Modelingtheconcentrationofwealthinpopulationusinga

Supervisor:ID:Modelingtheconcentrationofwealthinpopulationusingastochasticmodel用随机模型模拟人口财富的集中度Ⅰ.IntroductionⅡ.LiteratureReviewⅢ.Methodology3.1AKmodel3.2Markovprocess3.3DiffusionprocessandItolemma3.4StochasticwealthgrowthmodelⅣ.Projectplan4.1NumericalApproximation4.2Theplanandthetimeline4.3Thenewskillsandknowledge4.4MajorchallengesⅤ.ReferenceIntroductionEndogenouswithMalthuspopulationrepresentedbyclassicaltheoryofeconomicgrowth,duetotheassumptionthatpopulationgrowthisproportionaltothelevelofincome,theresultingeconomicandpopulationgrowthispessimisticconclusionstop.Neoclassicaltheoryofeconomicgrowthisanexogenousvariablepopulationwiththeeconomicgrowth,andfailtoexplainthegrowthrateofpopulation,whichhasappearedphenomenon.Thenewtheoryofeconomicgrowthappearsin90years,alsofailtoexplaintheinteractionbetweengoodpercapitaincomeandpopulationgrowth.Thesetheoriesregardpopulationgrowthasdetermined,anddon’ttreatthenumberofpeopleasadynamicrandomvariable.Inthispaper,thenumberofpopulationisafactorofuncertaintyintheeconomicgrowthmodel,constantreflectsthecommonrandomeffectoneveryoneinagiventimeandfertility-relatedsocialcustomsandpreferenceschange,naturaldisasters,nationaleconomicconditions,etc.Butthispaperdonotassumeconstantreturnstoscale,andneedn’tCobb-Douglasformsofproductionfunctions.Theestablishmentofadynamicstochasticmodelofeconomicgrowth,exportexistenceanduniquenessofthestationarydistributionpropertiesandsteady-statedensityfunctionshouldsatisfytheequationtoknow.Thepurposeofthisarticleistodescribetheeconomicgrowthmodel,andMarkovprocess.Andthenbasedondifferentialequationstheory,andassumptionofpopulationasarandomvariable,thispaperbuildastochasticwealthgrowthmodel.Besides,thisarticleusediffusionprocesstodiscussthedynamicpropertyofwealth-laborrate,andanalyzedtheuniqueexistenceofsolutionofthestochasticmodel.Normallyendogenousgrowththeoryassumesconstantmarginalproductofcapitalattheaggregatelevel,oratleastthatthelimitofthemarginalproductofcapitaldoesnottendtowardszero,especiallygiventhatthepopulationgrowthrateisfixedasconstant,whichisinconsistentwiththereality.Thispaperassumeamountofpopulationasasourceofrandomvariable,improveAKmodeltoderiveamorerealisticstochasticmodel.Thestructureofthispassageisorganizedasfollows:Section2summarizeresearchofdistributionofwealthbothathomeandabroad.Recallthetheoryofstochasticmodel,andrevealtheintensionofthisstudyistoexplorewealthgrowthmodelwithrandomvariableofpopulation.Section3willintroduceAKmodel,Markovprocessanddiffusionfirst,andthenmakeefforttoderivestochasticwealthgrowthmodelbasedontheknowledgebefore.Section4containsthenextplan.Sincestochasticwealthgrowthmodelhasbeingderived,nextplanisfocusedonnumericalapproximation.LiteratureReviewTherehavebeenanumberofempiricalstudiesshowthatdemographicchangescouldaffecttheeconomicdevelopment.VincenzoQuadriniandJosé-VíctorRíos-Rull(1997)hasdescribedthecurrentstateofeconomictheoryintendedtoexplaintheunequaldistributionofwealthamongU.S.households.Themodelsendogenouslygeneratedifferencesinassetholdingsasaresultofthehousehold'sdesiretosmoothconsumptionwhileearningsfluctuate.Thearticledescribesseveralfeaturesrecentlyproposedasadditionstothetheorybasedonchangesinearnings,includingbusinessownership,higherratesofreturnonhighassetlevels,randomcapitalgains,governmentprogramstoguaranteeaminimumlevelofconsumption,andchangesinhealthandmaritalstatus.LEIDong-xia,HUShi-geng(2007)employsanendogenousstochasticgrowthmodeltoanalyzetheroleofmaineconomicindicatorsinaffectingeconomicgrowthandwelfare.Thepaperderivetheexplicitsolutionoftheexpectedgrowthratesofconsumptionandsavings.Finally,ifgovernmentexpenditureimpactsdirectlyontheconsumptionandproductionoftheprivateagent,thenincreasingeconomicgrowthmeansdecreasingwelfareordecreasingeconomicgrowthmeansincreasingwelfare.PerKrusellandAnthony(1998)analyzedthedistributionofincomeandwealthandhowtoaffectthemacroeconomy,byusingacalibratedversionofthestochasticgrowthmodelwithpartiallyuninsurableidiosyncraticriskandmovementsinaggregateproductivity.Intheirbenchmarkmodel,theonlydifferencefromtherepresentative‐agentframeworkistheexistenceofuninsurableidiosyncraticrisk,displaysfarlesscross‐sectionaldispersionandskewnessinwealththanU.S.data.However,anextensionthatreliesonasmallamountofheterogeneityinthriftdoessucceedinreplicatingthekeyfeaturesofthewealthdata.Furthermore,thisextensionfeaturesaggregatetimeseriesthatdepartsignificantlyfrompermanentincomebehavior.MarcelF.Neutsgaveavarietyofresultsonqueuesandstochasticmodelswiththeunifyingfeatureofreadyalgorithmicimplementation,basedonaseriesoflecturestheauthorgavethisgraduate-leveltreatmentatTheJohnsHopkinsUniversity.Neutsdefinedastudyofstochasticmodelswithconcerningofalgorithmicfeasibilityoverawide,realisticrangeofparametervalues.Inthebook,theauthoralsodiscussedgeneralpropertiesofaclassofMarkovchains,andprocessesthatinthepositiverecurrentcasehaveamatrix-geometricinvariantprobabilityvector.Tokeepthevolumereasonablyconcise,theauthorhaslimiteddiscussiontooneclassofstructuredMarkovchains,andtoabroadselec