Question

John works for an antique store and sells rare and valuable artifacts. He is a rational person who maximizes his expected utility and he is only interested on the artifacts monetary value. His utility function is given by u(w) = w^½ = √w, where w is his wealth measured in million dollars. John has an initial wealth of w₀ = 4 million dollars. While on holiday in Paris he finds in antique shop a small statue of a falcon, which costs p=4 million dollars. The owner of the shop claims that it is the long-lost Maltese Falcon, an artifact for which certain collectors would pay John 100 million to get it. After a careful examination John gives a 15% probability to the item being indeed the genuine Maltese Falcon and an 85% probability to the item being a usual antique worth just 1 million dollars.

a. What will be John’s utility if he buys the item and it turns out to be the Maltese Falcon and what if it turns out to be just a usual antique? What will be his utility if he chooses not to buy it?

b. Calculate John’s expected utility from purchasing the item. Will he buy the item from the antique shop?

c. Given John’s probability distribution, what is the expected value of the item?

A man, called Adam, approaches John the day after he is back with the artifact. Adam is willing to offer John 7 million dollars to buy the artifact, before he finds out whether it is genuine or not.

d. Based om John’s expected value of the item, is this a fair offer? If not, does it favor John or Adam?

e. Will John accept Adam’s offer?

f. Is John a risk-averse, a risk-neutral or a risk-preferring person? Why?

g. Calculate John’s risk premium. What is the lowest amount Adam can offer to John and John still accept to sell him the artifact?

Answer 1

a. If he buys the item and it turns out to be Maltese Falcon, then his utility will be 100^1/2 or 10, since his utility is associated with the wealth he would finally have. If the item turns out to be a usual antique, then his utility will be 1^1/2 or 1. In case he will not choose to buy, then his utility will be 4^1/2 or 2.

b. The expected utility is the probability of one outcome(success) multiplied by the associated utility plus another outcome (failure) multiplied by the associated utility.

John's expected utility will be .15(10)+.85(1) or 2.35. Yes, he will buy it since the expected utility from buying the antique is greater than not having it which was 2 as calculated above.

c. the expected value of the item will be .15(100)+.85(1) which is 15.85 million dollars.

d.No this is not a fair offer. As the expected value of the item for John was 15.85 million dollars which were greater than what is offered by Adam.The offer is favourable to Adam.

e. No, he will not for the reasons given in d.