经济数学方法

1經濟數學方法壹、矩陣與行列式◎定義:mn階矩陣為一包括n列和m行的數字的方形排列，若以A代表此矩陣，則mnaaaaaaaaaaAijnmnnmm)(212222111211例:11133111,531321213102BA分別為43和24矩陣◎定義:若mnijmnijbBaA)(,)(則mnijmnijijCbaBA)()(=CmnijaA)(例:315212,112312BA則227520311152231122BA84513412315212551015510)1(55BABA2AAA21123122224624112312112312◎定義:若A=()ija為mn矩陣，B=()ijb為km矩陣，則A和B的乘積AB為kn矩陣C例:130112001,102210BA求AB及BA130112001012120AB=1.1)1(00.23.11.00.20.12.01212)1(10.03.21.10.00.22.11.0=132172BA無法計算3332◎行列式:Cramer'sRule已知1212111bXaXa2222121bXaXa2112221112222122211211222121*1aaaaababaaaaababX2112221112121122211211221111*2aaaababaaaaababaX例:解下列聯立方程式:025312121111321XXX3032225321321321XXXXXXXXX9439312121111310122115*1X9239302121151*2X1*39219012221511X貳、微分◎微分公式:)(XfYdXdYXxfXXfXfXYX)()(lim)(0)(2222XfdXYdXY◎若RXnXXfRXXXfnn,)(,)(1◎設)(Xf與)(Xg皆存在:dXXdgdXXdfXgXfdXd)()()()(dXXdfXgdXXdgXfXgXfdXd)()()()()()(乘法公式0)(,)()()()()()()(2XgXgXgXfXgXfXgXfdXd除法公式◎鏈鎖律(chainrule):設函數f與g皆可微分)())(())((XgXgfXgfdXd◎反函數(inversefunction):設函數f與g滿足f(g(Y))=Y函數g為f之反函數g(f(X)=X且g=f1XXffYYff))(())((114◎偏微分:),(),(211121XXfXyXXfy),(),(212221XXfXyXXfy例:XYXdXd6232◎全微分:),(21XXfy2211dXXydXXydy例:TE=PQPdQdPdTE2◎自然對數(e)與自然指數(ln):性質:(1)0lim1)0()(XXxefeXf、XXelimXfXXfXlnlim0)1(ln)(、XXlnlim(2)XXeedXd(3)設f存在)()()()(XfeedXdXfXf(4)RYXeeeYXYX,,(5)XXee1(6)0,1lnXXXdXdxyexlnx115(7)0,1XXXndxd(8))(()(XfXfXfndXd(9)YXYXlnlnln(10)YXYXlnlnln(11)XYXYlnln(12)XeXln且XeXln(13)YXXeYln◎切線與射線:給定切線上任一點(X,Y))()(00XfXXXfy射線角度值00tanXy◎函數的高階導數:dXdYdXddXYd22、2233dXYddXddXydXXfXXfXXXfXfXfXXX)()(lim)()(lim)(00000000◎函數的臨界點及反曲點:(一)若，不有在Xf或Xf，函數定義域DfX))((0)()(000則0XX為函數f之臨界點(二)函數f在ba,為嚴格遞增)()(2121XfX則fXX(x0,y0)y=f(x)αyxf/(x)0X1X2f(x2)f(x1)XYab6函數f在ba,為嚴格遞減)()(2121XfX則fXX(三)0)(Xf為上凹ba函數f在baX,,f/(x)0X1X2f(x2)f(x1)XYabconcaveupwardf/(x)0f//(x)0XY上凹f/(x)0f//(x)0XY上凹concavedownwardf/(x)0f//(x)0XY下凹f/(x)0f//(x)0XY上凹反曲點(inflectionpoint)上凹下凹xyC070)(Xf為下凹ba函數f在baX,,故f函數遞增遞減性，f函數凹性(四)第一導數檢驗定理:0)(Cf或不存在Cf)(XCXC切記f-+f(C)為局部極小值f+-f(C)為局部極大值f--f++f(C)為非局部極值第二導數檢驗定理:0)(Cf為局部極小值CfCf)(0)(為局部極大值CfCf)(0)(0)(Cf本定理失敗參、積分(一)不定積分(Indefiniteintegral)積分值積分函數、dXX積分符號、f::)(:dXXf)(:而)()(XfXFf為F之導函數、F為f之反導數故F為f之反導數)()()(常數KXFdXXf◎性質:dXXgdXXfdXXgXf)()()()(xyf(C1)f(C2)C2局部最小值C1局部最大值8CdXXfCdXXf)()()()(XfdXXfdXdCXfdXXfdXd)()(◎CXndXX1(二)定積分(definiteintegral)◎性質:Cba)(abCdXCbadXXfCdXXfba)()(dXXgdXXfdXXgXfbababa)()()()()()()()()(錯dXXgdXXfdXXgXfbabababaCdXXfdXXfdXXfbccaba,,)()()(f在X=a被定義0)(dXXfaadXXfdXXfabba)()(0)(0)(dXXfX設fbaxyf(x)abdxxfba)(9肆、齊次函數與尤拉定理(一)n階齊次函數(homogeneousfunctionofdegreen)◎定義:),(21XXfy若0),,(),(2121XXfXXfn則稱為nXXfy),(21階齊次函數(二)尤拉定理(EulerTheorem)◎定義:若nDO為HXXfy...),(21則211XfXXfny2X◎証明:),(),(2121XXfXXfn對入微分:),(2112211XXfnXXfXXfn令:1),(:212211XXfnXXfXXf(三)齊序函數(同位函數)(homotheticfunction)◎定義:(一階齊次函數的正單調上升轉換稱之)若),(21XXg為H.O.D1且0dgdff),()),((2121XXhxxgf則y稱之。例:若有齊次偏好，所得1000元，買40本書，60張CD,當所得為1500時，而書，CD價格不變，會買60本書，90張CD伍、古典規劃分析:最適化(Optimization)(一)未受限制下的極大與極小◎單變數函數(X)1.極大:Max)(XfyI.C.C.I=1500I=1000CDbook4060609010...0)(COFdXXfdy0)(XfdXdY判斷選一個CO由SX個解求得CO由FX...2...*2*1...02COSYdMaxYXXfdXYddXXfYd*1122220)(0)(2.極小:Min)(Xfy0)(...0)(...XfCOSXfCOF(二)多變數函數(),21XX1.),(21)(XXfYMaxMin...COF0dY0),(),(2211211dXXXfdXXXf*22122*121110),(00),(0XXXfXYXXXfXY...COS正定Min全為正MatrixHessionYd負定Max負正相間MatrixHessianYd)(0)(0220,0222112111122211211fffffffffH◎有限制條件下之極值分析:MaxmethodLagrangeXXfy),(21()Min..tSCXXg),(21MaxStep:1CXXgXXfXXL),(),(),,(212121...:2COFStep01XL0),(),(211211XXgXXf*1X02XL0),(),(212212XXgXXf*2X110L0),(21CXXg*...:3COSStepLd20BoarderHessianMatrix正負相間(Max)全為正(Min)021222221221112121111ggggfgfggfgfF陸、古典規劃分析應用:Optimization(1)max)()(QCPQQQ(2)minC=WkrLKL,3個主要問題類型),(..LKFQts(3)maxf(x)maxU(x,y)xoryx,0,0)(xxgs.tIypxpyx◎TheStructureofanOptimizationProblemMaxf(x)f(X)=objectivefunctionsxX:choicevariablesS:feasiblesetsolutions:*XSxxfxf)()(*Importantgeneralproblemsaboutthesolutionstoanyoptimizationproblem:(1)ExistenceofSolutionsPropositions:Anoptimizationproblemalwayshasasolutionif12(1)theobjectivefunctionis“continuous”(2)thefeasiblesetis“nonempty,closeandbounded”(2)LocalandGlobalOptima)(),()(:),()(:*****xBexxfxfSolutionLocalSxxfxfSolutionGlobalPrepositions:Alocalmaximumisalwaysaglobalmaximumif(1)theobjectivefunctionisquas