Demographic dynamics for the pay-as-you-go pension system

The present study focuses on certain problems of pay-as-you-go pen- sion systems. In particular, based on reproduction and survival rates, a Leslie type model is used for the analysis of the demographic dynamics of a population. The presented model is dierent from the classical Leslie model, since it considers a population divided into sexes, implying that the system matrix does not satisfy the irreducibility and primitivity conditions used in the Perron-Frobenius theory. Applying this theory to a submatrix corresponding to a subpopulation of females, the existence of an asymptotic age distribution is proved and then extended to the whole population. The obtained result is applied to the Italian population, providing projections on its demographic evolution. In the paper the impact of the asymptotic demographic equilibrium on the pay-as-you-go pension system, in terms of nancial sustainability, is also studied.