1. science
2. mathematics
3. algebra

Donghun Joo / Journal of Economic Research 19 (2014) 93{123
93
How to exit from zero interest rate
when there is a financial accelerator
Donghun Joo1
Hanyang University, Korea
Abstract
This paper extends a simple new Keynesian DSGE model with a financial accelerator (FA) to study an exit strategy from zero interest rate. The extension of the model is made in two steps to consider the role of uncertainty explicitly: first introducing the FA
mechanism that amplifies a shock and then investigating the financial shock uncertainty. The FA mechanism makes monetary policy to be less aggressive under the optimal discretionary policy regime. The introduction of the financial shock uncertainty however
overwhelms the FA effect and makes monetary policy to be more
aggressive in total. As a whole, the zero interest rate period is prolonged against the negative demand shock and the overshooting of
output and inflation gaps are allowed during the phase of recovery.
However, when the interest rate exits from the zero interest rate, it
should be raised more steeply to a higher level than in the case of
Nakov (2008) when the economy recovers.
Keywords: financial accelerator, zero lower bound, optimal discretionary policy, collocation method
JEL Classification: E5
1Assistant
Professor, Department of Economics, College of Business and Economics, Hanyang University; 55, Hanyangdaehak-Ro, Sangrok-Goo, Ansan, Kyoungki-Do,
426-791, Republic of Korea; e-mail: [email protected]
This work was supported by the research fund of Hanyang University (HY-2013-G).
First received, March 18, 2014; Revision received, April 23, 2014; Accepted, May 2,
2014.
94
How to exit from zero interest rate when there is a financial accelerator
1 Introduction
The zero interest rate, once considered as only a theoretical possibility,
has became a worldwide reality since the global financial crisis of 2008.
The fact that the interest rate reached its zero lower bound (ZLB), also
known as the liquidity trap, means that the conventional monetary policy
implemented with a short term interest rate cannot be used anymore to
boost the economy in the Great Recession. As a result, the so called quantitative easing was employed to make monetary policy still be effective.
Even though it is uncertain yet whether the economies of advanced countries have recovered from the recession so that they can retreat from these
unconventional policies,2 the economies are at least heading in that direction and the termination of the quantitative easing in a couple of years is
a viable possibility.
The termination of quantitative easing will imply the ensuing end of
zero interest rate policy. Then, how should the interest rate be raised
from the zero bound? From the theoretical point of view, an answer to
this question is not simple because the policy function is kinked when the
rate reaches ZLB which makes the problem to be non-linear. The first literature that studied this question is Jung et al. (2005). They investigated
both optimal discretionary and commitment policies under the non-negativity condition on nominal interest rate, i.e., i $ 0, with a linear quadratic
optimization problem of the monetary policy authority. With a model
setup of perfect foresight, they argued that the nominal short term interest rate should be kept at zero for a certain period even after the negative
demand shock3 turns up to the positive area under the optimal commitment policy (OCP). Under the optimal discretionary policy (ODP), however, the interest rate is supposed to rise from the zero bound immediately
when the demand shock turns out to be positive.
2Chang et al. (2010) is a worthwhile reference for identifying the state and length
of the global financial crisis.
3In the related literature, natural real rate represents the demand shock. The
steady state of the natural real rate is set as a positive value, like 3%. Without ZLB,
the optimal policy to demand shock is simple: the central bank can keep the output
and inflation gap at zero by changing the interest rate corresponding to the change of
the natural real rate proportionally.
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
95
Noting that the optimal policy of Jung et al. (2005) is obtained as a
solution of perfect foresight, Nakov (2008) studied optimal policies when
the demand shock is stochastic.4 Under this circumstance, an analytical
solution is not obtainable anymore. Furthermore, uncertainty itself plays
its role because of the non-linearity property of the optimization problem.
Nakov (2008) argued that, under ODP as well as OCP, the zero interest
rate should be kept for some period even after the demand shock turns up
to positive territory. This implies a more aggressive policy is required as
the private sector suspects the ability of the central bank in responding to
the negative demand shock under uncertainty because of ZLB.5
This paper extends these studies by adding the financial accelerator
(FA) mechanism to the model of Nakov (2008). The FA mechanism attracted a lot of attention after the global financial crisis as it was expected
to overcome the deficiency of the financial sector in the conventional new
Keynesian DSGE model. The implication of FA is that exogenous shocks
will be amplified through the FA mechanism and it is expected that ZLB
will be hit more frequently as Merola (2010) asserted. Hence, studying a
monetary policy in the model that includes both ZLB and FA is a worthwhile subject at present.
Before proceeding, a couple of caveats for this study should be noted.
First, we confine our interest to ODP. The reason is mainly technical. As
FA is going to be introduced to the model with a financial shock, the state
space needed to deal with OCP is three dimensional.6 Solving the optimization problem with three dimensional state space takes much time for
the computation7 and the presentation of the solution is also cumbersome,
even though it is not an impossible task. Narrowing our interest to ODP is
4Adam
and Billi (2004) and Adam and Billi (2006) also studied the similar subject.
(2008) explains this as following: \... when the natural real rate is close to
zero, private sector expectations reflect the asymmetry in the central bank's problem:
a positive shock in the following period is expected to be neutralized, while an equally
probable negative one is expected to take the economy into a liquidity trap. ... so it
is rational for the central bank to partially offset the depressing effect of expectations
about the future on today's outcome by more aggressive lowering of the nominal rate."
Sweidan ea al. (2012) also argued aggressive measures are necessary to prevent a deflationary trap.
6Policy functions of the optimal problem with commitment will be the functions
of the past state represented by the past Lagrange multiplier as well as demand and
financial shocks.
7This is known as the curse of dimension.
5Nakov
96
How to exit from zero interest rate when there is a financial accelerator
not so restrictive for deriving the policy implications in practice however,
considering the difficulty of manipulating private agents' expectations.
Second, the model is constructed without physical capital following
Jung et al. (2005) and Nakov (2008). Adopting this model strategy in
this paper makes the results comparable with previous studies. The strategy however casts the challenge of introducing FA to the model. Seminal
papers in FA literature such as Bernanke and Gertler (1989) and Kiyotaki and Moore (1997) incorporate FA into the model with a credit risk
premium or credit constraint which are related to the value of physical
capital or durable assets. In their modeling setups, the analysis of optimal
monetary policy becomes complicated.8 To our relief, there have recently
been some efforts of constructing the FA model without physical capital
recently. (Carlstrom et al. (2009), Curdia and Woodford (2008), Demirel
(2009), and Fiore and Tristani (2009)) The equilibrium equations in their
models appear as a simple extension of the basic new Keynesian DSGE
model. Although their modeling strategies of microeconomic foundation
are different, their equilibrium equations are in a very similar form. Specifically, a credit risk premium term which represents the financial friction
is added on both of the forward looking IS curve and the new Keynesian
Phillips curve (NKPC)9 or on the NKPC curve only.10
The model employed in this paper is an eclectic one. The model of Fiore and Tristani (2009) and Curdia and Woodford (2008) is consistent with
a conventional FA model in the sense that the financial factor affects both
the IS and NKPC curves. However, the responses of the credit risk premium to output change in their models are different from the conventional
FA model with physical capital. The model of Carlstrom et al. (2009) can
be an alternative but the direct effect of the financial factor to the output
which exists in the conventional FA model is omitted in their model. As it
is necessary to keep the characteristics of the conventional FA mechanism,
the model used here is set in an eclectic way by adding the credit risk premium term to the forward looking IS curve of the model of Carlstrom et
8For this reason, the monetary policy in these kind of models is usually set as the
Taylor rule type policy function.
9Fiore and Tristani (2009) and Curdia and Woodford (2008) are such examples.
10Demirel (2009) and Carlstrom et al. (2009) are such examples.
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
97
al. (2009).11
Incorporation of the FA mechanism into the model is proceeded in two
steps: first the FA mechanism itself and then the financial shock uncertainty in addition to the demand shock. In a way that the demand shock
uncertainty itself made a difference between Jung et al. (2005) and Nakov
(2008) in ODP, the financial shock uncertainty also plays its own role
other than that of a shock amplification mechanism that FA does. The
effect of FA mechanism to ODP is quantitatively minor but qualitatively
interesting. It makes the interest rate respond less aggressively to a shock.
This is counterintuitive considering the fact that FA amplifies the effect of
the shock. The reason can be induced by comparing ODP with the Taylor
rule as a monetary policy function. While the Taylor rule is a feedback
rule of which the change of the interest rate is widened proportionately
with the amplified output and inflation gaps, the ODP rule considers the
amplification of the effect of the interest rate as well as the output and
inflation gap. However, the effect of the FA mechanism is dwarfed by the
introduction of the financial shock uncertainty. It makes the response of
the policy interest rate to be more abrupt and allows the overshooting of
output and inflation gaps from the steady states.
The rest of the paper is organized as follows. In Section 2 we represent
the model. Section 3 explains the computational method and calibration
procedure. The computation results are given in Section 4. In Section 5 we
conclude. Sensitivity tests for the parameter values are given in the appendix.
2 Model
Before explaining the model, let us present the ODP functions of Jung
et al. (2005) and Nakov (2008) with Figure 1. The dashed lines show the
results of Jung et al. (2005) with policy functions of output gap, inflation
11Admittedly, this is an arbitrary modification of the model in the sense that the
IS curve with the credit risk premium term is not derived from the optimization problem solution. The credit risk premium term appears when the IS curve is expressed in
terms of marginal cost but is eliminated when the IS curve is expressed in output gap
in Carlstrom et al. (2009). The construction of the FA model without physical capital
that is consistent with the conventional FA model is a task in pursuit.
98
How to exit from zero interest rate when there is a financial accelerator
gap, and nominal interest rate12 against the demand shock, or the natural
rate of interest. The results are the same with optimal monetary policy
without a ZLB constraint13 when the natural rate of interest is greater
than zero: both the output and inflation gaps stay at zero by changing the
nominal interest rate with the same amount as the natural rate of interest
change. When the natural rate of interest is less than zero, ZLB constraint
binds and output and inflation gaps become negative. In other words, the
monetary policy effect of adjusting the short term policy interest rate is
restrictive when the demand shock is large enough to make the natural
rate of interest to be negative.
Figure 1: ODP with/without the uncertainty of demand shock
12Optimal discretionary policy functions are obtained by solving the quadratic welfare loss function of the central bank with the constraints of an intertemporal IS curve
and NKPC. Details are explained in the later section which explains the model.
13Without zero lower bound, output gap and inflation stays at zero regardless of
uncertainty. Note that the problem is in linear quadratic form. Hence, the theorem of
certainty equivalence is applied.
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
99
The uncertainty of demand shock14 changes the picture significantly.
The results of Nakov (2008) are presented with solid lines in Figure 1.
Note that keeping the output and inflation gaps at zero is no more an
optimal solution even when the natural rate of interest is greater than
zero. Uncertainty of the natural rate of interest causes the expectation of
deflation as the public know that the central bank cannot sufficiently respond to the negative natural rate of interest because of ZLB constraint.
To relieve the expectation of deflation, the central bank keeps the interest
rate lower than in the case of certainty, which makes the output gap to
be positive at around the steady state of the natural rate of interest that
is calibrated as 3% in their study. As the natural rate of interest becomes
larger, the probability of the negative natural rate of interest becomes
14Computationally, introduction of uncertainty is achieved by setting the variance
of the shock as a certain positive value.
100
How to exit from zero interest rate when there is a financial accelerator
smaller and the interest rate policy under uncertainty converges to that of
certainty. Similar results are also obtained by Adam and Billi (2006). The
results of our study will be compared with this result of Nakov (2008).
different,
linearized
appear
in similar
The model
used inequations
this paperofisthe
an model
extension
of Nakov
(2008)forms.
aug- Those equ
ifferent,
equations
of
the
model
appear
in
similar
forms.
Those
equations
can
ifferent, linearized
linearized
equations
of
the
model
appear
in
similar
forms.
Those
equations
can be
be
mented with FA mechanism and financial shock using the results of Fiore
written
as follow:
and Tristani (2009), Curdia and Woodford (2008), and Carlstrom et al.
ritten as
written
as follow:
follow:
(2009). Although microeconomic structures of their models are different,
linearized equations of the model appear in similar forms. Those equations
can be written as follow:
xxttt =
=
ππttt =
=
∆
=
∆ttt =
rˆrˆtnttnn =
=
nnttt =
=
−1
−1
n
n
xtt = Ettxt+1
t+1 − σ (itt − Ettπt+1
t+1 − rtt ) − ϕ(∆tt − Ett∆t+1
t+1)
−1
nn
−1
n
−1
σ
E
r
ϕ(∆
E
(1)
E
−
(i
−
π
−
)
−
−
∆
)
(1)
t
t
t+1
t
t
t+1
Etttxxt+1
−
σ
(i
−
E
π
−
r
)
−
ϕ(∆
−
E
∆
)
(1)
t+1
t
t
t+1
t
t
t+1
ttt
t+1
t
t t+1
t
t t+1
πtt = βEttπt+1
t+1 + κxtt + υ∆tt
βE
+
κx
+
υ∆
(2)
(2)
t
ttt
t+1
t
βEtttππt+1
+
κx
+
υ∆
(2)
t+1
t
)
−
ξE
∆
−
n
∆tt = −ς(xtt − Ettxt+1
t+1
tt t+1
t+1
tt
−ς(x
(3)
−ς(xttt −
−E
Etttxxt+1
−ξE
ξEttt∆
∆t+1
−nnttt
(3)
(3)
t+1))−
t+1 −
t+1
t+1
n
n
n
n
rr
rˆtt = ρrrrˆt−1
t−1 + εtt
nn
rrr
n
ρρrrrrˆrˆt−1
+
ε
(4)
+
ε
(4)
(4)
ttt
t−1
t−1
n
n
ntt = ρnnnt−1
t−1 + εtt
nn
n
ρρnnnnnt−1
+
ε
(5)
t−1
+
ε
(5)
ttt
(5)
t−1
n∗, rr
where rˆtntn ≡ rtntn − rn∗
, εtt ∼ N (0, σr2r2),, and εntnt ∼ N (0, σn2n2).. Equation
Equation (1) is the u
where
nn
nn
n∗
22
nn
22
n
n
n∗ rrr
2
n
2
n∗
−
,, εεtusual
∼
N
(0,
σ
),
and
ε
∼
N
(0,
σ
).
Equation
(1)
is
the
usual
here rˆrˆttt ≡
forward
looking
IS
curve
except
the
last
term
of
credit
≡ rrttt (1)
− ris
r the
∼
N
(0,
σ
),
and
ε
∼
N
(0,
σ
).
Equation
(1)
is
therisk
usual forward
forward
where
r
t
n
n
tt
rr
tt
n
.
x
looking
IS
curve
except
the
last
term
of
credit
risk
premium
∆
t
t
tt is outp
D
s
p
x
premium t. t is output gap, is the risk aversion coefficient, t is infla.
x
is
output
gap,
σσ isis
oking IS
except
the
last
term
of
credit
risk
premium
∆
n
t
t
t
t
.
x
is
output
gap,
ooking
IS curve
curve
except
the
last
term
of
credit
risk
premium
∆
t
t operator.
tion, rt is the natural rate of interest, and Et is the
expectation
the risk
aversion coefficient, πtt is inflation, rtntn is the natural rate of interest, a
The output gap πdecreases
when the
nn risk premium increases. The concrete
interest, and Ett is the
he
inflation, rrtttn isis the
the natural
natural rate
rate of
of
he risk
risk aversion
aversion coefficient,
coefficient, πttt isis inflation,
15 interest, and Et is the
mechanism is operator.
model dependent:
Fiore and
(2009)
explains
this premium in
expectation
The output
gapTristani
decreases
when
the risk
with increasing
default
Bernanke
et al.
(1999)
this
by
xpectation
The
gap
decreases
when
the
risk
premium
increases.
The
decreases
xpectation operator.
operator.
The output
outputrisk
gapwhile
when
the
risk explains
premium
increases.
The
14
expla
concrete
mechanism
is model
dependent:
Fiore andtax
Tristani
(2009)14
arguing that
the risk premium
plays
a role of distortionary
on invest14
14
14
explains
oncrete
model
dependent:
Tristani
(2009)
mechanism
model
Fiore
explains
this with
with
oncrete mechanism
dependent:
and
Tristani
(2009)
ment. Inisisthe
model
of CarlstromFiore
et al.and
(2009)
this term
depends
on the this
increasing
default
risk
while
Bernanke
et
al.
(1999)
explains
this
by
arguing
consumption of entrepreneurs. Equation (2) is also a usual new Keynesncreasing
risk
Bernanke et
al.
(1999)
explains
by
that
ncreasing default
default
risk while
while
etlast
al. term
(1999)
explains this
this The
by arguing
arguing
that the
the risk
risk
ian Phillips
curveBernanke
of risk
increase
premium
plays
aexcept
role ofthe
distortionary
tax premium.
on investment.
In theofmodel of Ca
premium
raises inflation
byon
affecting
the marginal
regardless
of
remium
aa role
of
tax
investment.
In
model
of
et
remium plays
playsrisk
role
of distortionary
distortionary
tax
on
investment.
In the
thecost
model
of Carlstrom
Carlstrom
et al.
al.
the underlying
model
specification.
(3) is contentious.
Fiore and
(2009)
this term
depends
on theEquation
consumption
of entrepreneurs.
Equation (2) i
2009)
depends
on
the
of
(2)
Tristani
(2009)
that the increase
of output raises Equation
the risk premium.
2009) this
this term
term
depends
onargue
the consumption
consumption
of entrepreneurs.
entrepreneurs.
Equation
(2) isis also
also aa usual
usual
new Keynesian Phillips curve except the last term of risk premium. The inc
in
ew
ew Keynesian
Keynesian Phillips
Phillips curve
curve except
except the
the last
last term
term of
of risk
risk premium.
premium. The
The increase
increase of
of risk
risk
15IS curve
of Fiore
and Tristani
also includes
nominal interest
rate term as of the unde
premium
raises
inflation
by (2009)
affecting
the marginal
cost regardless
a
cost
channel
factor.
We
abstract
this
term
to
focus
on
financial
factor.
remium
remium raises
raises inflation
inflation by
by affecting
affecting the
the marginal
marginal cost
cost regardless
regardless of
of the
the underlying
underlying model
model
specification. Equation (3) is contentious. Fiore and Tristani (2009) argue that
tha
pecification.
pecification. Equation
Equation (3)
(3) isis contentious.
contentious. Fiore
Fiore and
and Tristani
Tristani (2009)
(2009) argue
argue that
that the
the increase
increase
of output raises the risk premium. However, this is because they assume that th
f output
output raises
raises the
the risk
risk premium.
premium. However,
However, this
this isis because
because they
they assume
assume that
that the
the net
net worth
worth isis
rate
interest.
whichoffollow
the process of equation (4) and (5). Note that rn∗ is the steady state natur
rate of interest.
The
model
is completed
the interest
rate decision
rule. We
first
consider
rate
ofisinterest.
which
consistent
with the with
conventional
FA model
of Bernanke
et al.
(1999).
nt isthe
theOD
ne
The model is completed with the interest rate decision rule. We first consider the OD
under
and
FA
to compare
results
with
of previous
literature.
Th
TheZLB
is completed
with
the interest
decision
rule.worth
We
first
theshock
OD
worth
ofmodel
theconstraint
borrower.
The
natural
rate the
of rate
interest
and those
net
are consider
exogenous
Joo / and
Journal
Economic
Research
(2014) 93{123
101literature. Th
under ZLBDonghun
constraint
FAof to
compare
the19results
with those of previous
result
ODP
willprocess
be compared
thatand
of the
Taylor
rulethose
type
rule.
In the
cas
under
ZLB
constraint
andofFA
towith
compare
the
results
with
previous
literature.
Th
isfeedback
the
steady
state
natur
which of
follow
the
equation
(4)
(5).
Note
that
rn∗ of
result of ODP
will
be
compared
with
that
of
the
Taylor
rule
type
feedback
rule.
In
the
cas
However,
this is because
they
that the net worth
is given exogof
theofof
optimal
commitment
policy,
it that
isassume
history
another
dimensio
result
ODP will
be compared
with
of thedependent.
Taylor ruleThis
typeimplies
feedback
rule. In
the cas
rate
interest.
enously.
In
Curdia
and
Woodford
(2008),
output
does
not
the another
risk
of the optimal commitment policy, it is history dependent. Thisaffect
implies
dimensio
premium
the
riskthe
premium
is just
an
inefficiency
the first
financial
of the
state
variables
inbecause
addition
to
demand
and
financial
shock
dimensions
to derive
commitment
policy,
it interest
is history
dependent.
Thisof
implies
another
dimensio
Theoptimal
model
is
completed
with
rate
decision
rule.
We
consider
thepolic
OD
16
intermediary.
Here we
Carlstrom
et al. (2009)
so that
the increase
of state variables
in addition
tofollow
demand
and financial
shock
dimensions
to derive polic
functions
needs
toinbe
considered
andpremium,
causes
computational
difficulty
of to
the
well know
of
state
variables
addition
to risk
demand
andaresults
financial
shock
derive
polic
under
ZLB
and
FAthe
compare
the
thosedimensions
of previous
literature.
Th
ofconstraint
output
decreases
which
iswith
consistent
with
the confunctions needs
to
be
considered
and
causes
a
computational
difficulty
of
the
well
know
ventional FA model of Bernanke et al. (1999). nt is the net worth of the
curse
For
this reason,
we
defer
dealing
with
optimal
commitment
for
functions
needs
to be
be
considered
and
causes
a computational
difficulty
of rule.
the policy
well
know
result of
of dimension.
ODP
will
compared
with
that
of and
the
Taylor
rule
feedback
In the
cas
borrower.
The
natural rate
of interest
net
worth
aretype
exogenous
shocks
curse of dimension. For this reason, we defer dealing with optimal commitment
policy
for
rn* is the
which
follow
equation
(4)dependent.
and with
(5). Note
future
study.
curse
dimension.
Forthe
thisprocess
reason,ofitwe
dealing
optimal
commitment
for
of theofoptimal
commitment
policy,
is defer
history
Thisthat
implies
anotherpolicy
dimensio
steady state natural rate of interest.
future study.
Forstudy.
the
optimal
set
the welfare
loss dimensions
function
the
centralpolic
ban
future
The model
is completed
with we
the
interest
rate decision
rule. Weoffirst
of state
variables
indiscretionary
addition
to policy,
demand
and
financial
shock
to derive
For the consider
optimalthe
discretionary
policy,
we
set
the
welfare
loss
function
of
the
central
ban
ODP under ZLB constraint and FA to compare the results
as
For theneeds
optimal
discretionary
policy,
we
set
the
welfare
loss
function
of
the
central
ban
functions
to
be
considered
and
causes
a
computational
difficulty
of
the
well
know
with those of previous literature. The result of ODP will be compared with
as
that of the Taylor rule type feedback rule. In the case of the optimal comas
curse of dimension. For this reason, we defer
dealing with optimal commitment policy for
∞
mitment policy, it is history dependent.
This implies another dimension of
t
2
2
∞
state variables in addition
and
shock
λx2t + ψ∆
(6
min toE0demand
β (π
t +financial
t ). dimensions to
future study.
it ,πt ,xt ,∆t
2
2
∞ considered
β t (πt2 + λx
+
ψ∆
).
(6
t=0
derive policy functionsmin
needsEto0 be
and
causes
a
computational
t2
t2
t
2
it ,πt ,xt ,∆t
E
β
(π
+
λx
+
ψ∆
).
(6
min
For the difficulty
optimal of
discretionary
policy,
we
set
the
welfare
loss
function
of
the
central
ban
0
t=0
the welli ,πknown
curse of dimension.
t
tFor thist reason, we defer
t (2009),
t ,xt ,∆t
All of Fiore and Tristani
Curdia
and Woodford (2008), and Carlstrom et a
t=0
dealing with optimal commitment policy for a future study.
and Tristani (2009), Curdia and Woodford (2008), and Carlstrom et a
as All of Fiore
Forand
the
policy,
we set
the welfare
loss function
of premium et
(2009)
that
theoptimal
above discretionary
quadratic
loss
function
which
includes
the
riskCarlstrom
tera
All show
of Fiore
Tristani
(2009), Curdia
and
Woodford
(2008),
and
the
central
bank
as
(2009) show that the above quadratic loss function which includes the risk premium ter
can
be derived
from
theabove
utilityquadratic
function of
Solving
this lossthe
minimization
proble
∞consumers.
(2009)
show that
the
loss
function which
includes
risk premium
ter
t
2
2
2
can be derived from the utility min
function
this
loss
minimization
proble
E0 of consumers.
β (πt + λxSolving
+
ψ∆
).
(6
(6)
t
t
ias
,xtKuhn-Tucker
,∆t
t ,πfunction
ta
with
of (1)
problem,
we
obtain
the
following
first
ord
can beconstraints
derived from
the -(3)
utility
of
consumers.
Solving
this
loss
minimization
proble
t=0
with constraints
of
(1) -(3)
asTristani
a Kuhn-Tucker
problem,
we obtain(2008),
the following
first ord
All
of
Fiore
and
(2009),
Curdia
and Woodford
and
conditions:
withAllconstraints
of (1)
-(3) as(2009),
a Kuhn-Tucker
problem,
we obtain
theand
following
first et
orda
of Fiore
and
Tristani
Curdia
and
Woodford
(2008),
Carlstrom
conditions: Carlstrom et al. (2009) show that the above quadratic loss function which
includes
the above
risk premium
termloss
can function
be derivedwhich
from includes
the utilitythe
function
conditions:
(2009)
show
that the
quadratic
risk premium ter
of consumers. Solving this loss minimization problem with constraints of
(1)~(3)
a Kuhn-Tucker
problem,
we obtainSolving
the following
first
order
can be derived
fromasthe
utility function
of consumers.
this loss
minimization
proble
conditions:
(7
i (λxt + (κ − ςυ)πt − ςψ∆t ) = 0
with constraints of (1) -(3) ast a Kuhn-Tucker
problem, we obtain the following first ord
(7
it (λxt + (κ − ςυ)πt − ςψ∆t ) = 0
(7)
(λx
+
(κ
−
ςυ)π
−
ςψ∆
)
=
(7
i
≥
0
(8
i
t
t
t
t
t
conditions:
(8)
(8
it ≥ 0
≥
0
(8
i
(9)
(9
λxt + (κ − ςυ)πt − ςψ∆t ≤
(9
λxt + (κ − ςυ)πt − ςψ∆t ≤ 0
(9
λxt + (κ − ςυ)πt − ςψ∆t ≤ 0
(λx
− ςυ)πt −
ςψ∆t ) =to the
0 amount of cred16If the inefficiencyitof
financial
intermediary
is proportional
t + (κ
9
it, then the risk premium is determined endogenously.}
9
9
(7
it ≥ 0
(8
λxt + (κ − ςυ)πt − ςψ∆t ≤ 0
(9
102
How to exit from zero interest rate when there is a financial accelerator
On
On the
the other
other hand,
hand, the
the Taylor
Taylor rule
rule isis augmented
augmented with
with risk
risk premium.
premium. Taylor
Taylor (2008)
(2008)
and
andMcCulley
McCulleyand
andthe
Toloui
Toloui
(2008)
(2008)
proposed
proposed
this
thismodification
modification
sosothat
that
the
therisk
financial
financial
factor
factorcan
can
On
other
hand,
the Taylor
rule
is augmented
with
premium.
Taylor (2008) and McCulley and Toloui (2008) proposed this modification
be
beconsidered
consideredininthe
therule.
rule. The
TheTaylor
Taylorrule
rulewith
withZLB
ZLBtakes
takestruncated
truncatedforms
formsformulated
formulatedby
by
so that the financial factor can be considered in the rule. The Taylor rule
maximization
maximization
function.
function.
The
Thespecific
specificform
form
ofofthe
the
Taylor
Taylorrule
rule
asasfollowing:
following: function.
with
ZLB takes
truncated
forms
formulated
byisismaximization
The specific form of the Taylor rule is as following:
max{0, iit−1
+(1(1−−)(r
)(rn∗n∗++ηηππππt t++ηηxxxxt t−−ηη∆∆∆∆t )},
≤≤≤1.1. (10) (10)
(10)
itit==max{0,
t−1+
t )}, 00≤
3 Computational Method and Calibration
33 Computational
Computational Method
Method and
and Calibration
Calibration
16
The solution
method
for expectation
aexpectation
linear rational
expectation
model17
isour
notmodel
The
Thesolution
solutionmethod
method
for
foraalinear
linear
rational
rational
model
model16
isisnot
notapplicable
applicable
for
for
our
model
applicable for our model as the problem is nonlinear in the sense that the
asasthe
theproblem
problemisisnonlinear
nonlinearininthe
thesense
sensethat
thatthe
thepolicy
policyfunction
functionofofthe
thenominal
nominalinterest
interestrate
rate
policy function of the nominal interest rate has a kinked point when it hits
the
ZLB.
Hence,
wethe
tryZLB.
to obtain
a we
numerical
solution
using thesolution
colloca-using
has
hasaakinked
kinked
point
point
when
when
itithits
hits
the
ZLB.
Hence,
Hence,
wetry
trytotoobtain
obtain
aanumerical
numerical
solution
using
tion method. The collocation method is an extension of function approxi- 1717
the
thecollocation
collocationmethod.
method.
Thecollocation
collocationmethod
isisan
anextension
extensionofoffunction
function
approximation
18
19 approximation
mation
thatThe
is used
to solve method
a functional
equation
problem.
1818
our model
belongsproblem.
to a rational
expectation problem with an
that
thatisisused
usedtotoNote
solve
solvethat
aafunctional
functional
equation
equation
problem.
arbitrage-complementary condition of the form,
Note
Notethat
thatour
ourmodel
modelbelongs
belongstotoaarational
rationalexpectation
expectationproblem
problemwith
withan
anarbitrage-complementary
arbitrage-complementary
condition
conditionofofthe
theform,
form,
f(st, xt, Eth(st+1, xt+1)) = f t,
where st follows the state transition function,
, ,xxt+1
))))==φφt ,t ,
ff(s(st ,t ,xxt ,t ,EEt h(s
t h(s
t+1
t+1
t+1
st+1 = g(st, xt, e t+1),
where
wheresst tfollows
followsthe
thestate
statetransition
transitionfunction,
function,
and xt and f t satisfy the complementary conditions,
a(st) # xt # b(st), xjt > aj(st) Þ f jt # 0, xjt < bj(st) Þ f jt $ 0,
=g(s
g(st ,t ,xxt ,t ,εεt+1
),),
sst+1
t+1=
t+1
where f t is a vector whose jth element, f jt, measures the marginal loss
and
andxxt tand
andφφt tsatisfy
satisfythe
thecomplementary
complementaryconditions,
conditions,
from activity j. st is the vector of state variables, xt is the vector of endog17Blankchard
and Kahn (1980) method is such an example.
)≤
≤xx
≤
b(s
b(st ),
xjtjt>>aaj (s
(st )t )⇒
⇒φφjtjt≤≤0,
0, xare
xjtjt<interpolation
<bbj (s
⇒φand
φjtjt≥≥
0,0,
a(s
a(st )t18
t t≤
t ),of x
japproximation,
j (s
t )t )⇒
As
an
example
function
there
spline.
19Most of the explanation on this computational method is the repetition of Nakov
Blanchard
Blanchardand
andKahn
Kahn(1980)
(1980)method
methodisissuch
suchan
anexample.
example.
(2008). This is given here for the reader's convenience.
As
Asan
anexample
exampleofoffunction
functionapproximation,
approximation,there
thereare
areinterpolation
interpolationand
andspline.
spline.
1818
Most
Mostofofthe
theexplanation
explanationon
onthis
thiscomputational
computationalmethod
methodisisthe
therepetition
repetitionofofNakov
Nakov(2008).
(2008).This
Thisisisgiven
given
here
herefor
forthe
thereader’s
reader’sconvenience.
convenience.
1616
1717
19
vector of endogenous
and εt is the vecto
the
of state variables,
t is the
Thisvector
is approximated
using axlinear
combination
of n knownvariables,
basis functions,
of shocks to the state variables. The function that will be approximated is Et h(st+1 , xt+1
where φt is a vector whose j th element, φjt , measures
the marginal loss from activity j. st
n
This is approximated using a linearh(s,
combination
of
n
cj θjknown
(s). basis functions,
x(s)) ≈
19
is the Research
vector of
endogenous
the vector of
stateJoo
variables,
Donghun
/ Journal ofxtEconomic
19 (2014)
93{123 variables, and
103εt is the vecto
j=1
n
of
to theendogenous
state variables.
Theare
function
that
will be
approximated
is Ertnh(s
t+1 , xt+1
In shocks
our context,
variables
it , πof
variables
are
t, x
t , andto∆the
t , state
t and nt , an
e t ish(s,
enous variables,20 and
the x(s))
vector
shocks
state
variables.
The
cj θj (s).
≈
function thatusing
will be
approximated
is Ej=1
, xknown
is approximatth(sof
t+1n
t+1). This
This
is
approximated
a
linear
combination
basis
functions,
r
n
shocks to state variables are ε and ε .
ed using a linear combination of n known basis functions,
In our
endogenous
variables by
are the
it , πfollowing
are rtn value
and ntof, an
t , xt , and ∆
t , state variables
Thecontext,
coefficients
are determined
algorithm.
For a given
th
n
cj θj (s).
εnresponses
.x(s)) ≈ xi are
shocks
to state
variables
are εr andh(s,
computed at the n collocation nod
coefficient
vector
c, the equilibrium
j=1
The
coefficients
are determined
by thewhich
following
algorithm.into
Fora standard
a given value
of th
solving
the complementary
problem
is transformed
root findin
si by
D t, variables
i f t, x∆
In ourendogenous
context, endogenous
state variables
t, and
In our context,
variablesvariables
are it , πare
are rtn and nt , an
t , xtt, and
t , state
r
nthe
are rtngiven
and
shocks to responses
state
variables
are ecomputed
and e n. at the n collocation node
coefficient
vector
c, the
equilibrium
xi are
t, and
problem. Then,
equilibrium
r
nresponses xi at the collocation nodes si , the coefficien
and ε . by the following algorithm. For a given
shocks to state
variables
areare
ε determined
The
coefficients
the
complementary
problem
islinear
transformed
a standard root findin
svector
i by solving
c, thewhich
c is value
updated
bycoefficient
solving the
n-dimensional
systemxinto
of the
vector
equilibrium
responses
i are computed
The coefficients
are
determined
by
the
following
algorithm.
For
a given value of th
at the n collocation nodes si by solving the complementary problem which
problem. Then, given the equilibrium
responses xi at the collocation nodes si , the coefficien
is transformed
a standardresponses
root finding
problem. Then, given the equicoefficient vector
c, the into
equilibrium
nxi are computed at the n collocation node
x
s , the
librium
responses
at
the
collocation
nodes
coefficient vector c is
i
vector c is updated by solving the n-dimensional
linear
system
cj θji(s
h(si , xi ) =
i ).
the
complementary
problem
which
is
transformed
into a standard root findin
si by solving
n-dimensional
updated by solving the
linear
system
j=1
n x at the collocation nodes s , the coefficien
problem.
Then, given
the equilibrium
responses
i
This iterative
procedure
is repeated
until
thei distance between successive
values of
cj θj (si ).
h(si , xi ) =
vector
c issufficiently
updated by
solving
the n-dimensional
linear
system
j=1 to
becomes
small.
In addition,
we need
discretize
the shock to s to approxima
This procedure
iterative procedure
is repeated
untildistance
the distance
between
suciterative
is repeated
until
the
between
successive
values
of
theThis
expectation
functions. Here
the normal
shocks
to state variables
are
discretized
using
n
c
cessive values of becomes sufficiently
small. In addition, we need to dis= the to
cj θdiscretize
h(si , xwe
becomes
sufficiently
small.
Insaddition,
the shock
to the
s to approximat
i ) need
j (si ).
cretize the
shock to
to approximate
expectation
functions.
Here
K-node Gaussian
quadrature
scheme:
j=1
normal shocks to state variables are discretized
using a K-node Gaussian
the expectation
functions. Here the normal shocks to state variables are discretized using
quadrature scheme:
This iterative
procedure is repeated
Kuntil
n the distance between successive values of
K-node Gaussian quadrature
scheme:
ωk cj θj (g(si , x, εk )),
Eh(s, x(s)) ≈
becomes sufficiently small. In addition, we need
to discretize the shock to s to approxima
j=1
k=1
n
K the
expectation
Here
the normal
shocks
to state
aresodiscretized
using
e kare
w k are
where
andGaussian
Gaussian
quadrature
and variables
weights
chosen
so the discre
where
εk and
ωkfunctions.
quadrature
nodes nodes
and
weights
chosen
that
ωk cjthe
θj (g(s
x(s)) ≈approximates
i , x, εk )),
that the discreteEh(s,
distribution
continuous
normal distriK-node
Gaussian
quadrature
scheme:
j=1
k=1
distributionbution.
approximates
the
continuous
distribution.
For details,
refer
to Mirandanormal
and Fackler
(2004). For details, refer to Mirand
where
εk and(2004).
ωk are Gaussian quadrature nodes and weights chosen so that the discret
and Fackler
n
K 19
ωkdistribution.
cj θterm
)),using
Eh(s,
x(s))
≈
distribution
approximates
continuous
For
details,
to Mirand
j (g(sinstead
i , x, εk
Here we allow
a notational the
abuse
of using xt normal
as a general
of
it as refer
standing
for outp
gap.
k=1 j=1
and Fackler (2004).
where εk and ωk are Gaussian quadrature nodes and weights chosen so that the discre
19
we allowabuse
a notational
abuse
of using
general
term of
instead
it
xt as a
Here we allow20Here
a notational
of using
xt as
a general
term
instead
usingofitusing
as standing
for outpu
as
standing
for
output
gap.
11
gap.
distribution approximates the continuous normal distribution. For details, refer to Mirand
and Fackler (2004).
19
Here we allow a notational abuse of using xt as a11general term instead of using it as standing for outp
104
How to exit from zero interest rate when there is a financial accelerator
Table 1: Parameter Value & Explanation
Parameter
l
y
s
b
k
rn*
rr
sr
hp
hx
hD
_
j
u
V
x
rn
sn
Value
0.003
0.001
4
0.993
0.024
3%
0.65
3.72%
1.5
0.5
0, 1
0.8
3.2520
0.003
0.1212
0.0297
0.7806
0.6151%
Explanation
weight on output gap of the welfare loss of the central bank
weight on risk premium gap of the welfare loss of the central bank
risk aversion preference parameter
annual, discount factor
slope of the NKPC
annual, natural rate of interest
persistence of demand shock
standard deviation of demand shock
Taylor rule coefficient on inflation
Taylor rule coefficient on output gap
Taylor rule coefficient on risk premium gap
smoothed Taylor rule coefficient on the past interest rate
financial accelerator parameter of intertemporal IS curve
financial accelerator parameter of NKPC
financial accelerator parameter of risk premium equation
financial accelerator parameter of risk premium equation
financial shock persistency
financial shock standard deviation
Parameter values used for computation are presented in Table 1. Parameter values, other than those related to risk premium, are brought
from Nakov (2008) to compare our results with those of Nakov (2008).
They are presented above the middle line in Table 1. Note that the steady
state natural rate of interest is 3%. Parameter values of the central bank's
loss function (6), l and y , are determined with reference to related literature. The central bank's loss function can be derived from consumer's
welfare. Then the parameter values of the loss function can be computed
from deep parameter values of the model. The values of l and y that are
obtained from deep parameters in Carlstrom et al. (2009) are, for instance,
0.00364 and 0.00983. We set l as the same with Nakov (2008) and y as
0.001.
Parameter values related with he financial acceleration mechanism and
net worth shock are estimated with Bayesian estimation using Dynare.
Equations (1)~(5) and the smoothed Taylor rule (10) are used for the
estimation. Data used for the estimation are gross domestic product and
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
105
credit risk premium of the United States.21 The output gap is obtained as
a difference between the level of log GDP and its trend obtained by the
HP filter. Credit risk premium is defined as the difference between the
three year Baa corporate bond rate and the Treasury bond rate. The data
period is from 2000 to 2009. Note that the purpose of the estimation is to
obtain a fairly reasonable parameter value rather than to use it for a practical purpose such as forecasting.22 Detailed estimation results, including
priors used for the estimation, are given in Appendix A. Estimated parameter values are presented below the middle line of Table 1.
The estimation results show that the data have little information on
parameters of u and x . Hence, we implemented sensitivity tests not only
for these but also for other variables. The variations of most parameters
do not change the qualitative properties of our results. The results that
have some implications however are given in Appendix B. If u is large
enough, net worth shock might work as a cost shock. Following the result
of Walsh (2009), we focus on the case where the net worth shock works
as a demand shock. Implementing the sensitivity test, we found the net
worth shock persistency parameter, r n, has important policy implications,
as will be mentioned later.
4 Computation Results
We consider the ODP rule as an interest rate policy rule in a new
Keynesian DSGE model extended to include the ZLB constraint and FA
mechanism. Before presenting the results, it is necessary to note the reason that literature of ZLB constraint models focus on the demand shock.
There are two exogenous shocks in a conventional DSGE model: demand
shock attached to forward looking IS curve and cost shock attached to
NKPC. In a model without ZLB constraint, the demand shock poses no
21As there are only two stochastic variables, we can use just two data variables. By
adding monetary policy shock and cost shock to each of the Taylor rule and NKPC, we
could use inflation and federal fund rate data for the estimation. But they did not contribute much to the estimation of parameters related to the financial accelerator and
risk premium shock.
22Using our model for the purpose of practical forecast is impossible considering the
simplicity of the model.
106
How to exit from zero interest rate when there is a financial accelerator
significant policy choice problem while the cost shock makes the central
bank face the tradeoff between output and inflation gaps. However, a
large enough negative demand shock makes the natural rate of interest to
be negative and poses a substantial problem when there is ZLB constraint,
the liquidity trap. The cost shock does not cause the liquidity trap as the
shock makes the output and inflation gaps to move in opposite directions.
Hence, only the demand shock matters in a model with ZLB constraint.
When there is no ZLB constraint, demand shock is completely neutralized by the interest rate policy induced by the optimal discretionary policy
so that it has no effect on output and inflation gaps. Under the Taylor
rule, output and inflation gaps deviate from zero if the natural rate of interest deviates from its steady state level because of demand shock. In this
sense, the Taylor rule is a suboptimal rule. Incorporating FA mechanism
does not change this result.
With ZLB constraint, however, irrelevance of the demand shock to
output and inflation gaps under optimal discretionary policy is no longer
valid. As we have seen in Figure 1, ZLB constraint makes the output and
inflation gaps deviate from zero in response to the demand shock even under an optimal discretionary policy. In the following, we introduce the FA
mechanism to the model of Nakov (2008) and investigate the effects of the
FA mechanism under the optimal policy rule.
4.1 Effects of a Financial Accelerator
In Section 2, we saw that the uncertainty itself significantly changes
the shape of the optimal discretionary policy under ZLB constraint. With
this observation, we introduce the financial accelerator mechanism in two
steps. At first we introduce the FA mechanism only. The uncertainty related with financial shock, nt, will not be included at this phase. This is
implemented by setting the variance of financial shock to be zero. Still,
there is an uncertainty in the model related with the demand shock. In the
next step, we supplement the uncertainty of financial shock to the model.
The baseline will be the model of Nakov (2008) in which there is no financial accelerator and financial shock uncertainty.
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
Figure 2: Policy variable responses to demand shock under ODP(1)
107
108
How to exit from zero interest rate when there is a financial accelerator
Figure 2 presents policy functions against the natural rate of interest
under the optimal discretionary policy. In the figure, solid lines are the
baseline economy policy functions that are derived from the model without FA and dashed-dotted lines are the policy functions derived from the
model with FA. The third panel of the second column in the figure is the
policy function of credit risk premium that is available only for the model
with FA. The shape of this policy function is roughly the inverse of the
output gap policy function. It can be understood from equation (1) and (3).
Output and inflation gap policy functions rotates in a counter-clockwise
direction. This implies output and inflation gaps are amplified with an FA
mechanism. It should be noted that the amplification of output and inflation gaps comes from the effect of demand shock uncertainty23 enlarged by
23Of
course, here the uncertainty is that of the demand shock.
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
109
the FA mechanism, rather than the FA mechanism itself. If the problem
was merely a linear problem without ZLB, incorporating the FA mechanism would not change the optimal policies of zero output and inflation
gaps that could be achieved by the change of interest rate corresponding
to a demand shock. It is also interesting to note that the rotating axis
point of the output gap policy functions is about 0.5% of the natural rate
of interest, not 3%, the steady state natural rate of interest. This implies
that, under the optimal discretionary policy, FA has an effect of alleviating the negative demand shock when the natural rate of interest is still
greater than 0.5%.
Another interesting implication of incorporating FA into the model appears at the nominal interest rate policy function. The change of the interest rate policy function is shown in the left panel of Figure 2. The amount
of interest rate change caused by FA is not so conspicuous quantitatively.
At most, when there is a negative demand shock, the difference of the interest rate is 30 basis points at 1.7% of the natural rate of interest. This
implies that the introduction of an FA mechanism alone does not require
the interest rate policy to change significantly despite increased volatility of the output and inflation gaps. However, the change of interest rate
policy function under the optimal discretionary policy is somewhat counterintuitive in the sense that the interest rate policy function rotates in a
clockwise direction. As FA amplifies the variations of output and inflation
gaps, it seems likely that the interest rate should respond more sensitively
to the change of the natural rate of interest, which makes the slope of the
interest rate policy function steeper. Actually, this is what happens if we
set the monetary policy to follow the Taylor rule of equation (10), the
simple feedback rule.
Then, what causes the opposite to occur under the optimal discretionary policy? Note that not only the nominal interest rate but also the output, inflation, and risk premium gaps are choice variables in the optimal
monetary policy problem. In this setup, the FA can be used to reinforce
the effects of interest rate policy. When there is a positive demand shock,
the interest rate is raised to stabilize output and inflation but with less
amount because the financial accelerator amplifies the effect of the raised
interest rate. The opposite mechanism works when there is a negative demand shock. As a result, the policy function of the nominal interest rate
becomes flatter with FA under the optimal discretionary policy regime.
110
How to exit from zero interest rate when there is a financial accelerator
4.2 Effects of Financial Shock
Now we add financial shock uncertainty to the model in addition to
FA. In equation (3) we defined financial shock as an exogenous shock imposed on net worth. The net worth affects risk premium and, in turn, the
risk premium affects output and inflation gaps as noted in equations (1)
and (2).
Note that the policy functions are defined on state variables of natural
rate of interest, rtn, and net worth, nt. Proceeding the analysis with policy
functions defined on these two state variables is cumbersome as we have
to deal with 3 dimensional figures. To avoid this inconvenience, we first
look at the policy functions at nt = 0 so that we can compare them with
previous results. After that, we conduct the analysis with impulse response
functions (IRF) obtained from policy functions so that the demand shock
and the financial shock can be considered separately. IRF analysis also
gives more intuitive interpretation than the analysis with policy functions.
In Figure 3, dotted lines show responses of policy variables to a demand shock under the optimal discretionary policy when both demand
and net worth shock uncertainties are considered under nt = 0. Solid and
dashed-dotted lines are the same as in Figure 2. The figures show that the
effects of introducing uncertainty of a financial shock are more significant
than those of introducing the FA mechanism. First of all, adding financial
shock uncertainty strongly reinforces the aggressive response of monetary
policy by dropping the interest rate in a precipitous manner when the
economy confronts the ZLB constraint as shown in the left panel of Figure 3. This monetary policy implies, when ZLB is reached with a negative
demand shock, output and inflation to be over-boosted compared with the
economy without financial uncertainty.24
Another noteworthy result is the fact that the change of the nominal
interest rate should be bigger than that of the natural rate of interest even
when the natural rate of interest is higher than the steady state level of 3%.
The result contrasts with the case of demand shock uncertainty where the
nominal interest rate policy function converges to a certainty case when
the natural rate of interest is greater than 3%. In fact, the effect of financial shock uncertainty alone makes a parallel shift of Nakov (2008)'s nomi24These policy functions are not the description of empirical facts but the theoretical optimization results.
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
Figure 3: Policy variable responses to demand shock under ODP(2)
111
112
How to exit from zero interest rate when there is a financial accelerator
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
113
nal interest rate policy function.25 The dotted line representing the policy
function under both demand and financial shock uncertainty converges
to the parallelly shifted line, rather than infinitely diverging from the
policy function of the certainty case. In sum, the effect of financial shock
uncertainty dominates FA effect so that the nominal interest rate policy
function rotates counter-clockwise in the end. This steeper policy function
implies a more aggressive monetary policy response to the demand shock.
Responses of other policy variables are similar except the abrupt changes
in accordance with the precipitous drop of nominal interest rate at around
the point where the ZLB constraint starts to bind.
Based on the policy functions, we also can implement the analysis with
IRF. In the IRF analysis, one period in the model corresponds to one
quarter and the shock will be given with the size of {1.5 times one standard deviation shock so that the ZLB constraint binds for some period
of time under an optimal discretionary policy. IRFs are obtained in the
dimensions of both of the demand and financial shocks. We first describe
the effects of financial shock uncertainty on the IRFs to demand shock by
fixing net worth shock at zero. The IRFs to financial shock by fixing demand shock at zero will be explained later.
Figure 4 presents the IRFs to a demand shock. In this Figure, the solid
line is the case of Nakov (2008) where both the FA mechanism and the
financial shock are absent. The dotted lines are IRFs of policy variables
under the optimal discretionary policy with FA and the financial shock
uncertainty. The bold solid line in the third panel of the Figure shows the
process of natural rate of interest caused by the demand shock. Responding to this process, the nominal interest rate drops to the level of zero and
stays there for four quarters under the case of Nakov (2008). We can see
that the nominal interest rate stays at zero for two more quarters even after the negative natural rate of interest rises above zero under the baseline
economy.26
When the FA mechanism and financial shock uncertainty are intro25This can be shown with the policy function under financial shock uncertainty
alone by setting the variance of demand shock as zero.
26If there were no ZLB constraint, the nominal interest rate would have moved
along with the natural rate of interest. If there were no uncertainty at all, the nominal
interest rate would have risen above zero right after the natural rate of interest rises
above zero.
114
How to exit from zero interest rate when there is a financial accelerator
Figure 4: IRFs to a demand shock
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
115
duced under the optimal discretionary policy, the nominal interest rate
stays at the level of zero for one quarter longer and rises more quickly
compared with Nakov (2008)'s case. The steady state nominal interest
rate is also higher by 38 basis points than that of Nakov (2008)'s. An interesting observation is that the output gap overshoots at the end of the
zero interest rate period under optimal discretionary policy. Thanks to the
continued zero rate policy, the output gap reaches 0.8% in the fifth quarter after the shock occurs and subdues as the interest rate rises steeply.
The IRF of the inflation gap also shows a similar pattern of movements.
These overshootings are not observed without financial shock uncertainty.
Here, it should be noted that these results of aggressive nominal interest rate policy in response to demand shock and overshooting of IRFs
in output and inflation gaps critically depend on the persistency of net
worth shock. As the persistence parameter, r n, becomes lower, the added
financial shock uncertainty works in the same way with increased demand
shock uncertainty, i.e., the financial shock increases the range of negative
demand shock, or the natural rate of interest, that requires a zero nominal interest rate but the nominal interest rate policy function approaches
to that of Jung et al. (2005) without any jump, as in the same way that
Nakov (2008)'s nominal interest rate policy function did. This implies a
prolonged zero interest rate period in the IRF nominal interest rate but a
smoother rise of interest rate with the recovery and a lower steady state
interest rate than in the case of without financial shock uncertainty. For
more details, refer to Appendix B. Considering the persistence of the financial shock that was experienced during the past global financial crisis,
however, policy implications of aggressive nominal interest rate response is
more likely to be valid.
Now let us consider the effects of financial shock itself. Figure 5 presents the IRFs to a financial shock. As the financial shock cannot exist in
the model without FA, IRFs corresponding to Nakov (2008)'s case do not
exist and hence the solid lines that correspond to those of Figure 4 are not
present. Looking at the dotted lines, we can recognize that the IRFs show
the same pattern as those of a demand shock: they are just horizontally
stretched ones of the IRFs to the demand shock. While the size of the output response is not much different in both cases, the response of inflation
116
How to exit from zero interest rate when there is a financial accelerator
Figure 5: IRFs to a net worth shock
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
117
is much smaller in the case of financial shock.27 This is understandable
if we remember that the financial shock works as both cost and demand
shock. The negative financial shock decreases output unilaterally but it
has both direct and indirect effects that work in opposite ways. It raises
the price as the financial cost rises but the ensueing decrease of output
lowers the price. Hence, the effect of financial shock on inflation is a matter of parameter choice based on empirical data, as Walsh (2009) argued.
He showed that the financial shock works as a demand shock and the
IRFs to a financial shock confirms that parameter settings in our model
are consistent with his result, i.e., financial shock appears as a demand
shock.
5 Conclusion
This paper introduced an FA mechanism to the simple new Keynesian
DSGE model with ZLB in two steps: firstly, a shock amplifying the FA
mechanism and, then, the financial shock uncertainty, and studied their
theoretical effects on macroeconomic variables including especially the
nominal interest rate. The reason that the FA mechanism is introduced in
two steps is that the uncertainty itself has substantive meaning in a nonlinear rational expectation model.
The effect of the FA mechanism reveals the fundamental differences of
the two different monetary policy regimes: the optimal discretionary policy versus the Taylor rule. Under the Taylor rule, the introduction of an
FA mechanism amplifies the response of policy interest rate to a demand
shock as well as output and inflation gaps. This is natural as the Taylor
rule is just a feedback rule with which the interest rate is adjusted according to the change of output and inflation gaps. Interestingly, the response
of the policy interest rate is reduced under the optimal discretionary policy
regime, even though it is quantitatively small, despite the amplified output and inflation gaps. This is because the monetary authority should consider the amplified effect of interest rate policy under the optimal policy
regime unlike the simple feedback rule. This implies that monetary policy
becomes less aggressive when the FA mechanism is introduced under the
27Note
that the scale of vertical axes are different between Figure 4 and 5.
118
How to exit from zero interest rate when there is a financial accelerator
optimal discretionary policy regime.
Next we looked over the effect of the introduction of financial shock
uncertainty only under the optimal discretionary policy. From the IRF
analysis of the policy functions, we found that monetary policy becomes
more aggressive, overwhelming the previous smoothing effect of the FA
mechanism. When the financial shock uncertainty is added to the existing demand shock uncertainty, the zero interest rate period is prolonged
against the negative demand shock and the interest rate is raised more
steeply to a higher level than in the case of Nakov (2008) when the
economy recovers. Consequently, the overshooting of output and inflation
gap is allowed during the prolonged zero interest rate period. It should be
mentioned that this result critically depends on the persistence of the financial shock. If the financial shock persistency were weak, then it would
be just like increasing demand shock uncertainty and there would be no
overshooting in output and inflation gaps, although the zero interest rate
period is still prolonged.
The model in this paper is a simple extension of Nakov (2008)'s model
which incorporates an FA mechanism and related financial shock. The
policy implication of Nakov (2008) for the monetary policy was to keep
zero interest rate for some time even after the economy recovered from a
deep recession which required zero interest rate. The policy implication of
our results is to reinforce that policy implication. When the financial shock
uncertainty, in addition to demand shock uncertainty, is considered in the
model, the decision of escaping from the zero interest rate policy should be
made more cautiously so that even the overshooting of output and inflation gaps are allowed. Once the interest rate is decided to be raised from
ZLB, it will be raised more steeply to a higher level than suggested by Nakov (2008).
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
119
References
Adam, Klaus and Billi, Roberto M., \Optimal monetary policy under
discretion with a zero bound on nominal interest rates," Working
Paper Series 380, European Central Bank, 2004.
Adam, Klaus and Billi, Roberto M., \Optimal monetary policy under
commitment with a zero bound on nominal interest rates," Journal
of Money, Credit and Banking 38, 2006, 1877{1905.
Bernanke, Ben and Gertler, Mark, \Agency costs, net worth, and business uctuations," American Economic Review 79, 1989, 14{31.
Bernanke, Ben S., Gertler, Mark, and Gilchrist, Simon, \The nancial accelerator in a quantitative business cycle framework," In J.B. Taylor and M. Woodford, editors, Handbook of Macroeconomics, vol. 1,
Elsevier, 1999.
Blanchard, Olivier Jean and Kahn, Charles M., \The solution of linear dierence models under rational expectations," Econometrica 48, 1980,
14{31.
Carlstrom, Charles T., Fuerst, Timothy S., and Paustian, Matthias, \Optimal monetary policy in a model with agency costs," Working papers, 2009.
Chang, Kook-Hyun, Cho, Kyung Yup, and Hong, Min-Goo, \Stock volatility, foreign exchange rate volatility and the global nancial crisis,"
Journal of Economic Research 15, 2010, 14{31.
Curdia, Vasco and Woodford, Michael. \Credit frictions and optimal
monetary policy," Technical report, 2008.
Demirel, Ufuk D. \Optimal monetary policy in a nancially fragile economy," The B.E. Journal of Macroeconomics 9.
Fiore, Fiorella De and Tristani, Oreste, \Optimal monetary policy in a
model of the credit channel," Working Paper Series 1043, European
Central Bank, 2009.
Jung, Taehun, Teranishi, Yuki, and Watanabe, Tsutomu, \Optimal monetary policy at the zero-interest-rate bound," Journal of Money,
Credit and Banking 37, 2005, 813{35.
120
How to exit from zero interest rate when there is a financial accelerator
Kiyotaki, Nobuhiro and Moore, John. \Credit cycles," Journal of Political
Economy 105, 1997, 211{48.
McCulley, Paul and Toloui, Ramin. \Chasing the neutral rate down: Financial conditions, monetary policy, and the Taylor rule," Technical report, Global Central Bank Focus, PIMCO, 2008.
Merola, Ossana. \Financial accelerator and the zero lower bound on interest rates," mimeo.
Miranda, Mario J. and Fackler, Paul L., Applied Computational Economics and Finance, vol. 1 of MIT Press Books. The MIT Press, 2004.
Nakov, Anton. \Optimal and simple monetary policy rules with zero
floor on the nominal interest rate," International Journal of Central
Banking 4, 2008, 73{127.
Sweidan, Osma D., Raj, Mahendra, and Uddin, Md Hamid. \The fed's reprogramming strategy to re-energize the U.S. economy during the
great recession," Journal of Economic Research 17, 2012, 159{187.
Taylor, John B. \Monetary policy and the state of the economy," Technical report, testimony before the Committee on Financial Services,
U.S. House of Representatives, 2008.
Walsh, Carl E. \Using monetary policy to stabilize economic activity,"
Working paper, 2009.
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
121
Appendix A Parameter estimation
Bayesian estimation is implemented with Dynare to estimate parameters related with the financial accelerator (j , u , V, x , r n, s n) given the rest
of parameters are brought from Nakov (2008). Prior means are set based
on deep parameter values of Carlstrom et al. (2009). From Figure 6, we
can see that the data does not contain information for parameters of u
and x .
Table 2: Prior and posterior distributions of the estimated parameters
Parameter
j
u
V
x
rn
sn
Dist.
Gamma
Gamma
Gamma
Gamma
Gamma
Inv. Gamma
Prior
Mean
3
0.003
0.13
0.03
0.8
1
Std.
0.3
0.0001
0.03
0.005
0.1
{
Posterior
Mean
Low conf. High conf.
3.252
2.7901
3.6675
0.003
0.0028
0.0032
0.1212
0.0868
0.1543
0.0297
0.0216
0.0376
0.7806
0.6881
0.8702
0.6151
0.4913
0.7176
Figure 6: Prior and posterior distributions of the estimated parameters
122
How to exit from zero interest rate when there is a financial accelerator
Appendix B Sensitivity tests
The data used for parameter estimation have little information on the
parameters of u and x . This means that the values of these parameters are
calibrated, rather than estimated, as the values of prior mean. Hence, we
implemented sensitivity tests for these parameters. The results are given
in Figure 7 as changes of nominal interest rate policy function according
to the change of parameter values.
Figure 7: Sensitivity test for the parameters of u and x
Reducing the values of these two parameters toward zero has no significant effect on our results. On the other hand, increasing the values of
these parameters has some effects. First, increasing u up to three times of
the baseline value of 0.003 has little effect on the policy functions. However, if the value of u is increased more than that, the shape of the policy
functions start to change significantly. This might come from the change
of the financial shock characteristic: as u becomes larger, the financial
shock works as a cost shock, rather than a demand shock. Second, increasing x makes the nominal interest rate policy function to be more aggressive against the possibility of zero interest rate, even though the qualita-
Donghun Joo / Journal of Economic Research 19 (2014) 93{123
123
tive characteristic of the policy does not change up until the parameter
value reaches seven times the baseline value. If the value is raised beyond
that point, the solution for the problem does not exist.
We also implemented the sensitivity tests with other parameters of j ,
V, and r n. In case of j and V, the sensitivity tests showed similar results
with those of x , i.e., increasing the parameter values does not change the
qualitative characteristics of the policy functions but the solution does not
exist if the values of those parameters are raised to beyond certain points.
The change of the value of r n, the financial shock persistency, has substantive meaning on the policy function of nominal interest rate, however, as
shown in Figure 8.
Figure 8: Sensitivity test for the parameter r n
As r n is lowered, for instance, to the level of 0.65, the eect of nancial
shock uncertainty to the nominal interest rate policy function becomes
the same as that of demand shock uncertainty. It appears as just like an
added demand shock uncertainty, or the increase of the demand shock
standard deviation. In that case, the overshooting of output and in ation
gaps would have disappeared in their IRFs and the steady state interest
rate level would have been lower than in the case of Nakov (2008).