Public pension schemes serve as mechanisms for inter-temporal income smoothing and within-cohort redistribution. This paper examines the influence of income and lifespan inequalities on the structure of a democratically chosen tier-pension scheme. We use a probabilistic voting model where agents vote on the size and the degree of redistribution (i.e. the Beveridgean factor) of the pension scheme and can supplement it with voluntary contributions. Our analysis reveals that when all agents can supplement the public scheme with private contributions, their voting behavior depends solely on the share of total income redistributed through the pension system, referred to as the redistributive power of the pension. Income inequality positively correlates with the equilibrium redistributive power, while lifespan inequality exhibits the opposite effect, leading to a resource-time trade-off; particularly when both inequality measures are correlated. In scenarios where low earners are hand-to-mouth and unable to make voluntary contributions, the effects on pension size (through mandatory contributions) and degree of redistribution become disentangled. Income inequality diminishes pension size while augmenting redistribution, whereas lifespan inequality increases pension size while reducing redistribution. We provide empirical evidence from OECD countries supporting these theoretical findings and calibrate the model on French data to quantify the effects.