A classic result in dynamic public economics says that for a dynamically efficient overlapping-generations economy, there is no long-run welfare role for unfunded, pay-as-you-go (PAYG) pensions. Subsequently, the literature has shown that if agents are sufficiently myopic or present-biased, a welfare rationale arises only when agents wish to but cannot borrow (“borrowing constraint”) against future pensions—their private, voluntary retirement savings are zero. In this paper, we extend the scope of the results mentioned above. We prove that a positive optimal pension cannot co-exist with a positive private retirement saving under standard preferences without the borrowing constraint. The same is true under myopia. Co-existence may obtain under the self-control and temptation preferences popularized by Gul and Pesendorfer (2004).