1、1經濟數學方法壹、 矩陣與行列式定義: 階矩陣為一包括 列和 行的數字的方形排列，若以 A 代表mnnm此矩陣，則naaAijnmn)(212112 例:13,53120BA分別為 4 和 2 矩陣定義: 若 則mnijmnijbBaA)(,)(=CijijijCmnija)(例: 3152,3BA則 275031284513250)(5BA2AA213246132 定義:若 A=( 為 矩陣，B=( 為 矩陣，則 A 和 B 的)ijamn)ijbkm乘積 AB 為 矩陣 Ck例: 求 AB 及 BA1302,120BA13021= 1.)(0.2 2.= 1327BA 無法計算 3 行列式
2、: Cramers Rule已知 1211bXa221212121* aa21121*2 ababX例:解下列聯立方程式: 05321X30322521X943120125*1X92305*2X1*392015貳、微分 微分公式: )(XfYdXYxffX)(lim0)(22fd 若 RnfRXfn ,)(,)( 1 設 與 皆存在:f)(g dXgfdX)()( ffgf )()()( 、 0)(,)()(2ggfd 鏈鎖律(chain rule):設函數 f 與 g 皆可微分 )()()(XgfXfd 反函數 (inverse function):設函數 f 與 g 滿足 f(g(Y)=Y
3、 函數 g 為 f 之反函數4g(f(X)=X 且 g=f 1XfY)(1 偏微分: ),(, 21121 Xfyy,),(22ff例: XYdX632 全微分: ),(21fy21dXy例: TE=P QPdTE 自然對數(e)與自然指數(ln):性質: (1) 、0lim1)0()( XxefeXf Xeli、Xnln Xln(2) Xed(3)設 存在f )()()(fedXff (4) RYeXY,xy exlnx115(5) Xe1(6) 0,lnd(7) ,1x(8) )()(XffndX(9) Ylnl(10) Yl(11) ln(12) 且XeeXln(13) Yl 切線與射線
4、:給定切線上任一點(X, Y)()(0Xffy 射線角度值 0tany函數的高階導數:、 dXY2 23dXYyffffXf )()(lim)()lim)( 0000 00函數的臨界點及反曲點:(一 ) 若 、Xff、Df )()( 000 則 為函數 f 之臨界點X(x0,y0)y=f(x)yxf/(x)0X1 X2f(x2)f(x1)XYa b6(二) 函數 f 在 為嚴格遞增ba,)()(2121Xf、X函數 f 在 為嚴格遞減ba,)()(2121Xf、fX(三)f/(x)0f/(x)0XY上凹concave downwardf/(x)0f/(x)C 切記- + f(C)為局部極小值f
5、+ - f(C)為局部極大值- - f+ + f(C)為非局部極值第二導數檢驗定理: 0)(Cf 、Cf0)( f)( 本定理失敗f xy f(C1) f(C2)C2局部最小值C1局部最大值8參、積分(一) 不定積分(Indefinite integral) 、dX、f :)(:而 F 之導函數、F 為 f 之 dXf)(: )(fFf反導數故 F 為 f 之反導數 )()(Kdf 性質: dXgXfgf )( CdfdXf( )(ff CfdfX( nd1(二) 定積分 (definite integral) 性質: Cba)(adX dXffba) dXgfgf baba )()( ) 、
6、dfdXfbaxy f(x)a bdxfb)(9 baCdXfdXfdf bccaba ,)()()( X=a 被定義f、0f dfdfabba)()( 00XfXfa肆、齊次函數與尤拉定理(一) 階齊次函數 (homogeneous function of degree n)n 定義: ),(21Xfy若 0),(,21ffn則稱 階齊次函數、fy),(21(二) 尤拉定理 (Euler Theorem)定義:若 nDOHXf.),(21則 1fny2 証明: ),(),(212ffn對入微分: ),(211 XfnXX令 :),(:212fff(三) 齊序函數 (同位函數 ) (homot
7、hetic function) 定義: (一階齊次函數的正單調上升轉換稱之)若 為 H.O.D 1 且 ),21Xg0dgf稱之。),(2Xhxf、y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I.C.C.I=1500I=1000CDbook40 60609010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.0)(COFdffY、S.2*21.02Sd MaxYXfdXYfY *1122 0)()(2.
8、 極小: Minfy)(.fCOSF(二) 多變數函數( ),21X1. ,()(fYaxMin .COF0d0),(), 21121 dXfXf*22122 11),(0fY .COS11、MinatrixHesionYd ax)(020,02121 fff 有限制條件下之極值分析:Max methodLagrnXfy),(21( )in.tSCg),(21axtep:1CXgfL),(),(2121.2OF01X0),(),(2121f *12L,Xgf200),(21C*.:3COStepLd2BoardeHsinMatrix正負相間(Max)全為正 (Min)021221gffF陸、古
9、典規劃分析應用:Optimization(1) max )()(QCPQ(2) min C=W krL3 個主K,12要問題類型),(.LKFQts(3) max f(x)max U(x, y)x or yx,s.t 0,)(xg Iypx The Structure of an Optimization ProblemMax f(x) f(X)=objective functionX: choice variablessxS: feasible setsolutions: *XSxfxf)(Important general problems about the solutions to a
10、ny optimization problem:(1) Existence of SolutionsPropositions: An optimization problem always has a solution if(1) the objective function is “ continuous”(2) the feasible set is “nonempty, close and bounded”(2) Local and Global Optima)(),(: *xBexffSolutinLcaSGlbPrepositions: A local maximum is alwa
11、ys a global maximum if(1) the objective function is quasiconcave.(2) the feasible set is convex.(3) Uniqueness of SolutionPropositions: Given an optimization problems in which the feasible set is convex and the objective function is nonconstant and quasiconcave, a solution is unique if:(1) the feasi
12、ble set is strictly convex, or(2) the objective function is strictly quasiconcave, or 13(3) both(4) Interior and Boundary Optima(5) Location of the Optimumminmax f(x) F.O.C 0)(dxfX R S.O.C (ma)in2f(多變數) 1x Multivarial Case )(21fYF.O.C Gradient vector of f:/21fxfS.O.C Hessian of f nffH.1jiijxfnow, ma
13、x f( ),21xCOF.01xf21,x 2fS.O.C 021xf 01f(負定)2f( 0)()() 2121221 f、xffxf 21)(fff21)( Quadratic Forms and their Signs14naA1symmetric: jiijX A X=( mnnmXaX111.).=nitjjixa1(1) Negative SemidefiniteRXA,0(2) Negative definite,(3) Positive Semidefinite,0(4) Positive definite,XAex n=2 21211)(xa= 21xa= )()( 22
14、12121 XaX= 212121)(axa-Negative definiteand 01021- Positive definite:and 01a021a續 Hessian; 15H is negative definite if 211,0fff2H is positive definite if 21,fffGeneral Case A =(X nnxaXi .). 111Negative definite: 01a021MAX . nnia.)(1Positive definite: 01a021MIN .11nna Optimizations: The unconstrained
15、 caseI. may f( )xMinF.O.C Gradient Veotor)(1nffXffxfDS.O.C Hessian Matrix2112 nntffffH16Necessary conditions isHCOSDfF.0tesmidfnnegativposSufficient conditions Df=0H is definitenositvegaf is concave (dx) H(dx)0ex 1. )()(maxqcpqF.O.C MCPd0S.O.C 022q2. xwpfxq*)(aF.O.C 00*WiXfiiVMPS.O.C H is negative d
16、efinite f is concave.II. The Constrained Casexma)(xfs.t g( =b )、dxiXg.0Lagrangian Function:L (ax)()(, bfconstraint gualification: ,Uxig0Xi 0)(,21XiifnIF.O.C xjgijfi)(170)(xgbLD 22)()(xdLS.O.C 0Ts.t. Dg( 全微分0)xd Bordered Hessian)(),( 2222 1212122 xLDxxLxgxxLxLDHninn nS.O.C. for ma(i)The naturally ord
17、ered principled mincrs of the bordered(all be negative) guaslconcaveHessianmatrix alternate in sign, the sign of the first being positive i.e000 232321323 312121212122211 xLxxLgggxLgex min wxts.f)(Lagrangian funotion: )(),(*gxfwxL18xmaF.O.C. MRTSxjfiwjinIifwixiL ,.210)(),(*)(),(*xfg Nonlinear Progra
18、mmingMax f( inequality constraint)xxfs.)(gb0)(xgx0F.O.C )(*xifXi0i)(2xifMax f(x)xls.Langrangian Function:)()(,(1nixgxfLMax ,nXEx nxts.1b)(0,.0,21uxx),(2nnuxL= ).)(21nxbgf F.O.C19四種可能情況011uxgfxL2fnn0,*L因有 ineguediy, ,111xuxu. 所以要多考慮這些可能0,0nnnLex“” min 21xws.t y0122121221 )(),( XMyxxMXL F.O.C 檢查這些條件是否
19、都符合0212211,)3.()().(XMyxLwX 1U2L限制式中 共有四種組合0,2XCase 1 ( 代入 (2) 式)y0,2xCase 2 step2,22W( 1(,1、step2 012 wyx用第 2 種生產要素.1WCase 3 用第 1 種生產要素,0,221XCase 4 2020Ex yWyC2121,min),(2121WyWCifKuhn-Tucker Formulation0)(.0)(.minaxXgtsgtsff, bfXLKuhn-Tucker Conditions0,0(max)1(in1LLXii iii21inWX21,xs.t. y0,21)(2
20、1yXL(K-T conditions):0,111 LWX,0222XL21Y0LUtility Maximization Problemmax u(x, y)x, ys.t IyPxX0,0,yxComparative Statics21F.O.C ).(,0)(.,. 21*21211 mijnmnn a、xXequtosnxxFf Implicit FunctionsImplicit Functions TheoremIf D= =01nf*jX).(mih、aajxii.01*-totally differentiating the system0211111 mnnn adfdfx
21、fdxff 21. ffffnn .f mnmnnnn dfdfdafdxff 111111 .).(. D i ijDj22/xfDxjdxijjiii (無限制式) ex max 2121),(. XWXfPF.O.C0111wpfxTotally differentiate F.O.C with respect to :1W22 01*21*211wxfPxf01*21*2xpff21Pf01*wxBy Cramers Rule0)(02112112*1 ffpfpfwx0)(021121*2 ffpfpx同理 )(21*ffw0)12()(2112*1 fsignffpx(有限制式)m
22、ax 121)(XWf21xXs.t.)()( 2121XPLF.O.C23)(,0021*21211 XPWpfXLw20 ),(021*S.O.C 1XL2H2212 1211 210xxxgXg= 221100pff= (p 1pf )01p0 0.maxHC、OFfTLFrom0)(20*1211XWPfLX1.、WF011*21*211wxpffPtotally differentiating with respect to w1:0110*1*2wxpfBy the Cramers Rule:240/0/1121*pfwxf)(xf ),(),(), 021*021021* xpwxw把 算出，代入利潤函數中，即可得: *profit Function、x12But 此題中 可直接代入 為 one decision 的問題，不需如此02X麻煩。
