INTEREST AND PRICES

MICHAEL WOODFORD

FLEX-PRICE, COMPLETE-MARKETS MODEL

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tt

t

tt

t

Mc

cpTypWBM

ts

p

McuE

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..

;;(max0

0,

MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING

Complete Markets

1,

1,

111

11,

1

11

11

11

1

)(

);()(),(

);(

);()(),(

)(),()(

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sstts

t

N

ssttst

i

Q

DQE

zzprobzDzzQ

zzprob

zzprobzDzzq

zDzzqzq

= price kernel

Value of portfolio with payoff D

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tttt

BiA

QEi

)1(

)(1

1

1

1,

Interest coefficient for riskless asset

Riskless Portfolio

Budget Constraint

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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

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For all states in t+1

For all t, to prevent infinite c

The equivalent terminal condition

Lagrangian

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(1

1

);;(

);;(

1);;(

);;(

)1

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0,00

00

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icpQE

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McuE

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cpTypWBM

WQE

0)(lim ,

Transversality condition:

Flow budget constraint:

Market Equilibrium

st

t

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st

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st

stt

stt

tt

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1)(

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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:

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11

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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

)

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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:

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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)

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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 tt

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

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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*

);(

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ttt

ttt

ttt

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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

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)(

1

11

0

1

11

0

1

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diipP

diicC

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1

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1

0

1

0

));(();;(

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)()(

tttt

t

tt

t

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diihvP

McuEU

diidiihiwyP

diicipcP

cPTyPwBM

First Order Conditions:

t

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t

t

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tttc

t

t

tttc

tttm

P

iw

mcu

ihv

P

P

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mcu

i

i

mcu

mcu

)(

);;(

));((

);;(

);;(

1);;(

);;(

11,111

Firm’s Optimization:

))(

(1

);(

);(

1)

)((

)()(

)))(

(('

1)

)((

1)

)(()()(

))(

()()()(

1

1

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t

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A

iy

Ayu

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iy

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iyfiwihiwVC

Nominal

Real

1);;(

)()(

));(();(

););(())(

(

1

)()(

))(

(

~

1~~

~1

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nt

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tt

t

tt

dt

yys

yjyiy

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yiysy

iy

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P

ipyy

Natural Level of Output

Log-linearization of real mc:

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)()()(

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))(1

)(

)())()((

)(

1)log()log(

));((

);(

))(

();)(

(

);;(

11

_

_

_

_

Partial-equilibrium relationship?

‘where

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y

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y

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yA

y

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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

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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( tnttt 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

tttntt

nt YgEYgr

3-EQUATION EQUILIBRIUM SYSTEM:

1

Proportion offirm that prefixprices

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

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

Gtnt

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