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

Abstract

Modern research often requires the use of economic modelswith multiple agents that interact over time. In this paper we researchoverlapping generations models, hereinafter OLG. In these models, thephenomenon of the multiplicity of long-term equilibrium may arise. Thisfact proves to be important for the theoretical justification of some eco-nomic effects, such as the collapse of the market and others. However,there is little theoretical research on the possibility of multiple equilibriain these models. At the same time, the works that exist are devoted tomodels with only few periods. This is due to the fact that the complexityof algorithms that calculate all long-term equilibria grows too fast withrealistically selected lifespan values. However, solutions of some OLGmodels after the introduction of additional variables can become polyno-mial systems. Thus it is possible to represent many long-term equilibriaas an algebraic variety. In particular, the Gr¨obner basis method becamepopular. However, this approach can only be used effectively when thereare 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 system’s solutions and use it toinvestigate the presence of multiple solutions in realistically calibratedmodels with long-lived agents. We also examine these models for mul-tiple equilibria using the Monte Carlo method and replicate previouslyknown results using a new algorithm.

Keywords

• olg-models
• multiplicity of equilibrium
• grobner basis
• polynomial systems