In this action brought by former law students of the University of Georgia Law School against bar review companies, we affirm the district court’s ruling that the bar review companies did not violate the Sherman Act.
FACTS
Appellants, all 1985 graduates of the University of Georgia Law School in Athens, Georgia, brought this antitrust action against BRG of Georgia, Inc. (BRG), a Georgia corporation, Ronald 0. Pelletier, BRG’s owner, and Harcourt Brace Jovano-vich Legal and Professional Publications, Inc. (HBJ), a Delaware corporation.1 The appellants took a bar review course offered by BRG during January and February of 1985 in preparation for the Georgia bar examination.
In 1979, BRG offered a bar review course which covered the federal multi-state and Georgia law components of the Georgia bar exam. The BRG course included written materials plus lawyers and professors’ live and videotaped lectures. In 1979, BRG used West Publishing Company’s (West) free standardized multi-state materials which were being test marketed. BRG offered its bar review course at a lower price than it would have charged otherwise due to the availability of the free multi-state materials.
In 1973, HBJ began offering a review course for the Georgia bar exam and has offered bar review courses and materials nationwide under the trade name “Bar/Bri.” The record does not contain a precise description of HBJ’s Georgia course, but the parties state that it is similar to the format of the BRG course. HBJ developed its own standardized multi-state materials and sold these as a part of its Georgia course.
In 1979, BRG and HBJ competed in Georgia. In response to the low price of BRG’s course, HBJ reduced the price of its Georgia course and lost money as a result. Richard Conviser, chairman of HBJ’s board of directors, states in his affidavit that HBJ lost $45,000 on revenues of $120,000 without accounting for overhead.
At some point in 1979, West informed BRG that it planned to sell its multi-state materials through book stores, thus ending the test market arrangement. At about the same time, the lawyer who had been in charge of conducting HBJ’s Georgia course suffered a heart attack. Conviser’s affidavit states that HBJ decided to withdraw from the Georgia market at this time, but no documentation of this decision has been offered.
Conviser met with Pelletier in early 1980. On April 22, 1980, BRG and HBJ entered into a written agreement which gave BRG an exclusive license to use HBJ’s name “Bar/Bri” in Georgia. HBJ agreed that it would no longer offer a bar review course in Georgia and that it would not compete with BRG in Georgia. BRG agreed not to compete with HBJ outside of Georgia.
*1419Immediately after execution of the 1980 agreement, the price of BRG’s course rose from $150 to over $400. The record does not disclose what prices HBJ or other companies were charging for similar bar review courses. In February, 1982, a group of Georgia law students brought a class action lawsuit against BRG and HBJ alleging identical antitrust violations as are alleged in this case. See Edwards, et al. v. BRG of Georgia, Inc., et al., Middle District of Georgia, Circuit A, No. 82-13-Ath. The class included those students taking BRG’s course between April 22, 1980 and June 15, 1984. By offering partial refunds to the class members, BRG and HBJ settled that class.
During the pendency of the 1982 class action, BRG and HBJ executed a modified agreement. In this modified agreement, HBJ withdrew BRG’s exclusive right to market HBJ’s multi-state materials in Georgia. However, BRG retained the exclusive right to use “Bar/Bri” in connection with its course. Also, the modified agreement dropped the express covenant not to compete which had been contained in the initial agreement. Since the execution of the 1982 agreement, HBJ has not competed with BRG in Georgia nor licensed its multi-state materials for use by any other Georgia bar review course.
In 1979, the majority of University of Georgia law students who used a bar review course conducted in Athens, Georgia, took either the HBJ or the BRG courses. The record contains no similar data for subsequent years. The BRG course is currently marketed statewide, and is conducted at various locations in Georgia, including Athens and Atlanta. Other bar review courses are offered to Georgia bar applicants, namely, the “NORD” and “PMBR” courses, but these are not described in the record.
PROCEDURAL HISTORY
The appellants sought to represent a class consisting of those law students who attended a BRG course in Athens, Georgia, between June 15, 1984, and the present. Counts I through IV each alleges section 1 violations of the Sherman Act, 15 U.S.C.A. § l.2 A different theory of per se liability was advanced under each of these counts, namely, price-fixing cartel (Count I); market and customer allocation (Count II); boycott and concerted refusal to deal (Count III); and unreasonable joint venture (Count IV).3 Counts V through IX of the complaint each alleged section 2 violations of the Sherman Act, 15 U.S.C.A. § 2.4 The allegations were: conspiracy to monopolize in Georgia (Count V); attempt to monopolize in Georgia (Count VI); monopoly in Georgia (Count VII); conspiracy to monopolize nationwide (Count VIII); and attempt to monopolize nationwide (Count IX).
BRG and HBJ answered denying liability and denying that the class described by the appellants could be properly certified. On February 7, 1986, appellants moved for class certification and later moved for partial summary judgment on one of the conspiracy counts of their complaint. On May 2, 1986, BRG and HBJ moved for summary judgment. Appellants subsequently filed a motion for Fed.R.Civ.P. 11 sanctions against Pelletier.
On January 9, 1987, the district court denied appellants’ motion for summary judgment, but granted BRG’s and HBJ’s motion for summary judgment as to Counts II through IX of the complaint, and deferred ruling on Count I. The district court found that BRG Publications, Inc. *1420and Bar Review Group, Inc. were related corporations which have been defunct at relevant times and had no involvement in any allegedly anti-competitive activities. The district court granted summary judgment to BRG Publications, Inc. and Bar Review Group on all counts. In separate orders, the district court also denied appellants’ motions for class certification and Rule 11 sanctions.
On February 9, 1987, appellants filed a Motion for Reconsideration of the January 9 rulings, and a memorandum responding to the district court’s invitation to produce evidence to support a section 1 “rule of reason violation” of the Sherman Act. The district court declined to reconsider its previous rulings, held that the appellants had failed to support a rule of reason violation, and granted BRG’s and HBJ’s motion for summary judgment on Count I. On July 16, 1987, the district court entered judgment. Appellants filed a motion to alter or amend the judgment, and the district court denied that motion. Appellants then filed a timely notice of appeal.
ISSUES
The appellants raise five issues on appeal:
1) whether the district court’s determination that the appellants “may lack financial commitment” to this consumer class action constitutes a proper legal standard;
2) whether the district court abused its discretion in denying a motion for sanctions against Pelletier;
3) whether the district court erred in granting BRG and HBJ summary judgment;
4) whether evidence that BRG’s and HBJ’s combination enabled them to: increase price 800-percent; reduce output; and eliminate consumer choice for a product uniformly perceived to be indispensable, is sufficient proof of anticompetitive effect to preclude summary judgment under section 2 of the Sherman Act; and
5) whether the district court erred in granting BRG and HBJ summary judgment when the appellants produced sufficient probative evidence from which a jury could have reasonably inferred that any relevant product and geographic market existed; monopoly power existed or was dangerously likely to exist in that claimed market; and that BRG and HBJ took actions to secure or intended to secure such monopoly power as required for appellants to establish a violation of section 2 of the Sherman Antitrust Act.
DISCUSSION
I. Consumer Class Certification
The appellants contend that the district court applied an erroneous legal standard in applying Fed.R.Civ.P. 23(a)(4) and misread the record in denying their motion for class certification. Citing Kirkpatrick, et al. v. Bradford, et al., 827 F.2d 718 (11th Cir.1987), the appellants argue that consumer class certification should not be denied because of a lack of subjective interest on the part of the named plaintiffs, unless their participation is so minimal that they virtually have abdicated the conduct of their case to their lawyer.
BRG and HBJ cite Kirkpatrick to argue that the adequacy of class representation under Fed.R.Civ.P. 23(a) is primarily a factual issue best left for the district court’s determination. They argue that the trial court’s denial of certification should be upheld absent an abuse of discretion. BRG and HBJ further argue that the individual appellants had little knowledge of the class action when deposed, and that the named appellants were unaware of the extent of their obligation to pay expenses. Thus, BRG and HBJ assert that the record demonstrates the appellants’ lack of involvement and unwillingness to finance the case.
Whether a lawsuit may proceed as a class action is committed to the sound discretion of the district court, and its determination will not be overturned absent a showing that it has abused its discretion. In re Dennis Greenman Securities Litigation, 829 F.2d 1539, 1543-44 (11th Cir.1987). The general rule provides that the district court’s class certification is final unless an abuse of discretion exists, or the *1421court has applied impermissible legal criteria or standards. Lawler v. Alexander, 698 F.2d 439, 441 (11th Cir.1983).
The district court held:
At this juncture, plaintiffs only state that they are only capable of assuming the cost of mailing notice to approximately 280 absent class members.
Rule 23(a), Fed.R.Civ.P. requires that parties seeking to represent a class demonstrate that they will adequately protect the interest of the class. Because plaintiffs may lack financial commitment sufficient for adequate investigation and trial preparation in light of the court’s ruling, the court finds that certification of the class would be improper at this juncture. Accordingly, the motion for class certification is hereby denied.
We hold that the district court did not abuse its discretion, nor apply impermissible legal criteria or standards in denying the appellants’ request for class certification. At the point where certification became important, the district court had granted BRG’s and HBJ’s motion for summary judgment on all counts except one. The district court allowed the appellants thirty days to establish a rule of reason claim under Count I, because it had rejected their claims on per se liability. The district court properly exercised its discretion in denying the class certification because the court expected the rule of reason development to require a greater degree of financial commitment than the appellants’ depositions reflected they possessed. Accordingly, the district court’s denial of consumer class certification is affirmed.
II. Fed.R.Civ.P. 11 Sanctions
The appellants established that Pelletier submitted a false affidavit. BRG and HBJ first entered into a licensing agreement in April, 1980. Paragraph 27 of that agreement provided:
Because HBJ is providing its skill, expertise and special methods to Pelletier and BRG, if either Pelletier or BRG desires to conduct a bar review examination course in any state other than Georgia, it shall first offer HBJ the opportunity to license its courses and materials on the same terms and conditions as contained herein. Because HBJ or its licensees already conduct courses in certain states, BRG and Pelletier agree: (1) not to request a license from HBJ for such states; and (2) they will not directly or indirectly own, manage, operate, join, invest, control or participate in or be connected as an officer, employee, partner, director, independent contractor or otherwise, with any business which is operating or participating in the preparation of candidates in a state in which HBJ or its licensee is then operating a bar examination review course. Those certain states in which HBJ or its licensees are presently operating is attached hereto as Exhibit “A”.
Mr. Pelletier’s affidavit states in paragraphs 3:
At no time have I ever agreed with Harcourt Brace Jovanovich Legal Publications, Inc. (“HBJ Legal”), or Richard Conviser, or any agent or employee of HBJ Legal concerning:
a) the prices BRG would charge for bar review courses;
b) where BRG would offer courses or not offer courses; or
c) hiring only lecturers made available by HBJ Legal.
The appellants argued that Pelletier committed perjury in paragraph 3(b) of his affidavit and thereby violated Fed.R.Civ.P. 115, warranting sanctions. The district court found that the alleged inconsistency between the two statements did not establish perjury, and denied the motion for sanctions.
*1422The appellants contend that Rule 11 sanctions are mandatory, not discretionary. BRG and HBJ contend that the district court properly denied the appellants’ motion for sanctions and correctly ruled that no evidence in the record established that Pelletier had committed penury. BRG and HBJ further argue that the district court did not abuse its discretion in denying the appellants’ Rule 11 motion for sanctions.
Although the district court found that Pelletier submitted a false affidavit, the district court did not impose Rule 11 sanctions because it could not determine whether Pelletier was guilty of perjury, that is, whether his untrue statements were intentional. In reviewing the record, we find that the district court did not abuse its discretion in denying the appellants’ motion for Rule 11 sanctions. See Donaldson v. Clark, 819 F.2d 1551 (11th Cir.1987). Accordingly, we affirm the district court’s denial of Rule 11 sanctions.
III. Summary Judgment on Antitrust Claims
A. Standard of Review
Appellate review of the granting of a motion for summary judgment questions whether any genuine issue of material fact exists. Fed.R.Civ.P. 56(c). When reviewing a summary judgment decision, the reviewing court is bound by the same legal standards as those that control the district court in determining whether summary judgment is appropriate. Amey Inc. v. Gulf Abstract & Title, Inc., 758 F.2d 1486, 1502 (11th Cir.1985), cert. denied, 475 U.S. 1107, 106 S.Ct. 1513, 89 L.Ed.2d 912 (1986).
To survive BRG’s and HBJ’s motion for summary judgment, the appellants must establish a genuine issue of material fact as to whether BRG and HBJ entered into an illegal conspiracy that caused the appellants to suffer a cognizable injury. Fed.R. Civ.P. 56(e). This showing has two components: first, the appellants must show more than a conspiracy in violation of the antitrust laws existed; they must show an injury to them resulting from the illegal conduct; second, the issue of fact must be genuine. Matsushita Electric Industrial Co. v. Zenith Radio Corp., 475 U.S. 574, 585-86, 106 S.Ct. 1348, 1355-56, 89 L.Ed.2d 538 (1986); Fed.R.Civ.P. 56(c), (e).
When BRG and HBJ carry their burden under Fed.R.Civ.P. 56(c), the appellants must do more than simply show some metaphysical doubt as to the material facts. The appellants must come forward with specific facts demonstrating a genuine issue for trial. Rule 56(e). Where the record, taken as a whole, does not lead a rational trier of fact to find for the appellants, no genuine issue for trial exists. Matsushita, 475 U.S. at 586-87, 106 S.Ct. at 1356; Dunnivant v. Bi-State Auto Parts, 851 F.2d 1575, 1579-80 (11th Cir.1988).
In reviewing a grant of summary judgment, the inferences to be drawn from the underlying facts must be viewed in the light most favorable to the non-moving party. But antitrust law limits the range of permissible inferences from ambiguous evidence in a Sherman Act section 1 case. Conduct as consistent with permissible competition as well as illegal conspiracy does not, standing alone, support an inference of antitrust conspiracy. Matsushita, 475 U.S. at 587-88, 106 S.Ct. at 1356-57; Dunnivant, 851 F.2d at 1579-80.
To survive BRG’s and HBJ’s motion for summary judgment, the appellants must present evidence tending to exclude the possibility that BRG and HBJ acted independently. The appellants must demonstrate that the inference of conspiracy is reasonable in light of the competing inferences of independent action or collusive action that could not have harmed them. Matsushita, 475 U.S. at 588, 106 S.Ct. at 1357; Dunnivant, 851 F.2d at 1579-80.
In evaluating the appellants’ per se liability claims under section 1 of the Sherman Act, the district court found that generally, a plaintiff asserting a section 1 claim must prove that the defendant’s acts or conduct had an anti-competitive effect in relevant geographic and product markets. See Standard Oil Co. v. United States, 337 U.S. 293, 69 S.Ct. 1051, 93 L.Ed. 1371 (1949). Thus, the appellants bore the bur*1423den of proof and were required to make a factual showing in support of these elements when BRG and HBJ moved for summary judgment. See Celotex Corp. v. Catrett, 477 U.S. 317, 106 S.Ct. 2648, 91 L.Ed.2d 265 (1986).
The district court observed that where a defendant’s complained of acts and conduct fit certain judicially recognized categories, the plaintiffs need not establish that the defendants’ conduct actually had an anti-competitive effect, nor establish the relevant product or geographic markets. Thus, the defendant is per se liable for any damages suffered by the plaintiff. The rationale is that the designated conduct is so inherently and reliably anti-competitive that its anti-competitive effects may be presumed.
B. Sherman Act § 1 Analysis
1. Per Se Violation
Certain categories of concerted action which violate section 1 of the Sherman Act have been held to be per se illegal. Per se rules are appropriate only for conduct that is manifestly anti-competitive, that is, conduct that would almost always tend to restrict competition and decrease output. See Business Electronics Corporation v. Sharp Electronics Corp., 485 U.S. 717, 108 S.Ct. 1515, 1519, 99 L.Ed.2d 808 cert. denied, — U.S. -, 108 S.Ct. 1727, 100 L.Ed.2d 192 (1988). See also FTC v. Indiana Federation of Dentists, 476 U.S. 447, 458-59, 106 S.Ct. 2009, 2017-18, 90 L.Ed.2d 445 (1986) (the Court has “been slow ... to extend per se analysis to restraints imposed in the context of business relationships where the economic impact of certain practices is not immediately obvious.”); NCAA v. Board of Regents of University of Oklahoma, 468 U.S. 85, 103-04, 104 S.Ct. 2948, 2961, 82 L.Ed.2d 70 (1984) (“Per se rules are invoked when surrounding circumstances make the likelihood of anti-competitive conduct so great as to render unjustified further examination of the challenged conduct.”); and National Society of Professional Engineers v. United States, 435 U.S. 679, 692, 98 S.Ct. 1355, 1365, 55 L.Ed.2d 637 (1978) (agreements are per se illegal only if their “nature and necessary effect are so plainly anticompetitive that no elaborate study of the industry is needed to establish their illegality”).
The appellants contend that the district court erred in denying their motion for partial summary judgment based upon theories of per se liability under section 1 of the Sherman Act. The appellants argue that the existence of an agreement between HBJ and BRG providing for the withdrawal of HBJ from an existing market and its refusal to bid for customers in that market (Georgia), in exchange for the agreement that BRG not sell in other markets or bid for customers in other markets, constitutes per se illegal market allocation and bid rigging. The appellants argue that the district court erroneously held this agreement not to constitute a per se illegal market or customer allocation agreement. The appellants further argue that an agreement between competitors to eliminate price competition and preclude future competition through the exchange of mutual covenants not to compete, coupled with a provision to share profits, constitutes per se illegal price fixing.
The appellants further argue that the district court’s holding that BRG’s and HBJ’s conduct does not constitute per se illegal price fixing because neither BRG nor HBJ agreed upon the price each would continue to charge for its product is erroneous. According to appellants, BRG’s and HBJ’s conduct is precisely the kind of combination to which the per se rule is intended to apply.
BRG and HBJ contend that the district court properly granted summary judgment because of the appellants’ failure to present significant probative evidence indicating a violation of the antitrust laws. BRG and HBJ assert that they were in a vertical supplier-retailer relationship at the time of the first license agreement; had executed a second agreement; and have submitted affidavits that they were free to compete with each other.
The appellants seek to denominate the agreements and dealings between BRG and *1424HBJ as arrangements of the type to which per se liability has been recognized, namely, price fixing, United States v. Socony-Vacuum Oil Co., 310 U.S. 150, 60 S.Ct. 811, 84 L.Ed. 1129 (1940); market and customer allocation, United States v. Topco Associates, Inc., 405 U.S. 596, 92 S.Ct. 1126, 31 L.Ed.2d 515 (1972); boycott and concerted refusal to deal, Fashion Originators’ Guild of America v. Federal Trade Commission, 312 U.S. 457, 61 S.Ct. 703, 85 L.Ed. 949 (1941); and unreasonable joint venture, Citizens Publishing Co. v. United States, 394 U.S. 131, 89 S.Ct. 927, 22 L.Ed.2d 148 (1969).
The district court found that the arrangement between BRG and HBJ did not fit within any recognized category of per se liability. The district court further found that the 1982 agreement was not a classic form of price fixing where two competitors agreed what price they would charge for their products. On the contrary, the district court found that neither the 1980 nor the 1982 agreement explicitly addressed the factor of price, and that HBJ has never had any right under either agreement to be consulted about the price of the BRG course. See Socony-Vacuum Oil Co. 310 U.S. 150, 60 S.Ct. 811. The district court also found that although the agreement of two competitors to “pool” their products may exert an upward influence of price, such an agreement is not inherently anti-competitive.
The district court found that neither agreement between BRG and HBJ constituted the kind of market or customer allocation agreement which has been recognized as a basis for per se liability. See Topco, 405 U.S. 596, 92 S.Ct. 1126. The district court found that this was not a situation where competitors divided up a market in which both were doing business, each taking a portion of the market. The district court also found that BRG had never done business outside the state of Georgia, that nothing in the record suggested that it ever intended to do so, and that HBJ has done business nationwide, but withdrew from the Georgia market following the 1980 agreement between BRG and HBJ. Thus, the district court found the only market ever claimed by both defendants was the state of Georgia, which was not divided up under either the 1980 or 1982 agreements.
The district court ruled that BRG's and HBJ’s conduct did not constitute a boycott or concerted refusal to deal. See Zenith Radio Corp. v. Hazeltine Research, Inc., 395 U.S. 100, 89 S.Ct. 1562, 23 L.Ed.2d 129 (1969). The district court rejected the appellants’ attempt to invoke the per se theory based on the fact that the agreement prohibited BRG from hiring certain law lecturers because the appellants had no standing to raise that argument.
Finally, the district court found that the conduct described in the record differed from that recognized in Citizens Publishing Co., 394 U.S. 131, 89 S.Ct. 927, as an unreasonable joint venture which was per se illegal. The district court found that BRG and HBJ did not pool capital, and HBJ had no risk of loss on the Georgia bar review course. Thus, the arrangement created was not a joint venture, and the Citizens Publishing theory is inapplicable.
The district court denied the appellants’ motion for partial summary judgment and granted summary judgment to the defendants on Counts II through IV. The district court’s analyses of these issues are thorough and legally sound. Accordingly, we affirm the district court’s ruling that the appellants failed to establish per se liability under section 1 of the Sherman Act.
2. The Rule of Reason Violation
Because no theory of per se liability applied to the appellants’ claims under section 1 of the Sherman Act, the appellants may only prevail by demonstrating that BRG’s and HBJ’s arrangement had an anti-competitive effect in relevant geographic and product markets. Whether the action violates section 1 of the Sherman Act is determined through case-by-case application of the rule of reason when the action is not per se illegal; the fact-finder weighs all of the circumstances of a case in deciding whether a restrictive practice should be *1425prohibited as imposing an unreasonable restraint on competition. Business Electronics, 108 S.Ct. at 1519.
The district court required the appellants to submit significant probative evidence that the 1980 agreement between BRG and HBJ, as modified in 1982, had an anti-competitive effect in relevant product and geographic markets. As evidence of anti-competitive effect, the appellants offered the contract itself, the fact that the price went up after the parties signed the contract, and the affidavit of Leon Van Gelderen, a Georgia State School of Law graduate.
The appellants relied on an affidavit of Dr. William Henry, filed February 9, 1987, to address the issues of relevant product market and relevant geographic market. This affidavit stated that the relevant market is “comprehensive bar review courses that prepare students to take the Georgia bar examination” and that the relevant geographic market was “Athens, Georgia, during the winter academic term of law school.”
The appellants contend that BRG’s and HBJ’s 1982 contract modification and subsequent pricing and noncompetitive conduct are evidence of a continuing horizontal conspiracy.. The appellants argue that no material facts are genuinely disputed regarding Counts I and II of the complaint (price fixing and market allocation). They contest the district court’s denial of their motion for partial summary judgment on liability for those counts. The appellants argue that the district court erroneously applied structural and temporal tests in granting BRG’s and HBJ’s motion for summary judgment on the appellants’ section 1 rule of reason theory. The appellants further contend that their evidence was sufficient to create reasonably conflicting inferences regarding the unreasonableness of BRG’s and HBJ’s conduct.
BRG and HBJ contend that the appellants failed to show any basis for a section 1 rule of reason violation. They argue that the appellants’ rule of reason analysis suffers from its failure to specify exactly which agreement was unreasonable. BRG and HBJ contend that no significant probative evidence exists in the record from which a jury could reasonably infer that an unwritten agreement existed between BRG and HBJ not to compete against each other in selling bar review courses. BRG and HBJ argue that if the appellants are relying on this unwritten agreement, then the rule of reason analysis ends because no agreement exists to be evaluated.
The second possible agreement upon which the appellants may have relied was the license agreement between HBJ and BRG when the appellants took their bar review courses. In this agreement, HBJ promised to supply bar review materials to BRG on a nonexclusive basis. The district court found that the license agreement did not contain any anti-competitive provisions. Thus, BRG and HBJ contend that the appellants’ failure to prove the existence of an appropriate geographic and product market in which the effects of HBJ’s agreement with BRG could be evaluated defeated their rule of reason claim.
In analyzing the appellants’ rule of reason claim, the district court examined whether the appellants offered significant probative evidence that BRG’s and HBJ’s 1980 agreement as modified in 1982 had an anti-competitive effect in relevant geographic and product markets. The district court found that the possibility of anti-competitive effect existed in the contract between BRG and HBJ, which was amended in 1982, to make BRG’s license nonexclusive and to delete the covenants not to compete, but that such effect cannot be presumed from the contract itself. The district court found that although the price of BRG’s course went up significantly after the 1980 contract signing, this fact did not suffice to demonstrate anti-competitive effect, especially when the price increase came on the heels of West’s withdrawal of the free multi-state materials. The district court found that it was as safe to assume that the price increase reflected these additional costs as it was to assume that the higher prices were merely to gouge the customer. The district court found that the affidavit of Van Gelderen did not add *1426significant evidence on the issue of anti-competitive effect.
Finding flaws in Dr. Henry’s affidavit, the district court determined that it was not a substitute for expert product market analysis, and that the anti-competitive effect of the contract was more appropriately measured in a statewide market. The district court also ruled that “Athens, Georgia, during the winter academic term of law school” was not a relevant geographic market. The district court found that no significant probative evidence established that the modified contractual arrangement had a present anti-competitive effect in a relevant product and geographic market. Accordingly, the court granted BRG’s and HBJ’s motion for summary judgment on Count I. In reviewing the district court analysis, we affirm.
C. Sherman Act § 2 Analysis
In addressing Count V, conspiracy to monopolize in Georgia; Count VI, attempt to monopolize in Georgia; and Count VII, monopoly in Georgia, the district court examined the appellants’ evidence of relevant product and geographic markets and monopoly power or dangerous probability of monopoly. This circuit has held that “proof of relevant product and geographic market is absolutely essential to appellants’ Section 2 claims.” American Key Corp. v. Cole Nat. Corp, 762 F.2d 1569, 1579 (11th Cir.1985). Further, to prove an attempt to monopolize (Count VI), an antitrust plaintiff must prove the existence of “methods, means and practices which would, if successful, accomplish monopolization, and which, though falling short, nevertheless approach so close as to create a dangerous probability of it_” American Tobacco Co. v. United States, 328 U.S. 781, 785, 814-15, 66 S.Ct. 1125, 1127, 1141-42, 90 L.Ed. 1575 (1946). To prove a claim of monopoly (Count VII), an antitrust plaintiff must prove monopoly power in the relevant market, that is, the power to control prices or exclude competition. United States v. Grinnell Corp., 384 U.S. 563, 571, 86 S.Ct. 1698, 1704, 16 L.Ed.2d 778 (1966).
The appellants contend that they did not have to prove that BRG's and HBJ’s combination had an anti-competitive effect in a specifically defined product and geographic market, nor that a particular written contract has a present anti-competitive effect in order to establish the section 2 violation. The appellants argue that the structural standards for proof of unreasonableness contained in both the district court’s January 9 and July 8 orders are wrong for three reasons: (1) that proof of specific product and geographic markets is not required in a horizontal rule of reason case to prove anti-competitive effect; (2) the evidence of an anti-competitive purpose alone, even without anti-competitive effects, would prove unreasonableness; and (3) evidence that prices were intended to and did “skyrocket,” output was reduced, and consumer choice eliminated is sufficient to prove a rule of reason violation.
Because the district court found that the appellants presented no significant probative evidence that the modified contractual arrangement between BRG and HBJ had a present anti-competitive effect in a relevant product and geographic market, the appellants have no foundation for a section 2 claim. Even so, the appellants argue that evidence of an 800-percent price increase without loss of sales or new entry, two market surveys and unopposed expert opinion showing that defendants control at least 93-percent of any relevant market, and a reduction in output and elimination of consumer choice are sufficient to prove an anti-competitive effect in a relevant product and geographic market. The appellants argue that the district court transformed their rule of reason claim into a section 2 structural claim and then arbitrarily refused to acknowledge or consider their structural evidence.
The appellants further argue that the district court’s conceptions of product market definition, geographic market definition, and anti-competitive effect are contrary to precedent. They assert that the court misread the complaint in reference to their charge of statewide market allocation, ignored a deposition, market survey, and BRG’s and HBJ’s enrollment contracts *1427demonstrating increasing price without loss of sales statewide, and gave insufficient weight to other affidavits. The appellants also argue that BRG’s and HBJ’s product is perceived as unique by consumers, its price is significantly higher than theoretical alternatives, and it has the distinctive properties and discreet vendors. They assert that every criteria for product market identification cited in Brown Shoe v. United States, 370 U.S. 294, 82 S.Ct. 1502, 8 L.Ed.2d 510 (1962) is satisfied by the record in this case. Thus, the appellants contend that BRG and HBJ took actions to secure, or intended to secure monopoly power in violation of section 2 of the Sherman Act.
BRG and HBJ argue that the authorities cited by the appellants have little to do with the kind, quantum, or quality of proof that a plaintiff must offer to show that an agreement in place between BRG and HBJ in 1985 violated the rule of reason. BRG and HBJ argue that the appellants are simply wrong in suggesting that a restraint which is not per se illegal does not need to be evaluated within the context of a defined product and geographic market. BRG and HBJ contend that appellants’ failure to establish the existence of a relevant product and geographic market thus defeats their section 2 claims.
BRG and HBJ argue that the appellants failed to produce sufficient evidence to avoid summary judgment on their theories with respect to monopolization of alleged Georgia markets. BRG and HBJ contend that the appellants failed to establish the structural elements of the relevant product or geographic market. BRG and HBJ further contend that the section 2 offenses of conspiracy, attempt, and actual monopolization all require the same proof with respect to the structural attributes of the economic market in which the violations are alleged to have taken place. To prevail, BRG and HBJ argue that the appellants must show: (1) the existence of some economically meaningful market, defined by its product and geographic characteristics, and (2) the existence of monopoly power within the identified market. BRG and HBJ argue that the appellants failed on both accounts. Because the appellants failed to establish any significant probative evidence of conspiracy or agreement, BRG and HBJ contend that all section 2 theories must fail. Also, BRG and HBJ contend that no significant probative evidence has been produced from which a jury could reasonably infer that BRG and HBJ were parties in 1985 to an agreement to give BRG control over any Georgia market.
The district court held that proof of the relevant geographic and product market is essential to all section 2 claims. The district court found Dr. Henry’s affidavit to be insufficient to identify or justify a proper product or geographic market. The district court also found that the entirety of the appellants’ effort to identify proper product and geographic markets was contained in the affidavit of Dr. William Henry, filed in opposition to BRG’s and HBJ’s motion for summary judgment.6 The dis*1428trict court held Dr. Henry’s affidavit insufficient to define the relevant product market. Particularly, because Dr. Henry chose to define the product market in such a narrow and seemingly artificial manner, the district court found that the lack of justification for his conclusions rendered it valueless. The district court held the relevant geographic market to be the geographic area of “effective competition” for the product in question. American Key Corp., 762 F.2d at 1581.
Although the appellants’ failure to properly identify relevant product and geographic markets required the district court to grant summary judgment to BRG and HBJ on Counts V through VII, the district court found that Count VI (attempt to monopolize in Georgia) and Count VII (monopoly in Georgia) would have failed for additional reasons. Count VI would have failed under the American Tobacco requirement, and Count VII would have failed under the Grinnell requirement. See supra p. 1426. The district court found that the appellants’ only effort to show evidence of these requirements was the conclusory statements of the appellants that they had “no other choice” than the BRG course when they took it in January through February, 1984 in Athens, and the statement in Dr. Henry's first affidavit that “[f]urther, Ronald Pelletier testified that defendants accounted for at least eighty percent of the sales of bar review courses in the Athens area.” The district court found that these assertions were “clearly insufficient.”
CONCLUSION
Because of the district court's thorough, complete, and correct antitrust analysis regarding the appellants’ claims and BRG’s and HBJ’s motions for summary judgment, we affirm.
AFFIRMED.
. BRG includes Bar Review Group, Inc. and BRG publications, Inc., all Georgia corporations.
Pelletier formed BRG in 1979 for the sole purpose of offering bar review courses to Georgia bar applicants.
. Section 1 of the Sherman Act provides: “Every contract, combination in the form of trust or otherwise, or conspiracy, in restraint of trade or commerce among the several States, or with foreign nations, is declared to be illegal...."
. The district court noted that these theories were improperly pled as separate counts because the plaintiffs only have one cause of action for an alleged section 1 violation of the Sherman Act. However, for ease of reference, the district court referred to the designations chosen by plaintiffs.
.Section 2 of the Sherman Act provides: "Every person who shall monopolize, or attempt to monopolize, or combine or conspire with any other person or persons, to monopolize any part of the trade or commerce among the several States, or with foreign nations, shall be deemed [to have violated this Act],...”
. Fed.R.Civ.P. 11 provides, in pertinent part:
If a pleading, motion, or other paper is signed in violation of this rule, the court, upon motion or upon its own initiative, shall impose upon the person who signed it, a represented party, or both, an appropriate sanction, which may include an order to pay to the other party or parties the amount of the reasonable expenses incurred because of the filing of the pleading, motion, or other paper, including a reasonable attorney’s fee.
. The affidavit states:
Dr. William Henry, being duly sworn, deposes and says: j.
I am Professor of Finance in the College of Business Administration at Georgia State University with a Ph.D. degree in economics from North Carolina State University, and give this affidavit in the above-captioned matter. 2.
I have reviewed the depositions of the plaintiffs in this action, the deposition of Ronald Pelletier, the enrollment contracts produced by the defendants, and the deoposition of Professor David Kamershen taken in the case of Edwards et al. v. BRG of Georgia, Inc., C.A. No. 82-13-Ath together with the exhibits thereto. 3.
Upon review of the depositions of the named plaintiffs and the deposition of Ronald Pelletier it is apparent that nearly all students at University of Georgia School of Law take the defendants’ bar review course. Because the students taking the course are also enrolled as full-time students at the law school in Athens, it is not realistic to expect that they could reasonable [sic] take a course offered outside the Athens area. Further, Ronald Pel-letier testified that defendants accounted for at least eighty percent of the sales of bar review courses in the Athens area.
I conclude that a comprehensive bar review course which includes lectures and written material offered in Athens, Georgia to prepare students to take the Georgia bar examination is the relevant product market and that Ath*1428ens is a relevant geographic market for purposes of ecomonic analysis. I further conclude that defendants have and have had the power to control prices in the market, and that defendants therefore have monopoly power in the relevant geographic and product markets.
This the 12th day of May, 1986.
/s/ William R. Henry
Dr. William Henry