In an ad that has blanketed radio airwaves in the Washington region, a woman’s voice gently intones, “Imagine . . . one million new jobs.”

“One million new American jobs,” echoes a man. “One million new opportunities to build a career,” says the woman.

“Support a family.”

“Follow your dreams.”

And where will these “one million new jobs” come from? By expanding oil and gas drilling and building new pipelines, says the American Petroleum Institute, an industry lobbying group that paid for the ad campaign, which also has featured in newspapers, on television and on Metro platforms.

Oil companies aren’t the only ones promising jobs if Washington gives them their way. A wide array of businesses are saying they can help solve the country’s unemployment crisis if only the government would roll back some regulations, approve their big mergers or lower their taxes.

Yet the industry often touts debatable jobs numbers. Mergers between big companies, for instance, tend to result in layoffs rather than new positions overall. And a closer look shows that API’s ads exaggerate the effect that looser drilling policies would have on employment; more than half of its projected job growth would come between 2015 and 2030.

Nonetheless, some policymakers and presidential candidates have cited these statistics as they echo companies’ claims about creating jobs.

“We just learned today that if the federal government would pull back on all of the regulatory restrictions on American energy production, we could see 1.2 million jobs created in the United States,” Rep. Michele Bachmann (Minn.) said at a Sep. 7 Republican presidential debate.

In a letter last month to the Justice Department, 100 lawmakers defended the merger between AT&T and T-Mobile, repeating the companies’ argument that the government’s lawsuit to stop the deal on antitrust grounds would “thwart job creation and economic growth.”

And a central element in the economic plans of other Republican presidential candidates, such as Mitt Romney and Rick Perry, is to roll back “job-killing” regulations to spur hiring.

“It’s really hard if you’re against regulation to let a good crisis go to waste, and right now we have high unemployment,” said Roger Noll, an economics professor at Stanford University and co-director of the school’s program on regulatory policy. “You can use the current economic condition as a Trojan horse.”

The horses come in different shapes. The coal industry is running ads that show working people weighed down by regulation-filled briefcases being violently thrown from broncos in a rodeo. A new study commissioned by environmental groups, however, says that regulation of coal ash disposal would actually create jobs.

Other company numbers also become fuzzy upon examination.

Big company mergers are widely known to lead to job losses in the short term as firms seek savings, or “synergies” in merger jargon. But that has not stopped AT&T and Capital One, whose proposed mergers have raised antitrust scrutiny, from saying that their acquisitions will result in more jobs.

AT&T has been running a television ad showing its employees hard at work and consumers enjoying their wireless phones and tablets. The ad says that if the merger with T-Mobile is approved, AT&T will bring back 5,000 jobs that were outsourced overseas. It also says the merger will create investment in broadband that would create “as many as 96,000 jobs.”

The latter number comes from a study by the Economic Policy Institute looking at the employment benefits of the $8 billion AT&T has promised to plow into broadband investment should the deal be approved in court. EPI concluded that such an investment would yield 55,000 to 96,000 new jobs over seven years.

But the EPI study does not address whether the merger would ultimately result in more jobs — after factoring in potential layoffs.

AT&T has said it expects to save $3 billion annually from the deal but has not explained whether that would include job cuts. An AT&T spokesperson said that the company expects to keep all of its call-center jobs and that most of the reduction in the company’s workforce would come from attrition rather than layoffs.

In another big corporate merger the government is looking at, Capital One has proposed acquiring ING Group’s online banking unit for $9 billion. The company says the deal would create 500 new jobs in Delaware. It has also said, though, that it expects $90 million in cost savings from the merger. Capital One declined to say whether positions would be cut.

For more than a year, the API has been highlighting the number of jobs it says are linked to the oil and gas industry.

“Sure, you know oil and natural gas fuel transportation and comfort,” a blond-haired woman in a black pantsuit says in one API television ad. Behind her, silver trucks and cars glide by, an African American family shares a meal, people pour out of a rush-hour train. Then a crowd appears before giant letters that read: “9.2 million JOBS.”

But many economists say that the API has exaggerated the number of jobs linked to the oil and gas industry by including direct and indirect jobs (such as steel suppliers), and a seldom-used category known as “induced” jobs that API says covers everything from valets to day-care providers, from librarians to rocket scientists.

Moreover, the single biggest category of people working directly for the petroleum industry is cashiers at gasoline stations and stations with convenience stores — 533,830 of them, according to the Labor Department’s Bureau of Labor Statistics. Yet hardly any of those cashiers pump gas, check engines or inflate tires; mostly they ring up sales of snacks, not gasoline. According to the Labor Department, their median hourly wage is a meager $8.68.

“As the old saying goes, statistics do not lie but statisticians do,” said Philip K. Verleger, an economist, consultant and retired professor of management at the University of Calgary’s business school. “The API is the best there is at lying with statistics.”

“Anybody dismissing any kind of a job is silly,” responded John Felmy, chief economist of the API, adding that including cashiers as industry workers is “a matter of BLS accounting.”

He added that counting induced jobs was “completely appropriate methodology.”

The API has made new claims in its recent ads, asserting that a different energy policy — including a new pipeline to Canada’s oil sands, drilling on now-protected federal lands and waters, and unimpeded shale drilling — would add 1.4 million jobs. “Are you one in a million?” its latest ad reads, showing a man in a hard hat. (The ad, which credits API in small type, says it is the product of a more populist-sounding “The People of America’s Oil and Natural Gas Industry.”)

“It’s a triple win,” API president Jack Gerard said. He said that the industry could create jobs, generate hundreds of billions in tax revenue and double North American oil production — although the main production increase would occur in Canada.

The ad cites a study by Wood Mackenzie, a consulting firm hired by API. Scott Mitchell, who oversaw the study as head of North America upstream research at Wood Mackenzie, said that only a third of the 1.4 million positions created would go to people working directly for the petroleum industry and that the rest would be indirect and induced jobs.

“To be confident about the induced job effects of additional spending is incredibly complex,” said Mark Fulton, head of research at Deutsche Bank’s team of climate change advisers. Citing such figures involves “going another step toward lack of accuracy,” he said.

The Wood Mackenzie study also makes assumptions about current policy. For example, it assumes that current regulations limit the number of Gulf of Mexico exploratory wells to 20 a year. But the number of exploration wells being drilled now is already well above that. As a result, gulf exploration would have little effect on job creation.

Josh Bivens, an economist at EPI, said the amount of job creation from industry spending depends on economic conditions. “In today’s economy, if an oil company decided to do something new, that would create jobs,” he said. “If unemployment were 4 percent, it would suck people away from other employment and there would be no net increase in jobs.”

Felmy, the API economist, retorted: “You may be moving jobs around,” but wages would rise because companies would be bidding for skilled workers.

Drilling further into the API numbers of existing petroleum industry jobs shows just how murky these numbers and definitions can be.

According to the BLS, the number of people in the United States drilling wells, extracting oil and gas, refining petroleum and manning gasoline stations is about 1.1 million. If sole proprietors and business partners are included, the number rises to about 2 million, according to the Commerce Department’s Bureau of Economic Analysis.

Kurt Kunze, an official at the BEA, said, “The big discrepancy in oil and gas extraction is that there are some big master limited partnerships with lots of partners. There is no way to separate out the people digging holes in the ground from someone who is just a financial partner.” He said given the number of partnerships, he would tend to use the lower BLS figure.

API used the higher figure. If half a million people were taken out of its baseline projection, the final total would have been reduced by 1.8 million.

Most of the jobs provided by big oil companies are overseas. At Royal Dutch Shell, for example, 21 percent of its 97,000 employees worked in the United States; the rest work in 89 other countries. Exxon Mobil has 4,970 people working in its Canadian Imperial Oil subsidiary, 885 in Norway and about 2,000 in Malaysia.

A study by Democrats on the House Natural Resources Committee last month found that the four biggest oil companies combined — Exxon Mobil, Chevron, Shell and BP — reduced their U.S. workforce by 11,200 employees between 2005 and 2010. BLS statistics show that although the number of oil and gas extraction positions has risen, the number of gasoline station jobs has dropped. The Democratic lawmakers noted that the four companies earned $546 billion in profits during that period.

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