Module 3: Budget Constraint – Intermediate Microeconomics

The budget constraint is the set of all the bundles a consumer can afford given that consumer's income. We assume that the consumer has a budget – an amount of ... Skiptocontent MainBody Module3–BudgetConstraint ThePolicyQuestion:HybridCarPurchaseTaxCredit—IsittheGovernment’sBestChoicetoReduceFuelConsumptionandCarbonEmissions? TheU.S.governmentpolicyofextendingtaxcreditstowardthepurchaseofelectricandhybridcarscanhaveconsequencebeyonddecreasingcarbonemissions.Forinstance,aconsumerthatpurchasesahybridcarcouldspendlessmoneyongasandhavemoremoneytospendonotherthings.Thishasimplicationsforboththeindividualconsumerandthelargereconomy. Eventherichestpeople–fromBillGatestoOprahWinfrey–can’taffordtoowneverythingintheworld.Eachofushasabudgetthatlimitstheextentofourconsumption.Economistscallthislimitabudgetconstraint.Inourpolicyexample,anindividual’schoicebetweenconsuminggasolineandeverythingelseisconstrainedbyhisorhercurrentincome.Anyadditionalmoneyspentongasolineismoneythatisnotavailableforothergoodsandservicesandvice-versa.Thisiswhythebudgetconstraintiscalledaconstraint. Thebudgetconstraintisgovernedbyincomeontheonehand,howmuchmoneyaconsumerhasavailabletospendonconsumption,andthepricesofthegoodstheconsumerpurchasesontheother. ExploringthePolicyQuestion Whataresomeofthebudgetimplicationsforaconsumerwhoownsahybridcar?Whatpurchasedecisionsmightthisconsumermakegivenhisorhersavingsongas,andhowdoesthis,inturn,affectthegoalsofthetaxsubsidypolicy? 3.1DescriptionoftheBudgetConstraint LO1:Defineabudgetconstraint,conceptually,mathematically,andgraphically. 3.2TheSlopeoftheBudgetLine LO2:Interprettheslopeofthebudgetline. 3.3ChangesinPricesandIncome LO3:Illustratehowchangesinpricesandincomealterthebudgetconstraintandbudgetline. 3.4Coupons,Vouchers,andTaxes LO4:Illustratehowcoupons,vouchers,andtaxesalterthebudgetconstraintandbudgetline. 3.5PolicyExample:TheHybridCarSubsidyandConsumers’Budgets 3.1DescriptionoftheBudgetConstraint LO1:Defineabudgetconstraint,conceptually,mathematically,andgraphically. Thebudgetconstraintisthesetofallthebundlesaconsumercanaffordgiventhatconsumer’sincome.Weassumethattheconsumerhasabudget–anamountofmoneyavailabletospendonbundles.Fornow,wedonotworryaboutwherethismoneyorincomecomesfrom,wejustassumeaconsumerhasabudget. Sowhatcanaconsumerafford?Answeringthisdependsonthepricesofthegoodsinquestion.Supposeyougotothecampusstoretopurchaseenergybarsandvitaminwater.Ifyouhave$5tospend,energybarscostfiftycentseach,andvitaminwatercosts$1abottle,thenyoucouldbuy10bars,andnovitaminwater,nobarsand5bottlesofvitaminwater,4barsand2vitaminwatersandsoon. Thistableshowsthepossiblecombinationsofenergybarsandvitaminwaterthestudentcanbuyforexactly$5: EnergyBars BottlesofVitaminWater 10 0 8 1 6 2 4 3 2 4 0 5 Itisalsotruethatyoucouldspendlessthan$5andhavemoneyleftover.Sowehavetoconsiderallpossiblebundles−includingconsumingnoneatall. Notethatwearefocusingonbundlesoftwogoodssothatwemaintaintractability(asexplainedinmodule1),butitissimpletothinkbeyondtwogoodsbydefiningoneofthegoodsas“moneyspentoneverythingelse.” Mathematically,thetotalamounttheconsumerspendsontwogoods,AandB,is: (3.1),[latex]P_{A}A+P_{B}B[/latex] where[latex]P_{A}[/latex] isthepriceofgoodAand[latex]P_{B}[/latex] isthepriceofgoodB.Ifthemoneytheconsumerhastospendonthetwogoods,hisincome,isgivenasI,thenthebudgetconstraintis: (3.2)[latex]P_{A}A+P_{B}B\leqI[/latex] Notetheinequality:Thisequationstatesthattheconsumercannotspendmorethanhisincomebutcanspendless.Wecansimplifythisassumptionbyrestrictingtheconsumertospendingallofhisincomeonthetwogoods.Thiswillallowustofocusonthefrontierofthebudgetconstraint.AsweshallseeinModule4,thisassumptionisconsistentwiththemore-is-betterassumption–ifyoucanconsumemore(ifyourincomeallowsit)youshouldbecauseyouwillmakeyourselfbetteroff.Withthisassumptioninplace,wecanwritethebudgetconstraintas: (3.3)[latex]P_{A}A+P_{B}B=I[/latex] Graphically,wecanrepresentthisbudgetconstraintasinFigure3.1.Wecallthisthebudgetline:Thelinethatindicatesthepossiblebundlestheconsumercanbuywhenspendingallhisincome. Figure3.1 Thebudgetlineisthegraphofthebudgetconstraintequation(3.3). LO2:Interprettheslopeofthebudgetline Fromthegraphofthebudgetconstraintinsection3.1,wecanseethatthebudgetlineslopesdownwardandhasaconstantslopealongitsentirelength.Thismakesintuitivesense:Ifyoubuymoreofonegood,youaregoingtohavetobuylessoftheothergood.Therateatwhichyoucantradeonefortheotherisdeterminedbythepricesofthetwogoods,andtheydon’tchange. WecanseethesedetailsinFigure3.2 Figure3.2 Interceptsandslopeforthebudgetline   Wecanfindtheslopeofthebudgetlineeasilybyrearrangingequation(3.3)sothatweisolateBononeside.Notethatinourgraph,Bisthegoodontheverticalaxis,sowewillrearrangeourequationtolooklikeastandardfunctionwithBasthedependentvariable: (3.4)[latex]B=\frac{I}{P_{B}}-\frac{P_{A}}{P_{B}}A[/latex] Now,wehaveourbudgetlinerepresentedinpoint-slopeformwhere: Thefirstpart,[latex]\frac{I}{P_{B}}[/latex],istheverticalintercept. Thesecondpart,[latex]-\frac{P_{A}}{P_{B}}[/latex] ,istheslopecoefficientonA. Notethattheslopeofthebudgetlineissimplytheratiooftheprices,alsoknownasthepriceratio.Thisistherateatwhichyoucantradeonegoodfortheotherinthemarketplace.Toseethis,let’sreturntothecampusstorewith$5tospendonenergybarsandvitaminwater. Supposeyouoriginallydecidedtobuy5bottlesofvitaminwaterandplacedtheminabasket.Aftersomethought,youdecidedtotrade1bottlefor2energybars.Nowyouhave4bottlesofvitaminwaterand2energybarsinthebasket.Ifyouwantevenmorebars,thesametradeoffisavailable:2morebarscanbehadifyougiveuponebottleofvitaminwater,andsoon. Theslopeofthebudgetlineisalsocalledtheeconomicrateofsubstitution(ERS). TheslopeofthebudgetlinealsorepresentstheopportunitycostofconsumingmoreofgoodAbecauseitdescribeshowmuchofgoodBtheconsumerhastogiveuptoconsumeonemoreunitofgoodA.Theopportunitycostofsomethingisthevalueofthenext-bestalternativegivenupinordertodogetit.Forexample,ifyoudecidetobuyonemorebottleofvitaminwater,youhavetogiveuptwoenergybars.Notethatopportunitycostisnotlimitedtotheconsumptionofmaterialgoods.Forexample,theopportunitycostofanhour’snapmightbethehourofstudyingmicroeconomicsthatdidnothappenbecauseofit. ChangesinPricesandIncome LO3:Illustratehowchangesinpricesandincomealterthebudgetconstraintandbudgetline. Fromourmathematicaldescriptionofthebudgetline,wecaneasilyseehowchangesinpricesandincomeaffectthebudgetlineandaconsumer’schoiceset—thesetofallthebundlesavailabletoheratcurrentpricesandincome.Let’sgobacktoequation(3.3): (3.3)[latex]P_{A}A+P_{B}B=I[/latex] WeknowfromthepreviousFigure3.3thattheverticalinterceptforequation(3.3)is[latex]\frac{I}{P_{B}}[/latex] andthehorizontalinterceptis[latex]\frac{I}{P_{A}}[/latex]. NowconsideranincreaseinthepriceofgoodA.NoticeinFigure3.3thatthisincreasedoesnotaffecttheverticalintercept,onlythehorizontalintercept.As[latex]P_{A}[/latex] increases,[latex]\frac{I}{P_{A}}[/latex] decreases,movingclosertotheorigin.Thischangemakesthebudgetline‘steeper’ormorenegativelyslopedaswecanseefromtheslopecoefficient:[latex]-\frac{P_{A}}{P_{B}}[/latex].As[latex]P_{A}[/latex] increases,thisratioincreasesinabsolutevalue,sotheslopebecomesmorenegativeorsteeper.Whatthismeansintuitivelyisthatthetrade-offoropportunitycosthasrisen.Now,theconsumerhastogiveupmoreofgoodBtoconsumeonemoreunitofgoodA. Figure3.3 Changingthepriceofonegoodchangestheslopeofthebudgetline. Next,considerachangeinincome.Supposetheconsumergetsanadditionalamountofmoneytospend,soIincreases.Iaffectsbothintercepttermspositively,soasIincreasesboth[latex]\frac{I}{P_{B}}[/latex]  and[latex]\frac{I}{P_{A}}[/latex]increaseormoveawayfromtheorigin.ButIdoesnotaffecttheslope:[latex]-\frac{P_{A}}{P_{B}}[/latex].Thustheshiftinthebudgetlineisaparallelshiftoutward–theconsumerwiththeadditionalincomecanaffordmoreofboth. 4Coupons,Taxes,andVouchers LO4:Illustratehowcoupons,vouchersandtaxesalterthebudgetconstraintandbudgetline. Budgetconstraintscanchangeduetochangesinpricesandincome,butlet’snowconsiderothercommonfeaturesofthereal-worldmarketthatcanaffectthebudgetconstraint.Westartwithcouponsorothermethodsfirmsusetogivediscountstoconsumers. Consideracouponorasalethatgivesconsumersadiscountonthepriceofoneiteminourbudgetconstraintproblem.Acouponthatentitlesthebearertoapercentageoffinpriceisessentiallyareductioninpriceandhaspreciselythesameeffect.Forexample,a20%offcoupononagoodthatnormallycosts$10isthesameasreducingthepriceto$8. Morecomplicatedisacouponthatgivesapercentageofftheentirepurchase.Inthiscase,thepercentageistakenfromthepriceofbothitemsAandBinourbudgetconstraintproblem.Inthiscase,thepriceratio,ortheslopeofthebudgetconstraint,doesnotchange. Forexample,ifthepriceofAisregularly$10andthepriceofBisregularly$20thenwith20%offtheentirepurchase,thenewpricesare$8and$16respectively.Intuitively,wecanseethatthisisequivalenttoincreasingtheincome,andachievesthesameresult:byexpandingthebudgetset,theconsumercannowaffordbundleswithmoreofbothgoods. Product RegularPrice NewPricewith 20%discounton entirepurchase A $10 $8 B $20 $16 Anothercommondiscountisonamaximumnumberofitems.Forexample,youmightseeanadvertisementfor20%offuptothreeunitsofgoodA.ThisdiscountlowerstheopportunitycostofAintermsofBforthefirstthreeunits,butrevertsbacktotheoriginalopportunitycostthereafter.Figure3.4illustratesthis. Figure3.4Theeffectofa20%discountonthefirstA̅ unitsofA. Taxeshavethesameeffectsascouponsbutintheoppositedirection.Anadvaloremtaxisataxbasedonthevalueofagood,suchasapercentagesalestax.Intermsofthebudgetconstraint,anadvaloremtaxonaspecificgoodisequivalenttoanincreaseinprice,asshowninFigure3.5.Ageneralsalestaxonallgoodshastheeffectofaparallelshiftofthebudgetlineinward.Notealsothatincometaxesare,inthiscase,functionallyequivalenttoageneralsalestax,theycauseaparallelshiftinwardofthebudgetline. Figure3.5 Anadvaloremtaxchangestheslopeandhorizontalinterceptofthebudgetline. Vouchersthatentitlethebearertoacertainquantityofagood(eithervalueorquantity)areslightlymorecomplicated.Let’sreturntoyourpurchaseofvitaminwaterandenergybars.Supposeyouhaveavoucherfor2freeenergybars. Youhave$5 Thepriceof1energybaris$0.50 Thepriceof1bottleofvitaminwateris$1. Howwouldwenowdrawyourbudgetline? Oneplacetostartistoconsiderthesimplebundlethatcontains2energybarsand2bottlesofvitaminwater.Notethatgivingup1or2barsdoesnotallowthestudenttoconsumeanymorevitaminwater.Theopportunitycostofthese2barsis0,andsothebudgetlineinthisparthasa0slope.Afterusingthevoucher,ifthestudentwantsmorethan2barstheopportunitycostisthesameasbefore–.05abottleofvitaminwater–andsothebudgetlinefromthispointonisthesameasbefore.Thenewbudgetlinewiththevoucherhasakink. 5PolicyExample:TheHybridCarSubsidyandConsumers’Budgets Forseveralmodules,wehaveconsideredthepolicyofahybridcartaxcredit.InModule1,wethoughtaboutvariousdrivingpreferencesofatypicalconsumer.InModule2,wetranslatedthesepreferencesintoatypeofutilityfunctionandcorrespondingindifferencecurve.Now,let’sthinkabouttheappropriatebudgetlineforourpolicyexample. Tostart,let’susethesametwoaxesasweusedfortheindifferencecurvemapasshowninFigure3.6.Inotherwords,let’splace‘milesdriven’onthehorizontalaxisand$,whichisallthemoneyspentonotherconsumptionontheverticalaxis.Fornow,wewon’tspecifythepreciselevelofincome.. Nowwecanask,whatisthepriceof‘otherconsumption?’Sincewearetalkingaboutmoneyleftoverafterpayingformilesdriven,thepriceforotherconsumptionissimply1.Thisisbecausewearetalkingaboutmoneyitselfandthepriceofadollarisadollar.So,theinterceptonthisaxisissimplythevalueofI. Butwhatisthepriceofamiledriven?Thisquestionismorecomplicatedandincludesthecostofmaintenanceanddepreciation.However,becausewearefocusedontheeffectofincreasingthemilespergallonofgas,let’sconcentrateononlythecostasitrelatestothepurchaseofgasoline.Inthiscase,thecostofdrivingamileisthepriceofgasolinedividedbythecar’smilespergallon(MPG).Sinceweareagaininterestednotinanindividualbutagroup,wecanusetheaveragepriceofagallonofregulargasdividedbytheaverageMPGofcarsdrivenintheUnitedStatesasareasonableapproximationofthecostofamiledriveninanon-hybridcr.Nowwehavethe‘price’ofdrivingamile;dividingincomebythispricegivesustheinterceptonthe‘milesdriven’axis. Figure3.6 Aconsumer’sbudgetconstraintforthehybridcarpolicy Nowthatwehaveabudgetconstraintforourelectricandhybridcarsubsidypolicyexample,wecanseetheeffectofthepolicyontheconstraint.DoublingtheMPGfrom20,say,to40,dramaticallyreducesthepriceofdrivingamile.Thisreductioncausesthe‘milesdriven’intercepttomoveupwardsandtheentirebudgetconstrainttomoveoutward.Notethatnowthetypicalconsumercanaffordtoconsumebundleswithmoreofbothmilesdrivenandeverythingelse–bundlesthatwereunavailabletothempriortothepolicy. Equation(3.4)summarizesthebudgetconstraintformilesdrivenandothergoods. (3.4)Income=(PMilesDriven)(MilesDriven)+DollarsSpentonOtherConsumption I ExploringthePolicyQuestions Whatcanwesayabouttheavailabilityofbundlesafterthehybridcartaxcreditisenactedcomparedtobefore?Dothebundlesrepresentmoreconsumptionofonlymilesdrivenordotheyrepresentmoreofothergoodsaswell? Anothertypeofcarthatishighmileage(highMPG)isadieselcar.IntheUnitedStates,however,thepriceofdieselgasistypicallyhigherthanthepriceofregulargas.HowwouldonlyhigherMPGshiftthebudgetlineinFigure3.9?Howwouldonlyhigherpricedgasshiftthebudgetlineinfigure3.9?HowwouldthesetwofactorstogetheralterthebudgetlinefromFigure3.9? Ifthegovernmentsubsidizesthepurchaseofhybridcarsthrougharebatethataddstotheincomeofconsumers,whathappenstothebudgetlineinFigure3.9? SUMMARY Review:TopicsandRelatedLearningOutcomes 3.1DescriptionoftheBudgetConstraint LO1:Defineabudgetconstraint 3.2TheSlopeoftheBudgetLine LO2:Discusstheinterpretationoftheslopeofthebudgetline 3.3ChangesinPricesandIncome LO3:Illustratehowchangesinpricesandincomealterthebudgetconstraintandbudgetline 3.4Coupons,VouchersandTaxes LO4:Illustratehowcoupons,vouchersandtaxesalterthebudgetconstraintandbudgetline 3.5PolicyExample Learn:KeyTermsandGraphs Terms AdValoremTax BudgetConstraint BudgetLine EconomicRateofSubstitution OpportunityCost Graphs Normalbudgetconstraint Budgetconstraintwithcoupon Budgetconstraintwithvoucher Equations Budgetconstraint   Previous/nextnavigation Previous:Module2:Utility Next:Module4:ConsumerChoice Backtotop License IntermediateMicroeconomicsbyPatrickM.EmersonislicensedunderaCreativeCommonsAttribution-NonCommercial-ShareAlike4.0InternationalLicense,exceptwhereotherwisenoted. ShareThisBook ShareonTwitter