advanced microeconomic theory · – !(/,2) is the walrasian demand correspondence, which specifies...

AdvancedMicroeconomicTheory

Chapter2:DemandTheory

Outline

• Utilitymaximizationproblem(UMP)• Walrasiandemandandindirectutilityfunction• WARPandWalrasiandemand• Incomeandsubstitutioneffects(Slutskyequation)

• DualitybetweenUMPandexpenditureminimizationproblem(EMP)

• Hicksiandemandandexpenditurefunction• Connections

AdvancedMicroeconomicTheory 2

UtilityMaximizationProblem


UtilityMaximizationProblem• Consumermaximizeshisutilitylevelbyselectingabundle𝑥 (where𝑥 canbeavector)subjecttohisbudgetconstraint:

max%&'

𝑢(𝑥)s. t. 𝑝 0 𝑥 ≤ 𝑤

• Weierstrass Theorem: foroptimizationproblemsdefinedonthereals,iftheobjectivefunctioniscontinuousandconstraintsdefineaclosedandboundedset,thenthesolutiontosuchoptimizationproblemexists.


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.


UtilityMaximizationProblem• Walrasiandemand𝑥(𝑝,𝑤)

atbundleA isoptimal,astheconsumerreachesautilitylevelof𝑢B byexhaustingallhiswealth.

• BundlesB andC arenotoptimal,despiteexhaustingtheconsumer’swealth.Theyyieldalowerutilitylevel𝑢C,where𝑢C < 𝑢B.

• BundleD isunaffordableand,hence,itcannotbetheargmaxoftheUMPgivenawealthlevelof𝑤.


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.


PropertiesofWalrasianDemand


– 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.





– For𝑥 ∈ 𝑥(𝑝,𝑤),thereisabundle𝑦,ε–closeto𝑥,suchthat𝑦 ≻ 𝑥.Then,𝑥 ∉ 𝑥(𝑝,𝑤).

1x

2x

x x

y y


3)Convexity/Uniqueness:

a) Ifthepreferencesareconvex,thentheWalrasiandemandcorrespondence𝑥(𝑝, 𝑤)definesaconvexset,i.e.,acontinuumofbundlesareutilitymaximizing.

b) Ifthepreferencesarestrictlyconvex,thentheWalrasiandemandcorrespondence𝑥(𝑝, 𝑤)containsasingleelement.



Convexpreferences Strictlyconvexpreferences


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


UMP:SufficientCondition

• WhenareKuhn-Tucker(necessary)conditions,alsosufficient?– Thatis,whencanweguaranteethat𝑥(𝑝, 𝑤) isthemax oftheUMPandnotthemin?


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.


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.


UMP:ViolationsofSufficientCondition


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.


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


UMP:CornerSolution• Consumerseekstoconsume

good1alone.

• Atthecornersolution,theindifferencecurveissteeperthanthebudgetline,i.e.,

𝑀𝑅𝑆C,B >7o7p

orwxo7o

> wxp7p

• Intuitively,theconsumerwouldliketoconsumemoreofgood1,evenafterspendinghisentirewealthongood1alone.


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.


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)


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.


WalrasianDemand:WealthEffects

• Normalvs.InferiorgoodsW% 7,9W9

>< 0 normalinferior

• Examplesofinferiorgoods:– Two-buckchuck(areallycheapwine)–Walmartduringtheeconomiccrisis


WalrasianDemand:WealthEffects• Anincreaseinthewealthlevel

producesanoutwardshiftinthebudgetline.

• 𝑥B isnormalasW%p 7,9W9 > 0,while

𝑥C isinferiorasW% 7,9W9 < 0.

• Wealthexpansionpath:– connectstheoptimalconsumption

bundlefordifferentlevelsofwealth

– indicateshowtheconsumptionofagood changesasaconsequenceofchangesinthewealthlevel


WalrasianDemand:WealthEffects• Engelcurvedepictsthe

consumptionofaparticulargoodinthehorizontalaxisandwealthontheverticalaxis.

• TheslopeoftheEngelcurveis:– positiveifthegoodisnormal– negativeifthegoodisinferior

• Engelcurvecanbepositivelysloppedforlowwealthlevelsandbecomenegativelysloppedafterwards.


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.



• Ownpriceeffect


Usualgood Giffengood



• 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.


PropertiesofIndirectUtilityFunction

• IftheutilityfunctioniscontinuousandpreferencessatisfyLNSovertheconsumptionset𝑋 = ℝ@A , thentheindirectutilityfunction𝑣 𝑝, 𝑤 satisfies:

1) Homogenousofdegreezero:Increasing𝑝 and𝑤byacommonfactor𝛼 > 0 doesnotmodifytheconsumer’soptimalconsumptionbundle,𝑥(𝑝, 𝑤),norhismaximalutilitylevel,measuredby𝑣 𝑝,𝑤 .




x1

x2

wp2

αwαp2 =

wp1

αwαp1 =

x(p,w)=x(αp,αw)

v(p,w)=v(αp,αw)

Bp,w=Bαp,αw


2)Strictlyincreasingin𝑤:𝑣 𝑝,𝑤P > 𝑣(𝑝, 𝑤) for𝑤P > 𝑤.

3)non-increasing(i.e.,weaklydecreasing)in𝑝[




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.


W

P1

w*

p*

v(p,w) < v(p*,w*)

v(p,w) = v1


- 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�




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?


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′.


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�.


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”.


CheckingforWARP

- Step2: Checkifbundles𝑥 𝑝,𝑤 isaffordableunder𝐵7�,9�.§ Thatis,graphically𝑥 𝑝,𝑤 mustlieonorbelowbudge

line𝐵7�,9� .§ Ifstep2issatisfied,thenthisWalrasiandemandviolates

WARP.§ Otherwise,theWalrasiandemandsatisfiesWARP.


CheckingforWARP:Example1• First,𝑥 𝑝,𝑤 and𝑥 𝑝′, 𝑤′ arebothaffordableunder𝐵7,9.

• Second,𝑥 𝑝,𝑤 isnotaffordableunder𝐵7�,9�.

• Hence,WARPissatisfied!


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.




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)


ImplicationsofWARP• Adjusttheconsumer’swealthlevelsothathecanconsumehisinitial

demand𝑥 𝑝, 𝑤 atthenewprices.– shiftthefinalbudgetlineinwardsuntilthepointatwhichwereach

theinitialconsumptionbundle𝑥 𝑝, 𝑤 (compensatedpricechange)


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 .


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.



• DoesWARPrestrictbehaviorwhenweapplySlutskywealthcompensations?– Yes!

• WhatifwewerenotapplyingtheSlutskywealthcompensation,wouldWARPimposeanyrestrictiononallowablelocationsfor𝑥 𝑝P, 𝑤P ?– No!




• Seethediscussionsonthenextslide


• 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



• 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).



• 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?



• IsWARPsatisfiedundertheuncompensatedlawofdemand(ULD)?– 𝑥 𝑝, 𝑤 isaffordableunder

budgetline𝐵7,9,but𝑥(𝑝P, 𝑤) isnot.§ Hence,thepremiseofWARPis

notsatisfied.Asaresult,WARPisnotviolated.

– But,isthisresultimplyingsomethingaboutwhetherULDmusthold?§ No!§ AlthoughWARPisnotviolated,

ULDis:adecreaseinthepriceofgood1yieldsadecreaseinthequantitydemanded.



• Distinctionbetweentheuncompensatedandthecompensatedlawofdemand:– quantitydemandedandpricecanmoveinthesamedirection,whenwealthisleftuncompensated,i.e.,

∆𝑝C 0 ∆𝑥C > 0as∆𝑝C ⇒ ∆𝑥C

• Hence,WARPisnotsufficienttoyieldlawofdemandforpricechangesthatareuncompensated,i.e.,

WARP⇎ ULD,butWARP⇔ CLD


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.



• 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


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)



• 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!



• Proposition:IfpreferencessatisfyLNSandstrictconvexity,andtheyarerepresentedwithacontinuousutilityfunction,thentheWalrasiandemand𝑥 𝑝,𝑤generatesaSlutskymatrix,𝑆 𝑝,𝑤 ,whichissymmetric.

• Theaboveassumptionsarereallycommon.– Hence,theSlutskymatrixwillthenbesymmetric.

• However,theaboveassumptionsarenotsatisfiedinthecaseofpreferencesoverperfectsubstitutes(i.e.,preferencesareconvex,butnotstrictlyconvex).



• Non-positivesubstitutioneffect,𝑠ff ≤ 0:

𝑠ff 𝑝, 𝑤��( )

=𝜕𝑥f(𝑝, 𝑤)𝜕𝑝f

¡��¢£��: ��¢£¤��¥@ ¦��¤��¥


𝜕𝑤 𝑥f(𝑝, 𝑤)§��¨��:@ ��©¨¢£¤��¥ ��©��©¤��¥

• SubstitutionEffect=TotalEffect+IncomeEffect⇒ TotalEffect=SubstitutionEffect - IncomeEffect


ImplicationsofWARP:SlutskyEquation

𝑠ff 𝑝, 𝑤��( )

=𝜕𝑥f(𝑝, 𝑤)𝜕𝑝f

¡��¢£��


𝜕𝑤 𝑥f(𝑝, 𝑤)§��¨��

• TotalEffect:measureshowthequantitydemandedisaffectedbyachangeinthepriceofgood𝑙,whenweleavethewealthuncompensated.

• IncomeEffect:measuresthechangeinthequantitydemandedasaresultofthewealthadjustment.

• SubstitutionEffect:measureshowthequantitydemandedisaffectedbyachangeinthepriceofgood𝑙,afterthewealthadjustment.– Thatis,thesubstitutioneffectonlycapturesthechangeindemandduetovariation

inthepriceratio,butabstractsfromthelarger(smaller)purchasingpowerthattheconsumerexperiencesafteradecrease(increase,respectively)inprices.


• Reductioninthepriceof𝑥C.– Itenlargesconsumer’ssetoffeasible

bundles.– Hecanreachanindifferencecurve

furtherawayfromtheorigin.

• TheWalrasiandemandcurveindicatesthatadecreaseinthepriceof𝑥C leadstoanincreaseinthequantitydemanded.– Thisinducesanegativelysloped

Walrasiandemandcurve(sothegoodis“normal”).

• Theincreaseinthequantitydemandedof𝑥C asaresultofadecreaseinitspricerepresentsthetotaleffect(TE).


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 ).


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).


ImplicationsofWARP:SlutskyEquation

SubstitutionandIncomeEffects:NormalGoods

• Decreaseinthepriceofthegoodinthehorizontalaxis(i.e.,food).

• Thesubstitutioneffect(SE)movesintheoppositedirectionasthepricechange.– Areductioninthepriceof

foodimpliesapositivesubstitutioneffect.

• Theincomeeffect(IE)ispositive(thusitreinforcestheSE).– Thegoodisnormal.


Clothing

FoodSE IETE

A

B

C

I2

I1

BL2

BL1

BLd

SubstitutionandIncomeEffects:InferiorGoods


• TheSEstillmovesintheoppositedirectionasthepricechange.

• Theincomeeffect(IE)isnownegative(whichpartiallyoffsetstheincreaseinthequantitydemandedassociatedwiththeSE).– Thegoodisinferior.

• Note:theSEislargerthantheIE.


I2

I1BL1 BLd

SubstitutionandIncomeEffects:GiffenGoods


• TheSEstillmovesintheoppositedirectionasthepricechange.

• Theincomeeffect(IE)isstillnegativebutnowcompletelyoffsetstheincreaseinthequantitydemandedassociatedwiththeSE.– ThegoodisGiffengood.

• Note:theSEislessthantheIE.


I2

I1BL1

BL2

SubstitutionandIncomeEffectsSE IE TE

NormalGood + + +Inferior Good + - +GiffenGood + - -


• 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.


ExpenditureMinimizationProblem



• Expenditureminimizationproblem(EMP):

min%&'

𝑝 0 𝑥

s.t. 𝑢(𝑥) ≥ 𝑢

• Alternativetoutilitymaximizationproblem


ExpenditureMinimizationProblem• Consumerseeksautilitylevel

associatedwithaparticularindifferencecurve,whilespendingaslittleaspossible.

• Bundlesstrictlyabove𝑥∗ cannotbeasolutiontotheEMP:– Theyreachtheutilitylevel𝑢– But,theydonotminimizetotal

expenditure

• Bundlesonthebudgetlinestrictlybelow𝑥∗ cannotbethesolutiontotheEMPproblem:– Theyarecheaperthan𝑥∗– But,theydonotreachthe

utilitylevel𝑢


X2

X1

*x

*UCS( )x

u


• Lagrangian𝐿 = 𝑝 0 𝑥 + 𝜇 𝑢 − 𝑢(𝑥)

• FOCs(necessaryconditions)𝜕𝐿𝜕𝑥[

= 𝑝[ − 𝜇𝜕𝑢(𝑥∗)𝜕𝑥[

≤ 0

𝜕𝐿𝜕𝜇 = 𝑢 − 𝑢 𝑥∗ = 0


[=0forinteriorsolutions]

ExpenditureMinimizationProblem• Forinteriorsolutions,

𝑝[ = 𝜇 WY(%∗)

W%XorC

±=

^_(`∗)^`X7X

foranygood𝑘.Thisimplies,^_(`∗)^`X7X

=^_(`∗)^`a7a

or7X7a=

^_(`∗)^`X

^_(`∗)^`a

• Theconsumerallocateshisconsumptionacrossgoodsuntilthepointinwhichthemarginalutilityperdollarspentoneachgoodisequalacrossallgoods(i.e.,same“bangforthebuck”).

• Thatis,theslopeofindifferencecurveisequaltotheslopeofthebudgetline.


EMP:HicksianDemand

• Thebundle𝑥∗ ∈ argmin𝑝 0 𝑥 (theargumentthatsolvestheEMP)istheHicksiandemand,whichdependson𝑝 and𝑢,

𝑥∗ ∈ ℎ(𝑝, 𝑢)

• Recallthatifsuchbundle𝑥∗ isunique,wedenoteitas𝑥∗ = ℎ(𝑝, 𝑢).


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.


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)


x2

x1


2) Noexcessutility:foranyoptimalconsumptionbundle𝑥 ∈ ℎ 𝑝, 𝑢 ,utilitylevelsatisfies𝑢 𝑥 = 𝑢.


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.



3) Convexity:Ifthepreferencerelationisconvex,thenℎ 𝑝, 𝑢 isaconvexset.



4) Uniqueness:Ifthepreferencerelationisstrictlyconvex,thenℎ 𝑝, 𝑢containsasingleelement.



• 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.


TheExpenditureFunction

• PluggingtheresultfromtheEMP,ℎ 𝑝, 𝑢 ,intotheobjectivefunction,𝑝 0 𝑥,weobtainthevaluefunctionofthisoptimizationproblem,

𝑝 0 ℎ 𝑝, 𝑢 = 𝑒(𝑝, 𝑢)

where𝑒(𝑝, 𝑢) representstheminimalexpenditure thattheconsumerneedstoincurinordertoreachutilitylevel𝑢 whenpricesare𝑝.


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.


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


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.


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


x2

x11/ 'w p

2/w p

2/ 'w p

2/ ''w p

x’’

x’x

u

1/w p1/ ''w p

Connections


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)



• 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.



• 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.


RelationshipbetweenHicksianandWalrasianDemand

• Whenincomeeffectsarepositive(normalgoods),thentheWalrasiandemand𝑥(𝑝, 𝑤) isabove theHicksiandemandℎ(𝑝, 𝑢) .– TheHicksiandemandissteeper thantheWalrasiandemand.


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


• Whenincomeeffectsarenegative(inferiorgoods),thentheWalrasiandemand𝑥(𝑝, 𝑤) isbelowtheHicksiandemandℎ(𝑝, 𝑢) .– TheHicksiandemandisflatter thantheWalrasiandemand.


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


• 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[



• 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¿)



• Butthiscoincideswith𝑠f[ 𝑝, 𝑤 thatwediscussedintheSlutskyequation.– Hence,wehave𝐷7ℎ 𝑝, 𝑢

A×A

= 𝑆 𝑝,𝑤 .

– Or,morecompactly,𝑆𝐸 = 𝑇𝐸 + 𝐼𝐸.


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.


SummaryofRelationships

• TheWalrasiandemand,𝑥 𝑝, 𝑤 ,isthesolutionoftheUMP.– Itsvaluefunctionistheindirectutilityfunction,𝑣 𝑝,𝑤 .

• TheHicksiandemand,ℎ(𝑝, 𝑢),isthesolutionoftheEMP.– Itsvaluefunctionistheexpenditurefunction,𝑒(𝑝, 𝑢).




x(p,w)

v(p,w) e(p,u)

h(p,u)

The UMP The EMP

(1) (1)


• 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.




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)


• 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 .




x(p,w)

v(p,w) e(p,u)

h(p,u)

The UMP The EMP


(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.




x(p,w)

v(p,w) e(p,u)

h(p,u)

The UMP The EMP


(1) (1)

(2)

3(a) 3(b)

4(a) Slutsky equation(Using derivatives)



x(p,w)

v(p,w) e(p,u)

h(p,u)

The UMP The EMP


(1) (1)

(2)

3(a) 3(b)

4(a) Slutsky equation(Using derivatives)

4(b) Roy’s

Identity 4(c)

Appendix2.1:DualityinConsumption


DualityinConsumption

• Wediscussedtheutilitymaximizationproblem(UMP)– theso-calledprimal problemdescribingtheconsumer’schoiceofoptimalconsumptionbundles– anditsdual:theexpenditureminimizationproblem(EMP).

• Whencanweguaranteethatthesolution𝑥∗ tobothproblemscoincide?


DualityinConsumption• Themaximaldistancethata

turtlecantravelin𝑡∗ timeis𝑥∗.– Fromtime(𝑡∗) todistance(𝑥∗)

• Theminimaltimethataturtleneedstotravel𝑥∗ distanceis𝑡∗.– Fromdistance(𝑥∗) totime(𝑡∗)

• Inthiscasetheprimalanddualproblemswouldprovideuswiththesameanswertobothoftheabovequestions


DualityinConsumption• Inordertoobtainthesame

answerfrombothquestions,wecriticallyneedthatthefunction𝑀(𝑡)satisfiesmonotonicity.

• Otherwise:themaximaldistancetraveledatboth𝑡Cand𝑡B is𝑥∗,buttheminimaltimethataturtleneedstotravel𝑥∗ distanceis𝑡C.

• Hence,anon-monotonicfunctioncannotguaranteethattheanswersfrombothquestionsarecompatible.


DualityinConsumption• Similarly,wealsorequire

thatthefunction𝑀(𝑡)satisfiescontinuity.

• Otherwise:themaximaldistancethattheturtlecantravelintime𝑡∗ is𝑥B ,whereastheminimumtimerequiredtotraveldistance𝑥C and𝑥B is𝑡∗ forbothdistances.

• Hence,anon-continuousfunctiondoesnotguaranteethattheanswerstobothquestionsarecompatible.


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)


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


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.


Hyperplane Theorem• If𝐾 isaclosedandconvex

set,wecanthenconstructhalf-spacesforalltheelementsintheset(𝑥C, 𝑥B, … ) suchthattheirintersectionscoincides(“equivalentlydescribes”)set𝐾.– Weconstructacage(or

hull)aroundtheconvexset𝐾 thatexactlycoincideswithset𝐾.

• Bundlelike�̅� ∉ 𝐾 liesoutsidetheintersectionofhalf-spaces.


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.


Hyperplane Theorem

• Theconvexhullofset𝐾 isoftendenotedas𝐾¿,anditisconvex(unlikeset𝐾,whichmightnotbeconvex).

• Theconvexhull𝐾¿ isconvex,bothwhensetK isconvexandwhenitisnot.


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 .


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).


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.


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 �̅� = �̅�.


Appendix2.2:Relationshipbetweenthe

ExpenditureFunctionandHicksianDemand


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


ProofI(UsingDualityTheorem)

• UCSisaconvexandclosedset.

• Hyperplane 𝑝 0 𝑥∗ =𝑒 𝑝, 𝑢 providesuswiththeminimalexpenditurethatstillreachesutilitylevel𝑢.


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


Appendix2.3:GeneralizedAxiomofRevealed

Preference(GARP)


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.


GARP


• 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.


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

GARP

• 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

GARP

• ExampleA2.1 (continued):– LetuscheckGARP.– Assumethatbundle𝑥C isrevealedpreferredto𝑥B,𝑥C ≿ 𝑥B,andthat𝑥B isrevealedpreferredto𝑥ª,𝑥B ≿ 𝑥ª.

– Itiseasytoshowthat𝑥ª ⊁ 𝑥C asbundle𝑥ª isnotaffordableunderbundle𝑥C’sprices.

– Hence,𝑥ª cannotberevealedpreferredto𝑥C,– Thus,GARPisnotviolated.


GARP

• 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)


GARP

• 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.


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.


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.


GARP

• 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


GARP

• ExampleA2.2 (continued):– Inotherwords,both𝑥C and𝑥ª areaffordableat𝑡 = 3 butonly𝑥ª ischosen.

– Then,theconsumerisrevealingapreferencefor𝑥ª over𝑥C,i.e.,𝑥ª ≿ 𝑥C.

– ThisviolatesGARP.


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)


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


– 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.


GARP


– 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.


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.


GARP

• ExampleA2.4 (continued):– At𝑡 = 1,WARPwillbeviolatedbythesamemethodasinourfirstexample.

– At𝑡 = 3,GARPwillbeviolatedbythesamemethodasinoursecondexample.

– Hence,neitherWARPnorGARPholdinthiscase.


GARP

• GARPconstitutesasufficientconditionforutilitymaximization.

• Thatis,ifthesequenceofprice-bundlepairs(𝑝Ë, 𝑥Ë) satisfiesGARP,thenitmustoriginatefromautilitymaximizingconsumer.

• Wereferto(𝑝Ë, 𝑥Ë) asasetof“data.”


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.


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.


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


– Theutilityfunctionatpoint𝑥Ñ issteeperthantherayconnectingpointsA andB.

advanced microeconomic theory · – !(/,2) is the walrasian demand correspondence, which specifies...

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