Uncertainity and Insurance:

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

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

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

Hint: Use the fact that αln(x0)+βln(x1)=ln(x0αx1β) for any positive numbers x0 and x1.

(b) Find the risk premium of the gamble.

Now let δ = 0.1, xB = 10 and xG = 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)?

Hint: Refer to Section 17B.1.4 (p. 599-600) of the textbook or slide 20-22 of Chapter 17 for an illustration of choosing actuarily fair insurance.

(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?