Rent or buy?
The answer to this question is determined by two factors; instrument size, and player aptitude. Children grow, and subsequently do not play on smaller instruments for long enough to warrant the cost of a hand-made model. So, through the growing years, you will be dealing with “factory” made instruments, and for this level they are adequate.
Experience as a student and teacher has proved that if you buy an instrument smaller than a full size, you will likely not be able to re-sell it. Therefore, I recommend renting until at least the 3/4 size.
At that point, if your child is serious about his/her music study, buying could be considered. Most shops offer a trade-in policy that can be beneficial for acquiring a quality 3/4. Typically, if you buy an instrument from them, they will credit full purchase price of that instrument towards an upgrade. The nice thing is, you will get full return for your money, and not be stuck with something your child has out-grown, a strong consideration for this size. But caution, you are committed to dealing with that one vendor, which is like limiting yourself to only one car dealer - you could be taken advantage of, or have to settle for an upgrade that does not meet expectation. Due diligence will be needed for the best experience.
For a full size instrument, purchase is going to be most cost effective. There is an endless financial range at this point, hence, your child’s intent becomes a factor. Quality instruments are not cheap, it may be worthwhile to look ahead and do some financial planning.
For a list of vendors to contact, see Instruments on the resources page.