Chapter2BudgetConstraintCourse:MicroeconomicsText:Varian’sIntermediateMicroeconomics

1ConsumerTheoryEconomistsassumethatconsumerschoosethebestbundleofgoodstheycanafford.Thischapterfirstspecifiesindetailwhatconsumercanafford:thebudgetconstraintortheconsumptionpossibilityset.Whatisbestforconsumer,orthepreferenceonthepossibleconsumptionbundles,willbediscussedinthenextchapter.2BudgetConstraintsAconsumptionbundlecontainingx1unitsofcommodity1,x2unitsofcommodity2andsoonuptoxnunitsofcommoditynisdenotedbythevector(x1,x2,…,xn).Commoditypricesarep1,p2,…,pn.3BudgetConstraintsQ:Whenisaconsumptionbundle(x1,…,xn)affordableatpricesp1,…,pn?4BudgetConstraintsQ:Whenisaconsumptionbundle(x1,…,xn)affordableatpricesp1,…,pn?A:Whentotalexpenditureissmallerthanincome:

p1x1+…+pnxn

￡

m

wheremistheconsumer’s(disposable)income.Thatis,one’stotalexpenditureissmallerthanorequaltoone’sincome.5BudgetConstraintsThebundlesthatareonlyjustaffordablebytheconsumerisone’sbudgetconstraint.Thisistheset

{(x1,…,xn)|x130,…,xn

30and

p1x1+…+pnxn

=

m}.

6BudgetConstraintsTheconsumer’sbudgetset

isthesetofallaffordablebundles;

B(p1,…,pn,m)=

{(x1,…,xn)|x1

30,…,xn

30and

p1x1+…+pnxn

￡

m}Thebudgetconstraintistheupperboundaryofthebudgetset.7BudgetSetandConstraintforTwoCommoditiesx2x1Budgetconstraintisp1x1+p2x2=m.m/p1m/p28BudgetSetandConstraintforTwoCommoditiesx2x1Budgetconstraintisp1x1+p2x2=m.m/p2m/p19BudgetSetandConstraintforTwoCommoditiesx2x1Budgetconstraintisp1x1+p2x2=m.m/p1Justaffordablem/p210BudgetSetandConstraintforTwoCommoditiesx2x1Budgetconstraintisp1x1+p2x2=m.m/p1JustaffordableNotaffordablem/p211BudgetSetandConstraintforTwoCommoditiesx2x1Budgetconstraintisp1x1+p2x2=m.m/p1AffordableJustaffordableNotaffordablem/p212BudgetSetandConstraintforTwoCommoditiesx2x1Budgetconstraint(budgetline)isp1x1+p2x2=m.m/p1BudgetSetthecollection

ofallaffordablebundles.m/p213BudgetSetandConstraintforTwoCommoditiesx2x1p1x1+p2x2=misx2=-(p1/p2)x1+m/p2soslopeis-p1/p2.m/p1BudgetSetm/p214BudgetConstraintsForn=2andx1onthehorizontalaxis,theconstraint’sslopeis-p1/p2.Whatdoesitmean?

Note:Foralinearline:y=mx+c,mistheslopeoftheline,whilecisthey-intercept.15BudgetConstraintsForn=2andx1onthehorizontalaxis,theconstraint’sslopeis-p1/p2.Whatdoesitmean?

Toholdincomemconstant,increasingx1by1mustreducex2byp1/p2.16BudgetConstraintsx2x1Slopeis-p1/p2+1-p1/p217BudgetConstraintsx2x1+1-p1/p2Opp.costofanextraunitof

commodity1isp1/p2units

foregoneofcommodity2.18BudgetConstraintsx2x1Theopp.costofanextra

unitofcommodity2is

p2/p1unitsforegone

ofcommodity1.-p2/p1+119BudgetSets&Constraints;IncomeandPriceChangesThebudgetconstraintandbudgetsetdependuponpricesandincome.Whathappensaspricesorincomechange?20Howdothebudgetsetandbudgetconstraintchangeasincomemincreases?Originalbudgetsetx2x121HigherincomegivesmorechoiceOriginalbudgetsetNewaffordableconsumption

choicesx2x1Originalandnewbudgetconstraintsareparallel(sameslope).22Howdothebudgetsetandbudgetconstraintchangeasincomemdecreases?Originalbudgetsetx2x123Howdothebudgetsetandbudgetconstraintchangeasincomemdecreases?x2x1New,smallerbudgetsetConsumptionbundlesthatarenolongeraffordable.Oldandnewconstraintsareparallel.24BudgetConstraints-IncomeChangesIncreasesinincomemshifttheconstraintoutwardinaparallelmanner,therebyenlargingthebudgetsetandimprovingchoice.Decreasesinincomemshifttheconstraintinwardinaparallelmanner,therebyshrinkingthebudgetsetandreducingchoice.Theslope–p1/p2doesnotchange.25BudgetConstraints-IncomeChangesWhenincomeincreases,NOoriginalchoiceislostandnewchoicesareadded,sohigherincomecannotmakeaconsumerworseoff.Whenincomedecreases,theconsumermay(typicallywill)beworseoff,asonecannolongeraffordsomeofthebundlesanymore.26BudgetConstraints-PriceChangesWhathappensifjustonepricedecreases?Supposep1decreases.27Howdothebudgetsetandbudgetconstraintchangeasp1decreasesfromp1’top1〞?Originalbudgetsetx2x1m/p2m/p1’m/p1〞-p1’/p228Howdothebudgetsetandbudgetconstraintchangeasp1decreasesfromp1’top1〞?Originalbudgetsetx2x1m/p2m/p1’m/p1〞Newaffordablechoices-p1’/p229Howdothebudgetsetandbudgetconstraintchangeasp1decreasesfromp1’top1〞?Originalbudgetsetx2x1m/p2m/p1’m/p1〞NewaffordablechoicesBudgetconstraintpivots;slopeflattensfrom-p1’/p2to-p1〞/p2-p1’/p2-p1〞/p230BudgetConstraints-PriceChangesReducingthepriceofonecommoditypivotstheconstraintoutward.Nooldchoiceislostandnewchoicesareadded,soreducingonepricecannotmaketheconsumerworseoff.Similarly,increasingonepricepivotstheconstraintinwards(considerapricechangefromp1〞top1’),reduceschoiceandmay(typicallywill)maketheconsumerworseoff.31TaxandSubsidyQuantity/per-unittax:priceincreasesfromptop+t.Quantity/per-unitsubsidy:pricedecreasesfromptop-s.Advalorem/valuetax:priceincreasesfrompto(1+t)pAdvalorem/valuesubsidy:pricedecreasesfrompto(1-s)p32UniformAdValoremSalesTaxesAuniformsalestaxleviedatratetonallgoodschangestheconstraintfrom

p1x1+p2x2=m

to

(1+t)p1x1+(1+t)p2x2=m

i.e.

p1x1+p2x2=m/(1+t).33UniformAdValoremSalesTaxesx2x1Equivalentincomeloss

is34UniformAdValoremSalesTaxesx2x1Auniformadvalorem

salestaxleviedatratet

isequivalenttoanincome

taxleviedatrate35TaxandSubsidyLump-sumtax:governmenttaxafixedsumofmoney,T,regardlessofindividual’sbehavior.ThisisequivalenttoadecreaseinincomebyT,implyinganinwardparallelshiftofbudgetline.Similarly,lump-sumsubsidySimpliesanoutwardparallelshiftofbudgetlinecorrespondingtoanamountS.36TheFoodStampProgramFoodstampsarecouponsthatcanbelegallyexchangedonlyforfood.Howdoesacommodity-specificgiftsuchasafoodstampalterafamily’sbudgetconstraint?Hereweassumeoneofthetwogoodsisfood.37TheFoodStampProgramSupposem=$100,pF=$1andthepriceof“othergoods〞ispG=$1.Thebudgetconstraintisthen

F+G=100.38TheFoodStampProgramGF100100F+G=100:beforestamps.39TheFoodStampProgramGF100100F+G=100:beforestamps.Budgetsetafter$40food

stampsissued.1404040TheFoodStampProgramGF100100F+G=100:beforestamps.Budgetsetafter$40food

stampsissued.140Thefamily’sbudget

setisenlarged.4041BudgetConstraints-RelativePricesHowdoestheunitofaccountaffectthebudgetconstraintsandbudgetset?Supposepricesandincomearemeasuredindollars.Sayp1=$2,p2=$3,m=$12.Thentheconstraintis

2x1+3x2=12.42BudgetConstraints-RelativePricesIfpricesandincomearemeasuredincents,thenp1=200,p2=300,m=1200andtheconstraintis

200x1+300x2=1200,

uponsimplification,itisthesameas

2x1+3x2=12.Changingtheunitofaccountchangesneitherthebudgetconstraintnorthebudgetset.43BudgetConstraints-RelativePricesTheconstraintforp1=2,p2=3,m=12

2x1+3x2=12

isalso1x1+(3/2)x2=6,

theconstraintforp1=1,p2=3/2,m=6.Settingp1=1makescommodity1thenumeraireanddefinesallpricesrelativetop1;e.g.3/2isthepriceofcommodity2relativetothepriceofcommodity1.44BudgetConstraints-RelativePricesMultiplyingallpricesandincomebyanyconstantkdoesnotchangethebudgetconstraint.kp1x1+kp2x2=km

Anycommoditycanbechosenasthenumeraire(bytakingk=1/pi)withoutchangingthebudgetsetorthebudgetconstraint.Itisalsoclearfromthegraphthat,onlytheratiosp1/p2,m/p1andm/p2arerelevanttothebudgetlineandbudgetset.45ShapesofBudgetConstraintsQ:Whatmakesabudgetconstraintastraightline?A:Astraightlinehasaconstantslopeandtheconstraintis

p1x1+…+pnxn=m

soifpricesareconstantsthenaconstraintisastraightline.46ShapesofBudgetConstraintsButwhatifpricesarenotconstants?E.g.bulkbuyingdiscounts,orpricepenalties(ortax)forbuying“toomuch〞.Thenconstraintswillbecurvedorhavekinks.47ShapesofBudgetConstraints-QuantityDiscountsSupposep2isconstantat$1butthatp1=$2for0￡x1

￡20andp1=$1forx1>20.48ShapesofBudgetConstraints-QuantityDiscountsSupposep2isconstantat$1butthatp1=$2for0￡x1

￡20andp1=$1forx1>20.Thentheconstraint’sslopeis

-2,for0￡x1

￡20

-p1/p2=

-1,forx1>20

Alsoassumem=100.{49ShapesofBudgetConstraintswithaQuantityDiscountm=$10