-
Complex Systems WorkshopLecture II: Non-linear Cobweb Model
Cars Hommes
CeNDEF, University of Amsterdam
CEF 2013, July 9, Vancouver
Behavioral Rationality and Heterogeneous Expectations
in Complex Economic Systems
Cars Hommes
“Nosto ea facin ulput veros del utem zzrit duisseq uamet, si
esse vent am et euis
nonsectet ercidunt prat, consecte min eleniam zzrit essecte
feugue vel iusto odo
coreetu eraesequat ulla feui ea feuguer cillam zzrilla ad
doluptat ad te facillamet,
quat do consenim exer ipsummolore delis nulluptat. Lutat.
Feugait ulla cor
sequam, sequisl ullamcore feu feugiamet, velit aliqui blaorer
ostrud dit non ut at
ex et lum eugiate volore faccum nim estie velit dolore
magniscinit alit lum ex et,
quat.”
SOMEBODY, somewhere
“Nosto ea facin ulput veros del utem zzrit duisseq uamet, si
esse vent am et euis
nonsectet ercidunt prat, consecte min eleniam zzrit essecte
feugue vel iusto odo
coreetu eraesequat ulla feui ea feuguer cillam zzrilla ad
doluptat ad te facillamet,
quat do consenim exer ipsummolore delis nulluptat. Lutat.
Feugait ulla cor
sequam, sequisl ullamcore feu feugiamet, velit aliqui blaorer
ostrud dit non ut at
ex et lum eugiate volore faccum nim estie velit dolore
magniscinit alit lum ex et,
quat.”
SOMEBODY, somewhere
“Nosto ea facin ulput veros del utem zzrit duisseq uamet, si
esse vent am et euis
nonsectet ercidunt prat, consecte min eleniam zzrit essecte
feugue vel iusto odo
coreetu eraesequat ulla feui ea feuguer cillam zzrilla ad
doluptat ad te facillamet,
quat do consenim exer ipsummolore delis nulluptat. Lutat.
Feugait ulla cor
sequam, sequisl ullamcore feu feugiamet, velit aliqui blaorer
ostrud dit non ut at
ex et lum eugiate volore faccum nim estie velit dolore
magniscinit alit lum ex et,
quat.”
SOMEBODY, somewhere
Cover designed by Hart McLeod Ltd
Behavioral Rationality and Heterogeneous Expectations
in Complex Econom
ic Systems
Hom
mes
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1070
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 1
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Outline
1 Nonlinear Cobweb model with Homogeneous Beliefs (Chapter
4)Naive expectationsRational expectationsAdaptive expectations
2 Cobweb Model with Heterogeneous Beliefs (Chapter 5)Rational
versus naiveEvolutinoary Selection and Reinforcement
LearningFundamentalists versus naiveContrarians versus naiveSAC
learning versus naive
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Literature
Hommes, C.H., (2013), Behavioral Rationality and Heterogeneous
Expectations inComplex Economic Systems, Cambridge.Hommes, C.H.,
(1994), Dynamics of the cobweb model with adaptive expectationsand
nonlinear supply and demand, Journal of Economic Behavior and
Organization24, 315-335.Brock, W.A. and Hommes, C.H. (1997), A
rational route to randomness,Econometrica, 65, 1059-1095.
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Cobweb (‘hog cycle’) Model
market for non-storable consumption good (e.g. corn,
hogs)production lag; producers form price expectations one period
aheadpartial equilibrium; market clearing prices
pet : producers’ price expectation for period tpt : realized
market equilibrium price pt
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Cobweb (‘hog cycle’) Model (continued)
D(pt) = a − dpt(+�t) a ∈ R, d ≥ 0 demand (1)
Sλ(pet ) = tanh(λ(pet − 6)) + 1, λ > 0, supply (2)
D(pt) = Sλ(pet ) market clearing (3)
pet = H(pt−1, ..., pt−L), expectations (4)
Price dynamics: pt = D−1Sλ(H(pt−1, ..., pt−L))Expectations
Feedback System:dynamical behavior depends upon expectations
hypothesis;supply driven, negative feedback
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Demand and (nonlinear) Supply in Cobweb Model
4 5 6 7 8 90
0.5
1
1.5
2
2.5
3
S
c p*
q*
pe, p
q
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Expectations
naive expectations: pet = pt−1adaptive expectations; pet = wpt−1
+ (1− w)pet−1backward looking average expectations pet = w1pt−1 +
wpt−2rational expectations: pet = Et [pt ] = p∗
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Naive expectations
Naive Expectations Benchmark (pet = pt−1)
unstable steady state iff S ′(p∗)/D′(p∗) < −1
6
8
10
12
14
16
0 10 20 30 40 50
pt
t
-10
-5
0
5
10
0 10 20 30 40 50
pt-pte
t
Regular period 2 price cycle with systematic forecasting
errors
Agents will learn from their mistakes and adapt forecasting
behavior
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Rational expectations
Rational Expectations (Muth, 1961)
Expectations are model consistentall agents are rational and
compute expectations from market equilibriumequations
pet = Et [pt ] or pet = pt or pet = p∗
implied self-fulfilling RE price dynamics
pt = p∗ + δt
perfect foresight, no systematic forecasting errors
Important Note: this is impossible in complex, heterogeneous
world
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Rational expectations
Rational Expectations Benchmark (p∗ = 5.93)
5 10 15 20 25 30 35 40 45 500
2
4
6
8
10
pric
e
Problem: need perfect knowledge of “law of motion”and high
computing abilities
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Adaptive expectations
Adaptive Expectations (“error learning”)Nerlove 1958
pet = (1− w)pet−1 + wpt−1
= pet−1 + w(pt−1 − pet−1)
= wpt−1 + (1− w)wpt−2 + · · · (1− w)j−1wpt−j + · · ·
weighted average of past prices
1-D (expected) price dynamics: pet = wD−1S(pet−1) + (1−
w)pet−1
stable steady state if − 2w + 1 <S′(p∗)D′(p∗)(< 0)
more stabilizing in linear models, butpossibly low amplitude
chaos in nonlinear models
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Adaptive expectations
Adaptive Expectations may lead to Chaos
6
8
10
12
14
16
0 10 20 30 40 50
pt
t
5
6
7
8
9
10
11
0 20 40 60 80 100
pt
t
5
6
7
8
9
10
0 20 40 60 80 100
pt
t
w = 1 w = 0.5 w = 0.3weighted average of nonlinear monotonic D/S
curves
leads to non-monotonic chaotic map
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Adaptive expectations
Adaptive Expectations lead to Chaotic Forecast Errors
6
8
10
12
14
16
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
pt
w
-10
-5
0
5
10
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
pt-pte
w
-15
-10
-5
0
5
10
15
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
pt-pte
w
chaotic prices chaotic errors noisy errors
In a nonlinear world, adaptive expectations may lead to(small)
chaotic forecasting errors
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Adaptive expectations
Non-monotonic chaotic map for monotonic D and S
C.H. Hommes / Journal o~Eco~omic Behavior and Organization 24
(1994) 315-335 327
x Fig. 4. Graphs of the map f, for different values of the
expectations weight factor w, with a=0.8, b=0.25, and 1=4. For w
close to 0 fw has a globally stable equilibrium. For w close to 1
f, has a stable period 2 cycle. For w close to 0.5 the map f, is
chaotic.
the parameter a, in~niteIy many period doubling bifurcations
occur as w is increased from 0 to w2 and infinitely many period
halving bifurcations occur, as w is increased from wa to 1. The
parameter a has to be chosen in such a way that the supply and
demand curves intersect at some ‘suitable’ point between the steep
and the flat part of the S-shaped supply curve.
4.5. Geometric explanation of the occurrence of chaos
This subsection presents a geometric explanation how the
combination of adaptive expectations and nonlinear, monotonic
supply and demand curves can lead to erratic price fluctuations.
First consider the case of a linear demand and an S-shaped supply
curve. To stress the dependence on w, we write f, for the map
fn,b,w,l in (12). Fig. 4 shows the graphs of f,, for different
values of w. With the parameters a, b and 1 as in subsection 4.4,
the maps f, satisfy the following properties:
WI
(W2)
(W3)
For 0~ WC l/17 the map f, is increasing and has a globally
stable fixed point. For l/17 < w < 1, f, is non-monotonic and
has two critical points. For w= 1, f, is decreasing and has a
stable period 2 orbit. For w close to 1, wf 1, f, is non-monotonic
and has a stable period 2 orbit. All maps f, have the same fixed
point x_ The graph of fw lies
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Adaptive expectations
Adaptive Expectations in a Nonlinear World
adaptive expectations is stabilizing in the sense that it
reduces theamplitude of price fluctuations and forecast errorssmall
amplitude chaotic price fluctuations may arise around theunstable
steady stateforecast errors may be chaotic, highly irregular, with
little systematicstructurein a nonlinear world, adaptive
expectations may be abehaviorally rational strategy for boundedly
rational agents
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Adaptive expectations
Ken Arrow on Heterogeneous Expectations
“One of the things that microeconomics teaches you is that
individuals arenot alike. There is heterogeneity, and probably the
most importantheterogeneity here is heterogeneity of expectations.
If we didn’t haveheterogeneity, there would be no trade. But
developing an analytic modelwith heterogeneous agents is
difficult.”
(Ken Arrow, In: D. Colander, R.P.F. Holt and J. Barkley Rosser
(eds.),The Changing Face of Economics. Conversations with Cutting
EdgeEconomists. The University of Michigan Press, Ann Arbor, 2004,
p. 301.)
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Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)
Adaptive expectations
Cobweb Model with Homogeneous Expectations
Demand: D(pt) = a − dpt(+�t), a ∈ R, d ≥ 0.Supply: Sλ(pet ) =
spet , s > 0.Market clearing: D(pt) = Sλ(pet ).Expectations: pet
= H(pt−1, ..., pt−L).
Price dynamics: pt = D−1Sλ(H(pt−1, ..., pt−L)).Note: linear
supply curve derived from profit maximization withquadratic cost
function c(q) = q2/(2s).
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Rational
versus naive
Cobweb Model with Heterogeneous Beliefs I.
Market clearing:
a − dpt = n1tspe1t + n2tspe2t (+�t),
where n1t and n2t = 1− n1t are fractions of the two types.
Forecasting rules:1 rational: pe1t = pt ,2 naive: pe2t =
pt−1.
Information gathering costs: rational - C > 0; naive -
free.
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Rational
versus naive
Cobweb Model with Heterogeneous Beliefs II.
Market clearing becomes:
a − dpt = n1tspt + n2tspt−1 (+�t)
Price dynamics:pt =
a − n2tspt−1d + n1ts
.
How do fractions change over time?
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Evolutionary or Reinforcement Learning
Agents can choose between different types of forecasting
rules.
Sophisticated rules may come at information gathering costs C
> 0(Simon, 1957), simple rules are freely available.
Agents evaluate the net past performance of all rules, and tend
tofollow rules that have performed better in the recent past.
Evolutionary fitness measure ≡ past realized net profits.
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Discrete Choice Model
Fitness Measure: random utility
Ũht = Uht + �iht ,
Uht : deterministic part of fitness measure,�iht : idiosyncratic
noise, IID, extreme value distr.
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Fractions of Belief Types
Discrete choice or multi-nomial logit model:
nht = eβUh,t−1 / Zt−1,
where Zt−1 =∑
eβUh,t−1 is a normalization factor.
β is the intensity of choice, inversely related to SD
idiosyncraticnoise: β ∼ 1/σ.
β = 0: all types equal weight (random choice).β =∞:
“neoclassical limit”, i.e. all agents choose best predictor.
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Fitness Measure
Evolutionary Fitness Measure:weighted average of past realized
net profits
Uht = πht + wUh,t−1
πht net realized profit (minus costs) strategy h.
w measures memory strengthw = 1: infinite memory; fitness ≡
accumulated profits,w = 0: memory one lag; fitness most recently
realized net profit.
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Fitness Measure Profits
Profits of type h:
πht = ptspeht −(speht)22s .
Profits of rational agents:
π1t =s2p
2t − C
Profits of naive agents:
π2t =s2pt−1(2pt − pt−1)
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Profit difference
Difference in profits:
π1t − π2t =s2(pt − pt−1)
2 − C .
difference in fractions:
mt+1 = n1,t+1 − n2,t+1 = Tanh(β2 [π1t − π2t ])
= Tanh(β2 [s2(pt − pt−1)
2 − C ])
When the costs for rational expectations outweigh the
forecastingerrors of naive expectations, more agents will buy the
RE forecast.
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
2-D dynamic system
Pricing equation :
pt =a − n2tspt−1d + n1ts
=2a − (1−mt)spt−12d + (1+mt)s
.
Evolutionary selection
mt+1 = Tanh(β
2 [s2(pt − pt−1)
2 − C ]).
Note Timing:1 Old fractions determine market prices.2 Realized
market prices determine new fractions.
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
2-D dynamic system in deviations
deviation from RE fundamental price:
xt = pt − p∗
xt =−(1−mt)sxt−12d + (1+mt)s
mt+1 = Tanh(β
2 [s2(xt − xt−1)
2 − C ]).
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Properties of the 2-D Dynamics
If all agents are rational, then pt ≡ p∗ = a/(d + s).
If all agents are naive, then pt = a−spt−1d .
Unique steady state E = (p∗,m∗), m∗ = Tanh(−βC/2).
If pt−1 = p∗, then pt = p∗ and mt = m∗ :stable manifold contains
vertical line through steady state.
If s/d < 1, then globally stable steady state
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
"Neo-classical" limit case: β =∞
If s/d > 1, C > 0 and β = +∞ then
Steady state is locally unstable saddle point.Steady state is
globally stable.
Important note: homoclinic orbits!!
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Chaotic DynamicsIrregular switching between cheap destabilizing
free riding andcostly sophisticated stabilizing predictor
(a)
1
0.5
0 −0.5
−1
0 20 40 60 80 100
(b) 1
0.8
0.6
0.4
0.2
0 0 20 40 60 80 100
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Rational Route to Randomness
If s/d > 1 and C > 0, then
0 ≤ β < β∗1 : stable steady stateβ = β∗1 : period doubling
bifurcationβ∗1 ≤ β < β∗2 : stable 2-cycleβ = β∗2 : secondary
period doubling bifurcationsβ∗2 < β
∗ < β∗3 : two co-existing stable 4-cyclesβ > β∗3 :
complicated chaotic dynamics, strange attractors
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Rational versus naive: Rational Route to Randomness
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Rational versus naive: time series + attractors
0 20 40 60 80 100
−1
−0.5
0
0.5
1
(a)
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1(b)
−1 −0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
(c)
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Rational versus naive: unstable manifolds
�1.5 �1 �0.5 0 0.5 1 1.5 2pt
�1
�0.5
0
0.5
1
mt�
�e� � � 6
�1.5 �1 �0.5 0 0.5 1 1.5 2pt
�1
�0.5
0
0.5
1
mt�
�f� � � 10
�1.5 �1 �0.5 0 0.5 1 1.5 2pt
�1
�0.5
0
0.5
1
mt�
�c� � � 4
�1.5 �1 �0.5 0 0.5 1 1.5 2pt
�1
�0.5
0
0.5
1
mt�
�d� � � 5
�1.5 �1 �0.5 0 0.5 1 1.5 2
pt
�1
�0.5
0
0.5
1
mt�
�a� � � 2
�1.5 �1 �0.5 0 0.5 1 1.5 2
pt
�1
�0.5
0
0.5
1
mt�
�b� � � 3
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Rational versus naive: two co-existing stable 4-cycles
0 50 100 150 200−1.3
−1
−0.7
−0.4
−0.1
0.2
0.5
0.8
1.1
t
x t
0 50 100 150 200−1.3
−1
−0.7
−0.4
−0.1
0.2
0.5
0.8
1.1
t
x t
0 50 100 150 200−1.3
−1
−0.7
−0.4
−0.1
0.2
0.5
0.8
1.1
t
x t
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Evolutinoary
Selection and Reinforcement Learning
Basins of Attraction of two coexisting stable 4-cyclesfractal
basin boundaries
0
0.2
0.4
0.6
0.8
1
-1.5 -1 -0.5 0 0.5 1 1.5
n1
x
0
0.2
0.4
0.6
0.8
1
-1.5 -1 -0.5 0 0.5 1 1.5
n1
x
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Cobweb Model with Heterogeneous Beliefs (Chapter 5)
Fundamentalists versus naive
Discrete choice model, with asynchronous updating
nht = (1− δ)eβUh,t−1Zt−1
+ δnh,t−1,
where Zt−1 =∑
eβUh,t−1 is normalization factor,Uh,t−1 past strategy
performance, e.g. (weighted average) past profits
δ is probability of not updatingβ is the intensity of choice.β =
0: all types equal weight (in long run)β =∞: fraction 1− δ switches
to best predictor
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Cobweb Model with Heterogeneous Beliefs (Chapter 5)
Fundamentalists versus naive
Cobweb Model with Heterogeneous Beliefsmarket clearing a − dpt =
n1tspe1t + n2tspe2t(+�t)
n1t and n2t = 1− n1t fractions of two typesforecasting
rules:rational/fundamentalists/contrarians/SAC-learning at cost C
> 0versus free naive
pe1t = pt rational
= p∗ fundamentalist
= p∗ + β(pt−1 − p∗) contrarian,−1 < β < 0
= αt−1 + βt−1(pt−1 − αt−1) SAC-learning
pe2t = pt−1 naive
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Cobweb Model with Heterogeneous Beliefs (Chapter 5)
Fundamentalists versus naive
Fundamentalists versus naive
(xt , n1t) phase space price deviations fraction
fundamentalists
sample average sample autocorrelation
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Cobweb Model with Heterogeneous Beliefs (Chapter 5)
Fundamentalists versus naive
Fundamentalists versus naive (continued)
chaotic price fluctuations (when intensity of choice
large)sample average of prices close to fundamental pricestrong
negative first order autocorrelation in prices (βt → −0.85)
Question: will boundedly rational agents detect negative
AC?Replace fundamentalists by contrarians
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Contrarians
versus naive
Contrarians versus naive
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Contrarians
versus naive
Contrarians versus naive (continued)
chaotic price fluctuations (when intensity of choice
large)sample average of prices close to fundamental priceless
strong negative first order autocorrelation in prices (βt →
−0.57,with β = −0.85)
Question: can boundedly rational agents learn the correct
negative AC?Replace contrarians by SAC-learning
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Cobweb Model with Heterogeneous Beliefs (Chapter 5) Contrarians
versus naive
Contrarians versus naive: homoclinic intersections
−0.8 −0.4 0 0.4 0.8
0.2
0.3
0.4
0.5
−0.8 −0.4 0 0.4 0.8
0.2
0.3
0.4
0.5
−0.8 −0.4 0 0.4 0.80.15
0.25
0.35
0.45
0.55
−1.5 −1 −0.5 0 0.5 1 1.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 43
/ 47
-
Cobweb Model with Heterogeneous Beliefs (Chapter 5) SAC learning
versus naive
Behavioral Sample Auto-Correlation (SAC) LearningHommes and
Sorger, 1998
simple AR1 forecasting rule
pet = αt−1 + βt−1(pt−1 − αt−1)
sample average after t periods:
αt−1 =1t
t−1∑i=0
pi , t ≥ 2
the sample autocorrelation coefficient at the first lag, after t
periods:
βt−1 =
∑t−2i=0 (pi − αt−1)(pi+1 − αt−1)∑t−1
i=0 (pi − αt−1)2, t ≥ 2
Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 44
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-
Cobweb Model with Heterogeneous Beliefs (Chapter 5) SAC learning
versus naive
SAC-learning versus naive
agents learn to be contrarians, with first order AC βt →
−0.62part of the (linear) structure has been “arbitraged away”
fundamentalists: correct sample averagecontrarians: correct SAV
+ SAC
Note: adding more rules removes autocorrelations as in
experiments
Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 45
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-
Cobweb Model with Heterogeneous Beliefs (Chapter 5) SAC learning
versus naive
Summary Nonlinear Cobweb Model
in nonlinear cobweb model with monotonic demand and
supply,simple expectation rules may generate chaos in prices and
errors;simple rules in a nonlinear world may be behaviorally
rationalheterogeneous expectations driven by recent performance may
leadto homoclinic bifurcations and chaossimple rules survive
evolutionary competition, especially whenmore sophisticated rules
are costly
Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 46
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-
Cobweb Model with Heterogeneous Beliefs (Chapter 5) SAC learning
versus naive
Questions?
Read the bookor ask them now!!
Behavioral Rationality and Heterogeneous Expectations
in Complex Economic Systems
Cars Hommes
“Nosto ea facin ulput veros del utem zzrit duisseq uamet, si
esse vent am et euis
nonsectet ercidunt prat, consecte min eleniam zzrit essecte
feugue vel iusto odo
coreetu eraesequat ulla feui ea feuguer cillam zzrilla ad
doluptat ad te facillamet,
quat do consenim exer ipsummolore delis nulluptat. Lutat.
Feugait ulla cor
sequam, sequisl ullamcore feu feugiamet, velit aliqui blaorer
ostrud dit non ut at
ex et lum eugiate volore faccum nim estie velit dolore
magniscinit alit lum ex et,
quat.”
SOMEBODY, somewhere
“Nosto ea facin ulput veros del utem zzrit duisseq uamet, si
esse vent am et euis
nonsectet ercidunt prat, consecte min eleniam zzrit essecte
feugue vel iusto odo
coreetu eraesequat ulla feui ea feuguer cillam zzrilla ad
doluptat ad te facillamet,
quat do consenim exer ipsummolore delis nulluptat. Lutat.
Feugait ulla cor
sequam, sequisl ullamcore feu feugiamet, velit aliqui blaorer
ostrud dit non ut at
ex et lum eugiate volore faccum nim estie velit dolore
magniscinit alit lum ex et,
quat.”
SOMEBODY, somewhere
“Nosto ea facin ulput veros del utem zzrit duisseq uamet, si
esse vent am et euis
nonsectet ercidunt prat, consecte min eleniam zzrit essecte
feugue vel iusto odo
coreetu eraesequat ulla feui ea feuguer cillam zzrilla ad
doluptat ad te facillamet,
quat do consenim exer ipsummolore delis nulluptat. Lutat.
Feugait ulla cor
sequam, sequisl ullamcore feu feugiamet, velit aliqui blaorer
ostrud dit non ut at
ex et lum eugiate volore faccum nim estie velit dolore
magniscinit alit lum ex et,
quat.”
SOMEBODY, somewhere
Cover designed by Hart McLeod Ltd
Behavioral Rationality and Heterogeneous Expectations
in Complex Econom
ic Systems
Hom
mes
9781
1070
1929
4 H
OM
ME
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BE
HA
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L R
ATI
ON
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TY
AN
D H
ETE
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C M
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Thanks!
Cars Hommes (CeNDEF, UvA) Complex Systems CEF 2013, Vancouver 47
/ 47
Nonlinear Cobweb model with Homogeneous Beliefs (Chapter 4)Naive
expectationsRational expectationsAdaptive expectations
Cobweb Model with Heterogeneous Beliefs (Chapter 5)Rational
versus naiveEvolutinoary Selection and Reinforcement
LearningFundamentalists versus naiveContrarians versus naiveSAC
learning versus naive