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.

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