GSDP ABM Workshop 2011 Alan Kirman

7/30/2019 GSDP ABM Workshop 2011 Alan Kirman

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The Crisis in EconomicTheory

Alan Kirman

GREQAM, Universit PaulCzanne, EHESS,

Marseillepresentation at ABM conference

Paris sept. 9th 2011


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presentation at ABM conferenceParis sept. 9th 2011


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Todays Crisis

We have been faced with a virtual collapse of theworlds financial system which has had direconsequences for the real economy.

The system has just gone through another paroxysm

Most of the explanations given involve networks of

banks, trust and contagion at all levels These are not features of, nor characteristic of,

economic models

But they are typical of complex systems

Such systems can undergo sudden major changes



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A shift in approach in otherdisciplines

There has been a shift away from what Bob Maycalled the comfortable consensus in ecology forexample.

The standard view was that large systems in naturewere intrinsically stable if man did not interfere withthem

Furthermore the state of these systems was theresult of a long evolution towards optimality.

Both of these views have been challenged in otherdisciplines.



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Should this cause us to

fundamentally rethink ourtheory?



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The structure of this talk

1. Who is responsible for the crisis?

2. How sound is our basic theory?

3. General Equilibrium

4. The efficient markets hypothesis

5. Alternative approaches ABM models6. Two models

7. Fluctuating asset prices

8. Contagious information elimination

9. Conclusions



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Confidence in our theory

The central problem of depression-prevention has been solved, , Robert Lucas2003 presidential address to the AmericanEconomic Association.

In 2004, Ben Bernanke, chairman of theFederal Reserve Board, celebrated the Great Moderation in economicperformance over the previous two decades,which he attributed in part to improvedeconomic policy making.



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presentati on at ABM conference Paris

sept. 9th 2011

The responsibility ofscientists

This is a longstandingdebate with whichphysicists are familiar

It was brought intoparticular prominenceby the development ofnuclear weapons.

But what abouteconomists?


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Which side should we comedown on?

My basic claim is that we have been buildingunsound models which were the basis for manypolicies and practices.

This was not simply harmless academic research

Too many people developed and acted according to

a world view which was unjustified What are now referred to as excesses are an

intrinsic part of the economic system.

We were not guilty of not forecasting the onset of thecrisis but we were guilty of building models in which itcould not happen.



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Bourbaki

Why do applications [of mathematics] ever succeed?Why is a certain amount of logical reasoningoccasionally helpful in practical life? Why have someof the most intricate theories in mathematics becomean indispensable tool to the modern physicist, to theengineer, and to the manufacturer of atom-bombs?Fortunately for us, the mathematician does not feelcalled upon to answer such questions.

(Bourbaki Journal of Symbolic Logic 1949, 2)

Presentation au congres del'AFSE, Paris Septembre 8 2011


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The Governor of the EuropeanCentral Bank

When the crisis came, the serious limitations of existingeconomic and financial models immediately becameapparent. Arbitrage broke down in many market segments,as markets froze and market participants were gripped bypanic. Macro models failed to predict the crisis and seemed

incapable of explaining what was happening to theeconomy in a convincing manner. As a policy-makerduring the crisis, I found the available models of limitedhelp. In fact, I would go further: in the face of the crisis,we felt abandoned by conventional tools. In the absence ofclear guidance from existing analytical frameworks,policy-makers had to place particular reliance on ourexperience. Judgement and experience inevitably played akey role. Trichet (2010)presentation at ABM conference



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The View of those responsiblein the U.K.

But there is also a strong belief, which I share, thatbad or rather over-simplistic and overconfidenteconomics helped create the crisis. There was adominant conventional wisdom that markets werealways rational and self-equilibrating, that marketcompletion by itself could ensure economic efficiencyand stability, and that financial innovation andincreased trading activity were therefore axiomaticallybeneficial.

Adair Turner, Head of the U.K. Financial ServicesAuthority presentation at ABM conference



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An economic model is notscientific if it does nothaveSound Micro-

foundations By this we mean that we have a model based on the

rational optimising behaviour of the individuals in themarket or economy.This has been widely criticisedfrom Simon onwards.

In standard market models and in particular in macromodels we characterise aggregate behaviour asresulting from such an individual model.

This is at the heart of the General Equibrium Model

Yet much structure is lost under aggregation so thisis not legitimate theory.



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The scientific approach

There is something fascinating aboutscience. One gets such wholesalereturns of conjecture out of such a

trifling investment of fact

Mark Twain, Life on the Mississippi(1883)



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Rationality

Why are we economists so attached to our rationalindividuals?

Mathematical convenience or economic plausibility?

The assumptions are not testable they come from

introspection. (Pareto, Koopmans, Hicks..) They do not allow for development of preferences

over time

They do not allow for the influence of others



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The Easy Way Out

Macroeconomists make the assumption thatthe aggregate economy or market acts like anindividual.

They use the representative agent

This removes the problems raised by SMD

since an economy with one agent has aunique and stable equilibrium

But is this legitimate?

ABM explicitly remove this assumptionpresentation at ABM conference



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presentati on at ABM conference Paris

sept. 9th 2011

Explaining economic

phenomena Everyone wants to know how the economy can

suddenly go into a downturn like the current crisis.

Do economists build models which can explain this ordo they offer ad hoc explanations without reallyquestioning their models, (DSGE for example)?

In my view, we start with the wrong basis, we startfrom the isolated individual and build up to theaggregate without looking at the most importantfeature: the economy as a system of interactingagents.

I believe, that we should view the economy as a complex adaptive system and that a very usefulway of modelling such systems is provided by ABM.


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Correspondence with BobSolow April 1988

My view of the way economists actually do behavecoincides with yours , and most especially aboutmacroeconomists. I have become a sort of commonscold on this subject.

I wholeheartedly agree with the point that economics

self-destructs in part because we insist on supposingthat everywhere and always individuals maximizepurely individualistic preferences subject only totechnological, legal, and budget constraints.



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Correspondence continued

It is a transparently false assumption,and the brotherhood expends vastingenuity trying to account for facts

within that silly framework. There are at least two of us.

Robert M Solow



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The result of theinsistence on

scientific foundations

Modern macro-economists have built more andmore abstract and mathematically sophisticated

models (Dynamic Stochastic General EquilibriumModels) but continue to base these on the samefoundations.

These models do not contain the possibility of acrisis

They bear no perceptible relation to reality.



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Bob Solows View today

Maybe there is in human nature a deep-seated perverse pleasure in adopting and

defending a wholly counterintuitivedoctrine that leaves the uninitiated peasantwondering what planet he or she is on.

Robert M Solow 2009



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Our basic assumptions Trichetagain

First, we have to think about how to characterise the homoeconomicus at the heart of any model. The atomistic,optimising agents underlying existing models do notcapture behaviour during a crisis period. We need to dealbetter with heterogeneity across agents and the interaction

among those heterogeneous agents. We need to entertainalternative motivations for economic choices. Behaviouraleconomics draws onpsychology to explain decisions madein crisis circumstances. Agent-based modelling dispenseswith the optimisation assumption and allows for morecomplex interactionsbetween agents. Such approaches areworthy of our attention.



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Are there specific actors in the

economy who areresponsible?

In an avalanche no single snowflake

feels itself responsible

Voltaire



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Isaac Newton

I can calculate the motion of heavenlybodies, but not the madness of people



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Assumptions on Individuals

If we have agents who are different we canmake weaker assumptions on theirbehaviour, in particular on their preferences

and choices.

What looks at the aggregate level like thebehaviour of a very sophisticated agent may

be constructed from the aggregation ofsimple individuals, (Forni and Lippi).

We can use this insight in building ABM.



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Les abeilles



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Coordination v. Efficiency

Efficiency is the major concern of economists

We focus on efficient allocations of resources,yet perhaps the problem of coordination is themost important

How do collective outcomes emerge from theinteraction between individuals each of whomhas only a local vision of the situation?

If we think of the economic system as selforganising, will this result in a stable equilibrium ?



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Different dynamics

In the face of repeated, if infrequent crises should wenot question our models where sudden changes areattributed to exogenous shocks? Rather than trying toreturn to our basic assumptions perhaps we shouldrethink the whole structure.

Ben Bernanke The brief market plunge was just anexample of how complex and chaotic, in a formalsense, these systems have become Whathappened in the stock market is just a little exampleof how things can cascade, or how technology caninteract with market panic

Interview with the IHT May 17th 2010



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A Remark

We spent the twentieth centuryperfecting a model based on nineteenthcentury physics

Maybe in the twenty first century we canmake more use of twentieth centuryphysics



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Self Organisation

This idea that markets self organise wasespoused by Hayek

This has been used as a justification for notinterfering with markets.

Markets do clearly self organise but we haveno reason to believe that this is a stableprocess.

As the actors within them modify their rulesnew norms appear and these can gently leadthe system to major phase transitions.



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The weakness of ourfoundations

All the economists here are awareof the difficulties with GeneralEquilibrium Models, highlighted bySonnenschein, Mantel andDebreu

But financial economics is built onequally shaky foundations.



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A Link between financial and

real models: RationalExpectations

Originally a way of closing our modelsbut has become an article of faith

As several econometricians havepointed out it would be unreasonable forindividuals to have rationalexpectations if the underlying DGPhas structural breaks for example.



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A more modest view

Ben Bernanke again,

I just think it is not realistic to think that humanbeings can fully anticipate all possible interactionsand complex developments. The best approach fordealing with this uncertainty is to make sure that the

system is fundamentally resilient and that we have asmany fail-safes and back-up arrangements aspossible

Interview with the IHT May 17th 2010



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An Important Example:Financial Market Models

Models of financial markets share thesame basic building blocks.

Agents have a way of forecasting thefuture prices.

This determines how much the agentswish to buy and this in turn determinesthe price of the assets .

The prices will influence the forecasts.



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The Efficient MarketsHypothesis

This is very simple

All relevant information is contained inprices therefore there is no need to look

anywhere else: paradox This basic argument comes from the

work of Bachelier but his thesis advisersaid



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Un avertissement

Quand des hommes sont rapprochs, ils ne sedcident plus au hasard et indpendamment les unsdes autres ; ils ragissent les uns sur les autres. Descauses multiples entrent en action, et elles troublentles hommes, les entranent droite et gauche,mais il y a une chose qu'elles ne peuvent dtruire, cesont leurs habitudes de moutons de Panurge. Et c'estcela qui se conserve

Henri Poincar Report on Bacheliers thesis 1900



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But there were other clearwarnings

From the outset Poincar and others arguedthat the underlying Gaussian assumption wasflawed. The empirical evidence showed this

Keynes questioned Bacheliers assumptions,

Mandelbrot spent most of his life arguingagainst the efficient markets hypothesis

Yet, Markowitz developed his optimalportfolio theory on this basis

Worse, Black-Scholes is based on the sameassumption presentation at ABM conference



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The crucial role of information

Underlying the faith in the capacity of marketsto self organise is the efficient marketshypothesis But as Greenspan observed,

The whole intellectual edifice collapsedin the summer of last year

Alan Greenspan, testimony to House ofRepresentatives Committee on GovernmentOversight and Reform, October 23rd 2008



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Mencken cited by Krugman

H. L. Mencken: There is always an

easy solution to every human problem neat, plausible and wrong.



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Why then did we persist?

Because if we drop the Gaussianassumption we can no longer use thecentral limit theorem and we lose thefinite variance property

So we continued to look where therewas light

But Fama (1965) himself, pointed outthat diversification without thehypothesis is not justified!



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Inertia

The finance profession like theeconomics profession exhibited anenormous amount of inertia

Persist with a model you know how toanalyse even if it does not correspondto anything you might observe

In the economics case, even if majorcrises are not possible in the model.



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No Panic!



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Where does the efficientmarkets hypothesis go wrong?

Remember Poincars warning

Individuals do not only look at their owninformation they also observe theactions of others and infer informationfrom those actions.



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Where does the efficient marketshypothesis go wrong ?

The assumption is strongly related to that of RationalExpectations that is, individuals have a correct view of thedistribution of probabilities of futures states of the world.

As Trichet (2010) again said we may need to consider a richercharacterisation of expectation formation. Rational expectationstheory has brought macroeconomic analysis a long way over the

past four decades. But there is a clear need to re-examine thisassumption.

Again recent work has shown that once individuals are aware ofwhat other individuals forecast they tend to converge on acommon forecast which is not as good as the median forecast(the wisdom of crowds) but in which they feel much more

confident. (Lorenz et al. 2011)



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Where does the efficientmarkets hypothesis go wrong?

In a world with structural breaks in the underlyingstochastic process the RE hypothesis is unjustified.

As Hendry and Mizon (2010) point out

The mathematical derivations of dynamic stochastic

general equilibrium (DSGE) models and newKeynesian Phillips curves (NKPCs), both of whichincorporate rational expectations, fail to recognizethat when there are unanticipated changes,conditional expectations are neither unbiased norminimum mean-squared error (MMSE) predictors,and that better predictors can be provided by robustdevices presentation at ABM conference



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Looking into the sky quickly gets passers-by to follow.



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Informational Cascades 1

Here rational individuals, by theirinteraction, achieve an inefficient result

The restaurant example

Individuals have two signals about thequality of two restaurants A and B.

The private signal is 90% reliable andthe public signal is 55% reliable



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Informational Cascades 2

Suppose A is objectively better

The public signal says B is better

90% of the private signals say A is

better Everyone may wind up in B.

Collective influence eliminates privateinformation

Contradiction with efficient marketshypothesis



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What is the problem with the EfficientMarkets Hypothesis empirically?

What we have to explain is sudden

large movements without the arrival ofan exogenous shock or piece of news.



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Where did the switch comefrom?

Derive a more complicated stochasticprocess

Put it down to an exogenous shock, but

then you must be able to identify theshock

Find a micro model of interacting agentswhich generates this sort of shift



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Ants

Ants learn in an environment of whichthey have only very limited and local

knowledge. Yet they produce quite sophisticated

aggregate behavioiur.



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Ants learn to find the

route to food

Ants communicate witheach other

either through apheromone trail

or by tandemrecruiting.


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Ants learn to find the

route to food Ants communicate with

each other




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Ants learn to find asource of food

Ants communicate witheach other




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Acknowledgements for theants

Guy Theraulaz, CNRS, CRCA,Toulouse (for the vidos)More information at:

http://cognition.ups-tlse.fr/dynactom/dynactom.html

Jacques Gautrais, CNRS, CRCA,Toulouse (for the simulations)



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How should we model thislearning behaviour?

Think of the number of ants taking apath at time tas kt and suppose that

one ant meets another and is recruitedto the path of the other with probability

(1-and changes it path withprobability



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The recruiting process



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Limit distributions



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How might we use this idea tomodel financial markets?

Think of several types of gurus or forecastingrules

Two common examples are:

Fundamentalists who believe that prices will comeback to some fundamental level

Chartists who extrapolate from previous prices.

But one could add others or mixtures.

Success of one rule reinforces the recruitment to thatrule.



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Models in this spirit

With Hans Foellmer and Ulrich Horst,we have builtmodels of financial markets to help understand wherethese sudden changes come from

These models incorporate the idea that people followthe behaviour of others particularly when that

behaviour is successful The behaviour is not irrational. Horizons.

These models capture the contagion effects

There is structure in financial time series but noconvergence to equilibrium in the standard sense.



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A Microstructure Model for Financial Markets

Temporary equilibrium model for stock price dynamics.

Heterogeneous agents: fundamentalists and chartists.

Agents follow the recommendations of financial gurus.

Propensities to follow individual gurus depend on the gurusPerformances reinforcing learning effect.

Stock prices are driven by the fluctuations in the gurus marketshares and aggregate liquidity demand feedback effects.

Spontaneous herding generates temporary bubbles and crashes.

Prices temporarily deviate, but inevitably return to fundamentals.

We study a financial market model where temporary bubbles occur,But where the overall behavior of the asset price process is ergodic.



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Stopping the process fromexploding

Bound the probability that an individualcan become a chartist

If we do not do this the process maysimply explode

We do not put arbitrary limits on theprices that can be attained however



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Ergodicity of the PricePerformance Process

Theorem: Under certain regularityconditions on the probabilistic structureof the gurus recommendations the

price performance process is ergodic.

The presence of Chartists is clearlyrevealed by the nature of the limitdistribution.



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A New Idea of Equilibrium

The distribution of the time averages of prices converges.

If the probability of becoming a chartist is not too high.



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Systemic Risk and the Role ofthe financial network

As Haldane has pointed out thestructure of the financial network, thelinks between countries or financial

institution can play a major role inundermining the stability of the system.

Increased connectivity is not enough toguarantee stability, other features areimportant.



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The Bank of Englands View

When comparting the failure of Lehman bros and the epidemic of bird

flu, Haldane says,

These similarities are no coincidence. Both events were manifestationsof the behaviour under stress of a complex, adaptive network. Complexbecause these networks were a cats-cradle of interconnections,

financial and non-financial.Adaptive because behaviour in thesenetworks was driven by interactions between optimising, but confused,agents. Seizures in the electricity grid, degradation of ecosystems, thespread of epidemics and the disintegration of the financial system each is essentially a different branch of the same network family tree.

Andy Haldane, Director of the Bank of England responsible for financialstability.



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The evolution of theinternational financial

network1985-2005


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The danger signs

1. The scale and interconnectivity of the international financialnetwork has increased significantly over the past two decades.

2. Nodes have increased 14-fold and links have increased 6-fold.

3. The degree distribution has a long-tail. Measures of skew andkurtosis suggest significant asymmetry in the distribution.There is a small number of financial hubs with multiple spokes.

4. The average path length of the international financial networkhas shrunk over the past twenty years. Between the largestnation states, there are fewer than 1.4 degrees of separation.



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Result: Vulnerability

Such systems are vulnerable to thetransmission of problems, particularlythose originating in one of the large

nodes. But nobody planned that the system

should develop in this way, it is theresult of self organisation.



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Why is Greece of concern tothe U.S.?

U.S banks have very limited exposureto the Greek crisis. They hold very fewGreek bonds and are not linked with

Greek banks i.e. do not hold their bonds They are linked with German and

French banks

But the latter have extensive holdings ofGreek assets.



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Derivatives Warren Buffett Many people argue that derivatives reduce systemic problems,

in that participants who cant bear certain risks are able totransfer them to stronger hands. These people believe thatderivatives act to stabilize the economy, facilitate trade, andeliminate bumps for individual participants.On a micro level,

what they say is often true. I believe, however, that the macropicture is dangerous and getting more so. Large amounts of risk,particularly credit risk, have become concentrated in the hands

of relatively few derivatives dealers, who in addition tradeextensively with one other. The troubles of one could quicklyinfect the others.On top of that, these dealers are owed huge

amounts by non-dealer counter-parties. Some of these counter-parties, are linked in ways that could cause them to run into aproblem because of a single event, such as the implosion of the

telecom industry. Linkage, when it suddenly surfaces, cantrigger serious systemic problems.



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Derivatives Warren Buffett

The derivatives genie is now well out of the bottle, and these

instruments will almost certainly multiply in variety and numberuntil some event makes their toxicity clear. Central banks and

governments have so far found no effective way to control, oreven monitor, the risks posed by these contracts. In my view,derivatives are financial weapons of mass destruction, carryingdangers that, while now latent, are potentially lethal.

Warren Buffet 2002



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A Very Simple Example

What we must do is to build models whichcapture the role of the interaction betweenindividuals, their local rationality and theimpact of this on the aggregate evolution of

the market or economy.

The idea of our example is to show how thegradual but rational adoption of rules at theindividual level may lead to radical change atthe aggregate level



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Presentation at the Workshop for ChiefEconomists of Central Banks, Bank of

England May 2010



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Regulating this sort of system

My main argument in this context is that thesort of complex system I have described isintrinsically difficult to control

What we do not have in our model is an

essential feature that as the actors find itprofitable to make loans they will graduallylower their standards

This will, in turn, lead to an increase in p



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Regulating this sort of system If we put in place a set of constraints and

rules today they will have to be continuallyadapted as markets themselves adapt andself organise.

Individuals act rationally given the limited and

local information at their disposal but theymay engender major changes in theaggregate

We cannot simply design from scratch a new regulatory framework and then letthings run.



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How should we stabilise thesystem?

The view that we can set up a new moresophisticated set of rules and then everythingwill be under control is illusory.

It is based on the idea that there is a

correct model and that, and if only wecan find it we can establish the right rules andleave markets to sort things out.

But, in reality there is no reason to believethat self organisation is a stable process andfurthermore the economy is constantlyevolving and and therefore so must the rules.



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Conclusion

In a world where individuals interact witheach other locally and with limitedinformation, the collective behaviour of the

system may undergo sudden and large

changes without any exogenous shock .Asset markets, particularly derivativemarkets, are vulnerable to these. They arenot the result of individual irrationality but ofthe intrinsic fragility of such systems.



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How long will it take?

A new scientific truth does not triumph byconvincing its opponents and making them see thelight, but rather because its opponents eventually die,and a new generation grows up that is familiar withit

Max Planck, A Scientific Autobiography (1949).



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For those who wish to know

more



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Appendix

More details of the Foellmer,Horst Kirman model (2005)



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A Microstructure Model for Financial Markets

Temporary equilibrium model for stock price dynamics.

Heterogeneous agents: fundamentalists and chartists.

Agents follow the recommendations of financial gurus.

Propensities to follow individual gurus depend on the gurusPerformances reinforcing learning effect.

Stock prices are driven by the fluctuations in the gurus marketshares and aggregate liquidity demand feedback effects.

Spontaneous herding generates temporary bubbles and crashes.

Prices temporarily deviate, but inevitably return to fundamentals.

We study a financial market model where temporary bubbles occur,But where the overall behavior of the asset price process is ergodic.



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Specifying Individual Behavior

There is a finite set A of agents trading a single riskyasset.

The demand function of the agent a takes the log-linear form:

etap, : ct

aSt

a logp ta

where Staand t

a denote the agentscurrent referencelevel and liquidity demand, respectively.

The logarithmic equilibrium price St:= logPtis defined

through the market clearing condition of zero totalexcess demand:

St :1ct

cta

a

Sta t

Temporary equilibrium prices are given as a weightedaverage of individual price assessments and liquiditydemand.



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Choosing Individual Assessments

The choice of the reference level is based on therecommendations of some financial experts:

Sta Rt

1,...,Rtm

The fraction of agents following guru i in period tisgiven by

ti :

1

ctcta

aA

1 Sta Rti

The logarithmic equilibrium price for period t+ 1 takesthe form

St ti

i1

m

Rti t

Temporary equilibrium prices are given as a weightedaverage ofrecommendations and liquidity demand.



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The GurusRecommendations

The recommendation of guru i 1,...,m is based ona subjective assessmentFi of some fundamental valueand aprice trend:

Rti : St1

i Fi St1 i St1 St2

The dynamics of stock prices is governed by therecursive relation

St F St1,St2,t 1 t t St1 t

in the random environment t t,t

Unlike in Physics, the environment will be generatedendogenously.The dynamics of stock prices is described by a linearrecursive equation in a random environment of investorsentiment and liquidity demand.



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Fundamentalists

The recommendation of a fundamentalist conveys theidea that prices move closer to the fundamental value:

Rti : St1

iF

i St1 , i 0,1 If only fundamentalists are active on the market

St 1 t St1 t,t , i

i1

m

ti

and prices behave in a mean-reverting manner because

i 0,1 The sequence of temporary price equilibria may be

viewed as an Ornstein-Uhlenbeckprocess in a randomenvironment. Fundamentalists have a stabilizing effecton the dynamics of stock prices.



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Chartists

A chartistbases his prediction of the future evolutionof stock prices on past observations:

Rti : St1

iSt1 St2 ,

i 0,1

If only chartists are active in the market

St St1 t St1 St2 t, t it

i

i1

m

Returnsbehave in a mean-reverting manner, butprices

are highly transient. Chartists have a destabilizing effecton the dynamics of stock prices.



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The Interactive Effects of Chartists andFundamentalists

If both chartists and fundamentalists are active

Prices behave in a stable manner in periods where theimpact of chartists is weak enough.

Prices behave in an unstable manner in periods wherethe impact of chartists becomes too strong.

Temporary bubbles and crashes occur, due to trendchasing.The overall behavior of the price process turns out to beergodic if, on average, the impact of chartists is not toostrong.

St 1 t t St1 t St2 t,t ,



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Performance Measures

How do the agents decide what guru to follow?The agentspropensity to follow an individual guru

depends on the gurussperformance.We associatevirtualprofits with the gurustrading

strategies:Pt

i : Rt1i St1 eSt eSt1

Theperformance of the guru i in period tis given byUt

i : Ut1i Pt

i tj

j0

t

Pji

i.e., by a discounted sum of past profits.The agents adopt the gurusrecommendations withprobabilities related to their current performance.



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Performance Measures

Propensities to follow individual gurus depend onperformances:

t1 ~ Q Ut; where Ut Ut1,...,Ut

mThe better a gurusperformance, the more likely the

agents followshis recommendations.

The more agents follow a gurusrecommendation, thestronger his

impact on the dynamics of stock prices.The stronger a gurusimpact on the dynamics of stock

prices, thebetter his performance.The dependence of individual choices on performancesgenerates aself-reinforcing incentive to follow the currently mostsuccessful guru.



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Performance Measures and Feedback Effects

The dynamics of logarithmic stock prices are describedby a linear stochastic difference equation

St 1 t t St1 t St2 t,t

in a random environment t,t

Aggregate liquidity demand is modelled by anexogenousprocess.

The dynamics of{ t} is generated in an endogenousmanner.

The distribution of tdepends on all the prices up totime t-1.

The dependence of individual choices on performancesgenerates a feedbackfrom past prices into the randomenvironment.



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The Associated Markov Chain

Aggregate liquidity demand follows an iid dynamics.Stock prices are given by the first component of the

Markov chaint St,St1,Ut

The dynamics of the process t can be described by

t1 V t,t :F St,St1,t St

UtP St,St1,t

,t ~Z Ut; .

The map St,St1 P St,St1,t is non-linear.

The dynamics of the price-performance process t can be described by an iterated function system, butstandard methods do not appl .



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Stopping the process from exploding

Bound the probability that an individualcan become a chartist

If we do not do this the process maysimply explode

We do not put arbitrary limits on theprices that can be attained however



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Bounding the Impact of Chartists

We need a mean contraction condition for the priceprocess

St 1 t t St1 t St2 t,t

To this end, webound the impact of trend chasingassuming that

supu

1 u u supu

u 1

where u and u denotes the conditional expectedimpact of fundamentalists and chartists given Ut= u,respectively:

u : t1Ut u and u : t1Ut u

This mean contraction condition can be translated into anassumption on the behavior of an individual agent.



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Existence of Stationary Distributions

Theorem 1: Under our mean-contraction condition, theMarkov chain t is tight, i.e.,

limc

supt

P t c 0

The mean contraction condition prevents stock pricesfrom exploding.

Theorem 2: Under our mean-contraction condition, theMarkov chain

t has a unique stationary distribution_, and

limT

1

Tf t

t1

T

f t d P a.s.i.e., time averages converge to their expected valueunder.

If we bound the impact of trend chasing on stock pricedynamics a unique equili rium exists.



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Ergodicity of the PricePerformance Process

Theorem: Under certain regularityconditions on the probabilistic structureof the gurus recommendations the

price performance process is ergodic.

The presence of Chartists is clearlyrevealed by the nature of the limitdistribution.



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A New Idea of Equilibrium

The distribution of the time averages of prices converges.

If the probability of becoming a chartist is not too high.



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GSDP ABM Workshop 2011 Alan Kirman

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