【高级微观经济学】05年习题二

Q1.Thefollowingtableliststhepricesof2goods,andthequantitiesaconsumerchoseofthegoods,in5differentsituations.(Forexample,thesecondrowindicatesthattheconsumerchosethebundlex=(10,32)whenthepricevectorwasp=(4,2).)

Fromthesedata,whatcanbeconcludedabouttheconsumer’spreferences?Explainbriefly.

t12345

pt1321

pt212345

xt1510112025

xt240325113

A1.Giventhedatainthetableabove,thecostofeachbundleineachsituationcanbecomputed.Rowshererepresentthedifferentperiods,andcolumnsthedifferentbundles.

t12345

bundle165100135170205

bundle282104126148170

bundle360484236

bundle4111102938475

bundle5128106846240

(So,forexample,bundle2wouldcost148usingperiod–4prices.)

Ifanelementonthediagonaloftheabovematrixisgreaterthanorequaltoanelementinthesamerow,thenthebundlechoseninthatyearhasbeenrevealedpreferredtothebundlechosenintheotheryear.

So,forexample,thefirstrowofthematrixshowsthatx1waschosenwhenx3wasinthebudgetset,sothatx1isrevealedpreferredtox3.Usingtheshorthand“rpt”for“revealedpreferredto”(or“chosenover”),therowsshowthatx1rptx3;x2rptallofx1,x3andx4;x4rptx3andx5;x5rptx3.

Thisconsumer’spreferencesdonotviolatetheweakaxiomofrevealedpreference.Thedataalsoshowthatbundle2ishermostpreferredbundle,andbundle3herleastpreferred.Bundles1,4and5arerankedinthemiddle:theonlyinformationgivenbythetableconcerningtherelativerankingofthesethreebundlesisthat4isrankedhigherthan5.Butitwouldbeconsistentwiththesedataforhertorankthem1󰀄4󰀄5,or4󰀄1󰀄5,or4󰀄5󰀄1.

[Ifyoudidtheoriginal“mistaken”versionofthequestion,inwhichthepricesinsituation#5were(5,1),thentherewouldbe2violationsofWARP:bundles2and5,andbundles4and5.]

Q2.Findalltheviolationsofthestrongandweakaxiomsofrevealedpreferenceinthefollowingtable,whichindicatesthepricesptofthreedifferentcommoditiesatfourdifferenttimes,andthequantitiesxtofthe3goodschosenatthefourdifferenttimes.

t1234

pt158102

pt224210

pt37242

xt165108

xt210121012

xt312151010

A2..Againamatrixcanbeconstructedshowingthecostofeachbundleineachperiod.

t1234

bundle1134112128136

bundle21118134160

bundle3140140160140

bundle4134132144156

Letting“irpt”standfor“‘indirectlyrevealedpreferredto”,thentherowsshow:bundle1rptbundle4;bundle2rptbundle1;bundle3rptallotherbundles;bundle4rptbundles1and3.So:bundle1rptbundle4butbundle4rptbundle1;bundle3rptbundle4butbundle4rptbundle3;bundle1irptbundle2(sincebundle1rptbundle4rptbundle3rptbundle2)butbundle2rptbundle1;bundle1irptbundle3(sincebundle1rptbundle4rptbundle3)butbundle3rptbundle1;bundle2irptbundle3(sincebundle2rptbundle1rptbundle4rptbundle3)butbundle3rptbundle2;bundle2irptbundle4(sincebundle2rptbundle1rptbundle4)butbundle4irptbundle2(sincebundle4rptbundle3rptbundle2).

Thereare2pairsofconsumptionbundleswhichviolateWARP,andanother4whichviolateSARPbutnotWARP.InfacthereeverypairofconsumptionbundlesviolatesSARP.

Q3.Supposethataperson’sutility–of–wealthfunctioncouldbewritten

u(W)=A−e−βW

whereβ>0.

WhatwouldbetheriskpremiumassociatedwithaprojectwhichyieldedthepersonareturnofX>0withprobabilityπ,andapayoffofzerowithprobability1−π?Howdoesthepremiumvarywiththe“goodstate”returnX?

A3.ThecertaintyequivalentCtothisprojectisthesolutiontotheequation

e−β(W+C)=πe−β(W+X)+(1−π)e−βW

(3−1)

wheretheleftsideistheutitlityfromgetingCforsure,andtherightsideistheexpectedutilityfromtheproject.

Dividingbothsidesofequation(3−1)bye−βW,

e−βC=πe−βX+1−π

or

C=−

(3−2)

1

ln(πe−βX+1−π)β

(3−3)

TheriskpremiumPequalstheexpectedreturntotheproject,minusC.TheexpectedreturnhereisπX,sothat

1

P=πX+ln(πe−βX+1−π)(3−4)

βDifferentiating(3−4),

πe−βX∂P

=π−−βX∂Xπe+1−π

NowsinceX>0,then

e−βX<1

sothat

πe−βX+1−π>πe−βX+(1−π)e−βX=e−βX

implyingthat

πe−βX

<π

πe−βX+1−π

(3−7)(3−6)(3−5)

Equation(3−7)(substitutedintoequation(3−5))indicatesthatanincreaseinthereturnXwhentheinvestmentpaysoffwillincreasetheriskpremiumPassociatedwiththeinvestment.

Q4.Supposeaperson’sutility–of–wealthfunctioncouldbewritten

u(W)=Wa

where0Supposeaswellthatthepersonhadtochoosebetweeninvestingallherinitialwealthinabond,whichgaveacertainreturnofr0,andputtingallherinitialwealthinariskyasset,thegrossreturn1+rforwhichwasdistributeduniformlyovertheinterval[0,R].(Thatis,ifsheputallherwealthW0intheriskyasset,herend–of–periodwealthwouldbedistributeduniformlyover[0,RW0].)

WhatvalueofRwouldmakeherindifferentbetweenputtingallherwealthinthesafeasset,andallherwealthintheriskyasset?HowdoesthisRvarywithherinitialwealth,withherriskaversionparametera,andwiththegrossreturn1+r0onthesafeasset?

A4.Theexpectedutilitythatthepersongetsifsheputsallhermoneyinthebondis

[W0(1+r0)]a

whereasifsheputallhermoneyintheriskyasset,herexpectedutilitywouldbe

1R󰀁

0R

[W0X]adX

(sincethedensityfunctionofarandomvariablewhichisuniformlydistributedover[0,R]isf(x)=1/R).

Therefore,shewillbeindifferentbetweenputtingallhermoneyintheriskyasset,andputtingallhermoneyinthesafeassetifandonlyif

1

[W0(1+r0)]a=

R

or

1

(1+r0)a=

R

󰀁

0R

[W0X]adX

󰀁

0

R

XadX

(4−1)

Solvingtheintegralinequation(4−1),

(1+r0)a=

implyingthat

R=(1+a)1/a(1+r0)

(4−3)

11

R1+a

R1+a

(4−2)

ThemaximumreturnRwhichsatisfies(4−3)isproportionaltothegrossreturn1+r0onthesafeasset.Itisindependentoftheperson’sinitialwealthW0.

Thisperson’sutility–of–wealthfunctionimpliesthatshehasaconstantcoefficientofrelativeriskaversion,

u󰀁󰀁(W)W

=1−a(4−4)Rr(W)=−󰀁

u(W)Thehigherisaperson’scoefficientofrelativeriskaversion,thelesslikelysheistoacceptgambles.ThatmeansthatthevalueofRwhichsatisfiesequation(4−3)mustbeadecreasingfunctionofa:ifapersonwithutilityparameterawereindifferentbetweenthesafeassetandtheriskyasset,thenapersonwithalowervalueoftheparameter,a˜ToprovedirectlyhowRvarieswitha,note(from(4−3))thatRincreaseswithaifandonlyif(1+a)1/aincreaseswitha.Toseewhether(1+a)1/aincreaseswitha,takethelogarithmofthisfunction—(1+a)1/awillincreasewithaifandonlyifln[(1+a)1/a]increaseswitha.Thatlogarithmis

1

f(a)≡ln(1+a)

a

Differentiating,

f󰀁(a)=

Sof󰀁(a)>0ifandonlyif

a1[−ln(1+a)]a21+a

a

−ln(1+a)1+a

a

<0

(1+a)2(4−5)

g(a)≡

ispositive.Nowg(0)=0,and

g󰀁(a)=−

(4−6)

Sog(a)<0foralla≥0,implyingthatf(a)mustbeadecreasingfunctionofa(foralla>0).ThereforetherequiredmaximumgrossreturnRontheriskyassetisadecreasingfunctionofa,andanincreasingfunctionoftheperson’scoefficientofrelativeriskaversion.

NotethattheexpectedgrossreturnontheriskyassetisR/2.Soifa=1,equation(4−3)impliesthat1+r0=R/2,sothatbothassetshavethesameexpectedreturn:notsurprising,sincea=1impliesthatthepersonisriskneutral.SinceRisadecreasingfunctionofa,apositivecoefficientofrelativeriskaversionimpliesthatR/2>1+r0:theriskpremiumassociatedwiththeriskyassetispositive,andincreaseswiththeperson’scoefficientofrelativeriskaversion.

Q5.Ifaproductionfunctionf(x1,x2)hastheequation

f(x1,x2)=[x1ln(

x1+x2a

)]x1

where0A5.Bruteforcedifferentiationofthedefinitionoftheproductionfunctionyields:x1+x2x2x1+x2a−1

)][ln()−]x1x1x1+x2

x1+x2a−1x1

f2(x1,x2)=a[x1ln()]

x1x1+x2

asthemarginalproductsofthetwoinputs.

Therefore,themarginalrateoftechnicalsubstitutionis

f1(x1,x2)=a[x1ln(

MRTS=

f1(x1,x2)x1+x2x1+x2x2

=ln()−

f2(x1,x2)x1x1x1

(5−1)(5−2)

(5−3)

Toseehowthemarginalrateoftechnicalsubstitutionvariesaswemovealonganisoquant,let

g(z)≡(1+z)ln(1+z)−z

sothatequation(5−3)impliesthat

MRTS=g(

x2

)x1

(5−4)

Since

g󰀁(z)=ln(1+z)>0

thereforethemarginalrateofsubstitutionincreasesasweincreasez2andlowerz1:theisoquantsareconvextotheorigin,andtheproductionfunctionf(x1,x2)isquasi–concave.

Whatwouldhappenifbothx1andx2weretoincreasebysomefactork?Thedefintionoftheproductionfunctionimpliesthat

kx1+kx2ax1+x2ax1+x2a

)]=[kx1ln()]=ka[x1ln()]=kaf(x1,x2)

kx1x1x1

(5−5)

Therefore,theproductionfunctionishomogeneousofdegreea<1:theproductionfunctionexhibitsdecreasingreturnstoscale.

Sincetheproductionfunctionisquasi–concave,andexhibitsdecreasingreturnstoscale,thenitmustbeconcave.(ThisisreallyaconsequenceofTheorem3.1inthebook,thatquasi–concaveproductionfunctionsmustbeconcaveiftheyexhibitconstantreturnstoscale.)Concavefunctionsmusthavenon–negativeelementsalongthediagonaloftheHessianmatrix.Thereforef11≤0andf22<0,sothatthemarginalproductofeachinputdecreaseswiththequantityemployedofthatinput,apropertywhichmaynotbeimmediatelyapparentfromequations(5−1)and(5−2).f(kx1,kx2)=[kx1ln(