Question

You are planning to visit Timbuktu, and you expect to spend $10,000 (on travel, souvenirs, camel rides, and the like). Your utility function is given by U(Y) = log10Y, where Y is the amount you spend.

a. There is a 25% probability that you will lose $1,000 on the trip – your wallet falls out of your pocket while you are on the camel. What is the expected utility of the trip?

b. You can buy actuarially fair insurance against losing the money. Show that you will want to buy this insurance.

c. What is the maximum you would be willing to pay to insure against the possible loss of the $1,000?

d. Unfortunately, people who buy insurance may become careless; with insurance, the probability that you will lose the money rises to 30%. Will you buy actuarially fair insurance in this case? Explain.

Answer 1

a. To get the expected utility value we have to have a scale measuring the utility values we get from each circumstance.

If the person loses $1000 while riding a camel and then also could enjoy the rest of the trip with the remaining money, the utility if this scenario might be more than keeping the whole money and enjoying the trip. Expected utility captures a very important intuition that there is diminishing marginal utility of money. Hence, scenario 1 might give the person more utility than secnasce 2. Here, expected value and expected utility are considered two different things. Whole expected value can be calculated but expected insurance cannot be.

b. Acturially fair insurance policy = Probability of loss * Size of the loss = .25 * $1000 = $250

Buying an insurance here will give the person an insured amount of $10000, so it is better to buy an insurance.

c. The person will always buy an insurance amount of $1000 to secure his $1000 amount of money.

d. Actually fair insurance = Expected payout = 0.3 * (0) + 0.7(1000) = 700. Even here a person will buy actually fair insurance ad the insurance premium does not exceed the loss amount of money.