Bistability and Phase Transitions in Economics and Finance

1
Bistability and Phase Transitions in Economics
and Finance

• Gerald Silverberg
• UNU-MERIT and IIASA (DYN)

2
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

3
Time series of Industrial Capacity Utilization,
USA 1967-2009 (Federal Reserve, monthly data
seasonally detrended)
4
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

5
Bistability/Bifurcation Models
6
Cusp Catastrophe Derived from a Potential Function

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

7
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

8
Canonical form of cusp catastrophe

• PV
• Equilibrium condition
• Separatrix in parameter space

9
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,tT, calculate p(t) and q(t)
  from the regression

10
Estimated parameters for different window sizes
11
Time series of structural parameters
12
HWM85 Results for FRG and USA(five-year window)
13
HWM85 structural parameter time series
14
HWM85 Potential Function and Realized Path
15
The search for explanatory variables

• Menschs (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

16
Multiple regression analysis
I gross investment E expansionary investment R
replacement investment z(E-R)/(ER) O open
positions W working hours ind P inflation rate
17
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)?

18
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.

19
Segue to Bistability in Financial MarketsM.
Levy, 2008, Stock market crashes as social phase
transitions, JEDC, 32 137155

• 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 ltxgt

20
Bistability in Levy 2008 (cont)

• But since the average allocation
  isself-consistency in equilibrium requires
  that

21
Aggregating Heterogeneous Agents
22
Cusp Catastrophe in Aggregate Market Dynamics
23
Size of Crashes Depends on Degree of
Heterogeneity and Conformity
24
Simulated Time Series Volatility as Early Warning
25
(No Transcript)