The Basic New Keynesian Model
Jordi Galí
June 2015
Motivation and Outline
Evidence on Money, Output, and Prices:
Macro evidence on the 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: signi…cant price and wage rigidities
Figure 1. Estimated Dynamic Response to a Monetary Policy Shock
Federal funds rate
GDP
GDP deflator
M2
Source: Christiano, Eichenbaum and Evans (1999)
Source: Dhyne et al. (JEP, 2006)
Source: Álvarez et al. (JEEA, 2006)
Motivation and Outline
Evidence on Money, Output, and Prices:
Macro evidence on the 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: signi…cant price and wage rigidities
) in con‡ict with the predictions 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
max E0
1
X
t
U (Ct; Nt; Zt)
t=0
where
Ct
Z
1
1
1
Ct(i)1 di
0
subject to
Z
1
Pt(i)Ct(i)di + QtBt
Bt
1
+ WtNt + Dt
0
for t = 0; 1; 2; ::: plus solvency constraint.
Example:
lim Et
T !1
BT
PT
0
Optimality conditions
1. Optimal allocation of expenditures
Pt(i)
Pt
Ct(i) =
implying
Z
Ct
1
Pt(i)Ct(i)di = PtCt
0
where
Pt
Z
1
1
1
Pt(i)1 di
0
2. Other optimality conditions
Un;t Wt
=
Uc;t
Pt
Uc;t+1 Pt
Qt = E t
Uc;t Pt+1
Speci…cation of utility:
U (Ct; Nt; Zt) =
where
8
<
Ct1
1
: log Ct
zt =
Nt1+'
1+'
Nt1+'
1+'
1
z zt
1
Zt for
6= 1
Zt for
=1
+ "zt
Optimality conditions:
ct = Etfct+1g
where it
log Qt and
wt pt = ct + 'nt
1
1
(it Etf t+1g
) + (1
log
Ad-hoc money demand:
mt
pt = c t
it
z )zt
Firms
Continuum of …rms, indexed by i 2 [0; 1]
Each …rm produces a di¤erentiated good
Identical technology
Yt(i) = AtNt(i)1
where
at =
a at 1
+ "at
Probability of resetting price in any given period: 1
(Calvo (1983)).
2 [0; 1] : index of price stickiness
Implied average price duration
1
1
, independent across …rms
Aggregate Price Dynamics
Pt =
Dividing by Pt
1
(Pt 1)
1
+ (1
)(Pt )
:
1
t
= + (1
Pt
)
Pt 1
1
1
1
1
Log-linearization around zero in‡ation steady state
t
= (1
)(pt
pt 1 )
+ (1
)pt
or, equivalently
pt = p t
1
(1)
Optimal Price Setting
max
Pt
subject to:
1
X
k
t;t+k (1=Pt+k )
Et
Pt Yt+kjt
k=0
Yt+kjt =
for k = 0; 1; 2; :::where
t;t+k
Optimality condition:
1
X
k
Et
k
Pt
Pt+k
t+kjt
(2)
Ct+k
Uc;t+k =Uc;t
t;t+k Yt+kjt (1=Pt+k )
Pt
k=0
where
Ct+k (Yt+kjt)
0
Ct+k
(Yt+kjt) and M
1.
Flexible price case ( = 0):
Pt = M
tjt
M
t+kjt
=0
Zero in‡ation steady state
t;t+k
=
k
; Pt =Pt
1
= Pt=Pt+k = 1 ) Yt+kjt = Y ;
t+kjt
Linearized optimal price setting condition:
pt =
+ (1
)
1
X
k=0
where
t+kjt
log
t+kjt
and
log M
(
)k Etf
t+kjt g
=
t
; Pt = M
t
= 0 (constant returns)
Particular Case:
=)
t+kjt
=
t+k
Recursive form:
where
t
pt
t
pt =
and bt
Etfpt+1g + (1
)pt
(1
t
Combined with price dynamics equation yields:
t
where
= Et f
(1
t+1 g
)(1
bt
)
)bt
2 [0; 1)
General case:
t+kjt
= wt+k
= wt+k
mpnt+kjt
(at+k
nt+kjt + log(1
(at+k
wt+k
t+k
t+kjt
nt+k + log(1
=
t+k
+ (nt+kjt
=
t+k
+
=
t+k
1
1
))
))
nt+k )
(yt+kjt
(pt
yt+k )
pt+k )
Optimal price setting equation:
pt = (1
)
1
X
k=0
where
1
1
+
2 (0; 1].
(
)k Et fpt+k
bt+k g
Recursive form:
pt =
Etfpt+1g + (1
)pt
(1
Combined with price dynamics equation yields:
t
where
= Et f
(1
t+1 g
)(1
)
bt
) bt
Equilibrium
Goods markets clearing
Yt(i) = Ct(i)
for all i 2 [0; 1] and all t.
R1
1 1
di
Letting Yt
Y
(i)
0 t
1
:
Yt = C t
Combined with Euler equation:
yt = Etfyt+1g
1
(it
Etf
t+1 g
1
) + (1
z )zt
Labor market clearing:
Nt =
Z
1
Nt(i)di
0
=
Z
1
0
=
Yt
At
Up to a …rst order approximation:
nt =
Yt(i)
At
1 Z
1
1
1
di
1
0
1
1
(yt
Pt(i)
Pt
at )
1
di
Average price markup and output
pt
t
=
=
=
t
(wt pt) + (at
nt + log(1
))
( yt + 'nt) + (at
nt + log(1
))
'+
1+'
+
yt +
at + log(1
1
1
)
Under ‡exible prices:
'+
+
1
=
ytn
+
1+'
at + log(1
1
ya at
+
implying
ytn =
where
(1
y
)(
(1
log(1
)+'+
))
> 0 and
bt =
ya
+
y
1+'
(1 )+'+
'+
1
(yt
. Thus,
ytn)
)
New Keynesian Phillips Curve
t
where yet
yt
ytn and
= Et f
+ '+
.
1
t+1 g
+ yet
The Non-Policy Block of the Basic New Keynesian Model
New Keynesian Phillips Curve
t
= Et f
Dynamic IS equation
yet = Etfe
yt+1g
1
t+1 g
(it
where rtn is the natural rate of interest, given by
rtn =
Missing block:
(1
+ yet
Et f
a ) ya at
t+1 g
+ (1
rtn)
z )zt
description of monetary policy (determination of it).
Equilibrium under a Simple Interest Rate Rule
it =
where ybt
yt
+
t
+
v vt
1
y and
vt =
bt
yy
+ vt
+
btn
yy
+ "vt
Equivalently:
it =
where ybtn
ytn
y.
+
t
+
et
yy
+ vt
Equilibrium dynamics:
yet
= AT
t
where
ut
=
and
rbtn
n
y
b
vt
y t
ya ( y + (1
1
+ ( +
AT
with
1
+ y+
Etfe
yt+1g
+ BT ut
Etf t+1g
a ))at
+ (1
;
y)
z )zt
vt
BT
.
Uniqueness condition (Bullard and Mitra):
(
1) + (1
)
y
>0
Exercise: analytical solution (method of undetermined coe¢ cients).
1
Equilibrium under an Exogenous Money Growth Process
mt =
m
mt
1
+ "m
t
Money demand
lt mt
Substituting into dynamic IS equation
(1 +
Identity:
) yet =
pt = yet
it + ytn
Etfe
yt+1g + b
lt + Etf
b
lt
1
=b
lt +
t
mt
t+1 g
+ rbtn
ybtn
Equilibrium dynamics:
where
AM;0
2
4
2
yet
3
2
3
2
Etfe
yt+1g
AM;0 4 t 5 = AM;1 4 Etf t+1g 5 + BM 4
b
b
lt
lt 1
1+
0
3
0 0
1 05
1 1
;
AM;1
2
3
1
4 0
05
0 0 1
;
rbtn
ybtn
mt
BM
3
5
2
(3)
3
1 0
40 0 0 5
0 0
1
Uniqueness condition:
1
AM AM;0
AM;1 has two eigenvalues inside and one outside the unit circle.
Calibration
Households: = 1 ; ' = 5 ; = 0:99 ; = 9 ; = 4 ;
Firms: = 1=4 ; = 3=4 ; a = 0:9
Policy rules:
= 1:5, y = 0:125 ; v = m = 0:5
Dynamic Responses to Exogenous Shocks
Monetary policy, discount rate, technology
Interest rate rule vs. money growth rule
z
= 0:5
Dynamic responses to a monetary policy shock: Interest rate rule
Dynamic responses to a discount rate shock: Interest rate rule
Dynamic responses to a technology shock: Interest rate rule
Estimated Effects of Technology Shocks
Source: Galí (1999)
Estimated Effects of Technology Shocks
Source: Basu, Fernald and Kimball (2006)
Dynamic responses to a monetary policy shock: Money growth rule
Dynamic responses to a discount rate shock: Money growth rule
Dynamic responses to a technology shock: Money growth rule