  Motivated by school admissions, this paper studies screening in a population
with both advantaged and disadvantaged agents. A school is interested in
admitting the most skilled students, but relies on imperfect test scores that
reflect both skill and effort. Students are limited by a budget on effort, with
disadvantaged students having tighter budgets. This raises a challenge for the
principal: among agents with similar test scores, it is difficult to
distinguish between students with high skills and students with large budgets.
  Our main result is an optimal stochastic mechanism that maximizes the gains
achieved from admitting ``high-skill" students minus the costs incurred from
admitting ``low-skill" students when considering two skill types and $n$ budget
types. Our mechanism makes it possible to give higher probability of admission
to a high-skill student than to a low-skill, even when the low-skill student
can potentially get higher test-score due to a higher budget. Further, we
extend our admission problem to a setting in which students uniformly receive
an exogenous subsidy to increase their budget for effort. This extension can
only help the school's admission objective and we show that the optimal
mechanism with exogenous subsidies has the same characterization as optimal
mechanisms for the original problem.
