3.11. HISTORYDEPENDENTPUBLICPOLICIES
411
AnothersettinginwhichhistorydependentpoliciesnaturallyemergeiswhereinsteadofaRam-
seyplannerthereisasequenceofgovernmentadministratorswhosetimetmembertakesasgiven
thepoliciesusedbyitssuccessors
Westudytheseideasinthecontextofamodelinwhichabenevolenttaxauthorityisforced
• toraiseaprescribedpresentvalueofrevenues
• todosobyimposingadistortingflatratetaxontheoutputofacompetitiverepresentative
firm
Thefirmfacescostsofadjustmentandliveswithinacompetitiveequilibrium,whichinturnim-
posesrestrictionsonthetaxauthority
1
References ThepresentationbelowisbasedonarecentpaperbyEvansandSargent[ES13]
Regardingtechniques,wewillmakeuseofthemethodsdescribedin
1. thelinearregulatorlecture
2. thesolving LQ dynamic Stackelberg problems lecture
TwoSourcesofHistoryDependence
Wecomparetwotimingprotocols
1. AninfinitelylivedbenevolenttaxauthoritysolvesaRamseyproblem
2. Thereisasequenceoftaxauthorities,eachchoosingonlyatimettaxrate
Underbothtimingprotocols,optimaltaxpoliciesarehistory-dependent
Buthistorydependencecapturesdifferenteconomicforcesacrossthetwotimingprotocols
Inthefirsttimingprotocol,historydependenceexpressesthetime-inconsistencyoftheRamseyplan
Inthesecondtimingprotocol,historydependencereflectstheunfoldingofconstraintsthatassure
thatatimetgovernmentadministratorwantstoconfirmtherepresentativefirm’sexpectations
aboutgovernmentactions
Wedescriberecursiverepresentationsofhistory-dependenttaxpoliciesunderbothtimingproto-
cols
RamseyTimingProtocol Thefirsttimingprotocolmodelsapolicymakerwhocanbesaidto
‘commit’,choosingasequenceoftaxratesonce-and-for-allattime0
SequenceofGovernmentsTimingProtocol Forthesecondtimingprotocolweusethenotion
ofasustainableplanproposedin[CK90],alsoreferredtoasacrediblepublicpolicyin[Sto89]
1
Wecouldalsocallacompetitiveequilibriumarationalexpectationsequilibrium.
T
HOMAS
S
ARGENTAND
J
OHN
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TACHURSKI
April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
412
Akeyideahereisthathistory-dependentpoliciescanbearrangedsothat,whenregardedasa
representativefirm’sforecastingfunctions,theyconfrontpolicymakerswithincentivestoconfirm
them
WefollowChang[Cha98]inexpressingsuchhistory-dependentplansrecursively
Credibility considerations contribute an additional l auxiliary state e variable e in n the form of f a
promisedvaluetotheplanner
Itexpresseshowdecisionsmustunfoldtogivethegovernmenttheincentivetoconfirmprivate
sectorexpectationswhenthegovernmentchoosessequentially
Note: Weoccasionallyhearconfusionabouttheconsequencesofrecursiverepresentationsof
governmentpoliciesunderourtwotimingprotocols.Itisincorrecttoregardarecursiverepresen-
tationoftheRamseyplanasinanyway‘solvingatime-inconsistencyproblem’.Onthecontrary,
theevolutionoftheauxiliarystatevariablethataugmentstheauthenticonesunderourfirsttim-
ingprotocoloughttobeviewedasexpressingthetime-inconsistencyofaRamseyplan. Despite
that,inliteraturesaboutpracticalmonetarypolicyonesometimeshearsinterpretationsthatsell
Ramseyplansinsettingswhereoursequentialtimingprotocolistheonethatmoreaccurately
characterizesdecisionmaking. Pleasebewareofdiscussionsthattossaroundclaimsaboutcred-
ibilityifyoudon’t alsoseerecursiverepresentationsofpolicieswiththecompletelistofstate
variablesappearinginour[Cha98]-likeanalysisthatwepresentbelow.
Competitiveequilibrium
Arepresentativecompetitivefirmsellsoutputq
t
atpricep
t
whenmarket-wideoutputisQ
t
Themarketasawholefacesadownwardslopinginversedemandfunction
p
t
=A
0
A
1
Q
t
A
0
>0,A
1
>0
(3.117)
Therepresentativefirm
• hasgiveninitialconditionq
0
• enduresquadraticadjustmentcosts
d
2
(q
t+1
q
t
)2
• paysaflatratetaxt
t
perunitofoutput
• treatsfp
t
,t
t
g
¥
t=0
asexogenous
• choosesfq
t+1
g
¥
t=0
tomaximize
¥
å
t=0
b
t
p
t
q
t
d
2
(q
t+1
q
t
)
2
t
t
q
t
 
(3.118)
Letu
t
:=q
t+1
q
t
bethefirm’s‘controlvariable’attimet
First-orderconditionsfortherepresentativefirm’sproblemare
u
t
=
b
d
p
t+1
+bu
t+1
b
d
t
t+1
t=0,1,...
(3.119)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
413
Tocomputeacompetitiveequilibrium,itisappropriatetotake(3.119),eliminatep
t
infavorofQ
t
byusing(3.117),andthensetq
t
=Q
t
Thislaststepmakestherepresentativefirmberepresentative
2
Wearriveat
u
t
=
b
d
(A
0
A
1
Q
t+1
)+bu
t+1
b
d
t
t+1
(3.120)
Q
t+1
=Q
t
+u
t
(3.121)
Notation:Foranyscalarx
t
,let~x=fx
t
g¥
t=0
Givenataxsequenceft
t+1
g¥
t=0
,acompetitiveequilibriumisapricesequence~pandanoutput
sequence
~
Qthatsatisfy(3.117),(3.120),and(3.121)
Foranysequence~x=fx
t
g¥
t=0
,thesequence~x
1
:=fx
t
g¥
t=1
iscalledthecontinuationsequenceor
simplythecontinuation
Notethatacompetitiveequilibriumconsistsofafirstperiodvalueu
0
=Q
1
Q
0
andacontinua-
tioncompetitiveequilibriumwithinitialconditionQ
1
Also,acontinuationofacompetitiveequilibriumisacompetitiveequilibrium
Followingtheleadof[Cha98],weshallmakeextensiveuseofthefollowingproperty:
• Acontinuation~t
1
=ft
t
g
¥
t=1
ofataxpolicy~tinfluencesu
0
via(3.120)entirelythroughits
impactonu
1
Acontinuationcompetitiveequilibriumcanbeindexedbyau
1
thatsatisfies(3.120)
Inthespiritof[KP80],weshalluseu
t+1
todescribewhatweshallcallapromisedmarginalvalue
thatacompetitiveequilibriumofferstoarepresentativefirm
3
DefineQ
t
:=[Q
0
,...,Q
t
]
Ahistory-dependenttaxpolicyisasequenceoffunctionsfs
t
g¥
t=0
withs
t
mappingQ
t
intoa
choiceoft
t+1
Below,weshall
• Studyhistory-dependenttaxpoliciesthateithersolveaRamseyplanorarecredible
• Describerecursiverepresentationsofbothtypesofhistory-dependentpolicies
RamseyProblem
Theplanner’sobjectiveiscastintermsofconsumersurplusnetofthefirm’sadjustmentcosts
2
Itisimportantnottosetq
t
=Q
t
prematurely.Tomakethefirmapricetaker,thisequalityshouldbeimposedafter
andnotbeforesolvingthefirm’soptimizationproblem.
3
Wecouldinstead,perhapswithmoreaccuracy,defineapromisedmarginalvalueasb(A
0
A
1
Q
t+1
bt
t+1
+
u
t+1
/b,sincethisistheobjecttowhichthefirm’sfirst-orderconditioninstructsittoequatetothemarginalcostdu
t
ofu
t
q
t+1
q
t
. ThischoicewouldalignbetterwithhowChang[Cha98]chosetoexpresshiscompetitiveequilib-
riumrecursively. Butgiven(u
t
,Q
t
),therepresentativefirmknows(Q
t+1
,t
t+1
),soitisadequatetotakeu
t+1
asthe
intermediatevariablethatsummarizeshow~t
t+1
affectsthefirm’schoiceofu
t
.
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
414
Consumersurplusis
Z
Q
0
(A
0
A
1
x)dx=A
0
Q
A
1
2
Q
2
Hencetheplanner’sone-periodreturnfunctionis
A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
(3.122)
Attimet=0,aRamseyplannerfacestheintertemporalbudgetconstraint
¥
å
t=1
b
t
t
t
Q
t
=G
0
(3.123)
Notethat(3.123)forbidstaxationofinitialoutputQ
0
TheRamseyproblemistochooseataxsequence~t
1
andacompetitiveequilibriumoutcome(
~
Q,~u)
thatmaximize
¥
å
t=0
b
t
A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
(3.124)
subjectto(3.123)
Thus,theRamseytimingprotocolis:
1. Attime0,knowing(Q
0
,G
0
),theRamseyplannerchoosesft
t+1
g¥
t=0
2. Given
Q
0
,ft
t+1
g¥
t=0
,acompetitiveequilibriumoutcomefu
t
,Q
t+1
g¥
t=0
emerges
Note: InbringingoutthetimingprotocolassociatedwithaRamseyplan,werunheadoninto
asetofissuesanalyzedbyBassetto[Bas05]. ThisisbecauseourdefinitionoftheRamseytiming
protocoldoesn’tcompletelydescribeallconceivableactionsbythegovernmentandfirmsastime
unfolds. Forexample,thedefinitionissilentabouthowthegovernmentwouldrespondiffirms,
forsomeunspecifiedreason,weretochoosetodeviatefromthecompetitiveequilibriumassoci-
atedwiththeRamseyplan,possiblypromptingviolationofgovernmentbudgetbalance. Thisis
anexampleoftheissuesraisedby[Bas05],whoidentifiesaclassofgovernmentpolicyproblems
whoseproperformulationrequiressupplyingacompleteandcoherentdescriptionofallactors’
behavioracrossallpossiblehistories.Implicitly,weareassumingthatamorecompletedescription
ofagovernmentstrategycouldbespecifiedthat(a)agreeswithoursalongtheRamseyoutcome,
and(b)sufficesuniquelytoimplementtheRamseyplanbydeterringfirmsfromtakingactions
thatdeviatefromtheRamseyoutcomepath.
ComputingaRamseyPlan Theplannerchoosesfu
t
g¥
t=0
,ft
t
g¥
t=1
tomaximize(3.124)subjectto
(3.120),(3.121),and(3.123)
ToformulatethisproblemasaLagrangian,attachaLagrangemultipliermtothebudgetconstraint
(3.123)
Thentheplannerchoosesfu
t
g¥
t=0
,ft
t
g¥
t=1
tomaximizeandtheLagrangemultipliermtominimize
¥
å
t=0
b
t
(A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
)+m
"
¥
å
t=0
b
t
t
t
Q
t
G
0
t
0
Q
0
#
(3.125)
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S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
415
subjecttoand(3.120)and(3.121)
TheRamseyproblemisaspecialcaseofthelinearquadraticdynamicStackelbergproblemana-
lyzedinthis lecture
Thekeyimplementabilityconditionsare(3.120)fort0
HoldingfixedmandG
0
,theLagrangianfortheplanningproblemcanbeabbreviatedas
max
fu
t
,t
t+1
g
¥
å
t=0
b
t
A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
+mt
t
Q
t
Define
z
t
:=
2
4
1
Q
t
t
t
3
5
and y
t
:=
z
t
u
t
=
2
6
6
4
1
Q
t
t
t
u
t
3
7
7
5
Heretheelementsofz
t
arenaturalstatevariablesandu
t
isaforwardlookingvariablethatwe
treatasastatevariablefort1
Butu
0
isachoicevariablefortheRamseyplanner.
Weincludet
t
asastatevariableforbookkeepingpurposes: ithelpstomaptheproblemintoa
linearregulatorproblemwithnocrossproductsbetweenstatesandcontrols
However,itwillbearedundantstatevariableinthesensethattheoptimaltaxt
t+1
willnotdepend
ont
t
Thegovernmentchoosest
t+1
attimetasafunctionofthetimetstate
Thus,wecanrewritetheRamseyproblemas
max
fy
t
,t
t+1
g
¥
å
t=0
b
t
y
0
t
Ry
t
(3.126)
subjecttoz
0
givenandthelawofmotion
y
t+1
=Ay
t
+Bt
t+1
(3.127)
where
R=
2
6
6
6
4
0
A
0
2
0
0
A
0
2
A
1
2
m
2
0
0
m
2
0
0
0
0
0
d
2
3
7
7
7
5
A=
2
6
6
4
1
0
0
0
0
1
0
1
0
0
0
0
A
0
d
A
1
d
0
A
1
d
+
1
b
3
7
7
5
B=
2
6
6
4
0
0
1
1
d
3
7
7
5
TwoSubproblems
Workingbackwards,wefirstpresenttheBellmanequationforthevaluefunctionthattakesboth
z
t
andu
t
asgiven.Thenwepresentavaluefunctionthattakesonlyz
0
asgivenandistheindirect
utilityfunctionthatarisesfromchoosingu
0
optimally.
Letv(Q
t
,t
t
,u
t
)betheoptimumvaluefunctionforthetimet1governmentadministratorfacing
stateQ
t
,t
t
,u
t
.
Letw(Q
0
)bethevalueoftheRamseyplanstartingfromQ
0
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
416
Subproblem1 HeretheBellmanequationis
v(Q
t
,t
t
,u
t
)=max
t
t+1
A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
+mt
t
Q
t
+bv(Q
t+1
,t
t+1
,u
t+1
)
wherethemaximizationissubjecttotheconstraints
Q
t+1
=Q
t
+u
t
and
u
t+1
A
0
d
+
A
1
d
Q
t
+
A
1
d
+
1
b
u
t
+
1
d
t
t+1
Hereweregardu
t
asastate
Subproblem2 Thesubproblem2Bellmanequationis
w(z
0
)=max
u
0
v(Q
0
,0,u
0
)
Details Definethestatevectortobe
y
t
=
2
6
6
4
1
Q
t
t
t
u
t
3
7
7
5
=
z
t
u
t
,
wherez
t
=
Q
t
t
t
0
areauthenticstatevariablesandu
t
isavariablewhosetime0valueisa
‘jump’variablebutwhosevaluesfordatest1willbecomestatevariablesthatencodehistory
dependenceintheRamseyplan
v(y
t
)=max
t
t+1
y
0
t
Ry
t
+bv(y
t+1
)
 
(3.128)
wherethemaximizationissubjecttotheconstraint
y
t+1
=Ay
t
+Bt
t+1
andwhere
R=
2
6
6
6
4
0
A
0
2
0
0
A
0
2
A
1
2
m
2
0
0
m
2
0
0
0
0
0
d
2
3
7
7
7
5
A=
2
6
6
4
1
0 0
0
0
1 0
1
0
0 0
0
A
0
d
A
1
d
0
A
1
d
+
1
b
3
7
7
5
,andB=
2
6
6
4
0
0
1
1
d
3
7
7
5
.
Functionalequation(3.128)hassolution
v(y
t
)= y
0
t
Py
t
where
• PsolvesthealgebraicmatrixRiccatiequationP=R+bA
0
PA bA
0
PB(B
0
PB)
1
B
0
PA
• theoptimalpolicyfunctionisgivenbyt
t+1
Fy
t
forF=(B
0
PB)
1
B
0
PA
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
417
Nowweturntosubproblem1.
Evidentlytheoptimalchoiceofu
0
satisfies
¶v
¶u
0
=0
IfwepartitionPas
P=
P
11
P
12
P
21
P
22
thenwehave
0=
¶u
0
z
0
0
P
11
z
0
+z
0
0
P
12
u
0
+u
0
0
P
21
z
0
+u
0
0
P
22
u
0
=P
0
12
z
0
+P
21
u
0
+2P
22
u
0
whichimplies
u
0
P
1
22
P
21
z
0
(3.129)
Thus,theRamseyplanis
t
t+1
F
z
t
u
t
and
z
t+1
u
t+1
=(A BF)
z
t
u
t
withinitialstate
z
0
1
22
P
21
z
0
0
RecursiveRepresentation AnoutcomeoftheprecedingresultsisthattheRamseyplancanbe
representedrecursivelyasthechoiceofaninitialmarginalutility(orrateofgrowthofoutput)
accordingtoafunction
u
0
=u(Q
0
jm)
(3.130)
thatobeys(3.129)andthefollowingupdatingequationsfort0:
t
t+1
=t(Q
t
,u
t
jm)
(3.131)
Q
t+1
=Q
t
+u
t
(3.132)
u
t+1
=u(Q
t
,u
t
jm)
(3.133)
Wehaveconditionedthefunctionsu,t,andubymtoemphasizehowthedependenceofFonG
0
appearsindirectlythroughtheLagrangemultiplierm
AnExampleCalculation We’lldiscusshowtocomputembelowbutfirstconsiderthefollowing
numericalexample
Wetaketheparameterset[A
0
,A
1
,d,b,Q
0
]=[100,.05,.2,.95,100]andcomputetheRamseyplan
withthefollowingpieceofcode
#=
In the e following, uhat and tauhat are e what the e planner would
choose if f he e could d reset at t time t, uhatdif and d tauhatdif are
the difference between those e and d what the planner is constrained to
choose. The e variable mu is s the Lagrange multiplier r associated d with
T
HOMAS
S
ARGENTAND
J
OHN
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TACHURSKI
April20,2016
3.11. HISTORYDEPENDENTPUBLICPOLICIES
418
the constraint at time t.
For more complete description of inputs and d outputs s see the website.
@author : Spencer Lyon <spencer.lyon@nyu.edu>
@date: 2014-08-21
References
----------
Simple port of f the file examples/evans_sargent.py
http://quant-econ.net/hist_dep_policies.html
=#
using QuantEcon
using Optim
using PyPlot
type HistDepRamsey
# These are e the e parameters of the economy
A0::Real
A1::Real
d::Real
Q0::Real
tau0::Real
mu0::Real
bet::Real
# These are e the e LQ fields and stationary values
R::Matrix
A::Matrix
B::Matrix
Q::Real
P::Matrix
F::Matrix
lq::LQ
end
type RamseyPath
y::Matrix
uhat::Vector
uhatdif::Vector
tauhat::Vector
tauhatdif::Vector
mu::Vector
G::Vector
GPay::Vector
end
T
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ARGENTAND
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OHN
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TACHURSKI
April20,2016
3.11. HISTORYDEPENDENTPUBLICPOLICIES
419
function HistDepRamsey(A0, A1, d, Q0, tau0, mu, bet)
# Create Matrices for solving Ramsey y problem
[0.0 -A0/2 0.0
0.0
-A0/A1/2
-mu/2 0.0
0.0
-mu/2 0.0
0.0
0.0
0.0
0.0
d/2]
[1.0
0.0 0.0 0.0
0.0
1.0 0.0 1.0
0.0
0.0 0.0 0.0
-A0/d A1/0.0 A1/d+1.0/bet]
[0.0 0.0 1.0 1.0/d]'
0.0
# Use LQ to o solve e the Ramsey Problem.
lq LQ(Q, -R, , A, B, bet=bet)
P, F, _d stationary_values(lq)
HistDepRamsey(A0, A1, d, Q0, tau0, mu0, bet, R, A, B, Q, P, F, lq)
end
function compute_G(hdr::HistDepRamsey, mu)
# simplify y notation
Q0, tau0, P, F, d, A, B hdr.Q0, hdr.tau0, hdr.P, hdr.F, hdr.d, , hdr.A, , hdr.B
bet hdr.bet
# Need y_0 0 to o compute government tax x revenue.
u0 compute_u0(hdr, P)
y0 vcat([1.0 Q0 tau0]', u0)
# Define A_F and S matricies
AF F
[0.0 1.0 0.0 0]' * [0.0 0.0 1.0 0]
# Solves equation (25)
Omega solve_discrete_lyapunov(sqrt(bet) .* AF', bet .* AF' * AF)
T0 y0' * Omega y0
return T0[1], A, B, F, P
end
function compute_u0(hdr::HistDepRamsey, P::Matrix)
# simplify y notation
Q0, tau0 hdr.Q0, hdr.tau0
P21 P[41:3]
P22 P[44]
z0 [1.0 Q0 tau0]'
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.11. HISTORYDEPENDENTPUBLICPOLICIES
420
u0 = -P22^(-1.* P21*(z0)
return u0[1]
end
function init_path(hdr::HistDepRamsey, mu0, , T::Int=20)
# Construct t starting g values for the path of the Ramsey economy
G0, A, B, F, P compute_G(hdr, mu0)
# Compute the optimal u0
u0 compute_u0(hdr, P)
# Initialize vectors
Array(Float644, T)
uhat
Array(Float64, T)
uhatdif
Array(Float64, T)
tauhat
Array(Float64, T)
tauhatdif Array(Float64, T-1)
mu
Array(Float64, T)
G
Array(Float64, T)
GPay
Array(Float64, T)
# Initial conditions
G[1G0
mu[1mu0
uhatdif[10.0
uhat[1u0
y[:, 1vcat([1.0 hdr.Q0 hdr.tau0]', u0)
return RamseyPath(y, uhat, uhatdif, tauhat, tauhatdif, mu, G, GPay)
end
function compute_ramsey_path!(hdr::HistDepRamsey, rp::RamseyPath)
# simplify y notation
y, uhat, uhatdif, tauhat, rp.y, rp.uhat, rp.uhatdif, rp.tauhat
tauhatdif, mu, , G, GPay rp.tauhatdif, rp.mu, rp.G, rp.GPay
bet hdr.bet
G0, A, B, F, P compute_G(hdr, mu[1])
for t=2:T
# iterate government policy
y[:, t] (A F) y[:, t-1]
# update e G
G[t] (G[t-1bet*y[2, t]*y[3, t])/bet
GPay[t] bet.*y[2, t]*y[3, t]
#=
Compute the e mu if the government were able to reset t its s plan
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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