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Bistability and Phase Transitions in Economics and Finance Gerald Silverberg UNU-MERIT and IIASA (DYN) 1 Economic Systems Occasionally Seem to be Characterized by Rapid and Large Change without Apparent External Cause • Recent common descriptions in the business press: – ‘financial meltdown’ – ‘the economy is in free fall’ • The system may then remain in the new state for an indefinite length of time – The USA did not exit from the great depression until rearmament for WW2 began in earnest around 1939 2 Time series of Industrial Capacity Utilization, USA 1967-2009 (Federal Reserve, monthly data seasonally detrended) 95 90 85 80 cap_utl trend 75 70 65 60 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 3 What kinds of dynamical systems can describe this behavior? • Standard time-series econometrics (ARMA, VAR) posits a single stable equilibrium with fluctuations resulting from external shocks. Problems: – output fluctuations seem too large – persistance seems to high – exception: central driving role of energy prices (cf Hamilton 2009) • Limit cycle – necessitates prominent periodic component for which there is no empirical evidence • Deterministic chaos – requires too much data to establish empirically for real data – in finance, no evidence for returns but some for volatility 4 Bistability/Bifurcation Models 5 Cusp Catastrophe Derived from a Potential Function •Slow changes in parameters can push system between one and twostate regimes •perturbations can push system over barrier between regimes •hysteresis 6 Pedigree of Bistability Perspective in Macroeconomics • J. T. Schwartz, Theory of Money, 1961: the essence of Keynesianism is the assertion that there are coordination full and underemployment Nash equilibrium • Cooper, R. and John, A., 1988, “Coordinating Coordination Failures in Keynesian Models”, Quarterly Journal of Economics, 103: 441-461 • Durlauf, Steven N., 1991, “Multiple Equilibria and Persistence in Aggregate Fluctuations”, American Economic Review. Papers and Proceedings, 81: 70-74 7 Canonical form of cusp catastrophe PV: V ( y, p, q) 14 y 4 12 pt y 2 qt y Equilibrium condition: ye3 pye q 0 Separatrix in parameter space: D ( q2 ) 2 ( 3p ) 3 , D 0 unique equilibriu m, D 0 2 stable, one unstable equilibria 8 Constructing a (time-dependent) PV from a time series (Haag, Weidlich & Mensch 1985) Filtering structural from high-frequency, low-amplitude fluctuations: First calculate deviation from trend: x(t ) y(t ) ytrend (t ) The potential determines the dynamics as follows: dx dt V ( x,xpt ,qt ) t In a window [t-T,t+T], calculate p(t) and q(t) from the regression xt i xt i1 t x3 pt xt i qt , i T 1...T 9 Estimated parameters for different window sizes one-year window two-year window 500 200 0 0 -300 -250 -200 -150 -100 -50 0 50 -200 -150 -100 -50 0 50 -200 -500 q -400 q -1000 -600 -1500 -800 -2000 p -1000 p six-year window four-year window 100 200 50 100 0 -120 0 -120 q -100 -80 -60 -40 -20 0 -100 -80 -60 q -40 -20 -50 -100 -100 -150 -200 p -300 -200 p -250 10 0 Time series of structural parameters two-year window one-year window 200 200 1965 -300 1975 1985 1995 0 1965 2005 1975 1985 1995 2005 -200 p_1 -800 p_2 -400 q_2 q_1 -600 -1300 -800 -1000 -1800 four-year window six-year window 150 100 100 50 50 0 1965 -50 -100 -150 1975 1985 1995 2005 p_4 q_4 0 1965 -50 1975 1985 1995 2005 p_6 q_6 -100 -150 -200 -250 -300 -200 11 -250 HWM85 Results for FRG and USA (five-year window) 12 HWM85 structural parameter time series 13 HWM85 Potential Function and Realized Path 14 The search for explanatory variables • Mensch‘s (1979) original model assumed • where R(t) was replacement and modernization investment and E(t) was expansionary investment. • HWM85 generalize to multiple inputs with time delays: 15 Multiple regression analysis I: gross investment E: expansionary investment R: replacement investment z=(E-R)/(E+R) O: open positions W: working hours ind P: inflation rate 16 Explanations of Bistability Behavior • Investment coordination problem due to investment externalities in demand • Double-edged implications of composition of investment: modernization investment has both demand enhancing (multiplier) effects and employment-replacing effects • Herding behavior (informational externality)? 17 Implications of Bistability for Macrodynamics and Policy • Slowing varying structural variables can move the economy into or out of the bistability region, thus triggering or allowing for regime change • Once in the bistability region, small shocks can trigger rapid selfreinforcing movement ‘over the cliff’ into the other basin of attraction. Thus the relationship between size of causes and size of effects can break down • Hysteresis: reversing a regime transition can be more difficult and costly than triggering it. Implications for stimulous programs: until they induce a spontaneous return to the upper sheet, they are costly and relativelyineffectual. Once they do, the multiplier is very much higher. • If there are multiple (Nash) equilibria, the notion of ‘rationality’ loses its meaning except locally. Individual ‘rationality’ can be in conflict with social rationality. 18 Segue to Bistability in Financial Markets: M. Levy, 2008, “Stock market crashes as social phase transitions”, JEDC, 32: 137–155 • Heterogeneous agents with bounded rationality • Each agent has to make a portfolio decision: what percentage of her assets xi to allocate between a risky asset (shares) and a riskless one (gilts) • Each agent is influenced by idiosyncratic variables vi reflecting preferences, risk adversion, etc., plus publicly observable variables like interest rates, risk measures, etc. • Each agent is also subject (to different degrees) to a herding effect dependent on the average portfolio allocation <x>: f i xi f i (vi , x ), x 0 19 Bistability in Levy 2008 (con‘t) • But since the average allocation is x N 1 N x , i 1 i self-consistency in equilibrium requires that x 1 N f i (vi , x ) F ( x ), 0 F ( x ) 1 20 Aggregating Heterogeneous Agents 21 Cusp Catastrophe in Aggregate Market Dynamics 22 Size of Crashes Depends on Degree of Heterogeneity and Conformity 23 Simulated Time Series: Volatility as Early Warning 24 25