Is the insurance contract actuarily fair how much will you

Question

Suppose the function u(x) = x^0.5 where x is consumption represents your preference over gambles using an expected utility function.

You have a probability δ of getting consumption x_B (bad state) and a probability 1-δ of getting x_G (good state).

(a) Find the certainty equivalent x_CE of the gamble.

(b) Find the risk premium of the gamble.

For (c) to (e) below, let δ = 0.1, x_B = 10 and x_G = 100.

An insurance company allows you to choose an insurance contract (b, p), where b is the insurance benefit the company pays you if bad state occurs and p is the insurance premium you pay the company regardless of the state.

Suppose the company's offer is p = 0.2b. You may choose any combination of (b, p).

(c) Is the insurance contract actuarily fair? How much will you insure (i.e. find your optimal p and b)?

(d) What will your consumptions be in the two states with the insurance. Are you fully insured?

(e) What is the expected profit of the insurance company from the contract?