In the United States, the calculus of negligence, also known as the Hand rule, Hand formula, or BPL formula, is a term coined by Judge Learned Hand which describes a process for determining whether a legal duty of care has been breached (see negligence). The original description of the calculus was in United States v. Carroll Towing Co., in which an improperly secured barge had drifted away from a pier and caused damage to several other boats. Articulation of the rule Hand stated: This relationship has been formalized by the law and economics school as such: an act is in breach of the duty of care if: where B is the cost (burden) of taking precautions, and P is the probability of loss (L). L is the gravity of loss. The product of P x L must be a greater amount than B to create a duty of due care for the defendant. Rationale The calculus of negligence is based on the Coase theorem. The tort system acts as if, before the injury or damage, a contract had been made between the parties under the assumption that a rational, cost-minimizing individual will not spend money on taking precautions if those precautions are more expensive than the costs of the harm that they prevent. In other words, rather than spending money on safety, the individual will simply allow harm to occur and pay for the costs of that harm, because that will be more cost-efficient than taking precautions. This represents cases where B is greater than PL. If the harm could be avoided for less than the cost of the harm (B is less than PL), then the individual should take the precautions, rather than allowing the harm to occur. If precautions were not taken, we find that a legal duty of care has been breached, and we impose liability on the individual to pay for the harm. This approach, in theory, leads to an optimal allocation of resources; where harm can be cheaply avoided, the legal system requires precautions. Where precautions are prohibitively expensive, it does not. In marginal-cost terms, we require individuals to invest one unit of precautions up until the point that those precautions prevent exactly one unit of harm, and no less. Mathematical rationale The Hand rule attempts to formalize the intuitive notion that when the expected loss exceeds the cost of taking precautions, the duty of