Probably not. Pooling the resources is more likely to be efficient. If you do it at the individual level, everyone has to allocate resources for some kind of maximum likely cost versus allocating for the average need. That's also why it's more efficient to say, buy fire insurance than set aside the amount of money necessary to cover a fire - pooling risk.
I'd much rather be in a society where there are sufficient social services for the elderly and higher taxes than one where I have to depend on either my own savings or my children's earnings to cover my (very unpredictable) post-retirement needs. (Unpredictable because of the wide range of years I may live post-65.) Otherwise you have huge opportunity costs because of the need to allocate for a worst-case scenario.