Consider a simple example of moral hazard. Suppose that Bob goes into a casino to make one bet a day. The casino is very basic; it has two bets: a safe bet and a risky bet. In the safe bet, a nickel is flipped. If the nickel lands on heads, Bob wins $100 . If it lands on tails, Bob loses $100 . The risky bet is similar: a silver dollar is flipped. If the silver dollar lands on heads, Bob wins $5,000 . If it lands on tails, Bob loses $10,000 . Each coin has a 50% chance of landing on each side. What is the expected value of the safe bet?
What is the expected value of the risky bet?
Now suppose that an insurance company opens outside of the casino. They notice that Bob, like everyone else, always leaves the casino having played the safe bet, so they offer to sell Bob insurance for $50 that, if he loses, covers his losses. If Bob wins, he does not have to pay anything extra, having already paid the $50 . Note that as long as Bob does not change his behavior, the insurance company makes $0 in expectation.

Once Bob buys the insurance, what are his expected winnings from the safe bet? Ignore the cost of the insurance.

Once Bob buys the insurance, what are his expected winnings from the risky bet? Ignore the cost of insurance.