In an interview for "Axios on HBO," Housing and Urban Development Secretary Ben Carson told me that the $7.25-an-hour federal minimum wage would be "very difficult" to live on and that in his view it should be higher.

Driving the news: "I don't have any problem with raising the minimum wage," Carson said. "My personal opinion is that it should be indexed."

"You determine what the minimum wage should be, but when conditions change, it needs to change with it, it needs to be indexed," he added. "Then you don't keep having these arguments every 10 or 15 years."

Why it matters: This is the first time, as HUD secretary, that Carson has publicly admitted that America has a problem with its minimum wage.

Details: Carson, one of the most important but undercovered figures in national politics, is the top federal government official charged with housing the poorest and most vulnerable people in America.

The Trump administration has proposed sweeping cuts to his department's budget at a time when homelessness is on the rise.

Carson wants the federal government to play a smaller role in housing the poor.

He says local governments need to stop expecting more money from the federal government and should instead cut regulations, build smaller, cheaper homes, and encourage churches and the private sector to help people experiencing homelessness.

Between the lines: In our interview, Carson conceded the current minimum is too low after I cited a study by the National Low Income Housing Coalition that found that a worker earning $7.25 must work "103 hours per week (more than 2.5 full-time jobs) to afford a one-bedroom rental home at the national average fair market rent."

The big picture: "Axios on HBO" joined Carson on his recent bus tour to California, where he met with local officials to discuss a federal-state strategy for the state's homelessness and housing affordability crises.

During the tour, Carson said repeatedly that government programs have fostered "dependency" out of a misguided concept of compassion.

We followed Carson as he met with some formerly homeless youth who were getting back on their feet at the "Dream Center" in LA — a faith-based nonprofit serving the vulnerable, addicted and homeless.

Carson told young people there about the extraordinary potential of the human brain and recounted his rise-from-poverty story memorialized in books and on screen.

Key exchange: I asked Carson whether he could support a family and himself on the federal minimum wage.

"It would be very difficult," he replied.

"How would you do it," I asked.

"Probably the way my mother did it," Carson said. "Work three jobs at a time."

Yes, but: Carson made clear he still heavily favors market solutions.

Even the way Carson answered the question — carefully and with some equivocation — did not leave the impression that he would be spending political capital advocating for a minimum wage rise.

He didn't say what he thought the minimum wage should be, saying he would leave that to the economists.

Go deeper: More highlights from the Carson interview