Download Growth Theory from an Evolutionary Perspective

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 i1
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
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