GSDE-新凯恩斯模型-ch3

Lectures on Monetary Policy,In?ation and the Business Cycle

The Basic New Keynesian Model

by

Jordi Galí

February2007

Motivation and Outline

Evidence on Money,Output,and Prices:

The Long Run

Short Run E¤ects of Monetary Policy Shocks

(i)persistent e¤ects on real variables

(ii)slow adjustment of aggregate price level

(iii)liquidity e¤ect

Micro Evidence on Price-setting Behavior:signi…cant price and wage rigidities

Failure of Classical Monetary Models

A Baseline Model with Nominal Rigidities

monopolistic competition

sticky prices(staggered price setting)

competitive labor markets,closed economy,no capital accumulation

Households

Representative household solves

1X t=0 t U(C t;N t)

max E0

where

C t Z10C t(i)1 1 di 1 subject to Z10P t(i)C t(i)di+Q t B t B t 1+W t N t T t

for t=0;1;2;:::plus solvency constraint.

Optimality conditions

1.Optimal allocation of expenditures

C t(i)= P t(i)P t C t implying Z10P t(i)C t(i)di=P t C t where

P t Z10P t(i)1 di 11 2.Other optimality conditions

U n;t U

c;t =

W t

P t

Q t= E t U c;t+1U c;t P t P t+1

Speci…cation of utility:

U(C t;N t)=C1

t

1

N1+'

t

1+'

implied log-linear optimality conditions(aggregate variables)

w t p t= c t+'n t

c t=E t f c t+1g 1 (i t E t f t+1g )

where i t log Q t is the nominal interest rate and log is the discount rate.

Ad-hoc money demand

m t p t=y t i t

Firms

Continuum of…rms,indexed by i2[0;1]

Each…rm produces a di¤erentiated good

Identical technology

Y t(i)=A t N t(i)1

Probability of being able to reset price in any given period:1 , independent across…rms(Calvo(1983)).

2[0;1]:index of price stickiness

Implied average price duration11

Aggregate Price Dynamics

P t= (P t 1)1 +(1 )(P t)1 11

Dividing by P t 1:

= +(1 ) P t P t 1 1

1

t

Log-linearization around zero in?ation steady state

t=(1 )(p t p t 1)(1) or,equivalently

p t= p t 1+(1 )p t

Optimal Price Setting

1X k=0 k E t Q t;t+k P t Y t+k j t t+k(Y t+k j t) max

P t

subject to

Y t+k j t=(P t=P t+k) C t+k

for k=0;1;2;:::where

Q t;t+k k C t+k C t P t P t+k

Optimality condition:

1X k=0 k E t Q t;t+k Y t+k j t P t M t+k j t =0 where t+k j t 0t+k(Y t+k j t)and M 1

Equivalently,

1X k=0 k E t Q t;t+k Y t+k j t P t P t 1 M MC t+k j t t 1;t+k =0 where MC t+k j t t+k j t=P t+k and t 1;t+k P t+k=P t 1

Perfect Foresight,Zero In?ation Steady State:

P t P t 1=1; t 1;t+k=1;Y t+k j t=Y;Q t;t+k= k;MC=

1

M

Log-linearization around zero in?ation steady state:

1X k=0( )k E t f c mc t+k j t+p t+k p t 1g p t p t 1=(1 )

Equivalently,

1X k=0( )k E t f mc t+k+p t+k g

p t= +(1 )

Flexible prices( =0):

p t= +mc t+p t

=)mc t= (symmetric equilibrium)

Particular Case: =0(constant returns)

=)MC t+k j t=MC t+k

Log-linearization of optimality condition around zero in?ation steady state: p t p t 1=

1X k=0( )k E t f(1 )c mc t+k+ t+k g(2)

= E t f p t+1 p t g+(1 )c mc t+ t(3) where c

mc t mc t mc=mc t+ ,and log 1.

Combining(1)and(3):

t= E t f t+1g+ c mc t

where

(1 )(1 )

Generalization to 2(0;1)(decreasing returns)De…ne

mc t (w t p t ) mpn t

(w t p t ) 1

1

(a t y t ) log(1 )

Using mc t +k j t =(w t +k p t +k ) 1

1 (a t +k y t +k j t ) log(1 ),

mc t +k j t

=mc t +k +

1 (y t +k j t y t +k )=mc t +k

1

(p t p t +k )(4)Implied in?ation dynamics

t = E t f t +1g + c mc t

(5)

where

(1 )(1 )

1 1 +

Equilibrium

Goods markets clearing

Y t(i)=C t(i)

for all i2[0;1]and all t.

Letting Y t R10Y t(i)1 1 di 1,Y t=C t

for all https://www.360docs.net/doc/429793710.html,bined with the consumer’s Euler equation:

y t=E t f y t+1g 1 (i t E t f t+1g )(6)

Labor market clearing

N t=Z10N t(i)di

=Z10 Y t(i)A t 11 di

= Y t A t 11 Z10 P t(i)P t 1 di T aking logs,

(1 )n t=y t a t+d t where d t (1 )log R10(P t(i)=P t) 1 di(second order). Up to a…rst order approximation:

y t=a t+(1 )n t

Marginal Cost and Output

mc t=(w t p t) mpn t

=( y t+'n t) (y t n t) log(1 )

= +'+ 1 y t 1+'1 a t log(1 )(7) Under?exible prices

mc= +'+ 1 y n t 1+'1 a t log(1 )(8)

=)y n t= y+ya a t

where y ( log(1 ))(1 )

+'+ (1 )>0and ya 1+'

+'+ (1 )

.

=)c

mc t= +'+ 1 (y t y n t)(9) where y t y n t e y t is the output gap

New Keynesian Phillips Curve

t= E t f t+1g+ e y t(10) where +'+ 1 .

Dynamic IS equation

e y t=E t

f e y t+1

g 1 (i t E t f t+1g r n t)(11) where r n t is the natural rate of interest,given by

r n t + E t f y n t+1g

= + ya E t f a t+1g

Missing block:description of monetary policy(determination of i t).

Equilibrium under a Simple Interest Rate Rule

i t= + t+ y e y t+v t(12) where v t is exogenous(possibly stochastic)with zero mean. Equilibrium Dynamics:combining(10),(11),and(12)

e y t t =A T E t

f e y t+1

g E t f t+1g +B T(b r n t v t)(13) where

A T 1

+ ( + y) ;B T 1

and 1

+ y+

Uniqueness()A T has both eigenvalues within the unit circle Given 0and y 0,(Bullard and Mitra(2002)):

( 1)+(1 ) y>0

is necessary and su￠cient.

E¤ects of a Monetary Policy Shock

Set b r n t=0(no real shocks).

Let v t follow an AR(1)process

v t= v v t 1+"v t

Calibration:

v=0:5, =1:5, y=0:5=4, =0:99, ='=1, =2=3, =4. Dynamic e¤ects of an exogenous increase in the nominal rate(Figure1): Exercise:analytical solution