The Accelerator Multiplier Model

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1 TheAccelerator MultiplierModel AnnickAshley StephanieChapman MirInaamullah Math4700 Fall2009 Abstract: WeexaminetheAccelerator Multipliermodelintwodimensionstodeterminehowwellit conformstotrueempiricaldata.wethendevelopathirdequationtodescribethesavingsrateasa functionofbothincomeandtherateofchangeofincome.wefindthatthethree dimensionalmodel betterdescribestheperiodicmotionofthesavingsrate,whilemaintainingthecyclicnatureofincome. Themodelhassomeweakpoints,however,especiallytheregulartendencyofthesavingsratetobe negative,whichisonlyinfrequentlysointherealworld.thus,whileourmodeldescribesthevariation inthesavingsratesomewhatmoreaccuratelythanthetwo dimensionalaccelerator multipliermodel,it failstoaccuratelydepictthetruevaluestakenbyempiricaldata.

2 I.Background Wehaveresearchedthecombinationoftwoeconomictheories:theacceleratoreffectandthe multipliermodel.theacceleratoreffectmeasureshowmuchthegrowthofthemarketeconomyalters theamountofprivatefixedinvestment,(i.e.investmentintangiblecapitalgoodsorthereplacementof depreciatedones).itdoesthisthroughanalyzingthegrossdomesticproduct,(gdp),notingthatasit increasesbusinesseswill,intheory,seerisingprofits,highersalesandmorecashflow.allofthese trendswillallowbusinesstohavemoreconfidenceintheirabilitytoboostprofit;thustheywillmore likelyexpendmoremoneyonprivatefixedinvestment. Thereareafewimplicationsoftheacceleratoreffect,becauseeconomicgrowthmustbetakeninto consideration.oneisthateconomicgrowthislessprecariousthaninvestment.moreover,duringtimes ofstableeconomicgrowth,investmentwilllikewiseremainconstant.however,thismeansthatevenif GDPisrising,investmentrateswillactuallydropifeconomicgrowthdecreases. Wewillnowdiscussthemoreimportanttheorytoourmodelisthemultiplier.Amultiplier,in economics,isusedmoregenerallytomeasurehowanexogenousvariableaffectsanendogenousone. Westudiedthemultiplierprocessasisspecificallyinvolveshowone sspendingessentiallybecomes another sincome.ultimately,thereisamuchgreaterimpactontheequilibriumofnationalincome. Wecanseetheextenttowhichthemultiplieraffectsinvestmentthroughthegraphofaggregatesupply versusaggregatedemand: Theverticalarrowsrepresentaconstantriseindemand,whereasthehorizontalarrowsrepresentthe resultingshiftinsupplyfromthetwoequilibriumpoints.ascanbeseen,asmallchangeintheformer createsamuchgreaterdifferenceinthelatter.thisiswherethemultipliercomesintoplay.once aggregatedemandrises,expectedsalesandoutputincrease,which,inturn,causehigheremployment ratesandincome.then,theincreaseinincomewillproducemoreconsumption,raisingaggregate demandyetagain.thus,thecycleofthemultipliereffectwillcontinueinthisway. 2

3 Anexampleofthegreatimpactabusiness sspendingcanhavewouldbeifacompanyinvested200 milliondollarsintoanewmanufacturingplant.thiswillresultinawholechainofexpenditures.first, thebusinessessupplyingcapitalgoodstocreatethenewplantandthegooditproduceswillreap benefitsasfarastheirincomeisconcerned.thenthey,intheory,willspendaround3/5oftheirnew profits,whichmeansotherbusinesseswillgain120milliondollarscollectively. Thus,thetotaleconomyhasgained(200+(3/5)*200))dollarsinincomealready.Moreover,the producersofthenextgoodsandservicesonwhichthe120milliondollarswerespendwilllikewiseuse 60percentoftheirprofitsonexpenditures.Therefore,wenowhaveatotalamountof(200+((3/5)* 200)+((3/5)+120))dollarsworthofnewincomeintheeconomy. Thecyclewillcontinuethusly,butitisimportanttonotethattheriseinspendingdecreasesbya fractioneachstepoftheway.thecyclebecomesmostnotablewhenprofessionalsareattractedtoan industryinaparticularcityorregion.whenbusinessthrives,theflowofspendingwillhaveasignificant impactonthelocaleconomy. Themultipliermodelisespeciallyrelevantinregardstoemployment.Inthe1930sJohnMaynardKeyes recommendedtheadministrationuseittomeasurehowmuchgovernmentspendingwasnecessaryto ensurethepreciselevelofnationalincomethatwouldworkinavoidingunemployment.onewayforthe governmenttoencourageextraspendingistodecreaseincometax.indoingso,however,itmustalso stresstheimportanceofconsumingdomesticallymanufacturedgoods.thisisbecausewhenmoneyis spentonimportedgoods,incomeisnotcirculatingbackintothelocaleconomytobespentagain throughthemultipliercycles. Themultipliereffectalsomodelshowinvestmentintosavingsaccountscanincreasemonetary circulation.whenthefederalreservesetsthereserverequirementtoacertainamount,themoney supplyisaltered.forexample,ifthereserverequirementissettotwentypercent,abankcanloanupto 80dollarsofa100 dollarcustomerdeposit.the80dollarswillbe,then,depositedintoanotherbank thatcanthenloanoutanother64dollars.thecyclewillcontinuethiswaythroughthemultipliereffect. Itonlytrulyaffectscirculationwhenmoneyisdepositeddomestically,butitessentiallygivestheFed greatpowertoaffectmoneysupply.thus,ifthefednotesarecessionintheeconomyandraisesthe reserverequirementtosomewherearoundfortypercent,muchlessmoneywouldbecirculatinginthis samemultipliereffect. 3

4 II.BuildingtheModel Aswehaveexplained,thesetheMultiplierandAcceleratormodelsaccountforobservedphenomenain markets.however,inordertoexplainother,morecomplexpropertiesofbusinessdataitisnotenough examinethesetwomodelsindependently.inparticular,weseektoatleastpartiallyexplainthe existenceofbusinesscyclesbycreatingadynamicalsystemfromthebasisofthemultiplierand Acceleratormodels. Letuslookatanexampleofhowthesetwomodelstheoreticallyreinforceeachother,causingcyclical behaviorinthemarket.withanysortofincreaseinspending(oftendoneexogenouslybythe government),thereisanexactriseinconsumerincome.thisraisestheamountofmoneyspentby consumers,whichraisesaggregateoutputbythemultipliereffect.then,dependingonthehowquickly orslowlynetinvestmentincreases,theacceleratormodelwillkickin.thisinturnwilldictatean increaseordecreaseintotalspending,whichbeginsthecycleanew.thisverbalmodel,however,does notallowustoquantifytherelativeeffectsandstrengthsofthecompetingeffectsofthemultiplierand Acceleratormodels.Thus,wemustconstructadynamicalsystem,which,ifweuseaccurateparameters andconditions,shouldmodeltheeconomyofourchoice(whichinthiscase,meanstheuseconomy). OurmodelisafunctionofIncome,which,wemustrecall,issynonymouswithGDP.Ifthemodelis properlysetupandtheparametersareaccuratelychosen,weshouldseecyclicalfluctuationsinincome (I).Wewillexaminethereal worldimplicationsofourresultsbelowinthediscussion. Focusingonthemathematicsfornow,wealsoknowthatwemustincludeinourmodelparameters fromeachofthetwoparentmodels.fromthemultipliermodel,weporttheparameter s, the personalsavingsrate,andfromtheacceleratormodeltheparameter v, theratioofcapitolstock. WhatwehaveconstructedisasystemintheformofavanderPolequation. (*) ForeaseofanalysisanduseofXPPwerewritethefunctiona2 dimensionaldynamicalsystem: ( ) Whilewebelievethemodeltobeaccurate,wehavenowayofcheckinguntilwepluginestimatesof theparametersandexaminethebehaviorofthefunction,whichwewillachievethroughxpp.thefirst orderofbusinessisfindingaccuratevaluesfortheparameters s and v.since v isgenerally thoughttobeheldconstantfromtheacceleratormodel,wesimplyuseacommonlyagreeduponvalue ofv=1.1forourpurposes. Finding s however,isataskthatrequiresabitmorecaresincethesavingsrateisavolatileparameter. UsingdatafromtheU.S.BureauofEconomicAnalysis,weanalyzethebehaviorofthesavingsrateover thepast4years.inthisperiod,asevidencebythechart,thesavingsrateitselfhasseensome fluctuationsrangingfromroughly1% 5%.Itisinterestingtonotethatthedataindicateanaturallevel ofthesavingsrateatroughly2%(or0.02)overthepast4years,andthatonlyinthelastfewquarters havepersonalsavingsshotuptothehighof5%.this,too,shallbeaddressinthesimulationand Discussionportionsofthereport,butsufficeittosaythatwewillusethepresumednaturallevelof savingss=0.02astheinitialvalueoftheparameterforthepurposesofphaseplaneanalysis. 4

5 Usingtheseparameterestimateswearriveatthefinalformofourdynamicalsystem: ( ) Fromhere,weproceedtoouranalysisofthesystem. 5

6 III.AnalyzingtheModel Ouranalysisofthesystemmustproceedalongbothquantitativeandqualitativefronts.First,we analyzethesystemusingthecurrentparametersfrom( ),andanalyzeitsbehaviorusingxpptoseeif ourmodelsatisfactorilyfitsthecyclicalbusinessmodelasweexpected. Figure4.1 S=0.02 QualitativelyassessingthebehaviorinFigure1,itappearsthat,thesystemdoesindeedfollowcyclical behavior.itisworthnotingthatforthisparticularsystem,thexvalues(calledrinfigure1)represent therateofchangeofi(e.g.thehowquicklyincomechanges)andtheyvalues(callediinfigure1) representactualvaluesofincome.becauseofwhattheseaxesrepresent,itisimportanttonotethatin ordertoexhibitcyclicalbehaviorweareactuallylessinterestedinseeingastablelimitcycleinthe diagramasopposedtocyclicalmotioninonlythevaluesofi.inshort,toremovethepotentially confusingeffectorr,weexaminethechangeiniovertime. WeisolatedthegraphofIvs.TinXPP(asshowninFigure2)andfoundthatIncomelevelsveryquickly approachasteadyperiodicityandamplitude,whichleadsustobelievegivenreal worldsituations,the modeldoesexemplifycyclicalbehaviorofusaggregateincome.however,asitisunwisetotrustonly graphicalevidence,wealsoshowtheexistenceofalimitcycleinfigure1byanalyticalmeans.the equationin( )fitsthemoldofalienardequation.asproveninourtextbook,ithasbeenproventhat Lienardsystemshaveunique,stablelimitcycles.Thusweposittheexistenceofastablelimitcyclein ourmultiplier Acceleratormodel. 6

7 Figure4.2 S= 0.02 Oncewehavesolvedtheactualcase,however,wemustseehowtheMultiplier Acceleratormodelwill dealwithendogenousshocks.wehavealreadyestablishedthatbythenatureoftheparameters,the capitolstockrationwillbeheldlargelyconstant.evenifnotperfectlyso,theshockstovwillbeminor. Thesameisnot,however,trueforthesavingsrate.Asshownearlier,itisafactthatsvariesovertime. Letusthenexaminehowchangesinthesavingsrateaffectthelongtermbehaviorofthesystem. Toaidusinfindingcriticalpointswherethesystem sbehaviorwillchange,wecanreturntotheoriginal systemin(*).clearly,weseethatthefactorof changesthesignofthelinearterm,which dominatesforsmallvaluesof.noticethatthecoefficientofthecubicfactor,whichdominatesfor largevaluesof,isafunctionofv,andthuswillnotvarymuchduetoexogenousshocks.returningto thefactorof,wesuspectthatdifferentqualitativebehaviorwilloccurwhenthesignofthe termchanges.plugginginv=1.1,wegetthatthethreecriticalvaluesofsares<0.10,s=0.10,ands> 0.10.Letusexaminethesethreecases. First,examinethecasewheres<0.10.Fromtheoriginalmodel,weupdatestobeclosertothecritical value,increasingitto8%.thexppoutputfromrunningthissystemisdisplayedinfigure3. Qualitatively,itisinterestingtonoticethatthatlimitcycleismorecompressedinthedirectionof impliesthatwhenthesavingsrateisincreased,the speed atwhichtheinvestmentcyclesoccur increases.allinall,thegraphlookssimilartothereal worldsituationmodeledinitiallyinfigure1, whichistobeexpectedsincetheparameters=0.02fallsunderthedomainofs<0.10. This 7

8 Figure4.3 Figure4.4 S=0.08 S=0.10 Atthecriticalvalueofs=0.10,the systemseemstoapproachnearly standingcycles,asseeninfigure4. Thereasonstandingcirclesdonot formisthepresenceoftheextra factorof.contraryto expectation,thesecyclesarenot veryinterestingfortworeasons. Firstly,eveniftheinitialcondition andparametersledtothe economygetting caught inone ofthesestandingcircles,itwould instantaneouslyexhibitthe characteristiccyclicalmotionof Incomethatwedesiredinthefirst place.secondly,duetothe variationinsovertimecompared tothestaticnatureofv,weexpect thebalanceof =0tobe lostratherquickly. InFigure5,weseethelimitcycle thathadbeenshrinkingass increasedtowards0.10.oncethe thresholdwaspassed,theorigin becomesastablespiral.thisholds trueforallvaluesofsallthewayto 1,whichistheupperlimitonthe parameter(sinceitisapercentage). Figure4.5 S=0.30 Nowtakingastepback,wenotice wehavewitnessedthattheorigin,a fixedpoint,haschangedstabilities astheparametersofthesystem werealtered.moresignificantly,the originchangesfromastablenodeto anunstablenodesurroundedbya stablecycle.ifweweretoplotthe bifurcationdiagram,itisclearthat thesefeaturesarecharacteristicofa super criticalhopfbifurcation. 8

9 IV.ExpandingUpontheModel Inordertoextendthismodel,weconsiderhowthissystemmightbehaveifthesavingsratewerea variableaswell.thisadditiontothemodelmakessense,becauseasincomeandtherateofchangeof incomechangeovertime,onecanreasonablyexpectthesavingsratetochangeaswell.whilethe modeldescribesmacroeconomicconcepts,foreaseofcomposition,wewilldiscusstheincomeand savingsrateasiftheywereofanindividual. Thinkingaboutthepropertieswewouldwantthesavingsratetohavegivesusastartingpointfor thinkingaboutwhatsortoffunctionmaydescribetherateofchangeofsavings.consideringthecycles wehaveobservedalready,onemightexpectthatthesavingsratewillincreaseordecreasein accordancewithwhetheronesincomeisincreasingordecreasingatthetime,aswellaswhetherones incomeispositiveornegative(indebt).thus,wewouldlikethesignofthederivativeofstochangeas thecyclemovesfromquadranttoquadrant. Wemovefromthistothenconsiderwhatsignwewouldlikethederivativeofstohave.Inthefirst quadrant,incomeisincreasing,soonemightexpectthatthesavingsratewasincreasing,then decreasinginthesecondquadrant,increasinginthethirdquadrant,thendecreasinginthefourth quadrant.wealsowouldliketheabsolutevalueoftherateofchangeofthesavingsratetobelessthan one,becausethatwillinhibitthesavingsratefromgrowingtooquicklyaslongastheinitialconditionis between1and 1.Alloftheserequirementsleadustotheadoptionofthesinefunctionfors.Sincewe wants todependonbothincomeandtherateofchangeofincome,thenwewilltaketheargumentof sinetoberi.finally,sincewewouldliketobeabletocontroltheextentofvariationinthenewterm, wewilldividebyaconstantk,whichwewillseecanbeinterpretedastheresistanceofindividualsto changetheirsavingsrate,asitrestrictsthepossiblevariationofthesavingsrate.thus,wehaveanew system r = (v "1" s)r " vr 3 I = r s = sin(ri) k 3 " si (5.1) Asanalternativeoption,wewillalsoconsiderthesystemproducedbyusinganegativesineargument:! r = (v "1" s)r " vr 3 I = r s = sin("ri) k 3 " si (5.2) Bothsystemshavethepropertythataskapproachesinfinity,thatis,astheresistancetochangingthe savingsrateincreasestoinfinity,s goestozero,whichindicatesthatsbecomesaconstant.thus,ask! approachesinfinity,ourmodelreducedtotheoriginalsystemanalyzedinsectioniv. 9

10 SinceXPPhaslimitedcapabilitiesinthreedimensions,muchofoursubsequentanalysisisdoneinthe AppleapplicationGrapher.Byinputtingtheequationsandaninitialcondition(wewilluser=1,I=1and s=0.2,withparametervaluesk=2andv=1.1asinsectioniv),wecanobservethegeneralbehaviorof system5.1infigure5.1below: Wecanseethatthismodelfalls Fig5.1 intocyclicmotion(thoughnota limitcycle),whichisconsistentwith ourpreviousanalysis.also,the savingsratemovesinapredictable fashion,risingandfallingasit movesfromonequadrantoverthe xyplanetoanother.wecanseein Figure5.2thatthesavingsrate exhibitsperiodicbehavior,whichis consistentwithourpredictions aboutthisnewmodel.thismodel alsoallowsforthesavingsrateto becomenegativeinaregular, periodicfashionwithoutdisrupting thecyclicalnatureofthesystemasawhole.thisresultisencouragingforasocietysuchasoursthat frequentlyhasanegativesavingsrate,especiallysincenegativesavingsratesledtosignificantproblems inthetwodimensionalmodel. Notealsothatthesavingsrateapproacheszero,althoughitdoessoveryslowly. Fig5.2 10

11 Lookingatthesecondmodel,picturedinFigure5.3,wesee thatthesavingsratebehavessomewhatdifferently, oscillatingmuchmoretightlyaroundthes=0plane.aswe canseefromfigure5.4,theslightchangethatdifferentiates thismodelfromthepreviousleadstothesavingsrate approachingzeromuchmorequicklythantheother(note thatbothchartsarerepresentedonthesametimescale.) Thequestionremains:whichofthesemodelsisamore appropriatedescriptorofreality?thefirstmodelproduces muchsmootherbehavior,howeverthesavingsratesit predicts(withtheparametersused)areverylarge comparedtothetypicalsavingsratesinthiscountry. Fig5.3 However,thedampeningfactorkcaneasilybeincreasedto mitigatethisresult.bothmodelstheoreticallyallowfor savingsratesgreaterthanone,asituationthatmakesno practicalsense,andhasnotruephysicalmeaning.bothmodelspredictthatthesavingsratewillgo belowzeromuchmorefrequentlythanistrueintherealworld,whichindicatesthatadjustmentsmay needtobemadeinordertoadequatelyrepresentthetruecircumstancesofreality. Fig5.4 Themodelsarebeneficial,however,inthattheyaccountforsucheventsasnegativesavingsrates, unlikethetwodimensionalmodel.infact,givenanegativesavingsrate,thesemodelswill,aftera periodofeitherexplosivegrowthordebt,bringthesolutionsbacktothecyclicalmotioncharacteristic ofthesemodels. 11

12 V.Discussion Nowthatwehaveexaminedthebehaviorsofthegraphsatisfactorily,theultimatetestistoseehow wellourmultiplier Acceleratormodelcomparestoactualdata.Intheprocessofdeterminingthe savingsrateearlyonincomputingthemodel,weusedpubliclyreleasedgovernmentdatafromthelast fouryears.intheinterestofconsistency,wewillcheckoutmodelagainsteconomicdatafromthelast4 years.sincethedataispublishedquarterly,thismeanswemayusedatafromthelast16fiscalquarters tocheckourresults. HerewehaveachartshowingpercentagechangeinGDPoverthelast16quarters.Bearinginmindthat GDPisaggregateoutput,andfromandeconomicstandpointthetotaloutputatmarketequilibrium mustequaltotalincome,gdpmustbedirectlycorrelatedtoincome.thus,themovementof percentagechangeingdpiscorrelatedto.thus,bothourmodelandprevailingeconomictheory predictsinusoidalmotionofthecurve. AswecanseefromFigure6,theredoesappeartobeacyclicaltrend.Theglaringproblemwith correlatingourmodeltorealityistheunusuallylargedipinthecurve,whichlastedforthelast2 5 quarters(i.e.thelast15months).thisis,ofcourse,exactlythetimespan,whichcoveredtherecent globalrecession. Whywouldthisrecessionadverselyaffectourmodel?OurMultiplier Acceleratormodel,althoughmore powerfulthaneithermodelindependently,isstilllimitedinscopebytheparametersusedtocalculate it.whatthismeansisthatwhilethemodelshouldaccountforanypossibleendogenous,anyexogenous shockswillchangethemodebeyondtheexplanatorypowerofthemodel.inthiscasetherecession seemstohaveexacerbatedtheeffectsofthenaturaldropingdppredictedbythemodel. 12

13 Turningourattentiontothethreedimensionalextensionofthemodel,wecomparethepatternsof savingsratespredictedbythemodeltothetruepatternsofsavingsratesoverthepastseveralyears(fig 6.2)andfindthatwhilebothmodelsdomirrorthecyclicnatureofthesavingsrateoverthepast40 years,neithermodelcanaccountfortheexogenouseffectsofchangingsocieties,whichispresumably thecauseoftheconsistentdownwardtrendofthesavingsrate. Fig6.2 13

14 VI.Conclusion Theacceleratorandmultipliereffectsarewidelyacceptedasdescriptivemodelsofeconomics,asisthe interactionbetweenthetwo,themultiplier acceleratormodeldevelopedbypaulsamuelsonin1939 (Puu1997).Theyeffectivelypredictanddescribebusinesscyclesthatdeveloponamacroeconomic level.throughouranalysis,wehaveshownthatifwetaketheratioofcapitalstocktoincome(v)as constant(asiswidelyacceptedineconomics),asupercriticalhopfbifurcationoccursasthesavingsrate increases.thus,businesscyclesonlyexistforcertainrangesofs,afterwhichpointtheorigin(no income)becomesastablenode. Whenweextendouranalysisbeyondtwodimensionstotakesasvariable,wefindthatourmodels accuratelydescribetheperiodicityofthesavingsrateinreallife,thoughthevaluesourmodelpredicts forthesavingsratediffersomewhatfromtheobservedvalues.however,matchingthemovementof thesavingsrateismoresignificantthanmatchingtheobservedvalues,becauseitisarelativelysimple mattertoarrangeconstantsadjustingtheequationstothecorrectamplitudeandvalues.thus,our developedmodelhasachieveditsgoalofapproximatingthesavingsrateasafunctionoftimeanda variableinmotionwiththerestofthesystem,ratherthansimplyaconstant. VII.References FactsonPolicy:USSavingsRate. GrossDomesticProduct:ThirdQuarter2009(SecondEstimate). Multiplier AcceleratorModel. PersonalSavingsRate. JunhaiMaandQinGao.(2009) StabilityandHopfbifurcationsinabusinesscyclemodelwithdelay. AppliedMathematicsandComputation,Vol215(2): ( 4WGF0X4 3/2/e a764d01c69ac11511dd1aaa) Puu,Tönu.1997.NonlinearEconomicDynamics.NewYork:Springer. 14