From Mandelbrot to Chaos in Economic Theory* PHILIPMIROWSKI Universityof Notre Dame Notre Dame, Indiana
An interestin the historyof ideasis goodforthe scientist'ssoul. [49, 21]
I. Can Economic Theorists Admit That History Matters? We are living in the midst of a profoundrupturebetween older andemergentnotions of scientific explanation[67; 68; 24; 25; 51]. The very meanings of order and chaos, the deterministicand the stochastic, are being reconceptualizedin this decade, and it is fair to presumethat things will never be the same. Some economists, understandably,do not want to be left in the lurch, and have turnedtheirprodigiouseffortsto assimilationof some of the new doctrinesand mathematical techniques. But, so far, what this movementlacks is historicalperspective.While it is easy to be swept up in the enthusiasmof the momentfor what is unquestionablyone of the most significant intellectualinnovationsof the century,that still does not absolvethe economistfrom asking some very basic questions:Why should this particularset of ideas or mathematicalformalismsbe wellsuited to economic discourse?What is it aboutthis new mathematicsthatwill necessarilyimprove economics? What accounts for this rushto appropriatenew techniques?How will they change the way economics is done? Despite the appearanceof numeroussurveyarticlesthese questionshave not yet been adequatelyaddressed[5; 11; 33]. The reason these questionslanguishis thatthey are inherentlyhistoricalratherthan narrowly technical. Since most economists cannotbe presumedto have a firmbackgroundin the histories of physics or of economics, and such a backgroundcannotbe providedin the space of a journal article, most of the statementsmade in this section aboutthese historieswill not be documented here, althoughthe readeris directedto some of the author'swritings[56; 57; 58; 61; 62] for the corroboratingevidence. All that can be accomplishedhere is to set the stage for the narrativesin the subsequentsections of this paper, which will documentthe historicalrelationship(or, more to the point, the lack thereof) between Benoit Mandelbrot'swork in economics and the later work of such authorsas J. Grandmont,R. Day, J. Benhabib,W. Brock, J. Scheinkman,W. Barnett, P. Chen and others in attemptingto import the insights of chaos theory into economics. The purpose of this narrativeis to illustratethe major thesis of this paper, namely, that economists *The firstdraftof this paperwas writtenwith the help of a grantfrom the NationalEndowmentfor the Humanities. The authoris extremely grateful to William Barnett, William Brock and Ping Chen for supplyingunpublishedworking papers, and to Benoit Mandelbrotfor spendingsome time answeringwhatmusthave seemed to him to be very naive questions. I have been very fortunateto have received extensive commentson this paperfrom RandallBausor, John Burkett, Claude d'Aspremont, Roy Weintraub,Hal Varian,anonymousrefereesand Bruce McCulloch.
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have not thoughtthroughthe implicationsof the chaos literaturefor theirdisciplinewith sufficient depth and rigor, because they are not yet readyto admitfreely thathistorymattersat all levels of discourse. The intense attractionwhich the chaos literatureexerts upon modem mathematicaleconomics can be readily explained by the history of that discipline. Mathematicaleconomics is essentially co-extensive with the school of neoclassicaleconomictheory;and neoclassical theory was directlycopied from mid-19thcenturyenergy physics [58, chap. 5; 61]. The purposeof that transferof metaphor,which equated "preferences"or "utility"to a field of potentialenergy, was multi-levelled:at the grossest level, imitationof physics was thoughtto rendereconomic discourse intrinsically "scientific"; at another level, constrainedoptimizationover a conservativevector field was thoughtto embody the deterministicideal of all scientificexplanationprevalentin that era; at yet a third level, some believed that energy was the intermediateterm which allowed the reductionof the social to physical law from psychic energy down to mechanicalmotion. Hence mathematicaleconomics, and more specifically its neoclassical incarnation,has a long history of imitation of the physical sciences. What is more importantfor our present concerns is the problemswhich such imitationhas thrownup over the past century. First, while neoclassical economics attemptedto partakeof the ideal of deterministicexplanation,it was nowhere as successful at this endeavoras its exemplar,physics. This weakness can be directly traced to a persistent tergiversationover what, precisely, was conserved in the economic system [58, chaps. 5-7; 62; 64]. Without an analagousconservationprinciple, neoclassical economics was blocked from following physics into the realm of a serious formal dynamics, including the formal structureof Hamiltonians,'and insteadretreatedinto the spurious pseudo-dynamicsof ceteris paribus conditions.This inabilityto emulate the core of the ideal of deterministicexplanationtarnishedthe entireprogramof imitatingphysics. Secondly, the absence of a legitimate dynamics also compromisedthe ideal of a scientific empiricism [57; 59; 60]. What could it mean to attemptto fit neoclassicalrelationsto time series evidence when the fundamentaldeterminantsof neoclassicalequilibriumdisplayedno necessary stability from one moment to the next? Indeed, most prominentfirst- and second-generation neoclassicals were hostile to attemptsto importsuch techniquesas least-squaresestimationinto economics; and the earliest efforts in this area were pioneeredby individualsskeptical of neoclassical theory [59; 60]. Such disputesover the meaningof scientificactivity also compromised the claims of neoclassical theory to have attained"scientific"status. Third, there was the problem that physics continuedto evolve rapidly after the mid-19th century, whereas the neoclassical researchprogramtended to remainmired in its original 19th centuryorientation.In particular,fromJamesClerkMaxwellonwards,physics increasinglybegan to incorporatestochasticideas into physicalexplanations,whereasneoclassicaleconomics did not [66]. This progressiveabandonmentof the ideal of deterministicexplanationhas been summarized felicitously by Ilya Prigogine [67, 187]: In the nineteenthcentury,therewas a profusionof controversy between"energeticists" and "atomists",the formerclaimingthe secondlaw [of thermodynamics] destroysthe mechanical conceptionof the universe,the latterthatthe secondlawshouldbe reconciledwithdynamicsat 1. One may occasionally find an economist working with Hamiltonians;but it is clear that (a) they are not the favoredmode of conceptualizationof dynamics in neoclassicaltheory [75] and (b) there is absolutelyno agreementupon the properinterpretationsof Hamiltoniansdue to the problemsmentionedin [58, chap. 7; 62]. As a parentheticalremark, the favored tactic of resort to Liapunov techniques to discuss dynamics have never been plausibly linked to the core neoclassical tenet of optimizing behavior[6].
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Whatthis means suchas probabalistic the priceof some "additional arguments. assumptions" exactlycannowbe seenmoreclearly.The"price"is notsmallbecauseit involvesa far-reaching of the structure of dynamics. modification Now, it is true that neoclassical economics finally admittedsome aspects of stochastic concepts into its ambit with the rise of "econometrics"(and somewhatfurtherwith the "rational expectations hypothesis") [57; 63]; but the curious aspect of this developmentis that the stochastic terms were merely appendedto the existing constrainedoptimizationformulas, and were not part and parcel of a fundamentalreconceptualizationof economic theory, unlike the parallel events in physics. Moreover,as we shall observebelow, these stochastic"shocks" had little or no theoreticaljustification,but themselves seemed only an excuse to maintainthe pure deterministic ideal of explanationin the face of massive disconfirmingevidence. As one prominentneoclassical put the case: [neoclassicaleconomists]naturallytendedto thinkof modelsin whichthingssettledownto a of initialconditions.Technically uniquepositionindependently speaking,we theoristshopednot to introducehysteresisphenomena intoourmodel,as theBibledoeswhenit says, "Wepassthis wayonlyonce"and,in so saying,takesthesubjectoutof therealmof scienceandintotherealm of genuinehistory[71, 184-5]. Now, into this unsatisfactorysituation, we observe the intrusionof the furtherdevelopment of chaos theory in physics. The allureof this developmentfor neoclassicaleconomists is readily apparent:it seems to outsiders that there is now a "technical"solutionto many of the most irritating and endemic problems of neoclassical theory over the last century.Here (again we stress) it seems that the most rigid determinismis reconciled with the pervasive appearanceof random phenomena.Here again it seems thatthe previouslack of a substantivedynamicsmay be repaired with only a little more sophisticatedmathematics.Here it seems that the formalismof strange attractorsmay promise law-governedbehaviorindependentof historicallocation. Here, it seems there is a remedy for the palpable failure of a half-centuryof econometricendeavor.Further,it merely involves a simple extrapolationof the original tendenciesof neoclassical theory, since it involves furtherdirect imitationof theoriesgeneratedwithin the physics community.In summary, chaos theory looks like it just might be the salvationof the neoclassicalresearchprogram. Contraryto these impressions,nothingcould be furtherfromthe truth.
II. Mandelbrot on the Irrelevance of Modern Econometrics Benoit Mandelbrotwrote a number of stunninglyoriginal papers in economics from roughly 1962-1972; and then, afterwards,he went on to become famouselsewhere. AlthoughGleick [25, 81-118] writes as though Mandelbrot'seconomic ideas had a great impact upon the profession, the simple historical fact is that they have been by and large ignored, with some few exceptions [7; 19; 20; 53; 54] which seem to have been subsequentlyabandonedby those authors. Moreover, while it can be claimed that the work on economics was an importantinput into his later innovationsconcerning "fractals" which did make him famous, one can find no indication of how the connections might be made in the existing literature.Hence we observe the curious fact that economists have conceived of an enthusiasmfor the chaos literaturea decade or more after this literaturemay be said to have its roots in economics, and yet they exhibit no interest in or curiosity about those roots.
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Mandelbrotthe persona is at least as fascinatingas Mandelbrotthe theorist. His primary identity is as a mathematician,althoughon occasion mathematicianshave insisted that what he does is not really mathematics[25, 114]. He himself has said, "Veryoften when I listen to the list of my previousjobs I wonder if I exist. The intersectionof such sets is surely empty" [25, 86]. He is often voluble concerningthe view that he could neverhave accomplishedhis work in any conventionalacademic discipline or department,and thathe was extremelyfortunateto have worked for IBM, because they let him follow his instincts. And, as he admits, "Economics is very far from what I plannedto tackle as a scholar"[1, 214]. There are four special traitsof Mandelbrotand his work which arecentralto an understandof ing the episode from 1962 to 1972. The first is thatMandelbrothas been a perpetualoutsider in almost every intellectualcontext;and especially outsideof the hidebounddistinctionsbetween disciplines in the moder university.There is a clear connectionin this respectbetween his ideas and his life, as he himself has noted: "As I allowedmy self to drift, I soon came to view the normal unpredictabilityof life as contributinglayersor strataof experiencethatare valuable,demand no apology, and add up to a uniquecombination"[1, 208]. No rationalmaximizer,he. Second, in an attituderefreshinglyunorthodoxfor a mathematician,he dismisses Bourbakistformalism, and thinks the fetish for axiomatizationhas largely run its course. "To a student, the reduction to axioms is largely a matterof satisfying the teacher" [1, 216]. Clearly, such a person would not be taken with the work of a KennethArrow or a GerardDebreu. Third, he is a partisanof the revival of a specifically geometric intuitionin mathematics,to the extent of there being a profoundphenomenologicalapproachin all of his work (and not just the economics) [45, 261]. Any researchprogramthatregularlydisplaysa certainmethodologicaldisdainfor visual evidence would certainly find this attitudequaint, at best. And fourth, Mandelbrothas persistentlyridiculed the physics envy of neoclassical economists;and the ironyof much of statisticaltheory first originatingin social theory has not been lost on him. . . . it is not truethatdifferencesin development betweensciencesarerelatedto, andthereof the foreexcusedby, differencesof "age"as measured by theearliestsystematicinvestigation in differenttopics.Indeed,probability saw its first but it had been born theory triumphs physics, choice. . . Evenas late elsewhere,in the studyof problemsraisedby economic-psychological as 1912,statisticalsocialsciencecouldstillbe presented as a modelto be followedby statistical physics[51, 120]. Mandelbrot'swork in physics, geology, economics and meteorologyhave all been part of a largerpartiallyhidden agenda, one which he revealedin his 1964 "PrematureFractalManifesto," only recently published [51]. There he arguedthathis effortsshouldbe regardedin the vanguard of a "second stage of indeterminism,"one thathe suggestedwith his usual modesty, was a broad culturalphenomenon. The first stage of indeterminismwas an attemptto introduceprobability theory into such areas as physics and economics by the partitioningoff of the causal deterministic aspects from the stochastic disturbances,in effect subordinatingthe latterto the former.The key to the plausibilityof this first stage was a resortto the classical centrallimit theorem, which allowed those so inclined to cling to their previousdeterministicworld-views. The second stage would be markedby the explorationof those areaswherethe classicalcentrallimit theoremfailed to hold: these would be in the "less-developedsciences," the intellectualslums of deterministic science: weatherprediction,turbulence,tensile strengthand fractures,and of course, economics. Much of this resonatedwith the "holism vs. reductionism"debatewhich was endemic to social theory, which Mandelbrotthought was merely symptomaticof a larger problem: "in studying economic records, it may very well be preferableto avoidthe temptationto attackperiodsof crisis
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separately"[51, 124]. It didn't look like Kansas, Toto; and it certainlydidn't look like rational expectations. Mandelbrotwas drawn into the whole questionof prices in economics by a request from a co-worker as to whether "filter rules" wouldn't work better as an investmentstrategythan any other approachto stock-marketportfolio behavior.A filterrule calibratedat p% is a device that monitors a share price continuously,records all local maxima and minima, and activates a buy signal at the moment when the price first reaches a local minimumplus p%, and conversely activatesa sell signal when price first reaches a local maximumminusp%. Mandelbrotdecided that filter rules could not dominate any other strategy,and that the problem lay where no neoclassical economist would look for it: price changes were geometricallyconceptualizedas being continuous, generally by attemptingto model them by means of continuousfunctions, but in the real world they were not. "Even on days when price variationseems reasonablycontinuous, its continuityis the result of deliberateaction by a marketspecialist . . . The specialist creates bargains reservedto friends, while most customershave to buy at the next higherprice" [48, 102]. In many instances, even insider tradingdoes not experiencecontinuousprices, which is why it is not 100%foolproof. Recent rule changes such as stop-tradingorders, "circuit-breakers"and the like are simply de facto admissionsof this state of affairs. This set Mandelbrotdown the path of actually looking at time series of prices; and what he saw did not at all accord with the orthodoxeconomic stories. First, as just mentioned, price records were punctuatedby large discontinuouschanges. "The only reason for assuming continuity is that many sciences tend, knowingly or not, to copy the proceduresthatprove successful in Newtonian physics . . . But prices are different:mechanics involves nothing comparable" [49, 335]. Secondly, large changes tended to "bunch"together:this contradictedthe Gaussian smoothing propertywhich would be inherentin existing stochasticmodels of prices. Third, as a corollary of the first two attributes,root mean squaredeviationsdid not seem to stabilize as the record grew longer. All of this implied that prices should not be modelled as analagous to Brownianmotion, as had been (and generally still is) the practicesince Bachelier [15]. But further, price changes did exhibit the previouslyunnoticedattributethat their geometric appearance seemed unchangedby changes in time scale. In particular,for any arbitrarytime lag d, be it daily, weekly, monthly or yearly, log P (t + d) - log P (t) seemed to be distributedindependentlyof d, except for a scale factor. This implied a hyperbolicdistributionof price changes, at least for the tails of the distribution.One observes that Mandelbrotwas congenitallypredisposedto notice all of this, given his "geometric"intuitionand his phenomenologicalapproach. But there was another fortuitous element to Mandelbrot'spreparation.His teacher, Paul Levy, had demonstratedthatthe centrallimit theoremwas muchmore complicatedthanhad been previously suspected [35]. Briefly, he showed that the Gaussianor "normal"distributionwas only one of a family of "stable" distributions;the log of the characteristicfunctionfor the stable Levy (sometimes called "stableParetian"in the 1960s economicliterature)familyof distributions is [31]: logf(t)=
logf
-xoc
ist -
eiUtdP(u < u)= y Itla[1 -
i,3sgn(t)(t,
where 8 is any real number,y > 0, 1/31- 1, o(t, 'a),
tan(a7r/2) -(2/Tr)logIt
if ca = 1 if a = 1
a)]
(1)
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and sgn(t) =
if t >0
0
if t
-1
if t <0.
0
When the parameters a = 2, 8 = , and y = or2/2 (P is a skewness parameter; when P = 0, the
distributionis symmetric), then the resultingcharacteristicfunctionis Gaussian[26, chap. 7; 77]. Levy showed that the only possible limitingdistributionsfor sums of independentidenticallydistributedrandomvariableswas a stable Levy distribution.The conventionalcentrallimit theorem, which is a special case of the above, restrictsthe outcometo normalityby imposingthe condition that each of the constituentrandomvariableshas finite variance. Here, miraculously,a connection fell into place. Mandelbrothad been looking at distributions of price changes that were not Gaussianbecause they had too many outliers, which was referredto in some previousliteratureas the problemof "fat tails" [59]. It just so happenedthat the tails of all non-Gaussianstable Levy distributionsresembledan asymptoticform of "Pareto's Law," namely P(u > u*) -> [u*/V]-a
as u -- oo. Mandelbrot was familiar with the literature in
economics on the Pareto law for incomes, and had himself publishedon that topic [37; 38]. He now began to suspect that such hyperbolicdistributionswere endemic to economic variates, and that fact would have profoundconsequencesfor how one thoughtaboutthe economy. Levy Stable distributiontheory was, however, no piece of cake. For instance, explicit expressions (excluding series expansion approximations)for the density functionsof the family of distributionswere known only in three cases: the Gaussian,the Cauchy(a = 1,/ = 0) and what was sometimes explicitly called the Levy distribution(a = 1/2, 3 = 1, 8 = 0, y = 1). In the other cases, which were precisely the ones Mandelbrotsuspectedwere characteristicof economic variates(where 1 < a < 2), since there existed no analyticaldensity function(given the caveats above) one was preventedfrom makingany statementsaboutthe samplingbehaviorof estimators; nor could one write down an analytic expressionfor an estimator.Further,varianceswere infinite, so any estimatorcould not depend on any momentshigherthanthe first. Mandelbrottook to plottingthe cumulativesample density of {log P (t + d) - logP(t)} in orderto estimatethe magnitude of a by making use of the asymptoticParetolaw resultmentionedabove [40]. Although there was no way of gauging the confidenceintervals,a did seem to consistentlyclock in at less than two, a result seconded by Eugene Fama [18; 20]. Whateverelse one might say about this work, "It can be said without exaggerationthatthe problemof constructingstatisticalestimators of stable laws enteredinto mathematicalstatisticsdue to the workof Mandelbrot"[77, 217]. Any lesser intellect might rest content with this finding, perhapstesting for Levy stable distributionsin a thousanddifferentguises, improvisingsmall variationson this theme and padding the vita; but Mandelbrotwanted to push it further;to see how it connectedwith everythingelse, writ large. "When working in economics, I was similarlydying to be allowed to make it known in my researchpapersthatmy methodswere partof a generalphilosophy,of a certainapproachto irregularityand chaos, and thatthey also matteredin physics. Invariably,the refereesasked me to takethese statementsout" [ 1, 222]. Therewas, for instance, the problemof what the existence of infinitevariancemeant. It did not mean thatthe values of observedprices were infinite;nor did it mean that sample moments of all orderswere not themselvesfinite. All it meantwas that sample variances grew unpredictablyand without bound with increase in sample size. Then there was the problem of the widespreadhabit of enforcinga bound on samplevariancesby automatically
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regardingthe processes as non-stationary,and detrendingor otherwise "pre-whitening"in order to produce a finite variance.2 Here is where Mandelbrot'sphilosophicalposition began to set him apartfrom others. He regardedthis practice as backsliding, letting the old tropismaticpredispositionsfor a partitioning of a deterministic core and a subordinatestochastic explanation[42, 396-397] dominate the inquiry.More than once he suggested that a resortto the hypothesisof non-stationaritywas nonscientific [45, 266] in that it relinquishedthe search for truly general law; at the very least it violated the principle of parsimoniousexplanationin the context of a phenomenologicaldescription. "It is well known thatphotographyis simplest when an object is at an infinitedistance from the camera. Therefore,even if the actualdistanceis knownto be finite, the photographerought to set the distance at infinity if that distanceexceeds some finite threshold"[42, 399]. Increasingly, Mandelbrotwas led to think about what it would mean for somethingto appearnon-stationary, but in fact to have built-in low-frequencydependence. Thus Mandelbrotentered the next phase of his economics research,which he later distinguished as the differencebetween "Noah" and "Joseph"effects;it is my impressionno economist accompaniedhim beyond this point. The Noah effect was the extremenon-Gaussiancharacterof the marginaldistributionof prices; the name was intendedto conjureup the abruptflash floods that came out of nowhere and swept away all in their wake. The Josepheffect was the existence of very long-runtemporaldependence, independentof the natureof the marginaldistribution;the name was intended to refer to the Biblical seven lean years and the seven fat years. The Joseph effect was evident whenever observersthoughtthey saw "cycles" in economic time series which would fail to remain stable as the series lengthened. These two effects were separateand separable, but they did have one connection in Mandelbrot'sway of thinking:time series with long dependence would exhibit a characteristicspectrum,a hyperbolicspectraldensity S'(f) - f-2 with 1 < a < 2. By direct analogy with the previous case, a pure Gaussian spectral density would have a = 2 (thatis, the spectrumwould be flat, "whitenoise");a situationwith a < 2 was dubbed (perhaps unfortunately)"fractionalGaussiannoise". Again, note well that all of these processes are purely stationary. Infinite variance was certainly bad enough; but now the purely general case was mindnumbing. Wereall [economictime]seriesto havean infinitevariance,onemighthavebeenableto save by stableParetian[i.e., Levy]distributions. linearitywhilereplacingall Gaussiandistributions Butwe mustalsoallowforcoordinateserieswithfinitevarianceandanH-spectrum [of fractional Gaussiannoise] . . . No linearmodelI can thinkof allowsfor suchcoexistence,and I have becomeresignedto give up linearityfor the sakeof coexistence. . . [twotime series]can be identicalin the longrunevenif the structures arevastlydifferent,onehavinganH-spectrum and a finite variance, while the other has infinitevarianceand-in effect-has a white spectrum[43, 86, 88].
This was the first time in history that someone had developed a taxonomy for all of the possible cases which would cover the "typical spectral shape" of an economic time series. Yet it was not a taxonomy without rhyme or reason: the great preponderanceof cases encompassed 2. An even more involved (and epicyclic?) proceduretakes the formatof ARCH models, which allow variances of Gaussian disturbancesto themselves vary in an autoregressivescheme. See Engle [17]. A third way of building in epicycles is to treat the variance of the distributionof prices as if it were the result of a draw from a finite-variance distribution.See Mandelbrot'sreactionin [46].
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either infinite variance or infinite intertemporaldependence;Gaussianwhite noise was just the most special of cases. Anticipatingthe narrativein our next section, whereasthe Gaussiancase correspondedto the Euclidean world of integer dimension, the rest of the cases corresponded to fractal-dimensionedprocesses. But these were precisely the sorts of speculations regularly expurgatedby journal editors. While one can repairto the sequence of journalarticlesfrom 1962 to 1973 in orderto survey the technical issues, there is no text where Mandelbrotactuallydrew out all the implicationsof his programfor neoclassical economic theoryand for orthodoxeconometrics.Indeed, after 1972 one gets the impressionhe just lost interestin economics (or perhapsthe reactionsof neoclassical economists just grew tedious);3and in any event, insights he had gained into the relationships between fractaldimensions and randomprocesses were findingall sorts of fruitfulapplicationsin physics, meteorology and computergraphics. Nevertheless,one can glean from his papers a bill of indictmentsagainst orthodoxeconomics that is more profoundlycoherentthan any other such endeavorin the 20th century. To begin at the beginning: recall that the "marginalistrevolution"of the 1870s was derived from direct imitation of the energy formalismof 19th centuryphysics [56; 58]; the very name itself reveals that value was predicatedupon continuousand reversiblefields of force that gave rise to continuousand reversiblefunctionswith prices as theirmain arguments.The entireproject of smooth substitutionand constrainedmaximizationis compromisedby Mandelbrot'sinitial observation:empirical timeseries of prices are not continuousfunctions. "But prices are different: mechanics involves nothing comparable,and gives no guidance on this account" [49, 335]. In this context, one is not makingreferenceto a few discontinuitieshere and there, like thatdinosaur the "kinkeddemandcurve"; instead, the whole penchantfor differentiablefunctionsis where the programgoes off the rails. In orderto discuss manyof these randomprocesses, Mandelbrotoften finds he must resort to functionswhich are nowheredifferentiable,such as Weierstrassfunctions and Cantordusts. As usual, the neoclassicals were using the wrong mathematicsfor the wrong reasons [55; 61]. The Marshallian"law" of supply and demand is most certainlythe primaryvictim of this reconceptualization.In [56, chap. 4], I have arguedthatthe centralanalyticaldevice of the Marshallian system is the division of all economic phenomenainto differingtime frames, which are then ranked according to their relative inertia. Yet, as Mandelbrotobserves, "It is astonishing that the hypothesis of indepedence of weekly changes can be consistentlycarriedso far, showing no discernablediscontinuitybetween long-termadjustmentsto follow supply and demand, which would be the subjectmatterof economics, and the short-termfluctuations"[42, 406]. The primaryinsight of Mandelbrot'sempiricalwork is that distributionsof prices are approximately scale-invariant:it just doesn't matterfrom the stochasticpoint of view if you look at them minuteby-minuteor year-by-year.That scale invariancesuggeststhatthe Marshalliandistinctionbetween the short-runand long-runis an analyticalmistake. But the Noah and Joseph effects chamfereven deeper thanthat, cuttingto the very heartof the neoclassical world-view. The notion of efficient marketsis bound up with the possibility of arbitrage,that is, a class of rationaltradesthat serve to stabilizethe price by bringingit closer to the "fundamentals":to put it anotherway, perfectarbitrageshould "whiten"the spectraldensity, 3. In conversationsMandelbrothas expressedhis reticencewith regardsto protractedcontroversy.In thatinstance, a co-worker"hadgrown impatientwith my refusalto reopenold fightsthathadbeen won to an acceptabledegree, and for my deliberatepreferencefor seeking soft acceptance, with controversyonly when it is unavoidable,as opposed to hard acceptance, with unforgivingvictims."
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as often assertedby the rationalexpectationsschool. But in the presenceof the Josepheffect, that is, very long dependence in prices, "thereexists indeed a class of importantcases where useful implementationof arbitragingis impossible"[44, 225]. The idea is really quite simple, although it can be expressed as formally as one might wish. Suppose price changes are discontinuousand exhibit long dependence. The arbitrageuris tryingto "get rid" of sharpdiscontinuities(the Noah effect), but since changes are intrinsicallydiscontinuous,he can almost alwaysnever quite "catch up". With anythingless than a perfect infinite anticipatoryhorizon, his actions have systematic consequences far down the line (the Joseph effect) that no one can forsee, and so he ends up increasingthe variance of the overall time series. (Mandelbrotdoesn't mentionit, but Bayesian "learning"schemes are also compromisedin this context, since we have alreadynoted that long dependencecan give the impressionof periodic cycles whichjust aren'tthere.) Variouscombinations of the Noah and Joseph effects (or indeed some marketswherepurewhite noise reigns) will produce differentconsequences for the actions of the arbitrageurs."Those closest to efficiency are of two kinds: some in which anticipatoryhorizonis infinite, and otherswhere 'marketnoise' is so overwhelmingthat predictionis impossible and the assumptionof efficiency cannot be disproved!" [44, 233]. Ironically,incrementalimprovementsin anticipatoryforesightcould result in price variationthat is less smooth. A few perceptivesouls caughta glimpse of what this would mean for finance departments:"Since no rationalman with a quadraticutility function would invest in stocks, most normativework on utility approachesto speculativemarketsis obsolete" [15, 196-197]. (Notice the date on that quote-1964!) Of course, the main implicationsof Mandelbrot'swork revolve aroundissues of inference, information,and the meaningof rationalityin a worldthatdivergesfrom smoothNormality.Neoclassical theorists like to talk in terms of "economizing"with respect to information,to have inference governed by "loss functions" and the like. One of the profoundimplications of the Mandelbrotprogram is that the constrainedmaximizationversion of rationalbehavior is often meaninglesswhen confrontinga Levy stable stochasticenvironment.Take, for instance, a sample from a Cauchy distribution.The distributionof the sample mean (the expected value of the distributiondoes not exist) in this case is identicalto the distributionof each individualitem in the sample [22, 51]. Hence calculationof the sample averageis superfluous;and indeed, collecting more informationis futile [74, 23]. Since augmentationof the dataset neverimprovesthe performance of the estimator,the whole notion of "economizing"on informationis renderednugatory. To a great extent this is also the case with otherLevy stabledistributions,since augmentationof a sampledoes not resultin convergenceof samplestatisticsto highermoments;and manyconsistent estimatorsof first moments actuallydo not make use of all the componentsample elements [73]. The upshot is that almost every technique of orthodoxeconometricsis useless and would probablyhave to be discarded.First, and most obviously,one loses the Gauss-Markhovtheorem, and with it least squares;in the presence of Levy stable distributions,least squaresis a defective estimatorbecause it gives too much weight to outliers and is too heavily sample-dependent[73, 170; 40, 410]. ARIMA estimationwould also have to be jettisoned, because in the presence of the Josepheffect fractionalGaussiannoise is technicallyARMA (0, oo);and most sample statistics depend upon the assumptionsthat p < ooand q < oo. "When the Arma approachis viewed as analagousto fitting of curves by broken lines, and when it is recognizedthat such curve fitting does not warrantbeing called 'modelling', there is nothing in the approachto warrantcriticism (or interest?)" [52, 25]. Spectral analysis is also heavily compromised,because the technique is an attemptto decompose a time series into a sum of periodic harmoniccomponents. The thrust of the Joseph effect is that the separateperiodicitieshave no actualexistence, but are merely an
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artifactof infinite long dependence [45, 268]. Hence, while spectralanalysis might be useful in tryingto diagnose the presence of long dependence,theremay be nothingactuallyto estimate. If Mandelbrotis right, and all the econometricidols need to be smashed, then how is it possible that so many practitionershave been so grossly misled for so very long?Perhapshis most disquietingthesis is that their own flawedpracticesservedto deceive them: Oneverycommonapproachis to notethatlargepricechangesareusuallytraceableto welldetermined'causes'thatshouldbe eliminatedbeforeone attemptsa stochasticmodelof the remainder. Suchpreliminary closerto the Gaussian censorshipobviouslybringsanydistribution . . the distinctionbetweenthe causalandrandomareasis sharpin theGaussiancase andvery diffusein the stableParetiancase [40, 403, 415]. The very practice of fitting linear models, particularlythose involving trended variables, acted to filter out low frequencyvarianceand outliers, thus effectively "pre-whitening"the data. "Normal"distributionsweren't normal at all; they werejust artifactsof the shotgunwedding of deterministictheory with "randomshocks." In a stableLevy world, lineartransformsof stochastic processes could look deterministicand vice versa: in a sense, a preoccupationwith "theory" would obstruct the ability to "see" the full range of stochastic possibilities. This was a very wry twist on the orthodox interpretationof the victors in the "Measurementwithout Theory" controversy[59]. This was a project of the most thorough-goingindeterminism;one which could brook no compromise or make no peace with the neoclassical researchprogram, which, after all, was nothingmore than a bowdlerizednineteenth-centuryimitationof physics. significantandis felt to havechancesof being Broadlyspeaking,a patternis scientifically of havingoccurredby chanceis verysmall.... repeated,onlyif in somesenseits "likelihood" noiseis Paretian, onemustrealizethatone But,whenone worksin a fieldwherethebackground thanthatof physics. . . facesa burdenof proofthatis closerto thatof historyandautobiography andyet generate somestochasticmodels. . . dispensewithanykindof built-incausalstructure, the of thatis usually in which the unskilled and the skilled kind detail both eye distinguish paths associated with causal relations . . . these structuresshould be consideredas perceptualillusion
[39, 433-434].
III. Fractal Bridges over Troubled Waters The reactions of econometriciansand finance economists to Mandelbrot'scritique deserves a separatenarrative,which we intend to provide elsewhere [63]. However,there are some direct bridges between the work on Levy stable distributionsand fractalswhich may prove of interest to those concerned with the chaos literature.These areas of continuitybetween the earlier and later work may be found in the governingrole of chance, the geometryof the irregular,and the analyticimportanceof hyperbolicdistributions. The interestingcharacteristicof Levy stable distributions,as revealed by Levy's work on centrallimit theorems, was their self-similarityrelativeto scale. As Mandelbrothas often said in lecture, "A process that has no scale has the scale of the observer";or, as in TheFractal Geometry of Nature [49, 18], "The notion thata numericalresultshoulddependon the relationof object to observeris in the spirit of the physics of this century,"but not at all, we hasten to add, in the spirit of the 19th century. In that sense, a time series of prices looks like a coastline:it appears to be very irregular,but howeverclose you get, it also looks roughlythe same. A coastline also
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looks like it might be generatedby some sort of stochasticprocess;but the problem here is that standardGaussian or randomwalk processes are not able to adequatelysimulate the geometry.4 In an effort to describe the degree of irregularity,supposewe define a numberd(S) such that: lim lnM(E)/ln(1/e) d(S) = E---oc
(2)
where M(e) is the minimum numberof N dimensionalcubes needed to cover a subset S of a space containingsuch an irregularcurve. For small e, this impliesthatM(E) - Ke -d, whereK is some arbitraryconstant;Mandelbrotoriginallycalled d(S) the "fractaldimension"of the curve.5 If one is dealing with a self-similar curve, then a graphof n M / Ine against [ln E]-1 would be linear with slope d. This is precisely the estimationtechniqueMandelbrothad been using on his time series of prices; and here was the link which connected the geometry of the irregularto Levy stable stochastic processes. Just as the Gaussiandistributionwas only a special case of the more general Levy stable class of distributionswith an integerexponent, conventionalgeometry was only a special case with integer exponent of a much more general geometry, called fractal geometry. As Mandelbrothad earlier claimed that the stable distributionexponent a for price changes was a fractionbetween 1 and 2, he now claimedthatnatureaboundedwith the geometry of fractionaldimensions, far beyond anythingsuspectedin the heavenlycity of the Greek geometers. The norm of Nature was not the smooth celestial orbits of the planets, or the music of the spheres, but ratherthe roiling turbulenceof Heraclitus'brook. Even the distributionof intervals in music was closer to fractal noise than the Gaussian. Moreover, "the probabilitydistribution characteristicof fractalsis hyperbolic"[49, 422], just as one mightsay the probabilitydistribution characteristicof the Euclideanworld is the Gaussian. At this juncturewe leave the specific writingsof Mandelbrot,since the uses found for fractal geometry extended beyond his own interests in mathematics,hydrology, meteorology and geology. The applicationmost relevant to a modem neoclassical economist is in the area closest to his own heart: namely, the physics of energy and motion. More relevantto our overarching thesis is the impact of Mandelbrot'smarriageof geometryand probabilitytheory upon problems of Hamiltoniandynamics in physics, since it is that literature,and not Mandelbrot'seconomic writings, which has of late spurredthe mimetic impulseamongstmodem neoclassicaleconomists. The story of the breakdownof determinismin Hamiltoniandynamics has been well-told by Prigogine and others [67; 68; 24], and so we shall not repeat it here. While textbooks and classroom expositions of Hamiltoniansmade it seem as thoughin the generalcase Hamiltonians had determinateanalyticalsolutions, it was a growingembarrassment thatthey had an "incurable disease unmentionablein polite society" [24, 40], namely,no well-behavedconstantsof motion 4. We have sacrificeda certainmodicum of precisionhere in the interestof makingthe cross-disciplinaryconnections more transparent.In fact, exact self-similarityis a puregeometricconceptcharacterizingsuch abstractobjects as the Koch curve. Statisticalself-similarityis more characteristicof phenomenasuch as coastlines, mountains,etc. Finally, in a strict sense, fractionalBrownianmotion is self-affineratherthan self-similar:if the time axis were to be magnifiedby a factor6 > 1, the value of the self-affinevariablewould be magnifiedby a factorof OH,whereH > 1/2. 5. To be more precise, in [49, 15] Mandelbrotwrote, "A fractal is by definitiona set for which the Hausdorff Besicovich dimension strictly exceeds the topological dimension."But perusalof the appendices, and particularly[49, 357-66] makes one realize in retrospectthat the attemptto restrictfractalsto a metric conceptionwas leading to a proliferationof dimension definitions. This led in turn to a subsequentrealizationthat a single numberis not sufficient to characterizethe fractal objects pioneeredby Mandelbrot[50; 32]. But here we are concernedwith the recapitulationof the historicalroad from economics to fractals (and back again), which includes what would now be regardedas errors, detours, etc.
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other than H itself. The breakthrough came when physicists stopped looking for deterministic invariants and began looking at geometric patterns in phase space.6 What they found was a wholly different kind of order amidst the chaos, the phenomenon of self-similarity at different geometric scales. This suggested that many phase-space portraits of dynamical systems exhibited fractal geometries; and this was taken as an indication that a wholly different approach must be taken to the description of the evolution of mechanical systems. It is at just this point that economists tend to lose their way, so it is imperative to make it clear how all the strands of our narrative converge on just this point. Neoclassical microeconomics is predicated upon the metaphor and formalism of potential energy which is the heart and soul of the Laplacean daemon in physics, the deterministic world-view. However, neoclassicals never partook of the successes of that world-view, namely, a Hamiltonian dynamics in their core doctrine of Walrasian general equilibrium, however much they prognosticated that a viable dynamics was just around the corer.7 Mandelbrot essentially proposed to relinquish all hope of determinism by renouncing the quest for any mechanistic dynamics in favor of a thoroughgoing stochastic approach in economics, but since this inherently contradicted neoclassical theory, it was rejected by the majority of economists. In the 1980s, physicists have found the Hamiltonian approach wanting, but have used it as a benchmark and a springboard to reconceptualize the nature of dynamics. Now some neoclassical economists, with no sign of self-awareness that one can't persist in legitimate imitation of the physics if you never had the solid analytical structure of the Hamiltonian to start with, have rushed headlong to embrace the new mathematical technology largely, one fears, because of its popularity amongst the physicists. A brief consideration of the modem chaos literature will reveal that a little physics is a dangerous thing, indeed. Many introductory texts such as Devaney [16] and surveys for economists such as Kelsey [33] and Baumol and Benhabib [5] begin with the simple map xt+l = xt (1 xt) on the unit interval [0, 1], and show how raising the forcing parameter b causes period-doubling and the onset of chaos; then the reader is led to the more general mapf : IR -> IR and, by a detour through the Sarkovskii theorem,8 to the result that iff has a periodic point of period three, then f has periodic points of all other periods [16, 60-68]. While this is responsible mathematical pedagogy (and a great toy if you have access to computer graphics), it has been disastrous for economists, because it gives the impression that somehow this is what chaos is all about, not to mention giving 6. The technical terms are all explained at an incrediblynon-technicallevel in Gleick [25]. The rejectionof the whole Hamiltonianapproachto dynamics is clear from a quote from the physicistLibchaber:"A physicistwould ask me, how does this atom come here and stick there? . . . And can you write the Hamiltonianof the system? And I tell him, I don't care, what interestsme is the shape, the mathematicsof the shape and the evolution"[25, 210]. 7. One must acknowledge a numberof apparentlydynamicmodels in growththeory,capital theory, and the like, but sophisticatedneoclassical theoristswill admit that these achieve their seeming dynamicsby either resortingto single good models with only one consumer, no production,etc. etc., or else writingeverythingin terms of "utility units," hence obviating any general equilibriumstructure.For a surveyof such attempts,see [13; 76; 75]. The general problem of a full neoclassical dynamics is discussed in detail in [58; 62]. That is not to say that neoclassical theoristsappreciate having this century-longfailure pointed out to them: "Everyoneknows that economic dynamics isn't in a satisfactory state. Everyoneknows that errorterms shouldn'tjust be tackedon to deterministicequations.One more paper that says these things doesn't add anythingto the practiceof economics." Letterof Hal Varianto Philip Mirowski, June 17, 1989. 8. The Sarkovskiitheorem is based upon the following orderingof the naturalnumbers:
3 -5)- 7)- ...
2 - 3 )- 2 5 )- ...
)- 223
)-22 .5 -...
)-23 3 )-- 23 5
...
23) 22
2 - I
The theorem states that iff : IR-> IR is continuous, andf has a periodic point of prime period k, then if k )- e in the above ordering, thenf also has a periodic point of period e. See [16, 62 et seq.].
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TableI. TheCategories of a StochasticDynamics Degrees of Freedom
1,2
3 or more
conservative(integrable)system
resonance
mixing, c-systems Arnold diffusion
dissipativesystem
no stochastic behavior
strangeattractor true "chaos"
rise to the temptationto generate all sorts of single-variablematchbookmodels where a single recursivedifferenceequationis somehow supposedto "account"for the apparentlystochasticbehaviorof stock prices, macrofluctuations,monetarydisturbances,and every otherscourge known to mankind.The issue of conceptualizationof chaos in physics is much more complex than these little mathematicalexercises. More advancedtexts in physics acknowledgethatthe problemof the extractionof seemingly stochastic behaviorfrom apparentlydeterministicprocesses is much older than the presentchaos craze, dating back at least to the work of Henri Poincare[36; 67]. Over the last century,physicists have come to believe that there are many qualitativelydifferentkinds of stochastic output, and that they should be distinguishedby the sort of process that generatesthem. For instance, it has been long known that well-behavedHamiltonianscould themselvesgive rise to apparently stochastictrajectories,but this has nothingto do with the mathematicalniceties mentionedin the paragraphabove. In what follows, it may be useful for economists to be made more awareof the difference between fractal structurein maps and chaotic flows [70, 25]. In maps, as in the simple tent map above, a plot of the motion of the system is discrete, and there are no restrictionsupon the continuity of the motion described. However,the dynamicsof physicalsystems are generallycast as problemsof motion in phase space, which must conformto requirementsof continuityin motion [72, 107]. Althoughthe terminologyin this arenahas not yet stabilized,we shall reservethe terms "chaos" and "phase space" exclusively for flows because we arehere concernedwith the relation of neoclassicism to the treatmentof dynamics in physics. Thus the "degreesof freedom" of an economic model will be defined as the N parametersrequiredto fully specify the motion of the system minus the p independentconstraintson the system. This is extremely at variance with almost every other discussion of "chaos" in economics, which generallystartswith examples of maps and not flows [2; 3; 5; 33].9 The most importantdistinctionsto be made from the presentpoint of view are those which divide conservativefrom dissipative systems, and distinctionsbetween low and high numbersof degrees of freedom in the particulardynamicalproblem.The variouspermutationsof these categories are presentedbelow in Table I, in orderto stress the differingkinds of seemingly random behaviorthat may arise in a deterministiccontext. One can observe from this table that the terminologyof "chaos" has already been sadly abused in the literature, since it should only legitimatelybe attachedto dissipative systems of 9. This raises the possible objection that I have just now ignoredthe contentionof Mandelbrotmentionedearlier in this paper that the assumptionof continuous motion is part of the problem in orthodox economics. To prevent any confusion, I would like to suggest that the issues surroundingTable I are problemsof neoclassical economics, a theory which began by copying the physical metaphorof motion in a commodity space. In a truly alternativeeconomics, one which rejected imitationof physics, these confusionsover the meaningof "chaos"would not pose any obstacle [63].
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flows (and then, not to all of them): namely, systems that "lose availableenergy" or wind down due to friction, heat loss, etc. and hence evolve in an irreversiblemanner.The mere appearance of stochastic behavior in a deterministicsystemdoes not dependupon the presence of "chaos". Mandelbrot,as usual, was right: irregularityand randomnessare everywhere;it is only we who have chosen to avertour eyes from them. Already,the very notion of "determinism"has suffereda serioussetbackin these instances, since it is widely acknowledged that conservativesystems stand as the epitome of the ideal of determinism.But this is more than a terminologicalor philosophicalquibble;it cuts to the very quick of the neoclassical fervourto imitatethe physics. It is importantto recall that Hamiltonian dynamics only apply to conservativeor integrablesystems, whereastrue "chaos" is confined to those situationswhere Hamiltoniansare not applicableor solvable. The reasonfor this distinction is thatthe stochasticmotionthatone observesin Hamiltoniandynamicsis exhibitedover the entire parameterrange [36, 422], whereas the distinguishingcharacteristicof the phase space of dissipative systems is the simultaneouspresenceof regulartrajectoriesand regions of stochasticity.In other words, only dissipative systems display such phenomenaas a "transitionto chaos" conditional upon some forcing parameter.Only intrinsicallydissipativesystems can exhibit a strange attractorin phase space, because trajectoriescan't be "attracted"in a conservativesystem. It is the dissipation which serves to lower the "dimension"of the geometric characterizationof trajectoriesin phase space. Physicists needed somethingto groundtheir analysisin these situationswhere Hamiltonians couldn't work, and they found it in the entirely differentkind of invariantother than energy conservation:namely, the geometric self-similarityof the attractorsin phase space, i.e., in their "fractal"character.It was not just anotherversion of mechanismor a neat bit of math to get them over the hump (so to speak); it was an entirelydifferentway of attackinga long-standing problem.'0 It is critical to see that physicists have used their priortheory to divide the world up into regions of invarianceand regions of change, and thus know in what situationsone might expect to find a strangeattractor,and more importantly,have priortheoreticalexpectationsabout which variables should be governed by low-dimensionalattractors.Unfortunately,neoclassical microeconomistspossess no parallelexpertiseor theoreticalresources,and thereforearewhistlingin the darkwhen it comes to claiming any economic phenomenonis or even might be "chaotic". First, they possess no legitimateHamiltoniandynamics.The reason for this is that neoclassicals have never made up their mind about what precisely it is that shouldbe conservedin their theoretical system [58; 64]. It then follows that neoclassicals have no theoreticalrationalefor claiming that any economic system or phenomenonis dissipative,and hence, no rationalefor even looking for strange attractors.1Indeed, in the neoclassical case, one might argue that it is more likely that conservativepreference fields are the very paradigmof reversiblephenomenaand hence should give rise to mixing or Arnold diffusion, where stochastic behavioris pervasive and "order" is entirelyabsent. 10. Mitchell Feigenbaum:"The whole traditionof physics is thatyou isolate the mechanismand all the rest flows. That's completely falling apart. Here you know the right equationsbut they'rejust not helpful." Quoted in Gleick [25, 174]. 11. Kelsey [33, 25] mentions this in passing, but does not seem to realizehow it underminesthe rest of his article. Undoubtedlythis derives from his ignorance of the origins of neoclassical economics in copying energy physics. This is just one of a sequence of examples of the contretempswhich economic theoristscould avoid if they did not hold the historyof their discipline in such contempt.
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Secondly, neoclassical economists have no argumentsto place some bounds upon any expected degrees offreedom in any of their models. Since the numberof goods and actors in true general equilibriummodels is unbounded(especially if one resortsto the Arrow-Debreuruse of defining physical goods at differenttemporallocations as differentcommodities), they have no analogueto the physicists' a prioriknowledge of the relevantnumberof spatialdimensions, etc. Often this issue is finessed by arbitrarilyrestrictingmodels to a single actor, two goods, two periods, etc. as if degrees of freedom were not a centralcriterionfor model evaluation. Hence the neoclassicals have ignored the crucial distinctionsin TableI because they have no idea what restrictionsshould be placed on the sorts of randomphenomenato be detected in the social sphere. Do price fluctuationsmore resemble mixing sorts of turbulence,or are they rathermore like resonance? No matter how mathematicallysophisticatedthey might seem to an untutored outsider, neoclassical general equilibriummodels leave too many degrees of freedom dangling, and thereforeplace no intelligible restrictionsupon economic explanation. The prime implication of these lapses of explanatorypractice is that all claims that chaos theory will somehow "reconcile randomnessand determinismin economics" are spurious. This shows up in differentways, dependingon whetherthe statementsare made by a neoclassical economic theoristor else by an economist more concernedwith econometricpractices. In the former case, it often appearsin the context of a claim that some small recursivemodel "demonstrates" that randomoutput can be endogenous to the economic system, and then perhapssome further moralizing as to how this might disprove rationalexpectationsor reveal the legitimacy of governmentinterventionor Keynesian concerns. In fact such models have nothing coherent to say about the actual economy, since the incoherenceof a neoclassicaldynamics is neither rectified nor clarifiedby the introductionof nonlinearmathematics.If anything,chaotic models might in the futureserve to reveal the futility of the neoclassicalresearchproject, as we shall argue below, but this betokens the abandonmentof determinism,and not any sort of reconciliation. The case of the econometricallyorientedeconomist is much more interesting.Here we find researcherswho view the chaos literaturein a much more sophisticatedmanner,as a source of possible empirical techniques which may or may not have any direct connectionto the existing neoclassical researchprogram.These authorschampionthe use of such techniquesas estimation of Liapunovexponentsor the dimensionof strangeattractorsin economictimeseriesby means of the Grassberger-Procacciamethod [30; 72]. Yet even here the confusionover the correct meaning of "randomness"is evident. The crucial question,often skirtednervously,is what should we expect to find out from such procedures?What can we say about "intrinsic"versus "extrinsic" randomness?Some say we should expect both: In economics,however,it seemsinevitablethatwe will haverandomtermsin ourequations.If forno otherreason,thiswill be becauseeconomicactivityis affectedbybiologicalandmeteoroThesesystemswill almostcertainlybe chaoticin nature.As faras economics logicalphenomena. is concernedtheywill appearas randomtermsin ourequations[33, 12]. Otherssuspect that thatroad leads to perdition,and thereforetry to opt out of the problem: For the purposesof this articlewe will avoidthe deep questionsof philosophyand define "random process" to be a process whose [fractalstrangeattractor]dimensionis "high". A "deterministic process" is a process with "low dimension" [12, 74].
But this "deep philosophical" problem is not to be shuntedoff onto the despised band of philosophersand methodologistsso easily; thereis an impasseherebornof an unthinkingmimesis
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of the physicists. Both approachesto conceptualizingthe distinctionbetween orderand randomness are equally incoherent. Contraryto Brock [12], the dimensionalityof strangeattractorsdoes not exhaust the universe of forms of randomness,as explainedabove. Contraryto Kelsey [33], mixing chaotic processes with other stochasticprocesses rendersboth unrecoverable,and hence the distinctionmeaningless. The inability to divide the world up into types of stochastic phenomenadescribed above in Table I now comes home to roost, in that differentforms of randomnessbecome hopelessly tangled. As Grassbergerand Procaccia [30, 190] remindus, the estimatedLiapunovexponents may get at the "stretching"in phase space, but they have nothingto say about "folding," which is one of the main criteria of differentiationof types of stochastic behavior. And even worse, Liapunovexponents cannot even be defined in the presence of other sorts of "noise". As for the Grassberger-Procacciamethod of using the correlationintegralto arrive at an estimate of the dimension of a strange attractor,it is a techniqueseverely compromisedby the presence of "noise" precisely because the fine geometricdetail of a self-similarattractoris "fuzzed" out by extraneousshocks. All of this does not botherthe physicists much since their theories tell them when to expect each particulareventuality,and they can intervenein an experimentto take each into account. Economists wedded to finding an unspecific "deterministic"explanationadopting a quasi-phenomenologicalapproachto economic time series are not so well-equipped or wellprepared.All the possible "types" of randomnessare lumped together, and the net result is a wash. This, ratherthan any recalcitranceof the data, explainssuch disappointingresults as those in [10; 12].
IV. Chaos Comes to Economics It seems to me that the most sophisticatedeconomists concernedwith the chaos literatureare alreadybeginning to sour on the vauntedpromise that it would reconcile determinismand randomnessin neoclassical economics. For instance,the tone of Baumoland Benhabib[5] is notably muted. Or, in Brock and Malliaris [11, 322] we find the downbeatprognosticationthat "it is an irresolvablequestion whethermacroeconomicfluctuationsare generatedby high dimensional chaos or infinite dimensional, i.e., stochasticprocesses". Whathas been missed in the interimis thatthis was exactly the point of Mandelbrot'searlierwork in economics, and has been a theme of his work since his "PrematureFractalManifesto". Once one is open to the pervasivenessof seemingly randomphenomena,then it becomes possible to see the observeras inextricablybound up with the phenomena. Is economic life randomor deterministic?Is the glass half-full or halfempty?The much-trumpetedthesis that chaotic models "save" deterministiclaw is itself a bit of wishful thinking, when one observesfromsuch mathematicalworkas thatof Barnsleyand Demko [4] that Julia sets and other fractalshapes, while properlyviewed as the result of a deterministic process, have the equally valid interpretationas the limit of a randomprocess. It should be made clear that this paper does not argue that the chaos literaturehas no substantialimplicationsfor economics. Quite the contrary.The ubiquityof Levy stable distributions of economic variates such as prices and incomes suggests that fractal attractorsin phase space may ultimatelyhave some import for economic explanation.After all, as Mandelbrot[49, 422] has written, "the probabilitydistributioncharacteristicof fractalsis hyperbolic."Economic variates have often been noted to display forms of self-similarityas regardstime-scale, but no one has known what to make of the observation[69]. Further,as Mandelbrothas often claimed,
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the Grassberger-Procacciaprocedureof plottingthe log of the correlationintegralagainst log of "distance" really differs very little from Mandelbrot'soriginal procedureof plotting the log of price differences against the log of lags, so there is some hope for this procedureto be used as a taxonomic device, as long as it is subordinateto a coherenteconomictheory. This theory will have to allow for randomnessnot as erroror perturbation,but as constitutingthe very substrate of economic experience in a mannernever dreamedof by the advocatesof rationalexpectations. It will also affect the type of mathematicsused in economic explanations,in a mannerparallelto thatpredictedby a famous physicist: . . withoutthisinfiniteprecision,thecontinuum becomes,physicallyspeaking,meaningless onassumptions thatsomething wasinfinitewhenin ... Newtoniandynamicshastwicefoundered
fact it was not: the speed of light, c, and the reciprocalof Planck'sconstant1/h . . . Complexity
theorynowrevealsa thirdtacitlyassumedinfinityin classicaldynamics,namelytheassumption themoststrikingfeatureof this of infinitecomputational andobservational precision... Perhaps reformulation is thatall physicalvariableswillbe quantized[24, 46-47].
However, what the chaos literaturewill not do is augmentor save neoclassical economic theory. Mandelbrot'sheritage is not a mere matter of "stylized facts" that neoclassicals may choose to ignore when it suits them [11, 322]. On the contrary,if followed to their bitter conclusions, chaos models would renderorthodoxtheory meaningless, which is why I predict that it will prove as hot a potato as Mandelbrot'soriginalwork. Neoclassical theory exists to paint a portraitof a deterministic,lawlike orderin the economic sphereindependentof the machinations and beliefs of the economic actors. The chaos literatureinsteadrevealsthe curious symbiosis of randomnessand determinism [9], the blurringof the boundariesbetween order and chaos. We may finally have arrivedat the criticaljuncturein humanintellectualhistorywhere our images of the naturalworld are so very anomic and frighteningthateconomistscan no longerseek to imitate physics, since it would underminetheir role and functionin social life.
References 1. Albers, Donald and G. Alexanderson. "Interviewwith Benoit Mandelbrot"in MathematicalPeople, edited by Donald Albers and G. Alexanderson.Boston: Birkhauser,1987. 2. Barnett,William; ErnstBerndt;and HalbertWhite, editors.DynamicEconometricModelling.New York:CambridgeUniversityPress, 1988. 3. and Ping Chen. "DeterministicChaos and FractalAttractorsas Tools for NonparametricDynamical Economic Inference."MathematicalComputationand Modelling,June 1988, 275-96. 4. Barnsley,Michael and Stephen Demko, "IteratedFunctionSystems and the Global Constructionof Fractals." Proceedingsof the Royal Society of LondonA, 1985 v. 399, 243-75. 5. Baumol, William and Jess Benhabib, "Chaos:Significance,Mechanism,and Economic Applications."Journal of EconomicPerspectives,Winter 1989, 77-106. 6. Bausor, Randall. "LiapunovTechniquesin EconomicDynamics and ClassicalThermodynamics."Unpublished manuscript,Universityof Massachusetts, 1987. 7. Blattberg, Robert and Thomas Sargent, "Regressionwith Non-GaussianStable Distrubances."Econometrica, May 1971, 501-10. and Nicholas Gonedes, "A Comparisonof the Stable and StudentDistributionsas StatisticalModels 8. for Stock Prices." Journalof Business, Fall 1974, 244-80. 9. Bondi, Hermann. "Why Mournthe Passing of Determinacy?",in Oldand New Questionsin Physics, edited by A. van der Merwe. New York:Plenum, 1983. 10. Brock, William, "DistinguishingRandomand DeterministicSystems."Journalof EconomicTheory, October 1986, 168-95. and A. Malliaris.DifferentialEquations,Stabilityand Chaos in DynamicEconomics.New York:North11. Holland, 1989.
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