So obviously we can say "if you want to play louder, blow harder." However, is the connection really as simple as "more air means more volume," or is there a secondary effect which can permit louder sounds without having to expend more air.
I'm particularly thinking about how to play long fortissimo phrases where my instincts are to cut back on the loudness to complete the phrase, or long pianissimo phrases where so little air is used that I feel like I can't get enough oxygen unless I explicitly exhale before every breath (which there's often no time to do)