Question

11. In her last-minute preparations for final exams, a student has set aside five hours to split between studying for two subjects, finance and economics. Her goal is to maximize the average grade received in the two courses. (Note that maximizing the average grade and maximizing the sum of the grades are equivalent goals). According to her best guesses, grades vary with study as follows:

Study Hours 0/1/2/3/4/5

Finance Grade 70/75/83/88/90/92

Study Hours 0/1/2/3/4/5

Economics Grade 75/81/85/87/89/90

a. List the marginal values of additional hours worked for each subject

b. How much time should the student spend studying for each subject?

Answer 1

(a) The marginal values are as below.

Study Hours	MVF	MVE
1	75-70=5	81-75=6
2	83-75=8	85-81=4
3	88-83=5	87-85=2
4	90-88=2	89-87=2
5	92-90=2	90-89=1

The marginal value of each grade would be the ratio of difference between consecutive grades and difference between study hours (which is 1 for each case).

(b) The time spend on finance would be 3 hours and time spend on economics would be 2 hours. This is obtained as MV of finance and economics are closer where F=3 and E=2, subjected to the constraint. The reasoning behind this is that, increasing (by 1 study hour) F from 3 would increase the total grades by 2, but would also then reduce the total grades by 4, making the net increment of grade as -2 (which is loss of total grades by 2). Also, increasing (by 1 study hour) E from 2 would increase the total grades by 2 but then would also reduce the grade by 5, making the net increment of grades as -3 (loss in total grades by 3). Hence, at F=3 and E=2, the total grades will be maximized.

We have the constraint as , as there are 5 hours to allot. The maximization of grades can be confirmed be as below.

Distribution (F+E=5)	Total Grades
F=0,E=5	70+90=160
F=1,E=4	75+89=164
F=2,E=3	83+87=170
F=3,E=2	88+85=173
F=4,E=1	90+81=171
F=5,E=0	92+75=167

As can be seen, it can be confirmed that the total grades are maximum at where F=3 and E=2.