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Nonlinear and Chaotic Dynamics: an Economist's Guide Michael D

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Nonlinear and Chaotic Dynamics: an Economist's Guide Michael D Nonlinear and Chaotic Dynamics: An Economist's Guide Michael D. Weiss AbsiraLI 111 recellt 'lem, re~earch 111 both matheHla· state space RI' (where R IS the real number Ime) Sup­ tIC; alld the applied sCWllces ha\ plOdl/ced a revo{lI­ pose that the economy evolves determlmstlcally In tum l1/ the uudel "tcoullnq of llonll1leal dl/1wHllcal such a way that ItS state' at any time umquely deter­ "1st e 111 1 U,;ed wldehl111 eCOII01l"C\ and othe, d,sc1­ mmes ItS state at all later tImes Then, If the Imhal plwe, to model challge ovel tillie, the"e "I~ten" a,e pOSitIOn of the economy III R" at tIme 0 IS vo, the 1I0W kllOlV1I to be vIII"elable to a kmd oj "chaotIc," evolutIOn of the economy through tIme will be repre­ ''''/ll e,lIetable beh'lVlOr Till,; mi1de places tillS ,evoln­ sented by a path m R" startmg at Vo and traced out by t,lOll 111 1l1':Jturlcul c'olltext, dISCU,"lws som.e of Lts VI as tIme, t, moves forward ThiS path, called the I1HpllcatlOll':l tor econOHilC modell11g, and explalns orb,t generated by the ImtJal pOSItIOn Vo, represents a I/WIIII 0/ the I1l1pOltalll ",,,thematlcal Ideas all willch It "future history" of the system QuestIOns about the /I b",cd behaVIOr of the economy ovel time are really questIOns about ItS orbIts We are often mterested not so much Keywords Ll1mt, to predlctablhty, nonlmear and m the near-tenn behaVIOr of orbits as m theIr eventual chaotIc dvnamlcal systerns, strnctural stab1l1ty of behavlOl, as when we engage m long-range forecastmg econonl1c models, (ractals or study an economy's response to d new go vel nment pohcy 01' an unexpected shock after the ImtJal period of In the past two decades, the world of sCience has come adjustment has passed and the economy has settled to a fundamentally new unde1 standmg of the dynamiCS down of phenomena that vary over time Grounded m math­ ematlcdl discovery, yet given empIrIcal substance by Fractals, SensItive Dependence, and Chaos eVidence from a variety of diSCiplines, this new per­ spectIve has led to nothmg less than a re-exammatlOn SCIentIsts long have known that It IS pOSSIble for a sys­ of the concept of the predICtability of dynamiC be­ tem's state space to con tam an Isolated, unstable pomt hdvlOr OUI ImpliCIt confidence m the orderliness of p such that different Imtlal pomts near p can generate dynamlcdl systems, specifically of nontmear dynanucal orbIts WIth WIdely vdrylllg longrun behaVior (For sy,tems, has not, It turns out, been entirely Justified example, a marble bdlanced on the tiP of a cone IS Such systems are capable of behavmg m ways that are unstable m thiS sense) What WdS unexpected, how­ far more elTatlc and unpre(lIctable than once beheved ever, was the dIscovery that thiS type of mstdblhty FI ttmgly, the new Ideas are said to concern chaot,c can occur throughout the state space, sometImes actu­ dvnarrllco, or, SImply, chaos ally at every pOlllt, but often m'strangely pdtterned, fragmented subsets of the state space-subsets typ­ EconomIcs IS not Immune from the ImplicatIOns of thIs Ically of nomntegel dImenSIOn, called fractalo Once new understdndmg Aftel all, our subject IS replete mvestlgators knew what to look fOI, they found thiS WIth dynamIC phenomena I angmg from cattle cycles to phenomenon, termed 6cnsdwe dependence on l,n~i'wl stock mal ket cdtastrophes to the back-and-forth mter­ conchtlOns, to be Widespread among nonlmear dynam­ play of advertlsmg and product sales Ideas related to Ical systems, even among the Simplest ones Though the notIOn of chaotic behaVIOr are now part of the baSIC techmcal deflmtlOns vary, systems exhlbltlllg thiS mdthematlcal toolkIt needed for mSlghtful dynamIcal unstable behaVIOr have generally come to be called modeling Agncultural economIsts need to gam an "chaotIc" understandmg of these Ideas Just as they would any other slgmficant mathematICal contributIOn to their For chaotIC systems, any errOl m speclfYlllg an mltlal field ThiS article IS mtended to assist m thIS educa­ pomt, even the most mmute error (due to, say, com­ tIOnal process puter roundmg m the thousandth deCimal place), can g1Ve I1se to an orbit whose 10ng:lUn behaVIOr bears,no What exactly has chdos theory revealed? To address resemblance to that of the orbit of the {ntended Imtlal thIS questIOn, let us conSIder an economy, subject to pomt Smce, m the real world, we can never specify a change over time, whose state at time t can be pomt With mathematically perfect preCISIOn, It follows descl1bed by a vectOl, v" of (say) 14 numbers (money that practical longrun predictIOn of the state of a cha­ supply at time t, mflatlOn rate,at time t, and so on) otIc system IS ImpOSSIble Formally, thIS vector IS d pomt m the 14-dlmenslOnal Attractors WeISS IS an economist In the CommodIty ELOnomlcs DIVISion ERS The author thanks John McClelland and other partiCipants I~ the ERS Chaos Theon Semmar COl many stImulatmg dlSCllSSlon,> on For a dynamlCaJ system, perhaps the most baSIC ques­ chaotiC dynamiC'> Callos Arnade, Rlchclrd Heifner. and an anony­ tlOn IS "where does the system go, and what does It do mous lefelee furmshed helpful reView comments when It gets there?JJ In the earlIer view of dynamical 2 THE JOURNAL OF AGRICULTURAL ECONOMICS RESEARCHIYOL 4,1. NO 3. SUMMER 1991 systems, the place whel e the system went, the pomt nor the Imagmg techmques available at the tIme per­ set m the state space to whIch orbIts converged (called mItted him to explore h,S mtUltIOns fully an attraetor), was usually assumed to be a geo­ metrICally uncomphcated object such as a closed curve Followmg Pomcare's work and that of the AmerICan or a sIngle pomt EconomIc modelel s, for e,ample, mathematicIan G D Blrkhoff m the early part of thiS have often Imphcltly assumed that a dynamIC economIC century, and despIte contmumg mterest m the SovIet process WIll ultimately achIeve eIther an eqUlhbnum, a Umon, Lhe subject of dynamical systems fell mto lela­ cychc pattern, or some other orderly behaVIOr How­ tlve obSCUrity DUllng th,S perIod, there was some ever, another dIscovery of chaos theory has been that dwarene"s among mathemdtlclans, SCientIsts, and the attractor of a nonlInear system can be a b,zarre, engmeers that nonhnear systems were capable of erra­ fractal set wIthIn "h,ch the system's state can flIt tiC behaVIOr However, examples of such behaVIOr endlessly m a chaotIC, seemmgly random manner were Ignored, claSSified as "nOIse," or dIsmIssed as aberratIOns The Idea that these phenomena were ,Just as an economy can have two OJ more eqUlhbna, a chal actenstlc of nonhnear dynamICal systems and that dynamIcal system can have two 01 more attractors In It was the well-behaved, textbook examples that were such a case, the set of all Imtlal pomts whose orbIts the speCial cases had not yet taken root converge to a partIcular attractOl IS called a bas", of attractlOn A recent findmg has been that the bound­ Then, m the 1960's and 1970's, there was a flurry of ary between competmg basms of attractIOn can be a activity m dynamIcal systems by both mathematicIans fractal even when the attractors themselves are unex­ and SCIentists workmg entirely mdependently Mathe­ ceptIOnal sets A type of senSItIVIty to the mltlal comh­ matlClan Stephen Smale turned hiS attentIOn to the tIOn can operate here too the shghtest movement subject and used the techmques of modern differential away from an Imtlal pOInt Iymg m one basm of attIac­ topology to create ngorous theoretical models of cha­ tIOn may move the system to a new basm of attl actIOn otic dynamICS Meteorologist Edward Lorenz dIS­ and thus cause It to evolve toward a new attractor covered that a Simple system of equatIOns he had devI"ecl to Simulate the earth's weather on a pnmltlve ChaotIC beha vlOr wlthm a workmg model would be computer dIsplayed a surpnsmg type of senSItIvIty eaSIer to recognIze If all 01 bIts Imtlatmg neal an erra­ the shghtest change m the Imtlal condllIons,eventually tIC OJ b,t were also elratlc However, the potentIally would lead to weather patterns bearIng no resem­ fractal structure of the regIOn of senSItive dependence blance to those generated m the ongmalrun can allow ImtIaI pomts whose orbIts behave "sensIbly" and Imtlal pomts whose orb,t, are enallc to coeXIst mseparably m the state space hke two mtermmgled BIOlOgIst Robert May llsed the 10gIslic difference equa­ clouds of dust Thus, SImulatIOn of a model at a few tIOn xnT I = IXn (I-x,) to model populatIOn level, x, tI lal pOlnts cannot rule out the posslblhty of chaotIc over successive tIme penods He observed that for dynamICS Rather, we need a deeper understandmg of some chOIces of the growth rate parameter, r, the pop­ the mathematical propel ties of our models Nor can ulatIOn level would converge, for other chOices It chaotic dynamICS be d,sm,ssed as ansmg only m a few would cycle among a few values, and 'for slill others It qUIrky speCial cases As we shall see, It anses even would fluctuate seemmgly randomly, never achIevmg when the system's law of motIOn IS a Simple quadratIC either a steady state 01 any dlscermble repeatmg pat­ j tern When he attempted to graph the populatIOn level agambt the glowth 1ate parameter, he observed a The DIscovery of Chaos strangely patterned, fragmented set of pomts ) Recent years have Witnessed an explOSIOn of mterest and activity m the area of chaotiC dynamiCs What PhYSICISt MItchell FeIgenbaum Investigated the 1 behaVIOr of dynamical systems whose equatIOns of accounts for thiS new VISlblhty, WhICh extends even beyond the reseal ch commumty mto the pubhc medIa? molion anse from ummodal (hIll-shaped)
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