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.

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