Suppose the same problem as in the previous exercise, but assume in this case that the teacher must divide her time between three activities:

teaching preparation, mentoring, and lecturing. Assume further that the quality function for the students—or the utility function for the principal—is u = q = x · y · z where x, y, and z are, respectively, the time spent teaching, mentoring, and lecturing. Suppose that the principal can observe teaching activities and lecturing via teaching evaluations:

bad, good, or excellent, and pays the teacher accordingly:

The per hour costs for the teacher are as follows: teaching costs 10 Rs, mentoring costs four rupees, and lecturing seven rupees. Assume that a working day has ten hours. Compute the optimal time allocation for the teacher and for the principal. Briefly comment on your answer.

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Evaluations Teaching Time (hours/day) Teaching salary/day Lecturing (hours/day) 1-1.5 Salary for lecturing time/day 25 Rs Bad 1-2 30 Rs Good 2-2.6 50 Rs 1.5-2.5 45 Rs Excellent 3 or more 70 Rs 2.5 or more 65 Rs