Pay-as-you-go social security and educational subsidy in an overlapping generations model with endogenous fertility and endogenous retirement

This study analytically investigates the effects of pay-as-you-go social security and educational subsidies on the fertility rate, retirement age, and GDP per capita growth rate in an overlapping generations model, where parents invest resources toward their children's human capital. We find that an old agent retires fully when his or her labor productivity is low and retires later when the labor productivity is high. Under the unique balanced-growth-path (BGP) equilibrium, when an old agent is still engaged in work, tax rates are neutral to the fertility rate, higher tax rates encourage him or her to retire earlier, a higher social security tax rate depresses the GDP per capita growth rate, and a higher tax rate for educational subsidies can accelerate growth. However, when an old agent fully retires, higher tax rates increase the fertility rate, a higher social security tax rate lowers the GDP per capita growth rate, and a higher tax rate for educational subsidies boosts growth. Additionally, if an old agent's labor productivity increases, the fertility rate also increases. We also conduct numerical simulations and analyze how an old agent's labor productivity affects the retirement age, fertility rate, and GDP per capita growth rate under the BGP equilibrium