I originally thought that the changing of aspect ratios was done to cue the viewer to the different narratives. Three different aspect ratios (1.33, 1.85, 2.35:1) for three different timelines (1930s, 1960s, 1980s).
However, there is more at play. According to an interview, Wes Anderson filmed the movie in three different aspect ratios to reflect the history of cinema during those periods.
For the 1930s, everything is filmed in 1.33:1 (4:3). Or more precisely, 1.37:1. Wes Anderson mentions this,
1.37 might be the exact number. That’s pretty good. 1.33 is what I always thought it was, but then the German camera guys are very precise. You have to stop saying that (laughter). I was always told 1.33, but it’s not, apparently. It’s a tiny bit wider, I guess.
1932 is the year that this Academy Ratio was formally established. It is also the year in which Zero Moustafa's story starts and when he met Gustave H.
For the 1960s, everything is filmed in 2.35:1. This is a widescreen format that was introduced to distinguish film against the rise of 4:3 aspect ratios in television.
For the 1980s, everything is filmed in 1.85:1 (16:9), today's standard format.
What was your inspiration for changing ratios during the movie?
ANDERSON: It was, at first, just to do it all academy. And then, I thought, how are we gonna do it? Are we gonna do some of it in black and white? What choices are we gonna make to separate this time period from that time period? The thing we did with Dara is we used these technical anamorphic lenses, that were really old and strange... So, I thought, maybe we could use these lenses again, and we could make them different parts, like they’re different movies.
Aspect ratios for U.S. films rarely change. In this case, though, Wes Anderson used them to his advantage not only in distinguishing the different narratives from each other but also poke fun at how aspect ratios have changed in the past century.