In a recent article, Boucekkine et al. (2004) convincingly argue that the analysis of the relationship between demographic and economic trends should be built on a rigorous modeling of the specific decisions of the various cohorts that compose a population. Overlapping generations models, which feature an explicit age structure of the population, are hence the natural tool for studying the impact of demographic changes on individual decisions and aggregate economic variables.
The first model with overlapping generations developed in literature (Allais 1947, Samuelson, 1958 and Diamond, 1965) only involves two coexisting generations at each point of time, meaning that the length of a period is about thirty years.
The comparison with real world data is therefore difficult and, moreover, the heterogeneity yield by a demographic structure is implicitly eliminated. An alternative to this representation was proposed by Blanchard (1985). His approach has been celebrated since it allows for more than two periods within a life cycle and leads to simple analytics. However, this tractability hinges on the assumption of a constant hazard death rate, which is unable to capture the life cycle aspect of life. As a result, these two frameworks have been extensively used in theoretical works but not in applied ones. To measure the consequences of changes in the age structure of a population, Auerbach and Kotlikoff (1987) propose a deterministic frame work and Rios-Rull (1996) a stochastic one, which include many generations.
Recently, these models have been notably successful when applied to the evaluation of the impact of population aging on capital accumulation (Lau, 2009), public debt (Bullard and Russel, 1999), economic growth (Azoumahou et al., 2009), and pension systems (Bloom et al., 2007, Heijdra and Romp, 2009). From a theoretical point of view, recent contributions, such as those of Kehoe et al. (1991) and Azariadis et al. (2004), have thus been aimed at developing a fruitful analysis of large-square models and the setting out of specific conditions on the dynamic properties of equilibrium paths. However, a limitation with regard to the widespread application of such models springs from the difficulties that emerge in the course of their analytical resolution.
In the present work, we survey the latest developments of overlapping generations models with a realistic age structure of the population and introduce a simple method that allows for solving these models. We propose a complete resolution in a specific case that provides new results on the existence and the convergence properties of equilibrium paths. In models with many overlapping generations, the dynamics of endogenous variables depend on a finite number of their past and future realizations. This dependency is what gives rise to the analytical difficulty of these models.
In discrete-time frameworks, the analysis of the dynamics requires the study of polynomials, whose order increases with the number of generations, such an analysis is not tractable in general. We believe that this difficulty may however be circumvented using the continuous-time framework initially developed by Cass and Yaari (1967): it eases the mathematical resolution without affecting the qualitative analysis. However, in continuous-time frameworks, the dynamics are characterized by a functional differential equation of mixed type, i.e. the dynamics are affected by distributed delays and advances. In this work, we propose an integrated treatment of such a structure in a pure exchange economy, where preferences and endowments are stationary.
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