172 PART • Producers, Consumers, and Competitive Markets Why? The answer is implicit in our discussion of risk aversion Buying insurance assures a person of having the same income whether or not there is a loss Because the insurance cost is equal to the expected loss, this certain income is equal to the expected income from the risky situation For a risk-averse consumer, the guarantee of the same income regardless of the outcome generates more utility than would be the case if that person had a high income when there was no loss and a low income when a loss occurred To clarify this point, let’s suppose a homeowner faces a 10-percent probability that his house will be burglarized and he will suffer a $10,000 loss Let’s assume he has $50,000 worth of property Table 5.6 shows his wealth in two situations—with insurance costing $1000 and without insurance Note that expected wealth is the same ($49,000) in both situations The variability, however, is quite different As the table shows, with no insurance the standard deviation of wealth is $3000; with insurance, it is If there is no burglary, the uninsured homeowner gains $1000 relative to the insured homeowner But with a burglary, the uninsured homeowner loses $9000 relative to the insured homeowner Remember: for a risk-averse individual, losses count more (in terms of changes in utility) than gains A risk-averse homeowner, therefore, will enjoy higher utility by purchasing insurance THE LAW OF LARGE NUMBERS Consumers usually buy insurance from companies that specialize in selling it Insurance companies are firms that offer insurance because they know that when they sell a large number of policies, they face relatively little risk The ability to avoid risk by operating on a large scale is based on the law of large numbers, which tells us that although single events may be random and largely unpredictable, the average outcome of many similar events can be predicted For example, I may not be able to predict whether a coin toss will come out heads or tails, but I know that when many coins are flipped, approximately half will turn up heads and half tails Likewise, if I am selling automobile insurance, I cannot predict whether a particular driver will have an accident, but I can be reasonably sure, judging from past experience, what fraction of a large group of drivers will have accidents ACTUARIAL FAIRNESS By operating on a large scale, insurance companies can be sure that over a sufficiently large number of events, total premiums paid in will be equal to the total amount of money paid out Let’s return to our burglary example A man knows that there is a 10-percent probability that his house will be burgled; if it is, he will suffer a $10,000 loss Prior to facing this risk, he calculates the expected loss to be $1000 (.10 ϫ $10,000) The risk involved is considerable, however, because there is a 10-percent probability of TABLE 5.6 THE DECISION TO INSURE ($) INSURANCE BURGLARY (PR ؍.1) NO BURGLARY (PR ؍.9) EXPECTED WEALTH STANDARD DEVIATION No 40,000 50,000 49,000 3000 Yes 49,000 49,000 49,000