It depends what you consider costly. I switched to Japanese after college and moved to Japan for a year. Rather than learning anything about the writing system, I took a 6 week conversation crash course before leaving. When I got there I just mastered the accent and some of the cultural communication idosyncrasies like "active listening" rather than spending time reading and writing anything. I ended up going on far more dates with women than I ever have in my life prior (or after) and had two really cool girlfriends. In contrast my friend who was obsessed with mastering Japanese had no friends at all for about 3 years. Nobody wanted to talk to him because his accent was so grating. I feel like his approach was far more "costly" if you consider "social cost" a real factor.
In terms of real economic value I have come to the conclusion that a native english speaker mastering another language rarely pays off, in a financial sense. The main benefit is establishing relationships with people you wouldn't be able to meet, otherwise.
Accent seems to be way more important than vocabulary or grammar for social situations. It's easier to "fix" an imperfect sentence construction in your head and infer missing meanings from gestures and pictures than it is to decipher an extremely thick accent.
In terms of real economic value I have come to the conclusion that a native english speaker mastering another language rarely pays off, in a financial sense. The main benefit is establishing relationships with people you wouldn't be able to meet, otherwise.