the incentive effects of social policies on education and labor markets

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Curriculum Vitae

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iv

Acknowledgements

I would like to express my special thanks to Mark Bils, whose continuous encour-

agement constructive criticism, and excellent advice made me a better scientist

lam very grateful to Elizabeth Caucutt Gordon Dahl, Jeremy Greenwood, Hugo

Hopenhayn, and Per Krusell for very helpful discussions and comments on earlier drafts of my papers, on which this thesis is based I also thank the staff of the Economics Department, in particular Rosemary Dow and Lynn Enright for their

excellent work and support

My long-time friends Atila Abdulkadiroglu and Bahar Leventoglu deserve special

thanks for their friendship and support

I express my deepest gratitude to my parents and my sister for their absolute confidence in me Distance did not keep them from showing their endless support and love for me

Lastly, my sincere thanks are due to Kubilay to whom I dedicate this thesis |

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A social policy might have disincentive effects on its beneficiaries in the presence of asymmetric information This dissertation studies the incentive issues arising from the implementation of certain educational and labor market policies when informational asymmetry is present In particular, the first chapter deals with higher education subsidies and the second chapter studies unemployment insur-

ance

Chapter 1 analyzes the potential disincentive effects of higher education subsi-

dies on students’ performance A game-theoretical model is employed to analyze

the interaction between parents and their child prior to and during the college education The model is calibrated by using information from the High School and Bevond Sophomore Cohort: 1980-92 and the National Longitudinal Survey

of Youth 1979 data sets The experiments show that subsidizing tuition increases

enroilment rates and graduation rates Yet, there are two effects lowering student

effort First a low-tuition, high-subsidy strategy causes an increase in the ratio of less able and less highly-motivated students among college graduates Secondly,

all students, even the more highly-motivated ones respond to lower tuition levels by decreasing their effort levels

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long-term unemployment insurance plans (plans that depend on workers’ unemployment history) in economies with and without hidden savings The simulations

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Curriculum Vitae ee ee ili

Acknowledgements 0 002.20-020 0000 eee eee ee iv Abstract 2.0 0 0 V 31) 0 aaaaTTKc 1 1 The Incentive Effects of Higher Education Subsidies on Student Achievement 2 1.1 Introduction 2 0.0.20.02022.0 20.2002 2 0222000005 2

1.2 Differences in Study Time Across Students 7

1.3 The Basic Elements of Economy -.- - 11

1.4 A Simple Model with Complete Information 15

1.5 Dynamic Model with Incomplete Information 20

1.6 Commitment .0 0.02 0.0.0 0020020020008 26

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vill

1.8 Experiments 2.2 eee ee ee ee eee 35

1.8.1 How Do Students Respond to Different Educational Stan- dards? 2 Le ee 36 1.8.2 The Effect of Tuition Levels 2 - 0 0. 048 38

1.8.3 Human Capital 2.2 2 ee ee eee 4]

1.8.4 Comparison of Different Subsidy Schemes 42 185 7 Commitment cà cv ee eee ee 43 18.6 The Role of Grades 2 20-2 ee ee 44 1.9 Grade Infation Qua 46 1.10 Conclusion cu kh ko 47 Unemployment Insurance 49 21 Introduction or ee ee 49 2.2 The Economic Environment .- .-.-2-+0500- 55

22.1 The Model Economy with Savings - 55

292 The Model Economy without Savings - 60

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List of Figures 1.1 1.3 1.4 1.5 1.6

Distribution of the weekly study time for full-time college students from the NLSY79 Time Use Surveyin 1981

Game tree associated with the dynamic model with incomplete in-

formation cv KT ky Consumption pattern by educational attainment Generated ability distribution ch ee

Generated Family Income Distribution Pe Cự SA CC R8 C8 CN *

Effort choices of high-type and low-type students with different

innate ability, a = —0.35, a = 0 and a = 0.35, respectively

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Ll 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Ordinary least squares estimation Standard errors are in parentheses 9 Weighted least squares estimation Standard errors are in parentheses 2 0 Q Q Q Q Quà VN Tư kg kia Nà Ha 10 Total study time for full-time college students from the Time Use

Survey of NLSY79 in 1981 2 222 ee eee 30

Average tuition levels, HEGIS (1981-1982) 34

GPA of college graduates, HS&B Sophomore Cohort: 1982-1990 35 The effect of tuition on enrollment and college graduation rates; U.S Data from the Condition of Education, 1981 Note that T/w=0.18

is the baseline case 2 ee ee 38

The effect of tuition on enrollment by income quartiles 39 The summary statistics from the simulations for college graduates 39

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1.10 The effort choices of students who would have attended college for 1.11 1.12 1.13 1.14 1.15 1.16 1.17

all three tuition policies 2 2 ee ee ee ee ee The increase in the accumulation of human capital compared to

the baseline case where T/w=0.18 (Average Tuition=$4,200)

The effect of tuition on enrollment and college graduation rates for proportional subsidy and flat subsidy schemes For the proportional subsidy tuition to family income is changed from T/w=0.18

to T/w=0.14 and for the flat subsidy scheme tuition is decreased

by $1.000 for all students See ek

The effect of tuition on enrollment by income quartiles for proportional subsidy and flat subsidy schemes For the proportional

subsidy tuition to family income is changed from T/w=0.18 to

T/w==0.14 and for the flat subsidy scheme tuition is decreased by

$1,000 for all students 2 2 ee ee ee ee ee

The effect of tuition on enrollment and college graduation rates

with commitment eee ee em mw we

The effect of tuition on enrollment and college graduation rates for

ao, = 0.33 and o, = 0.2 ee ee em

The effect of tuition on effort and GPA for o, = 0.33 and o, = 0.2 The summary statistics without and with informational asymmetry

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The optimal UI plans and summary statistics for the benchmark

parameterization and 7 & (0,1), ™ = 1 ee em mm

The optimal UI plans and summary statistics for the benchmark parameterization and for 7 = 7; = 1 ee ee ewe

The optimal UI plans and summary statistics for the benchmark

eee eee

parameterization and for 1o = 0, 7; = 1

The optimal UT plans and the welfare gains for benchmark param- ee eee el eterization and 7; = 1, 7 € {0,0.1, 0.25, 0.5, 1} The optimal UI plans and the welfare gains for o = 0.5, 7 € [0,1) and 7, = 1 ˆ cơ NO mmm mmm ® The optimal UI plans and summary statistics for a = 0.5, 7 € [0.0.5], and 7, = 1 ema mw The optimal UI plans and the welfare gains for p = 10, 7 = 0 and eee maw www 8n Y A 6 U em Hmm — — —— — —— — —- ằ— —

The optimal UI plans derived in economies with and without sav-

ings and their effects on the economy with savings for benchmark

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Foreword

Chapter 2 is based on a joint work with Atila Abdulkadiroglu and Burhanet-

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The Incentive Effects of Higher

Education Subsidies on Student Achievement

1.1 Introduction

The major goal of higher education subsidies has been to promote college enroll-

ment bv reducing tuition costs Most studies, in fact, find that education subsidies

make college education more accessible by increasing families’ ability to pay for

college' What is less trivial, however, is how subsidizing higher education affects

students’ achievement When faced with lower educational costs, parents are likely to have less-demanding expectations about their child’s academic success First of all they might send their child to college even if the child’s benefit from college education is not high and similarly they might continue to pay for the child’s college

education even if they observe hardly satisfactory educational outcomes (grades)

Consequently, the child might reduce her engagement in academic activities and

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choose to study less This paper studies the potential disincentive effects of higher

education subsidies on students’ performance by analyzing the parental decision

to pay for college and the effort choices of students The results, indeed, indicate the presence of potentially important disincentive effects

Higher education subsidies take two basic forms: means-tested grant and loan programs and operating subsidies to public postsecondary institutions Oper- ating subsidies which are primarily funded by the state and local governments,

constitute the major part of higher educations subsidies For example, in 1997, the state and local governments provided $56.4 billion in subsidies to public insti-

tutions which is considerably more than the total amount offered by federal grant and loan programs Not surprisingly, at the average public four-year institution more than 50% of the educational expenditures are subsidized? Unlike means-

tested grant and loan programs which have certain eligibility criteria, operating

subsidies keep tuition low for all students who are admitted to college, thereby decreasing expenses to families from all income groups Accordingly, low tuition has the effect of increasing college enrollment On the other hand, what happens to parental expectations and student motivation is still an open question that

should be answered

Before turning to the body of the paper I briefly examine the relation between tuition, family income ability and study time of students based on the the National

Longitudinal Survey of Youth 1979 (NLSY79) data set There is a clear positive

relationship between the total time spent on academic activities and tuition paid by a student controlling for ability and family income I then look at how average

?See Kane (1999) and McPherson and Shapiro (1998) for a descriptive summary and empirical

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tuition students do study harder

I set up an economic framework to analyze the disincentive effects of tuition subsidies by using a game-theoretical model The parents’ and the child’s preferences are modeled as suggested by Becker (1974, 1981): parents are altruistic, i-e., they

care about the well-being of their offspring and the child is “rotten”, i.e., derives

utility only from her own consumption® In addition, parents assign no value to the child's utility from leisure

As in Becker and Tomes (1976 1979), altruism is the underlying reason for parental investment in the child’s human capital Specifically, parents invest in their child’s human capital by paying for her college education College education increases the human capital of the child and thus college educated children earn more Moreover the return to college education depends not only on the ability but also on the effort of the child Children differ both in their innate ability and motivation Parents know their child’s ability, however, they do not have perfect information about their child’s motivation’ This feature along with the assumption that parents and children do not share the same preferences creates a conflict of interest between the parents and the child

In this economic framework I develop a game-theoretical model to analyze the interaction between the parents and their child—the student—prior to and during the child’s college education*® College education is modeled to be two periods At

3] borrow the expression “the rotten kid” from Becker (1974)

4 Asymmetric information in the context of family has been an ongoing assumption in similar problems See, for example Loury (1981) Kotlikoff and Razin (1988}, Villanueva (1999) and Fernandes (2000)

“Students, unlike workers do not make decisions on effort independently but are, instead,

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the beginning of the first period, parents decide whether or not to send their child to college If they decide to do so, they pay for the first period of college and the

child chooses how much to study At the end of the first period, parents observe a noisy measure of the child’s effort-—grades—and update their beliefs about the child’s motivation Then they decide whether or not to keep the child in college Knowing that staying at college depends on her grades, the child studies harder to influence her parents’ decision However, when parents pay lower tuition, they tend to keep the child in college even if they observe low grades Hence, the child

is tempted to study less

The model is then calibrated by using the High School and Beyond (HS&B)

Sophomore Cohort: 1980-92 and the NLSY79 data sets The calibrated model is used to conduct experiments to compare college enrollment and effort choices of students under different tuition policies The experiments imply that subsidizing tuition increases enrollment rates and graduation rates Yet, there are two effects lowering student effort First a low-tuition, high-subsidy strategy causes an increase in the ratio of less able and less highly-motivated students among college graduates Secondly, all students, even the more highly-motivated ones, respond

to lower tuition levels by decreasing their effort levels

I also analyze the role of commitment in this economic framework As Laitner (1997) discusses, the commitment assumption is important in the analysis of inter- generational transfers The experiments show that if parents can actually commit to higher standards, there is no disincentive effect on students’ study time

The findings of this paper are complementary to Hanushek, Leung and Yilmaz

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studies are similar to this paper in the sense that they all analyze the role of education subsidies However, they concentrate on different aspects of education and

address related yet different issues Hanushek Leung and Yilmaz (2001) study

tuition subsidies as a redistribution device and conclude that education subsidies are effective redistribution mechanisrns only when there are externalities in pro- duction Caucutt and Kumar (2001) argue that a policy that aims to maximize the fraction of college-educated labor, by sending as many children as possible to college, results in little or no welfare gain They show that if the government subsidizes children without making the subsidy contingent on the child’s ability, the subsidy can actually decrease educational efficiency Blankenau and Camera (2001) study the role of education by separating human capital accumulation from educational investment decision In their framework, as in the case of this study, student effort is necessary for skill formation during education They show, by using a search-theoretic model that lowering educational costs does not necessarily increase skill formation unless incentives to student effort are provided

This study is also related to the educational standards literature The impact of educational standards on students’ achievements and earnings has received

considerable attention Costrell (1994) and Betts (1998) are recent examples

They both analyze how a policy maker would choose the educational standards and how students would respond to these standards Their main focus is how the policy maker sets standards; parents play no role in setting the standards On the contrary, | consider a framework where educational standards are implicitly set by the parents For an analysis of higher education which requires considerable

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Shudy Tine of Cofege Sustents oœ T v Y y T T ~ T Y + ` 0.08 1 eo7t | r1 = 80t _n 905† q0áƑ at 00t Gor a we 0 10 20 W 40 52 & 7ô SS 99 10 Hours

Figure 1.1: Distribution of the weekly study time for full-time college students

from the NLSY79 Time Use Survey in 1981

The plan of the paper is as follows In Section 1.2, I start with some evidence on how study time varies across students In Section 1.3, I describe the basic elements of the economy In particular 1 explain the development of the model

and motivate the choice of certain functional specifications Section 1.4 discusses

a simple model with complete information Section 1.5 explains the dynamic

model with incomplete information which is the baseline case for the simulations

Section 1.6 discusses a variant of the baseline model with commitment Section 1.7 explains the calibration of the model Section 1.8 discusses the results of the experiments, Section 1.9 studies grade inflation, and Section 1.10 concludes

1.2 Differences in Study Time Across Students

In order to examine how study time varies across students, I look at the NLSY79 Time Use Survey This survey was conducted in 1981 and it contains responses

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etc Figure 1.1 shows the distribution of weekly study time for full-time college

students® Study time is defined as the sum of total time spent on studying

and time spent at school, at classes, library, etc As Figure 1.1 suggests, study time varies considerably across students The average study time is 38.5 hours per week There is a drastic difference between the average study time of the bottom third and top third of the distribution The average weekly study time for the top third is 60.2 hours and it is only 18.7 hours for the bottom third This observation can be considered as an evidence of heterogeneity in student

motivation However, the empirical evidence should be evaluated carefully The

time use observations are available only for a certain week of a student’s college education Since a student’s study effort can change significantly throughout her college education part of the variation suggested by Figure 1.1 might result from the individual variations in study time Study time averaged over the whole course of college education would be a much better measure of study time However, data

limitations make it impossible to study the variation in this parameter

I also examine the relation between the tuition paid family income ability, and the study time for students I obtain the family income and the Armed Forces Qualification Test (AFQT) scores for 1476 students who reported attending college in 1981 from the NLSY79 The Federal Interagency Committee on Education (FICE) codes of the postsecondary institutions are used to identify the colleges that the respondents attended’ Then the FICE codes are merged with the Higher

®Some respondents report working at the time of enrollment I did not include these students in the sample The sample consists of full-time college students who did not work for pay during the time of the interview

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Education General Information Survey (HEGIS) data to obtain the tuition levels of the colleges Dependent Variable Independent Variables Study Time Constant 14.20 (9.16) log(Tuition) 3.64 (0.79) log(Family Income) -0.62 (0.90) AFQT Score 0.056 ( 0.034) Number of Obs 591 R? 0.05

Table 1.1: Ordinary least squares estimation Standard errors are in parentheses I estimate a relation between study time and variables of interest by OLS

S, = NB + Gj (1.1)

where S; is the total time spent on academic activities by the student z X; represents regressors such as parental income AFQT score and tuition level of the college that the student has attended As Table 1.1 shows there is a positive relationship between the total time spent on academic activities and the tuition level However one might argue that this effect is driven by the selection of highly-motivated students to more selective schools Since tuition levels at public

postsecondary institutions vary dramatically across states, a natural strategy for

the estimation is to analyze how the study time of college students differs across

states

In order to see how study time of students changes across states, [ estimate the

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Dependent Variable Independent Variables Study Time Constant 168.22 (49.04) log(State Public Tuition) 4.04 (1.61) log(Median Pamily Income) -17.28 (5.05) Average SAT Score 0.039 (0.016) Number of Obs 48 Rˆ 0.30

Table 1.2: Weighted least squares estimation Standard errors are in parentheses following relation by weighted least squares:

S; = ZY + &- (1.2)

S; is calculated from the NLSY79 Time Use Survey as in Table 1.1 and the weights reflect the number of observations in each state® The state-specific regressors are

average public tuition, median family income and average SAT score for that state Average public tuition is formed by using data from the Higher Education Coordinating Board’s Survey on Tuition and Fee Rates Median family income

for four-person families in 1981 is taken from the U.S Census Bureau The results

given in Table 1.2 imply a positive relationship between the total time spent on

academic activities and public tuition At the same time, there is a negative and

significant relation between the study time and median family income To sum up, in states with higher public tuition relative to family income, students study

harder

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11

The observation of wide differences in study time for full-time students and that

study time is positively related to tuition points to potential incentive effects of college costs Admittedly, these regressions do not necessarily identify the causal impact on study time A natural attack would be to ask how students react when tuition policies change However, the Time Use Data are available only for 1981, making it impossible to analyze how students’ study time changes as tuition policies change Cross-sectional IV estimation is difficult due to the small sample size and the lack of obvious instruments For these reasons | focus attention on a model calibrated as reasonably as possible to micro data

1.3 The Basic Elements of Economy

College graduates almost always tend to earn more than less-educated workers Yet, the return to college education varies considerably across individuals: more able students and more highly motivated students acquire more cognitive and

social skills in college® Whereas many factors could play some role in student

learning, the student’s inputs to education is often summarized as being the ability and the effort!® The separation of ability and effort may seem complicated, however from a practical point of view the main distinction is the fact that ability is an innate characteristic of an individual while effort is a choice variable Since every student acts as a utility-maximizing agent, it is crucial to consider the effort choices of students in the evaluation of educational policies

°Peer group effects reinforce this effect As Epple, Romano, and Sieg (2000) argue, there is substantial stratification of students by ability among postsecondary institutions So a more able and more highly-motivated student is more likely to be surrounded by high quality peers which will yield higher returns

19In this paper, I interpret effort as the quantity or amount of studying or other academic

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In order to add effort as a factor in the determination of lifetime earnings, I use a

Mincer type earnings specification!! The standard Mincer specification predicts a relation of the following form between one’s earnings, the years of education and the years of experience:

w(s.t) = exp(ag + ps + pot + pit” + €), (1.3)

where w(s.t) is the wage earnings for an individual with s years of schooling and

t years of work experience The coefficient yz is interpreted as the causal effect of

schooling In order to incorporate effort to the earnings function, | partition the

return to college into two parts: the return to ability and the return to effort IT also

assume that college education is two periods An individual who only completes the first period of college is a college dropout The completion of the two periods is necessary to be a college graduate This specification allows me to analyze both

the behavior of college graduates and the college drop-outs

Every student is assumed to be endowed with a certain level of innate ability and for all education groups future earnings depend on ability In particular a high-school graduate with ability a and years of experience t earns

w(a.t) = exp(a + a+ pot + pit’) (1.4)

Similarly, a college dropout’s earnings are

w(a.e;.t) = exp(a +a + pya + mei + pot + pit’), (1.5)

where p is the return to ability and 7 is the return to effort for the first period

of college education For a college graduate, earnings take the form of

w(a, €),€2,t) = exp(at+a+ ua + Me; + 2a + Tes + pot + mit’), (1.6)

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13

where ji is the return to ability and m is the return to effort for the second period

of college education

Now, I move on to look explicitly at how students choose their effort levels As Figure 1.1 suggests, effort choices vary considerably across students To capture the differences in the effort choices, I introduce an additional source of hetero-

geneity to the model: “motivation.” Specifically, there are two types of students:

high-motivation students and low-motivation students The high type is represented by 8, and the low type is represented by 6, The type of the student is orthogonal to her ability, ie there is no correlation between the student's type

and her ability High type and low type students differ in their assessment of the

disutility of effort

Parents have perfect information about their child’s ability, a On the contrary,

they do not have perfect information about their child’s motivation: they only know the prior distribution of @, i.e., 9 = 6, with probability 4 and @ = 6, with probability 1 — A

College graduates enter the labor market later than their peers who decide not to pursue a college degree As a result of this, those who attend college might enjoy more leisure compared to the others It is also possible that they might have to

work harder at college The disutility of effort for a college student is

5 log(1 — €), (1.7)

where e is the effort level at college Since 6; > 9, high-type students assign less weight to the disutility of effort compared to the low-type students

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which is the same for all workers Let 2 correspond to the effort level required for

a full-time job Then the the disutility of working is

5 logit — ê) (1.8)

[f e > ẽ, college education is associated with higher disutility compared to working and if e < é, the student experiences lower disutility at college compared to working

Parents are altruistic: they care about their child’s prosperity However, they do

not exactly share the child’s preferences Specifically, | assume that parents do

not assign any value to their child’s leisure Because of this feature of the model,

there is a conflict of interest between the parents and the child!?

Parents invest in their child’s human capital by paying for her college education This assumption has considerable empirical support Even though undergraduate borrowing has become more widespread and borrowing limits have increased for college students, the bulk of the burden is still on the parents’

Tuition is not the only cost of higher education If parents decide to send their

child to college, they also pay for her consumption during college years Particu-

'2Parents and their children might disagree about how the child should allocate her time or use her resources This type of preferences is labeled as “paternalistic” preferences by Pollak (1988) In this particular situation the child wants more leisure than her parents desire for her Parents generally try to influence the child’s decisions by making transfers of particular consumption goods rather than money For example paying for college is an example of tied transfers

In A Treatise on the Family Becker discusses the discrepancies between the utility functions of the parents and their child In the words of Becker, “Even altruistic parents do not merely accept the utility functions of young children who are too inexperienced to know what is good for them Parents may want children to study longer, or be more obedient than the children want to The conflict with older children is usually less severe, and altruistic parents are more willing to contribute dollars that the children can spend as they wish.”

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15

larly, they provide a certain level of consumption, which is denoted by é Hence,

the total cost of a 4-year college is

4 sa ~

Cost = 37 Tuition(t) + ẽ 2- Hư (1.9)

Parents cannot observe their child’s effort perfectly But they can observe her

grade point average (GPA), which is a noisy measure of her performance Grades

are assumed to be a function of ability as well as of effort This specification is supported by the study of Schuman, Walsh, Olson, and Etheridge (1985) which

argues that there are strong and monotonic relations between grades and both ap-

titude measures and class attendance In particular average grade, y/ is assumed to be

y =ayatmerté, (1.10)

where ¢ ~ N (0,02) is the random error term According to this specification

grades are normally distributed, i.e y’ ~ MN (aia + age, 02)

All the utility functions are in the form of log(-)

1.4 A Simple Model with Complete Information

I first analyze the parental college decision and students’ effort choices by using

a simple model where parents have perfect information about their child’s ability

and motivation The analysis of this problem is simple yet helpful in explaining

the economic environment Therefore, I find it useful to begin my analysis with

this simple model

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she graduates from high-school and they continue to pay for her college education

until she graduates!* If they choose not to send the child to college, the child

enters the labor market and starts working If they choose to send her to college,

she attends college for four years and chooses her effort level according to her

motivation types For consistency with the baseline model, the college education is assumed to be two periods: first two-years is the first period and the last two-

years is the second period

Let U,(w,.C) denote the discounted lifetime utility of a parent who has yearly

income of w and allocates a total amount of C to her child’s college education

Similarly, Uys(a.T) and Uc,({a.9,T) stand for the discounted lifetime utility of consumption for a high-school graduate and a college graduate child of type @ and ability a, respectively’® T is the worklife expectancy

At the beginning of the first period, parents make the decision that maximizes

their utilitv Note that parents’ utility depends on both their own consumption and their child’s consumption Parents’ optimization problem at the beginning of the first period is

ax, [Up(w.0) + 1 Uns(@.T), Up(w.C) + EUer(a.8,T —4))] (1)

where D is 1 if the decision is to send the child to college and 0 otherwise The first part of equation (1.11) corresponds to the utility of having a high-school

Since parents have perfect information about their child's type, grades do not provide any additional information in this model

College graduates enter the labor market four years later than the high-school graduates and they retire at the same age There is no labor supply decision Everybody works full-time For a college graduate entry to labor market takes place at age of 22 and she works for T-4 years, on the other hand a high-school graduate starts working at the age of 18 and she works for T years

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17

graduate child and the second part is the utility of sending the child to college

The degree of altruism is 0 < y < 1 and C is the total cost of college

The solution to the decision problem of the parents depends on the values of

Us(a,T) and Ucy(a,9,T) As mentioned before, a high-school graduate with

ability @ and t years of experience earns w(a,t) = exp(a + a@ + pot + p)t”) and

works for T periods The discounted lifetime utility of a high-school graduate is T gi!

Uns(a,T) + » “ log(1 — é) (1.12)

The discounted lifetime utility of consumption, Uys(a,7) can be computed by solving a straightforward optimal consumption problem which is

Uns(a,T) = max osu wel Ce) (1.13)

si

spat = 0,5; =0 (1.14)

Si = (+ 7)8: + Uy — ce,

where w, = w(a,t) is the earnings, u,{-) is the utility function of the child, s, is the savings, ¢; is the consumption, and r is the market interest rate, and ổ is the

discount factor

The discounted lifetime utility of a college-graduate depends on her ability and

effort at college The student chooses how much to study at the first and second

periods of college by solving

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S.f §T+1 = 0, s5 >= 0,

.l

S141 = (1 + T)S- + We — Ce ạ 5)

Note that tuy = w(a,e),€2,t) = exp(a +a + pya + Tiết + Hào + Taea + pạt + pit?) and @ is the amount of consumption that parents provide for their child during college education The first order conditions of the child’s problem are 1+ Ty w rap Eee t~5 "9 and 1+8) Fat 2 t=5

Let k= pt-'St For the particular choice of utility function in the form of

log{-), & is independent of ability So the variation in the effort choices comes solely from the child’s motivation type The optimal effort choices for high-type and low-type students are 1+ 1+) : =l————— €ỳ=l—-————, 1.18 Cth 84 Th 0yk ul đa m Aik ( ) L+6 “ 1+8 ` oe

Given e}, 5,, €j;, and e}, and @, parents solve their optimization problem stated in equation (1.11) So, parents’ decision is

pali €U,(w,0)-U,(w.C) < y(Uex(a,0,T ~4)-Uns.T)) 4 99)

0 if U,(w,0) — U,(w,C) > y(Ucr(a,8,T — 4) ~ Uns(a,T)) The following propositions summarize the implications of the model!’ :

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19

Proposition 1 High-type students study harder than low-type students, 1.e., eịn 2

ey and Can > Cại

Proposition 2 If the return to effort, m and 72 are higher, students study harder

Proposition 3 More able students and high-type students are more likely to enroll in college

Proposition 4 Students from high-income families are more likely to attend college

Proposition 5 When tuition levels are lower, the average ability of students who

enroll in college 1s lower

Proposition 6 When tuition levels are lower, the proportion of low-type students whe enroll in college increases

This simple model is quite successful in explaining several empirical observations, yet it overlooks the possible incentive effects embodied in the problem First of

all, the model analyzes the problem in a perfect information set-up However,

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analyze the behavior of college drop-outs since the model assumes that everybody who goes to college will graduate from college In order to address these issues, | develop a dynamic model in the following section

1.5 Dynamic Model with Incomplete Informa- tion

This section develops a dynamic model of parental college decision and student

effort decision In this model college decision is not final Parents can reconsider their decision about college at the end of the first period The total cost of college is C; + Cz where C; is the cost of the first period and Cy is the cost of the second period Nature Observe grades , Update Ato Student Figure 1.2: Game tree associated with the dynamic model with incomplete information

At the beginning of the first period, parents make their decision about sending the

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21 with probability A and @ = 6, with probability 1 — If they decide to send their child to college, they pay for the tuition costs and provide child’s consumption needs!® Then the student chooses how much time to allocate between leisure and

studying given her motivation type and ability At the end of the first period, the

educational outcome—average grade—is observed by the parents and the student Recall that y’ = aja + aoe +¢ Due to the random component of the grades, ¢,

the student does not know whether she will be allowed to complete college when she is choosing her effort level After the outcome is observed, parents update their beliefs about the type of the student Formally, they derive the posterior distribution of @ given the outcome and the prior distribution If the observed outcome is higher than a certain threshold, which I call “the cut-off grade level,” then the parents keep the child in college and the child chooses her second-period effort level according to her type If not, the student drops out and enters the

labor market

Figure 1.2 summarizes the timing of the decisions:

e Inthe first period parents decide whether or not to send their child to college;

the decision is D,(-)

e The student chooses the first period effort level, e,(-), given the fact that

parents will update their beliefs according to the observed outcome

e After observing first period’s outcome, parents update their beliefs about the type of their child and decide whether or not to keep the child in college;

the decision is D»(-) Equivalently, parents choose a cut-off grade level 7(-)

below which they no longer continue paying for the college

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College Drop-out High-School Graduate Age 3 58 wrt 18

Figure 1.3: Consumption pattern by educational attainment e The student chooses the second period effort level, e2(-)

These strategies and beliefs form a Perfect Bayesian Equilibrium iff

Nm

{e2(-)} is optimal for the student given that the student stays at college for

the last period

{4(-) Da(-)} is optimal for parents given e;{-) and the posterior probability

Nv

{e;(-)} is optimal for the student given 9(-) and the fact that parents’ second

period decision depends on {e;(-)}

D,(-) is optimal for parents given subsequent strategies

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Parents’ decision problem at the beginning of the first period is

pmax, {U,(w,0) + 7Uns(a,T), pU,(w, Cy + Co) + (1 - p)U,(w,C1) + yE(Ucnita)}

(1.21) where p = P(y’ > gy) and E(Ucaia) is the expected utility that the parents

derive from their college-attending child As equation (1.21) suggests, parents of

a high-school graduate child compare the utility of sending the child to college

with letting her enter the labor market as a high-school graduate The first part corresponds to the utility of having a high-school graduate child The second part

is the expected utility of having a child who attends college As the equation

shows parents do not know whether the child will stay in college for the second period If the average grade turns out to be greater than the cut-off grade level g, they will pay for the second period of college, otherwise they will not The probability that the child will stay in college, p, depends on the prior distribution

of # as well as the ability of the child Since y’ = aya + age +e and e ~ N(0, 02),

grades are normally distributed, i.e y’ ~ N(aia + age, a2)

Let ey, and e,, be the effort choices of high-type and low-type students Then the probability that the student will complete college is given by 1a + A2€1_ — Ù Os aya + Age — Y p= P(y > g) =A® )+a-A)#{ ), (1.22) €

where ®(-) is the cumulative distribution function for a normally distributed ran-

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where Uno(a 9, T ~2)is the discounted lifetime utility of a college drop-out Equa-

tion (1.21)states the parental college decision problem, yet there are still two important questions that should be answered: how do the parents set the cut-off grade level g, and how does the student choose first period effort level e,? These two decisions can be explained jointly For a given value of g, a student of type @ chooses her effort level according to 1+ TB max | , log(1 — e,)+ Ply’ < 9| 8) 5 tog(t ~ 8) + Unola.0.T ~ 2) , t=3 2) 32 T 3-1 +P(y' > §| 8) eee log( 1 — ea} +> “— la —é)+ Uer(a.0.7 -4)} t=5 (1.24)

The first part of equation (1.24) is the utility of effort associated with the first period of college The second and third parts of equation (1.24) correspond to the expected utility conditional on grades The probabilities are given by

Ply’ > gl@) = ø (“TT TE—") (1.28)

and

P <ø|9 =w(—“—”) (1.26)

A college student consumes é for four years and then starts working at the age of

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