04.02.27a. Rabin on Expected Utility and Small Risks. This came up in the law-and-econ lunch yesterday, and it's an important idea, easily mistinterpreted, including by Professor Rabin. The idea is this: the economist's standard interpretation of risk aversion as concave utility (which I discussed last September 17) fails to explain why people are averse to small risks-- on the order, say, of losing $100-- because utility functions are locally almost linear. Here are some examples from Rabin & Thaler (2001--references at the end-- and these are not all quotes--some are my rephrasings).
Example 1: John. Suppose John is a risk-averse expected utility maximizer, and that regardless of his initial wealth, he would always turn down the 50-50 gamble of losing $10 or gaining $11.

It can be shown that he would also turn down a 50-50 gamble of losing $100 or gaining $5 billion. Indeed, there is no gamble he would accept if it involved a 50% chance of losing $100.

Example 1 captures the Rabin Paradox, as I shall call it, perfectly. The catch is the phrase regardless of his initial wealth. It is perhaps reasonable that a very poor person would reject the 10-11 gamble. He might have only $10 to spend for the day's food, and if he loses $10, he is left hungry, whereas $21 merely buys him a more delicious meal than the one he would get without a gamble. But if the person starts with $1,000,000, then the gamble can be rewritten as a bet between a sure $1,000,000 and a 50-50 gamble between $999,990 and $1,000,011. Put this way, the gamble does not seem so unattractive. Let me rewrite Example 1, just to frame it differently.

Example 1': John Again. Suppose John is a risk-averse expected utility maximizer. He prefers a sure wealth of $1,000,000 to a 50-50 gamble between $999,990 and $1,000,011, and in fact would reject a 50-50 gamble of losing $10 versus winning $11 starting at any wealth level.

It can be shown that he would also turn down a 50-50 gamble between losing $100 or gaining $5 billion. Indeed, there is no gamble he would accept if it involved a 50% chance of losing $100.

Example 1' is still surprising, but it points us in the direction to resolve the Paradox: it is strange that John, when rich, would reject the 10-11 gamble. If he loses, the marginal utility of money will be trivially different from when he wins, so why not be almost risk neutral?

Thus,the puzzle here is not that expected utility theory doesn't work, generally. Rather, it is that concavity of the utility function implies that people should be indifferent to small risks, and sometimes it seems they are not.

My objection to Example 1 is that the phrase regardless of his initial wealth makes the utility function unreasonable. Professor Rabin has encountered that objection, and shows that this constraint can be relaxed and the paradox pretty much remains:

Example 2: Mary. Suppose Mary is risk averse. She would turn down a 50-50 bet of losing $100 or gaining $105 starting from any wealth level less than $350,000. We know nothing about her utility function above $350,000 except that it is concave.

It can be shown that starting with initial wealth of $340,000, Mary would turn down a 50-50 bet of losing $10,000 or gaining $5.5 million.

As he and Thaler explain,

The intuition is that the extreme concavity of the utility function between $340,000 and $350,000 assures that the marginal utility at $350,000 is tiny compared to the marginal utility at wealth levels below $340,000. Hence, even if the marginal utility does not diminish at all above $350,000, a person won't care nearly as much about money above $350,000 as she does below $340,000.

Of course, Mary is still being unreasonable in rejecting a gamble of 100 vs. 105 when she is so wealthy.

Rabin & Thaler want to go to mental accounting and loss aversion to explain why rich people sometimes *are* indifferent to small risks like this. I think I prefer different explanations. Such anomalies are best thought about using concrete examples, so here's one they present:

Ignoring calibration can even lead researchers to misidentify strong evidence against expected utility theory as support for the theory. Cicchetti and Dubin (1994), for instance, study the choice by consumers to purchase protection from their local telephone company against having to pay for repairs to their internal telephone wiring. The authors report that, on average, consumers faced a choice of paying 45 cents a month for the insurance against an average .005 probability in a given month of having to pay a total repair cost averaging $55. Hence, people were paying 45 cents a month to insure against a "risk" that will average 28 cents per month, with a small risk of having to pay $55, and a minuscule risk of having to pay more. Millions of Americans every year buy similar wiring protection, as did 57 percent of households in Cicchetti and Dubin's sample. If utility-maximizing customers had close to rational expectations about the probability of needing repair, or merely figured out that the phone company aimed to make money from them net of transactions costs, then it is implausible that they would buy the protection. In any event, it is easy to reject the joint hypothesis of approximate expected utility maximization and approximate rational expectations. The authors, however, draw precisely the opposite conclusion, and offer their analysis as a real-world confirmation of expected utility maximization. Their misinterpretation replicates our profession's grander-scale misinterpretation of risk aversion, because, like most economists, they fail to realize that expected utility theory does not permit risk aversion for so little money.

So why do people pay for this insurance against a trivial risk? I think (1) they don't want to have to think about it, and (2) they are worried about asymmetric information. Both of these reasons show up in a lot of contexts. In more detail:

1. Thinking costs. Having paid their few cents, people don't have to htink about the problem any more. If a repair problem occurs, they won't have to check bills. If there is no repair problem, they don't have to worry about being billed by mistake. Hassle is avoided.

2. Asymmetric information. The phone company knows the .005 probability and the customer does not. The customer may fear that having contracted to pay for repairs, they have committed themselves to paying large sums on a regular basis. On the other hand, there might *never* be a need for repairs, in which case they lose by buying insurance. The phone company, with its superior information, will win this game. But the customer can cut his losses by agreeing to the known-in-advance insurance cost instead of risking that the phone company was just trying to trick him into buying insurance and instead falling into the trap of having to pay for expensive repairs.

In either case, a simple reduced-form way to model small risks is as imposing a fixed cost on the person, in contrast to no cost being borne if avoids the gamble.

I'm running short on time, so I'll just quote some interesting stuff from the rest of the article:

Expected utility theory's presumption that attitudes towards moderate-scale and large-scale risks derive from the same utility-of-wealth function relates to a widely discussed implication of the theory: that people have approximately the same risk attitude towards an aggregation of independent, identical gambles as towards each of the independent gambles. This observation was introduced in a famous article by Paul Samuelson (1963), who reports that he once offered a colleague a bet in which he could flip a coin and either gain $200 or lose $100. The colleague declined the bet, but announced his willingness to accept 100 such bets together. Samuelson showed that this pair of choices was inconsistent with expected utility theory, which implies that if (for some range of wealth levels) a person turns down a particular gamble, then the person should also turn down an offer to play many of those gambles.

When Samuelson showed that his colleague's pair of choices was not consistent with expected utility theory, Samuelson thought that the mistake his colleague made was in accepting the aggregated bet, not in turning down the individual bet. This judgement is one we cannot share. The aggregated gamble of 100 50-50 lose $100/gain $200 bets has an expected return of $5,000, with only a 1/2,300 chance of losing any money and merely a 1/62,000 chance of losing more than $1,000. A good lawyer could have you declared legally insane for turning down this gamble.

... Samuelson and others have speculated as to the error his colleague was making, such as thinking that the variance of a repeated series of bets is lower than the variance of one bet (whereas, of course, the variance increases, though not proportionally, with repetition). Others have played off the fact that the equivalence theorem holds only approximately to explore the precise qualitative relationship that expected utility permits between risk attitudes over one draw and many independent draws of a bet. But our argument here reveals the irrelevance of these lines of reasoning. It does not matter what predictions expected utility theory makes about Samuelson's colleague, since the degree of risk aversion he exhibited proved he was not an expected utility maximizer. In fact, under exactly the same assumptions invoked by Samuelson, the theorem in Rabin (2000) implies that a risk-averse expected utility maximizer who turns down a 50-50 lose $100/gain $200 gamble will turn down a 50-50 lose $200/gain $20,000 gamble. This has an expected return of $9,900-with exactly zero chance of losing more than $200. Even a lousy lawyer could have you declared legally insane for turning down this gamble.

Arrow, Kenneth. 1971. Essays in the Theory of Risk Bearing. Markham Publishing Company.

Friedman, M. and L. Savage. 1948. "The Utility Analysis of Choice Involving Risk."Journal of Political Economy 56, pp. 279-304.

Samuelson, P. 1963. "Risk and Uncertainty: A Fallacy of Large Numbers." Scientia. 98, pp. 108-13.

Rabin, M. (2000) "Risk Aversion and Expected Utility Theory: A Calibration Theorem", Econometrica, 68, 1281-1290.

Rabin, M. and R. Thaler (2001), "Anomalies: Risk Aversion", Journal of Economic Perspectives, 15, 219-232.

Ariel Rubinstein's 2001 "Comments on the Risk and Time Preferences in Economics"

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