under/below/lower than/short of

I see no problem with using any one of those choices, either formally or informally.

'Short of' is often used to imply failure to meet a target

In formal writing you would avoid contractions and symbols. I do not know what currency you are thinking of, (let's assume euros) thus '50k/year' would be '50,000 euros per year' or '50,000 euros a year'. As I said before, use of 'below', 'under', 'lower than' or 'short of' emphasises the fact that the income is less than the figure mentioned, and might be more used when 50,000 per year is some kind of target or threshold: they cannot afford this mortgage, as their income is under 50,000 pounds per year. If you did not wish to imply that 50,000 is some kind of target, If the figure of 50,000 has no particular significance, e.g. it is just the nearest multiple of 10,000, you could say 'around', 'about', 'in the region of', 'nearly' or 'almost', 'just below', 'just under', 'just short of' etc.

My learned colleague is incorrect in averring that formal writing does not use contractions and symbols but he is correct in defining failure to meet a target as 'falling short'.

Family income below GBP50,000 p.a. is how I would describe it.