Consider a person selling “grade insurance” to a class of 100 students. Whoever buys this insurancegets ‘A’ in the class as the insurer makes the required cash payment to the instructor.The students know exactly the grade they are going to make in the class if they don’t buy theinsurance. Without any insurance, 10% of the students get A, 25% get B, 30% gets C, 25% getsD and 10% gets F. Assume that there is no moral hazard problem, i.e. each student works justas hard independent of their decision to buy insurance. If a student earned grade below A, theinsurer has to pay the instructor to overcome his scruples and raise the grade, with the size ofthe payment depending on how much the grade needs to be raised in order to get to A. If, onthe other hand, a student who bought the insurance, actually earned A, no grade adjustment isnecessary and no cost is incurred by the insurer. Lets suppose that the instructors scruples aresuch that it costs a minimum of c for him to raise a grade by one letter grade (and 2c to raise itby two letter grades, 3c to raise it three letter grades, etc.). Suppose that all students are willingto pay as much as 2c to raise their grade by one level and 0.5c for any additional increasein the grade by another level. Put differently, an F student is willing to pay 2c to raise his gradeto a D, 2.5c to raise his grade to a C, 3c to raise his grade to a B and 3.5c to raise his grade to an A.
(a) Assuming the students who know they are going to get an A on their own don’t wantto buy the insurance, what is the total cost for the insurer if everyone else buys insurance? Whatthe the average cost per student?
(b) Assuming the insurance company tries to prices the insurance at the average cost youfound in the previous part, who would be willing to buy the insurance? Explain.
(c) Given the students who would want to buy the insurance at this price (your answer inthe last part), what would be the revised total and average cost for the insurer now?
(d) Assuming the insurance company tries to price the insurance at the average cost youfound in the previous part, who would now be willing to buy the insurance? Explain.
(e) Repeat the steps in the last two parts and explain what will be the equilibrium price forA-insurance in this market. (For this step, repeat the calculations in parts c and d twice)
(f) What would be the equilibrium insurance premium if, in a system that forced all studentsto buy insurance, the only insurance policy offered were one that guarantees a B? What if the onlypolicy that were offered was one that guaranteed a C?
(g) In an efficient allocation of grade insurance (when only A-insurance is offered), whowould have A-insurance? (Hint: Compare the total cost of raising each student types grade to thetotal benefit that this would yield for each student type.)
(h) Would the insurer be able to sell A insurance if students were always willing to pay 2cfor every increase in their letter grade? Would the resulting equilibrium be efficient?