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ChapterOneTheMarket---AppreciatingEconomicModelingThePurposeofthisChapterTobegintounderstandtheartofbuildinganeconomicmodelTobegintounderstandthreebasicelementsofmodelingineconomics:PurposeSimplificationthroughassumptionsValuejudgmentThePurposeofanEconomicModel
Thepurposeofaneconomicmodelistohelpprovidepreciseinsights(精确的洞察力)onaspecificeconomicphenomenon.Thus:Differentphenomenaneedsdifferentmodel;SimplificationbyassumptionisnecessaryAnIllustration:ModelingtheApartmentMarketPurpose:Howareapartmentrentsdetermined?Arerents“desirable”?Simplifyingassumptions:apartmentsarecloseordistant,butotherwiseidenticaldistantapartmentsrentsareexogenous(外生变量)andknownmanypotentialrentersandlandlordsTwoVeryCommonModelingAssumptionsRationalChoice(理性选择):Eachpersontriestochoosethebestalternativeavailabletohimorher.Equilibrium(均衡):economicagentsinteractwitheachother,resultinginanequilibrium,inwhicheachpersonreachesanoptimaldecisiongivenothers’decisions.ModelingApartmentDemandDemand:Supposethemostanyonepersoniswillingtopaytorentacloseapartmentis$500/month.Then p=$500
QD=1.Supposethepricehastodropto$490beforea2ndpersonwouldrent.Then p=$490
QD=2.ModelingApartmentDemandTheloweristherentalratep,thelargeristhequantityofcloseapartmentsdemanded
p
QD
.Thequantitydemandedvs.pricegraphisthemarketdemandcurveforcloseapartments.MarketDemandCurveforApartmentspQDModelingApartmentSupplySupply:Ittakestimetobuildmorecloseapartmentssointhisshort-runthequantityavailableisfixed(atsay100).MarketSupplyCurveforApartmentspQS100CompetitiveMarketEquilibrium
(竞争性市场均衡)Quantitydemanded=quantityavailable
pricewillneitherrisenorfallsothemarketisatacompetitiveequilibrium.CompetitiveMarketEquilibriumpQD,QSpe100Peoplewillingtopaypeforcloseapartmentsgetcloseapartments.Peoplenotwillingtopay
peforcloseapartments
getdistantapartments.ComparativeStatics
(静态比较分析)Whatisexogenousinthemodel?priceofdistantapartmentsquantityofcloseapartmentsincomesofpotentialrenters.Whathappensiftheseexogenousvariableschange?Note:Wearenotanalyzingthetransitionprocessordynamicprocess.ComparativeStaticsSupposethepriceofdistantapartmentrises.Demandforcloseapartmentsincreases(rightwardshift),causingAhigherpriceforcloseapartments.MarketEquilibriumpQD,QSpe100Higherdemandcauseshigher
marketprice;samequantity
traded.MoreComparativeStatics
(DoThemYourself)Supposethereweremorecloseapartments.Or,renters’incomerises;ElaborationoftheBasicModel:TaxationPolicyAnalysisLocalgovernmenttaxesapartmentowners.Whathappenstopricequantityofcloseapartmentsrented?Isanyofthetax“passed”torenters?TaxationPolicyAnalysisMarketsupplyisunaffected.Marketdemandisunaffected.Sothecompetitivemarketequilibriumisunaffectedbythetax.Priceandthequantityofcloseapartmentsrentedarenotchanged.Landlordspayallofthetax.ImperfectlyCompetitiveMarket
Case1:AMonopolisticLandlordLandlordsetsarentalpricepherentsD(p)apartments.Revenue=pD(p).HechoosesptomaximizespD(p),subjecttoD(p)<=S(totalnumberofapartmentsinhishands)Typically,hisoptimalpissuchthat D(p)<S,thatis,therearevacantapartments.MonopolisticMarketEquilibriumpQD,QSMiddle
priceMiddleprice,mediumquantity
demanded,largerrevenue.
Monopolistdoesnotrentallthe
closeapartments.100Vacantcloseapartments.ImperfectlyCompetitiveMarketCase2:PerfectlyDiscriminatoryMonopolisticLandlordImaginethemonopolistknewwillingness-to-payofeverybody,Charge$500tothemostwilling-to-pay,charge$490tothe2ndmostwilling-to-pay,etc.DiscriminatoryMonopolisticMarketEquilibriumpQD,QS100p1=$5001DiscriminatoryMonopolisticMarketEquilibriumpQD,QS100p1=$500p2=$49012DiscriminatoryMonopolisticMarketEquilibriumpQD,QS100p1=$500p2=$49012p3=$4753DiscriminatoryMonopolisticMarketEquilibriumpQD,QS100p1=$500p2=$49012p3=$4753DiscriminatoryMonopolisticMarketEquilibriumpQD,QS100p1=$500p2=$49012p3=$4753peDiscriminatorymonopolist
chargesthecompetitivemarket
pricetothelastrenter,and
rentsthecompetitivequantity
ofcloseapartments.RentControl
(房租管制)Localgovernmentimposesamaximumlegalprice,pmax<pe,thecompetitiveprice.MarketEquilibriumpQD,QSpe100pmaxExcessdemandThe100closeapartmentsarenolongerallocatedby
willingness-to-pay(lottery,lines,
largefamiliesfirst?).ValueJudgement
(价值判断)Whichofthefollowingisbetter?RentcontrolPerfectcompetitionMonopolyDiscriminatorymonopolyBut,whatdoyoumeanby“better”?ParetoEfficiency
(帕累托效率)VilfredoPareto;1848-1923.AParetooutcomeallowsno“wastedwelfare”;i.e.theonlywayoneperson’swelfarecanbeimprovedistoloweranotherperson’swelfare.ParetoEfficiencyAParetoinefficientoutcomemeansthereremainunrealizedmutualgains-to-trade.Anymarketoutcomethatachievesallpossiblegains-to-trademustbeParetoefficient.ParetoEfficiencyCompetitiveequilibrium:allcloseapartmentrentersvaluethematthemarketpricepeormoreallothersvaluecloseapartmentsatlessthanpesonomutuallybeneficialtradesremainsotheoutcomeisParetoefficient.ParetoEfficiencyDiscriminatoryMonopoly:assignmentofapartmentsisthesameaswiththeperfectlycompetitivemarketsothediscriminatorymonopolyoutcomeisalsoParetoefficient.ParetoEfficiencyMonopoly:notallapartmentsareoccupiedsoadistantapartmentrentercouldbeassignedacloseapartmentandhavehigherwelfarewithoutloweringanybodyelse’swelfare.sothemonopolyoutcomeisParetoinefficient.ParetoEfficiencyRentControl:somecloseapartmentsareassignedtorentersvaluingthematbelowthecompetitivepricepesomerentersvaluingacloseapartmentabovepedon’tgetcloseapartmentsParetoinefficientoutcome.FurtherExtensionsoftheModelOvertime,willthesupplyofcloseapartmentsincrease?rentcontroldecreasethesupplyofapartments?amonopolistsupplymoreapartmentsthanacompetitiverentalmarket?AQuickSummaryEconomicmodelingservesaspecificpurpose;Forthepurpose,aneconomicmodelneedstomakesimplifyingassumptions;Economistsdohavevaluejudgments,whicharebasedonspecifiedcriterion.ChapterTwoBudgetConstraintWhereareWeintheCourse?
Weareworkingonthe1stofthe3componentsofmicroeconomics:Consumerbehavior,productiontheory,andmarket.Therearethreeelementsofconsumerbehavior:budgetconstraint,preference,andchoices.ConsumptionChoiceSetsAconsumptionchoicesetisthecollectionofallconsumptionchoicesavailabletotheconsumer.Whatconstrainsconsumptionchoice?Budgetary,timeandotherresourcelimitations.BudgetConstraintsQ:Whenisabundle(x1,…,xn)affordableatpricesp1,…,pn?A:When
p1x1+…+pnxn
£
m
wheremistheconsumer’s(disposable)income.BudgetConstraintsTheconsumer’sbudgetsetisthesetofallaffordablebundles;
B(p1,…,pn,m)=
{(x1,…,xn)|x1
³0,…,xn
³0and
p1x1+…+pnxn
£
m}Thebudgetconstraintistheupperboundaryofthebudgetset.BudgetSetandConstraintforTwoCommoditiesx2x1Budgetconstraintisp1x1+p2x2=m.m/p1AffordableJustaffordableNotaffordablem/p2BudgetSetandConstraintforTwoCommoditiesx2x1p1x1+p2x2=misx2=-(p1/p2)x1+m/p2soslopeis-p1/p2.m/p1BudgetSetm/p2BudgetConstraintsIfn=3whatdothebudgetconstraintandthebudgetsetlooklike?BudgetSetforThreeCommoditiesx2x1x3m/p2m/p1m/p3{(x1,x2,x3)|x1
³0,x2³0,x3
³
0andp1x1+p2x2+p3x3
£
m}BudgetConstraintsForn=2andx1onthehorizontalaxis,theconstraint’sslopeis-p1/p2.Whatdoesitmean?
Increasingx1by1mustreducex2byp1/p2.BudgetConstraintsx2x1Opp.costofanextraunitof
commodity1isp1/p2units
foregoneofcommodity2.And
theopp.costofanextra
unitofcommodity2is
p2/p1unitsforegone
ofcommodity1.-p2/p1+1BudgetSets&Constraints;IncomeandPriceChangesThebudgetconstraintandbudgetsetdependuponpricesandincome.Whathappensaspricesorincomechange?HigherincomegivesmorechoiceOriginalbudgetsetNewaffordableconsumption
choicesx2x1Originalandnewbudgetconstraintsareparallel(sameslope).BudgetConstraints-IncomeChangesNooriginalchoiceislostandnewchoicesareaddedwhenincomeincreases,sohigherincomecannotmakeaconsumerworseoff.Anincomedecreasemay(typicallywill)maketheconsumerworseoff.BudgetConstraints-PriceChangesWhathappensifjustonepricedecreases?Supposep1decreases.Howdothebudgetsetandbudgetconstraintchangeasp1
decreasesfromp1’top1”?Originalbudgetsetx2x1m/p2m/p1’m/p1”NewaffordablechoicesBudgetconstraintpivots;slopeflattensfrom-p1’/p2to-p1”/p2-p1’/p2-p1”/p2BudgetConstraints-PriceChangesReducingthepriceofonecommoditypivotstheconstraintoutward.Nooldchoiceislostandnewchoicesareadded,soreducingonepricecannotmaketheconsumerworseoff.UniformAdValoremSalesTaxes
(从价税)Anadvaloremsalestaxleviedatarateof5%increasesallpricesby5%,frompto(1+0×05)p=1×05p.Anadvaloremsalestaxleviedatarateoftincreasesallpricesbytpfrompto(1+t)p.Auniformsalestaxisapplieduniformlytoallcommodities.UniformAdValoremSalesTaxesAuniformsalestaxleviedatratetchangestheconstraintfrom
p1x1+p2x2=m
to
(1+t)p1x1+(1+t)p2x2=m
i.e.
p1x1+p2x2=m/(1+t).UniformAdValoremSalesTaxesx2x1Equivalentincomeloss
isTheFoodStampProgramFoodstampsarecouponsthatcanbelegallyexchangedonlyforfood.Howdoesacommodity-specificgiftsuchasafoodstampalterafamily’sbudgetconstraint?TheFoodStampProgramGF100100F+G=100;beforestamps.TheFoodStampProgramGF100100F+G=100:beforestamps.Budgetsetafter40food
stampsissued.140Thebudget
setisenlarged.40TheFoodStampProgramWhatiffoodstampscanbetradedonablackmarketfor$0.50each?TheFoodStampProgramGF100100F+G=100:beforestamps.Budgetconstraintafter40
foodstampsissued.140120Budgetconstraintwith
blackmarkettrading.40TheFoodStampProgramGF100100F+G=100:beforestamps.Budgetconstraintafter40
foodstampsissued.140120Blackmarkettrading
makesthebudget
setlargeragain.40BudgetConstraints-RelativePrices“Numeraire”means“unitofaccount”.Supposepricesandincomearemeasuredindollars.Sayp1=$2,p2=$3,m=$12.Thentheconstraintis
2x1+3x2=12.BudgetConstraints-RelativePricesTheconstraintforp1=2,p2=3,m=12
2x1+3x2=12
isalso1.x1+(3/2)x2=6,
theconstraintforp1=1,p2=3/2,m=6.Settingp1=1makescommodity1thenumeraireanddefinesallpricesrelativetop1;e.g.3/2isthepriceofcommodity2relativetothepriceofcommodity1.BudgetConstraints-RelativePricesAnycommoditycanbechosenasthenumerairewithoutchangingthebudgetsetorthebudgetconstraint.BudgetConstraints-RelativePricesp1=2,p2=3andp3=6
priceofcommodity2relativetocommodity1is3/2,priceofcommodity3relativetocommodity1is3.Relativepricesaretheratesofexchangeofcommodities2and3forunitsofcommodity1.ShapesofBudgetConstraintsButwhatifpricesarenotconstants?E.g.bulkbuyingdiscounts,orpricepenaltiesforbuying“toomuch”.Thenconstraintswillbecurved.ShapesofBudgetConstraints-QuantityDiscountsSupposep2isconstantat$1butthatp1=$2for0£x1
£20andp1=$1forx1>20.ShapesofBudgetConstraints-QuantityDiscountsSupposep2isconstantat$1butthatp1=$2for0£x1
£20andp1=$1forx1>20.Thentheconstraint’sslopeis
-2,for0£x1
£20
-p1/p2=
-1,forx1>20
andtheconstraintis{ShapesofBudgetConstraintswithaQuantityDiscountm=$1005010020Slope=-2/1=-2
(p1=2,p2=1)Slope=-1/1=-1
(p1=1,p2=1)80x2x1ShapesofBudgetConstraintswithaQuantityDiscountm=$1005010020Slope=-2/1=-2
(p1=2,p2=1)Slope=-1/1=-1
(p1=1,p2=1)80x2x1ShapesofBudgetConstraintswithaQuantityDiscountm=$100501002080x2x1BudgetSetBudgetConstraintShapesofBudgetConstraintswithaQuantityPenaltyx2x1BudgetSetBudgetConstraintShapesofBudgetConstraints-OnePriceNegativeCommodity1isstinkygarbage.Youarepaid$2perunittoacceptit;i.e.p1=-$2.p2=$1.Income,otherthanfromacceptingcommodity1,ism=$10.Thentheconstraintis
-2x1+x2=10orx2=2x1+10.ShapesofBudgetConstraints-OnePriceNegative10Budgetconstraint’sslopeis-p1/p2=-(-2)/1=+2x2x1x2=2x1+10ShapesofBudgetConstraints-OnePriceNegative10x2x1
Budgetsetis
allbundlesfor
whichx1
³0,
x2
³0andx2
£2x1+10.MoreGeneralChoiceSetsChoicesareusuallyconstrainedbymorethanabudget;e.g.timeconstraintsandotherresourcesconstraints.Abundleisavailableonlyifitmeetseveryconstraint.MoreGeneralChoiceSetsFoodOtherStuff10Atleast10unitsoffoodmustbeeatentosurviveMoreGeneralChoiceSetsFoodOtherStuff10BudgetSetChoiceisalsobudget
constrained.MoreGeneralChoiceSetsFoodOtherStuff10Choiceisfurtherrestrictedbyatimeconstraint.MoreGeneralChoiceSetsSowhatisthechoiceset?MoreGeneralChoiceSetsFoodOtherStuff10MoreGeneralChoiceSetsFoodOtherStuff10MoreGeneralChoiceSetsFoodOtherStuff10Thechoicesetistheintersectionofalloftheconstraintsets.SummaryThechoicesetisthesetconsistsofallpossiblechoicesofaconsumer;Inthesimplestcase,thechoicesetistypicallyboundedbyasinglebudgetconstraintwhoseslopeisthenegativeratioofprices;Ingeneralthechoicesetcanbeboundedbymultipleconstraints.ChapterThreePreferences消费者偏好WhereAreWeintheCourse?
Wearestudyingthe1stofthethreeblocksofmicroeconomics:Consumerbehavior,productiontheory,andmarketequilibriumWithinthe1stblock,weareworkingonthe2ndofthethreecomponents:choiceset,preference,andconsumerdemandWhatDoWeMeanbyPreference?(偏好)Itreferstotheorderedrelationshipamongalternativechoicesgivenbyaneconomicagent.Inmosteconomicliterature,consumerpreferenceistreatedastheultimateexogenouselement.PreferenceRelationsComparingtwodifferentconsumptionbundles,xandy:strictpreference:xismorepreferredthanisy.weakpreference:xisasatleastaspreferredasisy.Indifference:xisexactlyaspreferredasisy.Notationsdenotesstrictpreference;
~denotesindifference;denotesweakpreference;
p~fPreferenceRelationsxyandyximplyx~y.xyand(notyx)implyxy.~f~f~f~fpAssumptionsaboutPreferenceRelationsCompleteness:Foranytwobundlesxandyitisalwayspossibletomakethestatementthateither
xy
or
yx.~f~fAssumptionsaboutPreferenceRelationsReflexivity:Anybundlexisalwaysatleastaspreferredasitself;i.e.
xx.~fAssumptionsaboutPreferenceRelationsTransitivity:If
xisatleastaspreferredasy,and
yisatleastaspreferredasz,then
xisatleastaspreferredasz;i.e.
xyandyzxz.~f~f~fIndifferenceCurves
无差异曲线(或,无差异集)Takeareferencebundlex’.Thesetofallbundlesequallypreferredtox’istheindifferencecurvecontainingx’;thesetofallbundlesy~x’.Sinceanindifference“curve”isnotalwaysacurveabetternamemightbeanindifference“set”.IndifferenceCurvesx2x1x”x”’x’~x”~x”’x’IndifferenceCurvesx2x1z
x
yppxyzIndifferenceCurvesx2x1xAllbundlesinI1arestrictlypreferredtoallinI2.yzAllbundlesinI2arestrictlypreferredto
allinI3.I1I2I3IndifferenceCurvesx2x1WP(x),thesetof
bundlesweaklypreferredtox.WP(x)
includes
I(x).xI(x)IndifferenceCurvesx2x1SP(x),thesetof
bundlesstrictlypreferredtox,
doesnot
includeI(x).xI(x)IndifferenceCurvesCannotIntersect!(不相交!)x2x1xyzI1I2FromI1,x~y.FromI2,x~z.Thereforey~z.IndifferenceCurvesCannotIntersect
x2x1xyzI1I2FromI1,x~y.FromI2,x~z.Thereforey~z.ButfromI1andI2weseeyz,a
contradiction.pSlopesofIndifferenceCurvesWhenmoreofacommodityisalwayspreferred,thecommodityisagood.Ifeverycommodityisagoodthenindifferencecurvesarenegativelysloped.SlopesofIndifferenceCurvesBetterWorseGood2Good1Twogoods
anegativelyslopedindifferencecurve.SlopesofIndifferenceCurvesIflessofacommodityisalwayspreferredthenthecommodityisabad.SlopesofIndifferenceCurvesBetterWorseGood2Bad1Onegoodandone
badapositivelyslopedindifferencecurve.ExtremeCasesofIndifferenceCurves;PerfectSubstitutesIfaconsumeralwaysregardsunitsofcommodities1and2asequivalent,thenthecommoditiesareperfectsubstitutesandonlythesumofthetwocommoditiesinbundlesdeterminestheirpreferencerank-order.ExtremeCasesofIndifferenceCurves;PerfectSubstitutesx2x1881515Slopesareconstantat-1.I2I1BundlesinI2allhaveatotal
of15unitsandarestrictlypreferredtoallbundlesin
I1,whichhaveatotalof
only8unitsinthem.ExtremeCasesofIndifferenceCurves;PerfectComplementsIfaconsumeralwaysconsumescommodities1and2infixedproportion(e.g.one-to-one),thenthecommoditiesareperfectcomplementsandonlythenumberofpairsofunitsofthetwocommoditiesdeterminesthepreferencerank-orderofbundles.ExtremeCasesofIndifferenceCurves;PerfectComplementsx2x1I2I145o5959Sinceeachof(5,5),(5,9)and(9,5)contains5pairs,eachislesspreferredthanthebundle(9,9)
whichcontains9pairs.
PreferencesExhibitingSatiationAbundlestrictlypreferredtoanyotherisasatiationpointorablisspoint.Whatdoindifferencecurveslooklikeforpreferencesexhibitingsatiation?IndifferenceCurvesExhibitingSatiationx2x1BetterBetterBetterSatiation
(bliss)
pointIndifferenceCurvesforDiscreteCommoditiesAcommodityisinfinitelydivisibleifitcanbeacquiredinanyquantity;e.g.waterorcheese.Acommodityisdiscreteifitcomesinunitlumpsof1,2,3,…andsoon;e.g.aircraft,shipsandrefrigerators.IndifferenceCurvesforDiscreteCommodities
(studythisyourself)Supposecommodity2isaninfinitelydivisiblegood(gasoline)whilecommodity1isadiscretegood(aircraft).Whatdoindifference“curves”looklike?IndifferenceCurvesWithaDiscreteGood
(studythisyourself)Gas-olineAircraft01234Indifference“curves”
arecollectionsof
discretepoints.Well-BehavedPreferencesApreferencerelationis“well-behaved”ifitismonotonicandconvex.Monotonicity:Moreofanycommodityisalwayspreferred(i.e.nosatiationandeverycommodityisagood).Well-BehavedPreferencesConvexity(凸性):Mixturesofbundlesare(atleastweakly)preferredtothebundlesthemselves.E.g.,the50-50mixtureofthebundlesxandyis
z=(0.5)x+(0.5)y.
zisatleastaspreferredasxory.Well-BehavedPreferences--Convexity.x2y2x1y1xyz=(tx1+(1-t)y1,tx2+(1-t)y2)ispreferredtoxandyforall0<t<1.Well-BehavedPreferences--Convexity.x2y2x1y1xyPreferencesarestrictlyconvex
whenallmixturesz
arestrictly
preferredtotheir
component
bundlesxandy.zWell-BehavedPreferences--WeakConvexity.x’y’z’Preferencesareweaklyconvexifatleastonemixturezisequallypreferredtoacomponentbundle.xzyNon-ConvexPreferencesx2y2x1y1zBetterThemixturez
islesspreferredthanxory.MoreNon-ConvexPreferencesx2y2x1y1zBetterThemixturez
islesspreferredthanxory.MarginalRate-of-Substitution
(边际替代率)Theslopeofanindifferencecurveisitsmarginalrate-of-substitution(MRS)(边际替代率)HowcanaMRSbecalculated?MarginalRateofSubstitutionx2x1x’MRSatx’istheslopeofthe
indifferencecurveatx’MarginalRateofSubstitutionx2x1
MRSatx’is
lim{Dx2/Dx1}
Dx10
=dx2/dx1atx’Dx2Dx1x’MarginalRateofSubstitutionx2x1dx2dx1dx2=MRS´dx1so,atx’,MRSistherateatwhichtheconsumerisonlyjustwillingtoexchangecommodity2forasmallamountofcommodity1.x’MRS&Ind.CurvePropertiesBetterWorseGood2Good1Twogoods
anegativelyslopedindifferencecurveMRS<0.MRS&Ind.CurvePropertiesBetterWorseGood2Bad1Onegoodandone
badapositivelyslopedindifferencecurveMRS>0.MRS&Ind.CurvePropertiesGood2Good1MRS=-5MRS=-0.5MRSalwaysincreaseswithx1(becomeslessnegative)ifandonlyifpreferencesarestrictly
convex.MRS&Ind.CurvePropertiesx1x2MRS=-0.5MRS=-5MRSdecreases
(becomesmorenegative)
asx1increases
nonconvexpreferences
MRS&Ind.CurvePropertiesx2x1MRS
=-0.5MRS=-1MRS=-2MRSisnotalwaysincreasingasx1increasesnonconvex
preferences.Summary:
TheKeyConceptofthisChapterItistheindifferencecurve(IC)!Itconsistsofalltheconsumptionbundlesthatareindifferenttotheconsumer;TheshapeofanICtellsalotabouttheconsumer’spreference: ---Itsslopeiscalledthemarginalrate-of-substitution;---WecandefineconvexityofapreferencewhichcorrespondstotheshapeoftheIC.ChapterFourUtility效用WhatDoWeDoinThisChapter?Wecreateamathematicalmeasureofpreferenceinordertoadvanceouranalysis.UtilityFunctionsApreferencerelationthatiscomplete,reflexive,transitivecanberepresentedbyautilityfunction.UtilityFunctionsAutilityfunctionU(x)representsapreferencerelationifandonlyif:
x’x”U(x’)>U(x”)
x’x”U(x’)<U(x”)
x’~x”U(x’)=U(x”).~fppUtilityFunctionsUtilityisanordinal(i.e.ordering)concept.E.g.ifU(x)=6andU(y)=2thenbundlexisstrictlypreferredtobundley.Butxisnotpreferredthreetimesasmuchasisy.UtilityFunctions&Indiff.CurvesAllbundlesinanindifferencecurvehavethesameutilitylevel.U(x1,x2)=Constantistheequationofanindifferencecurve.UtilityFunctions&Indiff.CurvesUº6Uº4(2,3)
(2,2)
~
(4,1)x1x2pUtilityFunctions&Indiff.CurvesThecollectionofallindifferencecurvesforagivenpreferencerelationisanindifferencemap.Anindifferencemapisequivalenttoautilityfunction.UtilityFunctionsIfUisautilityfunctionthatrepresentsapreferencerelationandfisastrictlyincreasingfunction,thenV=f(U)isalsoautilityfunction
representing.~f~fGoods,BadsandNeutralsAgoodisacommodityunitwhichincreasesutility(givesamorepreferredbundle).Abadisacommodityunitwhichdecreasesutility(givesalesspreferredbundle).Aneutralisacommodityunitwhichdoesnotchangeutility(givesanequallypreferredbundle).Goods,BadsandNeutralsUtilityWaterx’Unitsof
waterare
goodsUnitsof
waterare
badsAroundx’units,alittleextrawaterisaneutral.Utility
functionSomeOtherUtilityFunctionsandTheirIndifferenceCurvesConsider
V(x1,x2)=x1+x2.
Whatdotheindifferencecurvesforthis“perfectsubstitution”utilityfunctionlooklike?PerfectSubstitutionIndifferenceCurves55991313x1x2x1+x2=5x1+x2=9x1+x2=13V(x1,x2)=x1+x2.PerfectSubstitutionIndifferenceCurves55991313x1x2x1+x2=5x1+x2=9x1+x2=13Allarelinearandparallel.V(x1,x2)=x1+x2.SomeOtherUtilityFunctionsandTheirIndifferenceCurvesConsider
W(x1,x2)=min{x1,x2}.
Whatdotheindifferencecurvesforthis“perfectcomplementarity”utilityfunctionlooklike?PerfectComplementarityIndifferenceCurvesx2x145omin{x1,x2}=8358358min{x1,x2}=5min{x1,x2}=3W(x1,x2)=min{x1,x2}PerfectComplementarityIndifferenceCurvesx2x145omin{x1,x2}=8358358min{x1,x2}=5min{x1,x2}=3Allareright-angledwithverticesonaray
fromtheorigin.W(x1,x2)=min{x1,x2}SomeOtherUtilityFunctionsandTheirIndifferenceCurvesAutilityfunctionoftheform
U(x1,x2)=f(x1)+x2
islinearinjustx2andiscalledquasi-linear.E.g.U(x1,x2)=2x11/2+x2.Quasi-linearIndifferenceCurvesx2x1Eachcurveisaverticallyshiftedcopyoftheothers.SomeOtherUtilityFunctionsandTheirIndifferenceCurvesAnyutilityfunctionoftheform
U(x1,x2)=x1a
x2b
witha>0andb>0iscalledaCobb-Douglasutilityfunction.E.g.U(x1,x2)=x11/2x21/2(a=b=1/2)
V(x1,x2)=x
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