Consider the case where a person has a utility function of where U signifies utility and W signifies wealth in £. The more wealth they have, the more utility they have so the happier they are.

Say they start with wealth of £2000. They own a machine that has a 10% chance of breaking down in a particular time period. If it breaks down it will cost £800 to repair it.

There are two possible outcomes here, a good outcome where the machine doesn’t break down and the person has wealth of £2000, or a bad outcome where the machine breaks down and they have £1200. We can use the utility function to calculate the amount of utility in the respective states:

Good outcome: U = 97.793

Bad outcome: U = 83.899

The machine has a 10% chance of breaking down, so that means a 90% chance of us having the good outcome and 10% chance of the bad outcome. We can use this to calculate the expected utility of this whole situation:

EU = (0.9 x 97.793) + (0.1 x 83.899) = 96.404

Now if we went back into the original utility function to find what wealth corresponds to a utility level of 96.404, we get .

In other words the expected utility of this situation is equivalent to having wealth of £1906.84. The difference between this and the starting wealth of £2000, £93.16, is the maximum **risk premium** the machine owner is willing to pay to remove the risk. If an insurer offers to sell insurance that will fully cover the costs of machine repair if it breaks down, then the machine owner is willing to pay up to £93.16 in order to buy that insurance.

Say the insurer offers to provide full insurance for a cost of £90. Then the machine owner has a choice – either he takes the insurance, which means he has guaranteed wealth of £1910, and a corresponding guaranteed utility of 96.452, or he doesn’t take insurance, which will mean he either has £2000 (and utility of 97.793) if it doesn’t break down, or £1200 (and utility of 83.899) if it does. In terms of expected utility, the expected utility from taking insurance is 96.452, and the expected utility from not taking the insurance is 96.404. As the expected utility from taking the insurance is higher than the expected utility from not taking insurance, then it is rational to take the insurance. If the insurance had cost more than £93.16, then the expected utility from taking the insurance would have been lower than the expected utility from not taking the insurance, so it would have been rational not to take out the insurance.

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A risk averse person will have a utility function relative to wealth, that looks similar to this:

A risk neutral person will have a utility function that looks like this:

A risk loving person will have a utility function that looks like this:

The shapes of the utility functions reflect the extent to which utility increases with wealth. The one of most interest to us is the risk averse one, because people are more likely to be risk-averse than anything else. This brings us to the concept of **risk premium** and **insurance**.

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