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 0.1 of getting consumption xB (bad state) and a probability 0.9 of getting xG (good state).

An insurance company allows you to choose an insurance contract (b, p), where b is the insurance benefit the company pays you if the 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) subject to this offer.

(i) 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. 601-602) of the textbook or slide 20-22 of Chapter 17 for an illustration. (But note that the example assumes actuarily fair insurance, so if the insurance contract here is not actuarily fair, you need to modify the variables accordingly.)

(ii) What will your consumption levels be in the two states with the insurance. Are you fully insured?

Slide 20-22

An insurance contract is a pair (b,p) where b is the insurance benefit that is paid in the bad state and p is the insurance premium that is paid regardless of the state. Consumption in good state: (G p) consumption in bad state: (y + b-p) The insurance company therefore receives the premium p with certainty and pays the benefit b with probability & This implies the expected cost is equal to the expected revenue if and only if Љ Any insurance contract (b,p) that satisfies this condition is called actuarily fair .e. The menu of actuarily fair insurance contracts is defined by the equation - - p or, equivalently, b 20

Answer 1

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