It’s official — the feds will pick up 100 percent of the cost of expanding Medicaid under President Obama’s health care law before scaling back their contribution to 90 percent by 2020, according to rules published Friday.

The Department of Health and Human Services announced it has proposed a final rule that describes a “simple and accurate method” for states to claim the matching rate for enrollees deemed “newly eligible” under the Affordable Care Act, which expands the program in 2014 to those making up to 133 percent of the federal poverty level.

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But deciding whether or not to expand the program, which the Supreme Court made optional in June, has been anything but simple for many states.

Republican governors and lawmakers are balancing the influx of federal funds to help their poorest residents with the political animosity and long-term budget calculations that surround the expansion under President Obama’s signature law.

GOP governors such as Jan Brewer of Arizona, Rick Scott of Florida, John Kasich of Ohio and Chris Christie of New Jersey have taken heat from conservatives for accepting the federal money offer to finance the expansion in their states, despite their previous opposition to Mr. Obama’s reforms.

GOP lawmakers in some of the states may put the kibosh on plans for an expansion, even if the governor supports it, because they see the Medicaid program as a broken system that will bust their budgets in the long run.

But Arkansas and other states may use the Medicaid dollars to purchase private insurance through their health exchanges under Mr. Obama’s law, an option that could find favor with Republican leaders.

Proponents of the Medicaid expansion say it would be foolish for states to pass up the federal funds that expand insurance coverage for their poorest residents.

“This is a great deal for states and great news for Americans,” HHS Secretary Kathleen Sebelius said. “Thanks to the Affordable Care Act, more Americans will have access to health coverage and the federal government will cover a vast majority of the cost. Treating people who don’t have insurance coverage raises health care costs for hospitals, people with insurance and state budgets.”