Productivity in this case is fairly simple to calculate = Cost/Duration.
You can factor in the time (presumably that’s paid for so you should have a real number) and cost of materials (On-site use historical or replacement cost) yourself.
Your points holds more generally though, especially when there aren’t clear prices paid.
Not trying to be argue for the sake of arguing, but is there some data to back this claim up? Anecdotally I'm aware of many parents who think music makes their kid well rounded, but I'm not aware of parents that specifically believe learning music makes their kid more intelligent in non-musical areas.
You can factor in the time (presumably that’s paid for so you should have a real number) and cost of materials (On-site use historical or replacement cost) yourself.
Your points holds more generally though, especially when there aren’t clear prices paid.