Peter Bogdanovich was on the most recent episode of Marc Maron’s WTF podcast and was asked why he made it in black and white.
As told by him, when he got the movie he discussed it with Orson Welles and said he wanted to get the same deep focus that had been used in Citizen Kane and Welles told him he’d never be able to get it in color. Bogdanovich said “but we have faster film now, can light it brighter” but Welles insisted it couldn’t be done. “I don’t think they’d let me do it in black and white” to which Welles said “have you asked?”
So he went to the studio and asked. And sold it as helping to get the period feel he wanted. The studio guy pretty quickly agreed and when Bogdanovich asked him later why he was so quick to agree, was told it was because it seemed like an interesting novelty.
When Maron asked if it was cheaper too, Bogdanovich said only marginally. So budget doesn’t seem to have played any role in the decision for anybody involved in making it. At least as told 45 years later.