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Transcript of Advanced Microeconomic Theory - UoA · PDF filebounded set, then the solution to such...
AdvancedMicroeconomicTheory
Chapter2:DemandTheory
Outline
• Utilitymaximizationproblem(UMP)• Walrasiandemandandindirectutilityfunction• WARPandWalrasiandemand• Incomeandsubstitutioneffects(Slutskyequation)
• DualitybetweenUMPandexpenditureminimizationproblem(EMP)
• Hicksiandemandandexpenditurefunction• Connections
AdvancedMicroeconomicTheory 2
UtilityMaximizationProblem
AdvancedMicroeconomicTheory 3
UtilityMaximizationProblem• Consumermaximizeshisutilitylevelbyselectingabundle𝑥 (where𝑥 canbeavector)subjecttohisbudgetconstraint:
max%&'
𝑢(𝑥)s. t. 𝑝 0 𝑥 ≤ 𝑤
• Weierstrass Theorem: foroptimizationproblemsdefinedonthereals,iftheobjectivefunctioniscontinuousandconstraintsdefineaclosedandboundedset,thenthesolutiontosuchoptimizationproblemexists.
AdvancedMicroeconomicTheory 4
UtilityMaximizationProblem
• Existence: if𝑝 ≫ 0 and𝑤 > 0 (i.e.,if𝐵7,9 isclosedandbounded),andif𝑢(0) iscontinuous,thenthereexistsatleastonesolutiontotheUMP.– If,inaddition,preferencesarestrictlyconvex,thenthesolutiontotheUMPisunique.
• WedenotethesolutionoftheUMPasthe𝐚𝐫𝐠𝐦𝐚𝐱oftheUMP(theargument,𝑥,thatsolvestheoptimizationproblem),andwedenoteitas𝑥(𝑝, 𝑤).– 𝑥(𝑝,𝑤) istheWalrasiandemandcorrespondence,whichspecifiesademandofeverygoodinℝ@A foreverypossiblepricevector,𝑝,andeverypossiblewealthlevel,𝑤.
AdvancedMicroeconomicTheory 5
UtilityMaximizationProblem• Walrasiandemand𝑥(𝑝,𝑤)
atbundleA isoptimal,astheconsumerreachesautilitylevelof𝑢B byexhaustingallhiswealth.
• BundlesB andC arenotoptimal,despiteexhaustingtheconsumer’swealth.Theyyieldalowerutilitylevel𝑢C,where𝑢C < 𝑢B.
• BundleD isunaffordableand,hence,itcannotbetheargmaxoftheUMPgivenawealthlevelof𝑤.
AdvancedMicroeconomicTheory 6
PropertiesofWalrasianDemand• IftheutilityfunctioniscontinuousandpreferencessatisfyLNSovertheconsumptionset𝑋 = ℝ@A ,thentheWalrasiandemand𝑥(𝑝, 𝑤) satisfies:
1)Homogeneityofdegreezero:𝑥 𝑝,𝑤 = 𝑥(𝛼𝑝, 𝛼𝑤) forall𝑝,𝑤,andforall𝛼 > 0
Thatis,thebudgetsetisunchanged!
𝑥 ∈ ℝ@A : 𝑝 0 𝑥 ≤ 𝑤 = 𝑥 ∈ ℝ@A : 𝛼𝑝 0 𝑥 ≤ 𝛼𝑤
Notethatwedon’tneedanyassumptiononthepreferencerelationtoshowthis.Weonlyrelyonthebudgetsetbeingaffected.
AdvancedMicroeconomicTheory 7
PropertiesofWalrasianDemand
AdvancedMicroeconomicTheory 8
– Notethatthepreferencerelationcanbelinear,andhomog(0) wouldstillhold.
x2
x1
2 2
w wp p=
1 1
w wp p=
x(p,w) is unaffected
2)Walras’Law:
𝑝 0 𝑥 = 𝑤 forall𝑥 = 𝑥(𝑝,𝑤)ItfollowsfromLNS:iftheconsumerselectsaWalrasiandemand𝑥 ∈ 𝑥(𝑝, 𝑤),where𝑝 0 𝑥 < 𝑤,thenitmeanswecanstillfindotherbundle𝑦,whichisε–closeto𝑥,whereconsumercanimprovehisutilitylevel.
Ifthebundletheconsumerchoosesliesonthebudgetline,i.e.,𝑝 0 𝑥P = 𝑤,wecouldthenidentifybundlesthatarestrictly preferredto𝑥P,butthesebundleswouldbeunaffordabletotheconsumer.
PropertiesofWalrasianDemand
AdvancedMicroeconomicTheory 9
PropertiesofWalrasianDemand
AdvancedMicroeconomicTheory 10
– For𝑥 ∈ 𝑥(𝑝,𝑤),thereisabundle𝑦,ε–closeto𝑥,suchthat𝑦 ≻ 𝑥.Then,𝑥 ∉ 𝑥(𝑝,𝑤).
1x
2x
x x
y y
PropertiesofWalrasianDemand
3)Convexity/Uniqueness:
a) Ifthepreferencesareconvex,thentheWalrasiandemandcorrespondence𝑥(𝑝, 𝑤)definesaconvexset,i.e.,acontinuumofbundlesareutilitymaximizing.
b) Ifthepreferencesarestrictlyconvex,thentheWalrasiandemandcorrespondence𝑥(𝑝, 𝑤)containsasingleelement.
AdvancedMicroeconomicTheory 11
PropertiesofWalrasianDemand
Convexpreferences Strictlyconvexpreferences
AdvancedMicroeconomicTheory 12
2
wp
1
wp
Unique x(p,w)
x2
2
wp
1
wp
( ),x x p w
{
x2
x1 x1
UMP:NecessaryConditionmax%&'
𝑢 𝑥 s. t. 𝑝 0 𝑥 ≤ 𝑤
• WesolveitusingKuhn-TuckerconditionsovertheLagrangian𝐿 = 𝑢 𝑥 + 𝜆(𝑤 − 𝑝 0 𝑥),
WAW%X
= WY(%∗)W%X
− 𝜆𝑝[ ≤ 0 forall𝑘,= 0 if𝑥[∗ > 0WAW] = 𝑤 − 𝑝 0 𝑥∗ = 0
• Thatis,inainterior optimum,WY(%∗)
W%X= 𝜆𝑝[ forevery
good𝑘,whichimplies^_(`∗)^`a
^_(`∗)^`X
= 7a7X⇔ 𝑀𝑅𝑆f,[ =
7a7X⇔
^_(`∗)^`a7a
=^_(`∗)^`X7X
AdvancedMicroeconomicTheory 13
UMP:SufficientCondition
• WhenareKuhn-Tucker(necessary)conditions,alsosufficient?– Thatis,whencanweguaranteethat𝑥(𝑝, 𝑤) isthemax oftheUMPandnotthemin?
AdvancedMicroeconomicTheory 14
UMP:SufficientCondition• Kuhn-Tuckerconditionsaresufficientforamax if:1) 𝑢(𝑥) isquasiconcave,
i.e.,convexuppercontourset(UCS).
2) 𝑢(𝑥) ismonotone.3) 𝛻𝑢(𝑥) ≠ 0 for𝑥 ∈ ℝ@A .– If𝛻𝑢 𝑥 = 0 forsome𝑥,thenwewouldbeatthe“topofthemountain”(i.e.,blissingpoint),whichviolatesbothLNSandmonotonicity.
AdvancedMicroeconomicTheory 15
x1
x2
wp2
wp1
x*∈x(p,w)
slope=- p2p1
Ind.Curve
UMP:ViolationsofSufficientCondition
1)𝒖(0) isnon-monotone:
– TheconsumerchoosesbundleA (atacorner)sinceityieldsthehighestutilitylevelgivenhisbudgetconstraint.
– AtpointA,however,thetangencycondition𝑀𝑅𝑆C,B =
7o7p
doesnothold.
AdvancedMicroeconomicTheory 16
UMP:ViolationsofSufficientCondition
AdvancedMicroeconomicTheory 17
x2
x1
A
B
C
Budget line
– Theuppercontoursets(UCS)arenotconvex.
– 𝑀𝑅𝑆C,B =7o7p isnota
sufficientconditionforamax.
– Apointoftangency(C)givesalowerutilitylevelthanapointofnon-tangency(B).
– TruemaximumisatpointA.
2)𝒖(0) isnotquasiconcave:
UMP:CornerSolution• Analyzingdifferentialchangesin𝑥f and𝑥f,thatkeepindividual’s
utilityunchanged,𝑑𝑢 = 0,
rY(%)r%a
𝑑𝑥f +rY(%)r%X
𝑑𝑥[ = 0 (totaldiff.)
• Rearranging,
𝑑𝑥[𝑑𝑥f
= −
𝑑𝑢 𝑥𝑑𝑥f𝑑𝑢 𝑥𝑑𝑥[
= −𝑀𝑅𝑆f,[
• CornerSolution:𝑀𝑅𝑆f,[ >7a7X,oralternatively,
s_ `∗
s`a7a
>s_ `∗
s`X7X
,i.e.,theconsumerpreferstoconsumemoreofgood𝑙.
AdvancedMicroeconomicTheory 18
UMP:CornerSolution• IntheFOCs,thisimplies:
a) WY(%∗)W%X
≤ 𝜆𝑝[ forthegoodswhoseconsumptioniszero,𝑥[∗= 0,and
b) WY(%∗)W%a
= 𝜆𝑝f forthegoodwhoseconsumptionispositive,𝑥f∗> 0.
• Intuition:themarginalutilityperdollarspentongood𝑙 isstilllargerthanthatongood𝑘.
^_(`∗)^`a7a
= 𝜆 ≥^_(`∗)^`X7X
AdvancedMicroeconomicTheory 19
UMP:CornerSolution• Consumerseekstoconsume
good1alone.
• Atthecornersolution,theindifferencecurveissteeperthanthebudgetline,i.e.,
𝑀𝑅𝑆C,B >7o7p
orwxo7o
> wxp7p
• Intuitively,theconsumerwouldliketoconsumemoreofgood1,evenafterspendinghisentirewealthongood1alone.
AdvancedMicroeconomicTheory 20
x1
x2
wp2
wp1
slopeoftheB.L.=- p2p1
slopeoftheI.C.=MRS1,2
UMP:LagrangeMultiplier• 𝜆 isreferredtoasthe
“marginalvaluesofrelaxingtheconstraint”intheUMP(a.k.a.“shadowpriceofwealth”).
• Ifweprovidemorewealthtotheconsumer,heiscapableofreachingahigherindifferencecurveand,asaconsequence,obtainingahigherutilitylevel.– Wewanttomeasurethe
changeinutilityresultingfromamarginalincreaseinwealth.
AdvancedMicroeconomicTheory 21
x1
x2
w0
p2{x∈ℝ+:u(x)=u1}2
{x∈ℝ+:u(x)=u0}2
w1
p2
w0
p1w1
p1
UMP:LagrangeMultiplier
• Letustake𝑢 𝑥(𝑝, 𝑤) ,andanalyzethechangeinutilityfromchangeinwealth.Usingthechainruleyields,
𝛻𝑢 𝑥(𝑝, 𝑤) 0 𝐷9𝑥(𝑝, 𝑤)
• Substituting𝛻𝑢 𝑥(𝑝, 𝑤) = 𝜆𝑝 (ininteriorsolutions),
𝜆𝑝 0 𝐷9𝑥(𝑝, 𝑤)
AdvancedMicroeconomicTheory 22
UMP:LagrangeMultiplier
• FromWalras’Law,𝑝 0 𝑥 𝑝, 𝑤 = 𝑤,thechangeinexpenditurefromanincreaseinwealthisgivenby
𝐷9 𝑝 0 𝑥 𝑝, 𝑤 = 𝑝 0 𝐷9𝑥 𝑝,𝑤 = 1
• Hence,𝛻𝑢 𝑥(𝑝, 𝑤) 0 𝐷9𝑥 𝑝,𝑤 = 𝜆 𝑝 0 𝐷9𝑥 𝑝,𝑤
C= 𝜆
• Intuition:If𝜆 = 5,thena$1increaseinwealthimpliesanincreasein5unitsofutility.
AdvancedMicroeconomicTheory 23
WalrasianDemand:WealthEffects
• Normalvs.InferiorgoodsW% 7,9W9
>< 0 normalinferior
• Examplesofinferiorgoods:– Two-buckchuck(areallycheapwine)–Walmartduringtheeconomiccrisis
AdvancedMicroeconomicTheory 24
WalrasianDemand:WealthEffects• Anincreaseinthewealthlevel
producesanoutwardshiftinthebudgetline.
• 𝑥B isnormalasW%p 7,9W9 > 0,while
𝑥C isinferiorasW% 7,9W9 < 0.
• Wealthexpansionpath:– connectstheoptimalconsumption
bundlefordifferentlevelsofwealth
– indicateshowtheconsumptionofagood changesasaconsequenceofchangesinthewealthlevel
AdvancedMicroeconomicTheory 25
WalrasianDemand:WealthEffects• Engelcurvedepictsthe
consumptionofaparticulargoodinthehorizontalaxisandwealthontheverticalaxis.
• TheslopeoftheEngelcurveis:– positiveifthegoodisnormal– negativeifthegoodisinferior
• Engelcurvecanbepositivelysloppedforlowwealthlevelsandbecomenegativelysloppedafterwards.
AdvancedMicroeconomicTheory 26
x1
Wealth,w
ŵ
Engelcurveforanormalgood(uptoincomeŵ),butinferior
goodforallw>ŵ
Engelcurveofanormalgood,forallw.
WalrasianDemand:PriceEffects
• Ownpriceeffect:𝜕𝑥[ 𝑝, 𝑤𝜕𝑝[
<> 0 UsualGiffen
• Cross-priceeffect:𝜕𝑥[ 𝑝, 𝑤
𝜕𝑝f>< 0 SubstitutesComplements
– ExamplesofSubstitutes:twobrandsofmineralwater,suchasAquafinavs.PolandSprings.
– ExamplesofComplements:carsandgasoline.
AdvancedMicroeconomicTheory 27
WalrasianDemand:PriceEffects
• Ownpriceeffect
AdvancedMicroeconomicTheory 28
Usualgood Giffengood
WalrasianDemand:PriceEffects
AdvancedMicroeconomicTheory 29
• Cross-priceeffect
Substitutes Complements
IndirectUtilityFunction
• TheWalrasiandemandfunction,𝑥 𝑝,𝑤 ,isthesolutiontotheUMP(i.e., argmax).
• WhatwouldbetheutilityfunctionevaluatedatthesolutionoftheUMP,i.e.,𝑥 𝑝,𝑤 ?– Thisistheindirectutilityfunction(i.e.,thehighestutilitylevel),𝑣 𝑝, 𝑤 ∈ ℝ,associatedwiththeUMP.
– Itisthe“valuefunction”ofthisoptimizationproblem.
AdvancedMicroeconomicTheory 30
PropertiesofIndirectUtilityFunction
• IftheutilityfunctioniscontinuousandpreferencessatisfyLNSovertheconsumptionset𝑋 = ℝ@A , thentheindirectutilityfunction𝑣 𝑝, 𝑤 satisfies:
1) Homogenousofdegreezero:Increasing𝑝 and𝑤byacommonfactor𝛼 > 0 doesnotmodifytheconsumer’soptimalconsumptionbundle,𝑥(𝑝, 𝑤),norhismaximalutilitylevel,measuredby𝑣 𝑝,𝑤 .
AdvancedMicroeconomicTheory 31
PropertiesofIndirectUtilityFunction
AdvancedMicroeconomicTheory 32
x1
x2
wp2
αwαp2 =
wp1
αwαp1 =
x(p,w)=x(αp,αw)
v(p,w)=v(αp,αw)
Bp,w=Bαp,αw
PropertiesofIndirectUtilityFunction
2)Strictlyincreasingin𝑤:𝑣 𝑝,𝑤P > 𝑣(𝑝, 𝑤) for𝑤P > 𝑤.
3)non-increasing(i.e.,weaklydecreasing)in𝑝[
AdvancedMicroeconomicTheory 33
PropertiesofIndirectUtilityFunction
AdvancedMicroeconomicTheory 34
W
P1
w2
w1
4)Quasiconvex:Theset 𝑝,𝑤 : 𝑣(𝑝, 𝑤) ≤ �̅� isconvexforany�̅�.
- InterpretationI:If(𝑝C, 𝑤C) ≿∗ (𝑝B, 𝑤B), then(𝑝C, 𝑤C) ≿∗ (𝜆𝑝C +(1 − 𝜆)𝑝B, 𝜆𝑤C + (1 − 𝜆)𝑤B); i.e.,if𝐴 ≿∗ 𝐵,then𝐴 ≿∗ 𝐶.
PropertiesofIndirectUtilityFunction- InterpretationII:𝑣(𝑝,𝑤) isquasiconvex ifthesetof(𝑝, 𝑤) pairsforwhich𝑣 𝑝,𝑤 < 𝑣(𝑝∗, 𝑤∗) isconvex.
AdvancedMicroeconomicTheory 35
W
P1
w*
p*
v(p,w) < v(p*,w*)
v(p,w) = v1
PropertiesofIndirectUtilityFunction
- InterpretationIII:Using𝑥C and𝑥B intheaxis,performfollowingsteps:1) When𝐵7,9,then𝑥 𝑝,𝑤2) When𝐵7�,9�,then𝑥 𝑝P, 𝑤P
3) Both𝑥 𝑝,𝑤 and𝑥 𝑝P, 𝑤P induceanindirectutilityof𝑣 𝑝,𝑤 = 𝑣 𝑝P, 𝑤P = 𝑢�
4) Constructalinearcombinationofpricesandwealth:𝑝PP = 𝛼𝑝 + (1 − 𝛼)𝑝P𝑤PP = 𝛼𝑤 + (1 − 𝛼)𝑤P�𝐵7��,9��
5) AnysolutiontotheUMPgiven𝐵7��,9�� mustlieonalowerindifferencecurve(i.e.,lowerutility)
𝑣 𝑝PP, 𝑤PP ≤ 𝑢�
AdvancedMicroeconomicTheory 36
PropertiesofIndirectUtilityFunction
AdvancedMicroeconomicTheory 37
x1
x2
Bp,wBp’,w’Bp’’,w’’
x(p,w)
x(p’,w’)
Ind.Curve{x∈ℝ2:u(x)=ū}
WARPandWalrasianDemand
• RelationbetweenWalrasiandemand𝑥 𝑝, 𝑤andWARP– HowdoestheWARPrestrictthesetofoptimalconsumptionbundlesthattheindividualdecision-makercanselectwhensolvingtheUMP?
AdvancedMicroeconomicTheory 38
WARPandWalrasianDemand• Taketwodifferentconsumptionbundles𝑥 𝑝,𝑤 and𝑥 𝑝′, 𝑤′ ,bothbeingaffordable 𝑝,𝑤 ,i.e.,
𝑝 0 𝑥 𝑝, 𝑤 ≤ 𝑤 and𝑝 0 𝑥 𝑝′, 𝑤′ ≤ 𝑤• Whenpricesandwealthare 𝑝,𝑤 ,theconsumerchooses𝑥 𝑝,𝑤 despite𝑥 𝑝′, 𝑤′ isalsoaffordable.
• Thenhe“reveals”apreferencefor𝑥 𝑝,𝑤 over𝑥 𝑝′, 𝑤′whenbothareaffordable.
• Hence,weshouldexpecthimtochoose𝑥 𝑝,𝑤 over𝑥 𝑝′, 𝑤′ whenbothareaffordable.(Consistency)
• Therefore,bundle𝑥 𝑝,𝑤 mustnotbeaffordableat𝑝′, 𝑤′ becausetheconsumerchooses𝑥 𝑝′, 𝑤′ .Thatis,𝑝′ 0 𝑥 𝑝, 𝑤 > 𝑤′.
AdvancedMicroeconomicTheory 39
WARPandWalrasianDemand
• Insummary,WalrasiandemandsatisfiesWARP,if,fortwodifferentconsumptionbundles,𝑥 𝑝,𝑤 ≠ 𝑥 𝑝′, 𝑤′ ,
𝑝 0 𝑥 𝑝′, 𝑤′ ≤ 𝑤 ⇒ 𝑝′ 0 𝑥 𝑝, 𝑤 > 𝑤′
• Intuition:if𝑥 𝑝′, 𝑤′ isaffordableunderbudgetset𝐵7,9,then𝑥 𝑝,𝑤 cannotbeaffordableunder𝐵7�,9�.
AdvancedMicroeconomicTheory 40
CheckingforWARP
• AsystematicproceduretocheckifWalrasiandemandsatisfiesWARP:– Step1: Checkifbundles𝑥 𝑝,𝑤 and𝑥 𝑝′, 𝑤′ arebothaffordableunder𝐵7,9.§ Thatis,graphically𝑥 𝑝,𝑤 and𝑥 𝑝′, 𝑤′ havetolieonorbelowbudgetline𝐵7,9.
§ Ifstep1issatisfied,thenmovetostep2.§ Otherwise,thepremiseofWARPdoesnothold,whichdoesnotallowustocontinuecheckingifWARPisviolatedornot.Inthiscase,wecanonlysaythat“WARPisnotviolated”.
AdvancedMicroeconomicTheory 41
CheckingforWARP
- Step2: Checkifbundles𝑥 𝑝,𝑤 isaffordableunder𝐵7�,9�.§ Thatis,graphically𝑥 𝑝,𝑤 mustlieonorbelowbudge
line𝐵7�,9� .§ Ifstep2issatisfied,thenthisWalrasiandemandviolates
WARP.§ Otherwise,theWalrasiandemandsatisfiesWARP.
AdvancedMicroeconomicTheory 42
CheckingforWARP:Example1• First,𝑥 𝑝,𝑤 and𝑥 𝑝′, 𝑤′ arebothaffordableunder𝐵7,9.
• Second,𝑥 𝑝,𝑤 isnotaffordableunder𝐵7�,9�.
• Hence,WARPissatisfied!
AdvancedMicroeconomicTheory 43
x1
x2
w’p2
wp1
’
Bp,w
Bp’,w’
x(p,w)
x(p’,w’)
CheckingforWARP:Example2
• Thedemand𝑥 𝑝′, 𝑤′underfinalpricesandwealthisnotaffordableunderinitialpricesandwealth,i.e.,𝑝 0 𝑥 𝑝P, 𝑤P > 𝑤.– ThepremiseofWARPdoesnothold.
– ViolationofStep1!– WARPisnotviolated.
AdvancedMicroeconomicTheory 44
CheckingforWARP:Example3
• Thedemand𝑥 𝑝′, 𝑤′underfinalpricesandwealthisnotaffordableunderinitialpricesandwealth,i.e.,𝑝 0 𝑥 𝑝P, 𝑤P > 𝑤.– ThepremiseofWARPdoesnothold.
– ViolationofStep1!– WARPisnotviolated.
AdvancedMicroeconomicTheory 45
CheckingforWARP:Example4
• Thedemand𝑥 𝑝′, 𝑤′underfinalpricesandwealthisnotaffordableunderinitialpricesandwealth,i.e.,𝑝 0 𝑥 𝑝P, 𝑤P > 𝑤.– ThepremiseofWARPdoesnothold.
– ViolationofStep1!– WARPisnotviolated.
AdvancedMicroeconomicTheory 46
CheckingforWARP:Example5
AdvancedMicroeconomicTheory 47
x1
x2
w’p2
wp1
’
Bp,w
Bp’,w’x(p,w)x(p’,w’)
• First,𝑥 𝑝,𝑤 and𝑥 𝑝′, 𝑤′ arebothaffordableunder𝐵7,9.
• Second,𝑥 𝑝,𝑤 isaffordableunder𝐵7�,9�,i.e.,𝑝P 0 𝑥 𝑝, 𝑤 < 𝑤′
• Hence,WARPisNOTsatisfied!
ImplicationsofWARP• HowdopricechangesaffecttheWARPpredictions?• Assumeareductionin𝑝C
– theconsumer’sbudgetlinesrotates(uncompensatedpricechange)
AdvancedMicroeconomicTheory 48
ImplicationsofWARP• Adjusttheconsumer’swealthlevelsothathecanconsumehisinitial
demand𝑥 𝑝, 𝑤 atthenewprices.– shiftthefinalbudgetlineinwardsuntilthepointatwhichwereach
theinitialconsumptionbundle𝑥 𝑝, 𝑤 (compensatedpricechange)
AdvancedMicroeconomicTheory 49
ImplicationsofWARP• Whatisthewealthadjustment?
𝑤 = 𝑝 0 𝑥 𝑝, 𝑤 under 𝑝,𝑤𝑤P = 𝑝P 0 𝑥 𝑝, 𝑤 under 𝑝P, 𝑤P
• Then,∆𝑤 = ∆𝑝 0 𝑥 𝑝, 𝑤
where∆𝑤 = 𝑤P − 𝑤 and∆𝑝 = 𝑝P − 𝑝.
• ThisistheSlutskywealthcompensation:– theincrease(decrease)inwealth,measuredby∆𝑤,thatwemust
providetotheconsumersothathecanaffordthesameconsumptionbundleasbeforethepriceincrease(decrease),𝑥 𝑝,𝑤 .
AdvancedMicroeconomicTheory 50
ImplicationsofWARP:LawofDemand
• SupposethattheWalrasiandemand𝑥 𝑝,𝑤 satisfieshomog(0) andWalras’Law.Then,𝑥 𝑝,𝑤 satisfiesWARPiff:
∆𝑝 0 ∆𝑥 ≤ 0where– ∆𝑝 = 𝑝P − 𝑝 and∆𝑥 = 𝑥 𝑝P, 𝑤P − 𝑥 𝑝,𝑤– 𝑤P isthewealthlevelthatallowstheconsumertobuy
theinitialdemandatthenewprices,𝑤P = 𝑝P 0 𝑥 𝑝, 𝑤
• ThisistheLawofDemand:quantitydemandedandpricemoveindifferentdirections.
AdvancedMicroeconomicTheory 51
ImplicationsofWARP:LawofDemand
• DoesWARPrestrictbehaviorwhenweapplySlutskywealthcompensations?– Yes!
• WhatifwewerenotapplyingtheSlutskywealthcompensation,wouldWARPimposeanyrestrictiononallowablelocationsfor𝑥 𝑝P, 𝑤P ?– No!
AdvancedMicroeconomicTheory 52
ImplicationsofWARP:LawofDemand
AdvancedMicroeconomicTheory 53
• Seethediscussionsonthenextslide
ImplicationsofWARP:LawofDemand
• Can𝑥 𝑝P, 𝑤P lieonsegmentA?1) 𝑥 𝑝, 𝑤 and𝑥 𝑝P, 𝑤P arebothaffordableunder
𝐵7,9.2) 𝑥 𝑝, 𝑤 isaffordableunder𝐵7�,9�.
⇒WARPisviolatedif𝑥 𝑝P, 𝑤P liesonsegment A
• Can𝑥 𝑝P, 𝑤P lieonsegmentB?1) 𝑥 𝑝, 𝑤 isaffordableunder𝐵7,9,but𝑥 𝑝P, 𝑤P is
not.⇒ThepremiseofWARPdoesnothold⇒WARPisnotviolatedif𝑥 𝑝P, 𝑤P liesonsegment B
AdvancedMicroeconomicTheory 54
ImplicationsofWARP:LawofDemand
• Whatdidwelearnfromthisfigure?1) Westartedfrom𝛻𝑝C,andcompensatedthewealthof
thisindividual(reducingitfrom𝑤 to𝑤P)sothathecouldaffordhisinitialbundle𝑥 𝑝,𝑤 underthenewprices.§ Fromthiswealthcompensation,weobtainedbudgetline𝐵7�,9�.
2) FromWARP,weknowthat𝑥 𝑝P, 𝑤P mustcontainmoreofgood1.§ Thatis,graphically,segmentB liestotheright-handsideof
𝑥 𝑝, 𝑤 .3) Then,apricereduction,𝛻𝑝C,whenfollowedbyan
appropriatewealthcompensation,leadstoanincreaseinthequantitydemandedofthisgood,∆𝑥C.§ Thisisthecompensatedlawofdemand(CLD).
AdvancedMicroeconomicTheory 55
ImplicationsofWARP:LawofDemand
• Practiceproblem:– Canyourepeatthisanalysisbutforanincreaseinthepriceofgood1?§ First,pivotbudgetline𝐵7,9 inwardstoobtain𝐵7�,9.§ Then,increasethewealthlevel(wealthcompensation)toobtain𝐵7�,9� .
§ Identifythetwosegmentsinbudgetline𝐵7�,9� ,onetotheright-handsideof𝑥 𝑝,𝑤 andothertotheleft.
§ Inwhichsegmentofbudgetline𝐵7�,9� cantheWalrasiandemand𝑥(𝑝P, 𝑤P) lie?
AdvancedMicroeconomicTheory 56
ImplicationsofWARP:LawofDemand
• IsWARPsatisfiedundertheuncompensatedlawofdemand(ULD)?– 𝑥 𝑝, 𝑤 isaffordableunder
budgetline𝐵7,9,but𝑥(𝑝P, 𝑤) isnot.§ Hence,thepremiseofWARPis
notsatisfied.Asaresult,WARPisnotviolated.
– But,isthisresultimplyingsomethingaboutwhetherULDmusthold?§ No!§ AlthoughWARPisnotviolated,
ULDis:adecreaseinthepriceofgood1yieldsadecreaseinthequantitydemanded.
AdvancedMicroeconomicTheory 57
ImplicationsofWARP:LawofDemand
• Distinctionbetweentheuncompensatedandthecompensatedlawofdemand:– quantitydemandedandpricecanmoveinthesamedirection,whenwealthisleftuncompensated,i.e.,
∆𝑝C 0 ∆𝑥C > 0as∆𝑝C ⇒ ∆𝑥C
• Hence,WARPisnotsufficienttoyieldlawofdemandforpricechangesthatareuncompensated,i.e.,
WARP⇎ ULD,butWARP⇔ CLD
AdvancedMicroeconomicTheory 58
ImplicationsofWARP:SlutskyMatrix
• Letusfocusnowonthecaseinwhich𝑥 𝑝,𝑤 isdifferentiable.
• First,notethatthelawofdemandis𝑑𝑝 0 𝑑𝑥 ≤ 0(equivalentto∆𝑝 0 ∆𝑥 ≤ 0).
• Totallydifferentiating𝑥 𝑝,𝑤 ,𝑑𝑥 = 𝐷7𝑥 𝑝,𝑤 𝑑𝑝 + 𝐷9𝑥 𝑝,𝑤 𝑑𝑤
• Andsincetheconsumer’swealthiscompensated,𝑑𝑤 = 𝑥(𝑝,𝑤) 0 𝑑𝑝 (thisisthedifferentialanalogof∆𝑤 = ∆𝑝 0 𝑥(𝑝, 𝑤)).– Recallthat∆𝑤 = ∆𝑝 0 𝑥(𝑝, 𝑤) wasobtainedfromtheSlutskywealthcompensation.
AdvancedMicroeconomicTheory 59
ImplicationsofWARP:SlutskyMatrix
• Substituting,
𝑑𝑥 = 𝐷7𝑥 𝑝, 𝑤 𝑑𝑝 + 𝐷9𝑥 𝑝,𝑤 𝑥(𝑝, 𝑤) 0 𝑑𝑝r9
orequivalently,
𝑑𝑥 = 𝐷7𝑥 𝑝, 𝑤 + 𝐷9𝑥 𝑝,𝑤 𝑥(𝑝, 𝑤)� 𝑑𝑝
AdvancedMicroeconomicTheory 60
ImplicationsofWARP:SlutskyMatrix• Hencethelawofdemand,𝑑𝑝 0 𝑑𝑥 ≤ 0,canbeexpressedas
𝑑𝑝 0 𝐷7𝑥 𝑝,𝑤 + 𝐷9𝑥 𝑝,𝑤 𝑥(𝑝,𝑤)� 𝑑𝑝r%
≤ 0
wheretheterminbracketsistheSlutsky(orsubstitution)matrix
𝑆 𝑝,𝑤 =𝑠CC 𝑝, 𝑤 ⋯ 𝑠CA 𝑝, 𝑤
⋮ ⋱ ⋮𝑠AC 𝑝, 𝑤 ⋯ 𝑠AA 𝑝, 𝑤
whereeachelementinthematrixis
𝑠f[ 𝑝, 𝑤 =𝜕𝑥f(𝑝, 𝑤)𝜕𝑝[
+𝜕𝑥f 𝑝, 𝑤
𝜕𝑤 𝑥[(𝑝, 𝑤)
AdvancedMicroeconomicTheory 61
ImplicationsofWARP:SlutskyMatrix
• Proposition: If𝑥 𝑝,𝑤 isdifferentiable,satisfiesWalras’law,homog(0),andWARP,then𝑆 𝑝,𝑤isnegativesemi-definite,
𝑣 0 𝑆 𝑝, 𝑤 𝑣 ≤ 0 forany𝑣 ∈ ℝA
• Implications:– 𝑠f[ 𝑝, 𝑤 :substitutioneffectofgood𝑙 withrespecttoitsownpriceisnon-positive(own-priceeffect)
– Negativesemi-definitenessdoesnotimplythat𝑆 𝑝, 𝑤 issymmetric(exceptwhen𝐿 = 2).§ Usualconfusion:“then𝑆 𝑝,𝑤 isnotsymmetric”,NO!
AdvancedMicroeconomicTheory 62
ImplicationsofWARP:SlutskyMatrix
• Proposition:IfpreferencessatisfyLNSandstrictconvexity,andtheyarerepresentedwithacontinuousutilityfunction,thentheWalrasiandemand𝑥 𝑝,𝑤generatesaSlutskymatrix,𝑆 𝑝,𝑤 ,whichissymmetric.
• Theaboveassumptionsarereallycommon.– Hence,theSlutskymatrixwillthenbesymmetric.
• However,theaboveassumptionsarenotsatisfiedinthecaseofpreferencesoverperfectsubstitutes(i.e.,preferencesareconvex,butnotstrictlyconvex).
AdvancedMicroeconomicTheory 63
ImplicationsofWARP:SlutskyMatrix
• Non-positivesubstitutioneffect,𝑠ff ≤ 0:
𝑠ff 𝑝, 𝑤������������������( )
=𝜕𝑥f(𝑝, 𝑤)𝜕𝑝f
¡��¢£������: ���¢£¤��¥@ ¦�����¤��¥
+𝜕𝑥f 𝑝, 𝑤
𝜕𝑤 𝑥f(𝑝, 𝑤)§���¨�������:@ ��©¨¢£¤��¥ ����©��©¤��¥
• SubstitutionEffect=TotalEffect+IncomeEffect⇒ TotalEffect=SubstitutionEffect - IncomeEffect
AdvancedMicroeconomicTheory 64
ImplicationsofWARP:SlutskyEquation
𝑠ff 𝑝, 𝑤������������������( )
=𝜕𝑥f(𝑝, 𝑤)𝜕𝑝f
¡��¢£������
+𝜕𝑥f 𝑝, 𝑤
𝜕𝑤 𝑥f(𝑝, 𝑤)§���¨�������
• TotalEffect:measureshowthequantitydemandedisaffectedbyachangeinthepriceofgood𝑙,whenweleavethewealthuncompensated.
• IncomeEffect:measuresthechangeinthequantitydemandedasaresultofthewealthadjustment.
• SubstitutionEffect:measureshowthequantitydemandedisaffectedbyachangeinthepriceofgood𝑙,afterthewealthadjustment.– Thatis,thesubstitutioneffectonlycapturesthechangeindemandduetovariation
inthepriceratio,butabstractsfromthelarger(smaller)purchasingpowerthattheconsumerexperiencesafteradecrease(increase,respectively)inprices.
AdvancedMicroeconomicTheory 65
• Reductioninthepriceof𝑥C.– Itenlargesconsumer’ssetoffeasible
bundles.– Hecanreachanindifferencecurve
furtherawayfromtheorigin.
• TheWalrasiandemandcurveindicatesthatadecreaseinthepriceof𝑥C leadstoanincreaseinthequantitydemanded.– Thisinducesanegativelysloped
Walrasiandemandcurve(sothegoodis“normal”).
• Theincreaseinthequantitydemandedof𝑥C asaresultofadecreaseinitspricerepresentsthetotaleffect(TE).
AdvancedMicroeconomicTheory 66
ImplicationsofWARP:SlutskyEquationX2
X1
I2
I1
p1
X1
p1
A
B
a
b
• Reductioninthepriceof𝑥C.– Disentanglethetotaleffectintothe
substitutionandincomeeffects– Slutskywealthcompensation?
• Reducetheconsumer’swealthsothathecanaffordthesameconsumptionbundleastheonebeforethepricechange(i.e.,A).– Shiftthebudgetlineafterthepricechange
inwardsuntilit“crosses”throughtheinitialbundleA.
– “Constantpurchasingpower”demandcurve(CPPcurve)resultsfromapplyingtheSlutskywealthcompensation.
– Thequantitydemandedfor𝑥C increasesfrom𝑥C' to𝑥Cª.
• Whenwedonotholdtheconsumer’spurchasingpowerconstant,weobserverelativelylargeincreaseinthequantitydemandedfor𝑥C (i.e.,from𝑥C' to𝑥CB ).
AdvancedMicroeconomicTheory 67
ImplicationsofWARP:SlutskyEquationx2
x1
I2
I1
p1
x1
p1
I3
CPP demand
A
BC
a
bc
• Reductioninthepriceof𝑥C.– Hicksianwealthcompensation(i.e.,
“constantutility”demandcurve)?
• Theconsumer’swealthlevelisadjustedsothathecanstillreachhisinitialutilitylevel(i.e.,thesameindifferencecurve𝐼Casbeforethepricechange).– Amoresignificantwealthreductionthan
whenweapplytheSlutskywealthcompensation.
– TheHicksiandemandcurvereflectsthat,foragivendecreaseinp1,theconsumerslightlyincreaseshisconsumptionofgoodone.
• Insummary,agivendecreasein𝑝Cproduces:– AsmallincreaseintheHicksiandemandfor
thegood,i.e.,from𝑥C' to𝑥CC.– AlargerincreaseintheCPPdemandforthe
good,i.e.,from𝑥C' to𝑥Cª.– AsubstantialincreaseintheWalrasian
demandfortheproduct,i.e.,from𝑥C' to𝑥CB.AdvancedMicroeconomicTheory 68
ImplicationsofWARP:SlutskyEquationx2
x1
I2
I1
p1
x1
p1
I3
CPP demand
Hicksian demand
• Adecreaseinpriceof𝑥C leadstheconsumertoincreasehisconsumptionofthisgood,∆𝑥C,but:– The∆𝑥C whichissolelyduetothepriceeffect(eithermeasuredbytheHicksiandemandcurveortheCPPdemandcurve)issmallerthanthe∆𝑥C measuredbytheWalrasiandemand,𝑥 𝑝,𝑤 ,whichalsocaptureswealtheffects.
– Thewealthcompensation(areductionintheconsumer’swealthinthiscase)thatmaintainstheoriginalutilitylevel(asrequiredbytheHicksiandemand)islargerthanthewealthcompensationthatmaintainshispurchasingpowerunaltered(asrequiredbytheSlutskywealthcompensation,intheCPPcurve).
AdvancedMicroeconomicTheory 69
ImplicationsofWARP:SlutskyEquation
SubstitutionandIncomeEffects:NormalGoods
• Decreaseinthepriceofthegoodinthehorizontalaxis(i.e.,food).
• Thesubstitutioneffect(SE)movesintheoppositedirectionasthepricechange.– Areductioninthepriceof
foodimpliesapositivesubstitutioneffect.
• Theincomeeffect(IE)ispositive(thusitreinforcestheSE).– Thegoodisnormal.
AdvancedMicroeconomicTheory 70
Clothing
FoodSE IETE
A
B
C
I2
I1
BL2
BL1
BLd
SubstitutionandIncomeEffects:InferiorGoods
• Decreaseinthepriceofthegoodinthehorizontalaxis(i.e.,food).
• TheSEstillmovesintheoppositedirectionasthepricechange.
• Theincomeeffect(IE)isnownegative(whichpartiallyoffsetstheincreaseinthequantitydemandedassociatedwiththeSE).– Thegoodisinferior.
• Note:theSEislargerthantheIE.
AdvancedMicroeconomicTheory 71
I2
I1BL1 BLd
SubstitutionandIncomeEffects:GiffenGoods
• Decreaseinthepriceofthegoodinthehorizontalaxis(i.e.,food).
• TheSEstillmovesintheoppositedirectionasthepricechange.
• Theincomeeffect(IE)isstillnegativebutnowcompletelyoffsetstheincreaseinthequantitydemandedassociatedwiththeSE.– ThegoodisGiffengood.
• Note:theSEislessthantheIE.
AdvancedMicroeconomicTheory 72
I2
I1BL1
BL2
SubstitutionandIncomeEffectsSE IE TE
NormalGood + + +Inferior Good + - +GiffenGood + - -
AdvancedMicroeconomicTheory 73
• NotGiffen:Demandcurveisnegativelysloped(asusual)• Giffen:Demandcurveispositivelysloped
SubstitutionandIncomeEffects• Summary:
1) SEisnegative(since↓ 𝑝C⇒↑ 𝑥C)§ SE< 0 doesnotimply↓ 𝑥C
2) Ifgoodisinferior,IE< 0.Then,TE = SE⏟
− IE⏟
̄@
⇒ if IE >< SE ,then
TE(−)TE(+)
Forapricedecrease,thisimpliesTE(−)TE(+) ⇒ ↓ 𝑥C
↑ 𝑥CGiffengood
Non−Giffengood
3) Hence,a) Agoodcanbeinferior,butnotnecessarilybeGiffenb) ButallGiffengoodsmustbeinferior.
AdvancedMicroeconomicTheory 74
ExpenditureMinimizationProblem
AdvancedMicroeconomicTheory 75
ExpenditureMinimizationProblem
• Expenditureminimizationproblem(EMP):
min%&'
𝑝 0 𝑥
s.t. 𝑢(𝑥) ≥ 𝑢
• Alternativetoutilitymaximizationproblem
AdvancedMicroeconomicTheory 76
ExpenditureMinimizationProblem• Consumerseeksautilitylevel
associatedwithaparticularindifferencecurve,whilespendingaslittleaspossible.
• Bundlesstrictlyabove𝑥∗ cannotbeasolutiontotheEMP:– Theyreachtheutilitylevel𝑢– But,theydonotminimizetotal
expenditure
• Bundlesonthebudgetlinestrictlybelow𝑥∗ cannotbethesolutiontotheEMPproblem:– Theyarecheaperthan𝑥∗– But,theydonotreachthe
utilitylevel𝑢
AdvancedMicroeconomicTheory 77
X2
X1
*x
*UCS( )x
u
ExpenditureMinimizationProblem
• Lagrangian𝐿 = 𝑝 0 𝑥 + 𝜇 𝑢 − 𝑢(𝑥)
• FOCs(necessaryconditions)𝜕𝐿𝜕𝑥[
= 𝑝[ − 𝜇𝜕𝑢(𝑥∗)𝜕𝑥[
≤ 0
𝜕𝐿𝜕𝜇 = 𝑢 − 𝑢 𝑥∗ = 0
AdvancedMicroeconomicTheory 78
[=0forinteriorsolutions]
ExpenditureMinimizationProblem• Forinteriorsolutions,
𝑝[ = 𝜇 WY(%∗)
W%XorC
±=
^_(`∗)^`X7X
foranygood𝑘.Thisimplies,^_(`∗)^`X7X
=^_(`∗)^`a7a
or7X7a=
^_(`∗)^`X
^_(`∗)^`a
• Theconsumerallocateshisconsumptionacrossgoodsuntilthepointinwhichthemarginalutilityperdollarspentoneachgoodisequalacrossallgoods(i.e.,same“bangforthebuck”).
• Thatis,theslopeofindifferencecurveisequaltotheslopeofthebudgetline.
AdvancedMicroeconomicTheory 79
EMP:HicksianDemand
• Thebundle𝑥∗ ∈ argmin𝑝 0 𝑥 (theargumentthatsolvestheEMP)istheHicksiandemand,whichdependson𝑝 and𝑢,
𝑥∗ ∈ ℎ(𝑝, 𝑢)
• Recallthatifsuchbundle𝑥∗ isunique,wedenoteitas𝑥∗ = ℎ(𝑝, 𝑢).
AdvancedMicroeconomicTheory 80
PropertiesofHicksianDemand
• Supposethat𝑢(0) isacontinuousfunction,satisfyingLNSdefinedon𝑋 = ℝ@A .Thenfor𝑝 ≫0,ℎ(𝑝, 𝑢) satisfies:1) Homog(0) in𝑝,i.e.,ℎ 𝑝, 𝑢 = ℎ(𝛼𝑝, 𝑢) forany𝑝,𝑢,
and𝛼 > 0.§ If𝑥∗ ∈ ℎ(𝑝, 𝑢) isasolutiontotheproblem
min%&'
𝑝 0 𝑥thenitisalsoasolutiontotheproblem
min%&'
𝛼𝑝 0 𝑥§ Intuition:acommonchangeinallpricesdoesnotalterthe
slopeoftheconsumer’sbudgetline.
AdvancedMicroeconomicTheory 81
PropertiesofHicksianDemand• 𝑥∗ isasolutiontotheEMPwhen
thepricevectoris𝑝 = (𝑝C, 𝑝B).• Increaseallpricesbyfactor𝛼
𝑝P = (𝑝CP , 𝑝BP ) = (𝛼𝑝C, 𝛼𝑝B)• Downward(parallel)shiftinthe
budgetline,i.e.,theslopeofthebudgetlineisunchanged.
• ButIhavetoreachutilitylevel𝑢 tosatisfytheconstraintoftheEMP!
• Spendmoretobuybundle𝑥∗(𝑥C∗, 𝑥B∗),i.e.,𝑝CP𝑥C∗ + 𝑝BP 𝑥B∗ > 𝑝C𝑥C∗ + 𝑝B𝑥B∗
• Hence,ℎ 𝑝, 𝑢 = ℎ(𝛼𝑝, 𝑢)
AdvancedMicroeconomicTheory 82
x2
x1
PropertiesofHicksianDemand
2) Noexcessutility:foranyoptimalconsumptionbundle𝑥 ∈ ℎ 𝑝, 𝑢 ,utilitylevelsatisfies𝑢 𝑥 = 𝑢.
AdvancedMicroeconomicTheory 83
x1
x2
wp2
wp1
x’
x∈h(p,u)
uu1
PropertiesofHicksianDemand• Intuition:Supposethereexistsabundle𝑥 ∈ ℎ 𝑝, 𝑢 forwhichthe
consumerobtainsautilitylevel𝑢 𝑥 = 𝑢C > 𝑢,whichishigherthantheutilitylevel𝑢 hemustreachwhensolvingEMP.
• Butwecanthenfindanotherbundle𝑥P = 𝑥𝛼,where𝛼 ∈ (0,1),verycloseto𝑥 (𝛼 → 1),forwhich𝑢(𝑥P) > 𝑢.
• Bundle𝑥P:– ischeaperthan𝑥 sinceitcontainsfewerunitsofallgoods;and– exceedstheminimalutilitylevel𝑢 thattheconsumermustreachinhis
EMP.• Wecanrepeatthatargumentuntilreachingbundle𝑥.
• Insummary,foragivenutilitylevel𝑢 thatyouseektoreachintheEMP,bundleℎ 𝑝, 𝑢 doesnotexceed𝑢.Otherwiseyoucanfindacheaperbundlethatexactlyreaches𝑢.
AdvancedMicroeconomicTheory 84
PropertiesofHicksianDemand
3) Convexity:Ifthepreferencerelationisconvex,thenℎ 𝑝, 𝑢 isaconvexset.
AdvancedMicroeconomicTheory 85
PropertiesofHicksianDemand
4) Uniqueness:Ifthepreferencerelationisstrictlyconvex,thenℎ 𝑝, 𝑢containsasingleelement.
AdvancedMicroeconomicTheory 86
PropertiesofHicksianDemand
• CompensatedLawofDemand:foranychangeinprices𝑝 and𝑝P,
(𝑝P−𝑝) 0 ℎ 𝑝P, 𝑢 − ℎ 𝑝, 𝑢 ≤ 0– Implication:foreverygood𝑘,
(𝑝[P − 𝑝[) 0 ℎ[ 𝑝P, 𝑢 − ℎ[ 𝑝, 𝑢 ≤ 0– Thisistrueforcompensateddemand,butnotnecessarilytrueforWalrasiandemand(whichisuncompensated):• RecallthefiguresonGiffengoods,whereadecreasein𝑝[ infactdecreases𝑥[ 𝑝, 𝑢 whenwealthwasleftuncompensated.
AdvancedMicroeconomicTheory 87
TheExpenditureFunction
• PluggingtheresultfromtheEMP,ℎ 𝑝, 𝑢 ,intotheobjectivefunction,𝑝 0 𝑥,weobtainthevaluefunctionofthisoptimizationproblem,
𝑝 0 ℎ 𝑝, 𝑢 = 𝑒(𝑝, 𝑢)
where𝑒(𝑝, 𝑢) representstheminimalexpenditure thattheconsumerneedstoincurinordertoreachutilitylevel𝑢 whenpricesare𝑝.
AdvancedMicroeconomicTheory 88
PropertiesofExpenditureFunction• Supposethat𝑢(0) isacontinuousfunction,satisfyingLNSdefinedon𝑋 = ℝ@A .Thenfor𝑝 ≫ 0,𝑒(𝑝, 𝑢)satisfies:
1) Homog(1) in𝑝,𝑒 𝛼𝑝, 𝑢 = 𝛼 𝑝 0 𝑥∗
µ(7,Y)= 𝛼 0 𝑒(𝑝, 𝑢)
forany𝑝,𝑢,and𝛼 > 0.
§ Weknowthattheoptimalbundleisnotchangedwhenallpriceschange,sincetheoptimalconsumptionbundleinℎ(𝑝, 𝑢) satisfieshomogeneityofdegreezero.
§ Suchapricechangejustmakesitmoreorlessexpensivetobuythesamebundle.
AdvancedMicroeconomicTheory 89
PropertiesofExpenditureFunction2) Strictlyincreasingin𝒖:
Foragivenpricevector,reachingahigherutilityrequireshigherexpenditure:
𝑝C𝑥CP + 𝑝B𝑥BP > 𝑝C𝑥C + 𝑝B𝑥B
where(𝑥C, 𝑥B) = ℎ(𝑝, 𝑢)and(𝑥CP , 𝑥BP ) = ℎ(𝑝, 𝑢P).
Then,𝑒 𝑝, 𝑢P > 𝑒 𝑝, 𝑢
AdvancedMicroeconomicTheory 90
x2
x1
x2
x1
PropertiesofExpenditureFunction3) Non-decreasingin𝒑𝒌 foreverygood𝒌:
Higherpricesmeanhigherexpendituretoreachagivenutilitylevel.• Let𝑝P = (𝑝C, 𝑝B, … , 𝑝[P , … , 𝑝A) and𝑝 =(𝑝C, 𝑝B, … , 𝑝[, … , 𝑝A),where𝑝[P > 𝑝[.
• Let𝑥P = ℎ 𝑝P, 𝑢 and𝑥 = ℎ 𝑝, 𝑢 fromEMPunderprices𝑝P and𝑝,respectively.
• Then,𝑝P 0 𝑥P = 𝑒(𝑝P, 𝑢) and𝑝 0 𝑥 = 𝑒(𝑝, 𝑢).
𝑒 𝑝P, 𝑢 = 𝑝P 0 𝑥P ≥ 𝑝 0 𝑥P≥ 𝑝 0 𝑥 = 𝑒(𝑝, 𝑢)
– 1st inequalitydueto𝑝P ≥ 𝑝– 2nd inequality:atprices𝑝,bundle𝑥minimizesEMP.
AdvancedMicroeconomicTheory 91
PropertiesofExpenditureFunction4) Concavein𝒑:
Let𝑥P ∈ ℎ 𝑝P, 𝑢 ⇒ 𝑝P𝑥P ≤ 𝑝P𝑥 ∀𝑥 ≠ 𝑥P,e.g.,𝑝P𝑥P ≤ 𝑝P�̅�and𝑥PP ∈ ℎ 𝑝PP, 𝑢 ⇒ 𝑝PP𝑥PP ≤ 𝑝PP𝑥 ∀𝑥 ≠ 𝑥PP,e.g.,𝑝PP𝑥PP ≤ 𝑝PP�̅�where�̅� = 𝛼𝑥P + 1 − 𝛼 𝑥′′
Thisimplies𝛼𝑝P𝑥P + 1 − 𝛼 𝑝PP𝑥PP ≤ 𝛼𝑝P�̅� + 1 − 𝛼 𝑝PP�̅�
𝛼 𝑝P𝑥Pºµ(7�,Y)
+ 1 − 𝛼 𝑝PP𝑥PPµ(7��,Y)
≤ [𝛼𝑝P + 1 − 𝛼 𝑝PP7̅
]�̅�
𝛼𝑒(𝑝P, 𝑢) + 1 − 𝛼 𝑒(𝑝PP, 𝑢) ≤ 𝑒(�̅�, 𝑢)
asrequiredbyconcavity
AdvancedMicroeconomicTheory 92
x2
x11/ 'w p
2/w p
2/ 'w p
2/ ''w p
x’’
x’x
u
1/w p1/ ''w p
Connections
AdvancedMicroeconomicTheory 93
RelationshipbetweentheExpenditureandHicksianDemand
• Let’sassumethat𝑢(0) isacontinuousfunction,representingpreferencesthatsatisfyLNSandarestrictlyconvexanddefinedon𝑋 = ℝ@A .Forall𝑝 and𝑢,
Wµ 7,YW7X
= ℎ[ 𝑝, 𝑢 foreverygood𝑘
Thisidentityis“Shepard’slemma”:ifwewanttofindℎ[ 𝑝, 𝑢 andweknow𝑒 𝑝, 𝑢 ,wejusthavetodifferentiate𝑒 𝑝, 𝑢 withrespecttoprices.
• Proof:threedifferentapproaches1) thesupportfunction2) first-orderconditions3) theenvelopetheorem(SeeAppendix2.2)
AdvancedMicroeconomicTheory 94
RelationshipbetweentheExpenditureandHicksianDemand
• TherelationshipbetweentheHicksiandemandandtheexpenditurefunctioncanbefurtherdevelopedbytakingfirstorderconditionsagain.Thatis,
𝜕B𝑒(𝑝, 𝑢)𝜕𝑝[B
=𝜕ℎ[(𝑝, 𝑢)𝜕𝑝[
or𝐷7B𝑒 𝑝, 𝑢 = 𝐷7ℎ(𝑝, 𝑢)
• Since𝐷7ℎ(𝑝, 𝑢) providestheSlutskymatrix,𝑆(𝑝, 𝑤),then
𝑆 𝑝,𝑤 = 𝐷7B𝑒 𝑝, 𝑢
wheretheSlutskymatrixcanbeobtainedfromtheobservableWalrasiandemand.
AdvancedMicroeconomicTheory 95
RelationshipbetweentheExpenditureandHicksianDemand
• Therearethreeotherimportantpropertiesof𝐷7ℎ(𝑝, 𝑢),where𝐷7ℎ(𝑝, 𝑢) is𝐿×𝐿 derivativematrixofthehicksiandemand,ℎ 𝑝, 𝑢 :1) 𝐷7ℎ(𝑝, 𝑢) isnegativesemidefinite
§ Hence,𝑆 𝑝,𝑤 isnegativesemidefinite.2) 𝐷7ℎ(𝑝, 𝑢) isasymmetricmatrix
§ Hence,𝑆 𝑝,𝑤 issymmetric.3) 𝐷7ℎ 𝑝, 𝑢 𝑝 = 0,whichimplies𝑆 𝑝, 𝑤 𝑝 = 0.
§ Notallgoodscanbenetsubstitutes W¾a(7,Y)W7X
> 0 ornetcomplements W¾a(7,Y)
W7X< 0 .Otherwise,themultiplication
ofthisvectorofderivativestimesthe(positive)pricevector𝑝 ≫ 0 wouldyieldanon-zeroresult.
AdvancedMicroeconomicTheory 96
RelationshipbetweenHicksianandWalrasianDemand
• Whenincomeeffectsarepositive(normalgoods),thentheWalrasiandemand𝑥(𝑝, 𝑤) isabove theHicksiandemandℎ(𝑝, 𝑢) .– TheHicksiandemandissteeper thantheWalrasiandemand.
AdvancedMicroeconomicTheory 97
x1
x2
wp2
wp1
x1
p1
wp1’
p1’
p1
x*1 x11 x21
h(p,ū)
x(p,w)
∇p1
I1I3
I2x1x3
x2
x*
SE IE
RelationshipbetweenHicksianandWalrasianDemand
• Whenincomeeffectsarenegative(inferiorgoods),thentheWalrasiandemand𝑥(𝑝, 𝑤) isbelowtheHicksiandemandℎ(𝑝, 𝑢) .– TheHicksiandemandisflatter thantheWalrasiandemand.
AdvancedMicroeconomicTheory 98
x1
x2
wp2
wp1
x1
p1
wp1’
p1’
p1
x*1 x11x21
h(p,ū)
x(p,w)
∇p1
I1
I2x1
x2x*
SE
IE
a
bc
RelationshipbetweenHicksianandWalrasianDemand
• WecanformallyrelatetheHicksianandWalrasiandemandasfollows:– Consider𝑢(0) isacontinuousfunction,representingpreferencesthatsatisfyLNSandarestrictlyconvexanddefinedon𝑋 = ℝ@A .
– Consideraconsumerfacing(�̅�, 𝑤¿) andattainingutilitylevel𝑢�.
– Notethat𝑤¿ = 𝑒(�̅�, 𝑢�).Inaddition,weknowthatforany(𝑝, 𝑢),ℎf 𝑝, 𝑢 = 𝑥f(𝑝, 𝑒 𝑝, 𝑢
9).Differentiatingthis
expressionwithrespectto𝑝[,andevaluatingitat(�̅�, 𝑢�),weget:
𝜕ℎf(�̅�, 𝑢�)𝜕𝑝[
=𝜕𝑥f(�̅�, 𝑒(�̅�, 𝑢�))
𝜕𝑝[+𝜕𝑥f(�̅�, 𝑒(�̅�, 𝑢�))
𝜕𝑤𝜕𝑒(�̅�, 𝑢�)𝜕𝑝[
AdvancedMicroeconomicTheory 99
RelationshipbetweenHicksianandWalrasianDemand
• UsingthefactthatWµ(7̅,Y¿)W7X
= ℎ[(�̅�, 𝑢�),W¾a(7̅,Y¿)W7X
= W%a(7̅,µ(7̅,Y¿))W7X
+ W%a(7̅,µ(7̅,Y¿))W9
ℎ[(�̅�, 𝑢�)
• Finally,since𝑤¿ = 𝑒(�̅�, 𝑢�) andℎ[ �̅�, 𝑢� =𝑥[ �̅�, 𝑒 �̅�, 𝑢� = 𝑥[ �̅�, 𝑤¿ ,then
W¾a(7̅,Y¿)W7X
= W%a(7̅,9¿)W7X
+ W%a(7̅,9¿)W9
𝑥[(�̅�, 𝑤¿)
AdvancedMicroeconomicTheory 100
RelationshipbetweenHicksianandWalrasianDemand
• Butthiscoincideswith𝑠f[ 𝑝, 𝑤 thatwediscussedintheSlutskyequation.– Hence,wehave𝐷7ℎ 𝑝, 𝑢
A×A
= 𝑆 𝑝,𝑤 .
– Or,morecompactly,𝑆𝐸 = 𝑇𝐸 + 𝐼𝐸.
AdvancedMicroeconomicTheory 101
RelationshipbetweenWalrasianDemandandIndirectUtilityFunction• Let’sassumethat𝑢(0) isacontinuousfunction,
representingpreferencesthatsatisfyLNSandarestrictlyconvexanddefinedon𝑋 = ℝ@A . Supposealsothat𝑣(𝑝, 𝑤) isdifferentiableatany(𝑝, 𝑤) ≫ 0.Then,
−^Â Ã,Ä^ÃX
^Â Ã,Ä^Ä
= 𝑥[(𝑝, 𝑤) foreverygood𝑘
• ThisisRoy’sidentity.• Powerfulresult,sinceinmanycasesitiseasierto
computethederivativesof𝑣 𝑝, 𝑤 thansolvingtheUMPwiththesystemofFOCs.
AdvancedMicroeconomicTheory 102
SummaryofRelationships
• TheWalrasiandemand,𝑥 𝑝, 𝑤 ,isthesolutionoftheUMP.– Itsvaluefunctionistheindirectutilityfunction,𝑣 𝑝,𝑤 .
• TheHicksiandemand,ℎ(𝑝, 𝑢),isthesolutionoftheEMP.– Itsvaluefunctionistheexpenditurefunction,𝑒(𝑝, 𝑢).
AdvancedMicroeconomicTheory 103
SummaryofRelationships
AdvancedMicroeconomicTheory 104
x(p,w)
v(p,w) e(p,u)
h(p,u)
The UMP The EMP
(1) (1)
SummaryofRelationships
• RelationshipbetweenthevaluefunctionsoftheUMPandtheEMP(lowerpartoffigure):– 𝑒 𝑝, 𝑣 𝑝, 𝑤 = 𝑤,i.e.,theminimalexpenditureneededinordertoreachautilitylevelequaltothemaximalutilitythattheindividualreachesathisUMP,𝑢 = 𝑣 𝑝,𝑤 ,mustbe𝑤.
– 𝑣 𝑝, 𝑒(𝑝, 𝑢) = 𝑢,i.e.,theindirectutilitythatcanbereachedwhentheconsumerisendowedwithawealthlevelw equaltotheminimalexpenditureheoptimallyusesintheEMP,i.e., 𝑤 = 𝑒(𝑝, 𝑢),isexactly𝑢.
AdvancedMicroeconomicTheory 105
SummaryofRelationships
AdvancedMicroeconomicTheory 106
x(p,w)
v(p,w) e(p,u)
h(p,u)
The UMP The EMP
e(p, v(p,w))=wv(p, e(p,w))=u
(1) (1)
(2)
SummaryofRelationships
• Relationshipbetweentheargmax oftheUMP(theWalrasiandemand)andtheargmin oftheEMP(theHicksiandemand):– 𝑥 𝑝, 𝑒(𝑝, 𝑢) = ℎ 𝑝, 𝑢 ,i.e.,the(uncompensated)Walrasiandemandofaconsumerendowedwithanadjustedwealthlevel𝑤 (equaltotheexpenditureheoptimallyusesintheEMP), 𝑤 = 𝑒 𝑝, 𝑢 ,coincideswithhisHicksiandemand,ℎ 𝑝, 𝑢 .
– ℎ 𝑝, 𝑣(𝑝, 𝑤) = 𝑥 𝑝,𝑤 ,i.e.,the(compensated)HicksiandemandofaconsumerreachingthemaximumutilityoftheUMP,𝑢 = 𝑣(𝑝, 𝑤),coincideswithhisWalrasiandemand,𝑥 𝑝, 𝑤 .
AdvancedMicroeconomicTheory 107
SummaryofRelationships
AdvancedMicroeconomicTheory 108
x(p,w)
v(p,w) e(p,u)
h(p,u)
The UMP The EMP
e(p, v(p,w))=wv(p, e(p,w))=u
(1) (1)
(2)
3(a) 3(b)
SummaryofRelationships• Finally,wecanalsouse:
– TheSlutskyequation:𝜕ℎf(𝑝, 𝑢)𝜕𝑝[
=𝜕𝑥f(𝑝, 𝑤)𝜕𝑝[
+𝜕𝑥f(𝑝, 𝑤)
𝜕𝑤 𝑥[(𝑝, 𝑤)
torelatethederivativesoftheHicksianandtheWalrasiandemand.– Shepard’slemma:
𝜕𝑒 𝑝, 𝑢𝜕𝑝[
= ℎ[ 𝑝, 𝑢
toobtaintheHicksiandemandfromtheexpenditurefunction.– Roy’sidentity:
−
𝜕𝑣 𝑝,𝑤𝜕𝑝[
𝜕𝑣 𝑝, 𝑤𝜕𝑤
= 𝑥[(𝑝, 𝑤)
toobtaintheWalrasiandemandfromtheindirectutilityfunction.
AdvancedMicroeconomicTheory 109
SummaryofRelationships
AdvancedMicroeconomicTheory 110
x(p,w)
v(p,w) e(p,u)
h(p,u)
The UMP The EMP
e(p, v(p,w))=wv(p, e(p,w))=u
(1) (1)
(2)
3(a) 3(b)
4(a) Slutsky equation(Using derivatives)
SummaryofRelationships
AdvancedMicroeconomicTheory 111
x(p,w)
v(p,w) e(p,u)
h(p,u)
The UMP The EMP
e(p, v(p,w))=wv(p, e(p,w))=u
(1) (1)
(2)
3(a) 3(b)
4(a) Slutsky equation(Using derivatives)
4(b) Roy’s
Identity 4(c)
Appendix2.1:DualityinConsumption
AdvancedMicroeconomicTheory 112
DualityinConsumption
• Wediscussedtheutilitymaximizationproblem(UMP)– theso-calledprimal problemdescribingtheconsumer’schoiceofoptimalconsumptionbundles– anditsdual:theexpenditureminimizationproblem(EMP).
• Whencanweguaranteethatthesolution𝑥∗ tobothproblemscoincide?
AdvancedMicroeconomicTheory 113
DualityinConsumption• Themaximaldistancethata
turtlecantravelin𝑡∗ timeis𝑥∗.– Fromtime(𝑡∗) todistance(𝑥∗)
• Theminimaltimethataturtleneedstotravel𝑥∗ distanceis𝑡∗.– Fromdistance(𝑥∗) totime(𝑡∗)
• Inthiscasetheprimalanddualproblemswouldprovideuswiththesameanswertobothoftheabovequestions
AdvancedMicroeconomicTheory 114
DualityinConsumption• Inordertoobtainthesame
answerfrombothquestions,wecriticallyneedthatthefunction𝑀(𝑡)satisfiesmonotonicity.
• Otherwise:themaximaldistancetraveledatboth𝑡Cand𝑡B is𝑥∗,buttheminimaltimethataturtleneedstotravel𝑥∗ distanceis𝑡C.
• Hence,anon-monotonicfunctioncannotguaranteethattheanswersfrombothquestionsarecompatible.
AdvancedMicroeconomicTheory 115
DualityinConsumption• Similarly,wealsorequire
thatthefunction𝑀(𝑡)satisfiescontinuity.
• Otherwise:themaximaldistancethattheturtlecantravelintime𝑡∗ is𝑥B ,whereastheminimumtimerequiredtotraveldistance𝑥C and𝑥B is𝑡∗ forbothdistances.
• Hence,anon-continuousfunctiondoesnotguaranteethattheanswerstobothquestionsarecompatible.
AdvancedMicroeconomicTheory 116
Distance
Time,t
M(t)
x2
t*
Jumpingturtlex1
DualityinConsumption
• Given𝑢(0) ismonotonicandcontinuous,thenif𝑥∗ isthesolutiontotheproblem
max%&'
𝑢(𝑥) s.t. 𝑝 0 𝑥 ≤ 𝑤 (UMP)
itmustalsobeasolutiontotheproblemmin%&'
𝑝 0 𝑥 s.t. 𝑢 𝑥 ≥ 𝑢 (EMP)
AdvancedMicroeconomicTheory 117
Hyperplane Theorem
• Hyperplane:forsome𝑝 ∈ ℝ@A and𝑐 ∈ ℝ,thesetofpointsinℝAsuchthat𝑥 ∈ ℝA: 𝑝 0 𝑥 = 𝑐
• Half-space:thesetofbundles𝑥 forwhich𝑝 0𝑥 ≥ 𝑐.Thatis,𝑥 ∈ ℝA: 𝑝 0 𝑥 ≥ 𝑐
AdvancedMicroeconomicTheory 118
Half-space
Hyperplane
Hyperplane Theorem• Separatinghyperplane theorem:Foreveryconvexandclosedset𝐾,thereisahalf-spacecontaining𝐾 andexcludinganypoint �̅� ∉ 𝐾outsideofthisset.– Thatis,thereexist𝑝 ∈ ℝ@A and𝑐 ∈ ℝ suchthat
𝑝 0 𝑥 ≥ 𝑐 forallelementsintheset,𝑥 ∈ 𝐾𝑝 0 �̅� < 𝑐 forallelementsoutsidetheset, �̅� ∉ 𝐾
– Intuition:everyconvexandclosedset𝐾 canbeequivalentlydescribedastheintersectionofthehalf-spacesthatcontainit.§ Aswedrawmoreandmorehalfspaces,theirintersectionbecomestheset𝐾.
AdvancedMicroeconomicTheory 119
Hyperplane Theorem• If𝐾 isaclosedandconvex
set,wecanthenconstructhalf-spacesforalltheelementsintheset(𝑥C, 𝑥B, … ) suchthattheirintersectionscoincides(“equivalentlydescribes”)set𝐾.– Weconstructacage(or
hull)aroundtheconvexset𝐾 thatexactlycoincideswithset𝐾.
• Bundlelike�̅� ∉ 𝐾 liesoutsidetheintersectionofhalf-spaces.
AdvancedMicroeconomicTheory 120
K
x
Hyperplane Theorem
• Whatifthesetwearetryingto“equivalentlydescribe”bytheuseofhalf-spacesisnon-convex?– Theintersectionofhalf-spacesdoesnotcoincidewithset𝐾 (itis,infact,larger,sinceitincludespointsoutsideset𝐾).Hence,wecannotuseseveralhalf-spacesto“equivalentlydescribe”set𝐾.
– Thentheintersectionofhalf-spacesthatcontain𝐾 isthesmallest,convexsetthatcontains𝐾,knownastheclosed,convexhullof𝐾.
AdvancedMicroeconomicTheory 121
Hyperplane Theorem
• Theconvexhullofset𝐾 isoftendenotedas𝐾¿,anditisconvex(unlikeset𝐾,whichmightnotbeconvex).
• Theconvexhull𝐾¿ isconvex,bothwhensetK isconvexandwhenitisnot.
AdvancedMicroeconomicTheory 122
K
Shadedareaistheconvexhull
ofsetK
SupportFunction• Foreverynonemptyclosedset𝐾 ⊂ ℝA,itssupportfunctionisdefinedby
𝜇É 𝑝 = inf% 𝑝 0 𝑥 forall𝑥 ∈ 𝐾 and𝑝 ∈ ℝA
thatis,thesupportfunctionfinds,foragivenpricevector𝑝,thebundle𝑥 thatminimizes𝑝 0 𝑥.– Recallthatinf coincideswiththeargmin whentheconstraintincludestheboundary.
• Fromthesupportfunctionof𝐾,wecanreconstruct𝐾.– Inparticular,forevery𝑝 ∈ ℝA ,wecandefinehalf-spaceswhoseboundaryisthesupportfunctionofset𝐾.§ Thatis,wedefinethesetofbundlesforwhich𝑝 0 𝑥 ≥ 𝜇É 𝑝 .Notethatallbundles𝑥 insuchhalf-spacecontainselementsintheset𝐾,butdoesnotcontainelementsoutside𝐾,i.e.,�̅� ∉ 𝐾 .
AdvancedMicroeconomicTheory 123
SupportFunction• Thus,theintersectionofthehalf-spacesgeneratedbyallpossiblevaluesof𝑝 describes(“reconstructs”)theset𝐾.Thatis,set𝐾 canbedescribedbyallthosebundles𝑥 ∈ ℝA suchthat
𝐾 = 𝑥 ∈ ℝA : 𝑝 0 𝑥 ≥ 𝜇É 𝑝 forevery𝑝
• Bythesamelogic,if𝐾 isnotconvex,thentheset
𝑥 ∈ ℝA : 𝑝 0 𝑥 ≥ 𝜇É 𝑝 forevery𝑝
definesthesmallestclosed,convexsetcontaining𝐾(i.e.,theconvexhullofset𝐾).
AdvancedMicroeconomicTheory 124
SupportFunction• Foragiven𝑝,thesupport𝜇É 𝑝 selectstheelementin𝐾 thatminimizes𝑝 0 𝑥 (i.e.,𝑥C inthisexample).
• Then,wecandefinethehalf-spaceofthathyperlaneas:
𝑝 0 𝑥 ≥ 𝑝 0 𝑥C±Ê 7
– Theaboveinequalityidentifiesallbundles𝑥 totheleftofhyperplane 𝑝 0 𝑥C.
• Wecanrepeatthesameprocedureforanyotherpricevector𝑝.
AdvancedMicroeconomicTheory 125
SupportFunction• Theabovedefinitionofthesupportfunctionprovidesuswithausefuldualitytheorem:– Let𝐾 beanonempty,closedset,andlet𝜇É 0 beitssupportfunction.Thenthereisauniqueelementin𝐾,�̅� ∈𝐾,suchthat,forallpricevector�̅�,�̅� 0 �̅� = 𝜇É �̅� ⇔ 𝜇É 0 isdifferentiableat�̅�𝑝 0 ℎ 𝑝, 𝑢 = 𝑒(𝑝, 𝑢) ⇔ 𝑒(0, 𝑢) isdifferentiableat𝑝
– Moreover,inthiscase,suchderivativeis𝜕(�̅� 0 �̅�)𝜕𝑝 = �̅�
orinmatrixnotation𝛻𝜇É �̅� = �̅�.
AdvancedMicroeconomicTheory 126
Appendix2.2:Relationshipbetweenthe
ExpenditureFunctionandHicksianDemand
AdvancedMicroeconomicTheory 127
ProofI(UsingDualityTheorem)• Theexpenditurefunctionisthesupportfunctionforthesetofallbundles
inℝ@A forwhichutilityreachesatleastalevelof𝑢.Thatis,
𝑥 ∈ ℝ@A : 𝑢(𝑥) ≥ 𝑢
UsingtheDualitytheorem,wecanthenstatethatthereisauniquebundleinthisset,ℎ 𝑝, 𝑢 ,suchthat
𝑝 0 ℎ 𝑝, 𝑢 = 𝑒 𝑝, 𝑢
wheretheright-handsideisthesupportfunctionofthisproblem.
• Moreover,fromthedualitytheorem,thederivativeofthesupportfunctioncoincideswiththisuniquebundle,i.e.,
Wµ 7,YW7X
= ℎ[ 𝑝, 𝑢 foreverygood𝑘or
𝛻7𝑒 𝑝, 𝑢 = ℎ 𝑝, 𝑢
AdvancedMicroeconomicTheory 128
ProofI(UsingDualityTheorem)
• UCSisaconvexandclosedset.
• Hyperplane 𝑝 0 𝑥∗ =𝑒 𝑝, 𝑢 providesuswiththeminimalexpenditurethatstillreachesutilitylevel𝑢.
AdvancedMicroeconomicTheory 129
ProofII(UsingFirstOrderConditions)
• Totallydifferentiatingtheexpenditurefunction𝑒 𝑝, 𝑢 ,
𝛻7𝑒 𝑝, 𝑢 = 𝛻7 𝑝 0 ℎ(𝑝, 𝑢) = ℎ 𝑝, 𝑢 + 𝑝 0 𝐷7ℎ(𝑝, 𝑢)�
• And,fromtheFOCsininteriorsolutionsoftheEMP,weknowthat𝑝 = 𝜆𝛻𝑢(ℎ 𝑝, 𝑢 ).Substituting,
𝛻7𝑒 𝑝, 𝑢 = ℎ 𝑝, 𝑢 + 𝜆[𝛻𝑢(ℎ 𝑝, 𝑢 ) 0 𝐷7ℎ(𝑝, 𝑢)]�
• But,since𝛻𝑢 ℎ 𝑝, 𝑢 = 𝑢 forallsolutionsoftheEMP,then𝛻𝑢 ℎ 𝑝, 𝑢 = 0,whichimplies
𝛻7𝑒 𝑝, 𝑢 = ℎ 𝑝, 𝑢
Thatis,Wµ 7,YW7X
= ℎ[ 𝑝, 𝑢 foreverygood𝑘.AdvancedMicroeconomicTheory 130
ProofIII(UsingtheEnvelopeTheorem)
• Usingtheenvelopetheoremintheexpenditurefunction,weobtain
𝜕𝑒 𝑝, 𝑢𝜕𝑝[
=𝜕 𝑝 0 ℎ 𝑝, 𝑢
𝜕𝑝[= ℎ 𝑝, 𝑢 + 𝑝
𝜕ℎ 𝑝, 𝑢𝜕𝑝[
• And,sincetheHicksiandemandisalreadyattheoptimum,indirecteffectsarenegligible,W¾ 7,Y
W7X= 0,
implyingWµ 7,YW7X
= ℎ 𝑝, 𝑢
AdvancedMicroeconomicTheory 131
Appendix2.3:GeneralizedAxiomofRevealed
Preference(GARP)
AdvancedMicroeconomicTheory 132
GARP• Considerasequenceofprices𝑝Ë where𝑡 =1,2, … , 𝑇 withanassociatedsequenceofchosenbundles𝑥Ë.
• GARP:Ifbundle𝑥Ë isrevealedpreferredto𝑥Ë@Cforall𝑡 = 1,2, … , 𝑇, i.e.,𝑥C ≿ 𝑥B, 𝑥B ≿𝑥ª, … , 𝑥� C ≿ 𝑥�, then𝑥� isnot strictlyrevealedpreferredto𝑥C,i.e.,𝑥� ⊁ 𝑥C.– Moregeneralaxiomofrevealedpreference– NeitherGARPnorWARPimpliesoneanother– SomechoicessatisfyGARP,someWARP,choicesforwhichbothaxiomshold,andsomeforwhichnonedo.
AdvancedMicroeconomicTheory 133
GARP
AdvancedMicroeconomicTheory 134
• ChoicessatisfyingGARP,WARP,both,ornone
GARP
• ExampleA2.1 (GARPholds,butWARPdoesnot):– Considerthefollowingsequenceofpricevectorsandtheircorrespondingdemandedbundles
𝑝C = (1,1,2) 𝑥C = (1,0,0)𝑝B = (1,2,1) 𝑥B = (0,1,0)𝑝ª = (1,3,1) 𝑥ª = (0,0,2)
– Thechangefrom𝑡 = 1 to𝑡 = 2 violatesWARP,butnotGARP.
AdvancedMicroeconomicTheory 135
GARP
• ExampleA2.1 (continued):– Letuscomparebundles𝑥C with𝑥B.– ForthepremiseofWARPtohold:
§ Sincebundle𝑥C isrevealedpreferredto𝑥B in𝑡 = 1,i.e.,𝑥C ≿ 𝑥B, itmustbethat𝑝C ⋅ 𝑥B ≤ 𝑤C = 𝑝C ⋅ 𝑥C.
§ Thatis,bundle𝑥B isaffordableunderbundle𝑥C’sprices.
– Substitutingourvalues,wefindthat1×0 + 1×1 + 2×0 = 1 ≤ 1= 1×1 + 1×0 + 2×0
– Hence,thepremiseofWARPholds.AdvancedMicroeconomicTheory 136
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• ExampleA2.1 (continued):– ForWARPtobesatisfied:
§ Wemustalsohavethat𝑝B ⋅ 𝑥C > 𝑤B = 𝑝B ⋅ 𝑥B.§ Thatis,bundle𝑥C isunaffordableunderthenewpricesandwealth.
– Pluggingourvaluesyields𝑝B ⋅ 𝑥C = 1×1 + 2×0 + 1×0 = 1𝑝B ⋅ 𝑥B = 𝑤B = 1×0 + 2×1 + 1×0 = 2
– Thatis,𝑝B ⋅ 𝑥C < 𝑤B,i.e.,bundle𝑥C isstillaffordableunderperiodtwo’sprices.
– ThusWARPisviolated.AdvancedMicroeconomicTheory 137
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• ExampleA2.1 (continued):– LetuscheckGARP.– Assumethatbundle𝑥C isrevealedpreferredto𝑥B,𝑥C ≿ 𝑥B,andthat𝑥B isrevealedpreferredto𝑥ª,𝑥B ≿ 𝑥ª.
– Itiseasytoshowthat𝑥ª ⊁ 𝑥C asbundle𝑥ª isnotaffordableunderbundle𝑥C’sprices.
– Hence,𝑥ª cannotberevealedpreferredto𝑥C,– Thus,GARPisnotviolated.
AdvancedMicroeconomicTheory 138
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• ExampleA2.2 (WARPholds,GARPdoesnot):– Considerthefollowingsequenceofpricevectorsandtheircorrespondingdemandedbundles
𝑝C = (1,1,2) 𝑥C = (1,0,0)𝑝B = (2,1,1) 𝑥B = (0,1,0)
𝑝ª = (1,2,1 + 𝜀) 𝑥ª = (0,0,1)
AdvancedMicroeconomicTheory 139
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• ExampleA2.2 (continued):– ForthepremiseofWARPtohold:
§ Sincebundle𝑥C isrevealedpreferredto𝑥B in𝑡 = 1,i.e.,𝑥C ≿ 𝑥B,itmustbethat𝑝C ⋅ 𝑥B ≤ 𝑤C = 𝑝C ⋅ 𝑥C.
§ Thatis,bundle𝑥B isaffordableunderbundle𝑥C’sprices.
– Substitutingourvalues,wefindthat1×0 + 1×1 + 2×0 = 1 ≤ 1= 1×1 + 1×0 + 2×0
– Hence,thepremiseofWARPholds.
AdvancedMicroeconomicTheory 140
GARP
• ExampleA2.2 (continued):– ForWARPtobesatisfied:
§ Wemustalsohavethat𝑝B ⋅ 𝑥C > 𝑤B = 𝑝B ⋅ 𝑥B.§ Thatis,bundle𝑥C isunaffordableunderthenewpricesandwealth.
– Pluggingourvaluesyields2×1 + 1×0 + 1×0 = 2 > 1= 2×0 + 1×1 + 1×0
– Thus,WARPissatisfied.
AdvancedMicroeconomicTheory 141
GARP
• ExampleA2.2 (continued):– Asimilarargumentappliestothecomparisonofthechoicesin𝑡 = 2 and𝑡 = 3.
– Bundle𝑥ª isaffordableunderthepricesat𝑡 = 2,𝑝B ⋅ 𝑥ª = 1 ≤ 1 = 𝑤B = 𝑝B ⋅ 𝑥B
– But𝑥B isunaffordableunderthepricesof𝑡 = 3,i.e.,𝑝ª ⋅ 𝑥B > 𝑤ª = 𝑝ª ⋅ 𝑥ª,since
1×0 + 2×1 + 1 + 𝜀×0 = 2 >1×0 + 2×0 + 1 + 𝜀×1 = 1 + 𝜀
– Hence,WARPalsoholdsinthiscase.AdvancedMicroeconomicTheory 142
GARP
• ExampleA2.2 (continued):– Furthermore,bundle𝑥ª isunaffordableunderbundle𝑥C’sprices,i.e.,𝑝C ⋅ 𝑥ª ≰ 𝑤C = 𝑝C ⋅ 𝑥C,
𝑝C ⋅ 𝑥ª = 1×0 + 1×0 + 2×1 = 2𝑝C ⋅ 𝑥C = 1×1 + 1×0 + 2×0 = 1
– Thus,thepremiseforWARPdoesnotholdwhencomparingbundles𝑥C and𝑥ª.
– Hence,WARPisnotviolated.
AdvancedMicroeconomicTheory 143
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• ExampleA2.2 (continued):– LetuscheckGARP.– Assumethatbundle𝑥C isrevealedpreferredto𝑥B,andthat𝑥B isrevealedpreferredto𝑥ª,i.e.,𝑥C ≿𝑥B and𝑥B ≿ 𝑥ª.
– Comparingbundles𝑥C and𝑥ª,wecanseethatbundle𝑥C isaffordableunderbundle𝑥ª’sprices,thatis
𝑝ª ⋅ 𝑥C = 1×1 + 2×0 + 1 + 𝜀×0 = 1 ≯ 1 + 𝜀 = 𝑤ª= 𝑝ª ⋅ 𝑥ª
AdvancedMicroeconomicTheory 144
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• ExampleA2.2 (continued):– Inotherwords,both𝑥C and𝑥ª areaffordableat𝑡 = 3 butonly𝑥ª ischosen.
– Then,theconsumerisrevealingapreferencefor𝑥ª over𝑥C,i.e.,𝑥ª ≿ 𝑥C.
– ThisviolatesGARP.
AdvancedMicroeconomicTheory 145
GARP
• ExampleA2.3 (BothWARPandGARPhold):– Considerthefollowingsequenceofpricevectorsandtheircorrespondingdemandedbundles
𝑝C = (1,1,2) 𝑥C = (1,0,0)𝑝B = (1,2,1) 𝑥B = (0,1,0)𝑝ª = (3,2,1) 𝑥ª = (0,0,1)
AdvancedMicroeconomicTheory 146
GARP
• ExampleA2.3 (continued):– LetuscheckWARP.– Notethatbundle𝑥B isaffordableunderthepricesat𝑡 = 1,i.e.,𝑝C ⋅ 𝑥B ≤ 𝑤C = 𝑝C ⋅ 𝑥C, since
1×0 + 1×1 + 1×0 = 1 ≤ 1= 1×1 + 1×0 + 1×0
– However,bundle𝑥C isunaffordableunderthepricesat𝑡 = 2,i.e.,𝑝B ⋅ 𝑥C > 𝑤B = 𝑝B ⋅ 𝑥B, since
2×1 + 1×0 + 1×0 = 2 > 1= 2×0 + 1×1 + 1×0
– ThusWARPissatisfied.AdvancedMicroeconomicTheory 147
GARP
• ExampleA2.3 (continued):– Asimilarargumentappliestothecomparisonofthechoicesin𝑡 = 2 and𝑡 = 3.
– Bundle𝑥ª isaffordableunderthepricesat𝑡 = 2,𝑝B ⋅ 𝑥ª = 1 ≤ 1 = 𝑤B = 𝑝B ⋅ 𝑥B
– But𝑥B isunaffordableunderthepricesof𝑡 = 3,i.e.,𝑝ª ⋅ 𝑥B > 𝑤ª = 𝑝ª ⋅ 𝑥ª,since
3×0 + 2×1 + 1×0 = 2 >3×0 + 2×0 + 1×1 = 1 + 𝜀
– Thus,WARPalsoholdsinthiscase.
AdvancedMicroeconomicTheory 148
GARP
• ExampleA2.3 (continued):– Asimilarargumentappliestothecomparisonofthechoicesin𝑡 = 1 and𝑡 = 3.
– Bundle𝑥ª isaffordableunderthepricesat𝑡 = 1,i.e.,𝑝C ⋅ 𝑥ª ≤ 𝑤C = 𝑝C ⋅ 𝑥C, since
𝑝C ⋅ 𝑥ª = 1 ≤ 1 = 𝑤B = 𝑝C ⋅ 𝑥C– Butbundle𝑥C isunaffordableunderthepricesof𝑡 = 3,i.e.,𝑝ª ⋅ 𝑥C > 𝑤ª = 𝑝ª ⋅ 𝑥ª,since𝑝ª ⋅ 𝑥C = 3×1 + 2×0 + 1×0 = 3𝑝ª ⋅ 𝑥ª = 3×0 + 2×0 + 1×1 = 1
– Hence,WARPisalsosatisfiedAdvancedMicroeconomicTheory 149
GARP
• ExampleA2.3 (continued):– LetuscheckGARP.–Weshowedabovethat𝑥C isunaffordableunderbundle𝑥ª’sprices,i.e.,𝑝ª ⋅ 𝑥C > 𝑤ª = 𝑝ª ⋅ 𝑥ª.
–Wecannotestablishbundle𝑥ª beingstrictlypreferredtobundle𝑥C,i.e.,𝑥ª ⊁ 𝑥C.
– Hence,GARPissatisfied.
AdvancedMicroeconomicTheory 150
GARP
• ExampleA2.4 (NeitherWARPnorGARPhold):– Considerthefollowingsequenceofpricevectorsandtheircorrespondingdemandedbundles
𝑝C = (1,1,1) 𝑥C = (1,0,0)𝑝B = (2,1,1) 𝑥B = (0,1,0)
𝑝ª = (1,2,1 + 𝜀) 𝑥ª = (0,0,1)
– Thisisactuallyacombinationofthefirsttwoexamples.
AdvancedMicroeconomicTheory 151
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• ExampleA2.4 (continued):– At𝑡 = 1,WARPwillbeviolatedbythesamemethodasinourfirstexample.
– At𝑡 = 3,GARPwillbeviolatedbythesamemethodasinoursecondexample.
– Hence,neitherWARPnorGARPholdinthiscase.
AdvancedMicroeconomicTheory 152
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• GARPconstitutesasufficientconditionforutilitymaximization.
• Thatis,ifthesequenceofprice-bundlepairs(𝑝Ë, 𝑥Ë) satisfiesGARP,thenitmustoriginatefromautilitymaximizingconsumer.
• Wereferto(𝑝Ë, 𝑥Ë) asasetof“data.”
AdvancedMicroeconomicTheory 153
GARP• Afriat’stheorem.Forasequenceofprice-bundlepairs(𝑝Ë, 𝑥Ë),thefollowingstatementsareequivalent:1) ThedatasatisfiesGARP.2) Thedatacanberationalizedbyautilityfunction
satisfyingLNS.3) Thereexistspositivenumbers(𝑢Ë, 𝜆Ë) forall𝑡 =
1,2, … , 𝑇 thatsatisfiestheAfriat inequalities
𝑢Ñ ≤ 𝑢Ë + 𝜆Ë𝑝Ë(𝑥Ñ − 𝑥Ë) forall𝑡, 𝑠4) Thedatacanrationalizedbyacontinuous,concave,
andstronglymonotoneutilityfunction.
AdvancedMicroeconomicTheory 154
GARP
• Adatasetis“rationalized”byautilityfunctionif:– foreverypair(𝑝Ë, 𝑥Ë),bundle𝑥Ë yieldsahigherutilitythananyotherfeasiblebundlex,i.e., 𝑢(𝑥Ë) ≥ 𝑢(𝑥)forall𝑥 inbudgetset 𝐵(𝑝Ë, 𝑝Ë𝑥Ë).
• Hence,ifadatasetsatisfiesGARP,thereexistsawell-behavedutilityfunctionrationalizingsuchdata.– Thatis,theutilityfunctionsatisfiesLNS,continuity,concavity,andstrongmonotonicity.
AdvancedMicroeconomicTheory 155
GARP
• Condition(3)inAfriat’s Theoremhasaconcavityinterpretation:– FromFOCs,wehave𝐷𝑢 𝑥Ë = 𝜆Ë𝑝Ë.– Byconcavity,
Y %Ò Y(%Ó)%Ò %Ó
≤ 𝑢P(𝑥Ñ)– Or,re-arranging,
𝑢 𝑥Ë ≤ 𝑢 𝑥Ñ + 𝑢P(𝑥Ñ)(𝑥Ë − 𝑥Ñ)– Since𝑢P 𝑥Ñ = 𝜆Ñ𝑝Ñ,itcanbeexpressedasAfriat’s inequality
𝑢 𝑥Ë ≤ 𝑢 𝑥Ñ + 𝜆Ñ𝑝Ñ(𝑥Ë − 𝑥Ñ)AdvancedMicroeconomicTheory 156
GARP
• Concaveutilityfunction
AdvancedMicroeconomicTheory 157
– Theutilityfunctionatpoint𝑥Ñ issteeperthantherayconnectingpointsA andB.