高级微观经济学

121„1„2„3„4„5„6---„731„1.1„1.2„1.3„1.4„1.5„1.6„1.7„1.8411.1„5„„1„2xyxxyff%%,xyyxxyfff%%6„A1completenees–––x,y,Xxyyx∈ff%%7„A2(reflexivity)–––,xXxx∈f%8„A3transitivity–––,,,,xyzXxyyzxz∈f%ff%%9A1—A310„A4continuity–––{}{}{}{},::::yXxxyxxyxxyxxy∈fp%%fp11–12221211121211122,,,;xxRxxxxxxxx+∈f12„A5(monotonicity)–:––,,,,xyXxyxyxyxyxy∈≥≥≠f%f13„A6(localnosatiation)–––0,xXyXxyyxεε∈∈−f14„A7convexity–––,,,,(1),,,,,,(1),xyzXxzyzttxtyzxyzXxyxzyzttxtyz∈≤≤+−∈≠+−ff%%f%f%ff%15A1---A716„“”12345617„„:nuRR+→181„1.1„1.2„1.3„1.4„1.5„1.6„1.7„1.81911.2„„„„„„2011.2(monotonictransformation)„'':,:,:()0()0nnfffRRFRRFfRRFFfFFf++++++→→→211.2(monotonictransformation)„„„221.2(monotonictransformation)„„MRS(1)[()]()0(2)[()]()(3)[()][()]0(4)[()]lg()nFuuFuuFuunFuuααα=⋅=+=≥=231.21231212uxxxx1212(,)uxxaxbx=+{}1212(,)min,uxxaxbx=241.245Cobb—Dauglas1212(,)()uxxvxx=+1212(,)uxxxxαβ=251.2„122(,),,(1,2),(,1,2)niijuxxxuinxuijnxx∂=⋅⋅⋅⋅⋅⋅∂∂=⋅⋅⋅⋅⋅⋅∂∂LL261.2„01111,01111,1,1,111,0(,)0(,,)(,,(,),)(,,)()iniiiiiniiiininiiiinuxxxuuxxxxxxxuxxxxxxxxxuxxxxxxuxu−+−−++−+=∂≠∂=≡==LLLLLLLLLL271.2„MRS••MRS012211,212(,),uxxudxMUMRSdxMU==−=281.21,()1((1))()(1)(),((1))()(1)(),uvyfxfuvfufvfconcavefuvfufvfconvexθθθθθθθθθ=≤≤+−≥+−+−≤+−291.21–((1))()(1)(),((1))()(1)(),fuvfufvffuvfufvfθθθθθθθθ+−+−+−+−301.21{}{}((1))min(),(),((1))max(),(),fuvfufvffuvfufvfθθθθ+−≥+−≤311.21–{}{}((1))min(),(),((1))max(),(),fuvfufvffuvfufvfθθθθ+−+−321.22„1„331.22„2f(x)„341.22„3f(x)-f(x)f(x)-f(x)„351.22„4f(x)g(x)f(x)g(x)f(x)g(x)f(x)g(x)„361.23„„371.24HessianMatrix„„14[]HH⎡⎤⎣⎦381.24HessianMatrix„1f(x)„2f(x)f(x)[]H[]H391.24HessianMatrix„3f(x)„4f(x)f(x)H⎡⎤⎣⎦H⎡⎤⎣⎦401.2separability&additivity1'''''',;,,,;,()(,)(,)(,)(,),mmmmmnmmmmmmmmmmmmmmmmmmSRxxSnnmSRxxSnSSSxxxxxxxxSS+−−+−−−−−−−−−⎧⎫⊂∈⎪⎪⎨⎬−⊂∈⎪⎪⎩⎭=⊗ff%%411.2separability&additivity1„x1,x2()(){}()%%(){}031212031212,:,;,2,:,;,nnxxxxxxxxxxxxxxLL421.2separability&additivity2„MRS1()[()]1,2niiiuFfxin===⋅⋅⋅⋅⋅⋅∑431.2separability&additivity2„201()()1,2niiiufxin===⋅⋅⋅⋅⋅⋅∑441.2separability&additivity3„MRS112212((,)(,))kkkkiuFfxxxfxxx+++=⋅⋅⋅⋅⋅⋅+⋅⋅⋅⋅⋅⋅+⋅⋅⋅⋅⋅⋅451.2separability&additivity3„20112212(,)(,)kkkkiufxxxfxxx+++=⋅⋅⋅⋅⋅⋅+⋅⋅⋅⋅⋅⋅+⋅⋅⋅⋅⋅⋅461.24Koopmans“”„„''22(,)(,)nnxyyxyyLLfLL%''22(,)(,)nnyyxyyxLLfLL%f%471.2„22(,)(,)nnxzzyzzLLfLL%22(,)(,)nnxywwyxwwLLfLL%f%481.2„KoopmansU(·)„Koopmans11()()0ntnttUxxuxαα==〉∑LL1α〈f%f%49''22'''22'11''22,(,)(1)()(),(,)(3)(3)()()()nnttttnnttttnnxyyxyyuyuyyyxyyxuyuyyyyy≥≥∑∑∑∑LfL%LfL%LfL%11()()()()(0)(5)ttttttuuuyuyαα∑∑501.212221222111()()()#ttttnnttttttntttuyuyuyuyuyuyuyuyUuyuyUxuxαααααααα===⋅===∑∑∑M511.2222()()()()()()(()())(()())()()01#uxuyuxuyuyuxuxuyuxuyuxuyααααααα++−−−∴Q521.2(homogeneous&homotheticutilityfunction)1„:0()()()nkfRRtftxtfxfxk++→=531.2„::()(())nnfRRgfRRfuxgfx++++→→=541.22„1kk-1551.22„2–MRS561.22„2–571.22„2–identical581.23()„„1()nbiiiUXaXb==∑1()lnniiiUXaX==∑591„1.1„1.2„1.3„1.4„1.5„1.6„1.7„1.8601.3„{}{}112211212,,(1)(2):(,),(,)(3):,,:,,nnniiinnnnpmpmnnpmpmpmpxpxpxmpxmPXmPpppXxxxHxRPXmmRPRHHxRPXmmRPRHH=++++++++++++⋅⋅⋅⋅⋅⋅+≤≤≤=⋅⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅⋅=∈=∈∈=∈∈∈=∑{},:,,nnpmxRPXmmRPRH+++++∈∈∈611.3„„„–––––621„1.1„1.2„1.3„1.4„1.5„1.6„1.7„1.8631.41„1,212,().(,)()()maxxxuxstpxmLxxuxmpxλλ==+−641.41„FOC1211122200(1)0(2)0(3)LLLxxLupxxLupxxLmpxλλλλ∂∂∂===∂∂∂∂∂=−=∂∂∂∂=−=∂∂∂=−=∂651.4FOC/(1)///(2)iiijjjjiijuxpMRSuxpuxuxppλ∂∂==∂∂∂∂∂∂==661.4„SOCSOCH⎡⎤⎣⎦H⎡⎤⎣⎦671.41„(),xxpm∗=(),pmλλ∗=681.41„1121221212312121212(,,;,,)0(,,;,,)0(,,;,,)0,,;,,FOCFxxppmFxxppmFxxppmxxppmλλλλ===691.4111122221233312*1112*2212*12//////0///(,,);(,,);(,,)FxFxFJFxFxFFxFxFxxppmxxppmppmJλλλλλ∂∂∂∂∂∂=∂∂∂∂∂∂≠∂∂∂∂∂∂===≠701.42„1„2711.42„312312721.43„„„„„„731.4„1x(p,m)pm–„2x(p,m)pm––.(,)(,)0iextptmxpmt=∀741.412„1v(pm)pm*()[(,)](,)(,)(,)uxuxpmvpmuvpmvpm===''''.:,(,)(,),(,)(,)ieppvpmvpmmmvpmvpm≥≤≥≥751.4„2v(pm)pm„3v(pm)pi.ek{p:v(p,m)k}.(,)(,),0ievtptmvpmt=∀761.424p0,m0,v(pm)771.43„FOC„λ()iiuxxpλ∂∂=umλ∂∂781.4Roy’sIdentity„––791.4„1111(,)((,))(,)((,))()(1)(,)(,)0(,)(2)(,)(1)(2)jjnniiiiijjijjjnniijiijiijjjpvpmuxpmpxxvpmuxpmuxpppxpppxpmmpxxxpmppxpmppvpmxpλλ====∂∂∂∂∂==⋅=⋅∂∂∂∂∂=∂∂+⋅=⇒⋅=−∂∂∂⇒=−∂∑∑∑∑(,)(3)jpm801.4111(,)((,))(,)((,))()(4)(,)1(5)(,)(4)(5)(6)(,)/(3)(6)(,)(,)/nniiiiiiniiiiimvpmuxpmmxxvpmuxpmuxpmmxmmpxpmmxpmvpmmvpmpxpmvpmλλ===∂∂∂∂∂==⋅=⋅∂∂∂∂∂=∂⋅=∂∂⇒=∂∂∂=−∂∑∑∑im∂81(,)(,)(,)(,)max().(,,)()()(,):(,)0(,)(1)xxpmpmiiixxpmixxpmiuxstpxmLxpmuxmpxvpmvpmLppxvpmpxλλλλλλλλλ=====+−∂∂=∂∂=−∂∂=„„„821.4(,)(,)(,)(,)0(,)(2)(1)(2)(,)/(,)(,)/xxpmpmpmiiivpmLmmvpmmvpmpxpmvpmmλλλλλλλλ===∂∂=∂∂=+∂=∂∂∂=−∂∂