經(jīng)濟數(shù)學方法

1、PAGE PAGE 33經(jīng)濟數(shù)學方法矩陣與行列式 定義義: 階矩陣為一一包括列和行的數(shù)字字的方形排列列，若以A代代表此矩陣，則則 例: 分別為44和2矩陣 定義: 若 則 =C 例: 則 定義:若若A=(為矩陣，B=(為矩陣，則AA和B的 乘積AB為為矩陣C 例: 求求AB及BAA = = BA無法法計算 行列式: Crammers Rulle 已知 例:解下列聯(lián)立立方程式: 貳、微分 微分公式式: 若 設(shè)與皆存在: 鏈鎖律(chainn rulee): 設(shè)設(shè)函數(shù)f與gg皆可微分 反函數(shù) (inverrse fuunctioon): 設(shè)函數(shù)f與與g滿足 f(gg(Y)=Y 函函數(shù)g為f之之反函

2、數(shù) g(f(XX)=X 且g=f 偏微分: 例: 全微分: 例例: TEE=PQ 自然對數(shù)數(shù)(e)與自自然指數(shù)(lln): xxyexlnx11 性質(zhì): (1) 、 、 (2) (33)設(shè)存在 (44) (5) (6) (7) (8) (99) (110) (11) (122) 且(x0,y0)y=f(x)yx (13) 切線與射射線: (x0,y0)y=f(x)yx給定切線上任一一點(X, Y) 射線角角度值 函數(shù)的高階導(dǎo)導(dǎo)數(shù): 、 函數(shù)的臨界界點及反曲點點: (一) 若若f/(x)0X1X2f(x2)f(x1)XYab則為函數(shù)f之臨臨界點 f/(x)0X1X2f(x2)f(x1)XYab

3、函數(shù)數(shù)f在為嚴格格遞增 f/(x)0X1X2f(x2)f(x1)XYab f/(x)0f/(x)0f/(x)0XY上凹f/(x)0XY上凹concave downwardf/(x)0f/(x)0f/(x)0XY上凹反曲點(inflection point)上凹反曲點(inflection point)上凹下凹xyC0 故函數(shù)遞增遞遞減性，函數(shù)數(shù)凹性 (四)第一導(dǎo)數(shù)檢檢驗定理:或或 XC 切記記 - + f(C)為局部極小小值 + - f(C)為局部部極大值 - - + + f(C)為非局局部極值xxyf(C1)f(C2)C2局部最小值C1局部最大值 第二二導(dǎo)數(shù)檢驗定定理: 本定理失失敗參、積分

4、 (一) 不定積分(Indeffinitee inteegral) 而F之導(dǎo)導(dǎo)函數(shù)、F為為f之 反導(dǎo)數(shù)故F為ff之反導(dǎo)數(shù) 性質(zhì): (二二) 定積分分 (deffinitee inteegral) xxyf(x)ab 性質(zhì): X=a被定義 肆、齊齊次函數(shù)與尤尤拉定理 (一) 階階齊次函數(shù) (homoogeneoous fuunctioon of degreee n) 定義: 若則稱階齊次函函數(shù) (二) 尤尤拉定理 (Eulerr Theoorem) 定義:若 則則 証明: 對入微微分: 令令 (三) 齊齊序函數(shù) (同位函數(shù)) (hommothettic fuunctioon) 定義: (一階齊次

5、次函數(shù)的正單單調(diào)上升轉(zhuǎn)換換稱之)若 為H.O.D 1 且且 稱之。I.C.C.I=1500I=1000CDbook40606090 例: 若有齊次偏偏好，所得11000元，買買40本書，660張CD, 當所得為為1500時時，而書，CCD價格不變變，會買600本書，9I.C.C.I=1500I=1000CDbook40606090 伍、古古典規(guī)劃分析析:最適化(Optimmizatiion) (一) 未受限制下下的極大與極極小 單變數(shù)函函數(shù)(X) 11. 極大: 2. 極小小: (二) 多變數(shù)函函數(shù)( 1. 有限制條條件下之極值值分析: ( 正正負相間(MMax) 全為正 (Min) 陸、古典

6、規(guī)劃分分析應(yīng)用: Optiimizattionmax Q (2) min CC=W 33個主要問題類型 (33) maxx f(x) max UU(x, yy) x orr s.t The SStructture oof an Optimmizatiion Prroblemm Maax f(x) f(X)=objeectivee funcction X: cchoicee variiabless S: feassible set ssolutiions: Impoortantt geneeral pprobleems abbout tthe soolutioons too any optimm

7、izatiion problemm: (1) Existtence of Soolutioons Propoositioons: AAn opttimizaation probllem allways has aa soluution if (1) the oobjecttive ffunctiion iss “ coontinuuous” (2) the ffeasibble seet is “noneempty, closse andd bounnded” (2) Loocal aand Gllobal Optimma Prreposiitionss: A llocal maximmum

8、iss alwaays a globaal maxximum if (11) thee objeectivee funcction is quuasicooncavee.(2) the ffeasibble seet is connvex. (33) Uniiqueneess off SoluutionPPropossitionns: Giiven aan opttimizaation probllems iin whiich thhe feaasiblee set is coonvex and tthe obbjectiive fuunctioon is noncoonstannt andd

9、 quassiconccave, a sollutionn is uuniquee if: (1) the ffeasibble seet is stricctly cconvexx, or (2) the oobjecttive ffunctiion iss striictly qquasicconcavve, orr (3) booth (44) Intteriorr and Bounddary OOptimaa (55) Loccationn of tthe Opptimumm miin maax f(x) F.O.CC XR S.OO.C (多變數(shù)) Multiivariaal Cas

10、se F.O.C Graadientt vecttor off f S.O.C Heessiann of ff noow, maax f( SS.O.C (負定) ( Quaadratiic Forrms annd theeir Siigns symmmetricc: X A XX=( = (1) Neegativve Semmidefiinite (2) Negattive ddefiniite (3) Poositivve Semmidefiinite (4) Posiitive definnite ex n=2 = = = -Neegativve deffinitee and - Poo

11、sitivve deffinitee: and 續(xù) HHessiaan; H iis neggativee defiinite if H iss posiitive definnite iif Generral Caase A =( Negaative definnite: MAAX . Posiitive definnite: MMIN Optimmizatiions: Thee uncoonstraained case I. maay ff( Min FF.O.C Graadientt Veottor SS.O.C Hesssian Matriix NNecesssary cconditt

12、ions Suffficiennt connditioons Df=00 H iis deefinitte f iss conccave (dx)H(dx)0 eex 1. F.O.C S.O.C 2. F.O.C S.O.C H iss negaative definnite ff is cconcavve. II. The Consttraineed Casse s.t g(=b Laagranggian FFunctiion: L ( connstraiint guualifiicatioon: F.O.CC D S.O.C s.t. DDg( 全微微分 Borddered Hessii

13、an S.O.C. forr The naturrally orderred prrincippled mmincrss of tthe boordereed (alll be nnegatiive)guuaslcooncavee Hessiaanmatrrix allternaate inn signn, thee signn of tthe fiirst beingg posiitive i.e ex miin Lagrrangiaan funnotionn: F.O.C. Nonllinearr Proggrammiing Max ff( innequallity cconstrrain

14、t F.OO.C Max f(x) Langranngian Functtion: Maax Exx = F.O.C 因有innegueddiy, . 所以以要多考慮這這些可能 ex “” min s.t F.O.C 檢查這這些條件是否否都符合 限制式中 共有四種種組合四種可能情況 Case 11 ( 代入入 (2) 式)四種可能情況 Casee 2 sstep2 ( stepp2 用第2種種生產(chǎn)要素 Casse 3 用用第1種生產(chǎn)產(chǎn)要素 Case 4 Ex Kuhnn-Tuckker Foormulaation Kuuhn-Tuucker Condiitionss s.t. (K-T ccondittions): UUtilitty Maxximizaation Probllem mmax uu(x, yy) x, y ss.t CComparrativee Stattics F.O.CC IImpliccit Fuunctioons IImpliccit Fuunctioons Thheoremm If D= = -totaally ddifferrentiaating the ssystemm D (無限制制式) exx max FF.O.C TTotallly diffferenntiatee F.O.C witth res