Since increasing attention is paid to consider the macroeconomic effects of the increasing longevity, we build on an overlapping generations model with endogenous population to investigate under changing longevity the steady-state and dynamical (under myopic expectations) effects of two historical alternatives as a means of old-age insurance, namely, voluntary intra-family transfers from young to old members versus pay-as-you-go public pensions. It is shown that the shift from a private system of old-age insurance to a public system of social security has favoured the rise in capital accumulation while also reducing the risk of cyclical instability in countries where longevity is large enough. In contrast, when adult mortality is high such a shift makes an economy with public PAYG pensions more prone to instability, and with a steady state stock of capital and GDP per worker lower in the long run. Moreover since an old age insurance motive seems prevalent also in highly developed as well long-lived countries as Italy, our results may also be of interest for pension policies.