1微观经济理论I教师信息微观经济理论微观经济理论LA数

1I1•–604–[e-mail]––•325•65902704•Email:[e-mail]2•–••–•NEBE•SPE3•–G.A.Jehle&P.Reny•AdvancedMicroeconomicTheory–A.Mas-Colell,M.D.Whinston&J.R.Green•MicroeconomicTheory–A.Rubinstein•MicroeconomicsLecturenote–Varian,H.R.,MicroeconomicAnalysis–4LA•–•–•–•*62•n–x=(x1,…,xn)•x1=(x11,…,xn1),x2=(x12,…,xn2)–x1≥x2iff(ifandonlyif)xi1≥xi2,i=1,…,n112iff(ifdlif)12i12–x1x2iff(ifandonlyif)xi1xi2,i=1,…,n2x2x12x11x21x22=x1x2x11x11x21x1x2x12x12x22x2•n\n•–x∈R,•12121212x,x--()()()dxxxxxx==−x2x1)x-x)(x-x()x,x(212121=d–x∈R2,•()()()121212121211221122x,x----()(,)(,)dxxxxxxxx=1221221122--()()xxxx=+x1x2•(ε-ε-ball)–x0∈Rn,ε0Bε(x0)={x⏐d(x,x0)ε}•–Bε(x0)={x⏐d(x,x0)≤ε}•n=2x0Bε(x0)εx0B*ε(x0)εx•A1.5:nS⊂Rn∀x∈S,∃ε0Bε(x)⊂S10x1x2x1x1x2x2•A1.5:nS⊂Rn∀x∈S,∃ε0Bε(x)⊂S11A1.6SScS•(M.F.3,P943)SSS,,{x1,x2,…,xn}1xn∈S∀n=1,2…2x∈Slimxxnn→∞=12x1x2x1x1x2x23•(BoundedSets)–nSε-–,∃ε0,x∈Rn,S⊂Bε(x)•(compactset)–x1x2xSε•–x,x′xt=tx+(1-t)x′∀t∈[0,1xx′xtx2x′2x′1x1xt2=tx2+(1-t)x′2xt1=tx1+(1-t)x′1A.•–Sx,x′SS–∀x,x′∈S,xt=tx+(1-t)x′∀t∈[0,1]xxtx′Sxxtx′SB.•A1.16DRf(x)DÆR16•A1.10–S⊆Rnf:SÆDx*,x#∈S,f(x*)≤f(x)≤f(x#)∀x∈SB.•–L(y)={x⏐f(x)=y,x∈D}–S(y)={x⏐f(x)≥y,x∈D}–I(y)={x⏐f(x)≤y,x∈D}•n=1n1–D=[-4,4]–L(2){-3,3.3}–S(2)=[-4,-3]∪[3.3,4]–I(2)=[-3,3.3]y=2y=f(x)-443.3-3B.•A1.22:f(x)DÆRD∀x1,x2,∈D,f(xt)≥tf(x1)+(1-t)f(x2)},t∈[0,1].18x2x1xttf(x1)+(1-t)f(x2)f(xt)4A2.4*:f(x)f(x)DÆR0000allx,(x)(x)(x)(x-x)z⇔∈≤+∇∀∈ForDfffDz(x)z0zHD⇔⋅⋅≤∀∈B.1n(x)(x)(x)(,...,)fffxx∂∂∇=∂∂(gradient)1111(x)nmmnffHff⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠…2(x)ijijffxx∂=∂∂B.•A1.24:f(x)DÆRD∀x1,x2,∈D,f(xt)≥min{f(x1),f(x2)},t∈[0,1].20x2x1xtf(x1)=min{f(x2),f(x2)f(xt)A1.15B.•21x2x1xtx2x1xtx2x1xt1.14f(x)DÆRS(y)y2S(y1)y1S(y)={x⏐f(x)≥y,x∈D}S(y2)1y3S(y3)y4S(y4)S(y4)•–S(y)⇒f(x)•∀x1,x2∈D•f(x1)≥f(x2)=y2=min{f(x1),f(x2)}1.14f(x)DÆRS(y)•⇒x1,x2∈S(y2)S(y2)•⇒xt∈S(y2)t∈[0,1]•⇒f(xt)≥f(x2)=min{f(x1),f(x2)}•⇒f(xt)≥min{f(x1),f(x2)}•⇒f(x)•–f(x)⇒S(y)•∀x1,x2∈S(y)y∈R,•f(x1)≥yf(x2)≥y1.14f(x)DÆRS(y)•f(x)•⇒f(xt)≥min{f(x1),f(x2)}•⇒f(xt)≥y•⇒xt∈S(y)5M.C.3-4:(MGW)f(x)iff(x)x-x(x)(x)fff′′⇔⋅≥∀≥1B.z(x)z0(x)z=0⇔⋅⋅≤∀∇⋅Hf2•A1.25:f(x)DÆRD∀x1,x2,∈D,f(xt)min{f(x1),f(x2)},t∈(0,1)B.26L(y1)S(y1)S(y1))={(x1,x2)⏐x1+x2=y1}B.•A1.25:f(x)DÆRD∀x1,x2,∈D,f(xt)≤max{f(x1),f(x2)},t∈(0,1)•A1.25:f(x)DÆRD∀x1,x2,∈D,f(xt)max{f(x1),f(x2)},t∈(0,1)27C•1.–•2.–•••3*.–-•Non-binding28C.1•P1Maxx∈Dy=f(x,a)–∂f(x*)⁄∂xi=0i=1,2,..,n–H=D2f(x*)•dxTH(x*)dx≤0,∀dx=(dx1,dx2,…,dxn)•(-1)i|Hi|0i=1,2,..,n•P2Minx∈Df(x,a)–∂f(x*)⁄∂xi=0i=1,2,..,n–D2f(x*)•|Hi|0i=1,2,..,n29C.1•n=1•P1Maxx∈Dy=f(x,a)–∂f(x*)⁄∂x=0•dy=f′dx=0∀dxB(x*,δ)∂2f(*)≤0–∂2f(x*)≤0•d2y=f″dx2≤0∀dxB(x*,δ)–306C.1•n=2•P1Maxx∈Dy=f(x,a)–∂f(x*)⁄∂xi=0i=1,2•dy=f1dx1+f2dx2=0∀dx=(dx1,dx2)–H=D2f(x*)•d2y=(f11dx1+f12dx2)dx1+(f21dx1+f22dx2)dx1•f(x,a)x*31[]1112121221222,0ffdxdydxdxffdx⎡⎤⎡⎤⇒=≤⎢⎥⎢⎥⎣⎦⎣⎦C.1•–f(x)x*f(x)x*f(x)32C.2•–P2Maxxf(x,a)s.t:x∈B={x:gj(x,a)=0i=1,2,..,m}gj(x,a)=0j=1,2,..,m33M.K.4(MWG-P962)P2BC.2•–P2Maxxf(x,a)s.t:x∈B={x:gj(x,a)=0i=1,2,..,m}gj(x,a)=0j=1,2,..,m•–L=f(x,a)+λ1g1(x,a)+…+λmgm(x,a)–•∂L⁄∂xi0,i=1,2,..,n•∂L⁄∂λj0,j=1,2,…,m34C.2•*–D2L(x)•dxTD2L(x)dx≤0,∀dxDgj(x)dx=0j=1,…,m–•Hm+i(-1)0,i=1,2,...,nmiH+•n2m1:352222122221111222222122LLLxxLLLHxxxxxLLLxxxxxλλλλλ⎛⎞∂∂∂⎜⎟∂∂∂∂∂⎜⎟⎜⎟∂∂∂=⎜⎟∂∂∂∂∂∂⎜⎟⎜⎟∂∂∂⎜⎟⎜⎟∂∂∂∂∂∂⎝⎠12221111222221220ggxxgLLxxxxxgLLxxxxx⎛⎞∂∂⎜⎟∂∂⎜⎟⎜⎟∂∂∂=⎜⎟∂∂∂∂∂⎜⎟⎜⎟∂∂∂⎜⎟⎜⎟∂∂∂∂∂⎝⎠(),,,,mi+230,0HHC.2•*–D2L(x)•dxTD2L(x)dx0,∀dxDgj(x)dx=0j=1,…,m–H•360,i=1,2,...,nmiH+7C.3•P3Maxxf(x,a)s.t:x∈B={x:gj(x,a)0i=1,2,..,m}gj(x,a)0j=1,2,..,m•P4Minxf(x,a)st:x∈B={x:g(xa)≤0i=12m}s.t:x∈B={x:gi(x,a)≤0i=1,2,..,m}gj(x,a)≤0j=1,2,..,m37M.K.4(MWG-P962)P3-4BC.3•g(x,a)=k-h(x)0–Bindingconstraint:k=k1•h(x*)=k1–Non-bindingconstraint:k=k2•h(x*)k2h(x)k238h(x,a)k1f(x,a)h(x,a)f(x,a)k2x*xmaxx*=xmaxC.3•g(x,a)=k-h(x)0•1f′(x,a)0–g(x*,a)=0bindingconstraint–•gj(x,a)0Ægj(x,a)=0gj(x,a)0Ægj(x,a)039k1f(x,a)h(x,a)k2x*x*C.3*Kuhn-Tucker•u(x)gi(x,a)•–L=u(x)+λ1g1(x,a)+…+λkgk(x,a)•Kuhn-Tucker–∂L⁄∂xi≤0,xi*(∂L⁄∂xi)=0,xi≥0,i=1,2,..,ni≤,i(i),i≥,,,,–∂L⁄∂λj0,λ*j(∂L⁄∂λj)=0,λj≥0,j=1,2,…,k•x*0–∂L⁄∂xi0,i=1,2,..,n–∂L⁄∂λj0j=1,2,…,k•D2L(x)–zTD2L(x)z≤0,∀zDg(x)z=040C.3*Kuhn-Tucker•u(x)gi(x,a)•–L=u(x)+λ1g1(x,a)+…+λkgk(x,a)•Kuhn-Tucker–∂L⁄∂xi≤0,xi*(∂L⁄∂xi)=0,xi≥0,i=1,2,..,ni≤,i(i),i≥,,,,–∂L⁄∂λj≤0,λ*j(∂L⁄∂λj)=0,λj≥0,j=1,2,…,k•x*0–∂L⁄∂xi0,i=1,2,..,n–∂L⁄∂λj0j=1,2,…,k•D2L(x)–zTD2L(x)z≤0,∀zDg(x)z=041D.1*•x*0–∂L(x,λ,a)⁄∂xi0,i=1,2,..,n(1)–∂L(x,λ,a)⁄∂λj0j=1,2,…,m(2)•(1-2)ak2//Lλλ⎛⎞∂∂∂∂∂⎛⎞4221122//(x,,a)*.......//kknknkaLaDLxaLxaλλλ⎛⎞∂∂−∂∂∂⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎜⎟∂∂−∂∂∂⎝⎠⎝⎠22(x,,a)/(x,,a)iikDLxaDLλλ∂∂=Ax=bxi=|Ai|/|A|8D.2•(maximum-valuefunction)–M(a)≡maxxf(x,a)s.t.g(x,a)=0,x≥0.–M(a)=f(x*(a),a)=L(x*(a),λ*(a),,a)=f(x*(a),a)+λ*(a)g(x*(a),a)•∂M(a)/∂aj≡∂[L(x*(a),λ*(a),a)]/∂aj=∂L(x*(a),λ*(a),a)/∂xi·∂xi*(a)/∂aj+∂L(x*(a),λ*(a),a)/∂λ·∂λ*(a)/∂aj+∂L(x,λ,a)/∂aj⏐(x*,λ*)=∂L(x,λ,a)/∂aj⏐(x*,λ*)=0[]D.2•A2.21:–fgaax(a)0x(a)aM(a)aλ(a)x(a)x(a)()(x,a)(x,a)(a)x(a),(a)jjjjMafgLaaaaλλ∂∂∂∂=−=∂∂∂∂