The Disaggregated Hand Formula

Commercial activities, like selling a car or serving hot coffee, can generate a risk of loss to which multiple individuals are exposed. Likewise, the burdens of avoiding such risks are rarely borne by a single person. When burdens and losses are distributed across multiple stakeholders, when should negligence law tolerate or condemn the risky choice? A famous answer at the center of the first-year curriculum invokes the Hand formula: The failure to avoid a risk is negligent when the sum of the burdens of risk-avoidance is less than the sum of the expected losses. However, it is rarely explained why the sum of burdens and losses ought to be compared when there are other ways of comparing interests. This modest observation can serve as the starting point for reconstructing and reclaiming a fundamental concept of American tort law.

This Article argues that the Hand formula should be applied to multiparty cases by, first, disaggregating burdens and losses and comparing them on a pairwise basis, starting with the individual who bears the highest burden and the one who bears the highest expected loss. Three reasons favor this reconstruction. First, the traditional interpretation of the Hand formula rests on a controversial ethics of aggregation. The “disaggregated” formula is more ethically defensible because it is informed by values like impartiality and equality, not just utility maximization or cost minimization. Second, famous jury verdicts as well as Learned Hand’s reasoning in United States v. Carroll Towing support the alternative interpretation. Students of tort law are usually taught that jurors repudiate the Hand formula, perhaps most famously in the “Ford Pinto” (Grimshaw v. Ford Motor Co.) and “McDonald’s coffee” cases (Liebeck v. McDonald’s Restaurant). But such cases are only inconsistent with the traditional purely aggregative interpretation, while lesser-known cases where the formula was successfully applied involved disaggregated reasoning. Third, the Hand formula, on the proposed construal, can be effectively applied to ongoing litigation, such as the Abbott/Reckitt class actions over the sale of contaminated infant formula, as well as formalized to reveal new economic models of risk-related decision-making. In short, moral, legal, and practical reasons favor disaggregating the Hand formula, with implications for the economic theory of accidents and for high-stakes lawsuits currently in the public spotlight.

Table of Contents Show

Introduction

Hand’s formula for negligence may not seem like it needs rehearsing. Its basic elements are exceedingly well-known: A risky choice is negligent if and only if the burdens of avoiding the risk (“B”) are outweighed by the losses due to the risk (“L”) discounted by their probability (“P”)—or B < PL.[1] Yet despite its enduring influence on how tort law is taught and applied, the formula remains controversial and is arguably misunderstood. Consider the following three cases:

Ford Pinto:	In the 1970s, Ford argued that its failure to implement an $11-per-car safety feature in its Pinto model to prevent deadly explosions was not negligent because the total cost of implementing the feature across all automobiles ($137 million) would have been greater than the total expected benefits to society in avoided damages across all potential plaintiffs ($49.5 million).[2] That is, Ford argued that the burdens of avoiding the risk of explosions outweighed the expected losses: B > PL. Nevertheless, the jury found Ford negligent.
McDonald’s coffee:	In the 1990s, McDonald’s defended serving its coffee at a scalding 180°F—far hotter than coffee from most restaurants and hot enough to cause third-degree burns in seconds—by arguing that lowering the temperature would have been more burdensome than beneficial for its average consumers, such as long-distance truck drivers.[3] Since the average burden of cooler coffee outweighed the average benefit, the total burdens would have likewise outweighed the total benefits: B > PL. The jury still deemed McDonald’s negligent.
Scripto lighters:	In the 2000s, Maria Calles argued that Scripto’s failure to childproof its lighters was negligent because the $50 million cost of childproofing was less than the $200–270 million in potential damages that would have been saved by avoiding fire-related deaths and injuries: B < PL. The Illinois Supreme Court was persuaded that a jury could infer from the facts that the lack of childproofing was negligent, and the parties settled.[4]

Why was Calles successful in her use of the formula when McDonald’s and Ford were not?

The answer turns on the different ways of comparing burdens and losses when multiple parties are affected by a risky choice. On the standard interpretation of the Hand formula, the question of negligence is settled by a single comparison: The sum of the burdens of risk-avoidance is compared against the sum of the expected risk of loss across all affected parties. That is how the formula is standardly conceived, taught, and applied by defendants, at least in cases like Ford Pinto and McDonald’s coffee. But the standard interpretation relies on a questionable principle for aggregating interests.

An alternative and more defensible approach involves comparing, first, burdens and risks at the individual level, starting with the pair of individuals who bear the highest burden of risk-avoidance and the highest expected injury. One reason this is worth doing is that it ensures that there are not significant disparities in stakes at the individual level. That is what Calles argued, based on studies and expert testimony. She did not just compare the aggregate cost of childproofing lighters against the aggregated expected loss from the lack of childproofing. She argued, additionally, that childproofing would have increased the cost per lighter for individual buyers by only “cent[s],” but the expected loss borne by a person with children due to a lack of childproofing was orders of magnitude higher.[5] Put differently, Calles disaggregated the burdens and losses and compared them on an individual level.

This Article makes the “disaggregated” Hand formula for multiparty cases explicit and explains its justification. In doing so, it goes against the way the Hand formula has been understood and taught for over fifty years, ever since Richard Posner’s landmark article, A Theory of Negligence, made the aggregative interpretation famous.[6] The reconstruction of a concept at the heart of American tort law is necessary, I argue, for three reasons: one, because the current formula is not aligned with a nuanced ethics of aggregation; two, because the reinterpretation renders the Hand formula more consistent with important jury verdicts and the broader case law; and three, because the formula, reconceived, can be easily translated into jury instructions and formalized to reveal new ways of modeling the regulation of risk. The real-world consequences include a framework for resolving cases currently in the national spotlight, such as the Abbott/Reckitt class actions alleging that the sale of dangerous and contaminated infant formula may have caused early death in infants.[7]

Before delving further, it is worth clarifying what this Article does not address. It does not defend the Hand formula against the many legitimate criticisms that have been levied against it by scholars over the course of several decades—e.g., that the formula presumes that the value of life and bodily integrity can be measured in monetary terms; that it ignores parties’ background income and has inegalitarian implications as a result; that it fails to account for risk-aversion; and that it ignores the rights of parties who never consented to the relevant risks.[8] Such criticisms are important and valid.[9] Nevertheless, the formula as a simplified heuristic continues to inform jury deliberations, how companies make decisions about risks, and the risk-utility calculus that regulators use to set policy.[10] Even if the formula’s use is suspect, the fact that plaintiffs, defendants, courts, and regulators sometimes rely on it is well-established.[11] And it is against this background that the Article’s core argument should be understood—namely, that the formula’s application can be improved even if its flaws cannot be wholly eliminated. It can be improved based on a disaggregated generalization for multiparty cases. Hence, the disaggregated Hand formula warrants precise articulation and defense. The reason it is not already in our collective consciousness—and only implicit, at best, in arguments that plaintiffs like Calles have made—is that one correctable flaw in the traditional interpretation of the formula has not been fully articulated.

The discussion is structured as follows. Part I clarifies the Hand formula’s aggregation problem. Simple aggregation results in morally questionable choices in several multiparty cases.[12] One such case is where the burden of risk-avoidance is shared by multiple agents, each of whom bears a negligible burden (a few cents), while a few victims bear the entire risk of a substantial injury (a burnt home). While scholars have previously criticized the conventional interpretation of the Hand formula on “egalitarian” grounds, and specifically for not taking into account “true” burdens based on an individual’s financial means and the diminishing marginal utility of income,[13] there is a further problem with the conventional interpretation, which persists even if we control for factors like income, background endowments, and diminishing marginal utility. The conventional interpretation promotes inequality and does not aggregate interests in an intuitively defensible way. Values like impartiality, according to which we should avoid treating others’ interests as if they mattered less than our own, militate in favor of subordinating the goal of utility maximization in some cases to avoid extreme disparities in interests at the individual level.[14]

Part II introduces and explains an alternative interpretation of the formula. Disaggregated BPL comparisons are a means of avoiding extreme disparities in interests at the individual level, and such comparisons can be combined with aggregate-level comparisons to conditionally minimize the total costs of accidents. A disaggregated BPL comparison involves comparing the highest burden associated with risk-avoidance borne by some person against the expected loss to the person most exposed to the risk. Additional pairwise comparisons can be made to ensure that the burdens and risk of loss measured at the individual level are roughly “on a par” before comparing total burdens and total losses.[15] In defending the disaggregated Hand formula, I rely on an approach to evaluating complex social choices defended by several moral philosophers.[16] It is somewhat surprising that though there have been major developments in recent years in the ethics of aggregation,[17] the application of this work to fundamental concepts in tort law has attracted no scholarly commentary as far as I can tell. Hence, the normative justification for pairwise BPL comparisons requires some discussion, though the actual comparisons are easy to implement and can be incorporated into simple jury instructions.[18]

Part III demonstrates that the disaggregated Hand formula is entirely compatible with Learned Hand’s own reasoning in United States v. Carroll Towing and subsequent cases. As several scholars have previously observed, Learned Hand never used the language of cost minimization or utility maximization to justify his influential formula.[19] What has been overlooked, however, is that in applying the formula, Hand considers individual-level burdens and losses and does not aggregate them when he could have. Moreover, famous jury verdicts that the literature portrays as inconsistent with the Hand formula—such as in Grimshaw v. Ford Motor Co. and Liebeck v. McDonald’s Restaurants—are consistent with and, indeed, explained by the disaggregated interpretation. Neither Ford nor McDonald’s conducted pairwise BPL comparisons. Such comparisons would have favored implementing the safety device in the Ford Pinto and lowering the temperature of McDonald’s coffee. Hence, had these actors applied the formula correctly from the start, significant harms would have been avoided. Furthermore, the disaggregated BPL comparisons intuitively explain the outcome in the Scripto lighter case and others like it.

Part IV discusses two applications of the suggested framework: one practical and urgent, and the other more theoretical. The practical application involves using disaggregated BPL comparisons to shed light on the ongoing Abbott/Reckitt litigation concerning dangerous and contaminated baby formula that may have exposed infants to a risk of diseases like necrotizing enterocolitis (NEC). Parents across the country have filed hundreds of lawsuits, and the cases have received extensive press coverage.[20] One important argument that some plaintiffs have made has been dubbed the “premium prices” argument—namely, that they paid nearly twice as much as generic brands for the Abbott products.[21] This argument has been standardly construed as a contractual claim: The unsafe products involved a breach of warranty that was implicit in the high price, and consumers were deprived of the “benefit of the bargain.”[22] But that may not be a winning strategy.[23] Instead, we can use the disaggregated Hand formula to explain why the plaintiffs’ point about a price premium can bolster the negligence theory. Reformulating infant milk to be safer may have resulted in higher prices charged by Abbott and Reckitt. However, if there were fifty percent cheaper generics available on the market, slightly more expensive reformulated Abbott/Reckitt products would not have prevented any parents from buying infant milk for their children. The consumer’s burden due to costly reformulation is especially worth worrying about in this context, unlike the burden on corporate shareholders, who are numerous and less likely to bear the highest per-person costs. Hence, the plaintiffs’ claim about price premiums bolsters the likelihood that the highest individual-level burden associated with reformulating the products would not have been very high whereas the highest expected loss due to the unimproved products was extreme—a point that ought to be made more explicitly. The disaggregated Hand formula is thus a crucial piece of the analysis, vital for motivating the negligence theory underpinning the lawsuits and for resolving the cases correctly.

A second application discussed in this final section is more theoretical. Disaggregated BPL comparisons can be incorporated into standard economic models for understanding the incentive effects of tort liability. Using a simple game-theoretic framework, we can show that some regulatory alternatives to tort liability, such as a no-fault accident compensation scheme of the kind employed in countries like New Zealand for car accidents, sometimes do better at realizing outcomes favored by the disaggregated Hand formula.[24] Put differently, the values underpinning the disaggregated BPL comparisons may, in some special cases, favor alternative regulatory schemes over tort liability, including those widely believed to be in tension with the principles underpinning the common law. This result is not meant to be a defense of such alternatives, but purely illustrative of a general point: the framework’s significance for questions of ideal institutional design.

I. The Hand Formula and the Ethics of Aggregation

Many cases of negligence involve a choice that exposes multiple potential victims to a risk of loss. Moreover, the safer alternative in such cases often entails burdens distributed across multiple parties. For example, partners in a joint enterprise might share the burdens of making a product safer or else risk injuries to dozens of customers. This fact—namely, that the interests implicated in the choice of risk vs. safety are often myriad—is sometimes obscured by the paradigmatic tort lawsuit that involves a single plaintiff and a single defendant, especially when it is a natural person on either side of the dispute. Nevertheless, simple cases, where the interests of only two natural persons are at stake, represent a helpful starting point for our analysis. Part I.A thus begins with the observation that in simple two-person cases, the application of the Hand formula can be motivated based on the value of utility-maximization (or cost-minimization). But there is another value that explains the formula’s appeal in such cases: impartiality. The connection is worth making salient for two reasons. First, the impartiality-based explanation for the Hand formula’s appeal has been largely overlooked in favor of the utility-based explanation; second, these two values do not have the same implications for multiparty cases. Part II.A explains why impartiality does not favor simple aggregation in multiparty cases. Part II.B explains why a purely aggregative version of the Hand formula entails choices that are morally contestable for other reasons having to do with the value of equality.

A few caveats before we begin. Throughout, I shall make simplifying assumptions in discussing cases, as interpreters of the Hand formula usually do. For example, I’ll be treating burdens and losses as roughly measurable in monetary terms,[25] and holding fixed across individuals such factors as background income and risk tolerance.[26] As indicated earlier, such simplifications are defensible at least in theory if not in practical application. The point of stylized cases is simply to test our moral or normative intuitions about risk and safety. The cases serve as a starting point for discussion rather than the final analysis. The framework can be refined once we have understood the moral intuitions that the stylized cases are effective at motivating. That is the spirit in which I hope the examples to follow will be understood: as a means of exploring the normative basis (if any) for BPL comparisons in negligence law.

A.     Two-Person Cases: Utility or Impartiality?

With the above caveats out of the way, consider a simple two-person scenario. Defendant, Sam, faces a choice of whether to use extra scaffolding to prevent damage to a neighbor’s property due to construction on his house. He knows that failing to use extra scaffolding entails a small risk of construction material falling on a shed next door, causing the neighbor, Susan, property damage. Hand tells us: Sam’s failure to properly secure scaffolding is negligent if and only if (iff) the burden on Sam of taking the precaution, B, is less than the expected harm to Susan, PL.[27] The claim calls out for a normative explanation: in virtue of what is it negligent or, to put it more plainly, wrong in a way that licenses moral or legal criticism, for Sam to fail to take the precaution when (and only when) B < PL?[28]

A famous explanation invokes the values of utility maximization or cost minimization:

Utility:

It is prima facie and pro tanto wrong for a person to choose in ways that fail to minimize aggregate costs.[29]

The utility-based explanation is straightforward. Taking the precaution of extra scaffolding minimizes costs when (and only when) B < PL.

However, the formula’s appeal in the simple two-person case might instead be explained by a different principle:

Impartiality:

It is prima facie and pro tanto wrong for a person to choose in ways that involve treating others’ interests as if they mattered less than the person’s own.

Impartiality differs from utility for reasons that will become clearer in a moment.[30] But first, consider the principle of impartiality on its own terms. Impartiality gives expression to a moral ideal of the equality of persons.[31] As Thomas Nagel observes, a set of values and corresponding call to action flow from “a point of view which abstracts from who we are, but which appreciates fully and takes to heart the value of every person’s life and welfare.”[32] What impartiality demands precisely is controversial in ethics, but it is also an extremely familiar and plausible idea that there are reasons not to favor any one person—ourselves especially—based solely on their identity (or other morally irrelevant characteristics like their skin color).[33] At the very least, partiality often seems like a pro tanto moral defect, one that calls for explanation or excuse. Of course, there are forms of partiality grounded in love and friendship that might be justified rather than simply excused. However, the form of partiality at issue in the cases I am about to discuss is self-regarding—that is, partiality in its most questionable form.

The connection between the Hand formula and impartiality is straightforward. Sam’s willingness to impose the risk on Susan when B < PL amounts to putting a thumb on the scale in favor of his own interests relative to Susan’s. By stipulation, the burden on Susan, as measured by the expected loss (PL), is greater than the burden on Sam of taking the precaution (B). So, in choosing not to take the precaution, Sam gives more weight to his own interests than Susan’s just because they are his own. Sam’s partiality is reflected in the fact that he would not inflict the same risk of loss on himself if given the choice to insure against it by taking the precaution.[34]

Impartiality also explains at least one reason that might be given in favor of the decision to inflict the risk on Susan when B > PL. In that case, it might be partial for Susan to insist that Sam take the precaution. Bear in mind our simplifying assumptions: equal background endowments, so no distributional concerns; a context where risk imposition does not clearly require advance consent; identically neutral attitudes towards risk. Given these assumptions, Susan might be treating her own interests as counting for more than Sam’s if she were to insist that Sam take on a burden that outweighs the costs associated with her own risk of loss, since she would not consent to insuring against the risk herself if she had to bear Sam’s burden. To be clear, this does not amount to a decisive argument in defense of Sam’s decision to risk inadequate scaffolding when B > PL. One could argue that Susan has a legal or moral right to be free of the risk as a property owner. But, again, our goal is simply to explain why the Hand formula might be appealing, not to show that it gets the all-things-considered correct verdict. The point is just that Sam’s decision to inflict the risk on Susan when B > PL is at least partly defensible based on the principle of impartiality. At a minimum, impartiality offers no worse a justification for the decision than utility.

So, two distinct values—impartiality and utility—can be invoked to explain the normative logic of the Hand formula for a simple two-party case. Yet only the utility-based interpretation of the formula is well-known. One reason might be that the utilitarian Hand formula easily generalizes to cases where there are multiple parties affected by the risk of injury and the burden of risk prevention, whereas the generalization of the impartial Hand formula is not as straightforward.

B.     Multi-Person Cases: The Error(s) in Simple Aggregation

Next, suppose that multiple parties are exposed to a risk while the cost of taking the precaution is, likewise, distributed across multiple persons. Perhaps the cost of extra scaffolding will be borne by multiple members of Sam’s family, and the minimal scaffolding exposes several houses in the neighborhood to a risk of property damage.[35] The utilitarian Hand formula easily generalizes to such cases. It invites us to sum up the burdens and expected risk of loss. We can state things more precisely: letting $$B\\_i, P\\_i,$$ and $$L\\_i$$ refer to the cost of precaution, the probability of the injury, and the cost of the injury respectively to person  of a set of n individuals, the failure to take the precaution is negligent when and only when  $$B_1 + P_1$$ … is less than  $$P_1L_1 + P_2L_2$$ … (that is, iff$$\sum_{i=1}^{n} B_i < \sum_{i=1}^{n} P_i L_i$$ ). This is the standard fully aggregative generalized Hand formula. It informs case law, as we will discuss in Part III, and serves as the foundation for the economic theory of negligence. It should be clear why it is consistent with the value of utility. Choosing safety minimizes aggregate expected costs when and only when $$\sum_{i=1}^{n} B_i < \sum_{i=1}^{n} P_i L_i$$.

But the value of impartiality does not imply the fully aggregative generalization. Impartiality invites us to treat others equally in weighing their interests. But treating others’ interests equally does not logically entail minimizing aggregate harm. As a starting observation, impartiality does not obviously entail any specific “identity-independent” scheme for weighing interests when there are multiple parties.[36] Accordingly, if the normative justification for the formula in the two-person case is impartiality, then we cannot assume without normative argument that the correct generalization of the formula for the multiparty case is the standard aggregative interpretation.

In fact, simple aggregation leads to outcomes that seem unfair for a familiar reason worth going over: It favors options that minimize total costs but may expose some individual to a disproportionately weighty burden, even when that individual’s costs significantly outweigh the minor costs to each of the many others.[37] We can illustrate the point with some cases. Consider the following:

Lone Resident:

A corporation can spend $1,001 of corporate funds to avoid a 10% risk of a $10,000 injury to a property owned by a single resident in a small town in which the company’s factories are based (assume there is only one permanent resident in the town who is affected). The corporation’s expense will fall on 100 shareholders at a $10.01 cost to each, all of whom treat the corporation as a purely profit-making venture.[38]

If we simply aggregate burdens and losses, then there is nothing wrong with the corporation inflicting the risk on the sole resident since the aggregate burden of risk avoidance ($1,001) is greater than the resident’s expected loss (10% x $10,000 = $1,000). But this result can be morally questioned. After all, if we look at the per-person burdens, it is the shareholders who bear the burden of safety in the morally relevant sense, not the corporate entity (by stipulation).[39] And the burden of safety on each shareholder is just $10. So, it may well be unreasonable for each shareholder to vote against the safety precaution when, at a minimal personal cost, they can prevent a much more serious risk of loss on a single resident. The lone resident’s claim in favor of the precautions, in order to avoid serious harm, should prevail even though the shareholders have a greater aggregate claim against the precautions.

Here is one way to motivate the moral intuition. Suppose one of the shareholders, let’s call him Khaled, had to swap positions with the town resident. Presumably, Khaled would have gladly paid $10.01 to insure against a 10% risk of a $10,000 injury, or an expected harm of $1,000. If we imagine that Khaled had to elect a policy before he finds out whether he will end up as a shareholder or the town resident, he would likely favor a policy requiring the shareholders to pay $10.01 to avoid the expected loss of $1,000 to the resident. Indeed, each shareholder should be able to reason in these terms behind the “veil of ignorance.”[40] It would thus be a form of partiality for each shareholder to vote to inflict the risk of injury, since it is partial to subject others to a form of treatment that one would object to if the roles were reversed.

Here is another way of motivating the moral intuition. Since the town resident bears the entire expected harm of $1,000, the choice to impose the risk will result in a less equal distribution of resources than the choice of safety, holding everything else the same. If the shareholders choose safety, they become $10 poorer than the town resident (bear in mind our ceteris paribus assumption concerning equal starting endowments). This is because they can share the cost of safety among themselves. But if they choose to impose the risk, the resident is $1,000 poorer in expected losses. We can vary the numbers to make the point more decisively: Sometimes pure aggregation results in choices relating to risk/safety that worsen inequality. So if we care about the distribution of resources, not just the total number of resources, we should find simple aggregative reasoning suspect. As I’ll discuss later using cases like Ford pinto and McDonald’s coffee, scenarios like lone resident are a familiar and recurring occurrence in the modern economy, and the intuition that it would be wrong for corporate actors to inflict the relevant risks is widely shared.

I realize that some readers might have conflicting moral intuitions about lone resident. That is understandable precisely because such cases implicate conflicting values. Some might think utility maximization or cost minimization matters much more than (a) an egalitarian distribution of resources or (b) a less determinate moral ideal like impartiality in decision-making. But the main argument does not depend on sharing moral intuitions about specific examples. The main point is that if we want Hand’s BPL analysis to be suitably neutral on hard questions of value and to leave room for values besides cost minimization, then we should not assume simple aggregation in multiparty cases is the only way to run the analysis.

Here is a different case that illustrates the same point:

Sharp Envelopes:

Kima sells envelopes that could have been designed to avoid a very minor injury, like a papercut. The safer design would have cost Kima $10,000, a cost she would not have been able to pass on to consumers. The sharp envelopes expose 1,000,001 buyers to a 1% risk of a small injury: a papercut that costs the buyer $1 (in band-aids and ointment).

This is a hard case in that it is not intuitively obvious whether Kima acts wrongfully, at least in a pre-legal moral sense. After all, none of the buyers affected would have insured against a 1% risk of a $1 injury at a cost of $10,000 to themselves. And so, if the 10,001 buyers who are expected to suffer the $1 injury (that is, 1% of 1,000,001) file a class action lawsuit demanding compensation on the grounds that Kima was negligent, their claim would be morally questionable. It might be unfair for each of the plaintiffs to maintain that Kima should—in a pre-legal moral sense of “should”—have taken on a $10,000 burden to avoid inflicting a negligible risk of an entirely miniscule injury on them. Yet on the standard aggregative interpretation of the Hand formula, the formula deems Kima negligent since her burden is less than the sum of the expected losses.

Scenarios like sharp envelopes may well invite some type of regulatory intervention. Perhaps regulators should require the safer design since it would be efficient or total-cost-minimizing. Indeed, there are many scenarios in which we expect individuals to take on large expenses to avoid small risks to many others.[41] But the question of interest is not the ex post question of what society should do, all things considered, in light of the social welfare loss due to Kima’s sharp envelopes. It is the ex ante moral question: Was it wrong for Kima to inflict the risk? And it is, at the very least, nonobvious that Kima acts wrongfully or unreasonably in being unwilling to suffer a $10,000 burden to avoid inflicting a 1% risk of a $1 injury on a large aggregate of others.

One explanation of this moral intuition is that a regime that obliges Kima to take on the burden entails a more unequal distribution of wellbeing than a regime that permits Kima to expose others to the risk, assuming (recall) that everyone starts out with the same background income.[42] If she takes on the burden, she is $10,000 poorer than everyone else. If she takes the risk, then 10,001 of her consumers are $1 poorer. In other words, the view that Kima clearly should have taken on the burden is straightforwardly inegalitarian. And it is inegalitarian even after we have eliminated the effects of unequal background endowments and diminishing marginal utility.[43]

Cases like lone resident and sharp envelopes illustrate that a utilitarian interpretation of the Hand formula endorses choices that privilege the total level of utility or costs in society, despite the potential for extreme disparities in burdens at the individual level. But when it comes to resources and burdens in society, it is reasonable to care not just about totals. We also care about the way resources and burdens are distributed across persons—that is, we care about features of the distribution. Hence, relative burdens on individuals should inform our judgments about when risk-imposition is morally acceptable, even if total utility or costs-in-aggregate remain important. These problems that result from unrestrained aggregation are well-discussed by moral philosophers.[44] But they are worth going over in this context to explain why the standard aggregative interpretation of the Hand formula is not logically entailed by the values that explain its appeal in simple two-person cases. Recall that a feature of the Hand formula in the simple two-person case that made it morally attractive was that one could avoid partiality by following its prescriptions. But the value of impartiality does not entail the utilitarian interpretation and is arguably inconsistent with it, for the utilitarian interpretation demands choices in relation to risk that many of us would find unfair.

Here is a more modest way of putting the point. The utilitarian interpretation of the Hand formula is, at best, a contestable interpretation of the demands of impartiality and a contestable interpretation of the values that should inform our decisions in relation to risk (setting aside, as I’ll argue in Part III, that it is also a contestable interpretation of Hand’s own reasoning). If the value of impartiality favors the Hand formula in the simple two-person case and calls for identity-independent concern for people’s wellbeing, then we should aspire for a generalization in multiparty cases that is appropriately neutral on contested moral questions about the limits of aggregation. And this sets up the Article’s positive contribution: Apart from a critique of the conventional generalization of the formula, the goal is to articulate an alternative generalization that captures values like impartiality and equality in addition to utility maximization.

II. Disaggregating the Formula in Multiparty Cases

If the Hand formula’s normative justification lies in values like impartiality, how should it inform decisions relating to risk/safety that affect multiple parties? Part II answers this question, drawing on a proposal for evaluating social choices and aggregating interests defended by several moral philosophers, including Thomas Nagel, Tim Scanlon, and Frances Kamm.[45] Part II.A explains why comparing the interests of persons on a pairwise basis is a way of avoiding extreme disparities in per-person costs. The process of pairwise BPL comparisons helps identify a choice—to take a precaution or not—that is impartially defensible. Part II.B explains how pairwise BPL comparisons can be combined with a comparison of total burdens and total losses in order to capture tradeoffs between total cost minimization and greater equality or impartiality.

A.     The Moral Importance of Pairwise Comparisons

The basic method of pairwise comparisons is simple. One way of “adding” up burdens and losses associated with safety/risk is to, first, order the affected parties in terms of their burdens and expected losses from highest to lowest; then, we can take pairs of individuals—starting with the one who bears the highest burden and the one who bears the highest expected loss—and run repeated BPL comparisons in sequence. If the highest individual burden is close enough to the highest expected loss to not raise equality-based scruples, we go on to the next highest pair and do another pairwise comparison. When the burden and expected loss for a pair diverge enough to raise egalitarian scruples, we discover a pro-tanto reason to make a choice that favors the person/side with the stronger interest. To use the example of sharp envelopes, since the highest individual-level burden, B, is $10,000 and the highest expected PL per person is less than $1, our pairwise comparison favors not opting for safer envelopes. When the highest per-person burden and highest per-person expected loss are roughly equal, the interests cancel each other out, and we look at the next highest, and we keep going until either one side wins or neither does from the point of view of equality.

The rationale behind the method can be explained as follows. The costs borne by a person associated with some choice captures the strength of their own personal interest in the choice. On what the philosopher Derek Parfit describes as the “Close Enough View,” costs that are “close enough” in moral gravity can be considered on a par when there are multiple affected parties.[46] Put differently, our monetary measure of burdens (a simplification of the Hand formula, recall) turns out to be “over-specified” in relation to the more coarse-grained phenomenon of moral interest—the moral weight of a person’s interest in some choice relative to another’s when there are multiple interests to consider. After all, there may be no significant moral difference between, say, $100 burden on Sam versus $80 in expected losses to Susan, when there are other people’s interests left to consider. The solution is to treat such interests of individuals as being in equipoise when the numbers are close enough, abstracting away from morally irrelevant differences in individual-level harms. We set the roughly equal claims aside, as approximate ties, and look to others’ interests in deciding what to do.[47]

So, we need a variable that captures when B and PL, as borne by a pair of individuals, are roughly equal. Let’s call it E. It represents the maximum difference between two harms that we are willing to treat as morally on a par—our tolerance for unequal interests at the individual level. So, e.g., if E = 10,000, it means that when individual burdens and losses are within $10,000, they will count as roughly equal ($$=_E$$). Shortly, I’ll suggest how E might be given a non-arbitrary value. For now, the point is just that whatever value we give it, it should reflect our intuitions about the moral proximity of claims. We can compare B and PL in terms of E-equality—e.g., in the case discussed earlier, $$B_{Sam}=_EP_{Susan}L_{Susan}$$ because Sam’s burden was within $10,000 of Susan’s expected loss.[48] We can state the overall proposal more formally to capture the nuances of the decision procedure.[49] Notice that apart from this “rough equality” parameter, E, the procedure relies on the same elements as Hand’s original formula. The BPL comparisons are performed on pairs of individuals rather than in aggregate, starting with the two with the strongest competing claims. Additional comparisons are performed as necessary to check whether the next highest claims are on a par as well, and so on, until the interests fall below E or there are ties all the way down.

When the pairwise comparisons favor a choice of risk or safety, the choice has the following morally important feature. Either the costliest prevented harm (per person) is significantly greater than the costliest unprevented harm (per person), or where the two are on a par, the next costliest prevented harm is significantly greater than the next costliest unprevented harm, and so on. We can say to the person most burdened by the recommended choice (either safety or risk) that the choice prevents a harm that all those affected would choose to prevent if they had to bear the harm, or else they would be indifferent.[50] The proposal is best viewed as capturing one category of pro tanto moral reasons for risk avoidance, where the reasons might be outweighed by other factors, including reasons related to the importance of total cost minimization, as I’ll argue shortly.[51]

The essence of the proposal so far might be captured by a jury instruction along the following lines:

If the highest expected loss suffered by any single person due to the risk was substantially greater than the highest burden that would have been borne by any single individual if the defendant had taken care to avoid the risk, that is a factor that counts in favor of the defendant’s negligence. If the reverse is true, it counts against the defendant’s negligence.

In Parts III and IV, I’ll explain the implications of this approach for famous as well as on-going cases involving potential negligence. But first we need to supplement it with the standard utilitarian interpretation.

B.     Combining the Utilitarian Hand Formula with Pairwise Comparisons

The pairwise comparisons proposed above give us a way to ensure that the choice of risk or safety recommended by the Hand formula does not entail gross disparities in burdens or losses of the sort we encountered in lone resident and sharp envelopes. But what happens if there are ties all the way down or the per-person interests fall below E? The traditional aggregate version of the Hand formula can then be used as a tie-breaker, at least if we are tempted by utilitarian considerations in negligence law. Rather than letting prospective injurers off the hook where there is a rough parity of interests on either side, the law might demand that defendants minimize aggregate harm. Recall that many find the principle of utility independently appealing as a criterion for reasonable risk-imposition.[52]

The full proposal can thus be stated as follows. If the pairwise comparisons entail a verdict regarding risk/safety (that is, if pairwise contests result in a clear winner), then we have our answer; if not, we follow the traditional utilitarian Hand formula (choose safety iff $$\sum_{i=1}^{n} B_1<\sum_{i=1}^{n} P_iL_i$$). The completed proposal requires the individual-focused choice where there are wide disparities in interests in pairwise contests, and the utility maximizing choice where the interests are in rough equipoise.[53]

Further refinements remain possible. Varying E—our measure of equality in pairwise contests—allows us to capture a broad range of attitudes concerning the importance of aggregate wellbeing promotion relative to fairness in the distribution of burdens. One approach might be to define E partly in terms of the difference in aggregate B and PL. The magnitude of the difference in aggregate costs (|$$\sum (B_i-P_iL_i)$$|) represents a measure of the strength of the utilitarian reasons favoring one choice. And so, there might be reasons to treat E as a variable directly related to that difference, on the plausible assumption that disparities in individual burdens become more tolerable the larger the aggregate welfare gain from the utilitarian choice. For example, if the total burdens associated with safety are hundreds of millions of dollars higher than the expected loss from a lack of safety, then there might be some reason to employ a larger E-value. The proposal thus provides a useful template for exploring different ways of balancing equality or impartiality versus utility.[54]

Our revised interpretation of the Hand formula for multiparty cases incorporates a broader range of values than just utility maximization—in particular, it also incorporates the values of equality and impartiality. Individuals are treated equally with no one person’s interests privileged based on their identity, and no one is asked to suffer an indefensibly large burden to avoid small burdens to many others. At the very least, our revised interpretation reflects a more neutral stance on the ethics of interest-aggregation than the traditional interpretation, since it does not purport to define E, which is a tolerance factor for disparities in individual-level interests. It also does not settle by stipulative fiat questions that Learned Hand himself left open when he introduced the formula, as I’ll argue shortly. Indeed, the next step in the argument involves showing that our revised interpretation explains famous legal verdicts: The failure to run pairwise comparisons explains questionable uses of the Hand formula by defendants.

III. Squaring the Case Law with the Disaggregated Formula

We turn now from moral theory to legal interpretation. The disaggregated generalization of the Hand formula is compatible with Learned Hand’s reasoning and original applications of the formula (Part III.A), as well as more recent cases widely portrayed as being inconsistent with the Formula conventionally construed (Parts III.B–D). Part IV extends this compatibility to ongoing cases.

A.     United States v. Carroll Towing

It is worth revisiting Hand’s original statement and application of his formula. We are now well-positioned to see why his discussion leaves open whether the correct generalization of the formula involves simple aggregation or a slightly more complex decision procedure of the kind introduced in Part II.

United States v. Carroll Towing involved multiple parties, injuries, and potential acts of negligence. Several barge-owners did not appropriately fasten their barges to the dock. And yet, despite the involvement of multiple parties, Learned Hand invokes his BPL formula only once and in non-aggregative form: To appraise whether the owner of the barge Anna C., one of the plaintiffs in the case, was contributorily negligent with respect to its own singular injury when the Anna C. sank after colliding with one of the other unfastened barges. The plaintiff and owner of the Anna C., Connors Company, was suing a tug company, Carroll Towing, for compensation for the injury. In deciding whether Connors Company was contributorily negligent for failing to ensure a bargee’s presence on the Anna C. despite the risk of collision, Hand considers a counterfactual scenario where the sinking injury is suffered by someone else:

It appears from the foregoing review that there is no general rule to determine when the absence of a bargee or other attendant will make the owner of the barge liable for injuries to other vessels if she breaks away from her moorings. However, in any cases where he would be so liable for injuries to others, obviously he must reduce his damages proportionately, if the injury is to his own barge.[55]

He then proceeds to apply his influential formula:

Since there are occasions when every vessel will break from her moorings, and since, if she does, she becomes a menace to those about her; the owner’s duty, as in other similar situations, to provide against resulting injuries is a function of three variables: (1) The probability that she will break away; (2) the gravity of the resulting injury, if she does; and (3) the burden of adequate precautions. Possibly it serves to bring this notion into relief to state it in algebraic terms: if the probability be called P; the injury, L; and the burden, B; liability depends upon whether B is less than L multiplied by P: i.e., whether B < PL. . . . In the case at bar the bargee left at five o’clock in the afternoon of January 3rd, and the flotilla broke away at about two o’clock in the afternoon of the following day, twenty-one hours afterwards . . . At the locus in quo—especially during the short January days and in the full tide of war activity—barges were being constantly ‘drilled’ in and out. Certainly, it was not beyond reasonable expectation that, with the inevitable haste and bustle, the work might not be done with adequate care. In such circumstances we hold—and it is all that we do hold—that it was a fair requirement that the Conners Company should have a bargee aboard (unless he had some excuse for his absence), during the working hours of daylight.[56]

There is no suggestion in the opinion that Hand is calculating “PL” based on multiple sinking injuries to multiple parties. He speaks of a single injury in stating the formula. And the actual injury at issue—the sinking of the Anna C.—was suffered by a single party: the barge-owner, the Connors Company. The analysis is, likewise, based on a single precautionary burden: the burden associated with ensuring the bargee’s presence on the Anna C. Hand finds Connors Company contributorily negligent because the burden of ensuring a bargee on board, B, was less than the expected disvalue of a sinking injury, PL, in an environment in which barges are “constantly ‘drilled’ in and out.”[57] In short, the formula is applied to a two-party scenario for the sole purpose of discerning whether the plaintiff’s own injury was the product of a negligent risk inflicted on others.

Moreover, Hand does not explain why it was negligent of the barge-owner to fail to have a bargee on board when the burden of the precaution was outweighed by the expected disvalue of the risk of injury to another barge. One possibility is that the choice is inconsistent with utility maximization or cost minimization. Another is that it fails to treat others’ interests impartially, as discussed in Part I. Since the moral basis for the formula is left unexplained, Hand leaves its correct generalization open. Put differently, an impartiality-based normative explanation—which would favor a disaggregated version of the formula for multiparty cases for reasons discussed in Part I—is perfectly compatible with Hand’s reasoning and application of the formula in Carroll Towing.

Hand next relies on his formula in Moisan v. Loftus, a case involving a truck driver who injures his passenger by driving at 50 mph during icy conditions.[58] The driver mistook a patch of ice for water, lost control of the vehicle when he braked suddenly, and hit a tree.[59] In applying the formula, Hand, again, focuses exclusively on the burden of care on the truck-driver and the probability of injuries to “the injured person”—that is, the sole passenger-plaintiff.[60] He does not, for example, aggregate the interests of others on the road who may have been exposed to a risk given the driver’s speed in icy conditions but avoided injury. Notably, this is a scenario in which Hand could have aggregated but does not—a fact of some interpretive significance. For instance, he acknowledges that “the injuries are always a variable within limits” and concedes that “a jury might find it negligent to drive at over fifty miles an hour in the night even on a straight road on which there was nothing ahead.”[61] Nevertheless, he dismisses the driver’s speed as a form of “carelessness which all of us fall into every day, and which does not condemn those guilty of it in . . . terms of the accepted definition [of negligence].”[62] As before, he analyzes the case from the point of view of only two affected parties, a scenario in which the aggregating and disaggregating Hand formulas are isomorphic.

B.     Grimshaw v. Ford Motor Co.

Hand may not have clarified the moral basis for his formula, yet famous jury verdicts on questions of negligence in more complex, contemporary cases support the disaggregated generalization in multiparty cases. These verdicts are widely portrayed as evidence that the Hand formula is inconsistent with the moral sensibilities of jurors on the nature of reasonable care; but, in fact, the verdicts only call into question the purely aggregative generalization.

Perhaps the most famous challenge to Hand’s formula is Grimshaw v. Ford Motor Co.[63] In the early 1970s, Ford Motor Company was under pressure to produce an affordable car that could compete with foreign vehicles. It quickly designed the Ford Pinto with a dangerous feature: The car’s fuel tank was positioned behind the rear axle in a way that made it vulnerable to even low-speed collisions. Even in 20–30 mph collisions, there was a risk that the Pinto’s fuel system would rupture and explode, which would potentially cause serious injury or death. Ford knew of the risk, could have corrected the design at a cost, but decided not to implement the correction. Its decision was informed by the Hand formula. As a result, families like the Grimshaws suffered tragedies: Lilly Gray was killed when her Pinto was rear-ended and exploded, and her passenger, thirteen-year-old Richard Grimshaw, suffered burns over 90 percent of his body, enduring extreme pain and multiple surgeries.[64]

At trial, an internal document revealed that Ford had argued before the National Highway Traffic Safety Administration (NHTSA) against regulations requiring all car manufacturers to correct the type of hazardous design at issue in the case.[65] Ford argued that the safety corrections would cost around $11 per car. Based on an estimated 11 million automobiles sold by the automobile industry, it calculated that the cost of implementing such safety features would be $137 million for the entire industry. It estimated that the safer design would result in 180 fewer deaths and avoid 180 cases of severe burn injuries across the entire population. Ford then used the NHTSA’s own “life-value” figure associated with the loss of life—$200,000 per person in the 1970s (roughly $1.6 million in purchasing power today)—and a corresponding measure of the cost of burn injuries—$67,000 per person—to calculate the total expected costs to society due to the uncorrected risk, which came out to $49.5 million.[66] It reasoned that since the total burden associated with the safety measures ($$ \sum B_i$$ = $137m) was greater than the expected disvalue of the risk in aggregate ($$\sum P_iL_i$$ = $49.5m), the safety features should not be required by NHTSA.

The jury disagreed and awarded $125 million in punitive damages against Ford for negligence. The verdict was a striking repudiation of Ford’s reasoning. One criticism was that Ford attached a monetary value to life and bodily integrity. It is extremely implausible that its valuation of serious injuries properly reflected the kind of harm suffered by victims like Richard Grimshaw. But there is at least one problem with that criticism: Ford was relying on measures regulators use to make such risk-utility decisions. The numbers were NHTSA’s own in the 1970s. There were surely other problems with Ford’s actions, including its willingness to make such consequential decisions unilaterally and non-publicly. But one major problem with Ford’s reasoning is that it did not apply the Hand formula in a disaggregated way. We can show, based on a back-of-the-envelope calculation, that had Ford disaggregated, it would have implemented the safety device and avoided deaths and serious injuries to its customers.

Recall that the disaggregated formula involves considering the highest per‑person costs associated with risk and with implementing the safety feature. In order to calculate the highest expected loss to an individual—that is, $$PL_{max}$$—one cannot rely on an average estimate of the probability of death or injury for the entire automobile industry—$$\frac{180 deaths + 180 injuries}{11,000,000 cars sold}$$—since different consumers will face different risks. Consumers face different risks based, for example, on how much they drive. Likewise, Ford plainly could not have relied on the “life value” measure of $200,000, which is a statistical average, albeit the NHTSA’s own. That measure averages out the potential victims in terms of life-years remaining, health, and earning capacity, and so averages out the loss in wellbeing due to the risk.[67] It does not consider, for example, the loss of life-years suffered by a thirteen-year-old. In order to run a pairwise comparison, we need individual-level probabilities and losses. So, while the average expected loss can be used to calculate $$\sum P_iL_i$$—that is, the total expected loss across the population—and is therefore useful for a utilitarian and purely aggregative BPL comparison,[68] the average expected loss cannot be used to calculate $$PL_{max}$$, the highest individual loss for a disaggregated comparison.

To properly disaggregate, Ford would have had to use the highest life-value and injury-value measures available from the NHTSA on grounds of impartiality. Suppose we modestly increase the life-value measure to $1,000,000 based on NHTSA’s guidelines and income levels in the 1970s. Even if we leave all the other values unchanged, such as injury-value and the probability of death or injury, we can already see that the expected loss to an individual customer would have been greater than $11: $$\frac{180}{11,000,000}\times{$67,000}+\frac{180}{11,000,000}\times{1,000,000}$$ = $17.5.[69] Accordingly, once we adjust the other values, especially the probabilities, the highest expected loss borne by the customer facing the highest risk would certainly have been far greater than $11.

As for individual burdens associated with the safety correction, some portion of the $11 cost per car may have been passed on to car buyers. Since Ford was petitioning NHTSA not to require the safer design, we can reasonably assume that it did not expect to pass on the full cost. Put differently, the burden would have been shared. It is thus overwhelmingly likely that $$PL_{max}$$ would have been greater than $$B_{max}$$. Alternatively, if the $137 million aggregate cost had been fully absorbed by the automobile industry rather than passed on to consumers, it would have impacted company profit margins and ultimately perhaps share prices. However, the impact on individual shareholders would have been negligible. For comparison, there were no unusual movements in Ford’s share price the year a jury awarded a $125 million penalty against it.[70] In other words, Ford presented no evidence that any individual’s share of the burden of absorbing the safety correction would have been greater than the highest expected loss suffered by an individual car buyer exposed to the risks associated with the Ford Pinto.

Accordingly, while the jury verdict in Grimshaw may have been an emphatic repudiation of a purely aggregative Hand formula, it was not a rejection of the disaggregated Hand formula. Quite the opposite; the disaggregated formula offers one explanation for why Ford decided wrongly—Ford failed to properly ascertain the potentially weighty claims of individuals in favor of the safety correction and the relatively less weighty claims of individuals who would have borne the cost of the correction (e.g., shareholders or customers). The jury likely intuited such reasons. While Ford cannot be fully faulted for using NHTSA’s own measures of life- and injury-value at the time, standard measures used in regulating risk, or even for using population averages given that it was computing aggregate loss in full compliance with the conventional utilitarian Hand formula, it can be faulted for being insufficiently sensitive to individual burdens. It can also be faulted for failing to consider, as it should have, the highest available measurement of loss in wellbeing due to severe bodily injuries and/or death.

Notably, our disaggregated analysis agrees with the verdict in Grimshaw despite several artificial assumptions built into the formula that make choices less likely to be classified as negligent. For one, we assumed risk neutrality, even though people are likely to be especially averse to extreme risk. Recall that the Hand formula is a simple heuristic with several false assumptions baked in. Such simplifications are useful in theory. But in practical applications of the Hand formula, we must adjust for risk aversion in running our pairwise BPL comparisons. To put the basic moral point less artificially, the question Ford should have considered is whether an individual shareholder or customer would have paid an entirely negligible amount associated with implementing a safety feature to avoid a small risk of extreme injury and possible death to themselves. The answer, in all likelihood, would have been yes, in which case it had impartial reasons to implement the feature.[71]

C.     Liebeck v. McDonald’s Restaurants

Another famous case that may be interpreted as a rejection of Hand’s formula is Liebeck v. McDonald’s Restaurants.[72] The seventy-nine-year-old plaintiff suffered severe burns when she spilled 180-degree coffee on her lap, causing second- and third-degree burns across her nether regions. Following skin grafts and after losing 20 percent of her body weight due to various medical procedures, she sued McDonald’s. A question in the case was whether McDonald’s failure to lower the temperature of the coffee it serves through its drive-through windows by twenty degrees was negligent, given that it would have taken five to seven seconds longer for the coffee to cause the plaintiff’s burns, giving her enough time to prevent serious injury.

McDonald’s defended its decision to not lower the temperature, despite having received seven hundred burn complaints over ten years, on the grounds that a large portion of its consumer base were those who drive a considerable distance with their coffee, such as long-distance truck drivers.[73] Those customers prefer a high initial temperature, which keeps the coffee hot during long-distance trips. Moreover, such customers are less likely to consume the coffee immediately, which lowers the risk of burns. A customer with such preferences was depicted as the “typical” or statistically average drive-through customer. In essence, McDonald’s was implying that the burden of lowering the temperature for the average consumer—perhaps even the majority of consumers—would have been greater than the benefit, which strongly implies that the total burdens across all consumers would have been greater as well. Nevertheless, the jury found McDonald’s negligent and awarded $2.7 million in punitive damages, an award that was ultimately reduced to $640,000.

McDonald’s aggregated interests in a way the jury found unpersuasive on the question of negligence. As one juror stated, “[T]here was a person behind every number and I don’t think the corporation was attaching enough importance to that.”[74] Aggregation may be morally required, but not at the risk of ignoring wide disparities in competing claims at the individual level. The disaggregated Hand formula requires a pairwise comparison of the burden of the lowered temperature on the individual most burdened by the extra precaution—the long-distance driver or, perhaps, a McDonald’s shareholder if the lowered temperature would have entailed a loss in profits—against the greatest risk of injury to customers like the plaintiff, who consume the coffee immediately and are at greater risk of burns. That comparison does not obviously entail that the choice was not negligent. In fact, it seems quite plausible that the highest individual-level expected loss to the short-distance driver must have been significantly greater than the highest individual-level burden from slightly cooler coffee. As a result, the jury’s punitive damages award served as a rejection of the purely utilitarian but not the disaggregated Hand formula.

D.     Calles v. Scripto-Tokai

Plaintiff Calles alleged that Tokai’s “Aim ’n Flame” utility lighter, used for lighting grills, candles, and fireplaces, among other things, should have been childproofed. She cited reports that children had used the Aim ‘n Flame or a similarly designed lighter to start approximately 250 fires since 1998, and more than 75 percent of these incidents involved children less than five years old.[75] Experts testified that:

From 1980 through 1985, an estimated 750 persons were injured and 120 people died each year in residential fires started by children playing with lighters. Estimated costs of these fires was between 300 and 375 million dollars. In 1988 through 1990, the number of injuries caused annually by children playing with lighters increased to 1,100, and the number of deaths rose to 150. Children three to four years old caused most of the fires. Prior to 1989, a child under the age of five was burned to death every day by a fire started with a disposable lighter.[76]

Calles was successful in persuading the court that a reasonable juror could find the lack of childproofing negligent. She presented studies indicating that the manufacturing costs of producing childproof lighters would be around “50 million dollars,” while “205–270 million dollars in potential damages would be saved by manufacturing childproof lighters.”[77] This is one broadly utilitarian reason for finding the lack of childproofing negligent. But crucially, Calles did not just present evidence of the aggregate burdens from childproofing and the aggregate expected loss. She argued, further, that “manufacturers were expected to see a one to five percent increase in production costs and a one to five cent increase in per-unit cost.”[78] Even assuming that such costs would have been passed on to customers, an added burden of only one to five cents is negligible when compared to the highest expected loss. Again, a back-of-the-envelope calculation of the expected loss confirms this: $300 million divided by 1,000 victims (of death or serious injury) gives us a $300,000 average per-person loss; and $$\frac {1,000} {5,000,000}$$ gives us the probability of being a victim, resulting in an average PL of $60. $$PL_{max}$$ would certainly have been much higher than $$PL_{ave}$$, and many orders of magnitude higher than a $$B_{max}$$ of $0.05. Hence, Calles’s utilitarian reasoning was bolstered by disaggregated reasoning. She reminded the court that the highest per-person burden due to childproofing was in all likelihood far lower than the highest per-person expected loss due to a lack of childproofing. This significant difference prevents the individual-level claims from being “morally on a par” for a range of values of E, which, recall, measures our tolerance for inequality in pairwise contests between claims.

IV. Practical and Theoretical Applications

A comprehensive discussion of the analytic framework’s applications would take us well beyond the scope of this Article. However, two applications in particular—one practical and the other theoretical—warrant discussion, not just to illustrate the framework’s usefulness, but because the practical implications are especially urgent. Part IV.A explains why the disaggregated Hand formula is necessary for understanding the basis for ongoing lawsuits against Abbott Laboratories (makers of Similac) and Reckitt Benckiser (makers of Enfamil). Parents across the country have sued these companies, alleging that their cow’s-milk-based infant formulas caused infants to develop various diseases, including necrotizing enterocolitis (NEC)—a severe, often lethal gastrointestinal disease that causes intestinal necrosis, leading to infection, organ failure, and sometimes death.[79] The plaintiffs’ negligence theory can be significantly bolstered with disaggregated reasoning. Indeed, I shall argue that the question of negligence in these high-stakes cases turns on pairwise BPL comparisons. This Article’s framework should thus figure more explicitly in the arguments plaintiffs and defendants are presently making before juries and courts.

Part IV.B describes a more theoretical application. The standard model that economists use to understand the incentive effects of tort liability can and should be updated to reflect the moral importance of disaggregation. Somewhat surprisingly, the updated models can be used to show that some radical and less familiar alternatives to tort liability, such as a “no-fault” taxpayer-funded accident compensation scheme, can, in certain limited contexts, realize outcomes that more closely conform to the requirements of Hand’s formula, impartially construed. This result is not meant to be a defense of such regulatory alternatives, but to illustrate a theoretical point: that the reinterpreted formula is useful in formal modeling and for comparing different institutional schemes for managing risk.

A.     The Abbott/Reckitt Lawsuits

For several decades, medical research has warned that premature infants who are fed cow’s milk-based formula are at a significantly higher risk of developing NEC compared to those fed human breast milk or fortified alternatives.[80] Despite these widely-discussed risks, Abbott and Reckitt continued to market their formulas as safe for premature infants, particularly to neonatal intensive care units (NICUs), without warning about the potential dangers. At the same time, Abbott faced numerous complaints related to serious contamination risks in its facilities that caused sepsis and meningitis, ultimately forcing them to recall their products.[81]

Parents across the country have sued Abbott and Reckitt, alleging several different claims. One theory of liability is that Abbott and Reckitt knew about the risks but failed to warn hospitals, doctors, or parents, as they should have. Indeed, many parents who fed their premature babies Similac and Enfamil were never informed of the risks. Other theories of liability include that the companies should have employed safer and reasonable alternative designs—such as a reformulation based on human milk-based fortifiers—and improved their production process to mitigate bacterial contamination. Tragic stories have emerged from these lawsuits. Parents have testified that their fragile newborns, fighting for survival in the NICU, developed NEC after being fed Similac or Enfamil, resulting in emergency surgeries, permanent intestinal damage, and in some cases, death. In 2024, juries had already awarded millions in damages to families.[82] While Abbott and Reckitt deny any wrongdoing, the lawsuits have triggered widespread scrutiny over how baby formula is marketed, regulated, and labeled, and reignited a national conversation about corporate responsibilities in relation to risk.[83]

Our present focus should be on the plaintiffs’ negligence theory based on safer alternative designs and defects in the manufacturing process.[84] The companies have defended principally on the grounds of causation and harm. That is, they claim that the plaintiffs cannot prove that their infants developed these diseases because of consuming Similac or Enfamil. In lawsuits stemming from the recalls, the defendants have also asserted that the risk of contamination was low.[85] Plaintiffs, in return, have pointed out that “the products were all produced in the same facility . . . and thus were ‘uniformly produced in unsafe and unsanitary conditions.’”[86] Moreover, some plaintiffs have emphasized that they would not have purchased the products at defendants’ “premium prices” had they known how the products were produced and the associated risks.[87] These plaintiffs have pointed out that they “paid premium prices – as much as twice (2x) the price of generic store brands.”[88]

At first glance, paying a “price premium” seems unrelated to the negligence theory. Indeed, the plaintiffs’ claims relating to price have been understood in contractual terms. The parties have portrayed the argument as based on a “benefit of the bargain” theory.[89] The idea is that the products’ high price impliedly warranted their safety. However, as the commentary to the Uniform Commercial Code points out, contractual concepts like “unmerchantable” are distinct from tort law concepts like “defective” or “negligently manufactured.”[90] Contractual concepts relate to foreseeability and consumer expectations, whereas tort law concepts relate to “risk/utility.” Hence, even if the “price premium” argument could succeed on contractual grounds, and it is questionable that it will, it is so far unclear how it connects to the tort claims.

However, the disaggregated Hand formula can help clarify the connection. To that end, we must first recognize that the disaggregated formula’s application is more nuanced than its aggregated counterpart. Indeed, the Abbott/Reckitt cases helpfully illustrate how disaggregated BPL comparisons depend on fine-grained, case-specific facts. Ultimately, we need to identify the individuals who would be most burdened by reformulating/decontaminating Similac or Enfamil, as well as the individuals who stand to benefit the most. This requires investigating several different questions of fact.

One question is whether reformulating/decontaminating would result in higher manufacturing costs that are passed on to consumers. If so, some low-income families might be priced out of the market, losing access to an essential source of nutrition. [91] In fact, some defenders of these companies have implied precisely that—namely, that the plaintiffs’ arguments “may significantly impact the supply or even the availability of preterm infant formula.”[92] If they are right, then the highest individual-level burden of reformulation/decontamination might have been quite high: Low-income families would have been unable to purchase reformulated products that are safer but more expensive, and their children would have suffered from malnutrition. Indeed, corporate defendants often defend against taking some costly precaution by alleging that the higher prices would deprive some buyers of needful goods and services.

The plaintiffs’ “price premium” theory, if correct, should be understood as a response to alleviate precisely this concern. If there are “generics” available on the market that cost 50 percent less than Similac and Enfamil, then it is not true that some parents would have been priced out of the market for infant formula if Abbott or Reckitt had opted for the safer, but more expensive, choice. Assuming that Similac and related formulas are not targeted at lower-income families, a cost increase from reformulation would likely not have resulted in the worst burdens: being unable to afford infant formula. Rather, the burden would have been measurable in dollars and cents. Of course, some parents who would have had to opt for cheaper generics may not have benefited from the safer products. But the reformulation would not have made such marginal consumers worse off on the assumption that the generics are not any worse than existing Similac and Enfamil in relation to the relevant risks.[93] The “price premium” argument is thus an essential piece of the disaggregated BPL analysis: It clarifies the highest individual burden. If they were not premium products, the marginal buyer affected by the safer choice would have suffered a very serious burden. But if Similac and Enfamil are marketed and purchased as premium products, then the highest individual level burdens would likely have only been higher prices for a segment of the market, a burden that is easier to overcome in pairwise contests than the burden of being unable to feed one’s child. And it seems likely that this per-person financial burden is significantly outweighed by the highest expected loss suffered by the most vulnerable infants.

A lot of “ifs” to be sure. But the point is precisely that the disaggregated BPL analysis turns on the facts and requires a lot more (discoverable) information than aggregate-level burdens and losses. The new framework thus provides greater clarity on the kinds of considerations that juries and courts should weigh in deciding these important cases. More details regarding individual-level burdens and expected losses are needed before any definitive conclusions can be reached about how these lawsuits should be resolved. Yet, even with the limited information available to us, we have seen that the disaggregated framework connects the dots between the plaintiffs’ negligence theory and their “price premium” argument, an argument that remains misunderstood in ways that could obscure the legally correct and just outcome.

B.     Rethinking the Economic Theory of Accidents: Tort Liability vs. No-Fault Accident Compensation Schemes

Throughout this Article, we have reflected principally on how the Hand formula is employed in litigation and corporate decision-making. These are real, high-stakes, and richly complicated contexts that warrant our attention. But the Hand formula is also employed in other contexts that are worth considering, especially since this Article’s framework has broader implications. One context we have not yet considered is economic theory: The Hand formula is often used as a yardstick for modeling and comparing different regulatory schemes for managing risk. In fact, the Hand formula is sometimes used to valorize the common law of torts and tort liability.[94] Standard economic models of decision-making under uncertainty have been leveraged to show, rigorously and formally, that imposing tort liability for sub-optimal care induces rational parties to take aggregate cost-minimizing levels of care.[95] One standard model is described in the Appendix, but the technical details are not essential for understanding the main point. The main point is that these models always use a purely aggregative conception of the Hand formula. That is, they assume that the sole purpose of the formula is to ensure total cost minimization. As a result, these models end up overlooking potential drawbacks of tort liability and potential virtues of alternative schemes. For instance, one alternative approach that is regularly criticized as being inconsistent with the Hand formula (and total cost minimization) is New Zealand’s no-fault accident compensation scheme. New Zealand’s approach to a range of injuries (e.g., related to car accidents) is to compensate victims through a taxpayer-funded social compensation fund.[96] There are virtually no inquiries into the injurer’s or victim’s failure to take appropriate care. This approach to regulating risk is widely believed to be in tension with the moral principles underpinning the common law.[97] It is, indeed, easy to show using the standard economic models that rational agents acting under this scheme will take less than “optimal” care defined as “aggregate cost-minimizing” care.

But we have seen that the Hand formula’s logic is not purely aggregative. The disaggregated interpretation reminds us that BPL comparisons can serve multiple functions. Disaggregated comparisons are a means of promoting equality and impartiality in the distribution of costs, not just cost minimization. Therefore, one question the overall analysis raises is whether we need to update the theoretical models we use to compare regulatory alternatives. And the answer is yes—at least insofar as these models rely on the Hand formula as a normative lens.

Disaggregated reasoning helps us recognize one crucial feature of the New Zealand approach that might otherwise escape notice. Here, I explain the point briefly, though it can be established more rigorously.[98] Disaggregated BPL analysis reminds us that we should consider not just whether an institution ensures that aggregate costs are minimized; we consider also whether the institution distributes costs across parties in a way that is broadly egalitarian and impartial. And a broadly egalitarian distribution of costs seems to be an express goal of the New Zealand approach. Tort liability, recall, shifts the entire expected loss (PL) from the victim to the injurer when the injurer was negligent (B < PL); otherwise, the victim suffers the entire loss.[99] By contrast, the New Zealand approach can be understood as an intervention designed to “redistribute” the expected losses across all taxpayers. The victim’s loss due to an accident is always compensated by the state, and the burden of compensation is redistributed to both the injurer and victim (along with everyone else) through higher taxes. One way to think about this is that a victim’s PL—say, $1,000 in expected car repair costs—is distributed equally across an entire society. The losses are “spread out” through the social insurance scheme. On the one hand, injurers under this scheme may cause greater accidents because they do not face any liability for suboptimal care. On the other, in situations where negligence involves injuries to one’s self and not just injuries to others, there might still be incentives to take care.[100] Indeed, in contexts such as driving, where everyone is subject to such natural incentives, the New Zealand approach to regulating risks may entail a distribution of costs (burdens of care and expected losses) that is more egalitarian, even if the total costs of accidents might be higher.

The point is not to make definitive claims about the incentives and outcomes entailed by any one regulatory scheme. Neither is it to defend or criticize any one approach. The point is simply to highlight a trade-off—between total cost minimization and relative equality—that is also at the heart of the Hand formula, presently construed. So, to properly evaluate whether a tort regime or a “no fault” social insurance scheme is more consistent with the disaggregated Hand formula, we need to measure and model not just how these different regulatory regimes create incentives for cost minimization, we also need to measure and model the distributional effects (for an illustration of how this can be feasibly done to derive some simple results, see the Appendix).

Hence, if this Article’s arguments are persuasive such that we should be positively inclined towards the disaggregated Hand formula, then we should be open to revising the way we model and evaluate the virtues and vices of different tools in the regulatory arsenal. My goal, in this final discussion, was not simply to reiterate the point, which I have now made in several different ways, that disaggregated BPL reasoning matters. Rather, it was also to help us see things, in legal practice as well as in theory, that we otherwise would not. The goal was to highlight a specific way in which the revised framework is important for addressing questions of ideal institutional design.

Conclusion

The exclusively aggregative interpretation of the Hand formula has had an outsized influence on ordinary and institutional actors charged with making high-stakes decisions about safety and risk management. This Article has defended a reinterpretation of that formula with the goal of improving the way it is used to guide theory and practice. Properly understood, the formula requires not just comparisons of total burdens and total expected losses, but also pairwise comparisons of highest burdens and highest expected losses. The disaggregated Hand formula favors risk/safety decisions that avoid imposing disproportionately high costs on a few persons in the interest of an aggregate with lower per-person costs. In cases where there are multiple stakeholders, we capture this notion of reasonableness by comparing burdens and losses in pairwise contests, to ensure that there are not great disparities in interests. And, with a suitably broad notion of what it means for individual-level burdens and expected losses to be in equipoise, we can combine a Hand formula grounded in considerations of impartiality with a Hand formula grounded in considerations of utility.

This modest yet consequential revision in our understanding of the formula has several virtues. First, it offers a normative defense of the formula based on a broader range of moral values and leaves open how evaluative trade-offs between, e.g., equality and utility, should ultimately be settled. That is, the proposal presents the formula as more neutral on contested questions concerning the ethics of aggregation. Second, the disaggregated Hand formula is perfectly compatible with Learned Hand’s original discussion and application of the formula, and closer, also, to the judgment of jurors in famous and important negligence cases. And third, the disaggregated Hand formula is vital for understanding some of the most challenging and high-stakes disputes concerning risk and safety facing the court system today, in addition to alternative regulatory schemes for managing risk besides tort liability. That is, the framework bears not just on the question of which risks are legally or morally defensible, but also on broader questions of ideal institutional design. While it does not presume to solve all the challenges associated with using the Hand formula in practice and in theory, the proposal represents a necessary step in the right direction.

Appendix

A standard setup in the economic analysis of negligence involves two persons, an injurer and victim, both of whom can take some degree of care to mitigate the risk of injury.[101] Let $$B_i\in [0,\infty)$$ for $$i = I, V$$ represent the amount spent on risk mitigation or “burdensome care” by the parties. Let L represent the injury cost. We can assume that the probability of the injury is a twice-differentiable, strictly-convex function of care levels: $$p(b_Ib_V)$$. Writing $$p_i$$ for $$\frac { \partial p }{ \partial b }$$ and  $$p_{ii} = \frac { \partial ^2p }{ \partial b_i^2}$$, we assume $$p_I<0$$,$$p_V<0$$, $$p_{II}>0$$, and $$p_{vv}>0$$. This is a standard assumption in economic theory, and in plain English, it means that the probability of the injury decreases continuously with increases in the care of injurer or victim, and the rate of decrease slows with additional care. It captures the intuitive thought that the ‘return’ from additional care is diminishing.

The generalization of the aggregative Hand formula for this standard setup is well known.[102] Utility demands individual care levels that minimize the aggregate cost function, $$b_i + b_v + p(b_ib_v)L$$, which is just the sum of the costs of care borne by the individuals and the risk-adjusted loss. Differentiating with respect to $$b_I$$ and $$b_v$$, we get first-order conditions for a minimum, $$p_I(b_I,b_v)L = -1$$ and $$p(b_I,b_v)L = -1$$, equations that aggregate-cost-minimizing levels of $$b_I$$  and $$b_V$$ must satisfy. Given our assumptions and for reasons that need not detain us, the second-order condition for a minimum $$p_{II}p_{VV} - (p_{IV})^2 > 0$$ is satisfied, and so solving the equations simultaneously gives us unique aggregate-cost-minimizing care levels, which we denote $$b_I^*$$ and $$b_V^*$$  and assume to be non-zero.

It is well established that a tort scheme that imposes liability on the injurer for failure to take the aggregate-cost-minimizing level of care and otherwise lets the loss due to the injury fall entirely on the victim ensures that rational parties take optimal care, whether they move simultaneously or sequentially, assuming they are fully rational and choose their care levels knowing others will behave rationally based on their own payoff structure.[103] The injurer’s ultimate burden ends up being $$b_I^2$$ while the victim’s burden is $$b_v^2$$ Here, as before, the outcome will sometimes involve grossly disproportionate burdens on injurer and victim.

By contrast, New Zealand’s approach to a range of injuries is to compensate victims through a taxpayer-funded social compensation fund. There are virtually no inquiries into the injurer’s or victim’s failure to take appropriate care. In modeling this scenario, I shall rely on the same setup as above, but with a few key differences. One way to represent the scheme is to suppose that the victim’s burden due to the injury is compensated by the state, and the burden of compensation is redistributed to the two parties equally through taxes based on the ability to pay.[104] In other words, the cost associated with the risk of injury ultimately falls equally on both sides, such that the injurer’s total cost function is $$b_I + \frac {p(b_I,b_v)L} {2}$$, the sum of their cost of care and share of the tax burden, while the victim’s is $$b_v + \frac {p(b_I,b_v)L} {2}$$.[105]

In the New Zealand scheme, there is no tort liability scheme to impose any incentives on either party to adopt any particular care level or minimize aggregate social costs. However, the injurer and victim each have an incentive to minimize their individual cost functions. This is the only incentive the parties are subject to, knowing that the other is subject to the same incentive. The choice of care levels can be modelled as a strategic game, where the injurer’s “best response function” for different values of $$b_v$$ is given by the derivative of the injurer’s individual cost function with respect to $$b_I$$ set to zero, holding $$b_v$$ constant: $$p_I((b_I,b_v)L = -2$$. Likewise, the victim’s best response function is given by $$p_v((b_I,b_v)L = -2$$. This game has a unique pure Nash equilibrium, found by solving the two equations simultaneously, giving us a pair of choices, $$b_I^\dagger $$ and $$b_v^\dagger$$, where neither the injurer nor the victim can do better through unilateral changes in care. The rationale for solving simultaneously is that we are looking for a pair of strategies that are reciprocally best responses, and a unique solution is guaranteed by the fact that the pair of equations solved simultaneously must give us the unique minimizers of the function $$b_I + b_v + \frac {p(b_I,b_v)L} {2}$$. The fact that there is a unique minimizer follows from our previous assumptions about the probability function.

We can now compare the New Zealand scheme, where we assume care levels converge to the Nash equilibrium, to a hypothetical U.S.-based tort scheme with liability rules designed to ensure the injurer and victim both take aggregate-cost-minimizing care levels $$b_I^*$$ and $$b_v^*$$ with the cost of injury falling entirely on the victim when they do.[106] By definition, the New Zealand scheme results in an increase in aggregate costs in violation of utility. But there is a gain in impartiality/equality, consistent with the disaggregated Hand formula, for an important range of cases. Impartiality/equality demands minimizing the difference in costs borne by the injurer and victim. The smaller the difference, the better the outcome from an impartial/egalitarian point of view. And the magnitude of the difference is given by the following equations:

United States: |$$b_I^*-(b_v^*+p(b_I^*,b_v^*)L)$$|

New Zealand: |$$(b_I^\dagger+\frac {p(b_I^\dagger ,b_v^\dagger )L} {2})-(b_v^\dagger+\frac {p(b_I^\dagger ,b_v^\dagger )L} {2})$$| = |$$b_I^\dagger-b_v^\dagger$$|

The outcome in New Zealand is impartially favored iff $$|b_I^\dagger-b_v^\dagger|<|b_I^*-(b_v^*+p(b_I^*,b_v^*)L)|$$.

We can now highlight an important result. Where p is symmetric in the injurer’s and victim’s care levels (that is, $$p(b_I,b_v)L=p(b_v,b_I)L$$ for all values of $$b_I$$ & $$b_v$$), there must be an impartiality gain in New Zealand since the injurer’s care level will be identical to the victim’s in both the United States and New Zealand. The claim that $$b_I^\dagger=b_v^\dagger$$ and $$b_I^*-b_v^*$$ when the probability function is symmetric follows from the fact that the pairs uniquely minimize $$b_I+b_v+\frac {p(b_I,b_v)L} {2}$$ and $$b_I+b_v+p(b_I,b_v)L$$ respectively. If (per impossibile) the optimum/equilibrium care levels were unequal, then, given symmetry, the functions would not have unique minima. To say more about the impartiality gain (and aggregate welfare loss), we would need to further specify the probability function. However, we can roughly glean from the model that the closer the equilibrium care levels and the higher the expected cost associated with the risk, the more likely it is that the outcome in New Zealand is favored by impartiality. In short, where I and V are more or less interchangeable in terms of the effectiveness of their care, their care levels will be similar in equilibrium, which, combined with a large $$p(b_I^*,b_v^*)L$$ value, entails outcomes in New Zealand that are more defensible from an impartial point of view than outcomes in the United States.

For a practical application, consider the case of automobile accidents, where the expected disvalue of the risk of injury may be ineliminably high, but relative efficiencies in taking care across drivers can be assumed to be comparable. Here, the New Zealand scheme seems more likely to distribute burdens impartially than the (hypothetical) U.S. scheme, assuming that parties behave rationally and liability in the United States is only imposed on drivers who opt for less-than-utilitarian care levels.

Copyright © 2026 Emad H. Atiq, Professor of Law and Philosophy, Cornell Law School & the Sage School of Philosophy. An earlier draft of this Article was presented at the North American Workshop on Private Law Theory held at UCLA Law School. I am grateful to questions and comments from Colin Bradley, Mala Chatterjee, Courtney Cox, Mark Gergen, Larissa Katz, Jeff Helmreich, Bob Hillman, Alexi Lahav, Seana Shiffrin, Manish Oza, Jed Stiglitz, Rebecca Stone, Brad Wendel, and Ben Zipursky. Thanks especially to Ken Simons, John Oberdiek, Steward Schwab, Jeff Rachlinski, Greg Keating, Rachel Schutz, and Lewis Kornhauser for their exceptionally helpful feedback. Thanks, finally, to the editors of the California Law Review for their thoughtful and incisive feedback on the piece.

             [1].      United States v. Carroll Towing, 159 F.2d 169, 173 (2d Cir. 1947).

             [2].      See generally Grimshaw v. Ford Motor Co., 174 Cal. Rptr. 348 (Ct. App. 1981). See also Gary T. Schwartz, The Myth of the Ford Pinto Case, 43 Rutgers L. Rev. 1013, 1023–26 (1991) (describing Ford’s calculations); Barbara H. Fried, Facing Up to Risk, 10 J. Legal Analysis 175, 176 (2018) (suggesting that the Ford Pinto case stands for “all that is wrong with cost/benefit analysis”); Edward C. Lyons, Balancing Acts: Intending Good and Foreseeing Harm—The Principle of Double Effect in the Law of Negligence, 3 Geo. J.L. & Pub. Pol’y 453, 472 (2005) (discussing the case).

             [3].      See generally Liebeck v. McDonald’s Rests., P.T.S., Inc., No. CV-93-02419, 1995 WL 360309 (D.N.M. Aug. 18, 1994). See also Lyons, supra note 2, at 472 (discussing the case).

             [4].      Calles v. Scripto-Tokai Corp., 832 N.E.2d 409, 415–16 (Ill. App. Ct. 2005), aff’d, 864 N.E.2d 249 (Ill. 2007). Calles drew on studies and expert testimony from related cases. See, e.g., Robins v. Kroger Co., 80 S.W.3d 641, 645–46 (Tex. App. 2002); see also $3.5 Million Settlement for Family of Girl Killed in Fire Started by Utility Lighter, Corboy & Demetrio (Aug. 2010), https://www.corboydemetrio.com/newsroom-news-Settlement-for-Family-Girl-Killed [https://perma.cc/AKQ9-M22M].

             [5].      Calles, 832 N.E.2d at 412.

             [6].      See Richard A. Posner, A Theory of Negligence, 1 J. Legal Stud. 29, 33 (1972); see also William M. Landes & Richard A. Posner, The Economic Structure of Tort law 1 (1987) (arguing that the common law of torts promotes efficiency and aggregate-cost minimization); Heidi M. Hurd, Is It Wrong to Do Right When Others Do Wrong? A Critique of American Tort Law, 7 Legal Theory 307, 307 (2001) (suggesting that it is “obvious . . . that the Hand Formula appears to allow rights violations in the name of utility or wealth maximization.” (footnote omitted)); id. at 312 (suggesting that “the Hand Formula is patently utilitarian”). For other interpretations of the Hand formula involving aggregation with the aim of cost minimization or utility maximization, see Allan M. Feldman & Jeonghyun Kim, The Hand Rule and United States v. Carroll Towing Co. Reconsidered, 7 Am. L. & Econ. Rev. 523, 530–31 (2005) (applying the formula in contexts where injurers and injured parties can be expected to act unreasonably).

             [7].      See generally In re Abbott Lab’ys, et al., Preterm Infant Nutrition Prods. Liab. Litig., No. 22 C 2011, 2023 WL 8527415 (N.D. Ill. Dec. 8, 2023). See also In re Recalled Abbott Infant Formula Prods. Liab. Litig., 97 F.4th 525, 532 (7th Cir. 2024) (affirming dismissal of plaintiffs’ claims against Abbott for “economic harm” caused by the sale of contaminated infant formula).

             [8].      For critical perspectives on the Hand formula, see generally Gregory Keating, Reasonableness and Rationality in Negligence Theory, 48 Stan. L. Rev. 311 (1996); Benjamin Zipursky, Sleight of Hand, 48 Wm. & Mary L. Rev. 1999 (2007); Patrick J. Kelley, Who Decides? Community Safety Conventions at the Heart of Tort Liability, 38 Clev. St. L. Rev. 315, 343–44 (1990); Patrick J. Kelley & Laurel A. Wendt, What Judges Tell Juries About Negligence: A Review of Pattern Jury Instructions, 77 Chi.-Kent L. Rev. 587, 591 (2002); Stephen Gilles, The Invisible Hand Formula, 80 Va. L. Rev. 1015 (1994); Richard W. Wright, The Standards of Care in Negligence Law, in Philosophical Foundations of Tort Law 249, 250 (David G. Owen ed., 1995); Richard W. Wright, Hand, Posner, and the Myth of the “Hand Formula,” 4 Theoretical Inquiries L. 145 (2003); Heidi Li Feldman, Prudence, Benevolence, and Negligence: Virtue Ethics and Tort Law, 74 Chi.-Kent L. Rev. 1431 (2000). For a general critique of the economic theory of law, see generally Jules L. Coleman, Economics and the Law: A Critical Review of the Foundations of the Economic Approach to Law, 94 Ethics 649 (1984). Most critics, apart from Ben Zipurksy, acknowledge that the Hand formula has a role to play in American tort law, even if it does not fully capture the law of negligence. While I agree with many of their criticisms, I shall argue that skepticism about the Hand formula is at least partly explained by the fact that the formula’s moral foundations and correct generalization have been misunderstood. See infra Parts I, II.

             [9].      One way to respond to such criticisms is to limit the application of Hand’s formula to appropriate cases—e.g., cases that involve measurable or purely financial losses. Additionally, there are bound to be cases where express consent is morally required before a risk is imposed—such as when potential victims of injury have a basic moral or legal right to deliberate for themselves whether to accept the risk. See generally David McCarthy, Rights, Explanation, and Risks, 107 Ethics 205, 215–17 (1997). Such cases cannot be decided using the Hand formula alone. That said, not all risks of injury inflicted on others are wrongful just because the injured party did not consent to the risk. Many socially beneficial activities that make modern life possible involve imposing risks without obtaining advance consent—for instance, driving a car. Presumably, the Hand formula has some application in such cases. Finally, the cases where I think the formula is most relevant are ones where a defendant adverts to the risk ex ante. We can distinguish such cases from entirely inadvertent risk imposition—e.g., Sam fails to advert to the need to check a monitor reading while operating on Susan and so risks Susan’s life, but his failure to advert is not traceable to a deliberate choice he made at some previous hour not to care. For discussion of such cases, see generally Gideon Rosen, The Problem of Pure Negligence, in Agency, Negligence and Responsibility 15 (Veronica Rodriguez-Blanco & George Pavlakos eds., 2021). In sum, this Article’s analysis is especially relevant to a specific class of negligence cases: where the risk is known or adverted to ex ante, where burdens and losses can be appraised at least roughly in monetary terms, and where fundamental rights are not at stake.

          [10].      See, e.g., Blackford v. Wal-Mart Stores, Inc., No. 07-437-GPM, 2008 WL 905912, at *3 (S.D. Ill. Apr. 2, 2008) (comparing “the cost to Wal-Mart to install a childproof starting mechanism on the buffer” against “the expected loss caused by an injury”); id. at *2 (“[I]t is clear that the cost to Wal-Mart to take all of the precautions that would be necessary to render a store . . . absolutely safe to a 2½ year old child is beyond the cost of the loss that could be expected from any given instrumentality on which a child may be injured, especially when that cost is discounted by the likelihood of injury.”); Nelson v. Sunbeam Prods., Inc., 579 F. Supp. 3d 857, 866 (E.D. Tex. 2022) (“Whether a design defect renders a product unreasonably dangerous depends on whether the product’s risk outweighs its utility . . . .”); id. at 871 (comparing “the cost of the alternative design” for the Sunbeam heater against the expected benefits in terms of safety); Genie Indus., Inc. v. Matak, 462 S.W.3d 1, 9–10 (Tex. 2015) (employing a “risk-utility” comparison to determine whether a product is “unreasonably dangerous”); Bugosh v. I.U.N. Am., Inc., 971 A.2d 1228, 1248 (Pa. 2009) (noting the importance of evaluating whether “the safety benefits of altering the product’s design in a particular manner would foreseeably have exceeded the cost of the alteration”); see also Stephen G. Gilles, On Determining Negligence: Hand Formula Balancing, the Reasonable Person Standard, and the Jury, 54 Vand. L. Rev. 813, 815 (2001) (“[T]he Hand Formula balancing approach is recognized as authoritative by judicial opinions in a majority of states, by the leading torts treatises, and by most contemporary tort scholars.”).

          [11].      See sources cited supra note 10.

          [12].      It is well-known that the traditional interpretation is based on a utilitarian logic of aggregation. See, e.g., Hurd, supra note 6, at 312; see also sources cited supra note 8. But the precise problems with aggregation have not been articulated in terms of the values of equality and impartiality. There have been major developments in moral philosophy on the ethics of aggregation in recent years. Yet none of this work has been applied to the Hand formula and related concepts in the economic theory of accidents. This Article draws on these developments in moral philosophy as well as the author’s own work on the nature of impartiality. See sources cited infra note 14. On the limits of aggregation, see, for example, Joe Horton, Aggregation, Complaints, and Risk, 45 Phil. & Pub. Affs. 54, 54 (2017) (“Aggregative views tend to have counterintuitive implications in cases in which we must choose between imposing large burdens on each of a few people and imposing small burdens on each of many.”); Alex Voorhoeve, How Should We Aggregate Competing Claims, 125 Ethics 64, 66 (2014) (defending a principle of aggregation that is sensitive to the relative strength of individual claims); Michael Otsuka, Risking Life and Limb: How to Discount Harms by Their Improbability, in Identified Versus Statistical Lives: An Interdisciplinary Perspective 77 (I. Glenn Cohen, Norman Daniels & Nir Eyal eds., 2015) (arguing against aggregation when the probabilistically discounted harms on one side are morally trivial); Johann Frick, Contractualism and Social Risk, 43 Phil. & Pub. Affs. 175, 175–78 (2015) (summarizing the problems with aggregation when harms and benefits are uncertain); Seth Lazar, Limited Aggregation and Risk, 46 Phil. & Pub. Affs. 117, 117–24 (2018) (defending a principle of limited aggregation based on the duty to rescue).

          [13].      See, e.g., Tsachi Keren-Paz, Egalitarianism as Justification: Why and How Should Egalitarian Considerations Reshape the Standard of Care in Negligence Law?, 4 Theoretical Inquiries L. 275, 289 (2003) (“The phenomenon of diminishing marginal utility clearly suggests that the parties’ prior holdings are relevant for purposes of determining loss-bearing capacity.”); Keating, supra note 8, at 384.

          [14].      On the nature of impartiality, see, for example, Thomas Nagel, Equality and Partiality 38 (1995). My own view is that impartiality is best understood as “identity-independent” and non-arbitrary concern for others’ wellbeing that is a requirement of epistemic virtue. See Emad H. Atiq, How to Be Impartial as a Subjectivist, 173 Phil. Stud. 757, 767 (2016) [hereinafter Atiq, How to Be Impartial]; Emad H. Atiq, Knowledge by Acquaintance and Impartial Virtue, 182 Phil. Stud. 911, 934 (2025). One reason to consider a variety of moral values in our interpretation of the formula is that accommodating a broad range of moral perspectives is legally desirable. On this point, see Kenneth W. Simons, The Hand Formula in the Draft Restatement (Third) of Torts: Encompassing Fairness as Well as Efficiency Values, 54 Vand. L. Rev. 901, 921–23 (2001) (noting that the Hand formula can be understood in terms of aggregate preference satisfaction as well as wealth-maximization); id. at 925 (suggesting that the Restatement’s characterization of the Hand formula should be “accommodating and general”).

          [15].      I discuss scenarios of special relevance to tort law. But for a general discussion of aggregation problems in situations of uncertainty, see, for example, Lazar, supra note 12, at 117–24. While I share Lazar’s view that simple aggregation in the relevant scenarios leads to choices that are intuitively unfair, my proposal for how we should think about risk in such contexts differs in important ways. See infra Part I. The proposal I defend is grounded in considerations of impartiality and equality and draws on my own work on the nature of these values. See Atiq, How to Be Impartial, supra note 14, at 767. For Lazar, the key consideration is the duty to rescue. Another difference is that I offer a precise account of how reasons of utility maximation might be weighed against more individualistic impartial reasons. See infra Part II.

          [16].      See generally, e.g., Nagel, supra note 14; T.M. Scanlon, What We Owe to Each Other (1998); F.M. Kamm, Aggregation and Two Moral Methods, 17 Utilitas 1 (2005).

          [17].      See sources cited supra note 12.

          [18].      A key part of the instruction might read (roughly): “If the highest expected loss suffered by any single person due to the risk was substantially greater than the highest burden that would have been borne by any single individual if the defendant had taken care to avoid the risk, that factor counts in favor of the defendant’s negligence. If the opposite is true, it counts against the defendant’s negligence.” Much of the Article is devoted to refining, supplementing, and defending this instruction.

          [19].      See, e.g., Keating, supra note 8, at 331; Zipursky, supra note 8, at 2006; Heidi M. Hurd & Michael S. Moore, Negligence in the Air, 3 Theoretical Inquires L. 333, 359–65 (2002); Gilles, supra note 8, at 848 (“[I]t seems all but impossible to reconcile Hand’s position . . . with Richard Posner’s wealth-maximization interpretation of the Hand Formula.”). Keating develops several important objections to the utilitarian Hand formula but does not offer an alternative moral framework for the formula. Keating, supra note 8, at 331 (“The real significance of the Hand Formula, then, is not technical, but conceptual: It isolates the elements of due care and the relations among them.”). Zipursky, likewise, rejects the need for an account of the formula based on an alternative moral theory. Zipursky, supra note 8, at 2006 (“I do not believe that the refutation of the assertion that the Hand Formula captures the meaning of ‘negligence’ requires some other theory.”).

          [20].      See In re Abbott Lab’ys, et al., Preterm Infant Nutrition Prods. Liab. Litig., No. 22 C 2011, 2023 WL 8527415, at *1 (N.D. Ill. Dec. 8, 2023) (“This case is one of hundreds consolidated by the Judicial Panel on Multidistrict Litigation (‘JPML’) and pending before this court, in which plaintiffs have alleged that infant formula manufactured by defendants caused necrotizing enterocolitis (‘NEC’) in babies born prematurely.”).

          [21].      Brief and Appendix of Plaintiffs-Appellants at 16, In re Recalled Abbott Infant Formula Prods. Liab. Litig., 97 F.4th 525 (7th Cir. 2024) (No. 23-2525). The “price premium” argument has been developed primarily by plaintiffs alleging contamination that may have caused sepsis and meningitis in infants. These particular lawsuits are associated with Abbott’s recalled baby formula.

          [22].      Id. at 17; see also Brief of Defendant-Appellee Abbott Laboratories at 3, In re Recalled Abbott Infant Formula Prods. Liab. Litig., 97 F.4th 525 (No. 23-2525).

          [23].      See, e.g., Abbott’s Memorandum of Law in Support of its Motion for Summary Judgment at 39, Willoughby v. Abbott Lab’ys, No. 22-cv-01322 (N.D. Ill. Feb. 26, 2025) (“The implied warranty of merchantability ‘does not mean that the product will fulfill a buyer’s every expectation but rather simply provides for a minimum level of quality.’” (quoting Rudy v. D.F. Stauffer Biscuit Co., 666 F. Supp. 3d 706, 721 (N.D. Ill. 2023))).

          [24].      The argument relies on some basic formal machinery standardly employed in economic models of negligence and tort liability. See generally Louis Kaplow & Steven Shavell, Fairness Versus Welfare, 114 Harv. L. Rev. 967 (2001); Jeonghyun Kim, Revisiting the Learned Hand Formula and Economic Analysis of Negligence, 169 J. Institutional & Theoretical Econ. 407 (2013); Samuel A. Rea, Jr., The Economics of Comparative Negligence, 7 Int’l Rev. L. & Econ. 149 (1987). On New Zealand’s approach to negligence, see Peter H. Schuck, Tort Reform, Kiwi-Style, 27 Yale L. & Pol’y Rev. 187, 194 (2008).

          [25].      This is a choice of convenience. No one supposes that Hand’s formula forces a monetary measure of harm. The choice is defensible where monetized costs and benefits are the best proxy for effects on individual wellbeing, and where they are not, one needs an alternative measure. Additionally, institutions that measure the moral significance of burdens in dollars and cents must make some effort to account for differences in background endowments, since the effect of a $1,000 loss on a person’s wellbeing varies with their income level. The Hand formula has been criticized on these grounds, and the points are well rehearsed. See, e.g., Keating, supra note 8, at 335–41. While I, too, adopt the convention of describing the burdens associated with risk in monetary terms, the choice should be construed as an artificial and simplifying assumption that wellbeing effects are measurable and interpersonally comparable on a ratio scale.

          [26].      Another simplifying assumption I’ll share with Hand is that parties are risk-neutral, so that in assessing outcomes under uncertainty, all we care about is the expected value of the outcome, not the associated level of risk. This is an extreme idealization that we know is false to the facts. People are not generally risk neutral, and they tend to be especially averse to extreme risks involving very bad consequences of low probability. Moreover, interpersonal variation in risk-tolerance gives rise to difficult and pressing questions of social policy, and yet we are abstracting away from this important complication. Still, such simplifications are defensible in theory. In practice, we must complicate the analysis as I explain in Part III.

          [27].      United States v. Carroll Towing Co., 159 F.2d 169, 173 (2d Cir. 1947) (“Since there are occasions when every vessel will break from her moorings, and since, if she does, she becomes a menace to those about her; the owner’s duty, as in other similar situations, to provide against resulting injuries is a function of three variables: (1) The probability that she will break away; (2) the gravity of the resulting injury, if she does; (3) the burden of adequate precautions. Possibly it serves to bring this notion into relief to state it in algebraic terms: if the probability be called P; the injury, L; and the burden, B; liability depends upon whether B is less than L multiplied by P: i.e., whether B < PL.”). For further discussion of Hand’s reasoning, see infra Part III.

          [28].      On why the question of “reasonable” care in negligence law is a fundamentally normative/moral question, see Emad H. Atiq, Legal vs. Factual Normative Questions & the True Scope of Ring, 38 Notre Dame J.L., Ethics & Pub. Pol’y 101, 128–33 (2018). On normative questions in law generally, see Emad H. Atiq, Reasonable Moral Doubt, 97 N.Y.U. L. Rev. 1373, 1382–401 (2022).

          [29].      On utility maximization as underlying the economist’s concern with minimizing the “costs of accidents,” see discussion in Richard A. Posner, Wealth Maximization and Tort Law: A Philosophical Inquiry, in Philosophical Foundations of Tort Law 99, 99–100 (David G. Owen ed., 1995).

          [30].      In a sense, the utilitarian treats all agents “equally.” But impartiality does not entail utility or single out any precise identity-independent scheme for promoting people’s interests. In fact, as I’ll argue shortly, utility entails choices that one would not choose if one were to suffer the negative consequences of the choice, and so there is a clear sense in which utility is inconsistent with the demands of impartiality.

          [31].      On the significance of the impartial point of view for moral theory, see, for example, Nagel, supra note 16, at 64–65 (“The impartial attitude is . . . strongly egalitarian both in itself and in its implications. As I have said, it comes from our capacity to take up a point of view which abstracts from who we are, but which appreciates fully and takes to heart the value of every person’s life and welfare.”); Bernard Gert, Moral Impartiality, 20 Midwest Stud. Phil. 102, 118 (1995) (“Since morality requires impartially obeying the moral rules, any justified violation of a moral rule must be one that counts as impartially obeying the rule.”); Atiq, How To Be Impartial, supra note 14, at 767 (explaining impartiality in terms of “identity-independent desires”); Roderick Firth, Ethical Absolutism and the Ideal Observer, 12 Phil. & Phenomenological Rsch. 317, 335–39 (1952) (explaining impartiality in terms of insensitivity to arbitrary features).

          [32].      Nagel, supra note 16, at 64–65.

          [33].      See sources cited supra note 31.

          [34].      Although they do not discuss the issue in terms of impartiality (or aggregation), several writers have discussed the usefulness of the “single owner” principle for figuring out what a reasonable person should do which involves assuming that the risk-imposer bears both the costs and the benefits of the risk-avoidance measure. The heuristic is useful as a mechanism for imagining how agents would act if they internalized externalities. See, e.g., Richard A. Epstein, Holdouts, Externalities, and the Single Owner: One More Salute to Ronald Coase, 36 J.L. & Econ. 553, 578 (1993) (“Thus, the strict liability standard forces the actor to treat the losses that are imposed as though they were his own. Therefore, it leads to an internalization of the losses and to the right set of incentives.”); Gilles, supra note 8, at 1032–37 (1994); Kenneth W. Simons, Deontology, Negligence, Tort, and Crime, 76 B.U. L. Rev. 279, 282 (1996). Another reason that the heuristic is useful is that it helps us identify a choice that does not involve unfair privileging of oneself over others.

          [35].      For the time being, we will continue to treat the choice of precaution as discrete as well as unilateral—those affected by the risk of injury are not able to take their own risk-eliminating measures.

          [36].      Cf. Nagel, supra note 16, at 65 (“The result [of the impartial point of view] is an enormous set of values deriving from individual lives, without as yet any method of combining them or weighing them against one another when they conflict, as they inevitably will in the real world.”).

          [37].      On the limits of aggregation, see sources cited supra note 12; John Rawls, A Theory of Justice 501 (original ed. 1971) (explaining that “the idea of maximizing the aggregate of well-being, or of attaining the greatest perfection, is vague and amorphous”).

          [38].      One might object that the assumption that the burdens fall entirely on the shareholders is unrealistic. In fact, even if we assume that some of the burden is borne by the corporation’s customers (through higher prices), it should not change our moral conclusions about the case so long as there are sufficiently many customers and the burdens are “spread out.”

          [39].      The law is often willing to treat corporate entities as individuals in their own right, a distraction we can avoid by modifying the case so that the 100 injurers are a mere plurality, collectively deciding whether to inflict the risk. Either way, the corporate form does not affect the moral dynamics. Note that concepts in corporate law do not legally constrain us in our application of the Hand formula or, for that matter, concepts of negligence law. One of the key points I will be pressing shortly is that an impartial generalization of Hand’s formula requires looking past the corporate form to the natural persons who bear the burdens associated with risk and risk avoidance. Stewart Schwab points out that treating corporate entities as equivalent to natural persons might be another simplification of the Hand formula. But it certainly is not entailed by anything Learned Hand or any of his defenders have ever said. Moreover, simplifying assumptions must be justified in terms of their practical payoff. Treating corporations as natural persons for purposes of applying Hand’s formula distorts our analysis on the question of negligence, at least if the animating principle is impartiality as I explain further below.

          [40].      On the importance of abstracting away contingent features of who we are in testing our fairness intuitions, see Rawls, supra note 37, at 118–92.

          [41].      One example might be requiring homeowners to improve a sidewalk to make it easier for pedestrians to walk comfortably. I think these cases are best analyzed as institutional expectations imposed not with an eye to preventing pre-legal moral wronging, but aggregate welfare. At the very least, the pre-legal moral wrongs in such cases are not obvious. Thanks to Jeff Rachlinski for discussion on this point.

          [42].      It would be helpful throughout to conceive of potential injurers not as necessarily malignant forces in the world, driven to expose others to risk for its own sake. Instead, we should conceive of them as persons whose pursuit of happiness creates risks for others for reasons over which they had no direct control and that they can only mitigate at a cost to themselves. Viewed from this lens, and even if the relevant form of luck is non-moral preference luck, I am not sure that Kima is necessarily unreasonable for being unwilling to bear the burden of risk avoidance. That said, I do not mean to suggest that the right moral verdict in such cases is clear. For example, one consideration that may pull in the direction of finding Kima negligent is the fact that she causes a physical injury (even if only a papercut), and perhaps one should always take on substantial financial burdens to avoid impinging on others’ physical integrity. Of course, that is not the reason why the utilitarian Hand formula convicts Kima of moral error. Moreover, we can change the case so that the $1 injury is purely financial. The main point stands that the utilitarian formula aggregates interests in a way that gets the wrong moral verdict in certain cases and, at the very least, occludes moral complexity. Thanks to Rebecca Stone for a helpful discussion on this point.

          [43].      For the relevance of background inequality, see discussion in Keren-Paz, supra note 13, at 286 (“From this perspective, the sufficiency of the precautions taken by the defendant should be evaluated against the background of the existing distribution of wealth and power since such distribution bears on the actor’s agency. The enormity of the burden imposed by a requirement to act in a certain way should affect the moral evaluation of non-compliance with the requirement.”).

          [44].      See, e.g., Scanlon, supra note 16, at 235 (“To claim that there is a tie in such a case [where one person suffers a much greater harm than many others experiencing minor inconvenience] would be already to claim that the fact that there are more people in one group makes it reasonable to reject a principle requiring one to help the smaller number, each of whom would suffer the greater harm.”); Kamm, supra note 16, at 1.

          [45].      See generally, e.g., Thomas Nagel, Equality, in Mortal Questions 106 (1979). Scanlon writes:

If one harm, though not as serious as another, is nonetheless serious enough to be morally “relevant” to it, then it is appropriate, in deciding whether to prevent more serious harms at the cost of not being able to prevent a greater number of less serious ones, to take into account the number of harms involved on each side. But if one harm is not only less serious than, but not even “relevant to,” some greater one, then we do not need to take into account in deciding which to prevent, but should always prevent the more serious harm.

Scanlon, supra note 16, at 239–40. For further discussion, see Kamm, supra note 16, at 1 (arguing that “it is not true that we produce a better outcome if the greater number survive, as there is no impartial perspective from which to judge this [in a case where one person and a group face equal harm and must be chosen between]”). Kamm writes:

[T]he Balancing Argument claims that in a conflict, justice demands that each person on one side should have her interests balanced against those of one person on the opposing side; those whose interests are not balanced out in the larger group help determine that the larger group should be saved.

Id. at 6. For critical commentary on the Scanlon-Kamm proposal, see Michael Otsuka, Saving Lives, Moral Theory, and the Claims of Individuals, 34 Phil. & Pub. Affs. 109, 111–18 (2006); see also Joe Horton, Aggregation, Risk, and Reductio, 130 Ethics 514, 516–20 (2020).

          [46].      See Derek Parfit, Justifiability to Each Person, 16 Ratio 368, 378–79 (2003).

          [47].      For further discussion on this point, see Voorhoeve, supra note 12, at 66; see also Lazar, supra note 12, at 123–25 (discussing the relevance of the “moral weight” of interests when comparing claims). As Lazar points out, there may be other factors that determine the weight of a person’s claim or interest (e.g., their moral desert), but we can hold these other factors fixed.

          [48].      We can write 𝐵𝑖<𝐸𝑃𝑗𝐿𝑗 for “the burden on i is less than the expected disvalue of the risk to j by more than E,” 𝐵𝑖=𝐸𝑃𝑗𝐿𝑗 for “the burden on i is neither greater nor less than the expected disvalue of the risk to j by more than E”, and  for “the burden on i is greater than the expected disvalue of the risk to j by more than E.” Instead of differences in burdens, we could instead use ratios, letting  represent a ratio such that person j’s expected loss morally outweighs person i’s burden when and only when .

          [49].      What follows is a more formal statement of the decision procedure. First, we partition the affected parties into two groups. One group is the pro-risk group consisting of individuals who are more burdened than benefited by the defendant’s decision to avoid a risk. The other group is the pro-safety group, with members who are more benefited than burdened by the defendant’s avoidance of the risk. The latter will be the primary bearers of the risk of loss. Let pro-risk  be the group of individuals who would be burdened by safety overall, and pro-safety  be the group who suffer overall from the risk. For ease of discussion, let us assume that the burden of risk-avoidance falls on a set of individuals entirely distinct from the ones who suffer an expected loss from the risk, though we can easily modify this simplifying assumption as explained below. Given our equality parameter $$\in[0,\infty)$$, person  $$i\in$$pro-risk, and person $$j\in$$pro-safety, we can write $$B_i<_EP_jL_j$$ for “the burden on i is less than the expected disvalue of the risk to j by more than E” and $$B_i=_EP_jL_j$$ for “the burden on i is neither greater nor less than the expected disvalue of the risk to j by more than E.” The decision procedure is as follows:

Pairwise Hand:

Pick i$$\in$$  pro-risk who suffers the highest burden (i.e. $$B_i=max$$ {$$B_1,B_2, . . . B_n$$}), and pick j$$\in$$ pro-safety for whom the expected harm is highest (i.e. $$P_jL_j=max{P_1L_1,P_2L_2, . . . P_mL_m}$$). If $$B_i<_EP_jL_j$$ choose safety. If $$B_i>_EP_jL_j$$, impose the risk. If $$B_i=_EP_jL_j$$, then proceed to the next step. If you’ve run out of members of one group (pro-risk or pro-safety) but not both, then there is a reason to take the choice that favors the remaining members (if their interest is $$>_E$$ than 0). Otherwise, pick the person who bears the next greatest burden associated with safety and the person who bears the next greatest expected loss from the risk and follow the procedure in Step 1 for them. Keep going until a choice is made or there are rough ties all the way down. In the latter case, the pairwise contests do not deliver a verdict.

Note that if we relax our simplifying assumption and allow the set of individuals who bear the burdens of risk-avoidance to overlap with the set who bear a risk of loss, we would need to measure the net expected costs borne by each person—that is, |$$𝐵_𝑖-𝑃_𝑖𝐿_𝑖$$| for person i—before allocating them to the pro-risk and pro-safety groups, and compare net expected costs in the pairwise contests. While the notation becomes slightly more cumbersome, the steps remain the same.

          [50].      All persons care about in our stylized cases is the expected harm. Of course, comparing harms across distinct individuals impartially once we build in risk aversion becomes more complicated (since individuals vary in their tolerance of risk). Arguably, our measure of each person’s burden would need to be adjusted based on their degree of risk of aversion. We are sidestepping this complication just as utilitarians and economists do in interpreting the formula. Bear in mind that this Article’s aim is not to perfect the Hand formula, but to offer a compelling alternative interpretation to the standard utilitarian one.

          [51].      Cf. Frick, supra note 15, at 212–23.

          [52].      It would be helpful to have a generalization of Hand’s formula informed by both utility and impartiality given plausible assumptions about moral uncertainty. It is tempting to suppose that when we are uncertain about which of two competing moral principles is correct, we should split our credence and consider inter-theoretic tradeoffs: for example, when there are strong reasons for choosing an outcome based on one principle but only weak disfavoring reasons based on the other. For the general conditions under which first-order moral theories allow for inter-theoretic comparison, see generally William MacAskill, Krister Bykvist & Toby Ord, Intertheoretic Comparisons of Choice-Worthiness, in Moral Uncertainty 112 (2020). Their framework is easily adapted to cases involving uncertainty about moral principles. I assume a degree of comparability between the choices recommended by aggregate comparisons and pairwise comparisons, as I go on to explain.

          [53].      Again, a simplified jury instruction can be developed: The following conditions favor a finding that the defendant was negligent:

the highest expected loss suffered by any single person was substantially greater than the highest burden due to risk prevention that would have been borne by any single individual had the defendant taken care;

or

the total burden of care was less than the total expected cost of the injury and the highest burden borne by any single individual due to risk-prevention had the defendant taken care would not have been substantially greater than the highest expected cost of injury suffered by any single person.

          [54].      Is my reliance on the utility principle, here, inconsistent with my earlier suggestion that impartiality is a more plausible moral principle than utility? Not quite, because the reasons I am incorporating utility are not “first-order”—that is, they aren’t based on my commitment to impartiality. My reasons are “second-order”—that is, they are based on some residual uncertainty about whether impartiality or utility is the right moral principle in this context. Any moral theory that acknowledges moral uncertainty needs to take such higher-order reasons into account. On this point, see MacAskill, Bykvist & Ord, supra note 52, at 127.

          [55].      United States v. Carroll Towing Co., 159 F.2d 169, 173 (2d Cir. 1947) (emphasis added).

          [56].      Id. at 173–74 (emphasis added).

          [57].      Id. At one point, Hand observes that “the likelihood that a barge will break from her fasts and the damage she will do, [will] vary with the place and time; for example, if a storm threatens, the danger is greater; so it is, if she is in a crowded harbor where moored barges are constantly being shifted about.” Id. at 173. But this does not constitute evidence that Hand was aggregating multiple injuries, since he was probably referring to the probability and size of a singular injury varying depending on “place and time.” The main point is that he did not aggregate the dangers in applying his formula.

          [58].      178 F.2d 148, 149 (2d Cir. 1949).

          [59].      Id.

          [60].      Id. (“It is indeed possible to state an equation for negligence in the form, C equals P times D, in which the C is the care required to avoid risk, D, the possible injuries, and P, the probability that the injuries will occur, if the requisite care is not taken. But of these factors care is the only one ever susceptible of quantitative estimate, and often that is not. . . . It assists us here to center on the factor of probability, because the difference between ‘gross’ and ‘ordinary’ negligence consists in the higher risks which the putatively wrongful conduct has imposed upon the injured person.” (emphasis added)).

          [61].      Id. It is possible that Hand did not aggregate in this case because the relevant risks and injuries are hard to estimate, as he acknowledges in the opinion. But even if that is true, the main point is that Hand was clearly focused on simple two-person cases in applying his formula. If he had a view on how the formula might be used in multiparty cases when the aggregate numbers are available, that view is not at all clear from his opinions.

          [62].      Id. at 150.

          [63].      174 Cal. Rptr. 348 (Ct. App. 1981). See generally Schwartz, supra note 2. See also Wright, supra note 8, at 263; Fried, supra note 2, at 176 (suggesting that the Ford Pinto case stands for “all that is wrong with cost/benefit analysis”).

          [64].      See Grimshaw v. Ford Motor Company, 1981, Am. Museum of Tort L. https://www.tortmuseum.org/ford-pinto/ [https://perma.cc/A8NX-8DP9].

          [65].      The report has been described as “possibly the most remarkable document ever produced in an American lawsuit.” See Schwartz, supra note 2, at 1020 (citation omitted).

          [66].      The calculation is: A common but entirely mistaken view among commentators is that Ford was using its own predicted “settlement costs” in its calculations. That is a serious misconception that distracts from the difficult moral questions raised by the case. Ford was using NHTSA’s own estimate of “life-value” that the agency uses to come up with safety standards. It was higher than what was recommended by the White House committee on auto regulation around that time. See Schwartz, supra note 2, at 1023 n.31, 1024 n.41, 1026.

          [67].      For information on how the average value of life is calculated, see, for example, U.S. Dep’t of Transp., Departmental Guidance: Treatment of the Value of Preventing Fatalities and Injuries in Preparing Economic Analyses 4 (2021) https://www.transportation.gov/sites/dot.gov/files/2021-03/DOT%20VSL%20Guidance%20-%202021%20Update.pdf [https://perma.cc/9W84-D7Z3] (“Prevention of an expected fatality is assigned a single, nationwide value in each year, regardless of the age, income, or other distinct characteristics of the affected population, the mode of travel, or the nature of the risk.”). The guidelines indicate how the life value measure varies based on different characteristics, information that is relevant for calculating $$PL_{max}$$.

          [68]. $$\sum P_iL_i = 180\times {$200,000} + 180 \times {$67,000}$$.

          [69].      The U.S. Department of Transportation reports a 0.55 percent to 2.24 percent increase in the “value of life” per 1 percent increase in earning capacity alone. U.S. Dep’t of Transp., supra note 67, at 8. That said, varying “life-value” based on earning capacity alone is morally questionable.

          [70].      Ford Motor Co. Stock Chart 1977–1979, S&P Capital IQ, https://www.capitaliq.spglobal.com/web/client?auth=inherit#company/stock?id=113873 [https://perma.cc/N49A-9EGT] (search “Ford Motor Company” in S&P Capital IQ; click [More] under Stock Price; select [Custom] to customize the date range; set the custom date range as 12/15/1977– 01/15/1979; select Column format for stock chart); Ford Motor Co. Stock Chart 1981–1982, S&P Capital IQ, https://www.capitaliq.spglobal.com/web/client?auth=inherit#company/stock?id=113873 [https://perma.cc/XY8R-EHDU] (search “Ford Motor Company” in S&P Capital IQ; click [More] under Stock Price; select [Custom] to customize the date range; set the custom date range as 01/01/1981– 01/04/1982; select Column format for stock chart); Dow Jones Industrial Average Index Chart 1977–1979, S&P Capital IQ, https://www.capitaliq.spglobal.com/web/client?auth=inherit#index/profile?id=1 [https://perma.cc/N95S-BZXB] (search “Dow Jones Industrial Average” in S&P Capital IQ; click the calendar icon in Index Chart; Set the custom date range as 12/15/1977–01/15/1979; set Mountain format for index chart); Dow Jones Industrial Average Index Charts, S&P Capital IQ, https://www.capitaliq.spglobal.com/web/client?auth=inherit#index/profile?id=1 [https://perma.cc/NG9P-FTJB] (search “Dow Jones Industrial Average” in S&P Capital IQ; click the calendar icon in Index Chart; Set the custom date range as 01/01/1981–01/04/1982; set Mountain format for index chart). Thanks to Isabel Boyer and the editors for a fruitful discussion on this point.

          [71].      To be clear, the argument does not entail that in making design decisions car manufacturers must always take the utmost precaution no matter the absolute cost. In many contexts, the extreme costs of extreme safety features will be passed on to customers, so not taking the precaution will simultaneously benefit customers (by reducing the price of the car) and burden them (by imposing a risk). It is entirely possible that a disaggregated comparison of claims in individual cases might reveal that the safety precaution is not required from the perspective of impartiality. Moreover, as I suggested in Part II, where the difference in aggregate burdens and losses is very substantial, we may be more tolerant of individual disparities in burdens.

          [72].      No. CV-93-02419, 1995 WL 360309 (D.N.M. Aug. 18, 1994); see also Lyons, supra note 2.

          [73].      See Andrea Gerlin, A Matter of Degree: How a Jury Decided that a Coffee Spill Is Worth $2.9 Million, Wall St. J., Sep. 1, 1994, at A1.

          [74].      Lyons, supra note 2, at 490 (citation omitted).

          [75].      See Calles v. Scripto-Tokai Corp., 832 N.E.2d 409, 412 (Ill. App. Ct. 2005).

          [76].      Id. (citations omitted).

          [77].      Id.

          [78].      Id. Calles drew on studies and expert testimony from related cases. See, e.g., Robins v. Kroger Co., 80 S.W.3d 641, 645–46 (Tex. App. 2002).

          [79].      In re Abbott Lab’ys, et al., Preterm Infant Nutrition Prods. Liab. Litig., No. 22 C 2011, 2023 WL 8527415, at *1 (N.D. Ill. Dec. 8, 2023).

          [80].      Plaintiffs’ Complaint and Jury Demand, R.J. v. Mead Johnson & Co., No. 1:22-cv-02011 (N.D. Ill. Mar. 4, 2022).

          [81].      In re Recalled Abbott Infant Formula Prods. Liab. Litig., No. 22 C 4148, 2023 WL 3585639, at *1 (N.D. Ill. May 22, 2023), appeal dismissed, No. 23-2233, 2023 WL 9020976 (7th Cir. July 6, 2023).

          [82].      Brendan Pierson & Dietrich Knauth, Abbott Must Pay $495 Million in Premature Infant Formula Trial, Jury Finds, Reuters (June 26, 2024), https://www.reuters.com/business/healthcare-pharmaceuticals/abbott-must-pay-95-million-premature-infant-formula-trial-jury-finds-2024-07-26/ [https://perma.cc/UZW9-MMWL].

[84].   Id.

          [84].      Plaintiffs’ Complaint and Jury Demand, supra note 80, at 17 (“The manufacturers of the premature infant formulas and fortifiers at issue in this suit owe the consuming public in general, and Plaintiffs in particular, a duty to exercise reasonable care in designing, testing, manufacturing, inspecting, and distributing a product free of unreasonable risk of harm to anyone who is likely to be exposed to the harm when the product is put to its intended use . . . .”).

          [85].      Brief of Defendant-Appellee Abbott Laboratories, supra note 22, at 30–31.

          [86].      Id. at 37 (citation omitted).

          [87].      Brief and Appendix of Plaintiffs-Appellants, supra note 21, at 16.

          [88].      Id.

          [89].      Id. at 17.

          [90].      See 1 James J. White, Robert S. Summers & Robert A. Hillman, Uniform Commercial Code: Practitioner Treatise Series § 10:29 (6th ed. 2012). Courts often point out that the terms “defective” as used in tort law and “unmerchantable” as used in contracts are not identical, and that a warranty claim may succeed where the tort claim fails, and vice versa. See generally Denny v. Ford Motor Co., 42 F.3d 106 (2d Cir. 1994). The tort standard is informed by a risk/utility calculus, whereas the contractual standard is governed by the buyer’s expectations and foreseeability.

          [91].      Nikki Multer, Sean Bond-Downey & Jennifer Cecil, Preterm Infant Formula Litigation: Key Takeaways from Recent Jury Verdicts, Hush Blackwell: Prod. Persp.: Complex Tort & Prod. L. (Sep. 11, 2024), https://www.productlawperspective.com/2024/09/preterm-infant-formula-litigation-key-takeaways-from-recent-jury-verdicts/ [https://perma.cc/9HPS-3V2S].

          [92].      Id.

          [93].      Of course, it is unfortunate that not all parents have access to safer food for their children. Income inequality has tragic consequences. But not all problems can be solved through negligence law.

          [94].      See, e.g., Feldman & Kim, supra note 6, at 542 (“The Learned Hand rule for determining negligence has fascinated students of law and economics for many decades, partly because Judge Hand wrote an algebraic expression . . . that is intuitive and attractive to economists.”).

          [95].      Id.; see also Thomas J. Miceli, The Economic Approach to Law 54 (2d ed. 2009) (“At the center of the theory is the model of precaution, which prescribes that injurers and victims should invest in accident-reducing activities up to the point where the last dollar spent on care equals the marginal savings in accident costs. The role of the law is to provide incentives in the form of liability rules for the parties to meet this standard. We argued that the law of negligence, as embodied by the Hand test, conforms well to this ideal.”); Appendix.

          [96].      See generally Geoffrey Palmer, Compensation for Incapacity (1979); Jeremy Waldron, Moments of Carelessness and Massive Loss, in Philosophical Foundations of Tort Law 387 (David G. Owen ed., 1995).

          [97].      See, e.g., Schuck, supra note 24, at 194; see also Miceli, supra note 95, at 72 (discussing the moral hazard that results from social insurance).

          [98].      See Appendix for a more formal proof based on a standard setup for modeling risk-related decision-making.

          [99].      In contexts where the risk of injury is ineliminably high even when parties take cost-minimizing care, tort liability will often have an “all or nothing” approach to distributing costs. There is no “cost-sharing” built into the scheme, setting aside doctrines like contributory negligence which are not always applicable.

        [100].      Although drivers may not take total cost-minimizing levels of care, that does not entail that rational parties who understand that a higher rate of accidents affects their own wellbeing, the public fisc, and their taxes, will not take some care.

        [101].      See sources cited supra note 24.

        [102].      See, e.g., Kim, supra note 24, at 426.

        [103].      Id.; see also Rea, supra note 24, at 150–53.

        [104].      For the purposes of modeling the New Zealand approach, I assume the injury is fully compensated. But it is not entirely clear whether awards in New Zealand (or, for that matter, in the United States through, say, worker’s compensation) are fully compensatory. But the model can be construed as an idealization. Thanks to Ken Simons for discussion on this point.

        [105].      Crucially, I assume, here, that our measure of burden is sensitive to background endowments, as it should be, which is why ability-to-pay taxation ensures equal burdens. We can generalize from the New Zealand scenario to cases where the tax burden, rather than being distributed equally, is imposed based on some such that the injurer’s costs are 𝑏𝐼+(1−𝛼)𝑝(𝑏𝐼,𝑏𝑣)𝐿 while the victim’s costs are 𝑏𝑣+𝛼𝑝(𝑏𝐼,𝑏𝑣)𝐿.

        [106].      In fact, it is almost certainly not the case that in reality U.S. tort liability is based on the utilitarian Hand formula. See sources cited supra note 8; see also supra Part III. Nevertheless, it is useful to compare New Zealand to the possible world where tort law wholly embraces the conventional aggregative approach to negligence.