A company plans to sell automobiles insurance policies that will cover the cost of repairing collision damage. Collision damage typically costs $5,000 to repair. The company plans to offer two versions of this policy. One version will have no deductible: if a collision occurs, the insurer will cover the full $5,000. The other version will have a $2,000 deductible: if a collision occurs, the insurer will pay $3,000 and the customer will pay $2,000. The insurer does not know how risky each individual customer might be, but it estimates that 1/3 of people are cautious drivers, 1/3 are typical drivers, and 1/3 are sporty drivers. Cautious drivers have a 10% chance of submitting a claim for damage repair each year; typical drivers have a 20% chance; and sporty drivers have a 35% chance. Drivers could instead buy insurance from a government-run program that has no deductible. Cautious drivers would pay $1100 per year for this coverage, typical drivers would pay $1400, and sporty drivers would pay $1700. Drivers will buy policies from the insurance company if their total cost is at least as low as the government rate: this includes the known cost of the premium plus the expected cost of the deductible amount (if any) if a collision occurs. The insurer wants to sell policies to all 3 types of drivers; given a choice, it would prefer to set its premiums as high as possible. Your task is to advise the insurance company regarding the best 2 annual premiums to charge.

Questions:

Treat this situation as a sequential-move game. The insurer moves first by specifying the premiums for its policies. Drivers move second by choosing whether to buy from the insurer; and if so, which policy to buy. Questions 1 and 2 below are simple warm-up exercises to get you thinking; you can solve them using participation conditions or simple logic. Question 3 requires you to model, analyze, and interpret the situation using decision variables, participation conditions, and incentive compatibility conditions.

1. Suppose the insurer only offers its zero deductible policy, and charges $1234 per year for it. i. Which drivers would buy this policy? ii. What would be the insurers average annual profit or loss per customer: i.e., the premium they receive minus the expected cost of damage they payout?

2. Suppose instead the insurer only offers its $2000 deductible policy, and charges $1234 per year for it. i. Which drivers would buy this policy? ii. What would be the insurers average annual profit or loss per customer: i.e., the premium they receive minus the expected cost of damage they payout?

3. Suppose instead that the insurer offers both types of policies, charging a high premium for the zero deductible policy and a low premium for the $2000 deductible policy. It wants only cautious drivers to choose the $2000 deductible policy, while typical and sporty drivers choose the zero deductible policy. i. What two annual premiums should it charge? ii. What would be the insurers average annual profit or loss per customer: i.e., the premium they receive minus the expected cost of damage they payout?