The obvious principle for determining fair compensation in injury suits is that the injuror must 'make good' the damage to the injuree; that is to say, that he must make him as well off as if the injury had not occurred.[1] This principle seems to fit both our intuitions about justice and the economic view of the legal process, according to which such suits provide potential injurers with appropriate incentives not to impose risks on others, by making the injurers bear the cost of their acts.[2]
Unfortunately, there are some injuries for which the application of that principle seems difficult or impossible. There may be no payment large enough to make a blinded man as well off as if the injury had not occurred. If the injury is fatal the problem is inherently insoluble, save for those altruistic victims who would have been willing to trade their lives for a sufficiently large post mortem donation to their favorite charity. Even where it is possible, by enormous payments, to just barely recompense the victim of personal injury, it is not immediately obvious that the creation of blind billionaires living in profligate luxury (and so compensating, with pleasures that large amounts of money will buy, for the lost pleasures that it cannot buy) is a good idea.
The purpose of this article is to argue that 'full compensation' in the sense just described would in fact be overcompensation, in terms both of justice and of economic efficiency, and that there is another criterion which leads to more appropriate levels of compensation and is better able to deal with the 'impossible' problem of fairly compensating the victim of death or serious injury. Part II is a verbal sketch of the arguments and the criterion they lead to; in Part III I introduce and employ the formal concepts of Von Neumann-Morgenstern utility and economic rationality to redo the argument in a more precise way. Part IV summarizes the results and discusses the relation of my arguments and conclusions to conventional legal principles.
I have tried to put my arguments in a form accessible to both economists and non- economists interested in the law; I apologize in advance to those economists who find my explanations of elementary economic concepts superfluous and to those non-economists who find them insufficient.
Compensation for death or bodily injury involves two quite different problems. The first is the problem of how much' damage there is to make up for. The second is the problem of in what coin damages can be paid. One might imagine that someone would be willing to give his life in exchange for a sufficiently high price--five years of ecstasy, perhaps. Faust, after all, traded not merely life but eternal bliss for a finite payment. More mundanely, we observe that people are willing to enter dangerous professions (driving dynamite trucks, for example) in exchange for somewhat higher pay, thus in effect trading life--a small increase in the probability of getting killed--for income. Both examples suggest that the reason it is impossible to 'fully compensate' someone for the loss of his life is not that the value of his life to him is infinite--it is not--but that the value of compensation to a corpse is in most cases small.[3] The same argument applies to less-than-lethal injuries. If an injury somehow made the victim blind half the time--on alternate weeks--it might be possible to compensate him by a payment of a million dollars. It does not follow that two million, or even three million, would be fair compensation for making someone blind all of the time. In part this is because being blind all of the time is more than twice as bad as being blind half the time. But in part it is because in the first case, some of the payment would be spent in forms of consumption (travel, opera) which require sight to be of much value; with the possibility of such forms of consumption eliminated, larger sums must be spent on other and less attractive forms of consumption in order to provide the same value to the victim. Another way of putting the same point is to say that bodily injury makes the victim worse off in two ways. It lowers his effective income by reducing his earning power and imposing costs (a seeing eye dog, hospital bills, etc.). In addition, it lowers the value to him of any given income by eliminating ways in which he can spend it. Death is the extreme case; not only does it lower the victim's income to zero, it simultaneously reduces to zero the benefit he can get by spending any form of income--including damage payments.
One thing this argument suggests is that 'full compensation'--a level of payment for damages which restores the victim to the level of welfare he had before the injury--is in a sense inefficient. To see this, imagine that the victim first receives compensation sufficient to make up for any loss of income from his injury, so that he can afford to consume exactly the same things as if he had not been injured. Since he is no longer able to consume some of those things (color television if he has been blinded), he spends what previously went for goods he can no longer use on the remaining sorts of consumption. Since he is spending more than before his accident on these (say, gourmet dinners) one would expect the value to him of additional expenditures on these goods to be lower than before. Before his injury, the last dollar spent on 'color television services' provided the same benefit as the last dollar spent on gourmet dinners; had that not been the case, he could (and would) have improved his welfare by spending more on the one form of consumption and less on the other. After his injury, he must transfer the money he previously would have spent buying a television to buying more (and increasingly less pleasurable) gourmet dinners; it is for this reason--because he is less able to make use of income--that he is worse off even if his income is not reduced. But this also implies that with the same income, the benefit he receives from the 'marginal' dollar is less. If his compensation is sufficient to both replace his lost income and provide enough additional income to compensate for the loss of his vision (supposing that to be possible), the conclusion holds a fortiori. Since in order to make up for all the pleasures he can no longer enjoy he must consume pleasures he can enjoy to a point of near satiation, the value to him of each additional dollar will be very low indeed. Hence 'full compensation' involves transferring income from uninjured persons, who can receive large benefits from each dollar, to injured (and already partly compensated) persons who receive very small benefits from each dollar.[4] Intuitively it seems that although this may be fair it is also wasteful; it is doubtful that actual courts and juries (at least in cases where the injurer is a visible human being and not a corporation or insurance company) make any attempt to push compensation that far.
The reader may suspect that I am now going to argue that the proper level of compensation is that which restores the injured party to his previous level of income without making up for the losses due to his lessened ability to make use of that income. If so he is wrong. In terms of the `efficiency' argument I have just sketched even that level seems too high, since even at that level the value of a dollar to the injured party is, I have argued, less than to a similar uninjured person with the same income. On the other hand, that level of compensation is inadequate as a deterrent to the injurer, since it understates the cost imposed by his actions.
In order to resolve this puzzle and find a level of compensation which is both efficient and adequate, we must move from the ex post situation, in which the injury has already occurred, to the ex ante situation, in
which the potential victims are subject to some probability of injury or death, but the damage has not yet occurred. Two things are worth noting from this perspective. The first is that if the risk is small, there is probably some sum of money such that the potential victims would be indifferent between receiving that sum and being subject to the risk, and having neither the money nor the risk. Hence although the damage may be enormous or even infinite (in the case of death) when viewed in terms of ex post compensation, it is finite and may even be small in terms of ex ante compensation. The second and closely connected observation is that if people are going to be compensated for a deadly risk, they would much prefer to receive their money when the risk does not eventuate and they are therefore alive to spend it.
One possible conclusion is that those who impose risks should be required to compensate ex ante those on whom the risks are imposed. The amount of compensation would be such as to just compensate for the risk (estimated, perhaps, from the risk premium on hazardous jobs), and the potential victims could then choose how to allocate the money among the different possible outcomes. If they believed that the money was more valuable to them if the accident did not occur they could consume the ex ante damage payments; if they believed it was more valuable after the accident, they could use them to buy insurance. If they believed that some but not all of it would be needed after an accident, they could divide their expenditures between consumption and insurance accordingly.
Such a solution is unworkable. It would require the courts to estimate in advance the risks imposed by an enormous variety of activities, including some for which the very nature of the risk could not be known until too late and many for which the estimation of probabilities and potential damages would be difficult for the parties concerned and virtually impossible for the courts. What we need instead is a system under which damages are paid when an accident occurs (at which point the damage done and the fact of the accident are known) but collected (at least in part) when it does not occur (that being when the money is of most use to the recipient). Such a system is by no means impossible.
Imagine that we have a system in which potential victims of accidents know that they (or their heirs) will be compensated in case of injury or death; further suppose for the moment that the formula is 'full compensation'. The potential victim knows that he is no worse off for being subject to risk; if injured he will be fully compensated. It might, however, occur to him that full compensation would involve payments of large amounts of money, much of which (in his hypothetical injured state) he could make little use of. If he were sufficiently ingenious and the society sufficiently well organized, he might then decide to sell insurance on himself. In exchange for some payment if he is not injured, he would agree to pay someone else part of the damage payment he receives if he is injured. Suppose, for example, that the chance of injury were one in a hundred and the 'full compensation' in case of injury were ten million dollars. He could agree, in exchange for ten thousand dollars now, to give the buyer a million dollars if he himself were injured. He would be left with nine million dollars in case of injury; while that would not be enough to fully 'make up for' the injury (if it happened he would wish it had not) the benefit he would expect to get if he 'won' his bet (i.e. was not injured) would more than make up for the loss if he 'lost' it, since a certainty of ten thousand dollars when he was uninjured was worth more to him than one chance in a hundred of having a million dollars when he was injured (and already compensated by a nine million dollar payment) .
This argument implies that if potential victims are able to sell fair insurance on themselves then 'full compensation' is actually overcompensation. Prior to such a sale, the potential victim is no worse off through being exposed to risk, since, by the definition of full compensation, any damage will be made good. After the sale, the potential victim is better off than before the sale (he has transferred income from an outcome where it had low marginal utility to one where it has high marginal utility) hence also better off than if he were not exposed to risk. But if the potential injurer is overcompensating, it follows that he will be overdeterred from imposing risk; he may fail to undertake some risky activities even though the net benefit more than makes up for the risk. It further follows from this argument that the correct level of compensation is that level such that the potential victim, after selling as much insurance on himself as he wishes, will be neither better nor worse off than if no risk had been imposed. This is the same result that would follow from the system of court imposed ex ante damages discussed above, provided the courts had the necessary knowledge. The difference is that the injurer pays off when and only when the injury actually occurs, thus allowing the risk to be measured directly by ex post outcomes in the real world instead of being estimated by the court. The potential victim may then transfer income from the outcome where he is an actual victim to the outcome where he is not by selling insurance, instead of transferring it the other way by buying insurance.
While the information problems under this system are reduced, they are not eliminated. In order for the potential injurer to decide what risky actions to undertake, he must first estimate the probability and seriousness of injuries in order to calculate the damages he may have to pay. He is, unlike the court, in the best possible position to make such estimates; he is the one taking the actions and presumably the one who knows most about their consequences. In addition his welfare depends on making correct estimates; that of the court may not. In addition to the estimate made by the potential injurer, a second estimate must be made by the insurance company which buys insurance on potential victims. Here again, the company has a private interest in doing a good job; if it overestimates the risk it will find itself paying more than the insurance is worth and losing money; if it underestimates it will be outbid by other companies with better estimates. Unlike the potential injurer, the insurance company may have no expertise in the particular area, although it is, unlike a court, expert in the general subject of risk. And even if the estimates of the insurance companies are wrong, the result will be only a redistribution between them and their customers; the actual payments made by the injuror, and hence his Incentive to avoid risky activities, will be determined by what happens, not by the insurance companies' estimates.
There remains the question of how much compensation is implied by this rule. A precise answer depends, as I will show in the next section, on the size of the risk, something which I have already argued that the court which must decide on the compensation cannot know. But as long as the risk is not very large there is an approximate answer (as I shall also show) which is independent of the size of the risk. To calculate it one simply takes the sum for which the potential victim would be willing to accept a very small probability of death (or injury) and divides by that probability. Hence if the victim were willing to accept a one in a thousand chance of death in exchange for a thousand dollars, the damages for actually killing him should be a million dollars.
Estimating the sum for which the average person would be willing to accept some small probability of death is difficult but not impossible. One approach is to estimate the wage premium on dangerous professions;[5] The problem with this is that those who enter such professions are presumably people with abnormally low objections to lethal risks; the figure calculated in that way will accordingly under-estimate the figure for the average victim. Other and more indirect ways might involve the observation of choices concerning amount and quality of medical care (where the purchaser has some grounds for estimating the effect on life expectancy), speed of driving, or other choice variables which affect probabilities of death or injury.
A utility function is a formal way in which economists describe how the attractiveness of alternative outcomes affects decisions. It may be thought of as a numerical measure of how much an individual values various alternatives, and expressed as a numerical function of variables such as health, goods consumed, etc. The function is so constructed that if the individual prefers one outcome (consuming 50 pounds of steak a year, working 40 hours a week, and living to 90) to another (40 pounds, 35 hours, 95), then the utility of the first set of values (for the variables 'consumption of steak', 'hours worked', 'lifetime') is higher than that of the second.
It is often convenient to think of a utility function as separated into several different parts, each depending on a different variable. Thus one can think of my total utility as being made up of the utility I get from reading books plus the utility I get from playing with my children, plus. . . . Such a description is a simplification of real utility functions, since preferences with regard to one form of consumption are likely to depend on how much I am consuming of something else; nonetheless the simplification is often a useful one. In considering the particular issue of injury, it may also be useful to separate effects on income from effects on consumption by imagining that the income you earn is itself a function of several inputs, among them the possession of certain innate abilities such as the ability to see. Your utility is then a function of your abilities and of other inputs, many of which can be purchased with income.[6] The individual seeks to maximize his utility subject to a budget constraint: expenditure on consumption goods equals income (I neglect, for purposes of simplicity, the fact that saving and borrowing may be used to reallocate income across time). The effect of injury may then be usefully divided into its effect on the victim's ability to produce income and its effect on his ability to use income to produce utility (for himself). The latter may be further simplified by supposing that the injury affects the victim's utility function in two ways.
1. It requires him to spend a certain amount of money to buy substitutes for things that he would have had at a lower cost or for free before (purchasing a seeing eye dog, for example). This can be considered as equivalent to a further reduction in his ability to produce income; some of the income he produces must be diverted to purchase these things, and only what is left is available for ordinary consumption goods.
2. It eliminates certain consumption possibilities, certain of the ways of converting income into utility (seeing movies or watching television--in the case of blinding) that went into the (assumed) additive utility function. By mak ing these assumptions we get a reasonably simple but not totally unrealistic model for illustrating the arguments of the previous section; it may be written as follows :
Here x is a vector of consumption goods; xl is the quantity of good 1 consumed, x2 is the quantity of good 2 consumed, and so forth. p is the corresponding vector of prices; hence xp, defined as xl times p1 plus x2 times p2 plus . . . , is the total amount spent on consumption goods (the amount spent on good 1 plus the amount spent on good 2 plus. . .). a is a vector of abilities (to see, to walk, to speak, etc.); each activity requires a corresponding ability. Y is income (net of any special expenditures required by an injury) and depends on abilities.
To analyze the effect of an injury[7] I suppose that there are only two abilities, al and a2, that the injury eliminates the first of them (al= 0) and so makes it impossible to get utility from good 1, and that the elimination of that ability also lowers income by an amount b (including both direct and indirect effects). In addition, I set p1 and p2 equal to one by defining my unit of quantity as `one dollar's worth'. More complicated situations (in which there are more abilities, in which several abilities enter each form of consumption, in which the individual's time is an input to both his income and utility functions and must be divided between the two, and in which the elimination of an ability alters the form of the income function) could of course be considered.
One more element is needed before this model can be used to repeat part of the argument of the previous section. We must suppose that as consumption of any good increases, the additional benefit received from an additional unit (`marginal utility of the good') declines. In other words, the greater xl is, the less the additional utility from an additional unit of xl, and similarly for x2. Prior to the injury, the individual adjusts the quantities he consumes so as to maximize his utility by shifting consumption between xl and x2 until the additional utility he receives from an additional dollar's worth of each good is the same; as long as the marginal utilities are different, he can increase his total utility by spending one more dollar on the good with the higher marginal utility, and one less on the good with the lower marginal utility.
Let the victim's original income be y. After the injury his income (assuming no compensation) is y' = y - b (primed variables are post-injury values). Since he can no longer produce utility via U1, he spends all of his income on buying x2. Hence x2'=y'. If this is less than x2 (in other words, if the reduction in his income plus the additional expenses imposed by the injury is more than what he used to spend on forms of consumption which are no longer available to him) the marginal utility he gets from a dollar after the injury is more than the marginal utility from a dollar before his injury. If x2' is greater than x2 the opposite is true; money is worth less to the victim after his injury than before since the injury has reduced his ability to use income more than it has reduced the income he has to spend.
Now suppose that the court compensates the victim by awarding him damages of b (since b is an income, one should think of the award as a sum large enough to yield an income of b dollars per year for the period of the injury--the rest of his life if it is permanent). His income, after paying for costs imposed by the injury, is now at its previous level; y'=y. Since he can only spend the income on one good, x2' =y' =y=xl +x2. His income is the same as before but his utility is lower. With his previous pattern of expenditure, the marginal utility of a dollar (and hence the marginal utility of one more unit of the good) was the same whichever good it was spent on. Imagine that he moves to his new pattern by reducing his consumption of xl by one unit and increasing his consumption of x2 by one unit, then doing the same thing again and again until xl is reduced to zero. The marginal utility of a unit of xl rises (as he consumes less units) and the marginal utility of a unit of x2 falls (as he consumes more units). Since for the first unit transferred the marginal utilities were the same, for each successive unit the utility gained by additional consumption of x2 is less than the utility lost by reduced consumption of xl. Hence his utility when using the same income as before to consume exclusively x2 is less than it was when that income was divided between the two goods. To put the argument differently, his previous pattern of expenditure maximized his utility, given that he had the option of getting utility from either good. Before the accident he could have consumed only good 2; he chose not to because he got more utility by consuming some of each good. After the accident he is forced to make the choice which he had rejected when he was free to choose. Hence his utility is lowered.
In the previous case (no compensation) utility was lowered, while the marginal utility of a dollar might either be raised (if he lost more income than he had spent on good 1) or lowered. In this case, since he is consuming more of good 2 than before, both his marginal utility and his total utility are lowered. To an economist this may at first seem paradoxical; we normally expect that the same change (an increase in income) which lowers marginal utility of income also raises total utility.
Is this level of compensation satisfactory? Not, it would appear, if our objective is to avoid giving money to people who have little utility for it. From that standpoint, it is too much compensation. Nor is it satisfactory if our objective is to adequately deter those who impose risks, by forcing them to make up for the damage they do. By that criterion it is too little compensation. We have here the same puzzle already presented in Part II.
To solve the puzzle we must again switch to the ex ante view, in which potential victims are seen as facing a lottery of outcomes which includes some probability of injury. We may then use an extension of the idea of utility due to Von Neumann and Morgenstern.
Von Neumann-Morgenstern utility not only allows us to consider behaviour under uncertainty, it also eliminates a weakness in the concept of the utility function as I have so far described it. I have spoken of utility in quantitative terms and described one change as producing a larger change in utility than another. But operationally, utility is defined in terms of choices; while we can observe that someone prefers A to B and B to C, we cannot tell by his choices whether his preference of A over B is more or less than his preference of B over C; in the language of utility, we cannot compare [U(A) - U(B)] to [U(B)- U(C)].
By expanding the idea of utility to cover behaviour under uncertainty, we can eliminate this ambiguity. Suppose I can choose either a certainty of outcome B or a lottery with equal chances of A and C. If I choose the lottery I am, in effect, saying that the chance of getting A instead of B (which I could have had for certain) more than makes up for an equal chance of getting C instead of B. In terms of utility, this translates into the statement that the utility gain from getting A instead of B is greater than the utility loss from getting C instead of B, or in other words that U(A) - U(B) > U(B) - U(C).[8]
More generally, Von Neumann and Morgenstern showed that if an individual's behaviour under uncertainty meets certain rather weak consistency requirements[9] it is possible to assign a utility to each outcome and describe his behaviour as choosing whatever lottery maximizes expected utility.[10] If we accept the idea that an individual is `equally well off' under either of two alternatives to which he is indifferent--if in other words we accept, as economists usually do, an individual's choices as a proper measure of the benefits he receives from different alternatives-- it seems natural to say that a lottery with a given expected utility is equivalent, ex ante, to a certain outcome with the same utility. In the case of compensation for damages, we can say that if an individual has imposed on him a probability p of some injury and receives a compensation c for this risk (whether he is injured or not) then if the expected utility of a lottery with probability 1-p of being uninjured and having c and probability p of being injured and having c is the same as the utility of being uninjured and not having c (his situation if the risk were not imposed on him) he is no worse off than before, and so (ex ante) has not been injured. Ex post, of course, either the injury does occur and he is worse off than if the risk was not imposed, or it does not occur and he is better off (since he still gets c).
We can now describe a fair ex ante compensation--that is to say, one which leaves the potential victim no worse off ex ante than if the risk had not been imposed. Using our previous utility function, letting b again be the loss of income resulting from the injury and c the compensation, we have, for a fair compensation c:
Here c1 and c2 are the amounts of the ex ante compensation which the individual chooses to allocate to goods 1 and 2 if he is not injured. Or, more generally:
Where utility is written as a function of income and ability.
This gives us a definition of fair compensation (although I have not yet discussed how it might be turned into real world numbers) but it is not yet an entirely satisfactory one. Looking at Eq. (1), there is no reason to expect that the marginal utility of income will be the same whether or not the injury occurs. Which marginal utility is larger will depend on whether x1+c1 is larger or smaller than b. But the individual can transfer income between the two outcomes by buying or selling insurance on himself; the result of his doing so will show the sense in which unequal marginal utility of income in different outcomes is inefficient within the context of Von Neumann-Morgenstern utility.
Suppose that the individual has access to a fair insurance market. By giving up a dollar with certainty, he can receive 1/p dollars in case of injury; alternatively, by giving up a dollar in case of injury, he can receive p dollars with certainty. Looking at Eq. (2), it is clear that if the additional utility he receives from a dollar if the injury does occur is greater than the additional utility he receives from a dollar if it does not occur, then his expected utility (the left hand side of Eq. (2)) is increased by buying insurance. If the inequality goes the other way, he benefits by selling insurance on himself. Hence a situation in which the two marginal utilities are unequal is inefficient; the individual, by buying or selling insurance, can benefit himself without imposing any cost on others. I assume in this argument that the number of such individuals is great enough so that the insurer can count on the law of large numbers to transform the lottery he buys into a nearly certain outcome with a value equal to the lottery's expected value. I also (and less plausibly) assume that the transaction costs of arranging such insurance can be ignored. If such an insurance market exists, a potential victim receiving the compensation specified by Eqs (1) and (2) will use it to transfer income from the outcome where it has the lower marginal utility to the outcome where it has the higher marginal utility; in doing so he raises his total utility. Hence that level of compensation, although lower than 'full compensation', is still too high; the imposition of the risk (plus the compensation) makes the potential victim better off, ex ante, than if it had not been imposed. Given such a market the correct rule for compensation becomes:
Here c is the (ex ante) compensation, c3 is the amount of it consumed if the injury does not occur (the full compensation minus the cost of any insurance bought or plus the amount of any insurance sold), and c4 is the amount consumed if the injury does occur (c3 plus the amount paid on insurance bought or minus the amount paid on insurance sold). For a given value of c, the potential victim is assumed to adjust c3 and c4 by buying or selling that amount of insurance which makes the marginal utility of income (the extra utility from having one more dollar to spend) equal for both of the utility functions on the left hand side of Eq. (3), and thus maximizes the left hand side, which is the expected utility given the probability of injury, the amount of compensation, and the amount of insurance bought or sold. The final step in the analysis consists of replacing the ex ante payment (which accompanied the imposition of risk, whether or not any injury occurred) with an ex post payment (which occurs only if there is an injury). Since the hypothetical insurance market permits the potential victim to freely transfer income between the two outcomes (injury and non-injury) at an "xchange rate' of p/(l-p) dollars if injured for each dollar if uninjured, a compensation of c/p if injured is equivalent to a certain payment of c. Hence our ex post rule is to compensate the injured party with a payment of c/p, c having the same value as in Eq. (3).
Before going on to discuss how c/p might be estimated it is worth seeing how this solution resolves the problems raised in Section I. First, it adequately compensates the victim and deters the imposer of the risk; ex ante the victim is no worse off if the risk is imposed than if it is not. If the injury actually occurs the victim may and probably will be worse off, but if so he will have transferred some of his damage payment to the outcome in which the injury did not occur (by selling insurance on himself) and will be better off if he is not injured by an amount which, given the probabilities, just compensates him for his loss if he is.
This solution solves the problem of giving large sums of money to individuals who cannot benefit from them. Assuming that the potential victim has correctly allocated the damage payment between alternative outcomes, the marginal utility of income will be the same to him whether he is or is not injured. It also deals with most, although not all, of the situations where 'full compensation' is impossible due to the inability of the injured (or dead) victim to get sufficient utility to compensate him from any payment however large. Under this system the damages may be consumed by potential victims when they do not become actual victims and are hence able to enjoy the consumption.
There is still a potential problem. Suppose that the utility of an
uninjured person is bounded; however great his income, he cannot
receive a utility from it of more than Umax=U([[infinity]]
, a). Further suppose that the utility to him of being dead is
U(Y',a')=O. Lastly let the utility of being uninjured be
U(Y,a). If (1 - P) < U(Y,a)/ Umax it is
easily seen that no possible compensation will prevent the potential
victim's expected utility from being lowered by the risk. If
empirical observations of the additional income which people require
in order to accept hazardous jobs are to be trusted, this is likely
to occur only if p is
large.[11] To estimate
c/p, the fair compensation for injury, we find some situation in
which we can observe the payment (c') in exchange for which the
individual is willing to accept a known probability (p') of the same
injury. As I will show below, c/p
c'/p', provided that
p and p' are both small.
To see why this is the case, we first note that as long as p is small, c3 in Eq. (3) must also be small; small amounts of money spent or received for insurance on a very unlikely event correspond to large amounts of insurance, and are therefore sufficient to cause large changes in c4 and correspondingly large adjustments in the marginal utility of income (given the injury). The equality of marginal utility of income across outcomes then gives us:
From this it follows that for small values ofp, c4 is approximately independent ofp. (U1 is the derivative of the utility function with regard to income.)
I now use Eq. (3) and the first order Taylor expansion of U about (Y,a) to get:
Solving for c3 yields:
Substituting this into Eq. (4) gives:
From which it follows that:
This is approximately independent of p for p small.[12] Hence the value of c'/p' which is deduced from observed behaviour (the wage premium on risky jobs, for example) may be used as an approximate value for c/p, the amount which should be awarded to someone who suffers the same injury.
My solution to the problem of fair compensation depends on the existence of an insurance market which allows individuals to shift compensation between different outcomes. No such market appears to exist; we see individuals purchasing insurance on themselves but not, as the argument suggests that they ought often to wish to do, selling it. One explanation is that this is a case of market failure, the transaction costs for such insurance being too great to allow the market to exist.[13] If so, that fact increases the damages that should be paid, substituting Eq. (2) for Eq. (3).
A second alternative is that the damages currently awarded are too small, with the result that nobody wants to sell insurance on himself. Certainly the formulae sometimes used by courts to set damages in terms of an estimate of the lost income from the injury could be expected to lead to undesirably low figures.
A third alternative is that there may be legal barriers which make it difficult or impossible for an individual to sell, in advance, the damages which he will receive if injured or killed. Since I am an economist and not a lawyer, that is a possibility about which my readers may know more than I do. Certainly if such barriers do exist, my analysis suggests that they should be eliminated.[14]
Even if my hypothetical insurance market existed, it might be argued that it would be unfair to limit damage payments to that sum which would be fair on the assumption that the victim had made use of that market to buy or sell insurance on himself in order to minimize the ex ante cost of the risk. I would reply that this corresponds to the familiar legal doctrine according to which the liability of the injuror is limited to the damage that would have occurred if the victim had taken reasonable steps to minimize it; if the injury caused by your negligence is multiplied by my refusal to accept treatment, you are not responsible for the result. I admit that my application of that doctrine is a somewhat unconventional one.
I will end this paper by saying what I believe that I have and have not done. I have provided some foundation for the idea (which is not original with me[15]) that damage payments for injury or death ought to be based on the payment which the individual would require to voluntarily accept a small probability of the same injury, with damage being set equal to that payment divided by that small probability. I have also shown that that formula is not precisely correct; even under the special circumstances I assumed for my analysis it somewhat understates the proper damages where the probability of the injury which has occurred is larger than the probability of the injury used to estimate the amount of damages; in the absence of those special circumstances it understates the proper damages substantially. I have therefore not clearly established what compensation ought to be in the real world of incomplete markets, but by showing what it should be and how it might be estimated in a wor of more complete markets I have, I believe, somewhat clarified the conceptual issues involved in determining fair levels of compensation.