Consider a teacher-student relationship, where the teacher sets up a grading rule that rewards the student with grades for learning efforts. If given grade r for effort level e, the student's utility is rce, where the parameter c is the student's privately known learning effort cost, while the teacher's utility is Ine. The range of grades is restricted to [0, 1], i.e., the highest grade is 1. Note that the teacher bears no cost of rewarding the student, i.e., grades are free for the teacher to give. It is common knowledge that the student's learning effort cost c is equal to ci with probability p and c2 with probability (1 - p), where c< c2. The student's reservation utility is 0. Find the individually rational and incentive-compatible grading rule {(1, 1), (r2, e2)} that maximizes the teacher's expected utility. (Hint: since grades are costless for the teacher to give, in the optimum the cost-efficient student type has to receive the highest grade, i.e., r = 1.)

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Given that grades are costless for the teacher to give and the costefficient student type has to rec... View the full answer