I have one authorized user on my Amex account. I haven't given them the card(daughter) but am considering letting her use it for small things. Can you set it to where certain dollar amounts are declined to keep the balance within a certain amount? I've always thought you could but haven't tried it.
I have my daughter on my bcp and will vary her credit limit from time to time. I usually set it at 1-2k but will up it to 5k or so for things like road trips and such (in case of emergencies.)
You can also set up to be notified by text or email for any purchase over x amount.