Lecture 3 Technique Preparation Discrete Time Dyna

Lecture3DiscreteTimeDynamicSystem:aTechnicalPreparationMacroeconomicsI.WhyDynamicAnalysisisImportant?Wehaveknownthatmacroeconomicsisabouthowmacroeconomicvariablesaredetermined.Inmacroeconomics,suchadeterminationcanoftenbedescribedbyadynamicsystemintermsofeitherdiscretetimeorcontinuoustime.II.TheDiscreteTimeDynamicSystemLetXt,Yt,…,Ztarerespectivelythedifferentvariablesatperiodt.Thentheirdeterminationmightbedescribedbythefollowingdynamicsystem,whichisindiscretetime:111111111(,,,)(,,,)(,,,)ttttttttttttXfXYZYgXYZZhXYZLLMLII.TheDiscreteTimeDynamicSystemIffunctionf(·),g(·),…,h(·)arealllinear,theabovesystemmaybewrittenaswhereaijandbi(i,j=1,2,…,n)arealltheparameters.11112111121122121211211tttnttttnttntntnntnXaXaYaZbYaXaYaZbZaXaYaZbLLMLII.TheDiscreteTimeDynamicSystemThestandardformofdynamicsystemNotethat(3.1)and(3.2)canberegardedasastandardformofdiscretedynamicsystem.Otherformsofdynamicsystemcanbetransformedintothestandardform(examplesareprovidedinthetextbook)Manytheorems(propositions)toresolvethedynamicsystemisbasedonthestandardform.III.TheSolutionPathandtheSteadyStateThesolutionpathThesolutionofasystemdescribehowthevariableschangeovertimegiventheinitialcondition(X0,Y0,…,Z0).III.TheSolutionPathandtheSteadyStateThesolutionpath(continued)ThegraphicrepresentationofsolutionpathstXXYIII.TheSolutionPathandtheSteadyStateThesteadystateThesteadystateofsystem(3.1)or(3.2),denotedas(X*,Y*,…,Z*),canbeobtainedbyposingtherestriction:*1*1*1ttttttXXXYYYZZZIII.TheSolutionPathandtheSteadyStateThesteadystate(continued)Thesteadystatehasthepropertythatifthesolutionpathofthesystemisconvergent,itmustbeconvergetothesteadystate(seethefollowinggraph)III.TheSolutionPathandtheSteadyStateConvergingtotheSteadyStatetXXYX*X*Y*III.TheSolutionPathandtheSteadyStateThesteadystate(continued)However,thereisnowarrantythatallsolutionwillconvergetothesteadystate(seethefollowinggraph)III.TheSolutionPathandtheSteadyStateNoConvergingtotheSteadyStatetXXYX*X*Y*III.TheSolutionPathandtheSteadyStateThesteadystate(continued)Thereisnowarrantythatthesteadystateisunique(multiplesteadystatescouldoccur)Thereisalsonowarrantythatthesteadystateevenexists(itcouldbecomplex).Bothoftheabovearemorelikelytooccurinanonlineardynamicsystem.IV.SolvingDynamicSystemItisnotalwayspossibletosolvedynamicsystem,thatis,obtainingthesolutionpath.However,inmanycases,itissufficienttodetectthestabilityofthedynamicsystem,thatis,whetherthesystemisconvergenttothesteadystateornot.Suchdetectioncanoftenrelyongraphictechniqueorsomewell-knownmathematicpropositions.IV.SolvingDynamicSystemUsinggraphictechniqueinonedimensionalsystemIV.SolvingDynamicSystemAnotherpossibilitytosolvedynamicsystemistousethecomputersimulation.AnexampleistouseExcelforsimulation(willbeintroducedintheclass)V.ImportantNotesInthiscourse,wegenerallyassumethesystemisstable,thatis,thesolutionwillconvergetothesteadystate.Sometimes,weevenignorethetimesubscript.Inthiscase,ouranalysiscanberegardedassteadystateanalysis,thatis,weassumethatallthevariablesbeattheirsteadystates.