the dynamic behavior of consump=on under uncertainty · prof george alogoskouﬁs, dynamic...

ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015

TheDynamicBehaviorofConsump=onunderUncertainty

TheDynamicBehaviorofConsump=onandPorAolioChoiceunderUncertainty


Op=miza=onofDiscountedU=lityofaRepresenta=veHousehold

maxU = B e−βt c(t)1−θ

1−θt=0

∞

∫ dt

k•(t) = r(t)k(t)+w(t)− c(t)− (n + g)k(t)

undertheconstraint

where β = ρ − n − (1−θ )g > 0

2


TheFirstOrderCondi=onsfortheMaximiza=onoftheIntertemporalU=lityoftheRepresenta=ve

Household

3

c•(t) = 1

θ(r(t)− ρ −θg)c(t)

Thefirstequa=onistheEulerequa=onforconsump=onandthesecondtheaccumula=onequa=on.

k•(t) = r(t)k(t)+w(t)− c(t)− (n + g)k(t)


TheIntertemporalBudgetConstraintoftheRepresenta=veHousehold

4

Theaccumula=onequa=onsisafirstorderdifferen=alequa=onwithvariablecoefficients.Asaresult,itssolu=onforeveryT≥0takestheform,

e− r(v)dv

v=0

T

∫ −(n−g)T⎛⎝⎜

⎞⎠⎟k(T )+ e

− r(v)dvv=0

t

∫ −(n−g)t⎛⎝⎜

⎞⎠⎟c(t)

t=0

T

∫ dt = k(0)+ e− r(v)dv

v=0

t

∫ −(n−g)t⎛⎝⎜

⎞⎠⎟w(t)

t=0

T

∫ dt

Thisinter-temporalbudgetconstraintimpliesthatat=me0,thepresentvalueoflaborincomebetween0andT,plustheini=alcapitalstockat0,isequaltothepresentvalueofconsump=onbetween0andT,plusthepresentvalueofthecapitalstockatT.Thetermthatincludestheintegralofinterestratesisatermthattransformsaunitofincome,consump=onorcapitalatinstantttoitspresentvalueatinstant0.Iftheinterestratewasconstant,thistermwouldsimplifyto-rt.


TheAverageInterestRateandtheInter-temporalBudgetConstraint

5

Wecandefinetheaverageinterestratebetween0andtas,

Withthisdefini=on,theinter-temporalbudgetconstraintcanbewriWenas,

r_(t) = 1

tr(v)dv

v=0

t

∫

e− r

_(T )−n−g⎛

⎝⎜⎞⎠⎟Tk(T )+ e

− r_(t )−n−g⎛

⎝⎜⎞⎠⎟tc(t)

t=0

T

∫ dt = k(0)+ e− r

_(t )−n−g⎛

⎝⎜⎞⎠⎟tw(t)

t=0

T

∫ dt


TheInter-temporalBudgetConstraintwithanInfiniteTimeHorizon

6

Asthe=mehorizonTtendstoinfinity,theinter-temporalbudgetconstraintcanbewriWenas,

limT→∞

e− r

_(T )−n−g⎛

⎝⎜⎞⎠⎟Tk(T )+ e

− r_(t )−n−g⎛

⎝⎜⎞⎠⎟tc(t)

t=0

∞

∫ dt = k(0)+ e− r

_(t )−n−g⎛

⎝⎜⎞⎠⎟tw(t)

t=0

∞

∫ dt

Asthe=mehorizontendstoinfinity,thetermontheleYmusttendtozero,asthehouseholdderivesu=lityonlyfromconsump=onandnotthecapitalstock.(transversalitycondi;on).Asaresult,theinter-temporalbudgetconstraintwithaninfinite=mehorizontakestheform,

e− r

_(t )−n−g⎛

⎝⎜⎞⎠⎟tc(t)

t=0

∞

∫ dt = k(0)+ e− r

_(t )−n−g⎛

⎝⎜⎞⎠⎟tw(t)

t=0

∞

∫ dt


TheTransversalityCondi=onandtheTimeHorizon

7

IfthehorizonofthehouseholdwasT,thentheop=malcapitalstockatinstantTwouldbeequaltozero.Asthe=mehorizonΤtendstoinfinity,thisimpliesthat,

Ifthiscondi=onisnotsa=sfied,forexampleiftheabovelimitisposi=ve,thenthehouseholdcouldalongtheop=malpathincreaseitsinter-temporalu=litybyconsumingalargerpartofitscapital.Iftheabovelimitisnega=ve,thenthehouseholdwouldbeaccumula=ngunsustainabledebts(nega=vecapital)alongtheop=malpath,whichisnotconsistentwithitsinter-temporalbudgetconstraint.Therefore,theonlyop=malpathconsistentwiththeinter-temporalbudgetconstraintoftherepresenta=vehouseholdistheonewhichsa=sfiesthecondi=onthatthepresentvalueofitscapitalstocktendstozeroas=metendstoinfinity.Thisistheinfinitehorizontransversalitycondi;on.Itissa=sfiedaslongasthecapitalstockperefficiencyunitoflabordoesnotincrease(ordecrease)ataratefasterthanr-n-g,whichisthesameassayingthattheaggregatecapitalstockdoesnotincrease(ordecrease)ataratefasterthanr.

limT→∞

e− r

_(T )−n−g⎛

⎝⎜⎞⎠⎟Tk(T ) = 0


TheConsump=onFunc=onoftheRepresenta=veHouseholdwithanInfiniteTimeHorizon

8

Ifwesolve(integrate)thedifferen=alequa=ondescribingtheEulerequa=onforconsump=on,thenwefindthatconsump=onat=metisdefinedby,

Subs=tu=ngforconsump=onc(t)intheinter-temporalbudgetconstraint,theconsump=onfunc=onatinstant0takestheform,

c(0) = γ (0) k(0)+ e− r

_(t )−n−g⎛

⎝⎜⎞⎠⎟tw(t)

t=0

∞

∫ dt⎛

⎝⎜

⎞

⎠⎟

c(t) = c(0)e1θ

r_(t )−ρ−θg⎛

⎝⎜⎞⎠⎟t

where γ (0) = er_(t )(1−θ )−ρ+θn

θ

⎛

⎝⎜⎜

⎞

⎠⎟⎟t

dtt=0

∞

∫⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1


TheInterpreta=onoftheConsump=onFunc=onoftheRepresenta=veHousehold• Therepresenta=vehouseholdconsumesashareofitstotalwealthγ(0),thatdependsontheevolu=onoftheaveragerealinterestrate,thepurerateof=mepreferencerateρ,theelas=cityofinter-temporalsubs=tu=onofconsump=on1/θ,andthepopula=ongrowthraten.

• Theimpactoftheaveragerealinterestrateonthepropor=onoftotalwealththatisconsumeddependsontheelas=cityofinter-temporalsubs=tu=onofconsump=on1/θ.Anincreaseinaveragerealinterestrateshastwokindsofeffectsontheaverageconsump=ontototalwealthra=o:aninter-temporalsubs;tu;oneffect,andanincomeeffect.First,itinducesthehouseholdtosubs=tutecurrentforfutureconsump=on,andincreasesthecostofcurrentconsump=onrela=vetofutureconsump=on.Thisistheinter-temporalsubs=tu=oneffectinconsump=on,whichtendstodecreasecurrentconsump=on.Second,anincreaseininterestratesincreasesincomefromcapital,andtendstoincreasebothcurrentandfutureconsump=on.Thisistheincomeeffect,whichtendstoincreasecurrentconsump=on.

• Iftheinter-temporalelas=cityofsubs=tu=oninconsump=onisgreaterthanone(θ<1),thenconsump=onasapropor=onoftotalwealthdecreaseswheninterestratesrise,becausethenega=vesubs=tu=oneffectisstrongerthantheposi=veincomeeffect,andthusprevailsontheincomeeffect.Iftheinter-temporalelas=cityofsubs=tu=onislessthanunity(θ>1),thenconsump=onasapropor=onoftotalwealthincreaseswheninterestratesrise,becausetheposi=veincomeeffectisstrongerthanthenega=vesubs=tu=oneffect.Finally,ifθ=1,whichisthecasewithlogarithmicpreferences,thetworesultscanceleachotherout,andconsump=onasapropor=onoftotalwealthisindependentofthepathofrealinterestrates.

9


TheConsump=onFunc=onoftheRepresenta=veHouseholdwithaConstantRealInterestRate

10

Iftherealinterestrateisconstantatr,thentheshareoftotalwealththatisconsumedisgivenby,

Forthistobeposi=ve,wemusthaver<(ρ-θn)/(1-θ).Onthebalancedgrowthpath,r=ρ+θg.Asaresult,theshareoftotalwealththatisconsumedonthebalancedgrowthpathisgivenby,

γ (0) = e1θr(1−θ )−ρ+θn( )t

t=0

∞

∫⎛⎝⎜

⎞⎠⎟

−1

= r(1−θ )− ρ +θn

θ limt→∞

e1θr(1−θ )−ρ+θn( )t

− limt→0

e 1θ

r(1−θ )−ρ+θn( )t⎛⎝⎜

⎞⎠⎟

= 1θ

ρ −θn − r(1−θ )( )

γ (0) = ρ − n − (1−θ )g( )

whichwehavealreadyassumedisposi=veinordertohaveawelldefinedinter-temporalop=miza=onproblemfortherepresenta=vehousehold.


TheConsump=onFunc=onoftheRepresenta=veHouseholdwithaUnitaryElas=cityofInter-

temporalSubs=tu=onofConsump=on

11

Theconsump=onfunc=onisconsiderablysimpleriftheelas=cityofinter-temporalsubs=tu=onofconsump=onisequaltounity(logarithmicpreferences).Inthiscase,

Giventhatwehaveassumedthatρ>n,withaunitaryelas=cityofinter-temporalsubs=tu=onofconsump=on,theshareofconsump=onintotalwealthisequaltothedifferencebetweenthepurerateof=mepreferenceandthepopula=ongrowthrate.

γ (0) = er_(t )(1−θ )−ρ+θn

θ

⎛

⎝⎜⎜

⎞

⎠⎟⎟t

dtt=0

∞

∫⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1

= e−(ρ−n)tt=0

∞

∫( )−1 = ρ − n− lim

t→∞e−(ρ−n)t + lim

t→0e−(ρ−n)t

= ρ − n


RealInterestRatesandAggregateConsump=onintheRepresenta=veHouseholdModel

• Finally,itisalsoworthno=ngthattheoverallimpactofrealinterestratesonconsump=onisnotlimitedtotheimpactonthepropensitytoconsumeoutoftotalwealth.

• Anincreaseinrealinterestratesleadstoadecreaseinthepresentvalueoffuturelaborincome,reducingtheoverallwealthoftherepresenta=vehousehold,andleadingtoareduc=oninconsump=on,eveniftheinter-temporalelas=cityofsubs=tu=onisequaltoone.

• Essen=ally,theeffectsofrealinterestratesonthepresentvalueofincomefromemployment,i.ethewealtheffectsofrealinterestrates,reinforcethesubs=tu=oneffectoncurrentconsump=on.

12


Consump=onunderUncertainty

• Wenextexaminedynamicmodelsofconsumerchoiceunderuncertainty.Wecon=nue,asintheRamseymodel,totakethedecisionofthehouseholdwithregardtolaborsupplyasgiven,assumingthateachhouseholdprovidesaunitoflaborperperiod.Wealsoassumethatthehouseholdcanborrowandlendfreelyincompe==vecapitalmarkets.

• Thechoiceofconsump=onundercondi=onsofuncertaintyislinkedtotheporAolioalloca=ondecisionsofthehousehold(Samuelson1969,Merton1969).

• Underuncertainty,consump=ongenerallydependsonthesamefactorsasundercertainty,onlyinthecaseofquadra=cpreferences,whichguaranteecertaintyequivalence.Inthecaseofquadra=cpreferenceswecanderivethepermanentincomemodelofconsump=on.

• Inallothercases,underuncertainty,wecannotgobeyondthefirstordercondi=onsandsolveexplicitlyforconsump=on,unlesswemakefurtherrestric=veassump=onsaboutthepreferencesofhouseholdsorthevariabilityoflaborincome.

• WithregardtoporAoliochoice,thismodelresultsintheconsump;oncapitalassetpricingmodel.Thissuggeststhat,underquadra=cpreferences,theexpectedreturnpremiumofariskyassetispropor=onaltothecovarianceofitsreturnwithconsump=on.Thisfactorofpropor=onalityissome=mesreferredtoasaconsump=onbeta,andcanbeusedtoexplainthevalua=onofriskyassets.

13


TheOp=miza=onofExpectedU=lityunderUncertainty

Weassumeahouseholdwitha=mehorizonT,whichisuncertainwithregardtoitsfuturelaborincomeandwithregardtothefuturereturnsonitsporAolioofassets.

Inperiod0thehouseholdmaximizes,

14

E01

1+ ρ⎛⎝⎜

⎞⎠⎟t=0

T −1

∑t

u(Ct )⎛

⎝⎜

⎞

⎠⎟

undertheassetaccumula=onconstraint,

At+1 = (At +Yt −Ct ) (1+ rt )ω t + (1+ xt )(1−ω t )[ ]


Defini=onsEtdenotesamathema=calexpecta=onbasedonthesetofavailableinforma=onavailableinperiodt.

ρisthepurerateof=mepreferenceofthehousehold.

uaperiodicu=lityfunc=on,dependingonconsump=on.

AtisthevalueoftheporAolioofassetsofthehouseholdatthebeginningofperiodt.

Ytislaborincome,whichisassumedtobearandomvariablewhosevalueisknowninperiodtbutnotbefore.

Grosssavingsofthehouseholdaredefinedby,At+Yt-Ct

Thehouseholdallocatesitsgrosssavingsbetweena“safe”asset,withacertainreturnrt,anda“risky”asset,withuncertainreturnxt.

TheporAolioalloca=ondecisionofthehouseholdisdeterminedbythepercentageωofitsassetsthatisinvestedinthe“safe”asset.Theterminbracketsthusdenotestheaveragerateofreturnofthehousehold’sporAolio.

15


SolvingtheProblemoftheHouseholdusingDynamicProgramming

Thehouseholdchoosesaconsump=onandporAolioalloca=onplanforperiod0,intheknowledgethatitwillbeabletochooseanewplaninthefollowingperiod1,anewplaninthefollowingperiod2,andsoon,un=lthepenul=mateperiodT-1.Theeasiestmethodofsolvingofdynamicproblemsunderuncertaintyisthemethodofstochas;cdynamicprogramming.

Dynamicprogrammingconvertsmul=-periodproblemsintoasequenceofsimplertwoperiodselec=onproblems.Thefirststepistheintroduc=onofavaluefunc=onVt(At),whichisdefinedas,

16

Vt (At ) = maxEt1

1+ ρ⎛⎝⎜

⎞⎠⎟s=t

T −1

∑s−t

u(Cs )⎛

⎝⎜

⎞

⎠⎟

Thevaluefunc=oninperiodtisthediscountedpresentvalueoftheexpectedu=lityofthehousehold,calculatedundertheassump=onthatthehouseholdfollowstheop=malprogramofconsump=onandporAolioalloca=on.Thisop=malvaluedependsonthevalueoftheporAolioofthehouseholdatthebeginningofperiodt,whichistheonlystatevariableaffec=ngthehousehold.


TheBellmanEqua=onandtheFirstOrderCondi=ons

Thevaluefunc=onsa=sfiesthefollowingrecursiveequa=on,whichisknownastheBellmanequa;on.

17

Vt (At ) = max{Ct ,ω t }u(Ct )+

11+ ρ

Et Vt+1(At+1)[ ]⎧⎨⎩

⎫⎬⎭

Thevaluefunc=oninperiodtisequaltothemaximumu=lityofconsump=oninperiodtplusthediscountedexpectedvaluefunc=oninperiodt+1.Thefirstordercondi=onsforthemaximiza=onofthevaluefunc=onundertheassetaccumula=onconstraintare,

′u (Ct ) = Et1

1+ ρ(1+ rt )ω t + (1+ xt )(1−ω t )( ) ′Vt+1(At+1)

⎡

⎣⎢

⎤

⎦⎥

Et ′Vt+1(At+1)(rt − xt )[ ]= 0


Interpreta=onoftheFirstOrderCondi=ons

ApplyingtheenvelopetheoremtotheBellmanEqua=on,i.etheeffectsofasmallchangeinthevalueoftheporAolioofassetsAtonbothofitssides,wegetthatthemarginalvalueofthehouseholdporAolioofassetsisequaltothemarginalu=lityofconsump=on.

18

′Vt (At ) = ′u (Ct )

AYersomesubs=tu=onsinthefirstordercondi=ons,wegetthat,

′u (Ct ) =1+ rt1+ ρ

Et ′u (Ct+1)[ ]

′u (Ct ) =1

1+ ρEt 1+ xt( ) ′u (Ct+1)⎡⎣ ⎤⎦

Thesecondi=onshaveasimpleinterpreta=on,whichisageneraliza=onoftheinterpreta=onoftheEulerequa=onforconsump=onintheRamseyproblem.Recall,thattheEulerequa=onforconsump=onintheRamseyproblemsuggeststhatthemarginalrateofsubs=tu=onbetweenthelevelsofconsump=oninthetwoperiodsmustbeequaltothemarginalrateoftransforma=onforboththe“safe”andthe“risky”asset.


Implica=onsoftheFirstOrderCondi=onsfortheDynamicEvolu=onofConsump=on

Thefirstordercondi=onsimplystrongrestric=onsforthedynamicbehaviorofconsump=on.Thefirstcondi=onimpliesthat,

19

Giventhemarginalu=lityofconsump=onu’(Ct),therenoaddi=onalinforma=onavailableinperiodtthatcouldhelppredictu’(Ct+1),thefuturemarginalu=lityofconsump=on.

Assumingthattheu=lityfunc=onisquadra=cinconsump=on,andthattherateofreturnofthe“safe”assetisequaltothepurerateof=mepreference,thenthiscondi=ontakestheform,

Consump=onfollowsa“randomwalk”.Giventhelevelofconsump=oninperiodt,noothervariableknowninperiodtcanhelppredictconsump=oninperiodt+1.Thispredic=onofthemodelwasfirsthighlightedbyHall(1978),whoinves=gateditempirically.Thishasgeneratedahostoftheore=calandempiricalfollowupstudiesofthispredic=on.

1+ rt1+ ρ

′u (Ct+1) = ′u (Ct )+ ε t+1 where, Et (ε t+1) = 0

Ct+1 = Ct + ε t+1


TheConsump=onCapitalAssetPricingModel(CAPM)

Thefirstordercondi=onscanalsobeusedtodeterminetherateofreturnoftheriskfreeassetandtheexpectedrateofreturn,andhencethepriceoftheriskyasset.Thisrequiresthatallindividualhouseholdsarealike,i.e.thatthereisarepresenta=vehousehold.

FromtheEulerequa=onfortheriskfreeasset,itsrateofreturnwillsa=sfy,

20

1+ rt = (1+ ρ) ′u (Ct )Et ′u (Ct+1)[ ]

FromtheEulerequa=onfortheriskyasset,itsrateofreturnwillsa=sfy,

Et 1+ xt( ) = (1+ ρ) ′u (Ct )Et ′u (Ct+1)( ) −

Covt 1+ xt , ′u (Ct+1)( )Et ′u (Ct+1)( )


TheExpectedReturnPremiumontheRiskyAsset

Takingthedifferenceofthefirstordercondi=onsfortheriskyandtheriskfreeasset,weget,

21

Theexpectedreturnpremiumoftheriskyassetdependsonthecovarianceoftherateofreturnoftheriskyassetwiththemarginalu=lityofconsump=on.Giventhatthemarginalu=lityofconsump=onisnega=velycorrelatedwithconsump=on,becauseofdecreasingmarginalu=lity,theexpectedreturnpremiumoftheriskyassetwilldependposi=velyonthecovarianceoftherateorreturnoftheriskyassetwithconsump=on.Riskyassetswhosereturnsareposi=velycorrelatedwithconsump=on,willtendtohaveahigherexpectedreturnrela=vetotheriskfreeasset.

Thismodelofthedetermina=onofexpectedassetreturnsisknownastheconsump;oncapitalassetpricingmodel,or,consump=onCAPM.

Et xt( )− rt = −Covt 1+ xt , ′u (Ct+1)( )

Et ′u (Ct+1)( )


FromtheFirstOrderCondi=onstotheFullAnalysisofConsump=on

• Fromthefirstordercondi=onswecannotfullydescribethebehaviorofconsump=onandsavings,apartfromspecificcases.

• Therearetwospecialcaseswherewecancomeupwithspecificsolu=ons.Thefirstisthecaseofinsurableincomerisk,andthelaWeristhecaseofquadra;cu;lityfunc;ons.

• AsdemonstratedbyMerton(1971),iflaborincomecanbeinsured,wecandeducespecificsolu=onsforconsump=onforabroadclassofu=lityfunc=ons,thesocalledhyperbolicabsoluteriskaversion,orHARA,u=lityfunc=ons.Thisclassincludesisoelas=cu=lityfunc=onwithconstantrela=veriskaversion(constantrela;veriskaversion,orCRRA),theexponen=alu=lityfunc=onwithconstantabsoluteriskaversion(constantabsoluteriskaversionorCARA)andquadra;cu;lityfunc;ons.

22


ThePrincipleofOp=malityandtheDeriva=onofSpecificSolu=ons

Onewaytoderivethespecificsolu=onistousetheBelmannprincipleofop;mality,whichsaysthatforanyvalueofthestatevariable(theporAolioofassetsinthiscase)atagiven=meperiod,thesolu=onforthefuturemustbeop=mal.

Usingthisprincipleandthevaluefunc=on,thesolu=oncanbefoundthroughbackwardinduc;on.Forexample,inperiodT-2,foranyvalueoftheporAolioofassetsAT-2,thehouseholdfacesatwo-periodproblem.Solvingthisproblem,takeastepback,andsolvetheproblemoftheperiodT-3,havingalreadyiden=fiedthevalueofthevaluefunc=onofT-2.Thenwemoveonandsolvethesameprobleminduc=velyfortheT-4periodandsoon.

Inthecaseofaninfinite=mehorizonwetakethelimitofthesolu=ontotheproblemofTperiods,asTtendstoinfinity.Alterna=vely,wecanalsosolvetheproblemofinfiniteperiodsdirectly.

23


TheSpecialCaseofHARAU=lityFunc=ons

• Iftheconsumerhasaninfinite=mehorizon,ifthe"safe"interestrateisfixedandifthe“uncertain”returnxtisdistributedaccordingtoanindependent,uniform,probabilitydistribu=on,thevaluefunc=onisindependentof=meandonlydependsonthestatevariableAt.Therefore,wecanpresumeitsform,derivetheconsump=onfunc=onandverifyifourpresump=onwascorrect.

• ThereasonthatHARAtypeu=lityfunc=onsallowustoinferanaly=calsolu=ons,isthatthevaluefunc=onbelongstothesamefamilyastheu=lityfunc=on,andallthatremainsistoinfertheparametersofthevaluefunc=on.

24


TheCaseofLogarithmicPreferences

Weconsiderthecaseinwhich,

25

u(Ct ) = lnCt

UsingtheresultofMertonthatthevaluefunc=onhasthesamefunc=onalformastheu=lityfunc=on,forHARAtypeu=lityfunc=ons,wepresumethatthevaluefunc=ontakestheform,

V (At ) = a ln(At )+ b

whereaandbareconstantparametersthatmustbedetermined.


TheProblemoftheConsumerwithLogarithmicPreferences

Thisconjectureallowsustoformulatethemaximiza=onprobleminperiodtas,

26

Wehavealsousedtheextraassump=onthatY=0.Thefirstordercondi=onsimplythat,

maxln(Ct )+1

1+ ρEt a ln(At+1)+ b[ ]

undertheconstraint,

At+1 = (At −Ct ) (1+ r)ω t + (1+ xt )(1−ω t )[ ]

Ct = 1+ a1+ ρ

⎛⎝⎜

⎞⎠⎟

−1

At Et (r − xt ) (1+ r)ω + (1+ xt )(1−ω )( )−1⎡⎣

⎤⎦ = 0


Interpreta=onoftheFirstOrderCondi=onsandDeriva=onoftheConsump=onFunc=on

Thefirstcondi=ondeterminesconsump=onasalinearfunc=onofthevalueofthetotalporAolioofassetsofthehousehold.Theseconddeterminesindirectlytheop=malpropor=oninvestedinthe“safe”assetωasaconstant,duetotheassump=onthatthereturnofthe“risky”assetxisdistributedaccordingtoauniform,independentprobabilitydistribu=on.Thepropor=onωisindependentofthetotalvalueoftheporAolioofassets.

Inordertofindaandbwesubs=tuteinthevaluefunc=onandcomparecoefficientswith.bisacomplexbutnoteconomicallyinteres=ngconstant,whichdependsonalltheparametersofthemodel.aisdeterminedas(1+ρ)/ρ.Subs=tu=ngthisvalueintheconsump=onfunc=on,weget,

27

a = 1+ ρρ

therefore, Ct =ρ

1+ ρAt

Consump=onisalinearfunc=onofthetotalvalueoftheporAolioofthehousehold,andthemarginalpropensitytoconsumeoutofwealthdependsonlyonthepurerateof=mepreferenceandnotontherealinterestrateortherateofreturnofthe“risky”asset.


TheCaseofQuadra=cPreferencesandCertaintyEquivalence

28

Thesecondcaseweshallexamineisthecaseofquadra=cpreferences.WeshallassumethattheporAolioconsistsonlyofthe“safe”asset.Themaximiza=onproblemofthehouseholdis,

undertheconstraint,

maxE01

1+ ρ⎛⎝⎜

⎞⎠⎟

t

aCt − bCt2( )t=0

T −1∑⎡

⎣⎢⎢

⎤

⎦⎥⎥

At+1 = (1+ r) At +Yt −Ct( )


FromtheFirstOrderCondi=onstotheFullAnalysisofConsump=on

29

Fromthefirstordercondi=ons,

Inwhatfollowsweshallassumethatr=ρ.Inthiscasethefirstordercondi=onstaketheform,

EtCt+1 =r − ρ1+ r

a2b

+ 1+ r1+ ρ

Ct

EtCt+1 = Ct

Therefore,itfollowsthat,

E0Ct = C0

fort=0,1,2,…,T-1.Expectedconsump=onisconstantforthedura=onoftheprogram(consump=onsmoothing).


TheInter-temporalBudgetConstraintandtheDetermina=onofConsump=on

30

Theinter-temporalbudgetconstraintisgivenby,

Asaresult,at=me0thehouseholdmustsa=sfy,

Subs=tu=ngforconsump=onandtakingthelimitasTtendstoinfinity,

11+ r

⎛⎝⎜

⎞⎠⎟t=0

T −1∑t

Ct = A0 +11+ r

⎛⎝⎜

⎞⎠⎟t=0

T −1∑t

Yt

E011+ r

⎛⎝⎜

⎞⎠⎟t=0

T −1∑t

Ct

⎡

⎣⎢

⎤

⎦⎥ = A0 + E0

11+ r

⎛⎝⎜

⎞⎠⎟t=0

T −1∑t

Yt⎡

⎣⎢

⎤

⎦⎥

C0 =r1+ r

A0 + E011+ r

⎛⎝⎜

⎞⎠⎟t=0

∞∑t

Yt⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜

⎞

⎠⎟


ThePermanentIncomeHypothesisProper=esoftheConsump=onFunc=onwith

Quadra=cPreferences• Consump=onisconstantandisafixedpercentageofthetotalwealthofthehousehold,includingthepresentvalueofexpectedlaborincome.

• Ineveryperiod,thehouseholdconsumesaconstantfrac=onofitstotalwealth,dependingontherealinterestrate(orthepurerateof=mepreference),sothatexpectedtotalwealthremainsconstant.

• Thechangeinconsump=onfromperiodtoperiodisdeterminedonlybytherevisionofexpecta=onsregardinglaborincome.

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ChangesinLaborIncomeandChangesinConsump=on

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Onlypreviouslyunan=cipatedchangesinlaborincomebringaboutchangesinconsump=on,

UnderthehypothesisthatincomefollowsanAR(1)processoftheform,

thechangeinconsump=onisdeterminedby,

Ct −Ct−1 =r1+ r

11+ r

⎛⎝⎜

⎞⎠⎟i=0

∞∑i

Et Yt+i( )− Et−1 Yt+i( )( )

Yt = Y0 + λYt−1 + ε t

Ct −Ct−1 =r

1+ r − λε t


TemporaryversusPermanentChangesinLaborIncomeandConsump=on

• Ifλ<1,disturbancesinlaborincomearetemporary,andthecoefficientoftheunan=cipatedchangeinlaborincomeεissmallerthanunity.Thiscaseincorporatesthepredic=onsofthe“permanentincome”hypothesisofFriedman(1957)andthe“lifecycle”hypothesisofModiglianiκαιBrumberg(1954),thatconsump=onsmoothsouttransitorychangesinincome.

• Ifλ=1,thendisturbancesinlaborincomeεareofapermanentnature,andthecoefficientofthechangeinconsump=onisalsoequaltounity.Permanentchangesinlaborincomeleadtoequivalentpermanentchangesinhouseholdconsump=on.

• Empiricalstudiesofthe“permanentincome”hypothesissuggestthataggregateconsump=ondisplays“excesssensi=vity”tochangesincurrentincome,indica=ngthatonemayhavetogobeyondthe“permanentincome”hypothesisinexplainingaggregateconsump=on.

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TheConsump=onCapitalAssetPricingModelwithQuadra=cPreferences

Assumingquadra=cpreferences,themarginalu=lityofconsump=onisgivenby,

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′u (Ct ) = a − bCt

Themarginalu=lityofconsump=onisthusanega=velinearfunc=onofconsump=on.Subs=tu=ngforthemarginalu=lityofconsump=oninconsump=onCAPM,weget,

Et xt( )− rt =2bCovt 1+ xt ,Ct+1( )

a − bEtCt+1

Underquadra=cpreferencestheexpectedreturnpremiumofariskyassetispropor=onaltothecovarianceofitsreturnwithconsump=on.Thisfactorofpropor=onalityissome=mesreferredtoasaconsump;onbeta,fromaregressionofconsump=ongrowthonassetreturns.


TheConsump=onCapitalAssetPricingModelandtheEquityPremiumPuzzle

• Theoriginalcapitalassetpricingmodel(CAPM)ofSharpe(1964)andLintner(1965)assumedthatinvestorsareconcernedwiththemeanandvarianceofthereturnoftheirporAolio,ratherthanthemeanandvarianceofconsump=on.Thatversionofthemodelthusfocusedonso-calledmarketbetas,thatiscoefficientsfromregressionsofassetreturnsonthereturnsofamarketporAolio.

• Acentralpredic=onoftheconsump=onCAPMisthatthereturnpremiumofariskyassetispropor=onaltoitsconsump;onbeta.

• FollowingMehraandPrescoW(1985),empiricalstudiessuggestasocalled“equitypremiumpuzzle”,i.eamuchbiggerdifferencebetweentheaveragereturnofequi=es(theriskyasset)andgovernmentbonds(thesafeasset)thanwouldbesuggestedbytheconsump=onCAPM.

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SomeConclusionsaboutConsump=onunderUncertainty

• Wehaveexaminedthedetermina=onofhouseholdconsump=onundercondi=onsofuncertainty,inconjunc=onwiththedetermina=onofthealloca=onoftheporAolioofthehouseholdamongalterna=veassets(Samuelson1969,Merton1969).

• Undercondi=onsofuncertainty,forahouseholdthatcanborrowandlendfreelyinthecapitalmarket,consump=ongenerallydependsonthesamefactorsasundercertainty.Thecurrentandexpectedfutureratesofassets,thecurrentandexpectedfuturelaborincomeandthetotalvalueoftheporAolioandhumanwealthofthehousehold.

• Consump=ondoesnotdependoncurrenthouseholdincomebutontotalwealth,whichconsistsofthevalueofitsporAolio,plusthepresentvalueofcurrentandexpectedfuturelaborincomeInthissense,consump=onsmoothsouttemporarychangesinincome,asitdependson“permanent”or“lifecycle”income.(Friedman1957,ModiglianiBrumberg1954).

• However,the“permanent”incomehypothesiscannotexplainmanyofthefeaturesofindividualoraggregateconsump=onpaWerns,especiallythe“excesssensi=vity”ofconsump=ontochangesincurrentincome.Inaddi=on,theconsump=oncapitalassetpricingmodel,whichisaassociatedpredic=onofstochas=cinter-temporalmodelsofconsump=on,seemstoberefutedbythe“equitypremium”puzzle.Theliteraturehasthusalsoexaminedmoregeneralmodelsthatresultinprecau=onarysavings,orinwhichmarketsareincompleteandhouseholdsarealsoboundbyborrowingconstraints(seeAWanasio1999).

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the dynamic behavior of consump=on under uncertainty · prof george alogoskouﬁs, dynamic...

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