The POA doesn't make you liable for your father's costs, it gives you responsibility for his care, financial decisions etc. to make the decisions that he would make if her were able and to fulfil those.
It doesn't change anything as regards whatever resources your father / family have available for his care, it sounds like this is something that needs to be considered though and a plan put in place if resources are limited as your post suggests. The POA when it is enforced will help you in that you can execute certain decisions, but it doesn't make you personally liable for costs. Hope I've written all that in a way that makes sense - 3am insomnia !