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The president said he personally likes the congressman, but has not been shy about attacking his proposals.

In choosing Mr. Ryan, Mr. Romney elevates the budget to a top issue, and highlights Senate Democrats’ failure to pass a budget for the last three years. As it spotlights Mr. Ryan’s plans, it will shed attention on Mr. Obama’s own budget, which was defeated by a vote of 99-0 in the Senate and 414-0 in the House — earning not a single Democratic vote of support.

Democrats argued the bills they were voting on weren’t actually Mr. Obama’s budget, though Republicans said the plans embodied all of the numbers and plans the president had proposed.

Mr. Ryan has been a major player in some of the big spending negotiations of recent years, including serving on — and voting against — the Bowles-Simpson commission report proposing tax increases and spending cuts to try to reduce the deficit.

But he was left out of last year’s deficit supercommittee. Wisconsin press reported that he asked to be left off the panel.