LOS ANGELES — The deficit reduction deal reached in Washington produced some relief across the country on Monday, as the nation appeared to have avoided default. But it also produced a sharp wave of anxiety among governors and mayors worried about how the cuts might hurt already beleaguered states and cities, and it sowed anger and cynicism among many Americans about leaders in Washington.

Few people — from government officials struggling to figure out the details of a moving target to Americans appalled at the partisan drama — seemed surprised that a deal had been reached at the last moment. But it came at what appears to be a high cost.

“This has eroded my faith in government, because there was such a high when Obama came in,” said Meredith Quill, 38, who lives in Los Angeles.

In Atlanta, Kathy Ashe, a Democratic state representative, said that the fight had undermined her belief in Washington’s basic competence. “It is no longer about the best and brightest minds steering the nation based on the best information available,” she said.