Some big U.S. companies, it seemed, have been giving generous raises. In a regulatory filing last year, the equipment manufacturer Caterpillar reported that median pay increased 12 percent over the prior year. It rose by 17 percent at Kellogg’s, the cereal-maker; by 10 percent at Procter & Gamble; and by 26 percent at Conagra.

Those official pay figures were not what they appeared, however. While Congress passed a law in 2010 requiring companies to report median pay, federal regulators adopted a corporate lobbyist’s proposal that allows firms to make a “shortcut” calculation that can render the figures meaningless.

What hundreds of companies have been reporting as “median pay” is not actual median pay, but the pay of a single worker who was at the median a year or two ago.

If that one median worker got a big raise, “median compensation” at the company shoots up.

If that worker took a pay cut, it plummets.

The law requiring companies to report median pay was part of the mammoth Dodd-Frank Wall Street Reform and Consumer Protection Act, a bill that was supposed to address the financial failures uncovered by the 2008 financial downturn.

Its advocates envisioned that reports of median worker pay would provide a measure of corporate transparency in a time of growing inequality. Comparable pay for chief executives was supposed to be reported as well, so investors and watchdogs could compare compensation of top executives and average workers.

But in the years that followed, as regulators at the Securities and Exchange Commission turned to the work of turning the idea into practice, U.S. business lobbyists prevailed at arguing that the law’s requirements were too expensive and logistically difficult for them to follow.

In doing so, they show how corporations can work with regulators to fundamentally refashion a law passed by Congress.

After the regulatory back-and-forth, companies began tracking median pay in 2017, and began disclosing it in 2018. Its third year is just beginning, but it’s clear from how some companies reported median pay in the first two years the numbers are cockeyed.

“A 17 percent increase for workers? I don’t know where in the world they would get numbers like that,” said Ron Baker, a representative of the bakers union that covers several Kellogg’s plants. He said union workers have seen annual wage increases of about 2 percent or less. “It sounds like fuzzy math.”

“There’s no way the pay rose 12 percent,” said an employee at a Caterpillar plant in Illinois, who asked that his name not be used out of fear of angering managers at his plant.

While the text of the 2010 law called for companies to report “the median of the annual total compensation of all employees,” the little-noticed 2015 decision by the SEC allows hundreds of companies to use the shortcut.

In drafting the rules, the SEC adopted a proposal from a business group that allowed companies to report the pay of a worker who was at the median a year or two previously, rather than calculating the actual median. This shortcut was supposed to be used only if a company “reasonably believes” that the method would not result in a “significant” change in the pay disclosure. “Significant” was not explicitly defined.

In a Deloitte survey of 331 companies in the S&P 500, more than half were using the shortcut.

Those large jumps in pay reported by Caterpillar, Kellogg’s, Procter & Gamble and Conagra are the product of the shortcut. They make the jumps in CEO pay at those companies seem less glaring.

By email, Kellogg’s acknowledged that wages at the company “did not increase by 17 percent” during the first two years of the law and that “we don’t think this is the best metric for evaluating overall pay practices for our people,” according to a statement from spokesperson Kris Bahner.

This month, in fact, the company reported its median pay for the third year, and this time, the company calculated the actual median. The median pay figure dropped significantly, and it’s now 3 percent lower than it was the first year.

A spokesman at Conagra, Dan Hare, said the 26 percent rise reported in median pay at that company “does not reflect a general increase.”

Procter & Gamble and Caterpillar declined requests to say whether their median pay reports reflect overall pay levels.

Other companies using the shortcut calculation reported large drops in median pay, and those changes don’t reflect actual median pay, either.

Pfizer, for example, reported median employee pay fell 10 percent. But the reported figures do not reflect a drop in overall company pay levels, according to a statement sent by Patricia Kelly, a Pfizer spokesperson.

Instead, the company’s reported median worker compensation is based on a single worker who was at the median one year but then went on disability. The pay of the median worker, who works outside the United States, also declined because the local currency fell in value, Kelly said.

As a result, the figures being reported to the federal government generally are not reliable as a measure of current median worker pay. It means that for anyone wondering whether a company’s prosperity is being shared with workers in the form of rising median pay, the mandated reports may provide few real answers.

“That’s unfortunate,” said Rosanna Landis Weaver of As You Sow, a nonprofit that lobbied for the disclosure rule. “There just seems to be a deep reluctance to disclose the inequalities that exist within corporations. Companies have resisted this rule for year and years and years — and they are still.”

“I fought for a strong CEO pay ratio rule to increase corporate transparency,” Sen. Robert Menendez (D-N.J.), who wrote the provision into Dodd-Frank, said in a statement. “However, information is only useful if it’s consistently and uniformly reported. Investors have a right to transparent and full reporting.”

Tim Bartl, the lobbyist who proposed the shortcut method to the SEC and met several times with SEC staff, said it was necessary to ease the task facing large international companies asked to calculate median pay.

“If you needed to take the total compensation for each employee globally and calculate the median, that endeavor is extremely difficult,” said Bartl of the HR Policy Association, which represents hundreds of large U.S. companies. “This was a way to say, ‘Okay, we understand it’s burdensome.’ . . . This is a shortcut to this.”

Despite some of the large swings in median pay reflected in the companies that use the method, Bartl said, he believes companies are calculating the numbers in good faith.

“Companies aren’t playing games with that, necessarily,” he said.

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Today, for leaders of both major parties, the issue of how much workers share in the nation’s economic prosperity has been at the forefront of voter appeals. Democratic presidential aspirants have argued that too little of the recent economic growth has reached workers. President Trump, meanwhile, has touted a “blue-collar boom,” noting that wages “are rising fastest for low-income workers.”

While the pay disclosure amounted to less than one page in the 849-page Dodd Frank legislation a decade ago, it has since provoked what may be one of the most contentious arguments arising from the landmark financial regulatory overhaul.

The measure was immediately attacked by lobbyists representing associations of big companies. The U.S. Chamber of Commerce and other groups argued that calculating median pay would be onerous for companies, especially those with global workforces, and its costs would outweigh any possible benefits. Enacting it, they said, would cost corporate America hundreds of millions of dollars.

At the core of the opponents’ lobbying effort was the Center on Executive Compensation, a division of the HR Policy Association, which represents more than 300 companies, most of them in the Fortune 500. Its directors include executives from major U.S. companies with familiar names — Boeing, Johnson & Johnson, Caterpillar, Marriott, General Electric and many others. The association lobbies Congress on a variety of employment issues and researches best practices; its budget runs about $10 million a year.

The center and other opponents soon were involved in an effort to repeal the requirement.

The following year, Rep. Nan A.S. Hayworth (R-N.Y.) introduced the Burdensome Data Collection Relief Act, which would have simply jettisoned the provision approved less than a year before.

The requirement, Hayworth said at the time, is “a burdensome regulation that provides no benefit and has substantial costs. Money spent complying with this provision is much better spent growing the economy and creating jobs.”

The bill was approved by a House committee but got no further. It was introduced again in 2013 and 2015 with similar results.

After the initial repeal efforts failed, opponents turned their attention to the Securities and Exchange Commission, the federal agency that was supposed to write the rules implementing the law.

The agency’s first draft of the implementing rules, issued in 2013, did not include the shortcut. But the debate over the requirement would soon become more political.

As the Securities and Exchange Commission considered the matter in September 2013, it became apparent that a majority of the five-member panel was less than enthusiastic about implementing the law Congress had passed.

Two Democrats on the panel, Luis A. Aguilar and Kara M. Stein, approved of the draft without the shortcut.

Two Republicans, Michael S. Piwowar and Daniel M. Gallagher, passionately opposed it, with Gallagher calling it a “rotten mandate” and Piwowar objecting to the commission “even considering it.”

“Shame on us for letting special interests distract us from our core mission,” Piwowar said.

In the middle was the commission’s chair, Mary Jo White, an independent appointed by President Barack Obama. Her voice would be critical.

Soon after the meeting, White revealed qualms about requiring firms to disclose information as a way of exerting “societal pressure on companies to change behavior.”

In a speech at Fordham University, White did not mention the pay disclosure rule specifically but said of some disclosure mandates that “I must question, as a policy matter, using the federal securities laws and the SEC’s powers of mandatory disclosure to accomplish these goals.”

While noting that “neither I nor the commission has the right to just say, ‘No,’ ” she said that the agency could “write the rule in a way that best comports with our view of our mission and tries to mitigate the costs.”

White and others on the commission invited the public to comment on the rule, and many people did. But what may have been the most consequential letter came from Bartl’s Center on Executive Compensation.

In December 2013, Bartl — who had already met with SEC officials four times on pay disclosure issues and sent letters — passed along his most detailed analysis of the rule. The CEC estimated it would cost companies upward of $180 million, more than twice what the commission had estimated, and Bartl said the information on median pay provided would be of little use to the public.

To ease the burden on companies, he made several suggestions. One of them was the shortcut.

Allowing companies to calculate a median employee once every three years, Bartl told the commission, “minimizes excessive burdens” in cases where the company’s employees haven’t changed enough to result in a “material” change in the pay level.

The purpose was to “create an ease of compliance,” Bartl said in an interview with The Post earlier this month. “It was a suggestion based on the fact that in many companies, you don’t see many changes. And if that was the case, do we need to go back out and come up with that median of all employees globally?”

He’d noted in his letter that a majority of the commission had expressed concerns about using disclosure “to further social objectives.”

The SEC agreed. Though the law passed by Congress said the disclosure would be annual, when the agency published rules for the pay disclosure in 2015, it adopted the proposal Bartl and the CEC advocated.

Under the SEC’s new version of the rule, companies would no longer be compelled to calculate and report median worker pay every year. Instead, companies could identify a median worker one year, and then, for the next two years, they could merely report that worker’s pay unless they “reasonably believe” there had been significant changes to their workforce.

“The rule provides companies with substantial flexibility in determining the pay ratio, while remaining true to the statutory requirements,” White said at the time.

White declined to comment to The Post on her role in crafting the pay rule.

What good are the figures? Bartl notes that the median pay figure probably offers little insight for workers wondering if they’re being paid fairly. For that purpose, he said, workers might be better off looking at websites such as Glassdoor that publish estimates of pay for specific jobs in specific markets.

On the other side, advocates say that the figures, if reported accurately, could help the public understand the gap between the executives and everyone else. By offering so much flexibility in calculating median pay, advocates argue, the SEC rule allows companies to obscure an important issue.

Aguilar, the former SEC commissioner, said the shortcut that was approved was intended to give information that was “current enough to be meaningful to investors.” He, too, voted for the pay rule.

“There were a lot of concerns that it would be expensive to calculate the median pay,” he said. “The thought was that if they didn’t have to calculate it every year, that would be helpful in easing the costs. . . . You have to hope that companies will make good faith efforts in identifying the appropriate median employee.”

Dan Keating contributed to this report.