A student has to decide whether to go to a party on the night before an exam. If they go to the party, then they have the probability of writing a good exam and the probability 1 of writing a bad exam. If they do not go to the party, then they have the probability > of writing a good exam. The student has the utility u = x^1/2 + H if they go to the party and get the mark x on the exam. But, if they do not go to the party they have the utility u = x^1/2. The professor gets to choose xb (the mark she gives to bad exams) and xg the mark she gives to good exams. Students also have the option of going to the party and then quitting the course. This gives them the utility U + H where U > 0.

(a) What is the student's expected utility from going to the party and her expected utility from not going to the party?

(b) Assuming students do not quit the course, what marks (x_g,x_b) can the professor set to stop students partying? Draw a picture of this set of marks with y_g = x_g on one axis and y_b = x_b on the other. Explain how this set changes as H and change.

(c) What marks can the professor set so the students prefer partying and taking the exam to quitting the course? What marks can the professor set so the students prefer not partying and taking the exam to quitting the course? Show these sets on a new picture.

(d) Suppose that the professor wants to promote equity and thus aims to make the dierence x_g x_b as small as possible while still stopping students from going to the party, but she ignores the possibility that students will quit the course. What marks (x_g,x_b) should she set to achieve her objective? Will this result in the outcome she planned?

(e) Suppose now the professor wants to minimise xg xb while still getting students to not party and attend the course. Plot the contours of her objective function on your (yg,yb) picture. What marks (xg,xb) should she set to achieve her objective now?

(f) The professor has other variables that she can control: she can make the exam harder, which decreases and , and she can make the party fail by calling the campus authorities, which decreases H. How does each of these variables aect her ability to achieve the objectives described above?