lecture4

DavidAutor14.03Fall2004

Agenda

1.Utilitymaximization2.IndirectUtilityfunction

3.Application:Giftgiving–Waldfogelpaper4.Expenditurefunction

5.RelationshipbetweenExpenditurefunctionandIndirectutilityfunction6.Demandfunctions

7.Application:Foodstamps–Whitmorepaper8.Incomeandsubstitutione¤ects9.Normalandinferiorgoods

10.Compensatedanduncompensateddemand(Hicksian,Marshallian)11.Application:Gi¤engoods–JensenandMillerpaper

Roadmap:

1

Axioms of consumerpreferencePrimalMaxU(x,y)s.t.pxx+pyy< IDualMinpxx+pyys.t.U(x,y) > UIndirect Utility functionU*=V(px,py, I)Expenditure functionE*=E(px,py, U)MarshalliandemandX =dx(px,py, I) =(by Roy’s identity)−∂V/∂px∂V/∂ISlutskyequationHicksiandemandX =hx(px,py, U) =(by Shepard’s lemma)∂E−∂px1

1.1

Theoryofconsumerchoice

Utilitymaximizationsubjecttobudgetconstraint

Ingredients:

󰀅Utilityfunction(preferences)󰀅Budgetconstraint󰀅PricevectorConsumer’sproblem

MaximizeutilitysubjettobudgetconstraintCharacteristicsofsolution:󰀅Budgetexhaustion(non-satiation)󰀅Formostsolutions:

psychictradeo¤=monetarypayo¤

󰀅Psychictradeo¤isMRS

󰀅Monetarytradeo¤isthepriceratio

2

Fromavisualpointofviewutilitymaximizationcorrespondstothefollowingpoint:

x

(Notethattheslopeofthebudgetsetisequalto󰀂ppy)

GryIC1IC2IC3BACDxWhat’swrongwithsomeofthesepoints?WecanseethatAPB,AID,C

P

A.WhyshouldonechooseA?

Theslopeoftheindi¤erencecurvesisgivenbytheMRS1.1.1

Interiorandcornersolutions[Optional]

Therearetwotypesofsolutiontothisproblem.1.Interiorsolution2.Cornersolution

3

GyTypical casexTheonebelowisanexampleofacornersolution.Inthisspeci…cexampletheshapeoftheindi¤erencecurvesmeansthattheconsumerisindi¤erenttotheconsumptionofgoody.Utilityincreasesonlywithconsumptionofx.

Graph 37yx4

Graph 38yxInthegraphabovepreferenceforyissu¢cientlystrongrelativetoxthatthethepsychictradeo¤isalwayslowerthanthemonetarytradeo¤.

Thismustbethecaseformanyproductsthatwedon’tbuy.Anothertypeof“corner”solutionresultsfromindivisibility.

Graph 39y10I = 500px= 450py= 501xWhycan’twedrawthisbudgetset,i.e.conectdots?

Thisisbecauseonly2pointscanbedrawn.Thisisasortof“integerconstraint”.Wenormallyabstractfromindivisibility.

5

Goingbacktothegeneralcase,howdoweknowasolutionexistsforconsumer,i.e.howdoweknowtheconsumercanchoose?

Weknowbecauseofthecompletenessaxiom.Everybundleisonsomeindi¤erencecurveandcanthereforeberanked:AIB,A󰀉B,B󰀉A.1.1.2

MathematicalsolutiontotheConsumer’sProblem

Mathematics:

maxU(x;y)

x;y

s:t:pxx+pyy

L

@L@x@L@y@L@󰀃

󰀇I

=U(x;y)+󰀃(I󰀂pxx󰀂pyy)=Ux󰀂󰀃px=0=Uy󰀂󰀃py=0=I󰀂pxx󰀂pyy=0

1:2:3:

Rearranging1:and2:

pxUx=Uypy

Thismeansthatthepsychictradeo¤isequaltothemonetarytradeo¤betweenthetwogoods.3:statesthatbudgetisexhausted(non-satiation).Alsonoticethat:

UxpxUypy

Whatisthemeaningof󰀃?

=󰀃=󰀃

1.1.3Interpretationof󰀃,theLagrangemultiplier

AtthesolutionoftheConsumer’sproblem(morespeci…cally,aninteriorsolution),thefollowingconditionswillhold:

@U=@x2@U=@xn@U=@x1

==:::==󰀃p1p2pn

6

Thisexpressionsaysthatattheutility-maximizingpoint,thenextdollarspentoneachgoodyieldsthesamemarginalutility.Sowhatis

@U@I?

ReturntoLagrangian:

L@L@x@L@y@L@󰀃@L@I

Bysubstituting󰀃=Weconcludethat:

Uxpx

=U(x;y)+󰀃(I󰀂pxx󰀂pyy)=Ux󰀂󰀃px=0=Uy󰀂󰀃py=0

=I󰀂pxx󰀂pyy=0

󰀆󰀇󰀆󰀇

@x@x@y@y

=Ux󰀂󰀃px+Uy󰀂󰀃py+󰀃

@I@I@I@I

Uy

py

and󰀃=bothexpressionsinparenthesisarezero.

@L

=󰀃@I

󰀃equalsthe“shadowprice”ofthebudgetconstraint,i.e.itexpressesthequantityofutilsthatcouldbeobtainedwiththenextdollarofconsumption.

Thisshadowpriceisnotuniquelyde…ned.Itisde…nedonlyuptoamonotonictransformation.

Whatdoestheshadowpricemean?It’sessentiallythe‘utilityvalue’ofrelaxingthebudgetconstraintbyoneunit(e.g.,onedollar).[Q:What’sthesignof@2U=@I2,andwhy?]

Wecouldalsohavedeterminedthat@L=@I=󰀃withoutcalculationsbyapplyingtheenvelopetheorem.Attheutilitymaximizingsolutiontothisproblem,x󰀃andy󰀃arealreadyoptimizedandsoanin…nitesimalchangeinIdoesnotalterthesechoices.Hence,thee¤ectofIonUdependsonlyonitsdirecte¤ectonthebudgetconstraintanddoesnotdependonitsindirecte¤ect(duetoreoptimization)onthechoicesofxandy.This‘envelope’resultisonlytrueinasmallneighboroodaroundthesolutiontotheoriginalproblem.CornerSolution:unusualcaseWhenatacornersolution,consumerbuyszeroofsomegoodandspendstheentirebudgetontherest.WhatproblemdoesthiscreatefortheLagrangian?

7

Graph 40yU0U1U2xTheproblemisthatapointoftangencydoens’texistforpositivevaluesofy.Hencewealsoneedtoimpose“non-negativityconstraints”:x󰀈0,y󰀈0.

Thiswillnotbeimportantforproblemsinthisclass,butit’seasytoaddtheseconstraintstothemaximiza-tionproblem.

1.1.4AnExampleProblem

Considerthefollowingexampleproblem:U(x;y)=

1

4lnx+

34lny

Noticethatthisutilityfunctionsatisifesallaxioms:

1.Completeness,transitivity,continuity[theseareprettyobvious]2.Non-satiation:Ux=

14x>0forallx>0.Uy=

34y>0forally>0.Inotherwords,utilityrises

continuallywithgreaterconsumptionofeithergood,thoughtherateatwhichitrisesdeclines(diminishingmarginalutilityofconsumption).3.Diminishingmarginalrateofsubstitution:

󰀂=Alonganindi¤erencecurveofthisutilityfunction:UTotallydi¤erentiate:0=

1

4x0dx

14lnx0+

UxUy

34lny0.

+

3

4y0dy.

Themarginalrateofsubstitutionofyforxisincreasingintheamountofyconsumedanddecreasingintheamountofxconsumed;holdingutilityconstant,themoreytheconsumerhas,themoreyhewouldgiveupforoneadditionalunitofx.

8

dy

Whichprovidesthemarginalrateofsubstitution󰀂dxjU󰀂=

=

4y0

12x0.

Examplevalues:px=1py=2I=12

WritetheLagrangianforthisutilityfunctiongivenpricesandincome:

maxU(x;y)

x;y

s:t:pxx+pyy

1:2:3:

Rearranging(1)and(2),wehave:

󰀇I

13

L=lnx+lny+󰀃(12󰀂x󰀂2y)

441@L=󰀂󰀃=0@x4x@L3

=󰀂2󰀃=0@y4y@L

=12󰀂x󰀂2y=0@󰀃

UxUy1=4x3=4y

==

pxpy12

TheinterpretationofthisexpressionisthattheMRS(psychictrade-o¤)isequaltothemarkettrade-o¤(price-ratio).What’s

@L

@I?

Asbefore,thisisequalto󰀃,whichfrom(1)and(2)isequalto:

󰀃=

13=:4x󰀃8y󰀃1

211

4x󰀂oritcouldbuy2additional3

simportantthat@L=@I4y󰀂).It’

Thenextdollarofincomecouldbuyoneadditionalx,whichhasmarginalutilityy0s,whichprovidemarginalutility

3

4y󰀂(so,themarginalutilityincrementis

=󰀃isde…nedintermsoftheoptimallychosenx󰀃;y󰀃.Unlessweareattheseoptima,theenvelopetheorem

󰀅󰀂@x󰀃󰀄@y

@y@x

doesnotapplySo,@L=@Iwouldalsodependonthecross-partialterms:Ux@I󰀂󰀃px@I+Uy@I󰀂󰀃py@I.

󰀃

9

1.1.5LagrangianwithNon-negativityConstraints[Optional]

maxU(x;y)

s:t:pxx+pyy

yL@L1:

@x@L2:

@y@L3:

@s

󰀇I󰀈0

=U(x;y)+󰀃(I󰀂pxx󰀂pyy)+󰀂(y󰀂s2)=Ux󰀂󰀃px=0=Uy󰀂󰀃py+󰀂=0=󰀂2s󰀂=0

Point3:impliesthat󰀂=0,s=0,orboth.

1.s=0;󰀂=0(since󰀂󰀈0thenitmustbethecasethat󰀂>0)

(a)

Uy󰀂󰀃py+󰀂

UypyUxpx

Combiningthelasttwoexpressions:

pxUx>Uypy

Thisconsumerwouldliketoconsumeevenmorexandlessy,butshecannot.2.s=0;󰀂=0

Uy󰀂󰀃py+󰀂Uypy

==

0󰀂!Uy󰀂󰀃py=0Ux

=󰀃px=<

0󰀂!Uy󰀂󰀃py<0󰀃

=󰀃

StandardFOC,herethenon-negativityconstraintisnotbinding.3.s=0;󰀂=0SameFOCasbefore:

Uxpx=pyUy

Herethenon-negativityconstraintissatis…edwithequalitysoitdoesn’tdistortconsumption.

10

1.2IndirectUtilityFunction

Forany:

󰀅Budgetconstraint󰀅Utilityfunction󰀅Setofprices

Weobtainasetofoptimallychosenquantities.

x󰀃1

=x1(p1;p2;:::;pn;I)

:::

x󰀃n

Sowhenwesay

=xn(p1;p2;:::;pn;I)

maxU(x1;:::;xn)s:t:PX󰀃󰀇I

wegetasaresult:

󰀃maxU(x󰀃1(p1;:::;pn;I);:::;xn(p1;:::;pn;I))

)U󰀃(p1;:::;pn;I)󰀆V(p1;:::;pn;I)

whichwecallthe“IndirectUtilityFunction”.Thisisthevalueofmaximizedutilityundergivenpricesandincome.

Sorememberthedistinction:

Directutility:utilityfromconsumptionofx1;:::;xnIndirectutility:utilityobtainedwhenfacingp1;:::;pn;IExample:

maxU(x;y)s:t:pxx+pyy

L

@L@x@L@y@L@󰀃

=x:5y:5󰀇I

=x:5y:5+󰀃(I󰀂pxx󰀂pyy)=:5x󰀂:5y:5󰀂󰀃px=0=:5x:5y󰀂:5󰀂󰀃py=0=I󰀂pxx󰀂pyy=0

11

Weobtainthefollowing:

:5x󰀂:5y:5:5x:5y󰀂:5

;󰀃==

pxpy

whichsimpliesto:

x=

Substitutingintothebudgetconstraintgivesus:

I󰀂px

pyy

󰀂pyypx

pyy

==

01I;2

pyy=

1I2

pyy

:px

x󰀃=

Halfofthebudgetgoestoeachgood.

II;y󰀃=2px2py

Let’sderivetheindirectutilityfunctioninthiscase:

U󰀆II;2px2py

󰀇=󰀆I2px

󰀇:5󰀆

I2py

󰀇:5

Whybothercalculatingtheindirectutilityfunction?Itsavesustime.Insteadofrecalculatingtheutilitylevelforeverysetofpricesandbudgetconstraints,wecanpluginpricesandincometogetconsumerutility.Thiscomesinhandywhenworkingwithindividualdemandfunctions.Demandfunctionsgivethequantityofgoodspurchasedbyagivenconsumerasafunctionofpricesandincome(orutility).

1.3TheCarteBlanchePrinciple

Oneimmediateimplicationofconsumertheoryisthatconsumersmakeoptimalchoicesforthemselvesgivenprices,constraints,andincome.[Generally,theonlyconstraintisthattheycan’tspendmoretheirincome,butwe’llseeexampleswherethereareadditionalconstraints.]

ThisobservationgivesrisetotheCarteBlancheprinciple:consumersarealwaysweaklybettero¤receivingacashtransferthananin-kindtransferofidenticalmonetaryvalue.[Weaklybettero¤inthattheymaybeindi¤erentbetweenthetwo.]

Withcash,consumershaveCarteBlanchetopurchasewhateverbundleorgoodsareservicestheycana¤ord–includingthegoodorservicethatalternativelycouldhavebeentransferedtothemin-kind.

Prominentexamplesofin-kindtransfersgiventoU.S.citizensincludeFoodStamps,housingvouchers,healthinsurance(Medicaid),subsidizededucationalloans,childcareservices,jobtraining,etc.[Anexhaustivelistwouldbelongindeed.]

Economictheorysuggeststhat,relativetotheequivalentcashtransfer,thesein-kindtransfersserveasconstraintsonconsumerchoice.

12

Ifconsumersarerational,constraintsonchoicecannotbebene…cial.

Forexample,consideraconsumerwhohasincomeI=100andfacesthechoiceoftwogoods,foodandhousing,atpricespf;ph,eachpricedat1perunit.Theconsumer’sproblemis

maxU(f;h)

f;h

s:t:f+h󰀇100

Thegovernmentdecidestoprovideahousingsubsidyof50.Thismeansthattheconsumercannowpurchaseupto150unitsofhousingbutnomorethan100unitsoffood.Theconsumer’sproblemis:

maxU(f;h)

f;h

s:t:f+h

h

󰀇150󰀈50:

Alternatively,ifthegovernmenthadprovided50dollarsincashinstead,theproblemwouldbe:

maxU(f;h)

f;h

s:t:f+h󰀇150:

Thegovernment’stransferthereforehastwocomponents:1.AnexpansionofthebudgetsetfromItoI0=I+50.2.Theimpositionoftheconstraintthath󰀈50.

Thecanonicaleconomist’squestionis:whydoboth(1)and(2)whenyoucanjustdo(1)andpotentiallyimproveconsumerwelfareatnoadditionalcosttothegovernment?1.3.1

ASimpleExample:TheDeadweightLossofChristmas

JoelWaldfogel’s1993AmericanEconomicReviewpaperprovidesastylized(andcontroversial)exampleoftheapplicationoftheCarteBlancheprinciple.

Waldfogelobservesthatgift-givingisequivalenttoanin-kindtransferandhenceshouldbelesse¢cientforconsumerwelfarethansimplygivingcash.

InJanuary,1993,hesurveyedapproximately150Yaleundergraduatesabouttheirholidaygiftsreceivedin1992:

1.Whatwerethegiftsworthincashvalue

2.Howmuchthestudentsbewillingtopayforthemiftheydidn’talreadyhavethem

13

Severalinterestingobservationsfromthearticle:

1.Value‘destruction’isgreaterfordistantrelatives,e.g.,grandparents.2.Value‘preservation’isnear-perfectforfriends

3.Groupsthattendto‘destroy’themostvaluearethemostlikelytogivecashinstead

It’susefultobeabletointerpretthebasicregressionresultgivenonthetopofpage1332:ln(valuei)=󰀂0:314+0:964ln(pricei)

.

(0:44)(0:08)

Thethingsinparenthesesarestandarderrors.Since0:964ismuchlargerthan2󰀄0:08,therelationshipbetweenvalueandpriceisstatisticallysigni…cant.

Thederivativeofvaluewithrespecttopriceis(recallthat@=@xoflnxis@x=x):

@valuei@valueipricei

=󰀃=0:964:

@priceivaluei@pricei

Thatis,a1percentriseinpricetranslatesintoa0:964percentriseinvalue.

But,thereisamajordiscrepancybetweenthelevelofvalueandprice.Rewritingtheequationandexpo-nentiating:

ln(valuei)

=󰀂ln(exp(0:314))+0:964ln(pricei)

󰀇󰀆:964

price0i

=ln

exp(0:314)

Exponentiatingbothsides:

valuei

:964

price0i

=

exp(0:314)

0:964

pricei=

1:37

0:964

=:73󰀄pricei

So,fora$100gift,theapproximaterecipientvaluationisabout$62.

Youcanseewhyit’shandytousenaturallogarithmstoexpresstheserelationships.Theyreadilyallowforproportionale¤ects.Theregressionequationabovesaysthatthevalueofagiftisapproximatelyequalto96%ofitspriceminus31percent.

TheWaldfogelarticlegeneratedasuprisingamountofcontroversy,evenamongeconomists,mostofwhomprobablysubscribetotheCarteBlancheprinciple.

Tomanyreaders,thisarticleseemstoexemplifythewell-worngripeabouteconomists,“Theyknowthepriceofeverythingandthevalueofnothing.”WhatisWaldfogelmissing?

15

1.4TheExpenditureFunction

Wearenextgoingtolookatapotentiallyricher(andbetter)applicationofconsumertheory:thevalueofFoodStamps.

Beforethat,weneedsomemoremachinery.

Sofar,we’veanalyzedproblemswhereincomewasheldconstantandpriceschanges.ThisgaveustheIndirectUtilityFunction.

Now,wewanttoanalyzeproblemswhereutilityisheldconstandandexpenditureschange.ThisgivesustheExpenditureFunction.

Thesetwoproblemsarecloselyrelated–infact,theyare‘duals.’

Mosteconomicproblemshaveadualproblem,whichmeansaninverseproblem.

Forexample,thedualofchoosingoutputinordertomaximizepro…tsisminimizingcostsatagivenoutputlevel:costminimizationisthedualofpro…tmaximization.

Similarly,thedualofmaximizingutilitysubjecttoabudgetconstraintisminimizingexpendituressubjecttoautilityconstraint.1.4.1

Expenditurefunction

Consumer’sproblem:maximizeutilitysubjecttoabudgetconstraint.

Dual:minimizingexpendituresubjecttoautilityconstraint(i.e.alevelofutilityyoumustachieve)Thisdualproblemyieldsthe“expenditurefunction”:theminimumexpenditurerequiredtoattainagivenutilitylevel.Setupofthedual1.Startwith:

maxU(x;y)

s:t:pxx+pyy

2.Solveforx󰀃,y󰀃)v󰀃=U(x󰀃;y󰀃)givenpx;py;I.V󰀃=V(px;py;I)

Vistheindirectutilityfunction.3.Nowsolvethefollowingproblem:

󰀇I

minpxx+pyy

s:t:U(x;y)󰀈v󰀃

16

givesE󰀃=pxx󰀃+pyy󰀃forU(x󰀃;y󰀃)=v󰀃.E󰀃=E(px;py;V󰀃)

1.4.2GraphicalrepresentationofdualproblemGraph 41yU = v*xThedualproblemconsistsinchoosingthelowestbudgetsettangenttoagivenindi¤erencecurve.Example:

minEs:t:x:5y:5

whereUpcomesfromtheprimalproblem.

󰀂󰀃

L=pxx+pyy+󰀃Up󰀂x:5y:5@L@x@L@y@L@󰀃

=px󰀂󰀃:5x󰀂:5y:5=0=py󰀂󰀃:5x:5y󰀂:5=0=Up󰀂x:5y:5=0

=pxx+pyy󰀈Up

17

The…rsttwooftheseequationssimplifyto:

x=

pyypx

WesubstituteintotheconstraintUp=x:5y:5toget

󰀆󰀇:5pyy

Up=y:5

px󰀆󰀇:5󰀆󰀇:5pypx󰀃󰀃

x=Up;y=Up

pxpy

󰀆󰀇:5󰀆󰀇:5

pxpy

Up+pyUpE󰀃=px

pxpy

5:5

=2p:xpyUp

HowdosolutionstoDualandPrimalcompare?1.4.3

RelationbetweenExpenditurefunctionandIndirectUtilityfunction

Let’slookattherelationbetweenexpenditurefunctionandindirectutilityfunction.

V(px;py;I0)E(px;py;U0)

V(px;py;E(px;py;U0))E(px;py;V(px;py;I0))

=U0=I0=U0=I0

ExpenditurefunctionandIndirectUtilityfunctionareinversesoneoftheother.Let’sverifythisintheexamplewesawabove.

󰀃

Recallthatprimalgaveusfactordemandsx󰀃p;ypasafunctionofpricesandincome(notutility).

Dualgaveusexpenditures(budgetrequirement)asafunctionofutilityandprices.II󰀃

x󰀃=;y=;U󰀃=pp

2px2py

Nowplugtheseintoexpediturefunction:

󰀆

󰀆I2px

󰀇:5󰀆

I2py

󰀇:5

󰀇:5󰀆󰀇:5II5:55:5

p:E󰀃=2Upp:xpy=Ixpy=2px2py

Finallynoticethatthemultipliersaresuchthatthemultiplierinthedualproblemistheinverseofthemultiplierintheprimalproblem.

UxUy

=pxpypxpy

=UxUy18

󰀃P󰀃D

==

1.5DemandFunctions

Now,let’susetheIndirectUtilityfunctionandtheExpenditurefunctiontogetDemandfunctions.Tonow,we’vebeensolvingfor:

󰀅Utilityasafunctionofpricesandbudget󰀅ExpenditureasafunctionofpricesandutilityImplicitlywehavealreadyfounddemandschedules.

Ademandscheduleisimmediatelyimpliedbyanindividualutilityfunction.

Foranyutilityfunction,wecansolveforthequantitydemandedofeachgoodasafunctionofitspricewiththepriceofallothergoodsheldconstantandeitherincomeheldconstantorutilityheldconstant.1.5.1

Marshalliandemand(‘Uncompensated’demand)

Inourpreviousexamplewhere:

U(x;y)=x:5y:5

wederived:

x(px;py;I)y(px;py;I)

=:5

I

pxI=:5

py

Ingeneralwewillwritethesedemandfunctions(forindividuals)as:

x󰀃1x󰀃2

=d1(p1;p2;:::;pn;I)=d2(p1;p2;:::;pn;I)

:::

x󰀃n

=dn(p1;p2;:::;pn;I)

Wecallthis“Marshallian”demandafterAlfredMarshallwho…rstdrewdemandcurves.1.5.2

Hicksiandemand(‘Compensated’demand)

Similarlywederivedthat:

19

x(px;py;U)y(px;py;U)

==

󰀆

󰀆

pypxpxpy

󰀇:5

UpUp

󰀇:5

Ingeneralwewillwritethesedemandfunctions(forindividual)as:

x󰀃1;cx󰀃2;c

=h1(p1;p2;:::;pn;U)=h2(p1;p2;:::;pn;U)

:::

x󰀃n;c

=hn(p1;p2;:::;pn;U)

Thisiscalled“Hicksian”orcompensateddemandafterJohnHicks.

Thisdemandfunctiontakesutilityasanargument,notincome.Thisturnsouttobeanimportantdistinc-tion.1.5.3

Graphicalderivationofdemandcurves

Ademandcurveforxasafunctionofpx

I/pyGraph 42dx(px,py,I)I/px20

Soademandfunctionisasetoftangencypointsbetweenindi¤erencecurvesandbudgetsetholdingIandpy(allotherprices)constant.

Whattypeofdemandcurveisthis?

Marshallian(dx(px;py;I).Utilityisnotheldconstant,butincomeis.Now,wehavethetoolstoanalyzetheFoodStampprogram.

21