川普先生的算术,给他带来了麻烦。

Donald J. Trump, the Republican presidential nominee, made his exit on Wednesday after speaking at a campaign event in Canton, Ohio. Credit Damon Winter/The New York Times

WASHINGTON — A few hours after Donald J. Trump publicly backed away on Thursday from a $1 trillion tax cut for small businesses, campaign aides privately assured a leading small-business group that Mr. Trump in fact remained committed to the proposal — winning the group’s endorsement.

The campaign then reiterated to the Tax Foundation, a conservative-leaning Washington think tank it asked to price the plan, that Mr. Trump had indeed eliminated the tax cut.

Call it the trillion-dollar lie: Both assertions cannot be true.

At issue is whether small businesses, partnerships and other “pass-through” entities would get a 15 percent tax rate under the Trump plan, as originally promised, or would continue to pay individual income tax rates as they do now — as high as 33 percent — under the latest Trump plan. The campaign’s conflicting accounts of its own proposal are particularly remarkable because the Republican presidential nominee and his advisers have taken months to refine the details.

But in this case, telling two versions of the same story has yielded some important benefits.

Dropping the tax cut is central to Mr. Trump’s optimistic claim that his plan would not increase the federal debt. But by simultaneously promising to keep the tax cut, the campaign won the support of the National Federation of Independent Business, an influential small-business lobbying group.

“We’re comfortable” that Mr. Trump is committed to preserving the tax break, said Jack Mozloom, a spokesman for the group. “We have it directly from his campaign.”

A spokeswoman for the Trump campaign did not immediately respond to emailed questions.

Mr. Trump on Thursday detailed a package of tax cuts that he claimed would cost $2.6 trillion, far less than the $10 trillion in cuts he proposed last year.

The estimated cost is based on an analysis by the Tax Foundation. Alan Cole, an economist at the foundation, said the Trump campaign provided him with the details of the plan, including the elimination of the small-business tax cut.

After learning about the statement by the business group, Mr. Cole said that he spoke again to the Trump campaign, and that he was assured that Mr. Trump still intended to eliminate the tax cut. The campaign told him it was making a more modest change in the plan that might reduce taxation for some small businesses.

“There is a disconnect between the plan as understood by us and the plan as understood by the N.F.I.B.,” Mr. Cole said on Friday. “And I think your inquiry into this is probably a productive one.”

Under current law, corporations are subject to one set of tax rates while income from other kinds of businesses entities, like partnerships, is taxed like any other personal income. This is known as pass-through income. In general, the largest businesses are organized as corporations while most smaller businesses are organized as various kinds of pass-throughs.

Mr. Trump has proposed a 15 percent tax rate for corporations. And last year, he proposed taxing pass-through income at the same rate. That would reduce federal tax revenues by around $1 trillion over 10 years, according to the Tax Foundation.

川普先生,因为兼得的算术问题,给自己带来聊大麻烦。

川普先生一要减税,二要花钱。他自己也不知道减税之后要花的钱从何而来。

The small-business federation said on Thursday that it was endorsing Mr. Trump’s tax plan because it preserves this specific proposal.

“We strongly support Mr. Trump’s proposal to create a single business tax rate that would create parity between small businesses and their larger competitors,” Juanita Duggan, the group’s president, said in a statement.

But if the group is right, Mr. Trump’s math is wrong.

Mr. Trump has said he would reduce taxation by $4.4 trillion, but the actual cost would be just $2.6 trillion, he claimed, because the cuts would generate explosive economic growth that would offset some of the lost revenue. He said he expected additional economic gains from changes in other aspects of federal policy, including less regulation and new trade agreements. The rest of the cost would be offset by spending cuts.

There is no precedent for a tax cut’s generating that kind of economic activity, nor for a similar boost from changes in regulation. And Mr. Trump has not detailed any specific cuts he would make in federal spending. Finally, the plan works only without the tax break the small-business group says it was promised.

If Mr. Trump wants to keep the small-business tax cut, and avoid increasing the debt, he would need to find $1 trillion somewhere else.

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