in
the data, making it comparable for the case of increased persistence of shocks.
80
the dispersion of the employment growth rate, which decreases in the data, and an
increase in the relative size of entrants which is relatively stable in the data.
3.5.2 Identification and Decomposition
The results from the previous section imply that higher entry cost κ and higher
persistence of shocks ρz should jointly account for the decline in new business creation
(in terms of both the entry rate and employment share of entrants). Since we can
observe neither κ nor ρz directly from the data, we can only infer the changes in
the two factors from the relevant moments that may be the consequences of them.
The moments we use for calibration give us some clue to separately identify and
quantify how changes in these two factors contribute to the observed declines in
the creation of new businesses. We thus conduct the following numerical exercise.
We jointly recalibrate the persistence of idiosyncratic entrepreneurial productivity
shocks ρz while fixing the unconditional distribution of shocks and fixed entry cost
κ to match a set of moments in the data for the 2010s, including the (1) entry rate
of entrepreneurs, (2) employment share of entrants, (3) relative size of entrants, (4)
dispersion of the employment growth rate, and (5) entrepreneurial employment share.
That is, we estimate two parameters to target five moments to exploit the power of
“over-identification.”
Table 3.4: Entry Cost and Persistence of Shocks
Model (1980s) Model (2010s) Data (2010s)
Parameter
ρz 0.915 0.940 –
κ 12.17 17.79 –
Moment
(1) Entry rate 0.125 0.079 0.077
(2) Entrants empl. share 0.021 0.012 0.013
(3) Entrants relative size 0.630 0.610 0.630
(4) SD empl. growth rate 0.634 0.592 0.535
(5) Entrep. empl. share 0.272 0.250 0.250
Next, we discuss how we choose these moments strategically. The first two
81
moments—i.e., entry rate and entrants’ employment share—are main indicators of the
creation of new businesses. Given the goal of this paper, we must include these two
moments into the set for re-calibration. However, since higher persistence of shocks ρz
and higher entry cost κ both lead to a lower employment share of entrants and a lower
entry rate, these two moments provide no information on the relative contribution of
higher κ and higher ρz to the decline in new business creation. Therefore, we add
(3) relative size of entrants and (4) dispersion of the employment growth rate, which
are informative moments in identifying κ and ρz in the baseline. To further discipline
the change of κ and ρz, we add (5) entrepreneurial employment share, because higher
ρz and higher κ have opposite impacts on this moment.
14 More specifically, a higher
ρz leads to a higher entrepreneurial employment share, since it favors large, highly
productive entrepreneurs—but higher κ leads to a lower entrepreneurial employment
share, since it makes it harder for potential entrants to become entrepreneurs. After
jointly recalibrating ρz and σz to match the chosen moments, we get ρz = 0.940 and
κ = 17.79. The results are summarized in Table 3.4.
Table 3.5: Entry Cost in Real Terms
Entry cost to Model (1980s) Model (2010s) Change
Entrants avg. profit 1.00 1.15 15%
Avg. entrep. profit 0.79 0.96 22%
Avg. labor income 6.00 8.80 47%
With the newly estimated persistence of shocks ρz and fixed entry cost κ, we find
that the entry cost now becomes around 1.15 times the average income of entrants,
96.3% of the average entrepreneurial income, and around 8.8 times the average labor
income. In contrast, for ρz and κ calibrated to the economy in the early 1980s, the
entry cost is only 1 times the average income of entrants, 78.9% of the entrepreneurial
income and 6 times the average labor income. We summarize the results in Table 3.5.
This means that the entry cost not only increases in nominal value, but also causes
potential entrants in the 2010s pay more, in real terms, to start a business than in
14Note that in the baseline calibration, we discipline the corporate productivity parameter A to
match the entrepreneurial employment share, but in the case of re-calibration, we fix A at the
baseline level.
82
the 1980s.
We then use the recalibrated parameters ρz = 0.940 and κ = 17.79 to perform
the following decomposition in order to ascertain which force plays a relatively more
important role in generating the decline in new business creation in terms of entry rate
and the employment share of startups. We first fix the entry cost κ = 12.17, which
is the calibration value in the initial steady state that captures the economy in the
early 1980s. We then let the parameter for the persistence of shocks ρz equal 0.940,
which is the newly calibrated value and captures the economy in the 2010s. This gives
us results that can be used to compute the relative contribution of higher ρz to the
decline in new business creation. Likewise, we follow the same procedure to compute
the relative contribution of higher κ. More specifically, denoting a variable of interest
by X, its value at time t when both channels move by X2t , and its hypothetical value
when channel i is shut down by X2−it , we can express the contribution of the channel
i to the total deviation over the three decades as follows:
contributioni =
X22010s −X2−i2010s
X22010s −X21980s
We summarize the paper’s main results in Table 3.6. We find that the relative con-
tribution of higher κ is more than 1.5 times that of higher ρz to the decline in the
entry rate of entrepreneurs, and around twice the decline of the employment share of
startups.
Table 3.6: Relative Contributions
Higher κ Higher ρ Both
Entry rate −2.85p.p. (62.6%) −1.72p.p. (38.0%) −4.60p.p.
Entrants empl. share −0.57p.p. (70.2%) −0.28p.p. (39.4%) −0.94p.p.
Note: Percentage values in parentheses measure the share of the contribution from the
specific channel to the total model-generated deviation between 1980s and 2010s.
3.5.3 TFP and Welfare
In this section, we examine the changes in entrepreneurial total factor productivity
(TFP) and consumption-equivalent welfare induced by higher entry cost and more
83
persistent entrepreneurial productivity shocks15 and summarize the results in Table
3.7. Overall, compared with our baseline estimation, the increased entry cost and
more persistent shocks faced by entrepreneurs in the 2010s lead to a 2.80% decline
in entrepreneurial TFP and a 1.97% reduction in consumption-equivalent welfare.
After decomposing the two factors that contribute to the decline in new business cre-
ation, we can see that higher entry cost alone reduces both entrepreneurial TFP and
consumption-equivalent welfare, while higher persistence of productivity shocks alone
generates a higher level of entrepreneurial TFP and consumption-equivalent welfare.
This is because entrepreneurs can undo capital misallocation via self-financing: With
persistent shocks, self-financing is an effective substitute for well-functioning capital
rental markets in terms of allocating production factors, as emphasized by Buera and
Shin (2011) and Moll (2014).
Table 3.7: TFP and Welfare
Higher κ Higher ρ Both
Entrep. TFP −5.04% 3.45% −2.80%
Welfare −1.76% 0.44% −1.97%
Our results, whereby a higher entry cost and higher persistence of productivity
shocks impact welfare and TFP differently, have important implications. If the decline
in new business creation is mainly driven by higher persistence of shocks, this decline
can be good in terms of welfare and TFP. If, on the other hand, the higher entry cost
is a more dominant reason for the decline in new business creation, this should be a
concern, since an increase in entry cost worsens aggregate productivity and overall
welfare. Since our key finding of this paper suggests that higher entry cost plays a
more important role in explaining the decline in new business creation, policymakers
should take action to effectively reduce the entry barriers faced by potential startups.
15The TFP of the entrepreneurial sector is computed as the Solow residual:
TFP =
∫
d(a,,z,d−)=1
y (a, , z, d−) dµ[(∫
d(a,,z,d−)=1
lt (a, , z, d−) dµ
)α (∫
dt(a,,z,d−)=1
kt (a, , z, d−) dµ
)1−α]γ
84
3.5.4 Robustness
In this section, we analyze the robustness of our main finding regarding how we
obtain the new estimation on entry cost κ and persistence of shocks ρz reported
in subsection 3.5.2. Generally speaking, we obtain new estimation on κ and ρz to
target five moments specified in Table 3.4 by searching an optimizer to minimize
the distance between the moments generated from the model and their empirical
counterparts given a specified weight matrix. Specifically, the vector of parameters Ψ
is chosen to minimize the minimum-distance-estimator criterion function
f(Ψ) = (mdata −mmodel(Ψ))′W (mdata−mmodel(Ψ)), (3.11)
where mdata,mmodel are the vectors of moments in the data and model, and W is a
diagonal weighting matrix.16 We give each moment an equal weight so that these
moments are balanced to each other to give us new estimation on entry cost κ and
persistence of shocks ρz jointly.
Next, we re-do this procedure by assuming alternative weighting matrices and
objective functions for minimization. The first case we consider is to give entry rate
and employment share of entrants a weight of five and give the rest three moments
a weight of one.17 The second case we consider is to use an alternative objective
function that has been used by the existing literature (e.g. Acemoglu et al. (2018);
Akcigit and Ates (2019b)) for calibration,18 which is defined as
N∑
k=1
|mdata −mmodel (k) |
1
2
|mdata|+ 12 |mmodel(k)|
(3.12)
16Since we only have a relatively small number of parameters calibrated within the model, we use
a local search method rather than a combination of global stage and local search used in quantitative
labor literature when the number of parameters calibrated within the model is large.
17We would always keep the weight equal for the two moments (1) dispersion of employment growth
rate, and (2) entrants relative size. Since these two moments are major governors’ of persistence of
shocks and entry cost respectively, we do not want to give either of them more weight. Otherwise,
the leading result on the relative contribution in the decline of new business creation will be biased
towards either higher persistent of shocks or higher entry cost.
18Compared to the one used in our paper, i.e. equation (3.11), the objection function specified in
equation (3.12) prioritizes the moments that are easier to match but sacrifices moments harder to
match.
85
where k denotes each moments and N is the number of targets. In this case, we
give each moment an equal weight. The third case we consider is to still use the
objection function defined in equation (3.12) but give entry rate and employment
share of entrants a weight of five and give the rest three moments a weight of one.
Table 3.8: Robustness Check
Data (2010s) Baseline Case I Case II Case III
Parameter
ρz – 0.940 0.939 0.938 0.938
κ – 17.79 17.81 17.42 17.24
Moment
(1) Entry rate 0.077 0.079 0.079 0.081 0.082
(2) Entrants empl. share 0.013 0.012 0.012 0.012 0.013
(3) Entrants relative size 0.630 0.610 0.614 0.613 0.611
(4) SD empl. growth rate 0.535 0.592 0.594 0.597 0.597
(5) Entrep. empl. share 0.250 0.250 0.249 0.250 0.251
We report the results on the estimation of κ and ρz in alternative ways in Table
3.8. The case that gives moment (1) and (2) more weight is labelled as ”Case I”,
the case that uses the objective function in (3.12) and give each moments an equal
weight is labelled as ”Case II”, and the case that uses the objective function in (3.12)
and give moment (1) and (2) more weight is labelled as ”Case III”. We can see that
the new estimation on ρz and κ in both cases are very close to the one used in this
paper, labelled as ”Main” in Table 3.8. In any case, our conclusion that the relative
contribution of higher entry cost is roughly 1.5 to 2 times as large as that of higher
persistence of shocks in explaining the observed declines in new business creation will
not be changed.
3.6 Impact of Credit Shocks
Our results in Section 3.5 implies that higher entry cost plays a relatively more im-
portant role in explaining the decline in firm entry as well as the employment share
of entrants in the recent three decades. In this section, we want to use the two sets of
parameters --- persistence of entrepreneurial productivity shocks ρz and fixed entry
86
cost κ --- to study how changes in persistence of shocks and entry cost affect the entry
and the stock of entrepreneurs in response to a credit shock that mimics a financial
crisis.
We simulate the aggregate dynamics of the model tightening the collateral con-
straint λt that is calibrated to generate a decline in the ratio of debt to the value-
added, ∫
max {kt (a, , z, d−)− a, 0} dµt∫
yt (a, , z, d−) dµt
.
We reduce the value of λt for the first three periods such that the largest decline in
the debt-to-value-added ratio the model is able to generate is around 25 percentage
points. This is consistent with the magnitude of the largest decline in the debt-to-
value-added ratio for the non-corporate business sector in the U.S. economy during
the Great Recession. After the three periods of negative credit shocks, we assume that
λt goes back to its pre-crisis level immediately.
19 To be clear, the initial contraction in
λt is a completely unexpected event, but its deterministic path after the initial drop is
perfectly known. The first two panels of Figure 1 shows the path of the credit shocks
fed into the model and the resulting evolution of the ratio of debt to value-added for
different sets of persistence of shocks and entry cost (ρz, κ) The first set (ρz = 0.915,
κ = 12.2) is our baseline calibration that targets moments in the early 1980s. The
second set (ρz = 0.940, κ = 17.8) is our re-calibration results that targets moments
in 2010s.
Since the results from previous section suggest that (1) entrepreneurs in 2010s
face higher entry cost than firms in 1980s, and (2) entrepreneurs’ productivity shocks
become more persistent in 2010s than 1980s, we would like to know if these changes are
going to lead to different results on the aggregate dynamics of firms in response to the
credit shocks defined above. Thus, we compare the results on transitional dynamics of
entry rate of entrepreneurs, the stock of entrepreneurs, and several important macro
variables for the two sets of parameters on persistence of shocks ρz and entry cost κ.
19We define the path of credit shocks in this way to simulate a financial crisis because we want
to show that the recovery speed of firm entry and the stock of firms depend on the values of ρz and
κ. Alternatively, we can let λt gradually recover and eventually converge to its pre-crisis level as
in Buera et al. (2015). In that case, the slow recovery of firm entry and the stock of firms will be
mostly driven by the slow recovery of λt.
87
Figure 3.1: Transition Dynamics: Entrepreneurs’ Dynamics
 = 0.915  = 12.2 (1980s)
 = 0.940  = 17.8 (2010s)
88
The one that captures 1980s is represented by the blue solid line, and the one that
captures 2010s is represented by the green dashed line. Our key finding is reported
in the third panel of Figure 3.1. When entrepreneurs in the economy face higher
persistence of shocks and higher entry cost that captures the conditions in 2010s, the
number of entrepreneurs declines less but recovers slower. The behavior of entry rate
is less volatile in the case of 2010s, meaning it declines less but also overshoots less.
To explain these patterns more clearly, we check how changes in ρz and κ sepa-
rately affect the transitional dynamics of entry and the stock of entrepreneurs. We
find that when only κ increases to the level of 2010s, both the entry rate and the
number of entrepreneurs decline less but recover slower. The reason is that when en-
try cost is very high, it is harder for incumbent entrepreneurs to exit the market since
they know if they exit, they are harder to re-enter the market. When only ρz increases
to the level of 2010s, both the entry rate and the number of entrepreneurs decline
more but recover faster. With more persistent shocks, entrepreneurs have a stronger
motivation to do self-financing thus saving more. This makes interest rate decline less
in response to the negative credit shock. Consequently, entrepreneurs’ profits drop a
lot, while the decline in equilibrium wage due to negative credit shocks is relatively
modest. Therefore, the decline in the entry and the stock of entrepreneurs is larger
when persistence of shocks is higher. Likewise, when shocks are more persistent,
agents with high productivity shocks that are previously exit the market due to the
negative credit shock will enter the market immediately as credit condition goes back
to the pre-crisis level. This renders a quicker recovery. When both ρz and κ increase
to our estimated level that captures the condition faced by firms in 2010s, the pattern
is more similar to the one when only κ rises. This implies that the increase in the
fixed entry cost is the dominant force to explain the different patterns of transitional
dynamics of entry rate and the stock of entrepreneurs in response to a negative credit
shock that mimics a financial crisis.
The results above provide some insights on the comparison of 1980-1982 recession
(”Double-Dip Recession”) and the Great recession in terms of the number of firms.
We know that both the Double-Dip Recession and the Great Recession experience
a dramatic drop in the number of firms, but the recovery in the stock of firms from
the Great Recession is slower. Our results indicate that higher entry cost faced by
89
potential startups may be an important reason for the phenomena.
Figure 3.2: Transition Dynamics: Macro Variables
 = 0.915 
 = 0.940 
We also check the aggregate dynamics (under perfect foresight) of several impor-
tant macro variables including output, total wealth (capital stock), and total factor
productivity (TFP) driven by credit shocks given different values of persistence shocks
ρz and entry cost κ. We report the results in Figure 3.2. We can see that changes in
ρz and κ do not make a significant difference on the transitional dynamics of these
macro variables. The output produced by entrepreneurs declines less and recovers
slightly slower in the case of higher persistence of shocks and higher entry cost. So
do the aggregate output, total wealth, and TFP of the entrepreneurial sector. Our
results on the decline in aggregate output resulting from credit crunch are consistent
with Shourideh and Zetlin-Jones (2017) that casts doubts on the ability of credit
90
shocks to generate significant economic fluctuations.20 Since our model features a
corporate sector which is not subject to a credit constraint,21 after calibrating our
model to match the employment share of entrepreneurs and corporates as well as the
debt-to-value-added ratio for non-corporate sector, our model also generates a modest
decline in total output, as the second panel of Figure 3.2 shows.
3.7 Conclusion
In this paper, we propose a general equilibrium model of entrepreneurship to sepa-
rately identify and quantify how changes in entry cost and persistence of idiosyncratic
productivity shocks contribute to the observed declines in the creation of new busi-
nesses. We show that entrepreneurs face more persistent productivity shocks and
higher entry cost to start up a business in the 2010s than in the 1980s. We find that
the relative contribution of higher entry cost is 1.5 times larger than that of higher
persistence of entrepreneurial productivity shocks in accounting for the decline in the
entry rate of entrepreneurs, and twice as large in accounting for the decline in the
employment share of new entrants. Our results suggest that higher entry barriers
potentially play a more important role in explaining the secular decline in the new
business creation experienced by the U.S. economy since the early 1980s. We also
find that higher persistence of shocks and higher entry cost have different impacts,
respectively, on TFP and welfare, which implies the importance of differentiating the
two types of shocks that contribute to the decline in firm creation.
Given the above results, we study the implications of changes in entry cost and
persistence of entrepreneurial productivity shocks on the aggregate dynamics of entry
and the stock of entrepreneurs following a credit crunch. The key finding is that
with higher persistence of entrepreneurial productivity shocks and the higher entry
20Shourideh and Zetlin-Jones (2017) develops a general equilibrium model of heterogeneous firms
with borrowing collateral constraints, which are the same as the ones in our paper. The authors find
that when the model is calibrated to match the observed financing patterns that roughly 80\% of
investment by private firms is financed externally compared to 20% for public firms, a large negative
credit shock which generates the decline in aggregate debt-to-assets observed following the Great
Recession can only lead to roughly a 1% decline in the aggregate GDP.
21Our assumption that corporate firms are not borrowing constrained is consistent with the existing
literature (see, for example, Dyrda and Pugsley (2019); Shourideh and Zetlin-Jones (2017) .
91
cost that captures the conditions faced by entrepreneurs in the 2010s, the number of
entrepreneurs declines less but recovers more slowly. This sheds light on the more
sluggish recovery in terms of the number of firms during and after the Great Recession
compared with the 1980-1982 recession: The entry barrier in the late 2000s is much
larger than that in the early 1980s which makes it harder for an entrepreneur who
previously exited the market due to a credit crunch to re-enter it.
The findings of this paper also present a direction for not only future research but
also policy design. Our results show that both the entry cost to start a firm and the
persistence of productivity shocks are higher than before, which are quantitatively
powerful in explaining the decline in new business creation. In our framework, we
assume the two factors are independent of each other. However, it is likely that higher
entry cost and higher persistence of shocks are in fact linked. For example, with
more persistent productivity shocks, the highly productive incumbents will gain more
market power, which may erect entry barriers for potential startups. Therefore, future
research should be devoted to understanding not only the underlying reasons for both
higher entry cost and higher persistence of shocks, but also the potential connections
between them. In terms of policy, our results suggest that the appropriate response
in terms of encouraging new business creation and reviving business dynamism in
the U.S. economy should focus more on reducing entry barriers. We leave for future
research the design of policies aimed at effectively reducing entry barriers.
92
References
Acemoglu, D., U. Akcigit, H. Alp, N. Bloom, and W. Kerr (2018): “Inno-
vation, Reallocation, and Growth,” American Economic Review, 108, 3450–3491.
Agostinelli, F. and M. Wiswall (2016): “Estimating the Technology of Chil-
dren’s Skill Formation,” Working paper.
Aiyagari, S. R. (1994): “Uninsured Idiosyncratic Risk and Aggregate Saving,” The
Quarterly Journal of Economics, 109, 659–684.
Akcigit, U. and S. T. Ates (2019a): “Ten Facts on Declining Business Dynamism
and Lessons from Endogenous Growth Theory,” Working Paper.
——— (2019b): “What Happened to U.S. Business Dynamism?” Working Paper.
Bai, C.-E., C.-T. Hsieh, and Y. Qian (2006): “The Return to Capital in China,”
Brookings Papers on Economic Activity, 2006, 61–88.
Bai, C.-E. and Z. Qian (2010): “The Factor Income Distribution in China: 1978–
2007,” China Economic Review, 21, 650–670.
Becker, G. S. and N. Tomes (1979): “An Equilibrium Theory of the Distribution
of Income and Intergenerational Mobility,” Journal of Political Economy, 87, 1153–
1189.
Bewley, T. (1986): “Stationary Monetary Equilibrium with a Continuum of Inde-
pendently Fluctuating Consumers,” Contributions to Mathematical Economics in
Honor of Ge´rard Debreu, 79.
93
Bhandari, A. and E. R. McGrattan (2019): “Sweat Equity in U.S. Private
Business,” Working Paper.
Boha´cek, R. and M. Kapicka (2016): “Tertiary Educational Choices: the United
States versus Europe,” Working Paper.
Buera, F. J., R. N. Fattal-Jaef, and Y. Shin (2015): “Anatomy of a credit
crunch: From capital to labor markets,” Review of Economic Dynamics, 18, 101–
117.
Buera, F. J., J. P. Kaboski, and Y. Shin (2011): “Finance and Development:
A Tale of Two Sectors,” The American Economic Review, 101, 1964–2002.
Buera, F. J. and Y. Shin (2011): “Self-insurance vs. self-financing: A welfare
analysis of the persistence of shocks,” Journal of Economic Theory, 146, 845–862.
——— (2013): “Financial Frictions and the Persistence of History: A Quantitative
Exploration,” Journal of Political Economy, 121, 221–272.
Cagetti, M. and M. DeNardi (2006): “Entrepreneurship, Frictions, and Wealth,”
Journal of Political Economy, 114.
Caucutt, E. and L. J. Lochner (forthcoming): “Early and Late Human Capital
Investments, Borrowing Constraints, and the Family,”Journal of Political Economy.
Choukhmane, T., N. Coeurdacier, and K. Jin (2017): “The One-child Policy
and Household Savings,” Working Paper.
Clementi, G. L., A. Khan, B. Palazzo, and J. K. Thomas (2014): “Entry,
Exit and the Shape of Aggregate Fluctuations in a General Equilibrium Model with
Capital Heterogeneity,” Working Paper.
Cunha, F. (2013): “Investments in Children When Markets are Incomplete,” Work-
ing Paper.
Cunha, F., J. J. Heckman, and S. M. Schennach (2010): “Estimating the
Technology of Cognitive and Noncognitive Skill Formation,” Econometrica, 78, 883–
931.
94
Daruich, D. (2018): “The Macroeconomic Consequences of Early Childhood Devel-
opment Policies,” FRB St. Louis Working Paper.
Decker, R., J. Haltiwanger, R. Jarmin, and J. Miranda (2014): “The Role
of Entrepreneurship in US Job Creation and Economic Dynamism,” Journal of
Economic Perspectives, 28, 3–24.
——— (2016): “Where Has All The Skewness Gone? The Decline in High-Growth
(Young) Firms in The US,” European Economic Review, 86, 4–23.
Ding, H. and H. He (2018): “A Tale of Transition: An Empirical Analysis of
Economic Inequality in Urban China, 1986–2009,” Review of Economic Dynamics,
29, 106–137.
Doesey, M., W. Li, and F. Yang (2019): “Demographic Aging, Industrial Policy,
and Chinese Economic Growth,” Working Paper.
Donovan, K. and C. Herrington (2019): “Factors Affecting College Attainment
and Student Ability in the US Since 1900,” Review of Economic Dynamics, 31,
224–244.
Dyrda, S. and B. Pugsley (2019): “Taxes, Private Equity, and Evolution of
Income Inequality in the US,” Working Paper.
Engbom, N. (2019): “Firm andWorker Dynamics in an Aging Labor Market,”Federal
Reserve Bank of Minneapolis Working Paper 756.
Evans, D. S. and B. Jovanovic (1989): “An Estimated Model of Entrepreneurial
Choice under Liquidity Constraints,” Journal of Political Economy, 97, 808–827.
Evans, D. S. and L. S. Leighton (1989): “Some Empirical Aspects of En-
trepreneurship,” The American Economic Review, 79.
Feng, Y. (2019): “The Labor Market Impact of China’s Higher Education Expansion
Reform,” Ph.D. thesis, UCLA.
95
Garriga, C. and M. P. Keightley (2007): “A General Equilibrium Theory of
College with Education Subsidies, In-School Labor Supply, and Borrowing Con-
straints,” Working Paper.
Gavazza, A., S. Mongey, and G. Violante (2018): “Aggregate Recruiting
Intensity,” American Economic Review, 108, 2088–2127.
Ge, S. and D. Yang (2014): “Changes in China’s Wage Structure,” Journal of the
European Economic Association, 12, 300–336.
Gentry, W. M. and R. G. Hubbard (2004): “Entrepreneurship and household
saving,” Advances in Economic Analysis & Policy, 4.
Gutierrez, G. and T. Philippon (2018): “How EU markets became more com-
petitive than U.S. markets,” NBER Working Paper.
Haltiwanger, J., R. Jarmin, and J. Miranda (2012): “Where Have All the
Young Firms Gone?” Business Dynamics Statistics Briefing.
Hendricks, L. and T. Schoellman (2014): “Student Abilities During the Ex-
pansion of US Education,” Journal of Monetary Economics, 63, 19–36.
Hopenhayn, H., J. Neira, and R. Singhania (2019): “The Rise and Fall of
Labor Force Growth: Implications for Firm Demographics and Aggregate Trends,”
Working Paper.
Huggett, M. (1993): “The Risk-free Rate in Heterogeneous-agent Incomplete-
insurance Economies,” Journal of Economic Dynamics and Control, 17, 953–969.
Hurst, E. and A. Lusardi (2004): “Liquidity Constraints, Household Wealth, and
Entrepreneurship,” Journal of Political Economy, 112.
I˙mrohorog˘lu, A. and K. Zhao (2018): “The Chinese Saving Rate: Long-term
Care Risks, Family Insurance, and Demographics,”Journal of Monetary Economics,
96, 33–52.
96
Jia, R. and H. Li (2016): “Access to Elite Education, Wage Premium, and So-
cial Mobility: the Truth and Illusion of China’s College Entrance Exam,” Stanford
Center for International Development, Working Paper 577.
Judd, K. L., L. Maliar, S. Maliar, and I. Tsener (2017): “How to solve
dynamic stochastic models computing expectations just once,” Quantitative Eco-
nomics, 3.
Katz, L. F. and K. M. Murphy (1992): “Changes in Relative Wages, 1963–1987:
Supply and Demand Factors,” The Quarterly Journal of Economics, 107, 35–78.
Khan, A. and J. K. Thomas (2013): “Credit Shocks and Aggregate Fluctuations
in an Economy with Production Heterogeneity,” Journal of Political Economy, 121,
1055–1107.
Kindermann, F. (2012): “Welfare Effects of Privatizing Public Education When
Human Capital Investments Are Risky,” Journal of Human Capital, 6, 87–123.
Kleiner, M. and A. Krueger (2013): “Analyzing the Extent and Influence of
Occupational Licensing on the Labor Market,” Journal of Labor Economics, 31,
137–202.
Kopecky, K. A. and R. M. Suen (2010): “Finite State Markov-chain Approxima-
tions to Highly Persistent Processes,” Review of Economic Dynamics, 13, 701–714.
Krueger, D. and A. Ludwig (2016): “On the Optimal Provision of Social In-
surance: Progressive Taxation versus Education Subsidies in General Equilibrium,”
Journal of Monetary Economics, 77, 72–98.
Lee, S. and B. A. Malin (2013): “Education’s Role in China’s Structural Trans-
formation,” Journal of Development Economics, 101, 148–166.
Lee, S. Y. and A. Seshadri (2019): “On the Intergenerational Transmission of
Economic Status,” Journal of Political Economy, 127, 855–921.
Li, H., P. W. Liu, and J. Zhang (2012): “Estimating Returns to Education Using
Twins in Urban China,” Journal of Development Economics, 97, 494–504.
97
Li, S., J. Whalley, and C. Xing (2014): “China’s Higher Education Expansion
and Unemployment of College Graduates,” China Economic Review, 30, 567–582.
Low, H., C. Meghir, and L. Pistaferri (2010): “Wage Risk and Employment
Risk over the Life Cycle,” The American Economic Review, 100, 1432–67.
Ma, S. (2014): “China’s College Expansion Policy: Ability Selection and Labor
Market Effects,” Job Market Paper.
Meghir, C., B. Abbott, G. Gallipoli, and G. L. Violante (forthcoming):
“Education Policy and Intergenerational Transfers in Equilibrium,” Journal of Po-
litical Economy.
Meng, X. (2012): “Labor market outcomes and reforms in China,” Journal of Eco-
nomic Perspectives, 26, 75–102.
Midrigan, V. and D. Y. Xu (2014): “Finance and Misallocation: Evidence from
Plant-Level Data,” American Economic Review, 104, 422–581.
Miranda, M. J. and P. L. Fackler (2002): Applied Computational Economics
and Finance, MIT Press.
Moll, B. (2014): “Productivity Losses from Financial Frictions: Can Self-Financing
Undo Capital Misallocation?” American Economic Review, 104, 3186–3221.
Mongey, S. (2015): “Lecture notes for Quantitative Macroeconomics,” Tech. rep.,
New York University.
Moschini, E. (2019): “Child Care Subsidies with One- and Two-Parent Families,”
Job Market Paper.
Mullins, J. (2019): “Designing Cash Transfers in the Presence of Children’s Human
Capital Formation,” Working Paper.
Pugsley, B. and A. Sahin (2019): “Grown-up Business Cycles,” The Review of
Financial Studies,, 32, 1102–1147.
98
Quadrini, V. (2000): “Entrepreneurship, Saving, and Social Mobility,” Review of
Economic Dynamics, 3, 1–40.
Rouwenhorst, G. (1995a): “Asset Pricing Implications of Equilibrium Business
Cycle Models,” Chapter 10, Frontiers of Business Cycle Research, ed. by T. Cooley.
Rouwenhorst, K. (1995b): Asset Pricing Implications of Equilibrium Business Cy-
cle Models. In: Cooley, T.F. (Ed.), Frontiers of Business Cycle Research., Princeton
University Press.
Salgado, S. (2018): “Technical Change and Entrepreneurship,” Working Paper.
Shourideh, A. and Zetlin-Jones (2017): “External Financing and the Role of
Financial Frictions over the Business Cycle: Measurement and Theory,” Journal of
Monetary Economics, 92.
Siemer, M. (2014): “Firm Entry and Employment Dynamics During the Great
Recession,” Working Paper.
Song, Z., K. Storesletten, and F. Zilibotti (2011): “Growing Like China,”
American Economic Review, 101, 196–233.
Tauchen, G. (1986): “Finite state markov-chain approximations to univariate and
vector autoregressions,” Economics Letters, 20, 177–181.
Young, E. R. (2010): “Solving the Incomplete Markets Model with Aggregate
Uncertainty Using the Krusell-Smith Algorithm and Non-stochastic Simulations,”
Journal of Economic Dynamics and Control, 34, 36–41.
Yu, J. and G. Zhu (2013): “How Uncertain is Household Income in China,” Eco-
nomics Letters, 120, 74–78.
Zhang, J., Y. Zhao, A. Park, and X. Song (2005): “Economic Returns to
Schooling in Urban China, 1988 to 2001,” Journal of Comparative Economics, 33,
730–752.
99
Appendix A
Appendix to Chapter 1
A.1 Data
A.1.1 Education Expenditure
The household-level data on education expenditure are used to construct empirical
evidence on how parental investments are associated with parents’ characteristics,
with children’s developmental stages, and with household income in Section 1.4.1.
The data comes from 2002-2009 waves of the Urban Household Survey (UHS).
Sample selection. First, I classify household members into three categories: child
(age 0 to 23), young adult (age 24 to 59), and old adult (above age 59). Given the
assumption on the household structure in the model, I start with all the households
with one child and one or more young adults. Given that I study education expendi-
ture for childhood development, I restrict the sample to households with a child whose
age is between 2 and 17. Since parents invest in their children’s education until the
end of the high-school stage in the model environment, I drop the households with
an employed child.
100
A.1.2 College Wage Premium
I use three survey data to estimate China’s college wage premium. They are 2002-
2009 waves of the Urban Household Survey (UHS), 1999, 2002, 2007, 2008, 2013 waves
of the Chinese Household Income Project (CHIP), and 2005 wave of 1% Population
Survey.
Sample selection. First, I only keep all individuals who currently have a job and
earn positive wages. Next, I drop all individuals whose age is above 59. Then, I drop
all observations whose working experience is either negative or above 50 years. I also
drop all observations whose years of education do not match their education level.
Finally, following standard literature, I restrict the sample to individuals working for
wages, which means self-employed individuals or business owners are excluded from
the sample.
A.2 Additional Results
The empirical evidence in Subsection 1.4.4 displays that the changes in children’s
four-year college attendance rate following college expansion vary significantly across
the parents’ income and education groups. Here, I re-define college as three- and
four-year institutions. Since three-year colleges in China mainly accept students with
low test scores, as discussed in Section 1.2, the college admission will be considerably
less competitive under the new definition. As a result, whether or not children attend
college mainly reflects their college choices instead of mirroring children’s human
capital.
Figure A.1 shows that, in contrast to the previous results, the changes in children’s
college attainments vary modestly across income and education groups. In particular,
for non-college-educated parents, their children are 25 percentage points more likely
to earn a college degree (vs. a 17 percentage point rise for college-educated parents).
These patterns suggest that college expansion has universally increased the three-
and four-year college attendance rates for children with different socioeconomic back-
grounds. One of the key driving forces is the decline in tuition-to-income ratio, which
makes college more affordable for children from low-income families.
101
Figure A.1: College Attainment of Children by Parents’ Characteristics
Note: Data source: UHS (2002, 2009). College is defined as three- and four-year institu-
tions. Sample restricted to urban households with an only child.
Nevertheless, this result does not contradict my main empirical conclusion. Al-
though children with low-income and non-college-educated parents are more likely to
go to college after college expansion, due to their low human capital, they will find
it difficult to be admitted by four-year institutions, which are more prestigious and
lead to much higher education returns than three-year institutions. Therefore, it is
still interesting to study the unequal education outcomes across socioeconomic groups
owing to college expansion.
102
Appendix B
Appendix to Chapter 2
B.1 Data
B.1.1 Test Scores
To inspect to what extent the probability of passing the College Entrance Examination
depends on children’s skill and to what extent wages are associated with human cap-
ital, I construct a dataset derived from the 2013 Chinese Household Income Project
(CHIP2013) survey. This cross-sectional survey follows a nationally representative
sample of over 18,000 households, and 64,000 individuals living in urban and rural
areas of China. It collects detailed information on a range of economic and demo-
graphic indicators. In particular, one crucial feature of the survey is the availability
of test score (in the college entrance exam) data.
Sample selection. I start with all the individuals for whom I can observe test
scores. I only keep the individuals who took either social science-oriented or natural
science-oriented exams.1 I drop individuals who took the College Entrance Exam-
1Some test takers took the uni-category exam or took the special exam for students with arts
or sports talent. Due to the complexity of interpreting the observed scores, I drop them from the
sample.
103
ination in Jiangsu province,2 and drop individuals who took the test before 1989.3
I restrict the sample to individuals who at least complete junior high school before
taking the College Entrance Examination. Furthermore, I restrict the sample to indi-
viduals who took the College Entrance Examination between the ages of 16 and 22.
For the individuals who took the College Entrance Examination between 1989 and
1992, I convert their raw scores to scaled scores consistent with the current scoring
system.4 Finally, I drop all the individuals whose test score is below 100 or above
700.5
Descriptive statistics. Table B.1 displays summary statistics on variables that
are useful for the empirical analysis of estimating the admission policy function and
return to skill. The average exam scores are almost identical before and after the
college expansion despite the rapid changes in the human capital distribution of test
takers.6 This fact suggests that the same test score does not necessarily imply the
same skill if test takers took the exam in different years, which may lead to a biased
estimation of return to human capital.
Additionally, although the dataset records information on rural households, the
rural test takers who took the exam between 1989 and 1998 are underrepresented
in the sample. Consequently, the actual national admission rate of four-year college
before college expansion is substantially below the number (34%) shown in the table.
2The College Entrance Examination in Jiangsu province uses a different scoring system from other
provinces. As a result, it is difficult to convert the raw scores to the scaled scores for individuals
who took the test in Jiangsu province.
3China resumed the National Higher Education Entrance Examination in 1977. However, the
scoring and admission system in the early years can be different from the status quo. So for estimating
the admission policy function before the reform, I only keep the individuals who took the exam
between 1989 and 1998.
4In the current scoring system, the full score is 750. However, between 1989 and 1992, the full
score could be either 640 or 710 depending on the orientation of test takers.
5These scores are extremely rare in the college entrance exam. Individuals are likely to misreport
their test scores in this case.
6There are at least two driving forces. First, since the tuition-to-income ratio is declining, and
admission possibility is rising, more individuals with low human capital may decide to take the test.
This force will lower the average student quality in the national admission pool. Second, the same
situations can also incentivize more intergenerational human capital investment, which can improve
the average student quality in the admission pool.
104
Table B.1: Descriptive Statistics (College Entrance Exam)
Variable All years 1989-1998 2008-2012
Raw test score (out of 750)
Mean 469.35 474.73 471.70
Standard deviation 85.05 88.73 79.57
Percentage of college admission 40.80 34.14 43.75
Variable All years 1989-1998 1999-2008
Annual disposable income (in RMB)
All 43,771.00 49,846.26 40,403.39
College graduates 51,797.95 61,141.90 47,118.22
No-college graduates 37,635.54 42,145.84 34,939.09
Working experience 11.52 18.67 7.56
Note: Data source: CHIP2013. This table shows unweighted averages of selected
characteristics. College is defined as four-year institutions. The fraction of test takers who
enters the four-year college shown in table only reflects the situation in sample.
B.1.2 Admission Probability
This appendix shows the admission policy function that maps the human capital of
children hc onto the probability of passing the College Entrance Examination χ(hc).
Note that the human capital shown in Table B.2 has been normalized by its log
difference from the average human capital of test takers.
B.2 Identifying Dynamic Complementarity Param-
eters
B.2.1 Description of Environment
Consider a single-parent and single-child problem. In the morning, the parent whose
human capital is (hp) chooses her consumption (cp) and human capital investment
(education expenditure) on her child (m). Parent’s income depends on human capital
as follows
y = hγ
105
Table B.2: Estimation of Exam Admission Probability
(1) (2) (3)
Baseline Existing Policy Fixed Capacity
hc χ(hc) hc χ(hc) hc χ(hc)
-0.80 0.00 -0.72 0.00 -0.72 0.00
-0.64 0.00 -0.59 0.00 -0.59 0.00
-0.54 0.00 -0.47 0.02 -0.47 0.00
-0.39 0.03 -0.37 0.05 -0.37 0.01
-0.26 0.08 -0.26 0.12 -0.26 0.05
-0.15 0.15 -0.15 0.16 -0.15 0.07
-0.01 0.21 -0.04 0.34 -0.04 0.08
0.11 0.35 0.07 0.55 0.07 0.14
0.23 0.59 0.17 0.82 0.17 0.36
0.35 0.84 0.27 0.87 0.27 0.80
Note: Data source: CHIP2013. This table presents the estimated results of admission
policy function. The test scores are normalized by taking their (log) difference from the
the average test score. The probability of admission is estimated using the observations on
individuals’ test scores and education outcomes (whether or not they have eared a four-year
college degree). Column (1) corresponds to the estimation for the baseline economy, which
reflects the pre-reform admission policy. Only individuals who took the College Entrance
Examination between 1989 and 1998 are in the sample to obtain the policy function. Column
(2) corresponds to the estimation for the post-reform admission policy. Only the individuals
who took the College Entrance Examination between 2008 and 2012 are in the sample to
obtain the policy function. Column (3) displays a counterfactual admission policy function
when the tuition-to-income ratio declines but the capacity constraint of college is not relaxed.
106
where γ controls the return to human capital. So the budget constraint of the parent
reads as
cp +m = h
γ
p .
Her child is endowed with human capital (h1) in the morning. The following
technology combines the human capital endowment (h1), the parent’s human capital
(hp), and parental investments (m) to produce new human capital h2.
h2 = h
ω
p [αh
ρ
1 + (1− α)mρ]
1−ω
ρ ,
where ω controls the share of the parent’s human capital in skill production, α controls
the share of the child’s current human capital in producing the new human capital,
and σ controls the elasticity of substitution between the child’s endowment of human
capital and monetary investment.
In the evening, the child earns income depending on her human capital (h2). The
child will consume everything she has. So the budget constraint reads as
cc = h
γ
2 .
The altruistic parent is the one who makes consumption and investment decisions.
She cares about her child’s consumption in the evening. The preference of the parent
is given by
log(cp) + νlog(cc)
where ν captures the degrees of altruism.
B.2.2 Household Problem
The household solves the following problem based on the description in the previous
subsection
max
cp,m
log(cp) + νlog(cc)
s.t. cc = [αh
ρ
1 + (1− α)(yp − cp)ρ]
γ
ρ .
I solve this problem using the standard algorithm. The solution can be character-
107
ized by the following Euler equation
1
cp︸︷︷︸
marginal
return to cp
= νγ(1− α)(1− ω) (h
γ
p − cp)ρ−1
αhρ1 + (1− α)(hγp − cp)ρ︸ ︷︷ ︸
marginal
return to m
,
which implies that the parent at optimality should equalize the marginal return to
her own consumption, and marginal return to human capital investment in her child.
B.2.3 Analytical Solution
Cobb-Douglas ρ = 0. There Euler equation is simplified as
1
hγp −m = νγ(1− α)(1− ω)
1
m
.
Then, the relationship between parent’s income hγp and human capital investment
m can be written as
m =
νγ(1− α)(1− ω)
1 + νγ(1− α)(1− ω)︸ ︷︷ ︸
slope
hγp .
Perfect substitute ρ = 1. There Euler equation can be simplified as
1
hγp −m = νγ(1− α)(1− ω)
1
αh1 + (1− α)m.
Then, the relationship between the parent’s income hγp and human capital investment
m can be written as
m =
νγ(1− α)(1− ω)
1 + νγ(1− α)(1− ω)− α︸ ︷︷ ︸
slope
hγp −
αh1
1 + νγ(1− α)(1− ω)− α.
Link to calibration. It is clear that if the current child’s human capital is more sub-
stitutable to parental investments, the household income (hγp) will have a stronger ef-
108
fect on education expenditure (m). The magnitude of difference (in effect of income on
education expenditure) between the Cobb-Douglas world and the perfect-substitute
world is determined by children’s human capital share (α). As a result, when I es-
timate ρ, I should first pin down α and other parameters. Next, I can simulate my
model and internally search for the parameter ρ through matching model-predicted
auxiliary coefficients to the data counterparts.
B.2.4 Link to the Full Model
In the quantitative life-cycle model, decisions on education expenditure will be com-
plicated by the heterogeneity in the current-period human capital of children and fi-
nancial wealth. However, the following numerical exercise shows that the full model’s
implications on the effects of income on parental investments are consistent with those
implied by the static problem.
Figure B.1 plots the data- and model-predicted effects of household income on
education expenditure on children. I plot the coefficients for each childhood develop-
mental stage, respectively. The green dashed lines are obtained from the estimation
results (first column) as displayed in Subsection 1.4.3. The blue solids lines are ob-
tained from repeatedly simulating the model by varying the dynamic complementarity
parameters ρj.
7 It is clear that a more substitutable relationship between current-
period children’s human capital and parental investments implies a stronger effect of
household income on education expenditure. With this feature in place, I can search
for the four parameters that minimize the weighted distance between model-predicted
coefficients and their data counterparts.
B.3 Additional Results
In Subsection 2.4.2, I have examined the macroeconomic effects of the existing college
expansion policy. In this exercise, I choose the parameter controlling the skill-biased
technical change Ac = 1.54 to ensure that the current policy generates the same college
7To obtain this figure, for each panel, I only vary one parameter each time and fix other param-
eters. In my calibration exercise, this set of parameters are jointly determined.
109
Figure B.1: Calibration of Dynamic Complementarity Parameters
Note: Each graph is associated to the calibration of a dynamic complementarity parameter,
ρj, where j corresponds to a child development stage. The solid blue line shows how effect
of household income on education expenditure is sensitive to changes in ρj in the model,
and the dashed green line shows the associated regression coefficient obtained from the data.
The intersection points give the calibrated value of ρj.
110
wage premium (0.50) at the final steady state as the baseline economy. This appendix
repeats the current college reform with two robustness experiments that include: (i)
raising the technology parameter to Ac = 1.67; (ii) lowering the technology parameter
to Ac = 1.33, and reports the aggregate and distributional results in Table B.3 and
Figure B.2, respectively.
Table B.3: Robustness: Skill-Biased Technological Change
(1) (2) (3) (4)
Baseline
Current High Low
Policy SBTC SBTC
Level Change Change Change
(a). Aggregate
Test taker share 17.19% 17.1pp 10.8pp 13.1pp
College share 4.73% 10.1pp 7.6pp 8.2pp
College wage premium 0.50 0.0% 30.0% -15.0%
Edu. expenditure 0.14 16.5% 21.5% 8.4%
Human capital, all 0.72 16.1% 21.3% 8.1%
Human capital, test takers 1.00 13.0% 35.4% 2.0%
Labor income 1.62 17.9% 20.8% 10.7%
Output 0.60 29.0% 37.6% 15.3%
Welfare -13.51 17.2% 18.5% 10.8%
(b). Std. Deviation
Edu. expenditure 0.63 3.6p 5.9p 1.6p
Human capital 0.28 5.2p 9.9p 1.3p
Labor income 0.53 1.2p 2.7p 0.3p
(c). Persistence
Human capital 0.80 5.3p 8.9p 1.4p
College education 28.91% 25.0pp 37.7pp 14.7pp
Note: Column (1) displays the macroeconomic variables generated from the baseline estima-
tion. Column (2) shows the (percentage point, percentage, or point) changes owing to the
existing college expansion policy. The two robustness experiments are as follows: Column
(3) reports the results from an economy where a skill-biased technological growth leads to a
30% increase in the college wage premium; Column (4) reports the results from an economy
where a skill-biased technological growth leads to a 15% drop in the college wage premium.
I highlight two particularly interesting results. First, although the education ex-
penditure and labor income rise more significantly than the standard case with a
111
high-level skill-biased technological change, the share of test takers as well as college
graduates are lower than the economy with a standard technical progress. This result
comes from the fact that the average human capital of test takers increases by 35.4%
with a high-level growth in skill-biased technology (vs. 13.0% with a standard growth
in technology). As I discussed in Subsection 2.3.1, the admission probability is deter-
mined by the difference between an individual’s test score and the average score of
test takers. Consequently, as shown in Panel (C), Figure B.2, the children of low-skill
parents face more difficulties getting into the college since their skill levels are more
distant from the average level. Since it is costly to take the College Entrance Exami-
nation, this situation will discourage low-skill parents from having their children take
the test.
Second, the changes in inequality and intergenerational persistence depend on
the degrees of skill-biased improvement in technology, which can lead to different
levels of college wage premium. With a higher college wage premium, as shown
in Panel (A), Figure B.2, only high-skill parents substantially spend more on their
children’s education. For low-skill parents, although the higher college wage premium
incentivizes parental investments, the marginal cost for them to do so is large due to
their budget limits. As a result, as shown in Panel (B) and (D), only the children from
rich families have significant rises in human capital and college attendance rate, which
implies more unequal outcomes owing to college expansion with a more substantial
improvement in skill-biased technology.
112
Figure B.2: Robustness: Impact of Skill-Biased Technical Changes on Distribution
Note: This figure presents how life-time education expenditure (Panel (A)), average hu-
man capital of children (Panel (B)), fraction of test takers (Panel (C)), and fraction of
college-educated children (Panel (D)) depend on parents’ human capital. The horizontal
axis corresponds to the human capital percentiles of parents. The blue solid line shows the
results in the benchmark (Ac = 1.54) , the green dashed line shows the results under the ex-
isting college expansion policy with a high-level skill-biased technological change (Ac = 1.67),
and the blacked dash-dot line shows the results under the existing college expansion policy
with a low-level skill-biased technological change (Ac = 1.33).
113
Appendix C
Appendix to Chapter 3
C.1 Computational Algorithm
In this computational appendix we first lay out the solution method for the household
problem described in Section 3.3. Then, we discuss the algorithm for computing the
transition dynamics.
C.1.1 Household Problem
In this subsection, we first describe the general solution algorithm for finding steady-
state equilibrium. Next, we lay out an efficient method for solving the recursive
equilibria.
Computing stationary equilibrium. We have three nested fixed point problems.
First, we have to solve market clearing wage (w) and interest (r¯) rate. Second,
we have to compute an approximation of the stationary distributions µWand µE
over financial asset a, working productivity , and entrepreneurial talent z for each
occupation. Third, we have to compute a fixed point in (expected) continuation
values v¯1 (containing entry option) and v¯2 (containing exit option).1 The iteration is
then as follows:
1We provide detailed information on what function to approximate in Appendix C.1.1.
114
1. Guess an initial wage rate, w0, initial distributions, µ
W
0 and µ
E
0 , and initial (ex-
pected) value functions v10 and v
2
0. The initial interest rate can be obtained by
the following equation:
r0 =
θ(A(1− θ)) 1θ
1− θ w
θ−1
θ
0 − δ
2. In price iteration i, given prices wi and ri, the loop as follows
(a) In iteration k, solving the household problem requires finding the fixed
point in value functions. We apply the collocation method to approximate
the expected continuation values, and use golden-search approach to solve
saving decisions. For each idiosyncratic state, entry decision can be ob-
tained by comparing the value of entry with the value of being a worker
and exit decision can be solved by comparing the value of exit with the
value of being an entrepreneur.
(b) We implement both value function iteration and Broyden’s algorithm to
update a finite set of coefficients that can define the value functions. The
root-finding problem stops if the convergence criterion max{‖ v1k+1 − v1k ‖
, ‖ v2k+1 − v2k ‖} < ζ is satisfied. Otherwise, repeat step 2(a) with the
updated coefficients.
(c) Solve saving, entry, and exit decision rules on a finer grid. Create big
transition matrix2 using saving policy functions and exogenous transitions
matrix P (z′|z) and P (′|).
(d) Start from the initial distribution µW0 and µ
E
0 . In iteration j, updating
distributions by applying big transition matrix, entry, and exit decisions
on existing ones.
(e) Iteration stops if the convergence criterion max{‖ µWj+1 − µWj ‖, ‖ µEj+1 −
µEj ‖} < ζ is satisfied. Otherwise, repeat step 2(d) with the updated
distributions.
2Here, we use Young (2010) method that is a discrete approximation to the law of motion of the
distribution of agents over states.
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(f) Aggregate across all households and compute aggregate labor supply, LW ,
asset supply, A, entrepreneur’s capital demand, KE. Capital for corporate
sector can be obtained by KC = A − KE. If KC < 0, then go to step 1
and guess a new price. Then, aggregate labor for corporate sector can be
calculated as follows:
LC =
1− θ
θ
ri + δ
wi
KC
3. Check if the convergence criterion ‖ LW − LC − LE ‖< ζ is satisfied. If yes,
STOP . Otherwise, update wage rate wi+1 = wi − φ(LW − LC − LE), where φ
controls the adjustment speed, and then go back to step 2.
Computing recursive equilibria. This subsection describes how to use Mongey
(2015) algorithm to compute the recursive equilibria. This approach accelerates com-
putation speed by adapting Judd et al. (2017) method on pre-computation of expec-
tation functions to collocation method for approximating value functions. Moreover,
Miranda and Fackler (2002) toolbox allows us to efficiently find optimal policies us-
ing vectorized golden-section search and solve fixed point problems using Broyden’s
algorithm.
The continuous and discrete choice problem described in the model section can be
solved by approximating two expected continuation values as follows
v1(s) =
∑
′,z′
P (′, z′|, z)max{vW (s′), vF (s′)},
v2(s) =
∑
′,z′
P (′, z′|, z)max{vW (s′), vE(s′)},
where v1(s) nests a worker’s entry decision and v2(s) nests an entrepreneur’s exit
decision,3 and the state vector is s = [a, , z].4 Note that we have Na = 100 asset grid
points, N = 9 working productivity shock grid points, and Nz = 9 entrepreneurial
3vW and vF in the continuation value v1 correspond to the options the individuals whose d− = 0
have: staying in the worker sector, or switching to the entrepreneur sector, respectively. vW and vE
in the continuation value v2 correspond to the options the individuals whose d− = 1 have: switching
to the worker sector, or staying in the entrepreneur sector, respectively.
4For computational purpose, s = [iNz×N ⊗ a, iNz ⊗ ⊗ iNa , z ⊗ iN×Na ], where ⊗ is a symbol of
tensor product and iN is a N-by-1 matrix of ones.
116
productivity shock grid points. In total, there are N = Na×N×Nz = 8, 100 states.
We now replace the functions we want to approximate with interpolants
v1(si) =
N∑
j=1
φ(si)c
1
j = Φ(s)c
1,
v2(si) =
N∑
j=1
φ(si)c
2
j = Φ(s)c
2,
where φ is a basis function, c1 and c2 are vectors of coefficients, and si ∈ s is a collo-
cation node. If we substitute these interpolants into the original system of functional
equations we have N system of equations with N unknowns as follows
Φ(s)c1 = (P ⊗ INa)[(1− If (s))Φ(s)cW + If (s)Φ(s)cF ],
Φ(s)c2 = (P ⊗ INa)[Ie(s)Φ(s)cW + (1− Ie(s))Φ(s)cF ],
where P ⊗ INa is a pre-computed expectation matrix, If (s) and Ie(s) captures the
entry and exit decisions at s, respectively, and Φ(s)cW , Φ(s)cE , and Φ(s)cF are values
associated with worker, entrepreneur, and entrant, respectively. Note that they can
be rewritten as follows
Φ(s)cW = max
{
u(w+ (1 + r)a− a′(s)) + βΦ([a′(s), , z])c1
}
,
Φ(s)cE = max
{
u(pi(s) + (1 + r)a− a′(s)) + βΦ([a′(s), , z])c2
}
,
Φ(s)cF = max
{
u(pi(s) + (1 + r)a− a′(s)− κ) + βΦ([a′(s), , z])c2
}
.
We can solve for c1 and c2 by applying either value function iteration or Broyden’s
(Quasi-Newton) algorithm. In practice, we start with two sets of initial guess, c10 and
c20. Then, we iterate on the following system for two times
c1k+1 = Φ(s)
−1(P ⊗ INa)[(1− If (s))Φ(s)cW + If (s)Φ(s)cF ],
c2k+1 = Φ(s)
−1(P ⊗ INa)[Ie(s)Φ(s)cW + (1− Ie(s))Φ(s)cF ].
117
Next, we rewrite the problem the system of equations as root-finding problem as
follows
g(c1, c2) = Φ(s)c1 − (P ⊗ INa)[(1− If (s))Φ(s)cW + If (s)Φ(s)cF ],
g(c1, c2) = Φ(s)c2 − (P ⊗ INa)[Ie(s)Φ(s)cW + (1− Ie(s))Φ(s)cF ].
The Jacobian of this problem is
D(c1, c2) =
[
Q11 Q12
Q21 Q22
]
,
where
Q11 = Φ(s)− β(P ⊗ INa)[(1− If (s))Φ([a′(s), , z])],
Q12 = −β(P ⊗ INa)[If (s)Φ([a′(s), , z])],
Q21 = −β(P ⊗ INa)[Ie(s)Φ([a′(s), , z])],
Q22 = Φ(s)− β(P ⊗ INa)[(1− Ie(s))Φ([a′(s), , z])].
Finally, we have the updating scheme[
c1k+1
c2k+1
]
=
[
c1k
c2k
]
−D(c1, c2)−1
[
g(c1k, c
2
k)
g(c1k, c
2
k)
]
,
where D(c1, c2)−1 is the inverse of the Jacobian.
C.1.2 Transition Dynamics
In this subsection, we describe the algorithm for computing the transition dynamics
discussed in Section 3.6.
1. Solve the steady-state equilibrium. Save the (expected) continuation values vWT ,
vET , v
F
T as well as stationary distribution µ
W
0 and µ
E
0 .
5
5Since the credit shocks only last for 3 periods and then recover to pre-crisis level immediately,
the initial and final value functions (as well as distributions) are identical as long as T is large.
118
2. Guess a sequence of wages w0 = {w1, w2, ..., wT} and compute the associated
sequence of interest rates r0 = {r1, r2, ..., rT}. Moreover, construct a sequence
of credit shocks Λ = {λ1, λ2, ..., λT}. We set T = 300.
3. Given the sequences of prices and credit shocks, we start from the final (ex-
pected) continuation values and iterate backward from t = T to t = 1. We also
solve the decision rules for optimal saving, entry (Ift ), and exit (I
e
t ).
4. For each period t, solve saving, entry, and exit decision rules on a finer grid and
construct big transition matrix QWt , Q
E
t , and Q
F
t using saving policy functions
and exogenous transitions matrices.
5. Starting from µW0 and µ
E
0 , the distribution of individuals over their asset, work-
ing productivity, entrepreneurial productivity, and occupation during transition
will evolve as follows
µEt+1 = Q
E′
t ((1− Iet )µEt ) +QF
′
t (I
f
t µ
W
t ),
µWt+1 = Q
W ′
t ((1− Ift )µWt ) +QW
′
t (I
e
t µ
E
t ).
6. For each t, aggregate across all individuals and compute aggregate labor sup-
ply, LWt , asset supply, At, entrepreneur’s capital demand, K
E
t , corporate sector
capital demand KCt , and corporate sector labor demand L
C
t .
7. Check if the convergence criterion ‖ LWt − LCt − LEt ‖< ζ is satisfied. If yes,
STOP . Otherwise, update wage rate wt+1 = wt− φt(LWt −LCt −LEt ) and their
associated rt for each t. Then, reconstruct wk+1 and rk+1 and go back to step
3.
119