Consider a risk-averse consumer who has an initial wealth of $1000. With probability 15%, this consumer incurs a loss of $900 due to illness. Suppose that this person is indifferent between their endowment and having $500 with certainty.

1. What is the actuarially fair rate for $1 of insurance?
2. How much insurance will the consumer purchase if they face actuarially fair rates?
3. What is the risk premium? Illustrate it.
4. Suppose that the consumer faced an insurance premium of $0.56 for a dollar of insurance.
5. Does the insurance company earn a profit at these rates?
6. Will the consumer purchase any insurance at these rates and, if so, will they fully insured?