RENEE MONTAGNE, HOST:

Let's look, now, at Mitt Romney's proposal to overhaul the tax code. The Republican presidential hopeful promises to lower everybody's tax rates. He says that would foster more economic growth, without adding to the deficit. Analysts have had a hard time testing Romney's claim because he hasn't offered many specifics. NPR's Scott Horsley reports.

SCOTT HORSLEY, BYLINE: Mitt Romney says it's possible to cut tax rates without driving the government deeper into the red. He says he can make up for the lost revenue by closing tax loopholes. But when he was pressed by NBC's David Gregory over the weekend, Romney did not offer a lot of detail.

(SOUNDBITE OF TV SHOW, "MEET THE PRESS")

DAVID GREGORY: Give me an example of a loophole that you will close.

MITT ROMNEY: Well, I can tell you that people at the high end - high-income taxpayers - are going to have fewer deductions and exemptions.

HORSLEY: That's not good enough for President Obama, who says Romney's plan lacks a basic ingredient: math.

(SOUNDBITE OF SPEECH)

PRESIDENT BARACK OBAMA: When you ask him, well, how are you going to do it? Well, they won't answer real clearly. But there are only a few ways of doing it, and most of them involve sticking it to the middle class.

HORSLEY: Mr. Obama often cites a study by the nonpartisan Tax Policy Center here in Washington, which found that under Romney's plan, middle-class families would have to give up some of their popular tax breaks - like the deduction for mortgage interest, for example - in order to offset the cost of cutting taxes for the rich. The author said anything else is not mathematically possible.

HARVEY ROSEN: To an economist, those are very strong words.

HORSLEY: Harvey Rosen is a Princeton economist, and a former adviser to George W. Bush. He set out to prove Romney's critics wrong.

ROSEN: Obviously, I'm a Republican. It kind of raised my hackles slightly, to see these - the proposal being characterized as mathematically impossible because that's different from ill-advised, unfair; you know, will blow the economy to smithereens.

HORSLEY: Rosen's study was similar to the Tax Policy Center's. He first looked at how much money the government would lose by lowering tax rates on the wealthy; then tried to figure out how much it could recoup by closing loopholes. And Rosen came to the opposite conclusion - that you can cut tax rates for wealthy Americans, without shifting costs onto the middle class.

ROSEN: So it's not mathematically impossible.

HORSLEY: Why the difference? Partly because the two studies make different assumptions about what tax loopholes might be on the chopping block. The Tax Policy Center assumes Romney would not want to eliminate tax breaks designed to encourage savings and investment - such as IRAs and tax-exempt government bonds - while Rosen assumes some of those breaks could be phased out. Of course, Rosen says, Romney could easily settle this debate simply by telling us which loopholes he'd do away with.

ROSEN: We wouldn't have to be guessing. But the kind of specificity that an academic might think is very cool, you know, might not make sense for a politician.

HORSLEY: In fact, Romney is following a well-worn political path. Bruce Bartlett is the author of a book on tax policy, called "The Benefit and the Burden." He says politicians are typically a lot more specific about the benefits of the tax cuts they're proposing, than the burden of the loopholes they want to close.

BRUCE BARTLETT: It's a general rule of politics - dating back to at least Cicero - that politicians promise all things to all people, and never promise pain unless their back is to a wall.

HORSLEY: In fairness, President Obama has been equally vague in discussing his own plans for tax reform. He agrees with most policy experts, that a tax code with lower rates and fewer loopholes would be more efficient. But like Romney, he has not spelled out which loopholes he'd cut. Bartlett says that omission is less glaring in the president's case.

BARTLETT: Obama has not made tax reform - by any stretch of the imagination - the centerpiece of his campaign; whereas I think it would be hard for people, off the top of their heads, to say what else Mitt Romney stands for.

HORSLEY: For now, tax analysts and voters will just have to keep guessing about how Romney's plan would work. And they'll have to decide whether they trust the candidate to fill in the details after the election.

Scott Horsley, NPR News, Washington.

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