The incessant demand for pornography, some have said, is an engine of technological development. John Tierney, Pom, the Low-Slung Engine of Progress, N.Y. Times, Jan. 9, 1994, § 2 (Arts & Leisure Desk), at 1 (noting as an example new pay-per-call technology). The telephonic system at dispute in this appeal is an example of that phenomenon — it was designed and implemented to ensure that consumers paid charges for accessing pornography and other adult entertainment. The system identified the user of an online adult-entertainment service by the telephone line used to access that service and then billed the telephone-line subscriber for the cost of that service as if it was a charge for an international phone call to Madagascar. *52This system had the benefit that the user’s credit card never had to be processed, but it had a problem as well: It was possible for someone to access an adult-entertainment service over a telephone line without authorization from the telephone-line subscriber who understood herself contractually bound to pay all telephone charges, including those that disguised fees for the adult entertainment.
The Federal Trade Commission (“FTC”) took a dim view of this billing system and brought suit to shut it down as a deceptive and unfair trade practice within the meaning of § 5(a)(1) of the Federal Trade Commission Act (“FTC Act”), 15 U.S.C. § 45(a)(1). The FTC sued Verity International, Ltd. (‘Verity”) and Automatic Communications, Ltd. (“ACL”), corporations that operated this billing system, as well as Robert Green and Marilyn Shein, who controlled these corporations during the relevant time period. These four defendants appeal from the district court’s decision and judgment finding them liable for violating § 5(a)(1). Green and Shein also appeal from a district court order holding them in contempt of court.
BACKGROUND
The district court found the following facts upon a bench trial.
I. Structure of the Billing System
The defendants-appellants’ billing system operated as follows: When a computer user visited a website providing adult-entertainment services, the website offered the user the ability to buy adult. content using a downloadable “dialer program.” The user downloaded the dialer program after clicking through a series of website disclosures containing the terms and conditions of use and an explanation that charges for the adult content would be billed to the telephone-line subscriber as the cost of an international phone call. The computer user then initiated the dialer program, and if the computer was connected by modem to a telephone line, the dialer program placed an international phone call to a Madagascar telephone number, bypassing the line subscriber’s designated carrier in favor of AT & T and later Sprint.
Either AT & T or Sprint carried the call to London where it handed off the call to a separate carrier, AT & T U.K. (later renamed Viatel). Instead of routing the call to Madagascar for completion, AT & T U.K/Viatel carried the call to a designated internet server in the United Kingdom, a practice known as “short-stopping” the call. That internet server finalized the connection between the user’s computer and the website providing the desired adult entertainment.
Charges for accessing the adult entertainment appeared on bills sent to the consumers whose telephone lines were used. AT & T and Sprint identified the telephone-line subscribers by the Automatic Number Identification (“ANI”) system, the standard means by which telephone companies bill for phone calls. These bills, at first telephone bills from AT & T and later separate bills designed by Verity and sent using information provided by Sprint, charged line subscribers for long-distance phone calls to Madagascar.
Notably, this billing system did not have a mechanism to ensure that a telephone-line subscriber authorized the computer user to access a given adult-entertainment service. The absence of such a mechanism allowed line subscribers to receive bills for adult-entertainment access about which they had no knowledge, which prompted the FTC to bring this lawsuit.
II. Creation and Operation of the Billing System
In May 1997, defendant-appellant ACL contracted with Telecom Malagasy, the na*53tional telecommunications carrier for Madagascar, for (1) the right to carry calls placed to certain international telephone numbers assigned to Madagascar, (2) the right to collect charges for these calls, and (3) the right to terminate these calls at any location of ACL’s choice, including locations outside Madagascar. The right to carry calls to these numbers was valuable because of the calls’ high per-minute tar-iffed rate under U.S. telecommunications law. Revenue generated from these calls would ultimately be divided between ACL, Telecom Malagasy, various phone-call carriers, ACL’s billing agents, a company that distributed the dialer program mentioned above, and various adult-website operators.
To exploit ACL’s right to carry calls to these Madagascar phone numbers, ACL contracted with Global Internet Billing, Inc. (“GIB”) for GIB to market the dialer program to adult-website operators and to use its best efforts to generate a minimum usage volume. ACL agreed to provide GIB with the Madagascar telephone numbers for inclusion in GIB’s dialer program. ACL paid a portion of call revenues to GIB, which in turn paid the adult-website operators, effectively making GIB a paid intermediary between ACL and the website operators.
ACL also needed to arrange for the carriage of calls from a computer’s modem to the U.K. internet servers that would connect the calling computer to an adult website in the United States. Accordingly, in January 1999, ACL contracted with two companies, AT & T and AT & T U.K., to carry the calls. AT & T agreed to carry calls placed to ACL’s Madagascar phone numbers to the London facilities of AT & T U.K. AT & T U.K. would then carry the calls to the designated U.K. internet servers. AT & T was responsible for billing and collection for these calls, and using ANI information, AT & T billed phone-line subscribers for the ACL calls on their regular monthly telephone statements.
The content of the telephone statements received by the subscribers is relevant here. AT & T charged subscribers only the tariffed rates for phone calls to Madagascar. It listed the charges in the “Long Distance” section of the bills, with Madagascar as the “Place Called.” Under the “Important Information” header, the bills stated that “nonpayment of toll charges may result in disconnection of local service, and other services may be restricted if not paid.”
In the roughly-seven-month period beginning in January 2000, when adult-website operators started using ACL’s system to provide adult-entertainment services to computer users, AT & T’s billings for traffic to ACL’s Madagascar numbers totaled $29 million, as compared to $1.6 million in total billings during the previous twelve months. At the same time, the percentage of total billings refunded to subscribers who contested their bills spiked from 8% in the previous year to 38% during this period.
ACL’s contract with AT & T, together with ACL’s other agreements, established a multitiered cascading-payment structure: AT & T sent to AT & T U.K. the amounts due both AT & T U.K. and ACL; AT & T U.K. then paid ACL from those funds. ACL then paid GIB, who in turn paid the adult-website operators. Each entity kept some of the money along the way. (Tele-com Malagasy was compensated separately by both AT & T and ACL for providing the phone numbers.) This arrangement was in effect from January 2000 until July 2000, when AT & T terminated the contract and stopped carrying calls for ACL. The district court deemed this the “AT & T Period.”
*54After AT & T terminated the agreement, ACL turned to Sprint as a replacement. ACL reached an agreement with Sprint which contemplated Sprint performing billing and collection functions, as AT & T did, but Sprint then quickly entered into a new agreement that released it from these duties. Under the new agreement, Sprint agreed to carry calls to the London facilities of AT & T U.K. (now renamed Viatel), but it would leave billing and collection to ACL by providing ACL with the ANI information identifying the subscribers whose telephone lines were used to call ACL’s Madagascar numbers. As it did in the AT & T agreement, ACL warranted that it would receive the calls and terminate them in Madagascar. ACL agreed to pay a per-minute fee to Sprint and AT & T U.K/Viatel for serving as carriers of the phone calls. This “Sprint Period” lasted from July 2000 through September 2000, when Sprint stopped carrying calls to ACL’s Madagascar phone numbers.
To handle billing and collection during the Sprint Period, ACL entered into an agreement, in Verity’s name, with eBillit, Inc., a subsidiary of Integretel, Inc. The agreement required eBillit to prepare and mail bills to line subscribers, collect payments from them, and handle their complaints. The eBillit bills were branded with the Verity logo and were separate from the line subscribers’ regular telephone bills. These Verity bills contained an invoice number, an account number, the subscriber’s telephone number, a summary of charges, a due date, and a statement in capital letters that “this bill accounts for international calls, from your modem to a Madagascar number, for website access.” The bills contained a “Detail of Charges” section in which the city called was listed as one of several cities within Madagascar. Defendant-appellant Robert Green approved the format of these bills.
The bills also contained a “1-800” number provided for line subscribers to call with questions about their bills. That number was widely used. During the Sprint period, 91,683 bills were sent to line subscribers and at least 24,986 subscribers contacted Verity about the bills. Calling the customer-service center was not a positive experience for many invoice recipients. The center was so understaffed that 72% of the calls placed to it were abandoned by callers. While waiting on hold for a customer-service representative, callers were played a recording warning that “[f]ailure to pay a Verity International bill may result in the blocking of your phone line to services of this nature from a variety of content providers and further collection activity of past due amounts.” Once connected to a customer-service representative, callers had to weather a “hard sustain” approach that involved the representative advising callers that the charges were valid, that the charges must be paid, and that nonpayment would subject the line subscriber to further collection activity. Robert Green and Marilyn Shein instructed the call center to maintain this hard-sustain approach, which did not change until the FTC brought the present lawsuit. During the Sprint Period, the Verity bills resulted in $1.6 million in collected billings and over 500 consumer complaints to the FTC. The billing system has not been resurrected after Sprint stopped carrying its calls.
III. Procedural History
A. The FTC’s complaint
The FTC’s complaint alleged that certain aspects of the defendants-appellants’ billing system were deceptive or unfair trade practices in violation of § 5(a)(1) of the FTC Act. That section provides that “unfair or deceptive acts or practices in or *55affecting commerce ... are hereby declared unlawful.” 15 U.S.C. § 45(a)(1). The FTC’s second amended complaint claimed relief on three grounds relevant on appeal. In Count I, the FTC alleged that the defendants-appellants engaged in a deceptive practice by falsely representing that a consumer could not legally avoid charges for website content accessed over the consumer’s telephone line, even if the consumer did not access the content or authorize others to do so. In Count II, the FTC alleged that the defendants-appellants engaged in an unfair practice by themselves or through others “billing and attempting to collect from line subscribers” whose telephone lines were used to access online adult content but who did not access or authorize others to access that content. In Count III, the FTC alleged that it was a deceptive practice for the defendants-appellants to cause billing statements to misrepresent the destination of outbound calls as Madagascar when in fact the calls did not terminate there.
B. Parties
ACL is a Bahamian corporation that operated the billing system in dispute. It was founded and controlled by Robert Green and Marilyn Shein, each of whom owned 40% of ACL’s shares until September 20, 2000, when an Australian corporation, Oriel Communications, Ltd., acquired half of ACL’s shares. The acquisition left Green and Shein each holding 20% of ACL’s shares and approximately 11% of Oriel’s shares. Green and Shein also founded and controlled Verity, a short-lived operation that was part of the billing system and was used for accounting purposes.
C. Preliminary injunction and contempt
On December 13, 2000, the district court entered a preliminary injunction that imposed an asset freeze on Verity, Green, and Shein to preserve funds for a possible monetary remedy. The preliminary injunction also required each of them to complete and return to the FTC a financial-disclosure form. The FTC proposed the financial-disclosure requirement as a way to evaluate the reasonableness of Green’s and Shein’s requests to unfreeze assets for living expenses, so Green and Shein did not contest the requirement at the time. Green and Shein contend that upon seeing the enormous amount of information requested by the disclosure form, they decided not to seek a release of their frozen assets and not to complete the disclosure form. But the court’s order to do so stood. Accordingly, the district court ordered Green and Shein held in contempt of court for their failure to comply with the preliminary injunction’s financial-disclosure requirement, an order from which Green and Shein appeal. The district court imposed a coercive per-day fine for their noncompliance and ordered their civil confinement, should they be found within the United States, until they complied with the financial-disclosure requirement. Soon thereafter, the district court denied their motion to lift the financial-disclosure requirement and held that disclosure was necessary “to assure enforcement of an asset freeze or to recover proceeds of wrongdoing that are the subject of an equitable claim for disgorgement.” To date, neither Green nor Shein has completed the financial-disclosure form. As of October 7, 2005, the date on which this appeal was argued, the coercive monetary fines totaled $16.1 million per contemnor.
D.Motion practice and bench trial
The defendants-appellants filed a motion for judgment on the pleadings, contending that the district court lacked subject-matter jurisdiction because (1) ACL was a *56common carrier outside of the FTC’s jurisdiction, (2) the filed-rate doctrine negated standing by precluding the FTC from contending that line subscribers could avoid the charges in question, and (3) the primary-jurisdiction doctrine required the FCC to first decide the case. The district court asked the FCC to brief, as amicus curiae, the merits of the defendants-appellants’ contentions, and the United States Attorney for the Southern District of New York submitted a letter brief on behalf of the FCC answering the district court’s questions. He concluded that ACL was not a common carrier under the Communications Act and that the primary-jurisdiction and filed-rate doctrines therefore did not apply. With the benefit of the FCC’s views, the district court denied the defendants-appellants’ motion for judgment on the pleadings and found that it had subject-matter jurisdiction to hear the case.
On September 17, 2004, following a bench trial on a record of stipulated facts, declarations, exhibits, and other evidence, the district court filed a memorandum opinion. The court incorporated the factual findings and legal holdings of its earlier opinion denying defendants-appellants’ motion for judgment on the pleadings, held that the FTC proved Counts I, II, and III of its second amended complaint, and held that individual as well as corporate liability was appropriate. Finding the restitution-ary remedy of disgorgement to be available and proper, the district court entered two money judgments against the defendants-appellants for a total of $17.9 million. The court also replaced the preliminary injunction with a permanent injunction, which did not contain a financial-disclosure requirement. The defendants-appellants timely appealed from the district court’s judgment.
DISCUSSION
Our review proceeds in multiple parts. In Parts I — III, we consider and find merit-less the defendants-appellants’ arguments that the district court lacked subject-matter jurisdiction. Next, in Part IV, we consider a challenge to the district court’s determination that the trade practices at issue violated § 5(a)(1) of the FTC Act and uphold the district court. Part V discusses why the district court erred in awarding monetary relief of $17.9 million. Finally, Part VI turns to the coercive contempt sanctions against Green and Shein, explaining why we must vacate them as moot.
I. The Common-Carrier Exception to the FTC’s Enforcement Power
The FTC Act limits the FTC’s enforcement power. Pertinent here is the FTC’s inability to enforce § 5(a)(1) of the FTC Act against “common carriers subject to the Acts to regulate commerce.” 15 U.S.C. § 45(a)(2) (providing this “common-carrier exception” to the FTC’s powers). One such act “to regulate commerce” is the Communications Act of 1934, along with its amendatory acts. Id. § 44 (citing the Communications Act of 1934, 47 U.S.C. § 151 et seq.). ACL argues that it is a telecommunications common carrier under the Communications Act and that therefore the common-carrier exception renders § 5(a)(1) of the FTC Act inapplicable.1
This contention raises the question whether the term “common carrier” under the FTC Act has the same meaning as the term “common carrier” under the Commu*57nications Act. As explained below, we determine that “common carrier” under the FTC Act is properly defined by reference to the common law of carriers and not to the Communications Act, even though the common law definition does not meaningfully differ from the Communications Act definition for the purposes of this appeal. Under both definitions, ACL is not a common carrier. A brief history of the two acts is helpful in explaining our conclusion.
The first federal regulation to impose duties on common carriers was the Interstate Commerce Act of 1887 (“ICA”), ch. 104, 24 Stat. 379 (1887), which applied to “any common carrier or carriers” engaged in the railroad transportation of people or property interstate. The ICA imposed on railroad common carriers traditional common-carrier requirements such as nondiscrimination, tariff-filing, and charging just and reasonable rates, id. §§ 1-7, and it created the Interstate Commerce Commission (“ICC”) to administer the provisions of the act, id. §§ 11-12. In 1910, Congress passed the Mann-Elkins Act, ch. 309, 36 Stat. 539 (1910), which amended the ICA to apply to interstate telephone companies and to deem such companies common carriers, id. § 7(1). Neither the ICA nor the Mann-Elkins Act contained a definition of “common carrier.”
In 1914, in the thick of the antitrust movement, Congress passed the Federal Trade Commission Act (the “FTC Act”), ch. 311, 38 Stat. 717 (1914), which created the Federal Trade Commission (“FTC”) as an enforcement agency. Congress did not intend the FTC to enforce unfair-competition law2 against common carriers because the ICC already regulated common carriers under the Interstate Commerce Act. Thus, for the purpose of preventing inter-agency conflict, the FTC Act common-carrier exception was created. See generally Marc Winerman, The Origins of the FTC, 71 Antitrust L.J. 1, 69 n.413 (2003) (mentioning the genesis of the common-carrier exception). Just as Congress had not provided a definition of “common carrier” in the Interstate Commerce Act, it did not provide a definition for that term in the FTC Act.
Regulation of telephone common carriers continued to rest with the ICC until 1934, when Congress passed the Communications Act of 1934, ch. 652, 48 Stat. 1064 (1934).3 That act created the Federal Communications Commission (“FCC”) and transferred to the FCC regulatory authority over telephone common carriers. Id. The Communications Act defined “common carrier” circularly, as “any person engaged as a common carrier for hire, in interstate or foreign communications by wire or radio ....” 47 U.S.C. § 153(10). As one researcher has noted, the conference report for the Communications Act of 1934 suggests that the phrase “common carrier” had an ordinary meaning at the time, explaining why the Interstate Commerce Act left the term undefined and why the Communications Act included only a circular definition. Phil Nichols, Redefining “Common Carrier,” 1987 Duke L.J. 501, 511. Congress did not modify the exemption of common carriers from FTC regulatory authority except to extend the exemp*58tion to common carriers now subject to the Communications Act. Wheeler-Lea Act of 1938, § 2, 52 Stat. Ill, 111 (1938).
The foregoing description brings us to the state of the relevant law today, against which we assess ACL’s assertion that the correct definition for “common carrier” under the FTC Act is found in the Communications Act. We find no statutory basis for so concluding. The term “common carrier” in § 5(a)(2) of the FTC Act is still undefined, both in the act itself and in FTC regulations. As did our sister circuit when interpreting the circular definition of “common carrier” under the Communications Act, we decide to give meaning to “common carrier” in the FTC Act according to the ordinary sense , of the word when Congress used it to create the exemption. See Nat’l Ass’n of Regulatory Util. Comm’rs v. FCC (“NARUC II), 533 F.2d 601, 608-09 (D.C.Cir.1976) (turning to the common law to define “common carrier” under the Communications Act); Nat’l Ass’n of Regulatory Util. Com’rs v. FCC (“NARUC I”), 525 F.2d 630, 640-42 (D.C.Cir.1976) (taking the same approach); cf. Sec. Indus. Ass’n v. Bd. of Governors of the Fed. Reserve Sys., 468 U.S. 137, 149-50, 104 S.Ct. 2979, 82 L.Ed.2d 107 (1984) (interpreting the undefined terms “security” and “note” by reference to their ordinary meaning to the 1933 Congress that used them).
The concept of a common carrier dates from the English common law and can be traced back to at least 1670 and the writings of Lord Chief Justice Hale. See Munn v. Illinois, 94 U.S. 113, 126, 4 Otto 113, 24 L.Ed. 77 (1876) (referencing Lord Hale’s treatise). Early common-carrier law applied to “almost all workers and tradesmen,” requiring them to “serve the public generally and to do so on just.and reasonable terms,” but over time, the common law of common carriers narrowed its focus to enterprises considered “public” in some way, such as by the government grant of a legal monopoly or their use of public funds. James B. Speta, A Common Carrier Approach to Internet Interconnection, 54 Fed. Comm. L.J. 225, 255-57 (2002); see also Nichols, supra, at 506-07 (describing Lord Hale’s concept of public interest in privately held business).
Eventually, the definition of a common carrier coalesced into two requirements: (1) the entity holds itself out as undertaking to carry for all people indifferently; and (2) the entity carries its cargo without modification. See NARUC I, 525 F.2d at 640-42 (describing how the common law imposed common-carrier regulation on entities that undertook to carry for all shippers or travelers indifferently); NARUC II, 533 F.2d at 608-09 (describing the “without modification” requirement); Nichols, supra, at 508-09 (quoting the formulation of an 1857 carriers treatise that “[t]o render a person liable as a common carrier, he must exercise the business of carrying as a ‘public employment,’ and must undertake to carry goods for all persons indiscriminately”). This definition does not differ meaningfully for our purposes from the definition of “common carrier” under the Communications Act — -both require that an entity provides carriage to the public. See 47 U.S.C. § 153(43), (44), (46) (providing that “[t]he term ‘telecommunications carrier’ means any provider” offering to the public “the transmission ... of information”).
Applying these definitions, we conclude that defendant-appellant ACL is not a common carrier subject to the Communications Act and therefore does not fit within the FTC Act common-carrier exemption. The carriage of the telephone calls in this case involved three carriers in concept and two carriers in fact. Conceptually, the calls were carried by an originating *59carrier, a transit carrier, and a destination carrier. AT & T, and later Sprint, served as the originating carrier, routing the calls from the United States to the United Kingdom. The transit carrier was AT & T U.K./Viatel, whose role as transit carrier was, in concept, “to route traffic [from an originating carrier] to a carrier in another country, the destination carrier.” In re AT&T Corp., 14 F.C.C.R. 19140, 19176 n. 168, 1999 WL 979615 (1999). Conceptually, Telecom Malagasy was the destination carrier, with ACL standing in its shoes by virtue of the agreement assigning ACL the right to terminate calls placed to Telecom Malagasy’s numbers. Even at this conceptual level, ACL is not exempt from the FTC Act because foreign terminating carriers are not carriers subject to the Communications Act, as contemplated by the FTC Act’s common-carrier exemption. 15 U.S.C. §§ 44, 45(a). An entity is subject to the Communications Act if it is “engaged within the United States” in “interstate and foreign communication by wire or radio [or] ... interstate and foreign transmission of energy by radio, which originates and/or is received within the United States.” 47 U.S.C. § 152(a). As indicated by the “engaged within the United States” limitation, the Communications Act does not apply to foreign terminating carriers. Cable & Wireless P.L.C. v. FCC, 166 F.3d 1224, 1229-30 (D.C.Cir.1999) (finding that because an FCC order applied to only domestic carriers, not foreign carriers, the FCC did not exceed its authority under the Communications Act in issuing it; noting that the FCC “claims no authority to directly regulate foreign carriers”). Thus, as a purely foreign terminating carrier at the conceptual level, ACL would not qualify for the FTC Act exemption.
Yet our determination that ACL was not a common carrier is even simpler because, in fact, the telephone calls at issue were carried by only two entities — the originating carrier and the transit carrier. AT & T, and later Sprint, carried the phone calls from the United States to the United Kingdom as originating carrier, and AT & T U.K./Viatel carried the phone calls to the U.K. internet servers where they were terminated. ACL simply brought together these carriers as part of its billing system; it never itself carried any calls. Thus, while AT & T and Sprint might be exempt from FTC enforcement, ACL is not.
On appeal, ACL presses the argument that the § 5(a)(2) common-carrier exemption applies to an entity with the “status of a common carrier” under the Communications Act,4 even if its activities relevant to a pending lawsuit are not common carriage. Assuming arguendo that common carrier “status” can exist and is determinative, this argument would aid ACL only if it had the status of a common carrier. ACL contends that it holds such status because the FCC granted it a license pursuant to 47 U.S.C. § 214 (the “ § 214 license”), yet this license simply authorizes ACL “to become a facilities-based international common carrier ... and/or to id-*60come a resale-based international common carrier” (emphasis added). The § 214 license does not purport to represent or determine that ACL is actually engaged in common carriage, nor did ACL’s application for the license so represent. Rather than rely on what an entity is authorized to do, courts must examine the actual conduct of an entity to , determine if it is a common carrier for purposes of the FTC Act exemption. Cf. Eagleview Techs., Inc. v. MBS Assocs., 190 F.3d 1195, 1197-98 (11th Cir.1999) (holding that an entity’s possession of an FCC license to provide services as a common carrier did not alone make the entity a common carrier under the Communications Act). Here, as explained above, none of ACL’s activities gave it the status of a common carrier subject to the Communications Act of 1934, and accordingly, the FTC Act common-carrier exception would not apply.
II. The Primary-Jurisdiction Doctrine
The doctrine of primary jurisdiction allows a federal court to refer issues “extending beyond the ‘conventional expertise of judges’ or ‘falling within the realm of administrative discretion’ ” to the appropriate administrative agency for resolution in the first instance. Nat’l Commc’ns Ass’n, Inc. v. AT & T Co., 46 F.3d 220, 222-23 (2d Cir.1995) (quoting Far East Conference v. United States, 342 U.S. 570, 574, 72 S.Ct. 492, 96 L.Ed. 576 (1952)). “Specifically, courts apply primary jurisdiction to cases involving technical and intricate questions of fact and policy that Congress has assigned to a specific agency.” Id. at 223. Although there is “[n]o fixed formula ... for determining whether an agency has primary jurisdiction,” courts typically consider four factors in this analysis:
(1)whether the question at issue is within the conventional experience of judges or whether it involves technical or policy considerations within the agency!s particular field of expertise;
(2) whether the question at issue is particularly within the agency’s discretion;
(3) whether there exists a substantial danger of inconsistent rulings; and
(4) whether a prior application to the agency has been made.
Id. at 222, 223.
The defendants-appellants contend that the district court should have referred this ease to the FCC because it raises complex questions of telecommunications policy. Of the questions raised in this case, the only issue that deserves close primary-jurisdiction analysis pertains to the interpretation of Communications Act terms. That issue, important to our discussion in Part III of the filed-rate doctrine, is whether the services offered by ACL are telecommunications services or, alternatively, information services.
The four primary-jurisdiction factors do not favor finding FCC primary jurisdiction over the characterization of the services provided by ACL. First, there are many precedents, including those of the FCC, on the meaning of the Communications Act terms “information service” and “telecommunications service.” See, e.g., Nat’l Cable & Telecommc’ns Ass’n v. Brand X Internet Servs., — U.S. -, 125 S.Ct. 2688, 2702-10, 162 L.Ed.2d 820 (2005); In re Federal-State Joint Bd. on Universal Serv. (“Universal Service Report”), 13 F.C.C.R. 11501, 11531, 1998 WL 166178 (1998); see generally Peter W. Huber et ah, Federal Telecommunications Law § 12.2, at 1077-86 (1999). The job of applying these reasonably settled definitions to the facts of this case is within the court’s competence. See Nat’l Commc’ns Ass’n, 46 F.3d at 223 (“This record does *61not present any issues involving intricate interpretations ... that might need the FCC’s technical or policy expertise.”). Second, while the classification issue is within the FCC’s discretion in the sense that the FCC is charged with administering the Communications Act, nothing about the terms invokes the FCC’s discretion in the same way that more abstract statutory terms such as “reasonable” or “public interest” do. See id. (“This case, however, does not involve the ■ statutory reasonableness of the tariff or other abstract concepts. Instead, it focuses on ... a rather simple factual question .... ”). Third, the defendants-appellants have pointed to no danger of inconsistent rulings on the classification of their service. Fourth, to the extent that the defendants-appellants contend that LO/AD Communications, B.V.I., Ltd. v. MCI WorldCom, No. 00 Civ. 3594, 2001 WL 64741 (S.D.N.Y. Jan.24, 2001), was a prior application to the FCC on the classification issue, they are incorrect. That ease concerned the abstract “reasonableness” standard of 47 U.S.C. § 201, not application of the descriptors that this case asks the court to apply. Id. at *2-*3. In sum, none of the four primary-jurisdiction factors weigh in favor of referring the case to the FCC on this issue.
The remaining issues in this case go to deceptiveness, unfairness, and common-carrier status under the FTC Act. Congress did not place the interpretation of these terms within the realm of FCC discretion, nor does the FCC have special expertise in interpreting these FTC Act terms. Although the defendants-appellants contend that “[ejvery other court presented with these issues has referred them to the FCC,” in each of the cited cases, the claimed violation was of the Communications Act, not the FTC Act. GTE.Net LLC v. Cox Commc’ns, Inc., 185 F.Supp.2d 1141, 1144 (S.D.Cal.2002); Au-diotext Int'l, Ltd. v. MCI WorldCom Commc’ns, Inc., No. Civ. A. 00-3982, 2001 WL 1580316, at *2-*3 (E.D.Pa. Dec.11, 2001); LO/AD Commc’ns, 2001 WL 64741, at *1. We note that the FCC filed an amicus submission stating that it had no particular interest in or expertise over the case so as to warrant declining jurisdiction. For the reasons given above, we conclude that the primary-jurisdiction doctrine does not require referring this case to the FCC.
III. The Filed-Rate Doctrine
The defendants-appellants also contend that the filed-rate doctrine, also known as the filed-tariff doctrine, deprives the FTC of standing and requires dismissal of its complaint. That doctrine is grounded statutorily in the Communications Act’s requirement that all common carriers file a schedule of their rates, i.e., a tariff, for FCC • approval. 47 U.S.C. § 203(a). Once the FCC approves the tariff, the rates filed for the carrier’s services are the only lawful charge. Id. § 203(c); AT & T Co. v. Cent. Office Tel., Inc., 524 U.S. 214, 222, 118 S.Ct. 1956, 141 L.Ed.2d 222 (1998). Under the filed-rate doctrine, customers of the carrier “are charged with notice of [the tariff], and they as well as the carrier must abide by it, unless it is found by the [FCC] to be unreasonable.” Cent. Office Tel., 524 U.S. at 222, 118 S.Ct. 1956 (quoting Louisville & Nashville Ry. Co. v. Maxwell, 237 U.S. 94, 97, 35 S.Ct. 494, 59 L.Ed. 853 (1915)). The doctrine serves the dual purposes of preventing carrier price discrimination and reserving the rate-making role to federal agencies by keeping courts from adjudicating the validity or reasonableness of carrier charges. Marcus v. AT & T Corp., 138 F.3d 46, 58 (2d Cir.1998). It is applied strictly to prevent a plaintiff from bringing a challenge to the validity of a filed tariff, “even in the face of apparent inequities.” Id. at 59.
*62The defendants-appellants contend that the filed-rate doctrine bars the present action. They reason that because the FCC approved AT & T’s and Sprint’s tariffs for the carriage of calls placed to Madagascar and because the bills sent to line subscribers charged only tariffed rates, any lawsuit founded on the premise that some subscribers were not required to pay for calls placed over their lines is attacking the validity of FCC-approved tariffed rates and is therefore barred by the filed-rate doctrine. Underlying this assertion, however, is the foundational assumption that the bills in dispute were for services covered by the FCC-approved tariffs. If, in fact, no filed tariff covered the service for which the defendants-appellants sought to charge line subscribers, there would be no filed rate to charge subscribers with notice of and no tariffs validity would be called into question by the FTC’s lawsuit.
We hold that the filed-rate doctrine does not apply in this case because the defendants-appellants point to no tariff that covers the actual service rendered to users of their billing system. The defendants-appellants contend that the tariffs filed by AT & T and Sprint apply, but those tariffs cover only telecommunications services, not the information services provided here. The Communications Act defines these two categories of service. A “telecommunications service” (for example, the carriage of a basic voice telephone call) is the offering of “the transmission, between or among points specified by the user, of information of the user’s choosing, without change in the form or content of the information as sent and received.” 47 U.S.C. § 153(43), (46). An “information service,” on the other hand, offers “a capability for generating, acquiring, storing, transforming, processing, retrieving, utilizing, or making available information via telecommunications.” Id. § 153(20). Examples include services allowing online browsing or electronic publishing. Id.; see generally Universal Service Report, 13 F.C.C.R. 11501, 1998 WL 166178 (1998) (discussing in depth the meaning of these statutory terms). The FCC has affirmed that the two types of service are mutually exclusive. Universal Service Report, 13 F.C.C.R. at 11507-08, ¶ 13. The FCC has further explained that “mixed or hybrid services,” in which the provider “offers a capability for generating, acquiring, ... or making available information via telecommunications, and as an inseparable part of that service transmits information supplied or requested by the user,” are information services and not telecommunications services. Id. at 11529, ¶¶ 56-57.
In applying these definitions, the defendants-appellants would have us focus on only part of their billing system. They contend, and it is certainly true, that the carriers handling transmission of computer users’ phone calls — AT & T, Sprint, and AT & T U.K./Viatel — did not change the form or content of the information transmitted. But examining only the service provided by these carriers misses the fundamental question in the filed-rate-doctrine analysis: the nature of the service for which consumers were billed. In this case, while the pure transmission of information — provided by AT & T, Sprint, and AT & T U.K./Viatel — was part of the service rendered to computer users, those users received more as part of their purchase, namely, adult content. It can hardly be denied that access to adult websites motivated computer users to run the GIB dialer program and incur charges via the defendants-appellants’ billing system. Indeed, the funds collected from paying line subscribers compensated both telecommunications carriers and adult-website operators. As explained above, and as undisputed by the parties, online adult entertainment is an information service and *63is therefore not covered by the AT & T or Sprint tariffs upon which the defendants-appellants rely. Because the defendants-appellants point to no other tariff covering the information service rendered to users of their billing system, the filed-rate doctrine does not apply. Accordingly, the FTC has standing to bring this action.
IV. Violation of the FTC Act
Section 5(a)(1) of the FTC Act declares unlawful “[u]nfair or deceptive acts or practices in or affecting commerce.” 15 U.S.C. § 45(a)(1). Following a bench trial, the district court ruled that the FTC had proved three bases for relief under § 5(a)(1), and the court entered judgment against the defendants-appellants accordingly. On appeal, the defendants-appellants contend that the district court erred because the FTC did not prove each of the elements required for relief. We review the district court’s conclusions of law de novo and its factual findings for clear error. United States v. Visa U.S.A., Inc., 344 F.3d 229, 238 (2d Cir.2003).
A. Count I
In Count I, the FTC alleged that the defendants-appellants engaged in a deceptive act or practice by falsely representing that a consumer could not successfully avoid charges for adult-website content accessed over the consumer’s telephone line, even if the consumer did not access the content or authorize others to do so. To prove a deceptive act or practice under § 5(a)(1), the FTC must show three elements: “[1] a representation, omission, or practice, that [2] is likely to mislead consumers acting reasonably under the circumstances, and [3], the representation, omission, or practice is material.” In re Cliffdale Assocs., Inc., 103 F.T.C. 110, 165 (1984); accord FTC v. Tashman, 318 F.3d 1273, 1277 (11th Cir. 2003); FTC v. World Travel Vacation Brokers, Inc., 861 F.2d 1020, 1029 (7th Cir. 1988). The deception need not be made with intent to deceive; it is enough that the representations or practices were likely to mislead consumers acting reasonably. World Travel, 861 F.2d at 1029.
The FTC contends that the first element is satisfied by proof that the defendants-appellants caused telephone-line subscribers to receive explicit and implicit representations that they could not successfully avoid paying charges for adult entertainment that had been accessed over their phone lines — what we call a “representation of uncontestability.” The district court found that during the AT & T period, the defendants-appellants caused charges for adult entertainment to appear on AT & T phone bills as telephone calls, thereby “capitalizing] on the common and well-founded perception held by consumers that they must pay their telephone bills, irrespective of whether they made or authorized the calls.” The district court found that this representation was also made during the Sprint Period by the format of the Verity bills and the call-center messages delivered to bill recipients. Upon reviewing the bills and call-center practices, we find that it was not clearly erroneous for the district court to find that they conveyed a representation of uncon-testability. See, e.g., Kemp v. AT & T Co., 393 F.3d 1354, 1360 (11th Cir.2004) (“It was clearly foreseeable that this [phone-bill] formatting[, which listed information-service purchases as long-distance-telephone-call charges,] would cause some customers to think that ... the charges had to be paid in order to maintain phone service.”).
The second requirement for § 5(a)(1) liability is that the defendants-appellants’ representation be likely to mislead consumers acting reasonably. The FTC con*64tends that the representation of uncontes-tabilty was false and therefore likely to mislead consumers who did not use or authorize others to use the adult entertainment in question; the defendants-appellants contend that the representation was rendered true by both the filed-rate doctrine and common law agency principles. We have already explained why the filed-rate doctrine does not apply, see supra Part III, so we turn now to the defendants-appellants’ agency argument.
Under common law agency principles, a person is liable to pay for services that she does not herself contract for if another person has actual, apparent, or implied authority to consent on her behalf to pay for the services. Merrill Lynch Interfunding, Inc. v. Argenti, 155 F.3d 113, 122 (2d Cir.1998); Restatement (Second) of Agency § 140 (1958). The defendants-appellants rely on apparent authority, contending that all calls made over a subscriber’s telephone line were necessarily made with the subscriber’s apparent authority because any user of a computer connected to that telephone line must have been given authority by the line subscriber to use the computer.
Apparent authority, “[u]nlike express or implied authority, ... exists entirely apart from the principal’s, manifestations of consent to the agent.” Towers World Airways, Inc. v. PHH Aviation Sys., Inc., 933 F.2d 174, 177 (2d Cir.1991). Rather, it would derive here either from manifestations of the principal (the line subscriber) to a third party (an entity involved in the billing system) or from the putative agent’s (the computer user’s) position, when justified by ordinary expectations and habits. Restatement (Second) of Agency § 49 cmts. a, b (1958). The defendants-appellants analogize the present case to Towers World Airways, supra, in which we held that purchases by a company’s employee made with a properly issued company credit card are made with the apparent authority of the company. 933 F.2d at 177-79. Notably, that case concerned a principal’s entrustment of a payment mechanism to its agent and relied upon specific customs of the aviation industry in finding apparent authority. Id. at 178. Here, in contrast, the computer is a multipurpose tool that is not primarily understood as a payment mechanism, and in the ordinary habits of human behavior, one does not reasonably infer that because a person is authorized to use a computer, the subscriber to the telephone line connected to that computer has authorized the computer user to purchase online content on the subscriber’s account. Apparent authority does not exist on these facts.
The representation of uncontestability is therefore false, unsupported by either the filed-rate doctrine or common law agency principles. Because the defendants-appellants offer no reason why this misrepresentation would not be likely to mislead consumers acting reasonably, we find that the district court did not err in finding that the FTC proved the second element of its Count I claim.
Finally, to establish a deceptive act or practice under § 5(a)(1), the FTC must prove that the misrepresentation was material to consumers. The district judge never expressly addressed whether the representation of uncontestability was material, but because he stated the correct legal standard for § 5(a)(1) deceptiveness liability and found liability upon labeling the representation “materially false,” we understand him to mean that the misrepresentation was material and false — a slightly different, but justifiable, conclusion. The FTC submitted evidence from which the district court could conclude that telephone-line subscribers found the representation material to their decision wheth*65er to pay the billed charges because of the worry of telephone-line disconnection, the perception of the futility of challenging the charges, the desire to avoid credit-score injury, or some combination of these factors. In any event, nowhere in the defendants-appellants’ briefing do they contend that the misrepresentation of uncontesta-bility was immaterial, so we deem the issue waived. Norton v. Sam’s Club, 145 F.3d 114, 117 (2d Cir.1998).
In sum, because the FTC proved all three elements of its § 5(a)(1) claim premised on the deceptive representation of uncontestability, the district court did not err in holding that the defendants-appellants violated the FTC Act. Because § 5(a)(1) is phrased in the disjunctive, prohibiting “unfair or deceptive” acts or practices, 15 U.S.C. § 45(a)(1), the FTC need not prove its other claims to obtain relief under the FTC Act. Liability is supported by the Count I claim alone. Nonetheless, we give brief attention below to Count II because it affects the scope of injunctive relief.
B. Count II
The district court held the defendants-appellants liable under Count II of the FTC’s complaint, which alleged that “billing line subscribers who did not use or authorize use of the Internet services offered by the defendants’ clients” was an unfair trade practice. We hold that the defendants-appellants waived their right to contest this unfair-practices determination by not raising it as an issue on appeal. The defendants-appellants presented several issues for review in their opening brief, and in the paragraph concerning FTC Act liability, they mention only the district court’s deceptiveness determination; indeed, in the four pages of their opening brief in which they discuss FTC Act liability, they focus only on deceptive-practice liability. Because the defendants-appellants did. not contest the district court’s unfair-practices determination until their reply brief, and then only cursorily, we deem it waived on appeal. Tischmann v. ITT/Sheraton Corp., 145 F.3d 561, 568 n. 4 (2d Cir.1998) (holding an argument waived because the appellant did not raise it until his reply brief); United States v. Gabriel, 125 F.3d 89, 100 n. 6 (2d Cir.1997) (same). For this reason, we affirm the district court’s determination that the act of “billing line subscribers who did not use or authorize use of the internet services offered by the defendants’ clients” is an unfair trade practice within the meaning of § 5(a)(1). Because the FTC prevails on the basis of waiver, we do not reach, and therefore express no opinion on, the district court’s rulings on the merits of the Count II claim.
C. Count III
In Count III, the FTC alleged that it was a deceptive practice for the defendants-appellants to cause consumers to be billed for calls to Madagascar when the calls actually terminated in the United Kingdom. Because liability under this count is not necessary to support the district court’s order of relief, we express no opinion on the district court’s determination that the FTC proved this claim for relief.
‡ ‡
We affirm the injunctive components of the district court’s October 26, 2004 final order for relief because they are supported by the defendants-appellants’ liability under Counts I and II of the FTC’s complaint.
V. Restitution
Two issues determine whether the district court’s award of disgorgement relief to the FTC should be affirmed. First, is *66restitution an available remedy under § 13(b) of the FTC Act, the provision under which the FTC seeks relief? Second, if so, did the district court correctly administer the restitution remedy?
A. Restitution under § 13(b) of the FTC Act
The FTC brought this action under the second proviso of § 13(b) of the FTC Act, which states that “in proper cases the [FTC] may seek, and after proper proof, the court may issue, a permanent injunction.” 15 U.S.C. § 53(b). Although this provision does not expressly provide for restitution, several courts, including the Fifth,5 Seventh,6 Eighth,7 Ninth,8 and Eleventh9 Circuits, have concluded that § 13(b) of the FTC Act allows restitution or other ancillary equitable relief. Cf. United States v. Lane Labs-USA, Inc., 427 F.3d 219, 223 (3d Cir.2005) (holding that even though a statute did not expressly allow courts to -award restitution, “such specifieity is not required where the government properly invokes a court’s equitable jurisdiction”).
The defendants-appellants do not contest on appeal the district court’s holding that restitution is available as ancillary equitable relief under § 13(b) of the FTC Act, so we assume without deciding that the district court’s holding is correct. See, e.g., United States v. Georgia, — U.S. -, 126 S.Ct. 877, 880, 163 L.Ed.2d 650 (2006) (applying this type of assumption).
The defendants-appellants do argue, however, that such restitution must be limited to so-called equitable restitution. We agree. This contention is based on the fact that two types of restitution are distinguishable: As Justice Scalia explained in Great-West Life & Annuity Insurance Co. v. Knudson, 534 U.S. 204, 122 S.Ct. 708, 151 L.Ed.2d 635 (2002), “In the days of the divided bench, restitution was available in certain cases at law, and in certain others in equity.” Id. at 212, 122 S.Ct. 708. Eq*67uitable restitution allowed the plaintiff to recover money or property in the defendant’s possession that could “clearly be traced” to money or property “identified as belonging in good conscience to the plaintiff.” Id. Legal restitution, on the other hand, was awarded when the plaintiff could not assert title to or the right to possession of particular property but nevertheless had some basis for recovering for some benefit that the defendant wrongly received from the plaintiff. Id. Here, because the availability of restitution under § 13(b) of the FTC Act, to the extent it exists, derives from the district court’s equitable jurisdiction, it follows that the district court may award only equitable restitution.10 The fact that only an equitable remedy is available eviscerates the defeñ-dants-appellants’ contention that the Seventh Amendment confers a right to a jury trial in this case. See Granfinanciera, S.A. v. Nordberg, 492 U.S. 33, 41, 109 S.Ct. 2782, 106 L.Ed.2d 26 (1989).
B. Administering restitutionary relief
The district court strayed off course in its application of the two-step burden-shifting framework for calculating the size of disgorgement relief. This framework requires the FTC to first “show that its calculations reasonably approximated” the amount of the defendant’s unjust gains, after which “the burden shifts to the defendants to show that those figures were inaccurate.” FTC v. Febre, 128 F.3d 530, 535 (7th Cir.1997) (citing SEC v. Lorin, 76 F.3d 458, 462 (2d Cir. 1996) (per curiam)). Two errors pervade the district court’s administration of this framework: (1) misidentifying the baseline for-restitutionary relief and (2) prematurely shifting the burden of proof to the defendants-appellants.
1. Misidentifying the restitutionary baseline
The district court measured the appropriate amount of restitution as “the full amount lost by consumers.” This was error. The appropriate measure for restitution is the benefit unjustly received by the defendants. See Pereira v. Farace, 413 F.3d 330, 340 (2d Cir.2005) (stating that “restitution is measured by a defendant’s unjust gain, rather than by a plaintiffs loss” (internal quotation and alteration marks omitted)) (citing Great-West Life & Annuity Ins. Co., 534 U.S. at 229, 122 S.Ct. 708 (Ginsburg, J., dissenting)); Restatement (Third) of Restitution § 2 (Discussion Draft 2000) (“Liability in restitution is based on and measured by the receipt of a benefit____”); Douglas Lay-cock, The Scope and Significance of Restitution, 67 Tex. L.Rev. 1277, 1279 (1989) (“[Rjestitution measures recovery by defendant’s gain rather than plaintiffs loss”). Labeling the remedy “consumer redress” or “disgorgement,” each a restitutionary remedy, does not alter the basic principle that restitution is measured by the defendant’s gain.
*68Undeniably, in many cases in which the FTC seeks restitution, the defendant’s gain will be equal to the consumer’s loss because the consumer buys goods or services directly from the defendant. Thus, in these cases it is not inaccurate to say that restitution is measured by the consumer’s loss. But it is incorrect to generalize this shorthand and apply it as a principle in cases where the two amounts differ — for example, when some middleman not party to the lawsuit takes some of the consumer’s money before it reaches a defendant’s hands. Both the district court and the FTC in its brief adopt this fallacy, relying on shorthand from cases in which only one who sold directly to consumers was sued. See FTC v. Febre, 128 F.3d 530, 536-37 (7th Cir.1997) (direct seller sued); FTC v. Gem Merch. Corp., 87 F.3d 466, 469-70 (11th Cir.1996) (direct seller sued; court held that “disgorgement, the purpose of which ‘is not to compensate the victims of fraud, but to deprive the wrongdoer of his ill-gotten gain,’ is appropriate”); FTC v. Sec. Rare Coin & Bullion Corp., 931 F.2d 1312, 1316 (8th Cir.1991) (direct sellers sued); FTC v. Amy Travel Serv., Inc., 875 F.2d 564, 573-75 (7th Cir.1989) (direct sellers sued); FTC v. Medicor, LLC, 217 F.Supp.2d 1048, 1058 (C.D.Cal.2002) (direct sellers sued); FTC v. Five-Star Auto Club, Inc., 97 F.Supp.2d 502, 534 (S.D.N.Y.2000) (measuring the “full amount lost by consumers” by the amount taken in by the defendant).11
This error manifested itself in the district court’s calculation of the sum to be disgorged. For the AT & T Period, the district court determined that consumers paid AT & T $16.3 million for the relevant services (so-called videotext services). We uphold this determination but instruct the district court on remand to further consider how much of this sum was in fact received by the defendants-appellants and is therefore subject to an order of restitution. The cascading payment structure during the AT & T Period indicates that AT & T, AT & T U.K./Viatel, and Telecom Malagasy received some fraction of the money paid by consumers before any payments were made to the defendants-appellants. Only the remaining fraction of total billings unjustly enriched the defendants-appellants and may be the basis for a disgorgement remedy.
For the Sprint Period, the cascading payment structure flowed differently. The defendants-appellants received consumers’ money through eBillit, and they paid Sprint, AT & T U.K./Viatel, Telecom Malagasy, and GIB from those unjustly received consumer funds. Thus, for the Sprint Period, the district ■ court should determine the amount of the $1.6 million in total billings that the defendants-appellants received from eBillit, without deducting monies paid by the defendants-appellants to other parties. For both periods, the focus of the district court’s restitution calculation should be on the defendants-appellants’ unjust gains.
2. Prematurely shifting the burden of proof to the defendants-appellants
The two-step burden-shifting framework for establishing the size of disgorgement relief requires the plaintiff to first “show that its calculations reasonably approximated” the amount of the defendant’s unjust gains. Febre, 128 F.3d at *69535 (citing Lorin, 76 F.3d at 462). Here, because some fraction of consumers who paid the bills incurred through the defem dants-appellants’ billing system actually used or authorized others to use the services at issue, the amount of the defendants-appellants’ unjust gains is only a fraction of the amount of their overall gains from the billing system. A reasonable approximation of the defendants-appellants’ unjust gains must take this into account.
Although the district court recognized that restitution is based on unjust payments, not just overall payments, it never explained its basis for concluding that the overall sum collected through the billing system reasonably approximated the amount of unjustly obtained funds. Indeed, for the AT & T Period, the consumer declarations offered by the FTC (although not all considered by the district court because it found them unnecessary) indicate that AT & T gave a one-time credit to any caller who complained to AT & T about the charges. See Pl.’s Exs. 1, 82-92. No consumer declaration indicates that AT & T refused to provide a credit or refund. Therefore, unlike during the Sprint Period, there is little basis to conclude that unjust gains were obtained from complaining customers during the AT & T Period (although this does not necessarily mean that every charge collected by AT & T was from a person who themselves accessed or authorized others to access an adult website). Accordingly, we do not think it reasonable to assume that total AT & T Period collections approximates the total amount paid by consumers who did not authorize use of the adult entertainment provided.12
Thus, because the district court did not first assess the reasonableness of the FTC’s approximation of unjust gain, the district court was premature in shifting the burden of proof to the defendants-appellants, which in turn allowed the district court to invoke the principle that “ ‘[t]he risk of uncertainty should fall on the wrongdoer whose illegal conduct created the uncertainty.’ ” Febre, 128 F.3d at 535 (quoting SEC v. First City Fin. Corp., 890 F.2d 1215, 1232 (D.C.Cir.1989)). This presumption against the wrongdoer should not have been invoked without first establishing a reasonable approximation of unjust gain because this presumption applies only in the second stage of the burden-shifting framework. See id. at 535 (invoking the presumption to hold that the defendants could not satisfy their burden of proving the inaccuracy of the FTC’s calculations); First City Fin., 890 F.2d at 1232 (holding that the government satisfied its burden of providing a reasonable approximation of unjust enrichment and that defendants could not meet their burden of rebutting the government’s calculations). If the law were otherwise, the FTC would be relieved at the first stage from submitting a reasonable approximation of unjust gain and could recover any amount that it chose to submit, however unreasonable, that fit within the presumption against the wrongdoer.
Of course, the reasonableness of an approximation varies with the degree of precision possible. But here, the district court required no precision in the FTC’s approximation even though precision could be had. The FTC’s investigatory power gives it the capacity to estimate with some degree of precision how many telephone-*70line subscribers who paid the bills for adult entertainment did so despite not using or authorizing others to use such services.
% * # ^ Hí #
For the foregoing reasons, we vacate the monetary award of the district court’s October 26, 2004 final order of relief and remand the case for further proceedings on the size of the disgorgement award. On remand, we direct the district court to revise its computations to focus on the benefits unjustly obtained by the defendants rather than the losses of consumers and to entertain only reasonable approximations of the defendants-appellants’ unjust gains, rather than their overall gains, before shifting the burden to the defendants-appellants to refute the approximation, handicapped by the presumption against wrongdoer-created uncertainty.
VI. Contempt Sanctions
The district court held defendants-appellants Green and Shein in contempt of court for failing to comply with a financial-disclosure requirement of the preliminary injunction and sanctioned them for the contempt by order filed May 2, 2001. The contempt sanctions imposed were coercive, intended to induce compliance with the financial-disclosure obligation by imposing per-day fines for noncompliance and ordering Green’s and Shein’s civil confinement until they completed the forms. See generally United States v. United Mine Workers of Am., 330 U.S. 258, 303, 67 S.Ct. 677, 91 L.Ed. 884 (1947) (“Judicial sanctions in civil contempt proceedings may, in a proper case, be employed for either or both of two purposes; to coerce the defendant into compliance with the court’s order, and to compensate the complainant for losses sustained.”).
Green and Shein appeal from the contempt order, which we now vacate. Our reasoning is straightforward. The permanent injunction that dissolved and replaced the preliminary injunction does not itself contain a financial-disclosure requirement. The district court therefore no longer requires Green and Shein to do the act that the contempt sanctions coerce them do to. Thus, the sanctions must be vacated. See Consol. Rail Corp. v. Yashinsky, 170 F.3d 591, 596 (6th Cir.1999) (holding that the expiration of a judgment mooted the coercive per-day fines of a contempt sanction imposed for not satisfying the judgment); see also Shillitani v. United States, 384 U.S. 364, 372, 86 S.Ct. 1531, 16 L.Ed.2d 622 (1966) (ordering a contempt sanction vacated as moot on appeal from the sanction). Green and Shein are relieved of all fines imposed by the order and are no longer subject to civil confinement under its terms. Of course, nothing here prevents the FTC from moving in the district court on remand for an appropriate order to obtain the financial disclosure desired and from seeking a new set of sanctions if the district court’s orders are subsequently disobeyed.
CONCLUSION
We affirm all components of the district court’s October 26, 2004 final order of relief except for the monetary judgment contained therein, which we vacate. We also vacate the district court’s May 2, 2001 contempt order. The case is remanded to the district court for further proceedings consistent with this opinion.
. Defendants-appellants Verity, Green, and Shein do not claim the benefit of this exception to the FTC Act.
. Section 5(a)(1) of the FTC Act originally prohibited only "[u]nfair methods of competition in or affecting commerce.” The Wheeler-Lea Act of 1938 expanded § 5(a)(1) to also prohibit “unfair or deceptive acts or practices in or affecting commerce.” § 3, 52 Stat. Ill, 111 (1938).
. See generally James B. Speta, A Common Carrier Approach to Internet Interconnection, 54 Fed. Comm. L.J. 225, 263 (2002) ("CT]he Act responded to the facts that most states had established commissions governing local telephone service and that the ICC felt overburdened by its regulation of railroads.”).
. The notion of some indelible common carrier "status” under the Communications Act is highly questionable. See Southwestern Bell Tel. Co. v. FCC, 19 F.3d 1475, 1481 (D.C.Cir. 1994) (explaining that "[wjhether an entity in a given case is to be considered a common carrier or a private carrier turns on the particular practice under surveillance” and that the FCC "is not at liberty to subject [an] entity to regulation as a common carrier” if the entity “is acting as a private carrier for a particular service”); see also NARUC II, 533 F.2d at 608 ("[I]t is at least logical to conclude that one can be a common carrier with regard to some activities but not others.”); In re Audio Commc'ns, Inc., 1993 WL 525815, 8 F.C.C.R. 8697, 8698-99, ¶ 12 (1993) (“[A] single firm that is a common carrier in some roles need not be a common carrier in other roles.”).
. FTC v. Southwest Sunsites, Inc., 665 F.2d 711, 718 (5th Cir.1982) (concluding, that "a grant of jurisdiction such as that contained in Section 13(b) carries with it the authorization for the district court to exercise the full range of equitable remedies traditionally available to it” and holding that § 13(b) "contains no express limitations on the otherwise full powers of the district court to mold appropriate decrees under its traditional equitable jurisdiction”).
. FTC v. World Travel Vacation Brokers, Inc., 861 F.2d 1020, 1026 (7th Cir.1988) (holding that the district court had power to grant a preliminary injunction under § 13(b) because Congress did not limit the court’s inherent equitable powers).
. FTC v. Sec. Rare Coin & Bullion Corp., 931 F.2d 1312, 1314 (8th Cir.1991) (upholding the district court's rescission remedy despite § 13(b)’s failure to expressly allow that remedy because ”[w]here Congress allows resort to equity for the enforcement of a statute, all the inherent equitable powers of the district court are available for the proper and complete exercise of the court’s equitable jurisdiction, unless the statute explicitly, or by a necessaty and inescapable inference, limits the scope of that jurisdiction”) (citing Porter v. Warner Holding Co., 328 U.S. 395, 397-98, 66 S.Ct. 1086, 90 L.Ed. 1332 (1946)).
. FTC v. H.N. Singer, Inc., 668 F.2d 1107, 1113 (9th Cir.1982) (reasoning that the district court has authority to grant an asset freeze and rescission in a case brought under § 13(b) of the FTC Act because "[ujnless otherwise provided by statute, all the inherent equitable powers of the [district [c]ourt are available for the proper and complete exercise of that jurisdiction”).
. FTC v. Gem Merch. Corp., 87 F.3d 466, 469-70 (11th Cir.1996) (holding that the district court could award consumer redress for violations of the FTC Act because § 13(b) invoked the court’s equitable jurisdiction and "absent a clear command to the contrary, the district court’s equitable powers are extensive” and include "the power to grant restitution and disgorgement”).
. We emphasize that equitable restitution is not limited to an award of the very funds that unjustly enriched the defendant and are still in the defendant's possession. Rather, tracing principles apply to allow a plaintiff to follow unjustly obtained funds into their product in the defendant's possession. See Great West, 534 U.S. at 204, 122 S.Ct. 708 (holding that equitable restitution allows recovery "where money or property identified as belonging in good conscience to the plaintiff could clearly be traced to particular funds or property in the defendant’s possession.” (emphasis added)). For a review of tracing rules, see generally Douglas Laycock, Modem American Remedies (3d ed.2002) (summarizing tracing rules), and 1 George E. Palmer, The Law of Restitution § 2.14-. 16, at 175-207 (1978) (describing tracing rules in more detail).
. In another case cited by the FTC to support the monetary award, the court explicitly relied on the authorization of damages in § 19 of the FTC Act, not just on the court's equitable power, to allow a remedy under the FTC Act of more than the defendant’s gains. See FTC v. Figgie Int'l, Inc., 994 F.2d 595, 606 (9th Cir.1993).
. Our objection does not translate to the Sprint Period, however, for which the reasonableness of the FTC's approximation is less doubtful. The evidence indicates that the customer-service center was a disaster during this period and that at least some consumers did pay for services for which they were improperly billed.