3.11. HISTORYDEPENDENTPUBLICPOLICIES
421
ff is the e tax x revenues the government would receive if they reset the
plan with Lagrange multiplier r mu u minus current G
=#
ff(mu) abs(compute_G(hdr, mu)[1]-G[t])
# find ff = 0
mu[t] optimize(ff, mu[t-1]-1e4, mu[t-1]+1e4).minimum
temp, Atemp, Btemp, Ftemp, Ptemp compute_G(hdr, mu[t])
# Compute alternative decisions
P21temp Ptemp[41:3]
P22temp P[44]
uhat[t] (-P22temp^(-1.* P21temp y[1:3, t])[1]
yhat (Atemp-Btemp Ftemp) [y[1:3, t-1], uhat[t-1]]
tauhat[t] yhat[3]
tauhatdif[t-1tauhat[t] y[3, t]
uhatdif[t] uhat[t] y[3, t]
end
return rp
end
function plot1(rp::RamseyPath)
tt 1:length(rp.mu) # tt is used to make the plot time index correct.
rp.y
n_rows 3
fig, axes subplots(n_rows, 1, figsize=(1012))
subplots_adjust(hspace=0.5)
for ax in axes
ax[:grid]()
ax[:set_xlim](015)
end
bbox (0.1.021..102)
legend_args {:bbox_to_anchor => bbox, :loc => 3, :mode => "expand"}
p_args {:lw => 2, :alpha => 0.7}
ax axes[1]
ax[:plot](tt, squeeze(y[2, :], 1), "b-", label="output"; p_args...)
ax[:set_ylabel](L"$Q$", fontsize=16)
ax[:legend](ncol=1; legend_args...)
ax axes[2]
ax[:plot](tt, squeeze(y[3, :], 1), "b-", label="tax rate"; p_args...)
ax[:set_ylabel](L"$\tau$", fontsize=16)
ax[:set_yticks]((0.00.20.40.60.8))
ax[:legend](ncol=1; legend_args...)
ax axes[3]
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ARGENTAND
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
422
ax[:plot](tt, squeeze(y[4, :], 1), "b-", label="first difference in output";
p_args...)
ax[:set_ylabel](L"$u$", fontsize=16)
ax[:set_yticks]((0100200300400))
ax[:legend](ncol=1; legend_args...)
ax[:set_xlabel](L"time", fontsize=16)
plt.show()
end
function plot2(rp::RamseyPath)
y, uhatdif, , tauhatdif, , mu rp.y, rp.uhatdif, rp.tauhatdif, rp.mu
G, GPay rp.G, rp.GPay
length(rp.mu)
tt 1:T # tt t is s used to make the plot time index correct.
tt2 1:T-1
n_rows 4
fig, axes subplots(n_rows, 1, figsize=(1016))
plt.subplots_adjust(hspace=0.5)
for ax in axes
ax[:grid](alpha=.5)
ax[:set_xlim](-0.515)
end
bbox (0.1.021..102)
legend_args {:bbox_to_anchor => bbox, :loc => 3, :mode => "expand"}
p_args {:lw => 2, :alpha => 0.7}
ax axes[1]
ax[:plot](tt2, tauhatdif,
label="time inconsistency differential for tax rate"; p_args...)
ax[:set_ylabel](L"$\Delta\tau$", fontsize=16)
ax[:set_yticks]((0.00.40.81.2))
ax[:legend](ncol=1; legend_args...)
ax axes[2]
ax[:plot](tt, uhatdif,
label=L"time inconsistency differential for $u$"; p_args...)
ax[:set_ylabel](L"$\Delta u$", fontsize=16)
ax[:set_yticks]((-3.0-2.0-1.00.0))
ax[:legend](ncol=1; legend_args...)
ax axes[3]
ax[:plot](tt, mu, label="Lagrange multiplier"; p_args...)
ax[:set_ylabel](L"$\mu$", fontsize=16)
ax[:set_yticks]((2.34e-32.43e-32.52e-3))
ax[:legend](ncol=1; legend_args...)
ax axes[4]
ax[:plot](tt, G, label="government revenue"; p_args...)
ax[:set_ylabel](L"$G$", fontsize=16)
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S
ARGENTAND
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OHN
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TACHURSKI
April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
423
ax[:set_yticks]((9200940096009800))
ax[:legend](ncol=1; legend_args...)
ax[:set_xlabel](L"time", fontsize=16)
plt.show()
end
# Primitives
T
20
A0
100.0
A1
0.05
d
0.20
bet 0.95
# Initial conditions
mu0 0.0025
Q0
1000.0
tau0 0.0
# Solve Ramsey y problem m and compute path
hdr HistDepRamsey(A0, A1, d, Q0, tau0, , mu0, , bet)
rp init_path(hdr, mu0, T)
compute_ramsey_path!(hdr, rp) # updates rp in place
plot1(rp)
plot2(rp)
TheprogramcanalsobefoundintheQuantEconGitHubrepository
ItcomputesanumberofsequencesbesidestheRamseyplan,someofwhichhavealreadybeen
discussed,whileotherswillbedescribedbelow
ThenextfigureusestheprogramtocomputeandshowtheRamseyplanfortandtheRamsey
outcomefor(Q
t
,u
t
)
Fromtoptobottom,thepanelsshowQ
t
,t
t
andu
t
:=Q
t+1
Q
t
overt=0,...,15
Theoptimaldecisionruleis
4
t
t+1
248.0624 0.1242Q
t
0.3347u
t
(3.134)
NoticehowtheRamseyplancallsforahightaxatt=1followedbyaperpetualstreamoflower
taxes
Taxingheavilyatfirst,lesslaterexpressestime-inconsistencyoftheoptimalplanforft
t+1
g¥
t=0
We’llcharacterizethisformallyafterfirstdiscussinghowtocomputem.
ComputingDefinetheselectorvectorse
t
=
0 0 0 1 1 0
0
ande
Q
=
0 1 0 0 0
0
andexpress
t
t
=e0
t
y
t
andQ
t
=e0
Q
y
t
4
Aspromised,t
t
doesnotappearintheRamseyplanner’sdecisionrulefort
t+1
.
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ARGENTAND
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
424
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
425
EvidentlyQ
t
t
t
=y
0
t
e
Q
e
0
t
y
t
=y
0
t
Sy
t
whereS:=e
Q
e
0
t
Wewanttocompute
T
0
=
¥
å
t=1
b
t
t
t
Q
t
=t
1
Q
1
+bT
1
whereT
1
=
å
¥
t=2
b
1
Q
t
t
t
ThepresentvaluesT
0
andT
1
areconnectedby
T
0
=by
0
0
A
0
F
SA
F
y
0
+bT
1
GuessasolutionthattakestheformT
t
=y0
t
Wy
t
,thenfindanWthatsatisfies
W=bA
0
F
SA
F
+bA
0
F
WA
F
(3.135)
Equation(3.135)isadiscreteLyapunovequationthatcanbesolvedfor WusingQuantEcon’s
solve_discrete_lyapunovfunction
ThematrixFandthereforethematrixA
F
=A BFdependonm
TofindamthatguaranteesthatT
0
=G
0
weproceedasfollows:
1. Guessaninitialm,computeatentativeRamseyplanandtheimpliedT
0
=y0
0
W(m)y
0
2. IfT
0
>G
0
,lowerm;ifT
0
<m,raisem
3. Continueiteratingonstep3untilT
0
=G
0
TimeInconsistency
RecallthattheRamseyplannerchoosesfu
t
g¥
t=0
,ft
t
g¥
t=1
tomaximize
¥
å
t=0
b
t
A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
subjectto(3.120),(3.121),and(3.123)
WeexpresstheoutcomethataRamseyplanistime-inconsistentthefollowingway
Proposition.AcontinuationofaRamseyplanisnotaRamseyplan
Let
w(Q
0
,u
0
jm
0
)=
¥
å
t=0
b
t
A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
(3.136)
where
• fQ
t
,u
t
g
¥
t=0
areevaluatedundertheRamseyplanwhoserecursiverepresentationisgivenby
(3.131),(3.132),(3.133)
• m
0
isthevalueoftheLagrangemultiplierthatassuresbudgetbalance,computedasde-
scribedabove
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ARGENTAND
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OHN
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TACHURSKI
April20,2016
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3.11. HISTORYDEPENDENTPUBLICPOLICIES
426
Evidently,thesecontinuationvaluessatisfytherecursion
w(Q
t
,u
t
jm
0
)=A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
+bw(Q
t+1
,u
t+1
jm
0
)
(3.137)
forallt0,whereQ
t+1
=Q
t
+u
t
UnderthetimingprotocolaffiliatedwiththeRamseyplan,theplanneriscommittedtotheout-
comeofiterationson(3.131),(3.132),(3.133)
Inparticular,whentimetcomes,theRamseyplanneriscommittedtothevalueofu
t
impliedby
theRamseyplanandreceivescontinuationvaluew(Q
t
,u
t
,m
0
)
ThattheRamseyplanistime-inconsistentcanbeseenbysubjectingittothefollowing‘revolu-
tionary’test
First,definecontinuationrevenuesG
t
thatthegovernmentraisesalongtheoriginalRamseyout-
comeby
G
t
=b
t
(G
0
t
å
s=1
b
s
t
s
Q
s
)
(3.138)
whereft
t
,Q
t
g¥
t=0
istheoriginalRamseyoutcome
5
Thenattimet1,
1. take(Q
t
,G
t
)inheritedfromtheoriginalRamseyplanasinitialconditions
2. inviteabrandnewRamseyplannertocomputeanewRamseyplan,solvingforanewu
t
,to
becalled
ˇ
u
t
,andforanewm,tobecalled
ˇ
m
t
TherevisedLagrangemultiplier
ˇ
m
t
ischosensothat,underthenewRamseyplan,thegovernment
isabletoraiseenoughcontinuationrevenuesG
t
givenby(3.138)
WouldthisnewRamseyplanbeacontinuationoftheoriginalplan?
TheanswerisnobecausealongaRamseyplan,fort1,ingeneralitistruethat
w
Q
t
,u(Q
t
jˇm)jˇm
>w(Q
t
,u
t
jm
0
)
(3.139)
Inequality(3.139)expressesacontinuationRamseyplanner’sincentivetodeviatefromatime0
Ramseyplanby
1. resettingu
t
accordingto(3.130)
2. adjustingtheLagrangemultiplieronthecontinuationappropriatelytoaccountfortaxrev-
enuesalreadycollected
6
Inequality(3.139)expressesthetime-inconsistencyofaRamseyplan
5
ThecontinuationrevenuesG
t
arethetimetpresentvalueofrevenuesthatmustberaisedtosatisfytheoriginal
time0governmentintertemporalbudgetconstraint,takingintoaccounttherevenuesalreadyraisedfroms=1,...,t
undertheoriginalRamseyplan.
6
Forexample,lettheRamseyplanyieldtime1revenuesQ
1
t
1
. Thenattime1,acontinuationRamseyplanner
wouldwanttoraise continuationrevenues,expressedinunitsoftime1goods,of
˜
G
1
:=
G bQ
1
t
1
b
. Tofinancethe
remainderrevenues,thecontinuationRamseyplannerwouldfindacontinuationLagrangemultipliermbyapplying
thethree-stepprocedurefromtheprevioussectiontorevenuerequirements
˜
G
1
.
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April20,2016
3.11. HISTORYDEPENDENTPUBLICPOLICIES
427
ASimulation TobringoutthetimeinconsistencyoftheRamseyplan,wecompare
• thetimetvaluesoft
t+1
undertheoriginalRamseyplanwith
• thevalue
ˇ
t
t+1
associatedwithanewRamseyplanbegunattimetwithinitialconditions
(Q
t
,G
t
)generatedbyfollowingtheoriginalRamseyplan
HereagainG
t
:=b
t(
G
0
å
t
s=1
b
s
t
s
Q
s
)
ThedifferenceDt
t
:=tˇ
t
t
t
isshowninthetoppanelofthefollowingfigure
Inthesecondpanelwecomparethetimetoutcomeforu
t
undertheoriginalRamseyplanwith
thetimetvalueofthisnewRamseyproblemstartingfrom(Q
t
,G
t
)
Tocomputeu
t
underthenewRamseyplan,weusethefollowingversionofformula(3.129):
uˇ
t
P
1
22
(mˇ
t
)P
21
(mˇ
t
)z
t
Herez
t
isevaluatedalongtheRamseyoutcomepath,wherewehaveincluded
ˇ
m
t
toemphasize
thedependenceofPontheLagrangemultiplierm
0
7
Tocomputeu
t
alongtheRamseypath,wejustiteratetherecursionstarting(??)fromtheinitialQ
0
withu
0
beinggivenbyformula(3.129)
Thusthesecondpanelindicateshowfarthereinitializedvalueuˇ
t
valuedepartsfromthetimet
outcomealongtheRamseyplan
Notethattherestartedplanraisesthetimet+1taxandconsequentlylowersthetimetvalueof
u
t
AssociatedwiththenewRamseyplanattisavalueoftheLagrangemultiplieronthecontinuation
governmentbudgetconstraint
Thisisthethirdpanelofthefigure
ThefourthpanelplotstherequiredcontinuationrevenuesG
t
impliedbytheoriginalRamseyplan
ThesefigureshelpusunderstandthetimeinconsistencyoftheRamseyplan
FurtherIntuition Onefeaturetonoteisthelargedifferencebetween ˇt
t+1
andt
t+1
inthetop
panelofthefigure
IfthegovernmentisabletoresettoanewRamseyplanattimet,itchoosesasignificantlyhigher
taxratethanifitwererequiredtomaintaintheoriginalRamseyplan
Theintuitionhereisthatthegovernmentisrequiredtofinanceagivenpresentvalueofexpendi-
tureswithdistortingtaxest
Thequadraticadjustmentcostspreventfirmsfromreactingstronglytovariationsinthetaxrate
fornextperiod,whichtiltsatimetRamseyplannertowardusingtimet+1taxes
Aswasnotedbefore,thisisevidentinthefirstfigure,wherethegovernmenttaxesthenextperiod
heavilyandthenfallsbacktoaconstanttaxfromthenon
7
Itcanbeverifiedthatthisformulaputsnon-zeroweightonlyonthecomponents1andQ
t
ofz
t
.
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.11. HISTORYDEPENDENTPUBLICPOLICIES
428
T
HOMAS
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ARGENTAND
J
OHN
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TACHURSKI
April20,2016
3.11. HISTORYDEPENDENTPUBLICPOLICIES
429
Thiscanalsobeenseeninthethirdpanelofthesecondfigure,wherethegovernmentpaysoffa
significantportionofthedebtusingthefirstperiodtaxrate
Thesimilaritiesbetweenthegraphsinthelasttwopanelsofthesecondfigurerevealsthatthereisa
one-to-onemappingbetweenGandm
TheRamseyplancanthenonlybetimeconsistentifG
t
remainsconstantovertime,whichwillnot
betrueingeneral
CrediblePolicy
Weexpressthethemeofthissectioninthefollowing:Ingeneral,acontinuationofaRamseyplan
isnotaRamseyplan
ThisissometimessummarizedbysayingthataRamseyplanisnotcredible
Ontheotherhand,acontinuationofacredibleplanisacredibleplan
Theliteratureonacrediblepublicpolicy([CK90]and[Sto89])arrangesstrategiesandincentivesso
thatpublicpoliciescanbeimplementedbyasequenceofgovernmentdecisionmakersinsteadofa
singleRamseyplannerwhochoosesanentiresequenceofhistory-dependentactionsonceandfor
allattimet=0
Hereweconfineourselvestosketchinghowrecursivemethodscanbeusedtocharacterizecredi-
blepoliciesinourmodel
Akeyreferenceonthesetopicsis[Cha98]
Acredibilityproblemarisesbecauseweassumethatthetimingofdecisionsdiffersfromthosefor
aRamseyproblem
Asequentialtimingprotocolisaprotocolsuchthat
1. Ateach0,givenQ
t
andexpectationsaboutacontinuationtaxpolicyft
s+1
g¥
s=t
anda
continuationpricesequencefp
s+1
g¥
s=t
,therepresentativefirmchoosesu
t
2. Ateacht,given(Q
t
,u
t
),agovernmentchoosest
t+1
Item(2)capturesthattaxesarenowsetsequentially,thetimet+1taxbeingsetafterthegovern-
menthasobservedu
t
Ofcourse,therepresentativefirmsetsu
t
inlightofitsexpectationsofhowthegovernmentwill
ultimatelychoosetosetfuturetaxes
Acredibletaxplanft
s+1
g¥
s=t
• isanticipatedbytherepresentativefirm,and
• isonethatatimetgovernmentchoosestoconfirm
Weusethefollowingrecursion,closelyrelatedtobutdifferentfrom(3.137),todefinethecontinu-
ationvaluefunctionforthegovernment:
J
t
=A
0
Q
t
A
1
2
Q
2
t
d
2
u
2
t
+bJ
t+1
(t
t+1
,G
t+1
)
(3.140)
Thisdiffersfrom(3.137)because
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April20,2016
3.11. HISTORYDEPENDENTPUBLICPOLICIES
430
• continuationvaluesarenowallowedtodependexplicitlyonvaluesofthechoicet
t+1
,and
• continuationgovernmentrevenuetoberaisedG
t+1
neednotbeonescalledforbythepre-
vailinggovernmentpolicy
Thus,deviationsfromthatpolicyareallowed,analterationthatrecognizesthatt
t
ischosense-
quentially
ExpressthegovernmentbudgetconstraintasrequiringthatG
0
solvesthedifferenceequation
G
t
=bt
t+1
Q
t+1
+bG
t+1
t0
(3.141)
subjecttotheterminalconditionlim
t!+¥
b
t
G
t
=0
Becausethegovernmentischoosingsequentially,itisconvenientto
• takeG
t
asastatevariableattand
• toregardthetimetgovernmentaschoosing(t
t+1
,G
t+1
)subjecttoconstraint(3.141)
Toexpressthenotionofacrediblegovernmentplanconcisely,weexpandthestrategyspaceby
alsoaddingJ
t
itselfasastatevariableandallowingpoliciestotakethefollowingrecursiveforms
8
RegardJ
0
asanadiscountedpresentvaluepromisedtotheRamseyplannerandtakeit asan
initialcondition.
Thenafterchoosingu
0
accordingto
u
0
=u(Q
0
,G
0
,J
0
),
(3.142)
choosesubsequenttaxes,outputs, andcontinuationvaluesaccordingtorecursionsthatcanbe
representedas
ˆ
t
t+1
=t(Q
t
,u
t
,G
t
,J
t
)
(3.143)
u
t+1
=x(Q
t
,u
t
,G
t
,J
t
,t
t+1
)
(3.144)
G
t+1
=b
1
G
t
t
t+1
Q
t+1
(3.145)
J
t+1
(t
t+1
,G
t+1
)=n(Q
t
,u
t
,G
t+1
,J
t
,t
t+1
)
(3.146)
Here
ˆ
t
t+1
isthetimet+1governmentactioncalledforbytheplan,while
• t
t+1
ispossiblysomeone-timedeviationthatthetimet+1governmentcontemplatesand
• G
t+1
istheassociatedcontinuationtaxcollections
8
Thischoiceisthekeytowhat[LS12]call‘dynamicprogrammingsquared’.
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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