linear models andmatrix algebra

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1、ChapterLinearModelsandMatrixAlgebraFortheone-commoditymodel(3.1),thesolutionsp~andQ~usexprc~sedin(3.4)and(3.5),respectively,aTCrelativelysimple,eventhoughanumberofparametersurcinvolved.Asmoreandmorecommoditiesun:-incorporatedintothem0dcl,suchsolutionformulasquick

2、lybecomecumbersomeandunwieldy.Thatwaswhywehadtoresorttoalittle,~horthand,evenforthetwo-cornmoditycase~inorderthatthesolutions(3.14)and(3.15)canstillbewritteninarelativelyconcisefashion.Wedidnotattempttotackleanythree-orfour-cOlnmoditymodels.eveninthelinearversion

3、,primarilybecausewedidMtyethaveatourdisposalamethodsuitableforhandlingalarget.ystemofsimultaneousequations.Suchamethodisfoundinmatrixalgebra.thesubjectofthischapterandthenext.Matrixalgebrac

4、equationsystem.evenanextremelylargeone.Second,itleadstoa'iayoftestingtheexistenceofasolutionbyevaluationofadeterminant-aconceptcloselyrelatedtothatofamatrix.Third,itgivesumethodoffindingthatsolution(ifitexish).Sinceequationsystemsareencounterednotonlyinstaticana

5、lysisbutalsoincomparative·staticanddynamicanalysesandinoptimizationproblems,youwilllindampleapplicationofmatrixalgebrainalmosteverychapterthatistofollow.Thisiswhyitisde~jrab!ctoin•troducematrixalgebraearly.Hov,levcr.oncslightcatchisthatmatrixalgebraisapplicableon

6、lytolinear~equationsystems.H(lwrealisticallylinearequationscandescribeactualeconomicrelati()nshipsde•pends,ofcourse,onthenatureoftherelationshipsinquestion~Tnmanycases,t:v~nifsomesacrificeofrealismisentailedbytheassumptionoflinearity,anassumedlincarrelation·shipc

7、anproduceasufficientlyclos.;;approximationtoanactualnonlinearrelationshiptowarrantitsusc.Inotherca~cs,whilepreservingthenonlinearityinthemodel,wecaneffectatransfor•mationofvariablessoastoobtainalinearrelationtow~lfkwith.Forexample,thenonlinearfunctionDy=axcanbere

8、adilytransformed,bytakingthelogarithmonbothsides,intothefunctionlogy=Joga+hlogx48Chapter4Lincar1l1oJdsum!Afulri;;Alglthru49whichislinearinthetwovariables(logy)