Bistability and Phase Transitions in Economics and Finance

Gerald SilverbergUNU-MERIT and IIASA (DYN)

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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

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Time series of Industrial Capacity Utilization, USA 1967-2009 (Federal Reserve, monthly data seasonally detrended)

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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

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Bistability/Bifurcation Models

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Cusp Catastrophe Derived from a Potential Function

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•Slow changes in parameters can push system between one and two-state regimes•perturbations can push system over barrier between regimes•hysteresis

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

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Canonical form of cusp catastrophe

PV:

Equilibrium condition:

Separatrix in parameter space:

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yqypyqpyV tt 2214

41),,(

03 qpyy ee

equilibria unstable one stable, 2 0

m,equilibriu unique 0

,)()( 33

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D

D

D pq

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:

The potential determines the dynamics as follows:

In a window [t-T,t+T], calculate p(t) and q(t) from the regression

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)()()( tytytx trend

txqpxV

dtdx tt

),,(

TTiqxpx titttxx itit ...1 ,31

Estimated parameters for different window sizes

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Time series of structural parameters

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HWM85 Results for FRG and USA(five-year window)

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HWM85 structural parameter time series

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HWM85 Potential Function and Realized Path

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The search for explanatory variables

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• 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:

Multiple regression analysis

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I: gross investmentE: expansionary investmentR: replacement investmentz=(E-R)/(E+R)O: open positionsW: working hours indP: inflation rate

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)?

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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 self-reinforcing 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.

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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>:

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0 ),,( xf

iiiixvfx

Bistability in Levy 2008 (con‘t)

• But since the average allocation is

self-consistency in equilibrium requires that

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,1

1

N

iiN xx

1)(0 ,)(),(1 xFxFxvfx iiN

Aggregating Heterogeneous Agents

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Cusp Catastrophe in Aggregate Market Dynamics

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Size of Crashes Depends on Degree of Heterogeneity and Conformity

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Simulated Time Series: Volatility as Early Warning

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