## Risk Preferences

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# Risk Preferences

Study Figure 6.1. We have a wealth of 5 plus an uncertain outcome of a lottery. Together, these give us an end wealth of either 0 or 10. We have indicated these values on the X-axis. The corresponding utilities we have indicated The expected utility of the wealth plus the lottery will be a point somewhere along a straight line from a to b, depending on the probability outcome. In the present case, the probability for each outcome is 50% and then the expected utility will be a point exactly half-way in between a and b, i.e. in point c where the utility is 1.6. (With other probabilities, we would have ended up in another point on the same straight line.) We have now illustrated the expected utility of an uncertain outcome of either 0 or 10.

What if we chose not participate in the lottery at all? In that case, we would keep a certain wealth of 5. The expected utility is unchanged: The expected value of a certain 5 is 5, and the expected value of a certain 5 plus the lottery is also 5. However, the utility of a certain wealth of 5 is 2.2, corresponding to point d in the figure. The outcome that, participating in the lottery gives less utility than not participating, depends on the fact that the utility function slopes less and less steep as wealth increases, i.e. that we have diminishing marginal utility. A person with such a utility function will always prefer not to participate in a fair lottery, and she is said to be risk averse. Depending on which type of marginal utility an individual has (compare to above), we can classify her attitude towards risk:

- Risk averse (diminishing marginal utility). Prefers not to participate in a fair lottery. Most people, if not all, are risk averse. Note, however, that this theory (at least in the basic version presented here) is unable to explain why many people in real life participate in lotteries.
- Risk neutral (constant marginal utility). Is indifferent.
- Risk loving (increasing marginal utility). Prefers to participate in the lottery. A very unusual property!

**Certainty Equivalence and the Risk Premium**

Look at Figure 6.1 again. We have already seen that an ordinary person (i.e. a risk averse person) prefers not to participate in the lottery. One may then ask which level of certain wealth she would value as much as participating in the lottery. As we saw before, her utility of participating is 1.6 (point c). The question is then which wealth would give her that same utility. Follow the line from 1.6 to the utility function and you will end up at point e, corresponding to a certain wealth of 2.6. This individual is, consequently, indifferent between participating in the lottery and having a certain wealth of 2.6. The value 2.6 is then said to be certainty equivalent to participating in this lottery.

Since she now has a wealth of 5, she is, in other words, prepared to pay 5 - 2.6 = 2.4 to avoid the lottery. Alternatively, if her wealth had been 2.6, how much would we have had to pay her in order to make her willing to participate in the lottery? The answer is the same: 2.4. This amount is called the risk premium.

**Risk Reduction**

Since most people are risk averse, they want to reduce risk. That is often achieved by pooling the risk and sharing it. This is, for instance, the idea behind insurance, where the risks are shared between many people.

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