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).

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