Resolving Aaron's Social Insurance Paradox

Abstract: This paper resolves Aaron's social insurance paradox, which suggests that introducing a pay-as-you-go (PAYG) pension system increases welfare when population growth plus average wage growth exceeds interest rates. Using a simplified overlapping generations model, we demonstrate this apparent advantage stems from asset reduction rather than inherent superiority. We analyze three pension systems - traditional PAYG, capital-funded, and capital-funded with bonus payments - and establish an equivalence between PAYG and the bonus-payment system. This equivalence reveals that systems with identical contributions and benefits differ only in accounting frameworks and asset positions, challenging the notion of PAYG superiority. Our analysis exposes a fundamental conceptual inconsistency in how sustainability is assessed across equivalent pension systems. As an alternative, we propose $\alpha$-stability, a framework using index shares to evaluate pension systems relative to economic indicators. These findings suggest that perceived advantages between pension systems often result from their formulation rather than substantive economic differences.