Rules for Maximizing Utility | Microeconomics - Lumen Learning

If a consumer wants to maximize total utility, for every dollar that they spend, they should spend it on the item which yields the greatest marginal utility per ... Skiptomaincontent Microeconomics Module6:Utility Searchfor: RulesforMaximizingUtility LearningObjectives Explainwhymaximizingutilityrequiresthatthelastunitofeachitempurchasedmusthavethesamemarginalutilityperdollar Calculatetheutility-maximizingchoice Theproblemoffindingconsumerequilibrium,thatis,thecombinationofgoodsandservicesthatwillmaximizeanindividual’stotalutility,comesdowntocomparingthetrade-offsbetweenoneaffordablecombination(shownbyapointonthebudgetlineinFigure1,below)withalltheotheraffordablecombinations. Mostpeopleapproachtheirutility-maximizingcombinationofchoicesinastep-by-stepway.Thisstep-by-stepapproachisbasedonlookingatthetradeoffs,measuredintermsofmarginalutility,ofconsuminglessofonegoodandmoreofanother.Youcanthinkofthisstep-by-stepapproachasthe“biggestbangforthebuck”principle.Forexample,saythatJoséstartsoffthinkingaboutspendingallhismoneyonT-shirtsandchoosingpointP,whichcorrespondstofourT-shirtsandnomovies,asillustratedinFigure1. Figure1.AChoicebetweenConsumptionGoods.Joséhasincomeof$56.Moviescost$7andT-shirtscost$14.ThepointsonthebudgetconstraintlineshowthecombinationsofmoviesandT-shirtsthatareaffordable. Joséchoosesthisstartingpointrandomly;hehastostartsomewhere.ThenheconsidersgivingupthelastT-shirt,theonethatprovideshimtheleastmarginalutility,andusingthemoneyhesavestobuytwomoviesinstead.Table1tracksthestep-by-stepseriesofdecisionsJoséneedstomake(Key:T-shirtscost$14,moviescost$7,andJose’s incomeis$56). Table1.AStep-by-StepApproachtoMaximizingUtility Try WhichHas TotalUtility MarginalGainandLossofUtility,ComparedwithPreviousChoice Conclusion Choice1:P 4T-shirtsand0movies 81from4T-shirts+0from0movies=81      –      – Choice2:Q 3T-shirtsand2movies 63from3T-shirts+31from0movies=94 Lossof18from1lessT-shirt,butgainof31from2moremovies,foranetutilitygainof13 QispreferredoverP Choice3:R 2T-shirtsand4movies 43from2T-shirts+58from4movies=101 Lossof20from1lessT-shirt,butgainof27fromtwomoremoviesforanetutilitygainof7 RispreferredoverQ Choice4:S 1T-shirtand6movies 22from1T-shirt+81from6movies=103 Lossof21from1lessT-shirt,butgainof23fromtwomoremovies,foranetutilitygainof2 SispreferredoverR Choice5:T 0T-shirtsand8movies 0from0T-shirts+100from8movies=100 Lossof22from1lessT-shirt,butgainof19fromtwomoremovies,foranetutilitylossof3 SispreferredoverT DECISIONMAKINGBYCOMPARINGMARGINALUTILITY Josécouldusethefollowingthoughtprocess(ifhethoughtinutils)tomakehisdecisionregardinghowmanyT-shirtsandmoviestopurchase: Step1.FromTable1,JosécanseethatthemarginalutilityofthefourthT-shirtis18.IfJoségivesupthefourthT-shirt,thenheloses18utils. Step2.GivingupthefourthT-shirt,however,freesup$14(thepriceofaT-shirt),allowingJosétobuythefirsttwomovies(at$7each). Step3.Joséknowsthatthemarginalutilityofthefirstmovieis16andthemarginalutilityofthesecondmovieis15.Thus,ifJosémovesfrompointPtopointQ,hegivesup18utils(fromtheT-shirt),butgains31utils(fromthemovies). Step4.Gaining31utilsandlosing18utilsisanetgainof13.ThisisjustanotherwayofsayingthatthetotalutilityatQ(94accordingtothelastcolumninTable1)is13morethanthetotalutilityatP(81). Step5.So,forJosé,itmakessensetogiveupthefourthT-shirtinordertobuytwomovies. JoséclearlypreferspointQtopointP.Nowrepeatthisstep-by-stepprocessofdecisionmakingwithmarginalutilities.JoséthinksaboutgivingupthethirdT-shirtandsurrenderingamarginalutilityof20,inexchangeforpurchasingtwomoremoviesthatpromiseacombinedmarginalutilityof27.JosépreferspointRtopointQ.WhatifJoséthinksaboutgoingbeyondRtopointS?GivingupthesecondT-shirtmeansamarginalutilitylossof21,andthemarginalutilitygainfromthefifthandsixthmovieswouldcombinetomakeamarginalutilitygainof23,soJosépreferspointStoR. However,ifJoséseekstogobeyondpointStopointT,hefindsthatthelossofmarginalutilityfromgivingupthefirstT-shirtis22,whilethemarginalutilitygainfromthelasttwomoviesisonlyatotalof19.IfJoséweretochoosepointT,hisutilitywouldfallto100.Throughthesestagesofthinkingaboutmarginaltradeoffs,JoséagainconcludesthatS,withoneT-shirtandsixmovies,isthechoicethatwillprovidehimwiththehighestleveloftotalutility.Thisstep-by-stepapproachwillreachthesameconclusionregardlessofJosé’sstartingpoint. Thisapproachtofindingconsumerequilibriumissomewhattedious.Next,we’llturntoaquickerandmoreintuitiveapproach. MaximizingUtilityRule Thisprocessofdecisionmakingdescribedpreviouslysuggestsaruletofollowwhen maximizingutility.SincethepriceofT-shirtsisnotthesameasthepriceofmovies,it’snotenoughtojustcomparethemarginalutilityofT-shirtswiththemarginalutilityofmovies.Instead,weneedtocontrolforthepricesofeachproduct.Wecandothisbycomputingandcomparingmarginalutilityperdollarofexpenditureforeachproduct. MarginalutilityperdollaristheamountofadditionalutilityJoséreceivesgiventhepriceoftheproduct. [latex]\begin{array}{rcl}\text{marginalutilityperdollar}&=&\frac{\text{marginalutility}}{\text{price}}\end{array}[/latex] ForJosé’sT-shirtsandmovies,themarginalutilityperdollarisshowninTable2. Table2.MarginalUtilityperDollar QuantityofT-Shirts TotalUtility MarginalUtility MarginalUtilityperDollar QuantityofMovies TotalUtility MarginalUtility MarginalUtilityperDollar 1 22 22 22/$14=1.6 1 16 16 16/$7=2.3 2 43 21 21/$14=1.5 2 31 15 15/$7=2.14 3 63 20 20/$14=1.4 3 45 14 14/$7=2 4 81 18 18/$14=1.3 4 58 13 13/$7=1.9 5 97 16 16/$14=1.1 5 70 12 12/$7=1.7 6 111 14 14/$14=1 6 81 11 11/$7=1.6 7 123 12 12/$14=0.86 7 91 10 10/$7=1.4 ARuleformaximizingUtility Ifaconsumerwantstomaximizetotalutility,foreverydollarthattheyspend,theyshouldspenditontheitemwhichyieldsthegreatestmarginalutilityperdollarofexpenditure. ApplyingtheRule José’sfirstpurchasewillbeamovie.Why?José’schoicesaretopurchaseeitheraT-shirtoramovie.Table1showsthatthemarginalutilityperdollarspentonthefirstT-shirtis1.6comparedwith2.5forthefirstmovie. BecausethefirstmoviegivesJosémoremarginalutilityperdollarthanthefirstT-shirt,andbecausethemovieiswithinhisbudget,hewillpurchaseamoviefirst. Joséwillcontinuetopurchasethegoodwhichgiveshimthehighestmarginalutilityperdollaruntilheexhauststhebudget.Joséwillkeeppurchasingmoviesbecausetheygivehimagreater“bangforthebuck”untilthesixthmovieisequivalenttoaT-shirtpurchase.JosécanaffordtopurchasethatT-shirt.SoJoséwillchoosetopurchasesixmoviesandoneT-shirt.Thatcombination,sixmoviesandoneT-shirt,ishisconsumerequilibrium AnotherRuleforMaximizingUtility SincethepriceofT-shirtsistwiceashighasthepriceofmovies,tomaximizeutilitythelastT-shirtchosenneedstoprovideexactlytwicethemarginalutility(MU)ofthelastmovie.IfthelastT-shirtprovideslessthantwicethemarginalutilityofthelastmovie,thentheT-shirtisprovidingless“bangforthebuck”(i.e.,marginalutilityperdollarspent)thanifthesamemoneywerespentonmovies.Ifthisisso,JoséshouldtradetheT-shirtformoremoviestoincreasehistotalutility.MarginalutilityperdollarmeasurestheadditionalutilitythatJoséwillenjoygivenwhathehastopayforthegood. IfthelastT-shirtprovidesmorethantwicethemarginalutilityofthelastmovie,thentheT-shirtisprovidingmore“bangforthebuck”ormarginalutilityperdollar,thanifthemoneywerespentonmovies.Asaresult,JoséshouldbuymoreT-shirts.NoticethatatJosé’soptimalchoicethemarginalutilityfromthefirstT-shirt,of22isexactlytwicethemarginalutilityofthesixthmovie,whichis11.Atthischoice,themarginalutilityperdollaristhesameforbothgoods.Thisisatell-talesignalthatJoséhasfoundthepointwithhighesttotalutility. Thisargumentcanbewrittenasanotherrule:theutility-maximizingchoicebetweenconsumptiongoodsoccurswherethemarginalutilityperdollaristhesameforbothgoods,andtheconsumerhasexhaustedhisorherbudget. [latex]\displaystyle\frac{MU_1}{P_1}=\frac{MU_2}{P_2}[/latex] Asensibleeconomizerwillpaytwiceasmuchforsomethingonlyif,inthemarginalcomparison,theitemconferstwiceasmuchutility.Noticethattheformulaforthetableaboveis [latex]\displaystyle\frac{22}{14}=\frac{11}{7}[/latex] [latex]1.6=1.6[/latex] Thefollowingfeatureprovidesstep-by-stepguidanceforthisconceptofutility-maximizingchoices. AnotherRuleforMaximizingUtility Therule,[latex]\displaystyle\frac{MU_1}{P_1}=\frac{MU_2}{P_2}[/latex],meansthatthelastdollarspentoneachgoodprovidesexactlythesamemarginalutility.So: Step1.IfwetradedadollarmoreofmoviesforadollarmoreofT-shirts,themarginalutilitygainedfromT-shirtswouldexactlyoffsetthemarginalutilitylostfromfewermovies.Inotherwords,thenetgainwouldbezero. Step2.Products,however,usuallycostmorethanadollar,sowecannottradeadollar’sworthofmovies.ThebestwecandoistradetwomoviesforanotherT-shirt,sinceinthisexampleT-shirtscosttwicewhatamoviedoes. Step3.IfwetradetwomoviesforoneT-shirt,wewouldendupatpointR(twoT-shirtsandfourmovies). Step4.Choice4inTable3showsthatifwemovetopointS,wewouldlose21utilsfromonelessT-shirt,butgain23utilsfromtwomoremovies,sowewouldendupwithmoretotalutilityatpointS. Table3.AStep-by-StepApproachtoMaximizingUtility Try WhichHas TotalUtility MarginalGainandLossofUtility,ComparedwithPreviousChoice Conclusion Choice1:P 4T-shirtsand0movies 81from4T-shirts+0from0movies=81      –      – Choice2:Q 3T-shirtsand2movies 63from3T-shirts+31from0movies=94 Lossof18from1lessT-shirt,butgainof31from2moremovies,foranetutilitygainof13 QispreferredoverP Choice3:R 2T-shirtsand4movies 43from2T-shirts+58from4movies=101 Lossof20from1lessT-shirt,butgainof27fromtwomoremoviesforanetutilitygainof7 RispreferredoverQ Choice4:S 1T-shirtand6movies 22from1T-shirt+81from6movies=103 Lossof21from1lessT-shirt,butgainof23fromtwomoremovies,foranetutilitygainof2 SispreferredoverR Choice5:T 0T-shirtsand8movies 0from0T-shirts+100from8movies=100 Lossof22from1lessT-shirt,butgainof19fromtwomoremovies,foranetutilitylossof3 SispreferredoverT Inshort,theruleshowsustheutility-maximizingchoice. Thereisanother,equivalentwaytothinkaboutthis.Therulecanalsobeexpressedas theratioofthepricesofthetwogoodsshouldbeequaltotheratioofthemarginalutilities.Whenthepriceofgood1isdividedbythepriceofgood2,attheutility-maximizingpointthiswillequalthemarginalutilityofgood1dividedbythemarginalutilityofgood2.Thisrulecanbewritteninalgebraicform: [latex]\displaystyle\frac{P_1}{P_2}=\frac{MU_1}{MU_2}[/latex] Alongthebudgetconstraint,thetotalpriceofthetwogoodsremainsthesame,sotheratioofthepricesdoesnotchange.However,themarginalutilityofthetwogoodschangeswiththequantitiesconsumed.AttheoptimalchoiceofoneT-shirtandsixmovies,pointS,theratioofmarginalutilitytopriceforT-shirts(22:14)matchestheratioofmarginalutilitytopriceformovies(of11:7). WatchIt Thisvideoappliesthesesameconceptstoagraph.Itdemonstrateshowtherewillcomeapointwhenthemarginalutilityperdollarofonegoodgoesdownenoughsothatitmakesmoresensetobuyanothergood. MeasuringUtilitywithNumbers Thisdiscussionofutilitystartedoffwithanassumptionthatitispossibletoplacenumericalvaluesonutility,anassumptionthatmayseemquestionable.Youcanbuyathermometerformeasuringtemperatureatthehardwarestore,butwhatstoresellsan“utilimometer”formeasuringutility?However,whilemeasuringutilitywithnumbersisaconvenientassumptiontoclarifytheexplanation,thekeyassumptionisnotthatutilitycanbemeasuredbyanoutsideparty,butonlythatindividualscandecidewhichoftwoalternativestheyprefer. Tounderstandthispoint,thinkbacktothestep-by-stepprocessoffindingthechoicewithhighesttotalutilitybycomparingthemarginalutilitythatisgainedandlostfromdifferentchoicesalongthebudgetconstraint.AsJosécompareseachchoicealonghisbudgetconstrainttothepreviouschoice,whatmattersisnotthespecificnumbersthatheplacesonhisutility—orwhetherheusesanynumbersatall—butonlythathepersonallycanidentifywhichchoicesheprefers. Inthisway,thestep-by-stepprocessofchoosingthehighestlevelofutilityresemblesrathercloselyhowmanypeoplemakeconsumptiondecisions.Wethinkaboutwhatwillmakeusthehappiest;wethinkaboutwhatthingscost;wethinkaboutbuyingalittlemoreofoneitemandgivingupalittleofsomethingelse;wechoosewhatprovidesuswiththegreatestlevelofsatisfaction.Thevocabularyofcomparingthepointsalongabudgetconstraintandtotalandmarginalutilityisjustasetoftoolsfordiscussingthiseverydayprocessinaclearandspecificmanner.Itiswelcomenewsthatspecificutilitynumbersarenotcentraltotheargument,sinceagoodutilimometerishardtofind.Donotworry—whilewecannotmeasureutils,bytheendofthenextmodule,wewillhavetransformedouranalysisintosomethingwecanmeasure—demand. TryIt TryIt Thesequestionsallowyoutogetasmuchpracticeasyouneed,asyoucanclickthelinkatthetopofthefirstquestion(“Tryanotherversionofthesequestions”)togetanewsetofquestions.Practiceuntilyoufeelcomfortabledoingthequestions. TryIt Thesequestionsallowyoutogetasmuchpracticeasyouneed,asyoucanclickthelinkatthetopofthefirstquestion(“Tryanotherversionofthesequestions”)togetanewsetofquestions.Practiceuntilyoufeelcomfortabledoingthequestions. Glossary consumerequilibrium: thecombinationofgoodsandservicesthatwillmaximizeanindividual’stotalutility marginalutilityperdollar: theadditionalsatisfactiongainedfrompurchasingagoodgiventhepriceoftheproduct;MU/Price Contribute! Didyouhaveanideaforimprovingthiscontent?We’dloveyourinput. 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