Suppose an individual earns income $600 when they are sick, and $1000 when they are healthy. Suppose this individual is sick with probability p = 0.5 and has a utility function over income, I, of U(I) = ln(I). Is this individual risk-averse, risk neutral or risk-loving? Suppose she is able to purchase insurance at any amount from at an actuarially fair price. Fully describe the amount she would purchase (payout, premium and final outcomes). Verify that she is better off with the contract in part b, as opposed to being uninsured. Suppose insurance company A offers a payout q = $400 (when she is sick) at a premium of r = $220 and insurance company B offers a payout of $200 at a premium of $100. Company A's contract is: Actuarially fair or unfair? Is it full or partial insurance? Company B's contract is: Actuarially fair or unfair? Is it full or partial insurance? Which contract does this individual prefer? Suppose contract A is unfair, but offers full coverage at price . Contract B is fair but only offers partial coverage at. The probability of being sick is . Derive a general expression for as a function of  and the other parameters, such that our individual would be indifferent between the two contracts.

Suppose an individual earns income $600 when they are sick, and $1000 when they are healthy. Suppose this individual is sick with probability p = 0.5 and has a utility function over income, I, of U(I) = ln(I).

1. Is this individual risk-averse, risk neutral or risk-loving?
2. Suppose she is able to purchase insurance at any amount from at an actuarially fair price. Fully describe the amount she would purchase (payout, premium and final outcomes).
3. Verify that she is better off with the contract in part b, as opposed to being uninsured.
4. Suppose insurance company A offers a payout q = $400 (when she is sick) at a premium of r = $220 and insurance company B offers a payout of $200 at a premium of $100.
   1. Company A's contract is: Actuarially fair or unfair? Is it full or partial insurance?
   2. Company B's contract is: Actuarially fair or unfair? Is it full or partial insurance?

Which contract does this individual prefer?

Suppose contract A is unfair, but offers full coverage at price . Contract B is fair but only offers partial coverage at. The probability of being sick is . Derive a general expression for as a function of  and the other parameters, such that our individual would be indifferent between the two contracts.

Hint: Just write down the system of equations, you do not need to solve for closed-form solutions.