• Syllabus
• Expected Value and Expected Utility
• Problems of Insurance
• Moral Hazard
• Risk Pooling
• Adverse Selection

Problems of Insurance

Consider the uncertain prospect we used to discuss insurance above. Suppose that the insured person's wealth before payment of the insurance premium is w₁, his/her wealth in the event of an uninsured loss is w₀, the premium is I, and the expected value of the prospect is w^*, so that I>w₁- w^*. As before, we will denote the certain wealth level at which the consumer is indifferent to the uncertain prospect as w´, and we assume that the insurance is purchased, so that I≤w₁- w´.

Now assume that at cost C, our insured person could reduce the probability of a loss from p to q < p. The expected loss without the action is p(w₁-w₀). If the action is taken, the expected loss is q(w₁-w₀). The action makes sense if (p-q)(w₁-w₀)≥C. If the action could be observed, competitive insurance providers would offer discounts to customers who took the action. But with an unobserveable action, the insured party gets no benefit from taking care.

How can this problem of obtaining desirable actions be dealt with? One way is to deny full coverage for losses, making the insured party bear some of the costs. Suppose that we use L to label the loss, (w₁-w₀). At a minimum we do not want the insured to be better off with a loss than without: if we cannot measure w₁ accurately, we will want to reduce the amount we pay in the event of the loss, implying partial coverage. Here the problem is one of preventing hidden actions to increase the probability of loss. Excessive fire insurance coverage can result in arson. Excessive life insurance coverage could induce suicide.

Is such an extreme outcome likely? Maybe not, but the case of “Nub City,” in Florida, is instructive:

In the late 1960s and early 1970s, the idea of trading body parts for cash on accident policies had just one name among insurance investigators around the country: Vernon, Florida. A backwoods town in the Florida Panhandle, Vernon had a general store, a combination post office/barber shop, one police car, and a main street that stretched only a block and a half. Selling reptiles from a roadside stand was a good business in Vernon, hunting turkeys, an obsession for some. For a time, though, losing limbs, fingers, arms, or legs in freak accidents became the town fashion. More than fifty such cases came out of Vernon in just a few years, a number that becomes all the more unusual when it is understood that the town's total population was less than 500. Investigators refused to name the town at the time, telling newspaper reporters only that they referred to it as Nub City. Self-amputees from the city, investigators said, were casually referred to as members of the Nub Club. “Somehow they always shoot off the parts they seem to need least,” one investigator remarked of the disproportionate number of left hands claimed lost as compared with right ones. Another investigator, John J. Healy of New York, worked cases in Vernon for a number of major insurers and later wrote about it at some length. “The second biggest occupation [in Vernon] seems to be the observation of hound dogs mating in the town square,” he noted back in 1975. “The biggest occupation was the deliberate maiming or severance of limbs to collect insurance money.” … To sit in your car on a sweltering summer evening on the main street of Nub City, watching anywhere from eight to a dozen cripples walking along the street, gives the place a ghoulish, eerie atmosphere.”

Most of the limbs lost in Nub City were shot off at close range with hunting rifles. The contrived accidents were all similar: triggers pulled unexpectedly as victims climbed fences; guns misfiring in the middle of being cleaned or after being dropped. And all of the mishaps involved men. (“Women never do dismemberments,” Healy later observed.) In the late 1950s, when the first claims came out of Nub City, a typical dismemberment was worth $1,500; by the early 1970s, the average claim was bringing tens of thousands of dollars. Over the years, Nub City got under the skin of investigator Healy, a nationally known expert in murder for insurance cases. “As inured as I am to all kinds of maimings and weird dismemberments, Nub City holds a morbid fascination for me,” he wrote. “I keep asking myself: How did it all start? What drove these people to sacrifice their limbs for money?” Healy tried to imagine the conversations that might have taken place between those who had already profited from losing a limb and those who were considering it. One man talks about the ten thousand dollars he got for his left hand and how he could use the money to buy a house or a car or a color television. “The other man looks at his own hand-the hand that probably has not earned him ten thousand dollars in five years,” Healy writes.

It is dirty, worn, the index finger crooked from a fracture that never was set straight. The fingers are tobacco-stained. He looks at it, turning it slowly, reflectively. Ten thousand dollars or more, he thinks. In one fell swoop. That's more money than he's ever seen in his life, probably more than he will ever see. He’s fifty years old and has been doing odd jobs for twenty years. He’s tired. His 1947 Plymouth may or may not start in the morning to get him to whatever job he may have.

In the end, the man completes a calculation that has been made by Americans at least since the depression of the 1890s, and probably earlier. “Does it hurt much?” he asks his friend, now talking more about strategy than principle. And with that, the deal is done, but for the bloody doing itself. A knowing look washes over the man’s face as he weighs the relative merits of knives, axes, or shotguns, and debates with friends the relative anesthetic properties of different brands of whiskey. “I hope I never have to go into that town again and see the mangled stump of an arm or a leg and listen to the old familiar story,” Healy concluded of Nub City. “But deep down inside me, I know that I will.”

From Ken Dornstein, Accidentally, on Purpose: The Making of a Personal Injury Underworld in America, (New York: St. Martin’s Press, 1996), pages 265–6, footnotes omitted.

The problem here is that the compensation offered for the loss of a body part exceeded the value of the part to the consumer in question. Coverage was more than full. The solution is to refuse to offer full, or in this case, excess coverage. Note that denying full coverage implies that some fraction of losses must be borne by the insured party. This can be done in several ways. A co-payment is a requirement that the insured pay for a given percentage of the loss in the event of a claim. A deductable is a fixed exclusion from the amount of the claim. In our case of known w₁, the two are equivalent. More generally, deductibles make most sense either when there are fixed costs of processing claims, whether large or small. These costs can include the cost of monitoring the actions of the insured. Co-payments are implied when the likelihood of detection of either an action increasing the possibility of claims or the absence of some desirable effort to reduce claims cannot be detected, even for large claims.

It is apparent that the actual processing costs for small claims, as opposed to costs associated with moral hazard, are not particularly significant in shaping coverage. Automotive collision insurance typically carries high deductibles, while comprehensive insurance does not. The likelihood of a collision claim is obviously sensitive to driver behavior. The likelihood of a car being hit by a falling tree in a windstorm (a claim falling under comprehensive coverage) is not.

Sometimes the threat of moral hazard can be so severe as to prevent any coverage from being available for the losses in question. Private disability insurance is often difficult to obtain, helping to explain why it is included under the Social Security system. Even Social Security did not cover disability initially, precisely because of the fear that persons certified as disabled would not choose to return to work even when recovered from their disabilities, preferring instead to collect benefits. The problem here is partially with unobserved actions that affect the probability of a disability, but more with the hidden actions that affect the size of the loss.

Risk Pooling

No. An insurance company is analogous to a waste disposal company. The insurer does not just take over the risk, but instead disposes of it. How? By pooling risks. If the chances that any given house in a community will burn down are one in one hundred in a give year, an insurer who provides insurance for 100,000 houses in the community can expect the number of claims files to be one thousand, give or take a few. Pooling risk does not change the risk for any particular home, but it does reduce the overall risk, at least partially disposing of it.

What can go wrong here? Suppose that the community in question is located on the prairie. Fires in this community are quite rare, but once in a while, say once in a hundred years, a fire rolls across the prairie, destroying the town entirely. Now there is no way for the citizens of the town, or any insurer limited to selling claims in the town, to pool the risk. It cannot be disposed of. To pool the risk, an insurer will have to cover lots of properties in other towns, towns that will be unaffected when disaster strikes our town.

When the disasters become more widespread, the ability to pool risk dwindles. In such cases, either the insurers will need to be very large, or, in extreme cases, insurance will not be available unless subsidized by the government. This helps to explain why earthquake insurance is hard to get and why the federal government subsidized flood insurance.

Adverse Selection

Suppose that only sellers of used automobiles know their quality. Potential buyers will correctly assume that the owners of inferior-quality automobiles will be especially eager to sell cars, so these potential buyers will offer only as much as the poorest quality car is worth—and that will be the only quality that sellers can offer for sale. (This vicious circle can be broken if sellers can acquire deserved reputations for reliability.)

George J.Stigler,
Memoirs of an Unregulated Economist
pp. 80–1

Suppose that we ignore the costs of providing insurance, so that it is possible to consider actuarially fair insurance. Suppose initially that such insurance is offered to a group of individuals, so that for the group as a whole, the premiums collected from the group are returned in the form of claims payments. Is all well? Not if the members of the group have private information about their own prospects for suffering the losses insured against. Consumers who know themselves to be at relatively high risk will earn rents from the insurance and will purchase even if risk neutral, but consumers who have low risk of the disease will opt out of the insurance plan, figuring that the premiums are too expensive for them relative to their comparatively low expected benefits from claims payment. As these people drop out of the insurance pool, the actuarially fair premium for those remaining in the pool rises, causing more members of the pool to drop out. The process repeats, possibly until only the most high risk individuals remain in the pool. The market can even collapse entirely, causing no insurance whatsoever to be offered.

Something like this may be happening in the United States, where on the order of 35 million individuals do not have health insurance. For them it is “too expensive”, meaning not that they cannot afford it, but that they do not choose to do so because of the high premiums demanded. These people are often between jobs, and therefore not a part of a group receiving coverage.

One defense against adverse selection is screening by the insurer. Screening is simply an attempt reduce the degree of asymmetry in information. Screening allows insurers to separate clients into different risk pools with correspondingly different premiums. Since screening often entails using observable proxies for underlying risk, proxies such as race and income, government can attempt to prevent insurers from using screens. This may reduce rates for some groups, but does run the risk of diminishing the number of policies offered.

Adverse selection can also be dealt with by preventing opting out by the members of a risk pool facing the lowest risk of claims. This is accomplished by offering group insurance coverage, where membership in the group is not optional. One of the arguments for national health insurance is that it makes the group as large as possible. The problem with such a solution, however, is that health insurance is subject to moral hazard as well. Medical facilities may be over-utilized with mandatory coverage, resulting in an explosion of health care costs.

Adverse selection differs from moral hazard in that in the former case the risks are not susceptible to modification by the actions of the insured. But there are costly actions that the insured can undertake to gain more information about risks. These costs, if incurred, serve to narrow the markets in question, and are not necessarily desirable, and therefore market makers may try to discourage such information gathering efforts. One interesting example of such a restriction comes from the Antwerp market for uncut diamonds. Diamond buyers are offered bags of uncut diamonds filled with unsorted stones. They are not permitted to sort through the bags in advance, choosing only the diamonds they want to buy.

Application to Law

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• Moral Hazard
• Risk Pooling
• Adverse Selection