
    GOODRICH CORP. (f/k/a B.F. Goodrich Co.), Crompton Manufacturing Co. (f/ k/a Uniroyal Chemical Co., Inc.), Reynolds Metal Co. (f/k/a Reynolds Aluminum Building Products Co.), Naugatuck Glass Co., Naugatuck Treatment Co., Cadbury Beverages, Inc., Coltec Industries, Inc., B.P. Amoco (f/k/a Ken Chas Reserve Co.), Risdon Corp., Unisys Corp., and Kerite Co., Plaintiffs-Appellants-Cross-Appellees, v. TOWN OF MIDDLEBURY, Town of Hamden, Town of Orange, Town of Seymour, Town of Westport, City of New Haven, Defendants-Appellees-Cross-Appellants, Borough of Naugatuck, Defendant-Appellee.
    Docket Nos. 01-6014, 01-6016, 01-6018.
    United States Court of Appeals, Second Circuit.
    Argued: April 12, 2002.
    Decided: Oct. 29, 2002.
    
      David E. Rosengren, Pepe & Hazard LLP, Hartford, CT, for Plaintiffs-Appellants-Cross-Appellees Goodrich Corp., Crompton Manufacturing Co., Inc. and Reynolds Metal Co.
    Christopher P. McCormack, Tyler Cooper & Acorn, LLP, New Haven, CT (Robert W. Allen, Robert B. Flynn, Timothy P. Jensen, Matthew A. Sokol, on the brief), for Plaintiffs-Appellants Members of the Laurel Park Coalition.
    Ann Catino, Halloran & Sage LLP, Hartford, Connecticut (Lori D. DiBella and John C. Huggins, Halloran & Sage LLP, Hartford Connecticut, and Nicholas J. Harding, Kosloff & Harding, West Hartford Connecticut, on the brief), for Defendants-Appellees-Cross-Appellants the Towns of Hamden, Middlebury, Orange, Seymour and Westport and the City of New Haven.
    Kevin McSherry, McSherry Law Office, Naugatuck, CT, for Defendant-Appellee Borough of Naugatuck.
    Before McLAUGHLIN, POOLER and B.D. PARKER, Circuit Judges.
   McLAUGHLIN, Circuit Judge.

The shades of Jamdyce & Jamdyce stalk this litigation. Charles Dickens, Bleak House. For the third time in a decade, we enter the legal morass created by a half-century of dumping hazardous waste at two Connecticut landfills, named Beacon Heights and Laurel Park, respectively. This time around, we consider, among other things, to what extent several municipalities must contribute to response costs that two coalitions of corporate polluters incurred cleaning up the mess.

BACKGROUND

We have set forth this case’s background in exhaustive detail in two prior opinions. See B.F. Goodrich v. Betkoski, 99 F.3d 505 (2d Cir.1996) (“Betkoski”); B.F. Goodrich Co. v. Murtha, 958 F.2d 1192 (2d Cir.1992) (“Murtha”). We presume some familiarity with these opinions and state only the facts pertinent to this appeal.

A. Procedural History

The Beacon Heights and Laurel Park landfills accepted industrial wastes, e.g., liquid chemical wastes, deposited by commercial entities and accepted municipal solid waste (“MSW”) deposited by several Connecticut municipalities. After the evidence confirmed that leachate was escaping from the landfills — a condition that threatened the local water supply — the Environmental Protection Agency (“EPA”) designated the landfills as “Superfund” sites on the National Priority List under the Comprehensive Environmental Response, Compensation, and Liability Act of 1980 (“CERCLA”), 42 U.S.C. §§ 9601-75, as amended by the Superfund Amendments and Reauthorization Act of 1986 (“SARA”), Pub.L. No. 99-499, 100 Stat. 1613 (1986).

The EPA began to identify the parties it believed were responsible for causing the mess. The EPA identified, among others, Terrance and Harold Murtha, and several corporations controlled by them (collectively, the “Murthas”), who owned and operated Laurel Park from 1961 until 1987 and owned Beacon Heights from 1970 to 1987. The EPA also identified several companies that had dumped waste at the landfills as potentially responsible parties (“PRPs”).

Two eponymous coalitions comprising corporate polluters, the Beacon Heights Coalition (the “BHC”) and the Laurel Park Coalition (the “LPC”), eventually emerged, entering into separate consent decrees with the EPA. Betkoski, 99 F.3d at 512. Under the consent decrees, the Coalitions agreed to “remediate” (i.e., clean up) the environmental damage to their namesake landfills. Id. Remediation at each landfill required placement of a cap over each landfill and extraction and treatment of dump leachate and groundwater from the landfills.

As we explained in Betkoski, the Mur-thas settled the claims brought against them by the United States, the State of Connecticut and the BHC, and assigned to the BHC any contribution claims they had against other third-party defendants. Id. Although the Laurel Park Coalition had not been formed when the Murtha consent decree was negotiated, the parties agreed that should an industrial coalition form within 18 months of the entry of the Mur-thas’ consent decree, the coalition would receive a portion of the funds. The LPC formed within that time period and received the allotted funds. Id.

Thereafter, the Coalitions pursued claims against hundreds of third party defendants, including several Connecticut municipalities, seeking contribution toward the costs of the cleanup, pursuant to CERCLA § 113(f), 42 U.S.C. § 9613(f). The Coalitions’ contribution claims against the municipalities are posited on the presence of hazardous substances, as constituent parts of consumer and industrial products, in the MSW that the municipalities dumped into the landfills.

In Murbha, we considered the municipalities’ appeal from the district court’s denial of their motion for summary judgment on the contribution claims brought against them, the BHC, and the Murthas. The municipalities argued that: (1) MSW did not fit within CERCLA’s definition of “hazardous substance”; (2) the exemption for household hazardous waste under the Resource Conservation and Recovery Act (“RCRA”), 42 U.S.C. §§ 6901-6992k, was incorporated into CERCLA; and (3) we should defer to the EPA’s interpretation of CERCLA, which purportedly excluded MSW from the definition of “hazardous substance.” Murbha, 958 F.2d at 1199, 1201, 1205. We rejected each of these contentions, holding that “the definition of hazardous substance under CERCLA includes municipal solid waste if that waste contains a hazardous substance, found in any amount, that is listed in any of the subsections” of 42 U.S.C. § 9601(14). Murbha, 958 F.2d at 1206.

On remand, the municipalities again moved for summary judgment, this time seeking dismissal of the contribution claims brought by both BHC and LPC. The district court granted the municipalities’ motion, holding that they bore no liability whatsoever for contribution under CERCLA because: (1) while products found in their MSW may have contained components or elements identified as hazardous substances under CERCLA, the products were not themselves listed as hazardous substances under the Act; (2) the hazardous substances contained in the products were not in releaseable form; and (3) they dumped only negligible amounts of hazardous substances. B.F. Goodrich Co. v. Murbha, 815 F.Supp. 539, 545-M6 (D.Conn.1993).

On appeal, we reversed the district court’s grant of summary judgment to the municipalities, holding that: (1) a mixture or waste solution containing any hazardous substance, even if contained only as a constituent element of a product, is deemed “hazardous” under CERCLA; (2) a defendant who disposes of hazardous substances that are not independently re-leaseable when the waste is dumped may still be held hable under CERCLA; (3) CERCLA’s “hazardous substance” definition includes even minuscule or nominal amounts; and (4) genuine issues of material fact precluded summary judgment on whether there were hazardous substances contained in the solid waste the municipalities deposited at the landfills. BetkosM, 99 F.3d at 515-16, 526-27. Accordingly, we remanded the case to the district court for further proceedings. .

By this juncture, the Coalitions had settled their claims against the vast majority of third-party defendants. On remand, the district court referred the Coalitions’ contribution claims against the remaining municipalities and other third-party defendants to a Special Master (the “Master”), for mediation and, if necessary, an eviden-tiary hearing followed by a report and recommendation (the “Report”) pursuant to Fed.R.Civ.P. 53(e). The Order of Reference charged the Master with “finding facts, drawing conclusions and recommending what, if any, contribution the involved Third-Party Defendant, or potentially responsible party, should be required to make under CERCLA as it has been interpreted by the Second Circuit Court of Appeals.” Moreover, the Order of Reference required the Master’s report to “recite the factors on which a recommended equitable allocation of the remediation cost as to a Third Party Defendant or potentially responsible party is based.”

B. Proceedings Before the Master

Through the Master’s mediation efforts, a few more third-party defendants settled the Coalitions’ contribution claims against them. Thus, when, the Master’s mediation efforts concluded, only a handful of third-party defendants remained in the contribution action, including defendants the Towns of Hamden, Middlebury, Orange, Seymour and Westport and the City of New Haven (collectively, the “Municipal Defendants”), and. defendant-appellee the Borough of Naugatuck. Accordingly, the Master proceeded to the merits of the Coalitions’ contribution claims against these parties.

The Master held an evidentiary hearing over a six-month period, during which 41 witnesses testified and over 500 evidentia-ry exhibits were introduced. We summarize the testimony of the parties’ key witnesses below.

The Coalitions relied extensively on the testimony of Dr. Kirk W. Brown, a Professor of Soil and Crop Sciences at Texas A & M University, to support their claim that the Municipal Defendants’ MSW contained hazardous substances that were eventually released into dump leachate. Dr. Brown testified that MSW typically contains partially-spent containers of common household and industrial products, the contents of which contain some hazardous substances. According to Dr. Brown, the hazardous components of such products are eventually released from their containers and find their way into leachate at landfills. He compared the “contaminants of concern” which the EPA said were in the dump leachate at'both landfills with the hazardous substances typically found in MSW. Dr. Brown concluded that hazardous substances released from the Municipal Defendants’ waste were likely a major source of hazardous substances found in leachate at the landfills. He did not undertake a site-specific analysis of the MSW dumped at each of the landfills. Rather, he relied on studies by the EPA and others, which he believed showed that the amount of hazardous substances present in a given town’s MSW generally will comport with a national profile.

Additionally, the LPC and BHC each presented allocation experts who submitted recommended allocation formulas for the court to apply in assessing contribution shares. The LPC elicited testimony from Austin Carey, Esq., an attorney who had participated in the formation of LPC. Carey explained that the LPC members, with the help of a neutral consultant, had created an allocation formula to apportion responsibility for response costs among its members. The LPC’s allocation model incorporated the volume of waste each party deposited, adjusted by: (1) a hazard factor designed to account for the relative toxicity of each party’s waste and the relative mobility of the hazardous substances contained therein; and (2) compaction and burning factors to account for changes in the waste’s volume after it was dumped. Using this model, Carey allocated percentage shares of response costs for Middle-bury, Orange and Seymour as 4.72%, 3.41% and 3.22%, respectively.

As its allocation witness, the BHC presented Richard Lane White, a director of the consulting firm Putnam, Hayes & Bartlett, Inc. (“Putnam Hayes”), who submitted an expert report allocating response costs among the parties alleged to be liable for response costs at Beacon Heights. The BHC’s “cost-causation” model was based on the volume of waste each party deposited, adjusted by: (1) compaction and burning factors; and (2) a cosLcausation factor designed to account for the specific response costs incurred by the party’s waste. Using this formula, White allocated the Municipal Defendants’ percentage share of response costs for Orange, Seymour, Westport, Hamden, New Haven and Middlebury as 8.9256%, 5.8077%, 32.2511%, 2.1040%, 2.8554% and 1.9640%, respectively.

Additionally, the Coalitions sought to establish the volume of waste disposed of by the Municipal Defendants. Among other evidence, they relied on the Municipal Defendants’ responses to discovery orders and the testimony of Terrance Murtha, who testified about the Murthas’ business practices in hauling and disposing of waste.

The Coalitions also presented extensive evidence concerning the • costs they incurred cleaning up the landfills. These costs were tremendous: the BHC claimed it spent over $50 million for remediation of Beacon Heights, while the LPC claimed it spent over $30 million for remediation of Laurel Park.

The Municipal Defendants countered Dr. Brown’s testimony with testimony from Dr. Robert K. Ham, Professor Emeritus of Civil & Environmental Engineering at the University of Wisconsin Madison. Dr. Ham testified generally that: (1) the hazardous substances found in MSW are far less releaseable and mobile than those found in industrial waste; (2) the most releaseable and mobile hazardous substances deposited at the landfills were liquid wastes deposited by Coalition members; (3) the hazardous constituents contained in MSW are released only after their containers decompose, which occurs over a long period of time; and (4) such hazardous constituents will not pose a threat to the environment if they bond with other chemicals present in the landfill and form new, nonhazardous substances before they escape from the landfill. Nevertheless, Dr. Ham conceded that MSW contributed to groundwater, contamination at the landfills.

Additionally, the Municipal Defendants presented an alternative to the Coalitions’ recommended allocation formulas. This they did through testimony from their allocation expert, Muriel Robinette, a Vice President of Haley & Aldrich, Inc. (“Haley Aldrich”). Not surprisingly, Ms. Robi-nette advocated apportionment of response costs to the Municipal Defendants in lower percentage than the Coalitions’ experts’ allocation. Her allocation models, which were substantially similar for each landfill, were based on the total volume of waste dumped by each party, adjusted by compaction, burning, cost-causation, toxicity, mobility and persistence factors. Ms. Robinette allocated percentage shares of response costs at Laurel Park for Middle-bury, Seymour and Orange as 1.29%, 0.62% and 0.54%, respectively. She allocated the percentage shares of response costs at Beacon Heights for Orange, Seymour, Westport and Middlebury as 0.3989%, 0.2639%, 1.3576% and 0.0149%, respectively. These lower percentages reflected her view that the hazardous substances in the Municipal Defendants’ wastes were not significant “drivers” of many of the response costs undertaken by the Coalitions. For example, she opined that had MSW alone been deposited in the landfills, a less costly cap would have been required to control leachate, and that groundwater and leachate extraction would not have been required. Nevertheless, she conceded that the amount of -waste dumped at a landfill will affect the size and attendant cost of the cap.

The Municipal Defendants countered the Coalitions’s response cost evidence through, among other things, testimony from Jeffrey A. Klaiber, also a Vice President of Haley & Aldrich. Mr. Klaiber focused his opinion on whether the response costs claimed by the Coalitions were necessary and whether they conformed to the national contingency plan (the “NCP”) under CERCLA § 107(a)(4)(B). According to him, the BHC’s and LPC’s recoverable response costs were limited to approximately $20 million and $19 million, respectively. Like Ms. Robinette, he opined that had Beacon Heights and Laurel Park accepted only the Municipal Defendants’ MSW, the landfills would have been covered with far less expensive caps.

C. The Master’s Report

After the parties submitted their proposed findings of fact, the Master submitted his Report.

The Master provided only scant details of the volume of waste dumped by the Coalitions, limiting his findings in this regard to the amount of waste dumped by BHC member Uniroyal Chemical Company (“Uniroyal”) at Beacon Heights from 1947 to 1976. For the better part of his Report, he did not differentiate between Beacon Heights and Laurel Park, and he spoke of the Coalitions and their wastes in general terms. He characterized the Coalition members as “major corporations” whose “wastes consist of by-products of their manufacturing process.” Throughout his Report, the Master referred to the Coalitions’ waste as “liquid and other chemical wastes,” and “highly mobile and toxic chemical waste.” In general, he found the Coalitions’ waste to be more toxic as a whole, and the hazardous substances found therein more mobile and more releaseable, as compared to the Municipal Defendants’ waste.

Curiously, the Master never mentioned the volume of waste dumped by the Municipal Defendants, focusing instead on its characteristics. In this regard, he rejected Dr. Brown’s assertion that MSW comports to a national profile. Rather, he found that MSW “has no definition which is applicable universally to the waste streams of each and every municipality. Its components vary with the demographic makeup of the community which generates it.” The Master criticized Dr. Brown for failing to undertake a site-specific analysis to determine whether the Municipal Defendants’ waste contained partially spent containers of the products he referred to in his expert report and testimony and for failing to analyze whether the Municipal Defendants had implemented sorting and separation policies for waste disposal whereby certain household products containing hazardous substances would have been excluded from the MSW the Municipal Defendants dumped at the landfills. The Master further noted that the sources and studies Dr. Brown relied on for his definition of “general” MSW showed that hazardous substances comprised only 0.1% to 0.4% of the MSW’s volume.

The Master also noted that the hazardous substances in MSW are not found in pure form; rather, they are constituent elements in consumer products that remain in the products’ containers after the products have been used. He credited Dr. Ham’s testimony, finding that such hazardous substances release slowly over time, as the containers and the substances contained therein decompose. The Master further noted that even if the hazardous constituent elements found in these products are released, they will not pose a threat to the environment if they bond with other chemicals and “form new and different compounds.”

Evaluating these and other factors, the Master recommended that the district court allocate no responsibility whatsoever to the Municipal Defendants. Having thus found that the Coalitions were not entitled to any recovery, he saw no need to make any fact findings regarding the amount of response costs the Coalitions incurred.

D. The District Court’s Allocation of Response Costs

Predictably, the Coalitions submitted extensive objections to the Report. And equally predictably, the Municipal Defendants filed submissions urging the district court to adopt the Report, albeit with minor modifications. Thereafter, the court issued its ruling (the “First Ruling”) on the parties’ objections to the Master’s Report. The Court later amended certain of its findings in response to the parties’ motions for reconsideration (the “Second Ruling”).

The court adopted the Master’s findings regarding the high toxicity of the Coalitions’ waste and the mobility and releasea-bility of the hazardous constituents found in their liquid waste. The court went on to supplement the Master’s Report with extensive findings describing the type and volume of waste dumped by the Coalitions at each landfill and the effect that burning the waste had on reducing the waste’s residual volume. In this regard, the court found that the Coalitions dumped both liquid and hazardous wastes at their respective landfills. Notably, however, the court found that while LPC members deposited large amounts of liquid hazardous wastes at Laurel Park, “the greater bulk was comparable to MSW.”

The court also supplemented the Master’s findings by determining the volume of MSW dumped by each Municipal Defendant, accepting, in large part, the amounts suggested by the Coalitions. The court found that the Municipal Defendants dumped MSW as follows: (1) Hamden— 4,770,009 gal.; (2) New Haven — 6,473,583 gal.; (3) Orange — 20,235,349 gal.; (4) Seymour — 13,166,688 gal.; (5) Westport — 73,-116,636 gal., reduced after burning to 286,-066 cu. yds.; and (6) Middlebury — 19,796 cu. yds.

Regarding Laurel Park, the district court accepted the LPC’s estimates and found that the Municipal Defendants dumped MSW at the landfill as follows: (1) Orange — 177,407 cu. yds.; (2) Seymour— 143,577 cu. yds. compacted and an additional 47,651 cu. yds. uncompacted; and (3) Middlebury — 228,854 cu. yds. compacted and an additional 47,651 cu. yds. uncom-pacted.

The court also assessed the Master’s findings regarding the quality of the Municipal Defendants’ MSW. Assessing evidence concerning the relative presence of hazardous wastes in the Municipal Defendants’ MSW, the court accepted the Master’s observations that: (1) a typical town’s MSW will not necessarily comport to a general national profile; and (2) the Coalitions failed to account for any sorting and separation procedures the Municipal Defendants may have implemented to reduce the presence of hazardous substances in MSW. While the court found that these concerns were equitable factors for its consideration, it was unimpressed with the Municipal Defendants’ sorting and separation procedures, finding them “neither uniform nor all-encompassing over the period of waste disposal in question.... ” In the end, the court affirmed the Master’s finding that the record reflected, at most, a concentration of hazardous substances in the Municipal Defendants’ MSW of 0.1% to 0.4%.

Ultimately, however, the court rejected the Report’s recommended allocation, finding that the Municipal Defendants’ “experts’ evidence suggests that some allocation was appropriate.” The court then set forth the list of equitable factors it applied in allocating the parties’ contribution shares. Among other factors, the court considered: (1) the total volume of waste each party disposed; (2) the amount of hazardous substances contained in each party’s waste; (3) the form of the hazardous substances — i.e., liquid or solid; (4) the releaseability of the hazardous substances, including the amount, if any, of decomposition required and the effect of the substances’ interaction with other matter such that new, non-hazardous, compounds form; and (5) the mobility of the hazardous substances in each party’s waste. The district court was also impressed by each Coalition’s decision to enter into the Consent Decrees and to remediate their respective landfills. The court believed that their cooperation with the Government entitled them “to some benefit of the doubt as to the equitable factors and factual uncertainty in allocating [response costs.]”

To determine each Municipal Defendant’s contribution share, the court fashioned an allocation model of its own creation, which incorporated: (1) the combined total volume of waste deposited by the Coalitions and .Municipal Defendants at the respective landfills; (2) the volume of MSW each Municipality dumped; and (3) a toxicity factor for each Coalition’s liquid and industrial waste as contrasted with its members’ and each Municipal Defendant’s MSW.

Applying this formula, the court allocated contribution shares among the Municipal Defendants as follows:

• For response costs incurred by the LPC: (1) Orange — 4.29%; (2) Seymour — 8.1%; and (3) Middlebury— 4.8%.
• For response costs incurred by the BHC: (1) Orange — 2.26%; (2) Seymour — 1.52%; (3) Westport — 6.51%; (4) Hamden — 0.27%; (5) New Haven— 0.45%; and (6) Middlebury — 0.45%.
• The court also allocated a 1.1% share to Naugatuck, but held, as an equitable factor, that its contribution payment would be offset by $325,000 to reflect a settlement payment the Nau-gatuck Treatment Company (“NTC”) made to the BHC using funds that belonged to Naugatuck.

The court also issued findings regarding the Coalitions’ response costs in its initial and subsequent decisions. All told, the court calculated BHC’s recoverable response costs as $39,818,648.26 and LPC’s recoverable response costs as $14,005,975.27. The court then subtracted from these figures the total settlement proceeds the Coalitions previously received from other third-party defendants. The resulting totals — $32,600,263.26 for BHC and $5,237,587.37 for LPC — were the amounts against which the Municipal Defendants’ percentage contribution shares were assessed.

In its Second Ruling, the court granted the Coalitions’ request for pre-judgment interest. The court rejected their claim of entitlement to compound interest, opting instead to apply simple interest. Thereafter, on August 2, 2000, the court entered its final judgment on the BHC’s contribution claims in the following amounts, inclusive of pre-judgment interest: (1) Middle-bury — $186,779.94; (2) Westport— $2,702,083.23; (3) New Haven— $186,779.94; (4) Orange — $938,050.39; (5) Seymour — $630,901.16; and (6) Nauga-tuck — $46,933.87. In the same judgment, the court awarded the LPC the following amounts, inclusive of pre-judgment interest: (1) Orange — $239,471.69; (2) Seymour — $204,507.50; and (3) Middlebury— $316,656.78. Later, however, the court amended its judgment regarding the LPC’s claims to include a declaratory judgment establishing the Municipal Defendants’ liability for LPC’s future response costs.

The Municipal Defendants appeal the district court’s allocation, arguing they should not be required to contribute any share of the response costs. They also contend the court erred by failing to hold a hearing prior to ruling on the Coalitions’ objections to the Master’s Report.

The LPC appeals from the district court’s denial of its claim to certain response costs and the court’s interest awards.

The BHC also appeals, asserting that the district court erred by allowing Nauga-tuck to offset its contribution share.

DISCUSSION

I. The District Court’s Allocation

1. Overview

CERCLA § 113(f) provides: “Any person may seek contribution from any other person who is liable or potentially liable under § 9607(a).... In resolving contribution claims, the court may allocate response costs among liable parties using-such equitable factors as the court determines are appropriate.” 42 U.S.C. § 9613(f)(1). Thus, the statute envisions a two-part inquiry: First, the court must determine whether the defendant is “liable” under CERCLA § 107(a); Second, the court must allocate response costs among liable parties in an equitable manner. See Kalamazoo River Study Group v. Menasha Corp., 228 F.3d 648, 656-57 (6th Cir.2000). The party seeking contribution bears the burden of proof at both prongs of the court’s inquiry. See, e.g., Minyard Enters., Inc. v. Southeastern Chem. & Solvent Co., 184 F.3d 373, 385 (4th Cir.1999); H.R.Rep. No. 99-253(111), at 19 (1985) (“Of course, the burden of proof is on the ... party seeking apportionment to establish that it should be granted.”).

As we explained in Betkoski, (the last time these parties were before us), a finding of liability under § 107(a) requires proof that:

(1) the defendant is within one of the four categories of responsible parties enumerated in § 9607(a); (2) the landfill site is a facility as defined in § 9601(9);
there is a release or threatened release of hazardous substances at the facility; (4) the plaintiff incurred costs responding to the release or threatened release; and (5) the costs and response actions conform to the national contingency plan.

99 F.3d at 514. The district court held that the Municipal Defendants are liable under this test, and they do not appear to contest that finding on appeal. In any case, the district court was clearly correct.

With regard to the first element of this test, those who arrange for disposal of hazardous substances are among the “responsible parties” covered under § 9607(a). And, as we first explained in Murtha, “[w]hen a mixture or waste solution contains hazardous substances that mixture is itself hazardous for purposes of determining CERCLA liability.” Murtha, 958 F.2d at 1201. The Municipal Defendants acknowledge that their MSW contained hazardous substances, conceding that “[fjrom the testimony of Dr. Ham ... this fact or conclusion was inescapable.” The Municipal Defendants also concede the remaining elements of the liability test, 1.e., that hazardous substances were released from the landfills, and that the Coalitions incurred response costs and undertook response actions that conform to the NCP. Accordingly, we turn to the Municipal Defendants’ objections to the district court’s allocation.

2. Standard of Review

We begin with our analysis by settling upon the standard of review, an issue that is hotly debated by the parties-and well it should because we find it determinative. As we have stated before, because “[Section 113(f)’s] expansive language ... affords a district court broad discretion to balance the equities in the interests of justice.... We will not overturn the district court’s [allocation of response costs] absent an abuse of that discretion.” Bedford Affiliates v. Sills, 156 F.3d 416, 429 (2d Cir.1998) (emphasis added) (citations omitted).

“A district court ‘abuses’ or ‘exceeds’ the discretion accorded to it when (1) its decision rests on an error of law (such as application of the wrong legal principle) or a clearly erroneous factual finding, or (2) its decision — though not necessarily the product of a legal error or a clearly erroneous factual finding — cannot be located within the range of permissible decisions.” Zervos v. Verizon New York, Inc., 252 F.3d 163, 169 (2d Cir.2001). The Municipal Defendants raise several arguments seeking to avoid this deferential standard of review. We address each argument in turn.

The Municipal Defendants’ first argument rests on a clause in the Order of Reference stating the standard the district court would apply when it reviewed the Master’s Report:

Any factual objection will be decided on whether the finding or recommendation is clearly erroneous based on the record before the Special Master.

(emphasis added). The Municipal Defendants assert that this clause extended to non-factual members and, accordingly, bound the district court to the Master’s recommended equitable factors and allocation of response costs among the parties unless there was clear error. This loose reading of the Order’s standard of review is inconsistent with the Order’s terms and even with the Master’s understanding of his role in the case.

A separate section of the Order of Reference discussing the mediation stage of the litigation states: “The success of mediation depends on the acknowledgment by each party that the decision after a hearing cannot be predicted. That uncertainty arises from the ‘equitable factors’ to be applied by the Court as ‘appropriate’ in accordance with CERCLA § 113[f].” (emphasis added). Thus, in referring the case to the Master, the district court clearly reserved to itself the power to allocate response costs among the parties according to the equitable factors it, not the Master, found appropriate to the case.

The Master himself acknowledged this on several occasions. For example, he advised the parties that “the ultimate question to be decided in this case, under § 113[f], is almost unpredictable, except to the extent that you can judge what Judge Dorsey believes to be a reasonable set of equitable factors_” (R. 1/29/1998 at 198:19-24.) Likewise, the Master stated “I don’t decide what equitable factors are equitable factors.... [T]he statute places the broadest discretion in the hands of Judge Dorsey and not me. And the statute specifically says equitable factors that he deems to be appropriate, not that I deem to be appropriate.” (R. 4/1/1998 at 204:11-16.) Thus, the Master clearly understood what the Order of Reference made apparent: that the district court reserved the right to determine the appropriate equitable factors to apply, and the appropriate allocation of response costs among the parties, unbridled by the clear error standard of review.

Next, the Municipal Defendants claim that, because allocation of liability under CERCLA is a question of fact, we must review the Master’s recommended allocation directly, for clear error. This is a none too subtle begging of the question. It is true that we review a master’s factual findings for clear error, whether or not the district court accepts them. Morris Plan Indus. Bank v. Henderson, 131 F.2d 975, 976-77 (2d Cir.1942); see also Fed. R.Civ.P. 53(e)(2). However, we are not persuaded that the allocation of response costs under § 113(f) is a question of fact. Rather, it is an equitable determination based on the district court’s discretionary selection of the appropriate equitable factors in a given case. See Bedford Affiliates, 156 F.3d at 429.

The Municipal Defendants also maintain that we are required to ignore the district court’s allocation of response costs and instead review directly the Master’s recommended allocation for abuse of discretion, citing Van Gemert v. Boeing Co., 739 F.2d 730 (2d Cir.1984). In Van Gemert, we reviewed the district court’s equitable distribution of the unclaimed balance of a judgment entered against the defendant. In determining whether the district court abused its discretion in returning the unclaimed funds to the defendant, we looked to the special master’s report and recommendation, which the district court had adopted in fall. Id. at 733, 736-38. Thus, Van Gemert stands for the unremarkable proposition that where a district court charged with making an equitable determination adopts a master’s recommendation on that issue, we review the master’s report and recommendation to determine whether the district court abused its discretion. Id. at 738. Of course, that is not the case here — the district court rejected the master’s recommended allocation and allocated response costs according to the equitable factors it determined were appropriate under the circumstances of the case. Accordingly, Van Gemert is inappo-site.

Finally, the Municipal Defendants claim that the district court was required to accept the Master’s recommended allocation absent the Master’s abuse of discretion, citing Gottlieb v. Barry, 43 F.3d 474 (10th Cir.1994). In Gottlieb, the court considered whether the district court abused its discretion in choosing the appropriate method for determining attorneys’. fees in a securities class action where the district court rejected the master’s recommendation on that issue. The court noted that because the way to calculate fees depended on the “particular circumstances of the case,” the district court should have “given some deference” to the master’s recommendation, which resulted from his special knowledge of the case. Gottlieb, 43 F.3d at 487. Finding that the district court had “simply rejected” the master’s recommendation “for reasons largely unrelated to the particular circumstances of [the] case,” the court reversed the district court’s decision and found that attorneys’ fees should be calculated as the master recommended. Id.

Consistent with Gottlieb, we hold that a district court must give “some deference” to a master’s recommendation where the master has “direct and extensive” knowledge about the particular circumstances of a given case. Id. at 487 & n. 10; see also Cook v. Niedert, 142 F.3d 1004, 1010 (7th Cir.1998). “A district court that extends some deference to the master will consider his recommendation and the factors influencing it, but will not regard the recommendation as the alpha and omega of the ... analysis.” Id.

3. Analysis

Turning to the merits of the Municipal Defendants’ claims, we believe the district court gave appropriate deference to the Master’s recommendation and did not abuse its discretion in allocating contribution shares among the Municipal Defendants. The court undertook a painstaking analysis of the Master’s Report, accepted most of the equitable factors suggested therein, and incorporated them into its allocation model. Moreover, to the extent the court supplemented the factors suggested by the Master or, conversely, did not apply factors recommended by the Master, we find that it acted within its discretion. We address the district court’s allocation and the Municipal Defendants’ objections thereto below.

The district court cited the volume of waste disposed of by the parties as an equitable factor and used his findings on the parties’ respective waste volumes as the starting point of his allocation formula. The Municipal Defendants do not deny that the volume of waste each party disposed of is a legitimate equitable factor. Indeed, their own allocation expert opined that it is a logical starting point in cases of this type.

Nevertheless, the Municipal Defendants claim the district court erred in supplementing the Master’s findings on the volume of MSW they disposed of. First, they argue that the district court is powerless to make supplemental findings of fact, even where the Master failed to make any findings on an issue. We reject this contention as contrary to the Federal Rules of Civil Procedure, which expressly permit a district court to “modify” a special master’s report. Fed.R.Civ.P. 53(e)(2); cf. United States v. Tampa Bay Garden Apartments, Inc., 294 F.2d 598, 603 (5th Cir.1961) (“We do not think the power to modify [under Rule 53(e)(2)] is to be narrowly construed or applied. The power to modify is a power to alter or change, and to enlarge and add to as well as to limit, restrict or reduce.”).

Next, we reject their claim that the Master’s failure to render any findings on the volume of waste they dumped is tantamount to a factual determination that the Coalitions did.not carry their burden of proof on this issue. We find nothing in the Master’s Report to support this suggestion. Moreover, the Municipal Defendants’ argument is specious, when it is recalled that they expressly admitted in their proposed findings of fact that they each dumped a significant volume of MSW at one or, in some cases, both of the landfills.

The Municipal Defendants also claim that the district court clearly erred by preferring the Coalitions’ estimates of the volume of waste each Municipal Defendant dumped over the municipalities’ own estimates. In this regard, the Municipal Defendants do not contend that the district court relied on inadmissible evidence; rather, they simply assert that the court should have accepted their own estimated waste volumes. However, “[t]he mere fact that there was evidence to support an inference contrary to that drawn by the trier of fact does not mean that the findings were clearly erroneous.” Cifra v. G.E. Co., 252 F.3d 205, 213 (2d Cir.2001); see also Anderson v. Bessemer City, 470 U.S. 564, 574, 105 S.Ct. 1504, 84 L.Ed.2d 518 (1985) (“Where there are two permissible views of the evidence, the factfinder’s choice between them cannot be clearly erroneous.”). Accordingly, we reject the Municipal Defendants’ contention that the district court’s volume findings resulted from clear error.

Next, the Municipal Defendants argue that the district court improperly ignored the Master’s finding that MSW will not necessarily comport with a national profile in terms of the hazardous waste contained therein. Yet their own expert witness, Dr. Ham, testified:

My point is that municipal solid waste is also very varied in composition. Yet as you look at the way we live, the way we all have a certain number of grocery stores per capita throughout the country and the grocery stores generally produce the same kinds of wastes and so forth, when you look at general MSW, it’s amazing how similar it is from one part of the country to the other, with the only real difference being the amount of yard waste and when that yard waste occurs. So even though municipal solid waste is heterogeneous, it is, in fact quite uniform in character and predictable.

(R. 1/23/1998 at 24:1-11.) (emphasis added).

The Municipal Defendants also argue that the district court failed to account for sorting and separation policies adopted by the Municipalities that supposedly reduced the amount of hazardous substances in their MSW. Although the Master chastised Dr. Brown for failing to take this factor into account, he did not render any specific findings regarding the Municipal Defendants’ implementation of such policies. And while the district court cited sorting and separation policies as an equitable factor to be considered, it found that the Municipal Defendants policies were neither “uniform nor all-encompassing over the period of waste disposal in question .... ” The record confirms this finding.

The Municipal Defendants also claim the district court did not give appropriate deference to the Master’s findings regarding the relative mobility and releaseability of the parties’ waste and abused its discretion by allocating response costs based on volume and toxicity alone. We reject this contention. Our review of the district court’s decision reveals that the court carefully considered the relevance of these factors in determining how to allocate costs among the parties. While the court’s ultimate methodology may have been imprecise, we cannot say it amounted to an abuse of discretion. Cf. Control Data Corp. v. S.C.S.C. Corp., 53 F.3d 930, 937 (8th Cir.1995) (“CERCLA, in the allocation stage, places the costs of response on those responsible for creating the hazardous condition. Allocating responsibility based partially on toxicity does just that because those who release substances that are more toxic are more responsible for the hazardous condition.”); see also Bancamerica Comm. Corp. v. Mosher Steel, 100 F.3d 792, 802-03 (10th Cir.1996) (affirming allocation based on volume and relative toxicity).

Finally, the Municipal Defendants argue that the district court ignored the Master’s concerns over the lack of direct proof that releases of hazardous substances from their MSW added to the costs incurred by the Coalitions. In the district court’s view, the Coalitions were entitled to some benefit of the doubt given their cooperation with the government. We are untroubled by this view. By entering into consent decrees with the government, the Coalitions ensured that the landfills would be remediated in an expeditious manner and relieved the government of future litigation costs. See Betkoski, 99 F.3d at 526 (CERCLA § 9622(a) urges the government to enter into “settlements in order to expedite effective remedial actions and minimize litigation.”). The district court did not abuse its discretion in taking this factor into account. Cf. United States v. R.W. Meyer, Inc., 932 F.2d 568, 573 (6th Cir.1991) (“I do not believe Congress intended to require meticulous findings of the precise causative contribution each of several hundred parties made to a hazardous site. In many cases, this would be literally impossible. Rather .... Congress intended the court to deal with these situations by creative means, considering all the equities and balancing them in the interests of justice.”).

In sum, we believe the court gave appropriate deference to the Master’s recommendation and did not abuse its discretion in allocating contribution shares to the Municipal Defendants.

II. Objections to Response Costs and Offsets

1. Response Costs

As we stated earlier, CERCLA § 113(f) provides: “Any person may seek contribution from any other person who is liable or potentially liable under § 9607(a).... In resolving contribution claims,' the court may allocate response costs among liable parties using such equitable factors as the court determines are appropriate.” 42 U.S.C. § 9613(f)(1). In turn, under CERCLA § 107(a)(4)(B), a plaintiff may recover “any ... necessary costs of response incurred ... consistent with the national contingency plan.” 42 U.S.C. § 9607(a)(4)(B); 40 C.F.R. § 300.700(c)(2).

A. The LPC’s Litigation Costs

The district court deducted $8,768,387.00 from LPC’s total response costs to arrive at the net recoverable response costs against which the Municipal Defendants’ percentage contribution shares were assessed. The LPC contends that the court should have excluded $3,592,258.12 — the amount the LPC expended in litigation costs settling those claims — from the total settlement proceeds to accurately reflect the amounts available to be expended on remediation. The district court rejected this argument as a back-door attempt to recover attorneys’ fees, which are generally not recoverable under § 113(f). We agree.

“Neither CERCLA § 107, the liabilities and defenses provision, nor § 113, which authorizes contribution claims, expressly mentions the recovery of attorney’s fees.” Key Tronic Corp. v. United States, 511 U.S. 809, 815, 114 S.Ct. 1960, 128 L.Ed.2d 797 (1994). Thus, “expenses incurred solely in preparation for litigation” cannot be recovered as response costs “unless they ‘significantly benefitted the entire cleanup effort and served a statutory purpose apart from the reallocation of costs.’ ” Gussack Realty Co. v. Xerox Corp., 224 F.3d 85, 91-92 (2d Cir.2000) (quoting id. at 820, 114 S.Ct. 1960). The LPC does not argue that the attorneys’ fees it incurred significantly benefitted the cleanup effort. Rather, it claims Key Tronic only precludes an “award” of litigation expenses, and should not apply where a party seeks to decrease an offset to its recoverable response costs. We fail to see any distinction. As the district court reasoned, if the litigation costs are not excluded from LPC’s total response cost, the Municipal Defendants will wind up footing part of the bill-which would be akin to an award of fees. Accordingly, we affirm the district court on this issue.

B. The BHC’s Response Costs

The Municipal Defendants claim the district court erred by failing to articulate its findings on the BHC’s response costs. In its First Ruling, the court held that the BHC incurred recoverable response costs totaling $40,664,785.77, despite the BHC’s claim to have spent more cleaning up the landfill. The court did not set forth a detailed list of each response cost it found to be recoverable; rather, it addressed only certain response costs, regarding the remaining costs as uncontested.

The Municipal Defendants moved for reconsideration of the First Ruling, seeking an articulation of the court’s findings as to each recoverable response cost. In response to the Municipal Defendants’ motion, the BHC alerted the court that the $40,664,785.77 figure overstated the court’s actual allowance of the BHC’s response costs. Accordingly, the BHC set forth a two-column table illustrating in one column each cost it claimed in its proposed findings of fact and, in the second column, the deductions, if any, to each cost as per the First Ruling. Using this table, the BHC calculated its recoverable response costs as $39,818,648.26.

In its Second Ruling, the court adopted this revised figure, specifically referring the parties to the table provided by the BHC. Additionally, the court appended an unannotated adding tape, intended to reflect the court’s correction of its mathematical error. While we agree with the Municipal Defendants that the district court’s adding tape was not particularly illuminating, we hold that its findings were sufficiently articulated to put the Municipal Defendants on notice of the response costs allowed.

We also note that, except as discussed below, the Municipal Defendants failed to identify on appeal any BHC response cost erroneously allowed by the district court. Instead, the Municipal Defendants invite us to ferret out their objections by sifting through their submissions to the district court. We decline their invitation.

Finally, the Municipal Defendants claim, and the BHC concedes, that the district court erred by awarding BHC a lump-sum payment of $4,314,485.00, allocated among the various Municipal Defendants, representing the future discounted contingent operations and maintenance costs for the Beacon Heights site. In Gus-sack Realty Co. v. Xerox Corp., we held the “proper remedy for future response costs is not a present lump-sum payment of anticipated expenses but instead a declaratory judgment award dividing future response costs among responsible parties.” 224 F.3d 85, 92 (2d Cir.2000) (per curiam)-, see also 42 U.S.C. § 9613(g)(2) (“[T]he court shall enter a declaratory judgment on liability for [future] response costs or damages-”). Here, as in Gus-sack Realty, “the district court provided a remedy not available under CERCLA.” Gussack Realty, 224 F.3d at 92. Accordingly, we remand the case to the district court for entry of a declaratory judgment consistent with § 9613(g)(2).

2. Naugatuck Treatment Company’s Settlement with the BHC

Naugatuck Treatment Company (“NTC”) — a wholly-owned subsidiary of BHC member Uniroyal — is one of the many third-party defendants who settled with the BHC prior to trial. NTC — in exchange for a complete release from any further liability — paid the BHC $325,000.00 in 1993.

Five years later, during the proceedings before the Master, the Borough of Nauga-tuck produced evidence showing that the $325,000.00 NTC gave to the BHC for the release was taken from a reserve fund held by NTC in escrow for the benefit of the Borough. The Borough’s witness explained that the reserve fund was earmarked for major capita! improvements at the Borough’s treatment facility, not for paying down NTC’s CERCLA liability.

After the hearings before the Master concluded, the Municipal Defendants argued in their papers that any contribution share attributable to the Borough for the BHC’s "response costs should be reduced by the NTC settlement. The Municipal Defendants claimed that NTC used the Borough’s funds “without any approval from the Borough’s governing body, its Mayor and Borough Burgesses.... ” Because the Master did not allocate any contribution share to the Borough, he saw no need to render any findings on the circumstances of NTC’s settlement.

In its First Ruling, the district court rejected the Master’s recommendation as to the Borough’s liability for contribution. Instead, the court allocated the Borough a 1.1% share of responsibility for BHC’s response costs, resulting in liability amounting to $361,862.92 “minus $325,000 that involved [NTC], a net of $46,465.05.” The district court offered no explanation for the offset.

Undeterred, the BHC asked the court to reconsider its ruling, contending that certain Borough officials knew, prior to NTC’s settlement with BHC, that the reserve fund would be tapped to pay the settlement. Yet they did nothing to stop the payment. The district court upheld the offset, explaining “[w]hether payment was made with knowledge of [Borough] officials does not resolve BHC’s argument. The question is not the propriety of the payment ... but rather whose money was used. As BHC has not shown to the contrary, [Borough] funds are found to have been used. Thus the credit is proper.”

BHC now appeals the court’s decision, urging us to hold as a matter of law that § 113(f)’s “equitable factors” are limited to “those factors directly related to the harm to the environment, the parties’ causal relationship to that harm, and the parties’ ability to contribute to ... remediation ....” As we held in Bedford Affiliates, however, Section 113(f) “does not limit courts to any particular list of factors.” 156 F.3d at 429. Instead, “[t]he statute’s expansive language ... affords a district court broad discretion to balance the equities in the interests of justice.” Id.; see also Envtl. Transp. Sys., Inc. v. ENSCO, Inc., 969 F.2d 503, 509 (7th Cir.1992) (declining to limit equitable factors); United States v. R.W. Meyer, Inc., 932 F.2d 568, 572 (6th Cir.1991) (“[T]he court may consider the state of mind of the parties, their economic status, any contracts between them bearing on the subject, any traditional equitable defenses as mitigating factors and any other factors deemed appropriate to balance the equities in the totality of the circumstances.”) (footnote omitted) (emphasis added). Accordingly, we reject the BHC’s invitation to curb the district court’s discretion to consider any equitable factors it deems appropriate.

The BHC also contends that the district court abused its discretion because the equities favor the BHC, not the Borough. As already noted, the BHC claims that certain Borough officials in charge of overseeing the Borough’s operating agreement with NTC knew about NTC’s planned use of the reserve fund but did nothing to stop it. It follows, BHC argues, that the Borough should not be rewarded for sitting on its rights.

Admittedly, the evidence .before the Master suggests that members of the Nau-gatuck Water Pollution Control Board (the “WPCB”) — i.e., the board that oversaw the operating agreement — did indeed know of NTC’s planned use of the reserve fund before the settlement. However, the evidence also showed that the WPCB was controlled at the time by Uniroyal employees. Thus, it is hardly shocking that the WPCB did nothing to stop the transaction, which enriched Uniroyal at the Borough’s taxpayers’ expense and permitted Uniroyal’s wholly-owned subsidiary to escape liability for any cleanup costs.

In short, we believe the district court acted within its discretion by applying the credit, notwithstanding the WPCB’s knowledge of NTC’s planned use of the fund. Accordingly, we affirm the district court’s decision to offset NTC’s settlement amount against the Borough’s allocated share of the BHC’s response costs.

III. The District Court’s Interest Calculations

1. Pre-Judgment Interest

The LPC argued to the district court that it was entitled under § 107(a)(4) to an award of compound prejudgment interest on the response costs for which the court held the Municipal Defendants liable. The district court disagreed, asserting that the right to pre-judgment interest, if any, is committed to the court’s equitable discretion under § 113(f). The court then awarded the LPC simple, rather than compound interest. LPC claims the district court erred by: (1) treating pre-judgment interest under § 113(f) as discretionary; and (2) failing to award compound pre-judgment interest. As explained below, we agree with LPC on both counts.

We review these questions' of statutory-interpretation de novo. See Gussack Realty, 224 F.3d at 91. This Court has never addressed the availability of pre-judgment interest under CERCLA § 113(f), j^et we find Bancamerica Comm. Corp. v. Mosher Steel, 100 F.3d 792 (10th Cir.1996), persuasive, and adopt its analysis. See id. at 799-803. Thus, we agree that “prejudgment interest must be awarded in § 113(f) cases just as it is in § 107(a) cases.” Id. at 800.

Under § 107(a), pre-judgment interest is mandatory. 42 U.S.C. § 9607(a)(4) (“The amounts recoverable in an action under this section shall include interest on the amounts recoverable ....”) (emphasis added). Thus, we agree with the LPC that the district court erred in treating pre-judgment interest as a discretionary award under § 113(f).

Next, we must determine whether the LPC is entitled to compound interest. Again, we look to § 107(a)(4), which provides:

The rate of interest on the outstanding unpaid balance of the amounts recoverable under this section shall be the same rate as is specified for interest on investments of the Hazardous Substance Superfund established under subchapter A of chapter 98 of Title 26.

Id. In turn, the rate of interest on investments of the Hazardous Substance Superfund:

[Sjhall be at a rate determined by the Secretary of the Treasury (as of the close of the calendar. month preceding the month in which the advance is made) to be equal to the current average market yield on outstanding marketable obligations of the United States with remaining periods to maturity comparable to the anticipated period during which the advance will be outstanding and shall be compounded annually.

26 U.S.C. § 9507(d)(3)(C) (emphasis added). A straightforward reading of these statutory provisions reveals that the district court was without discretion to grant anything other than compounded interest at the applicable rates set forth by the Secretary of the Treasury. Bancamerica, 100 F.3d at 802.

Thus, we direct the district court on remand to recalculate pre-judgment interest on LPC’s response costs, at the rates specified by the Secretary of the Treasury compounded annually, accruing from the latter of the date LPC made its demand on the Municipal Defendants or the date of the expenditure concerned. See 42 U.S.C. § 9607(a) (“Such interest shall accrue from the later of (i) the date payment of a specified amount is demanded in writing, or (ii) the date of the expenditure concerned.”).

2. Post-Judgment Interest

The district court initially entered judgment on the LPC’s contribution claims on August 2, 2000 and ordered that post-judgment interest “shall accrue from the date of judgment....” On January 5, 2001, the district court entered an amended judgment, which added a declaratory judgment regarding the Municipal Defendants’ liability for the LPC’s future response costs. The amended judgment was silent regarding post-judgment interest, though it stated that “[i]n all other aspects the [Initial] Judgment remains the same.” The LPC argues that the district court should have expressly provided that post-judgment interest runs from the date of its first judgment.

Under 28 U.S.C. § 1961(a), a plaintiff is entitled to post-judgment interest calculated as of the “date of the entry of the judgment....” Id. Such relief is “designed to compensate the plaintiff for the delay it suffers from the time damages are reduced to an enforceable judgment to the time the defendant pays the judgment.” Andrulonis v. United States, 26 F.3d 1224, 1280 (2d Cir.1994). Thus, “post-judgment interest commences from the date a judgment is ascertained in a meaningful way” and “supported by the evidence.” Greenway v. Buffalo Hilton Hotel, 143 F.3d 47, 55 (2d Cir.1998) (quoting id. at 1233) (internal quotation marks and alteration omitted).

Here, the district court’s judgment on the LPC’s past response costs was ascertained in a meaningful way on August 2, 2000. To the extent its amended judgment precluded interest from August 2, 2000, forward, the district court erred. Accordingly, we remand for appropriate recalculation of post-judgment interest on the LPC’s past response costs.

IV. Failure to Hold a Hearing

The Municipal Defendants complain that the district court should not have ruled on the Master’s Report without hearing oral argument on the parties’ submissions. We disagree.

Rule 53(e)(2) states, in relevant part:

Within 10 days after being served with notice of the filing of the report any party may serve written objections thereto upon the other parties. Application to the court for action upon the report and upon objections thereto shall be by motion and upon notice as prescribed in Rule 6(d). The court after hearing may adopt the report or may modify it or may reject it in whole or in part or may receive further evidence or may recommit it with instructions.

Fed.R.Civ.P. 53(e)(2) (emphasis added). The Municipal Defendants claim the above-underlined text mandates an oral hearing, an issue of first impression in this Court.

Several of our sister Circuits have weighed in on the issue and have uniformly found that there is a right to an oral hearing at the behest of the objecting party. See, e.g., Kieffer v. Sears, Roebuck & Co. 873 F.2d 954, 956 (6th Cir.1989) (“[W]e hold that when a district court is presented with objections to the report of a special master, the objector has a right to a hearing ....”) (emphasis added); In re Chicago, Milwaukee, St. Paul & Pac. R.R. Co., 739 F.2d 1169, 1172 (7th Cir.1984) (“One who files objections to the report of a Special Master thus has a right to be heard on those objections before the court acts on the report.”) (emphasis added); In re Wonderbowl, Inc., 424 F.2d 178, 180 (9th Cir.1970) (“[Ojbjectors to a referee-special master’s report have a right to be heard before the bankruptcy court acts on the report.”) (emphasis added). Even if we were to agree that the plain language of Rule 53(e)(2) entitles an objector to an oral hearing, this does not help the Municipal Defendants. As they readily admit in their appellate brief, they did not object to the Report; indeed, they requested its adoption. Accordingly, they had no right to oral argument on the Coalitions’ objections.

Additionally, counsel for the Municipal Defendants admitted at oral argument that the Municipal Defendants did not move for a hearing on the Master’s Report. Thus, even if they did possess any rights to a hearing, they waived such rights by fading to ask for one. Cf. Golden Door Jewelry Creations, Inc. v. Lloyds Underwriters Non-Marine Ass’n, 8 F.3d 760, 768 (11th Cir.1993) (“This court will not overturn the district court’s adoption of the special master’s report on a procedural technicality consignors have discovered only on appeal.”).

We have considered the remaining arguments raised by the Coalitions in then-appeals and the Municipal Defendants in their cross-appeal and find them to be without merit.

CONCLUSION

For the reasons stated herein, we AFFIRM the district court’s decision insofar as it allocated shares of the Coalitions’ response costs to the Municipal Defendants, but VACATE the district court’s:

• award of pre-judgment and post-judgment interest regarding the LPC’s claims; and
• award of a lump-sum for the BHC’s future recovery costs.

We REMAND the case for further proceedings not inconsistent with this opinion. 
      
      . The EPA defines leachate as "any liquid, including any suspended components in the liquid, that has percolated through or drained from hazardous waste." 40 C.F.R. § 260.10 (2002).
     
      
      . Only three of the BHC’s thirty-two members are participating in this appeal — Goodrich Corporation (fik/a B.F. Goodrich Company), Crompton Manufacturing Company (fik/a Uniroyal Chemical Company, Inc.) and Reynolds Metal Company (fik/a Reynolds Aluminum Building Products Company).
     
      
      .The following nine of the LPC’s nineteen members are participating in this appeal— Crompton Manufacturing Company, Kerite Company, Unisys Corporation, Risdon Corporation, Cadbury Beverages, Inc., Coltec Industries, Inc., Naugatuck Glass Company, Naugatuck Treatment Company and BP Amoco.
     
      
      . The LPC had not yet formed at this time.
     
      
      . The Town of Plymouth, Jacob Bros., Inc. George Clark, John Betkoski and Zoilo Drum Co. also remained as defendants, though they are not parties to this appeal. Accordingly, we omit discussion of issues that pertain solely to them.
     
      
      . He also referred to samples he had taken from seeps of other MSW — only landfills which he claimed confirmed the presence of hazardous substances in MSW.
     
      
      . As used by the parties in this case, “toxicity'' refers to the relative presence of a hazardous substance in a party's waste, whereas "mobility” refers to a hazardous substance’s propensity to find its way through the landfill into dump leachate.
     
      
      . The model did not incorporate the total volume of waste disposed by all polluters; rather, it used data limited to LPC members and the defendants who had not settled with the LPC at the time the model was applied.
     
      
      . For purposes of this factor, Mr. White separated the BHC's response costs into five categories: (1) excavation and capping of the landfill; (2) venting and treatment of methane; (3) leachate treatment; (4) leachate collection; and (5) offsite groundwater contamination. With respect to categories (1) through (4), Mr. White opined that because allocation based on adjusted volume adequately reflected cost-causation, all wastes should be treated similarly. Because Mr. White believed that liquid wastes were more likely to have incurred the response costs associated with category (5), he assigned a multiple of 15 to liquid waste shares.
     
      
      . Ms. Robinette used the term "persistence” to refer to the length of time a hazardous substance could continue tó exist in a recognizable state.
     
      
      . Ms. Robinette did not allocate percentage shares to Hamden and New Haven, claiming she lacked sufficient data concerning the volume of waste dumped by these defendants.
     
      
      . According to the district court’s conversion method, which was advocated by the BHC's allocation expert, 1 cu. yd. = 201.98 gal.
     
      
      . The district court did not allocate any of the BHC's response costs to Middlebury in its First Ruling, erroneously believing the BHC had settled its claims against that defendant. The court corrected its error in its Second Ruling.
     
      
      . The district court noted the absence of evidence in the record concerning the volume and type of waste disposed of by other parties that had settled with the Coalitions. Accordingly, the court assumed that the settlement amounts paid by these non-parties adequately reflected their allocable share of the Coalitions' response costs.
     
      
      . While the Municipal Defendants quarrel over the district court's determination of the amount of response costs incurred by the BHC, which we address in Part II, supra, they do not dispute that the BHC incurred at least some necessary response costs consistent with the NCP.
     
      
      . The Municipal Defendants cite our decision in United States v. Alcan Aluminum Corp., 990 F.2d 711 (2d Cir.1993), and the Fifth Circuit's decision in In Matter of Bell Petroleum Servs., Inc. (EPA v. Sequa Corp.), 3 F.3d 889 (5th Cir.1993), for the proposition that allocation of response costs is a question of fact. Both these cases, however, concerned apportionment of damages under the divisibility of harm defense available to defendants in a § 107(a) cost recovery action. See Bell Petroleum, 3 F.3d at 896; Alcan Aluminum, 990 F.2d at 722. In such cases, a potentially responsible party may avoid joint and severed liability if it can prove the environmental harm at a site is divisible. Upon such a showing, the party will be held liable only for the harm the court finds it caused. Bell Petroleum, 3 F.3d at 901-02; Alcan Aluminum, 990 F.2d at 722. Admittedly, the district court's apportionment of damages in such cases is a finding of fact. Bell Petroleum, 3 F.3d at 896; see also Alcan Aluminum, 990 F.2d at 722 ("[Ajpportionment itself is an intensely factual determination.”). However, unlike allocation of response costs under § 113(f), § 107(a)'s "divisibility of harm inquiry ... is guided not by equity ... but by principles of causation alone.” United States v. Hercules, Inc., 247 F.3d 706, 718 (8th Cir.2001). This difference renders Alcan Aluminum and Bell Petroleum inapposite.
      In Boeing v. Cascade Corp., 207 F.3d 1177 (9th Cir.2000), a § 113(f) case cited by the Municipal Defendants, the court held that while the district court's choice of equitable factors is reviewed for abuse of discretion, its "allocation according to those factors” is reviewed for clear error. Id. at 1187; see also id. at 1188 ("We review the district court's allocation as a finding of fact for clear error.”). As support for its holding, however, the court cited Alcan Aluminum and Bell Petroleum. See Boeing Co., 207 F.3d at 1187 & n. 33, 1188 & n. 39. As we explained above, we find the standard of review provided in Alcan Aluminum and Bell Petroleum inappo-site to the standard of review in a § 113(f) case. For the same reasons, we find Boeing unpersuasive.
     
      
      . The Master found, per Dr. Ham’s testimony, that the “preponderance of scientific evidence ... proves that such releases as may have occurred” of hazardous substances in the Municipal Defendants' MSW "formed new chemical compounds.” This finding overstates Dr. Ham's testimony. Dr. Ham testified that some hazardous substances found in MSW may decompose and join with other elements found in MSW to form a new, non-hazardous, compounds. But, significantly, he did not testify that this process would occur all of the time. Rather, he admitted on cross-examination that, in his belief, MSW contributed to groundwater contamination at the landfills. Thus, Dr. Ham's testimony, taken as a whole, leads at most to a conclusion that some, but not all, of the hazardous substances found in MSW decomposed to form other compounds. The Master clearly erred by extrapolating an impermissible inference from Dr. Ham's testimony.
     
      
      . We also affirm the district court's denial of ■ LPC's claim to response costs concerning: (1) drinking water supply line; (2) leachate alleviation; (3) "costs to complete”; and (4) EPA-directed construction supervisor.
     
      
      . The Borough's contribution share was later reduced to $36,862.92 to reflect the court’s recalculation of BHC's response costs.
     
      
      . The record reveals that on August 18, 1995, LPC issued a written demand on each Orange, Seymour and Middlebury. The court accepted this demand as valid in its Second Ruling and invited the LPC to submit detailed information on its post-August 18, 1995 response costs. In response to the district court's suggestion, the LPC submitted a list delineating its post-August 18, 1995 response costs. It is not clear whether the district court accepted the dates set forth therein in calculating LPC’s pre-judgment interest on a simple interest basis. We leave it to the district court to determine the dates from which the interest shall accrue and to ascertain the applicable Treasury rates.
     
      
      
        . Although the Municipal Defendants filed "exceptions” to the Master’s Report, they did not object in any meaningful way. Rather, they simply referred the court to their proposed findings of fact and urged the court to accept their findings to the extent the were not reflected in, or conflicted with, the Master's Report. Additionally, the Town of Seymour filed an "Application for Adoption of the Special Master’s Report with Modifications” urging the court to modify the report to specifically adopt the Municipal Defendants’ allocation model.
     