Class, 10/20

We talked about utilities. First we looked at the shapes of the curves that you all derived over the weekend. We learned that a straight line is neutral, a utility curve that curves up is risk-seeking and one that curves down is risk-averse.

We then discussed insurance on a house. We found that if h is the value of the house and m is the premium we pay for the insurance, and p is the probability of disaster (e.g., a fire burns the house down), then the insurance company will demand that p be less than m/h. On the other hand, the owner of the house (if her utility is neutral) will demand that p be greater than m/h, and no transaction can take place. But insurance is bought and sold, so there has to be an explanation for this. And there is an explanation, because although insurance companies have a nearly neutral utility curve except for truly huge amounts, people do not, and most people have risk-averse utility curves. They will demand that p be greater than u(-m)/u(-h) where u(-) means the value of the curve at the point in question (the quantities are negative because in both the case of the premium and the potential catastrophe, the person ends up with less assets). But, if you have a utility curve that curves down, that means that the ratio u(-m)/u(-h) will be less than m/h, so the person will be willing to buy the insurance after all. Therefore, the insurance company can now set a value of m that the consumer will be willing to pay and which will also give a profit to the company, thus keeping the stockholders happy.

I remarked that this is actually how all commerce works. There are two parties, a seller and a buyer. They are willing to make a transaction because their utility curves are different, and so it is a "win-win" situation where everyone, both the buyer and the seller, feel themselves better off (in terms of utilities) than they did before the transaction took place.

One student had remarked in class and in journals that this approach (decision theory) might not be adequate when considering lotteries, where there is a huge payoff of very low probability. Should someone wager to win the lottery, even if taxes and annuitization made it a positive payoff on expected return basis? My answer is, "Not Really." The reason is that we don't (or shouldn't) make decisions based on expected return. We should make decisions based on expected utility or expected loss. I posed the question, would you rather have $280 million with probability 1/2, or $10 million for sure. The overwhelming choice of the class was, take the $10 million. This means, that to most of the people in the class, having $280 million isn't that much better than having $10 million. This means that in the lottery problem, you probably won't want to use $280 million as the leaves on the ends of the decision tree. You probably will make as good a decision if you just put $10 million there. And if you did this, your decision would be just as rational, and would tell you that the lottery is not really a good place to invest your money (unless your only reward is the thrill of entering the lottery!) Final comment is that the student who raised this issue initially agreed that when utilities or losses were used as the payoff, then it would not be a problem.


I finally drew on the board a generally useful way to estimate utilities for any events whatsoever. I used the Monty Hall example of a car, a goat, and a trip to Hawaii. Presumably the car is the best and the goat is the worst, with the Hawaii trip in between. Draw a decision tree, put the car and the goat on the probability branches and the Hawaii trip on the "get for certain" branch. Then, pick a probability for getting the car that makes you neutral between the two branches of the decision tree. There should be a point where you are neutral, for if the probability of getting the car is 1, you'd take the car for sure, but if the probability of getting the car is 0, you'd pick the Hawaii trip for sure. One student volunteered p=0.8. That means that her utility for the Hawaii trip is 0.8, since at that point, both branches of the decision tree have exactly the same value.

I remarked finally that if you use this method for evaluating utilities, then the utilities so calculated are actually probabilities!