Story highlights President Donald Trump's plans for infrastructure have not been hashed out yet

Trump's budget director, Mick Mulvaney, has proposed deep cuts in agencies

Washington (CNN) President Donald Trump's promise to spend $1 trillion on infrastructure won applause Tuesday night in his first joint address to Congress, but has added to growing concerns on Capitol Hill about how he -- and lawmakers -- will pay for it.

"Budget-wise there's some real challenges there. I mean, he's talking about a huge infrastructure program, huge increases for defense, other huge expenditures, all while cutting taxes. All of which I like, but you've got to find a way to not blow the deficit to creation," said Rep. Hal Rogers, the former chairman of the House Appropriations Committee and a veteran spending hawk. "It's a big challenge."

The infrastructure plan carries an eye-popping price tag, but Republicans played down that cost Wednesday, saying that much of it would likely be covered by private investment and supported by measures like government-backed loans -- a favorite tool for energy programs and transportation projects.

The infrastructure details have not been hashed out yet in large part because congressional Republicans and the White House have been focused almost exclusively on the Obamacare replacement plan -- which is scheduled for a public crafting session in the House next week.

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