Prices, Sanctions, and Discontinuities

A city reduces parking congestion in the downtown by maintaining a parking lot where cars pay by the hour. A rational driver will stay parked so long as an extra hour is worth more to him than the price. As soon as staying another hour is worth less than the price, the driver will leave the parking lot. This example suggests that a price is usually a payment of money that is required in order to do what is permitted. A price allows the individual to choose what to do as long as he pays it. The rational individual equates his marginal benefit to the price. A fall in price increases demand (downward sloping demand curve).

To reduce parking congestion further, the city imposes a two-hour parking limit on downtown streets with a high fine for overstaying. This example suggests that a monetary sanction is a cost imposed for doing what is forbidden. If a sanction is attached to forbidden activities, there is a jump or discontinuity in the actor's costs at the partition between permitted and forbidden activities. Parking is free for two hours, so drivers will often stay for a second hour even when the extra hour has little value to them. However, rational drivers will be careful not to stay beyond two hours when costs jump up. For most people, the cost of overstaying easily exceeds the benefit. They are not on the margin where the cost of overstaying equals the benefit, so they do not respond to moderate changes in the sanction or the frequency of its application.

The price at the parking lot nudges customers to move along. The sanction for overstaying on the street compels drivers to do so. Prices nudge and sanctions compel. Is liability for causing an accident like a price or like a sanction? What about liability for breaking a contract? The law of torts and contracts sometimes nudges and sometimes compels, and economics explains why.

A. Distinguishing Prices from Sanctions

Unlike a price for permitted conduct, a sanction jumps up at the legal standard separating permitted and forbidden behavior. In a graph of liability costs, the jump is a discontinuity. The presence or absence of discontinuities is decisive for distinguishing between prices and sanctions in liability law, as some examples show.

Example 1: Leak. A firm hires a caterer to make a gala breakfast for 1,500 guests. The firm supplies the caterer with 500 liters of milk and 50 kilos of chocolate. The caterer supplies a vat and labor for making hot chocolate. The vat has a valve manufactured from the best available technology, but it cracks very occasionally when heated. After turning on the heat in the vat, the caterer goes to work in another room, and the valve immediately cracks. The caterer returns ten minutes later and half of the hot chocolate has gone down the drain.

With the valve cracked, the loss from the vat increased continuously with each second that the caterer was in another room rather than tending the vat. Figure 1.1 illustrates this graphically, with the vertical axis representing the loss and the horizontal axis representing the length of time that the vat was left unattended. The curve begins at the graph's origin (0,0) and increases continuously until the vat is drained.

In example 1, a rule of strict liability might hold the caterer liable for the entire loss from leaving the vat unattended. Alternatively, a negligence rule might hold the caterer liable for a fraction of the loss. If leaving the vat unattended for five minutes was reasonable, then the caterer's negligence did not cause the loss in the first five minutes. If leaving the vat unattended for more than five minutes was unreasonable, then the caterer's negligence caused the loss in the last five minutes. Under this application of the negligence rule, the caterer would be liable for half of the total loss.

In this scenario, the rational caterer's liability is twice as great under strict liability as under a negligence rule. Will a rule of strict liability cause the rational caterer to leave the vat unattended for longer than a rule of negligence? If you think the answer is "Yes," then you just made the most fundamental mistake in economics—confusing total and marginal values.

Assume that a rule of strict liability would cause a rational caterer to leave the vat unattended for seven minutes. So would a rule of negligence. Both rules induce the same behavior because the incremental liability from leaving the vat unattended for additional minutes after the seventh minute is the same under both rules. The rational caterer decides how long to leave the vat unattended by comparing the incremental loss and the opportunity cost of spending time watching the vat. A modest change in either one would cause a rational caterer to change how long he attends to the vat. The caterer is on the margin where even small changes in incremental costs cause changes in behavior.

Unlike example 1, the loss function in example 2 is naturally discontinuous.

Example 2: Boil. A firm hires a caterer to make a gala breakfast for 1,500 guests. The firm supplies the caterer with 500 liters of milk and 50 kilos of chocolate. The caterer supplies a vat and labor for making hot chocolate. The caterer pours the milk and chocolate into the vat and turns the dial to heat the vat. The caterer wants to heat the milk quickly to save time. The caterer negligently turns the dial too far, the vat heats to 110 degrees centigrade, and the milk boils and spoils.

Any temperature of 100 degrees or more would spoil the milk, and any temperature below 100 degrees would not spoil it. Harm is nil as the temperature rises to the boiling point and then the whole vat is abruptly ruined. In figure 1.2, with the vertical axis representing the loss and the horizontal axis representing the vat's temperature, the loss curve begins at the graph's origin (0,0) and remains 0 until the temperature reaches 100 degrees, but then the loss instantaneously jumps up to vat's entire value.

The caterer's overheating the milk caused its ruin, so the caterer is liable under a rule of strict liability. Overheating the milk was also negligent, and negligence caused the ruin, so the caterer is also liable under a negligence rule. Since liability is the same under both rules, so are the injurer's incentives. With either liability rule, a rational caterer would presumably aim to heat the milk a little below 100 degrees in order to leave a margin of error. A modest change in the amount of liability—say from 100 percent to 75 percent of the value of the vat—would have little or no effect on the rational caterer's precaution. The caterer is not on the margin where a small change in costs causes significant changes in behavior.

B. Tort Law

In example 2 discontinuity in nature causes discontinuity in liability. In many torts cases like example 3, however, a rule of negligence causes discontinuity in liability when nature is continuous.

Example 3: Probabilistic Cause. A firm hires a caterer to make a gala breakfast for 1,500 guests. The firm supplies the caterer with 500 liters of milk and 50 kilos of chocolate. The caterer supplies a vat and labor for making hot chocolate. The vat has a poorly maintained valve with a probability of cracking from heat equal to 3 in 1,000. With proper maintenance, the probability of cracking would be 1 in 1,000. The valve cracks as soon as the heat is turned on. The caterer immediately sees the crack but cannot save the hot chocolate.

The caterer is liable under a rule of strict liability because he owns the vat whose faulty valve caused the loss of hot chocolate. In contrast, liability under a negligence rule is unclear. Did the caterer's negligence cause this accident? The caterer's negligent maintenance of the valve increased the probability of its failure from 1 in 1,000 to 3 in 1,000. Presumably no one can tell whether this valve failure is one of 2 in 1,000 failures that reasonable maintenance would prevent, or the 1 in 1,000 incidence of failure that reasonable maintenance would not prevent. The plaintiff can prove that the defendant's negligence increased the probability of harm, but can he prove that the defendant's negligence caused this accident?

Perhaps the court decides that increasing harm's probability is not a proof of harm's cause. In that case, the caterer is not liable, which undermines the caterer's incentives for care. Or perhaps the court decides that increasing harm's probability is proof of harm's cause by a preponderance of the evidence. If the caterer had taken reasonable care in maintaining the valve, then two out of three accidents that occur could have been avoided. Perhaps this fact amounts to proof by the preponderance of the evidence. If so, the caterer will be liable for the valve failures that maintenance would have avoided, and he will also be liable for those that maintenance would not have avoided.

Liability for unavoidable harm causes a jump or discontinuity in the caterer's cost from not maintaining the valve. To see why, assume that the probability of valve failure increases continuously with the age of the valve, and reasonable maintenance of the vat requires replacing the valve annually. Replacing the valve more often would cost more than the accidents avoided by doing so, and replacing the valve less often would cost less than the accidents avoided by doing so. Under a rule of strict liability, the caterer's expected liability increases continuously as the delay between valve changes lengthens. A rational caterer would balance the cost of replacing the valve more often against the resulting savings in liability. By assumption, their marginal values equate at annual replacement of the valve. A small increase in liability results in a small increase in precaution, just like a small decrease in the market price of a good results in a small increase in its consumption. Liability resembles a price for catering.

Under a negligence rule, however, the caterer's liability jumps up at the legal standard of annual replacement. If the replacement rate falls a little below the legal standard, he immediately bears liability for accidents, including those that annual replacement would prevent and those that annual replacement would not prevent. If the replacement rate increases to the legal standard, he immediately avoids liability for accidents, including preventable and unpreventable accidents. To allow a margin of error, the caterer will probably replace the valve more than once per year. The caterer is not on the margin where a small change in liability causes significant changes in behavior. Liability resembles a sanction for failing to replace the valve annually.

In example 3, one interpretation of the causation requirement for negligence results in no liability for exposing others to risk. The injurer externalizes risk because it is unpriced. The other interpretation results in a sanction for failing to satisfy the legal standard. The legal rule of negligence induces a discontinuity, even though nature is continuous. Instead of an all-or-nothing approach, the court could divide costs according to probability. An example discussed for decades by economists illustrates such situations.

Example 4: Sparks. A railroad locomotive emits sparks that sometimes set fire to fields beside the tracks. Reasonable care requires installing a spark arrester, which cuts the emission of sparks and the resulting fires by 60 percent. It is impossible to distinguish between fires that a spark arrester would have prevented and fires caused by sparks that would have passed through a spark arrester. A farmer sues the railroad for destruction of crops worth $10,000 caused by a fire from a locomotive without a spark arrester. The farmer can prove that the probability that the railroad's negligence caused the fire is .6. The farmer cannot prove that the railroad's negligence caused this fire. The court decides that the railroad's negligence was more likely than not the cause of the fire and awards the farmer $10,000 in damages.

The injurer's negligence in example 4 caused harm of $10,000 with a probability of 60 percent. That is all that anyone knows or can know. However, the court holds the injurer liable for $10,000. In the long run when many such cases occur, injurers will be held liable for more than the actual harm that they cause. Therefore, once the railroad fails to comply with the legal standard, it is liable not only for harms caused by its negligence, but also for harms that would have occurred anyway. The railroad faces 67 percent more liability than the expected harm caused by its negligence. Liability for not installing a spark arrester is a sanction to deter wrongdoing, not a price to internalize the harm caused by negligence.

Alternatively, a small change in the facts will make the injurer liable for fewer fires than its negligence actually causes. If the probability that the railroad's negligence caused the fire is 30 percent, the court might decide that the farmer has not proved causation by the preponderance of the evidence, so the railroad is not liable. The railroad will be held liable for 0 percent of the fires, although it causes 30 percent of them in the long run. The railroad pays no price and bears no sanction in tort law for not installing a spark arrester.

Instead of an all-or-nothing approach, the court might reduce the damages according to the strength of the probabilistic evidence. Thus, the court might award damages of $6,000 if the spark arrester traps 60 percent of the sparks, and the court might award damages of $3,000 if the spark arrester traps 30 percent of the sparks.

Legal doctrine seldom allows probabilistic damages, although courts sometimes act as if they were allowed. With probabilistic recoveries, the railroad's expected liability equals the expected harm caused by its negligence and the discontinuity disappears. Liability for not installing a spark arrester is a price, not a sanction.

This problem pervades liability for automobile accidents.

Example 5: Driving. A driver fails to stop her car and hits a pedestrian crossing the street. The driver was traveling at 46 mph at the time of the accident. Given the risks of driving, the reasonable speed to drive was 45 mph. The court applying a negligence rule imposes liability on the driver for the harm suffered by the pedestrian. Did the court impose a sanction on the driver or charge her a price?

A court imposing liability in example 5 may reason that since speeding is the cause of the accident, the negligent driver should bear liability. If the driver drives 45 mph or slower, she will never bear liability for harms that slower driving will prevent; if, however, she drives 46 mph or faster she will bear liability for all harms that slower driving will prevent. There is a discontinuity in liability, so liability resembles a sanction.

Alternatively, the court may reason that the negligent driver is liable only for the incremental harm, namely the harm caused by her negligent behavior, and not for the harm that would have been caused even if she had driven reasonably. Under prevailing negligence law and causation principles, liability should be imposed only for harms caused by the injurer's negligence. Therefore, in example 5, reasoning compatible with prices better reflects the law than reasoning compatible with sanctions.

With a small change, example 5 (driving) can be converted to a probabilistic cause case, with discontinuity in liability. Assume that if the driver in example 5 had been traveling at 45 mph instead of 46 mph, the driver's probability of hitting the pedestrian, as determined by traffic authorities, would have fallen by 50 percent. Following prevailing practice, the court needs to know whether or not the accident would have occurred but for the driver's negligence, but the information is inconclusive. Either the court finds the driver not liable, an outcome that extracts no price from the driver for imposing negligent risk on others, or else the court finds the driver fully liable, in which case the court imposes a sanction.

The discontinuity problem pervades tort liability, as we show by some more examples. Example 6 illustrates discontinuity in a common medical malpractice case.

Example 6: Gout and Perfect Causal Attribution. A patient's big toe has gout (a painful, inflammatory arthritis). The doctor prescribes a powerful drug and tells him to stop drinking alcoholic beverages. The doctor prescribes the wrong drug for this type of gout and the patient continues drinking alcohol. Before these problems are detected and resolved, the patient's toe suffers permanent harm that makes walking painful. The patient sues the doctor and the court finds that compensation for permanent harm to the joint equals $10,000. It divides the permanent harm into four components: The mistaken prescription by the doctor caused $6,000; the interaction between drinking and the mistaken prescription caused $1,000; the patient's drinking caused $2,000; and $1,000 in harm would have occurred if the doctor had prescribed the right drug and the patient had stopped drinking. The court concludes that the doctor is liable to the patient for $7,000.