help with question 3

3. [40 points total] At Hulala University, NV classes are offered to students. Each student takes only one class, and the class size is the same for all classes. Let a; denote the fraction of students with grade A in class j. Assume for simplicity that a; is continuous between 0 and 1. Collective utility for class j is assumed as follows. Uj = (xj - 2x;) - 0.5X2 with X= 21 + 22 + + ON The first terms in parentheses represent benefits from generous grading in class j. As x; increases, more students get benefits from good grades in their career. The third term -0.5X2 represents the cost of generous grades. Too high proportion of A from Hulala reduces the validity of A, as people may attribute it to "grade inflation." Thus, they discount grades from Hulala, which hurts all students from Hulala. (a) [5 points] Why can we interpret this situation as a negative externality? (b) [7 points] Assume N = 2 for (b) and (c). Find (21, 22) that maximizes U1 + U2. (c) [10 points] The instructor of class 1 determines x1 to maximize U1, taking x2 as given. Similarly, the instructor of class 2 determines 2 to maximize U2 taking as 21 given. Find (21, 22) in the equilibrium. Interpret. (d) [8 points] Consider general N for (d) and (e). Find (x1, 22, . .., IN) that maxi- mizes U1 + U2 + . . . + UN. (e) [10 points] Assume that each instructor of class j determines x, to maximize Uj taking xi's from all other classes as given. Find (21, X2, . .. ,"'N) in the equilibrium. Interpret