The fast algorithm for computing all steady states in overlapping generations models

Modern research often requires the use of economic models with multiple agents that interact over time. In this paper we research overlapping generations models, hereinafter OLG. In these models, the phenomenon of the multiplicity of long-term equilibrium may arise. This fact proves to be important for the theoretical justification of some economic effects, such as the collapse of the market and others. However, there is little theoretical research on the possibility of multiple equilibria in these models. At the same time, the works that exist are devoted to models with only few periods. This is due to the fact that the complexity of algorithms that calculate all long-term equilibria grows too fast with realistically selected lifespan values. However, solutions of some OLG models after the introduction of additional variables can become polynomial systems. Thus it is possible to represent many long-term equilibria as an algebraic variety. In particular, the Gröbner basis method became popular. However, this approach can only be used effectively when there are few variables. In this paper we consider the task of finding long-term equilibrium in overlapping generations models with many periods. We offer an algorithm for finding the systems solutions and use it to investigate the presence of multiple solutions in realistically calibrated models with long-lived agents. We also examine these models for multiple equilibria using the Monte Carlo method and replicate previously known results using a new algorithm.