Monetary Policy Design
in the Basic New Keynesian Model
by
Jordi Galí
April 2015
The E¢ cient Allocation
max U (Ct; Nt; Zt)
where Ct
R1
0
1
1
Ct(i)1 di
subject to:
Ct(i) = AtNt(i)1 ; all i 2 [0; 1]
Z 1
Nt =
Nt(i)di
0
Optimality conditions:
Ct(i) = Ct, all i 2 [0; 1]
Nt(i) = Nt, all i 2 [0; 1]
Un;t
= M P Nt
Uc;t
where M P Nt
(1
)AtNt
Sources of Suboptimality of Equilibrium
1. Distortions unrelated to nominal rigidities: Market power
Optimal price setting: Pt = M MWP tNt , where M
"
" 1
>1
Un;t Wt M P Nt
=
=
< M P Nt
Uc;t
Pt
M
=)
t
Assume employment subsidy : Under ‡exible prices, Pt = M (1M P )W
Nt .
=)
Optimal subsidy: M(1
M P Nt
Un;t Wt
=
=
Uc;t
Pt
M(1
)
) = 1 or, equivalently,
= 1" .
2. Distortions associated with the presence of nominal rigidities:
Markup variations resulting from sticky prices (assuming optimal subsidy):
Pt
Pt M
Mt =
=
(1
)(Wt=M P Nt) Wt=M P Nt
Un;t Wt
M
=
= M P Nt
6= M P Nt
Uc;t
Pt
Mt
E¢ ciency requirement: average markup = desired markup, all t
=)
Relative price distortions resulting from staggered price setting: Ct(i) 6= Ct(j)
if Pt(i) 6= Pt(j). Optimal policy requires that prices and quantities (and hence
marginal costs) are equalized across goods.
Optimal Monetary Policy in the Basic NK Model
Assumptions:
optimal employment subsidy
=) ‡exible price equilibrium allocation is e¢ cient
no inherited relative price distortions, i.e. P 1(i) = P
1
for all i 2 [0; 1]
Optimal policy: replicate ‡exible price equilibrium allocation.
Implementation: stabilize marginal costs at a level consistent with …rms’desired
markup, given existing prices:
no …rm has an incentive to adjust its price, i.e. Pt = Pt
for t = 0; 1; 2; :::(aggregate price stability)
1
and, hence, Pt = Pt
equilibrium output and employment match their natural counterparts.
1
Equilibrium under the Optimal Policy
yt = ytn
)
=0
it = rtn
t
yet = 0
for all t.
Implementation: Some Candidate Interest Rate Rules
Non-Policy Block:
where rtn =
(1
yet =
1
(it
Etf
t+1 g
rtn) + Etfe
yt+1g
= Etf t+1g + yet
a ) ya at + (1
z )zt
t
An Exogenous Interest Rate Rule
it = rtn
Equilibrium dynamics:
yet
t
where
= AO
AO
1
Etfe
yt+1g
Etf t+1g
1
+
Shortcoming: the solution yet = t = 0 for all t is not unique: one eigenvalue of AO
is strictly greater than one. ! indeterminacy (real and nominal).
An Interest Rate Rule with Feedback from Target Variables
it = rtn +
Equilibrium dynamics:
yet
t
where
AT
= AT
1
+ y+
t
+
et
yy
Etfe
yt+1g
Etf t+1g
1
+ ( +
y)
Existence and uniqueness condition: (Bullard and Mitra (2002)):
(
1) + (1
)
y
>0
Taylor-principle interpretation (Woodford (2000)):
dit+k
=
k!1 d t
+
=
+
lim
de
yt+k
y
k!1 d t
)
y (1
lim
Figure 4.1 Determinacy and Indeterminacy Regions: Standard Taylor Rule
2
1.5
?:
Determinacy
1
0.5
Indeterminacy
0
0
0.5
1.5
1
?y
2
A Forward-Looking Interest Rate Rule
it = rtn +
Equilibrium dynamics:
yet
t
where
1
(1
AF
Etf
t+1 g
1
yt+1g
y Et fe
Etfe
yt+1g
Etf t+1g
= AF
1
+
1
y
(
1
y)
(
1)
1)
Existence and uniqueness conditions (Bullard and Mitra (2002):
(
(
1) + (1
)
1) + (1 + )
y
y
> 0
< 2 (1 + )
Figure 4.2 Determinacy and Indeterminacy Regions: Forward Looking Taylor Rule
25
20
Indeterminacy
?:
15
10
5
0
Determinacy
0
0.5
1.5
1
?y
2
Shortcomings of Optimal Rules
they assume observability of the natural rate of interest (in real time).
this requires, in turn, knowledge of:
(i) the true model
(ii) true parameter values
(iii) realized shocks
“Simple rules” :
the policy instrument depends on observable variables only,
do not require knowledge of the true parameter values
ideally, they approximate optimal rule across di¤erent models
Simple Monetary Policy Rules
Welfare-based evaluation:
1
X
Utn
t Ut
W
E0
UcC
t=0
=)
1
1 X
= E0
2 t=0
t
+
'+
1
expected average welfare loss per period:
1
'+
L=
+
var(e
yt) + var( t)
2
1
yet2 +
2
t
A Taylor Rule
it =
+
t
+
Equivalently:
it =
where vt
btn
yy
+
t
+
Equilibrium dynamics:
yet
t
where
AT
and
Exercise:
1
+ y+
at
= AT
et
yy
+ vt
Etfe
yt+1g
+ BT (b
rtn
Etf t+1g
1
+ ( +
: Note that rbtn
bt
yy
vt =
;
y)
ya (
(1
AR(1) + modi…ed Taylor rule it =
vt )
1
BT
a)
+
+
y )at
t
+
+ (1
y
z )zt
yt
Table 4.1
Evaluation of Simple Rules: Taylor Rule
Technology
Demand
1:5 1:5
5
1:5
1:5 1:5 5 1:5
0:125 0
0
1
0:125 0
0
1
y
(y)
(e
y)
( )
1:85 2:07 2:25 1:06
0:44 0:21 0:03 1:23
0:69 0:34 0:05 1:94
0:59 0:68 0:28 0:31
0:59 0:68 0:28 0:31
0:20 0:23 0:09 0:10
L
1:02 0:25 0:006 7:98
0:10 0:13 0:02 0:02
Money Growth Peg
mt = 0
Money demand:
lt = yt
where lt
mt
2 [0; 1).
Letting lt+
t
pt .
t
where
it
lt +
t
Imposig the assumed rule
=
t 1
b
lt = yet + ybtn
bit = 1 (e
yt + ybtn
+ "t
bit
t
b
lt+)
mt = 0, and clearing of the money market:
b
l+ = b
l+ + t
t
t 1
t
Equilibrium dynamics:
2
where
AM;0
2
4
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
Simulations and Evaluation of Simple Rules
;
rbtn
ybtn
t
BM
3
5
2
3
1 0
40 0 05
0 0 1
Table 4.2
Evaluation of Simple Rules:
Constant Money Growth
Technology Demand Money Demand
(y)
(e
y)
( )
1:72
0:92
0:35
0:59
0:59
0:12
1:07
1:07
0:55
L
0:29
0:04
0:69
The Taylor Rule (Taylor 1993)
it = 4 + 1.5(π t − 2) + 0.5 yt
Source: Taylor 1999
Source: Taylor 1999
Clarida, Galí and Gertler (QJE 2000)
it = ρ it −1 + (1 − ρ )[r + π * + β Et {π t +1 − π *} + γ Et { yt +1 − yt*+1}]
Orphanides (JME 2003)
Source: Orphanides (JME 2003 b)