Chaos and Bifurcation in 2007-08 Financial Crisis

The impact of increasing leverage in the economy produces hyperreaction of market participants to variations of their revenues. If the income of banks decreases, they mass-reduce their lendings; if corporations sales drop, and due to existing debt they cannot adjust their liquidities by further borrowings, then they must immediately reduce their expenses, lay off staff, and cancel investments. This hyperreaction produces a bifurcation mechanism, and eventually a strong dynamical instability in capital markets, commonly called systemic risk. In this article, we show that this instability can be monitored by measuring the highest eigenvalue of a matrix of elasticities. These elasticities measure the reaction of each sector of the economy to a drop in its revenues from another sector. This highest eigenvalue - also called the spectral radius - of the elasticity matrix, can be used as an early indicator of market instability and potential crisis. Grandmont [85] and subsequent research showed the possibility that the "invisible hand" of markets become chaotic, opening the door to uncontrolled swings. Our contribution is to provide an actual way of measuring how close to chaos the market is. Estimating elasticities and actually generating the indicators of instability will be the topic of forthcoming research. Part of this paper has spun off with more mathematical details and can be found on SSRN under the title "Financial Crisis Dynamics: Attempt to Define a Market Instability Indicator".