3.11. HISTORYDEPENDENTPUBLICPOLICIES
431
Theplanissaidtobecredibleif, foreachtandeachstate(Q
t
,u
t
,G
t
,J
t
),theplansatisfiesthe
incentiveconstraint
J
t
=A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
+bJ
t+1
(ˆt
t+1
,
ˆ
G
t+1
)
(3.147)
A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
+bJ
t+1
(t
t+1
,G
t+1
)
(3.148)
foralltaxratest
t+1
2
availabletothegovernment
Here
ˆ
G
t+1
=
G
t
ˆt
t+1
Q
t+1
b
• Inequalityexpressesthatcontinuationvaluesadjusttodeviationsinwaysthatdiscourage
thegovernmentfromdeviatingfromtheprescribed
ˆ
t
t+1
• Inequality(3.147)indicatesthattwocontinuationvaluesJ
t+1
contributetosustainingtimet
promisedvalueJ
t
– J
t+1
(ˆt
t+1
,
ˆ
G
t+1
)isthecontinuationvaluewhenthegovernmentchoosestoconfirmthe
privatesector’sexpectation,formedaccordingtothedecisionrule(3.143)
9
– J
t+1
(t
t+1
,G
t+1
)tellsthecontinuationconsequencesshouldthegovernmentdisappoint
theprivatesector’sexpectations
Theinternalstructureofacredibleplandetersdeviationsfromit
That(3.147)mapstwocontinuationvaluesJ
t+1
(t
t+1
,G
t+1
)andJ
t+1
(ˆt
t+1
,
ˆ
G
t+1
)intoonepromised
valueJ
t
reflectshowacredibleplanarrangesasystemofprivatesectorexpectationsthatinduces
thegovernmenttochoosetoconfirmthem
Chang[Cha98]buildsonhowinequality(3.147)mapstwocontinuationvaluesintoone
RemarkLetJbethesetofvaluesassociatedwithcredibleplans
EveryvalueJ2J canbeattainedbyacredibleplanthathasarecursiverepresentationofform
form(3.143),(3.144),(3.145)
Thesetofvaluescanbecomputedasthelargestfixedpoint ofanoperatorthatmapssetsof
candidatevaluesintosetsofvalues
Givenavaluewithinthisset,itispossibletoconstructagovernmentstrategyoftherecursive
form(3.143),(3.144),(3.145)thatattainsthatvalue
Inmanycases,thereissetaofvaluesandassociatedcredibleplans
InthosecaseswheretheRamseyoutcomeiscredible,amultiplicityofcredibleplansisakeypart
ofthestorybecause,aswehaveseenearlier,acontinuationofaRamseyplanisnotaRamseyplan
Forittobecredible,aRamseyoutcomemustbesupportedbyaworseoutcomeassociatedwith
anotherplan,theprospectofreversiontowhichsustainstheRamseyoutcome
9
Notethedoubleroleplayedby(3.143): asdecisionruleforthegovernmentandastheprivatesector’srulefor
forecastinggovernmentactions.
T
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ARGENTAND
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April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
432
Concludingremarks
Theterm‘optimalpolicy’,whichpervadesanimportantappliedmonetaryeconomicsliterature,
meansdifferentthingsunderdifferenttimingprotocols
Underthe‘static’Ramseytimingprotocol(i.e.,chooseasequenceonce-and-for-all),weobtaina
uniqueplan
Herethephrase‘optimalpolicy’seemstofitwell,sincetheRamseyplanneroptimallyreapsearly
benefitsfrominfluencingtheprivatesector’sbeliefsaboutthegovernment’slateractions
Whenweadoptthesequentialtimingprotocolassociatedwithcrediblepublicpolicies,‘optimal
policy’isamoreambiguousdescription
Thereisamultiplicityofcredibleplans
True, thetheoryexplainshowitisoptimalforthegovernmenttoconfirmtheprivatesector’s
expectationsaboutitsactionsalongacredibleplan
Butsomecredibleplanshaveverybadoutcomes
Thesebadoutcomesarecentraltothetheorybecauseitisthepresenceofbadcredibleplansthat
makespossiblebetteronesbysustainingthelowcontinuationvaluesthatappearinthesecond
lineofincentiveconstraint(3.147)
Recently,manyhavetakenforgrantedthat‘optimalpolicy’means‘followtheRamseyplan’
10
InpursuitofmoreattractivewaystodescribeaRamseyplanwhenpolicymakingisinpractice
donesequentially,somewritershaverepackagedaRamseyplaninthefollowingway
• TakeaRamseyoutcome-asequenceofendogenousvariablesunderaRamseyplan-and
reinterpretit(orperhapsonlyasubsetofitsvariables)asatargetpathofrelationshipsamong
outcomevariablestobeassignedtoasequenceofpolicymakers
11
• Ifappropriate(infinitedimensional)invertibilityconditionsaresatisfied,itcanhappenthat
followingtheRamseyplanistheonlywaytohitthetargetpath
12
• Thespiritofthisworkistosay,“inademocracyweareobligedtolivewiththesequential
timingprotocol,solet’sconstrainpolicymakers’objectivesinwaysthatwillforcethemto
followaRamseyplaninspiteoftheirbenevolence”
13
• Bythisslightofhand,weacquireatheoryofanoptimaloutcometargetpath
This‘invertibility’argumentleavesopentwoimportantlooseends:
1. implementation,and
2. timeconsistency
10
Itispossibletoread[Woo03]and[GW10]asmakingsomecarefullyqualifiedstatementsofthistype.Someofthe
qualificationscanbeinterpretedasadvice‘eventually’tofollowatailofaRamseyplan.
11
Inourmodel,theRamseyoutcomewouldbeapath(~p,
~
Q).
12
See[GW10].
13
SometimestheanalysisisframedintermsoffollowingtheRamseyplanonlyfromsomefuturedateTonwards.
T
HOMAS
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ARGENTAND
J
OHN
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TACHURSKI
April20,2016
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3.12. DEFAULTRISKANDINCOMEFLUCTUATIONS
433
Asfor(1),repackagingaRamseyplan(orthetailofaRamseyplan)asatargetoutcomesequence
doesnotconfrontthedelicateissueofhowthattargetpathistobeimplemented
14
Asfor(2),itisaninterestingquestionwhetherthe‘invertibility’logiccanrepackageandconceal
aRamseyplanwellenoughtomakepolicymakersforgetorignorethebenevolentintentionsthat
giverisetothetimeinconsistencyofaRamseyplaninthefirstplace
Toattainsuchanoptimaloutputpath,policymakersmustforgettheirbenevolentintentionsbe-
causetherewillinevitablyoccurtemptationstodeviatefromthattargetpath,andtheimplied
relationshipamongvariableslikeinflation,output,andinterestratesalongit
RemarkThecontinuationofsuchanoptimaltargetpathisnotanoptimaltargetpath
3.12 DefaultRiskandIncomeFluctuations
Contents
• DefaultRiskandIncomeFluctuations
– Overview
– Structure
– Equilibrium
– Computation
– Results
– Exercises
– Solutions
Overview
ThislecturecomputesversionsofArellano’s[Are08]modelofsovereigndefault
Themodeldescribesinteractionsamongdefaultrisk,output,andanequilibriuminterestratethat
includesapremiumforendogenousdefaultrisk
Thedecisionmakerisagovernmentofasmallopeneconomythatborrowsfromrisk-neutral
foreigncreditors
Theforeignlendersmustbecompensatedfordefaultrisk
Thegovernmentborrowsandlendsabroadinordertosmooththeconsumptionofitscitizens
Thegovernmentrepaysitsdebtonlyifitwantsto,butdecliningtopayhasadverseconsequences
Theinterestrateongovernmentdebtadjustsinresponsetothestate-dependentdefaultprobabil-
itychosenbygovernment
Themodelyieldsoutcomesthathelpinterpretsovereigndefaultexperiences,including
• countercyclicalinterestratesonsovereigndebt
14
See[Bas05]and[ACK10].
T
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ARGENTAND
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April20,2016
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3.12. DEFAULTRISKANDINCOMEFLUCTUATIONS
434
• countercyclicaltradebalances
• highvolatilityofconsumptionrelativetooutput
Notably,longrecessionscausedbybaddrawsintheincomeprocessincreasethegovernment’s
incentivetodefault
Thiscanleadto
• spikesininterestrates
• temporarylossesofaccesstointernationalcreditmarkets
• largedropsinoutput,consumption,andwelfare
• largecapitaloutflowsduringrecessions
Suchdynamicsareconsistentwithexperiencesofmanycountries
Structure
Inthissectionwedescribethemainfeaturesofthemodel
Output,ConsumptionandDebt Asmallopeneconomyisendowedwithanexogenousstochas-
ticallyfluctuatingpotentialoutputstreamfy
t
g
Potentialoutputisrealizedonlyinperiodsinwhichthegovernmenthonorsitssovereigndebt
Theoutputgoodcanbetradedorconsumed
Thesequencefy
t
gisdescribedbyaMarkovprocesswithstochasticdensitykernelp(y,y0)
Householdswithinthecountryareidenticalandrankstochasticconsumptionstreamsaccording
to
E
¥
å
t=0
b
t
u(c
t
)
(3.149)
Here
• 0<b<1isatimediscountfactor
• uisanincreasingandstrictlyconcaveutilityfunction
Consumptionsequencesenjoyedbyhouseholdsareaffectedbythegovernment’sdecisiontobor-
roworlendinternationally
Thegovernmentisbenevolentinthesensethatitsaimistomaximize(3.149)
Thegovernmentistheonlydomesticactorwithaccesstoforeigncredit
Becausehouseholdareaversetoconsumptionfluctuations,thegovernmentwilltrytosmooth
consumptionbyborrowingfrom(andlendingto)foreigncreditors
T
HOMAS
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ARGENTAND
J
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April20,2016
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3.12. DEFAULTRISKANDINCOMEFLUCTUATIONS
435
AssetMarkets Theonlycreditinstrumentavailabletothegovernment isaone-periodbond
tradedininternationalcreditmarkets
Thebondmarkethasthefollowingfeatures
• Thebondmaturesinoneperiodandisnotstatecontingent
• ApurchaseofabondwithfacevalueB
0
isaclaimtoB
0
unitsoftheconsumptiongoodnext
period
• TopurchaseB’nextperiodcostsqB
0
now
– ifB
<
0,then qB
0
unitsofthegoodarereceivedinthecurrentperiod,forapromise
torepay Bunitsnextperiod
– thereisanequilibriumpricefunctionq(B
0
,y)thatmakesqdependonbothB
0
andy
Earningsonthegovernmentportfolioaredistributed(or,ifnegative,taxed)lumpsumtohouse-
holds
Whenthegovernmentisnotexcludedfromfinancialmarkets, theone-periodnationalbudget
constraintis
c=y+B q(B
0
,y)B
0
(3.150)
Hereandbelow,aprimedenotesanextperiodvalueoraclaimmaturingnextperiod
ToruleoutPonzischemes,wealsorequirethatB Zineveryperiod
• Zischosentobesufficientlylargethattheconstraintneverbindsinequilibrium
FinancialMarkets Foreigncreditors
• areriskneutral
• knowthedomesticoutputstochasticprocessfy
t
gandobservey
t
,y
1
,...,attimet
• canborroworlendwithoutlimitinaninternationalcreditmarketataconstantinternational
interestrater
• receivefullpaymentifthegovernmentchoosestopay
• receivezeroifthegovernmentdefaultsonitsone-perioddebtdue
Whenagovernmentisexpectedtodefaultnextperiodwithprobabilityd,theexpectedvalueofa
promisetopayoneunitofconsumptionnextperiodis1 d.
Therefore,thediscountedexpectedvalueofapromisetopayBnextperiodis
q=
d
1+r
(3.151)
Government’sdecisions Ateachpointintimet,thegovernmentchoosesbetween
1. defaulting
2. meetingitscurrentobligationsandpurchasingorsellinganoptimalquantityofone-period
sovereigndebt
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.12. DEFAULTRISKANDINCOMEFLUCTUATIONS
436
Defaultingmeansdecliningtorepayallofitscurrentobligations
Ifthegovernmentdefaultsinthecurrentperiod,thenconsumptionequalscurrentoutput
Butasovereigndefaulthastwoconsequences:
1. Outputimmediatelyfallsfromytoh(y),where0h(y)y
• itreturnstoyonlyafterthecountryregainsaccesstointernationalcreditmarkets
2. Thecountrylosesaccesstoforeigncreditmarkets
Reenteringinternationalcreditmarket Whileinastateofdefault,theeconomyregainsaccess
toforeigncreditineachsubsequentperiodwithprobabilityq
Equilibrium
Informally,anequilibriumisasequenceofinterestratesonitssovereigndebt, astochasticse-
quenceofgovernmentdefaultdecisionsandanimpliedflowofhouseholdconsumptionsuchthat
1. Consumptionandassetssatisfythenationalbudgetconstraint
2. Thegovernmentmaximizeshouseholdutilitytakingintoaccount
• theresourceconstraint
• theeffectofitschoicesonthepriceofbonds
• consequencesofdefaultingnowforfuturenetoutputandfutureborrowingandlend-
ingopportunities
3. Theinterestrateonthegovernment’sdebtincludesarisk-premiumsufficienttomakefor-
eigncreditorsexpectonaveragetoearntheconstantrisk-freeinternationalinterestrate
Toexpresstheseideasmoreprecisely,considerfirstthechoicesofthegovernment,which
1. entersaperiodwithinitialassetsB,
2. observescurrentoutputy,and
3. chooseseitherto
(a) default,or
(b) tohonorBandsetnextperiod’sassetstoB
0
Inarecursiveformulation,
• statevariablesforthegovernmentcomprisethepair(B,y)
• v(B,y)istheoptimumvalueofthegovernment’sproblemwhenatthebeginningofaperiod
itfacesthechoiceofwhethertohonorordefault
• v
c
(B,y)isthevalueofchoosingtopayobligatonsfallingdue
• v
d
(y)isthevalueofchoosingtodefault
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.12. DEFAULTRISKANDINCOMEFLUCTUATIONS
437
v
d
(y)doesnotdependonBbecause, whenaccesstocreditiseventuallyregained, netforeign
assetsequal0
Expressedrecursively,thevalueofdefaultingis
v
d
(y)=u(h(y))+b
Z
qv(0,y
0
)+(1 q)v
d
(y
0
)
 
p(y,y
0
)dy
0
Thevalueofpayingis
v
c
(B,y)= max
B0 Z
u(y q(B
0
,y)B
0
+B)+b
Z
v(B
0
,y
0
)p(y,y
0
)dy
0
Thethreevaluefunctionsarelinkedby
v(B,y)=maxfv
c
(B,y),v
d
(y)g
Thegovernmentchoosestodefaultwhen
v
c
(B,y)<v
d
(y)
andhencegivenB
0
theprobabilityofdefaultnextperiodis
d(B
0
,y):=
Z
1fv
c
(B
0
,y
0
)<v
d
(y
0
)gp(y,y
0
)dy
0
(3.152)
Givenzeroprofitsforforeigncreditorsinequilibrium,wecancombine(3.151)and(3.152)topin
downthebondpricefunction:
q(B
0
,y)=
d(B
0
,y)
1+r
(3.153)
Definitionofequilibrium
Anequilibriumis
• apricingfunctionq(B
0
,y),
• atripleofvaluefunctions(v
c
(B,y),v
d
(y),v(B,y)),
• adecisionruletellingthegovernmentwhentodefaultandwhentopayasafunctionofthe
state(B,y),and
• anassetaccumulationrulethat,conditionalonchoosingnottodefault,maps(B,y)intoB
0
suchthat
• ThethreeBellmanequationsfor(v
c
(B,y),v
d
(y),v(B,y))‘aresatisfied
• Giventhepricefunctionq(B
0
,y),thedefaultdecisionruleandtheassetaccumulationdec-
sionruleattaintheoptimalvaluefunctionv(B,y),and
• Thepricefunctionq(B
0
,y)satisfiesequation(3.153)
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April20,2016
3.12. DEFAULTRISKANDINCOMEFLUCTUATIONS
438
Computation
Let’snowcomputeanequilibriumofArellano’smodel
Theequilibriumobjectsarethevaluefunctionv(B,y),theassociateddefaultdecisionrule,andthe
pricingfunctionq(B
0
,y)
We’lluseourcodetoreplicateArellano’sresults
Afterthatwe’llperformsomeadditionalsimulations
ThemajorityofthecodebelowwaswrittenbyChaseColeman
ItusesaslightlymodifiedversionofthealgorithmrecommendedbyArellano
• Theappendixto[Are08]recommendsvaluefunctioniterationuntilconvergence,updating
theprice,andthenrepeating
• Instead,weupdatethebondpriceateveryvaluefunctioniterationstep
Thesecondapproachisfasterandthetwodifferentproceduresdeliververysimilarresults
Hereisamoredetaileddescriptionofouralgorithm:
1. Guessavaluefunctionv(B,y)andpricefunctionq(B
0
,y)
2. Ateachpair(B,y),
• updatethevalueofdefaultingv
d
(y)
• updatethevalueofcontinuingv
c
(B,y)
3. Updatethevaluefunctionv(B,y),thedefaultrule,theimpliedexantedefaultprobability,
andthepricefunction
4. Checkforconvergence.Ifconverged,stop.Ifnot,gotostep2.
Weusesimplediscretizationonagridofassetholdingsandincomelevels
TheoutputprocessisdiscretizedusingTauchen’squadraturemethod
Thecodecanbefoundinthefilearellano_vfi.jlfromtheQuantEcon.applicationspackagebutwe
repeatithereforconvenience
(Resultsanddiscussionfollowthecode)
using QuantEcon: tauchen, MarkovChain, simulate
# ------------------------------------------------------------------- #
# Define the main Arellano Economy type
# ------------------------------------------------------------------- #
"""
Arellano 2008 8 deals s with a small open economy whose government
invests in foreign assets in order to smooth the consumption of
domestic households. Domestic households receive a stochastic
path of income.
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3.12. DEFAULTRISKANDINCOMEFLUCTUATIONS
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##### Fields
* b::Real: Time discounting parameter
* g::Real: Risk aversion parameter
* r::Real: World interest rate
* r::Real: Autoregressive coefficient on n income e process
* h::Real: Standard deviation of f noise e in income process
* q::Real: Probability of re-entering the world financial l sector r after default
* ny::Int: Number of points to use in approximation of income process
* nB::Int: Number of points to use in approximation of asset holdings
* ygrid::Vector{Float64}: This is the grid used to approximate income process
* ydefgrid::Vector{Float64}: When in default get less income than process
would otherwise dictate
* Bgrid::Vector{Float64}: This is grid used to o approximate e choices of asset
holdings
* P::Array{Float64, 2}: Transition probabilities between income levels
* vf::Array{Float64, 2}: Place to hold value e function
* vd::Array{Float64, 2}: Place to hold value e function n when in default
* vc::Array{Float64, 2}: Place to hold value e function n when choosing to
continue
* policy::Array{Float64, 2}: Place to hold asset policy function
* q::Array{Float64, 2}: Place to hold prices s at t different pairs of (y, B')
* defprob::Array{Float64, 2}: Place to hold the default probabilities for
pairs of (y, , B')
"""
immutable ArellanoEconomy
# Model Parameters
b::Float64
g::Float64
r::Float64
r::Float64
h::Float64
q::Float64
# Grid Parameters
ny::Int
nB::Int
ygrid::Array{Float641}
ydefgrid::Array{Float641}
Bgrid::Array{Float641}
P::Array{Float642}
# Value function
vf::Array{Float642}
vd::Array{Float642}
vc::Array{Float642}
policy::Array{Float642}
q::Array{Float642}
defprob::Array{Float642}
end
"""
This is the default constructor for building an economy as presented
in Arellano 2008.
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3.12. DEFAULTRISKANDINCOMEFLUCTUATIONS
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##### Arguments
* ;b::Real(0.953): Time discounting parameter
* ;g::Real(2.0): Risk aversion parameter
* ;r::Real(0.017): World interest rate
* ;r::Real(0.945): Autoregressive coefficient on income process
* ;h::Real(0.025): Standard deviation of noise in income process
* ;q::Real(0.282): Probability of f re-entering g the world financial sector
after default
* ;ny::Int(21): : Number r of points to use in n approximation n of income process
* ;nB::Int(251): : Number r of points to use in n approximation n of asset holdings
"""
function ArellanoEconomy(;b=.953g=2., r=0.017r=0.945h=0.025q=0.282,
ny=21, nB=251)
# Create grids
Bgrid collect(linspace(-.4.4, nB))
ly, P tauchen(ny, rh)
ygrid exp(ly)
ydefgrid min(.969 mean(ygrid), ygrid)
# Define value functions (Notice ordered different than Python n to o take
# advantage e of f column major layout of f Julia)
vf zeros(nB, ny)
vd zeros(1, ny)
vc zeros(nB, ny)
policy Array(Int, nB, ny)
ones(nB, ny) .* ((r))
defprob Array(Float64, nB, ny)
return ArellanoEconomy(bg, r, rhq, ny, , nB, ygrid, ydefgrid, , Bgrid, , P,
vf, vd, vc, policy, , q, , defprob)
end
u(ae::ArellanoEconomy, c) c^(ae.g(ae.g)
_unpack(ae::ArellanoEconomy) =
ae.b, ae.g, ae.r, ae.r, ae.h, ae.q, ae.ny, ae.nB
_unpackgrids(ae::ArellanoEconomy) =
ae.ygrid, ae.ydefgrid, ae.Bgrid, ae.P, ae.vf, ae.vd, ae.vc, ae.policy, ae.q, ae.defprob
# ------------------------------------------------------------------- #
# Write the value function iteration
# ------------------------------------------------------------------- #
"""
This function n performs s the one step update of the value function for the
Arellano model-- Using current value functions and their expected d value,
it updates the e value e function at every state by solving for the optimal
choice of savings
##### Arguments
* ae::ArellanoEconomy: This is the economy we e would d like to update the
value functions for
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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