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1、ChapterHigher-OrderDifferenceEquationsTheeconomicmodelsinChap_17involvedifferenceequationsthatrelatePIandPr-1toeachother.AsthePvalueinoneperiodcanuniquelydeterminethePvalueinthenext,thetimepathofPbecnmesfullydeleminuteonceaninitialvaluePoisspecified.[tmayhap·pen,however~thatth
2、evalueofaneconomicvariableinperiodt(saY,)'I)dependsnotonlyony,_1butalsoonYI2.SuchasituatIonwillgiverisetoadifTerenceequationofthesecondorder,Strictlyspeaking,asecond·orderdifferenceequationisonethatinvolves,Inexpression,6,2Yr'calledtheseconddifferenceof.h,butcontainsnodifferen
3、cesoforderhigherthan2.Thesymbol22L'..2,thediscrete·timecounterpartofthesymboldjdr,isaninstructionto"taketheseconddillereru,:c"asfollow~:t/rr=.1(.1.YI)=.1(Yi+l-y,)[bYII?I)]=CVI~2-.Fltl)-(Ylt1-J'I)[againby(/7.11]'=YI+2-2Yltl+YIThusaseconddifferenceafYIistransformableintoasumofte
4、nnsinvolvingatwo-periodtimelag,Sinceexpressionslike,6.2Ytand~)',arequitecllmbersometo-vNkwith,weshallsimplyredefineasecond-orderdifferenceequationasoneinvolvingatwo-periodtimeJaginthevariable,Similarly,athird~orderdifferenceequationisonethatinvolvesathree~periodtimelag,etc.Le
5、tusfirstconcentrateonthemethodofsolvingasecond-orderdifferenceequation,leavingthegen~ralization10higher-orderequationsinSection18.4.Tokeepthescopeofdiscu~~ionmanageable,weshallonlydealwithlineardifferenceequationswithconstantcoefficientsinthepresentchapter.However,boththeconst
6、ant·termandvariable-tefmvari•etje~willbe1;'"X3mined,tThatis,wefirstmovethesubscript,inthe(Vt+lVI)e;.:pressionforv"i1rdbyoneperiod,togetanewexpression(VC,,2-YI+I),andthenwe5ubtractfromthelattertheOriginalexpression.Notethat,sincetheresultingdifferencemaybewrittt'll~sil.VI+1-l1y
7、t,wemayinferthefollOWingruleofoper~tion:il.(Yttl)'t)=11)'1_1-AYt568Thisi~reminiscentoftheruleapplicabletothederivativeofasumordifference,Chapter18Higher·OrderDlfjin'nt:eEquationl'56918.1Second-OrderlinearDifferenceEquationswithConstantCoefficientsandConstantTermAsimplevarietyo
8、fsecond-ornerdifferenceequationstakestheformYrtZ+aU,'11j+(12)
