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Mr. Obama and House Republicans eyed a formula in the health care law known as modified adjusted gross income, saying it was out of kilter with how eligibility is calculated in other federal programs for low-income earners, such as food stamps and housing assistance.

Under the old formula for modified adjusted gross income, a couple with an annual income of $64,000 would be able to qualify for Medicaid because not all Social Security benefits were counted as income. That will change under the revised formula, meaning that fewer people will qualify for Medicaid, exchange subsidies and the program for the states subsidizing health insurance for children.

Changing the formula doesn’t affect anyone now because it does not kick in until 2014.

But the modification means that, in time, 500,000 to 1 million fewer Americans will be able to join Medicaid each year beginning in 2014. Most of those no longer eligible instead could qualify for subsidies on the exchanges, according to the Congressional Budget Office. In turn, some people would lose their eligibility for subsidies and be bumped off the exchanges.

Political magnet

In floor debates, Democrats argued that middle-class Americans also need help paying for insurance, while Republicans said the government should focus its resources on the poor. Rep. Diane Black, Tennessee Republican and sponsor of modified adjusted gross income, called the formula an “unintended consequence” of the Affordable Care Act.

“This is unacceptable, and I very strongly believe that it is our duty to ensure that the very scarce Medicaid resources will be there for the most in need,” Mrs. Black said.

Rep. Joseph Crowley, New York Democrat, countered that the original income rule was not a glitch.

“It was written into the law deliberately, and anyone who actually read the bill would have known that,” he said. “This language was deliberately put into the health care law to expand affordable health insurance and will particularly help early retirees between the ages of 62 and 64, as well as Americans on disability.”

He led Democrats in opposing the revision, though the bill passed by a vote of 262-157. While 27 Democrats joined Republicans in voting for the change, all 157 “no” votes came from Democrats.

The formula change eventually was rolled into a broader bill that included the withholding tax repeal and new tax credits for businesses to hire veterans. The broader bill passed unanimously, 95-0 in the Senate and 422-0 in the House.

Democrats who opposed the alteration were reluctant to talk about why their opposition softened.

“I don’t have any comment on that,” said Rep. Henry A. Waxman of California, the ranking Democrat on the Energy and Commerce Committee, when asked by The Washington Times about his votes.

Rep. Daniel Lipinski, an Illinois Democrat who voted for the revision both times, cited the administration’s support and said changing the formula wasn’t likely to affect many Americans.

“That was the pay-for that was offered on the floor,” he said. “I supported it because I think things should be paid for. Some people, it may limit what they get in subsidy, but I think most people agree and the president can agree that it’s not going to have that large of an impact.”