interest and prices michael woodford. flex-price, complete-markets model microfounded cagan-sargent...
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![Page 1: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/1.jpg)
INTEREST AND PRICES
MICHAEL WOODFORD
![Page 2: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/2.jpg)
FLEX-PRICE, COMPLETE-MARKETS MODEL
tttttttt
tt
t
tt
t
Mc
cpTypWBM
ts
p
McuE
tt
..
;;(max0
0,
MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING
![Page 3: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/3.jpg)
Complete Markets
1,
1,
111
11,
1
11
11
11
1
)(
);()(),(
);(
);()(),(
)(),()(
tt
tttt
ttt
N
ssttss
tt
ttt
N
sstts
t
N
ssttst
i
Q
DQE
zzprobzDzzQ
zzprob
zzprobzDzzq
zDzzqzq
= price kernel
Value of portfolio with payoff D
![Page 4: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/4.jpg)
ttt
tttt
BiA
QEi
)1(
)(1
1
1
1,
Interest coefficient for riskless asset
Riskless Portfolio
![Page 5: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/5.jpg)
Budget Constraint
tttt
tttttttttt
tttt
ttttttt
ttt
TypW
WQEQEcp
TypW
WQEMi
icp
)())(1(
)(1
1,1,1,
1,1,
Where T is the transfer payments based on theseignorage profits of the central bank, distributedin a lump sum to the representative consumer
![Page 6: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/6.jpg)
No Ponzi Games:
0)(lim
)((
)((
,
1,11
1,111
TTttT
tTTTTTtt
tTTTTTttt
WQE
TypQE
TypQEW
For all states in t+1
For all t, to prevent infinite c
The equivalent terminal condition
![Page 7: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/7.jpg)
Lagrangian
));;(
);;(
(1
1
);;(
);;(
1);;(
);;(
)1
((
)((();;((
1
11
11
11,
11
11
0,00
0,00
00
t
t
tt
ttc
tt
ttc
tt
t
t
ttt
t
ttc
tt
ttc
t
t
tt
ttc
tt
ttM
tt
t
tttt
ttttt
ttt
t
tt
t
p
p
pM
cu
pM
cu
Ei
or
p
p
QpM
cu
pM
cu
i
i
pM
cu
pM
cu
Mi
icpQE
TypQEwp
McuE
![Page 8: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/8.jpg)
ttttttt
TTttT
cpTypWBM
WQE
0)(lim ,
Transversality condition:
Flow budget constraint:
![Page 9: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/9.jpg)
Market Equilibrium
st
t
tt
st
sttttt
J
sttjttt
st
stt
stt
tt
Mi
iTWWQE
BQEA
AA
MM
yc
1)(
)(
1,1,
11,,111
1
Market solution for the transfers T
![Page 10: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/10.jpg)
Monetary Targeting: BC chooses a path for M
st
stt
st
st
st MMTMWB 110
Fiscal policy assumed to be:
Equilibrium is tt ip ; S.t.Euler-intertemporal conditionconditionFOC-itratemporal conditionTVCConstraint
For given sttt My ;;
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We study equilibrium around a zero-shock steady state:
___
11
___
11
1
1
_
1
_
_
1
_
111
1
ip
pi
mmmp
p
p
M
p
M
M
M
ii
p
p
mm
tt
tt
t
tttt
t
t
t
st
t
st
st
st
t
t
t
tt
t
![Page 12: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/12.jpg)
Derive the LM Curve
)0;;(
);;(
___
yLm
iyLp
Mttt
t
st
From the FOC:
At the steady state:
![Page 13: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/13.jpg)
);();();;( mvcumcu Separable utility :
_
_
_
log
log
log
i
ii
y
yy
m
mm
tt
tt
tt
Define:
The “hat” variables are proportional deviations from the steady state variables.
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tmt
i
y
L
m
i
L
m
i
y
L
m
y
_
_
_
_
_
1
1Similar to Cagan’ssemi-elasticity of money demand
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We log-linearize around zero inflation1_
define 1logloglog tttt PPLog-linearize the Euler Equation and transform it to a Fisher equation:
tc
cgt
cc
c
tttttt
tttt
u
ug
yu
u
gygyEr
Eri
_
111
1
)]()([
Elasticity of intertemporal substitution
g is the “twist” in MRS between m and c
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Add the identity
tttt mm
1
We look for solution
given exogenous shocks
ttt im ;;
ttt y ;;
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Solution of the system
))(1( 11
ttitttt EumEm This is a linear first-order stochastic difference equation ,where,
i
i
1
Exogenous disturbance (composite of all shocks):
)]()([ 111
tttttt
titymtt
gygyEr
ryu
![Page 18: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/18.jpg)
given
100 iThere exists a forward solution:
)()1(0
1
j
jtijttj
t uEm
From which we can get a unique equilibrium value for the price level:
0
_
log)(log)1(logj
jts
jttj
t muMEP
This is similar to the Cagan-Sargent-wallace formula for the pricelevel, but with the exception that the Lucas Critique is taken care of and it allows welfare analysis.
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I. Interest Rate Targeting based on exogenous shocks
Choose the path for i; specify fiscal policy which targets D:
st
st
st BMD Total end of period public sector liabilities.
Monetary policy affects the breakdown of D between M and B:
1,0
)1(
,
1,
JB
BrBs
Jtt
stt
stt
No multi-period bonds
Beginning of period valueof outsranding bonds
End of period, one-periodrisk-less bonds
![Page 20: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/20.jpg)
Steady state (around 1
tt
tD
t
tDt
m
endogenous
iy
exogenous
D
D
;;
:
;;;
:
11
fix
)
![Page 21: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/21.jpg)
tttt
tttttt
tmttityt
mm
EEi
miym
1
11
or,
Is unique
Can uniquely be determined!
PRICE LEVEL IS INDETERMINATE:
Real balances are unique
Future expected inflationis unique
But, neither
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To see the indeterminancy, let “*” denote solution value:
ttt
ttt
tt
v
v
mm
*
*
*
v is a shock, uncorrelated with(sunspot), the new triple is also a solution, thus:
ttt iy
,,
Price level is indeterminate under the interest rule!
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II. Wicksellian Rules: interest rate is a function of endogenous variables (feedback rule)
ttt
ttttt
t
tt
tt
MiP
DvPy
D
vP
Pi
;;
;*;;;
);*
(
V=control error of CB
Fiscal Policy
Exogenous
Endogenous
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Steady State:
0
1*
1
0
11
)0,1(
tt
t
D
t
v
yy
mm
i
Log-linearize:
)*
log(P
Pp tt
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1)2
);*
()1
)*
log(
);*
(
tttt
tttt
pt
tt
tt
t
Eri
vPvP
Pi
P
Pp
vP
Pi
We can find two processes
*log*
*)3
;
1
tt
tttt
tt
PP
iP
Add the identity
![Page 26: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/26.jpg)
1), 2) and 3) yield:
0
1*)1(
01
)1(
11
)((log)1(log
)*()1(
1)1(0
)*()()1(
jjtjtpjtt
jppt
jjtjtjtt
jpt
p
tttttttp
vrPEP
vrEP
vErPEP
P is not correlated to the path of M:money demand shocks affect M, butdo not affect P; the LM is not usedin the derivation of the solution to P.
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FEATURES:
• Forward looking
• Price is not a function of i; rather , a function of the feedback rule and the target
• suppose
p *tP
tttt
t
p
t
ryvv
iff
KP
KP
);(
0
*
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Additionally:
• If
tv
tt rv
Price level instability can be reduced by raising
p , an automatic response.
![Page 29: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/29.jpg)
Note, also that
• Big
• Smallp , reduces the need for accurate observation of tr
p , almost complete peg of interest rate
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The path of the money supply:
•
);;*;;( ttttt
s vyPPMM
By using LM, we can still express
But we must examine existence of a well-defineddemand for money. There’s possibly liquidity trap
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III. TAYLOR (feedback) RULE
*
*
1
*
)(
1)0,1(
0,0
0*1*
);(
ttt
ttt
ttt
ttt
tt
tt
vv
vi
i
vyy
vi
• Steady state
Assume:
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1
0
)1(
1
1
loglog
)(
1
ttt
jtjtj
tj
t
ttttt
tttt
PP
vrE
Erv
Eri
Taylor principle:
Is predetermined1tP
5.1
![Page 33: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/33.jpg)
Transitory fluctuations in
t
t
v
r
Create transitory fluctuations in t
Permanent shifts in the price level P.
![Page 34: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/34.jpg)
Optimizing models with nominal rigidities
Chapter 3
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))(()(
)(
)(
1
11
0
1
11
0
1
ihfAiy
diipP
diicC
ttt
tt
tt
![Page 36: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/36.jpg)
0
1
0
00
1
0
1
0
1
0
));(();;(
)()()(
)()(
tttt
t
tt
t
ttttt
tttt
tttttttt
diihvP
McuEU
diidiihiwyP
diicipcP
cPTyPwBM
![Page 37: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/37.jpg)
First Order Conditions:
t
t
tttc
tth
t
t
tttttc
tttc
t
t
tttc
tttm
P
iw
mcu
ihv
P
P
Qmcu
mcu
i
i
mcu
mcu
)(
);;(
));((
);;(
);;(
1);;(
);;(
11,111
![Page 38: INTEREST AND PRICES MICHAEL WOODFORD. FLEX-PRICE, COMPLETE-MARKETS MODEL MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d485503460f94a23ebe/html5/thumbnails/38.jpg)
Firm’s Optimization:
))(
(1
);(
);(
1)
)((
)()(
)))(
(('
1)
)((
1)
)(()()(
))(
()()()(
1
1
t
t
tttc
tth
tt
t
t
tt
t
tt
t
tt
ttt
t
tttt
A
iy
Ayu
hv
AA
iy
P
iwismc
Aiy
ffA
iy
AA
iyiwiSMC
A
iyfiwihiwVC
Nominal
Real
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1);;(
)()(
));(();(
););(())(
(
1
)()(
))(
(
~
1~~
~1
tnt
nt
ttt
tt
ttt
tttt
t
tt
t
tt
dt
yys
yjyiy
A
yfvyv
yiysy
iy
iSip
P
ipyy
Natural Level of Output
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Log-linearization of real mc:
n
tttt
ttssti
t
tttss
tit
tttss
tissii
tt
ttc
t
tt
t
th
ttt
yyiyis
AAA
F
sy
yyy
y
F
siy
iyiyiy
iy
F
sss
iyF
yu
Aiy
Aiy
v
yys
)()()(
)(1
))(1
)(
)())()((
)(
1)log()log(
));((
);(
))(
();)(
(
);;(
11
_
_
_
_
Partial-equilibrium relationship?
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‘where
yA
y
yA
y
AA
yfv
yA
y
A
yfv
yyu
yu
h
hh
cc
c
)(
)('
));((
)());((
);(
);(
1
1
Elasticity of wage demands, wrt to output holdingmarginal utility of incomeconstant
Elasticity of marginal product oflabor wrt output
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ONE-PERIOD NOMINAL RIDIGITY
0;;([);(
0)()(([0)(
)(
)])(
()()([
)(
~1
1
,11,11
111
,11
,11
ttttttct
tttttttt
tttt
t
tttttttttt
tttt
yysyyuE
iSipPyQEip
iQE
A
ipPyfiwipPyQMaxE
iQMaxE
Same as before, except for 1tE
Y need not be equal to the natural y
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1
11
111
);(
);(1
t
ttc
ttctt Cu
CuEi
Ct = consumption aggregate
11
t
tt P
P= = gross rate of increase in
the Dixit-Stiglitz price index Pt
A Neo-Wicksellian Framework
THE IS:
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1
11
111
);(
);(1
t
ttc
ttctt Yu
YuEi
Equilibrium condition:
A log-linear approximation arounda deterministic steady state yields the ISschedule:
)()ˆ(ˆ111 ttttttt EigYEgY
t
g=crowding out term due to fiscal shock
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)ˆ(lim TTtT
gYE
01)(ˆ
jjtjtttt iEgY
GCY
Yu
ug t
cc
ct
Equivalentto the fiscalshock
Effect on fiscalshock on C
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New Keynesian Phillips Curve:
1)ˆˆ( tnttt EYY
Taylor Rule:
tyttt Yii ˆ)( ** Inflation target
Deviation of natural outputdue to supply shock
Demand determinedoutput deviations
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Output gap:
nttt YYx ˆˆ
]ˆ)ˆ[( 11nttttttt rEixEx
1 tttt Ex
)()( *** xxii txttt
IS-curve involves an exogenous disturbance term:
)]ˆ()ˆ[(ˆ 111 n
tttntt
nt YgEYgr
3-EQUATION EQUILIBRIUM SYSTEM:
1
Proportion offirm that prefixprices
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INTEREST RULE AND PRICE STABILITY
1
111
);(
);(1
1
0
0
tntc
tntc
tnt
ntt
ntt
t
t
t
Yu
YuEr
where
ri
YY
x
THE NATURALRATE OF INTEREST
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log)1log(1
1logˆ
ntn
t
ntn
t rr
rr
Percentage deviation of the natural rate of interestfrom its steady-state value
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Inflation targeting at low, positive,inflation
*ˆˆ ntt ri
1
1
ˆ ttnt
qgY
Composite disturbances
ttt
tGtt
haq
csGg
11)1(
)1(ˆ
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])1()1)(1(
)1)(1(ˆ)1[(1
ˆ
))1((1
))1(ˆ(ˆ
11
1
1
1
1
thta
tcGtGnt
tt
Gtnt
ha
csGr
ha
csGY
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mt
ntity
st
mttityt
st
rYPM
iYPM
ˆˆ*loglog
ˆˆloglog
Evolution of money supply:
*t
nt
i
r
The only exogenous variables in the system are:
= the natural interest rate
=nominal rate consistent with inflation target