Stochastic Games for Intergenerational Equity in Mathematical Finance

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

Intergenerational conflicts of interest are inevitable to wealth and resource management in a society. This problem is more pressing now than ever as demographic change has reached a pivotal point. The combined working-age population in developed countries is projected to shrink continuously until 2050, while the retired population will grow rapidly. The tension between the two cohorts, working and retired (or, young and old), is almost certain to exacerbate, as the wealth of the former is used to support the latter, under Social Security or many other pension schemes. Yet, the precise economic consequences are not that apparent. This project aims to elucidate how an evolving demographic structure affects financial planning in a society. This will be done at two different levels: one from the eyes of the competing young and old cohorts, the other from the panoramic perspective of a social planner, such as a government. Graduate students are involved in the project. The first part of the project investigates the interplay between the young and the old cohorts. It is formulated as a two-player nonzero-sum game in a stochastic Ramsey model, with the dynamics of demographic structure explicitly modeled. The goal is to understand how the two cohorts interact through their consumption and saving decisions, and how an evolving demographic structure affects their decisions, and the resulting welfare. Nash equilibrium strategies of the two cohorts will be found and analyzed through a system of coupled Hamilton-Jacobi-Bellman equations with non-Lipschitz coefficients. This demands new developments in stochastic differential games and viscosity solution techniques. The second part of the project handles the social planner's dilemma: a financial planning strategy deemed optimal today by the current society may not be optimal from the view of the society in the future, as the evolving demographics continuously change the overall societal preference. This leads to a new form of time-inconsistent problems, with non-exponential and time-dependent discount functions, which will be treated under an intra-personal game between current and future societies. A new method for time-inconsistency, called the fixed-point approach, is developed in the project. It provides a new machinery for constructing time-consistent strategies, while possessing the potential of less mathematical technicalities.