经济数学方法

經濟數學方法壹、 矩陣與行列式◎定義: m n ⨯-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m n a a a a a a a a a a A ij nm n n m m ⨯=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=)(212222111211ΛM ΛM M ΛK例:⎥⎦⎤⎢⎣⎡--=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡---=11133111,531321213102B A 分別為43⨯和24⨯矩陣◎定義: 若m n ij m n ij b B a A ⨯⨯==)(,)( 則 m n ij m n ij ij C b a B A ⨯⨯=+=+)()( =C m n ij a A ⨯=)(αα例: ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡--=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=315212,112312B A 則⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-++++-=+227520311152231122B A⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡----+⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=-+=-84513412315212551015510)1(55B A B AA A A 21123122224624112312112312=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=+◎ 定義:若A=()ij a 為m n ⨯矩陣，B=()ij b 為k m ⨯矩陣，則A 和B 的 乘積AB 為k n ⨯矩陣C例: ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-=⎥⎦⎤⎢⎣⎡=130112001,102210B A 求AB 及BA ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-⎥⎦⎤⎢⎣⎡=130112*********AB =⎥⎦⎤⎢⎣⎡+-⋅+++++⋅⋅+-⋅+++++1.1)1(00.23.11.00.20.12.01212)1(10.03.21.10.00.22.11.0 =⎥⎦⎤⎢⎣⎡132172 BA 無法計算 33⨯Θ 32⨯◎ 行列式: Cramer's Rule 已知 1212111b X a X a =+ 2222121b X a X a =+⇒ 2112221112222122211211222121*1a a a a a b a b a a a a a b a b X --==2112221112121122211211221111*2a a a ab a b a a a a a b a b a X --==例:解下列聯立方程式: ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡--025312121111321X X X⎪⎩⎪⎨⎧=++=-+=+-⇒032225321321321X X X X X X X X X 9439312121111310122115*1==----=X 923932121151*2-=-=X 1*39219012221511±-=-=X貳、微分◎ 微分公式: )(X f Y =dXdY X x f X X f X f X Y X =∆-∆+='=∆∆→∆)()(lim )(0)(2222X f dX Yd X Y ''==∆∆ ◎ 若R X nX X f R X X X f n n ∈∀='⇒∈∀=-,)(,)(1 ◎ 設)(X f '與)(X g '皆存在:{}dXX dg dX X df X g X f dX d)()()()(±=± {}dXX df X g dX X dg X f X g X f dX d)()()()()()(+=⋅ []乘法公式0)(,)()()()()()()(2≠'-'=⎭⎬⎫⎩⎨⎧X g X g X g X f X g X f X g X f dX d []除法公式 ◎ 鏈鎖律(chain rule): 設函數f 與g 皆可微分)())(())((X g X g f X g f dXd'⨯'=⇒◎ 反函數 (inverse function):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=f 1-⇒ ⎩⎨⎧==--XX f f YY f f ))(())((11◎ 偏微分: ),(),(211121X X f X yX X f y =∂∂⇒= ),(),(212221X X f X yX X f y =∂∂⇒= 例:X Y X dXd6232=+ ◎ 全微分: ),(21X X f y =2211dX X y dX X y dy ∂∂+∂∂= 例: TE=P ⨯QP dQ dP dTE ⨯+⨯=⇒2 ◎ 自然對數(e)與自然指數(ln):性質: (1) 0lim 1)0()(=⇒=⇒=∞→X X x e f e X f 、∞=∞→X X e lim-∞=⇒=⇒=-∞→X f X X f X ln lim 0)1(ln )(、∞=∞→X X ln lim(2)X Xe e dXd = (3)設f '存在)()()()(X fe e dXdX f X f '⋅=⇒(4) R Y X e e e Y X Y X ∈∀⋅=+,, (5) X X ee 1=- (6)0,1ln >∀=X XX dX d x y e x lnx 1 1(7)0,1≠∀=X X X n dx d (8) )(()(X f X f X f n dX d '= (9) Y X Y X ln ln ln +=⋅ (10) Y X YXln ln ln-= (11) X Y X Y ln ln =(12) X e X =ln 且X e X =ln (13) Y X X e Y ln =◎ 切線與射線:給定切線上任一點(X, Y))()(00X f X X X f y '=--⇒射線角度值tan X y =α◎函數的高階導數:⎭⎬⎫⎩⎨⎧=dX dY dX d dX Y d 22、⎭⎬⎫⎩⎨⎧=2233dX Y d dX d dX y d XX f X X f X X X f X f X f X X X ∆'-∆+'=-'-'=''>∆→)()(lim )()(lim)(00000000◎函數的臨界點及反曲點:(一) 若，不有在X f 或X f ，函數定義域Df X ))((0)()(000'='∈ 則0X X =為函數f 之臨界點(二)函數f 在[]b a ,為嚴格遞增(x 0,y 0)y=f(x) αyxf /(x)>0 f(x 2) Y)()(2121X f X 則f X X <⇒函數f 在[]b a ,為嚴格遞減)()(2121X f X 則f X X >⇒(三)0)(>''X f [][]為上凹b a 函數f在b a X ,,⇔∈∀ 0)(<''X f [][]為下凹b a 函數f在b a X ,,⇔∈∀ ⇔故⇒'f 函數遞增遞減性，⇒''f 函數凹性(四)第一導數檢驗定理:0)(='C f 或不存在C f )(' X<C X>C 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)(='C f為局部極小值C f C f )(0)(⇒>'' 為局部極大值C f C f )(0)(⇒<''0)(=''C f 本定理失敗參、積分(一) 不定積分(Indefinite integral)⎰積分值積分函數、dX X 積分符號、f ::)(:⇒ ⎰dX X f )(: 而⇒=')()(X f X F f 為F 之導函數、F 為f 之xyf(C 1)f(C 2)C 2 局部 最小值C 1 局部 最大值反導數故F 為f 之反導數⇒⎰+=)()()(常數K X F dX X f ◎ 性質: {}⎰⎰⎰±=±dX X g dX X f dX X g X f )()()()(⎰C ⎰=dX X f C dX X f )()({})()(X f dX X f dXd=⎰C X f dX X f dXd+=⎰)()( ◎ C X n dX X+=⎰1(二) 定積分 (definite integral)◎ 性質:C b a ⎰ )(a b C dX -=C b a ⎰dX X f C dX X f ba )()(⎰={}dX X g dX X f dX X g X f ba b a b a )()()()(⎰⎰⎰±=± {})()()()()(錯dX X g dX X f dX X g X f b a b a b a ⎰⎰⎰⋅=⋅ []b a C dX X f dX X f dX X f bc c a b a ,,)()()(∈+=⎰⎰⎰f 在X=a 被定義0)(=⇒⎰dX X f a adX X f dX X f ab b a )()(⎰⎰-=0)(0)(≥⇒≥⎰dX X f X 設f b axy f(x)a b ⎰dx x f b a )(肆、齊次函數與尤拉定理(一) n 階齊次函數 (homogeneous function of degree n) ◎ 定義: ),(21X X f y =若0),,(),(2121>∀=λλλλX X f X X f n 則稱為n X X f y ),(21=階齊次函數(二) 尤拉定理 (Euler Theorem)◎定義:若n D O 為H X X f y ...),(21=則211XfX X f ny ∂∂+∂∂=2X ◎ 証明: ),(),(2121X X f X X f n λλλ= 對入微分: ),(2112211X X f n X X f X X f n -=∂∂∂∂+∂∂∂∂λλλλλλλ 令:1=λ),(:212211X X f n X XfX X f =∂∂+∂∂ (三) 齊序函數 (同位函數) (homothetic function)◎ 定義: (一階齊次函數的正單調上升轉換稱之)若 ),(21X X g 為H.O.D 1 且0>='dgdff ),()),((2121X X h x xg f 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 )(X f y =...0)(C O F dX X f dy →='=0)(='=→X f dX dY⇒ ⎩⎨⎧==判斷選一個C O 由S X 個解求得C O 由F X ...2...*2*1 ...02C O S Y d →<→MaxY X X f dXY d dX X f Y d =⇒<''=⇒<''=→*1122220)(0)( 2. 極小:Min )(X f y = ⎩⎨⎧>''='0)(...0)(...X f C O S X f C O F(二) 多變數函數(),21X X 1. ),(21)(X X f Y Max Min =...C O F 0=dY0),(),(2211211=+dX X X f dX X X f⎪⎪⎩⎪⎪⎨⎧===∂∂===∂∂*22122*12111),(00),(0X X X f X Y X X X f X Y...C O S正定Min 全為正Matrix Hession Y d 負定Max 負正相間MatrixHessianY d ⇒⇒<⇒⇒>)(0)(022 0,0222112111122211211>→=><f f f f f f f f f H◎ 有限制條件下之極值分析:Max method LagrangeX X f y →=),(21()Min..t S C X X g =),(21Max Step :1 []C X X g X X f X X L --=),(),(),,(212121λλ ...:2C O F Step01=∂∂X L0),(),(211211=-X X g X X f λ =*1X02=∂∂X L 0),(),(212212=-X X g X X f λ =*2X0=∂∂λL[]0),(21=--C X X g =*λ ...:3C O S Step → L d 2 0><Boarder ⇒ Hessian Matrix⇒正負相間(Max) 全為正 (Min)21222221221112121111g g g g f g f g g f g f F --------=λλλλ陸、古典規劃分析應用:Optimization(1) max )()(Q C PQ Q -=πQ(2) min C=W k r L ⋅+⋅[]K L , 3個主要問題類型),(..L K F Q t s = (3) max f(x)max U(x, y) x or {}y x ,0,0)(≥≥x x g s.t I y p x p y x =+◎ The Structure of an Optimization Problem Max f(x) f(X)=objective function s x ∈ X: choice variables S: feasible set solutions: *X S x x f x f ∈∀≥)()(*Important general problems about the solutions to any optimizationproblem:(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⎪⎩⎪⎨⎧∈∀≥∈∀≥)(),()(:),()(:*****x Be x x f x f Solution Local S x x f x f Solution GlobalPrepositions: A local maximum is always 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 feasible set is strictly convex, or (2) the objective function is strictly quasiconcave, or(3) both(4) Interior and Boundary Optima(5) Location of the Optimum minmax f(x) F.O.C0)(=dxx df X ∈R S.O.C (max )0(min)0)(22<>dx x f d(多變數) 21x x◎ Multivarial Case)(21x x f Y =F.O.C ://2121⎪⎪⎭⎫⎝⎛=⎪⎪⎭⎫ ⎝⎛∂∂∂∂=∇f f x f x f f Gradient vector of fS.O.C ⎪⎪⎭⎫ ⎝⎛=nn n f fn f f H ........1.........111 Hessian of f j iij x f f ∂∂= now, max f(),21x x C O F .. 01=∂∂x f{}21,x x 02=∂∂x fS.O.C 0212<∂∂x f011<f(負定)0222>∂∂x f(0)()()22211211212222212>⇒∂∂∂>∂∂⋅∂∂f f f f 即x x fx fx f212121*********)(0f f f f f f f >⇒>-⇒212)(f◎ Quadratic Forms and their Signs⎪⎪⎭⎫ ⎝⎛=nn n n a aa a A 1111 symmetric:ji ij a a =X A X=(⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛m nn n n m X X a a a a X X 111111..........................)........ =∑∑==ni nt j j i ij x x a 1(1) Negative SemidefiniteR X AX X ''∈∀≤',0 (2) Negative definite 0,0≠∀<'X AX X (3) Positive Semidefinite R X AX X ''∈∀≥',0(4) Positive definite0,0≠∀<'X AX X ex n=2⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛='212221121121)(x x a aa a X X AX X =2222211221112x a x x a x a ++=)()2(2222222112122221112122111122111X a X a a X a a x x a a x a +-++++=221122211211221112111)(X a a a a a x a ax a ++-Negative definite 011<a and022211211>a a a a- Positive definite: 011>a and 022211211>a a a a⇒續 Hessian;H is negative definite if 212121111,0f f f f >< 022<fH is positive definite if 212121111,0f f f f >> 022<fGeneral CaseX ' A X =(⎪⎪⎪⎭⎫ ⎝⎛⎪⎪⎪⎭⎫⎝⎛n nn n n n x x a aa a X X i ........................).. (11111)Negative definite:011<a022211211>a a a a…….nnninn a a a a ...............)1(111-Positive definite:011>a022211211>a a a a0................1111>nnn na a a a◎ Optimizations: The unconstrained case I. may f()x MinF.O.C ............)(11⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡∂∂∂∂=∂∂=n n f f X f X f x x f Df Gradient VeotorS.O.C ......... (22)21112112⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⋅⋅==nn ntn f f f ff f f f D H Hessian Matrix Necessary conditions ⎪⎩⎪⎨⎧=isH C O S Df C O F ..0..te semidefini negativepositiveSufficient conditions Df=0H is definitenositivenegative→f is concave (dx)1-H(dx)<0convex >0 ex 1. )()(max q c q p q q-=∆F.O.CMC P dpd =⇒=0∆S.O.C 002222>⇒<dq Cd dq d ∆ 2. x w x pf x q*)()(max -=∆F.O.C00*=-∂∂⇒=∂∂Wi Xifp xi ∆ Wi VMPi =⇒ S.O.C H is negative definite ⇒f is concave. II. The Constrained Casexmax )(x fs.t g()x =b ⇒ 在有限制下，求最大點dxi Xig.....0=∂∂∑⇒Lagrangian Function:max L ())(()(),b x g x f x --=λλ x ,λ constraint gualification: U xix g ≠∂∂)(0=∂∂Xi L 0)(,,,2,1=∂∂-∂∂⇒=XigXi x f n I λ F.O.C xj gxi gxj f xi f ∂∂∂∂-∂∂⇒)()(0)(=-=∂∂x g b LλD 222)()(x d x L d x L =S.O.C 0))(()(2≤dx x L D x d Tx d ∀ s.t. Dg(0)=x d x 全微分◎ Bordered Hessian)(...........................),(2222212212112212222x L D x L x x L x L x x L x L g x L x L x LL x L D H n in n nn →⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡∂∂∂∂∂∂∂∂∂∂∂∂∂-∂∂∂∂∂∂∂∂∂∂∂==λλλλλλ S.O.C. for max (min)The naturally ordered principled mincrs of the bordered(all be negative)⇒guaslconcaveHessianmatrix alternate in sign, the sign of the firstbeing positivei.e0000232232132332222212223122122121321222122221222121<∂∂∂∂∂∂∂∂-∂∂∂∂∂∂∂∂-∂∂∂∂∂∂∂∂---->∂∂∂∂∂-∂∂∂∂∂---x L x x Lx x Lg x x Lx L x x Lgx x Lx x Lx L g g g g x Lx x L g x x Lx L g g g ex min w xxt s .. g x f =)(⇒Lagrangian funotion: ))((),(*g x f x w x L --=λλ λx maxF.O.C. MRTS xj fxi fwj win I xi x f wi x xi L =∂∂∂∂=⇒=⇒=∂∂-=∂∂,...2,10)(),(***λλ0)(),(**=-=∂∂x f g x L λλ◎xf s .. )(xg b ≤ 0)(≤x g x 0≥ 0≤-xF.O.C 0)(***=∂∂xi x f X i0*≥i X0)(2≤∂∂xix f Max f(x)xl s .⇒Langrangian Function:)()(),(1∑=-=ni x igi x f x L λλMax ),,,(1n X X f Ex nx x t s ...1.. b x g ≤)(0,.......0,021≤-≤-≤-u x x x),,,,,,,,,,(2121n n u u u x x x L λ⇒=)...))()(2211n n x u x u x u b x g x f -++++--λ F.O.C01111=+∂∂-∂∂=∂∂u x gx f x L λ02=+∂∂-∂∂=∂∂u x gx f x L nn n λ 0,0**≥=∂∂λλλL0,0,011111≥≥=∂∂=x u u Lx u 因有ineguediy, …. 所以要多考慮這些可能,0,0≥≥=∂∂=nnnnnxuuLxuex“☆” min2211xwxw+s.t yxx=+211≥x02≥x22112122112121)(),,,,(XMXMyxxxwxwMMXXL---+-+=λλF.O.C⎪⎪⎪⎭⎪⎪⎪⎬⎫⎪⎪⎪⎩⎪⎪⎪⎨⎧==-+-=∂∂=--=∂∂=--=∂∂=022112122111,0)3.....()()2.....(2)1.(..........XMXMyxxLMwXLMwXLλλλ檢查這些條件是否都符合∥∥11=∂∂ULμ022=∂∂⋅ULμ,021≥≥XX∴共有四種組合,021=⇒==yXX (0,22=xμ代入 (2) 式),0,021>=XX step2222,0WW=→=-=⇒λλμ())1(,0,11式代入x=μstep211112,≥-=→+==μλμλWwyx.....21WW≥⇒用第2種生產要素Case 3 ....,0,01221WWXX≥⇒=>用第1種生產要素Case 4 λμμ==⇒==⇒>>212121,0,0WWXXEx {}yWyWyWWC2121,m in),,(=⇒⎪⎩⎪⎨⎧<<=><====212121122121WWyWyWCWWyWyWCWWifyWyWCKuhn-Tucker Formulation⎩⎨⎧≥≤0)(..)(.)(min)(maxXgt sXgt sXfXf))(()(),(b X g X f X L --=⇒λλ Kuhn-Tucker Conditions⎪⎪⎩⎪⎪⎨⎧=∂∂≥≥∂∂=∂∂≥≤∂∂≤≥0,0,00,0(max ),01(min)1λλλL L X L X X X Li i i i i2211m inX W X W + 21,x xs.t. y X X =+21 0,021≥≥X X)(212211y X X X W X W L -+-+=λ (K-T conditions):0,0,011111=∂∂≥≥-=∂∂X LX X W X L λ 0,0,022222=∂∂≥≥-=∂∂X LX X W X Lλ021=-+=∂∂Y X X L λ 0≥λ 0=∂∂λλL Utility Maximization Problem max u(x, y)x, ys.t I y P x P y X ≤⋅+⋅0,00,0≤≤-⇒≥≥y x y x Comparative StaticsF.O.C ).....(.....,,0)...................(0)....,........(21**2*1212121211m i j n m n n m n a a a 且x X X X 可求得equations n x x x x x x F x x x f ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧==M ααα → Implicit FunctionsImplicit Functions Theorem If D=01111≠nn nn f f f f *j X =)......(m i h αα之影響為正或負a 受即X aj xi m i .....0*1**α<>∂∂-totally differentiating the system0 (21)111112111=+++++++m m n n n n a d f d f dx f dx f dx f α11112211....αd f f dx f dx f dx f n nn n n n n n +++++++0...=+++am d mn f n⎪⎪⎭⎫⎝⎛⎪⎪⎭⎫ ⎝⎛---⎪⎪⎭⎫ ⎝⎛++-++--⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛++++++++m n m n n n m n n m n m n n n m m n n n n n n n d d f f f f d f da f d f d f dx dx f f f f ααααα1111111111111111.......).....()......(................... ∥ ∥ ∥ ∥ D i dx ij D j d α222//xf x f D D x j d D D dx ij j i iji ∂∂∂∂==∂∂⇒=⇒αααα <Cramer's Rule>(無限制式) ex max 221121),(.X W X W X X f P --=⨯π F.O.C⎪⎪⎩⎪⎪⎨⎧==∂∂===∂∂=0011111111w pf x w pf x ππππTotally differentiate F.O.C with respect to :1W⎪⎪⎩⎪⎪⎨⎧=-∂∂⋅∂∂⋅+∂∂⋅∂∂⋅=-∂∂⋅∂∂⋅+∂∂⋅∂∂⋅01011*2221*1121*2211*11w x x f P w x x f P w x x f P w x x f P ⎪⎪⎩⎪⎪⎨⎧=∂∂+∂∂=∂∂+∂∂⇒011*2221*1211*2121*111w x pf w x pf w x pf w x pf ⎪⎪⎭⎫ ⎝⎛⇒22211211Pf PfPf Pf ⎪⎪⎪⎭⎫ ⎝⎛=⎪⎪⎪⎪⎭⎫ ⎝⎛∂∂∂∂011*21*1w x w x By Cramer's Rule0)(012122211222221121122121*1<⊕Θ⊕-Θ==∂∂f f f p f pf pf pf pf pf pf w x 0)(012122211212221121112111*2<⊕Θ⊕-⊕-==∂∂f f f p f pf pf pf pf pf pf w x同理 0)(2122211112*2<⊕-Θ=∂∂f f f p f w x 0)12()(2122211122*1<-=⊕-Θ-=∂∂f sign f f f p f w x (有限制式)max 1121)(X W X X pf -=π 21x x22οX X =s.t.)()(21121X X X W X X P L -+-⋅=⇒ολ F.O.C)(),(000211*22211*1122111X P W X X X P W X X pf X L w pf X L ==⎪⎪⎭⎪⎪⎬⎫=-=∂∂=-=∂∂λ0220=-=∂∂X X L λ),(0211*X P W λλ= S.O.C 1X L ∂∂∂λ 2X L ∂∂∂λ=H 2222122212121210x x x x g X X XL xx gx gx g ∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂ππππ=222112110110010pf pf pf pf --- = 011>-pf (p )011<p >0 <000..011max >⇒⎭⎬⎫><∴H C 又又O F f TL ΘFrom⎝⎛=-=-=-=∂∂=∂∂=∂∂00)(0)(202**2*121*2*1121X X W X X Pf W X X Pf L X L X Lλ1..對W C O F ⎪⎪⎩⎪⎪⎨⎧=∂∂-∂∂+∂∂=-∂∂+∂∂0011*1*2221*1211*2121*111w w x pf w x pf w x pf w x Pf λtotally differentiating with respect to w1:⎪⎪⎪⎭⎫ ⎝⎛=⎪⎪⎪⎪⎪⎪⎭⎫ ⎝⎛∂∂∂∂∂∂⎪⎪⎪⎭⎫ ⎝⎛----0101010101001*11*11*22211211w x wx w pf pf pf pf λ By the Cramer's Rule:⎪⎪⎩⎪⎪⎨⎧>Θ-⊕=∂∂=-=∂∂<Θ-=∂∂0/0/00/111211*111*111*pf f w x pf w x pf w x1121)(x w x x pf -=π),,()),,(),,,((021*11021021*1*x p w x w x p w x p w x pf -=π 把和x *1*2x 算出，代入利潤函數中，即可得:…*profit FunctionBut 此題中022X X 可直接代入π為one decision 的問題，不需如此 麻煩。