metadata: authors: 0: author: Ferreyra,Maria Marta
1: author: Garriga,Carlos
2: author: Martin,Juan David
3: author: Sanchez Diaz,Angelica Maria
count: World
docna: 0: docna: Raising College Access and Completion : How
            Much Can Free College Help ?
docty: Policy Research Working Paper
subtopic: Educational Sciences,Tertiary Education,Public Sector Economics,Public Financial Management,Public Finance Decentralization and
            Poverty Reduction,Labor & Employment Law,Rural Labor Markets,Labor Markets
teratopic: Education,Public Sector Development,Law and Development,Rural Development,Social Protections and Labor
abstracts: cdata!: Free college proposals have become
            increasingly popular in many countries, yet cross-country
            evidence indicates that higher college subsidies raise
            enrollment but not graduation rates. To capture the evidence
            and evaluate proposals, this paper develops and estimates a
            dynamic model of college enrollment, performance, and
            graduation. A central piece of the model, student effort,
            has a direct effect on class completion and an indirect
            effect mitigating the risk of performing poorly or dropping
            out. The model is estimated  using rich student-level data
            from Colombia, and multiple free college programs are
            simulated. Among them, universal free college expands
            enrollment the most but does not affect graduation rates,
            thereby helping explain the evidence. Performance-based free
            college, in contrast, raises graduation rates yet has a
            smaller enrollment impact.
repnme: repnme: Raising College Access and Completion : How
            Much Can Free College Help ?
display_title: Raising College Access and
            Completion : How Much Can Free College Help
listing_relative_url: /research/2020/10/32466635/raising-college-access-completion-much-can-free-college-help
topicv3: Public Finance Management m1326316 716,Tertiary Education m1326349 482,Labor Markets m1326488 855,Education m1327048 470,Household Surveys m1504071 2761
publishtoextweb_dt: 2020-10-06T00:00:00Z
historic_topic: Education,Public Sector Development,Law and Development,Rural Development,Social Protections and Labor
origu: Chief Economist (LCRCE)
no_of_pages: 68
author_keys: 0: author_key: 1339928
1: author_key: 1276327
model_extractions: Policy Research Working Paper 9428 Raising College Access and Completion How Much Can Free College Help?

_Maria Marta Ferreyra Carlos Garriga Juan David Martin Angelica Maria Sanchez Diaz_ Latin America and the Caribbean Region Office of the Chief Economist October 2020
Policy Research Working Paper 9428

# **Abstract**

Free college proposals have become increasingly popular in many countries, yet cross-country evidence indicates that higher college subsidies raise enrollment but not graduation rates. To capture the evidence and evaluate proposals, this paper develops and estimates a dynamic model of college enrollment, performance, and graduation. A central piece of the model, student effort, has a direct effect on class completion and an indirect effect mitigating the risk of performing poorly or dropping out. The model is estimated using rich student-level data from Colombia, and multiple free college programs are simulated. Among them, universal free college expands enrollment the most but does not affect graduation rates, thereby helping explain the evidence. Performance-based free college, in contrast, raises graduation rates yet has a smaller enrollment impact.

This paper is a product of the Office of the Chief Economist, Latin America and the Caribbean Region. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at mferreyra@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Raising College Access and Completion: How Much Can Free College Help?<sup>∗</sup>

Maria Marta Ferreyra _The World Bank_ Carlos Garriga _Federal Reserve Bank of St. Louis_ Juan David Martin-Ocampo _Banco de la Rep ́ublica Colombia_ Angelica Maria Sanchez Diaz _Georgetown University_ Originally published in the Policy Research Working Paper Series on _October 2020._ This version is updated on _May 2021._ To obtain the originally published version, please email prwp@worldbank.org.

## _Keywords_ : Higher Education, free college, financial aid. _JEL codes_ : E24, I21

> ∗We thank Kartik Athreya, Raquel Bernal, Adriana Camacho, Stephanie Cellini, Doug Harris, Oksana Leukhina, CJ Libassi, Alexander Ludwig, Fabio Sanchez, James Thomas, and Susan Vroman for useful comments and suggestions. We thank Andrea Franco and Nicolas Torres for excellent research assistance. We benefited from seminars at The World Bank, the St. Louis Fed, the DC Economics of Education Working Group, and Universitat Autonoma de Barcelona, and by sessions at AEFP 2018, SED 2018, NASMES 2019, and LACEA 2018. Ferreyra acknowledges financial support from The World Bank. The views expressed herein do not necessarily reflect those of The World Bank, the Federal Reserve Bank of St. Louis, the Federal Reserve System, or of Banco de la Rep ́ublica Colombia. Email contacts: mferreyra@worldbank.org, carlos.garriga@stls.frb.org, jmartioc@banrep.gov.co, amsanchez456@gmail.com. Corresponding author: Carlos Garriga, Research Division, Federal Reserve Bank of St. Louis, P.O. Box 442, St. Louis, MO 63166-0442. Phone: (314) 444-7412, Fax: (314) 444-8731, Email: carlos.garriga@stls.frb.org.
# **1 Introduction**

In modern economies, higher education is crucial to the formation of skilled human capital. Not only can higher education raise a country’s productivity; it can also lower income inequality. By subsidizing access to higher education, policymakers can contribute to these two roles. The question, of course, is how large a subsidy they should provide. Advocates of free college argue that policymakers should provide a full subsidy, resulting in zero tuition for students. While free college has existed for years in a number of countries,<sup>1</sup> free college proposals have sprouted recently in other countries, including the United States, Chile, and Colombia.

Free college advocates claim that only when college is free can it realize its promise, particularly that of lowering inequality. Given Latin America’s distinction of being the most unequal region of the world (World Bank 2016), many in the region view free college as the ultimate solution to persistent inequality and lack of social mobility. The existing evidence on government spending in higher education in the region, however, does not bode well for free college. As Figure 1 shows, countries in the region that finance a greater share of higher education spending do have higher enrollment rates (panel a) yet fail to have higher graduation rates (panel b).

**Figure 1: Higher Education Enrollment and Graduation by Public Spending Share in Latin America.**

<!-- Start of picture text --> 100 70<br>Mexico<br>Chile Argentina<br>80 60 Peru<br>Peru Paraguay Chile<br>60<br>Colombia 50<br>Costa Rica<br>Colombia<br>40<br>Paraguay Mexico 40 Argentina<br>El Salvador<br>20 Honduras El Salvador Costa Rica Honduras<br>30<br>Guyana<br>20 40 60 80 100 20 40 60 80 100<br>Share of public spending (%) Share of public spending (%)<br>a. Enrollment b. Graduation<br>Enrollment rate (%) Graduation rate (%)<br><!-- End of picture text --> _Source_ : UNESCO for 2017 enrollment rates (panel a); own calculations based on SEDLAC (household surveys) for 2017 graduation rates (panel b) and on UNESCO for 2017 share of public spending in higher education. _Notes_ : Panel a shows gross enrollment rate, defined as the ratio between higher education enrollment and the number of individuals ages 18-24. Panel b shows graduation rate, defined as the ratio of number of individuals ages 25-29 who have graduated from higher education, and number of individuals ages 25-29 who have ever started higher education. Share of public spending is defined as government spending in higher education relative to total higher education spending.

To capture this evidence and investigate the potential effects of free college, in this paper we develop a dynamic model of college enrollment, performance, and graduation. In the model, a college student faces risks that may prevent her from completing a class or remaining in college,

> 1Among the countries surveyed by OECD (2018), college is free at public institutions in Argentina, Brazil, Cuba, Czech Republic, Denmark, Ecuador, Estonia, Egypt, Finland, Germany, Greece, Iceland, Mexico, Norway, Panama, Poland, Slovenia, Sweden, Turkey, and Uruguay.

2
and which she can mitigate through effort. We estimate the model with unique administrative data on the universe of higher education students in Colombia—a country highly representative of Latin America—and use the parameter estimates to simulate free college programs which differ in eligibility requirements. We study whether free college affects students’ risk and effort, and perform a simple cost-benefit analysis of free college.

Understanding the trade-offs in free college is critical, particularly in developing economies. In Latin America, higher education enrollment rates rose from 23 to 52 percent between 2000 and 2017. Since only 14 percent of the working age population has completed higher education, Mincerian returns are high—104 percent on average relative to a high school diploma (Ferreyra et al 2017). Although such high returns should attract many individuals to higher education, liquidity constraints severely limit access, as countries subsidize public but not private higher education and often lack credit markets. College subsidies are therefore a promising avenue to broader access. While students in several of these countries have recently taken to the streets in demand of full subsidies, the feasibility of free college has been called into question by these countries’ low growth and tight fiscal constraints in recent years—conditions, of course, that have been seriously aggravated by the COVID-19 pandemic.

Even if free college were implemented, the cross-country evidence presented above suggests that it might not raise the stock of skilled human capital. One possible explanation is that free college might attract academically weak students, yet another is that it might discourage student effort. Free college raises consumption during college, therefore enhancing the attractiveness of being a college student (the “college experience”) and potentially unleashing three effects on effort. First is the _loss-of-urgency effect_ , whereby the student wishes to enjoy the enhanced college experience and loses the urgency to graduate. Second is the _substitution effect_ , whereby the enhanced consumption compensates for greater effort. Third is the _risk effect_ , as the effort changes induced by the other two effects affect not only current but also future performance and graduation prospects. If a strong loss of urgency prevails for many students, graduation rates will not rise and may even fall.

Because of its key role in free college’s success, student effort is a fundamental piece in our study. In the model, heterogeneous high school graduates decide whether to enroll in college. Their labor market wage depends on their final educational attainment. To graduate, a student must complete a set number of classes in a given time. Each year she chooses the number of classes she expects to complete (henceforth, her target), which determines the effort she will make. The number of classes completed by the student in a year depends on her ability, effort, and a performance shock which depends on her cumulative performance. At the end of the year she receives another shock, also dependent on cumulative performance, which may force her to drop out. Effort, then, has a direct effect on class completion and an indirect effect mitigating the risk of performing poorly or dropping out.

Our model captures indivisibilities that limit policy impact. To affect class completion, a policy must induce a non-marginal effort change to complete at least one additional class. And, to affect graduation rates, the policy must induce a large enough effort increase to complete all the required classes. These effort changes may be simply too costly for some students.

Estimated with Simulated Method of Moments, our model captures the main features of the data. In Colombia’s high school class of 2005, a staggering 70 percent of students come from low-income families and only 32.3 percent of students enroll in college within five years. Among those who attend college, only 46 percent graduate—mostly late. Student ability plays 3
a large role in year 1-survival but a lesser role in subsequent performance. The speed at which students accumulate the completed classes required for graduation (henceforth, “cumulative performance” or “cumulative classes completed”) is highly persistent. As a result, early performance is a strong predictor of final outcomes.

According to our estimates, effort has a much greater impact than ability on the production of classes completed. If effort were not modeled, ability’s role would be two or three times larger. The model would not fit the variation in classes completed and college outcomes among students and over time, thereby affecting policy design particularly for students with lower income or ability. The model would also favor college subsidies promoting (positive) selection of high-ability students rather than incentivizing effort—based on an attribute outside students’ control rather than on a choice that they can control.

We simulate multiple free college programs with different eligibility requirements: 1) universal (all students), 2) need-based (low-income students), 3) ability-based (high-ability students), and 4) performance-based (all students eligible in the first year; eligibility conditional on past cumulative performance in subsequent years). We also simulate a need-based version of (3) and (4). In each counterfactual we distinguish between existing students (who enroll in the baseline and the counterfactual) and new students (who do not enroll in the baseline but enroll in the counterfactual) to assess the impact of free college on graduation.

In our simulations, free college programs expand enrollment. Relative to the baseline, the largest expansion is for universal free college (85 percent), followed by need-based and performance-based free college (72 and 66 percent, respectively). On average, new students are of lower income and ability than existing students except under ability-based programs, which induce positive selection of new students.

In contrast with these large enrollment rate effects, overall graduation rate effects are modest—between -3 and 6 percent relative to the baseline. New students pull down the overall graduation rate except under ability-based free college, in which case they pull it up. For existing students graduation rates change little except under performance-based programs, which raise them between 8 and 14 percent. By making free college contingent on performance, these programs eliminate the loss-of-urgency effect and induce greater effort. Our results provide a theoretical justification for the financial aid literature in the U.S., which finds—as we do— positive and large enrollment effects, small or null graduation effects, and larger graduation effects for performance-based than unconditional financial aid.<sup>2</sup> Aggregate free college effects mask great heterogeneity across students. Consider, for instance, universal free college. Enrollment effects are largest for low- and middle-income students, and for mid-ability students. Hence, universal free college subsidizes many students who already enroll in the baseline and do not need the subsidy. Graduation rate effects also vary among students. Graduation rates fall for high-ability or high-income students, who experience a strong loss of urgency, but rise for low-ability or low-income students, who experience a strong substitution effect. In contrast, performance-based free college induces greater effort on the part of _all_ students and leads them all to higher graduation rates.

Behind the graduation rate success from performance-based free college is the greater risk facing students. We develop a measure of anticipated risk and find that, as expected, anticipated risk falls when students exert greater effort or accumulate more classes completed. In the

> 2For recent reviews of this vast literature, see Avery et al (2019) and Dynarski and Scott-Clayton (2013).

4
early years, performance-based free college raises risk relative to the baseline and to universal free college because it makes the level of college consumption contingent on performance. By responding to greater risk with greater effort, students attain better outcomes. Providing students with partial—rather than full—insurance therefore leads to better outcomes.

Using a framework that controls for institutional and potentially differences among countries, our free college simulations rationalize the patterns in Figure 1. Greater college funding substantially raises enrollment rates when many high school graduates face severe financial constraints—as is the case in Latin America—but does not raise graduation rates unless it is performance-based, which is not the case in Latin America. And, even if performance-based programs were more widespread, they would have little effect on graduation rates due to the indivisibilities discussed above: free college fails to induce in _many_ students the _large_ effort increase needed to complete _all_ graduation requirements. This is reminiscent of Oreopoulos and Petronijevic (2019), who find that even when students realize that more effort is needed to improve outcomes, they adjust by lowering expectations rather than increasing effort.

For a policymaker committed to improving college access by providing free college to all or some students, we conduct a simple cost-benefit analysis to inform the choice among our simulated programs. They all raise the per-graduate cost relative to the baseline—if anything, because fewer students pay for college. For a policymaker seeking to raise the fraction of high school graduates that finish college while limiting costs, the best option is performance-based or even need-based free college—but not universal free college. Since the per-graduate cost is quite high in every scenario—ranging from one (baseline) to 2.5 (universal free college) times the per capita GDP—free college should be considered with great care given current fiscal constraints.

Between 2014 and 2018, Colombia enacted a tuition subsidy program, _Ser Pilo Paga_ (”Being diligent pays off”), providing performance-based free college to low-income, high-ability students. Its enrollment effects are quantitatively similar to our predictions (Londo ̃no et al 2020); no data is yet available to study its graduation effects. The program has been recently replaced by a similar one, _Generacion E,_ which costs less. The current clamor in Latin American countries, however, is not for this type of programs but rather for unconditional, universal programs—which, as explained, are not the most efficient use of public funds.

By construction, our counterfactuals assume the most favorable scenario for free college. We assume that colleges have no capacity constraints; that the average and marginal cost of educating new and existing students are the same; and that free college does not crowd out parental transfers to children in college. We do not model taxation (which might be required to pay for free college programs), and assume that the wage of college graduates relative to high school graduates (henceforth, the college premium) does not fall with more college graduates.<sup>3</sup> Relaxing any of these assumptions would lead to even less favorable free college outcomes.

The rest of the paper is organized as follows. Section 2 describes related literature, and Section 3 describes data sources and institutional environment. Section 4 presents stylized facts from the data. Section 5 presents the model, and Section 6 discusses its empirical implementation. Section 7 describes estimation strategy and results. Section 8 presents the free college counterfactuals, including analyses of anticipated risk, fiscal costs, and potential labor market general equilibrium effects. Section 9 concludes.

> 3In Section 8.8 we investigate potential general equilibrium effects from the greater supply of college graduates. Since we find them to be small even in the medium run, we abstract away from them in our analysis.

5
# **2 Related Literature**

Our paper relates to a large literature estimating sequential schooling models under uncertainty, with seminal contributions by Keane and Wolpin (2001), Eckstein and Wolpin (1998), and Keane (2002). This literature models college enrollment, performance, and college outcomes, and uncovers structural parameters based on students’ observed choices during college. In one strand of this literature, researchers model students as acquiring information (learning) throughout college—regarding, for instance, their ability and preferences for college or specific majors, and their expected labor market performance. This literature includes Arcidiacono (2004), Arcidiacono et al (2016), Ozdagli and Trachter (2011), Stinebrickner and Stinebrickner (2014), and Trachter (2015). As in these papers, students in our model adjust their expected graduation probability based on classes completed and choose effort accordingly.

The idea that higher education is risky is not new (Levhari and Weiss 1974, Altonji 1993, Akyol and Athreya 2005), but the recent availability of college transcript data in the U.S. has helped estimate the role of risk in students’ performance. These data reveal substantial and persistent heterogeneity in students’ credit accumulation rates, which are strongly related to graduation probability. According to Hendricks and Leukhina (2017, 2018), based on their credit accumulation rates more than 50 percent of college entrants should be able to forecast whether they are at least 80 percent likely to graduate. According to Stange (2012), the large uncertainty faced by students makes them place a high value on the ability to drop out at any point rather than pre-commit to completing all graduation requirements. Our paper is similar to these in the use of administrative data to track students’ performance, but different in that the risk associated to class completion or college continuity is not fully exogenous as in these papers but depends on an endogenous variable—student effort.

While the literature has placed much attention on the role of ability in performance and college outcomes, a growing line of research highlights the role of effort. Zamarro, Hitt, and Mendez (2019) use data from the Program for International Student Assessment (PISA) to show that different effort measures explain about a third of observed cross-country test score variation. Stinebrickner and Stinebrickner (2004) rely on time use surveys to estimate the effects of study time on grades. Ariely et al (2009) show that using incentives can help the average student improve her test performance, though the effect is limited on high-ability students. Beneito et al (2018) provide evidence that the tuition increase implemented by Spanish colleges in 2012 boosted student effort. Ahn et al (2019) model effort in response to grading policies. We contribute to this line of research by explicitly modeling the role of effort and embedding it in a dynamic setting, where it affects class accumulation and risk mitigation.

In an efficient and equitable world, college enrollment would depend on student ability rather than parental resources (Cameron and Heckman 1998 and 1999, Carneiro and Heckman 2002). In Colombia, as in other countries, parental resources matter greatly to college enrollment even controlling for ability. This provides strong evidence for credit constraints limiting college access, as discussed in a large literature. Lochner and Monge-Naranjo (2011) develop a model that helps explain the rising importance of family income for college attendance in the U.S. even in the presence of credit. Solis (2017) finds that relaxing credit constraints in Chile had a positive impact on enrollment and college years completed, particularly for low-income students. Parental resources and background, however, may be of limited importance. Hai and Heckman (2017) show that equalizing initial ability has larger effects on college outcomes and 6
inequality than equalizing parental background. The importance of credit access falls when students can supply work to pay for college (Garriga and Keightley 2007). Although Colombia is a large developing economy, the market for student loans is very limited, covering only 7 percent of students in 2003 (ICETEX 2010). Lack of family resources, limited opportunities to work during college, and missing credit markets for student loans are clear impediments to college access in countries such as Colombia.

In those countries, tuition subsidies may therefore expand college access. Our paper complements the literature on Chile’s recent free college policies (Bucarey 2018) and England’s free college elimination (Murphy et al 2019). It also joins in the vast literature of college financial aid in the U.S. (cited in Section 8.5), including the recent literature on free community college and the so-called “Promise” programs implemented in multiple U.S. states.

# **3 Data and institutional environment**

In this section we describe our data sources, Colombia’s higher education environment, and our study cohorts. Appendix A.1 contains further details.

Our data consists of student- and program-level information drawn from three administrative datasets: Saber 11, SPADIES, and SNIES. The Saber 11 dataset contains students’ test scores at the national mandatory high school exit exam (also named Saber 11) as well as selfreported socio-economic information. Saber 11 is a standardized test that measures a student’s preparedness for higher education—reflecting not only her innate ability but also her primary and secondary education quality. We use it as a measure of student ability, broadly understood as academic readiness for higher education. Family income is reported in brackets defined relative to the monthly legal minimum wage (MW), which is equal to 381,000 Colombian pesos (COP) in 2005 (US$ 1 = 2,321 COP in 2005).

SPADIES tracks every higher education student. For each semester, it records the number of classes attempted and passed by the student, as well as her graduation or dropout date. It does not record the specific classes attempted, number of attempts per class, or class grades. SNIES contains program-level information, including institution, field, and tuition. We focus on bachelor’s programs, which capture approximately 80 percent of the country’s total higher education enrollment. Enrollment is quite evenly split between public and private institutions; public institutions are heavily subsidized and charge a considerably lower tuition than private institutions.

To analyze college enrollment, we focus on the 2005 high school cohort. We calculate deciles and quintiles of their ability distribution; in what follows, deciles and quintiles always refer to this distribution. For consistency with the model, we classify students into “student types” defined by combinations of student ability quintiles and family income brackets. Appendix table A1 shows the distribution of student types in this cohort. While a remarkable 70 percent of high school graduates come from the lowest two income brackets, less than 5 percent come from the top one. Not surprisingly, income and ability and strongly and positively correlated.

For our analysis of final outcomes and academic progression we focus on students from the 2006 college entry cohort that enroll in five-year bachelor’s programs within five years of finishing high school. Since every program requires a different number of classes for graduation, we normalize the requirement to 100 classes for every program to facilitate exposition.We assume 7
that students are required to complete the same number of classes (20) every year. We use the term “classes completed” (or “performance”) to denote the number of classes completed _in a given year_ . We use “cumulative classes completed” (or “cumulative performance”) in a given year as the total number of classes completed _over all years_ up to (and including) that one. A student is on track for on-time graduation when she has completed her on-track requirements, equal to accumulating 20, 40, 60, 80, and 100 classes by the end of years 1 through 5 respectively.

It is useful to classify students into tiers based on their cumulative performance relative to the corresponding year’s on-track requirement. Tiers 1 through 4 correspond to students who complete the following percentage of the on-track requirements for the year: 95 percent or more for tier 1, (85, 95] percent for tier 2, (65, 85] percent for tier 3, and 65 percent or less for tier 4.<sup>4</sup> (See Appendix Table A2 for further details on tier classification). Importantly, a student can change tiers over time—catching up to a higher tier or falling behind to a lower one.

# **4 Stylized Facts**

The data has distinctive features that our model seeks to capture. We describe them below.

_Fact 1: Students of higher income or ability are more likely to enroll in college._ Although the overall enrollment rate is 32 percent,<sup>5</sup> enrollment rates vary widely among student types (see Table 1), and rise with income and ability. On average, the enrollment gap between the highest and lowest income brackets is equal to 55 percentage points (pp)—similar to the 50-pp gap between the highest and lowest ability. Free college may therefore have ample room to raise enrollment.

**Table 1: Enrollment Rates by Income and Ability.**

|Income||Abi|lity quin|tile|||
|---|---|---|---|---|---|---|
|Bracket|1|2|3|4|5|Total|
|5+ MW|32.85|44.14|58.87|69.23|83.85|73.38|
|3-5 MW|28.71|39.75|48.41|62.99|79.24|62.03|
|2-3 MW|20.34|28.72|36.96|48.03|67.88|43.50|
|1-2 MW|13.94|18.36|23.85|33.84|54.22|28.05|
|_<_1 MW|9.05|12.67|17.20|26.56|43.93|17.67|
|Total|13.43|19.15|26.20|38.93|63.74|32.29|

_Source_ : Calculations based on SPADIES and Saber 11 for 2005 high school graduates. _Notes_ : Each cell reports percent of high school graduates from a given income bracket and ability quintile who enrolled in a bachelor’s program between 2006 and 2010. Income reported in brackets; MW = monthly minimum wage. Ability reported in quintiles of standardized Saber 11 scores; quintile 1 is the lowest.

_Fact 2: Dropout rates are high, particularly for low-ability students in year 1._

> 4To exemplify, consider a student who accumulates 16, 35, 42, 50, and 60 classes by the end of years 1 through 5 respectively. This amounts to 80 (=16/20), 88 (=35/40), 70, 62.5, and 60 percent of the corresponding on-track requirements. Thus, the student falls in tiers 3, 2, 3, 4, and 4 in years 1 through 5 respectively.

> 5For comparison, in the US the enrollment rate of individuals ages 16-24 who finished high school in 2005 and started college right away (rather than within five years) is 44.6% (Source: Digest of Education Statistics).

8
Only 45.7 percent of students from our entry cohort graduate—15.1 percent graduates on time (in five years) and 30.6 percent graduates late (in 6-8 years).<sup>6</sup> Dropout rates are far from uniform over time. As Figure 2 shows, over a quarter of college students (half of all dropouts) leave in the first year, and the first two years account for about 70 percent of all dropouts. Similar to enrollment rates, dropout rates vary widely across student types (Table 2). Conditional on income, higher ability students have lower dropout rates; on average, the dropout rate gap between the highest and lowest ability quintiles is equal to 25 pp. In contrast, dropout rates do not vary much by income—suggesting that free college might affect enrollment more than dropout decisions.

Final college outcomes vary substantially by ability. High-ability students are more likely to graduate, whether on time or late (Figure 3), whereas low-ability students are more likely to drop out, particularly in year 1. After year 1, dropout rates vary little across abilities.

**Figure 2: Dropout Timing.**

<!-- Start of picture text --> 60<br>% relative to entry cohort<br>% relative to all dropouts<br>50<br>40<br>30<br>20<br>10<br>0<br>1 2 3 4 5 6 7 8<br>College year<br>Percent<br><!-- End of picture text --> _Source_ : Calculations based on SPADIES, for students from the 2006 entry cohort (first semester). _Notes_ : The blue line shows percent of students who drop out in a given college year. The red line shows, among all dropouts, percent of those who drop out in a given college year.

**Figure 3: College Outcomes by Ability.**

<!-- Start of picture text --> 50<br>40<br>30<br>20<br>10<br>0<br>1 2 3 4 5<br>Ability quintile<br>Dropout 1st year Dropout 2nd year<br>On−time graduate Late graduate<br>Dropout after 2nd year<br>Percent<br><!-- End of picture text --> _Source_ : Calculations based on SPADIES. Students belong to the 2006 entry cohort (first semester). _Notes_ : For each quintile, the graph shows the percent of students who attained each outcome. Percents add to 100 by quintile.

# _Fact 3: Cumulative performance varies more within than across abilities._

Figure 4’s panel a shows average number of classes completed by ability quintile in year 1. The thick black line depicts average over all students, whereas the color lines depict averages among the students with the following final outcomes: on-time graduation, late graduation, dropout in year 1, and dropout later. On average, high-ability students complete more classes than lowability students in year 1. However, the figure also shows a pattern repeated every year: average classes completed varies little _across_ abilities—overall and conditional on outcomes—but varies greatly _within_ abilities. For a given ability, on-time graduates complete more classes than late graduates, who complete more than dropouts. In other words, number of classes completed—as

> 6For comparison, in the U.S. 59.2 percent of students from the 2006 cohort graduated within six years—39 percent on time (in four years), and 20.2 percent late  ̇Source: Digest of Education Statistics.

9
**Table 2: Dropout Rates by Income and Ability.**

|||Abi|lity qui|ntile|||
|---|---|---|---|---|---|---|
||1|2|3|4|5|Total|
|5+ MW|81.36|65.83|61.48|52.13|39.04|44.73|
|3-5 MW|74.23|69.44|62.21|57.86|43.77|51.33|
|2-3 MW|68.54|67.58|63.73|57.68|46.53|55.09|
|1-2 MW|71.59|66.64|62.21|57.66|50.56|57.82|
|_<_1 MW|69.04|67.95|61.34|55.94|50.30|58.71|
|Total|70.99|67.44|62.37|56.96|45.84|54.36|

_Source_ : Calculations based on SPADIES for students from the 2006 entry cohort (first semester). _Notes_ : Each cell reports percent of students from a given income bracket and ability quintile who drop out of their bachelor’s program. A student is classified as a dropout if she does not graduate within eight years. Income is reported in brackets; MW = monthly minimum wage. Ability is reported in quintiles of standardized Saber 11 scores. Quintile 1 is the lowest.

early as in year 1— is a powerful predictor of final outcomes. This point is further illustrated in panel b, which classifies students into tiers by the end of year 1. Consistent with panel a, it shows the strong predictive power of early performance—as explained by the next stylized fact.

**Figure 4: First-year Classes Completed and College Outcomes.**

<!-- Start of picture text --> 23 60<br>21<br>19 45<br>17<br>15 30<br>13<br>15<br>11<br>9<br>1 2 3 4 5 0<br>Ability quintile Tier 1 Tier 2 Tier 3 Tier 4<br>On−time graduate Late graduate On−time graduate Late graduate<br>Dropout after 1st year Dropout 1st year Dropout 1st year Dropout 2nd year<br>All students Dropout after 2nd year<br>a. First-year classes completed b. College outcomes by first-year<br>by college outcome tier of classes completed<br>completed Percent<br>Avg. number of classes<br><!-- End of picture text --> _Source_ : Calculations based on SPADIES for students from the 2006 entry cohort (first semester). _Notes_ : In panel a, each color represents a college outcome. The green line, for instance, shows average number of classes completed by the end of year 1 by students of a given ability quintile who went on to graduate on time. The thick black line does the same for all students regardless of college outcome. In panel b, the graph shows the percent of students from a given performance tier by the end of year 1 that attain each college outcome. For the first year, tier 1 corresponds to 19+ classes completed; tier 2 to [17, 19); tier 3 to [13, 17); and tier 4 to [0, 13).

# _Fact 4: Cumulative performance is highly persistent over time._

To investigate whether a student’s performance changes over time, Figure 5 depicts a student’s probability of attaining each of four following outcomes—same-tier persistence, dropout, catch up, and fall behind—by the end of a year conditional on her previous year’s performance tier.<sup>7</sup>

> 7For example, a student who finished first year in tier 2 has second-year probabilities of persistence, dropout,

10
**Figure 5: Tiers of Cumulative Classes Completed: Transitions Throughout College.**

<!-- Start of picture text --> 100 100 Tier 1 100 Tier 2 to tier 1 100 Tier 1 to tier 2,3&4<br>Tier 2 Tier 3 to tier 1&2 Tier 2 to tier 3&4<br>90 90 Tier 3 90 Tier 4 to tier 1,2&3 90 Tier 3 to tier 4<br>Tier 4<br>80 80 80 80<br>70 70 70 70<br>60 60 60 60<br>50 50 50 50<br>40 40 40 40<br>30 30 30 30<br>20 20 20 20<br>Tier 1<br>10 Tier 2 10 10 10<br>Tier 3<br>0 Tier 4 0 0 0<br>2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5<br>Year Year Year Year<br>a. Persistence b. Dropout c. Catch−up d. Fall behind<br>Percent Percent Percent Percent<br><!-- End of picture text --> _Source_ : Calculations based on SPADIES for the 2006 entry cohort (first semester). _Notes_ : Each panel shows the probability that a student who ended the previous year in a given tier experiences the following outcomes in the current year: persist in the tier (panel a), drop out (panel b), catch up (rise) to a higher tier (panel c), or fall behind to a lower tier (panel d). For a given year and tier, probabilities add up to 100 across panels. For example, a student who finished year 1 in tier 3 is depicted in green. In year 2, she is 35, 23, 18, and 24 percent likely to persist in tier 3, drop out, catch up to tiers 1 or 2, and fall behind to tier 4 respectively.

Same-tier persistence rises over time (panel a), in part because dropout rates fall over time (panel b). Although low-performing students are more likely than others to drop out, students from all tiers face a non-zero dropout probability. Some students move across tiers by catching up or falling behind (panels c and d). Higher-performing students are more likely to catch up and less likely to fall behind than others, yet all students face a non-zero probability of falling behind. Despite transitions among tiers, the most likely outcome for a student in tier 1, 2, or 3 is to remain in it the following year.

Reaching year 5 does not guarantee graduation, but cumulative performance up to that point is strongly related to final outcomes. As Figure 7 shows, most students in the top three tiers graduate but most bottom-tier students do not. Further, most top-tier students graduate on time.

# _Fact 5: Within abilities, cumulative performance becomes more concentrated over time._

As low-performing students drop out over time, the distribution of cumulative performance becomes more concentrated at the top tiers. Panel a of Figure 6 shows the distribution across performance tiers at the end of year 1 by ability quintile. High-ability students are the most likely to belong to the top tier while low-ability students are the most likely to belong to the bottom tier. Nonetheless, performance varies greatly within ability quintiles, and a sizable catch up, and fall behind equal to 29, 14, 20, and 39 respectively. Beginning in year 5, students can ”transition” into graduation as well.

11
fraction of students from every quintile is concentrated in the middle tiers. Because of this “thick middle,” cumulative performance varies little, on average, across abilities. A similar picture holds for year 5 (panel b), although by then all the performance distributions are more concentrated in the top three tiers. Further, the performance distribution for low-ability students is more dispersed than that of high-ability students in both years.

**Figure 6: Tiers of Cumulative Classes Completed, by Ability.**

<!-- Start of picture text --> 60 60<br>50 50<br>40 40<br>30 30<br>20 20<br>10 10<br>0 0<br>1 2 3 4 5 1 2 3 4 5<br>Ability quintile Ability quintile<br>Tier 1 Tier 2&3 Tier 4 Tier 1 Tier 2&3 Tier 4<br>Percent Percent<br><!-- End of picture text --> <!-- Start of picture text --> a. Year 1 b. Year 5<br><!-- End of picture text --> _Source_ : Calculations based on SPADIES for students from the 2006 entry cohort (first semester). _Notes_ : For students of a given ability who start year 1, panel a shows their classification into tiers of cumulative classes completed by the end of year 1. Panel b does the same for students beginning year 5.

# _Fact 6: Higher-ability students attempt a higher number of classes._

Although a student cannot fully control her performance, she can control the number of classes she attempts (registers for) in a given year. While she may not pass all the classes attempted, the number of attempts is informative of her intended effort. As Figure 8 shows, on average students attempt fewer than the 20 required classes. In addition, on average high-ability students attempt more classes than low-ability students. We will return to these facts when discussing identification of our empirical model.

**Taking stock.** The data shows that students of higher income or ability are more likely to enroll in college. Conditional on enrolling, lower ability students are much more likely to drop out, particularly in year 1. Through this channel, ability serves a strong predictor of graduation. Ability, however, is not a strong predictor of cumulative performance. Although cumulative performance varies little, on average, across abilities, it varies greatly within abilities. Further, cumulative performance is highly persistent over time—students who start on track are more likely to remain on track, less likely to fall behind, and more likely to catch up should they fall behind. As a result, cumulative performance is highly predictive of final outcomes. Nonetheless, even on-track students face the risk of falling behind or dropping out. Our model seeks to capture these data features.

12
**Figure 7: Graduation by Cumulative Classes Completed in Year 5.**

<!-- Start of picture text --> 100<br>80<br>60<br>40<br>20<br>0<br>Tier 1 Tier 2 Tier 3 Tier 4<br>Graduation rate On−time graduation rate<br>Percent<br><!-- End of picture text --> _Source_ : SPADIES. Students belong to the 2006 entry cohort (first semester).

_Notes_ : The figure refers to students who start year 5 and classifies them into tiers based on their cumulative classes completed by the end of the year. For each tier, the blue bar shows the percent of students who graduate, and the green bar shows the percent of students who graduate on time, relative to all graduates from that tier.

**Figure 8: Average Number of Classes Attempted in Year 1, by Ability.**

<!-- Start of picture text --> 20<br>19.5<br>19<br>18.5<br>18<br>17.5<br>17<br>1 2 3 4 5 6 7 8 9 10<br>Ability decile<br>Avg. number of classes taken<br><!-- End of picture text --> _Source_ : Calculations based on SPADIES for students from the 2006 entry cohort (first semester). _Notes_ : Figure shows the average number of classes attempted by students in year 1 by ability decile.

# **5 Model**

We model a representative cohort of high school graduates who differ in ability, family income, and idiosyncratic preferences. They choose whether to enroll in college or enter the labor force as high school graduates. College graduation requires the completion of a set number of classes. The combination of ability, effort, and performance and dropout risks determines class completion and college outcomes. Students can mitigate risk by being on track in class completion. After college, students enter the labor market and remain there until retirement; wages depend on educational attainment (high school graduate, college graduate, or college dropout) and years of work experience.

## **5.1 Student endowments and preferences**

High school graduates differ in ability, _θ ∈{θ_ 1 _, θ_ 2 _, . . . , θnθ}_ , and parental resources received if they enroll in college, _y ∈{y_ 1 _, y_ 2 _, . . . , yny }_ . The combination of ability and parental resources defines the student’s type _j_ = 1 _,_ 2 _, . . . , J_ , with _J_ = _nθ × ny_ . For simplicity we refer to _y_ as “income.” Students receive _y_ only if enrolled in college; if not, they earn a labor market wage as explained below.

A period, _t_ , is equal to a year. For college students, instant utility depends on consumption, _c_ , and study effort, _e_ . Effort cost is heterogeneous across students as it depends on ability. The instant utility at _t_ is _U_ ( _ct, et, θ_ ) = _u_ ( _ct_ ) _− g_ ( _et, θ_ ). Function _U_ satisfies _u_<sup>_′_</sup> _>_ 0 _, u_<sup>_′′_</sup> _<_ 0 _, g_ 1 _, g_ 11 _>_ 0, _g_ 2 _<_ 0 _,_ and _g_ 12 _<_ 0 _._ Thus, effort cost is increasing and convex in effort, and total 13
and marginal effort costs are lower for higher-ability students.

For workers, instant utility depends only consumption. Both for college students and workers, the discount factor is _β_ .

# **5.2 College decisions**

We model college as a multi-period, risky investment. To graduate, students must complete a required number of classes, _h_<sup>_grad_</sup> . Students must spend at least five years in college in order to graduate, not to exceed eight. We use three concepts related to the number of classes per year. First is _x_ ̄, the annual number of classes required to graduate on time. Second is _qt_ , the expectation formed by the student, at the beginning of the year, of how many classes she will ̄ complete (henceforth, “target.”) The target can be greater or smaller than _x_ . Importantly, the student chooses effort for the year based on her target. Third is _xt_ , the actual number of classes she completes by the end of the year, which is a function of her effort, ability, and intervening luck. It can be greater or smaller than _qt_ because of “good” or “bad” luck, respectively.

Note that the model does not include the concept of “registering” for a class, and _qt_ should not be viewed as the number of classes for which the student registers. Instead, we assume that the student simply takes a class. This is because, in Colombia, there is no cost to add or withdraw from a class. Hence, students in Colombia register for the maximum number of classes they might complete, which may not be the number they expect to complete. For example, a student may start two classes but expect to complete only one ( _qt_ = 1), perhaps because the other class is poorly taught. She may find that the other class is better taught than expected and complete the two classes ( _xt_ = 2); or that both classes are poorly taught and complete none of them ( _xt_ = 0); or that her expectation was correct and complete just one ( _xt_ =1.) Since the student’s behavior is dictated by the number of classes she expects to pass rather than the number she attempts, we model the former. In estimation we use data on the number of classes attempted and view it is an upper bound for _qt._

## **5.2.1 College technology**

Let _ht_ denote the cumulative number of classes completed up to the end of _t_ , or _ht_ =<sup>�</sup><sup>_t_</sup> _n_ =1<sup>_xn_.</sup> Let _h_<sup> ̄</sup> _t_ denote the average number of classes completed per year up to end of _t_ , or _h_<sup> ̄</sup> _t_ = _ht/t_ . We assume that students start college with _h_ 0 = _h_<sup> ̄</sup> 0 = 0, and by the end of year 1 attain _h_ 1 = _h_<sup> ̄</sup> 1 = _x_ 1. While enrolled in college, students complete classes in year _t_ according to the following production function: The function describes completed classes, _xt_ , as a multiple of _<u>x</u>_ <u>.</u> The scalar _H_ ( _._ ) is a function of ability, effort, and a shock to classes completed (or performance shock), _zt >_ 0. We assume _H_ is non-negative and can be lower or greater than one. The shock, drawn from a continuous distribution known to the student, includes a random _i.i.d_ . component, as well as a component that depends on the student’s ability, cumulative classes completed up to beginning of _t_ , and year. The dependence of the shock on past cumulative classes completed seeks to capture the observed persistence of classes completed. Students can thus affect their “luck” next period by accumulating as many classes as possible this period. We allow the shock to depend on ability 14
to reflect that ability may be correlated with other elements, not modeled, that systematically “luck.”<sup>8</sup> If the student knew _zt_ when choosing her effort, then choosing _et_ would be equivalent to choosing classes completed, _xt_ . Since, as explained below, the student chooses _et_ before _zt_ is realized, choosing effort is equivalent to choosing a target, _qt_ , where _qt_ = _E_ ( _xt_ ). Thus, Assuming that _H_ ( _·_ ) is linear on _zt_ so that it can be expressed as _H_ ( _zt, θ, et_ ) = _ztH_<sup> ̃</sup> ( _θ, et_ ), target and effort are functions of _E_ ( _zt_ ): Meanwhile, the actual number of classes completed, _xt_ , is a function of the effort chosen given the target, and of the realized _zt_ . Cumulative classes completed by the end of the year, _ht_ , is Finally, we assume that the production function in (1) is such that, when the student supplies zero effort, she completes zero classes: _H_ ( _zt, θ,_ 0) = 0 _._ For every student type, there always exists a level of effort, _<u>et</u>_ , that allows her to complete _<u>x</u>_ <u>,</u> or _H_ ( _zt, θ,_ _<u>et</u>_ ) = 1. Also, students can exert high effort to compensate for low ability or expected “bad luck”.

# **5.2.2 The student’s optimization problem**

The student faces a sequential problem. We differentiate between the pre-graduation years (when she cannot yet graduate) and the graduation years (when she is eligible to graduate). We divide each year into two sub-periods. In the _first sub-period_ , the student chooses her target number of classes and hence effort. At the end of the first sub-period she receives the shock to the number of classes completed, which determines her actual (as opposed to target) number of classes completed. In the _second sub-period,_ she graduates if she has accumulated the required number of classes; otherwise she draws a shock that determines whether she will remain in college next year or drop out (“dropout shock”). Thus, as long as she has not completed her graduation requirements, the student draws two shocks per year—one to classes completed in the year and another to college continuity. The two shocks are endogenous in the sense that they depend on cumulative performance, which depends on effort. In a given year, a college student’s state vector is ( _t, ht−_ 1 _, θ, y_ ). Appendix Figure B1 summarizes the timing of events and decisions, described in detail below.

**Pre-Graduation Years (** **_t_ = 1,...,4).** During these years, students have not yet accumulated the required number of classes for graduation, or _ht < h_<sup>_grad_</sup> . In year 1, students start with zero cumulative classes completed, _h_ 0 = 0, and are heterogenous only in their type. Since students of a given type may vary in their first-year completed classes _h_ 1 (depending, as we will

> 8For example, lower ability students may choose less selective programs than others, or may have lower levels of the non-cognitive skills necessary to succeed in college. In the first case, _E_ ( _z_ ) would be higher for lower-ability students than for higher-ability ones; in the second case, it would be lower. In our estimation we let the data identify the sign of the relationship between _E_ ( _z_ ) and _θ._

15
see below, on their _z_ 1 shock), from year 2 onward students are heterogenous not only in their type but also in their cumulative classes completed at the beginning of the year, _ht−_ 1.

In each of these years, at the beginning of the _first sub-period_ the student chooses _et_ (and hence _qt_ ) before _zt_ is realized. As in (2), the chosen target is a function of the _E_ ( _zt_ ). At the end of the first sub-period _zt_ is realized and determines the number of cumulative classes completed, _ht ≥ ht−_ 1. In the _second sub-period_ the student receives her dropout shock, _d_<sup>_drop_</sup> _t_ = _{_ 0 _,_ 1 _}_ , which determines whether she will remain in college next year or drop out, respectively. The probability that this shock leads her to drop out is a function of her cumulative classes completed after the realization of _zt_ , that is _ht_ , as well as her type and the year: We assume that, by the end of _t_ , if the student has accumulated less than a pre-specified number ̃ of classes for the year, _h_<sup>_drop_</sup> _t_ , she must drop out: _p_<sup>_d_</sup> ( _t, ht < h_<sup>_drop_</sup> _t , θ, y_ ) = 1. If, in contrast, she completes _x_ ̄ classes each year and is on track for on-time graduation, her dropout probability is very low: _p_ ̃<sup>_d_</sup> ( _t, ht, θ, y_ ) _≈_ 0. In general, _p_ ̃<sup>_d_</sup> is decreasing in _ht_ . Importantly, the student can ̃ lower _p_<sup>_d_</sup> by exerting effort, which raises _ht_ .

If the student drops out, she will receive the market wage of a college dropout from the following year onward; the value of dropping out is _V_<sup>_drop_</sup> ( _t_ +1) _._ Meanwhile, the value of remaining in college is _V_<sup>_coll_</sup> ( _t_ + 1 _, ht, θ, y_ ).

**Graduation Years (** **_t_ = 5,...,7).** These years are different from the previous ones in that college students become eligible to graduate. Those who have fulfilled graduation requirements, _ht ≥ h_<sup>_grad_</sup> , will graduate and enter the labor market, whose value is _V_<sup>_grad_</sup> ( _t_ + 1). For them, additional classes beyond _h_<sup>_grad_</sup> yield zero marginal benefits. Remaining students draw the dropout shock to determine college continuity the following year.

**Terminal year (** **_t_ = 8).** This is the last year that a student is allowed in college. At the end of it, only two outcomes are possible: the student graduates if _h_ 8 _≥ h_<sup>_grad_</sup> , or drops out otherwise. Continuation values are equal to _V_<sup>_grad_</sup> (9) and _V_<sup>_drop_</sup> (9) respectively.

We can now present the student’s dynamic optimization problem from the _first subperiod_ of each college year: 16
Here, the argument of _Ez_ [ _·_ ] is the continuation value function. Variable _T_ ( _·_ ) is tuition, constant regardless of the target, _qt_ .<sup>9</sup> To accommodate our counterfactuals, we write _T_ ( _·_ ) in general form so that it can vary by year, cumulative classes completed, ability, or income. In our baseline it varies only by _y_ (see Section 6.2.1 below). For low-income students tuition might exceed income, which would violate the _ct >_ 0 constraint and make enrollment unfeasible. Note the severe credit constraint: students cannot borrow to pay for tuition nor can they save.<sup>10</sup> The policy function is the sequence of optimal efforts, _e_<sup>_∗_</sup> ( _t, ht, θ, y_ ), that solve the dynamic problem defined in (6).

# **5.3 Workers**

An individual can join the labor force after graduating from high school or college, or after dropping out from college.<sup>11</sup> The worker’s optimization problem, written in recursive form, is where _V_<sup>_m_</sup> ( _t_ ) is the value function of a worker with educational attainment _m_ = _{hs, grad, drop}_ , denoting high school graduate, college graduate, and college dropout respectively. The worker’s wage, _wt_<sup>_m_,isspecifictoeducationalattainmentandvarieswith</sup><sup>_t_toallowforreturnstoexpe-</sup> rience.

# **5.4 Enrollment decision**

To decide whether or not to enroll in college, a high school graduate compares the expected payoff of two choices—going to college, or joining the labor force as a high school graduate. The enrollment decision is a discrete choice problem, where the payoff associated to each option is the sum of three components. The first component is the expected value of going to college, _V_<sup>_coll_</sup> ( _t_ = 1 _, h_ 0 = 0 _, θ, y_ ) or of entering the labor force as a high school graduate, _V_<sup>_hs_</sup> . The second component is a type-specific preference for college enrollment, _ξj_ = _ξ_<sup> ̃</sup> ( _θj, yj_ ), which captures type-related unobserved factors (i.e., parental education) that affect enrollment. We normalize the unobserved preference for joining the labor force as a high school graduate to zero for all types. The third component is an idiosyncratic choice-specific shock for each individual, _ε_<sup>_hs_</sup> and _ε_<sup>_coll_</sup> , corresponding to working as a high school graduate or enrolling in college, respectively. Thus, all individuals face the same _V_<sup>_hs_</sup> and individuals of a given type face the same _V_<sup>_coll_</sup> and _ξj_ , yet individuals within and across types differ in their idiosyncratic shocks. We assume that

> 9This is in keeping with the Colombian context, where tuition is fixed regardless of the number of classes taken.

> 10We do not model student’s decision to work during college because our administrative data does not record this information. Further, data from Colombia’s National Survey of Time Use ( _ENUT_ ) reveals that high-income college students are more likely to work during college than their lower-income counterparts, suggesting that the primary motivation to work is not necessarily to pay for college (details available upon request). For a model of student workers, see Garriga and Keightley (2007).

> 11 We assume that workers consume all their earnings and do not have access to credit markets, which is an accurate representation of developing economies. Since wages rise with experience and workers discount the future at the interest rate, they have no incentives to save.

17
_ε_<sup>_hs_</sup> and _ε_<sup>_coll_</sup> are _iid_ and distributed Type I Extreme Value with a scaling factor of _σε_ . The individual chooses to attend college if As a result, the probability of college enrollment for an individual of type _j_ is Its complement, _P_<sup>_hs_</sup> ( _θ, y_ ) = 1 _− P_<sup>_coll_</sup> ( _θ, y_ ), is the probability of joining the labor force as a high school graduate.

# **6 Empirical implementation**

In this section we describe the parameterization and computational version of the model. We also describe the algorithm to compute model predicted values for a given parameter point.

## **6.1 Functional forms**

In the model, _t_ = 1 corresponds to age 18. Retirement age is 65, or _t_ = 48. Regardless of her educational attainment or when she joined the labor force, the individual accrues returns to experience (or becomes ”experienced”) from age 35 ( _t_ = 28) onward.

The utility of college students is given by where the need to meet the minimum consumption level, _<u>c,</u>_ might limit low-income students’ ability to enroll in college. To prevent this, we set _<u>c</u>_ equal to one million COP.<sup>12</sup> The utility of workers is given by We set _β_ = 0 _._ 04, which is consistent with an implicit discount rate of 4 percent, and assume _σε_ = 1.

The production function to complete classes has constant returns to scale in ability and where _α ∈_ (0 _,_ 1) is the elasticity of classes completed with respect to ability. Consistent with the model, we set _x_ ̄ = 20 classes. We set the minimum number of classes required to graduate,

> 12 Our chosen value for _<u>c</u>_ guarantees that, in our computational models, all students attain positive consumption if they enroll in college. We can think of _<u>c</u>_ as the minimum consumption guaranteed to college students through student subsidies, such as those for food and transportation in Colombia.

18
_h_<sup>_grad_</sup> , equal to 98.<sup>13</sup> The functional form for the _zt_ shock is as follows: where _h_<sup> ̃</sup> _t−_ 1 is a measure of past cumulative number of classes completed, with _h_<sup> ̃</sup> _t−_ 1 = ln( _h_<sup> ̄</sup> _t−_ 1) for every _t >_ 1, and _h_<sup> ̃</sup> 0 = 0 for _t_ = 1 _._ The terms associated with _d_ 1 allow the shock distribution to differ in year 1, when _d_ 1 = 1. The shock also depends on an _iid_ component, _νt_ , drawn from the uniform distribution _U_ (0 _,_ 1). The functional form in (13) ensures that _zt ∈_ (0 _,_ 1) for any combination of parameter values and for all _h, θ_<sup> ̃</sup> _∈_ R. Importantly, all the parameters in (13) affect the mean and variance of _zt_ . In Section 7.3 below we discuss the effect of _h_<sup> ̃</sup> _t−_ 1 and _κθ_ on this mean and variance at our specific parameter estimates.

We parameterize the probability of dropping out as where _δ_ ( _t, θ, y_ ) is a year-, ability- and income- specific fixed effect, and _h_<sup> ̃</sup> _t_ measures cumulative performance over all periods, including the current one. Evaluating _p_ ̃<sup>_d_</sup> ( _t, ht, θ, y_ ) at _π_ = 0 yields the “exogenous dropout probability”—the dropout probability for students of a given type in a given year if they had accumulated no classes. It is “exogenous” because it is independent of effort. For example, low-income, low-ability students may have a high exogenous dropout probability in year 1—perhaps due to lack of parental guidance to navigate college—yet a lower one in subsequent years.

The model’s full parameter vector is Θ = (Θ<sup> ̃</sup> _,_ **_ξ_** _,_ **_δ_** ), where is the vector of parameters common across individuals. Vector **_ξ_** _J×_ 1 contains type-specific unobserved preferences for college, _ξj_ (see (9)) and **_δ_** ( _J∗_ 8) _×_ 1 contains exogenous dropout probability fixed effects, _δ_ ( _t, θj, yj_ ), for the _J_ types and 8 years (see (14)).

# **6.2 Computational representation**

## **6.2.1 Student types**

To build the empirical distribution of ability and income for school graduates, Φ( _y, θ_ ), we start from Appendix Table A1, which classifies 2005 high school graduates by ability quintile and income bracket. We refine this table to work with ability deciles rather than quintiles, for a total of fifty student types. To construct values for _θ_ , we start from the distribution of standardized Saber 11 test scores and normalize them between 0 and 1.<sup>14</sup> Our _θ_ values are the 5th, 15th, ...95th percentiles from the normalized scores. We calculate the _y_ value corresponding to each

> 13We set this requirement to 98 rather than 100 because we observe students who graduate with slightly fewer than 100 classes, perhaps due to measurement error.

> 14 Let _sts_ denote the standardized test score. The normalized _sts_ is thus equal to ( _sts −_ min( _sts_ )) _/_ (max( _sts_ ) _−_ min( _sts_ )) _._

19
income bracket as the bracket’s average annual per capita income, computed from Colombia’s household survey data (SEDLAC). Lacking student-level data on tuition expenses, we estimate the tuition paid by students of a given _y_ as the average annual tuition paid by students from the corresponding income bracket at public institutions, calculated from SNIES and SPADIES.<sup>15</sup> Appendix Table A3 shows the resulting income and tuition by family income bracket. Income varies greatly across income brackets but tuition varies less. Although public institutions provide income-based tuition discounts, the highest income individuals do not pay proportionally to their income: their per capita income is about twenty times as large as that of the lowest-income individuals, yet their tuition is only 2.5 times as large.

### **6.2.2 Workers**

We use household surveys to compute the average wages earned by individuals with different educational attainment and experience in 2005. For workers aged 18-65, the average wage of a college graduate, a college dropout with at least one year of complete college, and a college dropout with less than one year of complete college is 160, 58, and 28 percent higher than the average wage of a high school graduate respectively (see Appendix Table A4).<sup>16</sup> Among college (high school) graduates, the average wage of experienced workers is 35 (29) percent higher than the average wage of inexperienced workers. Consistent with the data, we assume that the returns to experience of college dropouts are the same as those of high school graduates.

## **6.3 Computing predicted values**

Since the model does not have a closed-form solution, we use a numerical algorithm to solve students’ dynamic optimization problem for a given value of Θ. Appendix C provides a full description of the algorithm. The estimation of **_δ_** and **_ξ_** is nested within the model solution for a given of value of Θ, in the spirit of Berry, Levinsohn and Pakes (1995).

In anticipation of next section, a couple of remarks are in order. First, our model solution perfectly replicates observed enrollment rates by type by construction. Hence, enrollment rates are not a moment to match in the estimation. Second, our model solution attempts to replicate observed dropout rates at the (year, ability quintile, income) level, but only does it with mixed success (see Appendix C.3 for further details). Thus, dropout rates are a moment to match.

# **7 Estimation**

In this section we describe the estimation strategy and identification. We also present parameter estimates, describe the model’s fit, and address the role of effort given our estimates.

> 15 We use tuition at public institutions because there is always a public institution that the student can attend. Modelling the choice of college type (public or private) is beyond the scope of this paper.

> 16This creates, in effect, four college attainments: high school, college, some college (one year), some college (two or more years). The two “some college” categories correspond to college dropouts. We work with two rather than one dropout category because their wages are quite different and can hence affect dropout timing.

20
# **7.1 Estimation Strategy**

We estimate the model parameters using Simulated Method of Moments (SMM). Our estimation searches for the value of Θ whose predicted moments, **M**<sup>ˆ</sup> (Θ), best match the observed ones, **M** . The moments we match are listed in Table 3. They reflect the patterns of dropout, college outcomes, classes completed, persistence, and targets discussed in Section 4. Matching these 585 moments enables us to estimate our 13 parameters.

**Table 3: Moments Matched in Estimation.**

|||Number of|
|---|---|---|
|Data aspect|Moments|Moments|
|Dropout rates|Dropout rate by year.<br>Dropout rate by ability quintile and income.<br>Dropout rate by ability decile.|8<br>25<br>10|
|College outcomes|College outcomes by ability quintile.<br>Fraction of students that graduate by year (years 5-8).|25<br>4|
|Cumulative classes<br>completed|Average number of cumulative classes completed by<br>year, ability quintile, and college outcome.<br>Average number of cumulative classes completed by<br>year and ability decile (years 1-5).<br>Distribution of students into tiers of cumulative classes<br>completed, by year.<br>Distribution of students into tiers of cumulative classes<br>completed, by ability quintile and year (years 1-5).|140<br>50<br>24<br>75|
|Transition probabilities|Pr(tier _Y_ in _t_+ 1_|_tier _X_ in _t_) for years 1-7.<br>Pr(_d_<sup>_drop_</sup><br>_t_<br>= 1_|_tier _X_ in _t_) for years 1-8.|112<br>32|
|Target number of classes|Average target number of classes by ability decile and<br>year.|80|
|Total||585|

_Source_ : Own estimation.

_Notes_ : Moments per year are computed for years 1-8 unless otherwise specified. Tiers are 1-4, based on cumulative classes completed. For “College outcomes,” outcomes include on-time graduate, late graduate, drop out first year, drop out second year, drop out after second year. In “Cumulative classes completed,” which are calculated by year, outcomes include on-time graduate (until year 5), late graduate, drop out this year, drop out later (until year 7); “this year” and “later” refer to the year under consideration. In “Transition probabilities”, _t_ refers to year; tiers _X_ and _Y_ are 1,...,4. Observed data for target number of classes is average number of classes attempted by the corresponding students.

Formally, our SMM parameter estimates solve the following problem: where Θ is a 13 _×_ 1 vector of parameters, **M** and **M**<sup>ˆ</sup> are 585 _×_ 1 vectors of sample and predicted moments, respectively, and _W_ is a diagonal weighting matrix whose diagonal contains the standard error of the sample moments. We compute numerically the predicted values, **M**<sup>ˆ</sup> , for every value of Θ as explained in Appendix C.

21
# **7.2**

A critical challenge is identifying the role of ability, effort, and performance shocks in the production of classes completed. Below we provide intuition for identification by describing each parameter’s role in the model, and Section 7.5 below complements this discussion.

**Effort.** If effort had no role in the number of classes completed ( _α_ = 1), or if it were costless ( _μ_ = 0), then all students would take the required number of classes per year. The fact that students take, on average, a lower number of classes than required (see Section 4) indicates that effort does have a role in classes completed and helps identify _μ_ . An increase in _μ_ leads to lower targets, effort, and number of classes completed. An increase in _μ_ also leads to lower college enrollment—particularly for low-income students, who have lower consumption than others to compensate for effort. The speed of accumulation of classes completed, as well as the transitions among tiers over time, helps identify _γ_ . A high _γ_ penalizes high effort levels and makes it costly to catch up. The fact that higher ability students take more classes, on average, than their lower-ability counterparts indicates that their effort cost is lower and identifies _k_ . An increase in _k_ raises the variance of average target, effort, and classes completed across abilities.

**Performance shock.** Despite the low variation of average classes completed across abilities, number of classes completed varies widely within abilities. Conditional on ability, effort varies by income. This explains some, but not all, of the within-ability variation of classes completed. The remainder of this variation, then, is explained by the performance shock, _z_ . Since _θ_ is between 0 and 1, we restrict _z_ to be in this range as well. This helps us pin down the scale for effort. An increase in _κ_ 0 makes shocks more favorable to all students, thus raising the number of classes completed and lowering dropout rates across the board. Parameter _κ_ 1 is an intercept shifter that makes the scale of _z_ comparable across years. An increase in _κθ_ makes the expected shock relatively more favorable for high-ability students and raises the dispersion in average classes completed and college outcomes across abilities. Parameter _σ_ is identified by the overall variation of classes completed conditional on ability, whereas _σθ_ is identified by the greater variation of classes completed among lower ability students. The higher overall variation of classes completed in year 1 relative to other years identifies _σ_ 1. After year 1, _κh_ is identified by the persistence of students in their performance tiers.

**Ability.** Given the role of effort and performance shocks in the production of classes completed, the variation of average classes completed across abilities identifies _α_ . This variation rises with an increase in _α_ .

**Other parameters.** An increase in _ρ_ raises the aversion to consumption variations over time and decreases the propensity to college enrollment. It also lowers the speed of class accumulation and increases time-to-degree. Finally, the sensitivity of dropout rates with respect to current classes completed, conditional on student income and ability, identifies _π_ .

A sufficient condition for local identification is full rank for the matrix of first derivatives of the moments’ predicted values with respect to the parameter vector when evaluated at the true parameter point. Evaluated at our parameter estimates, this matrix has full column rank in our sample.

# **7.3 Parameter Estimates**

We now turn to our parameter estimates, shown in Table 4.

22
**Table 4: Parameter Estimates.**

|Parameter|Symbol|Estimate|
|---|---|---|
|Utility function|||
|Consumption curvature|_ρ_|0.882|
|Efort weight|_μ_|0.062|
|Efort curvature|_γ_|4.727|
|Efort cost w.r.t. ability|_k_|1.225|
|Number of classes completed|||
|Elasticity w.r.t. ability|_α_|0.085|
|Performance shock|||
|Constant|_κ_0|-4.207|
|Year 1 shifter|_κ_1|3.534|
|Persistence component|_κh_|1.304|
|Ability component|_κθ_|0.407|
|Std. dev. of _iid_ shock|_σ_|1.789|
|Std. dev. of _iid_ shock - Year 1 shifter|_σ_1|0.317|
|Std. dev. of _iid_ shock - Ability shifter|_σθ_|-1.282|
|Dropout shock|||
|Cumulative performance component|_π_|-2.951|

_Source_ : Own estimation.

The estimated _α_ is low, consistent with the observed low variation of average classes completed across abilities. As a result, the estimated elasticity of credits completed with respect to effort (equal to 1 _− α_ ) is high. We emphasize that our ability measure (Saber 11 score) captures college academic readiness and not necessarily ”true” ability. Hence, our estimate indicates that college academic readiness plays a small role in the accumulation of classes completed relative to effort. In order to affect classes completed, policies must therefore affect effort.

The estimated _ρ <_ 1 indicates that students have low risk aversion to consumption changes over time and implies an elasticity of intertemporal substitution (equal to 1 _/ρ_ ) greater than one. This makes students quite willing to attend college and graduate on time.

The estimated _γ_ , equal to 4.73, indicates a very high marginal cost of effort—exceeding typical quadratic costs—which prevents large catch-up efforts. The estimated _k_ indicates that effort is negatively and strongly related to ability. When _k_ = 0, effort cost is the same for all abilities, whereas when _k_ = 1 effort cost falls with ability, at a decreasing rate. Our estimated value of 1.23 yields the same qualitative pattern as _k_ = 1 but with an even greater gap in effort cost among high and low ability students.

Given estimates for the shock-related parameters, we find that _E_ ( _zt_ ) is higher for students with higher past performance, which creates persistence. In addition, low-ability students have higher _E_ ( _zt_ ) and _V ar_ ( _zt_ ) than high-ability students. Their (slightly) higher mean of _z_ is consistent with the observed fact that low-ability students enroll in less selective (and presumably less demanding) programs than their abler counterparts.<sup>17</sup> Together with effort, this higher mean helps us match the good performance and graduation of some low-ability

> 17We measure program selectivity as the average Saber 11 test score of the program’s students. We find a negative correlation between student ability and program selectivity.

23
students. The more dispersed shock for lower-ability students, in turn, helps us match their higher variance of classes completed.

To examine the relative impact of past performance and ability on _z_ given our estimates, consider students A and B. Student A is more able than B, with ∆ _θ_ = _θA − θB_ = 0 _._ 22 _._ This is a large ability difference, equal to the difference between the 55th and the 5th ability percentile, or between the 95th and the 75th percentile. Assume that, at the beginning of _t_ , A has completed one more class than B. Because A is abler than B, her _E_ ( _zt_ ) should be lower than B’s, yet because she has completed more classes, her _E_ ( _zt_ ) should be higher. Based on our estimates, just having completed that additional class gives her the same _E_ ( _zt_ ) as B’s, even though B is much less able. In other words, _E_ ( _zt_ ) is more sensitive to _ht−_ 1 than to _θ_ . This makes _z_ highly persistent and more dependent on something the student can control—her performance—than on ability, which she cannot control.

Finally, the estimated _π_ indicates that an additional class completed by the end of the year, ̃ on average, decreases the probability of dropping out by about 5 pp. Since _p_<sup>_d_</sup> ( _·_ ) has a logistic functional form, this marginal effect is stronger for students with intermediate values of the dropout probability rather than values close to zero or one.

ˆ Our full set of parameter estimates includes the dropoutˆ probability fixed effects in (14), _δ_ ( _t, θ, y_ ). To illustrate the relative magnitude of _π_ ˆ and _δ_ ( _t, θ, y_ ), consider the average number of additional classes that a student from the second ability quintile (“Q2 student”) must complete to attain the same dropout probability as a student from the top ability quintile (“Q5 student”). In year 1, she must complete more than 4 additional classes—20 percent of the annual requirements—reflecting a high exogenous dropout probability. In year 5 she only needs one additional class completed, as the Q2 students reaching year 5 are approximately on par with Q5 students. The important point is that, early on, low-ability students face a high exogenous dropout probability, which they can only counter through high initial effort or favorable performance shocks.

# **7.4 Goodness of**

The estimated model fits the data well and replicates the patterns described in Section 4. The observed graduation rate is 45.64 percent and the predicted one is 45.02 percent. The model captures the distribution of dropouts over time (Table 5) and across student types (Table 6). As Figure 9 shows, it also captures the distribution of college outcomes by ability. The on-time graduation rate (15.1 percent) is predicted perfectly. The observed fraction of 2005 high school graduates that complete college is 14.7 percent, and the predicted one is 14.5 percent.

As Appendix Tables D1-D8 show, we capture average classes completed by year and ability, and conditional on final outcome.<sup>18</sup> Therefore, the model replicates the wide variation in classes completed within abilities as well as the low variation across abilities (Appendix Figure D1). Further, it captures the qualitative patterns of persistence, drop out, catch-up and fall behind (Table 7), as well as the progressive concentration of students in the upper tiers over time (Figure 10). Finally, Appendix Figure D2 shows average predicted target and average number

> 18It should be kept in mind that the use of a weighting matrix in (16) implies that moments with a greater number of underlying observations (such as those from early years, or for higher-ability students) attain a better

24
of classes taken by the student. Recall that the latter is theoretically an upper bound for the former (see Section 5.2.) As such, our average predicted target is lower than the average observed number of classes.

**Figure 9: Goodness of Fit: College Outcomes.**

<!-- Start of picture text --> 100<br>On−time graduate Late graduate<br>90 Dropout 2nd year Dropout 1st year<br>Dropout after 2nd year<br>80<br>70<br>60<br>50<br>40<br>30<br>20<br>10<br>0<br>1 2 3 4 5<br>Ability quintile<br>Percent<br><!-- End of picture text --> _Source_ : SPADIES for observed data; fitted values for predicted data. _Notes_ : Whole lines show observed values; dashed lines show values.

**Figure 10: Goodness of Fit: Tiers of Cumulative Classes Completed, by Year.**

<!-- Start of picture text --> 100 Tier 1 Tier 2&3 Tier 4<br>90<br>80<br>70<br>60<br>50<br>40<br>30<br>20<br>10<br>0<br>1 2 3 4 5<br>Year<br>Percent<br><!-- End of picture text --> _Source_ : SPADIES for observed data; fitted values for predicted data.

_Notes_ : For each year, the figure shows the percent of students by tier of cumulative classes completed. Whole and dashed lines show observed and predicted percentages, respectively.

**Table 5: Goodness of Fit: Dropout Year.**

|Year|Observed|Predicted|
|---|---|---|
|1st|27.41|27.92|
|2nd|10.38|10.43|
|3rd|4.77|4.94|
|4th|2.7|2.83|
|5th|2.61|2.81|
|6th|2.74|2.92|
|7th|2.48|1.94|
|8th|1.28|1.19|
|Total|54.36|54.98|

_Source_ : SPADIES for observed data; fitted values for predicted data. _Notes_ : Table shows the observed and predicted percent of students who drop out each year.

25
**Table 6: Goodness of Fit: Dropout Rates by Income and Ability.**

|||||||Ability|quintiles||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|Income||O|bserv|ed valu|es|||P|redict|ed valu|es||
|Bracket|1|2|3|4|5|Total|1|2|3|4|5|Total|
|5+ MW|81.4|65.8|61.5|52.1|39.1|44.7|84.4|67.8|63.4|51.9|40.7|46.7|
|3-5 MW|74.2|69.4|62.2|57.9|43.8|51.3|81.6|68.3|62.2|57.3|44.4|52.5|
|2-3 MW|68.5|67.6|63.7|57.7|46.5|55.1|70.8|67.5|61.9|60.1|44.4|55.6|
|1-2 MW|71.6|66.6|62.2|57.7|50.6|57.8|69.9|66.5|60.9|56.2|46.9|56.7|
|_<_1 MW|69.0|67.9|61.3|55.9|50.3|58.7|69.6|66.3|58.6|53.7|46.2|57.5|
|Total|71.0|67.4|62.4|60.0|45.8|54.4|71.3|66.9|61.0|56.6|44.8|55.0|

_Source_ : Source: SPADIES for observed data; fitted values for predicted data. _Notes_ : Values are expressed in percentages (%). Income is reported in brackets; MW = monthly minimum wage. Ability is reported in quintiles of standardized Saber 11 scores. Quintile 1 is the lowest.

**Table 7: Goodness of Fit: Transitions Among Tiers of Cumulative Classes Completed.**

|||Observe|d values|||Predicte|d values||
|---|---|---|---|---|---|---|---|---|
||Year 2|Year 3|Year 4|Year 5|Year 2|Year 3|Year 4|Year 5|
|_Persistence_|||||||||
|Tier 1|64.33|79.78|86.98|79.89|63.70|75.80|75.10|74.30|
|Tier 2|28.67|39.23|51.47|57.76|22.80|38.40|53.70|60.10|
|Tier 3|34.94|49.44|60.94|69.04|35.10|59.20|71.20|78.80|
|Tier 4|29.67|45.75|57.87|58.59|37.10|42.50|60.70|76.00|
|_Dropout rate_|||||||||
|Tier 1|12.16|4.03|1.75|0.71|11.60|4.20|3.00|2.20|
|Tier 2|14.43|6.50|3.25|1.52|16.20|6.40|4.60|3.00|
|Tier 3|23.02|12.30|7.29|4.02|25.10|9.20|7.00|5.20|
|Tier 4|54.23|41.95|28.24|24.24|55.20|50.70|35.20|21.40|
|_Prob. of Catch up_|||||||||
|Tier 3 to Tiers 1 & 2|18.10|20.42|20.61|19.63|26.60|20.30|13.10|9.40|
|Tier 4 to Tiers 1 & 2|4.26|0.55|0.22|0.14|0.82|0.00|0.00|0.00|
|_Prob. of Fall behind_|||||||||
|Tier 1 to Tiers 3 & 4|11.01|5.41|2.54|1.66|4.30|1.20|0.30|0.00|
|Tier 2 to Tiers 3 & 4|36.8|28.77|17.70|18.40|26.00|25.60|22.80|24.40|

_Source_ : SPADIES for observed data; model simulations for predicted data. _Notes_ : Values are expressed in percentages (%).

# **7.5 The Role of**

Since our estimates show that class completion is much more responsive to effort than ability, we now examine the role of

## **7.5.1 Average effort over time.**

We begin by looking at average student effort by year (Figure 11). On average, effort is relatively high in years 1 and 4. In year 1, students work hard to complete classes and mitigate dropout 26
risk, which is highest that year. In year 4, they work hard in order to graduate. After year 5, average effort falls as the remaining students have few classes left.

**Figure 11: Predicted Average Effort by Year.**

<!-- Start of picture text --> 1<br>.8<br>.6<br>.4<br>.2<br>1 2 3 4 5 6 7 8<br>College year<br>Average effort<br><!-- End of picture text --> _Source_ : Model baseline predictions.

_Notes_ : The figure shows average student effort by year in the simulated baseline. Individual efforts are normalized by the 95th percentile of the year’s effort distribution.

# **7.5.2 What if the model did not include**

In order to highlight the role of effort, we re-estimate the model eliminating effort as an input in the production function. We do this by normalizing effort to a scaling constant. In this “exogenous effort” model, the student does not choose effort and thereby has no control over her performance. We run two variants of the exogenous effort model. In the first (“exogenous homogeneous effort”) we set the scaling constant to the same value for all students and years, equal to the overall effort average in the baseline. In the second (“exogeneous heterogeneous effort”) we set the constant for a given (student type, year) equal to the baseline average effort for that (student type, year). The first variant is similar to models that do not include effort at all (see Section 2). The second variant is akin to adding (student type, year) fixed effects to models without Appendix Table E1 compares parameter estimates and objective function values for our model and the exogenous effort models. When effort is exogenous, not surprisingly heterogeneous effort fits the data better than homogeneous effort. However, both exogenous effort models fit the data much worse than our model (the objective function rises by a factor of 5 to 8). In other words, the omission of effort cannot be solved by merely adding (student type, year) fixed effects to a model without effort.

Removing effort from performance places greater weight on ability and shocks. The elasticity of performance with respect to ability, _α_ , becomes 2.5 or 3 times larger. Since higher ability students have lower effort cost and exert more effort, other things equal, the exogenous effort model attributes to ability an effect that is actually mediated through effort. The implied _E_ ( _z_ ) rises steeply with ability, thereby amplifying ability’s role, and _z_ becomes more dispersed in an attempt to match within-ability performance variation (see Appendix Figure E1). Interestingly, the risk aversion parameter, _ρ_ , becomes twice as large to reflect that students, who can no longer mitigate risk through effort, behave as if they were more risk averse.

27
The exogenous effort model predicts the overall graduation rate quite well (at 45-47 percent), but relative to our model it predicts greater variation of college outcomes across abilities. As shown by Figure 12 panel a, it makes the dropout rate fall more steeply with ability and the ontime graduation rate rise much faster. It predicts higher performance variation across abilities but lower variation within abilities (compare Figure 12 panel b with Figure 6 panel a),<sup>19</sup> which leads high-ability students to complete classes and graduate faster than in our model, and conversely for low-ability students. Modeling effort, then, allows us to match the fact that some low-ability students perform better (due to effort) than predicted by their ability while some high-ability students perform worse (due to lack of effort) than predicted by their ability.

**Figure 12: Effort and Model Predictions.**

<!-- Start of picture text --> 80 60<br>50<br>60<br>40<br>40<br>30<br>20 20<br>10<br>0<br>1 2 3 4 5 0<br>Ability quintile 1 2 3 4 5<br>Ability quintile<br>Dropout On−time graduate Late graduate<br>Dropout On−time graduate Late graduate Tier 1 Tier 2&3 Tier 4<br>a. College Outcomes b. Tiers of Cumulative<br>Classes Completed (Year 1)<br>Percent<br>Percent<br><!-- End of picture text --> _Source_ : Own estimations.

_Notes_ : In panel a, for each ability quintile the graph shows the predicted percent of students who attain each outcome. Solid line indicates endogenous effort model and dotted line indicates exogenous (heterogeneous) effort model. In panel b, for each ability quintile the figure shows the predicted distribution into performance tiers by the end of year 1 for the exogenous (heterogeneous) effort model.

To summarize, a model without effort fits the aggregate graduation rate well but not the variation of outcomes across student types and over time. This has serious distributional implications—to raise the achievement of low-ability or low-income students, it is critical to model it correctly. It also has stark policy implications. When only ability and luck matter to performance, selection might be the policymaker’s only tool—subsidizing, for instance, the most able students. When effort matters, in contrast, the policymaker can affect student behavior—subsidizing students, for instance, based on performance. We return to this issue below.

# **7.5.3 Omitted bias**

Despite the importance of effort in performance, in general administrative datasets do not provide effort measures. When these datasets are used to estimate the relationship between

> 19Due to space limitations we compare the endogenous and exogenous _heterogenous_ effort model, but the conclusions hold for the exogeneous homogeneous model as well. Since our model fits the distribution of students across tiers well, we compare this prediction from the exogenous model directly to the observed data. Similar patterns arise when comparing predictions for subsequent college years.

28
classes completed and ability through linear regressions, lack of effort measures might bias the estimates. We investigate this issue in Table 8. We begin with the regression of log classes completed, ln( _xt_ ) _,_ on log ability using the observed data (column 1) and the simulated baseline (column 2). The coefficients on log ability in columns (1) and (2) are very close, which is additional evidence of our good fit. In both cases, log ability explains about 20 percent of the variation in log classes completed.

**Table 8: Classes Completed Per Year.**

||Actual data|Si|mulated Dat|a|
|---|---|---|---|---|
||(1)|(2)|(3)|(4)|
|ln(ability)|0_._166<sup>_∗∗∗_</sup>|0_._156<sup>_∗∗∗_</sup>|0_._090<sup>_∗∗∗_</sup>|0_._085<sup>_∗∗∗_</sup>|
||(0_._015)|(0_._005)|(0_._005)|(0_._000)|
|ln(efort)|||0_._854<sup>_∗∗∗_</sup>|0_._915<sup>_∗∗∗_</sup>|
||||(0_._004)|(0_._000)|
|ln(shock to classes completed)||||1_._000<sup>_∗∗∗_</sup><br>(0_._000)|
|Constant|2_._060<sup>_∗∗∗_</sup>|2_._748<sup>_∗∗∗_</sup>|2_._197<sup>_∗∗∗_</sup>|2_._996<sup>_∗∗∗_</sup>|
||(0_._012)|(0_._004)|(0_._004)|(0_._000)|
|_R_<sup>2</sup>|0.213|0.204|0.518|1.000|
|Num. Obs.|123,101|127,044|127,044|127,044|

_Source_ : OLS estimation using SPADIES for actual data and model’s baseline predictions for simulated data. _Notes_ : Dependent variable is ln(classes completed per year), or ln( _xt_ ) in the model. An observation is a student-year; years 1-8 are included. Ability is _θ_ , effort is _e_<sup>_∗_</sup> _t_<sup>,andshocktoclassescompletedis</sup><sup>_zt_.All</sup> regressions include year fixed effects (not shown). Standard errors (in parentheses) are clustered by student. * p _<_ 0.10, ** p _<_ 0.05, *** p _<_ 0.01.

One advantage of our structural model is that we can recover variables—effort and shocks— not observed in the administrative data. This allows us to expand regression (2) in order to gauge the relative roles of effort, performance shocks, and ability. Adding effort (column 3), we explain an additional 30 percent of the variation in classes completed. As expected, we explain all the variation in classes completed when we add the shock as well (column 4), with a coefficient on log ability equal to our point estimate for _α_ . Further, in column 4 effort and the shock account for 47 and 52 percent of the variation in classes completed, respectively, leaving a mere 1 percent explained by log ability. In other words, in our model almost the whole variation in classes completed is due to effort and shocks.

Consistent with estimates from the exogenous effort model, when effort is not controlled for in these regressions the coefficient on ability is overestimated by about 75 percent. This large upward bias should be taken into account not only when estimating achievement regressions with administrative datasets, which typically do not include effort measures, but also when designing policy. By ignoring or underestimating effort’s role, policy might place too much weight on a student trait—ability—that cannot be changed rather than on a student choice —effort—that could, in principle be changed.

# **7.5.4 Policy implications**

The previous findings elicit two important questions. The first is whether policies could actually raise the number of classes completed. Combining equations (3) and (12), we arrive at the effort 29
associated with a particular target: As the expression shows, effort is greater the higher the target, the lower the expected shock, and the lower the ability. If a policy aims at raising classes completed, it must raise a student’s optimal target, _qt_<sup>_∗_,anditscorrespondingeffort.</sup> Since the number of classes completed is discrete (for instance, a student can raise _qt_<sup>_∗_from14to15,butnotto14.5),atargetincrease</sup> requires a non-marginal effort increase. The answer to the first question, then, is that a policy can raise the number of classes completed only by inducing a non-marginal effort increase.

Assuming that a policy can accomplish this goal, the second question is which students will increase effort the most. A student chooses effort as the solution of the dynamic optimization problem in (6). In a given period, her chosen effort is related to her ability, income, target number of classes, cumulative classes completed (which may create the need to catch up), expected shock to classes completed, and graduation probability. In principle, the relationship between ability and effort is ambiguous. On the one hand, higher-ability students have lower effort costs, inducing them to greater effort. On the other hand, they need less effort to attain a given target, which induces them to lower effort.

Using the simulated data from our model, Table 9 shows effort correlates based on some reduced-form regressions. In addition to ability, column 1 controls for target number of classes, whereas columns 2, 3, and 4 control for other variables that determine target and effort—past cumulative classes completed (proxied by average classes completed per year in past years), expected dropout probability, and expected shock to classes completed. Column 1 shows that higher ability students exert lower effort, whereas column 2, 3 and 4 show the opposite. The reason for these opossing results is that column 1 shows the effect of ability controlling for target (indicated in equation 17), by which higher ability students need less effort for a given target, whereas the other columns show _total_ ability effects—including those on target choice—by which higher ability individuals choose higher targets and effort.

Column (2) shows that students with better past perfomance exert less effort. This is the outcome of a direct and indirect effect of past performance on effort, both moving in the same direction. Because of the direct effect, students with better past performance are closer to completing graduation requirements and need less effort. Because of the indirect effect, they expect better shocks to classes completed and a lower dropout probability, both of which make effort less necessary (columns 3 and 4). The effects of past performance illustrate the importance of a strong beginning: high-performing students in the early years can expect better ”luck” in the future, thereby eliminating the need for a costly catch-up.

We can now return to our second question. Policies that seek to affect effort might be more successful among high-ability students, whose effort cost is lower. But, importantly, they should aim at raising effort in the early college years to minimize the need for a costly catchup later on. As a result, free college targeted to specific students (e.g., of high ability) or contingent on student’s cumulative classes completed (e.g., performance-based policies) might be highly effective at raising effort and performance. These conclusions motivate some of the counterfactuals presented below.

30
**Table 9:**

||(1)|(2)|(3)|(4)|
|---|---|---|---|---|
|ln(ability)|_−_0_._015<sup>_∗∗∗_</sup>|0_._076<sup>_∗∗∗_</sup>|0_._107<sup>_∗∗∗_</sup>|0_._067<sup>_∗∗∗_</sup>|
||(0_._003)|(0_._003)|(0_._003)|(0_._003)|
|ln(target)|0_._787<sup>_∗∗∗_</sup>||||
||(0_._004)||||
|ln(average classes completed)||_−_0_._173<sup>_∗∗∗_</sup>|||
|||(0_._008)|||
|ln(expected dropout probability)|||0_._022<sup>_∗∗∗_</sup><br>(0_._000)||
|ln(expected shock to classes completed)||||_−_0_._159<sup>_∗∗∗_</sup><br>(0_._006)|
|Constant|_−_1_._537<sup>_∗∗∗_</sup>|1_._019<sup>_∗∗∗_</sup>|0_._764<sup>_∗∗∗_</sup>|0_._519<sup>_∗∗∗_</sup>|
||(0_._011)|(0_._018)|(0_._003)|(0_._006)|
|Adj. R<sup>2</sup>|0.766|0.285|0.557|0.286|
|Num. obs.|127,044|127,044|127,044|127,044|

_Source_ : OLS estimation using model’s simulated baseline values. _Notes_ : The dependent variable is ln(effort), or ln( _e_<sup>_∗_</sup> _t_<sup>) in the model.An observation is a student-year.Ability</sup> is _θ_ , target is _qt_ , and cumulative classes completed is _h_<sup> ̄</sup> _t−_ 1. Expected shock to classes completed is _E_ [ _zt_ ]; it varies across students and over time. Expected dropout probability is _E_ [ ̃ _p_<sup>_d_</sup> ( _t, ·_ )] as calculated by the individual prior to the realization of _zt_ . All regressions include year and income fixed effects (not shown). * ** *** Standard errors (in parentheses) are clustered by student. p _<_ 0.10, p _<_ 0.05, p _<_ 0.01.

# **8 Free college simulations**

In this section we present counterfactual simulations of alternative free college programs. We begin by describing their setup as well as the theoretical impact of free college on student decisions. Then we describe free college’s aggregate effects on enrollment and graduation and examine effort in order to understand graduation rate effects. We then turn to the heterogeneity of free college effects by student type, and relate our counterfactual results with other findings from the literature. We analyze the role of risk, both in the baseline and under free college. We conduct a simple cost-benefit analysis of free college and conclude with an examination of the possible long-run effects of free college.

## **8.1 Counterfactuals’ setup**

We simulate the following free college programs, which provide free tuition to the students listed below:

- **Universal free college:** all college students.

- **Need-based free college:** low-income students, defined as those from the lowest two income brackets.

- **Ability-based free college:** high-ability students, defined as those from the highest ability quintile.

31
- **Ability- and need-based free college** : students who are both low income and high ability—students from the lowest two income brackets and the highest ability quintile.

- **Performance-based free college:** all students in year 1 but only high-performing students afterwards. In a given year, a high-performing student is one who finished the previous year in tiers 1 or 2 based on her cumulative performance. Note that a student may be high-performing in one year but not in another.

- **Performance- and need-based free college:** all low-income students (from the lowest two income brackets) in year 1 but only students who are both low-income and highperforming afterwards.

Note that, by design, performance-based programs create a risk for the student, as the zero tuition is not guaranteed but conditional on performance. In contrast, the other free college programs guarantee zero tuition regardless of performance.

As we analyze these counterfactuals, it is convenient to distinguish between two groups of students— _existing_ and _new._ For a given counterfactual, we define existing students as those who start college both in the baseline and counterfactual. In contrast, new students do not start college in the baseline but they do in the counterfactual.

Three important assumptions hold in our counterfactuals. First, we assume a perfectly elastic higher education supply, whereby institutions can adjust capacity as needed in order to absorb additional students at a constant marginal cost, and tuition does not rise in response to greater demand. If there were capacity constraints, institutions would need to ration free college. Second, we assume that parents continue to transfer the same _y_ to their children in college even when college becomes free. In other words, public college funding does not crowd out private funding at all. If, in contrast, parents reduce their transfers one-for-one (full crowd-out), free college has no effects and is similar to the baseline. More generally, crowding out of any degree is analogous to a reduction in the tuition subsidy.<sup>20</sup> Third, we assume that the average cost of educating new and existing students is the same. This does not hold, for instance, when new students are more costly because they need additional services such as remedial education, in which case a government with fixed resources can only fund fewer students. Since these assumptions provide the most favorable setting possible for free college, the results presented below are best viewed as an upper bound on free college effects.

# **8.2 Free college and student choices**

Free college affects students on the extensive and intensive margins. On the extensive margin it raises college entry by adding new students, thereby affecting student body’s size and composition. On the intensive margin it has the following effects on student effort:

1. **Loss-of-urgency effect** . By raising consumption during college, free college enhances the ”college experience,” as it raises the value of being a student relative to joining the labor force and makes the student less eager to leave college. Other things equal, this effect

- 20There is a large literature endogenizing parental transfers as a function of education cost and labor market returns. See, for instance, Keane and Wolpin (2001), Restuccia and Urrutia (2004), and Abbot et al (2013).

32
lowers effort. This higher the college consumption relative to the after-college expected consumption, the stronger this effect.

2. **Substitution effect** . Since free college raises consumption while in college, it allows students to exert more effort without losing utility—in other words, it compensates for greater effort with higher consumption. Other things equal, this effect raises effort. The greater the effort cost or the lower the baseline consumption, the greater this effect.

3. **Risk effect.** The performance and dropout shocks, _zt_ and _d_<sup>_drop_</sup> _t_ respectively, depend on the number of cumulative classes completed, which in turn depends on effort. To the extent that free college raises (lowers) effort, it also lowers (raises) risk. Further, the longer (shorter) a student spends in college, the greater (lower) her exposure to risks. A student with a strong loss-of-urgency effect and a weak substitution effect, for example, is subject to a strong and negative risk effect.

At the extensive margin free college affects enrollment rates. At the intensive margin it affects graduation rates, which rise or fall depending on the net effort effect. Importantly, performance-based programs have a net positive effect on effort: by making tuition contingent on performance, they eliminate the loss-of-urgency effect and incentivize effort.

# **8.3 Aggregate Outcomes**

## **8.3.1 Enrollment and graduation**

Table 10 shows the aggregate effects of free college programs on enrollment and college outcomes. The magnitude of enrollment effects is related to the percent of high school students eligible for free college—equal to 100 for universal and performance-based free college, 71 percent for need-based and performance- and need-based free college, and only 20 percent for ability-based free college. All programs raise enrollment relative to the baseline. Universal free college delivers the largest enrollment increase (28 pp), followed by need-based free college (23 pp) and performance-based free college (21.3 pp). In contrast, ability-based free college (whether need-based or not) has the smallest effect on enrollment rates (3 or 4 pp).

New students account for almost half of the student body under universal free college but just about 10 percent under ability-based free college. The fraction of low-income students rises in all programs, particularly the need-based ones. The fraction of high-ability students rises with ability-based programs because these induce a positive selection of new students. In contrast, the other programs induce negative selection of new students.

While four out of six programs raise enrollment rates by more than 15 pp, no policy affects graduation rate by more than 4 pp—and some programs actually lower it. Aggregate graduation effects are small yet mask considerable differences between existing and new students. The graduation rate of existing students remains almost constant in all counterfactuals except for performance-based programs, which raise it by 4 to 6 pp. As for new students, their graduation rate is higher (by about 10 pp) than that of existing students under ability-based programs but is lower otherwise. Similar patterns hold for on-time graduation rate. Free college, therefore, can raise overall graduation and on-time graduation rates by inducing ability-based positive selection or by incentivizing effort—but will hardly raise them otherwise.

33
**Table 10: Free College Counterfactuals: Aggregate Outcomes.**

|Attribute|Data|Baseline|Universal|Need|Ability|Ability<br>& need|Perf.|Perf.<br>& need|
|---|---|---|---|---|---|---|---|---|
|Enrollment rate (%)|32.3|32.3|59.9|54.9|36.8|35.4|53.6|49.7|
|Eligible students (%)|||100.0|71.0|20.0|9.8|100.0|71.0|
|Student body|||||||||
|composition (%)|||||||||
|New students|||46.1|41.2|12.2|8.9|39.8|35.0|
|Low income|52.4|52.4|66.0|72.1|54.5|56.6|64.0|69.0|
|High ability|39.5|39.5|28.8|29.0|46.9|44.9|30.9|31.0|
|Graduation rate (%)|45.6|45.0|43.5|43.6|45.6|45.7|47.8|44.9|
|Existing students|||45.8|45.6|44.4|44.8|51.3|48.7|
|New students|||40.9|40.8|54.3|54.1|42.7|42.6|
|On-time|||||||||
|graduation rate (%)|15.1|15.1|13.9|14.4|15.2|15.6|15.9|16.1|
|Existing students|||14.0|14.5|14.3|14.8|16.0|15.9|
|New students|||13.8|14.3|22.2|24.2|15.7|16.4|
|High school|||||||||
|graduates that<br>complete college (%)|14.7|14.5|26.1|24.0|16.8|16.2|25.6|22.3|

_Source_ : Model’s predictions for baseline and counterfactuals. _Notes_ : In these simulations, college is free for the following: all students (universal); students from the two lowest income brackets (need-based); students from the top ability quintile (ability-based), students from the two lowest income brackets and the top ability quintile (ability- and need- based); students classified in Tiers 1 or 2 the previous period (performance-based); students classified in Tiers 1 or 2 the previous period from the two lowest income brackets (performance- and need- based). Eligible students are those who could, in principle, make use of free college. Existing students are those who enroll both in the baseline and counterfactual. New students are those who enroll in the counterfactual but not the baseline. Low income=two lowest income brackets; high ability=top ability quintile.

The ultimate policy goal of free college programs is raising the fraction of high school graduates that complete college, equal to the product of enrollment and graduation rates. All programs raise this fraction relative to the baseline. Universal free college raises it the most (by about 12 pp, or 80 percent relative to the baseline), followed closely by performancebased free college. Relative to universal free college, performance-based free college raises enrollment less because many students seek to avoid its risk. However, since performance-based free college graduates students at higher rates than universal free college, it delivers almost the same increase in the ultimate goal—the fraction of high school graduates completing college. This trade-off between risk and outcomes is a theme in the analysis that follows.

# **8.3.2 Graduation and**

To further understand the graduation rate difference between universal and performance-based free college, we focus on existing students and examine how their average effort and accumulation of classes completed differs between baseline and counterfactuals.

Under universal free college, on average students increase effort in the first two years but then lower it until year 6, when they ramp it up in order to graduate. Substitution and 34
risk effects prevail in years 1 and 2, leading students to work harder (Figure 13 panel a) and accumulate more classes (panel b) than in the baseline. Nonetheless, this greater accumulation of classes, together with the loss-of-urgency effect, leads students to work less and accumulate fewer classes than in the baseline in years 3-5. To compensate for their slower pace and graduate, students raise effort in years 6-8. Thus, students conduct an intertemporal re-allocation of effort which, on average, delivers almost the same graduation rate as the baseline yet a lower on-timegraduation rate.

**Figure 13: Free college: Changes in effort and cumulative classes completed**

<!-- Start of picture text --> 1<br>.04<br>.02 .5<br>0 0<br>−.02 −.5<br>−.04 −1<br>1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8<br>Year Year<br>Universal free college Performance−base free college Universal free college Performance−base free college<br>a. Change in effort w.r.t. to baseline b. Change in number of cumulative classes<br>Change in effort classes completed<br>Change in number of cumul.<br><!-- End of picture text --> _Source_ : Model’s simulations.

_Notes_ : For existing students in the counterfactuals, panel a shows average effort in the corresponding counterfactual minus average effort in the baseline. Panel b does the same for average number of cumulative classes completed.

In contrast, under performance-based free college students frontload effort. Absent the lossof-urgency effect, the substitution and risk effect produce greater effort than in the baseline in years 1-3. Since students accumulate more classes than in the baseline in those years, they can work less later while still accumulating more classes than in the baseline. These effort changes imply not just an intertemporal reallocation of effort but also a net effort increase that raises both graduation and on-time graduation rates relative to the baseline.

# **8.4 Outcomes by student type**

The aggregate outcomes shown above mask great heterogeneity among student types. For a policymaker interested in promoting enrollment or graduation for specific students, this heterogeneity is critical. To analyze it more easily, we aggregate our fifty student types into nine income- and ability-based groups. Income groups include high, middle and low-income— corresponding to the top, two middle, and two bottom income brackets respectively. Ability groups include high, middle, and low-ability—corresponding to the top, two middle, and two bottom ability quintiles respectively.

35
# **8.4.1 Enrollment**

Table 11 shows enrollment rate by student group in the baseline (panel a) and enrollment rate increase relative to the baseline under universal and performance-based free college (panels b and c, respectively). We focus on these two counterfactuals because the remaining four are special cases of these. For example, the “low income” row of panel b displays the effects of need-based free college, while the “high ability” column displays the effects of ability-based free college.

**Table 11: Free College Counterfactuals: Enrollment Effects.**

||a.|Baseli|ne|b|. Unive|rsal|c. Per|forman|ce-Based|
|---|---|---|---|---|---|---|---|---|---|
||Enrol|lment|Rate|Free C|ollege:|Change|Free|College:|Change|
||||||Abil|ity||||
|Income|High|Mid|Low|High|Mid|Low|High|Mid|Low|
|High|83.8|65.5|39.3|7.1|11.4|12.0|5.8|9.0|9.2|
|Mid|73.1|47.4|27.3|15.0|21.1|18.3|12.5|16.8|13.9|
|Low|51.4|26.0|13.6|32.2|36.5|27.8|27.1|28.5|20.1|

_Source_ : Model’s predictions for baseline and counterfactuals. _Notes_ : For each student type, panel a shows predicted enrollment rate (%) in the baseline; panel b shows the enrollment rate difference (pp) between universal free college and baseline; and panel c shows the enrollment rate difference (pp) between performance-based free college and baseline.

Both free college programs raise enrollment rates for all student groups. Under universal free college, the greatest effects are for low- and middle-income students, who are most budgetconstrained in the baseline. Their greater responsiveness justifies need-based free college as opposed to universal free college, which provides a subsidy to many students who do not need it. In terms of ability, effects are not greatest for high-ability students. Ability-based free college, therefore, cannot be justified based on enrollment considerations, but rather on the grounds that it raises overall graduation rate through new students’ positive selection and lowers the income-based enrollment gap among high-ability students.

As for performance-based free college, its enrollment effects are qualitatively similar to those of universal free college, albeit smaller. Effects are similar because performance-based free collegeprovides free college to all students in year 1, but smaller because some potential students do not enroll to avoid the effort necessary to remain in tiers 1 or 2 in subsequent years.

These enrollment results are consistent with the small free college program run in Colombia between 2014 and 2018, _Ser Pilo Paga_ (”being diligent pays off.”) This was an ability-, need-, and performance-based program that offered free tuition to high-ability, low-income students for the theoretical duration of their program; students who dropped out were required to pay the tuition back. Among the eligible population, the program raised college enrollment rates by 32 pp (Londo ̃no et al 2020), in line with the 27-32 pp increase depicted in Table 11 (panels b and c) for these students.

36
# **8.4.2 Graduation**

To compare graduation rates in the baseline and counterfactuals we focus on existing students, whose graduation rate changes are entirely driven by the incentives created by free college.<sup>21</sup> Following the same logic as with enrollment, Table 12 shows baseline graduation rates (panel a), as well as the graduation rate changes relative to baseline under universal and performancebased free college (panels b and c, respectively.)

**Table 12: Free College Counterfactuals: Graduation Rate Effects for Existing Students.**

||a.|Baseli|ne|b|. Unive|rsal|c. Per|forman|ce-Based|
|---|---|---|---|---|---|---|---|---|---|
||Grad|uation|Rate|Free C|ollege: <br>Abil|Change<br>ity|Free|College:|Change|
|Income|High|Mid|Low|High|Mid|Low|High|Mid|Low|
|High|59.3|44.4|26.3|-1.8|-1.9|3.9|4.4|3.2|5.5|
|Mid|55.6|39.9|29.9|-1.9|3.5|1.7|4.9|6.6|6.5|
|Low|53.3|42.5|32.1|-1.3|4.4|-1.5|8.1|8.8|2.6|

_Source_ : Model’s predictions for baseline and counterfactuals. _Notes_ : For existing students of each type, panel a shows predicted graduation rate (%) in the baseline; panel b shows graduation rate difference (pp) between universal free college and baseline; and panel c shows graduation rate difference (pp) between performance-based free college and baseline.

**Universal free college.** Although it raises graduation rates for some groups, it lowers them for others. The reason is that the prevailing effort effect—loss of urgency, substitution, or risk— varies among student groups. Loss of urgency is generally stronger among higher-income students, whose high baseline consumption only gets higher with free college. The substitution effect is stronger among lower-ability students, for whom effort is more costly and have greater need of compensation for additional effort. The risk effect is stronger for students with fewer cumulative classes completed, particularly when loss of urgency has slowed down their progress, exposing them to risk for a longer time.

As panel b shows, graduation rates fall in the upper triangle (students of high ability or high income) and generally rise in the lower triangle (students of low ability or low income). In the upper triangle, baseline effort is already high because effort cost is low and consumption high. Given the little room for additional effort, the substitution effect is weak. A strong loss-of-urgency effect prevails given the already high consumption, and leads to lower effort, greater risk, and lower graduation rates.

The story is reversed in the lower triangle. Baseline effort there is low because effort cost is high and consumption low, which gives room for a strong substitution effect. Since consumption remains low even with free college, the loss-of-urgency effect is weak and the substitution effect prevails, leading to higher effort, lower risk, and higher graduation rates.<sup>22</sup>

> 21Free college generates incentives both for existing and new students. We focus on existing students because they allow us to compare behavioral changes between the baseline and counterfactuals. New students do not allow for this comparison because they do not enroll in teh baseline.

> 22The exception in the lower triangle is the students with the lowest income and ability, for whom graduation rates fall. The combination of lowest ability (which renders effort very costly) and lowest income (which makes consumption as a college dropout higher than as a college student, even with free college) leads these students to exert lower effort and drop out at higher rates than in the baseline.

37
**Performance-based free college.** In contrast to universal free college, performancebased free college raises graduation rates for _all_ student groups because it eliminates the lossof-urgency effect. Substitution effect leads to greater effort and therefore less risk. In the aggregate, performance-based free college raises the graduation rate of existing students from 45 to 51.3 percent (see Table 10), which is the greatest graduation rate increase for existing students among all programs considered here. Even new students attain a higher graduation rate under performance-based than universal free college (42.7 v. 40.9 percent).

Back to the policymaker’s goal of raising the fraction of high school graduates who complete college, Table 13 shows this fraction in the baseline (panel a) as well as the differences with respect to it under universal and performance-based free college (panels b and c, respectively.) Relative to the baseline, both programs raise this fraction for every student group. Performancebased free college is at least as effective as universal free college for every student group except low-income students of middle or low ability, for whom the difficulty of meeting performance requirements acts as an enrollment deterrent. In other words, a combination of unconditional free college for those students and performance-based free college for others could meet the policymaker’s goal not only in the aggregate but also by student group.

**Table 13: Free College Counterfactuals: Percent of High School Graduates That Complete College.**

||a. <br>Perc<br>Schoo|Baseli<br>ent of <br>l Grad|ne<br>High<br>uates|b<br>Free C|. Unive<br>ollege:|rsal<br> Change|c. Per<br>Free|forman<br>College:|ce-Based<br> Change|
|---|---|---|---|---|---|---|---|---|---|
||||||Abil|ity||||
|Income|High|Mid|Low|High|Mid|Low|High|Mid|Low|
|High|49.7|29.1|10.3|2.9|3.1|5.1|7.1|5.9|5.3|
|Mid|40.7|18.9|8.2|6.7|10.1|6.4|10.7|10.1|6.4|
|Low|27.4|11.0|4.4|16.7|17.5|8.5|19.1|15.5|7.0|

_Source_ : Model’s predictions for baseline and counterfactuals. _Notes_ : For existing students of each type, panel a shows the percent of high school graduates that graduate from college; panel b shows difference (pp) in this variable between universal free college and baseline; and panel c shows difference (pp) in this variable between performance-based free college and baseline.

# **8.5 Discussion**

We can now return to the cross-country evidence depicted in Figure 1, which shows that, in Latin America, government funding for college has a large, positive relationship with enrollment rates but a weak (actually negative) relationship with graduation rates. The evidence is consistent with our counterfactuals, as these predict a greater impact of free college on enrollment than graduation rates. The graduation rate evidence is also consistent with the financial aid regimes prevailing in those countries, where universal or need-based aid is most common, ability-based aid is rare, and performance-based aid is almost non-existing (Ferreyra et al 2017). Indeed, for the most common regimes in Latin America our counterfactuals predict a (slight) decline in graduation rate (see Table 12), consistent with Figure 1.

Our findings are also consistent with the literature on U.S. higher education financial aid, 38
which has generally found positive enrollment and graduation effects.<sup>23</sup> In recent years, several states have implemented a variety of financial aid programs based on ability and/or need,<sup>24</sup> as well as “Promise” programs, which provide free tuition to eligible students for local community colleges or state four-year institutions.<sup>25</sup> Overall, these programs have had positive effects on enrollment and graduation. Also consistent with our counterfactuals, the literature has found greater effects of financial aid on enrollment than on graduation (the latter being in the modest range of 0 to 6 or 7 pp)<sup>26</sup> and greater effects for performance-based than unconditional aid (Dynarski and Scott-Clayton 2013).

Even though performance-based free college raises graduation rates the most in our simulations, it still delivers a relatively small graduation rate increase (approximately 10 percent of the baseline graduation rate). This begs the question of why free college (or financial aid, in general) fails to substantially raise graduation rates.

The answer emerging from our model is that free college fails to induce in many students a large, non-marginal effort increase. As Table 9 shows, students who need to catch up exert more effort. Particularly when the student has fallen far behind, catching up is difficult (Section 4), which limits the effectiveness of performance incentives. This suggests that additional supports for class completion—particularly in year 1—may be needed, including remedial education, advising, and tutoring.<sup>27</sup> Indeed, in their study of possible policies to raise graduation rates in the U.S., Avery et al (2019) conclude that need-based free college combined with institutional funding for these supports might be the most cost-effective policy. A cautionary tale, however, comes from Oreopoulos and Petronijevic (2019), who find that these supports help students realize the need for greater effort but have little effect on college continuity or graduation. The reason is that students respond not by raising effort but by expecting less of themselves.

Moreover, our counterfactuals have assumed some highly favorable—perhaps unrealistic— conditions, as discussed before. One of them is no capacity constraints in higher education. Free college, however, would likely run into capacity constraints and force institutions to ration access. Bucarey (2018) explores the potential effects of free college ( _gratuidad_ ) in Chile, where it was introduced in 2016 for the bottom 60 percent of the income distribution. Data prior to 2016 shows that when low-income students are given additional financial aid, institutions respond by becoming more selective, thereby leading to a _lower_ share of low-income students. Using his structural model, Bucarey (2018) predicts that capacity-constrained institutions would follow a similar strategy under free college. We have also implicitly assumed that college quality remains the same after college becomes free. This may not happen if the policymaker reimburses colleges for less than the per-student average cost. Indeed, Murphy et al (2019) provide evidence that

> 23See, for instance, Bettinger (2004), Dynarski (2003), Hoxby and Turner (2013) and the references therein, as well as the surveys by Avery et al (2019), Deming and Dynarski (2009), Dynarski and Scott-Clayton (2013), Long (2008), Page and Scott-Clayton (2016).

> 24See Bettinger et al (2019), Castleman and Long (2016), Cornwell et al (2006), Dynarski (2000, 2004, 2008), Scott-Clayton (2011), Scott-Clayton and Zafar (2016), and the references therein.

> 25These programs vary across states in terms of eligible institutions (community colleges v. four-year institutions) and students. See, for instance, Carruthers et al (2018), Dynarski et al (2018), Gurantz (2020).

> 26For recent, well-identified studies, see Bettinger et al (2019), Denning (2017), Mayer et al (2015), and ScottClayton (2011).

> 27In the context of our model, these policies would amount to changing parameters from the distribution of _z_ in order to reach higher _E_ ( _z_ ) and lower _V_ ( _z_ ) _._ Clotfelter et al (2018), Evans et al (2017), Scrivener et al (2015) and Sommo et al (2018) document positive effects of these supports on graduation rates.

39
higher education quality in England was lower when college was free because institutions were not receiving full funding. It might not happen either if the cost of educating new students is higher than that of existing students—for instance, because they need additional services.

# **8.6 Free college and anticipated risk**

The typical rationale for free college is that students face financial constraints which prevent them from undertaking an investment with positive net present value. Our work, however, highlights another important, novel rationale: college students face risks. They are uncertain about their performance and chances to remain enrolled due to shocks during college that might be particularly important in developing economies. Some students, then, might not attend college in order to avoid these risks—in which case free college might provide insurance and induce more students to enroll.

In this section we investigate whether free college mitigates risk. Free college affects risk through two channels. First, it raises the value of remaining in college relative to dropping out by raising consumption during college and enhancing the ”college experience.” Second, it affects effort through the loss-of-urgency, substitution, and risk effects. Effort changes affect the number of current classes completed which, in turn, affect the future shocks to classes completed and the dropout probability.

These consumption and effort changes are distinctive of our setting. In an economy with credit, consumption is closely related to lifetime income—as in standard Ben-Porath models. Since free college would not substantially alter lifetime income in that economy, consumption changes would be small. But in Colombia, where credit constraints are severe, consumption changes could be quite large. As for effort changes, these are not present in models with exogenous risk, unrelated to effort (e.g. Hendricks and Leukhina 2017). Our model, in contrast, views effort as the direct tool used by students to mitigate risk.

The consumption and effort changes in our model affect the distribution of the value of college, _V_<sup>_coll_</sup> ( _t, ht−_ 1 _, θ, y_ ) _,_ defined in (6). Whereas _V_<sup>_coll_</sup> ( _t, ht−_ 1 _, θ, y_ ) is calculated for the student’s optimal effort, _e_<sup>_∗_</sup> _t_<sup>,whiletakingexpectationover</sup><sup>_zt_,wenowdefinethecollegepayofffrom</sup><sup>_any_</sup> given effort and realization of _zt_ : From here, it follows that _Ez_ [ _V_<sup> ̃</sup><sup>_coll_</sup> ( _t, ht−_ 1 _, θ, y_ ; _zt, e_<sup>_∗_</sup> _t_<sup>)] =</sup><sup>_Vcoll_(</sup><sup>_t, ht−_1</sup><sup>_, θ, y_).</sup> Our proposed measure of anticipated risk for student _i_ , at the beginning of period _t_ under scenario _p_ (baseline or counterfactual) is the coefficient of variation of her college payoffs: 40
where the right-hand side is the ratio between the standard deviation and expected value of college payoffs, calculated for the student’s optimal effort, _e_<sup>_∗_</sup> _it_<sup>,chosenunderscenario</sup><sup>_p_.Given</sup> _e_<sup>_∗_</sup> _it_<sup>,therandomnessin</sup><sup>_V_ ̃</sup><sup>_coll_(</sup><sup>_·_)comesfromtheperformanceanddropoutshocks,</sup><sup>_z_and</sup><sup>_ddrop_</sup> respectively, which are associated with cumulative classes completed and therefore effort. Note that anticipated risk varies across students and programs, and over time.

Does effort lower anticipated risk? To investigate this matter, in principle we might think of using our simulated baseline data to regress anticipated risk on (optimal) effort, controlling for cumulative past performance. However, this regression would suffer from endogeneity because optimal effort would appear in both sides of the regression. Since _e_<sup>_∗_</sup> _it_<sup>isafunctionofthestate</sup> variables, _t, hit−_ 1 _, θi,_ and _yi_ , we instrument for effort using the state variables (see Appendix Table F1), and run the second-stage regression reported in Table 14. This regression indicates that, in our baseline model, effort indeed lowers anticipated risk. The elasticity of anticipated risk with respect to effort is negative and large at -2.34. Student effort is therefore highly effective at mitigating performance and dropout risk.

**Table 14: Anticipated Risk.**

||Dependent variable<br>ln(Anticipated risk)|
|---|---|
|ln(efort)|_−_2_._340<sup>_∗∗∗_</sup><br>(0_._089)|
|ln(average classes completed)|_−_1_._767<sup>_∗∗∗_</sup><br>(0_._043)|
|Constant|_−_0_._772<sup>_∗∗∗_</sup><br>(0_._068)|
|Num. obs.|116,761|

_Source_ : 2SLS estimation using model’s simulated baseline values. _Notes_ : The dependent variable is ln(anticipated risk), or ln( _CVit_ ) for the baseline. An observation is a student-year. Included students are “existing” in both counterfactuals; upper 5% tail of risk has been trimmed. Effort is _e_<sup>_∗_</sup> _it_<sup>,instrumentedwithbyclassescompleted,aswellasyear,income,andabilityfixed</sup> effects. ln(average classes completed) is _h_<sup> ̃</sup> _it−_ 1, as defined in the model. The regression includes year fixed * ** *** effects (not shown). Standard errors (in parentheses) are clustered by student. p _<_ 0.10, p _<_ 0.05, p _<_ 0.01.

Having established that effort indeed mitigates risk, the next question is whether free college raises or lowers risk relative to the baseline. Figure 14 shows average anticipated risk in the baseline and under universal and performance-based free college. To facilitate comparisons, we focus on the set of students who start college in all three scenarios and discuss average effects.

In the three scenarios, average anticipated risk is U-shaped with respect to time. In year 1, it is high for two reasons. The first is student body composition, which includes a large share of students with a high exogenous dropout probability. The second is that every student is uncertain about her future performance, and her exogenous dropout probability is highest in year 1. In years 2 through 5, anticipated risk falls as many high-risk students drop out while the remaining students settle on a performance path. For the students that remain after year 6, anticipated risk rises because they face the risk of not graduating.

Relative to the baseline, free college raises _Ez_ [ _V_<sup> ̃</sup><sup>_coll_</sup> ( _·_ )] by raising consumption while in college and, in some cases, effort (see Figure 13 a). Free college also raises Var _z_ [ _V_<sup> ̃</sup><sup>_coll_</sup> ( _·_ )] at least in year 41
## **Figure 14: Anticipated Risk.**

<!-- Start of picture text --> .1<br>.08<br>.06 Baseline<br>Univ. free coll.<br>Perf.−based free coll.<br>.04<br>.02<br>0<br>1 2 3 4 5 6 7 8<br>year<br>Average risk<br><!-- End of picture text --> _Source_ : Own calculations based on model’s predictions for baseline and counterfactuals. _Notes_ : The figure shows average anticipated risk for baseline and counterfactuals by year, calculated for students who begin college in year 1 in all three scenarios. Upper 5% tail has been trimmed.

1, although at some point Var _z_ [ _V_<sup> ̃</sup><sup>_coll_</sup> ( _·_ )] becomes lower than in the baseline. The net result is that, relative to the baseline, universal free college raises anticipated risk in year 1 but lowers it afterwards. Meanwhile, performance-based free college raises it in years 1 through 3 but lowers it afterwards. On average both programs have little effect on anticipated risk because they have little effect on effort, as discussed in Section 8.5.

It is interesting to compare anticipated risk under universal and performance-based free college. Performance-based free college makes the level of college consumption contingent on performance—thereby raising anticipated risk—but also elicits greater effort—thereby lowering anticipated risk. When the consumption force prevails (years 1-5), anticipated risk is higher under performance-based than universal free college, but the reverse happens when the effort force prevails. In other words, performance-based free college exposes students to greater risk than universal free college in the early years, but also elicits greater effort from them and therefore yields better outcomes.

To summarize, effort can substantially lower anticipated risk. On average, risk is initially high and decreases dramatically after the first two years. By introducing uncertainty in free college availability, performance-based free college subjects students to greater anticipated risk than the baseline or universal free college, but this very uncertainty leads them to higher effort and graduation rates.

# **8.7 A simple cost-benefit analysis of free college**

As we saw in Section 8.4, free college programs vary in their ultimate outcome, namely the fraction of high school graduates who finish college. Nonetheless, the policymaker would be mistaken in simply choosing the policy that maximizes this fraction because it might also be the most costly. Hence, in this section we conduct a simple cost-benefit analysis of free college 42
programs. The benefit is the fraction of high school graduates who finish college, and the cost is the net public spending incurred to produce them. We maintain all our previous assumptions.

Lacking data on the annual cost of educating a college student, _C_ , we assume it is equal to the tuition paid by students from the highest income bracket at public universities (equal to 2.2 million COP on average; see Appendix Table A3). This is a lower bound of the true cost, as public colleges subsidize tuition even for the highest income students, and private universities— whose tuition might be closer to actual costs—charge more than 6 million COP on average. To the policymaker, the average net cost per graduate under policy _p_ as follows: where _Nt_<sup>_p_isthenumberofstudentsenrolledunderpolicy</sup><sup>_p_inyear</sup><sup>_t_,</sup><sup>_C_istheannualcostof</sup> educating a college student, and _T_<sup>_p_</sup> ( _·_ ) is the tuition paid by student _i_ under policy _p_ . By adding in the numerator over all individuals enrolled, equation (20) shows that the per-graduate net cost accounts for the cost of dropouts as well.

For each policy, Figure 15 shows the average net cost per graduate in terms of per capita GDP (panel a). For convenience, it also shows the fraction of high school graduates who finish college (panel b), which provides a measure of _G_<sup>_p_</sup> . Net cost per graduate is already high in the baseline—about the same as per capita GDP. Even if enrollment did not rise, free college would raise net cost per graduate just because fewer students would pay tuition. The combination of lower tuition revenues and higher enrollment raises net cost per graduate under all free college regimes. The cost rises the least under ability-and-need based free college and the most under universal free college, mostly reflecting each policy’s enrollment expansion.

**Figure 15: Free College Cost**

<!-- Start of picture text --> 3 30<br>2.5 25<br>2 20<br>1.5 15<br>1 10<br>.5 5<br>0 0<br>Baseline Univ. Need Abil.Abil.+NeedPerf. Perf.+Need Baseline Univ. Need Abil.Abil.+NeedPerf. Perf.+Need<br>a. Average net cost per college graduate b. Percent of high school graduates<br>(relative to GDP per capita) that complete college<br>Percent<br>Avg. net cost per grad / GDP pc<br><!-- End of picture text --> _Source_ : Model simulations for enrollment and graduates. See the text for (gross) cost assumptions. GDP per capita is from Colombia’s National Administrative Department of Statistics (DANE). _Notes_ : Average net cost per college graduate is computed as described in the text.

Consider, now, a policymaker seeking to maximize the number of graduates subject to a budget constraint. Universal and performance-based free college maximize the number of 43
graduates, yet performance-based free college does it at a lower per-graduate cost. This is because the incentives embedded in performance-based free college produce fewer dropouts and graduate students faster than universal free college. By adopting performance-based rather than universal free college, the policymaker would save no less than 40 percent of per capita GDP per graduate.

Nonetheless, the policymaker might be reluctant to adopt performance-based free college, perhaps for political considerations. Indeed, students throughout Latin America have been clamoring for unconditional, universal free college ( _gratuidad_ ). As Figure 15 shows, universal free college raises the number of graduates only slightly more than need-based free college yet costs an additional 50 percent of per capita GDP per graduate. The reason is that universal free college provides a transfer to medium- and high-income students who would attend college anyway. If choosing between unconditional programs, then need-based is preferable to universal free college.

One caveat is in order. College dropouts seem to acquire some human capital during college given their wage premium relative to high school graduates. This may motivate the policymaker to raise enrollment regardless of graduation. The question, then, is whether starting but not completing a bachelor’s program is the most efficient way of acquiring post-secondary human capital. A short-cycle program, lasting two or three years, might be more cost-effective.<sup>28</sup> Subsidizing these programs—in the spirit of subsidies to community college students in the U.S.—might therefore be preferable.

# **8.8 Long-run effects**

We have assumed that, despite the greater supply of college graduates due to free college, the college premium does not change. However, the college premium might actually fall, thereby reducing the incentives to attend college. In addition, free college might be funded with taxes, further depressing after-tax wages for college graduates. In other words, free college might unleash general equilibrium effects that could undermine some of the free college effects previously described (Heckman et al 1998).

To analyze the equilibrium relationship between college premium and number of college graduates, we postulate a production function following Katz and Murphy (1992), Heckman et al (1998), and Card and Lemieux (2001). Ignoring for the moment the connection between capital and technology, consider a CES production function, _F_ ( _Nt_<sup>_h, N_</sup> _t_<sup>_g_) =</sup> � _A_<sup>_h_</sup> _t_<sup>(</sup><sup>_N h_</sup> _t_<sup>)</sup><sup>_λ_+</sup><sup>_A_</sup> _t_<sup>_g_(</sup><sup>_N g_</sup> _t_<sup>)</sup><sup>_λ_�</sup> _λ_<sup><u>1</u></sup> , that combines high school and college graduates, whose amounts are _Nt_<sup>_h_and</sup><sup>_N g_</sup> _t_<sup>respectively,</sup> to produce aggregate output. In this function, _A_<sup>_h_</sup> _t_<sup>and</sup><sup>_Ag_</sup> _t_<sup>areefficiencyparameters,and</sup> _ω_ = 1 _/_ (1 _− λ_ ) is the elasticity of substitution between college and high school graduates. In a competitive labor market, the college premium is equal to where _wt_<sup>_g_and</sup><sup>_w_</sup> _t_<sup>_h_arethewagesofcollegeandhighschoolgraduates,respectively.Takinglogs</sup>

> 28Ferreyra et al (2021) shows that students in short-cycle programs graduate at higher rates than those in bachelor’s programs, and that their graduates obtain better labor market outcomes than dropouts from bachelor’s programs.

44
on both sides and defining a time difference ( _△_ ) yields an expression commonly used to study changes in wage inequality: In this equation, changes in relative wages depend on two components. The first is technical changes in the relative productivity of the two labor inputs. Skill-biased technical change, or _△_<sup>_A_</sup> _<u>t</u>_<sup>_g_The second is changes in relative supply, adjusted</sup> ln( _A_<sup>_h_</sup> _t_<sup>)</sup><sup>_>_0, increases the college premium.</sup> by the elasticity of substitution. An increase in the supply of college graduates lowers the college premium by an amount inversely related to _ω_ . Over time, the evolution of the college premium depends on the relative strength of these two opposite forces.

To investigate the extent to which the increased supply of college graduates might depress the college premium, we focus on universal free college, which increases supply the most and provides an upper bound to the potential college premium decline. We examine short- and longrun effects. For the latter, we assume that universal free college becomes a permanent policy beginning with our 2006 cohort, and simulate the evolution of the share of college graduates (relative to all workers in the economy) and the college premium. We assume no skill-biased technical change and therefore estimate an upper bound to the college premium decline.

The critical parameter for this exercise is _ω_ , which we estimate as 3.2 for Colombia.<sup>29</sup> Using this estimate, the left and right panel of Figure 16 show the evolution of the share of college educated workers and the college premium, respectively. The former rises over time as young, more educated cohorts replace older, less educated ones. The replacement is complete around the year 2045, when the economy reaches steady state.

**Figure 16: Free College and and the College Premium**

<!-- Start of picture text --> 45 3<br>40 2.5<br>35 2<br>30 1.5 ω = 3.2<br>ω = 2.5<br>25 1 ω = 1.4<br>20 .5 ω = 1.2<br>2015 2020 2025 2030 2035 2040 2045 2050 2015 2020 2025 2030 2035 2040 2045 2050<br>Year Year<br>a. Share College in Labor Force b.College Premium<br>Percent<br>College premium<br><!-- End of picture text --> _Source_ : Own calculations based on estimates.

_Notes_ : Panel a shows predicted share of college graduates in the labor force, calculated as the share of workers (of all ages) with a college degree relative to all workers (all ages and educational attainment) in the economy. Panel b shows predicted college premium, calculated as the ratio between wage for college graduates and wage for high school graduates.

> 29There is a large number of estimates for this parameter in the literature. For example, Katz and Murphy (1992) estimate it to be 1.4 for the U.S. for the period 1963-1987, but the low range of estimates also include 0.5. Card and Lemieux (2001) estimate it for the United States (1959 to 1996) and the United Kingdom (1974-1996) and find estimates between 2 and 2.5. Following Card and Lemieux (2001), we estimate the production function parameters using Colombia’s household surveys (SEDLAC) for 2001-12. Due to data limitations, college graduates include those from 2- and 4-year programs.

45
As in our model, the 2006 entry cohort starts to graduate in 2010. In 2006, the initial share of college graduates in the workforce equals 27 percent.<sup>30</sup> Universal free college raises this share for a particular cohort by 12 pp, to a long-run value of 39 percent (panel a). In the short-run the college premium falls relatively little (panel b) because the increase in college graduates from the first few cohorts is small relative to the stock of college-educated workers, which includes individuals from 35 to 40 cohorts. Over time, the continuous inflow of a greater number of college graduates produces a larger college premium impact—but still small, given our estimated _ω_ , and unlikely to dissuade many high school graduates from college enrollment. As a robustness check, panel b depicts the implied college premium path for several _ω_ estimates for developed economies.<sup>31</sup> The college premium declines more for these values, yet even the greatest decline -down to 1.70- still leaves the economy with a substantial college premium, comparable to that in the U.S. Skill-biased technical change, of course, would lead to an even smaller decline for any value of _ω_ .

In sum, both the short- and long-run college premium decline induced by free college would likely be small in Colombia. As a result, abstracting away from the general equilibrium implications of free college is not likely to bias our predictions substantially. College, thus, would remain a worthy investment even if free college produced more college graduates.<sup>32</sup>

# **9 Conclusions**

In this paper we have developed and estimated a dynamic model of college enrollment, performance, and graduation. A central piece of the model, student effort, has a direct effect on the completion of classes and an indirect effect mitigating risks on class completion or college persistence. We have estimated the model using rich administrative data from Colombia. According to our estimates, effort has a much greater impact than ability on class completion. Failing to model effort as an input to class completion leads to overestimating the role of ability by two or three times, and to favoring policies that promote positive selection of new students instead of effort. We have used our parameter estimates to simulate multiple free college programs. According to our simulations, universal free college expands enrollment the most but has the highest per-graduate cost and does not raise graduation rates. Performance-based free college, in contrast, delivers a slightly lower enrollment expansion but has a greater graduation rate and a lower per-graduate cost. Performance-based free college faces students with greater risks than the current baseline or universal free college; precisely for this reason it elicits greater

> 30Notice that the initial share of 27 percent is higher than the initial fraction of high school graduates that finish college, equal to 14.5 in the baseline. The reason is that 27 percent includes graduates from all bachelor’s programs (rather than only five-year programs) as well as graduates from two-year programs. This is because the college premium is calculated for all higher education graduates, based on data that does not distinguish between graduates from bachelor’s and two-year programs.

> 31We use the point estimates of 1.4 from Katz and Murphy (1992) and Heckman et al (1998), 2.5 from Card and Lemieux (2001), and include lower value of 1.2. For each _ω_ value, the intercept has been adjusted to generate the same college premium for the year 2012.

> 32These results are consistent with Garriga and Keightley (2007), who conduct a similar analysis for the U.S. in a model that endogenizes the college premium, government budget constraint, after-tax earnings, and labor supply decisions. They show that tuition subsidies have small effects on the college premium even when accounting for taxes. Their results are partly due to the limited effects of the policy on graduation rates, and to the high elasticity of substitution between college and high school graduates.

46
effort and graduation rates.

For existing students, free college has little impact on effort and graduation rates, even when it is performance-based. This suggests that additional, complementary policies might be needed. Given the high persistence of performance, improving first-year performance and retention is critical. Supports such as tutoring, remedial education, mentoring, and advising might be helpful, particularly for the most disadvantaged students. Further, most college programs in Latin America start with classes specific to the major and rarely include general education requirements that transfer easily across majors. As a result, changing majors typically implies starting from scratch, which leads many students to abandon higher education. Shortening bachelor’s programs (which last five or six years) and connecting them with the labor market might raise effort and graduation rates as well.

Finally, there might be an additional reason why free college fails to elicit large effort changes: particularly for low-income students, free college might not be enough. For them, the effort necessary to complete college might require not only free tuition but also an additional, generous stipend for living expenses. In countries with limited fiscal resources, providing this additional stipend might require lowering the existing college subsidies for affluent students—a redistribution that, while politically costly, would nonetheless enhance equity and efficiency.

# **References**

- [1] Abbot, B., Gallipoli, G., Meghir, C., & Violante, G. (2013). Education decisions and intergenerational transfers in equilibrium (Working Paper). New York: New York University.

- [2] Ahn, T., Arcidiacono, P., Hopson, A., & Thomas, J. R. (2019). Equilibrium grade inflation with implications for female interest in stem majors. NBER Working Paper No. w26556.

- [3] Akyol, A. & Athreya, K. (2005). Risky higher education and subsidies. _Journal of Economic Dynamics and Control, 29(6)_ , pp. 979-1023.

- [4] Altonji, J. (1993). The demand for and return to education when education outcomes are uncertain. _Journal of Labor Economics, 11(1),_ pp. 48-83.

- [5] Arcidiacono, P. (2004). Ability sorting and the returns to college major. _Journal of Econometrics, 121(1-2),_ pp.343-375

- [6] Arcidiacono, P., E. Aucejo, A. Maurel, & T. Ransom (2016). College attrition and the dynamics of information revelation. NBER Working Paper No. 22325.

- [7] Ariely, D., U. Gneezy, G. Loewenstein, & N. Mazar (2009). Large stakes and big Mistakes. _Review of Economic Studies, 76(2),_ pp. 451-469.

- [8] Avery, C., Howell, J., Pender, M., & Sacerdote, B. (2019). Policies and payoffs to addressing America’s college graduation deficit. In BPEA Conference Draft, Fall.

- [9] Beneito, P., J. Bosca & J. Ferri (2018). Tuition fees and student effort at university. _Economics of Education Review, 64(C),_ pp. 114-128.

- [10] Berry, S., J. Levinsohn, A. Pakes (1995). Automobile prices in market equilibrium, _Econometrica,_ pp: 841–890

- [11] Bettinger, E. (2004). How financial aid affects persistence. In College choices: The economics of where to go, when to go, and how to pay for it (pp. 207-238). University of Chicago Press.

- [12] Bettinger, E., Gurantz, O., Kawano, L., Sacerdote, B., & Stevens, M. (2019). The long-

47
run impacts of financial aid: Evidence from California’s Cal Grant. _American Economic Journal: Economic Policy_ , 11(1), 64-94.

- [13] Bucarey, A. (2018). Who pays for college? Crowding out on campus. Working Paper, Massachusetts Institute of Technology.

- [14] Cameron S.V. & Heckman, J. J. (1998). Life cycle schooling and dynamic selection bias: Models and evidence for five cohorts of American males. _The Journal of Political Economy, 106(2),_ pp. 262-333.

- [15] Cameron S.V. & Heckman, J. J. (1999).The dynamics of educational attainment for blacks, hispanics and whites (Working Paper 7249). Cambridge, MA: National Bureau of Economic Research.

- [16] Card, D. & Lemieux, T. (2001). Can falling supply explain the rising return to college for younger men? A cohort-based analysis (Working Paper 7655). Cambridge, MA: National Bureau of Economic Research.

- [17] Carneiro, P. & Heckman, J. J. (2002). The Evidence on Credit Constraints in PostSecondary Schooling. _The Economic Journal, 112(482),_ pp. 705-734.

- [18] Carruthers, C. K., Fox, W. F., & Jepsen, C. (2018). Promise kept? Free community college and attainment in Tennessee. Working Paper.

- [19] Castleman, B. L., & Long, B. T. (2016). Looking beyond enrollment: The causal effect of need-based grants on college access, persistence, and graduation. _Journal of Labor Economics, 34(4),_ pp. 1023-1073.

- [20] Clotfelter, C. T., Hemelt, S. W., & Ladd, H. F. (2018). Multifaceted aid for low-income students and college outcomes: Evidence from North Carolina. _Economic Inquiry,_ 56(1), pp. 278-303.

- [21] Cornwell, C., Mustard, D. B., & Sridhar, D. J. (2006). The enrollment effects of meritbased financial aid: Evidence from Georgia’s HOPE program. _Journal of Labor Economics, 24(4),_ pp. 761-786.

- [22] Deming, D., & Dynarski, S. (2009). Into college, out of poverty? Policies to increase the postsecondary attainment of the poor. NBER Working Paper, 15387.

- [23] Deming, D., & Walters, C. (2017). The impacts of price and spending subsidies on US post-secondary attainment. NBER Working Paper, 23736.

- [24] Dynarski, S. (2000). Hope for whom? Financial aid for the middle class and its impact on college attendance (No. w7756). National Bureau of Economic Research.

- [25] Dynarski, S. M. (2003). Does aid matter? Measuring the effect of student aid on college attendance and completion. _American Economic Review, 93(1),_ pp. 279-288.

- [26] Dynarski, S. (2004). The new merit aid. In College choices: The economics of where to go, when to go, and how to pay for it (pp. 63-100). University of Chicago Press.

- [27] Dynarski, S. (2008). Building the stock of college-educated labor. _Journal of human resources, 43(3),_ pp. 576-610.

- [28] Dynarski, S., & Scott-Clayton, J. (2013). Financial aid policy: Lessons from research (No. w18710). National Bureau of Economic Research.

- [29] Dynarski, S., Libassi, C. J., Michelmore, K., & Owen, S. (2018). Closing the gap: The effect of a targeted, tuition-free promise on college choices of high-achieving, low-income students (No. w25349). National Bureau of Economic Research.

- [30] Eckstein, Z., & Wolpin, K. I. (1998). Youth employment and academic performance in high school. CEPR Discussion Paper Series No. 1861

- [31] Evans, W. N., Kearney, M. S., Perry, B. C., & Sullivan, J. X. (2017). Increasing community

48
college completion rates among low-income students: Evidence from a randomized controlled trial evaluation of a case management intervention (No. w24150). National Bureau of Economic Research.

- [32] Ferreyra, M., Dinarte, L., Urzua, S., & Bassi, M. (2021). _The fast track to new skills: short-cycle higher education programs in Latin America and the Caribbean_ . The World Bank.

- [33] Ferreyra, M., Avitabile, C., Botero Alvarez, J., Haimovich Paz, F., & Urzua, S. (2017). _At a crossroads: higher education in Latin America and the Caribbean_ . The World Bank.

- [34] Garriga, C. & M.P. Keightley (2007). A general equilibrium theory of college with education subsidies, in-school labor supply, and borrowing constraints. Federal Reserve Bank of St. Louis Working Paper 2007-051A.

- [35] Gurantz, O. (2020). What does free community college buy? Early impacts from the Oregon Promise. _Journal of Policy Analysis and Management, 39(1),_ pp. 11-35.

- [36] Hai, R., & J.J. Heckman (2017). Inequality in human capital and endogenous credit constraints. NBER Working Paper No. 22999.

- [37] Heckman, J. J., Lochner, L. & Taber, C. (1998). General equilibrium treatment effects: A study of tuition policy. _The American Economic Review, 88(2)_ , pp. 381-386.

- [38] Hendricks, L. & Leukhina, O. (2017). How risky is college investment? _Review of Economic Dynamics, 26,_ pp. 140-163.

- [39] Hendricks, L., & Leukhina, O. (2018). The return to college: selection and dropout risk. _International Economic Review, 59(3),_ pp. 1077-1102.

- [40] Hoxby, C., & Turner, S. (2013). Expanding college opportunities for high-achieving, low income students. Stanford Institute for Economic Policy Research Discussion Paper, 12, 014.

- [41] ICETEX (2010). Informe de gesti ́on 2003-2010. Instituto Colombiano de Cr ́edito Educativo y Estudios T ́ecnicos en el Exterior (ICETEX).

- [42] Katz, L. F., & Murphy, K. M. (1992). Changes in relative wages, 1963-1987: supply and demand factors. _The Quarterly Journal of Economics, 107(1),_ pp. 35-78.

- [43] Keane, M. P. & Wolpin, K. I. (2001). The effect of parental transfers and borrowing constraints on educational attainment. _International Economic Review_ , _42(4),_ pp. 10511103.

- [44] Keane, M. P. (2002). Financial aid, borrowing constraints, and college attendance: Evidence from structural estimates. _The American Economic Review_ , _92(2)_ , pp. 293-297.

- [45] Levhari, D. & Weiss, Y (1974). The effect of risk on the investment in human capital. _The American Economic Review, 64(6),_ pp. 950-963.

- [46] Lochner, L.J. & A. Monge-Naranjo (2011). The nature of credit constraints and human capital. _American Economic Review, 101(6),_ pp. 2487-2529.

- [47] Londo ̃no, J., Rodriguez, C., & Sanchez, F. (2020). Upstream and downstream impacts of college merit-based financial aid for low-income students: Ser Pilo Paga in Colombia. _American Economic Journal: Economic Policy, 12(2),_ pp. 193-227.

- [48] Long, B. T. (2008). What is known about the impact of financial aid? Implications for policy. An NCPR Working Paper. National Center for Post-secondary Research.

- [49] Mayer, A., Patel, R., Rudd, T., & Ratledge, A. (2015). Designing scholarships to improve college success: Final report on the Performance-based scholarship demonstration. New York: MDRC.

- [50] Murphy, R., S. Clayton, & G. Wyness (2019). The end of free college in England: Im-

49
plications for quality, enrollment, and equity. _Economics of Education Review, 71,_ pp. 7-22.

- [51] OECD (2018). Education at a glance 2018: OECD indicators. Organisation for Economic Co-operation and Development.

- [52] Page, L. C., & Scott-Clayton, J. (2016). Improving college access in the United States: Barriers and policy responses. Economics of Education Review, 51, 4-22.

- [53] Oreopoulos, P. & U. Petronijevic (2019). The remarkable unresponsiveness of college students to nudging and what we can learn from it. NBER Working Paper No. 26059.

- [54] Ozdagli, A. K., & Trachter, N. (2011). On the distribution of college dropouts: Household wealth and uninsurable idiosyncratic risk. Working Papers No. 11-8.

- [55] Restuccia, D. & Urrutia, C. (2004). Intergenerational persistence of earnings: The role of early and college education. _American Economic Review, 94(5)_ , pp. 1354-1378.

- [56] Scott-Clayton, J. (2011). On money and motivation a quasi-experimental analysis of financial incentives for college achievement. _Journal of Human resources, 46(3),_ pp. 614-646.

- [57] Scott-Clayton, J., & Zafar, B. (2016). Financial aid, debt management, and socioeconomic outcomes: Post-college effects of merit-based aid. Working Paper 22574, National Bureau of Economic Research.

- [58] Scrivener, S., Weiss, M. J., Ratledge, A., Rudd, T., Sommo, C., & Fresques, H. (2015). Doubling graduation rates: Three-year effects of CUNY’s Accelerated Study in Associate Programs (ASAP) for developmental education students. New York: MDRC.

- [59] Solis, A. (2017). Credit access and college enrollment. _Journal of Political Economy, 125(2),_ pp. 562-622.

- [60] Sommo, C., Cullinan, D., Manno, M., Blake, S., & Alonzo, E. (2018). Doubling graduation rates in a new state: Two-year findings from the ASAP Ohio demonstration. New York: MDRC, Policy Brief.

- [61] Stange, K.M. (2012). An empirical investigation of the option value of college enrollment. _American Economic Journal: Applied Economics 4 (1),_ pp. 49-84

- [62] Stinebrickner, R. & T. R. Stinebrickner (2004). Time-Use and college outcomes. _Journal of Econometrics, 121(1-2),_ pp 243-269.

- [63] Stinebrickner, R. & T. R. Stinebrickner (2014). Academic performance and college dropout: Using longitudinal expectations data to estimate a learning model. _Journal of Labor Economics, 32(3),_ pp. 601 - 644.

- [64] Trachter, N. (2015). Stepping stone and option value in a model of post-secondary education. _Quantitative Economics, 6(1),_ pp. 223-256.

- [65] World Bank (2016). Poverty and shared prosperity 2016: Taking on inequality. World Bank Publications.

- [66] Zamarro, G., C. Hitt, & I. Mendez (2019). When students don’t care: Reexamining international differences in achievement and student Effort. _Journal of Human Capital, 13(2),_ pp. 519-552.

50