Lectures on Monetary Policy, In‡ation and the Business Cycle A Model with Sticky Wages and Prices
by Jordi Galí
February 2007
Based on: Erceg, Henderson and Levin. (JME, 2000) Firms Technology Yt(i) = At Nt(i)1 Z 1 1 1w Nt(i) Nt(i; j) dj
w w 1
0
Cost minimization:
Nt(i; j) = for all i; j 2 [0; 1], where Wt
Z
w
Wt(j) Wt
Nt(i) 1
1
1
Wt(j)
1
w
w
dj
0
In addition,
Z
0
1
Wt(j)Nt(i; j) dj = WtNt(i):
(1)
Optimal price setting (as in baseline sticky price model) max Pt
1 X
k p
k=0
subject to
Yt+kjt = (Pt =Pt+k ) Aggregation: where bpt
t+k (Yt+kjt )
Et Qt;t+k Pt Yt+kjt
p t p t
p
=
=
mc ct ,
Et f p
p t+1 g
log
Ct+k
p p
p
p
bpt
1 , and
(2) (1 p
p )(1 p
p)
1 1
+
p
.
Households fraction of households/trade unions adjusting nominal wage: 1 w
w
: index of nominal wage rigidity
Optimal Wage Setting max Wt
subject to:
Pt+k
1 X k=0
(
w)
k
EtfU (Ct+kjt; Nt+kjt)g
Nt+kjt = (Wt =Wt+k ) w Nt+k Ct+kjt + Et+k fQt+k;t+k+1Dt+k+1jtg Dt+kjt + Wt Nt+kjt
where Nt
R1 0
Nt(i) di.
Tt+k
Optimality condition: 1 X
(
k w ) Nt+kjt Et Uc (Ct+kjt ; Nt+kjt )
k=0
where Mw
Wt + Mw Un(Ct+kjt; Nt+kjt) Pt+k
w w
1
Complete markets: Ct+kjt = Ct+k for k = 0; 1; 2; ::: Letting M RSt+kjt 1 X k=0
(
Un (Ct+k ;Nt+kjt ) Uc (Ct+k ;Nt+kjt )
k w ) Et Nt+kjt Uc (Ct+kjt ; Nt+kjt )
Wt Pt+k
Mw M RSt+kjt
=0 (3)
=0
Full wage ‡exibility (
w
= 0):
Wt Wt = = Mw M RStjt Pt Pt Zero in‡ation steady state: W = Mw M RS P
Log-linearization (after dividing (3) by Mw M RS): wt =
w
+ (1
w)
1 X
(
w)
k
Et mrst+kjt + pt+k
(4)
k=0
where
w
log
w w
1
.
With isoelastic separable utility =) mrst+kjt = Average marginal rate of substitution:
mrst+k
mrst+kjt = mrst+k + ' (nt+kjt = mrst+k w ' (wt
ct+k + ' nt+kjt . ct+k + ' nt+k
nt+k ) wt+k )
Hence, 1 wt = 1+ =
1 1+
where bwt
w w' w w' w t
1 X k=0 1 X k=0
(
w)
k
(
w)
k
Et f
w
+ mrst+k +
Et f(1 +
w ')
w'
wt+k
w
wt+k + pt+k g
bwt+k g
More compactly: wt =
w
Etfwt+1g + (1
w)
wt
(1 +
w ')
1
bwt
(5)
Wage In‡ation Dynamics 1 w w Wt 1
Wt =
+ (1
w )Wt
1
1 w
1
w
Log-linearization: wt =
w
wt
1
+ (1
w)
w t+1 g
w
(6)
wt
Combining (5) and (6): w t
where
w
(1
w )(1 w (1+ w ')
w)
= Et f
.
bwt
(7)
Additional Optimality Condition ct = Etfct+1g
1
(it
Etf
p t+1 g
)
Equilibrium De…ne real wage gap: ! et
! nt
!t
Price markups vs. output and real wage gaps: bpt = (mpnt ! t) = (e yt n et ) ! et =
yet
1
Combining (2) and (8): p t
where
p
p
1
.
=
Et f
p t+1 g
+
p
p
! et
yet +
(8)
p
! et
(9)
Wage markups vs. output and real wage gaps: bwt = ! t = ! et = ! et
w mrst ( yet + 'e nt ) ' + 1
Combining (7) and (10): w t
where
w
w
= Etf
+ 1' .
w t+1 g
+
w
yet
yet w
! et
(10)
(11)
Wage gap identity: ! et
w t
! et
1
p t
+
+ ! nt
(12)
Dynamic IS equation yet =
1
(it
Etf
p t+1 g
+
p t
w
rtn) + Etfe yt+1g
(13)
Interest Rate Rule: it =
p
+
w t
+
y
yet + vt
(14)
Dynamical system: xt = Aw Etfxt+1g + Bw zt where
Remark: yet =
xt [e yt; pt; wt ; ! e t 1]0 zt [b rtn vt; ! nt]0 p t
=
w t
= 0 cannot be solution, unless ! nt is constant.
Conditions for uniqueness of the equilibrium Particular case (
(15)
y
= 0): p
+
w
>1
Dynamic Responses to a Monetary Policy Shock Interest rate rule:
p
= 1:5 ;
y
=
Three calibrations: Baseline:
p
= 2=3,
w
= 3=4
Flexible wage:
p
= 2=3,
w
Flexible price:
p
= 0,
= 3=4
Figure 6.3
w
=0
w
= 0;
v
= 0:5
Monetary Policy Design with Sticky Wages and Prices Second Order Approximation to Welfare Losses 1
X 1 W = E0 2 t=0 L=
t
'+ + 1
'+ + 1
var(e yt ) +
yet2 p p
+
p
(
p
var(
p t)
p 2 t)
+
+
w (1
)
(
w
w (1
) w
var(
w 2 t )
+t:i:p:
w t )
Key policy issues replicating the natural equilibrium allocation is generally unfeasible. optimal monetary policy evaluation of alternative simple rules
Optimal Monetary Policy
min E0
1 X
t
+
t=0
subject to
'+ 1
p t
=
w t
= Etf
! et
Et f
1
! et
yet2 +
p t+1 g w t+1 g
+
+ w t
+
p
( pt)2 +
p
p
w
yet +
yet
p t
p
w
+ ! nt
w (1
! et
! et
) w
(
w 2 t )
Optimality conditions: '+ + 1 p p w (1
p t
) w
p
1;t
w
2;t
yet +
p
1;t
w t
+
1;t
+
+
3;t
2;t
=0
=0
2;t
3;t
w
3;t
Et f
(16) (17)
=0
3;t+1 g
(18) =0
(19)
Combined with (9), (11), and (12):
A0 xt = A1 Etfxt+1g + B where xt
[e yt ;
p t;
w t ;
! e t 1;
1;t 1 ;
2;t 1 ;
at
0 3;t ]
Dynamic Responses to a Technology Shock (Figure 6.4)
A Special Case with an Analytical Solution De…ne: (1
t
where #
p p+ w
#)
p t
+#
w t
(20)
+
yet
(21)
2 [0; 1]
Note that (9) and (11) imply: t
where
p w p+ w
=
Et f
t+1 g
+ '+ 1
no trade-o¤ ! when is it optimal to fully stabilize
t
(and the output gap)?
Assumptions:
p
=
;
w
p
=
w (1
)
Then, (16)-(18) simplify to:
for t = 1; 2; 3; ::: and
w
p t
+
p
w t
=
w
p 0
+
p
w 0
=
p
p
Equivalently, t
for t = 1; 2; 3; ::: , and
0
=
In levels: where qbt
qt
q
1
, and qt
=
#
#
yet
yet
ye0 for period 0
ye0 in period 0.
qbt = (1
#
yet
#) pt + # wt.
(22)
Combining (22) and (21) (using qbt = a qbt
for t = 0; 1; 2; :::where a
1
qbt
t
qbt 1):
+ a Etfb qt+1g = 0
# #(1+ )+
.
Stationary solution: where
1
Given that qb
p
1 4 a2 2a
1
qbt =
qbt
1
2 (0; 1) for t = 0; 1; 2; :::
= 0, the optimal policy requires:
for t = 0; 1; 2; :::
t
=0
yet = 0
Evaluation of Simple Rules under Sticky Wages and Prices Six rules: strict price in‡ation targeting ( strict wage in‡ation targeting (
p t = 0, all t) w t = 0, all t)
strict composite in‡ation targeting (
t
= 0, all t)
‡exible price in‡ation targeting (it =
+ 1:5
‡exible wage in‡ation targeting (it =
+ 1:5
‡exible composite in‡ation targeting (it =
p t) w t )
+ 1:5
Three scenarios baseline:
p
= 2=3 ;
w
= 3=4
low wage rigidities:
p
= 2=3 and
w
= 1=4
low price rigidities:
p
= 1=3 and
w
= 3=4
t)
Table 6.1: Evaluation of Simple Rules Optimal Strict Rules Flexible Rules Policy Price Wage Composite Price Wage Composite p
p
p
=
=
=
2 3
w
2 3
w
1 3
w
=
3 4
( p) ( w) (e y) L =
1 4
( p) ( w) (e y) L =
3 4
( p) ( w) (e y) L
0.64 0.22 0.04 0.023
0 0.82 0.98 0 2.38 0.52 0.184 0.034
0.66 0.19 0 0.023
1.50 1.05 0.75 0.221
1.08 0.30 1.16 0.081
1.12 0.42 0.01 0.089
0.29 1.24 0.19 0.010
0 0.82 2.91 0 0.61 0.52 0.038 0.034
0.21 1.63 0 0.012
1.40 1.49 0.29 0.097
1.45 0.98 0.68 0.104
1.30 1.25 0.32 0.083
1.64 0.11 0.17 0.016
0 1.91 0.98 0 2.38 0.27 0.184 0.021
1.75 0.06 0 0.017
2.58 1.47 0.87 0.271
2.10 0.07 0.60 0.030
2.10 0.10 0.58 0.031
