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Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent ModelsLecture 1
IntroductionRational vs. Agent Based Modelling
Heterogeneous Agent Modelling
Mikhail Anufriev
EDG, Faculty of Business, University of Technology Sydney (UTS)
July, 2013
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Outline
1 Overview
2 Financial Market Model
3 Heterogeneous Agent Models
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Overview
Economics as Expectation Feedback System
Expectations play utmost role in any human activitywhere and when to go to a vacation
choice of university degree and specific courses
when to buy a car, house, etc.
investment choice
Economics is an expectation feedback system
expectations affect people’s decisions
individual decisions are aggregated
aggregate variables affect expectations
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Overview
Economics as Expectation Feedback System
Expectations play utmost role in any human activitywhere and when to go to a vacation
choice of university degree and specific courses
when to buy a car, house, etc.
investment choice
Economics is an expectation feedback system
expectations affect people’s decisions
individual decisions are aggregated
aggregate variables affect expectations
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Overview
Cobweb (“hog cycle”) Model
market for a non-storable consumption good
production lag: producers form price expectations one periodahead
temporary equilibrium: market clearing prices at each time step
What will be the price at this market?
dynamics: prices evolve over time forming a trajectory
how does this trajectory look like? Is it “simple” or not?
dynamics depend on functional form of demand and supply andon the way how expectations are formed
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Overview
Cobweb (“hog cycle”) Model
market for a non-storable consumption good
production lag: producers form price expectations one periodahead
temporary equilibrium: market clearing prices at each time step
What will be the price at this market?
dynamics: prices evolve over time forming a trajectory
how does this trajectory look like? Is it “simple” or not?
dynamics depend on functional form of demand and supply andon the way how expectations are formed
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Overview
Naive Expectations in Cobweb Model
Naive expectationspe
t = pt−1
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Overview
Price Dynamics
pet : producers’ price expectation for period t
pt : realized market equilibrium price
temporary equilibrium
qdt = D(pt) , demand
qst = S(pe
t ) supplyqd
t = qst market clearing
pet = H(pt−1, pt−2, . . . ) expectations
Price dynamics
pt = D−1(S(pet ))
= D−1(S(H(pt−1, pt−2, . . . ))).
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Overview
Rational View
representative agent, who is perfectly rational
Rational Expectations: expectations are model consistent
Friedman argument: “irrational agents will lose money andwill be driven out the market by rational agents”
behavioral consistency: simple heuristics that work reasonablywell
evolutionary selection (‘survival of the fittest’) andreinforcement learning
laboratory experiments to test individual decision rules andaggregate macro behavior
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Overview
Overview
Lecture 1: Rational vs. agent-based vs. HAM approachesfinancial market model
Lecture 2: Bifurcation theory and Chaos theorychaos in economics, lessons of nonlinearity for economics
Lecture 3: Learning to Forecast Experimentsrole of market feedback
Lecture 4: Heuristic Switching Modelbehavioral model based on evolutionary switching betweenforecasting heuristics, which explain data very well
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
General setup
Market
Two alternatives for investors, two assets:
Risk-free asset gross riskless return on the asset R = 1 + r > 1
Risky asset dividend process yt
endogenous price pt per share
Demand zt for the risky asset is derived from myopicmean-variance utility maximization
Supply zs per investor is fixed (and assumed to be 0)
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
General setup
Mean-Variance DemandWealth evolution:
Wt+1 = R(Wt − ptzt) + (pt+1 + yt+1)zt
mWt+1 = RWt + (pt+1 + yt+1 − R pt)zt
Agent solves
maxzt
[Et Wt+1 −
a2
Vt Wt+1
],
where a > 0 is risk aversion.
Demand is
zt =Et[pt+1 + yt+1 − R pt]
a Vt[pt+1 + yt+1 − Rpt]
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Academic view of financial markets
Efficient Market HypothesisEugene Fama (1965), Paul Samuelson (1965)
Market is defined as informationally efficient if
price pt of an asset reflects all available (relevant) information
...that is price is an unbiased estimation of the aggregate beliefsabout the future perspectives (fundamental value p∗t )
Fundamental price is
p∗t =
∞∑k=1
yt+k
(1 + r)k
where yt is the dividend at time t
Formally: pt = Et[p∗t ]
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Academic view of financial markets
Efficient Market HypothesisEugene Fama (1965), Paul Samuelson (1965)
Market is defined as informationally efficient if
price pt of an asset reflects all available (relevant) information
...that is price is an unbiased estimation of the aggregate beliefsabout the future perspectives (fundamental value p∗t )
Fundamental price is
p∗t =
∞∑k=1
yt+k
(1 + r)k
where yt is the dividend at time t
Formally: pt = Et[p∗t ]
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Academic view of financial markets
Efficient Market Hypothesis
Eugene Fama (1965), Paul Samuelson (1965)
pt = Et[p∗t ] = p∗t + εt
errors εt are unpredictable on the basis of information availableat time t
the returns are unpredictable:
Et[pt+1 + yt+1 − (1 + r)pt] = 0
Question: Why are the markets efficient?
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Academic view of financial markets
Theoretical Underpinnings of Efficiency1st argument: Milton Friedman (1953)
arbitrage:people who argue that speculation is generally destabilizingseldom realize that this is largely equivalent to saying thatspeculators lose money, since speculation can be destabilizing ingeneral only if speculators on average sell when currency is lowin price and buy when it is high
Et[pt+1 + yt+1 − (1 + r)pt] = 0
m
1 + r =1pt
(Et[pt+1 + yt+1]) ⇔ pt =1
1 + rEt[pt+1 + yt+1]
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Academic view of financial markets
Theoretical Underpinnings of Efficiency2nd argument: Robert Lucas (1978)
fully rational behavior
optimizing behavior of representative investorabsence of systematic mistakes about price distribution
assume demand functions:
zt,n =Et,n[pt+1 + yt+1]− pt(1 + r)
an Vt,n[pt+1 + yt+1]
thenpt =
11 + r
∑n
wt,n Et,n[pt+1 + yt+1]
where wt,n is the relative weight on the group “n” of agents
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Academic view of financial markets
Theoretical Underpinnings of Efficiency2nd argument: Robert Lucas (1978)
fully rational behavior
optimizing behavior of representative investorabsence of systematic mistakes about price distribution
assume demand functions:
zt,n =Et,n[pt+1 + yt+1]− pt(1 + r)
an Vt,n[pt+1 + yt+1]
thenpt =
11 + r
∑n
wt,n Et,n[pt+1 + yt+1]
where wt,n is the relative weight on the group “n” of agents
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Academic view of financial markets
Theoretical Underpinnings of Efficiency2nd argument: Robert Lucas (1978)
Under Rational Expectations
pt =1
1 + r
∑n
wt,n Et,n[pt+1 + yt+1]
m
pt =1
1 + r
∑n
wt,n Et[pt+1 + yt+1]
m
pt =1
1 + rEt[pt+1 + yt+1]
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Empirical Consequences
Empirical Consequences
Both technical and fundamental analysis are useless (except by luck):
“blindfolded chimpanzee throwing darts at The Wall Street Journalcan select a portfolio that performs as well as those managed by theexperts”
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Empirical Consequences
Empirical Consequences
No-Excess Volatility:
pt = Et[p∗t ] = p∗t + εt
thereforeσpt ≤ σp∗t
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Empirical Consequences
Empirical ConsequencesNo-Excess Volatility:
Instead, we observe excess volatility:
Shiller, “Do Stock Prices Move Too Much to be Justified by Subsequent Changes in
Dividends?”, American Economic Review, 71, pp.421–436, 1981
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Empirical Consequences
Summary of rational view on financial market
all investors possess rational expectations; they instantaneouslydiscount all available information
no opportunities for speculative profit
no place for market psychology or herding
bubbles and crushes are either non-existent or “rational”
trading volume is almost zero
no autocorrelations of returns
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Implications of Heterogeneity
Heterogeneity of investorsassume that traders are not alike
pt =1
1 + r
∑n
wt,n Et,n[pt+1 + yt+1]
BUT:
Et,n[pt+1] =1
1 + rEt,n
[∑m
wt+1,m Et,m[pt+2 + yt+2]
]there is no way to form expectations about others’ expectationsabout dividends and – even more so – about others’ expectationsof price
consequently, complete knowledge is impossible and agents areboundedly rational
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Implications of Heterogeneity
Heterogeneity of investorsassume that traders are not alike
pt =1
1 + r
∑n
wt,n Et,n[pt+1 + yt+1]
BUT:
Et,n[pt+1] =1
1 + rEt,n
[∑m
wt+1,m Et,m[pt+2 + yt+2]
]there is no way to form expectations about others’ expectationsabout dividends and – even more so – about others’ expectationsof price
consequently, complete knowledge is impossible and agents areboundedly rational
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Implications of Heterogeneity
Heterogeneity of investors
Brian Arthur (1995)
we can think of the economy ultimately as a vast collection of beliefsor hypotheses, constantly being formulated, acted upon, changed anddiscarded; all interacting and competing and evolving and coevolving;forming an ocean of ever-changing, predictive models-of-the-world
Santa-Fe Artificial Stock MarketArthur, Holland, LeBaron, Palmer and Tayler (1997)”Asset Pricing under Endogenous Expectations in an Artificial StockMarket”
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Implications of Heterogeneity
Santa-Fe Artificial Stock Market: Setupeach agent is endowed with multiple forecasting models,M = 100forecasting model is a predictor
predictor checks the market condition and gets “active” ifcondition is satisfied
some of these conditions are “fundamental”some are “technical-trading”
predictor also provides two values a and b, which implyprediction E[pt+1 + yt+1] = a(pt + yt) + bfor each predictor the accuracy (squared forecasting error) istracked and updated, when the predictor is “active”
agent uses H most precise “active” predictors to combine themand get a forecaston a slower scale (on average every 250 or every 1000 periods)predictors get updated by Genetic Algorithm
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Implications of Heterogeneity
Santa-Fe Artificial Stock Market: Results
Two regimes:slow-learning-rate regime
GA is evoked every 1, 000 periods in averageaccuracy of predictors are slowly updatedmarket converges rapidly and then stays in evolutionary stable REregime, where agents use similar rules, trading volume is low,technical trading does not emerge
fast-learning-rate regimeGA is evoked every 250 periods in averageaccuracy of predictors are updated relatively fastmarket is often close to the RE solution, but temporary bubblesand crashes emerge systematically, technical trading emerged inthe market
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Implications of Heterogeneity
Santa-Fe Artificial Stock Market: Results
Two regimes:slow-learning-rate regime
GA is evoked every 1, 000 periods in averageaccuracy of predictors are slowly updatedmarket converges rapidly and then stays in evolutionary stable REregime, where agents use similar rules, trading volume is low,technical trading does not emerge
fast-learning-rate regimeGA is evoked every 250 periods in averageaccuracy of predictors are updated relatively fastmarket is often close to the RE solution, but temporary bubblesand crashes emerge systematically, technical trading emerged inthe market
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Financial Market Model
Implications of Heterogeneity
Santa-Fe Artificial Stock Market: Results
Two regimes:slow-learning-rate regime
GA is evoked every 1, 000 periods in averageaccuracy of predictors are slowly updatedmarket converges rapidly and then stays in evolutionary stable REregime, where agents use similar rules, trading volume is low,technical trading does not emerge
fast-learning-rate regimeGA is evoked every 250 periods in averageaccuracy of predictors are updated relatively fastmarket is often close to the RE solution, but temporary bubblesand crashes emerge systematically, technical trading emerged inthe market
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Heterogeneous Agent Modelling
Brock and Hommes (1998, JEDC) propose a simple model which canreach a similar conclusion.
It also answers the following questions:
Can heterogeneous agents destabilize markets?
Can “less-rational” traders survive against “more-rational”?
Is the market with heterogeneous agents efficient?
Is it possible to mimic “stylized facts” with a simple(low-dimensional) model?
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Model with Heterogeneous Agents
H = 2, 3, . . . types of traders
individual demand is
zh,t =Eh,t[pt+1 + yt+1 − R pt]
a Vh,t[pt+1 + yt+1 − Rpt]
Assume that agents have...
heterogeneous expectations about price
the same risk aversion
homogeneous expectations of the variance
complete knowledge of the dividend process (which is IID forsimplicity)
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Model with Heterogeneous Agents
H = 2, 3, . . . types of traders
individual demand is
zh,t =Eh,t[pt+1 + yt+1 − R pt]
a Vh,t[pt+1 + yt+1 − Rpt]
Assume that agents have...
heterogeneous expectations about price
the same risk aversion
homogeneous expectations of the variance
complete knowledge of the dividend process (which is IID forsimplicity)
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Fundamental SolutionEquilibrium pricing equation with homogeneous investors:
Rpt = Et[pt+1 + yt+1]
There exists an unique bounded fundamental solution p∗t (discountedsum expected future cash flow):
p∗t =Et[yt+1]
R+
Et[yt+2]
R2 + · · ·
For a special case of IID dividends, with Et[yt+1] = y:
p∗ =y
R− 1=
yr
Equilibrium pricing equation with heterogeneous investors:H∑
h=1
nh,tzh,t = 0 ⇔ pt =1
1 + r
H∑h=1
nh,t Eh,t[pt+1] +y
1 + r
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Dynamics in Deviations
write the equation in the deviations from the rationalexpectations benchmark xt = pt − p∗
xt =1
1 + r
H∑h=1
nh,t Eh,t[xt+1]
there are two homogeneous, self-fulfilling solutions:if Eh,t[xt+1] = 0 then xt = 0 fundamental solution.if Eh,t[xt+1] = R2xt−1 then xt = Rxt−1 bubble solution.
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Dynamics in Deviations
write the equation in the deviations from the rationalexpectations benchmark xt = pt − p∗
xt =1
1 + r
H∑h=1
nh,t Eh,t[xt+1]
there are two homogeneous, self-fulfilling solutions:if Eh,t[xt+1] = 0 then xt = 0 fundamental solution.if Eh,t[xt+1] = R2xt−1 then xt = Rxt−1 bubble solution.
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Forecasting rulesAssume that belief of type h on future prices has form
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Evolutionary selection of strategies4 define the performance measure as a (weighted sum of) realized
profitsUh,t = πh,t + wUh,t−1 − Ch
where Ch ≥ 0 are costs for predictor h, and w is memorystrength
w = 1: infinite memory; fitness ≡ accumulated wealthw = 0: memory is one lag; fitness ≡ most recently realized netprofit
5 update fractions of belief types according to the discrete choicemodel
nh,t =eβUh,t−1
Zt−1
where Zt−1 is normalization factor and β is intensity of choice.
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Asset Pricing Model with Heterogeneous Beliefs
pricing equation
R xt =
H∑h=1
nh,tfh(xt−1, . . . , xt−L) =
H∑h=1
nh,tfh,t
fraction of different investors’ types
nh,t =eβUh,t−1∑H
k=1 eβUk,t−1
performance of different types
Uh,t−1 = (xt−1 − Rxt−2)fh,t−2 − Rxt−2
aσ2 − Ch
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Example with linear predictors fh,t = ghxt−1 + bh
pricing equation
xt =
H∑h=1
nh,t(ghxt−1 + bh)
fraction of different investors’ types
nh,t =eβUh,t−1∑H
k=1 eβUk,t−1
performance of different types
Uh,t−1 = (xt−1 − Rxt−2)(ghxt−3 + bh − Rxt−2)
aσ2 − Ch
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Two Types Example
Two-types example: Fundamentalists versus trend-followers
Two typesfundamentalists, f1,t = 0, at cost C
trend-followers, f2,t = gxt−1 at cost 0
Define difference in fractions: mt = n1,t − n2,t
Derive 3-dimensional system
Rxt = n2,tgxt−1
mt+1 = tanh(β2
[− gxt−2
aσ2 (xt − Rxt−1)− C])
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Two Types Example
Fundamentalists versus trend-followersTheorem (Existence and stability of fixed points.)Let meq = tanh(−βC/2), m∗ = 1− 2R/g and x∗ be a positivesolution of
tanh(β
2[ g
aσ2 (R− 1)(x∗)2 − C])
= m∗
Then:
0 < g < R fundamental st-st E1 = (0,meq) is globally stable
g > 2R there are three st-st’s: E1 = (0,meq), E2 = (x∗,m∗) andE3 = (−x∗,m∗)
R < g < 2R E1 = (0,meq) is stable for β < β∗, E2 = (x∗,m∗) andE3 = (−x∗,m∗) are stable for β∗ < β < β∗∗.
system undergoes a pitchfork bifurcation for β = β∗
and Hopf bifurcation for β = β∗∗
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Two Types Example
Rational Route to Randomness(C = 1, g = 1.2, r = 0.1, y = 10)
Corollary: fundamentalists cannot drive out trend chasers.
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Two Types Example
Time series immediately after the secondary bifurcationβ = 2.81,C = 1, g = 1.2, r = 0.1, y = 10
98
99
100
101
102
103
0 100 200 300 400 500 600 700 800 900 1000
Pric
e
Time
initial price aboveinitial price below
noisy
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Two Types Example
Time series far from the secondary bifurcationβ = 4,C = 1, g = 1.2, r = 0.1, y = 10
94
96
98
100
102
104
106
0 100 200 300 400 500 600 700 800 900 1000
Pric
e
Time
initial price above noisy
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Other Examples
Fundamentalists versus two opposite biased beliefs
Three typesfundamentalists, f1,t = 0, at cost 0
positive bias, f2,t = b2 > 0 at cost 0
negative bias, f3,t = b3 < 0 at cost 0
Derive 3-dimensional system
R xt = n2,tb2 + n3,tb3
nj,t+1 = exp(
βaσ2 (bj − Rxt−1)(xt − Rxt−1)
)/Zt, j = 1, 2, 3
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Other Examples
Fundamentalists versus two opposite biased beliefs
Theorem. (Existence and stability steady state.)
The system has unique fixed point E, which equals to thefundamental fixed point when b2 = −b3.
E exhibits a Hopf bifurcation for some β = β∗, so that E isstable for 0 < β < β∗ and E is unstable for β > β∗.
Heterogeneous Agent Models Lecture 1 Introduction Rational vs. Agent Based Modelling Heterogeneous Agent Modelling
Heterogeneous Agent Models
Other Examples
Fundamentalists versus two opposite biased beliefs
Theorem. (Neoclassical limit, i.e. β =∞)When biased beliefs are exactly opposite, i.e. whenb2 = −b3 = b > 0, then the system has globally stable 4-cycle.For all three types, average profit along this 4-cycle is b2.
Corollary. Fundamentalists with zero costs and infinite memory cannot beat opposite biased beliefs!