Consumption
InLecturenote#3,weassumedthatconsumptiondependedonlyon
currentdisposable income,butitistoosimpletoprovideacomplete
explanationofconsumerbehavior.Inthisnoteweexaminethe
consumptionfunctioningreaterdetailanddevelopamorethorough
explanationofwhatdeterminesaggregateconsumptionbyshowing
diverseapproachestoexplainingconsumption.
IrvingFisherandIntertemporalChoice
Keynessconsumptionfunctionrelatescurrentconsumptionto
currentincome.However,whenpeopledecidehowmuchtoconsume
andhowmuchtosave,theyconsiderboththepresent andthe
future.Themoreconsumptiontheyenjoytoday,thelesstheywillbe
abletoenjoytomorrow.Makingthistradeoff enableshouseholdsto
lookaheadtotheincometheyexpecttoreceiveinthefutureandto
theconsumptionofgoodsandservicestheyhopetobeableto
afford.
TheIntertemporalBudgetConstraint
TheeconomistIrvingFisherdevelopedthemodelwithwhich
economistsanalyzehowrational,forwardlookingconsumersmake
intertemporalchoices thatis,choicesinvolvingdifferent periodsof
time.
Q:Whydopeopleconsumeless thantheydesire?
A:Becausetheirconsumptionisconstrainedbytheirincome.
Toputanotherway,consumersfacealimitonhowmuchtheycan
spend,calledabudget constraint.
TheIntertemporalBudgetConstraint
Def.(3B.1):Intertemporalbudgetconstraint
Thelinethatmeasuresthetotalresourcesavailablefor
consumptiontodayandinthefuture
Theconstraintconsumersfacewhendecidinghowmuchto
consumetodayversushowmuchtosaveforthefuture
DevelopmentofFishersModel:
Assumptions:(i)Aconsumerlivesfortwoperiods;periodone
representstheconsumersyouth,andperiodtworepresentsthe
consumersoldage.
(ii)Allvariablesarereal thatis,adjustedforinflation.
TheIntertemporalBudgetConstraint
(iii)Theinterestrateforborrowingisthesame astheinterestratefor
saving.
:Aconsumersincomeinperiod1
:Aconsumersconsumptioninperiod1
:Aconsumersincomeinperiod2
:Aconsumersconsumptioninperiod2
Q:Howdoestheconsumersincomeinthetwoperiodsconstrain
consumptioninthetwoperiods?
TheIntertemporalBudgetConstraint
Inthefirstperiod,
3 .1 :
,where canrepresent
eithersavingorborrowing.
Aconsumerissaving,and ispositive.
Aconsumerisborrowing,and isless thanzero.
Inthesecondperiod,
3 .2 :
1
,where isthe
realinterestrate.
Meaning:Consumptioninthesecondperiodequalstheaccumulated
saving,includingtheinterestearnedonthatsaving,plussecond
periodincome.
TheIntertemporal BudgetConstraint
From
3 .2 ,
1
1
1
Bydividingbothsidesby 1
since
;
3 .3 :
Meaning:Thepresent value ofconsumption mustequalthepresent
value ofincome.Itistheconsumersintertemporal budgetconstraint.
Notethat futureconsumptionandfutureincomearediscounted by
afactor 1
.Thisdiscounting arisesfromtheinterest earnedon
savings.
Present(Discounted)Value
Recall theconceptofpresentvalue:Ifyouput$100todayinabank
accountearning5percentinterestayear,in10yearsyouwouldhave
$ 1 0.05
$100 $162.89.
$162.89;
Thus,thepresentvalue of$162.89received10yearsfromnowis
$100.Formally,
where $ is the dollar amount received in year and is
theinterestrate.
TheIntertemporal BudgetConstraint
isthepriceof2ndperiodconsumptionmeasuredintermsof
1stperiodconsumption:itistheamountof1stperiodconsumption
thattheconsumermustgiveuptoobtain1unitof2ndperiod
consumption.
From
3 . 3 ,
Figure 3B.1:ConsumersBudgetConstraint
TheIntertemporal BudgetConstraint
Point :Theconsumerconsumesexactlyhis/herincomeineach
and
,sothereisneither saving nor
period
borrowing betweenthetwoperiods.
Point :Theconsumerconsumesnothinginthe1st period
0
andsaves allincome,sothe2nd periodconsumption is
1
.
Point :Theconsumerplanstoconsumenothinginthe2nd period
0 andborrows asmuchaspossibleagainst2ndperiodincome,
so1stperiodconsumption
is
ConsumerPreference
Theindifference curve representstheconsumerspreference
regardingconsumptioninthetwoperiods;i.e.,itshowsthe
combinationsof1stperiodand2ndperiodconsumptionthatmake
theconsumerequally happy.
Def.(3B.2):Indifferencecurve
Thecurvethatshowsdifferentcombinationsof and
tothesame levelofoverallhappiness(utility orwelfare)
thatlead
Figure3B.2:ConsumersPreferences
ThreeCharacteristicsofIndifferenceCurves
(i)Thefarther anindifferencecurveisfromtheorigin,thehigher its
utility.
Theconsumerisindifferent amongcombinations , , and
Becausetheyareallonthesame curve .
Becauseweareassumingthatmore consumptionisalwaysbetter,it
mustbetruethateverypointon
musthaveahigher utilitythan
everypointon .
(ii)Indifferencecurvesslopedown.
ThreeCharacteristicsofIndifferenceCurves
Tokeepaconsumerjustashappy,alossofhappinessfromlower
2ndperiodconsumptionmustbeoffsetbyhigherconsumptionin
period1,andviceversa.Indifferencecurvesarethereforedownward
sloping,illustratingthetradeoff betweenconsumingtodayversus
consumingtomorrow.
(iii)Indifferencecurvesarebowedin towardtotheorigin.
Thebowedinshapeoftheindifferencecurves,referredtoas
convexity,resultsfromthetypicalconsumersdislikeoflarge
fluctuationsinconsumptionfromoneperiodtothenext.Ingeneral,
peopleprefersmooth consumptionovertime.
ThreeCharacteristicsofIndifferenceCurves
The3rd characteristiccanbedescribedthroughtheconceptofthe
marginalrateofsubstitution (MRS)alsocalledtheintertemporal
marginalrateofsubstitution.
Def.(3B.2):Marginalrateofsubstitution
Therateatwhichaconsumeriswillingtogiveup(substitute)
consumptioninperiod2foradditionalconsumptioninperiod1
Theslopeoftheindifferencecurve
ThreeCharacteristicsofIndifferenceCurves
Sinceconsumersprefertosmooth consumptionovertime(andthus
becomesincreasinglyaversetogivingupconsumptioninperiod2),
themarginalrateofsubstitutionbecomessmaller as2ndperiod
consumptionfalls,givingtheindifferencecurveaconvex shape:
InFigure 3B.2,
issmaller than
,sothattheslopeofthe
indifferencecurvebecomesless negative.
Optimization
Usingbothanintertemporal budgetlineandasetofindifference
curves,wecanidentifytheoptimallevelofconsumptionforboth
periods.
Figure 3B.3:ConsumerOptimization
Optimization
Theoptimumindifferencecurvegiventheconsumersbudget
constraintis ,whichistangent totheintertemporal budgetlineat
pointO.AtpointO,consumptioninperiod1is andconsumption
inperiod2is .
1
1
From
3 . 3 ,
Theslopeoftheintertemporal budgetline:
AtpointO,theoptimumforaconsumeris:
TheslopeoftheIC =TheslopeoftheIBL
3 .4 :
Example (3B.1):Optimization
SupposeatypicalconsumernamedJenniferlivesfortwoconsumption
periods.Herrealincomeineachofperiods1and2, and ,equals
12.Designateconsumptionduringperiods1and2as and ,
respectively.Assumethatanysavingfromperiod1earnsinterestata
realinterestrateof50%,or0.50.SinceJenniferknowsshewilldieat
theendofperiod2,shedoesnotsaveinthesecondperiod.
a)WritedownJennifersintertemporal budgetconstraint,i.e.,the
budgetconstraintthatrelatesherlifetimeincomewithherlifetime
consumption.
(Ans.)
.
.
Example (3B.1):Optimization
Slope:
. ,indicatingthateachunitofconsumptionin
period1canbetransformedinto1.5unitsofconsumptioninperiod2
becauseanysavingearnsinterestattherateof50percent.
b)DrawJennifersintertemporal budgetconstraintandlocatethe
point onthegraphatwhichsheconsumesallofhercurrentincome
ineachofthetwoperiods,thepoint atwhichshesavesallofher
incomeinperiod1inordertomaximize ,andthepoint atwhich
shemaximizes byborrowingagainstallofherincomeinperiod2.
(Ans.)
Example (3B.1):Optimization
c)Furthermore,assumethattheamountofJenniferstotalsatisfaction
isgivenbythisutilityfunction:
.Findtheoptimal
consumptionof and ,locatethispointonthegraphinpart(b),
andlabelitpoint .Findherutilityattheoptimum.
(Ans.)
(i)Findthemarginalrateofsubstitution(MRS):
and
(ii)SetMRS equaltotheslopeofthebudgetline:
Example (3B.1):Optimization
1.5
.
.
By(1)and(2),
10
15 and
10
15 150.
HowChangesinIncomeorWealthAffectConsumption
Letsexaminehowconsumptionrespondstoanincreaseinincomeor
.Anincreaseineither or orwealthshifts
wealth
thebudgetconstraintoutward.
Figure 3B.4:ResponsetoaRiseinCurrentIncome,Future
Income,orWealth
HowChangesinIncomeorWealthAffectConsumption
TheindifferencecurvesinFigure 3B.4aredrawnunderthe
assumptionthatconsumptioninperiod1andconsumptioninperiod
2arebothnormal goods.Becausetheconsumercanborrowandlend
betweenperiods,thetimingoftheincomeorwealthisirrelevant to
howmuchisconsumedtoday(exceptthatfutureincomeorwealthis
3 . 3 ,consumption
discounted bytheinterestrate).From
dependsonthepresentvalueofcurrentandfuture
incomeorwealth,lifetimeresources,
initialwealth.
where
is
HowChangesinIncomeorWealthAffectConsumption
NoticethatthisisquitedifferentfromKeynes.Keynespositedthat
apersonscurrentconsumptiondependslargelyonthecurrent
income.Fishersmodelsaysthatconsumptionisbasedontheincome
theconsumerexpectsovertheentire lifetime.
Consumptionriseswhenthepresentvalueoflifetimeresources
increases,regardlessofitssource(currentincome,futureincome,or
wealth).
Consumerswillspreadoutanyincreaseinconsumptionovertoday
andtomorrow,eveniftheincreasestemssolelyfromanincreasein
current income.Thisbehaviorisknownasconsumption smoothing.
HowChangesinIncomeorWealthAffectConsumption
Consumptionsmoothingisalogicalconsequenceoftwoelementsof
theintertemporal choicemodel:theconvexity ofindifferencecurves
andtheabilityofconsumerstoborrow orsave.
Indeed,consumptionsmoothingisastrongfeatureofthedata
becauseconsumptionfluctuatesmuchless thanGDP.
HowChangesintheRealInterestRateAffectConsumption
LetsnowuseFishersmodeltoconsiderhowachangeinthereal
interestratealterstheconsumerschoices.
Therearetwocasestoconsider:
Case1:Theconsumerisinitiallysaving
Case2:Theconsumerisinitiallyborrowing
Case1:Seewhathappenswhentheinterestraterisefrom
to .
Figure3B.5:ResponsetoaRiseintheInterestRate
HowChangesintheRealInterestRateAffectConsumption
Atpoint ,thecurrentconsumptionisequaltoinitialwealthplus
.Inthisspecialcase,the
1stperiodincome;thatis,
consumerisneitherborrowing norsaving (lending),soconsumption
inthesecondperiodisequaltosecondperiodincome,
.
Whentheinterestraterisesto ,point isstillonthenew
because,withnoborrowing or
intertemporal budgetline
saving,thelevelofinterestratedoesnot matter,leaving
and
.
HowChangesintheRealInterestRateAffectConsumption
Whentheinterestraterisesto ,theslopeofthebudgetline
, sothe
becomessteeper.
becomesmorenegative 1
Becauseitstillgoesthroughpoint ,thismeansthatthenew
pivotsclockwise aroundpoint .
intertemporal budgetline
Sinceconsumersonaveragesavesomeoftheirincome,the
consumersinitialoptimalconsumptionoccursatpoint ,inwhich
.Whentheinterestraterisesto andthe
,thenewoptimallevelof
intertemporal budgetlinepivotsto
consumptionatpoint isonahigher indifferencecurve, .
HowChangesintheRealInterestRateAffectConsumption
Consumptioninperiod1hasfallen to
period2hasrisen to
,whileconsumptionin
Wecandecomposetheimpactofanincreaseintherealinterestrates
onconsumptionintotwoeffects:
Incomeeffect
Thechangeinconsumptionduetochangesinincome
Thechangeinconsumptionthatresultsfromthemovementto
ahigherindifferencecurve
HowChangesintheRealInterestRateAffectConsumption
Theconsumerhassavings,sohe/shelends money(probablyby
depositingthefundsinhis/herbankaccount)fromperiod1toperiod
2.Atahigher interestrate,theconsumerearnsmore interest,which
giveshim/hermore resourceswithwhichtoconsume.Thus,the
consumercanspendmore inbothperiods.Forsavers,theincome
effectincreases consumptioninboth periodswheninterestrates
rise.
Substitutioneffect
Thechangeinconsumptionthatoccursfromthechangeinthe
relative price ofconsumptioninthetwoperiods
HowChangesintheRealInterestRateAffectConsumption
Withahigher interestrate,thepresent(discounted)valueof
consumption
inthesecondperiodfalls,soconsumptioninthe
secondperiodbecomesless expensiverelativetoconsumptioninthe
firstperiod.Asaresult,theconsumerwilldecidetosubstituteaway
andreducefirstperiodconsumptiontoconsumemore inthesecond
period.Thesubstitutioneffectfromhigher interestratesleadsto
less consumptioninperiod1butmore consumptioninperiod2.
HowChangesintheRealInterestRateAffectConsumption
Wecanalsothinkofthesubstitutioneffectintermsofsaving.When
interestraterises,thereturntosavingishigher andsoaconsumer
willsavemore inthefirstperiodbyreducingconsumption,enabling
him/hertospendmore inthesecondperiod(consumptionrises).
Conclusion:Wheninterestratesincrease,bothincomeand
substitutioneffectsshiftsecondperiodconsumptionhigher.Butin
thefirstperiod,thesubstitutioneffecttampsdownconsumption
whiletheincomeeffectliftsconsumption.Hence,theultimate
directionofthechangeintodaysconsumptiondependsonthe
relative sizeofincomeandsubstitutioneffects.
HowChangesintheRealInterestRateAffectConsumption
Assumption:Inpractice,weusuallyassumethatthesubstitution
effectoutweighstheincomeeffect.
InFigure 3B.5,sincethesubstitution effectdominatestheincome
effect,arise ininterestrateslowers todaysconsumption(increases
saving),butboost futureconsumption.
Wemakethisassumptionbecauserealworlddatasupporttheview
thathigher interestratesareassociatedwithhigher savingandlower
consumptiontoday.However,theempiricalevidenceonthe
relationshipofinterestratestoconsumptionandsavingisnot
overwhelminglystrong.
BorrowingConstraints
Sofarwehaveassumedthatconsumerscanborrowandlendatthe
same interestrate.However,notallconsumersaregodcreditrisks.
Consumerswithlittleornowealthmayfindthattheycannot get
loans.
Iftheconsumerhasno wealthandcannot borrow,he/shecannot
spendmorethanhe/sheearns.Aconstraintonborrowingcan
,indicatingconsumptiontodaymust
thereforebewrittenas:
beless thanorequal toincometoday.Thisadditionalconstrainton
theconsumeriscalledaborrowing constraint (oraliquidity
constraint).
BorrowingConstraints
Toseetheimplicationforconsumptionchoices,letsfirstlookatthe
effectofaborrowingconstraintontheconsumersintertemporal
budgetline.
Intertemporalbudgetlinewithaborrowingconstraint
Supposethatwithoutaborrowingconstraint,theintertemporal
budgetlineisastraightlinewithaslopeof 1
.Witha
borrowingconstraint,theconsumercannot spendmore thanhis/her
currentincome.Thatis, cannot riseabove .Atpoint ,where
,theintertemporalbudgetlinebecomesvertical.
Figure 3B.6:ABorrowingConstraint
BorrowingConstraints
Q:Howwillaborrowingconstraintchangetheconsumers
consumptionchoices?
A:Weconsidertwocases.
Case1:Borrowingconstraintisnotbinding
Sincetheconsumersoptimallevelofconsumptionisless than ,
he/shehasno needtoborrow.Asaresult,he/shewillnot be
constrainedbyhis/herinabilitytoborrow,andwecansaythatthe
constraintisnot binding.
Figure 3B.7:OptimizationwithaBorrowingConstraint
BorrowingConstraints
Case2:Borrowingconstraintisbinding
Theconsumerwouldliketoborrowandchoosepoint .Butbecause
borrowingisnot allowed,thebestavailablechoiceispoint .When
theborrowingisbinding,firstperiodconsumptionequalsfirstperiod
and
.Theslopeoftheindifference
income,so
,sothat
.
curveatpoint issteeper than 1
Figure 3B.8:OptimizationwithaBorrowingConstraint
Example (3B.2):ContinuationofExample (3B.1)
Thecurvelabeled
representsJennifersinitialintertemporal
budgetconstraintorline.SupposethatJennifersincomeinperiod1
risesto12.78,while remainsequalto12.
a)WritedownJennifersnewintertemporal budgetlineorconstraint.
Graphthisnewintertemporal budgetconstraintandlabelit
(Ans.)
.
.
.
.
.
Example (3B.2):ContinuationofExample (3B.1)
b)Findthenewoptimalconsumptionof and ,locatethispointonthe
graphinpart(a),andlabelitpoint .Findherutilityatthenewoptimum.
(Ans.)
SetMRS equaltotheslopeofthebudgetline:
.
.
By(1)and(2),
15.59 and
10.39 and
10.39 15.59 162.
Consumptionduringeachperiodhasincreased.Thechangesin
consumptionbetweenpoint andpoint aretheresultofthe income
effect.
Example (3B.2):ContinuationofExample (3B.1)
c)Finally,supposethatJennifersincomeineachperiodreturnstoits
formerlevelof12andthattherealinterestraterisesto100percent,or
1.0.WritedownJennifersnewintertemporal budgetlineorconstraint.
Graphthisnewintertemporal budgetconstraintandlabelit
.
(Ans.)
d)Findtheoptimalconsumptionof and ,locatethispointonthe
graphinpart(a),andlabelitpoint .Findherutilityattheoptimum.
(Ans.)
SetMRS equaltotheslopeofthebudgetline:
Example (3B.2):ContinuationofExample (3B.1)
18 and
9 and
9 18 162.
By(1)and(2),
Thechangeinconsumptionfrompoint topoint reflectsthe
substitutioneffect.Jenniferrespondstothischangeintherelative
priceofconsumptioninthetwoperiodsbydecreasing and
increasing .Sincethesubstitutioneffectoutweighstheincome
effect inthiscase, obviouslyfallswhile increases.
