The Trump campaign called the findings "pure fiction," contending the analysis neglects a crucial benefit for low-income taxpayers — and insisting that Trump would instruct the congressional committees drafting his plan into law that taxes would not be allowed to rise for any low- or middle-income American.

"The fact that NYU didn't include in their model the $500 per-child match — a central element of our plan — demonstrates that their entire exercise is fatally flawed," Trump national policy director Stephen Miller said in a statement. "Nor did they model the effects of the tax-free spending on both children and elderly dependents that is additional to either the new deductions or those in current law. They modeled someone else's plan, but not ours."

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The analysis comes from Lily Batchelder, a professor at NYU's law school who focuses on tax policy and worked for President Obama's National Economic Council. She has friends and former colleagues on Democrat Hillary Clinton's campaign team but says she conducted this study entirely independent of the campaign.

Batchelder examined the likely effects of Trump's proposed changes to the income tax code on individuals and families to see whether their tax bills were likely to rise or fall based on his plan. That makes her analysis different from the broader economic analyses of groups such as the Tax Foundation, which has estimated Trump's plan would reduce federal revenue by up to $5.9 trillion over a decade. Those broader analyses predict the average size of tax cuts at various points on the income distribution, but they don't look at individuals.

What Batchelder discovered, for millions of individual Americans, was a math problem in Trump's tax plan as written. The plan eliminates some tax breaks while adding others. Notably, it eliminates what's called the personal deduction, which is currently $4,050 for every member of a household filing taxes. It also raises the standard deduction for all tax filers and creates new benefits to offset the cost of child care. It shuffles and consolidates tax brackets so that the first income to be taxed for anyone is taxed at a 12 percent rate instead of the current 10 percent.

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For 8 million families, Batchelder found, the Trump plan's tax breaks would add up to less money than the breaks they receive today. (She called that the "conservative" estimate; under a different set of assumptions about the provisions of Trump's plan, Batchelder found more than 10 million families would see tax increases.)

The math is straightforward: A middle-class family of five — two parents, three children — would gain $17,000 for their standard deduction but lose $20,250 in personal deductions. Their remaining income would be taxed at 12 percent, not 10 percent. Extra child-care benefits might not be enough to make up the difference, so they would pay more.

About 1 in 5 families would fall into that category, Batchelder estimated. “That becomes more and more of an issue the bigger the household, and if they have kids,” she said in an interview. “It is primarily going to raise taxes for low- and middle-income taxpayers.”

The analysis estimates that more than half of single parents would see tax increases, because Trump eliminates what is called "head of household" filing status, which gives single parents a higher standard deduction and lower rates than they otherwise would have had. Other researchers have also flagged that possibility.

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“Single parents get hit with all three of the tax increases under Trump’s plan” -- the loss of personal deductions, the loss of head-of-household status and higher rates on some income -- said Harry Stein, the director of fiscal policy for the liberal Center for American Progress Action Fund, who has written about the potential effects of the Trump plan.

The Tax Foundation estimates that middle-class taxpayers, on average, will see a nearly $500 a year boost in their incomes from Trump's plan, even before factoring in additional economic growth spurred by tax cuts. But an economist at the foundation who conducted that analysis, Alan Cole, said Friday that on a family-by-family basis, Batchelder's estimates sound "plausible," though the foundation has not done similar modeling itself.

“There are a lot of significant tax changes going on" with Trump's plan, Cole said. "If you were taking advantage of the personal exemption a lot before, and you don’t have that anymore, that would be a prime candidate for potentially having a tax increase under the plan right now.”

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Trump's campaign disputes the analysis in several ways. First, it notes that Batchelder did not estimate the benefits to families of one provision in Trump's proposal: a government match of up to $500 per year, per child, for parents who put money into a tax-preferred dependent care savings account. Any analysis without that provision is "invalid," Miller said.

(The Tax Foundation, whose estimates the Trump campaign touted to reporters in a news release Friday, also did not estimate the effects of the savings account provision, either on family incomes or on the overall cost of the plan. Batchelder and Stein both said it is difficult to predict how many families would take advantage of the provision, especially because low-income families are often hard-pressed to find enough money to put in savings accounts. The Trump campaign says low-income taxpayers will be able to direct their Earned Income Tax Credits to be deposited into the savings account, thus increasing the number of filers who receive the match.)

The Trump campaign also says the analysis makes mistaken assumptions on how many taxpayers will itemize their deductions and on how much income taxpayers earn from investments, which could change some calculations for individual filers. It also complains that the analysis does not project gains from additional economic growth that could result from tax cuts.

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Most importantly, Miller said Trump will instruct the committees writing his plan into law to make sure that it does not raise taxes on any low- or middle-income earners. "In sending our proposal to the tax-writing committees, we will include instructions to ensure all low- and middle-income households are protected," Miller said.