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2.12. RATIONALEXPECTATIONSEQUILIBRIUM
241
TheActualLawofMotionforfY
t
Aswe’veseen,agivenbelieftranslatesintoaparticular
decisionruleh
RecallingthatY
t
=ny
t
,theactuallawofmotionformarket-wideoutputisthen
Y
t+1
=nh(Y
t
/n,Y
t
)
(2.123)
Thus,whenfirmsbelievethatthelawofmotionformarket-wideoutputis(2.117),theiroptimizing
behaviormakestheactuallawofmotionbe(2.123)
DefinitionofRationalExpectationsEquilibrium
Arationalexpectationsequilibriumorrecursive
competitiveequilibriumofthemodelwithadjustmentcostsisadecisionrulehandanaggregate
lawofmotionHsuchthat
1. GivenbeliefH,themaphisthefirm’soptimalpolicyfunction
2. ThelawofmotionHsatisfiesH(Y)=nh(Y/n,Y)forallY
Thus,arationalexpectationsequilibriumequatestheperceivedandactuallawsofmotion(2.117)
and(2.123)
Fixedpointcharacterization Aswe’veseen,thefirm’soptimumprobleminducesamappingF
fromaperceivedlawofmotionHformarket-wideoutputtoanactuallawofmotionF(H)
ThemappingFisthecompositionoftwooperations,takingaperceivedlawofmotionintoa
decisionrulevia(2.118)–(2.120),andadecisionruleintoanactuallawvia(2.123)
TheHcomponentofarationalexpectationsequilibriumisafixedpointofF
ComputationofanEquilibrium
Nowlet’sconsidertheproblemofcomputingtherationalexpectationsequilibrium
MisbehaviorofF
Readersaccustomedtodynamicprogrammingargumentsmighttrytoad-
dressthisproblembychoosingsomeguessH
0
fortheaggregatelawofmotionandtheniterating
withF
Unfortunately,themappingFisnotacontraction
Inparticular,thereisnoguaranteethatdirectiterationsonFconverge
1
Fortunately,thereisanothermethodthatworkshere
ThemethodexploitsageneralconnectionbetweenequilibriumandParetooptimalityexpressed
inthefundamentaltheoremsofwelfareeconomics(see,e.g,[MCWG95])
1
Aliteraturethatstudieswhethermodelspopulatedwithagentswholearncanconvergetorationalexpectations
equilibriafeaturesiterationsonamodificationofthemappingFthatcanbeapproximatedasgF+(g)I.HereI
istheidentityoperatorandg2(0,1)isarelaxationparameter.See[MS89]and[EH01]forstatementsandapplications
ofthisapproachtoestablishconditionsunderwhichcollectionsofadaptiveagentswhouseleastsquareslearning
convergetoarationalexpectationsequilibrium.
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.12. RATIONALEXPECTATIONSEQUILIBRIUM
242
LucasandPrescott[LP71]usedthismethodtoconstructarationalexpectationsequilibrium
Thedetailsfollow
APlanningProblemApproach OurplanofattackistomatchtheEulerequationsofthemarket
problemwiththoseforasingle-agentchoiceproblem
Aswe’llsee,thisplanningproblemcanbesolvedbyLQcontrol(linearregulator)
Theoptimalquantitiesfromtheplanningproblemarerationalexpectationsequilibriumquantities
Therationalexpectationsequilibriumpricecanbeobtainedasashadowpriceintheplanning
problem
Forconvenience,inthissectionwesetn=1
Wefirstcomputeasumofconsumerandproducersurplusattimet
s(Y
t
,Y
t+1
):=
Z
Y
t
0
(a
0
a
1
x)dx
g(Y
t+1
Y
t
)2
2
(2.124)
Thefirsttermistheareaunderthedemandcurve,whilethesecondmeasuresthesocialcostsof
changingoutput
TheplanningproblemistochooseaproductionplanfY
t
gtomaximize
¥
å
t=0
b
t
s(Y
t
,Y
t+1
)
subjecttoaninitialconditionforY
0
SolutionofthePlanningProblem
Evaluatingtheintegralin(2.124)yieldsthequadraticform
a
0
Y
t
a
1
Y
2
t
/2
Asaresult,theBellmanequationfortheplanningproblemis
V(Y)=max
Y0
a
0
Y
a
1
2
Y
2
g(Y
0
Y)
2
2
+bV(Y
0
)
(2.125)
Theassociatedfirstorderconditionis
g(Y
0
Y)+bV
0
(Y
0
)=0
(2.126)
ApplyingthesameBenveniste-Scheinkmanformulagives
V
0
(Y)=a
0
a
1
Y+g(Y
0
Y)
Substitutingthisintoequation(2.126)andrearrangingleadstotheEulerequation
ba
0
+gY
t
[ba
1
+g(1+b)]Y
t+1
+gbY
t+2
=0
(2.127)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.12. RATIONALEXPECTATIONSEQUILIBRIUM
243
TheKeyInsight Returntoequation(2.122)andsety
t
=Y
t
forallt
(Recallthatforthissectionwe’vesetn=1tosimplifythecalculations)
Asmallamountofalgebrawillconvinceyouthatwheny
t
=Y
t
,equations(2.127)and(2.122)are
identical
Thus,theEulerequationfortheplanningproblemmatchesthesecond-orderdifferenceequation
thatwederivedby
1. findingtheEulerequationoftherepresentativefirmand
2. substitutingintoittheexpressionY
t
=ny
t
that“makestherepresentativefirmberepresen-
tative”
Ifitisappropriatetoapplythesameterminalconditionsforthesetwodifferenceequations,which
itis,thenwehaveverifiedthatasolutionoftheplanningproblemisalsoarationalexpectations
equilibriumquantitysequence
Itfollowsthatforthisexamplewecancomputeequilibriumquantitiesbyformingtheoptimal
linearregulatorproblemcorrespondingtotheBellmanequation(2.125)
TheoptimalpolicyfunctionfortheplanningproblemistheaggregatelawofmotionHthatthe
representativefirmfaceswithinarationalexpectationsequilibrium.
StructureoftheLawofMotion Asyouareaskedtoshowintheexercises, thefactthatthe
planner’sproblemisanLQproblemimpliesanoptimalpolicy—andhenceaggregatelawof
motion—takingtheform
Y
t+1
=k
0
+k
1
Y
t
(2.128)
forsomeparameterpairk
0
,k
1
Nowthatweknowtheaggregatelawofmotionislinear,wecanseefromthefirm’sBellman
equation(2.118)thatthefirm’sproblemcanalsobeframedasanLQproblem
Asyou’reaskedtoshowintheexercises,theLQformulationofthefirm’sproblemimpliesalaw
ofmotionthatlooksasfollows
y
t+1
=h
0
+h
1
y
t
+h
2
Y
t
(2.129)
Hencearationalexpectationsequilibriumwillbedefinedbytheparameters(k
0
,k
1
,h
0
,h
1
,h
2
)in
(2.128)–(2.129)
Exercises
Exercise1 Considerthefirmproblemdescribedabove
Letthefirm’sbelieffunctionHbeasgivenin(2.128)
Formulatethefirm’sproblemasadiscountedoptimallinearregulatorproblem,beingcarefulto
describealloftheobjectsneeded
UsethetypeLQfromtheQuantEcon.jlpackagetosolvethefirm’sproblemforthefollowingpa-
rametervalues:
a
0
=100,a
1
=0.05,b=0.95,g=10,k
0
=95.5,k
1
=0.95
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.12. RATIONALEXPECTATIONSEQUILIBRIUM
244
Expressthesolutionofthefirm’sproblemintheform(2.129)andgivethevaluesforeachh
j
Iftherewerenidenticalcompetitivefirmsallbehavingaccordingto(2.129),whatwould(2.129)
implyfortheactuallawofmotion(2.117)formarketsupply
Exercise2 Considerthefollowingk
0
,k
1
pairsascandidatesfortheaggregatelawofmotion
componentofarationalexpectationsequilibrium(see(2.128))
Extendingtheprogramthatyouwroteforexercise1,determinewhichifanysatisfythedefinition
ofarationalexpectationsequilibrium
• (94.0886298678,0.923409232937)
• (93.2119845412,0.984323478873)
• (95.0818452486,0.952459076301)
Describeaniterativealgorithmthatusestheprogramthatyouwroteforexercise1tocomputea
rationalexpectationsequilibrium
(Youarenotbeingaskedactuallytousethealgorithmyouaresuggesting)
Exercise3 Recalltheplanner’sproblemdescribedabove
1. Formulatetheplanner’sproblemasanLQproblem
2. Solveitusingthesameparametervaluesinexercise1
• a
0
=100,a
1
=0.05,b=0.95,g=10
3. RepresentthesolutionintheformY
t+1
=k
0
+k
1
Y
t
4. Compareyouranswerwiththeresultsfromexercise2
Exercise4 Amonopolistfacestheindustrydemandcurve(2.114)andchoosesfY
t
gtomaximize
å
¥
t=0
b
t
r
t
where
r
t
=p
t
Y
t
g(Y
t+1
Y
t
)2
2
FormulatethisproblemasanLQproblem
Computetheoptimalpolicyusingthesameparametersasthepreviousexercise
Inparticular,solvefortheparametersin
Y
t+1
=m
0
+m
1
Y
t
Compareyourresultswiththepreviousexercise.Comment.
Solutions
Solutionnotebook
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.13. MARKOVASSETPRICING
245
2.13 MarkovAssetPricing
Contents
• MarkovAssetPricing
– Overview
– PricingModels
– ClassesofAssets
– Implementation
– Exercises
– Solutions
“Alittleknowledgeofgeometricseriesgoesalongway”–RobertE.Lucas,Jr.
“Assetpricingisallaboutcovariances”–LarsPeterHansen
Overview
Anassetisaclaimonastreamofprospectivepayments
Thespotpriceofanassetdependsprimarilyon
• theanticipateddynamicsforthestreamofincomeaccruingtotheowners
• attitudestoriskandratesoftimepreference
Inthislectureweconsidersomestandardpricingmodelsanddividendstreamspecifications
Westudyhowpricesanddividend-priceratiosrespondinthesedifferentscenarios
Wealsolookatcreatingandpricingderivativeassetsbyrepackagingincomestreams
Keytoolsforthelectureare
• FormulasforpredictingfuturevaluesoffunctionsofaMarkovstate
• AformulaforpredictingthediscountedsumoffuturevaluesofaMarkovstate
PricingModels
Webeginwithsomenotationandthenproceedtofoundationalpricingmodels
Inwhatfollowsletfd
t
g
t0
beastreamofdividends
• Atime-tcum-dividendassetisaclaimtothestreamd
t
,d
t+1
,...
• Atime-tex-dividendassetisaclaimtothestreamd
t+1
,d
t+2
,...
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.13. MARKOVASSETPRICING
246
RiskNeutralPricing Letb=1/(1+r)beanintertemporaldiscountfactor
Inotherwords,ristherateatwhichagentsdiscountthefuture
Thebasicrisk-neutralassetpricingequationforpricingoneunitofacum-dividendassetis
p
t
=d
t
+b
t
[p
t+1
]
(2.130)
Thisisasimple“costequalsexpectedbenefit”relationship
Here
t
[y]denotesthebestforecastofy,conditionedoninformationavailableattimet
Inthepresentcasethisinformationsetconsistsofobservationsofdividendsupuntiltimet
Foranex-dividendasset,thebasicrisk-neutralassetpricingequationis
p
t
=b
t
[d
t+1
+p
t+1
]
(2.131)
PricingwithRandomDiscountFactor Whathappensifforsomereasontradersdiscountpay-
outsdifferentlydependingonthestateoftheworld?
Supposethatallagentsevaluatepayoffsaccordingtoastrictlyconcaveperiodutilityfunctionu
Michael HarrisonandDavidKreps[HK79]andLars Peter r Hansen andScott Richard[HR87]
showedthatinquitegeneralsettingsthepriceofanex-dividendassetobeys
p
t
=
t
[m
t+1
(d
t+1
+p
t+1
)]
(2.132)
wherem
t+1
isastochasticdiscountfactor
Comparing(2.131)and(2.132),thedifferenceisthatthefixeddiscountfactorbin(2.131)hasbeen
replacedbythestochasticdiscountfactorm
t+1
Wegiveexamplesofhowthestochasticdiscountfactorhasbeenmodeledbelow
AssetPricingandCovariances Firstareminder.Fromthedefinitionofaconditionalcovariance
cov
t
(x
t+1
y
t+1
)itfollowsthat
t
(x
t+1
y
t+1
)=cov
t
(x
t+1
,y
t+1
)+
t
x
t+1
t
y
t+1
(2.133)
Ifweapplythisdefinitiontotheassetpricingequation(2.132)weobtain
p
t
=
t
m
t+1
t
(d
t+1
+p
t+1
)+cov
t
(m
t+1
,d
t+1
+p
t+1
)
(2.134)
Itisusefultoregardequation(2.134)asageneralizationofequation(2.131)
• Inequation(2.131),thestochasticdiscountfactorm
t+1
=b,aconstant
• Inequation(2.131),thecovariancetermcov
t
(m
t+1
,d
t+1
+p
t+1
)iszerobecausem
t+1
=b
Equation(2.134)assertsthatthecovarianceofthestochasticdiscountfactorwiththeoneperiod
payoutd
t+1
+p
t+1
isanimportantdeterminantofthepricep
t
Wegiveexamplesofsomemodelsofstochasticdiscountfactorsthathavebeenproposedlaterin
thislectureandalsoinalaterlecture
Fornowlet’sstudysomeoftheimplicationsofequation(2.132)forassetprices
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.13. MARKOVASSETPRICING
247
SimpleExamples Whatpricedynamicsresultfromthesemodels?
Theanswertothisquestiondependson
1. theprocesswespecifyfordividends
2. thestochasticdiscountfactorandhowitiscorrelatedwithdividends
Let’slookatsomeexamplesthatillustratethisidea
Example 1: : Constantdividends, , riskneutralpricing Thesimplestcaseisa constant, non-
randomdividendstreamd
t
=d>0
Removingtheexpectationfrom(2.130)anditeratingforwardgives
p
t
=d+bp
t+1
=d+b(d+bp
t+2
)
.
.
.
=d+bd+b
2
d++b
1
d+b
k
p
t+k
Unlesspricesexplodeinthefuture,thissequenceconvergesto
¯p:=
1
b
d
(2.135)
Thispriceistheequilibriumpriceintheconstantdividendcase
Indeed,simplealgebrashowsthatsettingp
t
¯pforalltsatisfiestheequilibriumconditionp
t
=
d+bp
t+1
Theex-dividendequilibriumpriceis(b)
1
bd
Example2: Deterministicdividends,riskneutralpricing Consideragrowing,non-random
dividendprocessd
t
=ltd
0
where0<lb<1
Thecum-dividendpriceunderriskneutralpricingisthen
p
t
=
d
t
bl
=
l
t
d
0
bl
(2.136)
Toobtainthis,
1. setv
t
=p
t
/d
t
in(2.130)andthenv
t
=v
t+1
=vtosolveforconstantv
2. recoverthepriceasp
t
=vd
t
Theex-dividendpriceisp
t
=(1 bl1bld
t
If,inthisexample,wetakel=1+g,thentheex-dividendpricebecomes
p
t
=
1+g
r g
d
t
ThisiscalledtheGordonformula
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.13. MARKOVASSETPRICING
248
Example3: Markovgrowth,riskneutralpricing Nextweconsideradividendprocesswhere
thegrowthrateisMarkovian
Inparticular,
d
t+1
=l
t+1
d
t
where
fl
t+1
=s
j
jl
t
=s
i
g=P
ij
:=:P[i,j]
Thisnotationmeansthatfl
t
gisannstateMarkovchainwithtransitionmatrixPandstatespace
s=fs
1
,...,s
n
g
Toobtainassetpricesunderriskneutrality,recallthatin(2.136)thepricedividendratiop
t
/d
t
is
constantanddependsonl
Thisencouragesustoguessthat,inthecurrentcase,p
t
/d
t
isconstantgivenl
t
Thatisp
t
=v(l
t
)d
t
forsomeunknownfunctionvonthestatespace
Tosimplifynotation,letv
i
:=v(s
i
)
Foracum-dividendstockwefindthatv
i
=1+b
å
n
j=1
P
ij
s
j
v
j
Letting1beann1vectorofonesand
˜
P
ij
=P
ij
s
j
,wecanexpressthisinmatrixnotationas
v=(I b
˜
P)
1
1
Hereweareassuminginvertibility,whichrequiresthatthegrowthrateoftheMarkovchainisnot
toolargerelativetob
(Inparticular,thattheeigenvaluesof
˜
Pbestrictlylessthanb
1
inmodulus)
Similarreasoningyieldstheex-dividendprice-dividendratiow,whichsatisfies
w=b(I b
˜
P)
1
Ps
0
Example4:Deterministicdividends Consideranonrandomdividendstreamd
t
=l
t
dforsome
l>0
Furthermore,supposea(nonstochastic)discountfactorm
t+1
=bl forg1
Ourformulaforpricingacum-dividendclaimtothestreamd
t
=ltdbecomes
p
t
=d
t
+bl
g
p
t+1
Guessingagainthatthepriceobeysp
t
=vd
t
wherevisaconstantprice-dividendratio,wehave
vd
t
=d
t
+bl gvd
t+1
,or
v=
1
blg
Ifg=1,thentheprecedingformulafortheprice-dividendratiobecomesv=1/(b)
Heretheprice-dividendratioisconstantandindependentofthedividendgrowthratel
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.13. MARKOVASSETPRICING
249
Lucas’sModel WenowdescribeaversionofacelebratedassetpricingmodelofRobertE.Lucas,
Jr.[Luc78]
Wepresentonlyanoutline,withfullderivationslefttoalaterlecture
Inanutshell,Lucasmadetwokeyassumptionstospecializeourgeneralassetpricingequation
(2.132):
• Heassumedthatthestochasticdiscountfactortooktheform
m
t+1
=b
u
0(
c
t+1
)
u0(c
t
)
whereuisaconcaveutilityfunctionandc
t
istimetconsumptionofarepresentativeconsumer
• Heassumedthattheconsumptionprocessfc
t
g¥
t=0
oftherepresentativeconsumerisgov-
ernedbyaMarkovprocess
• Heassumedthattheassetbeingpricedisaclaimontheaggregateconsumptionprocess,so
thatd
t
=c
t
Itiscommontoassumethattheutilityfunctiontakesthefollowingform
u(c)=
c
g
g
withg>0
(2.137)
whereu(c)=lncwheng=1
Theconstantrelativeriskaversion(CRRA)utilityfunction(2.137)impliesthat
u
0
(c)=c
g
sothat
m
t+1
=b
c
t+1
c
t
g
or
m
t+1
=bl
g
t+1
where
l
t+1
c
t+1
c
t
(2.138)
Below,we’lloftenassumethatthegrossgrowthrateofconsumptionl
t+1
isgovernedbyaMarkov
process
ClassesofAssets
Fortheremainderofthislecturewefocusoncomputingassetpriceswhen
• endowmentsfollowafinitestateMarkovchain
• agentsareriskaverse,andpricesobey(2.132)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.13. MARKOVASSETPRICING
250
Asin[MP85],thereisanendowmentofaconsumptiongoodthatfollows
c
t+1
=l
t+1
c
t
(2.139)
Herel
t
isgovernedbythenstateMarkovchaindiscussedabove
ALucastreeisaclaimonthisendowmentstream
We’llpriceseveraldistinctassets,including
• TheLucastreeitself
• Aconsol(atypeofbondissuedbytheUKgovernmentinthe19thcentury)
• Finiteandinfinitehorizoncalloptionsonaconsol
PricingtheLucastree Using(2.132),thedefinitionofuand(2.139)leadsto
p
t
=
t
h
bl
g
t+1
(c
t+1
+p
t+1
)
i
(2.140)
DrawingintuitionfromourearlierdiscussiononpricingwithMarkovgrowth,weguessapricing
functionoftheformp
t
=v(l
t
)c
t
wherevisyettobedetermined
Ifwesubstitutethisguessinto(2.140)andrearrange,weobtain
v(l
t
)c
t
=
t
h
bl
g
t+1
(c
t+1
+c
t+1
v(l
t+1
))
i
Using(2.139)againandsimplifyinggives
v(l
t
)=
t
h
bl
g
t+1
(1+v(l
t+1
))
i
Asbeforeweletv(s
i
)=v
i
,sothatvismodeledasann1vector,and
v
i
=b
n
å
j=1
P
ij
s
g
j
(1+v
j
)
(2.141)
Letting
˜
P
ij
=P
ij
s
g
j
,wecanwrite(2.141)asv=b
˜
P1+b
˜
Pv
Assumingagainthattheeigenvaluesof
˜
Parestrictlylessthanb
1
inmodulus,wecansolvethis
toyield
v=b(I b
˜
P)
1
˜
P1
(2.142)
Withlogpreferences,g=1andhences
=
1,fromwhichweobtain
v=
b
b
1
Thus,withlogpreferences,theprice-dividendratioforaLucastreeisconstant
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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