Crises and Physical Phases of a Bipartite Market Model

We analyze the linear response of a market network to shocks based on the bipartite market model we introduced in an earlier paper, which we claimed to be able to identify the time-line of the 2009-2011 Eurozone crisis correctly. We show that this model has three distinct phases that can broadly be categorized as "stable" and "unstable". Based on the interpretation of our behavioral parameters, the stable phase describes periods where investors and traders have confidence in the market (e.g. predict that the market rebounds from a loss). We show that the unstable phase happens when there is a lack of confidence and seems to describe "boom-bust" periods in which changes in prices are exponential. We analytically derive these phases and where the phase transition happens using a mean field approximation of the model. We show that the condition for stability is $αβ<1$ with $α$ being the inverse of the "price elasticity" and $β$ the "income elasticity of demand", which measures how rash the investors make decisions. We also show that in the mean-field limit this model reduces to the Langevin model by Bouchaud et al. for price returns.