Since increasing attention is paid to consider the macroeconomic effects of the increasing longevity, we study an overlapping-generations model with endogenous fertility to investigate the steady-state and dynamic effects (with static expectations) of two historical alternatives as a means of old-age insurance: voluntary intra-family transfers from young to old members versus pay-as-you-go public pensions. We show that the change 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 cyclical instability in countries where longevity is large enough. In contrast, when adult mortality is high such a change makes an economy with public pensions more prone to be subject to fluctuations, while also reducing the steady-state stock of capital and GDP per worker. In addition, since the old-age insurance motive seems to prevail also in developed countries with long-lived individuals such as Italy, our results may also be of interest for pension policies.