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24 1. [][Rubinstein P.10]X()IxzX∈zx∼zXxy()()IxIy=()()IxIyφ∩= (){|,}IxzzxzX=∈∼(){|,}IyzzyzX=∈∼ xy∼∼ (){|,}{|,}()IxzzxzXzzyzXIy=∈=∈=∼∼ xy∼()()IxIyφ∩= ()()IxIyφ∩≠()wIx∈()wIy∈ ()()IxIywx∼wy∼∼xy∼ 2. [][Rubinstein P.50] ,xε∀121211212(,)(,)(2,)xxxxxxεδεδδ−+−++∼∼21δδ≥ 121211212(,)(,)(2,)xxxxxxεδεδδ−+−++∼∼⇒ 1212(2,)xxεδδ−++12121(,)(,)xxxxεδ−+∼⇒ 1212120.5(2,)0.5(,)xxxxεδδ−+++121(,)xxεδ−+⇒ 1212(,0.50.5)xxεδδ−++121(,)xxεδ−+⇒ 21δδ≥ 3. [][] xy≥xy≠xy xyxy yX∀∈0ε∀xX∃∈yxε−xy ()/1,/1,...,/1xynnnεεε=++++ xy/(1)yxnnεε−=⋅+ xy 4. [][2008II]xyxyxy≥,xy xyxy≥xyxyxyy'xy'y'yyxy∼'yx'xyx'yxy∼xy 5. [][2008II]25 demand correspondencex'x'xx≠''(1)'xxxαα=+− 'x x'x'xx∼ ''(1)'xxxαα=+−(0,1)α∈ 'xx∼x'x'xx ''xx'x x'x 6. [][2008II] x'x'xxx''x''xx(0,1)α∀∈''(1)xxxαα+−xα0x'''(1)xxxαα=+−'xx iipxpw≤iipxpw ix'ix'iixx'iipxpw ixiipxpw= ()0iipxw−=∑∑ 7. [][2007I]x'x'xx∼1t(')xxtxx+− 11/tα=−1t(0,1)α∈x'x'xx∼ ()(1)(')(')xxtxxxtxxαα+−+−+− (')xtxx+−()(1)(')xxtxxαα+−+−11/tα=− (')xtxx+−'x'xx∼(')xtxx+−x(')xxtxx+− 8. [][2009II] x'x(0,1)α∀∈(1)'xxαα+−'x()xp ()xp'()xp ''()()(1)'()xpxpxpαα=+−''()xp ()'()xpxp∼()xp'()xp''()xp'()xp ()xp'()xp''()xp'()xp''()xp ''()'()xpxp∼''()xp()xp 9. [][2006I] (1,2,..,)Nn=n12(,,...,)nnxxxxAR+=∈⊂ {}()max{1,2,...,}...ijixinstxxji=∈∀≠ 26 ,ijN∈ij≠ijxx≠()ix ,xyA∈xyxy ()()()()ixiyixiyxxyy+≥+ (1,2,3)x=(1,7,6)y=()ix()iyxy ()3ix=()2iy= ()()()()3256713ixiyixiyxxyy+=+=+=+=yx (8,2,8)z=−()1iz= ()()()()314880ixizixiyxxzz+=+=+=−=xz ()()()()8210178iziyiziyzzyy+=+=+=+=zy xzzy⇒xyyx 10. [][2007I] {,,}Xabc= {,}ab{,}bc{,}ac{,,}abc 2*2*2*3=24 11. [][2006I] A2AR⊂{|AxA=∈''22,}xAxx∈A AA()CAx='xA∈'11xxAthe Weak Axiom of Revealed Preference X1A2A12AAX⊂⊂21()CAA∈12()()CACA= 1{(3,3),(2,2),(4,1)}A=2{(3,3),(2,2),(4,1),(0,0)}A= 2()(4,1)CA=12()(3,3)()CACA=≠ 1. [][2009] (,)max{,}Uxyxy= uz'z()(')uzuz≥(')()uzuz≥z'z'zzz'z''z()(')uzuz≥(')('')uzuz≥()('')uzuz≥z'z'z''zz''z'zz≥'zz≠'zz(,)(2,2)zxy=='(',')(2,1)zxy==z'z'zz≥'zz≠'zz∼()2(')UzUz==27 z'z(1)'zzzλλλ=+−'z(0,1)λ∀∈(,)(4,2)xy=(',')(2,4)xy=0.5(,)0.5(',')(3,3)xyxy+=(4,2)(2,4) 2. [][2007I]WA (1) (2) (1) WA12max{,,...,}nuxxx=SARP (2) WA(0,1,2)p=(0,1,0)x='(1,0,2)p='(1,0,0)x=1'0pxpx⋅=≥⋅=x'xWAx'x'xx''1'0pxpx⋅=≥⋅='xxWA 3. [][Rubinstein P.51]()ux()(),...,()xuxux∼()ux xX∀∈{}|()()yXuyux∈≥{}|()()yXuyux∈≤xX∀∈{|yXy∈}x{|yXx∈}y()(),...,()xuxux∼{}|()()yXuyux∈≥{|yXy∈}x{}|()()yXuyux∈≤{|yXx∈}y ()ux 4. [][] ,nnxy1,2,...,n=∞nxnynxx→nyy→ ()()nnuxuy≥()ux()()uxuy≥xy x'x()(')uxux≥ 5. [][] 1X+= (0,1)x∈()0ux= 1x=()1ux= 28 1x()2ux= XxxX∀∈{|yXy∈}x 1x={|yXy∈1}(1,)=+∞ 6.[][]X=Ru(x)[x][4.6]=421/nxn=+31/nyn=−()2()2nnuxuy=≥=nxx→nyy→xy2nx→3ny→yxxy 7. []()ux (1) ()()1/33()()()()1Ifxuxuxux=+++ (2) ()2()()2()IIfxuxux=+ (3) 121212(,)(,)IIIfxxuxxxx=++ x∀yX∈xy()()uxuy⇔≥ (1) 1/33()1Iftttt=+++t()()uxuy≥()()IIfxfy≥()Ifx()ux (2) ()2()()11IIfxux=+−1−()()uxuy≥()()IIIIfxfy≥Xx()1ux≥−()IIfx()ux (3) ()ux31212(,)(,)uxxxx=()IIIfx()ux1212(,)uxxxx=(2,2)x=(1,3.9)y=1212(,)(,)uxxuyy1212(,)(,)IIIIIIfxxfyy 8. [][2007I] pxy⋅≤()ux11niip=∑yx'y'x'yy'xx 1x2x‐1212(,)uxxxx= ‐1x2x1212,,...,nnuxxxααα= 1/nijjαα=∑i1,2,...,in=21212(,)(/2)(/2)/(4)vpyypypypp==2''12(',')'/(4)vpyypp= 29 11/3p=22/3p=''121/2pp== 29(,)8vpyy=2(',')'vpyy='3122yy(',')(,)vpyvpy'yyx'x 9. [][2009] mfmwfw1 lnlnlniiiiUClnα=++ ,ifm= iCiliniαi (1) 0iα=,ifm=1 (2) (,,,,)ffmmClCln [lnlnln](1)[lnlnln]fffmmmClnClnθαθαΩ=+++−++ θ 0fmτττ==nnτ a) 1 b) (,,,,)ffmmClCln c) (1) {,}maxlnlniiiiiClUCl=+ s.t. (1)iiiiiiiCwlCwlw=−⇔+= 0.50.5iiiwlw⇒==0.5iiCw=ln2ln2iiUw=−,.ifm= (2) a) fffmmmffmCwlCwlwnwwτ++++=+ (*) b) {,,,,}maxffmmClClnΩ s.t. fffmmmffmCwlCwlwnwwτ++++=+ ⇔ (1)11{,,,,}maxfmffmmffmmClClnClClnθαθαθθθθ+−−− s.t. fffmmmffmCwlCwlwnwwτ++++=+ ‐Cobb‐Douglas (1)2(1)fmfmfmfwwnwθαθαθαθατ+−+=++− ()2(1)fmffmwwCθθαθα+=++− 30 ()12(1)fmffmfwwlwθθαθα+=++− (1)()2(1)fmmfmwwCθθαθα−+=++− (1)()12(1)fmmfmmwwlwθθαθα−+=++− c) ()22()02(1)fmfmfmffmwwdndwααααθτθαθα−+=⇒++− 11(,)wxpwpα=22(,)wxpwpα=(,)pw1122(,)(,)2pxpwpxpwwα+=0.5α≠22(,)(1)wxpwpα=−(,)pw213(,)pxpwp=123(,)pxpwp=−(,)xpw(,)xpw23(,)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′′=−(,)2pxpww′′==i(,)1pxpww′′==i31 (,)xpwpw1(,)(,)0niijjjxpwxpwpwpw=∂∂+=∂∂∑(,)(,)0pwDxpwpDxpww+=(,)xpy(,)(,)xtptwxpw=tt1(,)(,)0niijjjxtptwxtptwpwpw=∂∂+=∂∂∑1t=1(,)(,)0niijjjxpwxpwpwpw=∂∂+=∂∂∑(,)(,)0pwDxpwpDxpww+=(,)xpw(,)ppDxpwpw=−(,)xpw(,)xpwpw=w(,)1wpDxpw=(,)xpw(,)(,)xtptwxpw=tt(,)(,)0pwDxtptwpDxtptww+=1t=(,)(,)0pwDxpwpDxpww+=p(,)(,)0pwpDxpwppDxpww+=(,)1wpDxpw=(,)ppDxpwpw=−∈|≥||∈|≥||∈|≥||∩∈∈∩∈∈∈∈∈∈∈∈∈(,)xpw(,)pwwlk≠(,)/0lkxpwp∂∂=(,)/(,)(/)lklkklxpwxpwppη=lkηpw(,)xpww(,)(,)xpwxpwαα=0α(,)(,1)llxpwwxp=lk≠(,)