In Lucas, the Supreme Court indicated that the genesis of the concept of regulatory takings occurred in Pennsylvania Coal Co v Mahon, 260 US 393 (1922):

Prior to Justice Holmes’s exposition in [Pennsylvania Coal Co v Mahon, 260 US 393 (1922)] it was generally thought that the Takings Clause reached only a "direct appropriation" of property, [Legal Tender Cases, 12 Wall 457, 551 (1871)], or the functional equivalent of a "practical ouster of [the owner’s] possession," [Transportation Co v Chicago, 99 US 635, 642 (1879)]. See also [Gibson v United States, 166 US 269, 275-76 (1897)]. Justice Holmes recognized in Mahon, however, that if the protection against physical appropriations of private property was to be meaningfully enforced, the government’s power to redefine the range of interests included in the ownership of property was necessarily constrained by constitutional limits. [Mahon, 260 US at 414-15]. If, instead, the uses of private property were subject to unbridled, uncompensated qualification under the police power, "the natural tendency of human nature [would be] to extend the qualification more and more until at last private property disappear[ed]." [Id. at 415]. These considerations gave birth in that case to the oft-cited maxim that, "while property may be regulated to a certain extent, if regulation goes too far it will be recognized as a taking." [Id.]

Lucas, 505 US at 1014.

The Lucas court admitted that in considering the question of whether a particular regulation had gone "too far," the federal courts have generally engaged in "essentially ad hoc, factual inquiries." Id. at 1015 (citing Penn Central, 438 US at 124). Penn Central involved the question of whether the application of a New York City historical landmark law constituted a taking when that law was applied to prevent the construction of an office building atop Grand Central Terminal. The Supreme Court indicated that the following factors have "particular significance" in deciding whether there was a taking: (1) the economic impact of the regulation; (2) the extent to which the regulation interfered with investment-backed expectations; and (3) the character of the government action. Penn Central, 438 US at 124. The court rejected the claim that "diminution in property value, standing alone, can establish a ‘taking.’ " Id. at 131.

While noting that the majority of takings claims were decided on the Penn Central ad hoc basis, the Lucas court indicated it had "described at least two discrete categories of regulatory action as compensable without case-specific inquiry into the public interest advanced in support of the restraint." Lucas, 505 US at 1015. The first, which is not at issue here, involve regulations that require an owner to suffer even minute physical invasions, like requiring apartment owners to allow cable lines to be laid. The second, which is at issue here, is a categorical taking.

A categorical taking was described as occurring "where regulation denies all economically beneficial or productive use of the land." Id. But the Supreme Court admitted that it had not solved the denominator-parcel issue:

Regrettably, the rhetorical force of our "deprivation of all economically feasible use" rule is greater than its precision, since the rule does not make clear the "property interest" against which the loss of value is to be measured. When, for example, a regulation requires a developer to leave 90% of a rural tract in its natural state, it is unclear whether we would analyze the situation as one in which the owner has been deprived of all economically beneficial use of the burdened portion of the tract, or as one in which the owner has suffered a mere diminution in value of the tract as a whole. (For an extreme — and, we think, unsupportable — view of the relevant calculus, see [Penn Central Transportation Co v New York City, 366 NE2d 1271, 1276-77 (1977), aff’d, 438 US 104 (1978)], where the state court examined the diminution in a particular parcel’s value produced by a municipal ordinance in light of total value of the takings claimant’s other holdings in the vicinity.) Unsurprisingly, this uncertainty regarding the composition of the denominator in our "deprivation" fraction has produced inconsistent pronouncements by the Court. Compare [Pennsylvania Coal Co v Mahon, 260 US 393, 414 (1922)] (law restricting subsurface extraction of coal held to effect a taking), with [Keystone Bituminous Coal Assn v DeBenedictis, 480 US 470, 497-502 (1987)] (nearly identical law held not to effect a taking); see also [id., at 515-20 (Rehnquist, C.J., dissenting); Rose, Mahon Reconstructed: Why the Takings Issue is Still a Muddle, 57 SCalLRev 561, 566-69 (1984)]. The answer to this difficult question may lie in how the owner’s reasonable expectations have been shaped by the State’s law of property — i.e., whether and to what degree the State’s law has accorded legal recognition and protection to the particular interest in land with respect to which the takings claimant alleges a diminution in (or elimination of) value. In any event, we avoid this difficulty in the present case, since the "interest in land" that Lucas has pleaded (a fee simple interest) is an estate with a rich tradition of protection at common law, and since the South Carolina Court of Common Pleas found that the Beachfront Management Act left each of Lucas’s beachfront lots without economic value.

Lucas, 505 US at 1016 n. 7.

The majority in Lucas responded to dissenting Justice Stevens’ claim that the categorical rule was arbitrary by noting that much of takings law is arbitrary in regard to which owners qualify for compensation:

Justice STEVENS criticizes the "deprivation of all economically beneficial use" rule as "wholly arbitrary," in that "[the] landowner whose property is diminished in value 95% recovers nothing," while the landowner who suffers a complete elimination of value "recovers the land’s full value." Post, at 2919. This analysis errs in its assumption that the landowner whose deprivation is one step short of complete is not entitled to compensation. Such an owner might not be able to claim the benefit of our categorical formulation, but, as we have acknowledged time and again, "[t]he economic impact of the regulation on the claimant and . . . the extent to which the regulation has interfered with distinct investment-backed expectations" are keenly relevant to takings analysis generally. [Penn Central Transportation Co v New York City, 438 US 104 (1978)]. It is true that in at least some cases the landowner with 95% loss will get nothing, while the landowner with total loss will recover in full. But that occasional result is no more strange than the gross disparity between the landowner whose premises are taken for a highway (who recovers in full) and the landowner whose property is reduced to 5% of its former value by the highway (who recovers nothing). Takings law is full of these "all-or-nothing" situations.

Lucas, 505 US at 1019 n. 8.

In Lucas, Justice Stevens claimed that it was impossible to create a neutral system for defining the denominator parcel:

As the Court admits, whether the owner has been deprived of all economic value of his property will depend on how "property" is defined. The "composition of the denominator in our ‘deprivation’ fraction," ante, at 2894, n. 7, is the dispositive inquiry. Yet there is no "objective" way to define what that denominator should be. "We have long understood that any land-use regulation can be characterized as the ‘total’ deprivation of an aptly defined entitlement. . . . Alternatively, the same regulation can always be characterized as a mere ‘partial’ withdrawal from full, unencumbered ownership of the landholding affected by the regulation. . . ." Michelman, Takings, 1987, 88 ColumLRev 1600, 1614 (1988).

Lucas, 505 US at 1054 (Stevens, J., dissenting) (footnote omitted).

In 2001, the Supreme Court decided Palazzolo v Rhode Island, 533 US 606 (2001), which concerned a landowner’s claim that his property had been taken through the application of wetlands regulations. In that case, the Supreme Court discussed the purpose of the Takings Clause and the Lockean view of property. It also noted that it not been able to resolve the denominator-parcel issue.

The Supreme Court stated "the purpose of the Takings Clause. . . . is to prevent the government from ‘forcing some people alone to bear public burdens which, in all fairness and justice, should be borne by the public as a whole." Id. at 618 (citing Armstrong v United States, 364 US 40, 49 (1960)). The court then discussed whether a property owner who obtained a property after a potentially confiscatory regulation had been promulgated could maintain a takings claim, and it rejected a Hobbesian view in favor of a Lockean view:

The theory underlying the argument that postenactment purchasers cannot challenge a regulation under the Takings Clause seems to run on these lines: Property rights are created by the State. So, the argument goes, by prospective legislation the State can shape and define property rights and reasonable investment-backed expectations, and subsequent owners cannot claim any injury from lost value. After all, they purchased or took title with notice of the limitation.

The State may not put so potent a Hobbesian stick into the Lockean bundle. The right to improve property, of course, is subject to the reasonable exercise of state authority, including the enforcement of valid zoning and land-use restrictions. See [Pennsylvania Coal Co, 260 US at 413] ("Government hardly could go on if to some extent values incident to property could not be diminished without paying for every such change in the general law"). The Takings Clause, however, in certain circumstances allows a landowner to assert that a particular exercise of the State’s regulatory power is so unreasonable or onerous as to compel compensation. Just as a prospective enactment, such as a new zoning ordinance, can limit the value of land without effecting a taking because it can be understood as reasonable by all concerned, other enactments are unreasonable and do not become less so through passage of time or title. Were we to accept the State’s rule, the postenactment transfer of title would absolve the State of its obligation to defend any action restricting land use, no matter how extreme or unreasonable. A State would be allowed, in effect, to put an expiration date on the Takings Clause. This ought not to be the rule. Future generations, too, have a right to challenge unreasonable limitations on the use and value of land.

Palazzolo, 533 US at 626-27 (citation omitted and emphasis added).

The Supreme Court recognized the difficulty inherent in determining denominator parcels, but it held that the denominator-parcel issue would not be decided because the plaintiff had failed to preserve the issue:

In his brief submitted to us petitioner attempts to revive this part of his claim by reframing it. He argues, for the first time, that the upland parcel is distinct from the wetlands portions, so he should be permitted to assert a deprivation limited to the latter. This contention asks us to examine the difficult, persisting question of what is the proper denominator in the takings fraction. See Michelman, Property, Utility, and Fairness: Comments on the Ethical Foundations of "Just Compensation Law," 80 Harv LRev 1165, 1192 (1967). Some of our cases indicate that the extent of deprivation effected by a regulatory action is measured against the value of the parcel as a whole, see, e.g., [Keystone Bituminous Coal Assn v DeBenedictis, 480 US 470, 497 (1987)]; but we have at times expressed discomfort with the logic of this rule, see [Lucas, 505 US at 1016-17, n. 7] a sentiment echoed by some commentators, see, e.g., Epstein, Takings: Descent and Resurrection, 1987 SCt Rev 1, 16-17 (1987); Fee, Unearthing the Denominator in Regulatory Takings Claims, 61 U Chi LRev 1535 (1994). Whatever the merits of these criticisms, we will not explore the point here. Petitioner did not press the argument in the state courts, and the issue was not presented in the petition for certiorari.

Palazzolo, 533 US at 631.

Thus, under United States Supreme Court precedent, the concept of regulatory takings was recognized in 1924. The multifactored, ad hoc Penn Central test was created in 1978. In that same case, the U.S. Supreme Court held that a diminution of value alone could not constitute a compensable taking. The concept of a categorical taking was formalized in the 1992 Lucas case. In order to determine whether a categorical taking has occurred, the courts need to decide what constitutes a denominator parcel. The Lucas majority admitted this was a vexing question, but deemed the issue irrelevant in that case because the regulation at issue meant that none of the landowner’s property could be developed. In Palazollo, which was decided in 2001, the Supreme Court admitted that it still had not resolved the denominator issue. No United States Supreme Court case since then has solved the problem.

In Lucas, the Supreme Court recognized that it might be arbitrary that a person who lost 100% of the value of his or her property received compensation, while a person who lost 90% of the value of his or her property might receive nothing. But the Lucas majority noted that arbitrary results are a hallmark of takings law.

Thus, while paying lip service to the Lockean model, the federal courts have allowed a Hobbesian model to become ascendant. Under a Lockean model, the costs of a regulation to benefit the general public should not be borne by a select few individuals; rather, the costs should be distributed equally throughout society. But where arbitrary deprivations of individual rights are accepted, as in federal regulatory takings law, the Lockean model is not being followed, and the Hobbesian model predominates.