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
Differences in Study Time Across Students
The NLSY79 Time Use Survey, conducted in 1981, provides insights into how study time varies among college students This survey reveals that study time, which includes both studying and time spent in classes or libraries, shows significant disparities among students The average study time is 38.5 hours per week, with the top third of students averaging 60.2 hours, while the bottom third averages only 18.7 hours This stark contrast highlights the heterogeneity in student motivation However, it's important to interpret these findings cautiously, as the data reflects only a specific week of study, and individual study efforts can fluctuate throughout college Ideally, an average study time over the entire college experience would offer a clearer picture, but data limitations hinder this analysis.
This study explores the relationship between tuition fees, family income, academic ability, and study time among college students Utilizing data from the National Longitudinal Survey of Youth (NLSY79), I analyze information from 1,476 students who attended college in 1981, focusing on their family income and Armed Forces Qualification Test (AFQT) scores The Federal Interagency Committee on Education (FICE) codes are employed to identify the colleges attended by the respondents, which are then integrated with higher education data Notably, the sample is limited to full-time college students who did not engage in paid work during the interview period, excluding those who reported working at enrollment.
The NLSY79 dataset includes FICE codes beginning in 1984, when respondents were queried about their most recently attended college The sample is established by verifying whether students consistently reported the same institution between 1982 and 1984, which influences the overall sample size.
Education General Information Survey (HEGIS) data to obtain the tuition levels of the colleges
Dependent Variable Independent Variables Study Time
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
The equation S = NB + Gj (1.1) represents the total time a student spends on academic activities, denoted as S The variables X include factors such as parental income, AFQT scores, and the tuition level of the college attended As indicated in Table 1.1, there is a positive correlation between academic time investment and tuition levels However, it's important to consider that this relationship may be influenced by the tendency of highly motivated students to enroll in more selective institutions Given the significant variation in tuition rates among public postsecondary institutions across different states, a valuable approach for analysis is to examine how study time among college students varies by state.
In order to see how study time of students changes across states, [ estimate the small
Dependent Variable Independent Variables Study Time
Table 1.2: Weighted least squares estimation Standard errors are in parentheses following relation by weighted least squares:
The analysis presented in Table 1.1, derived from the NLSY79 Time Use Survey, utilizes state-specific regressors such as average public tuition, median family income, and average SAT scores Average public tuition data is sourced from the Higher Education Coordinating Board’s Survey on Tuition and Fee Rates, while median family income figures for four-person families in 1981 are obtained from the U.S Census Bureau The findings in Table 1.2 indicate a positive correlation between total academic time and public tuition, alongside a significant negative relationship between study time and median family income This suggests that in states with higher public tuition relative to family income, students tend to dedicate more time to their studies The sample comprises 1,476 observations grouped by state, excluding the District of Columbia and Maine due to a lack of data from these regions.
Research shows significant variations in study time among full-time students, indicating that study time is positively influenced by tuition costs, which may create incentive effects However, it's important to note that the regressions used do not definitively establish a causal relationship between tuition and study time A logical approach to further investigate this would be to analyze student responses to changes in tuition policies Unfortunately, the available Time Use Data limits this analysis.
In 1981, the challenge of analyzing the impact of changing tuition policies on students' study time arose due to limited data availability and insufficient instruments for cross-sectional IV estimation Consequently, this study emphasizes a model that is carefully calibrated using micro data to yield more reliable insights.
The Basic Elements of Economy
College graduates generally earn more than those with less education, but the financial return on a college degree can vary significantly among individuals Factors such as innate ability and personal effort play crucial roles in determining educational outcomes While ability is an inherent trait, effort is a choice that students make, impacting their learning experience Understanding these distinctions is vital for evaluating educational policies, as students are motivated to maximize their utility Additionally, peer group effects contribute to this dynamic; more capable and motivated students often associate with high-achieving peers, leading to greater benefits from their college experience.
19In this paper, I interpret effort as the quantity or amount of studying or other academic work that college students engage in.
To incorporate effort into the determination of lifetime earnings, I utilize a Mincer-type earnings specification, which establishes a relationship between earnings, years of education, and years of experience The model predicts that an individual's wage earnings, denoted as w(s.t), are influenced by their years of schooling (s) and work experience (t) The coefficient associated with schooling reflects its causal impact on earnings To account for effort, I divide the returns to college education into two components: the return to ability and the return to effort, while considering college education as a two-period process An individual who completes only the first period is classified as a college dropout, whereas completing both periods is necessary to achieve college graduate status This framework enables an analysis of the earnings dynamics for both college graduates and dropouts.
Every student possesses a certain level of innate ability, which significantly influences their future earnings across educational groups For instance, a high school graduate's earnings can be represented by the formula w(a, t) = exp(a + a + pot + pit’), while a college dropout's earnings are given by w(a, e, t) = exp(a + a + pya + mei + pot + pit’) In contrast, a college graduate's earnings are expressed as w(a, €), €2, t) = exp(at + a + ua + Mei + 2a + Tes + pot + mit’) These equations highlight the critical roles of both ability and effort in determining income levels based on educational attainment.
‘ISee Mincer (1974), Card (1995) and Heckman, Lochner and Todd (2001) for a thorough examination of Mincer regression and the return to schooling literature
13 where ji is the return to ability and m is the return to effort for the second period of college education
In examining how students determine their effort levels, it becomes clear that there is significant variation among individuals To better understand these differences, I introduce the concept of "motivation" as an additional factor influencing effort choices Students can be categorized into two groups: high-motivation students, represented by 8, and low-motivation students, represented by 6 Importantly, a student's motivation type is independent of their ability, meaning there is no correlation between the two This distinction highlights that high and low-motivation students perceive the disutility of effort differently.
Parents possess complete knowledge of their child's abilities, denoted as 'a' However, they lack full insight into their child's motivation, only being aware of the prior distribution of motivation levels, specifically with a probability of 4 for 'θ = 6' and a probability of 1 for 'θ = 6'.
College graduates typically enter the workforce later than those who forgo higher education, potentially leading to increased leisure time for graduates However, this advantage may come at the cost of greater effort and hard work during their college years.
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
Upon entering the labor market, a student contributes a consistent amount of labor that remains uniform across all workers For a full-time job, this labor corresponds to a required effort level of 2 Consequently, the disutility associated with working is established.
[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 inherently altruistic and prioritize their child's well-being, yet they often do not align with their child's preferences, particularly regarding leisure activities This disconnect can lead to a conflict of interest between parents and children, as parents may undervalue the importance of leisure in their child's life.
Parents play a crucial role in enhancing their child's future by financing their college education Research indicates that despite the growing prevalence of student borrowing and increased loan limits, the primary financial responsibility still rests with 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
In 1988, a child expressed a desire for more leisure time than her parents were willing to allow Typically, parents influence their children's choices through specific transfers of goods rather than providing cash A prime example of this is funding a child's college education, which represents a tied transfer aimed at shaping their future decisions.
In "A Treatise on the Family," Becker explores the differences in utility functions between parents and their children, highlighting that even altruistic parents do not fully align with their young children's preferences He notes that parents often desire their children to engage in longer study periods or exhibit greater obedience than the children themselves wish for This conflict tends to diminish with older children, as altruistic parents are generally more inclined to provide financial support that allows children to make their own spending choices.
13For example in 1997, a dependent student was allowed to borrow only up to $2,625 in her freshmen year through subsidized Stafford Loan program.
15 larly, they provide a certain level of consumption, which is denoted by é Hence, the total cost of a 4-year college is
Parents may not be able to perfectly assess their child's effort, but they can rely on the grade point average (GPA) as an indicator of performance GPA is influenced by both ability and effort, as supported by the research of Schuman, Walsh, Olson, and Etheridge (1985), which highlights a strong correlation between grades, aptitude, and class attendance The average grade can be expressed by the equation y = ay + at + ε, where ε represents a normally distributed random error term This indicates that grades are generally distributed, reinforcing the connection between academic performance and student engagement.
All the utility functions are in the form of log(-).
A Simple Model with Complete Information
In my analysis of parental college decision-making and students' effort choices, I utilize a straightforward model that assumes parents possess complete information regarding their child's abilities and motivation This fundamental approach not only simplifies the problem but also provides valuable insights into the economic context, making it an effective starting point for my examination.
In this model, parents determine whether to fund their child's college education after high school graduation If they opt not to send her to college, she enters the workforce immediately Conversely, if they choose to support her college journey, she will attend for four years, with her effort level influenced by her motivation types To align with the baseline model, the college experience is divided into two periods: the first two years and the final two years.
Let U(w, C) represent the discounted lifetime utility of a parent with an annual income of w who invests a total of C in their child's college education In parallel, Uys(a, T) and Uc(a, T) denote the discounted lifetime utility of consumption for high school and college graduates, respectively, of type θ and ability a, with T representing work-life expectancy.
At the start of the first period, parents aim to maximize their utility, which is influenced by both their own consumption and that of their child Their optimization problem can be expressed as ax, [Up(w.0) + 1 Uns(@.T), Up(w.C) + EUer(a.8,T —4)], where D equals 1 if they choose to send their child to college and 0 if they do not The initial segment of this equation reflects the utility derived from having a high school education.
Since parents have perfect information about their child's type, grades do not provide any additional information in this model
College graduates typically enter the labor market four years later than high school graduates, starting their careers at age 22, while high school graduates begin working at 18 Both groups work full-time and retire at the same age, with college graduates contributing to the workforce for T-4 years and high school graduates for T years.
The discounted lifetime utility incorporates the disutility of effort, represented by the term we, &*~} jog(1 — e}/9 However, this term is excluded from Uys(a,T) and Ucz(a,9,T) to streamline the formulation of the parents' decision-making process.
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 decision-making process for parents hinges on the values of Us(a,T) and Ucy(a,9,T) A high-school graduate with ability level @ and t years of experience earns a salary represented by the equation w(a,t) = exp(a + a@ + pot + p)t” Over T periods, the discounted lifetime utility for these graduates can be calculated, emphasizing the importance of educational attainment and experience in determining financial outcomes.
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)
The equation Si = (+7)8: + Uy — ce represents the relationship between earnings (w), the child's utility function (u), savings (s), consumption (¢), the market interest rate (r), and the discount factor (ổ) This formula illustrates how these variables interact to influence financial decision-making and resource allocation over time.
The lifetime utility of a college graduate is influenced by her abilities and effort during her studies The student strategically decides on the amount of study time to allocate in both the first and second periods of college.
Ucr(a,8,T — 4) = (1+ 8+ 6? + B)u(é) + max )7 6 'uc(cr), t+1,Ce t=5 T
S.f §T+1 = 0, s5 >= 0, S141 = (1 + T)S- + We — Ce ạ 5) l 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
Let k equal pt-'St, where the chosen utility function is in the form of log{-}, indicating that & remains unaffected by ability Consequently, the differences in effort choices arise exclusively from the child's motivation type The optimal effort levels for both high-type and low-type students are determined accordingly.
Cth 84 Th 0yk ul đa m Aik ( )
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)
The following propositions summarize the implications of the model!’ :
"The propositions follow directly from the first-order conditions of the child’s problem and parents’ decision problem.
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 col- lege
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
The existing model effectively explains various empirical observations but fails to account for incentive effects, particularly due to its reliance on a perfect information setup It inaccurately assumes that parents can fully observe their child's motivation type, a premise that requires relaxation Additionally, the model does not consider potential renegotiation between parents and their child, despite parents being able to gauge their child's motivation through observed grades This oversight extends to the behavior of college drop-outs, as the model presumes all students will graduate To address these shortcomings, a dynamic model will be developed in the following section.
Dynamic Model with Incomplete Information 20 1.6 Commitment 0 0.02 0.0.0 0020020020008 26 1.7 Calibration
This article presents a dynamic model that explores the relationship between parental college decisions and student effort In this model, college decisions are not fixed; parents have the option to reassess their choice at the end of the first period The overall cost of college is represented as C₁ + C₂, where C₁ refers to the expenses incurred during the first period and C₂ denotes the costs associated with the second period.
Observe grades , Update Ato Student
Figure 1.2: Game tree associated with the dynamic model with incomplete infor- mation
At the beginning of the first period, parents make their decision about sending the child to college given the child’s ability a and the prior distribution of ỉ, i-e., @ = @,
In this scenario, parents face a decision regarding their child's college education, factoring in tuition costs and consumption needs The student must balance leisure and study time based on their motivation and ability, ultimately leading to an observable educational outcome—average grades Due to the unpredictable nature of grades, the student is uncertain about their chances of continuing college while deciding on their effort level Once grades are revealed, parents reassess their beliefs about the student's capabilities, updating their understanding based on prior expectations and the observed results If the grades exceed a predetermined cut-off level, the child remains in college and adjusts their effort accordingly; otherwise, the student exits the academic path and enters the workforce.
Figure 1.2 illustrates the decision-making timeline for parents regarding their child's college education In the initial phase, parents determine whether to enroll their child in college, represented as decision D,(-) The student then selects their effort level, e,(-), knowing that parents will adjust their beliefs based on the observed results Following the first period's outcome, parents reassess their perceptions of their child's capabilities and make a decision on whether to continue funding their college education, indicated as decision Dằ(-) This process includes establishing a minimum grade threshold, 7(-), below which they will cease financial support for college.
'8In other words a college student is completely dependent on her parents until she graduates.
College Graduate College Drop-out
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
{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
{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
The posterior probability, 4’, is derived from the prior, the student’s strategy {e,(-)}, and the observed outcome by using Bayes’s rule.
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)}
In equation (1.21), p represents the probability that the utility derived from a college-attending child exceeds that of a high-school graduate Parents of high-school graduates weigh the benefits of sending their child to college against the option of entering the workforce immediately This comparison involves assessing the expected utility of a college education versus the utility of a high-school diploma Importantly, parents face uncertainty regarding their child's continuation in college, as they will only invest in the second year if the child's average grade surpasses a specified cut-off level, g The likelihood of the child remaining in college, denoted as p, is influenced by both the child's abilities and the prior distribution of grades The child's performance, represented by y’, follows a normal distribution, indicating that grades are expected to be normally distributed around a mean influenced by various factors.
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
€ where ®(-) is the cumulative distribution function for a normally distributed ran- dom variable The expected utility that the parents derive from their college- attending child is
The discounted lifetime utility of a college drop-out is represented by Uno(a 9, T ~2) The parental college decision problem, outlined in Equation (1.21), raises two critical questions: how do parents determine the cut-off grade level g, and how does the student decide on the initial effort level e? These two decisions are interrelated and can be analyzed together For a specified value of g, a student of type @ will select her effort level accordingly.
1+ TB max | , log(1 — e,)+ Ply’ < 9| 8) 5 tog(t ~ 8) + Unola.0.T ~ 2)
+P(y' > §| 8) eee log( 1 — ea} +> “— la —é)+ Uer(a.0.7 -4)}
Equation (1.24) outlines the utility of effort during the initial period of college, while its subsequent components reflect the expected utility based on academic performance The associated probabilities play a crucial role in determining these outcomes.
Ply’ > gl@) = ứ (“TT TE—") (1.28) and
A college student consumes é for four years and then starts working at the age of
22 as Figure 1.3 shows The expected lifetime utility of consumption for a college graduate is
Stay = (1+ r)sy + we — Gc (1.27) Similarly, for a college dropout, the expected lifetime utility of consumption is
Dpo(a.8,T — 2) = (1 + 8) ue() + max } 7B 'ue(cr), te ty &e t=3 T
Note that w, = w{a,e,,t) = exp(ata+puia+me:+pot+ pit”) and é is the amount of consumption that parents provide for their child during college education
The first order condition for the student's effort choice problem stated in equation
The equation P(y > 90)UG(e, T - 4) + Ply’ < §|@) Upola, 8, T - 2) illustrates the relationship between a student's effort and their lifetime utility, where ó(.) represents the normal probability density function The left side reflects the change in disutility from increased effort, while the right side captures the benefits of higher effort: the first term indicates the utility gained from a greater likelihood of remaining in college, and the second term represents the direct impact of increased effort on overall utility Ultimately, students adjust their effort levels until the disutility of their efforts outweighs the utility derived from those efforts.
Given e, and ey, after parents observe the first period’s outcome they solve
U,(w Cy + C2) + + [XUcy(a 0y, ẽ — 4) + (1 — A) Ucet(a, 6t, T ~ 4)}} : (1.30) where \’, the posterior probability that @ = 6, is given by'®
The choice of the cut-off grade level, 7, and the first period effort choice, e:, are dependent on each other When choosing her effort level, the student takes into
In the analysis of parental beliefs regarding their child's academic performance, it is essential to understand that parents adjust their expectations based on perceived actions of the student They evaluate the student's effort levels across various potential grade values, leading to equilibrium outcomes for key variables The decision-making process, as outlined in equation (1.30), illustrates how parents establish a cut-off grade level for continued financial support Notably, this solution may not always yield a middle-ground outcome; for instance, high tuition costs could prompt parents to withdraw financial support despite favorable grades, while low tuition may lead to unconditional support, allowing students to remain in college regardless of poor academic performance.
So the parents can no longer provide incentives by setting high cut-off grade levels, since the student knows that they will not commit to it
The assumption of commitment plays a crucial role in addressing incentive problems In this analysis, I examine a modified version of the baseline model that incorporates commitment At the start of the first period, parents establish a cut-off grade and commit to it after observing the grades The model's timing can be summarized as follows: parents finance the first period of college and determine the cut-off grade level, while the student selects her effort level based on her type.
3 Grades are realized after the first period’s effort is chosen and if y/ > %, parents keep the child in college, otherwise they do not
*0See Laffont and Tirole (1993) for a detailed study of the role of commitment in the analysis of incentive problems.
4, The student chooses her second-period effort level, e2, according to her type
This problem can be solved by using backward induction The decisions to be made are:
In the first period parents decide whether or not to send their child to college, D,(-) and the cut-off grade level, 7
The student chooses the first period effort level, e,(-) e Parents keep the child in college if y’ > 9 otherwise, the child drops out of college
The student chooses the second period effort level e{-)
These strategies and beliefs form a Subgame Perfect Equilibrium iff
1 {eo{-}} is optimal for the student given that she stays at college for the second period
2 {e;(-)} is optimal for the student given ỉ(-)
3 {D,{-),9(-)} is optimal for parents given subsequent strategies
In the current model, the formulation of the problem closely resembles previous versions, with a key distinction in the initial parental decision Parents establish a cut-off grade level in the first period and adhere to this choice in the second period, without considering the grades' revealed information This initial decision by parents can be articulated as follows.
Max p, €{0,1} [U,(w., 0) + 7 Uns(a, T), max, (pU,(w.Cy + C2) + (1p) Up(w, C1) +7E(Uenua))}, 2) where p= Ply >a) = ng (SEE One) +(1—dA)@ (—) (133)
The expected utility of enrolling a child in college hinges on the likelihood of their completion This can be expressed as E(Uchua) = P(y! 2 ÿ| 0) * Ucr(a, 0y, T — 3), reflecting the probability of success and the associated benefits of a college education.
The first order condition for the choice of @ is
2 cổ — nọ (“=1 Uplw, Cy) — Up(w.Cy + Ce)]
+ oS ~ 1) 9 (SEE SEE A) Uplw.C1) - Up(w.Ci + Cal]
+ yA [P(y’ = 919) Uor(a.O,.T — 4) + Ply’ < G1) Upola On, TF — 2)|
+3Ð~2) (êt _ pẹ =3) era 6,T = 3) ~ Upo(a, 8¡,T ~ 2)] a PS ; roe ' t+ết~À) [Ply > 916) Uˆ,(a.6;.T ~ 4) + P( < 816) Upbo(a.6,T — 2)
The first order condition presented is intuitive despite its complexity: an increase in 7 leads to a decreased probability of the child remaining in college, although it may also motivate the child to exert more effort to meet higher standards In equation (1.35), the left side reflects the increase in parents' utility from the reduced likelihood of funding the second period of college, while the right side represents the additional utility gained from raising the cut-off grade.
Parents' decisions regarding academic standards are crucial, as students' effort levels are influenced by the cut-off grade If the cut-off is set too high, students may feel discouraged and opt for minimal effort, believing they cannot meet the expectations Conversely, if the cut-off is too low, students might also choose to exert little effort, knowing they can remain in college despite poor grades.
This section provides a detailed explanation of the parameter choices made for the model calibration, utilizing the High School and Beyond Sophomore Cohort: 1980-92 and NLSY79 datasets The calibration specifically focuses on the U.S economy in 1981, as the NLSY79 Time Use Survey was only conducted during that year Each parameter's selection is thoroughly discussed throughout this section.
The average worklife for high school graduates is estimated at 40 years, while college dropouts have an average of 38 years, and college graduates typically enjoy 36 years of work These figures align with data from the U.S Bureau of Labor Statistics, which indicates that at age 25, male high school graduates can expect 33.4 years of paid work, compared to 34.5 years for those with some college education and 35.8 years for college graduates Additionally, the return to experience parameters are set at 0.05 and -0.0010, as established by Murphy and Welch (1992).