复旦管院2010高级微观试题

1.Aconsumerhasautilityfunctionu(x;y)=Axay1󰀂a;whereA=aa(1󰀂a)1󰀂a:Givenpricespxandpy;andincomeI;please…nd(1)theMarshalliandemandfunctions;(2)theindirectutilityfunc-tion;(3)theexpenditurefunction;(4)thecompensateddemandfunctions;and(5)verifytheSlutskyequation.Ans:(1)theMar-shalliandemandfunctions:

x=

aI(1󰀂a)I;y=:pxpy

(2)theindirectutilityfunction:

aa󰀂1v=p󰀂xpyI:

(3)theexpenditurefunction:

1󰀂ae=paxpyu

(4)thecompensateddemandfunctions

xh=a(

px1󰀂a

)u:yh=py

(5)verifytheSlutskyequation:

@x(p;I)@xh(p;u󰀄)@x(p;I)

=󰀂x(p;I)@px@px@IaIaaI2󰀂2=a(a󰀂1)p󰀂I󰀂xpxpxpx

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2.Acoinhasprobabilitypoflandingheads.Youareo¤eredabetinwhichyouwillbepaid$2jifthe…rstheadoccursonthejth‡ip.

(a)Whatistheexpectedvalueofthisbetwhenp=1=2?Ans:

Theprobabilityofheadsoccuringforthe…rsttimeonthe

j󰀂1

jthtossis(1󰀂p)p:HencetheexpectedvalueofthebetisP1

j󰀂1j

p2=1:j=1(1󰀂p)

(b)Supposethatyourutilityfunctionisu(x)=lnx:ExpressP1the

expectedutilityofthisgametoyouasasum.Ans:j=1(1󰀂

P1

j󰀂1j

p)pln(2)=pln(2)j=1j(1󰀂p)j󰀂1:

P1

1j

(c)Evaluatethesum.Ans:j=1(1󰀂p)=p:Di¤erentiateP1

bothsidesw.r.tptoobtainj=1j(1󰀂p)j󰀂1=p12:Hence

P2j󰀂1

=lnpln(2)1j=1j(1󰀂p)p

(d)Letw0betheamountofmoneythatwouldgiveyouthesame

utilityyouwouldhaveifyouplayedthisgame.Solveforw0:

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;thenw0=eln(2)=p:Ans:lnw0=lnp3.Considerarationalpreferencerelation%:Showthatifu(x)=u(y)impliesx󰀅yandifu(x)>u(y)implies󰀆;thenu(:)isautilityfunctionrepresenting%:Proof.Suppose…rstthatx%y:Iffurthermore,y%x;thenx󰀅yandhenceu(x)=u(y):Ifonthecontrary,wedonothavey%x;thenx󰀆y:Henceu(x)>u(y):Thus,ifx%y;thenu(x)󰀄u(y):Next,supposeconverselythatu(x)󰀄u(y):If,furthermore,u(x)=u(y);thenx󰀅yandhencex%y:If,onthecontrary,u(x)>u(y);thenx󰀆y;andhencex%y:Sou(:)represents%:

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