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ChapterTwo,BudgetConstraint,WhereareWeintheCourse?,Weareworkingonthe1stofthe3componentsofmicroeconomics:Consumerbehavior,productiontheory,andmarket.Therearethreeelementsofconsumerbehavior:budgetconstraint,preference,andchoices.,ConsumptionChoiceSets,Aconsumptionchoicesetisthecollectionofallconsumptionchoicesavailabletotheconsumer.Whatconstrainsconsumptionchoice?Budgetary,timeandotherresourcelimitations.,BudgetConstraints,Q:Whenisabundle(x1,…,xn)affordableatpricesp1,…,pn?A:Whenp1x1+…+pnxnmwheremistheconsumer’s(disposable)income.,BudgetConstraints,Theconsumer’sbudgetsetisthesetofallaffordablebundles;B(p1,…,pn,m)={(x1,…,xn)|x10,…,xn0andp1x1+…+pnxnm}Thebudgetconstraintistheupperboundaryofthebudgetset.,BudgetSetandConstraintforTwoCommodities,,,x2,x1,Budgetconstraintisp1x1+p2x2=m.,m/p1,,,,,Affordable,,,,Justaffordable,,Notaffordable,,,,,,m/p2,,,,BudgetSetandConstraintforTwoCommodities,,,x2,x1,p1x1+p2x2=misx2=-(p1/p2)x1+m/p2soslopeis-p1/p2.,m/p1,,BudgetSet,,,m/p2,BudgetConstraints,Ifn=3whatdothebudgetconstraintandthebudgetsetlooklike?,BudgetSetforThreeCommodities,,,,x2,x1,x3,m/p2,m/p1,m/p3,,{(x1,x2,x3)|x10,x20,x30andp1x1+p2x2+p3x3m},,,BudgetConstraints,Forn=2andx1onthehorizontalaxis,theconstraint’sslopeis-p1/p2.Whatdoesitmean?Increasingx1by1mustreducex2byp1/p2.,BudgetConstraints,x2,x1,,,,Opp.costofanextraunitofcommodity1isp1/p2unitsforegoneofcommodity2.Andtheopp.costofanextraunitofcommodity2isp2/p1unitsforegoneofcommodity1.,-p2/p1,+1,BudgetSetsIncomeandPriceChanges,Thebudgetconstraintandbudgetsetdependuponpricesandincome.Whathappensaspricesorincomechange?,Higherincomegivesmorechoice,,,Originalbudgetset,Newaffordableconsumptionchoices,,x2,x1,Originalandnewbudgetconstraintsareparallel(sameslope).,,,,,BudgetConstraints-IncomeChanges,Nooriginalchoiceislostandnewchoicesareaddedwhenincomeincreases,sohigherincomecannotmakeaconsumerworseoff.Anincomedecreasemay(typicallywill)maketheconsumerworseoff.,BudgetConstraints-PriceChanges,Whathappensifjustonepricedecreases?Supposep1decreases.,Howdothebudgetsetandbudgetconstraintchangeasp1decreasesfromp1’top1”?,,,,Originalbudgetset,x2,x1,m/p2,m/p1’,,m/p1”,Newaffordablechoices,Budgetconstraintpivots;slopeflattensfrom-p1’/p2to-p1”/p2,-p1’/p2,-p1”/p2,,,,BudgetConstraints-PriceChanges,Reducingthepriceofonecommoditypivotstheconstraintoutward.Nooldchoiceislostandnewchoicesareadded,soreducingonepricecannotmaketheconsumerworseoff.,UniformAdValoremSalesTaxes（從價稅）,Anadvaloremsalestaxleviedatarateof5%increasesallpricesby5%,frompto(1+005)p=105p.Anadvaloremsalestaxleviedatarateoftincreasesallpricesbytpfrompto(1+t)p.Auniformsalestaxisapplieduniformlytoallcommodities.,UniformAdValoremSalesTaxes,Auniformsalestaxleviedatratetchangestheconstraintfromp1x1+p2x2=mto(1+t)p1x1+(1+t)p2x2=mi.e.p1x1+p2x2=m/(1+t).,UniformAdValoremSalesTaxes,x2,x1,,,,Equivalentincomelossis,TheFoodStampProgram,Foodstampsarecouponsthatcanbelegallyexchangedonlyforfood.Howdoesacommodity-specificgiftsuchasafoodstampalterafamily’sbudgetconstraint?,TheFoodStampProgram,,,G,F,,100,100,F+G=100;beforestamps.,TheFoodStampProgram,G,F,100,100,F+G=100:beforestamps.,Budgetsetafter40foodstampsissued.,140,,,,,Thebudgetsetisenlarged.,,40,,,,TheFoodStampProgram,Whatiffoodstampscanbetradedonablackmarketfor$0.50each?,TheFoodStampProgram,G,F,100,100,F+G=100:beforestamps.,Budgetconstraintafter40foodstampsissued.,140,,,,,,120,,Budgetconstraintwithblackmarkettrading.,40,,,,,TheFoodStampProgram,G,F,100,100,F+G=100:beforestamps.,Budgetconstraintafter40foodstampsissued.,140,,,,120,,Blackmarkettradingmakesthebudgetsetlargeragain.,,,,,40,,,,BudgetConstraints-RelativePrices,“Numeraire”means“unitofaccount”.Supposepricesandincomearemeasuredindollars.Sayp1=$2,p2=$3,m=$12.Thentheconstraintis2x1+3x2=12.,BudgetConstraints-RelativePrices,Theconstraintforp1=2,p2=3,m=122x1+3x2=12isalso1.x1+(3/2)x2=6,theconstraintforp1=1,p2=3/2,m=6.Settingp1=1makescommodity1thenumeraireanddefinesallpricesrelativetop1;e.g.3/2isthepriceofcommodity2relativetothepriceofcommodity1.,BudgetConstraints-RelativePrices,Anycommoditycanbechosenasthenumerairewithoutchangingthebudgetsetorthebudgetconstraint.,BudgetConstraints-RelativePrices,p1=2,p2=3andp3=6?priceofcommodity2relativetocommodity1is3/2,priceofcommodity3relativetocommodity1is3.Relativepricesaretheratesofexchangeofcommodities2and3forunitsofcommodity1.,ShapesofBudgetConstraints,Butwhatifpricesarenotconstants?E.g.bulkbuyingdiscounts,orpricepenaltiesforbuying“toomuch”.Thenconstraintswillbecurved.,ShapesofBudgetConstraints-QuantityDiscounts,Supposep2isconstantat$1butthatp1=$2for0x120andp1=$1forx1>20.,ShapesofBudgetConstraints-QuantityDiscounts,Supposep2isconstantat$1butthatp1=$2for0x120andp1=$1forx1>20.Thentheconstraint’sslopeis-2,for0x120-p1/p2=-1,forx1>20andtheconstraintis,{,ShapesofBudgetConstraintswithaQuantityDiscount,m=$100,,,,50,,,100,,,20,,,Slope=-2/1=-2(p1=2,p2=1),Slope=-1/1=-1(p1=1,p2=1),,,80,x2,x1,ShapesofBudgetConstraintswithaQuantityDiscount,m=$100,,,,50,,,100,,,20,,,Slope=-2/1=-2(p1=2,p2=1),Slope=-1/1=-1(p1=1,p2=1),,,80,x2,x1,,,ShapesofBudgetConstraintswithaQuantityDiscount,m=$100,,,,50,,,100,,,20,,,,,80,x2,x1,,BudgetSet,BudgetConstraint,,,,ShapesofBudgetConstraintswithaQuantityPenalty,,,,,,,x2,x1,BudgetSet,BudgetConstraint,,,ShapesofBudgetConstraints-OnePriceNegative,Commodity1isstinkygarbage.Youarepaid$2perunittoacceptit;i.e.p1=-$2.p2=$1.Income,otherthanfromacceptingcommodity1,ism=$10.Thentheconstraintis-2x1+x2=10orx2=2x1+10.,ShapesofBudgetConstraints-OnePriceNegative,,,,10,,Budgetconstraint’sslopeis-p1/p2=-(-2)/1=+2,x2,x1,x2=2x1+10,ShapesofBudgetConstraints-OnePriceNegative,,,,10,x2,x1,,,Budgetsetisallbundlesforwhichx10,x20andx22x1+10.,MoreGeneralChoiceSets,Choicesareusuallyconstrainedbymorethanabudget;e.g.timeconstraintsandotherresourcesconstraints.Abundleisavailableonlyifitmeetseveryconstraint.,MoreGeneralChoiceSets,,,Food,OtherStuff,,,10,Atleast10unitsoffoodmustbeeatentosurvive,MoreGeneralChoiceSets,Food,OtherStuff,10,,BudgetSet,Choiceisalsobudgetconstrained.,,,MoreGeneralChoiceSets,Food,OtherStuff,10,,Choiceisfurtherrestrictedbyatimeconstraint.,,,MoreGeneralChoiceSets,Sowhatisthechoiceset?,MoreGeneralChoiceSets,,,Food,OtherStuff,,,10,,,MoreGeneralChoiceSets,,,Food,OtherStuff,,,10,,,,MoreGeneralChoiceSets,,,Food,OtherStuff,10,Thechoicesetistheintersectionofalloftheconstraintsets.,,Summary,Thechoicesetisthesetconsistsofallpossiblechoicesofaconsumer;Inthesimplestcase,thechoicesetistypicallyboundedbyasinglebudgetconstraintwhoseslopeisthenegativeratioofprices;Ingeneralthechoicesetcanbeboundedbymultipleconstraints.,