文稿講稿綜合ps4 answerkey

WISE, AdvancedMicroeconomicsI,Instructor:JacopoProblemsetQuestion1...............................................................10Showthatifthepreferencerelation%onLisrepresentedbyautilityU(·)thathastheexpectedutilityform,then%satisfiestheceLL%ForanythirdlotteryL00andanyα∈[0,1],considerthelotteriesU (1? (1?α)UU (1? (1?α)U (1?α)UU (1? (1?α)U (1?α)UU (1?U (1? U (1? (1?α)L00% (1?Question2...............................................................20Showthatiftheset esCisfiniteandtherationalpreferencerelationonLsatisfiestheceaxiom,thentherearebestandworstlotteriesL.Thatis,wecanfindlotteriesˉandLsuchthat:ˉ%L%LforallL∈Lee%Lnforalln∈{1,...,ifL%Lforalln∈{1,..., thenLp(4)we%%Thenwecanusethe ceaxiomandequation(7),mixingwithαˉα)ˉαˉ Thenequations(8)and(9)togetherˉα)ˉˉ%αL1+(1? Nowwechooseα=p1ands eP2 pn=1wealsohave1?α=p2.Thereforeweget:ˉ%p1L1+ orˉ% pnLn,withN=ShowingthelogicforthecaseN=2issufficienttogetfullgradeinthisProvingproposition(5)inthegeneralcasewithNcanbedoneinthefollowingFirst,wearegoingtoproveˉ

Pk?1

Ln?ˉ%

P

m=1

m=1

ˉ

m=1pm%LLandthisistheproofof(5)inthegeneralcasewithN110LU(·)oLL0LL0U()(1)U(L)>U(L0)L00α01]，αL+(1.α)L00αL0(1α)L00。由于U()的概率是線性的（作為預(yù)期效用函數(shù)），U(αL(1α)L00)=αU(L)(1α)U(L00)然后利用(2)U(αL(1α)L00αU(L(1α)U(L00)αU(L0(1α)U(L00)U()U(αL(1α)L00αU(L(1α)U(L00)αU(L0)(1α)U(L00)=U(αL0+(1.U()U(αL(1α)L00)U(αL0(1α)L00αL(1α)L00%αL0+(1.α)L00(3)(1)，條件(3220C.niteLLLLLLLLL。如果集合C是.nite，則有N個(gè)結(jié)果。因此，考慮N個(gè)退化彩票：L1,...,LN，其中=(0,10n（完全且可傳遞），Nˉ這N張彩票中至少有一張，L∈{L1,...,LN}，使得：ˉLLnn{1,N}可能不止一個(gè)，但這里我們只處理Lˉ唯一的情況（其他情況類似）ˉLLLLL=(p1,pNLp1L1pNLN接下來，我們需要證明對于任何概率集(p1,...,pN)使得PNn=1pn=1，NXˉLLnn{1,NˉLpnLn(5)N2L=(p1,p2)。從方程4）我們知道：ˉLL1(6)LˉL2(7)(6)L2αLˉ+(1%αL1+(1.α)L2αLˉ+(1.α)Lˉ%αLˉ+(1.α)L2(9)αLˉ+(1α)LˉαL11α)L2或ˉL%αL1+(1.α)L2(10)αp1n=1pn11αp2ˉL%p1L1+p2L2(11)ˉLn=1pnLnN2N2k.1kXXpnˉ.Pk.1LnPkLnpmpmn=1m=1n=1Pk.1Lnpmn=1PK.1pmL=Pmk=1L+PkL%pmpmm=1m=1PK.1pmPKˉPkm=1L+PkLk%pmpmm=1XpmpnpkPkm=1Pk.1L