DAVIE, Fla. — A crucial experiment in the future of Medicaid is playing out in Florida, where both houses of the Legislature are vying to find ways to drastically cut costs, manage care and reduce waste and fraud.

The cuts and changes being sought by the Republican-led Legislature and encouraged by the new Republican governor, Rick Scott, a wealthy former hospital company executive, are deeper than those in many other states.

In the past 11 years, the cost of Medicaid in Florida has grown to $21 billion from $9 billion and amounts to a third of the state budget. The federal government pays more than half the tab.

“There is a consensus that the Medicaid system is irretrievably broken,” said State Senator Joe Negron, a Republican who took the lead in writing the Senate bill, which is expected to come to a vote before the legislative session ends a week from Friday. The House approved its bill this month. The changes could go into effect as early as next year.