Monetary Policy under the Zero Lower Bound
by
Jordi Galí
April 2015
Source: Nakov (IJCB 2008)
The Zero Lower Bound: Issues
How to conduct monetary policy when the ZLB is reached?
How e¤ective are non-standard monetary policies when the ZLB is
binding?
How does the presence of the ZLB a¤ects the optimal conduct of
monetary policy in "normal" times? (response to shocks, in‡ation
target,...)
What policy rules are more likely to drive the interest rate to the
ZLB?
The Zero Lower Bound Literature: Overview
ZLB as an (unanticipated) initial condition
- Price-setting in advance: Krugman (1998)
- Basic NK: Jung-Teranishi-Watanabe (2005), Eggertsson-Woodford
(2013), Galí (2015, ch. 5)
- E¤ectiveness of …scal policy: Eggertsson (2010)
- NK with …nancial frictions: Eggertsson-Krugman (2012), EggertssonMehrotra (2014)
ZLB as an occasionally binding constraint
- Basic NK: Adam-Billi (2005, 2006), Nakov (2008)
Expectations-driven liquidity traps
- NK with Taylor rule: Benhabib et al. (2001)
- E¤ectiveness of …scal policy: Mertens and Ravn (2014)
Eggertsson and Woodford (BPEA 2003)
Basic NK model
Unanticipated drop in natural rate to rtn =
< 0, with constant
probability of (permanent) reversion to steady state rtn = > 0
Strict in‡ation target
+ rtn all t). Higher
it is costly itself.(Fig.)
may be unattainable if low (requires it =
reduces the costs of a liquidity trap, but
Optimal policy under commitment: lower current long-term real
rates, through commitment to lower future short-term rates (and
higher expected in‡ation) ) "forward guidance"
Jung, Teranishi and Watanabe (JMCB 2005): similar framework
t
with rtn =
+ for t = 0; 1; 2; :::where 2 [0; 1) and > .
Source: Eggertsson and Woodford (2003)
Source: Eggertsson and Woodford (2003)
Source: Eggertsson and Woodford (2003)
Additional result in EW: irrelevance of "quantitative easing" under
the ZLB (given unchanged future interest rates) in a basic NK
model with money in the utility function (non-separable and with
satiation) and simple interest rate rule.
Implementation of monetary policy:
mt
pt
L(it; yt; t)
When it > 0, it must hold with equality. But not if it = 0.
Does an "excess supply" of liquidity when it = 0 stimulate activity?
Answer: No, independently of the asset portfolio purchased by the CB
(no portfolio balance e¤ects).
The Monetary Policy Problem under the ZLB (Galí 2015)
1
X
t
2
2
min E0
t + #xt
t=0
subject to
t
xt =
1
(it
= Et f
Etf
t+1 g
t+1 g
+ x
rtn) + Etfxt+1g
it 0
for t = 0; 1; 2; :::where rtn follows the exogenous deterministic path:
rtn =
=
) optimal allocation (
) policy tradeo¤
<0
>0
t
for t = 0; 1; 2; :::; tZ
for t = tZ + 1; :::
= xt = 0 for all t) not attainable
Optimal Discretionary Policy
Each period the monetary authority chooses (xt;
2
t
+ #x2t
subject to:
t
= xt +
xt
1;t
0;t
for t = 0; 1; 2; :::with
0;t
1;t
taken as given.
t+1
xt+1 + (1= )(
t+1
+ rtn)
t)
to minimize
Lagrangean:
1
L=
2
2
t
+ #x2t +
1;t ( t
0;t )
xt
+
2;t (xt
Optimality conditions:
t
+
#xt
with slackness conditions:
2;t
0 ; it
=0
1;t + 2;t = 0
1;t
0;
2;t it
=0
The two optimality conditions can be combined as:
#xt =
t
2;t
1;t )
Conjecture
it = 0
> 0
for t = 0; 1; 2; :::; tZ
for tZ + 1; tZ + 2; :::
Equilibrium from tZ + 1 onward,
it =
xt =
>0
( =#)
implying
xt =
t
=0
t
Equilibrium for t = 0; 1; 2; :::tZ
Under it = 0:
xt
=A
t
xt+1
B
t+1
where
A
1
1
+
with terminal conditions xtZ +1 =
Note that xt < 0 and
as conjectured.
t
1
;
tZ +1
B
= 0.
< 0 for t = 0; 1; 2; ::; tz guaranteeing
2;t
> 0,
Optimal Policy with Commitment
State-contingent policy fxt; tg1
t=0 that minimizes
1
X
t
2
2
t + #xt
t=0
subject to the sequence of constraints
t
xt
=
t+1
+ xt
1
xt+1 + (
t+1
+ rtn)
Lagrangean:
L=
1
X
t=0
t
1
2
2
t
+ #x2t +
1;t (
t+1 ) +
xt
t
2;t (xt
xt+1
1
First order conditions
t
+
#xt
1
1;t
1;t 1
1;t
+
2;t 1
=0
(1)
2;t 1
=0
(2)
1
2;t
with slackness and initial conditions
2;t
0 ; it
1; 1
=
0;
2; 1
2;t it
=0
=0
(
t+1
+ rtn))
Conjecture:
it = 0
> 0
for t = 0; 1; 2; :::; tC
for tC + 1; tC + 2; :::
tZ
Equilibrium for t = tC + 2; tC + 3; ::::
(3)
t + 1;t
1;t 1 = 0
#xt
(4)
1;t = 0
(5)
t =
t+1 + xt
together with an initial condition for 1;tC +1 (to be determined). Combining (3) and (4):
(pt p )
(6)
xt =
#
for t = tC + 2; tC + 3; :: where p
ptC +1 + 1;tC +1. Combined with
(5):
pbt = pbt 1 + pbt+1
#
where pbt pt p and
#(1+ )+ 2 .
Stationary solution:
pbt = pbt 1
for t = tC + 2; tC p
+ 3; :: , with initial condition pbtC +1 =
and where
and
1
1 4
2
2
1;tC +1
< 0,
2 (0; 1). Thus:
pbtc+2+k =
xtc+2+k =
k+1
1;tC +1
k+1
1;tC +1
#
>0
(7)
for k = 0; 1; 2; ::as well as
tc +2+k
= (1
)
k
1;tC +1
>0
(8)
) in‡ation and the output gap converge to zero (only) asymptotically.
Equilibrium for tc + 1:
1
tc +1 +
1;tc +1
2;tc
1;tc
1
#xtc+1
1;tc +1
2;tc
=0
(9)
=0
(10)
= (1
) 1;tC +1 + xtc+1
Using (11) to substitute out 1;tC +1 from (9) and (10) yields:
tc +1
xtc+1
=M
tc +1
1;tc
(12)
2;tc
where
M
1 + (1
(1
)+
2
#
#
)
(11)
1
(1
)
0
1
1
#
Equilibrium for t = 0; 1; 2; ::: tC :
it = 0
xt
xt+1
=A
t
B
(13)
+B
(14)
t+1
for t = 0; 1; :::; tz , and
xt
xt+1
=A
t
t+1
for t = tz+1; :::; tc, where A is de…ned as above. In addition, (1) and
(2) imply:
x
1;t
= H 1;t 1
J t
(15)
2;t
with initial conditions
"
1
H
1; 1
=
1
1
t
2;t 1
1+
2; 1
#
= 0, where
;
J
0 1
#
Solution strategy:
1. Make guess on tc
2. Solve for (xt; t; 1;t; 2;t) for t = 0; 1; ::; tc + 1 using the system of
4(tc + 2) equations (12), (13), (14) and (15).
3. Combine implied
t = tc + 2; :::
1;tc +1
with (7) and (8) to determine (xt;
t)
for
4. Given the solution for f t; xtg solve for implied interest rate using
it = rtn +
t+1
+ (xt+1
xt )
and check whether it = 0 for t = 0; 1; ::; tc and it > 0 for t = tc + 1; tc +
2,...con…rming the initial guess. Otherwise the procedure is repeated
for a di¤erent tc guess.
Discretion vs. Commitment in the Presence of a ZLB (Galí 2015)
5
5
0
0
-5
-10
-5
-15
-10
discretion
commitment
-15
0
2
4
6
output gap
8
10
-20
12
6
-25
0
2
4
6
inflation
8
10
12
0
2
4
6
natural rate
8
10
12
6
4
4
2
2
0
-2
0
-4
-2
0
2
4
6
nominal rate
8
10
12
-6
Source: Galí (2015, ch. 5)
Expectations-Driven Liquidity Traps
Benhabib, Schmitt-Grohé and Uribe (2001)
- Taylor rule with ZLB: it = ( t) > 0 where 0 0, ( ) = +
and 0( ) > 1
- Note that ( ) = + will always have a second solution LT <
("liquidity trap")
- Multiplicity of equilibrium paths converging to the liquidity trap
steady state (Fig)
Source: Benhabib, Schmitt-Grohé and Uribe (2001)