上海财经大学经济学院《高级微观经济学》题库4

1/6213(,)pxpwp=123(,)pxpwp=−(,)xpw(,)xpw31iiipxw==∑33(,)wxpwp=221133(,)(,)ppxpwxpwppαααα===112233(,)(,)ppxpwxpwppαααα=−=−=3333(,)(,)wwxpwxpwppαααα===(,)xpw(1,2,1)p=1w=(1,1,1)p′=2w′=(,)(2,1,1)xpw=−(,)(1,1,2)xpw′′=−(,)2(,)pxpwwpxpw′′′′′===ii(,)xpw′′(,)xpw(,)1(,)pxpwwpxpw′′′===ii(,)xpw(,)xpw′′p'px'x'pxpx⋅≥⋅x'x'''pxpx⋅⋅'''pxpx⋅≥⋅'xx2.u(x1,x2)=x1(x2+c)c012312()xxcu+=x1x212{}12,(,m)axxxuxxs.t.1122pxpxy+=121211221122)(,)(()()uxxxxcLpxpxypxpxyλλ−++−=+−=−2/61/0Lx∂∂=2/0Lx∂∂=1122pxpxy+=⇒211*2ycpxp+=222*2ycpxp−=1*x2*x12(,)uxx(,)vpy221212()(,)(*,*)4ycpvpyuxxpp+==122(,)2epuppucp⇒=−2212()4ycpvpp+=1222yppvcp⇒=−3.uv0p0x(,)()vppxux≥p(,)()vppxux=(,)max()vppxux′=s.t.pxpx′≤x(,)()vppxux≥p(,)()vppxux=1()max(,...,)nuxxx=(1,...,1)x=0p11(,0,...,0)1()iippuuxpp==∑∑1(,0,...,0)ipp∑px12min{,}uxx=12xx≠-Cobb-Douglasxxixippx()(,)uxvppx=4.w1213331005ppwxpppβδ=−++122333ppwxpppαβγδ=+++3/631iiipxw==∑1221//hhxpxp∂∂=∂∂///hijijjixpxpxxw∂∂=∂∂+∂∂i12123333333333()(1005)ppppwwppppppppppβδβδαβγδβδ++++=+−++31p=221212()(100)5ppwppwβαδβδγδδβδδβδδ++++=+−++100α=5β=−5γ=−0δ=111213212223313233///(,)(,)//////hhhhhhhphhhxpxpxppuDxpuxpxpxpxpxpxpσ⎡⎤∂∂∂∂∂∂⎢⎥==∂∂∂∂∂∂⎢⎥⎢⎥∂∂∂∂∂∂⎣⎦(,)0pupσ⋅=i.2(,)(,)hppDxpuDepu=(,)epuii.222/(,)/(,)//hhijijjjixpepuppepuppxp∂∂=∂∂∂=∂∂∂=∂∂iii.(,)0pupσ⋅=(,)hxpupk()fx1()()niiifxxkfxx=∂=∂∑10(,)0hnhkikiixpxpup=∂=⋅=∂∑1,2,3.k=21212(,)()uxxfxx=+1()fx1x(quasi-linear)211()fx1x5.pσ(p.u)=0.p=(p1,p2,..,pn),σ(p.u)211p=22p=36p=10???4?3??−⎡⎤⎢⎥−⎢⎥⎢⎥⎣⎦pσ(p.u)=01,2,...,kn=10hniiikxpp=∂=∂∑4/6hikxp∂∂σ(p.u)ikhhikkixxpp∂∂=∂∂pσ(p.u)=01,2,...,kn=10hnkiiixpp=∂=∂∑minhpx⋅s.t.()huxu≥min()htpx⋅s.t.()huxu≥(,)(,)hhxpuxtpu=1,2,...,kn=10(,)0hnhkikiixpxpup=∂=⋅=∂∑1043abcdef−⎡⎤⎢⎥−⎢⎥⎢⎥⎣⎦ad=ce=10f=−(,)puσ1043442327/6−−⎡⎤⎢⎥−−⎢⎥⎢⎥−⎣⎦n()fxt()()kftxtfx≡()fxkA.()fxk1k−1212(,,...,)(,,...,)knnftxtxtxtfxxx≡ix()()kiiftxfxttxx∂∂=∂∂0t1()()kiiftxfxtxx−∂∂=∂∂B.k()fx1()()niiifxxkfxx=∂=∂∑t()()gtftx=t1212()(,,...,)(,,...,)knngtftxtxtxtfxxx==t11()'()()nkiiiftxgtxktfxx−=∂==∂∑1t=1()()niiifxxkfxx=∂=∂∑C.x1()()niiifxxkfxx=∂=∂∑()fxkBCJehleReny5/66.mfmwfw1lnlnlniiiiUClnα=++,ifm=iCiliniαi(1)0iα=,ifm=1(2)(,,,,)ffmmClCln[lnlnln](1)[lnlnln]fffmmmClnClnθαθαΩ=+++−++θ0fmτττ==nnτa)1b)(,,,,)ffmmClClnc)(1){,}maxlnlniiiiiClUCl=+s.t.(1)iiiiiiiCwlCwlw=−⇔+=0.50.5iiiwlw⇒==0.5iiCw=ln2ln2iiUw=−,.ifm=(2)a)fffmmmffmCwlCwlwnwwτ++++=+(*)b){,,,,}maxffmmClClnΩs.t.fffmmmffmCwlCwlwnwwτ++++=+⇔(1)11{,,,,}maxfmffmmffmmClClnClClnθαθαθθθθ+−−−s.t.fffmmmffmCwlCwlwnwwτ++++=+-Cobb-Douglas(1)2(1)fmfmfmfwwnwθαθαθαθατ+−+=++−()2(1)fmffmwwCθθαθα+=++−()12(1)fmffmfwwlwθθαθα+=++−(1)()2(1)fmmfmwwCθθαθα−+=++−6/6(1)()12(1)fmmfmmwwlwθθαθα−+=++−c)()22()02(1)fmfmfmffmwwdndwααααθτθαθα−+=⇒++−1-Cobb-Douglas1212...nnuxxxααα=iii11ln...lnnnuxxαα=++iα21212max...nnxxxαααiii11maxln...lnnnxxαα⇔++34CD11ln...lnnnuxxαα=++y()1iinjijyxpαα==∑1,2,...,in∀=/51212{}max...nnxxxxαααiiis.t.1122...nnpxpxpxy+++=12121122...(...)nnnnLxxxpxpxpxyαααλ=−+++−iii1212...0nniiiixxxLpxxααααλ∂=−=∂iii1,2,...,in∀=iiijjjpxpxαα=,1,2,...,ijn∀=1122...nnpxpxpxy+++=1iiinjjpxyαα==∑1,2,...,in∀=()1iinjijyxpαα==∑1,2,...,in∀=