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Kyle Kondik of the University of Virginia’s Center for Politics said the Ryan budget has proved to be a powerful weapon for Democrats.

“Their attacks on Republican Jane Corwin over her support of the Medicare changes in that budget seem to have gotten traction,” Mr. Kondik said. “This race is definitely a preview of the attacks that the Democrats are going to mount in next year’s campaign.”

If so, Mr. Davis said, the message could hinder efforts on Capitol Hill to make the spending cuts necessary to get a better handle on the nation’s trillion-dollar deficits and $14.3 trillion national debt.

“You can’t have a meaningful budget discussion over the long term without talking about the growth in Medicare,” he said.