Music instruments hold their resale value well (providing your child doesn't damage it
) Or you can hire in the short term. We hired a child's violin for about £45 for the first year. So, the killer is the cost of lessons as opposed to the cost of the instrument.
You ask whether it is worth investing in music lessons. I'm assuming that you mean in relation to getting your child in as opposed to from a personal development viewpoint.
Selective schools are, IMO, looking for well-rounded children who are able to show that they can focus, whether that be swimming, music sports etc. Some kids are natural stars but for others, they need something to distinguish them from all the other applicants and music is a good 'something'.
Looking at it from a personal development viewpoint, my kids are the better for it. It taught them discipline and to concentrate. And if the experts are to be believed, it actually exercises the mind and improves the memory (you try playing a 5 minute piece without music