Question: You have considered insuring a particular item of property (such as an expensive camera, your computer, or your Stradivarius violin), but after considering the risks and the insurance premium quoted, you have no clear preference for either purchasing the insurance or taking the risk. The insurance company then tells you about a new scheme called "probabilistic insurance." You pay half the insurance premium quoted but have coverage only in the sense that in the case of a claim there is a probability of one-half that you will be asked to pay the other half of the premium and will be completely covered, or that you will not be covered and will have your premium returned. The insurance company can be relied on to be fair in flipping the coin to determine whether or not you are covered.

a. Do you consider yourself to be risk-averse?

b. Would you purchase probabilistic insurance?

c. Draw a decision tree for this problem.

d. Show that a risk-averse individual always should prefer the probabilistic insurance.

(This is a difficult problem. To solve it, you must be sure to consider that you are indifferent between the regular insurance and no insurance. Write out the equation relating these two alternatives and see what it implies. Another strategy is to select a specific utility function- the log utility function U(x) = ln(x), say-and then find values for the probability of a claim, your wealth, the insurance premium, and the value of your piece of property so that the utility of paying the insurance premium is equal to the EU of no insurance. Now use these values to calculate the EU of the probabilistic insurance. What is the result?)