消费的负外部性理论及其在幸福经济学上的应用.pdf

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å`z¯Kff)"é?Ûi∈{1,2},j∈{1,2},i6=j,Maxui(x11,x21,xi2)x∈R4+++x21≤w1x12+x22≤w2()uj(x11,x21,xj2)≥ujx∗11,x∗21,x∗j2y²µ„Varian(1992,P330).Ïd§ÏéøX÷`ff˜(JffóŁÒŒ{z¦)±þ¯K"é/34–Â\0ƒff˜«)º3ž¤ffK Ü5ž§Ÿo7L¤Ü©] âU¢yøX÷k ffff˜(Jº )ºù‡¯K§·‚5w±eff~f" ü« ¬©O b½Â\¬ÚšÂ\¬" ¬1L«U^5 ï¢Ô ¬ffÂ\§2L«šÂ\5 ¬§) %nÆ[^5)ºI[ƒmÌ*34Y² Éff¤kšÂ\σ§'Xèx§° G §´Ó§[Ì)¹§Ò’G¹ " XEasterlin(1995,)¤Ñff@ §+Nënؽ K gnØ@§‡N4|Y²Ø Égfiž¤ffK …É Ù¦<Â\ff‡•K"¤k<ffÂ\O\¿Ø˜½¦z‡<ff4|O\"ϏpÂ\é4|Y²ff A¬ )¹IOJp‘5ffK A-"Ò´`§+Nënؽ K gnØ@‡NffÌ*4|µd´ÙgfiÂ\ffO¼ê§´Ù¦<Â\²þŁff~¼ê"ìù«b §·‚ ½ž¤öff ^¼êµ3:ui(x11,x21,xi2)=√xi1xi2−xj1,i∈{1,2},j∈{1,2},j6=idÚn1§øX÷k ff˜ fiŒ±ÏL¦)±e5y¯K Ñ"2 BA^¥ -©Ž½n§·‚b½ž¤8m83 ,3ù‡~f¥§ ^¼êflfi¿Ø‡¦ž¤8m8§ ´§XJ·‚±1˜–ž¤8§(J¿Ø¬ÉéŒK"„5º14 √Maxx11x12−x21x∈R+4++x21≤w(P1)1x12+x22≤w√2√x21x22−x11≥x∗21x∗22−x∗11†M√axx21x22−x11x∈R+4+(P2)11+x21≤w1x12+x22≤w2√√x11x12−x21≥x∗11x∗12−x∗21du8I¼êÚ å^‡Ñ´xˆ∈R4++þffëYŒ‡ff]¼ê§¿…Slater^‡4÷v§Ïd§øX÷k ff˜ fiŒ±d±þü‡`z¯K(P1)Ú(P2)ff˜ffi^‡‰Ñ"AOff§XJ√(P2)¥√x11x12−x21≥x∗11x∗12−x∗21ff¥ –©Ž¦f £ù˜:ò3e© y¤§K·‚IÄ(P2)ff˜ffi^‡" E.‚KF¼êXeµL=√(√)x21x22−x11+λ1(w1−x11−x21)+λ2(w2−x∗12−x22)+√µx11x12−x21−x∗11x∗12+x21.˜ffi^‡µ√xx1211:−1−λ1+µ=0,(1)√2x11x12:−λ2+µx11=0,(2)√2x121xx2221:−λ1−µ=0,(3)2√x211xx2122:−λ2x2=0,(4)22λ1:w1−x11−x21≥0,λ1≥0,λ1(w1−x11−x21)=0,(5)λ2:w2−x12−x22≥0,λ2≥0,λ2(w2−x12−x22)=0,(6)√√(√)µ∗:x11x12−x21−x∗11x∗12+x∗21≥0,µ≥√0,µx11x12−x21−x∗11x∗12+x21=0.(7)√d(2)Ú(4))ѵ=x12x21x>0"Ïd¤k11xÎÜ(£1¤-£7¤),ffff˜x∗∈R4++¬Óž22ÎܯK(P1)ff˜ffi^‡§´.‚KF¦fØÓ"du3`)?§(7)7,÷v§¤±øX÷`ffff˜8Œ± fid£1¤-£6¤‰Ñ"d(4),Œ λ2>0Ïd§Łâ(6),·‚k±eff; å^‡µx12+x22=w2.(8)4Slater^‡Ñ3˜:xˆ∈R+4+¦ 3T:?§¤k å^‡Ñ î‚Ø Ò"5 d£1¤-£4¤,ž xÚλ2,Œ ,[](1−µ)λ1(1+µ)+1+µ+µ2=0ùL²µ=1.Ϗµ=1…λ2>0,l(1)Ú(2)Œ µλ1=1−1.(9)4λ2d(2),(4),µ=1,λ2>0,kµx12=14λ2x11(10)2x22=1x21(11)4λ22ò£10¤†£11¤ƒ\§2Łâ£8¤§Œ µ√xλ11+x212=12w,(12)2ùL²,x12=x11w2(13)x11+x21x22=x21w2x11+(14)x21ò(12)‘\(9),Œ √λ1=√w21(15)2x11+−x21ùŒ± ^5(½] ¤ff .:"Ϗ3þï:?λ1≥0§¤±‡ÄXeü«œ¹"œ/1µλ1>0dž§d£15¤7,kx11+x21<w2/4§2Łâ(5),Œ µx11+x21=w1.(16)¤±§XJЩ Dw1<w2/4§KØI‡¤] §¿…d(13)!(14)x12=x11w2w(17)1x22=x21w2w(18)1œ/2µλ1=0K‡w1≥w2/4§d(15)Œ x11+x21=w2/4"6 Ïd§ Щ D÷vw1>w2/4ž§ÒI‡éÙ¢–¤§¤þw1−w2/4" w1=w2/4§ž§=¦džλ1=0§ØL¤"Ïd¤] ff .:w2/4" §l(13)!(14)±9x11+x21=w2/4§Œ µx12=4x11(19)x22=4x21,(20)…λ2=1/þ¤ã§·‚k±e·Kµ·K13äkfl!¤ ½ ^¼êffX †²L¥§ ˆ øX÷`ff˜7L?1] ¤ff¿©7‡^‡´µw1>w2/4"äNff§(1) Щ Dw1>w24ž§I¤] 5ˆ øX÷`§Ù¤þw1−w2/4§øX÷`ff˜ff/ªµ{}PO=x∈R4++:x12=4x11,x22=4x21,x12+x22=w2,w2/4=x11+x21<w1(2) w1≤w24ž§ØI¤] §øX÷`ff˜ff/ªµ{}PO=x∈R421w2++:x12=x11w2w,x22=x1w,x12+x22=w2,x11+x21=µ51XJ#N`)3>.:? §=x∈R4+§K·K1¥ffü:ãÒ¬C¤µ(1) Щ] Dw1>w2/4ž§ØL¤] 5ˆ k ff˜§Øš˜‡ž¤öž¤¤kffü« ¬§ ,˜‡ž¤öffž¤þ""?˜Úff§øX÷`ff˜ff/ªµ{}PO=x∈R4++:x12=4x11,x22=4x21,x12+x22=w2,w2/4=x11+x21<w1∪{x11=x12=0,x22=w2,w2/4≤x21≤w1}∪{x21=x22=0,x12=w2,w2/4≤x11≤w1}(2) w1≤w2/4ž§ØI‡¤] =Œ¢y`ff˜§øX÷k ff˜ff/ªµ{}PO=x∈R4+:x12=x11w2w,x22=x21w2w,x12+x22=w2,x11+x21=w111'uù«œ/ffy²' E,§k, ffÖöŒ±ëN¹"7 µ52þã~f5‰ffy² 34–Â\ƒ(Easterlin,1995)§)º Ÿo¤k<ffÂ\ӞO\ <‚ff34Y²%vkO\"§•·‚Ы §3·‚¤Äff²L¥§O\Â\ؘ½¬¦z‡<ff4| J,"äNff§ ˜I' «(ž§²LO´˜‡Ð¯§Ï§‘5ffãLO\¬¦ ¬‡Nff34aOr¶, §‘X²LffO§ ãLþ‡L,‡ .Ł§= w1>w2/4ž§²LO%™7´˜‡Ð¯§ÏXJdžò¤k] Ñ^už¤§@oé ‡ ¬5`34Y²‡ ¬k¤ü$§¤±¬ —øX÷à ff(J"Ïd§ù‡~féÐff)º 3L 25c¥§Ÿo{Iff4|Y²k¤eü§ =Iجff÷v§Ý XBlanchflowerÚOswald(2004)¤* ff@ § ±ØC"3TianÚYang(2005)¥§·‚é+NënØÚ34²LÆ?1 ¦ffn؆² ©Û"µ53l±þff~fŒ„§3˜‡3ž¤K Ü5ff²L¥§´ÄI‡¤K Ü5 ¬±ˆ ø\÷k ff˜(J§ù ûu²LgfiffA §=ž¤öff ÐÚ] ffЩ D"ŒNþù§‰½ ÐÚ,˜« ¬ff Dw2§ ¬1ffЩ Dþˆ ,˜Y²ž§ÒI‡¤"3·‚ffêŁ~f¥§é ¬1ffž¤w,¬ ) Kü« A§ AÏL4~ff>S ^¢y§ K AK‚5/ª"‘XЩ Dw1ffO\§ AØä~ § K A ±ØC"Ïd§ w1 Œž§o AK"ù)º Ÿow13 .Ł£fl~¥w2/4¤"] ¤ffA:y3·‚5Ääk˜„/ª ^¼êffX †²L§¿‰Ñù«œ/e] ¤ffA:"·‚ ^¼êff/ªµui(x11,x21,xi2)T¼êëYŒ‡§î‚[]…üN4O"=µ∂ui(x11,x21,xi2)/∂xil>0él=1,2"·‚?˜Úb Slater^‡÷v§3øX÷`:?ui(·)ffFÝ؏""Ïd§øX÷k ff˜x∗ fiŒ±d±e¯Kff˜ffi^‡û½µMaxu2(x11,x21,x22)x∈R4+++x21≤w1x12+x22≤w2u1(x11,x21,x12)≥u1(x∗11,x∗21,x∗12)8 ˜ffi^‡µ∂ux211:−λ1+∂uµ1=0(21)∂x11∂x11x12:−λ2+∂uµ1=0(22)∂x12∂ux2121:−λ1+∂uµ=0(23)∂x21∂x21∂ux222:−λ=0(24)∂x222λ1:w1−x11−x21≥0,λ1≥0,λ1(w1−x11−x21)=0(25)λ2:w2−x12−x22≥0,λ2≥0,λ2(w2−x12−x22)=0(26)µ:u1−u∗1≥0,µ≥0,µ(u1−u∗1)=0(27)d(24)§λ2=∂u2∂x>0§Ïdd(26),22x12+x22=w2(28)ùL²§éuØ ) Ü5ff ¬´ØI‡\±¤ff", §d(22)9(24)Œ µ∂u2µ=∂x22∂u.(29)1∂x12¤±§l(21)Ú(22)Œµ[]λ∂u1∂u21=∂x11+∂x11λ∂u21∂u(30)2∂x12∂x22d(23)!(24)Œ µ[]λ∂u2∂u11=∂x21∂x21λ∂u+2∂u(31)21∂x22∂x12Ïd§l(30)Ú(31)§·‚Œ± цIO ‰Ö£XVarian(1992,)¤¥ƒÓff>Sƒ ^‡"∂u1∂u2∂u2∂u1∂x11∂u+∂x111∂u=∂x212∂+∂x21u2∂u,(32)1∂x12∂x22∂x22∂x12þªL²3øX÷`:?§ü« ¬ff ¬>SO“ÇATƒ "l±þ^‡Œ„§ 3K Ü5ž§‡OŽ`^‡¥ff>SO“Ç7LĆ Úm ü« A"= ¢yøX÷`§ ˜‡ž¤öé ¬1ffž¤þO\ž§Ø I‡UCTž¤öé ¬2ffž¤§,˜∂u∂uij‡ž¤öé2ffž¤þAUC"Ïd鞤öi§ ¬1é2ff ¬>SO“Ç u∂xi1∂u+∂xi1i∂u"j∂xi2∂xjiÏL(21)Ú(23)¦)µÚλ1§ µ∂u2µ=∂x−∂u221∂x11∂u1u>0(33)1∂x−∂11∂x219 ∂u1∂u2λ∂x11∂x−∂u1∂u221∂x21∂x111=∂u(34)1∂∂x−11∂x21 ž¤äk Ü5ž§duλ2=∂u2∂x>0§d(30)½(31)§Œ± Ñλ1ð "Ó §22XJØ3 Ü5½3ü• Ü55ž§d(30)½(31)§½Œ λ1 "Ïd§ù œ¹efføX÷k ff˜Œ± fid>S^‡(32)Ú²ï^‡¤û½" ´§ ¤kž¤öÑ3K Ü5ž§d£30¤½£31¤m ‰Ñ½öd£34¤† ‰ÑffKuhn-Tucker¦fλ1´ Kü‘ƒÚ§ÙÎÒØU(½" ´§¿Ø–k ‰ÖXVarian(1992,438)¤ãff@ §÷v>S^‡Ú²ï^‡ÒU( øX÷k ff˜ff¢y"33K Ü5ffœ¹e§ yøX÷k ff˜ff¢y§7L‡¦3k :?λ1≥0§Ò´‡¦ ¬>SO“ÇšK"=µ∂u1∂u2∂u2∂u1∂x11111∂u+∂x1∂u=∂x22∂u+∂x212∂u≥0,(35)1∂x12∂x22∂x22∂x12½ö‡¦¤køX÷k :÷v(32)Ú∂u1∂u2≥∂u1∂u2(36)∂x11∂x21∂x21∂x11o>SÂÃ(o>S¤fl)(36)ªff†>§∂u1∂u2∂x11∂x§Œ± )º ¬1ff>SoÂç m>§∂u1∂u221∂x21∂x§Œn)11d ¬1ffž¤ Ü5‰ž¤ö‘5ff>So¤fl"Ïd§l±e^‡¥Œ± ÑøX÷k ff˜(J(PO)"∂u1∂u2∂u2∂u1∂x∂x1111∂x21x21∂u+1∂u=2∂u+∂2∂u≥01∂x12∂x22∂x22∂x12x(PO)12+x22=w2x11+x21≤w1∂u(1∂u2u1∂u2∂x11∂x−∂21∂x21∂x)(w111−x11−x21)=0∂u∂u2∂u2∂u5¿ §XJ1∂u1∂u2u1∂u2∂x21∂x11∂x>∂21∂x21∂x§½ö dff1∂x11+∂x1111∂u1∂u=∂x212∂u+2∂u>0§λ11>0K±∂x12∂x22∂x22∂xþff ü‡^‡Œ{zµ12x11+x21=w1"dž§ØI‡¤ ¬1"òx11+x21=w1Úx12+x22=w2“\>S^‡(32)§Œ x11Úx12ƒmff'X§ùU O(½øX÷`ff˜" >SoÂà u>So¤flžµ∂u1∂u2=∂u1∂u2,(37)∂x11∂x21∂x21∂x115k˜‡ž¤öffž¤¬é,˜‡ž¤ö ) Ü5"10 K∂u1∂u2∂u2∂u1∂x11∂x11∂x21∂x21∂u+1∂u=2∂u+2∂u=0(38)1∂x12∂x22∂x22∂x12Ïd§λ1=0"dž§ x11+x21≤w1ž§ØU(½´ÄAT¤] ", §=¦k7‡¤§·‚E,Œ±dx12+x22=w2Ú ¬>S¤fl"ff^‡£38¤(½øX÷k ff˜"¢Sþ§òx12+x22=w2“\£38¤=Œ^x125L«x11"XJé?Û÷v^‡x11+x21=w1,x12+x22=w2Ú>S^‡£32¤ffff˜§Ñk∂u1∂u2∂u1∂x¤á§K ¬>SO“Ç7,K"Ïd§Tff˜Ø´øX÷`ff"11∂x<∂u221∂x21∂x11dž§ ¢y`7L¤˜½þff ¬1§øX÷ff˜AT÷v£38¤ª"nþ¤ã§·‚ÏL±e·K‰ÑØӜ/eüa] ¤ff¿©^‡"·K2éu˜‡2×2ffX †²L§ Ù ^¼êui(x11,x21,xi2)´ëYŒ‡§î‚[]ff§…∂ui(x11,x21,xi2)∂x>0§l=1,2"il(1)XJ3øX÷k ff˜x∗?§>S ¬O“Ǐ 6§KØI‡¤w1=Œ¢yøX÷k ff˜"(2)XJé?Û÷v^‡x11+x21=w1,x12+x22=w2Ú^‡£32¤ffff˜(x1,x2)Ñk>S ¬O“ǏK§KI‡¤w15¢yøX÷k ff˜x∗"=x∗11+x∗21<w1¿…x∗dx12+x22=w2Ú(38)û½"Ïd§l±þ·KŒ ] ¤ff¿©^‡µé?Ûff˜(x11,x12,x21,x22)§∂u1∂u2∂u2∂u1∂x11∂x21∂u+∂x111∂u=2∂u+∂x212∂u1∂x12∂x22∂x22∂x12x11+x21=w1⇒∂u1∂u2<∂u1∂u2.(39)∂x11∂x21∂x21∂x11x12+x22=w2é?Ûff˜(x11,x12,x21,x22)ØI‡¤] ff¿©^‡´µ∂u1∂u2∂u2∂u1∂x11∂u+∂x111∂u=∂x212∂u+∂x212∂u1∂x12∂x22∂x22∂x12∂u2x(40)11+x21≤w1⇒∂u1∂u2>∂u1∂x11∂x21∂x21∂x11x12+x22=w2µ54·‚Œ±ÏLþã¿©^‡5­# !¥ffêi~fµd>S^‡(32),Œ µ(√)x2(√)212+1=x22+1(41)x11x216 Xþ©¤?Øff§ ž¤ Ü5 §Ã Ü5§½ökü• Ü5žœ¹(¢Xd"11 Ïdx12=x22.(42)x11x21-x11+x21≡x1.òx11+x21=x1Úx12+x22=w2“\(42)§Œ µx12=w2.(43)x11x1, §d(42)Ú(43)§Œ µ√√∂u1∂u2=1x12x22=x12=w2(44)∂x11∂x214x11x214x114x19∂u1∂u2=1.(45)∂x21∂x11∂uÏd§1∂u2∂u2∂u1x¯1=w∂u12/4´ y∂u1∂u2∂x11∂x−∂u2=0,½ dff∂x11∂u+∂x111∂u=∂x21∂x212∂u+2∂u=0ff 21∂x21∂x111∂x12∂x22∂x22∂x12.:"Ïd§XJw1>w24,K∂u1∂u2∂u2∂x11∂x−∂u121∂x39)§I‡¤] 5ˆ `21∂x<0§@od(11ff˜"XJw1<w24§K∂u1∂u2u1∂u2∂x11∂x−∂21∂x21∂x>0,d(40)Œ§ØI¤] Ҍ±¢y`"11 w1=w24ž§?Û÷v>S^‡(32)Ú²ï^‡x11+x21=w1x12+x22=w2ffff˜Óž÷v(36),Ïd§ù‡ff˜flfiÒ´øX÷k ff§ØL¤] "y3·‚5[?Ø'u] ¤ff¿©^‡"5¿ §XJ؝¤^‡ ÷v§K ªx11+x21=w1¤á"Ïd·‚Œ±l±en‡^‡ ѝ¤^‡Ú؝¤^‡µ∂u∂u12∂u2∂u1∂x11∂x11∂u+1∂u=∂x212∂u+∂x212∂u1∂x12∂x22∂x22∂x12x11+x21=x12+x22=w2XJ÷v±þn‡^‡ffff˜Óž÷v∂u1∂u2u1∂u2∂x11∂x≥∂21∂x˜´øX÷k ff§¤±21∂x§KTff11ØI‡¤] "ÄK§Tff˜ÒØ´øX÷k ff"XJ?Ûd±þn‡^‡¤û½ffff˜Ñ´šøX÷k ff§K¤køX÷k ff˜Ñ‡ÏL] ¤5¢y"5¿ §du∂u1∂x11Ú∂u2∂x“L>SÂçÏ~§‚´‘X ¬1ffž¤þO\ 4~ff"du∂u1∂xÚ∂u2´±>2121∂x11S¤fl/ªÑyff§¦‚ffýéŁAT‘X ¬1ž¤þffO\ 4O"Ïd§ Щ Dw1 ž§>S ¬ÂÃpu>S ¬¤fl§ÏdØI‡¤] =Œˆ `ff˜"‘X DþffO\§ öª¬‡Lcö§dž§AT¤Ü©] "l,˜« ݧÏL©Û ¬>SO“ǧ·‚Œ уÓff(Ø" ^¼êî‚[]ž§>SO“Ç4~"Ïd§3ž¤ffK Ü5^‡e§ ¬1ffž¤þv Œž§>S ¬O“Ǭ KŁ"ùžß§AT¤˜Ü© ¬1"‘X¤þffO\§ ¬1ffž¤þ~ § ¬>SO“ǬO\§dK= "12 33) Ü5žff] ¤3flÜ©¥§·‚5Ä3Ӟ3ž¤Ú) ffV­ Ü5ff) 5²L¥X۝¤] 5¢yøX÷`"Äk§·‚Äù ff²L‚¸§Ù¥kü« ¬!ü‡ž¤ö!ü‡‚û¶, §·‚ÄPkn« ¬£˜«Ý\¬ü«ž¤¬¤!ü‡ž¤ö!ü‡‚ûff²L"×2×2) 5²L¥ff] ¤3·‚¤Äff²L¥§kü« ¬!ü‡ž¤ö!ü‡‚û"z‡‚û) ˜« ¬§ ±,˜« ¬Ý\¬"·‚ ‚ûj) ¬j"3ù‡²L¥kA«/ªff Ü5"ÓcAÜ©?Øff˜ §˜‡<é ¬1ffž¤¬K,˜‡<ff ^Y²¶ü« ¬ff) L§ƒpK¶ ¬1ffž¤¬K 2ff þ¶ 2ff þ¬Kü‡ž¤öff ^"ù¡ff˜‡²;ff~f´U )˜íÀ/ffü«) L§"Ø UY^xL«ž¤þ§·‚„Ú\yL«ü‡‚ûff þ§v“L‚ûffÝ\§w=(w1,w2)L« D•þ" ¦^ü‡eIž§1˜‡L«‡N£ž¤ö½‚û¤§1 ‡L« ¬"'X§xijL«ž¤öié ¬jffž¤þ"vjlL«‚ûj3) ¬jffL§¥¤Ý\ff ¬lþ" ^¼êui(x11,x21,xi2,y2)½Â3R5++þffëYŒ‡§î‚[]ff¼ê§¿…‘XÙgfiž¤þffO\ 4O§=§∂ui(x11,x21,xi2,y2)∂x>0él=1,2"du̇ïÄff´ilž¤3K Ü5žff] ¤¯K§·‚„‡b½∂ui(x11,x21,xi2,y2)∂x≤0éj6=i") j1¼êy1=y1(v12,v21)Úy2=y2(v12,v21,x1)½Â3R2++ÚR3++þëYŒ‡ff]¼ê§Ù¥x1=x11+x21"·‚@ž¤ff Ü5´K] ¤ff'…󧤱§fl©‡¦ž¤äkK Ü5§ ) ff Ü5±9) ڞ¤ff p Ü5Œ ŒK"ü« ¬ff Dw1≥0§w2≥0" k¢S¿Â§·‚b½– ˜« D §ù ûu ÐÚEâ"3d²L¥§˜‡ff˜(x,y,v)≡(x11,x12,x21,x22,y1,y2,v12,v21)XJ÷v^‡(x,y,v)∈R8++,y1=y1(v12,v21),y2=y2(v12,v21,x1),x11+x21+v21≤y1+w1,…x12+x22+v12≤y2+w2K¡ÙŒ1"XJ˜ff˜(x,y,v)÷vx11+x21+v21=y1+w1,¿…x12+x22+v12=y2+w2K¡Ù²ïff˜"¡˜|Œ1ff˜(x,y,v)øX÷`ff§XJØ3?ÛÙ¦ffŒ1ff˜(x′,y′,v′)¦ éz‡i=1,2ui(x′11,x′21,x′i2,y′2)≥ui(x11,x21,xi2,y2)§¿…é,‡i§ui(x′11,x′21,x′i2,y′2)>ui(x11,x21,xi2,y2)"3±þ'u ^¼êÚ) ¼êffb½e§ …= ˜‡ff˜(x∗11,x∗12,x∗21,x∗22,v∗12,v∗21)7±e¯Kff)ž§§´øX÷`ff"Maxu2(x11,x21,x22,y2)(x,v)∈R+6+7Ϗ` þŒ±† ÏL) ¼ê¦Ñ§ OŽ{B§·‚3ff˜¥ŽÑ ÑCþ13 +x21+v21≤y1+w1x12+x22+v12≤y2+w2u1(x11,x21,x12,y2)≥u1(x∗11,x∗21,x∗12,y∗2).‚KF¼êµL=u2(x11,x21,x22,y2)+λ1(y1+w1−x11−x21−v21)+λ2(y2+w2−x12−x22−v12)+µ(u1(x11,x21,x12,y2)−u1(x∗11,x∗21,x∗12,y∗2))ӞÄy1=y1(v12,v21),y2=y2(v12,v21,x11+x21)§·‚Œ± ѱeff˜ffi^‡µ()∂uyx2211:+∂u2∂y2−∂y∂uλ211+λ2+µ+∂u1∂=0(46)∂x11∂y2∂x1∂x1∂x11∂y2∂x1ux112:−λ2+∂µ=0(47)∂x12()∂ux221:+∂u2∂y2−λ1+∂yλ22+∂uµ1+∂u1∂y2=0(48)∂x21∂y2∂x1∂x1∂x21∂y2∂x1∂ux222:−λ2=0(49)∂x22()∂u2∂y2v12:+∂yλ11+∂yλ22−1+∂uµ1∂y2=0(50)∂y2∂v12∂(v12)∂v12∂y2∂v12∂u2∂y2∂y∂yλ22v22+∂u1:+∂yλ11−1+µ1=0(51)∂y2∂v21∂v21∂v21∂y2∂v21λ1:λ1≥0,x11+x21+v21≤y1+w1,λ1(y1+w1−x11−x21−v21)=0(52)λ2:λ2≥0,x12+x22+v12≤y2+w2,λ2(y2+w2−x12−x22−v12)=0(53)µ:µ≥0,u1≥u∗1,µ(u1−u∗1)=0(54)∂ud2∂u2∂x(47)Ú(49),·‚EŒ Ñλ222=∂x>0±9µ=22∂∂x12(46)Ú(47)kµλ∂u1∂u2∂u2∂y2∂u1∂y21=∂x11λ∂u+∂x11∂y2∂x1∂u++∂y2+∂y2∂x1∂x∂u≡SMRS1(55)212∂u21∂x12∂x22∂x122∂x12d(48)Ú(49)kµλ∂u2∂u1∂u2∂y2∂u1∂y21=∂x21x21∂u+∂2∂u+∂y2∂x11∂u+∂y2+∂y2∂x12∂x∂u≡SMRS2.(56)λ21∂x22∂x12∂x122∂x12§(55)Ú(56)L«^ˆ« Ü A? ffü‡ž¤öff ¬>SO“Ç"14 , §l(50)Ú(51),Œ µ∂yλ211−∂u2∂y2∂u1∂y2=∂v12−∂y2∂v12∂y2∂v12λ∂y∂u21∂u2∂−y11∂≡SMRT,(57)y11∂v12∂x22∂v12∂x12∂v12∂yλ2∂u2∂y2∂u1∂y21=∂v21+∂(y2∂v21)+∂(y2∂v21)≡SMRT2(58)λ21−∂y1∂u2∂y1∂v21∂x1−∂u1122∂v21∂x1−∂y12∂v21ù©OL«ü‡‚ûff ¬>S=†Ç"Ïd§ò(55),(56),(57),Ú(58)Ü¿§·‚ ±eff>S^‡µλ1=SMRSλ1=SMRS2=SMRT1=SMRT2.(59)2ùL²3øX÷`ff˜¢yž§¤kž¤öڂûff ¬>SO“ÇÚ ¬>S=†ÇÑAT ±ƒ "8, § ¦ù ^‡( ff˜ff`5§ ¬>SO“ÇÚ ¬>S=†Ç„AӞšK"=µSMRS1=SMRS2=SMRT1=SMRT2≥0.(60)?˜Úff§XJ±þŁî‚ §½ö dffλ1>0§KØIé ¬1\±¤"nþ¤ã§·‚k±e·K:·K3阇2×2×2) 5²L§ ^¼êui(x11,x21,xi2,y2)ëYŒ‡§î‚[]§…∂ui(x11,x21,xi2,y2)∂x>0él=1,2,¶∂ui(x11,x21,xi2,y2)∂x≤0éj6=i"¿ ) ¼êëYŒ‡ffilj1î‚]¼ê§Kk±eff(ص(1)XJ3øX÷k ff˜(x∗,y∗,v∗)?§ ¬>SO“dž ¬>S=†Çя §KØI¤] ÒUˆ øX÷k ff˜(J"(2)éu?Û÷vx11+x21+v21=w1+y1,x12+x22+v12=w2+y2±9>S^‡ffff˜(x1,x2,y1,y2,v1,v2)§XJÑk ¬>SO“ÇÚ ¬>S=†ÇK§K‡ÏL¤] 5ˆ øX÷`ffff˜" ˜ ²wff(J§·‚ÄØ3) †ž¤ff Ü Ažffœ/§= ê∂y2∂ui∂x=0¶1∂y=0éi=1,‡˜ffi^‡†1 !¥ƒÓ§Ïd§d(21)-(24)§Œ µ2λ∂u1∂u2∂u2∂u11=∂x11∂x11u+.(61)λ∂21∂u=∂x212∂u+∂x212∂u1∂x12∂x22∂x22∂x12^‡(50)Ú(51)Œ{zµ()∂y∂yλ121+λ2−1=0(62)∂(v12)∂v12∂yyλ121−1+∂λ2=0(63)∂v21∂v218·‚b½z‡‚ûff>S=†ÇÑk½Â§†é{ù§b½3`:?(1−∂y1∂y1∂v)21∂v6=0"1215 ddµ∂yλ211−∂y2=∂v12λ∂=∂v21.(64)y21∂v1−∂y112∂v21Ïdd(61)Ú(64)§·‚ µ3±þff{zb e§3 Ü5ff) 5²L¥§>Sƒ ^‡µλ∂u1∂u2∂u2∂u11=∂x11λ∂u+∂x11∂u=∂x1−∂y2∂y221∂v12∂u+∂x212∂u=1∂=∂v21,(65)y2121∂x12∂x22∂x22∂x12∂v1−∂y112∂v21 ∂y1∂v<1,∂y221∂v<1,∂y212∂v>0§…∂y1ž§ ¬>S=†Ç §Ïλ21∂v>0§122=∂u2∂x>022¤±λ1>0§Ïd>S^‡(65)Úü‡] ²ï^‡ÒŒ± fi.½øX÷k ff˜"Ïd§XJ) äkK Ü5…‡ƒÝ\ff>S я §ÒØ3] ff¤¯K" ªff§·‚k±eíØ"íØ1é2×2×2ff) 5²L§ ^¼êui(x11,x21,xi2)ëYŒ‡§î‚[]§…´ž¤ögfiž¤þffO¼ê§) ¼êy1=y1(v12,v21)Úy2=y2(v12,v21)´ëYŒ‡ffî‚]¼ê§ ) ff Ü5K§‡ƒÝ\ff>S я §KØI¤] =Œˆ øX÷k ff˜") ffK Ü5)û ž¤K Ü5¤E¤ff] ¤¯K§ ´fl(ؿط^u) Ü5ffœ/"~1 ^¼ê†1 !¥‰ffêi~fƒÓ"ui(x11,x21,xi2)=√xi1xi2−xj1,j6=i,ü« ¬ff Dµ(w1,w2)=(1,10)"d1 !¥ff·K1§·‚ § Ã) ž§duw1=1<w2/4=ØI¤] ", § ) äk Ü5ž§7L¤˜Ü©] § L²ù˜:§ ) ¼êµy1=√2v12+2v21√y2=2v21−v21+√2v12,∂ydué?Û20<v∂12<190<v21<1kv√21√−∂y2∂v121−∂y=1−11v<09121∂y=v112−1<0§¤± Ý\ u1ž§ ¬>S=†ÇK∂v2"1Ïd§ ¢yùZ∂v1÷2`ff˜7L‡¦v12≥1…v21≥1" ¬2ffoøAþw2+√y2−v12=10+√2v21−v21+2v12−v123v21=1§v12=1ž§øAþˆ Œ§ÙŁ12§3v21=1v12=1:?§] ¤ .Ł(w2+y2−v12)/4ˆ Œ§TŁ3", §Ï3øX÷`ff˜?§ ¬2ff] å;§v12>19v21>1،U16 k Ý\"·‚y3y²(v12,v21)=(1,1)´˜ —øX÷k ff˜ffÝ\|Ü"¢Sþ§ (v12,v21)=(1,1)ž§y1=4§y2=3§ü‡Œ15^‡C¤x11+x21≤4Úx12+x22≤12"du ¬1ff¤ .:12/4=3<4§Ïdd·K1Œ§I‡¤˜ü ff ¬15¢yøX÷k ff˜§¤±øX÷k ff˜8µPO={(x,v,y)∈R8++:x12=4x11,x22=4x21,x12+x22=12,x11+x21=3,v21=1,v12=1,y1=4,y2=3.}.×2×2) 5²L¥ff] ¤líØ1¥·‚w §‡) ff Ü5K§KÚ\) ò)û] ¤¯K" ´§ù¿Ø´ÊH·^ff§e¡·‚òÑ§ ) ڞ¤ Ü5яKž§E‡ÏL¤] 5ˆ øX÷k (J"·‚Äù ffœ/§²L¥kn« ¬£Ù¥ü«ž¤¬˜«Ý\¬¤§ü‡ž¤ö§ü‡‚û"z‡ž¤öž¤ü« ¬§ ¬jd‚ûj) §ž¤¬ffЩ D""‚ûŒ±|^ á ½g,] £±rL«¤) ž¤¬§g,] ffЩ Dwr>0§Ù§^‡†c¡A! fiƒÓ§=˜‡ž¤öé ¬1ffž¤=¬é,˜‡ž¤ö ^Úü«ž¤¬ff þ ) ÜK"·‚ ½ž¤öff ÐڂûffEâXeµui=ui(x11,x21,xi2)yj=yj(r1,r2)Ù¥∂ui(x11,x21,xi2)∂x>0(l=1,2)§∂ui(x11,x21,xi2)∂yj∂yjil∂x≤0,j1∂r>0,…j∂r≤0éi6=j,j6=²L¥§XJ(x,y,r)∈R8++,y1=y1(r1,r2),y2=y2(r1,r2),x11+x21≤y1,x12+x22≤y2…r1+r2≤wr§K¡˜ff˜(x,y,r)≡(x11,x12,x21,x22,y1,y2,r1,r2)´Œ1ff"XJŒ15^‡Ñ Ò§K¡˜ff˜(x,y,r)´²ïff"·‚Œ±^Úþ©aqff{½ÂøX÷k ff˜"K3˜X ff K^‡§'X ÐÚEâffà5§ ÷vž§±e5y¯Kff)Ò´øX÷k ff˜µMaxu2(x11,x21,x22)(x,r)∈R6+++x21≤y1(r1,r2)x12+x22≤y2(r1,r2)r1+r2≤wru1(x11,x21,x12)≥u1(x∗11,x∗21,x∗12)17 3flfi.¥·‚¿3Ñ۞I‡é á ½öž¤¬\±¤" E.‚KF¼êXeµL=u2(x11,x21,x22)+λ1[y1(r1,r2)−x11−x21]+λ2[y2(r1,r2)−x12−x22]+δ(wr−r1−r2)+µ[u1(x11,x21,x12)−u1(x∗11,x∗21,x∗12)]˜ffi^‡µ∂ux211:−∂uλ11+µ=0(66)∂x11∂x11x12:−λ2+∂uµ1=0(67)∂x12∂ux221:−λ+∂uµ1=0(68)∂x121∂x21∂ux222:−λ2=0(69)∂x22∂yr1:λ11+∂yλ22−δ=0(70)∂r1∂r1∂y1r2:λ1+∂yλ22−δ=0(71)∂r2∂r2λ1:x11+x21≤y1,λ1≥0,λ1(y1−x11−x21)=0(72)λ2:x12+x22≤y2,λ2≥0,λ2(y2−x12−x22)=0(73)δ:r1+r2≤wr,δ≥0,δ(wr−r1−r2)=0(74)µ:u1−u∗1≥0,µ≥0,µ(u1−u∗1)=0(75)Óþ©˜ §¥ ©Ž¦f(λ1,λ2Úδ)ffÎÒE,´ ä] ¤†Äff'…^‡"o‡˜ffi^‡†1 !¥ƒÓ§Ïd·‚kµλ∂u1∂u2∂u2∂u11=∂x11∂u+∂x11=∂x21∂x21λ∂u212∂u+2∂u.(76)1∂x12∂x22∂x22∂x12d(70)9(71)Œ µ∂yλ221=∂r−∂y2∂r1.(77)λ∂y21y1∂r−∂1∂r2Ϗ(76)†(77)†àƒÓ§¤±k±eff>Sƒ ^‡µ∂u1∂u2∂u2∂u1∂y2∂x11∂x11∂u+1∂u=∂x212∂u+∂x212∂=∂r−∂y22∂r1u1∂(78)y1∂x−∂y1∂x12∂x2222∂x12∂r1∂r2TŁ §ÏdØLéü«ž¤¬\±¤§¤±x11+x21=y1(79)18 x12+x22=y2(80)ÏL(70)Ú(71)¦)䧌 µ∂y1∂yδ=∂u2∂y22∂r1∂r−∂y12∂r2∂r1∂x∂.(81)y221y1∂r−∂1∂r2δffÎÒØU(½§§ ûu∂y1∂y2y1∂y2∂r1∂rÚ∂2∂r2∂rffŒ "ùü‘©O¡o>SÂÃÚo>S1¤fl">Sƒ ^‡(78)§ü«ž¤¬ff²ï^‡§ëÓ(81)Ú(74)üª Óû½øX÷k ff˜8"∂y1∂y2y1∂y2∂r1∂r≥∂2∂r2∂r1∂u1∂u2∂u2∂u1∂y2y2∂x11∂u+∂x11∂u=∂x212∂u+∂x21∂r−∂2∂r12∂u=1∂y11y1∂x12∂x22∂x22∂x12∂r−∂1∂r2x11+x21=y1(r1,r2)(PPO)x12+x22=y2(r1,r2)r1+r2≤wr()∂y1∂y2y1∂y2∂r1∂r−∂2∂r2∂r(w1r−r1−r2)=0KÏL†1 !aqff?Ø·‚ Xe·Kµ·K43˜‡Xfl!¤ ½ff²L‚¸¥§-(x∗,y∗,r∗)L«øX÷k ffff˜§K·‚Œ ±e(Ø"(1)ØI‡ÏL¤ž¤¬5¢yk ff˜"(2) 3(x∗,r∗):?§∂y1∂y2∂y1∂y2∂r1∂r>2∂rKØI¤(x∗,2∂rr∗).=Œˆ øX÷`ff˜"1(3)é?Ûff˜§(x11,x12,x21,x22,r1,r2),XJ±eü‡^‡µr1+r2=wr,x11+x21=y1(r1,r2),x12+x22=y2(r1,r2),Ú>Sƒ ^‡(78) ÷vžÑk∂y1∂y2∂r1∂r<∂y1∂y22∂r2∂r¤á§K17LÏL¤˜Ü©wr5¢yøX÷k ffff˜(x∗,y∗,r∗)"µ55XJb ¬1ff) ¬éž¤öff ^ )K Ü5§=ui=ui(x11,x21,xi2,y1)§Ù¥∂ui∂y<0§KEŒ( λ11>0Ïd§ØI‡¤ ¬1"Ϗ3ù«b ƒe§(70)Ú(71)ªCµ∂u2∂y1r1:+∂y∂y∂yλ111+λ22−∂u1δ+µ=0∂y1∂r1∂r1∂r1∂y1∂r1∂u2∂y1r2:+∂yλ11+∂yλ22−∂uδ+µ1∂y1=0∂y1∂r2∂r2∂r2∂y1∂r2(∂yK2∂y)2λu2u1∂r−2∂r11=−∂∂y−µ∂1∂y+λ12∂y1∂y>0. ´§XJ·‚b ui=ui(x111,∂r−x21,xi2,y2),Ù1∂r2¥∂ui∂y<0§KØU d(J"219 4(Øfl©¥§·‚ïÄ 3ˆ« Ü5œ¹eXÛ¢yøX÷k ff˜ff¯K§ Ñ 3 Ü5ffX †Ú) 5²L¥øX÷k ff˜ff¢y^‡§¿‰Ñ I‡¤K Ü5ž¤¬ffäN^‡"ŒNþù§3üNÚà5ff Ð'Xe§XJ] 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˜|ff˜x′i1=x′i2=0,x′j1=x∗j1,x′j2=w2`ux∗"aqff§Œy²x∗i2=0"(²1Ï·‚ò>.øX÷`ff˜:¢½3˜‡é ff8ÜS§·‚I yÎÜ^‡(x∗i1=x∗i2=0,0<x∗j1≤w1,x∗j2=w2)ff:¥= ´øX÷`ff=Œ"(²2ÎÜ^‡x∗i1=x∗i2=0,0<x∗j1≤w1,x∗j2=w2ffff˜x∗´øX÷`ff …= §´e¡¯Kff)µM√axxj1xj2−xi1x∈R+4(Q1)+x21≤w1x12+x22≤w2√xi1xi2−xj1≥−x∗j1y²µ7‡5Œ† dÚn1 y"éu¿©5§dÚn1§·‚XUy²x∗´(Q1)ff)K§´√Maxxi1xi2−xj1x∈R+4+x21≤w1(Q2)ff)"x12+x22≤√w2√xj1xj2−xi1≥x∗j1w2√√b x∗Ø´(Q1)ff)§K3,˜‡ff˜x′¦ x′′j1xj2−x′i1≥x∗j1w2…√x′i1x′i2−x′j1>√x∗i1x∗i2−x∗j1=−x∗j1"dux∗j1>0§¤±x′j1A Ł"Ïd§Xx′i2>0KŒéx′i2‰21 √√‡þ~ §¿òùÜ©^uO\x′j2§ùò —x′j1x′j2−x′i1>x∗j1w2Ú√x′i1x′√i2−x′j1>√x∗i1x∗i2−x∗j√1.ù†x∗´(Q1)ff`)ff¯¢gñ"Xx′i2=0,Kx′j1<x∗j1.¤±x′j1x′j2−x′√√i1≤x′′j1xj2≤x′j1w2<x∗j1w2, Ñgñ"Ïd§d(²2Œ§e‡ >.`:§IįK(Q1)=Œ"(²3÷v^‡x∗i1=x∗i2=0,0<x∗j1≤w1,x∗j2=w2ffff˜x∗>.øX÷k ff˜ff¿‡^‡´é?ÛøX÷k S:x˜,e ^‡¥– ˜‡¤á"√√x˜j1x˜j2−x˜i1≤x∗j1w2(82)√x˜i1x˜i2−x˜j1<−x∗j1(83)3é(²3?1 ªy²ƒc§·‚I‡éى?˜Úff`²"d(²2§(²3ff dLã´µ÷v^‡x∗i1=x∗i2=0§0<x∗j1≤w1§x∗j2=w2ffff˜x∗Ø´¯K(Q1))ff¿‡^‡´3SÜøX÷k :x˜,¦ µ√√x˜j1x˜j2−x˜i1>x∗j1w2…√x˜i1x˜i2−x˜j1≥−x∗j1(84)du?ÛøX÷S):÷v] å§K(84)ff dLã´µXJx∗Ø´¯K(Q1)ff)§K3øX÷S):x˜3(Q1)¥`ux∗"¿©5Ü©´w,ff§·‚òÏLe˜‡/(²05y²7‡5Ü©"(²4XJ÷vx∗i1=x∗i2=0,0<x∗j1≤w1,x∗j2=w2ff?Ûff˜x∗ÑØ´(Q1)ff)§K3`ux∗ffS:x˜3Q1¥`ux∗"y²µÄk§·‚y²?Û>.ff˜x′3(Q1)¥ØŒU`ux∗"=XJ˜ff˜÷v(Q1)ff å^‡§KÙ،U`ux∗"e¡©A«œ/\±?Ø"œ√/1x′x′i1=√ å^‡√Œ µ√√x′i1x′i2−x′j1=−j1≥−x∗j1,¤±x′j1≤x∗§x′′j1x′j2−xx′i1=j1x′j2≤x∗j1x′j√2≤x∗j1w2”√√√œ/2x′√K§x′′j1x′j2−xi2=0Ó §x′j1≤x∗j1.√x′i1≤j1x′j2≤x∗j1x′j2≤x∗j1w2”œ/3x′j1=0”K√x′j1x′j2−x′i1=−x′i1≤0<x∗√j1w2.œ/4x′j2=′j1x′j2−x′i1=−x′i1≤0<x∗j1w2.¤±§XJff˜x∗Ø´¯K(Q1)ff)§K7,3`ffSÜff˜x˜′(ؘ½øX÷`)"XJx˜′flfiÒ´øX÷`ff§K¯K)û"ÄK§A3øX÷`ff˜x˜`ux˜′"Ä 3¯K(Q1)¥÷v] å^‡ff?Ûff˜x˜Ñ`ux∗ff¯¢§¿Łâ/øX÷`u0ff½Â§Œ3(Q1)¥x˜ØŒU`ux∗"(Ü1˜Ü©ffy²Œ§x˜ØŒU>.:"y."22 (²5>.øX÷k ff˜8(1) w1>w2/4žµPOB={x11=x12=0,x22=w2,w2/4≤x21≤w1}∪{x21=x22=0,x12=w2,w2/4≤x11≤w1}(2) w1≤w2/4žµPOB={x11=x12=0,x22=w2,x21=w1}∪{x21=x22=0,x12=w2,x11=w1}y²µd(²3§˜‡÷vx∗i1=x∗i2=0,0<x∗j1≤w1,x∗j2=w2ffff˜x∗´>.øX÷k ff˜§ …= éu?ÛøX÷k ffS:ff˜x˜§e¡ffØ ª– k˜‡¤áµ√√x˜j1x˜j2−x˜i1≤x∗j1w2√x˜i1x˜i2−x˜j1<−x∗j1·‚‡(½x∗j1ff Ł‰Œ§d·K1§·‚I‡?Øe¡ü«œ¹µœ/1µw1>w2/4d·K1§?ÛøX÷k ffS)x˜÷vx˜j1=14x˜j2§x˜i2=w2−x˜j2§x˜i1=14(w2−x˜j2)"Ïd§ êx∗j1ATäkù ff5Ÿµé?Û0<x˜j2<w2,e¡^‡¥– k˜‡¤á3√x˜j2−w2≤x∗4j1w2(85)4w2−3x˜j2<−x∗j1(86)24w,§éux˜j2∈(0w2,3]§Ø ª(85)¤á"é?Ûx˜j2∈w)2(23,w2)§Ak(x∗j1≥916wx˜2j2−w23½0<x∗j1<34x˜j2−w]22¤á§¤±§x∗j1˜½3«m[w24,w1þ Ł"œ/2µw1≤w2/4d·K1§x˜j1=w1wx˜2j2§x˜i2=w2−x˜j2,x˜i1=w1w(w22−x˜j2)"Ïd§(85)Ú(86)Cµ(√)w√11w+w2wx˜j2−w1≤x∗j1w2(87)2√(√)ww11w2−w+w1x˜j2<−x∗j1(882w)2ÏLaqff?Ø·‚Œ Ñx∗j1=w1"23 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[15]Salanie,B.,MicroeconomicsofMarketFailures,MITPress,2000.[16]Tian,G.,“ASolutiontotheProblemofConsumptionExternalities,”JournalofMathe-maticalEconomics,39(2003),831-847.[17]Tian,G.,“AUniqueInformationallyEfficientAllocationMechanisminEconomieswithConsumptionExternalities,”InternationalEconomicReview,45(2004),79-111.[18]Tian,,“AFormalTheoryofReferenceGroupforWell-beingStud-ies:ASolutiontotheHappiness-IncomePuzzle,”TexasA&MUniversity,2005,website: ff˜">.øX÷k ff˜´– ˜‡©þ"ffk ff˜•þ(x∗1,y∗1,x∗2,y∗2)"·‚ÏL±eA‡(² Ñ(Ø"(²6éu>.øX÷k ff˜(x∗1,y∗1,x∗2,y∗2),XJx∗=0½y∗=0,Ké?Ûi6=∗=iiiy∗=0…x∗ij>0,y∗j=2y²µ x∗=0"·‚Äky²x∗j>0"b T(Øؤá§Kx∗i()j=0"“\ ^¼êŒ µui(x∗1,x∗2,y∗)=ux∗ij1,x∗2,y∗j=ff˜x′=x′ij=ε,y′=y′ij=4εøX÷`()u(x∗1,y∗1,x∗2,y∗2),ù´Ïé?¿ êε§ui(x′1,x′2,y′)=uijx′1,x′2,y′j=ε>0", §·‚5yy∗=0"b y∗>0§Kd] å^‡§Œ y∗ii()√√j<2" ui(x∗1,x∗2,y∗)=i−x∗j=ui(x∗1,x∗2,0)§…Ïx∗j>0ujx∗1,x∗2,y∗j=x∗jy∗j<x∗j2=uj(x∗1,x∗2,2)"Ïd§·‚Œ ,˜‡`u(x∗1,y∗1,x∗2,y∗2)ffff˜§ù†(x∗1,y∗1,x∗2,y∗2)´øX÷`ff˜ff¯¢gñ" §b y∗j=2"Xy∗j<2,KŒuyff˜x′=y′ij=0,x′j=x∗j,y′j=2`u(x∗1,y∗1,x∗2,y∗2).éux∗=0ffœ/§(Ø y"iéuy∗=0ffœ/§y²{aq"i25 ÏL(²£6¤Œ±r>.øX÷k :›3 ff‰ŒƒS"·‚I y():x∗=y∗=0,0<x∗j≤2,y∗´Ä÷vøX÷^‡=Œ"iij=2()(²7 …= ˜ff˜x∗=y∗=0∗,0<xij≤2,y∗j=2´±e¯Kff)ž§ÙøX÷ki ff˜"M√axxjyj−xi(x1,y1,x2,y2)∈R+4+(Q1).x1+x2≤2y1+y2≤2√xiyi−xj≥−x∗jy²µ7‡5Ü©ŒdÚn1† Ñ"()éu¿©5Ü©§dÚn1§I‡y²µXJx∗=y∗=0,0<x∗j≤2,y∗(Q1)ffiij=2)§K§´(Q2)ff)"M√axxiyi−xj(x1,y1,x2,y2)∈R+4++x2≤wxy1+y2≤2√√xjyj−xi≥2x∗(j)√√b (Øؤá§K3,˜|ff˜′x′′′′,y,x′iij,yj¦ xjy′j−x≥2x∗ij…√x′y′−x′iij>√x∗y∗−x∗iij=−x∗j.Ϗx∗j>0√.¤±x′jA "Ïd§XJy′√>0§KŒ±òy′‰‡iiþ~ ¿^uO\√y′j"ùò —x′jy′j−x′>2x∗ijÚ√x′y′−x′iij>x∗y∗−x∗)iij.√Ïd§(†x∗√=y∗=0∗√,0<xij√≤2,y∗=2´(Q1))ff¯¢gñ"XJy′=0§Kx′j<x∗ijij, §x′jy′j−x′≤x′2x∗ijy′j≤x′j2<j,l Ñgñ"¤±§d(²7§ y˜ff˜fføX÷`5§IÄ5y¯K(Q1)=Œ"()(²8x∗=y∗=0,0<x∗iij≤2,y∗j=2´øX÷`ff˜ff¿‡^‡´µé?ÛøX÷`ffS:ff˜(x˜1,y˜1,x˜2,y˜2)∈POI,e ^‡¥– ˜‡¤áµ√√x˜jy˜j−x˜i≤2x∗j()√x˜iy˜i−x˜j<−x∗j()3‰Ñ(²8ff ªy²ƒc§·‚é§?1?˜Úff)º"d(²7§(²8ff dL()ã´µx∗=y∗=0,0<x∗j≤2,y∗Ø´(Q1)ff)§ …= 3,‡øX÷`ffSiij=2:(x˜1,y˜1,x˜2,y˜2)∈POI,¦ ±eü‡Ø ª¤á√√x˜jy˜j−x˜i>2x∗j§√x˜iy˜i−x˜j≥−x∗j()26 du?ÛøX÷`ffS:÷v] å^‡§¤±()ff dLã´µ˜‡ff()˜x∗=y∗=0,0<x∗iij≤2,y∗j=2Ø´(Q1)ff)ff¿‡^‡´µ3,‡øX÷`S():§(x˜1,y˜1,x˜2,y˜2)3¯K£Q1¤¥`ux∗=y∗=0,0<x∗iij≤2,y∗j=2"±þã¥§/¿©50w,¤á§e¡·‚ÏL(²95y²Ù7‡5"()(²9XJ˜‡ff˜x∗=y∗=0,0<x∗iij≤2,y∗j=2Ø´¯K(Q1)ff)§K3,‡øX()÷`ffS:(x˜1,y˜1,x˜2,y˜2)§3(Q1)¥`ux∗=y∗=0,0<x∗iij≤2,y∗j=2"()y²µÄk§·‚5y²3£Q1¤¥§?Û>.ff˜x′,y′,x′iij,y′j،U`u()x∗=y∗=0,0<x∗iij≤2,y∗j=2"Ò´`§XJ,‡>.ff˜÷v å^‡§KÙéAff8I¼êŁØpu()x∗=y∗=0,0<x∗j≤2,y∗j=2"iie¡ Ñ ¤kffœ/µœ/1µx′=0"d å^‡√x′y′−x′iiij≥−x∗j,Œx′j≤x∗jK§√√√√√x′jy′j−x′=x′ijy′j≤x∗jy′j≤2x∗j=x∗√jy∗j"√√√√œ/2µy′=0"aqff§x′ij≤x∗√j§K§√x′jy′j−x′√≤x′y′′ijj≤x∗jyj≤2x∗j=x∗jy∗j"œ/3µx′j=0"x′jy′j−x′=−x′√≤0<2x∗iij=x∗jy∗√√j"œ/4µy′j=0"x′jy′j−x′=−x′≤0<2x∗iij=x∗jy∗(j")¤±§XJff˜x∗=y∗=0,0<x∗iij≤2,y∗j=2Ø´¯K(Q1)ff)§K˜½3`ff,‡S:ff˜(x′1,y′1,x′2,y′2)XJ(x′1,y′1,x′2,y′2)flfiÒ´øX÷`ff§K(Ø y"ÄK§˜½3,‡øX÷k ff˜(x˜1,y˜1,x˜2,y˜2)`u(x′1,y′1,x′2,y′2).Ún1Œ( ù fføX÷k ff˜ff3(y²{†(²7aq)"Łâ/øX÷`u0ff½Â§Ä ?Û÷v å^‡ffff()˜(x˜1,y˜1,x˜2,y˜2)3¯K(Q1)¥`ux∗=y∗=0,0<x∗iij≤2,y∗j=2"(Ü1˜Ü©ffy²Œ µ(x˜1,y˜1,x˜2,y˜2)Ø´>.øX÷`ff˜"y."(²10>.øX÷k ff˜8µ{}{}11POB=y1=x1=0,y2=2,≤x2≤2∪y2=x2=0,y1=2,≤x1≤222()y²µd(²8§,‡ff˜x∗=y∗=0,0<x∗iij≤2,y∗j=2>.øX÷k ff˜ff¿‡^‡´é?ÛøX÷k S:(x˜1,y˜1,x˜2,y˜2)∈POI,e ü^‡¥– ˜‡¤áµ√√x˜jy˜j−x˜i≤2x∗j()27 √x˜iy˜i−x˜j<−x∗j()y3I‡(½x∗jff Ł‰Œ"d·K1ffy²Œµx˜j=14y˜j,y˜i=2−y˜j,x˜i=2−y˜j4"Ïd§ êx∗jA÷ve¡^‡µé?Û0<y˜j<2§e Ø ª– ˜‡¤á"31√y˜j−≤2x∗42j()1−3y˜j>x∗4j()éy˜j∈2(0,3]§Ø ª()¤á§¤±I‡Äy˜j∈2(3,2)§ffœ/"x∗j.é?Û()y˜2j∈(3,2)§ATkx∗j≥92232y˜j−3½x∗j<−34y˜j+1¤á§¤±x∗j7,3«m()12,2þ Ł"28