FRIEDMAN, Circuit Judge.
This is an appeal from a judgment of the United States District Court for the Eastern District of Michigan adopting the report of the United States Magistrate determining the damages in a patent infringement case. We affirm.
I.
The patent involved in this case was issued to the appellee Hanson in 1961. It covers a method and apparatus for making snow used in winter sports. Prior to Hanson’s invention, snow was made by mixing water and compressed air, and ejecting the combination under high pressure from a nozzle. The water froze and, by combining with water in the air, produced snow crystals. This method required a considerable amount of energy to compress the air, and the nozzles frequently froze.
Hanson’s patent disclosed a new method of making snow. As the magistrate explained, rather than relying on compressed air,
[t]he Hanson process discharges water into a hub mounted in the center of a spinning propeller. The water is then fragmented into droplets by the propeller blades generating spontaneous ice nuclei.... The efficiency of the [Hanson] snowmaking system [as opposed to the prior art method] is based upon the turbulence of the air created by the airstream which increased cooling capacity.
The magistrate found that “the airless snowmaking method of the Hanson patent is at least five to seven times as energy efficient as the prior art compressed air method.... ”
In 1969, Hanson licensed his patent to Snow Machines International, which subsequently assigned the license to Snow Machines Incorporated (both referred to as “SMI”), for a royalty of 2% percent of sales and 2V2 percent of the stock of SMI. Since 1969, about 1,500 SMI machines have been manufactured and sold, and SMI paid total royalties of approximately $26,000. The magistrate found that “[w]ithin the short span of twelve years, the airless snowmaking process has developed substantially and presently accounts for almost one half of all artificially produced snow.” The Hanson patent expired in 1978.
Hanson filed the present suit in February 1973. He alleged that appellant Alpine Valley Ski Area, Inc. (“Alpine”) had infringed his patent through the use of three snow-making machines manufactured by Hedco, Inc. Hedco defended the suit for Alpine.
After a trial' without a jury, the district court held that the Hanson patent was valid, that Alpine had infringed the patent by its use of the Hedco machines, and that Hanson was entitled to an accounting for damages. The court found that Alpine “has used the Hedco H-2d, Mark II and Mark III machines to produce snow” (finding 54). The court of appeals affirmed the determinations of validity and infringe[1077]*1077ment. Hanson v. Alpine Valley Ski Area, 611 F.2d 156, 204 USPQ 803 (6th Cir.1979).
The district court referred the issue of damages to the United States Magistrate as a special master. After a trial, the magistrate recommended that Hanson be awarded damages of $12,250 for the infringement. In its review of the magistrate’s report, the district court “considered the objections and arguments of counsel, and independently examined the legal basis for the Magistrate’s conclusions and recommendationfs].” The court concluded that “the Magistrate correctly decided this matter” and adopted the magistrate’s report as the opinion of the court.
The magistrate found that Alpine had used the three Hedco machines to produce snow and that Alpine had operated one of the machines during the 1972-73 and 1973-74 seasons and the two other machines during one season. He held that the evidence did not provide any basis for determining either the profits Alpine made or the profits Hanson lost through the infringement, and that damages therefore had to be determined on the basis of a reasonable royalty for the patent. In making that determination, the magistrate applied the “willing licensee-willing licensor rule. That is, at what royalty rate would a licensee accept a license and a licensor grant a license if both parties genuinely wish to execute a license in an arm’s length transaction.”
The magistrate accepted the testimony of Hanson’s witness, Sidney Alpert, whom he characterized as “a highly regarded expert in the field of negotiating patent licenses,” that “the royalty rate and licenses granted under the Hanson patent must be uniform.” The magistrate stated that “[pjerhaps, if Defendant had proffered its own expert in licensing and negotiating patents to refute the testimony elicited from Plaintiff’s expert, the Court would not be so inclined to accept these elements impelling a uniform license,” but that “[o]n the record before this Court the credible testimony of Plaintiff’s expert stands as the sole guidance germane to the question of licensing distinctions for manufacturers and users. The Court is limited to consideration of the record presented and as such Defendant’s analysis falls.”
Applying the pertinent factors for determining a reasonable royalty set forth in Georgia-Pacific Corp. v. United States Plywood Corp., 318 F.Supp. 1116, 166 USPQ 235 (S.D.N.Y.1970), modified and aff’d sub nom., Georgia-Pacific Corp. v. United States Plywood-Champion Papers, 446 F.2d 295, 170 USPQ 369 (2d Cir.1971), cert. denied, 404 U.S. 870, 92 S.Ct. 105, 30 L.Ed.2d 114 (1971), the magistrate concluded that
the reasonable royalty in this case must be based upon a portion of the annual cost savings attributable to use of the Hanson patent. Expert testimony on this record indicates that one-third of the cost savings would be deemed acceptable to both parties in an arm’s length license negotiation. Furthermore, based upon energy costs at the time of infringement in 1972-73, the airless snowmaking method of the Hanson patent generates a dollar savings of $75.00 per gallon (of water used to make snow) per minute. That is, under the Hanson method, the cost of producing snow using one gallon of water for one minute is $75.00 less than the cost of producing snow under the compressed air method using one gallon of water for one minute. Thus the cost savings for the Hanson method is a function of any machine’s capacity to make snow.
The magistrate concluded that a reasonable royalty would be one-third of the $75 savings per gallon of water that the Hanson method produced over the earlier compressed air method. Multiplying the $25 per gallon by the snowmaking capacity of the three Hedco machines Alpine used and the four years of use involved, the magistrate determined that Hanson was entitled to royalties of $3,000, $1,750, and $7,500 for each of the Hedco machines used, or a total of $12,250.
II.
The award of damages for patent infringement is governed by 35 U.S.C. § 284 (1976), which provides:
[1078]*1078Upon finding for the claimant the court shall award the claimant damages adequate to compensate for the infringement, but in no event less than a reasonable royalty for the use made of the invention by the infringer, together with interest and costs as fixed by the court.
The court may receive expert testimony as an aid to the determination of damages or of what royalty would be reasonable under the circumstances.
There are two methods by which damages may be calculated under this statute. If the record permits the determination of actual damages, namely, the profits the patentee lost from the infringement, that determination accurately measures the patentee’s loss. If actual damages cannot be ascertained, then a reasonable royalty must be determined. Panduit Corp. v. Stahlin Bros. Fibre Works, 575 F.2d 1152, 1157, 197 USPQ 726, 736 (6th Cir.1978).
The reasonable royalty may be based upon an established royalty, if there is one, or if not upon a hypothetical royalty resulting from arm’s length negotiations between a willing licensor and a willing licensee. As the Court of Claims stated with respect to the statute applicable to determining damages in infringement suits against the United States (28 U.S.C. § 1498 (1976)): “Where an established royalty rate for the patented inventions is shown to exist, the rate will usually be adopted as the best measure of reasonable and entire compensation.” Tektronix, Inc. v. United States, 213 Ct.Cl. 257, 552 F.2d 343, 347, 193 USPQ 385, 390 (1977), cert. denied, 439 U.S. 1048, 99 S.Ct. 724, 58 L.Ed.2d 707 (1978).
A. The magistrate held that
[t]he proofs offered in the case at hand do not suggest any basis for establishing profits experienced by the infringing Defendant in the use of the process patent nor do they establish a loss of income or loss of profit suffered by the patentee on any tangible basis by virtue of the nature of the interest the patentee has.
We have no basis for rejecting that factual determination.
B. The magistrate also ruled that the record did not show an established royalty. Alpine argues that Hanson’s license to SMI at a 2V2 percent royalty shows an established royalty. In rejecting this contention, the magistrate correctly pointed out that under the license SMI
paid Hanson 2.5% of the selling price and a 2.5% interest in a predecessor corporation SMI a New York corporation. Thus in granting this license Hanson speculated on the value of his interest in SMI as a corporate entity. The worth of that investment and the extent to which it enhanced the royalty is unknown. Such speculation hampers a finding of an established royalty....
Moreover, as the magistrate stated, “a single licensing agreement does not generally demonstrate uniformity nor acquiescence in the reasonableness of the royalty rate.” For a royalty to be “established,” it “must be paid by such a number of persons as to indicate a general acquiescence in its reasonableness by those who have occasion to use the invention.” Rude v. Westcott, 130 U.S. 152, 165, 9 S.Ct. 463, 468, 32 L.Ed. 888 (1889).
Finally, we cannot say that the magistrate erred in refusing to consider certain offers to license the Hanson patent at a 2V2 percent royalty as showing an established rate. The magistrate excluded evidence of those proposals because they were offers in compromise made in contemplation of infringement litigation and therefore inadmissible under Rule 408 of the Federal Rules of Evidence.
The fact that licenses were offered at a particular rate does not show that that rate was the “established” rate, since the latter requires actual licenses, not mere offers to license. (We discuss below the exclusionary ruling insofar as the excluded evidence related to the magistrate’s determination of a reasonable royalty under the willing licensor-willing licensee rule.) Moreover, since the offers were made after the infringement had begun and litigation was threat[1079]*1079ened or probable, their terms “should not be considered evidence of an ‘established royalty,’ ” since “Q]icense fees negotiated in the face of a threat of high litigation costs ‘may be strongly influenced by a desire to avoid full litigationPanduit, 575 F.2d at 1164, n. 11, 197 USPQ at 736, n. 11.
C. Since there was no established royalty for licensing the Hanson patent, the magistrate necessarily had to use “a willing-buyer/willing-seller concept, in which a suppositious meeting between the patent owner and the prospective [user] of the infringing [method] is held to negotiate a license agreement.” Tektronix, 552 F.2d at 349,193 USPQ at 391. “The key element in setting a reasonable royalty ... is the necessity for return to the date when the infringement began.” Panduit, 575 F.2d at 1158, 197 USPQ at 731. The infringement in the present case began in 1972. As 35 U.S.C. § 284 states, expert testimony may be considered “as an aid to the determination ... of what royalty would be reasonable under the circumstances.”
Each party presented two witnesses. Hanson’s witnesses were James L. Dilworth, an expert in snowmaking and the use of snowmaking machines, who had experience in designing and operating various types of snowmaking systems at a number of resorts; and Sidney Alpert, an expert in the negotiation of licenses for a broad range of patents. Alpine’s witnesses were Joseph F. Kosik, a 50-percent shareholder and the operator of Alpine’s resort in Michigan, who described the use of snowmaking machinery there; and Joseph J. Kohler, an expert who was the president and general manager of a ski resort in New York and who presented a study showing that use of the Hanson process at his resort would not have produced any savings over the cost of using the compressed air method. The magistrate credited the testimony of Hanson’s experts, which Alpine’s witnesses did not refute. That testimony formed the factual bases for the magistrate’s determination of a reasonable royalty.
Dilworth described the relative costs of using the Hanson and the compressed air methods of making snow. He concluded that by installing a Hanson method apparatus, a hypothetical but apparently representative ski resort would have saved in 1972 (the year infringement began) $75 per gallon per minute of the machine’s rated capacity over the cost of a compressed air system.
Alpert described a hypothetical license negotiation between Hanson and Alpine in 1972. He explained that, acting for Hanson, he would “need to set up a uniform standardized approach that will go across all potential licensees, all users of the patented method independent of the machine they use.... ” The magistrate summarized Alpert’s testimony on the need for a uniform license as follows:
[T]he royalty rate and licenses granted under the Hanson patent must be uniform, and the distinctions between manufacturer and user disregarded for purposes of negotiating licenses. Various factors coerce Hanson in this case to adhere to a reasonable licensing policy throughout the multitude of potential licensors [sic]. Integrity of Hanson as a licensor, exposure to antitrust violations, and lack of controls by Hanson over both manufacturers and users were elements cited by the expert as dictating a pervasive licensing pattern and for disregarding the distinctions between manufacturers and users.
Alpert further stated that the licensor would have insisted on a uniform license based on the gallons-per-minute rated capacity of the Hanson-method machines, and would have refused to grant a license based on actual use or other criteria relating to the situation of the particular licensee.
The magistrate’s acceptance of Alpert’s analysis and approach was fully justified by the record before him. As he pointed out, only Hanson introduced expert testimony regarding the basis upon which a willing licensor and a willing licensee would have negotiated a license of the Hanson patent in 1972, and the magistrate found this evidence credible and convincing. The magistrate pointed out that if Alpine had [1080]*1080presented “its own expert in licensing and negotiating patents,” he might have viewed the case differently, but that “[o]n the record before this court the credible testimony of Plaintiffs expert stands as the sole guidance germane to the question of licensing distinctions for manufacturers and users.”
On the basis of our review of the record in this case, we cannot reject as clearly erroneous the magistrate’s factual conclusions (a) that Hanson would have required uniform terms in all his licenses and would not have varied those terms to reflect the situation of a particular licensee, and (b) that a reasonable royalty would have been based upon one-third of the estimated cost savings resulting from use of the Hanson method instead of the compressed air method of making snow, calculated on the basis of savings of $75 per machine multiplied by the snowmaking capacity of the machine times the number of years used. To the contrary, these conclusions are fully supported by the record, and we affirm them.
D. Alpine challenges the magistrate’s determination of a reasonable royalty on a number of grounds. Most of these contentions founder upon the magistrate’s factual determination, which we have upheld, that the royalty rate would have been uniform for all licensees. They also are subject to the additional infirmities discussed below.
1. Alpine contends that because it acquired the Hedco machines for experimental purposes and did not use them much, the royalty should have been based upon actual use rather than upon estimated savings reflecting the snowmaking capacity of the machines.
Apart from the fact that this theory is inconsistent with the magistrate’s determination that Hanson would have granted licenses only at a uniform rate and not based on actual use, the record contains substantial evidence that actual use of the snowmaking machinery would not have been the basis upon which a willing licensor and a willing licensee would have established the royalty. As Alpert noted, it would have been extremely difficult to monitor actual use. Apparently no complete or accurate records were kept of the actual use of the Hedco machines at Alpine’s resort.
Equally or more important, a royalty based upon actual use would have been inconsistent with the function snowmaking equipment serves at a ski resort and the reasonable needs and expectations of both the licensor and the licensee. A resort has snowmaking machinery to enable it to function at times when there is no or insufficient natural snow. As the magistrate stated, the resort hopes that “natural snow will always be sufficient and that artificial snow will never be needed.” He noted that the “machines insure the business can function without natural snow” and that Hanson’s “expert likened the machine to an insurance policy.” The magistrate justifiably concluded that in these circumstances “[t]he number of hours a machine is used is irrelevant; the desire is never to use the machine. The machine’s utility simply does not depend upon its hours of operation.”
A royalty based on actual use would produce unsatisfactory results here for both the licensor and the licensee. If there were extensive snow during the season, there would be little use of the machine and the patentee would receive an inadequate return for the value of his invention. On the other hand, if there were little or no snow, the licensee would have to pay exceptionally large royalties.
We have no basis for rejecting the magistrate’s factual conclusion that in the circumstances the parties to the licensing negotiations would have agreed upon a royalty rate that would insure a fair and reasonable return to the patentee and avoid the payment of excessive royalties by the licensee, without regard to the size of the snowfall. The royalty rate the magistrate used, based upon the expert evidence, would have accomplished that objective.
2. The magistrate based his determination of a reasonable royalty upon the estimated cost savings resulting from Alpine’s use of the infringing Hedco machines. Reliance upon estimated cost savings from use of the infringing product is a well settled [1081]*1081method of determining a reasonable royalty. See, e.g., Trio Process Corp. v. L. Goldstein’s Sons, Inc., 533 F.2d 126, 189 USPQ 561 (3d Cir.1976), cert. denied, 449 U.S. 827, 101 S.Ct. 91, 66 L.Ed.2d 30 (1980); Decca Ltd. v. United States, 225 Ct.Cl. 326, 640 F.2d 1156, 1167 & n. 20, 209 USPQ 52, 60 & n. 20 (1980), cert. denied, 454 U.S. 819, 109 S.Ct. 99, 70 L.Ed. 89 (1981), and cases cited therein. The method is appropriate where other approaches (such as established royalty or lost profits) would be “difficult.” Leesona v. United States, 213 Ct.Cl. 722, 599 F.2d 958, 971, 202 USPQ 424, 436 (1979), cert. denied, 444 U.S. 991, 100 S.Ct. 522, 62 L.Ed.2d 420 (1979).
In determining Alpine’s savings from use of the Hedco machines, Dilworth (Hanson’s expert, whose testimony the magistrate accepted) based his calculations on the use of the machines for 800 hours a year. He explained that this constituted an average season’s use of snowmaking systems for resorts in the area where Alpine is located. Although Alpine challenges a number of the assumptions and calculations upon which Dilworth based his determinations, we cannot say that the magistrate’s conclusions, based upon Dilworth’s expert testimony, are clearly erroneous.
3. Alpine argues that the royalty the magistrate set is unreasonable because it would not have allowed it to make a profit. “[T]hat a reasonable royalty would leave an infringer with a reasonable profit ... is implicit....” Square Liner 360°, Inc. v. Chisum, 691 F.2d 362, 377, 216 USPQ 666, 677 (8th Cir.1982). See also Leesona, 599 F.2d at 970-71, 202 USPQ at 436.
Since the royalty the magistrate set was based upon one-third of Alpine’s estimated savings through use of the infringing Hedco machines, it is difficult to understand the basis of Alpine’s contention that the royalty would not have allowed it to make a profit. The issue of the infringer’s profit is to be determined not on the basis of a hindsight evaluation of what actually happened, but on the basis of what the parties to the hypothetical license negotiations would have considered at the time of the negotiations. “Whether, as events unfurled thereafter, [Alpine Valley] would have made an actual profit, while paying the royalty determined as of [1972], is irrelevant.” Panduit, 575 F.2d at 1164, 197 USPQ at 736. Alpine has not shown that the royalty the magistrate set would not have allowed it a reasonable profit.
4. Alpine further contends that no willing licensee would accept the “high” royalty awarded. But as the Court of Claims noted in Tektronix, supra, 552 F.2d at 349, 193 USPQ at 391-92: “[W]hether ... [the] defendant ... [was] never willing to pay a reasonable royalty[] is irrelevant.... The willing-buyer/willing-seller concept is ... employed by the court as a means of arriving at reasonable compensation and its validity does not depend on the actual willingness of the parties to the lawsuit to engage in such negotiations.” “There is, of course, no actual willingness on either side.... ” Panduit, 575 F.2d at 1159,197 USPQ at 732.
5. Alpine argues that the royalty rate is unreasonable because it amounts to 18 percent of the purchase price of an SMI machine (which would not have required a license to operate), so that over the six-year remaining life of the patent at the time of infringement in 1972, the total royalties paid would exceed the cost of the non-infringing machine.
The question, however, is whether the royalty the magistrate set, based on a hypothetical negotiation before infringement between a willing licensor and a willing licensee, is reasonable. We have held that it is. The reasonableness of the royalty does not depend upon or have any relationship to the cost of a machine which, because the manufacturer had a license from the patentee to make and sell, purchasers could operate without paying any royalty.
Alpine could have avoided infringement, and paying royalties therefor, by purchasing non-infringing machines from SMI. It chose, however, to purchase and use Hedco’s infringing machines. Having followed that course, it cannot invalidate an otherwise reasonable royalty on the claim that by hindsight it would have been better off if it [1082]*1082had purchased the non-infringing SMI machines.
6. During the trial, the magistrate excluded evidence Alpine proffered that in 1972 Hanson offered to three manufacturers of snowmaking machinery (including Hedco), whose machines Hanson believed infringed his patent, licenses at a royalty of 2% percent of the selling price of the machines. The magistrate excluded this evidence under Rule 408 of the Federal Rules of Evidence, which states that “[ejvidence of (1) furnishing or offering or promising to furnish, or (2) accepting or offering or promising to accept, a valuable consideration in compromising or attempting to compromise a claim which was disputed as to either validity or amount, is not admissible to prove liability for or invalidity of the claim or its amount. Evidence of conduct or statements made in compromise negotiations is likewise not admissible.” With respect to one of the license offers, the magistrate also questioned the relevance of the evidence, since the offer was made to a manufacturer, not to a user.
The parties dispute whether this evidence of offers to license at a specified royalty, made to alleged infringers before any infringement litigation against them had been instituted, was inadmissible under Rule 408. We find it unnecessary to resolve the question. Even assuming arguendo that the magistrate erred in holding the evidence inadmissible, we cannot say that the exclusion was prejudicial.
The uncontested expert testimony, which the magistrate accepted and upon which he based his findings that we have held are not clearly erroneous, was that in pre-infringement negotiations between a willing licensor and a willing licensee the royalty would have been based upon the snowmaking capacity of the machine and the estimated savings from use of the machine, and not upon the actual use of the machine. The royalty rate in the license offers was not based on estimated savings. It was based upon the licensee’s use of the patented process in the machines the licensee would make, calculated as a percentage of the selling price of each machine.
The fact that Hanson offered to license manufacturers based upon a percentage of the selling price of the machines they made is not inconsistent with and does not undermine the magistrate’s conclusion that any license Hanson would have granted a willing user would have been based upon the formula the magistrate employed and not upon actual use of the machine. There is nothing in the record to indicate or even suggest that, in view of the considerations involving a ski resort’s use of snowmaking machinery described above, Hanson would have been willing to license a user on the basis of a percentage of the manufacturer’s selling price of the machine. We accordingly conclude that if the magistrate had considered those offers, they would not have changed his decision. Any error in the exclusion of the evidence therefore was harmless. Cf. Deere & Co. v. International Harvester Co., 710 F.2d 1551, 218 USPQ 281 (C.A.Fed.Civ.1983).
III.
Alpine contends that Hanson is precluded from recovering for the infringing use, pri- or to the filing of the complaint, of two of the Hedco machines because Hanson did not prove that Hanson’s licensee SMI had marked the machines it sold, as 35 U.S.C. § 287 (1976) allegedly required. That section provides that if the patentee or persons making or selling any patented product fail to mark the article as patented, the patentee may recover damages only for infringement committed after the infringer was notified of the infringement, and that the filing of an infringement action constitutes such notice.
The magistrate rejected the contention on two grounds: (1) it was untimely raised, since Alpine had not presented it in any pretrial pleadings, and (2) the patent is a process patent, to which section 287 does not apply. We agree with the latter ground of decision, and therefore do not reach the former ruling.
[1083]*1083Alpine states that the Hanson patent also includes apparatus claims. The only claims that were found infringed in this case, however, were claims 1, 2, and 6 of the Hanson patent, which are drawn to “[t]he method of forming, distributing and depositing snow upon a surface.... ” See Hanson v. Alpine Valley Ski Area, 611 F.2d at 158, 204 USPQ at 805. In affirming the district court’s finding of infringement in this case, the court of appeals stated in the first sentence of its opinion that “the patent alleged to be infringed is [for] a process for making snow for winter sports.” 611 F.2d at 157, 204 USPQ at 803-04. It is “settled in the case law that the notice requirement of this statute does not apply where the patent is directed to a process or method.” Bandag, Inc. v. Gerrard Tire Co., 704 F.2d 1578, 1581, 217 USPQ 977, 979 (C.A.Fed.Cir.1983).
IV.
Our decision is a narrow one. We hold only that on the record in this case, the findings of the magistrate are not clearly erroneous, and his determination of a reasonable royalty based upon those findings has not been shown to be legally erroneous. On this basis, the judgment of the district court is affirmed.
AFFIRMED.