
569 F.3d 841 (2009)
UNITED STATES of America, Plaintiff-Appellee,
v.
Mustafa REDZIC, Defendant-Appellant.
Nos. 08-2418, 08-3662.
United States Court of Appeals, Eighth Circuit.
Submitted: May 13, 2009.
Filed: June 22, 2009.
*843 Mark G. Arnold, argued, Joseph G. Nassif and Patrick M. Flachs, on the brief, St. Louis, MO, for appellant.
Thomas Christian Albus, AUSA, argued, St. Louis, MO, for appellee.
Before WOLLMAN, JOHN R. GIBSON, and MURPHY, Circuit Judges.
MURPHY, Circuit Judge.
Mustafa Redzic was charged with mail fraud, wire fraud, bribery, and conspiracy to commit each of the substantive offenses in connection with a scheme to obtain Missouri commercial driver licenses for his students. He was convicted on all counts after a jury trial and sentenced by the district court[1] to concurrent terms of 75 months on each of the substantive counts and 60 months on the conspiracy. Redzic appeals, arguing that the government failed to prove he defrauded the state or bribed a state agent, requiring that his convictions all be reversed. We affirm.

I.
Redzic was the owner of Bosna Truck Driving School (Bosna) in St. Louis, Missouri, which trained students to operate commercial vehicles. Students who completed their studies at Bosna and wished to obtain a commercial driver license (CDL) were required to be tested at either a state run facility or a private company trained and licensed by the State of Missouri. The required test consisted of a written exam and a three part driving component which included a pre trip inspection, a skills test, and a road test.
Commercial Drivers Training Academy (CDTA) was a licensed third party tester located in Sikeston, Missouri. Redzic's codefendant, Troy Parr, oversaw testing at CDTA.[2] In order to generate more business and process more CDL applicants, Parr developed a system of "short testing" in which CDTA would omit entire portions of the driving exam or administer it in only a fraction of the time necessary for a proper evaluation. These short tests did not meet the minimum requirements set by the state for CDL testing. Parr regularly gave short tests to Redzic's students and submitted paperwork to the state falsely indicating that the students had successfully met the full requirements for their CDLs. On one occasion Redzic asked Parr to submit paperwork for two students who never even appeared at CDTA, and *844 Parr agreed to do so. A government wiretap also recorded Redzic promising the husband of a potential customer that he could "guarantee" his wife would pass the driving test despite her inability to operate properly a manual transmission.
CDTA charged $150 per student for each test, but Parr testified that Redzic included with his payments to CDTA between $50 to $200 in extra cash on a dozen or more occasions. Parr kept a portion of these overpayments and split the remainder with other CDTA examiners. Redzic also gave Parr two other cash payments of $600 and $2,500. Parr testified that he never requested any of these extra payments although he had been assisting Redzic in his attempt to obtain state certification for Bosna. Such a certification would qualify the school's students for state grant money and could produce another $1 million in business, according to Redzic's estimate. Parr said that he had been willing to consult without payment in exchange for future employment at Redzic's expanded school. He also testified that Redzic had offered him employment to be compensated at some 150 percent of his CDTA salary, a free apartment, bonuses, and free vacations. At Redzic's request Parr rejected a bribe from one of Bosna's competitors and reported the bribery attempt to state authorities. The next day Redzic personally delivered another $2,500 cash payment.
Redzic was charged with and convicted of mail fraud in violation of 18 U.S.C. § 1341, wire fraud in violation of 18 U.S.C. § 1343, bribery in violation of 18 U.S.C. § 666(a)(2), and conspiracy in violation of 18 U.S.C. § 371. Following the close of the government's case, Redzic filed a motion for judgment of acquittal based on an argument that the government's evidence was insufficient to sustain any conviction. The district court denied the motion. Redzic chose not to offer any evidence at trial and renewed his motion which was again denied. The jury returned a guilty verdict on all counts.
Redzic subsequently filed a post trial motion for new trial and an amended motion for new trial and for judgment of acquittal. In these post trial motions Redzic raised for the first time a number of issues, including the argument that CDLs are not property for purposes of the federal fraud statutes. The district court denied the motions in May 2008, and Redzic was sentenced the following month. In October 2008 the district court ordered the forfeiture of $25,272.48 in cash, seven trucks, and three trailers pursuant to 28 U.S.C. § 2461.

II.

A.
We first examine Redzic's convictions for mail and wire fraud. The mail fraud statute reads in part as follows:
Whoever, having devised ... any scheme or artifice to defraud, or for obtaining money or property by means of false or fraudulent pretenses, representations or promises ... [uses the mails in furtherance of the scheme shall be punished by imprisonment or fine or both].
18 U.S.C. § 1341. The wire fraud statute is, in pertinent part, identical. See 18 U.S.C. § 1343. The Supreme Court in McNally v. United States, 483 U.S. 350, 107 S.Ct. 2875, 97 L.Ed.2d 292 (1987), interpreted the statutory language to apply only to deprivations of property and noted in particular that it did not encompass "the right to have [a state's] affairs conducted honestly." Id. at 352, 107 S.Ct. 2875.
In response to McNally, Congress broadened the scope of the mail and wire *845 fraud statutes by enacting 18 U.S.C. § 1346. That section provides:
For the purposes of this chapter [including § 1341 and § 1343], the term "scheme or artifice to defraud" includes a scheme or artifice to deprive another of the intangible right to honest services.
18 U.S.C. § 1346. Thus, § 1346 brought schemes to deprive another of honest services within the scope of the mail and wire fraud statutes. Section 1346 does not create a separate offense but expands the definition of scheme or artifice to defraud for both mail and wire fraud. The substantive offenses of mail and wire fraud may therefore be committed by using either medium to deprive another of property or of the right to honest services.
Nevertheless, Redzic argues that the government failed to invoke this expanded scope of mail and wire fraud. He asserts that the indictment, the government's argument at trial, and the jury instructions all proceeded on the theory that he and Parr had defrauded the state of Missouri of property in the form of CDLs. He argues that as a matter of law "[s]tate and municipal licenses in general... do not rank as `property,' for purposes of § 1341, in the hands of the licensor." Cleveland v. United States, 531 U.S. 12, 15, 121 S.Ct. 365, 148 L.Ed.2d 221 (2000). As he points out, Cleveland dealt explicitly with mail fraud as defined in § 1341, but the Supreme Court explained in Carpenter v. United States, 484 U.S. 19, 25 n. 6, 108 S.Ct. 316, 98 L.Ed.2d 275 (1987), that "[t]he mail and wire fraud statutes share the same language in relevant part, and accordingly we apply the same analysis to both sets of offenses...." We agree with Redzic that the simple act of fraudulently obtaining a license from a government agency does not deprive that agency of property in the sense of §§ 1341 or 1343.
The government argues, however, that Redzic's fraud extended beyond the licenses themselves, for in order to obtain them for his students Redzic participated in a scheme in which Parr, a licensed agent of the state,[3] submitted false paperwork, thereby depriving Missouri of its right to Parr's honest services in violation of §§ 1341 and 1343. Redzic responds that the indictment charged him with defrauding the state of property, not honest services, and that to convict him on a honest services theory would amount to a constructive amendment of the indictment in violation of his Fifth Amendment rights. Constructive amendments are per se prejudicial. See United States v. Yeo, 739 F.2d 385, 387 (8th Cir.1984).
Where, as here, the sufficiency of an indictment is challenged after jeopardy has attached, the indictment is upheld "unless it is so defective that by no reasonable construction can it be said to charge the offense for which the defendants were convicted." United States v. Pennington, 168 F.3d 1060, 1064-65 (8th Cir.1999). (quoting United States v. Just, 74 F.3d 902, 904 (8th Cir.1996)). An indictment is insufficient if it fails to allege an essential element of the crime charged, but in determining whether an essential element has been omitted, "a court may not insist that a particular word or phrase appear in the indictment when the element is alleged `in a form' which substantially states the element." United States v. Mallen, 843 F.2d 1096, 1102 (8th Cir.1988). In this case the indictment fails to use the phrase "honest services," but its factual allegations clearly spell out the substance of Redzic and Parr's scheme. In particular, the indictment includes the following allegations:
(1) Parr was an agent of the State of Missouri in which capacity he served as a CDL examiner.

*846 (2) The proper certification and licensing of commercial truck drivers was "essential to the economic welfare and public safety of Missouri."
(3) Redzic and Parr developed a relationship in which Parr would administer inadequate short tests to Redzic's students.
(4) In one instance, Parr certified two of Redzic's students who had not even shown up at CDTA for testing.
(5) After administering inadequate short tests or failing to administer any test whatsoever, Parr and other CDTA examiners would "falsely complete paperwork to be mailed to the State of Missouri Department of Revenue stating that each student had submitted to and passed a full, complete three-part Missouri CDL test."
(6) Redzic benefitted from the scheme.
These allegations were more than sufficient to make out a case that Redzic and Parr intentionally devised a fraudulent scheme whereby the State of Missouri was deprived of its right to Parr's honest services. The indictment cites §§ 1341 and 1343, which establish the substantive offenses of mail and wire fraud. It does not specifically cite § 1346, but that section does not create a separate substantive offense; it merely defines a term contained in §§ 1341 and 1343. It was therefore not necessary to cite it in the charging instrument. See United States v. Gilkerson, 556 F.3d 854, 856 (8th Cir.2009) ("[A]ny alleged failure to define `migratory bird preservation facility' or to cite to where that definition could be found is immaterial.").
Redzic notes that the indictment refers to "a scheme and artifice to defraud the State of Missouri of money and other things of value including CDLs." Based on this phrase, he argues he was misled to believe that he would not be held to account for the deprivation of Parr's honest services. He cites the Second Circuit case of United States v. Cervone, 907 F.2d 332 (2d Cir.1990), in an attempt to distinguish "money and other things of value" from intangibles such as Parr's honest services. But Cervone hardly helps his argument. In that case the court noted that the phrase "money or other thing of value" "clearly indicat[es] ... tangible items or at least valuable services." Id. at 347 (emphasis added).
While we believe it would have been preferable in Redzic's case for the indictment to have included the term "honest services," its omission was not fatal. "An indictment is sufficient if it fairly informs the accused of the charges against him and allows him to plead double jeopardy as a bar to a future prosecution." Mallen, 843 F.2d at 1102. Here, the allegations clearly spelled out the course of conduct for which Redzic was on trial, including his role in Parr's submission of false and dishonest paperwork. Furthermore, it cited to the appropriate substantive statutes which incorporate the fraudulent deprivation of honest services. This was sufficient to satisfy Fed.R.Crim.P. 7(c)(1): "The indictment ... must be a plain, concise, and definite written statement of the essential facts constituting the offense charged...."
Redzic's contention that at trial the government relied exclusively on a loss of property theory is similarly misplaced. While the prosecutor neglected to use the phrase "deprivation of honest services," that is not determinative. For example, in his closing argument the prosecutor recounted the incident in which Parr, at Redzic's request, submitted CDL paperwork for two students who had completely failed to report for testing. He then noted pointedly, "that's what this scheme and artifice to defraud is all aboutgetting these guys tested, getting these guys short-tested, *847 sometimes getting these guys no tested." The government also argued that the scheme deprived Missouri of meaningful regulatory control over commercial truck drivers. In particular, Parr's malfeasance failed to implement the state's requirement that such drivers read, write, and speak English in order to understand road signs and, on one occasion, Redzic guaranteed favorable test results to a student who could not operate a manual transmission. In short, the thrust of the government's case was clearly not that Missouri had been damaged by losing the value of the physical pieces of paper on which the licenses had been printed. Rather, the government consistently referred to Parr's practice of administering inadequate short teststhat is, of withholding the honest services he owed the State of Missouri thereby undermining the state's efforts to promote highway safety.
Redzic also asserts that the district court instructed the jury exclusively on a deprivation of property theory and that any attempt to uphold his conviction on alternative grounds would deprive him of his Sixth Amendment right to jury trial. This argument also fails. The jury instructions correctly laid out the four elements of mail fraud:[4] (1) devising or participating in a scheme to defraud; (2) intent to defraud; (3) knowledge that it was reasonably foreseeable that the mails would be used; and (4) use of the mails in furtherance of the scheme. See, e.g., United States v. Onwumere, 530 F.3d 651, 653 (8th Cir.2008). The instructions further identified the scheme as follows: "Troy Parr and the defendant submitted by mail to the State of Missouri false test records for Commercial Drivers License tests purportedly given to the defendant's students." This description does not merely suggest that Redzic fraudulently obtained CDLs, but that he did so through the assistance of an agent of the state who withheld his honest services in order to obtain the desired licenses.
Elsewhere, the instructions define the term "scheme to defraud" to "include[] any plan or course of action intended to deceive or cheat another out of property by employing material falsehoods." According to this definition, the phrase "scheme to defraud" includes plans to deprive another of property but is not necessarily limited to such plans. It would have been preferable had the district court used the term "honest services" in place of "property" as the Model Jury Instructions allow, see Eighth Circuit Manual of Model Jury Instructions: Criminal, § 6.18.1341, at 309 (2007 ed.), but a district court's instructions will be upheld if, "when read as a whole, [they] fairly and adequately contain[] the law applicable to the case." United States v. Casas, 999 F.2d 1225, 1230 (8th Cir.1993). Here, the four elements of mail fraud were correctly identified and the charged scheme was characterized as one in which Parr, an agent of the state, submitted false information in dereliction of his duties. When the instructions are read as a whole, they adequately conveyed the law applicable to Redzic's case.
Redzic cites numerous cases in support of his argument that the indictment was insufficient to charge him with depriving the state of Parr's honest services. In particular, he relies on the Supreme Court's decision in Stirone v. United States, 361 U.S. 212, 80 S.Ct. 270, 4 L.Ed.2d 252 (1960). Stirone was tried and convicted of unlawful interference with interstate commerce. The indictment charged him with interfering with the importation *848 of sand into Pennsylvania, but the jury instructions allowed the jury to find that his conduct would have affected the future exportation of steel out of the state. The Court reversed Stirone's conviction, holding that the petit jury had impermissibly been given the option of convicting him for a form of interference not contemplated by the grand jury.
Redzic's case is easily distinguished. The facts and allegations essential to his conviction were clearly laid out in the indictment. Although the statutory phrase "honest services" was not itself used, it is obvious that the grand jury took note of Parr's short testing, his subsequent submission of false paperwork to the state, and Redzic's participation in this scheme. The petit jury was instructed to determine whether Redzic and Parr had "submitted... false test records for Commercial Drivers License tests purportedly given to the defendant's students," which corresponds to the conduct considered by the grand jury. Thus, unlike in Stirone, there was no invitation for the petit jury to wander beyond the allegations contained in the indictment. The acts which fraudulently deprived the state of licenses were the same as those which fraudulently deprived it of the honest services of examiner Parr.
Redzic's other authorities are similarly inapposite. United States v. Selwyn, 998 F.2d 556 (8th Cir.1993), reversed a conviction where the indictment charged embezzlement but the defendant's course of conduct actually amounted to stealing. Stealing and embezzlement are two distinct crimes with different elements. By contrast, mail and wire fraudwhether they deprive another of property or honest servicesare each a single substantive offense. Similarly, in United States v. Leisure, 844 F.2d 1347 (8th Cir.1988), the indictment cited to 18 U.S.C. § 1510 (obstruction of justice) but the case was submitted on the theory that the defendant had actually violated 18 U.S.C. § 1512 (witness tampering). In that case we noted that the two crimes were not the same and had different mens rea elements. In Yeo the indictment charged the defendant with extorting a specific individual, but the jury was permitted to convict upon finding he had extorted the named victim "or another." 739 F.2d at 387. In other words, the case submitted to the jury allowed it to convict Yeo for acts of extortion not alleged in the indictment. In Redzic's case, the course of conduct charged in the indictment corresponded with the government's proof at trial and the jury instructions. There was no danger that the jury might consider other, uncharged incidents in convicting him.
For the foregoing reasons, we conclude that the indictment, the government's case at trial, and the jury instructions were each sufficient to support a conviction for the fraudulent deprivation of honest services in violation of §§ 1341 and 1343.

B.
Redzic next argues that § 666(a)(2) only prohibits bribery and, in particular, does not criminalize the payment of "gratuities." Because the government produced sufficient evidence for a reasonable jury to conclude that Redzic had indeed bribed Parr, we need not consider his arguments concerning gratuities.[5]
*849 We review the district court's denial of a motion for judgment of acquittal de novo and consider all evidence in the light most favorable to the verdict, giving the verdict the benefit of all reasonable inferences. United States v. Birdine, 515 F.3d 842, 844 (8th Cir.2008).
Section 666(a)(2) reads in pertinent part as follows:
Whoever ... corruptly gives, offers, or agrees to give anything of value to any person, with intent to influence or reward an agent ... of a State ... government... in connection with any business, transaction, or series of transactions of such ... government ... involving anything of value of $5,000 or more[] shall be fined under this title, imprisoned not more than 10 years, or both.
18 U.S.C. § 666(a)(2). The statute applies if the identified government receives federal benefits in excess of $10,000 in any one year. 18 U.S.C. § 666(b).
By stipulation, the parties agreed that Parr was an agent of the State of Missouri, that a CDL is worth in excess of $5,000, and that the state had received in that year the requisite $10,000 in benefits from the federal government. In addition the government produced ample evidence that Redzic had given Parr substantial cash payments on multiple occasions as well as an offer of employment. Redzic argues, however, that the government failed to prove that he gave these cash payments "corruptly," that is, with the intent to "influence or reward" Parr.
To prove the payment of an illegal bribe, the government must present evidence of a quid pro quo, but an illegal bribe may be paid with the intent to influence a general course of conduct. It was not necessary for the government to link any particular payment to any particular action undertaken by Parr.
Rather, it is sufficient to show that the payor intended for each payment to induce the official to adopt a specific course of action.... The quid pro quo requirement is satisfied so long as the evidence shows a "course of conduct of favors and gifts flowing to a public official in exchange for a pattern of official actions favorable to the donor."
United States v. Jennings, 160 F.3d 1006, 1014 (4th Cir.1998) (internal citations omitted); see also United States v. Ganim, 510 F.3d 134, 148-49 (2d Cir.2007) (same). In this case the government was only required to prove that Redzic made payments to Parr with the intent of influencing or rewarding a pattern of behavior. It was not necessary to link any given payment to any given short test, for example.
In accordance with this circuit's model jury instructions, the district court allowed the jury to infer "that a person intends the natural and probable consequences of acts knowingly done." See Model Jury Instructions, § 7.05, at 582. There is no dispute that on multiple occasions Redzic gave Parr cash payments of $50 to $200 in addition to the standard examination fee charged by CDTA. Parr testified that he and other CDTA examiners would divide these payments among themselves. Although the government produced no direct evidence of an explicit agreement, a reasonable jury could find that these payments were intended to influence Parr to continue providing short tests to Bosna students or to reward him for having done so. The statute prohibits attempts either to influence or to reward.
Moreover, the evidence indicates that Redzic personally delivered a $2,500 cash payment to Parr one day after Parr had turned away the corrupt business of one of Redzic's competitors. Because CDTA could only process a limited number of students, a jury could infer that this payment *850 was intended to ensure that Bosna students would continue to have access to Parr's fraudulent scheme even at the expense of turning away other business.
Finally, Redzic made at least two other cash payments to Parr in the amounts of $600 and $2,500. Redzic argues that these payments were simply compensation for Parr's assistance in obtaining state certification for Bosna, a process which would allow the school's students to qualify for state grant money. Parr testified, however, that he never requested such compensation and was willing to assist in the certification process based only on his hope of future employment at Bosna.
Redzic was entitled to argue, as he did, that his various payments to Parr were part of his effort to expand his school's business through state certification. But the mere fact that he supplied an innocent explanation for his behavior does not negate the strength of the government's evidence. A reasonable jury could properly find that Redzic's actual intent was to induce Parr to continue offering substandard short tests to Bosna students. Construing the evidence in the light most favorable to the jury's guilty verdict, we conclude that there was sufficient evidence to convict Redzic of bribery under § 666(a)(2).

C.
Redzic seeks to overturn his conspiracy conviction based on the ground that each of the substantive charges against him was legally insufficient. Having determined that Redzic was properly convicted on each of the underlying substantive offenses, we find no defect in his related conviction for conspiring with Parr under 18 U.S.C. § 371, and we likewise find no reason to disturb the district court's order of forfeiture pursuant to 28 U.S.C. § 2461.

III.
For the foregoing reasons we conclude that the indictment, the government's case at trial, and the jury instructions adequately laid out a scheme to defraud the State of Missouri of Parr's honest services. It was not necessary for the government to use that exact phrase when the substance of the scheme was reasonably described. In addition, we conclude that the government produced sufficient evidence for a reasonable jury to convict Redzic for bribery. Consequently, the charge of conspiracy and the forfeiture of Redzic's assets were also proper. The judgment of conviction and the forfeiture are affirmed.
NOTES
[1]  The Honorable E. Richard Webber, United States District Judge for the Eastern District of Missouri.
[2]  Parr was also indicted on the fraud and conspiracy counts. He subsequently reached a plea agreement with the government in which he admitted to the conspiracy charge and agreed to cooperate with the government's prosecution of Redzic.
[3]  By stipulation, the parties agreed that Parr was an agent of the State of Missouri.
[4]  Since the jury instructions on wire fraud are in all relevant aspects identical to those on mail fraud, we do not discuss them separately.
[5]  These include observations about our decision in United States v. Zimmermann, 509 F.3d 920 (8th Cir.2007), where we held that a subsection of the same statute, 18 U.S.C. § 666(a)(1)(B), prohibits the payment of gratuities. Redzic argues, however, that the legislative history of the statutory provisions indicates that Congress did not intend to penalize the payment of gratuities. Since the evidence here is sufficient to sustain a conviction of bribery, we need not discuss the gratuity issues.
