Suppose students have to take an exam with N questions. The time for the exam is no more than T minutes. The score on each question, Si, depends on how good is the answer to that question, with a maximum score of Si*. Scores depend on a production function of performance for each question that depends on time put into that question, and the knowledge, K, brought to the exam.

1. Suppose the grade,G, depends on the sum of theSi, and that utility is equal toG. How much time would a person allocate to each question? How does that depend onKand the production function for each question?
2. Under what condition does a person walk out of the exam early? Assume the utility of each hour of his time outside the classroom equalsv. Do dumb or smart studentsdefined by theirK'swalk out early (assume they have the samev)?
3. How would your answers to parts (a) and (b) differ ifGdepends on the product of theSirather than their sum?
4. How would the time allocation differ from part (a) if the grade depends on the cube of the sum of theSion each question?
5. Do students have any incentive to collude on how much time they put into preparing for the exam? If so, do they have any incentive to cheat on such an agreement?
6. Suppose now that gradesGare based on a "curve", so that they depend on how the sum of all theSiof any individual ranks compared to the sum of the scores of other students in the class. Assume that everyone in top quarter gets an A, and everyone else gets a B. What rationale can you give for grading on a curve rather than on absolute scores?
7. Is there any incentive for students to collude on how much time they put into studying when they are graded on a curve? If so, would they have an incentive to cheat on the agreement?