The Standard New Keynesian Model:
A Quick Overview
Jordi Galí
CREI, UPF and Barcelona GSE
January 2013
Jordi Galí
(CREI, UPF and Barcelona GSE)
The NK Model: A Quick Overview
January 2013
1 / 13
Households and Wage Setting
Large number of identical in…nitely-lived households, each with a
continuum of members specialized in di¤erent types of labor services,
indexed by j 2 [0, 1].
Preferences
∞
E0
∑ βt U (Ct , fNt (j )g; Zt )
t =0
where β 2 (0, 1), Ct
R1
0
Ct ( i )
U (Ct , fNt (j )g; Zt ) =
1
ep
1
Ct1
1
di
σ
ep
ep 1
1
σ
and
Z 1
Nt (j )1 + ϕ
0
1+ϕ
dj
Zt
Budget constraint
Z 1
0
Jordi Galí
Pt (i )Ct (i )di + Qt Bt
(CREI, UPF and Barcelona GSE)
Bt
1+
Z 1
0
Wt (j )Nt (j )dj + Dt
The NK Model: A Quick Overview
January 2013
2 / 13
Optimal allocation of consumption:
Pt ( i )
Pt
Ct ( i ) =
R1
where Pt
0
Pt ( i ) 1
ep di
ep
Ct
1
1 ep
Intertemporal optimality condition:
Qt = βEt
Uc ,t +1 Pt
Uc ,t Pt +1
implying:
1
( it
σ
ct = E t f ct + 1 g
where π t +1
Jordi Galí
pt + 1
pt , it
(CREI, UPF and Barcelona GSE)
Et fπ pt+1 g
log Qt and ρ
The NK Model: A Quick Overview
ρ) +
1
(1
σ
ρz )zt
log β.
January 2013
3 / 13
Labor supply (under perfectly competitive labor markets):
Wt (j )
Un,t (j )
=
Uc ,t
Pt
wt (j )
pt = σct + ϕnt (j )
mrst (j )
Symmetric equilibrium:
wt
pt = σct + ϕnt
mrst
Optimal wage setting under monopolistic unions, ‡exible wages and
isoelastic labor demand:
wt (j )
where µw
Jordi Galí
log ewew 1
(CREI, UPF and Barcelona GSE)
wt
pt = µw + mrst (j )
pt = µw + mrst
The NK Model: A Quick Overview
January 2013
4 / 13
Staggered wage setting
The nominal wage for each labor type is reset with probability 1
each period (EHL)
θw
Average wage dynamics
wt = θ w wt
1
+ (1
θ w )wt
Optimal wage setting rule
wt = µw + (1
∞
βθ w )
∑ ( βθ w )k Et
k =0
where µw
log Mw and mrst +k jt
σct +k + ϕnt +k jt
Wage in‡ation equation
π wt = βEt fπ wt+1 g
where π wt
Jordi Galí
wt
wt
1,
(CREI, UPF and Barcelona GSE)
µwt
wt
mrst +k jt + pt +k
pt
λw (µwt
µw )
mrst and λw
The NK Model: A Quick Overview
(1 θ w )(1 βθ w )
.
θ w (1 + ew ϕ )
January 2013
5 / 13
Firms and Price Setting
Continuum of …rms, each producing a di¤erentiated good.
Technology
Yt = At Nt1
where Nt
Jordi Galí
R1
0
Nt (j )1
(CREI, UPF and Barcelona GSE)
1
ew
dj
α
ew
ew 1
The NK Model: A Quick Overview
January 2013
6 / 13
Optimality condition under perfectly competitive goods markets
Wt
= (1
Pt
wt
p t = at
α)At Nt
αnt + log(1
α
α)
mpnt
Optimal price setting under monopolistic competition, ‡exible prices and
isoelastic demand:
pt
where µp
Jordi Galí
= µp + ψt
= µp + (wt
mpnt )
e
log ep p 1 .
(CREI, UPF and Barcelona GSE)
The NK Model: A Quick Overview
January 2013
7 / 13
Staggered price setting
The price of each good reset with a probability 1
θ p each period
Average price dynamics
pt = θ p pt
1
+ (1
θ p ) pt
Opimal price setting rule
pt = µ p + ( 1
∞
βθ p )
∑ ( βθ p )k Et fψt +k jt g
k =0
where ψt +k jt
wt
mpnt +k jt and mpnt +k jt
Price in‡ation equation
π pt = βEt fπ pt+1 g
where µpt
Jordi Galí
pt
(wt
(CREI, UPF and Barcelona GSE)
mpnt ), and λp
λp (µpt
at
αnt +k jt + log(1
α)
µp )
(1 θ p )(1 βθ p )
1 α
θp
1 α+αep .
The NK Model: A Quick Overview
January 2013
8 / 13
Equilibrium
Aggregate demand, output and employment
yt = ct
yt = Et fyt +1 g
1
( it
σ
nt =
Jordi Galí
(CREI, UPF and Barcelona GSE)
Et fπ pt+1 g
1
1
α
(yt
ρ) +
1
(1
σ
ρz )zt
at )
The NK Model: A Quick Overview
January 2013
9 / 13
Prices, wages and economic activity
where, letting ω t
π pt = βEt fπ pt+1 g
λp (µpt
µp )
π wt = βEt fπ wt+1 g
λw (µwt
µw )
wt
µpt
pt
= pt
= at
µwt
(wt mpnt )
αnt + log(1 α)
= ωt
= ωt
ωt = ωt
Jordi Galí
(CREI, UPF and Barcelona GSE)
ωt
mrst
(σyt + ϕnt )
1
+ π wt
π pt
The NK Model: A Quick Overview
January 2013
10 / 13
Monetary policy rule
Example (Taylor rule):
it = ρ + φπ π pt + φy ybt + vt
Dynamic responses to a monetary policy shock
(i) baseline: θ p = θ w = 3/4
(ii) sticky prices: θ p = 3/4 and θ w = 0
(iii) sticky wages: θ w = 3/4 and θ p = 0
Jordi Galí
(CREI, UPF and Barcelona GSE)
The NK Model: A Quick Overview
January 2013
11 / 13
Figure 6.2 Dynamic Responses to a Monetary Policy Shock
0.1
0.2
0.1
0
0
-0.1
-0.1
-0.2
-0.2
-0.4
-0.3
baseline
flexible wages
flexible prices
-0.3
0
2
4
6
8
10
output gap
12
14
-0.4
16
4
-0.5
0
2
4
6
8
10
price inflation
12
14
16
0
2
4
6
12
14
16
0.5
2
0
0
-2
-0.5
-4
-1
-6
-1.5
-8
-10
0
2
4
6
8
10
wage inflation
12
14
16
-2
8
real wage
10
Monetary Policy Design: Some Key Results
Natural equilibrium allocation is no longer feasible (as long as it requires
real wage changes)
) tradeo¤ between output gap and in‡ation stabilization.
Welfare losses (second order approximation)
W=
∞
1
E0 ∑ βt
2 t =0
σ+
ϕ+α
1 α
or, in unconditional version:
L=
1
2
σ+
ϕ+α
1 α
var (yet ) +
yet2 +
ep p 2 ew (1 α ) w 2
(π ) +
(π t )
λp t
λw
ep
ew (1 α )
var (π pt ) +
var (π wt )
λp
λw
=) strict price in‡ation targeting is no longer optimal
Optimal monetary policy
Jordi Galí
(CREI, UPF and Barcelona GSE)
The NK Model: A Quick Overview
January 2013
12 / 13
Figure 6.3 Dynamic Responses to a Technology Shock under the Optimal Monetary Policy
1
0.1
0
0.05
-1
0
-2
baseline
flexible wages
flexible prices
-0.05
-0.1
0
2
4
10
8
6
output gap
12
14
-3
16
-4
4
1
3
0.8
2
0.6
1
0.4
0
0.2
-1
0
2
4
10
8
6
wage inflation
12
14
16
0
0
2
4
10
8
6
price inflation
12
14
16
0
2
4
6
10
8
real wage
12
14
16
Composite in‡ation and the output gap
where yet
yt
ytn and
π t = βEt fπ t +1 g + { yet
πt
(1
ϑ )π pt + ϑπ wt
λ
p
with ϑ
λp +λw 2 [0, 1].
) no tradeo¤
) stabilization of yet and π t optimal under a knife-edge parameter
con…guration
Evaluation of six simple rules
- strict in‡ation targeting rules
- ‡exible in‡ation targeting rules
Jordi Galí
(CREI, UPF and Barcelona GSE)
The NK Model: A Quick Overview
January 2013
13 / 13