微观经济学-数学基础

 1   ISlutsky 2   AbadirandMagnus2005MatrixalgebraCambridgeUniversityPressAtkesonandLucas,1992OnEfficientDistributionwithPrivateInformation,ReviewofEconomicStudies59(3)P.427-453BoydandVandenberghe2004ConvexoptimizationCambridgeUniversityPressJehleandReny2001Rubinstein2006LecturenotesinmicroeconomictheoryPrincetonUniversityPressSimonandBlume1994MathematicsforeconomistsW.W.NortonSydsaeterStromandBerck2005Economists'mathematicalmanualSpringerMas-ColellWhinstonandGreen,1995,MicroeconomicTheoryOxfordUniversityPressVarian,1992,MicroeconomicAnalysis.RobertGibbons,1992,GameTheoryforAppliedEconomists.ArielRubinstein,2007,LectureNotesinMicroeconomicTheory.WilliamSandholm,2006,LectureNotes,UniversityofWisconsinMadison.JimmyChan,2008,LectureNotes,SHUFE.BingyongZheng,2008,LectureNotes,SHUFE.LarrySamuelson,2008,LectureNotes,YaleUniversity.DebrajRay,2006,LectureNotes,NewYorkUniversity.DirkBergermann,2008,LectureNotes,YaleUniversity.YuliySannikov,2005,LectureNotes,UCBerkeley.FrancescoSquintani,2006,LectureNotes,UniversityofEssex. 3   1.[][JehleandReny]n()fxt()()kftxtfx≡()fxk()fxk1k−1212(,,...,)(,,...,)knnftxtxtxtfxxx≡ix()()kiiftxfxttxx∂∂=∂∂0t1()()kiiftxfxtxx−∂∂=∂∂1k−2.[][JehleandReny]k()fx1()()niiifxxkfxx=∂=∂∑t()()gtftx=()fxkt1212()(,,...,)(,,...,)knngtftxtxtxtfxxx==t11()'()()nkiiiftxgtxktfxx−=∂==∂∑t1t=1()()niiifxxkfxx=∂=∂∑3.[][JehleandReny]x1()()niiifxxkfxx=∂=∂∑()fxk1()()niiifxxkfxx=∂=∂∑xtx1nn+xx1()()niiiftxtxkftxx=∂=∂∑t()()gtftx='()()tgtkgt='()/()/gtgtkt=tln()lnlngtktC=+()kgtCt= 4   (1)()gfx=()Cfx=()()kgttfx=()()kftxtfx≡()fxk4.[][SydsaeterStromandBerck]k()fx''11()(1)()nnijijijxxfxkkfx===−∑∑t()()gtftx=()fxkt1212()(,,...,)(,,...,)knngtftxtxtxtfxxx==t11()'()()nkiiiftxgtxktfxx−=∂==∂∑t2211()''()(1)()nnkijiijjftxgtxxkktfxxx−==∂==−∂∂∑∑t1t=''11()(1)()nnijijijxxfxkkfx===−∑∑5.[][SimonandBlume]fgn1k2kfg+0t∀1()()kftxtfx=2()()kftxtfx=()()()xfxgxϕ=+()xϕ()()ktxtxϕϕ=12()()()()kkkktfxtgxtfxtfx+=+t1t=12()()()()kfxkgxkfxkgx+=+()fx()gx12kkk==12kk≠fg+6.[][SimonandBlume]n+()uxkn+()fxg()(())uxgfx=()uxn+()ux(1,2,...,in=iixy()()uxuy()ux)0t()()()()uxuyutxuty⇔()ux()()(())(())uxuygfxgfy⇔()(())uxgfx=()()fxfy⇔g()()kktfxtfy⇔0t()()ftxfty⇔()fxk(())(())gftxgfty⇔g()()utxuty⇔()(())uxgfx=0t()()()()uxuyutxuty⇔()()()()uxuyutxuty=⇔=(*) 5   n(1,1,...,1)e=()()gtute=()ux()()gtute=gvfvu=()()gfgvugvuu===gfuugf=()()()uxgfx=()()gtute=()()()uxufxe=()tfx=t()()uxute=(*)()()uxuteαα=()()gtute=()()uxgtαα=()()()uxgfx=()()()gfxgtαα=g()fxtαα=()()fxfxαα=f7.[][SimonandBlume]()uxn()()()()iijjuutxxxxuutxxxx∂∂∂∂=∂∂∂∂()uxnnk()fxg()(())uxgfx=11()()()()()'(())()()()()()'(())kiiiiikjjjjjuftxftxfxfxtxgftxtxxxxxuftxftxfxfxtxgftxtxxxxx−−∂∂∂∂∂∂∂∂∂∂====∂∂∂∂∂∂∂∂∂∂()()()'(())()()()'(())iiijjjufxfxxgfxxxxufxfxxgfxxxx∂∂∂∂∂∂==∂∂∂∂∂∂()()()()iijjuutxxxxuutxxxx∂∂∂∂=∂∂∂∂8.[]1(,,)fxyzAxyzαβγ=2(,,)max{,,}fxyzxyz=3(,,)min{,,}fxyzxyz=4(,,)lnlnlnfxyzxyzαβγ=++522(,)fxyxyxy=+1(,,)(,,)ftxtytztAxyztfxyzαβγαβγαβγ++++== 6   (,,)fxyzAxyzαβγ=2(,,)(',',')ftxtytzftxtytz(0t)max{,,}max{',','}txtytztxtytz⇔max{,,}max{',','}txyztxyz⇔max{,,}max{',','}xyzxyz⇔(,,)(',',')fxyzfxyz⇔(,,)max{,,}fxyzxyz=3(,,)min{,,}fxyzxyz=4(,,)gxyzxyzαβγ=(,,)ln(,,)fxyzgxyz=(,,)lnlnlnfxyzxyzαβγ=++5,xy2222fxyxyfyxxy∂∂+=∂∂+22(,)fxyxyxy=+1.[][SydsaeterStromandBerck]CC(nC⊆),xy(0,1)λ∈(1)xyCλλ+−∈Cf:C→,xyC∈(0,1)α∈((1))()(1)()fxyfxfyαααα+−≥+−f,xyC∈xy≠(0,1)α∈((1))()(1)()fxyfxfyαααα+−+−f,xyC∈(0,1)α∈((1))()(1)()fxyfxfyαααα+−≤+−f,xyC∈xy≠(0,1)α∈((1))()(1)()fxyfxfyαααα+−+−ff⇔f−f⇔f−,xyC∀∈(0,1)α∀∈((1))()(1)()fxyfxfyαααα+−≥+−((1)){()}(1){()}fxyfxfyαααα⇔−+−≤−+−−f⇔f−,xyC∀∈(0,1)α∀∈((1))()(1)()fxyfxfyαααα+−+−((1)){()}(1){()}fxyfxfyαααα⇔−+−−+−−f⇔f−2.[][SydsaeterStromandBerck]Cf:C→,xyC∈(0,1)α∈((1))min{(),()}fxyfxfyαα+−≥f,xyC∈xy≠(0,1)α∈((1))min{(),()}fxyfxfyαα+−f 7   ,xyC∈(0,1)α∈((1))max{(),()}fxyfxfyαα+−≤f,xyC∈xy≠(0,1)α∈((1))max{(),()}fxyfxfyαα+−f(1)f⇒f(2)f⇒f(3)f⇒f(4)f⇒f(1)f,xyC∀∈(0,1)α∀∈((1))()(1)()fxyfxfyαααα+−≥+−.()()fxfy≥()(1)()()fxfyfyαα+−≥.()()fyfx≥()(1)()()fxfyfxαα+−≥.()(1)()min{(),()}fxfyfxfyαα+−≥((1))min{(),()}fxyfxfyαα+−≥f(2)f,xyC∀∈(0,1)α∀∈((1))()(1)()fxyfxfyαααα+−≤+−.()()fxfy≥()(1)()()fxfyfxαα+−≤.()()fyfx≥()(1)()()fxfyfyαα+−≤.()(1)()max{(),()}fxfyfxfyαα+−≤((1))max{(),()}fxyfxfyαα+−≥f(3)f,xyC∀∈(xy≠)(0,1)α∀∈((1))()(1)()fxyfxfyαααα+−+−.()()fxfy≥()(1)()()fxfyfyαα+−≥.()()fyfx≥()(1)()()fxfyfxαα+−≥.()(1)()min{(),()}fxfyfxfyαα+−≥((1))min{(),()}fxyfxfyαα+−f(4)f,xyC∀∈(xy≠)(0,1)α∀∈((1))()(1)()fxyfxfyαααα+−+−.()()fxfy≥()(1)()()fxfyfxαα+−≤.()()fyfx≥()(1)()()fxfyfyαα+−≤.()(1)()max{(),()}fxfyfxfyαα+−≤((1))max{(),()}fxyfxfyαα+−f3.[][AtkesonandLucas](1)f12,,...,0nααα≥11niiα==∑11()()nniiiiiifxfxαα==≥∑∑()∗(2)exp()exp()1LLHHPwPw⋅+⋅=1LHPP+=01LP0LLHHPwPw⋅+⋅(1)2n=()∗nk=()∗1nk=+()1111111()(1)1kkiiikikkiikfxfxxααααα++++==+=−+−∑∑ 8   11111(1)()()1kikikkikfxfxαααα+++=+≥−+−∑f11111(1)()()1kikikkikfxfxαααα+++=+⎡⎤≥−+⎢⎥−⎣⎦∑11()kiiifxα+==∑fnk=()∗()∗f()∗ff12,,...,nααα1(2)()exp()''exp()0xx=exp()xexp()exp()exp()1LLHHLLHHPwPwPwPw⋅+⋅⋅+⋅=0LLHHPwPw⋅+⋅4.[][SydsaeterStromandBerck](1)fI(I∈)fI,xyI∈'()()()()'()()fyyxfyfxfxyx−≤−≤−(2)fI(I∈)fI'fI(3)fI(I∈)fI''()0f