Topic 2 Individual Preferences.docx

Topic 2 Individual Preferences.docx

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Topic2IndividualPreferences

E5650–MicroeconomicTheory

Topic2:

IndividualPreferences

PrimaryReadings:

DL–Chapter4;JR-Chapter3;Varian–Chapter7

Inmosteconomicmodels,westartwithanagent'sutilityfunction.Theutilityfunctionbasicallymapsfrombundlesthattheagentmightchoose,totherealline.Theutilityfunctionisquiteconvenient:

itcanbemaximizedandmanipulatedusingmathematicaltools.Butthequestionis:

Isitvalidtoreduceasimple,real-valuefunction,somethingascomplicatedasanagent'spreferencesoverawidevarietyofbundles?

Whatdoesitreallymeanabouttheagent'spreferences?

Areweimposingsomehiddenordesirableassumptionswhenwetakethisapproach?

Inthislecture,wewilltrytoanswerthesequestionsbyanalyzingtherelationshipsbetweenaxiomsaboutanagent'spreferences,andthenestablishingtheexistenceofautilityfunctionthatrepresentstheagent'spreferences.

2.1TheConsumer'sPreferences

2.1.1ConsumptionSet

Welettheconsumptionset,X,representthesetofallalternatives,orcompleteconsumptionplans,thattheconsumercanconceive-whethersomeofthemwillbeachievableinpracticeornot.EveryelementofXiscalledaconsumptionbundleoraconsumptionplan.

∙Xcapturestheuniverseofallpossiblechoicesaconsumermayhave.Forthisreason,theconsumptionsetisalsoknownasthechoiceset.

∙Normally,XRm+-theentirenonnegativeorthantoftherealspaceRm.

∙WewillalwaysassumethatXisaclosedandconvexset.

2.1.2BasicProperties&AxiomsofPreferences

∙Forx,yX,whenwewritex

y,wemeanthat"theconsumerthinksthatthebundlexisatleastasgoodasbundley."Wecall

apreferencerelationonX.

∙Wesay,"xis（weakly）preferredtoy".

∙Itisclearthat

isabinaryrelationdefinedonX.

Asthefinalpurposeofintroducingapreferencerelationistoorderthesetofconsumptionbundles,weneedtoassumeanumberofaxioms.Theseaxiomsofconsumerchoiceareintendedtogiveformalmathematicalexpressiontofundamentalaspectsofconsumerbehaviorandaltitudestowardtheobjectsofchoice.

AXIOM1:

（Completeness）x,yX,（x

y）（y

x）.（Note:

="or"）

∙Tosatisfythecompletenessaxiom,thepreference

cannotbedefinedsothatx

yxjyj,j.（Reason:

itisonlyapartialordering.）

（Note:

Whilethisaxiomappearsinnocuous,incombinationwiththeusualconfinementoftheconsumptionsettotheconsumptionoftheindividualonly,itrulesoutexternalitiesinconsumption.）

AXIOM2:

（Reflexivity）xX,x

x.

AXIOM3:

（Transitivity）（x

y）&（y

z）x

z.（Note:

&="and"）

Note:

∙Thefirstassumptionsaysthatanytwobundlescanbecompared,thesecondistrivial,andthethirdisnecessaryforanydiscussionsofpreferencemaximization:

forifpreferenceswerenottransitive,theremightbesetsofbundleswhichhadnobestelements.

Itisusefultoextendournotation:

∙Wewritexyandsaythatxisstrictlypreferredtoy.Wesometimealsowritenoty

x,meaningyisnotpreferredtox,whichisthesameasxy,givencompleteness.

∙Wewritex~yif（x

y）&（x

y）andsaythatxisindifferenttoy.

Examples:

（a）FiniteSet

∙IfXisafiniteset,thenapreferencerelationonXwillpartitionXintoafinitenumberofsubsetssuchthat

∙elementswithinasubsetareallindifferent;

∙Therewillbeastrictpreferenceforelementsfromdifferentsubsets.

（b）SummationOrdering:

∙LetX=Rm.

∙Definex

ytomeanthat

∙Itiseasytoshowthatthissummationorderingiscomplete,reflectiveandtransitive.

（c）LexicographicOrdering

∙LetX=Rm+.

∙x

yifandonlyif

∙either,thereexistssomejsuchthatxi=yiforiyj;

∙or,xi=yifor1im.

∙Essentially,thelexicographicorderingcomparesthecomponentsoneatatimebeginningwiththefirst,anddeterminestheorderingbasedonethefirstadifferenceisfound.

∙Thisimpliesthatthevectorwithgreatestcomponentisrakedthehighest.

Theabovethreeaxiomsarethebasicpropertiesofapreferencerelation.Anyrelationsatisfyingthese3axiomsiscalledanordering.Inordertohaveafunctionalrepresentation,wemayneedafewmoreaxioms（assumptions）.（IfXiscountable,noadditionalaxiomisneeded.）

AXIOM4:

（Continuity）ForallyinX,thesets{x:

x

y}and{x:

y

x}areclosedsets.Itfollowsthatthesets{x:

xy}and{x:

yx}areopensets.

∙Thisassumptionisnecessarytoruleoutcertaindiscontinuousbehavior.

∙Itsaysthat,if（xi）isasequenceofconsumptionbundlesthatareallatleastasgoodasyandifthissequenceconvergestosomebundlex*,thenx*isatleastasgoodasy.

∙Thekeyconsequenceofcontinuityisasfollows:

ifyisstrictlypreferredtozandifxisbundlethatiscloseenoughtoy,thenxmustbestrictlypreferredtoz.

Examples

∙Summationorderingiscontinuous.

∙Lexicographicorderisdiscontinuous（seethefollowingdiagramonR2+）

x2

{（x1,x2）（1,1）}

1

1x1

AXIOM4A:

（StrongMonotonicity）Ifxyandxy,thenxy.

AXIOM4B:

（WeakMonotonicity）Ifxiyiforalli,thenx

y.

∙Weakmonotonicitysaysthat"atleastasmuchofeverythingisatleastasgood."Iftheconsumercancostlesslydisposeofunwantedgoods,thisassumptionistrivial.

∙Strongmonotonicitysaysthatatleastasmuchofeverygood,andstrictlymoreofsomegood,isstrictlybetter.Thisissimplysaysassumingthatgoodsaregood.

∙Ifoneofthegoodsisa"bad",suchasgarbageorpollution,thenstrongmonotonicitywillnotbesatisfied.Butwecaneasilygetaroundthisproblembyrespecifyingthegoodtobeabsenceofgarbage,orabsenceofpollution,whichwillnormallyleadtostrongmonotonicity.

AXIOM5:

（Local?

Nonsatiation）GivenanyxinXand>0,thenthereissomebundleyinXwith||x-y||

（Analternativedefinition:

requiringthistoholdoversomesetthatcontainthesetdefinedbytherelevantbudgetconstraint.）

∙Localnonsatiationsaysthatonecanalwaysdoalittlebitbetter,evenifoneisrestrictedtoonlysmallchangeintheconsumptionbundle.

∙Itcanbeshownthatstrongmonotonicityimplieslocalnonsatiationbutnotviceversa.

∙Keyconsequenceoflocalnonsatiationrulesout"thick"indifferencecurves.

Thefollowingtwoassumptionsareoftenusedtoguaranteenicebehaviorofconsumerdemandfunctions.

AXIOM6A:

（Convexity）Givenx,y,zXsuchthatx

zandy

z,thentx+（1-t）y

zforall0t1.

AXIOM6B:

（StrictConvexity）Givenxy,zXsuchthatx

zandy

z,thentx+（1-t）yzforall0

∙Convexityimpliesthatanagentprefersaveragetoextremes.

∙Convexityisageneralizationoftheneoclassicalassumptionof"diminishingmarginalratesofsubstitution."

Beforewemoveonthefunctionalrepresentationofthepreferencerelation,wemustemphasizethattheapreferencerelationisanordinal,ratherthancardinal,concepteventhoughwehaveattemptedtoincorporateadditionalstructuresbyimposingsomeoftheaboveassumptions.

2.2UtilityFunctions

Autilityfunctionisareal-valuedfunctionudefinedontheconsumptionsetXsuchthatpreferencerankingsarepreservedbythemagnitudeofu.Thatis,autilityfunctionuhasthepropertythatgivenanytwoelementsx,yinX,u（x）u（y）ifandonlyifx

y.

Butnotallpreferencerelationscanberepresentedbyutilityfunctions.Arathergeneralresultisthatanycontinuouspreferenceorderingcanberepresentedbyacontinuousutilityfunction.Thisisaverydifficultresulttoprove（Debreu（1959））.（Moreover,whileanycontinuousorderingisalwaysrepresentable,continuityisnotnecessary.Thenecessaryandsufficientconditionsforrepresentationisrathertechnical;seeNg1979/83,App.1B.）Wewillfocusonasomewhatsimplerresult-theonethatcanbeprovedconstructively.Themainideasare:

∙Weselectarbitraryfixedlinethatcutsalloftheindifferencecurves（orsurfaces）.

∙Onceutilityisdefinedalongthisline,theutilityofanyotherpointisfoundbytracingtheappropriateindifferencecurvetothelineandusingtheutilityvaluethere.

∙Theassumptionofstrongmonotonicityguaranteesthattheindifferencecurvesexitandthatanylineoftheforme,>0ande>0,cutsthemall.

ExistenceofUtilityFunctions

∙SupposethatareferencerelationonX=Rm+iscomplete,reflexive,transitive,continuous,andstronglymonotone.Thenthereexistsacontinuousutilityfunctionu:

Rm+Rwhichrepresentsthepreferencerelation.

x2

e

x1

Proof

LetebethevectorinRm+consistingofallones.Thengivenanyvectorx,let

u（x）=suchthatx~e.

Wenowneedtoshowthatu（x）iswell-defined,i.e.,itexistsandunique.

Definethefollowingtwosets:

A={:

0,e

x}

B={:

0,x

e}

Thenbystrongmonotonicity,Aisnonempty.Biscertainlynonemptysince0B.BothAandBareclosedbythecontinuityassumption.Ontheotherhand,bythecompletenessassumption,weknowthatevery（0）mustbelongtoAB,thatis,AB=R+.

Notethatif*AB,then*e~xsothatwecanletu（x）=*.Therefore,weneedtoprovethatABisnonempty.

Bymonotonicity,itfollowsthatAimpliesthat'Aforall'.SinceAisclosedsubsetofR+,itmustbeinaformofclosedinterval[*,+）,whichimpliesthatB=[0,*]sinceBisanonemptyclosedsetsuchthatAB=R+.

Wenowhavetoprovethatthevalue*mustbeunique.Let1e~xand2e~x.Thenitisclearthat1e~1e（transitivitypropertyof"~"）.Bystrongmonotonicity,itmustbethecasethat1=2.

Letusprovethattheabove-definedutilityfunctionactually