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Approximate quadratic-linear optimization problem
Based on
Pierpaolo Benigno and Michael Woodford
The Quadratic Approximation to the Utility Function
•Consider the problem
)(
..
)},({,
yFx
ts
yxuMaxyx
The first-order condition
0
)}),(({max yyx
y
uFu
yyFu
The second-order approximation to the utility function
22 )(2
1)()(
2
1)( yyuyyuxxuxxu yyyxxx
The second-order approximation to the constraint
2)(2
1)( yyFyyFx yyy
•Substitute the second-order approximation to the constraint into the linear term of the second-order approximation to the utility function, using the FOC, yields a quadratic objective function
2
2
2
)(
)(
)(
yyu
xxu
yyFu
yy
xx
yyx
The approximate optimization problem
})(
)(
)({max
2
2
2
,
yyu
xxu
yyFu
yy
xx
yyxyx
Subject to:
)( yyFx y
0)(
)(
)(22
yyu
xFuyyFu
yyFu
yy
yxxyxx
yyx
Which is supposed to be(?) a first order approximation of
0 yyx uFu
A Linear-Quadratic Approximate Problem
•Begin by computing a Taylor-series approximation to the welfare measure, expanding around the steady state. As a second-order (logarithmic) approximation, BW get:
)log(
)(2
1
0
0
00
2
Y
YY
uuYYuYEuYconstU
tt
ttttyttyyt
tttct
The Quadratic Approximation to the Utility Function
•Consider the problem
)(
..
)},({,
yFx
ts
yxuMaxyx
The first-order condition
0
)}),(({max yyx
y
uFu
yyFu
The second-order approximation to the utility function
22 )(2
1)()(
2
1)( yyuyyuxxuxxu yyyxxx
The second-order approximation to the constraint
2)(2
1)( yyFyyFx yyy
Approximate optimization
•Substitute the second-order approximation to the constraint into the linear term of the second-order approximation to the linear term of the second-order approximation of the utility function, using the first-order conditions, yields a quadratic objective function.
•The approximate optimization is to maximize the quadratic objective function, subject to the first-order approximation of the constraint. The first-order condition is equal to the first order approximation of the FOC of the original problem.
222
22
)(2
1)()(
2
1))(
2
1)((
)(2
1)()(
2
1)(
yyuyyuxxuxyyFyyFu
yyuyyuxxuxxu
yyyxxyyyx
yyyxxx
The Micro-based Neo-Keynesian Quadratic-linear problem
Based on
Pierpaolo Benigno and Michael Woodford
The Micro-based Quadratic Loss Function
ttt
ttt
Htttt
ctttt
jtt
ttjtttt
tttt
GCY
jhAjy
jHjHv
CCu
djcC
djHvCuEU
1
1
1
1
11
0
1
1
0
)()(
)(1
));((
1);(
)(
));(();(
1
0
0
00
Welfare measure expressed as a function of equilibrium production
1
0
1
1
0
))(
(
);;(
));(();(
0
0
00
0
0
00
jt
tt
ttttt
tttt
ttjtttt
tttt
dP
jp
YUEU
djyvYuEU
Demand of differentiated product is a function of relative prices
The Deterministic (distorted) Steady State
1
0
1))(
(
)0;;(0
0
0
jt
tt
tttt
ttt
dP
jp
YUU
),,( tttY Maximize with respect to
Subject to constraints on
),,( tttY
•BW show that an alternative way of dealing with this problem is to use the a second-order approximation to the aggregate supply relation to eliminate the linear terms in the quadratic welfare function.
A Linear-Quadratic Approximate Problem
•Begin by computing a Taylor-series approximation to the welfare measure, expanding around the steady state. As a second-order (logarithmic) approximation, BW get:
)log(
)(2
1
0
0
00
2
Y
YY
uuYYuYEuYconstU
tt
ttttyttyyt
tttct
•There is a non-zero linear term in the approximate welfare measure, unless
•As in the case of no price distortions in the steady state (subsidies to producers that negate the monopolistic power). This means that we cannot expect to evaluate this expression to the second order using only the approximate solution for the path of aggregate output that is accurate only to the first order. Thus we cannot determine optimal policy, even up to first order, using this approximate objective together with the approximations to the structural equations that are accurate only to first order.
0
Welfare measure expressed as a function of equilibrium production
1
0
1
1
0
))(
(
);;(
));(();(
0
0
00
0
0
00
jt
tt
ttttt
tttt
ttjtttt
tttt
dP
jp
YUEU
djyvYuEU
Demand of differentiated product is a function of relative prices
The Micro-based Quadratic Loss Function of Benigno and Woodford
ttt
ttt
Htttt
ctttt
jtt
ttjtttt
tttt
GCY
jhAjy
jHjHv
CCu
djcC
djHvCuEU
1
1
1
1
11
0
1
1
0
)()(
)(1
));((
1);(
)(
));(();(
1
0
0
00
•There is a non-zero linear term in the approximate welfare measure, unless
•As in the case of no price distortions in the steady state (subsidies to producers that negate the monopolistic power). This means that we cannot expect to evaluate this expression to the second order using only the approximate solution for the path of aggregate output that is accurate only to the first order. Thus we cannot determine optimal policy, even up to first order, using this approximate objective together with the approximations to the structural equations that are accurate only to first order.
0
The Deterministic (distorted) Steady State
1
0
1))(
(
)0;;(0
0
0
jt
tt
tttt
ttt
dP
jp
YUU
),,( tttY Maximize with respect to
Subject to constraints on
),,( tttY
•BW show that an alternative way of dealing with this problem is to use the a second-order approximation to the aggregate supply relation to eliminate the linear terms in the quadratic welfare function.
MICROFOUNDED CAGAN-SARGENT
PRICE LEVEL DETERMINATION
UNDER MONETARY TARGETING
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
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
ttt
tttt
BiA
QEi
)1(
)(1
1
1
1,
Interest coefficient for riskless asset
Riskless Portfolio
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
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
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
ttttttt
TTttT
cpTypWBM
WQE
0)(lim ,
Transversality condition:
Flow budget constraint:
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
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 ;;
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
Derive the LM Curve
)0;;(
);;(
___
yLm
iyLp
Mttt
t
st
From the FOC:
At the steady state:
);();();;( 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.
tmt
i
y
L
m
i
L
m
i
y
L
m
y
_
_
_
_
_
1
1Similar to Cagan’ssemi-elasticity of money demand
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
Add the identity
tttt mm
1
We look for solution
given exogenous shocks
ttt im ;;
ttt y ;;
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
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.
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
Steady state (around 1
tt
tD
t
tDt
m
endogenous
iy
exogenous
D
D
;;
:
;;;
:
11
fix
)
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
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!
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
Steady State:
0
1*
1
0
11
)0,1(
tt
t
D
t
v
yy
mm
i
Log-linearize:
)*
log(P
Pp t
t
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
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.
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
*
Additionally:
• If
tv
tt rv
Price level instability can be reduced by raising
p , an automatic response.
Note, also that
• Big
• Smallp , reduces the need for accurate observation of tr
p , almost complete peg of interest rate
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
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:
1
0
)1(
1
1
loglog
)(
1
ttt
jtjtj
tj
t
ttttt
tttt
PP
vrE
Erv
Eri
Taylor principle:
Is predetermined1tP
5.1
Transitory fluctuations in
t
t
v
r
Create transitory fluctuations in t
Permanent shifts in the price level P.
Optimizing models with nominal rigidities
Chapter 3
))(()(
)(
)(
1
11
0
1
11
0
1
ihfAiy
diipP
diicC
ttt
tt
tt
0
1
0
00
1
0
1
0
1
0
));(();;(
)()()(
)()(
tttt
t
tt
t
ttttt
tttt
tttttttt
diihvP
McuEU
diidiihiwyP
diicipcP
cPTyPwBM
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
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
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
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?
‘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
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
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:
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
)ˆ(lim TTtT
gYE
01)(ˆ
jjtjtttt iEgY
GCY
Yu
ug t
cc
ct
Equivalentto the fiscalshock
Effect on fiscalshock on C
New Keynesian Phillips Curve:
1)ˆˆ( tn
ttt EYY Taylor Rule:
tyttt Yii ˆ)( ** Inflation target
Deviation of natural outputdue to supply shock
Demand determinedoutput deviations
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
tttn
ttnt YgEYgr
3-EQUATION EQUILIBRIUM SYSTEM:
1
Proportion offirm that prefixprices
INTEREST RULE AND PRICE STABILITY
1
111
);(
);(1
1
0
0
tn
tc
tn
tct
nt
ntt
ntt
t
t
t
Yu
YuEr
where
ri
YY
x
THE NATURALRATE OF INTEREST
log)1log(1
1logˆ
ntn
t
ntn
t rr
rr
Percentage deviation of the natural rate of interestfrom its steady-state value
Inflation targeting at low, positive,inflation
*ˆˆ ntt ri
1
1
ˆ ttnt
qgY
Composite disturbances
ttt
tGtt
haq
csGg
11)1(
)1(ˆ
])1()1)(1(
)1)(1(ˆ)1[(1
ˆ
))1((1
))1(ˆ(ˆ
11
1
1
1
1
thta
tcGtGnt
tt
Gtn
t
ha
csGr
ha
csGY
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
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
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
ttt
tttt
BiA
QEi
)1(
)(1
1
1
1,
Interest coefficient for riskless asset
Riskless Portfolio
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
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
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
ttttttt
TTttT
cpTypWBM
WQE
0)(lim ,
Transversality condition:
Flow budget constraint:
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
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 ;;
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
Derive the LM Curve
)0;;(
);;(
___
yLm
iyLp
Mttt
t
st
From the FOC:
At the steady state:
);();();;( 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.
tmt
i
y
L
m
i
L
m
i
y
L
m
y
_
_
_
_
_
1
1Similar to Cagan’ssemi-elasticity of money demand
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
Add the identity
tttt mm
1
We look for solution
given exogenous shocks
ttt im ;;
ttt y ;;
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
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.
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
Steady state (around 1
tt
tD
t
tDt
m
endogenous
iy
exogenous
D
D
;;
:
;;;
:
11
fix
)
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
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!
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
Steady State:
0
1*
1
0
11
)0,1(
tt
t
D
t
v
yy
mm
i
Log-linearize:
)*
log(P
Pp t
t
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
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.
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
*
Additionally:
• If
tv
tt rv
Price level instability can be reduced by raising
p , an automatic response.
Note, also that
• Big
• Smallp , reduces the need for accurate observation of tr
p , almost complete peg of interest rate
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
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:
1
0
)1(
1
1
loglog
)(
1
ttt
jtjtj
tj
t
ttttt
tttt
PP
vrE
Erv
Eri
Taylor principle:
Is predetermined1tP
5.1
Transitory fluctuations in
t
t
v
r
Create transitory fluctuations in t
Permanent shifts in the price level P.
Optimizing models with nominal rigidities
Chapter 3
))(()(
)(
)(
1
11
0
1
11
0
1
ihfAiy
diipP
diicC
ttt
tt
tt
0
1
0
00
1
0
1
0
1
0
));(();;(
)()()(
)()(
tttt
t
tt
t
ttttt
tttt
tttttttt
diihvP
McuEU
diidiihiwyP
diicipcP
cPTyPwBM
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
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
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
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?
‘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
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
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:
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
)ˆ(lim TTtT
gYE
01)(ˆ
jjtjtttt iEgY
GCY
Yu
ug t
cc
ct
Equivalentto the fiscalshock
Effect on fiscalshock on C
New Keynesian Phillips Curve:
1)ˆˆ( tn
ttt EYY Taylor Rule:
tyttt Yii ˆ)( ** Inflation target
Deviation of natural outputdue to supply shock
Demand determinedoutput deviations
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
tttn
ttnt YgEYgr
3-EQUATION EQUILIBRIUM SYSTEM:
1
Proportion offirm that prefixprices
INTEREST RULE AND PRICE STABILITY
1
111
);(
);(1
1
0
0
tn
tc
tn
tct
nt
ntt
ntt
t
t
t
Yu
YuEr
where
ri
YY
x
THE NATURALRATE OF INTEREST
log)1log(1
1logˆ
ntn
t
ntn
t rr
rr
Percentage deviation of the natural rate of interestfrom its steady-state value
Inflation targeting at low, positive,inflation
*ˆˆ ntt ri
1
1
ˆ ttnt
qgY
Composite disturbances
ttt
tGtt
haq
csGg
11)1(
)1(ˆ
])1()1)(1(
)1)(1(ˆ)1[(1
ˆ
))1((1
))1(ˆ(ˆ
11
1
1
1
1
thta
tcGtGnt
tt
Gtn
t
ha
csGr
ha
csGY
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
