A Tentative Consideration of the Three-Period Overlapping Generations Model

This paper considers an overlapping generations model with three periods using the AK model. In overlapping generations models, two-period models are generally more common. While two-period models are easier to analyze, three-period models are more complex and are usually examined through simulations. Needless to say, there are previous studies that have analytically analyzed three-period models. This study develops the methods of previous research by defining the ratio of variables as new variables, thereby reducing the number of variables and deriving the dynamic equations. Furthermore, it explicitly examines the dynamics of the birth rate when both the young and middle-aged generations engage in fertility behavior, offering a different perspective from the conventional assumption that only one generation engages in fertility behavior. Although this model setting can also be found in previous studies, this paper demonstrates that childcare support policies targeting a specific generation may reduce the birth rate of other generations. Additionally, it derives the equilibrium solution of a three-period overlapping generations model that incorporates endogenous fertility and capital accumulation.