104European Journalof Operational Research 30 (1987)104-109 North-Holland Nonlinearityinhigh-ordermodels ofsocialsystems JayW.FORRESTER SystemDynamicsGroup, Sloan School of Management,MassachusettsInstituteofTechnology,Cambridge, MA02139,U.S.A. Abstract:Historically limitationsofmathematicalanalysisforcedexclusionofmostnonlinearitiesfrom modelsofsocialsystems.Computersimulation removedthepressuretofocusonlinearrepresentations, butevenso,dataanalysisandmodelvalidationmethodshaveperpetuatedabiastowardlinearityin models.However,muchofreal-lifebehaviorarisesfromnonlinearities.Ifmodelsaretobegood representationsofsocialsystems,theremustbeunrestrictedwillingnesstoincorporatenonlinearity. Otherwise,weexcludeaccesstomuchoftheavailableinformationaboutthestructureandpoliciesthat causeobservedbehavior.Alargefractionofourknowledgeaboutsocialsystemsliesinwhatmust necessarily happenasextremeconditionsaregradually approached.Themajority offunctional relation- shipsare nonlinear, aswhentwo variablesaremultiplied, for example,salesratetimes pricetogeneratea payment stream.Takingadvantage of knowledge aboutreal-life nonlinearities andtheir crucialcontribu- tiontobehavior,leadstomodelsthatendogenously generatetheprincipalmodesofbehaviorthatare observed inactualsystems. Keywords:Nonlinearity, simulation, economics,social,behavior I.Nonlinearityinmodels Traditionally astrong bias has operatedagainst incorporatingnonlinearityintomodelsofsocial systems. Severalreasons seemto explain the reluc- tancetorepresentnonlinear relationships inmod- els: (a)Before modem computers and the availabil- ityofinexpensivesimulation,modelsofreal-life werelimitedtoequationsystemssimpleenough thatmathematical solutions couldbefound.Such solutions are elegant and comprehensive. They are desirablewhentheycanbeobtained.However, withminorexceptions,completemathematical solutionscannotbeobtainedfornonlinearsys- tems.Before computers,whenmathematical solu- tionsweretheonlyavailabletechnique,necessity imposedatraditionofignoringnonlinearity. The traditionoflinearthinkinghasbecamesofirmly ReceivedMarch1985 established that ithasdiverted most analysts from evenrecognizing theimportance of nonlinearities. (b)Many of themathematical proceduresused evenwithsimulationmodels,suchasstatistical analysisofdataandmethodsusedformodel validation, are severelyrestricted intheir ability to acceptnonlinearity.Therehasbeenareluctance togiveupthelinearmathematicalprocedures, withthe resultthat models have beenbiasedtofit thelinearproceduresattheexpenseoffaithful- nessinrepresentingtherealworld. (c)Therecognition of nonlinearity willusually destroythesimplicity,elegance,anduniversality bywhichresearchpapersarejudged.Juniorfa- cultymembersinuniversitiesenterthenonlinear worldattheriskoftheirpromotions.Whenone worksintherealmofnonlinearsystems,few universalanswersemerge.Nonlinearitiescause 'structuralshifts'inasystemthatcanalterthe relative importance of different partsof asystem. Behavioralinsightsobtainedforoneregionof system behavior can be markedly altered when the 0377-2217/87/$3.501987,Elsevier SciencePublishers B.V.(North-Holland) J. W.Forrester/Nonlinearity in high-ordermodelsof social systems105 internalbalanceofthesystemshifts.Theelegant anduniversalsolutionsobtainableforsimplelin- earsystemsgivewayinnonlinearmodelstoan experimentalevaluationofspecialcases.Accept- ingnonlinearitytendstoforceapersonoutofthe worldofthetheoristintotheworldofthepracti- tioner,ashiftthatdoesnotmatchthestandards forpromot i onintheacademiccommunity. (d)Apresumptionexiststhatnonlinearrepre- sentationsofsystemsaremoredifficulttowork withthanlinearapproximations.Iwillarguethat thereverseistrue,ifone' sobjectiveisseekingan understandingofreal-lifesystems. Ineconomics,realsystemsareespeciallynon- linear,whiletheirrepresentationinmodelsmostly ignoresnonlinearreality.Thecriticism byBlatt[1] statestheissue: "Ent i rel ytoomuchofeconometrictheoryis devotedtothestudyoflinearmodel s. . . Yet lin- earityisaverysevererestrictiononamodel. Thereisexcellentreasontobelievethatnolinear modelcanbeatalladequat e. . . Thetechniques usedforlinearmodel s. . ,donotextendeasily,or inmostcasesatall,tononlinearmodels." 2.Importanceofnonlinearity Weliveinahighlynonlinearworld.Froma mathematician' sviewpointKovachobserved[7]: "St rangethatthesenonlinearphenomenathat aboundsowidelyinnatureshouldbesointracta- ble.ItisalmostasifManistobedenieda completeknowledgeoftheuniverseunlesshe makesasuperhumaneffort tosolve itsnonlineari- ties . . . .Sofar,oureffortstoscalethenonlinear barrierhaveconsistedof chiselling afewfootholds whicharelowenoughsothatwecanalwayskeep onefootonlinearground . . . . Thereisnogeneral theoryfornonlinearproblems,sothatitisneces- sarytodevelopsolutionsbymeansofspecial techniquesforeachtype." Inhumandecisionmaking,theinputstoadeci- sionareperceivednonlinearily.Manyvariables whenintheirnormalrangesexertlittleinfluence, butthosesamevariablescandominateallothers whenthey move outsidenormalranges.Forexam- ple,cashposition(liquidity)haslittleeffecton hiringandproductiondecisionswhenthereisno liquiditydifficulty;however,underseverefinan- cialrestraints,financialconditioncanover-ride evendemandforproductandtheconditionof inventoriesindeterminingemploymentandpro- duction. Nonlinearitiescausewhateconomistsreferto as"structuralchanges"inasystem.But' struct- uralchange' isusuallylittlemorethanatermused tocoveranunexplainablebehavior.Ratherthan referringtostructuralchange,whichusuallyim- pliesanexogenous modificationinhowthesystem isorganized,wemightbetterspeakofshifting loopdominance.Byshiftingloopdominancewe meantheprocessbywhichcontrolofasystem movesfromonesetoffeedbackloopstoanother set,oftenwithdramaticchangesinbehavior.The severalcontrolloopswillallhavebeenpresentin thesystemfromthebeginningbutsomelieinac- tiveuntilconditionstriggerthemintooperation. Therelationshipofshiftingloopdominanceto nonlinearityhasbeendiscussedbyRichardson[9, p.502]: "Anot herconsequence of thechoice of representa- tionoffeedbacksystemsisthewaystructural changesarehandled.Theissueisveryimportant toscholarsinbot hfeedbackt hreads. . . Scholarsin thecybernetics threadhavetendedtocapturesuch structuralchangeslinguisticallyandsometimesdi- agrammatically,by' rewriting' or' redrawing' sys- temst ruct ure. . .Theissueofstructuralchangeis nolessimportanttoscholarsintheservomech- anismsthread,but intheirquantitativerepresen- tationsofsystemstructuresuchphenomenaare capturedinnonlinearities.Nonlinearmodelshave thepropertythattheycanshiftloopdominance andendogenouslychangethestructureoffeed- backloopsthatareactive over anygiven periodof simulatedtime.Indeed,fromafeedbackloop perspective,thisabilitytoshiftloopdominanceis thefundamentalreasonforadvocatingnonlinear modelsofsocialsystembehavior." TheprocessesbehindthetypicalS-shaped growthcurveserveasasimpleexampleofshifting loopdominance.Considerapopulationexpanding towardanupperlimittothecarryingcapacityof itsenvironment.Whenthepopulationiswellbe- lowthelimit,populationexpandsexponentially, drivenbyalinearpositivefeedbackloopinwhich additionstopopulationincreaseinproportionto populationitself.Thepositivefeedbacklooppro- 106J.W. Forrester /Nonlinearity in high-order models of social systems ducestheinitialupwardsweepingsectionofS- shapedgrowth.Butasthelimittopopulationis approached,apreviously dormantlinearnegative feedbackloopbecomesactive,interactsnonlin- earlywiththepositiveloop,reducesthegrowth rateofthepositivefeedbacklooptowardzero, andeventually takesfull controlto adjust popula- tiontowardthelimitwheneverpopulationde- viatesineitherdirectionfromthelimit.Thetwo loops come into operation at different times,First, thepositivefeedback loopof growthisincontrol duringtheearlyexponential growthphase.Later, thenegativefeedbackloopexertsincreasingcon- trolto neutralize the positive loop andconvert the system to agoal-seekingsearchfor anequilibrium atthepopulationlimit.Biologicalandsocialsys- temscontainnumerousstructuresthatmovein andoutof dominance asforcesshift. High-ordersystemsusuallycontainmultiple modesofbehavior.Forexampleinanational economy,onefindstheshort-termbusinesscycle with peaksthree tosevenyears apart,the Kuznets cyclewithpeaks15to25yearsapart,theeco- nomiclongwaveorKondratieff cyclewithpeaks separatedby 45to 60 years, andanoncyclic mode ofunrestrainedinflationarisingfromanincrease inmoneysupplycausedbymonetizinggovern- mentdeficits.Inalinearsystem,suchmodes wouldsimplysuperimposeandtheirseparateef- fectswouldbeaddedtoyieldthetotaleconomic behavior. Butinnonlinearsystems,different modesmay influenceoneanother.Thestructuresproducing theseparatemodesintersectthroughnonlinear coupling points that allow one mode to change the characteristicsofanothermode.Considerforex- amplethewaytheshort-termbusinesscycleand theeconomiclongwavecaninteract.Inthelast fewyears,bothbusinessmenandgovernment officialshavebeensurprisedbytheunexpected severityoftherecessionof1982andsurprised againbythevigorofthefollowingrecovery.In fact, recessionsandrecoveriesconsistently grewin amplitudeafter1965. Wefindthesamekindofchange inamplitude ofbusinesscyclesintheSystemDynamicsNa- tionalModel[5].TheNationalModeliscon- structedaccordingtotheprinciplesofthesystem dynamicsdisciplinethatwasfirstdevelopedto dealwithgrowthandstability ofindustrialenter- prises[2]. TheNationalModelisacomprehensive,high- order,very nonlinear representation of the policies inbusiness,labor,banking,households,and governmentthatinteracttoproduceeconomic behavior.Therearenoexogenousdrivingtime series.Thebehaviorisgeneratedentirelybythe internalinteractingpoliciesofthesystem justas theinteractionsoftheparticipantsgeneratereal- life economic behavior. TheModel exhibits allthe majormodesofbehaviorobservedinnational economies.Businesscyclesarisefromtheinterac- tionsofinventories,backlogs,employment,and production.Contrarytousualeconomictheory, theacceleratorprinciplerelatinginvestmentto salesdoesnotparticipatesignificantly inbusiness cyclesbutiscentraltoKuznetscyclesandthe economiclongwave[3,8].TheNationalModel also manifests stagflation, multiple modes of price change,andinflationfrommonetizinggovern- mentdebt. Theeconomiclongwaveandbusinesscycles interactintheNationalModeltocauseanin- creasingseverityofbusinesscyclesinthesame waythathasbeenobservedintherealeconomy since1965. The world'seconomies are now atand beyondapeakintheeconomiclongwaveas indicated by excess manufacturing capacity, exces- sivedebts,risingdefaults,risingunemployment, fallingreturnoninvestment,stagflation,andde- clining pricesofagricultural land.Atsuchapeak in the long wave in the National Model, short-term businesscyclesbecomemoresevere,asthey have alsobeendoingintheactualeconomy.Thein- creasingamplitudeofbusinesscyclesarisesfrom how the long wave affects business cycles. Nonlin- earitiesintheeconomicsystemcausethelong waveto change the conditions determining behav- ior of business cycle.Theeffect isexertedthrough theavailability of manufacturing capacity andun- employedlabor.Asapeakofthelongwaveis approached,bothexcessmanufacturingcapacity and unemployment increase.Underthe conditions of idle production inputs,ashort-term increase in demandcanbequicklyandaggressivelymetby businessengaging theidleresources;therecovery isvigorous;outputcanquicklyincreasebeyond demand;inventoriesriserapidlytolargerex- cesses;then,productionmustbesharplyreduced inasteeperthanexpectedrecession.By contrast, businesscyclesweremildduringthe1960s;there wasashortage of both factory capacity and labor; J. W.Forrester/NonlineariO, in high-ordermodelsof social .~vstems107 businesswasunabletooverproduceatthepeaks; andunsatisfieddemandliftedanytendencyto- wardrecessions.Theeconomiclongwavewidens andnarrowsthe range within which businesscycles canmove. Asweobserve behaviorintheNationalModel, andcorrespondingly interpretthehistoryof actual economicbehavior,mildrecessionsbetween1945 and1965werenotduetoKeynesianeconomicsor finetuningthroughmonetarypol i cy.Instead,the economiclongwavewasworkingthroughnonlin- earcouplingstoproduceweakbusinesscycles whilethelongwavewasexpanding,andthen shiftedtoproducegrowingbusinesscyclesasthe longwavereacheditspeakandstartedtocon- tract. Modesofbehaviorcanexistthatareunstable atsmallamplitudes,butboundedbynonlineari- tiesatlargeamplitudes.Byanunstablemodewe meananoscillationthattendstogrowinampli- tudefromcycletocycle.Inalinearsystem,the amplitudeofsuchamodewouldgrowwithout limit,butunlimitedexpansionisimpossibleinan actualsystem.Realsystemswithpersistentunsta- blemodescontainnonlinearitiesthatboundthe possibleexcursions.Weseetheeconomiclong wave,whichisresponsibleforthegreatdepres- sionsofthe1830s,1890s,and1930s,andforthe presentlyunfoldingeconomiccrosscurrents,as beingsuchanunstableoscillatorymode[10].The longwaveisavigorousandpersistentmodewith onlyasmall(some30percent)variationinperi- odicity.Anaggressive,sustainedoscillationwith smalldifferencesinintervalsbetweenpeakssug- gestsaboundedunstablemode,IntheNational Model,iftheModelissetinequilibriumandthen disturbed,thelong-wavemodebeginstodevelop andwillreachfullamplitudeaftertwoorthree cycles.Thelong-wavefluctuationintheNational Modelisunstable,thatis,itgrowsinamplitude fromasmallbeginning.Webelievethattheeco- nomiclongwaveinindustrialeconomiesislike- wiseanunstablemode.Astheamplitudegrows,it islimitedbyamultitudeofgraduallyincreasing nonlinearities.Therearenosharpdiscontinuities inthemodelorinreallifebutaseitherupperor lowerlimitsareapproached,furthermovement becomesmoreandmoredifficult. Thewiderangeofrealisticbehaviorinherentin theSystemDynamicsNationalModelarisesfrom itshigh-ordercomplexityandtheattentionthat hasbeengiventocorrectly incorporatingintothe Modelnonlinearitiesthatexistintherealeco- nomicsystem.Asanexampleofnonlinearcon- tent,thedecision functionthatorderscapitalplant foraproductionsector contains18 multiplications anddivisionsand8nonlineartablerelationships. Sucharichrepresentationofnonlinearrelation- shipsallowstheorderingprocessintheModelto respondsensiblytoawiderangeofconditions andtodifferingbalancesofforcesundervarious economiccircumstances. 3.Useofnonlinearities Bymakingextensiveuseofnonlinearities,the taskofconstructingrealisticmodelsofsocialsys- temsbecomeseasier.Useofinformationabout nonlinearitiesopensthewaytomanymoreinputs tomodelconstructionfromknowledgeaboutthe realsystem.Properinclusionofnonlinearities matchesamodelbettertothesystemitrepresents. Muchof theinformation we possessaboutreallife isinformationaboutnonlinearcontrolpolicies. Nonlinearmodelshavemorepointsofcontact withtherealsystemstheyrepresent,andlend themselvestoabroaderarrayofvalidationtests thandolinearmodels[6]. Knowledgeaboutnonlinearitiesistobefound primarilyinthementaldatabase[4].Theinfor- mationwepossessaboutasocialsystemcanbe dividedintothatstoredinpeople' sheads,thatin descriptivewrittenform,andthatwhichhasbeen measuredandforwhichnumericaldataexist.The mentalstoreof knowledge isoverwhelmingly more comprehensiveinrevealingsystemstructure,the informationavailableatdecision-makingpoints, andthenonlinearnatureofdecision-making criteria. Awealthofknowledgeexistsaboutthegeneral natureofnonlinearrelationshipsforwhichthere isnonumericaldata.Suchdescriptiveknowledge canbeusedinmodelbuilding.Theknowledgeis inpeople headsandcanbededucedfromthelogic ofhowaparticularpartofasystemmustneces- sarilyoperate.Inmanysituations,slopesofrela- tionships,intercepts,andasymptotesareade- quatelyknown,eventhoughtherealsystemmay neverhaveoperatedinthecorrespondingregions andnohistoricaldataexists.Itisimportanttouse suchknowledgeaboutextremeboundariesbe- 108J.W. Forrester /Nonfinearity in high-order models of social systems causetheysetlimitsonthedegreeofuncertainty thatcanexistinthemidrangeofafunction. Representationofsuchlimitingconditionsmakes amodelfar morerobustandpreparesittooperate properly under possible future conditionsthat have notbeenencounteredinthepast. Asanexampleof usingdescriptive information derivedfromboundaryconditions,considerthe relationshipbetweenthefractionofmaximum salesthatcanberealizedasthatfractionvaries withchangingdelay indeliveringtheproduct.The verticalaxisforsuchafunction,is"Percent of MaximumSales"(givenaparticularcombination ofotherconditions),andthehorizontalaxisis "DeliveryDelay(Timeunits)".The100percent pointforsalesisatthezerodeliverydelayposi- tion;asfarastheeffectofdelayonsalesis concerned,thatisthebestthatcanbedone.Whathappensasdeliverydelayincreases?Forsmall increasesindelay,salesdonotdrop,thecurveis horizontalaslongasdeliverydelayisbelowthe awarenesslevelofthecustomer.Butasdelay increasesfurther,salesbegintofall anddoso with increasingsteepnessasdelayrisestothepointof seriousconcerntothecustomer.Thenthecurve flaresouttobecomeasymptotictothehorizontal axisassalesapproachbutdonotreachabsolute zerowithincreasingdelay(foranylargedelivery delayatwhichasaleispossible,thereremains somesmallprobabilityofasaleatastillhigher delay).Oncetheintercept,theinitialslope,and theasymptoticendsectionhaveallbeende- terminedbylogicalconsiderationofthenatureof themarket,littlefreedom remainsforfilling inthe remainderofthefunctionalrelationship.Sensitiv- itytestsofalternativefunctionslyingwithinthe remainingrangeofplausiblechoicewillusually showlittleeffect onbehavior. Arichrepresentationofnonlinearitiesleadsto amodelthatisrelativelyinsensitivetoparameter values.Beinginsensitivetoparametervaluesis alsoacharacteristicofmostsocialsystems.Ifthe realsystemisinsensitive,soshouldbethemodel. Asanexampleofinsensitivitytoareasonable rangeofpolicies,atthepresenttimemanycoun- triesareexhibitingsimilareconomicpr obl ems- -risingunempl oyment , excessmanufact uri ng capacity,risingdebt,andinflation.Ifsucheco- nomicdifficultiesweresensitivetodifferinglaws, cultures,monet aryauthorities,andpolitical ideolgies,thentherewouldnotbesomuchover- lapinsymptomsofeconomic distress.Infact,the operatingpointofasystemtendstomovealong thechangingslopesofitsnonlinearitiesuntilit findsanoperatingregionthatisdeterminedmore bythestructureofthesystemthanbyplausible differencesinparametervalues.Inahigh-order nonlinearsystem,onecanmovemanyparameters withinaplausiblerangewithlittleeffect onessen- tialbehavior. 4.Summary Nonlinearitiesplayadominantroleinde- terminingthepuzzlingandtroublesomecharacter- isticsoftherealworldinwhichwefive.Social scientists,onewouldliketoassume,aredevoted tounderstandingwhytheworldbehavestheway itdoes,and,fromthatunderstanding,tocreating amorehumaneandsafersociety.Suchbetter under st andi ngwillprobablybeachievedonly throughmodelsthatcanbeusedforinvestigating theessentialcharacteristicsoftherealworld. Butisthepresentmodelingofsocialsystems closelyenoughalignedwiththenatureofthe systemsthatthemodelsaresupposedtorepre- sent?Itseemsthatmostmodelsdonotpassthe test.Modelshavetraditionallybeendominated morebythedesiretofollowtheeasyroadin modelbuildingthanbydedicationtocapt uri ng theessenceoftherealworldthatisbeingmod- eled. Theprevailingattitudeofignoringthefullsig- nificanceofnonlinearityisoneofseveralwaysin whichmodelsarecreatedatvariancewiththe structureandprocessesoftherealworld.Dealing withnonlinearsystemsislesselegantandless precisethancopingwithlinearsystems.Innonlin- earsystems, resultsare less generalizable, butmore relevant.Sweepingtheoriesarereplacedby boundedclassesofrulesofthumb.Ifthenonlin- earworldisforthrightlyattacked,theworkofthe socialscientistwillbecomemorelikethatofa professionalinengineeringormedicineandless likethatofatheoristinmathematicsorthephysi- calsciences. Onlythroughmoreacceptanceofthenatureof therealworld,andlessinsistenceonfollowing traditionalmathematicalandmodeling' methodol- ogies,willthesocialsciencesbegintocouplewith theconcerns,fears,andgoalsofthepublic. J.W.Forrester/Nonlinearity inhigh-ordermodelsof social systems109 References [1]Blatt,John,"Howeconomistsmisusemathematics",in: AlfredS.Eichner(eds.),WhyEconomicsIsNotYeta Science,M.E.Sharpe,Armonk,NY,1983. [2]Forrester,JayW.,IndustrialDynamics,TheMITPress, Cambridge,MA,1961. [3]Forrester,Jay W.,"Growthcycles",DeEconomist125(4) (1977)525-543. [4]Forrester,JayW.,"Informationsourcesformodelingthe nationaleconomy",JournaloftheAmericanStatistical Association75(371)(1980). [5]Forrester,JayW.,"Analternativeapproachtoeconomic policy:Macrobehavior from microstructure",in:NakeM. KamranyandRichardH.Day(eds.),EconomicIssuesof theEighties,TheJohnsHopkinsUniversityPress,Bal- timore-London,1979. 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