There are at least four key questions you need to ask about cars and teenagers at the outset, when they are as young as 13 or so. Then, there are at least four basic approaches to paying for the automobile.
The four questions
1 WANT VS. NEED
If there are school buses, then a teenager doesn't need a car. Yes, it would make it easier to get home from extracurricular and social activities, but if a teenager has been managing transport just fine without a car so far, then a personal vehicle isn't a need. And if it's a want, the child probably ought to pay for at least some of it.
Still, a teenager's car may be a want (or even a need) for the parents. You may want to rid yourself of chauffeuring duties. Car ownership could come with new responsibilities to ferry younger siblings too, which can take a load off an adult.
Not every teenager can maintain good grades while also working. (And how will a child get to the job without a car?) Even if it's possible to keep the grades up, does a job come at the expense of extracurricular activities? Is that okay with you? With the teenager?
3 CONTINUING EXPENSES
Paying for the car itself is just the beginning. Who is going to pay for the insurance? Gas? Maintenance? The nearly inevitable body work?
4 SAFETY BASELINE
Your child may end up with a used car. Perhaps very used. What sort of safety equipment is mandatory? Are there categories, like sports cars or sport utility vehicles, that are off-limits?
The four approaches
1 PAY NOTHING
If a car is purely a want, then teenagers can figure out whether they want one badly enough to pay for the whole thing themselves. It may be helpful to discuss this in the context of other large things that your family may be getting ready to pay for (and thus prioritizing), like college.
You could soften up a bit here by diverting birthday and holiday money toward a car fund for a few years before the big birthday.
2 PAY HALF
Dollar-for-dollar matching really gets children thinking. If they make $50 babysitting, that equals $100 for a car if they put the money away and wait for the match. Spend it, however, and they have nothing.
This also has the advantage of being realistic; it's going to be hard for most teenagers to scare up enough money to buy a safe car all by themselves. You may want to cap your match, however.
3 PAY IN FULL
If the car is truly a need, then perhaps it's worth parents paying for in full. But to avoid any entitlement issues, the terms of the deal and resulting expectations need to be crystal clear up front.
Some parents may see a new car as something that a teenager deserves, as a reward for good behavior or improved or consistent school performance. But that starts to feel more like compensation or a bribe for something a child ought to be doing anyway.
Then there is the toss-the-keys-to-the-2005-Dodge-Caravan approach. The parents have probably paid off any loan, so it feels free even if they have to buy themselves something else to drive. Teenagers probably won't hot rod around in a minivan either.
Still, when parents hand over any asset with this kind of value without asking teenagers to take some responsibility for it, it sends a message. It can be a good message or a rotten one, depending on the family conversations that surround it (or the lack of a conversation that sets financial and other expectations).