This is an excellent question. I do not think we need to answer this question with an either / or approach. The responsibility for caring for the elderly is a concern for the government and society. Let me explain.
While it would be great is all the elderly were taken care of by their families and friends, this is not going to happen. In fact, in the West this rarely happens. People are busy and have a hard enough time taking care of themselves. Moreover, in a tough economy, this will be harder. In light of this fact, elderly care becomes de facto an issue of the government.
The government has to make sure that the aging population is taken care of in terms of their health and social well-being. This is, of course, an enormous price to pay, but it is their responsibility when there is no one else. They seek to do this will social security and MEDICARE. Hopefully, all people will care for the elderly collectively.