This paper develops a quantitative Markovian overlapping generations model with altruistic individuals and incomplete financial markets in order to analyze the long-run distributional implications of two hypothetical public social security policy changes, made in response to impending future demographic shifts. The two policy changes considered are first, raising the tax rate while keeping the replacement rate constant and second, keeping the tax rate constant while lowering the replacement rate. Whereas this latter policy is detrimental to the relative situation of the retirees, the huge financial heterogeneity in the first scenario explains why the increase in the proportional labor tax is relatively badly absorbed by low-productivity workers, leading to an increase in welfare inequality. We show that the very popular idea that a more funded system would ineluctably lead to more inequalities in well-being can be justified only by focusing on the inequality of positions in case of general equilibrium.