Refusing to abide by the new law could cost governments and school districts up to 10 percent of their state aid payments. A local government, such as a city or county, could opt out of the health care payment requirements and avoid the funding cut only if two-thirds of its governing body agrees. School districts are not allowed to opt out of the requirements.

"The state has never been involved in negotiating our benefits before," said Ben Bodkin, legislative affairs director for the Michigan Association of Counties. "We believe helping counties specifically with additional tools to help control their costs themselves are a good idea across the board, but we do not support mandates."

While divisive laws curbing collective bargaining and other union rights pushed by Republican governors in Wisconsin and Ohio have wrought massive protests and legislative gridlock, Snyder has instead said since taking office in January that he wants to work with labor to reinvent Michigan's economy.

But unions say his actions have proved anything but conciliatory. They're trying to repeal a law pushed by Snyder earlier this year that lets state-appointed "financial managers" cancel union contracts and cut costs in financially troubled cities and school districts. State worker unions have banded together to try to fend off $145 million in concessions Snyder wants by Oct. 1, and state retiree groups have filed suit challenging laws taxing public pensions for the first time.

State workers also are concerned about pending bills that would require state workers with defined benefit pensions — about half of those now working — to pay 4 percent of their pension costs.

State budget spokesman Kurt Weiss said Michigan will now be one of a handful of states that have a say in how much of employees' health care costs local governments can cover. He noted that a similar law on the books in New York passed a court challenge by opponents.

Michigan Education Association spokesman Doug Pratt said the new requirements shift more costs onto workers who already have been forced to make concessions during the state's decade-long economic slump.

"The simple fact is that the Republican-led Legislature has put a target on the backs of public workers, school employees in particular," Pratt said. "This is just another phase of the attacks that they're waging against the middle class of this state."

But GOP Senate Majority Leader Randy Richardville said he would "guarantee there are a lot of hard-working, union, blue-collar men and women out there in the field right now that would love to pay 20 percent because they're paying considerably more."

Republican lawmakers had also sought to have the higher health care payments apply to state workers and university employees, but were unable to get the two-thirds vote needed in the House. Lawmakers, however, will pay a larger portion of their health care costs starting Jan. 1, however.

The caps won't affect existing union contracts until they're modified or expired.

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Associated Press writer Tim Martin contributed to this report.