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