The new policy was approved by the city, which owns the museum’s building. “Having a healthy Met is extremely important to New York City,” said Tom Finkelpearl, the city’s commissioner of cultural affairs. “The basic motivation was to help the Met balance its budget in a way that did not hurt New Yorkers.”

The Met currently receives about $26 million from the city. Under the new admissions policy, the $15 million that goes toward energy costs like heat and light will remain intact; the remaining $11 million which offsets the Met’s operating costs (for security and building staff) will reduce on a sliding scale after the first full year, depending on how much incremental revenue the new admissions policy generates, with a cap at $3 million.

The Met’s reduced portion of city funds will be redirected toward cultural institutions in underserved parts of the city, Mr. Finkelpearl said.

Fred Dixon, the chief executive of New York’s tourism agency, NYC & Company, said he did not believe the new policy would affect the flow of visitors to the city.

“Most folks expect to pay when they attend an attraction or museum,” he said. “When you look at the landscape of attraction pricing, the Met is an incredible value at $25.”

The admissions policy shift represents one of the ways in which the Met has been working to address a budget deficit that two years ago threatened to balloon to $40 million. While the museum now has a deficit of about $10 million, Mr. Weiss said it aims to balance its budget by 2020.