Copyrigth A.Rossato, 1994

Please mail to arossato@risc1.gelso.unitn.it your comments or suggestions.

This work will analyze the liability rules from an economic point of view. It is a theoretical study abstracted from the concrete application of liability rules as occurring in different legal systems. Its goal is to show how these rules work.

Since the economic reality is too complicated to be analyzed without simplifications, a number of assumptions is deemed necessary: the first is that the transaction costs to reach an agreement between the parties are prohibitive.

We also suppose that both victim and injurer are reasonable self-interested decision makers, whose goal is to minimize their expected loss and to maximize their expected utility.

The liability rules analyzed, that are supposed to be fully known by the parties, are : no liability; strict liability; negligence; negligence with the defense of contributory negligence; comparative negligence and strict liability. Both victim and injurer are supposed to have full knowledge of the liability rules under which they act.

The model is derived from: Shavell[1980], Shavell[1987] and Haddock[1985].

DUTY OF CARE
In the case United States v. Carroll Towing Co. (nt. 1)
Judge Learned Hand defined the duty of care as a function
of three variables: the probability of an injury (P), the
amount of harm (L) and the burden of precautions adequate
to prevent it (B). Given these variables, an injurer will
be found liable, for negligence, if the cost of precautions
is less than the expected loss due to the injurer's
conduct, that is if B

But "the formula is not explicit about whether accident costs and benefits are to be consider in the correct marginal rather than total terms (nt. 2)"; imagine this case: it's possible prevent a 100% probable loss of $100 spending $50, but it is also possible to reduce the probability of the accident to the 25% by an expense of $10 (in this case the expected loss is of $25 plus $10 of precautions).With an expected loss of 100, with a literal application of the Hand formula, the injurer will be found liable for not spending $50, even though the minimum expected total loss is $35 (that is $25 of expected loss plus $10 of precautions). In other words, the Hand Formula can produce an inefficient outcome.

It is thus possible to solve the problem more efficiently using marginal values instead of total ones. The function of the social costs due to an harmful conduct is composed by the costs of precaution and the expected loss, given a certain units of precautions.

It's possible to define the costs of precaution as a function of x, that is the level of care, measured in units of precautions: CP=A(x). A(x) is, in the domain x>0, a convex function or, in the simplest form, a function whose first derivative is greater the zero and the second derivative is always equal or greater than zero. That is, the units of precaution will increase their monetary value at a probably increasing rate. We do not analyze the case of economics of scale.

The expected cost of the harmful conduct will be P(x)D, where P(x) is the function of the probability of the harm given x units of precaution, and D is the monetary value of the loss caused by the injury. P(x) is a concave function or, in the simplest case, a function with a negative first derivative and second derivative always equal or minor than zero.

The following function is therefore the function of the social costs of a injurious conduct at the x point of care:

(1) L(x)=P(x)D + A(x).

This function is represented, together with its components, in figure 1.

The assumption in this case is that the level of care of the victim, or the injurer (the result doesn't change), is no taken into account since only one of them can use due care in order to avoid the accident, and not both: we are speaking about the unilateral case.

To find the minimum of this function we have to compare the marginal increase of costs due to caretaking and the relative marginal reduction of expected loss. Where the marginal increase of the expected cost begins to be greater than the marginal reduction of the expected loss, there the social costs will be the lowest. To find this minimum of the function we shall take the first derivative of L(x) and set it equal to zero:

(2) L'(x)= A'(x) + P'(x)D

A'(x)= -P'(x)D

We define x* as the value of x that minimizes the social costs, that is due care.

In figure 2 it's possible to see the expected liability
of an injurer in a situation where a negligence rule is
applied: if the injurer takes a level of care x

If also the victim, or vice versa the injurer, has some influence in the probability of an harmful event, influence due to his level of care, the function of the social costs depends now on two variables: the level of care of the injurer, x, and the level of care of the victim, y. This is the complete function:

(3) L(x,y)= P(x,y)D + A(x) + B(y)

To find the minimum of this function we have to set the first partial derivatives equal to zero:

(4) L'(x)= P'(x)D + A'(x)= 0

L'(y)= P'(y)D + B'(y)= 0

The values of x* and y* are the values where the marginal costs of care are equal to the marginal reduction of expected losses:

(5) A'(x*)= - P'(x*)D

B'(y*)= - P'(y*)D

so the min. L= P(x*,y*)D + A(x*) + B(y*)

Given the other party taking care, it's possible to represent graphically the level of x* and y* (see figure 3): the demand curves are the marginal reductions of expected losses and the supply curves are the costs of marginal units of care.

In this section we analyze the case in which the injury arises from an activity involving a single act. A will be the injurer and B the victim.

UNILATERAL CASEIn this subsection we will study the case in which precautions are deemed necessary to be taken only by one party and not by the other. It's possible to give an example of such a situation: suppose a water that breaks and floods the basement of a house, or a plane crashing into it. In such situations the victim can do nothing to avoid the accident.

Three liability rules will be analyzed: no liability, strict liability and negligence, the only rules that take account of the fact that only one party can choose a level of care that minimizes the total expected costs.

NO LIABILITYIn this case whatever the level of care chosen by the injurer is, he will not be liable and the losses will be borne completely by the victim. The injurer internalizes only the costs of the care he chooses, so he will minimize it by taking less units than optimal level (x < x*). If, in order to prevent the possibility of an accident, only the level of care of the injurer must be taken in account, the outcome will be inefficient. Instead, if for an efficient prevention of the injury the due level of care is deemed necessary to be taken by the victim, the outcome of this rule will be efficient (see (6) and (7)). That because a party will internalize the total expected costs and minimize them.

(6) if y*= 0 (only the injurer has to take care)

A's care A's costs B's costs x<x* A(x) P(x)D x* A(x*) P(x*)D

A will choose x<x* since A(x)<A(x*): the function is convex; since P(x) is concave, P(x)D>P(x*)D, that is, the expected costs of the injury will be greater.

(7) if x*= 0 (only the victim has to take care)

B's care A's costs B's costs y < y* / B(y)+P(y)D y* / B(y*)+P(y*)D

B will choose y* because it minimizes L, indeed B internalizes the social costs completely.

STRICT LIABILITYIn this case an efficient result will be reached only if the due care is to be taken by the injurer, who will internalize the total expected costs and minimize them. Indeed he will be found liable whenever an injury has occurred. Instead, if the victim is the party that can prevent the injury, he will recover damages whatever the his level care is. Therefore he will try to minimize his costs, which are only the precaution's costs. See (8) and (9).

(8) if y*= 0 (only the injurer has to take care)

A's care A's costs B's costs x < x* A(x)+ P(x)D / x* A(x*)+P(y*)D /

A will choose x* in order to minimize L. This is an efficient outcome due to the fact that A internalizes and minimizes the total expected costs.

(9) if x*= 0 (only the victim has to take care)

B's care A's costs B's costs y < y* P(y)D B(y) y* P(y*)D B(y*)

B will choose y

(10) y*= 0

A's care A's costs B's costs x < x* A(x)+P(x)D / x* A(x*) P(x*)D

A will choose x* if P(x)D+A(x)>A(x*).

By definition P(x)D+A(x)>L(x*)>A(x*) than
P(x)D+A(x)>A(x*).

In other words, there will be an efficient outcome if the due level of care has to be taken by the party that internalizes these costs, given the rule, that is the injurer. Indeed if he doesn't take the due care he will be found liable. Moreover the due care must be the optimal level of care, i.e. the level that minimizes the function of the social costs.

If the victim has to take care, the outcome will be efficient as well, since he will internalize the total social costs of the conduct of the injurer and will minimize it by taking due care (B(y*) is always minor than P(y)D).

COMPARISONWe have seen that, when the prevention of an accident can be taken only by one party and not by both, the outcome is efficient whatever the chosen liability rule is: when the injurer is the only party that can prevent the injury both strict liability and negligence, if due care is chosen at an optimal level, are efficient. Instead when the care is to be taken by the victim a no liability rule does work correctly as the negligence does.

We will now compare negligence and strict liability to find out eventual differences in the minimization of the total expected costs. This comparison is made taking into account that the injurer is the only party that must take the due care to prevent the injury.

Administrative costsStrict liability and negligence differ in the costs of administering the liability rule. These costs are principally information costs and claim costs.

The former are costs that the fact-finder has to spend to ascertain whether the level of care is equal to the optimal one. That is, if x was equal to x*. Under strict liability these costs are zero, since it is not necessary to find out whether the injurer did use care or not. Instead under negligence these costs are more than zero.

The claim costs are the costs of processing and collecting a legal claim, that is the costs of a judgment, the determination of damages, causation and so on. Under strict liability a claim arises every time an accident occurs, instead under negligence rule these costs arise only if the victim thinks that the injurer failed to use due care.

As Posner pointed out, "because information costs are higher under a negligence rule and claim costs are higher under strict liability, we would expect that, other things being equal, a fall in information costs would result in a shift away from strict liability toward negligence rule. Looking broadly at the history of the liability rules and the differences between liability rules in primitive and modern societies, we find that the secular decline in the costs of information associated with a growing literacy and knowledge of how the physical world operates has been accompanied by a movement away from strict liability and toward negligence as the dominant rule of liability.

Insurance effectsAnother aspect in which strict liability and negligence differ is the insurance component. Strict liability broadens the insurance component to the "unavoidable" accident. That is, it makes the injurer internalize the costs of the accidents that are unavoidable with the optimal due care. Instead the insurance effect of negligence is limited to those cases in which the injury is avoidable using the optimal level of care. The broader insurance component of strict liability provides for preferring strict liability to negligence.

Activity levelThe problem of the incentives to avoid accidents by reducing the level of activity, rather than by increasing the care with which the activity is conduced, must now be considered.

Indeed the probability of an injury increases if the level of activity increases: therefore, the injurer, that we suppose to be the only costs avoider, will not take into account the level of activity unless this level is part of x, that is not. But under strict liability the injurer will be found liable whenever an accident occurs, so he will interpret x* broadly, that means he will take in account the rising probability of an accident due to the activity level:

(11) UA= U(s) -{sA(x)+sP(x)D}

this is the function of net utility of A (the
injurer), where sA(x) is the costs of care at a level of
activity s with x units of care. U(s) will be the gross
utility; it depends on the level of activity s. U(s) is a
concave function, that is U'(s)>0, which means that the
utility increases at the increase of s, but U''(s)<0, that
means that the increase of gross utility due to the
increase of s occurs at a decreasing rate. Indeed the
utility function is supposed to be concave.

Suppose x=x* (the injurer is taking due care):

(12) U= U(s) - s{A(x*)+P(x*)D}

where U is the total expected utility

max. U= U'(s) - {A(x*)+P(x*)D}= 0

U'(s)= A(x*) + P(x*)D

That means that the maximum of net utility, given a due
level of care, is reached when the marginal gross utility,
due to an increase of s, is equal to the total expected
costs. The level of activity that maximizes U is s*.

Now we will analyze the case of negligence. We must take into account that the level of care chosen by the injurer is the optimal one. Since he's taking due care, is not liable and his expected net utility will be:

(13) UA= U(s) -sA(x*)

max. UA= U'(s) -A(x*)= 0 therefore he will choose
s=s# in order to set U'(s#)=A(x*)

but U'(s*)=A(x*)+P(x*)D; therefore U(s#)<U(s*) and since
U''(s)<0 ,that means that s#>s*.

The level of activity will be greater than the optimal one. In other words: since the injurer is not liable (we suppose he is using due care) he will not take into account the expected losses sP(x*)D and will choose a too high level of activity.

In the case of strict liability the injurer will be liable whatever the level of care is. We suppose he chooses due care, but he will internalize the expected costs of the injury and so he will minimize them too:

(14) UA= U(s) - s{A(x*)+P(x*)D}

max. UA= U'(s) - {A(x*) + P(x*)D}= 0

So he will choose s=s# in order to set
U'(s#)=A(x*)+P(x*)D and, since this is the max. total net
utility, s#=s* is the optimal level of activity.

The bilateral case is the case in which a due level of care in order to prevent the accident is to be taken by both the injurer and the victim. So the optimal level of care will be more than zero for both of them.

In this case there is an interdependence of the parties' behavior. That means that the injurer will act in certain way, given that the victim acts in the asserted way; and vice versa. A situation in which that happens will be called a situation of equilibrium, since neither the victim nor the injurer have reason to modify their behavior.

We have already defined the total expected costs (see (4)) and the minimum of them (see (5)) that is when the due care x* and y* is taken by both the parties.

In this subsection we will analyze four liability rules: negligence, negligence plus contributory negligence, strict liability plus contributory negligence and comparative negligence. After that a comparison is deemed necessary in order to investigate the capacity of the rules to influence the level of activity of both parties.

NEGLIGENCEIf the injurer takes at least due care he will not be found liable, otherwise he will bear the expected losses regardless of the victim's behavior. In the model (15) we can see the expected losses of the parties at the given level of care of each of them:

(15) care A's costs B's costs x*,y* A(x*) P(x*,y*)D+B(y*) x<x*,y* A(x)+P(x,y*)D B(y*) x<x*,y<y* A(x)+P(x,y)D B(y) x*,y<y* A(x*) P(x*,y)D+B(y)

Suppose A takes due care. B will take care if:

(16) P(x*,y*)D+B(y*)<P(x*,y)+B(y)

that is true from the definition of the minimum value of
L (see (5)).

The problem is whether A will take care or not. A will
if:

(17) A(x)+P(x,y*)D < A(x*);

remember that A(x)+B(y*)+P(x,y*)D < A(x*)+B(y*)+P(x*,y*)D
from the definition of expected total costs (5) therefore
A(x)+P(x,y*)D < A(x*) is true: A will take care because by
doing so he minimizes his expected costs. Since A takes
care, B will do the same. The outcome will be efficient:
indeed a situation of equilibrium, where neither the victim
nor the injurer have reason to change their behavior, is
reached.

In this case A will be found to be liable only if he failed to take due care and the victim did take; otherwise the victim will bear all the losses.

We analyze now the parties' expected costs in order to see if the allocation of these costs is chosen in a way that induces the parties to take care.

(18)

care A's costs B's costs x*,y* A(x*) P(x*,y*)D+B(y*) x<x*,y* A(x)+P(x,y*)D B(y*) x<x*,y<y* A(x) B(y)+P(x,y)D x*,y<y* A(x*) B(y)+P(x*,y)D

B takes care if:

(19) B(y*)<B(y)+P(x,y)D

and B(y*)<B(y)+P(x*,y)D

since B(y*)<P(x*,y*)D+B(y*) and
P(x*,y*)D+B(y*)<B(y)+P(x,y)D (nt. 5) and P(x*,y*)D+B(y*) <
B(y)+P(x*,y)D (nt. 6)

Therefore B will always take due care.

Since B takes care, A will take care if:

(20) A(x)+P(x,y*)D> A(x*)

but A(x)+P(x,y*)D= L(x,y*)-B(y*)

L(x,y*)-B(y*)>L(x*,y*)-B(y*) and L(x*,y*)-
B(y*)=P(x*,y*)D+A(x*)

P(x*,y*)D+A(x*)>A(x*) therefore A(x)+P(x,y*)D> A(x*) is
true and A will always take care.

We have seen that both parties minimize their expected costs by taking care, which is a situation of equilibrium, so that they minimize the total expected costs too.

STRICT LIABILITY WITH CONTRIBUTORY NEGLIGENCEIn this case the injurer will be found liable only if the victim has failed to take due care. In every case in which the victim takes due care the injurer will bear the costs of the injury.

We must study, in order to find if the allocation of the expected costs gives incentives to the parties to take due care, the parties' expected costs:

(21) care A's costs B's costs x*,y* A(x*)+P(x*,y*)D B(y*) x<x*,y* A(x)+P(x,y*)D B(y*) x<x*,y<y* A(x) B(y)+P(x,y)D x*,y<y* A(x*) B(y)+P(x*,y)D

If A uses care B will use care if:

(22) B(y*)< B(y)+P(x*,y)D

that is true: indeed B(y)+P(x*,y)D=L(x*,y)-A(x*)

and L(x*,y*)-A(x*)<L(x*,y)-A(x*)

and, since B(y*)<L(x*,y*)-A(x*)

therefore B(y*)<L(x*,y)-A(x*)

If A doesn't use care B will use care if:

(23) B(y*)<B(y)+P(x,y)D

that is true: indeed B(y*)<L(x*,y*)-A(x*)

since L(x,y)-A(x)=B(y) + P(x,y)D

and L(x*,y*)-A(x*)<L(x,y)-A(x) (nt. 5).

We have seen that B, in order to minimize his expected costs, takes the optimal due care. Given that chose of B we must see what A will do: he will take care if:

(24) A(x*)+P(x*,y*)D<A(x)+P(x,y*)D

that is true: indeed A(x*)+P(x*,y*)D= L(x*,y*)-B(y*)

and A(x)+P(x,y*)D=L(x,y*)-B(y*);

we know that L(x*,y*)-B(y*)<L(x,y*)-B(y*).

We have seen that both parties, in order to minimize their expected costs will choose to use due care, so the outcome is efficient. Indeed the situation in which both have to take care to minimize their costs is a situation of equilibrium.

COMPARATIVE NEGLIGENCEIn this case the injurer will bear the total expected costs only if he failed to use care but the victim used it. In the case in which both of them fail to take care the losses is shared between them according to their respective shares of damages. We will analyze this situation regardless the respective share of losses they have to bear. The only assumption that must be taken into account is that the sum of these shares is equal to 1 (Sa+Sb=1).

We must analyze the parties' expected costs in order to find an equilibrium situation in which the total expected costs are minimized, situation that occurs when both parties take due care.

(25) care A's costs B's costs x*,y* A(x*) B(y*)+P(x*,y*)D x<x*,y* A(x)+P(x,y*)D B(y*) x<x*,y<y* A(x)+SaP(x,y)D B(y)+SbP(x,y)D x*,y<y* A(x*) B(y)+P(x*,y)D

Let's suppose that A takes care. B will take care if:

(26) B(y*)+P(x*,y*)D< B(y)+P(x*,y)D

that is true: indeed B(y*)+P(x*,y*)D=L(x*,y*)-A(x*)

and B(y)+P(x*,y)D=L(x*,y)-A(x*)

and L(x*,y*)-A(x*)<L(x*,y)-A(x*)

If A doesn't take care B will take care if:

(27) B(y*)< B(y)+SbP(x,y)D

whose truth depends on the value of Sb

Now we must analyze the expected costs of A in relation
of B's behavior. If B takes care A will take care if:

(28) A(x*)<A(x)+P(x,y*)D

that is true: indeed A(x*)<A(x*)+P(x*,y*)D

and A(x*)+P(x*,y*)D=L(x*,y*)-B(y*)

and L(x*,y*)-B(y*)<L(x,y*)-B(y*)

and L(x,y*)-B(y*)=A(x)+P(x,y*)D

If B doesn't take care A will take care if:

(29) A(x*)<A(x)+SaP(x,y)D

whose truth depends on the value of Sa.

We have seen that a possible equilibrium can be reached when both parties take care. In this case the total expected costs are minimized.

The case in which only one party takes care is not a situation of equilibrium, that means tat if one party takes care the other will take care too.

The problem, here, is that we must understand if the situation in which both parties don't take care is an equilibrium situation. The equilibrium is reached when both (27) and (29) are true, in other words, when, for both injurer and victim the expected costs are cheaper than the costs of precaution.

(30) A(x*)>A(x)+SaP(x,y)D

B(y*)> B(y)+SbP(x,y)D

which means that:

(31) A(x*)+B(y*)>A(x)+B(y)+P(x,y)D

which means that L(x*,y*)-P(x*,y*)D>L(x,y) that cannot be
true.

So the situation in which both injurer and victim don't take care is not an equilibrium, which implies that for one of them will be better to take care. But if one takes care, also the other must, to minimize his expected costs.

"Suppose that an accident cost (after discounting) of $1,000 could be prevented by the defendant at a cost of $10, but by the plaintiff at a cost of only $50. The efficient solution is to make the plaintiff liable by refusing to allow him to recover damages from the defendant. If the defendant is liable, the plaintiff will have no incentive to take preventive measures (unless the damages to which he would be entitled would not fully compensate him for his injury), and the value-maximizing solution to the accident will not be obtained

"Several states have .. a comparative negligence standard, whereby the plaintiff damages are reduces by the percentage by which his own negligence contributed to the accident. This is not the correct economic standard ... In a case like this one, it would result in the parties' spending more than the efficient amount on accident prevention. If the defendant in the case was fully liable for the accident, he would spend $50 to prevent it and the plaintiff nothing, so the accident would be prevented at a cost of only $50. But suppose he were liable for only two third, say, of the accident cost, because the plaintiff was also negligent. Being liable for a judgment of $666.67, the defendant would still have an incentive to spend $50 on an accident prevention, while the plaintiff, since he must bear a cost of $333.33 if an accident occurs, would have an incentive to spend $100 to prevent the accident. The parties might therefore invest a total of $150 in accident prevention, resulting in a $100 increase in the cost of preventing the same accident; or they might invest nothing (either party, knowing that the other party had an incentive to prevent the accident, might, in reliance thereon, make on attempt to prevent it himself), resulting in an avoidable cost of $950."(nt. 7)

What Posner deals with is the problem of the least costs avoider: in this case a proper rule will set the standard of care equal to zero for the higher cost avoider who will therefore not have incentive to undertake redundant investment in safety.

In Calabresi's view the caretaking was seen as an either/or proposition: either the injurer or the victim should have taken due care. With such an approach, in a pure market, the "accident cost avoidance would require allocation of accident costs to those acts or activity which could avoid the accident costs more cheaply... The question for a pure market approach is, then, how we should determine who, in practice, is the cheapest cost avoider... In almost every area we can make some rough guesses, based on intuitive notions or on undifferentiated and unanalyzed experiences, as to who is clearly not the cheapest cost avoider and who may be... These are, however, guidelines which can be used for finding out who, in the absence of more information, is likely the cheapest cost avoider." (nt. 8)

A COMPARISONWe have seen that, when due care is the optimal level of care, whatever liability rule is chosen, the outcome will be efficient. But in the level of care is not taken into account the level of activity that has some influences in the function of the total expected costs.

The functions of the expected utility of injurer and victim are represented here:

(32) UA= U(s) -{sA(x)+sP(x)D}

(33) UB= U(t) -{tA(y)+stP(y)D}

where s and t are the level of activity of injurer and
victim.

We define the function of the total expected utility as
the sum of the utility functions of the injurer and the victim:

(34) U= U(s)+U(t)-s{A(x*)+P(x*)D}-t{B(y*)+sP(y*)D}

where
U is the total expected utility.

(35) max. U= U'(s) - {A(x*)+P(x*)D}= 0

U'(s*)= A(x*) + P(x*)D

max. U= U'(t) - {B(y*)+sP(y*)D}= 0

U'(t*)= B(y*) + sP(y*)D

That means that the maximum of net utility, given a due
level of care, is reached when the marginal gross utility
of the injurer and the victim, due to an increase of s and
t, is equal to the total expected costs. The level of
activity that maximizes U is s* and t*.

We must remember that the first derivative of the gross utility is greater than zero (U'(s)>0, U'(t)>0), but the second derivative is minor than zero (U'(s)<0, U'(t)<0) so the gross utility increases, at the increase of s and t, but at a decreasing rate.

We will now analyze the liability rules. Given that the level of care that both injurer and victim choose is the optimal one, we will study the expected utility of both in order to find out if the maximization of their expected utility creates a situation of equilibrium where the level of activity maximizes the total expected utility, that is s* for the injurer and t* for the victim.

First we take into account the negligence rule. These
are the expected utility:

(36) care UA UB x*,y* U(s)-sA(x*) U(t)- stP(x*,y*)D-tB(y*)

A will choose s in order to maximize U(s)-sA(x*):

(37) max. UA=U'(s)-A(x*)=0

U'(s)=A(x*) but U'(s*)= A(x*) + P(x*)D

since U''(s)<0 and U'(s)<U'(s*), therefore s>s*

B will choose t in order to maximize U(t)- stP(x*,y*)D-
tB(y*):

(38) max. UB=U'(t)- sP(x*,y*)D-B(y*)=0

that is U'(t)=sP(x*,y*)D+B(y*)

therefore U'(t)=U'(t*) and so t=t*

The negligence rule leads to an efficient outcome only in regard to the victim's level of activity. Indeed the victim, given the due care of both, will bear, and so will internalize, the expected losses and will chose to minimize them by reducing his level of activity until it's equal to t*.

The negligence with the defense of contributory negligence and the comparative negligence will have the same outcome of the negligence rule, indeed both the injurer and the victim will chose to take the due care and the victim will bear the expected losses in the same way of the negligence rule.

Under strict liability the parties expected utilities
will be:

(39) care UA UB x*,y* U(s)- sP(x*,y*)D-A(y*) U(t)-tB(y*)B will choose t in order to maximize U(t)-tB(y*):

(40) max.UB=U'(t)-B(y*)=0

U'(t)=B(y*) but U'(t*)= B(y*) +sP(y*)D

given U''(t)<0 and U'(t)<U'(t*) that means that t>t*

A will choose s in order to maximize U(s)- sP(x*,y*)D-
sA(x*):

(41) max. UA=U'(s)- P(x*,y*)D-A(x*)=0

that is U'(s)=P(x*,y*)D+A(x*)

therefore U'(s)=U'(s*) and so s=s*

In this case only the injurer, that internalizes the expected costs, given the chosen care as the due care, will minimize them by reducing his level of activity at the optimal level of s*.

"Because no rule that has been examined induces both injurers and victims to choose optimal level of their activities, one is led to ask whether there exists any conceivable liability rule that always result in optimal level of activities. The answer is no. The reason, in essence, is that for injurers to choose the correct level of their activities they must bear accident losses" and the same must happen to the victims. "Yet injurers and victims cannot each bear accident losses"! (nt. 7)

CONCLUSIONSWe have seen that, in the single act accident, whatever the liability rule is the outcome is efficient if the duty of care is chosen at a level in which the social costs are minimized. In other words the goal of the law is to induce the parties to take an optimal care. But the problem of the level of activity can be solved only through the use of a rule the charges on the better accident avoider all the losses his activity causes. This in order to make him internalize the correct cost of his actions, that involves social costs too. In the case when both parties have to take care, an efficient outcome cannot be reached.

The conclusion of this analysis is that, in a field where no rule has a complete efficient outcome, other goals must be taken into account, such as fairness in the distribution of risks and costs, goals that are possible to be identified as non economic: the economic analysis of law can not be useful in such a situation except to show how the rules work in a simplified economic world.

A possible solution could be the choice of different rules, as in many legal system happens, each applicable to situation for which its application leads to an optimal outcome. But the simplicity of the system, an assumption was made for which a full knowledge of the rules was implied in the analysis, is a per se economic goal.

NOTES
1) 159 F2d. 169 (2d Cir. 1947)

2) R.A. Posner[1987] pg 87.

3) R.A. Posner, idem, pg. 66.

4) R.A. Posner[1977].

5) P(x*,y*)D+B(y*) < B(y)+P(x,y)D is true:

indeed L(x*,y*)D<L(x,y)-{A(x)-A(x*)} is true, since {A(x)-
A(x*)}<0.
6) P(x*,y*)D+B(y*) < B(y)+P(x*,y)D is true, since

P(x*,y*)D+B(y*)=L(x*,y*)-A(x*)

and B(y)+P(x*,y)D= L(x*,y)D-A(x*)

and L(x*,y*)-A(x*)<L(x*,y)-A(x*) are true.

7) G. Calabresi[1970], pg. 139

8) S. Shavell[1987],pg. 29

1) R. Cooter - T. Ulen, Law and Economics, 1988,
Glenview, IL.

2) Posner [1987]: R.A. Posner, The Economic Structure of
Tort Law, 1987, Cambricge, MA.

3) Posner[1977]: R.A. Posner, Economic Analysis of Law,
2d ed. 1977, Boston.

4) Calabresi [1970]: G. Calabresi, The Costs of Accident:
a Legal and Economic Analysis, 1970, New Haven.

5) Shavell[1987]: S. Shavell, Economic Analysis of
Accident Law, 1987, Cambridge, Massachusetts.

6) Shavell[1980]: S. Shavell, Strict Liability v.
Negligence, in 9 Journal of Legal Studies, 1 (1980).

7) Shavell[1987b]: S. Shavell, Torts in Which Victim and
Injurer Act Sequentially, in 26 Journal of Law and
Economics, 589 (1987).

8) Schwartz[1978]: G.T. Schwartz, Contributory and
Comparative Negligence: A Reappraisal, in 87 Yale Law
Journal, 697 (1978).

9) Haddock[1985]: D. Haddock - C. Curran, An Economic
Theory of Comparative Negligence, in 14 Journal of Legal
Studies, 49 (1985).

10) Rubinfield[1987]: D.L. Rubinfeld, The Efficiency of
Comparative Negligence, in 16 Journal of Legal Studies, 375
(1987).

- In a single act accident either strict liability and
negligence have an efficient outcome; but:
- Negligence has lower claim costs; whereas:
- strict liability has a broader insurence effect, and has a deeper control in cases in which the activity level has an impact over the probability of an accident.