As you may remember, Nancy Lerner is taking an economics course in which her overall score is the minimum of the number of correct answers she gets on two examinations. For the first exam, each correct answer costs Nancy 10 minutes of study time. For the second exam, each correct answer costs her 20 minutes of study time. In the last chapter, you found the best way for her to allocate 1200 minutes between the two exams. Some people in Nancy’s class learn faster and some learn slower than Nancy. Some people will choose to study more than she does, and some will choose to study less than she does. In this section, we will find a general solution for a person’s choice of study times and exam scores as a function of the time costs of improving one’s score.

(a) Suppose that if a student does not study for an examination, he or she gets no correct answers. Every answer that the student gets right on the first examination costs P1 minutes of studying for the first exam. Every answer that he or she gets right on the second examination costs P2 minutes of studying for the second exam. Suppose that this student spends a total of M minutes studying for the two exams and allocates the time between the two exams in the most efficient possible way. Will the student have the same number of correct answers on both exams?

(b) Suppose that a student has the utility function

where S is the student’s overall score for the course, M is the number of minutes the student spends studying, and A is a variable that reflect show much the student dislikes studying. In Part (a) of this problem, you found that a student who studies for M minutes and allocates this time wisely between the two exams will get an overall score of S =

Substitute for S in the utility function and then differentiate with respect to M to find the amount of study time, M, that maximizes the student’s utility. M =

Your answer will be a function of the variables P1, P2, and A. If the student chooses the utility-maximizing amount of study time and allocates it wisely between the two exams, he or she will have an overall score for the course of S =_______

(c) Nancy Lerner has a utility function like the one presented above. She chose the utility-maximizing amount of study time for herself. For Nancy,

P1 = 10 and P2 = 20. She spent a total of M = 1, 200 minutes studying for the two exams. This gives us enough information to solve for the variable A in Nancy’s utility function. In fact, for Nancy, A =_______.

(d) Ed Fungus is a student in Nancy’s class. Ed’s utility function is just like Nancy’s, with the same value of A. But Ed learns more slowly than Nancy. In fact it takes Ed exactly twice as long to learn anything as it takes Nancy, so that for him, P1 = 20 and P2 = 40. Ed also chooses his amount of study time so as to maximize his utility. Find the ratio of the amount of time Ed spends studying to the amount of time Nancy spends studying. ___________. Will his score for the course be greater than half, equal to half, or less than half of Nancy’s?

__________.

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A. U(S, M) = S -