經(jīng)濟(jì)數(shù)學(xué)方法

1、經(jīng)濟(jì)數(shù)學(xué)方法擘方法壹、矩障典行列式aim a2manm(aj)n m定羲:n m P皆矩睡懸一包括n列和m行的數(shù)字的方形排列,若以A代表 此矩M JW TOC o 1-5 h z aiiai2a2ia22Aani an2例： HYPERLINK l bookmark30 o Current Document 0ii2Ai 23i 35分別懸4 3和2 4矩睡定羲:若 A (aj)nm，B (bj)nm U(aijbij) n m(Cij )n m =Caij ) n m TOC o 1-5 h z 22ii即J AB3225iii32ii2425i35i348i0 55A B 5A ( i)B

2、 i5 i055 HYPERLINK l bookmark4 o Current Document 4221 HYPERLINK l bookmark6 o Current Document 642322A HYPERLINK l bookmark8 o Current Document 2211止 HI :右 A=( a,j) nm矩彈，B=( bj) m k矩彈，JW A和B的乘稹AB n k矩睡C例：A0 1 2 ,B2 0 11 002 11 求 AB 及 BA0 31AB0 122 012.3 0.0 1 ( 1) 2 11.3 2.0 0 ( 1) 1.1_0,11.22.00.0

3、1.1-2 10.21.02.00.1_ 2 7 1- 231BA瓢法算3 3 2 3行列式：Cramers Rule已知a11Xa21Xa12Xa22Xb1b2*X1b1a12b2a22a11a12a21a22a11b1a22b2a12a11a22a12a21*X2a21a11b1b2a12anb2a21bla11a22a12a21a21a22 TOC o 1-5 h z HYPERLINK l bookmark47 o Current Document 111 X1例:解下列聊立方程式121 X2213 X3XiXi2X1X22X2X2X3X33X3*Xi*X2*X31211-212391

4、212j9熊微分微分公式:f(X)Y X 2Yf (X)則。f(X X)f(x)dYdXd2YdX7f (X)若 f (X) Xn, X R段f (X)典g (X)皆存在:f (X)ddXf(X) g(X) f dXdg(X)dXddXddXf(X) g(X) f(X)dg然 g(X)f(XI dXdXf(X) f(X)g(X) f(X)g(X)g(X)2 g(X),g(X)乘法公式除法公式 律(chain rule):IS函數(shù) f 典 g 皆可微分f(g(X) f (g(X) g (X)dXf(g(Y)=Y g(f(X)=X函數(shù)g融f之反函數(shù)且 g=f1反函數(shù)(inverse functio

5、n):鼓:函數(shù)f典g漏(足f(f 1(Y) Y1f(f 1(X) X偏微分：y f(X1,X2)/啟冬工)X1yf(X1,X2):f2(X1,X2)X22Y 6Xf (X1,X2)例：3X2dX全微分：ydy例：TE=P dTEy dYdXX1QdP工dX2 X2dQ P自然封數(shù)(e)典自然指數(shù)(In):性U: (1) f (X) exf(0)lim eXX0、limXf (X) InXf(1)lim ln XXlim ln XXd x x(2) e e dX(3成f存在ef(X) f (X)X,Y(5)(6)d . vln X dXdxnX7, X 0(8)ddXn|f(X)(10)lnX

6、YlnAln Xf (Xf(X)lnYY ln XY ln X eYXlnXInYYlnXX 且 ln eX XX lnY e給定切上任一黑占（X, Y）y f(X。)X X0射角度值tan(X)yX0函數(shù)的高P皆厚數(shù):2d Y d2dX2 dXdYdX3d ydX32d d2YdX dX2f(X0)XimX0f (X)(X0)X X0lim爭(zhēng)X0 0X)Xf (X0)函數(shù)的陶界黑占及反曲黑占：（一）若X0 Df （函數(shù)定羲域）f （X0）0（或f （X。）不有在）川JXX0懸函數(shù)f之陶界黑占函數(shù)f在a,b 格遮增X1X2 f (X1) f(X2)2 )f (X) 0Xa,b函數(shù)f在a,b懸上

7、凹f (X) 0Xa,b函數(shù)f在a,b懸下凹故f函數(shù)遮增遮減性，f函數(shù)凹性(四)第一醇嗷梅T瞬定理:f (C) 0或f (C)不存在XC 切言己f+-f+ f(C)懸局部趣小值f(C)懸局部趣大值f(C)懸非局部趣部第二醇嗷梅T瞬定理：f (C) 0f (C) 0f(C)懸局部趣小值f (C) 0f(C)懸局部趣大值f (C) 0本定理失敗：象稹分(一)不定稹分(Indefinite integral)：稹分符虢、f(X):稹分函數(shù)、dX :稹分值:f (X)dX 而F(X) f(X) 出F之厚函數(shù)、F懸f之反厚數(shù)故F懸f之反厚數(shù)f (X)dX F(X) K(常數(shù))性 it：f (X) g(X

8、) dX f (X)dX g(X)dXC f (X)dX C f (X)dXddXf(X)dXf(X)力(X)dXf(X) C1八一XnX CX(二)定稹分(definite integral)a f(x) dx性u(píng)：C dX C(a a)：C f (X)dX C a f (X)dXa f(X) g(X) dX bf(X)dXbg(X)dXa f(X) g(X)dX bf(X)dXbg(X)dX)：f(X)dX a f (X)dX bf (X)dX,C a,bf在X=a被定Ua f (X)dX 0：f(X)dXbf(X)dXtzf(X) 0：f(X)dX 0肆、膂次函數(shù)典尤拉定理(一)n 睞萍

9、次函數(shù) (homogeneous function of degree n)定羲： y f(XX2)若 f( X1, X2)nf(Xi,X2),0川師y f(X1,X2)懸n p皆膂次函數(shù)(二)尤拉定理(Euler Theorem)定羲:若 yf(X1,X2)H .O.D.n ffI J ny Xi 2X1 X2 tE明：f( X1, X2) n封入微分：X1Xi令 1: X1 -XiX2f(X1,X2)f-TX2 n n1f(X1,X2)Xf 2_、X2 n : f (X1,X2) X(三)膂序函數(shù)(同位函數(shù))(homothetic function)定羲：(一P皆膂次函數(shù)的正罩湖上升樽J用

10、S之)df若 g(X1,X2)懸 H.O.D 1 且 f 一 0dgJWyf(g(xhx2) h(X1,X2)耦之。例：若有膂次偏好,所彳# 1000元，40本善，60 CD,常 所得懸1500畤，而善，CD 格不建，1r60 本善，90 CD伍、古典規(guī)分析：最遹化(Optimization)(一)未受限制下的趣大典趣小10覃建數(shù)函數(shù)（X）1. S大：Maxf(X) dy(X)dX0 F.O.C.dY dX(X)*Xi*X2由F .O.C.求得洲解由S.O.C.判斷逐一他d2Y 0 S.O.C.2.趣?。篗in2-2_d Y f (X)dX 0y f(X)F.O.C.S.O.C.d2YdXf

11、(Xi) 0f (X) 0f (X) 0*Xi MaxY（二）多建數(shù)函數(shù)（Xi, X2）MaxY f(Xi，X2)F.O.C. dY 0fi(Xi,X2)dXi f (Xi,X2)dX2 0Y.*0fi(Xi,X2) 0 XiXiY,*0f2(Xi, X2) 0 X2X2S.O.C.d2Y 0 Hessian Matrix 負(fù)正相fW (Max)負(fù)定 d2Y 0 Hession Matrix 全卷正(Min) 正定 TOC o 1-5 h z ,fiifi2r八 fiifi2-Hfii0,0f2if22f2if22有限制僚件下之趣值分析：Max y f (Xi, X2) Lagrange me

12、thod(Min )S.t.g(Xi,X2) CStepi : Max L(Xi, X2, ) f(Xi, X2) g(Xi, X2) CStep2 : F.O.C.L 八，、， 、八* 0 fi(Xi，X2)gi(Xi，X2) 0XiXiiif2(Xi,X2)*X2g2(Xi,X2)0g(Xi,X2)c 02 .Step3: S.O.C. d L 0 Boarder Hessian Matrix正相IW (Max)全懸正 (Min)f11g11f12912g1f 21g12f22g22g2g1g20陛、古典烷副分析鷹用：Optimization max (Q) PQ C(Q)Q(2) min

13、 C=W L r kL,Ks.t.Q F(K,L) max f(x)maxxor x, yg(x) 0,x 0 s.tU(x, y)PxXPyy3(0主要 曲題IB型 The Structure of an Optimization Problem Max f(x) f f(X)=objective function x s j X: choice variablesS S: feasible set*solutions: X12_ * _ _f(x ) f (x) x S_ Important general problems about the solutions to any optim

14、ization problem:Existence of SolutionsPropositions: An optimization problem always has a solution ifthe objective function is “ continuous ”the feasible set is nonempty, close and boundedLocal and Global OptimaGlobal Solution: f (x_) f (x), x S*、Local Solution : f (x ) f (x), x Be(x )Prepositions: A

15、 local maximum is always a global maximum ifthe objective function is quasiconcave.the feasible set is convex.Uniqueness of SolutionPropositions: Given an optimization problems in which the feasible set is convex and the objective function is nonconstant 廣and quasiconcave, a solution is unique if:(1

16、) the feasible set is strictly convex, or(2) the objective function is strictlyquasiconcave, orbothInterior and Boundary OptimaLocation of the Optimum minmax f(x)ROC甯X R S.O.C(多m數(shù))xx2 Multivarial Cased2f (x)dx200(min)0(maR13F.O.Cf / Xif / x2f1:Gradient vector of fS.O.Cfnffn1f,1nHessian of ffijnnfiXj

17、now, maxf(Xi,X2)F.O.Cfxif xx2S.O.Cf11Xi,X2(Am)()E)1112x1 x2f11 f 22f12 f21(fj2122f11 f12(fj Quadratic Forms and their Signsa11a1nAan1annsymmetric: a。 ajiX A X=( Xi.Xm)a11anan1annn n=a., x Xij人i人ji 1 j tNegative SemidefiniteXAX0, XRNegative definiteXAX0, X 0Positive SemidefiniteXAX 0, X R14Positive d

18、efiniteexX AX 0, X 0n=2XAX(X1X2)ai2Xia2ia22X22= aiiXi2a12X1X22 a22 X2= aii(X22axX XiaiiX2i2ai22 aiiX；)2ai22-2 X 2 aiia22X；)aiia12= aii(xiai2 2一 X2)aiiNegative definiteaiia2ia22X20 andaii ai2a2ia22-Positive definite:aii0 andaiia2iai2a22罐 Hessian;2 TOC o 1-5 h z H is negative definite iffii0, fii fi2f

19、i2f2202H is positive definite iffii0, fii f12fi2f 220General Caseaii.ainXiX A X=(XiXn)ani. a nnXnNegative definite: aii 0aiia2iai2a22aii . ai nMAX I .(1)nani . ann15Positive definite: a110ai012a2i a22al1 alnMIN 0an1 . a nn Optimizations:LThe unconstrained casemay f(x)MinFQCDff(x)f XifiGradient Veoto

20、rfnSQCD2ffllf 21f12f22f1n. . Hessian MatrixNecessary conditionsfntf nnFOCDf 0SQCisnegativepositivesemidefini teSufficient conditionsDf=0H isdefinitenegativenositivef is concave (dx) 1 H(dx)0ex max (q)pq c(q)dFQC 0 P MC dp22SQC J 0 d 0dqdq16*2. max (x) px)wx q* fF.O.C 0 p Wi 0 xiXiVMPi WiS.O.C H is n

21、egative definite f is concave.II. The Constrained Casemaxxf(x)s.t g(x)=b0.在有限制下,求最大黑占(g(x) b)Lagrangian Function: max L ( x, ) f(x),x constraint gualification: g(x) UxiF.O.CXi 0f (x)I 1,2,nXigXif(xi)f(xj)g xig xjg(x) 0,2，、D2L(x) d/d x_S.O.C T _ 2s.t. Dg(x)dx 0 全微分(dx) D L(x)(dx) 0 dx Bordered Hessia

22、n172l2_2lH D2L(x, ). g1 2lxn2l2lxixn2l2lD2L(x)2xixi xn 2l2l xn X2xn(min)s.oc for maxThe naturally ordered principled miners of the bordered (all be negative) guaslconcaveHessianmatrix alternate in sign, the sign of the first being positive0gig2gi 2l 2 x2lg22lX2 xixi x22l2 x20gig2g3i.eex min w xxst.

23、f (x) gLagrangian funotion:*L(x, ) w x (f(x) g)maxx_ I *F.O.C. (x_, ) wixi*.f0 I i,2,.nxiwi wjMRTSL(x_, ) g f(x) 0Nonlinear Programming fMax f( x) inequality constrainti8s.f.g(x) b g(x) 0X 0 x 0_*F.O.C X X* * f (X)0 xiX Xi 0f(x2)xif(x)廣MaxLangrangian Function:nL(x,_) f (x) igi(x)i 1Max f(X1,Xn)x1.x1

24、Ex,s.t.g(x) bx1 0, X2 0,xu0L(xi,x2,xn, ,Ui,U2,Un)=f (x)g(x) b) UixiU2x2U n xn )xiLF.O.CfxifgxiUixnxngxnU2U1x10,0,Ui0, x1Ui因有 inegUediy,所以要多考ft適些可能UnxnQ UnUni9ex TOC o 1-5 h z 我 minw1 x1w2x2s.tx1x2yx10 x20(x1 x2 y) M1X1 M2X2L(X1, X2, ,M1,M2) w1x1w2x2WiF.O.CLX2 Lw2(XiM 1 0(1)M20.(2)檢查適些僚件是否都符合X2y)0.(3

25、)M1X10, M2X20限制式中Xi 0, X20共有四槿黜合X1X10,X20 y 0( 2,x20 代入式) TOC o 1-5 h z 0,X2 0, step2 2 0,W20W2(i,Xi 0,代入式)step2x2y, w11W1WiW.用第2槿生要素Case 3 X1 0,X20,Case 4 X1 0,X2 0,W2Wi.用第1槿生要素12 0WiW2Ex C(W1,W2,y) min W1y,W2yCWyW2 yifW1W2CW2yW1yW1W2CW1yW2yW1W2k Kuhn-Tucker Formulationmax f (X_)min f(X)s.tg(X) 0s.

26、t.g(X) 0L(X,_) f(X) (g(X b)20Kuhn-Tucker ConditionsLXi0(maX Xi0,Xi(min)i 0,1LXi0min X1 W2X2Xi, X2s.t. X1 X2 y TOC o 1-5 h z 二Xi 0,X20L W1X1 W2X2(X1 X2 y)(K-T conditions):W10, X1 0,X1 0X1X1L2 W20, X2 0,X2 0XX2X1 X2 Y 00 0Utility Maximization Problem max u(x, y) x, ys.tPx x Py y Ix 0, y 0 x Q y 0Compa

27、rative Staticsequations,可求得* * *X1,X2.Xn且 x；(a1a2.am)1 /F.O.Cf (x1x2xn, 1 2. m)0 nnF (為.僅2 xnx#2xm)0Implicit FunctionsImplicit Functions TheoremIf D=f1f1fn1*Xj = hi(m) *旦一0即Xi*受1.am之影警卷正或K aj21-totally differentiating the systemf ； dx1f 2 dx2fin mdamfiifinf 2 dxif 2 dx2fnndXnf 2ifn id i.fnn mdamfnnd

28、xidx1dxndxiD DidD(fniid i(fnnidaixiDj惻限制式）exmaxF.O.Cxipf1w1xipf1w1f n md m)f L1 n i.f,d m)2f / xP. f(Xi,X2) WiXiTotally differentiate F.O.C with respect tofixif2wi*xixiwi* HYPERLINK l bookmark148 o Current Document fix2X2Wi*道_xX2Wi HYPERLINK l bookmark44 o Current Document *，xi . x2 d HYPERLINK l bookmark245 o Current Document pfii一pfi2一iwiwi*